Breakthroughs in Physics

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SIGNET SCIENCE LIBRARY

T2673/75c

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J v

PETER WOLFF

„

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Digitized by the Internet Archive in

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a

THE PROGRESS OF SCIENCE from the third century

b.c. to

the twentieth century a.d.

Seven great scientists explain in their own words the work that has made their names immortal. Among others: Archimedes speaks on statics and hydrostatics (his theory of buoyancy has never been changed or improved); Newton treats the principles of physics in a section from the "Principia; Helmholtz discusses the first law of thermodynamics; and Einstein is represented by his

famous

The

treatise

on

relativity.

editor, Peter Wolff, has

chosen these are so diverse in time and place, because the endeavors of

particular scientists,

who

—

each man constitute a "breakthrough" major advance in the field of natural science. The goal of this unique volume is to acquaint the general reader with the thoughts, words, and works of these great men; the

commentaries explain how and why they were responsible for the continuing progress of science through the ages. editor's

"Breakthroughs

in Physics is

an interesting and

—

useful addition to scientific literature a worthy successor to the first work in this series."

—

Benjamin Bold, Board of Education of the City of New York

Other SIGNET SCIENCE

LIBRARY Books

Breakthroughs in Mathematics, Peter Wolff, editor These excerpts from the work of nine men who spurred mathematical revolutions represent the entire history of mathematics. With commentary by the editor.

(#T2389—75(J) Frontiers of Astronomy by Fred Hoyle An assessment of the remarkable increase in our knowledge of the universe. (#T2309—750) Relativity for the Layman by James A. Coleman An account of the history, theory and proofs of relativity, the basis of all atomic science. (#P2049 600)

—

The Universe and

Dr. Einstein (revised) by Lincoln Barnett A clear analysis of time-space-motion concepts and the structure of atoms. Foreword by Albert Einstein.

(#P2517—600) To Our Readers If your dealer does not have the Signet and Mentor books you want, you may order them by :

mail enclosing the list price plus 100 a copy to cover mailing. If you would like our free catalog, please request it by postcard. The New American Library of World Literature, Inc., P. 0. Box 2310, Grand Central Station, New York, N. Y., 10017.

Breakthroughs in

Physics

PETER WOLFF

(SJGNET)

A

SIGNET SCIENCE LIBRARY BOOK Published by

THE NEW AMERICAN LIBRARY, New Tor!t and Toronto THE NEW ENGLISH LIBRARY LIMITED, London

Copyright

© 1965 by Peter Wolff

All rights reserved

The author wishes to thank the various publishers and individuals who permitted the selections in this book to be reprinted. Copyright notices and credits are given on the first page of each selection. The author also wishes to thank his wife, without whose encouragement and tolerance this book could not have been written. First Printing, May, 1965

SIGNET TRADEMARK REG. U.S. PAT. OFF. AND FOREIGN COUNTRIES REGISTERED TRADEMARK MAROA REGISTRADA HEOHO EN OHIOAGO, U.S.A.

SIGNET SCIENCE LIBRARY BOOKS are published in the United States by The New American Library of World Literature, Inc., 1301

Avenue of the Americas,

New York, New York 10019, in Canada by The New American Library of Canada Limited, 156 Front Street West, Toronto 1, Ontario, United Kingdom by The New English Library Limited, Barnard's Inn, Holborn, London, E.G. 1, England in the

PRINTED IN THE UNITED STATES OF AMERICA

To

My

Children: Peter, Ted, and

Tom

Contents

PREFACE

CHAPTER ONE

—Simple

Archimedes

Machines

CHAPTER TWO

—The Theory of Buoyancy

Archimedes

CHAPTER THREE Galileo

—

Discoveries in the Heavens

CHAPTER FOUR Galileo

—Free

Fall

and Gravity

CHAPTER FIVE Pascal

—

Investigating the

Vacuum

CHAPTER SIX Newton

—Mathematical

Principles of Physics

CHAPTER SEVEN Huygens

—The Wave Theory of Light

CHAPTER EIGHT Von Helmholtz

—The Conservation of Energy

CHAPTER NINE

—

Einstein

Relativity

EPILOGUE SUGGESTIONS FOR FURTHER READING INDEX

PREFACE

On

the following pages

we

present to the reader nine selection describes and explains a "breakthrough," i.e., a major advance, in the field of physics. The lives of these seven authors span a period of more than 2,000 years from the third century b.c. to the twentieth century a.d. The fields in which these seven men worked are as diverse as the times and the places in which they lived. Archimedes (third century B.C.) lived and worked in Syracuse, Sicily; he is represented by works on statics and on hydrostatics. Galileo Galilei (early seventeenth century) did his work in northern Italy, in Pisa, Padua, Venice, and Florence; we have included selections by him dealing with discoveries in astronomy and in dynamics. Blaise Pascal, French philosopher, mathematician, and scientist (middle of the seventeenth century), lived in Paris; he also spent considerable time at Port Royal, the center of the Jansenist movement, with which Pascal sympathized. Here, of course, we are not concerned with Pascal's fascinating theologicalphilosophical writings, but present a treatise dealing with hydrostatics. Sir Isaac Newton was a Fellow of the English Royal Society; Christiaan Huygens was a member of the

works by seven

scientists.

Each

—

French Royal Society; Newton's and Huygens' works were published at the end of the seventeenth and the beginning of the eighteenth century. The former, of course, was English and a professor of mathematics at Trinity College, Cambridge, while the latter was Dutch and lived at The Hague, although he traveled extensively throughout Europe. Newton is repix

x

Preface

resented by a section of his Principia, treating of the principles of physics; the selection from Huygens comes from his Treatise on Light. Hermann Helmholtz, German natural philosopher of the nineteenth century, wrote on many parts of physics; we have chosen a treatise dealing with the conservation of energy and the first law of thermodynamics to represent him. The last selection in the book is by Albert

most famous physicist of the twentieth century at the time of his death; we reprint the first part of his popular treatise on relativity. Perhaps in a work of this kind, the choice of a particular selection would require an explanation and a justification. Why have we chosen these seven authors and these nine works? We do not expect much difficulty with the justification; hardly anyone is likely to doubt that these nine works are contributions of the highest importance. There might be room for disagreement if we claimed that these were the nine most important short pieces in physics. We make no such claim, of course. All we maintain is that these nine pieces are very important (certainly they are among the two dozen or so most important contributions to physics), that they are readable by the layman, and that anyone, layman or professional physicist, will benefit from reading them. Obviously, we make no claim that we have exhausted all the writings recording "breakthroughs" in physics. There are others, and some readers may regret the omission of some of them. Arbitrary considerations, such as limitations of space, had their say in making the selections; so did such considerations as whether an author had written a short and popular exposition of his work and whether this work had been translated into English. For example, Max Planck, the father of the Quantum Theory, certainly deserved to be represented by a selection. However, no work by Planck seemed quite suitable. In order not to slight so important a breakthrough as that represented by the Quantum Theory, we have added a brief epilogue explaining some of its aspects and consequences. Finally, we must admit that personal prejudice and preference played their part. Examination of this book will no doubt show that we have favorites among the authors; we admit to special fondness for two of the seven (Archimedes and Newton), while admiring all of them. Our main purpose with this book is to acquaint as many readers as possible with the thoughts and words of these great scientists. In this we are motivated by the belief that Einstein, easily the

Preface

xi

the best scientists are also the best teachers of science. Some readers, however, may find the subject matter in need of explication and amplification. For them, we have appended a commentary to each chapter. In these commentaries, we try to clear up points that may be a little obscure, to ease the reader over some of the mathematical shoals on which he might founder, and to make clear the relevance of what the author is saying to everyday experiences.

—

We hope that these commentaries will be helpful but they are not intended to take the place of the authors' original words. Certainly we would encourage any reader who prefers to read only the original texts. The greatest benefit and pleasure these selections can give will come to the reader who struggles with an author's own words and thoughts. Though giving a helping hand to the reader in our commentaries, we have tried not to deprive him of this joy of discovery and hope that at least in part each sense of achievement. reader will share with us the excitement and -thrill of following a great scientist as he explains the nature and importance

We

of his discovery.

Peter Wolff

CHAPTER ONE Archimedes

:

—Simple Machines

PART

I

One of the most extraordinary features of the famous buildings of antiquity the pyramids, the Parthenon, the Coli-

—

—

this add to their an intriguing question: how could these gigantic monuments have been built? Men and animals were the only power resources available to the build-

seum

is

their sheer size.

impressiveness, but

it

Not only does

also raises

ers for the execution of their plans. Their rather insignificant strength in relation to the enormity of the projects, was

multiplied by a number of devices known as "simple machines": levers, pulleys, inclined planes, screws, and many variations and combinations of these. These machines multiply a small force so that it can exert a powerful effect. They are called "simple machines" because they do not generate a force from some outside source but merely transform the

way in which an existing force acts. (A windmill, a water wheel, and a steam engine are not simple machines because they derive power from an outside source, such as wind, water, or steam.) The basic simple machine is the lever. In the following pages, Archimedes develops the fundamental theory of the lever and of all other simple machines. Although levers must have been used even before Archimedes' time, he was the first scientist to analyze the effects of the lever and derive these effects from basic principles. 13

Archimedes:

On

the Equilibrium of Planes,

OR,

w

The Centres

of Gravity of Planes*

BOOK

I

I

postulate the following?

1. Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater dis-

tance* 2.

If,

when

weights at certain distances are in equilibrium,

something be added to one of the weights, they are not in equilibrium but incline towards that weight to which the addition

was made.

3. Similarly, if anything be taken away from one of the weights, they are not in equilibrium but incline towards the

weight from which nothing was taken. 4.

When

plied to

equal and similar plane figures coincide if apone another, their centres of gravity similarly coin-

cide 5e In figures which are unequal but similar the centres of gravity will be similarly situated. By points similarly situated in relation to similar figures, I mean points such that, if straight lines be drawn from them to the equal angles, they make equal angles with the corresponding sides. 6. If magnitudes at certain distances be in equilibrium, (other) magnitudes equal to them will also be in equilibrium

at the

same

distances.

* From The Works of Archimedes, ed. in modern notation with introductory chapters by T. L. Heath. Cambridge: Cambridge University Press, 1897, pp. 189-194.

14

Simple Machines 7.

the

15

In any figure whose perimeter is concave in (one and) same direction the centre of gravity must be within the

figure.

Proposition

1.

Weights which balance at equal distances are equal. they are unequal, take away from the greater the between the two. The remainders will then not balance [Post. 3]; which is absurd. Therefore the weights cannot be unequal. For,

if

difference

Proposition

2.

Unequal weights

at equal distances will not balance but towards the greater weight.

will incline

For take away from the greater the two.

The equal remainders

Hence,

if

balance

difference

between the

will therefore balance [Post.

1],

we add

the difference again, the weights will not incline towards the greater [Post. 2],

but

Proposition Unequal weights

will

3.

balance at unequal distances,

the

greater weight being at the lesser distance.

B

Let A,

be two unequal weights (of which

greater) balancing about

A

C

at distances

C

AC,

BC

A

is

the

respectively.

B

Then shall AC be less than BC. For, if not, take away from A the weight (A - B). The remainders will then incline towards

B

AC = CB, AC > CB,

But this is impossible, for (1) if 3]. equal remainders will balance, or (2) if they will incline towards A at the greater distance [Post.

the

[Post. 1].

Hence

AC
B.

CB.

if

the weights balance, and

AC


ss

is the weight loss sustained by an amount of s in air. weighing Next, the weight loss sustained by the actual crown may is equal to the weight of water disbe called X. Thus placed by the crown. Consequently,

where

W

silver

X

X

zg + za This equals

W

W Then

Wg X X

g

+

W Xx s

WgX Xg+W XX WgX Xg~ WgXX S

Wg

=3

s

XX

s

=3 =3

=

(Xg-X)

w

(W,+ fF.) xxwg + xxw8 W s X x - Ws X xa W (X - Xs ) s

x-x

ca

g

s

Xg-X

W

W

Thus the ratio of to can be found, for the s g three quantities X, can each be found exand s g perimentally. All we need to do to find them is to weigh the jeweler's crown in air and in water, then weigh a crown

X

,

X

made

of pure gold in air and water, and finally weigh a silver in air and water. This will give us the three weight losses. To give a numerical example: Suppose the crown supplied by the jeweler weighs 100 oz. Submerge it in water, and let us suppose that the weight loss is 40 oz; that is, the crown weighs 60 oz in water. Therefore, 40. Next, let us take 100 oz of pure gold, submerge them in water, and weigh them again. The volume of 100 oz of pure gold will be smaller than the volume of the crown, and so the weight loss will be smaller. Let us say that the

crown made of pure

X=

weight loss is 10 oz. Again, let us take in water, and weigh of pure silver will be

Hence, X g = 10. 100 oz of pure silver, submerge them them again. The volume of 100 oz greater than the volume of the crown,

The Theory of Buoyancy

51

and so the weight loss will be greater. Let us say that the weight loss is 50 oz. Hence, 50. Then a

X =

W w

40- 50 10-40

9 8

ca

-10 -30

E=

1 —

3

In our example, therefore, the crown contains three times much silver as gold, or 75 oz of silver and 25 oz of gold. It may be presumed that the jeweler was cheating, since he obviously made a crown that was mostly silver. This could have been suspected from the initial figures. Since the finished crown lost 40 oz, while a pure gold crown would have lost just 10 oz, the finished crown obviously was much bigger, in volume, than the pure gold one. This would be an indication that the jeweler had to use much silver in order to make the crown come up to the required as

*

weight.

CHAPTER THREE Galileo:

—Discoveries PART

in the

Heavens

I

always due to great flashes of inof genius. Progress is generally due to hard and repetitious work. Of course, what would seem like drudgery to the uninquisitive would be intensely exciting to the person involved in the study. Thus, to look Scientific progress is not

sight that

come

to

a

man

may seem dull and laboriwas obviously a labor of love. For the first time in history, a man was able to see more of the sky and more in the sky than his ancestors. The at the

same

stars night after night

ous to many.

To

Galileo

—

it

—

invention of the telescope put Galileo in the unique position of being able to see stars that none of the great astronomers of antiquity or the Middle Ages Hipparchus, Ptolemy, Copernicus had even suspected were there. Galileo's patient efforts were well rewarded. As he tells us in the following pages, he discovered not just some new stars, but four very special bodies: the moons of Jupiter. Although seven other such moons have been discovered since, the original four honor their discoverer by being called "Galilean satellites."

—

—

Breakthroughs in Physics

52

Galileo Galilei:

The Starry Messenger*

Revealing great, unusual, and remarkable spectacles, open" ing these to the consideration of every

man, and

especially of

and astronomers; as Observed by Galileo Gentleman of Florence Professor of Mathematics in University of Padua, With the Aid of a Spyglass

philosophers Galilei

the

lately invented

In the surface of the T^ebulae,

Moon,

and above

all

in

by him,

in innumerable Fixed Stars, in

Four Planets

swiftly revolving

about Jupiter at differing distances and periods, and tynown to

no one before the Author recently perceived them and

decided that they should be

named

the

Medicean Stars

To the Most Serene Cosimo

II de'

Medici

Fourth Grand Duke of Tuscany Surely a distinguished public service has been rendered by those who have protected from envy the noble achievements of men who have excelled in virtue, and have thus preserved from oblivion and neglect those names which deserve immortality. In this way images sculptured in marble or cast in bronze have been handed down to posterity; to this we owe our statues, both pedestrian and equestrian; thus have we those columns and pyramids whose expense (as the poet 1

reaches to the stars; finally, thus cities have been names of men deemed worthy by posterity of commendation to all the ages. For the nature of the human mind is such that unless it is stimulated by images of things acting upon it from without, all remembrance of them passes easily away. says)

built to bear the

From

Discoveries and Opinions of Galileo, trans, with an inand notes by Stillman Drake. Garden City, N.Y.: Doubleday Anchor Books (Doubleday & Company, Inc.), 1957, pp. 27-58. *

troduction

1

Propertius

iii,

2, 17.

Discoveries in the Heavens

53

Looking to things even more stable and enduring, others have entrusted the immortal fame of illustrious men not to marble and metal but to the custody of the Muses and to imperishable literary monuments. But why dwell upon these things as though human wit were satisfied with earthly regions and had not dared advance beyond? For, seeking further, and well understanding that all human monuments ultimately perish through the violence of the elements or by old age, ingenuity has in fact found still more incorruptible monuments over which voracious time and envious age have been unable to assert any rights. Thus turning to the sky, man's wit has inscribed on the familiar and everlasting orbs of most bright stars the names of those whose eminent and godlike deeds have caused them to be accounted worthy of eternity in the

company of

the stars.

And

so the

fame of

Mars, of Mercury, Hercules, and other heroes by whose names the stars are called, will not fade before the

Jupiter, of

extinction of the stars themselves. Yet this invention of human ingenuity, noble, and admirable as it is, has for many centuries been out of style. Primeval

heroes are in possession of those bright abodes, and hold them in their own right. In vain did the piety of Augustus attempt to elect Julius Caesar into their number, for when he tried to give the name of "Julian" to a star which appeared in his time (one of those bodies which the Greeks call "comets" and which the Romans likewise named for their hairy appearance), it vanished in a brief time and his too ambitious wish. But we are able, most serene Prince, to read Your Highness in the heavens far more

mocked

accurately and auspiciously. For scarce have the immortal graces of your spirit begun to shine on earth when in the heavens bright stars appear as tongues to tell and celebrate your exceeding virtues to all time. Behold, then, four stars reserved to bear your famous name; bodies which belong not to the inconspicuous multitude of fixed stars, but to the bright ranks of the planets. Variously moving about most noble Jupiter as children of his own, they complete their orbits with marvelous velocity at the same time executing with one harmonious accord mighty revolutions every dozen 2 years about the center of the universe; that is, the sun.

—

2 This is the first published intimation by Galileo that he accepted the Copernican system. Tycho had made Jupiter revolve about the sun, but considered the earth to be the center of the universe. It was not until 1613, however, that Galileo unequivocally supported Copernicus in print.

Breakthroughs in Physics

54

Maker

of the stars himself has seemed by new planets Your Highness's famous name in preference to all others. For just as these stars, like children worthy of their sire, never leave the side of Jupiter by any appreciable distance, so (as indeed who does not know?) clemency, kindness of heart, gentleness of manner, splendor of royal blood, nobility in public affairs, and excellency of authority and rule have all fixed their abode and habitation in Your Highness. And who, I ask once more, does not know that all these virtues emanate from the benign star of Jupiter, next after God as the source of all things good? Jupiter; Jupiter, I say, at the instant of Your Highness's birth, having already emerged from the turbid mists of the horizon and occupied the midst of the heavens, illuminating the eastern sky from his own royal house, looked out from that exalted throne upon your auspicious birth and poured forth all his splendor and majesty in order that your tender body and your mind (already adorned by God with the most noble ornaments) might imbibe with their first breath that universal influence and power. But why should I employ mere plausible arguments, when I may prove my conclusion absolutely? It pleased Almighty

Indeed, the

clear indications to direct that I assign to these

God

that I should instruct

Your Highness

in mathematics,

which I did four years ago at that time of year when it is customary to rest from the most exacting studies. And since clearly it was mine by divine will to serve Your Highness and thus to receive from near at hand the rays of your surpassing clemency and beneficence, what wonder is it that my heart is so inflamed as to think both day and night of little else than how I, who am indeed your subject not only by choice but by birth and lineage, may become known to you as most grateful and most anxious for your glory? And so, most serene Cosimo, having discovered under your patronage these stars unknown to every astronomer before me, I have with good right decided to designate them by the august name of your family. And if I am first to have investigated them, who can justly blame me if I likewise name them, calling them the Medicean Stars, in the hope that this name will bring as much honor to them as the names of other heroes have bestowed on other stars? For, to say nothing of Your Highness's most serene ancestors, whose everlasting glory is testified by the monuments of all history, your virtue alone, most worthy Sire, can confer upon these stars an immortal name. will fulfill those expectations,

No one can doubt that you high though they are, which

Discoveries in the Heavens

55

you have aroused by the auspicious beginning of your reign, and will not only meet but far surpass them. Thus when you have conquered your equals you may still vie with yourself, and you and your greatness will become greater every day.

Accept then, most clement Prince, this gentle glory reserved by the stars for you. May you long enjoy those blessings which are sent to you not so much from the stars as

from God,

their

Maker and

their

Governor.

Your Highness's most devoted

servant,

Galileo Galilei Padua, March

12,

1610

Astronomical Message

Which

contains and explains recent observations

with the aid of a the

new

made

spyglass* concerning Xhe surface of

moon, the Mil\y Way, nebulous

stars,

and innumer*

able fixed stars, as well as four planets never before seen,

and now named The Medicean Stars Great indeed are the things which in this brief treatise I propose for observation and consideration by all students of nature. I say great, because of the excellence of the subject itself, the entirely unexpected and novel character of these things, and finally because of the instrument by means of which they have been revealed to our senses. Surely it is a great thing to increase the numerous host of fixed stars previously visible to the unaided vision, adding countless more which have never before been seen, exposing these plainly to the eye in numbers ten times exceeding the old and familiar stars. It is a very beautiful thing, and most gratifying to the sight, to behold the body of the moon, distant from us al-

A

s The word "telescope" was not coined until 1611. detailed account of its origin is given by Edward Rosen in The Naming of the Telescope (New York, 1947). In the present translation the modern term has been introduced for the sake of dignity and ease of reading, but only after the passage in which Galileo describes the circumstances which led him to construct the instrument (pp. 56-57).

Breakthroughs in Physics

56

most sixty earthly radii, 4 as if two such measures so that

—

times larger,

thirty

its

it

its

were no farther away than diameter

appears

almost

surface nearly nine hundred times,

and its volume twenty-seven thousand times as large as when viewed with the naked eye. In this way one may learn with all the certainty of sense evidence that the moon is not robed in a smooth and polished surface but is in fact rough and uneven, covered everywhere, just like the earth's surface, with huge prominences, deep valleys, and chasms. it seems to me a matter of no small importance have ended the dispute about the Milky Way by making

Again, to

nature manifest to the very senses as well as to the intelSimilarly it will be a pleasant and elegant thing to demonstrate that the nature of those stars which astronomers have previously called "nebulous" is far different from what has been believed hitherto. But what surpasses all wonits

lect.

and what particularly moves us to seek the astronomers and philosophers, is the discovery of four wandering stars not known or observed by any man before us. Like Venus and Mercury, which have their own periods about the sun, these have theirs about a certain star that is conspicuous among those already known, which they sometimes precede and sometimes follow, without ever departing from it beyond certain limits. All these facts were discovered and observed by me not many days ago with the aid of a spyglass which I devised, after first being illuminated by divine grace. Perhaps other things, still more remarkable, will in time be discovered by me or by other observers with the aid of such an instrument, the form and construction of which I shall first briefly explain, as well as the occasion of its having been devised. Afterwards I shall relate the story of the observations I have made. About ten months ago a report reached my ears that a ders

by

far,

attention of

4

The

That

all

original text reads "diameters" here and in another place. was Galileo's and not the printer's has been con-

this error

shown by Edward Rosen (Isis, 1952, pp. 344 ff.). The was a curious one, as astronomers of all schools had long agreed that the maximum distance of the moon was approximately sixty terrestrial radii. Still more curious is the fact that neither vincingly slip

nor any other correspondent appears to have called Galileo's attention to this error; not even a friend who ventured to criticize the calculations in this very passage. Kepler

Discoveries in the Heavens

57

had constructed a spyglass by means of which visible objects, though very distant from the eye of the observer, were distinctly seen as if nearby. Of this truly remarkable effect several experiences were related, to which some persons gave credence while others denied them. A few days later the report was confirmed to me in a letter from a noble Frenchman at Paris, Jacques Badovere, 6 which caused me to apply myself wholeheartedly to inquire into the means by which I might arrive at the invention of certain

Fleming

5

a similar instrument. This

I

did shortly afterwards,

being the theory of refraction. First

my

basis

prepared a tube of lead, at the ends of which I fitted two glass lenses, both plane on one side while on the other side one was spherically convex and the other concave. Then placing my eye near the concave lens I perceived objects satisfactorily large and I

near, for they appeared three times closer

and nine times

when

seen with the naked eye alone. Next I constructed another one, more accurate, which represented larger than

objects as enlarged

more than

sixty times.

Finally, sparing

neither labor nor expense, I succeeded in constructing for

myself so excellent an instrument that objects seen by means of it appeared nearly one thousand times larger and over thirty times closer than when regarded with our natural vision. It would be superfluous to enumerate the number and importance of the advantages of such an instrument at sea as well as on land. But forsaking terrestrial observations, I turned to celestial ones, and first I saw the moon from as near at hand as if it were scarcely two terrestrial radii away. After that I observed often with wondering delight both the planets and the fixed stars, and since I saw these

be very crowded, I began to seek (and eventually found) a method by which I might measure their distances latter to

apart.

5 Credit for the original invention is generally assigned to Hans Lipperhey, a lens grinder in Holland who chanced upon this property of combined lenses and applied for a patent on it in 1608. 6 Badovere studied in Italy toward the close of the sixteenth century and is said to have been a pupil of Galileo's about 1598. When he wrote concerning the new instrument in 1609 he was in the French diplomatic service at Paris, where he died in 1620.

Breakthroughs in Physics

58

it is appropriate to convey certain cautions to all intend to undertake observations of this sort, for in the place it is necessary to prepare quite a perfect telescope,

Here

who first

which will show all objects bright, distinct, and free from any haziness, while magnifying them at least four hundred times and thus showing them twenty times closer. Unless the instrument is of this kind it will be vain to attempt to observe all the things which I have seen in the heavens, and which will presently be set forth. Now in order to determine without much trouble the magnifying power of an instrument, trace on paper the contour of two circles or two squares of which one is four hundred times as large as the other, as it will be when the diameter of one is twenty times that of the other. Then, with both these figures attached to the same wall, observe them simultaneously from a distance, looking at the smaller one through the telescope and at the larger one with the other eye unaided. This may be done without inconvenience while holding both eyes open at the same time; the two figures will appear to be of the same size if the instrument magnifies objects in the desired proportion. Such an instrument having been prepared, we seek a method of measuring distances apart. This we shall accomplish by the following contrivance.

A

Let

I

ABCD

E

be the tube and be the eye of the observer. there were no lenses in the tube, the rays would reach the object and EDG. along the straight lines

Then

if

FG

ECF

But when the lenses have been inserted, the rays go along the refracted lines ECH and EDI; thus they are brought closer together, and those which were previously directed

FG now include only the portion of it HI. The ratio of the distance EH .to the line HI then being found, one may by means of a table of sines determine the size of the angle formed at the eye by the object HI, which we shall find to be but a few minutes of arc. Now, if to the freely to the object

Discoveries in the Heavens lens

CD we

fit

thin

some with smaller

plates,

59

some pierced with

larger and

now one plate and now we may form at pleasure

apertures, putting

another over the lens as required, angles subtending more or fewer minutes of arc,

different

and by this means we may easily measure the intervals between stars which are but a few minutes apart, with no greater error than one or two minutes. And for the present let it suffice that we have touched lightly on these matters and scarcely more than mentioned them, as on some other occa-

we

theory of this instrument. us review the observations made during the past two months, once more inviting the attention of all who are eager for true philosophy to the first steps of such important contemplations. Let us speak first of that surface of the moon

sion

Now

shall explain the entire

let

For greater clarity I distinguish two parts a lighter and a darker; the lighter part seems to surround and to pervade the whole hemisphere, while the darker part discolors the moon's surface like a kind of cloud, and makes it appear covered with spots. Now those spots which are fairly dark and rather large are plain to everyone and have been seen throughout the ages; these

which faces of this

I

shall

us.

surface,

call

the

them from others

"large"

or

"ancient"

spots,

distinguishing

that are smaller in size but so

numerous

over the lunar surface, and especially the lighter part. The latter spots had never been seen by anyone before me. From observations of these spots repeated many times I have been led to the opinion and conviction that the surface of the moon is not smooth, uniform, and precisely spherical as a great number of philosophers believe to be, but is uneven, it (and the other heavenly bodies) rough, and full of cavities and prominences, being not unas to occur

all

like the face of the earth, relieved

and deep

valleys.

The

by chains of mountains by which I was

things I have seen

enabled to draw this conclusion are as follows. On the fourth or fifth day after new moon, when the moon is seen with brilliant horns, the boundary which divides the dark part from the light does not extend uniformly in an oval line as would happen on a perfectly spherical solid, but traces out an uneven, rough, and very wavy line as shown in the figure below. Indeed, many luminous excrescences extend beyond the boundary into the darker portion, while on the other hand some dark patches invade the illuminated part. Moreover a great quantity of small blackish spots, entirely separated from the dark region, are scattered almost all over the area illuminated by the

60

Breakthroughs in Physics

sun with the exception only of that part which is occupied spots. Let us note, however, that the said small spots always agree in having their blackened parts directed toward the sun, while on the side opposite the sun they are crowned with bright contours, like shining summits. There is a similar sight on earth about sunrise, when we behold the valleys not yet flooded with light though the mountains surrounding them are already ablaze with glowing splendor on the side opposite the sun. And just as the shadows in the hollows on earth diminish in size as the sun rises higher, so these spots on the moon lose their blackness as the illuminated region grows larger and larger.

by the large and ancient

Again, not only are the boundaries of shadow and light moon seen to be uneven and wavy, but still more astonishingly many bright points appear within the darkened portion of the moon, completely divided and separated from in the

the illuminated part and at a considerable distance from it. After a time these gradually increase in size and brightness, and an hour or two later they become joined with the rest of the lighted part which has now increased in size. Meanwhile more and more peaks shoot up as if sprouting now here, now there, lighting up within the shadowed portion; these become larger, and finally they too are united with that

same luminous surface which extends ever

illustration of this

is

further.

to be seen in the figure above.

An And

on the earth, before the rising of the sun, are not the highest peaks of the mountains illuminated by the sun's rays while the plains remain in shadow? Does not the light go on spreading while the larger central parts of those moun-

Discoveries in the Heavens

61

tains are becoming illuminated? And when the sun has finally risen, does not the illumination of plains and hills finally become one? But on the moon the variety of elevations and depressions appears to surpass in every way the

roughness of the

terrestrial surface, as

we

shall

demonstrate

further on.

At present I cannot pass over in silence something worthy of consideration which I observed when the moon was approaching first quarter, as shown in the previous figure. Into the luminous part there extended a great dark gulf in the neighborhood of the lower cusp. When I had observed it for a long time and had seen it completely dark, a bright peak began to emerge, a little below its center, after about two hours. Gradually growing, this presented itself in a triangular shape, remaining completely detached and separated

from the lighted surface. Around it three other small points soon began to shine, and finally, when the moon was about to set, this triangular shape (which had meanwhile become more widely extended) joined with the rest of the illuminated region and suddenly burst into the gulf of shadow like a vast promontory of light, surrounded still by the three bright peaks already mentioned. Beyond the ends of the cusps, both above and below, certain bright points emerged which were quite detached from the remaining lighted part, as may be seen depicted in the same figure. There were also a great number of dark spots in both the horns, especially in the lower one; those nearest the boundary of light and shadow appeared larger and darker, while those more distant from the boundary were not so dark and distinct. But in all cases, as we have mentioned earlier, the blackish portion of each spot is turned toward the source of the sun's radiance, while a bright rim surrounds the spot on the side away from the sun in the direction of the shadowy region of the moon. This part of the moon's surface, where it is spotted as the tail of a peacock is sprinkled with azure eyes, resembles those glass vases which have been plunged while still hot into cold water and have thus acquired a crackled and wavy surface, from which they receive their common name of "ice-cups."

As

to the large lunar spots, these are not seen to be in the above manner and full of cavities and prominences; rather, they are even and uniform, and brighter

broken

patches crop up only here and there. Hence

if

anyone wished

Breakthroughs in Physics

62

Pythagorean 7 opinion that the moon is another earth, its brighter part might very fitly represent the surface of the land and its darker region that of the water. I have never doubted that if our globe were seen from afar when flooded with sunlight, the land regions would appear brighter and the watery regions darker. 8 The large spots in the moon are also seen to be less elevated than the brighter tracts, for whether the moon is waxing or waning there are always seen, here and there along its boundary of light and shadow, certain ridges of brighter hue around the large spots (and we have attended to this in preparing the diagrams); the edges of these spots are not only lower, but also more uniform, being uninterrupted by peaks or ruggedto revive the old

like

ness.

Near the

large spots the brighter part stands out particu-

way that before first quarter and toward last quarter, in the vicinity of a certain spot in the upper (or

larly in

such a

northern) region of the moon, some vast prominences arise both above and below as shown in the figures reproduced below. Before last quarter this same spot is seen to be walled about with certain blacker contours which, like the loftiest mountaintops, appear darker on the side away from the sun and brighter on that which faces the sun. (This is the opposite of what happens in the cavities, for there the part 7 Pythagoras was a mathematician and philosopher of the sixth century B.C., a semilegendary figure whose followers were credited at Galileo's time with having anticipated the Copernican system. This tradition was based upon a misunderstanding. The Pythagoreans made the earth revolve about a "central fire" whose light and heat were reflected to the earth by the sun. 8 Leonardo da Vinci had previously suggested that the dark and light regions of the moon were bodies of land and water, though Galileo probably did not know this. Da Vinci, however, had mistakenly supposed that the water would appear brighter than

the land.

Discoveries in the Heavens

63

away from the sun appears brilliant, while that which is turned toward the sun is dark and in shadow.) After a time, when the lighted portion of the moon's surface has diminished in size and when all (or nearly all) the said spot is covered with shadow, the brighter ridges of the mountains gradually emerge from the shade. This double aspect of the spot is illustrated in the ensuing figures.

There is another thing which I must not omit, for I beheld it not without a certain wonder; this is that almost in the center of the moon there is a cavity larger than all the rest, and perfectly round in shape. I have observed it near both first and last quarters, and have tried to represent it as correctly as possible in the second of the above figures. As to light and shade, it offers the same appearance as would a region like Bohemia 9 if that were enclosed on all sides by very lofty mountains arranged exactly in a circle. Indeed, this area on the moon is surrounded by such enormous peaks that the bounding edge adjacent to the dark portion of the moon is seen to be bathed in sunlight before the boundary of light and shadow reaches halfway across the same space. As in other spots, its shaded portion faces the sun while its toward the dark side of the moon; and for a draw attention to this as a very cogent proof of the ruggedness and unevenness that pervades all the bright region of the moon. Of these spots, moreover, those are always darkest which touch the boundary line between fight lighted part

is

third time I

9

moon and a much trouble for moon (or any other

This casual comparison between a part of the

specific region

Galileo.

Even

on earth was

later the basis of

in antiquity the idea that the

heavenly body) was of the same nature as the earth had been dangerous to hold. The Athenians banished the philosopher Anaxagoras for teaching such notions and charged Socrates with blasphemy for repeating them.

Breakthroughs in Physics

64

and shadow, while those farther

off appear both smaller and the moon ultimately becomes full (at opposition 10 to the sun) the shade of the cavities is distinguished from the light of the places in relief by a subdued and very tenuous separation. The things we have reviewed are to be seen in the brighter region of the moon. In the large spots, no such contrast of depressions and prominences is perceived as that which we are compelled to recognize in the brighter parts by the less dark, so that

when

,

changes of aspect that occur under varying illumination by the sun's rays throughout the multiplicity of positions from which the latter reach the moon. In the large spots there exist some holes rather darker than the rest, as we have shown in the illustrations. Yet these present always the same appearance, and their darkness is neither intensified nor diminished, although with some minute difference they appear sometimes a little more shaded and sometimes a little lighter according as the rays of the sun fall on them more or less obliquely. Moreover, they join with the neighboring regions of the spots in a gentle linkage, the boundaries mixing and mingling. It is quite different with the spots which occupy the brighter surface of the moon; these, like precipitous crags having rough and jagged peaks, stand out starkly in sharp contrasts of light and shade. And inside the large spots there are observed certain other zones that are brighter, some of them very bright indeed. Still, both these and the darker parts present always the same appearance; there is no change either of shape or of light and shadow; hence one may affirm beyond any doubt that they owe their appearance to some real dissimilarity of parts. They cannot be attributed merely to irregularity of shape, wherein shadows move in consequence of varied illuminations from the sun, as indeed is the case with the other, smaller, spots which occupy the brighter part of the moon and which change, grow, shrink, or disappear from one day to the next, as owing their origin only to shadows of prominences. But here I foresee that many persons will be assailed by uncertainty and drawn into a grave difficulty, feeling constrained to doubt a conclusion already explained and confirmed by many phenomena. If that part of the lunar sur10

Opposition of the sun and

line with the earth

conjunction,

when they

(new moon, or

moon

between them

are in line eclipse of the sun).

when they are in moon, or lunar eclipse); on the same side of the earth

(full

occurs

Discoveries in the Heavens

65

face which reflects sunlight more brightly is full of chasms (that is, of countless prominences and hollows), why is it that the western edge of the waxing moon, the eastern edge of the waning moon, and the entire periphery of the full

moon are not On the contrary

seen to be uneven, rough, and wavy? they look as precisely round as if they were drawn with a compass; and yet the whole periphery consists of that brighter lunar substance which we have declared to be filled with heights and chasms. In fact not a single one of the great spots extends to the extreme periphery of the raoo»\ but all are grouped together at a distance from the edge. Now let me explain the twofold reason for this troublesome fact, and in turn give a double solution to the difficulty. In the first place, if the protuberances and cavities in the lunar body existed only along the extreme edge of the circular periphery bounding the visible hemisphere, the moon might (indeed, would necessarily) look to us, almost like a toothed wheel, terminated by a warty or wavy edge. Imagine, however, that there is not a single series of prominences arranged only along the very circumference, but a great many ranges of mountains together with their valleys and canyons disposed in ranks near the edge of the moon, and not only in the hemisphere visible to us but everywhere near the boundary line of the two hemispheres. Then an eye viewing them from afar will not be able to detect the separation of prominences by cavities, because the intervals between the mountains located in a given circle or a given chain will be hidden by the interposition of other heights situated in yet other ranges. This will be especially true if the eye of the observer is placed in the same straight line with the summits of these elevations. Thus on earth the summits of several mountains close together appear to be situated in one plane if the spectator is a long way off and is placed at an equal elevation. Similarly in a rough sea the tops of the waves seem to lie in one plane, though between one high crest and another there are many gulfs and chasms of such depth as not only to hide the hulls but even the bulwarks, masts, and rigging of stately ships. Now since there are many chains of mountains and chasms on the moon in addition to those around its periphery, and since the eye, regarding these from a great distance, lies nearly in the plane of their summits, no one need wonder that they appear as arranged in a regular and

unbroken

To

line.

the above explanation another

may

be ad^ed; name-

Breakthroughs in Physics

66

that there exists

ly,

around the

around the body of the moon,

earth, a globe of

just

some substance denser than

as the

rest of the aether. 11 This may serve to receive and reflect the sun's radiations without being sufficiently opaque to prevent our seeing through it, especially when it is not illuminated. Such a globe, lighted by the sun's rays, makes the body of the moon appear larger than it really is, and if it were thicker it would be able to prevent our seeing the actual body of the moon. And it actually is thicker near the circumference of the moon; I do not mean in an absolute sense, but relatively to the rays of our vision, which cut it obliquely there. Thus it may obstruct our vision, especially when it is lighted, and cloak the lunar periphery that is ex-

posed to the sun. This may be more clearly understood from the figure below, in which the body of the moon, ABC, is surrounded by the vaporous globe DEG.

The at

eyesight

A

from

F

reaches the

moon

in the central region,

for example, through a lesser thickness of the vapors

DA,

while toward the extreme edges vapors, EB, limits and shuts out our of this is that the illuminated portion to be larger in circumference than the

a deeper stratum of One indication

sight.

of the

moon

appears

rest of the orb,

which

shadow. And perhaps this same cause will appeal to some as reasonably explaining why the larger spots on the moon are nowhere seen to reach the very edge, probable though it is that some should occur there. Possibly they are invisible by being hidden under a thicker and more luminous lies in

mass of vapors. That the lighter surface of the moon is everywhere dotted with protuberances and gaps has, I think, been made suf11

The

was the special substance of the heavenly bodies were supposed to be essentially different from all the earthly "eleyears Galileo abandoned his suggestion here

aether, or "ever-moving,"

which the sky and made, a substance

all

ments." In later moon has a vaporous atmosphere.

that the

Discoveries in the Heavens

67

from the appearances already explained. It reto speak of their dimensions, and to show that the earth's irregularities are far less than those of the moon. I mean that they are absolutely less, and not merely in relaficiently clear

mains for

me

tion to the sizes

of the respective globes. This

is

plainly

demonstrated as follows. I had often observed, in various situations of the moon with respect to the sun, that some summits within the shadowy portion appeared lighted, though lying some distance

from the boundary of the light. By comparing this separation whole diameter of the moon, I found that it some-

to the

times exceeded one-twentieth of the diameter. Accordingly, let CAF be a great circle of the lunar body, E its center, and CF a diameter, which is to the diameter of the earth as two is

to seven.

Since according to very precise observations the diameter of the earth is seven thousand miles, CF will be two thousand, CE one thousand, and one-twentieth of CF will be one hundred miles. Now let CF be the diameter of the great circle which divides the light part of the moon from the dark part (for because of the very great distance of the

sun from the moon,

this

does not differ appreciably from a

great circle), and let A be distant from C by one-twentieth of this. Draw the radius EA, which, when produced, cuts the tangent line (representing the illuminating ray) in the point D. Then the arc CA, or rather the straight line contains CD, will consist of one hundred units whereof

GCD

CE

one thousand, and the sum of the squares of DC and CE will be 1,010,000. This is equal to the square of DE; hence ED will exceed 1,004, and AD will be more than four of those units of which CE contains one thousand. Therefore the altitude AD on the moon, which represents a summit

Breakthroughs in Physics

68

GCD

reaching up to the solar ray and standing at the distance CD from C, exceeds four miles. But on the earth we have no mountains which reach to a perpendicular height of even one mile. 12 Hence it is quite clear that the prominences on the moon are loftier than those on the earth. Here I wish to assign the cause of another lunar phenomenon well worthy of notice. I observed this not just recently, but many years ago, and pointed it out to some of my friends and pupils, explaining it to them and giving its true cause. Yet since it is rendered more evident and easier to observe with the aid of the telescope, I think it not unsuitable for introduction in this place, especially as it shows more clearly the connection between the moon and the earth. When the moon is not far from the sun, just before or after new moon, its globe offers itself to view not only on the side where it is adorned with shining horns, but a certain faint light is also seen to mark out the periphery of the dark part which faces away from the sun, separating this from the darker background of the aether.

Now

if

we examine

the

matter more closely, we shall see that not only does the extreme limb of the shaded side glow with this uncertain light, but the entire face of the moon (including the side which does not receive the glare of the sun) is whitened by a not inconsiderable gleam. At first glance only a thin luminous circumference appears, contrasting with the darker sky coterminous with it; the rest of the surface appears darker from its contact with the shining horns which distract our vision. But if we place ourselves so as to interpose a roof or chimney or some other object at a considerable distance from the eye, the shining horns may be hidden while the rest of the lunar globe remains exposed to view. It is then found that this region of the moon, though deprived of sunlight, also shines not a little. The effect is heightened if the gloom of night has already deepened through departure of the sun, for in a darker field a given light appears brighter.

Moreover, (so to speak)

it is

is

sun. It diminishes 12 Galileo's

found that

this

light of the moon moon is closer to the the moon recedes from

secondary

greater according as the

more and more

as

estimate of four miles for the height of

some lunar

mountains was a very good one. His remark about the maximum height of mountains on the earth was, however, quite mistaken. An English propagandist for his views, John Wilkins, took pains to correct this error in his anonymous Discovery of a New World in the Moon (London, 1638), Prop. ix. . . .

Discoveries in the Heavens that is

body

until, after the first

quarter and before the

seen very weakly and uncertainly even

the darkest sky. But

of the sun

it

when

the

moon

is

69 last, it

when observed

in

within sixty degrees

shines remarkably, even in twilight; so brightly

indeed that with the aid of a good telescope one may distinguish the large spots. This remarkable gleam has afforded

no small perplexity to philosophers, and in order to assign a it some have offered one idea and some another. Some would say it is an inherent and natural light of the moon's own; others, that it is imparted by Venus; others yet, by all the stars together; and still others derive it from the sun, whose rays they would have permeate the thick solidity of the moon. But statements of this sort are refuted and their falsity evinced with little difficulty. For if this kind of light were the moon's own, or were contributed by the stars, the moon would retain it and would display it particularly during eclipses, when it is left in an unusually dark sky. This is contradicted by experience, for the brightness which is seen on the moon during eclipses is much fainter and is ruddy, almost copper-colored, while this is brighter and whitish. Moreover the other light is variable and movable, for it cause for

covers the face of the moon in such a way that the place near the edge of the earth's shadow is always seen to be brighter than the rest of the moon; this undoubtedly results from contact of the tangent solar rays with some denser zone which girds the moon about. 13 By this contact a sort of twi-

neighboring regions of the moon, on earth a sort of crepuscular light is spread both morning and evening; but with this I shall deal more fully in my book on the system of the world. 14 To assert that the moon's secondary light is imparted by Venus is so childish as to deserve no reply. Who is so igno-

light is diffused over the just as

Kepler had correctly accounted for the existence of this light ruddy color. It is caused by refraction of sunlight in the earth's atmosphere, and does not require a lunar atmosphere as supposed by Galileo. 13

and

its

thus promised was destined not to appear for decades. Events prevented its publication for many years, and then it had to be modified to present the arguments for both the Ptolemaic and Copernican systems instead of just the latter as Galileo here planned. Even then it was suppressed, and 14 The book more than two

the author was

condemned

to life imprisonment.

