Study of History of Mathematics

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■

THE STUDY OF THE HISTORY OF MATHEMATICS :ind

■ OF THE HISTORY THE STUDY OF SCIENCE (Two volumes bound as one.)

by

George Sarton

DOVER PUBLICATIONS,tl!ra@ NEW YORK

THE STUDY OF THE HISTORY OF MATHEMATICS

by

George Sarton

DOVER PUBLICATIONS, INC. NEW YORK

COPn.teld'•

1936

llY nnt Pl'l'.sro.tN"t Al\"D l"lt.l.1.0\\"S OF KA.1VAJU) 00U.Znd Th� Stauly o/ lbe HJSll"l'T o/ SttmN'. tht- tll"D '\'Olumc:s btulg bound as one

PREFACE TO THE DOVER EDITION

I am grateful to Dover Publications for issuing, at a rela­ tively low price, new editions of these two little books of mine which have long been out of print. Each of them con­ tains nvo parts: first, a text explaining the meaning of the history of science (or the history of mathematics) and, second, a bibliography facilitating additional studies. These bibliog� raphies are not up-to-date but contain the essential down to time of first publication. Readers who are sufficiently in­ quisitive will be able to complete them easily by referring to my book Horus, A Guide to the History of Science, published by Chronica Botanica, Waltham, Mass., in 1952. But even that is not absolutely up-to-date, for up-to-dateness in an evanescent quality in a world changing as fast as ours. Readers wanting the very latest information on the history of science should consult regularly in a public library, or better still at home, the current number of Isis, An International Quarterly Re.. view devoted to the History of Science> edited by I. Bernard Cohen, Widener Library 189, Cambridge 38, Mass. The main thing is to understand that in a world dominated by scientific methods and inventions the history of science should be the keystone of higher education.

Christmas 1954

GEORGE SARTON

CONTENTS THE STUDY OF THE HISTORY OF MATHEMATICS

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NOTE ON THE STUDY OF THE HISTORY OF MODERN

MATHEMATICS

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BIBLIOGRAPHY

I. General treatises

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II. Handbooks

III. Treatises devoted to the history of special branches of mathematics • . . . . . . . 46 IV. Mathematics in the nineteenth and twentieth centuries . . . . - . . . . . . . . . . 49 V. Philosophy and methodology . . . . . . . 53 VI. Bibliography . . . . . . . . . . . . . . 55 A. Guides. B. Encyclopaedias. C. Large catalogues. D. Journals VII. Journals on the history of mathematics

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VIII. Centres of research . . . . . . . . . . . 61 A. Academies of science. B. Mathematical societies. C. International congresses. D. In­ stitutes and libraries Biographies of moder n mathematicians .

APPENDIX.

INDEX

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THE STUDY OF THE HISTORY OF MATHEMATICS

HE remarks ,vhich I have made on another occasion ,vith reference to the history of science 1 would apply equally ,veil to the history of mathematics; there is no need of repeating them. Ho,vevcr, the history of any special science suggests ne,v considerations. As long as the history of science ,vas conceived as the sum of the histories of special sciences, the relationship bcrureen the former and the latter ,vas simple enough. The history of each special science ,vas simply a part of a ,vhole; a part which could easily be removed and isolated. Ho\vcvcr, that old Whewel­ lian conception had to be abandoned, as it ,vas borne in gradually upon scholars that one of the most valuable as-­ pccts of the history of science ,vas the study of the inter­ relationships bcnvecn different branches and their mutual enrichment. The history of science could no longer be conceived as a sum of particular histories, but rather as an organic in tcgra tion, whence no part could be abstracted ,vithout damage. Moreover the history of each science is necessarily more technical than that of science as a whole, more concerned ,vith scientific than with cultural continu­ ity. As it deals ,vith a more lillllted group of ideas it can hope to follow these more closely. To be sure, these differ­ ences are quantitative rather than qualitative, and would vary considerably from one historian to another. The one might write a history of science of a very abstract type; the other, a history, say, of chemistry, which would contain fewer technicalities than humanities. In general, however, we should expect the opposite. G. Sarton, The Study of the Hfrto,y of Scitnct1 with an Introductory Bibliography (Cambridge, Harvard University Press1 1936). See also Tiu History of Science Va.nu the History of Mtdicine (Isis, vol. 23, pp. 31 3-320, I 935). 1

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I

THE STUDY OF THE HISTORY OF MATHE�fATIC:S

The history of mathematics JS essentially different from the history of other sctenccs m its rclationshtp wtth the his­ tory of science, because 1t never was an integral part of the latter 1n the Wlte,-vcl11an sense The reason for thts is ob­ Vious mathemat!_CS being far more esoterJc_than_the 01hez:.. histo_ry can only be told to a .select group_oL !:!,ences, 1ts _It is mifia� true that there are 1n every science certain questions which are more difficult to explain than others, or cannot be explained without long prehmmaries, but those questions are almost exclusively recent ones, 1n the case of mathematics, on the contrary, the difficulties began very early There are problems wh.tch exercised the nunds of men 1n the fifth century B C and cannot be entirely ex­ plained to the non•mathemattcians of to-day, and it is lm• possible to make the latter realize the grandeur and beauty of Greek mathematics One might thus oppose the history of mathematics to the htstory of science, and thts s.s often done for praetteal rea• son! The teacher of the .htstory of �c.tence bcJng obbged to omit mathematical questions - cspcciaUy the most interest­ ing ones-, because only a part of hts audience could be ex:• pccted to understand them, 1t is natural enough to organize separate courses devoted to the history of mathematics There arc then at least two courses (or two series of courses) compieung one another, the history of science and the hIS­ tory of mathematics It 1S a p1ty that this should be so, for the history of mathe­ matics should really be the kernel of the history of culture Take the mathematical developments out of the history of science, and you suppress the skeleton which supported and kept together all the rest Mathematics g1ves to science its innermost uruty and cohes1on, rhlch can never be entirely replaced with props and butt« es or with roundabout con­ nectionst no matter how many of these may be introduced On the other hand, the histonan of mathematics, re­ membering that his activity 1S complementary to that of the

VARIOUS KINDS OF lfATHE�iATICAL HISTORY

5

historian of science, ,vill not attempt to do over again the latter's task, and he may even feel inclined to take of his O\\rn subject too technical and too narrow a view. There­ fore, it is \\'ell to insist that he should seize every occasion to indicate the relationships between mathematics and other sciences, and to insist that these relationships have always been reciprocal: mathematical problems being often the result of physical needs, vvhile mathematical elaboration gave physics, and, gradually, other sciences, not only means of discovery of almost miraculous potency, but also perfect models of analys is and synthesis. Some historians of mathematics, with a strong bent for hu�nism, are \villing to consider not only other scientific activities than the purely mathematical, but the whole gamu�. So much the better. Others, movfrig1n1he opposite direction, feel that the history of mathematics it­ self - not to speak of the history of science - is too com• plicated a subject, and, ,-vishing to avoid the endless intri­ cacies of the mathematical tree, they select one branch of it, and study its development in more or less complete iso­ lation from the others. Thus the historian may be led to investigate the development of algebra across the ages, or the amplifications of a single idea, like the idea of number, function, or group. Such abstraction in historical research, as opposed to the more natural procedure of considering each fact as it occurs in due chronological order, is very arbitrary. It is perhaps \vorth \vhile to examine the matter a little more carefully. The filiation of ideas is some\vhat like the filiation of in­ dividuals, except that the intricacy is even greater. Any individual A thinking only of his o\vn genealogy has a sim­ ple pattern in his mind like our figure 1, but that pattern is obviously a false one, from every point of view except that of his own unimportant personality. In reality, the pattern is enormously more complicated, for each couple may have had more than one child, each person may have married

J

THE STUDY OF THE lllSTOltY OF ?.tATHEt.tAncs 6 more than once, and marriages bcnveen cousins have in• troduced ne\V cross rclat1onsh1ps The complete picture of a man's family is hkc a net,.vork ,vhtch, 1f 1t be dra\vn com.. pletcly even for only a fc,v gcncrattons, 1s almost mex.. trzcabJe Of course the� JS nothing to prevent any ind1.. vidu.al from selecting 1n that net\vork and dra,-ving more

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heavily the hncs which concern him unmcdiately, the blood hnes, but the personal pattern thus abstracted from the whole ne tworJ.. 1s of no interest except to rum.self Now the fihat1on of ideas lS nccessar1ly more comph• catedt for the biological pattern JS rigorously luruted by the rule that each 1nd1vidual has nvo parents, neither more nor less, \vhdc each idea may cc::sult from the fus1on of more than nvo others, or on the contrary may be the fruit of a kind of parthenogenesis 'Whenever the rustorian tries to relate the hLStorv of a single group of 1deas he 1s obliged to abstract one pattern from a networl of endless complexity,

SECRET HISTORY

7

and such an abstraction, ho,vevcr interesting it may be, is ahvays arbitrary to a degree. The study of special branches of mathematics or of special mathematical ideas is very useful, for it helps one to under­ stand those particular ideas more deeply, but it should not be allo,ved to confuse our historical perspective. The his­ torian must try to keep in mind the chronological succes­ sion not of this or that idea, abstracted from the rest, but of the main ideas, all of them in their mutual relationships and in their diverse connections ,vi th the rest of life. Many times have I compared the history of science ,vith a �cret history, the account of a development taking place mysteriously in the darkness, ,vhile the majority of people are more interested and more immediately affected by the events happening on the battlefield or the forum, or by the vicissitudes of their o,rn selves and families. For societies, even as for individuals, one must make a sharp distinction bet\veen the things ,vhich are the most urgent and those which are the most important. These things are not by any means the same. The most urgent necessity is to live, to remain alive, that is, to eat, sleep, to be happy, to procreate children, and obtain security for one's family. That means physiology, business, and sport, and often enough war. Hov1ever, the most important things are not to satisfy one's physiological needs, but to increase the cultural heritage ,vhich has been bequeathed to us. The urgent things are obvious enough, and men's efforts to obtain them fill the whole historical picture; one can hardly see anything else. Yet all the time some men pursue in the darkness, secretly, the fulfilment of their intellectual desires and of humanity's highest purpose. If the history of science is a secret history, then the history of mathematics is doubly secret, a secret within a secret, for the growth of mathematics is unknown not only to the

Tim STUDY OF THE tIISTOR.Y OF l-1ATHEt.lATICS 8 general public, but even to scientific workers It is true, engineers may be found .from tune to time eri1ploying a new formula, but this docs not :amply a ny knowledge or under­ standing of the process whtch led to it Even so the average ctttzen uses every day more and more complicated and marvellous mac-Junes about which he knows less and less Yet that secret activity 1s fundamental, tt 1s aU the tune creating new theories, which sooner or later will set new wheels moving, new machines working, or, better still, 'h'lll enable us to obtain a deeper understanding of the mech­ anism of the universe The pract1cal man may neglect those secreta secretorum, but the phtlosopher cannot neglect them ,vithout loss and ,v1th� out disgrace The ' pracucal' and hard headed mathemat1 c1an, bent on his own 1n,est1gat1ons and noth1ng else, may neglect them too, but he will be a poorer man for do1ng so !naeed, one may etatm that the history of mathemat1cs provJdes for him the very best educatlon, the best humarus t1c 1n1t1at1on, one especially adapted to his own needs Let us contemplate for a moment the tnagmficent pano­ rama of mathematical history as it unfolds itself before us when we evoke the past First, miUenarics of preparation during '\Yh1ch some fundamental discoveries are already adumbrated the 1dca of number slowly emerges from the darkness, the idea of fractton, the idea of periodicity in geo• mctr1cal patterns, and others By the mJ ddle of the fourth millennium before Christ, the Egyptians were already ac­ quainted ,v1th large numbers of the order of mllhons, and Wlth a decimal system of numeration Uefore the mJddle of the second mJlJennium they had already attained suffi cient geometr1cal insight to deternune the area of any tr1angle as we do 1t ourselves, and to solve more difficult problem�, such as finding the volume of a frustum of a square pyramid To measure the area of a circle they squared eight-ninths of its diameter, which was a remark­ ably good approximation Dur1ng all that t1me the people

THE GREEK tllRACLE

9

of Mesopotamia had been developing a mathematics of their o,vw\vhich ,vas as admirable as the Egyp tian. In the fourth mil1ennium the Sumerians had already some kind of ' position' concept in the ,vriting of numbers, and had learned to treat submultiples in the same ,vay as multiples, an idea \'r?hlch the \\Testern \\·orld did not recapture until fifty centuries later. The geometry of the Babylonians did not reach the same level as that of the Egyptians, but on the other hand their resourcefulness in algebra ,vas as­ tounding,, for they succeeded in solving not only quadratic but even cubic equations. To appreciate the relative im­ portance of these achievements it is ,veil to remember that ,v·e are much closer to Euclid, often called ' the father of geometry,' than Euclid ,vas to the unkno,vn Egyptian and Mesopotamian mathematicians. In reality the ,vay for Euclidean mathematics ,vas very gradually and thoroughly prepared, not only by the mille­ nary efforts of Africans and Asiatics, but by three centuries of persistent investigations by the most gifted people among our ancestors, the Greeks of the golden age. The historian is made to vn.tness the building up, as it ,vere stone by stone, of that ,-,·onderful monument, geometry, as it ,vas finally transmitted to us in the Elemmts. The Greek ' miracle' con-­ tinned for at least six more centuries after Euclid, but ,vith less and less intensity and ,vith longer intervals of sleep be­ t\veen the periods of creation. In the mean,\rhile, the centre of mathematical light had moved from Athens for a brief intexval to Syracuse and then to the Greco-oriental city of Alexandria, ,vhere it remained for centuries. Thus ,vas their debt to Egypt abundantly repaid by the Greek mas­ ters and their Roman disciples. After the Romans came the barbarians, and ancient ,vis­ dom ,vas in danger of complete oblivion, \\1hen it ,vas unex­ pectedly rescued by the Arabs. These ,vere also barbarians, but barbarians redeemed by an intense faith and, for a few centuries at least, by an unquenchable curiosity. The

