UC-NRLF 
 
 B M 550 70E 
 
Euclid's Parallel Postulate: 
 
 Its Nature, Validity, and Place 
 
 In Geometrical Systems. 
 
 THESIS PRESENTED TO THE PHILOSOPHICAL FACULTY OF 
 
 YALE UNIVERSITV FOR THE DEGREE OF 
 
 DOCTOR OF PHILOSOPHY. 
 
 JOHN WILLIAM WITHERS, Ph.D. 
 
 Principal of the Yeatman High School, St. Louis, Mo. 
 
 CHICAGO 
 THE OPEN COURT PUBLISHING COMPANY 
 
 LONDON AGENTS 
 
 Kegan Paul, Trench, Trubner & Co., Ltd. 
 
 1905. 
 
^^ 
 
 Copyright 1905 
 
 THE OPEN COURT PUBLISHING CO. 
 Chicago 
 
PREFACE. 
 
 The parallel postulate is the only distinctive char- 
 acteristic of Euclid. To pronounce upon its valid- 
 ity and general philosophical significance without 
 endeavoring to know what Non-Euclideans have 
 done would be an inexcusable blunder. For this 
 reason I have given in the following pages what 
 might otherwise seem to be an undue prominence 
 to the historical aspect of my general problem. 
 
 In the last chapter, the positions taken are only 
 briefly defended, because they seem to flow directly 
 and naturally from results previously won. 
 
 I have included in the bibliography such works 
 as are mentioned in the body of the thesis, and 
 have not aimed at making a complete list. More 
 complete biographies of Hyperspace and non- 
 Euclidean Geometry are those of Halsted and Bon- 
 ola, which I have mentioned in my list. 
 
 My obligations not elsewhere explicitly acknowl- 
 edged are chiefly to Professor Geo. T. Ladd, at 
 whose suggestion this study was undertaken, and 
 under whose sympathetic direction it has attained 
 its present form. I am also indebted to Dr. E. B. 
 Wilson for light upon certain mathematical aspects 
 of the problem. 
 
 Neiv Haven, Connecticut, April, 1904. 
 
CONTENTS. 
 
 CHAPTER PAGE 
 
 I. The Pre-Lobatchewskian Struggle With 
 THE Parallel Postulate. 
 Historical origin of the Postulate. Attempts 
 to dispense with it. Substitution of different 
 definitions. Substitution of different postu- 
 lates. Efforts to prove the Parallel Postulate 
 on a basis of Euclid's other assumptions . . 1-20 
 
 II. The Discovery and Development of Non- 
 Euclidean Systems. 
 The clue. Invention of consistent geome- 
 tries independent of the Parallel Postulate. 
 The problem generafized. Attention directed 
 through projective methods to the qualitative 
 aspects "of spaCfe. Recent developments 23-60 
 
 III. General Orientation of the Problem. 
 General significance of the historical develop- 
 ment of non-Euclidean geometry. The motive. 
 Complex character of the problem and what 
 its solution requires. Delimitation of the pres- 
 ent study 63-76 
 
 IV. Psychology of the Parallel Postulate and 
 
 Its Kindred Conceptions. 
 General observations as to the development of 
 our space conception. Empirical sources of 
 special geometrical conceptiohs. Of measure- 
 ment and the consciousness of Homogeneity 
 and Free Mobility. Of the straight TTne" Of 
 Surfaces. Of the Parallel Postulate ; its em- 
 pirical sources ; its direct relations to sense- 
 
 V 
 
vi CONTENTS 
 
 CHAPTER PAGE 
 
 perception; its complex nature; its relations 
 to the conception of the straight line ; empirical 
 nature of the straight line. The validity of 
 the Parallel Postulate not to be settled in ab- 
 solute fashion by any empirical investigation 79-123 
 
 V. The Nature and Validity of the Parallel 
 Postulate. 
 The number and variety of possible geometries. 
 Space parameter not necessarily constant to 
 accord with the space of experience. Distinc- 
 tion between the facts of experience and the 
 intellectual constructs formed on a basis of 
 this experience. Conditions imposed upon 
 these constructs. Non-Euclidean systems log- 
 ically possible. The Parallel Postulate not a 
 priori necessary. How judge the validity of 
 Euclid. Difficulties with Poincare's doctrine 
 of convenience. Why is Euclid most con- 
 venient ? Kant's Argument. Its failure to es- 
 tablish the necessary validity of Euclid. Must 
 appeal to experience. Methods of scientific 
 testing and the difficulties which attend them. 
 The experimental method suggested in Ladd's 
 A Theory of Reality 127-152 
 
 VI. Resulting Implications as to the Nature 
 OF Space. 
 Distinctions drawn between conceptions of 
 space and of figures in space. The importance 
 of these distinctions pointed out. Varieties of 
 two-dimensional spaces and of tri-dimensional 
 spaces possible. Possibility of a fourth di- 
 mension.*^ Euclidean and Non-Euclidean tri- 
 dimensional spaces independent of a fourth 
 dimen'sion. What this implies. The space- 
 constant and what it means. The impossibility 
 of comparing different space-constants quan- 
 titatively in a strict geometrical sense. Log- 
 
CONTENTS 
 
 ical difficulties of the conception of variable 
 curvature quantitatively considered. Diffi- 
 culties due to abstract conception. Geometry 
 must fit experience ; not experience geometry. 
 Homogeneity does not mean mere logical any- 
 ness. Distinction between geometrical space 
 conceptions and the space category. The a 
 priori element. Summary and conclusion . 155-173 
 
 VII. Bibliography 177-192 
 
PRE-LOBATCHEWSKIAN 
 STRUGGLE 
 
 WITH THE 
 
 PARALLEL POSTULATE. 
 
CHAPTER I. 
 
 THE PRE-LOBATCHEWSKIAN STRUGGLE. 
 
 I. Historical Origin of the Parallel Postulate. — 
 Before the time of Euclid the science of geometry- 
 was already well advanced in Greece. Pythagoras 
 and his immediate followers, by perfecting a theory 
 of ratio and proportion, and by the study of areas 
 and the introduction of irrational quantities, had 
 brought the subject so prominently before the Greek 
 mind that no subsequent philosopher could afford 
 to neglect it. Accordingly, Zeno and Democritus, 
 Anaxagoras and Hippias, Plato and Aristotle, and 
 the great body of thinkers who were disciples of 
 these more or less extensively devoted themselves 
 to its study. 
 
 The duplication of the cube, the quadrature of 
 the circle, the trisection of the angle, were all vigor- 
 ously attacked and out of the struggle certain new 
 and important conceptions arose. Certain lines of 
 plane and double curvature were invented, impor- 
 tant properties of conies were discovered, and the 
 notion of infinity introduced. Methods of research 
 and geometrical exposition were also accurately 
 
2 THE PRE-LOBATCHEIVSKIAN STRUGGLE 
 
 Studied, among them the method of reduction of 
 Hippocrates, the analytical method of Plato, and 
 the method of exhaustions of Eudoxus. i\dded to 
 these the diorism of Leon, the determination by 
 Menjechmus of the necessary conditions for the 
 invertibility of a theorem which affords a fruitful 
 method of enlarging the number of propositions, 
 the introduction of formal logic and the powerful 
 influence of the dialectics of Socrates and the 
 Sophists, all contributed to make possible that re- 
 markable outburst of mathematical genius which 
 has been fittingly styled " The Golden Age of Greek 
 Geometry/' 
 
 It is obvious then in view of this remarkable de- 
 velopment that Euclid was by no means the author 
 of all the demonstrations contained in his Elements. 
 It is impossible to state exactly what he did con- 
 tribute. In the whole collection there is only one 
 proofs (I, 47) which is directly ascribed to him. 
 Of a few things, however, we are reasonably sure. 
 Euclid brought to irrefutable demonstration propo- 
 sitions which had been previously less rigorously 
 proved.^ The selection and arrangement of the 
 
 ^ Gow's History of Greek Geometry, Cambridge, 1884, p. 
 198. 
 
 - Proclus, at close of the Eudemian Summary. Eudemas, 
 a pupil of Aristotle, wrote a history of Geometry which has 
 been lost, but Proclus, in his commentaries on Euclid, gives 
 an abstract or summary of it. and this is the most trustworthy 
 information we have regarding early Greek Geometry. 
 
THE FRE-LOBATCHEllSKJAA' STRUGGLE 3 
 
 propositions is his.'' He chose the theorems and 
 demonstrations which should form a part of his 
 system. Many available demonstrations were cer- 
 tainly rejected."* We may attribute to his delib- 
 erate choice the distinctive characteristics of the 
 book as a whole. We owe to him that orderly 
 method of proof which proceeds by statement, con- 
 struction, proof, conclusion, even to the final Q. E. 
 D. (oTTcp 18a Set^oi) of the modern text. He is 
 responsible too for that peculiar logical design 
 of the book which proceeds always from a few 
 definitions, postulates and common notions, or 
 axioms, by sure steps which are always of precisely 
 the same kind until every link in the argument from 
 premises to conclusion is securely forged. 
 
 Euclid set at the beginning of his text certain 
 definitions, postulates and common notions ^ which 
 should serv^e as the foundation for his system. 
 Was the parallel postulate one of this number? 
 
 There is but one way to answer this question, 
 and that is to go back to the earliest editions of 
 Euclid at present accessible and observe whether 
 they contain it or not. In all these earlier editions 
 there is practical agreement in regard to the defini- 
 
 ' Proclus, Friedlein's Edition, p. 69. 
 * Compare Gow, op. cit. pp. iq8 fF. 
 
 '^ Euclid did not use the term axiom. This was introduced 
 by Proclus. 
 
4 THE PRE-LOBATCHEIVSKIAN STRUGGLE 
 
 tions of Euclid. As far back as lOO b. c. Heron's^ 
 " Definitions " appear in the same number and in 
 essentially the same form, though not in the same 
 order as we have them now. Among the postulates 
 and Common Notions,'^ however, there is consider- 
 able fluctuation. Many editions give three postu- 
 lates and twelve axioms. The first nine axioms 
 relate to all kinds of magnitudes; these remain in 
 all editions essentially unchanged ; but the last three 
 which relate to space only and are thus distinctively 
 geometrical fluctuate in a most interesting way. 
 They are as follows : lo, Two straight lines cannot 
 inclose a space; ii. All right angles are equal; 12, 
 If a straight line meet two straight lines, so as to 
 make the two interior angles on the same side of 
 it taken together less than two right angles, these 
 straight lines being continually produced, shall at 
 length meet on that side on which are the angles 
 which are less than two right angles.^ In nearly all 
 modern editions, except the most recent, these are 
 called " axioms " and classed, as here indicated, in 
 the same category with the other nine. In the older 
 manuscripts, however, no such blunder is commit- 
 ted. It seems hardly possible to account for this 
 
 ^ Simon's Euclid Sec. 6. Die Kommentatoren des Euclid. 
 
 '' Euclid's expression is Kotvai ivvowLL- 
 
 * In Clavius this is the 13th axiom. Robert Simson, on 
 whose edition modern English texts are usuallj' based, calls 
 it the I2th axiom; Bolyai and many others call it the nth 
 axiom. 
 
THE PRE-LOBATCHEWSKIAN STRUGGLE 5 
 
 unless we assume that Euclid himself drew a dis- 
 tinction between them. Of these older manuscripts 
 by far the greater number place these axioms 
 among the postulates where they rightly belong. 
 The Vatican Manuscript ^ gives "axiom" 10 as 
 postulate 6. Proclus omits it altogether with the 
 significant remark that it is really a theorem which 
 ought to be proved, and actually attempts to prove 
 it in book I, proposition 4. He also gives the 
 parallel " axiom " as postulate 5, but claims that it, 
 too, should be proved, and quotes Germinus ^^ in 
 support of this view. Thus it appears not only 
 that this famous postulate is at least of classical 
 origin, but also that the Greeks themselves better 
 understood its true nature than the majority of 
 modern writers have done. 
 
 But we have not yet answered our question; did 
 Euclid himself actually make use of this postulate? 
 If so, v/as it his own invention or did he borrow it 
 from another? The latter question we can not 
 definitely answer since we have no means of knowing 
 how many of the propositions which involve this 
 assumption were first demonstrated by Euclid hi 
 
 imi- 
 
 9 Discovered by Napoleon in Rome in the early part of last 
 century and brought to France. It was edited in Paris by 
 R Peyrard (1814-1818). who thought that he had here an 
 edition of Euclid more ancient than Theon's (about 380 A D ) 
 upon which many of the early MSS. which first came to light 
 "laimed to be based. 
 
 '" Germinus wrote about 60 B. C. 
 
6 THE PRE-LOBATCHEWSKIAN STRUGGLE 
 
 self; nothing like this postulate is mentioned how- 
 ever by any previous writer. 
 
 Regarding the first question we can speak with 
 more- confidence. The researches of Peyrard show 
 that in the earliest manuscripts now accessible, this 
 postulate does not appear among the other postulates 
 and common notions at the beginning of the text, 
 but is found in the demonstration of Proposition 29 
 where it is introduced to support the proof of the 
 equality of the alternate angles of parallel lines. 
 Euclid himself then not only employed this postulate 
 but the position in which he placed it seems to indi- 
 cate clearly that he appreciated the difficulties which 
 its use involves. Surely one who had formulated a 
 system so rigorous, who was master of a logic so 
 keen and true that the most critical efforts of modern | 
 thought have not destroyed but, on the contrary, 
 have only strengthened his claims to rigor, could not 
 have passed over such a manifest begging of the 
 question as appears upon, the very face of this postu- 
 late as he himself phrased it, without having first 
 made a desperate effort to prove it. Euclid makes 
 no attempt as did later waiters to conceal the diffi- 
 culty under the cloak of a subtle phraseology. He 
 states it frankly as a petitio principii of the baldest 
 type. It must then have appeared to him not as an 
 " axiomatic " truth, but as a theorem calling for 
 demonstration. Euclid proves propositions more 
 obvious bv far. He even demonstrates that two 
 
THE PRE-LOBArCIlEWSKIAN STRUGGLE 7 
 
 sides of a triangle are greater than the third side, 
 a proposition which the Epicureans derided as being 
 " manifest even to asses." '^ 
 
 The position of the postulate seems to indicate 
 that EucHd struggled on as far as possible without 
 it and postulated it finally only because he could 
 neither prove it nor' proceed any further without it. 
 Moreover, the astronomical system of Eudoxus and 
 the writings of x\ntolycus make it also apparent 
 that Euclid must have had some knowledge of sur- 
 face spherics and was therefore familiar with tri- 
 angles whose angle sum contradicts the truth of 
 this postulate. 
 
 II. Attempts to Dispense unth the Postulate. — 
 The intersection of two slowly converging straight 
 lines lies of course beyond the province of observa- 
 tion or construction. Hence it is obvious why the 
 successors of Euclid, habituated by him to strict 
 logical rigor, should have found fault with the 
 i:)arallel postulate and put forth their utmost en- 
 deavors to dispose of it in one way or another. In 
 1621 Sir Henry Saville^^ wrote: "In pnlcherrinw 
 Geouietricr eorpore duo sunt iiarvi, duce lobes nee 
 quod seiani plures in quibus elucendis et eniacu- 
 l end is euni veferum turn reeentioruni vigilavit in- 
 dustria." One of these "blemishes" was the par- 
 
 11 Proclus. op. cit.. also Cajori, History of Elementary 
 Mathematics, p. 74. 
 
 ^Lectures on Euclid, published at Oxford, 1621. 
 
8 THE PRE-LOBATCHEWSKIAN STRUGGLE 
 
 allel postulate, the other EucUd's theory of propor- 
 tion. Under the title " Parallel " in the " En- 
 cyclopadie der Wisscnschaften und Kiinste," pub- 
 lished at Leipzig in i^sS. Sohncke says that "in 
 Mathematics there is nothing over which so much 
 has been spoken, written and striven, and all so far 
 without reaching a definite result and decision." 
 Appended to this article there is a carefully pre- 
 pared list of ninety-two authors who had dealt with 
 the problem. These quotations show the extent to 
 which these earlier efforts were carried. Indeed it 
 appears that almost every writer on geometry ofj 
 any note from Euclid to Sohncke had given mor 
 or less attention to this difficult subject. 
 
 These earlier endeavors struck out in various 
 directions which we shall now briefly state and con 
 sider. Some attempted to avoid the difficulty 
 through a new definition of parallel lines ; by others 
 new assumptions which were considered less faulty 
 were substituted for Euclid's. These in reality only 
 concealed the difficulty ; they did not remove it. A 
 third class attempted ' to deduce the theory of 
 parallels from Euclid's other postulates, by reason- 
 ing upon the nature of the straight line and the 
 plane angle. These were by far the most desperate 
 attempts. Finally there were those who decided 
 that if this postulate is dependent upon the other 
 assumptions which constitute the foundations of 
 Euclid we shall by denying it and maintaining them, 
 
THE PRE-LOBATCHEWSKIAN STRUGGLE 9 
 
 become ultimately involved in contradiction. It 
 was this method of procedure which resulted in the 
 first establishment of a non-Euclidean geometry. 
 We shall consider these attempts in the order 
 named. 
 
 (i) The Substitution of Different Definitions. 
 
 Euclid's own definition was, that parallel lines 
 are straight lines which lie in the same plane and 
 will not meet however far produced. This defini- 
 tion is perhaps still best for elementary geometry. 
 In 1525 Albrecht Diirer/^ a German painter, pro- 
 posed the familiar definition that parallel lines are 
 straight lines which are everywhere equally distant. 
 Clavius^'* substituted for this the assumption that 
 a line which is everywhere equidistant from a given 
 straight line in the same plane is itself straight. 
 A.nother definition which is often preferred because 
 3f its apparent simplicity is. that parallel lines are 
 traight lines which have the same direction. This 
 definition possesses the peculiar advantage that 
 ;hose who adopt it have no further difficulty; for 
 :hey find no necessity to assume the parallel postu- 
 ate or anything equivalent to it. This is a great 
 idvantage, certainly ; but, as a matter of fact, any 
 me of these definitions, though apparently more 
 idvantageous than Euclid's, is in reality more com- 
 )lex and less satisfactory. The first two make use 
 
 13 Cajori, p. 266. 
 
 1* Edition of Euclid, 1574. 
 
10 THE PKE-LOBATCHEWSKIAN STRUGGLE 
 
 of the conception of distance. This of course in 
 volves measurement, which in turn embraces the L 
 whole theory of incommensurable quantities with I ^ 
 its entire outfit of necessary presuppositions and, 
 attendant difficulties. What is more, these defini- 
 tions only hold for Euclidean geometry; they are 
 not true for pseudo-spherical space where parallel 
 lines are still possible and where Euclid's definition 
 is still valid. The objection to the third definition 
 is its use of the term " direction," a word which 
 because of its apparent simplicity, but real obscurity 
 and vagueness, is exceedingly misleading and 
 troublesome. For example, the straight line is 
 often defined as one which does not change its 
 direction at any point, and yet this same line is said 
 to have opposite directions. Again, the angle is! 
 sometimes defined as a difference of direction. Mo- 
 tion in the circumference of a circle is said to be 
 in a clockwise or counter-clockwise direction, and 
 in this sense a point may move all round the cir- 
 cumference without changing its direction, and yet 
 we speak of this same circumference as a line which 
 changes its direction at every point. Killing has 
 shown that the word direction can only be defined 
 when the theory of parallels is already presup- 
 posed.^'' 
 
 Many other definitions have been proposed, but 
 
 '•'' Einfuehning in die Grundlagen der Geometric, Pader- 
 born, 1898. 
 
THE FRE-LOBATCHEIVSKIAN STRUGGLE ii 
 
 hey throw no light upon the problem, and with one 
 xception they may be omitted. This exception is 
 he definition proposed by Kepler ^^ and De- 
 argues/'^ which is that parallel lines are straight 
 nes which have a common point at infinity ; or " If 
 "v be a point without a given indefinite right 
 ne CD, the shortest line that can be drawn from 
 to it is perpendicular, and the longest line is 
 arallel to CD." ^'^ This definition is important for 
 rojective geometry. 
 
 (2) The substitution of different postulates 
 as been frequently made. Of these Playfair's ^® 
 )rmulation that " Two straight lines which cut 
 ne another cannot both be parallel to the same 
 :raight line," is perhaps the least objection- 
 Die. Cayley ^'^ considered this statement to be 
 Kiomatic. 
 Nasir Eddin (i 201 -1274), a gifted Persian 
 tronomer, in an edition of Euclid subsequently 
 rinted in Arabic and brought out in Rome in 1594, 
 akes the following assumption : " If AB is per- 
 ndicular to CD at C, and if another straight line 
 
 ^^ Kepler's Paralipomena, 1604. 
 
 ^ Brotiillon Proiect, 1639. 
 
 8 Stone's Neiv Mathematical Dictionary, London, 1743. 
 
 » Playfair credits this axiom to Ludlum. See Halsted's 
 tide in Science, N. S. Vol. XIII., No. 325, March 22, 1902, 
 t. 462-465. 
 
 His Presidential Address, Collected Math. Papers, Vol. 
 
 I., pp. 429-459. ^ ^---^-rn-^*^ 
 
 ■/ ^' or THE \ 
 
 I UNIVER81TYJ 
 
12 THE PRE-LOBATCHEVVSKIAN STRUGGLE 
 
 EUF makes the angle EDC acute, then the perpen- 
 diculars to AB comprehended between AB and EF, 
 and drawn on the side of CD toward E, are shorter 
 and shorter, the further they are from CD." Or 
 in general, two straight lines which cut a thuu 
 straight line, the one at right angles, the other at 
 some other angle, will converge on the side where 
 the angle is acute and diverge where it is obtuse. 
 Nothing is here said as to whether the two lines 
 will, or w^ill not, eventually meet ; the assumption 
 is therefore as valid for pseudo-spherical as it is 
 for Euclidean space. 
 
 The work of Nasir Eddin was taken up by John 
 Wallis and communicated in a Latin translation to 
 the mathematicians at Oxford ^^ in 1651; and on 
 the evening of July. 11, 1663, Wallis himself deliv- 
 ered a lecture at Oxford ^^ in w^iich he recom- 
 mended for Euclid's postulate the assumption of 
 the existence of similar figures of different sizes, or 
 to quote his own statement, " To any triangle an- 
 other triangle as large as you please can be drawn 
 which is similar to the given triangle." This is 
 easily shown to be equivalent to the Euclidean pos- 
 tulate. Such figures are impossible in any form of 
 non-Euclidean space. Saccheri proved that Euclid- 
 ean geometry can be rigidly developed if the exist- 
 
 2' Wallis. Ot^cra II., 669-673. 
 
 "- EiiRel and Staeckel, " Die Theorie der ParallcUinien von 
 Euclid his auf Gauss, Leipzig 1895. pp. 21-30. 
 
THE PRE-LOBATCHEWSKIAN STRUGGLE 13 
 
 ence of one such triangle, unequal but similar to 
 another, may be granted. Carnot and La Place and, 
 more recently. J. Delboeuf,^^ have proposed the 
 adoption of Wallis's postulate. 
 
 In 1833 1- Perronet Thompson of Cambridge 
 Dublished a book '^* in which he brilliantly demon- 
 
 trates the insufficiency of twenty-one different at- 
 tempts to dispdse of the Parallel postulate, and 
 :loses the volume with a demonstration of his own 
 
 n which he claims to " establish the theory of 
 parallel lines without recourse to any principle not 
 3Tounded on previous demonstration." ^"^ This en- 
 deavor, however, belongs to the third class of at- 
 tempted solutions which we must now very briefly 
 
 onsider. 
 
 (3) The first recorded attempt to prove the 
 parallel postulate on the basis of Euclid's other 
 assumptions was that of Ptolemy in his treatise on 
 pure geoiiicfry.-'' This proof assumes of course 
 the validity of postulate six,^'^ which does not hold 
 in elliptic s])ace and also involves the untenable 
 assertion that, in the case of parallelism, the sum 
 Df the interior angles on one side of the transversal 
 must be the same as that upon the other side. 
 
 23 Engel and .Stacckel op. cit. p. 19. 
 -* " Geometry without Axioms." 
 -■'■• Quoted from the title. 
 
 2fi Gow. p. 301. For this we are indebted to Proclus. 
 -' Of the Vatican MSS. : Two straight lines can not en- 
 close a space. 
 
14 THE PRE-LOBATCHEWSKIAN STRUGGLE 
 
 > One of the most scientific attempts of this class 
 was that of Girolamo Saccheri in his volume en- 
 titled. " Eiiclidis ab onini nccvo vindicatus, sive 
 conatus gcomctriciis quo stabiliuntur prima ipsa 
 univcrscc gcomatricE principia." ^^ This work only 
 recently came to light. As late as 1893 Professor 
 Klein, himself an able contributor to the knowledge 
 of Hyperspace and non-Euclidean geometry, had 
 not even heard of Saccheri. In 1889 E. Beltrami, 
 at the suggestion of the Italian Jesuit, P. Manga- 
 notti. published"" a note"^ in which he showed 
 that Saccheri had practically wrought out a non- 
 Euclidean geometry almost a century before Lo- 
 batchewsky and Bolyai. Apparently the only 
 thing which prevented Saccheri from perceiving 
 the significance of his discovery was his blinding 
 desire to " vindicate Euclid from every fault.'' His 
 statement of the problem shows clearly that he was 
 on the right road to discovery. If the parallel 
 axiom '" is involved in the remaining assumptions 
 
 -^ Published at Milan, where he was president of the Col- 
 Icgio di Brera, shortly before his death in 1733. This work 
 is now exceedingly rare, the only copy on the Western Conti- 
 nent, perhaps, is that of Professor Halsted, who has trans- 
 lated it into English. It has also been translated into German 
 and forms a part of Engel and Staeckel's History of Parallel 
 Theory previously referred to. 
 
 '-".'J//I dcUa Realc Accademia dci Lined. 
 
 ■■"' Un prcctirsorc italiane di Legendri c di Lobatchewski. 
 
 •'" Saccheri calls it an axiom. He studied Clavius's edition, 
 in which it appears as the i.nh axiom. 
 
THE PRE-LOBATCHEWSKIAN STRUGGLE 15 
 
 of Euclid, then it will be possible to prove without 
 its aid that in any quadrilateral ABCD having right 
 angles at A and B and the side XC equal to the 
 side DB, the angles C and D are also right angle? 
 and in that event the assumption that C and D are 
 either obtuse or acute will lead to contradiction. He 
 proves that these angles cannot be obtuse, for in 
 that case Euclid's axiom that two straight lines 
 cannot enclose a space is contradicted ; but when he 
 endeavors to prove that they cannot be acute he 
 fails of his purpose for in this case he does not 
 meet with any contradiction. 
 
 In regard to the angles C and D he distinguishes 
 three hypotheses as follows : ( i ) hypothesis anguli 
 recti, (2) hypothesis anguli ohtiisi, and (3) hy- 
 pothesis anguli aciiti. He then proves that if either 
 hypothesis is true in a single case it is always true,'^ 
 that in a right triangle the sum of the oblique angles 
 is equal to. greater than, or less than, a right angle 
 according as the hypothesis is anguli recti, anguli 
 obtusi, or anguli acutif''' He ne;xt shows that in 
 the first two hypotheses a perpendicular and an 
 oblique to the same straight line will meet if suffi- 
 ciently produced,^'* hence in these two cases Euclid's 
 postulate is not contradicted.^^ He then proceeds 
 
 3-' Propositions y., VI., and VII. 
 •■'•* Propositions VIII. and IX. 
 3* Propositions XI. and XII. 
 35 Proposition XIII. 
 
i6 THE PRE- LOB AT CHEW SKI AN STRUGGLE 
 
 to prove that according as the triangle's angle sum 
 is equal to, greater than, or less than, two right 
 angles we have hypothesis angiili recti, obtusi, or 
 acuti;^'^ and that with the hypothesis anguli aciiti 
 we can draw a perpendicular and an oblique to the 
 same straight line which will nowhere meet each 
 other.3^ 
 
 It is unnecessary to follow him further. We now 
 have enough to show that Saccheri understood the 
 close connection between the parallel postulate and 
 right angles. In his eager quest for contradictions 
 in pursuit of the hypothesis anguli acuti he prac- 
 tically attained without knowing it, those far reach- 
 ing conclusions which were disclosed a century 
 later. But on the very verge of discovery, being 
 blinded by an intellectual bias toward the tradi- 
 tional view, he rejects this hypothesis upon the un- 
 satisfactory ground that it is incompatible with the 
 nature of the straight line; " for." says he, " it per- 
 mits tl>e assumption of different kinds of straight 
 lines which meet at infinity and have there a com- 
 mon perpendicular." 
 