70

Breakthroughs in Physics

rant as not to understand that from new moon to a separation of sixty degrees between moon and sun, no part of the

moon which

is averted from the sun can possibly be seen from Venus? And it is likewise unthinkable that this light should depend upon the sun's rays penetrating the thick solid mass of the moon, for then this light would never dwindle, inasmuch as one hemisphere of the moon is always illuminated except during lunar eclipses. And the light does diminish as the moon approaches first quarter, becom-

ing completely obscured after that is passed. Now since the secondary light does not inherently belong to the moon, and is not received from any star or from the sun, and since in the whole universe there is no other body left but the earth, what must we conclude? What is to be proposed? Surely we must assert that the lunar body (or any other dark and sunless orb) is illuminated by the earth. Yet what is there so remarkable about this? The earth, in fair and grateful exchange, pays back to the moon an illumination similar to that which it receives from her throughout nearly all the darkest gloom of night. Let us explain this matter more fully. At conjunction the moon occupies a position between the sun and the earth; it is then illuminated by the sun's rays on the side which is turned away from the earth. The other hemisphere, which faces the earth, is covered with darkness; hence the moon does not illuminate the surface of the earth at all. Next, departing gradually from the sun, the moon comes to be lighted partly upon the side it turns toward us, and its whitish horns, still very thin, illuminate the earth with a faint light. The sun's illumination of the moon increasing now as the moon approaches first quarter, a reflection of that light to the earth

Soon the splendor on the moon extends into a and our nights grow brighter; at length the entire visible face of the moon is irradiated by the sun's resplendent rays, and at full moon the whole surface of the earth shines in a flood of moonlight. Now the moon, waning, sends us her beams more weakly, and the earth is less strongly also increases.

semicircle,

lighted; at length the

moon

returns to conjunction with the

and black night covers the earth. In this monthly period, then, the moonlight gives us alternations of brighter and fainter illumination; and the benefit is repaid by the earth in equal measure. For while the moon is between us and the sun (at new moon), there lies before it sun,

Discoveries in the Heavens

71

the entire surface of that hemisphere of the earth which is exposed to the sun and illuminated by vivid rays. The moon receives the light which this reflects, and thus the nearer hemisphere of the moon that is, the one deprived of sunlight appears by virtue of this illumination to be not a little

—

—

luminous. When the moon is ninety degrees away from the sun it sees but half the earth illuminated (the western half), for the other (the eastern half) is enveloped in night. Hence the moon itself is illuminated less brightly from the earth, and as a result its secondary light appears fainter to us. When the moon is in opposition to the sun, it faces a hemisphere of the earth that is steeped in the gloom of night, and if this position occurs in the plane of the ecliptic the moon will receive no light at all, being deprived of both the solar and the terrestrial rays. In its various other positions with respect to the earth and sun, the moon receives more or less light according as it faces a greater or smaller portion of the il-

luminated hemisphere of the earth. And between these two globes a relation is maintained such that whenever the earth is most brightly lighted by the moon, the moon is least lighted by the earth, and vice versa. Let these few remarks suffice us here concerning this matter, which will be more fully treated in our System of the world. In that book, by a multitude of arguments and experiences, the solar reflection from the earth will be shown against those who argue that the earth must to be quite real

—

be excluded from the dancing whirl of stars for the specific reason that it is devoid of motion and of light. We shall prove the earth to be a wandering body surpassing the moon in splendor, and not the sink of all dull refuse of the universe; this we shall support by an infinitude of arguments

drawn from nature. Thus far we have spoken of our observations concerning the body of the moon. Let us now set forth briefly what has thus far been observed regarding the fixed stars. And first of all, the following fact deserves consideration: The stars, whether fixed or wandering, 15 appear not to be enlarged by the telescope in the same proportion as that in which it magnifies other objects, and even the moon itself. In the stars 15 That is, planets. Among these bodies Galileo counted newly discovered satellites of Jupiter. The term "satellites" was troduced somewhat later by Kepler.

his in-

Breakthroughs in Physics

72

enlargement seems to be so much less that a telescope is sufficiently powerful to magnify other objects a hundredfold is scarcely able to enlarge the stars four or five times. The reason for this is as follows. When stars are viewed by means of unaided natural vision, they present themselves to us not as of their simple (and, so to speak, their physical) size, but as irradiated by a certain fulgor and as fringed with sparkling rays, especially when the night is far advanced. From this they appear larger than they would if stripped of those adventitious hairs of light, for the angle at the eye is determined not by the primary body of the star but by the brightness which extends so widely about it. This appears quite clearly from the fact that when stars first emerge from twilight at sunset they look very small, even if they are of the first magnitude; Venus itself, when visible in broad daylight, is so small as scarcely to appear equal to a star of the sixth magnitude. Things fall out differently with other objects, and even with the moon itself; these, whether seen in daylight or the deepest night, appear always of the same bulk. Therefore the stars are seen crowned among shadows, while daylight is able to remove their headgear; and not daylight alone, but any thin cloud that interposes itself between a star and the eye of the observer. The same effect is produced by black veils or colored glasses, through the interposition of which obstacles the stars telescope are abandoned by their surrounding brilliance. similarly accomplishes the same result. It removes from the stars their adventitious and accidental rays, and then it enlarges their simple globes (if indeed the stars are naturally globular) so that they seem to be magnified in a lesser ratio than other objects. In fact a star of the fifth or sixth magnitude when seen through a telescope presents itself as one of the first magnitude. Deserving of notice also is the difference between the appearances of the planets and of the fixed stars. 16 The planets show their globes perfectly round and definitely bounded, this

which

A

16

stars are so distant that their light reaches the earth as points. Hence their images are not enlarged by even the best telescopes, which serve only to gather more of their light and in that way increase their visibility. Galileo was never entirely clear about this distinction. Nevertheless, by applying his

Fixed

from dimensionless

knowledge of the effects described here, he greatly reduced the prevailing overestimation of visual dimensions of stars and planets.

Discoveries in the Heavens

73

moons, spherical and flooded all over with light; the fixed stars are never seen to be bounded by a circular periphery, but have rather the aspect of blazes whose rays vibrate about them and scintillate a great deal. Viewed with a telescope they appear of a shape similar to that which they present to the naked eye, but sufficiently enlarged so that a star of the fifth or sixth magnitude seems to equal the looking like

Dog

little

Star, largest of all the fixed stars.

Now,

in addition to

magnitude, a host of other stars are perceived through the telescope which escape the naked eye; these are so numerous as almost to surpass belief. One may, in fact, see more of them than all the stars included among the first six magnitudes. The largest of these, which we may call stars of the seventh magnitude, or the first magnitude of invisible stars, appear through the telescope as larger and brighter than stars of the second magnitude when the latter are viewed with the naked eye. In order to gife one or two proofs of their almost inconceivable number, I have adstars of the sixth

joined pictures of two constellations. you may judge of all the others.

With these

as samples,

had intended to depict the entire constelI was overwhelmed by the vast quantity of stars and by limitations of time, so I have deferred this to another occasion. There are more than five hundred new In the

first

I

lation of Orion, but

stars distributed

among

the old ones within limits of one or

two degrees of arc. Hence to the three stars in the Belt of Orion and the six in the Sword which were previously known, I have added eighty adjacent stars discovered recently, preserving the intervals between them as exactly as I could.

To

known or ancient stars, I have depicted and have outlined them doubly; the other (invisible) stars I have drawn smaller and without the extra line. I have also preserved differences of magnitude as well distinguish the

them

larger

as possible.

In the second example I have depicted the six stars of Taurus known as the Pleiades (I say six, inasmuch as the seventh is hardly ever visible) which lie within very narrow sky. Near them are more than forty others, no one of which is much more than half a degree away from the original six. I have shown thirty-six of these

limits in the invisible,

in the diagram;

as in the case of

and magnitudes, tween old stars and new.

their intervals

Orion

I

have preserved

as well as the distinction be-

i".

'**

•.*

•

:

*

*

*

O^JS> nature absolutely prevented Fig. 5-1 c a vacuum existing. Thus, Pascal realized that a vacuum can and does exist; but he agreed with conventional opinion in thinking that nature preferred not to have a vacuum and that this preference for the plenum (or abhorrence of a vacuum) causes many such effects as suction. At the time of writing to M. Perier in the year 1647, however, Pascal no longer believed in nature's abhorrence of a vacuum at all. He explains why very succinctly: nature's

^^ ^^

...

I can hardly admit that nature, which is not at all animated or sensible, can be capable of abhorrence,:. since the passions presuppose a soul capable of ex periencing them.

In Chapter Four,

we mentioned

that Salviati put a great

deal of stock in the principle that nature brings about her effects elegantly and simply. Such a principle is similar in

character to one which states that nature abhors a vacuum. Both principles are teleological, i.e., argue from the exist-

ence of an end for which nature acts. We remarked that teleological reasoning (i.e., reasoning by means of ends, purposes, or final causes) became suspect from about 1600 on; Bacon, for instance, vigorously argues against it in the

Investigating the

Novum Organum. Here in observe how one scientist

Vacuum

Pascal's treatise

145

we can

actually

switches from employing final causes to other, mechanical kinds of causes. Instead of telling us that the effects in question follow from nature's ab-

horrence of a vacuum, Pascal tells us that they follow from the weight of the mass of the air. "Weight of air" has been substituted for "abhorrence" as a cause. Pascal begins by recalling an experiment that, he says, he made in the presence of M. Perier; this experiment led him to the belief now to be tested that it is the weight 3f the air that produces the effects previously attributed to the abhorrence of a vacuum. Pascal does not describe the experiment in detail, beyond saying that it involved two tubes, one inside the other, and that it exhibited a vacuum jvithin a vacuum. We may guess that the experiment was

—

—

somewhat

A

as follows.

is completely filled with mercury and is closed both ends. It is then set upright in a dish of mercury (see Fig. 5-2). The lower end of the tube is now opened. This is

tube

at

—vacuum

i

—V

Fig.

sailed

rtudent

Toricelli's

of

experiment,

Galileo's

of

the

.he

result

(or

36 inches)

who

\

approx 30 inches

5-2

after first

Evangelista Toricelli,

performed

it.

What

a is

experiment? If the tube is, say, 3 ft then the mercury in the tube will

long,

call to "the usual height"; this height is somewhere between 29 and 30 inches. Many readers will recognize this is as the usual barometic pressure. This is no accident. vVhat the barometer measures is pressure of air, and what

146

Breakthroughs in Physics

holds up the column of mercury is also the pressure of air. Obviously, something is holding up the column of mercury, otherwise it would all run into the dish at the bottom. To show that it really is the air that counterbalances thei mercury, Pascal proceeded to enclose tube and dish in a vacuum, so that there would be no air to hold up the column. We must note that when the mercury in the tube falls to "the usual height," a vacuum is left at the top of the tube. This space had been filled with mercury, and nothing could have gone into it when the level of the mercury dropped. No air, for instance, could have got into it; for one thing, we see no bubble of air working its way up, for another, how would an air bubble first force its way down into the dish of mercury? Hence, in order to enclose the entire piece of apparatus in Fig. 5-2 in a vacuum, Pascal put it inside another, much and created a larger tube vacuum at the top of this tube.

Thus in Fig. 5-3, the little tube and dish have been placed in the upper part of the big tube. First, let us close the little tube at both ends. If the big tube is opened at the bottom, the mercury in it will drop to a Fig. 5-3 height of approximately 30 inches and leave a vacuum surrounding the little tube anc dish. If we now open up the little tube at the bottom, whai will happen? All of the mercury in the little tube will rur out into the dish; none will remain in the tube at i higher level than the level of the dish. This shows conclusively, says Pascal, that it was the weight of the air thai held up the column of mercury. For when the weight anc pressure of the air is removed, as in Fig. 5-4, the mercury ij no longer held up at all. The experiment does in fact, seem very convincing. Why then, does Pascal feel the need for a further experiment, the "Great Experiment" which he asks M. Perier to perform? The reason may simply be, of course, that Pascal wants more in-

Investigating the

Vacuum

147

furthermore true, as we shall see, that the "Great Experiment" not only shows that the air pressure holds up the mercury, but also estabformation and more assurance.

Fig.

lishes

It is

5-4

a quantitative relation between air pressure and mer-

We

shall see that as air pressure decreases, the height of the mercury column goes down proportionately. This precise quantitative relation could not be obtained simply from the earlier experiment. There may be yet a third reason for Pascal's desire for additional confirmation: perhaps he did not really perform the two-tube experiment. This, of course, is a mere conjecture. However, it gains some likelihood from the very cursory way in which Pascal refers to the experiment. If the details of the experiment are in fact as we have suggested, then it would seem that Pascal should have explained them with some care, since the procedures are by no means simple or obvious.

cury.

we have

correctly stated what remain considerable practical difficulties. For example, how is the little tube suspended in the big tube, and how is the lower end of the little tube opened, after the vacuum in the upper end of the big tube

Furthermore, assuming that

the experiment was, there

still

Breakthroughs in Physics

148

has been established? Obviously, apparatus

sophisticated

that

this

requires considerably

can maneuver inside the big

from the outside, without devacuum. (Probably it would be something like the remote-control apparatus that is used for working with radioactive substances behind a thick lead shield.) There is some doubt whether Pascal had such sophisticated apparatus; if he did, would he not have mentioned it? Thus, this experiment gives the appearance of being one that can easily be imagined but that it would be difficult to carry out. If Pascal himself performed the experiment only in his imagination, this would constitute a good reason for seeking another experiment that can actually be carried out. Leaving aside the question of whether the two-tube experiment was ever actually performed, let us now turn to the "Great Experiment." It involves merely a variation of the ordinary Toricelli experiment. Let us examine the latter in some detail. A glass tube about 36 inches in length tube, while being controlled

stroying the

is completely filled with mercury, closed at both ends, placed upright in a dish of mercury, and then opened at its lower end. Some of the mercury runs out of the tube into the dish, but not all of it does. Enough remains in the tube

column approximately 30 inches high (see Fig. does not all of the mercury run down? Something obviously must hold it up. The column of mercury in the tube presses down on the mercury in the dish; something else, therefore, must press in the opposite direction, otherwise there would be motion. This other pressure, we have already said, is provided by the air. But how can we to leave a

5-2).

Why

clearly convince ourselves that

it

is

air pressure that holds

up the column of mercury?

The

easiest

way

sider a dish of

to visualize what goes on is first to conmercury with no tube in it at all. Consider

any small portion of its surface, say the piece AB (see Fig. 55a). There is pressure on AB, namely, the weight of the entire column of air above it. Why does this pressure not cause AB to

move downward? The answer

obviously

is

that

if

AB

were to

A

B

/ Fig.

5-5a

Investigating the

move down, then some

Vacuum

149

other,

equal portion of mercury, such as

CD, would have to go up (see Or perhaps the level

Fig. 5-5b).

all the remaining mercury might move up just a little, so that the total volume of mercury would remain the same as before 5-5c). Why does (see Fig.

of

neither of the events sketched in Figs.

5-5b and 5-c take place?

Clearly,

CD

does not move upair pressure also

ward because pushes

down on

CD', similarly,

p °' _ ~ .

l

\^^ V"

\^

— ......./ = ~7

I

l

^

/

1 p-

'

5_< c

the entire level of the mercury

cannot be raised because each portion of the surface

is

subject

from above. The result is that AB cannot move downward; for the pressure that it exerts downward (on account of the column of air above it) cannot force any of the surrounding mercury upward because of counterbalancing to air pressure

air pressure.

Now what happens when we insert the full tube of mercury in the dish and then open it at the bottom? Because the tube is closed at the top, there is no air pressure on AB from the column of air above the glass tube. The force of that pressure is taken by the glass tube itself. To understand what happens, let us say that when we open the glass tube at the bottom, the mercury in the tube first runs out into the dish, thereby raising the level of the mercury in the dish just a little (see Fig. /^^ 5-6). Now consider the situation: Each little portion of the BC d e surface, such as CD, is subjected to air pressure from above. CD wants to move downward. It can only move downward if some other part of the mercury Fig. 5-6 moves upward. But portions of the surface of mercury like DE cannot move upward because there is just as much pressure on DE as there is on CD. But one bit of the surface can move upward, namely AB. There is no pressure on AB, for we have already seen that the weight of the

air

above

AB

is

supported by the glass tube

it-

Breakthroughs in Physics

150

AB

move upward,

The mercury on AB is just the same as that on CD or DE. But that will happen precisely at the moment when the weight of the mercury column above AB equals the weight of the air column above CD or DE. At that moment, therefore, equilibrium has been reached and the downward pressure on CD (or any other portion of the surface MN) can no longer push AB upward, because AB is being pushed downward just as strongly by the mercury above it. When we understand this, we can at once see what Pascal asked his brother-in-law to do. If the column of mercury inside the glass tube balances the weight of a column of air outside, then the height of the mercury must vary with the height of the air column. If we ascend a mountain with a Toricelli tube (i.e., a barometer) in our hand, the column of mercury should fall lower and lower as we climb higher and higher because the air pressure gradually decreases. This is what M. Perier did at the suggestion of Pascal. In the company of several respectable and reliable citizens as witnesses, M. Perier prepared two identical Toricelli tubes. He left one in the town of Clermont, and took the other one self.

Hence

will

into the tube.

in the glass tube will rise until the pressure

with him to the top of the nearby mountain, Puy de

(4806

ft).

At

Dome

the level of the town, both barometers gave a

reading of 26 inches and 3 1/2 lines above the level of the mercury in the dish. (1 line =1/12 inch.) This is a rather low reading, but is accounted for by the fact that the city of Clermont is itself not located at sea level but on a high plain. (The "usual" reading of the mercury level between 29 and 30 inches applies to sea level.) According to M. Perier, the difference in height between the top of the Puy de Dome and the town of Clermont is "some 500 fathoms," i.e., 3000 ft. This would make the elevation of Clermont approximately 1800 ft. At the top of the mountain, the level of the mercury stood at the height of 23 inches and 2 lines, a difference of 3 inches and 1 1/2 lines. An observer left behind in the town reported that the level of the mercury in the barometer there had not changed all day. Hence the only thing that could ac-

count for the lower level observed tain

was the lessened

air pressure.

at the top of the

moun-

Investigating the

Vacuum

151

Since M. Perier tells Pascal, near the end of his letter, that he is not too sure of the altitudes at which the various measurements of mercury level were taken, we cannot arrive at a precise figure for the difference in altitude that causes a difference of one inch in the height of the mercury. Approximately, however, the pressure exerted by 1000 ft of air is

equal to the pressure exerted by 1 inch of mercury. (Obviously, if a precise relationship could be established, the barometer could be and has been used as an altimeter.) There are other reasons besides M. Perier's uncertainty about the precise heights at which he made his measurements that would affect the mercury level in the tube. One obvious factor is temperature. Mercury expands with heat; unless the same temperature prevailed at Clermont and at the top of Puy de Dome, the measurement would be affected. Another factor is the vapor pressure of mercury. Mercury, like all liquids, has a tendency to change from the liquid to the gaseous state; thus water has a tendency to evaporate. In the Toricelli barometer, the mercury vapor is trapped in the vacuum above the liquid mercury; it thus exerts a downward pressure on the liquid mercury, causing a, lower reading than otherwise. Yet a third factor is climatic changes. As we know, the air pressure in a given area changes from day to day, as areas of low or high pressure move across the country. (Indeed, the barometer is employed by meteor-

—

—

ologists just in order to detect the existence of such highs

and lows.) If weather conditions caused any appreciable change in air pressure during the ascent of Puy de Dome, the readings of M. Perier would of course have been affected. Since the weight of the mass of the air can hold up a column of mercury, it clearly can also hold up a column of other liquids. If these liquids are lighter than mercury, it

column to balance the air pressure. For example, since mercury is 13.6 times as heavy as water, it should take a column of water 13.6 times as high as a mercury column to balance the air pressure. This means that if the Toricelli experiment were performed with water, the column of water in the tube would be approximately 34 ft high. This is also the theoretical height to which water can be raised in a suction pump; in fact, such factors as friction and the imperfections of the apparatus limit the height to about 30 ft. If water is to be drawn from a greater depth, some other kind of pump, say an electric one, must be emwill take a larger

ployed.

Breakthroughs in Physics

152

At the end of this little treatise, Pascal tells us, in a note addressed to the reader, that this experiment changed his mind about what causes the effects associated with a vacuum. He wished, he says, to hold on to traditional principles. Finally, he could no longer do so: In the end, however, the evidence of experiment compels me to lay aside the views which respect for antiquity induced me formerly to accept. I have departed from them indeed, only little by little, and have discarded them by degrees, for from the first of these three that nature has an unconquerable abhorprinciples rence of a vacuum, I passed on to the second that she does feel that abhorrence, but not insuperably; and

—

thence,

at

last,

have come

believe the

to

third

— —

that

nature has no such abhorrence at all. This is the position to which I was brought by this last experiment on the equilibrium of fluids. . . .

that marks him as a great scienPascal lays before us his old beliefs, his new beliefs, and his reasons for abandoning the first and adopting the second.

With the kind of candor

tist,

CHAPTER

SIX

Newton: Mathematical Principles of Physics

PART

I

The year 1687 is one that should be remembered by all educated people: this is the date of the publication of Newthe ton's Philosophiae Naturalis Principia Mathematica Mathematical Principles of Natural Philosophy. This is perhaps the most important and influential single book on physics ever written; it stands untarnished after more than

—

Mathematical Principles of Physics

153

almost was not published at all. taste for publishing the work, for he had had an earlier unpleasant experience when he published his thoughts on optics and found himself involved in lengthy controversy. Further, the Royal Society, to which Newton, as one of its members, presented the work, had difficulties in finding funds for the publication; fortunately, Edmund Halley, noted astronomer and friend of Newton's, himself bore the costs of printing and publishing the book. "We owe much to Halley," says the Encyclopaedia Britana masterful understatenica in referring to this incident ment. Halley was one of those who continually encouraged Newton and persuaded him to publish his work at all; Halley also prefaced the first edition of the Principia with a long ode to Newton in which he celebrates the author's genius and the importance of his discoveries. Our selection from the Definithe Principia consists of the introductory parts

250

years. Nevertheless,

Newton

himself had not

it

much

—

—

tions

and the Axioms.

Newton was born At one point early in

Isaac side.

in

1642 in the English countryhe was required to work

his life

on a farm in order to assist his mother. Fortunately for the world, however, he neglected his farm work for mathematics, for which he showed such aptitude that he eventually was sent to Trinity College in Cambridge. In 1667, he became a Fellow of Trinity College; two years later, he was appointed Lucasian Professor of Mathematics. It is pleasant to be able to report that Newton received recognition and his share of the world's honors during his lifetime. In addition to his position at Trinity, Newton also was a member of the Royal Sopresident in 1703 and remained in that Newton served the government with distinction as first Warden and then Master of the Mint; in 1705 he was knighted. Upon his death, he was ciety;

he became

its

position until his death in 1727.

buried in Westminster Abbey. As an indication of the esteem in which Newton was held by his friends and contemporaries, let us quote the last line of Edmund Halley's ode to Newton:

"Nearer

to the

gods no mortal

may

approach."

Breakthroughs in Physics

154

Sir Isaac

Newton: Mathematical

Principles

of J^atural Philosophy*

DEFINITIONS

Definition I The quantity of matter is the measure of the same, from its density and bulk conjointly.

arising

Thus air of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction, and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body, for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shown hereafter.

Definition

II

The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly. * From Mathematical Principles of Natural Philosophy and His System of the World, trans, by Andrew Motte in 1729. Translations revised by Florian Cajori. Berkeley, California: University

of California Press, 1946, pp. 1-28.

Mathematical Principles of Physics

155

of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple.

The motion

Definition

III

vis insita, or innate force of matter, is a power of resistby which every body, as much as in it lies, continues in its present state, whether it be of rest, or of moving uniformly forwards in a right line.

The

ing,

always proportional to the body whose force nothing from the inactivity of the mass, but body, from the inert nain our manner of conceiving it. ture of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita may, by a most significant name, be called inertia {vis inertice) or force of inactivity. But a body only exerts this force when another force, impressed upon it, endeavors to change its condition; and the exercise of this force may be considered as both resistance and impulse; it is resistance so far as the body, for maintaining its present state, opposes the force impressed; it is impulse so far as the body, by not easily giving way to the impressed force of another, endeavors to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished; nor are those bodies always truly at rest, which commonly are taken to be so. This force

it is

and

is

differs

A

Definition IV

An

impressed force is an action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a right line.

This force consists in the action only, and remains no longer in the body when the action is over. For a body maintains every new state it acquires, by its inertia only. But impressed forces are of different origins, as from percussion,

from

pressure,

from

centripetal force.

156

Breakthroughs in Physics Definition

V

A

centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre.

Of this sort is gravity, by which bodies tend to the centre of the earth; magnetism, by which iron tends to the loadand that force, whatever it is, by which the planets continually drawn aside from the rectilinear motions, which otherwise they would pursue, and made to revolve in stone;

are

A

stone, whirled about in a sling, encurvilinear orbits. deavors to recede from the hand that turns it; and by that endeavor, distends the sling, and that with so much the greater force, as it is revolved with the greater velocity, and as soon as it is let go, flies away. That force which opposes itself to this endeavor, and by which the sling continually draws back the stone towards the hand; and retains it in its orbit, because it is directed to the hand as the centre of

the orbit, I call the centripetal force. And the same thing is to be understood of all bodies, revolved in any orbits. They all endeavor to recede from the centres of their orbits; and were it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call centripetal,

A

would

fly off in right lines,

with

was not for the force of gravity, would not deviate towards the earth, but would go off from it in a right line, and that with an uniform motion, if the resistance of the air was taken away. It is by its gravity that it is drawn aside continually from its rectilinear course, and made to deviate towards the earth, more or less, according to the force of its gravity, and the an uniform motion.

projectile, if

it

its motion. The less its gravity is, or the quantity matter, or the greater the velocity with which it is projected, the less will it deviate from a rectilinear course, and the farther it will go. If a leaden ball, projected from the top of a mountain by the force of gunpowder, with a given velocity, and in a direction parallel to the horizon, is carried in a curved line to the distance of two miles before it falls to the ground; the same, if the resistance of the air were taken away, with a double or decuple velocity, would fly twice or ten times as far. And by increasing the velocity, we may at pleasure increase the distance to which it might be projected, and diminish the curvature of the line which it might describe, till at last it should fall at the distance of 10, 30, or 90 degrees, or even might go quite round the whole earth before it falls; or lastly, so that it might never

velocity of

of

its

Mathematical Principles of Phystcs

157

fall to the earth, but go forwards into the celestial spaces, and proceed in its motion in infinitum. And after the same

by the force of gravity, may be an orbit, and go round the whole earth, the moon also, either by the force of gravity, if it is endued with gravity, or by any other force, that impels it towards the earth, may be continually drawn aside towards the earth, out of the rectilinear way which by its innate force it would pursue; and would be made to revolve in the orbit which it now describes; nor could the moon without some such force be retained in its orbit. If this force was too

manner

made

that a projectile,

to revolve in

small,

it

would not

tilinear course; if

it

sufficiently turn the

was too

and draw down the

great,

moon from

it

its

moon

out of a recit too much,

would turn

orbit towards the earth.

necessary that the force be of a just quantity, and it belongs to the mathematicians to find the force that may serve exactly to retain a body in a given orbit with a given velocity; and vice versa, to determine the curvilinear way into which a body projected from a given place* with a given velocity, may be made to deviate from its natural rectilinear way, by means of a given force. It

is

The quantity of any centripetal force may be considered as of three kinds: absolute, accelerative, and motive. Definition VI The absolute quantity of a centripetal force is the measure of the same, proportional to the efficacy of the cause that propagates it from the centre, through the spaces round about. is greater in one loadstone and another, according to their sizes and strength of in-

Thus the magnetic force less in

tensity.

Definition VII The accelerative quantity of a centripetal force is the measure of the same, proportional to the velocity which it generates in a given time.

Thus the force of the same loadstone distance,

and

less

at

a greater:

greater in valleys, less

and yet

less

(as shall

is

greater at a less

also the force of gravity

is

on tops of exceeding high mountains; hereafter be shown), at greater dis-

158

Breakthroughs in Physics

tances from the body of the earth; but at equal distances, it is the same everywhere; because (taking away, or allowing for, the resistance of the air), it equally accelerates all falling bodies, whether heavy or light, great or small.

Definition VIII The motive quantity of a

centripetal force

is

the same, proportional to the motion which

the it

measure of generates in

a given time.

Thus the weight is greater in a greater body, less in a less body; and, in the same body, it is greater near to the earth, and less at remoter distances. This sort of quantity is the centripetency, or propension of the whole body towards the centre, or, as I may say, its weight; and it is always known by the quantity of an equal and contrary force just sufficient to hinder the descent of the body. These quantities of forces, we may, for the sake of brevity, by the names of motive, accelerative, and absolute forces; and, for the sake of distinction, consider them with respect to the bodies that tend to the centre, to the places of those bodies, and to the centre of force towards which they tend; that is to say, I refer the motive force to the body as an endeavor and propensity of the whole towards a centre, arising from the propensities of the several parts taken together; the accelerative force to the place of the body, as a certain power diffused from the centre to all places around to move the bodies that are in them; and the absolute force to the centre, as endued with some cause, without which those motive forces would not be propagated through the spaces round about; whether that cause be some central body (such as is the magnet in the centre of the magnetic force, or the earth in the centre of the gravitating force), or anything else that does not yet appear. For I here design only to give a mathematical notion of those forces, without considering their physical causes and seats. Wherefore the accelerative force will stand in the same relation to the motive, as celerity does to motion. For the quantity of motion arises from the celerity multiplied by the call

quantity of matter;

and the motive force arises from the by the same quantity of matter.

accelerative force multiplied

For the sum of the actions of the accelerative force, upon the several particles of the body, is the motive force of the whole. Hence it is, that near the surface of the earth, where

Mathematical Principles of Physics

159

the accelerative gravity, or force productive of gravity, in all bodies is the same, the motive gravity or the weight is as the body; but if we should ascend to higher regions, where the accelerative gravity is less, the weight would be equally diminished, and would always be as the product of the body, by the accelerative gravity. So in those regions, where the

gravity is diminished into one-half, the weight of a body two or three times less, will be four or six times

accelerative less.

and impulses, in the same sense, and motive; and use the words attraction, impulse, or propensity of any sort towards a centre, promiscuously, and indifferently, one for another; considering those I

likewise call attractions

accelerative,

forces not physically, but mathematically: wherefore the reader is not to imagine that by those words I anywhere take upon me to define the kind, or the manner of any action, the causes or the physical reason thereof, or that I attribute forces, in a true and physical sense, to certain centres (which are only mathematical points); when at any time I happen to speak of centres as attracting, or as endued with attractive powers.

Scholium Hitherto I have laid down the definitions of such words as known, and explained the sense in which I would have them to be understood in the following discourse. I do not define time, space, place, and motion, as being well known to all. Only I must observe, that the common people conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common. I. Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year. II. Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to are less

Breakthroughs in Physics

160

and which is commonly taken for immovable space; the dimension of a subterraneous, an aerial, or celesspace, determined by its position in respect of the earth.

bodies;

such tial

is

Absolute and relative space are the same in figure and magnitude; but they do not remain always numerically the same. For if the earth, for instance, moves, a space of our air, which relatively and in respect of the earth remains always the same, will at one time be one part of the absolute space into which the air passes; at another time it will be another part of the same, and so, absolutely understood, it will be continually changed. III. Place is a part of space which a body takes up, and is according to the space, either absolute or relative. I say, a part of space; not the situation, nor the external surface of the body. For the places of equal solids are always equal; but their surfaces, by reason of their dissimilar figures, are often unequal. Positions properly have no quantity, nor are they so much the places themselves, as the properties of places. The motion of the whole is the same with the sum of the motions of the parts; that is, the translation of the whole, out of its place, is the same thing with the sum of the translations of the parts out of their places; and therefore the place of the whole is the same as the sum of the places of and for that reason, it is internal, and in the whole

the parts,

body. IV. Absolute motion is the translation of a body from one absolute place into another; and relative motion, the translation from one relative place into another. Thus in a shipj under sail, the relative place of a body is that part of the) ship which the body possesses; or that part of the cavi which the body fills, and which therefore moves togethe with the ship: and relative rest is the continuance of the bod in the same part of the ship, or of its cavity. But real, ab solute rest, is the continuance of the body in the same pa of that immovable space, in which the ship itself, its cavity

and

moved. Wherefore, if the earth is which relatively rests in the ship,| will really and absolutely move with the same velocity! which the ship has on the earth. But if the earth also moves! the true and absolute motion of the body will arise, partly from the true motion of the earth, in immovable space partly from the relative motion of the ship on the earth: and if the body moves also relatively in the ship, its true) motion will arise, partly from the true motion of the earth in immovable space, and partly from the relative motions a: all

that

it

contains,

really at rest, the body,

is

Mathematical Principles of Physics

161

jvell of the ship on the earth, as of the body in the ship; and from these relative motions will arise the relative motion 3f the body on the earth. As if that part of the earth, where the ship is, was truly moved towards the east, with a velocity of 10010 parts; while the ship itself, with a fresh gale, and full sails, is carried towards the west, with a velocity expressed by 10 of those parts; but a sailor walks in the ship towards the east, with 1 part of the said velocity; then the sailor will be moved truly in immovable space towards the east, with a velocity of 10001 parts, and relatively on the earth towards the west, with a velocity of 9 of those

parts.

Absolute time, in astronomy, is distinguished from relaby the equation or correction of the apparent time. For the natural days are truly unequal, though they are commonly considered as equal, and used for a measure of time; astronomers correct this inequality that they may measure the celestial motions by a more accurate time. It may be, that there is no such thing as an equable motion, whereby time may be accurately measured. All motions may be accelerated and retarded, but the flowing of absolute time is not liable to any change. The duration or perseverance of the existence of things remains the same, whether the motive,

are swift or slow, or none at all: and therefore this duration ought to be distinguished from what are only sen-

tions

measures thereof; and from which we deduce it, by means of the astronomical equation. The necessity of this equation, for determining the times of a phenomenon, is evinced as well from the experiments of the pendulum clock, as by eclipses of the satellites of Jupiter. sible

As the order of the parts of time is immutable, so also is the order of the parts of space. Suppose those parts to be

moved

out of their places, and they will be moved (if the expression may be allowed) out of themselves. For times and spaces are, as it were, the places as well of themselves as of all other things. All things are placed in time as to order of succession; and in space as to order of situation. It is from their essence or nature that they are places; and that the primary places of things should be movable, is absurd. These are therefore the absolute places; and translations out of those places, are the only absolute motions.

But because the parts of space cannot be seen, or disfrom one another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body considered as

tinguished

Breakthroughs in Physics

162

we we

and then with respect to motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones; and that without any inconvenience in common affairs; but in philosophical disquisitions, we ought to abstract from our senses, and consider things themselves, distinct from what are only sensible measures of them. For it may be that there is no body really at rest, to which the places and motions of immovable, such places,

define

all

estimate

places;

all

may be referred. But we may distinguish rest and motion, absolute and relative, one from the other by their properties, causes, and effects. It is a property of rest, that bodies really at rest do rest in respect to one another. And therefore as it is possible, that in the remote regions of the fixed stars, or perhaps far beyond them, there may be some body absolutely at rest; but impossible to know, from the position of bodies to one another in our regions, whether any of these do keep the same position to that remote body, it follows that absolute rest cannot be determined from the position of bodies in our regions. It is a property of motion, that the parts, which retain given positions to their wholes, do partake of the motions of those wholes. For all the parts of revolving bodies endeavor to recede from the axis of motion; and the impetus of bodies moving forwards arises from the joint impetus of all the parts. Therefore, if surrounding bodies are moved, others

those that are relatively at rest within them will partake of motion. Upon which account, the true and absolute motion of a body cannot be determined by the translation of it from those which only seem to rest; for the external bodies ought not only to appear at rest, but to be really at rest. For otherwise, all included bodies, besides their translation from near the surrounding ones, partake likewise of their true motions; and though that translation were not made, they would not be really at rest, but only seem to be so. For the surrounding bodies stand in the like relation to the surrounded as the exterior part of a whole does to the interior, or as the shell does to the kernel; but if the shell moves, the kernel will also move, as being part of the whole, their

without any removal from near the shell. property, near akin to the preceding, is this, that if a place is moved, whatever is placed therein moves along with it; and therefore a body, which is moved from a place m motion, partakes also of the motion of its place. Upon which

A

Mathematical Principles of Physics account, all motions, from places in motion, are no than parts of entire and absolute motions; and every motion is composed of the motion of the body out first place, and the motion of this place out of its

163 other entire

of

its

place;

and so on, until we come to some immovable place, as in the before-mentioned example of the sailor. Wherefore, entire and absolute motions can be no otherwise determined than by immovable places; and for that reason I did before

immovable places, but relaones to movable places. Now no other places are immovable but those that, from infinity to infinity, do all retain the same given position one to another; and upon this account must ever remain unmoved; and do thereby constitute refer those absolute motions to tive

immovable space. The causes by which true and relative motions are distinguished, one from the other, are the forces impressed upon bodies to generate motion. True motion is neither generated nor altered, but by some force impressed upon the body moved; but relative motion may be generated or altered without any force impressed upon the body. For it is sufficient only to impress some force on other bodies with which the former is compared, that by their giving way, that relation may be changed, in which the relative rest or motion of this other body did consist. Again, true motion suffers always some change from any force impressed upon the moving body; but relative motion does not necessarily undergo any change by such forces. For if the same forces are likewise impressed on those other bodies, with which the comparison is made, that the relative position

may

be preserved, then that condi-

which the relative motion consists. And therefore any relative motion may be changed when the true motion remains unaltered, and the relative may be tion will be preserved in

when the true suffers some change. Thus, true moby no means consists in such relations. The effects which distinguish absolute from relative motion are, the forces of receding from the axis of circular motion. For there are no such forces in a circular motion purely relative, but in a true and absolute circular motion, they are greater or less, according to the quantity of the motion. If a vessel, hung by a long cord, is so often turned

preserved tion

about that the cord

is

strongly twisted, then

filled

with water,

and held at rest together with the water; thereupon, by the sudden action of another force, it is whirled about the contrary way, and while the cord is untwisting itself, the vessel continues for

some time

in this motion; the surface of the

Breakthroughs in Physics

164 water will at move; but after

first

be plain, as before the vessel began to

by gradually communicating motion to the water, will make it begin sensibly to revolve, and recede by little and little from the middle, and ascend to the sides of the vessel, forming itself into a concave figure (as I have experienced), and the swifter the motion becomes, the higher will the water rise, till at last, performing its revolutions in the same times with the vessel, it becomes relatively at rest in it. This ascent of the water shows its endeavor to recede from the axis of its motion; and the true and absolute circular motion of the water, which is here directly contrary to the relative, becomes known, and may be measured by this endeavor. At first, when the relative motion of the water in the vessel was greatest, it produced no endeavor to recede from the axis; the water showed no tendency to the circumference, nor any ascent towards the sides of the vessel, but remained of a plain surface, and therefore its true circular motion had not yet begun. But afterwards, when the relative motion of the water had decreased, the ascent thereof towards the sides of the vessel proved its endeavor to recede from the axis; and this endeavor showed the real circular motion of the water continually increasing, till it had acquired its greatest quantity, when the water rested relatively in the vessel. And therefore this endeavor does not depend upon any translation of the water in respect of the ambient bodies, nor can true circular motion be defined by such translation. There is only one real circular motion of any one revolving body, corresponding to only one power of endeavoring to recede from its axis of motion, as its proper and adequate effect; but relative motions, in one and the same body, are innumerable, according that, the vessel,

its

it bears to external bodies, and, like other relations, are altogether destitute of any real effect, any otherwise than they may perhaps partake of that one only true motion. And therefore in their system who suppose that our heavens, revolving below the sphere of the fixed stars, carry the planets along with them; the several parts of those heavens, and the planets, which are indeed relatively at rest in their heavens, do yet really move. For they change their position one to another (which never happens to bodies truly at rest), and being carried together with their heavens, partake of their motions, and as parts of revolving wholes, endeavor to recede from the axis of their motions. Wherefore relative quantities are not the quantities themselves, whose names they bear, but those sensible measures

to the various relations

Mathematical Principles of Physics

165

of them (either accurate or inaccurate), which are commonly used instead of the measured quantities themselves. And if the meaning of words is to be determined by their use, then by the names time, space, place, and motion, their [sensible] measures are properly to be understood; and the expression will be unusual, and purely mathematical, if the measured quantities themselves are meant. On this account, those violate the accuracy of language, which ought to be kept

who Nor do

interpret these words for the measured quantithose less defile the purity of mathematical and philosophical truths, who confound real quantities with their

precise, ties.

and sensible measures. indeed a matter of great difficulty to discover, and effectually to distinguish, the true motions of particular bodies from the apparent; because the parts of that immovable space, in which those motions are performed, do by no means come under the observation of our senses. Yet the thing is not altogether desperate; for we have some arguments to guide us, partly from the apparent motions, which are the differences of the true motions; partly from the forces, which s are the causes and effects of the true motions. For instance, if two globes, kept at a given distance one from the other by means of a cord that connects them, were revolved about their common centre of gravity, we might, from the tension of the cord, discover the endeavor of the globes to recede from the axis of their motion, and from thence we might compute the quantity of their circular motions. And then if any equal forces should be impressed at once on the alternate faces of the globes to augment or diminish their circular motions, from the increase or decrease of the tension of the cord, we might infer the increment or decrement of their motions; and thence would be found on what faces those forces ought to be impressed, that the motions of the globes might be most augmented; that is, we might discover their hindmost faces, or those which, in the circular motion, do follow. But the faces which follow being known, and consequently the opposite ones that precede, we should likewise know the determination of their motions. And thus we might find both the quantity and the determination of this circular motion, even in an immense vacuum, where there was nothing external or sensible with which the globes could be compared. But now, if in that space some remote bodies were placed that kept always a given position one to another, as the fixed stars do in our regions, we could not indeed determine relations It

is

from the

relative translation of the globes

among

those bodies,

166

Breakthroughs in Physics

whether the motion did belong to the globes or to the But if we observed the cord, and found that its tension was that very tension which the motions of the globes required, we might conclude the motion to be in the globes, and the bodies to be at rest; and then, lastly, from the transbodies.

lation of the globes among the bodies, we should find the determination of their motions. But how we are to obtain the true motions from their causes, effects, and apparent differences, and the converse, shall be explained more at large in the following treatise. For to this end it was that I composed it.

AXIOMS,

OR LAWS OF MOTION Law I

Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it. Projectiles continue in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. top, whose parts by their cohesion are continually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in freer spaces, preserve their motions both progressive and circular for a much longer time.

A

Law II The change of motion impressed; and is made which that force

is

is

proportional to the motive force

in the direction of the right line in

impressed.

any force generates a motion, a double force will generdouble the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subtracted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both. If

ate

Mathematical Principles of Physics

167

Law III To

every action there

is

always opposed an equal reaction:

the mutual actions of two bodies upon each other are always equal, and directed to contrary parts. or,

Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone; for the distended rope, by the same endeavor to relax or unbend itself, will draw the horse as much towards the stone as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinge upon another, and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are inversely proportional to the bodies. This law takes place also in attractions, as will be proved in the next Scholium.

Corollary I

A

body, acted on by two forces simultaneously, will describe the diagonal of a parallelogram in the same time as it would describe the sides by those forces separately.

M

impressed apart by the force A, should with an uniform motion be carried from A to B, and by the force N impressed apart in the same place, should be carried from A to C, let the parallelogram ABCD be completed, and, by both forces acting together, it will in the same time be carried in the diagonal from A to D. For since the force N acts in the direction of the line AC, parallel to BD, this force (by the second Law) If a

body

in the place

in a given time,

Breakthroughs in Physics

168

by the other force carried towards the line BD. The body therefore will arrive at the line in the same time, whether the force be impressed or not; and therefore at the end of that time it will be found somewhere in the line BD. will not at all alter the velocity generated

M, by which

the

body

is

BD

N

By the same argument, at the end of the same time it will be found somewhere in the line CD. Therefore it will be found in the point D, where both lines meet. But it will move in a right line from A to D, by Law I.

Corollary II

And

hence is explained the composition of any one direct force AD, out of any two oblique forces and CD; and, on the contrary, the resolution of any one direct force into two oblique forces and CD: which composition

AC

AD

AC

and

resolution are abundantly confirmed

OM

from mechanics.

ON

As if the unequal radii and drawn from the centre of any wheel, should sustain the weights and P by the cords and NP; and the forces of those weights to move the wheel were required. Through the centre

A

O

MA

O

draw the right line KOL, meeting the cords perpendicularly in K and L; and from the centre O, with OL the greater of the distances OK and OL, describe a circle, meeting the cord in D; and drawing OD, make AC parallel and DC perpendic-

MA

ular thereto. Now, it being indifferent whether the points K, L, D, of the cords be fixed to the plane of the wheel or not, the weights will have the same effect whether they are suspended from the points and L, or from and L. Let

K

D

Mathematical Principles of Physics

169

A

be represented by the line the whole force of the weight and CD, of AD, and let it be resolved into the forces directly from which the force AC, drawing the radius

AC OD

the centre, will have no effect to move the wheel; but the perpendicularly, will other force DC, drawing the radius have the same effect as if it drew perpendicularly the radius OL equal to OD; that is, it will have the same effect as the

DO

weight P,

if

P

:

A = DC

:

DA,

ADC and DOK DC DA = OK OD = OK

but because the triangles :

:

are similar, :

OL.

Therefore,

P As

these radii

pollent,

:

A=

lie

radius

in the

and so remain

known

OK

same in

:

radius

OL.

right line they will be equi-

equilibrium; which

is

the well-

property of the balance, the lever, and the wheel. If either weight is greater than in this ratio, its force to move the wheel will be so much greater. If the weight p P, is partly suspended by the cord Np, partly sustained by the oblique plane pG; draw pH, NH, the former perpendicular to the horizon, the latter to the plane pG; and if the force of the weight p tending downwards is represented by the line pH, it may be resolved into the forces pN, HN. If there was any plane pQ, perpendicular to the cord pN, cutting the other plane pG in a line parallel to the horizon, and the weight p was supported only by those

=

Breakthroughs in Physics

170

planes pQ, pG, it would press those planes perpendicularly with the forces pN, HN; to wit, the plane pQ with the force pN, and the plane pG with the force HN. And therefore if the plane pQ was taken away, so that the weight might stretch the cord, because the cord, now sustaining the weight, supplied the place of the plane that was removed, it would be strained by the same force pN which pressed upon the plane before. Therefore, the tension of

Therefore,

p

:

pN

tension of

:

PN =

line

pN

:

line

pH.

if

A = OK OL = :

line

pH

:

line

pN,

then the weights p and A will have the same effect towards moving the wheel, and will therefore sustain each other; as anyone may find by experiment. But the weight p pressing upon those two oblique planes, may be considered as a wedge between the two internal surfaces of a body split by it; and hence the forces of the wedge and the mallet may be determined: because the force with which the weight p presses the plane pQ is to the force with which the same, whether by its own gravity, or by the blow of a mallet, is impelled in the direction of the line pH

towards both the planes, as

pN pH; :

and to the force with which

it

presses the other plane

pG,

as

pN NH. :

thus the force of the screw may be deduced from a it being no other than a wedge im-j pelled with the force of a lever. Therefore the use of thisj Corollary spreads far and wide, and by that diffusive extent the truth thereof is further confirmed. For on what has been said depends the whole doctrine of mechanics variously demonstrated by different authors. For from hence are easily deduced the forces of machines, which are compounded ol wheels, pulleys, levers, cords, and weights, ascending directly or obliquely, and other mechanical powers; as also the force! of the tendons to move the bones of animals.

And

like resolution of forces;

Mathematical Principles of Physics

Corollary

171

III

The quantity of motion, which is obtained by taking the sum ?f the motions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers

from

no change

the action of the bodies

among

themselves.

