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Einstein’s Theories of Relativity and Gravitation: A Selection of Material from the Essays Submitted in the Competition for the Euge…

The Project Gutenberg eBook of Einstein's Theories of Relativity and Gravitation
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Title: Einstein's Theories of Relativity and Gravitation
Author: J. Malcolm Bird
Other: Albert Einstein
Release date: October 4, 2020 [eBook #63372]
Language: English
Original publication: United States: Scientific American Publishing Co., Munn & Co, 1921
Credits: Produced by Jeroen Hellingman and the Online Distributed Proofreading Team at https://www.pgdp.net/ for Project Gutenberg (This book was produced from scanned images of public domain material from the Google Books project.)
*** START OF THE PROJECT GUTENBERG EBOOK EINSTEIN'S THEORIES OF RELATIVITY AND GRAVITATION ***

[Contents]

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Dr. Albert Einstein, Originator of the Special and General Theories of Relativity
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Einstein’s Theories of Relativity and Gravitation
A SELECTION OF MATERIAL FROM THE ESSAYS SUBMITTED IN THE COMPETITION FOR THE
EUGENE HIGGINS PRIZE OF $5,000
COMPILED AND EDITED, AND INTRODUCTORY MATTER SUPPLIED
BY
J. MALCOLM BIRD,
Associate Editor, Scientific American
NEW YORK SCIENTIFIC AMERICAN PUBLISHING CO.,

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Einstein’s Theories of Relativity and Gravitation: A Selection of Material from the Essays Submitted in the Competition for the Euge…
MUNN & CO.
1921

Copyright 1921 by
Scientific American Publishing Company
All rights reserved
Great Britain copyright secured
The right of translation is reserved in all languages, including the Scandinavian
Swedish rights secured by Thall and Carlsson, Stockholm

[Contents] [iii]

PREFACE
The obstacles which the layman finds to understanding Einstein’s relativity theories lie not so much in the inherent difficulty of these theories themselves as in the difficulty of preparing the mind for their reception. The theory is no more difficult than any scientific development of comparable depth; it is not so difficult as some of these. But it is a fact that for a decent understanding of it, a large background of scientific knowledge and scientific habit of thought is essential. The bulk of the writers who have attempted to explain Einstein to the general reader have not realized the great gulf which lies between the mental processes of the trained mathematician and those of the man in the street. They have not perceived that the lay reader must be personally conducted for a long distance from
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the vestibule of the temple of science before he comes to Einstein, and that he cannot by any possibility make this journey unaided. The result has been to pitchfork the reader into the intricacies of the subject without adequate preparation.

The present volume avoids this mistake with the utmost care. It avoids it, in

fact, with such deliberation as to make it in order to say a word in

explanation of what will at first glance seem an extraordinary arrangement

of material. It was to be expected, doubtless, that this book would open

with a brief statement of the genesis and the outcome of the Einstein Prize

[iv]

Essay Contest for the $5,000 prize offered by Mr. Eugene Higgins. It was

doubtless to be expected that, after this had been dismissed, the winning

essay would be given the post of honor in advance of all other material

bearing actually on the Einstein theories. When the reader observes that this

has not been done, he will by all means expect a word of explanation; and it

is mainly for the purpose of giving this that we make these introductory

remarks.

The essays submitted in the contest, and in particular the comments of a

few disappointed readers upon Mr. Bolton’s prize essay, make quite plain

what might have been anticipated—that in the small compass of 3,000

words it is not possible both to prepare the reader’s mind for a discussion of

Relativity and to give a discussion that shall be adequate. Mr. Bolton

himself, in replying to a protest that he had not done all this, has used the

word “miracle”—we think it a well-advised one. No miracle was expected

as a result of the contest, and none has been achieved. But in awarding the

prize, the Judges had to decide whether it was the best preliminary

exposition or the best discussion that was wanted. They decided, and rightly

we believe, that the award should go to an actual statement of what the

Einstein theories are and what they do, rather than to a mere introduction,

however well conceived and well executed the latter might be.

Nevertheless, we should be closing our eyes to a very obvious fact if we did

not recognize that, without something in the way of preparation, the general

reader is not going to pursue Mr. Bolton’s essay, or any other essay on this

[v]

subject, with profit. It is in order the more forcefully to hold out

inducements to him to subject himself to this preparation that we place at

the head of the book the chapters designed to give it to him.

Chapter II. is intended so to bring the mind of the reader into contact with certain philosophical problems presented to us by our experiences with the external world and our efforts to learn the facts about it, that he may approach the subject of relativity with an appreciation of the place it occupies as a phase of human thought and a pillar of the scientific structure. Until the reader is aware of the existence of these problems and the directions taken by the efforts, successful and unsuccessful, to unravel them, he is not equipped to comprehend the doctrine of relativity at all; he is in much the same case as a child whose education had reached only the primer stage, if asked to read the masterpieces of literature. He lacks not alone the vocabulary, but equally the mental background on which the vocabulary is based.

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It will be noted that in this and the chapters immediately following it, the

Editor has supplied material freely. The obvious interpretation is that

satisfactory material covering the desired ground was not found in any of

the essays; for we are sure the scope and number of the credited excerpts

will make it clear that all contributions were adequately scrutinized in

search of available passages. This “inadequacy” of the competing essays

has been severely commented upon by several correspondents, and the

inference drawn that as a whole the offerings were not up to the mark. Such

[vi]

a viewpoint is wholly unjust to the contestants. The essays which paid

serious attention to the business of paving the way to relativity necessarily

did so at the expense of completeness in the later paragraphs where specific

explanation of the Einstein theories was in order. Mr. Law, whose essay

was by all means the best of those that gave much space to introductory

remarks, found himself left with only 600 words in which to tell what it

was that he had been introducing. The majority of the contestants appear to

have faced the same question as to subject matter which the Judges faced,

and to have reached the same decision. They accordingly devoted their

attention toward the prize, rather than toward the production of an essay

that would best supplement that of the winner. It is for this very reason that,

in these preliminary chapters, so large a proportion of the material has had

to be supplied by the Editor; and this very circumstance is a tribute to the

good judgment of the competitors, rather than ground for criticism of their

work.

The general introduction of Chapter II. out of the way, Chapters III. and IV.

take up the business of leading the reader into the actual subject of

relativity. The subject is here developed in what may be called the historical

order—the order in which it took form in Einstein’s own mind. Both in and

outside the contest of which this book is the outcome, a majority of those

who have written on relativity have followed this order, which is indeed a

very natural one and one well calculated to give to the rather surprising

assumptions of relativity a reasonableness which they might well appear to

[vii]

the lay mind to lack if laid down more arbitrarily. In these two chapters no

effort is made to carry the argument beyond the formulation of the Special

Principle of the relativity of uniform motion, but this principle is developed

in considerably more detail than would be the case if it were left entirely to

the competing essayists. The reason for this is again that we are dealing

with a phase of the subject which is of subordinate importance so far as a

complete statement of the General Theory of Relativity is concerned, but

which is of the greatest significance in connection with the effort of the

layman to acquire the proper preliminary orientation toward the larger

subject.

Chapter V. goes back again to general ground. Among the ideas which the competing essayists were forced to introduce into their text on a liberal scale is that of non-Euclidean geometry. The entire formulation of the General Theory of Relativity is in fact an exercise in this. The essayists— good, bad and indifferent alike—were quite unanimous in their decision that this was one thing which the reader would have to assume the responsibility of acquiring for himself. Certainly they were justified in this; for the Editor has been able to explain what non-Euclidean geometry is

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only by using up considerably more space than the contestants had for an

entire essay. No effort has been made to set forth any of the details of any

of the various non-Euclidean geometries; it has simply been the aim to

draw the dividing line between Euclidean and non-Euclidean, and to make

the existence of the latter appear reasonable, so that when the essayists

[viii]

come to talk about it the reader will not feel hopelessly at sea. In other

words, this is another case of providing the mental background, but on such

a scale that it has seemed necessary to give a separate chapter to it.

Chapter VI. completes the preliminary course in the fundamentals of relativity by tying up together the findings of Chapter V. and those of Chapters III. and IV. It represents more or less of a last-minute change of plan; for while it had been the Editor’s intent from the beginning to place the material of Chapters II.–V. in its present position, his preliminary impression would have been that the work of the present Chapter VI. would be adequately done by the essayists themselves. His reading of the essays, however, convinced him that it had not so been done—that with the possible exception of Mr. Francis, the essayists did not make either a serious or a successful effort to show the organic connection between the Special Theory of Relativity and the Minkowski space-time structure, or the utter futility of trying to reconcile ourselves to the results of the former without employing the ideas of the latter. So Chapter VI. was supplied to make good this deficiency, and to complete the mental equipment which the reader requires for his battle with the General Theory.

In laying down a set of general principles to govern the award of the prize,

one of the first things considered by the Judges was the relative importance

of the Special and the General Theories. It was their opinion that no essay

could possibly qualify for the prize which did not very distinctly give to the

[ix]

General Theory the center of the stage; and that in fact discussion of the

Special Theory was pertinent only so long as it contributed, in proportion to

the space assigned it, to the attack upon the main subject. The same

principle has been employed in selecting essays for complete or

substantially complete reproduction in this volume. Writers who dealt with

the Special Theory in any other sense than as a preliminary step toward the

General Theory have been relegated to the introductory chapters, where

such excerpts from their work have been used as were found usable. The

distinction of publication under name and title is reserved for those who

wrote consistently and specifically upon the larger subject—with the one

exception of Dr. Russell, whose exposition of the Special Theory is so far

the best of those submitted and at the same time so distinctive that we have

concluded it will appear to better advantage by itself than as a part of

Chapters III. and IV.

Following after Mr. Bolton’s essay we have tried to arrange the various contributions, not at all in any order of merit, but in the order that will make connected reading of the book most nearly possible and profitable. Each essay should be made easier of reading by the examination of those preceding it; at the same time each, by the choice of ground covered and by the emphasis on points not brought out sharply by its predecessors, should throw new light upon these predecessors.

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The reader will find that no two of the essays given thus in full duplicate or

even come close to duplicating one another. They have of course been

[x]

selected with this in view; each represents the best of several essays of

substantially the same character. Not all of them require comment here, but

concerning some of them a word may well be said.

Mr. Francis, we believe, has succeeded in packing more substance into his 3,000 words than any other competitor. Mr. Elliot has come closer than anybody else to really explaining relativity in terms familiar to everybody, without asking the reader to enlarge his vocabulary and with a minimum demand so far as enlarging his mental outlook is concerned. Were it not for certain conspicuous defects, his essay would probably have taken the prize. In justice to the Judges, we should state that we have taken the liberty of eliminating Mr. Elliot’s concluding paragraph, which was the most objectionable feature of his essay.

Dr. Dushman chose for his title the one which we adopted for this book. It became necessary, therefore, for us to find a new title for his essay; aside from this instance, the main titles appearing at the heads of the various complete essays are those of the authors. The subtitles have in practically every instance been supplied editorially.

Dr. Pickering submitted two essays, one written from the viewpoint of the

physicist, the other from that of the astronomer. To make each complete, he

naturally found it necessary to duplicate between them certain introductory

and general material. We have run the two essays together into a single

narrative, with the elimination of this duplicated material; aside from this

blue-penciling no alteration has been made in Dr. Pickering’s text. This text

however served as the basis of blue-penciling that of several other

[xi]

contestants, as indicated in the foot notes.

For the reader who is qualified or who can qualify to understand it, Dr. Murnaghan’s essay is perhaps the most illuminating of all. Even the reader who does not understand it all will realize that its author brings to the subject a freshness of viewpoint and an originality of treatment which are rather lacking in some of the published essays, and which it will readily be understood were conspicuously lacking in a good many of the unpublished ones. Dr. Murnaghan of all the competitors has come closest to making a contribution to science as well as to the semi-popular literature of science.

In the composite chapters, the brackets followed by reference numbers have been used as the most practicable means of identifying the various individual contributions. We believe that this part of the text can be read without allowing the frequent occurrence of these symbols to distract the eye. As to the references themselves, the asterisk marks the contributions of the Editor. The numbers are those attached to the essays in order of and at the time of their receipt; it has been more convenient to use these than to assign consecutive numbers to the quoted essays. The several numbers identify passages from the essays of the following contestants:

10: Frederick W. Shurlock, Derby, England.

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18: L. L. Whyte, Cambridge, England.

24: Prof. Moritz Schlick, University of Rostock, Germany.

30: C. E. Rose, M.E., Little Rock, Ark.

33: H. Gartelmann, Bremen, Germany.

35: Prof. Joseph S. Ames, Johns Hopkins University, Baltimore.

47: James O. G. Gibbons, East Orange. N. J.

82: Charles H. Burr, Philadelphia.

101: L. F. H. de Miffonis. B.A., C.E., Ottawa, Canada.

102: Charles A. Brunn, Kansas City.

[xii]

106: J. Elias Fries, Fellow A.I.E.E., Birmingham, Ala.

114: Dean W. P. Graham, Syracuse University, Syracuse, N. Y.

115: Rev. George Thomas Manley, London.

116: Prof. J. A. Schouten, Delft, Netherlands.

121: Elwyn F. Burrill, Berkeley, Cal.

125: Dorothy Burr, Bryn Mawr, Pa.

130: C. W. Kanolt, Bureau of Standards, Washington.

135: Robert Stevenson, New York.

139: Leopold Schorsch, New York.

141: Dr. M. C. Mott-Smith, Los Angeles, Calif.

147: Edward A. Clarke, Columbus, O.

149: Edward A. Partridge, Philadelphia.

150: Col. John Millis, U. S. A., Chicago.

152: George F. Marsteller, Detroit.

156: D. B. Hall, Cincinnati.

165: Francis Farquhar, York, Pa.

178: Dr. George de Bothezat, Dayton, O.

179: Professor A. E. Caswell, University of Oregon, Eugene, Ore.

182: C. E. Dimick, New London, Conn.

186: Earl R. Evans, Washington, D. C.

188: Norman E. Gilbert, Dartmouth College, Hanover, N. H.

192: A. d’Abro. New York.

194: L. M. Alexander, Cincinnati.

197: Kenneth W. Reed, East Cleveland, O.

198: Prof. E. N. da C. Andrade, Ordnance College, Woolwich, England.

216: Professor Andrew H. Patterson, University of North Carolina, Chapel Hill, N. C.

220: Prof. Arthur Gordon Webster, Clark College, Worcester, Mass.

221: Walter van B. Roberts, Princeton University, N. J.

223: Paul M. Batchelder, Austin, Tex.

227: Prof. R. W. Wood, Johns Hopkins University, Baltimore.

229: E. P. Fairbairn, M.C., B.Sc., Glasgow.

231: R. F. Deimel, Hoboken, N. J.

232: Lieut. W. Mark Angus, U. S. N., Philadelphia.

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235: Edward Adams Richardson, Kansas City. 263: Prof. William Benjamin Smith, Tulane University, New Orleans. 264: James Rice, University of London, London. 267: William Hemmenway Pratt, Lynn, Mass. 272: R. Bruce Lindsay, New Bedford, Mass. 283: Frank E. Law, Montclair, N. J.

In addition to the specific credit given by these references for specifically

quoted passages, the Editor feels that he ought to acknowledge his general

indebtedness to the competing essayists, collectively, for the many ideas

which he has taken away from their text to clothe in his own words. This

does not mean that the Editor has undertaken generally to improve upon the

language of the competitors, but merely that the reading of all their essays

has given him many ideas of such complex origin that he could not assign

credit if he would.

[xiii]

Table of Contents

I.—The Einstein $5,000 Prize: How the Contest Came to be Held, and

Some of the Details of Its Conduct. By the Editor

1

II.—The World—And Us: An Introductory Discussion of the Philosophy of

Relativity, and of the Mechanism of our Contact with Time and Space. By

various contributors and the Editor

19

III.—The Relativity of Uniform Motion: Classical Ideas on the Subject; the

Ether and the Apparent Possibility of Absolute Motion; the Michelson-

Morley Experiment and the Final Negation of this possibility. By various

contributors and the Editor

47

IV.—The Special Theory of Relativity: What Einstein’s Study of Uniform

Motion Tells Us About Time and Space and the Nature of the External

Reality. By various contributors and the Editor

76

V.—That Parallel Postulate: Modern Geometric Methods; the Dividing Line

Between Euclidean and Non-Euclidean; and the Significance of the Latter.

By the Editor

111

VI.—The Space-Time Continuum: Minkowski’s World of Events, and the

Way in Which It Fits Into Einstein’s Structure. By the Editor and a few

contributors

141

VII.—Relativity: The Winning Essay in the Contest for the Eugene Higgins

$5,000 Prize. By Lyndon Bolton, British Patent Office, London

169

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VIII.—The New Concepts of Time and Space: The Essay in Behalf of

Which the Greatest Number of Dissenting Opinions Have Been Recorded.

By Montgomery Francis, New York

181

IX.—The Principle of Relativity: A Statement of What it is All About, in

Ideas of One Syllable. By Hugh Elliot, Chislehurst, Kent, England

195

X.—Space, Time and Gravitation: An Outline of Einstein’s Theory of

General Relativity. By W. de Sitter, University of Leyden

206

XI.—The Principle of General Relativity: How Einstein, to a Degree Never

Before Equalled, Isolates the External Reality from the Observer’s

Contribution. By E. T. Bell, University of Seattle

218

[xiv]

XII.—Force Vs. Geometry: How Einstein Has Substituted the Second for

the First in Connection with the Cause of Gravitation. By Saul Dushman,

Schenectady

230

XIII.—An Introduction to Relativity: A Treatment in which the

Mathematical Connections of Einstein’s Work are Brought Out More

Strongly and More Successfully than Usual in a Popular Explanation. By

Harold T. Davis, University of Wisconsin

240

XIV.—New Concepts for Old: What the World Looks Like After Einstein

Has Had His Way with It. By John G. McHardy, Commander R. N.,

London

251

XV.—The New World: A Universe in Which Geometry Takes the Place of

Physics, and Curvature that of Force. By George Frederick Hemens, M.C.,

B.Sc., London

265

XVI.—The Quest of the Absolute: Modern Developments in Theoretical

Physics, and the Climax Supplied by Einstein. By Dr. Francis D.

Murnaghan, Johns Hopkins University, Baltimore

276

XVII.—The Physical Side of Relativity: The Immediate Contacts between Einstein’s Theories and Current Physics and Astronomy. By Professor William H. Pickering, Harvard College Observatory, Mandeville, Jamaica
287

XVIII.—The Practical Significance of Relativity: The Best Discussion of

the Special Theory Among All the Competing Essays. By Prof. Henry

Norris Russell, Princeton University

306

XIX.—Einstein’s Theory of Relativity: A Simple Explanation of His

Postulates and Their Consequences. By T. Royds, Kodaikanal Observatory,

India

318

XX.—Einstein’s Theory of Gravitation: The Discussion of the General

Theory and Its Most Important Application, from the Essay by Prof.

W. F. G. Swann, University of Minnesota, Minneapolis

327

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XXI.—The Equivalence Hypothesis: The Discussion of This, With Its

Difficulties and the Manner in Which Einstein Has Resolved Them, from

the Essay by Prof. E. N. da C. Andrade, Ordnance College, Woolwich,

England

334

XXII.—The General Theory: Fragments of Particular Merit on This Phase

of the Subject. By Various Contributors

338

Table of Contents

PREFACE

iii

Table of Contents

xiii

I. THE EINSTEIN $5,000 PRIZE

1

The Donor and the Prize

2

The Judges

5

Three Thousand Words

7

The Competing Essays

9

Looking for the Winner

12

The Winner of the Prize

16

II. THE WORLD—AND US

19

Getting Away from the Greek Ideas

21

Relativism and Reality

23

Laws of Nature

26

Concepts and Realities

29

The Concepts of Space and Time

33

The Reference Frame for Space

36

Time and the Coordinate System

38

The Choice of a Coordinate Frame

41

III. THE RELATIVITY OF UNIFORM MOTION

46

Who Is Moving?

48

Mechanical Relativity

50

The Search for the Absolute

52

The Ether and Absolute Motion

55

The Earth and the Ether

57

A Journey Upstream and Back

58

The Michelson-Morley Experiment

60

The Verdict

63

The “Contraction” Hypothesis

65

Taking the Bull by the Horns

68

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Questions of Common Sense

71

Shifting the Mental Gears

72

IV. THE SPECIAL THEORY OF RELATIVITY

76

Light and the Ether

78

The Measurement of Time and Space

80

The Problem of Communication

83

An Einsteinian Experiment

86

Who Is Right?

89

The Relativity of Time and Space

91

Relativity and Reality

95

Time and Space in a Single Package

98

Some Further Consequences

100

Assumption and Consequence

104

Relativity and the Layman

106

Physics vs. Metaphysics

109

V. THAT PARALLEL POSTULATE

111

Terms We Cannot Define

113

Laying the Foundation

115

The Rôle of Geometry

119

What May We Take for Granted?

122

And What Is It All About?

124

Euclid’s Geometry

126

Axioms Made to Order

128

Locating the Discrepancy

130

What the Postulate Really Does

132

The Geometry of Surfaces

133

Euclidean or Non-Euclidean

137

VI. THE SPACE-TIME CONTINUUM

141

The Four-Dimensional World of Events

144

A Continuum of Points

146

The Continuum in General

148

Euclidean and Non-Euclidean Continua

150

Our World of Four Dimensions

155

The Curvature of Space-Time

158

The Question of Visualization

162

What It All Leads To

165

VII. RELATIVITY

169

The Mechanical Principle of Relativity

170

The Special Principle of Relativity

171

The Four Dimensional Continuum

173

Gravitation and Acceleration

174

The General Principle of Relativity

177

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VIII. THE NEW CONCEPTS OF TIME AND SPACE

181

A World of Points

183

The Four-Dimensional World of Events

185

Successive Steps Toward Generality

187

Gravitation and Acceleration

189

Einstein’s Time-Space World

191

The Layman’s Last Doubt

193

IX. THE PRINCIPLE OF RELATIVITY

195

The Behavior of Light

197

Space and Time

198

The World of Reality

201

Accelerated Motion

203

X. SPACE, TIME AND GRAVITATION

206

The External World and its Geometry

208

Gravitation and its Place in the Universe

211

Gravitation and Space-Time

214

XI. THE PRINCIPLE OF GENERAL RELATIVITY

218

Gravitation and Acceleration

220

Paths Through the World of Four Dimensions

223

The Universe of Space-Time

225

XII. FORCE VS. GEOMETRY

230

The Relativity of Uniform Motion

233

Universal Relativity

235

The Geometry of Gravitation

237

XIII. AN INTRODUCTION TO RELATIVITY

240

The Electromagnetic Theory of Light

241

The Michelson-Morley Experiment

243

The Lorentz Transformation

245

The First Theory of Relativity

246

The Inclusion of Gravitation

248

XIV. NEW CONCEPTS FOR OLD

251

The World-Frame

253

The World-Fabric

257

Einstein’s Results

261

XV. THE NEW WORLD

265

The World Geometry

267

The Genesis of the Theory

270

The Time Diagram

273

XVI. THE QUEST OF THE ABSOLUTE

276

The Gravitational Hypothesis

281

The Special Relativity Theory

284

XVII. THE PHYSICAL SIDE OF RELATIVITY

287

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XVIII. THE PRACTICAL SIGNIFICANCE OF RELATIVITY 306

The Special Theory and Its Surprising Consequences 309

The Generalization

313

The Tests

315

XIX. EINSTEIN’S THEORY OF RELATIVITY

318

XX. EINSTEIN’S THEORY OF GRAVITATION

327

XXI. THE EQUIVALENCE HYPOTHESIS

334

XXII. THE GENERAL THEORY

338

[1]

I.
THE EINSTEIN $5,000 PRIZE
How the Contest Came to be Held, and Some of the Details of its Conduct
BY THE EDITOR
In January, 1909, an anonymous donor who was interested in the spread of correct scientific ideas offered through the Scientific American a prize of $500 for the best essay explaining, in simple non-technical language, that paradise of mathematicians and bugaboo of plain ordinary folk—the fourth dimension. Many essays were submitted in this competition, and in addition to that of the winner some twenty were adjudged worthy of ultimate publication. It was felt that the competition had added distinctly to the popular understanding of this significant subject; that it had done much to clear up popular misconception of just what the mathematician means when he talks of four or even more dimensions; and that it had therefore been as successful as it was unusual in character.
In November, 1919, the world was startled by the announcement from London that examination of the photographs taken during the total solar eclipse of May 29th had been concluded, and that predictions based upon the Einstein theories of relativity had been verified. In the reaction from the long surfeit of war news an item of this sort was a thoroughly journalistic one. Long cable dispatches were carried in the news columns all over the world; Einstein and his theories were given a prominent place on the front pages day after day; leading scientists in great number were called upon to tell the public through the reportorial medium just what the excitement was all about, just in what way the classical scientific structure had been overthrown.
Instead of being a mere nine days’ wonder, the Einstein theories held their place in the public mind. The more serious periodicals devoted space to
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them. First and last, a very notable group of scientific men attempted to explain to the general reader the scope and content of Einstein’s system. These efforts, well considered as they were, could be no more than partially successful on account of the very radical character of the revisions which the relativity doctrine demands in our fundamental concepts. Such revisions cannot be made in a day; the average person has not the viewpoint of the mathematician which permits a sudden and complete exchange of one set of fundamentals for another. But the whole subject had caught the popular attention so strongly, that even complete initial failure to discover what it was all about did not discourage the general reader from pursuing the matter with determination to come to some understanding of what had happened to Newton and Newtonian mechanics.