 Conclusions very like the foregoing w^re also 
 reached by John Henry Lambert, whom Kant calls 
 " dcr nnverglcichliche Mann." Lambert is free 
 from that strained reverence for Euclid which char- 
 acterized Saccheri. and consequently advances be- 
 
 3" Propositions XV. and XVI. 
 " Proposition XVII. 
 
THE PRE-LOBATCHEIVSKIAN STRUGGLE 17 
 
 yond him. He starts from the assumption of a 
 quadrilateral with three right angles and examines 
 the consequences that follow upon the hypothesis 
 that the fourth angle is right, obtuse, or acute. He 
 discovers that the second and third of these assump- 
 tions are incompatible with the existence of unequal 
 similar figures. The second assumption gives the 
 triangle's angle sum greater than two right angles, 
 is incompatible with the theory of parallels, but is 
 realized in the geometry of the sphere. From this 
 he was led to conjecture that the third hypothesis 
 might also be realized on the surface of a sphere of 
 imaginary radius. This is perhaps the first glimpse 
 of the conception of " pseudospherical " surfaces 
 afterwards developed and named by E. Beltrami. 
 Lambert also proves that the departure of the 
 triangle's angle sum from two right angles is a 
 quantity which is proportional to the area of the 
 triangle, the larger the triangle, the greater the 
 departure. Hence in the case of elliptic and hyper- 
 bolic surfaces this angle-sum is a variable quantity 
 which approaches two right angles as the sides of 
 the triangle become less and less. This of course 
 points to the fact that in the infinitesimal the spher- 
 ical and pseudo-spherical triangles approach the 
 Euclidean as a limit, and that in endeavoring to test 
 empirically the validity of the various forms of 
 geometry for the actual space world we must seek 
 
i8 rilL I'KE-LOBATCHEIVSKIAN STRUGGLE 
 
 among very large triangles for a measurable di- 
 vergence from the Euclidean conception. 
 
 Another important suggestion which we owe to 
 Lambert, is that in a space in which the triangle's 
 angle sum differs from two right angles there is an 
 absolute standard of measure, a natural unit of 
 length. 
 
 Gauss and Lcgendre also assumed that the theory 
 oi parallels is involved in Euclid's other assump- 
 tions. Gauss did not publish anything upon the 
 subject, and it was not known until after his death 
 that he had interested himself in it. His corre- 
 spondence, recently published,^^ shows that he was 
 possibly in possession of a non-Euclidean system, 
 but it does not make clear to what extent his in- 
 vestigations were actually carried. He announced 
 in 1/99 that Euclidean geometry would follow from 
 the assumption that a triangle can be drawn greater 
 than any given triangle. In 1804 he was still hop- 
 ing to prove the parallel postulate. In 1830 he an- 
 nounces that geometry is not an a priori science. 
 In 1 83 1 he .states that non-Euclidean geometry is 
 non-contradictory, that in it the angles of the tri- 
 angle diminish without limit when all the sides are 
 increased, and that the circumference of the circle 
 of radius r ~ ttk ( - -] where k is a constant 
 
 dependent upon the nature of space. It is clear 
 
 '■'^ EiiRcl and Staeckcl op. cit. 
 
THE PRE-LOBATCHEIVSKIAN STRUGGLE 19 
 
 from this that Gauss had in his possession the foun- 
 dations of pseudospherical or hyperboHc geometry, 
 and may have been the first to consider it as prob- 
 ably true ■'■^ of the actual world. He came to regard 
 geometry merely as a logically consistent system of 
 constructs, with the theory of parallels as a neces- 
 sary axiom ; he had reached the conviction that this 
 " axiom " could not be proved, though it is known 
 by experience to be approximately correct. Deny 
 this axiom and there results an independent geom- 
 etry which he calls anti-Euclidean.^^ An important 
 conception which Gauss introduced was his " Meas- 
 ure of Curvature," "** which expresses the condition 
 under which a surface has the property of free mo- 
 bility of figures. This " measure " is the reciprocal 
 of the product of the greatest and least radii of 
 curvature and remains unchanged if the surface is 
 bent without distension or contraction of its parts 
 into any position. For example, we can roll up a 
 sheet of paper into the form of a cylinder or cone 
 without changing the dimensions of figures drawn 
 upon it, and hence the geometry of the cylinder or 
 of the cone is the same as that of a limited plane. 
 
 In Legendre's work there is nothing new. His 
 results are of interest, however, because of Dehn's 
 
 ^" Compare Russell, Art. " Non-Euclidean Geometry." The 
 New Encyclopedia Britannica, Vol. XXVIII., pp. 674 ff. 
 ■*o Gauss Ztim Gedacchtiiis. Leipzig 1856. 
 *i IVerke Bd. IV., p. 215. 
 
20 THE PRE-LOBATCHEWSKIAN STRUGGLE 
 
 investigation which will come before us later. Le- 
 crendre was able to prove, by assuming the mfinity 
 or two-sidedness of the straight line and the Archi- 
 medes Axiom of Continuity, (i) that the sum of 
 the three angles of a triangle can never be greater 
 than two right angles, and (2) that if in any 
 triangle this sum is equal to two right angles, so 
 is it in every triangle. 
 
DISCOVERY AND DEVELOPMENT 
 
 OF 
 
 NON-EUCLIDEAN SYSTEMS. 
 
II 
 
CHAPTER 11. 
 
 THE DISCOVERY AND DEVELOPMENT OF NON- 
 EUCLIDEAN SYSTEMS. 
 
 VVe pass now to consider those attempts to solve 
 the riddle which proceed upon the hypothesis that 
 if the parallel postulate is dependent upon Euclid's 
 other assumptions we shall by denying it and affirm- 
 ing them be led into contradiction. This hy- 
 pothesis proved to be an exceedingly fruitful one. 
 The absolute necessity of the parallel postulate for 
 Euclidean geometry and. the possibility of many 
 other systems equally as rigorous and non-contra- 
 dictory as Euclid itself have resulted from it. 
 Several independent discoveries of non-Euclidean 
 geometry have come to light. Schweikart's was the 
 first to be published ; ^ Lobatchewsky's the first to 
 find its way into print. - 
 
 1 1812 in Charkow, communicated to Bessel, to Gerling and 
 afterward to Gauss in 1818. See Science N. S. Vol. XII., pp. 
 842-846. 
 
 " It is interesting to note that as late as Feb. 25, i860, a non- 
 Euclidean geometry was worked out by Prof. G. P. Young of 
 Canada, entirely without knowing that anything of the kind 
 had been previously done. See Canadian Journal of Industry, 
 Science and Art, Vol. V., pp. .341 -.•^S^. i860. 
 23 
 
jJ^/'2d, DISCOVERY OF NON-EUCLIDEAN SYSTEMS 
 
 Lobatchewsky 2 defines the straight Hne as one 
 which fits upon itself in all its positions so that if 
 we turn the surface containing it about two points 
 of the line the line does not move. 
 
 He proposes the following substitute for Euclid's 
 postulate.-* All lines which go out from a point in 
 a plane may with reference to a given line in the 
 plane, be divided into two classes — cutting and 
 non-cut ting lines. If we start in either class and 
 move in the direction of the other we shall event- 
 ually come upon a line which is the bounding 
 position between the two classes. This line is of 
 course unique and is defined as being parallel to 
 the given line. In the following figure, A i§ the 
 given point in the plane and CD the given straight 
 line. AD is perpendicular to CD and EA to AD. 
 In the uncertainty wdiether the perpendicular AE 
 
 3 Lobatchewsky first communicated his results in a lecture 
 before the Physical and Mathematical faculty of the Univer- 
 sity of Kasan, of which he was rector, in 1826. They were 
 printed in Russian in the Kasan Messenger in 1829 and, in 
 ni"ch more complete and extended form, in the Gclchrte 
 Schriften der Unirersitdt Kasan 1836-1838, under the title 
 " New Elements of Geometry, with a Complete Theory of 
 PTrallels." A briefer presentation appeared in German in 
 Berlin in 1840. Houel translated this into French at the 
 suggestion of Baltzer in 1866, and there is an excellent Eng- 
 lish translation by Halsted, 1896. The " Elements " is by far 
 Lobatchewsky's greatest w^ork. This and the paper of 1829 
 are now both accessible in German in Prof. Engel's vol., Leip- 
 zig. 1899. 
 
 * Proposition 16. Paper of 1840. See Halsted's Translation. 
 
DISCOVERY OF NON-EUCLIDEAN SYSTEMS 25 
 
 is the only line which does not meet CD, we may 
 assume it to be possible that there are other lines, 
 for example AG, which do not cut DC. how far 
 soever they may be prolonged. In passing over 
 then from the cutting lines, as AF, to the non- 
 cutting, as AG, we must come upon a line AH 
 parallel to DC, a boundary line upon one side of 
 which all lines, AG, etc., are such as do not meet 
 
 the line DC, while upon the other side every 
 straight line, AF,.etc.. cuts the line DC. Now the 
 angle HAD between the parallel HA and the per- 
 pendicular AD is called the parallel angle. H this 
 be a right angle the prolongation AE' of the perpen- 
 dicular will be parallel to the prolongation DB of 
 CD. In that event every straight line which goes 
 out from A, either itself or its prolongation, lies 
 
j6 UISCOIERY OF NON-EUCLIDEAN SYSTEMS 
 
 in one of the two right angles, made by EE' upon 
 DD'. which are turned toward BC, so that all lines, 
 except the parallel EE', must intersect BC if they 
 are sufficiently produced. In this case, which is 
 Euclid's, there is but one line in the plane parallel 
 to CB.f But if the "parallel angle" be less than a 
 right angle, and such an assumption is perfectly 
 legitimate, there will then lie upon the other side 
 of AD another line AK parallel to DB and making 
 DAK equal to the " parallel angle." 
 
 Upon this latter assumption Lobatchewsky con- 
 structs his geometry, proving in subsequent propo- 
 sitions that a straight line maintains the character- 
 istic of parallelism at every point,^ that two lines 
 are always mutually parallel,*' that in a rectilinear 
 triangle the sum of the angles cannot be greater 
 than two right angles,' and that if in any triangle 
 this sum is two right angles the same is true for all 
 triangles.^ He styles this system " Imaginary 
 Geometry," because its trigonometrical formulae are 
 those of the spherical triangle if its sides are im- 
 aginary, or if the radius of the sphere be taken 
 equal to r\^ — i. 
 
 Thus, by proving the possibility of other systems 
 of geometry. Lobatchewsky destroys the traditional 
 
 f- Proposition XVII. 
 8 Proposition XVIII. 
 ^ Proposition XIX. 
 * Proposition XX. 
 
DISCOVERY OF NON-EUCLIDEAN SYSTEMS 27 
 
 trust in Euclid as absolute truth, and opens up a 
 vista of new and suggestive problems; nor was he 
 wholly unaware of the epistemological import of 
 his discovery. He remarks, " We cognize directly 
 in nature only motion, without which the impres- 
 sions w^hich our senses receive are impossible. Con- 
 sequently all remaining ideas, for example, the 
 geometric, are created artificially by the mind since 
 they are taken from the properties of motion, and 
 therefore space, in and for itself alone, does not 
 exist for us." 
 
 John Bolyai obtained results so closely resem- 
 bling those of Lobatchewsky that Russell. Klein 
 and other distinguished writers have regarded the 
 tw^o as having had a common inspiration in the 
 person of Gauss.^ History, however, does not sup- 
 port this conjecture.^ "^ Bolyai's investigations 
 were published as a twenty-four page appendix to 
 
 » Russell does this in his Foundations of Geometry, p. 8, 
 and Klein, in his Goettingen lectures published in 1893, p. 175, 
 says, " Kein Zzveifel bestchen kann, dass Lobatcheffsky so 
 luohl wie Bolyai die Fragestellung iltrer untcrsuchttngcn der 
 Gaussischen Anregung verdankcn." 
 
 1" See Halsted's Article in Science N. S. Vol. IX., pp. 813- 
 817, June 9, 1899. Schmidt of Budapest has recently found a 
 letter of Gauss to W. Bolyai dated Nov. 25, 1804, and accom- 
 panied by a Latin treatise on parallel lines. This communica- 
 tion shows that neither Gauss nor Bolyai had solved the 
 problem. Both believed the parallel postulate to be demon- 
 strable and were " racing to prove it." 
 
28 DISCOVERY OF NON-EUCLIDEAN SYSTEMS 
 
 the " Tentamcn," ^^ a work of his father, W. 
 Bolyai, in 1831, but its conception dates from 1823. 
 He styles his new geometry the " Science Absolute 
 of Space." The theorems necessary and sufficient 
 for plane trigonometry in this new conception of 
 space are the following: 
 
 a' h' 
 
 ( 1 ) sinh-j^= sin x"-^"^ ^• 
 
 (2) cosh i^ = cosh 4-- cosh-^, in which 
 
 k k k 
 
 a' and b' are the legs of the right triangle; h', the 
 hypothenuse; A, the angle opposite a; and k, an 
 arbitrary constant which is presumed to be uniform 
 throughout space. When k is infinite, finite or im- 
 aginary, these formulas give results which are true 
 for Euclidean, Lobatchewskian. or spherical geom- 
 etry respectively. K then is a form of the space 
 constant. 
 
 Bolyai shows a profounder appreciation of the 
 importance of the new geometry than Lobatchew- 
 sky. The latter never explicitly treats the problems 
 of construction of the old geometry in the changed 
 form which they must take in the new; such, for 
 example, as, " To square the circle," " To draw 
 through a given point a perpendicular to a given 
 straight line," and the problem which in the new 
 
 *' For full title see Bibliography. This work is very rare. 
 There are only two volumes in the United States. These are 
 in possession of Professor Halsted, to whom I am indebted 
 for what knowledge I have of it. 
 
DISCOVERY OF NON-EUCLIDEAN SYSTEMS 29 
 
 geometry grows out of this : " To draw to one side 
 of an acute angle a perpendicular parallel to the 
 other side." All these are elegantly solved by 
 Bolyai. He also shows that the area of the greatest 
 possible triangle which, in this new space, has all 
 its sides parallel and its angles zero is "^i^ where i is 
 what we should now call the space constant. 
 
 Like Lobatchewsky he points out that Euclid is 
 but a limiting case of his own more general system ; 
 that geometry of very small spaces is always Eu- 
 clidean; that no a priori grounds exist for a deci- 
 sion; and that observation can only give an ap- 
 proximate answer as to which geometry is vaild 
 for reality. Thus the new geometry casts no man- 
 ner of doubt upon the geometry of perspective in 
 so far as this deals merely with incidence and coin- 
 cidence. Several propositions are equally true in 
 all these geometries, including that of Riemann, 
 which we are next to consider. It is mainly in the 
 measurement of distances and angles that differ- 
 ences arise. In the case of Euclidean geometry the 
 infinitely distant parts of an unbounded plane would 
 be represented in perspective by a straight horizon 
 or vanishing line, but according to this new 
 geometry we cannot hold that this line would be 
 straight; on the contrary it would be an hyperbola 
 as in the perspective of the terrestrial horizon. If 
 we accept Riemann's hypothesis we cannot be sure 
 that there will be any such line at all. for we do not 
 
30 DISCOIERV OF NON-EUCLIDEAN SYSTEMS 
 
 know that space has any infinitely distant parts. It 
 
 is possible that if we were to move off in any direc- 
 
 , tion in a straight line, we might find that after 
 
 \ traversing a sufficient distance we had arrived at 
 
 our starting point. 1^ rc'^j^t^e M<^ '-^i a. ,u^\ - ■^ir' 
 
 It was not the purpose, however, either of Lo- 
 l)atchewsky or of Bolyai to discuss the validity of 
 their own or of Euclidean geometry. Their motive 
 was logical and mathematical, not epistemological 
 or ontological. Is the result of denying the parallel 
 postulate contradictory or non-contradictory? 
 That was their problem. Nor did they solve it com- 
 pletely. The number of possible theorems in either 
 system is practically infinite. As far as they had 
 gone they were justified in saying there were no 
 contradictions, but in nothing more. That latent 
 contradictions might be revealed by further de- 
 velopments was perfectly possible. For this reason 
 logical dependence or independence of any group 
 of fundamental assumptions can never be com- 
 pletely tested by the method which they employed. 
 The purpose of Ricmann and Hclmholtz was a 
 'very different one. Their motive was philosophical. 
 The attack was no longer confined to the parallel 
 postulate. The problem was generalized. The old 
 synthetic method of Euclid, still adhered to by 
 Lobatchewsky and Bolyai. w'as now^ abandoned, 
 and the properties of space were couched not in 
 terms of intuition, but of algebra. As a result the 
 
, UNIVERSITY / 
 
 DISCOVERY OF NON-EUCLlt&i^&iJ^kS^ffs 31 
 
 subsequent history of non-Euclidean geometry took 
 on an analytical rather than a synthetic character. 
 
 Riemann and Helmholtz both sought to show 
 that all the so-called geometrical axioms of Euclid 
 are not a priori, but empirical in character. In- his 
 most remarkable dissertation ^^ Riemann expresses 
 this conviction in the following language : " The 
 properties which distinguish space from other 
 triply extended magnitudes are only to be deduced 
 from experience.. Thus arises the problem to dis- 
 cover the simplest matters of fact from wliich the 
 measure relations of space may be determined. 
 These matters of fact are, like all mat- 
 ters of fact, not necessary but only of empirical 
 certainty." 
 
 Riemann introduces into the problem the gen- 
 eral conception of a manifold of which space is but 
 a specialization arrived at through considerations 
 of measurement. A manifold is continuous or dis- 
 crete, according as there does or does not exist 
 among its specializations a continuous path from 
 one to another. When in the case of a continuous 
 manifold we can pass in a definite way from one 
 specialization to another, i. e.. where continuous 
 progress is possible only forward or backward, the 
 
 ^' Die Hypothesen zvelchc dcr Geometric su Gntndc licgcu 
 This was his inaugural dissertation before the Philosophical 
 faculty of the University of Goettingen in 1854. It was not 
 published until after his death in 1867. 
 
32 DISfOlERV OF NON-EUCLIDEAN SYSTEMS 
 
 manifold is one-dimensional. If this group of 
 specializations pass over into another entirely dif- 
 ferent, and again, in a definite way so that each 
 specialization of the one passes into a definite spe- 
 cialization of the other, all the specializations thus 
 formed constitute a doubly extended manifold. 
 Similarly we may get a triply extended manifold, 
 by supposing a doubly extended one to pass over 
 in a definite way into another entirely different. 
 This process may in general be repeated as often 
 as we please, since from the analytical point of view 
 there is nothing to limit the number of dimensions. | 
 Space, however, as known to us is a manifold of 
 only three dimensions whose specializations are 
 points. This we know not a priori, but as a matter 
 of experience. The line is a point aggregate in 
 which definite progress only forward or backward 
 is possible; a surface is a one-dimensional manifold 
 of line specializations in which progress always into 
 new specializations is possible only forward or 
 backward. It is therefore a two dimensional point 
 aggregate. In a similar manner we obtain the solid 
 as a one-dimensional surface aggregate, a two 
 dimensional line aggregate, or a three dimensional 
 point aggregate. But here according to experience 
 the process stops; further progress is not possible, 
 for we cannot in any way visualize the result. 
 
 Now to measure a continuous manifold certain 
 postulates are necessary. Definite portions of it are 
 
DISCOVERY OF NON-EUCLIDEAN SYSTEMS zz 
 
 called Quanta, and comparison of these is accom- 
 plished by measuring, which requires the possibility 
 of superposing the magnitudes compared. There- 
 fore at least one standard magnitude must be inde- 
 pendent of position, i. e., capable of being moved 
 about freely without altering its value. Let us 
 suppose that it is the length of lines which is thus 
 independent of position and that every line is cap- 
 able of measurement by means of every other. In 
 this way position fixing becomes a matter of quan- 
 tity fixing and consequently the position of a point 
 being expressed by means of n variables Xi, Xo, Xj, 
 
 Xj^, the determination of the line comes in 
 
 part to be a matter of giving these quantities as 
 functions of one variable. The problem then is to 
 establish a mathematical expression for the length 
 of a line and to this end the quantities x must be 
 regarded as expressible in terms of certain units. 
 To accomplish this let it be assumed that the ele- 
 ment of length ds is unchanged (to the first order) 
 when all the points undergo the same infinitesimal 
 motion; ds will then become a homogeneous func- 
 tion of the first degree of the increments of dx and 
 remains unchanged when all the dxs, change signs. 
 The simplest case obviously is that in which ds is 
 the square root of a quadratic function and this is 
 ;he only one which Riemann especially considers. 
 
 We must now consider Riemann's conception of 
 :he " Measure of Curvature " of space which is a 
 
34 DlSCOl'ERV OF S ON -EL' C LI DEAN SYSTEMS 
 
 somewhat obscure and misleading" extension of the 
 Gaussian conception. Since Riemann's use of this 
 conception, especially as popularized by Helmholtz, 
 has led to much confusion, it is necessary to pause 
 for a moment and endeavor to understand what it 
 really means. The conception goes back logically 
 as well as historically to our notion of the straight 
 line as a measure of length. Since exact congru- 
 ence is essential to geometrical measurement, 
 strictly speaking, only a straight line can be meas-: 
 ured by a straight line. If then the stretch (definite 
 portion of a straight line) is to be the standard of) 
 all linear measurement, it is evident that we cannotj 
 measure the circle except by passing to the consid 
 eration of infinitesimal arcs, which are to be re- 
 garded as straight. Similarly our notion of curva- 
 ture is referred to the circle as standard of meas- 
 urement. The curvature of the circle at any point 
 means, of course, its amount of bending or depart- 
 ure from the tangent line, and since this amount is! 
 constant for the same circle and is equal to th(! 
 square of the reciprocal of the radius, it becomes eji 
 convenient standard for the measurement of lineai 
 curvature in general. Now since the curvature o 
 other curves varies from point to point we agaii 
 pass to the infinitesimal and measure the amount o 
 this curvature by determining the circle which mos 
 nearly coincides with the curve at the point consid 
 ered. This circle w^ill pass through three consecu' 
 
DISCOIERV OF XON-ELCLJDEAX SyS'lEMS 35 
 
 :ive ix)ints of the curve and hence its construction 
 s always possible theoretically,^-' for any curve. 
 Diane or tortuous. In an analogous way Gauss 
 determined the curvature of surfaces by their 
 imount of departure from the plane. In the case 
 Df curved surfaces we can draw through any point 
 \n unlimited number of geodesic lines whose curva- 
 ;ure, generally speaking, will not be the same. If 
 :hen we draw all the geodesies from the point at 
 ivhich the curvature is tested to the neighboring 
 points on the surface they will form a singly in- 
 finite manifold of arcs" among which there will be 
 one of maximum and one of minimum curvature. 
 Representing the radii of the osculatory circles by 
 r and ri we shall-have for the " measure of curva- 
 ture " of the surface at the point considered the 
 expression — — In case of the sphere, all the geo- 
 desies going out from a point have the same curva- 
 ture, and hence the surface has as its measure of 
 curvature Jt-. For the Euclidean plane the radii 
 are all infinite and the curvature is therefore zero. 
 For the cone and the cylinder, since straight lines 
 can be drawn in each, the measure of curvature is 
 zero and any figure not too large which can be 
 drawn upon the plane may also be drawn without 
 distortion of parts upon the surface of any cylinder 
 
 13 It is interesting to note, however, that the assumption 
 that this construction is possible is equivalent to assuming 
 
 :he truth of the parallel postulate. 
 
36 DISCOVERY OF NON-EUCLIDEAN SYSTEMS 
 
 or cone. It is therefore an obvious corollary that 
 for any surface to possess the property of free mo- 
 bility its measure of curvature must have a constant 
 value ^"^ at every point. 
 
 But how is it possible to extend this conception 
 further? In what sense is it permissible to speak of 
 the curvature of space? Thus far it has been pos- 
 sible to visualize results, and in extending succes- 
 sively to wider and wider applications the notion 
 of curvature we have not departed at all essentially 
 from its original meaning. All along it has signi- 
 fied some degree of departure from a standard of 
 straightness or evenness of lines or of surfaces. 
 Does the word then still retain any vestige of this 
 original meaning when applied by Riemann to the 
 conception of space? In the consideration of sur- 
 face curvature as given above, a third dimension 
 of space was involved ; does the curvature then of 
 three dimensional space require a fourth dimen- 
 sion? These are important questions, and the fail 
 ure to comprehend their meaning has led to numer 
 ous vagaries on the part both of mathematician 
 and philosophers. 
 
 In seeking an answer, let us note, first of all, that 
 in the above consideration of curvature as a prop- 
 erty of surfaces no necessary reference is made to 
 a third dimension in the analytic development fro 
 
 1* See Minding's proof, Crelle's Journal, Vols. XIX., XX-j 
 1839-40. 
 
DISCOVERY OF NON-EUCLIDEAN SYSTEMS 3? 
 
 which it results. It is only when one tries to pic- 
 ture to himself what it may mean for sense that he 
 seems to find need of this extra dimension. Gauss 
 assumes that points on the surface may be deter- 
 mined by the two co-ordinates, u, v, and finds that 
 small arcs of the surface wjU be given by the 
 formula ( i ) ds- = Edu- + 2Fdu.dv + Gdv'-, in 
 which E, F, and G are functions of u, v. But u, 
 V may be lengths of lines on the surface or angles 
 between geodesic lines and therefore we do not need 
 to go outside the surface itself for their analytical 
 determination. Hence the idea of a third dimen- 
 sion may be dropped and the measure of curvature 
 be analytically regarded as an inherent property of 
 the surface itself, provided of course we are dealing 
 with a surface whose curvature is constant.'^ 
 
 Xow it was this formula of Gauss which Rie- 
 mann desired to make use of in determining a mean- 
 ing for his " measure of curvature " as applied to 
 space of more than tw-o dimensions. We can see . 
 already how this conception may be given an analyt- 
 ical meaning. As in the case of a surface we find an 
 unlimited number of radii of curvature at any point 
 corresponding to the geodesies going out from the 
 same, so in regard to space as a manifold of three 
 
 '■"' Otherwise a precise co-ordinate system which is required 
 n the development of our formula would appear to be logic- 
 illy impossible. Consult Russell's Foundations of Geometry, 
 ;hap. III. 
 
38 DISCOVERY OF NON-EUCLIDEAN SYSTEMS 
 
 or of ;/ dimensions' we may have an unlimited num- 
 ber of measures of curvature at any point corres- 
 ponding to the surfaces that may be passed through 
 it. If then we Hmit ourselves to a three dimensional 
 manifold and represent its measure of curvature at 
 any point by K we shall have K=f (XiX^XsJ ^15253; 
 77,^)where (X1X2X3) give the position of the point, 
 (91S2S3 ) the orientation of the surface and {r O) 
 the curvature at any point on the surface. Express- 
 ing K in terms of the quantities involved in formula 
 ( I ) for ds, we have, 
 
 (2)K 
 
 I 
 
 
 1 "[ 
 
 ( i/u 
 
 
 2v/ E G -F 2 
 
 dG' 
 
 d 
 
 + 
 
 2 
 
 du 
 
 dt> 
 
 LyEG-F 
 
 F d^ 
 
 ] 
 
 El/ 
 
 
 d7> 
 
 c 
 
 EG-F2 
 
 ) 
 
 I F 
 
 Eo/EG-F^ 
 
 rt'E I 
 
 ^7' V eg-f 
 
 d¥ 
 d%> 
 
 v/'EG-F: 
 
 ^E 
 
 dv 
 
 from which it_is j)lain that K may have any value 
 depending upoiTthe meaning which we give to our] 
 linear element ds. An unlimited number of geome- 
 tries is therefore possible, the essential foundatioi 
 of any one of them being the expression which i1 
 gives for the distance between two points takei 
 anywhere and lying in any direction from each^ 
 other, beginning with the interval between them as 
 infinitesimal. If we take ds^rrrdu^-f-dv^ which \\ 
 equivalent to assuming the existence of a rectangl( 
 
DISCOVERY OF NON-EUCLJDEAX SYSTEMS 39 
 
 of which ds is the diagonal and du and dv the ad- 
 jacent sides, F in ( i ) becomes equal to zero and E 
 and G each equal to unity. Substituting these val- 
 ues in (2) K becomes zero, its true value for 
 Euclidean space.^ 
 
 Riemann's measure of space curvature is then 1 
 simply a quantity obtained by purely analytical cal- ) 
 culation and so far forth does not necessarily in- . 
 volve any relations that would have a meaning for 
 sense-perception. What is the true meaning of ds? 
 That is the important epistemological question 
 which his treatment suggests, but does not suf- 
 ficiently answer. 
 
