LIBRARY 
 
 OF THE 
 
 University of California. 
 
 RECEIVED BY EXCHANGE 
 
 Class 
 
 
 
 
 
\ 
 
 THE FOUNDATIONS 
 
 EUCLIDIAN GEOMETRY 
 
 AS VIEWED FEOM THE 
 
 STANDPOINT OF KINEMATICS 
 
 DISSEETATIOK 
 
 SUBMITTED TO THE 
 
 BOARD OF UNIVERSITY STUDIES 
 
 OF THE 
 
 JOHNS HOPKINS UNIVERSITY 
 
 IN CONFORMITY WITH THE REQUIREMENTS FOR THE DEGREE OF 
 
 DOCTOR OF PHILOSOPHY 
 
 ISRAEL EUCLID RABINOVITCH 
 
 April 29, 1901 
 
 NEW YORK: 
 PUBLISHED BY THE AUTHOR 
 
 1903 
 
THE FOUNDATIONS 
 
 OF THE 
 
 EUCLIDIAN GEOMETRY 
 
 AS VIEWED FBOM THE 
 
 STANDPOINT OF KINEMATICS 
 
 ERRATA. 
 
 Page X, next to bottom line read " Bolyai '' instead of Bolyi. 
 Page xi, under Lobatchevski, close the quotation with ". 
 Page 26, last line, read " Matematiche " instead of Mathe- 
 matiche. 
 
 ^ET 
 
 ISRAEL EUCLID RABINOVITCH 
 
 Apeil 29, 1901 
 
 OF THE 
 
 UNIVERSITY 
 
 OF 
 
 PUBLISHED BY THE AUTHOR 
 1903 
 
THE FOUNDATIONS 
 
 OP THE 
 
 EUCLIDIAN GEOMETRY 
 
 AS VIEWED FEOM THE 
 
 STANDPOINT OF KINEMATICS 
 
 BISSERTATION 
 
 SUBMITTKD TO THE 
 
 BOARD OF UNIVERSITY STUDIES 
 
 OF THE 
 
 JOHNS HOPKINS UNIVERSITY 
 
 IN CONFOBMITY WITH THE BEQUIBEMENTS FOB THE DEGBEB OF 
 
 DOCTOR OF PHILOSOPHY 
 
 BY 
 
 ISRAEL EUCLID RABINOVITCH 
 
 Apbil 29, 1901 
 
 PUBLISHED BY THE AUTHOR 
 1903 
 
^* 
 
 w^ 
 
 Copyright 1903 
 By ISRAEL EUCLID RABINOVITCH 
 
 All rights reserved 
 
 PRtSIOF 
 
 The Nn e«a PRinTme Conpaiv 
 
 LAiCASTER, Pa. 
 
SDetitcateti 
 
 TO 
 
 FABIAN FEANKLIN, Ph.D., 
 
 FORMERLY PROFESSOR OP MATHEMATICS IN THE JOHNS HOPKINS UNIVERSITY, 
 
 3rn Grateful ^cfenotoleiffment 
 
 OF BENEFITS CONFEBRED UPON THE AUTHOR DURING HIS RESIDENCE IN 
 
 BALTIMORE AS A GRADUATE STUDENT OF THE 
 
 JOHNS HOPKINS UNIVERSITY. 
 
 lU 
 
 166146 
 
Digitized by the Internet Archive 
 
 in 2008 with funding from 
 
 IVIicrosoft Corporation 
 
 http://www.archive.org/details/foundationsofeucOOrabirich 
 
PREFACE. 
 
 The present work is the result of long meditations and of an 
 earnest search for truth. A conviction that truth in mathe- 
 matics must be absolute, not admitting of any compromises, 
 and an inmost feeling that Nature is not deceiving us and that 
 She reveals Herself to us in Her true appearance, unmutilated 
 by false logic, have guided me in my endeavors to solve one of 
 the hardest mathematical problems, so intimately connected 
 with the problem of the origin of our ideas, — namely, the 
 problem of the Foundations of Geometry. These meditations 
 were finally written up, in April, 1899, at the prompting of 
 my excellent and highly esteemed friend and benefactor. Dr. 
 Fabian Franklin, formerly Professor of Mathematics in the 
 Johns Hopkins University, to whom I have availed myself of 
 the present opportunity of expressing my gratitude, by inscrib- 
 ing this work to him. 
 
 Another name I ought to mention with gratitude is that of 
 Dr. Alexander S. Chessin, Professor of Mathematics in Wash- 
 ington University, who was the first to appreciate the value of 
 this work and to talk to me unreservedly about it, and also to 
 urge me to use it as a thesis for the Ph.D. degree. 
 
 And, finally, I owe a duty of gratefulness to my distinguished 
 professor. Dr. Frank Morley, for his guidance in the work of 
 reading up the literature of the subject, for discussing with me 
 points of difficulty in the literature, and also for allowing me to 
 present some of my theorems before the conference of the 
 mathematical seminary of the university, where the general 
 discussion by the audience helped me in improving the mode 
 of presentation of these theorems. To this discussion I owe, 
 in particular, the analytical presentation of my proof that, with 
 the point as its element, space must be three-dimensional. The 
 whole of the Introduction was undertaken and carried out, dur- 
 ing the academic year of 1900-1901, at the suggestion and under 
 the direction of Professor Morley, and it is intended as a critical 
 review of some of the most important results obtained by modern 
 
VI PREFACE. 
 
 mathematicians in the subject (Riemann, Beltrami, Lie and 
 Poincar6), so that in the light of these an adequate estimate of 
 the results achieved in this Dissertation might become possible. 
 And last, but not least, my thanks are due to Professor 
 Edwin R. A. Seligman and Professor Felix Adler of Columbia 
 University for the generous interest they have taken in the 
 publication of my work in full ; and also to Mr. Isador Goetz, 
 A.B., of New York City, for his valuable assistance in the re- 
 vision of the proof-sheets, and to the gentlemen of The New 
 Era Printing Company for the care and efficiency with which 
 the printing of this volume has been executed. 
 
 130 Heney Street, New York City, 
 November, 1903. 
 
CONTENTS. 
 
 Page. 
 INTRODUCTION 1-37 
 
 An infinity of mutnally contradictory geometries postulated by 
 
 Klein, Killing, and others — and its criticism 1-5 
 
 Sophns Lie's opinion as to the possibility of an efficient system 
 
 of postulates and axioms of geometry 5-6 
 
 Lie's treatment of the Riemann-Helmholtz space-problem by 
 the theory of transformation-groups, and the axioms which 
 he assumes for this purpose. A comparison of this set of 
 axioms with those assumed in the Dissertation 6-9 
 
 The interpretation of the non-Euclidian groups of displacements 
 
 represents but a partially solved problem 10-11 
 
 Poincare's opinions of the relation of pure reason and experi- 
 ence to the formation of our geometrical notions ; his opin- 
 ions as to the number of dimensions of space 11-16 
 
 An account of Riemann's Inaugural Dissertation on the Foun- 
 dations of geometry and multiply-extended manifolds. — 
 Resume 16-20 
 
 Importance of the problem from a purely mathematical point of 
 view. Theory of proportion and similar figures impossible 
 in the non-Euclidian geometry ; the squaring of the circle 
 possible 20-21 
 
 A summary and justification of the author's views in his treat- 
 ment of the subject. The conception of space as a number 
 manifold of three dimensions — to be sought in the properties 
 of rigidity, impenetrability, and infinite divisibility of mate- 
 rial substance. The notion of distance derived from rigidity. 
 The distance-line, or straight line, must be constructed in 
 space, not in a plane, and deduced from the notion of dis- 
 tance. The plane should be constructed from the straight 
 line, and both should be demonstrated to have all properties 
 commonly attributed to them without justification. For 
 instance, — the legitimacy of the assumption, of the plane 
 being capable to move upon itself in a triply-infinite number 
 of ways, and of its coincidence with itself when its two sides 
 are interchanged — must follow from its construction 21-25 
 
 An account of Beltrami's views of the interpretability of the 
 geometry of Lobatchevski upon the pseudosphere, based upon 
 the analogy of the straight line and a geodesic upon that sur- 
 face, when bending without stretching is allowed 26-27 
 
 Distinction emphasized between the straight line, as a figure 
 capable of lying in a plane, and its notion as a geodesic in a 
 
 vii 
 
Vlll 
 
 Page. 
 plane, — and reason given why in the later case the Euclidian 
 
 postulate could not be proved 27-29 
 
 Quotations from Bianchi, " Lezioni di geometria dififerenziale," 
 concerning the futility of all attempts of proving the Eu- 
 clidian postulate, and a justification of the author's attempt 
 to do it by means of the immaterial quadrilateral, not bound 
 
 to lie in a plane 29-32 
 
 The identity of the author's postulates with those necessary for 
 defining the group of displacements in general. A comparison 
 
 with the postulates assumed by Helmholtz 32 
 
 The interpretation of the non-Euclidian geometries again, and 
 Beltrami's opinion that the non-Euclidian geometry of three 
 dimensions is interpretable only analytically. — An allusion 
 to a probable interpretation in line-space, with certain special 
 conventions as to the meaning of metrical terms 32-37 
 
 DISSERTATION. 
 
 THE FOUNDATIONS OF THE EUCLIDIAN GEOMETRY. 38-115 
 CHAPTER I. 
 (inteoductoey.) 
 Space and Its Dimensions. 
 
 Definitions 1-6 : — Geometry ; space, extension ; impenetra- 
 bility, geometrical place ; vacant space, infinite divisibility ; 
 magnitude and form ; geometrical solid, volume 38-39 
 
 Postulate 1, — Motion without distortion. Definitions 7-8 : — 
 Geometrical equality, coincidence ; non-equality. Scholium 
 and R^sum6 : — A rational justification of the definitions and 
 postulate 40-43 
 
 Definitions 9-12 and R^sum^: — Distance and contact between 
 bodies; Surface physical and geometrical; Portions and 
 equality, or coincidence, of surfaces; Homogeneous and non- 
 homogeneous surfaces 44-47 
 
 Definitions 13-15: — Line; Portions, equality of lines, homoge- 
 neous lines; The point 48-50 
 
 Resume : — Discussion of the tri-dimensionality of space 50-53 
 
 Thirteen paragraphs purporting to give an analytical discussion 
 of the dimensions of space 54-62 
 
 CHAPTER IL 
 
 The Spheee, Ciecle, Steaight Line, Angle, Teiangle, Plane, Etc. 
 