The Donor and the Prize
In May, 1920, Mr. Eugene Higgins, an American citizen long resident in Paris, a liberal patron of the arts and sciences, and a lifelong friend of the Scientific American and its proprietors, suggested that the success of the Fourth Dimension Prize Contest of 1910 had been so great that it might be desirable to offer another prize in similar fashion for the best popular essay on the Einstein theories. He stated that if in the opinion of the Scientific American these theories were of sufficient importance, and the probability of getting a good number of meritorious essays were sufficiently great, and the public need and desire for enlightenment were sufficiently present, he would feel inclined to offer such a prize, leaving the conduct of the contest to the Scientific American as in the former event. It was the judgment of the editors of the Scientific American that all these provisos should be met with an affirmative, and that Mr. Higgins accordingly could with propriety be encouraged to offer the prize.
In his preliminary letter Mr. Higgins had suggested that in view of the apparent greater importance of the subject to be proposed for discussion by the contestants of 1920, the prize offered should probably be more liberal than in the former instance. This view met with the approval of the editors as well; but they were totally unprepared for the receipt, late in June, of a cablegram from Mr. Higgins stating that he had decided to go ahead with the matter, and that he was forwarding a draft for $5,000 to represent the amount of the prize. Such a sum, exceeding any award open to a professional man with the single exception of the Nobel Prize, for which he cannot specifically compete, fairly took the breath of the Editors, and made it immediately clear that the contest would attract the widest attention, and that it should score the most conspicuous success. It also made it clear that the handling of the contest would be a more serious matter than had been anticipated.
In spite of the fact that it would not for some time be possible to announce the identity of the Judges, it was felt that the prospective contestants should
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have every opportunity for extensive preparation; so the contest was announced, and the rules governing it printed as far as they could be determined on such short shrift, in the Scientific American for July 10, 1920. Several points of ambiguity had to be cleared up after this initial publication. In particular, it had been Mr. Higgins’ suggestion that in the very probable event of the Judges’ inability to agree upon the winning essay, the prize might, at their discretion, be divided between the contributors of the best two essays. This condition was actually printed in the first announcement, but the Post Office Department insisted upon its withdrawal, on the ground that with it in force the contestant would not know whether he were competing for $5,000 or for $2,500, and that this would introduce the “element of chance” which alone was necessary, under the Federal statutes, to make the contest a lottery. So this provision was replaced by one to the effect that in the event the Judges were not able to agree, the Einstein Editor should cast the deciding vote between the essays respectively favored by them.

The announcement attracted the widest attention, and was copied in

newspapers and magazines all over the world. Inquiries poured in from all

quarters, and the Einstein Editor found it almost impossible to keep himself

[5]

supplied with proofs of the conditions and rules to mail in response to these

inquiries. It was immediately clear that there was going to be a large

number of essays submitted, and that many distinguished names would be

listed among the competitors.

The Judges
In the Scientific American for September 18, announcement was carried in the following words:
“We are assured with complete certainty that the competition for the fivethousand-dollar prize will be very keen, and that many essays will be submitted which, if they bore the names of their authors, would pass anywhere as authoritative statements. The judges will confront a task of extraordinary difficulty in the effort to determine which of these efforts is the best; and we believe the difficulties are such that multiplication of judges would merely multiply the obstacles to an agreement. It is altogether likely that the initial impressions of two or three or five judges would incline toward two or three or five essays, and that any final decision would be attainable only after much consultation and discussion. It seems to us that by making the committee as small as possible while still preserving the necessary feature that its decision represent a consensus, we shall simplify both the mental and the physical problem of coming to an agreement. We believe that the award should if possible represent a unanimous decision, without any minority report, and that such a requirement is far more likely to be met among two men than among three or five. At the same time, the bringing together of two men and the details of general administration of their work together are far simpler than if there were three or five. So we
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have finally decided to have but two judges, and in this we have the endorsement of all the competent opinion that we have consulted.

“The gentlemen who have consented to act as Judges are Professors Leigh Page and Edwin Plimpton Adams, of the departments of physics of Yale and Princeton Universities, respectively. Both are of the younger generation of physicists that has paid special attention to those phases of mathematics and physics involved in the Einstein theories, and both have paid special attention to these theories themselves. We are gratified to be able to put forward as Judges two men so eminently qualified to act. We feel that we may here appropriately quote Professor Page, who says in his acceptance: ‘As the large prize offers a great inducement, I had thought of entering the contest. However I realize that not many people in this country have made a considerable study of Einstein’s theory, and if all who have should enter the contest, it would be difficult to secure suitable Judges.’ Without any desire to put the gentleman in the position of pleading for himself, we think this suggests very well the extent to which the Scientific American, the contestants, and the public at large, are indebted to Professors Page and Adams for their willingness to serve in the difficult capacity of Judges.”

It might appropriately have been added to this announcement that it was

[7]

altogether to the credit of science and the scientific spirit that the first two

gentlemen approached with the invitation to act as Judges were willing to

forego their prospects as contestants in order thus to contribute to the

success of the contest.

Three Thousand Words
Of the conditions, the one which evoked most comment was that stating the word limit. This limit was decided upon after the most careful discussion of the possibilities of the situation. It was not imagined for a moment that any contestant would succeed in getting within 3,000 words a complete discussion of all aspects of the Special and the General Theories of Relativity. It was however felt that for popular reading a single essay should not be much if any longer than this. Moreover, I will say quite frankly that we should never have encouraged Mr. Higgins to offer such a prize if we had supposed that the winning essay was the only thing of value that would come from the contest, or if we had not expected to find in many of the other essays material which would be altogether deserving of the light. From the beginning we had in view the present volume, and the severe restriction in length was deliberately imposed for the purpose of forcing every contestant to stick to what he considered the most significant viewpoints, and to give his best skill to displaying the theories of Einstein to the utmost advantage from these viewpoints. We felt that divergent viewpoints would be more advantageously treated in this manner than if we gave each contestant enough space to discuss the subject from all sides; and that the award of the prize to the essay which, among other requirements, seemed to the Judges to embody the best choice of material, would greatly
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simplify the working of the contest without effecting any injustice against those contestants who displayed with equal skill less happily chosen material. Perhaps on this point I may again quote with profit the editorial page of the Scientific American:

“An essay of three thousand words is not long enough to lose a reader more than once; if it does lose him it is a failure, and if it doesn’t it is a competitor that will go into the final elimination trials for the prize. If we can present, as a result of the contest, six or a dozen essays of this length that will not lose the lay reader at all, we shall have produced something amply worth the expenditure of Mr. Higgins’ money and our time. For such a number of essays of such character will of necessity present many different aspects of the Einstein theories, and in many different ways, and in doing so will contribute greatly to the popular enlightenment.

“Really the significant part of what has already appeared is not the part that

is intelligible, but rather the part that, being unintelligible, casts the shadow

of doubt and suspicion on the whole. The successful competitor for the

prize and his close contestants will have written essays that, without any

claim to completeness, will emphasize what seems to each author the big

outstanding feature; and every one of them will be intelligible. Together

they will in all probability be reasonably complete, and will retain the

[9]

individual characteristic of intelligibility. They will approach the various

parts of the field from various directions—we could fill this page with

suggestions as to how the one item of the four-dimensional character of

Einstein’s time-space might be set forth for the general reader. And when a

man must say in three thousand words as much as he can of what eminent

scientists have said in whole volumes—well, the result in some cases will

be sheer failure, and in others a product of the first water. The best of the

essays will shine through intelligent selection of what is to be said, and

brilliant success in saying it. It is to get a group of essays of this character,

not to get the single essay which will earn the palm, that the prize is

offered.”

The Competing Essays
At all times after the first announcement the Einstein Editor had a heavy correspondence; but the first real evidence that the contest was under way came with the arrival of the first essay, which wandered into our office in the middle of September. About a week later they began to filter in at the rate of one or two per day—mostly from foreign contestants who were taking no chances on the mails. Heavy returns did not commence until about ten days before the closing date. The great avalanche, however, was reserved for the morning of Monday, November 1st. Here we had the benefit of three days’ mail; there were about 120 essays. Among those which were thrown out on the ground of lateness the honors should no doubt go to the man who mailed his offering in The Hague on October 31st.
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Essays were received in greater quantity from Germany than from any other foreign country, doubtless because of the staggering value of $5,000 when converted into marks at late 1920 rates. England stood next on the list; and one or more essays were received from Austria, Czechoslovakia, Jugoslavia, France, Switzerland, the Netherlands, Denmark, Italy, Chile, Cuba, Mexico, India, Jamaica, South Africa and the Fiji Islands. Canada, of course, contributed her fair share; and few of our own states were missing on the roll-call.

The general level of English composition among the essays from nonEnglish-speaking sources was about what might have been expected. A man may have a thorough utilitarian knowledge of a foreign tongue, but when he attempts intensive literary competition with a man who was brought up in that tongue he is at a disadvantage. We read French and German with ease and Spanish and Italian without too much difficulty, ourselves; we should never undertake serious writing in any of these languages. Not many of the foreign contributions, of course, were as ludicrous as the one we quote to some extent in our concluding chapter, but most of them were distinctly below par as literary compositions. Drs. De Sitter and Schlick were the notable exceptions to this; both showed the ability to compete on a footing of absolute equality with the best of the native product.

We dare say it was a foregone conclusion that many essays should have

[11]

been over the limit, and that a few should have been over it to the point of

absurdity. The winning essay contains 2,919 words, plus or minus a

reasonable allowance for error in counting; that it should come so far from

being on the ragged edge should be sufficient answer to those who

protested against the severity of the limitation. One inquirer, by the way,

wanted to know if 3,000 words was not a misprint for 30,000. Another

contestant suggested that instead of disqualifying any essay that was over

the line, we amputate the superfluous words at the end. This was a plausible

enough suggestion, since any essay able to compete after such amputation

must necessarily have been one of extreme worth; but fortunately we did

not have to decide whether we should follow the scheme. Perhaps twenty of

the essays submitted were so seriously in excess of the limit that it was not

even necessary to count their words in detail; most of these offenders ran to

3,500 words or thereabouts, and one—a good one, too, from which we use

a good deal of material in this volume—actually had 4,700. On the other

extreme were a few competitors who seemed to think that the shortest essay

was necessarily the best, and who tried to dismiss the subject with 500 or

1,000 words.

By a curious trick of chance there were submitted in competition for the prize exactly 300 essays. Of course a few of these did not require serious consideration—this is inevitable in a contest of such magnitude. But after excluding all the essays that were admittedly not about the Einstein theories at all, and all those whose English was so execrable as to make them quite out of the question, and all those which took the subject so lightly as not to write reasonably close to the limit of 3,000 words, and all those which were given over to explanation of the manner in which Einstein’s theories verify those of the writer, and all those in which the writer attempted to substitute
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his own cosmic scheme for Einstein’s—after all this, there remained some 275 essays which were serious efforts to explain in simple terms the nature and content and consequences of Special and General Relativity.

Looking for the Winner
The Einstein Editor was in sufficiently close touch with the details of the adjudication of the essays to have every realization of the difficulty of this work. The caliber of the essays submitted was on the whole high. There were many which would have been well worthy of the prize in the absence of others that were distinctly better—many which it was not possible to eliminate on the ground of specific faults, and which could only be adjudged “not the best” by detailed comparison with specific other essays. It was this detailed comparison which took time, and which so delayed the award that we were not able to publish the winning essay any sooner than February 5th. Especially difficult was this process of elimination after the number of surviving essays had been reduced to twenty or less. The advantages of plan possessed by one essay had to be weighed against those of execution exhibited in another. A certain essay had to be critically compared with another so like it in plan that the two might have been written from a common outline, and at the same time with a third as unlike it in scope and content as day and night. And all the time there was present in the background the consciousness that a prize of $5,000 hung upon the decision to be reached. For anyone who regards this as an easy task we have no worse wish than that he may some day have to attack a similar one.
We had anticipated that the bulk of the superior essays would be among those received during the last day or two of the contest; for we felt that the men best equipped to attack the subject would be the most impressed with its seriousness. Here we were quite off the track. The seventeen essays which withstood most stubbornly the Judges’ efforts at elimination were, in order of receipt, numbers 8, 18, 28, 40, 41, 43, 92, 95, 97, 130, 181, 194, 198, 223, 267, 270, 275: a fairly even distribution. The winner was the 92nd essay received.
The Judges held their final meeting in the editorial office on January 18, 1921. The four essays which were before the committee at the start of the session were speedily cut to three, and then to two; and after an all-day session the Judges found themselves conscientiously able to agree on one of these as the best. This unanimity was especially gratifying, the more so since it by no means was to be confidently expected, on a priori grounds, that it would be possible of attainment. Even the Einstein Editor, who might have been called upon for a final decision but wasn’t, can hardly be classed as a dissenter; for with some slight mental reservations in favor of the essay by Mr. Francis which did not enter the Judges’ final discussion at all, and which he rather suspects appeals more to his personal taste than to his soundest judgment, he is entirely in accord with the verdict rendered.
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The fact that the prize went to England was no surprise to those acquainted with the history of Einstein’s theories. The Special Theory, promulgated fifteen years ago, received its fair share of attention from mathematicians all over the world, and is doubtless as well known and as fully appreciated here as elsewhere. But it has never been elevated to a position of any great importance in mathematical theory, simply because of itself, in the absence of its extension to the general case, it deserves little importance. It is merely an interesting bit of abstract speculation.

The General Theory was put out by Einstein in finished form during the

war. Owing to the scientific moratorium, his paper, and hence a clear

understanding of the new methods and results and of the sweeping

consequences if the General Theory should prevail, did not attain general

circulation outside Germany until some time in 1918 or even later. Had it

not been for Eddington it is doubtful that the British astronomers would

have realized that the eclipse expeditions were of particular consequence.

Therefore at the time of these expeditions, and even as late as the

November announcement of the findings, the general body of scientific

men in America had not adequately realized the immense distinction

between the Special and the General Theories, had not adequately

appreciated that the latter led to distinctive consequences of any import, and

[15]

we fear in many cases had not even realized explicitly that the deflection of

light and the behavior of Mercury were matters strictly of the General and

in no sense of the Special Theory. Certainly when the American newspapers

were searching frantically for somebody to interpret to their public the great

stir made by the British announcement that Einstein’s predictions had been

verified, they found no one to do this decently; nor were our magazines

much more successful in spite of the greater time they had to devote to the

search. In a word, there is not the slightest room for doubt that American

science was in large measure caught asleep at the switch—perhaps for no

reason within its control; and that American writers were in no such

favorable case to write convincingly on the subject as were their British and

continental contemporaries.

So it was quite in accord with what might have been expected to find, on

opening the identifying envelopes, that not alone the winning essay, but its

two most immediate rivals, come from members of that school of British

thought which had been in contact with the Einstein theories in their

entirety for two years longer than the average American of equal

competence. This riper familiarity with the subject was bound to yield riper

fruit. Indeed, had it not been for the handicap of writing in a strange

language, it is reasonable to assume that the scientists of Germany would

have made a showing superior to that of either Americans or British—and

for the same reason that Britain showed to better advantage than America.

[16]

The Winner of the Prize
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Mr. Bolton, the winner of the big prize, we suppose may fairly be referred to as unknown in a strict scientific sense. Indeed, at the time of the publication of his essay in the Scientific American nothing could be learned about him on the American side of the water beyond the bare facts that he was not a young man, and that he had for a good many years occupied a position of rank in the British Patent Office. (It will be recalled that Einstein himself was in the Swiss Patent Office for some time.) In response to the request of the Scientific American for a brief biographical sketch that would serve to introduce him better to our readers, Mr. Bolton supplied such a concise and apparently such a characteristic statement that we can do no better than quote it verbatim.

“I was born in Dublin in 1860, but I have lived in England since 1869. My

family belonged to the landed gentry class, but I owe nothing to wealth or

position. I was in fact put through school and college on an income which a

workman would despise nowadays. After attending sundry small schools, I

entered Clifton College in 1873. My career there was checkered, but it

ended well. I was always fairly good at natural science and very fond of all

sorts of mechanical things. I was an honest worker but no use at classics,

and as I did practically nothing else for the first four years at Clifton, I

came to consider myself something of a dunce. But a big public school is a

little world. Everyone gets an opportunity, often seemingly by accident, and

it is up to him to take it. Mine did not come till I was nearly 17. As I was

[17]

intended for the engineering profession, I was sent to the military side of

the school in order to learn some mathematics, at which subject I was then

considered very weak. This was certainly true, as at that time I barely knew

how to solve a quadratic, I was only about halfway through the third book

of Euclid, and I knew no trigonometry. But the teaching was inspiring, and

I took readily to mathematics. One day it came out that I had been making

quite a good start with the differential calculus on my own without telling

anybody. After that all was well. I left Clifton in 1880 with a School

Exhibition and a mathematical scholarship at Clare College, Cambridge.

“After taking my degree in 1883 as a Wrangler, I taught science and mathematics at Wellington College, but I was attracted by what I had heard of the Patent Office and I entered it through open competition in 1885. During my official career I have been one of the Comptroller’s private secretaries and I am now a Senior Examiner. During the war I was attached to the Inventions Department of the Ministry of Munitions, where my work related mainly to anti-aircraft gunnery. I have contributed, and am still contributing to official publications on the subject.

“I have written a fair number of essays on various subjects, even on

literature, but my only extra-official publications relate to stereoscopic

photography. I read a paper on this subject before the Royal Photographic

Society in 1903 which was favorably noticed by Dr. von Rohr of Messrs.

Zeiss of Jena. I have also written in the Amateur Photographer.

[18]

“I have been fairly successful at athletics, and I am a member of the Leander Club.”

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That Mr. Bolton did not take the prize through default of serious competition should be plain to any reader who examines the text from competing essays which is to be found in this volume. The reference list of these competitors, too, supplemented by the names that appear at the heads of complete essays, shows a notable array of distinguished personalities, and I could mention perhaps a dozen more very well known men of science whose excellent essays have seemed a trifle too advanced for our immediate use, but to whom I am under a good deal of obligation for some of the ideas which I have attempted to clothe in my own language.

Before leaving the subject, we wish to say here a word of appreciation for

the manner in which the Judges have discharged their duties. The reader

will have difficulty in realizing what it means to read such a number of

essays on such a subject. We were fortunate beyond all expectation in

finding Judges who combined a thorough scientific grasp of the

mathematical and physical and philosophical aspects of the matter with an

extremely human viewpoint which precluded any possibility of an award to

an essay that was not properly a popular discussion, and with a willingness

to go to meet each other’s opinions that is rare, even among those with less

ground for confidence in their own views than is possessed by Drs. Page

and Adams.

[19]

II.
THE WORLD—AND US
An Introductory Discussion of the Philosophy of Relativity, and of the Mechanism of Our Contact with Time and Space
BY VARIOUS CONTRIBUTORS AND THE EDITOR
From a time beyond the dawn of history, mankind has been seeking to explain the universe. At first the effort did not concern itself further probably than to make a supposition as to what were the causes of the various phenomena presented to the senses. As knowledge increased, first by observation and later by experiment also, the ideas as to these causes passed progressively through three stages—the theological (the causes were thought to be spirits or gods); the metaphysical (the causes were thought in this secondary or intermediate stage to be some inherent, animating, energizing principles); and the scientific (the causes were finally thought of as simply mechanical, chemical, and magneto-electrical attractions and repulsions, qualities or characteristics of matter itself, or of the thing of which matter is itself composed.)
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With increase of knowledge, and along with the inquiry as to the nature of

causes, there arose an inquiry also as to what reality was. What was the

essential nature of the stuff of which the universe was made, what was

matter, what were things in themselves, what were the noumena (the

realities), lying back of the phenomena (the appearances)? Gradually ideas

[20]

explaining motion, force, and energy were developed. At the same time

inquiry was made as to the nature of man, the working of his mind, the

nature of thought, the relation of his concepts (ideas) to his perceptions

(knowledge gained through the sense) and the relations of both to the noumena (realities).]283

[The general direction taken by this inquiry has been that of a conflict

between two schools of thought which we may characterize as those of

absolutism and of relativism.]* [The ancient Greek philosophers believed

that they could tap a source of knowledge pure and absolute by sitting

down in a chair and reasoning about the nature of time and space, and the mechanism of the physical world.]221 [They maintained that the mind holds

in its own right certain concepts than which nothing is more fundamental.

They considered it proper to conceive of time and space and matter and the

other things presented to their senses by the world as having a real

existence in the mind, regardless of whether any external reality could be

identified with the concept as ultimately put forth. They could even dispute

with significance the qualities which were to be ascribed to this abstract

conceptual time and space and matter. All this was done without reference

to the external reality, often in defiance of that reality. The mind could

picture the world as it ought to be; if the recalcitrant facts refused to fit into

the picture, so much the worse for them. We all have heard the tale of how

generation after generation of Greek philosophers disputed learnedly why

and how it was that a live fish could be added to a brimming pail of water

[21]

without raising the level of the fluid or increasing the weight; until one day

some common person conceived the troublesome idea of trying it out

experimentally to learn whether it were so—and found that it was not. True

or false, the anecdote admirably illustrates the subordinate place which the

externals held in the absolutist system of Greek thought.]*

[Under this system a single observer is competent to examine a single phenomenon, and to write down the absolute law of nature by referring the results to his innate ideas of absolute qualities and states. The root of the word absolute signifies “taking away,” and in its philosophical sense the word implies the ability of the mind to subtract away the properties or qualities from things, and to consider these abstract qualities detached from the things; for example, to take away the coldness from ice, and to consider pure or abstract coldness apart from anything that is cold; or to take away motion from a moving body, and to consider pure motion apart from anything that moves. This assumed power is based upon the Socratic theory of innate ideas. According to this theory the mind is endowed by nature with the absolute ideas of hardness, coldness, roundness, equality, motion, and all other absolute qualities and states, and so does not have to learn them. Thus a Socratic philosopher could discuss pure or absolute being, absolute space and absolute time.]121

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Getting Away from the Greek Ideas
[This Greek mode of thought persisted into the late Middle Ages, at which time it was still altogether in order to dispose of a troublesome fact of the external world by quoting Aristotle against it. During the Renaissance, which intellectually at least marks the transition from ancient to modern, there came into being another type of absolutism, equally extreme, equally arbitrary, equally unjustified. The revolt against the mental slavery to Greek ideas carried the pendulum too far to the other side, and early modern science as a consequence is disfigured by what we must now recognize as gross materialism. The human mind was relegated to the position of a mere innocent bystander. The external reality was everything, and aside from his function as a recorder the observer did not in the least matter. The whole aim of science was to isolate and classify the elusive external fact. The rôle of the observer was in every possible way minimized. It was of course his duty to get the facts right, but so far as any contribution to these was concerned he did not count—he was definitely disqualified. He really played the part of an intruder; from his position outside the phenomena he was searching for the absolute truth about these phenomena. The only difference between his viewpoint and that of Aristotle was that the latter looked entirely inside himself for the elusive “truth,” while the “classical” scientist, as we call him now, looked for it entirely outside himself.
Let me illustrate the difference between the two viewpoints which I have discussed, and the third one which I am about to outline, by another concrete instance. The Greeks, and the medievals as well, were fond of discussing a question which embodies the whole of what I have been saying. This question involved, on the part of one who attempted to answer it, a choice between the observer and the external world as the seat of reality. It was put in many forms; a familiar one is the following: “If the wind blew down a great tree at a time and place where there was no conscious being to hear, would there be any noise?” The Greek had to answer this question in the negative because to him the noise was entirely a phenomenon of the listener. The classical scientist had to answer it in the affirmative because to him the noise was entirely a phenomenon of the tree and the air and the ground. Today we answer it in the negative, but for a very different reason from that which swayed the Greek. We believe that the noise is a joint phenomenon of the observer and the externals, so that in the absence of either it must fail to take existence. We believe there are sound waves produced, and all that; but what of it? There is no noise in the presence of the falling tree and the absence of the observer, any more than there would be in the presence of the observer and the absence of the tree and the wind; the noise, a joint phenomenon of the observer and the externals, exists only in their joint presence.
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Relativism and Reality
This is the viewpoint of relativism. The statue is golden for one observer and silver to the other. The sun is rising here and setting in another part of the world. It is raining here and clear in Chicago. The observer in Delft hears the bombardment of Antwerp and the observer in London does not. If they were to be consistent, both the Greek and the medieval-modern absolutist would have to dispute whether the statue were “really” golden or silver, whether the sun were “really” rising or setting, whether the weather were “really” fair or foul, whether the bombardment were “really” accompanied by loud noises or not; and on each of these questions they would have to come to an agreement or confess their methods inadequate. But to the relativist the answer is simple—whether this or that be true depends upon the observer. In simple cases we understand this full well, as we have always realized it. In less simple cases we recognize it less easily or not at all, so that some of our thought is absolutist in its tendencies while the rest is relativistic. Einstein is the first ever to realize this fully—or if not this, then the first ever to realize it so fully as to be moved toward a studied effort to free human thought from the mixture of relativism and absolutism and make it consistently the one or the other.
This brings it about that the observed fact occupies a position of unexpected significance. For when we discuss matters of physical science under a strictly relativistic philosophy, we must put away as metaphysical everything that smacks of a “reality” partly concealed behind our observations. We must focus attention upon the reports of our senses and of the instruments that supplement them. These observations, which join our perceptions to their external objects, afford us our only objective manifestations; them we must accept as final—subject always to such correction as more refined observations may suggest. The question whether a “true” length or area or mass or velocity or duration or temperature exists back of the numerical determination, or in the presence of a determination that is subject to correction, or in the absence of any determination at all, is a metaphysical one and one that the physicist must not ask. Length, area, mass, velocity, duration, temperature—none of these has any meaning other than the number obtained by measurement.]* [If several different determinations are checked over and no error can be found in any of them, the fault must lie not with the observers but with the object, which we must conclude presents different values to different observers.]33
[We are after all accustomed to this viewpoint; we do not demand that Pittsburgh shall present the same distance from New York and from Philadelphia, or that the New Yorker and the Philadelphian come to any agreement as to the “real” distance of Pittsburgh. The distance of Pittsburgh depends upon the position of the observer. Nor do we demand that the man who locates the magnetic pole in one spot in 1900 and in another in 1921
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come to a decision as to where it “really” is; we accept his statement that its position depends upon the time of the observation.