 Riemann himself works out one form of Elliptic 
 geometry. It is called spherical because it holds for 
 the surface of the sphere. If the measure of cur- 
 vature has in actual space a positive value howso- 
 ever small tlrere are in reality no such things as 
 parallel lines, for all lines meet if sufificiently pro- 
 duced, and space itself is limited though still un- 
 bounded. 
 
 It is clear that Riemann's whole discussion in- 
 volves a profound mathematical and philosophical 
 significance, the utmost reaches of which have not 
 yet been fully explored. 
 
 Helmholtz and Riemann reached practically the 
 same conclusions in regard to the nature of our 
 5pace conception and the empirical origin of 
 geometric axioms, but their methods of investiga- 
 
40 DISCOVERY OF NON-EUCLIDEAN SYSTEMS 
 tion were quite different. Helmholtz was a phys- 
 icist and a physiologist, and was led to consider 
 the problem, as he says, by attempting to represent 
 spatially the color manifold and also by his inquiries 
 into the origin of our ocular measure of distance m 
 the field of vision. His method of approach was 
 from the standpoint of natural science rather than 
 from that of mathematics as in the case of Rie- 
 
 mann 
 
 While Riemann begins with an algebraic expres- 
 sion which represents in the most general form 
 the distance between two infinitely near points and 
 deduces therefrom the conditions of the free mobil- 
 ity of rigid figures; Helmholtz starts from free 
 mobility as an observed fact in nature and derives 
 from it the necessity of Riemann's algebraic ex- 
 pression. He also proves mathematically the lat- 
 ter's arc-formula, but his proof was not strictly 
 rigid and is now superseded by that of Sophus 
 Lie whose attention was called to the problem by 
 Professor Klein. Helmholtz starts with congruence, 
 without which measurement is impossible, and ther 
 affirms that congruence is proved by experience.^ 
 - We measure distance between points by applying 
 to them the rule or chain, we measure angles by 
 bringing the divided circle or the theodolite to the 
 vertex of the angle." " Thus all geometrical meas- 
 
 i6"Ueber die thatsaechlichen Grundlagen der Geometrie," 
 Wissenschaftliche Abhandhmgen, Vol. II.. 1866 (p. 610 ff). 
 
DISCOVERY OF NON-EUCLIDEAN SYSTEMS 41 
 
 urement depends on our instruments being really 
 as we consider them, invariable in form, or at least 
 as undergoing no other than the small changes we 
 know of as arising from variations of temperature 
 or from gravity acting differently at different 
 places." In all our measuring then we can only 
 make use of the surest means we have to determine 
 what we are otherwise in the habit of making out 
 by sight or touch. These statements of Helmholtz 
 though very suggestive, do not penetrate to the 
 heart of the matter. He is certainly right, however, 
 in starting as he does with the facts of experience 
 out of which our geometrical conceptions, whatever 
 we may say as to their ultimate source and justifi- 
 cation, have come to assume the forms in which 
 we now have them. 
 
 It is interesting to take note of the axioms which 
 Helmholtz comes to regard as being a necessary and 
 sufficient basis for geometry. They are briefly as 
 follows : ( I ) In a space of // dimensions, a point is 
 uniquely determined by the measurement of // con- 
 tinuous variables or co-ordinates. 
 
 (2) Between the 211 co-ordinates of any point 
 pair of a rigid body, there exists an equation which 
 Is the same for all congruent point-pairs. By tak- 
 ing a sufficient number of these point-pairs, we can 
 E^et more equations than we have unknown quanti- 
 ties, and thus the form of the equations may be 
 determined and all of them satisfied. 
 
42 DISCOVERY OF NON-EUCLlDEAN SYSTEMS 
 
 (3) Every point can pass freely and continuously 
 from one position to another. Hence by (2) and 
 (3) if two systems of points m and n can be 
 brought into congruence in any position they may 
 also be made congruent in every other position, 
 
 (4) If (n — i) points of a body remain fixed, 
 so that every other point can only describe a certain 
 curve then that curve is closed. 
 
 If now we limit ourselves to three dimensions 
 Helmholtz claims that these four axioms will be 
 sufficient to give us the Euclidean and non-Euclid- 
 ean systems as the only alternatives. This conclu- 
 sion, however, is open to criticism. Sophiis Lie, 
 for instance, has shown ^^ that the fourth axiom is 
 unnecessary. It is included in the axiom of con- 
 gruence when properly formulated. In fact, Con- 
 gruence and Free Mobility are both involved in the 
 conception of the homogeneity of space. Russell 
 has also pointed out^^ that these four axioms flow 
 directly from the fundamental assumption of the 
 relativity of position. 
 
 In a second paper/'' which from a mathe- 
 matical view-point constitutes his greatest contribu- 
 tion to the subject, Helmholtz adds two more 
 
 1^ " Grundlagen der Geometric," Leipsigcr Berichte, 1890. 
 
 '■'*" Foundations of Geometry," pp. 128 ff, 1897. 
 
 ^^ " Ueber die Thatsachen, die der Geometric zum Grund 
 liegen." Wissenschaftliche Abhandlungen, Vol. II., p. 618 ff., 
 1868. 
 
 I 
 
DISCUrERY OF NON-EUCLIDEAN SYSTEMS 43 
 
 axioms, namely, that space has three dimensions, 
 and that space is infinite. 
 
 The results of Riemann, Helmholtz, and Lie, had 
 in a measure been anticipated by Wolfgang Bolyai/i 
 in his famous " Tentamen. " to which we have 
 already alluded. Bolyai starts his analysis with the 
 principle of continuity,-'^ then postulates the prin- 
 ciples of congruence and free mobility of rigid 
 bodies -^ and finally adds to these the following 
 postulates, (i) If any point remains at rest any 
 region in which it is, may be moved about in 
 Innumerable ways so that any other point than the 
 one at rest may recur to its former position. If 
 two points are fixed, motion is still possible in a 
 specific way. (2) Three points not co-straight pre- 
 vent all motion.^- From these assumptions he de- 
 duces both Euclid and the non-Euclidean system , 
 of his son. John Bolyai. He also observes that the 
 measurements of astronomy show that the parallel 
 postulate is not suffieiently in error to interfere in 
 practieeP By casting off the assum])tion of the 
 infinity of space Riemann and Helmholtz obtained 
 as a third possibility for the universe, an elliptic 
 geometry; but even this is suggested- by John 
 
 -^ " Spatium est quantitas, est contimiiim."' P. 44-- 
 
 21 pp. 444, Sec. 3 : " Corpus idem in alio quoque loco vi- 
 
 Jenti quaestio succurrit : mim loca siiisdem divcrsa aequalia 
 
 unt?" Intiiitus ostendit, aequalia esse." 
 -- p. 446. Sec. 5- 
 
 23 p. 489. 
 
44 DlSCOl'ERY OF NON-EUCLIDEAN SYSTEMS 
 
 Bolyai in his proof that spherics are independent of 
 Euclid's assumption.^ ^ 
 
 Beltrami was the first to become clearly conscious 
 of the fact that the theorems of Lobatchewskian 
 geometry may be realized in ordinary Euclidean 
 space on surfaces of constant negative curvature.^^ 
 Minding had already shown^*^ that the geometry 
 of such surfaces so far as geodesic triangles 
 are concerned could be deduced from that of the 
 sphere by giving the radius an imaginary value, 
 Beltrami, however, generalized the problem first for 
 plane geometry in the Saggio, and in a subsequent 
 paper ^^ for ;/-dimensional manifolds of constant 
 negative curvature. He proves that a pseudo- 
 spherical space of any number of dimensions can 
 be considered as a locus in Euclidean space of 
 higher dimensions. This idea of space of one type 
 as a locus in space of another type and of dimen- 
 sions higher by one, as Whitehead says.^'^ is partly 
 due to John Bolyai. It is an exceedingly important 
 conception because it brings into relation the geom- 
 etries of Lobatchewsky, Euclid and Riemann, and 
 
 -* See Halsted : " Report on Progress of Non-Euclidean 
 Geometry " in Proceedings of the American Association for 
 the Advancement of Science, Vol. 48, 1899, pp. 53-68. 
 
 • 25 " Saggio di Interpretazione della Geometria non-Eu- 
 clidea," Giornali di Matcmatische, Vol. VI., 1868. 
 
 28 Crelle's Journal, Vol. XIX. 
 
 2T " Teoria fondamentale degli spazzii di curvatura Con- 
 stanta " Aiuiali di Matematica II., Vol II., 1868-9. 
 
 28 Universal Algebra, Cambridge 1898, Sec 262. 
 
DISCOVERY OF NON-EUCLIDEAN SYSTEMS 45 
 
 by establishing the fact that demonstrations of one 
 geometry may b^ appropriate adaptation, be made 
 to hold good for corresponding theorems in the 
 other two, it afforded, for the first time, a conclusive 
 proof that the geometries of Lobatchewsky and 
 Riemann are no more contradictory than is Euclid 
 itself, and thus gave to all three geometries a co- 
 ordinate rank. 
 
 We must now consider another very striking 
 change, both as to method and purpose, in the his- 
 torical development of Meta-geometry. Thus far 
 we have dealt altogether with metrical conceptions. 
 The quest has been to understand the necessary pre- 
 conditions to the possibility of spatial measurement 
 and to this end space itself has been regarded as a 
 species of magnitude whose peculiar properties 
 need to be defined. Starting with a doubt as to 
 the apodeictic truth of a single Euclidean axiom, 
 others have been called in question and the convic- 
 tion has grown that they are all of an empirical 
 and somewhat arbitrary character. One after an- 
 other of those sacred postulates which for two thou- 
 sand years even the best minds had considered as 
 eternally true, was denied, and new systems of 
 geometry sprang into existence as non-contradic- 
 tory and, so far as empirical observation could go, 
 as valid for reality as Euclid itself. Consequently 
 that same human desire for truth, logically pure 
 ind indubitable, which for more than twenty cen- 
 
46 DISCOVERY OF NON-EUCLIDEAN SYSTEMS 
 
 turies had reposed with perfect confidence upon 
 Euclid, gradually drove the geometer to seek the 
 longed for necessity and logical purity wholly out- 
 side the realm of metrical considerations. Though 
 avowedly mathematical and technical in its aims 
 and purposes this new movement has attained cer- 
 tain results that are of far-reaching philosophical 
 importance. \ Magnitude, superposition and congru- 
 ence are now dispensed with and the • attention is 
 directed to the purely qualitative, as opposed to 
 the quantitative, aspects of space?, 
 
 Perhaps the greatest names connected with this 
 movement are those of Cayley and Klein. By 
 treating the surface of the pseudo-sphere as a plane 
 and its geodesies (corresponding to great circles on 
 a sphere) as straight lines Beltrami had shown that 
 all the theorems of Lobatchewskian geometry can 
 be developed upon this surface ; but in Cayley's new 
 Theory of Distance we seem to have a much simpler 
 explanation, and one which requires no modification 
 of the ordinary conceptions of space or of Euclidean 
 planes and straight lines, but only an extension of 
 the customary ideas of measurement. Cayley was 
 throughout a staunch defender of Euclid. 
 
 In 1858 ^^ he states that " in any system of geom- 
 etry of two dimensions the notion of distance can 
 be arrived at from descriptive principles alone by 
 means of a conic called the Absolute and which in 
 -" See Collected Mathematical Papers, Vol. V., p. 550. 
 
DISCOVERY OF NON-EUCLIDEAN SYSTEMS 47 
 
 ordinary geometry degenerates into a pair of 
 points." He sets himself the task of estabHshing 
 this position mathematically in his *' Sixth Me- 
 moir upon Quantics" in 1859. In determining the 
 analytic expression for the distance of two points 
 Cayley first introduces the inverse sine or cosine of a 
 certain function of the co-ordinates and shows that 
 metrical properties become projective wath reference 
 to the degenerate conic called the Absolute ; ^^ but 
 later he recognizes and adopts Klein's definition as 
 an improvement upon his o\vn.^^ This definition is 
 expressed by the formula : distance PO = c log 
 AP.BQ 
 
 where A and B are the fixed points 
 
 AQ.BP 
 
 which determine the /Absolute. This formula pre- 
 serves the fundamental additive relation character- 
 istic of distance; viz., distance PQ + distance QR 
 = distance PR. 
 
 We can not here enter into the details of this 
 mathematical discussion but must be content with 
 the popular exposition which C^ley himself sup- 
 plies in his Presidential Address •"'^ of 1883. In 
 condensed form his conception is this.^^ Consider 
 an ordinary indefinitely extended plane; and let us 
 
 ^" Grundgebild, see Klein. 
 
 31 Collected Math. Papers. Vol. II., p. 604 
 
 =*-' Collected Math. Papers, Vol. XL, pp. 429-459- 
 
 33 Collected Matl^. Papers, Vol. XI., pp. 435 ff. 
 
48 DISCOVERY OF NON-EUCLIDEAN SYSTEMS 
 
 modify only the notion of distance. We measure 
 distance with a foot rule, let us say. Imagine then 
 the length of this rule constantly changing (as it 
 might do by an alteration of temperature) but 
 under the condition that its actual length shall de- 
 pend only on its situation in the plane and on its 
 direction. In other words, if its length is a certain 
 amount for a given situation and direction it will 
 be the same whenever it returns to this position, no 
 matter how, or from what direction it comes. Now 
 it is plain that the distance along any given straight 
 or curved line between any two points could be 
 measured with this rule, and always with the same 
 determinate result, no matter from what point in 
 the line we begin. Of course this distance will not 
 be what we usually mean by the term, for we do not 
 ordinarily regard our standards as varying quan- 
 tities. But for aught we know experimentally this 
 may be what actually occurs. Suppose then that 
 as this rule moves away from a fixed central point 
 in the plane it becomes shorter and shorter; if this 
 shortening takes place with sufficient rapidity, it is 
 clear that a distance which in the ordinary sense of 
 the word is finite, will in this new sense be infinite, 
 for no number of repetitions of the length of the 
 ever-shortening rule will be sufficient to cover it. 
 There will be then surrounding the central point a 
 certain finite area every point of whose boundary 
 will be, according to this theory, at an infinite dis- 
 
DISCO I EK y OF NON-E L CLIDEA N S ) \S- 1 EMS 
 
 49 
 
 tance from the central point. Beyond this boundary 
 there is an unkno^vable land or in mathematical lan- 
 guage an imaginary or impossible space. 
 
 By attaching to this variable standard of Cayley's 
 suitable laws of change, the various forms of non- 
 Euclidean plane geometry may be had upon this 
 Euclidean plane ; and the idea may be extended so 
 as to obtain non-Euclidean systems of solid geom- 
 etry in Euclidean space. 
 
 This connection of Cayley's Theory of Distance 
 with the various forms of Metageometry was first 
 pointed out by Felix Klein.^^ Klein showed that 
 if Cayley's Absolute be taken as real we get Lo- 
 batchewskian, or what he calls hyperbolic, geom- 
 etry ; if it be imaginary we get two forms of elliptic 
 geometry; the double elliptic, spherical, or Rieman- 
 nian already considered, in which all geodesies have 
 two points in common ; and the single elliptic which 
 we owe to Klein ^^ and in wdiich all geodesies are 
 closed cur\^es having only one point in common. 
 This is doubtless logically the simplest of all sys- 
 tems. ' Tf the Absolute be an imaginary point pair 
 we get parabolic geometry, and if this point pair 
 be what is known as circular points the result is 
 the ordinary Euclidean system. 
 
 The natural result of all this was that both Cay- 
 
 34 Nicht-EuclicI, Bonk T.. Chapters I. and II. 
 
 35 Also independently discovered by Simon Newcomb. See 
 his article in Crelle's Journal, Vol. 8,3. 
 
50 DISCOVERY OF NON-EUCLIDEAN SYSTEMS 
 
 ley and Klein came to regard the whole question 
 of non-Euclidean systems as having no philosoph- 
 ical importance, since it seemed to them that it in 
 no way concerns the nature of space, but only the 
 definition of distance, which in their view is per- 
 fectly arbitrary, being merely a question of conven- 
 ience. 
 
 Whether we accept this conclusion or not it must 
 be admitted that the projective method, employed 
 by them, is independent of metrical presuppositions 
 and deals directly with that qualitative likeness of 
 geometrical figures which is a necessary prerequisite 
 to quantitative comparison. The distinction be- 
 tween Euclidean and non-Euclidean geometry is a 
 metrical one; it essentially disappears altogether in 
 projective geometry. Hence projective geometry 
 deals with the conception of space from a higher 
 point of view, which includes within its scope every 
 variety of metrical space and whose defining adjec- 
 tives must in consequence possess very great philo- 
 sophical interest. Furthermore, these adjectives 
 may also be regarded as the simplest indispensable 
 requisites of geometrical reasoning. 
 
 In this connection the important work of Sophus 
 Lie, which was awarded the Lobatchewsky prize 
 Nov. 3, 1897. must briefly claim attention. ^^ Felix 
 Klein declared that this work excelled all others 
 
 ,ia " Theorie der Transformations griippen," Vol. III. Pub- 
 lished at Leipzig, 1893. 
 
DISCOVERY OF NON-EiCLlDEAS SYSTEMS 51 
 
 SO absolutely that no possible doubt could be enter- 
 tained as to the justice of this award. Helmholtz 
 had already originated the idea of studying the 
 essential characteristics of space by a- consideration 
 of the movements possible in it. Klein called the 
 attention of Lie to this problem of Helmholtz and 
 encouraged him to undertake an investigation of 
 it by means of his Theory of Groups. We can but 
 meagerly indicate the outcome of this investigation. 
 As stated by Lie his problem is : " To determine all 
 finite continuous groups of transformations in three 
 dimensional space in which two points have a single 
 invariant and more than two points have no essen- 
 tial invariant"; meaning by invariant the distance. 
 D, between the two points and by the statement 
 that more than two points have no essential in- 
 variant, no invariant which is not expressible in 
 terms of D. He finds that under the conditions of 
 the problem the group must be six-parametered and 
 transitive and cannot contain two infinitesimal 
 transformations whose path curves coincide.'*" 
 Two solutions of the problem are given. He first 
 investigates a group in space, possessing free mobil- 
 ity in the infinitesimal in the sense that if a point 
 and any line element through it be fixed, continuous 
 motion shall still be possible; but if. in addition, 
 any surface element through this point and the line 
 
 3^ Transformations-Gnippcn. \'ol. III., p. 405 f., 1893. 
 
52 DISCOVERY OF NON-EUCLIDEAN SYSTEMS 
 
 element be fixed, no continuous motion shall be 
 possible. The groups which in tri-dimensional 
 space harmonize with these conditions Lie finds to 
 be only those which are characteristic of the Euclid- 
 ean and non-Euclidean geometries, but strange to 
 say he also discovers that for the apparently an- 
 alogous but simpler case of the plane or two dimen- 
 sional space there are, besides these, certain other 
 groups where the paths of the infinitesimal trans- 
 formations are spirals. In his second demonstra- 
 tion, starting from transformation-equations with 
 Helmholtz's first three postulates he proves that for 
 a space of three dimensions the fourth postulate is 
 entirely superfluous.^^ 
 
 From an analytical point of view Professor Hil- 
 bert in a recent article in the Mathematische Anna- 
 len,^^ which may be mentioned here, has advanced 
 beyond these results of Lie by showing that it is 
 possible to do away with the dififerentiability of 
 functions which Lie's discussion requires. From 
 the intuitional standpoint his article ofifers no im- 
 provement and is open to certain criticisms which 
 Dr. Wilson ""^ has pointed out. 
 
 •■'^ For a brief statement of what is essentially Lie's method 
 in English, see Halsted's " Columbus Report," Proc. A. A. A. 
 S., Vol. 48, 1899. Also Poincare's Art. in Nature previously 
 cited. 
 
 ■■'s Bd. 56, Heft 3, pp. 381-422, October, 1902. 
 
 *f> Archiv der Mathematik und Physik III. Rcihe VI. 1. u. 2. 
 Heft, Jan. 1903. 
 
DISCOVERY OF NON-EUCLIDEAN SYSTEMS 53 
 
 In his famous Festschrift,^^ however, Professor 
 Hilbert has done more perhaps than any one else 
 except certain ItaHans to determine the precise num- 
 ber, meaning and relations of the postulates essential 
 to geometry. In this older work Hilbert followed 
 essentially the Euclidean method with a logic so 
 keen and pure and a result so simple that many 
 have even expressed the opinion that it will ulti- 
 mately supersede Euclid in the elementary schools. 
 Certain defects, however, have been pointed out by 
 Schur,"*- Moore, ■^-^ and others, showing that Hil- 
 bert's postulates are not independent as he had sup- 
 posed they were, and also illustrating how difficult 
 it is to satisfy logic when one seeks to determine 
 the foundations of geometry by the intuitional 
 method. The discovery of this fact has led Hilbert 
 in his recent article to abandon this method for the 
 more strictly logical one. He starts from the ideas 
 of Manifoldncsses and Groups as Lie had done, but 
 uses the new conception of Manifoldncsses intro- 
 duced by Georg Cantor thus dispensing with any 
 special reference to a system of co-ordinates in a 
 geometric space. 
 
 When the Lobatchewsky prize was awarded to 
 Lie. the thesis of M. L. Gerard, of Lyons, also re- 
 
 *i Grundlagen der Geometric, Leipzig 1899. 
 *2 Mathematische Annalcn Bd. 55. p. 265 flF. 
 *3 Transactions of the American Mathematical Societj-. Vol. 
 III., pp. 142 ff. 
 
54 DISCOVERY OF NON-EUCLIDEAN SYSTEMS 
 
 ceived honorable mention. In this thesis Gerard 
 endeavors to estabhsh the fundamental propositions 
 of non-Euclidean geometry without any hypothet- 
 ical constructions except the two which are assumed 
 by Euclid."*-^ ( i ) Through any two points a 
 straight line can be drawn. (2) A circle may be 
 described about any center with any given sect as 
 radius. But in order to establish the relations be- 
 tween the elements of a triangle in a thorough- 
 going manner he adds to these, two other assump- 
 tions as follows : ( I ) A straight line which inter- 
 sects the perimeter of a polygon in some other point 
 than one of its vertices intersects it again, and (2) 
 two straight lines, or two circles, or a straight line 
 and a circle, intersect if there are points of one on 
 both sides of the other. One of the most important 
 considerations for the advocates of non-Euclidean 
 geometry is the requirement that all its figures shall 
 be rigorously constructed. It was to meet this re- 
 quirement that Gerard's investigation was under- 
 taken and it is in this fact that its significance 
 mainly lies. 
 
 When the Commission of the Physico-Mathe- 
 matical Society of Kazan met in 1900 for the pur- 
 pose of awarding again the Lobatchewsky prize, 
 they found before them two new treatises on non- 
 
 ■** TTiis idea was suggested and partially developed by G. 
 Battaglini in his " Sulla Geometria Imaginaria di Lobatchew- 
 sky," Giornale di Mat. Anno V., pp. 217-231, 1867. 
 
DISCOVERY OF NON-EUCLIDEAN SYSTEMS 55 
 
 Euclidean geometry, the merits of which were so 
 nearly equal that the decision between them was 
 finally made by the casting of lots. These were 
 A. N. Whitehead's investigations in his " Universal 
 Algebra " "^^ and Wilhelm Killing's ** Grundlagen 
 der Geometric." '**^ 
 
 In the opinion of Sir Robert Ball, Whitehead's 
 investigation excels anything previously done in 
 two important particulars. In the first place he can 
 treat n-dimensions by practically the same formulae 
 as those used for two or three dimensions; and 
 secondly, the various kinds of space, parabolic, 
 hyperbolic and elliptic, present themselves in White- 
 head's methods quite naturally in the course of the 
 work, where they appear as the only alternatives 
 under certain definite assumptions. Perhaps the 
 most significant portion of Killing's effort is his 
 treatment of the " Clifford-Klein space- forms." 
 whose importance lies in the fact that they show 
 what a difference it makes whether we assume the 
 validity of our fundamental axioms for space as a 
 whole or only for a completely bounded portion of 
 space. The first assumption yields the Euclidean 
 and three non-Euclidean space-forms already men- 
 tioned, but the second gives a " manifoldness. at 
 
 *^ Cambridge, England, I? 
 4« Paderborn, 1898. 
 
56 DISCOVERY OF NON-EUCLIDEAN SYSTEMS 
 
 present not yet dominated, of different space 
 forms." ^'' 
 
 We must now call attention to the remarkable 
 results of Max Dehn's investigation, which was 
 undertaken at the suggestion of Professor Hilbert 
 and published in 1900.^^ 
 
 As early as 1898 Friedrich Schur had reached 
 the conviction that elementary geometry can be built 
 up without the use of the Archimedes axiom of 
 ! continuity, ''^ and proved Pascal's theorem without 
 ^the use either of this axiom or of the parallel 
 postulate. He constructs a sect Calculus in which 
 he shows that the theory of proportion can be 
 founded without the introduction of irrational 
 numbers and indicates that this might also be done 
 without the Archimedes Axiom. Hilbert accom- 
 plished this proof in 1899 and demonstrated thai 
 this axiom need no longer be regarded as necessar} 
 to elementary geometry. As we have already stated 
 Legendre, by assuming this axiom and also tha' 
 the straight line is of infinite length, demonstratec 
 ( I ) that the angle sum of any plane triangle canno 
 be greater than two right angles, and (2) that if ii 
 
 *' From Professor Engel of Leipzig, in a Russian pamphle 
 printed at Kazan, taken from Halsted's translation. 
 
 *8 Dehn was a pupil of Hilbert. He was 21 years old whei; 
 this investigation was completed. It is printed in Mat. Ami 
 53 Band, pp. 404-439- 
 
 *^ See Preface to his Lehrbuch der Analytischen Geometii 
 Leipzig, 1898. 
 
 i 
 
DISCOVERY OF NON-EUCLIDEAN SySTEMS 5; 
 
 any triangle this sum is equal to two right angles, 
 the same is true of every triangle. Hence the ques- 
 tion arose, Do these two theorems actually hold 
 good in Euclidean geometry? And the problem 
 suggested for Dehn was, Can these theorems of 
 Legendre be proved without the Archimedes 
 Axiom ? ^^ The results of his investigation are very 
 remarkable. He demonstrates the second of Le- 
 gendre's theorems without any postulate of continu- 
 ity, and shows that the first theorem cannot be dem- 
 onstrated without the Archimedes Axiom. This is 
 done by constructing a new geometry in which an 
 infinite number of lines can be drawn through a 
 point parallel to a given straight line, but in which 
 also the triangle's angle sum is greater than two 
 right angles. By assuming the Archimedes Axiom 
 and also that an infinity of parallels can be drawn to 
 a given straight line, through a given point. Lo- 
 batchewsky's geometry, in t\^hich the triangle's angle 
 sum is less than two right angles, results ; but Dehn 
 now shows that by denying the Archimedes Axiom 
 
 50 This axiom as stated by Dehn at the beginning of his 
 thesis is as follows : If A^ be any point upon a straight h'ne 
 between any given points, A and B, then we can construct the 
 points A„ A.J A^ ... so that Aj lies between A and 
 A^, A, between A^ and A^, A, between A„ and A^ and so 
 forth; "and moreover the sects AA^, AjA^, A,A,. A^.\^. 
 . . . are equal to each other; then there is in the scries 
 of points A, A^ A^ . . . a point A^ such that B lies 
 between A and An- 
 
58 DISCOVERY OF NON-EUCLIDEAN SYSTEMS 
 
 this angle sum is either greater than two right 
 angles or else equal to two right angles, it cannot 
 be less than two. He proves the former, as just 
 stated, by his non-Legendrian geometry, and the 
 latter case, namely, that this angle sum is equal to 
 two right angles, by constructing another geometry 
 in which the parallel postulate does not hold, but in 
 which nevertheless all the theorems of Euclid are 
 shown still to be true. He proves that the sum of 
 the angles of the triangle is two right angles, and 
 that various other theorems previously held to be 
 exactly equivalent to the parallel postulate are still 
 valid in this new geometry in which the parallel 
 postulate is thus contradicted. 
 