 Definition I, — Distance. Axiom 1, — Continuity in congruence. 
 Lemma 1, — A fixed body 63-64 
 
IX 
 
 Page. 
 Theorem 1, — Construction of a sphere. Cor. I, — Spherical 
 
 surface divides all space into two regions. Definition II, — 
 
 Greater and smaller distances. Cor. II, — Unique center. 
 
 Cor. Ill, — Two distinct spheres cannot have a finite portion 
 
 of surface in common 64-66 
 
 Theorem 2: — Measurement of distances from a fixed origin; 
 addition and subtraction of distances 67-72 
 
 Theorems 3-5, Operation of addition of distances obeys the 
 
 associative and the commutative laws 72-78 
 
 Scholium, Definition III and corollaries I-IX: — The homo- 
 geneous distance-line, or straight line, and its properties 78-83 
 
 Theorem 6, Definition IV: — Infinity of straight line; Angle. 
 Theorem 7, Definition V: — A right angle is one of four equal 
 ones formed by two intersecting straight lines in space 84-85 
 
 Theorem 8, Definition VI, and corollaries I-V : — The construc- 
 tion of a plane from a fixed origin; the properties of a plane 
 as an infinite homogeneous surface symmetrical with respect 
 to space. Theorem 9, — All right angles are equal 85-89 
 
 Scholiums I and II: — Every pair of crossing lines is applicable 
 to a pair of crossing rays in a plane; measurement of angles 
 by arcs of the circumference of a circle. 
 
 Theorem 10, — Adjacent supplementary, and vertical angles... 89-92 
 
 Definition VII and cor.. Theorem 11 and cor.: — Simplest 
 conditions of equality of triangles 92-93 
 
 Theorem 12, Definition VIII and cor., — A straight line, an 
 angle, and a triangle are plane figures. Scholium, — The 
 segment of a transversal of a variable angle increases with 
 the increase of the angle 93-95 
 
 Definition IX, Theorem 13 and corollaries: — A perpendicular 
 to a plane. A plane can be shifted upon itself in a triply- 
 infinite number of ways, and also turned upside down with- 
 out deformation 95-96 
 
 CHAPTER III. 
 
 The Quadbilateral, the Immaterial Quadrilateral. 
 Parallel Straight Lines. 
 
 An enumeration of the theorems of the first book of Euclid 
 which can easily be established by means of the principles 
 laid down earlier in this Dissertation 97 
 
 Theorem 14 and corollaries I-IV: — Sum of two interior angles 
 of a triangle <2 rt. /'s; Two straight lines making with a 
 third one two interior ^'s = 2rt. Z's> cannot meet; Exterior 
 angle of a triangle, etc. ; A perpendicular is the shortest dis- 
 tance from a point to a line 98-99 
 
Page. 
 
 Definition X, Theorems 15-16 and corollaries, — The quadrilat- 
 eral, the immaterial quadrilateral, and some of their proper- 
 ties 99-101 
 
 Theorem 17 and cor. : — In a plane quadrilateral two of whose 
 opposite sides are equal, and in which two interior angles 
 adjacent to the third side are supplementary, this third side 
 is not greater than the one opposite to it ; The sum of the 
 three angles of a triangle ^2rt. ^'s 101-102 
 
 Theorem 18, — The possibility of a certain deformation of an 
 immaterial quadrilateral with equal opposite sides established 102-106 
 
 Theorem 19, — A lemma from three-dimensional geometry. 
 Theorem 20 and corollaries : — The motion considered in 
 Theorem 18 is a plane motion ; Conclusion as to the sum of 
 the interior angles adjacent to the same side in an immaterial 
 quadrilateral subject to such a motion 106-108 
 
 Theorem 21 and corollaries : — The sum of the interior angles 
 adjacent to the same side of a plane quadrilateral with equal 
 opposite sides, equals two right angles ; The sum of the three 
 angles of a triangle, and the sum of the four angles of any 
 plane quadrilateral 108 
 
 Theorem 22, — Properties of two lines perpendicular to a third in 
 the same plane 108-110 
 
 Definition XI, — Provisional definition of parallel lines and cor- 
 ollaries 110-111 
 
 Theorem 23 and corollaries : — Three collinear points at equal 
 distances from a given straight line, must be in the same plane 
 with the latter, — and a consequent further definition of par- 
 allel lines 111-113 
 
 Theorem 24, — A proof of Euclid's Eleventh Axiom, and corol- 
 laries extending the definition of parallels to that given by 
 Euclid 113-115 
 
 Conclusion 115 
 
 Autobiography 116 
 
 LIST OF WORKS QUOTED IN THE INTRODUCTION OR CONSULTED 
 BY THE AUTHOR IN PREPARING THE DISSERTATION. 
 
 Ball, R. S., — "Measurement," article Encyclop. Brit., vol. XV. 
 
 Beltrami: — "Saggio di interpretazione della geometria non-Euclidea, " 
 Giornale di Matematiche, 1868, t. VI ; "Teoria fondamentale degli 
 spazii di curvatura costante, " Annali di Matematica, Ser. 2, II, 1868. 
 
 Bianchi, — "Lezioni di geometria difEerenziale," German translation, 1898, 
 
 t. II. 
 Bolyi, John, — "Science Absolute of Space," translated into English by 
 
 Professor George Bruce Halstead. 
 
XI 
 
 Cantor, — Mathematisohe Annalen, t. V, pp. 123-128. 
 
 Chrystal, Geo.,— "Parallels," article Ed cyclop. Brit., vol. XVIII. 
 
 Clifford, — Articles on the Axioms of Geometry in his Mathematical Papers. 
 
 Frischauf, — " Eleraente der absoluten Geometric," 1876. 
 
 Helmholtz, — " Ueber die Thatsachen die der Geometric zum Grunde liegen," 
 Konigliche Gesellsohaft der Wissenschaften zu Gottingen. 
 
 Henrici, CI., — *' Geometry," article Encyclop. Brit., vol. X. 
 
 Killing, — "Ueber die Grundlagen der Geometric," Crelle's Journal, Bd. 
 86, 109. 
 
 Klein, Felix: — Memoirs, " Zur Nicht-Euklidischen Geometric," Mathe- 
 matisohe Annalen, Bd. IV, VI, XXXVII; " Nicht-Euklidische Geo- 
 metric," lithographed lectures, Gottingen, 1893. 
 
 Lie, Sophus: — " Transformationsgruppen, " vol. Ill; " Continuierliche 
 Gruppen " ; " Die Grundlagen der Geometric," Leipziger Beriohte. 
 
 Lindemann, — Clebsch, "Vorlesungen iiber Geometric," Bd. II. 
 
 Lobatchevski : — "Theory of Parallels," translated by Halstead ; F. EngePs 
 ' ' Urkunden zur Geschichtc der Nicht-Euklidischen Geometric. 
 
 Playfair's Euclid. 
 
 Poincare : — Memoirs on the foundations of geometry in the : Bulletin de la 
 Soci^t^ Math^matiquc de France, t. 16, 1887 ; Revue Generale des 
 Sciences Pures et Appliquees ; Revue de M^taphysique et de Morale ; 
 "The Foundations of Geometry," Monist, vol. IX, 1888,— translated 
 into English by McCormack. 
 
 Riemann, — "Ueber die Hypothesen wclche der Geometric zu Grunde 
 liegen," Mathematisohe Werkc, pp. 254-269. 
 
INTRODUCTION. 
 
 A SURVEY OF THE MOST IMPORTANT VIEWS OF MODERN MATHE- 
 MATICIANS ON THE FOUNDATIONS OF GEOMETRY. 
 
 Both mathematicians and philosophers at present agree that — 
 although the science of mathematics as a whole is undoubtedly 
 the most exact of sciences, one of her most important branches, 
 at once the oldest and the most fascinating, namely geome- 
 try, has to some extent lost in its prestige and can no longer be 
 quoted by epistemologists as the prototype of purely deductive, 
 a priori science, and as a proof of the existence of certain in- 
 nate ideas, having a purely transcendental origin, wholly inde- 
 pendent of experience and partly conditioning it. This change 
 of view upon geometry has taken place during the past cen- 
 tury of all-pervading doubt and criticism, and, strange to say, 
 it did not originate with men outside the profession of mathe- 
 matics, but with those who had the greatest interest in pre- 
 serving her sanctity, in keeping up the halo of her alleged 
 transcendental origin. The very priests who worship at her 
 shrine, the greatest mathematicians who contributed most to 
 her fabulous growth and development in the nineteenth cen- 
 tury, — Gauss, Riemann, Helmholtz, Beltrami, and Clifford 
 among the immortal dead, and many prominent names still 
 among us, — have done most to cause this change. At present, 
 it is almost regarded as a heresy to attempt to restore some of 
 the old prestige to the science which was considered by the 
 Greeks to be the prototype of all science and all philosophy. 
 
 All this change of view has occurred, of course, not with re- 
 gard to the method employed by geometry, the soundness and 
 legitimacy of which have never been seriously doubted, but with 
 regard to the very foundations upon which geometry rests. It 
 is the body of axioms and postulates, the definitions and common 
 notions, — propositions and assumptions, both implicit and ex- 
 
 1 
 
plicit, which had, for a very long time, been considered as self- 
 evident, intuitive, and independent of all elaborate proof, — prop- 
 ositions that need only be stated in order to elicit unconditional 
 consent, — it is this body of, so-called, self-evident truths which 
 at present are questioned and doubted, and by many relegated 
 to the realm of empiricism, true only with a certain degree of 
 approximation, and capable of being modified in an infinity of 
 ways and, hence, of giving rise to a corresponding multitude of 
 geometries, each consistent in itself but in contradiction with 
 the others, each as perfect in theory as any other, and all very 
 nearly agreeing with our limited experience.* It is, however, 
 admitted on all sides, that the old system, namely, the Euclidian 
 system, more than all others, seems to agree with the results of 
 our experience, as far as this last goes ; and if we were able to 
 extend our experience considerably beyond the limits of the 
 fixed stars, and if even then we should find its norms to remain 
 unaltered and not needing revision, it would to a certain extent 
 prove the physical reality of the Euclidian geometry and the 
 unreality of the other systems, although the others would still 
 be theoretically admissible and would form a body of imaginary 
 geometries. 
 
