Essays on Galileo and the History and Philosophy of Science: Volume 3 9781487572020

Citation preview

ESSAYS ON GALILEO AND THE HISTORY AND PHILOSOPHY OF SCIENCE VOLUME III Stillman Drake Selected and introduced by N.M. Swerdlow and T.H. Levere

For forty years, beginning with the publication of the first modem English translation of the Dialogue Concerning the Two Chief World Systems, Stillman Drake was the most original and productive scholar of Galileo's scientific work. During those years, Drake published sixteen books on Galileo, including translations of almost all the major writings, and Galileo at Work, the most comprehensive study of Galileo's life and works ever written. Drake also published about I 30 papers, of which nearly I oo are on Galileo and the rest on related aspects of the history and philosophy of science. The three-volume collection Essays on Galileo and the History and Philosophy of Science includes 80 of those papers. In the papers included in Volume III, Drake explores some of the more technical and practical aspects of Galileo's work, focusing on his contributions to scientific instrumentation. The essays then tum to the history of science, demonstrating the breadth of Drake's interests both beyond and relating to the work of Galileo. These interests are again evident in the final papers in the collection, in which Drake writes on the philosophy of science and language. This volume draws to a conclusion this collection of Drake's writings on Galileo and the influences and themes in his life and work. is Professor, Department of Astronomy and Astrophysics, University of Chicago. T. H. LEVERE is Professor and Director, Institute for the History and Philosophy of Science and Technology, University of Toronto. N . M. SWERDLOW

Stillman Drake. Photo courtesy of Mrs Florence Drake.

Essays on Galileo and the History and Philosophy of Science VOLUME III Selected and introduced by N.M. Swerdlow and T.H. Levere

STILLMAN DRAKE

UNIVERSITY OF TORONTO PRESS

Toronto Buffalo London

© University of Toronto Press Incorporated 1999

Toronto Buffalo London Printed in Canada Reprinted in 2018 ISBN 0-8020-4344-5 (cloth)

ISBN 978-0-8020-8165-0 (paper)

Printed on acid-free paper

Canadian Cataloguing in Publication Data Drake, Stillman Essays on Galileo and the history and philosophy of science Includes bibliographical references and index. (v. 3 : bound) ISBN 978-0-8020-8165-0 (paper)

ISBN 0-8020-4344-5 1.

Galilei, Galileo, 1564-1642. I. Swerdlow, N.M . (Noel M.). II. Levere, Trevor H. (Trevor Harvey). Ill . Title QBJ6.G2D667 1999

520'.92

University ofToronlo Press acknowledges the financial assislance to its publishing program of the Canada Council for the Arts and the Ontario Arts Council. University of Toronto Press acknowledges the financial support for its publishing activities of the Government of Canada through the Book Publishing Industry Development Program (BPIDP).

Canada

Contents

Part VII

Galileo: Instruments

Galileo and the First Mechanical Computing Device 5 Tartaglia's Squadra and Galileo's Compasso 15 3 I Galileo Gleanings VI : Galileo's First Telescopes at Padua and Venice 33 4 / Galileo Gleanings XII: An Unpublished Letter of Galileo to Peiresc 1 /

2 /

Part VIII

45

History of Science: Ancient, Medieval, Renaissance, Seventeenth Century 57

Euclid Book V from Eudoxus to Dedekind 61 2 / Hipparchus - Geminus - Galileo 76 3 I Bradwardine's Function, Mediate Denomination, and Multiple Continua 87 4 / Medieval Ratio Theory vs Compound Medicines in the Origins of Bradwardine's Rule 105 5 / Early Science and the Printed Book: The Spread of Science beyond the Universities 1 18 6 I The Pseudo-Aristotelian Questions of Mechanics in Renaissance Culture (with Paul L. Rose) I 3 I 7 / An Agricultural Economist of the Late Renaissance 170 8 / Renaissance Music and Experimental Science 190 9 / Music and Philosophy in Early Modem Science 208 1 /

vi 1o

/ Impetus Theory and Quanta of Speed before and after Galileo 22 1 1 1 / Free Fall from Albert of Saxony to Honore Fabri 239 12 / Impetus Theory Reappraised 258 13 / A Further Reappraisal of Impetus Theory: Buridan, Benedetti, and Galileo 279 14 I The Rule behind "Mersenne's Numbers" 297 15 / Newton's Apple and Galileo's Dialogue 302

Part IX 1 /

Contents

Philosophy of Science and Language

31 1

Back from Limbo: The Rediscovery of Alexander Bryan Johnson 2 / A.B . Johnson and His Works on Language 327 3 I Literacy and Scientific Notations 345 4 I J.B. Stallo and the Critique of Classical Physics 364

A BIBLIOGRAPHY OF THE WRITINGS OF STILLMAN DRAKE INDEX ILLUSTRATIONS

387 follow page 220

379

315

PART VII GALILEO: INSTRUMENTS

Galileo's most well-known contribution to scientific instrumentation was of course the telescope, which he did not invent but did improve, and for some period he himself made the best available telescopes for astronomical observation. His first concern with the design and manufacture of an instrument, however, was in the early 1590s, when he made a ruled sector to be used for setting the angle of elevation of cannon. He later added more scales to the sector to solve problems in gunnery, computation, and surveying, the instrument reaching its final form in 1598. He also wrote descriptions of the use of the sector, and finally in 1606 published The Operations of the Geometric and Military Compass, his first publication under his own name. "Galileo and the First Mechanical Computing Device" ( 1) is a description of the development and application of the sector and "Tartaglia's Squadra and Galileo's Compasso" (2) is a more detailed examination of the manuscript sources showing its evolution. In 1978 Drake published a translation of the Geometric and Military Compass of 16o6 with an extensive introduction on its background. "Galileo's First Telescopes at Padua and Venice" (3) contains a translation of a letter from Galileo to Benedetto Landucci in Florence, dated 29 August 16o9, in which he reports on his first learning of the telescope, his making one that he demonstrated to the Signoria and Senate in Venice, and the increase of his salary at Padua as a reward. The letter seems to have been a way of communicating his good fortune to Cosimo de' Medici, from whom he was seeking an appointment in Florence. The paper also contains a reconstruction of the chronology of Galileo's first construction and demonstration of the telescope. The last paper in this section, "An Unpublished Letter of Galileo to Peiresc" (4), is not so much on an instrument as on the exposure of a fraud. On I April 1635 Galileo received a letter from Nicole Fabri de Peiresc, who had been attempting to have the restrictions following Galileo's sentencing removed or moderated. The letter concerned a clock invented by one Francis Linus, a Jesuit, consisting of a wax globe immersed in water that turned around every twenty-four hours, as though driven by an occult, magnetic force from the rotating heavens. Galileo damped Peiresc's enthusiasm - he seems to have thought it provided evidence for the rotation of the earth - by explaining that he had made something similar years ago in which a magnet was contained within the globe and the globe made to rotate by a second magnet carried around by a clockwork mechanism hidden in the base. It is also worth noting that one of Galileo's last inventions was of an escapement for a pendulum clock, which he dictated to his son Vincenzio in 1641, as described in Galileo at Work, pp. 419-2 I.

1 Galileo and the First Mechanical Computing Device

People have resorted to many kinds of mechanical devices to avoid the trouble of doing arithmetical calculations. The oldest of them is the abacus, with its sets of beads strung on parallel wires; by pushing beads back and forth according to set rules one can get the same result as one does when working with written numbers. One now pushes buttons on a small electronic calculator according to set rules. Since mathematicians long ago found ways to reduce geometry and physics to numbers, every practical problem involving mathematics can now be solved without any tedious computations on paper. Up to a few years ago people used the slower mechanical calculating machine, pushing buttons that set in motion numbered wheels. Half a century earlier, before the electric motor was applied, similar machines were operated by turning a crank and shifting a carriage. The basic mechanism goes back to the 17th century, when Blaise Pascal put digits around successive wheels that he connected to provide for carry-over. In the same century Gottfried Wilhelm von Leibniz introduced the movable carriage to speed up multiplication. Credit for the basic idea of a calculating machine goes back before Pascal to John Napier, who carved numbers on movable rods of bone rather than on wheels. Napier is deservedly more famous for his invention of logarithms in 161 I. That invention makes him ultimately responsible for the basic idea behind the slide rule, the chief alternative device for avoiding arithmetical calculations in modem times. It soon occurred to Edmund Gunter of England to lay out Napier's logarithms along a line and measure along it with an ordinary ruler. "Gunter's line" gave quick approximations without the nuisance of adding and Reprinted from Scientific American 234 no. 4 (April 1976), 104-13. Reprinted with permission. Copyright© 1976 by Scientific American, Inc. All rights reserved.

6

Instruments

subtracting numbers taken from tables. Slide rules quickly followed, first in circular shape and then in the more familiar slip-stick fonn. Now the small electronic calculator is rapidly converting the slide rule and the mechanical calculating machine into museum pieces. Yet it is less than a century since the slide rule and the mechanical calculator themselves converted into a museum piece another mechanical calculating device that had been invented a bit earlier and that competed with them successfully until motors were put on calculating machines and the slide rule came into wide use as a result of modem technical education. The earlier device was the sector. The fonn in which it first excited wide interest was invented about 1 597 by Galileo (see Plates I and 2). In its basic design the sector consists of two anns joined at one end by a pivot. The anns are of equal length (from four to 12 inches) and bear identical numerical scales. A geometrical problem that can be worked out with such an instrument calls for inscribing a polygon of equal sides in a circle of a given diameter. One sets a point at the free end of each ann of the sector so that the separation of the two points equals the diameter of the circle. Then, supposing that a pentagon is the figure to be inscribed in the circle, one measures the distance from the 5 on one ann of the sector to the 5 on the other arm. That distance is the length of each side of the pentagon [see illustration on page II]. Such a problem, however, lacks the feature of calculation that was added later. One advantage of the sector was that it could be easily understood and used by people of little education. Another was that for many common practical problems the user did not even need to think in tenns of numbers. Indeed, one of the influences that led Galileo to conceive of the sector as a universal calculator for all practical purposes was a problem he confronted that was beyond the mathematics of his time. The sector of old was a handsome instrument, usually made of brass or silver. At least 1 ,ooo such instruments survive in museums and private collections. During the past century a less ornate version of the sector became part of the standard kit of the carpenter, the craftsman and the draftsman. These instruments, made of wood or bone in mass production, can still be encountered in commerce. Nevertheless, few people today have even heard of the sector, and the story of its invention has remained more the subject of guesswork and controversy than of serious historical investigation. The story I shall tell is based on an examination of manuscript instructions that Galileo wrote before 16o6, when he first printed a revised version of them, and on my inspection of many early sectors. My interest was recently reawakened when Anahid lskian of New York, an expert on rare drawings and prints, called to my attention (and enabled me to acquire) a manuscript

Galileo and the First Mechanical Computing Device

7

copy of Galileo's instructions made in 1605. The instructions differed from those in Galileo's book, Operations of the Geometric and Military Compass, published in 1606. Previously I had found a different manuscript version, dating from about 1599, in the Rocco-Watson collection at the California Institute of Technology. These two manuscripts, together with the five other known versions (all in the Ambrosiana Library in Milan and dating from 1597 to 1600), have made it possible to reconstruct the evolution of Galileo's instrument. Its history turns out to be quite different from what was supposed. The use of a sector as a mechanical calculating device was first described in print by Thomas Hood. His account, written in English, was published in 1598. At the time Hood knew nothing of an Italian sector, and in 1597 Galileo had not heard of the English one, which probably had been in service for some years before Hood's book appeared. Independent simultaneous discoveries and inventions are not uncommon in science and technology. What is unusual here is that the two inventors had quite different backgrounds, Galileo being a professor of mathematics and Hood a practical scientist. In arriving at the sector they had different starting points and different objectives. Hood's sector bore three scales. Galileo's 1597 model had seven, only one of which (for constructing regular polygons) was on Hood's. Galileo eliminated this scale from his sector a year later, when he first included a scale for obtaining ordinary ratios, although that was the simplest scale of all and had from the outset played the main role on Hood's sector. The accessories of Hood's sector suggest that it originated as a surveying instrument. They included pairs of removable sights, a plumb line and a graduated quadrant attached to one ann. The scale of equal divisions on each ann was probably provided originally for mapping to any given scale. The variable opening, by altering the measurements between corresponding points along the two anns, was a simple mechanical aid in the solution of all problems of proportionality. A noncalculating type of sector was illustrated in a book on instruments printed in Venice in 1598. The instrument had been devised not long before by the Marchese Guidobaldo del Monte, Galileo's friend of many years, as a simple and inexpensive aid in two common problems of drafting and design. One problem was dividing a circle into a given number of equal arcs or constructing a regular polygon in a circle. The other was dividing a straight line of given length into an exact number of equal parts. Guidobaldo's sector was a consolidation and an improvement of two drafting instruments that had been in use in Italy since the 156o's. One was the ordinary proportional compass, which is still in use today; it had points at both ends and a movable pivot. The other was the reduction compass, with a fixed pivot, two

8

Instruments

fixed points and two sliding points at right angles to the arms. Later models were made so that all four points met the paper at right angles. All such instruments were expensive and required frequent resetting of the movable parts. Guidobaldo made a simple hinge the only movable part of his sector and provided permanent scales that gave direct readings for the number of parts of a circle or a line (Plate 3). The origin of Galileo's "military compass" has been conjecturally related in various ways to the proportional compass, the reduction compass and Guidobaldo's sector. It has always been supposed that Galileo took over a calculating device already in service and added to it more complex scales. One trouble with this supposition is that no calculating sector has been found that was in use in Italy before Galileo devised his "military compass" in 1597. Another objection can now be added: Galileo's sector was already quite complex before it bore the simplest scale of all. Let us start with its earlier history. The ancestors of Galileo's calculating sector were two instruments of a quite different kind, both invented 60 years earlier for military applications by Niccolo Tartaglia, a practical mathematician. They were combined into one by Galileo, with improvements, before he thought of the idea of mechanical calculation. In 1537 Tartaglia published in Venice a book, The New Science, in which mathematics was applied to artillery practice. The book introduced a gunner's elevation gauge consisting of a kind of carpenter's square that had one long leg, which was placed in the mouth of the cannon, and a fixed quadrant arc divided into 12 equal parts called points (Plate 4). A plumb line that was hung from the vertex indicated the elevation of the cannon, so that a dead-level shot was called "point-blank" and a shot made at 45 degrees was said to be "at six points." The device was quickly adopted throughout Europe. Tartaglia also discussed the determination of the height and distance of targets by sighting and triangulation. For this purpose he invented a second instrument, which was also based on the square (Plate 5). Between his time and Galileo's several men proposed more appropriate and convenient instruments for triangulation in the field. Many of Galileo's students at Padua from 1592 on were young noblemen destined for military careers. He tutored them privately in military architecture and fortification, and that work led him to improve on and consolidate Taraglia 's two instruments. First he remarked that it is not without peril to stand in front of a cannon, exposed to enemy fire, while adjusting the elevation. For that reason and others it would be best to gauge the elevation near the breech, which could be done by putting the ends of the arms of the instrument on top of the

Galileo and the First Mechanical Computing Device

9

barrel and reading points of elevation from the center of the quadrant rather than from one end. Galileo made the arms of the instrument equal in length. Since a cannon thickens toward the breech, a compensation had to be provided by somewhat lengthening the forward arm of the instrument. For this adjustment Galileo fitted his instrument with a "movable foot" mounted on a cursor and held in place by a setscrew. Next Galileo added graduations of the quadrant so that they extended for 90 degrees. The change made the instrument useful for astronomical sightings during long marches. He also added a clinometer scale reading in units of I oo percent of gradient; it enabled military architects to determine the slope of escarpments. That scale, giving the units of vertical drop per unit of horizontal advance, in tum suggested a simplification in the determination of height and distance by sighting. Galileo divided his quadrant into 200 equal parts, reading from zero at each end to 100 in the center, that is, at 45 degrees. Because such units are in modem terms I percent of grade, the scale dispensed with certain common calculations and made others a matter of simple mental arithmetic. The resulting instrument not only eliminated the need for two separate instruments for gunners but also was of value for civilian mappers and surveyors. It was probably in 1595 that Galileo wrote a brief untitled treatise on the uses of the combined instrument. The last part, which concerns triangulation, was copied at the end of one early manuscript of his first instructions ( 1597) for the use of the sector. Later additions to this appendix on triangulation show that it was composed before Galileo had perfected his calculating sector. The stage that Galileo's instrument had reached in 1595 or 1596 is illustrated in a drawing made by a German student and inserted in the recently discovered manuscript (Plate 6). The quadrant was drawn exactly the same size as that of Galileo's own sector, which is now preserved in Florence. The cursor and the setscrew for the "movable foot" are also shown, along with a bracket and a universal joint to mount the instrument for surveying work. This is the only known drawing of these accessories, but Galileo's account books show that he manufactured to order a nocella (universal joint) to fit the instrument on a tripod. Tartaglia's quadrant made an integral part of his elevation gauge. As long as the instrument was used only by gunners its large size and awkward shape did not matter. When Galileo incorporated the scale for triangulation, however, making the instrument useful also for surveyors, he detached the quadrant and hinged the arms so that the instrument could be carried more easily. This modification automatically created a sector. It was a natural step then to mark near its inside edges the two scales of Guidobaldo ' s sector, since both were helpful to mappers as well as to military architects.

1o

Instruments

No surviving example of Galileo's sector of 1597 is known, but the instrument can be easily reconstructed from his earliest instructions on how to use it. I have drawn a schematic diagram of the instrument with the scales labeled; the drawing is the basis for the illustration in Plates 7 and 8 and will be useful in comparing Galileo's earlier version with his later one. The two scales employed by Guidobaldo (G-1 for constructing a regular polygon and G-2 for obtaining equal sections of a line) were on the 1597 instrument but not on later models. Galileo added a scale, which I have designated V, that was even more useful than G-1 to military architects because it facilitated the construction of a given regular polygon on a line of given length. Fortifications often had parts of regular polygons in their design, and the length of one side was likely to be dictated by some feature of the terrain or by part of an older fort. Between V and G-1 Galileo placed another line (V/), which he called the "tetragonic." This scale gives directly, for any regular polygon, the side of any other such polygon having an equal area. The same lines served for giving the approximate quadrature of any circle and for comparing areas in square measure. (By the reduction of any linear figure to triangles its total area could be equated to a square.) Scales V and VI mark the emergence of a specialized mechanical computing device, still confined to basically geometric comparisons. Galileo's next step was to solve mechanically an important practical problem in artillery for which no solution had been known. It involved arithmetic, geometry and physics. Even when he had done so, however, his sector was still incapable of solving simple rule-of-three proportionality problems encountered in daily life by ordinary men. The idea of a single instrument for all purposes had not yet occurred to him. We shall see how the first general mechanical calculator evolved in his hands. As Galileo remarked, the same names were often applied in different places to weights and measures when the quantities described were quite different. A captain of artillery must know how to charge a gun of any bore for a ball of any material without relying on anything but his own precise knowledge of the proper charge for a specific bore and a ball of known material. The knowledge was necessary because captains were often called suddenly to foreign places and because when enemy guns were captured, the captains needed to know how to tum them against their former owners. Wasteful and dangerous errors could result from mistaken units of measure even when charging data were marked on unfamiliar guns. The only sure safeguard against burst cannon, killed or injured gunners and wasted ranging shots was to be able to swiftly solve in the field the problem Galileo called "making the caliber." The solution of the problem required scales I and II, which respectively gave

Galileo and the First Mechanical Computing Device

11

POLYGON IN CIRCLE shows a typical use of Guidobaldo's instrument and the later sector. Here the problem is to inscribe a pentagon with sides of equal length in a circle that has as its diameter the distance between the two endpoints of the instrument. With that distance established (top) one measures the distance (dashed line) between the 5 on one scale and the 5 on the opposite scale. That distance is the length of one side of the desired pentagon. The inscribed polygon (bottom) also divides the circle into equal arcs of 72 degrees.

12

Instruments

the relative volumes of equal weights of various metals and stones and the spherical volume relations corresponding to equal increments of radius. With their aid even a gunner without mathematical training could solve any problem of calibration in a few seconds. In 1597 algebra had not yet been applied to geometry, let alone to physics, so that Galileo himself could not have written a practical formula for the problem. Even if he could have, it would have been of no use to gunners or even to most captains because of their limited mathematical knowledge. Galileo's mechanical solution of this problem inspired him to exploit the sector for other problems. Its priority in his mind is clear from the fact that in every version of his instructions before 1600 the first problem taken up was calibration. In 1600, however, he rewrote his instructions to begin with a different scale (IV), which provided matching equal linear divisions, as if on two rulers. This scale had not even appeared on the 1597 model of the sector; it was added about a year later. By 1600 Galileo had found so many uses for the sector that thereafter the problem of calibration was not explained until his 20th chapter. The manuscripts show how mistaken historians have been in relying on common sense and the form of Galileo's final printed instruction manual to reveal his methods of invention. Common sense suggested that he started with something simple (the scale of equal divisions as on Hood's sector) and then saw how it could be applied to more complex problems, just as he later arranged his instructions. Actually he had previously devised a mechanical means of solving a problem that was not susceptible to the mathematics of his time: the calibration problem, which was a function of two independent variables. Only later did it occur to him that simple proportionality problems could also be solved mechanically. Why should a professor of mathematics be concerned about problems that gave him and his students no trouble? In time, however, Galileo came to care about such problems for the benefit of ordinary people, who could not do square roots and even had trouble with multiplication and division. Mathematically untrained gunners had been enabled to do precision work; now, for the first time, mathematically untrained civilians could learn even such occupations as surveying. Insertions made in the appendix on triangulation after I 600 reflect the evolution of Galileo's interest in dealing mechanically with simple mathematical problems. Originally (in I 595 or I 596) he gave detailed arithmetical calculations for triangulations. In the manuscript of 1605 and in his book each such example was followed by a passage showing how the same answer could be approximated quickly on the sector without using arithmetic. He introduced the new passages with such phrases as "But for those who cannot manage arithmetical calculations ... " or "If we wish to avoid tedious calculations ... "

Galileo and the First Mechanical Computing Device

I3

Evidence from affidavits filed in 16o7, from the dating of manuscripts preserved in Milan and from the form and content of the two other manuscripts of Galileo's instructions indicates that the final model of his sector was designed within a year or so of the 1597 instrument. He made two fundamental additions and some minor changes. One of the major additions resulted from the fact that Galileo regarded his instrument as being imperfect as long as it did not enable him to determine the area of every figure bounded by straight lines and circular arcs in any combination. For this problem he devised a scale that would give the area of any segment of a circle. It is the scale I have designated as VII. It requires two sets of numbers, one on each side of the line. Hence for legibility it had to be placed near the outer edges, where scale V had been. Since V was directly useful in fortification, Galileo abandoned G-1 and put V in its place, adding an instruction that showed how the inscribed polygons of G-1 could easily be obtained from V instead. The other major addition in 1598 replaced scale G-2 with the simple scale (/V) of equal divisions, starting from the pivot. Identical rulers hinged at a common point make possible the immediate solution of all problems of proportionality, which is equivalent to solving all linear equations. Together with scales III and II, which in effect extended this capability to certain quadratic and cubic equations, scale JV turned Galileo's final sector into an instrument that could solve mechanically many algebraic problems. In fact, it conquered all the practical mathematical problems of the time. By 1606, when Galileo printed his book on the "geometric and military compass," about 100 of the instruments had been sold to students or given to friends and dignitaries in Italy and abroad. Galileo's account books show that at least 20 were in Germany, Austria, France and Poland. His book was printed in Italian for the benefit of the general reader. Within months a Paduan student plagiarized the book in Latin and implied that Galileo had taken the invention from him. Since Galileo had dedicated his book to Prince Cosimo de' Medici, with whom he was seeking employment, this was a serious matter, and he took legal action against the plagiarist. A proper Latin translation was published in Germany in 161 3, by which time sectors were in use all over Europe. A rash of other claims appeared in other languages and other countries up to the 163o's. Meanwhile sectors were made with a great variety of combinations of scales. For many years I have collected copies of the books and examples of the instruments. Apart from Hood's independent invention of a calculating sector of less general applicability, only one other claim appears to me to have any merit. That is the assertion in 1610 by the Belgian mathematician Michel Coignet, who corresponded with both Guidobaldo and Galileo in the 158o's, that he had had since that time his "pantometric scale," which consisted of

14

Instruments

graduated lines similar to those on the later sector. The lines were engraved on a metal plate without moving parts. Coignet's instrument was used, together with an ordinary pair of dividers, to solve similar problems by construction and measurement, not by mechanical computation as Galileo's was. Apparently in 161 2 Coignet transferred his pantometric scales to a pair of sectors. This modification was the basis, after his death in 1623, for a claim by his French editor that Coignel had invented the sector in the 158o's. Since Coignet himself made no protest against the books of Galileo and other rivals, it seems doubtful that he meant to claim the calculating sector when he asserted his priority in the "pantometric rule." Coignet's pantometric instrument in its original form was in fact more like a set of portable tables than a mechanical computing device, at least in the usual sense of the term. Mechanical computers characteristically do several things: they abbreviate lengthy calculations, enable trained mathematicians to solve problems otherwise beyond their powers and put at the disposal of the mathematically untrained the power of calculating methods otherwise unavailable to them. It is significant that Galileo's sector did all three things from its inception, as did the later slide rule. Hood's sector and the calculating machine that evolved in the 17th century from the work of Pascal and Leibniz seem to me to be somewhat different in that they were limited to problems that could already be solved by older methods (although less conveniently). Galileo attacked mechanically a problem he could not solve in any other way. In the course of doing so he came to see that mechanical means could be made available for solving all the practical mathematical problems of the day, much as our practical problems in mathematics are now solved by electronic devices.

2

Tartaglia's Squadra and Galileo's Compasso

Shortly after Galileo became professor of mathematics at the University of Padua he began giving private instruction in addition to his public lectures. Records of the courses he offered privately do not exist for the earliest years, but military architecture and fortification appear to have been among the first. In publishing Galileo's syllabuses on those subjects, Antonio Favaro remarked that their origin appears to date from 1593, the same year to which the earliest version of Galileo's treatise on mechanics also belongs. His Mechanics exists in three successive forms, having probably been completed about 1600. Even more richly documented is the evolution of Galileo's development of the instrument that became the subject of his first acknowledged book, The Operations of the Geometric and Military Compass, printed at Padua in 16o6. Galileo began his work on that instrument in 1597, made improvements on it during the academic year 1598--99, and by 1601 had brought his instructions for its use to virtually the final form in which they were published in 1606. About one hundred of the instruments had then been made for him by Marcantonio Mazzoleni since 1599, when Galileo first employed him as his instrument maker. More than sixty of these are identifiable from Galileo's incomplete account books, at least twenty having been sold to foreign students or sent to distinguished men abroad, mainly in Germany, Austria, Poland, and France. Hence by the time Galileo's book was printed his "military compass" was very widely known (Plate 9). Claims to its invention proliferated after 1607, beginning with an impudent plagiarism by Baldessar Capra, a student at Padua (who in 1597 had been but a youth of seventeen). Affidavits secured in Galileo 's Reprinted from Annali dell' lstituto di storia de/la scienze di Firenze 2 ( 1977). 35-54, by permission.

16

Instruments

legal action against Capra, which establish the main dates, are supported by many other kinds of evidence. That the instruction manual had reached virtually its final form by 1601 is shown by a manuscript copy (herein designated/) owned by Gian Vincenzio Pinelli, who died at Padua in that year. Pinelli also owned four other manuscript copies of instructions for a somewhat different form of Galileo's instrument (designated a, b, c, and d by Favaro), preserved with fat the Biblioteca Ambrosiana at Milan. Those five documents were the only contemporary copies known to Favaro, who published in the second volume of Galileo's Opere the contents ofa and b, the parts of c and d relating to scales engraved on the arms of the instrument, and three short sections from f Favaro established beyond doubt the chronological order of these documents but did not attempt to assign dates to them within the period from 1597 to I 60 I. Emil Wohlwill, one of the few historians to recognize the true importance of Galileo's instrument in the history of science, believed it impossible to establish its precise evolution. 1 It is true that from the published material alone it would be impossible to reconstruct in detail Galileo's steps in inventing and developing this remarkable instrument, which was capable of solving mechanically any mathematical problem likely to arise in any useful occupation of the time. Yet from the five manuscripts owned by Pinelli that would have been possible, though very difficult. The essential clues are found in two other contemporary copies, not known to Favaro. Essential to the reconstruction is the dating of a supplemental treatise on the measurement of heights and distances by sighting and triangulation, not present inf and not published by Favaro in the form given in c and d. Counting the printed text of 1606, we can now distinguish five versions each of the instruction manual and the supplemental treatise, from which can be reconstructed not only the order and dates of composition, but also the manner in which Galileo was led to devise and develop his "military compass." One of the additional texts, here designated e, was found in the Rocco-Bauer collection at California Institute of Technology and a description of it was published in I 960.2 That manuscript follows the same order of presentation as the first four Pinelli copies, but had been greatly expanded and applied to the instrument as altered during 1598-99. At the end it contains the supplemental treatise omitted from f, which presented the instruction manual in a totally different order. At the time I described e, misunderstanding Favaro's brief allusions to the material he left unpublished, I thought that e contained the earliest known version of the supplemental treatise. In fact it contains a slightly expanded version of that treatise as found earlier in c and d, which in tum had been somewhat extended from the first text, written before 1597 for a still earlier instrument devised by Galileo.

Tartaglia's Squadra and Galileo's Compasso

17

The key to the entire reconstruction is found in still another manuscript, to be called g, which has recently come into my possession. Since many of the essential clues to Galileo's work in this matter are found in oddities of the manuscripts as such, I shall describe this briefly. The volume consists of about 300 pages, all uniformly watermarked except for one inserted drawing to be reproduced below. Most of the works contained are in Italian and are scribal copies of writings of Ostilio Ricci, Galileo's mathematics teacher. One, in the hand of the principal (or only) scribe, is a copy of the treatise of Levinus Hulsius on the proportional compass as published in German in 1604 (Plate 10). None of the items in the volume were paginated by the scribe except the Galilean treatise on measuring by sight, which begins on a fresh right-hand page after the unpaginated manual of instructions for use of the "military compass," and of which the final leaf is now missing. As to the manual of instructions, it is virtually word-for-word the same as f, though the year in which g was copied was almost certainly 1605 - at least four (and probably six) years after the composition off In what follows it is my principal purpose only to give an account of Galileo's own work on the "military compass." A proper description of other instruments that have been believed by others to have influenced that work, and of various conflicting claims to priority, must be deferred to another occasion. In place of that I shall speak briefly here of three relatively neglected instruments that are of prime importance in understanding the origin of Galileo's work in creating the first modem mechanical computing device. II

It is advisable to start with the establishment of an unambiguous terminology, lack of which has caused much confusion in discussions of the genesis of the instrument in question. The proper name for this in English is the "sector," given in I 598 by Thomas Hood to a pair of flat rules hinged stiffly at one end and bearing identical scales engraved on the two arms, different on the two faces (Plate 11). Hood's sector was the first mechanical calculating device of general practical use to be published since the abacus of remote antiquity. It had three scales and was fitted with removable sights and a graduated quadrant, plumb line, and accessory graduated arm. Its principal use was in surveying and mapping, though it had an earlier English ancestor designed for quite different purposes. Hood's principal scale was one of equal linear divisions from pivot to end of either arm. On the other face he provided a scale which gave the side of various regular polygons inscribed in a circle of diameter equal to the separation of the ends, and another which gave the side of a square having an area which was an integral multiple of the area of a unit square. The enormous value

18

Instruments

of Hood's sector for speedy mechanical approximation to a wide variety of the commonest practical mathematical problems is obvious, and he explained these at great length in his book. 3 In the same year (1598) there appeared at Venice a book on mathematical instruments written by G.P. Gallucci in which was illustrated a different kind of sector, having in common with Hood's only the scales for construction of regular polygons.4 I shall call this "Guidobaldo's sector," for though Gallucci did not name the inventor, other evidence points to Guidobaldo del Monte, Galileo's friend and patron. It had but two scales, one on either face, the second (similar in appearance to that for the sides of regular polygons) being designed to permit the division of a line into equal segments, just as the first was used to divide a circle into equal arcs (Plate 3). Guidobaldo 's sector was in no significant sense a calculating instrument; it simply gave direct mechanical solutions to two very common problems in drafting, designing, and instrument construction. Since the 1560' s those problems had been dealt with by the reduction compass of Fabricio Mordente, consisting at first of dividers with a pair of movable points at right angles to the plane of the instrument, and later of fixed and movable points all at right angles to the plane of the legs. Soon afterwards the proportional compass was devised, probably by Federico Commandino (see Plate 11 ), having four points in the plane of the instrument and a movable pivot passing through slots in the two legs. Both instruments were expensive and required frequent settings of points or pivot. Guidobaldo's sector was a cheaper and speedier means of solving the two basic problems, sufficiently accurate for most practical purposes. It is evident that when Gallucci published his book at Venice in 1598, no calculating sector of the English type was widely in use in Italy, since its value as a mathematical instrument is outstanding and Gallucci would hardly have omitted it if he had known of it. The simple scale of equal divisions could indeed have been added to his illustration of Guidobaldo's sector without interfering with its two original scales. Though the advantages of such a device seem selfevident to us, I have a dated Italian sector of the Guidobaldo type made in 1637, fully thirty years after publication of the Galilean sector and still without the scale of equal divisions. These facts cast grave doubt on the usual statement that Galileo simply added to a common mechanical calculating device a number of more complex scales, his object being only to augment his income by its manufacture and sale. We shall see how different was his actual procedure, while his account books show that he sold his instrument nearly at cost. 5 He made a good deal of income from teaching its use, but that can hardly have been in his mind before he invented it.

Tartaglia's Squadra and Galileo's Compasso

19

III

The pioneer application of mathematics to artillery practice was Niccolo Tartaglia 's Nova Scientia published at Venice in 1537. Two instruments introduced in that book are of prime importance to our story. One of these quickly passed into universal use; this was a gunners' elevation gauge for cannons (Plate 4). It consisted of a rigid right angle having one leg much longer than the other and a quadrant arc graduated into twelve equal parts, called "points." The longer leg was partly thrust into the mouth of the cannon, and a plumb line suspended from the vertex showed the elevation; a dead level shot was called "point blank," and one of 45° (for example) was said to be "at six points." In one of the Galileo syllabuses on military architecture and fortification there is a passing reference to this device, which he called the squadra de i Bombardiere, or gunners' square.6 Tartaglia discussed at some length the determination of the height and distance of a remote target, for which he introduced another instrument (Plate 5). This again consisted of a rigid right angle, but with equal arms, along one of which sights were provided. The square was completed by another right angle, of which each arm (ombra, or "shadow") was divided into twelve parts representing increments of 3¾ degrees at the vertex. Rules for use of the instrument were given separately, according as the plumb line from the vertex fell in one or the other ombra; that is, according as the sighting angle was more or less than 45°. This system had certain practical advantages, but Tartaglia's device had many rivals by Galileo's time, introduced by numerous writers on topographical surveying. Among these was the archimetro of Ostillio Ricci, Galileo's teacher of mathematics, having a pair of hinged sighting rods on a straight staff, one fixed and one moving on a cursor. This had the advantage of folding into a compact form when not in use, among other things. Just what instruction Galileo gave to young men destined for military careers in addition to his course on fortification is not recorded. It is evident, however, that he interested himself in problems of artillery captains, as Tartaglia had done and as was doubtless desired by some of Galileo's pupils. In any event he remarked that it is not without peril to stand in front of a cannon, exposed to enemy fire, while adjusting its elevation. For that and other reasons this had better be done near the breech, where the gunner ordinarily stood. This could be accomplished by making the legs ofTartaglia's elevation gauge equal and placing its feet atop the barrel of the gun, reading the points of elevation from the center of the quadrant rather than from one end. But since the bore is not exactly parallel to the barrel, which becomes thicker from mouth to breech, a "movable foot" must be provided to lengthen the forward leg appropriately for the given cannon. 7

20

Instruments

To this modified instrument, Galileo added three other quadrant scales. The first of these was a simple graduation into degrees, useful for astronomical sightings during long marches. The second was a clinometer scale of no little interest. Its purpose was the measurement of slope of an escarpment by military architects, the traditional expression of which slope had always been in terms of the "parts" of horizontal advance along a square equal in height to the escarpment. Galileo provided a projection at one end of the quadrant from which a plumb line was suspended; later on, a hole was drilled in one arm for this. The other arm was held against the slope, and ten slanting lines engraved on the quadrant permitted comparison with the string. These lines were numbered from I to Io, representing what we call the tangent of the base angle; thus I was 45°, 2 meant a rise of 2 units per unit of horizontal advance, and so on. The traditional expression of slope was not thereby changed, but the units employed were of great advantage in mental arithmetic. In effect, Galileo's units were what we should call "wo% grade." Probably it was this scale for military architects that suggested to Galileo, in conjunction with Tartaglia's use of two ombre for triangulation problems, the final scale placed at the outside edge of the quadrant. This divided the 90° arc into 200 equal parts, numbered from zero at either end to I oo in the middle, at 45°. This preserved the practical advantages of Tartaglia's system while greatly simplifying the arithmetic of triangulation, both as against Tartaglia's and as against the trigonometric system favored by trained mathematicians. Even today, military transits are graduated in mils rather than degrees for similar reasons. Tartaglia's two instruments had now been combined in one, with useful modifications and with added scales of use not only to military men but also to civil topographical surveyors. For the latter, however, it was inconvenient to carry a large heavy rigid square in the field (not a problem for cannoneers). Talcing his clue perhaps from Ricci's archimetro, Galileo made his squadra a folding instrument, detaching the quadrant and hinging the arms. In the recently discovered manuscript, g, there is a drawing made by (or for) a German student which illustrates Galileo's instrument at this stage and also shows certain accessories named in his account books but not elsewhere described or illustrated (Plate 6). The original drawing was probably made from an actual instrument, since the quadrant was drawn in exactly the size of that on Galileo's own "military compass" preserved in the Science Museum at Florence. It was not drawn from one of Mazzoleni 's manufacture, however, for the design of the cursor for the "movable foot" is different, as is also the shape of the instrument when closed, shown at lower left. This terminates in sharp points, whereas the "military compass" ends in smooth scroll work; moreover, the hinge is of different design,

Tartaglia's Squadra and Galileo's Compasso

21

pennitting the closed anns to overlap instead of meeting as in the standard later model. Both the points and the overlap were useful, the fonner for the gunners' elevation gauge and for mappers in the field and the latter for maximum compactness in carrying, but both were useless for the sector used as a mechanical calculator. The detail with which the quadrant was drawn makes it unlikely that the same artist would have omitted to indicate presence on the anns of several scales, if he had seen any. No drawings accompany any other copy of Galileo's instructions, and no illustration was provided in his 1606 book, owners of which had the instrument itself, for Galileo was not anxious to show others how to make it. Hence it is probable that this drawing was made from a Paduan instrument of Galileo's which had preceded his calculating sector. Of special interest are the bracket and universal joint for mounting the instrument for use in surveying. Galileo's account books record the joint (nocella or pal/a) which was fitted with a hollow conical member to slip onto the head of a tripod (piede, also listed in Galileo's account books) provided with steel points. 8 It was for such an instrument that Galileo first composed his treatise on the detennination of heights and distances by sighting and triangulation which later became an appendix to his manual of instructions for use in the calculating sector. For convenience of reference I shall regard this treatise in its original fonn as dating from 1 596; its text can be reconstructed from the later versions in the manner set forth in section V, below. Here it is sufficient to remark that the 1 596 treatise made no reference whatever to any scale that appeared on the later sector, all the calculations for triangulation being worked out there in detailed numerical examples. This implies that when Galileo first composed the treatise he was not yet aware of any mechanical device by which the results of calculations could be quickly approximated, which makes it certain that he did not have in his hands the instrument that was destined to play that role in the later versions. IV

Having modified Tartaglia's military instruments, combined and in folding fonn, Galileo's new squadra was the same shape as Guidobaldo's sector, with which he was undoubtedly familiar some time before its appearance in Gallucci 's book of 1598. Since Guidobaldo's two scales were very useful both to military architects and to surveyors and mappers, it was natural for Galileo to engrave them near the inner edges of the two faces of his squadra. That they were in those positions on his I 597 "military compass" is clear from documents a, b, c, and d. Both those scales, however, are lacking from Galileo's revised

22

Instruments

instrument of 1598-99, as shown by documents e,f, g, and the printed text of 1606. 9 The 1597 sector can in fact be completely reconstructed from the instructions in the first four documents, at least as to the positions and characters of the scales. This has been done here; the scales are labeled in a manner to facilitate comparison with the later model represented schematically in section VI, below (Plates 7 and 8). The two Guidobaldo scales are designated G- 1 and G-2 .

The probable order in which further scales had been added by Galileo in 1597 is as follows. For military architects it was more convenient to determine the construction of a regular polygon on a fixed line than to inscribe a polygon in a given circle. This was because parts of polygons were much used in fortification, and the length of a side was often fixed by a feature of natural terrain or by part of an existing fort. Scale V gives this construction directly. Between V and G- 1 is YI, which gives for any regular polygon any other of equal area, permitting the reduction of such figures to squares and hence the direct comparison of areas. Since a circle may be considered as a many-sided polygon, VI gives also the approximate quadrature of the circle. And since any rectilinear figure, however irregular, may be reduced to a collection of triangles, VI permitted the determination of the area of any such figure. Thus with the creation of scale VI the 1597 sector became a true mechanical calculating instrument, lacking the versatility of Hood's sector but capable of giving swift solutions to many problems more complex than those which the latter was designed to handle directly. The manipulation of area problems probably put Galileo to considering a certain problem relating to volumes that was of practical concern to artillery captains. As he pointed out in the opening part of his first instruction, which was devoted to this problem, the same names for weights and measures were applied in many places to different physical quantities. This practice created a hazard in properly charging a cannon with powder when an artilleryman was called to defend a foreign city, or when a foreign piece was captured and should be turned at once against the enemy. To avoid burst cannons, injured gunners, shortfalls, or waste of time and materials in ranging shots, the only sure remedy was to be able to calculate for any bore and any material of ball the charge equivalent to that known to be proper for some particular bore and material. No general mathematical solution of this problem of "caliber" (colibro or calibro, from qua libra, "by what weight") was possible at the time, forty years before algebra was applied even to geometry, let alone to physical expressions. Moreover, algebraic notations were still quite clumsy in 1597; few people dealt easily with cubic equations, and a table of constants would have to accompany any formula, making it cumbersome to apply in the field. Galileo solved the problem by scale I, which gave the relative diameters of balls of the same

Tartaglia's Squadra and Galileo's Compasso

23

weight but different material, together with scale II, which gave the volume relations for equal increments of diameter. By these scales, following simple instructions, not only captains of artillery but ordinary gunners were enabled to solve any problem of caliber quickly in the field . Next to the scale of volume relations Galileo placed scale III, giving the area relations for equal increments of diameter (or side of square). Probably, adopting the same analogy, he would have followed this with the scale of linear divisions, had G-2 not already been put along the inner edge for use in dividing lines into given numbers of equal parts. In a way, G-2 was simpler and more direct for that purpose, requiring one less operation than by using the scale of equal divisions as on Galileo's later sector. Absence of this scale prevented Galileo's 1597 model from becoming a universal mechanical calculating device; yet it is less surprising than it may seem at first glance that he neglected to include a scale for the mechanical solution of common rule-of-three problems. As a professor of mathematics, Galileo was accustomed to making the necessary arithmetical calculations, as were also his university students. It was for this instrument with its original seven scales that Galileo wrote his first manual of instructions, which commenced with the problem of caliber and proceeded in order of the physical arrangement of the scales on either face (rather than in the order of complexity). Copies of the first manual constitute documents a and b. In b there are also symbols in the margins to indicate the places for insertions written later, in 1597 or early 1598; for convenience of reference I assign b to the latter year. Document c includes the insertions in the places indicated in b, without further changes. In a second hand, c was continued to include the treatise on measurement by sight written in 1596 for the squadra, to which a short section was then added in still another hand. Document dis a copy of c, written in a single hand. As mentioned earlier, e was written for the revised sector, late in 1598 or early in 1599, much expanded from c and d but still arranged in the original order. Ordering from simple to complex operations as in the printed text of 1606 was first adopted inf, late in 1599, and that text was repeated almost verbatim in g, copied in 1605. V

The treatise on measuring by sight as it appeared in c (except for the final section added to c in another hand) had been composed before Galileo began work on his sector; it applied to the quite different squadra already described above. For convenience of reference I shall regard this treatise as composed in 1596, though it may have been earlier. The topic was a customary one in the instruction of students destined for military careers, and Galileo's first treatment of it differed from others before it only in its adaptation to a particular form of

24

Instruments

instrument devised by him. Evidence that it had preceded Galileo's work on the sector is quite compelling, as will be seen from the following and from the later discussion of document g. Documents a and b, and the part of c written in a neat scribal hand, all ended with the unfinished sentence Restafinalmente I' estrema circonferenza divisa in parti 200 .. . If this had happened only in a single manuscript in Galileo's own hand, it might be explained by supposing that he had simply tired of writing and intended at another time to compose a description of this quadrant scale and its uses. Such an explanation is hardly appropriate, however, for three different scribal copies made over a period of several months at least. It is much more reasonable to suppose that material beginning with the above words had been composed previously and was already in circulation, the above words being intended to indicate the place to insert it. That supposition further explains several other facts about the existing documents. The second scribe of c continued uninterruptedly the above sentence with the words ... quale euna sea/a per misurare le altezze ... and so on, evidently copying from the same original as that from which the first few words had previously been transcribed. Several pages of c are occupied by the balance of the 1596 treatise with its many diagrams and calculations, as outlined in section VII below. Favaro did not publish the text of the treatise as it appeared inc and d, deeming the subject of little intrinsic interest and the text as essentially similar to that of 1606 except for various additions of no great importance. The result was that one highly significant fact with regard to the reconstruction of Galileo's work was lost to view; namely, that as originally written, the treatise on measuring by sight included no reference of any kind to any scales on the sector, but dealt only with the outside scale on the accessory quadrant arc. References to scale IV were added in g and in the printed text of 16o6. Until revision of the sector in 1598---99, the treatise on measuring by sight was not an integral part of the instructions for use of the sector, but remained on optional addition to it. The original 1596 treatise probably comprised essentially the following material:

Opere 2, p. 357, 412 417 418 419 420

line 16 2 13 23 32

top. 358, 415 417 418 420 422

line 30 2 30 33 9 5

The various revisions of this material down to the printed text of 1606 will be reconstructed in section VII below.

Tartaglia's Squadra and Galileo's Compasso

25

VI

Galileo's 1597 sector was a very useful mechanical calculating device, but he was not entirely satisfied with it. His reason was that it was limited in application to the general problem of determination and comparison of areas, serving only for figures bounded by straight lines or for complete circles. Galileo wished to extend its use to areas bounded by straight lines and circular arcs in any combination whatever; that is, to all ruler-and-compass constructions. This he succeeded in doing, late in 1598 or early in 1599, by devising scale VII for determination of the area of any circular segment (Plate 8). Two sets of numbers were placed along the line of scale VII, one on either side, for the half-chord and for the altitude of the segment. Hence for legibility of both sets of numbers it was necessary to place scale VII near the outer edges of the arms, where scale V had been placed in 1597. Since the latter scale was more useful to military architects than was G-1, Galileo eliminated Guidobaldo's scale and moved scale V to its place along the inner edges, adding to his instructions a procedure by which scale V could be used to construct inscribed polygons as easily (though not quite as naturally) as G-1. At the same time G-2 was eliminated in favor of scale IV with its equal linear divisions from pivot to ends of arms. This converted Galileo's sector from a special-purpose mechanical calculator into a universal instrument capable of solving mechanically any mathematical problem likely to arise in useful occupations. Galileo at once saw the applicability of scale IV to arithmetical as well as geometrical problems, and in writing the expanded instructions of e, early in 1599, his discussion of these "Arithmetical lines" became the longest section of all. At this time he also began to insert section headings in his manual and assigned names to the particular scales on his sector. Affidavits of Jacques Badovere and Marcantonio Mazzoleni leave no doubt that the period of revising the 1597 sector was the academic year 1598-99. 10 No further changes were made in the instrument except for minor extensions of numbering for two scales before July 1599. A drastic revision of the order of presentation in the manual of instructions, however, was made in/, which was probably given to Pinelli in October I 599 together with one of the Mazzo Jeni sectors, to replace the 1 597 model corresponding to the earlier instructions in his possession. 11 Most of the contents off were already in e, but inf these were presented in the order used later in the printed text; that is, from the simplest scale (IV) to the more complex scales on that face, and then from simple to complex on the other face. All sections were also given headings and numbers; inf (and g) these were called "chapters," while in the later printed text they are called "operations."

26

Instruments

Except for one transposition the order remained unchanged, though in 1606 four new sections were added, two sections were entirely rewritten, and various minor changes and additions were introduced. The treatise on measuring by sight was entirely omitted from f, because in revising the order of the instruction manual Galileo decided to restore to the treatise its earlier optional status. From the foregoing discussion it is apparent that when he composed a, b, c, and d, Galileo did not consider the treatise as of necessary concern to owners of the sector, and in fact at least some examples of that instrument were made without the quadrant, one of which was for Paolo Sarpi. After I 599, however, military men and civilian surveyors who wanted the squadra might be very much interested in the sector with scale IV, especially those who were not proficient in arithmetical calculations. Hence the independence of the two sets of instructions was no longer complete, as it had been at first, and it was desirable to have them either separately or jointly available, according to the wishes of purchasers of the corresponding instruments. This fact is crucial to the story of Galileo 's work as reflected in the extant documents. So far as the manual for use of the sector without the quadrant is concerned, the text of g is virtually word-for-word that off The supplemental treatise on measuring by sight and triangulation, however, differs from that of c and d (the only places it appeared among the Pinelli copies) in a very striking way. The earlier text appears again in g with virtually no change. But in g seven insertions were made, of which six are all of one kind, and the scribe of g preserved in the margins the conventional symbols used by Galileo to indicate in the lost original early text the places for these insertions, 12 originally written on separate sheets as in the case of earlier additions to b. Each of the six insertions follows the detailed arithmetical calculation of a triangulation given as an example in the earlier text, and each shows the operations on the revised sector by which the same result can be quickly approximated mechanically. The six insertions begin with the following words: But since one may not be able to manage the numbers, we can find the same accounting on the instrument with the aid of the Arithmetical lines, and the rule will be ... 2 / And here also to escape the operations with numbers we can find our accounting on the Arithmetical lines ... 3 / Here likewise we can dodge the numerical accounting ... 4 / And anyone who wants to escape such operations with numbers ... 5 I The numerical operation can be escaped in this way ... 6 / In order that the present operation may be resolved without numbers ... 1/

Tartaglia's Squadra and Galileo's Compasso

27

In the printed text of 16o6, two of these explanations were omitted and the wording of the others was modified in such a way that the striking effect of the above insertions was lost. Hence Favaro did not recognize the significance of the absence from the text of the treatise in c and d of any reference to mechanical approximations in lieu of laborious arithmetical calculations. Since Favaro did not publish the earlier text, others had no way of knowing that the treatise on measuring by sight was composed before Galileo had any mechanical means of effecting mathematical computations. The 1597 sector had no scale applicable to simple proportionality problems. It was only with the substitution of scale IV for G-2 that this could be effected. Even with that change, Galileo did not at once note the versatility of the operations thus made possible; for in e, though many uses of scale IV were added, the above insertions were not introduced into the supplemental treatise. It is probable that the crucial insertions were added to the treatise late in 1 599, when the manual of instructions for the sector was entirely rewritten in the order of/, beginning with scale IV and emphasizing from the start the manifold uses of the instrument for numerical as well as geometrical problems. The original opening (the problem of calibration) was then moved to chapters 17-21 , and in the printed text became operations 21-25. At the same time, late in 1599, Galileo explained the mechanical extraction of square and cube roots and determination of mean proportionals by the use of scales II and Ill, which were already present but had previously been applied only to geometrical problems. Since all the mathematical problems arising in useful occupations came down ultimately to root extraction and the rule of three, the revised Galilean sector put into the hands of men untrained in mathematics the means of solving quickly and with all the accuracy required in practice virtually every mathematical problem that arose. The later slide rule and PascalLeibnitz digital calculators require for use a certain amount of mathematical knowledge to translate into numbers geometrical problems which the sector solved directly, whence the Galilean sector continued to be much more widely used than either of them almost down to the present century. It is still not hard to find early sectors, usually of brass, or nineteenth-century sectors mass-produced in bone or wood, while early slide rules and calculators are rarely found outside of museums. There is little doubt that Galileo's mechanical calculating device profoundly affected not only the society in which he lived, but also his own attitudes about the pursuit of science and toward mechanics and philosophers. Systematic conquest of everyday problems by correct application of mathematics greatly accelerated technological progress as compared with the previous ages of accumulation of improvements by trial and error, with consequent waste of time and materials. Galileo began his scientific career with writings like De motu in

28

Instruments

which, though disagreeing with some conclusions of Peripatetic physics, he sought in the traditional way to establish principles from which the causes of things could be deduced. As time went on, he became more distrustful of such procedures and more interested in the quest for reasonable certainty in observed phenomena. Good approximations for practical purposes came to interest him more than perfect and all-embracing theoretical solutions of abstract problems of the universe. The "military compass" was partly a result of this changing outlook, and partly a factor in creating Galileo ' s open breach with the philosophers over scientific principles and methods in 1604-05. It is not possible to explain here the close connection, evidenced in the controversy over the new star of 1604, 1 3 but it is perhaps worth mentioning that Galileo's later indifference to Kepler's elliptical planetary orbits is part of the same psychological orientation. The Copernican system was an enormous advance over the Ptolemaic as a reasonable practical approximation to modem astronomy, and that interested Galileo much more than the Keplerian improvements which solved problems of a totally different character. VII

Few readers today will care to retrace Galileo's actual work on the first universal mechanical calculating instrument. Most historians prefer to search for profound philosophical tenets supposedly embedded in Galileo's thought, instead of following its formation in his mere practical activities. For the benefit of those who may wish to follow the evolution of Galileo's 1606 book, however, utilizing material already published as a close approximation, I have provided the following list of presently known documents and their contents as reflected in published sources. DOC.

LOCATION

MANUAL FOR SECTOR

TREATISE

a b

Ambrosiana S.81 Sup. Ambrosiana D.95 Inf. Ambrosiana S.83 Sup., ff. 15(>...174 Ambrosiana S.99 Sup. California Institute of Technology Ambrosiana S.83 Sup., ff. 177-190 Drake Collection, Toronto Printed text, Padua 1606

IA IA [IBI

[I [I-II I, II

C

d

e

I

g

p

1B 1B IC

IIA IIA 11B

1597 1598 1598 1598 1599 16oo (16o5) 16o6

II III Absent IV V

15961 1596--98] 1596, 1598 1598 1599 [IV?] 16oo (?) 16o6

Material enclosed in square brackets is implied, but not actually present; the date in parentheses represents that of copying rather than ·of original composition. Each year shown is the latest that may reasonably be assigned to the document described.

Tartaglia's Squadra and Galileo's Compasso

29

The texts of a and b were published in Opere, 2, pp. 345-358, above the line. With respect to the manual for the sector, the texts of c and d were published in those same pages, material below the line being read into the places indicated. For the text of the treatise in c and d, one may read: p. 412, 417 418 419 420 422

line 2 13 23 32 27

top.415, 417 418 420 422 423

line 5 30 33 9 5 13

The text of e was described in my 196o paper, both as to the manual and the treatise. It should be noted that I was mistaken at that time in thinking that e might present the earliest extant form of the treatise. The erroneous calculation at the end of the treatise introduced in e was repeated in g and was not noticed and corrected until 16o6. As previously mentioned, the treatise is not present inf. Neglecting minor variants, the manual of instructions as it appeared inf (anding) reads as follows. p. 373, 375 378 380 384 385 386 386

line 1 24 3 19 4 I

21

top. 374, 377 379 380 384 385 386 387

391 392

391 393

395 397 400 4o6

395 399 404 406

407

IO

41 I

line 6

16 (to ... FG al/a AB.)

4 31 (Operations VI and VII belong to 1606) 24 25 IO (lines 11 to 20 rewritten in 1606) 14 (Operation XII was Ch. X inf-g; cf. p. 359, line I to p. 360, line 4. Operation XIII was added in 16o6) 17 20 (Operation XVI was Chapter XIV, and Operation XVII was Chapter XIII. Operation XVIII was Chapter XV, for which see p. 360, line 5 to p. 361, line 15) 20 (Operation XX was added in 16o6) 24 7 20 (Operation XXVII was Chapter XXIII, for which seep. 361, lines 16 to 27) 35

30

Instruments

Document f ends here, while g continues with material resembling p. 41 2, line 2 to p. 413, line 9, though worded more like the corresponding part of e than the printed version of 16o6. The text of the treatise in g, probably composed late in 1599 at the time when the manual was completely reordered, is as follows, beginning from p. 412, line 2. Resla finalmente l'eslrema circonferenza divisa in parti 200, quale e una scala per misurare altezze, distanze, et profondita col solo aiulo della vista ... (to p. 415, line 5; lines 6-21 were inserted in 16o6 for lhis reading in g): Ma perche non sapesse maneggiare Ii numeri, potremo sopra lo strumenlo ritrovare ii medesimo conto con l'aiuto delle linee Arithmetiche, et la regola sara ques1a: che prendiamo sempre rettamente [i.e., along the scale] 100, et questo applicheremo transversalmente [i.e., across the separalion of identically numbered points along the 1wo arms] al numero lagliato dal perpendicolo; dipoi, non movendo lo slrumento, prenderemo I oo trasversalmente et lo misuraremo rettamente, et tanla sara l'ahezza cercala, come per essempio ii perpendicolo havesse tagliato al punto 80, preso I oo rettamente et applicala a11'8o trasversalmente, el di nuovo preso 100 trasversalmenle, el misurato rettamente, ci mostrera 125, el tanta e l'altezza cercala. In altra maniera potremo ... (p. 415, line 22 top. 417, line 2; lines 3-12 were as follows in g): E per fuggire qui ancora l'operazione dei numeri, potremo ritrovare ii noslro conto sopra le linee Arithmetiche, come nell ' infrascritto essempio si fara manifesto. Habbia per essempio ii perpendicolo tagliato nella prima statione al punto 50 et nella seconda 80, prendasi rettamenle 50 et si applichi lrasversalmente alla differenza delli due numeri 80 et 50, cioe a 30, ii che fatto, pigliasi trasversalmente l'altro numero, cioe 80, et misuri rettamente, et troveremo 133 in circa, che sara I' altezza cercata. Possiamo in oltre ... (top. 417, line 30; lines 31-37 were as follows in g): Qui parimente polremo sfuggire ii conto numerale, valendosi delle medesime linee Arithmetiche, peroche stando nell medesimo essempio, prenderemo rettamente ii numero tagliato dal perpendicolo, cioe 18, et questo applicaremo trasversalmente sempre al I oo, ii che fatto, prenderemo pur trasversalmente ii numero de i punti, che fu 130, ii quale misurato rettamente ci dara 23 in circa per l'ahezza cercata, come fu per numeri ancora ritrovata. Venendo poi al misurare le distanze, prima mostreremo come si prenda la distanza Ira noi el qualsivoglia altro luogo; come per ... (lop. 419, line 23. The text of p. 418, line 34 top. 419, line 7 differed in g, reading: Puossi in altra maniera misurare una simile distanza, come saria la larghezza di un fiume . Venendo sopra la riva, oaltro luogo eminente si come nell'essempio si vede, nel quale volendo noi misurare la larghezza CB, venendo nel punto A traguarderemo con la costa AF l'estremita B, notando i punti D, E tagliati dal perpendicolo, quali siano v.g. 5; et quante volte questo numero entra in 100, tante

Tartaglia's Squadra and Galileo's Compasso

31

volte diremo l'altezza AC entrare nella larghezza CB . Misurando dunque quanta sia tale altezza AC, et pigliandola 20 volte, haveremo la larghezza cercata CB.) Et chi volesse fuggire tal operatione di numeri, prendendo rettamente dalle linee Arithmetiche 100, et applicandoli trasversalmente al numero dei i punti tagliati dal raggio [della vista], ed di nuovo pigliando too transversalmente, misurandolo poi rettamente, trovera la medesima distanza cercata. (This insertion, which would have come after p. 419, line 23, was omitted in 16o6.) Ma quando non ... (top. 420, line 9). L'operatione numerale si potro fuggire in tal modo. Habbia per essempio tagliato ii raggio della vista nella prima statione al numero 70, et nella seconda al numero 30, et preso lo strumento si aprira sin che le linee Arithmetiche facciano angolo retto, ii che faremo tutta volta che l'intervallo di 100 punti presi rettamente sara applicato trasversalmente da una banda al punto 80 dall 'altra al punto 6o. Et agguistato in tat modo lo strumento, prenderemo trasversalmente l'intervallo tra' I punto 100 el ii punto 70, che fu ii numero tagliato dal raggio nella prima statione, et tale intervalo si misurera rettamente, et lo troveremo essere 122 in circa, ii qual numero salveremo. Dipoi prenderemo sempre 1oo rettamente, et applicheremo trasversalmente alla differenza delli sue numeri tagliati da i raggi, che i questo essempio e 60, et poi piglieremo pur trasversalmente ii numero che si salvo, cioe 122 , et questo misureremo rettamente, el lo lroveremo essere 203 in circa, quanta e la distanza cercata. Segui ta che veggiamo ... (to p. 421, line 9 ). Le presente operatione, accio venga risoluta senza numeri, non ricerca altro se non che nel modo, non della precendente operalione ma nell'altra avanti dichiaralo, ritroviamo separatamente le due distanze A, B et C, A, et tratta l'una dell'altra haveremo quello che si cerca. (Deleted in 1606 printed text.) Ma se volendo ... (to p. 422 , line 5).

It is probable that after p. 422, lines 6-26, found only in the 1606 text, g contained the material from p. 422 , line 27 top. 423, line 27, though the last page is missing and the text breaks off at p. 423, line 6. The material from p. 423, line 28 top. 424, line 9 was certainly written only in 1606 for the printed edition. The material added to the manual of instructions for the Galilean sector in the 1606 final revision can be identified by comparing the text in Opere, 2, with that off and g as outlined above. It is of interest that the final additions were not of more complex uses of the instrument, but of uses for common arithmetical problems such as monetary exchange and compound interest calculations. This bears out the overall pattern of evolution of Galileo 's work on the sector set forth herein; namely, that he did not begin from a mechanical computing device already in use for numerical problems, but rather came to see the wide utility of such a device among persons untrained in mathematical operations after he had created it for the solution of particular practical problems that were difficult or

32

Instruments

impossible of general solution in the mathematical notations which existed at the time. NOTES I

2

3 4 5

6 7 8

9 IO

11 12

I3

Emil Wohl will, Galilei und sein Kampffiir die copernicanische Lehre (Hamburg and Leipzig, 1909), p. 137, n. 2: " ... Wieviel ihm von den Erfindungen seiner Vorgiinger bekannt geworden isl, welche Linien er denen der iilteren Instrumente hinzugefiigt hat, ist mit Sicherheit nicht zu entscheiden gewesen." Stillman Drake, "Galileo Gleanings IX. An Unrecorded Manuscript Copy of Galileo's Use of the Compass," Physis 4, II (1960), pp. 281-290. Thomas Hood, The Making and Use of the Geometrical Instrument, called a SECTOR ... (London, 1598). Gio. Paolo Gallucci, Della Fabrica et Uso di diversi stromenti ..., (Venetia, 1598). The first instruments made for Galileo by a craftsman employed in 1599 were sold for 42 lire; Galileo supplied the materials, furnished lodging to the craftsman, and paid him 25 lire for his work on each sector plus a small annual wage. Thereafter the price was reduced to 35 lire and the payments were increased to 30 lire. Cfr. Opere di Galileo Gali/ei, Ed. Naz. (Firenze, Barbera, 1921r-1939), vol. 19, pp. 131 ff. Opere di Galileo Galilei, vol. 2, p. 93. Opere di Galileo Galilei, vol. 2, pp. 357-358. Cfr. Opere di Galileo Gali/ei, vol. 19, p. 132, lines 36, 46-48; p. 138, lines 231r-240; p. 145, lines 491-2; p. 147, line 6; p. 148, line 39; p. 149, line 61. In my collection there is a sector of the Galilean type, probably of the seventeenth century, which is fitted with removable sights for surveying and has a friction joint attachable to one arm, provided with a similar hollow cone for mounting on a tripod. See section VII, below, for identification of the manuscripts. Cfr. Opere di Galileo Galilei, vol. 2, pp. 534-535. Opere di Galileo Galilei, vol. 19, p. 147, line 18. For the full original texts of these insertions, see section VII, below. It is of interest that the symbols used to identify them were the conventional signs for the sun and planets, and that Galileo used them in the Ptolemaic rather than the Copernican order, though this was two or three years after he had written to Kepler expressing his preference for the Copernican arrangement. Document g, where they appear, was probably not written until 1605, a year in which there is also other evidence that Galileo was still not firmly committed to the Copernican system, which he vigorously supported only after his telescopic observations in 1610. See Stillman Drake, Galileo Against the Philosophers (Los Angeles, 1976).

3

Galileo Gleanings VI: Galileo's First Telescopes at Padua and Venice

It was my intention early in this series of essays to pay tribute to the monumental labors of Antonio Favaro, who published in twenty magnificent volumes every scrap of material left by Galileo, as well as much previously unpublished manuscript material relating to him. In so doing, Professor Favaro reaped an extraordinarily rich field which he was never permitted to glean (as he had hoped to do in his later years) because of the insistence by his government that he embark upon another great task which he would have preferred to leave to others. This arbitrary assignment must be forever lamented by the world of scholarship, which was thus deprived of the truly definitive biography of Galileo contemplated by a man uniquely prepared and equipped to carry out such a work. Various circumstances affecting the scope and order of the papers presented in these gleanings have delayed until this time the acknowledgment of my obvious and enormous debt to Favaro. In paying tribute now to that patient and cautious scholar and great editor, I should add that when in this and future papers some of his opinions or conclusions are contested, or errors in the National Edition pointed out, such things are possible only because of the very precision and exhaustiveness of Favaro' s work, and are necessary only because of the governmental meddling that prevented him from personally completing its final critical analysis. Doubtless to the eyes of anyone but a specialist there appear to be occasional piles of chaff among the heaps of grain harvested by Favaro. Thus one might at Reprinted from Isis 50 ( 1959), 245-54, by permission of the University of Chicago Press. Copyright 1959 by the History of Science Society, Inc.

34

Instruments

first be inclined to regret, for example, that for lack of space he chose to exclude some two thousand additional short biographical sketches of persons mentioned in the twenty noble tomes, after he had devoted many pages of the nineteenth to various scraps in Galileo's handwriting containing accounts of grocery bills, payments to servants, and dealings with instrument makers, copyists, boarders and private students. Yet even such seeming trivia are not devoid of interest to historians of science, to say nothing of Galileo's future biographers. For it is fairly safe to say that the 700-odd entries made in these journals by Galileo during the period 1599-1610, during which time he was professor of mathematics at the University of Padua, supply us in every instance with a date on which he was physically present in that city. A careful comparison with data independently available from letters and published works discloses but two entries that are in conflict with this assumption, which is a sufficiently reasonable one in any case. Two errors in seven hundred entries might well occur even among accounts kept by a bookkeeper, let alone by a professor. One of the two, moreover, is easily rectified. This is the entry of a transaction as having occurred on 30 February 1610, implied by Galileo's repetition of the word detto, whereas both this entry and at least the one preceding it belong obviously to March of that year.' The other discrepancy is of a different sort: it concerns an entry which reads A di 20 di 7mbre (1605), a date on which Galileo was certainly at Florence and not at Padua. 2 In all probability this was a mere slip for 9mbre, since Galileo was not in the habit of back-dating entries upon his return from journeys; this is well attested by other parts of the journals. Assuming that the journal entries provide accurate dates of Galileo's presence in Padua, and making further use of the Galilean correspondence, it is possible to throw some new light upon the chronology of the production of Galileo's first good telescope and of its exhibition at Venice. Professor Edward Rosen has already dealt with this question in several places, notably in two valuable papers published in 1951. 3 But since previously reached conclusions may be modified if the above assumptions are accepted, it appears worthwhile to re-examine the question in their light. 2

Two published accounts by Galileo of the circumstances connected with the production of his first telescopes are already available in English translation.4 A third account, the authenticity of which had been questioned by Favaro, was thoroughly vindicated by Dr. Rosen in 1951. It occurs in a letter addressed to Benedetto Landucci at Florence, and reads as follows :5

Galileo's First Telescopes at Padua and Venice

35

Dear and Honored Brother-in-Law: I did not write after receiving the wine you sent me, for lack of anything to say. Now I write to you because I have something to tell you which makes me question whether the news will give you more pleasure or displeasure, since all my hope of my returning home is taken away, but by a useful and honorable event. You must know, then, that it is nearly two months since news was spread here that in Flanders there had been presented to Count Maurice a spyglass, made in such a way that very distant things are made by it to look quite close, so that a man two miles away can be distinctly seen. This seemed to me so marvellous an effect that it gave me occasion for thought: and as it appeared to me that it must be founded on the science of perspective, I undertook to think about its fabrication; which I finally found, and so perfectly that one which I made far surpassed the reputation of the Flemish one. And word having reached Venice that I had made one, it is six days since I was called by the Signoria, to which I had to show it together with the entire Senate, to the infinite amazement of all : and there have been numerous gentlemen and senators who, though old, have more than once scaled the stairs of the highest campaniles in Venice to observe at sea sails and vessels so far away that, coming under full sail to port, two hours or more were required before they could be seen without my spyglass. For in fact the effect of this instrument is to represent an object that is, for example, fifty miles away, as large and near as if it were only five. Now having known how useful this would be for maritime as well as land affairs, and seeing it desired by this Serene Ruler, I resolved on the 25th of this month to appear in the College and make a free gift of it to His Lordship. And having been ordered in the name 6 of the College to wait in the room of the Pregadi, there appeared presently the Procurator Priuli, 7 who is one of the Governors of the University. Coming out of the College, he took my hand and told me how that body, knowing the manner in which I had served for seventeen years in Padua, and moreover recognizing my courtesy in making such an acceptable gift, had immediately ordered the Honorable Governors [of the University] that, if I were content, they should renew my appointment for life and with a salary of one thousand florins per year; and that since a year remained before the expiration of my term, they desired that the salary should begin to run immediately in the current year, making me a gift of the increase for one year, which is 480 florins at 6 lire 4 soldi per florin. I, knowing that hope has feeble wings and fortune swift ones, said I would be content with whatever pleased His Lordship. Then Signor Priuli, embracing me, said: "Since I am chairman this week, and can command as I please, I wish after dinner to convene the Pregadi, that is the Senate, and your reappointment shall be read to you and voted on." And so it was, winning with all the votes. 8 Thus I find myself here, held for life, and I shall have to be satisfied to enjoy my native land sometimes during the vacation months. Well, that is all I have for now to tell you. Do not fail to send me news of you and

36

Instruments

your work, and greet all my friends for me, remembering me to Virginia and the family. God prosper you. From Venice, 29 August 1609. Your affectionate brother-in-law GALil.EO GALil.EI

The authenticity of this letter, which exists only in a contemporary copy, may be regarded as having been established beyond question by Dr. Rosen, though his interpretation of certain aspects of it may still be debated. And while it is true that Favaro neglected to state the reasons for which he felt the style of the letter to be not Galilean at certain points, this seems to me evident in the opening and closing paragraphs. These are quite extraordinarily awkward, so much so that at first sight they seem to exclude Galileo as the writer. On the other hand they may be readily reconciled with his authorship if considered as having been hurriedly tacked on to the body of the letter, which itself had been very carefully drafted, simply to supply Galileo with a pretext for sending it to Landucci and having its contents conveyed to friends . I cannot agree with Dr. Rosen that Galileo was ever on good terms with Landucci, even though he had exerted himself to obtain a minor government post for him a short time before. Certainly Landucci did not care whether Galileo ever returned to Florence, and (as Favaro observed) Galileo had many correspondents at Florence who would have been more suitable recipients of this stirring news. Now when one examines the body of the letter, it becomes very plausible to suppose that that was written for other eyes than those of Galileo's brother-inlaw. To Landucci's mind, the parade of dignitaries and the high salary offered would have been an intolerable display of boasting. As Galileo well knew, Landucci 's own job carried no salary, and brought him fees amounting to no more than sixty florins a year. What, then, could be the sense in Galileo's telling him that when offered a salary of one thousand florins a year for life, he had accepted it only because "hope has feeble wings and fortune swift ones"? What more, Landucci might exclaim, could a man possibly want? But this phrase would certainly have a very real significance to the Grand Duke, who had delayed in acting upon Galileo's appeals for employment, and for whose eyes I believe the letter was really intended. If my reconstruction of events is correct, Galileo was embarrassed to admit to Cosimo that he had suddenly committed himself to remain in the service of the Venetian Republic for life while he had been negotiating for a post at the Court of Tuscany. Nevertheless he felt a need to have this news reach the Florentine court from himself, before word got there from others. Accordingly he sent his message as a family letter to his brother-in-law, who held a minor government

Galileo's First Telescopes at Padua and Venice

37

post, with specific instructions to greet all his friends for him. Landucci did convey the news promptly, and the interest that was immediately shown at the court in the topic doubtless accounts for the survival of this letter in a contemporary manuscript copy whereas any other letters Galileo may have written to Landucci are lost. The phrase "if I were content," coupled with Galileo's insistence upon the impossibility of his ever returning pennanently to Florence, implies that he had accepted as a condition of the increase in salary the stipulation that he remain for life. If we accept the letter as genuine, and consider how soon after the events it was written, we are obliged to believe that such a promise was exacted. But if so, a great deal of light is thrown upon some subsequent events, and especially upon the bitterness that was created at Venice by Galileo's later departure.9 The letter further implies that Galileo had been given to understand by Priuli that the increase in salary would take place immediately, and that Galileo was not told that there would be a restriction against possible future increases. In the official award, the increase was set to begin the following year, and an absolute prohibition of future increases was made. Favaro (rather unfairly) condemned the letter as a fabrication on the assumption that the final award coincided with the original tenns proposed to Galileo by Priuli. Neither Favaro nor Rosen appears to have considered that once we assume the authenticity of this letter, Galileo's mounting irritation and subsequent departure from Venice become quite comprehensible, and that certain other problems outside the scope of the present paper are more easily solved. 3

In an account written by Galileo more than a decade after these events (published in The Assayer), he asserted that the first news of the Flemish invention came to him at Venice, where he happened to be visiting, and that he immediately returned to his residence at Padua. The day after his return, he says, he sent word of his success in constructing a crude instrument to the friends he had been visiting, and a few days later he returned to Venice with a ten-power telescope. There, he said, he exhibited it for more than a month on end, and only then received his reward for service to the Republic. The period of more than a month is demonstrably an exaggeration if the account books previously mentioned are reliable. Galileo was in Padua on the 23rd, 28th, and 29th of June; 10 on the third of July he wrote to Florence from Padua mentioning an illness; 11 on the 1 1th and 18th of July he again made entries in his account books at Padua, as he did on the third and the 20th of August and on the first and third of September. 12

38

Instruments

It was on the 2 I st of August that Galileo showed his telescope from the Tower of St. Mark to a distinguished group of Venetian gentlemen, including Priuli. There can be no doubt that this was Galileo's first showing of his telescope in Venice, and that he had barely arrived back there. Circumstances existed which required the utmost speed on his part; a foreigner had arrived about this time with a mediocre instrument, probably from the Netherlands, and was attempting to sell its "secret" to the Venetian government. 1 3 But Galileo was still in Padua on the 20th of August. Hence the period he spent at Venice exhibiting his telescope was not "a month on end," but less than two weeks, that is, from the 2oth or 21st to the 30th or 31st of August, and the presentation to the Senate occurred within the first of those two weeks. The foreigner who appeared at Venice with a telescope during August was almost certainly the same who had arrived in Padua late in July, and of whom Lorenzo Pignoria wrote on the first of August to his friend Paolo Gualdo at Rome. It is hardly possible that Galileo was in Padua when this man arrived, as both Pignoria and Gualdo were friends of Galileo's, and if he had been present at these exhibitions, or could have been consulted, his name would in all probability have been mentioned in Pignoria's letter. It would appear that the stranger arrived during the first period in which Galileo had gone to Venice to visit friends and there heard of the Belgian invention. Galileo's departure for Venice on this visit may have occurred at any time commencing I 8 July, and the date of his return may be fairly certainly set as 3 August. 14 Rumors of the telescope were general toward the end of July, and by the first of August news would probably have reached Venice that the mysterious stranger was showing one of the instruments at Padua. Galileo says that he left Venice immediately after discussing the stories of the invention with his friends, and his motive may very well have been to find the foreigner and see his instrument. But upon his arrival he must have learned that the stranger had already left for Venice, where he attempted to sell the "secret" to the govemment. 15 Not long before, toward the end of June, Galileo had talked to Piero Duodo at Padua about the possibility of improving his position, presumably with regard to increased salary. Duodo, on his return to Venice, wrote to s1 that he would take the matter up but was dubious about its speedy success. 1 From this it is clear that Galileo was not at the time on the track of any new and wonderful discovery, but that he was in a frame of mind to do anything he could to secure favorable attention. The golden opportunity presented itself while he was at Venice late in July, and he hastened back to Padua to pursue it. Perhaps it was there that a letter from Jacques Badovere at Paris awaited him, in which the reality of the invention was certified. 17 At any rate, Galileo set to work at once; as he tells us in The Assayer (and had implied in the first draft of the Starry Messenger) he succeeded overnight in conceiving and testing a combination of

Galileo's First Telescopes at Padua and Venice

39

concave and convex lenses for the purpose. During the next several days he busied himself in obtaining or grinding more suitable lenses and constructing a tube for them. 18 On 20 August he made a payment of 21 lire to his instrument maker, and (I believe) set out for Venice. The results of this second visit to Venice in 1609 brought him unprecedented fame. To sum up the results of the foregoing reconstruction, the following tentative calendar of events is offered:

ca. 20

19 July July ff.

ca.

26

ca.

1

2

July

August

or 3 August

4 August

5-20 21

August

August

Galileo leaves Padua to visit friends at Venice. He hears rumors of the Holland instrument for the first time and listens to discussions pro and con. He visits Sarpi to ask his opinion and is shown corroborating letters, perhaps including one from Badoer (see section 5). He hears that a foreigner has arrived at Padua with one of the instruments and is exhibiting it there. He returns to Padua, but learns that the stranger has already departed for Venice to sell the "secret." He attempts to deduce the construction of the instrument, using information from letters and descriptions by those who have seen it. He verifies by trial that suitably separated convex and concave lenses will enlarge distant objects. He sends word to Venice (probably to Sarpi) that he has the "secret." He succeeds in constructing an instrument of about ten diameters magnification, and sets out again for Venice. He exhibits this instrument to officials from the Tower of St.

Mark. August 29 August 31 August 24-25

He exhibits it to the Signoria and to the Senate. He dispatches the letter to Landucci. He returns to Padua and prepares for the journey to Florence. 4

The mysterious stranger who appears in the letters of Pignoria and of the Tuscan Ambassador at Venice (for it is hard to conceive that this was not the same man) played a very large part in Galileo's career, though it is virtually impossible that they ever met face to face. Had it not been for the physical presence in Italy of an instrument that was causing excitement wherever it was shown, Galileo would most probably have carried his first telescope to Florence and used it to further his attempts to secure employment there. As things stood, he would not have dared to take the risk of such a delay. For all he knew, others would soon duplicate his own achievement. 19 And at best, by the time he could

40

Instruments

reach Florence it would be known there that a similar instrument was being shown in Venice, so that his claim to superiority would be hard to establish. As a result, from the moment he learned that the stranger had left for Venice, he may be presumed to have worked only to beat him there at his own game, as indeed he succeeded in doing. Having succeeded, however, the next thing he had to do was to communicate the news to the Grand Duke, his natural prince and former pupil. This was embarrassing. To the Duke, he had to present the circumstances in a light that would explain his failure to return to Florence and would justify his gift of the new and important device to a foreign government. The recital of events he prepared is plausible, if not precise in all regards. He says it is nearly two months since the rumors spread (circa a 2 mesi; had he meant "about two months," his words would probably have been intorno a 2 mesi). He does not say that he himself had heard the rumors at that time. 20 He says that word reached Venice of his having succeeded; so it did, but according to the later account, it was he himself who had promptly sent news of his success to friends. The manner in which he recounted events to the Florentines was designed to make it appear that he was the victim of circumstances, and had acted from that time under orders of the government which employed him. "It is six days since I was called by the Signoria," he says, making the context imply that they called him from Padua, whereas on the 23rd he was already in Venice, and he had shown the instrument to others before he was called by the rulers. 21 On the two successive days it was shown to the select Signoria and then to the whole Senate. The offer made to him was generous; had he refused it, he could not have been sure of doing so well at Florence. Such was the story written for the eyes of Cosimo. It may, in the light of later events, have been an oblique reopening of Galileo's application for a court position. If Galileo tried by his letter to be first to get the news to Florence, however, he did not succeed. On the same day that his letter was posted at Venice, Eneas Piccolomini wrote from Florence at the request of the Grand Duke to inquire about the instrument and solicit one or instructions for making one. Galileo did more than comply with this request; he personally made a hurried trip to Florence the following month. There he repaired any damage that had been done, and paved the way for negotiations the following spring which eventually culminated in his long-desired appointment by Cosimo. 5

It is appropriate here to call attention to a document which, though not unpublished, appears to have escaped attention in the various reconstructions of the

Galileo's First Telescopes at Padua and Venice

41

events discussed above. In 1926, after Favaro's death, a letter of Fra Paolo Sarpi to Jacques Badovere was discovered at Paris by Manlio Duilio Busnelli, and was published the following year. 22 It is dated 30 March [1609), and reads in part as follows : ... I have given you my opinion of the Holland spectacles. There may be something further; if you know more about them, I should like to learn what is thought there. I have practically abandoned thinking of physical and mathematical matters, and to tell the truth my mind has become, either through age or habit, a bit dense for such contemplations. You would not be able to believe how much I have lost both in health and composure by attention to politics ...2 3

Sarpi, who later was requested by the Venetian government to report on the foreigner's instrument and was probably the pivotal figure in its rejection, had been the first man in Italy to learn of the Flemish invention. His information came from Francesco Castrino in November of 1608, only a month after Lipperhey applied for the patent. In a letter to Castrino dated 9 December 1608, Sarpi acknowledged receiving "a month ago" the report of the embassy of the King of Siara 24 to Count Maurice and of the new spectacles. 2 s Writing to Jerome Groslot de L'lsle on 6 January 1609, Sarpi said: I have had word of the new spectacles more than a month, and believe it sufficiently not to seek further, Socrates forbidding us to philosophize about experiences not seen by ourselves. When I was young I thought of such a thing, and it occurred to me that a glass parabolically shaped could produce such an effect. I had a demonstration, but since these are abstract matters and do not take into account the fractiousness of matter, I sensed some difficulty. Hence I was not much inclined to the labor, which would have been very tiresome, so I did not confirm or refute my idea by experiment. I do not know whether perhaps that [Flemish] artisan has hit upon my idea - if indeed the matter has not been swelled by report, as usual, in the course of its journeys. 26

Probably a similar account of Sarpi's own speculations had been sent to Badovere, followed by the inquiry of 30 March for further information if any was to be had. It is very likely that Badovere replied to Sarpi's letter, confirming the effectiveness of the instruments and perhaps describing the two lenses used (for Sarpi appears to have thought a single parabolic glass to have been sufficient). Now though Sarpi had long been informed of the new device, there is no evidence that Galileo had yet heard of it, or that he was at this time in correspondence with Badovere. And considering the importance of the letter from Badovere that Galileo saw, in conjunction with the relatively complete preser-

42

Instruments

vation of his own correspondence around this time, it is truly remarkable that the letter was not kept if it was indeed addressed to Galileo. Quite possibly what Galileo had seen was Badovere's reply to Sarpi, shown to him at Venice when he asked Sarpi's opinion of the rumors that were being discussed among his other friends. Galileo's words in the Sidereus Nuncius admist of such a possibility; he says "confirmed to me from Paris in a letter from the noble Frenchman Jacques Badovere," and not "confirmed in a letter to me from ... Jacques Badovere."27 Nor is it anywhere clear whether this confirmation came to him while he was still at Venice, or upon his return to Padua. Finally it may be mentioned that among the many curious circumstances of this entire episode is Galileo's reference to Badovere as a "noble Frenchman." Badovere (or Badoer) had been a pupil of Galileo's, residing at his house in I 598, and Galileo certainly ought to have known that he was not a Frenchman but the son of a rich Venetian merchant. Originally a Protestant, Badovere had migrated to France; there he forsook his religion and became closely linked with the Jesuits, for whom he undertook some rather risky enterprises. His appointment to a diplomatic post under the French government about this time was abruptly terminated as a result of vigorous protests from Sully and other ministers of importance. Whether his difficulties were due to nationality, religion, or the persistent and scandalous rumors of perversion which circulated against him, is still a subject of conjecture. NOTES

1 Opere, xix, 146. 2 Opere, xix, 142. Cf. x, 146--147, from which it is evident that Galileo's return to Padua via Venice was in the latter part of October. The debit side of the accounts cited suggests also that his return to Padua was just prior to I November. 3 The Authenticity of Galileo's Leuer to Landucci, Modern Language Quarterly, 1951, 12: 473-486; When Did Galileo Make His First Telescope? Centaurus, 1951, 2 : 44-51. 4 Discoveries and Opinions of Galileo (N.Y., 1957), pp. 28-29 and 244. The leller to Landucci was translated by J.J. Fahie in Galileo: his life and works (London, 1903), pp. 77-78, but not in its entirety. 5 Opere, x, 253-254. 6 Here the letter is damaged, and the reading is conjectural. 7 Antonio Priuli, who accompanied Galileo on the occasion of the first exhibition of the telescope from the Tower of St. Mark and has left us a description of the instrument (xix, 587 and 588). 8 The vote was not unanimous, and this was one of the points that made Favaro doubt

Galileo's First Telescopes at Padua and Venice

9 10 11 12 13 14

15

16 17 18

43

the authenticity of the letter; but the phrase may simply mean "winning when all the votes had been counted." The words used are not "winning all the votes," but "winning with all the votes." Priuli in particular turned bitterly against Galileo; xi, 503. Opere, xix, 158, 145, 174. Opere, X, 254. Opere, xix, 145, 166. Opere, x, 253, 255. The payments from Count Montalban, who stayed at Galileo's house, through this summer to complete his degree, were collected near the end of April and June, but-at the beginning of August and September. Montalban's presence through the summer would have made it difficult for Galileo to be away for an entire month at a time in this year. Montalban had been a pupil of Galileo's since November, 16o4, but had never before remained in summer. Various stories were spread laler that Galileo had stolen the invenlion from the stranger, but if they had met at all it could not fail to have been known to many at Padua or Venice. Yet Galileo denied having seen the instrument, and published without challenge this account of the matter. A rather plausible story was that Fra Paolo Sarpi had been asked by the government to look into the stranger's claims, that he had passed word along to Galileo about the device, and that he then used his influence to get the stranger to depart and leave the field for Galileo (x, 255). Probably it was to Sarpi that Galileo communicated news of this first success early in August, and Sarpi may have advised the government to withhold action until Galileo should arrive with his instrument. Sarpi had heard of such instruments in November of the previous year, but had declined to express any opinion without seeing one. A foreigner was at Milan in May of 1609 with one of the instruments, which was seen by Sirturi. One was in the hands of Cardinal Borghese at Rome, and Federico Cesi (who had not yet met Galileo or heard of his telescope) wrote 10 G.B. Porta about the "secret" late in August. It is interesting that no surviving documents dated before August 16o9 mention the instruments as having been seen in Italy, whereas during that month they are mentioned often and in various parts of Italy. Opere, x, 247; Doudo's visit to Padua was probably on 23 June (xix, 158). This letter, mentioned in the Stany Messenger, is not extant and may not have been addressed to Galileo; see further under section 5 below. In the Assayer Galileo says "six days," but he frequently used this phrase (or "eight days") in contexts which seem to have meant merely "several days," just as "four days" is often used to mean "a few days." Anyone who has constructed a telescope in modem times, when component parts are readily available, will be inclined to doubt that even Galileo could have made his second improved telescope at Padua in less than two weeks.

44

Instruments

19 Ironically enough, no one in Europe was able to make telescopes as good as Galileo's for a long time after he had explained the general principle in his book and molds had been surreptitiously taken of his lenses. 20 Probably the rumors were current in Venice when he arrived about 20 July, and he discussed them with various friends as he tells us. This would justify the phrase "nearly two months" when writing at the end of August. The sudden return to Padua of which Galileo speaks was probably made when word arrived that such an instrument was being shown in Padua itself, and not when he first heard the vaguer rumors. 21 Again the phrase "six days" may mean merely "several days," but assuming it to be precise, there is not a contradiction as Favaro asserts. The day on which Galileo was called may well be believed to have been the day before he was supposed to appear. Priuli says he appeared before the Signoria on the 24th, and that on the 25th the Senate approved the reappointment. The day of which Galileo speaks is only the latter, but this does not preclude his having presented the telescope in the Signoria the previous day, as Priuli reported (xix, 588). It appears that the College met on the morning of the 25th and required Priuli to get Galileo's assent to the terms before the entire Senate took official action later that day, which seems eminently reasonable and prudent. 22 Atti de/ R. lstituto Veneto de Scienze. Lettere ed Arri, 1927, 87: pt. 2, 1025 ff. 23 Ibid., p. 116o. 24 This embassy has got into the literature as being from the King of Siam; see, for example, V. Ronchi, Galileo e ii Cannochiale (Udine, 1942), p. 138. Prior to Busnelli's publication, the usual source was Pierre de l'Estoile's Memoires-Journaux. to which I do not have access; perhaps the error arose from a misprint in that journal. Siara (Ceara) is a province of northern Brazil which submitted to the Dutch in 1637 (Busnelli, op. cit., p. 1069, p. 1). 25 Ibid., p. 1o69. 26 Busnelli, ed.: Sarpi, Lettere ai Protestanti (Bari, 1931 ), I, 58. 27 mihi per literas a nobili Gallo Jacobo Badovere ex lutetia confirmatum est (Opere, iii, 6o).

4 Galileo Gleanings XII: An Unpublished Letter of Galileo to Peiresc

During a visit to England in July, I 96 I, I was privileged to meet Mr. Kenneth K. Knight and to see his distinguished collection of early scientific books, instruments, and manuscripts. Among these there is an autograph Jettier of Galileo which Mr. Knight has graciously pennitted me to publish. The letter, in an excellent state of preservation, is written on the first two sides of a single folded sheet measuring 40 cm x 27.7 cm when opened. On its face, in the upper lefthand margin, is the single word, "Galilei," in an early script; with this exception, the entire document is in Galileo's own writing. The letter is without direction or cover sheet, but is easily identifiable as a reply transmitted to Peiresc with the aid of Roberto Galilei at Lyons. 1 Contemporary references exist which mention this reply by date, but hitherto it has had to be presumed lost. 2 The correspondence of which this letter fonns an essential part began on 26 January I 634, when Peiresc wrote to Galileo recalling his brief residence in Padua "more than thirty years ago," and his presence at some of Galileo's public lectures there. 3 At Gassendi's request, Peiresc sent along to Galileo a letter from Gassendi and a letter and book by Martin Hortensio. Peiresc himself wrote to second the request made in Gassendi's letter for the use of one of Galileo's telescopes, their own instruments being unsatisfactory for observations which they wished to pursue. All these letters and the book were transmitted by Peiresc to G.G. Bouchard at Rome. Bouchard in due course added certain other letters and one of his own, forwarding the lot to Galileo on 18 March. 4 From Arcetri, on 25 July, Galileo wrote at length to Elia Diodati in Paris. At Reprinted from Isis 53 ( 1962), 201-11, by pennission of the University of Chicago Press. Copyright I 962 by the History of Science Society, Inc.