What this really means is that the distance to Pittsburgh and the position of

the magnetic pole are joint properties of the observer and the observed—

relations between them, as we might put it. This is obvious enough in the

case of the distance of Pittsburgh; it is hardly so obvious in the case of the

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position of the magnetic pole, varying with the lapse of time. But if we

reflect that the observation of 1900 and that of 1921 were both valid, and

both represented the true position of the pole for the observer of the date in

question, we must see that this is the only explanation that shows us the

way out.

I do not wish to speak too definitely of the Einstein theories in these introductory remarks, and so shall refrain from mentioning explicitly in this place the situation which they bring up and upon which what I have just said has direct bearing. It will be recognized when it arises. What must be pointed out here, however, is that we are putting the thing which the scientist calls the “observed value” on a footing of vastly greater consequence than we should have been willing offhand to concede to it. So far as any single observer is concerned, his own best observed values are themselves the external world; he cannot properly go behind the conditions surrounding his observations and speak of a real external world beyond these observations. Any world which he may think of as so existing is purely a conceptual world, one which for some reason he infers to exist behind the deceptive observations. Provided he makes this reservation he is quite privileged to speculate about this concealed world, to bestow upon it any characteristics that he pleases; but it can have no real existence for him until he becomes able to observe it. The only reality he knows is the one he can directly observe.

Laws of Nature
The observations which we have been discussing, and which we have been trying to endow with characteristics of “reality” which they are frequently not realized to possess, are the raw material of physical science. The finished product is the result of bringing together a large number of these observations.]* [The whole underlying thought behind the making of observations, in fact, is to correlate as many as possible of them, to obtain some generalization, and finally to express this in some simple mathematical form. This formulation is then called a “law of nature.”]35
[Much confusion exists because of a misunderstanding in the lay mind of what is meant by a “law of nature.” It is perhaps not a well chosen term. One is accustomed to associate the word law with the idea of necessity or compulsion. In the realm of nature the term carries no such meaning. The laws of nature are man’s imperfect attempts to explain natural phenomena; they are not inherent in matter and the universe, not an iron bar of necessity
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running through worlds, systems and suns. Laws of nature are little more than working hypotheses, subject to change or alteration or enlargement or even abandonment, as man’s vision widens and deepens. No sanctity attaches to them, and if any one, or all, of them fail to account for any part, or all, of the phenomena of the universe, then it or they must be supplemented or abandoned.]102

[The test of one of these laws is that it can be shown to include all the

related phenomena hitherto known and that it enables us to predict new

phenomena which can then be verified. If new facts are discovered that are

not in agreement with one of these generalized statements, the assumptions

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on which the latter is based are examined, those which are not in

accordance with the new facts are given up, and the statement is modified so as to include the new facts.]10 [And if one remembers that the laws of

physics were formerly based on a range of observations much narrower

than at present available, it seems natural that in the light of this widening

knowledge one law or another may be seen to be narrow and insufficient.

New theories and laws do not necessarily disprove old ones, but explain

certain discrepancies in them and penetrate more deeply into their

underlying principles, thereby broadening our ideas of the universe. To

follow the new reasoning we must rid ourselves of the prejudice behind the

old, not because it is wrong but because it is insufficient. The universe will not be distorted to fit our rules, but will teach us the rules of existence.]125

[Always, however, we must guard against the too easy error of attributing

to these rules anything like absolute truth.]* [The modern scientist has

attained a very business-like point of view toward his “laws of nature.” To

him a law is fundamentally nothing but a short-hand way of expressing the

results of a large number of experiments in a single statement. And it is

important to remember that this mere shortening of the description of a lot

of diverse occurrences is by no means any real explanation of how and why

they happened. In other words, the aim of science is not ultimately to

explain but only to discover the relations that hold good among physical

quantities and to embody all these relations in as few and as simple

physical laws as possible.]221 [This is inherently the method of

[29]

relativism.]* [Under it a set of phenomena is observed. There are two or

many observers, and they write down their several findings. These are

reviewed by a final observer or judge, who strains out the bias due to the

different viewpoints of the original observers. He then writes down, not any

absolute law of nature governing the observed phenomena, but a law as general as possible expressing their interrelations.]121 [And through this

procedure modern science and philosophy reveal with increasing emphasis

that we superimpose our human qualities on external nature to such an extent that]106 [we have at once the strongest practical justification, in

addition to the arguments of reason, for our insistence that the contact

between objective and subjective represented by the observation is the only

thing which we shall ever be able to recognize as real. We may indulge in

abstract metaphysical speculation to our heart’s content, if we be

metaphysically inclined; we may not attempt to impose the dicta of

metaphysics upon the physical scientist.]*

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Concepts and Realities
[From the inquiry and criticism which have gone on for centuries has emerged the following present-day attitude of mind toward the sum total of our knowledge. The conceptual universe in our minds in some mysterious way parallels the real universe, but is totally unlike it. Our conceptions (ideas) of matter, molecules, atoms, corpuscles, electrons, the ether, motion, force, energy, space, and time stand in the same or similar relation to reality as the x’s and y’s of the mathematician do to the entities of his problem. Matter, molecules, atoms, corpuscles, electrons, the ether, motion, force, energy, space, and time do not exist actually and really as we conceive them, nor do they have actually and really the qualities and characteristics with which we endow them. The concepts are simply representations of things outside ourselves; things which, while real, have an essential nature not known to us. Matter, molecules, atoms, corpuscles, electrons, the ether, motion, force, energy, space and time are merely devices, symbols, which enable us to reason about reality. They are parts of a conceptual mechanism in our minds which operates, or enables our minds to operate, in the same sequence of events as the sequence of phenomena in the external universe, so that when we perceive by our senses a group of phenomena in the external universe, we can reason out what result will flow from the interaction of the realities involved, and thus predict what the situation will be at a given stage in the sequence.
But while our conceptual universe has thus a mechanical aspect, we do not regard the real universe as mechanical in its nature.]283 [This may be illustrated by a little story. Entering his friend’s house, a gentleman is seized unawares from behind. He turns his head but sees nothing. His hat and coat are removed and deposited in their proper places by some invisible agent, seats and tables and refreshments appear in due time where they are required, all without any apparent cause. The visitor shivers with horror and asks his host for an explanation. He is then told that the ideas “order” and “regularity” are at work, and that it is they who acquit themselves so well of their tasks. These ideas cannot be seen nor felt nor seized nor weighed; they reveal their existence only by their thoughtful care for the welfare of mankind. I think the guest, coming home, will relate that his friend’s house is haunted. The ghosts may be kind, benevolent, even useful; yet ghosts they are. Now in Newtonian mechanics, absolute space and absolute time and force and inertia and all the other apparatus, altogether imperceptible, appearing only at the proper time to make possible a proper building up of the theory, play the same mysterious part as the ideas “order” and “regularity” in my story. Classical mechanics is haunted.]116
[As a matter of fact, we realize this and do not allow ourselves to be imposed upon with regard to the true nature of these agencies.]* [We use a mechanistic terminology and a mechanistic mode of reasoning only because we have found by experience that they facilitate our reasoning.
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They are the tools which we find produce results. They are adapted to our

minds, but perhaps it would be better to say that our minds are so

constructed as to render our conceptual universe necessarily mechanical in

its aspect in order that our minds may reason at all. Two things antithetic

are involved—subject (our perceiving mind which builds up concepts) and

object (the external reality); and having neither complete nor absolute

knowledge of either, we cannot affirm which is more truly to be said to be

mechanistic in its nature, though we may suspect that really neither is. We

no longer think of cause and effect as dictated by inherent necessity, we

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simply regard them as sequences in the routine of our sense-impressions of

phenomena. In a word, we have at length grasped the idea that our notions

of reality, at present at least, whatever they may become ultimately, are not

absolute, but simply relative. We see, too, that we do not explain the

universe, but only describe our perceptions of its contents.

The so-called laws of nature are simply statements of formulæ which resume or sum up the relationships and sequences of phenomena. Our effort is constantly to find formulæ which will describe the widest possible range of phenomena. As our knowledge increases, that is, as we perceive new phenomena, our laws or formulæ break down, that is, they fail to afford a
description in brief terms of all of our perceptions. It is not that the old laws are untrue, but simply that they are not comprehensive enough to include all of our perceptions. The old laws are often particular or limiting instances of the new laws.]283

[From what we have said of the reality of observations it follows that we

must support that school of psychology, and the parallel school of

philosophy, which hold that concepts originate in perceptions. But this does

not impose so strong a restriction upon conceptions as might appear. The

elements of all our concepts do come to us from outside; we manufacture

nothing out of whole cloth. But when perception has supplied a sufficient

volume of raw material, we may group its elements in ways foreign to

actual occurrence in the perceptual world, and in so doing get conceptual

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results so entirely different from what we have consciously perceived that

we are strongly tempted to look upon them as having certainly been

manufactured in our minds without reference to the externals. Of even more

significance is our ability to abstract from concrete objects and concrete

incidents the essential features which make them alike and different. But

unlike the Greeks, we see that our concept of coldness is not something

with which we were endowed from the beginning, but merely an

abstraction from concrete experiences with concrete objects that have been

cold.

The Concepts of Space and Time
When we have formed the abstract ideas of coldness and warmth, and have had experience indicating that the occurrence of these properties varies in degree, we are in a position to form the secondary abstract notion covered
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by the word “temperature.” When we have formed the abstract ideas of size

and position and separation, we are similarly in a position to form a

secondary abstraction to which we give the name “space.” Not quite so

easy to trace to its definite source but none the less clearly an abstraction

based on experience, is our idea of what we call “time.” None of us are

deceived as to the reality of these abstractions.]* [We do not regard space

as real in the sense that we regard a chair as real; it is merely an abstract idea convenient for the location of material objects like the chair.]198 [Nor

do we regard time as real in this sense. Things occupy space, events occupy

time; space and time themselves we realize are immaterial and unreal;

[34]

space does not exist and time does not happen in the same sense that

material objects exist and events occur. But we find it absolutely necessary

to have, among the mental machinery mentioned above as the apparatus by

aid of which we keep track of the external world, these vessels for that

world to exist in and move in.

Space and time, then, are concepts.]* [It is not strange, however, that when confronted with the vast and bewildering complexity of the universe and the difficulty of keeping separate and distinct in our minds our perceptions and conceptions, we should at times and as respects certain things project our conceptions illegitimately into the perpetual universe and mistake them for perceptions. The most notable example perhaps of this projection has occurred in the very case of space and time, most fundamental of all of our concepts. We got to think of these as absolute, as independent of each other and of all other things, and as always existing and continuing to exist whether or not we or anything else existed—space as a three-dimensional, uniform continuum, having the same properties in all directions; time as a one-dimensional, irreversible continuum, flowing in one direction. It is difficult to get back to the idea that space and time so described and defined are concepts merely, for the idea of their absolute existence is ingrained in us as the result probably of long ancestral experience.]283

[Newton’s definitions of course represent the classical idea of time and

space. He tells us that “absolute, true and mathematical time flows in virtue

of its own nature, uniformly and without reference to any external object;”

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and that “absolute space, by virtue of its own nature and without reference

to any external object, always remains the same and is immovable.” Of

course from modern standpoints it is absurd to call either of these

pronouncements a definition; but they represent about as well as any words

can the ideas which Newton had about time and space, and they make it

clear enough that he regarded both as having real existence in the external

world.

If space and time are to be the vessels of our universe, and if the only thing that really matters is measured results, it is plain enough that we must have, from the very beginning, means of measuring space and time. Whether we believe space and time to have real existence or not, it is obvious that we can measure neither directly. We shall have to measure space by measuring from one material object to another; we shall have to measure time by some similar convention based on events. We shall later have something further to say about the measurement of time; for the present we need only point
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out that]* [Newtonian time is measured independently of space; and the existence is presupposed of a suitable timekeeper.]10

[The space of Galileo and Newton was conceived of as empty, except in so

far as certain parts of it were occupied by matter. Positions of bodies in this space were in general determined by reference to]283 [a “coordinate

system” of some kind. This is again something that demands a certain

amount of discussion.

[36]

The Reference Frame for Space
The mathematician, following the lead of the great French all-around genius, Descartes, shows us very clearly how to set up, for the measurement of space, the framework known as the Cartesian coordinate system. The person of most ordinary mathematical attainments will realize that to locate a point in a plane we must have two measurements; and we could probably show this person, without too serious difficulty, that we can locate a point in any surface by two measurements. An example of this is the location of points on the earth’s surface by means of their latitude and longitude. It is equally clear that if we add a third dimension and attempt to locate points in space, we must add a third measurement. In the case of points on the earth’s surface, this might be the elevation above sea level, which would define the point not as part of the spherical surface of the earth but as part of the solid sphere. Or we may fall back on Dr. Slosson’s suggestion that in order to define completely the position of his laboratory, we must make a statement about Broadway, and one about 116th Street, and one telling how many flights of stairs there are to climb. In any event, it should be clear enough that the complete definition of a point in space calls for three measurements.
The mathematician formulates all this with the utmost precision. He asks us to]* [pick out any point whatever in space and call it O. We then draw or conceive to be drawn through this point three mutually perpendicular lines called coordinate axes, which we may designate OX, OY and OZ, respectively. Finally, we consider the three planes also mutually perpendicular like the two walls and the floor of a room that meet in one common corner, which are formed by the lines OX and OY, OY and OZ, and OZ and OX, respectively. These three planes are called coordinate planes. And then any other point P in space can be represented with respect to O by its perpendicular distances from each of the three coordinate planes—the distances x, y, z in the figure. These quantities are called the coordinates of the point.]272

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[To the layman there seems something altogether naive in this notion of the scientist’s setting up the three sides of a box in space and using them as the basis of all his work. The layman somehow feels that while it is perfectly all right for him to tell us that he lives at 1065 (one coordinate) 156th Street (two coordinates) on the third floor (three coordinates), it is rather trivial business for the serious-minded scientist to consider the up-and-down, the forward-and-back, the right-and-left of every point with which he has occasion to deal. There seems to the layman something particularly inane and foolish and altogether puerile about a set of coordinate axes, and you simply can’t make him believe that the serious-minded scientist has to monkey with any such funny business. He can’t be induced to take this coordinate-axis business seriously. Nevertheless, the fact is that the scientist takes it with the utmost seriousness. It is necessary for him to define the positions of points; and he does do it by means of a set of coordinate axes.
The scientist, however, is not interested in points of empty space. The point is to him merely part again of the conceptual machinery which he uses in his effort to run along with the external world. He knows there are no real points, but it suits his convenience to keep track of certain things that are real by representing them as points. But these things are in practically every instance material bodies; and in practically every instance, instead of staying put in one spot, they insist upon moving about through space. The scientist has to use his coordinate system, not merely to define a single position of such a “point,” but to keep track of the path over which it moves and to define its position in that path at given moments.
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Time and the Coordinate System
This introduces the concept of time into intimate relationship with the spatial coordinate system. And at once we feel the lack of a concrete, visualized fourth dimension.]* [If we want to fix objects in the floor alone, the edge of the room running toward the ceiling would become unnecessary and could be dropped from our coordinate system. That is, we need only two coordinates to fix the position of a point in a plane. Suppose instead of discarding the third coordinate, we use it to represent units of time. It then enables us to record the time it took a moving point in the floor to pass from position to position. Certain points in the room would be vertically above the corresponding points occupied by the moving point in its path across the floor; and the vertical height above the floor of such points corresponds to a value of the time-coordinate which indicates the time it took the point to move from position to position.]152 [Just as the path of the point across the floor is a continuous curve (for the mathematician, it should be understood, this term “curve” includes the straight line, as a special case in which the curvature happens to be zero); so the series of points above these in the room forms a continuous curve which records for us, not merely the path of the point across the floor, but in addition the time of its arrival at each of its successive positions. In the algebraic work connected with such a problem, the third coordinate behaves exactly the same, regardless of whether we consider it to represent time or a third spatial dimension; we cannot even tell from the algebra what it does represent.
When we come to the more general case of a point moving freely through space, we have but three coordinates at our disposal; there is not a fourth one by aid of which we can actually diagram its time-space record. Nevertheless, we can write down the numerical and algebraic relations between its three space-coordinates and the time which it takes to pass from one position to another; and by this means we can make all necessary calculations. Its motion is completely defined with regard both to space and to time. We are very apt to call attention to the fact that if we did have at our disposal a fourth, space-coordinate, we could use it to represent the time graphically, as before, and actually construct a geometric picture of the path of our moving point with regard to space and time. And on this account we are very apt to speak as though the time measurements constituted a fourth coordinate, regardless of any question of our ability to construct a picture of this coordinate. The arrival of a point in a given position constitutes an event; and this event is completely defined by means of four coordinates—three in space, which we can picture on our coordinate axes, and one in time which we cannot.
The set of coordinate axes in space, together with the zero point from which we measure time, constitute what we call a frame of reference. If we are not going to pay any attention to time, we can think of the space coordinate system alone as constituting our reference frame. This expression appears
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freely throughout the subsequent text, and always with one or the other of these interpretations.

We see, then, how we can keep track of a moving point by keeping track of

the successive positions which it occupies in our reference frame.]* [Now

we have implied that these coordinate axes are fixed in space; but there is

[41]

nothing to prevent us from supposing that they move.]272 [If they do, they

carry with them all their points; and any motion of these points which we

may speak about will be merely motion with reference to the coordinate

system. If we find something outside our coordinate system that is not

moving, the motion of points in our system with regard to those outside it

will be a combination of their motion with regard to our coordinate axes

and that of these axes with regard to the external points. This will be a great

nuisance; and it represents a state of affairs which we shall try to avoid. We

shall avoid it, if at all, by selecting a coordinate system with reference to

which we, ourselves, are not moving; one which partakes of any motion

which we may have. Or perhaps we shall sometimes wish to reverse the

process, in studying the behavior of some group of bodies, and seek a set of

axes which is at rest with respect to these bodies; one which partakes of any

motion they may have.

The Choice of a Coordinate Frame
All this emphasizes the fact that our coordinate axes are not picked out for us in advance by nature, and set down in some one particular spot. We select them for ourselves, and we select them in the most convenient way. But different observers, or perhaps the same observer studying different problems, will find it advantageous to utilize different coordinate systems.]* [The astronomer has found it possible, and highly convenient, to select a coordinate frame such that the great majority of the stars have, on the whole, no motion with respect to it.]283 [Such a system would be most unsuited for investigations confined to the earth; for these we naturally select a framework attached to the earth, with its origin O at the earth’s center if our investigation covers the entire globe and at some more convenient point if it does not, and in either event accompanying the earth in its rotation and revolution. But such a framework, as well as the one attached to the fixed stars, would be highly inconvenient for an investigator of the motions of the planets; he would doubtless attach his reference frame to the sun.]101
[In this connection a vital question suggests itself. Is the expression of natural law independent of or dependent upon the choice of a system of coordinates? And to what extent shall we be able to reconcile the results of one observer using one reference frame, and a second observer using a different one? The answer to the second question is obvious.]* [True, if any series of events is described using two different sets of axes, the descriptions will be different, depending upon the time system adopted and the relative motion of the axes. But if the connection between the reference
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systems is known, it is possible by mathematical processes to deduce the

quantities observed in one system if those observed in the other are known.]35 [This process of translating the results of one observer into those

of another is known as a transformation; and the mathematical statement of

the rule governing the transformation is called the equation or the equations

(there are usually several of them) of the transformation.]*

[Transformations of this character constitute a well-developed branch of

[43]

mathematics.]35

[When we inquire about the invariance of natural law it is necessary to be rather sure of just what we mean by this expression. The statement that a given body is moving with a velocity of 75 miles per hour is of course not a natural law; it is a mere numerical observation. But aside from such numerical results, we have a large number of mathematical relations which give us a more or less general statement of the relations that exist between velocities, accelerations, masses, forces, times, lengths, temperatures, pressures, etc., etc. There are some of these which we would be prepared to state at once as universally valid—distance travelled equals velocity multiplied by time, for instance. We do not believe that any conceivable change of reference systems could bring about a condition in which the product of velocity and time, as measured from a certain framework, would fail to equal distance as measured from this same framework. There are other relations more or less of the same sort which we probably believe to be in the same invariant category; there are others, perhaps, of which we might be doubtful; and presumably there are still others which we should suspect of restricted validity, holding in certain reference systems only and not in others.

The question of invariance of natural law, then, may turn out to be one

which may be answered in the large by a single statement; it may equally

turn out to be one that has to be answered in the small, by considering

particular laws in connection with particular transformations between

particular reference systems. Or, perhaps, we may find ourselves justified in

[44]

taking the stand that an alleged “law of nature” is truly such a law only in

the event that it is independent of the change from one reference system to

another. In any event, the question may be formulated as follows:

Observer A, using the reference system R, measures certain quantities t, w, x, y, z. Observer B, using the reference system S, measures the same items and gets the values t′, w′, x′, y′, z′. The appropriate transformation equations for calculating the one set of values from the other is found. If a mathematical relation of any sort is found to exist between the values t, w, x, y, z, will the same relation exist between the values t′, w′, x′, y′, z′? If it does not, are we justified in still calling it a law of nature? And if it does not, and we refrain from calling it such a law, may we expect in every case to find some relation that will be invariant under the transformation, and that may therefore be recognized as the natural law connecting t, w, x, y and z?

I have found it advisable to discuss this point in such detail because here more than in any other single place the competing essayists betray

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uncertainty of thought and sloppiness of expression. It doesn’t amount to

much to talk about the invariance of natural laws and their persistence as

we pass from one coordinate system to another, unless we are fairly well

fortified with respect to just what we mean by invariance and by natural

law. We don’t expect the velocity of a train to be 60 miles per hour alike

when we measure it with respect to a signal tower along the line and with

respect to a moving train on the other track. We don’t expect the angular

[45]

displacement of Mars to change as rapidly when he is on the other side of

the sun as when he is on our side. But we do, I think, rather expect that in

any phenomenon which we may observe, we shall find a natural law of

some sort which is dependent for its validity neither upon the units we

employ, nor the place from which we make our measurements, nor

anything else external to the phenomenon itself. We shall see, later, whether

this expectation is justified, or whether it will have to be discarded in the

final unravelling of the absolutist from the relativistic philosophy which,

with Einstein, we are to undertake.]*

[46]

III
THE RELATIVITY OF UNIFORM MOTION
Classical Ideas on the Subject; the Ether and the Apparent Possibility of Absolute Motion; the Michelson-Morley Experiment and the Final Negation of This Possibility
BY VARIOUS CONTRIBUTORS AND THE EDITOR
When we speak of a body as being “in motion,” we mean that this body is changing its position “in space.” Now it is clear that the position of an object can only be determined with reference to other objects: in order to describe the place of a material thing we must, for example, state its distances from other things. If there were no such bodies of reference, the words “position in space” would have no definite meaning for us.]24 [The number of such external bodies of reference which it is necessary to cite in order to define completely the position of a given body in space depends upon the character of the space dealt with. We have seen that when we visualize the space of our experience as a surface of any character, two citations are sufficient; and that when we conceive of it as surrounding us in three dimensions we require three. It will be realized that the mathematician is merely meeting this requirement when he sets up his system of coordinate axes to serve as a reference frame.]*
[What is true of “place” must be true also of “motion,” since the latter is nothing but change of place. In fact, it would be impossible to ascribe a state of motion or of rest to a body poised all alone in empty space.
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Whether a body is to be regarded as resting or as moving, and if the latter at

what speed, depends entirely upon the objects to which we refer its positions in space.]24 [As Einstein sits at his desk he appears to us to be at

rest; but we know that he is moving with the rotation of the earth on its

axis, with the earth in its orbit about the sun, and with the solar system in its

path through space—a complex motion of which the parts or the whole can

be detected only by reference to appropriately chosen ones of the heavenly

bodies. No mechanical test has ever been devised which will detect this motion,]182 [if we reserve for discussion in its proper place the Foucault

pendulum experiment which will reveal the axial rotation of our globe.]*

[No savage, if he were to “stand still,” could be convinced that he was moving with a very high velocity or in fact that he was moving at all.]30

[You drop a coin straight down a ship’s side: from the land its path appears

parabolic; to a polar onlooker it whirls circle-wise; to dwellers on Mars it

darts spirally about the sun; to a stellar observer it gyrates through the sky]263 [in a path of many complications. To you it drops in a straight line

from the deck to the sea.]* [Yet its various tracks in ship-space, sea-space,

earth-space, sun-space, star-space, are all equally real,]263 [and the one

[48]

which will be singled out for attention depends entirely upon the observer,

and the objects to which he refers the motion.]* [The earth moves in the

solar system, which is itself approaching a distant star-cluster. But we

cannot say whether we are moving toward the cluster, or the cluster toward us,]18 [or both, or whether we are conducting a successful stern chase of it,

or it of us,]* [unless we have in mind some third body with reference to which the motions of earth and star-cluster are measured.]18 [And if we

have this, the measurements made with reference to it are of significance

with regard to it, rather than with regard to the earth and the star-cluster

alone.]*

[We can express all this by saying “All motions are relative; there is no such thing as absolute motion.” This line of argument has in fact been followed by many natural philosophers. But is its result in agreement with actual experience? Is it really impossible to distinguish between rest and motion of a body if we do not take into consideration its relations to other objects? In fact it can easily be seen that, at least in many cases, no such distinction is possible.