 His results are well summarized by the following 
 table : 
 
 The an- 
 gle sum 
 in the tri- 
 angle is: 
 
 Through a given point we can draw to a 
 straight : 
 
 
 No parallel. 
 
 One parallel. 
 
 An infinity of 
 parallels. 
 
 >2R 
 
 = 2R ^ 
 <2R 
 
 Elliptic 
 geometry 
 
 (Impossible) 
 (Impossible) 
 
 (Impossible) 
 
 Euclidean 
 geometry 
 
 (Impossible) 
 
 Non-Legendrian 
 geometry 
 
 Semi-Euclidean 
 geometry 
 Hyperbolic 
 geometry 
 
 If these results of Dehn should withstand future 
 criticism and prove to be logically unimpeachable 
 
DISCOyERV OF NON-EUCLIDEAN SySTEMS 59 
 
 they will vindicate Euclid in a remarkable manner, 
 for not one of the proposed substitutes for his 
 parallel postulate is, after all, its exact equivalent, 
 except when the axiom of Archimedes is already 
 assumed. 
 
 It is impossible in this brief historical survey to 
 do justice to certain important contributions that 
 have very recently appeared and which from differ- 
 
 V 
 
 ent pomts of startmg have throwrT-:ji new light upon 
 the foundations of mathematics in general. We 
 refer to the contributions of Dedekind/'^ Cantor, 
 Peano, Fieri, Padoa, Poincare, Vailati, Russell. 
 Frege, and others on number, continuity, series, and 
 other topics which come to be involved in any thor- 
 ough consideration of the fundamentals of descrip- 
 tive, projective, and metrical geometry. Collectively 
 considered these contributions reveal a very decided 
 movement to carry the whole of so-called pure 
 mathematics over to a final grounding in formal or 
 symbolic logic. We shall refer to certain features 
 of this movement in subsequent chapters. 
 
 It now remains to notice with a word, in closing 
 this chapter, the efforts that have been made to deal 
 with the philosophical problems created by meta- 
 geometry. These efforts have proceeded sometimes 
 
 SI Some of these writings are in reality not so recent as some 
 of those which we have already considered, but they all belong 
 to the one general movement to which we wish to call at- 
 tention. 
 
6o DISCOVERY OF NON-EUCLIDEAN SYSTEMS 
 
 from the mathematicians themselves, as in the case 
 of Riemann and Helmholtz, and sometimes from 
 students of philosophy. With one notable excep- 
 tion they have all suffered more or less from the 
 writer's inability to take the point of view of the 
 philosopher on the one hand or of the mathematician 
 on the other — an incapacity due to lack of special 
 training, to see the problem clearly and steadily in 
 both its mathematical and its philosophical relations. 
 The exception referred to is that found in the con- 
 tributions of B. A. W. Russell, especially in his 
 Principles of Mathematics,^^ the first volume of 
 which only recently appeared. Mr. Russell brings 
 to his study of the problem the training both of a 
 mathematician and a philosopher ; the result is a con- 
 tribution of remarkable and permanent value. 
 
 In view of the historical development thus some- 
 what imperfectly presented, we shall endeavor in the 
 chapters which follow, ( i ) to orient the problem 
 and point out its complex relations; (2) to trace 
 the parallel postulate and its closely allied concep- 
 tions to their psychological sources; (3) to deter- 
 mine the nature and validity of this postulate and its 
 place in geometrical systems; and finally (4) to in- 
 dicate the conclusions which seem to follov/ from 
 this discussion as to the nature of space. 
 
 "'- Published at Cambridge, England, 1903. This is beyond 
 question the best work on the Philosophy of Mathematics yet 
 published. 
 
GENERAL ORIENTATION 
 
 OF THE 
 
 PROBLEM. 
 
CHAPTER III. 
 
 A GENERAL ORIENTATION OF THE PROBLEM. 
 
 The foregoing historical sketch brings promi- 
 nently to view certain in^tters of great philosophical 
 interest^ First of all it has certainly become clear 
 that in so far, at least, as any system of geometry 
 has professed scientific value, or has claimed to be 
 in any sense valid for reality, the whole history of 
 meta-geometry has been, as a matter of fact, one 
 long and very fruitful search for the philosophic 
 foundations of mathematics in general. -The same 
 spirit which through the centuries endeavored so 
 earnestly to justify Euclid as an orderly system of 
 necessary and indubitable knowledge by removing 
 the objectionable theory of parallel lines, has finally 
 subjected the fundamentals of arithmetic as well as 
 those of geometry, to the most searching critical 
 testing. The sufficiency, independence, and mutual 
 compatibility of the various adjectives which pre- 
 sumably define our notions of number and space. 
 have become problems of absorbing interest and 
 promise. As is usual in every marked intellectual 
 advance, every existing difficulty removed has 
 63 
 
64 ORIENT AT ION OF THE PROBLEM 
 
 opened up new fields of research, new tendencies of 
 thought and methods of investigation, and conse- 
 quently new and more difficult problems calling for 
 solution. 
 
 The light thus thrown both directly and indirectly 
 upon the space problem has led to a very great re- 
 finement of the space conception which has resulted 
 more and more in restricting the a priori realm and 
 in handing over to the empirical, as possibly contin- 
 gent and depending ultimately upon the peculiar 
 nature of experience, certain matters which were 
 previously thought to be apodeictically true. Prior 
 to Lobatchewsky " geometry upon the plane at in- 
 finity " was considered as being just as well known 
 as the geometry of any portion of the table upon 
 which I am writing, but today the geometer " knows 
 nothing about the nature of actually existing space 
 at an infinite distance; he knows nothing about the 
 properties of this present space in a past or a future 
 eternity." ^ He does know, however, that, within 
 the limits of the utmost refinements of instrumenta- 
 tion and observation thus far attained, the assump- 
 tions of Euclid are true for small portions of space 
 and perhaps, when all due allowance for probable 
 error is made, even for that immense region which 
 
 1 W. K. Clifford : Lectures and Essays, Vol. I., p. 359. Lon- 
 don, 1901. We do not subscribe to the naive space-realism 
 latent in these words of Clifford ; the passage is quoted because 
 it indicates very clearly the changed point of view regarding 
 the nature of geometry. 
 
ORIENTATION OF THE PROBLEM 65 
 
 is swept by telescopic vision. Hence the important 
 question as to what are the necessary and sufticient 
 marks of the category of space, once regarded as 
 settled, takes on decidedly a new interest for specu- 
 lative thought. 
 
 Glancing back for a moment over the history of 
 this movement, one can easily trace from a psycho- 
 logical view-point the predominant intellectual and 
 practical interests out of which it has grown. It is 
 often contended that geometry is concerned with 
 ideal objects. At present, this is certainly true; but 
 as a matter of history it has not always been true. 
 Geometry is in reality a complex product of two 
 factors; the one empirical or, if you please, intu- 
 itional, and the other logical. Both appear to be 
 necessary to any geometry which would validate its 
 claims to be a bona fide body of systematic knowl- 
 edge. Our historical sketch shows that these factors 
 have been variable quantities, the intuitional element 
 having been steadily reduced until at present so far 
 as space is concerned it is entirely rejected. Geom- 
 etry as a part of "pure" mathematics is coming to 
 be regarded as merely a branch of symbolic Logic, 
 which no longer claims to throw direct light upon 
 the nature of space. 
 
 Xevertheless, in contemplating the " pure " ab- 
 stract science which thus claims to be free from all 
 intuitional bias, we should not forget its humbler 
 
1« 
 
 66 ORIENTATION OF THE PROBLEM 
 
 origin. Even among the Egyptians ^ and the early 
 Greeks,^ where authentic history first finds the sub- 
 ject already somewhat advanced, it is almost wholly 
 an empirical matter. In the hands of the later 
 Greeks, however, the treatment of geometry under- 
 went essential modifications. This naive conception 
 of things disappears. Geometry passes from a 
 purely technical to a scientific state and becomes 
 the subject of professional and scholarly contempla- 
 tion. For the first time a conscious effort is made 
 to separate the directly cognizable from what is 
 logically deducible and to throw into distinct relief" 
 the thread of deduction. For purposes of instruc- 
 tion the principles which are simplest, most easily 
 gained and apparently freest from doubt and contra- 
 diction are placed at the beginning, and the re- 
 mainder based ypon them. The motive now arises 
 to reduce the number of these principles as far as 
 possible. In this respect the superiority of Euclid 
 was recognized, as we have seen, for more than 
 twenty centuries. His selection of fundamental con- 
 ceptions seemed to withstand every opposition and 
 all efforts at a further reduction. 
 
 Having at length found, however, that the denial 
 of Euclid's parallel postulate led to a different sys- 
 
 - Compare Eisenlohr : Bin Mathematischen Handbuch dcr 
 alten Aegypter: Papyrus Rhind, Leipzig, 1877. 
 
 ^ James Gow : A Short History of Greek Mathematics. 
 Cambridge, 1884. 
 
ORIENTATION OF THE PROBLEM 67 
 
 tern which was self-consistent and possibly true of 
 the actual world, a new motive arose. " Mathe- 
 maticians "* became interested in developing the con- 
 sequences flowing from other sets of axioms more 
 or less resembling Euclid's. Hence arose a large 
 number of geometries inconsistent as a rule with 
 each other but each internally consistent." Even 
 the resemblance to Euclid at first required in any set 
 of axioms which it was desired to investigate was 
 gradually disregarded. Possible systems were in- 
 vestigated on their own account, and thus, as we 
 have said, the intuitional aspect of geometry became 
 altogether a matter of indifference. Geometrical 
 propositions are no longer assertorical in character. 
 They do not claim to state what actually is, but! 
 merely assert that certain consequences flow from 1 
 given premises. Whereas Euclid asserted not only 
 that certain geometrical inferences were logically 
 sound but also that both the premises and the con- 
 clusion were actually true, the new geometry pro- 
 nounces upon the inference merely, and leaves 
 premises and conclusion both as matters of doubt. 
 The implications alone belong to geometry; with 
 axioms and propositions it is not concerned. The 
 geometer deals with certain entities to be sure, but 
 these are carefully defined and guaranteed to exist 
 only in the sense that they are logically compatible. 
 
 ♦ Hon. B. A. W. Russell's " Principles of Mathematics," 
 Vol. I., p. 373. Cambridge, 1903. 
 
68 
 
 ORIENTAriON OF THE PROBLEM 
 
 They are not even necessarily points, or lines, or 
 any of those objects usually regarded as the legiti- 
 mate subject-matter of geometry, but may be any] 
 mental constructs which harmonize with certain con- 
 ditions arbitrarily chosen. The question as toj 
 whether any set of axioms and propositions hold of] 
 actual space or not, is then a problem of appliec 
 mathematics, to be decided, so far as decision ii 
 possible, by experiment and observation. Purf 
 mathematics contents itself with merely asserting 
 that if any space has such and such properties it wil 
 also have such and such other properties. 
 
 Riemann's generalization through the intro 
 tion of analytical conceptions so extensively employee 
 by subsequent writers and the important works 
 Dedekind,^ Cantor,^ and others, on the nature o! 
 continuity, have given rise to new interests in lin< 
 with the general demand for logical rigor and havj 
 shown the necessity of subjecting the prerequisite 
 of analytical geometry to a careful investigation. 
 
 In the employment of the analytical method space 
 was regarded as a manifold of points referred to 
 system of co-ordinates and each capable of bein^ 
 definitely determined by means of numbers. Line 
 were defined by establishing a one to one correspond- 
 ence between the ensemble of numbers and a certaii 
 
 5 " Was sind und was sollen die Zahlen." Brunswick, 1893.1 
 6 " Ein Beitrag zur Mannigfaltigkeitslehre," Borchardt'sl 
 Journal. Band 84, pp. 242-258, Dec, 1877. 
 
ORIENTATION Of THE I'KOBLEM 69 
 
 series of points; surfaces, by setting up a similar 
 correspondence between the ensemble of numbers 
 and a series of lines; solids, by establishing the same 
 correspondence between the ensemble of numbers 
 and a certain series of surfaces. Thus the geometri- 
 cal continuum came to be regarded as generated by 
 the number of continuum, and the question naturally 
 arose as to how this course of procedure may be 
 justified. Dare we include all numbers in the en- 
 semble spoken of, imaginary and real, and rational 
 and irrational? In the development of geometry 
 this has actually been done. It is indeed " impos- 
 sible to exaggerate the importance even of imaginary 
 numbers, for without them the fabric of modern 
 geometry could not stand for a moment."* Hence 
 in investigating the relations of Euclid to the mod- 
 ern system, especially if we regard the latter as the 
 more ultimate and " pure," the question of right 
 becomes an interesting one. Is there not really, 
 though perhaps not so patently, as much room for 
 doubt here as in the case of the parallel postulate? 
 Given a co-ordinate system, we may readily admit 
 that if any set of quantities actually determine a 
 point, they determine it uniquely, but hoiv do ur 
 know that they determine it at all? That is the 
 interesting question. To say that to each of a cer- 
 tain series of objects there corresponds a number is 
 
 'Professor Edwin S. Crawley. Popular Sci. Mon., Jan.. lOOi. 
 
70 ORIENTATION OF THE PROBLEM 
 
 surely quite different from saying that to every pos- 
 sible number there corresponds an object in the given 
 series. Cayley's Absolute requires for Euclidean 
 geometry circular points at infinity. Dare we as- 
 sume that there is anything in reality corresponding 
 to these mythical entities? Not pausing to reply 
 but only to indicate the general course of this inter- 
 esting development, we need only say that if this 
 question be answered affirmatively there arises an- 
 other no less interesting and difficult, which has 
 contributed its share to the quest for mathematical 
 rigor resulting in the general thought movement 
 characterized by Professors Pierpont ^ and Klein as 
 the " Arithmetization of Mathematics." Rational 
 numbers apparently give no trouble. Their arith- 
 metic is easy, but when it comes to interpolating 
 among these that infinity of irrational numbers, such 
 as radicals, logarithms and others, met with in the 
 development of mathematics, it has been customary 
 to proceed upon the tacit assumption that the arith- 
 metic of these is the same as that of rational num- 
 bers and that whatever operations may be performed 
 upon the one class may also be performed upon the 
 other. Thus the Kantian '' Quid Juris? " again 
 confronts us. Certain eminent mathematicians have 
 replied that no rigor is possible except upon a basis 
 of rational numbers. 
 
 8 Bull. Amer. Math. Soc, 2d series, Vol. V., No. 8, pp. 379- 
 385. May, 1899. 
 
ORlENTATIOy OF THE I'KOBLEM 71 
 
 To meet the demands of such reasoning the old 
 Aristotehan logic was, of course, not altogether ade- 
 quate, it, too, like Euclid, stood in need of critical 
 renovation and extension. To get rid of the want 
 of accuracy which creeps in unnoticed through the 
 association of ideas and is therefore not allowed for 
 when ordinary language is employed, symbols for 
 different logical processes were introduced. It was 
 found that all deduction is not syllogistic as the 
 scholastics had thought. Asyllogistic inferences 
 must also be recognized. A new logic was created, 
 for which Boole, C. S. Peirce, and Peano have 
 been largely responsible. To this final court of ap- 
 peal all mathematical difiiculties are henceforth to 
 be brought. Russell's latest work is an elaborate 
 and thorough-going effort to establish the thesis that 
 all pure mathematics, geometry included, is merely 
 a branch of this symbolic logic. 
 
 All this then, as inspired by one supreme motive, 
 an age-long struggle for w^hat men have seen fit to 
 call absolute rigor. If we turn to the cautious 
 mathematician and ask what he means by this word, 
 how he shall know^ when he has attained it. and 
 what is his standard, we do not find him at all ready 
 to reply. Indeed it is wise to be silent, for much 
 that was once thought to be rigorous is now no 
 longer so regarded; a large part of the reasoning 
 of the last century would be rejected today. An 
 English or American treatise on Calculus twenty 
 
^2 ORIENTATION OF THE PROBLEM 
 
 years old is now almost as obsolete as a work on 
 chemistry of the same date would be. Geometrical 
 rigor is a variable quantity approaching a limit 
 which can scarcely be reached except as mathematics 
 becomes wholly divorced from actual sense experi- 
 ence. But geometry refuses to be thus a mere -mat- 
 ter of logic. So regarded its territory can only be 
 arbitrarily defined. It becomes a mere study of 
 multiple series, so Russell has actually defined it, 
 and as such it includes complex numbers as a legiti- 
 mate part of its subject-matter. But why restrict 
 it to multiple series, why not also include series of 
 only one dimension, and thus do away with the name 
 geometry altogether. Indeed it is only when we 
 introduce some notion of applied geometry, some 
 conception so defined as to resemble more or less 
 approximately what we know to be true of our 
 actual space, that our employment of this term 
 seems to be anything more than an unjustifiable mis- 
 use of words. Geometry as logic may indeed care 
 very little as to the particularized existence, either 
 actual or possible, of the entities with which it deals, 
 but geometry, as geometry, in any justifiable mean- 
 ing of this word, is certainly something more than 
 this. And even as pure logic it is in a sense still 
 subject to the space category; its entities are con- 
 ceived as delimited, externalized, and otherwise 
 spatially related. They are supposed to be capable 
 of certain spatial transformations. How, then, do 
 
 I 
 
 I 
 
ORIENTATION OF THE PROBLEM 
 
 73 
 
 we know that certain positions and transformations 
 of these entities are allowable, except as we fall back 
 upon those more ultimate axioms which though they 
 have a wider application are still in an important 
 sense geometrical. They are logical necessities 
 which cannot even be thought, or regarded as hav- 
 ing any meaning whatever, without reference to 
 some sort of co-existent realities actual or abstractly 
 possible which are conceived of as entering into re- 
 lations with each other which are only partially 
 defined by these "axioms " and which inevitably 
 imply the space category. 
 
 But upon the consideration of the logical and 
 epistemological questions thus brought to view it is 
 not our purpose at present to enter. The aim here 
 is simply to point out the motive dominating tills 
 movement and to show that the general problem 
 which lies before us is one of extreme complexity, 
 which has been by no means generally understood 
 by those who have essayed its solution. 
 
 Certainly the profound questions growing out of 
 the parallel postulate, its validity for reality, tiie 
 general nature of space which this involves and the 
 fundamental relations of Euclid to other systems 'if 
 geometry can never be settled by pure mathematics 
 alone, remarkable as the contributions fmm this 
 source have certainly been. The non-Knclidcnn 
 movement, as such, has proliably produced already 
 rlmost all the modifications it is likely to produce in 
 
74 ORIENTATION OP THE PROBLEM 
 
 the foundations of geometry.'-^ In the final answer, 
 if indeed a final answer is possible, the psychologist, 
 the philosopher, the physicist, and the physiologist 
 must each have something to say. The claims of 
 individual investigators to have solved the problem 
 completely, and such claims even yet occasionally 
 appear, can only be looked upon with a measure of 
 suspicion. 
 
 On the philosophical side of the problem the old 
 question as to the a priori nature of the postulates of 
 Euclid affirmed by Kant and denied by Mill, is still 
 in debate. The psychological analysis necessary to 
 a just and fruitful treatment of this question has 
 at times been entirely wanting ; at others, its signifi- 
 cance has been utterly ignored. The term a priori 
 itself has been used in widely dififerent senses by 
 different writers and sometimes even by the same 
 writer. The word " curvature," as applied to space, 
 unfortunately introduced by Riemann, Helmholtz, 
 and Beltrami, has also been a source of perennial 
 confusion for both mathematicians and philosophers 
 alike. The essential meaning of the straight line 
 as used in the various forms of geometry, and of the 
 subordinate conceptions of distance, direction, and 
 motion by which this line is usually defined, has not 
 been clearly determined. And finally the abstract 
 spaces of geometry demand further study as regards 
 
 ^ Consult Russell's ." Principles of Mathematics," p. 381. 
 
ORIENTATION OF THE PROBLEM 75 
 
 their relations to each other and to the space of 
 actual experience. They require to be more thor- 
 oughly tested by that conception which must arise 
 when all bona iide human experiences of a spatial 
 order are taken into the account. 
 
 In the next chapter we enter upon a psychological 
 study of the parallel postulate and certain closely 
 allied conceptions without which this postulate 
 would have no meaning at all. Our purpose will 
 be tw^ofold. First, starting with experience, we 
 shall endeavor to trace the genesis and development 
 of these conceptions as they appeal to the ordinary 
 consciousness of man; and secondly, starting with 
 these conceptions in the highly abstract forms de- 
 manded by modern geometry, we shall try to see 
 how far these forms may be made to have meaning 
 for actual experience. 
 
I 
 
 THE PSYCHOLOGY OF THE PAR 
 ALLEL POSTULATE 
 
 AND ITS 
 
 KINDRED CONCEPTIONS. 
 
CHAPTER IV. 
 
 PSYCHOLOGICAL SOURCES OF THE PARALLEL POSTU- 
 LATE AND ITS CLOSELY ALLIED CONCEPTIONS. 
 
 There can be little doubt that geometry sjjrang 
 originally from man's interest in the spatial rela- 
 tions of physical bodies. It bears in every part un- 
 mistakable evidence of this empirical origin, and 
 the course of its development can be rendered fully 
 intelligible only on consideration of these traces. 
 Various forms of sensory experience contributed 
 the data. By virtue, it may be. of the peculiar 
 structure of the body with its pairs of sense-organs 
 symmetrically located, whose conscious deliverances 
 possess in each case a remarkable similarity as con- 
 trasted wfth those of other sense-organs, the power 
 of orienting these organs themselves and finally of 
 the whole body with reference to presented stimuli, 
 has at last been acquired.^ This power of orienta- 
 
 1 Consult Loeb's Physiology of the Brain (N. Y., lOOoV 
 Also Royce's Outlines of Psychology, pp. I39-I47- It '« P*"''- 
 haps due to this cause that the lower animals know how to 
 strike in and to hold more or less steadily a straipht direction 
 in movement. Consult Von Cyon's articles in Pflncger's 
 Archiv fner Physiologie. 
 
 79 
 
8o PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 tion, taken with sensations of movement,^ sight, and 
 touch,^ and of the so-called statical sense of the semi- 
 circular canals, furnishes the empirical basis for the 
 perception of space as a continuous whole. 
 
 The unitary perception of space thus arising is, 
 of course, complex in its nature and is determined 
 in each case by the character of the sensory factors 
 which it actually involves. So-called psychological 
 spaces corresponding to different senses are not 
 wholly identical. The space-perception of a man 
 born blind ^ is unitary in character, but quite differ- 
 ent from that of a man whose vision remains un- 
 impaired. These sensory differences have been un- 
 consciously carried over into the foundations of 
 geometry, so that the different forms of this science 
 can be classified as motor, visual, etc., according as 
 special emphasis has been laid in their construction, 
 now on one, now on another, of these sensory fac- 
 tors. Projective geometry is entirely visual, while 
 Euclid is largely motor. These psychological dis- 
 tinctions are important when questions of validity ' 
 are raised regarding the foundations of competitive 
 systems. Had man's spatial experience been con- 
 
 - Poincare holds that without sensations of movement and 
 the actual ability to move geometry could never arise. The 
 Monist. See also Professor Ladd's A Theory of Reality, pp. 
 229-230. 
 
 3 Ladd's Psychology Descriptive and Explanatory, pp. 323 f. 
 
 * Consult Dr. Alexander Cameron's Thesis on " Tactual Per- 
 ception." Yale, 1900. 
 
 il 
 
PSYCHOLOGIC ASPECTS OF THE PROBLEM 8i 
 
 fined, for example, to vision'^ alone, the struggle 
 between Euclid and Lobatchewsky could never have 
 been, since for vision alone there are no such things 
 as parallel lines. Through a point in a plane there 
 is neitiier one nor a pencil of lines which do not cut 
 a given line in the plane ; every such line is seen to 
 converge toward the given line in one or in both 
 directions. 
 
 When, therefore, we come to inquire what are 
 the sensory factors that enter into the subordinate 
 conceptions which define the spaces o'f the different 
 geometries, where in each the emphasis is actually 
 laid, in the choice of conceptions and in the charac- 
 ter of the demonstrations which follow upon them, 
 and finally what is the true balance to be maintained 
 among these sensory factors in determining the na- 
 ture of space as it appears to be actualized in the 
 world of reality, it is plain that the psychological 
 difficulties involved strike deeper than mere ques- 
 tions of accommodation and convergence in visual 
 perception. Nevertheless, investigations of this sort 
 have thrown, and no doubt will continue to throw, 
 light upon the general problem. We organize our 
 experiences to harmonize with our own and with 
 the movements of physical bodies; consequently 
 those changes in the environment of an object which 
 necessitate changes in the character of the move- 
 
 5 This is perhaps an impossible hypothesis, but it scn-es our 
 purpose in this connection. 
 
82 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 ments by which it is perceived occasion differences 
 in the perception of its spatial properties. In this 
 way optical illusions arise. With these so-called 
 false perceptions, eye-movements ^ are now known to 
 be closely related if not indeed a determining cause. 
 The fact that long-continued practice dispels these 
 illusions, the perceived object remaining constant 
 while the perception itself gradually but uncon- 
 sciously changes, indicates that the peculiar marks 
 of any concept of space founded simply upon visual 
 perceptions can hardly be called a priori in Kant's 
 sense of the word. Photographs taken at intervals 
 during the presence of these optical illusions and 
 after they have finally disappeared show quite 
 clearly that changes in eye-movements correspond- 
 ing to those in the perception itself successively oc- 
 cur. Increasing accuracy of movement and correct- 
 ness of perception develop together. Whether the 
 movement or the perception itself is to be regarded 
 as the determining cause of this improvement is an 
 interesting but difficult question. It seems very 
 probable, however, that cerebral organization and 
 accurate motor adjustment must first be secured 
 before correct perception becomes at all possible. 
 ]f the mere recognition of error were all that is 
 needed, no long-continued process of perceptual edu- 
 
 •^ Based upon some very interesting but as yet unpublished 
 experiments in photographing eye-movements in the Yale 
 Laboratory by Assistant Professor Judd and Dr. McAllister. 
 
PSYCHOLOGIC ASPECTS OF THE PROBLEM s-. 
 
 cation would seem to be required. Correct percep- 
 tion would then take place as quickly and completely 
 as when, through a false perception, we have mis- 
 taken some stranger for a friend. ^^ 
 
 Turning now from these general considerations 
 to study those special facts from which geometry 
 as a science has actually developed, we lind that the 
 first geometrical knowledge which can be strictly 
 so-called was acquired accidentally and without de- 
 sign through practical experiences in varied employ- 
 ments. This peculiar empirical origin is shown in 
 W'hat history w^e have of the beginnings of geometry 
 among the ancient Egyptians "^ when as yet the scien- 
 tific spirit in its search for the logical interconnec- 
 tions of the experiences in question had not arisen. 
 It appears also even more clearly in the history of 
 primitive civilizations at large, as may be seen in 
 the rise of metrical conceptions and in those facts of 
 experience out of which the conception of the 
 straight line and Euclid's theory of parallels have 
 developed. We shall now study these developments 
 in the order here mentioned. 
 