For action and its opposite reaction are equal, by Law in, and therefore, by Law n, they produce in the motions equal changes towards opposite parts. Therefore if the motions are directed towards the same parts, whatever is added to the motion of the preceding body will be subtracted from the motion of that which follows; so that the sum will be the same as before. If the bodies meet, with contrary motions, there will be an equal deduction from the motions of both; and therefore the difference of the motions directed towards opposite parts will remain the same. is 3 times greater than the Thus, if a spherical body spherical body B, and has a velocity = 2, and B follows in the same direction with a velocity =10, then the

A

motion of

A

:

motion of B

=

6

:

10.

Suppose, then, their motions to be of 6 parts and of 10 parts, and the sum will be 16 parts. Therefore, upon the meeting of the bodies, if A acquire 3, 4, or 5 parts of motion, B will

many; and therefore after reflection A will proceed 10, or 11 parts, and B with 7, 6, or 5 parts; the sum remaining always of 16 parts as before. If the body A lose as

with

9,

acquire 9, 10, 11, or 12 parts of motion, and therefore after 15, 16, 17, or 18 parts, the body B, losing so many parts as has got, will either proceed with 1 part, having lost 9, or stop and remain at rest, as having

meeting proceed with

A

whole progressive motion of 10 parts; or it will go back with 1 part, having not only lost its whole motion, but (if I may so say) one part more; or it will go back with 2 parts, because a progressive motion of 12 parts is taken off. lost its

And

so the

15

sums of the conspiring motions,

+

1

or

16

+

0,

and the differences of the contrary motions, 17

—

1

and

18

—

2,

always be equal to 16 parts, as they were before the meeting and reflection of the bodies. But the motions being known with which the bodies proceed after reflection, the will

Breakthroughs

172

in Physics

known, by taking the velocity motion after is to the last case, where the

velocity of either will be also

after to the velocity before reflection, as the

motion before. As in

the

motion of

=

A

that

A

before reflection (6) motion of after (18) before (2): velocity of after (jc); :

A

velocity of

A

is,

6

But

:

18

=

2

x,

:

x

=

6.

the bodies are either not spherical, or, moving in different right lines, impinge obliquely one upon the other,

and

if

motions after reflection are required, in those cases to determine the position of the plane that touches the bodies in the point of impact, then the motion of each body (by Cor. II) is to be resolved into two, one perpendicular to that plane, and the other parallel to it. This done, because the bodies act upon each other in the direction of a line perpendicular to this plane, the parallel motions are to be re-

we

their

are

tained

first

the

same

after

reflection

as

before;

and

to

the

perpendicular motions we are to assign equal changes towards the contrary parts; in such manner that the sum of the conspiring and the difference of the contrary motions may remain the same as before. From such kind of reflections sometimes arise also the circular motions of bodies about their own centres. But these are cases which I do not consider in what follows; and it would be too tedious to demonstrate every particular case that relates to this subject.

Corollary IV The common centre of gravity of two or more bodies does not motion or rest by the actions of the bodies themselves; and therefore the common centre of gravity of all bodies acting upon each other (excluding external actions and impediments) is either at rest, or moves alter its state of

among

uniformly in a right

line.

For lines,

if two points proceed with an uniform motion in right and their distance be divided in a given ratio, the divid-

ing point will be either at rest, or proceed uniformly in a is demonstrated hereafter in Lem. xxiii and Corollary, when the points are moved in the same plane; right line. This

and by a like way of arguing, it may be demonstrated when the points are not moved in the same plane. Therefore if any number of bodies move uniformly in right lines, the common centre of gravity of any two of them is either at rest, or

P

Mathematical Principles of Physics

173

proceeds uniformly in a right line; because the line which connects the centres of those two bodies so moving is divided it that common centre in a given ratio. In like manner the jommon centre of those two and that of a third body will be ;ither at rest or moving uniformly in a right line; because at :hat centre the distance between the common centre of the

:wo bodies, and the centre of this last, is divided in a given ratio. In like manner the common centre of these three, and of a fourth body, is either at rest, or moves uniformly in a -ight line; because the distance between the common centre 3f the three bodies, and the centre of the fourth, is there also divided in a given ratio, and so on in infinitum. Therefore, in i system of bodies where there is neither any mutual action imong themselves, nor any foreign force impressed upon them from without, and which consequently move uniformly in right lines, the common centre of gravity of them all is sither at rest or moves uniformly forwards in a right line. Moreover, in a system of two bodies acting upon each other, since the distances between their centres and the common centre of gravity of both are reciprocally as the bodies,

motions of those bodies, whether of approaching or of receding from that centre, will be equal among themselves. Therefore since the changes which happen to motions are equal and directed to contrary parts, the common centre of those bodies, by their mutual action between themselves, the relative to

is

to

neither accelerated nor retarded, nor suffers any change as its state of motion or rest. But in a system of several

common

bodies, because the

centre of gravity of any two

no change in its state by and much less the common centre of gravity of the others with which that action does not intervene; but the distance between those two centres is divided by the com-

acting

upon each other

suffers

that action;

mon

centre of gravity of all the bodies into parts inversely proportional to the total sums of those bodies whose centres they are; and therefore while those two centres retain their the common centre of all does also manifest that the common centre of all never suffers any change in the state of its motion or rest from the actions of any two bodies between themselves. But in such a system all the actions of the bodies among themselves either happen between two bodies, or are composed of

state of

retain

motion or

its

state:

it

rest,

is

actions interchanged between

some two

bodies; and there-

fore they do never produce any alteration in the

centre of

all

as to

its

state of

motion or

rest.

common

Wherefore since

Breakthroughs in Physics

174 that centre, either

is

when

at rest or

do not act one upon another, moves uniformly forwards in some right

the bodies

notwithstanding the mutual actions of the bodies themselves, always continue in its state, either of rest, or of proceeding uniformly in a right line, unless it is forced out of this state by the action of some power impressed from without upon the whole system. And therefore the same law takes place in a system consisting of many bodies as in one single body, with regard to their persevering in their state of motion or of rest. For the progressive motion, whether of one single body, or of a whole system of bodies, is always to be estimated from the motion of the centre of line, it will,

among

gravity.

Corollary V The motions of bodies included

among

same moves without any circular mo-

in a given space are the

themselves, whether that space

uniformly forwards in a right

line

is

at rest, or

tion.

For the differences of the motions tending towards the same and the sums of those that tend towards contrary parts, are, at first (by supposition), in both cases the same; and it is from those sums and differences that the collisions and impulses do arise with which the bodies impinge one upon another. Wherefore (by Law n), the effects of those collisions will be equal in both cases; and therefore the mutual motions parts,

among themselves in the one case will remain equal to the motions of the bodies among themselves in the other. clear proof of this we have from the experiment of of the bodies

A

where all motions happen after the same manner whether the ship is at rest, or is carried uniformly forwards m a ship;

a right

line.

Corollary VI bodies,

//

moved

in

any manner among themselves,

art

urged in the direction of parallel lines by equal accelerativ* forces, they will all continue to move among themselves, afte the same manner as if they had not been urged by thos forces.

For these forces acting equally (with respect to the quanti of the bodies to be moved), and in the direction of paralle

ties

175

Mathematical Principles of Physics

(by Law n) move all the bodies equally (as to velocity), and therefore will never produce any change in the positions or motions of the bodies among themselves. lines, will

Scholium Hitherto I have laid down such principles as have been received by mathematicians, and are confirmed by abundance of experiments. By the first two Laws and the first two Corollaries, Galileo discovered that the descent of bodies varied as the square of the time (in duplicata ratione temporis) and that the motion of projectiles was in the curve of a parabola; experience agreeing with both, unless so far as these motions are a little retarded by the resistance of the air. When a body is falling, the uniform force of its gravity acting equally, impresses, in equal intervals of time, equal forces

upon

and therefore generates equal velocities; and whole time impresses a whole force, and generates a

that body,

in the

And the spaces deproportional times are as the product of the velocities and the times; that is, as the squares of the times. And when a body is thrown upwards, its uniform gravity impresses forces and reduces velocities proportional to the times; and the times of ascending to the greatest heights are as the velocities to be taken away, and those heights are as the whole velocity proportional to the time. scribed

in

product of the velocities and the times, or as the squares of the velocities. And if a body be projected in any direction, the motion arising from its projection is compounded with the by its motion arising from its gravity. Thus, if the body motion of projection alone could describe in a given time the right line AB, and with its motion of falling alone could describe in the same time the altitude AC; complete the parallelogram ABCD, and the body by that compounded motion will at the end of the time be found in the place D; and the curved line AED, which that body describes, will be a parabola, to which the right line AB will be a tangent at A; and whose ordinate BD will be as the square of the line AB. On the

A

same Laws and Corollaries depend those which have been demonstrated con-

things

cerning the times of the vibration of pendulums, and are confirmed by the daily experiments of pendulum clocks. By the same, together with Law m, Sir Christopher Wren, Dr. Wallis, and

Breakthroughs in Physics

176

greatest geometers of our times, did severaldetermine the rules of the impact and reflection of hard bodies, and about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was somewhat earlier in the publication; then followed Sir Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiments on pendulums, which M. Mariotte soon after thought fit to ex-

Mr. Huygens, the ly

upon that subject. But to bring this experiment to an accurate agreement with the theory, we are to have due regard as well to the resistance of the air as to

plain in a treatise entirely

the elastic force of the concurring bodies. Let the spherical bodies A, B be suspended by the equal and parallel strings

|

g

F

t

H

AC, BD, from

the centres C, D. About with centres, these those lengths as radii, describe the semicircles EAF, GBH, bisected

A

to any by the radii CA, DB. Bring the body of the arc EAF, and (withdrawing the body B) let from thence, and after one oscillation suppose it to rewill be retardation arising to the point V: then let ST be a fourth the resistance of the air. Of this situated in the middle, namely, so that

respectively

point it

go

turn

from part,

R

RV

RV

RS = TV, and

RS ST = :

3

:

2,

then will ST represent very nearly the retardation during the descent from S to A. Restore the body B to its place; to be let fall from the point S, and, supposing the body the velocity thereof in the place of reflection A, without sensible error, will be the same as if it had descended in vacuo from the point T. Upon which account this velocity may be represented by the chord of the arc TA. For it is a proposition well known to geometers, that the velocity of a pendulous body in the lowest point is as the chord of the arc which it has described in its descent. After reflection, comes to the place s, and the body B suppose the body

A

A

Mathematical Principles of Physics

|

from which

177

Withdraw the body B, and find the place v, the body A, being let go, should after one

to the place k. if

the place r, st may be a fourth part of rv, so placed in the middle thereof as to leave rs equal to fv, and let the chord of the arc tA represent the velocity had in the place which the body immediately after reflection. For t will be the true and correct place to should have ascended, if the resistance of which the body the air had been taken off. In the same way we are to correct the place k to which the body B ascends, by finding the place / to which it should have ascended in vacuo. And oscillation return to

A

A

A

may be subjected to experiment, in the same we were really placed in vacuo. These things we are to take the product (if I may so say)

thus everything

manner

as if

being done, of the body A, by the chord of the arc

TA (which repremotion in the place A immediately before reflection; and then by the chord of the arc tA, that we may have its motion in the place A immediately after reflection. And so we are to take the product of the body B by the chord of the arc B/, that we may have the motion of the same immediately after reflection. And in like manner, when two bodies are let go together from different places, we are to find the motion of each, as well before as after reflection; and then we may compare the motions between themselves, and collect the effects of the reflection. Thus trying the thing with pendulums of 10 feet, in unequal as well as equal bodies, and making the bodies to concur after a descent through large spaces, as of 8, 12, or 16 feet, I found always, without an error of 3 inches, that when the bodies concurred together directly, equal changes towards the contrary parts were produced in their motions, and, of consequence, that the action and reaction were always equal. As if the body A impinged upon the body B at rest with 9 parts of motion, and losing 7, proceeded after reflection with 2, the body B was carried backwards with those 7 parts. If the bodies concurred with contrary motions, A with 12 parts of motion, and B with sents

its

velocity), that

A

we may have

its

receded with 2, B receded with 8; namely, with 6, then if a deduction of 14 parts of motion on each side. For from the motion of subtracting 12 parts, nothing will remain; but subtracting 2 parts more, a motion will be generated of 2 parts towards the contrary way; and so, from the motion of the body B of 6 parts, subtracting 14 parts, a motion is generated of 8 parts towards the contrary way. But if the bodies were made both to move towards the same way, A,

A

Breakthroughs in Physics

178

the swifter, with 14 parts of motion, B, the slower, with 5, and after reflection went on with 5, B likewise went on with 14 parts; 9 parts being transferred from to B. And so in other cases. By the meeting and collision of bodies, the quantity of motion, obtained from the sum of the mo-

A

A

same way, or from the difference of those that were directed towards contrary ways, was never changed. For the error of an inch or two in measures may be easily ascribed to the difficulty of executing everything with accuracy. It was not easy to let go the two pendulums so exactly together that the bodies should impinge one upon the other in the lowermost place AB; nor to mark the places s , and k, to which the bodies ascended after impact. Nay, tions directed towards the

1

and some

might have happened from the unequal of the pendulous bodies themselves, and from the irregularity of the texture proceeding from errors, too,

density of the

parts

other causes. But to prevent an objection that may perhaps be alleged against the rule, for the proof of which this experiment was made, as if this rule did suppose that the bodies were either absolutely hard, or at least perfectly elastic (whereas no such bodies are to be found in Nature), I must add, that the experiments we have been describing, by no means depending upon that quality of hardness, do succeed as well in soft as in hard bodies. For if the rule is to be tried in bodies not perfectly hard, we are only to diminish the reflection in such a certain proportion as the quantity of the elastic force requires. By the theory of Wren and Huygens, bodies absolutely hard return one from another with the same velocity with which they meet. But this may be affirmed with more certainty of bodies perfectly elastic. In bodies imperfectly elastic the velocity of the return is to be diminished together with the elastic force; because that force (except when the parts of bodies are bruised by their impact, or suffer some such extension as happens under the strokes of a hammer) is (as far as I can perceive) certain and determined, and makes the bodies to return one from the other with a

which is in a given ratio to that relative which they met. This I tried in balls of wool, made up tightly, and strongly compressed. For, first, by letting go the pendulous bodies, and measuring their reflection, I determined the quantity of their elastic force; and relative velocity,

velocity with

then, according to this force, estimated the reflections that

ought to happen in other cases of impact. And with this computation other experiments made afterwards did accord-

j

Mathematical Principles of Physics

179

ingly agree; the balls always receding one from the other with a relative velocity, which was to the relative velocity with which they met as about 5 to 9. Balls of steel returned with almost the same velocity; those of cork with a velocity something less; but in balls of glass the proportion was as about 15 to 16. And thus the third Law, so far as it regards percussions and reflections, is proved by a theory exactly agreeing with experience. In attractions, I briefly demonstrate the thing after this manner. Suppose an obstacle is interposed to hinder the meeting of any two bodies A, B, attracting one the other: then if either body, as A, is more attracted towards the other body B, than that other body B is towards the first body A, the obstacle will be more strongly urged by the pressure of than by the pressure of the body B, and therethe body fore will not remain in equilibrium: but the stronger pressure will prevail, and will make the system of the two bodies, together with the obstacle, to move directly towards the parts on which B lies; and in free spaces, to go forwards in infinitum with a motion continually accelerated; which is absurd and contrary to the first Law. For, by the first Law, the system ought to continue in its state of rest, or of moving uniformly forwards in a right line; and therefore the bodies must equally press the obstacle, and be equally attracted one by the other. I made the experiment on the loadstone and iron. If these, placed apart in proper vessels, are made to float by one another in standing water, neither of them will propel the other; but, by being equally attracted, they will sustain each other's pressure, and rest at

A

an equilibrium. So the gravitation between the earth and its parts is mutual. Let the earth FI be cut by any plane EG into two parts EGF and EGI, and their weights one towards the other will be mutually equal. For if by another plane HK, parallel to the former EG, the greater part EGI is cut into two parts EGKH and HKI, whereof HKI is equal to the part last in

first cut off, it is evident that the middle part EGKH have no propension by its proper weight towards either side, but will hang as it were, and rest in an equilibrium between both. But the one extreme part HKI will with its whole weight bear upon and press the middle part towards the other extreme part EGF; and therefore the force with which EGI, the sum of the parts HKI and EGKH, tends towards the third part EGF, is equal to the weight of the

EFG,

will

part

HKI,

that

is,

to

the weight of the third part

EGF.

Breakthroughs in Physics

180

And the

weights

of

EGI and EGF,

one

therefore

two parts

the

towards the other, are equal, as was to prove. And indeed if those weights were not equal, the whole earth floating in the nonI

would give way

resisting ether

to the greater weight, and, retiring

from

it,

would be carried

off in infinitum.

And

as those bodies are equi-

pollent in flection,

whose

the impact and re-

velocities are inversely as their innate forces,

of mechanic instruments those agents are and mutually sustain each the contrary pressure of the other, whose velocities, estimated according to the so

in

the use

equipollent,

determination of the forces, are inversely as the forces. So those weights are of equal force to move the arms of a balance, which during the play of the balance are inversely as their velocities upwards and downwards; that is, if the ascent or descent is direct, those weights are of equal force, which are inversely as the distances of the points at which they are suspended from the axis of the balance; but if they are turned aside by the interposition of oblique planes, or other obstacles, and made to ascend or descend obliquely, those bodies will be equipollent, which are inversely as the heights of their ascent and descent taken according to the perpendicular; and that on account of the determination of gravity downwards.

And in like manner in the pulley, or in a combination of pulleys, the force of a hand drawing the rope directly, which is to the weight, whether ascending directly or obliquely, as the velocity of the perpendicular ascent of the weight to the velocity of the hand that draws the rope, will sustain the weight. In clocks and such like instruments, made up from a combination of wheels, the contrary forces that promote and impede the motion of the wheels, if they are inversely as the velocities of the parts of the wheel on which they are impressed, will mutually sustain each other. The force of the screw to press a body is to the force of the hand that turns the handles by which it is moved as the circular velocity of the handle in that part where it is impelled by the hand is to the progressive velocity of the screw towards the pressed body.

Mathematical Principles of Physics

181

The

forces by which the wedge presses or drives the two parts of the wood it cleaves are to the force of the mallet upon the wedge as the progress of the wedge in the direc-

upon it by the mallet is to the which the parts of the wood yield to the wedge,

tion of the force impressed

velocity with

in the direction of lines perpendicular to

wedge.

And

the like account

is

the sides of the

to be given of

all

machines.

The power and use of machines consist only in this, that by diminishing the velocity we may augment the force, and the contrary; from whence, in all sorts of proper machines, we have the solution of this problem: To move a given weight with a given power, or with a given force to overother given resistance. For if machines are so contrived that the velocities of the agent and resistant are inversely as their forces, the agent will just sustain the resistant, but with a greater disparity of velocity will overcome

come any

it.

So that

come

all

if

the disparity of velocities

that resistance

is

so great as to over-

which commonly

arises either

from

the friction of contiguous bodies as they slide by one another, or from the cohesion of continuous bodies that are to be separated, or from the weights of bodies to be raised, the excess of the force remaining, after all those resistances are overcome, will produce an acceleration of motion proportional thereto, as well in the parts of the machine as in is not my present business. I was aiming only to show by those examples the great extent and certainty of the third Law of Motion. For if we estimate the action of the agent from the product of its force and velocity, and likewise the reaction of the

the resisting body. But to treat of mechanics

impediment from the product of the velocities of its several and the forces of resistance arising from the friction, cohesion, weight, and acceleration of those parts, the action and reaction in the use of all sorts of machines will be found always equal to one another. And so far as the action is propagated by the intervening instruments, and at last imparts,

upon the resisting body, the ultimate action will be always contrary to the reaction.

pressed

Breakthroughs in Physics

182

PART

II

In several previous chapters we have noted that the important achievement of an author did not consist so much in discovering some new phenomenon or even some new law of nature; many times, the big contribution lay in finding the principles of a branch of science and in then deriving important propositions from those principles. Thus we commented in Chapter One that Archimedes' greatness lay not simply in discovering the law of the lever (which probably had been known before his time), but in establishing the principles of the science of statics and deriving the law of the lever from them. Similarly, in Chapter Five, we praised Pascal for discerning the principles underlying the effects associated with a vacuum and deriving these effects from those principles rather than from an old-fashioned and discredited principle (the horror of the vacuum). Sir Isaac Newton is even more concerned than any of the earlier scientists with discovering and clarifying principles. The very name of his book indicates this. The work is called Mathematical Principles, thus indicating that Newton's treatment and interest is mathematical. Not that mathematical treatment of physical problems was new; had Archimedes so chosen, he might have called the work we studied in Chapter One the "Mathematical Principles of Statics," for that is certainly an accurate description of it. Newton's work is remarkable, however, for the fact that (as the title indicates) its subject matter is not just one branch of science like statics or hydrostatics but all of natural science. Its full title in English is Mathematical Principles of Natural Philosophy; this long name is often abbreviated simply to Principia, the Latin word for "principles." If we take the title literally, we would expect the Principia to contain the mathematical principles of statics (the subject of On the Equilibrium of Planes), of dynamics (Galileo's new science in the "Third Day" of his Two New Sciences), of hydrostatics (discussed in Archimedes' On Floating Bodies and in Pascal's Great Experiment), of astronomy (discussed in the Starry Messenger) in other words, the principles of everything we have so far read about in this book. In addition, the Principia should contain the principles of those branches of

—

—

—

— Mathematical Principles of Physics

183

—

we have

not yet discussed such sciences as hydrodynamics, and many others. Indeed, e Principia does contain all of these principles and even ore. Sciences that were not even thought of in Newton's ty, such as electrostatics and electrodynamics, were later veloped on the analogy of the sciences of statics and namics in the Newtonian manner. Small wonder that the ime of Newton dominated eighteenth-century science and at Alexander Pope could write: iysics •tics,

that

acoustics,

Nature and Nature's law lay hid in night: God said, Let Newton be! and all was light. Before we investigate what makes Newton's Principia such powerful book, we should notice one additional fact. Its :le proclaims that it deals with natural philosophy rather an with physics. Is there a difference between physics and itural philosophy, or is the latter merely a quaint and oldshioned name for the former? The answer is rather strange: ere used to be a difference between these two disciplines; Dwever, Newton was perhaps more responsible than anyone se for the fact that the distinction

between the areas be-

ime blurred. In terms of the distinction, the Principia is isnamed; although Newton titles his work as though it jals with natural philosophy, in fact it deals with physics. "Natural philosophy," as its very name indicates, is a and we can see why ranch of philosophy. This means ewton liked the name that it is concerned with principles, ecause it is natural philosophy, it is concerned with the rinciples of nature and especially with the principles of lange. "Physics," on the other hand, deals with the phe-

—

—

—

of bodies falling in air, of of light particles traversing a me-

amena and causes of motion lanets circling the sun,

ium, and so on. Natural philosophy, we have said, treats of the principles (i.e., of If all change. For the basic phenomena of nature atural as opposed to artificial things) are those of change aange in size (i.e., growth and diminution), change in qualy (i.e., alteration of attributes like color, taste, and other

change in place (i.e., locomotion, as free fall), ad finally the basic change, from being into not-being and ice versa (i.e., generation and passing away, as in birth ad death). These four kinds of change are enumerated by jristotle in a book entitled, in translation, Physics. This tie multiplies the confusion that also inheres in Newton's inse data),

j

Breakthroughs in Physics

184

book should be called "Natural PhiNewton's should be called "Mathematical Principles of Physics." For Aristotle's work deals with the principles of all change; Newton's deals with the mathematical principles of only one kind of change, namely, locoactually Aristotle's

title;

losophy"

and

motion.

What makes Newton's work

—being mathematical —although only

so powerful

—

is

the fact that

very precise, and deals with locomotion it maintains that it that all other kinds of change should be reduced to locomotion. Newton's Principia, in other words, begins a reduction of physics in two ways: the method of physics, after Newton, has been exclusively mathematical; and its content, after Newton, has been exclusively the change of place of bodies, both large and small. Reductionism is often dangerous; indeed, it may be dangerous for physics, too. Whatever the perils, however, it is also true that the reduction of all change to change in place accounts at least in part for the power of Newton's book: since it contains the mathematical principles of change in place in a far more precise and profound fashion than any in character

it

is

—

—

—

—

—

book

and since it assumes that all other change is change in place (for instance, that change in color is caused by the motion of certain kinds of particles), therefore, everything we need to know about all change is contained in Newton's Principia. It is this which accounts for Newton's role as the prince of physics and for such words as those of Pope we quoted above; and although Newtonian physics has come in for its share of reexamination and reearlier

basically

evaluation, especially in the twentieth century, nevertheless there is no doubt that Newton more than any other man

has shaped the course of science for centuries.

Our

selection

from the Principia

is

brief.

It

comes from

the very beginning of the book and contains, so to speak,] the principles of the Principia. One of the astonishing things i is that all of its many propositions applications to every conceivable branch of physics should stem from these terse beginnings. Indeed, Newton

about Newton's work

and

its

himself is aware of this; in his preface to the first edition, he notes that his work is geometrical (or mathematical) in nature and that "it is the glory of geometry that from few principles it is able to produce so many things." Newton's basic tools, then, are just these: eight definitions, followed by a scholium; and three axioms, or laws of motion, followed by six corollaries and again a scholium. .

.

.

.

.

.

185

Mathematical Principles of Physics >ince the corollaries

and scholia obviously contain supple-

mentary rather than basic material, we can say that the most eleven in number: eight definitions

principles are at

ind three laws. It is unfortunate, of course, that the reader have to take for granted that these few principles are is powerful as we say they are. Still, we hope to be able it least to give a glimpse of Newton's powerful method. Let us begin with a look at the first two definitions. Their statement sounds somewhat archaic, at least in our translation. Actually, nothing very obscure or mysterious hides belind the phrase "the quantity of matter is the measure of ihe same." Newton simply notes that the quantity of matter iivill

i

£ measured by the product of a body's density and its i/olume. Similarly, the quantity of motion (or momentum) of a body is measured by the product of the body's velocity |ind its quantity of matter. These definitions are quite simple;

simple things they are somewhat deceptive. In I, the deception lies in the fact that |ive are not told how the density of a body is to be obtained. In practical fact, of course, a body's density is determined by the ratio of weight to volume; the more weight [per volume a body has, the denser it is. Weight, however, is [a function of mass; i.e., the more mass a body has, the more jand like

all

(the case of Definition

I

it also has. Thus, Definition I seems to be circular: mass is defined in terms of density, while density is determined in terms of mass. But though Definition I does not tell us how to measure [density, the omission is not merely a trick to hide circularity. [The mass of a body quite aside from how the quantity of tmass is to be measured is the amount of "stuff" it contains. Intuitively, it should be clear that the mass (or "stuff") of a body remains unaffected by the location of the body. The mass of one and the same body is the same here and on the moon, if the body should happen to be transported to the moon. But, as every schoolboy knows in this day of spaceships and spacemen, one and the same body has considerably less weight on the moon than it has on earth. (That we know this is, of course, due to Newton.) Thus, what Newton defines in his first definition is not weight, but mass as something independent of weight (while weight is dependent on mass though not solely dependent on mass). This mass, or i

weight

i

—

—

—

"stuff,"

is

measured by how much of

thickly or thinly

body.

it

is

it

is and how volume of the

there

distributed over the

Breakthroughs in Physics

186

An

mass is the which the description of mass atomic terms. Nothing seems more natural than

interesting sidelight to this description of

easy, almost necessary, is

made

in

way

in

to think of the "stuff" of bodies as being small atoms; the density of a body then is measured by the number of atoms

per volume. And Newton, indeed, considered matter to be made up of just such hard little atoms, as he tells us elsewhere (in the Opticks). If we adopt a sort of simple-minded atomism, viz., a view that considers atoms as hard little balls, then density would be simply determined by the number of atoms per unit volume and thus could at least theoretically be determined without weighing the body in question. It is, perhaps, not stretching things too far to say that the first definition of the Principia is an indication of Newton's thoroughgoing adoption of the principle of atomism, an adoption he does not openly acknowledge in the Principia, although he makes it explicit in the Opticks, published in 1704. Now let us turn to the Axioms, or Laws of Motion. There is nothing more important in the entire work.

—

—

Law

I: Every body continues in its state of rest, or of uniform motion in a right [straight] line, unless it is compelled to change that state by forces impressed upon it.

This law states a fundamental difference between Newton's view of the world and that of the earlier natural philosophers (such as Aristotle). body, Newton says, will continue in whether that condition be one of rest its present condition or one of uniform motion in a straight line unless some force causes the body to change from this state. Another way of putting the first law is this: If no force acts on a body, then it must either be at rest or move uniformly in a straight line. For example, if a force was acting on a body

A

—

—

and forced it to move in all sorts of ways (now up, now down, now fast, now slow) but then stopped acting, the body must continue to move, uniformly, in the last direction given it by the force and with the last speed given it by the force. Let us try yet a third manner of stating the first law: A body does not have only one condition which is "natural" to it, i.e., a condition to which it reverts when left alone; rather there are two equally natural conditions that a body may have: rest, or uniform straight line motion. According to Aristotle, there is only one condition which is natural to a body, and that is rest. All motion must terminate in rest, and all motion is forcibly impressed on a body, according to

— Mathematical Principles of Physics

187

the Aristotelian view. But for Newton, that which rests, rests, and that which moves, moves. Neither condition is more preferable or more natural than the other. Most of us, living in this age of technology and discovery,

accept the Newtonian principles unhesitatingly. Lest we conearlier natural philosophers too quickly, it is well to remind ourselves that many a child will adhere to the naive view that motion must necessarily stop (and also to remind ourselves that naivete in philosophy is often an advantage). Furthermore, we should remember that no one has ever seen a motion that did not stop. Of course, the somewhat sophisticated reader will point out that motion stops because of additional forces being brought to bear, such as the forces of air resistance and friction. But to extrapolate from these forces to the conclusion that if they were absent, then the motion would never stop, is quite a daring act of the mind. With the coming of the space age and the actual observation of artificial satellites that circle the earth for weeks, months, and even years, the experimental evidence for the ability of motion to continue indefinitely has become considerable. Here, however, it is well to remember that the motion of artificial satellites (and of natural ones, such as the moon) is not inertial motion of the kind Newton describes; the satellites' motion though uniform is circular and not straight. (Of course, the fact that the satellites continue indefinitely in their circular or elliptical paths is connected with Newton's First Law; but the connection is indirect and has to be demonstrated.) With all this, we do not mean to say that Newton's First Law is not true. On the contrary, it is true and it is a most useful and fruitful law. What we do mean to indicate is that it is not necessarily a self-evident law, that there is at least some evidence against it, that to a certain extent common sense rebels against it, and that therefore it is all the more remarkable that Newton was able to state it so clearly and unequivocally. What in Law I is stated as an axiom of motion is stated in Definition III as an innate property or power of bodies. There, the fact that bodies continue either at rest (if they are at rest) or in a uniform-straight-line motion (if they are in such motion) is ascribed to a power called inertia. Inertia is what makes a body at rest remain at rest; inertia is also what makes a body in uniform-straight-line motion remain in such uniform-straight-line motion. While Definition III merely describes and names a power (without asserting its existence the common failing of all definitions), Law I asserts that such

demn

—

—

Breakthroughs in Physics

188

a power exists and tells us what this power does. Law I is therefore the more important statement of the two; it actually tells us what, according to Newton, is the case with

moving

bodies.

Law

II: The change of motion is proportional to the motive force impressed; and is made in the direction of

the right line in which that force

is

impressed.

This law tells us what happens when a force is applied to a body: a change of motion. The more force is applied, the more change of motion is produced; this is what is meant by the change of motion being proportional to the force. In a way, the Second Law is almost a consequence of the First

Law. For state, i.e., it

since

may

follows that

inertial state

uniform-straight-line motion

exist

when no

when

a force

must take

force whatever is

applied,

is

is

an

inertial

applied, then

some change in the was rest,

place. If the inertial state

then application of force produces a condition other than namely motion; if the inertial state was uniform motion in a straight line, then application of force produces a condition other than this, namely, an accelerated motion, or a motion in a different direction, or even rest. Quantity of motion, we have seen, is measured by X v (mass X velocity) the effects of an impressed force, therefore, are proportional to a change in (the X v. Clearly, mass) does not change when a force is applied; hence all the change is expressed in terms of v, the velocity. Consequently, we can say that

rest,

m

;

m

m

the amount of force of motion

is

proportional to change in quantity

the

amount of force

is

proportional to change in

the

amount of force

or

(m X

v)

or is

proportional to

mX

(change of ve-

locity).

Change of velocity, however, is colloquially called "acceleration" (though sometimes a change of velocity may actually be a deceleration). Hence it is usual to write that the

amount of force

the

amount of force

is

proportional to mass

X

acceleration

or

Though

is

proportional to

(m X

a).

Galileo does not have such a neat set of definitions

Mathematical Principles of Physics and axioms as Newton, there Galileo was aware of this law.

is

good reason

189

to think that

We

noted earlier that a good part of Galileo's language is difficult and confused because he is trying to clarify some very difficult concepts; hence we must not expect him to make a neat statement that force is proportional to change in motion. If we look closely at the text of the Two New Sciences, however, we can find evidence that Galileo believed that a body in motion stays in motion and that, therefore, when a force is applied to a body, it does not produce motion but, rather, change of motion. a definition that describes and names the axiom asserting its existence, so there are a number of definitions that describe and name different kinds of force, in addition to the axiom that Just as there

power of

is

inertia, in addition to the

actually exists such impressed force and change of motion or acceleration. Definition IV distinguishes an impressed force a force coming from asserts that there

that

it

results in

—

—from

V

deinnate force, or inertia. Definition fines centripetal force as force directed toward a center; and Definitions VI, VII, and VIII distinguish among three kinds of centripetal force. The remarks Newton makes in connection with Definition are especially interesting. "A centripetal force," he writes, "is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre." He immediately notes that gravity is such a centripetal force, for it tends toward Newthe center of the earth. His next example is the force ton does not name it that holds the planets in their orbits the outside

V

—

—

around the sun. Finally, he gives a good everyday example: we attach a stone to a string and whirl it around and around, then the stone wants to fly off. We commonly give this endeavor the name "centrifugal force." Why doesn't the stone, in fact, fly off? Because the centrifugal force is counteracted by another force, equally strong, that acts in if

the opposite direction. This force is the centripetal force. The tautness of the string to which the stone is attached is also evidence of the existence of these forces: if only one force existed, the stone

would

either fly off or else

move toward

the center; since, however, there is both a centrifugal and a centripetal force, the string is stretched tautly and the stone

revolves around the center.

That force ... by which the sling continually draws back the stone towards the hand and retains it in its orbit, because it is directed to the hand as the centre of the orbit, I call the centripetal force.

Breakthroughs

190

Then Newton adds two

in Physics

significant sentences:

And

the same thing is to be understood of all bodies, revolved in any orbits. They all endeavor to recede from the centres of their orbits; and were it not for the opposition of a contrary force which restrains them to, and detains them in their orbits, which I therefore call centripetal, would fly off in right lines, with an uniform motion.

This means that there is no basic difference between the stone attached to the string and whirling about, and the moon revolving around the earth, or the planets, including the earth, revolving

around the sun. Both centrifugal and cen-

tripetal forces are present,

ly bodies there

is

no

although in the case of the heaven-

string or other bodily connection.

On

account of the centrifugal force, the various heavenly bodies want to leave their orbits and shoot off into space; but on account of the centripetal force, exerted by the sun or by the earth, they cannot fly off but are required to remain in their orbits. Even though the sun is not directly attached to the planets (as the string),

it still

hand acts

—

is to

the whirling stone by

exerts a force

means of the

—on them. In other words,

the sun acts on the planets at a distance, and action-at-adistance, though full of philosophical difficulties, is a physical reality. Basically, Newton implies, there is no difference between direct action (such as the pushes and pulls that exist in the whirling sling)

and action-at-a-distance.

stop there. He goes on to make a connecbetween direct action and action-at-a-distance by means of projectiles. Imagine, he tells us, a projectile that is shot off in a horizontal direction from the top of a mountain. If there were no gravity, the projectile would proceed in a straight line with uniform motion. On account of gravity, it deviates from its straight-line motion and curves so that eventually it hits the ground. The greater the speed with which

Newton does not

tion

the projectile

ground.

is

Newton

And by

shot

off,

the longer

it

will take to hit the

clearly describes the process:

increasing the velocity, we may at pleasure increase the distance to which it might be projected, and diminish the curvature of the line which it might describe, till at last it should fall at the distance of 10, 30, or 90 degrees, or even might go quite round the whole earth before it falls; or lastly, so that it might

Mathematical Principles of Physics never

fall to

spaces,

191

the earth, but go forwards into the celestial in its motion in infinitum.

and proceed

A

projectile derives its motion from two sources: from the direct action of whatever it is that gives it its initial impetus, and from the indirect (at-a-distance) action of gravity. After the initial impetus ceases, the motion of the projectile is due to the combination of its inertia (tendency to maintain whatever movement it possesses) and gravity. This same combination of inertia and gravity also brings about planetary and satellite motion (such as that of the

moon). Newton between

explicitly tells us that there

projectile

and

motion:

satellite

if

is

no difference

the initial impetus

on a projectile is made sufficiently strong, it can be made to revolve completely around the earth, just as the moon revolves around the earth. Newton,

had the notion of an arwhich would come into being by a projectile

in other words, clearly

tificial satellite,

being shot off with sufficiently great speed. Of course, this is what happens when a rocket launches an artificial satellite. The rocket, to be sure, is not launched in a horizontal direction, but vertically; after the rocket has attained considerable altitude and speed, it is then made to turn toward the horizontal. The reason for launching a rocket vertically has nothing to do with the theory of satellites; it lies in the fact that a large rocket is stable only in the vertical

precisely

position at the initial low speeds.

In the words of

Newton

possibilities are envisaged:

sufficient

speed so that

it

around the earth before ficial

earth satellite)

;

if

is

just above,

two

that of giving a projectile

once or several times (thereby becoming an arti-

will revolve

falling

the other

an even greater speed so that the force of gravity.

we quoted

that

one

is it

The second

that of giving the projectile

from must be realized the moon and other

frees itself completely possibility

rockets and spaceships are to be sent to

celestial bodies.

How

can a body be hurled beyond the pull of earth's gravdiscovered, as we can read in a part of the Principia not here reprinted, that the force of gravity is inversely proportional to the square of the distance from the center of the earth. That is, if we designate the force at a distance R x by f x and the force at a distance R 2 by /2 , then the following proportion obtains: ity?

Newton

Breakthroughs in Physics

192

U Thus,

Ri

the force of gravity at the distance of 8000 miles (i.e., at the surface of the earth) is 32 ft/ sec 2 (as is in fact the case), then at double that distance, 16,000 miles from the center (or 8000 miles up from the surface), the force of gravity will be 8 ft/ sec 2 . Obviously, this force diminishes very rapidly. if

from the center of the earth

If the acceleration due to gravity remained constant, no body thrown up could escape from it. Let us give an example. What happens if a body is thrown upward near the surface of the earth, where gravity is in fact for all practical purposes constant? The force of the throw imparts a certain uniform velocity to such a body. As an example, let us say that the uniform velocity is 64 ft/ sec. Then in 4 seconds, if no other force were acting, the body would have traveled 4 X 64 ft, or 256 ft up. However, at the same time that the inertial force makes it go up, the force of gravity makes it go down. In 4 seconds, a body that is only under the pull of gravity would fall a distance equal to Vi X 32 X 16 ft, or also 256 ft. This means that at the end of 4 seconds, our body will be down on the earth again. At the end of 1 second, the body would have gone up 64 ft, but will have fallen back 16 ft on account of gravity, and so its actual height will be 48 ft. At the end of 2 seconds, the body would have gone up 2 X 64, or 128 ft; gravity will make it fall downward 64 ft, so its actual height will be 64 ft. At the end of 3 seconds, it would have reached 3 X 64 or 192 ft; gravity will have made it also fall downwards 144 ft, so that the body will be at 48 ft. And at the end of 4 seconds, the rise of 256 ft due to the upward throw will have been completely canceled out by the downward fall of 256 ft, so that the body will be back on the

ground.

The same

happens when we thrust a rocket escape the earth's gravity. But endeavor as compared to the example

sort of thing

into space, hoping that

two things favor

we

this

it

will

all, it is possible to give a modern rocket a fantastically powerful thrust. This by itself would not suffice, however, were it not for the second fact: as the rocket travels away from the earth, the force of gravity rapidly diminishes. The whole trick in getting the rocket to escape from the earth is to give it such a great speed that the effects produced by gravity diminish faster than the effects produced by speed diminish.

just gave.

First of

Mathematical Principles of Physics

193

To

understand this we must again introduce the notion of work. (See Chapter One, p. 34.) Work, the reader will remember, is measured by the product of force and distance. Thus, to raise a body to the height of 64 ft, we must do work in the amount of 64 x gravity, since the force to be overcome is gravity. If the body has been raised to the height of 64 ft, i.e., if a certain amount of work has been expended to raise it to this position, then the body has acquired potential energy, or energy of position, equal to the amount of work. For, having been raised up, the body can descend again and therefore spend the potential energy stored up in it. If we throw a body up into the air and it rises higher than 64 ft, this must be due to the fact that the body had more energy than necessary to do the work of raising it 64 ft against the force of gravity. What gave this body the energy was the speed with which it was thrown up. This energy, derived from speed, is called kinetic energy. (See Chapter Four, where we discussed kinetic energy in connection with the pendulum, p. 129.) In other words, if it takes a certain amount of (potential) energy to raise a body to the height h, against the force /, this energy may be supplied in kinetic form, by imparting speed to a body. In the pendulum, we noticed that at the top of the swing all the energy was potential, whereas at the bottom of the swing all the energy

was

kinetic.

In order to overcome the force of gravity, we must give a body such a speed that its kinetic energy is greater than the potential energy needed to lift a body infinitely far above the earth. Even though we are to lift the body infinitely high, this is not to say that we must necessarily expend an infinite amount of work to do this. (If it did, we could not, of course, supply a body with sufficient kinetic energy to overcome the potential energy, except by giving it infinite speed, an obvious impossibility.) For the force of gravity becomes continuously smaller and smaller; to calculate the amount of work needed to raise a body to a certain height, we must first calculate the amount of work needed to raise it a short distance short enough so that the force of gravity may be considered as constant for that distance. We must do this again and again and finally add up all the amounts of work. It turns out, fortunately, that the amount of work needed to raise a body to an infinite distance from the earth (i.e., the amount of potential energy that we would store in such a body) is finite. Consequently, it is possible to find a velocity, called the escape velocity, that is just enough to raise a body

—

194

Breakthroughs in Physics

to this height. Practically speaking, we would, of course, give our rocket a velocity somewhat higher than the precise escape velocity in order to have a reserve of energy to cope with friction and similar added forces. Calculations show that the escape velocity for the earth's gravity is just under 7 miles per second. But now let us return to Newton's words. In addition to envisaging that a projectile might be given a speed that would enable it to escape altogether from the earth's gravity, he also anticipated the idea of a projectile being given a somewhat lower speed and being turned into an artificial satellite. The speed necessary for such a projectile depends on the height of the orbit. Obviously, the higher the orbit, the less rapidly the satellite needs to go in order to counteract gravity. The precise speed needed will be determined by the height of the orbit. In practice, it will work the other way around: the satellite will settle into whatever orbit its speed permits. The approximate launching speed for the various satellites that have been put into orbit by the U.S.A. and the U.S.S.R. is 5 miles per second. Let us return once more to Definition V and Newton's text. After envisaging artificial satellites, he writes:

And

after the same manner that a projectile, by the force of gravity, may be made to revolve in an orbit, and go around the whole earth, the moon also, either by the force of gravity, if it is endued with gravity, or by any other force, that impels it towards the earth, may be continually drawn aside towards the earth, out of the rectilinear way which by its innate force it would pursue; and would be made to revolve in the orbit which it now describes; nor could the moon without some such force be retained in its orbit.

Here is a basic, if abbreviated, statement of Newton's law of universal gravitation: the same force that turns a projectile toward the ground and that makes an artificial satellite of the earth revolve also makes the moon revolve in its orbit. Given the inverse square law (see above, p. 167), we can show that it is the same gravity that acts on the moon as acts on bodies on earth. (Incidentally, the belief that it is the same gravity that acts at the surface of the earth and on the moon is, of course, what connects the moon and "Newton's apple": the force making the apple fall from the tree is the same force that makes the moon "fall" toward the earth and therefore remain in its orbit.)

195

Mathematical Principles of Physics

In one of the early propositions of the Principia (Book

Prop. 4), Newton derives the mathematical formula for he amount of centripetal force needed to hold a body in a ircular orbit. He finds the accelerative force to be given by 'Vr, where v is the orbital velocity of the body and r is he distance from the body to the center of force. In the case

moon, we can easily calculate the orbital velocity. The from the moon to the earth is 60 times the earth's adius; in other words, 240,000 miles. The length of the

-f

the

listance

therefore 2 X tt X 240,000 = 1,500,000 the moon 28 days to complete a revoluion. In 28 days, there are 28 X 24 X 3600 seconds, or 2,419,:00 seconds. In order to obtain the speed in miles per second, /e need only divide the distance by the time: loon's ailes.

orbit

is

takes

It

,500,000

1,500,000

,,419,200

2,420,000

,

(appr.) v rr '

=

1.5 -

=

.

,.

-

.

0.62 miles/ sec.

2.42

Now in order to find the force acting on the moon, we must quare 0.62; this gives us 0.3844. This must now be divided ty the distance from the moon to the earth, namely, 240,000 ailes.

0.3844 miles 2 / sec 2

=

ftnnnnA1 v miles/ sec _2 0.0000016 ..

,

.

240,000 miles

The

acceleration due to gravity at the surface of the earth been found by experiments such as those of Galileo with he inclined plane; it is approximately 32 ft/ sec 2 If the orce of gravity diminishes as the square of the distance from he center of the earth, then we need only divide 3600 into las

.

he value of gravity

on

the surface of the earth in order to

on the moon. (For since the 60 times farther from the center of the earth than is he surface of the earth, gravity on the moon should be only /60 2 or 1/3600, of what it is at the earth's surface.) First we must change the value of 32 ft/ sec 2 into miles er second 2 ; then we must divide by 3600. Thus we have ind the force of gravity acting

aoon

is

,

5280

X

3600

=

0.0000016 miles/ sec 2.

Breakthroughs in Physics

196

But

this is precisely

centripetal force

the

same value

needed to keep the

as

that given for the ||

moon

in orbit.

Hence,

if

gravity diminishes with the square of the distance from the center of force, then it is the earth's gravity that holds the

moon

And

there are good reasons that were availindicating that gravity does so diminish: one from the shape of the planetary orbits. If such

in orbit.

able to

Newton

reason comes

orbits are elliptical (as they are), this is an indication of a centripetal force diminishing as the square of the distance

from the center of force (located at one of the foci of the ellipse), This is proved by Newton in Book I, Prop. 11. Since we have already mentioned rockets and rocket travel, it is appropriate that we now turn to Newton's Third Law, for all rocket and jet travel is based on this law.

Law

III: To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to con-

trary parts.