 Now, this is rather a peculiar state of the geometrical science, 
 singular in its kind. For, while in other branches of science 
 two contradictory systems of thought would hardly be allowed 
 to stand side by side, both claiming to represent the truth simul- 
 taneously, — while, for instance, the Ptolemaic and Copernican 
 systems of astronomy could not avowedly coexist — the latter 
 having superseded the former as soon as it was found to agree 
 better with astronomical observations and with the abstract 
 laws of mechanics, — while no quarter was given to the Aristo- 
 telian theory of the fixity of species by the new evolutionary 
 systems of Lamarck and Darwin, or to the ancient doctrine 
 of the Four Elements by modern chemistry and physics, — the 
 contradictory systems of geometry, according to some mathe- 
 maticians of note, could be allowed to stand together, side by 
 side, and be of equal theoretical (if not practical) value and 
 importance. So, for instance, F. Klein, in his memoirs in the 
 
 * Professor F. Klein in many places in his memoirs on the non-Euclidian 
 geometry, Math. Ann., Bd. IV, VI, XXXVII, and in his ''Nicht-Euklidische 
 Geometric," lithographed impression, Gott., 1893, forcibly presents and de- 
 fends this opinion. See lithogr, lectures, I, pp. 298-365; Math. Ann., 
 XXXVII, p. 570. 
 
Mathematische Annalerij vols. 4, 6, 7 and 37, and in his " Nicht- 
 Euklidische Geometrie," second impression, Gottingen, 1893, 
 develops [from the projective point of view three systems of 
 geometry, — the Elliptic, the Hyperbolic, and the Parabolic 
 systems (which in broad features had been drawn already by 
 Riemann), corresponding to the three possible hypotheses which 
 can be made concerning our space, namely, as possessing con- 
 stant positive, negative, or zero, curvature. 
 
 Giving no theoretical preference to any of these systems, he 
 even goes so far as to think that the question, whether one 
 of these systems is to be preferred as expressing the real rela- 
 tions of our space, is unanswerable, since by allowing the 
 radius of curvature to be sufficiently great, the elliptic or hy- 
 perbolic geometry would give results approximating, with as 
 great a degree of accuracy as we please, to the results obtain- 
 able by the most exact measurements, performed with the most 
 powerful telescopes upon distances such as are involved in 
 the determination of the annual parallax of a fixed star. He 
 prefers the parabolic geometry, however, on account of its pre- 
 senting the simplest hypothesis in the theory of measurement. 
 So he says in his lectures on the non-Euclidian geometry, 
 referred to above, first part, page 277:* "There is, how- 
 ever, on the other hand, no lack of enthusiasts, who do not 
 answer the question in the way we have done, by asserting that 
 to our conception and experience of space could with sufficient 
 precision correspond alike the hyperbolic or the elliptic, as well as 
 the parabolic system of measurement, and that we decide in 
 favor of the parabolic, solely on account of its offering the 
 simplest hypothesis (just as in physics, among hypotheses of 
 equal probability, the simplest is always allowed to prevail)." 
 Each of these geometries, according to Klein, in another place in 
 the same work (p. 295), admits of an infinity of different space- 
 forms, — in which respect he differs from Killing, who thinks 
 that only in case of the elliptic geometry an infinite variety of 
 space-forms — Raumformen — is possible. He says : " We thus 
 expressly contradict the remark of Killing that in case of the 
 hyperbolic or parabolic metrics there exists the possibility only 
 of one space-form ; we say, on the contrary, that also in these 
 cases there exists an infinity of space-forms." — It has come to 
 
 * See also pp. 161-170 of same work, where this idea is presented with 
 especial force and elegance. 
 
pass, indeed, that some mathematicians are vying with one 
 another in devising new space-forms, as they call them, for 
 which new systems of geometry are supposed to hold. Clif- 
 ford, Klein, Lindemann, Killing, and others have contributed 
 much to this field of investigation. The enumeration and de- 
 scription of some of these geometries would lead us too far, 
 and the reader interested is referred to the numerous memoirs 
 in the Math. Ann. and in Crelle^s Journal and to separate 
 books and reprints from mathematical and philosophical peri- 
 odicals, which have appeared since the beginning of the seven- 
 ties up to the end of the past century. 
 
 It may, however, be said without exaggeration, that most of 
 these space-forms impress the reader rather with the ingenuity 
 of their inventors than with their actual value in bearing upon 
 the question of the foundations of geometry. There seems to 
 be rather too much license given to the imaginative faculty of 
 the human mind ; and while the origin of almost all investiga- 
 tions of this nature is to be sought in the impetus given to the 
 non-Euclidian geometry by the deep-searching criticism of 
 Riemann's inaugural dissertation on the foundations of geome- 
 try,* — where, it may be said, he formulated questions without 
 giving final answers to some of them, — the new systems devised 
 hardly ever carry conviction with them, and it may be stated 
 as a certainty, that many of them would not stand a scrutiniz- 
 ing criticism and would have to be relegated to the realm of 
 fancy rather than be classed with such an exact science as 
 mathematics. To quote an instance, the elliptic space, i. e., one 
 of positive curvature with two geodesies meeting only in one 
 point, described by Klein, Lindemann f and Killing, | is one of 
 such systems. Beltrami in his "Teoria fondamentale degli 
 spazii di curvatura costante '^ § makes the express statement 
 that any two geodesies in a space of positive curvature meet in 
 two antipodal points, through which a whole pencil of (an 
 
 *"Ueber die Hypothesen, welche der Geometrie zu Grunde liegen," 
 Math. Werke, pp. 254-269. It was not intended for publication by the 
 author in the form it appeared after his death in the Gottinger Abhand- 
 lungen. See "Nioht-Euklid.," I, p. 206 ; Lie, " Transformationsgruppen, " 
 III, pp. 485-486. 
 
 t Clebsoh, Vorlesungen iiber Geometrie, t. II. 
 
 X Crelle, t. 86, p. 72. 
 
 \AnnnH di Matematica, ser. 2, II, 1868 (The French translation of this 
 work of Beltrami and also of his famous *'Saggio," by Hoiiel, appeared in 
 the Jour, de VEcole Normale, t. VI). 
 
infinity of) similar geodesies must pass ; and to this statement 
 Klein takes exception in his " Nicht-Euklid. Geometric," pp. 
 240-242 and in other places of the same work, and in the Math, 
 Ann., t. 6, p. 125 and t. 37, p. 554 et seq. Another instance 
 is the spiral space-form, in which a rotation of a rigid body is 
 accompanied by an increase or decrease in its volume, so that 
 by a continuous rotation about a fixed point, any arbitrarily 
 chosen body can be made to enclose any given point of space, 
 and by an inverse rotation the body can be made to shrink 
 down to an arbitrarily small portion of space around the fixed 
 point. (Killing, " Ueber die Grundlagen der Geometric," 
 Crelle's Jour., t. 109, 1891, pp. 185-266). 
 
 There are, however, other mathematicians, who, — agreeing in 
 the main that the foundations of geometry have thus far not 
 been laid down with any degree of certitude and that they are, 
 therefore, open to considerable differences of opinion, — think, 
 nevertheless, that there ought to be some objective truth con- 
 cerning the nature of these foundations, and that it is not at all 
 unlikely that some day a satisfactory body of axioms and pos- 
 tulates may be found, which will prove undebatable. The 
 question is only in finding the minimum of simple truths, de- 
 rived from experience as an original source and formulated by 
 abstraction into a body of definitions and propositions which, 
 — on account of their incontestable efficiency as a basis for 
 geometry, on the one hand, and by their unquestionable real- 
 ity, on the other, as well as by their being irreducible to a 
 smaller number with equal efficiency, — should carry conviction 
 into the mind of the mathematician, whose taste is especially 
 fastidious in this relation, and should satisfy him that the basis 
 is a unified whole, without leaks, and that it is capable of stand- 
 ing the test of a scientific scepticism (of course, not a meta- 
 physical scepticism putting questions of the nature of whether 
 space and time or even matter and mind, the ego, the universe, 
 and so on, have any reality, objective or subjective, phenomenal 
 or noumenal, etc.). Among these mathematicians is especially 
 to be mentioned Sophus Lie, who, it seems, has contributed 
 more than any contemporary mathematician to sound views in 
 this matter, by treating the so-called Riemann-Helmholtz prob- 
 lem in a masterly way, which won for him the Lobatchevski 
 Prize in 1897. So Lie says in his " Transformationsgruppen," 
 Vol. Ill, p. 398, " We wish, however, to express the opinion, 
 
6 
 
 which is a conviction with us (wollen wir doch als unsere Ueber- 
 zeugung die Auffassung aussprechen), that it is in no wise im- 
 possible to establish a system of geometrical axioms which at once 
 shall be sufficient and shall contain nothing superfluous. It is 
 distressingly certain, however, that there are very few investiga- 
 tions which have actually furthered the problem as to the foun- 
 dations of geometry." 
 
 In another place in the same work (p. 536) he says : " Ge- 
 ometry in its different stages ought, as much as possible, to be 
 founded on a purely geometrical basis ; this is a demand with 
 which everybody undoubtedly will agree. 
 
 " For the first stage of geometry are necessary, in the first 
 place, certain fundamental conceptions, like space, curve, point, 
 and surface ; second come certain axioms concerning, for in- 
 stance, the properties of the right line, the existence of a sphere, 
 and so on. Every new conceivable stage is characterized by 
 the introduction of new axioms, — one stage, for instance, by the 
 axiom of parallels, another by the Cantor-axiom." — In my own 
 work, this last axiom, i. e., that the straight line is a number- 
 manifold, will be proved to be one of the fundamental properties 
 of the straight line, from which its construction and all its other 
 properties are obtained. — " Upon this axiom it is possible to 
 establish rationally the conceptions of area, length of arc, etc., 
 while Euclid virtually needs a separate axiom in each case. 
 
 " The great question is now, what axioms in each stage are 
 not only sufficient but also necessary, in other words, are in- 
 dispensable. In the answer to this question the whole problem 
 of the foundations of geometry would find its solution. 
 