Who Is Moving?
Imagine yourself sitting in a railroad car with veiled windows and running on a perfectly straight track with unchanging velocity: you would find it absolutely impossible to ascertain by any mechanical means whether the car were moving or not. All mechanical instruments behave exactly the same, whether the car be standing still or in motion.]24 [If you drop a ball you will see it fall to the floor in a straight line, just as though you had dropped it while standing on the station platform. Furthermore, if you drop the ball from the same height in the two cases, and measure the velocities
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with which it strikes the car floor and the station platform, or the times which it requires for the descent, you will find these identical in the two cases.]182

[Any changes of speed or of direction (as when the car speeds up or slows down or rounds a curve) can be detected by observing the behavior of bodies in the car, without apparent reference to any outside objects. This becomes particularly obvious with sudden irregularities of motion, which manifest themselves by shaking everything in the car. But a uniform motion
in a straight line does not reveal itself by any phenomenon within the vehicle.]24

[Moreover, if we remove the veil from our window to the extent that we

may observe the train on the adjoining track, we shall be able to make no

decision as to whether we or it be moving. This is indeed an experience

which we have all had.]* [Often when seated in a train about to leave the

station, we have thought ourselves under way, only to perceive as the

motion becomes no longer uniform that another train has been backing into

the station on the adjoining track. Again, as we were hurried on our journey,

we have, raising suddenly our eyes, been puzzled to say whether the

passing train were moving with us or against us or indeed standing still; or

more rarely we have had the impression that both it and we seemed to be at

rest, when in truth both were moving rapidly with the same speed.]82 [Even

[50]

this phrase “in truth” is a relative one, for it arises through using the earth

as an absolute reference body. We are indeed naive if we cannot appreciate

that there is no reason for doing this beyond convenience, and that to an

observer detached from the earth it were just as reasonable to say that the

rails are sliding under the train as that the train is advancing along the rails.

One of my own most vivid childhood recollections is of the terror with

which, riding on a train that passed through a narrow cut, I hid my head in

the maternal lap to shut out the horrid sight of the earth rushing past my

window. The absence of a background in relatively slow retrograde motion

was sufficient to prevent my consciousness from drawing the accustomed

conclusion that after all it was really the train that was moving.]*

Mechanical Relativity
[So we can enunciate the following principle: When a body is in uniform rectilinear motion relatively to a second body, then all phenomena take place on the first in exactly the same manner as on the second; the physical laws for the happenings on both bodies are identical.]24 [And between a system of bodies, nothing but relative motion may be detected by any mechanical means whatever; any attempt to discuss absolute motion presupposes a super-observer on some body external to the system. Even then, the “absolute” motion is nothing but motion relative to this superobserver. By no mechanical means is uniform straight-line motion of any other than relative character to be detected. This is the Principle of Mechanical Relativity.
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There is nothing new in this. It was known to Galileo, it was known to Newton, it has been known ever since. But the curious persistence of the human mind in habits of thought which confuse relativity with absolutism brought about a state of affairs where we attempted to know this and to ignore it at the same time. We shall have to return to the mathematical mode of reasoning to see how this happened. The mathematician has a way all his own of putting the statement of relativity which we have made. He recalls, what we have already seen, that the observer on the earth who is measuring his “absolute” motion with respect to the earth has merely attached his reference framework to the earth; that the passenger in the train who measures all motion naively with respect to his train is merely carrying his coordinate axes along with his baggage, instead of leaving them on the solid ground; that the astronomer who deals with the motion of the earth about the sun, or with that of the “fixed” stars against one another, does so simply by the artifice of hitching his frame of reference to the sun or to one of the fixed stars. So the mathematician points out that dispute as to which of two bodies is in motion comes right down to dispute as to which of two sets of coordinate axes is the better one, the more nearly “natural” or “absolute.” He therefore phrases the mechanical principle of relativity as follows:

Among all coordinate systems that are merely in uniform straight-line

[52]

motion to one another, no one occupies any position of unique natural

advantage; all such systems are equivalent for the investigation of natural

laws; all systems lead to the same laws and the same results.

The mathematician has thus removed the statement of relativity from its intimate association with the external observed phenomena, and transferred it to the observer and his reference frame. We must either accept the principle of relativity, or seek a set of coordinate axes that have been singled out by nature as an absolute reference frame. These axes must be in some way unique, so that when we refer phenomena to them, the laws of nature take a form of exceptional simplicity not attained through reference to ordinary axes. Where shall we look for such a preferred coordinate system?]*

The Search for the Absolute
[Older theory clung to the belief that there was such a thing as absolute motion in space.]197 [As the body of scientific law developed from the sixteenth century onward, the not unnatural hypothesis crept in, that these laws (that is to say, their mathematical formulations rather than their verbal statements) would reveal themselves in especially simple forms, were it possible for experimenters to make their observations from some absolute standpoint; from an absolutely fixed position in space rather than from the moving earth.]264 [Somewhere a set of coordinate axes incapable of motion was to be found,]197 [a fixed set of axes for measuring absolute motion; and for two hundred years the world of science strove to find it,]147 [in
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spite of what should have been assurance that it did not exist. But the search failed, and gradually the universal applicability of the principle of relativity, so far as it concerned mechanical phenomena, grew into general acceptance.]* [And after the development, by the great mathematicians of the eighteenth century, of Newton’s laws of motion into their most complete mathematical form, it was seen that so far as these laws are concerned the absolutist hypothesis mentioned is quite unsupported. No complication is introduced into Newton’s laws if the observer has to make his measurements in a frame of reference moving uniformly through space; and for measurements in a frame like the earth, which moves with changing speed and direction about the sun and rotates on its axis at the same time, the complication is not of so decisive a nature as to give us any clue to the earth’s absolute motion in space.

But mechanics, albeit the oldest, is yet only one of the physical sciences.

The great advance made in the mathematical formulation of optical and

electromagnetic theory during the nineteenth century revived the hope of

discovering absolute motion in space by means of the laws derived from this theory.]264 [Newton had supposed light to be a material emanation, and

if it were so, its passage across “empty space” from sun and stars to the

earth raised no problem. But against Newton’s theory Huyghens, the Dutch

astronomer, advanced the idea that light was a wave motion of some sort.

During the Newtonian period and for many years after, the corpuscular

[54]

theory prevailed; but eventually the tables were turned.]* [Men made rays

of light interfere, producing darkness (see page 61). From this, and from

other phenomena like polarization, they had deduced that light was a form

of wave motion similar to water ripples; for these interfere, producing level

surfaces, or reinforce each other, producing waves of abnormal height. But

if light were to be regarded as a form of wave motion—and the phenomena

could apparently be explained on no other basis—then there must be some medium capable of undergoing this form of motion.]135 [Transmission of

waves across empty space without the aid of an intermediary material

medium would be “action at a distance,” an idea repugnant to us.

Trammeled by our tactual, wire-pulling conceptions of a material universe,

we could not accustom ourselves to the idea of something—even so

immaterial a something as a wave—being transmitted by nothing. We

needed a word—ether—to carry light if not to shed it; just as we need a word—inertia—to carry a projectile in its flight.]231 [It was necessary to

invest this medium with properties to account for the observed facts. On the whole it was regarded as the perfect fluid.]235 [The ether was imagined as

an all-pervading, imponderable substance filling the vast emptiness through

which light reaches us, and as well the intermolecular spaces of all matter.

Nothing more was known definitely, yet this much served as a good

working hypothesis on the basis of which Maxwell was enabled to predict

the possibility of radio communication. By its fruits the ether hypothesis

justified itself; but does the ether exist?]231

[55]

The Ether and Absolute Motion
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[If it does exist, it seems quite necessary, on mere philosophical grounds, that it shall be eligible to serve as the long-sought reference frame for absolute motion. Surely it does not make sense to speak of a homogeneous medium filling all space, sufficiently material to serve as a means of communication between remote worlds, and in the next breath to deny that motion with respect to this medium is a concept of significance.]* [Such a system of reference as was offered by the ether, coextensive with the entire known region of the universe, must necessarily serve for all motions within our perceptions.]186 [The conclusion seems inescapable that motion with respect to the ether ought to be of a sufficiently unique character to stand out above all other motion. In particular, we ought to be able to use the ether to define, somewhere, a system of axes fixed with respect to the ether, the use of which would lead to natural laws of a uniquely simply description.

Maxwell’s work added fuel to this hope.]* [During the last century, after

the units of electricity had been defined, one set for static electrical

calculations and one for electromagnetic calculations, it was found that the

ratio of the metric units of capacity for the two systems was numerically

equal to what had already been found as the velocity with which light is

transmitted through the hypothetical ether. One definition refers to

electricity at rest, the other to electricity in motion. Maxwell, with little

[56]

more working basis than this, undertook to prove that electrical and optical phenomena were merely two aspects of a common cause,]235 [to which the

general designation of “electromagnetic waves” was applied. Maxwell

treated this topic in great fullness and with complete success. In particular,

he derived certain equations giving the relations between the various

electrical quantities involved in a given phenomenon. But it was found,

extraordinarily enough, that these relations were of such character that,

when we subject the quantities involved to a change of coordinate axes, the

transformed quantities did not preserve these relations if the new axes

happened to be in motion with respect to the original ones. This, of course,

was taken to indicate that motion really is absolute when we come to deal

with electromagnetic phenomena, and that the ether which carries the

electromagnetic waves really may be looked to to display the properties of

an absolute reference frame.

Reference to the phenomenon of aberration, which Dr. Pickering has

discussed adequately in his essay and which I need therefore mention here

only by name, indicated that the ether was not dragged along by material

bodies over and through which it might pass. It seemed that it must filter

through such bodies, presumably via the molecular interstices, without

appreciable opposition. Were this not the case, we should be in some doubt

as to the possibility of observing the velocity through the ether of material

bodies; if the ether adjacent to such bodies is not dragged along or thrown

into eddies, but “stands still” while the bodies pass, there seems no

[57]

imaginable reason for anything other than the complete success of such

observations. And of course these are of the utmost importance, the

moment we assign to the ether the rôle of absolute reference frame.

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The Earth and the Ether

One body in motion with respect to the ether is our earth itself. We do not know in advance in what direction to expect this motion or what magnitude to anticipate that it will have. But one thing is clear.]* [In its motion around
the sun, the earth has, at opposite points on its orbit, a difference in velocity with respect to the surrounding medium which is double its orbital velocity with respect to the sun. This difference comes to 37 miles per second. The earth should therefore, at some time in the year, show a velocity equal to or greater than 18½ miles per second, with reference to the universal medium.
The famous Michelson-Morley experiment of 1887 was carried out with the expectation of observing this velocity.]267

[The ether, of course, and hence velocities through it, cannot be observed

directly. But it acts as the medium for the transmission of light.]* [If the

velocity of light through the ether is C and that of the earth through the

ether is v, then the velocity of light past the earth, so the argument runs,

must vary from

to

, according as the light is moving exactly in

the same direction as the earth, or in the opposite direction,]182 [or

diagonally across the earth’s path so as to get the influence only of a part of

the earth’s motion. This of course assumes that C has always the same

value; an assumption that impresses one as inherently probable, and one

that is at the same time in accord with ordinary astronomical observation.

It is not possible to measure directly the velocity of light (186,330 miles per second, more or less) with sufficient accuracy to give any meaning to the variation in this velocity which might be effected by adding or subtracting that of the earth in its orbit (a mere 18½ miles per second). It is, however, possible to play a trick on the light by sending it back and forth over several paths, and comparing (not measuring absolutely, but merely comparing) with great minuteness the times consumed in these several round trips.

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A Journey Upstream and Back
The number of letters the Scientific American has received questioning the Michelson-Morley experiment indicates that many people are not acquainted with the fundamental principle on which it is based. So let us look at a simple analogous case. Suppose a swimmer or a rower make a return trip upstream and down, contending with the current as he goes up and getting its benefit when he comes down. Obviously, says snap judgment, since the two legs of the journey are equal, he derives exactly as much benefit from the current when he goes with it as he suffers handicap from it when he goes against it; so the round trip must take exactly the
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same time as a journey of the same length in still water, the argument applying equally in the case where the “swimmer” is a wave of light in the ether stream.

But let us look now at a numerical case. A man can row in still water at four miles per hour. He rows twelve miles upstream and back, in a current of two miles per hour. At a net speed of two miles per hour he arrives at his turning point in six hours. At a net speed of six miles per hour he makes the down-stream leg in two hours. The elapsed time for the journey is eight hours; in still water he would row the twenty-four miles in six hours.

If we were to attempt an explanation of this result in words we should say that by virtue of the very fact that it does delay him, the adverse current prolongs the time during which it operates; while by virtue of the very fact that it accelerates his progress, the favoring current shortens its venue. The careless observer realizes that distances are equal between the two legs of the journey, and unconsciously assumes that times are equal.

If the journey be made directly with and directly against the stream of water or ether or what not, retardation is effected to its fullest extent. If the course be a diagonal one, retardation is felt to an extent measurable as a component, and depending for its exact value upon the exact angle of the path. Felt, however, it must always be.

Here is where we begin to get a grip on the problem of the earth and the

ether. In any problem involving the return-trip principle, there will enter

[60]

two velocities—that of the swimmer and that of the medium; and the time

of retardation. If we know any two of these items we can calculate the third.

When the swimmer is a ray of light and the velocity of the medium is that

of the ether as it flows past the earth, we know the first of these two; we

hope to observe the retardation so that we may calculate the second

velocity. The apparatus for the experiment is ingenious and demands

description.

The Michelson-Morley Experiment
The machine is of structural steel, weighing 1,900 pounds. It has two arms which form a Greek cross. Each arm is 14 feet in length. The whole apparatus is floated in a trough containing 800 pounds of mercury.

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Four mirrors are arranged on the end of each arm, sixteen in all, with a seventeenth mirror, M, set at one of the inside corners of the cross, as diagrammed. A source of light (in this case a calcium flame) is provided, and its rays directed by a lens toward the mirror M. Part of the light is allowed to pass straight through M to the opposite arm of the cross, where it strikes mirror 1. It is reflected back across the arm to mirror 2, thence to 3, and so on until it reaches mirror 8. Thence it is reflected back to mirror 7, to 6, and so on, retracing its former path, and finally is caught by the reverse side of the mirror M and is sent to an observer at O. In retracing its path the light sets up an interference phenomenon (see below) and the interference bands are visible to the observer, who is provided with a telescope to magnify the results.
A second part of the original light-beam is reflected off at right angles by the mirror M, and is passed to and fro on the adjacent arms of the machine, in exactly the same manner and over a similar path, by means of the mirrors I, II, III, … VIII. This light finally reaches the observer at the telescope, setting up a second set of interference bands, parallel to the first.
A word now about this business of light interference. Light is a wave motion. The length of a wave is but a few millionths of an inch, and the amplitude is correspondingly minute; but none the less, these waves behave in a thoroughly wave-like manner. In particular, if the crests of two waves
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are superposed, there is a double effect; while if a crest of one wave falls with a trough of another, there is a killing-off or “interference”.

Under ordinary circumstances interference of light waves does not occur.

[62]

This is simply because under ordinary circumstances light waves are not

piled up on one another. But sometimes this piling up occurs; and then, just

so sure as the piled-up waves are in the same phase they reinforce one

another, while if they are in opposite phase they interfere. And the

conditions which we have outlined above, with the telescope and the

mirrors and the ray of light retracing the path over which it went out, are

conditions under which interference does occur. If the returning wave is in

exact phase with the outgoing one, the effect is that of uniform double

illumination; if it is in exactly opposite phase the effect is that of complete

extinguishing of the light, the reversed wave exactly cancelling out the

original one. If the two rays are partly in phase, there is partial

reinforcement or partial cancelling out, according to whether they are

nearly in phase or nearly out of phase. Finally, if the mirrors are not set

absolutely parallel—as must in practice be the case when we attempt to

measure their parallelism in terms of the wave-length of light—adjacent

parts of the light ray will vary in the extent to which they are out of phase,

since they will have travelled a fraction of a wave-length further to get to

and from this, that or the other mirror. There will then appear in the

telescope alternate bands of illumination and darkness, whose width and

spacing depend upon all the factors entering into the problem.

If it were possible for us to make the apparatus with such a degree of

refinement that the path from mirror M via mirrors 1, 2, 3, etc., back

through M and into the telescope, were exactly the same length as that from

[63]

flame to telescope by way of the mirrors I, II, III, etc.—exactly the same to

a margin of error materially less than a single wave-length of light—why,

then, the two sets of interference fringes would come out exactly

superposed provided the motion of the earth through the “ether” turn out to

have no influence upon the velocity of light; or, if such influence exist,

these fringes would be displaced from one another to an extent measuring

the influence in question. But our ability to set up this complicated pattern

of mirrors at predetermined distances falls far short of the wave-length as a

measure of error. So in practice all that we can say is that having once set

the instrument up, and passed a beam of light through it, there will be

produced two sets of parallel interference fringes. These sets will fail of

superposition—each fringe of one set will be removed from the

corresponding fringe of the other set—by some definite distance. Then, any

subsequent variation in the speed of light along the two arms will at once be

detected by a shifting of the interference bands through a distance which we

shall be able to measure.

The Verdict
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Under the theories and assumptions governing at the time of the original

performance of this experiment, it will be readily seen that if this machine

be set up in an “ether stream” with one arm parallel to the direction of the

stream and the other at right angles thereto, there will be a difference in the

speed of the light along the two arms. Then if the apparatus be shifted to a

[64]

position oblique to the ether stream, the excess velocity of the light in the

one arm would be diminished, and gradually come to zero at the 45-degree

angle, after which the light traveling along the other arm would assume the

greater speed. In making observations, therefore, the entire apparatus was

slowly rotated, the observers walking with it, so that changes of the sort

anticipated would be observed.

The investigators were, however, ignorant of the position in which the apparatus ought to be set to insure that one of the arms lie across the ether drift; and they were ignorant of the time of year at which the earth’s maximum velocity through the ether was to be looked for. In particular, it is plain that if the solar system as a whole is moving through the ether at a rate less than the earth’s orbital velocity, there is a point in our orbit where our velocity through the ether and that around the sun just cancel out and leave us temporarily in a state of “absolute rest.” So it was anticipated that the experiment might have to be repeated in many orientations of the machine and at many seasons of the year in order to give a series of readings from which the true motion of the earth through the ether might be deduced.

For those who have a little algebra the demonstration which Dr. Russell

gives on a subsequent page will be interesting as showing the situation in

perfectly general terms. It will be realized that the more complicated

arrangement of mirrors in the experiment as just described is simply an

eightfold repetition of the simple experiment as outlined by Dr. Russell, and

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that it was done so for the mere sake of multiplying by eight the distances

travelled and hence the difference in time and in phase.

And now for the grand climax. The experiment was repeated many times, with the original and with other apparatus, indoors and outdoors, at all seasons of the year, with variation of every condition that could imaginably affect the result. The apparatus was ordinarily such that a shift in the fringes of anywhere from one-tenth to one one-hundredth of that which would have followed from any reasonable value for the earth’s motion through the ether would have been systematically apparent. The result was uniformly negative. At all times and in all directions the velocity of light past the earth-bound observer was the same. The earth has no motion with reference to the ether!

[The amazing character of this result is not by any possibility to be
exaggerated.]* [According to one experiment the ether was carried along
by a rapidly moving body and according to another equally well-planned
and well-executed experiment a rapidly moving body did not disturb the ether at all. This was the blind alley into which science had been led.]232

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The “Contraction” Hypothesis
[Numerous efforts were made to explain the contradiction.]* [It is indeed a very puzzling one, and it gave physicists no end of trouble. However Lorentz and Fitzgerald finally put forward an ingenious explanation, to the effect that the actual motion of the earth through the ether is balanced, as far as the ability of our measuring instruments is concerned, by a contraction of these same instruments in the direction of their motion. This contraction obviously cannot be observed directly because all bodies, including the measuring instruments themselves (which after all are only arbitrary guides), will suffer the contraction equally. According to this theory, called the Lorentz-Fitzgerald contraction theory,]272 [all bodies in motion suffer such contraction of their length in the direction of their motion;]283 [the contraction being made evident by our inability to observe the absolute motion of the earth, which it is assumed must exist.]272 [This would suffice to show why the Michelson-Morley experiment gave a negative result, and would preserve the concept of absolute motion with reference to the ether.]283
[This proposal of Lorentz and Fitzgerald loses its startling aspect when we consider that all matter appears to be an electrical structure, and that the dimensions of the electric and magnetic fields which accompany the electrons of which it is constituted change with the velocity of motion.]267 [The forces of cohesion which determine the form of a rigid body are held to be electromagnetic in nature; the contraction may be regarded as due to a change in the electromagnetic forces between the molecules.]10 [As one writer has put it, the orientation, in the electromagnetic medium, of a body depending for its very existence upon electromagnetic forces is not necessarily a matter of indifference.]*
[Granting the plausibility of all this, on the basis of an electromagnetic theory of matter, it leaves us in an unsatisfactory position. We are left with a fixed ether with reference to which absolute motion has a meaning, but that motion remains undetected and apparently undetectable. Further, if we on shore measure the length of a moving ship, using a yard-stick which is stationary on shore, we shall obtain one result. If we take our stick aboard it contracts, and so we obtain a greater length for the ship. Not knowing our “real” motion through the ether, we cannot say which is the “true” length. Is it not, then, more satisfactory to discard all notion of true length as an inherent quality of bodies, and, by regarding length as the measure of a relation between a particular object and a particular observer, to make one length as true as the other?]182 [The opponents of such a viewpoint contend that Michelson’s result was due to a fluke; some mysterious counterbalancing influence was for some reason at work, concealing the result which should normally have been expected. Einstein refuses to accept this explanation;]192 [he refuses to believe that all nature is in a contemptible conspiracy to delude us.]*
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[The Fitzgerald suggestion is further unsatisfactory because it assumes all

substances, of whatever density, to undergo the same contraction; and above all for the reason that it sheds no light upon other phenomena.]194 [It

is indeed a very special explanation; that is, it applies only to the particular

experiment in question. And indeed it is only one of many possible

explanations. Einstein conceived the notion that it might be infinitely more

valuable to take the most general explanation possible, and then try to find

from this its logical consequences. This “most general explanation” is, of

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course, simply that it is impossible in any way whatever to measure the absolute motion of a body in space.]272 [Accordingly Einstein enunciated,

first the Special Theory of Relativity, and later the General Theory of

Relativity. The special theory was so called because it was, limited to

uniform rectilinear and non-rotary motions. The general theory, on the other

hand, dealt not only with uniform rectilinear motions, but with any arbitrary

motion whatever.

Taking the Bull by the Horns
The hypothesis of relativity asserts that there can be no such concept as absolute position, absolute motion, absolute time; that space and time are inter-dependent, not independent; that everything is relative to something else. It thus accords with the philosophical notion of the relativity of all knowledge.]283 [Knowledge is based, ultimately, upon measurement; and clearly all measurement is relative, consisting merely in the application of a standard to the magnitude measured. All metric numbers are relative; dividing the unit multiplies the metric number. Moreover, if measure and measured change proportionately, the measuring number is unchanged. Should space with all its contents swell in fixed ratio throughout, no measurement could detect this; nor even should it pulse uniformly throughout. Furthermore, were space and space-contents in any way systematically transformed (as by reflection in curved mirrors) point for point, continuously, without rending, no measurement could reveal this distortion; experience would proceed undisturbed.]263
[Mark Twain said that the street in Damascus “which is called straight,” is so called because while it is not as straight as a rainbow it is straighter than a corkscrew. This expresses the basic idea of relativity—the idea of comparison. All our knowledge is relative, not absolute. Things are big or little, long or short, light or heavy, fast or slow, only by comparison. An atom may be as large, compared to an electron, as is a cathedral compared to a fly. The relativity theory of Einstein emphasizes two cases of relative knowledge; our knowledge of time and space, and our knowledge of motion.]216 [And in each case, instead of allowing the notions of relativity to guide us only so far as it pleases us to follow them, there abandoning them for ideas more in accord with what we find it easy to take for granted, Einstein builds his structure on the thesis that relativity must be admitted, must be followed out to the bitter end, in spite of anything that it may do to our preconceived notions. If relativity is to be admitted at all, it must be
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admitted in toto; no matter what else it contradicts, we have no appeal from its conclusions so long as it refrains from contradicting itself.]*

[The hypothesis of relativity was developed by Einstein through a priori

methods, not the more usual a posteriori ones. That is, certain principles

were enunciated as probably true, the consequences of these were

developed, and these deductions tested by comparison of the predicted and

the observed phenomena. It was in no sense attained by the more usual

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procedure of observing groups of phenomena and formulating a law or

formula which would embrace them and correctly describe the routine or

sequence of phenomena.

The first principle thus enunciated is that it is impossible to measure or
detect absolute translatory motion through space, under any circumstances
or by any means. The second is that the velocity of light in free space
appears the same to all observers regardless of the relative motion of the
source of light and the observer. This velocity is not affected by motion of the source toward or away from the observer,]283 [if we may for the
moment use this expression with its implication of absolute motion.]* [But
universal relativity insists that motion of the source toward the observer is identical with motion of the observer toward the source.]283

[It will be seen that we are at once on the horns of a dilemma. Either we

must give up relativity before we get fairly started on it, or we must

overturn the foundations of common sense by admitting that time and space

are so constituted that when we go to meet an advancing light-impulse, or

when we retreat from it, it still reaches us with the same velocity as though

we stood still waiting for it. We shall find when we are through with our

investigation that common sense is at fault; that our fixed impression of the

absurdity of the state of affairs just outlined springs from a confusion

between relativism and absolutism which has heretofore dominated our

thought and gone unquestioned. The impression of absurdity will vanish

when we have resolved this confusion.]*

[71]

Questions of Common Sense
[But it is obvious from what has just been said that if we are to adopt Einstein’s theory, we must make very radical changes in some of our fundamental notions, changes that seem in violent conflict with common sense. It is unfortunate that many popularizers of relativity have been more concerned to astonish their readers with incredible paradoxes than to give an account such as would appeal to sound judgment. Many of these paradoxes do not belong essentially to the theory at all. There is nothing in the latter that an enlarged and enlightened common sense would not readily endorse. But common sense must be educated up to the necessary level.]141
[There was a time when it was believed, as a result of centuries of experience, that the world was flat. This belief checked up with the known
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facts, and it could be used as the basis for a system of science which would account for things that had happened and that were to happen. It was entirely sufficient for the time in which it prevailed.