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THE STUDY OF THE HISTORY OF 1.fATIIEMATI�

masterpieces of Greek mathemat1cs were translated into Arabic and thus transmitted to the West If we call the Greek astounding rat1onahzaoon of geometrical thought a Dllracle (by means of which word we simply mean to convey that we cannot account for the ach.tcvement but only marvel at it), then the Arabic rescue and renaissance was another miracle, that 1s, a series of events whlch nobody could have foreseen and which nobody can completely explam The Arabs were mainly transnutters and brokers, but theu brokerage Jn a period of cru1s ,vas almos t providential They brought together Hindu and Greek 1deas, fertilizing the ones Wlth the others, and rcvolutsonwng anthmeuc, algebra, and trigonometry Their own contnbuttons m thtse brancilts of mathcmaucs were cf Isis,. vol 23 ;p;p 278--'.28o s Ett01'C Bortoloth, • La propagauon de la science l traven le, 11�cle, ., in Scitntia, vol 52, supplement,, pp 13,-146 (1932)

NECESSITY AND CAPRICE

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Our mathematical practice of to-day is still littered \vith the fossils of earlier times, such as Roman numerals, sexagesimal fractions, the English weights and measures, etc. ; on the other hand, other relics have been abandoned, the redis­ covery of ,vhich delights the historian, even as obsolete curios delight the archaeologist. When ,ve compare the ,vhole of mankind with a single man growing in knowledge and wisdom, we may stretch the comparison a little further: no man remembers equally ,vell everything; even the best memory experiences lapses, betrayals, and preferences. The ,vhole of mankind is like a man ,vith a memory that is good but not perfect. The deterministic theory of mathematical progress re­ mains insufficient even when one has corrected and tem­ pered it as we have done. It is not al,vays possible to ac­ count for the development of mathematical ideas by a combination of external events ,vith personal impulses on the one hand and personal inhibitions on the other, great as is the flexibility of such a method_ There are many facts ,vhich one cannot account for in a general ,vay, and this applies to mathematical inventions as ,veil as to any other details of human behavior. Many mathematical develop­ ments are capricious in the extreme, and it is a waste of time to try to find a rational explanation of them. Strangely enough, in the same text ,vherein Galois expressed so strongly his belief in mathematical fatality, he also called attention to the great irregularity and disorder of our mathematical knowledge. An orderly development would only be possible for a godlike mathematician knowing in advance all the possible mathematics. Here is really the crux of the matter. Mathematicians and other scientists, however great they may be, do not know the future_ Their genius may enable them to project their purpose ahead of them; it is as if they had a special lamp, unavailable to lesser men, illuminating their path; but even in the most favorable cases the lamp sends only a

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THE STUDY OF THE HISTORY OF ltATIIE!.{ATICS

very small cone of hght mto the mfirute darkness Enthu­ siastic admirers of great men often make the nustake of giv• 1ng them credit for the endless consequences of their dis­ coveries, consequences which they could not possibly foresee To credit Galois \\l'lth all the results of the theory of groups zs as foolish as to a-edit Faraday ,v.ith all the wonders of electtotecbrucs1 or to hold C.Olumbus responsible for all the good and eV1l done 1n the New World since 1492 The founder of a ne,v theory or of a new science deserves full credit for the discoV'cnes wluch he actually made, less credit for those which he adumbrated, and still less fot' those wluch he made possible but dtd not realize 'Wlule ,ve honor h1rn as a founder, we must remember that he could not pos• sibly anticipate all the consequences of his ideas and all the fn11ts of hJ.s deeds He is the spiritual lord of the domain which lus unagmation could encompass, neither more nor less We often call bun the father of this or that, and such a term ,s appropnate enough to express our respect, even our veneration, ,1 we bear 1n mmd that parents should not be praLSCd or blamed too much for their chtldrcn, though they made them, not to speak of thell' more distant descend.. ants who sprang from other loins The eapr1ciousness of mathematical development cannot be emphasized too much \¥by were the early Greeks so interested 1n the theory of numbersy and so httle 1n plam arithmetic' The latter was highly needed Every reason of economic necessity should have caused the development of arithmetic, and discouraged as a ltL�ury the gro,vth of fanciful ideas on the properties of numbers \\'hy did magic squares 1ntercst so many peoples East and West) Why' Why) The student of history should not ask such childish queries Hts purpose cannot be to give a completely Jogir.al account of the past, for such account 1s obV1ously out of the question It is only here and there that a few logical knots can be tJed, for the rest, ,ve must be satisfied Mth a faithful descr1pt1on of the possib1/itn::s- whreb. m-.rrenai\zed among-

WAYS OF DISCOVERY

an infinity of others which did not. The shortest distance from one point to another is a geodetic line, but such a line can only be followed if one kno\VS one's destination, in which case there would be no discovery. The ways of dis­ covery must necessarily be very different from the shortest way, indirect and circuitous, with many windings and re­ treats. It is only at a later stage of knowledge, when a new domain has been sufficiently explored, that it becomes pos­ sible to reconstruct the whole theory on a logical basis, and to sho\V how it might have been discovered by an omniscient being, that is, how it might have been discovered if there had been no real need of discovering it ! Galois's impatience \vith the textbooks of his day was inconsistent. It is as if an explorer of an unkno\m territory complained of the absence of maps, or the student of an unknown language of the lack of grammars and dictionaries. To conclude, capriciousness is of the essence of discovery, because we can only know where we are going, and whether it is worth going to, when we are there. Accordingly we cannot help following many false trails, and going astray in many ways. Moreover, caprice is of the essence of life in general, and of human life in particular, because of life's very complexity and indetermination. *** Nevertheless, the development of mathematics is perhaps less capricious than that of other sciences, more completely determined (or less undetermined), if not by external fac­ tors, at least by internal ones, for each theory presses forward as it were, and the mathematicians who are playing with it must needs perceive some of its consequences. The desire to follow them to the limit is then likely to prove irresistible, whether these consequences be useful or not. The m..ncat�­ natiops of mathematical ideas are not divorced from life, far fr�� it, but they are less influenced than other scientific ideas by accidents, and it is perhaps more possible, and

�o THE SlUDY OF THE :WSIOltY OF lL\TIIDtATICS more permissible, for a mathematician than for any other roan to .secrete himsdf in a to""U of i,-ory·.

• * ..

The bistory of mathematics is thus a good fie1d for the investi gation of theories ronccming the progress of science in general, and the possibilities of logic.a.I development in particular. It is concdvabfe that the capriciousness is only rclati,·e after � that it aH'ttts the dt"tails of the picture rather than the main outline.. Such a conception is attrac­ tive enough to be fully in\·cstigated, and only the lustorian can do it. The \-irusitudcs of history might be o�ttiooked in a .first approximation. One might assume that �fan (not this man or that, ,,·hate\'cr be bis genius} follo,\'Cd un• erringly the geodetic hne from A to B (A and B bcing ti.\'O mawematical dlSCO,-cries) instead of meandering and beat• ing about the bush (fig. �).

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f i g .. 2. This assumption, 1n fact, the historian of mathematics is often obhgcd to male; if only for the sale of brevity and simphcity. Reality is far too complc.'t to be represented to the last detail, hut the l1i.,;ton.an•s simphfication must TC­ main sufficiently close to it; it must rcp�t the main outline in chronological sequentc. It is thus C."Cceedingly different from the iynthetic reconstruction made by teach­ ers, wherein the chronological :sequence is necessarily dis•

HISTORICAL VS. hfATHEMATICAL SYNTHESIS

21

regarded as irrelevant. The mathematician whose privilege it is to give to a theory its :final, ' classical' shape, is likely to define the function he is dealing vvith by means of the property which \\7as perhaps the last one to be discovered . That is all right: the synthesis thus created, ho,vever dis­ tant it may be from historical contingencies, is closer to the deeper mathematical realities. We need equally the t\vo kinds of synthesis: the historical and the purely mathematical. The latter is the shortest if not always the easiest path to knowledge, but it fails to ex­ plain the human implications; it may satisfy the matter­ of-fact and hurried mathematician; it cannot satisfy the philosopher and the humanist. As to the pure mathematician, even he should not be too easily satisfied ,vith the latest synthesis. To begin with, that synthesis may be incomplete. Some elements which were not deemed essential for it may have other values, they may prove to be essential for other structures, or the one from ,vhich they ,vere eliminated may not be as final as it seems. Indeed, no theory is ever final. A ne,v discovery, a new point of vie,v may cause its abandonment and its super­ sedure by another, and the facts neglected in one shuffling may be considered invaluable in another. Every synthesis implies sacrifices; it is not merely a simplification but also and unavoidably a betrayal of reality, a distortion of the trnth, and the mathematician ,vho takes the trouble of con­ sidering the origin and evolution of ideas, as well as their final shape, vml improve his understanding of them and enrich his mind. The study of history may, or may not, help the mathema­ tician to make new discoveries by suggesting new connec­ tions benveen old ideas or ne,v applications of old methods; in any case it will complete his mastery of the subject, and provide him ,vith new opportunities for a deeper and more intuitive grasp of it.

***

!22

THE STIJDY OF THE HISTORY OF MATHEMATICS

The main reason for studying tbc history of mathematics, .,,. or the history of any science, is purelyJlumanistic. Being men, we are interested in other men, and especially in such men as have helped us to fulfil our highest destiny. As soon as we realize the great part played by individual men in mathematical discoveries -for, hmvevcr these may be de­ termined, they cannot be brought about except by means of human brains -, we arc anxious to know all their cir• cumstanccs 1 How did it happen that this man or that man, among others, wa! devoted to mathematics? Was he thus consecrated before being conscious of it, or did he con­ secrate himself.I How did his mathematical genius assert itself, how did it blossom out? Was it hard for him or easy? Did he succeed 1n tstabltshiog his theories and convincing his contemporaries of their importance? All these ques­ tions and many others are deeply interesting, especially for other mathematicians: if they are young, because of their dreams of the future and their hopes and doubts; if they arc older, becau1e of their memories of the past, and also, though in a different way, because of their hopes and doubts One soon realizes that mathematicians arc much ltke other me.n, except z.n the single respect of their special gcruus, and that that genius its�lf has many shapes and as• pects. I remember reading in John Addington Symonds•s biography this portrait of a great musician� Here was a man Handel, a Cat native of Halle, ,n the Duchy of Magde­ burg. arucled at eight years old to an organut. and from that moment given up to music - a man who never loved a woman. who (to use the Suc:h 1nqu1s1uvene� may 1ttm. idle to thc:,se t1:lat1vely few men who arc too engrossed 10 their own thoughts to care for anydung else. but tt repre• sents one of the oldest human 11utn1cts The 1amc 1rutlnct reveals itself on a lower level through the muncll'SC cunosaty concerrung murde� News­ papers ace slul(ul 1n pandenng to such cun011ty to their own profit Even as the mass of the people aceuisatrablc UI their de11ce to know every det.ul of a murder case_. ro those who .re more thoughtful wish to 1nvest1gatc every dewl of scu:ntdic ducovent;t or other crcabve aclllevemcnts. l

1'{ATHEMATICAL GENIUS

,,,ords of his enthusiastic biographer) continued irritable, greedy, fond of solitude, persevering, unaffectionate, coarse and garrulous in con­ versation, benevolent, independent, fond of beer, religious, without pas• sions, and ,vithout a single intellectual taste. He had never received any education except in counterpoint. He had had no experience. Yet he could interpret the deepest psychological secrets; he could sing dithy­ rambs to God, or pttach moral sermons; he could express the feelings of mighty nations, and speak with the voice of angels more effectually than even Milton ; he could give life to passion, and in a few changes of his melody lead love through all its variations from despair to triumph there ,va.s nothing that he did not know·.. The whole world had become for him music, and his chords ,vere co-extensive ,vith the universe. Raphael's capability to paint the school of Athens, after coming from the workshop of Perugino, was perhaps less marvellous than Handel's to delineate the length and breadth and height and depth of human nature in his choruses. We shall never comprehend, noru autru, the mysteries of genius. It is a God-sent clairvoyance, inexplicable, and different in kind from intcllect.1