 And then further (p. 537) Lie sketches a programme for the 
 mathematician who would undertake to establish the necessary 
 axioms, which perhaps might prove fewer in number than those 
 which Lie assumed provisionally for the purpose of solving the 
 Riemann-Helmholtz problem. — " First one would have to estab- 
 lish certain fundamental conceptions, like space, point, curve, 
 surface, and also the conception of motion. . . . 
 
 " As a first axiom one would have to establish the following : 
 If a point P is fixed, every other point can still describe a sur- 
 face which does not pass through the point F. In this may be 
 found the reason that two points in all rigid motions (Bewegun- 
 gen) remain separated." [These two conceptions are certainly 
 also connected in my treatment (see pp. 63-64, definition of 
 
distance and the proof following of the existence of a sphere), 
 except that the second is coming first, as the simplest one, and 
 following at once from the conception of rigidity.] 
 
 " As a second axiom it would be necessary to assume that, — 
 When two points P^ and P^ are fixed, there are still an infinity 
 of other points which remain fixed simultaneously, and these 
 points form one and only one line passing through Pj and Pg." — 
 I think Lie would certainly not object to a proof of this prop- 
 osition, which at once makes space a number-manifold and, 
 thus, supplies also the Cantor-axiom. He says that these prop- 
 ositions are not yet sufficient for the first stage of geometry. 
 Now, I think, that by means of only one additional axiom (viz, 
 axiom 1, p. 63), which, according to my mind, happens to 
 coincide with Lie's fundamental notion of a continuous group 
 of displacements, I have succeeded in establishing all that is 
 necessary for the first stage together with the most important 
 postulate of the second stage, namely, the postulate of parallels, 
 — in a sense, however, that does not exclude the spherical and 
 pseudospherical geometries, proving only the necessity of a 
 plane geometry in the Euclidian sense. 
 
 Lie himself treated this subject from the point of view of 
 continuous group-transformations. Starting with the Riemann- 
 Helmholtz postulate that space is a manifold of three dimen- 
 sions, in which the position of the single element, the point, is 
 determined by three coordinates, and adding a few very simple 
 postulates, characterizing the group of continuous motions of 
 rigid bodies (i, e., transformations in which every two points 
 have one essential invariant), he showed that there remains 
 only the possibility of the Euclidian and the two non-Eucli- 
 dian systems of motions.* The latter two are such as leave 
 invariant respectively the imaginary surface x? -f- icl + icf -f 
 1 = (Eiemannian group), or the real surface xl -{- xl -}- x^ -- 
 1 = (Lobatchevski group of motions). The group of Eucli- 
 dian motions together with the group of transformations by 
 similar figures are characterized by their leaving invariant the 
 absolute, 
 
 x' + f + z'^O, <=0.t 
 
 * Transformationsgruppen, III, pp. 464-479. The whole of the fifth 
 chapter is devoted to the Riemann-Helmholtz problem. See also Leipziger 
 Benchte, 1890, pp. 356-418, 284-321, and 1892, pp. 297-305. 
 
 t TransformatioDsgruppen, III, p. 218. 
 
8 
 
 The following are the axioms * which Lie considers suffi- 
 cient for his investigation : 
 
 I. Space is a number-manifold of three dimensions, R^, 
 
 II. The displacements or motions of B^ form a real continu- 
 ous group of point-transformations. 
 
 III. If any real point of general position, 3/ J, y^, 3/3, is held 
 fixed, all the real points x^^ x^y x^ into which another real point 
 tcj, xlf xl can be moved, satisfy an equation with real coeffi- 
 cients, of the form 
 
 ^{Vv 2/2? Vs » ^V ^2> ^3 J ^V ^2' ^3) ~ ^f 
 
 which is not fulfilled for 0;^ = ^J, X2 = yl, x^ = 1/3, and which, 
 in general, represents a real surface passing through x^, x^^ xl. 
 (A synthetic proof of this proposition is given in Theorem 1, 
 p. 64, of my Dissertation.) 
 
 IV. About the point 2/J, yl, yl a finite triply-extended region 
 may be so bounded that, after fixing the point y^, y\^ yl, every 
 other real point x\j x\, xl of the region can still pass by con- 
 tinuous motion into the position of every other real point of the 
 region which satisfies the equation, TF= 0, and which is joined 
 to the point y\y yl, yl by an irreducible continuous series of 
 points. (This proposition is also proved in the theorem referred 
 to above.) 
 
 In the Leipziger Berichte, 1890, pp. 357-358, he character- 
 izes the axioms necessary and sufficient to define the Euclidian 
 and the two kinds of non-Euclidian motions in a somewhat 
 different way, which, however, amounts to the same thing. 
 The analytical formulae are interesting, and I repeat them here. 
 The assumptions are : — 
 
 An infinite aggregate of real transformations of the points ol 
 JR3 (xy y, z) is given by the equations : — 
 
 ^1 =/(^> y> h «!> «2? • • •)> Vi = *(^? y^ h ^v ^v ' ' ')> 
 
 z^^y\r{x,y,z,a^,a,^, . . .). 
 
 These equations have to satisfy the following conditions : 
 A. The functions /, </>, -^ are analytical functions of the co- 
 ordinates X, y, z and of the parameters a^, a^, a^, ■ • • 
 
 * J64U, pp. 506-507. 
 
B. Any two points x^^ y^, z, ; x^^ y^, z^ have one essential 
 invariant under all transformations of the group, of the form 
 
 ^{^v Vv \ ^ ^2^ Vv h) = const. 
 Hence, 
 
 ^(^v Vv h > ^21 2/2? ^2) = ^«y y'v < y ^'v Vv ^2% 
 
 where x[j y[, z[ ; x'^, y'^, z'^ are the new positions of the original 
 pair of points x^, y^, z^ ; x^, y^, z^, which these obtain in virtue 
 of any transformation of the group. 
 
 C. The group is transitive, i. e., any point of the -R3 can be 
 transformed into any other point. If, however, one point 
 x^, y^y z^ is iixed, every other point x.^, y^, z^ can assume 00^ 
 different positions, which are defined by the equation : 
 
 ^(^1, yv h ; ^v Vv = ^{^v Vv ^i y ^v Vi^ ^2)' 
 
 If two points x^f y^y z^ and x^y y^y z^ are fixed, any third point 
 of general position, x^y 3/3, z^y can assume 00^ different positions 
 SC3, y'^y z'^ defined by the equations : 
 
 ^ (^1? Vv ^i > ^3^ 2/3J h) = ^ (^v Vv h y ^v Vv ^3) 
 
 ^ (^2> Vv \ ; %y y'v ^D = ^ (^2> yt> ^ > ^3> 2/3^ ^3)- 
 
 [The point x^y y^y z.^ must be of general position, in order to be 
 able to assume oo^ different positions, since there exists a singly- 
 infinite number of points defined by the last two equations II, 
 such that x'^y 2/3, z[^ = x^y y^y Z3, namely the 00^ points collinear 
 with x^y y^y z^ ; x^, y^y z^.] ^ If three points x,^y y^y 2, ; x^y y^, z^ ; 
 x^y 3/3, z^ are fixed, all points of space remain fixed, and three 
 similar equations will be satisfied only for x'^ — x^y y'^ — y^y z\ 
 
 The interpretation of the results obtained by Lie with regard 
 to the possibility of the two groups of non-Euclidian motions, 
 according to my mind, represents still an unsolved mathemat- 
 ical problem, alongside with the interpretation of similar re- 
 sults obtained by him for n-dimensional manifolds. A concrete 
 interpretation of these must be found in our empirical space, 
 for which the Euclidian axioms hold, in order to appreciate 
 their full geometrical significance. 
 
10 
 
 Not considering myself competent at present to undertake even 
 a partial solution of the problem, I wish, however, to indicate 
 that for two-dimensional manifolds an interpretation seems to 
 be near at hand, not at all in conflict with our Euclidian con- 
 ceptions of space. The non-Euclidian groups of transforma- 
 tions, assuming an invariant between two elements, only of 
 the most general kind, and presenting projective relations to 
 certain special forms of the fundamental quadric, must repre- 
 sent the real metric relations of our space, as imaged by pro- 
 jection upon certain surfaces of the second degree. In such a 
 projective image the anharmonic ratio of four points, or some 
 function of it, will remain unaltered, which will have to be 
 taken for a definition of distance or angle in these transformed 
 metrics; so that these metrics will coincide with the generalized 
 Cayleyan metrics, developed by Klein. For an interpretation 
 of this nature, we may refer to Poincar^'s paper on the founda- 
 tions of geometry in the Bull, de la Soc. Math, de France, t. 16, 
 Nov., 1887. Another interpretation for these groups has been 
 found in the metrics upon surfaces of positive and negative 
 curvature, when the straight lines are replaced by geodesies on 
 these surfaces, — an interpretation which has been fully justi- 
 fied by the works of Beltrami, to which reference will be made 
 later. Klein, in his " Nicht-Euklid. Geometric,'^ has shown, it 
 seems to me, conclusively (although he intended his procedure 
 to illustrate and to show in a concrete manner the possibility 
 of the plane having elliptic or hyperbolic metrics) that what he 
 calls an elliptic plane is actually the central projection upon a 
 Euclidian plane of the metrics upon a sphere,* and what he 
 calls the hyperbolic plane is the orthographic projection of a 
 system of metrics upon a sphere touching the plane, in which 
 the straight lines are represented by circles orthogonal to the 
 equator of the sphere which is parallel to the plane. At any 
 rate, this also shows that by certain processes of projection and 
 by making certain conventions as to the meaning of " distance " 
 and "angle,'' we may be able to account for the two non- 
 Euclidian groups of motions, as well as for the non-Euclidian 
 metrics deduced by Klein from the Cayleyan metrics. The 
 non-Euclidian groups of displacements for three dimensions, 
 
 * The radius of the sphere is taken to be = 2k, and ± l/4fc2 is the measure 
 of curvature of the elliptic or hyperbolic plane, resp. See " Nicht- 
 Euklid.," pp. 94-97 and pp. 220-237. 
 
11 
 
 however, as well as the Euclidian and the non-Euclidian 
 groups of displacements for manifolds of a higher number of 
 dimensions than three, which Lie derives from postulates simi- 
 lar to those he assumed for R^, still need an interpretation, and 
 this interpretation seems to lie in the change of element from a 
 point to a figure depending upon any number of parameters, 
 which is Pliicker's idea of making our space n-dimensional. 
 (To this I shall yet have occasion to refer later.) 
 