This law, perhaps more than anything Principia,

Newton's own and

is

We

else

in the entire

his contribution to physics.

have already noted that Galileo had the notion of inertia and that therefore he also had at least implicitly the con cept of the Second Law. But no one before Newton had ever thought of the Third Law. It is his own, and it is of the greatest importance. Without it, no motion that we actually

—

—

see could be explained.

And

yet,

when we

first

read

it,

this

law seems self-contradictory: how can there be any motion at all, if any force is opposed by another force acting in the opposite direction?

To understand what

this

law means, we must carefully body

tinguish between agent and patient, between the

and the body that receives the body A exerts a force

exerts the force

Third

Law

asserts that if

B, then body

B

on

A

direction)

dis that|

The on body

force. fj

exerts an equal force /2 (but in the opposite acts on B 9 while (see Fig. 6-1). Because

A

Fig. 6-1

B

acts

and

on A, the

is not lack of motion; rather, both A put into motion by the force f lf whicl

result

B move. For B

is

Mathematical Principles of Physics

197

4 exerts; A is put into motion by the force / 2 which B ;xerts. If we think of the forces involved as being pushes, then he Third Law asserts that if A pushes B, then B pushes A vith the same strength, but backward. Both A and B, there,

lore,

move, away from one another.

Let us give a more specific example. Imagine two icecaters standing near one another. (We choose the example of skating because the amount of friction is reduced under these circumstances. ) If skater A pushes against skater 2?, then B is put into motion, i.e., begins to glide away. However, at the iame time, B exerts a counterpush on A, and therefore A ilso begins to glide away. Naturally, therefore, the two

two skaters, uniform motion. Thus, there can never be just one body with motion that is not uniform and in a straight line; if one body is in a condition of accelerated motion, another body must also be in a similar condition. For the condition of the first body must be due to some force; consequently, another force, gqual in quantity and opposite in direction to the first one, must also have been in existence. Another good example of the Third Law is the stone being whirled around the hand by means of a string. We noted earlier that what keeps the string taut is the simultaneous action of two forces: the centrifugal force pulls outward on skaters separate

and the forces cease to

act; the

.herefore, will continue in a state of inertial

the string, while the centripetal force pulls inward. As the result of both of these equal and opposite forces, the stone

keeps whirling and the string remains taut. If only one force were present, the stone would either fly off or collapse inward toward the hand, and the string would not be tautly stretched.

Other instances of the Third Law at work are all around us every day. For instance, what happens when we walk? We exert a force with our leg against the ground. Nothing noticeable happens to the ground, but the ground does exert a force back against the leg, i.e., pushes it upward. This propels us forward. Then the other leg pushes the ground, the ground pushes the leg, and so forth. Similarly, in driving a car, it is important that the car, through its wheels, pushes against the pavement of the street. The pavement then pushes forward against the wheels (i.e., turns them), and the car goes. The truth of this can be seen when a car spins its wheels on ice: the wheels turn rapidly, but because of the ice no push against the ground is exerted. Consequently, the pavement cannot push back, and the car fails to get moving.

!

Breakthroughs in Physics

198

The most dramatic and obvious

Law

illustration of the

Third

occurs in jet-propelled or rocket-propelled motions.

The

backward on the fuel and indifferent whether the oxygen is

plane, or rocket, exerts a push

oxygen mixture

(it

is

"breathed in" as in a jet engine, or vehicle's load in a rocket), expelling

is it

carried as part of the at a

high rate of speed.

In accordance with the Third Law, the material being expelled i.e., being thrust backward exerts an equal and op-

—

—

push forward on the plane. The plane therefore goes forward. A similar thing happens in a propeller plane, but it is perhaps a little less obvious. Here, the propellers thrust back the air; the air in turn exerts a forward thrust on the propellers and therefore on the entire plane. Obviously, a propeller-driven vehicle needs to be surrounded by air (or aj posite

similar

medium); otherwise the propeller has nothing to (The propeller or screw of a ship similarly

thrust against.

back the water, which, in turn, thrusts the propeller and ship forward.) A jet plane or rocket does not need any air to go forward (although the jet engine requires air to supply its oxygen) A jet or rocket goes forward because it forces something else the material escaping at the exhaust to go backward. Jet and rocket engines are often called "reaction engines" because of the nearly perfect way in which they illustrate the law of equal and opposite reaction. thrusts

.

—

After the corollaries.

Laws

—

of Motion,

Here we

shall

Newton

puts

examine only the

down first

a

number of

two. Corollary'

I is a very important one;

it introduces the "parallelogram of forces," or the concept of the "independence of forces.'

1

Most of the time when we deal with actual situations in! we have more than one force acting on a body; for|

nature,

example, in the case of a projectile, there

is

both the

initial;

which gives it its starting velocity, and the force of; gravity, which causes the projectile ultimately to fall down,! Thus the question naturally arises: What happens when two!! forces simultaneously act on one and the same body? Newforce,

ton's answer, in Corollary

for "adding"

two

forces.

I,

We

consists in giving us directions^

represent the

rection and magnitude, by a line

first

AB; we then

force, in

di-j 111

represent the

second force in direction and magnitude by the line AC; and finally we must complete the parallelogram ABCD (see Fig.] 6-2). The combined force of AB and AC will be represented]

J

Mathematical Principles of Physics

199

AD, in both dimagnitude. probably best to accept

by the diagonal •ection It is

and

Corollary

in

of

postulate

a

as

I

rather mechanics Jian to try to prove it. Corollary tells us how to add up forces;

Newtonian [

it is basically the same kind thing as the postulate that tells lis, when we deal with numbers,

lius )f

hat a

+

b

=

b

+

a.

Forces

Fig.

6-2

dif-

'er from things like numbers in that they possess not only a mantitative aspect (thus one force may be double or triple mother force) but also a directional aspect (thus two forces nay act in the same direction, opposite directions, or at an mgle to one another). Quantities like numbers that do not possess a directional aspect are often called scalars; quantities .hat also possess a directional aspect, such as forces, are often called vectors. Corollary I then provides us with a rule for carrying out the addition of vector quantities. Just as there can be no proof of the postulate that

a

+

b

=

b

+

a

[beyond the undoubted utility of this relation), so there can )e no proof of the postulate of vector addition. If we examine Newton's "proof," we discover at least two flaws in it. One representing the effects of forces rather all clear how distances are to be unlerstood as effects of forces. The more serious problem with he proof lies in the fact that it is based on the assumption )f the independence of forces. The effect of the first force 'in the direction AB) is not at all affected, says Newton, by he second force (in the direction AC). This may well be true, nit it is not necessarily obvious. Indeed, there is good realifficulty lies in his

han

forces;

it is

not at

independence of the forces as a postulate. a postulate in order to prove Corollary be just as easy to consider the corollary itself as a Furthermore, the analogy with the postulates of

son to treat this

3ut i,

it

if

we must make

may

>ostulate.

for scalar quantities suggests that the rule for idding vectors should be a postulate also. In Corollary II, Newton applies the postulate of the addiion of forces. He imagines a lever with two unequal arms,

iddition

OL and OM. Not only

is

irms are not in a straight

OL

OM, but the two Consequently, what we must

greater than

line.

Breakthroughs in Physics

200 imagine

is

not

only

a

weightless lever but also a weightless wheel, with as its center and OL as its radius (see Fig. 6-3). This wheel will cut the line (indicating the direction in

O

MA

which the weight suspended from acts) in some point such as D. Now draw the radius OD, which therefore is equal to OL. Next draw a line AD to

M

represent

the

W

weight

VW,

x

(Newton calls it A), which acts at the point M. By means of Corollary I, resolve AD into two other forces that together add up to the force AD. There Fig. 6-3 is an infinite number of ways in which the force AD can be resolved; Newton chooses to resolve AD into two such weights that one of them, DC,

DO

acts in a direction perpendicular to (i.e., tangentially to the wheel), while the other force, AC, acts in a direction parallel to (i.e., radially to the wheel). As far as equilibrium of the wheel is concerned, has no influence on it at all; cannot turn the wheel. DC,

DO

AC

AC

however,

affects the equilibrium; it can turn the wheel. If are equilibrium, it must be because and 2 equal. For both are acting at equal distances from the center or fulcrum; furthermore, both weights act perpendicularly to a radius of the wheel, so that both are equally effective in bringing about a turning effect. The turning effects of are equal; for 1 is x and of resolved into two forces has no and AC, of which turning effect. Hence, if the weight represented by bal-

there

W

is

W

W

DC

DC

W

DC

W

W

AC DC

ances then also balances But we have also 2 2 x seen that the weight represented by is equal to 2 (since they balance at equal distances). Hence .

DC

W W x

:

Because of the similarity of

2

= DA DC.

triangles,

W :W = DO 1

:

2

:

KO.

W

:

Mathematical Principles of Physics Since

DO =

W :W = 1

But

2

OL: KO.

of course, is exactly Archimedes' law of the lever. of equilibrium is that the two weights r and be to one another inversely as the distances from the this,

W

The condition

W

2

201

OL,

fulcrum.

Newton

gives another

example of the resolution of forces

and then concludes this Corollary spreads far and wide, and by that diffusive extent the truth thereof is further confirmed. For on what has been said depends the whole doctrine of mechanics variously demonstrated by different authors. For from hence are easily deduced the forces of machines, which are compounded of wheels, pullies, levers, cords, and weights, ascending directly or obliquely, and other mechanical powers; as also the force of the tendons to move the bones of animals.

Therefore the use of

Once more we can well as Newton's

see the

power of Newtonian physics

own awareness

as

of this power: the whole

doctrine of mechanics can be reduced to these two corollaries. should note, now leave Newton and the Principia. however, that we have omitted any mention of two of the

We

We

most important sections included in our reading: the scholium following the Definitions and the scholium following the Axioms. The first of these deals with the notions of absolute and relative time, space, place, and motion. The second of them deals with the experimental evidence for the existence of various kinds of forces. Our omission in treating of them is not to be interpreted as a sign that these scholia are unimportant. On the contrary, they are among the most important sections of the entire Principia. However, they are more properly discussed in connection with Einstein's theory of relativity, which we do in the last chapter.

CHAPTER SEVEN Huygens:

—The Wave Theory of Light PART

I

Huygens

(1629-1695) was the most brilof a brilliant family. His grandfather was a Dutch government official, his father an eminent literary figure, his brother a scientist like himself. Huygens himself Christiaan

liant

member

never married and left no offspring. Like his father, Huygens led a cosmopolitan life; he became a member of the French Royal Society (as he tells us in the preface to the book before us) and lived in France as well as in Holland. In the field of dynamics, Huygens made many discoveries; of special interest are those relating to the pendulum. His chief work in this field is entitled Horologium oscillatorium (pendulum clock). He showed, among other things, that a simple pendulum, in order to be perfectly tautochronomous (i.e., in order always to have the same period for a swing), must follow the path of a cycloid. The work on the pendulum, published in 1673, is said to have influenced Newton's thought.

Huygens

is most famous for his Treatise on Light, pubFrench in 1690; the selection before us consists of the first three chapters. In this portion, Huygens develops the wave theory of light. In later chapters, he also deals with the special double refraction of Iceland spar and with the shapes of lenses. Though Huygens' work in dynamics was overshadowed by Newton's, in optics it was Huygens' theory that prevailed over Newton's for many years.

lished in

202

The Wave Theory of Light

Christiaan

Huygens

203

Treatise on Light*

PREFACE wrote this treatise during my visit in France twelve years and I communicated it in the year 1678 to the learned persons who then composed the Royal Society, to which I bad had the honor of being appointed by the King. Several of this body, who are still alive, can remember having been present when I finished that lecture, and especially those I

ago,

among them who particularly applied themselves to the study of mathematics; among these I need cite only the celebrated Messrs. Cassini, Romer, and De la Hire. And although I have made corrections and changes in several places since that time, the copies which I have made since that time could serve to prove that I have not added anything to it, except some conjectures concerning the formation of Iceland Spar,

and a new remark about the refraction of rock crystal. I have wanted to set down these details in order to make known how long I have been meditating about these things which I am now publishing and not in order to take away credit from those who, without having seen anything of what I had written, might be found to treat of similar matters: as in fact happened to those two excellent geometers, Messrs. Newton and Leibniz as regards the problem of the figure of a lens for gathering the rays of light when one of the surfaces is

given.

One may ask why I have delayed so long to bring this work to light. The reason is that I had written it quite carelessly in the language in which you now see it [French], with the intent of translating

it

into Latin, being thus able

pay more attention to the subject matter. After which I intended to combine it with another treatise on dioptrics, in which I explain telescopes and other things which belong to that science. However, the pleasure of novelty being past, I have delayed time and again the execution of this design, and I do not know when I shall be able to finish it, being so often diverted either by business or by some new study. Considering this, I have finally judged that it would be better to let this work appear such as it is, rather than run the risk, by waiting still longer, of letting it perish altogether. to

*

From

Treatise

on Light,

trans,

by Peter Wolff.

Breakthroughs

204

in Physics

You will see here those kinds of demonstrations which do! not produce a certitude as great as those of Geometry and which even differ greatly from them. For while the geome-i tricians prove their propositions by means of certain and incontestable principles, here the principles are verified by the conclusions which one draws from them. The nature of things does not permit that this could be otherwise. It is always possible to arrive here at a degree of likelihood which often differs but little from certainty. For when the things which have been demonstrated by the assumed principles correspond perfectly to the phenomena which experience has! made us notice, above all, when they are large in number and even more importantly, when one can describe and prediet new phenomena which ought to follow from the hypotheses employed and when one finds that in this the effect 1

—

corresponds to our expectation

hood are found

which

if

all

these proofs of

likeli-l

have proposed in this treatise (as it seems to me they are), then it must be a great confirmation of the success of my research and it would be in that

I

if things are not more or less as I repwould believe, therefore, that those who love; to know causes and who admire the miracle of light will find some satisfaction in these diverse speculations concerning it, and in the new explanation of its intrinsic property,! which constitutes the main foundation of the construction of! our eyes and of those great inventions which extend their! use. I hope also that in the future those who pursue these!

pretty strange indeed

resent them. I

beginnings will penetrate these matters farther than I wasj able to, since there is still much which has not been solved.!

This will be apparent by the places I have marked where I! leave difficulties without resolving them, and even more by! those things which I have not touched on at all, such as!

luminous bodies of various

and everything which perno one until now can boast!

sorts,

tains to color. In these matters

of having succeeded. Finally, there remains much to be investigated concerning the nature of light, which I do not! pretend to have discovered; I will be indebted greatly to him!

who can

supply those things which

I

do not yet understand.!

At

the Hague, 8 Jan. 169C

The Wave Theory of Light

CHAPTER Of Rays In a The demonstrations

in

205

I

Straight Line

optics,

as

in

all

the

sciences

in

which geometry is applied to matter, are founded on truths drawn from experience; such as that the rays of light travel in a straight line, that the angles of incidence and reflection are equal, and that in refraction a ray is bent according to the law of sines, which is now well known and not any less certain

than the preceding truths.

Most of those who have written concerning the

different

parts of optics have been content to presuppose these truths. But some, who were more curious than others, have wanted

to investigate the origin of

them and

their causes, consider-

to be admirable effects of nature. Since some ingenious things have been put forth but which, nevertheless, were such that the most intelligent men wished for explana-

ing

them

which would satisfy them more, I want to propose here I have thought about this subject, in order to contribute as much as I can to the clarification of this part of not without reason is reputed to natural science, which be one of the most difficult ones. I know myself to be in debt to those who were the first ones to begin dissipating the strange obscurity in which these things were wrapped and to give hope that they could be explained by intelligible reasons. But on the other hand I am also astonished that these same persons very often have been willing to accept reasoning with little evidence as quite certain and demonstrative. Thus, I have not found as yet that anybody has explained in a probable fashion those first and most notable phenomena of light, viz. why it always travels in straight lines, and how it happens that the visible rays, coming from an infinity of different directions, cross one another without in any way hindering one another. Thus, by means of principles that are nowadays accepted in philosophy, I shall attempt in this book to give clearer and more probable reasons, first, concerning the properties of light propagated in straight lines and secondly, concerning that property which makes it be reflected when it encounters other bodies. Then I shall explain the symptoms of rays which are said to suffer refraction in passing through tions

what

—

—

transparent bodies of different kinds: there I shall also treat the effects of reflection of air by means of the different densities of the atmosphere.

206

Breakthroughs in Physics

Following

this I shall

examine the causes of the strange rewhich is brought from Iceland.

fraction of a certain crystal

And

finally, I shall treat

the different figures of transparent

and reflecting bodies, by means of which rays are collected in one point or turned in different manners. There one will see from our new theory, with what ease are found not only ellipses, hyperbolas and the other curved lines which M. Descartes has invented with such subtlety, but also others which must form the surface of a glass when the other surface is given as spherical, flat, or any other figure whatever. One cannot doubt that light consists in the motion of a certain matter. For if one considers its production, one finds that here on earth it is primarily fire and flame which engender it; without doubt these contain bodies which are in rapid motion, since they dissolve and melt other more solid bodies. But if one considers its effects, one sees that when light is collected, as by concave mirrors, it has the power of boiling just as fire does, that is to say, that it breaks up the particles of bodies. This is surely the mark of movement, at least in the true philosophy, in which one conceives the cause of all natural effects by means of mechanical reasons. This is what must be done, in my opinion, or else we must renounce all hope of ever understanding anything in physics. And since, following this philosophy, one holds as certain that the sensation of sight is excited by the impression of some motion of matter which acts upon the nerves at the base of the eyes, this is still one more reason for believing that light consists in the movement of matter between us and the luminous body. Moreover, when we consider the extreme speed with which

and that, when it comes even quite opposite ones, the light rays cross one another without hindrance, we easily understand that when we see a luminous object, this could not be due to the travel of some matter which from the object comes to us, as a ball or an arrow travels through air. For surely this is repugnant to both qualities of light, and espelight is

from

propagated in

all

j


y means of the travel of a body which passes from one to ily

on

all

he other.

sides,

If,

it

then, light uses time for

—

its

travel

—

as

we

shall

follows that this movement imparted o matter is successive and that consequently it is propagated, List as sound is, by means of spherical surfaces and waves. 7 or I call them waves from their resemblance to those which me sees in water when a stone is thrown into it; these epresent such a successive circular propagation, although hey proceed from another cause and take place only in a >resently

examine

it

)lane surface.

In order to see, therefore, whether the propagation of light akes time, let us first consider if there are any experiments vhich could persuade us of the opposite. As for those which me can make here on the Earth, with fires placed at great listances, although these prove that light does not use any sensible time in order to travel over these distances, one ;an with reason say that the distances are too small and hat one can only conclude from them that the travel of ight is extremely rapid. Mr. Descartes, who held the opinion hat light travels instantaneously, based

it,

not without rea-

upon a better experiment drawn from the eclipses of :he Moon. Nevertheless, as I shall show, this experiment is lot at all convincing. I shall expound it a little differently :han he, in order to make the conclusion more comprehenjon,

sible.

Let A be the place of the sun, BD a part of the orbit or annual route of the Earth, ABC a. straight line, which I suppose meets the path of the Moon, represented by the circle CD, at C.

"D

Breakthroughs

208

in Physics

Then, if light needs some time, for example one hour, in order to traverse the distance between the Earth and the Moon, it follows that if the Earth has reached B, its shadow or the interruption of light which it causes, has not yet reached the point C, but will not arrive there until an hour later. Thus it will be an hour later, counting from the time that the Earth was at B, that the Moon, arriving at C, will be darkened. But this darkening or interruption of light will not reach the Earth for another hour. Let us suppose then that in these two hours the Earth would have arrived at E. Then, the Earth being at E, we will see the Moon eclipsed at C which the Moon left an hour earlier, and at the same time we will see the Sun at A. For since the Sun is unmoved, as I assume with Copernicus, and since light is propagated in straight lines, the Sun must always appear where it is. But it has always been observed, they say, that when the Moon is eclipsed, it appears to be directly opposite the Sun on the ecliptic. According to the diagram, however, it should appear behind this place by the angle GEC, which

ABC

makes two right angles. Thus this is contogether with would be quite sensible, trary to experience, since angle and be around 33 degrees. For according to our assumptions (contained in the treatise of the causes of the phenomena of Saturn) the distance BA between the Earth and the Sun is around twelve thousand terrestrial diameters and is therefore four hundred times greater than BC, the distance of the Moon, which is 30 diameters. Thus the angle ECB will be nearly four hundred times greater than BAE, which is five minutes (the path traversed by the Earth in two hours in its orbit). And thus the angle BCE is almost 33 degrees, and than which it is greater by five minutes. so is the angle

GEC

CEG

But it must be noted that the speed of light in this reasoning has been supposed to be such that it would take one hour of time in order to traverse the distance from here to the Moon. If one then supposes that it only takes one minute would be of time, then it is manifest that the angle only 33 minutes, and if it were to take only 10 seconds of time, this angle would only be six minutes. And then it would not be easy to perceive this angle in the observations of an eclipse, nor consequently would it be allowable to conclude anything on the basis of this concerning the instantaneous

CEG

movement It is

of light. true that this

is

to suppose a strange speed

which

The Wave Theory of Light

209

would be 1000 times greater than that of sound. For sound, according to what I have observed, travels about 180 toises in the time of one second or one heartbeat. But this supposition need not appear to have anything impossible about it: for it is not a question here of the travel of a body with this speed, but rather of a successive movement, which passes from one body to the next. Hence I have not hesitated, in thinking about these things, to suppose that the emanation of fashion all its phenomena can be explained, and that by following the contrary opinion they all are quite incomprehensible. For it has always light takes time, seeing that in this

seemed to me and also to many others besides me, that the same Mr. Descartes, who has had as his goal to treat intelligibly of all topics in physics and who has assuredly had better success in this than anyone before him, has never said anything which was not full of difficulties or even inconceivable concerning light and its properties. But what I merely stated as a hypothesis, achieved recently the grand appearance of unshakable truth, by means of the ingenious demonstration of

Mr. Romer, swhich

I will

report here, until he himself gives everything that serves to confirm it. Just like the preceding demonstration, it is based on celestial observations and proves not only that light needs

time for its travel, but also lets us see how much time it needs and that its speed is at least six times greater than that

which

I stated.

the eclipses undergone

The demonstration makes use of

planets which circle about Jupiter and which often enter into its shadow. Here is his reasoning: Let A be the annual orbit of the Earth, F Jupiter, the Sun, the orbit of the nearest of its satellites, for this is the one which is more suitable for this investigation than any of

by the

little

GN

BCDE

the three others, because of the speed of its revolution. Let then be the satellite entering the shadow of Jupiter, and

G H the same satellite, leaving the shadow. If

we suppose

then that the Earth

we would have

is

at

B

(sometime before

seen the said satellite leave the shadow. Necessarily then, if the Earth remained in the same place, after 42 1/2 hours we would again see the satellite emerge. For this is the time in which the satellite makes a complete orbit and returns to opposition with the Sun. And if during 30 revolutions of this satellite the Earth always remained at B, for example, it would again see it the

last

quarter),

Breakthroughs in Physics

210

h|g

shadow

after 30 times 42 1/2 hours. But since the time travels to C, thereby increasing its distance from Jupiter, it follows that if light needs time for its passage, the illumination of the little planet will be perceived later at C than at B, and it will be necessary to add to this time of 30 times 42 1/2 hours the time needed by the

leave the

Earth during

this

MC, the difference between the and BH. Similarly on the side toward the other quadrature, when the Earth from D has come to E in its approach to Jupiter, the entries of the satellite G into the shadow ought to be observed earlier at E than they would have appeared if the Earth had remained at D. Thus, by means of many observations of eclipses, made during ten consecutive years, these differences are found to be quite considerable, such as six minutes and more, and one can conclude from this that in order to traverse the entire diameter of the annual orbit KL which is double the dislight to travel the distance

distances

CH

The Wave Theory of Light

211

tance from here to the Sun, light needs about 22 minutes of time.

The motion of Jupiter in its orbit, while the Earth moves from B to C, or from D to E, is included in this calculation, and lets us see that we cannot attribute the retardation of the illuminations or the acceleration of the eclipses to the which is found in the motion of this little planet

irregularity

or to

its

eccentricity.

Thus if we consider the vast extent of the diameter KL, which according to me is some 24 thousand diameters of the Earth, we can understand the extreme speed of light. For suppose that KL were only 22 thousand of those diameters; evidently being passed through in 22 minutes, this would make a speed of a thousand diameters in a minute, and 16 2/3 diameters in a second or one heartbeat; this amounts to more than eleven hundred times a hundred thousand toises, since the diameter of the Earth contains 2865 leagues (of 25 per degree) and each league is 2282 toises, according to the exact measurement which Mr. Picard made by the order of the King in 1669. But sound, as I have said above, only makes 180 toises in the same time of one second. Thus the speed of light is more than six hundred thousand times greater than that of sound: nevertheless, this from being instantaneous, since

ferent thing

is

quite a dif-

it

is

the

same

between a finite and an infinite thing. The successive motion of fight being confirmed in this fashion, it follows, as I have already said, that it is propagated by means of spherical waves, just as the motion of sound. But if light and sound resemble one another in this respect, difference

as

they differ in several others: in the origin of the movement which causes them, in the matter in which this movement is propagated, and in the manner in which it is communicated. For as far as the production of sound goes, we know that this comes about through the sudden shaking of an entire body or of a considerable part of it, which then agitates all the contiguous air. But the movement of light must arise as though it comes from each point of the luminous object, in order to make perceivable all the different parts of the object, as will be seen better later on. And I do not believe that this motion could be better explained than by supposing that

among

the luminous bodies which are liquid, like and apparently the sun and the stars, are composed of particles which float in a much more subtle matter, which agitates them with a great rapidity and makes them hit against the particles of the ether which surround them and

those

flame,

— 212

Breakthroughs in Physics

which are far fewer than they. But in luminous solids such as carbon or red-hot metal, this same movement is caused by the violent shaking of the particles of metal or wood; those among them that are at the surface hit the same ethereal matter. Furthermore, the agitation of the particles light must be much more prompt and rapid than that of the bodies which cause sound, since we do not see that the trembling of a body which makes a sound is capable of causing light to come to be, just as the movement of a hand in air is not capable of producing sound. Now if we examine what may be this matter in which the movement that comes from luminous bodies is propagated which I call Ether we see that it is not the same as that which serves for the propagation of sound. For we find that the latter is indeed the very air which we feel and which we breathe; when this is removed, the other matter which serves for light is still to be found there. This can be proved by enclosing a sounding body in a glass vessel from which the air is then withdrawn by means of the machine which Mr. Boyle has given us and with which he made so many beautiful experiments. But in doing this, it is necessary to place the sounding body on cotton or on feathers so that it cannot communicate its vibrations to the glass vessel enclosing it nor to the machine a matter which has been neglected so far. Then, after all the air has been evacuated, we no longer hear any sound of the metal although it is struck. We see from this not only that our air, which does not penetrate through glass, is the matter in which sound is propagated; but also that it is not the same air, but some other matter in which light is propagated, since when the air has been removed from the vessel, the light does not stop to travel through it as before.

which engender

—

—

And this last point is also shown more clearly in the celebrated experiment of Toricelli: in this experiment a glass tube from which quicksilver is withdrawn remains empty of air, but transmits light just the same as when it had air. This proves that a matter different from air is found in this tube and that this matter must have penetrated the glass or the quicksilver or both, although both are impenetrable to air. And since in this same experiment one brings about the vacuum by putting a little water above the quicksilver, one can conclude similarly that the said matter passes through glass, or water, or both. Now as to the different fashions in which I say that the successive movements of sound and of air are communi-

j

The Wave Theory of Light

213

easy to understand how this takes place in the case of sound, when we consider that the air is of such a nature that it can be compressed and reduced to a space cated:

it

is

smaller than that which it ordinarily occupies. And to it has been compressed it makes an effort to restore its size again. This together with its penetrability, which remains in spite of its compression, seems to prove that it is made up of little bodies which float and which are agitated more rapidly than the ethereal matter composed of much smaller particles. Thus the cause of the propagation of

much

the extent that

sound waves is the effort made by the little bodies which hit one another to restore their size, since they are packed a little more tightly in the course of the waves than elsewhere. But the extreme velocity of light and the other properties which it has, do not permit such a propagation of motion and I am going to show here how I believe it must take place. It

erty

is

necessary to explain, for this purpose, the prop-

which makes hard bodies transmit movement from one

to the other. If we take a number of balls of equal size, made of some material that is very hard, and arrange them In a straight line so that they touch one another, we will find that, if we hit with a similar ball against the first of the balls, the movement passes as if in an instant to the last one, which separates itself from the line-up without our being able to

perceive that the others have moved. And even the one which did the striking remains immobile with the others. Here we see a passage of movement that is extremely rapid and which is greater in proportion as the matter of the balls is of a greater hardness. But it is still clear that this progress of motion is not instantaneous but successive and that it therefore takes time. if the motion, or if you will, the inclination to movement, did not pass successively through all the balls, all

For

it at the same time, and all together would forward. Yet this does not happen, but the last ball leaves the line-up and acquires the speed of the one which we pushed. Besides, there are experiments which show that all those bodies which we rank among the hardest, such as tempered steel, glass, and jasper, have some spring and bend in some fashion, not only when they are extended in rods, but also when they are in the form of balls or otherwise. That is to say, that they are dented a little at the point where they are hit, and that they then restore themselves to their previous shape. For I have found that if a

would acquire

move

214

Breakthroughs in Physics

is struck against a large and thick piece of the same kind, which has a flat surface and is ever so slightly tarnished with breath or otherwise, round marks are left, more or less large, according as to whether the blow was strong or feeble. Which lets us see that these kinds of bodies yield at their encounter and then are restored; and this necessarily takes time.

ball of glass or jasper

Hence, in order to apply this sort of motion to that which produces light, nothing prevents us from assuming the particles of ether to be of a matter that approaches as closely as we wish to perfect hardness and elasticity. It is not necessary here to examine the cause of this hardness nor that of for this consideration would take us too far from I will say, nevertheless, in passing that we can conceive that the particles of ether, notwithstanding their smallness, are again composed of other parts and that their elasticity consists in the very rapid movement of a subtle elasticity,

our subject.

matter which travels through them from all sides and constrains their texture to dispose itself in such a fashion that it gives to this fluid matter the most open and easy passage that there can be. This agrees with the reason which Mr. Descartes gives for elasticity, except that I do not suppose holes in the form of round and hollow canals, as he does. And it is not necessary to imagine that there is anything absurd in this or impossible; on the contrary it is very believable that there

ferent sizes

is

and of

an

infinite progress of corpuscles of dif-

different degrees of speed,

which nature

uses in order to bring about so many marvellous effects. But while we are ignorant of the true cause of elasticity, we see always that there are many bodies which have this property; and therefore there is nothing strange in supposing it also in little invisible bodies such as those of the ether. So that if one wants to seek out some other fashion in which the movement of light is successively communicated, one will not find anything which fits better than elasticity together with equal progression. This seems to be necessary, because if the movement diminishes to the extent that it is distributed among more matter (as it goes out from the source of the light), it could not preserve this great velocity over such large distances. But if we suppose elasticity in the ethereal matter, the particles will have the property of rebounding equally quickly, whether they are strongly or feebly pushed; and therefore the progress of light will always continue with equal speed. And it is necessary to recognize that although the particles

The Wave Theory of Light

215

as our array of

of ether are not arrayed in straight lines balls, but confusedly so that one particle touches several others, this does not prevent them from transmitting their motion and always propagating it forward. For this, we must note a law of motion which serves for this propagation and which is verified by experiment. This is that if there is a ball such as A, touching several other similar ones such as C, is struck by another ball B so that C, C, and if this ball

A

all the other balls C which it touches, of its motion to them and remains afterwards immobile as does the ball B. And without supposing that the ethereal particles are spherical (for I do not see that it is necessary to suppose them such), we easily understand that this property of impact contributes to the said propagait

I

makes an impact on

then

it

transmits

all

tion of motion.

more neceswould have to be a backward reflection of motion when the motion passes from a smaller to a larger particle, according to the laws of impact which Equality of size (of the particles) seems to be

sary, because otherwise there

I

published several years ago.

However, we will see later on that we do not have need to suppose this equality for the sake of the propagation of light, but only in order to make it easier and stronger. It is only an appearance that the particles of ether have been made equal for such a considerable effect as that of light, at least in that vast extension which exists about the atmosphere, which seems to serve only for transmitting the light of the Sun and the stars. Thus I have demonstrated in what manner one can conceive light to be propagated successively by means of spherical waves, and how it is possible that this propagation takes place with as great a speed as experiments and celestial observations demand. Here we must once more remark that although the particles of ether are supposed to be in continual

motion (for there are good reasons for

that), the sue-

Breakthroughs in Physics

216

waves does not need to be hinbecause this does not consist in the transportation of particles, but only in a little disturbance which the particles are not prevented from communicating to those that surround them, in spite of the motion which agitates them and makes them change place among themselves. But it is necessary again to consider more particularly the origin of these waves, and the manner in which they are propagated. And first it follows from what has been said about the production of light that each little spot of a luminous body such as the Sun, a candle, or a hot coal, engenders its waves, of which the spot is the center. Thus in the flame of a candle, let there be marked the points A, B, C. Thie concentric circles described around each of these points represent the waves which proceed from them. And the same must be understood concerning each of the points of the surface and of a particle within that flame. cessive propagation of the

dered by

this,

But since the percussions at the center of these waves do not follow any rule, we must therefore not imagine that the waves themselves follow one another at equal distances. And

The Wave Theory of Light

217

these distances appear to be so in that figure, that is more mark the progress of one and the same wave in equal times than in order to represent several waves that if

in order to

have proceeded from the same center. Finally

it

is

not necessary that this prodigious quantity of

waves which cross one another without confusion or without cancelling one another, should seem inconceivable. It is certain that one and the same particle of matter can serve for several waves, coming from different sides or even from opposite sides, not only if it is hit by blows which succeed one another quickly, but even by those which act upon it at the same instant. This is so because the motion is propagated successively. This can be proved by means of an array of equal balls, of hard material, such as we spoke of earlier. If

0OO00O© same time from opposite and D, then one will see each bounce back with the same speed with which it approached, and the whole array will remain in its place, although the motion has passed through its length, and twice. And if these contrary movements meet at the ball in the middle, B, or at some other one, such as C, it must yield and spring back, in two directions, and thus serve in the same instant for transmitting both of these movements. But what may seem at first most strange and even inconceivable is that the wave motions produced by motions and by against this array one pushes at the sides the equal balls

'

A

I

i

I

(

I

•

i

bodies so small can extend themselves to distances that are so immense, as for example from the Sun or from the stars to us. For the force of the waves must become weaker in proportion to their distance from the origin, so that the action of a particular one becomes, without doubt, incapable of being sensed by our sight. But we will cease to be astonished by this if we consider that at a great distance from the luminous body an infinity of waves, although originating at different points of the body, are united so that apparently they constitute only one wave, which consequently must have enough strength to be sensed. Thus the infinite number of waves which arises in the same instant from all the points of a fixed star, perhaps as great as the Sun, produces in ap-

Breakthroughs in Physics

218

pearance only one single wave, which can easily have enough strength to make an impression on our eyes. Besides, from each luminous point there can come forth several thousands of waves in the least time imaginable, by means of the frequent percussion of the particles which strike the ether at these points; this again contributes to make their action more sensible.

We

must

emanation of these waves, which a wave is propagated

also consider, in the

that each particle of matter in

its movement not only to the next parthe straight line drawn from the luminous body, but that it also must necessarily give (its motion) to all the others which touch it and which are opposed to its movement.

must communicate ticle in

Thus, around each particle a wave particle

is

the center.

Thus

if

DCF

is

is

made

of which this

a wave emanating from

the luminous point A which is its center, the particle B (one of those contained in the sphere DCF) will produce its at C, particular wave KCL, which touches the wave at the same moment when the principal wave, coming from point A, has reached DCF; and it is clear that only the point C of the wave will touch the wave DCF, the point C

DCF

KCL

namely being on the straight line which is drawn through AB. In the same way, the other particles of the sphere DCF, such as bb, dd, etc. will each make its own wave. But each of these waves can only be infinitely weak compared to the wave DCF, to the composition of which all the others contribute by means of that part of their surface which is in a straight line with the center A,

The Wave Theory of Light

DCF

We

see furthermore that the wave the limit of the motion which came from

A

is

219 determined by

in a certain period

of time, there being no movement outside of this wave, although there is a good deal of it in the space which it encloses, namely in those parts of particular waves which do not touch the sphere DCF at all. And all this need not appear to have been searched out with too much care of subtlety; for as will be seen later on, all the properties of light and all those things which appertain to its reflection and refraction are principally explained by this means. This was not known to those who before this began to consider the waves of light, among whom are Mr. Hook in his Micrographie and Father Pardies, who, in a treatise of which he let me see a part, and which he could not finish because he died a short time afterwards, undertook to prove the effects of reflection and refraction by means of waves. But the principal foundation,

which

remark I have just made, was lacking in and he held, for the rest, opinions quite from mine, as perhaps will be seen some day if his

consists in the

his demonstrations,

different

writings are preserved.

To come now each

to the properties of light, let us' first

remark

of

a wave must be propagated in such fashion

that

part

*\

^E,

that the extremities are

always straight

from

between the

the

drawn

lines

luminous

Thus the part of the wave BG, having

point.

the

luminous point

for

its

center,

will

A be

propagated in the arc CE, limited by the straight lines ABC, AGE, For although the particular waves produced by the particles which enclose the space CAE also spill over outside this space, these never come together at the same moment in order to compose one wave which limits their movement, except precisely in the circumference CE, which is

their

And

common

tangent.

here one sees the reason why light, at least that whose rays are not reflected or refracted, only travels in straight lines, so that it does not illuminate any object except when the path from the source to the object is unobstructed in the direction of a straight line. For if, for example, there is an

220

Breakthroughs in Physics

opening BG, bordered by opaque bodies BH, GI, the lightwave which leaves the point A will always be limited by th< straight lines AC, AE as has been demonstrated: the parts o the little waves which are propagated outside the space ACl being too feeble to produce any light there. Thus, however small we may make the opening BG, then is always the same reason for letting the light pass betweei the straight lines. For this opening is always large enough t( contain a great number of particles of ethereal matter, which are of an inconceivable smallness. Thus it seems that each little part of the wave advances necessarily in the direction o the straight line which comes from the shining point. Anc thus it is that one can take the light rays as though they were straight lines. It would appear, finally, because of what has been r marked concerning the weakness of the little waves, that i

not necessary that all the particles of the Ether are equa themselves, although equality would be more prope for the propagation of the motion. For it is true that the in equality will bring it about that one particle, in pushing an other one that is greater, would make an effort to recoil wit! one part of its movement, but there will not result from thi anything except some little waves going backwards towan the luminous point, incapable of producing light, and no is

among

one wave composed of several, as is CE. Another one of the marvellous properties of light is this that when it comes from several directions, or even from op posite ones, each ray produces its effect without the ray hindering one another. On account of this it happens tha through one and the same opening several spectators can se different objects at the same time, and that two persons cai see each other's eyes at the same instant. Following what ha been explained concerning the action of light and how thi waves do not destroy one another, nor interrupt one anothe when they cross, the effects concerning which I have spoken are easy to conceive. Which they are not, at all, in my view, according to the opinion of Descartes, who makes light consist in a continuous pressure which only tends toward motion. For this pressure, since it cannot act at the same time in two opposite directions, toward bodies which do not have

any inclination to approach one another, makes it impossible to understand what I have said about two persons who mutually see each other's eyes, nor how two torches can illuminate one another.

The Wave Theory of Light

CHAPTER

221

II

Concerning Reflection

Having explained the

effects of light

waves when they are

propagated in a homogeneous medium, we shall now examine what happens to them when they encounter other bodies.

We

how by means of these same waves the reof light is explained and how this preserves the equality of angles. Let there be a plane and polished surface, made of some metal, glass, or other body, AB, which first of all I shall consider as perfectly smooth (reserving to speak shall first see

flection

of the inequalities from which it cannot possibly be exempted end of this demonstration) and let the line AC, inclined to AB, represent a part of a light wave of which may be considered as the center is so far that this part a straight line. For I consider everything here as though in one plane, imagining that the plane in which the figure lies cuts until the

AC

AB

the sphere of the wave through its center and the plane at right angles; and it is sufficient to remark this once for all.

The point C of the wave AC will have advanced, in a certain time, to the plane AB at B, following the straight line \CB, which one must imagine as coming from the luminous I center and which consequently is perpendicular to AC. But

Breakthroughs in Physics

222

same amount of time, the point A of the same wave, is prevented from transmitting its movement by the plane AB, at least in part, must have continued its movement in the matter which is above the plane and this for a length equal to CB, making its own particular wave, according to what we said above. This wave is here represented by the circumference SNR whose center is A, and whose radius AN is equal to CB. If we then consider the other points H of the wave AC, it in the

which

appears that they will not only have reached the surface parallel to CB, but that by means of straight lines furthermore they will have engendered, from the centers K, spherical wavelets in the transparent medium, represented here by circumferences whose radii are equal to KM, that to the straight line is to say to the continuations of BG parallel to AC. But all these circumferences have for their common tangent the straight line BN, namely that same one which is drawn from B tangent to the first of these equal to BC, as it circles, whose center is A and radius is easy to see. Thus it is that the line (taken between B and the point N, where the perpendicular from A falls on it) is as it were formed by all the circumferences and terminates the movement which is brought about by the reflection of the wave

HK

AB

HK

AN

BN

AC. And

this is also

where the movement

is

found

in

much

greater quantity than anywhere else. That is why, according is the propagation of the to what has been explained,

BN

wave AC For there

at the

moment when

its

point

C

has arrived at B.

which like BN is the common tangent to all the circles mentioned except BG, below the plane AB; this BG would be the propagation of the wave if the movement could have been extended in a matter homogeneous with that which is above the plane. If one wants to see how the wave AC has successively come to BN, one needs only to draw in the same figure the straight lines KO parallel to BN and the straight lines KL parallel to AC, Thus one will see that the wave AC from being straight has been broken up into all the bits OKL successively and that it has again become straight at NB. Thus it appears from this that the angle of reflection is equal to the angle of incidence. For the triangles ACB, BNA being right-angled, and having the side AB in common, and is

no other

line

the side CB equal to NA, it follows that the angles opposite these sides are also equal, and therefore also the angles CBA, NAB. But since CB, perpendicular to CA, marks the

The Wave Theory of Light direction of the incident ray, so

wave BN, marks

AN,

223

perpendicular to the

the direction of the reflected ray; these rays,

therefore, are equally inclined to the plane

AB.

one could is the common tangent of say that it is indeed true that the circular waves in the plane of the figure, but that the waves, being in truth spherical, have still an infinity of similar tangents, namely all the straight lines which from the point B are drawn on the surface of a cone engendered by the straight line BN, around the axis BA. It remains, then, to show that there is no difficulty in this. And for the same reason one will see how the incident and the reflected ray are always in the same plane, perpendicular to the reflecting plane. I say then that the wave AC, being considered only as a line does not produce any light For a visible ray of light, however thin it may be, always has some thickness, and hence in order to represent the wave whose progress to produces this ray, it is necessary instead of a^ line put a plane figure, as in the figure below the circle HC, But

in considering the preceding demonstration,

BN

AC

luminous point is infinitely easy to see, following the preceding demonstration that each little point of this wave HC, when it reaches the plane AB will engender there its own little wavelet and that these will all have (when C has arrived at B) a. common plane which touches them, namely a circle BN, equal to CH, and which will be cut in the middle, and at right angles, by the same plane which cuts also the circle

if

we suppose

far removed.

as before that the

Then

it

is

CH and the ellipse AB. We

also see that the said spherical wavelets

cannot have

plane touching them beside the circle BN, so that it will be this plane which will have much more reflected movement than any other and which therefore will carry the light continued from the wave CH.

another

common

224

Breakthroughs in Physics

have also said in the preceding demonstration that the motion of the point A of the incident wave cannot be transmitted beyond the plane AB, at least not entirely. Here it is necessary to remark that, although the movement of ethereal matter communicates itself in part to that of the reflecting body, this cannot in any respect alter the speed of the progress of the waves on which the angle of reflection depends. For a slight impact must engender waves just as swift as a strong I

one, in the same matter. This comes from the property of bodies which possess elasticity concerning which we have already spoken above: namely, that whether they are pressed little or much they restore themselves in the same time. Consequently, in all reflection of light, no matter by means of which body, the angles of reflection and incidence must be equal, notwithstanding that the body may be of such a nature that it absorbs a portion of the motion which makes the incident light. And experience shows that in fact there is no polished body which does not follow this rule. But beyond this what needs to be noted in our demonstra* tion is that it does not require that the reflecting surface be considered as one plane, as has been supposed by all those who have undertaken to explain the effects of reflection. It need only be of such an evenness as the particles of the reflecting body can bring about if they are placed close to one another. These particles are larger than those of the ethereal matter, as will be apparent by what we are going to say in treating of transparency and opacity of bodies. For, the surface consisting thus of particles put together, and the ethereal particles being above and being smaller, it is evident that one would not know how to demonstrate the equality of the angles of incidence and reflection from the resemblance to what happens when a ball is thrown against a wall, which analogy has always been made use of. But in our fashion the matter is explained without difficulty. The smallness of the particles of quicksilver, for instance, being such that we must conceive that there are millions of them in the least visible surface imaginable, arranged as in a heap of sand which one has smoothed off as much as possible this surface then becomes equal to a polished glass in our view. And although it always remains uneven with respect to the particles of Ether, it is evident that the centers of the little waves of reflection, of which we have spoken, are very nearly in the same smooth plane, and that therefore the common tangent can well enough encompass them for the purposes of the production of light. And this only is required in our manner »

—

The Wave Theory of Light

225

demonstration, in order to bring about the equality of the ;aid angles, without the remainder of the reflected movenent of all parts being able to produce an opposite effect. >f

CHAPTER

III

Concerning Refraction

were explained by waves of from the surface of polished bodies, so we hall explain transparency and the phenomena of refraction )y means of waves which are propagated within or through Just as the effects of reflection

light

reflected

whether these be solids like glass, or But in order that it may not seem itrange to suppose such a passage of waves within bodies I ihall begin by showing that this can be understood in more han one way. ransparent bodies,

iquids like water,

oil, etc.

matter did not penetrate transbodies at all, their particles themselves could succesrively transmit the movement of the waves just like the ^articles of ether, if they be supposed, like those, elastic n nature. And it can be easily conceived in the case of water md other transparent liquids that they are composed of nonFirst, then, if the ethereal

ient

But it may seem more difficult as reand other bodies that are transparent and also lard, because their solidity does not seem to permit that they :ould receive any motion except in all of their mass at once. ouching

particles.

gards glass

Nevertheless, this conclusion

is

not necessary, because the

of these bodies is not such as it seems to us. It s probable that these bodies are rather composed of particles vhich only are placed close to one another and are kept together by the pressure of some other outside matter and by lie irregularity of shapes. First, this non-solidity is apparent jolidity

irom the ease with which the matter of the magnetic vortices ind that which causes weight passes through. Furthermore, ve cannot say that these bodies are of a texture similar to :hat of a sponge or of light bread, because the heat of fire :an make them flow and change thereby the relative position It remains, therefore, that they are, as has :>f the particles. 3een said, collections of particles which touch one another, vithout constituting a continuous solid. This being so, the notion which these particles receive in order to continue the ivaves of light cannot but be communicated from one to the >ther. This happens without the bodies' leaving their place or their being disarranged among themselves and thus can

Breakthroughs in Physics

226

well bring about this effect without in any

way

prejudicing

the solidity of composition which is apparent to us. By the pressure from without, of which I have spoken, we need not understand that of air which would not be sufficient for this, but that of another more subtle matter. This|

pressure manifests itself in that experiment which by chance I encountered a long time ago, namely, that of water cleansed, of air, which remains suspended in a tube of glass open at, the lower end, notwithstanding that the air has been removed from the vessel in which the tube is enclosed. One can in this manner conceive of transparency without it being necessary that the ethereal matter, which serves forj light, passes through other matter nor that it finds pores^ through which to insinuate itself. But the truth is that thisj matter not only does pass through, but even does so with/ the greatest facility, concerning which the experiment of; Toricelli, alluded to above, is already a proof. Because wheni the quicksilver and water leave the upper part of the glass? tube, it appears that it is filled at once with ethereal matter,/

because light travels through it. But here is another argument] this easy penetrability, not only in transparent bodies but also in all others. When light passes through a concave sphere of glass, closed^

which proves

on

it is full of the ethereal matter,: spaces outside the sphere. And this ethereal matter, as has been shown before, consists of little,' particles which very nearly touch one another. If this matter were enclosed in such a way in the sphere that it could notj leave through the holes of glass, it would be obliged to follows the motion of the sphere when one makes it change place, Consequently it would very nearly require the same force to; impart to this sphere a certain speed, when it is at rest on a horizontal plane, as if it were full of water or perhaps quick-?: silver, because any body resists the speed which one wants! to give it according to the quantity of matter which it conn tains and which must follow the motion. But one finds, on the; contrary, that the sphere does not resist the imparting ofi motion except according to the quantity of glass of which it; is made: hence it is necessary that the ethereal matter, which) is within, is not shut in but it flows through with the greatest: freedom. shall see afterwards that the same penetrability also obtains, by this means, for opaque bodies. The second way of explaining transparency (and the one which seems more likely) consists in saying that the light! waves continue in the ethereal matter which continuously

just

all sides, it is

as

much

as

clear that

the

(

i!