'This account has recurred to me almost every time I have been introduced to an original scientist, for it applies to him as well as to his musical brethren. The great mathematician may be a man of very limited experience and wisdom outside his own field and his advice in non-mathematical matters may be of very little value; he may be burdened with all kinds of passions and weak­ nesses; in short, he is like the rest of us except in one es­ sential respect. When we write his biography it is clear that it is that essential thing, his genius, which must remain in the centre of the picture, but our curiosity does not stop there. We may be so deeply interested in his personality that we desire to know everything, whether good or bad, which concerns it. That is all right. Full and honest biog­ raphies should be encouraged by all means, they help us to kno,v our fellow men and ourselves better, but that loathsome fashion of our time, which is called ' debunk­ ing ' - the dragging down of great men to the level of their John Addington Symonds, as quoted by Horatio F. 'Brown in the latter's biography of him (London, I 895, 2 vols.), vol. I, PP· 343-3 44· J

24

THE STUDY OF THE HISTOR.Y OF MATHE}..tATICS

meretricious biographers - should be discouraged It is a matter of measure It ts very well to sho,-v all the weak­ nesses of a hero, but tlus should be subordinated to the main purpose, the descr1pt1on of hts genius, the explana­ tion of the discovcr1es which 1t made possible, the contem­ plation of the truth and beauty which it revealed For example, a great mathcmat1c1an may be a drunkard, for his m1nd's obsession with mathematical ideas may some­ times become unendurable, a real torture from ,-vh1ch he may wish to es a fC'\, £,·CI"\ anoent mathemanaan \\hose name has come don'll to us must be dealt ,,,th, ,,hue for modem ones need spca1.. only ()f a sm.all mmonl) of those ,,. horn L.no,v To use another Cb portrait (1926)

BIBLIOGRAPHY

43

180. Memoires scientifiques, edited by his widow since 1 91 2 ; thus far (1 936), 1 3 vols. published (see Isis, passim) . It was necessary to mention Tannery in a general way, though there is no danger of his being overlooked, as his papers, all anterior to 1905, are often referred to by Cantor, Heath, Sarton, and others. I shall quote now a series of books completing Cantor's monu­ mental history, and constituting ,vith it a primary shelf of refer­ ence. Otto Neugebauer, Vorlesungen iibtr Geschichte der antiken mathe­ matisclzen J,Vissenschajten, vol. 1 . Vorgriechische Mathematik (Ber­ lin, 1 934) ; cf. Isis, vol. 24, pp. 151-1 53. The main texts of Egyptian mathematics have been edited as follows: The Rhind Mathematical Papyrus, ed. by Arnold Buffum Chace, Ludlow Bull, Henry Parker Manning and Raymond Clare Archi­ bald, Oberlin, Ohio (2 vols., 1927-29) ; cf. Isis, vol. 14, pp. 25 1255. Includes an elaborate bibliography of Egyptian (and Baby­ lonian) mathematics by Archibald. Mathematischer Papyrus des staatlichen Museums dtr schiinen Kiinste in Moskau, ed. by W. W. Struve and B. A. Turajeff (Berlin, 1930) ;

cf. Isfs, vol. 1 6, pp. r 48-155· •

1n

The texts of Mesopotamian mathematics have just been printed Mathtmatische Keilschriftlexte,

ed. by O. Neugebauer (Berlin, 1 935, 2 vols.). Including a bibliography of Babylonian mathe­ matics.

The standard work for the history of Greek mathematics is Sir Thomas Heath, A History of Greek Mathematics (Oxford, 192 1 , 2 vols.); cf. Isis, vol. 4, pp. 532-535. There is an abbrevi­ ated edition in a single volume, A Manual of Greek Mathematics (Oxford, 193 1 ) ; cf. Isis, vol. 16, pp. 450-451 . . . no, Mila ed., (2d a Greci antica nell' Gino Loria Le scienze esatte 19 14) ; cf. Isi;, vol. 1 , pp. 714-716. Covers a wider field than Heath, and is less elaborate, yet very valuable. For Hindu mathematics, see George Rushy Kaye (1866-1 9 29), " Indian Mathematics," in Isis, vol. 2, pp. 326-3 56 ( 1 9 1 9).

44

THE STUD\:• OF THE HISTORY OF .M'.ATHEl\fATICS

Sarad�kanta Ganguli, "Notes on Jnd1an Mathematics A Cr1tic1sm of George Rushy Kaye•:s Intcrpretat1on,u 1n Ins, vol 12, pp 132-145 (1929) Bibhut1bhusan Datta, Tlit &unu of the $ul6a A Stuay in Early Hindu Gtometry (Calcutta, 1932) 1 cl Im, vol 22, pp 272-1277 B1bhutibhusan Datta and Avadhcsh Narayan Singh, History of Hinau Mat!It:matics A Source Boole Part 1 1 Numeral Notat1ons and ArJthmetic (Lahore, 193.5 282 pp ) For mathemat1cs in Eastern Asta, sec Yoshio Mikami, Tht Dtvtlopment of J,,falhtmatics tn Cluna and Japan (Leipzig 1:913) Davtd Eugene Sm1th and Yosh,o Mikami, A History of Japan• tse Mathematies (Ch1cago, 1914) 1 cf Ins, vol l2 1 pp 410-41:3 For general reference, cspeetally with regard to med1aeval and or,ental ma.thematics George Sarton, lnlro tk Hrslo')' of Science (2 vols 1n 3, Washington, 1927-st) Vol t, From Homer to Omar Khayyam, 1927, wl 21 From Rabbt ben Ezra to Roger Bacon, 1931, vol 3, Fourteenth Century, an preparatton Errata and addenda to these volumes are pubhshed per1od1cally ,n the crtt1cal b1bhograph1es of Ins, bcginn1ng W1th the ni.neteenth 1n vol 8 for the first volume, and with. the thi.rty first ,n vol 16 for the second

II

HANbBO()KS

There are a number of small trcattSCs on the history of mathe matics, most of which arc sufficient The average level ts far superior to that of books on the hutory of science Indeed, some of these boob, hke Smith s and CaJor1's> contain valuable ma tcr1als not available clse,vhcrc I can mention only a fc,v of these treatises Hieronymus Georg Zeuthcn (1839-19�0)> Gtsr:lnclitt der Jfatlit matik im Altertum und Mttttlalt« (Cpcnhagcn, 1896) French translation by Jean Mascart1 ttvlkd by the author (Pans, 1902) Gtsch.ic!ite dtr Math.ematii. im ,\"VI und AYll Jalirlzundert (Leipzig, 1903) , Zcuthcn s books are 1mp0rtant because of his mathematical interpretations, he was lumsclf a creative mathematician and had a keener tense of mathematical subtleties than Cantor

BIBLIOGRAPHY

45

:Valter William �ouse Ball (1850-1 925), A Short Account of the History of Mathematics (London, 1 888; sixth edition, 1 915); cf. Isis, vol. 1 , p. 561 . This history has enjoyed more popularity than any other; though obsolescent, it may still prove of use. See Fl?rian Cajori, " Vv. W. R. Ball," in Isis, vol. 8, pp. 321�24, portrait (1 926). Siegmund Gunther (1 848-1 923), Gtschfchte der Matlumatik. I. Tell, Bis Cartesius (Leipzig, 1 908) . Heinrich Wieleitner (1 874-1 93 1 ), Gtschichte der Mathematik. II. Tei], Von Cartesius bis zur ,vende des 18. Jahrhunderts (Leipzig, 191 1 --2 1 , 2 parts). See Julius Ruska, " Heinrich Wieleitner," in Isis, vol. 1 8, pp. 1 50-165, portrait (1 932). Florian Cajori ( 1 859-1930), A History of Mathematics (Ne\'1 York, 1 895; revised and enlarged edition, 1 9 1 9). See R. C. Archibald, " Florian Cajori,'' in Isis, vol. 1 7, pp. 384-407, portrait (1 932). Cajori is especially valuable for the modern period. He deals, very briefly it is true, ,vith a large number of mathematicians of the nineteenth and nveotieth centuries. David Eugene Smith, History qf J.1a thematics (Boston, 1 923--25, 2 vols.); cf. Isis, vol. 6, pp. 440-444-; vol. 8, pp. 221 -225- This is restricted to elementary mathematics, but contains an abund­ ance of ne,v and out of the ,vay information, and many illustra­ tions. Smith's books constitute the best introduction to the subject. Gino Loria, Storia ddlt matematiche (Torino, 1929-33, 3 vols.). Vol. 1 , to the Renaissance, 1 929; vol. 2, sixteenth and seven­ teenth centuries, 1 93 1 ; vol. 3, eighteenth and nineteenth cen­ turies, 1 933 ; cf. Isis, vol. 1 3, p. 228; vol. 1 9, p. 23 1 ; vol. 22, p. 598. Includes a good general account of modern mathematics. To these handbooks may be added t\\"O so-called source-books. Heinrich Wieleitner, Mathtmatisclze Quelltnbiicher (Berlin, 1 92729, 4 small vols.). These volumes, containing a series of annotated extracts from mathematical classics, deal respectively "ith (1) arithmetic and algebra, (2) geometry and trigonometry, (3) an­ alytical and synthetic geometry, (4) infinitesimal calculus. Cf. Isis, vol. 1 1 , p. 240 ; vol. 12, p. 413. David Eugene Smith, A Source Book in Afathematics (Ne,,· York, 1 929); cf. Isis, vol. 14, pp. 268--270. Arranged in topical order: number, algebra, geometry1 probability, calculus. Unfortunatelr

46

THE STUDY OF ntE HISTORY OF 1-IATHEAIATI�

the seleet1on begins only ,nth the end of the .fifteenth century, thas 1.1 not the author's fault, but lS due to the stupid programme of the eolleet1on 1n ,\lhteh 1t u mcluded On that account, if on no other, Wieleitner's sourcc•book 1s preferable

III

TREATISES DEVOTED TO THE HISTORY OP SPECIAL BRANCHES OF l{ATIIEUATlC:S

Before dcahng ,v1th the special branches t,vo more gentral books must be rnentioned Johannes Tropflce, Ges,h.i,h.te dt1 E11mtntar-J..fal/umatik (first ed , Leipzig 1902-03, 2 vols ) This first edition 1.1 cued only for the sake of curiosity Soon after •� pubbeataon the author began th� prcparat\o. n of a second �d\t\Ont \vh,ch was pubhshcd U\ seven volumes {rom 1921 to 1914 The pubhcatton of a third cd1t1on began 1n 1930 I 1nd1eate the contents of each volume, wsth the date, of the .second and (so tar as Jt " pubhshed) thll'd editions o( each Vol 1 , computauon, 1921, 1930 Vol 2, general arsthmeuc, 1921, 1933 Vol 3 , proportions, equations, 19:22 Vol 4, plane geometryt 1923 Vol 5, trigonometry, 1923 Vol 6, analysas (1 c t series, computation of interest, combinations, probab1httes, conunued fractions, ma:x1ma and m,n,ma), analyucal geometry, 1924 Vol 7, stereometry, 1nd1ccs, 1924 Sec lsir, vol 5, pp 18�­ i 86, and pasnm The third edition is completely ready in manu,aipt > but its pubbeatton has been temporarily d1scont1nued because of Ger many's eeonomic d1fficult1cs Thas 1.1 very unfortunate, for Tropfke's history 1s a mane ot 1nfonnat1on on every branch of elementary mathemaua, and the hutortan must ahva� con• suit 1t Florian Ca1or1 (185g-1930), A Hulo'J of }.falkrnatical NolatioM (Chicago, 1928-29, !l vols ), cf Int, vol 12, pp 332-336, vol 13, pp •�9-130 Arithmetic David .Eugene Smith, Rora Arithmetica {Boston, 1908) D E Smith and L C Karpinski, Tht Hindu Arabic .Nu mnals (Boston, 1911) LouIS Chat-lcs Karp1nsk1, The History if Ar,thmelle (Chicago, 19�5}, cf Ins. vol 8, pp 231-232 Tobias Dantz1g, Numbn, the LangU1Jge of Sanu:.e {Nc\i Yot'k, 1930, cf

BIBLIOGRAPHY

47

Isis, _vol. 16, pp. 455-45�; 2d ed. , 1 933 ; cf. Isis, vol. 20, p. 592) . Richard Brown, A History of Account£ng and Accountants (Edin­ burgh, 1 905). Thtory of Numbers. Leonard Eugene Dickson, History of tk Theory of Numbers (Washington, 191 9-23, 3 vols.). Vol. 1, divisibility and primality, 1919; cf. Isis, vol. 3, pp. 446448. Vol. 2 , Diophantine analysis, 1920 ; cf. Isis, vol. 4, pp. 107108. Vol. 3, quadratic and higher forms, 1 923; cf. Isis, vol. 6, pp. 96-g8. These volumes do not contain a history in the usual sense of the word, but very elaborate collections of facts with the dates and references relative to each* The work is a magnificent intro­ duction to the history of the subject. Algebra. Pietro Cossali (1 748-1815), Origine, trasporto in Italia, primi progress£ in essa dell' algebra. Storia critz'ca dz' nuove dz'squi"sz'zioni analz"ticht o metajisiche arrz"chita (Parma, I 797-gg, 2 vols.) . Pietro Franchini (I 768-1 837 ), La storz"a dell' algebra e de' szzoi principali scrittori sz·no al secolo XIX rettificata, illustrata ed estesa col mezt.o degli orz·gz·nali documenti (Lucca, 182 7). Though obsolete in many respects, those two volumes must still be consulted. H. G. Zeuthen, Sur l'origine -de l'algebre (Danske Videnska­ bernes Selskab, Matkmatisk-Fysiske Meddelelstr, ii, 4) (Copen­ hagen, 1919, 70 pp.). Determ£nants. Sir Thomas Muir (1844-1 934), The Tkory of Determinants in the Historical Order of Development (London, 1 906-23, 4 vols.), with supplement, Contrz'butions to the History of Determinants, I900-I920 (London, 1 930). The same remark applies to these books as to Dickson's ab��e (cf. Isis, vol. 4, p . 199; vol. 7, p. 3 1 2 ; vol. 16, p. 5 10} .. Mu�r s work is really an analytic catalogue of all papers deahng with determinants, but he has often overlooked very important mat• ters because these were included in papers specifically devoted to other subjects. This illustrates a fundamental weakness of such undertaki,igs. Trigonometry. Anton van Braunmi.ihl (185 3-1908), Vorlesungen iiber Geschichte der Trigonometrie (Leipzig, 1 900-03, 2 v�ls.): Geometry. Michel Chasles (1 793-1 880) , AperfU hzstonque sur forigine et le developpement des methodes en glomltrie (Bruxelles, rB37 ;