 Another eminent thinker on the subject who, in the main, 
 holds the same opinions and who has written some expositions of 
 the ideas of Lie and of his own views on the subject, is the illus- 
 trious French mathematician Poincar6. The first publication of 
 his on this subject is the paper in the Bull, de la Soc. Math, de 
 France^ quoted above ; then papers of his on the same subject 
 appeared in the Revue Generale des Sciences Pures et Appliquies, 
 t. Ill, 1892, and in the Revue de Metaphysique et de Morale. 
 Another paper, in which he further develops and complements 
 his views in the previous papers, appeared in the Monist, Vol. 
 IX, 1898, translated into English by McCormack. I quote 
 extensively from this paper, as I find in it so many points of 
 agreement with some of my own views — to which I have ar- 
 rived independently — about the relation of experience and 
 pure reasoning to the formation of our geometrical notions, also 
 in relation to the number of dimensions of space, and other 
 points. 
 
 He begins thus : " Our sensations cannot give us the notion 
 of space. That notion is built up by the mind from elements 
 preexisting in it, and external experience is simply the occa- 
 sion for its exercising this power ....'' 
 
 He maintains further that variations in our sensations give 
 rise to our notions of space. We observe two kinds of changes 
 in our impressions, which we thus separate into two classes : 
 
 1) External changes, independent of our will, and 
 
 2) Internal changes, accompanied by voluntary muscular 
 exertions. 
 
 The external changes again fall into two subdivisions : 
 
 1) Displacements y capable of being corrected by an internal 
 change, and 
 
 2) Altei-ationSy or physical changes not having this property. 
 Only Displacements are the Object of Geanieiry. 
 
 An identical displacement can be repeated a number of times. 
 Hence the introduction of number. 
 
12 
 
 The ensemble or aggregate of displacements form a group, 
 since the combination of any number of these is one of the 
 aggregate. 
 
 The notion of group could not be formed by a priori reas- 
 oning, but by experience together with reasoning. We ah- 
 stract from the concrete alterations which may accompany dis- 
 placements, so that geometry is safe from all revision. 
 
 "When experience teaches us that a certain phenomenon 
 does not correspond to the laws of the group, we strike it from 
 the list of displacements. When it obeys these laws only ap- 
 proximately, we consider the change by an artificial convention 
 as the resultant of two compound changes. One is regarded 
 as a displacement, rigorously satisfying the laws of the group, 
 while the second is regarded as a qualitative alteration. Thus 
 we say that solids undergo not only great changes of position, 
 but also small thermal alterations.'^ (Compare with this Pos- 
 tulate 1 and Scholium to Definition 8 in my Dissertation.) 
 
 " The fact that the displacements form a group contains in a 
 germ a host of important consequences. Space must be homo- 
 geneous ; that is, all points are capable of playing the same 
 part . . . 
 
 "Being homogeneous, it will be unlimited, for a category 
 that is limited cannot be homogeneous, seeing that the boun- 
 daries cannot play the same part as the center. But this does 
 not say that it is infinite, for a sphere is an unlimited surface, 
 and yet it is finite." 
 
 [To this reasoning I should object, for I should ask : Is not 
 a sphere a bounded body, and therefore non-homogeneous in 
 the third dimension ? If space were finite, it would not be 
 homogeneous in some dimension ; but as all dimensions belong 
 to space, it would be non-homogeneous in some of its own 
 dimensions ; hence, we could not say without limitation that 
 any two displacements form a new displacement. 
 
 In my proof of the infinite extent of the straight line, I do 
 not, however, assume the infinity of space. I only postulate 
 that " each point is capable of playing the same part as any 
 other,'' which is postulated by Poincar6 also, as being involved 
 in the notion of the group. Hence, from any point we can de- 
 scribe a sphere with a distance actually given by a previous 
 construction, if it is possible to do it for one point and for one 
 given distance. (See Theorem 2, p. 67.)] 
 
13 
 
 After postulating continuity for the group of displacements, 
 and defining what is meant by subgroups, isomorphism, inva- 
 riant subgroups, etc., into which I cannot go in detail without 
 transcribing the paper as a whole, Poincare goes on to say 
 that by experience combined with abstraction we arrive at the 
 notion of a rotative subgroup^ or the ensemble of displacements 
 which conserve a certain system of sensations. Then he says : 
 
 " By new experiences, always very crude, it is then shown : 
 
 "1. That any two rotative subgroups have common dis- 
 placements. 
 
 " 2. That these common displacements, all interchangeable 
 among one another, form a sheaf, which may be called a rotative 
 sheaf (rotations about a fixed axis). 
 
 "3. That any rotative sheaf forms part not only of two 
 rotative subgroups but of an infinity of them. There is the 
 origin of the notion of the straight line, as the rotative sub- 
 group was the origin of the notion of the point." 
 
 [In these few sentences one may discover a somewhat crude 
 empirical statement of the facts of which I availed myself in 
 constructing the straight line in Theorem 2 (see pp. 67-83).] 
 
 He then goes on to say that the existence of an invariant stib- 
 groupj namely, the subgroup of translations, in which all dis- 
 placements are interchangeable, is the only fact " that determines 
 our choice in favor of the geometry of Euclid, as against that of 
 Lobatchevshiy because the group that corresponds to the geometry of 
 Lobatchevski does not contain such an invariant subgroup.'^ 
 
 When he comes to the discussion of dimensions, Poincar^ 
 points out the distinction, from the point of view of the theory 
 of groups, between the order k and the degree n of a group, and 
 states that the order k is the more important characteristic of a 
 group. So that two groups can be isomorphic (i. e., their opera- 
 tions obey the same laws of combination and hence have the 
 same number of subgroups etc.), and still be of different de- 
 gree, provided their order k is the same. In continuous groups, 
 in general, and in the group of displacements, in particular, the 
 object of operations ''is the ensemble of a certain number n of 
 quantities susceptible of being varied in a continuous manner, 
 which quantities are called coordinates." — " Then, every infini- 
 tesimal operation of the group can be decomposed into k other 
 operations belonging to k given sheaves. The number n of the 
 coordinates (or of the dimensions) is then the degree, and the 
 
14 
 
 number h of the components of an infinitesimal operation is the 
 order J ^ — " The degree is an element relatively material and sec- 
 ondary, and the order a formal element/' — The study of the 
 group is mainly the study of its formal properties. The order 
 k corresponds to the number of essential parameters of a group 
 of transformations. — " The group of displacements is of the 
 sixth order." — The order k in case of R^ is 6, since B^ can 
 have 00^ displacements. — As to the degree n, it depends upon 
 the choice of the element. 
 
 If we choose the different transformations of a rotative sub- 
 group, we get a triple infinity of elements. " The degree of 
 the group is three. We have chosen the point as the element 
 of space and given to space three dimensions. 
 
 '* Choosing the different transformations of a helicoidal sub- 
 group, we obtain a quadruple infinity of elements. We have 
 chosen the straight line as the element of space, — which gives 
 to space four dimensions. 
 
 " Suppose, finally, that we choose the different transformations 
 of a rotative sheaf. The degree would then be five. We have 
 chosen as the element of space the figure formed by a straight 
 line and a point on that straight line. Space would have five 
 dimensions. 
 
 "The introduction of a group more or less complicated, ap- 
 pears to be absolutely necessary. Every purely statical theory 
 of the number of dimensions will give rise to many difficulties, 
 and it will always be necessary to fall back upon a dynamical 
 theory." 
 
 [I wish to observe here that the deduction of the number of 
 dimensions in my Dissertation is based upon kinematical prin- 
 ciples.] 
 
 " When I pronounce the word ^ length/ a word which we 
 frequently do not think necessary to define, I implicitly assume 
 that the figure formed by two points is not always superposable 
 upon that which is formed by two other points ; for, otherwise, 
 any two lengths whatever would be equal to each other. Now, 
 this is an important property of our group. 
 
 "I implicitly enunciate a similar hypothesis when I pronounce 
 the word * angle.' " 
 
 I have still to quote his ideas concerning contradictions in 
 geometry, as I think they are of cardinal importance. Here is 
 what he says : 
 
15 
 
 " In following up all the consequences of the different 
 geometrical axioms are we never led to contradictions ? . . . 
 The axioms are conventions. Is it certain that all these con- 
 ventions are compatible? 
 
 " These conventions, it is true, have all been suggested to us 
 by experience, but by crude experience. We discover that 
 certain laws are approximately verified, and we decompose the 
 observed phenomenon conventionally into two others : a purely 
 geometrical phenomenon, which exactly obeys these laws ; and 
 a very minute disturbing phenomenon. 
 
 "Is it certain that this decomposition is always permissible ? 
 It is certain that these laws are approximately compatible, for 
 experience shows that they are all approximately realized at 
 one and the same time in nature. But is it certain that they 
 would be compatible if they were absolutely rigorous ? 
 
 " Fo7' us the question is no longer doubtful. Analytical geom- 
 etry has been securely established, and all the axioms have been 
 introduced into the equations which serve u^ as its point of depar- 
 ture; we could not have written these equations if the axioms Jiad 
 been contradictory. Now that the equations are written, they can 
 be combined in all possible manners ; analysis is the guarantee 
 that contradictions shall not be introduced.'^ 
 
 We see thus that both Lie and Poincar^ are of the opinion, 
 that the question about the foundations of geometry represents 
 a more concrete and, therefore, more easily manageable prob- 
 lem than Klein and Killing and some others are willing to 
 grant. Neither of the former mathematicians allows an infin- 
 ity of contradictory geometries, and Poincar^ even gives some 
 reasons for our choice in favor of the Euclidian geometry. He 
 thinks only that this is solely due to the mode of experience 
 we have of space, and when he speaks of hypothetical beings, 
 whose experience might have led them to a predilection for the 
 geometry of Lobatchevski, he certainly is right, in the sense 
 that our geometrical notions, siich as they are, are not altogether 
 independent of the mode of experience we have, and the nature 
 of the universe we live in. In Poincar^'s own words : 
 
 " It is our mind that furnishes a category for nature. But 
 this category is not a bed of Procrustes into which we viole^Uly 
 force nature, mutilating her as our needs require. We offer to 
 
16 
 
 nature a choice of beds, among which we choose the couch best 
 suited to her stature/' 
 
 Now, this view of the matter is certainly more encouraging 
 to the one who would venture to find, by methods more ele- 
 mentary and, consequently, more legitimate for the given pur- 
 pose than those based upon the laws of continuous groups, ex- 
 actly which couch is the most suitable for nature's stature, and 
 even under what conditions the other couches may become just 
 as suitable. 
 