We

1

The Wave Theory of Light

227

Occupies the interstices or. pores of transparent bodies. For ;ince light passes through them continually and with ease, it lollows that they are always found to be full of this matter. \nd one can even demonstrate that the interstices occupy

nuch more space than the cohering particles which constiute the body. For if it is true, as we have said, that one leeds force for imparting a certain horizontal speed to bodies n proportion to the coherent matter which they contain, and

proportion of this force follows the ratio of weights, experience confirms, then the quantity of the constitutive natter of bodies also follows the proportion of weights. Thus ve see that water weighs only one-fourteenth as much as an jqual portion of quicksilver. Hence the matter of water only )ccupies the fourteenth part of the space which holds its nass. But it should occupy even less, since quicksilver is ess heavy than gold, and is much less dense than the matter )f gold, as follows from the fact that the matter of the magletic vortices and of that which causes weight passes through mite freely. But one can object here that, if the body of- water is of :o great a rareness and its particles occupy so small a portion )f the space of its apparent extent, it is very strange how n spite of this it resists so strongly to being compressed, vithout letting itself be condensed by any force that one has ;ver tried to employ so far, preserving even all of its f the is

when it suffers such pressure. This difficulty is not small. Nevertheless we can resolve it >y saying that the very violent and rapid motion of subtle natter which makes water liquid, by shaking the particles >f which it is composed, maintains this liquidity despite any >ressure which so far has been devised for application there. The rareness of transparent bodies being such as we have aid, we can easily conceive that the waves can be continued Q the ethereal matter which fills the interstices of the paricles. Furthermore, we can believe that the progress of these /aves must be a little slower within bodies, on account of iquidity

tie

little

how

detours which these same particles cause.

I

shall

that the cause of refraction consists in these different

peeds of light. I have indicated earlier the third and last way in which re can understand transparency; this consists in supposing aat the motion of the light waves is transmitted indifferently ither in the particles of ethereal matter which occupy the iterstices

of

or in the particles which constitute motion passes from one to the other. We

bodies,

odies, so that the

Breakthroughs in Physics

228

on

well to exnlait that this hypothesis serves well explah the double refraction of certain transparent bodies. 1 If one should object that the particles of ether are smalle than those of transparent bodies, since they pass througl their interstices, and that it follows that they could onl]

shall see later

1

transmit a little of their motion, then it can be answere< that the particles of bodies are themselves composed o other particles that are smaller; and thus it will be thes< second particles which will receive the motion from the par tides of ether. Finally,

if

the

particles

of

transparent

bodies

posses;

prompt than that of etherea particles and nothing prevents us from supposing this follows once more that the progress of the light waves wil be slower within the body than outside in the ethereal matter This then is everything which I have found most likeli concerning the way in which light waves pass through trans parent bodies. To this we must still add in what respec these bodies are different from those that are opaque; anc| this is all the more necessary because it may seem, on aoj count of the easy way in which the ethereal matter penetrates bodies (of which we have spoken) that there are nc bodies which are not transparent. For in the same way ai elasticity that is

—

a

little

less

used the hollow sphere to prove the small density of glasi in which ethereal matter penetrates it, one can also prove that this same ease of penetration belongs tc! metals and to every other kind of body. If the sphere h made of silver for example, it is certain that it contain!*" ethereal matter which serves for light, since this matter wa!| there as well as air, when the opening of the sphere wai closed. Now, being closed and placed on a horizontal plane it offers resistance to motion which we wish to impart to i only in proportion to the quantity of silver of which it H I

and the easy way

made, so that we must necessarily conclude, as above, tha the ethereal matter which is enclosed does not follow the motion of the sphere, and that therefore silver, as well as glass is quite easily penetrated by this matter. Thus ether is founc continuously and in quantity between the particles of silvei and of all other opaque bodies; and since ether serves fo] the transmission of light, it seems that these bodies mus also be transparent like glass. Nevertheless, this is not th< case.

1

The portion of

this is

the Treatise on Light in which

not included here.

Huygens

doe*i

ji

The Wave Theory of Light One

will

ask,

whence comes opacity? Are

229 the

particles

which compose these bodies soft, that is, are the particles, ijeing themselves composed of lesser ones, capable of changing their shape on receiving impact from ethereal particles, whose motion they thereby blunt and thus hinder the continuation Df the light waves? That cannot be; for if the particles of

how

and mercury have found most likely in this is to say that the bodies of metals, which are almost the only ones truly opaque, have mixed in with their hard particles some soft ones. Thus the ones serve to cause reflection, and the others to hinder transparency. On the other aand, transparent bodies contain only hard particles, which have the property of being elastic, and which serve together with those of the ethereal matter, as has been said, for the netals are soft,

is

^o strongly reflect light?

it

that polished silver

What

I

transmission of the light waves. Let us now pass on to the explanation of the effects of refraction, by supposing as we have done, that. light waves travel through transparent bodies and that thereby their speed is diminished.

The

first

AB, which

property of refraction

is

that a ray of light, as

and falls obliquely on the polished surface of a transparent body like FG, is broken at the point 3f incidence B, so that it makes an angle CBE, with the straight line DBE which cuts the surface at right angles, svhich is less than angle ABD, which it makes with the same is

in air

perpendicular in the air. And the amount of these angles za.il be found by describing a circle around the point B as :enter, which cuts the rays and BC. The perpendiculars AD, CE, drawn from the points of intersection to the straight line (which are called the sines of the angles

AB

DE

ABD

and CBE) have a certain

ratio to

one another which

230 is

Breakthroughs in Physics

always the same for

all

inclinations of the incident ray,

in the case of a given transparent body. In the case of glass

the ratio is very nearly 3 2, in the case of water very nearly 4:3, and there is a different ratio for other transparent bodies. Another property, similar to this one, is that refractions are reciprocal between rays which enter a transparent body and those which leave it. That is to say, if the ray AB upon entering into the transparent body is broken in the direction BC, then CB, if it be taken as a ray within the body, will be broken, upon leaving, in the direction BA. Now, in order to explain the reasons for these phenomena] in accordance with our principles, let there be the straight line AB, which represents a plane surface separating the the transparent bodies which are toward C and toward N, When I say "plane," this does not indicate a perfect equality but such a one as we understood when we spoke of reflec tion, and for the same reason. Let the line represent a part of a light wave whose center is supposed to be so fart :

AC

:

away, that this part can be considered as a straight line, Then, in a certain amount of time the point C of the wave! AC will have advanced as far as the plane AB, following the straight line CB, which we must imagine as coming from the luminous center and which consequently cuts AC at right; angles. Then in the same time the point A would have come! to G along the straight line AG, equal and parallel to CB, and the entire part of the wave AC would be at GB, if the!

The Wave Theory of Light

231

matter of the transparent body were to transmit the motion of the wave as quickly as that of ether. But let us suppose that it transmits this motion less quickly, for example, by a

Motion will have been distributed, therefore, from the point A inside the matter of the transparent body, through a length equal to two thirds of CB, making its own spherical wave, according to what was said above. This wave is therefore represented by the circumference SNR, whose center is A, and whose radius is equal to 2/3 of CB. If we afterof the wave AC, it seems wards consider the other points that in the same time in which the point C has come to B, they have not only arrived at the surface AB, by means parallel to CB, but that besides they of straight lines will have engendered, around the centers K, their own waves in the transparent body, represented here by the circumferences whose radii are equal to 2/3 of the lines KM, that is to the straight line to say, 2/3 of the continuations of third.

H

HK

HK

BG. For the radii would have been equal to the if the two media had the same penetrability.

entire

KM,

All of these circumferences have as their common tangent BN, namely, the same line which is the tangent to the circumference SNR from the point B, which we considered first. For it is easy to see that all the other circumferences are going to touch the same line BN, between B and the point of tangency N, which is also the falls perpendicularly on BN. same point where the line Thus BN, which is as it were formed by the little arcs of circles, limits the motion which the wave has communicated in the transparent body, and is where this motion is found in much greater quantity than anywhere else. And hence this line, following what has been said more than once, is the propagation of the wave at the moment when its point C has arrived at B. For there is no other line below the plane AB which is, like BN, the common tangent of all the said little waves. And if one wishes to know how the ivave successively came to BN, it is only necessary to draw, in the same figure, the straight lines parallel to BN, and all the lines parallel to AC. Then one will see :hat the wave CA, from being straight has become bent into ill the lines successively, and that it has again become straight at BN. Since this is evident from what has already Deen shown, it is not necessary to make it clear further. Then in the same figure, if one takes which cuts the Diane AB at right angles at the point while is perK the straight line

AN

AC

AC

AC

KO

KL

LKO

EAF A

AD

°

Breakthroughs in Physics

232

DA

pendicular to the wave AC, it will be which marks the] ray of incident light, and AN, which is perpendicular to| BN, which marks the broken ray. For the rays are nothing else but the straight lines in the direction of which the parts! of the waves are extended. From this it is easy to understand the principal property! of refractions, namely that the sine of the angle has! always the same ratio to the sine of the angle NAF, ncj matter what may be the inclination of the angle DA, andt that this ratio is the same as that of the speed of the waves in the medium toward to their speed in the medium toward AF. For considering AB as the radius of a circle, the sine of angle BAC is BC, and the sine of the anglej is AN. But the angle BAC is equal to DAE, since each of them, added to angle CAE, makes a right angle! And the angle is equal to angle NAF, since each eg them together with makes a right angle. Thus the sine of angle is to the sine of angle as BC is to AN': But the ratio of BC to is the same as that of the speeds) of light in the matter which is toward and in that which;; is toward AF. Hence also the sine of the angle tci the sine of the angle will be as the said speeds of lightl In order now to see what the refraction must be whei, waves of light pass into a body where the motion is prop! agated more rapidly than in that from which they come (li us assume this time the ratio of 3 to 2), we need onljj repeat the same construction and demonstration which wej have just made, merely substituting 3/2 for 2/3. And wj will find by means of the same reasoning, in this othei figure, that when the point C of the wave has reaches to the surface AB at B, the entire part of the wave hi advanced to NB, in such a way that BC, perpendicular t AC, is to AN, perpendicular to BN, as 2 is to 3. And thi: same ratio of 2 to 3 will also be the ratio between the sin«. of the angle and the sine of the angle FAN. From this we see the reciprocity in the refractions of ray entering and leaving the same medium. If, namely, N; falling on AB is refracted into AD, then also the ray D/ will be refracted, upon leaving the transparent body, int»;

DAE

AE

ABN

ABN BAN

DAE

NAF

AN

AE

DAE

NAF

AC

AC

EAD

AN.

We

also see the reason for a notable event that happen is that, from a certain obliquity c f

in this refraction. This

DA

the incident ray on, it can no longer penetrate at a into the other transparent medium. For if the angle DAt or CBA is such that in the triangle ACB, CB is equal t

r,

I I i

r

The Wave Theory of Light

AB

233

AN

cannot make a even greater, then ANB, because it becomes equal to AB or greater. For this reason, the part BN of the wave will be found nowhere, nor consequently AN, which must be per[pendicular to it. And the incident ray DA then does not at all 2/3 of

or

is

side of the triangle

,

I

pierce the surface ]

AB.

When

the ratio of speeds of the waves is as 2 to 3, as in our example, which is that which applies to water and air, must be greater than 48° 11', in order that (the angle [the ray be able to pass through without breaking. And 'Iwhen the ratio of speeds is as 3 to 4, as it is very nearly in must exceed 41° 24'. Water and in air, this angle [And this agrees perfectly with experience. But here one may ask, since the encounter of the wave with the surface must produce motion in the matter which is on the other side, why no light passes through. The |

DAQ DA

DAQ

l

AB

AC

has been said waves is engendered in the matter which is on the other side of AB, these waves do not happen to have a common tangent (either straight or curved) at the same instant; and thus there is no line which terminates the propagation of the

answer to

this is easy, if

For although an

jjabove.

AC

we remember what

infinity of particular

below the plane AB, nor any place where the moin sufficiently large quantity to produce flight. And we shall easily see the truth of this, namely that when CB is greater than 2/3 of AB, the waves excited | below the plane AB have no common tangent, if around the

{waves tion

is

collected

Breakthroughs in Physics

234

K

we describe circles having their radii equal to 3/2 center of the LB's that correspond to them. For all these circles are enclosed within one another and all of them pass below the point B. Then it is to be noted that, when the angle is smaller to than it has to be in order to permit the broken ray pass into the other medium, we find that the interior reflection, which takes place at the surface AB, is much increased in clarity. This can easily be done experimentally with a triangular prism. The reason for this can be given by our is still large enough to make theory. When the angle be able to pass, it is manifest that the light of the ray is collected in a smaller length, the part of the wave bewhen it reaches BN. It is also clear that the wave is made smaller; comes smaller as the angle CBA or until this angle having become diminished to the limit noted

DAQ

DA

DAQ

DA

AC

BN

DAQ

a

earlier,

little

point.

That

is

has arrived at of

AC,

is

wave BN becomes collected all in one when the point C of the wave AC B, the wave BN, which is the propagation this

to say,

altogether reduced to the very point B; in the has arrived at K, the as, when the point

H

same fashion

AH

has become altogether reduced to the very point K. This lets us see that in proportion to the encounter of the wave CA with the surface AB, a great quantity of motion is found along this surface. This motion must also be spread out within the transparent body, and to have strongly reinforced the particular waves, which produce the interior reflection against the surface AB, following the laws of reflection explained above. And a little diminution of the angle of incidence become nothing from being still quite makes the wave 11', angle large: for when this angle in water is 49° 21'; but when the same angle is is still 11° is rediminished by only one degree, then the angle part

DAQ

BN

DAQ BAN

BAN

From this it comes that the interior refrom being obscure becomes suddenly quite clear, when the angle of incidence is such that it no longer allows duced

to nothing.

flection

passage for refraction. As for the ordinary exterior reflection, that is for that; is still which takes place when the angle of incidence large enough to let the refracted ray penetrate below the surface AB, this reflection must be made from particles of matter which touch the transparent body on the outside. And the reflection is apparently from particles of air and

DAQ

others that are

mixed among the ethereal matter, and are

The Wave Theory of Light more gross than

it,

since,

235

on the other hand, the exterior made from the particles which

reflection of these bodies is

ompose them and which are

also larger than those of matter, since this flows in their interstices. It is true that there remains in this some difficulty with the experiments where this interior reflection is made without the particles of air being able to contribute, as in the vessels ethereal

tubes from which air has been drawn. For the rest experience teaches us that these two reflections ire pretty nearly of equal force and that different transparent bodies have more of it just to the extent as their refraction is greater. Thus we can clearly see that the reflection of glass is stronger than that of water, and that of diamond is stronger than that of glass. I shall finish this theory of refraction by demonstrating i remarkable proposition which depends on it: A ray of ight, in going from one point to another, when these points are in two different media, is broken at the plane surface which joins the two media in such a fashion that it uses >r

time possible; just as is also the case, in reflection from one plane surface. Mr. Fermat has advanced the first of these properties of refraction, holding as we do (and directly contrary to the opinion of Mr. Descartes) that light travels more slowly through glass and water than through air. But he supposed, besides this, the constant proportion Df sines, which we have proved by means of these diverse degrees of speed: or, what comes to the same thing, he

the least

supposed, besides the different speeds, that light employed in its passage the least time possible, in order to derive from this the constant proportion of sines. His demonstration, which can be seen in his printed works and in the book of letters of Mr. Descartes, is very long; that is why I here give another one that is simpler and easier. Let there be a surface KF; let there be a point A in the

medium

in

which

there be a point

light travels

C

more

in the other

easily,

such as

air;

let

medium more difficult to a ray have come from A,

penetrate, such as water; and let through B, to C, being broken at B according to the law demonstrated a little earlier. (That is to say, having taken PBQ which cuts the plane at right angles, the sine of the angle ABB has the same ratio to the sine of the angle CBQ, as the speed of light in the medium wherein A is has to its speed in the medium wherein C is.) We are to demonstrate that the times for the passage of the light through AB and BC, taken together, are the shortest that they could be. Let

1

Breakthroughs in Physics

236

us take some other route, whatever it may be, and first by; means of lines AF, FC, of such a sort that the point of refraction F is more distant than B from point A. Let be perpendicular to AB, FO parallel to AB, perpendicular to FO, and FG perpendicular to BC.

AO

BH

Since then the angle HBF is equal to PBA, and angle is equal to QBC, it follows that the sine of angle will also be to the sine of angle BFG in the same ratio as the speed of light in the medium A to its speed in the medium C. But these sines are the straight lines HF, (if we take BF as the radius of a circle). Then the lines HF, have to one another the said ratio of speeds. And therefore the time light would take to travel through (supposing that the ray were OF) would be equal to the within the medium C. time it takes to travel through is equal to the time But the time of travel through through OH; hence the time of travel through OF is equal, to the times through AB, BG. Again, the time of travel through FC is greater than that through GC; thus the time;

BFG HBF

BG

BG

HF

BG AB

CFO

will be longer than that through greater than OF; hence the time through exceed even more the time through ABC.

through

AF A

is

Now to C

closer

let

us take the case

when

the ray has

ABC. But

AFC

will

come from;

by way of AK, KC, the point of refraction K being; to A than the point B. Let CN be perpendicular to

The Wave Theory of Light BC;

KN

let

BC, and Here

KBM,

let

BM

be parallel to BC, let be perpendicular to BA.

KL

BL

that

and

KM

are

the

angles

BKL,

PBA, QBC. And

there-

sines

to say, of the angles

is

fore they are to

medium A

237

be perpendicular to

of

the

one another as the speed of

light

in

the

speed in the medium C. Thus the time of travel through LB is equal to the time through KM; and since the time through BC is equal to the time through MN, the time through LBC will be equal to the time through is longer than that through KMN. But the time through is longer than that AL; and therefore the time through is longer than KN, the time through ABC. And since will surpass even more that time through through ABC. And thus it appears that the time through ABC is the shortest there can be: which was to be demonstrated. is

to the

AK

AKN

KC

AKC

PART

II

Based on the brilliance of his mind, the universality of his and the importance of his discoveries, Christiaan Huygens could surely have been expected to be the most famous physicist of the generation following Galileo. If this is not the case, it is due simply to the fact that Huygens had the "misfortune" to be a contemporary of Newton. No fame and reputation could compete with that of Sir Isaac. All the more remarkable, therefore, that Huygens' researches attracted as much attention and praise as they did, even in his lifetime. Both Rene Descartes and Sir Isaac himself interests,

spoke favorably of his achievements. Indeed, in his treatise on the pendulum clock, Huygens made important additions to Newton's work in the field of gravity and the motions resulting therefrom, and in the Treatise on Light of which we have a portion here before us Huygens presented

—

—

many years eclipsed those of Newton. statement obviously calls for an explanation. This, however, must wait until we understand a little more of Huygens' theory of light. Suffice it to say that the area of conflict between Huygens and Newton in the field of optics has to do with the manner in which light is propagated and with the speed of light. We shall discuss this controversy later and see how it was resolved (there is some doubt that it was resolved at all). We must begin by understanding what Huygens says about light. If we consider light from a merely geometric point of theories that for

The

last

Breakthroughs

238 view,

we would

in Physics

say that light travels in straight

lines.

This

seems to be evident from occasions when we can see the rays of the sun (as in a cloud of dust); the very fact that we use the word "ray" with respect to light indicates our natural feeling that light in traveling from point A to point is best represented by the straight line AB. Other evidence

B

for this straight-line travel seems to

lie

in the fact that

we

cannot see objects behind other objects; i.e., light does not bend around things, and we cannot see around corners. Furthermore, "geometrical optics," i.e., optics based on the assumption that a light ray may be represented by a straight line, leads to a number of highly useful and verifiable conclusions. In our chapter dealing with Galileo's Starry Messenger we had an example of geometrical optics. In the explanation of how different kinds of telescopes work, we

drew straight lines to represent light rays, and showed how these rays (or lines) were bent by lenses so that magnification resulted. These and similar results can be verified by experiment and therefore confirm to a very high degree of probability that light travels in straight lines.

None is

of this, however, touches the question of how light produced physically or how we are to understand its

propagation in terms of physical causes. When a light source at A is seen by an eye at B, we may still ask: What happens between A and B to produce this appearance? To comprehend what we mean by the question, it is only necessary to ask the same question concerning sound: When a sound, produced by a source at A, is heard at B, what happens between A and B to produce this appearance? In the case is that the air or other medium between compressed in a certain way, and that this compression (and the variations therein) is felt by the eardrum. Nerves from the inner ear relate the pressures to the brain, which somehow interprets them as sounds. What is the evidence that sound is in fact propagated this way? Huygens himself tells us that experiments have been made showing that when a vacuum (or near-vacuum) is produced around a sounding apparatus, the sound can no longer be heard (see p. 212). That sound is actually produced by a series of compressions and rarefactions of air is a little more difficult to prove in detail. Experiments can, however, be performed that leave no doubt about the propagation of sound. Collateral evidence comes from familiar phenomena such as "sonic boom": When an airplane flies at nearly the same speed as sound, the noise produced by its engines occasionally

of sound the answer

A

and

B

is

!

The Wave Theory of Light

239

reaches the ground in one large pressure wave, causing a boom that may be so strong as to shatter windows and jar juildings.

Now let us again ask the question about light: What happens between A and B, when a light source at A is seen at B? Light travels from A to B, but how? Light does not, like sound, require air for its propagation. The same experiment vhich shows that a vacuum prevents sound from being propigated (for example, an alarm clock in a glass jar from which he air has been almost completely removed cannot be heard), Uso shows that light traverses a vacuum (for the alarm clock emains visible even after the vacuum has been established), furthermore, light from the stars, millions of miles away, eaches us, traveling through "empty" space, thus indicating he superfluity of air for the travel of light. If light does not equire air, how does light travel from A to Bl One answer which was Newton's is that light consists of ninute particles that are sent out from the light source. When ight travels from A to B, a whole series of tiny corpuscles hoots out from A to B. This would make the straight-line notion of light understandable: inertia would keep the little j>odies from traveling any other way. One objection at once irises: if light consists in the actual travel of bodies, how can jhese corpuscles penetrate so-called "transparent" bodies? liow can light particles pass through glass and similar substances? The answer does not seem too difficult: obviously he light particles must be of such small size that they can iniite easily pass through the interstices of grosser bodies. Notice again how Newton's thought is facilitated by atomism; ill bodies consist of atoms, placed more or less closely together. Transparent bodies are those whose atoms are sufficiently far apart to permit the passage of light corpuscles; >paque bodies are those which do not permit this passage. Huygens, however, did not think that light could be propagated in this fashion. His argument was based on the speed >f light. Light, Huygens believed, took time to travel from i to B; but the speed with which light travels is so tremendous hat Huygens did not believe any body, no matter how small, ould possibly travel with this speed. Consequently, Huygens >roposed a different mode of propagation. First, let us see how Huygens convinced himself that light loes not travel instantaneously (i.e., with infinite speed), but akes time. Earlier natural philosophers, such as Rene )escartes, had believed in the instantaneous propagation of ight, but their reasoning, according to Huygens, was based on

—

|

I

—

Breakthroughs in Physics

240

insufficient evidence.

He

begins by referring to

phenomena

based on eclipses of the moon, which might be thought to constitute evidence for instantaneous propagation (see p. 207). These phenomena, Huygens tells us, at most prove that light travels very rapidly more than a thousand times as rapidly as sound, for instance but they still do not prove that light takes no time whatsoever to travel from the moon to the earth. All the evidence drawn from eclipses of the moon is quite compatible with a motion of light that consumes time, although very little. But while these earlier experiments based on eclipses of the moon merely showed that the speed of light though great might still be finite, this matter was settled once and for all in an experiment designed by Ole Romer. Since this matter is of great importance to Huygens, he reproduces the reasoning of Romer. It is based on eclipses of the satellites of Jupiter discovered by Galileo. (Notice how each scientific achievement depends on preceding achievements: the determination of the speed of light required the careful observa-

—

of

tions

—

Romer; these observations,

in

turn,

depended on

Galileo's discovery of the satellites of Jupiter; this discovery

depended on the perfection of the telescope.) Romer's demonstration depends on the existence of an observable

phenomenon

that

is

repeated a number of times

and that can be observed while the earth is at several different positions in its annual orbit around the sun. Since the radius of this orbit is quite large approximately 94 million

—

—

miles there may be detectable differences in the time it takes for light to travel from the phenomenon to the earth. The first three satellites of Jupiter present us with suitable phe-

nomena

Because these satellites are almost as that in which Jupiter travels around the sun, they are each eclipsed once during every revolution. (The fourth Galilean satellite travels in a plane slightly inclined to Jupiter's orbit and therefore sometimes is not eclipsed even though it is in opposition to the sun.) Romer based his argument on the innermost satellite, since it revolves most rapidly around Jupiter, once every 42 Vi in their eclipses.

exactly in the

same plane

hours.

Every A2Vi hours, therefore, the satellite emerges from the shadow of Jupiter and can be seen. For observational purposes this means that the satellite is "turned on" every 42^ hours. This "turning on" can, of course, be observed from the earth. During each 42Vi-hour revolution, the earth also moves. If we take as the location from which the first "turn-

The Wave Theory of Light

241

a point such as B (see p. 210), then the earth its distance from Jupiter and from the atellite until the earth reaches L. Now, suppose that durng 30 revolutions of the satellite the earth has traveled from

ng on" vill

is

made

steadily increase

passage of light requires time, then the time first "turning on" to the last "turning on" hirty revolutions later will be greater than 30 X 42V6 by the

? to C. If the

;lapsed

ime

from the

through the distance by which

light requires to travel

HC

exceeds HB. If, however, light travels instantaneously, hen the "turning on" at C will be observed exactly 30 X 42 Vi lours later. Furthermore, if light does not travel instantaleously, its speed can be calculated from these observations, rluygens reports that these calculations lead to a speed such hat light takes 22 minutes to traverse the diameter of the take half as long to travel half as far, to the earth. According to ac:urate modern calculations, this figure actually should be a ittle more than 8 minutes. The first determination of the jarth's orbit. It will e.,

11 minutes

from the sun

.peed of light by terrestrial

methods was made

3. L. Fizeau.

in

1849 by

'

Whether we use Huygens'

figure or the

more accurate

later

tremendous. No body, Huygens naintains, could actually travel with such speed. It seemed o him, for reasons that he is not very clear about, that such /elocity was not compatible with a real, material particle. For this reason, Newton's hypothesis that light consists in he motion of small bodies (advanced in the Opticks) seems :o Huygens absurd. But if light does not consist in the moion of actual bodies, what does it consist of? And how are »ve to explain the speed of light? Huygens' proposal is that light is essentially a disturbance n the medium in which it travels. This medium is not any inown, visible, or detectable material; it is a new medium to which Huygens gives the name ether. The disturbance that igure, the speed of light

is

produces the phenomenon we call light consists in the moion of the little particles that make up the ether. These paricles must be imagined as being perfectly hard and elastic. [f one of the particles of the ether moves, it will soon hit igainst its neighboring particle. Because of the assumed perfect elasticity of these particles, all of the motion of the first ^article will be communicated to the second particle. This iecond particle, being set in motion, soon bounces against a hird particle, communicating all of the motion to it, and so If we label the successive particles A, B, C, and so on, Ne see that first A moves, but that A soon communicates its

m.

Breakthroughs

242

in Physics

its motion| motion to B. B A, after transmitting the motion to B, will rest again; similarly, B, after transmitting the motion to C, will also be; at rest again. A, therefore, moves only a very short distance, as does B, and C, and so forth. But since each particle transmits the motion it received to its next neighbor, the motion, i.e., the disturbance, continues to move. At first, the motion is in the vicinity of A, next the motion is around B, then it isj in the neighborhood of C, and so on. No one particle moves from A to Z, but the motion moves from A to Z. And! since, by assumption, all of the particles are perfectly elastic, none of the motion is lost in the process of transmission; as much motion arrives at Z as had left A. Since no actual body moves from A to Z, but only the motion that first was at A moves to Z, Huygens sees no difficulty in saying that this motion moves with the great speed

in turn moves, but communicates

to C.

1

:

required to propagate light. We see a light source when the disturbance in the ether started by the source reaches our eye. Since the particles of ether are not arrayed in a straight line, but rather are distributed through all of space, whenever a particle begins to move, it hits not only one particle but many, for it has many neighbors. Each of the particles put in motion by A in turn transmits its B lt B 2 B 3 motion to many of its neighbors. Thus B x transmits its moB 2 transmits its motion to tion to C llt C 12 C 13 . and so on. The total effect of this is that #2i> #22> #23> the disturbance started by A will be transmitted in all directions. This is similar to what happens when a stone is thrown into a calm pond. The stone makes a disturbance, which is also transmitted in all directions. Just as the disturbance in the water moves continually outward, so the disturbance in the ether which we call light moves continually outward. Both water and ether, therefore, transmit their disturbances in the ,

.

,

.

.

,

•

•

•

,

.

.

;

;

form of waves. But if the wave motion of light solves the difficulty arising from the speed of light, it immediately creates another one:

How

are

we

to explain that light seems to travel in straight waves in the ether all

lines? If a light source generates light

around

why

it,

well? This

is

a light bulb, light it

waves

not light transmitted in all directions if a light source, say in the middle of a room, it does send out is

really a false problem; for is

in all directions, as

we know from the fact that some one spot on the

illuminates the entire room, not just

Nor, on the other hand, is this any argument against a corpuscular theory of light; such a theory would quite readily

wall.

The Wave Theory of Light agree that a light source sends out rays of light in tions. These rays, together with their reflections,

243 all

direc-

would

il-

luminate the entire room.

What lines?

meant when we say

is

We mean

that light travels in straight

we have a light source (such as the sun) and if we shade this source (by means of a a pin prick or a similar device) in such a way that

if

bulb or the screen with that only a small opening remains, then the light that comes through this opening will travel in a straight line. But this phenomenon, Huygens tells us, can as well be explained by the wave theory as by the corpuscular theory. Such an open-

we

ing,

Now ing,

But, ing

wave of light through (see p. 216). Huygens admits, that after passing the opensome of the wave will spill over into the "dark" region. he adds, the initial wave that comes through the open-

is

read, will let a

it is

true,

followed by other waves. Furthermore, each of the parwas agitated by the initial impulse

deles of the ether that

wave) becomes, in turn, the secondary wave, because the particle keeps viDrating for some little time. These secondary waves will also Dass through the opening. Furthermore, Huygens shows that :hese secondary waves will reinforce the primary waves, but 3nly in the direction of the primary wave. In the diagram,

(thus giving rise to the initial origin of a

secondary waves will reinforce the wave CE, which is primary wave that came through the opening BG. All :he parts of the primary wave that spilled over into the "dark" region and similar parts of the secondary waves become lost because they do not reinforce one another. (They are out of Dhase and interfere with one another.) This explanation ac:ually accords with the observed phenomena: a tiny pin prick n a screen in front of a light source will produce a straightine ray, but there will be just a little illumination outside the ?ath of the ray, because no pin prick can be made small mough to prevent some of the waves from spilling into the

:he :he

'dark" region.

Let us add up the score as between the corpuscular and the vave theory of light. What are the points favorable and mfavorable to each one? So far it would seem that the corpuscular theory has slightly the better of it. It is a simpler heory and assimilates the motion of light to the motion of >odies in general. It explains the straight-line propagation of ight by simply referring to the inertia of bodies, rather than

more complicated reasoning of the wave theory. It need to imagine the existence of an all-pervading lew medium which has to be invented by the the ether

jiving the

loes not

—

—

Breakthroughs

244

in Physics

wave theory. The tremendous speed of light seems to be the main point in favor of the wave theory: we do not have to suppose that material bodies are actually transported 186,000 miles every second, but merely that a sort of "ripple" in the medium moves with this speed. There are some phenomena, however, that can only be explained on the basis of the wave character of light. Fore-

most of these a ray of light

is

the

falls

phenomenon known

as interference.

on a screen with two narrow

slits

When,

A

and

C

Fig. 7-1

B

(close to one another), light emerges from each slit. Ac^ cording to Huygens' theory, a light wave is started at each opening (see Fig. 7-1). If the two slits are very close to each; other, the two waves will soon begin to cross one another. If now a translucent screen C is placed so as to catch the image' of the light coming through the two slits, we find that we do: 1

We do not see this,

The Wave Theory of Light

245

two illuminated rectangles, but rather a series of light and dark bands (see Fig. 7-2). The light areas are those where the two light waves reinforce each other, because two svave crests come together. The dark areas, on the other hand, are those places where the crest of one wave and the trough of the other one come together and cancel each other out. These dark bands can only be explained if light is propagated in wave form and therefore has crests and troughs. Otherwise, it is hard to see how the coming together of two light sources can produce anything but greater quantity of lot see

light.

The wave theory has still another important point in its if we assume that light is propagated in waves, then

favor:

two basic laws of optics can immediately be derived as consequences of the wave motion. These are the laws of reflection and of refraction. For the corpuscular theory, as Newton develops it in the Opticks, the laws of reflection and refraction are simply postulates. That is, we must simply accept as an experimental fact that the angle of incidence and the angle of reflection are equal. In the case of refraction, we must similarly accept as an experimental fact that for any two given media there exists a constant ratio between the ,

sine of the angle of incidence

and the sine of the angle of

refraction (see Fig. 7-3).

= constant angle

A = angle B

Fig.

7-3

Huygens, however, can derive both of these laws as consewave theory. In reflection (see Chapter Two), tie shows that the light wave "bounces" off the reflecting sur-

quences of the

Breakthroughs in Physics

246

face in such a way that the angle of reflection is necessarily equal to the angle of incidence. In Chapter Three, Huygens similarly shows that the constancy of the ratio between the two sines (known as Snell's law) is a consequence of the wave theory. Huygens' reasoning is as follows: Let be a. wave front. (See the diagram on p. 230.) can be considered as a straight line because the origin of the wave is quite far away. Let be the surface that separates the two media; the medium above is the rarer one, while the medium below is the denser one. What happens when a portion of the wave front, such as A, reaches the surface AB? If the wave had remained in the upper medium, then in a given period of time, the portion would have reached point (where is a straight line and a continuation of AD). Because the lower medium is denser, the wave front will not have got as far as G. If we assume that light travels 3/4 as fast in the lower medium as in the upper one, the wave front will have traversed a distance 3/4 as long as AG. Therefore, describe a circle with A as center and 3/4 of as the radius; let it be SNR. Then the portion A of the wave front will have reached somewhere on that circle. Now consider some other point of the wave front, such as H. It will reach the separating surface at a point K. If there had been only one medium, when A reached G, then would have reached M. Because, after getting as far as K, the wave front was slowed down, the portion at will not reach as far as M, but only penetrate a distance equal to 3/4 of KM. Drawing a circle with as center and 3/4 of as radius, we conclude that this portion of the wave front

AC

AC

AB

AB

AB

A

G

AG

AG

H

K

KM

K

be found somewhere on this circle. The entire wave AC, therefore, will be found along the line BN, which is the tangent common to all the circles drawn around with radius of 3/4 KM. Based on this construction, it is easy to prove that the sine will

front

K

of the angle of incidence fraction as

is

to the sine of the angle of re-

1:3/4. From the construction

it

is

also

clear

the ratio of the speeds of light in the two different media. And the reason why the ratio of the sines is constant, no matter what the angle of incidence may be, is that the ratio of the speeds is constant, being in

that the ratio of the sines

is

fact independent of the angle of incidence.

This derivation certainly gives a great deal of plausibility wave theory. However, Newton was too good a scientist not to try to derive the laws of reflection and refraction

to the

The Wave Theory of Light imilarly

from the assumption There is very

aaterial particles. if

reflection.

that light

247

consists

little difficulty

of

little

with the law

We know

that this law holds for large bodies. for instance, observe the law that the angle of equal to the angle of reflection; this law can be

Jilliard balls,

ncidence is imply derived from Newton's Axioms (especially Corollary the parallelogram of forces). Things are a little more dificult with the law of refraction, however. Newton derives his law not in the Opticks, but deep in the body of the D rincipia. There, when he is discussing the motion of bodies raveling in different media, he is able to derive the law of efraction for the case where the bodies traveling are very :mall (as is the case with light particles), while the bodies iround them (such as the media within which they travel) ire very large. Newton derives the law of refraction by assuming that the light particles are attracted by the refracting nedium; hence their path is bent. In order to arrive correctly it Snell's law in this fashion, Newton has to make an assumption opposite to the one made by Huygens: For Newton's proof, it is necessary that light travel more rapidly in a ienser medium than it does in a rarer one. Here is a direct confrontation of two theories. According light must travel more rapidly in air the well-known phenomena of refraction are to be explained. According to Newton, light must travel more rapidly in water than in air, if these same phenomena :o

Huygens' theory,

chan in water,

if

are to be explained. Here, then, seems to be a perfect opportunity for a crucial experiment: let the speed of light be

measured in

;

and then

and see which is greater. Huygens was right; if the speed is greater in water, then Newton was right. The experimental apparatus available at the time of Newton and Huygens did not permit such a measurement to be made. Indeed, we have seen that the measurement of the speed of light was made by means of astronomical phenomena; these phenomena obviously could give no indication of the speed of light in some medium such as water. Accordingly, the crucial experiment had to wait for the perfection of apparatus. Finally, the day came when the speed of light could be measured, and the experiment was performed by Jean Foucault in 1850. The result? Huygens was right; light travels faster in air than in water. Thus the wave theory appeared to have If

the speed

won

air

is

greater in

in water,

air,

a decisive victory over the corpuscular theory.

Breakthroughs in Physics

248

But this is not the end of the story. Toward the beginning of the twentieth century, phenomena were discovered that! were not at all compatible with the wave character of light;] they could only be understood if light was assumed to be corpuscular in nature. These phenomena principally have to do with the emission and absorption of light by matter. In 1900, Max Planck made the supposition (in order to explain some puzzling phenomena of radiation) that when I light is emitted, the energy sent out is always a whole-number multiple of the quantity / X h, where / is the frequency of the light and h called Planck's constant is a tiny but perfectly definite constant, numerically equal to 6.62 X 10-27 . Again, when light strikes a metal plate, electrons are given, off. It was found that each electron takes on the same amount of energy and that this energy is independent of the intensity of the light. This was now explained by the supposition that light consists of discrete packets of energy, called photons. When the light striking the plate is not very intense,) there are only a few photons, but each photon has the same! amount of energy and hence can liberate electrons. If the! light were a wave front, the small amount of energy in the! i

—

—

dim light would either be insufficient to liberate any electrons or if it did liberate a few, the energy would be distributed: over the entire wave and would give to each liberated electron a smaller addition of energy than is in fact found to be' the case. The discovery, or better, supposition of this photoelectric effect was made by Einstein in 1905. It is, of course,; based on Planck's earlier theory. Einstein received the Nobel! Prize for his work.

We

now

avoid the question: What is the nature of! essentially the character of waves or oi bodies? Both answers seem to be required, and yet the two answers seem incompatible. There is no satisfactory answer that we can give here. The question is dealt with in quantum! mechanics. Here we can only give some hints as to how the! light?

cannot

Does

it

have

answer might be constructed. One answer consists in saying that light is corpuscular in nature, but the light particles have certain wave properties.; and each light particle has a certain wave associated with it Another answer is just the opposite: light is basically of £ wave character, but each wave has a particle associated witt it. Perhaps the more satisfactory answer is the one tha maintains that light is corpuscular in nature. Because there h a basic uncertainty and indeterminacy in nature, this viev

The Conservation of Energy

249

continues, the wave associated with the particle is needed to give an indication of how probable it is that a given particle of light will be found in a given place.

CHAPTER EIGHT Von Helmholtz:

—The Conservation of Energy PART

I

Hermann Ludwig Ferdinand von Helmholtz (1821-1894) another one of those scientists whose genius could not be contained in any one field. The work we here reprint is a popular lecture On the Conservation of Force, delivered in 1862; Helmholtz's thoughts on the subject had first been presented to the Physical Society of Berlin in 1847. In addition to this work, which we may classify as belonging to theoretical physics, or perhaps to the theory of heat, Helmholtz is

in optics, especially physiological that part of optics dealing with the problems of the human eye. He also did research in physiologi-

worked with great success optics,

sight cal

i.e.,

and

i.e., the problems of hearing. others in which Helmholtz did brilliant

acoustics,

fields,

Besides

these

work included

hydrodynamics, electricity, and meteorology. Helmholtz's eminence is attested to by his academic honors and positions. In 1871 he became professor of physics at the University of Berlin and in 1887 he also became director of the physicotechnical institute at Charlottenburg; he held both positions until his death. In view of his extraordinary distinctions and fame, it is significant that Helmholtz was one of the first physicists to devote himself to popular writing and lecturing. The tradition has grown since his time; for instance, Einstein's

work on

popular treatment of a physicist.

relativity in the next chapter

difficult subject

is

a

by another eminent

250

Herman von

Breakthroughs in Physics

Helmholtz;:

On

the Conservation of Force*

Introduction to a Series of Lectures delivered at Carlsruhe in the Winter of 1862-1863.

As I have undertaken to deliver here a series of lectures, I think the best way in which I can discharge that duty will be to bring before you, by means of a suitable example, some view of the special character of those sciences to the study of which I have devoted myself. The natural sciences, partly in consequence of their practical applications, and partly from their intellectual influence on the last four centuries, have so profoundly, and with such increasing rapidity, transformed all the relations of the life of civilised nations; they have given these nations such increase of riches, of enjoyment of life, of the preservation of health, of means of industrial and of social intercourse, and even such increase of political power, that every educated man who tries to understand the forces at work in the world in which he is living, even if he does not wish to enter upon the study of a special science, must have some interest in that peculiar kind of mental labour which works and acts in the sciences in question. On a former occasion I have already discussed the characteristic differences which exist between the natural and the mental sciences as regards the kind of scientific work. I then endeavoured to show that it is more especially in the thorough conformity with law which natural phenomena and natural products exhibit, and in the comparative ease with which laws can be stated, that this difference exists. Not that I wish by any means to deny, that the mental life of individuals and peoples is also in conformity with law, as is the object of philosophical, philological, historical, moral, and social sciences to establish. But in mental life, the influences are so interwoven, that any definite sequence can but seldom be demonstrated. In Nature the converse is the case. * From Popular Lectures on Scientific Subjects, trans, by E. Atkinson, Vol. I. New York and Bombay: Longmans, Green, and Co., 1898, pp. 277-317.

!

The Conservation of Energy

251

[t has been possible to discover the law of the origin and progress of many enormously extended series of natural phe-

aomena with such accuracy and completeness

that

we can

predict their future occurrence with the greatest certainty; or

cases in which we have power over the conditions under which they occur, we can direct them just according to our

in

The greatest of all instances of what the human mind can effect by means of a well-recognised law of natural phenomena is that afforded by modern astronomy. The one simple law of gravitation regulates the motions of the heavenly bodies not only of our own planetary system, but also of the far more distant double stars; from which, even the ray of light, the quickest of all messengers, needs years to reach our eye; and, just on account of this simple conformity with law, the motions of the bodies in question can be accurately predicted and determined both for the past and for future years and centuries to a fraction of a minute. On this exact conformity with law depends also the certainty with which we know how to tame the impetuous force of steam, and to make it the obedient servant of our wants. On this conformity depends, moreover, the intellectual fascination which chains the physicist to his subjects. It is an interest of quite a different kind to that which mental and moral sciences afford. In the latter it is man in the various phases of his intellectual activity who chains us. Every great deed of which history tells us, every mighty passion which art can represent, every picture of manners, of civic arrangements, of the culture of peoples of distant lands or of remote times, seizes and interests us, even if there is no exact scientific connection among them. We continually find points of contact and comparison in our own conceptions and feelings; we get to know the hidden capacities and desires of the mind, which in the ordinary peaceful course of will.

civilised life

remain unawakened.

not to be denied that, in the natural sciences, this kind of interest is wanting. Each individual fact, taken by itself, can indeed arouse our curiosity or our astonishment, or be useful to us in its practical applications. But intellectual satisfaction we obtain only from a connection of the whole, just from its conformity with law. Reason we call that faculty innate in us of discovering laws and applying them with thought. For the unfolding of the peculiar forces of pure reason in their entire certainty and in their entire bearing, there It is

252 is

no more

Breakthroughs in Physics suitable arena than inquiry into

wider sense, the mathematics included.