48

Tim STUDY OF THE HISTORY OF MAnrElfATICS

2d ed , Paris, 1875) The same, &pp111t sur Its P,t>grts de la gl()tn!# trie (Pans, 1870 388 pp ) Gino Lona, II passato t ii J,restnte tfelle prsnr:ipali teone geometriche (Torino, 1887, 2d ed , 18g7, 3d, 1907, 4th, 1931), cf Ins, vol 19, Pl> 229-231 Max Simon, Uber die Entwullung lk, Elementar-Geometrie im XIX Jalzrlzundtrl (Le1pz1g, 1go6 �86 pp ) Ana[ytual Geometry Gino Loria, Da Descartes e Fermat a Monge

e Lagrange

Contributo alla stona della geometria artalit,ca

{Roma,

1924), cf lnr, vol 8, p 6o6 Gino Lor1a, Curve p,ane sptaalt olgihr,c!u e trascendenti (Milano, 1930, � vols ) , tf Isis, vol t4, p 542, vol t 5, J> 467 Prcvt­ ousl)' published 1n German (Leipzig, 1910-r 1, � vols ) Con•

tain$ considerable historical ,nformauon "latlve to each curve Dt$'1tJ,titJe Geomtlry Noel Germ1nal Poudra (1 794-1894), Hit• 1,11,, de /a P"'pe,hllt 12ntu1u·1.t 1I mcdtrM (Par,s, 1864) Gtno Loria, Slorta della geomttrta dtserittiva dal/1 origin, sino a, giorn\ nostn (M1la.no, 19�1), e{ Im, vol 5, J>P 181-181 SJ11t!uht Gtcmtlf)' Ernst Kotter, "Dte Entwzc.kelung der ,yn­ thetuchen Geometrlc, I Theil Von l\1onge bis aufStaudt, 1847," in the Jahresbtricht o( the Deutsche Mathemati.ker-Verein1gung, vol 5, p t l:r (�1pzJg, 1898-1901 514 pages) No more pub• lished Non•Euelidean Geometry Fr1edr1cn Engel and Paul Stackc� Die Tlut>111 dt'f Pa-ra/111/Jrntn llc>n Eukl1d 1n.t IJUj Gau.u, nM Urkuruknsamm• lung �ur Vorgeschicht� der ntckltu>.lid1tekn Geometne (LeiJ>zig, 18g5 336 pp), and Ur�utulen J;tlt Gttclucklt der nttfiteuldadisckn Geome• Int, vol t 1n 2 parts, deahng with Lohachevsku (Le1p11g, 189899), vol 2 \n !2 parts, dealing Wllh B6lya1 (1913) Rober10 Bonofa (1 874-191 1) ) " Index Operum ad Gcometr1am Absolutam spec1ant1um,>> pp 81-154 in 1he memorial volume published by 1he Hungarian University of Kolozsvar for B6lya1's centenary in 1902 Also La gtometna non-t11.clirita Esponttone stonco•critica del suo :viiuppo (Bologna., 1906 22() pages) German translation by He1nr1ch Liebmann (L1:1pz1g, 1908 252 PJ> , 76 fig , 2d ed , 1919, reprinted 1n 19�1) English 1ranslatton by H S Carslaw (Chtcago., 1921 28o pp ) Duncan M Y Sommerville, Bibho.�aph.1 of Non-Euclidean Geome­ try, including tk Tlu:e1t;1 ef Parallels, tire Foundalton ef Gtamet,y, and

BIBLIOGRAPHY

49

Space of n Dimensions (London, 1 91 1 : 4 1 5 pp .) . In chronological order, ,vith subject and author indices. Anarysis. The early history of the differential and integral calculus (do,vn to 1800) is naturally dealt with in vols. 3 and 4 of Cantor's Vorlesungen, but there is no history of modern an­ alysis. There is an outline of the history of functions of complex varia­ bles in Felice Casorati (1 835--go), Teorica delle funzioni di variabili complesse (vol. 1 , Pavia, 1 868; no more published) . The history down to 1865 covers 1 43 pages. Calculus of Variations. Isaac Todhunter (1 820�84), History of the Calculus of Variations during the Ninetunth Century (Cambridge, 1 86 1 : 544 pp.). Theory of Probabilities. Isaac Todhunter, History of the Mathe­ mat£cal Theory of Probabilityfrom the Time of Pascal to that of Laplace (Cambridge, 1865 : 640 pp.). Helen M� Walker, Studies in the History of Statistical Methods (Baltimore, 1929: 237 pp., 1 2 ills.) ; cf. Isis, vol. 13, pp. 382-383.

IV.

MAntEMATzcs IN THE NINETEENTH AND TWENTmTH CENTURIES

Many of the books named in Section III deal with the modern history of mathematics, sometimes almost or quite exclusively. For example, in Muir's elaborate history of determinants we find mention of only eleven papers anterior to 1 800: there were no others. The best general view of nineteenth-century mathematics is the one contained in Cajori's History ef Mathematics (2d ed.> 19 19). Almost half of the volume (pp. 278-5 1 6) is devoted to the nine­ teenth century and after. While it is only a bird's eye view, it is the most complete I know of. The information given is often so brief as to be unintelligible except to one well acquainted with the subject. However, it indicates the mutual relationship of many hundreds of mathematicians. It is a modest but excellent reference book. The third volume of Loria's Storia (1 933) contains also a good account, dealing with fewer mathematicians, but giving more space to each.

50

THE STUDY OF THE HISTORY OF }fATHEMATICS

The most ambitious attempt to outline the history of modern mathematla was made by Fehx Klein {184g-r925) in the form. of lectures given Jn his own bome during the war years K1e1n was eminently qualtfied for this task1 because he combined h1s• torical learning with mathemaucal depth to an unusual d egree These lectures were unfortunately interrupted by tllne53 in 1919 They have been posthumously published. as far as they go they arc very prec1ou, Fehx Klein, Vorltsung,n uber drt Entwuldung dtr Mathematik im T9 Jahrh.undert Part I edited by R Courant and O Neugebauer (Berlin, 1926 400 pp ) , cf lsrr. vol 9. pp 447-449 Patt 2 ed1ted by R Courant and Stephan Cohn-Vossen (Berlin, 1927 :208 pp ), c:f /r,1, voJ 10, p 505 These two volumes arc so ,mportant that lt JS worth wh1le to indicate their content! with some dctatl I give for each chapter not only the title but the names of the pnneipal mathemaHctans dealt with Pa.rt r Chapter I GauS3 a Fran� and che Ecole Poly.. tcchnJque at the beginning of the century {Powon, FourJet, Cauchy, Sadi Carnot, Ponc�let, Cor1ol1$, Monge, Galo-is) 3 Foundat1on of Crelle's Journal and blossom1ng of the German school (Crelle, D1r1chlet, Abel,Jacob1, MO and Otto Toeplitz (Berlin, 1 929, etc.) and published in t,vo sections: A. Quellen (vol. 1 , 1 930; vol. 3, in t\Y0 parts, in 1 935) . B. Studien (vol. 1 , 1 92g--3 1 ; vol. 3 , in 1 935). 8. 1 932. Saipta Mathematica. Edited by Jekuthiel Ginsburg (vol. 1 , Ne,v York, 1 932-33; vol. 3 in 1 935). 9. I 936. Osiris. Studies on the history and philosophy of science and on the history of learning and culture. Edited by George Sarton. Vol. 1 (778 pp., 24 figs., 35 facs., 22 pls.) published in January, 1 936, and dedicated to David Eugene Smith, is a collection of 38 papers on the history of ma thematics. �1ore information on some of these journals and on other journals ,vill be found in G. Sarton, " Soixantc-deux revues et collections relatives a l'histoire des sciences," in Isis, vol. 2, pp. 132-161 (1914) . The nine journals above mentioned might be divided into three groups according to their bibliographical interests. I. Journals attempting to give a complete bibliography of the history of mathematics: nos. 1 , 3 , 5. IL Journals containing bibliographi­ cal information "'ithout system and ,Yithout attempt at com­ pleteness: nos. 4, 6, 8. III. No bibliography of current publica­ tions: nos. 2, 7, 9· For the making of any bibliography of the history of mathematics, it ,vill generally suffice to consult nos. 1 , 3, 5, that is, Boncompagni for the period 1 868-87, Encstrom for the period 1 884-1915, Sarton for the period beginning in 1 9 1 3 or a little before . VI II.

CENTRES OF RESEARCH

A. Academies of Science Every academy of science has a mathematical section or de­ votes some attention to mathematics in its publications or other.. wise. Academies organize competitions and promote mathe-

62

THE STUDY OF THE HISTORY OF l-fATHEt.fATl�

maucal research 1n various wa>s They have occM1onally shown some interest 1n the history of mathematics

B

Matlumal1'al Socithts

By the beg1nn1ng of the nineteenth century science had al ready deveJapcd to .such an extent that acadcmJes had become somewhat obsolete They preserved some kind of administra• uve unity, but they lost theu- organ•c unity, as it became generally impossible for each member to be interested 1n the acttv1t1cs of the majority of the other members Hence 1t was necessary to create new societies devoted to the promotion of special sciences The mathematical soc1ct1es we� ttlat,vely :dow 1n appearing 1

1865 1872 1883 1684 1888

London Mathcmaucal s«:iety Soeict6 ?,,fathemaoque de France Edinburgh Mathematical Society C1rcolo ?,,fatemauoo d• Palermo American Mathematical Society (first called New York

1890 1907 1911 1915

Deutsche Mathemattket Vercinigung Indian Mathemaucal Society Soc»cd.ad Matemauca E.,patie>Ja Mathematical Assoetauon of America

M S)

These societies, and other samilar ones cxi.sung tn many other co1Jntr1es, encourage the study and the ttach1ng of mathemaucst but they pay relatively little attention ta 1u hutary

C

Internation11l Congrtssts

In -;p11e of the fact that many mathematical sac1et1es have an international membership, each of them u necessarily most con• cerncd with the problems of mathemallcal study and teaching • That u ,nflucnual soacbcs orgaru� bke academics and supportlllg sp�c1al pubhcattons Lacal groul)I of matbcmattc.tam were organized much earher. and perhaps 1n more ccnttts {c g • umven1t1es) than we can realu:e Two interesting examples are the Mathcmausche Gt!ctlschaft 2u Hamburg founded 1n t6go (for wluch see the F,sml,nft it pubh:shed two centuncs latetl and the Mathcmauc:al Sooc.ty or St>1taffieitls (in. I.nrui'lD.l wundi:d w 1717 (Nature, vol 64, p -478. 1901)

BIBLIOGRAPHY

,vithin its national boundaries. By the end of the century this ,vas felt to be insufficient, and efforts were made to supplement these national organizations by international ones. The first t\vo international congresses ,vere: I . 1889. Paris. Congres International de Bibliographie des Sciences Mathcmatiques. 2 . 1 893. Chicago. International Mathematical Congress. Ho,vever, these hvo congresses are not counted in the official list of international congresses. I . 1 897. Zurich. 2 . 1900. Paris. 3· 1 904. Heidelberg. 4· 1 908. Rome. 5. 1 9 1 2 . Cambridge. The sixth meeting, Stockholm, 1 9 1 6, could not take place be­ cause of the \Var. After the War, a ne,v series of congresses ,vas begun, the internationality of \vhich ,vas at first incomplete. r .. 1 920. Strasbourg. 2. 1 924. Toronto. 3. 1 928. Bologna. 4. 1 932. Zurich. 5. 1936. Oslo. The Oslo congress is thus the fifth congress of the ne,v series, or the tenth or t\velfth congress of the ,vholc series. Each of these congresses has devoted a part of its attention to the history and philosophy of mathematics, a special section be.. ing generally reserved for these t,vo subjects. It may be ,vorth ,vhile for the historian to examine the publications of these con­ gresses ,vith special reference to their historical and philosophical questions. In addition, the literature of those congresses ,vill be of great value to the later historian, because they ,vill help him to de­ termine the fluctuations of mathematical interests from year to year and the mathematical atmosphere of each definite period. Therefore the publications of the international congresses should be kept in good order in our historico-mathematical libraries. Questions relative to the history and philosophy of mathematics

64

THE STCJDY OF THE IDSTORY OF MATHEMATIC-5

have also been pcnod1cally ducussed 1n other 1nternat1onal con• gresses, to w1t, the congresses or philosophy, or history, and those of the history of se1ence and the plulosophy of sc1enee, for all or wh1ch see my Study of tlie Hislor.J of &t.mce D ln.stctutts and L!Jranes Every mathematical l1brary JS a natural centre for h1Stoneal research a! well a! for mathematical research proper For m• vcst1gations on modern mathematics necessnate the ava1lab1hty or set:! or mathematical pcr1od1cals, wh1ch are lcgson, or at least of sets of the most important onC$, which arc numerous enough Such 1nvest1gat1ons can generally be conducted 1n every one of the largest hbranes, and 1n the mathematical departments or the main un1�rs1t1cs One European lJbrary mu.st be $tngled out, namely the one founded by the Swedish mathemaucian Gosta Mittag-Leffler and his wife 1n 1916 for the speaal benefit or mathemaue1ans 0£ the Scand1navzan 'ir.o p".f, � Bnifwechsel ;:,wtsckn W Olbas ruul F W Bessel, ed by Adolf

MODERN MATHEMATICIANS

Erman (Leipzig, 1852, 2 vols.).. both.