46

Instruments

the same time he sent lenses for a telescope to be forwarded to Gassendi, asking that Peiresc also be allowed to use them, and apologizing for his not having replied directly to Peiresc in view of a host of troubles that forced him to abstain from all but the most necessary activities.S The package containing the lenses was in fact long delayed in transit, and did not reach Diodati until about the beginning of November. On the tenth of that month he sent them on to Gassendi with a copy of Galileo's letter, and suggested that Peiresc might use his good offices with the powerful Francesco Cardinal Barberini to obtain some moderation of the restrictions imposed upon Galileo. 6 At the same time he wrote directly to Peiresc, mentioning the letter of Galileo that he had sent to Gassendi. 7 Peiresc, who had previously obtained a copy of the sentence pronounced against Galileo, wrote early in December to Cardinal Barberini, remonstrating in the strongest terms against the treatment of his former friend and teacher. Among other things, he said that it would be a blemish upon the reputation of Pope Urban VIII (uncle to the cardinal) if he should fail to give Galileo his special protection and some particular assistance. 8 This courageous appeal, however, elicited from the cardinal only the rather cool response: "I shall not fail to convey to His Holiness what you write me about Galileo; but as I am one, albeit the least, of the cardinals who attend meetings of the Holy Office, you will excuse me if I do not extend myself to reply to you in more detail."9 Early in February I 635, Roberto Gali lei obtained and forwarded to Galileo copies both of Peiresc's plea and the cardinal's reply. 10 Galileo promptly wrote to thank Peiresc in an eloquent letter, remarking that he expected no relief, precisely because he had committed no crime, whereas clemency derives from forgiveness and not from admission of juridical error. 1 1 Meanwhile the dauntless Peiresc had already replied to the cardinal, predicting that posterity would regard the persecution of Galileo just as men looked upon that of Socrates. 12 Somehow a copy of this letter also reached Galileo, and he wrote again to Peiresc ( 16 March) in admiration of his persistence in beating against a rock that gave no sign of yielding to any blows, even to those so well aimed. 13 To Galileo's first letter, Peiresc had replied at great length on the first of April. After giving reasons for his persistence and outlining his future strategy, he wrote as follows: But to keep the matter alive, His Eminence having written me that Father Sylvester Pietrasancta had presented to him a book of his, De symholicis heroicis, 14 which the Father had shown me here when passing through at Christmas with Monsignor [Pier' 1uigi) Caraffa, Nuncio of Cologne, I took occasion to remind His Eminence that if the press of other and more worthy affairs had not permitted him to read or scan that book, he might deign to read in Book IV, Chapter V, what the author says about a water-clock devised by Father Linus, 15 of which you will see here the diagram and description

An Unpublished Letter of Galileo to Peiresc

47

[Plate 12), which is a marvelous thing, if indeed ii works; and since the author of the book says nothing of having seen the machine itself, nor does he name anyone who has seen it, I begged His Eminence to call in Father Sylvester and ask him as 10 the real truth of this machine and lo get also the opinion of Msgr. Cara ffa, who ought lo know about it, not only from having seen something of it, but perhaps also by having penetrated its secret. And I also wrote, under the same cover. not only 10 Father Sylvester, who is now at the Collegio Romano, but lo the said Nuncio (who, passing through here incognito, had wished 10 spend a couple of hours in my study with Father Sylvester). telling them both of my regret, after they had left, that I had forgotten to speak with them about that machine of Father Linus, in order lo learn from them directly what could be believed of ii; and thus they are under obligation not only 10 give an account of this to His Eminence, but also to give me some share and part in their dealings with him. Whence I hope at the proper time to take occasion 10 reopen your Excellency's case with greater vigor and effectiveness than before, inasmuch as if the machine truly works (as Peter Paul Rubens writes me from Anversa in a letter of 16 March, which I received yesterday afternoon, he having heard the testimony of Father Sylvester and others who affirm it to be as represented, Father Sylvester having added 10 him that he had seen it at his leisure, and that Msgr. Caraffa had taken it to his house to examine ii al his ease, and having observed it for some days, found it to be most exact). ii seems that this is a proof and testimony fallen from heaven into the hands of a Jesuit Father rather than anyone of another profession, to leave no suspicion against the testimony of the father who invented ii and that other who published it, in order lo overthrow the error of those who find so much repugnance in the Copernican doctrine and in what you have proposed in sport and as a puzzle. Signor Rubens, a great admirer of your genius, even promised me to ride post to Liege 10 visit Father Linus and see his machine, which he will not do without giving me an account of it; and I have urged him to do so as soon as possible; and I shall seek some dealings and correspondence with Father Linus through Msgr. Caraffa and Father Sylvester or others, since they knew him; rather, I shall seek to have him called to Rome, and arrange that he take the road through these countries, to enjoy his passage and draw from him the most I can by word of mouth, if he does not bring with him the water-clock so that we can see it here in his hands; and all this is to have always new devices to remind those of you who can aid you better than I can. 16 II

Here we may well pause to consider the reasons for Peiresc ' s great excitement over this reported device, which he evidently believed to be of such importance that it might convince even the authorities at Rome of the earth's diurnal rotation, and thus induce them to relax the sentence against Galileo and the ban on the Copernican theory. A summary of the state of affairs at that time has already been

48

Instruments

written by Comelis de Waard in a note on a letter sent by Godfrey Wendelin to Mersenne in 1633, which we cannot do better than to quote, particularly as it contains the very passage from Pietrasancta's book alluded to by Peiresc: Projects for such magnetic clocks may be traced back to Roger Bacon and Peter de Marincourt, whose booklet was in print after r 588. Cardan and Jean Taisnier also made mention of these supposed perpetual motions. Gilbert had assumed, to explain the diurnal motion of the earth, that a suspended magnetic sphere could set itself in rotation, but this opinion was refuted by Galileo in his work of 1632. Among the first scholars who had knowledge of Father Linus's magnetic clock we find the nuncio Caraffa and his Jesuit confessor, Father Sylvester Pietrasancta. The former had taken the clock to his house, and after observing it for a day or two found it accurate. The latter gave the ensuing picture and description of it in a book published in 1634, though the licenses were dated the previous year:

I know the effects of the magnetic stone are utterly remarkable, and from its power something new is always being produced. Thus at Liege recently, in the English College of our Society, Father Franciscus Linus, teacher of mathematics, invented a most pleasing globe, which placed at the center of surrounding fluid in a vessel (as the earth is surrounded with air) preserves a mysterious balancing of its mass. But the rotation of the heavens from east to west nevertheless, by an occult force and as if it were lovingly followed, drives it completely about in the space of twentyjour hours. A little fish is placed inside as indicator, and like an expert swimmer, its weight poised, watches the fleeting hours and designates them with its snout, its eyes gazing intently on them. If by motion of the vessel the water takes away the impetus , the globe soon regains its course by its own accord, and the ratio of the time is thoroughly consistent after tranquility is restored. Also the indicator placed in the vessel in like manner shows the hours. And it imitates the sun in his sphere, and indeed follows the sun in the east, on the meridian, in the west; and what is more.from the seat of its driving out, at that place it requires and seeks again agreement with the stations of the stars. So much the more will it be hastened, because love does not know delay; nevertheless, while it leaps ahead sometimes and springs back, at length it gets to a place in which it again unerringly accompanies the sun .... Perhaps Mersenne, influenced by theories that attributed a magnetic action to the sun which moved the planets, saw in this (as Peiresc did later) an argument for the Copernican hypothesis. But the water-clock of Father Linus, which did not tum consistently during twenty-four hours, has nothing to do with the diurnal movement of the earth; moreover, when one compares it with the apparatus of the same kind set up by Father Kircher, its operation seems evident. 17

An Unpublished Letter of Galileo to Peiresc

49

Two kinds of odd "clocks" had made their appearance about this time, one kind purporting to be governed by magnetic force, and the other to be influenced by the occult properties of seeds or roots of plants which tend to face the sun. One of the latter, exhibited by Kircher, had been mentioned by Bouchard to Galileo in a Ieuer previously cited, and was more fully described to him in a letter from Rafael Magiotti wriuen on the same day: There is now at Rome a Jesuit, long in the Orient, who, besides knowing twelve languages and being a good mathematician etc., has with him many lovely things, among them a root which turns as the sun turns, and serves as a most perfect clock. This is affixed by him in a piece of cork, which holds it freely on the water, and on this cork there is a needle of iron that shows the hours, with a scale for knowing what hour it is in other parts of the world. He has two roots which attract each other as a magnet attracts iron. ' 8

The story of Kircher' s "botanical clock," Linus's magnetic clock, and Kircher's later plagiarism of the latter device, was given at length by Georges Monchamps. 19 That there was some kind of device similar to that described by Magioui can hardly be doubted, as Kircher exhibited it to many persons at Aix, Avignon, and Rome, after obtaining a supply of the mysterious root from an Arab merchant at Marseilles in 1633. Monchamps appears to have accepted the belief of Kircher's contemporaries that it was sunflower root or seed, with an innate property of following the sun (even at night!). Possibly it was some aromatic root, and in any case it probably acted (as will chips of camphor) by affecting the surface tension of the water in a restricted area. The cork may be presumed to have been fairly large and flat, and to have been prevented from moving laterally by some sort of pivot or by a rod inserted through a central hole. Linus's clock was a very different matter. He did not divulge its secret, but Kircher was able to divine this, and published it as his own in the Magnes (Plate 13). The reconstruction by Monchamps assumes the globe to have been suspended by a fine silk thread, invisible to the naked eye, which he deduces from the experimental difficulty of maintaining exact equilibrium, and from Pietrasancta 's observation that when disturbed, the globe overshot the mark somewhat in returning to its original course, a phenomenon attributed by Monchamps to torsion of a thread. To account for the rotation of the suspended globe, Monc~amps assumes that a bit of iron, attached to the inside of the globe, was acted upon by a magnet hidden in the hollow wooden base and driven by some sort of clockwork, possibly clepsydral. A very similar account, he says, is given by Kircher. He notes, however, that in an earlier section of the Magnes, Kircher had described a different method of suspending a solid mass in a fluid; namely, by utilizing two layers of fluid similar in color but differing in density.20

50

Instruments

The explanations offered by Kircher, Monchamps, and (as we shall see) by Galileo appear to be correct in principle but to require some refinement. The suggestion of a thread-suspension to account for certain observed oscillations of the globe is liable to the objection that a thread sufficiently strong to produce them would quickly become wound to a tension capable of checking the progress of the globe. Moreover, it would be difficult to secure the two ends of the thread in a glass sphere without permitting leakage of the fluid . Equilibrium must therefore have been maintained by the use of two fluids; but then, if a piece of iron or loadstone were affixed near the perimeter of the wax globe, it would always tend to approach the concealed attracting magnet, thus preventing any effective horizontal rotation. In all probability, the hollow wax globe was pierced horizontally by a compass needle or its equivalent, and the concealed mechanism simply rotated a bar magnet or elongated piece of loadstone centered beneath it. Such a system assures horizontal stability of the globe. Assuming the magnet to move jerkily, this device would exhibit periodically a notable oscillation of its own when one end of the needle approached and passed the earth's magnetic pole. If this happened to occur about the time of sunset and sunrise, it would produce the curious effect which contemporary observers attributed to an exchange of solar for stellar influence. 21 Ill

To return to Peiresc's long letter of I April 1635: this was duly sent to Roberto Galilei, who forwarded it to Galileo on 16 April. The next day, Peiresc received Galileo's second letter of thanks (dated 16 March), and immediately wrote again at considerable length, referring several times to the device displayed by Linus, as well as to the (thermometric) circular tubes containing liquid which had been constructed by Cornelius Drebbel as a tidal model. Thence he went on to observations of his own about tides, and to tentative conjectures about the mechanism of the water-clock and its value in proof of the Copernican system. 22 Peiresc had in fact already become so confident of securing amelioration of Galileo's sentence that he took steps to delay publication by Diodati of Bernegger's Latin translation of the Dia logo lest this damage his chances of success. 23 To all this good-hearted but mistaken enthusiasm, Galileo responded in the following words, transcribed from the original letter in the possession of Mr. Kenneth Knight: 111.mo et Ecc.mo Sig:re et Pad:" mio Col.mo La l[ette]ra di V. S.111:ma et Ecc:ma sparsa tutta d'affetti di cortesia, e benignita, con-

An Unpublished Letter of Galileo to Peiresc

51

tinua di fanni parer sempre piu soave la fortuna del mio infortunio, et in certo modo benedir le persecuzioni de' miei nimici; senza le quali mi sarebbe sempre restata occulta la parte piu da stimarsi dell 'humanita, e benigna propensione di molti miei Sig:mi e Pad:"i e sopra tutti I' Amore di V .E: ii quale non meritando d 'esser promosso da talento alcuno di virtu che Ia Natura habbia riposto in me, ha in vece di lei supplito Ia Sorte con accender nelle tor menti ii fuoco eldla Carita, con la quale vanno compassianando Io stato mio; net quale oltre alla ragion detta me e di non piccolo sollevamento ii creder che non un'animo che sempre piu si vada inasprendo sia quello che continui di tenenni oppresso, ma piu presto una quasi diro ragion di stato di quelli che voglion ricop[r]ir ii primo errore d'haver attorto offeso un'innocente, col continuar l'offesi, e i torti, accio l'universale si fonni concetto che possano altri gravi demeriti non fatti palesi aggravar la colpa del Reo. Hor sia quel che piace a chi e conceduta Ia potesta di fare ii suo arbitrio, che in tutti gl'eventi restero io perpetuam[ente] obbligato alla somma bonta di V.E. la quale con tanta premura si appassiona net mio interesse, e con tanta industria, e vigilanza indefessam[ente] va specolando i mezzi che possano essenni de sollevamento. L'Horologio hydraulico sara veramente cosa di estrema maraviglia, quando sia vero che ii Globo pendente net mezo dell' Acqua vadia naturalm[ente] volgendosi per occulta virtu magnetica. Io feci gia molti anni sono una simile invenzione; ma con l'aiuto d'un ingannevole artifizio; e Ia Machina era tale. II Globetto diviso con I 2. meridiani per le 24. hore era di rame, voto dentro, e con un pezzetto di Calamita, postogli net fondo, equilibrato quasi alla gravita dell' Acqua, siche posta net vaso una parte d'acqua salata, e poi sopra quella altra dolce, ii Globo si fennava Ira le due acque, cioe net mezo del vaso: ii qual vaso posava sopra un piede di legno dentor al quale stava ascosto un horologio fabbricato aposta con tal'arte che girava un pezzo di Calamita, che sopra vi era accomodata, facendogli fare una revoluzione i 24. hore, al cui moto ubidiva l'altra calamita posta net Globetto facendogli girare, e mostrar le hore. Sin qui arrivo gia la mia specolazione: ma se questa del P. Lino senz'altro artifizio fa che ii suo Globo ubbidisca al moto del Cielo, sara veram[ente] cosa celeste, e divina, et haremo un Moto perpetuo, V. E. con quei mezi che va nominando potra facilm[ente] venire in cognizione del tutto, io tra tanto ne l'ho voluto significare ii mio pensiero per havere un testimonio omni exceptione maius che non ho usurpata l'invenzione al P. Lino; se pero la sua machina non havesse altro di piu che la mia. Non devo nasconder a V. E. come sentendo un Principe grande l'ordine mandato dal S.10 Off:0 a tutti gl'Inquisitori di non dar licenza non solam[ente] che si ristampi alcuna delle opere mie gia molti anni fa pubblicate, ma che non si Iicenzi alcuna di nuove che io, ed altri volesse stampare, si che Ia proibizione e de omnibus editis, et edendis; si e preso assunto di fare stampare ii resto delle mie fatiche non publicare ancora, e forse si e mosso per curiosita di veder I'esito di questa impresa, e che fortuna correranno tali materie lontaniss: da proposiz.e attenente a religione piu che none ii Cielo dalla Terra. Io contro a mia voglia sono stato forzato a concedeme copia a S. A. sicuro che a me non ne

52

Instruments

possa succeder se non qualche travaglio; se bene non mi e stata fatta, ne accennata proibizione alcuna; per lo che non devo ne anco haver notizia del divieto fatto a gl 'Inquisitori; per lo che questo che conferisco a V. E. sia detto in confidenza. Da questo, e dall'esser state raccolte in Firenze, et in Roma tulle l'opere mie si che piu non se ne trovare per le librarie, apertam[ente) si scorge che si fa ogni opera per levar dal mondo la mia memoria; nella qual vanita, se sapessero i miei avversarii quanto poco io premo, forse non si mostrerebbero tanto ansiosi d'opprimenni. Io non finiro di parlar con lei senza di nuovo ringraziarla della sua infinita benignita, e del fervore col quale inviglia ne miei interessi, e se ii sollevare chi fuor di tulle sue colpe viene travagliato e alto meritorio, puo V. E. viver sicura che ne ricevera guiderdone dalla divina bonta. E qui con reverente affetto gli bacio le mani, e nella sua bonta gra[zia] mi ral:do Dalla Villa d' Arcetri le di V. S. 111:ma et Eccma

12

Maggio I 635 Dev:mo Et Obblig:mo Ser:'e Galileo Galilei Translation

Your Excellency's letter, filled throughout with feelings of courtesy and goodwill, continues to make the fortune of my misfortune appear sweeter to me, and in a certain way to bless the persecutions of my enemies, without which there would have remained concealed from me that which is most to be admired in humanity, and the benign inclination of many of my noble patrons, and above all your Excellency's love; all which, since it does not deserve to be aroused by any worthy talent that Nature might have reposed in me, Fortune has instead made up for through you by kindling in their minds the fire of charity, by which they are moved to compassion for my situation, in which, in addition to the reason mentioned, there is for me no little comfort in believing that it is not a spirit of ever-increasing cruelty that continues to hold me under oppression, 24 but rather, as I shall say, a sort of official policy on the part of those who want to cover up the original error of having wronged an innocent man, by continuing their offenses and wrongs so that people will conceive that other grave demerits, not made public, may exist to aggravate the guilt of the culprit. Well, let it be as pleases him to whom is granted the power to do what he will; for in all events I shall remain perpetually obliged to the consummate goodness of your Excellency, who is so energetically aroused in my interest, and with such industry and vigilance goes dauntlessly thinking up means to be of assistance to me. The water-clock will truly be a thing of extreme marvel if it is true that the globe suspended in the middle of the water goes naturally turning by an occult magnetic force. Many years ago I made a similar invention, but with the aid of a deceptive artifice, and the machine was this. The little globe with I 2 meridians for the 24 hours was of copper,

An Unpublished Letter of Galileo to Peiresc

53

hollow within, with a little piece of magnet placed at the bottom, and almost in balance with the density of water; so that placing in the vessel some salt water, and then on that some sweet water, the globe stayed between the two waters, that is, in the middle of the vessel, which vessel had a wooden base in which there was concealed a clock made expressly in such a way as to rotate a piece of magnet that was fitted upon it, making one revolution in 24 hours, which motion the other magnet placed in the little globe obeyed, making it tum and show the hour. Thus far went my speculation; but if this one of Father Linus without any trickery makes his globe obey the motion of the heavens, truly it will be a celestial and divine thing, and we shall have a perpetual motion. Your Excellency, by those means which you recite, will easily be able to come to a knowledge of the whole matter; I, meanwhile, have wished to indicate my thought in order to have a witness beyond all exception that I have not usurped the invention from Father Linus - if indeed his machine does not have any more to it than mine. I should not hide from your Excellency that a great prince, hearing of the order sent by the Holy Office to all the Inquisitors not to license not only the reprinting of any work of mine published many years ago, but not to license anything new that I and others wish to publish, so that the prohibition is de omnibus editis et edendis, has undertaken the task of having the balance of my still unpublished labors printed, and perhaps he has been moved by curiosity to see the outcome of this undertaking, and the fortune encountered by such materials, more remote from propositions pertaining to religion than are the heavens from the earth. I, against my will, was constrained to grant a copy to his Highness, certain that for my part only some travail must ensue, though I had not been given any prohibition whatever, nor the hint of one, since I might not even be notified of the ban given to the Inquisitors; wherefore let this that I impart to your Excellency be said in confidence. From this, and from the fact that in Florence and in Rome all my works have been called in, so that no more are to be found in the bookstores, it is easy to see that every effort is being made to remove all memory of me from the world, but if my adversaries knew how little I strive for such a vanity, perhaps they would not show themselves so anxious to oppress me. I cannot conclude speaking to you without again thanking you for your infinite kindness, and for the fervor with which you keep watch over my interests; and if it is a meritorious act to comfort one who is in trouble through no faults of his own, your Excellency may live in certainty of receiving reward from divine Providence. And here with reverent affection I kiss your hands, and in your good grace I am rejoiced. From the Villa of Arcetri, the 12th of May, 1635 from your Illustrious and Excellent Lordship's Most devoted and obligated servitor, Galileo Galilei

This letter was sent to Roberto Galilei, who on 28 May wrote to say:

54

Instruments

I have read and reread many times the letter your Excellency has written to the distinguished Councillor of Peiresc. Truly it is a golden letter, not only for the polish of its style, but for your Excellency's having exposed the stratagem of Father Linus, which I do not believe to be other than that which you describe in this letter. I sent your letter immediately to the said gentleman, and for my part I hope you shall have an answer to it. ... Shortly after writing yours of the 12th, you will have received the others, as I have a letter of that dale from my brother Girolamo saying that he had them ready to deliver to you. 2 s Peiresc received Galileo's letter on 25 May, and wrote to Gassendi the next day that "if my man had transcribed the letter, as I instructed him to, you should now see the copy, for I am sending the original unsealed to Mons. Rossi 26 and (thence] to Mons. Diodati tomorrow, God willing . ... " 2 7 It appears that God was willing, but the secretary was weak, and thus the letter passed from Peiresc's hands without a copy having been preserved among his papers, or Gassendi ' s. The original probably found its way safely to Diodati, but has wandered ever since without finding its way into the published correspondence either of its celebrated writer or its distinguished recipient. It may be noted that Peiresc put off answering Galileo's letter because of an expected visit of Henry Donnalius, who on his behalf had carefully observed the device of Father Linus. When he finally made his report (about 18 June), Peiresc found the evidence unsatisfactory, and concluded that some artifice similar to that described by Galileo must have been employed by the Jesuit, as he then wrote to Gassendi. 28 The only other known letter from Peiresc to Galileo is dated 24 February 1637, shortly before the death of Peiresc. In it he recounted the fortunes of his various pleas to Cardinal Barberini on Galileo's behalf, and described some observations that had been made with the excellent lenses sent by Galileo to Gassendi; but he did not again refer to the affair of the magnetic water-clock. 29 NOTES

Roberto was a distant relative of Galileo's, about thirty years his junior, who served as intermediary for much of Galileo's foreign correspondence after the trial of 1633. His brother Girolamo resided in Florence. By the use of these family connections, Galileo probably managed to escape rigorous censorship. 2 The letter was received by Peiresc on 25 May (Opere XVI, 268). Its supposed loss was noted as recently as 1959 by Comelis de Waard in the Correspondance du P. Marin Mersenne, V, 243, n. 3 and 222, n. I. 3 Peiresc had dwelt twice at Padua during his student days, first in the winter of 15991600, and again very briefly in the summer of 1601. I

An Unpublished Letter of Galileo to Peiresc

55

4 Opere XVI, 64. By coincidence, Bouchard's letter also announced to Galileo the arrival at Rome of the German Jesuit Father Athanasius Kircher with a mysterious clock which is further discussed below. 5 Ibid., 115-119. 6 Ibid., 153. 7 Ibid., 154. A copy of Galileo's letter, probably that which was sent to Gassendi, was forwarded to Peiresc; a notation in his hand appears on the copy published by Favaro, the original being lost. 8 Ibid., 169-171. 9 Ibid., 187 (2 January I 635). to Ibid., 2o6-207 (7 February). 11 Ibid., 215-216 (21 or 22 February). 12 Ibid., 202. 13 Ibid., 215-216. 14 Antwerp, 1634. The frontispiece of this book was designed by Rubens, which explains the fact that Peiresc turned to him, among others, for light on this device. A copy of the relevant diagram and passage from the book was found among Peiresc' s papers, and one was probably sent to Galileo with this letter. 15 Francis Linus ( vere Hall) was born at London in 1595 and entered the Society of Jesus in 1623. He taught mathematics at Liege for 22 years, after which he resided principally in England. Linus is remembered principally for his criticisms of Robert Boyle, which slimulated lhe laner to publish a precise formulalion of his gas law. 16 Opere XVI, 245 ff. 17 De Waard, op. cit., III, 435-436. 18 Opere XVI, 65. Probably the anraction of the two roots was also observed on the surface of water, and was similarly a surface-tension phenomenon, as explained below. 19 Galilee et la Belgique (Saint-Troud, 1892), pp. 127-141. 20 This suggestion, as will be seen, brings Kircher 's explanation into complete conformity with that of Galileo. It is not impossible that word of the contents of Galileo' s letter to Peiresc had reached Kircher before 1641, when the first edition of the Magnes was published. Nor is it entirely beyond belief that Galileo had described (or even built) such a device many years earlier, as he says, in which case both Linus and Kircher may have heard accounts of it during the period of widespread intense interest in the lodestone and its properties. On the whole, however, it appears most likely that Galileo and Kircher each independently deduced the nature of the mechanism that must have been employed by Linus, and that the English Jesuit was the only one actually to have built a model. 21 For most of the suggestions in this reconstruction, as well as for the interpretation of Pietrasancta 's florid prose, I am indebted to conversations with my son Daniel.

56

Instruments

22 Opere XVI, 25~262. 23 Ibid., 249. 24 This view had been previously expressed by Galileo in his letter to Diodati; cf. Opere XVI, I 16, lines 44-45. 25 Ibid., 269. 26 Sieur de Rossy, superintendent of posts at Lyons. 27 Opere XVI, 268. 28 Ibid., 280. 29 Opere XVII, 33.

Although concentrating most of his effort on Galileo, Drake was interested, and read widely, in the history of the physical and mathematical sciences from antiquity to the twentieth century. Nevertheless, most of his published papers on other aspects of the history of science are in one way or another related to his work on Galileo. Thus, he referred often in his papers on motion and mechanics to Galileo's proportional treatment of physical quantities for distance, speed, and time in the Two New Sciences. "Euclid Book V from Eudoxus to Dedekind" (I) is a consideration of the significance of Definition IV, omitted in the standard Latin translation by Campanus, for the theory of proportion of continuous magnitudes, used by Galileo, rather than of numbers. His interest in proportional theory can also be found in "Bradwardine's Function, Mediate Denomination, and Multiple Continua" (3) and "Medieval Ratio Theory vs Compound Medicines in the Origins of Bradwardine's Rule" (4), both concerned with the correct meaning and expression of Bradwardine's principle that the ratios of speed in motion are proportional to the ratios of the motive powers to resistances. "Hipparchus - Geminus - Galileo" (2) concerns an explanation by Hipparchus of natural acceleration in fall as a loss of upward motion, referred to by Galileo in his De motu, in which he put forth the same explanation. The importance of the recovery of ancient science in the Renaissance is considered in two papers. The more general, "Early Science and the Printed Book: The Spread of Science beyond the Universities" (5), very neatly described by its title, is about the function of printing both in the recovery of the science of antiquity, displacing the scholastic science of the universities, and in disseminating the new science produced outside of the universities, of which the work of Tartaglia is taken as a particularly important example. "The Pseudo-Aristotelian Questions of Mechanics and Renaissance Culture" (6), written with Paul Lawrence Rose, is an examination of the history of this very important text from the late fifteenth to the late sixteenth century as it was treated by humanists, vernacular writers, mathematicians, including university professors, and writers on engineering and mechanical devices. The paper is an entirely original study that can stand as a model of the kind of historical analysis that could be given to other works of ancient science in the Renaissance. Printing of vernacular books is again of importance in "An Agricultural Economist of the Late Renaissance" (7), which concerns a treatise by Giuseppe Ceredi on the water screw for irrigation, showing how practical ingenuity improved upon the inventions of antiquity. Early music was always one of Drake's keenest interests, as all his friends knew, and not surprisingly his interest extended beyond playing the viola da gamba to the place of music in early science. Galileo's use of music for timing descent on an inclined plane is treated in "The Role of Music in Galileo's

60

History of Science

Experiments" (Volume II, Part VI, chapter 13). "Renaissance Music and Experimental Science" (8), published five years earlier, proposes that experiments by G.B. Benedetti and Vincenzio Galilei to refute G. Zarlino's purely arithmetic theory of musical intervals and consonance can be taken as the beginning of experimental physics, a lesson in method not lost on Vincenzio's son. The subject is again taken up in "Music and Philosophy in Early Modem Science" (9), one of Drake's last published papers, which extends the discussion of experiment to Galileo's inclined plane and pendulum experiments (Volume II, Part VI, chapters 16 to 19). The paper is also notable for a clear and forceful statement of his views on the absence of relation between physics as practised by Galileo and natural philosophy. Acceleration in free fall apart from Galileo's law of fall is treated in a series of four related papers, which cover a variety of topics concerned with impetus and free fall. "Impetus Theory and Quanta of Speed before and after Galileo" (10) considers Buridan's impetus theory of fall, a causal explanation of acceleration, as a discontinuous rather than continuous acquisition of speed, and traces the development of the theory through Honore Fabri and Giovanni Battista Baliani. Both in this paper and more so in the following "Free Fall from Albert of Saxony to Honore Fabri" (I I), it is emphasized that this is not a description of fall from rest, but of discontinuous intervals of time and distance. "Impetus Theory Reappraised" (12) is a less technical discussion of the subjects of the preceding papers, with particular attention to definition of technical terms; it is probably advisable to read it first. Finally, "A Further Reappraisal of Impetus Theory: Buridan, Benedetti, and Galileo" ( I 3) considers principally Benedetti' s restriction of impetus to straight motion and the relation of this to Galileo's neutral motion in his early De motu. If papers (2) and (3) on Bradwardine's function are considered not quite pure mathematics, then Drake's only paper on pure mathematics, number theory in fact, is "The Rule behind •Mersenne 's Numbers'" ( I 4). These are primes of the form 2n - 1, originally of interest because of their relation to perfect numbers, 2,._ 1(2n - 1 ), equal to the sum of their factors. Mersenne 's list of such numbers contains errors of both omission and inclusion, and the paper is devoted to showing the rule that accounts for the list. The last paper in this section, "Newton's Apple and Galileo's Dialogue" ( 15), which is admittedly speculative, suggests that Newton's original insight concerning the descent of bodies and the acceleration of the moon toward the earth may have been suggested by the discussion and diagram in the Second Day of the Dialogue concerning the impossibility of the projection of bodies due to the rotation of the earth.

1

Euclid Book V from Eudoxus to Dedekind

I

MAGNITUDE AND RA TIO

The fifth book of Euclid's Elements set forth a general theory of ratios and proportionality among magnitudes. Creation of the theory is usually credited to Eudoxus of Cnidos, who lived at the time of Plato and Aristotle about a halfcentury before Euclid's Elements were compiled. Until Eudoxus the formal theory of ratios and proportion had dealt exclusively with relations between numbers, which (as then defined) were capable of division only as far as their prime factors. The concept of magnitude, discussed by Aristotle, included capability of indefinite division into smaller magnitudes and implied impossibility of a least magnitude. Treatment of ratios, if granted to exist between magnitudes, must accordingly differ essentially from treatment of number ratios. The established theory of numerical proportion was presented in Book VII, the first of Euclid 's "arithmetical books." The younger theory of ratios among magnitudes appeared, understandably, among his introductory six geometrical books. The basic concepts employed by Eudoxus in founding the theory of ratios among magnitudes were that of measure, and that of greater or less. Two definitions open Book V: I / 2 /

A magnitude is part of a magnitude, the less of the greater, when it measures the greater. The greater is a multiple of the less when it is measured by the less. The concept of measure was not logically necessary for the number concept;

Reprinted from History in Mathematics Education, ed. I. Granan-Guiness, Cahiers d' histoire et de Phi/osophie des sciences, n.s. 21 ( 1987), 52--04, by permission.

62

History of Science

the notion of order sufficed. Measure was nevertheless the key concept in numerical ratio theory. A "part" of a number had been defined (VII, Def. 3) exactly as was a part of a magnitude, above. In numerical ratio theory, "parts" of a number were further defined, to include any smaller number that did not measure a given number. That was possible because the unit (what we call the number one) had been distinguished and excluded from numbers by defining number as "multitude of units" (VII, Def. 2). No analogous "parts" of a magnitude could be usefully defined, by reason of the non-existence of any unit magnitude. Instead, the next step was to define ratio as applicable to magnitudes. In that definition Eudoxus used the word pelikotes, not found elsewhere in Euclid. Heath rendered it in English as "size," recounting other proposals and the reasons for which several commentators rejected the whole definition as spurious (Euclid, vol. 2, 116-117). The Greek word was used in comparing ages, of which we would not say they have size, though of any two there is a greater and a lesser. The same is true of offences, for example, or of loves, and various other things that are compared without assigning sizes. In keeping with these things and the preceding definition, I would give this one as: 3 / A ratio is a sort of relation in respect of greater or less between two magnitudes of the same kind. The third definition was not used in any theorem, which made it seem superfluous to commentators who rejected it. But it would not have seemed superfluous to anyone who intended next to define "have a ratio," as did Eudoxus: 4 / Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another. This definition is of particular interest and importance, both intrinsically and with respect to the subsequent history of Book V. The property named in the definition held automatically for numbers, by the definition of "multiply" (VII, Def. 15). Accordingly it was unnecessary for Euclid to mention numbers in Definition 4 or to repeat that definition in Book VII with "number" substituted for "magnitude." These things should be remembered when I attempt later to account historically for omission of Definition 4 from the standard Latin version of the Elements until the sixteenth century. Readers should also remember them when reading Heath (Euclid, vol. 2, 113, 126, etc.) and other commentators who suppose that number was a particular case of magnitude, concerning Euclid's two treatments of proportion. Classification of what is not indefinitely divisible as a particular case of what is indefinitely divisible cannot safely be

Euclid Book V from Eudoxus to Dedekind

63

attributed to Euclid, in my view, though anyone now is free to generalize however he pleases. 2 TEACHING OLD CONCEPTS

However interesting what has been said thus far may be to historians, it may appear to be of no use to teachers of mathematics. Theories of ratio and proportionality lost importance after the emergence of algebra in equation fonn with conventional rules and symbols as now taught to students. Simple transfers of tenns from one side of an equation to the other have long replaced the tedious reasoning of early Greek mathematicians. Likewise we might smile patronizingly at the Greek practice of representing numbers by letters of the alphabet instead of developing a place-system of notation, treating the unit as a number, and advancing to decimal factions in which the ancient distinction between number and ratio is quickly lost to view. (Is I .4142 a number or a ratio?) In fact those things came much later, after quite a lot of respectable mathematics had been thought out without them and was not always exactly translated into the new fonns. It is not self-evident how far teachers should go in discarding apparently excess baggage from mathematics. I now press buttons on a pocket calculator instead of solving problems in the way I was taught, but I seriously doubt that mathematical education should be replaced by instruction in the use of electronic devices. Mathematically as well as historically there was great merit in Euclid, Book V, despite the "failure" of its author to recognize numbers as a particular case of magnitudes. That was quite clear to Richard Dedekind a century ago when he created the real number system and laid a foundation for modem understanding of the continuum. The fifth definition in Book V, he said, had been the source of his theory as well as of several more or less complete attempts to introduce logically irrational numbers into arithmetic. (Dedekind's statement will be cited more fully and in context in §6 below). He himself embarked on his investigations, Dedekind said, when as a teacher of mathematics "I found myself for the first time obliged to lecture on the elements of the differential calculus and felt more keenly than ever before the lack of a really scientific foundation for arithmetic" (1872, 1). Teachers not infrequently become sensitive to difficulties that do not occur to their students. By the same token, students of mathematics often feel uneasy for reasons they find it hard to express, or that they fear ridicule of any attempt to express. As Dedekind remarked: "It happens that many very complicated notions (as for example that of [cardinal) number) are erroneously regarded as simple" (1888, 34).

64

History of Science

Historical understanding of the origin and evolution of concepts as basic as those of number and ratio might enhance a teacher's appreciation of real difficulties felt by thoughtful students whose defection from the study of mathematics would be most regrettable. Also, the fact that Euclid's definition of magnitudes in the same ratio turned out to be useful again two millennia later is not my only reason for choosing the present topic. Effective Joss to Europe of Book V until the sixteenth century had positive as well as negative effects in mathematics. Its recovery and exploitation then created mathematical physics in recognizably modem form, and simultaneously opened one of the main roads to invention of the calculus. In tum its prompt replacement by algebraic manipulations Jed also to further effects, both positive and negative. Above all, it is only recently that the overall history of Book V has begun to make sense. Though I may be mistaken in thinking the story will be of practical use to teachers of mathematics, I am confident that most will find parts of it novel and interesting. 3

EUDOXUS AND EUCLID

Researches by Professor Wilbur Kriorr in recent years, as a specialist in early Greek mathematics, call into question a historical tradition of very Jong standing. It has been customary to credit Eudoxus with the creation in toto of the theory of proportion presented in Euclid, Book V. Knorr 1978 found and presented strong reasons to believe that an evolutionary process rather than a sudden revolution took place. For pre-Eudoxian extension of the old numerical ratio theory, I refer readers to his illuminating account. My view of the contribution by Eudoxus differs from Knorr's, though it fits in with the same evolutionary perspective. It appears to me to have been made incidentally to another achievement of Eudoxus, rather than specifically to advance the theory of proportion. We have it on the authority of Archimedes that it was Eudoxus who first offered proofs of results found by the so-called method of exhaustion (Archimedes, 2). Assuming, as I have done, that the first four definitions in Book V were Eudoxian in origin, only one more seems to me to have been required for proofs by the method of exhaustion. In that method identity or equality is never reached; hence the conditions of "same ratio" did not have to be determined. Definition of "have a greater ratio" would suffice, inasmuch as that would entail the meaning of "have a lesser ratio." A definition of "have a greater ratio" is of course given in Book V. There it appears as Definition 7 and is phrased in such a way as to refer back to wording in the celebrated fifth definition. That is quite understandable, but it was not necessary for the purpose I ascribe to Eudoxus. Originally it could have

Euclid Book V from Eudoxus to Dedekind

65

been simpler; if I am _not mistaken, the fifth Eudoxian definition could have read: [5 Eud.) When, of four magnitudes, the first is greater than the second but the third is not greater than the fourth, the first magnitude is said to have a greater ratio to the second than the third to the fourth.

I am inclined to credit Euclid himself with completion of the theory of ratios among magnitudes by supplying the definition now numbered fifth, which begins "Magnitudes are said to be in the same ratio ..." Eudoxus would have adhered to the verb used in his definitions of "have a ratio" and of "have a greater ratio." In fact Archimedes, who was the peerless master of the method of exhaustion, used "have the same ratio" and not the wording found in Euclid's Elements, though Archimedes came after Euclid. Concerning Euclid's definition of "same ratio" I shall offer one comment that I have not found elsewhere, though it must have occurred to others. I do this because certain kinds of perplexity hunt in pairs, so to speak. Heath, thinking that Euclid must have regarded number as a kind of magnitude, believed it remarkable that he dealt separately with ratios of magnitudes and ratios of numbers. Heath explained that as excessive devotion to tradition. But then he was again puzzled (as Simpson and others had been) by the absence of any proof in the Elements that the two separate treatments were free of conflict (Euclid, vol. 3, 25). My comment is as follows. The phrase "are alike equal to" in Heath's translation of Definition 5 was not required in defining "same ratio" for magnitudes capable of indefinite division. Those words took care of any case in which division came to an end; that is, precisely the case of numerical ratios that were treated of in Book VII. Hence the proof supplied by Simpson for the Elements is superfluous unless we suppose that numbers, however defined, were recognizable to the Greeks as necessarily some kind of magnitudes. Why that ought not to be supposed has already been explained above. 4 MEDIEVAL RATIO THEORIES

The Elements were correctly translated into Latin from Greek in southern Italy during the twelfth century, but that translation had no detectable influence on European mathematics. Later in the same century Adelard of Bath, while residing in Spain, made a Latin translation from an Arabic version. It was, on the whole, fairly reliable and complete except for two serious defects in the definitions of Book V. Soon afterwards the founding of universities in Europe began.

66

History of Science

Another Latin translation, also from an Arabic text, was made by Campanus of Novara and embellished with extensive commentaries (Molland 1983). The same two defects in Book V definitions occurred in this annotated version, which remained the standard university text for several centuries. The defects were omission of Definition 4 and insertion of a spurious definition (of "continuously proportional"). The wording being virtually the same for two translators separated in space and time, it is probable that the alteration of Book V was of Arabic rather than European origin, though as far as I know the corresponding Arabic text has not been found. I wish to use the terrn "number-dogmatic" to characterize a certain attitude on the part of some mathematicians in every age and every land. It denotes a temperamental predilection that is uncorrelated with mathematical ability. Some outstanding mathematicians have been number-dogmatic in my sense, as well as some mediocre and some poor mathematicians. The well-known dictum of Leopold Kronecker - that God made the natural numbers and all the rest is the work of man - expresses quite well the attitude meant. It is frequently associated with impatience toward other ways of working than beginning always from the counting-numbers. Dedekind's view was rather different (1888, 35-36): ... it appears as something self-evident and not new that every theorem of algebra and higher analysis, no matter how remote, can be expressed as a theorem about natural numbers - a declaration I have heard repeatedly from the lips of Dirichlet. But I see nothing meritorious - and this was just as far from Dirichlet's thought - in actually performing this wearisome circumlocution and insisting on the use and recognition of no other than rational numbers. On the contrary, the greatest and most fruitful advances in mathematics and other sciences have invariably been made by the creation and introduction of new concepts.

Omission of the Eudoxian fourth definition from the text of Book V at any time seems curious, and its continued omission for centuries even more so. Some number-dogmatic editor might have deliberately removed it as selfevident, superfluous, and even noxious. It is indeed superfluous if all ratios are regarded as ultimately ratios between numbers under the definitions of Book VII. Now, supposing that the Eudoxian fourth definition was removed as a blemish by someone of that temperament, it becomes easier to account for the presence of the spurious definition that appears historically to have been associated with the omission. Without Definition 4, the question why Book V was written at all would arise. Campanus contributed an answer in his commentary on the spurious definition, which he took as genuine (Molland 1983). Euclid, he said, distinguished proportionalities into two kinds, the continuous and the sep-

Euclid Book V from Eudoxus to Dedekind

67

arated. The continuously proportional arose from repeated application of any ratio, and isolated instances of sameness of ratio fell under the fifth definition. Campanus probably divined correctly the intention of the spurious definition, though nothing can justify his attributing it to Euclid; it was circular and no definition at all. Its mathematical idiocy lends support to the idea that both alterations of the Book V definitions were the work of an editor incapable of understanding even the Pythagorean discovery of incommensurability. And whatever his misunderstanding, that of Campanus is inexcusable. Campanus had in hand an ancient comment on Definition 5 (now attributed to Theon of Alexandria) that he placed, unidentified, directly before his own long explanation that flatly contradicted it. This reduced the fifth definition to a circularity as bad as that of the spurious definition. The overall effect was to degrade Book V into a largely pointless excursion on continued proportion, leaving medieval mathematicians with only Book VII as a basis for proportion theory. Medieval theory remained numerical in outlook. Knowing as we do the elegance and power of Book V, it is natural to suppose that its effective loss to Europe was a tragedy for mathematics. It was, however, not an unmitigated tragedy. An extremely interesting extension of numerical ratio theory originated from Thomas Bradwardine's speculations about velocities in his I 328. By the end of that century this had flowered into a development by Nicole Oresme equivalent to a treatment of fractional exponents. Oresme 1 J6o recognized that the corresponding magnitudes probably could not exhaust the continuum, and he made a very respectable approach toward the Dedekind cut. The special terminology of medieval proportion theory continued to be taught long after recovery of the authentic definitions of Book V. For my present purpose it will suffice to explain the term "denomination," numerical expression of a ratio according to conventions of medieval proportion theory. It may be useful also to add that the word "irrational" was the Latin chosen for Greek alogos, the sense of which was not at all "unreasonable," but rather inexpressible, or nameless, in ordinary words and numbers. The following passage from Oresme's "On ratios of ratios" (meaning roughly "On rational exponents of rationals") should now serve to illustrate what I have said above (Oresme I J6o, 289-291 ): The denominations of some ratios are knowable; namely, those of all rationals and some irrationals, while the denominations of some irrationals are not knowable ... Thus, if there be some velocity arising from a ratio whose denomination is not knowable, it is impossible to make known its ratio. Nevertheless we can investigate by the second proposition [of Chapter 4] whether any ratio given (or to be given) to us is greater or less than [some] such irrational, unknowable, and unnameable ratio. Finally, in this way we can

68

History of Science

find two ratios sufficiently close to such an unknowable that that will be greater than the lesser ratio, and less than the greater one. And that ought to suffice.

5

THE RECOVERY OF BOOK V

Printed books eventually exposed to general scrutiny the conclusions that had been established among the learned. The standard Elements with commentaries by Campanus had twice been printed when Bartolomeo Zamberti, in 1505, published at Venice his Latin translation from a Greek manuscript. He was primarily a humanist whose purpose was to restore classical Greek texts previously known only through retranslations of Arabic versions. For the first time the correct definitions of Book V appeared in print, but Zamberti did not write explanatory commentaries, being content to excoriate Campanus in a long preface for his many errors now revealed. The hold that medieval proportion theory had gained by centuries of university acceptance is amusingly illustrated by the reception at Venice of Zamberti 'sedition. Luca Pacioli, whose reputation as a mathematician had been established by his huge volume on proportion (Pacioli 1 494), lectured against the correct definitions. In 1509 Pacioli produced a new Venetian edition of the medieval Euclid with the text of his lecture inserted before Book V. Understanding of the treasure unearthed by Zamberti appears to have begun in 1543, when the first complete translation of the Elements into modem language appeared, also at Venice. Its title page promised to place in the hands of any man of average intelligence the whole body of mathematics. It is the most entertaining and among the most informative editions of the Elements known to me. Presented in colloquial Italian are the full texts of both the medieval and the Zamberti versions, together with the Campanus commentaries, critiques of them, and a great deal of additional information. I regard it as the first printed Euclid with comments by a real mathematician. The author was Nicolo Tartaglia, remembered mainly for his solution of cubic equations (called by scholars "Cardan 's solution"); the edition cited here is Tartaglia 1569. Tartaglia was born at Brescia about 1500 and was as nearly self-educated as anyone ever was. From youth he had been teaching mathematics to practical men. Respect for academic tradition and authority did not cloud his mathematical judgment. Tartaglia recognized two things accomplished by the Eudoxian fourth definition, on which his own was the first modem commentary. First, it banished from the domain of ratios any unbounded magnitude. Second, it implied that a ratio must exist between the circumference and diameter of a circle, though that ratio remained unknown. That is hardly electrifying today, but it was then. Aristotle, whose philosophy dominated the universities, had built

Euclid Book V from Eudoxus to Dedekind

69

his science on the principle that no ratio could exist between the curved and the straight. That principle was doomed by Definition 4. In my opinion Aristotle would never have adopted it had he known the Eudoxian theory of ratios among magnitudes, as Heath believed that Aristotle did (Heath 1949, 223). Following his translation of the Campanus explanation of the spurious medieval definition, Tartaglia commented (1569, 84v): This definition is found only in the first translation, which definition I think, nay I hold for certain, is not Euclid's, and for three reasons. First, there is no reason for it, since neither by the definition nor by the commentary can we know or prove that three continuous magnitudes are continuously proportional ; and I much marvel that the commentator wants to define three continuously proportional quantities by three continuously proportional quantities; that is, by their multiples ... The second reason is that Euclid does not use this definition at any place in his whole work ... and it was not his habit to put in anything useless. The third reason is its absence from the second translation, from which I suppose it had been added by some [later] pretender to knowledge .... Had it been necessary, Euclid would have known how to define and where to place this, as I shall explain after the next commentary.

In the next commentary the bald internal contradiction was evident to Tartaglia, who inserted a dagger to separate the Theonine from the Campanus part. At the end he wrote (p. 85v): This exposition is doubtless a mixture of two different commentators, for which reason I divided it in two parts . ... Now, I say that whoever wrote the first part truly understood Euclid, because in it he explained very well and sufficiently the true sense of the [fifth) definition, nor is it necessary to impose ... any of the conditions set forth in the rest. ... The second part, which I believe is an addition by Campanus, not only muddies the sense of the definition, but totally confounds the student, who does not know which side is up in the welter of conditions and asides of liule truth ...

Similar treatment was given to the Campanus commentary on the definition of "have a greater ratio." Pointing out the error of Cam pan us concerning the definition of "doubled" (squared) ratio, Tartaglia made the valuable observation that when Euclid did deal with continued proportion, he dealt with magnitudes rather than with numbers, so that the latter should not be used to exemplify propositions in Book V (p. 88). That had been the practice in the Middle Ages, when all ratios were regarded as governed by Book VII, and Latin editions (as well as the first English edition of 1570) long continued to be so cluttered,

70

History of Science

"confounding the student" in Tartaglia's vivid phrase. The Euclid most used in universities half a century after Tartaglia, edited by Christopher Clavius, explained medieval proportion tenninology in Book V, where it was completely out of place (Clavius 1574). Tartaglia had more properly reserved that in his edition to Book VII (and Book X), where (if anywhere) it belonged. From 1543 on, Euclid Book V was again alive, and well, but living mainly in Italy and entirely outside the universities, to the best of my knowledge. Galileo's introduction to the Elements came not in a university but from a practical mathematician, Ostilio Ricci, who is said to have been a pupil of Tartaglia. Whether he was or not, Ricci taught from Tartaglia's Italian text in lectures to the Tuscan court, where Galileo heard him in 1 583. In an early treatise on motion of around 1588, Galileo proved a theorem relating to hydrostatics by using Euclid's definition of"same ratio." 6

BOOK V AND DEDEKIND

The context of Dedekind's allusion to Book V paraphrased earlier was discussion of differences between his theory of irrationals and those of some contemporaries. In 1872 he had published his fonnulation in Stetigkeit und irrationale Zahlen (after 14 years of reflection). As usual in such matters, claims of priority on behalf of others arose. In his preface to Was sind und was sollen die Zahlen? of 1888, Dedekind remarked that there was more to his theory than the conviction that an irrational number is defined by specification of all rationals that are less, and of all that are greater, than the irrational to be defined, adding (1888, 39-40): If one regards the irrational number as the ratio of two measurable quantities, then this manner of determining it was already set forth in the clearest possible way in the celebrated definition which Euclid gave of the equality [sameness] of ratios ... The same most ancient conviction has been the source of my theory as well as those of Bertrand and many more or less complete attempts to lay foundations for the introduction of irrational numbers into arithmetic ... Bertrand's presentation, in which the phenomenon of the cut in its logical purity is not even mentioned, has no similarity whatever to mine, inasmuch as he resorts at once to the existence of measurable quantity, a notion which for reasons outlined above I wholly reject.

As noted at the outset, the concept of measure was basic to both treatments of proportionality in the Elements. Dedekind dispensed with it and founded his work on the concept of order, the only thing logically required in establishing number. I shall return to the role of measure in Galileo's exploitation of Book V

Euclid Book V from Eudoxus to Dedekind

71

presently. As to number, it is widely assumed that what early Greek mathematicians meant by that was what we call the natural numbers; yet that is not quite so. In classical Greek mathematics, two was the least number, because one is not a multitude. It was not until 1585 that anyone proposed treatment of one as a number. This was Simon Stevin, who in L' arithmetique ( 1585), in the section on practical arithmetic, introduced decimal fractions. On grounds of strict rigor there were cogent objections against regarding one as a number, which of course could have been met by redefining "number" formally, in some logically satisfactory way. No one did, so far as I know, and the overwhelming practical advantages of Stevin's proposal and practice led quickly to universal adoption. The almost simultaneous emergence of systematic algebraic analysis in equation form, and its prompt application (by Descartes) to geometrical problems, opened a period of unprecedentedly rapid discovery. Ultimately, in my opinion, that was the source of the uneasiness felt by Dedekind in 1858. The concept of measure, on which Eudoxus had founded the theory of ratios among continuous magnitudes, became also the basis of mathematical physics when the authentic definitions of Book V were restored by Zamberti, explained by Tartaglia, and exploited by Galileo. To measurements of actual motions he applied the Euclidean doctrine of proportion among continuous magnitudes, finding the laws of fall and of the pendulum. Those were first stated only in proportionality form, for Galileo never wrote an equation in his life. At the end of his life, when blind, Galileo dictated an essay on the definitions of Book V, his purpose being to make the Euclidean theory easier for students (Drake 1978, 422-436). For that he proposed an assumption based on the defi nition of "greater ratio" from which the conditions of "same ratio" could be derived. If my account of the roles of Eudoxus and Euclid in originating and completing Book V is correct, Galileo's essay repeated the order of its historical evolution, which of course remained unknown at that time. The essay was published three decades after Galileo's death, but by that time algebra had replaced proportion theory in practice. Newton warned against confusion of space and time with their measures, a rapidly growing tendency with the tum to equations (1687, 11). 7 BOOK V, THE CONTINUUM, AND NUMBER

Bonaventura Cavalieri studied mathematics under Galileo's ablest pupil, Benedetto Castelli, who introduced him to Galileo. In 1621- 1622 Cavalieri fell to thinking about a kind of paradox arising from Book V and wrote to Galileo for assistance. It happened that the same puzzle had arisen in connection with instantaneous speeds during fall quite early in Galileo's studies of motion and

72

History of Science

had not been satisfactorily resolved by him for nearly five years (1604-1609). His solution of the physical problem had been a bit more radical in its implications than Cavalieri' s eventual resolution of the purely mathematical puzzle, which opened one road to the later invention of the calculus. That, however, has been somewhat obscured by reason of Cavalieri 's choice of the word "indivisibles" to characterize elements of one less dimension than the magnitude under consideration. The same word was used by Roberval for a very different concept, and it had long had a bad name among philosophers as a synonym for atoms, which were detested by Aristotelians. In his book "Geometry advanced in a certain way by indivisibles of continua," Cavalieri 1635 did not define indivisibles as such, and used the word only incidentally. What he defined were the phrases "all the lines of a plane figure," "all the planes of a solid," and "all the points of a line" (Andersen 1983). A passage from his first letter to Galileo indicates the origin and implies the outcome ofCavalieri's investigations (Drake 1978, 282):

If in a plane figure any straight line be drawn, and then all the lines parallel to it, I call these "all the lines of the figure"; and if in a solid are drawn all the planes parallel to a certain plane, I call those "all the planes of the solid." Now, I want to know if all the lines in one plane figure have a ratio to all the lines of another one. It appears that all the lines are indefinitely many and therefore fall outside the definition [V, Def. 4) of magnitudes that have a ratio to another. But since, when the figure is enlarged, the lines are also enlarged by the excess of the new figure over the old, it appears that they do not fall outside that definition. Years later, in his published book, Cavalieri did his best to make it clear that he did not assert, and had no need to assert, that a plane figure is composed of lines. Immediately after his first principal proposition (that "all the lines" are magnitudes having ratios, as are "all the planes") he appended a scholium (1635, 205-206): Someone may question this proposition, not well understanding how indefinitely many lines, or planes, as may be deemed what I call "all the lines" or "all the planes" of this or that entity, can be mutually compared. For that reason it seems to me that I should say that when I consider all the lines or all the planes of a figure I do not compare their number, which we do not know, but only their magnitude, which equals the space occupied by the said lines, they being congruent with it. Since that space is bounded, their magnitude is also enclosed within the same bounds, and can be added or subtracted without our knowing their number. That suffices, I say, for their mutual comparison; for otherwise not even the spaces of the figures would be comparable. In a word, the continuum is

Euclid Book V from Eudoxus to Dedekind

73

either nothing but these very indivisibles, or it is something else. If nothing but these very indivisibles, doubtless unless their aggregates can be compared the space or continuum cannot be compared. But if the continuum is something else, it is fair to assume that whatever this is lies between the indivisibles, so that the continuum is resolvable into something-elses which compose the continuum and are similarly indefinitely numerous. Cavalieri's caution was probably partly inspired by the antipathy of most intellectuals against atomism of any kind, but it represented chiefly his own mathematical conservatism. Euclid had similarly exercised great care to keep paradoxes of the infinite out of mathematics. An example is the axiom in Book I, "The whole is greater than the part." Galileo showed that that did not hold true of infinite aggregates (1638, 40-41 ); by establishing a one-to-one correspondence between members of such a whole and its half, he was able to derive the law of fall from the definition of uniform acceleration. The introduction of infinite aggregates into mathematics dates from the exploitation of Euclid, Book V, by Galileo and Cavalieri (pp. 157, 160, 165-166 ). Introduction of the equation form, conventional symbolisms, and rules for transposition soon displaced proportion theory in mathematical practice. The shift from a mathematics conducted largely in ordinary language to a mathematics conducted largely in symbols inaugurated a long period during which rapid advances distracted attention from linguistic expressions. In the end that inattention was remedied, and Dedekind was able to write ( 1888, 33): This memoir can be understood by anyone possessing what is usually called good common sense; no technical philosophical, or mathematical, knowledge is in the least degree required. But I feel conscious that many a reader will scarcely recognize in the shadowy forms I bring before him his numbers, which all his life long have accompanied him as faithful and familiar friends.

8

BIBLIOGRAPHY

"E" indicates edition used. The list contains some items not cited in the text. Andersen, K. 1983 Cavalieri' s method of indivisibles, Aarhus (Universitet). Archimedes E: The works of Archimedes, ed. T.L. Heath, 1897, Cambridge (University Press). [Reprinted New York (Dover).] Bradwardine, T. 1328 Tractatus de proportionibus. E: ed. H.L. Crosby, 1955, Madison (University of Wisconsin Press). Cavalieri, 8. 1635 Geometria indivisibilibus continuorum nova quadam ratione pro-

74

History of Science

mota. Bologna. E: Geometria degli indfrisibili, ed. L. Lombardo-Radice, 1966, Torino (U.T.E.T.). Clavius, C. 1574 Euc/idis e/ementorium libri XV, 2 vols., Rome. Dedekind, R. 1872 Stetigkeit und irrationa/e Zah/en, Braunschweig (Vieweg). E: 1901 , 1-27. - 1888 Was sind und was sol/en die Zahlen? , Braunschweig (Vieweg). E: 1901, 29115. - 1901 Essays on the theory of number, trans. W.W. Beman, Chicago (Open Court). [Reprinted 1963, New York (Dover).] Drake, S. 1970 "Bradwardine's function, mediate denomination, and multiple continua," Physis, 12, 51-68. - 1973 "Velocity and Eudoxian proportion theory," Physis, 15, 49-64. - 1974 "Impetus theory and quanta of speed," Physis, 16, 46-75. - 1975 " Free fall from Albert of Saxony to Honore Fabri," Stud. hist. phi/. sci .. 5, 347386. - 1978 Galileo at work, Chicago (University of Chicago Press). Euclid E: The thirteen books of Euclid's Elements, ed. and trans. T.L. Healh, 2nd ed., 1926, Cambridge (University Press). [Reprinted in 2 vols., New York (Dover).] Frajese, A. 1964 Galileo matematico, Rome (Editrice Studium). Galilei, G. 1638 Discorsi e dimostrazioni matematiche intorno a due nuove scienze, Leiden, E: Two new sciences, ed. S. Drake, 1974, Madison (University of Wisconsin Press) Heath, T.L. 1921 A history of Greek mathematics, vol. I, Oxford (Clarendon Press). - 1949 Mathematics in Aristotle, Oxford (Clarendon Press). Jones, C.V. 1978 "The concept of One as a number," Ph.D. thesis (University ofToronlo). Knorr, W.R. 1975 The evolution of the Euclidean elements, Dordrecht (Reidel). - 1978 "Archimedes and the pre-Euclidean proportion theory," Arch. int. hist. sci .. 28, 183-244. Le Tenneur, A.J. 1640 Traite des quantitez incommensurables, Paris (Boullenger). [Pseudonym I.N.T.Q.L.: against Slevin.] Molland, A.G. 1983 "Campanus and Eudoxus; or, trouble with texts and quantifiers," Physis, 25, 213-225. Murdoch, J. 1963 "The medieval language of proportion," in Scientific change, ed. A.C. Crombie, London (Heinemann), 237-271. - 1971 "Transmission of the Elements," in "Euclid" in Dictionary of scientific biography, vol. 4, 437-456. Newton, I. 1687 Phi/osophiae naturalis principia mathematica, London (Royal Society), E: Mathematical principles ... , ed. F. Cajori, 2 vols., 1934, Berkeley (University of California Press).

Euclid Book V from Eudoxus to Dedekind

75

Oresme, N. 1 J6o De proportionibus proportionum. E: ed. and trans. E. Grant, 1966, Madison (University of Wisconsin Press). [Original date is approximate.] - Questiones super geometriam Euclidis. E: ed. H.L.L. Busard, 1961, Leiden (Brill). Pacioli, L. 1494 Summa de arithmetica, geometrica. proportioni et proportionalita, Venice (Paganini). Saccheri, G. 1733 Euc/idis ab omni naevo vindicatus, Milan (Montani), I 02-131. Stevin, S. 1585 L'arithmetique, Leiden. E: The principal works of Simon Stevin, ed. E.J. Dijksterhuis, vol. 2a, 1958, Amsterdam (Swets and Zeitlinger). Tartaglia, N. I 569 Euclide ... secondo le due tradottione con una ample esponsitione ... , Venice (Rossinelli). [The first edition, 1543, has a different pagination.] Viviani, V. 1674 Quinto Ubro degli elementi d' Euclide ... col/a dottrina def Galileo, Florence (Condotta). [Reprinted 1690, and later with the other books of the Elements.]

2

Hipparchus - Geminus - Galileo

Ever since Pierre Duhem's rich discoveries among medieval manuscripts relating to natural philosophy, vigorous attempts have been made to link Galileo's work on motion to developments in the 14th century. Underlying those attempts is a conviction that no advance in science has occurred which was not dependent upon previous developments of no great age. At Galileo's time the most recent epoch of advance in physical thought had been the 14th century, to which Duhem and later historians turned for sources of the 17th-century revolution in physical science. In that quest it was understandably overlooked that there had been an unsuccessful attempt, in Greek antiquity, to direct physical science along the lines it took in the 17th century. Galileo himself started out in that direction before he learned that his first steps had been anticipated around 150-125 BC. Galileo's first steps had been taken in 1586-1587, soon after the end of his student days. They can be retraced and ordered from his early manuscripts and were recalled in two autobiographical passages of the Pisan De motu of I 590- I 59 I . In a chapter on the cause of acceleration in fall , after remarking that the cause assigned by Aristotle had never appealed to him, Galileo wrote: When I discovered an explanation that was completely sound (at least in my own judgment), I at first rejoiced. But when I examined it more carefully, I mistrusted its apparent freedom from any difficulty. And now, having finally ironed out every difficulty with the passage of time, I shall publish it in its exact and fully proved form. 1 This account is consistent with other documents in Galileo's hand. His rejecReprinted from Studies in History and Philosophy of Science 20, 47-56, copyright 1989, with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB, UK.

Hipparchus - Geminus - Galileo

77

tion of Aristotle's doctrine that heaviness, the cause of fall, increases in the process of fall, dated from his student days and originated in his having observed hailstones of different sizes striking the ground together after the very long fall. His own theory of fall began from an analogy between fall and sinking in water, set forth in a dialogue written in 1586--1587. Some subsequent memoranda indicate the nature of the problems to be ironed out. In De motu, having stated his basic explanation of acceleration in fall as the loss of a finite residual upward inclination to motion, he went on to say: After I had thought it out, and happened two months later to be reading what Alexander2 says on the subject, I learned from him that this had also been the view of the very able philosopher Hipparchus, who is cited by the teamed Ptolemy. Hipparchus is, in fact, greatly esteemed, and he is extolled with the highest praises by Ptolemy throughout the whole text of his Almagest. Now according to Alexander, Hipparchus too believed that this was the cause of the acceleration of natural motion. But, since he added nothing beyond what we said above, the view seemed imperfect and was thought to deserve rejection by philosophers. For it seemed to apply only to those cases of natural motion which were preceded by forced motion, and not to be applicable to that motion which does not follow forced motion-3 Indeed, philosophers were not content to reject the view as imperfect; they considered it actually false, and not true even in the case where the motion was preceded by forced motion. But we shall add the matters not explained by Hipparchus, and shall show that the same cause holds good also in the case of motion not preceded by forced motion, attempting to free our explanation from every fallacy. I would not, however, say that Hipparchus was wholly underserving of criticism, for he left undetected a difficulty of great importance. 4

It will be seen in due course that these clues to Galileo's sources are sufficient to show how things that seem to have been taken by him from medieval sources were already present in what had been preserved from the writings on physics of the astronomer Hipparchus around I 50- I 2 5 BC. The one thing lacking has been a knowledge of why that astronomer had concerned himself with the theory of fall in the first place. In approaching that question it is best to begin from the time, soon after Hipparchus, when a well-known and extremely important statement about differences between astronomy and natural philosophy, ascribed to Gem in us, was set forth. In part, that statement was as follows: Alexander carefully quoted a certain explanation of Geminus taken from his summary of the Meteorologica of Posidonius. Geminus's comment, which is inspired by the views of Aristotle, is as follows .... The only things of which astronomy gives an account, it can

78

History of Science

establish by means of arithmetic and geometry. In many cases the astronomer and the physicist [i.e. natural philosopher] will propose to prove the same point, e.g. that the sun is of great size or that the earth is spherical; but they will not proceed by the same road. The physicist will prove each fact by considerations of essence or substance, of force, of its being best that things should be as they are, or of coming into being, and of change. The astronomer will prove things by the properties of figures or of magnitudes, or by the amount of motion and the time that is appropriate to it. Again, the physicist will in many cases reach the cause, by looking to creative force; but the astronomer, when he proves facts from external conditions, is not qualified to judge of the cause, as when for instance he declares the earth or the stars to be spherical. And sometimes he does not even desire to ascertain the cause, as when he reasons about an eclipse; and at other times he invents, by way of hypothesis, and states certain expedients by the assumption of which the phenomena will be saved.5 ... For it is no part of the business of an astronomer to know what is by its nature suited to a position of rest, and what sort of bodies are apt to move, so he introduces hypotheses under which some bodies remain fixed while others move, and then he considers which hypotheses will correspond to the phenomena actually observed in the heavens. But he must go to the physicist [i.e. the natural philosopher] for his first principles - namely, that the movements of the stars are simple, and uniform, and ordered; and by means of these principles he will then prove that the rhythmic motion of all alike is in circles, some being turned in parallel circles, others in oblique circles. 6

Before the year 100 BC it had become necessary to clarify the boundary line between the domains of astronomy and natural philosophy. The reason was that the astronomer Hipparchus had antagonized the philosophers by proving mathematically, from measurements extending back some four centuries, that the earth is not at the centre of the sun's apparent motion through the zodiac. The cosmology of Aristotle's De caelo placed the earth at the exact centre of the universe and of all the celestial motions. In his Metaphysics Aristotle had adopted the system of homocentric spheres, invented by Eudoxus to show how all the apparent irregularities of solar, lunar, and planetary motions could be produced by uniform revolutions of spheres concentric with the earth, the axis of each sphere being carried by the next sphere, held at a fixed angle. No actual measurements, of course, were necessary for the devising of this scheme, nor had any been needed for Aristotle's logical and normative proofs that the earth must be fixed at the centre of the universe. The useful work of astronomers, as for example the prediction of eclipses, depended on reasoning from careful measurements over long periods of time. Now, natural philosophers could not allow a mere mathematician like Hipparchus to contradict them in the higher knowledge of essences and causes, but neither could astronomers yield to natural philosophers in the carrying out of the useful work of astronomy.

Hipparchus - Geminus - Galileo

79

Geminus provided the compromise that was accepted on both sides until the time of Copemius and Kepler. The key to this was the statement emphasized above - the phenomena will be saved. Cosmologists, who knew the why of the heavenly motions, treated the task of astronomers as nothing more than to save appearances (for Greek "phenomenon" simply means "appearance"). Astronomical theories came to be viewed by later philosophers as mathematical fictions, devoid of any physical truth. Now, cosmologists were also physicists in the Aristotelian sense, so under the Geminus compromise they remained the final authorities on matters of physics. But Hipparchus had not only challenged Aristotle's cosmology; he had also written a book in contradiction of Aristotle's theory of fall and his conception of heaviness, striking at the very heart of accepted physics. In Aristotle's theory of fall, heaviness was treated as the cause of fall, while the natural striving of heavy things toward their natural place at the centre of the universe was the cause which always directed fall straight toward that centre; that is, the earth. Of bodies falling in the same medium, the heavier was the faster, while descent of a body through different media was swifter in the less corporeal medium. It was necessary for fall, as natural motion, Aristotle said, to be faster at the end than in its middle part; motion faster in the middle part was forced. To account for observed acceleration during fall, Aristotle said that the heaviness of a body increases in the process of fall. The contrary accounts advanced by Hipparchus are preserved only in part, likewise in the commentaries of Simplicius: ... there is general agreement that bodies move more swiftly as they approach their natural places, but various explanations are adduced. Aristotle holds that as bodies approach the totality of their own element they acquire a greater perfection; and thus by added heaviness earth is carried faster to the center. Hipparchus, in the book he wrote On bodies carried down by heaviness,? declares that in the case of earth thrown upward by a certain power [virtutem] which projects it, the projecting power is the cause 8 of the upward motion as long as the projecting power overcomes the downward tendency of the projectile, and that to the extent that this projecting power predominates, the object moves the more swiftly 9 upward. Then as this power diminishes (1) the upward motion continues, but no longer at the same rate; (2), the body then moves downward under the influence of its own internal impulse, even though the original projecting power lingers in some measure; (3), as this power continues to diminish, the object moves downward always more swiftly; and (4), most swiftly when this power is entirely lost. Now, Hipparchus asserts that the same cause operates in the case of bodies let fall from on high. For, he says, the power which held them back remains with them up to a

80

History of Science

certain point, and this opposition of contrarieties becomes the cause for the slower movement at the start of the fall. Alexander replies: "This may be true in the case of bodies moved or kept in place by force [vi] against their nature," rightly saying "but what was said would not fit when something upon coming into being is moved in accordance with its own proper nature to its proper place." 10 On the subject of heaviness, also, Hipparchus contradicts Aristotle, as he holds that bodies are heavier the further removed they are from their natural places. This, too, appears improbable to Alexander. "For," he says, "it is far more reasonable to suppose that when there is transmutation to another nature, as when the light becomes heavy, this still retains something of its former nature while it is yet at the very beginning of the downward fall and is just changing to that form by virtue of which it is carried downward, and that it becomes heavier as it goes along, than to suppose it still to be carried by the upward power which at the beginning detained it, and prohibited its being carried downward .... For if these bodies were to be moved downward more swiftly in proportion to their distance from above, it would be unreasonable to suppose that they exhibit this [same] property in proportion as they are less heavy .... For to hold such a view is to deny that these bodies move downward because of weight. Now, also in things projected upward by force, and in those held up, and in those changing place upward when by nature they go down, the s.ime is seen to be done." These things, Alexander says against Hipparchus; and he says well (as I deem), especially since if on account of heaviness the speed becomes less, as in weighs thrown [upward?], 11 it is clearly impossible that they be heavier by detraction, the heavier being what is more remote and is moved more swiftly than the speed existing when nearer. "The reason given by Aristotle for acceleration in natural motion, namely an addition of weight (or of lightness)," says Alexander, "is a sounder reason and more in accordance with nature. Aristotle would hold that acceleration is due to the fact that as the body approaches its natural place it attains its form in a purer degree; that is, if it is a heavy body, it becomes heavier, and if light, lighter."' 2

No Greek manuscript or Arabic translation of the book named above is known to survive. It is uncertain that even Simplicius himself knew more about it than Alexander of Aphrodisias had cited, two or three centuries earlier, in order to raise certain objections against it. Yet the title, On bodies carried down by heaviness, suggests a general treatise on heavy bodies, and not just on the matter of acceleration in fall. Indeed, that alone would not normally have called forth a whole book; and if it did, the author would certainly have adduced concrete evidence against the accepted theory of Aristotle. Inasmuch as Simplicius did not mention any reason for the composition of the book,• 3 I propose the following historical reconstruction of the circumstances. Almost as easily observable as acceleration in fall is the equality or near-

Hipparchus - Geminus - Galileo

8I

equality of times of fall by two bodies differing considerably in weight. Nevertheless, Aristotle cannot have made that simple observation when he wrote, quite explicitly: If a certain weight move a certain distance in a certain time, a greater weight will move the same distance in a shorter time, and the proportion which the weights bear to one another, the times too will bear to one another; e.g. if the half weight cover the distance in x, the whole weight will cover it in x/2. 14

That it was not difficult to refute Aristotle on this matter is shown by the observation of another 6th-century commentator, Philoponus (though the phenomenon was overlooked by Simplicius): If you let fall from the same height two weights, of which one is many times as heavy as the other, you will see that the ratio of the times required for the motion does not depend on the ratio of the weights, but that the difference in time is a very small one. 13

It is curious historically that this observation seems not to have been adduced until the 6th century, but neither does this near-equality of times appear to have been mentioned again until 1553. It then appeared not as an observation, but as a theorem published by Galileo's predecessor G.B. Benedetti. No reason is apparent why Hipparchus should not have observed this phenomenon in antiquity. As an astronomer he was a keen observer - and not one to hesitate about challenging Aristotle's science whenever it conflicted with observation. There is no reason to question that Hipparchus accepted the doctrine that heaviness is the cause of fall, but rejected its increase and a consequent acceleration throughout the fall of two bodies differing in weight, from the same height. For in such a case, the two would strike the earth at markedly different times in long falls. Observing that they do not, Hipparchus proposed only to limit the duration of acceleration from rest. The fact of acceleration was beyond doubt, as an observed phenomenon. In other words, that pioneer among systematic astronomers ventured also an improvement in physics, by "saving the phenomena" of natural motions by heavy bodies from the "physics" of Aristotle. Before the compromise of Geminus, not just astronomy but all of natural philosophy was open to criticism and improvement on the basis of observed phenomena of nature. Galileo placed himself in that same position with respect to natural philosophy as soon as he left the University of Pisa. He saw physical inquiry as an enterprise similar to astronomy, to be guided by principles taken from natural philosophers but required to respect the phenomena, not to "save" them in the

82

History of Science

old sense of preserving old principles at all costs. In the matter of fall, Galileo was at once led to the same conclusion as Hipparchus - that the duration of acceleration must be limited - before he learned that he was not the first to arrive at that theory from the phenomena. Now, the theory of fall proposed by Hipparchus did not perish with his book. The concept of an incorporeal motive power is also found in the commentaries of Philoponus on Aristotle's Physics, in a passage suggesting that he was unaware that anyone else had already proposed it: ... It is necessary to assume that some incorporeal motive force is imparted by the projector to the projectile, and that the air contributes either nothing at all or else very little to this motion of the projectile. If, then, forced motion is produced as 116 have suggested, it is quite evident that if one imparts motion contrary to nature, or forced motion, to an arrow or a stone ... there will be no need of any [material] agency external to the projector ... 17

The concept of an incorporeal force imparted to a body seems to have survived in the Arabic world mai/ 18 and to have passed into European medieval natural philosophy as the vis derelicta'9 of Francesco di Marchia in the 1320s. There is little doubt that this, in tum, inspired the impetus of Jean Buridan two decades later, seemingly a great conceptual advance in that the impressed force thereby became a thing not evanescent, but of a permanent 20 nature, reduced only by conflict with an external resistance or an internal contrary tendency to motion. For that reason it is customary to view impetus as a step toward the inertial concept. It has long been regarded as a great puzzle that the young Galileo adhered in his Pisan De motu to the seemingly primitive notion of self-expending impetus, as it has been called, after the concept of impetus as a kind of enduring impressed force had become generally accepted among all but the most stubbornly orthodox Aristotelians. This no longer remains puzzling in the light of the fact that Galileo was concerned with phenomena of natural motions of heavy bodies, and in particular those of fall. Like Aristotle, Galileo regarded forced motions as of secondary interest in his early writings. In contrast, Buridan' s impetus theory was primarily directed toward filling the hiatus left in physics by Aristotle's relative neglect of continued motion by a projectile after contact is lost with the inaugurating force. The conviction that no advance in science is made without dependence upon a previous development is justified, but the added condition "of no great age" is less clearly valid. In fact, the search for Galileo's sources in writings two centuries old entailed a certain improbability on the face of things. Had he been able

Hipparchus - Geminus - Galileo

83

to find inspiration for a new science of motion in Buridan, or Heytesbury, or Oresme, so would others before him have been. But no gradual progression from the 14th century to the 17th has been found. Galileo's first steps to his new science of motion were taken along the much older road that the truly original medieval natural philosophers had abandoned. What can be found is a source for the self-expending vis derelicta of Marchia in the Arabic mail, and a source for that in the incorporeal force of Philoponus, or in the work of Hipparchus as preserved by Simplicius. What happened in the 14th century was a "revolution," as Duhem perceived, when the ancient concept of an evanescent force gave way to the truly original medieval concept of an abiding force imparted to a heavy body. Because that concept appeared to anticipate in certain ways the concept of Newtonian inertia, historians looked for something like that in the writings of Galileo. But they looked in vain, for inertia is a dynamic concept, whereas Galileo's mature physics - to which Newton alluded with respect - was purely kinematic. Even in Galileo 's early writings, before he moved to Padua in 1592, no trace of an inertial concept can be found, though De motu attempted a kind of dynamics. In it the terms "impressed force" and "impetus" were used, interchangeably, but nothing of lasting value to Galileo emerged. Not until 1598 did Galileo find, in the Questions of mechanics then ascribed to Aristotle, a statement which put into his hands the principle of conservation of motion that took the place of inertia in his mature science of motion. The ancient Greek author's phrase rev ainov 1) may be doubled in addition or diminution ad inflnitum. 3

If the concept of square-root extraction, by which the velocity is halved ("doubled in diminution"), were already implicit in Bradwardine's general rule (let alone root extraction ad infinitum) - that is, if n in expression (2) were seen by him as capable of taking fractional values, or even the particular value 1/2 then there would be no need of any further assumption in addition to Theorem I. Hence it is reasonable to suppose that Bradwardine did not originally intend his general rule to include fractional values for n, but that he subsequently introduced values of the form 1/2n into a later theorem by a separate assumption, known independently of Theorem I. The form of expression he used in stating Theorem X tends to bear this out, with the phrase "twice as slow" where he might have put "half as fast." The same supposition is consistent with the verbal formulation of Bradwardine' s general rule, which permits the ratio of the two velocities to be arranged

90

History of Science

so as to have the larger come first; that is, to have the ratio of velocities greater than unity. This in tum would make the treatment of F.fR 1 easily understandable by his readers; namely, its self-multiplication an integral number of times. The absence of any explanation of fractional self-multiplication suggests that Bradwardine did not consider such a possibility. And in fact he suggests no root other than square root. Something should here be said concerning Bradwardine's conception of the square root. This was for him not a kind of number, but a kind of proportionality; the square root of a quantity was the geometric mean between that quantity and unity. The mean might happen to coincide with a number, in which case it was called a "rational ratio"; more often it did not, and was called an "irrational ratio." Every F/R > I was associated uniquely with a particular velocity V; associated with the velocity V/2 was the geometric mean between F/R and unity. For F = R, V = o; and F < R was excluded from motion (see Theorem VIII below). Our habit of considering all irrationals as numbers in a single continuum with the rational numbers makes it difficult for us to remember that to Bradwardine, irrationals were ratios, each dependent for its existence on two other ratios or quantities. On the assumption that Bradwardine began with those cases in which V2 was an integral multiple of V., it follows that the associated F 1/R I and F 2 /R 2 were necessarily "commensurable" in the sense of Bradwardine's theory of proportion when applied to powers; that is, both were members of the same power series.4 There is nothing in his text that conflicts with this assumption. Yet he certainly recognized that many velocities were incapable of being expressed as integral multiples of one another, as in the case V 1 = 2 and V2 =3. What, if anything, did he mean to say about the ratios of velocities in such cases? The answer to this question is crucial to any transition from expression (2), on the propriety of which all agree, to formula ( 1), the form of "Bradwardine's function." The prevailing view is that Bradwardine intended to express a single general rule governing all possible ratios of velocities, taken in any pairing whatever. This view is embodied in our customary unrestricted equation. But it is perfectly possible that Bradwardine meant rather to give a rule linking velocities in pairs, without asserting that all conceivable pairs existed. The repeated singular, "ratio" in his first statement certainly suggests this: "The ratio of speeds in motions follows the ratio of the motive powers to resistances, and conversely." Such a statement guarantees that wherever a pair of motive powers and resistances exists, a pair of speeds having a ratio dependent on them also exists. But it does not guarantee that every conceivable F-R couple forms a ratio, or that every conceivable speed exists for any given body. Likewise, Bradwardine's immediately following alternative statement makes only the first

Bradwardine's Function

91

guarantee: "The ratios of motive powers to resistances, and the speeds in motion, exist in the same order of proportion, and conversely." When we read these statements as representative of continuous functions in the modem sense, we make a gratuitous assumption. The assertions may have been intended in something like the sense in which we say: "The squares of prime numbers follow the prime numbers themselves, and conversely." No continuous function exists here, and no rule of proportionality is chosen because consideration of the medieval concept of "commensurability" - which was neither the Euclidean concept nor our own, nor is ours that of Euclid - seems to me to reveal an aspect of the matter that is not mentioned by historians of medieval mathematics. That is, the speeds associated with a particular integral FIR ratio through repeated squaring and square-root extraction - the only operations specifically mentioned by Bradwardine - were, in the medieval sense, entirely "incommensurable" (or non-communicating) with the speeds similarly associated with any other integral FIR ratio prime to it. In this sense, Bradwardine' s rule certainly says something quite general about the association of velocities in pairs, but it is far from offering any general formula that could relate all FIR couples to all pairs of velocities, in the sense of formula (I). In support of this interpretation of Bradwardine's general rule, I may cite his Theorem VIII, calling particular attention to the independent assumption that he requires for its proof: Theorem VIII. No motion follows from either a proportion of equality or one of lesser inequality between mover and moved [i.e., V = o if F/R ~ 1 ]. This may be proved from Theorems 7 and 8 of Chapter 1, and Theorem 1 [Bradwardine 's General Rule], with the addition of this assumption, known by itself: "All motions of the same species according to slowness or speed may be compared to each other." 5

The words "of the same species according to slowness or speed" appear to me to mean "commensurable with a particular FIR ratio," for that ratio was the underlying cause of speed in Bradwardine 's (and in Aristotle's) theory of motion. This interpretation is perfectly consistent with expression (2). If Bradwardine had had in mind the basic conception embodied in formula ( 1 ), it is hard to account for his words "of the same species." According to formula ( 1 ) , as we read it today, any two motions whatever may be compared. Pairs of FIR ratios not "commensurable" in Bradwardine's sense are made comparable, in our sense, by formula ( 1 ). Or rather, the idea of "commensurability" in Bradwardine 's sense is abandoned today and made quite pointless, and that is one way of exhibiting the precise anachronism inherent in formula ( 1) and in the expression, "Bradwardine's function ."

92

History of Science

We seldom think of the concept of commensurability today; hence we are likely to forget its special sense and the large role it played in medieval theory of proportion. Misunderstanding of the Eudoxian definition of ratio equality, given by Euclid in Book V but misinterpreted in the commentary of Campanus, compelled medieval mathematicians to construct a theory of proportion more arithmetical than geometrical, and hence far removed from our idea of the algebraic number continuum. As a result, mathematical rigour demanded their close attention to questions of "commensurability" in equating ratios. In my view, Bradwardine (and later Oresme) began by conceiving that the integral powers of the ratio F/R = 2/1 gave rise to an infinite set of velocities, the respective integral multiples of a particular V that was uniquely associated with F/R = 2/I. Powers of F/R = 3/I give rise to another such set, uniquely associated with a different V; powers of 3/2 to still another; and so on. Each such set constituted a "species of motion with regard to speed." Associated with each such species there was also a species of motion with regard to slowness, obtained from the relevant F/R by taking successive mean proportionals back toward unity; or, as we should put it, by successive integral root-extraction. All motions of the same species (same basic F/R) with regard to speed and slowness were mutually commensurable, and this, I take it, was Bradwardine's supplemental assumption for Theorem VIII. But no more general statement could be made, since the particular velocity generated by a specified F/R ratio was not detenninable, any more than the area of a particular circle could be detennined by Euclid, though he could fonnulate the rule relating it to the area of another given circle. Only the conditions for doubling and halving velocities were given by Bradwardine. A more general treatment was given by Oresme, equivalent to the manipulation of general fractional powers, but still under the same conception of "commensurability" and hence not properly expressed by fonnula (I). Before developing this point, another possible approach to the meaning of Bradwardine's "motions of the same species" is worth mentioning, though I think it implausible. The standard English translation of Bradwardine's treatise renders the passage differently, reading "All motions of the same species may be compared with each other with regard to slowness and fastness," where I read" ... species with regard to slowness or speed."6 Hence it might be argued that not F/R, but the classes of ratios, may detennine "species of motion." Ratios were indeed classified into three species, those of equality, greater inequality, and lesser inequality. But motions are not ratios, they are quantities; furthennore, no motion results from F/R ratios of equality or lesser inequality, as may be seen from Theorem VIII itself. Hence "all motions of the same species" cannot very well mean "all motions arising from ratios of greater inequal-

Bradwardine's Function

93

ity," which would be mere tautology at best. Nor can it mean " ... ratios respectively of greater inequality, lesser inequality, or equality," the two last-named being null classes. Either interpretation means no more than "all motions," and thus fails to explain the phrase "of the same species." I do not see how that phrase can reasonably be interpreted except by taking it to mean "commensurable" in Bradwardine's sense of arising within the same power series. 7 The essential restrictions, if I am correct, may be derived in modem notation thus, where F, R, V and n are restricted to positive integers and V 2 > V 1: (F 2/R 2) = (F 1/R 1 )", making F2/R 2 "commensurable" with F 1/R 1 (F2/R2)v' = [(F,JR,)"]v, = (F,JR,)°v, Hence

3/

(F /R )v' =(F /R )v2 {V2 =nV 1 ,F,R, Vandnpositiveintegers 2 2 I I F2/R2 = (F,JR,)" > I

Formula (3) is by no means equivalent to formula (I). It is true that to the modem eye (3) is immediately convertible into (I) by taking the V I th root of both sides. But obviously, such a demand is very different from a simple appeal to the meaning of the symbol"= ." In any case, the restrictions are the important thing here. Other reasons for preferring to avoid the general fractional form of exponent in formula ( 1 ), for Oresme as well as for Bradwardine, will be given later. When we are asked to believe that Bradwardine had in mind the concept of extracting any root whatever, and of equating what to him would be incommensurable quantities, the essentially anachronistic aspect of the conception contained in "Bradwardine's function" becomes apparent. In his chapter on motion, Bradwardine does not so much as mention even the cube root, let alone roots in general, although in the preceding discussion of proportion, geometric means corresponding to these had been mentioned. Significantly, the one theorem in which they were mentioned there is the only theorem of the associated group that is not used in any of the proofs of Bradwardine's six specific theorems regarding F/R ratios. 8 Oresme, on the other hand, did develop the conception of fractional powers, and he discussed at length the theory of a process which Bradwardine did not suggest by so much as the phrase "and so on. " 9 If Bradwardine's general rule is properly reflected by formula (3), he may have perceived that any two ratios belonging to the same "species" - that is, the same power series in integral exponents - could be brought into equality by the further raising of each to a compensating power. This conception is easily seen

94

History of Science

in formula (3). What is not reflected there, and is conspicuous in the unrestricted formula (I), is the concept that compensating powers must similarly exist for incommensurable values of FifR 2 and F ifR., an assertion that for Bradwardine (and even later for Oresme) would have appeared, if not a patent contradiction, no more than a wild surmise unworthy of a rigorous mathematician. Indeed, for Oresme's primary purpose in writing De proportionibus proportionum, the concept of incommensurability had to be emphasized, whereas formula (I) conceals this concept from us by introducing a single unified continuum. Oresme dealt with what may be called multiple continua, exploring their separations and connections; Bradwardine, in this Treatise at any rate, dealt with a single and highly specialized "continuum." The idea of many distinct numerical continua is as unfamiliar to us as the idea of a single unified continuum would have been to medieval writers. They considered as continuous any set of quantities in which any two are connected by a mean. For us, such a set is dense, but not necessarily continuous; that is, it guarantees only that between any two members, there exists another member. This is necessary, but not sufficient, to define our continuum. The medieval idea of "incommensurability" may be illustrated in terms of "multiple continua." A line segment bisected and rebisected ad infinitum would be cut at an infinite number of points, and those points would be considered by medieval writers as forming a continuum, because the (arithmetic) mean between any two of them would again be represented by a point belonging to the same set. But the points trisecting the same line-segment would not be among those points, nor would any of the infinite points corresponding to successive further trisections coincide with any of the points obtained by bisection. Even if each set were expanded by including all the geometric means between the pairs of points in either set, respectively, the two sets would still have not a single point in common. Repeated division of the same segment into six equal parts would cut the segment at all the points obtained by repeated bisection and trisection, and some additional ones, for example, the point I /36. Yet this would not conned the two previous "continua" together; it would only create a new "continuum" in which both the others participated without exhausting it. Bradwardine adumbrated these concepts and Oresme pursued them, correctly conjecturing that still other points might exist which would not be reached by any finite process of the same kind. Bradwardine, however, did not pursue his analysis of velocities beyond the case of bisection and the single geometric mean, or square root, taken repeatedly. The unique importance of the number 2 to Bradwardine, illustrated by his emphasis on bisection, square root, and the FIR ratio of 2/I as reference points in the analysis of motion, has not been sufficiently recognized in discussions of

Bradwardine's Function

95

his work, and of the work of medieval mathematicians generally. It has ramifications of the most interesting sort, both mathematical and philosophical. The arithmetic mean, central in importance to the Merton School in the investigation of uniform acceleration, is ultimately related to bisection. Arithmetic and geometric means, which latter (unlike harmonic means) defined "continuous" proportion, depended on the existence of two extremes. The idea of connection through a middle term (mean) played a large role in Aristotelian philosophy. The square and square root are of special significance with respect to commensurability in Euclidean geometry. Thus a special medieval interest in the number 2 is not surprising, and it should caution us against gratuitous generalizations of results reached by medieval mathematicians that are specifically linked only to this number in their writings. If I understand Bradwardine and Oresme correctly, there was simply no reason for either of them to investigate motions having any other base than FIR == 2/1. All the relations that could exist within a set of motions having any other base existed within that one, and there was no way, in general, to pass from any base to any other. The favoured base contained, moreover, the only case in which a "ratio of ratios" could be the same as the "ratio of denominations" (because 2 2 == 2 x 2), a phenomenon remarked on by Oresme. 10 Since the puzzles concerning the idea of "mediate denomination" seem to me to be directly related to this dominant role of the number 2 in medieval mathematics, and to the associated concept of a "mean," this is an appropriate place to digress in order to consider those puzzles. The idea of mediate denomination is believed to have been introduced by Bradwardine, though it has been conjectured that he spoke for a previously existing tradition. The reason for that conjecture lies in his apparently anomalous introduction of a new concept without any definition or explanation, certainly an unusual procedure for any mathematician, let alone one of Bradwardine's caliber. The key passage, as I would translate it, is as follows: [Ratio] is twofold; for that which is of rationals and in the first degree of proportionality is that which is immediately denominated by some number, as is the double ratio, and the triple [i.e. 2/1, 3/I] and so on. But the second degree has those that are called irrationals, which are not immediately denominated by any number, but only mediately (being immediately denominated by some ratio, which is immediately denominated by a number), as is the mean of the double ratio, which is the ratio of diagonal to side [of a square], and the mean of the ratio 9/8, which constitutes the [major] semitone. 11

John Murdoch, remarking on this passage, observed that "this extension of the role of denomination would certainly not have satisfied all medieval mathe-

96

History of Science

maticians. Thus Campanus ... [speaking of] irrationals ... asserts their denominations are not knowable." 12 Murdoch suggested three possible interpretations of Bradwardine's "mediate" denominations, concluding that "none of these possible interpretations are without problems; problems connected with an inconsistency in the meaning of denomination, and secondly with the equality criterion for proportions," for in a later chapter Bradwardine took it as an axiom that all ratios having the same or equal denomination are equal. In summing up, Murdoch said: "Whatever one's interpretation, it must at least satisfy the following: there are three 'entities' involved; the first is immediately denominated by the second, the second is immediately denominated by the third, therefore the first is only mediately denominated by the third. The problem is, basing oneself on Bradwardine 's words, to discover precisely what three 'entities' are intended. The most one can gain with any degree of assurance from the text in question is that the first is an irrational proportion and the third is a number; but precisely which rational proportion and number are the second and third designed to signify?" 1 3 The problem has been further discussed by Edward Grant, whose solution of it with special reference to the work of Oresme is very convincingly argued. 14 I am concerned here, however, with the origin of the concept in Bradwardine 's treatise rather than with its later development. Murdoch and Grant both formulated the question of mediate denomination of irrational ratios in terms of the most general irrational, (A/B/1Y. Since this may be a rather more sophisticated problem than Bradwardine had in mind, I shall instead begin with the examples he actually gave, which involve only the square root of 2 and the square root of 9/8. Considering the relative completeness of Bradwardine's exposition of medieval proportion theory in his treatise, and recognizing its didactic purpose, I cannot assume that he took for granted his readers' acquaintance with the concept of "mediate denomination," and neglected to be more specific for that reason. Nor is it plausible that as a mathematician he could be so neglectful as to omit any explanation of a new term. I think we are therefore obliged to examine the possibility that the trouble lies not in something Bradwardine omitted to write, but in something we fail to read. It is quite possible that his contemporaries, reading his words in a certain medieval context, recognized something that the modem context in which we read them has concealed from us, though it is right before our eyes. We make distinctions they did not, and they made distinctions we do not. For example, we hardly bother to distinguish proper from improper fractions, while their entire nomenclature for ratios was built on the distinction of those of lesser inequality from those of greater inequality. To the latter, they assigned names by a particular kind of classification, and then

Bradwardine's Function

97

these names were modified by a prefix (sub) in order to designate their inversions. We think of fractions and roots as numbers; they thought of fractions as designations (denominations) of ratios, and of roots as ratios. We usually think of a certain number-pair as being a ratio, or of a ratio as being a fraction; they never did. To them, a relation was not identifiable with a quantity. Again, when we read Bradwardine's text and encounter the word "immediately," we take it to mean "directly; with nothing in between." It happens that this is a word that has long since passed into everyday language, and we fail to ask whether it might not be a technical term that proclaimed its nature to medieval readers, much as "transcendental" proclaims itself in a modem mathematical text to have something other than its ordinary use. And having ignored this possibility, we next encounter the word "mediate" and take it to be a word coined from "immediate" and meaning "indirectly, with something in between." We are then puzzled that a mathematician should be so vague as not to tell us what is in between. If we forget to ask, "in between what?", that is because we assume that Bradwardine must have meant something between the "entity" and the "denomination." But obviously "mediate" is related to medium, the mean, and this can hardly be an accident in a treatise on proportion. Mediate does not appear in Calepinus 's sixteenth-century glossary of words in seven languages, nor in the modem dictionary of classical and medieval Latin by Andrews and Freund, nor in Latham 's Revised Medieval Latin Word-List. Thus it seems to have been a coined word, not in common usage. But it seems not to have been coined from "immediate," a word given by neither Calepinus nor Andrews-Freund, though it is in Latham as dating from the twelfth century. Mediate denomination - or rather, to be literal "denomination mediately" - is, for Bradwardine and his readers, simply denomination through the [geometric] mean. Immediate denomination is denomination without reference to the mean, already accomplished for rationals by the ancient classification system. As remarked earlier, Bradwardine does not conceive of roots except as ratios given by means. Now, in a later chapter, Bradwardine informs us that Boethius called proportionality medietas, and thereafter Bradwardine himself adopts that term; thus: Geometrica vero medietas continua est similitudo proportionum per communem terminum medium, vel per communes terminos medios, copulata ... "Continuous geometric proportion [medietas] consists in a similarity of ratios united by a mean term, or mean terms ... " Medietas in is a sense an old word, used by Cicero as the translation of a Greek term (µeO"oTqc;) and defined by Andrews-Freund as "middle, mean." But it was not a word in common use in classical times, nor is it given by Latham. (He does, however, give immedietas as meaning, amusingly enough, "continuity," dating this usage after 1394.

98

History of Science

Medietas is the word used in Bradwardine's two examples of mediate denomination, translated "square root" in the standard English version. I have used "mean" in my translation above, partly because it is closer to medietas, and partly because it stresses the ratio-character of square root as conceived by Bradwardine. But the important thing is the nature of these two examples, for which his words are medietas duplae proportionis and medietas sesquioctavae proportionis. Everyone, it seems, has taken these as examples of irrationals. But I believe that they are examples of mediate denomination. Thus what Bradwardine has done, far from being vague and neglectful, is to coin a term for the denomination possible for irrationals, "denomination by the [geometric] mean," and to give examples of such denomination in which medietas is used as a prefix attached to the usual ratio name, just as sub was already used as a prefix to indicate inversion of the ratio named. It appears that Bradwardine intended no more than to suggest a means of denominating square roots, and the necessity of his doing that will be immediately apparent when one reflects that neither the use of exponents nor the use of radicals was then current. For the purposes of his treatise, it was sufficient to indicate a means of denominating square roots, since no other roots were specifically referred to or even hinted at in his chapter on motion, as mentioned previously, and since the number 2 played a special role in his entire analysis. But two further possibilities are worthy of mention. First, considering that medietas is later used by Bradwardine (following Boethius) to mean "proportionality," such a phrase as medietas duplae proportionis may well have given rise to Oresme 's phrase "ratio of ratios" in the sense of "ratio of exponents." It is by no means impossible that Bradwardine himself chose his terminology merely to avoid confusing repetition, and had in mind not merely "square root" or even "mean proportion," but "proportionality of the double proportion." His addition of specific identifications - the ratio of side to diagonal, and of semitone to tone - may not have been a mere redundant explanation of what had just been expressed, but rather the supplying of single examples for a more general concept. Thus medietas duplae proportionis may have signified not just the square root of two, but the entire series of ratios generated by taking successive mean proportionals, starting at 2 and progressing back toward unity. These would be successively related to each other just as 2 to its square root, or as that square root to unity. In that case, denominations for whole classes of irrationals would be indicated, and the precise denomination for the 1/2"th root would be easily established by conventions. Second, if medietas was restricted to square root, being its specific denomination, as is likely both from Bradwardine's ultimate purpose in the treatise and from the examples he gave here, it would still have been easy to extend the

Bradwardine's Function

99

scheme to other roots. Since the number of means was always one less than the designation of the power, bimedietatis would serve for cube roots, and so on. Thus, with very few conventions, a denomination plan for all irrationals was within easy reach. The fact that Bradwardine did not extend "denomination mediately" beyond square root does not imply inability to do so, but merely lack of motive. His analysis of motion did not require it, as he was concerned primarily with rectifying a certain text of Aristotle, in which only doubling and halving were specifically mentioned. Oresme preferred to pursue his further investigation by the use of pars rather than medietas, and thus a promising start on generalized mediate denomination was never worked out in detail. It is therefore in vain that we shall seek the three entities mediately denominating the general irrational (A/B)x/y by a number. For the square root, the word medietas takes the place of the radical (or of the exponent 1/2); together with A/B, it "mediately" denominates that square root. For other roots, no convention was named, but the form of their mediate denomination would surely have been also that of a prefix combining some number-word with medietas, just as we use the unadorned radical for square root, and add an appropriate number to it for any other root. As for the raising of irrational roots to powers our fractional exponent in general - I do not see why this concept should have occurred to Bradwardine, or what use it could have been to him for the particular problem that he undertook to solve, and did successfully solve, without any more general rule than that which linked the doubling and halving of speed for F 1/R 1 = 2/I, making use of no powers more general than those of the form 2±n_ The general rule he formulated seems to me more likely to have been a byproduct of that specific analysis, than a rule of superlatively general form from which he developed only one minutely specific case. In the light of all the foregoing discussion, the trouble with formula ( 1) and its name, "Bradwardine's function" is that they direct our attention to an artificial historical problem and divert it from real ones. When we see the formula, it is hard not to read into it a description of continuously varying velocity in a given body under changing force-resistance relationships. But that is unlikely to have been a central concept for Bradwardine. Only when R 1 = 1 could it be said that while the force was varied, the moving body remained the same for successive integral powers of F/R. Bradwardine took one such case for analysis; namely F 1/R 1 = 2/I. He concluded that the ratios applying to any other case must be different from those which held for this one; for example, although doubling the force would double the speed in this case, it would fall short of doing so for F 1/R 1 = 3/1. He then applied his analysis to the classification of F/R ratios with respect to F/R = 2/I , ratherthan to the study of a motion V = 2n, or V = (F/Rt.