And

Nature it

is

in the

not only

the pleasure at the successful activity of one of our most esmental powers; and the victorious subjections to the power of our thought and will of an external world, partly unfamiliar, and partly hostile, which is the reward of this labour; but there is a kind, I might almost say, of artistic satisfaction, when we are able to survey the enormous wealth of Nature as a regularly-ordered whole a kosmos, an image of the logical thought of our own mind. The last decades of scientific development have led us to the recognition of a new universal law of all natural phenomena, which, from its extraordinarily extended range, and from the connection which it constitutes between natural phenomena of all kinds, even of the remotest times and the most distant places, is especially fitted to give us an idea of what I have described as the character of the natural sciences, which I have chosen as the subject of this lecture. This law is the Law of the Conservation of Force, a term the meaning of which I must first explain. It is not absolutely new; for individual domains of natural phenomena it was enunciated by Newton and Daniel Bernoulli; and Rumford and Humphry Davy had recognised distinct features of its presence in the laws of heat. The possibility that it was of universal application was first stated by Dr. Julius Robert Mayer, a Schwabian physician (now living in Heilbronn), in the year 1842, while almost simultaneously with, and independently of him, James Prescott Joule, an English manufacturer, made a series of important and difficult experiments on the relation of heat to mechanical force, which supplied the chief points in which the comparison of the new theory with experience was still wanting. The law in question asserts, that the quantity of force which can be brought into action in the whole of Nature is unchangeable, and can neither be increased nor diminished. first object will be to explain to you what is understood by quantity of force; or, as the same idea is more popularly sential

—

My

expressed with reference to its technical application, what we call amount of work in the mechanical sense of the word. The idea of work for machines, or natural processes, is taken from comparison with the working power of man; and we can therefore best illustrate from human labour the most important features of the question with which we are con-

The Conservation of Energy

253

cerned. In speaking of the work of machines and of natural forces we must, of course, in this comparison eliminate anything in which activity of intelligence comes into play. The latter is also capable of the hard and intense work of thinking, which tries a man just as muscular exertion does. But whatever of the actions of intelligence is met with in the work of machines, of course is due to the mind of the con-

and cannot be assigned

to the instrument at work. the external work of man is of the most varied kind as regards the force or ease, the form and rapidity, of the motions used on it, and the kind of work produced. But both the arm of the blacksmith who delivers his powerful blows with the heavy hammer, and that of the violinist who produces the most delicate variations in sound, and the hand of the lacemaker who works with threads so fine that they are on the verge of the invisible, all these acquire the force which moves them in the same manner and by the same organs, namely, the muscles of the arm. ^An arm the muscles of which are lamed is incapable of doing any work; the moving force of the muscle must be at work in it, and these must obey the nerves, which bring to them orders from the brain. That member is then capable of the greatest variety of motions; it can compel the most varied instruments to execute the most diverse tasks. Just so is it with machines: they are used for the most

structor

Now,

diversified

arrangements. We produce by their agency an of movements, with the most various degrees

infinite variety

of force and rapidity, from powerful steam-hammers and where gigantic masses of iron are cut and shaped like butter, to spinning and weaving-frames, the work

rolling-mills,

of which rivals that of the spider. Modern mechanism has the richest choice of means of transferring the motion of one set of rolling wheels to another with greater or less velocity;

of changing the rotating motion of wheels into the up-anddown motion of the piston-rod, of the shuttle, of falling hammers and stamps; or, conversely, of changing the latter into the former; or it can, on the other hand, change movements of uniform into those of varying velocity, and so forth. Hence this extraordinarily rich utility of machines for so extremely varied branches of industry. But one thing is common to all these differences; they all need a moving force, which sets and keeps them in motion, just as the works of the human hand all need the moving force of the muscles.

254

Breakthroughs in Physics

Now, the work of more intense exertion

the smith requires a far greater and of the muscles than that of the violin-

and there are in machines corresponding differences power and duration of the moving force required. These differences, which correspond to the different degree of exertion of the muscles in human labour, are alone what we have to think of when we speak of the amount of work of a machine. We have nothing to do here with the manifold character of the actions and arrangements which the maplayer; in the

chines produce; we are only concerned with an expenditure of force. This very expression which we use so fluently, *expenditure of force,' which indicates that the force applied has been expended and lost, leads us to a further characteristic analogy between the effects of the human arm and those of machines. The greater the exertion, and the longer it lasts, the more is the arm tired, and the more is the store of its moving force for the time exhausted. shall see that this peculiarity of becoming exhausted by work is also met with in the moving forces of inorganic nature; indeed, that this capacity of the human arm of being tired is only one of the consequences of the law with which we are now concerned. When fatigue sets in, recovery is needed, and this can only be effected by rest and nourishment. We shall find that also in the inorganic moving forces, when their capacity for work is spent, there is a possibility of reproduction, although in general other means must be used to this end than in the case of the

We

human arm. From the we can form

feeling of exertion

and fatigue

a general idea of what

in our muscles,

we understand by amount

of work; but we must endeavour, instead of the indefinite estimate afforded by this comparison, to form a clear and precise idea of the standard by which we have to measure the amount of work. This we can do better by the simplest inorganic moving forces than by the actions of our muscles,

which are a very complicated apparatus, acting in an extremely intricate manner. Let us now consider that moving force which we know best, and which is simplest gravity. It acts, for example, as such in those clocks which are driven by a weight. This weight, fastened to a string, which is wound round a pulley connected with the first toothed wheel of the clock, cannot obey the pull of gravity without setting the whole clockwork

—

The Conservation of Energy

255

Now

I must beg you to pay special attention to in motion. the following points: the weight cannot put the clock in motion without itself sinking; did the weight not move, it could

not move the clock, and its motion can only be such a one as obeys the action of gravity. Hence, if the clock is to go, the weight must continually sink lower and lower, and must at length sink so far that the string which supports it is run out. The clock then stops. The useful effect of its weight is for the present exhausted. Its gravity is not lost or diminished; it is attracted by the earth as before, but the capacity of this gravity to produce the motion of the clockwork is lost. It can only keep the weight at rest in the lowest point its path, it cannot farther put it in motion. But we can wind up the clock by the power of the arm, by which the weight is again raised. When this has been done, it has regained its former capacity, and can again set the

of

clock in motion.

We

this that a raised weight possesses a moving must necessarily sink if this force is to act; that by sinking, this moving force is exhausted, but by using another extraneous moving force that of the arm its ac-

learn

from

force, but that

it

—

—

can be restored. The work which the weight has to perform in driving the clock is not indeed great. It has continually to overcome the small resistances which the friction of the axles and teeth, as well as the resistance of the air, oppose to the motion of the wheels, and it has to furnish the force for the small impulses and sounds which the pendulum produces at each oscillation. If the weight is detached from the clock, the

tivity

pendulum swings for a while before coming to rest, but its motion becomes each moment feebler, and ultimately ceases entirely, being gradually used up by the small hindrances I have mentioned. Hence, to keep the clock going, there must be a moving force, which, though small, must be continually at work. Such a one is the weight. We get, moreover, from this example, a measure for the amount of work. Let us assume that a clock is driven by a weight of a pound, which falls five feet in twenty-four hours. If we fix ten such clocks, each with a weight of one pound, then ten clocks will be driven twenty-four hours; hence, as each has to overcome the same resistances in the same time as the others, ten times as much work is performed for ten pounds fall through five feet. Hence, we conclude that the

Breakthroughs in Physics

256

height of the fall being the same, the work increases directly as the weight. Now, if we increase the length of the string so that the; weight runs down ten feet, the clock will go two days instead of one; and, with double the height of fall, the weight! will overcome on the second day the same resistances as on the first, and will therefore do twice as much work as when it can only run down five feet. The weight being the same,] the work increases as the height of fall. Hence, we may take the product of the weight into the height of fall as a measure of work, at any rate, in the present case. The applica-j tion of this measure is, in fact, not limited to the individual! case, but the universal standard adopted in manufactures fori measuring magnitude of work is a foot pound that is, the amount of work which a pound raised through a foot cani

—

produce. 1

We may

apply this measure of work to

all kinds of ma^l able to set them all in motion by means of a weight sufficient to turn a pulley. could thus] always express the magnitude of any driving force, for any! given machine, by the magnitude and height of fall of suchi a weight as would be necessary to keep the machine going with its arrangements until it had performed a certain work.

chines, for

we should be

We

Hence it is that the measurement of work by foot pounds is: universally applicable. The use of such a weight as a drivings force would not indeed be practically advantageous in those* which we were compelled to raise it by the power) of our own arm; it would in that case be simpler to work the machine by the direct action of the arm. In the clock! we use a weight so that we need not stand the whole day at< the clockwork, as we should have to do to move it directly.; By winding up the clock we accumulate a store of working capacity in it, which is sufficient for the expenditure of the next twenty-four hours. The case is somewhat different when Nature herself raises the weight, which then works for us. She does not do this with solid bodies, at least not with such regularity as to be utilised; but she does it abundantly with water, which, being raised to the tops of mountains by meteorological processes, returns in streams from them. The gravity of water we use as moving force, the most direct application being in whal are called overshot wheels, one of which is representee in Fig. 8-1. Along the circumference of such a wheel are a cases in

1

1

This

scientific

is

the

technical measure of work; to convert it into it must be multiplied by the intensity of gravity

measure

!

The Conservation of Energy

257

Fig. 8-1

series

of buckets, which act as receptacles for the water, and,

on the side turned to the observer, have the tops uppermost; on the opposite side the tops of the buckets are upside'down. The water flows at into the buckets of the front of the wheel, and at F, where the mouth begins to incline downwards, it flows out. The buckets on the circumference are filled on the side turned to the observer, and empty on the other side. Thus the former are weighted by the water contained in them, the latter not; the weight of the water acts continuously on only one side of the wheel, draws this down, and thereby turns the wheel; the other side of the wheel offers no resistance, for it contains no water. It is thus the weight of the falling water which turns the wheel, and furnishes the motive power. But you will at once see that the mass of water which turns the wheel must necessarily fall in order to do so, and that though, when it has reached the bottom, it has lost none of its gravity, it is no

|

M

longer in a position to drive the wheel, to

its

original position,

either

if it

is

not restored

by the power of the human

arm or by means of some other natural force. If it can flow from the mill-stream to still lower levels, it may be used to

j

258

Breakthroughs in Physics

wheels. But when it has reached its lowest level, the sea, the last remainder of the moving force is used up 5| which is due to gravity that is, to the attraction of the earthy and it cannot act by its weight until it has been again raised to a high level. As this is actually effected by me-j teorological processes, you will at once observe that these

work other

—

are to be considered as sources of moving force. Water-power was the first inorganic force which man learnt to use instead of his own labour or of that of domestic animals. According to Strabo, it was known to King Mithridates of Pontus, who was also otherwise celebrated for his knowledge of Nature; near his palace there was a water-!

wheel. Its use was first introduced among the Romans in the time of the first Emperors. Even now we find water-mills in all mountains, valleys, or wherever there are rapidlyflowing regularly-filled brooks and streams. We find water power used for all purposes which can possibly be effected by machines. It drives mills which grind corn, saw-mills, hammers and oil-presses, spinning-frames and looms, and| so forth. It is the cheapest of all motive powers, it flows spontaneously from the inexhaustible stores of Nature; but it is restricted to a particular place, and only in moun tainous countries is it present in any quantity; in level coun tries extensive reservoirs are necessary for damming the rivers to produce any amount of water-power. Before passing to the discussion of other motive forces 1 must answer an objection which may readily suggest itself We all know that there are numerous machines, systems ol pulleys, levers and cranes, by the aid of which heavy burdens may be lifted by a comparatively small expenditure of force We have all of us often seen one or two workmen hoisi heavy masses of stones to great heights, which they wouk be quite unable to do directly; in like manner, one or twc men, by means of a crane, can transfer the largest an heaviest chests from a ship to the quay. Now, it may b asked, If a large, heavy weight had been used for driving machine, would it not be very easy, by means of a crane or system of pulleys, to raise it anew, so that it could agar be used as a motor, and thus acquire motive power, withouj being compelled to use a corresponding exertion in raising the weight?

The answer to this is, that all these machines, in that de gree in which for the moment they facilitate the exertiorj also prolong it, so that by their help no motive power ultimately gained. Let us assume that four labourers hav i

The Conservation of Energy

259

hundred-weight by means of a rope assing over a single pulley. Every time the rope is pulled own through four feet, the load is also raised through four set, But now, for the sake of comparison, let us suppose le same load hung to a block of four pulleys, as represented )

raise a load of four

Fig.

8-2

A

8-2. single labourer would now be able to raise by the same exertion of force as each one of the four Hit forth. But when he pulls the rope through four feet, the Dad only rises one foot, for the length through which he pulls 'tie rope, at a, is uniformly distributed in the block over four opes, so that each of these is only shortened by a foot. To *aise the load, therefore, to the same height, the one man lust necessarily work four times as long as the four together i

Fig.

ie load

id. But the total expenditure of work is the same, whether Dur labourers work for a quarter of an hour or one works for n hour. If, instead of human labour, we introduce the work of a

260

Breakthroughs in Physics

weight, and hang to the block a load of 400, and at a, where; otherwise the labourer works, a weight of 100 pounds, the block is then in equilibrium, and, without any appreciable exertion of the arm, may be set in motion. The weight ol

100 pounds sinks, that of 400 rises. Without any measurable! expenditure of force, the heavy weight has been raised bj; the sinking of the smaller one. But observe that the smalleii weight will have sunk through four times the distance thai the greater one has risen. But a fall of 100 pounds througtj four feet is just as much 400 foot-pounds as a fall of 400 pounds through one foot. The action of levers in all their various modifications h precisely similar. Let a b, Fig. 8-3, be a simple lever, sup-

Fig.

8-3

ported at c, the arm c b being four times as long as th4 other arm a c. Let a weight of one pound be hung at b, ani a weight of four pounds at a, the lever is then in equilibrium and the least pressure of the finger is sufficient, without an; appreciable exertion of force, to place it in the position ! b\ in which the heavy weight of four pounds has beei raised, while the one-pound weight has sunk. But here, alsq you will observe no work has been gained, for while thj heavy weight has been raised through one inch, the lightej one has fallen through four inches; and four pounds througl one inch is, as work, equivalent to the product of one pounl through four inches. Most other fixed parts of machines may be regarded 4 modified and compound levers; a toothed- wheel, for instanc as a series of levers, the ends of which are represented bj the individual teeth, and one after the other of which is pif in activity in the degree in which the tooth in questid seizes or is seized by the adjacent pinion. Take, for instancl

The Conservation of Energy

261

the crabwinch represented in Fig. 8-4. Suppose the pinion

Fig.

on

8-4

he axis of the barrel of the winch has twelve teeth, and he toothed-wheel, H H, seventy-two teeth, that is six times is many as the former. The winch must now be turned ound six times before the toothed-wheel, H, and the barrel, D, have made one turn, and before the rope which raises he load has been lifted by a length equal to the circumerence of the barrel. The workman thus requires six times he time, though to ^e sure only one-sixth of the exertion, vhich he would have to use if the handle were directly ipplied to the barrel, D. In all these machines, and parts of nachines, we find it confirmed that in proportion as the veocity of the motion increases its power diminishes, and that ,vhen the power increases the velocity diminishes, but that he amount of work is never thereby increased. In the overshot mill-wheel, described above, water acts by ts weight. But there is another form of mill-wheel, what is galled the undershot wheel, in which it only acts by its impact, as represented in Fig. 8-5. These are used where the

:

Breakthroughs in Physics

262

Fig.

8-5

height from which the water comes is not great enough to flow on the upper part of the wheel. The lower part of undershot wheels dips in the flowing water which strikes against their float-boards and carries them along. Such wheels are used in swift-flowing streams which have a scarcely perceptible fall, as, for instance, on the Rhine. In the immediate neighbourhood of such a wheel, the water need not necessarily have a great fall if it only strikes with considerable velocityj It is the velocity of the water, exerting an impact against the float-boards, which acts in this case, and which produces the

motive power. Windmills, which are used in the great plains of Holland and North Germany to supply the want of falling water, afford another instance of the action of velocity. The sails are driven by air in motion by wind. Air at rest could just as little drive a windmill as water at rest a water-wheel. The

—

driving force depends here on the velocity of moving masses, bullet resting in the hand is the most harmless thing in the world; by its gravity it can exert no great effect; but

A

when all

and endowed with great velocity it drives through most tremendous force. lay the head of a hammer gently on a nail, neither its

fired

obstacles with the If I

small weight nor the pressure of my arm is quite sufficient to drive the nail into the wood; but if I swing the hammer anc allow it to fall with great velocity, it acquires a new force, which can overcome far greater hindrances. These examples teach us that the velocity of a moving

mass can act as motive force. In mechanics, velocity in far as it is motive force, and can produce work, is calleq vis viva.

The name

is

not well chosen;

it is

too apt to suggest

The Conservation of Energy

263

us the force of living beings. Also in this case you will e, from the instances of the hammer and of the bullet, that jlocity is lost, as such, when it produces working power, the case of the water-mill, or of the windmill, a more careinvestigation of the moving masses of water and air is tl ;cessary to prove that part of their velocity has been lost i

t

the

i

The

work which they have performed.

working power is most simply pendulum, such as can be conructed by any weight which we suspend to a cord. Let M, ig. 8-6, be such a weight, of a spherical form; A B, a horimtal line drawn through the centre of the sphere; P, the )int at which the cord is fastened. If now I draw the eight on one side towards A, it moves in the arc a, e end of which, a, is somewhat higher than the point A relation of velocity to

id clearly seen in a simple

M

M

the horizontal line. ight

A

a.

Hence

The weight

is

thereby raised to the

my arm

must exert a certain force to Gravity resists this motion, and en-

ing the weight to 0. avours to bring back the weight to M, the lowest point hich it can reach. Now, if after I have brought the weight to a I let it go, obeys this force of gravity and returns to M, arrives there ith a certain velocity, and no longer remains quietly hanging as it did before, but swings beyond towards b, where 5 motion stops as soon as it has traversed on the side of B

M

M

Fig. H

8-6

arc equal in length to that on the side of A, and after it a distance B b above the horizontal line, which

is risen to

Breakthroughs in Physics

264

A

equal to the height a, to which my arm had previously it. In h the pendulum returns, swings the same way back through towards a, and so on, until its oscillations are gradually diminished, and ultimately annulled by the resistance of the air and by friction. You see here that the reason why the weight, when it comes from a to M, and does not stop there, but ascends to b, W§ opposition to the action of gravity, is only to be sought in its velocity. The velocity which it has acquired in moving from the height a is capable of again raising it to an equal height, B b. The velocity of the moving mass, M, is thus capable of raising this mass; that is to say, in the language of mechanics, of performing work. This would also be the case if we had imparted such a velocity to the suspended weight by a blow. From this we learn further how to measure the working power of velocity or, what is the same thing, the vis viva of the moving mass. It is equal to the work, expressed in foot is

raised

M

A

—

pounds, which the same mass can exert after its velocity has been used to raise it, under the most favourable circumstances, to as great a height as possible. 2 This does not depend on the direction of the velocity; for if we swing a weight attached to a thread in a circle, we can even change a downward motion into an upward one. The motion of the pendulum shows us very distinctly how the forms of working power hitherto considered that of a raised weight and that of a moving mass may merge into one another. In the points a and b, Fig. 8-6, the mass has no velocity; at the point it has fallen as far as possible, but possesses velocity. As the weight goes from a to the work of the raised weight is changed into vis viva; as the weight goes further from to b the vis viva is changed into the

—

—

M

m

m

work of a

raised weight.

Thus the work which the arm

imparted to the pendulum is not lost in these oscillations, provided we may leave out of consideration thei influence of the resistance of the air and of friction. Neither I does it increase, but it continually changes the form of its originally

manifestation.

2 The measure of vis viva in theoretical mechanics is half the product of the weight into the square of the velocity. To reduce it to the technical measure of the work we must divide it by the

intensity of gravity; that

is,

by the velocity

second of a freely falling body.

at the

end of the

first

The Conservation of Energy now

265

other mechanical forces, those of elastic bodies. Instead of the weights which drive our clocks, we find in time-pieces and in watches, steel springs which are coiled in winding up the clock, and are uncoiled by

Let us

pass

to

the working of the clock. To coil up the spring we consume the force of the arm; this has to overcome the resisting elastic force of the spring as we wind it up, just as in the clock we have to overcome the force of gravity which the

weight exerts. The coiled spring can, however, perform work; gradually expends this acquired capability in driving the clockwork. If I stretch a crossbow and afterwards let it go, the stretched string moves the arrow; it imparts to it force in the form of velocity. To stretch the cord my arm must work for a few seconds; this work is imparted to the arrow at the moment it is shot off. Thus the crossbow concentrates into an extremely short time the entire work which the arm had communicated in the operation of stretching; the clock, on the contrary, spreads it over one or several days. In both cases no work is produced which my arm did not originally impart to the instrument, it is only expended more conven-

it

iently.

The case is somewhat different if by any other natural process I can place an elastic body in a state of tension without having to exert my arm. This is possible and is most easily observed in the case of gases. If, for instance, I discharge a firearm loaded with gunpowder, the greater part of the mass of the powder is converted into gases at a very high temperature, which have a powerful tendency to expand, and can only be retained in the narrow space in which they are formed, by the exercise of the most powerful pressure. In expanding with enormous force they propel the bullet, and impart to it a great velocity, which already seen is a form of work. In this case, then, I have gained work which my arm has not performed. Something, however, has been lost the gun-

we have

—

powder, that is to say, whose constituents have changed into other chemical compounds, from which they cannot, without further ado, be restored to their original condition. Here, then, a chemical change has taken place, under the influence of which work has been gained. Elastic forces are produced in gases by the aid of heat, on a far greater scale. Let us take, as the most simple instance, atmospheric air. In Fig. 8-7 an apparatus is represented such as Regnault used

266

Breakthroughs in Physics

W\ SYi S SA

vXVf

air.

course, by insulating the boiler, but

we cannot

avoid

it

Breakthroughs in Physics

290

completely. (That there

mined by

is

loss in the engine

from the number of

calculating,

can be deter-

calories that are

used in heating the boiler, how many mechanical equivalents of heat are put in; we can then calculate how much mechanical energy we in fact extract from the engine. The difference between the energy input and the energy output is the heat loss.) If we imagine that the heat used to heat the boiler of the steam engine had been produced by mechanical means i.e., had been transformed into heat from a form then there has been an overall loss of mechanical energy of mechanical energy in the process. Even if all the original mechanical energy was turned into heat, there was a loss;; for in the reverse process of going from heat to mechanical energy, some of the heat remained as heat and did not be-j come converted. The best that can be theoretically expected

—

—

j

(and never reached practically) is that the same amount of mechanical energy reappears at the end of the two transformations. The reason for the unavoidable heat loss seems perfectly apparent: when the boiler has been thoroughly heated, it is! then considerably hotter than its surroundings. It seems nat-; ural and obvious to us that heat by itself flows from hotter regions (such as the boiler) to cooler regions (such as the! surrounding air). More interesting even if it also seems obj

the reverse of this: heat does not ever by itself flowj to hotter regions. To stay with our ex-, in the steam engine for one more timeij the air surrounding the boiler, though cooler than the boiler, is still hotter than a lot of other bodies. For instance, thej surrounding air is hotter than ocean water. Since the sur-;

vious

is

from cooler regions ample of the boiler

i

contains some heat, does it ever hapspontaneously transferred to a body thatj is already hotter than it? The answer is that this never hap-; pens. Consider for a moment the consequences if it did hap-; pen. If a certain portion of the air, by itself, gave heat to) something that was hotter, then the law of the conservation of energy requires that the heat loss be made up by a gain.) in other forms of energy, i.e., by a gain in mechanical en-i ergy. Thus, if heat could spontaneously flow from cooler to hotter bodies, there would be a spontaneous gain of me-!

rounding

pen that

air therefore

this

heat

is

chanical energy.

Such spontaneous gain of mechanical energy does not take This is stated as an axiom in the Second Law of\ Thermodynamics: Heat does not spontaneously, or by itself,; flow from cold to hot regions. Emphasis must be placed on? place.

The Conservation of Energy

291

lie words "spontaneously, or by itself." Heat does flow from older to hotter regions when we force it to do so. For example, a refrigerator makes heat flow from a cooler region (inside the box) to a hotter region (the surrounding air). The iow is not spontaneous, however. It can only be accomplished 3y means of the expenditure of mechanical energy. Furthermore, more mechanical energy is expended (in the pumping mechanism of the refrigerator) to produce the cooling than is gained as the result of the cooling. Thus we may also look jpon the Second Law of Thermodynamics as stating that there is no overall gain of mechanical energy in the transformations from mechanical energy to heat and back; at best there is even exchange, and in practical fact there is loss of mechanical energy accompanied by gain in heat. The steam sngine, unlike the refrigerator, results in a gain in mechanical nergy: for here we raise a body (the water in the boiler) to temperature higher than that of the surrounding region. Mechanical energy is produced when this water or steam is pooled in the process of doing work. (Whether or not there is a real gain in mechanical energy depends on how we look at the heating of the water: energy is of course expended in this process, but the energy used is usually stored up in a form such as coal or oil. The stored-up energy in turn has ultimately been produced by mechanical means, such as the pressure of land masses on fossils to produce carbon; if we take this into account, there is no overall gain of mechanical energy. If, however, we look at the energy we obtain from fossil fuels like coal or oil as "free," there is at least an ap,

parent gain of mechanical energy.) When heat was seen to be a form of energy, as stated in the First Law of Thermodynamics, it became apparent that older theories concerning the nature of heat could no longer be maintained. Just what heat is had puzzled natural philosophers for a long time. The ruling hypothesis during the eighteenth century was that heat is a substance, called caloric (see p. 270). This substance had, of course, to be invisible (since no one had ever seen it) and imponderable, or weightless (for no one had been able to detect that hot bodies weigh more than cold bodies). The transfer of heat from one body to another could be explained by the caloric substance flowing

from one body

to another.

It

was further assumed

each other. This accounted for the expansion of bodies as they become hotter; that the particles of caloric strongly repel

with the addition of caloric, there are more and more particles inside the hot body, repelling one another and thereby

expanding the body.

Breakthroughs in Physics

292

There are some immediate objections to the caloric theory. For one thing, there is something inherently implausible about a substance of this sort. (To be sure, it is no more implausible than the ether Huygens assumed for the propagation of light waves.) Furthermore, there are some phenomena that it fails to explain. Water contracts as it becomes colder; however, unlike other substances, water reaches

maximum density at 4°C. If water is cooled further, it begins to expand again. (Evidence for this is that ice water below 0°C often cracks during very cold winters.) Since the cooling below 4° must be interpreted as the removal of caloric, it is hard to see why water behaves so anomalously. The third and most serious objection to the caloric theory, however, comes from the generation of heat by means of friction (see p. 271). When we cause friction between two bodies, we are apparently able to generate heat unendingly. As long as we rub two pieces of brass together, for instance, we keep generating heat. The caloric theory cannot explain whence this heat comes; in fact, to an adherent of this theory the heat of friction appears to be created ex nihilo. On the other hand, if heat is understood as a form of energy, it is not hard to see where the heat of friction comes from: when the two pieces of brass are rubbed together, energy is obviously spent in making the two pieces move; the heat generated is merely the energy of motion transmuted into energy of heat (see p. 275). But if the First Law of Thermodynamics sounded the death knell for the caloric theory of heat, it nevertheless still leaves unanswered the question, What is heat? The First Law itself suggests an answer, arising from the fact that there is an equivalence between heat and kinetic energy. Why should this equivalence exist? The suggested answer is that heat energy and kinetic energy are really the same; heat energy is merely the gross appearance of the kinetic energy of the particles of a body. What we here put forth as merely an answer suggested by the First Law of Thermodynamics has in fact been fully developed into the kinetic theory of gases. It is there assumed; that all gases consist of molecules in random motion. The impingement of these molecules on the wall of the container causes the measurable pressure of the gas. The temperature of the gas is caused by the motion of its molecules. Heat is, in fact, molecular motion. Since the fact that the molecules are in motion means that they are possessed of kinetic en ergy, it is clear that the energy of heat and the kinetic its

—

—

The Conservation of Energy

293

jnergy of the molecules are the same thing. There is good jvidence for this molecular-kinetic theory of gases. Many of

he experimental facts concerning gases can be explained by instance, Boyle's Law and Charles' Law, which relate temperature, pressure, and volume of a gas, become consequences of this theory. The same theory also provides us for the first time with the idea of an absolute zero temperature; i.e., a temperature below which it is not possible to go. This would be that condition when all molecular motion ceases. In general, the more molecular motion there is, the more volatile a gas is. As the molecular motion is decreased, by suitable adjustments of temperature and pressure, a gas can be changed into a liquid or even into a solid (steam-water-ice is an obvious example). But when there is no longer any molecular motion, it is clear that we have reached the limit of coldness. Since molecular motion equals heat, when there is no more molecular motion, there is also t.

For

che

no more heat. We have neglected to say anything about chemical and electrical forms of energy. Both can also be transformed into mechanical and heat forms of energy. Helmholtz briefly mentions them, and it is easy to see that here, too, there is no gain or loss of energy, but only a transformation of energy to another. Thus we say that Helmholtz's thesis is that the total energy in the universe neither increases nor decreases but always remains constant.

from one form

Though it may seem hard to imagine how any law could have greater applicability than this, we must look ahead from the last half of the nineteenth century to the

first half of the of the conservation of energy was overthrown, at least in the narrow sense in which Helmholtz used it. Energy can be added to the store available in the universe, but at the cost of destroying matter. Einstein and his relativity theory introduced the notion of the equivalence of matter and energy. That in fact matter can be transformed into energy has been dramatically illustrated by the explosion of nuclear devices. Until Einstein's time, it had been believed that there were two quantities in the universe that were conserved: energy and matter. After Einstein, the two laws of the conservation of energy and of matter became one: the total quantity of matter and energy in the universe is constant. Or perhaps it would be more elegant to say simply that the total quantity of energy in the universe remains constant, but that matter is one of the forms of energy. And this brings

twentieth.

The law

us logically to the next chapter.

CHAPTER NINE Einstein:

—Relativity

PART

I

Important

scientific discoveries and the applications defrom them are usually discussed in scientific and professional journals, not in daily newspapers. But when Einstein's theory of relativity was verified and applied in the explosion of a nuclear device over the city of Hiroshima in 1945, relativity and its author became front-page news overnight. Not that the theory of relativity had lacked verifica-

rived

tion before; but other verifications, such as the "red shift"

of light and the aberrations in the orbit of the planet Mercury, had not been nearly so spectacular. All of these phenomena are actually explained by the general theory of relativity; but the special theory of relativity (and we have before us only that portion of Einstein's book dealing with the special theory) is a simpler and easier version that contains almost all of Einstein's basic thinking. An interesting and often forgotten sidelight on the theory of relativity is its date. Einstein was born in 1879. At the age of 26, in 1905, he published four papers that established his reputation. They dealt with the special theory of relativity, the equivalence of energy and mass, Brownian movement, and the photon theory of light. In 1922 Einstein received the Nobel Prize for his work. In 1905, Einstein was not even in an academic post; he was employed in the Swiss Patent Ofiice. After 1905, Einstein's academic career was, of course, secure. In 1916, while he was a professor at the University of Berlin, he published his formulation of the general theory of relativity.

From

the 1920's until his death, Einstein continued to

work

being especially concerned with the development of a "unified field theory," i.e., a theory that would be equally applicable to gravitational and to electromagnetic fields. Einstein left his native Germany in 1933 after the establishment of the Nazi government, and worked at the Institute for Advanced Study in Princeton. letter by Einstein to President Roosevelt during World War II was instrumental in in theoretical physics,

A

294

— 295

Relativity

up the "Manhattan Project," which ultimately led to the development of the atomic bomb. Einstein died in Princeton in 1955, at the age of 76. setting

Albert Einstein: Relativity* The

Special Theory of Relativity I

Physical

Meaning of Geometrical

Propositions

In your schooldays most of you who read this book made acquaintance with the noble building of Euclid's geometry, and you remember perhaps with more respect than love the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers. By reason of your past experience, you would certainly regard every one with disdain who should pronounce even the most out-of-the-way proposition of this science to be untrue. But perhaps this feeling of proud certainty would leave you immediately if some one were to ask you: "What,

—

do you mean by the assertion that these propositions are true?" Let us proceed to give this question a little con-

then,

sideration.

Geometry sets out from certain conceptions such as "plane," "point," and "straight line," with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as "true." Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to proposition follow from those axioms, i.e. they are proven. is then correct ("true") when it has been derived in the recognised manner from the axioms. The question of the "truth" of the individual geometrical propositions is thus reduced to one of the "truth" of the axioms. Now it has long been known that the last question is not only unanswerable by

A

the methods of geometry, but that

it

is

in itself entirely

We

cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called "straight

without meaning.

*

From Relativity: The Special and General Theory, W. Lawson. New York: Crown Publishers, 1931,

Robert

trans,

by

pp. 1-68.

Breakthroughs in Physics

296

each of which is ascribed the property of being uniquely determined by two points situated on it. The concept "true" does not tally with the assertions of pure geometry, because by the word "true" we are eventually in the habit of designating always the correspondence with a "real" object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves. It is not difficult to understand why, in spite of this, we feel constrained to call the propositions of geometry "true." Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of those ideas. Geometry ought to refrain from such a course, in order to give to its structure the largest possible logical unity. The practice, for example, of seeing in a "distance" two marked positions on a practically rigid body is something which is lodged deeply in our habit of thought. We are accustomed further to regard three points as being situated on a straight line, if their apparent positions can be made to coincide for observation with one eye, under suitable choice of our place of observation. If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies. 1 Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately ask as to the "truth" of geometrical propositions interpreted in this way, since we are justified in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the "truth" of a geometrical proposition in this sense we understand its validity for a construction with ruler and compasses. lines," to

1 It

follows that a natural object

is

associated

C

also with

a

straight line. Three points A, B and on a rigid body thus lie in being given, B is chosen a straight line when, the points and is as short as such that the sum of the distances and possible. This incomplete suggestion will suffice for our present

A

purpose.

C AB

BC

j

j

!

297

Relativity

Of course the conviction of the "truth" of geometrical propositions in this sense is founded exclusively on rather incomplete experience. For the present we shall assume the "truth" of the geometrical propositions, then at a later stage (in the general theory of relativity) we shall see that this "truth" is limited, and we shall consider the extent of its limitation.

II

The System

On

of Coordinates

the basis of the physical

interpretation

of

distance

which has been indicated, we are also in a position to establish the distance between two points on a rigid body by means of measurements. For this purpose we require a "distance" (rod S) which is to be used once and for all, and which we employ as a standard measure. If, now, A and B are two points on a rigid body, we can construct the line

them according to the rules of geometry; then, startfrom A, we can mark off the distance S time after time until we reach B. The number of these operations required is the numerical measure of the distance AB. This is the basis of all measurement of length. 2 joining

ing

Every description of the scene of an event or of the posiis based on the specification of the point on a rigid body (body of reference) with which that event or object coincides. This applies not only to scientific description, but also to everyday life. If I analyse the place specification "Trafalgar Square, London," 3 I arrive at the following result. The earth is the rigid body to which the specification of place refers; "Trafalgar Square, London" is a

tion of an object in space

well-defined point, to which a name has been assigned, and with which the event coincides in space. 4 2 Here we have assumed that there is nothing left over, i.e. that the measurement gives a whole number. This difficulty is got oyer by the use of divided measuring-rods, the introduction of which

does not

demand any fundamentally new method.

have chosen this as being more familiar to the English reader than the "Potsdamer Platz, Berlin," which is referred to in 3 1

the original. (R.

W.

L.)

4 It is not necessary here to investigate further the significance of the expression "coincidence in space." This conception is sufficiently obvious to ensure that differences of opinion are scarcely likely to arise as to its applicability in practice.

Breakthroughs in Physics

298

This primitive method of place specification deals only with places on the surface of rigid bodies, and is dependent

on the existence of points on this surface tinguishable from each other. But we can

which are

dis-

ourselves from both of these limitations without altering the nature of our specification of position. If, for instance, a cloud is hovering over Trafalgar Square, then we can determine its position relative to the surface of the earth by erecting a pole perpendicularly on the Square, so that it reaches the cloud. The length of the pole measured with the standard measuring-rod, combined with the specification of the position of the foot of the pole, supplies us with a complete place specification. On the basis of this illustration, we are able to see the manner in which a refinement of the conception of position has been developed. imagine the rigid body, to which the place speci(a) fication is referred, supplemented in such a manner that the object whose position we require is reached by the completed free

We

rigid body.

(b) In locating the position of the object, we make use of a number (here the length of the pole measured with the measuring-rod) instead of designated points of reference. (c) We speak of the height of the cloud even when the pole which reaches the cloud has not been erected. By means of optical observations of the cloud from different positions on the ground, and taking into account the properties of the propagation of light, we determine the length of the pole we should have required in order to reach the cloud. From this consideration we see that it will be advantageous if, in the description of position, it should be possible by means of numerical measures to make ourselves independent

of the existence of marked positions (possessing names) on the rigid body of reference. In the physics of measurement this is attained by the application of the Cartesian system of co-ordinates. This consists of three plane surfaces perpendicular to each other and rigidly attached to a rigid body. Referred to a system of co-ordinates, the scene of any event will be determined (for the main part) by the specification of the lengths of the three perpendiculars or co-ordinates (x, y, z) which can be dropped from the scene of the event to those three plane surfaces. The lengths of these three perpendiculars can be determined by a series of manipulations with rigid measuring-rods performed according to the rules and methods laid down by Euclidean geometry.

299

Relativity

In practice, the rigid surfaces which constitute the system of co-ordinates are generally not available; furthermore, the magnitudes of the co-ordinates are not actually determined by constructions with rigid rods, but by indirect means. If the results of physics and astronomy are to maintain their clearness, the physical meaning of specifications of position must always be sought in accordance with the above considerations. 5

We

thus obtain the following result: Every description of events in space involves the use of a rigid body to which such events have to be referred. The resulting relationship takes for granted that the laws of Euclidean geometry hold for "distances," the "distance" being represented physically

by means of the convention of two marks on a

rigid body.

Ill

Space and Time in Classical Mechanics

"The purpose of mechanics is to describe how bodies change their position in space with time." I should load my conscience with grave sins against the sacred spirit of lucidity were I to formulate the aims of mechanics in this way, without serious reflection and detailed explanations. Let us proceed to disclose these sins. It is not clear what is to be understood here by "position" and "space." I stand at the window of a railway carriage which is travelling uniformly, and drop a stone on the embankment, without throwing it. Then, disregarding the influence of the air resistance, I see the stone descend in a straight line. pedestrian who observes the misdeed from the footpath notices that the stone falls to earth in a parabolic curve. I now ask: Do the "positions" traversed by the stone lie "in reality" on a straight line or on a parabola? Moreover, what is meant here by motion "in space"? From the considerations of the previous section the answer is selfevident. In the first place, we entirely shun the vague word "space," of which, we must honestly acknowledge, we cannot form the slightest conception, and we replace it by "motion relative to a practically rigid body of reference." The positions relative to the body of reference (railway carriage or embankment) have already been defined in detail in the preceding section. If instead of "body of reference" we

A

5

A

refinement and modification of these views does not beuntil we come to deal with the general theory of relativity, treated in the second part of this book.

come necessary

Breakthroughs

300

in Physics

which is a useful idea for are in a position to say: The stone traverses a straight line relative to a system of coordinates rigidly attached to the carriage, but relative to a system of co-ordinates rigidly attached to the ground (eminsert "system of co-ordinates,"

mathematical description,

bankment) ample it is

it

we

describes a parabola.

clearly seen that there is

With the aid of this no such thing as an

exin-

dependently existing trajectory (lit. "path-curve" 6 ), but only a trajectory relative to a particular body of reference. In order to have a complete description of the motion, we must specify how the body alters its position with time; i.e. for every point on the trajectory it must be stated at what time the body is situated there. These data must be supplemented by such a definition of time that, in virtue of this definition, these time-values can be regarded essentially as magnitudes (results of measurements) capable of observation. If we take our stand on the ground of classical mechanics, we can satisfy this requirement for our illustration in the following manner. We imagine two clocks of identical construction; the man at the railway-carriage window is holding one of them, and the man on the footpath the other. Each of the observers determines the position on his own reference-body occupied by the stone at each tick of the clock he is holding in his hand. In this connection we have not taken account of the inaccuracy involved by the finiteness of the velocity of propagation of light. With this and with a second difficulty prevailing here we shall have to deal in detail later.

IV The Galileian System of Coordinates As is well known, the fundamental law of the mechanics of Galilei-Newton, which is known as the law of inertia,

A

body removed sufficiently far from other bodies continues in a state of rest or of uniform motion in a straight line. This law not only says something about the motion of the bodies, but it also indicates the reference-bodies or systems of co-ordinates, permissible in can be stated thus:

mechanics, which can be used in mechanical description.! visible fixed stars are bodies for which the law of in-] ertia certainly holds to a high degree of approximation. Now if we use a system of co-ordinates which is rigidly attached to the earth, then, relative to this system, every fixed

The

6

That

is,

a curve along which the body moves.

Relativity

301

a circle of immense radius in the course of an astronomical day, a result which is opposed to the statement of the law of inertia. So that if we adhere to this law we must refer these motions only to systems of co-ordinates relative to which the fixed stars do not move in a circle. system of co-ordinates of which the state of motion is such that the law of inertia holds relative to it is called a "Galileian system of co-ordinates." The laws of the mechanics of GalileiNewton can be regarded as valid only for a Galileian system of co-ordinates. star describes

A

V The

Principle of Relativity (in the Restricted Sense)

In order to attain the greatest possible clearness, let us return to our example of the railway carriage supposed to be travelling uniformly. call its motion a uniform translation ("uniform" because it is of constant velocity and direction, "translation" because although the carriage changes its position relative to the embankment yet it does not rotate in so doing). Let us imagine a raven flying through the air in such a manner that its motion, as observed from the

We

straight line. If we were from the moving railway carriage, we should find that the motion of the raven would be one of different velocity and direction, but that it would still be uniform and in a straight line. Expressed in an abstract manner we may say: If a mass m is moving uniformly in a straight

embankment,

is

uniform and in a

to observe the flying raven

with respect to a co-ordinate system K, then it will also be moving uniformly and in a straight line relative to a second co-ordinate system K', provided that the latter is executing a uniform translatory motion with respect to K. In accordance with the discussion contained in the preceding

line

section,

follows that: a Galileian co-ordinate system, then every other co-ordinate system K' is a Galileian one, when, in relation to K, it is in a condition of uniform motion of translaIf

K

it

is

tion. Relative to K! the mechanical laws of Galilei-Newton hold good exactly as they do with respect to K. We advance a step farther in our generalisation when we express the tenet thus: If, relative to K, is a uniformly moving co-ordinate system devoid of rotation, then natural phenomena run their course with respect to K' according to exactly the same general laws as with respect to K. This statement is called the principle of relativity (in the re-

K

stricted sense).

Breakthroughs in Physics

302

As long as one was convinced that all natural phenomena were capable of representation with the help of classical mechanics, there was no need to doubt the validity of this principle of relativity. But in view of the more recent development of electrodynamics and optics it became more and more evident that classical mechanics affords an insufficient foundation for the physical description of all natural phenomena. At this juncture the question of the validity of the principle of relativity

became

ripe for discussion,

and

it

did

not appear impossible that the answer to this question might be in the negative. Nevertheless, there are two general facts which at the outset speak very much in favour of the validity of the prin-

Even though classical mechanics does not supply us with a sufficiently broad basis for the theoretical ciple of relativity.

presentation of all physical phenomena, still we must grant it a considerable measure of "truth," since it supplies us with the actual motions of the heavenly bodies with a delicacy of detail little short of wonderful. The principle of relativity must therefore apply with great accuracy in the domain of mechanics. But that a principle of such broad generality should hold with such exactness in one domain of phenomena, and yet should be invalid for another, is a priori not very probable.

We now over,

we

proceed to the second argument, to which, more-

shall return later. If the principle of relativity

(in

the restricted sense) does not hold, then the Galileian coordinate systems K, K', K", etc., which are moving uniformly relative to each other, will not be equivalent for the description of natural phenomena. In this case we should

be constrained to believe that natural laws are capable of being formulated in a particularly simple manner, and of course only on condition that, from amongst all possible Galileian co-ordinate systems, we should have chosen one (K ) of a particular state of motion as our body of reference. We should then be justified (because of its merits for the description of natural phenomena) in calling this system

K

"in "absolutely at rest," and all other Galileian systems motion." If, for instance, our embankment were the system then our railway carriage would be a system K, relative to which less simple laws would hold than with respect to This diminished simplicity would be due to the fact that the carriage would be in motion {i.e. "really") with respect to In the general laws of nature which have been formulated with reference to K, the magnitude and di-

K

,

K

.

K

K

.

i

303

Relativity

rection of the velocity of the carriage would necessarily play should expect, for instance, that the note emitted a part. by an organ-pipe placed with its axis parallel to the direction of travel would be different from that emitted if the axis

We

of the pipe were placed perpendicular to this direction. Now in virtue of its motion in an orbit round the sun, our earth is comparable with a railway carriage travelling with a velocity of about 30 kilometres per second. If the principle of relativity were not valid we should therefore expect that the direction of motion of the earth at any moment would enter

and also that physical systems in behaviour would be dependent on the orientation in space with respect to the earth. For owing to the alteration

into the laws of nature, their

in direction of the velocity of revolution of the earth in the course of a year, the earth cannot be at rest relative to the

K

throughout the whole year. Howhypothetical system ever, the most careful observations have never revealed such anisotropic properties in terrestrial physical space, i.e. a physical non-equivalence of different directions. This is a very powerful argument in favour of the principle of relativity.

VI The Theorem of the Addition of

Velocities Employed in

Classical Mechanics Let us suppose our old friend the railway carriage to be travelling along the rails with a constant velocity v,

a

man

and that

traverses the length of the carriage in the direction

of travel with a velocity w.

How

quickly, or, in other words,

W

with what velocity does the man advance relative to the embankment during the process? The only possible answer seems to result from the following consideration: If the man were to stand still for a second, he would advance relative to the embankment through a distance v equal numerically to the velocity of the carriage. As a consequence of his walking, however, he traverses an additional distance w relative to the carriage,

and hence

also relative to the

embank-

ment, in this second, the distance w being numerically equal to the velocity with which he is walking. Thus in total he

W= +

covers the distance v w relative to the embankment shall see later that this result, in the second considered. which expresses the theorem of the addition of velocities employed in classical mechanics, cannot be maintained; in other

We

words, the law that we have just written down does not hold in reality. For the time being, however, we shall as-

sume

its

correctness.