73

Including short biographies of

BETTI, ENRICO (1 823-g2). Algebraic equations, elliptic functions, mathematical physics, analys is situs. Vito Volterra, Saggi scientijici (Bologna, 1 920), pp. 35, 55 ff. F. Enriques in Enciclopedia italiana, vol. 6, p. 834 (1 930) . Opere matematiche (Milano, I 903-1 3, 2 vols.) . With bibliography and portrait. BOLTZMANN, LUDWIG (1844-1 906) . Mathematical physics, prin­ ciples of mechanics. Festschrift L. B. gewidmet z.um sechtigsten Geburtstage, ed . by Stefan Meyer (Leipzig, 1904: 942 pp., portr., 101 fig., 2 pl.). Anton Lampa, in Biographisches Jahrbuch, vol. I 1 , pp. 96-1 04 (1908) . Johannes Classen, Vorlesungen iiber moderne Naturphilosophen, pp. 1 08-128 (Hamburg, 1 908). Wissenschqftliche Abhandlungen, ed. by Fritz Hasenohrl (Leipzig, 1 909, 3 vols.). Portrait in vol. 3. Populare Schrijten (Leipzig, 1 905: 446 pp. }. BoLYAt DE BoLYA, FARKAS (1 775-1 856), and his son B6LYAI, Jmos (1 802-60) . Non-Euclidean geometry. Franz Schmidt, "Aus dem Leben zweier ungariscber Mathe.. matiker Johann und Wolfgang Bolyai van Bolya," in Grunert's Archiv der Mathematik und Physik, vol. 48, pp. 2 I 7-228 (1 868). In French in the Memoires of the Societe des Sciences Physiques et Naturelles de Bordeaux, vol. 5, pp. 191-205 ( 1 867), and in Janos B6lyai, La science absolue de l'espace (Paris, 1 868), pp. 7--2 I . Libellus post Saeculum quam I. B. . • . Claudiopoli natus est ad celebrandam Memoriam eius lmmortalem, ed . by Ludwig Schlesinger (Cluj, 1 902; 1 70 pp., facs.). Ludwig Schlesinger, " Neue Beitrage zur Bio­ graphic van Wolfgang und Johann Bolyai," in Bibliotheca Mathe­ matica, 3d series, vol. 4, pp. 260-270 (1 903). Ludwig Schlesinger, "Johann Bolyai : Festrede," in Deutsche Mathematiker-Vereini.. gung, Jahresberi"cht, vol. 12, pp. 1 65-194 (1 903). Wolfgang and Johann Bolyai, Geometrische Untersuchungen, ed. by Paul Stackel (Leipzig, 1 9 1 3). 1. Leben und Schriften der beiden B. II. Stucke aus den Schriften der beiden B.

74

THE STUDY OF THE HISTORY OF ifATHEitATICS

Briefwe,hstl �w:schen Corl Fntlnck Gauss und 1Vo!fgang Bo{Jar,

ed by Franz Schmtdt and Paul Stackel (Lc1pz1g, 1899 220 pp, 2 pl , 14 facs ) BoLZANo, BER.NARO (1781-1848) Pr1ne1ple! of analysl.S, theory of funetJons Hugo Bergmann, Das plnlosoJlusclte J-Vtrk B B (Halle a S , 1909 244 pp ) Ruth Stn»k and D J Stro1k, "Cauchy and B 1n Prague," 1n Isis, vol 1 1 , pp 364-:366 (1928) Heinrich Fels, B B , .urn ubtn und strn WnJ: (Lc1pz1g, 1929 1 1 9 pp ) The pubhcat1on of has complete work! (Schrifitn) was begun by the Konaghehe Bohmische Gcsellschaft der W1ssenschaften 1n Prague 1n r930 (l1ts, vol r5, pp 353-355), vol 2, 193c (lnr, vol 19, p 405), vot 3, 193� (1111, vol 19, pp 404-405), vol 4, 1935 Wuseiucheftsleh,, (Lc1pz1g, 19�9-31, 4- vols , ef Ins, vol t8, p 470) Booue, GEOP.OE (1815-64) One of the founders of mathematical logac John Venn, 1n DNB, vol 5, pp 36g-370 (1886) �facfarlane, Ltttures on T,n British J.fathtmatu,ans cf th, Nirutttnth Century, PP 5o-63 Collected Logical JVorkr Vol I not yet published, vot 2, The Laws of Thought (Chaeago, 1916, O\lt of print) BokcnAJU>T, KARL \VII.HELM (181 7-8o) Theory f determinants, the ta functions Note by M Cantor, 1n ADB, vol 47, p 1 1 2 (1 903) Gtsamme/Je JYtrkt, edsted by G Hettner (Berlin, 1888 512 PP , portr ) BRtOSCHt, FkANCEsco (1 824--9:,) Theory of 1nvar1ants, elbpuc and hypc:rclhpt1c functions, theory of surfaces, mcehan1cs Eugenio Beltrami, 14 F B :' 1n Annals di rnattmatl(a, ser1e 2, vol 26, pp 343""-347 {1 897) E Pascal, Poclu cenni su F B (Milano, 1898) Giulio V1vantJ,Jn EnnclDpulm1lol1ana, vol i, p 868 (1930) Prasad (1934), pp 94-11 5 Optrt maltmatzclte (Milano. 1901-09, 5 vols )

MODERN MATHEtfATICIANS 75 CANTOR, GEORG (1845-1 918). Theory of aggregates. Adolf Fraenkel, C( G. C.," in Deutsche Mathematiker Vereini­ gung, Jahresbericht, vol. 39, pp. 189--266 (1 930), with portrait, also in Enc;,clopaedia Judaica, vol. 5, pp. 28-29 (1930). Prasad ( 1934), PP· I 83--2 1 I .

Gesamme//e Abhandlungen: Mit Anmerkungen sowie mil Ergiinzungen aus dem Briifwechse/ Cantor-Dedekind, ed. by Ernst Zermelo (Berlin,

193'2.: 494 pp., portr.) . Includes a biography by Adolf Fraenkel. See also Isis, vol. 3, p. 343.

CARNOT, LAZARE (1 753-1823). Geometry of position (1803), principles of analysis and mechanics. Obituary by Arago; English translation in Biographies of Dis­ tinguished Scientific A1en (1 857), pp. 287-36 1 . Mathieu Noel Rioust, C. (Gand, r B r 7). First printed in Paris, r Br 7, and sup-­ pressed by the police, reprinted in Brussels, then in Ghent, in the same year. Wilhelm Korte, Das Leben L. N. M. Carnots: Mit einem Anhange enthaltend die ungedruckten Poes£en Carnots (Leipzig, 1820: 490 pp.). Hippolyte Carnot (1801-88), Memoires sur C. par son.ftls (Paris, 1 861-63 , 2 vols.). Includes a bibliography; the covers are dated 1869 ! Ne,v edition (Paris, 1893, 2 vols.). Cen-­ tenaire de L. C., notes el documents inedits (Paris, 1 923). Includes a bibliography and portraits. Cf. Isis, vol. 3, p. 1 16 ; vol. 4 , p. 594. Correspondance genlrale (1 792-g5], ed. by f:tienne Charavay (1892- 1907, 4 vols.). This, of course, is chiefly political. CAUCHY, AUGUSTIN (1789-1857). Reorganization of analysis. Claude Alphonse Valson, La vie et /es travaux du baron C. (Paris, 1 868, 2 vols.). F. J. Studnicka, C. o/s Jorma/er Begriindtr der De­ terminantentheorie (Prag, 1876: 40 pp.). Notice by Joseph Ber­ trand, read at the Academic des Sciences, 1898, and reprinted in his E/oges acadlmiques, nouvelle serie, pp. 101-120 (Paris, 1902). Philip E. B. Jourdain, "The Origin of C.'s Conceptions of a Defi­ nite Integral and of the Continuity of a Function," in Isis, vol. r, pp. 661 -703 (19 1 4). Ruth Struik and D . J. Stroik, " C. and Bol-­ , zano in Prague, , in Isis, vol. I t , pp. 364-366 (1928). Maurice d'Ocagne, Hommes el choses de science (Paris, 1 930), pp. I 1 1-125. Prasad (1 933), pp. 68--1 10.

76

THE STUDY OF THE lUSTORY OF }.fA':rHE?.tATIC:S

t �4 vols to date Three OeuDres compl?tes (Paris, 1882more volumes are scheduled to appear)

CAYUY, ARTHUR (1821 -95) Theory or invariants A R Forsyth, 1n DNB, supp t vol I t pp 401-402 (1901) Macfarlane, Bntisk Matkmahaans, pp 64-77 Prasad (1 934), pp 1-33 Colletted Mathtmahcal Papas (Cambr1dge, 1889--98, 14- vols ) lncludmg a biography by A R Forsyth 1n vol 8, and portraits 1n vols 6, 7, 1 1 Bnefwechstl von L Sc!Jafti m:t A Cayley, cd by J H Craf, 1n Naturforschcnde C�cllschaft, Bern, Afitt!itilungen, 1905 CHASLES, MICHEL (1793-188o) Synthetic geometry Notice by Bertrand read at the Acadcm,e des Se1ences m 1892 and reprinted 1n his ilog,1 '1(4dlmt1tJ1.1, nouvel1e ,6r1e, pp 27-58 (1 902) Paul Tannery, Mlm":rts scuntifiques1 vol 6, pp .517-521 (1 926) Gino .Lor1a, "M C e Ja teoria deUe sentone coniehe," 1n O.szr1.r, vol 1, pp 412-.w1 (1916), wath portrait Chasles's paper1 arc preserved 1n the archives of the Acad6m1e

des se1enees

CHE!3VSffEV, P.AF.NUTll L'vov1CH (18�1--94) Theory of number, A Vass1het, "P.afnutu Lvovuch Tchebychef ct son oeuvre se1cnt1fique," 1n Loria's B"llett,n" di bibliograjia t storia detlt satn.te matematick, vol 1, pp 33-45, 81-92, 1 13-139 (Torino, 1898) Also 1n Cerman (Le1pzlg, 1900) 0(tJt'f(.t, cd by A Marlcolf and N Sontn (St -P6tersbourg1 1899-1907, 2 vols ) With a btography by C A Posse and por­ traits CLAusrus, RUDOLF ( 1 822-88) Mathemattcal physics, chiefly thermodynamics, potenttal Eduard Riecke, R C {Cothngcn, 1888 40 pp ) Includes a b1bJtography �orgc Franas Fitzgecald ,n Ro}·al Society of London, Proccedtn.gs, vol 48, pp 1---'Vlll (1890) F Fohc, cc R C , sa , v1e, ses travaux ct Ieur porttc mttaphyinque/ 1n Revue des ques� tions scientifiques, vol 27, pp -419-487 (Bru:icclles, 1890) Noucc by Josiah Wtllard C1bbs, Samttfe Paper� {London, 1906, vol 2, pp 261-267) Max Re1nganu.m, m ADB, vol 55, pp 720-729 (1910)