Then one day a man arose to point out that all the known facts were equally

accounted for on the theory that the earth was a sphere. It was in order for

his contemporaries to admit this, to say that so far as the facts in hand were

concerned they could not tell whether the earth was flat or round—that new

facts would have to be sought that would contradict one or the other

hypothesis. Instead of this the world laughed and insisted that the earth

could not be round because it was flat; that it could not be round because

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then the people would fall off the other side.

But the field of experimentation widened, and men were able to observe facts that had been hidden from them. Presently a man sailed west and arrived east; and it became clear that in spite of previously accepted “facts” to the contrary, the earth was really round. The previously accepted “facts” were then revised to fit the newly discovered truth; and finally a new system of science came into being, which accounted for all the old facts and all the new ones.

At intervals this sort of thing has been repeated. A Galileo shows that preconceived ideas with regard to the heavens are wrong, and must be revised to accord with his newly promulgated principles. A Newton does the same for physics—and people unlearn the “fact” that motion has to be supported by continued application of force, substituting the new idea that it actually requires force to stop a moving body. A Harvey shows that the things which have been “known” for generations about the human body are not so. A Lyell and a Darwin force men to throw overboard the things they have always believed about the way in which the earth and its creatures came into being. Every science we possess has passed through one or more of these periods of readjustment to new facts.

Shifting the Mental Gears
Now we are apt to lose sight of the true significance of this. It is not alone our opinions that are altered; it is our fundamental concepts. We get concepts wholly from our perceptions, making them to fit those perceptions. Whenever a new vista is opened to our perceptions, we find facts that we never could have suspected from the restricted viewpoint. We must then actually alter our concepts to make the new facts fit in with the greatest degree of harmony. And we must not hesitate to undertake this alteration, through any feeling that fundamental concepts are more sacred and less freely to be tampered with than derived facts.]* [We do, to be sure, want fundamental concepts that are easy for a human mind to conceive; but we also want our laws of nature to be simple. If the laws begin to become, intricate, why not reshape, somewhat, the fundamental concepts, in order to
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simplify the scientific laws? Ultimately it is the simplicity of the scientific system as a whole that is our principal aim.]178

[As a fair example, see what the acceptance of the earth’s sphericity did to

the idea represented by the word “down.” With a flat earth, “down” is a

single direction, the same throughout the universe; with a round earth,

“down” becomes merely the direction leading toward the center of the

particular heavenly body on which we happen to be located. It is so with

every concept we have. No matter how intrinsic a part of nature and of our

being a certain notion may seem, we can never know that new facts will not

develop which will show it to be a mistaken one. Today we are merely

confronted by a gigantic example of this sort of thing. Einstein tells us that

when velocities are attained which have just now come within the range of

our close investigation, extraordinary things happen—things quite

[74]

irreconcilable with our present concepts of time and space and mass and

dimension. We are tempted to laugh at him, to tell him that the phenomena

he suggests are absurd because they contradict these concepts. Nothing

could be more rash than this.

When we consider the results which follow from physical velocities comparable with that of light, we must confess that here are conditions which have never before been carefully investigated. We must be quite as well prepared to have these conditions reveal some epoch-making fact as was Galileo when he turned the first telescope upon the skies. And if this fact requires that we discard present ideas of time and space and mass and dimension, we must be prepared to do so quite as thoroughly as our medieval fathers had to discard their notions of celestial “perfection” which demanded that there be but seven major heavenly bodies and that everything center about the earth as a common universal hub. We must be prepared to revise our concepts of these or any other fundamentals quite as severely as did the first philosopher who realized that “down” in London was not parallel to “down” in Bagdad or on Mars.]*

[In all ordinary terrestrial matters we take the earth as a fixed body, light as

instantaneous. This is perfectly proper, for such matters. But we carry our

earth-acquired habits with us into the celestial regions. Though we have no

longer the earth to stand on, yet we assume, as on the earth, that all

measurements and movements must be referred to some fixed body, and are

only then valid. We cling to our earth-bound notion that there is an absolute

[75]

up-and-down, back-and-forth, right-and-left, in space. We may admit that

we can never find it, but we still think it is there, and seek to approach it as

nearly as possible. And similarly from our earth experiences, which are

sufficiently in a single place to make possible this simplifying assumption,

we get the idea that there is one universal time, applicable at once to the entire universe.]141 [The difficulty in accepting Einstein is entirely the

difficulty in getting away from these earth-bound habits of thought.]*

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IV
THE SPECIAL THEORY OF RELATIVITY
What Einstein’s Study of Uniform Motion Tells Us About Time and Space and the Nature of the External Reality
BY VARIOUS CONTRIBUTORS AND THE EDITOR
Whatever the explanation adopted for the negative result of the MichelsonMorley experiment, one thing stands out clearly: the attempt to isolate absolute motion has again failed.]* [Einstein generalizes this with all the other and older negative results of similar sort into a negative deduction to the effect that no experiment is possible upon two systems which will determine that one of them is in motion and the other at rest.]121 [He elevates the repeated failure to detect absolute motion through space into the principle that experiment will never reveal anything in the nature of absolute velocities. He postulates that all laws of nature can and should be enunciated in such forms that they are as true in these forms for one observer as for another, even though these observers with their frames of reference be in motion relative to one another.]264
[There are various ways of stating the principle of the relativity of uniform motion which has been thus arrived at, and which forms the basis of the Special Theory of Einstein. If we care to emphasize the rôle of mathematics and the reference frame we may say that]* [any coordinate system having a uniform rectilinear motion with respect to the bodies under observation may be interchangeably used with any other such system in describing their motions;]232 [or that the unaccelerated motion of a system of reference cannot be detected by observations made on this system alone.]194 [Or we can let this aspect of the matter go, and state the relativity postulate in a form more intelligible to the non-mathematician by simply insisting that it is impossible by any means whatever to distinguish any other than the relative motion between two systems that are moving uniformly. As Dr. Russell puts it on a later page, we can assume boldly that the universe is so constituted that uniform straight-ahead motion of an observer and all his apparatus will not produce any difference whatever in the result of any physical process or experiment of any kind.
As we have seen, this is entirely reasonable, on philosophical grounds, until we come to consider the assumptions of the past century with regard to light and its propagation. On the basis of these assumptions we had expected the Michelson-Morley experiment to produce a result negativing the notion of universal relativity. It refused to do this, and we agree with
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Einstein that the best explanation is to return to the notion of relativity,

rather than to invent a forced and special hypothesis to account for the

experiment’s failure. But we must now investigate the assumptions

[78]

underlying the theory of light, and remove the one that requires the ether to

serve as a universal standard of absolute motion.

Light and the Ether
It is among the possibilities that the wave theory of light itself will in the end be more or less seriously modified. It is even more definitely among the possibilities that the ether will be discarded.]* [Certainly when Lord Kelvin estimates that its mass per cubic centimeter is .000,000,000,000,000,001 gram, while Sir Oliver Lodge insists that the correct figure is 1,000,000,000,000,000 grams, it is quite evident that we know so little about it that it is better to get along without it if we can.]216 [But to avoid confusion we must emphasize that Einstein makes no mention whatsoever of the ether; his theory is absolutely independent of any theory of the ether.]139 [Save as he forbids us to employ the ether as a standard of absolute motion, Einstein does not in the least care what qualities we assign to it, or whether we retain it at all. His demands are going to be made upon light itself, not upon the alleged medium of light transmission.
When two observers in relative motion to one another measure their velocities with respect to a third material object, they expect to get different results. Their velocities with regard to this object properly differ, for it is no more to be taken as a universal super-observer than either of them. But if they get different results when they come to measure the velocity with which light passes their respective systems, relativity is challenged. Light is with some propriety to be regarded as a universal observer; and if it will measure our velocities against each other we cannot deny it rank as an absolute standard. If we are not prepared to abandon universal relativity, and adopt one of the “fluke” explanations for the Michelson-Morley result, we must boldly postulate that in free space light presents the same velocity C to all observers—whatever the source of the light, whatever the relative motion between source and observer, whatever the relative motion between the several observers. The departure here from the old assumption lies in the circumstance that the old physics with its ether assigned to light a velocity universally constant in this ether; we have stopped talking about the medium and have made the constant C refer to the observer’s measured value of the velocity of light with regard to himself.
We are fortified in this assumption by the Michelson-Morley result and by all other observations bearing directly upon the matter. Nevertheless, as Mr. Francis says in his essay, we feel instinctively that space and time are not so constituted as to make it possible, if I pass you at 100 miles per hour, for the same light-impulse to pass us both at the same speed C.]* [The implicit assumptions underlying this feeling, be they true or false, are now so
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interwoven with the commonly received notions of space and time that any

theory which questions them has all the appearance of a fantastic and unthinkable thing.]115 [We cannot, however, go back on our relativity; so

when]* [Einstein shows us that an entirely new set of time and space

[80]

concepts is necessary to reconcile universe relativity with this fundamental fact of the absolute constancy of the observed velocity of light in vacuo,]18

[all that is left for us to do is to inquire what revisions are necessary, and

submit to them.]*

[The conceptual difficulties of the theory arise principally from attributing to space and time the properties of things. No portion of space can be compared with another, save by convention; it is things which we compare. No interval of time can be compared with another, save by convention. The first has gone when the second becomes “now”.]149 [It is events that we compare, through the intervention of things. Our measurements are never of space or of time, but only of the things and the events that occupy space and time. And since the measurements which we deal with as though they were of space and of time lie at the foundation of all physical science, while at the same time themselves constituting, as we have seen, the only reality of which we are entitled to speak, it is in order to examine with the utmost care the assumptions underlying them. That there are such assumptions is clear—the very possibility of making measurements is itself an assumption, and every technique for carrying them out rests on an assumption. Let us inquire which of these it is that relativity asks us to revise.]*

The Measurement of Time and Space
[Time is generally conceived as perfectly uniform. How do we judge about it? What tells us that the second just elapsed is equal to the one following? By the very nature of time the superposition of its successive intervals is impossible. How then can we talk about the relative duration of these intervals? It is clear that any relationship between them can only be conventional.]178 [As a matter of fact, we habitually measure time in terms of moving bodies. The simplest method is to agree that some entity moves with uniform velocity. It will be considered as travelling equal distances in equal intervals of time, the distances to be measured as may be specified by our assumptions governing this department of investigation.]179 [The motions of the earth through which we ultimately define the length of day and year, the division of the former into 86,400 “equal” intervals as defined by the motions of pendulum or balance wheel through equal distances, are examples of this convention of time measurement. Even when we correct the motions of the earth, on the basis of what our clocks tell us of these motions, we are following this lead; the earth and the clocks fall out, it is plain that one of them does not satisfy our assumption of equal lengths in equal times, and we decide to believe the clock.]*
[The foregoing concerning time may be accepted as inherent in time itself. But concerning lengths it may be thought that we are able to verify
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absolutely their equality and especially their invariability. Let us have the

audacity to verify this statement. We have two lengths, in the shape of two

rods, which coincide perfectly when brought together. What may we

conclude from this coincidence? Only that the two rods so considered have

[82]

equal lengths at the same place in space and at the same moment. It may

very well be that each rod has a different length at different locations in

space and at different times; that their equality is purely a local matter. Such

changes could never be detected if they affected all objects in the universe.

We cannot even ascertain that both rods remain straight when we transport

them to another location, for both can very well take the same curvature

and we shall have no means of detecting it.

Euclidean geometry assumes that geometrical objects have sizes and shapes independent of position and of orientation in space, and equally invariable in time. But the properties thus presupposed are only conventional and in
no way subject to direct verification. We cannot even ascertain space to be independent of time, because when comparing geometrical objects we have to conceive them as brought to the same place in space and in time.]178 [Even the statement that when they are made to coincide their lengths are equal is, after all, itself an assumption inherent in our ideas of what
constitutes length. And certainly the notion that we can shift them from place to place and from moment to moment, for purposes of comparison, is an assumption; even Euclid, loose as he was from modern standards in this business of “axioms,” knew this and included a superposition axiom among his assumptions.

As a matter of fact, this procedure for determining equality of lengths is not

always available. It assumes, it will be noted, that we have free access to

the object which is to be measured—which is to say, it assumes that this

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object is at rest with respect to us. If it is not so at rest, we must employ at

least a modification of this method; a modification that will in some manner

involve the sending of signals. Even when we employ the Euclidean

method of superposition directly, we must be assured that the respective

ends of the lengths under comparison coincide at the same time. The

observer cannot be present at both ends simultaneously; at best he can only

be present at one end and receive a signal from the other end.

The Problem of Communication
Accordingly, in making the necessary assumptions to cover the matter of measuring lengths, we must make one with regard to the character of the signals which are to be employed for this purpose. If we could assume a system of signalling that would consume no time in transmission all would be simple enough. But we have no experience with such a system. Even if we believe that it ought to be possible thus to transmit signals at infinite velocity, we may not, in the absence of our present ability to do this, assume that it is possible. So we may only assume, with Einstein, that for our signals we shall employ the speediest messenger with which we are at
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present acquainted. This of course is light, the term including any of the electromagnetic impulses that travel at the speed C.

Of course in the vast majority of cases the distance that any light signal in

which we are interested must go to reach us is so small that the time taken

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by its transmission can by no means be measured. We are then, to all intents

and purposes, at both places—the point of origin of the signal and the point

of receipt—simultaneously. But this is not the question at all. Waiving the

fact that in astronomical investigations this approximation no longer holds,

the fact remains that it is, in every case, merely an approximation.

Approximations are all right in observations, where we know that they are

approximations and act accordingly. But in the conceptual universe that

parallels the external reality, computation is as good an agent of observation

as visual or auditory or tactile sensation; if we can compute the error

involved in a wrong procedure the error is there, regardless of whether we

can see it or not. We must have methods which are conceptually free from

error; and if we attempt to ignore the velocity of our light signals we do not

meet this condition.

The measurement of lengths demands that we have a criterion of

simultaneity between two remote points—remote in inches or remote in

light-years, it does not matter which. There is no difficulty in defining

simultaneity of two events that fall in the same point—or rather, in agreeing

that we know what we mean by such simultaneity. But with regard to two

events that occur in remote places there may be a question. A scientific

definition differs from a mere description in that it must afford us a means

of testing whether a given item comes under the definition or not. There is

some difficulty in setting up a definition of simultaneity between distant

events that satisfies this requirement. If we try simply to fall back upon our

[85]

inherent ideas of what we mean by “the same instant” we see that this is not

adequate. We must lay down a procedure for determining whether two

events at remote points occur at “the same instant,” and check up alleged

simultaneity by means of this procedure.

Einstein says, and we must agree with him, that he can find but one reasonable definition to cover this ground. An observer can tell whether he is located half way between two points of his observation; he can have mirrors set up at these points, send out light-signals, and note the time at which he gets back the reflection. He knows that the velocity of both signals, going and coming, is the same; if he observes that they return to him together so that their time of transit for the round trip is the same, he must accept the distances as equal. He is then at the mid-point of the line joining the two points under observation; and he may define simultaneity as follows, without introducing anything new or indeterminate: Two events are simultaneous if an observer midway between them sees them at the same instant, by means, of course, of light originating at the points of occurrence.]*

[It is this definition of simultaneity, coupled with the assumption that all observers, on whatever uniformly moving systems, would obtain the same experimental value for the velocity of light, that leads to the apparent

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paradoxes of the Special Theory of Relativity. If it be asked why we adopt

it, we must in turn ask the inquirer to propose a better system for defining

simultaneous events on different moving bodies.]198

[86]

[There is nothing in this definition to indicate, directly, whether simultaneity persists for all observers, or whether it is relative, so that events simultaneous to one observer are not so to another. The question must then be investigated; and the answer, of course, will hinge upon the possibility of making proper allowances for the time of transit of the light signals that may be involved. It seems as though this ought to be possible; but a simple experiment will indicate that it is not, unless the observers involved are at rest with respect to one another.

An Einsteinian Experiment
Let us imagine an indefinitely long, straight railroad track, with an observer located somewhere along it at the point M. According to the convention suggested above, he has determined points A and B in opposite directions from him along the

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track, and equally distant from him. We shall imagine, further, than a beneficent Providence supplies two lightning flashes, one striking at A and one at B, in such a way that observer M finds them to be simultaneous.
While all this is going on, a train is passing—a very long train, amply long enough to overlap the section AMB of the track. Among the passengers there is one, whom we may call M′, who is directly opposite M at the instant when, according to M, the lightning strikes. Observe he is not opposite M when M sees the flashes, but a brief time earlier—at the instant when, according to M’s computation, the simultaneous flashes occurred. At this instant there are definitely determined the points A′ and B′, on the train; and since we may quite well think of the two systems—train-system and track-system—as in coincidence at this instant, M′ is midway between A′ and B′, and likewise is midway between A and B.
Now if we think of the train as moving over the track in the direction of the arrow, we see very easily that M′ is running away from the light from A and toward that from B, and that, despite—or if you prefer because of—the uniform velocity of these light signals, the one from B reaches him, over a slightly shorter course, sooner than the one from A, over the slightly longer course. When the light signals reach M, M′ is no longer abreast of him but
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has moved along a wee bit, so that at this instant when M has the two signals, one of these has passed M′ and the other has yet to reach him. The upshot is that the events which were simultaneous to M are not so to M′.

It will probably be felt that this result is due to our having, somewhat

unjustifiably and inconsistently, localized on the train the relative motion

between train and track. But if we think of the track as sliding back under

the train in the direction opposite to the arrow, and carrying with it the

points A and B; and if we remember that this in no way affects M’s

[88]

observed velocity of light or the distances AM and BM as he observes them:

we can still accept his claim that the flashes were simultaneous. Then we

have again the same situation: when the flashes from A and from B reach M

at the same moment, in his new position a trifle to the left of his initial

position of the diagram, the flash from A has not yet reached M′ in his

original position while that from B has passed him. Regardless of what

assumption we make concerning the motion between train-system and

track-system, or more elegantly regardless of what coordinate system we

use to define that motion, the event at B precedes that at A in the

observation of M′. If we introduce a second train moving on the other track

in the opposite direction, the observer on it will of course find that the flash

at A precedes that at B—a disagreement not merely as to simultaneity but

actually as to the order of two events! If we conceive the lightning as

striking at the points A′ and B′ on the train, these points travel with M′

instead of with M; they are fixed to his coordinate system instead of to the

other. If you carry out the argument now, you will find that when the

flashes are simultaneous to M′, the one at A precedes that at B in M’s

observation.

A large number of experiments more or less similar in outline to this one

can be set up to demonstrate the consequences, with regard to measured

values of time and space, of relative motion between two observers. I do

not believe that a multiplicity of such demonstrations contributes to the

intelligibility of the subject, and it is for this reason that I have cut loose

from immediate dependence upon the essayists in this part of the

[89]

discussion, concentrating upon the single experiment to which Einstein

himself gives the place of importance.

Who Is Right?
We may permit Mr. Francis to remind us here that neither M nor M′ may correct his observation to make it accord with the other fellow’s. The one who does this is admitting that the other is at absolute rest and that he is himself in absolute motion; and this cannot be. They are simply in disagreement as to the simultaneity of two events, just as two observers might be in disagreement about the distance or the direction of a single event. This can mean nothing else than that, under the assumptions we have made, simultaneity is not an absolute characteristic as we had supposed it to be, but, like distance and direction, is in fact merely a relation between
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observer and objective, and therefore depends upon the particular observer who happens to be operating and upon the reference frame he is using.

But this is serious. My time measurements depend ultimately upon my

space measurements; the latter, and hence both, depend closely upon my

ideas of simultaneity. Yours depend upon your reading of simultaneity in

precisely the same way.]* [Suppose the observer on the track, in the above

experiment, wants to measure the length of something on the car, or the

observer on the car something on the track. The observer, or his assistant,

must be at both ends of the length to be measured at the same time, or get

simultaneous reports in some way from these ends; else they will obtain

[90]

false results. It is plain, then, that with different criteria of what the “same

time” is, the observers in the two systems may get different values for the measured lengths in question.]220

[Who is right? According to the principle of relativity a decision on this question is absolutely impossible. Both parties are right from their own points of view; and we must admit that two events in two different places may be simultaneous for certain observers, and yet not simultaneous for other observers who move with respect to the first ones. There is no contradiction in this statement, although it is not in accordance with common opinion, which believes simultaneousness to be something absolute. But this common opinion lacks foundation. It cannot be proved by direct perception, for simultaneity of events can be perceived directly,]24 [and in a manner involving none of our arbitrary assumptions,]* [only if they happen at the same place; if the events are distant from each other, their simultaneity or succession can be stated only through some method of communicating by signals. There is no logical reason why such a method should not lead to different results for observers who move with regard to one another.

From what we have said, it follows immediately that in the new theory not

only the concept of simultaneousness but also that of duration is revealed as dependent on the motion of the observer.]24 [Demonstration of this should

be superfluous; it ought to be plain without argument that if two observers

cannot agree whether two instants are the same instant or not, they cannot

[91]

agree on the interval of time between instants. In the very example which

we have already examined, one observer says that a certain time-interval is

zero, and another gives it a value different from zero. The same thing

happens whenever the observers are in relative motion.]* [Two physicists

who measure the duration of a physical process will not obtain the same

result if they are in relative motion with regard to one another.

They will also find different results for the length of a body. An observer who wants to measure the length of a body which is moving past him must in one way or another hold a measuring rod parallel to its motion and mark those points on his rod with which the ends of the body come into simultaneous coincidence. The distance between the two marks will then indicate the length of the body. But if the two markings are simultaneous for one observer, they will not be so for another one who moves with a different velocity, or who is at rest, with regard to the body under

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observation. He will have to ascribe a different length to it. And there will be no sense in asking which of them is right: length is a purely relative concept, just as well as duration.]24

The Relativity of Time and Space

[The degree to which distance and time become relative instead of absolute quantities under the Special Theory of Relativity can be stated very definitely. In the first place, we must point out that the relativity of lengths applies with full force only to lengths that lie parallel to the direction of relative motion. Those that lie exactly perpendicular to that direction come out the same for both observers; those that lie obliquely to it show an effect, depending upon the angle, which of course becomes greater and greater as the direction of parallelism is approached.

The magnitude of the effect is easily demonstrated, but with this

demonstration we do not need to be concerned here. It turns out that if an

observer moving with a system finds that a certain time interval in the

system is T seconds and that a certain length in the system is L inches, then

an observer moving parallel with L and with a velocity v relative to the

system will find for these the respective values

and

, where

[Contents] [92]

C in this expression of course represents the velocity of light. It will be

noted that the fraction

is ordinarily very small; that the expression

under the radical is therefore less than 1 but by a very slight margin; and

that the entire expression K is itself therefore less than 1 but by an even

slighter margin. This means, then, that the observer outside the system finds

the lengths in the system to be a wee bit shorter and the time intervals a wee

bit longer than does the observer in the system. Another way of putting the

matter is based, ultimately, upon the fact that in order for the observer in the

system to get the larger value for distance and the smaller value for time, his

measuring rod must go into the distance under measurement more times

than that of the moving observer, while his clock must beat a longer second

in order that less of them shall be recorded in a given interval between two

events. So it is often said that the measuring rod as observed from without

is contracted and the clock runs slow. This does not impress me as a happy

statement, either in form or in content.]*

[The argument that these formulae are contradicted by human experience can be refuted by examining a concrete instance. If a train is 1,000 feet long at rest, how long will it be when running a mile a minute?]232 [I have quoted this question exactly as it appears in the essay from which it is taken, because it is such a capital example of the objectionable way in
which this business is customarily put. For the statement that lengths decrease and time-intervals increase “with velocity” is not true in just this

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form. The velocity, to have meaning, must be relative to some external

system; and it is the observations from that external system that are

affected. So long as we confine ourselves to the system in which the alleged

modifications of size are stated as having taken place, there is nothing to

observe that is any different from what is usual; there is no way to establish

that we are enjoying a velocity, and in fact within the intent of the relativity

theory we are not enjoying a velocity, for we are moving with the objects

which we are observing. It is inter-systemic observations, and these alone,

that show the effect. When we travel with the system under observation, we

get the same results as any other observer on this system; when we do not

[94]

so travel, we must conduct our observations from our own system, in

relative motion to the other, and refer our results to our system.