 Primitive man was already well advanced in cer- 
 tain geometrical ideas before measurement, strictly 
 so-called, began. He had acquired a considerable 
 practical knowledge of physical bodies and their 
 grosser spatial relations without taking advantage 
 
 7 Consult Gow's A Short History of Greek M.ithematirs. 
 Cambridge, 1884. 
 
84 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 of this artificial assistance. Through the compari- 
 son of various kinds of sensory experiences he had 
 come to attribute to bodies a certain spatial con- 
 stancy; he had learned to locate them, to estimate 
 their form, size, and distance, with considerable 
 accuracy and to govern his actions accordingly. But 
 it was in the objects themselves, in their capacity 
 to satisfy his needs and not in their spatial relations 
 that his interests primarily centered. It was only 
 when he had so far triumphed over his foes in the 
 struggle for self-preservation as to be able to reflect 
 that the place, form, and size of objects have almost 
 everything to do with determining the character of 
 those activities by which his wants are best satisfied 
 and his enemies overcome or avoided, that the desire 
 for a more accurate determination of these quanti- 
 ties by means of measurement arose. His first esti- 
 mates were, of course, obtained by the comparison 
 of memory images with present perceptions. This 
 mode of estimation, however, depends upon certain 
 physiological and psychological conditions which 
 are difficult to control and is therefore unsatisfac- 
 tory when exact measurement is required. This is 
 especially true when the interval of time between the 
 experiences compared is large and the memory image 
 has, as a consequence, considerably faded. Hence 
 it becomes necessary to provide characteristics which 
 depend as little as possible upon these conditions, and 
 this can only be done by removing the time interval 
 
PSYCHOLOGIC .ISPBCTS OF Tllli PROBLEM 85 
 
 between the remembered and the perceived experi- 
 ences altogether, or by rendering it as brief as pos- 
 sible, — the ideal being the substitution of direct 
 perception in the place of memory. . This can l)c 
 done only by securing the exact congruence of the 
 -bodies compared, and hence it is always theoretically 
 impossible, because of the inevitable limitations of 
 sense-perception, and the want of absolute rigidity 
 in all natural objects. Nevertheless this is the prin- 
 ciple of measurement, and it remains the same for all 
 spatial measurements whether i)er formed by the low- 
 est savage or the most exact geometer, as a direct 
 perception of physical congruence or as a purely ab- 
 stract visualization. 
 
 For all physical measurement, then, a convenient 
 portable standard of some sort is needed, and one 
 whose want of appreciable variation during trans- 
 portation may be directly perceived. Naturally the 
 first objects of this sort to appeal to the primitive 
 man would be various parts of his own body. The 
 names of the oldest measures, and various other 
 facts, indicate clearly that these were the standards 
 actually employed.,^ Among these names, for exam- 
 ple, are the hand, nail, ell, span, cubit, foot, fathom, 
 pace, and mile. These names have, of course, lost 
 much of their original meaning, and have now come 
 to be associated with standard measures which they 
 happen to represent with only tolerable accuracy. 
 
 It w^as a significant forward step in civilization 
 
86 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 when nations like the Egyptians and Babylonians 
 abandoned these physiological for more exact phys- 
 ical standards. But the evolution of this form of 
 measurement has also taken place in accordance 
 with known psychological laws. In harmony with 
 the general principle that it is in the material rather 
 than in the more distinctly spatial properties of ob- 
 jects that human interests first become centered the 
 first measurements of this sort were doubtless meas- 
 urements of volume,^ not measurements by means 
 of definitely chosen standards of volume, however, 
 but merely inaccurate estimates of the capacity of 
 vessels and storerooms by determining the quantity 
 or number of similarly shaped bodies which they 
 would contain.^ Similarly the first estimates of 
 area were probably made by the number of fruit- 
 bearing trees which a field would grow, the amount 
 
 s It is interesting to note that those who have given most 
 attention to the best methods of introducing geometry to be- 
 ginners have generally agreed that it is best to begin with 
 solids and considerations of volume, rather than with points, 
 lines, and angles, and there is a decided movement now on in 
 elementary education in America to follow with the individual 
 a method very similar to that by which as we here find the race 
 has been educated. Note some recent American publications 
 such as Campbell's Observational Geometry, New York, 1899; 
 Spear's Advanced Arithmetic, Boston, 1899; Hanus's Geometry 
 in the Grammar School, Boston, 1898. Tlie Harvard Catalogue 
 for 1901-1902, p. 307: Row's Geometric Exercises in Paper- 
 Folding, Chicago, Open Court, 1901 — Tr. 
 
 ^ E. Mach : The Development of Geometry, TTie Monist, 
 Vol. XII., p. 486. Also Eisenlohr: Papyrus Rhind, etc., pre- 
 viously cited. 
 
PSYCHOLOGIC ASPECTS OP THE PROBLEM 87 
 
 of labor necessary to cultivate it. or the number of 
 animals that could be grazed upon it. The meas- 
 urement of one surface by another may have been 
 suggested by estimating in this way the relative 
 value of fields which lay near one another. 
 
 Herodotus ^° states that when Xerxes wished to 
 count the army which he led against the Greeks he 
 adopted the device of drawing up 10.000 menxlosely 
 packed together within an enclosure which was then 
 made to serve- as a standard, and each succeeding 
 division that filled it was counted as another 10,000. 
 This is an example of another type of operations 
 which naturally led to the measurement of one sur- 
 face by another. By abstracting at first instinct- 
 ively and then consciously from the height of the 
 practically identical bodies thus covering a surface 
 the notion of a surface unit w^ould finally be reached. 
 The fact that among the Egyptians.^' the early 
 Greeks, and even as late as the Roman '- sTrrreyo^u- 
 rules for the measurements of surfaces of irregular 
 figure were often grossly inaccurate, seems in gen- 
 eral to favor this view\^'' 
 
 But what needs to be emphasized most perhaps in 
 
 1" Herodotus VII., 22, 56, 103, 223. 
 
 ^^ Eisenlohr : op. cit. 
 
 ^-' M. Cantor: Die romischcn Agriiiicnsoren. Leipzif?. 1875. 
 
 " Thucydides VI., i states that surfaces havinR equal per- 
 imeters have equal areas. And Ahmcs in Papyrus Rhind Rcts 
 the area of the triangle by multiplying together two of its 
 sides. 
 
88 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 connection with measurement is the fact that all acts 
 of measurement exhibit the free-mobility of approxi- 
 mately rigid bodies, and that these facts when con- 
 ceptualized bring to clear consciousness the corre- 
 sponding postulate which lies at the foundation of 
 every system of metrical geometry. And further- 
 more the recognized possibility of constructing simi- 
 lar solids of different sizes leads by a similar path- 
 way to the postulate of homogeneity. 
 
 The modern method of defining points, lines, and 
 surfaces as boundary conceptions, though logically 
 necessary and presenting little difficulty to a mind 
 skilled in abstract thinking, nevertheless conceals 
 rather than exhibits the process whereby these con- 
 ceptions have developed. The straight line still 
 bears in its name "a suggestion of its origin. 
 Straight is the participle of the old verb to stretch 
 and line is from line^t, which signifies a thread, hence 
 the straight line literally means a stretched cord or ; 
 thread. By decreasing the thickness of such an ob- 
 ject until it becomes vanishingly small the concep- 
 tion of the line as a geometrical magnitude of only 
 one dimension is reached by an easy abstraction. 
 
 By making fast one end of a string and drawing 
 the other through a hole or a ring and observing 
 how more and more of it passes out until the whole 
 becomes stretched or straight we have an example 
 of that type of experiences from which the notion of 
 
PSYCHOLOGIC ASPECTS OF THE PROBLEM 89 
 
 the straight Hne as the shortest distance between 
 two poyits came to be clearly discerned. 
 
 Another class of simple experiences supplies that 
 peculiar property of the straight line by which it may 
 be defined as a unique figure determined by any two 
 of its points. If we slide a crooked material line 
 betw'een any two fixed points it will be observed 
 that the portion between the two points will con- 
 tinually alter as regards the form and position of 
 its parts. The more uniform the object, the less 
 this variation becomes until the limit of perceptual 
 uniformity is reached/ when the object will be seen 
 to slide within itself. We now have a property 
 which belongs obviously as much to the circle or the 
 __S£iral as it does to the straight line, for these figures 
 also when thus operated upon are seen to possess it. 
 But if we rotate these objects about the two points 
 in question a difference is at once detected, and we 
 come upon a peculiar property of the straight line. 
 It rotates unthin itself. 
 
 Now^ when we consider these two properties of 
 the straight line. tJwt it rotates unthin itself ami is 
 also the shortest distance hetnren t7i'o points, it be- 
 comes perfectly obvious why this figure has been 
 made fundamental in all systems of geometry. // 
 is the only one-dimensional magnitude that is physio- 
 logically simple and perceptually constant when 
 vieived from any point not in it. The same is true 
 of the plane, and explains its unique position among 
 
90 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 two dimensional magnitudes. It is the perceptual 
 simplicity and constancy of these two figures that 
 has forced them upon us as the only invariable, and 
 hence the only satisfactory, metrical standards. 
 Among tridimensional objects the cube and the 
 sphere hold a similar relation, and for a similar 
 reason. Of these two figures the sphere is of course 
 the simpler from a perceptual point of view. It 
 alone of all solids appears constant, in form, from 
 whatever external point we view it ; nevertheless the 
 law of simplicity and economy holds in the selection 
 of the cube as the fundamental standard of volume 
 because of its obvious relations to the one and two 
 dimensional standards which we have just consid- 
 ered. 
 
 Any theory of parallel lines must obviously in- 
 volve the idea of surfaces, hence before passing on 
 to a direct study of Euclid's conception of parallels 
 it is necessary to take some note of the origin of 
 this idea. The conception of surfaces as geomet- 
 rical magnitudes absolutely without thickness was 
 certainly not reached at a bound. Nor is it a notion 
 which is rendered necessary by the essential nature 
 of a perceiving mind. It is essentially a metrical 
 conception, and is therefore conditioned by those 
 characteristic'-' of things and our experiences with 
 them which render measurements possible. Like 
 the conception of spatial equality, it could have no 
 meaning in a homogeneous fluid world, and in such 
 
PSYCHOLOGIC ASPECTS OF THE PROBLEM 91 
 
 a world would doubtless never arise. The empirical 
 origin of this conception is also shown by the fact 
 that even the adult mind already schooled in abstract 
 thinking- has the greatest difficulty in clearly con- 
 ceiving it. Perhaps not one in a thousand correctly 
 represents to himself the surface as he has been 
 taught to define it. It is usually imagined as a c(jr- 
 poreal sheet of constant thickness which can be 
 made small as we please. 
 
 Suppose we consider for a moment the surface of 
 the ink in the bottle before me. This surface, we 
 say, is the boundary which separates the ink from 
 the air that is above it. But what is this boundary ? 
 Is it ink? Certainly not, for then we should still 
 need a boundary to separate that ink from the air 
 above it. For the same reason we cannot say that 
 it is air. If, then, we should magnify the two as 
 much as we please and should find that they remain 
 always homogeneous, each filling up the space adja- 
 cent to the other, we should still have to say that the 
 surface between them is neither the one nor the 
 other ; it is not a layer of air. of ink. of ether, nor of 
 space; it is simply a boundary or geometrical sur- 
 face, and as such occupies no space of all. And yet 
 as we reach this conclusion how difficult it is to 
 avoid falling back upon that class of experiences by 
 abstraction from which this conception was orig- 
 inally obtained, and thus picturing to ourselves an 
 exceedingly thin material partition, or at least some 
 
92 PSYCHOLOGIC ASPECTS OP THE PROBLEM 
 
 small portion of space which might serve to keep 
 the two substances separate and distinct from each 
 other. 
 
 This, of course, does not satisfy reason. To put 
 a space boundary between two adjacent portions of a 
 continuous quantity is not only to regard space as 
 a spread-out, empty reality which, while providing 
 " room " for things and permitting them to approach 
 or recede from each other, nevertheless keeps them 
 asunder; but it is also to propose the old problem 
 over again while doubling its difficulty, for we now 
 want a boundary between the space in question and 
 each portion of the quantity separated by it. It is 
 the same difficulty which is met with in all limiting 
 conceptions. It arises from the disparity always to 
 be felt between actual experience and those intel- 
 lectual ideals the peculiar character of which this 
 experience itself suggests and determines. 
 
 Coming now directly to Euclid's theory of parallel 
 lines, we note that it was also determined to be what 
 it is by a variety of empirical data. If these data 
 had been decidedly different from what they are 
 some Lobatchewsky might ultimately have given us 
 the Euclidean system, but this system would cer- 
 tainly not have been first in the order of develop- 
 ment. The data in question are many and simple, 
 and were certainly familiar even to the most ancient 
 civilizations. The ornamental designs of savage 
 tribes in weaving, drawing, wood-carving, and the 
 
PSYCHOLOGIC ASPECTS OF THE PROBLEM. 
 
 93 
 
 like often suggest that the triangle's angle sum is 
 two right angles. 
 
 By folding a triangular piece of cloth or paper as 
 shown in the figure, this truth is directly perceived. 
 The angles of the triangle are seen to form a straight 
 angle by bringing their vertices together at C'.*-* 
 
 The same truth doubtless suggested itself over 
 and over again to the workmen in clay and stone of 
 Babylonia, Egypt, and Greece in the mosaics and 
 pavements which they are known to have made from 
 differently colored stones of the same shape. In 
 this way, too, it was easily found that the plane 
 " space " about a point can be completely filled only 
 by three kinds of regular polygons, that is, by six 
 equilateral triangles, by four squares and by three 
 regular hexagons," '^ and hence that this space is 
 always equal to four right angles. 
 
 In this ancient paving, triangular stones of the 
 same shape and size were frequently used by placing 
 
 i*Tylor suggests this as the probable origin of this theo- 
 rem. Anthropology, New York, 1896, p. 320. 
 
 15 Proclus attributes this theorem to the Pythagoreans. 
 Gow, op. cit. foot note p. 143. 
 
94 
 
 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 their bases on the same straight line as in the accom- 
 panying figure.^ ^ 
 
 Here the system of equally distant lines is a strik- 
 ing feature. It is also made clear by this figure 
 that the sum of the interior angles on the same side 
 made by the intersection of any two of these equally 
 distant lines by a third line is two right angles. 
 The obvious fact that this style of paving may be 
 extended without limit leads naturally to the convic- 
 tion that parallel lines will not meet however far 
 produced, for such lines are seen to be ideally unlim- 
 ited and yet everywhere equally distant. Thus we 
 have suggested, at least, in this simple figure all the 
 empirical data necessary to the formation of the 
 Euclidean theory of parallel lines. 
 
 But by whatever fact or class of facts this theory 
 was originally suggested, the mechanical arts furnish 
 
 16 See E. Mach's valuable article in The Monist, Vol. XTL, 
 pp. 481-515. 
 
PSYCHOLOGIC ASPECTS OF THU. PROBLEM. ys 
 
 today innumerable examples of its practical truth. 
 The existence of similar figures of unequal sizes ami 
 the actual construction of rectangles whose angles all 
 remain right angles and \vh(jse opposite sides con- 
 tinue to be equal when tested l)y the most accurate 
 measuring instruments are constantly recurring 
 proofs of Euclid's validity. If. however, in the 
 construction of quadrilaterals with angles all right 
 angles, and sides practically the straightest possible. 
 it had uniformly been found that the opposite sides 
 are unequal, as actually happens in surveying such 
 figures of large size upon the surface of the earth, 
 we should then doubtless have reached with equal 
 confidence a different conclusion, and our sciences 
 of mechanics, physics and astronomy would have 
 been quite different from what they are; but as a 
 matter of history this has not been true. From the 
 crudest measurements to the most refined ; from the 
 ancient " Harpedonaptai " *^ of Egypt to the present 
 day in the constantly increasing refinement of the 
 powers of accurate observation, man's ability to 
 measure slight deviations from right angles and 
 from the equality of distances has remained rela- 
 tively equal. There are today no observed phe- 
 
 iT Egyptian " Rope Stretchers " mentioned by Democritus. 
 Their principal duty was the construction of right angles by 
 means of ropes divided into 3, 4- and S equa' ""•''^ ^^ IciiRth 
 See Cantor's Vorlesungen iibcr Ccschiclilr dcr Mathcmotik. 
 Vol. I., p. 64, Leipzig, 1894. 
 
96 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 nomena coming within tlie scope of the physical 
 sciences that seem to contradict in any appreciable 
 degree the Euclidean postulate. Therefore even 
 when we set aside, as we must, Kant's contention 
 for the a priori synthetic nature of the parallel postu- 
 late, it may still be claimed that there is no law of 
 nature reached by scientific induction that can be 
 said to have so good a right to be called funda- 
 mentally and universally true of the world as we 
 know it. 
 
 As just stated, the .parallel postulate is rooted 
 psychologically, in so far as direct perception is con- 
 cerned, in the ability to discriminate slight varia- 
 tions in the size of angles, in the length of lines, and 
 in the departure of the latter from " absolute 
 straightness." It requires that this ability shall re- 
 main as a rule, relatively speaking, precisely equal in 
 these difTerent directions. Now it is obvious that 
 '* the relations between the least perceptible differ- 
 ence of the angles of a parallelogram and the least 
 perceptible differences of the lines forming its sides 
 are exceedingly complex and variable." ^^ It is 
 therefore possible that the law of these relations 
 when determined by the most refined experimental 
 analysis ^^ w\\\ prove to be in essential agreement 
 
 i» Ladd : A Theory of Reality, New York, 1899, p. 315. 
 
 1^ The writer has already begun, at Professor Ladd's sug- 
 gestion, the experimental problem here referred to. So far, 
 however, the results obtained are not of a very positive char- 
 acter. 
 
PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 97 
 
 with the EucHdean postulate. Indeed this may be 
 even confidently expected in view of the general 
 experiences of the race to which we have referred. 
 
 This conclusion, however, is by no means to be 
 regarded as a necessary one. In fact, that ideal 
 exactitude which is required to establish beyond 
 doubt the validity of this postulate is no more to be 
 expected in this than in any other form of empirical 
 testing. The absolute validity of Euclid can never 
 be established by any mere appeal to perceptual ex- 
 perience, however refined. Our space may possibly 
 be proved in this way to be non-Euclidean, but it can 
 never be shown to be exactly Euclidean. 
 
 The analysis, then, of the experiences involvetl in 
 the theory of parallel lines requires the considera- 
 tion of a special relation between angles and dis- 
 tances, and the problem is to determine the peculiar 
 character of this relation. 
 
 We have already seen (Chap. I., pp. 9 and 10) 
 that in defining angles as differences of direction, 
 straight lines, as those which do not change their 
 direction, and parallels, as straight lines having the 
 same direction, we overleap at a bound the difficul- 
 ties involved in the parallel postulate. It is there- 
 fore evident that the peculiar properties of parallel 
 lines are somehow bound up in the word direction, 
 and could be distinguished if the meaning of this 
 word were subjected to a careful analysis. Through 
 the recognition of the importance, for Euclid, of the 
 
98 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 idea of direction. Von Cyon -" was led to believe 
 that, in discovering, as he thought, a special sense- 
 organ in the semicircular canals for the perception of 
 three fundamental directions corresponding to the 
 co-ordinates of the Cartesian system, he had solved 
 the whole space problem completely. But the fact 
 already pointed out (Chap. I., p. lo) that direction 
 as ordinarily understood can only be completely de- 
 fined when the parallel postulate is already assumed 
 shows quite clearly that such a solution is only ap- 
 parent. It proceeds, in fact, upon the customary 
 assumption expressed in the above definition that 
 direction is the one essential, sufficient, and peculiar 
 property of the straight line; in other vv^ords, that 
 straightness and direction are exactly equivalent in 
 meaning, and that the latter word is used in pre- 
 cisely the same sense in all three of the above defini- 
 tions. Now we wish to show that there is uncon- 
 sciously introduced into this word as it is used in 
 the last two definitions, in addition to the idea of 
 straightness which it always carries, an element of 
 meaning which ultimately arises from the peculiar 
 structure and symmetry of our bodily organism. 
 
 It will be remembered that in the early part of the 
 present chapter we called attention to the peculiar 
 symmetrical arrangement of corresponding sensory 
 organs and the influence of this arrangement upon 
 
 20 PHuegc/s Archk' fucr Physiologic, 1901. 
 
FSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 99 
 
 the orientation -^ of the body with reference to any 
 stimulus appeaHng to the senses. This is no doubt 
 largely due to the remarkable similarity of the 
 sensory experiences of corresponding organs, as. for 
 example, the eyes or the ears. These sensations, 
 however, are not wholly identical, and their differ- 
 ences, combined with other sensation factors, mainly 
 those of movement brought out in repeated acts oi 
 adjustment, give rise to characteristic distinctions in 
 sensations of direction. For example, the direc- 
 tions, before and behind, up and down, right and 
 left, as actually experienced, involve sensory differ- 
 ences somewhat analogous to those of color.-^ 
 These differences attach themselves to our notion of 
 direction, and when unconsciously carried over with 
 this notion into the abstract space of mathematics 
 they lead to confusion. There is no such thing as a 
 difference of direction in geometrical space. In such 
 space tzco points are always required to determine 
 
 -1 This orientation may even be wholly involuntary, as in the 
 case of the moth when it helplessly flies into the flame and 
 is burned. 
 
 -- It is an interesting fact that Helmholtz was led to inves- 
 tigate the problem of space by the analogy which he perceived 
 between space and the color system as tridimensional manifolds. 
 We shall see when we come to consider the impossibility of 
 carrying to one space the same metrical standard which was 
 employed in another that the so-called space-constants arc 
 qualitatively different and yet serially related to each other in 
 a manner which is in all essential respects similar to what wc 
 experience in the perception of differences of color. 
 
100 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 a line. If a number of lines are conceived of as 
 going out from a point they can only be distin- 
 guished by an actually perceived or else an imagined 
 relation to our bodily selves. We can transport 
 ourselves in imagination to the vertex of any angle 
 formed in this way, and by the use of a distinction 
 not inherent in the figure itself we can represent the 
 angle to ourselves as a difference of direction, and 
 thus distinguish the lines from each other. In no 
 other way can we do this ; we picture ourselves as it 
 were standing at the vertex of the angle, looking 
 alternately down its sides, and thus in imagination 
 sensing their directions as different. 
 
 We carry, then, into our abstract space-world with 
 this notion of direction a misleading subjective dis- 
 tinction. The bodily self enters into spatial rela- 
 tions with the other objects of this world in a pe- 
 culiar way ; it is not allowed to take its place, as this 
 abstract conception of space requires that it should, 
 merely as one among many objects of whose distin- 
 guishing peculiarities except in so far as those quali- 
 tative similarities are concerned which measurement 
 requires, geometry can take no account whatever. 
 
 It is also easy to see that the same " physiolog- 
 ical " element of meaning attaches to the word direc- 
 tion as employed in the definition of parallel lines. 
 For what else can be meant by saying that such lines 
 have the same direction and that even to infinity? 
 This is clearly apparent if we follow closely the Ian- 
 
PSYCHOLOGIC ASPECTS OF THE PROBLEM loi 
 
 guage of Mr. J. S. Mill in the following passage : -^ 
 " Though, in order actually to see that two given 
 lines never meet, it would be necessary to follow 
 them to infinity ; yet without doing so we may know 
 that if they ever do meet, or if, after diverging from 
 one another, they begin again to approach, this must 
 take place not at an infinite but at a finite distance. 
 Supposing, therefore, such to be the case, we can 
 transport ourselves thither in imagination, and can 
 frame a mental image of the appearance which one 
 or both of the lines must present at that point, which 
 we may rely on as being precisely similar to the 
 reality." ^'^ 
 
 But the mere straightncss of tivo lines lying in the 
 some plane cannot, of itself, justify any statement 
 as to whether they will or 7cill not meet zchen pro- 
 longed without limit. If it could, there would then 
 
 23 Logic. Book ii., Chap. V., § 5. 
 
 -^ It is interesting to note while this passage is before ii'' 
 that its closing statement is open to criticism. On a preced- 
 ing page Mr. Mill has stated that " we should not be author- 
 ized to substitute observation of the image for observation of 
 the reality, if we had not learnt by long continued experience 
 that the properties of the reality are faithfully represented in 
 the image." Now it is evident that experience can only tell 
 us this in the case of realities and images, both of which have 
 been experienced : both must be known before we have a right 
 to say that the one faithfully represents the other. Hence in 
 admitting, as Mill here does, the universality of the truth of 
 the parallel postulate, he introduces a factor of knowledge 
 which can not, strictly speaking, be gotten from experience 
 alone as he conceives it. 
 
102 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 be no need whatever of the parallel postulate, for 
 Euclid becomes established at once when the exist- 
 ence of straight lines in this sense is granted. ' It is 
 difficult to grasp the truth of this statement, so 
 natural has it become through long-continued asso- 
 ciation of these ideas to look upon " straightness " 
 and " direction " in their adjective relations as per- 
 fectly synonymous terms. 
 
 Thus it has come to appear that if only the lines 
 are actually straight the parallel postulate must fol- 
 low of necessity, and that it is only when we have 
 unconsciously or purposely introduced some sort of 
 bending or curvature into the lines themselves that 
 this postulate fails. But careful reflection upon the 
 essential meaning of straight lines when carried 
 back to the source in experience whence this concep- 
 tion has sprung reveals that this a priori necessity 
 of the parallel postulate simply does not exist. 
 |: Some distinction between the idea of straightness 
 and that of direction must obviously be maintained 
 : if any non-Euclidean system is to be taken seriously 
 jas having anything to do with reality. If these two 
 / words are, in fact, identical in meaning and the word 
 " direction " as used in the definitions of straight 
 I lines, angles, and parallels as given above holds pre- 
 cisely the same significance in each definition, it is 
 easy to see that any serious struggle between Euclid 
 and his modern rivals is out of the question, for 
 Euclid alone is left on the field. 
 
FSVCHOLOGIC ASPUC'IS OF THE PROBLEM lo.^ 
 
 A familiar illustration will assist in making this 
 distinction clear. As everyone knows, a curve is 
 frequently defined in elementary geometry as a line 
 which changes its direction at every point. Now if 
 we substitute straightness for direction in this defini- 
 tion it becomes at once absurd. For even if we 
 could give the resulting expression, " changes its 
 straightness," etc., an intelligible meaning by re- 
 garding it as signifying the amount of angular devia- 
 tion from the tangent to the curve at the point 
 considered, the definition would still break down 
 when applied to the circle, for in this case we have 
 a figure whose angular deviation from the tangent 
 line is a constant quantity, and therefore, according 
 to this definition, the circle could not be a curve 
 at all. 
 
 We see, then, that the word " direction " names a 
 -complex idea which is altogether too inclusive and 
 variable in meaning to specify accurately what is 
 meant by the straight line. It leads to confusion 
 where precision of statement and accuracy of mean- 
 ing are urgently demanded, and should therefore be 
 avoided if possible. 
 
 Generally speaking, the confusion so frequently 
 met with in discussions on the philosophic founda- 
 tions of geometry arises from the general tendency 
 to regard as simple and without the need or the i)OS- 
 sibility of further analysis certain ideas which m 
 reality are complex in character and therefore can- 
 
104 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 not be regarded as ultimate. Emphasis is laid now 
 upon one phase, now another, of these complex 
 ideas, and the whole problem is solved by overlook- 
 ing entirely the real question at issue, 
 -^ Of all the fundamentals of geometry, that which 
 stands most in need of satisfactory analysis is the 
 conception of the straight line. What is there in 
 this word, let us ask, that makes it not only univer- 
 sally applicable, but even indispensable to all forms 
 of geometry? What do we and what ought we to 
 mean by " straightness " as applied to lines in all 
 these systems? 
 
 This conception, by virtue of its complication with 
 certain metrical ideas that have always been asso- 
 ciated with it, gives philosophically the greatest 
 trouble in any attempted critical estimation of differ- 
 ent geometrical systems. Angles present no diffi- 
 culties of so serious a character. The old Euclid- 
 ean postulate that all right angles are equal is in 
 reality a theorem which has lately been rigorously 
 proved,^^ and is just as true of non-Euclidean as it is 
 of Euclidean geometry. The angular magnitude 
 about a point is equal to four right angles in any un- 
 bounded geometrical surface whose curvature is con- 
 stant, and is therefore the same in all forms of 
 
 25 Killing, Gnmdlagen der Geometrie (Paderborn, 1898), 
 Vol. II., p. 171 ; and especially Hilbert, Gnmdlagen der Geom- 
 etrie, Leipzig, 1899, p. 16. 
 