Breakthroughs in Physics

304

VII Incompatibility of the Law of Propagation of Light with the Principle of Relativity There is hardly a simpler law in physics than that according to which light is propagated in empty space. Every child at school knows, or believes he knows, that this propagation takes place in straight lines with a velocity c = 300,000 km./ sec. At all events we know with great exactness that this velocity is the same for all colours, because if this were not the case, the minimum of emission would not be ob-

The Apparent

served simultaneously for different colours during the eclipse of a fixed star by its dark neighbour. By means of similar considerations based on observations of double stars, the Dutch astronomer De Sitter was also able to show that the velocity of propagation of light cannot depend on the velocity of motion of the body emitting the light. The assumption that this velocity of propagation is dependent on the direction "in space" is in itself improbable. In short, let us assume that the simple law of the constancy of the velocity of light c (in vacuum) is justifiably believed by the child at school. Who would imagine that this simple law has plunged the conscientiously thoughtful physicist into the greatest intellectual difficulties? Let us consider

how

these difficulties arise.

Of course we must

refer the process of the propagation of (and indeed every other process) to a rigid referencebody (co-ordinate system). As such a system let us again choose our embankment. We shall imagine the air above it to have been removed. If a ray of light be sent along the embankment, we see from the above that the tip of the ray will be transmitted with the velocity c relative to the embankment. Now let us suppose that our railway carriage is again travelling along the railway lines with the velocity v, and that its direction is the same as that of the ray of light, but its velocity of course much less. Let us inquire about the velocity of propagation of the ray of light relative to the carriage. It is obvious that we can here apply the conlight

sideration of the previous section, since the ray of light plays the part of the man walking along relatively to the carriage.

W

velocity of the man relative to the embankment is here replaced by the velocity of light relative to the embankment, w is the required velocity of light with respect to the

The

carriage,

and we have

w=

c

— v.

Relativity

305

The

velocity of propagation of a ray of light relative to the carriage thus comes out smaller than c.

But

this result

comes

into conflict with the principle of V. For, like every other general

relativity set forth in Section

law of nature, the law of the transmission of

light in

vacuo

must, according to the principle of relativity, be the same for the railway carriage as reference-body as when the rails are the body of reference. But, from our above consideration, this would appear to be impossible. If every ray of light is propagated relative to the embankment with the velocity c, then for this reason it would appear that another law of propagation of light must necessarily hold with respect to the carriage a result contradictory to the principle of relativity. In view of this dilemma there appears to be nothing else for it than to abandon either the principle of relativity or the simple law of the propagation of light in vacuo. Those of you who have carefully followed the preceding discussion are almost sure to expect that we should retain the principle of relativity, which appeals so convincingly to the intellect because it is so natural and simple. The law of the propagation of light in vacuo would then have to be replaced by a more complicated law conformable to the principle of relativity. The development of theoretical physics shows, however, that we cannot pursue this course. The epoch-making theoretical investigations of H. A. Lorentz on the electrodynamical and optical phenomena connected with moving bodies show that experience in this domain leads conclusively to a theory of electromagnetic phenomena, of which the law of the constancy of the velocity of light in vacuo is a necessary consequence. Prominent theoretical physicists were therefore more inclined to reject the principle of relativity, in spite of the fact that no empirical data had been found which were contradictory to this principle. At this juncture the theory of relativity entered the arena. As a result of an analysis of the physical conceptions of time and space, it became evident that in reality there is not

—

the least incompatibility between the principle of relativity

and the law of propagation of

light,

and that by systemati-

cally holding fast to both these laws a logically rigid theory

could be arrived at. This theory has been called the special theory of relativity to distinguish it from the extended theory. In the following pages we shall present the fundamental ideas of the special theory of relativity.

Breakthroughs in Physics

306

VIII

On

the Idea of Time in Physics

on our railway embankment from each other. I make the additional assertion that these two lightning flashes occurred simultaneously. If I ask you whether there is sense in this statement, you will answer my question with a decided "Yes." But if I now approach you with the request to explain to me the sense of the statement more precisely, you find after some consideration that the answer to this question is not so Lightning has struck the

at

two places

A

and

B

rails

far distant

easy as it appears at first sight. After some time perhaps the following answer would occur to you: "The significance of the statement is clear in itself and needs no further explanation; of course it would require some consideration if I were to be commissioned to determine by observations whether in the actual case the two events took place simultaneously or not." I cannot be satisfied with this answer for the following reason. Supposing that as a result of ingenious considerations an able meteorologist were to discover that the lightning must always strike the places A and B simultaneously, then we should be faced with the task of testing whether or not this theoretical result is in accordance with the reality. We encounter the same difficulty with all physical statements in which the conception "simultaneous" plays a part. The concept does not exist for the physicist until he has the possibility of discovering whether or not it is fulfilled in an actual case. We thus require a definition of simultaneity such that this definition supplies us with the method by means of which, in the present case, he can decide by experiment whether or not both the lightning strokes occurred simultaneously. As long as this requirement is not satisfied, I allow myself to be deceived as a physicist (and of course the same applies if I am not a physicist), when I imagine that I am able to attach a meaning to the statement of simultaneity. (I would ask the reader not to proceed farther until he is fully convinced on this point.) After thinking the matter over for some time you then offer the following suggestion with

which

to test simultaneity.

By measuring along the rails, the connecting line AB should be measured up and an observer placed at the mid-point M\ of the distance AB. This observer should be supplied with anj

i

Relativity

307

two mirrors inclined at 90°) which allows him visually to observe both places A and B at the same time. If the observer perceives the two flashes of lightning at the same time, then they are simultaneous. arrangement

(e.g.

I am very pleased with this suggestion, but for all that I cannot regard the matter as quite settled, because I feel

constrained to raise the following objection: "Your definition would certainly be right, if I only knew that the light by means of which the observer at perceives the lightning > flashes travels along the length A with the same > M. But an examination of velocity as along the length B this supposition would only be possible if we already had at our disposal the means of measuring time. It would thus appear as though we were moving here in a logical circle." After further consideration you cast a somewhat disdainful glance at me and rightly so and you declare: "I maintain

—

M—

M

—

—

my

previous definition nevertheless, because in reality it assumes absolutely nothing about light. There is only one demand to be made of the definition of simultaneity, namely, it must supply us with an empirical decision as to whether or not the conception that has to be defined is fulfilled. That my definition satisfies this demand

that in every real case

indisputable. That light requires the same time to traverse the path A is in reality as for the path B neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own

is

^

M

>

M

free will in order to arrive at a definition of simultaneity." It is clear that this definition can be used to give an exact meaning not only to two events, but to as many events as we care to choose, and independently of the positions of the scenes of the events with respect to the body of reference 7 (here the railway embankment). We are thus led also to a definition of "time" in physics. For this purpose we suppose

that clocks of identical construction are placed at the points

A,

B

and

C

of the railway line (co-ordinate system), and manner that the positions of their

that they are set in such a

We

7 suppose further that, when three events A, B and C take place in different places in such a manner that, if A is simultaneous with B, and B is simultaneous with C (simultaneous in the sense of the above definition), then the criterion for the simultaneity of the pair of events A, C is also satisfied. This assumption is a physical hypothesis about the law of propagation of light; it must certainly be fulfilled if we are to maintain the law of the constancy of the velocity of light in vacuo.

.

Breakthroughs in Physics

308

pointers are simultaneously (in the above sense) the same. Under these conditions we understand by the "time" of an event the reading (position of the hands) of that one of these

clocks which

is

event. In this

in the immediate vicinity (in space) of the is associated with every

manner a time-value

event which is essentially capable of observation. This stipulation contains a further physical hypothesis, the validity of which will hardly be doubted without empirical evidence to the contrary. It has been assumed that all these clocks go at the same rate if they are of identical construction. Stated more exactly: When two clocks arranged at rest in different places of a reference-body are set in such a manner that a particular position of the pointers of the one clock is simultaneous (in the above sense) with the same position of the pointers of the other clock, then identical "settings" are always simultaneous (in the sense of the above definition)

IX

The

Up

to

now

Relativity of Simultaneity

our considerations have been referred to a

body of reference, which we have styled a "railway embankment." We suppose a very long train travelling along the rails with the constant velocity v and in the direction inparticular

dicated in Fig. 9-1. People traveling in this train will with advantage use the train as a rigid reference-body (co-ordinate system) ; they regard all events in reference to the train. Then train 9

__j

*y

embankment Fig. 9-1

every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises: Are two events {e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. When we say that the lightning strokes A and B are simul-

Relativity

309

taneous with respect to the embankment, we mean: the rays of light emitted at the places A and B, where the lightning of the length occurs, meet each other at the mid-point

M

of the embankment. But the events A and B and B on the train. Let M' also correspond to positions B on the travelling be the mid-point of the distance A 8 of lightning occur, this point train. Just when the flashes M' naturally coincides with the point M, but it moves towards the right in the diagram with the velocity v of the train. If an observer sitting in the position M' in the train did not possess this velocity, then he would remain permanently at M, and the light rays emitted by the flashes of lightning and B would reach him simultaneously, i.e. they would meet just where he is situated. Now in reality (considered with reference to the railway embankment) he is hastening towards the beam of light coming from B, whilst he is riding on ahead of the beam of light coming from A. Hence the observer will see the beam of light emitted from B earlier than he will see that emitted from A. Observers who take the railway train as their reference-body must therefore come to the conclusion that the lightning flash B took place earlier than the lightning flash A. thus arrive at the important

A

B

A

>

A

We

result:

Events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa (relativity of simultaneity). Every reference-body (co-ordinate system) has its own particular time; unless we

are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event. before the advent of the theory of relativity it had always tacitly been assumed in physics that the statement of time had an absolute significance, i.e. that it is independent of the state of motion of the body of reference. But we have just seen that this assumption is incompatible with the most natural definition of simultaneity; if we discard this assumption, then the conflict between the law of the propagation of light in vacuo and the principle of relativity (developed in Section VII) disappears. were led to that conflict by the considerations of Section VI, which are now no longer tenable. In that section we concluded that the man in the carriage, who traverses the distance w per second relative to the carriage, traverses the same distance also with respect to the embankment in each

Now

We

8

As judged from

the

embankment

Breakthroughs in Physics

310

second of time. But, according to the foregoing considerations, the time required by a particular occurrence with respect to the carriage must not be considered equal to the duration of the same occurrence as judged from the embankment (as reference-body). Hence it cannot be contended that the man in walking travels the distance w relative to the railway line in a time which is equal to one second as judged from

embankment.

the

Moreover, the considerations of Section VI are based on yet a second assumption, which, in the light of a strict consideration, appears to be arbitrary, although it was always tacitly made even before the introduction of the theory of relativity.

X On

the Relativity of the Conception of Distance

Let us consider two particular points on the train 9 travelembankment with the velocity v, and inquire

ling along the

as to their distance apart.

We

already

know

that

it is

neces-

body of reference for the measurement of a with respect to which body the distance can be

sary to have a distance,

measured up.

It is the simplest plan to use the train itself as the reference-body (co-ordinate system). An observer in the train measures the interval by marking off his measuring-rod in a straight line (e.g. along the floor of the carriage) as many times as is necessary to take him from the one marked point to the other. Then the number which tells us how often

down is the required distance. a different matter when the distance has to be judged from the railway line. Here the following method suggests itself. If we call A' and B' the two points on the train whose distance apart is required, then both of these points are moving with the velocity v along the embankment. In the first place we require to determine the points A and B of the embankment which are just being passed by the two points A' and B' at a particular time t judged from the embankment. These points A and B of the embankment can be determined by applying the definition of time given in Section VIII. The distance between these points A and B is then measured by repeated application of the measuring-rod along the embankment. the rod has to be laid It is

—

A ment

priori

it

is

by no means

will supply

9 e.g.

certain that this last measure-

us with the same result as the

the middle of the

first

and of the hundredth

first.

carriage.

Thus

—

—

Relativity

311

the length of the train as measured from the embankment may be different from that obtained by measuring in the train itself. This circumstance leads us to a second objection which must be raised against the apparently obvious consideration of Section VI. Namely, if the man in the carriage measured from the covers the distance w in a unit of time then this distance as measured from the embank' train, ment is not necessarily also equal to w.

— —

XI

The Lorentz Transformation The results of the last three sections show that the apparent incompatibility of the law of propagation of light with the principle of relativity (Section VII) has been derived by means of a consideration which borrowed two unjustifiable hypotheses from classical mechanics; these are as follows: (1)

The

(2)

The

time-interval (time) between two events is independent of the condition of motion of the body of

reference. space-interval (distance) between

rigid

body

tion of the

is

two points of a independent of the condition of mo-

body of

reference.

If we drop these hypotheses, then the dilemma of Section VII disappears, because the theorem of the addition of velocities derived in Section VI becomes invalid. The possibility presents itself that the law of the propagation of light in vacuo may be compatible with the principle of relativity, and the question arises: How have we to modify the considerations of Section VI in order to remove the apparent disagreement between these two fundamental results of experience? This

question leads to a general one. In the discussion of Section to do with places and times relative both to the train and to the embankment. How are we to find the place and time of an event in relation to the train, when we know the place and time of the event with respect to the railway embankment? Is there a thinkable answer to this question of such a nature that the law of transmission of light in vacuo does not contradict the principle of relativity? In other words: Can we conceive of a relation between place and time

VI we have

of the individual events relative to both reference-bodies, such that every ray of light possesses the velocity of transmission c relative to the embankment and relative to the train? This question leads to a quite definite positive answer,

Breakthroughs in Physics

312

and to a perfectly definite transformation law for the spacetime magnitudes of an event when changing over from one body of reference to another. Before we deal with this, we shall introduce the following incidental consideration. Up to the present we have only considered events taking place along the embankment, which had mathematically to assume the function of a straight line. In the manner indicated in Section II we can imagine this reference-body supplemented laterally and in a vertical direction by means of a framework of rods, so that an event which takes place anywhere can be localised with reference to this framework. Similarly, we can imagine the train travelling with the velocity v to be continued across the whole of space, so that every event, no matter how far off it may be, could also be localised with respect to the second framework. Without committing any fundamental error, we can disregard the fact that in reality these frameworks would continually interfere with each other, owing to the impenetrability of solid bodies. In every such framework we imagine three surfaces perpendicular to each other marked out, and designated as "co-ordinate planes" ("co-ordinate system").

A

K

co-ordinate system then corresponds to the embankment, and a co-ordinate system K' to the train. An event, wherever it may have taken place, would be fixed in space with respect to by the three perpendiculars x, y, z on the coordinate planes, and with regard to time by a time-value U Relative to K', the same event would be fixed in respect of space and time by corresponding values x', y', z', ?,

K

which of course are not identical with x, y, z, t. It has already been set forth in detail how these magnitudes are to be regarded as results of physical measurements. Obviously our problem can be exactly formulated in the following manner. What are the values x', y', z', f of an event with respect to K', when the magnitudes x, y, z, t, of the same event with respect to are given? The relations must be so chosen that the law of the transmission of light in vacuo is satisfied for one and the same ray

K

Fig.

9-2

Relativity

313

K

of light (and of course for every ray) with respect to and the relative orientation in space of the co-ordinate systems indicated in the diagram (Fig. 9-2), this problem is solved by means of the equations:

K\ For

x

—

vt

Vy=

y

z'

z

=

V

X

t

x

c2

f

v. This system of equations is known as the "Lorentz transformation." If in place of the law of transmission of light we had taken as our basis the tacit assumptions of the older mechanics as to the absolute character of times and lengths, then instead of the above we should have obtained the following equations:

— = z' = f =

X*

X

y'

y

—

vt

z t.

This system of equations is often termed the "Galilei transformation." The Galilei transformation can be obtained from the Lorentz transformation by substituting an infinitely large value for the velocity of light c in the latter transformation. illustration, we can readily see accordance with the Lorentz transformation, the law of the transmission of light in vacuo is satisfied both for the reference-body and for the reference-body K\ light-signal is sent along the positive *-axis, and this lightstimulus advances in accordance with the equation

Aided by the following

that,

in

K

A

x

=

ct,

i.e. with the velocity c. According to the equations of the Lorentz transformation, this simple relation between x and t

Breakthroughs in Physics

314

involves a relation between x' and f. In point of fact, if we substitute for x the value ct in the first and fourth equations of the Lorentz transformation, we obtain:

x*

(c

=

— v)t

Vv

2

c2

f

=

Vv

2

c2

from which, by

division, the expression

xf

=

cf

immediately follows. If referred to the system K', the propagation of light takes place according to this equation. We thus see that the velocity of transmission relative to the refis also equal to c. The same result is obtained for rays of light advancing in any other direction whatsoever. Of course this is not surprising, since the equations of the Lorentz transformation were derived conformably to this point of view.

erence-body K'

XII

The Behaviour of MeasurinG'Rods and Clocks

in

Motion

place a metre-rod in the Jt'-axis of K' in such a manner that one end (the beginning) coincides with the point x' — 0, whilst the other end (the end of the rod) coincides with the point x' = i. What is the length of the metre-rod relatively to the system K? In order to learn this, we need only ask where the beginning of the rod and the end of the rod he with respect to at a particular time t of the system K. By means of the first equation of the Lorentz transformation the values of these two points at the time t o can be shown I

K

=

to be

^(beginning of rod)

Vv i

2

Relativity

:

(end of rod)

=

315 V2

I.

V

the distance between the points being

V--.

But the metre-rod is moving with the velocity v relative to K. It therefore follows that the length of a rigid metre-rod moving in the direction of its length with a velocity v is y/ in is

— v 2 /c 2 of a metre. The rigid rod is thus shorter when motion than when at rest, and the more quickly it moving, the shorter is the rod. For the velocity v = c we i

should have y/ i — v 2 /c 2 = o, and for still greater velocithe square-root becomes imaginary. From this we conclude that in the theory of relativity the velocity c plays the part of a limiting velocity, which can neither be reached nor exceeded by any real body. Of course this feature of the velocity c as a limiting velocity also clearly follows from the equations of the Lorentz transformation, for these become meaningless if we choose values of v greater than c. If, on the contrary, we had considered a metre-rod at rest in the *-axis with respect to K, then we should have found that the length of the rod as judged from K' would have ties

—

been y/ I v 2 /c 2 ; this is quite in accordance with the principle of relativity which forms the basis of our considerations.

A priori it is quite clear that we must be able to learn something about the physical behaviour of measuring-rods and clocks from the equations of transformation, for the magnitudes x, y, z, t, are nothing more nor less than the results of measurements obtainable by means of measuringrods and clocks. If we had based our considerations on the Galilei transformation we should not have obtained a contraction of the rod as a consequence of

Let us

now

its

motion.

consider a seconds-clock which

is

ticks:

at

the origin

(*'

=

o)

of K'.

f

=

permanently

o and f = i are two successive ticks of this clock. The first and fourth equations of the Lorentz transformation give for these two situated

Breakthroughs in Physics

316

t

=

o

and

t

=

V

c2

As judged from K,

the clock is moving with the velocity v; from this reference-body, the time which elapses between two strokes of the clock is not one second, but as judged

i

—

seconds,

i.e.

a somewhat larger time.

As a

conse-

Vx-c 2

quence of rest.

Here

its

motion the clock goes more slowly than when at an unattainable

also the velocity c plays the part of

limiting velocity.

XIII

Theorem of the Addition of Velocities. The Experiment of Fizeau

Now in practice we can move clocks and measuring-rods only with velocities that are small compared with the velocity of light; hence we shall hardly be able to compare the results of the previous section directly with the reality. But, on the other hand, these results must strike you as being very singular, and for that reason I shall now draw another conclusion from the theory, one which can easily be derived from the foregoing considerations, and which has been most elegantly confirmed by experiment. In Section VI we derived the theorem of the addition of velocities

in

one direction in the form which

from the hypotheses of

also

results

mechanics. This theorem can also be deduced readily from the Galilei transformation (Section XI). In place of the man walking inside the carriage, we introduce a point moving relatively to the co-ordinate system K' in accordance with the equation classical

317

Relativity

=

x'

wf.

of the first and fourth equations of the Galilei transformation we can express *' and t in terms of x, and t, and we then obtain

By means

x

=

(v

+ w)t.

This equation expresses nothing else than the law of motion (of the man with of the point with reference to the system reference to the embankment). We denote this velocity by the symbol W, and we then obtain, as in Section VI,

K

W= But we can carry out

v

+w

(A).

this consideration just as well

on the

basis of the theory of relativity. In the equation x>

= wf

we must

then express x' and f in terms of x and t, making use of the first and fourth equations of the Lorentz transformation. Instead of the equation (A) we then obtain the equation

v

W

w

4-

ooooeeeeeoooooa

^^

\-*^/ 9

vw

1+

—

which corresponds to the theorem of addition for velocities in one direction according to the theory of relativity. The question now arises as to which of these two theorems is the better in accord with experience. On this point we are enlightened by a most important experiment which the brilliant physicist Fizeau performed more than half a century ago, and which has been repeated since then by some of the best experimental physicists, so that there can be no doubt about its result. The experiment is concerned with the follow-

ing question. Light travels in a motionless liquid with a particular velocity w. quickly does it travel in the direction of the arrow in the tube T (see the accompanying dia-

How

gram, Fig. 9-3) when the liquid above mentioned is flowing through the tube with a velocity v? In accordance with the principle of relativity we shall certainly have to take for granted that the propagation of light always takes place with the same velocity w with respect to the liquid, whether the latter is in motion with reference to other bodies or not. The velocity of light relative

and the velocity of the latter relative to the tube are thus known, and we require the velocity of light relative to the tube* to the liquid

Breakthroughs in Physics

318 It is

clear that

before us.

we have

The tube plays

the problem of Section VI again the part of the railway embankment

± Fig.

9-3

or of the co-ordinate system K, the liquid plays the part of the carriage or of the co-ordinate system K', and finally, the light plays the part of the man walking along the carriage, or of the moving point in the present section. If we denote the velocity of the light relative to the tube by W, then this is given by the equation (A) or (B), according as the Galilei transformation or the Lorentz transformation corresponds to the facts. Experiment 10 decides in favour of equation (B) derived from the theory of relativity, and the agreement is, indeed, very exact. According to recent and most excellent measurements by Zeeman, the influence of the velocity of flow v on the propagation of light is represented by formula (B) to within one per cent. Nevertheless we must now draw attention to the fact that a theory of this phenomenon was given by H. A. Lorentz long before the statement of the theory of relativity. This theory was of a purely electrodynamical nature, and was obtained by the use of particular hypotheses as to the electromagnetic structure of matter. This circumstance, however, does not in the least diminish the conclusiveness of the experiment as a crucial test in favour of the theory of relativity, for the electrodynamics of Maxwell-Lorentz, on which the original theory was based, in no way opposes the theory of relativity. Rather has the latter been developed from electrodynamics as an astoundingly simple combination and generalisation of the hypotheses, formerly independent of

each other, on which electrodynamics was

built.

XIV The

Heuristic

Value of the Theory of Relativity

Our train of thought in the foregoing pages can be epitomised in the following manner. Experience has led to the conviction that, on the one hand, the principle of relativity holds true, and that on the other hand the velocity of transmission of light in vacuo has to be considered equal to a con-

319

Relativity

c. By uniting these two postulates we obtained the law of transformation for the rectangular co-ordinates x, y, z and the time t of the events which constitute the processes of

stant

(-i) J

the index of refraction of the liquid.

On

where n

,

=

—

is

w

the other hand, owing

vw to the smallness of

—

as

compared with

c2

I,

(vw\ —— i

order of approximation by

we can

w+

v

]

( i

,

,

replace (B)

or to the same

which agrees with

J

Fizeau's result.

nature. In this connection

we

did not obtain the Galilei trans-

formation, but, differing from classical mechanics, the Lorentz transformation.

The law of transmission of is

justified

light, the acceptance of which by our actual knowledge, played an important part

in this process of thought.

Once

in possession of the Lorentz

we can combine and sum up the theory

with the prin-

transformation, however,

this

ciple of relativity,

thus:

Every general law of nature must be so constituted that transformed into a law of exactly the same form when,

it is

instead of the space-time variables x, y, z, t of the original co-ordinate system K, we introduce new space-time variables f y', x', z , f of a co-ordinate system K'. In this connection the relation between the ordinary and the accented

magnitudes

is given by the Lorentz transformation. Or, in General laws of nature are co-variant with respect to Lorentz transformations. This is a definite mathematical condition that the theory

brief:

demands of a natural law, and becomes a valuable heuristic aid

of relativity

in virtue of this,

the theory

in the search for

general laws of nature. If a general law of nature were to be found which did not satisfy this condition, then at least one of the two fundamental assumptions of the theory would have been disproved. Let us now examine what general results the latter

theory has hitherto evinced.

—

,

Breakthroughs in Physics

320

XV General Results of the Theory It

is

clear

from our previous considerations that the grown out of electrodynam-

(special) theory of relativity has ics and optics. In these fields the predictions of theory, but

has not appreciably altered has considerably simplified the theoretical structure, i.e. the derivation of laws, and what is incomparably more important it has considerably reduced the number of independent hypotheses forming the basis of theory. The special theory of relativity has rendered the Maxwell-Lorentz theory so plausible, that the latter would have been generally accepted by physicists even if experiment had decided less unequivocally in its favour. Classical mechanics required to be modified before it could come into line with the demands of the special theory of relativity. For the main part, however, this modification affects only the laws for rapid motions, in which the velocities of matter v are not very small as compared with the velocity of light. We have experience of such rapid motions only in the case of electrons and ions; for other motions the variations from the laws of classical mechanics are too small to make themselves evident in practice. We shall not consider the motion of stars until we come to speak of the general theory of relativity. In accordance with the theory of relativis no ity the kinetic energy of a material point of mass longer given by the well-known expression it

it

—

m

v2

m— 2 but by the expression

V This expression approaches infinity as the velocity v approaches the velocity of light c. The velocity must therefore always remain less than c, however great may be the energies used to produce the acceleration. If we develop the expression for the kinetic energy in the form of a series, we obtain

mc2

4-

v2

3

v4

2

8

c2

m — + — m — + ....

Relativity

321

v2

When

—

is

small compared with unity, the third of these

c2

always small in comparison with the second, which alone considered in classical mechanics. The first term mc2 does not contain the velocity, and requires no consideration if we are only dealing with the question as to how the energy of a point-mass depends on the velocity. We shall speak of its essential significance later. The most important result of a general character to which the special theory of relativity has led is concerned with the conception of mass. Before the advent of relativity, physics recognised two conservation laws of fundamental importance, namely, the law of the conservation of energy and the law of the conservation of mass; these two fundamental laws appeared to be quite independent of each other. By means of the theory of relativity they have been united into one law. We shall now briefly consider how this unification came about, and what meaning is to be attached to it. The principle of relativity requires that the law of the conservation of energy should hold not only with reference to a co-ordinate system K, but also with respect to every coordinate system K' which is in a state of uniform motion of translation relative to K, or^ briefly, relative to every "Galileian" system of co-ordinates. In contrast to classical mechanics, the Lorentz transformation is the deciding factor in the transition from one such system to another. By means of comparatively simple considerations we are led to draw the following conclusion from these premises, in conjunction with the fundamental equations of the electrobody moving with the velocity v, dynamics of Maxwell: which absorbs u an amount of energy E in the form of radiation without suffering an alteration in velocity in the process, has, as a consequence, its energy increased by an

terms

is

last is

A

amount

Vv — 2

In consideration of the expression given above for the kinetic energy of the body, the required energy of the

comes out 11

En

is

to

body

be

the energy taken up, as judged

tem moving with the body.

from a co-ordinate

sys-

Breakthroughs in Physics

322

c2

I

c2

Thus the body has the same energy

m)

If

moving with the

body of mass

as a

velocity v.

a body takes up an amount of energy

Hence we can

E

,

then

say:

its inertia!

£ mass increases by an amount

—

;

the inertial mass of a body

c2

not a constant, but varies according to the change in the energy of the body. The inertial mass of a system of bodies can even be regarded as a measure of its energy. The law of the conservation of the mass of a system becomes identical with the law of the conservation of energy, and is only valid provided that the system neither takes up nor sends out energy. Writing the expression for the energy in the form

is

mc2 + E

1-c2

term mc which has hitherto attracted nothing else than the energy possessed by the body 12 before it absorbed the energy EQ direct comparison of this relation with experiment is not possible at the present time, owing to the fact that the changes in energy E to which we can subject a system are not large enough to make themselves perceptible as a

we

see

that

our attention,

2

the

,

is

.

A

E

change in the

mass of the system.

—

is too small c2 in comparison with the mass m, which was present before the alteration of the energy. It is owing to this circum-

inertial

was able to establish sucmass as a law of independent

stance that classical mechanics cessfully the conservation of validity. 12

As judged from a

co-ordinate system

moving with the body.

Relativity

323

me add

a final remark of a fundamental nature. The success of the Faraday-Maxwell interpretation of electromagnetic action at a distance resulted in physicists becoming convinced that there are no such things as instantaneous actions at a distance (not involving an intermediary medium) of the type of Newton's law of gravitation. According to

Let

the theory of relativity, action at a distance with the velocity of light always takes the place of instantaneous action at a distance or of action at a distance with an infinite velocity of transmission. This is connected with the fact that the velocity c plays a fundamental role in this theory.

XVI Experience

and the

Special

Theory of Relativity

To what extent is the special theory of relativity supported by experience? This question is not easily answered for the reason already mentioned in connection with the fundamenexperiment of Fizeau. The special theory of relativity has crystallised out from the Maxwell-Lorentz theory of electromagnetic phenomena. Thus all facts of experience which support the electromagnetic theory also support the theory of relativity. As being of particular importance, I mention here the fact that the theory of relativity enables us to predict the effects produced on the light reaching us from the fixed stars. These results are obtained in an exceedingly simple manner, and the effects indicated, which are due to the relative motion of the earth with reference to those fixed, stars, are found to be in accord with experience. We refer to the yearly movement of the apparent posital

from the motion of the earth round the sun (aberration), and to the influence of the radial components of the relative motions of the fixed stars with respect to the earth on the colour of the light reaching us from them. The latter effect manifests itself in a slight distion of the fixed stars resulting

placement of the spectral

lines of the light transmitted to us

compared with the

position of the they are produced by a terrestrial source of light (Doppler principle). The experimental arguments in favour of the Maxwell-Lorentz theory, which are at the same time arguments in favour of the theory of relativity, are too numerous to be set forth here. In reality they limit the theoretical possibilities to such an extent, that no

from a fixed same spectral

star,

lines

as

when

— Breakthroughs in Physics

324

other theory than that of Maxwell and Lorentz has been able to hold its own when tested by experience. But there are two classes of experimental facts hitherto obtained which can be represented in the Maxwell-Lorentz theory only by the introduction of an auxiliary hypothesis, which in itself i.e. without making use of the theory of appears extraneous. relativity It is known that cathode rays and the so-called /3-rays emitted by radioactive substances consist of negatively electrified particles (electrons) of very small inertia and large velocity. By examining the deflection of these rays under the influence of electric and magnetic fields, we can study the law of motion of these particles very exactly. In the theoretical treatment of these electrons, we are faced with the difficulty that electrodynamic theory of itself is unable to give an account of their nature. For since electrical masses of one sign repel each other, the negative electrical masses constituting the electron would necessarily be scattered under the influence of their mutual repulsions, unless there are forces of another kind operating between them, the nature of which has hitherto remained obscure to us. 13

—

we now assume

that the relative distances between the masses constituting the electron remain unchanged during the motion of the electron (rigid connection in the sense of classical mechanics), we arrive at a law of motion of the electron which does not agree with experience. Guided by purely formal points of view, H. A. Lorentz was the first

If

electrical

to introduce the hypothesis that the particles constituting the electron experience a contraction in the direction of mo-

tion in consequence of that motion, the

amount of I

this

con-

^

traction being proportional to the expression "Wj

m '

This

c2 hypothesis, which

is

not justifiable by any electrodynamical

motion which has been confirmed with great precision in recent years. The theory of relativity leads to the same law of motion, without requiring any special hypothesis whatsoever as to the structure and the behaviour of the electron. We arrived facts,

supplies us then with that particular law of

at a similar conclusion in Section XIII in connection with the experiment of Fizeau, the result of which is foretold 13

The general theory of

trical

forces.

relativity renders

it

likely that the elec-

masses of an electron are held together by gravitational

Relativity

by the theory of on hypotheses as

relativity

325

without the necessity of drawing

to the physical nature of the liquid.

The second class of facts to which we have alluded has reference to the question whether or not the motion of the earth in space can be made perceptible in terrestrial experiments. We have already remarked in Section that all attempts of this nature led to a negative result. Before the theory of relativity was put forward, it was difficult to become reconciled to this negative result, for reasons now to be discussed. The inherited prejudices about time and space did not allow any doubt to arise as to the prime importance of the Galilei transformation for changing over from one body of reference to another. Now assuming that the Maxwell-Lorentz equations hold for a reference-body K, we then find that they do not hold for a reference-body K' moving uniformly with respect to K, if we assume that the relations of the Galileian transformation exist between the co-ordinates of and K'. It thus appears that of, all Galileian co-ordinate systems one (K) corresponding to a particular state of motion is physically unique. This result was interpreted physically by regarding as at rest with respect to a hypothetical aether of space. On the other hand, all coordinate systems K' moving relatively to were to be regarded as in motion with respect to the aether. To this motion of K' against the aether ("aether-drift" relative f to were assigned the more complicated laws which ) were supposed to hold relative to K'. Strictly speaking, such an aether-drift ought also to be assumed relative to the earth, and for a long time the efforts of physicists were devoted to attempts to detect the existence of an aetherdrift at the earth's surface. In one of the most notable of these attempts Michelson devised a method which appears as though it must be decisive. Imagine two mirrors so arranged on a rigid body that the reflecting surfaces face each other. ray of light requires a perfectly definite time T to pass from one mirror to the other and back again, if the whole system be at rest with respect to the aether. It is found by calculation, however, that a slightly different time is required for this process, if the body, together with the mirrors, be moving relatively to the aether. And yet another point: it is shown by calculation that for a given velocity v with reference to the aether, this time is different when the body is moving perpendicularly to the planes of the mirrors from that resulting when the motion is parallel to these planes. Although

V

K

K

K

K

A

T

T

Breakthroughs in Physics

326

the estimated difference between these two times is exceedingly small, Michelson and Morley performed an experiment involving interference in which this difference should have

been clearly detectable. But the experiment gave a negative result a fact very perplexing to physicists. Lorentz and FitzGerald rescued the theory from this difficulty by assuming that the motion of the body relative to the aether produces a contraction of the body in the direction of motion, the amount of contraction being just sufficient to compensate for the difference in time mentioned above. Comparison with the discussion in Section XII shows that also from the stand-

—

point of the theory of relativity this solution of the difficulty was the right one. But on the basis of the theory of relativity the method of interpretation is incomparably more satisfactory. According to this theory there is no such thing as a "specially favoured" (unique) co-ordinate system to occasion the introduction of the aether-idea, and hence there can be no aether-drift, nor any experiment with which to demonstrate it. Here the contraction of moving bodies follows from the two fundamental principles of the theory without the introduction of particular hypotheses; and as the prime factor involved in this contraction we find, not the motion in itself, to which we cannot attach any meaning, but the motion with respect to the body of reference chosen in the particular case in point. Thus for a co-ordinate system moving with the earth the mirror system of Michelson and Morley is not shortened, but it is shortened for a coordinate system which is at rest relatively to the sun.

XVII Minkowski's Four-dimensional Space

The non-mathematician is seized by a mysterious shudderwhen he hears of "four-dimensional" things, by a feeling not unlike that awakened by thoughts of the occult. And yet there is no more common-place statement than that the world ing

in

which we live is a four-dimensional space-time continuum. Space is a three-dimensional continuum. By this we mean

that

it

is

possible to

describe the position of a point

(at

by means of three numbers (co-ordinates) x, v, z, and that there is an indefinite number of points in the neighbourhood of this one, the position of which can be described by co-ordinates such as x x y lf z lf which may

rest)

,

be as near as we choose to the respective values of the coordinates x, y, z of the first point. In virtue of the latter property we speak of a "continuum," and owing to the fact

Relativity

we speak

that there are three co-ordinates

"three-dimensional." Similarly, the world of physical briefly

called

327 of

it

as being

phenomena which was

"world" by Minkowski

is

naturally

four-di-

mensional in the space-time sense. For it is composed of individual events, each of which is described by four numbers, namely, three space co-ordinates x, y, z and a time co-ordinate, the time-value t. The "world" is in this sense also a continuum; for to every event there are as many "neighbouring" events (realised or at least thinkable) as we care to choose, the co-ordinates x lf y 19 zlf *i> °f which differ by an indefinitely small amount from those of the event x, y, z, t originally considered. That we have not been accustomed to regard the world in this sense as a fourdimensional continuum is due to the fact that in physics, before the advent of the theory of relativity, time played a different and more independent role, as compared with the space co-ordinates. It is for this reason that we have been in the habit of treating time as an independent continuum. As a matter of fact, according to classical mechanics, time is absolute, i.e. it is independent of the position and the condition of motion of the system of co-ordinates. We see this expressed in the last equation of the Galileian transformation (f = t). four-dimensional mode of consideration of the is natural on the theory of relativity, since according to this theory time is robbed of its independence. This is shown by the fourth equation of the Lorentz transforma-

The

"world"

tion:

v t

x

v2

V

c2

Moreover, according to this equation the time difference Af' of two events with respect to K' does not in general vanish, even when the time difference A* of the same events vanishes. Pure "space-distance" of two with reference to

K

K

results in "time-distance" of the events with respect to same events with respect to K\ But the discovery of Minkowski, which was of importance for the formal development of the theory of relativity, does not lie here. It is to be found rather in the fact of his recognition that the four-

Breakthroughs in Physics

328

continuum of the theory of relaformal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude y/ — I. ct prodimensional

tivity,

in

its

space-time

most

essential

portional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity as-

sume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space co-ordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry. It must be clear even to the

non-mathematician that, as a consequence of this purely formal addition to our knowledge, the theory perforce gained clearness in no mean measure.

PART

II

"The non-mathematician is seized by a mysterious shuddering when he hears of 'four-dimensional' things" these are Einstein's words at the beginning of the last chapter of the present selection. He might just as well have begun the first chapter with similar words: "The non-physicist is seized by a mysterious shuddering when he hears of 'the theory of relativity.' " Einstein goes on to assure his readers that there is nothing very mysterious or occult about the four-dimensionality of events; we may similarly say that there is nothing very mysterious or occult about the theory of relativity

—

—

especially about the special theory of relativity, with

alone

which

we

are here concerned. It is true that things become a little more difficult when Einstein turns to the general theory of relativity which is one reason why we are omitting consideration of it here. However, we do want to point out that the sense in which Einstein uses the word "special" is not that of "extraordinary" or "out-of-the-common"; rather, he means to indicate that this first consideration of relativity has to do with only a limited situation; the general theory deals with the

—

more generalized is

case.

Thus the

easier to understand because

situation tions

—

a situation that

have been omitted.

is

special theory of relativity applies only to a special

it

special because

some complica-

Relativity

329

To give the reader still more assurance that what Einstein has written can be understood by the layman, we should point out that this is not the work in which Einstein's original discoveries and theories were announced.

What we have

before us is Einstein's own popularization of his technical work, written (in German) in the nineteen-twenties. If this is not reassurance enough, let the reader realize that we here are engaged in popularizing a popular treatment. Further than this it is hardly possible to go; all that is required is that the reader meet us halfway with an open mind and discard any prejudgments about the difficulty of what is to follow.

Einstein is doing battle in this book; he is attacking an opposing theory. But whom is he attacking? Whose theory is he finding fault with? It is Newton, and Newton's theories of space and time, that are Einstein's targets, although Einstein for the most part ascribes the views he is opposing to Galileo. Indeed, Galileo and Newton held similar, or perhaps even the same, views on these matters; but Newton makes his positions explicit while Galileo does not. Newton's views are stated in the Scholium to Definition VIII. The reader may remember that in Chapter Six we deferred consideration of this Scholium until we could compare it with Einstein's point of view. In the Scholium, Newton speaks of absolute space and absolute time. Here is what he says concerning them:

and mathematical time, of itself, and nature, flows equably without relation to anything external, and by another name is called duration . . .

Absolute, true,

from

its

own

Absolute space, in its own nature, without relation to anything external, remains always similar and immovable .. . (p. 159)

Absolute time, according to this statement, is something; it "flows," it has "its own nature," and there are things "external" to it. There is, so to speak, an entity called absolute time, by reference to which other things are measured. Whether or not there are motions to be measured, absolute time flows on; it evidently has gone on from the beginning of the universe and will continue to flow forever and ever. There clearly is only one absolute time; there may be a time of one motion and another time of another motion, but for

there

is

only one absolute time for both motions. For in-

Breakthroughs in Physics

330

may

motion that is measured in solar is measured in celestial time; yet the absolute one of these motions. If there

be a time, while another one stance,

there

neither time is are several kinds of time (such as solar, celestial, standard), absolute time is nevertheless a special and specially favored time. It is the time to which, above all others, events and motions in the universe are to be referred. For it alone flows on without being affected in any way whatsoever by

anything else. This Newtonian view, that time is an entity, that it moves, and that it is independent of things around it, seems quite ingrained in most people today. Such phrases as "time marches on" reflect this concept of time; so does the notion, often expressed in physics, that time is an "independent" variable in motion. The science-fiction notion of a "timemachine," i.e., of a device that can reverse the direction of time or that can speed up the flow of time, is also an indication of how deeply committed our civilization is to this concept of time. This is not the place to enter into a complete philosophical discussion of the concept of time; we are merely concerned with Newton's and with Einstein's concepts of it. But it may be well at least to remind the reader that there are other concepts of time. Theologians, for instance, are much concerned with time. According to Christian doctrine, time was created by God, just as everything else was created. And time, like everything else, will be destroyed at the end of the world. In this doctrine, God Himself is eternal (which does not mean that He endures through unending time, but rather that He is outside of time). God, unlike time, does not come to an end. The Christian view of time is reflected, for example, by Shakespeare in many places; one of the clearest occasions occurs in the First Part of King Henry IV. When Prince Hal has fatally wounded his rival Harry Percy, the dying Harry speaks:

But thought's the slave of

And

life,

time, that takes survey of

Must have a

and

all

life

time's fool;

the world,

stop.

Newtonian absolute time never has a stop. It flows equably on and on, because of what it is. Newton's view of absolute space is perfectly commensurate with his view of time. Space thing external as time

is.

is

as

Because of

independent of everyits own nature, space

Relativity

331

always remains the same, similar everywhere, infinitely extended, and immovable. It extends in all directions, and any portion of it is exactly the same as any other portion of it. Why did Einstein attack these concepts of space and time? The answer is simple: because he had to, because flaws had been discovered in the Newtonian system. Since the Newtonian mechanics had worked beautifully for two hundred years, we can easily imagine that Einstein's rebellion was no light undertaking. It was forced on him, he felt, by the facts. The basic fact that made the theory of relativity necessary was the constancy of the speed of light. Einstein reminds us that the speed of light is 3 X 10 10 cm/ sec (186,000 miles per second) and that this speed is the same for light of all colors. Furthermore, he tells us "the Dutch astronomer De Sitter was also able to show that the velocity of propagation of light cannot depend on the velocity of little later he adds the body emitting the light" (p. 304). that H. A. Lorentz found that there are reasons based on electrodynamics to indicate that "the law of the constancy of the velocity of light in vacuo is a necessary consequence" (p. 305). This means that electrodynamic theory provides grounds for maintaining that the speed of light is always the same, whether or not the light-emitting body is moving. The constancy of the speed of light, together with the Galilean theorem concerning the addition of velocities, seemed to overthrow, Einstein tells us, the principle of relativity.* The theorem about the addition of velocities simply such as a railroad train moves in tells us that if a body a straight line with a uniform velocity v, while another body on the train such as a man walking moves with a uniform velocity w in the same direction as the train, then

A

—

—

—

— W

the velocity

with which the

man moves

relative to

given by

some-

W=v+

w. again the train moves in a straight line with uniform velocity v, while a car moves parallel to the railroad track with the velocity w, then the speed of the train thing stationary outside the train Similarly,

if

relative to the

to the

is

moving

moving car

is

car, the train

W = v — w.

That

is,

relative

moves more slowly than

rela-

such as a house. In a general way, familiar with this fact from personal experience

tive to stationary objects

we *

are

all

The reader should note

that Einstein speaks both of the

pnn-

ciple of relativity and of the theory of relativity. The principle, in this usage, is something old, handed down from Galileo and Newton; the theory is something new, propounded by Einstein.

Breakthroughs

332

in Physics

when one car passes another, the passing car quite slowly relative to the one being passed. Con-

with cars:

moves

when two cars approach one another, they move very rapidly relative to one another (their relative speed being the sum of their individual speeds). The latter fact is also dramatically illustrated when two jet planes pass one another going in opposite directions. Though each plane is flying at approximately 600 miles per hour, there is little sensation of speed. When, suddenly, another jet plane going the opposite direction comes into view, the tremendous speeds involved can be recognized: relative to the other plane, each plane travels at 1200 miles per hour, or 20 miles per minute. Thus, the two planes first approach and then separate at the rate of 1 mile every 3 seconds; a fact made apparent because the passing plane is visible to passengers in the other plane for only a few seconds. versely,

So much for the addition of uniform what is the principle (not the theory) of

velocities.

Now,

relativity? Stated

it maintains that in the vast expanse of Newtonian space and in the everlasting flow of Newtonian time, there is no favored place or time. If there is some law of nature, then that law is the same, relative to whatever co-ordinate system (reference frame) it is expressed in, as long as the reference frames move only with uniform, straight-line motions relative to one another. Under those restrictions, if Q is a law of nature in one reference frame, then Q is also a law of nature in any other reference frame that moves with a uniform straight-line velocity relative to the first one. Already the problem facing Einstein should be obvious: if the constancy of the speed of light is to be accepted, then the principle of relativity seems to be false. For if the speed of light, traveling in the same direction as a railroad car, is c relative to stationary objects while the speed of the train itself relative to stationary objects is v, then the speed of light relative to the train would seem to be c v (according to the law of the addition of speeds). If, however, we were to insist that the principle of relativity is true, then the law of the speed of light would apparently have to be different, so as to conform to the principle of

broadly,

—

—

—

—

—

relativity.

theory of relativity avoids both of these unHe finds a way for adhering consistently to the constancy of the speed of light and to the principle of relativity. Obviously, something must be changed, if this Einstein's

desirable results.

Relativity

333

be achieved. What must be changed, it turns out, is the concept of time, the concept of space, and with these, the theorem of the addition of velocities. No longer can we say, as Newton did, that time flows equably without relation to anything external. No, we must now say that time flows inequably and that its flow is precisely so adjusted to motion (or, so affected by motion) that its change suffices to make the principle of relativity true result

is

to

again.