 Let us now turn to Riemann, whose paper on the founda- 
 tions of geometry seems to have been the occasion (apparently 
 unintended by the author),* of many a misconception sanc- 
 tioned by his name. Riemann himself, as well as can be gath- 
 ered from the fact that he tried to find the laws of free mobility 
 in manifolds of n dimensions, considering space as a special 
 CQjSe of such manifolds, where n=: S, seems to have been of the 
 opinion that geometry, as a science of space and spacial magni- 
 tudes alone, must be one and only one, although he did not de- 
 cide in favor of any of the three possible systems. He thought, 
 at any rate, that the discovery of the truth concerning the natwe 
 of the geometry of our space repr^esents a concrete and not unsolv- 
 ahle scientific p)rohlem. He stated expressly that the solution of 
 this problem was not to be found in investigations of such a 
 general character as was his own about manifolds in general — 
 a thing which his followers have not always heeded sufficiently 
 — since space was for him a manifold of special character, 
 whose science alone he called geometry, as distinguished from 
 the science of the general laws of manifolds, which, accord- 
 ing to him, belongs to analysis and the theory of functions. 
 So he says on p. 258 of his " Math. Werke " : " These magni- 
 tudinal relations " (of multiply-extended manifolds in general) 
 " admit of investigation only in terms of abstract quantity, their 
 natural connection being representable by formulce ; under certain 
 assumptions they can, however, be decomposed into relations, 
 which, taken separately, are capable of geometrical represeittation, 
 and through this it becomes possible to express geometrically the 
 
 * See above, note to p. 4. 
 
17 
 
 remit of the, calculation. So that, although an abstract investi- 
 gation, by means of formulae, remains unavoidable, it will still 
 be possible to invest its final results in geometrical attire." 
 And on page 256 he says, " More frequent occasions for cre- 
 ating and developing these conceptions (of multiply-extended 
 manifolds) we find only in the higher mathematics." 
 
 His own investigation he considered useful only in so much 
 as it threw light upon the extent and nature of the implicit a«- 
 sumptions of geometry and upon the many questions of measure- 
 ment in the wider region of multiply-extended manifolds, upon 
 which these assumptions touch, and which, apparently, have 
 escaped the attention of his predecessors. He says on p. 268 : 
 " Such investigations which, like the one here carried through, 
 start from general conceptions, can serve only the end that 
 this work (the investigation of the real facts underlying our 
 notions of space and of its magnitudinal relations) shall not be 
 hampered by too narrow conceptions, and that the progress of 
 discovery of the connection of things should not be impeded by 
 the burden of inherited prejudice." — He looked, however, for 
 the solution of this problem in the wrong direction, when he 
 thought that some physical hypothesis which may in time prove 
 necessary, to account for certain, as yet unexplained, physical 
 phenomena in the realm of the infinitely small, might also 
 throw some light upon the true nature of geometry. He cer- 
 tainly erred in respect of this physical hypothesis of the geo- 
 metrical properties of our space.* They could lead to no better 
 results than astronomical observations upon the stellar paral- 
 laxes, instituted with the purpose of finding some testimony in 
 the immensely large triangles concerning the amount by which 
 the sum of the three angles of a triangle is less than two right 
 angles, — as is very evident from the truly philosophical treat- 
 ment of this subject by Poincar^. But be this as it may, the 
 fact still remains that Riemann, in the first place, regarded 
 space as an unbounded manifold of three dimensions, and spoke 
 of it as being an empirical certainty greater than any other we 
 have, and, secondly, thought that the problem as to the admissi- 
 bility of the propositions of the Euclidian geometry beyond the 
 
 *'*It must be, therefore, either that the realities which lie at the basis 
 of space form a discrete manifold, or that the fonndation of its magnitudinal 
 relations ought to be looked for outside, in the binding forces working upon 
 it." (p. 268). 
 
18 
 
 bounds of observation was still an unsolved, but not unsolvahlcy 
 
 scientifixi problem. 
 
 Some of the results arrived at by Riemann are : 
 
 1. The simplest form of the linear element in any n-fold- 
 
 extended manifold admitting of measurement, is 
 
 ds = V^OL^jdx.dXj^, 
 
 where the a's are continuous functions of the ic's ; of these, n func- 
 tions can be taken arbitrarily, and n{n — l)/2 are fixed by the 
 nature of the manifold. In space, for instance, even if it were 
 curved, three of the a's could be taken = 0, each, and the rest would 
 have to take their chances, which would depend upon the nature 
 of the curvature. At this stage of his investigation, Riemann 
 seems to assume that space and the plane are flat manifolds, so 
 that their linear elements can be brought to the form of V^dx^ 
 (see p. 200, Werke). It would seem, therefore, that when later 
 he speaks of the possibility of space-curvature, and of a physical 
 investigation in the realm of the infinitely small, he means 
 rather that, since he does not see any logical, a 'priori necessity 
 of the necessary and sufficient assumptions of the Euclidian 
 geometry, which he establishes iu § 1 of Art. Ill, p. 205, he 
 hopes to find an explanation of their necessity in physics, as 
 he does not hope to find light on this subject in geometry 
 proper, her realm being only the finite. 
 
 2. In an n-manifold we can construct at each point oc"~^ 
 geodesies ; then a surface-element is determined by any two of 
 these given by their linear elements, when these are prolonged 
 until they become finite geodesies. In other words, as Klein 
 puts it in his " Nicht-Euklid. Geom.,^' p. 211, we have to con 
 sider the collectivity of geodesies whose linear elements 
 
 ds.= \'d's.^ \"d"s., 
 
 or such whose initial directions are in the same linear manifold 
 with the two given ones. Each of the surfaces thus obtained 
 will have its own initial Gaussian curvature, which Riemann 
 defines as the curvature of the n-manifold at the given point in 
 the given surface-direction. 
 
 3. A manifold of constant curvature is such as has the Gauss- 
 ian curvature in its surface-element the same at all points and 
 in all surface-directions. But the nature of the manifold at a 
 
19 
 
 point will be completely determined as soon as the surface-cur- 
 vature is given in 7i(n — l)/2 surface-directions. 
 
 4. Only a manifold of constant curvature allows free mobility 
 of figures, and if the Gaussian curvature be denoted by ot, the 
 linear element of such a manifold can be reduced to the form of 
 
 V^dx'. 
 
 5. All metrical relations of the manifold depend upon the 
 value of the curvature. The number of ways in which an 
 n-manifold can move in itself without deformation is 
 
 n{n H- 1) _ 2 _ K^ ~ ^) _ 
 
 number of coordinates minus number of distances between n 
 points. 
 
 Rieraann then gives three possible forms of the conditions 
 necessary and sufficient to determine the measure-relations of 
 space, as distinguished from all other three-dimensional mani- 
 folds admitting of measurements and flat in their smallest 
 parts, i. e.y such whose line-length is independent of position 
 and in which the linear element is expressible as the square 
 root of a positive differential expression of the second degree. 
 
 1°. The Gaussian curvature in three surface-directions is zero 
 at each point ; or, otherwise, the metric relations of space are 
 completely determined, if the sum of the three angles of a tri- 
 angle is always equal to two right angles. 
 
 2°. Besides the independence of line-length from position, we 
 may assume with Euclid the existence of rigid bodies, inde- 
 pendent of position, — which is equivalent to postulating constant 
 curvature. The sum of the three angles in all triangles is then 
 determined, when it is known in one triangle. 
 
 3°. We may assume not only the independence of line- 
 length from position, but also the independence of length and 
 direction of lines from position. 
 
 Each of these three alternatives adds something to the prop- 
 erties of a manifold, flat in its smallest portions. The first 
 and last lead at once to the Euclidian geometry ; the middle 
 one allows the possibility of all three different geometries, 
 
20 
 
 according as the sum of the angles of a triangle is greater 
 than, equal to, or less than, two right angles.* 
 
 Now, then, the position and the opinions of this second cate- 
 gory of mathematicians (E-iemann, Lie, and Poincare), and 
 especially those of Lie, seem to indicate that the task of estab- 
 lishing an efficient system of axioms is not perfectly hopeless. 
 And if the reader admit with Lie that but few investigations 
 ,have materially furthered the problem concerning the founda- 
 tions of geometry, he may find it of interest to read through 
 the present memoir, even if its purpose is disclosed at the very 
 beginning to be — the establishment of such a system and the 
 restoration of a goodly part of the old prestige to the origin 
 and foundations of the geometrical science. 
 
 It may not be amiss here, in this connection, to remind the 
 generously disposed and impartial reader that this problem, 
 besides its philosophic interest, is also of importance from a 
 purely mathematical point of view. The fact which will sub- 
 stantiate my statement is, indeed, very well known to mathe- 
 maticians who have familiarized themselves, at least superficially, 
 with the non-Euclidian geometry, although little stress is put 
 upon its bearings by those mathematical writers on the subject, 
 who, having satisfied themselves that Avithin the bounds of our 
 limited experience the Euclidian geometry holds, concluded 
 that beyond these limits actual deviations of the metrical rela- 
 tions of space may take place, of which we are not bound to 
 take heed in our analytical geometry, as long as we intend to 
 avail ourselves of its results in actual practice only. Already 
 the earliest non-Euclidians, and among them the two great 
 founders of the hyperbolic geometry, Lobatchevski and Bolyai, 
 have made it clear that the theory of proportion and similar 
 figures is based upon the parabolic system of measurement, and 
 that it has no meaning when the Euclidian postulate of paral- 
 lels does not hold. In fact, they have both given formulae for 
 the solution of rectilinear triangles, perfectly analogous to those 
 of the spherical trigonometry, f Further, Bolyai has shown 
 
 *See Lie, "Transformationsgruppen," Vol. Ill, p. 497, where he finds 
 this paragraph in Kiemann's paper (§ 1 of art. Ill) not clear. 
 
 fSee Lobatchevski, "Theory of Parallels," translated by Halstead, pp. 
 35-45. Also, " Urkunden zur Geschichte der Nioht-Euklid. Geom." F. 
 Engel, pp. 216-235. 
 