PSYCHOLOGIC ASPECTS OP THE PROBLEM 105 
 
 geometry.^*' Hence the creation of new geometries 
 does not essentially modify the difficulties to be met 
 with in the treatment of angles. Angular magni- 
 tudes can always be expressed as ratios of linear 
 magnitudes, and are therefore easily determined 
 when the latter are known. Upon the possibility of 
 such ratios the whole science of trigonometry cor- 
 responding to any conception of space is erected. 
 
 Let us, then, focus our thought upon the straight 
 line and try by careful analysis to answer our ques- 
 tions. Having already traced the growth of this 
 conception as it presents itself to the consciousness 
 of the ordinary man, let us now turn to the mathe- 
 matician and endeavor to learn from him what he 
 considers to be essential to the straight line as shown 
 by his definitions. When the results of this analysis 
 have been attained we shall then try to relate them. 
 if possible, to actual experience. 
 
 We have seen that Riemann makes room for an 
 unlimited number of geometries differing from each 
 other fundamentally in the meaning assigned to the 
 linear element ds, and that in the further working 
 out of his system emphasis was laid upon cun'a- 
 ture^'^ as an important conception. Cayley makes 
 
 28 There may appear to be an exception to this statement in- 
 the case of the apex of the cone where this anprular magnitude 
 is always less than four right angles, but I have attempted to 
 remove this objection by the word unbounded. 
 
 27 We have already noted the development of this conccp- 
 
io6 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 distance fundamental, and with the aid of Klein I 
 shows that a variation of the laws by which distance 
 is measured is all that is necessary to distinguish 
 Euclidean and non-Euclidean systems. From this 
 it appears that geometries are to be distinguished ■, 
 by the number and character of the postulates em- 1 
 ployed to determine certain conceptions which for 
 the ordinary man define the straight line. For him, 
 as we have said, this line is at all times simply the 
 shortest distance between two points or else a line 
 which does not change its direction at any of its 
 points. But these words are not simple in meaning, 
 and therefore stand in need of definition themselves. 
 Direction has already been considered and need no 
 longer detain us. Let us now examine the phrase 
 " shortest distance " and endeavor to determine its 
 meaning. Reflection shows that this, too, is far 
 from being a simple conception. It presupposes, in 
 fact, all the assumptions necessary for measurement. 
 It presumes beforehand the possibility of measuring 
 all kinds of lines that can be drawn anywhere in 
 space, else how could it be said that a certain one of 
 these lines is the shortest distance between two given 
 points. This is certainly a tremendous assumption. 
 All the postulates demanded by metrical geometry 
 are wrapped up in this definition. Before measure- 
 
 tion in the historical treatment of Riemann's geometry in 
 Chapter I.*, we shall take it up for more thorough treatment in 
 our last chapter in the discussion of space conceptions. 
 
FSYCHOLUGIC ASPECTS OF THE PROBLEM 107 
 
 tnent is possible we must have a standard of meas- 
 urement, and this standard is itself the straight line. 
 Measurement, then, presupposes the straight line as 
 a necessary pre-condition of its own possibility, and 
 therefore cannot be taken as a simpler and more 
 ultimate notion with which to define the straight 
 line. Furthermore, the existence of a minimum is 
 itself an assumption which involves certain logical 
 consequences of an interesting character, which can- 
 not be dealt with here. Finally, this definition fails 
 in the case of straight lines which join antipodal 
 points of double elliptic space. Between such points 
 there is an infinity of shortest lines. 
 
 These are difficulties which are certainly of a seri- 
 ous character. Nevertheless it seems hardly possi- 
 ble wholly to avoid some conception of distance 
 when we talk of straight lines. Long ago Leibnitz 
 made distance fundamental,^^ and the same point of 
 starting has recently been taken by Frischauf and 
 Peano. Peano defines the straight line ab as a class 
 of points ,v, such that any point y, whose distances 
 from a and b are respectively equal to the distances 
 of .r from a and b, must be coincident with x. But 
 Peano fails to prove either that such a line exists 
 or that, if it does exist, it is determined by any two 
 of its points.2» This, of course, is impossible with- 
 out the use of certain special axioms. According to 
 
 2" Russell, Principles of Mathewatics, Vol. I., p. 410- 
 2» Russell, Principles of Mathematics. Vol T.. p. 410 flF. 
 
io8 PSYCHOLOGIC ASPEC'TS OF THE PROBLEM 
 
 Peano, five such axioms are needed. The group 
 which he has selected is an interesting one, because 
 it defines distance by the use of hetn'cen as an in- 
 definable notion. His axioms for the straight line 
 ab are as follows : ( i ) Points hetzveen which and b 
 the point a lies; (2) the point a; (3) points betzvcen 
 a and 6; (4) the point b; (5) points between which 
 and a the point b lies.'"' Just what is meant by be- 
 tzveen is nowhere clearly explained. 
 
 Vailati attempts an explanation which is rejected 
 by Peano ^^ on the ground that betzveen is a relation 
 of three points and not of two only. As a matter 
 of fact if we confine ourselves to projective geometry 
 even three points on a line are not so related that 
 any one of them can be said to be betzvcen the other 
 two. The word, between, involves a certain order- 
 ing of the points which in projective geometry de- 
 pends upon the nature of the quadrilateral construc- 
 tion which requires for its proof a point outside its 
 own plane and hence is not possible without three 
 dimensions. It also requires four perspective tri- 
 angles. The generation of order by this method is 
 therefore considerably complicated and the simplest 
 proposition involving betzveen which remains unal- 
 tered by projection is one which requires four points. 
 
 It is obvious then that if '' between " is to be estab- 
 lished at all as a unique relation of any tzifo points 
 
 3" Rivista di Matematica, Vol. IV., p. 62. 
 ^'^Rh'isfa di Matematica, Vol. I., p. 393. 
 
PSYCHOLOGIC ASPECTS OF THE PROBLEM 109 
 
 it cannot be done by any appeal to projective con- 
 siderations. But in spite of this Russell ^- sets 
 Peano's criticism aside as inadequate and adopts 
 what is practically Vailati's position. He avoids the 
 word " between," however, and introduces in its 
 stead a new indefinable. He posits a class of 
 asymmetrical transitive relations ^^ which he calls K 
 and assumes that betw'een any two points there is 
 one and only one relation of this class. Eight 
 axioms which Russell shows to be distinct and mu- 
 tually independent are required, as he thinks, to 
 define what is meant by this class of relations.'*^ 
 
 Avoiding his symbolism these axioms may be 
 stated as follow^s: (i) There is a class of asym- 
 metrical transitive relations K; (2) there is at least 
 one point, and if R be any term of K we have; (3) 
 R is an aliorelative, that is, a relation which no term 
 has to itself; (4) the converse of R is a term of K; 
 (5) R2 = R, (6) the domain of the converse of R 
 
 32 Principles of Mathematics, Vol. I., pp. 394 ff- 
 ^^ In the sense in which these terms are employed by Mr. 
 Russell they may be explained as follows : When the converse 
 of a relation is the same as the relation itself, the relation is 
 said to be symmetrical ; but when the converse and the orip- 
 inal relation are incompatible the relation is said to be asym- 
 metrical. Examples of the former are such as identity, equal- 
 ity, and inequality; and of the latter such as better and worse, 
 greater and less. A relation is transitive when any such con- 
 dition as the following holds : viz.. if a be similar to b and b 
 similar to c. then a is similar to c. 
 ^* Princip. Math., Vol. I., pp. .WS-.^Q^- 
 
no PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 is contained in the domain of R; (7) between any 
 two points there is one and only one relation of the 
 class K; (8) if a, h be points of the domain of R, 
 then either a holds the relation R to & or t holds 
 this relation to a. The seventh axiom is obviously 
 double. It asserts ( i ) that there is one such relation 
 between any two points, and (2) that there is only 
 one. The first member of the group is not an axiom 
 at all but only the assumption of the indefinable class 
 of relations K. 
 
 With this outfit of assumptions the various forms 
 of geometry may be constructed. The mutual inde- 
 pendence of the entire group is shown in the usual 
 way. Any one of them may be denied and a logic- 
 ally consistent system built upen what remains. 
 The fifth and seventh axioms are the most interest- 
 ing here. In ( 5 ) we may deny either that R is con- 
 tained in R2 or that R^ is contained in R. If we 
 deny the former the resulting straight line becomes 
 a series of points which does not possess the 
 " density " or "compactness " which the construc- 
 tions of Euclid and non-Euclid require. The gen- 
 eral result, however, is logically sound, the series in 
 question simply lacks the degree of continuity which 
 geometry requires. If we deny the latter, the result 
 proves to be untrue of angles which otherwise may 
 be made to satisfy all the conditions expressed by 
 this group of axioms. If a Euclidean and a hyper- 
 bolic space be considered together all the axioms 
 
PSYCHOLOGIC ASPECTS OP THE PROBLEM in 
 
 Still hold except the first part of (7). The whole 
 group with the exception of (7) is shown to be 
 necessary, but for (7) Mr. Russell is only able to 
 maintain a high degree of probability. 
 
 We have now to consider another interesting 
 effort to reach the " minimum essential " to the 
 notion of " straightness." This lays the emphasis 
 upon a different factor of experience from those 
 just considered, and consequently appeals very 
 strongly to those who prefer to approach geometry 
 from the motor side, rather than from that statical 
 conception of an " empty " space which may be 
 reached by abstraction from the materials furnished 
 by vision alone without reference to bodily move- 
 ments actual or imagined. 
 
 The effort to w^hich we refer is that of Pieri.^"^ 
 who proceeds to build geometry upon the two in- 
 definables. point and motion. He defines the straight 
 line joining two given points as the class of points 
 whose internal relations remain unchanged by any 
 motion which leaves the two points fixed. His 
 system is simple and logically unimpeachable.'" But 
 here again we are dealing with a complex idea. 
 Motion as used by Fieri is not simply the motion <if 
 a single point, but a certain law of motion is assumed 
 along with this simpler idea. We have the motion 
 
 ^■' Delia gcometria elcmcntarc como sisfcma ipotitico dcdut- 
 tivo, Turin, 1899. 
 
 36 Russell, Principles of Math., p. 410. 
 
112 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 of a number of points taking place in such a way 
 as to leave certain relations unchanged. In other 
 words it is the motion of a rigid body which is here 
 assumed. 
 
 If then we ask how motion as thus understood is 
 to be distinguished from other transformations it 
 becomes readily apparent why Fieri was able to con- 
 struct metrical geometry by considering such a no- 
 tion as indefinable. For when we attempt to answer 
 this question we see that in this idea of motion all 
 the conditions necessary to metrical geometry are 
 already assumed. Fieri means simply a transforma- 
 tion which leaves all essential metrical properties 
 and conditions unchanged. Having thus presup- 
 posed metrical properties in his conception of mo- 
 tion, he then turns about and defines these same 
 properties in terms of this conception. Little wonder 
 that he succeeds. 
 
 It must be observed also that Riemann's famous 
 dissertation involves the same difficulty. He as- 
 sumed that the linear element ds remains unaltered 
 by the same infinitesimal motion of all of its points, 
 and bases his system upon this assumption. He 
 thus presupposes the existence of equal spatial quan- 
 tities in different places which is equivalent to the 
 assumption of spatial homogeneity, or the free mo- 
 bility of rigid bodies. In other words by assuming 
 metrical properties in his ds and then proceeding to i 
 determine these properties upon the basis of this 
 
PSYCHOLOGIC ASPECTS OI- THE PROBLEM iij 
 
 assumption, he easily draws out at the faucet what 
 he has already poured in at the bung. 
 
 Glancing back now over the definitions which we 
 have just considered in the light of the critical ex- 
 amination to which we have subjected them let us 
 briefly summarize the resulting facts. We have 
 found that in each case certain metrical properties 
 at some stage or other in the process were either 
 unconsciously assumed or else openly postulated. 
 Peano starts with distance and closes his discussion 
 by grounding this conception upon " between " as 
 an indefinable notion. Russell, though holding es- 
 sentially the same meaning, dispenses with this word 
 and substitutes in its place a particular class of 
 asymmetrical transitive relations by means of which 
 the order of points on tlie straight line may be gen- 
 erated in a perfectly definite way. And finally we 
 found Fieri employing in his definition the idea of 
 motion which he assumed to take place in such a 
 way as to allow the unique character of the straight 
 line as well as certain metrical properties to remain 
 wholly unchanged. 
 
 If now we sever these words from their abstract 
 logical surroundings and endeavor to carry them 
 back to a meaning in experience, we find that each 
 word in its turn bears with it what is formally essen- 
 tial to the idea of the straight line. All the way 
 through, this conception of the line as reached by 
 abstraction from certain forms of experience already 
 
114 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 pointed out, and defined as the shortest distance 
 between two points, or as a geometrical object which 
 when rotated about any two of its points remains 
 always within itself, has really been present deter- 
 mining the result. It is this conception undoubt- 
 edly which has guided Peano and Russell in select- 
 ing the postulates by which their conceptions are at 
 last defined. It is this conception and naught else 
 which has determined the number and character of 
 these postulates to be what they are. There are cer- 
 tainly no a priori reasons which independently of 
 those actual experiences in which the notion of the 
 straight line is ultimately grounded, could have de- 
 cided this matter. Why select this peculiar con- 
 ception at all as the one most convenient and there- 
 fore best suited to serve as the point of starting in 
 all systems of geometry, and why ascribe to it just 
 those peculiar marks that we do? It takes experi- 
 ence to answer this question. No amount of mere 
 passive contemplation of blank space, if such a thing 
 were possible, could ever lead to a direct intuition 
 of the peculiar properties of the straight line. We 
 can reproduce such an object in imagination, it is 
 true, but in reality this is but an " experiment in 
 thought '] which could not take place except upon 
 a basis of past experience. 
 
 We do not wish to be misunderstood. What we 
 are here contending for is the empirical origin of 
 the special properties of the straight line which dis- 
 
PSYCHOLOGIC ASPECTS OF TUli PROBLEM ns 
 
 tinguish it as a peculiar relation between two points. 
 We do not mean to assert that there is nothing a 
 priori about this conception. The straight line is a 
 peculiar relation between points. It both separates 
 and unites these points in a relational way. The 
 points can not coalesce, and become one point, they 
 must be kept distinct from each other, else the line 
 which they are supposed to determine vanishes away. 
 So much is true of every straight line however its 
 peculiar characteristics may otherwise be deter- 
 mined, w-hether we represent it to ourselves by 
 means of a visual, a tactual, or a motor image. So 
 much at least appears to be involved in the very 
 notion of space as an externalizing principle which 
 renders possible any system of distinct coexisting 
 entities. Whatever is more than this has a basis in 
 experience, and w henever we conceive such a line to 
 be something which admits of definition it becomes 
 much more than a mere abstract relation between 
 points, it becomes a straight relation, the question of 
 the economy of effort represented as being somehow 
 felt in passing from one point to the other now 
 enters into the conception and carries us back at 
 once to data which we can only represent to our- 
 selves as being furnished by some form of sensory 
 experience, real or imagine'd. How shall I proceed 
 in the shortest time possible and with a minimum 
 of effort from one point to the other? That path- 
 way if you please which answers this f|uestion is 
 
ii6 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 the straight Hne. If we now ask just how this 
 pathway is to be determined, we fall back upon cer- 
 tain special experiences such as we have already de- 
 scribed. The line which best satisfies this require- 
 ment of a minimum of efifort for us tridimensional 
 beings, is one which is congruent with itself, in all 
 positions, which is the shortest distance between two 
 points which rotates within itself, which looks the 
 same from all points not in it, or one whose internal 
 relations remain unchanged by any motion which 
 leaves two of its points fixed. If we neglect the 
 question as to the number of space cKmensions 
 which are to be taken account of, these definitions 
 all mean essentially the same thing, but they empha- 
 size somewhat different aspects of the complex ex- 
 perience involved. 
 
 This conception then is a generalization from 
 facts which come within the limits of experience 
 and must therefore fit these facts when carried back 
 to them. It is only when we go beyond the facts 
 and, by following the lead of suggestions offered 
 by experience or by investigating the logical possi- 
 bilities involved, pronounce upon such matters as 
 the infinity or two sidedness of the straight line that 
 the roads leading to Euclid and non-Euclid begin 
 to diverge. ) This may be made clear by the follow- 
 ing Tllusfration. Suppose we have two perfectly 
 straight lines lying in any position in the same plane 
 and extending without limit in both directions ; let 
 
PSYCHOLOGIC ASPECTS OF THE PROBLEM 117 
 
 one of these lines rotate about a given point not on 
 the other Hne, starting the rotation with the lines 
 intersecting. The point of intersection moves along 
 the fixed line either away from or toward the fixed 
 point of rotation according to the direction in which 
 the line is rotated. For convenience let it move 
 away from this point, /rhree different results are 
 logically possible. When the rotating line ceases to 
 intersect the fixed line in one direction it will imme- 
 diately intersect in the opposite direction,'*' or it will 
 continue to rotate for a time before intersection 
 takes place, or else there will be a period of time 
 during which the two lines intersect in both direc- 
 tions. The first of these possibilities gives Euclid's, 
 the second Lobatchewsky's. and the third Riemann's 
 geometry. 
 
 The mind's attitude toward these three possibili- 
 ties taken successively illustrates in a curious way 
 the essentially empirical nature of the straight line 
 as we conceive it. Logically one of these possibili- 
 ties is just as acceptable as the other. From this 
 point of view strictly taken there is certainly no 
 reason for preferring one of them to another. Psy- 
 chologically, however. Riemann's hypothesis seems 
 absolutely contradictory, and even Euclid's is not 
 quite so acceptable as that of Lobatchewsky. The 
 lines cannot intersect in both directions, we say. for 
 
 37 We can not say the opposite end for the lines in each case 
 and in both directions are supposed to be unbounded. 
 
ii8 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 in that event they would necessarily be curved or at 
 any rate not continuous straight lines. Even the 
 assumption which leads to Euclid seems not to be in 
 accord with the lines of experience which we call 
 parallel; for all such lines must turn a little way 
 after ceasing to intersect at one end before they 
 begin to intersect at the other. The logical require- 
 ments which in this case lead to Euclid, seem to be 
 at variance with those very experiences which have 
 produced this geometry. The lines seem to be as 
 pictured by the imagination, necessarily curved. 
 Suppose that the point about which the rotating 
 line turns is at a great distance from the fixed line, 
 it then appears wholly impossible to our picturate 
 thinking to represent the two lines as straight under 
 the given conditions. They appear to be geodesies 
 joining antipodal points on the surface of an im- 
 mense sphere. The reason for this appearance is 
 obvious. It is because of our familiarity with finite 
 spherical surfaces whose straightest lines are curved 
 and are known to intersect in the manner assumed 
 by Riemann. Hence in harmony with the mind's 
 general tendency to settle its abstract ideas upon 
 some convenient concrete model perceived or im- 
 agined, Riemann's straight lines become at once 
 attached to the sphere and are thus pictured as 
 curved. In Lobatchewsky's assumption the case is 
 quite different, for it seems to contradict none of 
 our ideas of the straight line. Beltrami's pseudo- 
 
PSYCHOLOGIC ASPECTS OF THE PROBLEM ng 
 
 Spherical surfaces upon which all the theorems of 
 Lobatchewsky may be projected and there visualized 
 as figures whose sides are curved, are not at all 
 familiar to us. But suppose these saddled shaped 
 surfaces had been the ones with which our notion of 
 curved lines had always been associated and that 
 pseudo-spheres and spheres had simply exchanged 
 places in our experience, would not Lobatchewsky's 
 assumption in that event have appeared to be just as 
 incompatible w4th our notion of straight lines as 
 Riemann's does now ? 
 
 I Again there is logically no reason why the geom- 
 etry of Lobatchewsky should have appeared in his- 
 tory before that of Riemann. We have seen, 
 however, that the former assumption worried Sac- 
 cheri very profoundly in his heroic efforts to 
 " vindicate Euclid," more than a century and a 
 quarter prior to the publication of Riemann's dis- 
 sertation. It was not a logical but a psychological 
 or experiential difficulty which caused Saccheri to 
 reject the logical conclusions to which his own labors 
 clearly and inevitably pointed ; and it was certainly 
 
 , the same sort of difficulty which caused the imme- 
 diate rejection, by himself and by subsequent 
 mathematicians, of the assumption upon which Rie- 
 mann's geometry is grounded. 
 
 We must now briefly summarize the results of the 
 present chapter and indicate the direction which m 
 
120 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 consequence of the positions here taken, our further 
 treatment of the problem must follow. 
 
 We have endeavored to show that all strictly 
 metrical conceptions, including the straight line and 
 the theory of parallels, grow out of experience with 
 the objects of our environment, and that the particu- 
 lar form which these conceptions have assumed is 
 I determined to be what it is by the peculiar character 
 1 of this experience. This is true not only of the 
 special conceptions by which the different forms of 
 geometry are defined, but it is also true of the 
 space conceptions which lie at the foundation of 
 each. Geometrical spaces are abstract ideal concep- 
 tions which arise out of experience in much the 
 same way and by means of essentially the same 
 faculties as those which are employed in the develop- 
 ment of other conceptions. There is nothing espe- 
 cially mysterious about them. They differ funda- 
 mentally as to the logical possibilities which the 
 same general experience suggests in regard to what 
 may conceivably happen beyond its actual realm, 
 and also as to the particular aspects of this experi- 
 ence upon which emphasis is especially laid. 
 From somewhat heterogeneous psychological spaces 
 through the physical and geometrical on up to that 
 higher conception which in a sense must include 
 and harmonize them all. the process is a unitary 
 one and essentially unbroken. By fusing together 
 the spatial data derived from' the various senses 
 
PSYCHOLOGIC ASPECTS OF THE PROBLEM ui 
 
 through ignoring- the differences between their sev- 
 eral dehverances and by correcting the appearances 
 to one sense through those of another, in such a 
 manner as to give the most complete and trust- 
 worthy perception of the objects of our environment 
 in their relations to ourselves and to each other, we 
 derive our notion of space in which the physical 
 world is apparently set. We simply manipulate the 
 data of sense as if always with an end in view, 
 namely that of perceiving the things (^f our |)hysical 
 world successfully and adjusting ourselves comfort- 
 ably to them. At a higher stage the same purposive 
 process yields those adjectives which define the 
 various forms of geometrical space. Just as the 
 varying appearances of things to the different senses 
 for example were ignored in order to arrive at their 
 real place in our mental picture of. the world, so the 
 varying and irregular deformities, to which they are 
 actually subjected in different places by virtue of 
 their relations to each other, when abstracted from 
 lead directly to the conception of relativity and 
 homogeneity of space. In a similar fashion notions 
 of unboundedness and infinite divisibility arise. . 
 
 We have pointed out the indispensableness of the 
 straight line and indicated its relation to the parallel , 
 postulate. The two are bound together by no j 
 logical necessity but by a relation which is assumed 
 to hold in a region which lies beyond all experiences 
 actual or possible. This conception r,f the straight 
 
122 PSYCHOLOGIC ASPECTS OF THE PROBLEM 
 
 line when freed from certain ideas whicl\ by con- 
 tingent, though long continued and practically un- 
 interrupted associations have served to disguise its 
 essential meaning, is, so far as it is grounded in 
 actual experience, the most fundamental figure in 
 all forms of geometry. Straight lines are straight 
 lines in non-Euclid as well as in Euclid and 
 although we may, to aid the imagination and for 
 convenience of study, conceive the figures of these 
 newer geometries to be projected upon surfaces in 
 finite portions of Euclidean spaces of higher dimen- 
 sions — surfaces whose lines are all curved ; yet we 
 must not confuse these props of the imagination 
 with the realities for which these geometries as 
 logical systems may be supposed to stand. 
 
 When we go beyond the experiences which have 
 given us this conception of the straight line, as in 
 thought we may, and consider the different logical 
 possibilities which lie open to us there, we get by 
 postulating these possibilities in succession in con- 
 junction always of course with those idealizations 
 which actual experience affords, the different sys- 
 tems of geometry. 
 
 There is then, and must always remain a gap 
 between empirical and mathematical exactness, be- 
 tween experiences actual and the ideals which ex- 
 perience suggests. In the case before us there is 
 such a gap between the conditions imposed by sense- 
 perception and the mathematical precision which 
 
PSYCHOLOGIC ASPECTS OP PHP PROBLEM 123 
 
 results when these conditions are supposed to be 
 withdrawn. In this opening, as we have said, 
 logical possibilities lie which are seized upon and 
 woven into systems by the different geometries. 
 Under the growing refinement of methods and 
 means of observation this opening is a decreasing 
 variable; it is therefore not only conceivable but 
 empirically possible that an appreciable deviatioi-Lof 
 the parallel postulate from the space of experience 
 may eventually be found. In other words, as tested 
 by the court of perceptual experience alone the case 
 may some day go against Euclid but it can never be 
 decided in this way absolutely in his favor. 
 
 We reach then at the close of this chapter the 
 important conclusion that Euclid's validity must be 
 established, if established absolutely, upon some 
 other basis 'than that of mere perceptual experience. | 
 Whether such a basis exists remains to be seen. 
 
THE NATURE AND VALIDITY 
 
 OF THE 
 
 PARALLEL POSTULATE. 
 
CHAPTER V. 
 
 THE NATURE AND VALIDITY OF THE PARALLEL 
 
 POSTULATE. / 
 
 We have reached the conclusion that geometrical 
 spaces, merely as such, are all of them abstract c(jn- 
 ceptions. They are grounded on and grow out of 
 the same general experience which they interpret 
 differently while seeking to simplify and to system- 
 atize it by means of the peculiar postulates which 
 define them. We can also see that the quantitative 
 concepts which underlie the different geometries 
 have been chosen somewhat arbitrarily so that when 
 we carry them back to the facts of spatial experi- 
 ence they do not reproduce these facts in any case 
 with absolute precision. Different groups of ideas 
 may therefore serve to express all the facts with 
 equal exactitude within the region accessible to 
 observation. 
 
 The space-world as we know it is not riuantita- 
 tively infinite nor does the assumption that it is 
 seem to be a necessary one. although Euclid requires 
 it. If then we refuse to call it infinite in this sense 
 but still accept the present laws of optics and astron 
 127 
 
128 THE VALIDITY OF EUCLID • 
 
 omy which presuppose EucHd's vahdity and if we 
 also admit the postulate of free mobility which is 
 the same as to assume that the parameter of space 
 (K)^ is a constant quantity, we find that even 
 under these restrictions it is still possible to repre- 
 sent all the facts with equal accuracy by the 
 geometries of Euclid, Lobatchewsky, and Riemann. 
 So far then as experience goes at present, or can 
 ever go for that matter, there is no necessary reason 
 for starting any physical inquiry with the Euclidean 
 assumption that K is infinite. All we need is to 
 take K sufficiently large to make the deviation from 
 Euclid fall within the limits of astronomical obser- 
 vation. As this observation becomes more refined 
 and exact, one of two things must inevitably hap- 
 pen. Either the facts will ultimately appear to go 
 against Euclid or else it will be shown that the 
 actually known quantity of space, though always 
 finite, will transcend successively certain increas- 
 ingly large amounts which the new approximations 
 to the value of K will give us the right to affirm. 
 
 But to go so far as to assert even that K is con- 
 stant seems, on the surface at least, to be an arbi- 
 trary matter which is not demanded either by 
 experience or by logic. It means the same thing as 
 to assert an absolutely rigid standard which may 
 be transported unchanged to any part of space. 
 
 1 See Chapter II. under the discussion of Riemann's idea of 
 curvature. 
 
I HE VALlUlTy Of EUCLID 
 
 ijy 
 
 But we certainly do not meet with any such rigid 
 measures among the objects of experience from 
 which as we have seen the idea of a metrical stand- 
 ard has actually been derived. What we really 
 know is that some of these objects are less subject 
 to quantitative variations than others are. and that 
 consequently a series of them may be arranged 
 whose members as we ascend the scale approximate 
 more and more nearly a quantitative invariability. 
 Hence here as in the case of the straight line the 
 intellect may form the pure abstract conception of 
 a rigid body which, when thought of as independ- 
 ent of position, furnishes also the conception of 
 spatial homogeneity. But between the actual facts 
 of experience and the essential proi)erties of such 
 an intellectual construct there exists, as stated, a 
 gulf which sense-perception of itself cannot bridge. 
 It is then both logically and empirically permis- 
 sible to assume that the actual parameter of space 
 is a variable quantity which oscillates within cer- 
 tain narrow limits.^ By assuming this oscilla- 
 tion to occur in accordance with law other geome- 
 tries compatible with experience could also be 
 obtained. 
 