1'!0DERN MATHEMATICIANS

77

CLEBSCH, ALFRED (1 833-72). Theory of curves and surfaces, geometrical applications of Abelian functions, invariants, representation of one surface upon another, mathematical physics. Obituary by C. Neumann in the Gottingen Gesellschaft der Wissenschaftcn, Naclirichten, 1 872, pp. 550-559. Vorlesungen iiber Geometr£e, ed. by Ferdinand Lindemann (Leip­ zig, 1 876-g 1 , 2 vols.; revised ed., 1 906-32). French translation by Adolphe Benoist (Paris, 1 879-83, 3 vols.}. CLERK-MAXWELL, see MAX\VELL. CLIFFORD, WILLIAM KINGDON ( 1 845-79) . Synthetic geometry, metaphysics, philosophy of science. Leslie Stephen, in DNB, vol. 1 1 , pp. 82-85 (1 887). Macfar­ lane, British A1athtmaticians, pp. 78-g I . Lectures and Essays, ed. by Leslie Stephen and Frederick Pollock (London, 1 879, 2 vols., portr ..). Including a short biographical sketch. Mathtmatical Fragments, being Facsimiles of his Unfinished Papers Relating to the Theory of Graphs (lithographed, London, 1 881). Mathematical Papers, ed. by Robert Tucker (London, 1 882: 728 pp.). ,,Vith introduction by H. J. S. Smith. CouRNOT, ANTOINE AuousnN (1801-77). Probabilities, pioneer application of mathematics to political economy, history and philosophy of mathematics and of science. Souvenirs d'A. Cournot, 176o-1860; precedes d'un introduction par E. P. Bottinelli (Paris, 1 9 1 3 : 302 pp.). Special number of the Revue de metaplzysique et de morale (May, I 905) devoted to him. Gaston Milhaud, " Le h,asard chez Aristote et chez C.," " La raison chez C.," in his Etudes sur la pensle scientifique clzez !es Crees et chez Jes modernes (Paris, I 906), pp. 1 37-1 76 . Fran!;ois Mentre, C. et la renaissance du probabilisme au X[Xe siecle (Paris, I 908: 658 pp.). J. Segond, C. et la psychologie vitalisle (Paris, 1 9 I o). Georges Loiseau, Les doctrines economiques de C. (Paris, 1 9 1 3). E. P. Bottinelli, A. C., metaphysicien de la connaissance (Paris, 1 9 1 3). F. Y. Edgeworth, in Palgrave's Dic­ tionary of Political Economy, vol. I , pp. 445-447 ( 1925). F. Mentre, Pour qu'on lise Cournot (Paris, 1 927). Lilly Hecht, A. C. und L. Walras (Heidelberg, 1 930: 93 pp.). R. Ruyer, L'humanite de

78

THE S'fUDY OF THE IDSTORY OF

l'avenir d'apres C (Paris, 1930)

MATHEMATICS

Fran�ois Bompa1re, Du prinapt dt /1/Jerlt ltonom,qut dan.s ft)lU1/Tt IU C el Jans £tllt tlt ftCQ/t dt Lau­ sanne (Walra.r, Pardo) (Part!, 193r 740 pp ) Rene Roy, " C ct J'ecole mathematique," in &a1t1Jm 2 vols ) With b1bhography and portrait ERtc lvAR (1866-1927) Integral equat1ons Nils Ze1low, "I F ," m Atta MalhLmalzea, vol 54, pp 1-xv1, w1th facsJmilc (1 930) Ugo .Amald1, m .Etuuloped,a 1taliana, vol 16, p 49 (1932) FREtDHOLJf,

FREG.E, GoTTLOB (1848-1925} Mathemattcal log1c, foundauons of ar1thme11c Wilma Papst, G F ob Plu.lo.soph (Berlin. 1932) 51 pp )

81 FRESNEL, AUGUSTIN (1 788-1827). ,,\Tave theory of light, optical surfaces. Notice by Arago, read in 1830; in English in Biographies of Dislinguis!ud Scientific Mtn (1 857), pp. 399-471 . Emile Picard, Clrlmonie du centenaire d� la mor/ de Freme/ (Paris, 1 927: 35 pp.). Louis de Broglie, Recuez'l d'txposls sur /es ondes et corpuscules (Paris, 1 930), pp. 1--26. Oeuvres completes, ed. by Henri de Senarmont, Emile Verdet, and Leonor Fresnel (Paris. 1866-70, 3 vols.). Isis, vol. 4, p. 155; voL 9, p. I 70. ?.fODERN lfATHElfATICIANS

Fucns, LAZARUS (1833-1 902). Linear differential equations. Gesammtlte mathematische 1-Vake, ed. by Richard Fuchs and Lud­ ,vig Schlesinger (Berlin, 1904-09, 3 vols., portr.). , GALOIS, EvARISTE (181 1--32). Groups of substitution. Paul Dupuy, " La vie d'E. G.," in Anna/es de !'Ecole Normale Supbieure, 3 c serie, vol. 13, pp. 197�66 (1896), ,vith portrait. Reprinted in Charles Peguy's Cahiers de la quinzaine, 5c serie, 2e cahier, 104 pp., ,,.,ith portrait (Paris, 1903). Abbreviated in Eng­ lish by G.. Sarton in the Scientific Monthly, vol. 1 3, pp. 363-375 (Ne,v York, 192 1). , Oeuvres mathlmatiques, ed. by Emile Picard (Paris, 1897: 74 pp.). " Manuscrits et papiers inedits de G.," ed. by Jules Tannery, in Bulletin des sdences mathbnaliques, deuxieme serie, vol. 30, pt. 1, pp. 226-2-{.8 (1906); vol. 3 1 , pt. 1 , pp. 2 75-308 ( 1 907). GAUSS, CARL FRIEDRICH (1 777�i855). Wolfgang Sartorius, G. �um Gediichtniss (Leipzig, 1 856). Adolphe Quetelet, Sciences mathlmatiques et physiques tMZ /es Beiges (Bruxelles, 1866), pp. 643-655. F. A. T. Winnecke, G., ein Umriss seines Ltbens und Wirkens (Braunschvveig, 1 877). Theodor Witt­ stein, G. (Hannover, 1877). Ernst Schering, G's Geburtstag nach hundertjiihriger Wiederkehr (Gottingen, 1 877: 40 pp.). Ludwig Hanselmann, K. F. G., zwolj Kapitel aus seirum Ltben (Leipzig, 1 878: 1 10 pp.). Festschrift �ur Feiu da Enthiillung des Gauss- Weber­ _ _ Dmkmals in Giittingen (Leipzig, 1899: 204 pp.). Felix Klem and others, Afataialien ftir eine wissenschaftliche Biographie von_ G. (Leip• _ Mack, zig, 1 91 1 -20, 8 vols.; cf. Isis, vol. 4, p. 154). Heinrich

82

TIIE STUDY OF THE HISTORY OF llATHEllATICS

C F G untl drt Stinen Ftstsdrnft zy. snntm 150 Geburtstait (Braun seh'"cig, 1927 162 pp , 12 pl ) Prasad (1933), pp 1-67 JVe,ke (Leipzig, 1866-1933, 12 \.ols ) Ins, vol 20, p 559 "Gauss' wissensehafthehes Tagcbuch, 1 796-1 814," ed by F Klein, 1n Gott1ngen Gcsellschaft der \Vwenschaften, Festschnjl (Bcrhn, 1901), pp 1-44, ,vtth portr and fac Brrefur:chstl �tsclwt C F G wul H C Sc-ltumackr, ed by C A Peters (Altona, 186o-65, 6 vols ) Bneft �uchtn A ton Hum boldl untl G (Leipzig, 1877 79 pp ) Brteft r,on C F G an B Ni,olat (Karlsruhe, 1877 36 pp ) Brtefu.ec/ise/ �isclun G rmd Bessel (Berbn, 188o 623 pp ) Britju.eclisel .tUtschm C F G untl 1Vo!fgang Bo!)•ar, ed by Franz Schmidt und Paul Stackel (Lc1p-, zig, 1899 224 pp ) Bntju,-echstl r:u:tschn,. O/hns und G , ed by C Sehuhng and I Kramer (Berltn, 1900) Britjueclise/ .tU,1u!zm C F G und Christian Ludwit Gtrlint, cd by Clemens Schaefer (Bcrbn, 1927 840 pp ) Ins, vol It, p 197 GER"AtN, SOPHIE (1776-183c) surfaces, thc()ry of numbers

Elasttc surfaces, curvature ot

Cinq ltttrts dt S G "C H , " 1n R,w, scientifigut, 4 • �ettc, vol 15, pp l�g-13 1 (1901) M Nocthcr, ••c H ," in }.fath4matisclu Annalen, vol 55, pp 337-:385 (1901) E Picard, "L'ocuvre sc1ent1fiquc de C H , " m Ecole Normale Supertcure, A.nna!,t Jfll g/qm1t,u1ue1 ttt nus� ti m /t(Jnra1; (Kasan, 1883-86 t 2 vols ) G((fnfttru:at Rettartlr.tr on tkt Tlttor,1 ef Ptrrallelr, translated by G B Halsted (Ausnn, Tcxa!, 189t • new cd , with portrait and b1bhography, Ch1cago1 1914} LoREN"TZ, H£NDtUK AN'TooN (t853-1928)

itathemaucaf physics,

relativity Rtcuol de lrar,a� offeris par lei aulturt a H A Lormtz (Archir,ts neerlandaises dts scitntt.s txacltJ tt noturelles, 2d series, vol 5, La Haye, 1900) Ethel Truman, Jn Encyclopaedia :Br11ann1ca, 14th ed , vol 14, p 393 (19z9) Collected Papers, vols c (wuh portt ), ;, 8 {The Hague, 1935, , , 34- 35) A61rarullungen ulur t!teo1ttuclu Pnysik, vol t (Le1pz1g, 1 907 490 pp ) No more pubJubcd Vorltsungtn u6tt tlttMet,sclte Pl!Jnk (Lc1pz1g, 19!27--:; 1, 5 voh ) MAcCuLLACH, JAMES

(1809-47) Analyt1e geometry, quadr1cs,

attraction of efhpso1ds, wave thcry Charles Platts, in DNB, vol 35, p 15 (18g3) Colltcted works, ed by John Hcwittjellett and Samuel Haugh­ ton (Dublm1 1880 390 pp )

tfODERN MATHEMATICIANS

91

MAXWELL, JAMES CLERK ( 1831-79). Mathematical phys ics, electromagnetic theory. Lewis Campbell and William Garnett, The Life of J. C. M. {London, 1 882, 678 pp., 3 portr.; ne,v ed., abridged and re­ vised, London, 1884, 436 pp.}. Sir Richard Tetley Glazebrook, J. C. M. and Modern Physics (Ne,v York, 1896 : 224 pp.). A Com­ mtmoration Volume, J. C. M. 1831-1931 {Cambridge, 1 93 1 : 152 pp., portr.) ; cf. Isis, vol. 2 1 , p. 400. sa·entific Papa-s, ed . by Sir William Davidson Niven ( Cambridge, 1 890, 2 vols. ; photographic reprint, 1 927). Includes a brief biographical sketch and a portrait. MERAY, CHARLES (1 835-1 9 1 1 ). Theory of irrationals. Joseph Pionchon, '' Notice sur la vie e t les travaux de C. M.,» in Revue bourguzgnonne, voL 22, pp. 1-1 58 (Dijon, 1 9 1 2). MINKOWSKI, HERMANN ( 1864-1909) . Relativity, mathematical physics. David Hilbert, « H. M. : Ged achtnisrede,» in Gottingen Gesellschaft der Wissenschaften, Nachrichten, Geschaftliche Mit­ teilungen, 1 909, pp. 72-101 . Gesammelte Abhandlungen, ed. by David Hilbert {Leipzig, 1 9 1 1, 2 vols.). MrrrAc-LEFFLER, GosTA {1846-1 927). Theory of functions. Prasad ( 1934), pp. 2 12-244, portr. For the Mittag-Leffler Institute, see above at the end of the Bibliography, Section VIII, D. MoBWS, AUGUST FERDINAND (1 790-1 868). Barycentrical cal­ culus� , Heinrich Gretschel, '' A. F. M., , in Archiv der Mathematik und Physik, vol. 49, Literarischer Bericht clxxxxv, pp. 1-g ( 1 869). M. Cantor, in ADB, vol. 22, pp. 38-43 ( 1885). Gesammelte Wake (Leipzig, 1 885-87, 4 vols.). There is a short biographical sketch by R. Baltzer in vol. I . NEUMANN, FRANZ (1 798-1895). Spherical functions, mathemati­ cal physics, crystallography. Paul Volkmann, F. N. (Leipzig, 1896: 74 pp., portr.). Luise Neumann, F. }l.: Erinnerungsbliitter von seiner Tochter (Tubingen,

92

THE STUDY OF THE HISTORY OF !.fATltE.?.!ATICS

1904 475 pp , portr ) Robert Knott, in ADB, vol 52, pp 68o-

684 (1906) Albert \Vangttzn, F A' und ,an JY,rken als ForscMT und ukrn {Braunschwc1g, 1907 195 pp , portr ) Gesammtllt lf'nkt {Le1p:i:ig, 1go6-28, 3 vols ) V0, ,-ol 2, Samt \'enant to Lo--d Kcl,'1!1.

n.

Scm,ARZ, HnllA..,, AltA....,'l>cs (13.t:,1921) Confonn �"' tatton of mnunal ffll'f.?ccs. 1/t:JJ_-.:..':S:k 4.'tf.t:::,-21�-tr- H A. S .cz set:!'-D 1:t..-v,z!rJ.z::rt gt:r0-.rt {Berka, 1914 -46o P?, po-1r ) Qm.. st.mun Cantheod oi, , 10 D�1!.!s l, pt:f":i.s:� Jt:.."-f--1-., vot 3, PP 23HJ8 (192i) Qse--•!.L r-u.n-.::!ut:1.t A'-"A.':"'2!��- (Be.rl.:l, I8go, !: wk)

run.aces.,

fr/�-��

S'ffllt, lh:..,"llY Jo� STE.PRE:.'\ (1826-83) ThC'O'r) er numbers. J L Gt:w"aer, m Royal Astm:1om.cal &x:iety, -,.fc::!.1.!., .,, ,trar, ,-cl 44 (1�) A \f Oerl.e, JD D,B, ,�l 53, pp 5o--s3 (tSoS) '\t:aew-lane., Bn:U.'i \fc.1.r-.::.�.:, pp g:,-1oG o,r�.1d \14..JL""".mtd Pa:,ets, ed bv J ,\ L GI:--w':if!!!' (Ch!ord, 1894, � ,-ols.) ,,1th bio:stap!uea! uetchcs aod a pottxa.tt.