21 
 
 that, by assuming the hyperbolic geometry to be true, the prob- 
 lem of the squaring of the circle presents no difficulty.* It is, 
 therefore, evident that the establishment of the Euclidian 
 geometry on a basis more rational than mere empiricism even 
 if very accurate, still remains a desideratum. 
 
 It will, however, become incumbent upon me to explain my 
 own point of view in this matter, and give in outline the re- 
 sults at which I have arrived, and also to throw some light 
 upon the methods pursued in this dissertation, as well as to 
 present the reasons which, according to the best of my judg- 
 ment, can be assigned to the final success with which these 
 methods have been rewarded. 
 
 To give my own views upon the r6le of experience and reason 
 in the formation of our geometrical conceptions would, I think, 
 only be a repetition of what is stated more or less explicitly in 
 my introductory chapter on dimensions, as well as a repetition 
 of many excellent remarks of Poincar^ in his paper in the 
 Monist, which I have allowed myself to quote so extensively. 
 In a few words these views may, however, be summarized 
 thus : — 
 
 As in all pure sciences, our fundamental conceptions in 
 geometry are formed by experience helped on by pure reasoning^ 
 which abstracts from certain unessential irregularities in the rough 
 data of experience, by reducing certain general norms to ideal 
 forms, not admitting of exception. The exceptions, indeed, are 
 purposely eliminated by ascribing to them some other causes, 
 which are not the subject of the given investigation. So, for 
 instance, in mechanics, the fact that no body in actual experi- 
 ence, possessing a certain momentum, can go on and move for- 
 ever, does not bother the physicist, who postulates the first law 
 of Newton, and ascribes the stopping of the body or the retar- 
 dation of its motion to external causes, like frictional resistance, 
 etc. Similarly, if ideal solids are postulated in geometry, the 
 deformation which natural solids undergo in motion is ascribed 
 to physical causes, and not to properties of space.! 
 
 Further, I think that we cannot start simply with an axiom 
 — that space is a number-manifold, i. e., each point in it can 
 
 *See " Science Absolute of Space " by John Bolyai, Halstead's transla- 
 tion, p. 47. 
 
 t See postulate 1, Definition 6, and Scholium to Definition 8, of the intro- 
 ductory chapter of my Dissertation (pp. 40, 41). 
 
22 
 
 be determined by three coordinates, or three numbers which 
 can be made to vary continuously. I think rather that space 
 ought to be proved capable of being made a number-manifold, 
 and the best starting point in this direction is, according to my 
 opinion, to be found in the simple physical facts of impenetra- 
 bility, rigidity, and divisibility of bodies, each in geometry be- 
 ing idealized. The bulk of a given portion of space, bounded 
 on all sides, can certainly be represented by a number, showing 
 how many times it will contain a smaller bulk of definite shape. 
 By considering the smaller bulk as rigid, or such in which 
 internal motion or rearrangement of parts is excluded, and 
 making this smaller bulk take up all possible positions within 
 the larger bounded portion of space, we observe that no matter 
 whence it he placed within the larger one, it always occupies or 
 Jills up the same numerical portion of the larger bulk. The num- 
 ber of other bulks like the smaller, necessary to fill up the 
 larger completely, besides the smaller one itself, or the num- 
 ber of places the smaller can be made to occupy within the 
 larger, such that no two have any portion in common, is always 
 the same. And this is true also when the smaller bulk is 
 broken up into infinitely small portions, free to change position 
 with respect to one another, but still capable of filling up com- 
 pletely the same space ; or, in other words, when the smaller 
 bulk is allowed to change its form in all possible ways, so as to 
 retain only impenetrability. Equal bulks are then measured 
 by equal spaces of same shape, which they are capable of filling.* 
 We postulate that this be true for any bounded space and 
 for any small bulk which is placed in the larger one, in any 
 position, — that there should always be the same numerical re- 
 lation between the smaller and the larger bulk as soon as these 
 are given, as a rigid solid, on the one hand, and a bounded 
 vacuum in which the first is to lie in any position, on the other 
 hand. We arrive at the notion of congruent portions within 
 the bounded space, meaning such which the same solid fills up 
 to the exclusion of others, — and by considering, besides, very 
 small portions of the smaller bulk, their number and disposi- 
 tion with respect to one another are seen not to change as long 
 as rigidity of form is postulated for the whole. So that, after 
 
 * For a complete and rigorous treatment of this question, the reader is 
 referred to the introductory chapter, Scholium to Definition 8, pp. 41-44. 
 Here is possible only a short indication of the procedure. 
 
23 
 
 we have worked ourselves up to the notion of surface, line, and 
 point, as done in the introductory chapter, rigidity is seen to 
 imply that no two points, separated in one position of a rigid 
 body, become ever coincident on account of change of position 
 of the solid. And, moreover, the same continuous series of 
 separated points of the solid must be capable of being con- 
 structed between any two given points of the solid in any one 
 of its positions as in any other. 
 
 But this is only a starting point. We must further make 
 clear to ourselves what we understand when we say that space 
 is a three-dimensional manifold, considering a point as its ele- 
 ment, and whether there is any sense in looking for a fourth 
 dimension, not directly given by experience. It will appear 
 from the treatment of the question in the introductory chapter 
 that the tridimensionality of space is actually postulated by the 
 definition of a point, and that, therefore, to look for a fourth 
 dimension, without changing its element from that which lies at 
 the basis of the metrical geometry of Euclid to some other 
 geometrical object (which is, in fact, a figure in the Euclidian 
 sense, depending upon a certain number of parameters), as is 
 done, for instance, in Pliicker's line-geometry, — is a contradic- 
 tion in terms. Finally, from the same principle of rigidity, 
 the notion of distance as an invariable relation between two points 
 in rigid connection or in fixed space, is easily derived by a defi- 
 nition which makes use of the principle of superposition. 
 
 Next, continuity must be postulated,* and then we must show 
 how distances can be added and subtracted, and whether there 
 is a line, or a one-dimensional manifold, in space, capable of rep- 
 resenting by the actual distances of its points all possible dis- 
 tances arrived at by addition and subtraction, and whether this 
 can be done in a unique way. It appears, that from this property 
 alone a notion of the straight line can be deduced, which will 
 have all other properties of the straight line, commonly postu- 
 lated for it in the Euclidian geometry ; the construction, more- 
 over, based upon this property, will make it a number-manifold f 
 of infinite extent, such as can in no way be mixed up with a 
 geodesic returning into itself at a finite distance. This, in a 
 
 * See axiom 1, p. 63. 
 
 fThe construction makes the straight line a number-manifold in the 
 Cantor-sense, since, as we can construct all possible sums of all possible ra- 
 tional numbers, we can construct the irrational numbers by sequences, in the 
 way it is done by Cantor for pure numbers. Math. Ann.f t. 5, pp. 123-128. 
 
24 
 
 certain way, disposes of the so-called elliptic geometry of the 
 straight line. Further, the line so constructed will prove to 
 be both an axis of rotation and an axis of helicoidal motion, or, 
 in Poincar6 and Lie's language, admitting either the transfor- 
 mations of a rotative sheaf, or those of a helicoidal subgroup, ac- 
 cording as one point, at least, upon the line is taken as an in- 
 variant point, or none, at a finite distance, is taken as an inva- 
 riant point — the line being able to slide upon itself, while 
 all points in rigid connection with the line, but outside it, 
 being free only to twist, i. e., to move with a screw-like motion. 
 The notion of distance, as thus defined, is, of course, of a 
 very abstract nature, and does not depend upon the paths 
 which either of two non-coincident points can be made to pass 
 in a given surface from its own position to the position of 
 the second, nor upon any given position of the point-couple in 
 space, but only upon the relative position of the two points 
 themselves. It is simply a fact of experience that a pair of 
 points in a solid are capable of coincidence only mth certain de- 
 terminate couples of points in other solids or vacant space, and 
 this fact of congruence or non-congruence is the only factor 
 determining equality or non-equality of distances. It is only 
 after it has been proved that this geometrical magnitude is 
 representable uniquely and perfectly by some line (a priori, a 
 surface or a volume might perhaps have been found more 
 capable to represent this magnitude, as happens, for instance, 
 with the angular magnitude, which is equally well represented 
 by a portion of a circular arc as by the area of a sector of the 
 circle of radius unity, and as, for instance, the solid angle, 
 considered as a geometrical magnitude, may be measured equally 
 by the area of a spherical surface of radius unity whose center 
 is the vertex of the angle, or by the corresponding spherical 
 sector ; so that it is only an accident, having, of course, its rea- 
 sons in the nature of things a posteriori, that distance as a 
 geometrical magnitude has for its representation a line), it is 
 only after this fact has been established, that any curve can be 
 broken up into linear elements ds, each of which is comparable 
 with the elements of three given straight lines dx, dy, dz, and 
 can be expressed in terms of these. So that any complicated 
 expression for a linear element of some curve in space, in terms 
 of dx, dy, dzy say, ds —f(dx, dy, dz), must necessarily be based, 
 in the first instance, upon a certain relation, which could be 
 
25 
 
 considered the simplest and which would be formed exactly in 
 the same way as a finite distance is expressed in terms of three 
 other finite distances, which must be taken as parameters in the 
 case of tridimensional space. 
 
 I wish here to call attention to the fact that the deduction of 
 the existence of the straight line in space, not as a line in the 
 plane, as far as I am aware, seems never to have constituted a 
 serious problem with mathematicians. This is, perhaps, the 
 only reason why the straight line has always been regarded as 
 a geodesic which is determined by two points in the surface to 
 which it belongs, i. e., as a geodesic in some plane. The plane 
 is postulated or constructed before the straight line, and angles 
 and triangles and circles are regarded not only as plane figures, 
 that iSy such as can lie in a plane, but also as figures constructed 
 in a plane. The distinction, according to my mind, is not at 
 all trivial, since the existence of a plane can be proved with 
 perfect rigor only after a number of theorems concerning 
 angles, triangles, and circles, have been established for these 
 figures in space. Then only, according to my opinion, we 
 ought to prove that these simple figures, as well as the straight 
 line, are plane figures. 
 