 It is such considerations as these that force upon 
 
 '^ When we assume the possibility of measurement in the 
 exact sense required by geometry, and the facts of experience 
 make this of course a legitimate inference, this assumed varia- 
 bility of K must be ruled out on purely logical grounds, as we 
 shall later attempt to show. 
 
130 THE VALIDITY OF EUCLID 
 
 US the necessity of making and maintaining a care- 
 ful distinction between the facts of experience and 
 those intellectual constructs whose formation these 
 facts have suggested. Upon these constructs two 
 conditions are imposed : they must fit the facts to 
 which they relate, and must also meet the logical 
 requirements of mutual non-contradiction. When 
 these conditions are fulfilled, that is, when it is 
 shown that different systems of geometry, Euclid- 
 ean and non-Euclidean, are equally permissible 
 under both requirements, the question of validity 
 assumes its most interesting and difficult form. 
 
 We have shown that these geometries within cer- 
 tain limitations already pointed out, are empirically 
 indistinguishable ; it therefore remains to consider 
 the other half of the problem. Are these geometries 
 when considered strictly from the logical point of 
 view equally tenable? Fortunately, as our his- 
 torical chapter has abundantly shown, we already 
 have all that seems to be desired for a satisfactory 
 answer. Granting the groups of assumptions from 
 which they set out, accepting the condition merely 
 that these assumptions be so defined as to be mu- 
 tually independent and logically consistent, and 
 finally, disregarding the question as to their easy 
 compatibility with the known facts of reality ; 
 almost an unlimited number of geometries can be. 
 and very many indeed have been, actually built up 
 
THE VALIDITY OF EUCLID 131 
 
 in such a manner as to satisfy the strictest demands 
 for internal consistency. 
 
 When we contine ourselves then to those consid- 
 erations which the idea of exact measurement re- 
 quires, and hence to those assumptions only which 
 are in all respects very similar to Euclid's, the 
 demonstrated fact that the proofs of the resulting 
 non-Euclidean systems hold good for corresponding- 
 theorems of Euclid's geometry when the appro- 
 priate substitutions have been made, shows beyond 
 question that any logical defects which may possibly 
 hereafter be discovered in any non-Euclidean sys- 
 tems must also apply with equal force against Euclid. 
 A revelation of contradiction in one must prove to 
 be a revelation of contradiction in the other. 
 
 Therefore the apparently inevitable conclusion 
 which we are forced to face is that Euclid is neither 
 empirically nor even logically necessary to the world 
 of reality. It is based upon a space conception 
 derived by abstraction from that world as we know 
 it and all its theorems and constructions are conse- 
 quently in perfect logical harmony with that con- 
 ception. But there are other conceptions derived 
 in essentially the same way from the same world 
 and defined by slightly different postulates. From 
 these conceptions also other geometries are known 
 to flow with equal harmony and necessity. What 
 one then of these opposing systems is the true geom- 
 etry? It seems almost absurd to ask this question. 
 
132 THE VALIDITY OF EUCLID 
 
 They are all, of course, true — true logically and, 
 within Hmits, true to the facts; therefore the deci- 
 sion between them, if made at all, must be made 
 not upon a basis of truth but simply as a matter of 
 convenience. It is not a question of necessity but 
 one of utility. When therefore it is generally ad- 
 mitted that Euclid is the most convenient of them 
 all the obvious conclusion is that Euclid will con- 
 tinue in favor though robbed of that peculiar maj- 
 esty as a system of absolute truth which it formerly 
 seemed to possess. 
 
 But we cannot dispose of the difficulty in this 
 easy fashion. Again we have solved the problem by 
 simply hiding its meaning. The word convenient 
 here introduced is employed by M. Poincare as 
 though it ended the matter. The axioms of geom- 
 etry are for him all of them mere conventions, they 
 are all true, but some groups are simply more con- 
 venient than others ; but in reality the central philo- 
 sophical puzzle originating in metageometry lies 
 concealed in Poincare's use of this word. We 
 cannot dismiss the matter on the mere ground of 
 convenience. Granted that Euclid is the most con- 
 venient, how, why, and in what sense is it so ? We 
 certainly cannot say that it is. because of any ob- 
 served matters of fact to which geometry relates, 
 nor of any requirement of logical consisteney. Nor 
 can we say that Euclid is logically the most simple. 
 
 I 
 
THE VALlUny Of EUCLID ij3 
 
 for from this point of view the single elliptic system 
 is much more beautiful and attractive. 
 
 It is true that from the standpoint of algebra the 
 constructions of Euclid do not involve the use of 
 such complicated equations as are required by his 
 rivals. For these constructions nothing more diffi- 
 cult is involved than a general equation of the 
 quadratic form. Hence the Euclidean lines and 
 planes do not require so high a degree of continuity 
 as that which is demanded by Lobatchewsky and 
 Riemann. The Euclidean plane may be regarded 
 in fact as a continuous two-difnensional manifold 
 of points, some of which have been dropped out at 
 regular intervals. All the constructions of Euclid 
 can be made in such a manifold without anywhere 
 falling into a hole. To sense, it would appear as a 
 sieve, and the straight 'line as a picket fence where 
 the pickets and the distances between them corre- 
 spond to the points on the line and their relations 
 to each other.'' This sort of simplicity, however. 
 results from the application of numbers to luiclidean 
 conceptions and can hardly be regarded a? neces- 
 sarily inherent in these conceptions themselves."' To 
 
 3 Compare Dedekind : IVas sitid uud zcas sollcn die Zahleu. 
 Brunswick. 1893- Vonvort s. XII. 
 
 * Professor G. B. Halsted states in a letter to the present 
 writer under date of Jan. 4. 1904 : " I build up my Rational 
 Geometry wholly without number or arithmetic. usinR no 
 ratios, irrationals, or complexes, as you will see when it ap- 
 pears in a month or two:" 
 
134 THE VALIDITY OF EUCLID 
 
 say that Euclid is the most convenient in this sense 
 is but to say that his conceptions happen to be most 
 amenable to easy algebraic interpretation. But this 
 is certainly no mere accident. Like every other fact 
 of human experience it demands philosophic recog- 
 nition. Why indeed should it be that the very con- 
 ception of space which is historically first and at the 
 same time most natural, should also be the most 
 easily interpretable by means of algebraic formulae 
 which have been derived from a widely different 
 and apparently independent source in experience? 
 Why does that which is spatially the most simple and 
 which can be successfully handled without regard 
 to number, thus prove to be at the same time most 
 convenient for numerical interpretation? 
 i In attempting to answer this question the influence 
 )of custom should not be ignored. Relations which 
 have been established by long and uninterrupted asso- 
 ciations are not easily distinguished from those which 
 are logically necessary. Hence it must not be for- 
 gotten that civilization has been steeped in Euclid 
 for more than two thousand years so that today this 
 geometry underlies all our physical sciences and we 
 are giving expression to Euclidean forms in all the 
 mechanical arts. 
 
 During this long period the reasoning and even 
 the form of Euclid have been generally regarded as 
 the most perfect model of scientific and even of philo- 
 
IHE rAUDITY OF EUCLID 135 
 
 sopliic '"' thought and expression. (Jur intellectual 
 life has become attuned both to Euclid's doctrine of 
 space and to his peculiar form of expression. Little 
 wonder then that it seems almost sacrilege to depart 
 from them or even to call them in question ; we come 
 to feel with Kant and with thousands of others, tiiat 
 if there is an a priori necessity anywhere in human 
 knowledge we find it in Euclid. But suppose all this 
 had not been, and there is surely nothing violent in 
 this supposition, could we still say that Euclid is the 
 most convenient geometry ? 
 
 We have seen how the peculiar facts which have 
 pointed unmistakably to Euclid have been the most 
 patent, universal, and familiar at all stages of the 
 race's development. They have not needed pro- 
 longed meditation, observation, and experiment to 
 bring them into prominence, they lie open and every- 
 where ready at hand. In consequence, Euclid has 
 gained a long start in advance of his competitors in 
 the race for general acceptance, and popular favor. 
 
 But here again we must face the question which 
 seems to confront us withersoever we turn. Why 
 
 '' Spinoza's Ethics is a good illustration of Euclid's inflncnce 
 in this direction. Even the fundamental ontolopical concep- 
 tions have often been borrowed directly or indirectly from 
 Euclid. This veneration of Euclid as a body of ideal knowl- 
 edge has proved exceedingly mischievous in many ways. 
 YMathematics and formal logic instead of being the very ideal 
 of truth are in an important sense the farthest removed from 
 truth. 
 
136 THE VALIDITY OF EUCLID 
 
 this ancient and more favorable start ? It is certainly 
 not a mere matter of chance that our world of experi- 
 ence should be so favorably disposed for the sugges- 
 tion of Euclid. Though the mere antiquity of this 
 science, in itself, is no proof of its absolute and ex- 
 clusive necessity; no sure demonstration that it, as 
 against other logically and empirically justifiable 
 possibilities, is alone a priori and native to the con- 
 stitution of the mind; nevertheless it does, when 
 taken in conjunction with the matter of algebraic 
 simplicity already pointed out, compel us to consider 
 Kant's doctrine. There is certainly something, 
 either in the knowing mind or the world that is 
 known, which makes for Euclid, and philosophy is 
 called upon to locate it and to determine satisfac- 
 torily its essential nature. 
 
 Is Euclid then, as Kant maintained, based on an 
 a priori form of our sense intuition ? Must we see 
 things whether we will or no, through Euclidean 
 glasses? And are the non-Euclidean systems after 
 all merely ingenious and interesting intellectual con- 
 structs which cannot even be thought of as realized 
 except in Euclidean spaces? 
 
 The first of these questions, in the light of our 
 previous discussion, is now comparatively easy to 
 answer; the last will be reserved for the following 
 chapter. At present then we shall have nothing to * 
 do with those spatial characteristics which Euclid 
 and non-Euclid hold in common with each other. 
 
THE VALIDITY OF EUCLID 
 
 ii7 
 
 We are concerned merely with that pecuHar feature 
 which distinguishes EucHd. Dehn's investigation 
 has put beyond question what Euchd's one essential 
 pecuHarity is; our present task therefore concerns 
 merely the parallel postulate. Can it be maintained 
 that this postulate is a priori in Kant's meaning of 
 that term? If it cannot, then Euclid's last claim of 
 necessary supremacy must be rejected. 
 
 By space Kant meant, of course, a universal and 
 necessary form of sense-intuition which is native to 
 the mind. By establishing this thesis as a firm 
 ground of standing he sought to explain how it is 
 possible to have a knowledge of objects not only 
 prior to all experience with them but which tran- 
 scends the very bounds of all possible experience. 
 
 Kant never raises the question as to whether a 
 priori synthetic knowledge of any sort is possible. 
 " Of course it is possible," he would say. " in fact, 
 it is actual, for we have it already in Euclid." In- 
 deed Euclid in Kant's day was, and had been for 
 centuries, universally accepted. Hence to prove be- 
 yond cavil that certain knowledge of the real world 
 was possible independent of experience Kant had 
 but to point to Euclid. There geometry as a beauti- 
 ful science stopd, that it had apodeictic certainty 
 could not be doubted, for " none but a fool could 
 doubt its validity or deny its objective reference." 
 An a priori synthetic body of knowledge is therefore 
 possible: but how is it possible? That was Kant's 
 
138 THE VALIDITY OF EUCLID 
 
 problem. If, as he thought, Euchd has apodeictic 
 certainty then space must be a priori and purely sub- 
 jective; conversely, if space is subjective geometry 
 must have apodeictic certainty. Hence his argument 
 assumes a twofold form. On the one hand geometry 
 exists as a science and is known to have apodeictic 
 certainty; therefore it follows that space is a priori 
 and subjective. On the other hand, it follows from 
 considerations which are independent of geometry 
 that space is subjective and a priori, therefore geom- 
 etry must have apodeictic certainty. 
 
 We now purpose to show that, in the light of our 
 previous discussion, the first of these two arguments, 
 taken by itself, falls short of the mark, and makes 
 against Euclid ; and that the second is just as valid 
 for non-Euclidean space-forms as it is for Euclid's 
 and applies in so far as it can be accepted at all only 
 to what these space-forms possess in common. The 
 position which we shall thus aim to establish is that 
 the parallel postulate is not a priori but empirical in 
 character. , 
 
 Kant's first argument, in so far as it is distinct 
 from the second, infers from the mere existence of 
 Euclid the a priori and subjective character of Eu- 
 clidean space. Prior to the introduction of non- 
 Euclidean geometry this argument seemed of course 
 to be a forcible one, but now the case is quite dififer- 
 ent. For if we do not beg the whole question at the 
 outset by dogmatically asserting that Euclid has 
 
THE VALIDITY OF EUCLID 
 
 U9 
 
 intuitive certainty while non-Euclid has not, and if 
 we also accept the position already established that 
 these geometries are otherwise on a par with each 
 other, it must be admitted that Kant's argument 
 applies as well to the modern systems as it does to 
 Euclid. But when the argument is thus extended 
 its invalidity becomes at once apparent. For if we 
 assume three a priori space forms corresponding to 
 these three geometries, and we dare not now make 
 this assumption with regard to one without also 
 making it with regard to the others, they enter into 
 a hopeless conflict with each other. For if it must 
 be universally and necessarily true that the sum of 
 the angles of a triangle is exactly equal to two right 
 angles it cannot also be, at the same time, uni- 
 versally and necessarily true that this same sum is 
 also greater and less than two right angles accord- 
 ing as we happen to be speaking of Euclidean, Lo- 
 batchewskian, or Riemannian space. 
 
 The mere existence of Euclidean geometry is 
 therefore not in itself sufificient to prove the possi- 
 bility of an a priori intuition of Euclidean space. 
 Logically this is a distinction which can no more be 
 claimed for Euclidean than for non-Euclidean 
 geometry. We now see that it- cannot be claimed 
 for both without contradiction, therefore it cannot 
 be claimed for either, and this division of Kant's 
 argument falls to the ground. 
 
 But in proving the invalidity of this one argu- 
 
140 THE VALIDITY OF EUCLID 
 
 ment we have not done with Kant. In the fore- 
 going refutation we have made use of the principle 
 of contradiction to establish our position ; but Kant 
 denies that this principle applies. He claims that 
 we can frame an intuition of Euclidean space and 
 that a priori. On page 20, for instance, of Muller's 
 Translation of the Critique of Pure Reason, Kant 
 affirms that all geometrical principles as, for ex- 
 ample, " that in every triangle two sides are to- 
 gether greater than the third side," are never to he 
 derived from general concepts of side and triangle 
 but from an intuition, and that a priori, with apo- 
 deictic certainty. Kant here sets up the claim that 
 geometrical reasoning is not fundamentally a mere 
 matter of logical consistency, but by virtue of our 
 supposed intuition of space, it is synthetic and can- 
 not, though a priori, be upheld by the principle of 
 contradiction alone. 
 
 Admitting the soundness of this claim for the 
 present, let us examine the arguments which Kant 
 advances to support it. These are embodied in his 
 general doctrine of the a priori synthetic nature of 
 geometrical judgments and also in the five argu- 
 ments given on pages 18-20 of Muller's translation. 
 The latter may be omitted here with the simple 
 remark that if admitted at their face value they 
 prove nothing peculiar to the special nature of 
 Euclidean space. When taken into full confidence 
 they only establish the necessity of space as a form 
 
THE VALIDITY OF EUCLID 141 
 
 of externality, a sort of differentiating principle 
 whose special nature is left undetermined. For our 
 purpose then we need only consider the argument 
 from the a priori synthetic nature of geometrical 
 judgments. Kant held that these judgments are 
 not deducible from logic; their contradictories are 
 not self-contradictory. They combine subjects with 
 predicates which cannot be shown by logic to have 
 any connection whatever, and yet these judgments 
 have apodeictic certainty. They are apodeictic not 
 simply because we have a subjective conviction that 
 they are, but because witliout them experience itself 
 would be impossible. Let us examine this claim. 
 
 Since Kant's day attention has been called to the 
 fact that any judgment may be either analytic or 
 synthetic according to the point of view from which 
 we consider it.^ While, therefore, this distinction 
 so sharply drawn and so greatly emphasized by 
 Kant has a logical value as marking predominant 
 aspects rather than exclusive characters in the classi- 
 fication of judgments it is really without value for 
 a theory of knowledge. For if every judgment is 
 analytic in one aspect, at least, the principle of con- 
 tradiction may be applied to all judgments without 
 exception, and our previous argument hold-. 
 
 To make our meaning clear let us take Kant's 
 own example of a synthetic judgment : " All bodies 
 
 « Consult Bosanquet's Logic. Vol. I., pp. Q7 to 103 
 
142 THE VALIDITY OF EUCLID 
 
 are heavy." This judgment he claims is purely 
 synthetic, but is it not also analytic? If the subject 
 signifies one's developed, every-day "motion of body, 
 the judgment is obviously analytic, for weight is as 
 much an inseparable attribute of this notion of body 
 as is extension itself. If then this judgment is to 
 be regarded as synthetic we must go back to the 
 origin of the concept body and ask how it was that 
 the attribute weight came to be joined to it. If 
 one had no muscular sense, which alone acquaints 
 one with facts of resistance, his conception of body 
 would never involve the attribute weight; but if he 
 came somehow to be suddenly possessed of this 
 sense and through actual experience with bodies 
 became conscious of weight, the judgment, " All 
 bodies are heavy," would for him be synthetic, for 
 it adds a new predicate to his concept body. 
 
 But on the other hand let us suppose a being born 
 into the world in possession of all the special senses 
 except sight and touch. For such an hypothetical 
 being the judgment " all bodies are heavy " would 
 certainly be analytic for weight must be for him an 
 inseparable attribute of body. If now we restore 
 the lost' senses of sight and touch and allow him to 
 see and handle objects until through experience he 
 is able to form the judgment which Kant calls an- 
 alytic, " All bodies are extended," it is plain that 
 for him such a judgment would appear to be syn- 
 
THE VALlDnV OF EUCLID 143 
 
 thetic, for as before it adds in the predicate some- 
 thing not previously found to be in the subject. 
 
 But it may be objected that Kant's distinction 
 between analytic and synthetic judgments was not 
 based upon the origin, but upon the nature, of the 
 connection between subject and predicate. In reply 
 we need to discriminate between the essential nature 
 of the judgment and its symbolic expression. Fail- 
 ure to distinguish the judgment itself from one's 
 mode of apprehending it often occasions the error 
 of regarding as purely synthetic a judgment which 
 is in fact analytic. If we but attend to the nature 
 of the judgment itself regardless of its external 
 expression and how we came to know it. the rela- 
 tion between subject and predicate is seen to be an 
 identical one. This we believe to be true of geo- 
 metrical judgments in general;^ the principle of 
 identity rules throughout. This becomes evident 
 when we consider the way in which such judgments 
 are actually extended. This extension does not in- 
 volve d necessary synthesis ; nor does it depend upon 
 the universality and necessity of space as an a priori 
 form of sense-intuition, but rather upon the iden- 
 tity and homogeneity of space as an abstract con- 
 ception. It is therefore analytic and follows as a 
 natural consequence from this homogeneous nature 
 of space. All the judgments of physics are hypo- 
 
 7 Consult Lacld's A Thcnr\ of Reality, pp. 3.^1 ff 
 
144 THE VALIDITY OF EUCLID 
 
 thetically a priori, that is to say, the connection be- 
 tween subject and predicate is strictly necessary, 
 provided we may be allowed to assume that the sub- 
 ject remains always identically the same. But in 
 this science the subject as a rule does not meet this 
 requirement, for as a result of further investigations 
 it is liable to change. In space, however, we have 
 something which is homogeneous and identical 
 always and throughout; so that in it, because of 
 this peculiar character, such a thing as difference 
 or change cannot occur. Consequently all geometri- 
 cal judgments, which do but exploit the nature 
 of space, flow directly from this concept by the 
 principle of identity. But this identity or homo- 
 geneity of space as we have already shown is not an 
 intuition but a conceptual property. Geometrical 
 judgments are dependent on the nature of the space 
 concept, and simply define its content. These judg- 
 ments are therefore analytic in nature and conse- 
 quently, here as elsewhere, the principle of contra- 
 diction may be applied. 
 
 Euclid's judgments, therefore, are not something 
 sui generis, nor is this particular space-form proved 
 by Kant to be an intuitive necessity presupposed 
 in the very possibility of sense-experience; nor is 
 Euclidean reasoning possessed of any peculiar cer- 
 tainty that can be denied to non-Euclidean systems. 
 Geometrical certainty of any sort, Euclidean or non- 
 Euclidean, is but the certainty with which any con- 
 
THE VALIDITY OF EUCLID 145 
 
 elusions follow from non-contradictory premises. 
 It is logical certainty only, and in each case flows 
 directly from definition. The certainty with which 
 the sum of the angles of any triangle may be as- 
 serted to equal two right angles in Euclidean geom- 
 etry is exactly the same sort of certainty as that by 
 which it may be shown to be more or less than two 
 right angles in the other two geometries. In each 
 case it is but the certainty of intrinsic consistency. 
 Kant has certainly failed to prove the a priori 
 necessity of Euclidean space. Therefore the objec- 
 tion often raised that we have no intuition of non- 
 Euclidean space-forms comes without force against 
 these systems until some one has shown that Euclid 
 is in any special sense intuitively certain. 
 / Our conclusion therefore is that the parallel pos-j 
 Ltulate is not a subjective necessity ; it is not essential 
 to the constitution of man's intellectual nature in 
 the strict sense in w^hich Kant claimed it to be so. 
 There remains nothing, so far as we can see. on 
 which such a claim may be grounded except the 
 mere subjective conviction that it must be so; ob- 
 viously this conviction cannot count for much in 
 the case of one who simply does not feel it. Fur- 
 thermore any such extreme doctrine of the exclu- 
 sive subjectivity of space defeats its own ends.* 
 For if space is thus merely a private form of intu 
 
 8 Consult Professor Ladd's A Theory of Reality: the chap- 
 ter on "Space and Motion," especially pages 235-240. 
 
146 THE VALIDITY OF EUCLID 
 
 ition, having no correlate in objective reality, it is 
 certainly possible that other beings may have other 
 space-forms without suspecting any difference in 
 the world which they arrange under them. There- 
 fore Euclid's validity, instead of being shown in 
 this way to be a necessity, could only be estab- 
 lished by an empirical investigation of the nature 
 of the actual forms of space-intuition of a large 
 number of individuals.*^ 
 
 Let us go back now to answer the question which 
 led to this long consideration. The position is now 
 certainly justified that the comparative simplicity 
 and convenience of Euclid is not due to anything 
 essential to intelligence as such, nor is the fact that 
 Euclid is more amenable to algebraic interpretation 
 than his modern rivals to be regarded as a necessity 
 of any rational nature. That certain algebraic for- 
 mulae possess greater simplicity than others and that 
 ^ Euclid as a system builds itself up in the mind more 
 easily and readily from certain points of starting 
 than from others are facts which seem native to our 
 minds in a much truer sense than the ordinary facts 
 of sense-perception ; but that these facts are a priori 
 in the sense of being necessary presuppositions of 
 rationality itself and therefore essential to the very 
 possibility of any world whatever must certainly be 
 denied. For from this purely rational point of view 
 
 9 Compare Lotze's Metaphysics, Book II., Chap. 2. 
 
THE lALlDlTV 01- EUCLID 
 
 H7 
 
 Strictly taken there exists no reason why one point 
 of setting out should be regarded as better or more 
 convenient than another. The rational goal in each 
 case is simply to construct the entire system, and 
 when a sufficient number of conditions are postu- 
 lated to define completely what is logically involved, 
 no room for a preference is left. No difference 
 being admitted in the nature of the entities em- 
 ployed none could be allowed in the relations be- 
 tween them, and it ought to be just as simple and 
 convenient to trace these relations from one point 
 of starting as from another. 
 
 It is not meant, of course, that no more than this 
 is essential to a world as actually known, nor is it 
 maintained that a merely formal world could really 
 exist or be truly known if it did exist. That a 
 world of real beings each with peculiar forms and 
 laws of its own must be postulated in the interests 
 of a genuine cognition will be readily granted ; that 
 they are is a presupposition necessary to the possi- 
 bility of experience, but precisely z^hat they are in 
 particular, is a matter which experience itself must 
 reveal. Hence just what algebraic formulae, what 
 geometry, and what points of starting in any par- 
 ticular geometry, are really most convenient and 
 simple cannot be foretold by any mere analysis of 
 what it means to be rational. These are truths 
 which are essentially a postcriorc in their character: 
 they come from experience, individual and racial. 
 
148 THE VALIDITY OF EUCLID 
 
 and hence to experience itself the ultimate appeal 
 must be made to tell what they are. 
 
 We may certainly conclude then that the parallel 
 postulate and the corresponding assumptions of the 
 other geometries are competitive possibilities aris- 
 ing out of the difference between the exactitude 
 demanded by mathematics and the approximate pre- 
 cision which the limitations of sense-perception in- 
 evitably impose. The question as to which 
 possibility, if any one of the group, exactly accords 
 with reality, is one which can be answered only 
 when the nature of reality is itself more- accurately 
 known. The validity of Euclid is therefore, at bot- 
 tom, an empirical matter which can be decided, so 
 far as decision is possible, only by a more refined 
 appeal to actual sense-experience. 
 
 The scientific question as to how this appeal can 
 best be made has frequently come to the surface in 
 what has been said, but a fuller reply to it must now 
 be attempted. And first let account be taken of 
 the peculiar difficulties which beset even the best 
 evidence we have concerning the nature of actual 
 space. The Euclidean conception is approximately 
 correct. This we know. The facts thus far do not 
 necessitate the least departure from it, and yet it 
 is possible, though not very probable, that such 
 anomalous facts may at length be found. But if 
 they should be found, there is every consideration 
 to dispose one to interpret them, if possible, in har- 
 
THE VALIDITY Of EUCLID 149 
 
 mony with Euclid. In the actual nieasurenients 
 of astronomical distances, and in the formulation 
 of the laws of optics, physics and astronomy the 
 parallel postulate has been confidently accepted. It 
 is furthermore a custom with physicists resulting 
 from long and thoroughly tried practical experi- 
 ence, to adhere steadfastly to the simplest assump- 
 tions until the facts have so complicated them as 
 to force their rejection or their modification. Now 
 of all the concepts which any physical inquiry must 
 employ, the simplest pf course are those of space 
 and time. We can imagine any sort of figure or 
 complication of figures to be realized in Euclidean 
 space without doing violence to this conception. 
 Other concepts are more narrowly restricted by the 
 nature of the facts, A perfect gas, or a perfectly 
 elastic body, for example, does not exist. The 
 physicist is perfectly conscious that these are 
 fictions which conform only approximately and by 
 arbitrary simplifications to the actual facts. There 
 are deviations which cannot be removed. Hence. 
 among the conceptions of physics there is a recog- 
 nized scale of perfection : and when facts are dis- 
 covered that render a modification of some sort 
 necessary it is naturally the less perfect conceptions 
 which must suffer the change. Consefpiently if 
 facts should be discovered which seemingly go 
 against Euclid, it would be simpler and easier to 
 maintain that the laws of physics, astronomy or 
 
150 THE VALIDITY OF EUCLID 
 
 optics are slightly incorrect than to -sacrifice Euclid 
 and reorganize the whole of physical science upon 
 the basis of a different geometry. 
 
 It is therefore obvious that mere measurements 
 of stellar parallax, however refined, can never be 
 regarded as conclusive evidence either for or against 
 Euclid. Under present instrumental limitations 
 and since the base line cannot in any event be greater 
 than the diameter of the earth's orbit, the probable 
 error of the best astronomical observations admits 
 the possibility of a departure of a triangle's angle 
 sum from two right angles, amounting to ten de- 
 grees or more in the case of stellar triangles whose 
 sides all equal the distance from the earth to Sirius. 
 Such a measurement, assuming as it does that rays 
 of light from the most distant stars are Euclidean 
 straight lines, is far from proving that K is infinite. 
 It only shows that the actual space constant is very 
 large as compared even with stellar distances. 
 