Now when no particular observer is specified, we must of course assume an observer connected with the train, or with whatever the body mentioned. To that observer it doesn’t make the slightest difference what the train does; it may stand at rest with respect to some external system or it may move at any velocity whatsoever; its length remains always 1,000 feet. In order for this question to have the significance which its propounder means it to have, I must restate it as follows: A train is 1,000 feet long as measured by an observer travelling with it. If it passes a second observer at 60 miles per hour, what is its length as observed by him? The answer is now easy.]* [According to the formula the length of the moving train as seen from the ground will be

feet, a change entirely too small for detection by the most delicate instruments. Examination of the expression K shows that in so far as terrestrial movements of material objects are concerned it is equal to 1]232 [within a far smaller margin than we can ever hope to make our observations. Even the diameter of the earth, as many of the essayists point
out, will be shortened only 2½ inches for an outside observer past whom it rushes with its orbital speed of 18.5 miles per second. But slight as the difference may be in these familiar cases, its scientific importance remains the same.]*

Relativity and Reality

[A simple computation shows that this effect is exactly the amount suggested by Lorentz and Fitzgerald to explain the Michelson-Morley experiment.]188 [This ought not to surprise us, since both that explanation and the present one are got up with the same purpose. If they both achieve that purpose they must, numerically, come to the same thing in any numerical case. It is, however, most emphatically to be insisted that the present “shortening” of lengths]* [no longer appears as a “physical”
shortening caused by absolute motion through the ether but is simply a

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188

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result of our methods of measuring space and time.]188 [Where Fitzgerald

and Lorentz had assumed that a body in motion has its dimensions shortened in the direction of its motion,]220 [this very form of statement

ceases to possess significance under the relativity assumption.]* [For if we

cannot tell which of two bodies is moving, which one is shortened? The

answer is, both—for the other fellow. For each frame of reference there is a

scale of length and a scale of time, and these scales for different frames are related in a manner involving both the length and the time.]220 [But we

must not yield to the temptation to say that all this is not real; the

confinement of a certain scale of length and of time to a single observation

system does not in the least make it unreal.]* [The situation is real—as real

[96]

as any other physical event.]165

[The word physical is used in two senses in the above paragraph. It is

denied that the observed variability in lengths indicates any “physical”

contraction or shrinkage; and on the heels of this it is asserted that this

observed variability is of itself an actual “physical” event. It is difficult to

express in words the distinction between the two senses in which the term

physical is employed in these two statements, but I think this distinction

ought to be clear once its existence is emphasized. There is no material

contraction; it is not right to say that objects in motion contract or are

shorter; they are not shorter to an observer in motion with them. The whole

thing is a phenomenon of observation. The definitions which we are

obliged to lay down and the assumptions which we are obliged to make in

order, first, that we shall be able to measure at all, and second, that we shall

be able to escape the inadmissible concept of absolute motion, are such that

certain realities which we had supposed ought to be the same for all

observers turn out not to be the same for observers who are in relative

motion with respect to one another. We have found this out, and we have

found out the numerical relation which holds between the reality of the one

observer and that of the other. We have found that this relation depends

upon nothing save the relative velocity of the two observers. As good a way

of emphasizing this as any is to point out that two observers who have the

same velocity with respect to the system under examination (and whose

mutual relative velocity is therefore zero) will always get the same results

[97]

when measuring lengths and times on that system. The object does not go

through any process of contraction; it is simply shorter because it is

observed from a station with respect to which it is moving. Similar remarks

might be made about the time effect; but the time-interval is not so easily

visualized as a concrete thing and hence does not offer such temptation for

loose statement.

The purely relative aspect of the matter is further brought out if we consider a single example both backwards and forwards. Systems S and S′ are in relative motion. An object in S which to an observer in S is L units long, is shorter for an observer in S′—shorter by an amount indicated through the “correction factor” K. Now if we have, in the first instance, made the objectionable statement that objects are shorter in system S′ than they are in S, it will be quite natural for us to infer from this that objects in S must be longer than those in S′; and from this to assert that when the observer in S measures objects lying in S′, he gets for them greater lengths than does the
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home observer in S′. But if we have, in the first instance, avoided the objectionable statement referred to, we shall be much better able to realize that the whole business is quite reciprocal; that the phenomena are symmetric with respect to the two systems, to the extent that we can interchange the systems in any of our statements without modifying the statements in any other way.

Objects in S appear shorter and times in S appear longer to the external

“moving” observer in S′ than they do to the domestic observer in S. Exactly

in the same way, objects in S′ appear shorter to observers in the foreign

[98]

system S than to the home observer in S′, who remains at rest with respect

to them. I think that when we get the right angle upon this situation, it loses

the alleged startling character which has been imposed upon it by many

writers. The “apparent size” of the astronomer is an analogy in point.

Objects on the moon, by virtue of their great distance, look smaller to

observers on the earth than to observers on the moon. Do objects on the

earth, on this account, look larger to a moon observer than they do to us?

They do not; any suggestion that they do we should receive with

appropriate scorn. The variation in size introduced by distance is reciprocal,

and this reciprocity does not in the least puzzle us. Why, then, should that

introduced by relative motion puzzle us?

Time and Space in a Single Package
Our old, accustomed concepts of time and space, which have grown up through countless generations of our ancestors, and been handed down to us in the form in which we are familiar with them, leave no room for a condition where time intervals and space intervals are not universally fixed and invariant. They leave no room for us to say that]* [one cannot know the time until he knows where he is, nor where he is until he knows the time,]220 [nor either time or place until he knows something about velocity. But in this concise formulation of the difference between what we have always believed and what we have seen to be among the consequences of Einstein’s postulates of the universal relativity of uniform motion, we may at once locate the assumption which, underlying all the old ideas, is the root of all the trouble. The fact is we have always supposed time and space to be absolutely distinct and independent entities.]*
[The concept of time has ever been one of the most absolute of all the categories. It is true that there is much of the mysterious about time; and philosophers have spent much effort trying to clear up the mystery—with unsatisfactory results. However, to most persons it has seemed possible to adopt an arbitrary measure or unit of duration and to say that this is absolute, independent of the state of the body or bodies on which it is used for practical purposes.]272 [Time has thus been regarded as something which of itself flows on regularly and continuously, regardless of physical events concerning matter.]150 [In other words, according to this view, time is not affected by conditions or motions in space.]272 [We have deliberately
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chosen to ignore the obvious fact that time can never appear to us, be

measured by us, or have the least significance for us, save as a measure of

something that is closely tied up with space and with material space-

dimensions. Not merely have we supposed that time and space are

separated in nature as in our easiest perceptions, but we have supposed that

they are of such fundamentally distinct character that they can never be tied

up together. In no way whatever, assumes the Euclidean and Newtonian

intellect, may space ever depend upon time or time upon space. This is the

assumption which we must remove in order to attain universal relativity;

and while it may come hard, it will not come so hard as the alternative. For

[100]

this alternative is nothing other than to abandon universal relativity. This

course would leave us with logical contradictions and discrepancies that

could not be resolved by any revision of fundamental concepts or by any

cleaning out of the Augean stables of old assumptions; whereas the

relativity doctrine as built up by Einstein requires only such a cleaning out

in order to leave us with a strictly logical and consistent whole. The rôle of

Hercules is a very difficult one for us to play. Einstein has played it for the

race at large, but each of us must follow him in playing it for himself.

Some Further Consequences

I need not trespass upon the subject matter of those essays which appear in

full by going here into any details with regard to the manner in which time

and space are finally found to depend upon one another and to form the

parts of a single universal whole. But I may appropriately point out that if

time and space are found to be relative, we may surely expect some of the

less fundamental concepts that depend upon them to be relative also. In this

expectation we are not disappointed. For one thing,]* [mass has always

been assumed to be a constant, independent of any motion or energy which

it might possess. Just as lengths and times depend upon relative motion,

however, it is found that mass, which is the remaining factor in the

expression for energy due to motion, also depends upon relative velocities.

The dependence is such that if a body takes up an amount of energy E with

respect to a certain system, the body behaves, to measurements made from

that system, as though its mass had been increased by an amount

,

where C is as usual the velocity of light.]194

[This should not startle us. The key to the situation lies in the italicized words above, which indicate that the answer to the query whether a body has taken up energy or not depends upon the seat of observation. If I take up my location on the system S, and you on the system S′, and if we find that we are in relative motion, we must make some assumption about the energy which was necessary, initially, to get us into this condition. Suppose
we are on two passing trains.]* [The chances are that either of us will assume that he is at rest and that it is the other train which moves, although if sufficiently sophisticated one of us may assume that he is moving and that the other train is at rest.]272 [Whatever our assumption, whatever the

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system, the localization of the energy that is carried in latent form by our

systems depends upon this assumption. Indeed, if our systems are of

differing mass, our assumptions will even govern our ideas of the amount

of energy which is represented by our relative motion; if your system be the

more massive, more energy would have to be localized in it than in mine to

produce our relative motion. If we did not have the universal principle of

relativity to forbid, we might make an arbitrary assumption about our

motions and hence about our respective latent energies; in the presence of

this veto, the only chance of adjustment lies in our masses, which must

differ according to whether you or I observe them.]*

[102]

[For most of the velocities with which we are familiar

is, like the

difference between K and unity, such an extremely small quantity that the

most delicate measurements fail to detect it. But the electrons in a highly

evacuated tube and the particles shot out from radioactive materials attain

in some cases velocities as high as eight-tenths that of light. When we

measure the mass of such particles at different velocities we find that it

actually increases with the velocity, and in accordance with the foregoing law.]194 [This observation, in fact, antedates Einstein’s explanation, which

is far more satisfactory than the earlier differentiation between “normal

mass” and “electrical mass” which was called upon to account for the

increase.]*

[But if the quantity

is to be considered as an actual increase in mass,

may it not be possible that all mass is energy? This would lead to the

conclusion that the energy stored up in any mass is . The value is very

great, since C is so large; but it is in good agreement with the internal

energy of the atom as calculated from other considerations. It is obvious that conservation of mass and of momentum cannot both hold good under a

theory that translates the one into the other. Mass is then not considered by Einstein as conservative in the ordinary sense, but it is the total quantity of

mass plus energy in any closed system that remains constant. Small amounts of energy may be transformed into mass, and vice versa.]194

[Other features of the theory which are often displayed as consequences are

really more in the nature of assumptions. It will be recalled that when we

[103]

had agreed upon the necessity of employing signals of some sort, we

selected as the means of signalling the speediest messenger with which we

happened to be acquainted. Our subsequent difficulties were largely due to

the impossibility of making a proper allowance for this messenger’s speed,

even though we knew its numerical value; and as a consequence, this speed

enters into our formulae. Now we have not said in so many words that C is

the greatest speed attainable, but we have tacitly assumed that it is. We need

not, therefore, be surprised if our formulae give us absurd results for speeds

higher than C, and indicate the impossibility of ever attaining these.

Whatever we put into a problem the algebra is bound to give us back. If we

look at our formula for K, we see that in the event of v equalling C, lengths

become zero and times infinite. The light messenger itself, then, has no

dimension; and for it time stands still.

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If we suppose v to be greater than C, we get even more bizarre results, for then the factor K is the square root of a negative number, or as the mathematician calls it an “imaginary” quantity; and with it, lengths and times become imaginary too.

The fact that time stops for it, and the fact that it is the limiting velocity,

give to C certain of the attributes of the mathematician’s infinity. Certainly

if it can never be exceeded, we must have a new formula for the

composition of velocities. Otherwise when my system passes yours at a

speed of 100,000 miles per second, while yours passes a third in the same

direction at the same velocity, I shall be passing this third framework at the

[104]

forbidden velocity of 200,000 miles per second—greater than C. In fact

Einstein is able to show that an old formula, which had already been found

to connect the speed of light in a material medium with the speed of that

medium, will now serve universally for the composition of velocities.

When we combine the velocities v and u, instead of getting the resultant

as we would have supposed, we get the resultant

or

This need not surprise us either, if we will but reflect that the second velocity effects a second revision of length and time measurements between the systems involved. And now, if we let either v, or u, or even both of them, take the value C, the resultant still is C. In another way we have found C to behave like the mathematician’s infinity, to which, in the words of the blind poet, if we add untold thousands, we effect no real increment.

Assumption and Consequence
A good many correspondents who have given the subject sufficient thought to realize that the limiting character of the velocity C is really read into Einstein’s system by assumption have written, in more or less perturbed inquiry, to know whether this does not invalidate the whole structure. The answer, of course, is yes—provided you can show this assumption to be invalid. The same answer may be made of any scientific doctrine whatever, and in reference to any one of the multitudinous assumptions underlying it. If we were to discover, tomorrow, a way of sending signals absolutely instantaneously, Einstein’s whole structure would collapse as soon as we had agreed to use this new method. If we were to discover a signalling agent with finite velocity greater than that of light, relativity would persist with this velocity written in its formulae in the place of C.
It is a mistake to quote Einstein’s theory in support of the statement that such a velocity can never be. An assumption proves its consequences, but
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never can prove itself; it must remain always an assumption. But in the

presence of long human experience supporting Einstein’s assumption that

no velocity in excess of C can be found, it is fair to demand that it be

disputed not with argument but with demonstration. The one line of

argument that would hold out a priori hope of reducing the assumption to

an absurdity would be one based on the familiar idea of adding velocities;

but Einstein has spiked this argument before it is started by replacing the

direct addition of velocities with another method of combining them that

fits in with his assumption and as well with the observed facts. The burden

of proof is then on the prosecution; anyone who would contradict our

assertion that C is the greatest velocity attainable may do so only by

showing us a greater one. Until this has been done, the admission that it

may properly be attempted can in no way be construed as a confession of

weakness on the part of Einstein.

[106]

It may be well to point out that in no event may analogy be drawn with sound, as many have tried to do. In the first place sound requires a material medium and its velocity with regard to this rather than relative to the observer we know to be fixed; in the second place, requiring a material medium, sound is not a universal signalling agent; in the third place, we know definitely that its velocity can be exceeded, and are therefore barred from making the assumption necessary to establish the analogy. The very extraordinary behavior of light in presenting a velocity that is the same for all observers, and in refusing to betray the least material evidence of any medium for its transmission, rather fortifies us in believing that Einstein’s assumption regarding the ultimate character of this velocity is in accord with the nature of things.

Relativity and the Layman
A great deal can be said in the direction of general comment making the Special Theory and its surprising accompaniments easier of acceptance, and we shall conclude the present discussion by saying some of these things.]* [It has been objected that the various effects catalogued above are only apparent, due to the finite velocity of light—that the real shape and size of a body or the real time of an event cannot be affected by the point of view or the motion of an observer. This argument would be perfectly valid, if there were real times and distances; but there are not. These are earth-bound notions, due to our experience on an apparently motionless platform, with slow-moving bodies. Under these circumstances different observations of the same thing or of the same event agree. But when we no longer have the solid earth to stand on, and are dealing with velocities so high that the relativity effects become appreciable, there is no standard by which to resolve the disagreements. No one of the observations can claim to be nearer reality than any other. To demand the real size of a thing is to demand a stationary observer or an instantaneous means of information. Both are impossible.
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When relativity asks us to give up our earth-bound notions of absolute space and absolute time the sensation, at first, is that we have nothing left to stand on. So must the contemporaries of Columbus have felt when told that the earth rested on—nothing. The remedy too is similar. Just as they had to be taught that falling is a local affair, that the earth is self-contained, and needs no external support—so we must be taught that space and time standards are local affairs. Each moving body carries its own space and time standards with it; it is self-contained. It does not need to reach out for eternal support, for an absolute space and time that can never quite be attained. All we ever need to know is the relation of the other fellow’s space and time standards to our own. This is the first thing relativity teaches us.]141

[The consequences of Einstein’s assumptions have led many to reject the

theory of relativity, on the ground that its conclusions are contrary to

common sense—as they undoubtedly are. But to the contemporaries of

Copernicus and Galileo the theory that the earth rotates on its axis and

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revolves around the sun was contrary to common sense; yet this theory

prevailed. There is nothing sacred about common sense; in the last analysis

its judgments are based on the accumulated experience of the human race.

From the beginning of the world up to the present generation, no bodies

were known whose velocities were not extremely small compared with that

of light. The development of modern physics has led to discovery of very

much larger velocities, some as high as 165,000 miles per second. It is not

to be wondered at that such an enlargement of our experience requires a

corresponding enlargement or generalization of the concepts of space and

time. Just as the presupposition of primitive man that the earth was flat had

to be given up in the light of advancing knowledge, so we are now called

upon to give up our presupposition that space and time are absolute and

independent in their nature.

The reader must not expect to understand the theory of relativity in the sense of making it fit in with his previous ideas. If the theory be right these ideas are wrong and must be modified, a process apt to be painful.]223 [All the reader can do is to become familiar with the new concepts, just as a child gets used to the simple relations and quantities he meets until he “understands” them.]221 [Mr. Francis has said something of the utmost significance when he points out that “understanding” really means nothing in the world except familiarity and accustomedness.]* [The one thing about the relativity doctrine that we can hope thus to understand at once and without pain is the logical process used in arriving at our results.]221 [Particularly is it hard to give a satisfactory explanation of the theory in popular language, because the language itself is based on the old concepts; the only language which is really adequate is that of mathematics.]223 [Unless we have, in addition to the terms of our ordinary knowledge, a set of definitions that comes with a wide knowledge of mathematics and a lively sense of the reality of mathematical constructions, we are likely to view the theory of relativity through a fog of familiar terms suddenly become self-contradictory and deceptive. Not that we are unfamiliar with the idea that some of our habitual notions may be wrong; but knowledge of their illusory nature arises and becomes convincing only with time. We may
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now be ready to grant that the earth, seemingly so solid, is really a whirling globe rushing through space; but we are no more ready immediately to accept the bald assertion that this space is not what it seems than our ancestors were to accept the idea that the earth was round or that it moved.]156 [What we must have, if we are to comprehend relativity with any degree of thoroughness, is the mathematician’s attitude toward his assumptions, and his complete readiness to swap one set of assumptions for another as a mere part of the day’s work, the spirit of which I have endeavored to convey in the chapter on non-Euclidean geometry.]*

Physics vs. Metaphysics
[The ideas of relativity may seem, at first sight, to be giving us a new and metaphysical theory of time and space. New, doubtless; but certainly the theory was meant by its author to be quite the opposite of metaphysical. Our actual perception of space is by measurement, real and imagined, of distances between objects, just as our actual perception of time is by measurement. Is it not less metaphysical to accept space and time as our measurements present them to us, than to invent hypotheses to force our perceptual space into an absolute space that is forever hidden from us?]182 [In order not to be metaphysical, we must eliminate our preconceived notions of space and time and motion, and focus our attention upon the indications of our instruments of observation, as affording the only objective manifestations of these qualities and therefore the only attributes which we can consider as functions of observed phenomena.]47 [Einstein has consistently followed the teachings of experience, and completely freed himself from metaphysics.]114 [That this is not always easy to do is clear, I think, if we will recall the highly metaphysical character often taken by the objections to action-at-a-distance theories and concepts; and if we will remind ourselves that it was on purely metaphysical grounds that Newton refused to countenance Huyghens’ wave theory of light. Whether, as in the one case, it leads us to valid conclusions, or, as in the other, to false ones, metaphysical reasoning is something to avoid. Einstein, I think, has avoided it about as thoroughly as anyone ever did.]*

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V
THAT PARALLEL POSTULATE
Modern Geometric Methods; the Dividing Line Between Euclidean and Non-Euclidean; and the Significance of the Latter
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BY THE EDITOR

The science of geometry has undergone a revolution of which the outsider

is not informed, but which it is necessary to understand if we are to attain

any comprehension of the geometric formulation of Einstein’s results; and

especially if we are to appreciate why it is proper and desirable to formulate

these results geometrically at all. The classical geometer regarded his

science from a narrow viewpoint, as the study of a certain set of observed

phenomena—those of the space about us, considered as an entity in itself

and divorced from everything in it. It is clear that some things about that

space are not as they appear (optical illusions), and that other things about it

are true but by no means apparent (the sum-of-squares property of a right

triangle, the formulæ for surface and volume of a sphere, etc.). While many

things about space are “obvious,” these need in the one case disproof and in

the other discovery and proof. With all their love of mental processes for

their own sake, it is then not surprising that the Greeks should have set

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themselves the task of proving by logical process the properties of space,

which a less thoughtful folk would have regarded as a subject only for

observational and experimental determination.

But, abstract or concrete, the logical structure must have a starting point; and it is fair to demand that this consist in a statement of the terms we are going to use and the meanings we are going to attach to them. In other words, the first thing on the program will be a definition, or more probably, several definitions.

Now the modern scientist has a somewhat iconoclastic viewpoint toward definitions, and especially toward the definition of his very most fundamental ideas.

We do not speak here in terms of dictionary definitions. These have for

object the eminent necessity of explaining the meaning and use of a word to

some one who has just met it for the first time. It is easy enough to do this,

if the doer possesses a good command of the language. It is not even a

matter of grave concern that the words used in the definition be themselves

known to the reader; if they are not, he must make their acquaintance too.

Dr. Johnson’s celebrated definition of a needle stands as perpetual evidence

that when he cannot define a simple thing in terms of things still simpler,

the lexicographer is forced to define it in terms of things more complex. Or

we might demonstrate this by noting that the best dictionaries are driven to

define such words as “and” and “but” by using such complex notions as are

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embodied in “connective,” “continuative,” “adversative,” and “particle.”

It is otherwise with the scientist who undertakes to lay down a definition as the basis of further procedure in building up the tissue of his science. Here a degree of rigorous logic is called for which would be as superfluous in the dictionary as the effort there to attain it would be out of place. The scientist, in building up a logical structure that will withstand every assault, must define everything, not in terms of something which he is more or less warranted in supposing his audience to know about, but actually in terms of things that have already been defined. This really means that he must

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explain what he is talking about in terms of simpler ideas and simpler things, which is precisely what the lexicographer does not have to worry about. This is why it is quite trivial to quote a dictionary definition of time or space or matter or force or motion in settlement of a controversy of scientific or semi-scientific nature.

Terms We Cannot Define
But the scientist who attempts to carry out this ideal system of defining everything in terms of what precedes meets one obstacle which he cannot surmount directly. Even a layman can construct a passable definition of a complex thing like a parallelopiped, in terms of simpler concepts like point, line, plane and parallel. But who shall define point in terms of something simpler and something which precedes point in the formulation of geometry? The scientist is embarrassed, not in handling the complicated later parts of his work, but in the very beginning, in dealing with the simplest concepts with which he has to deal.
Suppose a dictionary were to be compiled with the definitions arranged in logical rather than alphabetical order: every word to be defined by the use only of words that have already been defined. The further back toward the beginning we push this project, the harder it gets. Obviously we can never define the first word, or the second, save as synonymous with the first. In fact we should need a dozen words, more or less, to start with—God-given words which we cannot define and shall not try to define, but of which we must agree that we know the significance. Then we have tools for further procedure; we can start with, say, the thirteenth word and define all the rest of the words in the language, in strictly logical fashion.
What we have said about definitions applies equally to statements of fact, of the sort which are going to constitute the body of our science. In the absence of simpler facts to cite as authority, we shall never be able to prove anything, however simple this may itself be; and in fact the simpler it be, the harder it is to find something simpler to underlie it. If we are to have a logical structure of any sort, we must begin by laying down certain terms which we shall not attempt to define, and certain statements which we shall not try to prove. Mathematics, physics, chemistry—in the large and in all their many minor fields—all these must start somewhere. Instead of deceiving ourselves as to the circumstances surrounding their start, we prefer to be quite frank in recognizing that they start where we decide to start them. If we don’t like one set of undefined terms as the foundation, by all means let us try another. But always we must have such a set.
The classical geometer sensed the difficulty of defining his first terms. But he supposed that he had met it when he defined these in words free of technical significance. “A point is that which has position without size” seemed to him an adequate definition, because “position” and “size” are
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words of the ordinary language with which we may all be assumed familiar. But today we feel that “position” and “size” represent ideas that are not necessarily more fundamental than those of “line” and “point,” and that such a definition begs the question. We get nowhere by replacing the undefined terms “point” and “line” and “plane,” which really everybody understands, by other undefined terms which nobody understands any better.
In handling the facts that it was inconvenient to prove, the classical geometer came closer to modern practice. He laid down at the beginning a few statements which he called “axioms,” and which he considered to be so self-evident that demonstration was superfluous. That the term “selfevident” left room for a vast amount of ambiguity appears to have escaped him altogether. His axioms were axioms solely because they were obviously true.

Laying the Foundation
The modern geometer falls in with Euclid when he writes an elementary text, satisfying the beginner’s demand for apparent rigor by defining point and line in some fashion. But when he addresses to his peers an effort to clarify the foundations of geometry to a further degree of rigor and lucidity than has ever before been attained, he meets these difficulties from another quarter. In the first place he is always in search of the utmost possible generality, for he has found this to be his most effective tool, enabling him as it does to make a single general statement take the place and do the work of many particular statements. The classical geometer attained generality of a sort, for all his statements were of any point or line or plane. But the modern geometer, confronted with a relation that holds among points or between points and lines, at once goes to speculating whether there are not other elements among or between which it holds. The classical geometer isn’t interested in this question at all, because he is seeking the absolute truth about the points and lines and planes which he sees as the elements of space; to him it is actually an object so to circumscribe his statements that they may by no possibility refer to anything other than these elements. Whereas the modern geometer feels that his primary concern is with the fabric of logical propositions that he is building up, and not at all with the elements about which those propositions revolve.
It is of obvious value if the mathematician can lay down a proposition true of points, lines and planes. But he would much rather lay down a proposition true at once of these and of numerous other things; for such a proposition will group more phenomena under a single principle. He feels that on pure scientific grounds there is quite as much interest in any one set of elements to which his proposition applies as there is in any other; that if any person is to confine his attention to the set that stands for the physicist’s space, that person ought to be the physicist, not the geometer. If he has produced a tool which the physicist can use, the physicist is welcome to use
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it; but the geometer cannot understand why, on that ground, he should be asked to confine his attention to the materials on which the physicist employs that tool.

It will be alleged that points and lines and planes lie in the mathematician’s domain, and that the other things to which his propositions may apply may not so lie—and especially that if he will not name them in advance he cannot expect that they will so lie. But the mathematician will not admit this. If mathematics is defined on narrow grounds as the science of number, even the point and line and plane may be excluded from its field. If any wider definition be sought—and of course one must be—there is just one definition that the mathematician will accept: Dr. Keyser’s statement that “mathematics is the art or science of rigorous thinking.”

The immediate concern of this science is the means of rigorous thinking—

undefined terms and definitions, axioms and propositions. Its collateral

concern is the things to which these may apply, the things which may be

thought about rigorously—everything. But now the mathematician’s

domain is so vastly extended that it becomes more than ever important for

him to attain the utmost generality in all his pronouncements.