Let us say the same thing again in a different way. What Einstein proposes is to keep both the principle of relativity and the constancy of the speed of light. If, therefore, the velocity of light is to be equal to c, relative to stationary objects and relative to the moving train, then we must adjust the rate at which we say that time flows. Relative to stationary objects, we said, the speed of light is c. Relative to the train moving with the velocity v, it seemed to us that

—

v (i.e., as the velocity of light should have come out as c less than before). If, now, we want the speed of light to come out as equal to c nevertheless, then this can be accomplished if we say that on the train (where the speed is measured) the time by which we measure the velocity of light is different from what it is on stationary ground. Although the speed of light relative to the train seems to be only c v, according to the older ways of thinking, nevertheless, according to Einstein, it is c relative to the moving

—

train,

because the time on the train has slowed

down

just

compensate for the lessening of speed. Thus suppose that the speed of light is 186,000 miles per second. Let the train move with the speed of 1 mile per second (still a fantastically high speed: 3600 miles per hour). Based only on the Galilean theorem of the addition of

sufficiently

to

we would say that if light moves with the speed of 186,000 miles per second relative to stationary objects, then relative to the train it moves with the speed of 185,999 miles per second. Let us say, then, there is a clock on the ground (i.e., stationary). Let one tick of this clock (1 second) elapse. During this time, the light ray under consideration will have traveled 186,000 miles. During this same time (as

velocities,

measured from the stationary ground), the train will have moved 1 mile. Let us also assume that train and light ray began their motions from the same point. Now during the time that the clock on the ground has made one tick, the light ray has gone 186,000 miles and the train 1 mile in the same direction. Relative to the train, therefore, the light ray

Breakthroughs in Physics

334

has moved 185,999 miles. But, on the train, 1 second has not yet elapsed, as measured by clocks on the train. Rather, when the ray of light has, relative to the train, traveled 185,999 miles, a fraction of time less than 1 second has the fraction being just enough so that 185,999 -H elapsed time (as measured on the train) = 186,000 miles per second. While 1 second has elapsed as measured on the stationary ground, less than 1 second has elapsed as measured on the

—

Time as measured on the moving train, therefore, moves more slowly than it does on the stationary ground. train.

What we said above provides an intuitive way of understanding what Einstein is proposing: In order to maintain the consistency of two principles that physics cherishes but that seem to be in conflict, he proposes to change our concept of something that hitherto had not been thought of as capable of change: time. "Before the advent of the theory of relativity," Einstein writes, had always tacitly been assumed in physics that the statement of time had an absolute significance, i.e. that it is independent of the state of motion of the body of it

reference, (p. 309)

With the coming of the theory of

relativity,

every reference-body (co-ordinate system) has its own particular time; unless we are told the reference-body to which the statement of time refers, there is no meaning in a statement of the time of an event, (p. 309)

Let us go over the same ground a third time. Where bewe relied mainly on intuition, we shall this time try to be more precise. Let us assume that there is some one event, A. Let us characterize this event by indicating both where it takes place and when it takes place. Let the where be indicated by the three space coordinates x, v, z of an ordinary Cartesian system. Let the when of A be given by t, where t is measured by a clock that is at rest relative to the coordinate system K. fore

Now imagine that there is a second coordinate system K\ Let K' be in uniform motion relative to K, along the jcaxis, with speed v. Thus, for any event the values of the ^-coordinate in are different from those in K'; but the values of the y-coordinate and z-coordinate are the same. The same event A may now also be referred to the coordi-

K

—

.

Relativity

K

335

r

In this system, where is given by three coy', when is given by f, where f is z'\ measured on a clock that is at rest in K! (and therefore

nate system ordinates x\ in

motion

.

relative to

K)

we know

the values of x, y, z, and t, can we then deTo help us find x\ termine the values of x' t y', z', and y\ z\ and f, we have two bits of knowledge: (1) the velocities v of the two coordinate systems are relative to one another, (2) the velocity of light is the same value (usually designated by "c") when measured in either system. Because of the simplifying assumption we have made i.e., that the system K! moves in the direction of the xaxis we can say immediately that for any event A, If

H

—

y=

y

t= But

z.

not of course be equal to x. If we did not must be the same, whether referred to or then we should simply be able to , x*

will

postulate that the speed of light

we

K

K

f

say that tf

=

x

f

=

— vt

while t.

These four equations, which tell us how to go from a systo a system K' on the Galilean-Newtonian assumptem

K

tions of absolute time, Einstein calls the "Galilean transfor-

mation." If, however, the speed of light is to be c, no matter in which reference system it is measured, then the following equations must hold. They constitute the "Lorentz transformation":

y= z' =

y z

X

—

vt

V*

V

1

c2

V

X

t

c2

f

=

V

1

)

Breakthroughs in Physics

336

(The derivation of these equations is not difficult, only timeconsuming, and we shall here omit it. Both the space and the time coordinate, therefore, involve the value of c, the speed of light. Einstein points out that these equations do in fact satisfy the condition that if x = ct (i.e., if a light ray goes a distance x in the time t as measured in K), then also x' = cf (i.e., the light ray goes the distance x' in the time f as measured in K'). In both systems, then, the speed of light is c (see pp. 313-314). What is the meaning of the fact that not only is f different from t, but jc' is also different from x? We have been prepared to find that time would be different if referred to the moving system K! from what it would be if referred to the stationary system K. In fact, in his Chapter XII, Einstein shows that just as we said earlier clocks in the moving system K' move more slowly than clocks in the stationary system K. But now he also adds in this same chapter the observation that the size of bodies is different according to whether they are measured in a stationary or in a moving

—

—

reference system. That is the meaning of the fact that x' vt, but rather the more complicated exdoes not equal x pression of the Lorentz transformation. If an observer in K' and an observer in were to measure the same length (which is at rest relative to K'), the result obtained by the second observer would be less than the result obtained by the first observer. It follows from all this that if there are two coordinate systems that are in uniform motion relative to one another, then the space coordinate and the time coordinate of an event will be different in the two systems. (This much is also true in the Galileo-Newton systems.) The two sets of coordinates will be related to one another by the Lorentz transformation rather than by the Galilean transformation.

—

K

The Lorentz transformation special event,

such that

if

we

light,

the same, judged

from

that the speed of light

ordinate

is

namely the propagation of

system.

is

The Lorentz transformation

apply

it to a turns out whatever coit

also

shows

judged from one system, the clocks in the other (relatively moving) system go more slowly and that, judged from that,

one system, the measuring sticks in the other (relatively moving) system are shorter. Because both the space and the time coordinate of an event are affected in this way, it should not be surprising that the theorem of the addition of velocities (involving as it does both space and time) is not

Relativity

337

the simple one of Galileo, but a more complicated one (see p. 317). In a general way, then, the most important and the simplest thing that can be said about the theory of relativity is that it manages to keep everything (or more correctly, almost everything) the same as before. All laws of nature remain the same, whether referred to one or to another reference frame, as long as those reference frames (or coordinate systems) are either at rest or move uniformly with respect to one another. This is just what the situation was in Galilean-Newtonian physics. But one additional condition has been added: the speed of light has to be the same in all coordinate systems. Consequently, when a law of nature is expressed first in one and then in another coordinate system, the transformation is made by means of the Lorentz transformation rather than by the Galilean transformation. What we have said in the last paragraph is the equivalent of what Einstein says in Chapter XIV. In the constancy or invariance of all laws of nature (together with the fact that the Lorentz transformation is to be employed in going from one coordinate system to another) lies the heuristic value of the theory of relativity. That is, we use the invariance of natural laws and the Lorentz transformation to discover new and more accurate formulations for old laws of nature. The most striking exhibition of the heuristic value of the theory of relativity comes in connection with the concepts of mass and energy. Einstein points out the discoveries that have been able to be made on account of the theory of relativity in Chapter XV. First, he notes that the simple expression Vi mv 2 no longer suffices to indicate accurately the kinetic energy of a body with speed v. Instead, we must

mc z use the expression

.

Next he notes

that

if

this

v2

V

1

c2

to absorb (say by radiation) an amount of then the kinetic energy of the body must be

same body were energy

Eoi

written as follows:

K) v2

i

1

c2

.

Comparing

this

with

,

Breakthroughs in Physics

338

the earlier expression for kinetic energy (before the energy was absorbed), we notice that the two expressions are has been replaced by the the same, except that the term 2 Thus the absorption of energy resulted term E /c2 ) can now in an increase of mass, for the term (m be seen as the new "mass term" in the expression of kinetic energy. Thus we could write

E

m

m + EJc

.

+

m+—=m+m

1

c2

where

m

1 is

the additional mass.

Hence

E — = mx c2 or

E =m

xc

2

.

This suggests that the original mass may also be considered as having an energy equivalent. That is,

E

m=

— c2

or

mc2 = This, of course, is the clear age possible.

Looking back

we

E. (seep. 322.)

famous equation

at the previous chapter,

that

we

made

the nu-

recall that there

coming of relativity theory, another equivalent form of energy was found. Whereas the First Law of Thermodynamics added heat as a form of energy to potential and kinetic energy, so now mass, or matter, is added as yet another form of energy. And thus the laws of the conservation of matter and the conservation of energy turn out to be one and the same. There can be no better place, no more important breakthrough in physics, at which to leave the reader. said that with the

EPILOGUE No work

that chronicles "breakthroughs" in physics can

at least some attention to Max Planck and to quantum theory. We would have liked to include a selection by Planck explaining quantum theory, just as we have included Einstein's explanation of the special theory of relativity. Unfortunately, there is no brief essay by Planck that expounds quantum theory in language easily comprehensible by the layman. Planck did write many essays for

be complete without

laymen, but they are not specifically concerned with the details of his theory, but rather with its philosophical consequences. Planck was concerned with the implications of his theory for our understanding of the universe, of such concepts as causality and freedom, and of similar matters. Some of the implications of the quantum theory by no means pleased the man who was responsible for them. Indeed, it sometimes appears as though Planck was the one physicist who most deplored the consequences of the quantum theory and who attempted most strenuously (if unsuccessfully) to avoid the break it necessitated with the physics of the past. In Chapters Seven and Nine we mentioned the rudiments of the early quantum theory. Many modifications and additions have been made to Planck's original theory as the result of experimental and theoretical work by many physicists. What started out as an ad hoc theory to explain heat radiation has become a crucial part of the physicists' new understanding of the structure of matter. Toward the end of the nineteenth century, the "old," or "classical," or "Newtonian," physics reigned supreme. Phenomenon after phenomenon was explained by the principles of Newton. In the theory of light, of course, the wave theory (originally due to Huygens) made a brilliant comeback, but the forces involved in the propagation of light were still calculated according to Newton's Laws of Motions. Wave theory, in fact, became a most successful and widespread part of physics, largely because of the work of James Clerk Maxwell, who formulated the theory of electricity in terms of wave equation. (The experimental portion of the work on electricity made tremendous advances through the work of many experimenters, but especially that of Michael Faraday.) As far as the understanding of the structure of matter

339

Breakthroughs in Physics

340

it was still for the most part based on Dalatomic theory, which envisaged truly indivisible atoms one kind for each element. Mendeleev's of different kinds work on the periodic table of the elements came toward the end of the century and gave an inkling that there were still many things to be learned about the elements that make

was concerned,

ton's

up

—

matter.

1900, there were only a few hints of the trethat the "new physics" was about to make in man's understanding of the world. The discovery of Xrays by Roentgen was one such hint, especially when it was followed shortly by the discovery of yet another kind of radiation, that associated with certain elements like uranium, thorium, and radium. Another important hint was given by J. J. Thomson's discovery of the divisibility of the atom and of its electric nature. After 1900, a whole host of new discoveries and theories appeared on the scene; Planck's quantum theory was the first, and perhaps most important,

Before

mendous changes

such theory. Planck's theory had to do with the emission of light and heat energy from a hot body. Light, it had become clear through the work of Clerk Maxwell, was a form of energy. It also seemed clear that light was propagated in the form of waves, being in fact but a kind of electromagnetic wave. (Since light is a form of electromagnetic radiation, it is easy to understand why the speed of light and the speed of electric current are the same.) However, experimental facts did not in all respects agree with the view that energy was radiated in waves from a hot body. In order to explain these experimental facts, Planck put forth his theory that energy is not radiated continuously (as in a wave) but discontinuousin a series of discrete packets that Planck named ly, quanta. The size of these energy quanta is determined by the frequency of the light or other form of energy being radiated (frequency being proportional to the inverse of wavelength) and by a new constant, called Planck's constant. If we denote Planck's constant by h, as is customary, and the frequency of radiation by /, then the quantum theory maintains that energy can be emitted only in packets that are equal to hf or multiples of it, such as 2 hf, 3 hf, The constant h has the dimensions of action (work X time) and is numerically equal to 6.62 X 10~27 Obviously, /* is a very small quantity; it makes itself importantly noticed only when the other quantities involved in a physical process are also of similar magnitude. Thus, in ordinary light radiation, where a great deal of energy is being radiated^ .

.

.

.

Epilogue light

seems to be propagated

still

in

341 waves, for

all

the energy

packets together form a wave front. At first it seemed that the quantum theory, having been invented for the purpose of explaining just one set of phenomena, might remain an isolated part of physics isolated and anomalous but perhaps not very important. Planck himself may have hoped so. However, events moved swiftly. In 1905, Einstein published his famous paper on the photoelectric effect (see Chapter Nine), which utilized Planck's concept of the energy quantum and therefore established the quantum theory still more firmly as a crucial part of

—

theoretical physics.

had to do with the absorption of energy When light falls on a metal plate, electrons are liberated from the metal. Light of a certain Einstein's paper

—

specifically, light energy.

frequency) always liberates electrons with the an increase in the intensity of the light does not increase the velocity but does increase the number of the electrons being liberated. This phenomenon was explained by Einstein as being due to the fact that light consists of energy quanta (which he called photons). Light of a certain frequency consists of photons whose energy is some multiple of hf. Greater intensity of light of the same color (i.e., of the same frequency /) produces more photons, each with the same amount of energy. When such a photon strikes the metal, it gives up all of its energy to an electron, which is thereby liberated. The photon then ceases to be. If light were propagated as an electromagnetic wave, an increase in the intensity of light would put more energy at color

same

(i.e.,

velocity;

wave

and this would give greater velocity to the But according to Einstein's explanation, although there is greater total energy, any given photon does not have greater energy when the light is more intense; consequently, the energy transferred by light of a given color to an electron is always the same, no matter what the in-

the

front,

liberated electrons.

theory of the explaining the observed phenomena rather than Maxwell's theory of the wave nature of light, here is another success of the quantum theory of the emission of energy. But this is not all. The quantum of energy h, which was first introduced by Planck in 1900 and which was shown to play such an important role by Einstein in 1905, showed up even more importantly in 1913, when the Danish physicist Nils Bohr published a paper on the structure of the atom. tensity

of

the

light.

Since

it

is

Einstein's

discrete character of light that succeeds

in

Breakthroughs

342

in Physics

By

this time, it had become apparent that the atom consists of a positively charged nucleus and of negatively charged electrons. But the internal structure of the atom was first correctly stated by Bohr. He assumed that the electrons revolve in orbits around the nucleus, but that only some orbits are "permitted" to the electrons. The permitted orbits are those in which the angular momentum is equal to a multiple of a quantity that involves h, namely nh/2. An electron revolving in one of these "permitted" orbits radiates no energy. If an electron absorbs enough energy from the outside, it may jump to a higher orbit; when it returns to its previous orbit, it gives up the energy it had absorbed. This energy turns out to be a photon, i.e., the amount of energy given up is hf. In giving up this energy, therefore, the electron produces radiation of frequency /. It is apparent from this that electrons and light have many similarities. From this it is an easy step to the assertion, first made by Louis de Broglie in 1924, that electrons,

have both particle and wave characteristics. From electron de Broglie was able to deto assume: namely, that electrons can move around the nucleus only in certain "permitted" orbits. The wave property of electrons was later verified by the fact that experimenters were able to show that electrons

like light,

wave character of an rive what Bohr had had the

suffer diffraction.

The wave character of an electron means that matter has a wave character as well as a particle character, for electrons

among

the major constituents

of matter. Usually, of is far overshadowed consequence of the wave char-> by the particle character. acter of matter or of electrons is that Heisenberg's Uncertainty Principle applies: we can never simultaneously obtain completely accurate values for both the position and the velocity of an electron. The greater the accuracy of the value for the position, the less accurate the value for the velocity, and vice versa. This uncertainty is not due to any imperfection of the measuring instruments but is a basic limitation inherent in the nature of matter. The amount of uncertainty, i.e., the limit of simultaneous accuracy for position and velocity, is again a consequence of quantum theory are

course, the

wave character of matter

A

and involves the constant Planck's

h.

quantum theory therefore had far-reaching

changed

ef-

views of the nature of light, of electrons, of matter, of the structure of the atom, and of the degree of certainty obtainable in physics.

fects: It

physicists'

Suggestions for Further Reading

The reader who has become interested in the authors and achievements represented in this book may want to examine the complete treatises from which the preceding selections have been taken. All are available in paperback form, except Pascal's Physical Treatises. Newton's Opticks, to which we refer several times in the chapter on Huygens, has also been reprinted as a paperback. Listed below are some other paperbound books that the reader may want to consult. The choice is both arbitrary and personal; in large part it is dictated by what is currently available in inexpensive form. Because new books are constantly being added to publishers' lists, while older books often go out of print, the interested reader may do best by visiting one of the large and well-stocked paperback stores that can be found in metropolitan areas and university communities.

D'ABRO, A.

Evolution of Scientific Thought from Newton New York: Dover, 1950. ANDRADE, E. N. da C. Rutherford and the Nature of the Atom. New York: Doubleday, 1964. BARNETT, LINCOLN. The Universe and Dr. Einstein. New York: New American Library, 1952. BELL, ERIC TEMPLE. Mathematics: Queen and Servant of Science. New York: McGraw-Hill, 1951. BRIDGMAN, PERCY W. The Logic of Modern Physics. New York: Macmillan, 1946. BURTT, EDWIN A. Metaphysical Foundations of Modern Science; New York: Doubleday, 1954. CALDER, RITCHIE. Science in Our Lives. New York: New American Library, 1955. CAMPBELL, R. Foundations of Experimental Science ("Physics: The Elements"). New York: Dover, to Einstein.

NORMAN

1957

COHEN,

BERNARD.

I.

The Birth of a

New

Physics.

New

York: Doubleday, 1960. COLEMAN, JAMES A. Modern Theories of the Universe. New York: New American Library, 1963. Relativity for the Layman. New York: New American Library, 1958. CONANT, JAMES B. Modern Science and Modern Man. New York: Doubleday, 1959. 343 .

Breakthroughs in Physics

344

Science and Conn.: Yale, 1951. CROMBIE, ALISTAIR C. .

Common

New

Sense.

Haven,

Medieval and Early Modern

New York: Doubleday, 1959. DAMPIER, WILLIAM CECIL. A Shorter History of Science. New York: World, 1957. DE SANTILLANA, GIORGIO. The Crime of Galileo. Science. 2 vols.

Chicago,

University of Chicago, 1955. Introduction to Space. New York: Columbia, 1962. EDDINGTON, SIR ARTHUR. Space, Time and Gravitation. New York: Harper & Row, 1959.

DU

111.:

BRIDGE, LEE A.

ALBERT

EINSTEIN,

lution of Physics.

GAMOW, GEORGE. wood

Cliffs,

New

LEOPOLD INFELD.

The EvoYork: Simon & Schuster, 1938. The Atom and Its Nucleus. Engle-

and

New

Jersey: Prentice-Hall, 1961.

The Birth and Death of American Library, 1945. .

New

the Sun.

New

York;

Gravity. New York: Doubleday, 1962. Frontiers of Astronomy. New York: American Library, 1963. -.

HOYLE, FRED.

KNICKERBOCKER, WILLIAM Science. Boston, Mass.:

Classics

of

Modern

Beacon Press, 1962. The Sleepwalkers. New

KOESTLER, ARTHUR. Grosset & Dunlap, 1963. MACH, ERNST. Popular Scientific Open

S.

Lectures.

La

New

York:

Salle, HI.:

Court, 1903.

MAXWELL, JAMES CLERK.

Matter and Motion.

New

York: Dover, 1952.

NEWMAN, JAMES

R. The World of Laws and the World of Chance (Vol. II of The World of Mathematics). New York: Simon & Schuster, 1956. PEIRCE, CHARLES S. Essays in the Philosophy of Science* Indianapolis, Ind.: Bobbs-Merrill, 1957. POINCARfi, HENRI. Science and Hypothesis. New York: Dover, 1952.

REICHENBACH, HANS. Philosophy of Space and Time. New York: Dover. RUSSELL, BERTRAND. ABC of Relativity. New York: New American Library, 1959. SANDFORT, JOHN F. Heat Engines: Thermodynamics in Theory and Practice. New York: Doubleday, 1962. WHITEHEAD, ALFRED NORTH. Science and the Modem World. New York: New American Library, 1948*

INDEX Acoustics, physiological, 249 Action, direct, and action-at-a-

(Galileo),

Aether. See Ether Air: elastic forces produced by heat, 265; (and) mercury, equilibrium, 130-52 Air pressure: (and) mercury, 145-52; Great Experiment, 147-52; Toricelli experiment, 145-48, 150-52 Altimeter, 151 Anaxagoras, fn 63 Angle of incidence, and angle of reflection, 222 Antiperistasis, 141 Antiquity, simple machines, 13 Archimedes, 182; "crown problem," 19, 49-51; death, 19; Earth, Moving The, 19, 31-32; (On the) Equilibrium of Planes, or, The Centres of Gravity of Planes, 14-18, commentary , 19-34; (On) Floating Bodies, 36-42, commentary, 42-51; legends, 1920; lever, law of the, 13-34, 199-201, 260; machines, sim-

lem, 286; time, absolute, 161 Dalton's theory, 340; structure,

75 Astronomy, centrifugal 251; and centripetal forces, 190; Copernican doctrine, 84, 110; Copernican system, fn 53, fn 69, 82; distance measurement, 58-59; eclipses, 207 //; Aselli,

Galilean satellites, 51; Galilean system of coordinates, 300-301; Kepler, 91; lawfulness, 285; Marius, 91; Ptolemaic system, fn 69; regularMessenger ity, 285; Starry

341-42

Atomism, 239 Attractions, 159, 179

Augustus, 53

Bacon,

Francis,

Novum

Or-

ganum, 142-45, Badovere, Jacques, 57 Bannier, Very Rev. Father, 135 Barometer (Toricelli), 145-48, 150-51 Battery: galvanic, 279; voltaic, 279, 281

Begon, Monsieur, 135

252 change of

Bernoulli, Daniel, v

Body(ies):

place,

184; falling, accelerated motion of, 112 //; (On) Floating Bodies (Archimedes), 36-42, commentary, 42-51; hard, impact and reflection, 175; in-

nate property or power of, 187; material, strength of, 111; opaque, and refraction, 226, 228; polished, reflection, 221-24; rigid (body of reference), 296 //; three-body problem, 286; transparent,

13-18, commentary, 1934; reasoning, mathematical, 48; Works of Archimedes,

187; Physics, 183-84

commen-

Atom:

ple,

14, 36 Aristotle, 44, 89, 111, 113, 186,

52-83,

tary, 84-91; three-body prob-

distance, 190

and refraction, 225-37 Bohr, Nils, 341-42 Boyle, Robert, 212 Boyle's law, 293

Brownian movement, 294 Buildings, antiquity, 13

273 Buoyancy, Archimedes' theory Bullets, of,

35-51

Caesar, Julius, 53 Caloric theory, 291-92

Cancer (constellation), fn 75 Carbon, 276-77 Carbonic acid, 276-77

345

346

IrJDEX

Carnot, Sadi, 271, 288 Cartesian coordinates, 122-23,

298 Cassini, Monsieur, 203 gravity, of Center

172-74; (two) meanings, 25-27, 3031; planes, 14-34 Centripetal force, 189, 190; absolute, 157; accelerative, 157; motive, 158-59 Certainty, and likelihood, 204 of and principles Change,

mathematics, 152-201 Charles' law, 293 Chastin, Rev. Father, 136, 137

Chemical work, 276-77 Chromatic aberration, 88 Clausius, 271, 275 Clocks, 129, 175, 180-81, 202, 237, 254-55, 265, 314-16 Coal, burning of, 276

Coincidence in space, 297 Color, and chromatic aberration, 88 Combustion, 276-78, 279 Compression, in gases, 274 Conformity with law, 250-52 Conservation of energy. See Thermodynamics, first law of Conservation of Force, On the (von Helmholtz), 250-85, commentary, 285-93 Constant (Planck's), 340 Constellations, 75 Coordinates, 297-99, 304; Cartesian system, 298; Galilean system, 300-301 Copernican doctrine, 84, 110 Copernican system, fn 53, fn 69, 82 Copernicus, 51, fn 53, 208 Crabwinch, 261 "Crown problem" (Archimedes), 49-51 Dalton's atomic theory, 340 da Vinci, Leonardo, fn 62 Davy, Sir Humphry, 252, 272 de Broglie, Louis, 342 De la Hire, 203 De la Mare, Very Rev. Father,

137 de Santillana, Giorgio, 84 Descartes, Rene, 206-208, 214, 220, 235, 237, 239; Geometry, 123 De Sitter, Willem, 304, 331

Dialogue on the Great World Systems (Galileo), 84, 110, 111 Dialogues

Two

Concerning

New

Sciences (Galileo), Naturally Accelerated Motion, 92-110, commentary, 110-29 Dini, Piero, fn 63 Distance, and relativity, 310-11

Dog

Star,

73

Doppler principle, 323 Drake, Stillman, 84 Dynamics, 111, 112, 183; definition, 23; gravity and free fall, 92-129; pendulum, 202;

wave theory of

light,

202-48

Earth: (and) moon, fn 53, 5963, 66-71; (a) sphere, 44-45; (and) water in equilibrium, 43-45 Eclipses: Galilean satellites, 209-10, 240; moon, 69, 207208, 240 Einstein, Albert, 248, 293, 294; and Galileo, 329; and Newton's theories of space and time, 329-37; photoelectricity, 341; Special Theory of Relativity, 295-328, commentary, 328-38 Elasticity, 214-16, 224, 228,

241-42 Electricity, 249, 339; telegraph,

281, 282; work, 277 // Electrodynamics, 183, 302, 305, 321, 331 Electromagnetism, 305, 322 Electrons: ticle light,

characteristics,

par-

and wave, 342; (and) 342

183 Elements, periodic table, 340 Energy: chemical, 293; electrical, 293; (as) heat, 338; kinetic, 121, 193, 275, 287-88, mass, 337-38; 338; 292, (and) matter, equivalence, 293; mechanical, 252-85; (due to) motion, 287; (due to) position, 287; potential, 193, 287-88; quanta, 340; quantum theory, 339-42; Special Electrostatics,

Theory

of

Relativity

stein), 295-328,

328-38

(Ein-

commentary,

Index Engines: high-pressure, 267; reaction, 198 Equilibrium: air and mercury, 130-52; buoyancy theory, 3551; fluids, 130-52; planes, 1434. See also Statics Equilibrium of Planes, or, Centres of Gravity of Planes

(Archimedes), 14-18, mentary, 19-34 Escape Velocity, 193-94

com-

Dialogue on the Great World Systems, 84; Dialogues Concerning Two New Sciences, 84, 92-110, 189, commentary, 110-29; Discoveries and Opinions of, 52; (and) Einstein, 329; free fall, 92; gravity, 92; Inquisition, 84; Naturally Accelerated Motion, 92-110, commentary, 110-29;

Ether, 75, 83, 89, 212, 214-18, 224-30, 235, 241-43, 292; and the moon, 66, 68 Euclidean geometry, and Einstein's Special Theory of Relativity,

295-338

Experience, and relativity, 323-

26 Experiment, 247

crucial,

142-43,

pendulums, and their isochronism, 92; Starry Messenger, 52-83, 110, commentary,

Faraday, Michael, 339 Fermat, 235

84-91; telescope, 51, 52, 55, 85-89, 92, construction, 56-59, Jupiter satellites (discovery of), 76-83, stars, fixed (observations of), 71-75, moon (observations of), 59-71; velocities, addition of (theorem), 303, 316-18, 331-33 Gases: buoyancy, 43-44; compression, 274; heated, elastic force of, 265 //; molecularkinetic theory, 292-93; per-

FitzGerald, 326 Fizeau, H. L., 241 Fizeau experiment, 316-19, 323,

324 (On) Floating Bodies (Archimedes),

36-42,

commentary,

42-51 Fluids: buoyancy theory, 35-51; defining characteristic, 42-43; in equilibrium, 42-43, 130-52 Foot pound, 256, 264 Force (or weight): and buoyancy theory, 35-51; centrifugal, 190; centripetal, 15659, 189-90; counterbalance, 20; elastic, 265 //; experimental evidence for existence of, 174-81; impressed, 155; moving, 253 //; parallelogram, 198; planes, equilibrium and centers of gravity, 14-34; quantity of (amount of work), 252 //; theory,

249-93 Foucault, Jean, 247 Free fall (or accelerated motion), 111; definition, 112 //; Galileo, 92; (and) gravity,

92-129

Friction:

heat,

272, 275, 292;

liquids, 272, 289; mechanical effects of, 272; solids, 289

347

Galilean satellites, 56, 76-83, 90-91, 240; eclipses, 209-10 Galilean system of coordinates, 300-301 Galilean telescope (or opera glass), 85-86 Galilean transformation, 319, 325, 327, 335-37 Galileo, 84, 174, 188-89, 195;

v

manent, 274; phenomena of, 275 Geometry, 184, 204; coordinate, 123, 297-99; Euclidean, and Einstein's special theory

of relativity, 295-338; propositions, physical meaning of,

295-96

Geometry (Descartes), 123 Glass,

God:

refraction,

need time, 330

for,

225-30, 235 286; (and)

Gravitation, law of, 251, 285,

286 Gravity, 189-94, 198, 237, 254, 263; (and) free fall, 92-129; Galileo, 92-129; water, 256-

58 Great Experiment on the Equilibrium of Fluids (Pascal), 130-43, commentary, 143-52

Gunpowder, 276

Index

348

Halley, Edmund, 153 Heat, 250-93; caloric theory, 291-92; (and) force, 252-85; (a) form of energy, 288, 289, 338; friction, 272, 292; Joule, J. P., 252, 273-74, 288(and) kinetic energy, 89; 292; latent, 270-72; mechanical equivalent of, 289; mechanical power, 265-73; (a) motion, 275; (a) substance, 271; unit of, 273; (and)

work, relations between, 274 uncertainty

Heisenberg's ciple,

prin-

342

Helmholtz, von, Herman Ludwig Ferdinand, 249; On the Conservation of Force, 25085, commentary, 285-93; thesis, 293 Hevelius,

90 Hiero

map

II,

the

of

King

moon,

Syracuse

of

49

(Sicily), 19,

High-pressure engine, 267 Hipparchus, 51 Hook, Mr., Micrographie, 219

Huygens, Christiaan, 339; Horologium

175-78, oscillato-

rium, 202; Treatise on Light, 202-37, commentary, 237-48

Hydrodynamics, 249 Hydrogen, 277, 278 20;

Hydrostatics,

On

Floating

Bodies (Archimedes), 36-42,

commentary, 42-51; Pascal's investigation,

Iceland spar, 202, 203

130-52

and

refraction,

Immersion, and theory of buoyancy, 35-51 Impact, 215, 272, 275; inelastic,

272

Impulse, 159 155, 187-89, 191, 239, 243; law of, definition, 300

Inertia,

Inquisition,

and Galileo, 84

Instance of the cross, 142-43 Integral calculus, 124 Jet planes, 196-98, 332 Joule, James Prescott, 252, 27374, 288-89 Jupiter (planet): Galilean satellites, 51-54, 240 ff, eclipses of,

209-10

Kennedy, John Fitzgerald, 90 Kepler, Johannes, fn 56, fn 69, fn 71, fn 83, 90; "circumjovial planets" (moons encircling Jupiter), 91; telescope, 85-86; third law, 91 Kinetic energy, 121, 193, 275, 287-88, 292, 337-38 Kronig, 275

La La

Porte, Monsieur, 135 Ville,

Monsieur, 135

Law, conformity with, and ural phenomena, 250-52 Lawfulness, 285 Legends, 30-31

Archimedes,

nat-

19-20,

Baron Gottfried Wilhelm, 203 Length, measurement, 297 Lenses: achromatic, 88-89; and chromatic aberration, 88; eyepiece in refracting telescope, 85; shapes, 202; telescopes, 57-59, 85-89 Lever, law of (Archimedes), 13-34, 199-201, 260; definiLeibniz,

tion,

34

Light: corpuscular theory, 24248; electrons and, 342; emission and absorption of, by matter, 247; (form of) energy, 340; (and) heat energy, emission of, 340; interference, 244; nature of, 248, 339-42; propagation, law of, relativity, 304-305; propagation in terms of physical causes, 238; properties, 219 ff; rays in a straight line, 205-21; reflection, 22124; refraction, 86-88, 225-37; (and) sound, 211-13, 238; speed constancy, 331-34; determination, 240; speed speed measured in air and in water, 247; wave theory, 20248, 339; wave theory, and corpuscular theory compared,

and

242-48 Light, Treatise

on (Huygens),

202-37, commentary, 237-48 Likelihood, and certainty, 204 Lipperhey, Hans, fn 57, 85 Liquids: buoyancy, theory of, 35-51; friction, 272, 289;

Index transparency and refraction, 225-37

Locomotion, Newton's principles of mathematics, 152-201 Lorentz, H. A., 305, 324, 326, 331 Lorentz transformation, 311-18, 319, 320, 335-37 Machines: //;

simple,

258; water-power, 258; work, measurement by foot pounds,

256 Magneto-electrical

machine,

280-82 Magnet(s), 281

Manhattan

ure, fn

Medicean

Project,

295

Mariotte, M.,

176 Marius (Dutch astronomer), 91 Mass, 154; energy, 293, 338; (and) relativity theory, 33738; (and) weight, 185-86 Mathematical Principles of Natural Philosophy (Newton): definitions, 154-59, scholium, 159-66; axioms, or laws of motion, 166-74, 186= 201, scholium, 174-81; commentary, 182-201 Mathematics: Einstein's special theory of relativity, 295-328, commentary, 328-38; lever, law of (Archimedes), 13-34; Newton's principles, 152-201; statics, 13-24 Matter: particle-wave character, 342; structure, 339 Maxwell, James Clerk, 275, 321, 339, 340 Maxwell-Lorentz theory, 318 Maxwell's theory, 341 Mayer, Dr. Julius Robert, 252, 288 Mayr, Simon, fn 77 Measurement: Cartesian system

of coordinates, 298; distance, astronomical, 58-59; length, 297; time, 129 Measuring-rods, motion, 314-16 Mechanical energy: and force, 252-85; heat, 288 Mechanics, 168-70; doctrine of, 201; law of, 300; quantum, 248; relativity, principle of, 302; space and time, 299300; velocities, addition of

264 stars,

52,

54,

55,

II, 52 Mendeleev, D. I., 340 Mercury, 229, 274; (and)

air,

76-83 Medici, de, Cosimo

130-52; Toricelli experiment,

moving

force, 253 13-34, 168-70,

349 (theorem), 303, 316-18, 33133; work, theoretical meas-

145-48

Mercury

(planet), 53, 56, 91 Mersenne, Rev. Father, 134 Metals, refraction, 228 Meteorology, 249, 256, 285;

barometer, 151 Michelson, 325-26 Micrographie (Hook), 219

Milky Way,

56,

75

Mill-wheels: overshoot, 256-58, 261; undershoot, 261-62 Minkowski, four-dimensional space, 326-28 Mithridates, King of Pontus,

258 Molecular-kinetic

theory

of

gases, 292-93

Momentum, 120 Moon: ether, 66, 68; cavities and prominences, 61-71; v

cusps, 61; (and) earth, fn 53, 59-63, 66-71; eclipses, 69, 207-208; horns, 61, 68; icecups, 61; (and) sun, 62-71; surface, 55-71; velocity, orbital, 195 Morley, 326 Morse's telegraph, 281 Mosnier, Monsieur, 135, 137

Motion (or momentum) abso160-61, and relative, lute, :

accelerated (or free 92-129, definition, 112 definition, //; change of, 188; 94, 98, 100; heat, 275; measuring-rods and clocks, 31416; naturally accelerated, 921 1 0-29 ; commentary, 1 1 0, axioms, 166-81; Newton's projectile and satellite, 191; quantity, 170-72, 185, 188; (and) rest, 162; (and) time, 92-129, 329-30; uniform, in distance and time, 122; uni163;

fall),

form translation, 301 Motion, Naturally Accelerated (Galileo),

92-110,

tary, 110-29

commen-

Index

350

Motive power, 268, 269 force, 253 //; meteorological processes, 256-58

Moving

Natural

philosophy, 111-12; change, principles of, 152201; gravity and free fall, 92-129; (and) physics, 183 Natural sciences, character of, 252, 285 Naturally Accelerated Motion (Galileo), 92-110, commentary, 110-29 Nature: law, conformity with, 250-52; phenomena, basic, 183; processes, meteorological, 256-58; purpose(s), 113; vacuum, abhors a, 113, 13033, 140, 143-45 Newton, Sir Isaac, 203, 237, 252, 285; gravitation, universal, law of, 194; locomotion, 152-201; Mathematical Principles of Natural Philosophy (Principia) 152-201, 247, 285, 286; Opticks, 88, 186, 241,

245-247; scope, 182-83; (theories of) space and time, 329-37; three-body problem, 286; white light, refraction of, 87

Nitric acid, 279

Novum

Organum

(Bacon),

142-45 Opacity, 229

Opera

(Galilean teleglass scope), 85-86 Opticks (Newton), 88, 186, 241, 245, 247 Optics, 302, 305; geometrical, 238; light, wave theory of, 202-48; physiological, 249; reflection

and

refraction,

245-47 Orbits, 156-57 (nebula): Orion laws

of,

Sword

of,

73-74;

Belt

Head

and of,

75 Oxygen, 276-79 Parallelogram of forces, 198 Pardies, Father, 219 Pascal, Blaise, 130, 182; Great Experiment on the Equilibrium of Fluids, 130-43, com-

mentary, 143-52, conse-

quences, 140; letter to Perier, Treatises, 131-34; Physical 130; Provincial Letters, 130 Pendulum experiment, 118-22 Pendulum(s), 175, 176, 202, 237, 255, 263, 264, 273, 202; 287, 288; dynamics, isochronism, 92; kinetic energy, 193 Pensees (Pascal), 130 Perier, M., 130, 131, 144, 145; "Great Experiment," 135-39, 146-52; Pascal, letter to, 134-

35 Perpetual motion, 284 Phenomena, 339; reduced

to

one law, 285 Philosophy,

natural,

96, 98, 130; mathematical principles (Newton), 152-201; nature abhors a vacuum, 131-33; (and) physics, 183 Photoelectricity, 248; Einstein,

341 Photon(s), 248, 341, 342; theory of light, 294 Physics (Aristotle), 183-84 Picard, Mr., 211 Place: change in, 183; (and) space, 160 Planck, Max, 248, 340, 341; quantum theory, 339-42 Planck's constant, 248, 340 Planes: Equilibrium of Planes, or, The Centres of Gravity of Planes (Archimedes), 1418, commentary, 19-34; inclined, 13, 121, 127-29 Pleiades, The, 73-74

Plenum, 144 Pope, Alexander, 183 Postulates, 23-24 Potential energy, 193, 287-88

Praesepe (nebula), 75-76 Predictability, 285 Principia (Newton). See Math' ematical Principles of Natural Philosophy 109,

Principles,

182, 183, 204,

205 Prism 44 234 156-57, 194, 198

Projectiles,

190

//,

Propensity, 159 Letters Provincial

130

166,

174,

(Pascal),

Index

351

Ptolemaic system, fn 69 Ptolemy, 44, 51

Salviati,

Pulleys, 13, 19, 20, 180, 258-60 Pump, suction, 143-44, 151

Satellites,

Pythagoras, fn 62 Quality, change in, 183

time, 330

See Mercury

Radiation, electromagnetic, 340 Reaction engines, 198 Reason, 251; teleology, 145

Reductionism, 184 Reference, body of, 296 // Refraction: effects of, 229-37; 225, 226, 228, 230, 235; Iceland spar, 202, 203; metals, 228; polished bodies, 221-24; principal property, 232; reciprocity, 232; speed of light, 227; telescopes, 8588; (and) transparency, 22537; water, 227, 230 Regnault, 265-66, 275 Regularity, 285 Relativity (principle), 301-303,

Snell's law, 246,

//

247

Socrates, fn 63 Solids: buoyancy,

35-51; fric289; submersion, 45-48; transparency and refraction, tion,

glass,

v

225-37 Sonic boom, 238-39 Sound: (and) light, 238; motion of, 206 //; propagation, 238-39, 240 Space: absolute, 159-60; coincidence, 297; Einstein's rela295-328; imtivity theory, movable, 163, 165; Minkow326ski's four-dimensional, 28; place, 160; relative, 160;

Special Theory of (Einstein), 293, 295-328, commentary, 328-38; (conception of) distance, 310-11; experience and, 323-26; heuristic value, 318-19, 337-38; (and) propagation of light,

Relativity,

304-305;

(and)

relativity

(gen(principle), fn 331; eral) results of, 319-23; si-

multaneity, 308-10 Rest,

and motion, 162

Rock

crystal, refraction,

203

295 Rosen, Edward, fn 55, fn 56

Rumford, 252 Giovanfrancesco,

HI, 115

three-dimensional continuum, 326; (and) time, 161; (and) time in classical mechanics, 299-300; (and) time theories (Newton), 329-37 Special Theory of Relativity com295-328, (Einstein), mentary, 328-38 Spheres, 36-38, 44-45 Springs, 265, 282 Spyglass. See Telescope Star(s): fixed, 71-76; nebulous, 56, 75

Rockets, 191, 192, 196-98 Roentgen, W. C, 340 Romer, Ole, 202, 208, 240 Delano, Roosevelt, Franklin

//,

Subjects, Popular

Shape, and buoyancy, 35-51 Silver, and refraction, 228, 229 Simplicio, 96 //, 111 Simultaneity, 306-308; relativity of, 308-10 Size, change in, 183

42

Sagredo,

110,

Lectures on (Helmholtz), 250 Screws, 13, 181 Shakespeare, William, and

Quantity: scalars, 199; vectors, 199; work, 252 // Quantum mechanics, 248 Quantum theory (Planck), 339-

332

ff,

187, 191, 194; Galilean, 56, 76-83, 90-91 Saturn, phenomena of, 208 Scalars, 199 Scientific

Quanta, 340

Quicksilver.

94

Filippo,

144

94

(Galileo), Messenger Starry 52-83, 110, commentary, 84-

91 definition, 23; 182; Galileo's Two New Sciences, 92-129; lever, law of the (Archimedes), 13-34; NewMathematical Princiton's

Statics,

ples,

154-201

Index

352

Steam-engines, 266-68, 274-76,

289-90 Strabo, 258 Submersion,

solids, 45-48 Suction pump, 143-44, 152 Sun, and moon, 62-71 System of coordinates, 297-99

Telegraph, 281, 282 Teleology, 113, 145, 286 Telescope, 51-91, 203; chromatic aberration, 88; eyepiece, 85; Galileo's, construction of, 56-59; Keplerian, 85; lenses, 57-59, 85-89; optical, 85; reflecting, 85; refracting, 85; U.S., 85 Temperature, absolute zero,

293 Theology, time,

113,

130, 286;

and

330

Thermal equivalents, 275 Thermodynamics first law, :

249-93, 338, meaning, 28992; second law, 290-91 Thermo-electric elements, 281

Thermometer, 270 Thomson, J. J., 340 Time: absolute, 159, 161; (and) distance in uniform motion, 122; Einstein's con330-38; measurement, 129; (and) motion, 92-129, 329-30; Newton's concept, 330-38; (in) physics, 306308; relativity theory (Einstein), 295-328; (and) space, 161, 299-300, 329-37 Toricelli, Evangelista, 145 Toricelli experiment, 145-48, cept,

150-52, 212, 226

Transparency, 225-37

and

refraction,

Travel, rocket and jet, 196-98 Two New Sciences, Dialogues Concerning (Galileo), 92, 110, 111, 115, 123, 189 Tycho, fn 53

Unit of heat, 273 United States, satellite launching speed, 194

Vacuum: tion,

238

Pascal's

130-52;

investiga-

(and)

sound,

Vectors, 199 Velocity: action of, 262; addition of (Galilean theorem), 303, 316-18, 331-33; change of (acceleration), 188; escape velocity, 193-94; gravity and free fall, 92-129; moving mass, 262-63; working

power, relation

Venus

to,

263-64

(planet), 56, 69-70, 72,

90, 91 vis viva.

See Kinetic energy

Volume, and buoyancy, 35-51 Wallis, Dr., 175, 176

Watches, 265 Water: (and)

caloric

theory,

292; (in) equilibrium, and the earth, 43-45; (heated by) friction, 274; gravity, use as moving force, 256-58; refraction, 230; Toricelli experiment, 152 Water-clocks, 129 Watermill, 263 Water-power, 258

Water wheels, 258

Wave

theory of

light,

202-48

Wedge, 181 Weight (or force), 20; (theory of) buoyancy, 35-51; (and) mass, 185-86; planes, equilibrium and centers of gravity, 14-34; raised, 281-82 Wheel, 272-73; overshot millwheel, 256-58, 261; toothedundershot 260-61; wheel, mill-wheel, 261-62; weightless,

200

Wilkins, John, fn 68 Windmills, 262, 263 Work, 34, 193; chemical, 27677; electricity, 277 //; foot pound, 256; (and) force, relationship, -heat 252-85; 274; mechanical forces, 280; mechanical power, 265-75; moving mass, 264; raised weight, 264 Wren, Sir Christopher, 175, 176, 178

X

rays,

340

Yerkes Observatory, 85

Zeno paradox, 112 Zinc, 279-80

Archimedes Galileo

Pascal

Newton Huygens Helmholtz Einstein

Breakthroughs

in

Physics presents nine works by seven

scientists who provided major contributions to the scientific knowledge of their times. Each essay is prefaced by the editors commentary, explaining the historical significance of these contributions, which span twentythree centuries of progress and discovery. is, in my opinion, performing an enormous servthose members of the general public who are interested in science. Few laymen — few scientists even — read many of the classics of science. Yet these classics are worth reading, not only for the insight they give into the fundamentals of science and the course of scientific development, but for their own sake — their charm of expression and clarity of thought. Now, with Wolff as cicerone and interpreter, there is no reason to fear difficulties. One can read with comfort and security — and, — Isaac Asimov most of ail, pleasure."

"Wolff ice to

PETER WOLFF is Executive Editor of The Great Ideas of Today an annual supplement to Great Books of the Western World published by Encyclopaedia Britannica, ,

,

A graduate

of St. John's College, he taught mathematics there while completing his master's thesis. He worked on the Syntopicoia a two-volume "idea index" for the Great Books of t he Western Worl d, and was Assistant Director and a Fellow of San Francisco's institute of Philosophical Research, set up under a grant from the Ford Foundation. Mr. Wolff's Breakthroughs in Mathem atics is published in a Signet Science Library Inc.

,

edition.

PUBLISHED BY THE HEW AMERICAN LIBRARY

DESIGN: HENRY WOLF