100

History of Science

It is only by interpretation of formula ( 1), and not by considering the intention of Bradwardine, that we see his law as a rule of acceleration rather than as a kind of classification. For he does not discuss acceleration as such, but only pairs of velocities. Father William Wallace has shown in his recent papers that the study of motion in the Middle Ages centered in an essential way on problems of classification.15 The motion of a body as a whole was distinguished from the motion of its parts, and this led to various subclasses. The Merton Rule may be regarded as part of a classificatory scheme in which uniform and difform motions were distinguished. Likewise, Bradwardine's rule forms part of a common activity, historically recognizable, when viewed as a mathematical scheme for classifying the velocities of motions in terms of their cause - that is, the force-resistance relation. His six specific theorems classify velocities in this way:

Not three, but six theorems result, because Bradwardine distinguished two ways of satisfying the stated condition; namely, by doubling F, or by halving R 1. These six theorems gave him a complete solution to the special problem he had attacked; that is, to justify a particular passage in Aristotle's Physics. In order to deduce these six theorems, Bradwardine first adduced a general rule, but nothing compels us to see that rule as differing from formula (2) above with the values of all its terms restricted to positive integers. It is in formula (2) that I believe Bradwardine's rule should be expressed. For the later work of Oresme, formula (3) above seems to me the best expression in modem notation. The use of any single equation for Bradwardine's rule is likely to be misleading because it suggests to us that any two velocities whatever can always be expressed as a ratio. It is true that appropriate FIR values must exist for any two possible velocities, but that did not mean (to medieval writers) that any two velocities must have a ratio. Velocities corresponding to "incommensurable" FIR ratios would themselves have no ratio - a key point to Oresme - for a ratio of incommensurables was to them (but not to Euclid) a contradiction in terms. We must not confuse this concept of"no ratio" with that of an "irrational ratio," which was perfectly acceptable to them, though it sounds strange to us - just as our idea of "irrational numbers" would have seemed ridiculous to them. The square root of 2 is irrational, but it "communicates" with 2 and with all multiples of 2, so that they all have ratios to Y2 - irrational ratios. But (for them) V2 did not communicate with 3, and hence there could be no ratio of 3 to the

Bradwardine's Function

101

square root of 2. (Medieval translations of Euclid omitted the Eudoxian definition of ratio as possessed by all magnitudes capable of exceeding one another by multiplication.) Nor could there be a ratio between any "ratio of greater inequality" and a "ratio of lesser inequality," because (roughly speaking) no power of a proper fraction can ever become a mixed number; this was in fact the mathematical basis of Bradwardine's theory of motion. Thus our assumption that any two velocities whatever may be expressed as a ratio, an idea implicit in formula ( 1), was totally unacceptable to medieval writers. Another way of putting this. would be to say that we ought really to regard Bradwardine's rule as a test rather than as a functional relationship. In the medieval theory, a unique one-to-one correspondence exists between velocities and FIR ratios. This postulate was unequivocal, since the FIR ratio was seen as the cause of speed. A certain velocity v3 is associated with FIR = 3/1, and it is greater than another velocity v2 that is uniquely linked with FIR= 2/I. From the powers and roots of those FIR ratios arise sets of velocities, the commensurability or incommensurability of which, and hence the possibility of ratios existing between pairs of which, was governed by the respective FIR ratios. Whatever v 2 and v 3 might be, they were entirely incommensurable; otherwise some root or power of 2/1 would be equal to some root or power of 3/1. All multiples and fractional values of v2 were incommensurable with any of those of v3. Given the postulate above, this consequence is obvious to us, and we see it as totally irrelevant to ideas such as that of continuous acceleration from rest. But that was not the case for medieval writers. The question of continuous acceleration is unlikely to have occupied Bradwardine 's attention, but if it had concerned him, he would probably have exemplified it by the motion of a body having an initial FIR ratio of 2/I and subjected to successive doublings of force in successive instants of time. To us, this would not be continuous acceleration, but acceleration by quantum-jumps, so to speak. There is no place among the infinite set of values of v 2 for the velocity v3. But to such an objection, Bradwardine could reply that continuity is governed by continuous geometric proportion, and that a discontinuity would arise only if some "incommensurable" speed such as v 3 were introduced. If we can detach our idea of continuity from the reasoning thus adduced, we shall see that "Bradwardinian acceleration" was highly suitable to a theory of motion in which a fixed speed was associated with the application of a constant force. (Resistance is here irrelevant, since the only way of keeping the "same body" for all powers of FIR is to have R = 1.) In that view, a falling body was supposed to have a certain natural speed with which it travelled uniformly from the moment of its release from rest. Granted such a concept, it was quite reasonable to regard continuous acceleration as produced by rapidly succeeding

102

History of Science

increments of force; and in fact, Buridan had thus explained acceleration in fall in terms of successive forces accumulating in the body. If Buridan conceived of continuous acceleration in this way, then it would be clear to him that different rates of fall would inhere in different bodies, depending on their initial F/R ratios. It would also be clear that this form of acceleration would not remain uniform difform motion beyond the second impulse, even for F/R = 2/J. This conjectural "Bradwardinian acceleration" is, however, mathematically compatible with the belief that the speed of falling bodies increases with the space traversed. It is quite possible that the long delay in linking increases of speed in fall with time rather than with space was partly the result of considerations of this kind. Still more interesting is the conceptual kinship of the Merton Rule with the above view of continuous acceleration. The Merton Rule depends on the notion that uniform speed from rest is conceivable and can be compared with accelerated motion from rest. Even the best medieval mathematicians seem to have seen no incongruity in the idea of "uniform speed from rest," and offered no apology for it, say as a mere mathematical abstraction. Indeed, it was for the idea of uniform acceleration from rest that Heytesbury offered a proof or justification. 16 These and other interesting ramifications of the thought behind Bradwardine 's rule throughout the pattern of medieval speculations on motion deserve study. They seem to be presently neglected in favour of the pursuit of connections between separate parts of that theory and parts of later conceptions. We shall better understand the clues leading from medieval to modem concepts when we have seen the interweaving of threads in the medieval pattern, for there were perhaps fewer loose ends than we now suppose. NOTES

Thomas Bradwardine, Tractatus de Proportionibus, ed. and tr. H.L. Crosby, Jr. (Madison 1955), p. 113. This work is hereinafter cited as "Bradwardine"; the translations are modified in some instances. 2 Use of our exponential notation and of the equality sign came much later, though here they introduce no conceptual novelties so long as each letter is taken to represent a positive integer. 3 Bradwardine, p. I 15. 4 Bradwardine's text is consistent with the more limited concept that FifR 2 must be an integral power of F 1/R 1; it was Oresme who explicitly made it a sufficient condition for commensurability that the two be members of a single power series with integral exponents, as noted further below. I

Bradwardine's Function

103

5 Bradwardine, p. I I 5. 6 Loe. cit. : the Latin (p. I 14) is: Omnis motus eiusdem speciei secundem velox vet tardum possunt adinviciem comparari. 7 Commensurability as demanded by Bradwardine's investigations of F/R required a common measure of powers rather than a common measure of multiples. His general rule stated that the multiples of one ratio (velocities) followed the powers of another ratio (force-resistance). That is why it is possible to express his thought by expression (2) with an added restriction to positive integers, but not by formula ( 1) without the proper restriction on commensurability with respect to powers of F/R. 8 The theorem on proportion which mentions cubes and higher powers is Theorem 2 of Chapter 1, Part 3 (Bradwardine, p. 79). Theorems of the same group invoked in proofs of the six specific theorems on motion are, in order, Theorems 1, 1, 4, 3, 6, 5; the omission of Theorem 2 is thus very striking. 9 Although Oresme's work falls outside the scope of this paper, a remark is in order. Oresme allowed n in formula (3) to have fractional values, but not in the sense of making F2/R 2 a free variable as is implied when we read formula ( 1 ). He recognized that there are integers such that a 2 = b3; for example, 8 2 = 4 3 but that they are always related by a multiple power, in this case the sixth power, 2 6 = 64. The rule of commensurability remained binding on Oresme, as on Bradwardine, and constituted a condition that forbade such cases as 3 = 2"; see note IO, below. Any correct formulation of the ir conceptions must impose that condition, and not appear as a general functional relationship implying such cases and suggesting their implicit solution. 10 Nicole Oresme, De proportionibus proportionum, ed. and tr. by Edward Grant (Madison, 1966), p. 229: "Only quadruple and double ratios, and no others, are so privileged that their ratio of ratios is just the same as their ratio of denominations." Oresme's extension of Bradwardine' s concept of commensurability as to powers is stated thus: "Any ratio which is denominated by any of these numbers [2"] is commensurable to any denominated by others of the same numbers ... And any ratio which is not denominated by any of these numbers is incommensurable to any of them, as a sesquialterate, and a triple, etc." (op. cit., p. 231 ). That is, (3/2)" and (3/1 )" are incommensurable to one another and to ( 2/ 1 )" . 11 Bradwardine, p. 67. 12 John Murdoch, "The Medieval Language of Proportions," Scientific Change, ed. A. Crombie (New York, 1963), p. 259. 13 Op. cit., p. 340. 14 Oresme, op. cit., pp. 31 ff. 15 William Wallace, O.P., "The Concept of Motion in the Sixteenth Century." Proceedings of the American Catholic Philosophical Association ..., 1967, pp. I 84-195; "The Enigma of Domingo De Soto," Isis , 59 ( 1968) pp. 384-401. 16 M. Clagett, The Science of Mechanics in the Middle Ages (Madison, 1959), p. 237:

104

History of Science

"And since any degree of velocity whatsoever differs by a finite amount from zero velocity ... therefore any mobile body may be uniformly accelerated from rest to any assigned degree of velocity." Probably the English mathematicians thought of uniform acceleration as a series of equal steps, for which the Menon Rule worked just as well as for the triangular diagram later introduced by Oresme, since the steps were always taken in pairs by repeated bisection.

4 Medieval Ratio Theory vs Compound Medicines in the Origins of Bradwardine 's Rule

Marshall Clagett suggested in 1959 that Bradwardine's rule for speeds, forces, and resistances may have been inspired by a rule concerning the potencies of compound medicines, traceable to al-Kindi. 1 Subsequently, Michael McVaugh detected a plausible linkage between the two rules in the work of Arnald of Villanova. Presenting his researches, Mc Vaugh said: The search for sources of Bradwardine's law has been neglected by modem critics, who prefer to study its importance for subsequent European science. It may well be that those who picture Thomas Bradwardine as a fourteenth-century Newton would suppose him to have independently worked out a formulation to satisfy the inadequacies of the Aristotelian rule. There is, however, good reason to believe that his law is not a piece of creative mathematics, but is instead the adaptation of an expression having considerable currency among another group of contemporary scientists - the medical theorists of Montpellier. 2

Creative mathematics in explaining the Aristotelian laws of motion must in my opinion be credited to Bradwardine, whether or not he was familiar with the compound-medicine theory expounded by Arnald. Yet I certainly agree with Mc Vaugh that historians have neglected to set forth a clear account of the mathematical source of Bradwardine's rule. To do so is my principal concern in the present paper. It may then be weighed against the source proposed by Clagett as "certainly within the realm of probability" and supported with equal caution by Reprinled from Isis 64 ( 1973), 67-77, by permission of the University of Chicago Press. Copyright 1973 by lhe Hislory of Science Sociely, Inc.

106

History of Science

Mc Vaugh. Nothing conclusive may emerge, but historians will in this way have alternative probabilities to evaluate. That Bradwardine 's approach to his laws of motion lay in a theory of proportion is evident from the full title of the work in which he set them forth.3 Everyone familiar with that treatise is aware that the medieval theory of proportion made use of an elaborate terminology that was not derived from Euclid as we know the Elements. Though Bradwardine refers to Euclid, Book V, which contains the theory of ratios and proportionality of magnitudes in general, one such reference is disparaging, 4 and the definition given in the Treatise for equality of ratios is not the Eudoxian definition of Book V but a definition in terms of the "denominations" of ratios. This term, though it is not in our Euclid, had made its appearance among the definitions of Book VII in the standard medieval Euclid, a translation from an Arabic text with commentaries by Campanus of Novara. Book VII begins the special treatment of numerical ratios and proportionality apart from the general theory of magnitudes given in Book V. It is the theory of proportion embodied in Book VII, as embellished in the medieval version and supplemented by ancient arithmetical ratio terminology, that lies at the basis of Bradwardine 's Treatise. Today we have, from Euclid, Book V, definition 4, a sharp and clear concept of ratio: 'Two magnitudes are said to have a ratio if either, by multiplication, can be made to exceed the other." This definition was absent from the medieval Euclid, which retained only the rather vague wording of definition 3: "Ratio is a disposition (or relation) of two magnitudes of the same kind in respect of size." The wording in the medieval Euclid was modified so as to speak of quantities rather than magnitudes. The commentary of Campanus, however, went on to confer ratio on things as well as on quantities: "Ratio is the mutual disposition of two things of the same kind in respect of that by which one is greater or less than the other, or equal. Ratio is found not only in quantities, but in weights, powers, and sounds .... "5 The concept of ratio was thus the key to any application of mathematics to physics, and before the development of algebraic equations into something like their modern form, ratio remained the only available key to such applications. Proportionality, defined as equality of ratios, was the lock turned by this key, and its definition has always been the same. It occurs in Book V and applies to all magnitudes of whatever kind. But in Book VII Euclid gave another, special, definition for proportionality among numbers: VII, def. 20. Numbers are proportional when the first is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth .

The words multiple, part, and parts had of course been previously defined in

Medieval Ratio Theory vs Compound Medicines

107

tenns of the operation of multiplication, which was a well-defined operation for numbers. But multiplication was an operation unknown to geometry, 6 so that this definition of numerical proportionality hardly applied outside of arithmetic. Definition 20 did not appear in the above fonn in the medieval Euclid; its place was taken by three consecutive definitions whose history is obscure. Their basic concepts originated in the arithmetic transmitted through Nicomachus and Boethius to the Latin West. Equivalent concepts entered geometry through Eutocius and Theon, becoming incorporated in Arabic and thence in Latin texts of Book VII. Their form in Campanus' edition is this: VII def. 20. The denomination of a ratio of a lesser to greater number will be said to be the part, or parts of this lesser [number] existing in the greater, and [the denomination of a ratio of) greater to lesser [number] will be said to be the whole, or the whole and part, or the whole and parts, by which the greater exceeds the lesser. 21. Ratios that have the same denomination are called similar, one, or the same; and those that have a greater [denomination] are called greater [ratios], and those that have a lesser, smaller. 22 . Numbers that have one ratio are said to be proportional.

The principal purpose of this use of the concept of "denomination" was simply that of reduction of a ratio to its (lowest) tenns: 23. Numbers are called terms or roots of a ratio when smaller numbers than those cannot be taken in the same ratio.

These modifications in the original Euclidean apparatus of definitions for the arithmetical books, VII-IX, facilitated the treatment of ratio entirely by numbers. Numbers did not appear as such even in the arithmetical books of Euclid, where they were represented by line segments. The ancient nomenclature for ratios, transmitted by Boethius, includes numerical indications and can be translated into the fractions that we use lo represent ratios, but the system of classification of ratios displayed their numerical relations quite differently from the fractional notation we use. This ancient tenninology did not place all ratios in a single order with respect to size, as was done by medieval definition 21 cited above, which in effect gave to unity - the "ratio of equality" - a unique position somewhat analogous to that which we give to zero. Ratios were thus made greater as they departed from unity and smaller as they approached it from either side. Unity itself, what we call the "number" one, was not a number at all so far as ancient and medieval mathematicians were concerned, since number was defined as a multitude of units, and the unit was not a multitude itself. Somewhat similarly, we do not think of zero as the smallest (positive) number

108

History of Science

but as a positional benchmark among numbers, having certain curious properties of its own that are not shared with (other) numbers. Most definitions of Book VII deal with number and numerical operations. Those dealing with numerical ratio and numerical proportionality have been cited previously, with one exception. Book V introduced a special terminology for certain ratios arising in continued, or geometric, proportionality : these were the duplicate and triplicate, or doubled and tripled (not double and triple), ratios. This terminology was also reintroduced into the medieval (but not the authentic) Book VII : Def. 18. When there are any number of numbers continually proportional, the ratio of the first to the third is said to be that of the first to the second duplicated (doubled) ; and [that of the first] to the fourth [is said to be] that of the first to the second triplicated (tripled) .

Thus Campanus' Book VII embodied a complete definitional apparatus and terminology for the theory of proportion characteristic of medieval mathematics and was given its definitional base without any conceptual need for references to Book V, 7 but likewise without the possibility of rigorous application to magnitude in general in the Eudoxian sense. Ratio and proportionality were made essentially arithmetical, and the medieval theory of proportion became a doctrine concerned with the relations of positive integers. Into this doctrine the concept of irrationals was to enter, not as a kind of number, but as a kind of ratio dependent on continued proportionality of numbers. We now think of the square root of 2 as a number, subject to the usual arithmetical operations; but medieval mathematicians thought of it as a mean term, the geometric mean between unity and 2, having no existence apart from the continued proportionality in which it participated. Ratios are not automatically subject to all the usual arithmetic operations defined for numbers, any more than relations of any kind are subject to all the ordinary treatments given to quantities. Multiplication of a ratio by a number is not defined in Euclid. The idea of "compounding" ratios does indeed make its appearance in Euclid VI, 23, and in some Greek manuscripts a marginal additional definition was added to Book VI in an attempt to make specific the idea of a ratio compounded from two ratios. But this spurious addition was not present in the medieval Euclid, nor was any attempt made in the additional medieval definitions in Book VII to define the addition or multiplication of ratios. The only lead in that direction was the terminology introduced in the previously cited definition 18 for two specific cases of terms in continued proportion (and of course its equivalents in definitions 9 and Io of Book V). This did

Medieval Ratio Theory vs Compound Medicines

109

not indicate a general rule even for the compounding of equal ratios, or, as we would look upon it, the raising of ratios to powers. Campanus, in very long commentaries on the relevant definitions of Book V, 8 emphasized that the dimensionality of geometric entities, line, surface, and bulk, restricted the significant cases to duplication and triplication. Departure from this ancient geometrical limitation appears to be a distinctively fourteenth-century contribution, and it owes much to the arithmetical character of medieval theory of proportion. The idea of "denomination" achieved essentially the recognition of simple numerical fractions as representatives of ratios, present neither in the authentic Euclid nor in Boethian ratio nomenclature. Multiplication of fractions together, in our general sense, did not suggest itself either as justified or as serving any purpose; nevertheless, the effect of duplication and triplication of ratios, a terminology sanctioned by Euclid, was easily recognizable from their numerical denominations as identical with the squaring and cubing of their terms. Between these two special operations, however, there was an enormous difference, in that a single geometric mean could always be found between two numbers by geometric construction, whereas a pair of geometric means could not. And since geometric proportionality was associated in the Middle Ages with the mean term that formed the denominator of one ratio and the numerator of the next in continued proportion, the "doubled ratio" lent itself to further exploration in a way that the "tripled ratio" did not. II

It was Bradwardine who inaugurated a series of investigations of the "doubled ratio" that led, in the fourteenth century, to developments in a theory of proportion that had no counterparts in antiquity. That theory went on to anticipate, in Nicole Oresme's work, the essential mathematics of logarithms and of fractional exponents that emerged in the seventeenth century. Bradwardine's own investigations enabled him to rationalize a passage in Aristotle that had long been subjected to criticism. He himself felt that he had successfully recaptured Aristotle's own meaning, and Oresme supported that possibility; but this notion is presently rejected by historians, largely because they see in Bradwardine's work the creation of the idea of a mathematical function in the modem sense of an expression of relation between continuous variables. There is, however, another way of looking at it that is more plausible mathematically, psychologically, and historically. The passage in Aristotle reads as follows: If A is the mover, B the thing in motion, S the length over which motion has occurred,

1 1o

History of Science

and T the time taken, in a time T a force equal to that of A will cause a thing which is half of B to move over length 2S, and it will cause it to move over the length S in half the time of T, for thus there will be a proportion. And if the force of A causes B to move over half the length S in time T, it also causes a thing equal to half of B to move over half of S in half the time of T, and a force equal to half of A causes a thing equal to half of B to move over a length S in time T. For example, let E be half of A and F be half of B; then the strengths are similarly and proportionately related to the weights, so in equal times they will cause motion over equal lengths. But if E causes F to move over length S in time T, it does not follow that in time T it will cause 2F to move over S/2. Thus, if A can cause B to move over a distance Sin a time T, then half of A, which is E, may not cause B to move, either for a part of time T over a part of S, or over a part X of S so related that X:S::E:A. For it may happen that E will not cause B to move at all, for if a given strength causes a quantity of motion, half that strength may not cause any quantity of motion, or may not cause motion in any given time .... 9

Aristotle's sole support of these rules lies in the clause "for thus the proportions are preserved." 10 Aristotle's word for proportion, analogia, is Euclid's word for geometrical proportion. In the Middle Ages the word "proportion" was broader in sense than it is now: Nichomachus and Boethius had indicated ten or eleven kinds of proportion, three kinds being of special importance which continued to be treated in general works. These were arithmetic proportion, which we do not call proportion at all, but arithmetic progression; geometric proportion, which we simply call ratio; and harmonic proportion, a relationship limited to four terms such that a:d::a-b:a-c. Thus what Aristotle asserted was, to Bradwardine's ears, that the relations of space, time, power, and body moved must be such that the geometric proportions remained the same. What Bradwardine did was to treat velocity as embracing space and time relations in a single concept, and to form a ratio of the force of the mover and the force of the body. He was careful to avoid external resistances in his general rule and to use only the force of opposition to motion resident in the body itself (the later Keplerian concept of "inertia," a supposed tendency to come to rest the instant the moving force ceases to act). This conception gives a quantity to be related to a ratio. The quantity is velocity, measured in degrees and thus capable of being reduced to pure number. The ratio, represented by historians as FIR, is in our physics a fictitious one, as if every body had a specific internal resistance to any force imposed on it. But real or imaginary, to Bradwardine it was irreducibly a ratio, not capable of being expressed as a quantity or degree. It was a ratio that was different for different degrees of force applied to the same or to different bodies.

Medieval Ratio Theory vs Compound Medicines

11 I

III

Now, how does a quantity follow a ratio? Had Bradwardine been confronted with this question in its full generality, he might not have attempted to answer it. But Aristotle, as it happened, had spoken only of doubling and halving. The relation between doubling a quantity and doubling a ratio was already specified in the definitions of Book VII; a "doubled ratio" was that of the first and third tenns in a continued geometric proportionality, whereas a "double ratio" was simply the ratio 2/1. We may consider this a mere accident of tenninology, but psychologically it is a very plausible source of Bradwardine 's inspiration for his rule of motion that the ratio of the speeds follows the geometric proportionality of the forces and resistances. The plausibility is heightened when we note that in Bradwardine's ensuing discussion of the rule and its implication, everything is done in tenns of the ratio 2/1, its integral powers, and its successive roots of the fonn 2n; that is, its square root, fourth root, eighth root, and so on. 11 Bradwardine's rule successfully answered the objections that had been raised against the purely arithmetical interpretation of Aristotle's suggestion. The main objection was that the ratios could not be preserved in reverse, since in that case any body, however great, could be moved by any force, however small, which is observed not to be the case. Aristotle takes care of that, seemingly by an arbitrary cutoff, in denying that motion always occurs, and it seems strange that neither his opponents nor Bradwardine fully acknowledged this fact. This objection disappeared with the substitution of geometric proportionality for arithmetic progression; if dividing the speed by 2 meant taking geometric means between FIR and unity, some excess of F over R always remained to explain motion, whereas if it meant halving F (or doubling R), cases were implied in which motion would take place despite an excess of resistance over force. Bradwardine's conviction that he had not only solved the objections but had in fact recovered Aristotle's original meaning, is regarded as merely mistaken by most historians. Some express surprise that Bradwardine did not recognize, in the third of the four positions he opposed, the position of Aristotle himself. I do not think that Bradwardine's conviction is as unfounded as may appear. Aristotle's word analogia had a special meaning to Bradwardine; it referred specifically to geometric proportion, which was only one of the three kinds of proportion discussed in Bradwardine's Treatise. Bradwardine had found a rule that saved Aristotle, and he added at its end "and this means geometric proportionality. "12 There was no reason that I can see why Bradwardine should not have been persuaded that Aristotle had in mind exactly the rule that he had been able to rediscover.

1 12

History of Science

Such a statement is absurd to those who would say: "Surely you do not believe that Aristotle held that F2 /R 2 = (F .fR 1 ) Vi/V 1, for this is a highly sophisticated concept mathematically and is quite out of keeping with his epoch and his own philosophy." Indeed not; I should say that in the sense they give to that equation, neither Aristotle nor Bradwardine ever thought of any such thing. In short, I do not agree with the sense they give to it. What Bradwardine did think of was perfectly comprehensible to his contemporaries, and he would certainly not have put it past Aristotle's superb comprehension. It involved, not the perfectly general functional concept based on a modem notion of the continuum of the real number system that we read into our unrestricted algebraic equation, but a very simple mathematical and physical conception limited to halving and doubling velocities, and taking the integral multiples and repeated square roots of numerical ratios; or rather, so far as Bradwardine was concerned, of analyzing those operations on the ratio 2/1 alone; and indicating their extension to other "ratios of greater inequality" only in tenns of greater-than and less-than doubling and halving. •3 Whether Bradwardine was right about crediting Aristotle with his own reasoning is another question. I think he was mistaken, but not unreasonably mistaken. It may seem strange that in expressing my interpretation of Bradwardine's rule I mention all the integral powers of a ratio but only those of its roots which are themselves powers of 2. If the cube, why not the cube root? And if that, why not all fractional powers, rational and irrational, and consequently all possible powers, rational and irrational? In a word, why not our real number system and a continuous function in the modem sense? Well, confining myself to Bradwardine, who in my opinion made the first restricted suggestion that in tum led to some remarkably astute subsequent developments at the hands of others, notably Swineshead and Oresme, I have noted that only in one of his theorems on proportion does he ever mention the cube, and that this is the only one (of six) theorems on proportion that he avoids using in his theorems on motion. This does not sound 10· me like a man given to rapid and speculative extrapolations, but like a conservative mathematician content to demonstrate just enough to accomplish his original purpose - that of justifying a text of Aristotle. Even more cogently, Bradwardine states that irrational and noncommunicating and incommensurable ratios exist between quantities that have no exact common measure; such, for example, would be 3 and V2. I think that this excludes the idea that a generalized continuous function, in which, say, 3 = 2n, could have had any meaning to medieval writers. In summary of this section, I suggest that Bradwardine 's specific purpose was to justify mathematically a particular statement of Aristotle's, and that the source of his solution needs no more explanation than the existence in medieval

Medieval Ratio Theory vs Compound Medicines

1 13

proportion theory of two senses of the word "doubling," one as applied to a quantity and the other as applied to a ratio. IV

With this background, let us return to the suggestion that a source, or perhaps the source, of Bradwardine 's law lay in the rule for mixtures of heat and cold in compound medicines. This laller rule is summarized by Mc Vaugh as follows: ... by adding up the number of hot parts and of cold contained in the simple constituents (each of known degree [of intensity]) of a compound medicine, and determining their ratio, the degree/ of the compound can easily be found , since HIC = 2degree, or/= log 2 (H/C). Such a system is indeed a perfectly possible precursor of Bradwardine's law.

Then Mc Vaugh cites Clagell: " ... Bradwardine was one of the first to make the analogy between velocity as intensity of motions and the intensity of permanent qualities. It is not then much of a step to go from (I) an exponential relationship between qualitative degrees of intensity and qualitative powers to (2) an exponential relationship between degrees of motion and the ratio of motive and resistive powers. It is certainly within the realm of probability that Bradwardine took precisely this step." 14

The plausibility of Clagett's parallel depends heavily on our previous familiarity with the modem notion of "exponential relationship." In an age before any such grouping had been defined, it might be a very large step to relate one member of what we include in that class to another member. Clagett describes the step as one from an exponential relationship between two quantities (entities capable of expression by single numbers) to one between a quantity and a ratio (the Jailer entity implying two numbers). Since a quantity and a ratio could not be equated under the Euclidean definitions (or any existing definitions much before the twentieth century), the relationship had to be some kind of correlation, mathematical or physical, and not a simple equation. This in itself was a large step, but it is one that had already been taken by al-Kindi and Amald. If correlating a ratio and quantity were all that we were concerned with, it would be true that no great conceptual advance was represented in Bradwardine's law as against the compound-medicine rule. But the correlation of a quantity with a ratio is not all that is involved in Bradwardine's law, and Amald's rule does nothing to explain the taking of the next big step by Bradwardine. In the foregoing symbolic expression of the compound-medicine rule (which

1 14

History of Science

will be criticized later), the exponent is associated with a certain numerical constant, the number 2 , and not with a variable of any kind, even one capable of taking on only discrete values. It is the exponent itself that can have only integral values, the highest of which is 4, as will be seen below. Nothing of this sort appears in the usual formula for Bradwardine ' s law. The basic ratio FIR is there subjected to self-multiplication, as the basic ratio H!C here is not. The emphasis in Bradwardine's law is on a ratio of exponents VifV., whereas the exponent/ in Amald's rule is not related to another exponent, and nothing in that rule suggests that a ratio of exponents would have any further significance. The propriety of the formula HIC = i1 is extremely dubious, as are all algebraic equations used as shorthand for mathematics in which the question of implied continuity has not first been settled. In the present case, it is pretty certain that/ was originally intended to take on only the discrete values 1, 2 , 3, and 4 . Moreover, it is at least questionable whether H and C were continuous variables, or whether they were controlled by discrete minima naturalia. As it stands without restrictions, the formula implies H = i1c, and written in that form it has not even the distinction of relating a quantity to a ratio, which is the one element that entitles it to comparison with Bradwardine ' s law. Great care ought to be taken when employing anachronistic modem symbolization to supply the appropriate restrictions. In this case, for example, it is part of the theory that H plus C represents the whole compound, and this is left out in the symbolism. The ratio HIC as it stands might be less than unity, in which case fractional values for / are implied in complete generality; yet even Bradwardine limited himself to fractional exponents of the form ( 1/2l, and no more general concept is known before Oresme. In the compound-medicine rule, the concepts of HIC < 1 or> 4 seem to be ruled out: From the time of Galen to that of Bradwardine and beyond, medical theory recognized four intensive degrees in any medicinal quality, degrees corresponding to four different levels in the action of drugs on the human body; these ranged from the first degree, which could barely be sensed, to the fourth, which eventually destroyed the body to which it was given . ... A temperate, qualitatively balanced medicine has, according to th is [al-Kindi's] theory, equal parts of hot and cold; ... one hot in the second degree four times as much; and, ultimately, one in the fourth degree has sixteen times the hotness of a temperate medicine. 1s

These degrees sound much like the degrees medically ascribed to bums today, which are not customarily distinguished in a continuous scale but classified by discrete numbers. The formula implied may not be properly expressible

Medieval Ratio Theory vs Compound Medicines

I I5

as a continuous function at all but rather as a correlation between certain positive integers and tenns of the binary geometric progression. The degrees of hotness I, 2, 3, 4 correspond to the HIC ratios 2/1, 4/1 , 8/1, and 16/1. Higher values are not explicitly pennitted, nor are intennediate degrees. That discrete values without intennediates were meant seems also to be implied by this reasoning of Amald 's: ... it is apparent in the case of the objects of other senses that if one of them cannot affect someone's sense, it is necessary that the force with which it has to affect that sense be at least doubled . ... For if a flame set in a given place cannot be seen by the eye, hindered by the complete unsuitability of the medium, in order that it be seen by the same at the same distance it must be at least doubled; likewise with other things visible . ... If a sound cannot be heard at all by someone at a given distance, though medium and organ be well disposed, it must be at least twice as loud in order to be heard by the same. 16

McVaugh 's comment on this does not touch on the phrase "at least doubled," which seems to me crucial to the mathematical basis of Amald's rule. He says: Given the similarity between Amaldian pharmacy and Mertonian mechanics, it now becomes of considerable interest to discover how general this law was meant to be: does the reference to "quantity" mean that Amald believed a change in quantity of any measurable force to be geometrically related to the change in its intensity? Specifically, would he have applied this physical law to the forces involved in local motion? Apparently he would have . ... 17

Let us suppose, however, that the similarity mentioned is not given but remains to be explored; and in this light let us consider the phrase "at least doubled." If continuity were implied, and thence the existence of intennediate "degrees" as well as intennediate values of HIC, the phrase "at least doubled" would be mathematically inappropriate. Say that a light would carry threequarters of the way to the eye but no further. Such a light would not need to be at least doubled in order to reach the eye - except in the case that there was no smaller unit of light than the original. It is true that if the least unit of light were, say, I candle, then at least 2 candles would be needed for visibility in the case cited. But then within the next degree, from bare visibility to double intensity, it might be that 4 candles would not be required; 3 might suffice. Now, we have only Amald's discussion of the case from zero perception to degree one, in which the concept is clearly not one of continuity but of a minimal quantum. The most natural extension of this gives us a quantum interpretation of his general view.

I 16

History of Science

Another extension, perhaps not quite so natural, would be the idea that excluding the cases of degree Jess than the first, any intermediate force ratio produces that integral degree of effect which lies nearest to it. Thus a force ratio of 7/4 would produce an effect of the second degree, while a force ratio of 5/4 would produce only the first degree. This extension would also explain Amald ' s phrase "at least doubled." But it likewise involves the idea that the degrees are discrete and integral, not continuous. The concept of continuity in degrees is irreconcilable under any reasonable theory of proportion with the "at least doubled" rule for passage from one degree to the next. Even if Amald "believed a change in quantity of a measurable force to be geometrically related to the change in its intensity,'' 18 that idea would still not be the key notion ascribed to Bradwardine. Bradwardine's rule says that each ratio of force to resistance gives rise to a specific velocity in such a way that any ratio of velocities implies a certain "ratio" of force ratios ~ssociated with it. This essentially new mathematical concept of a "ratio of ratios" was not in fact explicitly signalled or defined by Bradwardine, but its significance and power were perceived by Oresme, who wrote a book by that title and developed the new concept mathematically. The concept that Bradwardine applied implicitly to a specific problem in Aristotle's Physics was not a concept implicit in the compound-medicine rule, with its set of four discrete integral degrees and its exponential treatment of a constant. Yet that rule and Bradwardine's have a great deal in common besides the use of an exponent. Both are concerned with powers of 2, exclusively in the case of Amald, and at least primarily (if not exclusively, as I believe) in the case of Bradwardine. Both deal with ratios which tend toward the production of zero effect as the (causal) ratios approach unity, the "ratio of equality." Both deal with discrete comparisons of correlated values and not with continuous variables in the modem sense - despite the symbolism by which we confer on them both a concept of continuity. All the essential points of contact between Amald and Bradwardine belong to the medieval theory of proportion and need no further explanation in terms of direct transmission of ideas from Amald to Bradwardine. That there probably was such a transmission is an interesting fact, but it is not one that throws light on the origin of Bradwardine's law, which definitely added a piece of creative mathematics not in and not clearly implied by the medical rule. NOTES 1

2

Marshall Clagett, The Science of Mechanics in the Middle Ages (Madison: University of Wisconsin Press, 1959), p. 439, n. 35. Michael Mc Vaugh, "Amald of Villanova and Bradwardine's Law," Isis, 1967, 58: 56--64, on p. 56.

Medieval Ratio Theory vs Compound Medicines

1 17

3 Thomae Bradwardini tractatus proportionum seu de proportionibus velocitatem in

4

5

6 7 8 9 10 11

12 13

14 15 16 17 18

motibus, usually cited as "Treatise on proportions" without attention 10 the alternative title, "On the ratios of speeds in motions." Thomas Bradwardine, His "Tractatus de proportionibus." Its Significance for the Development of Mathematical Physics, ed. and trans. H. Lamar Crosby (Madison: University of Wisconsin Press, 1955), pp. 76, 77. Proportio: est habitudo duarum rerum eiusdem generis adinvicem in eo quod earum altera maior aut minor est reliqua ve/ sibi aequalis. Non enim so/um in quantitatibus reperitur proportio: sed in ponderibus, potentiis et sonis ... . Direct citations from the Campanus Euclid were taken from the Paris edition of 1516; citations in English from this are my own translations. The accepted modem English Euclid used for comparison is that of Heath (The Thirteen Books of Euclid's Elements, trans. and commentary by Sir Thomas L. Heath, 2nd ed., Cambridge: Cambridge University Press, 1908; Dover reprint, 1956). Page references are omitted, the definitions being easily found at the beginning of Bk. V or VII, as indicated. See Heath, Vol. 11, p. 190. References for certain terminological additions, however, did occur. What we call definitions 9 and Io in Bk. V were called Io and 11 in Campanus. Aristotle, Physics, trans. with commentaries and gloss. by Hippocrates G. Apostle (Bloomington: Indiana University Press, 1969), p. 146. Apostle's translation puts ii not so strongly. The seeming asymmetry of this treatment, mentioned later in another connection, calls for some explanation, especially since ii is not the prevailing view of Bradwardine 's concept. For Bradwardine, the "proportionality of the ratio 2/ 1" (medietas duplae proportionis) resided in the geometric mean between 2 and 1, which we call "the square root of 2." Thus he recognized as an "irrational ratio" what we should write as 2/½ (and also recognized its equality with what is written Y2/1 ). The geometric progression formed with 2/½, proceeding upward from unity, includes all the integral powers of 2, and they are the only rational ratios found . Bui proceeding downward, we have the powers of Y2/2, in which the only rational ratios are of the form 1/2n. Thus in the proportionality envisioned by Bradwardine the cube of 2 appears, but not the cube root of 2; and so on. Bradwardine, Tractatus, ed. Crosby, p. 112, lines 4'}-50. Necessary restrictions on the usual equation for Bradwardine •s law are suggested in my paper "Bradwardine's Function, Mediate Denominations, and Multiple Continua," Physis, 1970, 12: 51-68, esp. pp. 57-58. Mc Vaugh, "Amald of Villanova," p. 57, citing Clagett, loc. cit. Mc Vaugh, p. 57. Ibid., p. 61, citing Aphorismi de gradibus, MS Merton College 230, fol. 88v A. McVaugh, p. 62. Ibid., italics added.

5

Early Science and the Printed Book: The Spread of Science beyond the Universities

The history of science as a separate branch of cultural and intellectual history is not very old; not much older than the concept of the Renaissance itself. That concept, though it has been subjected to attack in recent years, is particularly applicable to the study of history of science, at least in its strictest and most literal sense of a rebirth. In my opinion this is best seen in relation to the advent of the printed book, and the picture of this relation that I wish to put before you can best be introduced by a brief statement about the case of physical science, to which my remarks will in any case be essentially limited. Physical science has a special relation to mathematics. Now, there is a sense in which mathematics is the only subject in which antiquity possessed exact knowledge of permanent value that had become lost to the official scholarship of the Latin West long before the invention of printing. Obviously I do not mean to say that all knowledge of ancient mathematics had been lost. Clearly it had not; the case of Euclid makes that clear. But there were important segments of ancient mathematics that were capable of altering the whole face of physical science that had long dropped from sight. Moreover, the standard medieval Latin translation of Euclid and the commentary that accompanied it had failed to give a correct account of ancient refinements in at least one fundamental branch of mathematics, the theory of proportion. A study of medieval mathematics has convinced me that certain ingenious substitutes for the classic theory of proportion had made it most unlikely that that would be again hit upon in any way other than by the restoration and study of old texts, a specific activity associated with the concept of the Renaissance. As to debates in our time over the validity of the Renaissance concept, these appear to me to have been conducted in a particularly unfruitful manner. It is Reprinted from Renaissance and Reformation Journal 6 ( 1970). 43-52, by permission.

Early Science and the Printed Book

1 19

obviously useful for historians to have a descriptive name for a period that was characteristically neither medieval nor modem. The fact that no precise date can be set for the beginning and end of such a period in general does not destroy the utility of the concept. In any branch of intellectual and cultural history, specialists can define a Renaissance period according to criteria appropriate in that field, and those criteria will then serve also to clarify the meanings of "medieval" and "modem" with respect to its development. An overlapping of periods defined in that way, by a variety of specialists in many fields, enables us to say something about the general state of culture within any particular range of dates. This seems to me a perfectly defensible way of going about our business as historians. Generally, when the Renaissance concept is attacked, that is done not on the basis of utility but on grounds of propriety. Should we use one word, "Renaissance," to denote a wide variety of periods? Confronted with this question, excessively scrupulous historians tend to disparage the word as a name for something that had no clear existence, and to disown the concept as a capricious nineteenth-century creation corresponding to no demonstrated historical facts. Hardier souls have in effect adopted in its defence the position of Humpty Dumpty; at any rate, they seem to me to have said little more than "when I use a word, it means whatever I want it to mean, nothing more and nothing less." The proper procedure, it seems to me, would be not to engage in ethical disputes over the right to coin words, or in ontological disputes over the existence or non-existence of historical patterns, nor yet to support uncritically the right of everyone to use any word, especially a term of art, in any manner he pleases. Rather, we might better make clear the criteria by which words like "medieval," "Renaissance" and "modem" are to be applied in a designated area, and then go boldly ahead and apply them, to see whether the patterns of historical fact that thus emerge are interesting, or instructive, or of heuristic value. If in some fields they are not, or if it is impossible to make the criteria clear, then not only the word "Renaissance" but also the word "medieval" should probably be abandoned in those fields, but without prejudice to the terms themselves; though in such cases there is not much point in retaining the word "modem," either, unless that is to be contrasted with "ancient." For however hard it may be to define a period between medieval and modem ideas and activities in a given cultural field, it must be still more difficult to fix a date in that field on which medieval ideas and pursuits went out of date and were replaced by modem ones. Hence it seems to me that wherever we can agree to retain the word "medieval," we are likely to find the Renaissance concept also useful. Now, the history of science is very definitely one field in which both terms are useful. As I have said, the word Renaissance is particularly apt, in its most

120

History of Science

literal sense, when applied to one of the main developments in sixteenthcentury physical science. It was then that ancient Greek mathematical and scientific works were rediscovered that had been unknown to, or utterly neglected by, medieval writers. Correct texts were established, proper translations were made, and appropriate commentaries were written also for other ancient works that had previously been known only in part or only in corrupt versions. Now, the rectification of mathematical texts has an essentially different role from that of the restoration of authentic literary or philosophical works, the interest of the latter being mainly, if not entirely, antiquarian in character. But the works of Euclid, Archimedes, Apollonius of Perga and Pappus of Alexandria were replete not with ancient opinions, but with extremely valuable knowledge, most of it as valid in the sixteenth century, and today, as when it was written. It is at least open to question whether the retrieval of the authentic texts of Plato and Aristotle represented anything like the same acquisition of permanently valuable knowledge, putting antiquarian interest aside, as those of Archimedes. Indeed, a defective philosophical text could easily be more inspiring than a correct one, through what Aldous Huxley called "lapses into lucidity." In any case, the restoration of classical mathematics in the sixteenth century did make possible some immediate scientific advances that would have required a great deal of time to achieve in any other way. Given unprecedentedly wide circulation in printed books, it had a direct and far-reaching effect on both the form and content of physical science. This was a true rebirth of ancient scientific knowledge, most aptly described by the word "Renaissance." If there had been no Renaissance, historians of science would have had to invent it. Immediately preceding this, and therefore independently of it, a reawakening of interest had begun in direct inquiries into particular phenomena, as against the speculative assignment of causes to things that characterized medieval science. The most prominent representatives of this aspect of Renaissance science are well known - Leonardo, Copernicus, Paracelsus and Vesalius. Indeed, an exaggerated attention paid to a few such men has rendered the phrases "Renaissance mechanics," "Renaissance astronomy," "Renaissance chemistry" and "Renaissance anatomy" significant of little more to most people than the work of those four men respectively. Even among historians of science generally, the adjective "Renaissance" seldom carries with it the kind of thematic characterization that accompanies the words "ancient," "medieval," and "modern," when applied to specific scientific disciplines. Yet Renaissance physics, at least, does have a character of its own - or rather a split personality - that has escaped notice for rather interesting reasons. The first historians of science naturally relied almost exclusively on printed books as their sources of information. Now, very few medieval works of recog-

Early Science and the Printed Book

121

nizably scientific interest ever passed into book fonn, whereas most of the ancient scientific works known in the sixteenth century, as I have already mentioned, were widely circulated then for the first time. Hence medieval science tended to be overlooked by early historians. Until quite recent times, the period from the close of the Alexandrian era to the opening of the sixteenth century was dismissed as of no historical importance for the exact sciences. Thus the question of a special character for Renaissance science did not immediately arise; the 16th century was simply regarded as the initial stage of modem science, inspired by ancient models. That view, associated with the name of William Whewell and the 19th century, could no longer be maintained after Charles Thurot and Pierre Duhem turned at the end of that century to the serious examination of medieval manuscripts. In particular, Duhem brought to light a conscious program of applying mathematics to physics in writings of the 13th and 14th centuries. Since Galileo and Descartes in the r 7th century mathematicized physics, Du hem assumed a line of transmission from the Middle Ages through the Renaissance to the pioneers of modem science. The assumption is so plausible that it pervades history of science today, though still undocumented through printed books. Thus again the question of a special character for Renaissance science was slighted off, and the sixteenth century was regarded as merely a period of transmission of much older works, medieval as well as ancient. From this it is evident that the study of printed books as against manuscript sources has always been fundamental to the historiographical question of Renaissance science. That being the case, one might expect the differences as well as the resemblances between books and their predecessor manuscripts to have been brought to the fore and carefully analyzed. But the advent of the printed book seems instead to have been treated by historians of science, and perhaps by other intellectual historians, not as a fundamentally important event, but as a mere replacement of the scriptoria by printing-houses. This seems to me as absurd as if social historians in the future were to regard the replacement of the horse by the automobile as having introduced no fundamental change. Transportation was made faster and more generally used, but that is not all. The printed book made it easier for a scholar to acquaint himself with the work of other scholars, and if that had been all it did, it might reasonably be regarded as relatively unimportant. The nature of scholarship would not have been affected, but only its rate of progress. But in science, at least, printed books did change the nature of scholarship. When we recognize that fact, we also see that the older historical view of Renaissance science was not entirely wrong, nor is the view that has replaced it entirely right. The printed book changed the nature of scholarship by effecting a fundamen-

12 2

History of Science

tal alteration in the composition of the world of learning. Perhaps this is more evident, as it is certainly more significant, in mathematics and physical science than anywhere else. The point may be illustrated by naming half a dozen medieval writers whose works are now recognized as of paramount importance in the history of physical science, and seeking their counterparts in the sixteenth century. The medieval writers that occur to me are Jordanus Nemorarius, Robert Grosseteste, Thomas Bradwardine, Jean Buridan, Albert of Saxony and Nicole Oresme. Of these men, all but Jordanus were certainly university teachers, and even Jordanus, a shadowy figure, is known to have composed at least one work specifically for university instruction. Physical science through the sixteenth century may be represented by Leonardo, Niccolo Tartaglia, Girolamo Cardano, Giovanni Battista Benedetti, Guidobaldo de Monte and Simon Stevin. Of these men, only Cardano taught in a university, and his contributions to physics are the least important of any man in the list. It js safe to say that during the Middle Ages, physical science was a university monopoly, whereas after the printed book it was far from being so. Obviously other lists could be drawn up, and perhaps my last statement could be shown to be an exaggeration. But the basic idea contained in it would not be changed. It may be debated whether purely technical treaties on the one hand, and purely philosophical discussions of physics on the other hand, should or should not be included in the lists. I believe the result will be pretty much the same, so long as the criterion is the same for both periods. If spokesmen for medieval continuity in science should produce a list to show that medieval science was not a university monopoly, I think they would find that in that list the unity of medieval science - that special character by which they recognize it in later periods - will have slipped from their grasp. That may be precisely what some of them desire, but in that case they are merely asking us to abandon a valuable working concept in intellectual history when they question the idea of a Renaissance. The change in composition of scientific scholarship that followed the advent of the printed book consisted essentially in the addition of self-educated men and talented amateurs of liberal education to the ranks of those who made substantial contributions to science. Their actual contributions differed in an essential way from those that continued to be made by official scholarship; that is, university professors. I shall illustrate the difference presently, but first I wish to speak of the half-century lag between the invention of printing and the effect on physical science that I attribute to the printed book. Obviously the invention itself could not produce the effect immediately. First it was necessary that books having some relevance to science appear, which brings us to about 1480. Mathematics was especially relevant to the effect in physics, and so far as geometry is

Early Science and the Printed Book

123

concerned the crucial date is 1482; for algebra, it is 1494. By 1500, wide and relatively inexpensive production of books had become a reality throughout Europe, which means that the process of self-education and the discovery by amateurs of their own particular talents in fields previously unfamiliar to them had begun. The number of such persons in the general population was certainly very large as compared with the university population; but not everyone, even in the universities, was destined to make a lasting contribution to his field. The half-century lag between the invention of printing and the appearance of significant scientific books written by men outside the universities is therefore no reason for denying an intimate connection between the two events. A rapid multiplication of treaties on scientific topics took place after I 500 that contrasts with the relatively even production of manuscripts on such subject in preceding centuries. This phenomenon probably owes a good deal to the difference between the printer and the professional scribe whose work he forced into obsolescence. The scribe generally copied only those works he was commissioned to copy, and his methods of work were unrelated to the contents of his manuscripts. The situation of the early printer was different. Although some books were printed to order or by commission, for the most part they were commercial ventures attended with risk and involving considerable capital. That printers took risks in such untested markets as physical science suggests a special interest on their part. Printers were concerned with the lever and the screw as essential devices of their art. They were also concerned with alloys of metals and methods of working them with tools, with the composition of inks and the properties of paper, and with other technical problems in a way in which the scribe had not been concerned at all. It is therefore not surprising if some printers took both a practical and theoretical interest in mechanical principles and devices, as well as in commercial arithmetic and the geometry of design, so that books dealing with mechanics, mathematics, chemistry or metallurgy appeared to them likely to find a market among men who had similar interests. In some instances, scientific pioneers were themselves printers. Rudolf Hirsch remarked, in a recent book on printing and reading, that printers generally relied on advisers in selecting titles, and went on to remark that: "The earliest event in 'new science' was an exception: Regiomontanus printed in Numberg between 1472 and 1476 writings of his teacher, George Peurbach, and his own, which according to well-informed critics heralded the beginning of modem astronomy and mathematics. Printers and publishers especially of the 16th century seem to have been quite willing to produce contemporary works which contained novel theories or presented the results of investigations in a new way; they would not have done so unless they could sell these products successfully." Professor Hirsch's remark about novelties here has particular

124

History of Science

significance with respect to the role of university science vis-a-vis the new science typical of the sixteenth century, of which I shall speak presently. In the 16th century, universities were conservative institutions, whether or not they still are today. Their job was to preserve learning, examine it critically, and impart it; not to seek or create new knowledge, or at least not that primarily. Printers of books also served that purpose, but did not confine themselves to it. Their first productions were heavily weighted in the direction of classical texts and learned commentaries, for most of their assured markets lay in the universities and among the clergy. But in seeking new markets, in which the competetion would be less direct, they did welcome writers for whose works they could claim novelty, and this was of no small assistance to the spread of unconventional science. In saying that the age-old university monopoly on science was broken in the sixteenth century, as a direct result of printed books, I do not mean that science passed out of the universities and into the hands of other men. Rather, there were now two streams of scientific thought where before there had been but one. Professors of philosophy continued to write on physics throughout the sixteenth century, and well into the seventeenth, in the same style and with the same objectives as their medieval and early Renaissance predecessors. Fashions changed in the particular topics on which they wrote, but physics remained in their domain so far as official science was concerned. Indeed, they were the men who first attacked the new physics of Galileo in the seventeenth century, long before the church became concerned about his views. He himself remarked in his Dialogue, written after a lifetime of experience, that philosophers would never be swayed by "one or two of us who bluster a bit." Thus one phase of Renaissance science was quite literally a continuation of medieval science. Centered in the universities, this phase remained Aristotelian to the core, and defined science as the knowledge of things through their causes. It was also influential outside the universities; even Descartes was to criticize Galileo's science on the basis that by trying to investigate particular motions without first determining the cause of motion, he had built without foundation. But the continued existence of official science, and even its spread beyond the universities through the printed book, does not give Renaissance science the unity that had existed in medieval science. There is no getting around the independent origin of a totally different set of inquiries outside the universities. It is this non-U science, to use Miss Mitford's apt term, that is really a distinctive Renaissance entity, and in my opinion it could not have come into and remained in existence before the printed book. I can think of nothing that could have kept such inquiries alive and separate from the centers of learning except its inexpensive and widespread circulation in durable form.

Early Science and the Printed Book

125

The medieval university community was clearly relatively small and homogeneous as compared with the rest of society in the Middle Ages. A large number of copies of any given work was never necessary within that community, whether we think of this as a single university, of several universities in a geographical area, or even as all the universities in Europe. It was unlikely that a person living in the university community would long remain unaware of the existence of a significant work related to his particular interests. If a copy was not immediately at hand, it was probable that the scholar who told him of the existence of the work would also be able to outline the nature of its contents. But outside the university community, this situation was reversed. It was there highly unlikely that a person would know of all the works in manuscript pertaining to his particular interests. If told of the existence of such works, it was improbable that his informant could also tell him the nature of the contents. For these and similar reasons, the invention of printing was potentially much more important to society outside the universities than to men within them. In a large and heterogeneous community there is never any lack of special talents or of varied interests, but there is always the problem of keeping alive from one generation to the next the fruits of any application of a particular talent to the appropriate interest. Before printing, this problem found no practicable solution. These and similar considerations of statistical probability seem to me sufficient to account for the relative absence of non-U science up to the closing years of the fifteenth century, and for its swift emergence thereafter. In every period, excellent contributions to knowledge may have been made by men outside the official world of scholarship. Before printing, however, their contributions would be expected to languish unless noticed by a member of the learned community. Those so noticed would be brought into the official stream of science. A manuscript not thus brought into the mainstream, either for lack of merit or lack of notice, would be most unlikely to fall under the eyes of another non-U person (outside the scholarly world) who would be interested in it and capable of appreciating it. The probability was always that a work would be either lost, or incorporated into the main body of doctrine. It is in these terms that I should account for both the unity of medieval science and the university monopoly on science before the printed book. The conservative character of the university in the sixteenth century is further attested to by a curious fact that seems not to have been mentioned by others, if it is indeed a fact. Not only does it appear that meritorious scientific ideas propounded in sixteenth-century books by authors outside the universities aroused no interest or attention inside their walls, but even the new editions and translations of classic Greek mathematics were not incorporated in university studies, so far as I have been able to discern. It is well known that the astronomy of

126

History of Science

Copernicus was not take up by the universities generally. The one exception known to me is Tiibingen, where Kepler's teacher, Michael Maestlin, was favourable to Copernicus; but even in this instance he seems not to have gone further than to teach it together with the Ptolemaic and Tychonic systems. The same thing seems to be true of new ideas in mechanics published by Tartaglia, Guidobaldo and Benedetti. Their books appeared in more than one edition, and some of them in more than one language during the sixteenth century, but I have not yet found their ideas taken up in any book published by a professor of the period. On the other hand, each of these non-university men mentions the work of the others who had written before him, either critically or approvingly. Multiple editions and cross-referrals show that a viable body of mechanical science existed that simply did not impress the university community. Two or three professors of mathematics at the University of Padua in the sixteenth century did lecture on mechanics, but only as a commentary on the pseudoAristotelian work on that subject that had come to light for the first time late in the fifteenth century. So a neglect in university circles of contemporary writers on mechanics cannot be excused on the grounds that this was not a university subject. Rather, it supports the view that the doors of sixteenth-century universities were closed to new scientific ideas from the outside. It has been mentioned that the theme of official science was the knowledge of things through their causes. In general, that is a different theme from the mathematical description of physical events. Mathematics could not be admitted as a cause of anything under the Aristotelian view, and descriptions failed to disclose causes in the Aristotelian sense. Nevertheless, as Duhem long ago pointed out, medieval philosophers of the 14th century had formulated some ingenious and elaborate methods of introducing mathematics into Aristotelian physics. Some of the best of those medieval works had reappeared in printed books from about 1480 to 1530. Many historians today believe that the same doctrines continued to be discussed in the universities throughout the 16th century, and that Galileo was inspired by them to undertake his own program of mathematicizing physics. The evidence of printers' production seems to me to be against this. The number of such books was greater before 1500 than after, and the medieval treaties in question were not reprinted after 1530. About that time, the universities (except Paris) seem to have taken a greater interest in going back to premedieval commentaries on Aristotle, of which a great number were printed and reprinted from about 1525 on into the seventeenth century. More important from the standpoint of science is the popularity of debates on method in sixteenth-century universities, particularly in Italy. The certainty of mathematical knowledge was a topic explored by several prominent philosophers inside and outside the universities. These debates on method and on mathematical cer-

Early Science and the Printed Book

127

tainty are just now coming to be seen as important to the history of science, in view of the later development of physics. But since the philosophers who wrote on these subjects did not also contribute to scientific knowledge, it is small wonder that they were not recognized by the earliest historians of science. Critiques of method were also written by anti-Aristotelian philosophers of the late Renaissance; among these we may count Bernardino Telesio, Pierre de la Ramee, and Francis Bacon. It seems to me likely that these critiques would have been made in any case, whether or not a new astronomy and a new mechanics had been proposed by sixteenth-century scientists, because neither the debates on method and mathematical certainty within the universities nor the anti-Aristotelian programs of other philosophers were based on the idea that new discoveries had to be accommodated, or that new sciences ought to be admitted. Indeed, Bacon was outspoken against the Copernican astronomy, which at that time had made more headway in England than anywhere else, and as I have previously mentioned, Descartes opposed the new sciences of Galileo. The ferment of ideas on method and of opposition to the authority of Aristotle seems to me to be a product of the general broadening of discussion that followed the printed book, rather than a by-product of the new ideas in science itself. The two are linked together by later developments in science, but perhaps not by common interests of the authors who put them forward. It is time now to tum our attention away from official science, which in effect continued in the universities after the invention of printing much as it had before, but which spread out from them into a more general literature, and to consider the two movements in science that arose outside the universities and that give to Renaissance science its own characteristic content, neither medieval nor modern. The first of these is the editing, translation, and writing of commentaries on classic Greek mathematical and scientific works. Euclid was of course first in the field, and least removed from the medieval tradition. In fact the first Euclid printed was in the medieval translation (Latin from Arabic) by Cam pan us of Novara. This was followed in 1505 by the Renaissance translation of Zamberti directly from the Greek. Both were reprinted many times in the sixteenth century, chiefly in France and Italy. The first vernacular translation was into Italian and appeared in 1 543, the same year as the great works of Copernicus and Vesalius. An English translation with much improved commentaries was published in 1570; partial French and German translations also belong to the sixteenth century. Vernacular translations put rigorous mathematics into the hands of a much larger population for the first time, while the first really competent mathematical and textual commentary, written by Federico Commandino, set new technical standards. The first mathematical works of Archimedes to be

128

History of Science

printed came in the first decade of the sixteenth century, followed in 1543 by his two works dealing with physics; and the following year by the Greek text of most of his works and a Renaissance Latin translation. Ptolemy's astronomy was printed in 1515 from an Arabic text; better translations followed, and the Greek text, by mid-century. The mechanical and mathematical works of Hero and Pappus of Alexandria appeared in Latin, and some of them also in Italian. None of these Renaissance translations, so far as I know, was the work of a university professor, though most of them were by men who had been educated in universities. Others were the work of mathematicians and engineers whose efforts seem to have no explanation other than sheer admiration for the mathematical excellence of the works, and a desire to make them accessible to a wide public. The original work of Renaissance scientists outside the universities cannot be better illustrated than by the case of Niccolo Tartaglia, who was born at Brescia in 1500. His family was very poor, and the death of his father when he was still a young child meant that he had almost no schooling. According to his own account, he had learned only half the alphabet from a writing master when funds for the tutor ran out, and he completed his education alone. Even allowing for some exaggeration in this story, it is evident that Tartaglia was largely selfeducated, for his Italian style was anything but cultured, and occasioned derision from some of his better-educated contemporaries. His only publications in Latin were editions of medieval translations in which · he did not venture to make textual corrections. But Tartaglia had a great talent for mathematics, particularly algebra, and when he reached maturity he began to offer private instruction in that field. Instances of self-education to such an extreme degree are not common even in the Renaissance, and it is hard to see how they would even be possible before the existence of printed books. In 153 1, when Tartaglia was tutoring private pupils at Verona, he was asked by an artilleryman how to point a cannon in such a way as to attain the maximum distance for a shot with a given charge. This problem he solved correctly, and then became interested in the possibility of working out a general mathematical treatment of projectile paths. Having done this to his own satisfaction, he says that he decided not to publish it because it would be sinful to teach Christians how better to slaughter one another. However, when the Turks threatened Venice in 1537, he overcame his scruples and published it in a book called Nova Scientia, written in Italian, dedicated to a military commander, and intended for the practical use of soldiers. This is the first book I know of that proclaims novelty of a science as a recommendation. Such titles were not uncommon thereafter, as in his rival Girolamo Cardano's Opus novum de proportionibus and Galileo 's Due nuove

Early Science and the Printed Book

129

scienze. Tartaglia's new science of ballistics was presented in mathematical fonn, with definitions, axioms, postulates and theorems. So far as I can tell, it is devoid of influence by medieval physics; its principles are drawn from Aristotle and Euclid and its fonn is neither syllogistic nor argumentative, but that of mathematical deduction. Perhaps the work of Archimedes on floating bodies suggested the pattern to him. Several traits of this first significant attempt at a new physics strike me as characteristic of the age: use of the vernacular, stress on practical utility in the dedication, pride in novelty, and the stress on mathematical rigour in scientific argument. This first book of Tartaglia's went through half a dozen Italian editions during the sixteenth century and was translated into Gennan and English. Its results were adopted by many writers on military matters, who continued to cite it in the 17th century even after Galileo's correct theoretical analysis of projectile motions, which was not of practicable application to gunnery. But despite this widespread acceptance, I have never run across any reference to Tartaglia's theory that would connect it with any university course, or any citation of it by a professor writing on motion or on physics. Tartaglia published the first vernacular translation of Euclid in 1543, working from the Latin of Campanus and Zamberti, whose commentaries he also translated, adding further commentaries of his own which contain some interesting textual conjectures. He probably had very little knowledge of Greek, and seems not to have consulted manuscript sources. In the same year he published some medieval Latin translations of works by Archimedes, including the two which dealt with mathematical physics. All these works are of interest in their departure from ordinary scholarly practices. No professor would have bothered to make a vernacular translation, and still less to publish uncritically a Latin translation from the Greek. Tartaglia's motivation can hardly have been anything other than zeal to provide for others an easier road to self-education than that which he himself had followed. In 1546 he published his second original work, a collection of scientific questions with which he had dealt in his tutoring. In his book he made what was perhaps the first open challenge to the scientific authority of Aristotle, directed against the treatment of certain problems related to the balance. As a remedy to this he published in Italian an improved version of the medieval science of statics due to Jordanus Nemorarius. In 1551 he followed this with a treatise on the raising of sunken ships, accompanying it with an Italian translation of and explanatory commentary on the work of Archimedes on floating bodies. With this publication, Tartaglia had set forth in Italian virtually everything of importance that the past had to offer on theoretical mechanics, whether .ancient or medieval, and had contributed a good bit of his own. His final work was a vast

130

History of Science

treatise on mathematics, most of which was published posthumously about the time Galileo was born. Tartaglia's publications, and a certain notoriety he had obtained as a mathematician by winning challenge contests and by a celebrated dispute with Cardano, failed to win for him any university chair in mathematics, though he made at least one serious attempt to obtain one. He was certainly a much better mathematician than the incumbent at Padua, Catena, who is remembered mostly for his mystical interpretations of mathematics. It is clear, however, that Tartaglia's work had a wider effect, and did more good for science, than it was likely to have done from within a university. He enonnously widened the access of Italian readers in every walk of life to mathematics and its application to practical problems. One of his private pupils was Benedetti, who was in many ways Galileo's most important Italian predecessor, and another of his pupils is supposed to have been Ostilio Ricci, Galileo's own teacher of mathematics and also unconnected with a university. As I remarked, Tartaglia affords a striking illustration of the things that distinguish Renaissance from medieval science, but he is not thereby typical of it. Many important contributors to Renaissance science through original works, translations of classic mathematics, or scientific commentaries were universityeducated men who did not remain to teach. Such men were Commandino, Guidobaldo, and Bernardino Baldi. In England they had counterparts in Robert Recorde, Thomas Digges, Henry Billingsley and Thomas Harriot; in Switzerland, Walter Ryff and Michael Varro; in Holland, Simon Stevin. In France, no second stream of science outside the universities seems to have flourished, perhaps because the University of Paris continued much longer than other universities to pursue the medieval tradition that linked mathematics to physics. Possibly there is also a connection with the fact that printing in Paris was much more closely linked to the University than in any other city. If in conclusion I were to try to characterize Renaissance physical science, I should say that it was the period during which the unity of science was lost by its passage outside the universities, a period that opened with the inexpensive printed book and that closed with the work of Galileo, which re-established a unity of science that depended not on its institutionalization, as it had in the Middle Ages, but on its removal from the tyranny of any authority other than those of mathematics and of nature herself.

6

The Pseudo-Aristotelian Questions of Mechanics in Renaissance Culture 1

Historians often assert that the origins of modem science lay in a conscious revolt against the authority of Aristotle, a revolt that was openly proclaimed by Pierre de la Ramee, Francis Bacon, William Gilbert, and Galileo Galilei in the late sixteenth and early seventeenth centuries. There is little agreement about the reasons for the revolt. Some hold that the essential characteristic of the new science was an increased attention to observation and experiment; others, that an emphasis on mathematics transformed the character of scientific inquiry. Those who emphasize the role of experiment have generally tended to favor what may be called social explanations of the rise of science, including technological, economic, religious, and political developments. In contrast, the rise of mathematical inquiries has been customarily linked with philosophical explanations of the new science, primarily in terms of Renaissance currents of orthodox Platonism and of esoteric Pythagoreanism. More recently, the source of modem scientific thought has been sought in the Renaissance debates on method conducted by philosophers in the universities, and especially at the University of Padua. It would be foolhardy to deny that modem science symbolized a revolt against authority, and particularly that of Aristotle; that social factors favored its emergence; that concern with mathematical laws goes hand in hand with Platonist speculation and mysticism; or that concepts of method formulated by philosophers have a clear and direct bearing on the validity of scientific inquiry. Granting all the evidence that previous research has amassed concerning these things, however, we are still somewhat short of a clear conception of the way in Co-wriuen wirh Paul Lawrence Rose. Reprinred from Studies in the Renaissance 18 ( 1971 ). 65-104, by permission.

1 32

History of Science

which the various contributing factors came together in the work of any one man, or any series of men having interests sufficiently in common to give rise to the pursuit of recognizably modem science. Social and philosophical accounts do give us a milieu in which the new developments no longer surprise us when they come, but do not show us just how a historical event took place, or why it took place at one time rather than any of several others. In order to add something to our knowledge of the evolution of modem physical science, then, it would be helpful to find some single work that engaged the attention of several men from different standpoints; men of different backgrounds and interests who nevertheless are recognizably in the chain that leads back from Galileo and the beginning of the new science. Works such as the Physica of Aristotle or the Timaeus of Plato are of little use for this purpose, since they attracted the attention of too many persons not in that chain, and since Renaissance interpretations of them were too diverse. Neither are the works of Archimedes of much use, since they attracted the attention of too few, and their interpretation is too direct and uncontroversial. But such a work as is desired exists, and moreover it is in a sense exclusively the property of Renaissance science, neglected in Europe before the mid-fifteenth century and put aside by the mid-seventeenth century. This is the pseudo-Aristotelian Questions of Mechanics (Quaestiones Mechanicae, Mechanica Problemata, Mechanica), on which Galileo lectured publicly at Padua and on which he wrote a commentary, now lost, before he began the publications that inaugurated a new physics - publications in which he acknowledged his debt to the ancient work, and at the same time criticized parts of it. We believe that exhaustive studies of the introduction and fate of the Mechanica will throw new light on the evolution of modem physics in a variety of ways, and will make possible a further step in the ordering of Renaissance factors with respect to that important development. The same studies can throw further light on philosophical differences between the old and new sciences. Above all, they may reveal characteristic Renaissance reactions to medieval and to classic traditions in physical science, responses previously noted by historians in many other fields. Thereby, they may tell us still more about the influence of printing and about the spread of intellectual activities beyond the walls of the universities. It is with a view to the inauguration of such exhaustive studies that the present paper is offered. Its objectives are limited to the bibliographical aspects of the problem and to the links that existed (or in our opinion probably existed) between certain Renaissance scholars and translators who dealt with the work chosen for study. Even in these regards, it cannot hope to be complete in every important respect. It is our hope to make it accurate, so far as it goes, in order

The Pseudo-Aristotelian Questions of Mechanics

I 33

that on the one hand it may serve as a basis for further studies in which it may be referred to for bibliographical and personal identifications, and on the other hand that others may complete it in those respects without the necessity of revising what has here been said in any truly essential regard. No attempt is made in the present paper to describe or classify the textual and diagrammatic errors, omissions, emendations, and conjectures made by scribes, translators, and commentators, nor are their particular commentaries and additions analyzed from the standpoint of either textual understanding or scientific correctness. Those matters, highly important to the history of science as such, are left for separate treatment in subsequent studies, in which we hope that others will soon join us. In an attempt to suggest an orderly pattern for further studies, we have indicated some chronological divisions of activity and some classifications of principal interest among persons concerned during the Renaissance with the Mechanica. These divisions and classifications are not intended as anything more than provisional aids to the perception of differences and relationships that seem to us important at the present stage of investigation. Our rough categories are certainly not pennanent and rigid boundary lines. Indeed, we are not in agreement ourselves on all of them; for example, on whether Bernardino Baldi belongs most properly among the humanists, the literati, or the mathematicians. But in a sense, it is precisely the existence of such disagreements at this stage that most persuades us that exhaustive studies of the Mechanica during the Renaissance will ultimately shed much light on many significant historical problems. II CLASSICAL TRADITIONS OF MECHANICS IN THE RENAISSANCE

The postclassical tradition of the Aristotelian Mechanica is largely a Renaissance phenomenon. Apparently unknown in the Middle Ages, the tradition was effectively born with the extensive copying in Italy of the Greek text by Quattrocento humanists. The inclusion of the Mechanica in the corpus Aristotelicum published by Aldus in 1495-1498 (and reprinted in other editions) first gave the text wide circulation during the sixteenth century. Continuing and expanding this humanist interest were two Latin translations and a Latin paraphrase published between I 5 I 7 and 1547 by three Italian men of letters, Vittore Fausto, Niccolo Leonico Tomeo, and Alessandro Piccolomini. Almost simultaneously with the last of these publications, the Mechanica began to attract mathematicians as well as humanists and men of letters, and it was for a time invariably cited in works dealing with either theoretical or practical mechanics. Among the mathematicians who cited or made use of the work were Niccolo Tartaglia, Girolamo Cardano, G.B. Benedetti, and Galileo. Its practical mechanical inter-

1 34

History of Science

est induced Ramus to promote the Mechanica in France, where one of his pupils published the text and a new Latin translation with commentary at the end of the century. The Mechanica became known in Portugal and Gennany through writers on navigation and architecture, and in England through John Dee. Essentially, however, the sixteenth-century tradition may be regarded as an Italian one. This flourishing tradition was dependent on two conditions. One was the scholarly interest in Greek texts and the philosophical concern with the corpus Aristotelicum for its own sake. The other was the stimulus given by the scientific ideas of the Mechanica to mathematicians and mechanical writers. With the end of the Renaissance, these two conditions vanished. Humanist and philosophical interest in the text (which was, after all , only pseudo-Aristotelian) waned, and at the same time the rise of a new science exhausted the scientific interest of the work. By the time Galileo published his Two New Sciences in 1638, the Mechanica was virtually dead. Its lifetime had coincided with the two centuries of the Renaissance. This background of the Mechanica serves to illustrate a central fact of Renaissance culture: namely, the linkage of science with other elements of that culture. The Renaissance program for the revival of classical texts extended to scientific texts as to literary ones. 2 But the men who inspired, patronized, or carried out this revival and its accompaniment of translation, emendation, and commentary were of the most diverse background - humanists (Bessarion, Leonico ), astronomers (Regiomontanus, Maurolico), mathematicians (Tartaglia, Guidobaldo, Benedetti), men of letters (Piccolomini, Baldi) and practicing engineers (Aleotti, Guarino) - to mention but a few . These men and others produced a mathematical-mechani.cal renaissance as consciously programmed as the philosophical renaissance had been planned by Ficino and others. To understand Renaissance science one must also understand Renaissance culture in general. Some research has already been devoted to the classical traditions of mechanics in the Renaissance; it may be useful identify these traditions. At the beginning of this century Pierre Duhem distinguished two classical approaches to statics and attempted to trace their influence from Antiquity through the Middle Ages and the Renaissance to Galileo and the scientific revolution of the seventeenth century.3 One school was the Aristotelian, characterized by an approach to statics through dynamic principles; the other was the Archimedean, which was exclusively static and rigorously mathematical in its approach. A third school, the Alexandrian, combined elements of dynamic and static approaches in technological and empirical mechanics. Following the collapse of the ancient world, different fates befell these three schools. Few of the Alexandrian writings reached Europe; the principal theoret-

The Pseudo-Aristotelian Questions of Mechanics

135

ical treatise, the Mechanics (or Elevator) of Hero of Alexandria, disappeared until it was rediscovered in an Arabic translation late in the nineteenth century. The treatise thus remained unknown to the Latin West. Renaissance mathematicians had access only to excerpts from it which Pappus had recorded in his Mathematical Collections, 4 and even these remained in manuscript until r 588. A different phase of the Alexandrian tradition existed in the Renaissance period thanks to the survival of two other works of Hero, the Pneumatica and the Automata. Excerpts translated from the former appeared in Giorgio Valla's De Expetendis et Fugiendis Rebus of I 501. Translations of both works followed during the sixteenth century.5 Hero's works inspired the engineers who designed toys and fountains for Renaissance courts, but contained no systematic treatment of mechanical laws. 6 Authentic Archimedean works dealing with mechanical investigations were but two in number, most of his writings being purely mathematical. Although most of the authentic books of Archimedes were extant in Greek texts in Italy during the Middle Ages, and were put into Latin late in the thirteenth century, they appear to have been neglected by medieval writers on physics and mechanics. In their place, a spurious treatise attributed to Archimedes preserved his treatment of specific gravities but not his concept of centers of gravity or his proof of the law of the lever. Publication by Tartaglia in 1543 of the two authentic works by Archimedes on mechanical subjects, in an anonymous thirteenth-century Latin translation (now known to be that of William of Moerbeke), represented the first widespread Archimedean revival. Followed in 1544 by a printed edition of text and Latin translation, including the mathematical works, the full impact of authentic Archimedean writings was a phenomenon of the latter half of the sixteenth century, though his work was by no means unknown long before that time. 7 The Aristotelian tradition in mechanics, prior to the sixteenth century, had no evident connection with the text of the Questions of Mechanics. Scholarly attempts to connect medieval writings on statics with this work have thus far been unavailing. At best, any connection is limited to a few possible sections. The opening section of the Mechanica deals with the behavior of balances and levers; and so, of course, do virtually all treaties on statics of whatever period. 8 Truly characteristic questions raised in the Mechanica - why large balances are more sensitive than small ones, and why balances do not return to a level position when the support is from below - were ignored in medieval treatises, 9 as were all the questions dealing with oars, rudders, sails, vortices, fracture, and so on, and also the important theoretical principles of motion, such as those virtual velocities and the parallelogram of velocities, that are adumbrated in the pseudo-Aristotelian work.

136

History of Science

Nevertheless, the statics of the Middle Ages was built on the idea of motion as a means of analyzing equilibrium, just as motion dominated the problems dealt with in the Mechanica. It is not hard to account for this, since the disciple of Aristotle who composed the Mechanica had recourse for his principles to the Physica and other authentic texts of Aristotle, just as did writers in the Middle Ages who wrote on statics. In order that the reader may form an idea of the problems discussed in the Mechanica, the "questions" are listed below in summary form. All deal with motion rather than with statics as such.

2

3 4 5 6 7 8 9 10 11

12

13 14 15 16 17 18 19

20 21

22 23

24

Why larger balances are more accurate (i.e., sensitive) than smaller ones. Why the balance seeks the level position when supported from above, but not when supported from below. Why small forces can move great weights by means of the lever, despite the added weight of the lever. Why rowers in the middle of ships contribute most to their movement. Why the rudder, though small, moves the huge mass of the ship. Why ships go faster when the sailyard is raised higher. Why in unfavorable winds the sail is reefed aft and slackened afore. Why round and circular bodies are most easy to move. Why things are drawn more easily and quickly by means of greater circles. Why a balance is more easily moved when without weights than when weighted. Why heavy weights are more easily carried on rollers than in carts. Why a missile thrown from a sling moves farther than one thrown by the hand. Why larger (i.e., longer) handles move windlasses more easily. Why a stick is more easily broken over the knee when the hands are equally spaced, and farther apart. Why seashore pebbles are rounded. Why timbers are weaker the longer they are, and bend more when raised. Why a wedge exerts great force and splits great bodies. Why two pulleys in blocks reduce effort in raising or dragging. Why a resting axe does not cut wood, and a striking axe splits it. How a steelyard can weigh heavy objects with a small weight. Why dentists use forceps rather than the hand for extraction. Why nutcrackers operate without a blow. Why the lines traced by points of a rhombus are not of equal length. Why concentric circles trace paths of different length when rolled jointly on this or that circumference. (Wheel of Aristotle.)

The Pseudo-Aristotelian Questions of Mechanics 25 26 27 28 29 30 31 32 33 34 35

137

Why beds are made in length double the width. Why long timbers are most easily carried by their centers. Why longer timbers are harder to raise to the shoulder. Why swing beams at wells are counterweighted. Why of two men carrying a beam, the man nearer the center of the beam feels more of the weight. Why men move feet back and shoulders forward to rise from sitting. Why objects in motion are more easily moved than those at rest. Why objects thrown ever stop moving. Why objects move at all when not accompanied by the moving power. Why bodies thrown cannot move far, but [a distance] related to the thrower. Why objects in a vortex finish at the center.

The questions are thus challenging, specific, and sufficiently varied to permit reasonably certain identification of later writings by authors who had access to the text and who themselves attempted to deal with like problems. The solutions offered by the text, and the amendments or alternative solutions offered by commentators, fall outside the scope of the present paper. III THE GREEK TEXT AND ITS SOURCES

The Mechanica (or Mechanica Problemata) consists of an introduction followed by the discussion of thirty-five problems in statics and dynamics. It dates from the third century B.c. and as such it is the earliest surviving treatise on mechanics. Although it is no longer thought to be the work of Artistotle himself, it can be assigned to the Lyceum, possibly to Strato. 10 During the Renaissance, opinions on its authorship were divided. The translators and commentators of the Mechanica, as well as various editors of the corpus Aristotelicum, credited it to Aristotle, but Cardano and Patrizi were somewhat sceptical. 1 1 Bibliographies and histories of the text may be found in Pauly-Wissowa, in Moraux, and in the modem editions. 12 A useful list of Reinaissance Italian editions and translations was compiled in 1 886 by Riccardi and recently supplemented by Drake and Drabkin. '3 There are three main modem editions of the text: Cappelle (1812), Bekker ( 1831 ), and Apelt ( 1888). 14 None of these yet gives a fully satisfactory critical version of the text which, as the sixteenth-century commentators and translators had already noted, is a highly corrupt one. English translations of the text are available in the Oxford 15 and Loeb 16 series, the latter providing a convenient modem edition of the Greek, though the diagrams supplied are not always to be trusted, particularly those in Question I.

138

History of Science

Elipdio Mioni has recently drawn attention to the need for a new edition of the Mechanica based upon the archetype codex, now in the Biblioteca Marciana at Venice. 17 This manuscript (designated "Ha,, by Bekker) dates from the twelfth century and was the source of the two Vatican manuscripts upon which Bekker largely based his edition (Mss. Vat. Graec. 1339 and Urb. Graec. 44). Table I, a chronological list of these and the other extant manuscripts stemming from Marciana Z.Gr.214, is drawn from Wartelle, with our own addition of the Escorial and Naples codices and some dates. 18 A precise year of copying is given where possible. Excluding the London partial text and the Laurenziana copy of the Aldine edition, there are twenty-eight manuscript texts of the Mechanica dating before 1650. The earliest belongs to the twelfth century; only two more copies are known to have been made in the ensuing two centuries. Only three possible allusions to the work before 1413 have been found. 19 Then, in the fifteenth century, a sudden flurry of activity produced another seventeen extant mllnuscripts. The activity lessened in the sixteenth century, with only seven known codices after the first printed text. Between 1600 and 1650 there is but a single new manuscript copy. The main impulse which lay behind the sudden multiplication of copies of the Mechanica during the latter half of the fifteenth century was the general humanist ideal of reviving works of Antiquity. But certain circumstances affecting copies in Italy reveal an unexpected interest by some prominent Quattrocento humanists in mechanics from the standpoint of technology. The history of two known Mechanica manuscripts will illustrate this point. Marciana MS. Gr.lV,57 was copied at some time before 1446 by the scribe Joannes Symeonachis of Crete and was dedicated to the Venetian noble, Marco Lippomano, a noted jurist who was duke of Candia from 1435 to 1437. 21 Lippomano seems to have been particularly interested in the mechanical sciences, for he is mentioned in the writings of the contemporary Venetian engineer Giovanni de' Fontana. In connection with an experiment to detennine whether deep sea water was sweet, Fontana says that he constructed an instrument for Lippomano to drop over a ship's side. 22 Thus it is quite possible that this copying of a Greek manuscript was carried out not from a humanist, but from a technological interest. Marciana MS. Z.Gr.200 was copied in 1457 by the scribe Rhosos for the great humanist Cardinal Bessarion. It contains nearly all the works of Aristotle, apart from the books on logic. 23 This manuscript is especially important in that it was to fonn the basis of the printed corpus Aristotelicum in the Renaissance. Significantly this is the first known manuscript in which the Mechanica appears together with the genuine works of Aristotle; previously it had accompanied

The Pseudo-Aristotelian Questions of Mechanics

139

TABLE 1

Biblioteca Marciana Biblioteca Vaticana

1100-1200 MS. Z.Gr.214 (479) 1200-1400 MS. Vat.Gr.253. Sec. xiv, possibly xm MS. Vat.Gr.1339. Sec. xiv

1400-1500 MS. Z.Gr.216 (404). Anno 1445 MS. Gr.iv, 57 (1428). Before 1446 MS. Z.Gr.200 (327). Anno 1457 MS. Z.Gr.215 (752) MS. 245 (Phillipps 7488) . Anno ca. 1450 Yale University Library MS. Phil.Gr.231 . Anno 1458 Nationalbibliothek, Vienna MS. 4653 (N26). Anno 1470 Biblioteca Nacional, Madrid MS. xxvm,45. Anno 1445, Milan Biblioteca Laurenziana, Florence MS. Acquisti e Doni 65 MS. LXXXV, 26 is in fact a printed Aldine tex1 of 1497.) MS. Urb.Gr.44 Biblioteca Vaticana MS. Pal.Gr.162 MS. Urb.Gr.76 MS. Est.Gr.76 (Alpha T.9.21) Biblioteca Estense, Modena Bibliotheque Nationale, Paris MS. Gree 2507 MS. Suppl.Gree 541 MS. Graec. 402 Burgerbibliothek, Bern Bibliotheck der Rijksuniversiteit, Leyden MS. Voss.Gr.O .25 Biblioteca Marciana

Biblioteca de El Escorial Biblioteca Nazionale, Naples Staatsbibliothek, Berlin Biblioteca Vaticana Bibliotheque Nationale, Paris

Library of the Synod, Moscow (British Museum, London

1500-1600 MS. Phi 1, 10. Anno 1542 MS. Graec.266 MS. Graec.103 (Phillipps 1507) MS. Barb.Gr.22 MS. Reg.Gr.124 MS. Gree 2115 MS. Suppl.Gree 333 1600-1650 MS.453 MS. Burney 67 - excerpts)

N.8.: (1) Many diagrams are supplied by the scribes of Marcianus Iv, 57, Laur. Acquisti 65, and the Bern and Yale manuscripts.20 (2) Two unidentifiable manuscripts at Venice in the seventeenth century are listed by J.P. Tomasini, Bibliothecae Venetae Manuscriptae Publicae et Privatae (Udine, 1650), pp. 17, 59. The manuscripts were formerly in the libraries of Sant'Antonio and San Giorgio Maggiore.

140

History of Science

other pseudonymous works or had been separately preserved. In assembling the corpus Aristotelicum (in Z.Gr.200) Bessarion, it seems, had taken special care to have the Mechanica copied into it, probably directly from the archetype Z.Gr.214 which he also owned and bequeathed to the Venetians. Very likely a technological interest lay behind Bessarion 's special concern for the inclusion of the Mechanica; Bessarion was fascinated by the economic importance of machines in Italy, and as early as 1444 had urged the importation of European mechanical skills and techniques into Byzantium. 24 The fifteenth century in Italy was fruitful of engineering treatises by such men as Da Vinci, Brunelleschi, Taccola, and Martini. It is quite possible that their range of interests and achievements attracted the sympathetic attention of humanists and stimulated among them a revival of neglected classic texts such as the Mechanica. On the other hand, the lack of evidence of any fifteenthcentury Latin translation of the work suggests that the relationship was not reciprocal, and that the humanists did not make accessible to the engineers much knowledge of ancient mechanical science. The role of Cardinal Bessarion in stimulating the revival of Platonism is well known. Less widely recognized, but perhaps of more importance to the origins of modem science, was his role in bringing to Italy Greek manuscript sources from which a knowledge of ancient mathematics and physics was to be drawn in the sixteenth century. Not least in importance among these was the Mechanica, from which Galileo ultimately extracted and refined the principle of virtual velocities, the idea of composition of motions, and probably the limited concept of conservation that is found in his own treatise on mechanics. IV THE HUMANIST EDITIONS AND TRANSLATIONS,

1497-1531

The Bessarion manuscript corpus Aristotelicum, mentioned in the previous section, has special significance because it was used as the basis for the Aldine editio princeps of the complete works of Aristotle, published at Venice in four volumes 1495-1498. The Mechanica, in virtue of its inclusion in the Bessarion manuscript, appeared in Greek (without diagrams) in the second volume of the Opera, published in 1497. The second and third volumes were planned by Aldus as an encyclopaedia of the natural sciences. 2 5 Publication of the Aldine text marked the beginning of the Mechanica's century of scientific significance. For twenty years it remained the only printed text of or reference to the Mechanica, despite two bibliographical "ghosts." The first of these is erroneously listed by Riccardi as published at Florence in 1484. The work, by Raffaello Francesco, Verificatio Universalis in Ref?ulas Aristotelis de motu non recedens a communi mathematicarum doctrina, does

The Pseudo-Aristotelian Questions of Mechanics

141

exist, though it is extremely rare. The only two copies known are in the Biblioteca Comunale Augusta, Perugia, and the Columbia University Library, New York. Riccardi indicated a copy in the Marciana, but we were not able to trace this. The known copies of the Verificatio (which carries no date in the colophon) are assigned by the respective libraries to Bologna, 1490, and Florence, 1519. 26 Riccardi's dating of 1484 seems to have been based on an error by Hain, who described the Verijicatio with another work bound with it: namely, Bernardinus Tornius, Dicta supra cap. de motu /ocali Hentisberi, Pisis, 1484 (Hain 7351 ). 27 In any event, the Verijicatio has not the slightest connection with the Mechanica, despite its inclusion in the Riccardi list of editions concerned with the Mechanica. Rather, it bears mainly on the doctrine of motion and the "Calculatores." The second "ghost" appears in Panzer as a purported Aldine edition of 1507 "Aristotelis Mechanica, Venetiis, apud Aldum, MDVII, 8°."28 There is no trace of this edition in the Renouard Aldine bibliography nor in the best-known Aldine collections. Like the first printed Greek text, the first Latin translation was produced by a humanist, though one deeply committed to mechanical technology. This was the Venetian Vittore Fausto (1480-1551?), public lecturer in classics at Venice from 1518. Fausto was praised for his umanita by Ariosto 29 and described by a later mathematical commentator on the Mechanica as peripateticus Mathematicus insignis ac linguae Graecae professor eximius. 30 In 1526 he put to practical use his knowledge of ship and oar mechanics (topics discussed in Questions 4 to 7 of the Mechanica) when he designed a quinquereme on the classical model for the Venetian government.JI In 1517, after he had spent some years travelling around Europe, learning techniques of ship construction and also studying with Erasmus, 32 Fausto published his Latin translation at Paris. It was entitled Aristote/is Mechanica Victoris Fausti industria in pristinum habitum restituta ac latinitate donata, in Aedibus Iodoci Badii, Paris, 1517. 33 The edition, which is unpaginated, consists simply of the preface, a literal translation, and several diagrams. The text on which Fausto based his translation is not specified, but was probably the Aldine editio princeps. But though that printed text reduced the dependence of Renaissance scholars on manuscripts, many still sought to make better sense of difficult passages by recourse to old manuscripts. As an Aristotelian, Fausto accepted the concept of "mixed sciences" by which the Mechanica related mathematical and physical problems. Their unification was a problem which exercised the minds of succeeding commentators on the Mechanica, philosophical as well as mathematicaJ.34 Fausto dedicated

142

History of Science

his translation to the Venetian ambassador to France, Giovanni Badoer, who had previously received the dedication of a mathematical treatment of astronomy, Giorgio Valla's translation of Aristarchus' De Magnitudinibus , Venice, 1498 (Hain I I 748). Through his humanist and philological activities, Fausto became very well known at Venice and Padua, drawing the attention of other humanists to his technical interests. Cardinal Pietro Bembo was an enthusiastic admirer of Fausto' s quinquereme.35 Bembo in tum was friendly with the translator of the Mechanica, Niccolo Leonico Tomeo (1456--1531).3 6 As librarian of the Marciana, Bembo had lent Leonico several Greek manuscripts. Leonico, a protege of Bessarion's, had introduced reforms at the University of Padua in 1497 by encouraging the teaching of medicine and philosophy from Greek texts rather than Latin translations. This interest in pure Greek texts may have attracted him to the Mechanica. The first of many editions of Leonico's Latin translation of the Mechanica was accompanied by his diagrams and commentaries. Entitled Conversio Mechanicarum Questionum Aristotelis cum Figuris, et Annotationibus quibusdam, it appeared in 1525 with some other works under the general title Nicolai Leonici Thomaei Opuscula nuper in lucem edita, Bemardinus Vitalis, Venice, 1525, between his treatments of the Salemitan Quaestiones Natura/es and Proclus on the Timaeus . The preface of the Opuscula addressed to Johannes Burgarinus is dated Padua, 1524 , and states that the Latin translation had been finished some years before. 3? A second edition of the entire Opuscula was printed in 1530 along with another work of Leonico's, Aristotelis Stagiritae Parva quae vocantur Naturalia in Latinum conversa . .. . explicata a Nicolao Leonico Thomaeo .. . Ejusdem Opuscula nuper in lucem edita, Paris, Simon Colinaeus, 1530, of which there are copies in the Library of Congress and the British Museum. Leoni co' s translation of the Mechanica went through a good many subsequent reprintings in the sixteenth century, but without diagrams and his commentaries. It was also included in the Bekker edition (m, 409-415). Among the reprintings are the Giunti Opera Aristotelis, Venice, 1552, 1562; the Laemarius, Geneva, I 597; the La Roviere, 16o6; and the Duvalle editions, 161 9-1629, I 639, 1654. It also appears in the Compendium Mathematicum Michaelis Pselli, Leyden, 1647, By achieving this wide circulation with printings in Italy, France, and Holland, it was Leonico's work rather than that of Fausto that became the standard translation of the sixteenth century. Galileo's copy was in the original and complete edition of 1525.38 Later sixteenth-century commentators (Piccolomini, Mendoza, and Baldi) refer to the Leonico translation , claiming to have improved both it and its "adnotatiunculae." Baldi went so far as to speak of its commentaries as mere marginal notes, whereas they were in fact fairly exten-

The Pseudo-Aristotelian Questions of Mechanics

143

sive paragraphs of an explanatory character. These commentators objected mainly to Leonico's lack of mathematical understanding. The high period of the humanist tradition (colored by technological interests) of the Mechanica opened with the publication by Aldus in 1497 of the text established by Bessarion. It continued with the Fausto and Leonico translations. This period may be said to have closed in 1531 with the reappearance of the Greek text in the Erasmus-Grynaeus edition of the Aristotelis Opera Omnia at Basie. Having been thus firmly established by the humanists, the Mechanica entered a period of cultivation among the literati generally and among mathematicians. V THE LITERATI AND THE MECHANICA AT PADUA,

1538-1547

After Bessarion and Lippomano in the fifteenth century, Venetian humanists continued their interest in machines. Giorgio Valla, a great collector of Greek manuscripts, published the first (partial) translation of the Pneumatica of Hero in 1501, and also designed some machines. 39 Fausto was active at the Venetian Arsenal. But by the late 153Os, philological humanism was giving way to a new humanism based on the vernacular. 40 One of the main centers of the new movement was the university city of Padua, where Cardinal Bembo, the guiding force behind the Paduan literati, had been prominent in the older humanism and also a patron of Fausto. A manifestation of interest in classical technology among the literati was the Vitruvian revival, which accompanied that of the Mechanica. The Tenth Book of Vitruvius' De Architectura deals with machines, and this was associated by sixteenth-century commentators with the Mechanica. Daniele Barbaro, a leading figure in Paduan intellectual circles of the 1540s and afterwards the leading commentator on Vitruvius, followed the Mechanica in associating mechanics with unique properties of the circle and of circular motion.4 1 Another widely read commentator, Philander, frequently adduced the Mechanica to explain passages of the Tenth Book.42 Vitruvius was so popular among the literati that two academies were set up, one at Rome and one at Padua, to study and correct De Architectura. At Rome, before 1542, Claudio Tolomei founded the Accademia delle Vertu, also known as the Accademia Vitruviana, with the intention of publishing a definitive edition of Vitruvius along with supplementary works explaining, inter alia, the machines in the Tenth Book.43 Some years later at Padua, Sperone Speroni, the noted letterato and friend of Daniele Barbaro, made a proposal for an Accademia dei Gimnosofisti which was to embrace both men of letters and men of arms. Most of the curriculum was to concern mathematics, architecture, and

144

History of Science

fortification, and as lecture subjects Speroni specified two works, De Architectura and the Mechanica .44 Out of this fruitful Paduan ambience came two important works dealing directly with the Mechanica, written by visitors to Padua in the early 1540s Alessandro Piccolomini and Diego Hurtado de Mendoza. Alessandro Piccolomini of Siena ( 1508-1579), author of many philosophical and literary works, frequented the university and academies at Padua from 1538 to 1542. 45 There, some time before 1540, Piccolomini met Diego Hurtado de Mendoza, Imperial ambassador to Venice from 1539 to 1546. Probably as a result of this meeting, Piccolomini extended his interest to mechanics and in a work published in 1542 he spoke enthusiastically of machines and the Mechanica. Noblemen ought to study the science of machines, he says, inasmuch as "Aristotle himself has written of them in his short though excellent book [the Mechanica] . Of this book, which by reason of the great corruption of its text is very obscure and has been explained by no one, I have made, at the persuasion of Don Diego de Mendoza, a Paraphrase."46 Ultimately Piccolomini published his Latin paraphrase at Rome, and it went on to achieve a second edition and an Italian translation. The first edition was entitled In Mechanicas Quaestiones Aristotelis Paraphrasis paulo quidem plenior, Antonius Bladus, Rome, 1547. A second edition was published under the same title by Traianus Curtius at Venice in 1565. The Italian translation was made by Oreste Vannoccio Biringucci: Parafrasi di Monsignor Alessandro Piccolomini . ... sopr'1 le Mechaniche d' Aristotile, Francesco Zanetti, Rome, 1582. 47 Piccolomini published a commentary on the Meteorology of Aristotle and wrote on the Poetics and the Ethics. His Mechanica paraphrase may be treated as falling within this series of expositions of Aristotle. On the question of authorship Piccolomini commented: I shall not discuss whether Aristotle is the author, so as not to prejudice the Questions in so manifest an argument, but a comparison of the phrases and vocabulary with those in the other works (such as the De /ride), where Aristotle uses mathematical demonstrations, would make it plain that these are the phrases peculiar and proper to Aristotle. I will merely add that although this work is short, it can be thought of as boundless since it allows one to understand the force of intellect and incredible learning of Aristotle, in that it investigates the true reasons for almost all the wonderful machines which not only have already been discovered but which will be invented in the future. 48

Philosophical interests permeated Piccolomini's paraphrase. As an Aristotelian, Piccolomini supported in his paraphrase the concept of mechanics as a

The Pseudo-Aristotelian Questions of Mechanics

145

mixed mathematical and physical science, while insisting on the basic Aristotelian distinction between mathematics and physics. To develop this topic further, Piccolomini appended a treatise De Certitudine Mathematicarum which set forth orthodox Aristotelian teachings on the differing certainties of demonstration in mathematics and physics, and also on the validity of the "mixed sciences. "49 This treatise was omitted in the Italian translation. Piccolomini was careful to include some practical aspects of mechanics, referring in various places to machines which he had seen in use at Venice, Padua, and elsewhere.5° He also used Vitruvius' Tenth Book and an actual machine to illustrate the principle of the pulley proposed in Question 18 (noi habbiamo seguitato I' uso di Vitruvio e quel/o de i nostri tempi, accio che la nostra descrittione fusse piu familiare all' occhio).51 Piccolomini indeed encouraged Biringucci to translate the Paraphrasis into Italian so that it might be of practical benefit to engineers ignorant of Latin. Piccolomini expressed surprise that the work had not been repeatedly translated into Latin. He did not mention the Fausto translation, and was dissatisfied with Leoni co. 52 On account of the alleged obscurity of the book, there has been no one who has attempted before now to expound it, except Leonico alone, by whom some brief annotations are to be found. But this alleged obscurity arises out of the great corruption and garbling of the words also (and especially) from ignorance of mathematics, a discipline much cultivated in the times of Aristotle but today much neglected, so that it is no marvel that comparing our times with his, philosophy has languished. But since this is not the place to mourn the mathematical disciplines, let that suffice that I have said of it. ... I have, however, pitted all the learning and zeal that I possess against the said obscurity of the work, newly finding many texts which I was permitted to examine in the most famous libraries of Venice, Padua, Bologna and Florence. I have corrected for the most part this truly golden book and clarified it with this very long and complete paraphrase, or rather commentary.53

Some of the codices alluded to, especially those in Padua, were probably in private libraries and would be difficult to identify with extant codices.5 4 Piccolomini does give, however, two specific references to texts located in what were then, and still are, essentially public libraries. Commenting on the incomprehensibly garbled wording of Question 25, Piccolomini remarks: This demonstration occurs in a very corrupt passage. Although I have looked through a great many libraries in Italy and emended many bad readings in the Mechanica by the collation of texts, nevertheless I have not seen any text which gives an accurate wording

146

History of Science

of this passage. I must admit, however, that I have been somewhat enlightened by a very old text [quodam satis antiquo in the 1547 edition, f. 55] at the Marciana in Venice. I read this carefully, and if it has not been sufficient to establish and reconstruct the exact text, at least it enabled me to follow the sense and true intent of the author.55 This could, of course, refer to any of the fifteenth-century Marciana codices, which may have appeared old to Piccolomini a century later. One would prefer to believe, however, that Piccolomini had made use of the twelfth-century archetype, Marciana MS. Z.Gr.214. The second reference is more precise. In discussing Question 20 Piccolomini says: "I have not yet seen any text of the Mechanica which does not have throughout it bad readings, particularly in Questions 2, 25, and 30. It is true, though, that a very old codex which is in the Laurenziana at Florence was much less garbled than the others."56 The only manuscript of the Mechanica in the Laurenziana in Piccolomini's time was MS. Pl.xxvm, 45, written at Milan in 1445, which may be presumed to be the manuscript he used. In addition to textual collation, Piccolomini stressed the importance of experimental verification, in accord with the Aristotelian theory of scientific demonstration. By thus subjecting mechanics to sense-verification, Piccolomini pointed up the physical content of mechanics as a mixed mathematical-physical science.57 A concern of Piccolomini' s, shared with the Paduan literati, was the use of the Italian language in scientific works, in order to extend their practical utility.5 8 This concern found belated expression in his encouraging Biringucci to prepare an Italian version (published after Piccolomini's death) that would be useful to engineers. Engineering interest is further evidenced in two additions by the translator to the text: namely, the preface and a final chapter. According to Biringucci: Piccolomini greatly regrelled having translated it solely into Latin, since he thereby deprived of its use those who would most have profited therefrom. For this reason he chose me, though perhaps the least competent of his disciples, to translate it into our language. Although I recognised my weakness and lack of knowledge of mathematics, he showed me how I ought to proceed and pointed out several passages which required some correction or addition. He explained to me also how useful it would be to engineers and architects. To these few instances which he indicated, I have seen fit to add a great many of my own. But although I have been asked by certain persons to include in this edition a compendium of all the machines and instruments now in common use, I wish to defer that project and to make of it a separate book.59 Biringucci's most important addition was a final chapter in which he dis-

The Pseudo-Aristotelian Questions of Mechanics

147

cussed the screw, omitted by Aristotle. The discussion is taken largely from the recent Uber Mechanicorum (1577) of Guidobaldo del Monte, which had itself been translated into Italian in 1581. There was in a fact a widespread attempt in Italy to bring scientific works on mechanics to the notice of practical engineers in the later sixteenth century.6o The next translator of the Mechanica, Diego Hurtado de Mendoza ( I 5031575), Imperial ambassador to Venice from 1539 to 1546, was a frequent visitor to the nearby Paduan academies during those years and was friendly with Bembo and Piccolomini.61 At Venice he studied the Mechanica with Tartaglia as discussed below. It was Mendoza who first encouraged Piccolomini to write his paraphrase of the Mechanica. While at the Council of Trent in 1545, Mendoza made his own Spanish translation of the Mechanica and discussed it with other delegates to the Council between sessions. The translation, entitled La Mechanica de Aristotiles, remained in manuscript until 1898, when it was printed for the first time by Foulche-Delbosc from MS.f.m, 15, the most complete of three manuscript drafts in the Biblioteca de El Escorial. 62 The manuscript is not autograph, but contains numerous corrections in Mendoza's own hand. A less complete manuscript, MS.f.m, 27 incorporates many of the autograph corrections in the first manuscript. The third manuscript, M.S.K. m, 8, contains only the first section of the translation; but it does name the duke of Alva as the patron to whom the translation was dedicated. Mendoza's translation comprises a long dedicatory letter on the principles and utility of the work. In the dedication Mendoza remarks that "Niccolo Leonico was a great and most learned man of Greek and Latin letters, but he had a poor understanding of the mathematical sciences and made mistakes both in the sense and in the letter of Mechanica.',6 3 As might be expected from his studies with Tartaglia, Mendoza was much concerned not only with mathematical precision but also with practical utility: "The intention of Aristotle was to introduce to the mathematician the practice and use of his subject, especially of geometry .... What moved our author to write was the difficulty which, he is aware, lies in joining practice with theory in the mathematical sciences.',64 Mendoza had an extensive knowledge of Greek and was among the greatest collectors of Greek manuscripts in his day, his library enjoying a European reputation.65 Of the three hundred Greek manuscripts that he owned, two hundred ninety-four are now in the Escorial. Many of these are mathematical and mechanical in subject, among them texts of Hero and Archimedes.66 Mendoza also owned a manuscript of the Mechanica (Escorial MS. Phi 1, IO, fs. 128159).67 At Venice he had ready access to the Marciana library and is known to have borrowed several codices from it. 68

I 48

History of Science

As a friend of Bembo and Piccolomini and pupil of Tartaglia, Mendoza acted as a bridge between principal phases of the Renaissance tradition of the Mechanica.69 Until the 1540s. humanists and literati had promoted the work. For the remainder of the century, the Mechanica was mainly in the hands of mathematicians. The republication of Piccolomini's literary paraphrase by Tartaglia's predominantly mathematical publisher Traianus Curtius is symbolic of the link. VI THE MATHEMATICIANS AND THE MECHANICA

At Venice, Mendoza was a friend and pupil of the mathematician Niccolo Tartaglia (I 500-1557). In Tartaglia's Quesiti ed lnventioni Diverse, published at Venice in 1546, Mendoza appears as the pupil to whom the science of statics is expounded. Tartaglia opens his discussion in dialogue with the treatment of the balance in the Mechanica, omitting its other questions, as an introduction to the medieval treatises: MENDOZA:

TARTAGLIA:

MENDOZA: TARTAGLIA:

MENDOZA: TARTAGLIA :

MENDOZA :

TARTAGLIA :

Tartaglia, since we took a vacation from the reading of Euclid I have found some new things relating to mathematics. And what has your Excellency found? Aristotle's Questions of Mechanics in Greek and Latin. It is quite a while since I saw these, particularly the Latin. What did you think of them? "They are very good, and certainly most subtle and profound in learning. I, too, have run through them and I understood most of them; yet many questions remained with me, which I should like to have more fully explained. Sir, should you wish me to explain them to you properly, many of the problems would require .that I first explain to your Excellency the principles of the Science of Weights [scientia de ponderibusJ.7°

Tartaglia then shows the inadequacy of the Aristotelian treatment of the questions on the balance, this being perhaps the first open attack on Aristotle's scientific accuracy to appear in print. Tartaglia's was the first extended mathematical commentary on any part of the Mechanica, and was instrumental in bringing the Mechanica under the scrutiny of other mathematicians. That his Quesiti succeeded in stimulating a market among mathematicians both for the Mechanica and for medieval statics is evident from the fact that in 1565 Tartaglia's publisher, Traianus Curtius, issued both a second edition of the Piccolomini Paraphrasis and an edition of a previously unprinted work ascribed to Jordanus.