\\

SrAti,r, .KARL Gr.o:;u:: CR:Ju:sn.A.., ,.o, (1 jg8-1SS;) S)'D�be gcometr), G--'tnl ,,, u:t,t. grom-tnc mterprct:3tl0:l or mu� dcments. Carl Fnednch Philipp ,-on '.fa.rtn1� " C. G C. ,-on Staudt," tn hcb en- \fc!f..-� rr.i P".,nl., vol -t9 Lit�er Bcncht tlxn x,n, pp 1-s (186) '\f C:wtar, m .J..DB, ,-ol 35, p 5� (18g3) \fax 'l\octhe:r,'" Zur Ennnaung an K G C. wn S ;• 1.n Um,-emt1 of Erlan� Frs1.st:kf: L-: Pn--.:::,ttr".!r l::c,,../1! 1:J. b­ trl-r::...1-.J (Erlancc.:, 1901), ,-oJ .f-t pt. �, pp 63-86 Sn:t.\"E.R; JACOB (1 ;g&-186:3) Si-nthettc gro:::netr}" Otto Hcsse, "J S ,..,. m JC"!!1'r.dfo � rtr:!'..e 'Ir!'.:! c.;.r::r.a-..!.e \f4-.L ..... ,....a.'il:.. '\-Ol 6�, pp 1 � (1 663) C. F Get.c:o". Z,1:r Er::.-:;;-:,:,� c:. J S {SchaHhau..:rn, 18;4 3; pp) '\! Canto.., 10 A.DB, -roJ 3::,, pp �03 (18g3} Johann Hcmnch Graf. D,.,- \f.::!�..d'.1:.t'1' J S l'Cn U�rr.ref (Bern, 189; s.;. pp ,, porb." ) Juhus Lan� Jc.c'!> S.n:r.ns u1-rr.s;cbt V"' Bnlv:,18::!r 1853 r..::..h sr-un Pus--L�

JwfODERN MATHEMATICIANS

97

dargestellt (Berlin, 1899: 70 pp., portr.) . Emil Lampe, " Zur Biographie von J. S.," in Bibliothtca Mathtmatica, 3d series, vol. 1, pp. 1 29- 141 (1900). Gesammelte 1-Verke, ed. by Karl WeierstraM (Berlin, 1881-82, 2 vols.). " Brief,vechsel z,vischen J. S. und L. Schlafli" (208 pp.), ed. by J. H. Graf, in Naturforschende Gesellschaft, Bern, Millluilungen (1896) .

STOKES, S1R GEORGE GABRIEL (18 1 9-1 903). Mathematical physics. Sir Joseph Larmor, in DNB, 2d sup., vol. 3, pp. 421-424 ( 1 9 1 2). Macfarlane, Brt'tish Physicists, pp. 94-105. Mathanatical and PJv,s-ical Papa-s (Cambridge, 1880-1 905, 5 vols.) . Includes obituary by Lord Rayleigh in vol. 5 and portraits. Me­ mofr and Scientific Correspondence, ed. by Sir Joseph Larmor (Cam­ bridge, 1907, 2 vols.). Biographical notes and appreciations by various hands, portraits. Snow, LUDVIG (1832-19 1 8). Substitution groups. Helge Bergh Kragemo, Ludvig Sylow (Oslo, 1933, in Nonvegian : 27 pp.). Skrijter (1933), including biography, bibliography, portrait. SYLVESTER, JAMES JoSEPH (18 14-97). Invariants, theory of equa­ tions, theory of numbers, multiple algebra. Fabian Franklin, An Addrtss Commemorative of :J. J. S. (Balti­ more, 1897: 15 pp.). P. A. MacMahon, in Royal Society of London, Proceedings, vol. 63, pp. ix-xxv, with portr. (1898). P. E. Matheson and E. B. Elliott, in DNB, vol. 55, pp. 258-260 (1898). Macfarlane, Brit£sh Matlumalicians, pp. 107-1 2 1 . David S. Blond­ heim, A Brilliant and Eccentric Mathematician (Baltimore, 1 921 : 22 pp.). Emile Picard, Melanges de mathematiques et de physique, pp. 29-34 (1924). R. C. Archibald, " Unpublished Letters ofJ. J. S. and other New Information concerning his Life and Work," in Osiris 1 , pp. 85-154 (1 936). D. E. Smith, in DAB, vol. 18, pp. 256-257 (1936) . Collected J.1athematical Papers, ed. by H. F. Baker (Cambridge, 1904-12, 4 vols.). \,Vith biographical notice and portrait in vol. 4.

g8

TilE STUDY OF TilE BISIORY OF lf.ATii:E�lATICS

TAIT, hTER GUTHIUE (1831-1go1)

�fechamQs, mathematical

physi� quaternions Alcyandcr l\tacfarlane, "P G T .,,. m Bibliot-,..l'ta J.fct,..rrUJhta, 3d sene.s, ,.-ol 4, pp 18.5�00, portt (1 903) Cargill Gtlston Knott, Lzf� mu! Snmtif..c- JVn! '!/ P C T (Cambndge, 1911 388 pp , -t portt ) J H Harntlton D•cbo", in D.A'h, 2d supp , ,.-ol 3, pp 471-474 (1912) �facfarlane. Bntur Ph,Jnmts, pp 38--s,t Sntnhfic Paf-ers (Caml>ndgc. 18g8-1goo, 2 ,.-Ols ) Lrct:rrrs on tcmt Rttrnt Ad.;-tn"..crs 11t Pr:,mal Same� (London. 18;6 350 pp )

Tcm:sYCHEv, sec Cm:s\'!HEV THoMSO�, Sm ,vn11,1c, see KEt.\T. \VEI£RS'I'RAss, K,..Jt.L (181$�7) Theory of funct:ions \\..uhclm Killmg, E JV' (\{u.ruter 1 \\' , 1897 21 pp) Emd Lampe� K JV {Leipzig, 18g7 �4 pp ) Karl '"On Vost, "Karl Theodor \\'°tlhclm ,v ,'' m ?-{wuch Akadenue der\\'lsscn.«:Cbaft.cn, &J..�JMUJ:k, math -ph), Cb� ,-oJ �;, FF -.02-409 (18g1}

H Pomca:re, 0 L'oeuvrc mathematiquc de \\' ,''

in A,ta

J.to!k­

maliea, vol 22, pp 1-18 (18g8), Stnmls tt lmra:,ns (1910) 1 pp �01�12 G �1ltt2g..Leffiet, "\\' ,.. m Arta J.latl-.t1'1'a�, ,-ol �1, pp 7�2 (18g7), "Zur B1ogtaphte ,-an \\' :• ilru:lnn, ,-cl 35, pp 2g-65 (1911), "Die ttstcn 4 Jahre des kbcns voa \\' ," 11/tt!mt, ,-ol 39, pp 1-5; (r923) Emtle P,c.t.td. ,\lll�rr it malJ.bnah�t ti /h,Jtt(['.it (19:2-1), pp 23-28 Prasad (1933), PP 22o--2go J.fatkmatmf..r 1''trit (Bcrim, 1'9-l-1927, 7 vots ) Portrait m '\"QI 3

MODERN l!ATHEMATICIANS

99

It w·ould be interesting to name these 1 1 8 mathematicians in chronological order. A rough ,vay of doing this is to name them in order of their death years ; their main activities may not have occurred exactly in the same order, but at any rate the death year is a superior limit of the activity of each person.

1 822 1 823 1 825 1 827 1 829 1830 1831 1832 1 833 1840 1 841 1 842

1 846 1 847

1 848 1851 1 852 1 855 1 856

1860 1863 1 864 1 865 -1866

Ruffini L. Carnot Pfaff Fresnel Laplace Abel Fourier Germain Galois Legendre Olbers Poisson Green Ivory Bessel Gopel MacCullagh Balzano Jacobi Eisenstein Gauss F. B6lyai Lobachevskii Cauchy Peacock Dirichlet Poinsot J. B6lyai Steiner Boole Encke Hamilton Riemann

1 867 Poncelet 1868

1879 1 880

1882 1883 1884 1 886

1888 1889 1891

Staudt Mobius Plucker Lame Babbage De Morgan Clebsch Rankine Hesse Cournot Grassmann Leverrier Clifford Maxwell Bellavitis Chasles Borchardt B. Peirce Jevons Liouville H. J. S. Smith Aronhold Laguerre Oppolzer Saint-Venant Kirchhoff Rosenhain Clausius Halphen Kovalevskaia Kronecker

1 00

THE STIJDY OF nlE l-llS'lORY OF P.CATHE\lATI�

1892 Adams 1893 1894 1895 1897

Atry Betti Kummer Cheb}"Shev Helmholtz Hertz Caylcy Neumann Brioschi Syl,cstcr \Veicrstrass

1899 Lte 1goo Beltrami Bertrand 1901 Hermite 1902

1 903

1go6 1907 1909 1911

Tatt Fuchs Cremona Gibbs Stokes Boltzmann KclV1n 1{1nlcow-sk1 Newcomb }.{&ay

1912 Danvm

1913 1914 1916 1917

t918

1919

1920

Fiedler Gordan Lemoine Poincarc Ball C.Outurat Hill Dedek1nd Darbou.'C G Det.unbn::. J B J • I 2. 1 3 De 1'{organ. A.. 52, 79 De l,{organ. S E • 79 Dcsargucs. 10 Descartes. IO• 45• 48 Dack.sou, J H H • g8 Did.Jon, L. &.. ,4 7 Duhon. I • 94 Diopha.ntos. 47 Dancbkt. G L... 50 79• 86 Doonan. F G • th

.«

Drach, J . 94

•• 93 Droru:e, A Duhamel, �f C • 53 J>uhffll, P • 8:i Dupuy, P , 8 , Dvek, \V v • 5'7

J

Edtcworth, F Y • 77, 79 Eanst�111, A , r 6, 6� 912 Eueohart, L. � , 78 �nsteio. G • 19 the de Buumont, L., 88 I.!bott, £. B • 97 E.n:llnanucl. C., 8g Esn�oc:lcs. 4!t Eo.cke J F • 7�. 8o Enatrorn, G , 42. 57• 6o. 6t, 68 Engel F • 48. 83. Engelmann. R.., 72 Eunqucs. F • S5, 73 Erman., A., 73 Em.rt. W • 93 Euchd. 9. I 3• aru. 1839}, vol 1, p .f77

SCIENTIFIC AND 1-llSTORICAL PREPARATION

7

longer but n1orc systematic; i t must be carried on in a cer­ tain order. One cannot study analysis before algebra, nor phrsiology before chen1istry and ph)sics. On the other hand, it is possible, though pcrhap$ unadvisablc, to study history in almost any order, and the n1ajority of scholars ha,·c obtained their historical kno\\·lcdgc in the most hap­ hazard ,,�ay. One may be an expert in American history and kno\v nothing ,,·hatevcr of the Sumerians or the Hit­ tites. Some time ago I \\'as obliged to examine the ,vork of a Transyh·anian chemist vd10 flourished a t the end of the eighteenth century. To appreciate his ,,·ork it ,,·as neces­ sary to consider on the one hand the historical milieu, and on the other the contemporary chemical traditions. I kncv; nothing of the historv of Transylvania at that time, but it did not eoc;t me mueh trouble to obtain the information ,,·hieh I needed , or, \,;th the aid of my kno\,•lcdge of com­ parable conditions clsc\\·hcre, to understand it. Iiappily I ,,·as ,,·ell acquainted ,,ith chemistry and the chemical kno\,·lcdge of that period in '"'cstcrn Europe and ,vith the so-called ' ehcmieal revolution, , subjects in \\fhich it \\'Ould have been impo$Siblc for me to remedy my ignorance so readily. I should have been obliged to study chemistry and the complete history of chemistry do\,71 to that time ! Ho\•,cver, the most fundamental difference bcn,·een his­ torical k.no,vlcdgc and scientific kno\vlcdge is revealed by the ,,·ay they gro,v. 1-Iistorieal kno\,·ledgc gro\VS slov;ly and precariously; precariously, because of the constant recur­ rence of discredited errors; slo\vly, because of increasing difficulty in obtaining nc"' material. Though our kno,vl-­ cdgc tends tO\\l'ard completeness, it is asymptotic, and never reaches the goal. For example, consider our kno,vledgc of ancient Greece. It continues to improve, to be sure, but ,vith smaller and smaller increments of truth. It ,vill not be more difficult, it ,,,ill probably be simpler, to study it a fe\v centuries hence than no,v. On the contrary, any branch of science may be completely revolutionized at any time by a

8

TIIE STUDY OF TIU: JJISTOR\' OF' SCIENCE

discovery necessitating a radically ne\V approach to the sub• jeet. Chemistry to-day is essentially different from chem• istry in the eighteenth century. The fundamental notions are different, the methods arc different, the scope is in• credibly larger, and the contents infinitely ntore varied. We may safely assume that the chemistry of the twenty­ fifth century will be as unlike that of the present as that, in turn, is unlike that of the fifteenth century. On the other hand, in the twenty-fifth century or in the thirtieth it ,vill take about the same pains and time a! to-day to study Latin grammar, Greek literature, or the history of the eighteenth century. The literary subjects, we may say, tend to be closed subjects, whose expansion after a certain point is so slow ns to be imperceptible. In strong contrast ,vith them, the growth of scientific subjects is unpredictable, lu:icuriant, and sometimes explosive in its intensity nnd destructiveness. The scientists then, ean never rcla:ic in his efforts nnd enjoy h1mself, like a genial and sensible grammarian, but must be prepared to learn new things every day, nnd, what is worse, unlearn others with which he has grown intimate, and change the tenets of a life time on fundnmental points. No wonder that such a harassed individual is generally un.. wllling to contemplate the past, or, should he have any vclleities to do so, unable to do it ,veil. He innocently be• licves, it may be, that he knows ho,v to do lt. Historical work, he seems to thtnk, consists in taking a fc\V old books and copying from them this and that. He may be ,vell trained and fastidious in his own exacting technique, yet not realize that the technique of establishing the truth, or the maximum probabdity, of past event! has its own compli­ cated rules and methods Historical work, as he conceives it in his candor, is cxcccding)y easy; almost all that is needed, he think!, is to kno,v how to read and write, and he despises it accordingly. He docs not realize that he is merely despis­ ing hi5 perverted image of it. The historian whom he scorns and ridicules is nobody f>ut himself.