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One barrier to such generalization is the very name “geometry,” with the restricted significance which its derivation and long usage carry. The geometer therefore must have it distinctly understood that for him “geometry” means simply the process of deducing a set of propositions from a set of undefined primitive terms and axioms; and that when he speaks of “a geometry” he means some particular set of propositions so deduced, together with the axioms, etc., on which they are based. If you take a new set of axioms you get a new geometry.

The geometer will, if you insist, go on calling his undefined terms by the

familiar names “point,” “line,” “plane.” But you must distinctly understand

that this is a concession to usage, and that you are not for a moment to

restrict the application of his statements in any way. He would much prefer,

however, to be allowed new names for his elements, to say “We start with

three elements of different sorts, which we assume to exist, and to which

we attach the names A, B and C—or if you prefer, primary, secondary and

tertiary elements—or yet again, names possessing no intrinsic significance

at all, such as ching, chang and chung.” He will then lay down whatever

statements he requires to serve the purposes of the ancient axioms, all of

these referring to some one or more of his elements. Then he is ready for

the serious business of proving that, all his hypotheses being granted, his

elements A, B and C, or I, II and III, or ching, chang and chung, are subject

to this and that and the other propositions.

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The objection will be urged that the mathematician who does all this usurps the place of the logician. A little reflection will show this not to be the case. The logician in fact occupies the same position with reference to the geometer that the geometer occupies with reference to the physicist, the chemist, the arithmetician, the engineer, or anybody else whose primary interest lies with some particular set of elements to which the geometer’s

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system applies. The mathematician is the tool-maker of all science, but he does not make his own tools—these the logician supplies. The logician in turn never descends to the actual practice of rigorous thinking, save as he must necessarily do this in laying down the general procedures which govern rigorous thinking. He is interested in processes, not in their application. He tells us that if a proposition is true its converse may be true or false or ambiguous, but its contrapositive is always true, while its negative is always false. But he never, from a particular proposition “If A is B then C is D,” draws the particular contrapositive inference “If C is not D then A is not B.” That is the mathematician’s business.

The Rôle of Geometry
The mathematician is the quantity-production man of science. In his absence, the worker in each narrower field where the elements under discussion take particular concrete forms could work out, for himself, the propositions of the logical structure that applies to those elements. But it would then be found that the engineer had duplicated the work of the physicist, and so for many other cases; for the whole trend of modern science is toward showing that the same background of principles lies at the root of all things. So the mathematician develops the fabric of propositions that follows from this, that and the other group of assumptions, and does this without in the least concerning himself as to the nature of the elements of which these propositions may be true. He knows only that they are true for any elements of which his assumptions are true, and that is all he needs to know. Whenever the worker in some particular field finds that a certain group of the geometer’s assumptions is true for his elements, the geometry of those elements is ready at hand for him to use.
Now it is all right purposely to avoid knowing what it is that we are talking about, so that the names of these things shall constitute mere blank forms which may be filled in, when and if we wish, by the names of any things in the universe of which our “axioms” turn out to be true. But what about these axioms themselves? When we lay them down in ignorance of the identity of the elements to which they may eventually apply, they cannot by any possibility be “self-evident.” We may, at pleasure, accept as selfevident a statement about points and lines and planes; or one about electrons, centimeters and seconds; or one about integers, fractions, and irrational numbers; or one about any other concrete thing or things whatever. But we cannot accept as self-evident a statement about chings, changs and chungs. So we must base our “axioms” on some other ground than this; and our modern geometer has his ground ready and waiting. He accepts his axioms on the ground that it pleases him to do so. To avoid all suggestion that they are supposed to be self-evident, or even necessarily true, he drops the term “axiom” and substitutes for it the more color-less word “postulate.” A postulate is merely something that we agreed to accept, for the time being, as a basis of further argument. If it turns out to be true, or if we can find circumstances under which and elements to which it
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applies, any conclusions which we deduce from it by trustworthy processes are valid within the same limitations. And the propositions which tell us that, if our postulates are true, such and such conclusions are true—they, too, are valid, but without any reservation at all!

Perhaps an illustration of just what this means will not be out of place. Let

it be admitted, as a postulate, that

is greater, by 1, than

. Let

us then consider the statement: “If

, then

.” We

know—at least we are quite certain—that

is not equal to 65, if by

“7” and “19” and “65” we mean what you think we mean. We are equally

sure, on the same grounds, that

is not equal to 66. But, under the

one assumption that we have permitted ourselves, it is unquestionable that

if

were equal to 65, then

certainly would be equal to 66. So,

while the conclusion of the proposition which I have put in quotation marks

is altogether false, the proposition itself, under our assumption, is entirely

true. I have taken an illustration designed to be striking rather than to

possess scientific interest; I could just as easily have shown a true

proposition leading to a false conclusion, but of such sort that it would be

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of decided scientific interest as telling us one of the consequences of a

certain assumption.

What May We Take for Granted?
This is all very fine; but how does the geometer know what postulates to lay down? One is tempted to say that he is at liberty to postulate anything he pleases, and investigate the results; and that whether or not his postulate ever be realized, the propositions that he deduces from it, being true, are of scientific interest. Actually, however, it is not quite as simple as all that. If it were sufficient to make a single postulate it would be as simple as all that; but it turns out that this is not sufficient any more than it is sufficient to have a single undefined term. We must have several postulates; and they must be such, as a whole, that a geometry flows out of them. The requirements are three.
In the first place, the system of postulates must be “categorical” or complete—there must be enough of them, and they must cover enough ground, for the support of a complete system of geometry. In practice the test for this is direct. If we got to a point in the building up of a geometry where we could not prove whether a certain thing was one way always, or always the other way, or sometimes one way and sometimes the other, we should conclude that we needed an additional postulate covering this ground directly or indirectly. And we should make that postulate—because it is precisely the things that we can’t prove which, in practical work, we agree to assume. Even Euclid had to adopt this philosophy.
In the second place, the system of postulates must be consistent—no one or more of them may lead, individually or collectively, to consequences that contradict the results or any other or others. If in the course of building up a
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geometry we find we have proved two propositions that deny one another, we search out the implied contradiction in our postulates and remedy it.

Finally, the postulates ought to be independent. It should not be possible to prove any one of them as a consequence of the others. If this property fails, the geometry does not fail with it; but it is seriously disfigured by the superfluity of assumptions, and one of them should be eliminated. If we are to assume anything unnecessarily, we may as well assume the whole geometry and be done with it.

The geometer’s business then is to draw up a set of postulates. This he may

do on any basis whatever. They may be suggested to him by the behavior of

points, lines and planes, or by some other concrete phenomena; they may

with equal propriety be the product of an inventive imagination. On

proceeding to deduce their consequences, he will discover and remedy any

lack of categoricity or consistence or independence which his original

system of postulates may have lacked. In the end he will have so large a

body of propositions without contradiction or failure that he will conclude

the propriety of his postulates to have been established, and the geometry

based on them to be a valid one.

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And What Is It All About?
Is this geometry ever realized? Strictly it is not the geometer’s business to ask or answer this question. But research develops two viewpoints. There is always the man who indulges in the pursuit of facts for their sake alone, and equally the man who wants to see his new facts lead to something else. One great mathematician is quoted as enunciating a new theory of surpassing mathematical beauty with the climacteric remark “And, thank God, no one will ever be able to find any use for it.” An equally distinguished contemporary, on being interrogated concerning possible applications for one of his most abstruse theorems, replied that he knew no present use for it; but that long experience had made him confident that the mathematician would never develop any tool, however remote from immediate utility, for which the delvers in other fields would not presently find some use.
If we wish, however, we may inquire with perfect propriety, from the side lines, whether a given geometry is ever realized. We may learn that so far as has yet been discovered there are no elements for which all its postulates are verified, and that there is therefore no realization known. On the other hand, we may more likely find that many different sets of elements are such that the postulates can be interpreted as applying to them, and that we therefore have numerous realizations of the geometry. As a human being the geometer may be interested in all this, but as a geometer it really makes little difference to him.
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When we look at space about us, we see it, for some reason grounded in the

psychological history of the human race, as made up in the small of points,

which go to make up lines, which in turn constitute planes. Or we can start

at the other end and break space down first into planes, then into lines,

finally into points. Our perceptions and conceptions of these points, lines

and planes are very definite indeed; it seems indeed, as the Greeks thought,

that certain things about them are self-evident. If we wish to take these self-

evident properties of point, line and plane, and combine with them enough

additional hair-splitting specifications to assure the modern geometer that

we have really a categorical system of assumptions, we shall have the basis

of a perfectly good system of geometry. This will be what we unavoidably

think of as the absolute truth with regard to the space about us; but you

mustn’t say so in the presence of the geometer. It will also be what we call

the Euclidean geometry. It has been satisfactory in the last degree, because

not only space, but pretty much every other system of two or three elements

bearing any relations to one another can be made, by employing as a means

of interpretation the Cartesian scheme of plotting, to fit into the framework

of Euclidean geometry. But it is not the only thing in the world of

conceptual possibilities, and it begins to appear that it may not even be the

only thing in the world of cold hard fact that surrounds us. To see just how

this is so we must return to Euclid, and survey the historical development of

geometry from his day to the present time.

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Euclid’s Geometry
Point, line and plane Euclid attempts to define. Modern objection to these efforts was made clear above. Against Euclid’s specific performance we urge the further specific fault that his “definitions” are really assumptions bestowing certain properties upon points, lines and planes. These assumptions Euclid supplements in his axioms; and in the process of proving propositions he unconsciously supplements them still further. This is to be expected from one whose justification for laying down an axiom was the alleged obvious character of the statement made. If some things are too obvious to require demonstration, others may be admitted as too obvious to demand explicit statement at all.

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Thus, if Euclid has two points A and B in a plane, on opposite sides of a line M, he will draw the line AB and without further formality speak of the point C in which it intersects M. That it does so intercept M, rather than in some way dodges it, is really an assumption as to the nature of lines and planes. Or again, Euclid will speak of a point D on the line AB, between or outside the points A and B, without making the formal assumption necessary to insure that the line is “full” of points so that such a point as D must exist. That such assumptions as these are necessary follows from our previous remarks. If we think of our geometry as dealing with “chings,” “changs,” and “chungs,” or with elements I, II and III, it is no longer in the least degree obvious that the simplest property in the world applies to these elements. If we wish any property to prevail we must state it explicitly.
With the postulates embodied in his definitions, those stated in his axioms, and those which he reads into his structure by his methods of proof, Euclid has a categorical set—enough to serve as foundation for a geometry. We may then climb into Euclid’s shoes and take the next step with him. We follow him while he proves a number of things about intersecting lines and about triangles. To be sure, when he proves that two triangles are identically constituted by moving one of them over on top of the other, we may protest on the ground that the admission of motion, especially of motion thus imposed from without, into a geometry of things is not beyond dispute. If Euclid has caught our modern viewpoint, he will rejoin that if we have any doubts as to the admissibility of motion he will lay down a postulate admitting it, and we shall be silenced.
Having exhausted for the present the interest of intersecting lines, our guide now passes to a consideration of lines in the same plane that never meet. He defines such lines as parallel. If we object that he should show the existence of a derived concept like this before laying down a definition that calls for it to exist, he can show that two lines drawn perpendicular to the same line never meet. He will execute this proof by a special sort of superposition, which requires that the plane be folded over on itself, through the third dimension of surrounding space, rather than merely slid along upon itself.
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We remain quiet while Euclid demonstrates that if two lines are cut by any transversal in such a way as to make corresponding angles at the two intersections equal, the lines are parallel. It is then in order to investigate the converse: if the lines are parallel to begin with, are the angles equal?
Axioms Made to Order
This sounds innocent enough; but in no way was Euclid able to devise a proof—or, for that matter, a disproof. So he took the only way out, and said that if the lines were parallel, obviously they extended in the same direction and made the angles equal. The thing was so obvious, he argued, that it was really an axiom and he didn’t have to prove it; so he stated it as an axiom and proceeded. He didn’t state it in precisely the form I have used; he apparently cast about for the form in which it would appear most obvious, and found a statement that suited him better than this one, and that comes to the same thing. This statement tells us that if the transversal makes two corresponding angles unequal, the lines that it cuts are not parallel and do meet if sufficiently prolonged. But wisely enough, he did not transplant this axiom, once he had arrived at it, to the beginning of the book where the other axioms were grouped; he left it right where it was, following the proposition that if the angles were equal the lines were parallel. This of course was so that it might appeal back, for its claim to obviousness, to its demonstrated converse of the proposition.
Euclid must have been dissatisfied with this cutting of the Gordian knot; his successors were acutely so. For twenty centuries the parallel axiom was regarded as the one blemish in an otherwise perfect work; every respectable mathematician had his shot at removing the defect by “proving” the objectionable axiom. The procedure was always the same: expunge the parallel axiom, in its place write another more or less “obvious” assumption, and from this derive the parallel statement more or less directly. Thus if we may assume that the sum of the angles of a triangle is always exactly 180 degrees, or that there can be drawn only one line through a given point parallel to a given line, we can prove Euclid’s axiom. Sometimes the substitute assumption was openly made and stated, as in the two instances cited; as often it was admitted into the demonstration implicitly, as when it is quietly assumed that we can draw a triangle similar to any given triangle and with area as great as we please, or when parallels
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are “defined” as everywhere equidistant. But such “proofs” never satisfied anyone other than the man who made them; the search went merrily on for a valid “proof” that should not in substance assume the thing to be proved.

Locating the Discrepancy
Saccheri, an Italian Jesuit, would have struck bottom if he had had a little more imagination. He gave an exhaustive reductio ad absurdum, on the basis of the angle-sum theorem. This sum must be (a) greater than or (b) equal to or (c) less than 180 degrees. Saccheri showed that if one of these alternatives occurs in a single triangle, it must occur in every triangle. The first case gave little trouble; admitting the possibility of superposing in the special manner mentioned above, which he did implicitly, he showed that this “obtuse-angled hypothesis” contradicted itself. He pursued the “acuteangled hypothesis” for a long time before he satisfied himself that he had caught it, too, in an inconsistency. This left only the “right-angled hypothesis,” proving the Euclidean angle-sum theory and through it the parallel postulate. But Saccheri was wrong: he had found no actual contradiction in the acute-angled hypothesis—for none exists therein.
The full facts were probably first known to Gauss, who had a finger in every mathematical pie that had to do with the transition to modern times. They were first published by Lobatchewsky, the Russian, who anticipated the Hungarian John Bolyai by a narrow margin. All three worked independently of Saccheri, whose book, though theoretically available in Italian libraries, was actually lost to sight and had to be rediscovered in recent years.
Like Saccheri, Lobatchewsky investigated alternative possibilities. But he chose another point of attack: through a given point it must be possible to draw, in the same plane with a given line (a) no lines or (b) one line or (c) a plurality of lines, which shall not meet the given line. The word parallel is defined only in terms of the second of these hypotheses, so we avoid it here. These three cases correspond, respectively, to those of Saccheri.
The first case Lobatchewsky ruled out just as did Saccheri, but accepting consciously the proviso attached to its elimination; the third he could not rule out. He developed the consequences of this hypothesis as far as Euclid develops those of the second one, sketching in a full outline for a system of geometry and trigonometry based on a plurality of “non-cutters.” This geometry constitutes a coherent whole, without a logical flaw.
This made it plain what was the matter with Euclid’s parallel axiom. Nobody could prove it from his other assumptions because it is not a consequence of these. True or false, it is independent of them. Trinity Church is in New York, Faneuil Hall is in Boston, but Faneuil Hall is not in Boston because Trinity is in New York; and we could not prove that Faneuil Hall was in Boston if we knew nothing about America save that Trinity is in
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New York. The mathematicians of 2,000 years had been pursuing, on a gigantic scale, a delusion of post hoc, ergo propter hoc.

What the Postulate Really Does
Moreover, in the absence of an assumption covering the ground, we shall not know which of the alternatives (a), (b), (c) holds. But when one holds in a single case it holds permanently, as Saccheri and Lobatchewsky both showed. So we cannot proceed on this indefinite basis; we must know which one is to hold. Without the parallel postulate or a substitute therefor that shall tell us the same thing or tell us something different, we have not got a categorical set of assumptions—we cannot build a geometry at all. That is why Euclid had to have his parallel postulate before he could proceed. That is why his successors had to have an assumption equivalent to his.
The reason why it took so long for this to percolate into the understanding of the mathematicians was that they were thinking, not in terms of the modern geometry and about undefined elements; but in terms of the old geometry and about strictly defined and circumscribed elements. If we understand what is meant by Euclidean line and plane, of course the parallel postulate, to use the old geometer’s word, is true—of course, to adopt the modern viewpoint, if we agree to employ an element to which that assumption applies, the assumption is realized. The very fact of accepting the “straight” line and the “flat” plane of Euclid constitutes acceptance of his parallel postulate—the only thing that can separate his geometry from other geometries. But of course we can’t prove it; the prior postulates which we would have to use in such an attempt apply where it does not apply, and hence it cannot possibly be consequences of.
To all this the classical Euclidean rejoins that we seem to have in mind elements of some sort to which, with one reservation, his postulates apply. He wants to know what these elements look like. We can, and must, produce them—else our talk about generality is mere drivel. But we must take care that the Euclidean geometer does not try to apply to our elements the notions of straightness and flatness which inhere in the parallel postulate. We cannot satisfy and defy that postulate at the same time. If we do not insist on this point, we shall find that we are reading non-Euclidean properties into Euclidean geometry, and interpreting the elements of the latter as straight lines that are not straight, flat planes that are not flat. It is not the mission of non-Euclidean geometry thus to deny the possibility of Euclidean geometry; it merely demands a place of equal honor.

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The Geometry of Surfaces
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Let us ask the Euclidean geometer whether he can recognize his plane after

we have crumpled it up like a piece of paper en route to the waste basket.

He will hesitate only long enough to recall that in the special case of

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superposition he has reserved for himself the privilege of deforming his

own plane, and to realize that he can always iron his plane out smooth

again after we are through with it. This emphasizes the true nature of the

two-dimensionality which is the fundamental characteristic of the plane

(and of other things, as we shall directly see). The plane is two-dimensional

in points not because two sets of mutually perpendicular Euclidean straight

lines can be drawn in it defining directions of north-south and east-west,

but because a point in it can be located by means of two measures. The

same statement may be made of anything whatever to which the term

“surface” is applicable; anything, however crumpled or irregular it be, that

possesses length and breadth without thickness. The surface of a sphere, of

a cylinder, of an ellipsoid, of a cone, of a doughnut (mathematically known

as a torus), of a gear wheel, of a French horn, all these possess two-

dimensionality in points; on all of them we can draw lines and curves and

derive a geometry of these figures. If we get away from the notion that

geometry of two dimensions must deal with planes, and adopt in place of

this idea the broader restriction that it shall deal with surfaces, we shall

have the generalization which the Euclidean has demanded that we

produce, and the one which in the hands of the modern geometer has shown

results.

In this two-dimensional geometry of surfaces in general, that of the plane is

merely one special case. Certain of the features met in that case are general.

If we agree that we know what we mean by distance, we find that on every

surface there is a shortest distance between two points, together with a

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series of lines or curves along which such distances are taken. These lines

or curves we call geodesics. On the plane the geodesic is the straight line.

On surfaces in general the geodesic, whatever its particular and peculiar

shape, plays the same rôle that is played by the straight line in the plane; it

is the secondary element of the geometry, the surface itself and all other

surfaces of its type are the tertiary elements. And it is a fact that we can

take all the possible spheres, or all the possible French-horn surfaces, and

conceive of space as we know it being broken down by analysis into these

surfaces instead of into planes. The only reason we habitually decompose

space into planes is because it comes natural to us to think that way. But

geometric points, lines and surfaces must be recognized as abstractions

without actual existence, for all of them lack one or more of the three

dimensions which such existence implies. These figures exist in our minds

but not in the external world about us. So any decomposition of space into

geometric elements is a phenomenon of the mind only; it has no parallel

and no significance in the external world, and is made in one way or in

another purely at our pleasure. There isn’t a true, honest-to-goodness

geometrical plane in existence any more than there is an honest-to-

goodness spherical surface: so on intrinsic grounds one decomposition is as

reasonable as another.

Certain of the most fundamental postulates are obeyed by all surfaces. As we attempt to discriminate between surfaces of different types, and get, for
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instance, a geometry that shall be valid for spheres and ellipsoids but not

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for conicoids in general, we must do so by bringing in additional postulates

that embody the necessary restrictions. A characteristic shared by planes,

spheres, and various other surfaces is that the geodesics can be freely slid

along upon themselves and will coincide with themselves in all positions

when thus slid; with a similar arrangement for the surface itself. But the

plane stands almost unique among surfaces in that it does not force us to

distinguish between its two sides; we can turn it over and still it will

coincide with itself; and this property belongs also to the straight line. It

does not belong to the sphere, or to the great circles which are the geodesics

of spherical geometry; when we turn one of these over, through the three-

dimensional space that surrounds it, we find that the curvature lies in the

wrong way to make superposition possible. If we postulate that

superposition be possible under such treatment, we throw out the sphere

and spherical geometry; if we postulate that superposition be only by

sliding the surface upon itself we admit that geometry—as Saccheri failed

to see, as Lobatchewsky realized, and as Riemann showed at great length in

rehabilitating the “obtuse-angled hypothesis.” Lobatchewsky’s acute-angled

geometry is realized on a surface of the proper sort, which admits of

unrestricted superposition; but it is not the sort of a surface that I care to

discuss in an article of this scope.

Euclidean geometry is the natural and easy one, I suppose, because it makes

it easy to stop with three dimensions. If we take a secondary element, a

geodesic, which is “curved” in the Euclidean sense, we get a tertiary

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element, a surface, which is likewise curved. Then unless we are to make

an altogether abrupt and unreasonable break, we shall find that just as the

curved geodesic generated a curved surface, the curved surface must give

rise to a “curved space”; and just as the curved geodesic needed a second

dimension to curve into, and the curved surface a third, so the curved three-

space requires a fourth. Once started on this sort of thing, there doesn’t

really seem to be any end.

Euclidean or Non-Euclidean
Nevertheless, we must face the possibility that the space we live in, or any other manifold of any sort whatever with which we deal on geometric principles, may turn out to be non-Euclidean. How shall we finally determine this? By measures—the Euclidean measures the angles of an actual triangle and finds the sum to be exactly 180 degrees; or he draws parallel lines of indefinite extent and finds them to be everywhere equally distant; and from these data he concludes that our space is really Euclidean. But he is not necessarily right.
We ask him to level off a plot of ground by means of a plumb line. Since the line always points to the earth’s center, the “level” plot is actually a very small piece of a spherical surface. Any test conducted on this plot will exhibit the numerical characteristics of the Euclidean geometry; yet we
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know the geometry of this surface is Riemannian. The angle-sum is really

greater than 180 degrees; lines that are everywhere equidistant are not both

geodesics.

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The trouble, of course, is that on this plot we deal with so minute a fraction of the whole sphere that we cannot make measurements sufficiently refined to detect the departure from Euclidean standards. So it is altogether sensible for us to ask: “Is the universe of space about us really Euclidean in whatever of realized geometry it presents to us? Or is it really nonEuclidean, but so vast in size that we have never yet been able to extend our measures to a sufficiently large portion of it to make the divergence from the Euclidean standard discernible to us?”

This discussion is necessarily fragmentary, leaving out much that the writer would prefer to include. But it is hoped that it will nevertheless make it clear that when the contestants in the Einstein competition speak of a nonEuclidean universe as apparently having been revealed by Einstein, they mean simply that to Einstein has occurred a happy expedient for testing Euclideanism on a smaller scale than has heretofore been supposed possible. He has devised a new and ingenious sort of measure which, if his results be valid, enables us to operate in a smaller region while yet anticipating that any non-Euclidean characteristics of the manifold with which we deal will rise above the threshold of measurement. This does not mean that Euclidean lines and planes, as we picture them in our mind, are no longer non-Euclidean, but merely that these concepts do not quite so closely correspond with the external reality as we had supposed.

As to the precise character of the non-Euclideanism which is revealed, we

may leave this to later chapters and to the competing essayists. We need

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only point out here that it will not necessarily be restricted to the matter of

parallelism. The parallel postulate is of extreme interest to us for two

reasons; first because historically it was the means by which the

possibilities and the importance of non-Euclidean geometry were forced

upon our attention; and second because it happens to be the immediate

ground of distinction between Euclidean geometry and two of the most

interesting alternatives. But Euclidean geometry is characterized, not by a

single postulate, but by a considerable number of postulates. We may

attempt to omit any one of these so that its ground is not specifically

covered at all, or to replace any one of them by a direct alternative. We

might conceivably do away with the superposition postulate entirely, and

demand that figures be proved equivalent, if at all, by some more drastic

test. We might do away with the postulate, first properly formulated by

Hilbert, on which our ideas of the property represented in the word

“between” depend. We might do away with any single one of the Euclidean

postulates, or with any combination of two or more of them. In some cases

this would lead to a lack of categoricity and we should get no geometry at

all; in most cases, provided we brought a proper degree of astuteness to the

formulation of alternatives for the rejected postulates, we should get a

perfectly good system of non-Euclidean geometry: one realized, if at all, by

other elements than the Euclidean point, line and plane, and one whose

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elements behave toward one another differently from the Euclidean point,

line and plane.

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Merely to add definiteness to this chapter, I annex here the statement that in

the geometry which Einstein builds up as more nearly representing the true

external world than does Euclid’s, we shall dispense with Euclid’s

(implicit) assumption, underlying his (explicitly stated) superposition

postulate, to the effect that the act of moving things about does not affect

their lengths. We shall at the same time dispense with his parallel postulate.