The Pseudo-Aristotelian Questions of Mechanics

149

Because Renaissance culture was highly unified, it is no surprise to find that renaissances in literature, philosophy, and art were accompanied by a renaissance in mathematics. Such a renaissance is evident in the work of several mathematicians in the systematic revival of classical mathematical texts. Outstanding in this regard was Federico Commandino, who translated and edited a series of Greek mathematical texts in the middle years of the sixteenth century. The Mechanica was not among the works published by Commandino, who concerned himself more with pure mathematics. But it was given attention in a commentary published by Commandino's pupil Bernardino Baldi, who was both letterato and mathematician, and who served also as historian of the mathematical renaissance. The Mechanica figured explicitly in the programs of some predecessors of Commandino who failed to realize all their projects. Thus it was named in the Tradelist of projected editions published by the Gennan mathematician Regiomontanus (Johann Muller, 1446-1476) at Nuremberg in 1474. Nothing came of this project for the Mechanica, nor is there a text or translation of the work among the Regiomontanus manuscripts in the Nuremberg Stadtbibliothek. 71 Seventy years later, a similar program was put forward by the Sicilian mathematician Francesco Maurolico ( I 494-1575). In the course of a dedicatory letter of 1 540, addressed to his friend Cardinal Bembo, Maurolico printed a long, thematically arranged list of works to be translated and edited, including those of Hero, Vitruvius, Archimedes, and the Mechanica.72 Maurolico made an abridged Latin translation of the M echanica to which he added further questions and comments of his own in 1569, but it was not published until 161 3, together with some other papers edited by Silvestro Maurolico. A copy is preserved at the Bibliotheque Nationale in Paris, entitled Problemata Mechanica cum appendice, et ad Magnetem, et ad Pixidem Nauticam pertinentia (Messina, 1613). The earlier Renaissance mathematicians, as direct heirs of medieval statics, seem to have felt that both the Mechanica and the medieval tradition of Jordanus were quite compatible with the rigorous statics of Archimedes. Leonardo da Vinci, Cardano, and Tartaglia were among these conciliatores.73Admiring Archimedes, they believed deeply in the mathematical analysis of mechanics. At the same time, like Jordanus and the author of the Mechanica, they uncritically accepted in statics intuitive dynamic postulates such as that of virtual displacement. In their enthusiasm, they failed to supply the rigor needed in order to unify statics and dynamics into the one science of mechanics. Tartaglia, for example, considered the Mechanica as capable of being rectified by the similarly based statics of Jordanus. He did not question the fundamental idea of the Aristotelians that mathematics and physics met in the "mixed science" of mechanics, where the problems were physical but the methods of

150

History of Science

demonstration were mathematical. Tartaglia's views may have been formed before he came across the authentic works of Archimedes, and he seems not to have perceived the difference between the two opposed views of the role of mathematics. Tartaglia was as tolerant of intuitive principles based on motion as of rigorous postulates of equilibrium in the discussion of static problems. Later in the sixteenth century, the conciliatory viewpoint was attacked vigorously by the strict Archmedeans, notably by Guidobaldo de Monte ( 1 5451607), a disciple of Commandino who advocated a rigorously mathematical analysis of statics from which dynamic postulates were excluded. 74 Guidobaldo accompanied these scientific objections with a devastating attack on the source to which Tartaglia had appealed in his attempted reform: namely, the medieval author Jordanus, for whom he expressed contempt as a mechanician in terms strongly reminiscent of the literary humanists' contempt for the centuries of barbarian darkness. 75 Yet Guidobaldo was not hostile to the Mechanica; on the contrary, he defended it against certain misinterpretations by which the others had attempted to reinforce their views. Indeed, he considered Aristotle's principles to have been the starting point for Archimedes. This view of Guidobaldo's, taken up by Bernardino Baldi, is discussed below. Bernardino Baldi of Urbino (1553-1617) was active as a translator, mathematician, historian of mathematics, and man of letters.76 He was well acquainted with the main traditions in mechanics through his studies with Guidabaldo under Commandino. Baldi knew Piccolomini's paraphrase of the Mechanica and had been a friend of Piccolomini himself, of whom he wrote a biography.77 While at Padua in 1573-1575, Baldi attended Pietro Catena's lectures on the Mechanica. Returning to Urbino, he became Commandino's last pupil. Like Guidobaldo, Baldi attempted to show that Archimedes, far from contradicting Aristotle, had merely given to the physical conclusions of the Mechanica a quantitative expression. In his Life of Archimedes Baldi explained it thus: Mathematics being a physical subject but nevertheless demonstrable by mathematical reasoning, it seems that Aristotle, leaving aside the mathematical aspects, preferred to draw his demonstrations from physical principles which are of such force that, if accompanied by mathematical reasoning, they can provide a complete and general theory for the actuality of machines .... Guidobaldo saw that this book of Aristotle's was quite sound in its principles (though only implicity so and not evidently) and that the adding of mathematical proofs to the physical principles would bring it to specific conclusions. It was Archimedes who, assuming the lever principle of Aristotle, went on beyond him ... As Guidobaldo notes, Archimedes followed entirely in the footsteps of Aristotle as to the principles, but added to these the exquisite beauty of his proofs.7 8

The Pseudo-Aristotelian Questions of Mechanics

151

Thus he attempted to explain away the obvious inadequacies of the Mechanica. For Baldi, as for Guidolbaldo, Aristotle gave the physical principles that Archimedes later expressed mathematically. The mathematical analysis had been done once and forever by Archimedes, and the medieval regression to Aristotelian dynamic postulates was fruitless, in their view. This attitude influenced the commentary on the Mechanica which Baldi wrote but left unpublished in his lifetime.79 Entitled In Mechanica Aristotelis Prohlemata Exercitationes, it appeared posthumously at Mainz in 162 1. The printed text was thus too late to have any influence on Renaissance mechanics, as was its paraphrase in German published by Moegling in 1629. Baldi was well informed on other aspects of the Mechanica besides its philosophical and mechanical import. In his Praefatio Authoris to the 162 1 edition he gives a balanced account of the history of the text and of the debate over its authorship. He inclines to Piccolomini's view that it is a genuine Aristotelian work. As to the commentators, Baldi is critical of Leonico but admiring of Piccolomini. Baldi ' s Exercitationes is grouped here with the commentaries of mathematicians on the Mechanica , though it is strictly speaking neither exclusively mathematical nor limited to a commentary. Baldi used the various questions of the Mechanica as pretexts for the introduction of a large number of his own related speculations, in which his wide-ranging interests and acute observations are apparent. Elsewhere he points out the relation of the Mechanica to Vitruvius' mechanical theories.80 He seems to have been the first, prior to Galileo, to perceive the importance of the questions concerning fracture and their relation to architectural problems and practices. Of a very different character is the commentary of Giovanni Battista Benedetti (1530-1590), which occupies Chapters 11-25 of the section headed Mechanics in his Diversarum Speculationum Mathematicarum et Physicarum Uber. Originally published at Turin in 1585, the sheets of this book were reissued with slightly varying titles at Venice in 1586 and 1599. Benedetti directed his attention only to selected questions in the Mechanica: those of the balance, the sail as lever, the motion of circular bodies, projection by slings, fracture, the wedge, the pulley, the "wheel of Aristotle," centers of gravity, and bodies in vortices. On all other questions, he asserts the correctness of Aristotle. Benedetti 's relative isolation at Turin probably accounts for the fact that he did not specifically refer to any previous commentator except Tartaglia, who had been briefly his own teacher at Venice, whereas others writing at this time and afterward (Baldi, Monantheuil, Blancanus) show familiarity with virtually all their predecessors except Tartaglia. On the whole, Benedetti's criticisms are specific and cogent, though with respect to the reduction of simple machines to the lever

152

History of Science

they fall short of Guidobaldo's earlier analysis, with which he seems not to have been familiar. Benedetti 's reflections on the rectilinear character of projection from circular motion constitute the most valuable of his commentaries on the Mechanica. Early seventeenth-century commentaries on the Mechanica by Joseph Blancanus ( 1615) and Giovanni di Guevara ( 1627) are also of mathematical interest, but because of their late appearance are merely mentioned here without further comment. VII THE MECHANICA AT THE UNIVERSITY OF PADUA,

1548-1610

During the latter half of the sixteenth century, the Mechanica was expounded from the mathematical chair of the University of Padua by three successive professors of mathematics - Catena, Moleto, and Galileo. Pietro Catena was professor of mathematics at Padua during the years 15481576. Bernardino Baldi (who was there from 1573 to 1575) states that Catena lectured on the Mechanica, adding that he lectured so badly that his audiences came mainly to display their derision. 81 Baldi's low opinion of Catena was shared by another visitor to Padua in the 1570s, Guidobaldo del Monte. 82 Catena's analysis of the Mechanica has not survived. The next mathematician at Padua was Giuseppe Moleto ( 1577-1 588), who delivered a course of lectures on the Mechanica in 1581. The text of the first fourteen of these lectures is preserved in a manuscript in the Biblioteca Ambrosiana at Milan (Ms. s. 100 sup., fs. 1-99 passim). Their main emphasis was on the status of mechanics as an intennediary science between mathematics and physics. Moleto also seems to have been interested in the practical aspects of mechanics; he stands alone among the Italian commentators in mentioning the Mechanica in the translation of Vittore Fausto, whose ship-building activities he praised. Paduan mathematicians were much interested in discussing the relation of mathematics to logic and philosophy. On this subject Catena wrote two works, in one of which, Universa loca in logica Aristotelis in Mathematicas Disciplinas, Venice, 1556, he tried to supply the lost mathematical basis for Aristotle's theory of demonstration as explained in the Posteriora Analytica. It may have been Catena who started Baldi off to reflecting on the mathematical expression of physical postulates. Certainly as early as I 576 Baldi was interested in the relation between mathematical and physical demonstrations, and referred in this context to the Mechanica. 8 3 The discussion at Padua was continued by Moleto, who left manuscript treatises on the nature and essence of the mathematical disciplines. 84

The Pseudo-Aristotelian Questions of Mechanics

153

Another man who served briefly as Paduan professor of mathematics, Francesco Barozzi, though he did not lecture on the Mechanica, published an Oratio de Certitudine Mathematicarum, Padua, 1560, giving the text of his inaugural lecture of 1559. This Oratio is of particular interest in the present context since it was written in refutation of the treatise, also named De Certitudine Mathematicarum, which Alessandro Piccolomini had published with his Mechanica paraphrase of 1547. The juxtaposition of the two works seems to bear out the thesis that a close relation existed between the tradition of the Mechanica and the sixteenth-century debates on the certainty of mathematics. Moleto's successor in the chair of mathematics at Padua was Galileo, who served from 1592 to 161 o. The university rolls show that in 1598 he lectured on the Questions of Mechanics publicly, as a part of his official duties.85 Now, prior to that year, he had been offering to private pupils a course in mechanics of his own, for which various versions of his syllabus exist. In its final form, in which it was widely circulated in manuscript, and was ultimately published posthumously as Le meccaniche, it refers to another treatise, on the "problems of mechanics." There is little doubt that the latter, now lost, comprised essentially Galileo's public lectures on the Aristotelian text. Although surviving manuscripts give no hint of Galileo's views on the Mechanica, they do afford reason to believe that he had not been greatly influenced by it until 1594, after he had moved to Padua. There is some corroborating evidence for this surprising possibility. In March I 593, Galileo had written to Giacomo Contarini on the placement of oars in galleys. He noted that the water is the fulcrum in this case of the lever, and stressed that the handling of the oar to minimize movement of the water maximizes the impulse given to the ship. This treatment differs sharply from that given in the Mechanica, where the oarlock was treated as the fulcrum, and movement of the water (considered as the weight to be moved) was treated as the objective. Yet Galileo did not mention the Aristotelian work or criticize its opposing view (which was probably generally accepted at the time), suggesting that he was not yet impressed by the book. Galileo's syllabus for his course given to private students at Padua exists in three forms. The most primitive form, which probably belongs to 1593, had no preamble and contains nothing to indicate that the Mechanica had yet entered into Galileo's consideration. This original syllabus was based on Archimedes and on Guidobaldo. It had a short appendix on the force of percussion, in which Aristotle was not mentioned by name, and which made no reference to the existence of Galileo's own treatise on problems of mechanics. The version of I 594, however, of which copies exist at Regensburg and Hamburg, began with a preamble introducing the idea of miraculous effects in mechanics in the mov-

I 54

History of Science

ing of large weights with small forces, characteristic of the Mechanica. The appendix on the force of percussion remained unchanged in this intermediate form of Galileo's treatise. The final version, dating probably from about 1600 and much expanded, dealt quite differently with the nature of mechanical effects. In contradiction of the Mechanica, it opened with a denial that anything miraculous exists in mechanical effects, setting forth the rule that whatever is gained in power is lost in time required or space traversed. The advantage of mechanical devices thus resided solely in the human advantages they offer in manners of utilizing available force for tasks in which direct application would not succeed, or would be inconvenient. The Mechanica had mentioned this as only one kind of advantage, while Galileo made it exclusive of all others. His appendix on the force of percussion was also expanded; Aristotle was now mentioned by name, and a separate treatise of Galileo's on the problems of mechanics was specifically designated. Galileo's separate treatise on problems of mechanics is again mentioned in a letter of 16 IO as one of his finished works, ready for publication. Its contents may be guessed at from various comments in his books, particularly the posthumous Mechanics, the Discourse on Bodies in Water (1612), the Dialogue (1632), and the Two New Sciences (1638). Galileo's first published mention of the Mechanica, in 1612, credited it explicitly with the principle of virtual velocities, of which Galileo made extensive use. In crediting the principle to Aristotle, he drew criticism from Ludovico delle Colombe, who pointed out that the work was of doubtful authenticity. 86 Galileo replied that this was certainly irrelevant, since the book was cited by good Peripatetics as sound Aristotelian doctrine. 87 Thus Galileo made no attempt to avoid crediting Aristotle for his inspiration in the matter. Another acknowledged debt of Galileo to the Mechanica concerns the socalled "wheel of Aristotle," discussed in the Two New Sciences. 88 It is highly probable that Galileo's interest in many other matters was stimulated by the same work: for example, in projectiles thrown by slings, in questions of the breaking strength of materials, in the composition of motions, and perhaps in the continuance of motion in bodies thrown, all of which problems were raised in the Mechanica. The Aristotelian element in Galileo's mature mechanics is much larger than is generally recognized. This matter cannot be dealt with here, except to point out that the idea of velocities rather than displacements is repeatedly stressed in the Mechanica by reference to the element of time, whereas in the medieval works the emphasis was on displacements alone. In Archimedean statics, of course, time had no place at all.

The Pseudo-Aristotelian Questions of Mechanics

I 55

VIII THE ENGINEERS AND THE MECHANICA

The foregoing sections have outlined the growing influence of the Mechanica on sixteenth-century thought through the humanists, the literati, and the mathematicians. This influence was intended to be not only philological and scientific but also technological. But what of the men most concerned with technology: namely, the engineers? It appears that for them generally the Mechanica remained inaccessible in Greek, nearly so in Latin, and only belatedly became known in vernacular translation. There is no overt trace of the M echanica in the manuscripts of Taccola, who was among the earliest of the fifteenth-century Italian engineers to have left documents. Taccola's pupil Francesco di Giorgio Martini may have known and applied the principles of the Mechanica to machines. For example, in one of his drawings of a catapult, Francesco di Giorgio has superimposed two circles which suggest a Mechanica-type analysis of the operation of the instrument. 89 In a manuscript at Florence, an anonymous contemporary depicts the lever law similarly in tenns of areal displacements, 90 but this is as likely to represent the medieval tradition as that of the Mechanica. During the later sixteenth century there is good evidence of the dissemination of the Mechanica among engineers. 91 Aristotle was cited in explanation of particular machines by Giuseppe Ceredi and Vittorio Zonca.9 2 A military engineer, Buonaiuto Lorini, discussing the balance, referred to the Mechanica . Giambattista Alcotti, the engineer of the Este, referred to the lever and pulley propositions of Aristotle, Archimedes, and Guidobaldo in his translation of Hero and in his dissertation on sluice gates.93 Alessandro Giorgi of Urbino, translator of Hero 's Pneumatics into Italian, discusses the philosophical significance of the Mechanica in his preface.94 In 1588 Agostino Ramelli wrote that the science of mechanics was "useful in peace and war, as can be seen in the Mechanica of Aristotle. In the Mechanica are included all the principles of the many machines and instruments which have been made and will be made in the future. "95 Giovanni Branca, writing in 1628, was of a similar mind: "Underlying all these machines are those principles which Aristotle discussed and proposed in his Mechanica , a work of which man will always be able to avail himself as the foundation for the invention of machines suitable for meeting any needs that may arise."96 It was an engineer who first produced an Italian translation of and commentary on the Mechanica . This was Antonio Guarino ( 1504-1590), inspector of fortifications for the duke Alfonso II d ' Este of Modena, who published Le Mechanice d' Aristotile trasportate di Greco in Vo/Rare ldioma , con le sue Dechiarationi nel

I 56

History of Science

fine ... in partico/ar volume da se, Andrea Gadaldino, Modena, 1573.97 This translation is now extremely rare, the only known copies being in the Biblioteca Marciana at Venice and the Biblioteca Estense at Modena. Le Mechanice d' Aristotile is in two parts, the first being a translation of the Mechanica with some added diagrams, and the second a seventeen-page commentary with diagrams. In the dedication to Cornelio Bentivoglio, Guarino gives an account of its composition: In the past days the extreme cold has obliged me by reason of my advanced years to suspend my building of the fortifications of Modena. Thinking of something with which to while away such a bitter winter, it occurred to me to translate Aristotle's Mechanica from Greek into our own language. It is in this work that the Philosopher explains clearly the causes of the effects of motions together with all the operations which, through the efforts of ingenious inventors, bring such miracles to men. But I have not been content only to translate this work and to abide by the intention and arrangement of the author (though this is difficult enough in view of the many and peculiar meanings of the Greek words, compared with the few and ill-fitting meanings of our Italian words, a fact which renders the senses of the text somewhat obscure). Rather, in addition to this I have compiled several explanations for the more difficult passages, and these explanations will be found in a separate fascicule at the end of the translation. Moreover, in addition to adding figures for the proofs, I have used in these figures the Greek letters of Aristotle so that my work may serve not only the layman, but also those who wish to apply themselves to the study of the Greek text. These figures, which until now have been wanting in the Greek text, I leave to your judgment.

Guarino, by translating from the original Greek, evinced a literacy which may seem surprising in an engineer. Duke Alfonso II d'Este, however, seems to have had a penchant for employing highly literate engineers, for a few years later his inspector of fortifications at Ferrara, Giovanni Battista Aleotti, translated Hero's Pneumatics from the Greek and dedicated it to Alfonso. The circumstances in which the two translations were conceived were also similar. Like Guarino, Aleotti decided to make a translation while recovering from an illness and may well have been inspired by Guarino's example. 98 Guarino would have had no difficulty in finding a Greek text. In addition to the Aldine edition, he would have had access to the manuscript text in the Estense Library at Modena. IX THE MECHANICA OUTSIDE ITALY

The foregoing account may have given the impression that the tradition of the Mechanica was an Italian monopoly , and indeed this is almost true: except for

The Pseudo-Aristotelian Questions of Mechanics

157

the first Latin translation, the most important work in reviving the Mechanica was done in Italy. There was, however, an accompanying revival of the Mechanica in several other countries. In Portugal, Pedro Nunes associated the Mechanica with navigation. His comments on the oar and rudder questions were published in I 566 and 1573, subtitled In Problema Mechanicorum Aristotelis de Motu Navigii ex Remis Annotatio Una.99 In England, John Dee owned the Fausto translation of 1517 and the Piccolomini paraphrase of 1547. 100 The influence of the Mechanica can be discerned in Dee's remarks on circular motion in machines in his "Mathematical Preface" to the English Euclid of 1570. The Swiss polymath Walter Ryff (Rivius) published a German translation of much of the Quesiti of Tartaglia in his Der fiirnembsten , notwendigsten der gantzen Architectur angehorigen Mathematischen und Mechanischen Kunst ... in drey fiirneme Bucher abgeteilt, Nuremberg, I 558, and perhaps also in an earlier edition. Ryff's mechanics was reedited by Daniel Moegling in a littleknown volume, Mechanische Kunst-Kammer, Erster Theil , Matthaeus Merianus, Frankfurt, 1629. Moegling also added German translations from Guidobaldo's Uber Mechanicorum, the Aristotelian Mechanica, and Baldi's commentary. The only copy presently known to us of the Moegling book is in the New York Public Library. A curious example of neglect of the Mechanica by humanists in France is afforded by Lazare de BaiJ's book on classical ships, De Re Navali, published at Basie in 1537 and often reprinted. Baff had been French ambassador to Venice at the time of Fausto' s construction of the quinquereme for the Venetians; in his preface, he recommends that the French follow the example of Fausto and revive the marine technology of Antiquity. A natural concomitant of this would seem to be revival of the Mechanica·, which discusses the placement of oars and problems concerning sails of ships; yet BaiJ made no mention whatever of the work, which Fausto had translated thirty years earlier. Revival of interest in the Mechanica was instigated by Ramus (Pierre de la Ramee ), who encouraged the study of mathematics, though it was a subject for which he himself had little aptitude. Ramus corresponded with Italian mathematicians, including Commandino. Being preeminently interested in the practical applications of any academic discipline, it is not surprising that he became an enthusiast for the Mechanica. Ramus even overcame in this matter his aversion to Aristotle, whom he actually praised for having written a mathematical work such as the Mechanica. 101 Moreover, he had a high opinion of the thoroughly Aristotelian paraphrase of Piccolomini. 102 In his Scholae Mathematicae of 1569 Ramus remarks that he lectured in a previous year ( I 564-1565?) on the Mechanica. It has been suggested that the

1 58

History of Science

Greek text of the Mechanica (with diagrams) printed by Ramus ' publisher, A. Wechel (at Paris in I 566), was connected with these lectures. 103 (The text was reprinted by Friedrich Sylburg in his Aristotelis Opera, A. Wechel Haeredes, Frankfurt, 1587, VI, 43-75. Copies of the I 566 edition are in the British Museum and the New York Public Library.) A pupil of Ramus, Henri de Monantheuil (1536-1606), published at Paris in 1599 the Greek text with his own Latin translation and commentary : Henricus Monantholius, Aristote/is Mechanica Graeca, emendata, Latinafacta, et Commentariis illustrata, Paris, Jeremia Perier, 1599. A second edition of this work appeared at Lyons in 1600. Monantheuil was also the author of several mathematical works. 104 In addition to a thorough acquaintance with the ancient writers on mechanics, Monantheuil cites medical and literary works of Antiquity in his commentary. He favored the work of Guidobaldo in explanation of the simple machines. For the windlass he cited the placing of recent foundation stones of a Paris bridge; and for compound pulleys, the raising of the Roman obelisk for Sixtus V by Domenico Fontana. The analysis of oars by Nunes is included in part, and references occur to all the previous Latin translations or paraphrases. On the whole, Monantheuil's is the most complete and erudite of the sixteenth-century commentaries on the Mechanica, though not the most original in outlook. SUMMARY AND CONCLUSION

As stated in the opening section, the objectives of the present study are limited to the bibliographical aspects of the pseudo-Aristotelian Mechanica and to the links between Renaissance scholars and translators concerned with that work. With respect to the first of those objectives, it now seems safe to say that the Mechanica as such remained without direct influence from the decline of Alexandrian science until the Greek revival of the fifteenth century. Latin writers of the Middle Ages who did encounter the Greek text were insufficiently impressed by it to continue the discussion of its problems or to make and preserve even a partial translation. From the mid-thirteenth century on, a corpus of writings on mechanics inspired by works on the balance, the lever, and the behavior of bodies immersed in water, or newly created in the Latin West, replaced the Mechanica as a focus of interest for students of physical phenomena. There was, nevertheless, a Greek manuscript text dating from the twelfth century which led to two other surviving manuscripts in the ensuing two centuries, so that the text was never entirely inaccessible to medieval scholars. The fifteenth century saw the rapid multiplication of Greek copies, seventeen of which are extant, but there was still no Latin translation, and very few refer-

The Pseudo-Aristotelian Questions of Mechanics

159

ences to the work have been found, even counting questionable titles. The end of the fifteenth century saw the first printed Greek text, followed early in the sixteenth century by two published Latin translations. There ensued a considerable number of commentaries and translations into Italian and Spanish. Humanist and philosophical movements account for the reawakening of interest in the Greek text, but for its Latin translation and the beginning of extensive commentaries on the Mechanica we find a strong indication of technical and mathematical concern, first among humanists and literary men themselves, and then among engineers and men of practical rather than purely scholarly bent. The background of this merging of humanist with practical concerns is traceable primarily to the emergence of architecture as a distinguished profession in the fifteenth century, and particularly with the revival of interest in the text of Vitruvius and the writings of Leon Battista Alberti, associated in tum with engineering treaties by Taccola and Martini that circulated widely in manuscript. The role of Cardinal Bessarion seems decisive in making the Renaissance revival of the Mechanica primarily an Italian affair. Deposit of his Greek manuscripts at Venice put them at the easy disposal on the one hand of scholars associated with the University of Padua, and on the other hand of technical men and nonuniversity mathematicians such as Niccolo Tartaglia. From Venice, the spread of interest to Spain, France, and Germany was naturally rapid, but the concentration of study remained largely Italian. Later in the sixteenth century, as a natural consequence of the multiplication of printed texts and commentaries, the problems of the Mechanica were pursued more extensively by engineers and mathematicians than by philosophers and men of letters whose interest had first been attracted to them by their Aristotelian and classic character. In the course of this later development, the Mechanica gave way to more sophisticated mathematical treatment of the same problems that were discussed qualitatively in the Mechanica itself. With the advent of Galilean mechanics, which was more deeply indebted to suggestions from the ancient treatise than is yet generally realized, the Mechanica passed into obsolescence early in the seventeenth century. Turning to the second of our present objectives - the linkage between Renaissance scholars and translators concerned with the Mechanica - a rather remarkably close-knit fellowship emerges. Diego Hurtado de Mendoza, who studied with Tartaglia at Venice, is closely associated with Alessandro Piccolomini, whom he met at Padua and in whose city, Siena, he served later as military governor. All three men are responsible for vernacular commentaries on the Mechanica, in Italian and Spanish, and for attempts to link the doctrines of the ancient work with practical pursuits and with the independent medieval tradition in mechanics. Benedetti, also a student of Tartaglia's, provided mathe-

160

History of Science

TABLE2 Chronology of the Main Renaissance Editions, Translations, and Commentaries on the Mechanica For full titles, see the earlier Sections cited in brackets. Greek text, in Opera Aristotelis, II, Venice, 1497. [Section IV.) Reprinted in collected editions throughout the sixteenth century. Latin translation, Aristotelis Mechanica, Paris, 1517. MS VITTORE FAUSTO copies in Vienna and the Vatican. [Section IV.) Latin translation and commentary, in Opuscula, Venice, 1525; NICCOLO LEONICO Paris, 1530. Translation reprinted in several later editions of Aristotle. MS copies in Vatican. [Section IV.) DIEGO HURTADO DE MENDOZA Spanish translation, Mechanica de Aristotiles, 1545. MSS in the Escorial. Published by Foulche-Delbosc, Revue Hispanique, 1898. [Section V.) NICCOLO TARTAGLIA Partial commentary in Ouesiti ... , Venice, 1546, 1554. [Section VI. I ALESSANDRO PICCOLOMINI Commentary, In Mechanicas .. . Paraphrasis, Rome, 1547; Venice, 1565. Italian translation, Rome, 1582. MS copy in Florence. [Section V.) Abridged Latin translation and commentary, in Problemata FRANCESCO MAUROLICO Mechanica ... , Messina, 1613, with dedication dated 1569. [Section VI.) Greek text, Paris, 1566. Reprinted Frankfurt, 1587. AWECHEL (ed. Pierre de la Ramee?) PEDRO NUNES Partial commentary in De arte ... navigandi, Basie, 1566; Coimbra, 1573. [Section IX.) ANTONIO GUARINO Italian translation and commentary, Le Mechanice d'Aristotile, Modena, 1573. [Section VIII.) GIUSEPPE MOLETO Lectures on the Mechanica, Padua, 1581 . MS in Ambrosiana. [Section VII.) G1ovANN1 BATTISTA BENEDETTI Partial commentary in Diversarum Speculationum Uber, Turin 1585; Venice, 1586, 1599. [Section VI.) BERNARDINO BALDI Commentary, In Mechanica .. . Exercitationes, Mainz, 1621 . German paraphrase, Frankfurt, 1629. [Section VI.) Probably completed before 1590. MS at Florence lost. HENRI DE MONANTHEUIL Greek text, Latin translation and commentary, Aristotelis Mechanica ... 11/ustrata, Paris, 1599; Lyons, 1600. [Section IX.) ALDUS MANUTIUS

JOSEPH BLANCANUS GIOVANNI 01 GUEVARA

Partial commentary in Aristotelis Loca Mathematica, Bologna, 1615. Commentary, In Aristotelis Mechanicas Commentarii, Rome, 1627.

The Pseudo-Aristotelian Questions of Mechanics

161

TABLES Relations Among Sixteenth-Century Commentators on the Mechanica Arranged by date of composition of work (Read down and across)

.. 0

Fausto (1517)

~

:!

·g8

.g

i

C

j

.

~ I!

~

·a

·e ] 0

i:i,.

0

~

~

~

~z ·EC=. 0

0 "'O

;; ..c 0 "'O

'3 C

·3 j!

::: .,

:a ., ;;

~

p:i

i=I

Leonico (1525)

Mendoza (1545)

s

M

Tartaglia (1546)

p

T,M

Piccolomini (1547)

T

p

Maurolico (1569) Nuiies (1573)

M

Guarino ( I 573) Guidobaldo (1577) Bcncdctti (I 58 5)

S.M

B3ld.i (a. r 590) Morunthcuil ( r 599) M

s

mentions student of

p

M

M

M iM

P

r

M

I

I

P,M IM

personal acquaintance with teacher

M,P M

M

~C

"'O

0

~

I

162

History of Science

matical critiques of the work, as did Moleto, Galileo's precedessor in the chair at Padua. Highly original additions were offered by Bernardino Baldi, pupil of Guidobaldo del Monte - Galileo's patron - who, however, stressed the incompatibility of medieval statics with that of Archimedes and associated the latter with the doctrines of Aristotle. Without repeating more of what has previously been recounted, it seems evident to us that the closest of links existed directly between the men who studied the Mechanica in the sixteenth century and those who gave birth to modem mechanics. So much may be said with regard to the broad outlines of the present study. It remains to examine the text of the Mechanica, the early attempts to restore sense to corrupt passages and to supply proper diagrams, and the course of discussion of particular problems of special importance on which disagreement persisted during the Renaissance: For example, the problems of oars and levers, of the "wheel of Aristotle," of vortex motions and of the persistence of motion in projectiles. As such problems - some purely practical, others fundamentally mathematical, and still others philosophical in character - received attention after having escaped discussion for a millennium, the integrated science of mechanics as we know it began to emerge. We urge the study of these more detailed problems on others as we continue to pursue it ourselves. NOTES I

2

The research for this paper was undertaken at the University of Toronto from funds provided by a grant from The Canada Council. For advice on particular manuscripts, thanks are due to Professor Elpidio Mioni, Dr. Chr. v. Steiger, and Dr. Giorgio E. Ferrari. George Sarton, Appreciation of Ancient and Medieval Science during the Renaissance ( 1450-1600) (Philadelphia, 1955). Also R.R. Bolgar, The Classical Heritage and its Beneficiaries from the Carolingian Age to the End of the Renaissance (Cam-

bridge, 1954). 3 Pierre Duhem, Les Origines de la Statique (Paris, 1905-19()6), 2 vols.; Etudes sur Leonard de Vinci (Paris, 19~1913), 3 vols. See also S. Drake and I.E. Drabkin, Mechanics in Sixteenth-Century Italy (Madison, 1969) pp. 3--6o. 4 For the Mechanics of Hero in English translation, see A.G. Drachmann, The Mechanical Technology of Greek and Roman Antiquity (Copenhagen, 1963). An account of the Pappus tradition is currently being prepared by Marjorie Boyer for the Catalogus Translationum. 5 Marie Boas, "Hero's Pneumatica, a Study of its Transmission and Influence," Isis, XL ( I 949), 38-48. The article on Hero for the Catalogus Translationum is being prepared by Charles B. Schmitt.

The Pseudo-Aristotelian Questions of Mechanics

163

6 For accounts of the engineering importance of the Hero tradition see 8. Gille, The Renaissance Engineers (London, 1966, tr. from Les lngenieurs de la Renaissance, Paris, 1964). Also A. Keller, " Pneumatics, Automata and the Vacuum in the Works of Giambattista Aleotti," British Journal for the History of Science, m ( 1967), 338-347. 7 Marshall Clagett, Archimedes in the Middle Ages: I - The Arabo-latin Tradition (Madison, 1964). The history will continue up to the year I 565. 8 A recent discussion of the treatment of balances in the Mechanica is given by M. Schramm, "The Mechanical Problems of the 'Corpus Aristotelicum,' the ' Elementa lordani super Demonstrationes Ponderum,' and the Mechanics of the Sixteenth Century," in Atti de/ Primo Convegno lnternazionale di Ricognizione de/le Fonti per la Storia def/a Scienza Italiano : I Secoli XIV-XV/ (Pisa 14-16 Settembre 11)66) (Florence, 1967), pp. 151-163. Schramm is in error when he states that Fausto's translation " includes no figures" (p. I 54). He does not mention Fausto's important interest and ability in technology. 9 These treatises are printed and translated into English in Ernest Moody and Marshall Clagett, The Medieval Science of Weights (Madison, 1952). 10 Cf. Marshall Clagett, Greek Science in Antiquity (New York, 1963), pp. 93~4. 11 For Piccolomini's views see below. Against him see Girolamo Cardano, Opus Novum de Proportionibus (Basie, 1570) in Cardano, Opera (Lyons, 1662), IV, 515; Francesco Patrizi, Discussiones Peripateticae (Venice, I 571 ), I, Lib. 3. Cf. Henri de Monantheuil, Aristotelis Mechanica ... (Paris, 1599), p. 1. 12 Paulys Real-Encyc/opiidie der classischen Altertumswissenschaft, herausgegeben von Georg Wissowa (Stuttgart, [1894]), n, cols. !044-1045, and "Supplementband," XI (1968), col. 315. Paul Moraux, Les listes anciennes des Ouvrages d'Aristote (Louvain, 1951), p. 120. 13 Pietro Riccardi, Biblioteca Matematica ltaliana (Modena, I 886 ), n, Parte Seconda, 89. Drake and Drabkin, Mechanics, p. 391. 14 J.P. van Cappelle, Aristotelis Quaestiones Mechanicae (Amsterdam, 1812). I. Bekker, Aristoteles Graece (Berlin, [ I 831 ]), 11, 847-858. 0 . Apelt, Aristotelis quae feruntur ... Mechanica (Teubner) (Leipzig, 1888). 15 E.S. Forster, tr., in The Works of Aristotle translated into English, ed. W.D. Ross (Oxford, 1913), VI. 16 W.S. Hett, tr., Aristotle: Minor Works (Loeb) (London, 1936), pp. 327-41 I. 17 MS. Z. Graecus 214 (479), fs. 203-2!0, described in E. Mioni, Aristotelis Codices Graeci qui in Bibliothecis Venetis adservantur (Padua, 1958), pp. 70-71 , !03, 130. I 8 Andre Wartelle, /nventaire des Manuscrits Grecs d' Aristote et de ses Commentateurs (Paris, 1963). 19 M. Claggett, Science of Mechanics in the Middle Ages (Madison, 1959), pp. 71-72 . 20 A.G. Drachmann, Mechanical Technology (p. 13), says that no figures appear in the manuscripts, but diagrams were added in some later copies.

164

History of Science

21 The manuscript is described in Mioni, Aristotelis Codices, pp. 148-149. For Symeonachis see G.S. Mercati, "Di Giovanni Simeonachis, protopapa di Candia," in Miscellanea G. Mercati, m, Studi e Testi, 123 (Vatican City, 1946), p. 314. For Lippomano see D.J. Geanakoplos, Greek Scholars in Venice (Harvard, 1962), p. 50. 22 Fontana's work was later published as Pompilius Azalus, Liber de Omnibus Rebus Naturalibus (Venice, 1544). Cf. L. Thorndike, "An Unidentified Work by Giovanni de' Fontana," Isis, xv (1931), 31-46. 23 Described by Mioni, Aristotelis Codices, pp. 113-1 15. 24 A.G. Keller, "A Byzantine Admirer of ' Western' Progress: Cardinal Bessarion," Cambridge Historical Journal, u ( I 955), 343-348. 25 L. Minio-Paluello, "Attivita Filosofico-Editoriale dell'Umanesimo," Umanesimo Europeo e Umanesimo Veneziano, ed. Vittore Branca (Venice, 1963), pp. 245-262. 26 Cf. A.C. Klebs, "Incunabula Scientifica et Medica," Osiris, 4 ( 1938), 1-359, especially p. 144. 27 Ibid., p. 322. 28 W. Panzer, Anna/es Typographici (Nuremberg, 1800), VJD, 387, no. 397. Taken from A.C. Burgasso and J. Morelli, Serie dell' Edizioni A/dine, 2d ed. (Padua, 1790), p. 28. 29 Ariosto, Orlando Furioso , canto 46, stanza 19. 30 Giuseppe Moleto, "Lectures on the 'Mechanica' ," Biblioteca Ambrosiana, Milan, MS. s. 1 oo supra. 31 Giovanni degli Agostini, Notizie lstorico-Critiche intorno la Vitae le Opere degli Scrittori Viniziani (Venice, 1752-1745), n, 448- 472, gives a biography of Fausto. One of the present authors is preparing for publicaton an extended account of Fausto. 32 According to Ramusio's biography at the beginning of Victor Faustus, Orationes Quinque (Venice, 1551 ). Copies of the 1517 edition are in the British Museum, Bodleian, Bibliotheque Nationale, Marciana, and New York Public Libraries. 33 Ph. Renouard, Bib/iographie de I' lmprimerie et des Oeuvres de Josse Badius Ascensius (Paris, 1908), n, 47. There are copies in the British Museum, Bodleian, Bibliotheque Nationale, and New York Public Libraries. Two manuscript copies are known: Biblioteca Vaticana MS. Urb. Lat. 1321, and Nationalbibliothek, Vienna, MS. 10849. 34 On this topic see Paul Lawrence Rose, '" Certitudo Mathematicarum • from Leonardo to Galileo," to appear in Atti de/ Simposio lnternaziona/e "Leonardo da Vinci nella Scienza e nella Tecnica " (Firenze, Giugno 1969). 35 Pietro Bembo, Opere (Venice, 1729). 1v, 221. 36 Niccolo Leonico Tomeo, who is not to be confused (as was done by Baldi) with Niccolo Leoniceno, a contemporary humanist. See C. Castellani, " II Prestito dei Codici Manoscritti della Biblioteca di San Marco in Venezia," Atti de/ Reale lstituto Veneto di Scienze, lettere ed Arti, 55, i (1896-1897), 311-377, especially p. 312.

The Pseudo-Aristotelian Questions of Mechanics

37

38 39

40

41

42 43

44

45

I 65

Mr. C.H. Talbot of the Wellcome Museum, who is preparing an edition ofLeonico's correspondence (Biblioteca Vaticana MS. Rossianus 997), informs us that he knows of no material bearing on the translation of the Mechanica. For Leonico see Paolo Giovio, £logia Virorum litteris lllustrium (Basie, 1577), pp. 170 ff., and A. Serena, Appunti Letterari (Rome, 1903), pp. 3-32. Cf. the bibliographical account by Conrad Gesner, Bibliotheca Universalis (Zurich, 1545), fs. 520-521v. The translation is at fs. 23-54v (misfoliated as 52). Copies of this edition with some sixteenth-century manuscript notes are in the libraries at the University of Toronto and Columbia University. Manuscript copies of the translations, along with a preface to Gasparo Contarini not in the printed editions, are included in Vaticana MS. Reg. Lat. 1291. Professor William F. Edwards kindly confirmed this fact. A. Favaro, "La Libreria di Galileo Galilei," Bullettino di Bibliografia e di Storia delle Scienze Matematiche e Fisiche, XIX (1886), 210-290 at p. 243. Giuseppe Ceredi, Tre Discorsi sopra ii Modo d' Alzar Acque da' Luoghi Bassi (Parma, 1567), pp. I 1-12. Several of Valla's codices came to the Biblioteca Estense in Modena. Although it is not mentioned in Heiberg' s printed catalogues of the Valla manuscripts, the Estense manuscript of the Mechanica, MS. Estense Gr. 76, is from Valla's library. An excellent treatment of the relation between science and Renaissance culture is given by L. Olschki, Geschichte der neusprachlichen wissenschaftlichen literatur, 3 vols. (Leipzig, 191~1922). Daniele Barbaro, / Dieci libri dell'Architettura di M. Vitruvio (Venice, 1556), p. 254 (Proemio to Book x). For a general account of the Vitruvian tradition see V.P. Zoubov, "Vitruve et ses Commentateurs du XVIe Siecle," in la Science au XV/e Siecle (Colloque de Royaumont 1957) (Paris, 196o), pp. 674Jo. On Barbaro and Vitruvius see Frances Yates, Theatre of the World (Chicago, 1969). Philander's edition of Vitruvius was first published at Rome in 1544. There were many subsequent editions in the sixteenth century. Claudio Tolomei, lettere (Venice, 1558), fs. 81-85. For machines see f. 84. Other contemporary mentions of the Accademia Vitruviaila are Luca Contile, lettere (Pavia, I 564), I, 20, 53-54; Contile, Ragionamento de/le /mprese (Pavia, I 574), f. 42; Ceredi, Tre Discorsi, pp. 64-66. Sperone Speroni, "Discorso circa ii fare un' Accademia," in Opere (Venice, 1740), m, 456--46o. S. May lender, Storia de lie Accademie d' Italia (Bologna, 1928), m, 99, assigns the founding of the Gimnosofisti to 1564, but that seems a rather late dating. Piccolomini, Archbishop of Patras, was later coadjutor of Siena. On Piccolomini at Padua see F. V. Cerreta, Alessandro Piccolomini (Siena, 1960), pp. 1~48, especially pp. 45-46; also Rufus Suter, "The Scientific Work of Alessandro Piccolomini," Isis, LX (1969), 210-222. A biography of Piccolomini by his contemporary and fellowcommentator on the Mechanica, Bernardino Baldi, was printed by E. Narducci,

166

46 47

48

49 50 51 52

53 54

55 56 57 58

59 60 61

History of Science

"Vite lnedite di Matematici Italiani scritte da Bernardino Baldi," Bullettino di Bibliografia e di Storia de/le Scienze Matematiche e Fisiche, XIX ( 1886), 625-633. Piccolomini, Della Jnstitutione di Tutta la Vita de/ Homo Nobile (Venice, 1542), c. 59v. The preface is dated Padua, January 1540. In the Biblioteca Nazionale at Florence there is a sixteenth-century manuscript copy of the 1547 edition: MS. o, IV, 214. We are grateful to Prof. Charles Schmitt for supplying a microfilm thereof. Piccolomini, Parafrasi ( I 582), Proemio, pp. 11-12. Piccolomini had published a work, De /ride, at Venice in 1540 with his translation of Alexander Aphrodisiensis ' commentary on the Meteora. For the relevance of this subject to the Mechanica, see Rose, "Certitudo Mathematicarum," toe. cit. For instances see Piccolomini's comments on Questions 11, 13, 28 (caps. 16, 18, 33). Piccolomini, Parafrasi (I 582), p. 71. He did not reproach the translator, however, having noted the erroneous text: "It is no wonder, therefore, that in Question 20 the true sense has been corrupted by a certain translator," ibid., p. 75. See below for Piccolomini's preceding comment. Ibid., Proemio, p. 12. The only text of the Mechanica known for certain to have been at Bologna in the sixteenth century was a (printed?) copy at San Salvatore of the Aldine text. Cf. M.H. Laurent, Fabio Vigili et /es Bibliotheques de Bologne au Debut du XV/e Siecle, Studi e Testi, 105 (Vatican City, 1943), p. 267 list (from MS. Vat.Barb.Lat. 3185). Before leaving Padua at the end of 1542 Piccolomini may have seen Mendoza's Greek manuscript text which dates from earlier that year. See below. Piccolomini, Parafrasi ( I 582), pp. 93-94. Ibid., pp. 74-76. The remark cited in n. 52 follows this comment. Piccolomini (1547), f. 22v. For Piccolomini's views on science and the vernacular, see Olschki, Geschichte der neusprachlichen wissenschaftlichen Literatur (Leipzig, 1922), II, 223-240. An interesting letter by Piccolomini on the subject is printed in Lettere scritte a Pietro Aretino, ed. G.B. del Monte, "Scelta di Curiosita Letterarie e Rare dal Secolo XIII al XVIII," 132 (Bologna, 1874), 11, parte 1, 229 ff. The letter is dated 1541. Piccolomini, Parafrasi (1582), translator's preface, pp. 5-6. Some drawings of machines by Biringucci are in the Biblioteca Comunale, Siena, MS. s.1v.1. See Section 8. For a general survey see Paolo Rossi,/ Filosofi e le Macchine ( 14~ 1700) (Milan, 1962), pp. 11-67. Angel Gonzalez Palencia and Eugenio Mele, Vida y Obras de Don Diego Hurtado de Mendoza (Madrid, 1941-1943), 3 vols. Cf. 1, 20(}-2IO, 282, 286-288, 291-292, 313-315; m, 139.

The Pseudo-Aristotelian Questions of Mechanics

167

62 R. Foulche-Delbosc, "Mechanica de Aristotiles," Revue Hispanique, 5 (1898), 365405. For the dating of the translation as 1545, see ibid., 366. Facsimiles of pages ofMsS f. III, 15 and 27 are reproduced in Charles Graux, Essai sur /es Origines du Fonds Gree de L' Escurial (Paris, 1880), q.v., pp. 168, 357-358. Like Piccolomini's work, Mendoza's translation was part of a projected series of Aristotelian translations. 63 Foulche-Delbosc, "Mechanica de Aristotiles," p. 368. 64 Ibid., p. 368. 65 Mendoza's collection was praised by Conrad Gesner, Bihliotheca Universalis (Zurich, 1545), f. 205v. 66 For the Mendoza manuscripts now in the Escorial see Graux, Essai, pp. 162-273. A partial catalogue of Mendoza's printed books is in Palencia and Mele, Vida, m, 481 572. 67 B.E. Miller, Catalogue des Manuscrits Grecs de la Bibliotheque de L' Escurial (Paris, 1848), p. 145. Gesner ( 1545) had wrongly attributed 1he work to Georgi us Pachymeres; see Graux, Essai, pp. 386-400, 462-464. 68 Castellani, "Prestito dei Codici," loc. cit., records several loans by Mendoza from the Marciana. 69 For Mendoza's friendship with Bembo see Palencia and Mele, Vida, 1, 200-201 . He was for a time military governor of Siena. 70 The translation is from Drake and Drabkin, Mechanics , pp. 104-105. A facsimile of the 1554 edition of the Quesiti with an introduction by Arnaldo Masotti was issued by the Ateneo di Brescia in 1959. See in this edition, fs. 78 ss. 71 One of the two existing copies of the tradelist is the British Museum, shelfmark 1, C.7881. The title is Haec Opera Fient in oppido Nuremberga Germaniae ductu /oannis de Monteregio. See the item, "Aristoteles, Prohlemata Mechanica." For a facsimile see George Sarton, "The Scientific Literature transmitted through the Incunabula," Osiris, 5 ( 1938), 11 5; 162-163. Regiomontanus' Greek manuscript, though included in the 1512 and 1522 lists of his books, did not reach the Nuremberg library. 72 The letter, dated 1540, is printed at the beginning of Maurolico, Cosmographia (Venice, 1543). 73 On these authors see the works of Duhem, Origines and Etudes, cited in n. 3; also Drake and Drabkin, Mechanics, Introduction. On Leonardo see J.B. Hart, The Mechanical Investigations of Leonardo da Vinci, 2d ed. (Berkeley, 1963), pp. 56 ss. 74 For Guidobaldo's opinions on the Mechanica see his discussion of the balance in the Liber Mechanicorum (translated by Drake and Drabkin, Mechanics, pp. 285 ff.) and the preface of his Archimedis .... Paraphrasis of I 588. Cf. the preface of his translator, Filippo Pigafetta, to the Italian version of the former work, Le Mechaniche (Venice, 1581 ). 75 Guidobaldo del Monte, In duos Archimedis Aequeponderantium lihros Paraphrasis (Pesaro, 1588), pp. 18-19.

168

History of Science

76 See I. Affo, Vita di Monsignor Bernardino Baldi (Panna, 1783). P. Zaccagnini, Bernardino Baldi nella Vitae nelle Opere, 2d ed. (Pistoia, 1908). Duhem, Etudes, 1, 87 ff. 77 Enrico Narducci, "Vite Inedite di Matematici Italiani scritte da Bernardino Baldi," Bullettino di Bibliographia e di Storia de/le Scienze Matematiche e Fisiche, XIX (1886), 625-633. 78 The complete Vita di Archimede is printed by Narducci, "Vite," Bul/ettino, XIX (1886), 388-4o6; 437-453. This passage is on pp. 438-439; translation from Drake and Drabkin, Mechanics, pp. 14-15. Similar views are given in Guidobaldo's preface to Archimedis ... Paraphrasis. 79 Written in 1582, according to Fabritius Scharlonicinus in his remarks at the beginning of the posthumous publication of the Baldi commentary, but more likely completed after I 586, to judge from a reference by Baldi to Simon Stevin. Paul Oskar Kristeller, lter lta/icum, 1 (Leyden, 1963), 227, notes that a manuscript copy was at Florence, in the Accademia Colombaria, but this is now missing. 80 Baldi, "De Verborum Vitruvianorum Significatione," in Giovanni Polen, Exercitationes Vitruvianae (Padua, I 739), p. 168. 81 Bernardino Baldi, Cronica de' Matematici (Urbino, 1707), pp. 135-136. 82 Affo, Baldi, p. 9. 83 Baldi, Erone Alessandrino Degli Automati (Venice, 1589), fs. 4v, 6v. The preface is dated 1576. 84 Biblioteca Ambrosiana MSS. 0.422 inf.; s.103 sup. 85 Galileo, Ope re (ed. A. Favaro), XIX, 120. 86 Ibid., IV, 364. 87 Ibid., IV, 674 88 Galileo, Two New Sciences, tr. Crew and De Salvio, pp. 2~50, passim. 89 The drawing is in two Martini manuscripts; Biblioteca Estense, Modena, MS. Alpha a.4 .21 (Estense Ital.421); British Museum MS. Harleian 3281. 90 Biblioteca Nazionale Centrale, Florence MS. Palatino 1417, c.55. 91 P. Garzoni, la Piazza Universale di tutte le Professioni de/ Mondo (Venice, 1586), pp. 768-776, testifies to the good standing of practical mechanics in Italy. He adduces the Mechanica as proof of Aristotle's regard for mechanics (p. 773) and lists the leading mechanical writers of the modems, including Alberti, Guidobaldo, Tartaglia, and Fausto (p. 776). 92 Ceredi, Tre Discorsi, p. 76; Vittorio Zonca, Novo Teatro di Machine et Edijicii (Padua, 1607), p. 22. 93 Lorini, le F ortificationi (Venice, 1609), p. I 99; Aleotti, Spiritali (Ferrara, I 589), p. IOI; also his engineering treatise in Biblioteca Estense, Modena, MS. Gamma e.2.8., f.282. 94 Giorgi, Spirita/i (Urbino, 1592), Introduction.

The Pseudo-Aristotelian Questions of Mechanics

I 69

95 Ramelli, Le Diverse et Artificiose Machine (Paris, I 588), "Alli Lettori Benigni." 96 Branca, Le Machine (Rome, 1628), "All i Lettori ." 97 For Guarino see G. Tiraboschi, Biblioteca Modenese (Modena, 1783), m, 34-36; F. Argelati, Biblioteca dei Vo/garizzatori (Milan, 1747), 1, 104-105; Riccardi, Biblioteca Matematica Italiano , 1, 638. Correspondence between Guarino and Alfonso II d'Este is in the Archivio di Stato, Modena. 98 Aleotti, Spiritali (1589), Preface to Alfonso II. 99 In Nunes, De Arte atque Ratione Navigandi Libri Duo (Coimbra, 1573), pp. 121126. Previously printed with another work in his Opera ... (Basie, 1566). Oarage questions were discussed by many writers including Cardano, Opus Novum de Proportionibus (Basie, 1570), Propositions 77, 78, 81, 82, 111, and I 19; also by Galileo in his letter to Contarini. I oo Bodleian Library MS. Additional c. 194, fs. 17, 20v, records Dee' s copies of Piccolomini and Fausto. See also "Mathematical Preface" to Billingsley's translation of Euclid (London, 1570), f. c.iii.v. For Dee's mechanical interests see Frances A. Yates, Theatre of the World (Chicago, 1969). 101 Ramus, Scholarum Mathematicarum Libri XXXI (Basie, 1569), p. 21 . 102 Ibid., p. 1o6. 103 J.J. Verdonk, Petrus Ramus en de Wiskunde (Assen, 1966), pp. 43 and 403. Ramus, Scholae Mathematicae, p. 21 . Cf. ibid., p. 219, for a quotation from the Mechanica . For the lectures see Ramus, Col/ectaneae Praefationes (Paris, 1577), p. 197. I 04 Oratio pro mathematicis artibus (Paris, I 574); De angu/o contactus (Paris, I 58 I); Ludus iatromathematicus mus is foetus (Paris, I 597); Problematis ... demonstratio (Paris, 1600); Tractatus de puncto (Lyons, 16oo).

7

An Agricultural Economist of the Late Renaissance

The advent of inexpensive printed books about the beginning of the sixteenth century had a marked effect on the pursuit of science. Virtually a university monopoly during the Middle Ages, occasioned at least partly by the concentration of manuscripts in and around centers of learning, the study of science was no longer so restricted after 1 500. The same printing presses that made it possible for men to educate themselves also gave to those with special talents and unconventional interests a means of conveying their contributions to others. Where science in the universities had been - and continued throughout the sixteenth century to be - an integral pllrt of philosophy, science in the wider republic of letters began to be linked in various ways with practical concerns, with mathematics, with technology, and with broad problems of a changing society. Thus for about a century and a half there were two effective streams of science - one within the universities, hardly distinguishable from medieval science, and one outside the universities, hardly recognizable as science under the older criteria. By the time of Francis Bacon and Galileo, sciences concerned with utility had found spokesmen capable of winning attention and commanding respect even from a few professors. Galileo himself entitled his little syllabus on mechanics "On the utilities to be drawn from mechanical science and its instruments" and not, as an earlier professor would have done, "On the causes underlying mechanics and its instruments." Utility had traditionally not been a consideration in science; but only in the arts and crafts. How this transformation Reprinted from On Pre-Modern Technology and Science: A Volume of Studies in Honor of Lynn White , Jr., edited by 8.S. Hall and D.C. West (Malibu: Undena Publications, 1976), 53-73, by permission.

An Agricultural Economist of the Late Renaissance

171

of the concept of "science" came about during the sixteenth century will never be discovered by studying science in the universities of that time. Nor will it be discovered by studying the history of technology alone, for technology had always got along quite nicely without assistance from science. The physics of Aristotle was designed to explain the causes of things, and not to be of use to the engineer, the architect, or the builder. To discover the origins of the sciences of utility (as contrasted with useful practices), we must attend to some men who were not satisfied either by reason or by experience alone, but who both thought and tinkered. The number of such men has probably always been large, but the number who wrote down what they had thought and also what they had done, in a way designed to show others how the two activities were related, seems to have remained small until the seventeenth century (when this had become respectable). One such man was Giuseppe Ceredi, whose only known book set forth what was perhaps the first published proposal for a systematic scheme of irrigation using a standardized mechanical device. That alone would deserve attention; but Ceredi also analyzed the economic feasibility of his scheme and went on to consider certain of its implications with respect to public health, land utilization, navigable streams, and the effect on the poor of a competitive export advantage. It is further of interest that the mechanical device in question was a very ancient one, and was not entirely unknown to Ceredi's contemporaries, though its transformation into a practicable form for use in irrigation had been neglected. Ceredi's contribution was to give it an optimum form by the construction and testing of models, taking into consideration the strength of avail able materials and the source of power to be applied, and this approach seems not to have been previously explained in print. Finally, Ceredi had a certain conception of science which he ably set forth - one that seems to me very close to that of the later seventeenth century (at least outside the universities), and to our own. Ceredi 's book as a whole has not yet attracted the attention of .historians of science, even in Italy, but neither has it gone entirely unnoticed. About the time that I first happened on a copy and became interested in its pioneering agricultural economics (being at that time engaged in problems of financing irrigation projects in California), Professor Lynn White published his Medieval Technology and Social Change, in which he wrote: The mechanical crank is extraordinary not only for its late invention, or arrival from China, but also for the unbelievable delay, once it was known, in its assimilation to technological thinking ... The earliest evidence of compound crank and connecting rod in Italy is found in a manuscript of Mariano di Jacopo Taccola which is not earlier than

I 72

History of Science

1441 ... The enthusiasm of Renaissance engineers for the flywheel-crank combination was such that they tried lo assimilate the two by frequently bending the central section, or key-seal, of their cranks into half circles. In 1567 Giuseppe Ceredi, in the first theoretical discussion of the crank which I have found, points oul that this is mechanically useless; nevertheless, gracefully curved cranks continued lo be common deep into the nineteenth cenlury. 1

The strange neglect by later designers of the elementary mechanical consideration to which Ceredi had called attention is no more surprising than the neglect by historians of science of Ceredi ' s book as a whole. Renaissance craftsmen may have been influenced by the idea that circular motion will best be imparted by a device having a circular shape, in accord with the all-pervading Aristotelian dichotomy between straight and circular motions. Historians of science may be influenced by an all-pervading notion that continuity must be found in the progress of ideas, and since in sixteenth-century science continuity par excellence is found within the universities, there is perhaps some reason for them to neglect books that never entered within university walls. Alternatively, we may adopt the second-stream view already suggested above, and assume that writers who addressed the new republic of science created by cheap books broke the intellectual continuity by looking on the one hand back to classical antiquity, by-passing their immediate predecessors, and on the other hand to actual experience and to tinkering with things themselves. Doubtless it will tum out, when Ceredi's work is carefully studied, that everything he set forth had other precursors than those whom he himself was careful to acknowledge in his book. Before those others can be discovered, however, and before Ceredi can be fitted into his place in that historical continuity that abhors original ideas more than Nature abhors a vacuum, it will be necessary to call attention to his book. That is the principal purpose of this paper, and the best way I know to go about it is to translate some selected passages. All that is known about Ceredi at present must be taken from his Tre Discorsi sopra ii Modo d' Alzar Acque da' Luoghi Bassi, published at Panna in I 567. He was certainly of Piacenza, and appears to have been educated as a physician. During the years I 565-66 he obtained patents at Milan, Venice, and Panna for the manufacture and installation of Archimedean water-screws for the irrigation of lands and the drainage of swamps. Professor Carlo Poni of the University of Bologna has recently sought at Piacenza in vain for records of his birth or residence there. At present, then, we can only sunnise that Ceredi's dates were probably about 1520-1570. It is highly probably that he met and conversed with Giovanni Battista Benedetti at Panna on the occasion of his application for

An Agricultural Economist of the Late Renaissance

173

a patent from Ottavio Farnese, at whose court Benedetti was employed as engineering adviser before he moved to Torino in 1567. (Ceredi 's patent was granted in December, I 566.) Benedetti had a wide correspondence from Torino with men interested in science and engineering; the absence of Ceredi's name from it tends to corroborate the idea that he did not live long after 1567. II

At first glance, it may seem odd that Ceredi or anyone else should have had the gall to seek a patent on the Archimedean water-screw as late as I 565, and still more so that sophisticated and prudent governments like those of Milan, Venice, and Parma should grant such patents. The device itself was by no means new, and even its application to irrigation was ancient according to tradition. Its invention was attributed to Archimedes in the third century before Christ; some said while he was in Egypt, and for the purpose of irrigation, while another story had him invent it in Syracuse to remove water from the hold of a ship of Hiero's. Its design and construction were discussed by Vitruvius, whose book On Architecture had a great vogue in the sixteenth century, and (as will be seen) its use in Italy for the drainage of swamps was not uncommon then. On the other hand, Professor Alex Keller states quite positively that the device was not known in the Middle Ages. That it was still not in use for irrigation up to the very year of Ceredi 's book is evident from the commentary of Daniel Barbaro on Vitruvius, in the second Italian edition published in 1567, cited below. There was a reason for this, and it in tum explains why Ceredi was granted his patents. The reason lies in the various forms that the device could take, in their very different problems of construction, and in the respect for the authority of Vitruvius that prevailed. When I think of the water-screw or the auger conveyor, I think of it as a movable screw turning in a fixed casing. Such a device is easy today to build, especially of metal. Obviously it would not have been so easy to build of wood, and if a wooden device of this form had been built for use in water it would not have been successful. Unless the screw had very consistent and narrow clearance from the casing, either excessive friction or leakage past the screw would seriously impair its efficiency. The swelling of wood in water and its shrinkage when dried would present real problems for an irrigation device which was intended to be used only three months in each year and removed when not in use, to protect it against damage from floods and floating logs. A much simpler alternative form was, I believe, the only one actually in use at the time Barbaro wrote his commentary on Vitruvius. In this form, a pipe or pipes are wound spirally round a central shaft; lead pipes would lend them-

I 74

History of Science

selves nicely to this. Such a screw would be very practicable for the drainage of swamps, where a constant but relatively small amount of water is to be lifted continuously for long periods of time from a sump into which marshy land would slowly drain . This would pay only where a water wheel supplied the power, and both Barbaro's account and the illustrations I have seen tend to support the idea that this form of screw was in use. But such a device would be of no use for irrigation, or for the removal of water from deep mine shafts, where large quantities of water must be moved in short periods of time. Agricola, who devoted considerable space to devices for removing water from mines, did not mention the screw at all, whence I think it unlikely that screws of the first type found in Spanish mines were used as early as the sixteenth century. The third form of water screw was described by Vitruvius, and this appears to be the form to which the name is most properly applied. The manuscripts referred to a diagram, which unfortunately was missing, and the standard sixteenth-century illustration supplied by some early editor is quite ridiculous. Modem translations supply correct illustrations. 2 Neither a moving auger nor a twisted pipe was used; eight similar spiral channels wound round the central shaft, and the casing was affixed to the spiral partitions; the whole apparatus was then revolved as a unit. This form of screw is identical with Ceredi 's, except that he used only two worms rather than eight, and built the machine in proportions vastly different from those prescribed by Vitruvius. The result was a practicable apparatus capable of being worked by manpower and of producing water in quantities useful for irrigation, which accounts for the granting of patents. The commentary on Vitruvius by Barbaro that was printed in the same year as Ceredi's book is as follows: I have seen this instrument marvellously proved in our swamps for drying water from them; and in particular in swamps close to the river Brenta, where the wheel that turned the screw was placed above the water [level] in such a way that the water which turned the [water] wheel moved other wheels and gears beside its pivot and turned the screw, which, drawing water from the swamp, made it fall into a receptacle beneath, from which it flowed through a wooden flume and back into the river. Some people will have it that with this same water, motion could be given to the wheel that turns the screw, after the initial motion; so that this would be almost a perpetual motion. But I think that we want other considerations in order to make use of the screw to irrigate fields, as the Egyptians did according to what Diodorus says in his first book; and he says that it was the invention of Archimedes. The construction of this instrument as given by Vitruvius is no less pretty than easy, and no less easy than useful, and will be understood through our translation and from the figure drawn by us.3

An Agricultural Economist of the Late Renaissance

I 75

But if Barbaro drew a proper figure, it was not reproduced in the printed book, which gives the usual absurd illustration (see Plate 14). The construction is not explained, and not even a cross-section is shown. As a matter of fact, the procedures described by Vitruvius in the text are anything but easy; the eight wonns are built up out of many lamina of twisted strips, very precisely applied, and to very considerable heights. In Barbaro's illustration, the spiral banding shown makes little sense unless taken to be a twisted pipe; iron bands to hold the casing would be parallel, as on barrels. 4 The driving mechanism described by Barbaro is not shown, and though a wheel is placed centrally on the screw that is partitioned like a water wheel high above the water, it could serve no purpose other than that of a flywheel as drawn. The proportions of the device are in line with the rules given by Vitruvius, which would make it monstrously heavy; judging by the figures in the landscape, it would weigh two or three tons when full of water. It was to such practical considerations as the source of power, strength of materials, weight of water, and angle of lift that Ceredi first applied himself, before considering the cost of operating the machine and its value to the owner. It was in this way that science entered as the companion of technology, a matter on which Ceredi's views are set forth near the beginning: But although many subtle and learned men, Greeks, Latins and barbarians, have long labored with the guide of mathematical and physical reasoning, and with various experiences, to give assistance to such good and useful enterprises [as the raising of water), yet there has been no one up to our time, so far as is known, who has offered a way to raise great quantities of water to any desired height in order to inigate lands, dry out vales, work mills, and other such things in which copious water is sought, and in such a way that the expense of the moving power and that of manufacturing and maintaining the machines shall not tum out to be greater than, or nearly equal to, the utility that could ensue from such an invention. And although one of those good ancient Greeks may have left rules by which some industrious man, knowing them, might perhaps anive successfully at these results, they have remained in shadows through the injury of time and the little diligence of less ancient writers (as I shall make clear later on); for these effects have almost always been judged, by many otherwise prudent men, to be impossible to put into actual operation. That judgment is the more confirmed when it is seen that many large and rich cities of Europe have the best of rivers and abundant water that passes beside them; thus Rome, the Tiber. Ferrara, the Po; Toledo in Spain has the Tago; and there are many lands along the Reno, and the Danube, and others in other provinces; and still they have never been able to draw water from those within their own walls, or over the banks, for the necessary use and convenience of their citizens; but rather, at great expense, they have this [water] canied by men or by beasts of burden into the cisterns of

176

History of Science

private houses. Without having great need [of water supply], the Germans, people who (as everyone knows) are of supreme excellence in working ingenious machines to raise water from the mines in which they are very rich, (because it is necessary to dry out the waters that flow from subterranean veins in great quantities and with the greatest impediment of all [to mining] into the hollows of mines ... ), still have never yet put into use any machine, among the many that have been seen, which in quantity of water lifted, and with greater utility than expense, would be apt to satisfy the previously mentioned needs of irrigation, and such things ... 5 I can only declare with some reason that it was almost by accident, and outside my principal concern, that there came together for me all the factors that usually give birth to such results; for rarely are they found all put together in [the works of] any one of those who have been the authors of very many and very beautiful inventions. After my most serious studies, in which (as we of Piacenza do) I usually apply not only my mind but also my hands; and toying often in the field of certain mathematical demonstrations, now simple and again mixed with various kinds of sciences, and among other things in the rules of proportionality and the science of weights, it occurred to me (as indeed recorded by Aristotle and Galen) that no science or art whose end is in operation can be perfectly possessed in such a way that whoever has the precepts of these need not then confirm them with various experiences often repeated and succeeding with certainty. Whence I considered my wish to get some pleasure also regarding such a useful subject, by putting into actual effect those standards which (against their true end) are commonly understood only in the abstract by scientific men ...6 Almost by chance there had been sold to me by one who did not understand them certain writings of Hero, of Pappus, and of Dionisidorus, taken from the library that once belonged to the learned Giorgio Valla of our Piacenza. He, by his merit, was elevated through the liberality of the illustrious Signor Giacomo Trivulzi, at that time governor of the state of Milan, and was in a position to collect all the most worthy Greek writings which at that time were brought from Greece (then oppressed by its greatest ruination) and were carried in various ways to our Italy. In those works, never printed or translated that I know of, I confess that I found many of those things that I am about to set forth below, after [reading] many propositions of Euclid, of Archimedes, of Apollonius of Perga, and many others who came later. These were already known to those who wished them, since they must have had these operations in hand, and they shed no little light on the road for me, for I believe I have arrived at the determination of this machine. By the liberality of the excellent duke, our lord Signor Ottavio Farnese ... and without abandoning my principal professions, I was able to fabricate a great many models, small and large; adding, changing, and removing various things according to the condition of the material, or the grouping of many primary and secondary causes, or the variety of the mediums, or the proportions, or the force of the movers, or many other obstacles that hinder the thing sought. For it is well known by scientists [scientiati] that when things

An Agricultural Economist of the Late Renaissance

177

are put in operation, so numerous and great a heap of observations need to be kept in mind all together to hit on any new and important effect that it is almost impossible to fit them all properly together and arrange them securely in the orderly work, except after many errors, recognized at various times from experience and somehow corrected by reason, so that at the end no one comes to perfection of the art and to the definite production of the effect that was expected ... For ingenious men who are not scientists never do anything good except by chance; and, not knowing how interlaced are the causes of error that happen in their labors, but finally frightened off by the difficulty of the successful outcome, they abandon the enterprise and allege a thousand fables and falsehoods by way of excuse. 7 [Here I make] known openly all the true rules for constructing and positioning [the water-screw] so that no difficulty will enter into anyone's putting it into the most useful operation. Nor do I want chains of avarice to bind my breast, for the patents conceded to me by almost all the princes of Italy for this industry are to me more pleasing than [even] the reasonable courtesy and recognition of those who will enjoy its vast utility ... Let anyone who likes make merchandise of the fruits of this science and of the offsprings of the divine substance of the mind; to me, it suffices to have become in this a mechanic like Hippocrates, Plato, Aristotle and Galen, and so to have produced some benefit to men of the times to come - without my being held to be a merchant through selling, or holding concealed, things that belong to the common utility, just in order to draw more money from it ... In the first [discourse] I tell of all the machines used to raise water up to the present, making comparison of the water-screw with them, and I shall give the true rules for making this and putting it properly in place. In the second [discourse] I shall treat of the ways of driving it, where there will be occasion for some elegant and easy notices about the science of weights. In the third [discourse] I balance up the use and the harm of this instrument as to health and as to agriculture, for the benefit of the economists ... 8 III

Since my concern here is mainly with the Third Discourse, I shall but summarize - with small justice to Ceredi's wide reading and assiduous practice - the gist of the first two. In consultation with engineers, Ceredi had found that no unifonn rules were to be had concerning the optimum construction of waterscrews. Ultimately he detennined that the best procedure would be to use a screw about 25 feet long, to raise water about 15 feet (see Plate 15); if greater lift were required, a second stage with an intennediate sump was recommended (see Plate 16). With this device, a man of ordinary strength could produce onefifth the volume of water flowing continuously in an ordinary meadow-canal. A battery of five standard screws, worked by ten men in alternate four-hour shifts,

I 78

History of Science

would continually irrigate I ,500 pertiche, or nearly ing this, Ceredi commented:

250

acres. Before explain-

But if this has not been put into use up to now, I am sure that the great authority of Vitruvius (who alone of all known writers has treated of it), and the imperfections of his rules, must be the special and powerful reason ... Above all there is the power of the mover, which [under Vitruvius's rules) would have to be enormous. The distinguished Signor Pandolfo Contareni told me that several noble Venetians, of whom he was one, had built these, always without success, and had found them wanting in many respects, but especially in the great difficulty of moving them; so finally they gave up, after much expense, and preferred other means. From which I marvel even more that Mons. Barbaro never tried the experiment .... of making perpetual motion by having the water fall on a wheel fixed in the middle .. .9 [Using the rules of Vitruvius), take a central shaft one-half braccio in diameter; according to the first rule its length must be eight braccia. The channel, if single, will be 18 inches wide by the second rule ... By the third rule, the entire thickness of the instrument will be one braccio. Put in operation, its slope is to be as 5 to 3. Clearly it must be submerged in water by as much as its thickness; otherwise reason and experience show that it will not fill up. Thus about 20 inches are lost under the water; the remainder is 6 braccia 4 inches, which will lift the water little more than 3½ braccia ... 10 As to the hope of Mons. Barbaro - I mean, his strong reason to believe that with this instrument better than any other there might be made a perpetual motion ... I think that, as is usual with most learned scientists, and especially those who occupy themselves in profounder matters, he never put this to the test; for if he had (as I have done more than once, and in many ways, though rather to test in practice mathematical ratios and to know these better through experience, than for any other reason), and if he had seen that the hoped-for effect never happened, he would no doubt have clung to his "true cause" and found from it another, opposed to it, that interfered with the operation; otherwise, knowing wherein his original idea was defective, he would more easily have judged of the truth ... 1 1 The [central) support may be made as long and as thin as one wishes: the [spiral) channels may be made parallel and as wide, straight, and deep as you like [in theory). But it is necessary that all these things be proportioned to the power of the mover and to the height [of lift) required. This ... will not be difficult to understand. But to put this into execution, and have it based firmly on experience as guided by reason, it was necessary to make a large number of models, both small and large, now with one length and height of channels and now with another, in order to be able to proportion the whole to the mover and to its organ [the crank). In the end, it was concluded that the central shaft should be thirteen braccia long and four inches thick; being made of fir, a wood of very strong grain, this will not bend in those proportions and will be quite light.

An Agricultural Economist of the Late Renaissance

I 79

The channels [will be] six inches apart and two inches wide or a little more, with willow worms fastened with small nails one [section] to another, for no other wood will permit the bending into spirals; and being daubed with pitch it will endure and resist water for a long time. Likewise the casing is to be of fir, nailed on the worms and daubed inside and out, and skilfully bound with iron bands which need not be described, since they may be seen in the diagram. Thus made, it will be inserted with the end about one braccio under water. The remainder will be 12 braccia, which will rise [at an angle] taking away three parts in five [to a vertical height above the river of] seven braccia and about 3½. And since the tilt takes away 2/5" in every 6", or 48/5" inches, that is, 9 3/5", a rise of more than three parts in five can be obtained, and the lift outside the water will be at least 7½ braccia. 12

The "tilt" referred to here is the inclination of the screw to the surface of the water to be raised, and not the angle (about 37°) indicated by a slope of three parts in five. In order for the screw to operate, it must be placed in the water at a slightly smaller angle than that of the screw. Ceredi's wording indicates that although the screw is designed to give little more than seven braccia of lift at the designated slope, there is sufficient leeway in the tilt to allow a lift up to 7½ braccia. I presume that the angle of his screw was about 40°. In this manner, since the channel is double, twelve inches of water, Piacenza style, will be let out and taken in [during one full tum], and with flow from a tilt of about nine inches; or, with the alternative slope, the revolution of the worms gives a gradient of about six inches in 18 braccia; and that gradient, as we shall say in the Third Discourse, is much greater than in ordinary [meadow-canals] and carries more [water] than in these. 13

Here the reference to the channel as double implies that Ceredi used two similar augers starting 90° apart; the advantage over a single auger is not clear, but perhaps the spiral sections of willow would otherwise have had to be too thick in order to bear the weight of water. The meaning of "inches of water, Piacenza style" had previously been explained; these represented square inches of crosssection in a rectangular ditch, the water flowing continuously at whatever gradient was standard for irrigation; that is, quite slowly. An ordinary meadow-canal was taken as carrying sixty inches of water, Piacenza style; for example, six inches of depth in a ten-inch square ditch. The Milanese usage appears to have been different, as noted later below. Ceredi indicated further that a horse of average strength could draw 36 inches of water continually to a height of seven braccia, or more than one-half enough to supply one meadow-canal, and he reckoned the power of a man at one-third that of a horse.

I 80

History of Science

Among the concluding remarks of the Second Discourse, Ceredi mentioned that some people wanted a flywheel placed on the screw to keep the flow steady, though he considered this unimportant for irrigation water (see Plate 17). He also noted that use of a horse with his standard screws meant the introduction of gears to drive them. This he regarded as undesirable because a gear might occasionally break and take the system out of operation for days, whereas single screws are virtually accident-proof. Thus he concluded that it is better to pay a bit more for laborers than to risk loss of service of the system during the dry season. It is a pity not to include in full Ceredi 's comments on the history of irrigation, the hydrography of the Po valley, and on various matters of public health and of medicine. But we must get on to the final discourse, which seems to me to be highly original and of great interest with respect to the relation of science and technology to public affairs. IV

Ceredi's Third Discourse opened as follows: I know that most of those who read these discourses of mine have been awaiting this last part, as the final goal of the structure and of those many reasonings that have gone before. For few are they who, in reasonings about things whose end is operation, appreciate the wonders of their causes unless they see this followed at once by some manifest utility that is guaranteed by experience. For the most part, men follow instinctively the advice of Socrates and leave [pure] speculation to those minds which, not distracted by the necessities of conserving civil life, try in a way to transform themselves into the nature of celestial minds; and who, rather than basing themselves on operations, seek to discern as quickly as possible the benefit that they proposed to themselves at the outset. But I should not like to give occasion to practical men of this world to say to me, in almost the same way (though on a different subject), that which was replied to the great philosopher Aristotle by a Spartan totally ignorant of science, when he was astonished that there was no one in all Sparta able to give reasons for political science, or who knew the causes of moral virtues, with which it is fitting to ornament our minds. The Spartan replied to him: "I am even more astonished that in Sparta, where no one debates virtues and vices, there are for the most part men of good customs and who live generally according to the standards of good laws; while in Athens, where science and the causes of good action are taught every day, nearly all men, and especially Philosophers, are wicked, and live strangely dissolute lives." From this reply it is concluded that it is one thing to argue with reasons about actions, and quite another to carry them out. Sometimes, as Aristotle says, the arts are better served by good practice, and the habit of oper-

An Agricultural Economist of the Late Renaissance

181

ating well, and long usage confinned by many experiences, than by mere speculation, untested by the various effects that take place, and following the order of reason (alone]. Hence I, after discoursing about reasons, come now at this point, (as I did in the construction of the water-screw and (the jlnalysis] of its motive power) to their test, and to the effects promised therefrom. I shall therefore in this Third Discourse give account of the utility and of the hann that may follow on the use of the water-screw, through all those events that are seen by experience; and then I shall say in what places it will be seen to work on these principles, and to produce the utilities that we shall have concluded should follow from it. But since [its] utility may be of three kinds - one with regard to agriculture, net of all the losses and expenses; another with regard to our bodily health; and the third with regard to the public benefit, which pertains to governmental regulations - so, one at a time, we shall show succinctly the truth in each of these. It was already concluded above that our water-screw which raises at least twelve inches of water, in the measure of Piacenza, [continually] to a height of seven braccia, can be moved by a man of average strength; that it can be situated at any place and in any water, running or still; that its moving [of water] can be doubled or tripled [in stages], up to the height of twenty braccia; and that it can be moved from place to place, and secured against every peril. We now pass on to examine further certain other topics that either arise necessarily in the generation of profit, or at any rate that bring very great facility. I say first that the twelve inches of water that rise and pass through the water-screw when it is moved with the speed that can be imparted by a man of average strength put to maintain this [speed) have a swifter tlow than geometers and architects ordinarily want to assign to the motion of waters. The least drop, according to them (as mentioned above) is one foot in every mile (8 furlongs]; the greatest [drop] in navigable streams is not more than six feet in every mile; for if more, ships could not conveniently be driven against the current of the river. Vitruvius gives to aqueducts at least half a foot [of drop] to each hundred feet, which rule exceeds all ordinary experiences; and if it were observed, water would seldom be conducted any great distance from one place to another, since locations of that slope are rarely to be found. The Milanese ship-canal could never have been made with that [steepness]; as Cesariese writes, who was present when its drop was surveyed with spirit-level [si compassava ... col cherobare], this had three parts in four less than what Vitruvius advised, and it went very well, and in such a way that if it had been wished to give it more (gradient], the route [Milan to Pavia] would not have pennitted this. But the water-screw far exceeds in gradient the excessive rule of Vitruvius, for being about thirteen braccia long, it has in a distance of twenty braccia [of advance] more than five inches of descent, which is almost as much as a distance of eighty braccia should have according to the teaching of Vitruvius. 14

A comment on this passage seems advisable. All that Ceredi wished to establish

182

History of Science

was that the water delivered at the upper end of the screw does not flow more slowly than water in an ordinary meadow-canal. Since he did not have measures of velocity of flow, and did not think in tenns of quantities delivered per unit of time as we do, he seems to have reasoned as follows. If the water of the river were merely lifted parallel to itself, the flow delivered would be the same as the flow of the source. But it is lifted at a certain angle to the river, and flows off at the upper end more or less at that angle. The water flows off between two sections of screw, six inches apart, so that when one of these sections is just at the lip, the height of water opposite the lip is six inches, or would be if the flow were intennittent and the screw were parallel to the river. But in fact, for the screw to operate, its angle with the river must be somewhat less than that of the screw; Ceredi had previously indicated the difference as 2/5" in every 6". Hence he says that the water emptying from the screw "has more than five inches of slope," and consequently runs off much faster than the river runs. At reasonably fast speeds of the screw, it does run faster; and if the reasoning behind Ceredi's conclusion seems a bit odd, he nevertheless deserves credit for recognizing that some consideration of rate of flow is required. The manner in which he compares this supposed gradient with that of an irrigation ditch is itself interesting. He thinks of the twenty braccia of horizontal advance in raising water 7½ braccia through a I 3-braccia screw as somehow relevant, treating this as if a unifonn gradient of six inches through twenty braccia were the same as a steep initial drop of six inches followed by level flow for the balance of the twenty braccia. The speed at the end would be the same for a frictionless body, but not for a fluid, especially in an irrigation ditch. Having reviewed the capabilities of his water-screw, Ceredi now proceeds to the economics of its use: Further, I make it known (because I have taken measures of this) that the banks of the Po, when its water is lowest, are not ordinarily much higher than thirteen braccia, Piacenza style, and [this height would] require two balleries of water-screws. But the banks of other rivers, like the Tanaro, the Ambro, the Adda, the Oglio, and others, do not in many places exceed a height of seven braccia, for which a single battery of waterscrews will suffice. This is also [true] in some still or slowly running waters, as watered fields and the like. Moreover, everyone who has had any experience in agriculture knows that a meadowcanal [flowing) at sixty inches of water, Piacenza style (which is 3¾ Milanese style) even when this is not Po water, which is very dense - will in one day (that is, during 24 hours), irrigate about sixteen acres [one hundred pertiche] of land that is not naturally very moist, since it will usually irrigate about 5¾ acres every six hours. It is also manifest that meadows are commonly given water only once every fifteen

An Agricultural Economist of the Late Renaissance

183

days, and do not suffer any significant dryness provided that they are irrigated twice a month. From these two positions it is concluded that a meadow-canal of water running continuously will irrigate about 250 acres [ I ,500 perticheJ of land; that is, sixteen acres every day, and, returning again and again, will never leave them lacking water for all needs. Likewise it is known that ordinarily our meadows, especially those that lie along the Po where the land is quite moist, do not begin to be given water until the middle of May, and [then only] up until the middle of August, which is the space of three months, since often the first hay is cut without irrigation, and dryness is felt only in the second and the third [harvestings] . One can then easily figure that if one water-screw draws twelve inches of water, five water-screws will draw, with a single lift (that is, lo a height of about seven braccia), sixty inches, which is the same as the meadow-canal mentioned above, and five men are needed to tum these. But since they need time to eat, to sleep, and to refresh themselves, it will be good to have another five, who in four-hour shifts enter into this moderate labor and make the water run day and night continuously, and [will do this] with greater ease than there is in culling hay and performing the other rustic operations. These ten men being given work for three months on end, there is no doubt that paying their own expenses, they can be paid three scudi per month each, to be generous. This gives an expense of thirty scudi per month, which comes in three months to ninety scudi, or, at six lire per scudo, 540 lire. It is true that for the most part, gentlemen manage at much lower price by making use of those laborers that around Piacenza are called braceros [brazzentiJ. Likewise it is clear that every pertica of land that never lacks water produces, through all three cullings of the grass, at least one-third of a wagonload [carro] of hay each year (more than pastures of four cullings [de/Ii quartiruoli]), so that 1,500 pertiche will produce at least 500 wagonloads of hay. It is also proved that, for many years now, hay is never sold for less than twenty lire per wagonload, and much is sold for twenty-five or thirty lire; yet in this accounting let us take the smallest price we can, and let this be 16 lire per load, so that 500 loads are to be sold at the stacks for at least 8,000 lire, and that much will be got al least from 1,500 pertiche of meadowland. Now it is necessary to subtract first that which was taken from the land before it became meadow, which al most, by common usage, is two lire per pertica- though most land that is leased, when entirely without water, goes lillle beyond three reali per pertica. Thus, 1,500 pertiche produce 3,000 lire [unirrigated]. Add to this the expense of the men who tum the water-screws, spoken of above, which for a single ballery is 540 lire, so that in all there is 3,540 lire; or for two balleries, 1,080 lire, or in all 4,080.

184

History of Science

The agricultural expense in preparing the land, cutting the hay, carrying it to the barns

[cassine], allowing for manuring [dare ii /edame], and such things, must also be subtracted; and in the meadows that are called prime [prata] - as if readied [parata] - this can hardly be more than seven soldi to the pertica, or thereabouts. In 1,500 pertiche this would be 525 lire; and in all (the costs are] 4,505 [lire; probably in error for 4,605 assuming two batteries of screws]. For the expenses of preparing and placing the water screws every year we may put at most one scudo for each, because they are very strong and so well made that neither by long use nor because of any great shock will they be greatly damaged. The pivots, which wear out more than any other part, are tempered in such a way, and so well placed, that they can be removed with no great trouble and expense, it sufficing that they be adjusted once a year before being put in use. The casing also, though artfully and strongly made, can nevertheless be easily lifted [off], and thus maintenance can also be made to the shaft, and as to the turning of the worm, with no great labor; any ordinary master [mechanic] will be able to do this. And it will be seen by experience that a well-cared-for water-screw will give good service for twenty years - something that has not yet been seen in some of the other machines used up to the present time. Therefore the expense of repairs and maintenance being ten scudi [each], the whole [cost of this and the above] will be 4,565 [lire). Allow also for the cost of cleaning the canals and paying the foreman [camparo] the amount of 150 lire, bringing the whole to 4,715 lire per year. But the expense of water canals will be inconsiderable for those who have them close to their lands or right on them, and may [otherwise] be had as I shall say in certain notices below. The 285 lire added to make a round sum of 5,000 lire can be put to the account of royalties for the invention according to the patent obtained; though as I have already said, it is not my intention to make merchandise of this. Everything from 5,000 to 8,000 lire is profit. And finally it is understood that land which [without irrigation] would yield at most 3,000 lire, will yield 6,000 lire with the help of the water-screw, after deducting every cost and expense. (I omit the price of public water to him who requires it, since it is not my job to discuss this.) And even where it is necessary to have two batteries [of screws), they will at least double the [owner's] profit. Where only one battery is needed, the profit will be [still) higher in the amount of the expense of ten men for three months, plus five scudi_saved in expense of maintenance and repair of the screws, which will be, in all, 570 lire more. And additional gain will ensue if the hay is sold at twenty or twenty-five lire per load, as very often happens. So let him make [this] accounting who will; it suffices that I have taken the lowest price and the greatest expenses. Nor do I wish to leave out, after the foregoing accounting, certain useful notices about the present matter; and they are as follows: ... '5 Rainwater and flood waters can be taken with this instrument from fields that cannot

An Agricultural Economist of the Late Renaissance

185

naturally drain, and hence that remain uncultivated, which happens in many places around Padua, Ferrara, Mantua and Ravenna, and nearly all the lowest places, with grave harm to their inhabitants. Yet with little expense in terms of the utility, by means of our water-screw, these can be dried, for the most part; and the more so in all those places (and I have seen and measured many of them) where the greatest height to which water must be raised does not exceed seven braccia, which takes only one bank of waterscrews. But in some terrains the rainwater is little, and could be given exit with so little lifting and in various ways, that now that this simple apparatus, secure in every way, is made manifest, those landlords will be accused of the grossest ignorance, or of negligence, or wantonness, for not making provision for this, rather than be permitted to excuse themselves on grounds of unimportance or difficulty. Yet I should not like someone to think [here] of those depressions near the sea, where there are certain continuous large stretches of land that just rise above the water, called Cori: or where some nearby river, that seems almost supported in the air by ridges in order that it may more easily empty into the sea, sends flood waters in great quantity into fields below [their banks]; for neither this instrument nor any other can lift away the water that abounds continually in such quantity, so as to remove it. For in such places the water does not seem much, nor very high, since it must stay at the level of its source. If that were not the case, it would always become quite high, through that said source which sends over a greater quantity to it; whence however much is taken away, that much re-enters to the same level ... Finally, with these instruments water can be sent for Rome from the Tiber, for Ferrara from the Po, and for other cities from other rivers; or water of any kind for use in private houses, and gardens, with much less expense than when carried by beasts of burden, or by people; because a very few of the men, or horses in their place, that are now needed, would suffice to tum the water screws to raise the water which would then run by itself to the ordained places. And so much for the expenses and the profits. Next, as to damage to health. I should not like it if because of the convenience of the water-screw, and through the greed of men, these should be made too numerous and adapted to the production of rice. It is true that one water-screw alone, worked continuously, would provide all the water required for any large field of that crop. But if that were done in too many places, public damage should no doubt be put before private profit. For the cultivation of rice requires that the fields be always very soft and almost covered with water, in the manner of shallow swamps; and it would be to be feared that such heated humidity, not dissipated by the rays of the sun, would cause putrefaction in the air, and would be to neighboring inhabitants the source of some general and dangerous illness, especially in those places not exposed to northerly winds - for in those [places] governed by these [winds] the humidity of the air is consumed, and no damage of consequence would follow, as is read in Strabo, a very serious author, of Alexandria in Egypt, and of Ravenna in Italy, to which we may add Venice. For though built in the

I 86

History of Science

midst of swamps, for which reason their cities should be subject to very dangerous illnesses, they are nevertheless preserved by the agency of cold and dry north winds that frequently blow in those regions . ... Thus also it comes about that the air of our Piacenza is very healthy, since it is defended from the pernicious southern wind by the mountains and by the woods planted in the terrace of the city wall; and from the north, in very high places toward the Po, it is disencumbered of all humidity and corruption by the north wind that blows almost continually on it ... But awaiting some better occasion I shall say no more about this here, except that the greed of some men is so great that even though no subsidy were proposed, I am sure that they would not rest on that account (provided that for other reasons they held the water-screw suitable to their use), from employing it and making the greatest accumulation of damage possible to them. Two other damaging oppositions were made to me a year ago, and this is that part I spoke of which falls under the charge of government. One [objection] was made by the illustrious Signor Alessandro Archinto in his Council of extraordinary imports into Milan; the other, at Ferrara by a learned gentleman of his Highness. Archinto, who had perfectly understood and considered the mastery of this machine, said: "By my faith, this gives hope that it may be found very useful to the [creation of] meadows; but as to grains, with which this our Magistracy is especially concerned, abundance [of water] will do the opposite. For the terrain that will be converted into meadow may easily be such that grains will be exported with this ease of their cultivation, and what is left will perhaps not suffice for abundance in the State. Yet it would be better, especially for the poor, that they should lack meat and cheese rather than bread. Before Lodigiano had water drawn from the Muzza, it always had an abundance of every sort of grains. Then, when most of it was converted to meadows, because of the provision of water, the [grain] harvest could not be controlled, so that its cities had no bread at all for the whole year. Thus the richer this land is made by imports, the less abundant it has remained in the primary aliment of its own inhabitants." Nevertheless, the resolution of this Council, as I heard later, was that since water is usually much more useful to millet, to beans, and to other March sowings, that are the main staple of life to the poor, so useful an instrument should be allowed to be put in use; and later, at the proper time, remedies should be applied in order that with its aid not too many meadows should be created, out of accord with the condition of other places and the number of subjects of that State. Thus bit by bit there should be a list in writing of everything, as needed, [pertaining to land utilization]. The Ferrara gentleman argued otherwise - that my information being right, so much water might well be raised, not only from the Po, both openly and through subterranean channels, but from all the tributary rivers which increase it and make it very easily navigable, that (since close to Ferrara it divides into many branches which at certain times can hardly be navigated), it would not surprise him if these [branches] would refuse pas-

An Agricultural Economist of the Late Renaissance

187

sage to loaded ships for much longer periods of time, with great inconvenience to the cities above, which make use of the Venetian markets. To which it was replied that this was not much to be feared in a river which by reason of its size is called the King of all others; and the more so since the larger branch of Loredo, which leads to Venice, had never dropped to that lowness of which he spoke. Further, it is only in the autumn and the winter, and when it does not rain much, that the waters in this river are partly lacking. In the spring and summer they are always very abundant, and many are more so than desirable, since the snows and the caverns full of subterranean waters, combined with the cold that retires into caves in the summer (and is the reason that air thickening turns into water), usually maintain the river strong almost always through August; that is, as long as occasion to irrigate the meadows endures ... [Thales made a fortune by cornering the oil market]; this shows us that Philosophers do not lack ways to enrich themselves, provided that they want to. So I hope that the proof [of my machines] will undeceive some people who now have been scoffing at mathematical works, and will reveal to them that scientific men (among whom I confess myself to be the le;ist), if they wish to apply their thought, and use their knowledge with prudence, are in a position to draw honest profit for themselves, and with great benefit to others, from the practice of the sciences; just as ordinary men, with so much more avarice, customarily amass substance which their heirs then do not spend - as perhaps is needful [for scientists to do] according to right reason. 16 V

The passages translated above afford a fair sample of parts of Ceredi's book. I hope they will suffice to attract attention to it on the part of economic historians, as well as historians of science. Ceredi may have been the first author to advocate in print the building of models on different scales as a means of optimizing practical efficiency. Certainly the history of technology must abound in examples of that practice, the most notable being that of Leonardo da Vinci. How much influence Leonardo's unpublished notebooks may have had on writers of the sixteenth century remains a matter of debate, however; and it would be of no little interest to explore the origin and spread of the model-building technique among authors of printed books. Considerations of scale in models are among the many interesting points discussed by Ceredi which had to be omitted here. Of scarcely less interest is Ceredi's account of his own steps to the reconstruction of the ancient irrigating device in a practicable form. He says that it was almost by chance that he was led to this, finding useful clues in various ancient writers, and he remarks that it is usually the case that not everything relevant to a problem will be found in the writings of any one particular man, even

188

History of Science

in those of noted inventors. The implication is that if everything needed were to be found in one place, then the problem would have been solved much earlier. In this instance, there was evidence that Ceredi's particular problem had indeed been solved in antiquity, but that the solution had got lost; hence it was natural for him to seek its parts in ancient writings. As he says, the illusion that Vitruvius had given precise rules for the water-screw doubtless operated to prevent others in the sixteenth century (when there was a great revival of interest in Vitruvius) from experimenting with other designs. Ceredi realized soon enough that the formula of Vitruvius (length to central shaft diameter as 16: 1) was both pointless and impracticable; but he did not conclude, as others probably did, that this meant there was nothing more to be done, and that the water-screw was limited in the height to which water could be raised with it (as Vitruvius said). Rather, he says he found that Pappus suggested the use of a central shaft of any circumference that would not bend under its load, this in tum being the weight of water that the proposed motive power could move, and that Dionisidorus added that the amount of lift was proportional to the pitch of the screw. What texts he may have seen that could justify these attributions, I do not know; but Ceredi comments: "Good God, how briefly and how clearly these two worthy Greeks understood the whole mastery of this useful instrument!" Ceredi ' s book should throw new light on the forms of water-screw used in Italy before his publication, and on the purposes to which they were actually applied. This in tum may help to date the old screws found in Spanish mines, where a copper auger turned within a fixed casing, for it is clear that neither Agricola nor Ceredi had heard of any such device in the mid-sixteenth century, and both had talked to many travelling engineers to learn from them of practices abroad. These investigations belong to the historian of technology, who will find much else of interest in this book that I have not touched on here. Whether Ceredi's patents were put to any extensive use remains open to question. He clearly intended to manufacture a standard screw for sale, and to depend on this rather than on royalties from use for his own income. Whether he lived Jong enough after I 567 to carry out such a plan is not presently known; I am inclined to doubt it, from the mere fact that neither lawsuits nor further publications by him have come to light. The water-screw was, however, used for irrigation along the Po well into the nineteenth century, as shown by records compiled then for the Austrian government during the reign of Maria Theresa and before the liberation of northern Italy from Austrian domination. Quite possibly they were first introduced toward the end of the sixteenth century. They were certainly not created spontaneously by peasants, nor are they likely to have been purchased by landlords and put into operation at their own expense before someone had persuaded them that this would produce a net profit. Ceredi

An Agricultural Economist of the Late Renaissance

189

would hardly have been granted patents if that had been done before I 567. Whether or not his arguments were effective, it hardly seems possible that systematic mechanical irrigation in Italy already existed in that year, though the Archimedean water-screw had long been employed for drainage of swamps, and fanciful diagrams for its application to irrigation were not uncommon. NOTES I 2

3

4

5 6 7 8 9 to

11 12 13 14 15 16

Lynn White, Jr., Medieval Technology and Social Change (Oxford, 1962) pp. 110, 114,116. See Vitruvius, The Ten Books on Architecture, Ir. M.H. Morgan (Cambridge, Mass., 1914; Dover ed. New York, n.d.), p. 295. The standard sixteenth-century illustration is shown on p. 296. See also A.G. Drachmann, The Mechanical Technology of Greek and Roman Antiquity (Copenhagen, 1963), pp. 152-3 for schematic diagrams and a literal translation of the text. / dieci libri del/'Achitettura di M. Vitruvio tradotti et commentati da Monsig. Daniel Barbaro ... (Venice, 1584), p. 462. This edition is a reprint of that of 1567. I have not seen the first Italian edition of 1556, which may have had the same commentary on this passage. The commentary in the Latin edition of 1567 omits the references lo observed screws and to perpetual motion, showing that it was the Italian version that Ceredi had in mind below. A drawing of Leonardo's shows this device with a series of similar spiral pipes wound round the shaft. Ceredi's own illustrations show parallel iron bands around the casing. G. Ceredi, Tre discorsi ... (Panna, 1567), pp. 3-4. Tre discorsi, pp. 5-6. Tre discorsi, pp. 6--7. Tre discorsi, p. 9. Tre discorsi, p. 20. Tre discorsi, p. 21. The braccio was roughly two feet in length. Tre discorsi, p. 26. Tre discorsi, p. 36. Tre discorsi, p . 36. Tre discorsi, pp. 83-85. Tre discorsi, pp. 85-88. Tre discorsi, pp. 9--97.