NATURAL VS. HISTORICAL LAWS

***

9

The difference between the historical and the scientific points of view has been amusingly illustrated by Henri Poincare as follows : 1 Carlyle has written somewhere something like this: " Nothing but facts are of importance. John Lackland passed by here. Here is something that is admirable. Here is a reality for which I would give all the theories in the ,vorld." Carlyle was a countryman of Bacon • • �, but Bacon would not have said that. That is the language of the historian. The physicist would say rather: ''John Lackland passed by here; I don, t care, for he will not pass this ,vay again. ,,

Physical sciences deal with the ' laws of nature, ' with the repetition of facts under given circumstances, not only in the past but also in the future; history deals with isolated facts of the past, facts which cannot be repeated and hence can­ not be thoroughly verified. At first view it seems impossible to bridge that abyss. And yet the difference is perhaps quantitative rather than qualitative. For, on the one hand, historical facts are more or less repeated. When a tyrannical rule is introduced into a country, one or more of certain well known series of events are bound to happen in con­ sequence of it. The repetition is not complete and detailed, as in the case of physical or chemical facts, yet there is a repetition of patterns which deserves to be taken into ac­ count. The trouble with the Carlyle-Poincare example is that it is too particular; it would be too particular even for the physicist. John Lackland will never come again; but there may be others like him, and patterns of a definite kind will entail a succession of other definite patterns. On the other hand, on account of the infinite complexity of causes and of the dissipation of energy, physical events never re­ peat themselves exactly. The planets do not follow twice the same trajectories. The old saying of Heraclitus is truer than ever: Ila.vra ptl, 1

La Science el l'Hypothese, p. 168.

IO

THE STUl)Y OF THE IIISTORY OF SCIENCE

everything flows The physical world is less regular and the social world more regular than one generally admits, and thus the two are not so Widely apart as we imagine As opposed to the more exact mathematical sciences, the lustor1ca] 'sciences' seem to usurp their name, but it is not fair to compare the extremes of a scnes A compar1Son with the natural sciences is more adequate The lustonan of science, to return to him, is a collector of .scientific ideas in the same way that the cntomolog1St is a collector of insects, , the 'collect1on in both cases being only the first step along the road to knowledge The point is that both \rul use sum Jar methods to make sure that the items of their collections arc as unequivocally and completely determined as possible, and when the facts are duly established they must needs use similar methods to draw their conclus1ons and to build up progress1ve1y a system of knowledge The companson of the lustorian with th� naturalist nught be pursued retrospec uvely at different stages of their growth There was a time of innoeence when their methods were equally immature and inconclusive, both have learned gradual1y, very gradu allyJ to make the most of the available evtdence, the most but not more, and even to measure to some extent their approx1mat1on to the truth Under the healthful influence of geological and prdustor1c r�search, some h1stonans have now become full brothers to natural1Sts The prelustor1ans and other archacolog1Sts have btult a sohd bridge between history and sacnce, and ,ve, historians of sctence, arc now procecdmg to build another one, even more substant1al, and thus to help span the chasm which IS cutting our culture asunder and threatening to destroy it The scientific spirit is as much unproved and purified by the adnuxturc of hlstor1cal considerations as IS humanism itself by the introduction of scientific methods

.. * .

The main point to emphasize - and 1f this 1s properly understood all the rest follows without difficulty - is that

ACCURACY

ll

accurac;, is as Jundamental in the historical field as in the scientific one, and that it has the same meaning in both fields. Experienced historians may find it strange that I should trouble to ex­ plain what to them is obvious, but it is necessary to do so in order to eradicate the prejudices of scientists against us, and enable them to come and meet the historians of science half ,vay, instead of thro,ving spokes into our wheels. Let us suppose that a physicist has to measure the length or distance AB. He may state that AB is 3 m. long, or 300 cm., or 3000 mm. These are three different statements, for they imply different degrees of accuracy: in the last case, for example, that the length is correct ,vi.thin a millimetre (2999 mm. < AB < 3001 mm.) . The degree of accuracy obtainable or desirable varies vvith the circumstances, but one must be accurate ,vithin the limits which are appro .. priate and ,vhich are suggested by the choice of units or the number of decimals. The situation is exactly the same ,vi.th regard to dates. If you state that a certain event happened on October 20, 1 495 (Gregorian), you must be reasonably sure that it is the 20th, not the I 9th or the 2 1st. A scientist, however meticu­ lous he may be in his own field, will shrug his shoulders and grumble: " What do I care vvhether it is the 20th or the 25th� . . . " Very ,vell. It may indeed be of no importance, but then why state the day? Why not say " October, 1495," or " 1495," or " toward the end of the fifteenth century"? The last statement might be the best one. To say " Oct. 20, 1495," ,-vhen you are not sure of the day, is nothing but a lie, just as if you said that AB was 3000 mm. long after a perfunctory measurement with a draper's yardstick. To affect a higher degree of precision than one can vouch for is just as reprehensible in the one case as in the other. Dates can be determined with reference to a definite calendar by means of authentic documents, or by means of coincidences with other events duly dated, or by a more complicated system of deductions. In every case they are determined

I :?

THE SIUD\' OF nt£ liiSIORY OF' SCIB?-:CE

,,ithin certain limits, and \\e arc bound to state them as cu11c:ctlr as is possible 1\ithin these limits, the limits them­ selves being indicated by the choice of units or more c.�hcitly. I have selected these tl\-O cx:amplcs, lengths and dates, because they arc C."tcecdingly simple and }Ct fundamental Physical me.asurcmcnts arc grn.aall)• reduced to measure­ ments of length (linear or angular; on the object itself or on our instruments); as to the dates:, thc)y arc to the historian \\hat spherical coordmatcs arc to the geographer or the astronomer. Nothing in either case could be more funda­ mental. ?,.tore technical examples could not pro,idc better illustrations of the argument, for these hit the root of the matter: precision has the s:,me meaning in histo11· as in science, and cntaiJs the same obligations. It is true the errors of historians may rcx:nain unnoticed; they cannot be found out as easily as the scientific ones (some of the latter ,,-ould be almost automatically detected sooner or later) ; but this does not decease the historian's ttSponsibihty; 1t ma-cases it. In ph) sical measurements, the re.adings of our instru• ments need oor1ect1on. If measure the ICJlgth of a bar- of must talc the temperature into aca>unt, and re-­ metal, duce our sundr)• .rc3cbngs to the same temperature. If measure the coordmates of stars» ,,c must talc into account the abcn-ation of light, the nutation ()f the earth's a- t914, cf lsu, vol g, p 368) The author 1$ a .mathematletan and the head of the 1ruututc for the history of sc1c:nce attached to the Un1vcrs1ty of Rome Henn Po1ncar� (1854-191�), La termte �, rnypt,thltt (Paris, 1908), La valeur dt la 1eunce (1909), Seunet ti mithixk {t 909) Thc�c books have been often reprinted and translated into .many languages ihc English trans• Jatton of them by George Bruce Halsted wnh a ,pec,al preface by the author and an 1ntroduct1on by Josiah Royce 1s ava1lable 1n a i1nglc vol wnc (New York, 1913, many times rcpnn�) Frederic William Westaway, &smtifie Method Its Phtlosophieal Basu attd ib Modes ef Appltcat,on (London, i912, later editions 1919, 1924, 1931 , cf Isu, voJ 41 pp , rg-r:1�) On a much lower Jeve1 than the pceccd1ng books, and thus more accessible to the average student The author 1S an inspector of the Enghsh schools Arthur Davtd Ritchie, S-1900 (Cambrtdge. 1867-19'.l5. 19 vols ) Sub1cct tnclex: (t9o8-,4, 4 vols ) Thts WC>rk ts so tmportant that we must paust a moment to describe lt Its cC>mp1latton was first ,uggcstcd at the Glasgow meeting or the B A A S tn 1855 by Joseph Henry (1797-1878). secr�tary or the Smtth�oruan lnstitubon. and the plan was drawl\ up 1n After many yean of preparat,on and considerable expenditure, the first volume appeared in 186,, and the publicauon (Onttnucd a� follWI Ftrst series Vols 1-va, c.:atalogwng the pa.pen of 180-63, t867-77 Second stnes Vols vu vui. htcraturc 0£ 1864-73, 1877-79 Third sc.r1d VoJs ix ,n, Ltt'J'atlll'e of 1874--83, 1891-96 Vol xu Supplement to the prevJous volumes, J 91>.2 Fourth se.ne.,_ Vol:, XU!-x1x, .hteratutt of 1884-:900. 19J4-25 To g1ve an idea of the ssu of thJ.S catalogue u wJl suffice to remark that the papers catalogued 1n the fow-th senes alone, for the penod 18841900, number 384,478. b)' 68.577 authors 'I'he compilation o[ a sttbJcct andeic, without wh,ch the work losc.s much of ?ts value, was already contcmplat«I 1n the first p}an (1857) It Wa.5 finally decided to arrange 1t 111 accordance wuh the lnterMtra11al Cata/ggu1 of Seterrttfic Ltterature (see below) 'I1us me.ant that n would include sc:vcn­ \ttn ·�·.:1u·�, er,� ft:rr cath ,;;J.. the � ��\ent�s rtcogn1zcd 1n that catalogue The fu-st volume, Pure l\,fathemaucs. appeared in 1908, the

•es,

BIBLIOGRAPHY

59

second, Mechanics, in 1 909, the third, Physics, in two instalments, Gen­ eralities, Heat, Light, Sound in r 9 1 2, Electricity and Magnetism in 1 91 4. The publication seems to have been finally discontinued, which is a great pity. Whatever the fate of the International Catalogue may be, there is no justification for leaving the Royal Society Catalogue essentially in­ complete, and thus nullifying a large part of the past labor and expendi­ ture. International Catalogue of Sdmtific Littraturt. Published for the Inter­ national Council by the Royal Society of London. This is an outgrowth of the Royal Society Catalogue, as it ,vas felt that the scientific literature of our century was too extensive to be dealt with by a single scientific society. Its organization was arranged at the initia­ tive of the Royal Society, by an international conference which met in London in 1 896, then again in 1 898, in 1 900, etc. It was decided to divide science into seventeen branches: A- Mathematics. B. Mechanics. C. Physics. D. Chemistry. E. Astronomy. F. Meteorology (incl. Terrestrial magnetism). G. Mineralogy (incl. Petrology and Crystallography). H. Geology. J. Geography (mathematical and physical). K. Palaeontology. L. General biology. M. Botany. N. Zoology. 0. Human anatomy. P. Physical anthropology. Q. Physiology (incl. experimental Psychology, Pharmacology, and experimenta1 Pathology). R. Bacteriology. A large number of volumes were actually published from 1902 to t9t6, but the gigantic undertaking was a victim of the World War and of the national selfishness and loss of idealism ,vhich the War induced. The volumes published cover the scientific literature for the period from 1 901 to about 1 913.1

The publication includes 254 octavo volumes, varying in thickness from half an inch to two inches, and the original price was about £260. The stock has been sold to William Dawson and Sons, London, who offer a complete set for the price of £60 unbound, or £100 bound (November, 1935). 1

6o

THE STUDY OF Tim HlSTO.R.Y OF SCI£NCE C. Umtm Lm "./Snnl,fa Pmtxluab

The two most unportant latsof that bnd arc t The U,uc,, Last of SmtJ, m Lr6rtsn1s ef tu Umttt! Su,11s tmtl Cmuz/a (Nr:w York, 19�7. one ,-e,y 1.a.rgc qumo volume of •588 pp} Rcgutenng some 70,000 J