And we shall add a fourth dimension to his three—not, of course, anything

in the nature of a fourth Euclidean straight line perpendicular, in Euclidean

space, to three lines that are already perpendicular to each other, but

something quite distinct from this, whose nature we shall see more exactly

in the next chapter. If the present chapter has made it clear that it is proper

for us to do this, and has prevented anyone from supposing that the results

of doing it must be visualized in a Euclidean space of three dimensions or

of any number of dimensions, it will have served its purpose.

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VI
THE SPACE-TIME CONTINUUM
Minkowski’s World of Events, and the Way It Fits Into Einstein’s Structure
BY THE EDITOR, EXCEPT AS NOTED
Seeking a basis for the secure formulation of his results, and especially a means for expressing mathematically the facts of the dependence which he had found to exist between time and space, Einstein fell back upon the prior work of Minkowski. It may be stated right here that the idea of time as a fourth dimension is not particularly a new one. It has been a topic of abstract speculation for the best part of a century, even on the part of those whose notions of the fourth dimension were pretty closely tied down to the idea of a fourth dimension of Euclidean point-space, which would be marked by a fourth real line, perpendicular to the other three, and visible to us if we were only able to see it. Moreover, every mathematician, whether or not he be inclined to this sort of mental exercise, knows well that whenever time enters his equations at all, it does so on an absolutely equal footing with each of his space coordinates, so that as far as his algebra is concerned he could never distinguish between them. When the variables x, y, z, t come to the mathematician in connection with some physical investigation, he knows before he starts that the first three represent the dimensions of Euclidean three-space and that the last stands for time. But if the algebraic expressions of such a problem were handed to him
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independently of all physical tie-up, he would never be able to tell, from them alone, whether one of the four variables represented time, or if so, which one to pick out for this distinction.

It was Minkowski who first formulated all this in a form susceptible of use

in connection with the theory of relativity. His starting point lies in the

distinction between the point and the event. Mr. Francis has brought this out

rather well in his essay, being the only competitor to present the Euclidean

geometry as a real predecessor of Newtonian science, rather than as a mere

part of the Newtonian system. I think his point here is very well taken. As

he says, Euclid looked into the world about him and saw it composed of

points. Ignoring all dynamic considerations, he built up in his mind a static

world of points, and constructed his geometry as a scientific machine for

dealing with this world in which motion played no part. It could to be sure

be introduced by the observer for his own purposes, but when so introduced

it was specifically postulated to be a matter of no moment at all to the

points or lines or figures that were moved. It was purely an observational

device, intended for the observer’s convenience, and in the bargain a mental

device, calling for no physical action and the play of no force. So far as

Euclid in his daily life was obliged to take cognizance of the fact that in the

world of work-a-day realities motion existed, he must, as a true Greek, have

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looked upon this as a most unfortunate deviation of the reality from his

beautiful world of intellectual abstraction, and as something to be deplored

and ignored. Even in their statuary the Greeks clung to this idea. A group of

marvelous action, like the Laocoon, they held to be distinctly a second rate

production, a prostitution of the noble art; their ideal was a figure like the

majestic Zeus—not necessarily a mere bust, be it understood, but always a

figure in repose without action. Their statuary stood for things, not for

action, just as their geometry stood for points, not for events.

Galileo and Newton took a different viewpoint. They were interested in the world as it is, not as it ought to be; and if motion appears to be a fundamental part of that world, they were bound to include it in their scheme. This made it necessary for them to pay much more attention to the concept of time and its place in the world than did the Greeks. In the superposition process, and even when he allowed a curve to be generated by a moving point, the sole interest which Euclid had in the motion was the effect which was to be observed upon his static figures after its completion. In this effect the rate of the motion did not enter. So all questions of velocity and time are completely ignored, and we have in fact the curious spectacle of motion without time.

To Galileo and Newton, on the other hand, the time which it took a body to pass from one point of its path to another was of paramount importance. The motion itself was the object of their study, and they recognized the part played by velocity. But Galileo and Newton were still sufficiently under the influence of Euclid to fit the observed phenomena of motion, so far as they could, upon Euclid’s static world of points. This they effected by falling in with the age-old procedure of regarding time and space as something entirely disassociated and distinct. The motion of an object—in theory, of a point—was to be recorded by observing its successive positions. With each
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of these positions a time was to be associated, marking the instant at which the point attained that position. But in the face of this association, space and time were to be maintained as entirely separate entities.

The Four-Dimensional World of Events
This severe separation of time and space Minkowski has now questioned, with the statement that the elements of which the external world is composed, and which we observe, are not points at all, but are events. This calls for a revision of our whole habit of thought. It means that the perceptual world is four-dimensioned, not three-dimensioned as we have always supposed; and it means, at the very least, that the distinction between time and space is not so fundamental as we had supposed.
[This should not impress us as strange or incomprehensible. What do we mean when we say that a plane is two-dimensional? Simply that two coordinates, two numbers, must be given to specify the position of any point of the plane. Similarly for a point in the space of our accustomed concepts we must give three numbers to fix the position—as by giving the latitude and longitude of a point on the earth and its height above sea-level. So we say this space is three-dimensional. But a material body is not merely somewhere; it is somewhere now,]182 or was somewhere yesterday, or will be somewhere tomorrow. The statement of position for a material object is meaningless unless we at the same time specify the time at which it held that position. [If I am considering the life-history of an object on a moving train, I must give three space-coordinates and one time-coordinate to fix each of its positions.]182 And each of its positions, with the time pertaining to that position, constitutes an event. The dynamic, everchanging world about us, that shows the same aspect at no two different moments, is a world of events; and since four measures or coordinates are required to fix an event, we say this world of events is four-dimensional. If we wish to test out the soundness of this viewpoint, we may well do so by asking whether the naming of values for the four coordinates fixes the event uniquely, as the naming of three under the old system fixes the point uniquely.
Suppose we take some particular event as the one from which to measure, and agree upon the directions to be taken by our space axes, and make any convention about our time-axis which subsequent investigation may show to be necessary. Certainly then the act of measuring so many miles north, and so many west, and so many down, and so many seconds backward, brings us to a definite time and place—which is to say, to a definite event. Perhaps nothing “happened” there, in the sense in which we usually employ the word; but that is no more serious than if we were to locate a point with reference to our familiar space coordinate system, and find it to lie in the empty void of interstellar space, with no material body occupying it. In this second case we still have a point, which requires, to insure its existence and
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location, three coordinates and nothing more; in the first case we still have an event, which requires for its existence and definition four coordinates and nothing more. It is not an event about which the headline writers are likely to get greatly excited; but what of that? It is there, ready and waiting to define any physical happening that falls upon it, just as the geometer’s point is ready and waiting to define any physical body that chances to fall upon it.

A Continuum of Points
It is now in order to introduce a word, which I shall have to confess the great majority of the essayists introduce, somewhat improperly, without explanation. But when I attempt to explain it, I realize quite well why they did this. They had to have it; and they didn’t have space in their three thousand words to talk adequately about it and about anything else besides. The mathematician knows very well indeed what he means by a continuum; but it is far from easy to explain it in ordinary language. I think I may do best by talking first at some length about a straight line, and the points on it.
If the line contains only the points corresponding to the integral distances 1, 2, 3, etc., from the starting point, it is obviously not continuous—there are gaps in it vastly more inclusive than the few (comparatively speaking) points that are present. If we extend the limitations so that the line includes all points corresponding to ordinary proper and improper fractions like ¼ and 17⁄29 and 1633⁄7—​ what the mathematician calls the rational numbers—we shall apparently fill in these gaps; and I think the layman’s first impulse would be to say that the line is now continuous. Certainly we cannot stand now at one point on the line and name the “next” point, as we could a moment ago. There is no “next” rational number to 116⁄125​, for instance; 115⁄124 comes before it and 117⁄126 comes after it, but between it and either, or between it and any other rational number we might name, lie many others of the same sort. Yet in spite of the fact that the line containing all these rational points is now “dense” (the technical term for the property I have just indicated), it is still not continuous; for I can easily define numbers that are not contained in it—irrational numbers in infinite variety like ; or, even worse, the number pi = 3.141592 … which defines the ratio of the circumference of a circle to the diameter, and many other numbers of similar sort.
If the line is to be continuous, there may be no holes in it at all; it must have a point corresponding to every number I can possibly name. Similarly for the plane, and for our three-space; if they are to be continuous, the one must contain a point for every possible pair of numbers x and y, and the other for every possible set of three numbers x, y and z, that I can name. There may be no holes in them at all.
A line is a continuum of points. A plane is a continuum of points. A threespace is a continuum of points. These three cases differ only in their
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dimensionality; it requires but one number to determine a point of the first continuum, two and three respectively in the second and third cases. But the essential feature is not that a continuum shall consist of points, or that we shall be able to visualize a pseudo-real existence for it of just the sort that we can visualize in the case of line, plane and point. The essential thing is merely that it shall be an aggregate of elements numerically determined in such a way as to leave no holes, but to be just as continuous as the real number system itself. Examples, however, aside from the three which I have used, are difficult to construct of such sort that the layman shall grasp them readily; so perhaps, fortified with the background of example already presented, I may venture first upon a general statement.

The Continuum in General
Suppose we have a set of “elements” of some sort—any sort. Suppose that these elements possess one or more fundamental identifying characteristics, analogous to the coordinates of a point, and which, like these coordinates, are capable of being given numerical values. Suppose we find that no two elements of the set possess identically the same set of defining values. Suppose finally—and this is the critical test—that the elements of the set are such that, no matter what numerical values we may specify, it we do specify the proper number of defining magnitudes we define by these an actual element of the set, that corresponds to this particular collection of values. Our elements then share with the real number system the property of leaving no holes, of constituting a continuous succession in every dimension which they possess. We have then a continuum. Whatever its elements, whatever the character of their numerical identifiers, whatever the number n of these which stands for its dimension, there may be no holes or we have no continuum. There must be an element for every possible combination of n numbers we can name, and no two of these combinations may give the same element. Granted this condition, our elements constitute a continuum.
As I have remarked, it is not easy to cite examples of continua which shall mean anything to the person unaccustomed to the term. The totality of carbon-oxygen-nitrogen-hydrogen compounds suggested by one essayist as an example is not a continuum at all, for the set contains elements corresponding only to integer values of the numbers which tell us how many atoms of each substance occur in the molecule. We cannot have a compound containing carbon atoms, or oxygen atoms. Perhaps the most satisfactory of the continua, outside the three Euclidean spacecontinua already cited, [is the manifold of music notes. This is fourdimensional; each note has four distinctions—length, pitch, intensity, timbre—to distinguish it perfectly, to tell how long, how high, how loud, how rich.]263 We might have a little difficulty in reducing the characteristic of richness to numerical expression, but presumably it could be done; and we should then be satisfied that every possible combination of four values,
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l, p, i, t for these four identifying characteristics would give us a musical effect, and one to be confused with no other.
There is in the physical world a vast quantity of continua of one sort or another. The music-note continuum brings attention to the fact that not all of these are such that their elements make their appeal to the visual sense. This remark is a pertinent one; for we are by every right of heritage an eyeminded race, and it is frequently necessary for us to be reminded that so far as the external world is concerned, the verdict of every other sense is entirely on a par with that of sight. The things which we really see, like matter, and the things which we abstract from these visual impressions, like space, are by no means all there is to the world.

Euclidean and Non-Euclidean Continua
If we are dealing with a continuum of any sort whatever having one or two or three dimensions, we are able to represent it graphically by means of the line, the plane, or the three-space. The same set of numbers that defines an element of the given continuum likewise defines an element of the Euclidean continuum of the same dimensionality; so the one continuum corresponds to the other, element for element, and either may stand for the other. But if we have a continuum of four or more dimensions, this representation breaks down in the absence of a real, four-dimensional Euclidean point-space to serve as a picture. This does not in the least detract from the reality of the continuum which we are thus prevented from representing graphically in the accustomed fashion.
The Euclidean representation, in fact, may in some cases be unfortunate—it may be so entirely without significance as to be actually misleading. For in the Euclidean continuum of points, be it line, plane or three-space, there are certain things which we ordinarily regard as secondary derived properties, but which possess a great deal of significance none the less.
In particular, in the Euclidean plane and in Euclidean three-space, there is the distance between two points. I have indicated, in the chapter on nonEuclidean geometry, that the parallel postulate of Euclid, which distinguishes his geometry from others, could be replaced by any one of numerous other postulates. Grant Euclid’s postulate and you can prove any of these substitutes; grant any of the substitutes and you can prove Euclid’s postulate. Now it happens that there is one of these substitutes to which modern analysis has given a position of considerable importance. It is merely our good old friend the Pythagorean theorem, that the square on the hypotenuse equals the sum of the squares on the sides; but it is dressed in new clothes for the present occasion.
Mr. Francis’ discussion of this part of the subject, and especially his figure, ought to make it clear that this theorem can be considered as dealing with the distance between any two points. When we so consider it, and take it as
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the fundamental, defining postulate of Euclidean geometry which

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distinguishes this geometry from others, we have a statement of

considerable content. We have, first, that the characteristic property of

Euclidean space is that the distance between two points is given by the

square root of the sum of the squares of the coordinate-differences for these

points—by the expression

where the large letters represent the coordinates of the one point and the small ones those of the other. We have more than this, however; we have that this distance is the same for all observers, no matter how different their values for the individual coordinates of the individual points. And we have, finally, as a direct result of looking upon the thing from this viewpoint, that the expression for D is an “invariant”; which simply means that every observer may use the same expression in calculating the value of D in terms of his own values for the coordinates involved. The distance between two points in our space is given numerically by the square root of the sum of the squares of my coordinate-differences for the two points involved; it is given equally by the square root of the sum of the squares of your coordinatedifferences, or those of any other observer whatsoever. We have then a natural law—the fundamental natural law characterizing Euclidean space. If we wish to apply it to the Euclidean two-space (the plane) we have only to drop out the superfluous coordinate-difference; if we wish to see by analogy what would be the fundamental natural law for a four-dimensional Euclidean space, we have only to introduce under the radical a fourth coordinate-difference for the fourth dimension.
If we were not able to attach any concrete meaning to the expression for D the value of all this would be materially lessened. Consider, for instance, the continuum of music notes. There is no distance between different notes. There is of course significance in talking about the difference in pitch, in intensity, in duration, in timbre, between two notes; but there is none in a mode of speech that implies a composite expression indicating how far one note escapes being identical with another in all four respects at once. The trouble, of course, is that the four dimensions of the music-note continuum are not measurable in terms of a common unit. If they were, we should expect to measure their combination more or less absolutely in terms of this same unit. We can make measurements in all three dimensions of Euclidean space with the same unit, with the same measuring rod in fact. [This presents a peculiarity of our three-space which is not possessed by all threedimensional manifolds. Riemann has given another illustration in the system of all possible colors, composed of arbitrary proportions of the three primaries, red, green and violet. This system forms a three-dimensional continuum; but we cannot measure the “distance” or difference between two colors in terms of the difference between two others.]130
Accordingly, in spite of the fact that the Euclidean three-space gives us a formal representation of the color continuum, and in spite of the fact that the hypothetical four-dimensional Euclidean space would perform a like office for the music-note continuum, this representation would be without
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significance. We should not say that the geometry of these two manifolds is Euclidean. We should realize that any set of numerical elements can be plotted in a Euclidean space of the appropriate dimensionality; and that accordingly, before allowing such a plot to influence us to classify the geometry of the given manifold as Euclidean, we must pause long enough to ask whether the rest of the Euclidean system fits into the picture. If the square root of the sum of the squares of the coordinate-differences between two elements possesses significance in the given continuum, and if it is invariant between observers of that continuum who employ different bases of reference, then and only then may we allege the Euclidean character of the given continuum.

If under this test the given continuum fails of Euclideanism, it is in order to ask what type of geometry it does present. If it is of such character that the “distance” between two elements possesses significance, we should answer this question by investigating that distance in the hope of discovering a non-Euclidean expression for it which will be invariant. If it is not of such character, we should seek some other characteristic of single elements or groups of elements, of real physical significance and of such sort that the numerical expression for it would be invariant.

If the continuum with which we have to do is one in which the “distance”

between two elements possesses significance, and if it turns out that the

invariant expression for this distance is not the Pythagorean one, but one

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indicating the non-Euclideanism of our continuum, we say that this

continuum has a “curvature.” This means that, if we interpret the elements

of our continuum as points in space (which of course we may properly do)

and if we then try to superpose this point-continuum upon a Euclidean

continuum, it will not “go”; we shall be caught in some such absurdity as

trying to force a sphere into coincidence with a plane. And of course if it

won’t go, the only possible reason is that it is curved or distorted, like the

sphere, in such a way as to prevent its going. It is unfortunate that the

visualizing of such curvature requires the visualizing of an additional

dimension for the curved continuum to curve into; so that while we can

picture a curved surface easily enough, we can’t picture a curved three-

space or four-space. But that is a barrier to visualization alone, and in no

sense to understanding.

Our World of Four Dimensions
It will be observed that we have now a much broader definition of nonEuclideanism than the one which served us for the investigation of Euclid’s parallel postulate. If we may at pleasure accept this postulate or replace it by another and different one, we may presumably do the same for any other or any others of Euclid’s postulates. The very statement that the distance between elements of the continuum shall possess significance, and shall be measurable by considering a path in the continuum which involves other elements, is an assumption. If we discard it altogether, or replace it by one
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postulating that some other joint property of the elements than their

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distance be the center of interest, we get a non-Euclidean geometry. So for

any other of Euclid’s postulates; they are all necessary for a Euclidean

system, and in the absence of any one of them we get a non-Euclidean

system.

Now the four-dimensional time-space continuum of Minkowski is plainly of a sort which ought to make susceptible of measurement the separation between two of its events. We can pass from one element to another in this continuum—from one event to another—by traversing a path involving “successive” events. Our very lives consist in doing just this: we pass from the initial event of our career to the final event by traversing a path leading us from event to event, changing our time and space coordinates continuously and simultaneously in the process. And while we have not been in the habit of measuring anything except the space interval between two events and the time interval between two events, separately, I think it is clear enough that, considered as events, as elements in the world of four dimensions, there is a less separation between two events that occur in my office on the same day than between two which occur in my office a year apart; or between two events occurring 10 minutes apart when both take place in my office than when one takes place there and one in London or on Betelgeuse.

It is not at all unreasonable, a priori, then, to seek a numerical measure for

the separation, in space-time of four-dimensions, of two events. If we find

it, we shall doubtless be asked just what its subjective significance to us is.

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This must be answered with some circumspection. It will presumably be

something which we cannot observe with the visual sense alone, or it would

have forced itself upon our attention thousands of years ago. It ought, I

should think, to be something that we would sense by employing at the

same time the visual sense and the sense of time-passage. In fact, I might

very plausibly insist that, by my very remarks about it in the above

paragraph, I have sensed it.

Minkowski, however, was not worried about this phase of the matter. He

had only to identify the invariant expression for distance; sensing it could

wait. He found, of course, that this expression was not the Euclidean

expression for a four-dimensional interval. He had discarded several of the

Euclidean assumptions and could not expect that the postulate governing

the metric properties of Euclid’s space would persist. Especially had he

violated the Euclidean canons in discarding, with Einstein, the notion that

nothing which may happen to a measuring rod in the way of uniform

translation at high velocity can affect its measures. So he had to be prepared

to find that his geometry was non-Euclidean; yet it is surprising to learn

how slightly it deviates from that of Euclid. Without any extended

discussion to support the statement, we may say that he found that when

two observers measure the time- and the space-coordinates of two events,

using the assumptions and therefore the methods of Einstein and hence

subjecting themselves to the condition that their measures of the pure time-

interval and of the pure space-interval between these events will not

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necessarily be the same, they will discover that they both get the same value for the expression

If our acceptance of this as the numerical measure of the separation in space-time between the two events should lead to contradiction we could not so accept it. No contradiction arises however and we may therefore accept it. And at once the mathematician is ready with some interpretative remarks.
The Curvature of Space-Time
The invariant expression for separation, it will be seen, is in the same form as that of the Euclidean four-dimensional invariant save for the minus sign before the time-difference (the appearance of the constant C in connection with the time coordinate t is merely an adjustment of units; see page 153). This tells us that not alone is the geometry of the time-space continuum non-Euclidean in its methods of measurement, but also in its results, to the extent that it possesses a curvature. It compares with the Euclidean fourdimensional continuum in much the same way that a spherical surface compares with a plane. As a matter of fact, a more illuminating analogy here would be that between the cylindrical surface and the plane, though neither is quite exact. To make this clear requires a little discussion of an elementary notion which we have not yet had to consider.
Our three-dimensional existence often reduces, for all practical purposes, to a two-dimensional one. The objects and the events of a certain room may quite satisfactorily be defined by thinking of them, not as located in space, but as lying in the floor of the room. Mathematically the justification for this viewpoint is got by saying that we have elected to consider a slice of our three-dimensional world of the sort which we know as a plane. When we consider this plane and the points in it, we find that we have taken a cross-section of the three-dimensional world. A line in that world is now reduced, for us, to a single point—the point where it cuts our plane; a plane is reduced to a line—the line where it cuts our plane; the three-dimensional world itself is reduced to our plane itself. Everything three-dimensional falls down into its shadow in our plane, losing in the process that one of the three dimensions which is not present in our plane.
For simplicity’s sake it is usual to take a cross-section of space parallel to one of our coordinate axes. We think of our three dimensions as extending in the directions of those axes; and it is easier to take a horizontal or vertical section which shall simply wipe out one of these dimensions than to take an oblique section which shall wipe out a dimension that consists partly of our original length, and partly of our original width, and partly of our original height.
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If we have a four-dimensional manifold to begin with, we may equally

shake out one of the four dimensions, one of the four coordinates, and

consider the three-dimensional result of this process as a cross-section of

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the original four-dimensional continuum. And where, in cross-sectioning a

three-dimensioned world, we have but three choices of a coordinate to

eliminate, in cross-sectioning a world of four dimensions we have four

choices. By dropping out either the x, or the y, or the z, or the t, we get a

three-dimensioned cross-section.

Now our accustomed three-dimensional space is strictly Euclidean. When we cross-section it, we get a Euclidean plane no matter what the direction in which we make the cut. Likewise the Euclidean plane is wholly Euclidean, because when we cross-section it in any direction whatever we get a Euclidean line. A cylindrical surface, on the other hand, is neither wholly Euclidean nor wholly non-Euclidean in this matter of crosssectioning. If we take a section in one direction we get a Euclidean line and if we take a section in the other direction we get a circle (if the cylindrical surface be a circular one). And of course if we take an oblique section of any sort, it is neither line nor circle, but a compromise between the two— the significant thing being that it is still not a Euclidean line.

The space-time continuum presents an analogous situation. When we cross-

section it by dropping out any one of the three space dimensions, we get a

three-dimensional complex in which the distance formula is still non-

Euclidean, retaining the minus sign before the time-difference and therefore

retaining the geometric character of its parent. But if we take our cross-

section in such a way as to eliminate the time coordinate, this peculiarity

disappears. The signs in the invariant expression are then all plus, and the

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cross-section is in fact our familiar Euclidean three-space.

If we set up a surface geometry on a sphere, we find that the elimination of one dimension leaves us with a line-geometry that is still non-Euclidean since it pertains to the great circles of the sphere rather than to Euclidean straight lines. In shaking Minkowski’s continuum down into a threedimensional one by eliminating any one of his coordinates, if we eliminate either the x, the y or the z, we have left a three-dimensional geometry in which the disturbing minus sign still occurs in the distance-formula, and which is therefore still non-Euclidean. If we omit the t, this does not occur. We see, then, that the time dimension is the disturbing factor, the one which gives to space-time its non-Euclidean character so far as the possession of curvature is concerned. And we see that this curvature is not the same in all directions, and in one direction is actually zero—whence the attempted analogy with a cylinder instead of with a sphere.

Many writers on relativity try to give the space-time continuum an appeal to our reason and a character of inevitableness by insisting on the lack of any fundamental distinction between space and time. The very expression for the space-time invariant denies this. Time is distinguishable from space. The three dimensions of space are quite indistinguishable—we can interchange them without affecting the formula, we can drop one out and never know which is gone. But the very formula singles out time as distinct

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from space, as inherently different in some way. It is not so inherently

different as we have always supposed; it is not sufficiently different to offer

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any obstacle to our thinking in terms of the four-dimensional continuum.

But while we can group space and time together in this way, [this does not

mean at all that space and time cease to differ. A cook may combine meat

with potatoes and call the product hash, but meat and potatoes do not thereby become identical.]223

The Question of Visualization
To the layman there is a great temptation to say that while, mathematically speaking, the space-time continuum may be a great simplification, it does not really represent the external world. To be sure, you can’t see the spacetime continuum in precisely the same way that you can the threedimensional space continuum, but this is only because Einstein finds the time dimension to be not quite freely interchangeable with the space dimension. Yet you do perceive this space-time continuum, in the manner appropriate for its perception; and it would be just as sensible to throw out the space continuum itself on the ground that perception of the two is not of exactly the same sort, as to throw out the space-time continuum on this ground. With appropriate conventions, either may stand as the mental picture of the external world; it is for us to choose which is the more convenient and useful image. Einstein tells us that his image is the better, and tells us why.
Before we look into this, we must let him tell us something more about the geometry of his continuum. What he tells us is, in its essentials, just this. The observer in a pure space continuum of three dimensions finds that as he changes his position, his right-and-left, his backward-and-forward, and his up-and-down are not fixed directions inherent in nature, but are fully interchangeable. The observers, in the adjoined sketch, whose verticals are as indicated by the arrows, find very different vertical and horizontal components for the distance between the points O and P; a similar situation would prevail if we used all three space directions. The statement analogous to this for Einstein’s four-dimensional continuum of space and time combined

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