8

Renaissance Music and Experimental Science

The influence of early modem science on musical theory and practice has not gone unnoticed by historians of music. Professor Claude Palisca has called particular attention to the impact of scientific thought on the music of the 'late sixteenth and early seventeenth centuries. His "Scientific Empiricism and Musical Thought" concludes with this summary: "It would be wrong to conclude from this exposition that the sensualism and freedom of early Baroque music can be ascribed mainly to the liberalizing force of scientific investigation .. .. Scientific thought did reveal, however, the falsity of some of the premises on which existing rationalizations of artistic procedures had rested .... By creating a favorable climate for experiment and the acceptance of new ideas, the scientific revolution greatly encouraged and accelerated a direction that musical art had already taken. Finally, the new acoustics replaced that elaborate conglomeration of myth, scholastic dogma, mysticism and numerology that had been the foundation of the older musical theory, giving it a less monumental but more pennanent base." 1 The reciprocal influence of musical theory and practice upon early modem science, however, has been neglected by historians of science. This is somewhat surprising when we consider that music in medieval education held an equal place in the quadrivium with arithmetic, geometry, and astronomy as the recognized mathematical disciplines. For that reason alone, music might be expected to have remained rather closely associated with mathematics and science, and to have shared in their sudden transfonnations during the late Renaissance. I am convinced that that was indeed the case, and that the origins of the experimental aspect of modem science are to be sought in sixteenth-century Reprinted from the Journal/ the History of Ideas, 31 ( 1970), 483-500, by pennission of the Johns Hopkins University Press.

Renaissance Music and Experimental Science

191

music, just as its mathematical origins have been traced to the ancient Greek astronomers and to Archimedes. In saying this, I have in mind the emergence of physics as a distinct separate study, no longer an integral part of philosophy, after the work of Galileo. The new physics, in which mathematics and experiment replaced logic and authority, was so successful as to become a sort of model for other sciences, at least so far as method was concerned. It is, of course, possible to trace the lineage of physics as far back as one wishes. Any form of intellectual activity that exists in one generation can be found to have some counterpart in the preceding generation, and it is instructive to detect and study these. On the other hand, it is no less instructive to recognize the occurrence from time to time of fundamental changes in patterns of intellectual activity. One way to do this is to select some pattern in modem times, and to move back until an essential element in the pattern disappears, or at least becomes of negligible importance in the writings of men who pursued the activity under study. It is easy to trace modem physical science back in an unbroken line to Sir Isaac Newton, who unquestionably linked experimental evidence in his work with mathematical laws, in a way that is highly characteristic of science today. Many historians of science feel that it is proper to trace this same unbroken line further back, to Galileo. Others, who would pursue it yet further, are soon confronted with definite evidence of discontinuities, to say the least. The most influential historians of science (e.g. Pierre Duhem, Alexandre Koyre) contend that the work of Galileo was merely a continuation of lines of thought laid down in the Middle Ages by men whose work subsequently fell into obscurity until very recent times. In that way a link with classical antiquity is found, through commentators on Euclid and Aristotle who flourished in the Middle Ages. But if Galileo's work was a continuation of medieval thought, there seems to have been some interruption, for it is hard to find any sixteenth-century writers on mechanics whose conception of the relation of mathematics to experimental evidence even resembles that of Galileo. Hence, if we want to go farther back than Galileo in an unbroken line, we have to proceed with caution. Obviously the unbroken line reaching back from the present will end wherever we discover the disappearance from physics of either mathematics or experiment. Now, the sixteenth century marked a notable increase of emphasis on and development of mathematics, ushered in by the publication and translation of ancient Greek works such as those of Euclid, Archimedes, and Pappus of Alexandria. And since mathematics pervades all typically sixteenth-century writings on mechanics, our quest turns automatically into a search for the origins of experiment. But here we shall need some kind of definition of experiment in its characteristically scientific sense.

192

History of Science

For such a definition we can eliminate at once the concept of experiment that is associated with the name of Sir Francis Bacon; that is, the idea that we can arrive at the true explanation of things by the systematic and patient accumulation of observed facts, without any preconceived theory. In Bacon's view, the explanation would ultimately force itself upon us as a result of this procedure. That may be true, though in many mailers it is open to doubt; but whether it is true or not, it has really nothing to do with science. The mere accumulation of data, useful though its results may be, does not constitute scientific knowledge as such. I shall cite a single example in illustration of this. One of Galileo's "Two New Sciences," published in 1638, was the science of strength of materials. There had been an impressive accumulation of information on that subject by earlier builders and engineers. But it was not such knowledge that gave rise to that science, or ever would have given rise to it, though that knowledge can be seen to fit into Galileo's new science. The new science was mathematical in character, and it developed not out of engineering, but out of the theoretical science of mechanics, and it was derived by Galileo from the Archimedean law of the lever: So in eliminating the Baconian definition of experiment, I am not denying the existence from time immemorial of accumulated observational data. Still less am I denying the usefulness of such activities. I mean only that it is not the Baconian type of experimentation that did, or could, account for the rise of modem science. The kind of experimentation which interests historians of physics is the deliberate manipulation of physical objects for the purpose of corroborating by their behavior a definitely preconceived mathematical rule, or for the purpose of discovering a mathematical rule applicable to their behavior. Some physical objects are not amenable to manipulation - e.g., the heavenly bodies - so in their case deliberate selections of precise observations are made, and comparisons of them serve in place of manipulation. Thus some kind of tinkering with external measurements, directly or indirectly, is involved. Such an investigation, with an exact rule in question, for the purpose of seeing whether that rule is obeyed, or of discovering some exact rule, constitutes scientific experimentation. In physics and astronomy, "exact rule" means some mathematical expression; in other sciences, a rule may be considered precise without its being mathematically expressed. But it is always the testing of exact rules by some kind of external activity that distinguishes experimental method in science from various other possible approaches to knowledge and truth. It is therefore natural for historians of science to be interested in discovering when, and under what circumstances, this approach began its uninterrupted prosecution down to the present time. It is equally natural for them to suppose that, so far as physics is concerned, experiment in the sense of manipulations to confirm or establish a law began in mechanics.

Renaissance Music and Experimental Science

193

Proceeding thus, it is tempting to suppose that the first physical experiments were concerned with relatively simple measurements of equilibrium and of motion: for example, with the balancing of weights and the measuring of speeds by comparing distances and times. Suppositions of that kind are so easily made, and so intrinsically plausible that they tend to go unquestioned. It is thus that people reconstruct the invention of the wheel by supposing that it must have evolved from the use of logs or rollers in the moving of heavy objects. Fortunately for those who enjoy that kind of speculation, the invention of the wheel goes so far back that they need not fear contradiction, though they cannot adduce the slightest positive evidence in favor of their unanimous conjecture. A historian who may try to discover the origin of scientific experimentation, however, is less fortunate . In unbroken succession, this really does not go very far back in time; not even as far back as the invention of printing, let alone of writing. Hence he may expect to see any assumption or conjecture he makes about the origin of scientific experimentation challenged, and he is obliged to look for something that can be supported by some kind of recorded evidence. With these things in mind, let us take a new look at the idea that scientific experimentation began in mechanics. A very simple and fundamental law in mechanics is the law of the lever. Of course, that law may have been discovered by a Baconian accumulation of observations, but then again it may not. The law is a mathematical statement, and it is perfectly possible that nobody ever knew it in that form before Archimedes. Levers may have been used long and widely without their exact law having been known, and that law may have been deduced by someone who never used a lever, or at least never experimented with it. Certainly Archimedes made no use of experimental induction in his proof of the law of the lever, nor do his postulates suggest an experimental origin. I say this not to suggest that there was no scientific experimentation in antiquity, but to show that the most natural conjectures about its role in the origin of mechanics may be quite misleading. In any event, the law relating distances and weights for the lever was both discovered and mathematically formulated so long ago that it would have been simply silly to perform any experiments to verify it as late as the sixteenth century. Not only that, but the law of the lever is so simple and so far-reaching, once it is known, that there was also little point in performing actual experiments to determine the laws of any of the other simple machines. They could be much more effectively discovered and demonstrated mathematically from the law of the lever. And that, I am quite certain, is exactly the attitude that was taken by writers on mechanics in the sixteenth century. Reliance on mathematical reasoning had by then driven out of physics any feeling of need for experimentation that may once have existed. We shall see

194

History of Science

presently what the analogous situation was in music. The reason I am quite certain of the situation in mechanics is the following. Pappus of Alexandria, writing in the fourth century of our era, had derived mathematically a law for the force required to drive or draw a heavy body up an inclined plane. His law was quite mistaken, because he had employed a false assumption, but the mathematical derivation was very ingenious and complicated, and it had the lovely aura of remote antiquity to recommend it to sixteenth-century mechanicians. From the theorem of Pappus, there easily followed a law for the equilibrium of bodies suspended on different inclined planes, also false, of course. Now, it happened that the correct law of equilibrium on inclined planes had been stated in the thirteenth century by Jordanus Nemorarius, who did not know the work of Pappus. In 1546, Niccolo Tartaglia published the medieval theorem, with some improvements in its purported proof. So all later writers were confronted with two different laws, each of which was accompanied by an attempted mathematical demonstration. If ever there was an occasion for experimental test, this was it; moreover, the deciding experiment would have been very easy to perform. But that is not what happened. In 1570, Girolamo Cardano, who was certainly familiar with the correct medieval theorem and who in all probability also knew the work of Pappus, ignored them both and published a brand-new law for inclined planes, which had to be in error, since the medieval theorem was correct. Seven years later Guido Ubaldo del Monte published the first comprehensive work on mechanics. An astute critic of his predecessors, he certainly knew the correct theorem of Jordanus, and probably also knew the erroneous theorem of Cardano; yet he adopted in his own book the false (but ancient and elegant) theorem of Pappus. These events are hardly explicable if the idea of experimentation in mechanics, to test a preconceived mathematical rule or to discover a new one, is assumed to have been prevalent in the sixteenth century. I might add that even in the later work of Galileo we find evidence of the use of experiment only to confirm a preconceived mathematical law, and not of its systematic use to discover new laws. That step came after his time, as a logical extension of his work. But we are interested in tracing the probable origin of Galileo's application of experiment (in the scientific sense) to mechanics, which means that it is time to tum to the history of music. In the sixteenth century, music and mechanics were more obviously closely related sciences than they are today. Tartaglia wrote in his revised preface to the first vernacular translation of Euclid ever published: "We know that all the other sciences, arts and disciplines need mathematics; not only the liberal arts, but all the mechanical arts as well .... And it is also certain that these mathematical sciences or disciplines are the nurses and mothers of musical sciences,

Renaissance Music and Experimental Science

I 95

since it is with numbers and their properties, ratios, and proportions that we know the octave, or double ratio, to be made up of the ratios 4:3 and 3:2, and it is similarly that we know the former [that is, the interval of the fourth] to be composed of two tones and a [minor] semitone, and the latter [that is, the perfect fifth] to be composed of three tones and a minor semitone. And thus the octave (or double) is composed of five tones and two minor semitones; that is, a comma less than six tones; and likewise we know a tone to be more than 8 commas and less than 9. Also, by virtue of those [mathematical] disciplines, we know it to be impossible to divide the tone, or any other superparticular ratio, into two equal [rational] part [in geometric proportion], which our Euclid demonstrates in the eighth proposition of Book VIII." 2 But if, about the year 1550, the sciences of music and mechanics were alike in their purely mathematical character, the relations of those two sciences to the practical arts that bore the same names were totally different. Musical theorists were in possession of mathematical rules of harmony that they believed must be very strictly followed in practice. It would be unthinkable for musicians to depart from those rules, lest the very basis of music be destroyed. Musical theorists, moreover, received a good deal of respectful attention from musical practitioners and even had a certain authority over them. This was certainly not the case with theorists in mechanics. Musicians quite frequently studied under musical theorists, but engineers did not study under mechanical theoreticians. And if there were any theorists of mechanics who believed that their rules must be strictly followed in practice, this would have been in the sense that failure to observe the rules would result in wasteful use of materials or in the collapse of buildings, not in the ruin of architecture. There was a further difference between the arts of music and mechanics in the sixteenth century, a difference that probably has a bearing on the events which (in my opinion) led to the origin of experimental physics. This was the fact that commencing about 1450, and quite markedly after 1550, musical practice underwent fundamental changes, while mechanical practice did not. Those changes in musical practice brought about a real need to expand or alter musical theory, and with it a need for critical examination of its basis and its claims to correctness. No such need was felt by engineers. It was just as easy to test the mathematical rules of music in practice as it would have been to test those of mechanics; the difference in applying or neglecting such a test lay only in the feeling of need. Furthermore, the tests could be carried out in music with a great deal more accuracy, for no sixteenth-century mechanical measurement came even close to the precision of the trained ear of a musician. I believe that that may very well still be true today, if we restrict ourselves to mechanical measurement and do not bring in electronic devices.

196

History of Science

Finally, one might go so far as to say that the only possible means of detecting errors in, and ultimately of overthrowing, incorrect musical theories, was the appearance of experimental evidence against them. This was perhaps not equally true of errors in theoretical mechanics, which were in fact corrected by the detection and elimination of certain false assumptions, rather than by the revision of the whole theoretical structure of the science. But a discussion of that would lead us too far afield. The heart of ancient musical doctrine was an arithmetical theory of proportion credited to the Pythagoreans. The very existence of musical consonances and dissonances was ascribed in it to certain numbers, related as ratios. That is, the cause of consonance itself was thought to be the so-called sonorous numbers. According to the Pythagorean rule, the musical intervals of the octave, fifth, and fourth were governed by the ratios 2: 1, 3:2, and 4:3. Music owed the possibility of its existence to these ratios, which are made up of the smallest integers. Here we have a marvelous example of the use and abuse of mathematics and experiment. It is said that Pythagoras discovered these ratios as a result of his having noted, in good Baconian fashion, the different tones given out by hammers of different weight when striking an anvil. The story is certainly apocryphal, but observations of string-lengths would have led to the ratios. Having obtained their simple numbers in this or some other empirical way, ancient theorists put them in the place of any further experimentation, and derived from them an elaborate theory of musical intonation. In time, the arithmetical theory was seen as transcending in authority its original source in pleasant sensation. The consequences were not serious for a very long time, but by the late sixteenth century it was no longer possible to maintain the ancient idea of simple numerical ratios as the cause of harmony. We must pause to see how that came about. Ancient Greek music, though it bore the name "harmony," was largely innocent of harmony in its present musical sense. Singing was chiefly homophonic, and when it was accompanied by instruments, they were played in unison with the voice, or, at most, in the octave. The fitting-together that was implied by the word "harmony" was not a fitting-together of different melodies or of parts sung simultaneously, as with us, but a fitting-together of succeeding notes so as to preserve consonance within the recognized tonoi, ancestral to the medieval modes, which differed in an essential way from our keys. These facts, coupled with the limited range of the human voice, and the practice of remaining within the selected mode during a given song, made it possible to decide on the intervals that were to divide the octave without imposing limitations on the composition of music. By the thirteenth century, however, the simultaneous singing of two or three airs had long been in vogue, and thereafter much ingenuity was exercised in the

Renaissance Music and Experimental Science

197

composition and arrangement of motets in such a way that separate voices might interweave without clashing. The use of perfect fourths and fifths exch,1sively as chief accents, or points of repose, fitted in very nicely with this purpose, and the momentary appearance of imperfect consonances, as thirds and sixths had come to be regarded, did not disturb the ear so Jong as they were not carried to excess. In time, however, deliberate violation of the ancient rules began to be attractive; the ears of singers and of listeners came to recognize a sort of general harmonic flow, though the form of composition remained polyphonic. Along with this came the development and multiplication of instruments, used at first to accompany voices, but later coming to be played together with or without voices. Instruments introduced a new element, because unlike voices, most instruments are incapable of producing whatever note the player desires with the exactitude of the human voice. Lutes, recorders, and viols, for example, unlike trombones and the later violins, are limited to definite sets of notes by fretting or by the positions of wind-holes. Organ pipes emit specific notes that cannot be varied in pitch by the organist, just as the harp and most of the keyboard instruments are governed by fixed string-lengths. Inevitably, problems arose over the proper tuning of instruments, particularly when different kinds of instruments began to be played together, which brings us to the sixteenth century. The fretting of early stringed instruments had led quickly and naturally to tempered scales for them . Octaves simply must be in tune; even the untrained ear cannot tolerate much variation for the octave, which is heard instinctively much the same as unison. Intervals of the fourth and fifth together must also make up the octave, and while it is nearly as easy to tune perfect fourths or fifths as it is perfect octaves, you cannot make several strings agree over a range of two or three octaves by tuning them in perfect fourths or fifths. There is that plaguey "comma" that Tartaglia mentioned, and it has to be divided up in some way. For Jutes and viols, this is accomplished by tempering, usually by making each successive fret interval 17/ I 8ths the length of the preceding lower fret, a rule advocated by Vincenzo Galilei late in the sixteenth century. Organs, however, seem to have been tuned by the mean-tone system from an early period, certainly by the fifteenth century. Recorders, on the other hand, were generally tuned in C or F, depending on size, in just intonation. But temperament, meantone tuning, and just intonation do not make use of precisely the same intervals. The nature of the difficulty that gave rise to those various intervals is purely mathematical in a sense; the numbers 1, 2, 3 ... are very useful for counting, but they do not form a continuum. Musical sounds do form a continuum, and hence the whole numbers, even when they are formed into fractions, have a limited application to musical sounds. The set of fractions that will divide a given

198

History of Science

octave into seven different tones will obviously divide a different octave into seven analogous tones; but if we give them names in the ordinary way, those which bear the same names will not all be in unison or separated by exact octaves. As Tartaglia remarked, and as the ancients knew, the ratio 2:1 for the octave and the ratio 3:2 for the fifth implied the ratio of 9:8 for the whole tone between the fifth and the fourth . This in tum implied too small a value for the exact semitone. As Simon Stevin, the Flemish counterpart of Galileo, wrote, "The natural notes are not correctly hit off by such a division. And although the ancients perceived this fact, nevertheless they took this division to be correct and perfect, and preferred to think that the defect was in our singing - [which is] as if one should say the sun may be wrong, but not the clock. They even considered the sweet and lovely sounds of the minor and major third and sixth, which sounded unpleasantly in their misdivided melodic line, to be wrong, the more so because a dislike for inappropriate numbers moved them to do so. But when Ptolemy afterwards wanted to amend this imperfection, he divided the syntonic diatonic in a different way, making a distinction between a major whole tone in the ratio 9:8 and a minor whole tone in the ratio to:9, a difference that does not exist in nature, for it is obvious that all whole tones are sung as equaJ."3 By "sung," Stevin meant "sounded" in general, and his final remark is rather an exaggeration. Regarding the musical string as a continuum, Stevin announced flatly that rational proportions had nothing whatever to do with music, and declared that the proper division of the octave into equal semitones was at the twelfth root of two. It is in this way that we tune pianos today; it enables us to play in all keys without seriously disturbing the ear in any. His conclusion was, however, not experimental but purely mathematical; one might say that it is only accidental that it opened the door to modem harmony. I shall say no more about it here except to mention that Stevin, in a characteristic aside, considered the whole problem to have arisen from an inadequacy of the Greek language: " ... the Greeks were of the most intelligent that Nature produces, but they lacked a good tool, that is, the Dutch language, without which in the most profound matters one can accomplish as little as a skilled carpenter without good tempered tools can carry on in his trade."4 For Galileo, the book of Nature was written in mathematics; for Stevin, the book of mathematics was written in Dutch. And so, finally , we come to the sixteenth-century controversy over musical tuning that created modem music, with experimental physics as a by-product. In one comer stood the mathematical theory of antiquity, which took sonorous number as the cause of concord and asserted that number must govern stringlength, or the placement of wind-holes, or the like. In the other was the human ear, with that curious taste for pleasant sounds that goes along with - or at least

Renaissance Music and Experimental Science

199

once went along with - the composition and performance of music. They were in conflict, as Slevin observed, and with or without the Dutch language, that conflict had to be resolved. The question was, which was to be master, numbers or sounds? The conduct of the debate is instructive, because it is symbolic in many ways of the debates that created the modern world, .)nd of many that are still going on in it. For it was at once a battle of authority versus freedom; of theory versus practice; of purity versus beauty, and so on. Curiously enough, the issue was partly decided by antiquity; that is, by the recovery of an ancient treatise that was largely neglected by theorists before the sixteenth century. Nothing helped a good cause in that century like the discovery that some ancient writer had already thought of it. Copernicus was careful to mention some ancient writers, e.g., some Pythagoreans, who were said to have subscribed to the motion of the earth. It was of great help to musicians in their struggle for emancipation from the syntonic diatonic tuning of Ptolemy to be able to cite the view of Aristoxenus that when all was said and done, the ear of the musician must prevail. I should like to remark at this point that in a paradoxical way, the struggle to free and broaden music in our own time is a mirror image of the struggle that freed and broadened it in the seventeenth century. Electronic computer music has much in common with the program of the late Renaissance conservatives who fought to preserve the mathematical beauty of sonorous number and superparticular ratios against the mere pleasure of the human ear. The argument of Gioseffo Zarlino, roughly paraphrased for its philosophical content, was that there could be only one correct tuning, established by the mystic properties of numbers, and if that put limitations on instrumental music, then so much the worse for instruments. The voice must govern instruments, and must in tum be governed by divine proportions of numbers. It is amusing that the argument for the divine right of theory, used at one time to impose narrow bounds on music, is now used to support the abandonment of all bounds, and with them all mere sensory criteria for music. Theory before pleasure, once conservative, is now avant-garde. On purely theoretical grounds, Zarlino advocated the extension of the Pythagorean ratios up to the number 6, from the ancient 4. This would allow as consonances, in addition to the fourth and fifth, the major sixth as 5:3, the major third as 5:4, and the minor third as 6:5. He was also willing to allow the minor sixth as 8:5, but that finished the list; no other consonances were admissible. The number 6 was, after all, the first perfect number - that is, the lowest number that was equal to the sum of its factors - so the sonorous numbers could safely be extended that far, but no further. Zarlino's Harmonic Institutions, published in I 558, did away with all rivals of the syntonic diatonic tuning, such

200

History of Science

as temperament and the mean-tone system, as theoretically unjustifiable. His treatise on composition thus limited music to polyphony as before, though this was given a somewhat larger range. Method had been perfected. Mathematics was saved by the senario, as Zarlino called his six-based ratio system, but harmony was stillborn if his system prevailed. About five years later, Zarlino's scheme was subjected to criticism, both mathematical and experimental, by G.B. Benedetti. Benedetti has long been recognized among historians of science as the most important by far of the Italian precursors of Galileo in mechanics, but only recently Professor Palisca has called attention to his achievements in the physics of music. These are contained in two letters written to Cipriano da Rore at Venice, probably in 1563, and published in I 585 together with Benedetti 's more famous contributions to mathematics and physics. First, he showed by means of examples that a strict adherence by singers to Zarlino's ratios for consonances could result in a considerable change in pitch within a few bars. Benedetti argued from this that singers must in fact make use of some kind of tempered scale in order to preserve even the most elemental musical orthodoxy with regard to pitch of opening and closing notes. Next, Benedetti proceeded to examine the problem from the standpoint of physics. Instead of relating pitch and consonance directly to numbers, that is to string-lengths and special ratios, he related them to rates of vibration of the source of sound. Consonance, he asserted, was heard when air waves produced by different notes concurred or recurred frequently in agreement: dissonance, when they recurred infrequently or broke in on one another. Thus strings vibrating in the ratio 2: 1 would concur on every other vibration; in the ratio 3:2, on every sixth vibration, and so on. The direct cause of consonance was thus related to a physical phenomenon, which in tum bore a relation to certain numbers; but the numbers as such were not regarded by Benedetti as the cause of the phenomenon of consonance. Pursuing this reasoning, he suggested an index of consonance, formed by multiplying together the terms of the ratio. The smaller this product, the greater the consonance. Viewed in this light, consonance and dissonance were not two separate and contradictory qualities of sounds, but rather they were terms in a continuous series without sharp divisions. A similar view was soon to emerge for heat and cold, speed and slowness, and other ancient physical concepts. Also, by Benedetti's index, it followed that the traditionally abhorred subminor fifth, with the ratio TS, was in fact a better consonance than the minor sixth, with the ratio 8:5, which Zarlino himself had allowed even though it lay outside his senario. Obviously this was not just a further modification of the old theory, but a fundamental attack on the very basis of that theory.

Renaissance Music and Experimental Science

201

In framing these ideas, Benedetti had had recourse to experiment; that is, to the deliberate manipulation of physical equipment in order to test a preexisting mathematical theory. The physical equipment consisted of a monochord, on which a movable bridge permitted the division of a single stretched string into any desired ratio, and allowed the measurement of the ratio to a fair degree of accuracy by measurement of the lengths of the two sections, which could then be sounded simultaneously by plucking in order to determine the tonal effect. Benedetti 's basis for asserting that string-lengths were proportional to frequencies of vibration is not stated; it may or may not have been experimental. But the appeal to physical results over the authority of a mathematical theory that had been developed over many centuries was in any case a novel event. Experiments similar to Benedetti 's, though probably not directly inspired by him because he did not publish them, will presently be shown to have been intimately connected with Galileo's work in mechanics. First, though, I want to comment on the step-by-step nature of the process, for Benedetti 's results were in an important sense only a small start toward the modem physical science of musical acoustics. Benedetti 's brief writings on music did not remove mathematics from musical theory; they merely changed its role. For his contemporaries, as for the ancients, numbers were the cause of harmonious sound in a totally different way from that in which Benedetti saw them as related to that cause. Others regarded numbers as ruling the nature of sound; Benedetti formulated a physical theory of sound that was capable of explaining the association of certain numbers or ratios with certain tonal effects. But Benedetti himself was not a composer, and therefore did not concern himself with the possibility or desirability of expanding the range of acceptable tonal effects. He seems not even to have noticed the phenomenon of partial vibrations. These things were soon to come. What Benedetti contributed was in the nature of a new explanation of known effects, rather than a basis for the exploration of new ground. Thus to the question, "Why is the fifth more harmonious than the major third?" the classical answer was of this form: "Because the ratio 3:2 contains two sonorous numbers, and the ratio 5:4 does not." Zarlino's reply might have been: "The ancients were mistaken: both are equally harmonious, for all numbers within the senario are sonorous numbers. But since sonorous numbers are the cause of harmony, the slightest departure from the ratios named must be avoided, or dissonance will result." Benedetti's reply would have been, "Because harmony proceeds from agreement of vibrations, which occurs every sixth time for the fifth, and only ever twentieth time for the major third." But to the question, "What can we do to widen the scope of harmonious music?" no one gave an answer. To Benedetti ' s opponents, the question was unthinkable; and to Benedetti, the first man

202

History of Science

who would have been able to answer, the question never occurred; he was a scientist and not a musician. It is virtually certain that Zarlino, who opposed any tempering of the vocal scales, never heard of Benedetti's demonstration that such tempering was necessary in practice and theoretically justifiable. But Zarlino was opposed in print by a former pupil of his, Vincenzo Galilei, father of Galileo. In 1578 Galilei sent to Zarlino a discourse in which the departures of practicing musicians from the tuning recommended by Zarlino were stated and defended; there was no escape from a tempered scale in the music of the late sixteenth century. Zarlino paid no attention to this attempt of Galilei's to refute his theory; indeed, he appears to have attempted to suppress the printing of his former pupil's book, which nevertheless appeared much expanded, in 1581. Galilei, who had long followed Zarlino's teaching, began to question it only after he had learned from Girolamo Mei, the best informed man in Italy on the music of the ancients, that among the ancients themselves there had been musicians who questioned the absolute authority of mathematical theory over sense. It was Mei who invited Galilei to put to actual test certain doctrines of the old theory. The result was Galilei's abandonment ofZarlino's teaching and his publication of the contrary view in favor of tempered scales. Zarlino, far from accepting these criticisms, counterattacked in 1588 with his final book on music theory, the Sopplementi musicali. Though he did not mention Galilei by name, he quoted from his book and identified him as a former pupil. Galilei lost no time in replying; in I 589 he published a little volume which he dedicated, with obvious sarcasm, to Zarlino. The first part of this book is merely polemical and personal, but the balance of it is of the greatest interest with respect to the beginnings of experimental physics. Galilei was, first of all, outspokenly against the acceptance of authority in matters that can be investigated directly. He did not even accept the idea that some musical intervals were natural, or that any were consonant by reason of their being capable of representation by simple proportions. Any sound was as natural as any other. Whether it pleased the ear was quite another matter, and the way to determine this was to use the ear, not the number system. Zarlino had extolled the human voice as the greatest musical instrument, being the natural one, and concluded that musical instruments, as artificial devices, were bound by the laws of the natural instrument. Galilei replied that instruments had nothing to do with the voice, and made no attempt to imitate it; they were devised for certain purposes, and their excellence could be determined only by the degree to which they successfully carried out those purposes. Mathematics had no power over the senses, which in tum were the final criterion of excellence in colors, tastes, smells, and sounds.

Renaissance Music and Experimental Science

io3

For a tuning system, Galilei advocated an approximate equal temperament as detennined by the trained ear. Here we should recall that Stevin, about the same time, went still further; he boldly declared that ratios and proportions had nothing to do with music and that the proper and ideal intervals were those given by the twelfth root of two. Stevin, like Benedetti, failed to publish his discussion of music, but it is likely that he was influential in his own country. It is in Galilei's final refutation of Zarlino that we find for the first time the specific experimental rejection of sonorous number as the cause of consonance. Referring to the celebrated opinion of Pythagoras that the small-number fractions are always associated with agreeable tones, Galilei stated first that he had long believed this, and then said that he had detennined its error by means of experiment. The ratios 2: I, 3:2, 4:3 will give octaves, fifths, and fourths for strings of like material and equal tension but of lengths in these ratios, or for columns of air of similar lengths. But if the lengths are equal and the tensions are varied, then the weights required to produce the tensions are as the squares of these numbers. In a later, unpublished manuscript, he added that the cubes would have to be calculated where volumes detennined the sounds. Thus, he said, the ratio 9:4 was just as closely associated with the fifth as the ratio 3:2; and by implication, so was the ratio 27:8 - ratios which were simple abominations to the Pythagorean and Ptolemaic numerologists who believed in sonorous number as a cause. In addition, Galilei experimented with strings of different materials and weights, discovering that unison cannot be consistently obtained between two strings if they differ in any respect whatever. Stringing a Jute with strings of steel and gut, he found that if these were brought into agreement as open strings, they would not be in perfect agreement when stopped at the frets. The removal of number magic from musical theory perfonned a doubly significant service. First, it deprived number of causal properties; second, and perhaps more important, it called attention to the real significance of number as it referred to a specific dimension, such as length, surface, and volume. The trouble with the older musical theory was not only that it failed to provide anything fruitful for expanding musical practice, but also that by purporting to give an explanation in mystical numerical tenns, it tended to prevent the direct study of the actual application of number to the material instruments of music. A similar double error pervaded most of physics; not only did the erroneous physical principles of Aristotle fail to agree with observation, for example with regard to falling bodies, but their very existence discouraged the search for correct principles.5 The experiments of Vincenzo Galilei, like those of Benedetti, were true scientific experiments in the sense in which we have defined that tenn: the manip-

204

History of Science

ulation of physical objects for the purpose of verifying a mathematical rule preconceived as applicable to their behavior, or for the purpose of discovering a rule involved in that behavior. Whether music inspired the first such experiments in an unbroken line to the present remains to be seen. If others preceded them, it would seem that they must have been independent of them, for it is hard to see how Benedetti and Galilei might have attacked musical theory experimentally under the influence of some earlier experimental attack on some other mathematical theory. And to establish the probable continuity of their work with the application of experiment to mechanics, which has had such profound consequences for all science, we may now tum briefly to Galileo himself. Galileo was enrolled at the University of Pisa in I 58 I, his father having selected for him a medical career. Two years later he became deeply interested in mathematics under the guidance of a family friend, Ostilio Ricci. The chair of mathematics at the University appears to have been vacant when Galileo was there. His new interest distracted him from the study of medicine, and in I 585 he left the University without a degree. During the next five years he lived mainly in Florence, giving some private instruction in mathematics and commencing on researches that obtained for him the chair of mathematics at Pisa in I 589. Now it was precisely during those years, and particularly in I 588-89, that Vincenzo Galilei is most likely to have carried out many of the experiments in refutation of Pythagorean music theory that he never published, but that survive in manuscripts among Galileo's papers. It seems to me extremely likely that Galileo was himself involved in his father's experiments, some of which he appropriated and published many years later in his Two New Sciences. Galileo was an accomplished amateur musician, instructed by his father, and as a young mathematician he could hardly have remained indifferent to what his father was doing in the measurement of tuned strings and the examination of Pythagorean musical numerology. Thus the conception of experimental verification of mathematical laws in physics, which is often illustrated in Galileo's books, may very well have been inspired by his father's work during the years in which he had just left the university and was developing his own mathematical skills. It was precisely at this time that Galileo applied experiment to an ancient theory; he devised a hydrostatic balance and reconstructed the reasoning of Archimedes in the famed detection of the goldsmith's fraud in making the crown of Hiero of Syracuse. If we consider the nature of Vincenzo Galilei's experiments, we gain also a possible clue to Galileo's early interest in the pendulum, a device with which his future work in physics was to be intimately connected. One of the most interesting of his father's experiments concerned the determination of the numbers associated with particular musical intervals under different conditions of

Renaissance Music and Experimental Science

205

the production of tones. You will recall that the celebrated ratios 3/2 for the fifth, 4/3 for the fourth, and 2/1 for the octave were related to string lengths, given strings of the same material and diameter. It was Vincenzo Galilei who remarked for the first time, in his final diatribe against Zarlino, that the same ratios did not hold at all for the weights that must be used to stretch a given string to the equivalent pitches; here, the inverse squares of the ratios would hold, and to raise the pitch an octave, where half the length would suffice, quadruple the weight was required. The experiment is not particularly difficult, but however it is carried out, an observer can hardly escape the phenomena of the pendulum. This is obvious if one thinks of suspending two strings of equal length and size, and weighting one with four times the weight attached to the other. In order to set up the experiment, let alone to elicit a tone, say by plucking, the strings and their weights are bound to be set in at least a slight swinging motion. If Vincenzo Galilei used instead the apparatus commonly illustrated for the false Pythagorean rule, the pendulum effect would also occur; here, parallel strings are stretched over a flat bed, weights being applied to the ends hanging down from a tenninal bridge. The application of different weights to these relatively short vertical strings would set them swinging and would invite attention to the lengths and periods of oscillation. Galileo's observations of the pendulum were already reflected in his earliest contribution to the analysis of motion and of the law of falling bodies. About 1590 he composed a treatise on motion, which was never published. In this treatise he attempted to analyze the motions of bodies on inclined planes, and this was done in a manner that treated inclined planes as tangents to a circle. By 1602, we find him discussing the motions of bodies along arcs and chords of circles, and it is quite evident that he and his patron, the Marquis Guido Ubaldo del Monte, were doing some experimental observations of those motions. Galileo's decision not to publish his first treatise on motion seems to be related to his discovery, by experimental test, that the rules he had deduced for speeds on inclined planes were incorrect. By 1609, after various false starts that can be traced in his letters and notes, he had got to a point where he planned to publish a systematic treatise on motion, having ironed out the earlier false assumptions by means of a combination of mathematical reasoning and simple experimental tests. But in 1609 his attention was diverted to the newly invented telescope, and as a result his science of motion was not published until many years later, in 1638. As one would expect, Galileo's first use of experiment in physics was limited to rough and simple tests to see whether the rules he had worked out mathematically were actually followed. As time went on, he took a greater interest in devising experiments to corroborate or illustrate his science of motion. The

206

History of Science

extent to which he actually conducted experiments such as those he described late in life is a matter of debate, but there is no question that they served as models to his pupils and his readers. Such I believe to have been the origin of experimental method in physics in the sense of the unbroken thread leading from the late sixteenth century to the present. The first conscious experiments to test a preexisting mathematical theory were probably the musical experiments of Benedetti and Vincenzo Galilei. They were extended into mechanics by Galileo, whose pupils Castelli and Torricelli carried them on over into hydraulics and phenomena of air pressure; refined by Pascal and Boyle, experiment led to the gas law. Boyle's law is said to have been the first scientific law to be experimentally discovered. Yet Vincenzo Galilei's discovery that the weights required for producing tensions corresponding to given pitches are as the inverse squares of lengths must have been empirical. In any event, the manipulation of physical equipment set up to test a mathematical law had come much earlier than Newton, even earlier than Galileo; and it came because of the conflict between numerology and physics in the field of music. The desire for ever-increasing precision in experiment, a necessary counterpart of the new method in science that sought certainty in the reconciliation of sense experience and mathematics, was strikingly evidenced by Marin Mersenne, the friend of Descartes and the spokesman of Galileo in France. Mersenne is best known for his Harmonie Universe/le of 1636-37, a monumental work on the theory and practice of music in which the role of the science of mechanics is much emphasized. It was Mersenne who carried out with great care the experiments on falling bodies descending along inclined planes which Galileo mentioned but only roughly described in his books. 6 The intimate linkage of music to mechanics, mathematics, and experiment that is made in Mersenne's work tends to increase the probability of my general thesis concerning the origin of experimental physics. I should like to conclude with a remark about a totally different relationship between Renaissance music and modem science, in an area where experiment in the sense of deliberate physical manipulation is not possible; that is, in astronomy. This second musical approach to science found expression in the work of Johannes Kepler, an almost exact contemporary of Galileo. Kepler was deeply motivated by precisely the kind of faith in the domination of the universe by numerical harmonies in astronomy that Galileo's father fought to destroy in music. Little interested in mechanics, Kepler devoted his life to the discovery of that Pythagorean music of the spheres that ought to be produced by the celestial motions. His goal turned out to be as illusory as that of the rule of musical harmony by exclusively small-number fractions. In the course of his

Renaissance Music and Experimental Science

207

work, however, Kepler discovered the mathematical laws that do describe the planetary motions. Those laws destroyed the heavenly spheres themselves, replacing circles with ellipses and uniform motions with varying speeds, much as physical laws destroyed the Pythagorean musical ratios. With varying distances from the sun, the planets were freed from monotony; but they could only grind out, on Kepler's most favorable calculations, a dreary succession of uninteresting scale-passages. Nevertheless, in his unending quest for better results and his devotion to precision of measurement, Kepler established a new physics of the heavens as firmly in mathematics and observation as Galileo founded a new terrestrial physics on the same solid base. The fountainhead of Renaissance music was thus at least partly responsible for the emergence not of experimental science alone, but of a whole new approach to theoretical science that we now know as mathematical physics. It is well known that Sir Isaac Newton accomplished the synthesis of Kepler's astronomical laws and Galileo's science of motion, setting the pattern of modem physics that unifies terrestrial with cosmic events. In retrospect, it seems not to have mattered greatly which side the scientist took, provided only that he was deeply interested in the musical controversies of the Renaissance. What did matter was that the older mathematical theories of music were capable of exact test by experiment and precise observation. That was a key which, in the hands of Galileo and of Kepler, opened the door to modem science. NOTES

C. Palisca's article in Seventeenth Century Science and the Arts, ed. H.H. Rhys (Princeton, 1()61), 91-137. 2 Euc/ide ... diligentemente rasettato ... per ... Nicolo Tarra/ea (Venice, 1543), has a shorter and incorrect reference to musical proportions. 3 The Principal Works of Simon Stevin. 5, Music, ed. A.O. Fokker (Amsterdam, 1966), I

431-33. 4 Ibid., 433. 5 C. Palisca, 130: cf. V. Galilei, Discorso intorno a/le opere di Gioseffo Zarlino (Venice, I 589; facs. repr. Milan, 1933), 127-28 6 Mersenne, Harmonie Universe/le (Paris, 1636--37), I. 112; Les nouvel/es pensees de Galilee (Paris, 1639), 188.

9

Music and Philosophy in Early Modern Science

Those acquainted with my writings may be surprised that I have included philosophy in the title of this article, since I have neglected it in the past. That was because I preferred to leave philosophical analyses of the work of Galileo, and of the science of his time, to colleagues who regard philosophy as the very basis of the Scientific Revolution of the seventeenth century. It is widely held that all roots of that event are to be found in the writings of ancient and of medieval natural philosophers. The present volume makes it advisable for me to say why I dissent. From antiquity until the Scientific Revolution, science remained only one branch of philosophy, defined by Aristotle as the understanding of natural phenomena in terms of causes hidden from our senses. Greek philosophers had begun that enterprise earlier, in ways examined critically by Aristotle before he coined the word "physics" to designate the science of nature. He later investigated its principles in a book he called "first philosophy," but which was renamed "metaphysics" by his later editors. Aristotle's books on physics, on the heavens, and on meteorology constituted the essential basis of all natural philosophy throughout the Middle Ages, and until nearly the end of the Renaissance. From the very beginnings of universities, mainly in the twelfth and thirteenth centuries, Aristotelian natural philosophy dominated all education in science. Astronomy was usually taught by professors of mathematics, rather than by natural philosophers, probably because Aristotle's book on the heavens, De cae/o, dealt not with astronomy but only with cosmology. Whether astronomy was a science under Aristotle's definition is questionable, for about 150 B.C. the Greek astronomer Hipparchus showed that the earth cannot be at V. Coelho (ed.), Music and Science in the Age o/Galileo, 3-16. © 1992 Kluwer Academic Publishers. Printed in the Netherlands. Reprinted by permission of Kluwer Academic Publishers.

Music and Philosophy in Early Modem Science · 209 the exact center of the sun's apparent motion, as required by all cosmologists, Aristotelians and Platonists alike. A compromise, attributed to Geminus, was soon reached by which astronomers would refrain from considering causes of celestial motions, contenting themselves with framing mathematical hypotheses in accord with the measurements and leaving causal explanations to philosophers. The compromise went unchallenged until the Copemician revolution. 1 Astronomy without causal explanations was philosophically a merely practical discipline; it was not, strictly speaking, a science at all. While recognizing the existence of knowledge gained through practice, Aristotle explicitly excluded it from truly scientific understanding, or episteme. For practical knowledge he reserved the distinguishing name, tekne. Astronomical knowledge among the Greeks at the time of Aristotle was pitifully meager. But by the time of Hipparchus it had been enlarged by records of Babylonian observations extending back some four centuries or more. Greek astronomers systematized those records, putting themselves in a position to challenge old cosmological speculations, on the solid ground of careful measurements. That threat to philosophers was nipped in the bud by the compromise of Geminus; and thus the first of sciences in the modem sense was expelled from the domain of true science as defined by Aristotle. But beginning from the time of Kepler and Galileo, physics and astronomy soon became a single unified science in the modem sense of the word, while the ancient philosophical separation of physics from useful knowledge vanished from the scientific scene. Hence, to look for roots of modem science in natural philosophy before the Copernican period is semantically an idle enterprise, the meaning of the word "science" being different before and after the seventeenth century. Yet roots of Galileo's physics did exist; those are to be found not in past philosophy but in the practices of musicians of his own time, just as roots of Kepler's cosmology are to be found in music theory. The exploration of musical roots of the Scientific Revolution, hitherto relatively neglected, falls naturally within the province of the present collection of essays. Of course, natural philosophy by no means ceased to dominate science in the universities, which were conservative institutions from the very beginning. Even outside the universities, where most of the action took place in the Scientific Revolution, a kind of counter-revolution was led by Rene Descartes. He offered a new natural philosophy to replace that of Aristotle, and at the same time to remedy a serious defect in Galileo's physics, as Descartes saw it, because it neglected causal explanations. Writing about Galileo's new science of motion, Descartes held that to be built without foundations, because it did not start from the cause of the motion. 2 That opinion would have been endorsed by

21 o

History of Science

every previous philosopher, but it hardly reflects the spirit of science in the age of Galileo, which immediately preceded that of Descartes. Music, in sharp contrast with philosophy, not only reflected the spirit of Galileo's science, but it had made possible the rise of his new physics. Later I shall explain how, in some detail, because the story is still not widely known. But during the age of Galileo there was not a single academic professor of philosophy who published in support of his new science, whereas a dozen or more of them published books against it. Because of that fact, it is curious that historians of science now debate whether Galileo was inspired by the philosophy of Aristotle or by that of Plato, and argue as if he had owed little or nothing to music, or to any other activity than the reading of books by previous philosophers. The historical fact requiring explanation is not which philosophy anticipated Galileo's sciences, if any did, but why philosophers of his time opposed the rise of modem science. The answer lies in knowing how Galileo discovered some laws of physics, and who if any, did not oppose the rise of modem science but welcomed it. Like Galileo, Johannes Kepler, who founded modem astronomy, received no support from any recognized philosopher of his time, which coincided very nearly with the age of Galileo. And though (unlike Galileo) Kepler was not born into a family of musicians, he was unusually well informed in the classical theories of music as those bore on pure mathematics. Kepler became an enthusiast for possible applications of harmonic theory to the Copemician astronomy. His very first book embodied a scheme of the celestial spheres circumscribed around the five Platonic solids, nested in a certain order around the central Sun.3 His later revolutionary discovery that planetary orbits are not circular, but elliptical, marking the veritable beginning of modem astronomy, failed to dim Kepler's earlier enthusiasm. He saw elliptical orbits as relieving the music of the spheres from dull monotony. Ellipses produced scale passages and chords to replace the sustained tones that would inevitably result from perfectly circular motions. That Kepler's debt to music in science was different in kind from that of Galileo resulted from the fact that it stemmed from theories of music, and Galileo's came from musical practice. That best shows why, in my opinion, the birth of modem science cannot be fully explained without considering the role of music in it. One conspicuous difference between natural philosophy and modem science is that modem science embraces both theory and practice. Premodem science had been definitively separated from practice, as from any utilitarian aspirations, by Aristotle himself. The origin of modem science can therefore not be adequately explained without taking into account disciplines like music, in which both theory and practice existed side by side, as was also

Music and Philosophy in Early Modem Science

2 11

the case in medicine and in architecture. All three fields contributed to the rise of modem science, welcomed by musicians, doctors, and engineers. Although Kepler was indebted to music for his cosmological schemes, he was hardly less deeply influenced by philosophy, and particularly by the Platonism which conferred on mathematics the highest rank of all among the sciences. Galileo differed. That is hardly surprising when we recall that Galileo's contributions to astronomy were chiefly observational, whereas Kepler's were entirely theoretical. Observation does not required a philosophy, as theorizing does. Theoreticians classified music as one branch of mathematics, rooted in arithmetic. In classical Greek mathematics there exists an unbridgeable gulf between arithmetic, which involves only the discrete, and geometry, which involves also continuous magnitudes. Astronomy being the branch of pure mathematics that in classical times belonged with geometry, Kepler's linkage of it through music with arithmetic contradicted the ancient separation between that and geometry. Like musical practice, observational astronomy was hampered by an ancient tradition - that the heavens, being perfect, could have no motions that were not perfectly circular motions, and that celestial bodies must likewise be perfectly spherical in shape. In 1609 Kepler published his discovery that planetary orbits are ellipitcal, and the next year Galileo announced his new telescopic discoveries. Discovery of mountains and craters on the moon met with more open hostility from philosophers than even the finding of new planets, as Galileo called Jupiter's satellites. After the two-pronged attack of 1609-10 by Kepler and Galileo, the ancient worldview was doomed to collapse, though not without a struggle. While Galileo was completing his final book, a monumental treatise on music which incorporated critical discussions of the newly emerging physics was just being published at Paris - Marin Mersenne's Harmonie Universe/le. In that treatise, and in his later works, Mersenne did more to propagate emerging new sciences of acoustics, pneumatics, and ballistics than anyone else of his time, though he is remembered mainly as a musical theorist. Indeed, Mersenne's own original contributions to science were modest. It was chiefly as spokesman and translator of Galileo in France, and as friend and loyal supporter of Descartes, that Mersenne furthered the spread of modem sciences, not only through books, but through his voluminous correspondence with savants all over Europe. Lacking the flair for mathematics shared by Stevin, Kepler, Galileo, Descartes and Huygens, Mersenne instead brought to the advancement of physical sciences a flair of his own. He was, above all, a tireless and resourceful experimentalist, at first in the field of musical acoustics and subsequently in physics, a leader in the early days of modem experimental measure-

212

History of Science

ments. Skill in the design and conduct of experiments was replacing speculative philosophy as a guarantee of correct analyses of nature. The Dutch engineer Simon Stevin was first to test G.B. Benedetti's proposition that speeds in fall are not governed by the weights of the falling bodies. 4 Stevin's tests, from a height of thirty feet, were conducted in 1585-86 and published in Dutch four years before equal speeds in fall were exhibited by Galileo from the Leaning Tower of Pisa. It is an illuminating fact about the state of physics in the latter half of the sixteenth century that neither Benedetti, who first published his demonstration of equal speeds in fall in I 553, nor any of his supporters or adversaries in this matter over the next three decades, appears to have put his innovative conclusion to actual test, which was an easy and seemingly obvious thing to do. The question was put to nature not by the challenged Aristotelian natural philosophers, but by the mathematical physicist Stevin. Galileo, then a young man who had just completed his years as a student at the University of Pisa, was probably still unaware of Benedetti's proposition. He had reached the same conclusion from the same book that had inspired Benedetti long before, a work on raising sunken ships by the mathematician Niccolo Tartaglia, first printed at Venice in I 55 I . Tartaglia had published in I 537 a book titled Nova scientia, though innovation in science had always been reprehensible in the view of orthodox natural philosophers. Mersenne took up experiments described by Galileo and added observations and measurements of his own. Actual measurements of motion had no place in Aristotelian natural philosophy, since they could not reveal hidden causes behind the phenomena. Still less could careful measurement have had any place in the philosophy of Plato, who forbade careful attention to sensible phenomena as a potentially misleading distraction from the archetypal world that he believed superior to the changing world of sensible experience. Archetypes became the favored study of Kepler, while Galileo used that word only once, in a letter written in 1633, to reject them.5 Like Kepler, Mersenne had a lifelong interest in philosophy, hoping by that to explain the source of truth in science. Unlike Kepler, however, Mersenne was less impressed by speculations of ancient philosophers than he was by some novel ideas of his own contemporaries, especially those of Descartes, with whom he often corresponded on matters concerning science and philosophy. Stevin's manuscript treatise on music was the first European work to venture the bold conclusion, against tradition as old as the ancient Pythagoreans, that exactly equalized tuning is possible but requires use of the 12th root of 2. 6 It is said that Stevin had been anticipated in this by a Chinese, though it is left unexplained why an equal division of the twelve-note scale should have been of

Music and Philosophy in Early Modem Science

213

interest to any sixteenth-century Oriental musician. Certainly the sixteenth-century algebraic concept of roots higher than the cube, and actual techniques for determining 12th roots to any needed degree of approximation, were Stevin's own contributions to mathematics, along with the decimal fractions required for their expression.7 Of even greater fundamental importance was Slevin' s title for the first chapter of his L'Arithmetique, published in 1585. There he stated that one is a number, contradicting the definition of number by Euclid as "multitude of units." The unit itself could not be a number under that definition, though Stevin did not offer a new definition of number to replace it. The supposed irreconcilability of any discrete and countable quantities with all continuous and infinitely divisible magnitudes and their ratios was, of course, theoretical, arid of no practical concern. That is why this tradition holds the key to the musical dispute between Vincenzo Galilei and Gioseffo Zarlino, a quarrel anticipated in Greek antiquity by the position of Aristoxenus in opposition to classical arithmetical musical theory. No matter what the mathematicians said, the ear of a musician can accurately divide musical intervals in ratios that cannot be expressed in terms only of the numbers by which things are in fact counted. The practical inadequacy of arithmetic alone was also the key to the new science of motion created by Galileo. For that reason I have stressed this close relation between the dead hand of theory that held back certain developments in music until the age of Galileo 's father and that which delayed the birth of modem physical science until the age of Galileo. Stevin, along with Niccolo Tartaglia, Benedetti, and Galileo, was a principal founder of modem hydrostatics and of theoretical as well as practical mechanics. Had it not been for his publishing chiefly in Dutch, Stevin would doubtless have become much more widely known as a pioneer modem scientist than is presently the case. Curiously enough, Stevin took the ~sition that Dutch was the only language fully suited to the science of nature, because it allowed the coining of new words whose precise meanings would be clear at once to others. In his treatise on music, it was to lack of the Dutch language that Stevin ascribed the failure of all ancient Greek writers to arrive at a fully correct musical theory. But Stevin himself had also a preconception - that all mathematics must in principle be ultimately reducible to the numbers that are use in counting, assumed by Arabs who garbled in translation the Euclidean general theory of proportion for continuous magnitudes. Whether mathematics is in fact so reducible is completely irrelevant to the practice of music, and to useful science, though until the age of Galileo that was not perceived. Even today this preconception tends to cloak the refutation of medieval impetus theory that was brought about by Galileo's mathematical physics. His new physics owed its origin to two Euclidean definitions, those of

2 14

History of Science

"having a ratio to one another," and of "same ratio" as applied to mathematically-continuous magnitudes. The first of these had been omitted, and the second became hopelessly garbled, in the standard medieval Latin transation of Euclid's Elements taken from Arabic (and not authentic Greek) texts. Neither definition was entirely reestablished until 1543, and at first was limited to the Italian translation of Euclid's Elements by Tartaglia. 9 As a result, the Italians enjoyed a halfcentury head start over the rest of Europe in the creation of recognizably modem mathematical physics, most especially Italians who could not read Greek or Latin; for in the universities no attention whatever was paid to Tartaglia because he had not published in an academically respectable language. It is clearly as a result of overlooking the mid-sixteenth-century revival of Euclidean proportion theory that historians of science still imagine that recognizably modem science must have come from speculative philosophy. As to that, Galileo sarcastically asked, "What has philosophy got to do with measuring anything?" 10 His use of precise measurements as the main basis of his new science required such measurements to be subjected to a mathematically rigorous theory of ratios and proportionality, and that had been nonexistent in Europe from the fall of Rome until 1543. As Tartaglia said on the title-page of his translation, it was made in order to put into the hands of any person of average intelligence the whole body of mathematical knowledge. Nothing like that was ever the intention of ancient or medieval natural philosophers, whose monopoly on science ended with the invention of printing from movable type and its early sixteenth-century sequel, the first appearance of inexpensive books in living languages. Astronomy already had a two-millenium history of accurate measurements of actually observed motions before the first known measurements of pendulums, falling bodies, descents on inclined planes, and projectile motions were made by Galileo in 1604-08. Now, by that time, a profound revolution in musical practice and theory was already well under way, one that seems to have originated mainly in resentment of restraints put upon the practice of music by Jong accepted theories of musical consonance. Ancient tradition decreed consonance to depend only on ratios of the smallest numbers, a metaphysical conception unduly limiting practice that was utterly rejected by Vincenzo Galilei. A closely parallel conception still delayed the rise of modem science, but was soon to be thoroughly refuted by his son Galileo. The revolution in music found voice in the books of Vincenzo against Zarlino. It is to the writings of Claude Palisca that I owe my interest in the musical theories of the late Renaissance and I apologize to him for invading the same territory briefly, in order to exhibit the direct role of music in Galileo's main discoveries in physics. By doing so, I hope to add something to what

Music and Philosophy in Early Modem Science

215

Palisca alluded to when he wrote, in I 961 : "By creating a favorable climate for experiment and the acceptance of new ideas, the scientific revolution greatly encouraged and accelerated a direction that musical art had already taken. " 11 It is the other side of that coin of which I am about to speak. Vincenzo Gali lei appears to me to have been the first person ever to have discovered a law of physics by experimental measurements involving motion. Late in his long controversy with Zarlino he found that the ratio 3:2 does not hold for the perfect fifth when sounds are produced by tensions in strings, rather than by their lengths. He published an account of his experiments in 1589 and various circumstances support my belief that those were carried out in 1588. 12 In that year Vincenzo's son Galileo, t.hen teaching mathematics privately at Florence, was probably residing with his parents. In his notes for a treatise on motion written in 1588, Galileo alluded in passing to the motion of a pendulum, a fonn of "natural motion," as spontaneous descent was called at that time, that had generally escaped attention by natural philosophers. Vincenzo's study of tensions in strings required weights to be attached to them, whether hanging freely or suspended over the bed of a monochord, and in either case a pendular motion would be observably imparted to them. It is thus probable that the young Galileo was present at Vincenzo's experimental measurements. In those years, though Galileo was already in disagreement with some fundamental propositions about motion that were then taught as being Aristotle's whether or not they were, in fact - he did not yet doubt that physics must concern itself mainly with causal inquiry. Years later, in 1602, Galileo 's working papers show him to have been making careful experiments with very long pendulums, which led him to a correct and important theorem about motions along inclined planes, and an incorrect conjecture about their relation to motions of pendulums. Within two years he was to discover first, the law of the pendulum; from that, the law of falling bodies; and next, that this same law applied to descents along inclined planes. Galileo's physics from then on concerned only laws of nature, not causal inquiries of the kind dominating physics for the past 2,000 years. No such revolutionary change in the very nature of science itself would have occurred to Galileo had the musical measurements of his father not first interested him in the motions of pendulums. Galileo's working papers on motion from 16o2 to 1637 still survive nearly complete at the Biblioteca Nazionale Centrale in Florence, though now chaotically bound together in Volume 72 of the Galilean manuscripts. Those that bear theorems, solutions of problems, or enough other words to fonn one complete sentence or more, were transcribed and published in the definitive Edizione Nazionale of Galileo 's works around 1900. It happened, however, that Galileo's experimental measurements, being recorded on pages with few or no

2 16

History of Science

words, had gone unnoticed by historians of science until very recently. Without talcing them into account, it was not possible to reconstruct the experiments underlying these papers, and it remained mere speculation to debate how Galileo discovered the law of falling bodies, opening the road to modem physics. The first page of those notes to be identified and dated was associated with Galileo's discovery of the parabolic trajectory of a horizontally-launched projectile, in 1608. That left still unknown the manner of his discovering the law of fall, achieved no later than 1604. It did, however, give the name that Galileo used for his unit of length in making measurements, the punto. A note on another page, written probably in 1605 or 1606, made it possible to convert the punto into metric units. It was, to my surprise, less than one millimeter; to be exact, it was 0.94 mm. Knowing this unit of length made it possible to reconstruct the uses made of it. In 1975 I published my analysis and reconstruction of a page numbered folio 107v,' 3 and for a decade or so I regarded that page as the discovery document for Galileo's law of fall (see Plate 18). A set of calculations in the middle of folio 107v shows how Galileo had arrived at the eight distances he tabulated there. In every case he had first multiplied a number by sixty and then had added a number less than sixty to it, showing that Galileo owned a ruler divided accurately into sixty equal parts, which he called punti. His measurements were made along an inclined plane, grooved to guide a rolling ball, and they represented the places of the ball at the end of each of eight equal intervals of time. It was not difficult then to reconstruct the experimental setup behind the measurements. The plane was tilted by raising one end sixty punti above the horizontal. Because it was about 2,000 punti long, its slope was 1.7°. At that slope, a ball rolling the full length of the plane will talce four seconds, permitting eight halfsecond marks. Calculation shows Galileo's accuracy to have been within 1/64th of a second for every mark except the last, when the ball was moving about a thousand punti per second. Interestingly, that was the only measurement that he subsequently altered. His final entry for it was almost exactly correct, as calculated from modem physical equations. I think that musicians will be less reluctant than historians of science have been in granting Galileo's ability to have timed half-second intervals accurately to 1/64th of a second. As I reconstruct his procedure, he tied frets around the plane, so that the ball would make audible bumping sounds as it passed over the frets, which were then adjusted patiently until every bump coincided with a note of some song of rhythmic regularity. When Galileo readjusted the lowest fret, he also placed a plus or a minus sign on four other measurements. It is a great nuisance to adjust any fret but the last, because that requires moving all the frets below it; and in this case, the differences were not worth the bother.

Music and Philosophy in Early Modem Science

2 17

Galileo being a good amateur musician, my reconstruction plausibly accounted for everything on the page. Nevertheless, as I eventually found out, it was not folio 107v that was the discovery document for the law of falling bodies. Into the narrow left-hand margin. Galileo had squeezed the first eight square numbers, in a slightly smeared bluish ink. That was odd, as everything else on the page is in black ink and in small, neat writing. If Galileo already knew the times-squared law of fall from his work on folio 107v, as I supposed, then it was a puzzle why the square numbers had not been entered at once. In fact, all that Galileo found from his first experimental measurements was the rule that speeds grow from rest, in equal times, as do the odd numbers 1, 3, 5, 7, and so on. Since it had been only a rule for speeds that he was looking for, he laid folio 107v aside before he came to discover the times-squared law of fall. But at that time he recognized that if a precise rule could be found by equalizing eight times musically, much more might be learned by accurately measuring brief times, and not just equalizing them. At the end of the page he drew a preliminary sketch for a timing device that he described years later, in Two New Sciences of 1638. A bucket of water with a tube through its bottom was hung up. The water flowing during a fall or pendulum swing was collected; that water was weighed, and these weights became his measures of time. His first recorded weighing was I ,337 grains during fall of 4,000 punti, about twelve feet. That was Galileo's poorest timing, high by 1/30 of a second. Next he timed half this fall, at 903 grains weight of flow, correctly within 1/100 second. He then adjusted the length of a five-foot pendulum until its swing to the vertical accompanied the flow of water to the previous mark on his collection vessel. His measured length for this pendulum, 1590 punti, was exactly correct, as shown by modem calculation. Galileo next concentrated his attention on pendulums, and found the rule that doubling the length quadruples the time. Choosing sixteen grains of flow for a new unit, to fit with proportion theory, he named this the tempo (= 1/92 second). He then calculated length and time for a very long pendulum, about ten meters, and verified his result by hanging and timing such a pendulum, implying the general pendulum law that the times are always as the square roots of the lengths. From that, he found the law of fall, as was seen on the discovery document when that was finally identified as folio 189v. 14 Using his fall law, Galileo corrected his one poor timing and turned back to his data on folio 107v to test whether the law of fall held true also for descents on inclined planes. Writing the square numbers on it, he multiplied each one by the distance to his first mark, and saw that those products were almost identical with the eight distances he had previously measured. Hence the seeming puzzle (that had been pointed out in 1975) vanished. The

2 18

History of Science

reconstruction of folio I 07v was correct, but that was not the discovery document for the law of free fall. On the actual discovery page folio 189v, Galileo next found the rule for descents along two planes differing in both slope and length, and verified numerically a theorem he had found experimentally in 1602 - that although the distance is greater, the time is less for motion along two conjugate chords to the low point of a vertical circle than along the single chord joining the same endpoints. Thus was modem mathematical physics born - not of metaphysical principles, or philosophical speculations - but of accurate measurements inspired by those which had already refuted the ancient philosophical theory of musical consonance. Without Galileo's having been present at his father's musical experiments in 1588, he probably would not have gone on to his own study of pendulum motion. Without musical training, Galileo would hardly have been able to make his very first timings nearly exact. Music played not only a unique, but an essential role in leading Galileo to his new physics, a science of precise measurements, for music is an art demanding precise measurement and exact divisions. Galileo never composed a treatise on music, but he did once set forth a novel theory of consonance, following his statement of the pendulum law in the first section of his Two New Sciences. Confining himself to the octave and the perfect fifth, Galileo considered amplitudes of vibration of two strings whose lengths are as 3:2, supposing that a pulse is emitted only when a string reverses its direction. The pattern of pulses reaching the ear will then be: both-together, upper, lower, upper, both-together, and so on, as long as vibrations continue. Each pulse as such is a toneless brief pressure; only the pattern of single pulses that separates pulses of double strength is responsible for the tonal sensation associated with the perfect fifth . The simplest such pattern is that of strings sounding the octave; pulses from the lower string are always doubly strong, each being accompanied by a pulse from the upper, while between them is interposed a beat of single strength coming from the upper string alone. The tonal sensation of dissonance will occur when the pattern departs from equality in time-intervals betwe~n pulses, and is most marked when adjacent half-tones are involved. Such was the physics of consonance proposed in Two New Sciences. Galileo's account resembled one by G.B. Benedetti found in letters not published until 1586, though written years earlier. Galileo did not make the frequencies alone directly responsible for sensations of consonance and dissonance, as Benedetti did, but rather indirectly, through pulses of air reaching the ear with a temporal pattern of pressures having single and double strengths. As a physicist, Galileo would not assume that pitch was an occult quality of the vibrating string, in the terminology of the natural philosophers. But as a musi-

Music and Philosophy in Early Modem Science

219

cian, Galileo added this picturesque description of the consonances of the octave and fifth: In the octave, pulses of the lower string are always accompanied by pulses of the upper, but between the latter there is interposed a solitary pulse at equal intervals. [ ... ] Such a harmony is too bland, and lacks fire. The fifth, however, is characterized by its displaced beats, that is, by the interposition of two solitary beats of the upper, and one of the lower, between each case of simultaneous pulses; moreover, these three are separated by timeintervals one-half of that which separates simultaneous pulses from pulses of the upper string. Thus the effect of the fifth is to produce a tickling of the eardrum, so that its gentleness is modified by sprightliness, giving the impression simultaneously of a gentle kiss and a bite. 15

That suffices to answer scholars who complain that Galileo's new science dehumanized nature when he relegated our sensations to the category of "secondary qualities." It would be more accurate to credit him with directing attention precisely to the very human quality of the sensation of musical consonance by distinguishing that sharply from a mere succession of mechanical pulses of air. NOTES 1

2 3

4

5

6 7

Details are given in my "Hipparchus-Geminus-Galileo," Studies in the History and Philosophy of Science 20 (1989), pp. 47-56. Letter from Descartes to Marin Mersenne, 1 1 October 1638. Translated in Stillman Drake, Galileo at Work (Chicago, 1978), pp. 387-88. Johannes Kepler, Mysterium Cosmographicum (Tiibingen, 1596), trans. A.M . Duncan as Mysterium Comsographicum: Secret of the Universe (New York, 1981), esp. pp. 85-to5. Benedetti 's proposition is translated in I.E. Drabkin and Stillman Drake, Mechanics in Sixteenth-Century Italy (Madison, 1960), pp. 147-53, as it first appeared in Venice in 1553. Translated in Stillman Drake, "A Neglected Galilean Letter," Journal of the History of Astronomy 17 (1987), pp. 93-105. "On the Theory of the Art of Singing," trans. A.D. Fokker in The Principal Works of Simon Stevin, ed. E.J. Dijksterhuis, vol. 5 (Amsterdam, 1966), pp. 422-64. Stevin 's original contributions to mathematics, both pure and applied, are less known but no less important to science than the analytical geometry of Descartes. It was Stevin who in 1585 first narrowed the classical gulf between the discrete and the continuous in mathematics by his invention of decimal fractions.

220

History of Science

8 The Principal Works of Simon Stevin, vol. 1 (Amsterdam, 1955), pp. 58-65. 9 Niccolo Tartaglia, Euclide ... diligentemente rassettato, et al/a integrita ridotto ... talamente chiara , che ogni mediocre ingegno, senza la notitia over suffragio di alcuna altra scienza confaci/ita sara capace a poterio intendere (Venice, 1543). Io Stillman Drake, Galileo Against the Philosphers (Los Angeles, I 976), p. 38. 11 Claude Palisca, "Scientific Empiricism in Musical Thought," in SeventeenthCentury Science and the Arts, ed. H.H. Rhys (Princeton, 196 I), p. 137. 12 Discorso intorno all' opera di messer Gioseffo Zarlino da Chioggia (Florence, 1589/ rpt. Milan, 1933). 13 Stillman Drake, "The Role of Music in Galileo's Experiments," Scientific American 232(June, 1975),pp.98-104. I 4 Details are given in the second edition of my translation of Galieo 's Two New Sciences (Toronto, 1989). 15 See my translation of Two New Sciences, p. 107.

Plate 1: Galileo's "geometric and military compass" is shown in the form in which it was made from 1598 onward. Its two sides are shown; the numerals are indistinct because of wear and the fact that the instrument appears at less than full size. This instrument, which was probably made in Florence, is from the author's collection. Galileo designed his compass, which later became known as the sector, to solve a problem he called "making the calibre." The problem was to find the appropriate charge for an artillery weapon of a given bore, calibrated according to the material the ball was made of. Scales of instrument are shown in Plates 7 and 8.

Plate 2: Compass with quadrant is portrayed. The quadrant made the instrument useful for astronomical sightings and for surveying. Galileo designed the quadrant to be detachable so that the entire instrument could be easily carried. A scale that reads from zero at each end to 100 in the centre was also his idea; in modem terms each division is I per cent of grade. With the quadrant the geometric and military compass could be employed in the determination of heights, distances, and slopes, all of which were important military problems. Photograph by Ben Rose. © The Estate of Ben Rose. Every effort has been made to obtain permission to reproduce this illustration.

Plate 3: Forerunner of sector was devised by Galileo's friend the Marchese Guidobaldo del Monte. These two drawings appeared in a book that was printed in Venice in I 598. Ink from the reverse page shows through in each case; the faint lines in the drawing at right are therefore extraneous. On one side of the instrument (left) were scales that gave the sides of regular polygons to be inscribed in a circle with a diameter equal to the distance by which the two points at the bottom of the instruments were separated. With the scales on the other side (right) the user could divide into various equal parts a line of a length equal to the separation of the endpoints of the instrument.

Plate 4: Gunner's gauge invented early in the sixteenth century by the Italian mathematician Niccolo Tartaglia was one of two instruments modified by Galileo to make his military compass. Tartaglia's gauge is shown here as it was illustrated in a book published in 1537. The longer leg (left) was put in the mouth of the cannon; the elevation of the gun could then be read in "points" on the quadrant by means of the plumb line. A level gun was said to be at "point-blank," marked F.

J.

"E .\.I

k.

I>

G

0

"'

I ~

. . 'P

~

0

Plate 5: Triangulation instrument, the second of Tartaglia's inventions adapted by Galileo, shown as it appeared in another illustration in Tartaglia's book The New Science. The instrument's function was to assist artillerymen in determining the height and distance of targets by sighting and triangulation. Galileo made his military compass by combining the two instruments devised by Tartaglia, changing the square to a quadrant and adding a clinometer.

Plate 6: Student's drawing of accessories for Galileo's military compass, found in a seventeenthcentury manuscript recently acquired by the author. The manuscript contained Galileo's instructions on the use of the sector. The drawing was probably made by one of Galileo's many German students before sector scales were added. At the top is the quadrant. Below it to the right is the cursor, which was fitted to one arm of the compass to carry a "movable foot" that Galileo designed to compensate for the taper of a cannon so that the gun's elevation could be read at the breech instead of the mouth. Below that are a bracket that mounted the instrument on a universal joint (bottom centre), whose conical base could fit on a tripod, and (bottom left) the arms of the instrument in the folded position in which the compass was carried.

Plates 7 and 8: Galileo's scales are portrayed schematically as they appeared on the 1597 model of his sector (left) and on the sector as it was made after 1598 (right). Each scale was inscribed on both the left and the right arm of the sector; here the numerals appear only once and the representation of the scale on the other arm is indicated for interior scales by broken lines. Numerals at bottom that identify the scales are the author's. Scales I and II dealt with the artillery man's problem of "making the calibre." I gave the volumes for equal weights of various metals and stones. II provided a means of obtaining cube roots, ill of obtaining square roots. IV, which first appeared in I 598, gave equal divisions of linear measure; the H signifies that a sector devised independently by Thomas Hood had a similar scale. V was for constructing a given regular polygon on a line of any given length. VI gives for any polygon the side of any other polygon of equal area. G- I and G-2 are like Guidobaldo's scales: G-1 is for fitting a regular polygon in a circle, G-2 for obtaining equal sections of a line.

hic Lines

Metals I - Lines of tic Lines et m eo er - St

V - Polygrapnic Lines V I - Tetrago Polygons ar G-1 - Regul

II etric Lines II I - Geomar Parts ne Li G-2

10

9 8

7

l I I

6

I

3

I 12 I 4 I 11 t 10

5 6

/ 9

¢I 8

I

7

I

2

I I I 6 I

20

I

I I

5

I

I

ioI

1 ~

I

7\

I

I

51

I I

I I

5I I I I I I

I

I

4

10 /

I I

t

11

13 12 11

I

I I I I

4

I

I I

I

t

12

I

I I I I I I

I

I I

13

3

I

I

I I I JJ

G-2

G -2

Ill

VJ

3

G-7

G-1

VI

I I I V

ls I - Lines of Meta nes

c Li II - Stercometri Lines II I - Geometric Lines JV - Arithmetic

II

s

Line V - Polygraphic Li nes VI - Tetragonic nes VI I - Adjoined Li

VI

VII

Plate 9: Galilean sector owned by the author. Somewhat larger than Galileo's own Paduan instrument preserved at the Museo di Storia della Scienza in Florence, this sector was probably one of several authorized by Galileo to be made at Florence (note fleur-de-lys on screwheads and in scrollwork). The linee aggiunte bear a small square mark for altitude scales, described in mss. e and g but altered in the printed text of 16o6, suggesting earlier manufacture. The quadrant arc is a modem replica, the original being lost.

\

':Oritt,r tractat t,er ,ned}antf~cn ]n~rumentcn

•I

LEVINI HVLSlf.

l1cfcQrtt6ung l'nt> ~n.. tcrrt~t be~ Joo~ Q5urgt ~roVQft10,~

§trctds/b4tburd) tt1ft fonbctridjnn ucrt~fl dn teg, fi02( ~ffl2ftOM €arcfd Unf/4rlt ~dcf)t/~aubcattc/4U8ntf~tu f11/ tn funff .reg1daribus, -ault j 41k lrtt~aria coreora, &t:. -ficqumilld) fJnntn Jcrtt,.uft/ Sffl

lo) n11 • :t,~l

-1rr:--_1.cn-•

.~n1Wt11t~n,crgr~mrr-.,no Mj6!facct 'nhff ~ M O t . JU\'OfflUI

t "

Cn,cf.9~g.

, I.IS

'OfJ)

Plate JO: The proportional compass (not a sector) as illustrated by Levinus Husius, who attributed its invention to Joost Burgi though it probably originated at Urbino about 1560.

'

Plate II: Thomas Hood's sector as published at London in 1598. Courtesy of The British Library.

I

Plate 12: The water-clock of Father Linus as depicted in Pietrasancta's De Symbolis Heroicis. Reproduced by permission of the Dibner Institute for the History of Science and Technology.

Plate 13: Following the design shown by Pietrasancta in 1634 the Jesuit scientist Athanasius Kircher included a similar design in his book on the magnet that first appeared in 1641. The engraving shows the driving clepsydral mechanism for actuating the magnet N; drapes hid the mechanism from view. Reproduced by permission of the Dibner Institute for the History of Science and Technology.

C/_;p!Ydra

AB .£_fistomiun ijrrius £ F Incubus E_;;istomium S1:fmi1s D M Cjft'ndrus C ~1raculum

MCJJ,nts Pondus D~hr~ma

.N

G.

DC

Plate 14: Water-screw as illustrated in Daniel Barbaro's Vitruvius (1567).

Plate 15: Single battery of Ceredi's standard water-screws, to be used along low-banked rivers.

Plate 16: Ceredi's illustration of two batteries of standard water-screws with intermediate sump, for irrigation from the river Po.

Plate IT Ceredi 's illustration of the water-screw provided with a flywheel, turned by a crank, to provide irrigation.

,...... ,J.

,.

-~, -·

,.r ,,. .. ,,.,. ,. 'l

'\. .. t ;' ,-

J

(

✓

.

1. • .,..

....

. .. .-~

f· /1.1

f . 6°,

,.

,

..

I I f

~

,_,

., 0

t "·

.._

"L

,·...:,_._

I ;.~,-

I .- "'\. •

/ '. 1- I

.

-~ - ~· .. j ,~l ..~ : . ·-.~ ~

J'!.•

f I"1 J'•

f'"

_::...:J

-· •1

...

'1 . -t-1

~

,,

,~,

+1.

•·

.,.

1

-~. __

. •: .. ·7

;,-.

~, i

I

. . . :;,

·

,·

~

Ii

,

•! .'

I-

• ·Cl.-•

··.. ·. .

-~

.-:~

. -~.

:.j !

..-,.,.

. ~. ;

. ... •.•..:.-- ,

Plate 18: Folio I07v. Reproduced by permission of the Biblioteca Nazionale Centrale, Florence.

j I

I

;

Plate 19: Japanese print of Newton watching the apple was part of a series of prints that depicted great men of the Western world. The artist, Hosai, chose to represent science by the story of Newton's apple. The characters are translated, "Issac Newton, very great head of school but not pompous." The print, which was made in about 1869, is part of the author's collection.

p C

Plate 20: Figure 4 of the "Dialogue" may have inspired Newton to ask himself why the moon did not move away from the earth or fall to it as the apple did. Galileo's diagram was supposed to demonstrate that an object could not be flung off the earth (represented by the arc AP) by the earth's rotation. Whether or not the demonstration is ultimately convincing scarcely matters. An invalid proof could still have interested Newton, particularly because the case of a weightless body (which Galileo did not consider) would have resembled the case of the moon.

10

Impetus Theory and Quanta of Speed before and after Galileo

It is universally believed by historians of science that from the time of Albert of Saxony in the late fourteenth century to that of Galileo in the seventeenth, the speeds acquired by a body in free fall were proportional to the distances fallen rather than to the times of fall from rest. That rule of proportionality was explicitely rejected by Galileo in his Two New Sciences of 1638, and the manner of his introducing it for discussion suggests that he considered it to be widely held in his day and therefore to be in need of express refutation. Indeed, he seems to say that he had for a long time believed it to be true, though in fact he said rather that he had once supposed that it made no difference whether the speeds acquired were regarded as proportional to distances fallen or to times elapsed in fall. 1 Acting probably on the evidence from this passage of Galileo's, Vincenzio Riccati wrote about a century later: It was the opinion of some that a body driven by its heaviness (which is assumed to be always preserved in the descent of that body or, as mathematicians ordinarily say, is constant) newly receives at each little distance an equal degree of speed, in such a way that the speed acquired shall always be proportional to the distance passed. 2

Riccati was writing in defence of G.B. Baliani, a contemporary of Galileo's, whose name had become attached to the incorrect rule of proportionality. This misunderstanding, which will be clarified below, had originated with Jacob Hermann in I 709 and was perpetuated in his P horonomia and in the popular course of physics written by Christian Wolf. Baliani had in fact never contended that Reprinted from Physis t 6 ( 1974) 47-65, by permission.

222

History of Science

the rule applied to measurable distances. But the emoneous proportionality had been stated in so many words by Michael Varro in his De motu tractatus, published at Geneva in I 584. Thus it is quite possible that it was widely accepted a half-century later, when Galileo published its definitive refutation. It does not follow by any means, however, that the proportionality of speeds to distances in free fall had prevailed generally among writers from the time of Albert of Saxony to that of Varro. Indeed, Domingo de Soto had associated free fall with uniformly difform motion - that is, with uniform acceleration - in the mid-sixteenth century, and that implied the correct proportionality of speeds to times from rest. It is a curious historical fact that so far as is known, Soto was the first to have explicitly assered this association, though the mean-speed analysis of uniform acceleration had been carried to a high degree of sophistication during the fourteenth century. The failure of others to have made the association long before Soto's time is one of the current puzzles in the history of physics that finds an easy solution, highly creditable to early mathematical physicists, under the analysis to be offered herein. The immediate question, however, is whether any explicit statement is to be found, among writers before Varro, asserting that speeds in free fall are proportional to the distances fallen from rest. Among the documents that have been published thus far, I have been unable to find any explicit statement of that kind before I 584, nor have medieval historians whom I have consulted yet provided any unambiguous formulation of the kind from manuscripts known to them but not yet published. I accordingly assume, for the purposes of this paper, that the prevailing view is ultimately based on a single source; that is, on a mathematical formulation given by Albert of Saxony in his commentary on Aristotle's De caelo. If that formulation has been misconstrued, no matter how plausibly, several puzzles besides the one just mentioned may tum out to have been entirely illusory. Thus, for example, it seems strange that no one appears to have raised the question, in any of the numerous surviving commentaries or treatises on physics before I 6 8, whether speeds are proportional to times or to distances. That would certainly have been a reasonable question to raise if there were in fact two competing theories. We know that mean-speed analysts always assumed speeds proportional to times in uniform acceleration, and perhaps no alternative proposal was ever actually made before Varro's. It is true that Niccolo Tartaglia, in 15 7, emphasized the increase of speed with the height of fall and the approach to the ground, but he did not treat of this in terms of proportionality. Likewise G.B. Benedetti, writing in I 585, left that matter open and said only that the speed increases the more the falling body moves. It has been customary for historians to say, Jaure de mieux, that there was a general carelessness about the application of the word "proportionality,"

Impetus Theory and Quanta of Speed

223

and I myself subscribed to this explanation in the past. But I now find, on examination, that the term seems not to have been applied at all, and there are very good reasons that it should not have been. If I am correct about the thought of Buridan and its mathematization by Albert of Saxony and Nicole Oresme, there was an inherent indeterminacy in the impetus theory of free fall which made it impossible to assign a unique time or place in free fall from rest for any one "degree of speed," whence proportionality in Galileo's sense, and ours, remained inapplicable to this physical phenomenon. According to the analysis herein presented, medieval physics was far more complete and consistent than is now generally thought. The impetus theory of fall was not, as is frequently asserted, incapable of being completely formulated mathematically. The most logical and simplest formulation was promptly offered by Albert of Saxony and was universally accepted. It was not further worked out in detail until after Galileo's published experimental confirmation of an apparently quite different rule. The well known seeming contradictions in Leonardo's writings on the subject are internally as faultless as Albert's, and depended on the same reasoning. Nor was Jean Buridan's account of acceleration in free fall a vague philosophical suggestion; it remained the best causal account ever given up to the time of Isaac Newton, and was reached by acute physical reasoning that can now be exactly reconstructed. What was important to medieval physicists was that impetus theory gave a causal explanation of acceleration in free fall. The later resistance to Galileo's law was directly connected with his rejection of any search for causal explanation of natural acceleration, and it was on that account that the mathematics of impetus theory was eventually worked out in great detail. This was done with the same logical precision that had characterized medieval mathematical physics, by a very astute critic of Galileo. II

In 1638 Galileo published his analysis of acceleration in free fall, giving the times-squared law, the odd-number rule, and the double-distance rule, applying these to descent along inclined planes and broken lines as well as to the motions of projectiles. He offered no physical explanation of free fall or of acceleration in it, but argued hypothetcally from a definition of uniform acceleration and a certain postulate. These he believed to be true mainly because they implied results that were borne out by actual experiement. Several of Galileo's contemporaries accepted the new law of free fall, notably Pierre Gassendi and Marin Mersenne. Others opposed it and offered alternative rules. Baliani accepted it but later believed that he had found a more

224

History of Science

subtle law that lay behind Galileo's. In 1646, the same year in which Baliani proposed this idea, Honore Fabri published a large quarto treatise in which the same underlying rule was developed at great length. There is no reason to assume that either Baliani or Fabri owed his idea to the other, though both were correspondents of Mersenne's. Their subtle law was in fact that which Albert of Saxony had given in his commentary on De caelo two centuries earlier, mistakenly implying (or at lease not denying) that it would hold for measurable times and distances. It was the most natural, and perhaps the only consistent, mathematization of Buridan's impetus theory of fall; hence it is not even necessary to assume that Baliani or Fabri borrowed it directly from Albert. Since the two men lived far apart, at Genoa and Lyons, and since they did not use the same means of relating this rule to Galileo' s, I believe their work to have been dependent only on a correct understanding of Buridan's widely accepted theory of free fall : One must imagine that a heavy body not only acquires motion unto itself from its principal mover; i.e., its heaviness, but that it also acquires unto itself a cenain impetus with that motion. This impetus has the power of moving the heavy body in conjunction with its permanent natural heaviness. And because that impetus is acquired in common with the motion, hence the swifter it is, the greater and stronger the impetus is. So, therefore, the heavy body is moved from the beginning [of motion] by its natural heaviness only; hence it is moved slowly. Afterward it is [still] moved by that same heaviness, and [also] by the impetus acquired at the same time; consequently it is moved more swiftly. And because the movement becomes swifter, therefore the impetus also becomes greater and stronger, and thus the body is moved by its natural heaviness and by the greater impetus, and so again it will be moved faster; and thus it will always and continually be accelerated to the end.3

This acceleration is clearly successive, and not continuous, since at first only a single cause acts, and then afterward two causes act, one of them constantly and the other successively. Albert of Saxony, after examining and discarding some rules that would lead to infinite speed in a finite time or distance, gave this quantitative rule: When some space has been traversed, [the speed] is some amount; and when double space is traversed, it is faster by double; and when triple space is traversed, it is faster by triple; and so on beyond ....4

Historians viewing this statement with modem eyes have taken it to be meant to assert that the speeds in fall are proportional to the distance from rest, though

Impetus Theory and Quanta of Speed

225

that would have been easier to say if it had been what Albert had in mind. Such a rule is, however, contradictory of the idea of rest, as Galileo later showed by a simple argument that would by no means have been beyond Albert's powers. But Albert's words are susceptible of a different interpretation, which is also compatible with Buridan's explanation, and this is almost certainly what Albert had in mind; namely, that at the end of some distances, the speed is v; next, throughout a second distance 2s, the speed is 2v; then thoughout a third distance 3s, the speed is 3v; and so on. In this way, infinite speed could never be reached in a finite time, which was all that Albert was worried about in this passage. The rule he gave happens not to be correct, but there is nothing internally wrong with it, nor does it conflict with the notion of initial rest. In a sense, though not quite in the modem (or the Galilean) sense, Albert's rule makes the speeds in fall proportional to the times elapsed from rest, since the nth distance ns is traversed at the speed nv. But we must be careful about the idea of "proportionality," which means an identity (or equality) of ratios. As we look at things, the use of this word is justified only if a unique and determinate speed must exist for each time during fall and for each distance fallen, and conversely. But under Albert's rule as I interpret it, this converse would not hold; since each separate speed was (I believe) assumed to be constant throughout a certain distance, and those distances were bound to become greater during equal successive times. This was a simple consequence of the successive (and hence mathematically discontinuous, in our sense) character of speeds in acceleration under Buridan's rule, adopted for causal reasons. No temporal discontinuity was implied, and of course no discontinuity in space; but the speeds were discontinuous in magnitude. The falling body had a speed at all times, but it jumped from a speed of one degree to a speed of two degrees without passing through any intermediate speed. Thus in the medieval arithmetic theory of proportion, applying only to discrete quantity, Albert's rule did indeed make the speeds proportional to distances - but to distances traversed in successive times, and not to cumulative distances from rest. The speeds are proportional under this rule to the distances fallen during specified instants of time, which is in fact why the times themselves come out to be equal. Yet Albert's rule was not a mere tautology, asserting only that distances are proportional to speeds when times are equal, which is part of the meaning of "speed." His rule asserted in addition a definite law of growth in distances traversed, a law that happens not to agree with experiment. Albert's rule was thus a rule having quantum-jumps of speed between successive "physical instants" (to use Fabri 'slater term): that is, in very brief durations rather than at mathematical instants without duration. And this is the physical picture of free fall that we should attribute to fourteenth-century writ-

226

History of Science

ers (except perhaps to Oresme), as well as to Baliani and Fabri in the seventeenth, when impetus theory was further developed. It was opposed to Galileo's non-causal continuity theory of acceleration, rejected even by Descartes. Galileo's treatment of the same problem was in terms of mathematical instants, and it did assign a unique speed to each point in time and to each point in space during fall. No two different times or distances corresponded to any given speed. That treatment makes the distances from rest proportional to the squares of the times. Experiment bore out the odd-number rule for successive distances (and hence for successive average speeds) in equal times. So far as Galileo was concerned, there was no reason to treat physical distances, times, or speeds as having any finite size, however small, and hence his mathematics of free fall was essentially indistinguishable from ours. Given any distance, any time from rest, or any acquired speed, the other two of these three things was uniquely determined for Galileo. The same was not true under the medieval causal theory, in which any speed must be maintained over some duration and through some interval. Thus while it was true that under Albert's rule, given any distance or time from rest, a definite speed was determined, the converse of this was not true. Given a certain speed during fall, no unique temporal instant or postion was thereby determined. Each "physical instant" determined a unique spatial interval of different length, so that increasing speeds advanced as it were by quantum-jumps having the same ordinal number.5 Such a description was adequate, and it was in accord with the principle of causation that was fundamental to medieval and Aristotelian physics. But it differed in two essential ways from Galileo's conception in that ( 1) proportionality of continuous magnitudes was excluded, and (2) a certain indeterminacy was inescapable with regard to exact positions and speeds. The first of these differences has special importance, since it shows us why we should not expect any medieval writer to have even considered the possibility that speeds in fall are proportional to (that is, have the same ratios as) distances traversed from rest. That possibility is excluded by conceiving of speeds in fall as individually constant, successive, and increasing by integral multiples of the undeterminable (and to them characteristic) speed of a given body in the first physical instant after rest. Here I should qualify my previous statement that Galileo's mathematics of fall is essentially indistinguishable from ours, which is true only in a sense. Because Galileo did not compare magnitudes unlike in kind, as we do, he assigned only ratios (and not individual values) to speeds in fall. That restriction, however, did not invalidate his results.6 Medieval physicists would not feel the indeterminacy mentioned above as an

Impetus Theory and Quanta of Speed

227

unsatisfactory element. Their objective was to give a causal account, which had been done. Impetus as a cause in acceleration had been quantified by the number of the instant. Weight, the other cause, acted uniformly in all instants. Since bodies of different weight were supposed to fall at different speeds, it followed that their speeds were different in the first instant. 7 Such a difference could not be causally accounted for if (as Galileo later assumed) every body were to pass through every possible speed, whether or not these were integral multiples of the first. And in fact Galileo's assertion was countered shortly after his death, in the true spirit of medieval impetus theory, as follows: "A body falling freely will go faster in the vertical than it will along an inclined plane. Therefore it must start faster. Hence there must be some speeds along the plane that are not present in straight fall. " 8 III

The above argument appeared in Honore Fabri 's comments on Theorem 6 1, Book II, of his "Physical treatise on local motion, in which are explained all the effects pertaining to impetus, natural motion, and mixed motion," published at Lyons in 1646. The theorem itself went counter to Galileo's assertion that a body falling from rest and acquiring any speed must first have passed through every possible smaller speed. Descartes also had found this entirely unacceptable, writing to Mersenne that although that might sometimes happen, it was certainly not usually the case.9 Fabri 's explanation helps us to understand the prevailing objection, and I think we are safe in taking this as representative of the view that Albert of Saxony would have adopted if anyone had proposed Galileo's position in his day . Fabri wrote: Theorem 61 . Naturally accelerated motion is not propagated through every degree of slowness. Since there are as many of these degrees of propagation [of motion; as many degrees of speed] as there are [physical] instants through which the motion endures, because new impetus is made in single instants (as shown in our Metaphysics) , then [even] if infinite instants were pennitted, the propagation would not be through every degree of slowness that this [particular] series of degrees did not include. For it clearly begins to move more slowly on an inclined plane than straight down in a free medium, and in a dense medium [more slowly] as against a rare one (that is, more slowly in water than in air). Therefore this [particular] degree of slowness with which it begins to move on a plane slightly inclined is not contained among those [degrees] with which it moves straight down. 10

Let us interrupt Fabri at this point, because a better example could not be

228

History of Science

found of the medieval arithmetical approach in applying mathematics to physics. This approach is totally ignored in most modem historical accounts of medieval physics, because of the prevailing notion today that our algebraic equations are an acceptable shorthand device, albeit anachronistic, for summarizing the long-winded verbal reasoning of pre-algebraic authors. Although medieval writers did not command the Eudoxian theory of proportion for continuous magnitude, they were perfectly aware of the existence of incommensurable magnitudes and of the irrationality of the diagonal of a square. Hence it would have been clear to them that if the speed of some body in the first physical instant happened to be, say, V2, then all its infinite subsequent speeds obtained by taking all the integral multiples of V2 would not contain any of the speeds of some other body that happened to start with a speed of 2. This example is only meant to illustrate Fabri's point, and it is not damaged by the fact that a speed of V2 in free fall was not likely to come into serious consideration by medieval writers. 1 1 The very fact that such speeds were implied by the times-squared law should make clear the immediate objection to Galileo's "all possible speeds." But back to Fabri: By what is said here, Galileo is rejected on two counts. First, it is in vain that he assumes infinite instants without necessity; and second, the ratio he gives is not convincing. Indeed, he calls rest "infinite slowness," from which the moveable then recedes; and no doubt his motion would then proceed to be propagated through all degrees of slowness. But against him, first, rest is in fact not slowness, as it [rest] can have no motion. Second, fast motion ensues immediately from rest as well as from slow motion, as is evident in projectiles. Third, motion does begin, and therefore with something, whence the initial motion is not infinitely removed from rest. And there are other reasons, proposed above, bearing on the same.

Galileo was ably defended against Fabri three years later by J.A. Le Tenneur, who on this particular point simply remarked that it was a debate over terrninology .13 But Fabri's reference to his Metaphysics, published at Lyons also, in 1648, shows that he would not have been able to see it that way. To Fabri, how we choose to name things is not arbitrary, but depends on how things really are. Galileo's fault, in Fabri's eyes, was that he could provide no way to explain continuous change, in speed or anything else, on causal principles; hence to assume continuity in acceleration was to exclude cause. Of course, that is precisely what Galileo had done - excluded cause; hence Galileo would not have considered this a fault. To him, "the way things really are" was indistinguishable from his experimental results. Fabri contended that something different from those must underlie them.