 For short measurements on the earth rays of 
 light are perhaps the best examples we have of 
 actual straight lines, and we naturally assume that 
 they maintain this characteristic even in the most 
 remote regions of space. We must make some 
 assumption regarding the course of light in these 
 far off regions, otherwise astronomical measure- 
 ments are altogether impossible. In doing so we 
 are obliged to infer that what holds true of light 
 here also holds true of it there, but we have seen 
 
THE VALIDITY OF EUCLID 
 
 J51 
 
 that within the hmits of fact which must serve as 
 a basis for this inference Euclidean and non-Euclid- 
 ean straight lines are not to be distinguished from 
 each other. Hence with K finite but very large 
 as compared with terrestrial measurements and 
 with different laws of optics the same facts might 
 be accounted for in a manner quite as simple as they 
 are at present. 
 ^ On the other hand, the discovery of a parallax 
 in the case of the most distant star or of a negative 
 parallax for any star cannot be regarded as disprov- 
 ing Euclid. Such anomalies, if not too numerous, 
 could be accounted for in a manner which is much 
 less expensive by making suitable changes in the 
 received laws of optics; or even by assuming a 
 slight strain in the ether itself in those particular 
 regions of space. While therefore an actual dis- 
 covery of any such facts would certainly prove 
 interesting and suggestive this alone could not be 
 accepted as a conclusive proof or disproof of 
 Euclid's validity; for the very conditions which 
 would render this discovery possible must be found 
 to involve certain assumptions which might very 
 well be withdrawn. The facts, therefore, which 
 count most either for or against Euclid must be 
 found if possible within the realm of direct obser- 
 vation where such assumptions as to what takes 
 place beyond this realm do not need to be made. 
 The problem then is at bottom a psychological one 
 
iSa THE VALIDITY OF EUCLID 
 
 which must be decided for the most part by experi- 
 ment. What are the sensory contributions which 
 should be taken into account in the formation of 
 the abstract conception of space? If we confine our- 
 selves to the contributions of vision alone our geom- 
 etry is not Euclidean ; it is projective, and so far as 
 its space conception is concerned is not to be dis- 
 tinguished from the double or single elliptic systems 
 of metrical geometry. But if we consider the larger 
 realm of spatial experiences in which sensations of 
 motion, direction and touch are also involved, all 
 the known facts inevitably suggest the parallel pos- 
 tulate. Whether they shall continue to do so as 
 tested by the most refined experimental analysis of 
 man's ability to discriminate under the most favor- 
 able conditions slight variations in the size of angles, 
 in the length of lines, and in the latter' s departure 
 from ideal straightness remains to be determined. 
 
 As already stated, the author has undertaken an 
 experimental investigation of this sort, at the sug- 
 gestion of Professor Ladd. Unfortunately the 
 data thus far obtained are not sufficient to justify 
 any positive statement as to what the outcome will 
 probably be. Obviously a very great number and 
 variety of experiments need to be performed by a 
 large number of individuals to obtain results that 
 can be regarded as significant. 
 
 I 
 
CONCLUSIONS 
 
 AS TO THE 
 
 NATURE OF SPACE. 
 
CHAPTER VL 
 
 RESULTING IMPLICATIONS AS TO THE NATURE OF 
 SPACE. 
 
 In the present chapter it shall be our purpose 
 to point out certain imphcations as to the nature 
 of space which seem to result from denying the 
 necessary validity of the parallel postulate and from 
 the consequent possibility of non-Euclidean geome- 
 tries. As a preparatory step in this direction it is 
 necessary to get a clear understanding of precisely 
 what is meant by non-Euclidean spaces. The 
 question therefore recurs, are these so-called spaces 
 after all anything more than ingenious logical con- 
 structs which one cannot even think of except as 
 tinite forms in Euclidean spaces? 
 
 In framing a reply to this question we must 
 guard against certain errors which almost inevit- 
 ably result from the introduction of Euclidean 
 analogies of corresponding non-Euclidean concep- 
 tions. Spherical and pseudo-spherical surfaces 
 upon which non-Euclidean straight lines arc repre- 
 sented as curves in ordinary Euclidean space of 
 three dimensions are. in reality, very diflfcrent from 
 155 
 
156 THE NATURE OF SPACE 
 
 the corresponding non-Euclidean two-dimensional 
 spaces which they are designed to represent. These 
 surfaces are not non-Euchdean spaces, although the 
 analytical treatment is the same in both cases. 
 They are merely analogies to aid the imagination. 
 Failure to understand this distinction has often led 
 to the error of regarding non-Euclidean two-dimen- 
 sional spaces as curved surfaces in tri-dimensional 
 Euclidean space, and non-Euclidean spaces of three 
 dimensions as constructs requiring a Euclidean 
 space of four dimensions for their possible realiza- 
 tion. 
 
 If this view were correct it would be very easy 
 to solve the special philosophical problem by reduc- 
 ing everything to Euclid. A non-Euclidean space 
 of any number of dimensions gets itself realized in 
 a Euclidean space of higher dimensions, non-Eu- 
 clidean planes are Euclidean curved surfaces, and 
 non-Euclidean straight lines are Euclidean curves; 
 hence all we need is a vocabulary whereby the lan- 
 guage of one system may be translated into that of 
 another, and the problem is very beautifully solved. 
 Non-Euclidean geometry thus becomes a mere play 
 upon words ; it is simply Euclid with a change of 
 names. The next step is to argue that Euclidean 
 spaces of four or more dimensions are inconceivable, 
 and consequently that non-Euclidean spaces of even 
 three dimensions are impossible. This done, the 
 refutation of non-Euclid would stand complete. 
 
^ 
 
 UNIVERSITY 
 _ or 
 
 THE NATURE OF SPACE 157 
 
 But this solution of the clifficuky would be alto- 
 gether too easily won. There seems to be an error 
 involved, and we need to discover its source. The 
 confusion springs, it seems to me, from a misappre- 
 hension of the proper meaning of space and of curva- 
 ture as applied to space. To dispel the fog we must 
 clear up the meaning of these two conceptions. 
 
 Space, properly speaking, always means a total- 
 ity. There is considerable difference between a Eu- 
 clidean tw^o-dimensional space, for example, and the 
 Euclidean plane as ordinarily conceived. Their in- 
 ternal relations are exactly the same ; their analytical 
 treatment is also the same. But the plane has in 
 addition to these internal relations certain external 
 relations ; it is two-sided, for instance, and has posi- 
 tion. A two-dimensional space, on the other hand. 
 is a totality, it contains everything within itself. It 
 therefore has no position and is not two-sided. 
 
 In the ordinary study of Euclidean surfaces we 
 frequently make use of certain points and figures 
 wdiich are not on these surfaces at all. We talk of 
 normals, of tangent lines and planes; we say that 
 surfaces may be flexed, shoved about, turned over, 
 or displaced in any direction or into any position. 
 In the proof of equality by the principle of congruent 
 superposition, it is i^ermitted to take a figure fn^n off 
 the plane into a third dimension, obvert it and bring 
 it back again into congruetice with another figure in 
 the plane. In a two-dimensional space none of the^e 
 
158 THE NATURE OF SPACE 
 
 things are possible ; space, as we have said, must be 
 complete in itself and independent of everything else. 
 But the self-dependence of any space can only be 
 guaranteed when it is shown that measurements can 
 be effected and a complete geometry constructed 
 without leaving it at all or even appealing to any- 
 thing external to it. In other words, it must be pos- 
 sible for intelligent beings inhabiting the space in 
 question, capable of moving about in it and con- 
 scious of nothing at all outside, to construct a geome- 
 try true for any and every part of it. 
 
 As we have seen, a Euclidean plane may be 
 wrapped about a cylinder or a cone, it may be trans- 
 formed in a thousand ways so far as its relations to 
 a third dimension are concerned, it may even be 
 v^^added into a confused mass and yet so long as no 
 straining or distortion of its internal relations occurs 
 its own proper geometry remains unchanged. Hence 
 considered as a two-dimensional space it is the same 
 throughout. For intelligent beings whom we have 
 supposed to inhabit it, who cannot leave it and who 
 know nothing external to it, it is just the same space ; 
 its metrical properties have not changed, all its fig- 
 ures remain the same, and its straight lines are al- 
 ways and everywhere the same straight lines. 
 
 That the construction of such a geometry is possi- 
 ble must be admitted when Riemann's formula for 
 ds comes to be clearly understood. The simple con- 
 ditions which we have imposed, that the curvature of 
 
THE NATURE OE SPACE 159 
 
 the space in question shall be constant and that its 
 internal relations shall remain intact, admit the ex- 
 istence of equal spatial quantities in different places, 
 and this is all that Riemann's arc formula requires. 
 By selecting co-ordinates which shall have meaning 
 in the space with whicii we are dealing, and this 
 can always be done, we shall have all the conditions 
 necessary to the analytical construction of the re- 
 quired geometry. 
 
 The same truth obviously holds for non-Euclid- 
 ean spaces of two dimensions, for in these also the 
 necessary conditions of free mobility and of un- 
 changed internal relations are fully supplied. Two- 
 dimensional manifolds, both Euclidean and non- 
 Euclidean, considered individually as self-dependent 
 totalities, are therefore possible, and we may also 
 have many varieties of these provided we think of 
 them as unbounded totalities without external re- 
 lations. We need, then, simply to keep in mind the 
 fact that a surface in a space of any variety of curva- 
 ture is quite a different thing from a two-dimen- 
 sional space of that same variety. The surface 
 always requires a third dimension; the two-dimen- 
 sional space does not. 
 
 In the case of tri-dimensional spaces the same prin- 
 ciple holds good. Here, too, more than one variety 
 of space is possible ; but in this case the problem for 
 imagination is an impossible one. As tri-dinicn 
 sional beings, we may easily represent to ourselves 
 
i6o THE MATURE Of SPACE 
 
 ordinary surfaces as two-dimensional spaces for ra- 
 tional beings supposed to inhabit them and to be con- 
 scious only of their internal relations. We can then 
 see from the standpoint of our third dimension the 
 numerous modifications which can be made to take 
 place in the external relations of these surfaces with- 
 out in any sense disturbing their internal relations. 
 Under the conditions imposed the two-dimensional 
 beings inhabiting these spaces, obviously, could not 
 become aware of any of these external changes ; but 
 supposing them to be endowed with intellects like 
 our own, some Lobatchewsky among them would 
 sooner or later work out two-dimensional geometries 
 which would not hold for the particular space inhab- 
 ited by him. Three-dimensional geometries might 
 also be constructed by this same two-dimensional 
 genius. And yet such a being could not have the 
 necessary sensory experience to represent to himself 
 what these spaces really are as judged by their ex- 
 ternal relations. 
 
 As we stand externally related to these two-dimen- 
 sional beings, so it is conceivable a four-dimensional 
 being might stand related to us. For such a being 
 various modifications in the external relations of the 
 space-world known to us and of the objects in it 
 might occur without our being at all conscious of 
 the change. 
 
 We cannot in any a priori fashion dogmatically 
 denv the existence of a four-dimensional space-world 
 
THE NATURE OF STAGE 16 1 
 
 any more than our two-dimensional beings could 
 deny that our world exists. That such a world is 
 inconceivable in the sense that it is impossible for us 
 to imagine it must be admitted on the ground of 
 actual experience. We have not had the peculiar 
 sense-experience in a fourth dimension which could 
 render any sort of picture of it possible. We can. 
 of course, say with confidence that our universe as 
 we know it and every known agency in it is confined 
 by some law of its being or at least of our knowing 
 to three dimensions. Our space, so far as actual 
 sense-experience goes, is undoubtedly tri-dimen- 
 sional. But we know this not as an a priori neces- 
 sity, as Kant contended, but as an empirical fact. 
 We cannot, therefore, rightly affirm that spaces of 
 higher dimensions than ours are objectively impos- 
 sible or even inconceivable in the sense that they are 
 not rationally construablcv^ 
 
 Did a four-dimensional world exist externally re- 
 lated to our own as ours, in turn, stands related to 
 the two-dimensional world just considered, then, as 
 we have said, it would be possible for an inhabitant 
 of that world to witness a vast number of external 
 modifications of our world without our being at all 
 conscious of any change, for no intrinsic trans- 
 formation would actually occur. The known world 
 would remain to us the same spatially unchanging 
 reality that it now appears to be. 
 
 But if we deny the actuality of a fourth dimen 
 
162 THE NATURE OF SPACE 
 
 sion, the modifications which we have supposed to 
 take place in that dimension, of course, could not 
 occur. In that event would more than one variety 
 of tri-dimensional space be possible? This question 
 is a crucial one. Its answer determines whether 
 or not non-Euclidean geometry is to be regarded as 
 having any peculiar philosophical significance. The 
 world, as our actual experience reveals it, is certainly 
 tri-dimensional ; judged by the same standard, it is 
 also Euclidean. If, then, only one variety of tri- 
 dimensional space is possible, if non-Euclidean tri- 
 dimensional geometry really demands a fourth di- 
 mension, the so-called non-Euclidean spaces are in 
 reality not spaces at all, for they are not self-depend- 
 ent totalities. It is not, then, a question as to 
 whether non-Euclidean geometries are possible, but 
 a question as to whether non-Euclidean tri-dimen- 
 sional spaces are possible. It is, of course, possible 
 to construct such geometries by making use of the 
 idea of a fourth dimension, just as we ordinarily 
 build up our plane geometry by frequently referring 
 to figures which are only possible in a third dimen- 
 sion ; but this, of course, is very dififerent from estab- 
 lishing the possibility of non-Euclidean tri-dimen- 
 sional spaces. 
 
 The question, then, simply reduces to this : Are 
 tri-dimensional space-worlds rationally possible 
 whose internal relations considered as totalities are 
 essentially different from each other? And it is an- 
 
THE NATURE OF SPACE 163 
 
 swered by showing that the geometries of such spaces 
 can be constructed without appeahng to a fourth 
 dimension. This can be done. As in the case of 
 two-dimensional spaces, we have here also all the 
 conditions necessary to render such geometries pos- 
 sible. Indeed, the most interesting and significant 
 feature of non-Euclidean solid geometries lies in the 
 fact that they are just as independent of a fourth 
 dimension as is, Euclid itself. There are, to be sure, 
 certain facts in all these geometries that make us 
 wish sometimes for a fourth dimension and the 
 power of moving into it. but they do not necessarily 
 imply this dimension. The simple principle of con- 
 gruence fails, for example, if we attempt to apply it 
 directly in proving the equality of two Euclidean 
 pyramids wiiose corresponding parts are mutually 
 equal but arranged in reverse order. As we have 
 said, the analogous theorem in plane geometry is 
 proved by obverting one of the triangles in the third 
 dimension. Were there a fourth dimension and had 
 we the power of moving into it, it is conceivable 
 that this might also be done for the pyramids. What 
 would happen is simply this : By obverting one of 
 the pyramids in the fourth dimension and then re- 
 turning it to its own tri-dimensional world, its rela- 
 tions to the other objects af this world are changed 
 in a way that is wholly impossible so long as we 
 confine it to three dimensions. But the internal rela- 
 tions of the pyramid itself, as in the observed case of 
 
i64 THE NATURE OF SPACE 
 
 the triangle, remain entirely unaltered. The self- 
 identity of the figure is retained. But as we have 
 said, these facts cannot be regarded as implying the 
 logical dependence of Euclid, or of non-Euclid, upon 
 a fourth dimension. 
 
 Granting, then, as we must, the possibility of in- 
 ternally different tri-dimensional spaces, we need to 
 inquire briefly into their mutual relations and what 
 they imply. 
 
 Their differences, as already indicated, are chiefly 
 metrical. Hence in all of them those qualitative 
 characteristics which are presupposed in measure- 
 ment are implied. It is these characteristics which 
 render possible in each individual case those peculiar 
 internal relations which constitute the essential na- 
 ture of space as a totality. Whatever intrinsic qual- 
 ities are necessary to all measurement, as such, these 
 different spaces must possess in common. Just 
 what their common elements are we shall postpone 
 for the moment and consider here their differences 
 merely. These differences are bound up in the 
 modifications of meaning which are possible in the 
 peculiar conception of curvature to which we have 
 often referred. 
 
 As already pointed out,^ curvature as applied to 
 space represents an intrinsic property of the particu- 
 lar space in question and does not necessarily imply 
 
 Chap. II., p. 33' -3^ ff-^ I'j 
 
I THE NATLRt Ul- SPACE 165 
 
 a higher dimension. Measurement ni the exact 
 sense required by geometry is not possible unless the 
 space in which it occurs possesses the property of 
 perfect homogeneity. It must permit the free mo- 
 biHty of a rigid standard, and its metrical peculiari- 
 ties, whatever they are, must be true of it as a whole. 
 This is what is meant by saying that its curvature is 
 constant. 
 
 If. then, geometrical spaces of the same number 
 of dimensions may be of different varieties, it follows 
 that their space-constants must also be different, and 
 it is this difference that we need to examine. It is 
 almost universally regarded as a quantitative one. 
 Men talk as though space constants might be ar- 
 ranged as a series of quantities whose ratios could 
 be exactly determined by the use of a common stand- 
 ard of measurement. This is wrong. The law of 
 exact homogeneity forbids it. Strictly speaking, 
 quantitative differences can only exist in the same 
 space, for here alone the exact measurement of all 
 figures by the same rigid standard is possible. Dif- 
 ferent spaces of the same number of dimensions do 
 not admit of the free mobility throughout their whole 
 extent of the same rigid body ; if they did they 
 could not be metrically distinguished. But. as we 
 have said, it is in metrical properties that Euclidean 
 and non-Euclidean space-conceptions are different ; 
 projectively treated these differences disa|)pear. 
 Space-constants must therefore be regardefl as stand- 
 
i66 THE NATURE OF SPACE 
 
 ing related to each other in a manner similar to what 
 obtains in different shades of the same color. They 
 are qualitatively different. They have much in com- 
 mon, but are at the same time so unlike as to make 
 exact geometrical measurement impossible. There 
 exists no common standard ; hence, strictly speaking, 
 we cannot say that any space-constant is greater or 
 less than another. Logical difficulties of a similar 
 nature obviously follow the attempt to regard the 
 curvature of any geometrical space as a variable.^ 
 This involves the assumption of an absolute position 
 and consequently denies the relativity and homogene- 
 ity which strict geometrical measurement always 
 requires. Such a variable curvature is represented 
 for example by the surface of an tgg, in which, gen- 
 erally speaking, it is not possible to make one piece 
 fit upon another without distortion. Regarded as a 
 space, it is not homogeneous. 
 
 Here again we are forced to observe the necessary 
 disparity between the facts of experience and the ab- 
 
 2 It is not necessary to enter into an extended argument in 
 support of this position. Tlie curvature of space might be 
 regarded as variable if the law of its change were known. 
 This is practically what is involved in Cayley's idea of con- 
 structing non-Euclidean geometries in Euclidean spaces by 
 simply modifying the notion of distance. It is also suggested 
 by Erdmann (^Die Axiome der Geometric, Leipzig, 1877). 
 But while such a law of change may be assumed we are left 
 absolutely without any means of determining what it is or even 
 of detecting its presence. Diflferences of magnitude where 
 comparison can not reveal them are manifestly at variance 
 with the very notion of magnitude. 
 
THE NATURE OF SPACE 167 
 
 solute exactitude demanded by geometry. Geomet- 
 rical measurement is of course impossible without 
 the assumption of absolute homogeneity; but prac- 
 tical measurement, the only kind possible in the world 
 as we know it, can be performed. It is also clear, in 
 support of our general position, that the question 
 whether the actual space-parameter of our world, 
 when assumed to be constant, is Euclidean or non- 
 Euclidean, cannot be settled by any mere appeal to 
 a rigorous geometry. For, as we have just seen, 
 the metrical comparison of different space-parame- 
 ters upon such a basis cannot occur. We cannot 
 travel, so to speak, with our metrical standard from 
 one space to another. 
 
 We must remember, however, that spatial homo- 
 geneity is merely a conception ; as to its nature and 
 origin it does not differ from other conceptions. It- 
 is simply an idealization of certain familiar facts of 
 our spatial experience. It is the facts that determine 
 the meaning of homogeneity, and not homogeneity 
 that determines the facts. Geometry must fit experi- 
 ence, not experience geometry. 
 
 To reverse this order is a crime against knowledge 
 which has been too often committed. The homo- 
 geneity of space is in fact a complex idea and admits 
 of being variously interpreted. This is shown by 
 the existence of different metrical geometries in 
 which homogeneity is so conceived as not to admit 
 of the free mol)ility of the same rigid body in the 
 
i68 THE NATURE OF SPACE 
 
 different cases. It is not, then, a question as to 
 which of these conceptions is to be accepted as the 
 absolute truth, but as to which one accords, and will 
 continue to accord, best with man's actual experi- 
 ence. Experience itself must decide this matter, and 
 even its decision must always be regarded as only 
 approximate. Homogeneity, then, whatever par- 
 ticular view we take of it, is a different conception 
 from mere logical anyness with which it is often con- 
 fused. It bears, in some sense, the stamp of actual 
 experience as is seen in the fact that the geometrical 
 figures which it admits appear to have a distinct 
 character and reality of their own. If one does not 
 believe this let him take any material body and en- 
 deavor to cut from it a Euclidean regular poly- 
 hedron of seven plane faces and his doubts will 
 vanish. But why is this figure impossible? It is 
 certainly not because of any obstacles which the ma- 
 terial itself can offer, for this may be sawed through 
 in one direction as easily as another. The truth is 
 the lines, angles, and planes enter into the structure 
 of such geometrical figures with all the meaning 
 which our actual tri-dimensional experience has 
 given them. Indeed, it ought now to be perfectly 
 clear that lines and planes as ordinarily conceived 
 are not, in reality, one and two-dimensional objects 
 respectively. They are this and more. To give 
 them their full meaning they require a spatial experi- 
 ence of a tri-dimensional order. The straight' line 
 
THE NATL' HE UF SPACE Kxj 
 
 as it exists in the average man's consciousness is per- 
 haps best defined as a Hne which looks the same from 
 every point not in it, but this definition clearly impli- 
 cates three dimensions. If, however, we define it as 
 the shortest distance between two points, and it is 
 then pointed out that arcs of great circles on a sphere 
 meet this requirement, we are not satisfied; for this 
 is evidently not what we mean. It is the chord, we 
 say, not the arc, that stands as a true representative 
 of our idea of the straight line. Thus again we 
 must enter the third dimension before we can decide 
 the matter. If an ordinary plane be conceived as 
 wrapped about a cone, it remains the same two- 
 dimensional object that it was before, but it is for 
 us no longer a plane. To be a plane it must sustain 
 certain definite relations to a third dimension. 
 
 It is therefore obvious that the peculiar form of 
 our tri-dimensional experience has given us these 
 conceptions of the straight line and the plane. The 
 moment we begin to deal with definite geometrical 
 figures of any sort we are considering combinations 
 of relations whose peculiar character as combinations 
 is determined by certain conceptions which are thus 
 grounded in experience. They are certainly not ra- 
 tional necessities. Homogeneity of some sort is an 
 essential characteristic of any space-conception which 
 will admit of the exact quantitative determinations 
 of metrical geometry. The particular combinations 
 of this conception and those assumptions which are 
 
170 THE NATURE OF SPACE 
 
 needed in each case to define accurately and com- 
 pletely the peculiar foundations of the different 
 geometries must, of course, be regarded as necessary 
 to the particular space-conceptions upon which these 
 geometries are founded; but necessities of thought, 
 presupposed in the very possibility of any spatial ex- 
 perience whatever or even of experience as we actu- 
 ally have it, they certainly are not. 
 
 Underneath these conceptions and implicit in them 
 the true category of space is to be found. As a 
 " differentiating principle " ^ of some sort, both sub- 
 jectively and transsubjectively operative, externaliz- 
 ing and at the same time uniting a world of co-exist- 
 ing selves and things, this category is necessary to all 
 human cognition or mental representation of the 
 world of reality. As such we cannot escape it, for 
 we can think of no entity to which it does not apply. 
 It was the unavoidable presence of space in this 
 meaning of the word that seems to contradict flatly 
 much that was said at the beginning of this chapter, 
 where we represented geometrical spaces as totalities 
 and yet spoke of them as though they stood related 
 somehow to realities to which as spaces they could 
 not apply. The confusion at that stage of the dis- 
 cussion seemed necessary. We could not avoid it. 
 It is now possible to make clear what was meant. 
 The space-conceptions of geometry are not catego- 
 
 3 See Professor Ladd's admirable chapter on " Space and 
 Motion," in A Theory of Reality, New York, 1899, pp. 214-252. 
 
THE NATURE OF SPACE 
 
 171 
 
 ries, not necessities of thought or reahty ; they imply 
 the space category, it is true, but they also imply 
 more. Stamped upon them are certain marks of an 
 empirical origin which are grounded in the peculiar 
 nature of our human spatial experience. To deter- 
 mine these marks which, so far as we can see, might 
 have been different ; to separate them out and if pos- 
 sible to get directly at the peculiar nature of the cate- 
 gory itself, was the difficult problem which we there 
 had in mind. When we limit the number of dimen- 
 sions and introduce particular figures and special 
 metrical considerations we no longer deal with the 
 " pure " category of space, as it must lie implicit in 
 any rational being. 
 
 We conclude, then, that the actual characters of 
 our space experience so far as they involve tri-dimen- 
 sionality, the parallel postulate, &c.,'are empirical in 
 the same sense in which any notable feature of our 
 present human experience is empirical. As it is an 
 empirical fact that there are many different men 
 with different minds or that we can grasp in one act 
 of attention only a narrow range of facts, so it is 
 an empirical fact that the above-mentioned charac- 
 ters belong to our space-consciousness. Other be- 
 ings may have other space experiences. We our- 
 selves may some day acquire other such experiences. 
 There is no demonstrable necessity accessible 
 regarding the matter, so far as the details are con- 
 cerned. 
 
172 THE NATURE OF SPACE 
 
 The only a priori manifold at present definable in 
 Kant's sense of a priori seems to be a manifold con- 
 stituted by a totality of logical classes or distinctions 
 of any similar sort. The constitution of such a com- 
 plete system of logical entities must be implicitly 
 known to any rational being. It depends upon the 
 fundamental illative relation expressed by such state- 
 ments as " a implies b," or that " a is subsumed un- 
 der b." A world in which this relation has full sway 
 contains the " negation " of any of its own terms. 
 It contains series of subsumptions and also certain 
 combinations whereby its terms are grouped into 
 wholes. Such a system has a formal structure 
 which Kempe ^ has shown includes all the types of 
 order which appear in every sort of geometry of n 
 dimensions. Any space you please may be viewed 
 as a selection from amongst the entities of such a 
 system made by adding certain arbitrary postulates 
 to these fundamental logical ones. 
 
 The connection between this a priori logical mani- 
 fold and the empirical space of our own experience 
 lies in the fact that the space-aspect of experience 
 is the one which jnost definitely implies and is im- 
 plied by our power to co-ordinate our activities so 
 that " a leads to b leads to c," &c. It is that aspect 
 which enables us to introduce illative relations 
 
 ■* " Relation Between the Logical Theory of Classes and the 
 Geometrical Theory of Points." Proceedings of the London 
 Mathematical Society for 1890, Vol. 21. 
 
THE NATURE OF SPACE 173 
 
 among acts and systems of acts of our own (acts 
 actual and acts possible). 
 
 That this aspect of experience exists is an em- 
 pirical fact. What correlations of acts it permits 
 and how it permits them are also empirical. All the 
 details are empirical. But if it is to permit such a 
 system at all, it has to conform to the general type 
 of the illative relation and its parts viewed as co- 
 existent must be related to each other in accordance 
 with the general type of an illative relation. " No 
 actualization of the space-principle is. therefore, pos- 
 sible either from the point of view of its subjective 
 origin or of its trans-subjective applicability, unless 
 this principle itself is conceived of as the mode of 
 the action of one all-differentiating and yet all unify- 
 ing Force," •' functioning in a way that is essentially 
 law-abiding and rational. 
 
 3 Ladd's A Theory of Reality, p. 252. 
 
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