Impetus Theory and Quanta of Speed

229

What underlay the odd-number rule of increase of distances in equal times was, in the eyes of Baliani and of Fabri, the natural-number rule in the realm of physical instants That is, continuity in the macroscopic world was an appearance generated by the reality of discontinuity in the submicroscopic world I shall give Baliani 's argument rather than Fabri 's, as being the simpler to follow Baliani supposes a body to fall a certain distance in a certain time, and he divides this motion into three equal times He also divides it into spaces that are not equal, but that progress in length as the natural numbers 1, 2, 3, 4, 5, Suppose these divisions are such that the first ten spaces fall in the first onethird of the time, the second ten in the second, and the third ten in the last of the three equal times Then the length traversed in the first time is the sum of the first ten digits, or 55; that covered in the second equal time will be 155 (that is, the sum of the numbers 1 1 to 20, or 21 ~55), and that covered in the last time will be 255 But these distances, 55-155-255, are about in the ratio of I, 3, 5; and if we had made I oo progressive distances fall in each equal time, instead of Io, the numbers would be 5050, I 5050, 25050, which are still closer to 1, 3, 5 For extremely large numbers, corresponding to a very tiny initial space, Galileo's odd-number rule would be borne out by any possible laboratory investigation, though in reality the progression of spaces in physical instants would always differ from it, no matter how small we made the first quantum-jump of distance in the first instant of motion 14 Fabri's analysis is extremely lengthy and detailed; he gives 164 theorems on impetus alone, followed by 132 theorems ( I 37 if he had not made a mistake in numbering theorems 4~44) on naturally accelerated motion, following these 300 theorems with many more on motion upward, mixed motion, circular motion, and so on The arguments are medieval in flavor and tenninology, and though Fabri does not name any medieval writers, I think it is safe to see in his arguments the kind of logical development that would naturally have been made if anyone in the Middle Ages had wished to pursue the matter after Albert of Saxony Obviously, this was not the development followed by Galileo Fabri 's work shows us that such a quantum theory of impetus could have led to the odd-number rule and the times-squared law, at the hands of anybody between Albert of Saxony and Galileo But Galileo's work shows us that the law of free fall made its appearance in quite a difference way No amount of logical reconciliation of his results with those derived logically from medieval impetus theory helps one iota in reconstructing Galileo's thought That can best be reconstructed from his own works and his manuscript notes, in which vestiges of medieval impetus theory stop no later than mid-1604, with Galileo's discovery of the law of free fall and his abandonment of all causal arguments relating to natural acceleration

230

History of Science IV

In order that the reader may have some idea of the medieval quantum theory of accelerated motion as reflected and developed further after Galileo abandoned it - for I do not think it possible that he was unaware of its traditional mathematical interpretation - I shall summarize the first two sections of Fabri's treatise on motion. (The book appears to be quite rare, and is ignored in recent historical works, though Fabri ' s name was frequently mentioned by seventeenth-century physicists of great stature.) In the first book Fabri defines impetus, without asserting its existence, as a quality requiring motion or local flow of its subject, saying that if impetus exists, there is no doubt that all that is required in the subject of this quality is motion itself. Later, in a scholium to Th. 14 of Book II , he distinguishes three kinds of impetus: innate natural impetus, acquired natural impetus, and violent impetus. The first belongs to the heaviness of a body, and is never changed or destroyed except with the body itself. The second arises in natural motions apart from acceleration; I believe it is meant to apply to bouncing, and perhaps to the imaginary case of a body falling through a tunneled earth and passing beyond the center. The third arises from an external mover such as the hand, or gunpowder. A few selected theorems of the first book are of interest; thus "No motion caused by impetus occurs in the first instant in which there is impetus" (Th. 34); this indicates the essential discontinuity of speeds in acceleration. Theorems 143-151 say that impetus produced at one time is conserved as long as the motion, but is not conserved by the first productive cause (which I take to mean by the heaviness of the body itself), for if so conserved it would never be destroyed. Impetus is conserved as long as nothing requires its destruction, but innate natural impetus requires the destruction of other impetus impressed in a different direction than straight down. The unchanging natural innate impetus fights every external impetus. A scholium remarks that impetus is not perennial, as that would lead to absurdities; also that there are no forces unassociated with bodies. Innate natural impetus is never destroyed, because it is never in vain, having always the effect of gravitation; but all other kinds are ultimately destroyed, as in a resting body they would be causes without effects. The second book deals with naturally accelerated motion, which was Galileo's topic. Fabri's definition is deliberately vague, saying only that in such acceleration, more distance is traversed in each successive equal time. Fabri objects to Galileo's definition - equal speeds acquired in equal times - on the ground that this does not hold for pendulums, for example. An important postulate for Fabri 's purposes is that not every time, nor every distance, is sensible (Hyp. 4), and one of his axioms states that "The effect increases in the same

Impetus Theory and Quanta of Speed

23 t

ratio as the cause, and conversely, if applied in the same way to the same subject." Theorem 5 asserts that natural motion is from impetus, and Th. 8 identifies heaviness with innate impetus. Impetus produced in the first instant endures in the second instant, but is conserved by a different cause from that of its production (Ths. 9--1 o ). Theorem 12 asserts that new impetus is produced in the second instant, and the same in the third, etc. It follows (if each new impetus is equal), that the motion of a heavy body grows as 1, 2, 3 ... , the impetus growing in proportion to the effect. Theorems 17-19 culminate in Galileo 's definition, which Fabri believes to be proved as follows: equal impetus acquired in equal times; motion proportional to impetus; therefore equal speeds acquired in equal times. But having arrived at Galileo's definition, Fabri asserts that impetus grows in single (discrete) instants (Ths. 33-36), noting that physical time cannot be explained except by finite instants. By "instant" he says he means "all that time in which some thing is produced all at once," and all that time in which the first acquired impetus is produced he calls "the first instant of motion" - during which, as noted above, no motion is produced by that acquired impetus. Theorems 37-43 assert that speeds and distances traversed, as well as impetuses, grow as the numbers 1, 2, 3, 4 ... There is no reference here to distances from rest; everything is treated as in succession, just as I believe Albert of Saxony intended. This leads to Fabri 's misinterpretation of Galileo's rejection of spaceproportionality of speeds, in the Two New Sciences: Theorem 44. From this, I may say that the speeds grow in any instants according to the ratio of spaces run in those instants. This is surely true when the legitimate senses of the words are understood, despite Salviati's crying out in Galileo's Third Day when the progression of increments [of speed) in single instants is assumed - as in fact it was - "Why then in one [proportionality) rather than the other?" Certainly if the speed in one instant is compared with the speed in another, both will clearly have the same ratio as that of the spaces; if therefore the speed in one instant is compared with the speed in another, they will clearly have the same ratio as that of the spaces: if therefore in one instant one space is run through with one degree of speed, surely in an [additional) equal instant there is acquired double the space with two degrees of speed. Nor does it hurt, as Galileo objects, that then the motions are made uniform [sic; Galileo objected that if both began from rest, the times would be equal), since motion that is made in an instant should be considered as uniform. Indeed, I call "an instant" all that time in which some impetus is acquired all at once; and it is necessary that some [impetus) be acquired all at once. Nor is it any bar [to this) when he says that instantaneous motion cannot exist; doubtless many would deny the same. [But) I explained in my Metaphysics the extent to which instantaneous motion

232

History of Science

can exist, and in fact occurs actually, and not [just] potentially, since for any duration [N.B.) there can be a smaller, whence for any given motion a lesser can occur. '5 A more complete misconstruction of Galileo's position can hardly be imagined. Galileo spoke of a single and perceptible motion from rest; Fabri speaks of motions in two imperceptible instants taken successively. Galileo attacked the hypothesis that speeds at two mathematical points were proportional to their distances from rest; Fabri defended the rule that if in the first instant the first stretch is covered at some uniform speed, then the ensuing stretch, double the first, is covered at double that uniform speed, and hence in a time equal to the first instant. No such rule was discussed by Galileo. The fact that Fabri seems to have thought it was is itself of interest. It is evident that in the Two New Sciences, Galileo wished to refute not a merely imaginary theory, but a widespread error of his time; that speeds in fall are proportional to distances from rest. The error was a plausible one to nonmathematicians, and it had appeared in print in Varro's treatise of 1584. Quite possibly by the time Galileo published his book in 1638, some people may already have construed Albert of Saxony's rule as one of space-proportionality of. speeds, precisely as modem historians do. But the scholastically trained Jesuit Fabri could make no such mistake, having a very firm grip on both the mathematics and the physics of medieval impetus theory with its quantum analysis of natural acceleration. In those circumstances it is easy to see how it just would not occur to him that Galileo was attacking a totally different conception, and his opening words imply that Galileo simply had not understood the legitimate sense of space-proportionality. The rest of Fabri's argument is then devoted to clarifying that sense against a supposed misunderstanding of it by Galileo, and not to a consideration of the quite different sense of space-proportionality that Galieo put into the mouth of Sagredo. Fabri went on to point out that the ratio in Theorem 44 applies strictly only to physical instants, and that when larger parts of time are taken, the ratio of spaces is a bit greater than the ratio of speeds, adding that: ... the [physical] instant cannot be perceived by sense, nor even the part of time in a thousand [such] instants combined; nor [can sense perceive] the space acquired in the first instant. Many instants are required to make clear the ratio of these accelerations existing in single instants. But this matter we leave to practice and to the senses, and we [are going to] assume that other, sensible, ratio [of Galileo's] which comes close to the truth, and will not be sensibly wrong ... . 16 We have already seen Baliani ' s explanation of this point, and may neglect

Impetus Theory and Quanta of Speed

233

Fabri 's calculations in his "Discussion of Naturally Accelerated Motion" which followed Theorem 61 . In that discussion, Fabri described the experiments that seem to confirm Galileo's law and that very closely approximated his own rule, interpreted as above. He then said: There is a most certain rule that no philosopher denies: [that] whenever some experiment is such that two contrary hypotheses can stand with it, neutrality [of the experiment] can surely be deduced. Therefore Galileo cannot legitimately deduce his hypothesis from the experiments propounded, as I shall clearly show. When it is said [as by Galileo] that the second distance is triple the first, assuming equal [successive] times [from rest], this is not geometrically certain and accurate. I say that on this assertion, taking some [physical] instant in time ended or run through, if in fact it is said that the space is triple the first minus 100000 points, or that the second time is greater by 100000 instants, how is this difference either in space or time to be sensibly perceived? For every physical experiment must be subject to the senses; and no matter who says that however often observed, at many places and [different] times, it is the same and no error at all can be attributed, yet since the distances are small and insensible, as are the differences in greater, less, or equal times, spaces more or less equal to triple can be taken as triples when there is no sensible discrimination .... 17 I ask Galileo if he or anyone else can say whether one space is [exactly] triple another; and, if someone contends that it is off by 1/100000, whether this experiment is convincing?18

Since the criterion of experiment is indecisive between continuous acceleration and the quantum jumps of impetus theory, Fabri asserts that the choice must be decided by reason. Now, Galileo adduced in support of his definition the assumption that Nature always acts in the simplest way. But, says Fabri, the series 1, 2, 3, 4 .. . is simpler than 1, 3, 5, 7 ... ; whence his hypothesis is the better on Galileo's own grounds. Further, Galileo merely assumed his rule, whereas Fabri believed that he had proved his rule demonstratively; hence it was the better. 19 In another place, Fabri gave the reason for perferring impetus theory that was most telling to philosophers; namely, that he gave a physical cause, and Galileo did not. In the third corollary to Th. 6 I he put it this way: "Here is the key to the problem. The simple progression has its principle physically, not experimentally, and the odd-number progression [has its basis in] experiment, and not principle. We combine the two, principle and experiment; and in fact if sensible parts of time are taken, the former [i.e., principle] goes to the latter [i.e. the experimental results], and if ultimate [physical] instants are assumed, the latter goes to the former. " 20 For our present purposes, this summary will suffice, though there is a great

234

History of Science

deal more in Fabri's book that merits attention. He goes on to show that if equal spaces are considered, then the corresponding physical instants are in decreasing series; that this series is irrational; that the impetus decreases correspondingly; and that if sensible distances alone are considered, the times are nearly as the square roots of the distances. He remarks that if fall took place in a vacuum, this ratio would be observed, and goes on to work out a theory of fall in a resisting medium, concluding against Galileo that no terminal uniform speed would be reached. Other sections of the book deal with motion upward, mixed motion, circular motion, and such miscellaneous problems as the center of percussion. V

Fabri's book shows that a consistent and complete theory of motion could be worked out on the assumption of acceleration by quantum-jumps of speed, with its accompanying indeterminacy as to position at any given speed. The relative amount of this indeterminacy is in effect worked out arithmetically in Fabri's "Discussion," using a step-function diagram in place of Galileo's (or Oresme 's) triangular diagram of continuous change. Such a step-function, and not a true hypotenuse , was indeed implied in the earliest mean-speed analyses, which did not use any diagram. Fabri calls it a "denticulated" line 21 ; his older contemporary Libertus Fromondus, writing of paradoxes of the continuum, had called discrete steps a "serpentine" line. 22 A clear understanding that such lines never became truly straight, however small the steps, prevailed in the Middle Ages as in antiquity (the cone of Democritus) and today (standard first lecture in elementary calculus). Quantum-jumps were quite naturally associated with the medieval arithmetical theory of proportion, and with regard to speeds they were essential in order that physical causes might be assigned to all changes. Fabri's book shows further that, contrary to the view of modem historians, impetus theory did lend itself to detailed mathematical treatment. The explicit working out of all the implications of Albert of Saxony's simple rule was neglected until after Galileo announced his law of free fall and its experimental confirmation. That neglect is not surprising, since for the main purpose of medieval physics (causal explanation), mere facts were of no significance. As Galileo had Simplicio remark in the Dialogue, such details were left by philosophers to mere mechanics. 23 And as Aristotle had remarked, the precision of mathematics seems to many people mean and petty, like haggling. But in the wake of the Two New Sciences, mathematical details of impetus theory could no longer be neglected. A rival account of free fall, based on a rival mathematics, had at last been put in the field. Fabri 's less sophisticated contem-

Impetus Theory and Quanta of Speed

235

porary, Pierre Cazre, tried to refute Galileo by adducing different experiments which supported increase of speeds according to the series of natural numbers (impact experiments; the reasoning is quite similar to that temporarily adopted by Galileo in 1604 as a basis for redefining "speed"). Baliani and Fabri preferred to present Galileo's law as a mere approximation to the underlying causal law. Ever since the revival of interest in medieval physics about the beginning of this century it has been clear that it contained remarkably modem elements. It now appears that medieval writers had already adumbrated quantum analysis, indeterminacy, and the inherently approximative character of even the most carefully carried out experiments, as compared with mathematical theory . The attempt to relate medieval and Galilean physics has concealed this interesting fact. Historians write algebraic equations to represent medieval theories without regard to the essential role of integers therein, and of the theory of incommensurables. Algebraic equations are also written to represent Galileo's laws without regard to his restriction of ratios to magnitudes of the same kind. Then, the interpretative equations being the same, the same physical views are ascribed to both. When our modem equations coincide with them, as for free fall, a wholly fictitious unification of medieval, Galilean, and modem physics is made to appear historically real. The worst of this it is that it hides the medieval genius that it was designed to extol. VI

The genius of Buridan as a physicist was indeed great; but it was not that of foreshadowing the concept of inertia. The course of his reasoning about free fall is only obscured by identifying impetus and inertia. Impetus was a force, indissolubly linked with motion. As Buridan wrote, "It is also probable that just as that quality [impetus] is impressed in the moving body along with the motion, by the mover, so with the motion it is diminished, destroyed, or impeded by resistance or by a contrary inclination to motion ." 24 Inertia is not impressed in things, nor is it diminished by resistance. But the important point is the concept of struggle between contrary inclinations to motion. Diminution of impetus by a contrary inclination to motion implied that even in the absence of resistance, the violent motion of a projectile must eventually be overcome by its innate and unchanging tendency straight downward. (As Fabri later put it, contrary inclinations fight within the same body, and since the inclination downward due to heavinesss is never diminished, it eventually wins out over an initially impressed impetus, however strong.)25 This accounted satisfactorily for observed projectiles, and it agreed with Aristotle's principle that nothing violent can long endure.

236

History of Science

Now, Buridan had the genius to see that there is one single exceptional case in which there can be no contrary tendency even in terrestrial projectiles (he had already recognized this in the celestial motions). This unique case is that of a projectile hurled straight down. In this case, the impetus impressed will be completely conserved, barring external resistance, for it is not opposed by any contrary inclination at all. Hence, by using a single additional postulate, impetus could be made to give an account of acceleration in free fall. It was necessary only to assume in addition that impetus could be impressed on a body not only by an external mover, but also by speed itself, with which impetus was in any case inextricably linked. (Even with an external source, the more impetus impressed the more speed imparted.) Reasoning thus, Buridan would be led to the passage cited earlier, which begins: "One must imagine that a heavy body acquires ... a certain impetus with motion." That the resulting acceleration would be one of successive steps was also made clear by Buridan, who explained that to assume continuous action of impetus in accelerated motion would in tum require a cause for the changing impetus, just as impetus was taken as the cause for changing speed; and this would lead to an infinite regress of causes. All this was clear to Buridan's contemporaries and to classic impetus traditionalists later, like Fabri. It has become fogged only in modem times, and only through an excessive solicitude in providing Galileo with reputable sources for his physics. The puzzles mentioned earlier in this paper that have been created as by-products need trouble us no longer. Finally, it need not be surprising that no one before Domingo de Soto, about 1 550, appears to have adduced free fall as an example of uniform acceleration and related it to medieval mean-speed analysis. There is an inherent discrepancy, though it may be made as small as you wish, between the hypotenuse of a triangle and the broken line of the step-function. 26 This alone should have sufficed to keep strict logicians and mathematical physicists who adopted impetus theory from identifying natural acceleration with truly uniform acceleration. All agreed that heaviness alone acts in the first instant, and that heaviness is constant; but a constant cause must have a uniform effect, and a single uniform speed at the outset would destroy mean-speed analysis. That analysis, moreover, impled a tripling of distance in the second half of any time, whereas impetus theory impled a doubling of distance in the second instant. There was no incentive to identify (or reconcile) the two until after Galileo's Two New Sciences. Or rather, no general incentive; Father William Wallace has suggested that Soto may have exemplified uniform acceleration by free fall in order to escape the charge of idle speculation or of nominalism at a time when purely mathematical physics was out of favor in Spain, where his books were first published. 27

Impetus Theory and Quanta of Speed

237

NOTES 1

2

3 4 5

6

7 8

9 1o

11

I2

13 14 15

Cf. S. Drake, "Galileo's Discovery of the Law of Free Fall," Scientific American, 228 (May, 1973) pp. 84-92. In order to avoid a certain mathematical complexity, Galileo adopted for about four years a definition of "speed" that was equivalent to our v 2 , using this only for instantaneous speeds in accelerated motion. This implies v ex: t. Cf. le Scienze (July 1973), pp. 36 ff. Letter of Vincenzio Riccati lo his brother, Giordano; cited in Ottaviano Cametti, lettera criticomeccanica ... (Rome, 1758) p. 12. M. Clagett, The Science of Mechanics in the Middle Ages (Madison, 1959), pp. 5512, 660-01. Cited below as "Clagett." Clagett, p. 566. The speeds here in question are speeds through tiny distances during "physical instants," and not mathematically instantaneous speeds in the modem sense. They are speeds "in an instant" rather than "at an instant." We consider a particular speed as the "ratio" of a distance lo a time. Under Euclidean rules of proportionality, no true ratio exists between magnitudes of different kinds. Galileo's expressions accordingly did not assign "absolute" values to single speeds, but always paired them in proper ratios. The supposed difference in speed due to weight did not exist, as Galileo pointed out later, but such a difference was a basic postulate of Aristotelian physics. CF. Honore Fabri (ed. P. Mousnier), Tractatus physicus de motu /ocali ... (Lyons, 1646), p. 96 (cited below as Fabri). This is an abridgment of the passage cited below under Theorem 61. Letter of 11 October 1638; Correspondance du P. Marin Mersenne (ed. P. Tannery and C. De Waard), vol. 8 (Paris, 1963), p. 114. Fabri, p. 96. What were called "irrational ratios" were much discussed by fourteenth-century writers, but I doubt that they would have been considered as relevant to actual free fall by impetus theorists. Fabri, p. 96. J.A. Le Tenneur, De motu naturaliter accelerato ... (Paris, 1649), pp. 77-9 (and elsewhere). Cf. G.B. Baliani, De motu naturali gravium ... (Genoa, 1646), pp. 11~12. (This argument does not appear in the first edition, 1638.) Fabri, p. 89. Whether Fabri overlooked the specific nature of the hypothesis attacked by Galileo, or whether he believed that Galileo was arguing improperly against Albert of Saxony's rule, is not entirely clear. He began by asserting the truth of the latter "when the legitimate senses of the words are understood," and seems to believe that impetus as a causal principle guarantees the relation 1, 2, 3, 4 ... for successive

238

16 I7 18 19 20

2I

22

History of Science

spaces, as if Albert's rule were a pure tautology. But the tautologous part (spaces proportional to speeds for equal times) holds equally well for Galileo's odd-number rule, or any other. There is no tautology in asserting that the distance covered in the second equal time is double that covered in the first, though Fabri begins that assertion with "surely .... " Whether this ratio is to be detennined by experiment, or by insistence that a causal law different from observed law is the essence of true physics, constitutes the whole point in dispute. Fabri, p. 89. Fabri, p. I 00. Fabri, p. IOI. Fabri, p. I 04. Fabri, p. 97. E.g., Fabri, p. I 08. Libertus Fromondus, labyrinthus sive de compositione continui (Antwerp, 163 I), p.

45. Galileo, Dialogue (tr. Drake) (Berkeley, 1953), p. 164. Clagett, p. 537. Fabri, pp. 68-70. The problem lies in the initial unifonn motion due to heaviness alone in the first physical instant. Even if one were to allow that very tiny steps came lo the same thing as a straight line, the representative "triangle" would still have a minute rectangle at its very beginning, small as you like, but different from the rest of the "hypotenuse." This fact would have been enough to induce Oresme, at least, lo have denied that free fall can be truly unifonn diffonn motion under the impetus explanation, if he had been asked. For he says quite clearly that any configuration differing at all from a right triangle cannot represent "unifonnly diffonn quality." 27 W.A. Wallace, "Mechanics from Bradwardine to Galileo," Journal of the History of Ideas 32 ( I 97 I), p. 26. 23 24 25 26

11

Free Fall from Albert of Saxony to Honore Fabri

In summarizing the history of free fall from the fourteenth century to the seventeenth, N.R. Hanson wrote as follows: In the first moment after the removal of a body's support the impetus was held to result from an external cause - gravitational attraction. During later instants, however, the impetus itself was taken to cause movement. Temporal and spatial aspects of impetus were clearly distinguished by da Vinci: "The freely falling body acquires with each degree of time a degree of motion, and with each degree of motion a degree of velocity." Why did Leonardo, Benedetti, and Varro assert the proportionality of the falling body's velocities to the spaces traversed, and not to the times? Doubtless they regarded these as equivalent .... Given the alternatives, (a) velocities are proportional to the times, and (b) velocities are proportional to the spaces traversed [from rest], Leonardo, Benedetti, Varro, Galileo (1604), and Descartes all chose (b).'

Until quite recently I considered Hanson's question legitimate, and his answer appeared to me the only plausible one. I now believe the question itself to be based on a misapprehension that is widely shared by historians of science. It is true that Michel Varro, in 1584, supposed the speeds in fall to be proportional to the distances fallen from rest, in the ordinary sense of those words today. 2 It may be that others before him had also suggested this; but if so, I have not yet run across any unequivocal statement of it. Earlier views of the mathematical rule for speeds of falling bodies seem to me ultimately traceable to Reprinted from Studies in History and Philosophy of Science 5 ( I 975), 347-366, with kind permission from Elsevier Science Ltd. , The Boulevard, Langford Lane, Kidlington OX5 1GB, UK .

240

History of Science

Albert of Saxony and to be based on a totally different conception of proportionality. In order to have the main point as clear as possible from the outset, I shall substitute the word "unit" for "degree" in Leonardo ' s statement cited by Hanson. This will not distort his meaning. Leonardo's word, grado, originally meant a step, as on a ladder or staircase; that is, a distinct and separate unit in a discrete scale. Velocities were traditionally numbered by degrees, integrally and without fractions. Let us therefore read Leonardo as having said: "The freely falling body acquires with each unit of time a unit of motion, and with each unit of motion a unity of velocity." It we take a "unit of motion" as a unit of space or distance traversed, a kind of proportionality between times, distances, and speeds is implied in which these units are taken successively, but that will not be the same as a proportionality between the same factors taken cumulatively from rest. Under Leonardo's conception, then, the data for a freely falling body might be tabulated thus: During time

1,

it falls a distance of

I

with a speed of

1

2

2

2

3

3

3, etc.

In such a scheme the acceleration would be uniform, but it would not be continuous. Equal speeds are added in equal times, but they are added discretely, at the beginning of each new period of time, and the speed remains the same during that period. A proportionality might be said to exist between such speeds and the corresponding times from rest, though not a proportionality in the modem algebraic sense, in which we consider the time and the speed as both undergoing continuous change. But a proportionality between speeds and distances could not be said to exist in the same way as between speeds and times, even in unitary arithmetical terms, because although the numbers representing times and distances are the same, they do not accumulate in the same way. In medieval terminology, time is a successive thing while distance is a permanent thing; that is, the latter is capable of being present all together at the same time. Thus by the end of time 2, the distance has become 3; and by the end of time 3, the distance has become 6; and in short no proportionality - which was defined as "equality of ratios" - could exist between speeds and distances, as it could exist between speeds and times. There are good reasons, both mathematical and physical, for believing that writers before Varro thought of free fall in the way suggested by the foregoing tabulation. Hence it is very doubtful that any of them thought of speeds in fall

Free Fall from Albert of Saxony to Honore Fabri

241

as proportional to distances fallen from rest. It is most likely that although they regarded time and distance as continuous, they regarded speeds in physical fall as discrete, uniform, and successive. That such a view found intelligent defence and elaborate mathematical development after Galileo's analysis of speeds as continuously changing lends strong support to this view. As to the mathematical reasons, these relate to the history of the theory of proportion. Euclid treated this separately for continuous magnitudes (Book V) and for arithmetical quantities (Book VII). Because of an omitted definition and a spurious definition introduced by Arabic writers into Book V, and carried over into the standard Latin translation, medieval writers turned to Book VII for their theory of proportion. This was consequently arithmetical and unitary in character; and though it was carried to elaborate development in the fourteenth century, it did not represent the continuum as we conceive it now, or as Eudoxus had presented it in antiquity. Physical entities such as times and distances could be taken in units as small as one pleased, but they always remained durations or stretches and could not be treated as mathematical points without magnitude. It happened that this mathematical framework was also in remarkable accord with certain physical conceptions of Aristotle. For example, it was impossible in Aristotelian physics for the same mathematical instant to be both the last instant of rest of a body and its first instant of motion, since there would then be an instant at which the same body was both at rest and in motion, violating Aristotle's logical principle of contradiction. On the other hand, a first "physical instant" of motion, having a duration no matter how small, could exist and would be distinguishable from the last instant of rest. Although the above phrase emerged only later, it throws much light on medieval physics conceptually, as it does on early discussions of incipit et desinit and of the latitudes of forms. In that light we shall examine the mathematics of free fall proposed by Albert of Saxony on the basis of Jean Buridan 's impetus theory. We shall see that impetus theory did lend itself to mathematicization, contrary to the prevailing view that Galileo turned from it because it did not, and that the issues which divided him from medieval physicists were of quite a different kind, very subtle, and still undecided. II

Albert of Saxony reasoned that regular increase of speed might take place by multiplication or addition, in either case according to the natural numbers. Increase by multiplication through successive proportional parts of either a given time or a given distance would lead to infinite speed in a finite time or space, inadmissible on principle; hence he concluded:

242

History of Science

It is understood that the speed is increased by double, triple, etc. in such a fashion that when some space has been traversed by this, it has a certain velocity, and when a double space has been traversed by it, it is twice as fast, and when a triple space has been traversed, it is three times as fast, and so on.3

All modem historians have assumed that all spaces and speeds must be measured from the beginning of motion, and hence that Albert's rule displayed continuous uniform acceleration from rest, and made speeds continuously proportional to distance traversed. But another interpretation is also open; and this is that the spaces and velocities to which Albert referred were successive. In that case his rule was that the body goes a certain distance at unit speed, and then twice as far at speed of 2, and then three times as far at speed 3, and so on. Since motion twice as far at double speed implies the same time, as does motion three times as far at triple speed, this would lead to the above tabulation. Now, this is a very simple rule to grasp, and I believe that it was so understood by others, including Leonardo, as the mathematics of speed in free fall. In short, I believe that each separate speed in free fall was thought of as proportional (in the sense of discrete arithmetical proportion) to a separate interval of time. Continuous proportionality of speeds to distances from rest was indeed suggested late in the sixteenth century, as previously mentioned. But that suggestion was untenable, as Galileo demonstrated in his Two New Sciences, and its untenability would problably have been quickly discovered by medieval mathematicians if it had been proposed in their time. Galileo's refutation of it cannot be used against Albert's rule as I understand it; it holds only against continuous acceleration as a function of distance. Indeed, it is quite possible that Albert's rule does, in fact, hold for actual falling bodies, if the "physical instant" used as the unit of time is sufficiently small. At any rate, tiny quantum jumps of speed cannot be disproved experimentally. In order to explain this statement, I shall revive an argument devised by G.B. Baliani in 1646. 4 Baliani never doubted the fact that measured distances in fall, during measured equal times, increase as do the odd numbers from unity. But he suggested that behind this rule may lie another, in which insensible distances increase as do the natural numbers, as in the rule which I ascribe to Albert of Saxony. Baliani reasoned thus. Take any three successive measured distances of fall in successive equal times, starting from rest. Suppose each of these to be divided into ten distances, but not equal distances; these smaller distances are to progress as the natural numbers, assumed to be passed in equal times. Then the first ten such distances are coverd in the same measurable times as the second ten, and as the third; but the sums of small distances so covered, expressed in the new units, are respectively 55, 155, 255, which distances are very nearly in

Free Fall from Albert of Saxony to Honore Fabri

243

the ratio I :3:5. Now, if we had used one hundred divisions instead of ten, the corresponding numbers would become 5050, I 5050, 25050. Clearly the ratio I :3:5 would be borne out to any desired degree of accuracy, for distances covered in physical instants milliions of times shorter than any we can measure, assuming that such distances covered in equal times progressed as do the natural numbers. Thus a quantum theory of free fall, with a succession of extremely short but increasing uniform speeds succeeding one another contiguously, is just as tenable as Galileo's assumption of continuity of speeds in free fall. We adopt the latter assumption because it facilitates application of our mathematics - that of the infinitesimal calculus - to free fall. And why should not Albert have made the former assumption, which made his mathematics - that of arithmetical proportion - easier to apply? Thus Albert's rule (as I understand it) does not involve an internal contradiction, as does Varro's rule of continuous proportionality of speeds to distance fallen . All that was wrong with Albert's rule is that it does not apply to actual measurements of freely falling bodies. Albert's rule implies that in the second of two equal times, the falling body goes twice as far as in the first of those times; but, in fact, for measurable distances, an actual falling body goes three times as far. This faults Albert's rule for measurable phenomena, but it does not fault Albert, who did not appeal to experimental facts. Measurements formed no part of medieval mathematical physics, which was concerned only with causal explanation. Nor would it have been easy for anyone to have judged, by merely observing instances of free fall, whether the ratio of distances in the first of two equal times was 2: I or 3: I. One of the principal virtues of this interpretation is that it does not have to attribute to Albert of Saxony, Oresme, and Leonardo any simple confusions or elementary oversights. When anything forces the historian to impute mistakes of that sort to such men, the chances are that the historian himself has failed to grasp fully the ideas they expressed. This might be a dangerous principle to apply in the history of philosophy, but it is reasonable one in the history of mathematics or of mathematical physics. For (as Galileo remarked) it is the nature of mathematics to reveal mistakes to us in so glaring a manner that we hasten to correct them. The conflict between the ratios 2: I and 3: I for distances in the first two equal times of fall is so sharp that we need not wonder any longer why medieval mathematicians never suggested that free fall is an instance of uniform acceleration. They knew well enough that the 3: 1 ratio would hold for truly continuous uniform acceleration. When Domingo de Soto suggested, much later, that free fall was amenable to the mean-speed rule, he may have achieved a truly

244

History of Science

remarkable insight, which he simply did not bother to develop mathematically. But it is more likely, as Father William Wallace has suggested, that Soto merely did not wish to expose himself to the charge of idle speculation by discussing motion without giving concrete examples, at a time when the Church in Spain was much opposed to vain theorizing. 5 Ill

Historians of science have engaged during recent decades in a search for links connecting Galileo with the Middle Ages, instead of searching for historical events that could account for a new approach to the science of motion. The crucial event, in this instance, was the restoration of the authentic Eudoxian theory of proportion in Euclid, Book V. Not only a new science of motion arose from this, but also a new branch of mathematics, the infinitesimal calculus. Since the preparatory events have been strangely neglected even by historians of mathematics, a brief account of them is justified here. The most ancient theory of proportion was based on number, which was placed by the Pythagoreans at the basis of all nature. The discovery that no ratio of integers could express the relation of the diagonal of a square to its side shook the foundations of that view of nature; magnitudes that were incommensurable could not be ruled by number. But so long as they were magnitudes of the same kind, they were capable of having ratios to one another. The means of working with such ratios were provided in due course of Eudoxus, who defined two essential concepts for the new theory: that of "having a ratio to one another," and that of "same ratio." The first of those definitions stated simply that two magnitudes of the same kind have a ratio to one another if the smaller may be made to exceed the larger by simple multiplication. This definition, however, was omitted from the particular Arabic text from which the standard medieval Latin Euclid was translated. The second Eudoxian definition stated in effect that ratios are the same if multiplication of their first terms by any number, and multiplication of their second terms by any number, always produced ratios that were both greater than unity, both equal to unity, or both less than unity. This definition was properly translated in the Latin Euclid of the Middle Ages, but in the absence of the other it was interpreted incorrectly to define what we would call continued proportion, rather than proportionality of continuous magnitude. Book V was thus reduced to triviality. As a result, medieval theory of proportion was based on the pre-Eudoxian arithmetical theory of Book VII. Despite the fact that arithmetical theory was carried in the fourteenth century to a peak of extremely subtle development, medieval writers did not modify the ancient definition of "number" as "multi-

Free Fall from Albert of Saxony to Honore Fabri

245

tude of units," as was ultimately done late in the nineteenth century; hence they did not quite attain the arithmetization of the continuum on which algebra and function theory were finally to be rigorously based. Applications of medieval proportion theory to physics were therefore linked to the idea of multiplicity of units, rather than to the concept of the continuum. Thus physical units might be infinitesimal in size, but physical entities were not treated as mathematical points, even though limits of infinite series were correctly understood as numbers in many instances.6 The concept of"velocity" may appear to be an exception, since a clear meaning was given in the fourteenth century to the concept of a mean speed as existing at a mathematical point. This ideal case was analysed mathematically, but it was not then applied to the physical phenomenon of free fall. Motion at a point was physically contradictory, and velocity was a property of motion alone. The Italian Renaissance was characterized by a certain revulsion against works transmitted by the Arabs, and by a return to Greek texts. The medieval translation of Euclid from Arabic had been in print for about two decades when, in 1505, an Italian humanist, Bartolomeo Zamberti, published a new Latin translation from a Greek manuscript at Venice. The crucial Eudoxian definition was thus restored, though Zamberti did not fully expound the classical theory, being content merely to castigate the errors of Campanus of Novara. Luca Pacioli soon came to the defence of Campanus with a "corrected" edition of the old text and commentary, (1509), ignoring the restored definition. And, in fact, the elaborate terminology of medieval proportion theory had so long been built into calculational methods that professors of mathematics paid no attention to this dispute, and arithmetics teaching the old methods continued to be printed throughout the sixteenth century. Zamberti's translation was frequently reprinted together with that of Campanus during the sixteenth century. But in those dual editions, the erroneous commentary of Campanus continued to be published without any explanation of the true sense of Book V. It was only in I 543 that Niccolo Tartaglia, in his Italian translation of Euclid, commented specifically on the errors of the medieval interpretation and explained the real significance of the two key Eudoxian definitions. He pointed out that although zero and infinity were excluded from ratios by them, ratios of incommensurables were permitted, and means were provided for dealing with them rigorously even though such ratios could not be represented by integral numbers. The way was thus reopened to the mathematical treatment of continuous magnitude, and publication of the authentic works of Archimedes in 1543 and I 544, simultaneously provided examples of the power of those methods. In fact, Archimedes had derived the proportionality of times and distances in uniform motion by applying the Eudoxian definitions to the Aristotelian defini-

246

History of Science

tions of equal and greater speed. It remained only to extend this mathematics to accelerated motion, which was the contribution of Galileo. It may be wondered why, if the reintroduction of Eudoxian proportion theory was all that was needed, the new science of motion was not produced very soon after 1543. Various factors enter into the answer. First, it should be remembered that university instruction was given in Latin, and it was not until the 1570s that Latin commentaries on Euclid began to explain Book V correctly. 7 Tartaglia's vernacular translation was intended for self-educated people like himself, and though it may have influenced his pupil, G.B. Benedetti, and Galileo's teacher, Ostilio Ricci, it had no known effect on university courses. Second, Pacioli's defence of the erroneous medieval interpretation reflects the power of a longentrenched conventional terminology and calculatory technique that took time to be superseded algebraically. Pacioli himself had written the earliest and most elaborate work on proportion to have been printed, a work of great importance to the development of algebra. Yet even he remained oblivious to the importance of Book V when that was freed from dependence on numerical proportion theory . And finally, the new science of motion had to await the attention of someone who was interested both in mathematics and in physics, against the orthodox Aristotelian tradition that separated the two disciplines. IV

Galileo's use of the Eudoxian theory of proportion began no later than 1588, when he applied it to a lemma in a preliminary version of his treatise De motu. 8 In the finished treatise, probably in 1590, the notion of continuity of changing speeds was repeatedly applied. An example is Galileo's argument against the Aristotelian dictum that in reversal of motion, rest intervenes. Galileo reasoned that a body thrown upward and falling back could never be in uniform upward motion, or a rest, for any time whatever, or through any distance whatever; the instant of reversal of direction was a mathematical point and not a duration. But it was a long time before he discovered the mathematical law of free fall; nearly a decade and a half, in fact. The recognition that acceleration is not just a temporary event at the beginning of motion, but acts uniformly and continuously in fall, took a long time, and it required the abandonment of entrenched causal ideas in the analysis of motion. It was still longer before Galileo found a rigorous logical basis for the establishment of his law of free fall. Even in 1632, in his Dialogue, he gave only a probable reason for its acceptance, and it was only in 1638 that he published a full discussion and derivation. It is thus evident that although Eudoxian proportion theory was a necessary condition for Galileo's new science of motion, it was not also a sufficient condition.

Free Fall from Albert of Saxony to Honore Fabri

247

In presenting his analysis of free fall in his final book, Galileo appealed (as he had previously done in the Dialogue) to the principle of sufficient reason, arguing that for a body to pass from one speed to a higher speed, there was no reason to suppose that any possible intervening speed had been skipped. Knowing that this would puzzle his readers, he had his interlocutors question whether a body could pass through an infinitude of different speeds in a finite time, to which he replied that the possibility existed so long as no speed was held for more than a mathematical instant of time. This argument was not accepted by many of Galileo's immediate successors; thus Descartes wrote to Mersenne that, though this might happen sometimes, it was not the usual way in which bodies fell. 9 Remarks of that sort are hard for us today to understand, but that is because we do not see them against the background of medieval tradition, properly understood. In their anxiety to provide Galileo with medieval predecessors, historians have attributed to the young Galileo and to Albert of Saxony an error that neither of them made, but that appears to belong to the late sixteenth-century amateur Michel Varro. This same error is then read into the thinking of Descartes himself, and in place of a proper history of subtle differences and fundamental principles, we have a misch-masch of elementary confusions attributed to writers of great intelligence who in fact differed on basic issues of both physics and mathematics. This is what Father Joseph Clark called writing history von oben bis unten, attributing errors to earlier scientists in the light of present day science and mathematics, rather than that of their own times. ' 0 The notes of Leonardo da Vinci on accelerated motion are amenable to the same interpretation already given for Albert's rule. Discrete increments of distance, time, and speed are to be assumed whenever Leonardo speaks of the falling body as acquiring one additional unit of speed and one additional unit of distance with each unit of time. On that basis, his notes are both intelligible and intelligent. Benedetti, so far as I know, did not discuss the question of speeds as such, but rather that of forces. Judged by effective impact after fall from a height, forces do increase proportionally with the distances fallen, though speeds, as we define them, increase only as the square roots of distances fallen from rest. It was natural enough in Benedetti's day to reason that since the body remains the same, only the "speed" can account for the greater blow, and hence that the "speed" is proportional to the distance fallen. Benedetti did not specifically say this, as Galileo did in his letter to Paolo Sarpi in 1604. The important thing to note here is that such a statement is an error only if the same word "speed" is simultaneously used in its modem sense. Anyone who simply defines "speed" in a way different from ours, as whatever it is that does increase proportionally

248

History of Science

with distance fallen (that is, our v2 ), and does not also, during the same period, use the word "speed" in our sense, need not be guilty of inconsistency .1 1 It is merely misleading for historians to call this treatment of "speed" a mistake. Benedetti's treatment elsewhere of speed in fall adduced the same causal concept as Buridan's, and there is no reason to suppose that he would have dealt with it mathematically in any way different from Albert of Saxony or Leonardo, especially since that rule is borne out by the action of piledrivers. Benedetti's new insight in this matter had to do with equal speed of fall for similar bodies in a given medium, regardless of weight; he did not investigate the mathematical rule governing a given body in acceleration from rest. This brings us to the discussions of Descartes and Beeckman in 161 8, long after Galileo had arrived at all the correct rules but before he had published them. Beeckman proposed to Descartes the problem of finding the distance traversed in a second equal time from rest, given the distance in unit time, and assuming that the same force acts perpetually and no speed is lost. Descartes' solution, which has been much criticized, was as follows: In triangulo isocelo rectangulo ABC spatium (motum) repraesentat; inequalitas spatii a puncto A ad basim BC. motus inequalitatem. lgitur AD percurritur tempore, quod ADE repraesentat; DB vero tempore quod DEBC repraesentat: ubi est notandum minus spatium tardiorem motum reprasentare. Est autem AED tertia pars DEBC: ergo triplo tardius percurret AD quam DB. 12

In translations of this passage, spatium is universally taken as in the accusative; thus Koyre gives Dans le triangle isoce/e rectangle, ABC represente I' espace (le mouvement) ... , and Hanson gives: "In the right angled isoceles triangle ABC represents the space (the movement) .... " But this seems very dubious; the body is twice spoken of as traversing AD and DB, and never as traversing the space ADB or the space ABC. Let us therefore read spatium in the nominative: In the right-angled isoceles triangle, space ABC represents [the motion]; the inequality of the space [ABC] from point A to the base BC [represents] the inequalities of the motion. Therefore AD is run through in the time that ADE represents; but DB [is run through) in the time that DEBC represents; where it is to be noted that the lesser space represents the slower motion. Further, AED is one-third of DEBC; therefore it runs through AD three times as slowly as [through] BD.

This does indeed solve the problem presented by Beeckman, and it arrives at a ratio well known to medieval mean-speed writers, though it makes no use

Free Fall from Albert of Saxony to Honore Fabri

249

whatever of the Merton rule. It was perhaps a bit arbitrary on the part of Descartes to associate the ratio of times with the ratio of "motions," but he was certainly entitled to let areas represent the "motions" associated with distances of fall AD and DB and to assume that constant action of the same force produced a uniform inequality of those "motions." On the other hand, Descartes was not obliged to explain what he meant by "motion," as distinguished from the ordinary meaning of "motion through AD, DB, or AB." The answer to Beeckman was that in the second hour, the body would fall three times as far, as in the first. Yet it would be no more true to say that Descartes' solution implied the times-squared law than it is to say (as is frequently said) that the medieval ration of 1:3 for distances in two halves of a given uniformly accelerated motion from rest implied that law.• 3 Descartes' other writings show in fact that he rejected Galileo's deduction of the times-squared law for actual falling bodies, and that he rejected it precisely because he did not regard speeds in fall as rigorously continuous. The question asked of Descartes in 1618 concerned only a particular case, that of doubled time, and that is the only question that his solution specifically covered. Had Beeckman asked how far a body falls in the third hour, Descartes would problably have answered "five times as far as in the first hour, given your assumptions"; but that is still not equivalent to his stating the times-squared law. Descartes, in fact, went on to propose different assumptions, similar to those used in a passage of Leonardo's, whence it is not clear that he accepted Beeckman 's. Even Beeckman's own solution, which ended with the use of a square root and which is generally seen as a complete and correct analysis, was still limited to the case of fall in a given time, and in its double. It rested on the "double proportion" of area to line; and while it is true that this phrase meant the "squared ratio," it does not follow that Beeckman saw his "double proportion" as applying to any case except that of doubled time. I do not mean to detract from the high regard that is properly felt for the acumen of Beeckman as a mathematical physicist, but I do mean to point out and to stress again and again that he did not take the final step, which could be so easily put in a single sentence, of saying "and what has been shown for this case holds equally for all cases; that is, that whatever the ratio of times from rest, the spaces traversed are as their double proportion or squared ratio." The reason for the absence of this step, in Beeckman as well as in Descartes, lies in the physical conception behind the mathematical representation. Mathematically, Beeckman' s little triangles can be diminished until they are reduced to zero, and, as he remarks, "such is the moment of space traversed by the body." But physically, the vanishing of all the little triangles removes any

250

History of Science

meaning from the concept of "moment" itself. The justification for the macroscopic diagonal in Beeckman's diagram is lost when all the microscopic diagonals vanish - unless one is willing to abandon the idea of tiny moments, and regard the entire change as truly continuous. But then the little triangles are redundant to begin with. It is not their size, but their shape that makes the argument physically plausible. After Galileo did abandon finite increments of speed, and published a probable argument in the Dialogue (having not yet found the means of presenting it as rigorously deduced), Descartes commented on it in a way that is now widely taken as embodying a mistaken recollection of his own previous solution. But that interpretation also seems to me to be wrong. Descartes wrote: But I must declare to you that I have found in his book some of my thoughts; among others, two that I think I have written to you before. The first is that the spaces through which heavy bodies pass when they fall are to one another as the squares of the times they take in descending; that is to say that if a ball takes three moments to descend from A to B, it will take only one [moment] to continue on to C, [twice as far], etc. This is what I said, with many restrictions, since in fact it is never entirely true, as I am thinking of proving ... 14

Certainly Descartes had said earlier that the speed in the first time was onethird of that in the second; and Galileo had presented this as a corollary of his times-squared law; hence, for Descartes to associate the two in referring to the past was not unreasonable. What is important, it seems to me, is not Descartes' seeming claim to have thought of the times-squared law first; such a claim, while not strictly true, would conform to Descartes' habits of talking. What is important is that in this later reference, he implies not only that the timessquared law is not entirely true, but also that in his earlier statement, not even the special case of triple time for the first half of the distance should have been taken too seriously. As he remembered it, he had given this solution only "with many restrictions"; and that is true. If we look back, we see that Descartes presented it only as the solution of a problem proposed to him by someone else, under assumptions of constant force and conserved speed that he himself did not necessarily accept as physically true. With such restrictions, he was willing to give a purely mathematical solution. I have already stressed that Descartes' solution should also be looked on as having another restriction, which may or may not have been present in Descartes' mind when he later wrote the above; namely, that only a single special case was ever discussed. This was the same special case that was always discussed in the Middle Ages in connection with uniform acceleration; namely the

Free Fall from Albert of Saxony lo Honore Fabri

251

bisection of a bounded motion, and not its division into any number of equal parts. The most that was implied by this would be the result of repeated bisections (the medieval "proportional parts" toward rest), all of which taken together would still leave an infinitude of other cases completely undecided. Those cases are decided only by the assumption of continuity of change in speeds, in the fully modem sense; and that assumption is not likely to be found among writers seeking the cause of acceleration. But that had been the main objective of all writers on the subject of free fall up to 1638, when Galileo openly rejected any such quest. V

The quest for cause as the main objective of physics was not abandoned by others merely because Galileo ridiculed it. Hence it should be no surprise that Descartes was not alone in rejecting Galileo's law of free fall. Pierre Cazre rejected it on one basis, and Honore Fabri on another. Fermat gave a proof against it before 1638, and a proof in its support afterwards. Gassendi supported it with reasoning that Galileo would certainly have repudiated, while Mersenne supported it with the remark that experiments did not fully bear it out, and that a different rule was equally acceptable. Even Torricelli's vindication of it by a theory of accumulating instantaneous impacts was much more Cartesian than Galilean in spirit. In short, even those who did not reject Galileo's law of free fall did reject the physics in which he embedded it, a physics as free as possible from the ancient concept of causal explanation. If the attacks on and defences of Galileo's law of free fall after 1638 had been studied by historians as assiduously as they have sought anticipations of it in the Middle Ages, it would have been noticed long ago that the essential novelty in Galileo's theory of free fall was the passage from a quantum theory of speeds to a continuity theory, or from infinitesimal assumptions to the use of indivisibles in the sense of Cavalieri. In the last analysis, this comes down lo the forsaking of all arilhemetical theories of proportion in favour of the Eudoxian theory. Viewed in that light, medieval physics would be found far more complete and consistent than is presently believed. It did indeed have a strong influence on physicists of the seventeenth century. About the only one it did not influence was Galileo, who is precisely the one that historians have been most anxious to portray as having been so influenced. The most illuminating book with regard to medieval traditions in the seventeenth century is Honore Fabri's Tractatus physicus de motu locafi in quo effectus omnes, qui ad impetum ... pertinet, explicantur, published at Lyons in 1646, Book II, De motu naturali, employs the hypothesis that greater and lesser arcs

252

History of Science

of a pendulum are swung in equal times (Hyp. 2) and that not every time, or every distance, is sensible (Hyp. 4). Fabri's axioms are that impetus is additive; that the ratio of increase in cause is the same as the ratio of increase in effect; that the same necessary cause produces equal effects in equal times; that any effect not produced by the primary cause or by any external cause is produced from within; and that a cause acts in proportion as it meets with less resistance. These hypotheses and axioms, quite different from Galileo's, would have been acceptable in the main to medieval physicists. On the basis of these assumptions Fabri developed a vast number of theorems. Many agree with Galileo's, and many contradict positions of his. Theorem 19 proves that equal times of fall result in equal speeds. Fabri notes this to have been assumed by Galileo, when, in Fabri's view, it needed proof. His proof is that motion grows proportionally to impetus, while equal degrees of impetus are acquired in equal times (Theorems 17 and 18). Theorems 20 to 32 prove many of Galileo's propositions; Fabri notes that Galileo gave them, but neglected to supply any reason in terms of physical causes. In the immediately ensuing theorems, basic physical concepts emerge that are fundamental to Fabri's anti-Galilean position. Those concepts, I believe, reflect the unspoken assumptions of most of Fabri's contemporaries and of all his medieval precursors. Theorems 33 to 36 declare that impetus grows in single instants which are all equal, and a scholium adds: Note above equal instants, since physical time cannot be explained otherwise than by finite instants, as shown in our Metaphysics [Lyons, 1647). But however that may be, I call "instant" all that time in which some thing is produced all at once; whether [this is] larger or smaller, or [whether there exists] a larger or smaller part, does not matter for our purposes, since given any finite time you can have a larger and a smaller [time]; that is certain. Therefore all that time in which the first acquired impetus is produced, I call "the first instant of motion," in which there then succeed equal times thereafter. 1 5

Thus a first "physical instant" (as Fabri later calls these) exists in which a first degree of impetus is produced, and there is not a uniform and continuous growth of that first degree of impetus from rest. On this, Fabri's physics departs from the physics of Galileo, just as Galileo's physics had departed from all previous views of free fall. Once this notion of "first instant" is clarified, the rest follows easily. Theorem 37 states that impetus grows in arithmetic progression as single instants add equal impetus; Theorem 38, that speeds grow in the same way, as do the natural numbers. Theorems 40 to 43 apply the same to successive distances traversed. Theorem 44 and its scholium explain the sense in which proportionality can be said to exist between speeds and dis-

Free Fall from Albert of Saxony to Honore Fabri

253

tances, which is not at all the sense of continuity theory, but is strictly discrete and arithmetical: From this I may say that the speeds grow in any instant according to the ratio of spaces run in those instants. This is certainly true, when the legitimate senses of the words are understood, though Salviati cries out in Galileo's Third Day when the progression of growth by individual instants is assumed - as in fact it was - "Why therefore at one [instant] rather than another?" [Salviati spoke of speed at one mathematical instant, not in one "physical" instant.] Certainly if the speed in one instant is compared with the speed in another, both will be in the same ratio as that of the spaces [traversed in those "instants"] ; if therefore in one instant one space is run through with one degree of speed, surely in [two] equal instants there is acquired double space, at two degrees of speed; nor does it hurt, as Galileo objects, that then these motions are uniform; for motion that is made in an instant should be considered as uniform. Indeed, I call "an instant" all that time in which some impetus is acquired all together. And it is necessary that some [impetus] be acquired all together, nor is it any bar when he says that instantaneous motion cannot exist, which many people would doubtless deny. I explained in my Metaphysics how far instantaneous motion can exist, which in fact is given actually and not just potentially, since for any duration there can be a smaller, and thus for a given motion, a lesser can be given. Observe that this ratio of spaces is the same as the ratio of the speeds assumed to exist in these spaces, [and to be] acquired in single instants. If we take larger parts of time, wherein many instants flow together, doubtless the ratio of spaces is greater than that of speeds; for example, if the first [unit) space is acquired in the first instant, 2 in the second, 3 in the third, and 4 in the fourth, then if we compare the speeds of the first instant and that of the fourth, they will have the same ratio as that of spaces, that is, 1 :4. But if parts of time are taken consisting of two instants, then of these four instants there are made up two equal parts of time. In the first, a space of 3 is acquired, and in the second, 7; ... and if the speed in the second instant is compared with the speed in the fourth, this is as 1 :2 , but the first space is to the second in the ratio 3:7 ... 16

Theorem 61 again flatly contradicts Galileo: "Naturally accelerated motion is not propagated through every degree of slowness." ... Galileo is rejected by what is here said, on two grounds. First, because he unnecessarily assumes infinitely many instants; and second, because the ratio he gives is not convincing. He calls rest "infinite slowness", and no doubt his [kind of] motion would be propagated through every degree of slowness. But against this, first, rest is not in fact slowness, since rest can have no motion. Second, fast motion follows immediately on

254

History of Science

rest as well as on slow motion, as is evident from projectiles. Third, motion begins, and hence with something, whence the first motion is not infinitely remote from rest ... I reject the opinion of those who want the acceleration of natural motion to be made so that the spaces acquired in equal times follow the odd numbers t, 3, 5, 7, etc., with the spaces as the squares of the times ... which is obviously false from what has been said, and certainly [is false] if in equal times there are acquired equal moments of speed. 17

The last sentence quoted shows the abyss between Fabri and Galileo. The latter deduces the odd-number rule from the proposition that equal increases of speed are acquired in equal times, while the former deduces from the same proposition the negation of that rule, holding that it implies instead the rule of the natural numbers for distances traversed in successive equal times. That was the rule I believe to have been formulated by Albert of Saxony and accepted up to the time of Galileo; its sophisticated defence by Fabri (and independently by G.B. Baliani in the same year, 1646) shows the vitality and continuity of medieval physics, which no one denies. At the same time it shows the independence of Galilean physics from medieval tradition, which virtually all modem historians have denied. One mathematical rule follows from the application of arithmetical, discrete, analysis, and the other from geometrical, continuous, analysis. Since the birth of quantum theory in 1900, conflicting physical consequences of the two approaches have again become notorious - on a higher plane, indeed, but nevertheless in a way that makes it easier for us now to see the basic conceptual issues between Galileo and his immediate critics, between medieval physicists and Galileo, and between Pythagorus and Eudoxus. But it is very important to recognize that Fabri did not challenge Galileo on the results of his careful experiments, nor indeed on any measurable and testable phenomena; his challenge was solely on the conceptual basis of motion itself. What was at issue was the manner in which physicists should imagine events that lie beyond the reach of experiment. Some of Fabri 's contemporaries appear not to have understood that this was the issue, and to have supposed him antagonistic to Galileo's results; for in a later book, written in dialogue form, Fabri took pains to correct that misapprehension. 18 The real issue will be clear from the ensuing extracts from Discussion of Naturally Accelerated Motion, inserted immediately after Theorem 61: The opinion thought out by Galileo is confinned by his supporters on two heads; first, by experience, and second, by reason. As to experience, three very powerful experiments are offered. The first is by motion vertically downward ... for indeed many grave writers on philosophical matters have often tested this sensibly, and repeated their experiments ad nauseam . ... The second experiment is on the inclined plane ... and Galileo says in

Free Fall from Albert of Saxony to Honore Fabri

255

express words that he himself tested this often and found no discrepancy ... . The third experiment is taken from pendulums, in which it is observed that the lengths ... are as the squares of the times .... On these three very powerful experiments is founded the hypothesis of Galileo, which in my opinion could not be more clearly or more sincerely expounded [than above] . Before adducing the reasons behind this opinion and refuting them, I shall show first how our hypothesis can stand with these experiments. Certainly there is a rule concerning this that no philosopher denies; namely, that whenever some experiment is such that two contrary hypotheses can stand with it, the experiment can be deduced to be neutral [to both]; therefore I shall show that Galileo cannot legitimately deduce his hypothesis from the experiments mentioned . ... Taking either some instant in time or some distance completed or run through, if one says that in fact the second distance is triple the first minus one million points, or that the second time is greater than the first by a hundred thousand instants, how is the difference in either space or time to be sensibly perceived? For every physical experiment must be subjected to the senses . ... I ask Galileo whether he, or anybody else, can say whether one space is [actually] triple another; and whether if someone says that it is off by one one-millionth, this experience carries conviction? The first reason Galileo gives in favour of his law is that since Nature always uses the simplest means in her operations, and since acceleration in natural motion cannot be made more easily and more simply than in this progression of squares [of times], there is no doubt that natural acceleration is made in that ratio, especially since all experiments agree with it and it can explain all the phenomena. I reply that the first arithmetical progression, in these simple numbers 1, 2 , 3, 4, is much simpler than that by I , 3, 5, 7 as no one would deny. Next since two hypotheses happen to agree with all the experiments or phenomena, there must be some reason why one should be chosen over the other; but there is no reason why Galileo takes his, as we shall see. But we prove our ratio demonstratively; therefore ours is to be preferred as the true theory of things. Yet since the other approximates, for sensible times, to the true one, it may be taken in practice and for the ordinary measure of these motions . ... Galileo's hypothesis is false under the hypothesis of finite instants, since new increments of speed are made in single (physical] instants; but speaking physically. it comes out the same as if it were true, since it can be tested only in sensible parts of time. Surely. since any sensible part contains almost innumerable instants in which the progression takes place, the difference between the two [hypotheses] cannot be made sensible . ... Thus, in the common opinion, in which it is said that time consists of actual infinite parts, Galileo's progression can stand. Therefore, here is the key to the difficulty; the simple progression has its principle physically, not experimentally; the progression by odd numbers [has its basis] by experiment and not in principle. We compound the two, by principle and experiment ... '9

256

History of Science

Elsewhere, Fabri puts his point in this way: Galileo's theory and Fabri's agree in observed results, but the fonner cannot give the cause of acceleration, while the latter can and does; hence it is superior. Now it is quite true that on the assumption of continuity, the quest for cause is vain (as Galileo himself recognized), whereas the assumption of intennittent action pennits construction of a causal apparatus. Which physics is superior, however, remains a matter of taste. Likewise, the manner in which a heavy body near the earth actually does fall, in tenns of microseconds, can hardly be detennined by experiment (or if it can, then its behaviour in micromicroseconds cannot). But again, whether this is a matter of proper concern to physicists or belongs rather to metaphysics remains a matter of taste. Galileo's physics was thus not real physics at all by medieval standards, or by the standards of intelligent traditionalists like Fabri. Mere proportionalities of free fall answered none of the questions that had always interested academic physicists, precisely for the reason that Aristotle had given in the first place against the utilization of mathematics in physical inquiries: The minute accuracy of mathematics is not to be demanded in all cases, but only in the case of things which have no matter. Hence its method is not that of natural science; for presumably the whole of nature has matter. 20 Fabri's standard was also that of Descartes, who remarked that Galileo had built without foundation because he had merely examined the ratios of some motions, without first having detennined the cause of motion. In a way, the same objection was to be brought against Newton's law of universal gravitation, and Newton himself felt the need to look further for a cause. Galileo's idea of physics was, in short, a mere anachronism that has no place in the history of philosophy. NOTES

Hanson, Patterns of Discovery (Cambridge, 1958), 39; hereinafter cited as Hanson. Michael Varro, De motu tractatus (Geneva, I 584), 14. Albert of Saxony, Questiones in libros de celo et mundi; see Document 9.2 in Marshall Clagett, The Science of Mechanics in the Middle Ages (Madison, 1959), 566, 568; hereinafter cited as Clagett. G.B. Baliani, De motu naturali gravium solidorum et liquidorum (Genoa, 1646), 111-13. W.A. Wallace, "On the Concept of Motion in the 16th Century," Proc. Am. Cath. Phil. Assoc. (Washington, 1967), 194.

I N.R.

2 3

4 5

Free Fall from Albert of Saxony to Honore Fabri

257

6 This statement is meant to be limited in application to the best-known medieval physical treatises as distinguished from mathematical treatises. A statement that for the physical entity, heat, degrees are like points on a line is found in a Renaissance summary on motion prefixed in 1505 to Bradwardine's Treatise on Proponions; see Clagett, Document 7.1, p. 450. This was probably written by Bassano Politi for his edition of several medieval treatises. Among the latter was that of Giovanni de Casali, in which "degrees" were identified with parts of latitudes rather than with points, and were said to be either uniform or difform, which seems to have been the ordinary medieval viewpoint. Cf. verso of third unnumbered leaf from beginning of Casali's treatise, ed. Politi, Venice, 1505: Gradus in /atitudine nihil a/iud est quam pars latitudinis . ... ; Gradus uniformiter difformis est caliditas difformi.~ ... , etc.

But see Clagett, 38~7. for other readings in manuscripts. 7 The Latin edition of Flussates de Candalla (de Foix) in 1566, which strongly influenced the English Euclid of 1570, did give the correct definitions, but it then explained them arithmetically, replacing Euclid's line diagrams with numerical examples. 8 A. Favaro (ed.), Opere di Galileo, I (Florence, 1929), 348-g. 9 A. Fararo, note 8, XVII, 391 (Descartes to Mersenne, 11 October 1638). 1o M. Clagett (ed.), Critical Problems in the History of Science (Madison, 1959), 103-4. 11 S. Drake, "Galileo's Discovery of the Law of Free Fall," Scientific American (May, 1973), esp. 90-2. 12 Cf. A. Koyre, Etudes Galileennes (Paris, 1939), 1o6. 13 Cf. E. Grant, Physical Science in the Middle Ages (New York, 1971), 5~7. 14 Hanson, 49. 15 H. Fabri, Tractatus physic us de motu /oca/i ... (Lyons 1646), 87-8; hereinafter cited as Fabri. 16 Fabri, 89. 17 Fabri, 9~7. 18 H. Fabri, Dialogi Physici ... (Lyons, 1665), 66. 19 Fabri, 9~ io8, passim. 20 Aristotle, Metaphysics, 995a, I 5ff.

12 Impetus Theory Reappraised

The importance of medieval impetus theory in the history of physics is unquestioned and unquestionable. Whether or not it represented an entirely new departure in the explanation of motions continued by a body in the absence of contact with any visible mover, the development of impetus theory in the fourteenth century carried its explanatory value far beyond anything that had been suggested by earlier Greek and Arab physicists. That medieval development produced the first fully rational account of a mechanism by which acceleration in free fall could be explained, an account which remained the best until Isaac Newton propounded the law of universal gravitation more than three centuries later. It is also believed by most historians of science that medieval impetus theory led on logically and almost inexorably to inertial concepts in the works of Galileo, Descartes, and Newton. When we consider the fundamental status of inertia and of acceleration in free fall with respect to modem physics, it is evident that time spent in the analysis of medieval impetus theory in relation to those claims cannot be entirely wasted. To the best of my knowledge, certain ideas set forth in the present paper have not previously been advanced. If they are borne out by further investigations, they will greatly modify the prevailing historical interpretation of impetus theory. Such investigations, however, would require the expertise of medievalist historians of science who are familiar with manuscript sources still unpublished. Venturing on my analysis largely because of the bearing of impetus theory on the work of Galileo, I am limited to the study of printed materials, from which I have drawn tentative conclusions that may not accord with unpublished documents known only to specialists. On the other hand, my analysis seems to I. -

Reprinted from the Journal of the History of Ideas 36 ( t 975), 27-46. Reprinted by permission of the Johns Hopkins Universi1y Press.

Impetus Theory Reappraised

259

be supported by a number of books printed in the mid-seventeenth century by later spokesmen for impetus theory who adduced it sometimes in opposition to, and sometimes in confirmation of, Galileo's law of free fall. These works are perhaps as unfamiliar to medievalists as unpublished medieval manuscripts are to me, and they may throw light on some phases of the thought of earlier impetus theorists that have remained puzzling to those who have devoted the most study to them. Prominent among such puzzles is the fact that concurrently with the best fourteenth-century expositions of impetus theory, the mean-speed analysis of uniform acceleration was also introduced and vigorously pursued. This can hardly have remained unnoticed by impetus theorists, nor their work by meanspeed analysts; yet application of uniform acceleration to the case of free fall is strangely absent in the fourteenth and even the fifteenth century. Indeed, it has appeared that the basic concept used in mean-speed analysis - the proportionality of speeds in acceleration to times elapsed from rest - simply did not occur to impetus theorists in their discussions of free fall, and it has seemed that instead they adopted quite arbitrarily the rule that in natural acceleration, speeds were proportional to distances traversed from rest. All historians, including the present writer in earlier papers, have adopted the view that no one before Galileo so much as suspected that there might be a difference between the two concepts. Galileo himself, in his last book, acknowledged that he had not questioned this for a long time. Given the sophistication of medieval mathematical physicists, the puzzle is real; but if my analysis is correct, its solution lies precisely in their sophistication, and not in any oversight or carelessness on their part. Central to the analysis presented herein is the thesis that medieval writers generally, and especially those who applied impetus theory to the explanation of acceleration in free fall, did not regard the speeds as continuous in the modem mathematical sense of that word. Galileo did, and therein lies one of the paramount differences between his physics and that of his medieval predecessors. It will perhaps be thought that in saying this, I wish to press a certain value judgment and to say that Galileo's physics was better than medieval physics by being more modem, not only in point of time but in soundness of mathematical basis. That is not my intention here, though I may argue that way elsewhere. So far as modernity is concerned, medieval discontinuities and discrete quanta of speed have the upper hand over Galileo's continuous-motion hypothesis, which incidentally was promptly rejected by Descartes. Likewise, Galileo's mathematics was far more ancient in this context than was that of the Middles Ages; for he relied on the Eudoxian theory of proportion that was not completely preserved in the standard Latin Euclid of the Middle Ages. Medieval writers devel-

260

History of Science

oped their own arithmetical theory of proportion, which far transcended the Pythagorean theory embodied in the seventh book of Euclid's Elements. There was nothing wrong with any of those theories of proportion, but they were far from being all the same thing; nor (in my opinion and that of the ancient Greeks) would any of them have been strictly deducible from any other.' Each theory of proportion, when applied to physics, yields a characteristic set of ideas concerning the underlying physical entities, and that is about all that can be usefully said about them. Which one is "really true" is, in a sense, still under debate, though many of us now reject such questions as illusory. Granting, then, that my analysis of medieval impetus theory may be overthrown by researches among still unpublished manuscripts, I believe it deserves attention meanwhile for a variety of reasons. It offers some basis for understanding Descartes' seemingly curious rejection of Galileo's assertion that in attaining any speed from rest, a body must first have passed through every lesser speed, such speeds being infinite in number. It also throws light on the efforts of Pierre Gassendi, G.B. Baliani, and Honore Fabri to reconcile Galileo's law of free fall with a causal explanation which Galileo did not even deign to mention, though it was adhered to by such eminent contemporaries of his as G.B. Benedetti, Isaac Beeckman, Baliani (and perhaps Descartes himself at one stage), and was favored in a new form by Galileo's disciple Evangelista Torricelli. My view offers a possible new construction of certain ideas common to Albert of Saxony, Nicole Oresme, Leonardo da Vinci, and Galileo at an early stage of his work, ideas taken up again by Pierre Cazre in direct opposition to Galileo's mature conclusions. Most important of all, it identifies a specific event in the history of mathematics, which occurred in I 543, as the explanation of an otherwise puzzling lag in the development of mathematical physics between Oresme and Galileo. Translators know only too well that more pitfalls exist in dealing with ordinary words used in technical senses than occur in assigning suitable senses to technical neologisms. A clear example is to be found in the universal choice of the ordinary word "motion" or its equivalent as the translation of Aristotle's kinesis. We now think of motion only as change of place with respect to time, whereas kinesis in Aristotle's physics embraced not only local motion, but any kind of change that took place in time. Only those changes that occurred instantaneously were excluded; these were called not "motions" but "mutations," and were dealt with not in Aristotle's Physics but in his book on creation and destruction. Thus kinesis included the growth of a tree, change of color in heated iron, or the aging of an individual, none of which are thought of by us as "motions." But if the translators had chosen "change" to render kinesis then 11 -

Impetus Theory Reappraised

26 I

other problems would have arisen; for example, most of the things we call chemical changes would have been regarded by Aristotle not as changes, but as mutations. Historians of science dealing with old translations are confronted with problems that make the above example trivial. Not only the original text, but the translation in use at a certain period, must be taken into account; how, then should we translate an old translation? Historical questions relate not only to the ideas in the original, but to their understanding in some other version. Meanwhile, medieval commentators frequently introduced terms of their own that may have had an ordinary usage at the time but were intended in a technical sense; and in some cases the terms may since have acquired ordinary usages. There is one word in particular, used in the discussion of impetus, that deeply affects the prevailing account of its historical relation to inertia. The word is permanens, usually treated now as if it could only have meant "permanent" as opposed to "temporary"; that is, enduring unchanged and forever, as if derived form permaneo. Now, it happens that this word had also quite a different sense in the thirteenth and fourteenth centuries, a technical sense of "being present all at once" or "all being there at the same time," from permano (penetrate, flow through). In this sense it was opposed not to "temporary" but to "successive," the latter being said of things which, like motion, could not be present all at the same time. Speed, or intensity of motion, on the other hand, could be present all at once. We can see the point, a bit anachronistically, by considering the way in which we now use a phrase like "sixty miles an hour." A one-hour journey cannot be present all at the same time, but somehow the whole sixty miles is present at the instant of our saying that we are going sixty miles an hour. Medieval writers did not use such phrases (speaking rather of "degrees of speed"), but that did not keep them from carefully distinguishing entities that had to be spread out from those that could be present at one instant. This brings us in tum to the idea of an instant. In modem usage, an instant is a mathematical point in time without duration or extension. That concept was by no means unknown to medieval physicists; it had been used by Aristotle, for example, in his refutation of the existence of the void. He argued this from the impossibility of instantaneous motion, and here the instant was a durationless point in time, since Aristotle was not concerned with denying extremely swift motion. He meant that motion could not occur except in time; otherwise it would be mutation. It follows that for physical motions, physical instants rather than mathematical instants were required, and a physical instant had necessarily a tiny duration, a certain latitude. During a physical instant a tiny distance was traversed, and hence a definite degree of speed, which might be very large,

262

History of Science

could be present. But it is doubtful that "instant" in physical discussions of the Middle Ages referred always, or even frequently, to mathematical instants in our sense. Medieval physicists were much concerned with the concept of a "first instant" of motion. The last instant of rest, if identical with the first instant of motion, would violate the principle of non-contradiction, since it would imply an instant at which a body was both at rest and in motion. There was no way around this with mathematical instants; but a physical instant having any duration at all, however small, offered no problem. It is easy to distinguish the last physical instant in which no motion was present from the first physical instant in which motion is present. Indeed, Aristotle's celebrated discouragement of the excessive use of mathematics in physical problems (often exaggerated today by Platonist historians) was firmly based by Aristotle himself on the presence of matter in all things concerning physics, and its entire absence in all strictly mathematical considerations. Perfectly consonant with this would be the tacit assumption by medieval physicists of a sort of atomic structure of physical time, made up of physical instants divisible only potentially. Here I am not trying to assert anything about the various positions taken in the Middle Ages, but rather am searching for a coherent and consistent view in which Aristotelian principles were sustained in the process of attempting to extend and improve (rather than to contradict, as Galileo later did) Aristotle's physical conclusions. The phrase "degree of speed" which came into use in the Middle Ages is also deserving of attention. The Latin gradus very definitely has the meaning of "step," in such senses as that of a pace forward or of the step of a ladder. In adopting this word for the distinction of velocities, as likewise for coldness, fever, and the like, there is little doubt that medieval writers had in mind discrete gradations. The assignment of numbers to degrees was thus natural, and it was also natural to think of these units as being smaller rather than to think of them as being divided. Later on, for Galileo, the Latin gradus gave way to Italian grado, and he came to see the number of these gradi as infinite in all naturally accelerated motion, such as free fall. The opposition raised against that conception suggests, however, that the traditional notion was one of finite steps of speed toward the terminal velocity of a given motion. Much more could be said along these lines, and equally much in opposition to them. But enough has been said, I believe, to justify provisionally the analysis of medieval impetus theory that follows. Neglecting precursor theories in antiquity, it is generally agreed that impetus theory began to take shape in the 132o's with the proposal by Fran-

111 -

Impetus Theory Reappraised

263

ciscus de Marchia of a "leftover force" (vis derelicta) operating on a projectile after it had left the effective source of violent motion. This force was thought of as residing in the projectile (rather than in the medium, as Aristotle had suggested), and as weakening with the progress of the motion until the projectile came to rest. The leftover force became a force impressed on the projectile by the mover, and it acquired the specific name of "impetus," with Jean Buridan in the midfourteenth century. Buridan's use of the word "permanent" as applied to impetus has generally been taken as meaning that he attached to impetus what was to become the essential kernel of the later concept of inertia, because the impetus would never leave the projectile except as a result of some external force counter to it, such as resistance of the medium or the striking of an obstacle to motion. I believe that this view is mistaken and that it at present greatly hampers our understanding historically of the introduction of an inertial concept. Even more, it prevents a proper historical appreciation of many writings of the mid-seventeenth century, in which impetus theory was adduced sometimes to counter and sometimes to support the initial results obtained in inertial physics. That the permanence assigned to impetus by Buridan was meant by him in the sense opposed to successiveness, and not temporariness, seems evident from the passage most commonly chosen to illustrate the contrary. In the translation by Marshall Clagett, this reads as follows: The first [conclusion] is that that impetus is not the very local motion (itself) in which the projectile is moved, because that impetus moves the projectile and the mover produces motion. Therefore, the impetus produces that motion, and the same thing cannot produce itself. Therefore, etc .... The second conclusion is that that impetus is not a purely successive thing (res), because motion is just such a thing and the definition of motion [as a successive thing] is fitting to it, as was stated elsewhere. And now it has just been affinned that that impetus is not the local motion (itself). Also, since a purely successive thing is continually corrupted (destroyed) and produced (created), it continually demands a producer. But there cannot be assigned a producer of that impetus which would continue to be (always) simultaneous with it. The third conclusion is that that impetus is a thing of pennanent nature (res nature permanentis), distinct from the local motion in which the projectile is moved. This is evident from the two aforesaid conclusions and from the preceding [statements]. And it is probable (verisimile) that that impetus is a quality naturally [i.e., physically ] present and predisposed for moving a body in which it is impressed, just as it is said that a quality impressed in iron by a magnet moves the iron to the magnet. And it is also probable that just as that quality - the impetus - is impressed in the moving body along with the

264

History of Science

motion by the motor, so with the motion it is remitted, corrupted (destroyed), or impeded by resistance or by a contrary inclination.2

It is the third conclusion that particularly interests us here. If the words "of permanent nature" (or "naturally permanent," as I should translate the phrase) are read in the normal modem sense, the first sentence indeed suggests some inertia-like property; that is, it sounds as if the projectile had a certain impetus independently of the particular local motion undergone. Yet permanence in this sense is hardly reconcilable with the concluding clause, which says that along with the motion itself, the impetus "is remitted, destroyed, or impeded by resistance or by a contrary inclination." Unlike inertia, impetus - though distinct from the motion - behaves just like the motion and is weakened or destroyed along with it. Hence impetus does not, in any modem sense, reside permanently in the projectile whether it is in motion or at rest; it dwells there temporarily, only when there is motion, and in an amount proportioned in some way to the motion. Impetus is a property imparted by a mover, as magnetic attraction is a property imparted by a magnet. It is a kind of force (impressed force), and forces are not normally permanent in the modem sense of that word-3 On the other hand, if Buridan's words "naturally permanent" imply "as opposed to successive," then the entire passage is free of difficulty. It says that impetus is not motion; that motion is successive by nature, and that impetus is permanent by nature; that is, it is all there at a given time, though it changes in amount from one time to another and even from one instant to another. In any physical instant, the amount of impetus present is all the impetus there is in the projectile at that instant. Such is the meaning I take to be Buridan's. We may have difficulty in conceiving an instantaneous permanent quality, but medieval physicists did not; thus the distinguished writers of the article on Walter Burley in the Dictionary of Scientific Biography note that "intrinsic limits are naturally appropriate when a given permanent thing has only instantaneous existence."4 Elsewhere, Buridan remarks that "the impetus would last indefinitely (ad infinitum) if it were not diminished by a contrary resistance or by an inclination to a contrary motion," and goes on to say that such is the case with the celestial motions.5 This is indeed a case of permanence in the modem sense, and it is to be noted that here Buridan does not say permanet, but in.infinitum duraret, supporting the view that "permanent" was not intended in its modem sense elsewhere in Buridan 's discussion. The same passage also introduces another point of central significance to the analysis proposed in the present paper; this is his clear distinction between (1) a resistance, and (2) an inclination to some contrary motion. Neither exists with regard to celestial motions; no resistance is offered by the aetherial medium, and heavenly bodies have no tendency to any

Impetus Theory Reappraised

265

motion other than circular. The case is different with terrestrial heavy bodies, in a way that is frequently neglected. A heavy body near the earth meets with resistance in many forms; it is resisted by the air, a contrary wind, a solid obstacle, a yielding obstacle, a body moving in the opposite direction, or a body moving more slowly in the same direction. All these may be lumped together as resistances, and thought of as external forces or impediments, in Buridan's day as in ours. Such things, he says, may reduce, destroy, or hinder the impetus of a projectile along with its motion. But separate and distinct from all such forces (if not to our minds, to those of all medieval physicists) was the existence of any inclination on the part of the projectile to some contrary motion. Whatever its direction of flight, every projectile was seen as endowed with a natural inclination to move straight downward to the center of the earth. We have Buridan 's express word for it that impetus is reduced, destroyed, or impeded by a contrary inclination; and in terrestrial bodies, unlike celestial ones, there is always present an internal inclination to go downward, whatever other motion may be followed by it at a given time. Hence - with a single important exception to be discussed later - the impetus in Buridan 's projectiles was always in the process of being reduced by the natural contrary inclination downward; and thus in point of fact, as he saw things, terrestrial projectiles were always undergoing a reduction of impetus that would eventually bring them to rest, even in the absence of external resistances such as those mentioned above. Buridan appears not to have discussed the paths of projectiles, but the common view at his time - and well into the sixteenth century - was that only one motion could prevail at a given time in a given body. If the force of projection carried the body horizontally or at any angle, that was because it was stronger than the natural inclination of the body downward. But the unnatural force had to lessen with the motion, until eventually the natural inclination became the stronger. At that time, the natural inclination took over and the body dropped to earth. There is no reason to believe that Buridan held a different view, or that he conceived of a composition of motions during the flight of a projectile. This concept of a contrary inclination, capable of weakening impetus wherever both were present, is treated by some recent historians as simply another external force and is lumped together by them with the resistances previously listed. In this way, Buridan's impetus can be made to appear as coming very close to the later inertial concept; indeed, to become identical with Newton's vis inertiae. But such a treatment contradicts the very basis of Aristotle's physics, in which all motions were classified as either natural or violent. 6 Natural motions were cause by an internal or intrinsic property, which in the case ofterIV -

266

History of Science

restrial heavy bodies carried them straight down, and to speak of this property as an external force (our "force of gravity") is an unpardonable anachronism. If Buridan had intended to challenge such a fundamental tenet of Aristotle's physics as the distinction between external and internal causes of motion, he would certainly have devoted many pages of argument to the task. But there was no reason for him to challenge this. The distinction between natural motions, undertaken by a body merely freed from restraint, and violent motions induced by external forces, was a very useful one. Even Galileo, who was not notably influenced by the authority of Aristotle, habitually made use of that same distinction and its terminology, though he early rejected the completeness of the ancient dichotomy. For Galileo there were also neutral motions, neither natural nor violent, and there were in addition preternatural motions at speeds incapable of being reached by any actual falling body. But Buridan's classifications must be regarded as Aristotelian, and as distinguishing external resistance from internal inclination. v - Up to this point, reasons have been given for believing that ( 1) the term "permanent" applied by Buridan to impetus was meant only in opposition to "successive," and not to "temporary" or even "instantaneous"; and (2) that so far as terrestrial projectiles were concerned, impetus was bound to waste away with the violent motion associated with it, just as it was bound to do so under the concept of leftover force as previously introduced by Franciscus de Marchia. Buridan offered an explanation of the wasting process in terms of resistance or contrary inclination, or both; this made better physics of it than his predecessor's simple assertion of such wasting away. It also opened to Buridan the possible explanation of the eternal circulation of heavenly bodies, in the absence of his specified conditions of wasting; and that was a very substantial contribution to physics. It now remains to comment on Buridan's explanation of acceleration in free fall, which I have already called the first fully rational and the best explanation of that phenomenon known up to the time of Newton. In my opinion, Buridan arrived at it in a way that is highly creditable to him as a physicist. To describe this brings us to the important exceptional case of projectile motion mentioned in the previous section. Under Buridan's theory, a projectile moving in any direction would necessarily Jose impetus and come to rest because of the universal "natural" inclination of heavy bodies to go straight down. This is most obvious in the case of a body hurled straight upward, which is the only case that Buridan seems to have specificalJy discussed. But granting his reasoning to be quite general, it implied one special case in which the rule of wasting would not hold, barring external

Impetus Theory Reappraised

267

resistances; and this would be the case of a projectile hurled straight down. In that event, contrary inclination to the imparted motion would be. absent, as in the case of the heavenly bodies; hence, barring external impediments, and the fact that the projectile must eventually strike the earth, the impetus would be pennanent in the modem sense of that word. I believe that Buridan promptly saw in this a means of explaining acceleration in free fall. All that was needed was for him to take a single further step, and though that step may seem simple in retrospect, it was profoundly significant to physics. Thus far, impetus had been thought of as associated only with violent motion; that is, as always imparted to the projectile by an external mover. But by virtue of Buridan 's close association of impetus with speed, why should impetus not likewise be imparted by natural motion? Taking this momentous step, Buridan could explain acceleration in free fall by an accumulation of impetus, which would be conserved without wasting in one (and only one) direction; namely, that in which there could be no contrary inclination. This condition is fulfilled in downward motion, and Buridan wrote: One must imagine that a heavy body not only acquires motion unto itself from its principal mover, i.e., its heaviness, but that it also acquires unto itself a certain impetus with that motion. This impetus has the power of moving the heavy body in conjunction with the permanent natural heaviness. And because that impetus is acquired in common with the motion, hence the swifter the motion is, the greater and stronger the impetus is. So, therefore, the heavy body is moved from the beginning [of motion] by its natural heaviness only; hence it is moved slowly. Afterward it is moved [still) by that same heaviness and [also] by the impetus acquired at the same time; consequently it is moved more swiftly. And because the movement becomes swifter, therefore the impetus also becomes greater and stronger, and thus the body is moved by its natural heaviness and by that greater impetus and so again it will be moved faster; and thus it will always and continually be accelerated to the end.7

This explanation is an elegant example of good physical thinking. The consequences of a previous concept of impetus, together with a single additional postulate, are brought to bear on a fonnerly unsatisfactorily explained phenomenon of nature. The new element is an implicit notion of conservation, and in this respect it is inertia-like. But it is not the "pennanence" of impetus that counts in this conservation, which occurs only in motion in one special direction; and unlike inertia, impetus is still a force-like entity imparted to the body rather than a property inherent in it. VI -

The principal point of this paper depends heavily on the passage just cited.

268

History of Science

From the beginning of motion, the heavy body in free fall is moved by its natural and internal heaviness alone; according to Aristotle, this motion is induced by the desire of every body to be at rest in its natural place, which for heavy bodies is as near to the center of the earth as they can get. The same cause continues to operate at all times, and to operate uniformly. But, according to Buridan, at some place other than the beginning of motion a second cause begins to operate along with the constant heaviness, this cause being the impetus acquired along with motion. No impetus exists at the beginning of motion, where there is only the natural inclination downward. Now, this treatment introduced a necessary discontinuity into acceleration, overlooked by historians, that is the key to medieval impetus theory and to its much later development. It is clear that from the beginning of motion to a certain other point, which may indeed by extremely close to the initial point of rest, Buridan assumes only a single cause of motion; namely, natural heaviness. At a certain other point, impetus is first added, and thereafter two causes act conjointly. We may call this second point "the beginning of impetus," and we may place this as close to the "beginning of motion" as we like; but we cannot make the beginning of impetus and the beginning of motion quite coincide, or we should have a body at rest and yet already having a certain speed. Now, in the first physical instant of motion, the body must have been considered as having a certain uniform motion downward, dependent only on its weight if we are faithful to Aristotle. The speed of this motion, dependent on a uniform cause, would remain uniform if it were not for the subsequent intervention of impetus. The motion induced by heaviness alone, however, becomes accelerated by virtue of impetus added at the end of the first physical instant of motion; this impetus operates uniformly through the next physical instant, together with the natural heaviness, which continues to act uniformly as before. At the end of this physical instant, impetus is again added, acting uniformly together with heaviness through the next instant; and so on. Any other scheme that I can think of conflicts either with Buridan' s description or with the fundamental principles of Aristotelian physics. But in this scheme the addition of impetus (and hence of speed) takes place in successive quantum-jumps, and not continuously. If represented graphically, such acceleration would give a step-function rather than a triangle; and no matter how small the steps, the "hypotenuse" of such a figure will never become a straight line. In fact, the length of this broken line would be the sum of the sides of the "triangle," and not the square root of the sum of their squares as in a genuine triangle. 8 The view that medieval writers thought of free fall as a series of tiny quantumjumps in speed gives a key to the little they wrote about the mathematics of natural acceleration. Before discussing this, I might point out that although literal

Impetus Theory Reappraised

269

continuity of acceleration was perfectly understood mathematically at the time, it was physically inadmissible in free fall because it would require both impetus and natural heaviness to be present at the beginning of motion. Even if Buridan 's description had not excluded this, it would lead to unifonn speed rather than acceleration - unless some new cause of motion were added a bit later; but that would only bring back the same problem in infinite regress. In short, the difficulties concerning quantum-jumps of speed in successive physical instants were not as serious as the difficulties concerning mathematically continuous analysis as applied to Aristotelian notions of physical cause. This situation was to change only with the restoration of Eudoxian proportion theory in the sixteenth century. Now, the simplest available way of representing discrete additions of speed, numerically, would be to assign one degree of speed to the (unifonn) motion in the first physical instant, and to assume that this produced one unit of impetus. The (unifonn) motion in the second physical instant is then at 2 degrees of speed - one from the (constant) natural heaviness and one from the added impetus. These 2 degrees of speed in tum give two units of impetus to be added to natural heaviness in the third physical instant, producing 3 degrees of speed in that instant, and so on. Thus the speed during any physical instant would simply be the ordinal number of that instant. There is a sense in which this scheme would make the speeds "proportional" to the times, but it is not quite the same sense as that in which we now speak of speeds in continuous uniform acceleration as proportional to times, since each physical instant would have a certain duration. This would be the simplest way of mathematicizing the impetus theory of acceleration, and is like the first way that Galileo later thought of when he undertook to establish proportionalities between speed, distance, and time in accelerated motion.9 There is a good reason to think that most other people around the time of Galileo also thought of the process as consisting of little spurts of unifonnly growing unifonn motions, each lasting only a physical instant. But there seems to be a difficulty about attributing this simple mathematical scheme to the original impetus theorists. If the times are equal physical instants, and the speeds grow as the numbers 1, 2, 3 ... , then the successive distances will also be as those same numbers; but the cumulative distances from rest would then be as 1, 3, 6, 1o, ... ; that is, as the triangular numbers and not as the square numbers. Even neglecting the fact that proportionality of speeds to times implies distances from rest proportional to the squares of the times, which clearly was not obvious even to the mean-speed analysts of the fourteenth century, we must deal with the view, now universally accepted among historians; that speeds in free fall were regarded as proportional to the distances traversed

270

History of Science

from rest by everyone who discussed the matter up to Galileo, and by Galileo himself in the early stages of his attack on the problem. We are forced to admit, then, that any writer who declared that speeds in free fall are proportional to distances from rest could not possibly have thought in tenns of the above scheme, where the speeds are as 1, 2, 3, ... and the distances from rest are as 1, 3, 6, .... But we are not forced to admit that any fourteenthcentury writer did in fact believe the speeds in fall to be proportional to the distances from rest, unless we can find such a declaration in so many words. It is not a difficult idea to express, and was certainly within the powers, say, of Albert of Saxony. But I am not aware that he ever expressed it in so many words. What he did say was this: When some space has been traversed, [the speed] is some amount; and when double space is traversed, it is faster by double; and when triple space is traversed, it is faster by triple; and so on beyond .... 10

Certainly these words appear to mean the same thing as "the speeds are proportional to the spaces traversed," and they are now universally so construed by historians. But another interpretation can be put on them, and it is to this that I want to call attention. If Albert of Saxony thought of speed as being increased successively rather than continuously, the above fonnula may mean simply that in the initial distance, the speed is V; and then during the next distance equal to double the first, the speed is 2 V; and then during the next distance equal to triple the first, the speed is 3 V, and so on for further distances and speeds. Reasons for believing that this may very well have been his intention are given below. Assuming that it was, then it is clear that the times of the successive motions described above would necessarily be equal, and hence the mathematicization outlined previously would apply here. It would follow that nothing was intended to be said about proportionality of speeds to overall distances from rest. As to the idea that Albert and his contemporaries thought of acceleration in free fall as made up of successive discrete unifonn speeds - a concept rather hard for us to accept because of our habitual use of notions later advanced by Galileo - we may consider this earlier passage by Albert in the same section: Natural motion does not accelerate by double, triple, and so on in such a way that in the first proportional part [one-half] of the hour it is a certain speed and in the second proportional part [one-quarter] of the hour [one] twice as fast, and so on .... 11

In its context, this is not a denial of discrete accelerations, but rather it is part

Impetus Theory Reappraised

271

of an argument against rules that would produce infinite speed in a bounded motion. But the phrases "at a certain speed," and "twice as fast," suggest to me that Albert did think of each speed as continuing uniformly over a given time, and then being replaced suddenly by a higher speed, uniform over the next specified interval of time, and so on. At no place do I find such a phrase as "because physical speed is always changing with each mathematical point in time," and I do not believe that this idea was ever part of the outlook of impetus theorists. As I presently view the matter, then, the mathematicization indicated by Albert of Saxony went as far as it could in the direction of quantifying speeds in free fall on the theory of discrete increments of impetus. There was really nothing that could be said about overall speeds from rest through any interval except the first. The mean-speed rule is simply not applicable to this kind of acceleration. Much later, the two analyses were very ingeniously reconciled by Baliani and Fabri, but only after the times-squared law had been shown by Galileo to hold for actual experimental results. It would be too much to expect earlier theorists to have pursued the matter so far; and by the same token, it should be no surprise that Galileo reached his results by an entirely different mathematicization of free fall, employing continuity concepts physically different from those of impetus theory, and using a theory of proportion unfamiliar to medieval Latin writers (through no fault of their own). It suffices to have shown the possibility (and I believe the great probability) that Albert of Saxony, Nicole Oresme (whose analysis in this matter closely paralleled that of Albert), and Leonardo da Vinci did not overtly and clearly state an incorrect rule of spaceproportionality of speeds in the sense now attributed to them, that is, to distances from rest; and that there was nothing logically or mathematically incorrect in what they did state, though that happens not to correspond on the macroscopic level to what we observe in the laboratory. 12 To conclude this section, it may be added that the implied cumulative distances from rest in the unending series I, 3, 6, IO, ... do not appear in medieval analyses for a very good reason. A motion, properly speaking, had to be closed at both ends by a terminus a quo and a terminus ad quem; hence all medieval analyses of physical motion were made by subdivision within a bounded line. A profound difference between these analyses and that of Galileo, widely overlooked, was introduced when Galileo left his triangular diagrams of increasing speed open at the bottom; this is true not only of Theorem II on accelerated motion in the Two New Sciences, but as early as 1604, in the discovery document and in the subsequent demonstration written out for Paolo Sarpi. The medieval analysis sufficed for any bounded motion; but the odd-number rule and the times-squared law are not easy to discover by the division of bounded

272

History of Science

motions into proportional parts, as medieval analysts customarily divided them. VII - Medieval mathematicians did not generally make use of the correct and complete definitions of the Eudoxian theory of proportion set forth in Euclid, Book V, not in Campanus, though translated by Gerard of Cremona. Instead, they developed a very ingenious arithmetical theory of proportion from Euclid, Book VII and from other ancient sources plus their own acute additions, brought by Oresme to a point almost equivalent to our own arithmetization of the continuum that was hardly perfected before Dedekind and Cantor. Nor did Aristotelian physicists need the continuum for their analysis of motion, because Aristotle himself had declared that projectile motions may appear continuous, but the only truly continuous motion is that induced by the unmoved mover. •3 The proper text of Euclid, Book V was restored in the Latin translation from Greek by Bartolomeo Zamberti, printed at Venice in 1505. But no clear commentary in which the entrenched errors of the standard medieval text and commentary on this book were corrected appears to have been printed until I 543. Then Niccolo Tartaglia did this in a vernacular edition, with his own derisive remarks as a first-rate mathematician and practicing teacher of hard-headed practical men. Tartaglia' s Italian Euclid was probably not used in any university, and the importance of the Eudoxian theory to mathematical physics was hardly recognized until Galileo applied it to the analysis of motion. Neglecting causes and forces, he published near the end of his life a rigorous kinematical treatment of falling bodies. This was diluted in a posthumous edition ( 1655) by an attempt to introduce a dynamical proof of his earlier postulate that speed at the end of a fall is independent of the path taken, at least along straight lines. Galileo's basic view of accelerated motion was that it was continuous; that is, that in order to reach any speed from rest in uniform acceleration, every lesser speed must have been passed through. This position implies some things quite repugnant to common sense, to which Galileo replied in dialogue form. But even a mathematical physicist of the caliber of Descartes rejected the concept outright, saying in letters to Marin Mersenne that although this might sometimes happen, it certainly was not the usual case with falling bodies. 14 It is hard today to understand such an attitude, though the writings of Galileo's immediate successors throw some light on it. In the year of Galileo's death, his stout supporter Pierre Gassendi published a book on the motion impressed by a moving mover, and in this he attempted to derive Galileo's law of free fall from a mixture of considerations of impetus and influence of the medium. Gassedi ' s reasoning was not very convincing, and drew some letters from a Jesuit, Pierre Cazre, who believed Galileo 's rejection of space-proportionality of speeds to be equivocal and fallacious, and who also

Impetus Theory Reappraised

273

believed his times-squared rule for distances to be wrong. Cazre held - as I believe Albert of Saxony had implied - that the distances traversed in successive equal times during fall were not as the odd numbers, but as the natural numbers 1, 2, 3, 4, ... He believed Galileo's experiments with inclined planes worthless because no one could measure times accurately enough to conclude anything. (Cazre, like some of Galileo's recent critics, failed to perceive that Galileo's device of weighing flowed water was designed to measure ratios of times, for which purpose it is extremely sensitive.) Cazre declared that his own experiments conclusively proved speeds in fall to be proportional to distances, and expressed surprise that Galileo had not hit on such experiments for himself. 1 5 Ironically, Cazre's experiments published in 1645 were essentially similar to observations on which Galileo, in 1604, had grounded his demonstration of the times-squared law for Paolo Sarpi. At that time, as I have explained elsewhere, Galileo adopted a definition of velocity that corresponds to our v 2 , doing this in order to escape a certain clumsiness of expression that he at first thought was forced by the now ordinary definition that he adopted four years later. 16 Galileo at first relied on observations of pile drivers, which produce double effect when dropped from doubled height, reasoning that since the weight is the same and only the "speed" changes, the "speed" was proportional to the height of fall. Cazre arranged a very strong balance so that one pan rested on a table, heavily weighted, and then dropped a fixed weight from different heights into the other pan, noting the height required to move the weighted pan. Gassendi's lengthy reply to Cazre in 1646 threw little new light on the dispute. But in that same year G.B. Baliani and Honore Fabri published books (at Genoa and Lyons respectively) in which the incorrect rule of Cazre (who was mentioned by neither of them) was related on the one hand to Galileo's correct law, and on the other hand to classical impetus theory. These books are not as well known to historians as they should be. Fabri's work in particular brilliantly illuminates the impetus doctrines that had always been around since the fourteenth century, and were generally accepted, but which were not developed mathematically from the time of Albert of Saxony and Oresme until after Galileo had announced his experimental results and his view of accelerated motion in free fall. Baliani had published in 1638, simultaneously with Galileo's Two New Sciences and independently of it, a book on motion in which the times-squared rule was derived from certain assumptions about the motions of pendulums. 17 His expanded edition of this, published in 1646, has been very unjustly treated. Baliani had obtained the odd-number rule from Galileo in 161 5, and he never abandoned it or its correlative, the times-squared law. But in the 1646 edition he set forth a conjecture that was misconstrued by later writers and resulted in the

274

History of Science

incorrect rule of Cazre receiving the name of "Baliani 's law," widely subjected to scorn. What Baliani in fact suggested was not that the odd-number rule was wrong, but that behind it there lay the simpler and "real" rule of the natural numbers for spaces traversed in successive equal times on a very minute scale. Thus, he said, take a distance actually traversed in fall and divide it into little distances progressing in length as the numbers 1, 2, 3, .... Let the first ten of these be traversed in a given time, the next ten in a second equal time, and so on. Then the total distance traversed in the first time is 55, the sum of the first ten spaces; that in the second time is I 55, the sum of the 11th to 20th; and that in the third, 255. Now, 55, 155, and 255 are pretty nearly in the ratio 1:3:5; and if we had divided the line so that I oo parts instead of ten were found in the first time, and so on, then the approximation to the ratio I :3:5: ... would be still closer (as 5050: 15050:25050). Hence the spaces passed in very tiny equal times could really be as the natural numbers 1, 2, 3, ... although for measurable times the distances would be as 1, 3, 5, .... Baliani thought he had thus discovered the true and simpler behavior of distances and times during physical instants. The reader will note that Baliani 's idea squares with the simplest mathematics of the classic impetus theory in which quantum-jumps of speed were assumed; and of course, for equal minimal times, the successive speeds would also increase as the distances traversed. The same idea was developed still more ingeniously and at much greater length, with many theorems of accelerated motion based on quantum-impetus additions, by Honore Fabri. 18 In one scholium he even outlined the method of determining the theoretical departure of observation from "real" motion in terms of the size of a physical instant, of which only a finite number could be contained in a given motion. One of his theorems throws light on the objection of Descartes; Fabri reasons that a body dropped vertically goes faster than the same body dropped along an inclined plane; therefore, it must start faster; therefore, there must be some degrees of speed acquired along the inclined plane that will be absent from those passed through by the vertically falling body. I adduce this theorem not in order to be sarcastic about it, by any means; rather, it shows that a very elaborate body of self-consistent theorems can be (and were) built up on a quantum-view of acceleration with discrete increments of impetus and intervening uniform motions of brief duration. Baliani and Fabri did not deny any of Galileo's experimental results (as had Cazre), but contended that they could only approximate to the real underlying nature of motion. That the approximation was extremely close only showed that physical instants were very small indeed. Fabri contended that his impetus theory was better than Galileo's continuity theory because his gave not only the expected results, but their true cause, while Galileo could not give any cause,

Impetus Theory Reappraised

275

but could merely report dull facts. Neglecting Fabri's value judgment about which was better physics, this was a very accurate statement of the principal difference between Galileo and his medieval forebears. It should perhaps be added that Pierre Fermat once offered a mistaken proof that Galileo's notion of continuous motion from rest led to a contradiction, before he had seen the Two New Sciences, and knew of Galileo's ideas only from the Dialogue. ' 9 But in 1646 Fermat sent to Gassendi a valid Archimedean proof that Galileo's rejection of proportionality of speeds to distances was perfectly sound, and he urged Gassendi to abandon his verbose dispute with Cazre. Finally, Jacques Alexandre Le Tenneur replied to both Cazre and Fabri in 1649, vindicating Galileo's arguments and conclusions mathematically and posing very difficult problems for his Jesuit adversaries. 20

vm - In conclusion, it seems to me to be of the greatest interest that impetus theory as set forth in the fourteenth century neither died out nor received any ingenious or useful development until after a totally different view of acceleration in free fall had been published by Galileo. Then, within a few years, the older theory was used to account for the facts he had disclosed and had confirmed by careful experiment as fitting observed phenomena. On the face of it, this shows that the law of free fall might have been discovered by the adaptation of Buridan 's impetus explanation to the mean-speed rule, just as most historians today contend was done by Galileo. It is true that before Galileo announced the law of free fall, such a development would have been much more difficult to carry out than it was to produce after that event, but the task would not have been impossible. Galileo himself was thoroughly familiar with classical impetus theory, and had argued stoutly for it in his De motu of 1590, which he withheld from publication. He did not, however, apply it to the explanation of acceleration in free fall, which at that time he believed to be only an event of short duration, quickly turning into uniform speed. His own early explanation of natural acceleration was quite different, and of course it was mistaken. The germ of his later inertial idea is found in the same book, but it had nothing to do with impetus theory. It started from certain "neutral motions" that he offered in contradiction to Aristotle's conception that all motions must be either natural or violent. As long as Galileo held to the mistaken idea that acceleration was an evanescent event at the beginning of fall, he made no attempt to investigate its laws. Late in 1602 he began to question whether acceleration could be ignored in motion along inclined planes, and soon afterwards began to seek ratios of speed, distance, and time in uniform acceleration. These he succeeded in finding not later than mid-1604; the discovery document survives, and shows that

276

History of Science

his success was due to the adoption of a strictly mathematical device that had nothing to do with any theory of the cause or mechanism of acceleration. So although it would have been possible for his restricted inertial concept to have emerged either from a mean-speed analysis or from a theory of impetus, that is not in fact what actually happened. Even Galileo's temporary adoption of a concept of velocity that required its measure by the square of the entity he later identified as velocity was suggested not by impetus theory, but by a certain mathematical convenience which he later discarded. The revival of impetus theory as a competitor of, and then as a true cause behind, Galileo's law of continuous uniform acceleration, has a certain analogy to the introduction of the quantum and the older corpuscular theory of light as a competitor of, or as a companion to, the wave theory. At the bottom of both lay the irreconcilable difference between the discrete and the continuous, already recognized by Greek mathematicians and enshrined in Euclid 's two separate treatments of proportion. No one yet knows whether Galileo or Fabri was right about the motion of an actual falling body; indeed, Heisenberg 's · uncertainty principle shows that that question makes no sense for bodies of the size we ordinarily handle. Fabri's scholium would have come close to showing the same thing in 1646 with regard to this particular problem, had he handled it a bit differently. 21 The history of science will be improved when it learns to respect equally the discrete and the continuous as approaches to the analysis of motion, and when it stops pretending that one is right and the other wrong, or that one approach ever led inevitably to the other. The two approaches have always been poles apart , and always will be, as are atomism and plenism, and for the same basic reason. The fact that with intelligent application they can usually be reconciled with the same physical observations does not unify the two approaches, any more than modem arithmetization of the continuum has put a stop to discussions of Zeno's paradoxes. NOTES

The rigorous arithmetization of the continuum by a complete revision of the classical concept of number enables us now to deduce the Pythagorean theory of proportion from the Eudoxian; but to Euclid, the former by no means appeared as a mere special case of the latter; see Section VII, below. 2 Marshall Clagett, The Science of Mechanics in the Middle Ages (Madison, 1959), 536-37; hereafter "Clagett." The glosses in square brackets are Clagett 's, as are Latin words in parentheses. English glosses in parentheses are mine. 3 Even what we now call the "force of gravity" changes with the distance separating 1

Impetus Theory Reappraised

4 5 6

7

8

9

1o 11 12

13 14

277

objects moved by it, and hence changes with their motion. I find it hard to conceive of a "permanent force" acting at all. Aristotle identified force with violence, and held that nothing violent could long endure. Cf. II, 610, col. 2, op. cit. (New York, 1970). Clagett, 524 Buridan himself always mentioned both contrary inclination and resistance in passages relating to the weakening of impetus; clearly he did not think of them as things of the same kind. Clagett, 56o-61; for convenience of reading I have transposed one phrase, added one gloss, and altered "gravity" to "heaviness." With regard to the occurrence here of the word "permanent," it should be noted that all the heaviness of a body is present at the same time, whence the technical sense of this word is applicable as well as the modem meaning. The earliest writers on uniform acceleration did not employ triangular diagrams, but reasoned arithmetically. Oresme made use of triangles as well as other figures in connection with his doctrine of configurations of qualities, and if he did not think of actual acceleration of physical bodies as mathematically continuous in the modem sense, he was certainly capable of having done so. The case of Oresme, however, appears to me exceptional. At the top of the sheet on which Galileo appears first to have correctly derived ratios in acceleration, he first entered examples derived from a hypothesis of discrete integral accumulations of speed. It was in testing this hypothesis and discovering contradictory ratios that he substituted the notion of continuous acceleration in actual fall; see S. Drake, "Galileo's Discovery of the Law of Free Fall," Scientific American, 228 (May 1973), 84--92 . Clagett, 566. Clagett, 565 It may be asked why, if the medieval rule implied equal increments of speed in equal times - as did Galileo's rule, and ours - the implications for distance were not the same in all cases. Basically the answer is that proportionality (sameness of ratio, as Euclid defined it) does not apply similarly to continuous magnitude and to discontinuous quantity. The question "where is the body at a particular instant?" can be answered unambiguously only for mathematical instants; during a physical instant having a certain duration (however small), the body covers an appreciable distance and is therefore not situated at a single definite distance from rest. The conceptual problem is related to that of fixing place and speed simultaneously in quantum theory. Aristotle, Physics, 267a, 12-15; 267b, 17. The final paragraphs of this book are essential to an understanding of impetus theory. P. Tannery and C. De Waard, eds., Correspondance du P. Marin Mersenne (Paris, 1963), IV, 114.

278

History of Science

15 [Pie.rre Cazre] Physica demonstratio qua ratio ... accelerationis motus in naturali descensu gravium determinatur ... (Paris, 1645), 7-8. 16 Cf. paper cited in note 9, above. Subsequently I have found some sketches of parabolas on another manuscript of the same period which strongly support my conjectural reconstruction of Galileo's thought in changing his definition of "velocity" as presented in that paper. The manuscript is analyzed in Historia Mathematica. 1 ( 1974), 139-43. 17 G.B. Baliani, De motu gravium solidorum (Genoa, 1638). The second edition (1646) has the additional words et liquidorum. 18 P. Mousnier, ed., Tractatus physicus de motu /ocali ... cuncta exerpta ex prae/ectionibus R. P. Honorati Fabry SJ. (Lyons, 1646). 19 P. Tannery and C. Henry, eds., Oeuvres de Fermat (Paris, 1896), III, 302--09; V, 36ff. 20 J.A. Le Tenneur, De motu naturaliter accelerato tractatus physicomathematicus ... (Paris, 1649). 21 Cf. note 12, above. The basic concepts that differed between medieval impetustheory explanations of acceleration in free fall and Galileo's account of this phenomenon foreshadow differences between quantum- and continuity-concepts of energy emission in modem physics. Fabri happened to be on the losing side in his day, and has been neglected, to the detriment of historical understanding in view of the attention paid to him by leading physicists of the latter part of the seventeenth century.

13 A Further Reappraisal of Impetus Theory: Buridan, Benedetti, and Galileo

The impetus concept introduced by Jean Buridan in the fourteenth century was intended primarily to provide a better explanation of motion continued by a heavy body after loss of contact with a mover than given by either of the two theories mentioned by Aristotle. Both the older explanations relied on a supposed action of the medium surrounding the body; impetus theory substituted the action of an impressed force imparted to the body by its mover. Consideration of a special case enabled Buridan to account also for the acceleration of a freely falling body in tenns of increments of impetus occasioned not by external force but by speeds successively acquired. A mathematical rule for this process put forth by Albert of Saxony was accepted widely until the end of the sixteenth century. Certain historical puzzles arising from Albert's statement of that rule become pseudo-problems under a reappraisal of impetus theory in my recent papers dealing with Buridan's physical concepts and medieval proportion theory. 1 The present paper concerns a different aspect of impetus physics, no less important historically, which appears to be at least equally puzzling if one accepts the presentation of Professor Dudley Shapere in a recent book. There a passage from Buridan is cited which I would translate as follows : It seems to me that it should be said that the mover in moving a body impresses in it a certain impetus or a certain motive force of the moving body in the direction in which the mover was moving it, either up, or down, or laterally, or circularly ...2 Reprinted from Studies in History and Philosophy of Science 7 (1976), 31~36. with kind permission from Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington OX5 1GB. UK.

280

History of Science

To this, Professor Shapere appends the comment: "It is important to notice that, for Buridan, the impetus may be such as to direct the body along a rectilinear or along a circular path. " 3 This would indeed be a strange idea for Buridan to have introduced, considering that his intention was to explain commonly observed instances of continued motion after separation from the mover. No experience could have suggested that a body may be moved so as to continue in a circular path in the ordinary sense of those words. The puzzle disappears if instead we say: " ... to direct the body along a rectilinear path or in a circular motion." Common observation does provide examples of bodies hurled straight and of wheels set in rotation, both of which continue in motion after we have ceased to move them in the respective directions, straight or circular. This appears to be the plain meaning of Buridan's words, though it would not serve for Professor Shapere's purposes. If we examine the entire discussion from which the above passage of Buridan 's was extracted, we find that it in fact begins with mention of continued motion by a mo/a fabri, meaning a millstone or a grindstone, which cannot be accounted for by action of the surrounding air as in either of the old Aristotelian explanations. After suggesting the impetus idea, Buridan reverts to the mo/a fabri as something harder to stop (and hence the more endowed with impetus) the larger it is. This is circular motion, but hardly motion along a circular path. His only other examples of circular motion are a spinning top and the rotations of the celestial orbs, which at Buridan 's time were conceived as rigid transparent spheres carrying the lights we call stars and planets. Stones, arrows, lances, and boats after rowing has stopped provide Buridan's examples of continued straight motion. It seems to me evident that Buridan recognized two different directions in which impetus acted, which he saw no way to reduce to one. The same two kinds of impetus were mentioned by Henry of Hessia a few decades later, with the same example of the mo/a fabri for circular impetus.4 Hence the puzzling suggestion of a "circular path" following the impression of impetus, as if medieval physicists had supposed that a stone released from a whirling sling ought to go on moving around, appears to be no more than a modem romantic notion created by a careless paraphrase. Its purpose in Professor Shapere's book is quite evident; he believes that Galileo was obsessed by such a notion and is reluctant to tax him with the pure invention of such a nonsensical bit of physics. But it is quite unjust to saddle this on the superb physicist Buridan, nor can any error of Galileo 's be plausibly ascribed to excessive respect for the authority of earlier writers. We still distinguish motions of translation and motions of rotation about an axis. An attempt to give the former privileged status over the latter was made by G.B. Benedetti in 1585, and this has been seen by some historians as a step

A Further Reappraisal of Impetus Theory

281

toward the modem inertial concept, though I believe that is a mistake. Among them is Analiese Maier, whose comment implies a view similar to that of Professor Shapere in the matter previously discussed: A first trace of the principle of inertia is early found in Benedetti, but already with him in a form that cannot be brought into agreement with impetus theory and in fact cuts across it. Where that occurs is in the problem of the direction in which the imparted impetus operates. It is very important that Benedetti first brought to the impetus concept a further development insofar as he recognized only a rectilinearly working impetus; that is, insofar as he allowed to every impetus necessarily a rectilinear, and no circular, motion.5

It is true that Benedetti rejected circular impetus in the sense in which Buridan had opined that a millstone, if it could be freed from all kinds of external resistance, might well go on rotating forever. It is not true, however, that Benedetti attributed perpetual continuance of motion to straight impetus, which would indeed have been a step toward the inertial principle. I say "a step toward," and not the inertial principle itself, because the latter principle as stated by Isaac Newton embraced also the phenomena of perpetual rotation about an axis in the sense of Buridan. This is not a rigorous logical implication of the law of inertia as stated by Newton, but Newton included such rotation in the illustrations he gave for his first law of motion, and it was necessary for parts of his physics. 6 Hence the exclusively rectilinear operation of impetus presumed by Benedetti, even if it had included the notion of eternal continuation, would not have led to inertial physics in the Newtonian sense so long as it denied to rotations the same eternal character. On the other hand there was a matter of great importance in which Benedetti did cut across the medieval ideas embedded in impetus theory, which Miss Maier might have singled out for mention but did not. In order to understand both classical impetus theory and Benedetti's departures from it, we must examine some of his particular statements, after which it will be easier to see the great differences in the approach of Galileo, which did lead to an inertial idea. II

The case that was not discussed by Buridan and that has doubtless led to misapprehension on the part of Professor Shapere and others is that of a stone released from a whirling sling. It is in such a case, if any , that a physicist might conceivably postulate a tendency to continue along a circular path as "the direction in which the mover was moving the body," though I do not believe that any physicist ever did. Benedetti discussed the case of the sling, and was perhaps the first

282

History of Science

to have done so, though he did not claim originality for his analysis with regard to the part that immediately concerns us here - that the stone departs along the straight line tangent to the circular motion at the point of release. An identical analysis was given later by Galileo in his Dialogue, drawn under questioning from the spokesman for Aristotelian principles in that book and treated as an obvious conclusion from common observation. Benedetti wrote: When the projectile is released from the sling it takes its path, with nature as its guide, from the point where it has quit [the sling] along a line tangent to the circle which it last made ... Now it is true that the impressed impetus gradually and continuously decreases ... 7

The second sentence suffices to show that Benedetti, like Buridan, did not consider the rectilinear motion to be uniform and perpetual but to be subject to immediate and continuing diminution upon loss of contact with the mover. Any doubt on this score will be removed later, when it is seen how Benedetti treated the action of heaviness of the stone in this case, and the action of the straight tendency in the case of the heavy rotating wheel. Buridan believed that imparted impetus must be weakened not only by any external resistance, but also by any natural tendency in the body to move in a direction contrary to that of the impetus. Hence in oblique motion upwards of a heavy body, supposing no air resistance, the imparted impetus drove the body straight only so long as it remained greater than the natural tendency downward inherent in the body because of its weight. This tendency remained always constant, just as the weight remained unchanged, whence by reason of a supposed conflict between the downward tendency and the impressed impetus, the latter must be weakened until the two were of equal strength, at which point the downward tendency brought the body to earth. The matter on which Benedetti differed from Buridan is made clear in the continuation of the above passage: ... Hence the downward tendency of the body, caused by its heaviness, enters at once and, mingling itself with the impressed force, does not permit the line to remain straight for long, but causes it quickly to become curved. For the body is moved by two powers, one impressed by force, the other by nature. 8

Benedetti went on here to criticize his former teacher, Niccolo Tartaglia, for having supposedly denied this composition of forces in determining the path of the projectile. The criticism was unfair; Tartaglia had in fact mentioned the necessary curvature in his Nova Scientia of 1 537, though since in that book he was concerned in particular with high-speed cannonball trajectories he preferred to

A Further Reappraisal of Impetus Theory

283

treat the initial motion as straight for practical reasons.9 In his next book, the Quesiti of 1546, Tartaglia made a special point of discussing this matter of practical as against theoretical analysis. 10 Hence the recognition of composition of motions in the trajectory of a free projectile is properly credited to Tartaglia, though Benedetti was the first to signal its importance in the modification of impetus theory. Tartaglia, so far as I know, never mentioned impetus and probably remained unaware of the work of Buridan, though it had been printed at Paris in 1509. Tartaglia was self taught and little concerned with scholastic disputations. The recognition of composition of motions was of paramount importance to the development of mathematical physics. The idea was as old as the pseudoAristotelian Questions of Mechanics, a treatise which appears to have remained unknown in the Middle Ages since there is neither an Arabic nor a Latin translation of it before 1517. During the sixteenth century it was the subject of several commentaries on many interesting questions raised in it of which there is no trace of discussion by medieval writers. Among these was the question why a stone is hurled farther by a sling than by the hand, a problem that lent itself naturally to solution by impetus theory. Since Benedetti seems to have been the first to supply that solution, it is unlikely that medieval writers were familiar with the little ancient treatise. The importance of Benedetti' s solution rests, however, not in the rectilinear character of the path of the stone under impressed impetus, but in the recognition of composition rather than conflict between that impetus and the natural downward tendency of the heavy projectile. Benedetti 's perception of this makes even more curious his denial of circular impetus, as we shall see. First, however, we should consider a bridge passage that led from the case of the sling to that of the rotating wheel: But if you wish to see this truth more clearly, imagine that while the body, i.e., the top, is spinning rapidly around, it is cut or split into many parts. You will observe that those parts do not immediately fall toward the centre of the universe, but that they move in straight lines, horizontally so to speak. No one, so far as I know, has previously made this observation on the subject of the top. 1 1

The claim of originality here is doubtless correct, and the observation led Benedetti to an important consideration relating to centrifugal force. Doubtless many people had been aware of this phenomenon, which Copernicus had mistakenly imputed to Ptolemy, that a body may fly apart in very rapid rotation. No physicist before Benedetti, however, appears to have commented on the compulsion necessary to hold the parts together, and Benedetti drew from this a strange conclusion:

284

History of Science

Suppose a millstone rested on a virtually mathematical point and was set in circular motion; could that circular motion continue without end, it being assumed that the millstone is perfectly round and smooth? I answer that this kind of motion [which was Buridan ' s circular impetus] will certainly not be perpetual [as Buridan conjectured] and will not even last long. For apart from the fact that the wheel is constrained by the air which surrounds it and offers it resistance, there is also the resistance from the parts of the moving body itself. When those parts are in motion they have by nature an impetus to move along a straight path. Hence since all the parts are joined, and every one of them is continuous with another, they suffer constraint in moving circularly and remain joined only under compulsion. For the more [swiftly] they move, the more there grows in them the natural tendency to move in a straight line, and therefore the more contrary to their nature is their circular motion. And so they come to rest naturally; for since it is natural for them when they are in motion to move in a straight line, it follows that the more they rotate under compulsion, the more one part resists the next one, and so to speak holds back the one in front of it. 12

If one allows two kinds of impetus, with Buridan and Henry of Hessia, Benedetti 's conclusion does not follow; circular impetus, imparted by rotation of a heavy object about a central axis internal to it, may continue indefinitely in the absence of external resistance. For there is no natural tendency in the wheel to any contrary motion, or to come to rest of its own accord, whence by the principle of sufficient reason the rotation must continue. In this respect medieval circular impetus was truly inertial, unlike straight impetus in which conflict with a contrary natural tendency was necessarily present in any heavy body. In giving preferential status to rectilinear impetus alone, which Miss Maier considered to be Benedetti 's claim to traces of the inertial principle, Benedetti in fact surrendered the one trace of such a principle that was genuinely present in medieval impetus theory. Why he preferred the rectilinear impetus which by his own statement (in agreement with Buridan) Benedetti believed must gradually and continuously decrease, is beside the point; the fact is that in making this choice he gave up any chance to conceive of inertia or the conservation of motion as such. His correct observation that some compulsion must be present to hold the parts of a rotating body together was vitiated by the medieval notion of a necessary conflict between two tendencies to different motions within the same body, with a consequent weakening of at least one such tendency. To me it is the more surprising that having recognized the composition of tendencies in a projectile, Benedetti should have insisted on their conflict in a rotating body. For his own diagram drawn for the stone in a sling might equally well have suggested to him the action of diametrically opposite straight pulls as so many couples, keeping the wheel in rotation without loss of motion . But that is not what Benedetti did.

A Further Reappraisal of Impetus Theory

285

He made the straight tendency natural, a move in which Buridan would never have joined him (since it was forcibly imparted), and he considered it to exist by virtue of motion as such, no matter how imparted, whereas Buridan regarded the mode of imparting impetus as governing the character of the impetus imparted. Thus Buridan had admitted only the second and third of the four points made by Benedetti in his more detailed analysis elsewhere: If we revolve [a potter's wheel] with all our strength and then let it go, it does not rotate forever. There are four causes for this. First, such a motion is not a natural motion of the wheel. [For Buridan, the only important fact was that the wheel had no tendency to contrary motion.] Second, even if the wheel rested over a mathematical point it would still have to have a second point of support above to keep it horizontal, and this support would require some corporeal embodiment. Third, the contiguous air at all times constrains the wheel and resists its motion. Fourth, any portion of corporeal matter which moves by itself when an impetus has been impressed on it by an external motive force has a natural tendency to move on a rectilinear, not a curved, path. 13

There is no question that Buridan would have rejected this analysis, since to admit a natural straight tendency due to motion as such would destroy his suggestion that the celestial spheres move uniformly and perpetually under an initially received impetus. In this respect Buridan's impetus theory was more nearly an inertial theory than was Benedetti's modified impetus theory in which preferential treatment was given to straight motions. It is now usually said that Galileo made exactly the opposite mistake, giving preferential treatment lo circular motions. There is of course an element of truth in this, or that conclusion would not have been reached by able historians. Yet the sense in which it is true is presently badly misunderstood, just as the sense of Buridan's words" ... or circularly" has been misunderstood, and as the sense of Benedetti's preference for straight impetus has been misconstrued as a step toward the inertial principle. It remains to retrace Galileo's first steps toward such a principle, which proceeded very differently from those of Buridan and those of Benedetti. III

In a valuable paper published several years ago, Dr. J .A. Coffa remarked that "a careful analysis of Galileo's work shows that he was looking for the properties of what in post-Galilean terminology can be described as force-free systems." 14 The quest to which Dr. Coffa referred was evidenced by citations from Galileo's latest books - the Dialogue, Two New Sciences, and a posthumously published treatise on the force of percussion. Dr. Coffa admitted that the terminology

286

History of Science

"force-free systems" was post-Galilean, and he did not mean to claim that Galileo had consciously undertaken such a search from the very beginning of his work forty or fifty years earlier. Dr. Coffa was concerned with a conceptual analysis of Galileo's terrestrial physics and the refutation of the common notion that it somehow involved, or was based on, a concept of "circular inertia." At the same time I published a companion paper in which historical, rather than conceptual, considerations were applied to the same common notion which I likewise believe to be quite mistaken. I did not then see Dr. Coffa's paper as bearing literally in any way on the question of Galileo's original researches as presented in the De motu of 1590, though I had earlier called attention to the roots there of his later restricted inertial principle for motions of heavy bodies near the surface of the earth. 15 It was only in the course of my re-examination of Buridan 's impetus theory as set forth in the preceding section that a connection between Dr. Coffa's acute analysis and Galileo's first investigations emerged. For I had not previously recognized the limited and explicit character of Buridan's circular impetus and its interesting resemblance to and difference from Galileo's discussion of rotating spheres in De motu, to which a separate chapter was devoted. The organization of Galileo's treatise was exhaustively discussed by Dr. Raymond Fredette, who noted that a reference to "the first book" in chapter 19 shows that Galileo intended a separation of De motu before that place. 16 It is my opinion that the division Galileo had in mind related to the distinction between "natural" and "forced" motions made by Aristotle and continued in all later discussions, not excluding Galileo's own. It is noteworthy that the first fourteen chapters of De motu relate to natural motions in the old sense; that is, to motions undertaken by a body simply as a result of the removal of any constraint. Chapter 17 is entitled "By what agency projectiles are moved," and the idea of force plays a much larger role in the ensuing chapters than in the first fourteen . The theory of acceleration in free fall which Galileo had decided to adopt in De motu depended on a concept of "residual force" existing at the beginning of fall, whence it would not have been unreasonable to include fall, though it is a natural motion, in the section devoted to forced motions. Between the two sections of De motu, if Galileo's division into "books" was intended on the basis I suppose, there is chapter 16, entitled "On the question whether circular motion is natural or forced." The circular motion referred to is the rotation of a solid sphere, not the path of a free body. For the purpose of this chapter, Galileo defines natural motion as motion in which bodies approach their natural places, and forced motion as that in which they recede from their natural place. Under that definition the rotation of a heavy sphere having its centre at the centre of the universe would be a motion neither natural nor

A Further Reappraisal of Impetus Theory

287

forced. Various other cases, differing as to location of the centre and homogeneity of the sphere, were classified by Galileo, yielding a number of motions that were "neither natural nor forced" and a number that were "sometimes natural and sometimes forced." The question which (if any) of these would necessarily be perpetual was not exhaustively discussed, but a paradox was offered: "If its motion is not contrary to nature, it seems that it should move perpetually; but if its motion is not according to nature, it seems that it should finally come to rest" after receiving a start from an external mover. Now, Galileo had previously given a mathematical proof that a body on a perfectly level plane would be set in motion by a force smaller than any previously assigned force. To this he added in the margin, presumably after writing chapter 16, that motions which neither moved heavy bodies toward the centre to which they naturally tend, nor away from that centre, should be called "neutral motions," as partaking of neither the natural nor the forced. These neutral motions certainly represent the first clue from which Galileo proceeded over a period of years to his restricted inertial principle for terrestrial physics. It can be fairly safely dated to the time of writing De motu, which was pretty certainly I 590, and the order of events was (I) proof that motion on the horizontal plane may result from any force, however small, (2) consideration of rotation of a heavy homogenous sphere situated at the centre of gravity of all heavy things, (3) consideration of such a sphere wherever located, provided all external resistances are removed, and (4) classification of "neutral motions," neither natural nor forced. The relevance of Dr. Coffa's terminology to Galileo's first clue to the inertial principle, despite the anachronistic character of that terminology, may now be seen. A motion that was neither natural nor forced in Galileo's sense must be force-free in our modem sense, because by virtue of its not being natural, it is not affected by what we call the "force of gravity," but which Galileo and all his precursors called a "natural tendency" and not a force of any kind. It is of the utmost importance to remember that the distinctions natural-forced and internal-external were built into all previous physics, and that those who today pretend that Buridan regarded downward motion as produced by any kind of external force are simply mistaken; it was seen as an internal tendency essential to the heavy body and situated in the body. External forces could affect a body only by direct contact; heaviness, or gravity, was in no sense the product of any such force. Hence what we would call "force-free" motion required the removal not only of what they called "force," but also of what they called "natural motion." So far as I know, the first exploration of such a situation was Galileo's, and it began in his De motu about 1590. A long time elapsed before he saw its relevance to projectile motions, which he dealt with in De motu in terms

288

History of Science

of impressed impetus in the sense of Buridan, necessarily diminished at every instant (as with Benedetti), but curved by composition of impetus with downward motion, as in Benedetti's analysis. In saying above that a motion neither natural nor forced is not affected by what we call the "force of gravity," I did not mean that the body in motion is not so affected, but only the motion as such. Since only heavy bodies were in question, they were always affected by our "force of gravity" or Galileo's "natural tendency downward." In order for the motion to remain unaffected while the body was so affected, it was necessary to shield the body from action of that natural tendency, and the only way to do that was to place it on a surface capable of supporting it. In this respect Galileo's discussions introduced a whole class of cases which neither the medieval writers nor Benedetti had considered; namely, the motions of bodies supported on surfaces. 17 Galileo had begun with motions on inclined planes, which are natural in one direction and forced in the other. This had led him to the unique case of the level plane, and thence to the steps numbered above. But it was clear to Galileo even in De motu, and even before he had finished his chapter on inclined planes, that actual motion along the horizontal plane is not quite neutral motion. Since the plane touches the earth at only one point, any motion away from that point increases the distance from the centre; hence, he said, it is not remarkable that in actual practice a large sphere cannot be moved on the level plane by a very small force. It is a direct consequence of this that motion along the horizontal plane, if started by an external force, would not remain perfectly unifonn but would be slowed for the same reason as on a plane tilted upward; that is, increasing distance from the centre. For reasonable distances along the plane the slowing might be inappreciable, but it must necessarily occur. It followed that only when supported on a sphere concentric with the earth could there exist truly neutral motions of translation under Galileo's definitions, and in all his subsequent writings he was careful to impose this condition. It is that fact which has led historians to say that Galileo never got closer to an inertial principle than to assert that certain circular motions could continue unifonnly forever, and never under any circumstances any straight motion. Such a statement is unobjectionable, though it should be followed by remarking that for ordinary bodies and over reasonable distances, this principle is indistinguishable from our inertial principle. The tangent horizontal plane is indistinguishable from the sphere concentric with the earth for a very considerable distance, far greater than any which Galileo considered in his physics. But instead of this the statement of historians is usually followed by quite a different commentary; namely, that Galileo was haunted by the perfection of circular motions, that he drew his terrestrial physics from analogies to the heavens, that

A Further Reappraisal of Impetus Theory

289

he accounted for the motions of heavenly bodies by circular inertia, that he failed to apply his own principles in deriving the parabolic trajectory, and in short that Galileo's physical conceptions were remote from those which later guided Newton and gave us our inertial physics. Now, it is of interest that although Galileo's marginal addition concerning neutral motions in De motu was added at the place where he had previously proved his proposition about motion supported on the horizontal plane, the wording of his note does not strictly apply to that or any other straight motion, but only to motions in which the distance from the centre remains unaltered. His reason for adding it there, and not in chapter 16 where he first conceived of motions neither natural nor forced, is fairly evident. That chapter dealt with rotations, not translations to which the same concept might also apply, and he went back to chapter 14 so to apply it. It was in writing this note that the word "neutral" occurred to him, and he placed it at the end. The rotations discussed in chapter 16 were no less neutral in the sense of being force-free, once started, and hence uniform and perpetual. Galileo's ensuing discussion in chapter 14, already mentioned, shows that he had recognized that although motion could be started on the horizontal plane by any minimal force, continuation of that motion would be neither uniform nor perpetual. No fair-minded critic can fail to see that from the beginning, Galileo regarded the actual horizontal plane as very nearly inertial for bodies free of external impediments, to be replaced by a surface concentric with the earth for truly uniform and perpetual translation. It is, then, hardly a fair description of Galileo's serious physics to say that it remained remote from inertial considerations in the modem sense. Nor is it reasonable to portray it as the direct descendant of medieval impetus theory, in which the whole point was to provide a force capable of driving the body on in continued motion, whereas Galileo's earliest analysis consisted in eliminating the need for any such force. Certainly Galileo's ideas recited above did not derive from Buridan via Benedetti, who rejected the perpetual rotations from which Galileo first derived his class of motions neither forced nor natural. The prevalent modem criticisms of Galileo's physics are often based on an assumption that is pretty clearly mistaken; namely, that its goal was to support the Copernican astronomy and that the motions Galileo wanted most to explain were those of the planets. It is understandable that anyone who had read only his Dialogue of 1632 would suppose this, but not that an objective reader of the Two New Sciences would call that a Copernican book as most historians now do, following Koyre. Everything essential to that book had been done by Galileo before 161 o, when the telescope attracted his attention to astronomy. Earlier than that there is nothing to indicate that Galileo had even a normal interest in astronomy for professors of mathematics of his time who had to teach it. His

290

History of Science

preference for the Copernican system is attested only by two letters written in 1 597, one of which implies that the preference was grounded mainly on a mechanical theory of the tides. Mechanics and motion, along with practical matters, took up most of his time until I 6 Io. Thereafter his contributions to astronomy were entirely observational and not theoretical. His surviving extensive correspondence and working papers reveal little trace of any attempted celestial physics. The few comments offered in the Dialogue, mainly in the early pages, hardly justify the assumption that celestial mechanics dominated Galileo 's thought, as it did Kepler's. The evidence from Galileo 's papers points in exactly the opposite direction; namely, that long study of mechanics and the motions of heavy bodies near the earth's surface had put him in possession of a useful body of terrestrial physics that he was able to apply later to refute many common objections to the earth 's motions. The findings presented in Two New Sciences fit perfectly with the assumption that Galileo was interested mainly in a useful physics of ordinary objects on earth, and fit badly with the assumption that he sought a science of universal generality, or even believed such a thing humanly possible. Both in that book and in letters answering objections to passages in the Dialogue, he made it clear that his physics was not supposed to apply to bodies at extreme distances or speeds; thus his parabola of fall could not be valid near the centre of the earth, nor true of trajectories for cannon - as opposed to mortar - shots. 18 The horizontal plane was not truly a plane, nor was the surface of the earth infinitely distant from its centre as he assumed for practical purposes and thought that Archimedes had likewise assumed for similar reasons. Those who attempt to convert such candid admissions by Galileo into evidence that he was confused, contradictory, or inconsistent do little service to the history of science. Indeed, I think it would be good to tum the tables on them in a way they have perhaps not anticipated. They think Galileo was not a good physicist because he derived the parabolic trajectory on the assumption that horizontal motion is uniform when he believed that only circular motion around the centre of the earth could truly be uniform for heavy bodies, and they deride his remark that in falling from a ship's mast, an object has impressed on it the circular motion around the earth. Let us suppose them to say to Galileo that in the first instance he happened to be right, but only because he ignored his own false principle of circular inertia around the earth, and that in the second instance he was simply wrong because he relied on that same false principle. I shall attempt to answer them on Galileo 's behalf. As to your first point, gentlemen, it seems to me that in deriving the parabolic trajectory neither you nor I pretend that it applies to bodies beginning or ending their fall very far

A Further Reappraisal of Impetus Theory

291

from the surface of the earth. Your demand that I justify my substitution of the horizontal plane for the surface of a sphere concentric with the earth therefore appears to me unreasonable. With equal justice I might demand that in your derivation, based on the estimable physics of Mr. Newton, you must explain why you do not take into account the attraction of the moon, which though very small for a terrestrial cannon ball is nevertheless part of your theory. So is equidistance from the centre of the earth as a condition of truly uniform translatory motion a part of my theory. The horizontal plane meets that condition sufficiently well for bodies beginning and ending their motion not far from the surface of the earth. Just so, the earth's attraction suffices for such bodies in your theory, without bothering about the moon's position, distance, and consequent influence on the parabolic path. Your second point concerns the fall of a body from a ship's mast to its deck, say a distance of thirty feet. We agree that it strikes at sensibly the same point whether the ship is in motion or at rest. I say that its uniform advance during fall when the ship moves is measured by the advance of the top of the mast along an arc concentric with the earth; you say that advance along the horizontal tangent through that top at the start of fall should be used. Very well; I shall make this "correction" (or rather, pedantic change) if you, with equal courtesy, will hereafter take into account the increase in what you are pleased to call the "force of gravity" during a fall of thirty feet by reason of increasing proximity to the centre of the earth. When we have both performed the tedious calculations required by our respective theories, we shall again agree in our original conclusion. I consider it a part of useful physics to know what can be left out of consideration in a given specific problem. I believe your Mr. Newton would take my side on this, and not yours, just as I believe that Aristotle would be more receptive to my discoveries than were his self-styled followers in my day. IV

Something should perhaps be said about Galileo's comments on the motions of the heavenly bodies, particularly in the opening pages of the Dialogue. These "integral bodies of the universe," he said, must either remain at rest or move circularly if their order is to be preserved. That order is preserved in the heavens is accepted as a postulate by the three parties to the dialogue. Circular motion or rest remain to be assigned to particular heavenly bodies, this assignment being the principal point of disagreement between Ptolemy and Copernicus. The preservation of perfect order would in tum require perfectly circular motions. This follows from another postulate, that any approach toward or recession from the centre would occasion change of speed. Several of Galileo's statements entail this consequence; but none, so far as I know, assert that perfect order in this sense exists in the heavens. The absence of such an assertion by Galileo, instead of suggesting to recent historians of science that he may

292

History of Science

have considered the precise movements of the planets undiscoverable, has led them to postulate some metaphysical belief on his part about "this great book of Nature that stands forever open to our gaze"; namely, that the language of mathematics which enables us to read it at all must be really all there is to it. Thus although in the Dialogue itself he remarked that the speed of the earth about the sun could not in fact be uniform, 19 some say he really believed the orbits to be perfectly circular and the motions in them to be perfectly uniform. From this it is easy for them to deduce that he accounted for planetary motions by "circular inertia," despite the fact that he expressly denied that he could explain the planetary motions.20 In this and similar ways Galileo is being transformed into a philosopher and his science into a branch of philosophy, in defiance of the opinion of his contemporaries and of earlier historians that Galileo's science was a revolt against philosophy. It is this trend, I believe, which accounts for the uncritical attribution of views to Buridan, Benedetti, and Galileo that no man of common sense, and certainly no physicist at any epoch, would have attempted to defend. The result can be damaging to historical understanding of the roots of modem physical science. Speaking as a biographer of Galileo, it is my view that from the time of his controversy with Cesare Cremonini in I 604-05 over the place of a supernova and the proper manner of deciding this, Galileo undertook to establish if possible a science limited in scope and objectives, independent of the pursuit of philosophy. Without denying the existence of loftier goals, he was willing to leave them to others: SIMP. I believe that some great mystery may perhaps be contained in those true and admirable conclusions [about the generation of concentric or internally tangent sets of circles by straight uniform and straight accelerated motions] - I mean a mystery that relates to the creation of the universe ... SALV. I feel no repugnance to that same belief. But such profound contemplations belong to doctrines much higher than ours [as scientists], and we must be content to remain the less worthy artificers who discover and extract from quarries that marble in which industrious sculptors later cause marvellous figures to appear that were lying hidden under those rough and formless exteriors. 21

The fact that other scientists, before and after Galileo, were unwilling to restrict themselves to so mean a role and arrogated to themselves the tasks of philosophy constitutes, in my opinion, insufficient grounds for supposing that Galileo did not have in mind quite a different programme for the science of physics. That he did have is evidenced by such remarks as these:

A Further Reappraisal of Impetus Theory

293

The introduction of such [irregular] lines is in no way superior to the sympathy, antipathy, occult properties, influences and other terms employed by some philosophers as a cloak for the correct reply, which would be "I do not know."22 There is not a single effect in nature, not even the least that exists, such that the most ingenious theorists can ever arrive at a complete understanding of it. 2 3 We must find and demonstrate conclusions abstracted from the impediments, in order to make use of them in practice under those limitations that experience will teach us.24

It is in such remarks actually published by Galileo, rather than in an experientially groundless addiction to "circular inertia," that the metaphysical foundations of Galileo's science may reasonably be sought. That it had few if any followers is an interesting historical fact, but not one that would have surprised him: To say plainly what I am trying to hint, and dealing with science as a method of demonstration and human reasoning capable of pursuit by mankind, I hold that the more it shall partake of perfection, the smaller the number of conclusions it will promise to teach, and the fewer yet will it demonstrate; hence the less attractive it will be, and the smaller will be the number of its followers. On the other hand, magnificence of [book] titles and grandeur and number of promises attract the natural curiosity of men and hold them perpetually involved in fallacies and chimeras, without ever offering them one single sample of the sharpness of demonstration by which the taste might be awakened to the insipidity of their ordinary fare. 2 5 NOTES

1 "Impetus Theory Reappraised," J. Hist. Ideas 26 (1) (1975), pp. 27-46; "Free Fall from Albert of Saxony to Honore Fabri," Stud. Hist. Phil. Sci. 5 (4) (1975), 347-366; "Impetus Theory and Quanta of Speed Before and After Galileo," Physis 16 (1) (1974), 47~5. 2 ldeo videtur mihi dicendum, quod motor movendo mobile imprimit sibi quendam impetum vet quondam vim motivam illius mobilis ad illam partem ad quam motor movebat ipsum, sive sursum sive deorsum sive tateraliter vet circutariter ... cited by Annaliese Maier, Zwei Grundprobteme der schotastischen Naturphi/osophie (2nd edn, Rome 1951), p. 211. (Hereinafter called Maier.) 3 D. Shapere, Galileo: A Philosophical Study (Chicago, 1974), p. 49. 4 Consurgunt diverse species motivarum qualitatum, quas vocant impetus motionis, quorum quidam est motionis circularis, ut apparet in molafabri, et quidam recta. Cited by M. Clagett, The Science of Mechanics in the Middle Ages (Madison, I 959), p. 637, note 16. It is perhaps worth mentioning that "circle" in early usage more often referred to the enclosed area than just to the perimenter.

294

History of Science

5 Maier, pp. 306-7 6 Sir Isaac Newton's Mathematical Principles ... (trans. by Cajori) (Berkeley, 1947), p. 13: "Law 1. Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it. Projectiles continue in their motions ... A top ... does not cease its rotation ... The planets ... preserve their motions both progressive and circular for a much longer time." This last example of Newton's under his inertial law has perhaps escaped the allention of Professor Shapere, who might like to include Newton along with Galileo as a superstitious believer in "circular inertia" as the basis of celestial mechanics. The same method would serve; that is, selection of an isolated remark to prove a thesis. 7 S. Drake and I.E. Drabkin, Mechanics in Sixteenth-Centwy Italy (Madison, 1969), p. 189. (Hereinafter called Mechanics.) 8 Mechanics, p. I 89. 9 Mechanics, p. 84. 10 Mechanics, pp. 101-104. 11 Mechanics, p. 229. 12 Mechanics, pp. 228-9. 13 Mechanics, p. 186. 14 A. Coffa, "Galileo's Concept of Inertia," Physis 10 (4) (1968), p. 261. 15 "Galileo and the Law of Inertia," Am. J. Phys. 32 (1964), 6o1-6o8. 16 "Galileo's De motu antiquiora," Physis 14 ( 1972), 335--6. My own account of the evolution of Galileo's De Motu will appear in Isis for June 1976. 17 See appendix for a summary of the motions considered by Buridan, Benedetti and Galileo with the opinions of each, stated or conjectured, concerning all cases. 18 See, for example, Galileo's letter to Pierre Carcavi, 5th June 1637 (Opere, Ed. Naz. of A. Favaro, xvii, pp. 90-91 ); Two New Sciences (trans by Drake) (Madison, 1974), pp 223-5, 227-8. 19 Galileo, Dialogue (trans by Drake) (Berkeley, 1953), pp. 453,455. Much earlier, in notes on a philosophical attack on Copernicus by J.C. Lagalla ( 1612), Galileo had noted that perfectly concentric planetary motions were absolutely false and impossible; cf. Opere 3, p. 338. 20 Ibid., p. 234 21 Two New Sciences (trans by S. Drake) (Madison, 1974), pp. 182-3. 22 S. Drake and C.D. O'Malley, The Controversy on the Comets of 1618 (Philadelphia, 1960), p. 197. (Hereafter called Contro1•ersy.) 23 Dialogue (trans. by S. Drake) (Berkeley, 1953), p. IOI. 24 Two New Sciences, p. 225. 25 Controversy, p. 189.

A Further Reappraisal of Impetus Theory

295

APPENDIX

In the course of writing the foregoing paper I compiled a list of motions dealt with by Buridan, Benedetti, and Galileo that may be useful to others in comparing their agreements and differences. Entries in square brackets represent cases not explicitly discussed by the writer named; the treatment indicated represents my own opinion of his probable view in accordance with the principles he appears to have adopted. The views taken from or ascribed to Galileo refer only to the period of composition of De motu, around 1590 and for the next decade during which we have no evidence of his activities in the study of motion. All three writers dealt with external resistances to motion in essentially the same way, as actually preserit but as acting separately and independently of natural tendency and impressed impetus, both seated in the body itself. Hence external impediments to motion were no source of differences between these men concerning the motions listed, and have been assumed throughout to be considered as non-existent. 1.

Stone thrown vertically upward

BURIDAN:

BENEDETI1: GALILEO:

2.

Impetus in conflict with natural tendency downward. Violent impetus overpowered in this conflict, after which motion downward ensues, accelerated by increments of natural impetus. Same as Buridan, above. Same except that acceleration down is due to loss of residual upward impetus remaining in the stone when exact balance is reached with the natural tendency downward.

Stone hurled up at an angle less than vertical

[BURIDAN:

BENEDETI1: GALILEO:

Impetus maintains straight motion until overpowered as in Buridan 1 ; then natural motion down ensues.] Violent impetus combined with natural tendency down curves the path. Same as Benedetti.

3. Stone released from sling [BURIDAN:

BENEDETI1: [GALILEO:

Same as Buridan 2, above] Same as Benedetti 2, above. Same as Benedetti.]

4. Rotating millstone or homogeneous sphere BURIDAN:

Impetus imparted acts circularly to maintain motion.

296 BENEDETTI: GALILEO:

History of Science

Impetus of each part acts straight, slowing the motion. Motion is unifonn, neither forced nor natural.

5. Celestial spheres Impetus acts circularly and perpetually. [BENEDETil: If spheres postulated, impetus would not maintain perpetual rotations.] [GALil..EO: If spheres postulated, same as Galileo 4.)

BURIDAN:

6. Body supported on inclined plane and freed from restraint [BURIDAN: Motion along plane accelerated by naturally acquired impetus.] [BENEDETil: Same as Buridan.) GALILEO: Speed detennined by slope; acceleration ignored. 7. Body supported on level plane and moved [BURIDAN: Horizontal impetus weakened by conflict with natural tendency downward.] [BENEDETil: For sliding object, same as Buridan; for ball, same as Benedetti 4.) GALILEO: Motion unifonn over any ordinary distance.

8. Body supported on surface concentric with the earth [BURIDAN: Impetus weakened by conflict with natural tendency downward.] [BENEDETil: Motion slowed by conflict of straight impetus and circular path.] GALILEO: Motion unifonn over any distance; perpetual circulation.

14

The Rule behind "Mersenne's Numbers"

The basis on which Marin Mersenne selected the numbers known by his name has long been a puzzle to students of the theory of numbers. Thus a good recent summary of studies in the identification of perfect numbers reports that: W.W.R. Ball gave the name "Mersenne's numbers" to integers of the form (2n - 1) where n is a prime, because the French mathematician [Marin] Mersenne made a statement in his Cogitata in 1644 regarding numbers of this type. The conjectures as to Mersenne's source of information have given this investigation such a stimulus that Herculean labors of calculation have been performed to verify or contradict his statements ... Peter Bungus, who lived in the seventeenth century ... in a book entitled Numerorum Mysteria ... listed 24 numbers said to be perfect, of which Mersenne stated that only 8 were correct ... [and] added on his own account the three values n = 67; 127; 257 .... Subsequent investigations showed that he was wrong in admitting the values 67 and 257 and that he omitted [61 ], 89 and 107 for which the Mersenne numbers are primes. But it took 303 years, from 1644 to 1947, completely to check and correct Mersenne's statement .... There have been many conjectures as to how Mersenne arrived at his results, and after a lapse of over 300 years no answer has been found. He must have discovered or had available some theorem not yet rediscovered, since empirical methods could hardly have been used - the Mersenne number for n = 257 has 78 digits. Some have supposed that the talented mathematician [Pierre] Fermat communicated these results to him. 1

The historical problem of Mersenne's source and of the nature of the basis on which the numbers were selected is capable of solution with a very high degree of probability by reason of a further reference to the matter in Reprinled from Physis I 3 ( I 971 ), 421-4, by permission.

298

History of Science

Mersenne's own supplement to the Cogitata of 1644, the Novarum observationum physico-mathematicorum ... published at Paris in 1647. Page 182 of that work mentions a three-volume manuscript on the theory of numbers by B. Frenicle de Bessy, to whom Mersenne says he owes his progress in the subject. That it is probably to Frenicle rather than Fermat that Mersenne owed the list of numbers bearing his name, or the hints from which it was derived, will be apparent from the fact that on the same page is given a rule closely related to that list. The text reads: Sequens Regula numeris primis agnoscendis admodum utilis: vidilicet numerum (sic; numerus?) binarii analogicum unitate decurtatum, cuius exponens primus, temario, vet minore numero ab aliquo binarii analogo, cuius exponens sit par, est numerus primus. Verbi gratia, 7 est exponens 128, nam 7 differt temario a 4, binarii analogo, cuius exponens est par, ideoque 127, est primus. This passage was noted in 1935 by R.C. Archibald, who conjectured that the verb obviously missing from the first sentence had the sense of "exceeds." Archibald was thus able to account for Mersenne's omission of n = 61; but since no light was thrown in this way on his correct inclusion of n = 13, the passage was not seen as the key to the rule behind Mersenne's numbers.2 As a matter of fact, the missing verb was differt, and on the verso of p. 235, in the MEN DA ante libri lectionem emendanda, that correction is given, so that the passage would read: ... ternario, vel minore numero differt ab .... With this amendment, the statement may be freely translated thus: A useful rule for knowing prime numbers ,is this: Any number is prime which is of the form zP - 1, where p is prime and differs by three or less from some even power of two. Using this rule to form a table, neglecting those even differences which could not be prime, and all composite numbers (here enclosed in square brackets), we have: 22n

-3

4 16 64 256

13 61 (253)

-2 2

-I 3 (15) [63] (255)

+I 5 17 (65) 257

+3 7 19 67 (259)

The agreement between the numbers yielded by the "useful rule for knowing

The Rule behind "Mersenne's Numbers"

299

primes" and the list of Mersenne's numbers implied in the Cogitata of 1644 is striking: "Rule" numbers: 2, 3, 5, 7, 13,

17,

19,

61 ,

67,

Cogitata numbers: 2, 3, 5, 7, 13,

17,

19,

3 I,

67,

257

127,

257

The inclusion of 31 and 127 in the 1644 list, as well as the incorrect 257, is easily accounted for; they are recognizable primes of the form 2n - 1, for which n happens to be even only in the case of the incorrect 257. That Mersenne had by no means rejected 231 -1 as a factor of a perfect number in his 1647 publication is evidenced by the following passage on the same page, which also enables us to reconstruct with high probability the entire rule by which Mersenne formulated his conjecture concerning perfect numbers. Praetera si 64, temarius addatur, surget primus 67, atque adeo 67, potestas plus (sic) t , erit numerus, qui sequitur, primus, 147573952896776412927: quorum haec est proprietas, ut in sui medium ducti numeros perfectos (sic) generent: quod intellige de solis numeris primis, qui sunt unitate minores numero binarii analogo, eapropter non convenit haec proprietas numero primo 5, sed numeris 3, 7, 31, 127, 8191, 131071 , 524287, 2147483647 et omnibus aliis eiusmodi generis.3

This passage entirely eliminates, as Archibald pointed out, any idea that 67 was a typographical error for 61 in the 1644 work; Mersenne believed that 267 - 1 was indeed prime, which it is not, and he seems to have failed to recognize 2 61 - 1 as prime, which it is, since the list given at the end of the passage gives the Mersenne numbers through the exponent 31, and that at the beginning gives the value for 67, while nothing whatever is said about 61 , for which it would have been equally easy to calculate the Mersenne number. Failure to mention 6 I as an exponent in both lists is puzzling; but if this single trouble is put aside, it is possible to construct Mersenne's own rule exactly, in a way moreover that accounts for the fact that he completely missed the exponents 89 and 107, which have since been found to yield primes, and which are the only other numbers missed by Mersenne that are valid and are less than 521 , far beyond his own (incorrect) upper limit of 257. Mersenne's rule for exponents, then, was that they must differ by not more than one from a value of 2n, or by not more than three from a value of 2 211 • This rule explains his entire conjecture and accounts for all errors, whether of inclu-

300

History of Science

sion or omission, with the sole exception of the omitted exponent 61. It is likely that 61 was omitted because of some mistaken calculation in which Mersenne (or Frenicle) believed he had found a factorization of 2 61 - 1. 4 I wish to add a further conjecture of my own, that Mersenne may have written out this rule for the I 647 work, and that the fault was that of the printer. Making use of the idea from the second citation ... numeris primis, qui sunt unitate minores numero binarii analogo ... , recast in the language of the first citation, the entire rule may have been sent by Mersenne to the printer in this fonn, in which the words in square brackets are supposed to have been omitted in printing: Sequens Regula numeris primis agnoscendis admodum utilis: vidilicet numeris binarii analogicus unitate decurtatus, cuius exponens primus [differt primario a numero binarii analogo, seu cuius exponens primus] temario, vel minore numero, ab aliquo binarii analogo, cuius exponens sit par, est numerus primus. The commonest fault in copying, whether by a scribe or a typesetter, is known as haplography; that is, the dropping of a group of words that has been preceded by an unusual word or by a group of identical words that recur within a line or two. In this case the words cuius exponens primus having been set in type, the continuation is taken up again after their second occurrence in the copy. The verb dijfert is lost in the process. This loss is obvious in reading the printed text, and as mentioned previously, Archibald supposed it to be "exceeds." Quite possibly Mersenne in sending the errata noted only that the verb was missing, which he knew to be dijfert, and supplied it after numero without noting that an entire clause had been dropped. That the printer of this book was not very reliable is indicated not only by the existence of two pages of errata, but by the places marked sic. In the second passage cited, "minus" is certainly called for rather than "plus," and the plural is absurd, there being only one perfect number in question. Mersenne's writing is very difficult to read, and confusion by the printer between hiss and mas tenninals would account not only for the odd plural mentioned, but also for the accusative numerum in the first citation, which leaves the final est numerus primus without a subject. NOTES I

A.H. Beiler, Recreations in the Theory of Numbers (New York, 1964), pp. 13-14. Beiler's omission of 61 before 89 has been supplied in brackets. That Mersenne actu-

The Rule behind "Mersenne's Numbers"

301

ally utilized a lost theorem rather than just an ingenious but quite mistaken conjecture is rendered improbable in the present paper. 2 R.C. Archibald, "Mersenne's numbers," Scripta Mathematica, 3 (1935), p. 113. 3 "Further, if three is added to 64, the prime 67 arises, and thus the 67th power, plus [really minus] 1, will be the prime [really composite] number that follows, 147573952589676412927, of which this is a property, that multiplied into its half, perfect numbers [clearly a single number] are generated; which is assumed only of prime numbers that are less by one than a number of the binary powers, whence this property does not fit the number 5, but the numbers 3, 7, 31, 127, 8191, 131971, 524287, 2147483647 and all others of this genus." 4 The prime exponents he eliminated from Bungus' s list must have involved some actual factoring of 2P - 1.

15 Newton's Apple and Galileo's Dialogue

The final section of Isaac Newton's Phi/osophiae Natura/is Principia Mathematica develops the law of universal gravitation and shows how it explains the falling of objects to the earth, the orbiting of the moon, the motions of the planets and the phenomenon of tides. How did Newton discover this powerful law? He told a friend that the sight of an apple falling from a tree started him thinking in the right direction. Perhaps the moon happened to be in the daylight sky as he watched the apple. In any event he asked himself why the moon did not move away from the earth or fall to it as the apple did. He knew that the force of gravity extends without sensible diminution to the peaks of high mountains. Since it must extend farther, why not to the moon? If the familiar story of the apple is true, it still does not fully explain how Newton came to formulate his question about the moon. I believe what might have inspired his extension of gravity to the moon was a section he had read in Galileo Galilei's Dialogue on the Two Chief Systems of the World. The question about the moon started Newton on a chain of reflections in which a chief consideration was the weakening of gravity with distance. He made a calculation based on the weakening's being proportional to the square of the distance and found that it worked "pretty nearly." That was in 1666, when the University of Cambridge was closed because of an outbreak of the plague and Newton was back on the family farm in Lincolnshire. The data he had were inexact, and he put the matter aside for the time being. It was 20 years before he developed his complete theory of gravitation. It became the crowning achievement of the last section of the Principia, called "The System of the World," which unified terrestrial and celestial physics. Reprinted from the Scientific American 243 no. 2 (August 1980). 122-8. Reprinted with permission. Copyright© 1980 by Scientific American, Inc. All rights reserved.

Newton's Apple and Galileo's Dialogue

303

The story of Newton and the apple is well known all over the world for what I think are two main reasons. First, it makes something that has always been mysterious seem understandable in tenns of everyday experience. Second, and perhaps more appealing, it shows that even the greatest achievements of profound genius can have small beginnings accessible to ordinary minds. This encourages us even as it heightens our admiration of genuinely innovative human beings. Galileo, who died in 1642 about a year before Newton was born, recognized that great ideas often spring from simple phenomena. He praised William Gilbert, who in 1600 treated the earth as a huge magnet, because he had "framed such a stupendous concept by regarding an object that innumerable men of splendid intellect had handled without paying attention to it." Galileo went on to say (in translation, of course): "I do not doubt that in the course of time this new science will be improved with still further observations, and even more by true and conclusive demonstrations. But that need not diminish the glory of the first observer. I do not have smaller regard for the original inventor of the harp because of the certainty that his instrument was very crudely constructed and even more crudely played; rather, I admire him much more than I do a hundred artists who in ensuing centuries have brought this profession to the highest perfection .... To apply oneself to great inventions, starting from the smallest beginnings, and to judge that wonderful arts lie hidden behind trivial and childish things, is not for ordinary minds; these are concepts and ideas for superhman souls." What Galileo wrote about Gilbert captures what people feel when they first learn about Newton, the apple and the gravitational law that unites the earth and the heavens. It explains why a Japanese artist in about 1869, soon after interchanges between Occidental and Oriental cultures first began in earnest, chose the story of Newton's apple to symbolize Western science. Yet everyone realizes that the simplicity of Newton's great discovery is only apparent. The actual events must have been much more complex. Something was in his mind when the famed apple fell that was different from anything that had occurred to other people in similar circumstances. It would be good to know what the something could have been. I do not say "what it was," because there is no way to be certain of it, but I can suggest something it could well have been. Newton's early notes on gravitation and dynamics were published together only recently with indications of when they had been written. In 1965 John Herivel of Queen's University in Belfast published Background to Newton's Principia and I. Bernard Cohen of Harvard University presented a detailed study of the sources from which Newton learned of Galileo's work. Herivel showed conclusively that Newton took the Jaw of inertia from the writings of

304

History of Science

Rene Descartes. And Cohen presented convincing evidence that before Newton had published the Principia in 1687 he had not read anything of Galileo's except an English translation of the Dialogue. The law of inertia was fundamental to Newton's first investigation of the moon's orbit because he assumed that a large body such as the moon would move uniformly in a straight line in the absence of an external force. He sought the force that prevented the moon from doing so. Because the law of inertia was never stated by Galileo historians of science have understandably emphasized Descartes's influence on Newton. Nevertheless, both Herivel and Cohen acknowledge Newton's early reading of the Dialogue. By 1666 Newton had taken notes on the English translation published in 1661 in London by Thomas Salusbury. In the book is a demonstration that could scarcely have failed to interest Newton. Moreover, the diagram that accompanies the demonstration suggests the kind of analysis Newton first applied to the orbit of the moon. What Galileo had set out to prove had nothing to do with the moon. In fact, Galileo never tried to apply his physics to celestial bodies. He only discussed heavy bodies moving through measurable distances near the surface of the earth. The demonstration was supposed to prove that no heavy body resting on the earth would be flung off by the earth 's rotation regardless of the weight of the body or the angular velocity of the earth. Herivel characterized the proof as "unconvincing," and until recently I agreed with other historians that it was ingenious but fallacious . That a dubious proof about heavy bodies resting on the earth had been ignored in all discussions of Newton and the apple is not at all surprising. Let me emphasize, however, that what may have set a particular man on a given line of thought at a given time is not necessarily something that is clearly and easily related to it, any more than it is something that would naturally and logically give rise to it. A passage Galileo wrote and particularly a diagram he drew may have been applied by Newton to a purpose that never occurred to Galileo. I think this was probably the case with Newton's first analysis of the motion of the moon about the earth. The illustration in Plate 20 shows Galileo 's diagram designed to prove that a heavy body at rest could not be sent off along a tangent by the earth's rotation. He made the diagram to refute his contemporaries who thought this would happen if the earth really had a rotational velocity as high as 24,000 miles per day, or 1 ,ooo miles per hour. Galileo assumed that if a heavy body resting at A in the diagram were flung off by the earth's rotation, it would move uniformly along the tangent AB. How far the body would move horizontally in equal times would be proportional to

Newton's Apple and Galileo's Dialogue

305

the lines designated AF, FH and HK. The speeds of its fall in those times would be proportional to the verticals FG, HI and KL in the right triangle AKL. Galileo had shown that the vertical distance the body falls depends not on the time but on the square of the time. Indeed, the time-squared rule is his famous law of free fall. Here, however, he preferred to explain his reasoning about bodies in terms of the speeds with which they would fall in assigned times. Galileo relied on Euclid's proof that a "mixed" angle (consisting of a line and a circular arc) is always smaller than a rectilineal angle between that line and any other straight line whatsoever. In the diagram this means that the (mixed) angle of tangency between AB and the surface of the earth (represented by the arc AP) is smaller than any rectilineal angle that can be constructed. Therefore if the body ever left the earth, its downward speed would be more than enough to bring it back to the surface. Such was the burden of Galileo's demonstration, although he phrased it differently. Consider F to be arbitrarily close to A. Since the resting heavy body at A would not have left the earth where FG intersects its surface, a new tangent could be drawn at that point. The same analysis could then be applied to that point to show that the body would remain on the earth, still another tangent could be drawn and so on. Any central tendency giving rise to uniform acceleration would serve to hold down the resting body forever, regardless of the earth's speed or the body's weight as long as it weighed something. Galileo carefully specified that he spoke only of heavy bodies. Galileo's Aristotelian adversaries believed the speed of fall was proportional to the weight of the body, and so he included for their benefit the line AD representing a lighter body. He went on to point out that in fact the weight did not affect the speed of fall and that he had included AD merely to convince his opponents that even if the speed of descent did depend on the body's weight, the body would never be cast off along the tangent as they supposed. The fact that Galileo included this superfluous consideration makes his demonstration even more interesting in the light of Newtonian dynamics. If the rotational speed of the earth were to increase, the resting body would lose some weight. At a certain rotational speed it would become completely weightless (the one possibility Galileo had excluded). Nevertheless, it would not be flung off. It would begin to orbit the surface of the earth, having reached the appropriate orbital speed for that radius. If the earth were then to rotate faster, it would not be able to impart any new speed to the orbiting body because the body would no longer be resting on it. As a result the body would begin to move opposite to the projection (in the direction BA). Yet as Galileo correctly said, in no case would a heavy body be

306

History of Science

thrown off along the tangent solely by the earth's rotation. Whether this demonstration is adequate is a matter of opinion. It is now known that Galileo's conclusion was correct because bodies cannot by their own motion change their common center of gravity (Newton's fourth corollary to his laws of motion). In any case, the physics as well as the mathematics of Galileo's demonstration would have fascinated Newton, particularly because comparing the angle of tangency with a rectilineal angle suggests in a way the concept of infinitesimals of higher order in the calculus Newton invented. The case Newton investigated was the special one Galileo had excluded: a weightless body. What made the moon weightless was precisely a certain orbital radius and a certain speed. Newton's notes show that at the time he first worked on the orbit of the moon he was also investigating centrifugal force. In the Dialogue Galileo touches on centrifugal force only a few pages after the demonstration. Therefore it is highly plausible that Newton's basic question about the apple and the moon was inspired by his recent reading of the Dialogue. I want to return now to the origin of the law of inertia, which Herivel showed was taken by Newton not from Galileo but from Descartes. Newton could not have got the law from the Dialogue because not only is it not stated there but also it seems to have been denied in certain passages. Nevertheless, in the section of the Dialogue before the demonstration Galileo added three successive marginal notes expressing the three conditions that would later come to underlie Newton's law of inertia. The notes, which were in the English translation of the Dialogue Newton had read, are as follows: The motion impressed by the projicient is onely in a right line. The project[ile] moveth by the Tangent of the circle of the motion precedent in the point of separation. A grave project[ile], as soon as it is separated from the projicient, beginneth to decline.

The notes mean respectively that if a body is projected, the motion imparted to it is exclusively straight. If a body is released from whirling, it proceeds straight along the tangent at the point of release. And if the body is heavy, it starts dropping as soon as it is free. Since Galileo's diagram shows he considered the imparted motion to be uniform in speed, his discussion gave all three conditions of Newton 's law of inertia: uniform speed, rectilinearity and the effect of deviation by another force (in this case weight). Yet it is true that Galileo did not state any universal law of inertia as Descartes did and that Newton took the form of his law from Descartes. All these points deserve more explanation.

Newton's Apple and Galileo's Dialogue

307

Galileo never stated any kind of universal physical law. He rejected the traditional physics of his time chiefly because of its tendency to postulate universal laws without paying close attention to the actual physical world. In the Dialogue he explicitly denied that a body could actually move uniformly and perpetually in a straight line, because in that case it could leave the universe. Descartes ignored Galileo's law of free fall and offered a universal principle of uniform straight motion but went on to posit vortexes of "subtle matter" that prevented any body from moving in accordance with his law. Newton's law of universal gravitation excluded in principle the perpetual uniform straight motion of a massive body, since every body was continually subject to attraction by other bodies. With Einstein the Euclidean straight line simply disappeared from the physical universe. It could be said that in the realm of inertia Galileo stayed out of trouble, Newton got Descartes out of trouble and Einstein showed where the trouble had always been. When I try to guess what was on Newton's mind when he asked his pregnant question about the moon, I need not speculate about what was on Galileo's mind when he forged the physics of the Dialogue. It is enough to read with care what he published in the Dialogue, because that is what Newton did. Newton regarded Galileo not as a confused philosopher who frequently contradicted himself and made physical assumptions unworthy of an intelligent child but as a competent mathematical physicist. Newton would not have been side-tracked by statements in the Dialogue that may seem paradoxical. Instead he would have critically examined any interesting physical demonstrations in the book I believe it is a mistake to suppose that embedded in Galileo's world view was the conviction that all impressed motion must be circular, not straight. He certainly recognized exceptions, as the marginal notes I quoted above make clear. He did say that the particular motion arising in heaving bodies from the rotation of the earth is circular. "Keeping up with the earth," Galileo wrote, is a motion "indelibly impressed" in bodies that are or have been in contact with the earth. For example, he pointed out that a bird beats its wings only to move with respect to the earth, not to keep up with its rotation. Galileo's demonstration that objects cannot be flung off the earth by the earth's own rotation indicates only what he thought about the particular circular motion of "keeping up with the earth." To him the motion was not purely inertial in the sense of that term in modem physics but was the result of two straight motions, one inertial and one directed toward the earth's center. There is now a tendency to generalize this compound motion beyond the range to which Galileo limited it. It was not a simple "natural tendency" to move circularly but a combined motion. Some historians have postulated "circular inertia" as Galileo's fundamental conviction without specifying what circle (or circles) they

308

History of Science

mean. Newton probably did not read the Dialogue in the way these historians have. In fact, in the Principia Newton credited Galileo with having relied on the law of inertia, even though Newton had taken the statement of the law from Descartes, who was the first to make it universal. There is another passage in the Dialogue that must have intrigued Newton. In the passage the imaginary discussant Salviati speaks for Galileo and Simplicio speaks for the contemporary Aristotelians. Salvia ti: I did not say that the earth has neither an external nor an internal principle of moving circularly; I say that I do not know which of the two it has. My not knowing that does not have the power to remove it. But if this author [a German anti-Copernican) knows by which [kind of) principle other world bodies are moved around, as they certainly are moved, then I say that what makes the earth move is a thing similar to whatever moves Mars and Jupiter, and which he believes also moves the stellar sphere. If he will advise me as to the motive power of one of these movable bodies, I promise I shall be able to tell him what makes the earth move [around the sun]. Moreover, I shall do the same if he can teach me what it is that moves earthly things downward."

The last comment is certainly a remarkable prophecy for Galileo to have made, because it is the promise Newton ultimately carried out with his law of universal gravitation. Whatever Galileo may have had in mind, the words he published might have suggested to Newton his question about the apple and the moon. No less interesting are the lines that follow in the Dialogue. Simplicio: The cause of this effect is well known; everyone is aware that it is gravity. Salviati: You are wrong, Simplicio; what you ought to say is that everyone knows it is

called "gravity." What I am asking you for is not the name of the thing but its essence, of which essence you know not a bit more than you know about the essence of whatever moves the stars around. I except the name that has been attached to it and that has become a familiar household word through the continual experience we have of it daily. But we do not really understand what principle or what force it is that moves stones downward, any more than we understand what moves them upward after they leave the thrower's hand, or what moves the moon around. For Galileo science differed from philosophy in concentrating less on finding principles or forces of nature than on learning all it could from "sensate experiences and necessary demonstrations." From such experiences he had discovered the law of free fall and the parabolic trajectory of projectiles. For Descartes, on the other hand, anything of consequence in science came from

Newton's Apple and Galileo's Dialogue

309

principles, and he called his major work Principles of Philosophy. Galileo names his chief book Two New Sciences. The English title of Newton's magnum opus, Mathematical Principles of Natural Philosophy, falls neatly in the middle because in his day "natural philosophy" meant "physical science." There Newton set forth his celebrated three laws of motion: the law of inertia, the law of force and the law of action and reaction. Then he remarked in a scholium: By the first two laws and the first two corollaries, Galileo found that the descent of heavy bodies is in the squared ratio of the times, and the motion of projectiles is parabolic, experience agreeing except to the extent that their motion is somewhat retarded by resistance of the air.

If it was unfair of Newton not to add that Descartes was the first to state the inertial law as a universal principle, he was just as unfair to himself by not mentioning that the first statement of the force law was his own. Galileo had spoken only of acceleration, not of force, and only in connection with the fall of heavy bodies. When Newton identified force with acceleration in his second law and made acceleration a universal measure of force, he went far beyond anything Galileo had ever had in mind. Newton of course knew that. Before he developed the science of dynamics the concept of forces in nature was considered to be self-contradictory: what was forced was not natural by definition. I suppose Newton felt that Galileo, with his mathematical treatment of the acceleration of falling objects, had taken the first small step toward the dynamics that would come to link terrestrial and celestial physics. If Newton began to develop the science of dynamics when he contemplated the apple and the moon, and if he remembered that his reflections were inspired by Galileo's Dialogue, it is no wonder he paid exaggerated tribute to him. He may have felt about his Italian predecessor as Galileo did about his English predecessor, Gilbert.

"Nothing I have ever done gives me more satisfaction than my rediscovery of A.B. Johnson," Drake wrote in "Back from Limbo: The Rediscovery of Alexander Bryan Johnson" ( 1 ) . This paper, published in a collection of papers on Johnson, is in part an account of his own discovery of Johnson in 1938 when he found a copy of the Treatise on Language: or the Relation which Words Bear to Things ( 1836), which he called "perhaps the greatest original American contribution to philosophy - or rather to the theory of knowledge," and which he himself edited and printed in 1940. Johnson ( 1786- I 867), a banker who lived in Utica, New York, wrote on the philosophy of language and meaning, taking the view that language refers to our perception of things and that the greatest source of error is mistaking our description of nature for nature itself. He sought "to make nature the expositor of words, instead of making words the expositor of nature." "A.B. Johnson and his Works on Language" (2), published in 1944, is an exposition of Johnson's theory of language based upon extensive quotation from the Treatise, and was Drake's first historical publication. Johnson, to whom Drake occasionally referred, continued to be of importance in Drake's own work, much of which is concerned, after all, with the difficulties of reaching a correct scientific description of nature within the limitations of language, including mathematics. For example, the next paper, "Literacy and Scientific Notations" (3), although not mentioning Johnson specifically, is concerned with the way in which forms of notation, as proportional or algebraic or chemical, just as language, contribute to determining our concepts of nature. Drake referred to Johnson 's book as "an antimetaphysical bombshell," and the same is true of the work of the one philosopher of science, of course aside from Galileo, on whom he wrote, in "J.B. Stallo and the Critique of Classical Physics" (4). John Bernard Stallo (1823-1900), at times a professor of physical sciences and philosophy, a lawyer, and a diplomat, published in 1882 The Concepts and Theories of Modern Physics, a criticism of the implied metaphysics of current physics. His work has much in common with Mach, who corresponded with Stallo following the publication of his book and wrote a preface to the German edition. Like Johnson, Stallo also began with the limitations of language and used it to criticize the assumption that things correspond to our way of conceiving them. He turned this on the atomic theory of matter, the kinetic theory of gases, and the ether, which presumably is why his work was of interest to Mach, and its value lies not so much in the effectiveness of its criticism, which was at times misdirected, as in its uncovering of metaphysical assumptions taking the place of scientific knowledge. In a sense, Stallo was in the tradition of Galileo's criticism of Aristotelianism as being about words rather than things, a world on paper rather than the world of nature, which may have something to do with why Drake found him of particular interest.

1 Back from Limbo: The Rediscovery of Alexander Bryan Johnson

For nearly thirty years I have looked forward to this occasion. We arc here to honor the memory of Alexander Bryan Johnson, one of Utica's first citizens in every sense of the word. I am happy to take part and proud to do so in the company of distinguished scholars drawn from many fields, whose very presence shows the pennanent place that Johnson has earned in the history of ideas. For this I must thank Hamilton College, the Munson-Williams-Proctor Institute, and all the persons and organizations that have cooperated to make the occasion possible. Many aspects of Johnson's life and works are about to be discussed by experts in various phases of his activity and by specialists in the history of Utica. It is fitting that the professional appraisal of his life and work should fall to them. But first in order of time, if not of importance, is the story of events which brought Johnson's work, after a century of neglect, to its proper place on the American intellectual scene. It was in 1938 that, as Professor David Rynin said in the Introduction to a modem edition of Johnson's Treatise on language, I "by chance found a copy" of that book which I "luckily recognized for what it is, a philosophical classic."1 Apart from the fact that in what Professor Rynin acclaimed as a philosophical classic, I recognized only an antimetaphysical bombshell, that account is correct. In a sense it gives the whole story and gives it very succinctly. The discovery was certainly a lucky chance. But it is now generally agreed that every event has causes, and that when we speak of "chance" and "luck," we do not mean to deny that causes exist; we mean only that they are obscure, or Reprinled from Langua!ie and Value, eds. Charles L. Todd and Russell Blackwood (Wes1pon: Greenwood Publishing Group, Inc., 1970), 3-15. Copyright© 1970 by Greenwood Publishing Group, Inc. Reproduced with permission of Greenwood Publishing Group. Inc .. Westpon. CT.

316

Philosophy of Science and Language

unknown to us. It is not always possible to trace such causes accurately and completely, and they may not all be of equal significance. But to the historian of ideas they are all of possible interest. In the present instance the long neglect of Johnson 's thought is no less surprising than the resemblance between it and that of other men who came after him but knew nothing of his work. This story of its rediscovery may throw light on both those puzzles. The history of ideas presents many examples in which an important concept has been announced before its time, so to speak, and has been recognized as valuable only later. Sometimes the later writer has found the idea in the writings of an earlier man, and then he may have hailed the genius of his predecessor, or he may have tried to conceal his ancient source. The laller possibility is ever present, and gives special trouble to historians of science. Again, the later writer may have developed the idea quite independently, and its discovery in the works of an earlier writer may occur only much later. To trace the actual facts in all cases is not always possible; the historian may be obliged to make conjectures. It therefore seems to me that where an actual detailed account of events can be given in one instance - as in that of A.B. Johnson's critique of language - it should be given in full, as a possible guide to the reconstruction of other instances far in the past. No one has yet demonstrated an indebtedness of Johnson to earlier writers for his fundamental method of attack on philosophical problems. It also appears that writers in the present century who developed a similar method of auack owe no debt to Johnson, directly or through any philosophical tradition. Such a situation is rare in the history or ideas; perhaps it is unique. Most instances of the anticipation of later ideas have been found through the discovery of manuscript of other unpublished material, or occasionally where a publication was obscured by reason of its place or language. But Johnson's important work was published in English in New York, and not once but twice, eight years apart. It was also published in diluted form in some of his numerous other pamphlets and books. I can think of no truly parallel case of neglect and belated recognition since the invention of printing, five centuries ago. An isolated idea may indeed be put forth in a lengthy book and escape notice as original and valuable. But Johnson's contribution is not an isolated idea; it is a truly new line of thought concerning a very much agitated question - the nature of philosophical problems. Johnson's line of thought is not merely suggested, but is worked out consistently and in considerable detail. No one before him seems to have hit on it, let alone to have worked it out. Yet for centuries men of high intellect had concerned themselves with the same problems. Among themselves, they had expressed differences of philosophical opinion or that is how I should describe the differences between John Locke, George

Back from Limbo: The Rediscovery of Alexander Bryan Johnson

317

Berkeley, and David Hume. Johnson was able to show that each of those three philosophers had on occasion slipped into a linguistic trap, in the belief that some truth about nature was being shown. Johnson's contribution was thus no mere difference of opinion, but a promising and novel method of semantic analysis. It was published repeatedly. But it attracted no attention from recognized philosophers. The normal explanation would be that Johnson's proposed analysis was worthless. But the modem, independent development of linguistic analysis in philosophy obviates that explanation. Johnson's method of investigation, if not all of his conclusions, has been entirely vindicated. Another possible explanation would be that Johnson's method was not clearly set forth and illustrated. But no one has taken or is likely to take that position. Indeed, the opening lecture of the Treatise makes it unmistakably clear that Johnson realized the novelty of what he wished to say, as well as the difficulty of preventing his listeners or readers from confusing it with quite ordinary critiques of language. "I allude to no defects of language that you ever heard of or conceived," he said. "I also allude to none that can be obviated. The most that I hope to perform is to make them known, as we erect a beacon to denote the presence of a shoal which we cannot remove." He spared no effort to clarify his view of the manner in which words get their meanings and to show its relevance to the problems of philosophy. I hope I have sufficiently convinced you that the problems presented to the historian of ideas by the neglect of Johnson's Treatise on Lan8uage are both interesting and unusual. Their solutions must be sought in a variety of circumstances that must have combined to push the book into limbo. It is easy to suggest a few of these: Johnson was an American venturing on abstruse matters at a time when America was considered (and considered itself) a rude and uncultured land; Johnson's views, if understood, might threaten religion in an intensely religious age; Johnson lived far to the west of the American universities, and had no connection with them; Johnson was a banker, and therefore a mere amateur in the eyes of philosophers. There is always a great surplus of amateur productions in philosophy to which philosophers can scarcely afford to pay attention. The effect that each of these factors had on the neglect of Johnson 's Treatise by philosophers is an interesting question for historians. But the rediscovery of Johnson ' s Treatise presents a further challenge to historians of ideas. The phenomenon of simultaneous discovery is one of the most illuminating reflections of a cultural epoch. Now the Treatise was rediscovered independently within a year or so by two men, unknown to one another, in widely separated places, under totally different circumstances. That fact needs explanation in terms of cause and effect. Chance and luck might pass as expla-

318

Philosophy of Science and Language

nations for one incident, but not for two so closely related and yet so different. What was the intellectual climate that suddenly made the Treatise of interest to a businessman in one city and a scholar in another? A young German philosopher, inspired by the creation of symbolic logic, had turned his attention to language, with devastating effects for conventional philosophy. The English translator of his work had ventured a work of his own, a semipopular book called The Meaning of Meaning. A Polish engineer had found in linguistics the basis for a visionary reform of the human race, set forth in two books, The Manhood of Humanity and Science and Sanity. A great depression and the threat of an unscrupulous dictator had inspired an American popularizer to apply the ideas to politics in a book called The Tyranny of Words. 2 All these are demonstrably factors in the double rediscovery of Johnson's Treatise, and goodness knows how many other things may have been involved. But was Johnson really important? Was the rediscovery of his Treatise important? My opinion is affirmative on both questions, but my opinion is obviously biased. It is up to historians of ideas to explore the mines and determine whether they will repay further working. From what I have learned of Utica, it is no mere accident that the first and perhaps the greatest original American contribution to philosophy - or rather to the theory of knowledge - originated here. I should not wonder if many other cultural aspects of this early western outpost now begin to receive the attention they deserve from American historians of ideas as a result of this conference on A.B. Johnson. For in my opinion his pioneer examination of our most fundamental means of communication, language, is destined to take a place in the history of Western culture far above that which anyone else imagines at present. It is only when all the questions I have mentioned above have been explored, and their ramifications begin to be perceived, that Johnson, in his restless and versatile contemplations and activities, will be recognized as a prophet of our own time; a lonely, self-taught thinker whose voice cried out in the wilderness, and who - had he been heard might have changed the intellectual history of mankind. Because I believe that, I shall try to present in as much detail and as accurately as I can the story of Johnson's return from limbo. If my view is correct, no scrap of data should be left out. I do no know which of the anecdotes I shall relate may tum out to be enlightening to the historian, which ones merely amusing, and which ones wrong despite my best efforts. Of the many people who deserve credit for their contributions to the rediscovery of A.B. Johnson and to the awakening of interest in his life and works, I should like first to mention three of my closest friends at the University of California in 1930: Daniel Belmont, Mark Eudey, and Henry Ralston. Belmont and

Back from Limbo: The Rediscovery of Alexander Bryan Johnson

319

I were then fellow students in the department of philosophy. After my graduation in 1932 he won a fellowship which enabled him to study at Cambridge under Professor Ludwig Wittgenstein. That good fortune of his turned out to be still luckier for me, and ultimately for A.B. Johnson . For it was from Belmont, after his return from England, that I was able to learn much about the nature of philosophical problems and about their frequent, if not invariable, origin in misapprehensions of the nature of language. But by that time we had both left Berkeley and were at work outside the university. About the beginning of 1938 I organized a sort of cooperative seminar with my friends in order that I might take advantage of their special knowledge in divergent fields. The idea was that each of us would serve as instructor at two or three meetings of the group, as the price of receiving instruction at the others. Meeting weekly in the evening at one another's house, we thus heard from Belmont about modem trends in philosophy, from Mark Eudey about astrophysics and cosmogony, and from Henry Ralston about evolution and genetics. Other friends dealt with such a variety of topics as electronics, history, and naval strategy, while I undertook to talk on comparative philology. In order to brush up on that subject, I went one Saturday afternoon to a San Francisco bookstore in which I had lately noticed a number of works on linguistics. Among them I saw a book labeled Johnson's Treatise on Langua,f?e. The book was obviously too old to contain any scientific philology, but I took it off the shelf anyway, wondering if it might be some rare work by Dr. Samuel Johnson of dictionary fame. The title page quickly showed that idea to be mistaken; the book was by one A.B. Johnson, whose name was new to me. But I did not put it back on the shelf, because my attention was arrested by its subtitle: "The Relation Which Words Bear to Things." Now, that had nothing to do with my principal interest at the moment, but it chanced to bear directly on the matters that Belmont had studied under Wittgenstein at Cambridge and that he had just been discussing in our seminar. I therefore looked further into the book, and found it to be composed in numbered paragraphs, beginning with the sentence, "Man exists in a world of his own creation." Curiously enough, Wittgenstein's one published book at that time was known as the Tractatus, or treatise; it was composed in numbered sentences, and began with the sentence, "The world is everything that is the case." 3 Struck by those coincidences, I read Johnson's preface and glanced through his table of contents, noting at once other similarities in style and intention on the part of the two writers. Reading then at random a few paragraphs of the Treatise, I became the first modem Johnsonian in a matter of minutes. Clearly, whoever the man might have been, he knew what he was talking about, was aware of the special difficulties that would attend its communication to others, and had a peculiar gift of expression.

320

Philosophy of Science and Language

The book was priced at two dollars, which in those times of economic recession was exorbitant for an old book by an unknown author. But the dealer, Mr. Malcolm McNeill, was not inclined to bargain with me. He contended that any book more than a century old, particularly an American book so venerable, was worth two dollars or it was worth nothing. As a bookseller, he would not consider the latter alternative. I urged it on the grounds that if the book had any real merit, I should have heard of it at the university. I lost the argument, but I did happen to have two dollars with me, and luckily I decided to let comparative philology go for that day and to buy the Treatise instead. I started to read it on the streetcar home, and with only an interruption for dinner I continued reading it far into the night, finishing it on Sunday morning. Since that memorable weekend nearly three decades ago I have read the Treatise through many times, never without renewed admiration for its neglected author. I soon showed my new find to Mark Eudey, who of course had also been recently awakened to the importance of the topic at our seminars. He read it, and we then concocted a practical joke on Daniel Belmont. Each of us prepared a paper arranged in the style of Wittgenstein's Tractatus, but composed simply by taking a selection of sentences from the list of paragraph headings that served as table of contents of Johnson's original book (and which has unfortunately been omitted from all modern editions). These papers we showed to Belmont at lunch one day, saying that each of us had tried to set down the most essential points we had learned from his seminar discourses, amplified by some thoughts of our own, and since we were not in complete agreement, we wanted him to referee the matter. This he obligingly did, writing out his comments on our two papers and giving them to us a week later, without ever suspecting that they were a hoax, cribbed word for word from an author who had written more than a century before. When we revealed that fact, he was amazed to the point of incredulity, so close were Johnson's insights to the best contemporary thought. I then lent him the book and began to check the principal bibliographies of philosophy, as well as a very long bibliography of semantics which had recently been published, but I could find no mention of Johnson anywhere. It may seem odd, in retrospect, that Johnson's Treatise has gone entirely unnoticed by historians of philosophy. But there are reasons for their silence, of which I shall mention only one. On the last page of the Treatise, summing up his work, Johnson said: Our misapprehension of the nature of language has occasioned a greater waste of time, and effort, and genius, than all the other mistakes and delusions with which humanity has been afflicted ... and though metaphysicks [sic], a rank branch of the error, is fallen

Back from Limbo: The Rediscovery of Alexander Bryan Johnson

321

into disrepute, it is abandoned like a mine which will not repay the expense of working, rather than like a process of mining which we have discovered to be constitutionally incapable of producing gold.

Professional philosophers could scarcely be expected to admit to their ranks the writer of those words, especially in Johnson's day. Thus I believe that the amateur status of the commillee that welcomed Johnson back from limbo was not entirely a mailer of chance. Though I am gelling ahead of my story, I shall say now that the lukewarm reception of the Treatise by philosophers who reviewed its republication in 1947 lent support to that view. So did the fact that an independent rediscovery of the Treatise, not long after I had found my copy, was made by another ma_n whose interests were far from those of recognized philosophers.4 Nor does it seem irrelevant to the same point that Henry Ralston , who was next to borrow my copy, promptly became the most enthusiastic Johnsonian of us all. Ralston, a biologist, was even less familiar with formal philosophy than the rest of us, and perhaps for that reason he endorsed the entire Treatise without any reservation. By this time quite a number of my acquaintances wanted to read the Treatise, while I quite naturally became reluctant to lend my copy further and risk its loss, for none was to be found in any library on the Pacific Coast. I made the most determined effort to find another copy through book dealers, but without success. Finally I agreed to print some copies for my friends if enough money could be raised to buy paper and a press. Thus it was that in May 1939 I embarked on a totally unfamiliar project, which I had ample opportunity to regret before it was done. Working evenings and weekends, I needed more than a year to complete it. When the printing was finished in August 1940, Belmont undertook the job of binding the forty-two numbered copies, which were distributed to the subscribers at the end of that year. I sent copies to Aldous Huxley and to Professor Eric Temple Bell, from who I received thanks and comments. Huxley saw in it, as I did, some remarkable anticipations of the operational isl philosophy of the late Professor Percy Bridgman. Later on I sent a copy to Bridgman, who confirmed our opinion, though he remarked that Johnson was clearly unacquainted with the actual procedures of the physical scientist, procedures that lay at the heart of his operationalism. I sent a copy also to Albert Einstein, who recognized in Johnson a truly original thinker capable of working out his ideas in a clear and uncompromising way. In my opinion, that tribute alone justifies all Johnson's lonely labors and our gathering here today to do him honor. By systematic searching and correspondence I was able to find out a lillle about Johnson, to compile a reasonably complete bibliography of his writings,

322

Philosophy of Science and Language

and to identify many of his sources. In all these efforts Miss Alice Dodge of the Utica Public Library was immensely helpful. We used to supply one another with typewritten copies of rare Johnson items in our respective collections. And over a period of years I was able to acquire, with the help of book dealers scattered from California to Vennont, copies of all of Johnson 's printed books and many of his pamphlets. In that way I became aware of the broad scope of his genius, which is no less astonishing than the highly original quality of his mind as shown in his analysis of the nature of philosophical problems. Early in 1943 my friend Robert P. Willson called to my attention a book in which several quotations from Johnson's Treatise were given. Called Language Habits in Human Affairs, it had been published in 1941 by Professor Irving J. Lee of Northwestern University. Lee 's book marked an epoch in the revival of interest in Johnson's work, constituting as it did the first publication of citations from the Treatise in the present century.5 Early in 1944 I wrote to Lee, being curious to know whether a copy of my privately printed edition had come to his attention. His reply is interesting, for it shows that Johnson ' s return from limbo would not have been delayed much longer even if chance had not put a copy of the Treatise in my hands at a time when other circumstances had made me capable of appreciating it. Lee, then a captain in the anny, wrote in reply: I am happy to discover some one else who found A.B. Johnson a remarkable, even though unknown figure. I came upon the Treatise on Language while doing some early work in semantics. In the usual routine of going through the Union Catalogue, etc., I found a copy in the John Crerar Library in Chicago. There was nothing exciting about the physical discovery, but on several occasions during the writing of my book I was disposed lo stop everything and do an edition of Johnson.

Lee suggested that I write a paper on Johnson for a new journal called ETC., A Review of General Semantics, edited by Professor S.I. Hayakawa. In the same letter, he urged me to prepare a modem edition of the Treatise such as he himself had projected. I replied that such a task would require extensive historical and bibliographical infonnation that I lacked, and that I had already tried to induce Daniel Belmont to undertake such an edition, but that he had declined for the same reason . For us to study all the books concerning similar problems that had been written around Johnson's time (with some of which he was familiar) was out of the question; we had neither the time nor the facilities to do so. I did, however, submit a paper to ETC., outlining in Johnson's life and his linguistic analysis. That paper appeared in the summer issue of the journal in

Back from Limbo: The Rediscovery of Alexander Bryan Johnson

323

1944. I distributed a number of reprints of it, of which one was sent to Professor Wittgenstein, who sent me a courteous acknowledgment. As I have said, Lee's independent rediscovery of the Treatise shows that the time was at last ripe for the emergence of A.B. Johnson in the modem world. Lee's deep interest in the General Semantics of Alfred Korzybski (an enthusiasm which I have never shared) led him to the Treatise in Chicago not long after my acquaintance with Wittgenstein's writings enabled me to recognize its merit when I ran across it in San Francisco. It would be hard to imagine two more divergent philosophical backgrounds leading to the same discovery. But the history of ideas, like the history of science, provides other instances of simultaneous discovery when a particular problem is "in the air." Very often such discoveries are made by people well outside the mainstream of conventional thought as in the present case. It is much to be regretted that Professor Lee is no longer with us, for I should have liked to meet him. Before his untimely death, however, he contributed further to the spread of Johnson's ideas. In 1948 he persuaded Mr. John Chamberlin of Milwaukee to publish Johnson's Meaning of Words in an edition of five hundred copies, and in 1949 he cited Johnson at some length in another book of his own. 6 About the time I first wrote to Lee in the spring of 1944, Daniel Belmont was invited to speak before the Philosophy Club on the Berkeley campus and he chose as his topic Johnson's Treatise. Professor Rynin was present and became deeply interested. Having bought a copy of my privately printed edition, he asked whether my copy of the original would be available if he could persuade the University of California Press to publish a modem edition. I readily agreed to assist in any way to make Johnson's work more generally accessible, and offered not only the book but my bibliography of Johnson and my notes identifying his probable sources. Professor Rynin was privileged to consult Johnson's manuscript autobiography through the courtesy of Mr. Alexander Bryan Johnson III, and wrote a biographical introduction, as well as a critical essay which discussed Johnson's work in modem philosophical perspective. To the text of the Treatise he added the variant readings and additional passages found in the edition of 1828, Johnson's Philosophy of Human Knowledge, and some material from The Meaning of Words. All this took time, and it was not until February 1947 that Professor Rynin's scholarly modem edition of the Treatise issued from the press. Though it was reviewed in many places both in America and in England, Johnson's name still continued to be omitted from histories of philosophy, and even from a later book specifically concerned with the history of philosophy in America. In 1945, while the new edition of the Treatise was in preparation, my friend

324

Philosophy of Science and Language

Henry Ralston had become associate professor of physiology at the University of Texas in Galveston. There he contributed to the spread of Johnson's views by issuing in mimeographed form for the use of his students and friends another scarce book of Johnson's, The Physiology of the Senses. Professor Ralston remarked to me that he found the book particularly interesting with regard to the problem of referred pain, which appears not to have received special attention among physiologists until a later time. Quite independently of all the activities I have mentioned thus far, Johnson was called to public attention in 1946 by Professor Joseph Dorfman in the first volume of his The Economic Mind in American Civilization, 16o6-1865.7 I wrote to Professor Dorfman, questioning the basis for a statement of his concerning Johnson's political position during the Civil War. His courteous reply showed that he knew a great deal more about many phases of Johnson's thought than I did, though by that time I had made a fairly extensive study of Johnson's writings in every field . It seems to me sufficiently curious to deserve mention that although I have spend most of my life in the field of investment banking, I have found Johnson's critique of language much more interesting than his successful banking career - as perhaps Johnson himself did. It would be interesting to know whether his attention was turned upon language by problems of law and finance rather than by philosophy. That would be by no means impossible, and well deserving of investigation in the context of surviving letters and unpublished manuscripts. 8 In 1948 the distinguished antiquarian bookseller Jacob Zeitlin of Los Angeles, who had become interested in Johnson through personal conversations with me in 1942, instigated the printing, in a fine limited edition of two hundred copies, of a letter of A.B. Johnson to the Honorable Nathan Williams (dated October 6, 1834, and printed in the Albany Argus Extra of October 17, 1834) to which Mr. Zeitlin added a brief preface. Professor Rynin's edition of the Treatise was sold out long before 1959, when it was reprinted in paperback form. That new issue received an enthusiastic review in England by Lancelot Law Whyte, the author of several important critiques of science and of society. Mr. Whyte had visited me when he was in California around 1955, to discuss Johnson's work and to see my collection of his books. The new edition of the Treatise also received an extensive and favorable review in Italy, where the operationalist philosophy is more highly esteemed than here in the land of its birth. An Italian translation of the Treatise was even projected at one time, though it did not materialize. More recently, Johnson has been honored in Sweden and in Germany through the labors of Dr. Lars Gustafsson. Thus there can be no doubt that A.B. Johnson is at last safely back from obscurity, to which he will never again be

Back from Limbo: The Rediscovery of Alexander Bryan Johnson

325

relegated. For he is no longer merely a local, nor even a national, but an international figure. Early in 1966 Professor Charles L. Todd was in San Francisco, where Professor S.I. Hayakawa spoke to him enthusiastically of Johnson 's pioneer place in semantics. He put Professor Todd in touch with me, and on a later occasion we all dined together. It was at that dinner in San Francisco that the idea of the present centennial celebration was first discussed. Today, thanks to Professor Todd's tireless energy, it is a reality. To me it is a dream come true, and my only regret is that not all the people I have mentioned could be here to share in it. Nothing I have ever done gives me more satisfaction than my rediscovery of A.B. Johnson . Your presence here to honor him makes me feel that I have partly repaid my enormous debt to him for having freed me from many errors and prejudices, and having shown me how to avoid many delusions into which one may falLthrough a misunderstanding of language. But scarcely less is my debt to those friends who put me in a position to benefit from the happy chance that first placed the Treatise in my hands, and to others whose support made possible its reprinting. Only a few of those friends have been mentioned by name today, but all took part in welcoming back one of America's most profound and original thinkers, the Philosophical Banker of Utica. NOTES

A.B. Johnson, A Treatise on Language ... , edited by David Rynin (Berkeley: University of California Press, 1959), p. 3. 2 Ludwig Wiugenstein, C.K. Ogden, Alfred Korzybski, and Stuart Chase. Any discussion of these mailers here would go far beyond the scope of this paper. 3 Ludwig Wiugenstein, Tractatus Logico-Phi/osophicus (London, 1922). Through Belmont's kindness, I also knew Wittgenstein's then unpublished "Blue Book," which was still more helpful in the recognition of Johnson's genius. 4 Professor Irving J. Lee, who wrote in 1947: "For almost a year before the winter of 1940, I had been systematically going through everything in the catalogue listings under 'Language' in the John Crerar Library in Chicago. The going had been slow and dusty. And then a subtitle stood out clear and clean: The Relation which Words Bear to Things. The book was entitled A Treatise on Langual:e by A.B. Johnson, published in 1836 by Harper & Brothers ... " 5 My privately printed edition did not constitute publication, as did Lee's book. Earlier in the century, I am infonned by Professor Joseph Dorfman, Johnson's economi theories had received auention in publications by Professor Wesley C. Mitchell, but his studies in language and meaning remained unnoticed. I

326

Philosophy of Science and Language

6 The Langua.~e of Wisdom and Folly, edited and with an Introduction by Irving J. Lee (New York, 1949), pp. 113-123, 188-198. 7 New York, 1946; reprinted, t 966. 8 Both Daniel Belmont and Mark Eudey, who shared my early interest in Johnson's work, are presently engaged in San Francisco in work related 10 investment banking, as I was until the present year. In the same field is Johnson's direct descendant and namesake, Mr. Alexander Bryan Johnson, who is partner in the finn of Cyrus J. Lawrence and Sons, New York.

2

A.B. Johnson and His Works on Language

At the Utica Lyceum in 1825, Mr. Alexander Bryan Johnson, a locally prominent banker, expounded to his fellow townsmen some novel views on the nature of language. "My lectures," he said, "will endeavour to subordinate language to nature; to make nature the expositor of words, instead of making words the expositors of nature. If I succeed, the success will ultimately accomplish a great revolution in every branch of learning .... That language will eventually receive the interpretation for which I contend, I cannot doubt; but that I possess the ability to make existing errours [sic] perceived even, I much question."(2) 1 His lectures were published three years later under the title, The Philosophy of Human Knowledge, or A Treatise on Language. This work "excited no attention except many gratifying letters ... from strangers in various states of our Union, and one extensive review." (2) Nevertheless, he received so many requests for copies after the first edition had been exhausted that he revised the work and republished it in 1836 under the less pretentious title of A Treatise on Language, or The Relation which Words Bear to Things. The intervening years had apparently strengthened his doubts in his own ability to accomplish the revolution he had predicted, for in the preface to this second edition he says: ... all that the book contains is the elucidation of but one precept; namely, to interpret language by nature . ... The precept itself which I have sought to illustrate, I profoundly respect; but whether I have demonstrated its importance, the publick must determine. Amid active and extensive employments, and with no external stimulus to literary pursuits, I shall be satisfied if the succeeding discourses shall commend the doctrine to the Reprinted from ETC: A Rt•i•iew o/Generul Semuntin I (Summer 1944). 238-52, by permission.

328

Philosophy of Science and Language

efforts of men whose understandings are more comprehensive than mine, and whose labours the world is accustomed to respect. Even this modest hope was never realized. The two authors among his contemporaries in whose work I have been able to find mention of Johnson's views did nothing to further them. The first, an English surgeon named Edward Johnson,2 prefaced his Nuces Phi/osophae (London, 1842) with an extensive quotation from the Treatise, but devoted the book to a defense and extension of Home Tooke's thesis that etymology affords the true key to the meaning of language. The second author was one whose labors were certainly widely respected, but not one who was likely to devote them to the program outlined by Johnson. It was Dr. Horace Bushnell, who said, in the introductory dissertation on language included in his God in Christ (Hartford, 1849): There are no words, in the physical department of language, that are exact representatives of particular physical things. For whether we take the theory of the Nominalists or the Realists, words are, in fact, and practically, names only of genera, not of individuals or species. To be even more exact, they represent only certain sensations of sight, touch, taste, smell, hearing - one or all. Hence the opportunity in language for endless mistakes and false reasonings, in reference to matters purely physical. This subject was labored some years ago with much acuteness and industry by one of our countrymen, Mr. A.B. Johnson, in a "Treatise on Language, or the Relation of Words to Things." The latter part of his title, however, is all that is justified; for to language in its more comprehensive sense, as a vehicle of spirit, thought, sentiment, he appears to have scarcely directed his inquiries. The next reference to Johnson's views on language (so far as I can find) was made nearly one hundred years later, in Irving Lee's language Habits in Human Affairs (New York, 1941). Meanwhile, the revolution that Johnson forecast appears to have begun in at least some branches of learning, without the slightest influence being traceable to him. The similarity of his works to those of the semanticists, although astonishingly close, seems to have escaped comment - perhaps because of the scarcity of Johnson's books after a century of neglect. 3 The selection of quotations from them for this article was guided by a desire to enable the reader to make the comparison. II

A few words about the man himself may be in order first. A.B. Johnson was born in Gosport, England, in 1786. He remained there until the completion of

A.B. Johnson and His Works on Language

329

his formal education, in 1801. His father having proceeded to America some years previously, the son joined him and went to work in his general store at Utica, New York. After a few years of this work, Johnson established a glass factory at Geneva. This he soon sold, going to New York City, where he became familiar with banking. Returning to Utica at the outbreak of the War of 1812, he participated in the organization there of a branch bank. Soon afterward he contrived to get through the State Legislature a charter for an insurance company, which implied banking powers - a device also used by Aaron Burr. Under this charter he operated his own bank until 1818, when the courts and legislature intervened. Johnson studied law, and was admitted to the bar, but never practiced. Until I 852 he was president of an important bank in Utica; in that year he resigned and made a tour of Europe, as the result of a "severe domestic affliction." The source of these biographical remarks, M.M. Bagg's Pioneers of Utica, does not go into details. Johnson himself speaks of his life as being "marked with sorrows ofno ordinary magnitude." (3) When he returned from Europe, Johnson organized a bank of his own. This was an unfortunate venture, for the bank collapsed inside of eighteen months, apparently as a result of important information being withheld from him. Although clearly not to blame for the disaster, he worked unremittingly to save what he could from the wreck, and succeeded in paying all creditors in full. He died in 1867. Bagg says that Johnson had the reputation of being intensely scrupulous, but harsh and severe, and of being interested only in money-making. He himself contended that he followed his profession only to attain financial independence, which he desired for the purpose of being free to write. He published a number of books covering a wide range of subjects - economics, politics, religion, and language - and a quantity of periodical articles, mostly on political matters. He issued in 1830 a proposal to compile a "Collated Dictionary, or a Complete Index of the English Language: designed to exhibit together all words which relate to the same subject, for the benefit of persons who are not acquainted with the whole compass of the language, and to assist the memory of persons who are acquainted." He worked for several years on this project, but never completed it. Roget's Thesaurus appeared in London in 1852, fulfilling the stated object. In the arrangement of material below, I have followed the order used by Johnson in developing his subject in the original lectures. Each quotation is given in its earliest form, except where a subsequent alteration seems clearer, or a later example more apt. Italics have been used where another author is quoted by Johnson. One more prefatory remark: In attempting an analysis of language, it should

330

Philosophy of Science and Language

not matter where one begins, or in what direction one strikes out, since the whole ground must eventually be covered, and a foothold is difficult to obtain anywhere. But an author must start somewhere, and each has certain predilections which induce him to choose one starting point as essentially the simplest or most fundamental. This is likely to have the effect of emphasizing excessively one phase of his analysis, and detracting from others of equal importance. In each of Johnson's books on language, the analysis is begun by pointing out the individuality of the data received through each of the senses. The reader is cautioned that in doing this, Johnson's concern was with language, and not with epistemology; this warning appears frequently in each of his books. So also does the disavowal of any intention to subvert any philosophical or theological system, or to refute any scientific position. "Many of the scientific tenets on which I comment are probably no longer authoritative, and kindred errors may be numerous; but if the reader shall collect from my comments the views of language that I entertain, he will collect all that I seek to accomplish, and I plead guilty in advance of all the scientific errors of which he may be able to convict me." (3) III

The opening lecture begins with a definition of terms: To avoid an ambiguity which is inherent in language, I will apply the term sights, to all the information that we derive from seeing; the term sounds, to all the information of hearing; and the terms feels, tastes, and smells, to all the information of the other senses. Hence, instead of saying that an orange is one existence, endued with several qualities, I shall estimate it as several existences, associated under one name, orange. Its appearance, I shall denominate the sight orange; its flavour, the taste orange; its odour, the smell orange; and its consistence, the feel orange. I shall adopt this phraseology not to build thereon a theory, but to discriminate between the information of different senses. (I) To investigate the sights, sounds, feels, tastes, and smells, which separately, and in various associations, constitute the external universe, is not my present object; nor shall I discuss whether sights, sounds, tastes, feels, and smells, are words which appropriately designate external existences. I adopt the phraseology as a means of investigating the nature of language; and if I shall establish the utility of the adoption, I trust you will tolerate the expressions, how much soever they may offend against euphony and custom. (2)

Johnson then sets forth his first general observation on language; namely, that the identity of an object as implied by language may or may not correspond to an identity in nature, and most frequently does not.

A.B. Johnson and His Works on Language

331

The simple property which pennils a word to name phenomena of different senses, has enabled theorists to convert the realities of life into a fairy tale. A universally admitted speculation of this character is, that distance, magnitude, figure, and extension, are not visible. This was originally suggested by Bishop Berkeley. He perceived that there are in roundness two phenomena - a sight and a feel; while there is but one name - roundness. The unity which exists in the name he attributed to nature; hence, he decided that the feel is the true roundness, and that sight possesses only an imaginary significance . ... Berkeley never imagined that invisibility was predicable of roundness by means of a latent ambiguity in language; but he accused vision with the production of a delusion. ( 1) When/ look at a book, says Professor Reid, it seems to possess thickness, as well as length and breadth; but we are certain that the visible appearance has no thickness.for it can be represented exactly on a flat piece of canvas. I am as certain as Mr. Reid that paintings possess not the feel thickness; still, this is no contradiction of what I see. The picture proves only that the sight thickness and the feel are not always associated .... From the frequency with which the sight and the feel are associated, we have given them but one name, and suppose them identical; but pictures would always have taught us the contrary, if we had not preferred the construction of a paradox. (I)

If you place a grain of musk in a phial, and hennetically seal the phial, you will soon smell the musk notwithstanding its confinement. The intellect conceives an analogy between the passage of the odour through the glass, and the passage of water through a sieve; hence we apply the word passage to both cases, though the passage of the odour is a smell and the passage of water is a sight and a feel. The intellect of a child will see an analogy between the two passages, and will therefore understand when you tell him that the odour passes through the phail, and he will become surprised thereat only when you stultify him by the belief that the passage of water and the passage of the odour are not merely analogous intellectually, but physically identical. A better use of the experiment is to elucidate therewith the nature of language, and for that purpose I adduce it. (4)4

The almost irresistible tendency to impute to nature the identities that exist in language is illustrated by a wealth of examples. Johnson asserts that many metaphysical positions owe their lives to this confusion. IV

His next principle is stated in these words: Every word is a sound, which had no signification before it was employed to name some phenomenon, and which even now has no signification apart from the phenomena to which it is applied. (I)

332

Philosophy of Science and Language

This principle, when thus expressed, seems obvious; still, in practice, it has escaped the vigilance of the most acute, and supplied metaphysicks with its most perplexing doctrines. Thus, take the word weight - it names a feel. The feel is abundantly familiar. It is discoverable in a feather, in a piece of lead, and in nearly every object. The word possessed no significancy before its introduction into language, and it now possesses none apart from the feel that it designates. Admit then that weight is the name of a feel, and observe how speciously I can employ the word after I divest it of all signification: thus, many objects are too small to be seen with the unassisted eye; and some the most powerful microscope can render but just visible; we may therefore well believe that numerous atoms are so small that no microscope can reveal them: still each must possess ... weight. Now observe, if weight names a feel, how has the word any signification when we predicate it of an atom, in which confessedly the feel cannot be experienced? What feel is that which cannot be felt? We have subtracted from the word all its significancy, and left nothing but a vacated sound. It becomes weight minus weight. ( 1) Because a cubick inch of air weighs the third part of a grain, we calculate the number of cubick inches of air which rest on a man in a column of forty or fifty miles in altitude; and by calling every inch the third part of a grain, we conclude that every man supports fourteen tons of air. Is not this divesting the phrase fourteen tons of its significance? Weight (and especially fourteen tons) is the name of a feel; and to use the word where no feel is discoverable, is like talking of a toothache which cannot be felt, or of an inaudible melody. (2) V

The third property of language to which Johnson turns his attention is the variety of applications which each word possesses in actual use: "Snow is white, paper is white, silver is white, the air is white, glass is white, you are white, and the floor is white; hence, after you are satisfied of the propriety of calling an object white, I shall know but little of its appearance, without I take an actual view of the object." (2) Though we suppose generally that external objects cause in other persons similar sights, tastes.feels, sounds, and smells, to those which they produce in us; yet, say metaphysicians, no man can possibly know this with certainty. Apparently there is a mysterious contradiction in the above metaphysical assertion; for while we wonder at the alleged impossibility, we are confident of its practical inefficiency. But the difficulty proceeds from not knowing that the word similar has several meanings, and that it is used diversely in the above positions. When I say, that the heat

A.B. Johnson and His Works on Language

333

which I am feeling is similar to what I felt yesterday, the word similar refers to the antecedent feel and the present. So long as I restrict thus its meaning, I cannot know that fire produces in you a similar feeling to what it produces in me. I cannot feel with your organs. But we intend a different meaning when we affirm that the feel which you experience is similar to mine. The word similar means now that you display, under the operation of heat, appearances like those which I exhibit, or that you describe your feelings in the same language, &c .... The assertions refer to different phenomena. ( 1) An ignorance of the principle which we are now considering occasions also much admiration; thus Professor Stewart, in his Philosophy, says, an expert accountant can enumerate, almost at a glance, a long column , though he may be unable to recollect any of the figures which compose the sum. Thus far the statement... creates no perplexity, but when he adds, nobody doubts but each of these figures has passed through the accountant's mind, the case seems altered. The accountant begins to wonder that he does not recollect the several figures. Passing through the mind, he supposes to mean something different from what he experiences in addition. He does not know that words mean, in every case, the phenomena to which they refer. He supposes rather that the passage of the figures through the mind signifies the same as the passage of an army through the gate of a city. ( 1) Our natural bias is to conceive white, blue, and yellow, which exist in the mind only, as something spread over the surface of bodies. (Stewart) A painter might startle if he should be informed that white, blue, and yellow are not spread over the surface of bodies. Has he suffered a delusion which you are about to dispel? No; you are using the phrase "spread over the surface" as no man ever used it when applied to colours. You insist that the phrase has but one signification, and because that signification is undiscoverable in colours, you conclude that mankind are suffering an egregious errour. The errour is, however, in language, which has not a peculiar term to express every phenomenon, but employs the same term to name several phenomena. You transfer a defect which exists in language, to our senses, where it exists not. What is the spreading over the surface to which Mr. Stewart refers? He will admit that baize can be spread over the surface of a table - this affords an elucidation of his meaning. It "is" where we can feel the body that is covered and the body that covers. All, then, which Mr. Stewart means is, that colour cannot be felt - a sight cannot be felt ... Mr. Stewart will admit that the oil and lead which compose colour can be spread over the surface of bodies. It is the sight colour which produces the difficulty. The sight can never be spread over the surface of a body, so long as we confine the signification of the phrase to the phenomena of feeling. ( t)

334

Philosophy of Science and Language VI

Scientists as well as philosophers are reproached for the deliberate choice of phrases designed to excite astonishment and admiration. "After a moment's exposure a drop of the otto of roses will fill with odour many rooms, while the drop will exhibit no diminution of size. This phenomenon is too common to excite admiration, but much may be excited if you exhibit the experiment to teach a person the expansiveness of matter. He will now snuff the odour with astonishment. Bless me! how wonderfully a little matter may be expanded! A dozen rooms are full of it! The person is evidently interpreting the smell by the phrase 'expansiveness of matter.' He knows not that the phrase should be interpreted by the smell." (2) But if he is astonished at the preceding, what will he say of the particles of light? They fall, says Natural Philosophy, millions of miles, and with a velocity so wonde,ful as to accomplish the descent in an instant; still they do not hurt even the eye, though they alight immediately on that susceptible organ. Many a man grown old under the rays of the sun is astonished at this recital. The astonishment does not proceed from the phenomena, but from the language. Minuteness, he supposes, is the only difference between a particle of light and a particle of stone. That he cannot feel the particles of light he attributes to the grossness of his senses, and not to the non-existence of a tangible object: hence, if he is informed farther that philosophers have in vain endeavoured with the nicest balances, to discover weight in sunbeams, even when the number of particles thrown into a scale has been multiplied by a powerful lens, the experiment increases his wonder at the smallness of the particles; though it ought to teach him that the mystery is nothing but a latent sophistry of language. The word particle, when applied to light, means only the phenomenon to which it is applied. It names a sight. To wonder that the eye cannot feel the particles of light is to wonder that they cannot feel a sight. We may as well wonder that we cannot taste sounds and hear smells. ( 1) That light, itself a body, should, says Mr. Brown, pass freely through solid crystal, is regarded by us as a physical wonder. Why? ... No man was ever surprised at finding light enter his room when he threw open his window shutters. The wonder is produced by our interpretation of the words in which this common phenomenon is expressed. When we suppose that the passage of light through crystal is the same as the passage of my hand through crystal, we are necessarily astonished; but when we find that the phrase means only what crystal is continually exhibiting, our surprise vanishes with the delusion that created it. It is instructive to observe how insidiously language enables us to infer that light ought to encounter opposition in its passage through crystal. If Mr. Brown had merely stated that light passes through crystal, there would have appeared no reason why it should not

A.B. Johnson and His Works on Language

335

pass through. But the addition of one word seems to show that there is, in the passage of light, a wonder which, if not so miraculous as the passage of Moses through the Red Sea, is more inconceivable: I allude to the word body .... Body is generally the name of a feel; hence, when we say that light is a body, we do not consider that the signification of the word body is governed by the phenomena to which it is applied. We suppose rather that the name regulates the character of the phenomenon, and that to apply the term body to light, determines that light is a feel : and hence the wonder that light should pass through crystal. The wonder is not that the sight which we witness should occur, but that something else should happen - a something which is purely a delusion of language. ( 1) VII

Having established to his satisfaction that every word has as many meanings as it has different applications (that the meaning of a word is the phenomenon to which it refers in any given case), Johnson proceeds with a similar analysis of general propositions. "Every general proposition has as many significations as it possesses different particulars to which it refers." ( 1) To say that the earth is a sphere, that it revolves round the sun, and round its own axis; that the moon influences the tides, and that there are antipodes, are truths so long as we consider the expressions significant of cenain phenomena to which the propositions refer. If you inquire of an astronomer whether the earth is a sphere, he will immediately refer you to various phenomena. He will desire you to notice what he terms the earth's shadow in an eclipse of the moon, the gradual disappearance of the hull of a ship as it recedes from the shore, &c. After hearing all that he can adduce in proof of the earth's sphericity, consider his proposition significant of these phenomena. If you deem it significant beyond them, you are deceived by the forms of language. ( I ) I heard a man contend that no degree of heat could melt diamonds; whilst another was positive that they would melt. I discovered that he who asserted their fusibility, referred to nothing but an article which he had read in a Cyclopedia; and he who maintained their infusibility, referred to an assertion of his father. Both persons were positive, because they intended no more than the above facts. If, however, each had discovered the other's meaning, the controversy would probably not have terminated. It would unconsciously have changed to another question, whether the Cyclopedia was entitled to more credence than the father; the discussion of which would have produced an altercation as virulent as the former, and with as little understanding by each disputant of the facts referred to by the other. ( 1) The more, says St. Pierre, temples are multiplied in a state, the more is religion enfeebled. What did St. Pierre mean? You will find it in his succeeding paragraph. Look, says he, at Italy, covered with churches, yet Constantinople is crowded with Italian renega-

336

Philosophy of Science and Language

does; while the Jews, who had but one temple, are so strongly attached to their religion , that the loss of their temple excites, to this day, their regret. This general proposition means but the above particulars, therefore you need not controvert the position, and show that in your country the increase of temples increases the number and zeal of worshippers. If you argue with St. Pierre, place the contest on its proper basis; blame him for using words in a way which you do not approve, but not for denying facts to which he never alluded. Malebranche, in accounting for the phenomena of memory, says, in childhood the fibres of the brain are soft and flexible; but time dries and hardens them , so that in old age they are gross and inflexible. Malebranche is not enumerating any phenomena discoverable by inspection of the brain. What then does he mean? It follows in his own words : we see that flesh hardens by time, and that a young partridge is more tender than an old one. You may wonder what this has to do with memory. I know not. It has, however, to do with his theory; and it probably constitutes all he means by the hardness and inflexibility which he makes age inflict upon the brain. ( 1) From an ignorance of the principle which I have now endeavoured to illustrate, ... we are prone to award unmerited commendation to the authors of general propositions: thus, the assertion attributed to Pythagoras, that the earth revolves round the sun, is supposed to imply a knowledge by him of the Newtonian theory; while probably no feature of it was ever imagined by Pythagoras. He may have intended some particulars that have long been exploded from science. (I) Speculative writers are particularly fruitful of general propositions, and usually without subjoining any objective example by way of key to the respective riddles which they thus construct ... Politicians find such propositions a convenient mode of expressing opinions that will suit everybody; for instance, a "judicious tariff' is precisely what every man desires, though one man may mean thereby an import duty of a hundred per cent. ad valorem, and another man may mean a total exemption from any duty . ... "To give to every private individual the utmost personal liberty that is useful to the individual himself as well as the nation," will meet the approbation of the most liberal democrat and the most tyrannical autocrat. ... When we consider the character of general propositions, we see a propriety in refraining from speech if we have nothing to communicate but general propositions. ... "Fellow citizens!" might a Fourth of July orator say ... "The destiny of our beloved country is in our keeping; let us be faithful to our sacred trust, and support no man for any public station who will not advocate principles that will bear the scrutiny of virtue; for, without virtue, all seeming prosperity is baseless, and all seeming good but evil in disguise . ... Finally, fellow citizens, let us not be deceived by false demagogues who speak what they mean not, but let us believe honest patriots who mean what they speak, though they tell us unpalatable truths."

A.B. Johnson and His Works on Language

337

Propositions like the above assimilate to poetry or music. They speak to the feelings rather than to the intellect, and hence probably their power to please. (3) VIII

Johnson's view of general propositions had a marked influence on his own writings. No author has been more generous in supplying specific examples to illustrate his meaning. In fact, the examples constitute the excellence of his work. He himself repeatedly reminds us that such general propositions as he can frame to sum up his critique of language are hopelessly obvious, and that it is only from their continued application that any benefit can be derived. He knew that no linguistic reform could overcome the difficulties he spoke of: "Indeed, I can afford no better guide to lead you ultimately to a correct understanding of the defects of language, than to say, at a hazard, that I allude to no defects that you ever head of or conceived. I also allude to none that can be obviated. The most that I hope to perform is to make them known; as we erect a beacon, to denote the presence of a shoal which we cannot remove." (2) And, far from advocating some reform to improve language, he said: " A better philosophy floats in our colloquial phraseology than phraseology receives credit for." (3) Johnson later suggested, for the use of philosophers (who insist on making statements never made in colloquial usage), the indication of the sense of certain words by affixing an appropriate index; as S for sight, F for feel , etc. For an example, he thus translates a sentence of Hume ' s: The table (S) which we see, seems to diminish (S) as we recede from it, but the real table (F) suffers no diminution (F). The whole zest of the proposition consists in the sensible duality of each of the nominal units table and diminution. That the sight table exhibits a visible diminution (S), while the feel table suffers no tactile diminution (F), are no contradiction of one another. That the sight diminution and the feel undiminution can exist thus together, is a physical fact of much interest; but we can make a mystery of it only when we play at bo-peep with words. (3) IX

After discussing the limited significance of general propositions, the argument of the original lectures proceeds thus: A principle as fundamental as any of the former, and more essential ... to a just apprehension of human knowledge ... is this - language can effect no more than to refer us to

338

Philosophy of Science and Language

phenomena. In painting we are forced not only to delineate objects with the colours, how incongruous soever, that we possess; but there are numerous existences to the delineation of which all colours are inapplicable. Even to intimate that colours are unable to represent sounds, tastes, and smells, seems absurd from the obviousness of the fact. The boundary which separates the phenomena that may be represented by colours from those to which colours are inadequate, is, therefore, sufficiently defined; but no writer has imagined that there is a limit beyond which words also cannot discourse. Nor is the latter position easily conceived, for we can no more exemplify with words that there is a limit to their applicability, than a painter can demonstrate with colours that there are phenomena which colours cannot delineate. (I) If I have never heard a cataract, you may inform me what the sound is like; and if I have heard the similar sound, I shall be instructed; but language, nor any other agent, can effect no more than such an approximation. Should you wish to acquaint a child with the sound of a cataract, his conception of it will probably be very erroneous; not because his faculties are less acute than yours, or language less operative on him than on you; but because his experience is less than yours, and language can be significant to him of his experience only. If he has heard no sound more consonant, you must refer to even the lowing of an ox. You may qualify the comparison, by saying the cataract is awfully louder; but if he has heard nothing louder, the qualification will not add to his instruction, except that it may teach him he is still ignorant of the correct sound of a cataract. ( 1) We possess words which signify phenomena only, as white, sour, pain, loud, &c, and ... other words which sometimes signify phenomena and sometimes words; as estuary, shipwreck, murder, &c. We have still another class: words that never signify phenomena, but words only ... Infinity is an example of this class. It is never a sight, feel, taste, or smell; nor is it a sound, except as it names other words. (I) When Locke says that the meaning of rainbow can be revealed to a person who never saw the phenomenon, provided he has seen red, violet, and green, &c., Locke is alluding to the verbal meaning of rainbow. This meaning can be known to the blind .... But it may be thought, we are differently circumstanced from the blind; and that an enumeration of the colours of a rainbow, and of their figure, size, position, and arrangement, to us who know the phenomena which the words signify severally, would reveal to us a rainbow, not verbally merely, but visibly. The premises are, however, impossible. No person can have experienced the colours which compose a rainbow, and their figure, position, and arrangement, without having seen a rainbow. Take any one of the colous, say red; it names not one sight only, but numerous sights. Fire is red, blood is red, my hand is red, bricks are red, and an Indian is red; which of these is he to imagine, when you speak of the red of a rainbow? The same remark will apply to the other colours and to their figure, positions, and arrangement. But admit the possibility of the premises, and that a person who has never seen a rain-

A.8. Johnson and His Works on Language

339

bow, shall still have seen all its colours. Admit further, that when you enumerate the colours, he shall guess the precise red, orange, yellow, &c., to which you refer; and admit the like of their figure, size, and position; yet it will be impossible for the person to know how the colours will look when they are combined; much less, how they will appear, when drawn into the shape, size and position of a rainbow. If he has seen such a combination, he has seen a rainbow; but if he has not seen the combination, language is inadequate to reveal it. After the most copious definition ... a person will be conscious of a new sight the moment he sees a rainbow. ( 1) X

In the original lectures, the development of the four principles outlined above completed Johnson's presentation of what he regarded as the fundamental properties of language. He followed them with an examination of the conditions which govern our assent to certain propositions, stating his views on logic. "We assent to a proposition when we find that the premises affirm the conclusion. The most elaborate reasoning, and the most recondite, can effect no more than to show us that the conclusion is admitted by the premises . ... When a writer finds that his conclusions are not obviously admitted by his premises, he will so explain the premises as to show that they do embrace his conclusions." ( 1) Every object. how gorgeous soever its colour in the light, is void of colour in the dark. Perhaps you will not assent 10 this proposition, though you will admit that colour is invisible in the dark. Natural philosophy proceeds, therefore, as follows: colour is the reflection of certain coloured rays of light. Admit this, and objects become remedilessly void of colour in the dark. If you cannot apprehend this consequence, the following arguments may convince you, for they will show you that the consequence is included in the premises: - thus, colour is nothing but the reflection of certain coloured rays of light; hence, where no light exists, no reflection of coloured rays can exist; therefore, all objects are void of colour in the dark, however they may be endued with the conformation of parts that adapts them to reflect in the light its most gorgeous rays. (2)

The above example is followed by an excellent footnote, which I offer in lieu of further extracts on the present topic: When a tradesman brings me an account which asserts that I am his debtor, say a hundred dollars, I may be sure that the aggregate is fairly slated, for few men are careless enough 10 commit an errour in addition. The items of the bill may required examination.

340

Philosophy of Science and Language

So, when a logician tells me the conclusion to which he is arrived by any process of argumentation, I seldom care to investigate his arguments. I assume that he will not make a false deduction, any more than the tradesman will make a false addition. The part which requires examination are the logician's premises; these are like the tradesman's items. Most people, however, waste all their attention on a logician's arguments, and let him assume what premises he pleases. This is analogous to permitting a tradesman to charge you without restraint, provided he will be honest in his addition of the items. (2) XI

The necessity of our assent to implied propositions, Johnson goes on to say, is founded upon our experience. "Why cannot the same spot be, at the same time, both white and black? Because the word white implies that the spot is not black. But how came white by this implication? Was it arbitrarily imposed by the framers of our language? No. They called one sight white and another black, merely to name what they saw. That proposition is a result of expertence. If I assert that the same spot cannot be both white and hard, the proposition will be untrue. Why? Because my senses can discover such a coincidence. There is no other reason." (I) But once we have, by sensible experience, established a relation of implication between certain words or phrases, we can go on applying it in cases where no sensible phenomena are present. This, in Johnson's opinion, is the basis of nearly every metaphysical speculation. "Such a use of language is like the trick of a juggler who, having adroitly conveyed a shilling from under a candlestick, talks of the money as still under the candlestick." ( 1) What is the meaning of created? It is a name applied to certain phenomena. Any of my senses will teach me a signification of the word. I can see a brickmaker create bricks. I can hear sounds created. You can tell me to place a piece of sugar in my mouth, and it will create a taste; or to press my hand against a needle, and it will create pain. Each of these processes furnishes a meaning of the word created. But what do I mean by applying the word created to the sun? When I apply it to bricks, I refer to the process by which I have seen bricks produced; but when I apply it to the sun, I refer to nothing but the sun itself. But do I not see that the sun exists, and must not every existence have been created? Here again the necessity is verbal, and language is a connivance of men, and relates to their operations only. Why must every brick have been created? Try to cause the existence of a brick, and you will discover.... When we apply the same language to the sun, the necessity is merely verbal. It refers to nothing, and signifies nothing. To persons who have never esteemed language as a collection of mere sounds

A.B. Johnson and His Works on Language

341

employed to designate men's operations and experience, I am aware that this doctrine must be abstruse. That nothing can exist without a previous creation, is, besides, a proposition which applies significantly to so many objects, that there is but little wonder it should be deemed universally applicable .... If all tactile objects possessed a sweet taste, we should consider sweetness essential to the sun, in the same manner as we consider a commencement essential. We even now attribute to the sun temperature, gravity, density , and every other property that is constantly associated with the bodies which we can handle. But it may be asked whether I mean to assert that the sun's existence never had a commencement? No. I mean only, that commenced has no signification but as the name of some phenomenon; and when applied to the sun, there is no phenomenon to which the word refers: hence it is used insignificantly. We should err equally, if we were to assert that the sun never had a commencement; for we must remember that the meaning of a word is governed by the phenomenon to which the word refers. To apply the word bitter to the sun will not affect the sun, but it will affect the word. It will render the word insignificant .... I am aware that this doctrine is so novel, that I may be accused of saying that the sun had no creator. I hold, however, that such an assertion is no more significant than its converse. Language is impertinent to the whole subject. The phenomena to which words refer give them significancy; and when we employ a phrase without referring to any phenomenon, the words are divested of signification. That the sun was created is highly significant, when we refer to the declarations of scripture; but when we refer to nothing, our assertion signifies nothing. ( 1) XII

Johnson's books on language abound in examples of the use of words insignificantly (in the above sense) by philosophers, theists, and scientists. It is hard to resist the temptation to include a dozen of them, but it would be equally difficult to select the best from the scores that he gives. Instead, I have selected some of Johnson's remarks on the construction of theories to close this series of extracts from his books. In a new colony the various necessary utensils are framed of such articles as the region yields. From the absence of more suitable materials, I have seen wooden latches, wooden wash-bowls, wooden candlesticks, and even wooden wicks. Theorists are similarly limited in the agents which they employ. Where language is a scanty vocabulary of spontaneous phenomena, the rude philosopher must theorize with the gross agents which surround him. The earth is then supported on the back of an elephant, and the elephant on the back of a tortoise. But why not a butterfly? Because he refers to his experience of the strength of an elephant, and the endurance of a tortoise ....

342

Philosophy of Science and Language

We smile at theories in which the agents are so crude; and from the numerous phenomena that industry has accumulated for us, we select instruments more subtle .... If a philosopher were to account for the fall of bodies by saying that matter has an inherent love for matter, we might estimate this a very rational exposition. We experience that love produces a desire of contraction, to which the fall of bodies is sufficiently congruous. I wrote thus far without recollecting, that love has been an agent in theories. Chymists employed it in the composition of bodies. Nitric acid and copper combined, because they had a strong affinity for each other. The acid would leave the copper and unite with iron, because its love for iron is stronger. ( 1) Combustion ... was formerly produced by phlogiston, a very subtile and insensible agent which combustible bodies emitted when heated to a certain temperature. This phlogiston was so light, that some bodies became heavier by losing it. When a theory is driven to conclusions so repugnant to our operations, its dissolution is near; accordingly, phlogiston had soon to relinquish its agency in combustion to a more accommodating instrument. Combustion is now performed by means of oxygen. When combustible bodies arrive at a certain temperature, the oxygen loves to unite with them; and as it thus passes from the form of air, to a fixed state, it liberates the caloric which distended it, and for which it has no longer any occasion. The deserted caloric scatters indignantly, and is the heat which we experience. This theory is congruous to a great number of phenomena, and may never be superseded; still, like every other theory, it is significant of nothing but the phenomena which are adduced in proof of the theory. That combustibles will not bum without oxygen, that the residuum &c. of burnt phosphorus will acquire by combustion as much weight as is lost by the air in which the phosphorus is burnt, and that the remaining air will be devoid of oxygen, are truths which I do not dispute. Still, that the oxygen unites with the phosphorus, and that the heat which ensues is the discarded caloric of the oxygen, are the mere verbal machinery by which we reconcile to our own operations the phenomena which we discover. Why must the heat which ensues have existed in the oxygen? Because no other source accords so well with our experience. This is a good reason while it lasts, but a similar reason may induce us tomorrow to attribute the heat to another cause. The language is truly significant of the phenomena only to which it refers; and with this limitation we can never err, adopt what phraseology we please. ( t) XIII

Such, then, are the views which A.B. Johnson first fonnulated on the nature, structure, and meaning of language. Naturally, his later works expanded them in many respects and modified them in others. To present them critically, or

A.B. Johnson and His Works on Language

343

even to discuss their further development at his hand in later years, would require very extensive study. Whether he had been anticipated by other writers. I do not know; but whatever else may have been written on the subject of language, Johnson's contributions display every mark of originality and profound insight. That they were not correspondingly well received is no surprise. Far from having any academic standing. Johnson was without fonnal education beyond the age of fifteen. Philosophical studies in American universities at the time were practically limited to theology. a field in which some of Johnson's remarks were more likely to raise eyebrows than hats. Dr. Timothy Flint praised Johnson's work in a review of American literature which appeared in the London Athenaeum in 1835. and suggested that its publication in England instead of America might have gained it the attention that it merited. Johnson believed that a knowledge of the nature and structure of language must underlie the pursuit of all other sciences. He hoped to point the way to a non-metaphysical philosophy of human knowledge. Whatever his success or failure, his intentions were clear: Speculative researches are accommodating to human weakness. From geology, which teaches us what exists in the center of the earth, to astronomy, which reveals what is transpiring in the empyrean; and from physicks, which discourse about the body, to metaphysicks, which treat of the mind; the mass of verbal doctrine assumes any shape which ingenuity strives to create - like the pebbles of Rockaway, that change their position as every wave, rising on the ruins of its predecessor, rushes, (lord of the moment,) proudly over the beach. To fix the fluctuating mass of theories, no man has suggested any other expedient than the construction of some new theory, to whose authority, [like to Johnson's orthography] all persons shall submit. The remedy is constantly augmenting the disease. I shall not imitate so unsuccessful a procedure; but as theories are the means by which we attempt to discourse of external existences that our senses cannot discover; and as the desire for such discourse originates a large portion of our theories; I will teach you the capacity of language for such an employment, and thereby enable you to judge more understandingly than you can at present, the utility of most theories, and the signification of all. (2) Our misapprehension of the nature of language has occasioned a greater waste of time, and effort, and genius, than all the other mistakes and delusions with which humanity has been afflicted. It has retarded immeasurably our physical knowledge of every kind, and vitiated what it could not retard. The misapprehension exists still in unmitigated virulence; and though metaphysicks, a rank branch of the errour, is fallen into disrepute, it is abandoned like a mine which will not repay working, rather than like

344

Philosophy of Science and Language

a process of mining which we have discovered to be constitutionally incapable of producing gold. (2) NOTES 1

Following are the principal works; quotations from each are identified by the corresponding number. The Philosophy of Human Knowledge, or a Treatise on Language. New York, 1828. 2 / A Treatise on Language, or the Relation which Words Bear to Things. New York, 1836. 3 / The Meaning of Words : Analysed into Words and Unverbal Things. New York, 1854. 4 / The Physiology of the Senses ; or, How and What we See, Hear, Taste, Feel, and Smell. New York and Cincinnati, 1856. 1/

2 Whether or not he was a relative of A.B. Johnson, I am unable to determine. 3 Copies of the books are practically unobtainable. Of the four quoted here, the Library of Congress has all but the last. There is a complete set of Johnson's works at the Utica Public Library. 4 The Westminster Review for October 1856, reviewing The Physiology of the Senses, described it as "distinguished by an agreeable raciness of style."

3 Literacy and Scientific Notations

The rise and spread of notation systems in science is a very large area, in which I can touch on only a few examples here. These have been chosen for their value in representing certain basic features of families of special notations that historically proved to be remarkably useful in basic sciences such as physics and chemistry. As a historian of science, not as a scientists and as one who specializes in the early modem period, I stress the rise of special scientific notation systems rather than their spread, attempting to acquaint you with some reasons for which they were introduced, bearing on their extraordinary success. If I succeed in that. the spread of such notation systems will hardly remain puzzling, or their spread beyond the basic sciences into derivative ones and even beyond science itself into other areas of modem culture. Hence it should be clear that my approach is not meant to limit or restrict the discussion but to suggest, by considering the earliest adoption of some fundamental special notations in science, a few basic considerations affecting special notations in general. Everyone today is conscious of the great impact of science on society, but few people are aware of the essential role of literacy in the origin of science itself in recognizably modem fonn. That was a recent event in long-range historical terms, dating roughly from the time of Copernicus in the mid-sixteenth century to that of Newton at the end of the seventeenth century. Before that period, science existed as philosophical speculation, and technology existed quite separately from it; in its modem fonn, science tends to embrace technology and to regard philosophical speculation as a subordinate activity. Had it not been for a certain abrupt rise in general literacy, it is not at all likely that science Reprinted from Towards a New U11derstandi11!( of literacy, eds. Merald E. Wrolslad and Dennis F. Fisher (New York: Praeger 1986), 135-55. Copyright© 1986 by Praeger Publishers. Reproduced with permission of Greenwood Publishing Group, Inc., Westport. CT.

346

Philosophy of Science and Language

would have acquired its modem fonn when it did. I accordingly sketch the role of literacy in the creation of modem science, hoping in that way to bring out a neglected aspect of our general theme and at the same time throw light on the way in which the rise and spread of special notation systems have resulted in a distinctive kind of scientific literacy that has tended to isolate science from other human activities in which the general literate public feels more at home. Basically, as you will see, that problem is rooted in the difficulty of expressing in ordinary language the full import of conclusions reached by employing special notations. That difficulty is of two kinds, one of which has to do with human convenience and excessive demands on accurate memory, the other has to do with a fact of our internal organization that makes it possible for us to grasp certain relations visually at a glance but not to describe them in words with anything like equal precision. Since literacy in the ordinary sense concerns things written in words, the proliferation of special notations requires a new understanding of literacy, as vividly illustrated by the rise of modem science. The primary concern in this chapter is not with symbolic representation as such but with systems of notation, that is, with interwoven symbols that are ordered and manipulated according to clearly understood rules. For instance, the symbols used by astrologers to designate the sun, moon, planets, and signs of the zodiac did not constitute a system of notation that solved problems in new ways, though it is true that a certain view of the universe was created when alchemists adopted some of the same symbols to designate certain metals. Again, the mixture of abbreviations and symbols used by physicians and pharmacists was adopted for convenience in writing rather than to solve problems, and it also shared with the symbolism of astrologers and alchemists the advantage of concealing the mysteries of an art from the uninitiated. In sharp contrast with these stands the first special notation system to appear in physics, that of algebra, and especially the use of algebraic equations in solving problems. Ramifications of algebraic notation, both conceptual and historical, are so extensive and easily recognized that their exploration should throw much light on the essential character of many successful notation systems in science. When algebraic equations were first applied in physics, they already contained genns of important physical principles that were not fully perceived and exploited until the nineteenth century. To say that algebra was the first special notation system to appear in recognizably modem physics is to imply that a previous general notation system existed, to which algebra contributed new ways of solving problems that had been solved before in another way. The primordial notation system was, of course, ordinary language, which was applied to physics differently at two distinct stages. For some 2,000 years physics, as the science of nature, was dealt

Literacy and Scientific Notations

347

with speculatively in Greek, in Arabic, and then in Latin. The problems thus solved, with few exceptions, would not have been capable of solution algebraically, being causal inquiries of the greatest generality. Soon after 1500 a new kind of physics emerged, separately from the verbal enterprise known as natural philosophy, which had remained virtually a university monopoly and had been written almost exclusively in Greek and Latin. The newly emergent physics belonged to useful science; it originated outside the universities, was written mainly in living languages, and applied arithmetic and geometry to practical problems. Its appearance marked the beginning of the modem era in science, yet without any special notation system. The beginning of general literacy dates from the same period, and not by mere coincidence. Without that, it seems possible that the kind of science with which we are familiar might not yet have come into being. Three centuries of university scholarship had not produced anything like it, and there is no reason to believe that another four centuries would have done so. Useful science was a contradiction in terms under the Aristotelian definition accepted by scholars, in which science (episteme) was divorced from practical knowledge (techne). What happened about 1500 was that throughout Europe printers had become sufficiently numerous, and their investment sufficiently great, to induce them to widen the market for books. Until then, books had been published mainly in Latin on subjects of interest to clerics and scholars. They did not circulate widely, were generally large and expensive, and accumulated in places not accessible to the public. Small books, easily portable and cheap, began to appear in large numbers, printed in living languages and on subjects of general interest; they circulated far from centers of learning. By standardizing the forms of letters, printing had made reading easier to learn; books on popular subjects in national languages provided a new incentive to do so. Protestant revolts a few years later produced innumerable pamphlets, replied to by Catholics, and all churches began to expand their schools. By the middle of the sixteenth century, according to recent studies of notarial and legal documents in large Italian cities, over half the residents could write at least their names. That degree of general literacy was new, and the number of books having little or no interest to scholars was by then very large. In any population, people having special skills and knowledge far exceed those with advanced education. Artisans, architects, craftsmen, and master builders were among them; having learned orally and by apprenticeship, they were able to instruct the public in books solicited by printers, as practical arithmeticians had earlier begun to do. Niccolo Tartaglia, born in 1500 and as nearly self-educated as anyone could be, played a vital role in the origin of modem science. He was an exceptional mathematician who became interested in practical

348

Philosophy of Science and Language

problems through his private tutoring at Verona. His New Science of 1537 applied mathematics to artillery problems; later he wrote in Italian on statics and hydrostatics, applying that to the diving bell and a method of raising sunken ships. In 1543 Tartaglia translated Euclid's Elements into Italian, saying on the title page that this put into the hands of anyone of average intelligence the whole science of mathematics. In fact it did because no special notation was used by Euclid; every term was defined in ordinary words, and every proof of a theorem or solution of a problem was written out in full. The Elements of Euclid had been used in universities for centuries, but in a defective Latin version with a misleading commentary that deprived Europe of the general theory of ratio and proportionality. Tartaglia restored that, and on it Galileo founded his new physics. Like Euclid, Galileo introduced no special notation; any literate person could determine the precise meaning and logical correctness of proof of any physical proposition written by Galileo. The modem era in physics thus opened with theorems and solutions of problems that were fully understandable, but in a form that was anything but compact, remained inconvenient for calculations, and was limited in the power of suggesting roads to new discoveries. Those disadvantages vanished when algebraic notations were introduced soon afterward; but along with that came a different disadvantage, namely, that the physical meanings of algebraic expressions were not easy to reduce accurately to ordinary language. You may wonder why Galileo himself did not use the algebraic notation that was to prove so fertile of new discoveries in physics. Partly, the answer lies in the fact that although algebra had been known in Europe since the thirteenth century, it did not attain a form useful for anything except numerical calculations until about 1600. Even the familiar equal sign did not appear in many books on algebra until the 1630s, when Galileo's work was virtually completed. Still more important is the fact that Galileo would not have accepted algebraic equations as properly applicable in physics because they implied more than he could prove to be rigorously true. What algebraic equations implied in physics beyond what could be proved by their early users generally did tum out later to be true, and that is why the new notation system was of such remarkable power in leading to scientific discoveries. Along with this, however, went an inevitable departure from the possibility of specifying clearly in ordinary language precisely what was being asserted about the physical realities of the universe around us. Perhaps it would be better to say that grounds for disagreement among physical scientists themselves were being laid as they went along. They knew the correctness of the algebraic equations and developed further consequences from them, but without stopping to make sure that all agreed on their physical interpretation.

Literacy and Scientific Notations

349

This situation was not fully recognized until it had gone on for a very long time. Eventually, after mathematical physics had become very highly developed, two great revolutions occurred in it, one on the heels of the other, brought about by the theory of relativity and quantum theory. Both were inaugurated at the beginning of the present century. Many of us regard them as beyond our grasp, except in a general way. Nevertheless, there is a sense in which each of those revolutionary developments in science hinged on a problem stated and a question asked in ordinary language. It will be best to postpone consideration of that curious fact, or at any rate of that way of looking at relativity and quantum theory, until later. At present my point is that the situation which did arise at a very advanced stage of physics would not have arisen except for the introduction of special notations, especially the unrestricted use of algebraic equations in place of strict application of the general theory of ratios and proportionalities. One might say that the prudence of Galileo limited the progress he could make in physics, but it protected what he did prove from later refutation. The Euclidean theory of ratio and proportionality restored by Tartaglia defined ratio as a relation between two magnitudes of the same kind, and proportionality as sameness of ratio. By very careful measurements, Galileo was able to establish the basic law of falling bodies: "If a movable body descends from rest in uniformly accelerated motion, the spaces run through in any times are proportional to the squares of the times." That can be properly written in the form S 1 :S 2 ::T::T;. In that form, which is not algebraic, a test requires the measurement of two distances and two times; there is no way to calculate a distance if given only a time. Galileo's statement could be verified at sea level or on the top of Mt. Everest, at the equator or the north pole. Now we write, in algebraic notation, S = 1/2gt2. Given any time, we can calculate the corresponding distance, provided that we know the appropriate value of g and use the same units of distance and time as embodied in it. We can calculate the speed acquired at the end of time, and the average speed through the distance, whereas Galileo could prove only that one was double the other, stating a relation, or ratio, between two speeds. We might be able to explain the meaning of S = 1/2gt2 fully in ordinary language, but that would be very wordy; it is only a slight exaggeration to say that in explaining what g means we should have to outline Newton's law of universal gravitation. To explain fully what Galileo wrote requires no more than to state three definitions from Euclid, Book V. Algebraic notation did not enter directly into physics until after the invention of analytic geometry by Descartes in 1637. Hardly had the equational form of algebraic notation appeared in books before it was applied to geometry, with problem solving and heuristic uses in mathematics so great as to stagger the

350

Philosophy of Science and Language

imagination of anyone who, knowing only Euclid's Elements, should first see it. (Analytic geometry said infinitely more than Descartes, or anyone before 1878, could rigorously prove. It is an example of the general rule that whatever is gained in heuristic power is lost in rigor of proof, and conversely.) Analytic geometry was promptly adopted in physics because the early problems in physics had mainly to do with motion and were most naturally represented by geometric diagrams. That is how algebraic notation came into physics, but it remained clumsy for a long time. Various special conventions of symbolism and rules for manipulation of equations that seem to us obvious, through familiarity, were not fully developed until perhaps a century later. Through conventions and rules of manipulation, algebraic equational notation became a language in itself, in a very real sense. Just as we learn a foreign language by at first translating its expressions into English, and then gradually stop bothering, so mathematical physicists stopped long ago translating into ordinary language the things expressed by their equations, except occasionally or when introducing new symbols or new operations. In effect they created a new language; indeed, Etienne de Condillac in the mid-eighteenth century declared that science is nothing more than language well arranged, and in developing that idea he took algebra as the paradigm of language itself. Just as semantic shifts of meaning have occurred in ordinary languages, algebra (as the language of physics, so to speak) has not been immune from a similar phenomenon. To the extent that any notation comes to be used as a language, its symbols and structure exert power over the concepts represented, much as words and grammar do in ordinary language. Only in the present century has that power been recognized fully and found not easy to analyze. The semantics of scientific notations is even more difficult to study, especially as they become ever more remote from ordinary language, in which alone we can approach that study. The point may be illustrated by the opening pages of Newton's Principia , half a century later, when notations of proportionality still appeared side by side with algebraic equations. Newton was keenly aware of the difference - and of the dangers to physics in confusion between the concepts represented. Following his definitions he wrote a long and famous scholium discussing the concepts of absolute and relative time and space, noting that all measures of motion are necessarily relative. Newton (1947) then cautioned that relative quantities are not the entities themselves whose names they bear, but sensible measures thereof, either exact or approximate, that are commonly used instead of the entities measured. And if the meaning of words is to be determined by their use, then by the names time, space, and motion, their sensible measures are to be understood . ... On this account, people violate the accuracy of language, which ought to be kept precise,

Literacy and Scientific Notations

351

when they interpret those words for the measured entities. Nor do people less defile the purity of mathematical and philosophical truths, who confuse real quantities with their ratios and sensible measures. (p. 11)

The role that algebraic notation systems came to play in such confusion was not Newton's fault, for he wrote the three laws of motion that lie at the base of useful physics not algebraically, but in ordinary language. First came Newton's inertial Jaw with the qualification: "unless the body is compelled to change its state by forces impressed on it." (1947, p. 13). Then came the force Jaw, which we are taught as F = kma, whereas Newton's words were, "The change of motion is proportional to the motive force, and is made in the direction of the straight line in which that force is impressed." Obviously our algebraic equation F = kma is not a symbolic translation of Newton's words, which included neither mass nor acceleration. We adopt a paraphrase that, in a different language, serves equivalent purposes. Newton stated only a proportionality, as we are supposed to know from the constant k that follows the sign of equality. We do not in practice notice this; rather, we take the equality as literal, as existing between terms, not between ratios, and as applying to physical entities, not merely to measures that are by their nature relations. At least most of us do, and in that way we come to regard force as the product of mass times acceleration, uncritically. No harm seems to result; we even feel that here is the key to elementary physics. But what can it mean to multiply a mass by an acceleration? Could we explain that in ordinary language? We know how to multiply numbers, an operation defined by Euclid, but for Euclid even multiplication of ratios made no sense. They are relations, and a rule is lacking for multiplying one relation by another. Euclid recognized a certain kind of addition of ratios, but that is of even less help. We are warned not to add apples and oranges, so surely we could not add a mass and an acceleration. Yet multiplication is just successive addition, as we are taught in arithmetic. So there is something very strange about a notation multiplying mass and acceleration if we try to explain the idea in ordinary language. Yet these strange operations are justifiable in physics because their use advances our knowledge, and Newton himself declared - in curiously Wittgensteinian tenns - that the meanings of words are detennined by their use. Wittgenstein also held that whatever can be said, can be said clearly, and Newton remarked that the accuracy of language is violated by those who confuse entities and relations. The fonnulation F = kma did not appear until long after the death of Newton, who would not have approved of it. Like Galileo, Newton wished to limit physical statements to what could be said in accurate language. As physics expanded, the prolixity required for complete explanation of special

352

Philosophy of Science and Language

notations became excessive, and some were adopted for which it would perhaps have been impossible. Nothing better illustrates this than the special notation created by Newton for solving many essential problems in physics, and a rival notation that replaced it. Newton called his procedure "the method of fluxions" and took pains to circumvent misunderstanding of its basis. His notation was used in England for a century and then was abandoned in exchange for a different one devised by Leibnitz about the same time as Newton's, for the infinitesimal calculus. Both produced the same solutions of problems, but they entailed very different understandings of the universe. The source of that difference deserves attention. The notation of Leibnitz implied the existence of real infinitesimals, immeasurably small parts of mathematically continuous magnitudes. That conception is impossible to justify in ordinary language, though it has recently been vindicated mathematically by an arithmetical approach called non-standard analysis. It is as difficult to explain clearly as is the monadology of Leibnitz that peopled the universe with indivisible souls, each of which mirrored the whole. Nevertheless, the notation of infinitesimal calculus is easier to teach and is better adapted to calculations than Newton's fluxion notation. The tall, slender S that is our integral sign stood for "summation" of infinitely many actual parts to reach a finite magnitude, an unclear idea that made the differentiating sign dyldx appear to be a ratio with separable terms. Just as Newton had warned against confusion of ratios with physical entities, he cautioned early in the Principia against imagining tiny indivisble particles as true parts of continuous magnitudes: If hereafter I should happen to consider quantities as made up of particles, or should use little curved lines for straight ones, I would not be understood to mean indivisibles, but evanescent divisible quantities; not sums and ratios of determinate parts, but always the limits of sums and ratios. (1947, p. 38)

Again, later in the Principia, he forbade consideration of continuous variables in any other way than as increasing or decreasing, as it were, by a continual flow; I call their instantaneous increments by the name of moments . ... Take care not to look upon finite particles as such. Those are not moments, but quantities generated by moments. We do not consider the magnitudes of moments, but their first ratio. Instead of moments, we may use speeds of increase, which may be called the fluxions of quantities. (Newton, 1947, p. 249).

Denoting mathematically instantaneous change by a dot placed over the

Literacy and Scientific Notations

353

symbol for the variable, Newton avoided any misleading analogies with finite parts, as implied in the calculus notation subsequently adopted universally. It would lead far afield to discuss the mathematical background and physical implications of the two notations, but they are very like those which in our time made possible the radically different outlooks of quantum mechanics and wave mechanics. Those, like the notations of calculus and of fluxions, led to like solutions of problems but with very different implications. Systems of notation may serve equally in solving problems but conflict profoundly in interpretations, and that fact has now become a part of scientific understanding of the universe. I hinted earlier that the extraordinary fertility of the algebraic-equation system of notation resulted from the fact that it happened to embody the seeds of physical principles not fully recognized and exploited until the nineteenth century. I had in mind general conservation principles. Nothing is lost or added during the manipulation of algebraic equations; everything is conserved though not necessarily in the same form. The same is true in the physical universe: matter is conserved in chemical transformations; motion is conserved in some form; work is conserved in mechanical processes, for as Galileo noted of simple machines, whatever is gained in power is lost in speed, distance, or time consumed. Here I wish to cite a remarkable statement of Galileo's directed at the removal of an Aristotelian objection to use of mathematics in physics but curiously prophetic of a nineteenth-century use of equations to unify seemingly separate branches of physics. To the argument that a physical sphere does not touch an actual plane at a single point, Galileo replied: It would be news indeed if computations made in abstract numbers should not thereafter correspond to concrete coins and merchandise. Just as the accountant who wants his calculations to deal with sugar, silk, and wool must discount the boxes, bales, and packings, so the mathematical physicist, when he wants to recognize in the concrete those effects which he has proved in the abstract, must deduct the material hindrances; and if he knows how to do that, I assure you that things are in no less agreement than arithmetical computations. The errors, then, lie not in abstractness or concreteness, but in an accountant who does not know how to balance his books. ( I 953, pp. 207-08)

In the nineteenth century the early principles of conservation of motion and of work led on to the principle of conservation of energy in general, a discovery that revolutionized scientific understanding of the universe. Balancing the books of nature involved unifying the equations of mechanics with those of heat, of electricity, and of acoustics. The discovery followed a series of ingenious and precise experimental measurements and was not just a product of

354

Philosophy of Science and Language

equational notations, but all were related to the concept of force. Newton's second law of motion would have been incapable of suggesting all that flowed from its compact algebraic-equation form, suggestive of innumerable uses wherever the concept of force entered, having been devised when that did not extend beyond ordinary mechanical action. The fact that even in that restricted domain it would be difficult to explain and justify F = kma in ordinary language only made it easier to perceive and pursue new and general implications when electromotive force was named and made measurable. Just as the original problems of early modem physics centered on motion and had yielded to solution rapidly under equational representation, in which the concept of force emerged and was given in time a compact general expression, so force became the center of the new set of problems that occupied physicists in the nineteenth century. The correlation of forces, as it was called in the mid-19th century, led to the principle of conservation of energy. It is not quite accurate to say above that the concept of force did not extend beyond ordinary mechanical action when Newton created the science of dynamics. One supremely important exception emerged at the outset, and that was gravitational force. Great resistance was aroused by Newton's law of universal gravitation because of its implication that bodies acted on one another at enormous distances, without contact or connection as in ordinary mechanical actions. That appeared to many physicists as introducing a spooky element into science. The situation was aggravated when electric and magnetic phenomena became precisely measurable. Not just active forces, but fields of force appeared to exist in the vicinity of magnets, and in certain circumstances around conductors of electricity. The concept of force fields existing somehow in something, even in the absence of anything tangible on which the forces acted, was disturbing. Their representation at first was by literal depiction, in the forms suggested by the behavior of visible objects in magnetic and electrical force fields. That was a kind of notation, but not as satisfactory as equational notation, in which quantitative relations could be established. The achievement required novel mathematical expressions capable of dealing adequately with electromagnetic phenomena, and those were created by James Clerk-Maxwell in his Treatise on Electricity and Magnetism. Maxwell's treatise was swiftly recognized by physicists as of extraordinary importance, but it was found extremely difficult to understand because of unfamiliar notations employed in Maxwell 's equations. Not only were they incapable of explanation in ordinary language, but they resisted reduction to familiar concepts and operations in the already very advanced mathematical language of physics. It seemed that a general theory of electromagnetic phenomena had been set forth, but that the theory eluded any statement in ordinary language

Literacy and Scientific Notations

355

comprehensible even to scientists. In the ensuing efforts to fonnulate such a statement, the eminent physicist Heinrich Hertz simply declared, "Maxwell's theory is Maxwell's equations." That was not a witticism, but a profound judgment. Physics, which had begun its modem career without any special notation and could at first be made clear to any literate person, had reached a point at which it was entirely a special notation in its most fundamental inquiries and could be fully comprehended only by mastery of that notation. Maxwell's equations dealt with light as an electromagnetic phenomenon, and in virtue of the continuous-function notation in which they were stated, they implied the existence of a range of wave frequencies, thus far unsuspected, between radiant heat and light. Hertz set out to detect experimentally electric waves of those intennediate frequencies, using methods suggested to him by Maxwell's equations, and he soon found them to accompany the discharge of electricity through sparks. Maxwell had implied more than could have been known from existing physical data, and it turned out to have been true. Maxwell's equations had not yet yielded a theory in the usual sense, from which his special notation system would follow, but his equations implied physical phenomena that can be described in ordinary language. At the end of his brief life, Hertz composed his Principles of Mechanics ( I 893), in the introduction to which he had this to say of the processes of science: We form for ourselves pictures or symbols of external objects, and the form which we give them is such that the necessary consequents of the pictures in thought are always pictures of the necessary consequents in nature of the things pictured. In order that the requirement may be satisfied, there must be a cenain conformity between nature and thought. Experience shows that the requirement can be satisfied, whence such a conformity does exist. (p. 1

The pictures we fonn in science may be ordinary grammatical statements or they may be special notation systems or they may be quite literally pictures drawn to represent structural relations among external objects, actual or hypothetical. Structural relations are frequently perceptible at a glance when they would be very cumbersome to describe in words, and might not be as efficiently conveyed by equational or other mathematical notations. Pictorial notations are often valuable in physics, as for instance in crystallography. They are still more useful in chemistry, which in its beginnings in modem fonn was faced with problems different in kind from those of early modem physics - problems of structure and combination rather than of motion and force. When physics reached the stage at which its most fundamental inquiries

356

Philosophy of Science and Language

required notations that resisted interpretation in ordinary language, it was a problem and a question stated in ordinary language that resulted in a most profound revolution in our understanding of the universe, as remarked earlier. The problem was stated at the opening of Einstein's 1905 paper, titled "On the Electrodynamics of Moving Bodies," but famed for its special theory of relativity. It was well known, he said, that Maxwell's equations implied that it makes a difference whether a magnet is moved in an electric force field, or a current-carrying conductor is moved in a magnetic field. Since we can measure only relative motions, that should not make a difference. You know how others, led by Lorentz, had previously dealt with the motions of particles in electromagnetic fields, and certainly there is something spooky about the Lorentz-Fitzgerald contraction as described in ordinary language. Einstein adopted a very different approach, asking and answering a question that seemed limpidly clear, so much so that no one had bothered to examine it: "What do we mean when we say that two events happen at the same time?" Einstein's question, or at any rate his answer to it, struck at the very roots of the algebraic-equation notation system that had replaced proportionality language in physics in its early modem form, converting relations to quantities. Newton pointed out, before he stated his three laws of motion, that sensible measures are of relative quantities and cannot be more than that. This is true by the very nature of measurement, which requires the employment of units. We measure relations to arbitrary units, stating a ratio, though we make it look like a quantity. The way in which we do that is by using the notation system of decimal fractions, which was not introduced until 1585. The decimal notation system for simple numbers, written in arabic numerals, had been introduced into Europe in the thirteenth century, but decimal fractions, though they seem natural and obvious to us, defied the very definition of number in terms of multitude of discrete units that had always been used in rigorous mathematics. It was not until 1878 that rigorous foundations were given to algebra by a new definition of number, compatible with its uses in expressions for continuous functions. Decimal fractions, algebraic equations, and special notations in physics all emerged about the same time, around 1600, half a century after Tartaglia restored the rigorous Greek treatment of ratios and proportionality for continuous magnitudes. That was in tum to become the basis of giving rigor to those subsequent notations after they had served for two centuries as marvelous heuristic devices, partly at least because they were not hampered by mathematical rigor. There is such a thing as rigor morris, as the historian of mathematics Kenneth May liked to point out. To have life and vitality, the notations of science must be free from that, as their history plainly shows. From these things it is clear that notation systems in science that have provided

Literacy and Scientific Notations

357

new ways of solving problems and of understanding the universe do indeed have semantic aspects of a very fundamental kind. Freedom from rigor pennits and even leads to discovery, but at the same time it leaves the interpretation of discoveries somewhat loose within the notations employed. Excessive freedom reduces the probability of success under the Hertzian requirement, so reducibility to ordinary language is a useful guide in restraining overconfidence in any notation for its own sake. Yet ordinary language itself is not free of semantic pitfalls. For example, it tempts us to suppose we know what "at the same time" means in every possible case, and it was this that Einstein questioned. Measurement of time requires that we be able to say whether two events separated in space occur at the same time. Certain distances are "measured" by knowing speeds and times. At such distances as cannot be directly measured, independently, the elementary algebraic equations that replaced proportionality notations cannot guarantee their own applicability. Einstein's clear question in ordinary language challenged them; and five years later Minowski made the celebrated statement that space and time as separate physical entities had vanished, to be replaced by the new and revolutionary concept of space-time that totally transfonned scientific understanding of the universe. Or we might say, with equal justice, it gave us for the first time a clear understanding of notation systems in science that had become a language in themselves, subject to the inherent semantic problems of ordinary language but concealing those more effectively from notice. In that regard, Einstein's simple question was rather like the child's remark about the emperor's new clothes: it called attention to an unreality in the way everyone had been talking. The family of notations discussed here, equational in fonn whether we apply algebraic or differential equations, has one essential characteristic: it is best suited to the treatment of magnitudes that are, or may be assumed to be, mathematically continuous. Until the present century the notations most suitable for dealing with discrete quantities were little used in physics. That is no longer the case; quantum theory revealed the extreme importance of small, exact, discrete numbers in our understanding of the fine structure of the universe, at atomic and subatomic levels, and of the nature of light and radiation in general. The mention of atoms brings to mind chemistry, and it suggests why notations most suitable for dealing with discrete quantities began to predominate in chemistry soon after it acquired its modem fonn. Before turning to that, I wish to touch on quantum theory as the other revolution in physics that hinges on a question in ordinary language that was first seriously asked at a very advanced stage. Sir Arthur Eddington characterized relativity as the outlook leading to the conclusion, "We only observe relations between physical entities." Behind that, he emphasized the question, "What is it we really observe?" He then went on:

358

Philosophy of Science and Language

It is common to describe the state of theoretical physics in this century as a succession of revolutions, but it was all one revolutionary movement started by that simple question. Heisenberg repeated it in 1925 when he asked "What is it we really observe in an atom?" The result was the new quantum mechanics. (Eddington, 1939, p. 31)

Heisenberg's name is best known in connection with the celebrated principle of uncertainty that declares it impossible to establish the exactness both of the position and the velocity of a moving particle. The more exactly we measure one, the more uncertainty attends our measure of the other. This does not matter with baseballs or express trains, but when we get down to elementary particles it matters a great deal, for certainty becomes large with respect to the size of the things to be measured. That is why it was only at an advanced stage of science that the problem was noticed. Physics had been constructed to fit reasonably large-scale phenomena, and as Eddington (1939) remarked: Most physicists have a tendency to treat the mathematical development of a theory as the only part which deserves serious attention. But in physics everything depends on the insight with which the ideas are handled before they reach the mathematical stage. (p. 55)

In a way, the uncertainty principle might have been expected to appear when actual measurements became extremely precise. In ordinary language we speak of the position of a thing at any time as if it were at rest, and we do not speak of its velocity unless it is moving. When its velocity is changing, still another complication enters, over and above the difficulty of treating something as if it were at rest and moving at the same time. The difficulties remain concealed until very small particles at very high speeds need to be considered; this is so because for the usual objects of physics neither position nor velocity need be measured with absolute precision, and nothing is changed when it is discovered that they cannot be, in actual observations. The special notations of chemistry began to enter about a century after Newton. Because chemistry was dominated by problems of structure and combination of parts, rather than by problems of continuous motion, the notations most useful in solving problems of chemistry were characteristically adapted to identifying and enumerating discrete things and describing relations among them. The notations that had been successful in physics, namely algebraic and differential equations, were of little help to chemists in the early days of its recognizably modem development. Structure and combination are most easily thought of as stationary patterns, and chemical change as the replacement of one such pattern by another, for which the notations of physics that had developed by a century after Newton were not convenient.

Literacy and Scientific Notations

359

What long delayed emergence of chemistry as an exact science was an inherited nomenclature of great complexity and no discernible system. Names of chemical substances had been assigned inconsistently, often on the basis of colors or romantic ideas of hidden powers. To clear away redundancies and introduce system, Lavoisier identified some forty elements and devised a system of naming compounds in tenns of the elements contained in them. In the introduction of his treatise on elementary chemistry, Lavoisier cited Condillac's statement that science is no more than language well arranged, adding that when he set out to improve the language of chemistry, he discovered that that was impossible without improving the science itself, and conversely. In the process of detennining what elements combined to fonn compounds, and what elements could be extracted from compounds, Lavoisier soon noted the recurrence of fixed proportions of ingredients, in ratios represented by small numbers. Condillac's further statement, that algebra was the best model of language well arranged, was not mentioned by Lavoisier. Recurring small-number ratios of ingredients suggested to John Dalton an atomic theory in which the smallest particles of one elementary substance would combine with those of another, in various ratios disclosed by measurements of larger quantities. Dalton pictured atoms as small circles in 181 o, and arranged them in groups representing the smallest parts of compounds (see Figure 1 ). Each element was designated by special marking of the circle representing its atom, the first four shown being hydrogen, nitrogen, carbon, and oxygen. Dalton's notation required the memorizing of some forty different markings, and except in the smallest diagrams for compounds, it was necessary to count the number of atoms of each kind. Soon afterward, Berzelius replaced the specially marked circle with the initial letter in the Latin name of the element and eliminated the need for counting by placing an appropriate number beside the initial. Chemical combinations were indicated by using the plus sign. Thus a part of the special chemical notation we still use was quickly developed, but - as had happened long before with algebra - the equal sign was not introduced or the chemical equation put to use until a good deal later. Basically the reason was that although combining proportions were noted quite early, it was half a century before chemists could agree on the atomic weights to be assigned - even though Avogadro had proposed the key hypothesis only a year after Dalton's book. Use of initial letters to represent elements led to structural diagrams for complex organic compounds, much more helpful than Dalton's schematic grouping of circles. Substances had been found with the same chemical fonnula in the Berzelius notation but with quite different properties. Possible different placements of the same atoms within molecules offered the most plausible explanation. Such structural diagrams also pennitted easy representation of differences

360

Philosophy of Science and Language

-

Figure

1

a. . •.. 0. e. •. e.., E>. 0 - 0 0- - E> • .. aL&Na•T ■

CJ)

(0 (i)

(i)

• (i)

0

-

0

E>

. oe. . CDC>.. ee

00

(;JG>

d?b ~~I. .

.

.

~·.

..

.

~. .

.,

Figure

2

II

I

II I

C

/"-, H-C 'c:-11 I I C-H H-C

'-/ C I

H

II I

H-C-0-H

H-C--0-H

H-C-0-H

H--C-0-H

I I H--C-0-H I H-C-0-H I H-C-0-H I 11-C-O

t

I I H-C-0-H I I H-C...,.>-H

H-C-0-H

C=-0

I

H

""'1c.n.1

rn(C1H1.01l

F-

(Coll,.C>ol

Literacy and Scientific Notations

361

in bonding between particular atoms, as seen in the celebrated benzene ring and in the diagrams for the two sugars shown in Figure 2. Even rather complex structural diagrams of the kind shown in Figure 2 can be taken in at a glance because there are symmetries and repeated groupings. These are not literal structural diagrams in the sense that ordinary geographical maps are, but it is easy to learn to read them and find one's way about in a molecule, so to speak. Ability of the eye to apprehend structural relations at a glance lies behind the utility of many kinds of graphic representation, in science as elsewhere, in contrast with verbal descriptions that would be not only excessively wordy but perhaps incapable of simultaneously indicating spatial relations between objects grouped together in a plane. Chemical equations, though algebraic in appearance, were not subject to all usual rules of algebraic manipulation. More important to chemistry was the periodic arrangement of elements in order of weights, which confirmed in a new way the great importance of exact small numbers in the structure of matter. The periodic table is itself a notation system in which the recurring properties of elements in each group made possible the assignment of atomic numbers, detection of missing elements, and their subsequent discovery in nature. Just as Maxwell's continuous-function equations disclosed unsuspected wave frequencies that Hertz was able to confinn, so the discrete grouping of elements disclosed unsuspected elementary substances that were sought and found. Structure is generally more effectively depicted than described. The example of crystallography in physics was of great use to chemistry, which also developed rather early the schematic notation for representing the nature of attachments between atoms in compounds and the linking of molecules, or radicals, with atoms or with one another. Sµch notations led to the very complex threedimensional models required in the solution of the double-helix structure of DNA, linking biology, chemistry, and physics. The line between notations and models is no longer sharp, nor is the line between theories and models. Thus we see how special notations may become equivalent to theories and serve science in the place of words and sentences. I shall now attempt to tie together various threads in this chapter, deliberately left disconnected until now. To have emphasized their relationships as I went along would, I think, have distracted your attention from the historical fact that special notations in science which turned out to be valuable in solving problems in new ways and which resulted in our present view of the universe, grew like Topsy. They were not rationally designed first and then logically applied to the scientific infonnation gained from observation and experiment. That is the key lesson that we can learn from the history of science about successful special notations. There is a great temptation now to design special notations based on

362

Philosophy of Science and Language

patterns taken from the exact sciences and then attempt to apply them rigorously to other uses, as if the facts in every field were obliged to conform to those same patterns. That temptation may be weakened if we see that successful notations grow out of interactions between the subject matter and the manner in which we represent it. The safest, if not the only, common denominator between useful representations and the things to be represented is ordinary language that remains within the reach of every intelligent literate person. Special notations permit us to represent things much more efficiently than we can by words and sentences, but it will pay to doubt any representation to the extent that we ourselves, personally, are unable to reduce it to ordinary words and sentences. The exact sciences learned this healthy skepticism only slowly, with difficulty, and at the cost of having eventually to scrap ideas that had seemed established forever. That is the second lesson about special notations taught by history - not to suppose them absolutely trustworthy just because they lead to some discoveries that might be missed if we used nothing but ordinary language. We saw how it happened that algebraic equations led to physical discoveries because that notation implied conservation laws that were only later established. Notations can be particularly appropriate to certain phenomena they represent, for reasons not known at the time they are adopted. When their success suggests unlimited applicability, there is reason to examine the notations with the same care that is normally used in examining the things represented. Notations cannot govern phenomena, but only reveal aspects of their structure. We also saw why graphic notations were of more use to early chemistry than algebraic equations, even though the first steps in both physics and chemistry were based on recognition of fixed proportionalities. The best notation for continuous change is not likely to be even a good notation for definite structure; motion is best represented one way, and substitution or replacement in a different way. Phenomena cannot govern our notations, but they can reveal in them structures inappropriate to their proposed uses. Finally, the history of exact sciences - in which special notations played conspicuous roles - shows that those sciences originated when literacy became general. Science advanced spectacularly by departing from some nonns of general literacy, but not without generating illusory confidence in conclusions expressible only in special notations. The value to us of that knowledge as we tum to the study of special notations in other areas, broadening our understanding of literacy, may best be expressed by paraphrasing the remark of Sir Arthur Eddington (1939) about physics, saying, "Everything depends on the insight with which the ideas are handled in ordinary language, before they reach the stage of special notations."

Literacy and Scientific Notations

363

REFERENCES

Cajori, Aorian. 1974. History of Mathematical Notations. La Salle: Open Court. Cipolla, Carlo, 1969. literacy and Development in the West. London: Penguin Books. Condillac, Etienne de. 1801. la langue des Ca/cuts. Paris. Drake, Stillman, n.d. Early science and the printed book. Renaissance and Reformation 6 (3), 43-52. Eddington, Arthur. 1939. The Philosophy of Physical Science. Cambridge: Cambridge University Press. Galilei, Galileo. 1953. Dialogue Concerning the Two Chief World Systems. Berkeley, Calif.: University of California Press. Hertz, Heinrich. 1899. Elements of Chemistry. New York. Lavoisier, Antoine. 18o6. The Principles of Mechanics. London: MacMillan. Newton, Isaac. 1947. Mathematical Principles of Natural Philosophy. Berkeley: University of California Press. Wittgenstein, Ludwig. 1959. The Blue and Brown Books. Oxford: Blackwell.

Note: Most of the publication dates cited here are of translations or recent editions.

4

J.B. Stallo and the Critique of Classical Physics

"He is the author of the profoundest and most original work in the philosophy of science that has appeared in this country - a work which is on a par with anything that has been produced in Europe .... It is, in fine, safe to say not only that the influence of Stallo' s work will be a permanent one, but that it will also steadily increase." 1 Such was the appraisal of Stallo's Concepts and Theories of Modern Physics in an obituary sketch of its author written by Thomas J. McCormack, translator of Ernst Mach 's principal works into English and probably, in his period, the most competent American judge in matters related to criticism of scientific theories. McCormack's verdict in his first statement was correct, even if the prediction he ventured in the second was not. Stallo's profundity and originality cannot be questioned. That his influence remained small among scientists was not remarkable, since he had opened his book with the words: "The following pages are designed as a contribution, not to physics, nor, certainly, to metaphysics, but to the theory of cognition." 2 Traditionally the theory of knowledge was a concern of philosophers and not of scientists. Events of the present century no longer justify a continued neglect of Stallo, especially from the historical standpoint. The revolution in physics that commenced about the time of his death in I 900, originating in philosophical no less than in strictly physical considerations, has vindicated Stallo to a great extent. We are by now accustomed to the fact that epistemological postulates may be no less important in shaping a new scientific concept or theory than the newly discovered data that are to be incorporated in it. Such a state of affairs had already been clearly envisioned by Stallo a full quarter century before it began Reprinted from Men and Moments in the History of Science , ed. H.M. Evans (Seattle: University of Washington Press, 1969, c. 1959), 23-37, by permission.

J.B. Stallo and the Critique of Classical Physics

365

to intrude itself into the work of professional physicists, and he had even been able to indicate certain directions in which alterations would have to be made in classical theories in the interests of further progress. In his own day, much of Stallo's reasoning rested upon grounds that lay outside the accepted boundaries of physical thought; subsequently those boundaries have been widened to include the epistemological considerations Stallo utilized. To those interested in the evolution of scientific thought, a brief account of this almost forgotten man and his work may therefore be welcome, despite his apparent failure to influence directly the course of events in physics. In the Concepts, Stallo undertook to reveal the degree to which outmoded philosophical doctrines still permeated science in spite of all previous efforts to root them out. Stallo was no longer a professional philosopher or scientist during the years in which the book was being thought out, written, and published. Yet the reader of the work cannot fail to be struck by the single-mindedness with which the author restricted himself to the subject matter of physical science and by his entire familiarity with its literature (including even those journals that ordinarily fall only under the attention of men active in the profession) as well as with that of philosophy. In view of the technical nature and scope of his book, the great number of editions and translations through which it passed between 1882 and 1911 can hardly be accounted for except by its wide dissemination among scholars, however little they may have cited or credited it, during that crucial epoch in the history of physics. The first edition of the Concepts was published by D. Appleton and Sons in New York as the thirty-eighth volume of that memorable set of well-made duodecimos that collectively bore the name of "The International Scientific Series." The title page bears the date 1882, though the copyright date is 1881 and the book is said to have made its first appearance in November of that year. 3 Simultaneous publication in England (London: Kegan Paul, Trench and Triibner) was the rule for these volumes, and a French translation followed immediately under the title la matiere et la physique moderne (Paris: Felix Akan). H.A. Rattermann states that a German translation was published at Leipzig simultaneously with the American and English editions. But despite Rattermann's having had access to Stallo's own collection of books (until 1885 at any rate), he seems to be in error on this point. The first German version was the translation prepared by Hans Kleinpeter with the encouragement of Ernst Mach and published by Barth at Leipzig in 1901. 4 Mach's preface to this volume, and Kleinpeter's article on Stallo in connection with its publication, made it evident that the man and his work had been previously unknown in Germany. Rattermann states further that translations were made into Italian (Bologna), Spanish (Madrid), and Russian (St. Petersburg). A second American edition

366

Philosophy of Science and Language

appeared in 1884 and a third in 1891; the third English edition had appeared in the previous year. The fourth printing of the French translation is dated 1905, and a second German edition was published in 191 1. Thus it appears that no less than fifteen printings of the work were made, in six different languages, within thirty years of its original appearance. The American editions of 1884 and later are of particular interest, for they contain a long introduction prepared by Stallo in answer to his American and English critics and reviewers of the first edition. At the same time he made a few minor alterations in and additions to the text, but no substantially revised or expanded edition was ever produced. Stallo's illuminating and often sprightly reply to his critics was included only in these American editions, though the textual alterations were carried over into the English republications and thence into the German translation. In this regard, the French title pages are misleading; what purports to be the quatrieme edition is merely a reprint of the original text, containing neither the 1884 introduction nor the revisions made at that time. The preface to the French translation was supplied by C. Friedel; of this, Mach said (quite justly): "That the book is properly esteemed by experts may well be questioned from any indications. The French edition is even provided with a preface to which one can scarcely attribute any other purpose than that of weakening the effect of the book. "5 Mach became acquainted with the book and its author only a short time before Stallo's death. He first learned of its existence through a citation by Bertrand Russell; 6 and, he says, "I naturally took a lively interest in the man whose scientific aims so closely approximated my own. No one in England could give me any information about him, Professor A. Schuster of Manchester merely gave me his surmise that he might be an American." 7 With the assistance of Paul Carus, Mach at length established contact with Stallo, who was then residing in Florence. The short-lived correspondence into which they entered was soon interrupted by a serious illness of Mach's and was hardly resumed when terminated by the death of Stallo. Mach's personal judgment of Stallo, from whom he had meanwhile received a brief biographical sketch and copies of his two other books, is set forth in his preface to Kleinpeter's translation of the Concepts, from which the following excerpts are taken: From his own account of his life, Stallo may be considered essentially self-taught, allowing himself to be led in his scientific studies only by the writings of the great discoverers of ancient and modem times. Without the personal direction of a teacher he was forced to resolve his own doubts through quiet and continual reflection. Thus he achieved an originality and independence such as is puzzling to orthodox youngsters of the modem school of physics, and toward which they are rather hostile ....

J .8. Stallo and the Critique of Classical Physics

367

Through his philosophical and historical studies, Stallo was placed in a position to recognize in the presently received views of physics traces and elements of the outlook of past times which modem physicists in general take to have been long since vanquished, and which in undisguised form they would hardly recognize as their own .... Since Stallo examined modem physics under the influence of this viewpoint, he necessarily perceived the scholastic-metaphysical elements which pervade it throughout. After this recognition, the gradual complete emancipation of science from these traditional, often primitive and barbaric, modes of thought appears as merely a necessary consequence of the further development, strengthening, and critical clarification of physics. I cannot completely agree with Stallo on all points; thus, I cannot join with him in his sharp and all-inclusive opposition to so-called metageometrical researches. But in the battle to eliminate from science the latent metaphysical elements I agree with him completely, and his works offer to me a valuable and welcome complement to my own. As a more important point of agreement I may adduce especially the rejection of the mechanic-atomic theory - not as a means of assistance to physical discovery and representation, but as the general foundation of physics and as a world view. We share also the apprehension of physical concepts such as mass, force, etc., not as peculiar realities, but as mere relations; connections of certain elements with the appearances of other elements. Through the assumption of the relativity of all physical properties and definitions, including those of space and time, there necessarily follows finally an agreement in the rejection of all expressions about the universe as a whole. My writings are directed, as conditioned by my training, my capacities, and my calling, to those physicists who are not averse from the logical clarification and philosophical deepening of their science. Accordingly I search primarily for scientific confusions and irrelevancies in particulars, in order to go on from here to a more general viewpoint. Stallo, on the other hand, takes the opposite path. Starting from very general observations, he brings to bear upon physics the propositions thus discovered. Both paths lead to almost invariably agreeing insights. Here I can but repeat what I have already said elsewhere: It would have been very heartening and beneficial to me, when I commenced my critical works about the middle of the 186o's, to have known of the related exertions of such a comrade as Stallo. 8

In point of time, Stallo's first researches leading to the Concepts very nearly coincided with Mach's. Although not published in book fonn until several years later, the Concepts must have been written in large part by I 873, for in October of that year, Stallo published the first of a series of four articles in the Popular Science Monthly under the title "Primary Concepts of Modem Science"; these articles embody much of the central theme of the Concepts in

368

Philosophy of Science and Language

words that recur almost without alteration in the text of that book. That these ideas had begun to take shape in his mind long previously is shown by passages in his paper, "Materialismus," first published in 1855 and reprinted in 1893.9 Furthennore, it appears that he was prepared to continue the four articles mentioned above, since the final words in the last of these (January, 1874) promise a further article dealing with the application of the principle of conservation of energy to the field of theoretical chemistry. The promised article did not appear, but the indicated subject matter was published some eight years later in the concluding chapter of the Concepts. Mach's contrast between his own approach to the critique of physics and that of Stano is certainly correct so far as it goes. There were, however, other differences both in their aims and in their outlooks, which bore essentiany upon their methods and results. One such difference is instanced by the more marked radicalism in Stano's application of the principle of relativity of physical data. Mach was never willing to accept what might be called, for want of a less paradoxical tenn, an absolute relativism; later this became quite apparent in his opposition to Einstein and more particularly to some of the latter's disciples. Stano, had he lived, would undoubtedly have taken keen delight in at least the special theory of relativity and would have hailed it ·as the next essential step in physical theory. Mach, as a professional physicist, to some extent had a vested interest in the data of physics and thus was inclined to temper his criticism when he could not perceive any alternative means of preserving that body of knowledge intact. Stano, on the other hand, had no stake in the fortunes of those data; this appears repeatedly from his suggestions as to possible modifications in the concept of mass, in the fonn of conservation principles, and the like. A striking illustration of this contrast between the men is to be found in their respective treatments of an argument that had been advanced in 1869 by Professor C. Neumann, as demonstrating the necessity of some absolute body of reference. Mach says: The most captivating reasons for the assumption of absolute motion were given thirty years ago by C. Neumann. If a heavenly body be conceived rotating about its axis and consequently subject to centrifugal forces and therefore oblate, nothing, so far as we can judge, can possibly be altered in its condition by the removal of all the remaining heavenly bodies. The body in question will continue to rotate and will continue to remain oblate. But if the motion be relative only, then the case of rotation will not be distinguishable from the state of rest. All the parts of the heavenly body are at rest with respect to one another, and the oblateness would necessarily disappear with the disappearance of the rest of the universe. I have two objections to make here. Nothing appears to me to be

J.B. Stallo and the Critique of Classical Physics

369

gained by making a meaningless assumption for the purpose of eliminating a contradiction. Secondly, the celebrated mathematician appears to me to have made here too free a use of intellectual experiment, the fruitfulness and value of which cannot be denied. When experimenting in thought, it is permissible to modify unimportant circumstances in order to bring out new features in a given case; but it is not to be antecedently assumed that the universe is without influence on the phenomenon in question. In fact the provoking paradoxes of Neumann only disappear with the elimination of absolute space. 10

Stallo, several years previously, had treated the subject as follows: The reasoning of Professor Neumann is irrefutable, if we concede the inadmissibility of his hypothesis of the destruction of all bodies in space but one. But the very principle of relativity forbids such a hypothesis. The annihilation of all bodies but one would not only destroy the motion of this one remaining body and bring it to rest, as Professor Neumann sees, but it would also destroy its very existence and bring it to naught, as he does not see. A body cannot survive the system of relations in which alone it has being; its presence or position in space is no more possible without reference to other bodies than its change of position or presence is possible without such a reference; and, as I have abundantly shown, all properties of a body are in their nature relations, and imply terms beyond the body itself. The case put by Professor Neumann is thus an attestation of the truth that the essential relativity of all physical reality implies the persistence both of force and of matter, so that his argument is a demonstration, not of the falsity, but of the truth of the principle of relativity. 11

When Mach calls Neumann's hypothesis of annihilation a "meaningless assumption" he refers to no more than the fact that it is never realized in experience; here Mach is asserting his sensationalistic philosophical views, from which his belief in the relativity of physical data is derived and by which it is limited. "Elimination of a contradiction" by such means appears to Mach pointless because a hypothetical contradiction in the above sense is subordinate in his eyes to his principle of economy of thought in the construction of physical theories, which program demands only the avoidance of concrete contradictions. It is only upon such assumptions that he is enabled to assert next that the circumstances modified in Neumann's thought-experiment are unimportant and thus to reject Neumann's assumption of the noninfluence of remote bodies. (And in view of Mach's previously cited statement, that all propositions about the universe as a whole are to be rejected, his appeal here to the possible influence of the universe upon the phenomenon in question is at best a desperate expedient.) In effect Mach simply avoids, and does not refute, the whole logical fabric of Neumann's argument.

370

Philosophy of Science and Language

Now Stallo also calls Neumann's hypothesis meaningless, but without any qualification as to "what is to be gained" or what is or what is not "important"; he takes his stand firmly upon the principle of relativity of all physical properties, not excluding the property of existence itself. Thus Stallo exposes the futility from a logical standpoint of Neumann's position, for Neumann has attempted to address the relativists with an argument that must be and remain meaningless to them. The disparity between Mach 's approach and Stallo' s, as illustrated in this single example, may perhaps be accounted for in the following way: Mach was primarily concerned with describing the occurrences, persistences, and reappearances of metaphysical biases among scientists, and with accounting for their gradual historical elimination in accordance with his principle of economy of thought. Stallo, on the other hand, addressed himself to the task of revealing the origins of such ideas in scientific contexts and of explaining how they got there in the first place and how they might be avoided. In this sense it is especially true, as Mach says, that the works of the two men admirably complement one another; it is almost as if in stocktaking late in the nineteenth century, Mach showed the scientists where they had been and Stallo pointed out to them where they were likely to be going. "Although the founders of modem physics at the outset of their labors were animated by a spirit of declared hostility to the teachings of mediaeval scholasticism," says Stallo in the introduction to the second edition of the Concepts, " ... when they entered upon the theoretical discussion of the results of their experiments and observations, they unconsciously proceeded upon the old assumptions of the very ontology which they openly repudiated ... founded upon the inveterate habit of searching for 'essences' ... before the relations of words to thoughts and of thoughts to things were properly understood." His object, he declares, has been ... to consider current physical theories and the assumptions which underlie them in the light of the modem theory of cognition - a theory which has taken its rise in very recent times, and is founded upon the investigation, by scientific methods analogous to those employed in the physical sciences, of the laws governing the evolution of thought and speech. Among the important truths developed by the sciences of comparative linguistics and psychology are such as these: that the thoughts of men at any particular period are limited and controlled by their forms of expression; viz., by language; that the language spoken and "thought in" by a given generation is to a certain extent a record of the intellectual activity of preceding generations, and thus embodies and serves to perpetuate its errors as well as its truths; that this is the fact hinted at, if not accurately expressed, in the old observation that every system of speech involves a distinct metaphysical sys-

J.B. Stallo and the Critique of Classical Physics

371

tem ... that philosophers as well as ordinary men are subject to the thralldom of the intellectual prepossessions embodied in their speech. 12

Proceeding with this program, Stallo identified four metaphysical assumptions so intimately linked with traditional views of the uses of language that he tenned them "structural fallacies of the intellect." These are the fallacious assumptions that: ( 1) every concept is the counterpart of a distinct objective reality; (2) general concepts and their counterparts pre-exist to the less general; (3) the genetic order of concepts is identical with the genetic order of things; and (4) things exist independently of and antecedently to their relations. Examples of these fallacious assumptions in physical science he found on all sides, but particularly in the declared intention of scientists to reduce all natural phenomena to mass and motion while insisting upon the absolute disparity of the two and upon their separate conservation. That mass remains the same whether at rest or in motion, Stallo recognized as an unwarranted assumption; that the conservation of mass had any independent meaning apart from consideration of energy, he saw to be equally so. But his chief target of attack was the atomic theory of his day. To attribute indestructibility to atoms in order to explain it in gross matter struck him as particularly ridiculous: "A phenomenon is not explained by being dwarfed. A fact is not transfonned into a theory by being looked at through an inverted telescope." 13 "A valid hypothesis reduces the number of the uncomprehended elements by at least one," he remarks, citing Zoellner. 14 But "some of the uses made of the atomic hypothesis, both in physics and in chemistry ... replace a single assumption by a number of arbitrary assumptions among which is the fact itself" - that is, the fact which was to be explained by means of that hypothesis. 15 The kinetic theory of gases and the wave theory of light are subjected to searching criticism in the Concepts; the disparities between the atoms demanded by physicists and those required by chemists are displayed; the variety of ethers provided on order for the various purposes they were to fulfill is ridiculed, and the contradiction between the accepted phenomena of gravitation and the avowed rejection of action-at-a-distance is exhibited. Many supposed attempts to reduce the complex to the simple are shown to have been merely reductions of the unknown to the familiar, which is a very different thing. Stallo's relentless relativism broke down, however, at one point; he was unable to admit the relevancy of non-Euclidean geometries to physical questions. In two chapters devoted to this subject he argues, much as did Poincare in his popular writings on this matter, that the mutual translatability of all geometries precludes the necessity of abandoning Euclidean geometry for any scientific purpose, and that the endowment of space with properties capable of

372

Philosophy of Science and Language

rendering any portion of it distinguishable from any other amounts to depriving space of that very essence which renders it distinguishable from matter. Since Stallo's arguments on this point are inadequate in any case to accomplish his purpose in raising them, he cannot be entirely forgiven for his essential departure here from his own basic tenets. The speculations of W.K. Clifford upon possible uses of the new geometries in physical theory were familiar to Stallo, but he rejected them as imputing material properties to space. It is curious, in view of his critique as a whole, that Stallo should have failed to recognize the inacceptability of the only alternatives to granting this sort of relativity between space and matter - either the abandonment of the concept of space entirely, or the assignment to space of a separate and independent existence. The latter alternative is, in effect, actually accepted and defended by Stallo; thus the geometrical section of the Concepts is completely out of accord with modem physical views. The concluding chapters of the Concepts are devoted to a critique of the then prevailing cosmogonies and cosmological speculations, and to a commentary upon attempts to reduce all chemical phenomena to explanation by the principle of conservation of energy - attempts which the author felt were certain to succeed eventually, with the appropriate recognition of the interrelationship of energy and mass. Such, in brief, was the program of Stallo's Concepts. The book was not well received by the critics of his time; it did not obtain a highly favorable evaluation from a scholar in the field until after his death, when Kleinpeter (who at the time was preparing a German translation of the book) wrote an article entitled "J.B. Stallo als Erkenntsnisk.ritiker," which appeared late in 1901. 16 Subsequently Stallo's work received attention from Emile Meyerson, who referred to it often in his ldentite et Realite ( 1908); more recently Stallo has been cited by Rudolf Carnap in Der Raum ( 1922) and in the Physikalische Begriffsbildung (1926), and by Percy Bridgman in The Logic of Modern Physics (1927). Perhaps it is only in the light of the hard-won physical knowledge of the twentieth century that an approach like Stallo's to the domain of theory construction can be truly appreciated. II

The author of the Concepts, John Bernard Stallo, was born March 16, 1823, in the town of Damme, parish of Sierhausen, in southern Oldenburg, Germany. His father was a schoolmaster in that town, as had been his grandfather; so far as he could trace his family tree, all his forebears on both his father's and his mother's side were country schoolteachers.

J.B. Stallo and the Critique of Classical Physics

373

My grandfather [Stallo told Ratterrnann] was my first teacher. He was a venerable old Frisian (Stallo is not an Italian but a Frisian name, meaning "forester"), who to his dying day wore the three-cornered hat, knee-breeches, and buckle shoes [of his profession]. He undertook my upbringing despite his more than seventy years of age, and was not a little pleased when I had learned to read and to do all sorts of problems in arithmetic before the end of my fourth year. To him I am indebted for my education in the ancient languages and in English; but French I learned from my father behind my grandfather's back, for he hated "Frenchness" with all his soul.

Stallo's education was well advanced when he emerged from school at his first communion at the age of thirteen, and he was then entered in the teachers' college at Vechta, which he could attend free of charge. There he had the opportunity of studying in the neighboring Gymnasium as well and in two years was ready for admission to a university. But his father lacked the means for that step, and "there remained to me only the choice of following the family profession of teaching school, or emigrating to America. This lay close to my heart, since my father's brother, Franz Joseph Stallo, had opened the way for emigrants from Oldenburg by going there in the early 183o's." This uncle had fled to America after having antagonized the authorities in Germany with his liberal political views. After an ill-fated venture in founding a community of his own, he had settled in Cincinnati where he was a successful inventor and printer. Stallo arrived in Cincinnati in the spring of 1839, armed with letters of recommendation to eminent Catholic clerics in that area. Once there, it became a grave problem what he should do, since he was too well educated to become a laborer and yet he knew no trade. On the strength of his recommendations, he was given a post of instruction in a parish school; having emigrated from Germany to avoid becoming a schoolteacher, he promptly became one in America. His principal assignment was to teach German, and he soon set to work on a book to which in his later years he was wont ironically to refer as "my most brilliant literary success" - an elementary language text called A. B. C. Buchstabir und Lesebuchfiir die deutschen Schulen Amerikas. Published anonymously in I 840, it went through numerous editions and was widely adopted by schools in all parts of this country. About this time the Athenaeum High School in Cincinnati was converted by some Belgian and French Jesuits into St. Xavier's College, and Stallo's abilities having come to their attention, they put him, at age eighteen, in charge of instruction in German. During the first semester he was asked to assist in Greek and Latin as well as in mathematics, and students flocked to his classes. By arrangement with the authorities he was allowed to carry on his studies while teaching. The college had a fine library in physics and chemistry and a reason-

374

Philosophy of Science and Language

ably satisfactory laboratory for the time, and Stallo accordingly made use of his spare time for the study of those sciences under the Jesuit instructor. He managed so well that in the autumn of 1844 he was called as professor of physics, chemistry, and mathematics to St. John's College in New York (now Fordham University), where he remained until 1847. Shortly after resigning this post he published his first serious work under the title General Principles of the Philosophy of Nature (Boston, 1848). The book had some merit in acquainting American scholars with the philosophical systems of several Germans, notably Oken and Hegel, but despite Rattermann's attempts to find passages of scientific importance in this work (which he adduces as anticipations of Darwin's evolution theory), it was best described by Stallo himself in the preface to the Concepts: I deem it important to have it understood at the outset that this treatise is in no sense a further exposition of the doctrine of a book ... which I published more than a third of a century ago. That book was written while I was under the spell of Hegel's ontological reveries - at a time when I was barely of age and still seriously affected with the metaphysical malady which seems to be one of the unavoidable disorders of intellectual infancy. The labor expended in writing it was not, perhaps, wholly wasted, and there are things in it of which I am not ashamed, even at this day; but I sincerely regret its publication, which is in some degree atoned for, I hope, by the contents of the present volume. 17

Rattermann once asked Stallo whether the failure of the earlier book to gain acceptance had been his reason for forsaking the teaching profession, and says that Stallo replied: See here, my friend, I had rather not discuss that. I found out that the American spirit was not yet ready for philosophy. Only its superficial growths flourish here; those which have deep roots and bear fruit cannot yet be so much as planted, for they will not grow to ripeness. I desired primarily to make sure of a secure living for the future, so I came back to Cincinnati. I wanted to become practical, as the Americans are.

Upon his return to Cincinnati, Stallo hesitated between Jaw and medicine, deciding upon the former on the advice of a fellow German who had found the latter neither a lucrative nor a pleasant calling among his compatriots in that city. Admitted to the bar in 1849, Stallo commenced a practice which was to continue uninterruptedly with the exception of a brief term on the bench ( 185355), until his departure from American in 1885. His most famous court case was the erudite and skillful defense of the Cincinnati public school board for dropping from the curriculum both Bible reading and hymn singing. Such practices

J.B. Stallo and the Critique of Classical Physics

375

had been offensive to Catholics, German Evangelists, Jews, and agnostics alike, since they had had to bear the burden of taxes for schools to which they could not conscientiously send their children, and which they were accordingly duplicating with various parochial schools. Stallo lost the case before the Cincinnati Superior Court (and "only the theologico-political Sanhedrin of Cincinnati" could so have acted, as he told his fellow townsmen), but was later vindicated by the Ohio Supreme Court. Stallo served for seventeen years as an examiner of teachers for the publics schools, and during a part of this time served on the Board of the University of Cincinnati. A great admirer of Jefferson, Stallo was a staunch Democrat in his earlier years, but as the slavery question became paramount, he forsook that affiliation and became one of the founders of the Republican party and campaigned for Fremont. At the outbreak of the Civil War he called upon the Germans of Cincinnati to form a regiment of their own, the Ohio Ninth, which was one of the first in the field and became known as "Stallo's Turnier Regiment." After the war he became increasingly dissatisfied with the selfishness and corruption of the Republican machine and in 1872 was a leader in the ill-fated Liberal Republican party, which attempted to nominate Charles Francis Adams. The disastrous "politicking" of Carl Schurz, his houseguest during the Cincinnati convention, resulted in the ridiculous nomination of Horace Greeley and utterly disgusted Stallo. In 1876 he rejoined the Democratic party and assisted the Tilden campaign; in 1880 he denounced the Republican party for having outlived its usefulness and becoming a mere machine for the furtherance of the ends of industrial politicians, monopolists, and speculators. In 1884 he supported Cleveland in the hot battle against high tariffs, and following the Democratic victory of that year, he was rewarded with the post of minister to Italy. Four years later the Republicans regained power and Stallo's assignment at Rome was abruptly ended, but he did not return to the United States. Instead, he removed to Florence, where in accordance with a long cherished dream he settled down to the exclusive pursuit of his scientific and cultural interests. William J. Youmans, then editor of the Popular Science Monthly, seems to have expected this to result in a supplemental volume to the Concepts, 18 but Stallo's only further publication was a collection of earlier periodical articles, addresses, and letters, which appeared under the title Reden, Abhandlungen und Briefe (New York, 1893). This volume, compiled at the request of friends, gives us a fairly complete picture of Stallo's political views and contains in addition such characteristic pieces as biographies of Jefferson and von Humboldt, short popular articles on science and philosophy, a commentary on the English language, and an account of the case against Bible reading in the public schools. The book was

376

Philosophy of Science and Language

highly praised by Ernst Mach both for its style and for its content. Stallo died at Florence on January 6, I 900. Stallo's character was marked by the depth of his convictions, his love of knowledge, and his passion for freedom; he had the reputation of being somewhat brusque and was not lacking in personal courage. During 1862, when there was widespread hostility toward Wendell Phillips, Stallo was asked to preside at a meeting where Phillips was to speak. He was most unsympathetic with Phillips' disunionist views and at first declined, but when he heard that others whom he had recommended as more suitable for the chair had refused either from pressure or from fear of the crowd, he consented to preside. The meeting had no sooner begun than a shower of missiles greeted both Phillips and Stallo, who nevertheless stood beside him throughout the ensuing turmoil. Stallo's interests in music and literature were broad and deep, and his home was an established calling place for distinguished visitors to Cincinnati. He had built up, in addition to his law library, a collection of some five thousand volumes relating to literature and science. His chief treasure was a single volume in which were bound copies of Kepler's Astronomia nova, Stereometria, Harmonices mundi, and Tabulae Rudolphinae; these were heavily annotated in the margins by a hand which was, to the best of Stallo's knowledge and belief, that of Kepler himself. Teacher, lawyer, judge, and diplomat - such was the man whom Kleinpeter astutely declared to have "made a juridical approach to physics, first establishing the crime and then prosecuting the offenders .... Mach, Clifford and Pearson are indeed in the first rank of epistemologists, but ... Stallo is the only one who has rigorously and systematically pursued a critique of knowledge in mathematics and the sciences." 19 Perhaps the only other nineteenth-century scientific amateur of equal theoretical insight was the British jurist to whom E.L. Youmans referred when he first introduced Stallo to the readers of a scientific journal in America: "It has long been the honor and boast of the British bar that Mr. Justice Grove, the author of The Correlation of Forces, belonged to it; it is equally to the credit of the legal profession in this country that a member of it has cultivated scientific philosophy to such excellent purpose." 20 NOTES 1 2

Thomas J. McCormack, "John Bernard Stallo," Open Court, XIV, No. 5 (May, 1900), 276, 283. J.B. Stallo, The Concepts and Theories of Modern Physics (New York: D. Appleton and Co., 1882), p. 7.

J.B. Stallo and the Critique of Classical Physics

377

3 H.A. Rattermann, Johann Bernard Sta/lo, deutsch-amerikanischer Phi/osoph , Jurist und Staatsmann: Denkrede gehalten im Deutschen Literarischen Klub von Cincinnati am 6 November I