 The plane, as constructed from a certain origin, must then he 
 shown to he capahle of moving upon itself in a triply-infinite num- 
 ber of ways, and also of coincidence with itself when its two sides 
 are interchanged. The first will establish the legitimacy of 
 what is called in analytic geometry change of origin and change 
 of axes ; the second, a revolution of the plane through an angle 
 TT, which appears in the theory of groups to need special sub- 
 groups of displacements. (See Lie, " Continuierliche Gruppen,^' 
 p. 101. The first kind of displacements form the group of con- 
 gruent figures, the second, the group of symmetrical figures.*) 
 In the Euclidian geometry both processes are invariably used 
 in superposition of figures for demonstrations. In spherical 
 and pseudospherical geometry this is also practised, with the 
 understanding that bending without stretching is postulated. 
 In spherical geometry bending is necessary only for making the 
 inner side of a portion of a spherical surface coincide with the 
 outer side of the same surface, or for the purpose of applying 
 
 * Group of congruent figures, a?i = a; cos a — y sin a + a, yi = « sin a -{- 
 
 y cos a + 6 ; group of symmetrical figures, a^i =^ a cos a + y sin a + «i yi = 
 
 X sin a — y cos (x-\-h. 
 
26 
 
 its figures to figures fi^rmed upon other surfaces of equal con- 
 stant curvature ; in case of pseudospherical geometry, i, e., sur- 
 faces of constant negative curvature, no superposition of parts 
 of different regions, even on the same side of the same surface, 
 would be possible without bending. And it was just by ab- 
 stracting from rigidity, in so much as bending without stretch- 
 ing was allowed, that Beltrami, in his "Saggio^^* and in his 
 " Teoria fondamentale,'' was able to prove that Lobatchevski's 
 geometry holds good, in our Euclidian space, upon surfaces of 
 constant negative curvature, which he first named pseudo- 
 spherical surfaces. — " The fundamental criterion of the demon- 
 strations in the elementary (Euclidian) geometry,'' Beltrami 
 begins his investigation in his " Saggio,'' " consists in superpo- 
 sition of figures. The criterion is applicable not only to the 
 plane, but also to all surfaces upon which there can exist equal 
 figures in different positions, that is to say, to all surfaces whose 
 any portion can by means of simple flexion be applied to any 
 other portion of the same surface. We see, in fact, that the 
 rigidity of the surfaces upon which the figures are traced, is 
 not an essential condition for the application of this criterion ; 
 for instance, the exactitude of the plane Euclidian geometry 
 would not become deteriorated, if we should begin by conceiv- 
 ing the figures traced upon the surface of a cylinder or a cone, 
 instead of a plane." 
 
 Stating then that the surfaces whose figures have a structure 
 independent of position, and hence allowing the principle of 
 superposition without restriction, are those of constant curvature 
 only, he goes on to say : 
 
 " The most important element of figures is the straight line. 
 The specific characteristic of this line is that it is completely 
 determined by two of its points, so that two straight lines can- 
 not pass through two points in space without their coinciding 
 in all their extent. In 'plane geometry , however y this principle is 
 used only in the following form : 
 
 " In making coincide two planes, in each of which there is a 
 straight line, it is sufficient that the two lines coincide in two points, 
 in order that they coincide in the whole of their extent, 
 
 " Now, this propefi^ty the plane has in common with all surfaces 
 of constant curvature, whetx, instead of the straight lines, we take 
 
 ***Saggiodi interpretazione della geometria non-Euolidea, " Oiomale di 
 Mathematiche, 1868, t. VI (see note above, p. 4). 
 
27 
 
 the geodesies, , . ,' If we make coincide two surfaces of constant 
 and equal curvature, so that two of their geodesic lines have two 
 points in common, these lines will coincide in all their extent, 
 
 " It follows that, excluding the cases where this property is sub- 
 ject to exceptions, the theorems of planimetry whieh are proved by 
 the principle of superposition and the postulate of the straight line 
 for plane figures, are true also for figures formed in an analogous 
 way, upon surfaces of constant curvature, by geodesic lines, 
 
 " TJpmi this are based the many analogies between the geometry 
 on a plane and that on a sphere, the straight lines corresponding 
 to geodesies, i. e., to arcs of great circles. For a sphere, haW' 
 ever, there exist exceptions, for any two poi7its diametrically op' 
 posite, or antipodal points, do not determine a geodesic without 
 ambiguity, since through such points an infinity of great circles 
 will pass. This is a reason why certain theorefins in plane geom- 
 etry are not true for the sphere, as, for instance, the theorem that 
 two perpendiculars to the same line do not meet,'' 
 
 Beltrami further makes clear that the basis of investigation in 
 plane geometry is too general, if, as usually done, the only facts 
 lying at this basis are taken to be the principle of superposition 
 and the postulate of the straight line. The results of the demon- 
 strations must exist whenever this principle and this postulate, are 
 true. They must, evidently, be true for surfaces of constant cur- 
 vature, in which the postulate of the straight line holds without 
 exception. 
 
 Now, the purpose of Beltrami's investigation was precisely to 
 show that this postulate does not admit of exceptions in case of 
 surfoMes of constant negative curvature. And, in his own words, — 
 " If we can prove that such exceptions do not exist for these sur- 
 faces, it becomes evident that the theorems of the non-Euclidian 
 planimetry hold without restriction upon such surfaces. And then 
 certain results which seem incompatible with the hypothesis of a 
 plane, may become corweivable upon such a surface and obtain 
 ther-eby an explanation, not less simple than satisfactory. At the 
 same time the determinations which produce the transition from 
 the non-Euclidian to the Euclidian j^lanimetry, are shown to be 
 identical with those which specify the surfaces of zero curvature in 
 the sanies of surfaces of constant negative curvature," 
 
 Lobatchevski and Bolyai, who, together with Legendre, were 
 aware of the defects of the Euclidian geometry in respect to 
 all the postulates regarding the straight line and the plane, have 
 
28 
 
 both tried to construct them and to deduce their properties 
 from the construction.* But, as it seems to me, they did not 
 succeed in doiTig it with sufficierd rigor ; and, besides, neither of 
 them freed himself from the idea that a plane is given in concep- 
 tion previous to a straight line, and, therefore, they constructed 
 first the plane and then the straight line in it. The straight line, 
 therefore, again has the same properties as a geodesic upon a 
 surface, which will coincide with another geodesic in a similar 
 surface, of same constant curvature, as soon as the two surfaces 
 are superposed so that two congruent pairs of points in the two 
 geodesies are made to coincide. At least, neither of the two 
 mathematicians separated the straight line from the plane suffi- 
 ciently, to come to the clear idea, that figures of straight lines in 
 space can he considered without considering the planes in which 
 they lie. This is one of the reasons why they could not prove the 
 postulate of parallels, which, in fact, distinguishes the plane from 
 all other surfaces of constant curvature. For, it is evident, that 
 since the curvature of a surface is an extrinsic property of the 
 surface, i. e., it is a parameter by varying which we can obtain 
 all surfaces of constant positive and negative curvature, the 
 limit between the two being the plane (of zero curvature)t, — 
 those properties of the geodesies which depend upon any parti<M- 
 lar value of the curvature, could not he found, except hy leaving 
 the surface and going out into space. Of course, the expression 
 of the linear element can, certainly, give the true metrical prop- 
 erties of the corresponding surface, and hence, also of the plane. 
 J3ut this very expression, as Riemann has shoicn, depends upon 
 the curvature. (In fact, for a surface of Gaussian curvature, 
 Ijh^ = a, the linear element is reducible to the form 
 
 i + |2- 
 
 V^dc\ 
 
 so that in case of positive curvature, a is -f , of negative curva- 
 ture, a is — , and in case of the plane, if the ^* parallel postulate " 
 
 *See "Urkunden znr Gesohichte der Nioht-Euklidischen Geometric, 
 Nikolaj Iwanowitsch Lobatschefskij, " by F. Engel, 1898 (pp. 93-109), and 
 Frisohanf, " Elemente der absoluten Geometric," 1876, pp. 8-18. 
 
 t dt Ijk^ gives curvature of spheres and pseudospheres, with all their, 
 varieties resulting from bending, and k= co gives the plane. See Biemann, 
 Ueber die Hypothesen, etc.," II, 5. 
 
29 
 
 holds, a= Oj and, conversely, if a = 0, then the postulate of par ' 
 allels holds.) 
 
 Hence, as long a^ we remain in the surface itself, and suppos- 
 ing we do not know the form of the linear element, we could in no 
 way prove or disprove the ^^ parallel postulated' And this is 
 exactly the reasoning of those who think that the " parallel postu- 
 late " cannot be proved, I could not do better than refer 
 again to the last quotations from Beltrami, who, although 
 he, as far as seen from his " Saggio " and " Teoria fonda- 
 mentale,'' may never have expressed himself directly on the 
 possibility or impossibility of proving the postulate of paral- 
 lels, showed, however, the reason why Lobatchevski and 
 Bolyai had arrived at a geometry for the plane which is actu- 
 ally true only for pseudospheres. In another place in his 
 " Saggio" he says : "We see, then, that two points of the sur- 
 face (pseudospherical surface), chosen in any manner whatever, 
 always determine uniquely a geodesic line. . . . Thus, sur- 
 faces of constant negative curvature are not subject to excep- 
 tions which, in this respect, happen in the case of surfaces of 
 positive curvature, and, therefore, we can apply to them the 
 theorems of the non-Euclidian planimetry. Moreover, these 
 theorems, in their greatest part, are not susceptible of a conc7^ete 
 interpretation, unless they are referred precisely to these surfaces, 
 instead of the plane, as we are going to prove presently in detaiV 
 
 It is interesting to compare this sober view upon the non- 
 Euclidian geometry of the great Italian mathematician, whose 
 works more than those of any other succeeded in putting this 
 geometry upon a respectable footing, with a few quotations 
 from Bianchi, " Lezioni di geometria differenziale," German 
 translation, 1898, t. II, p. 434. The quotations referred 
 to are headed by the superscription, " A Bev^iew of the JVon- 
 Euclidian Geometry, '^ and follow an excellent account of pseu- 
 dospherical geometry treated by the method of conform repre- 
 sentation, and they read as follows : — 
 
 " In the principal theorems of the pseudospherical geometry, 
 which we have deduced in the preceding paragraphs, a close 
 analogy is observable to the propositions of the plane and the 
 spherical geometry. The basis for these analogies, as well as 
 for the differences, we can foresee a priori. If, in fact, we 
 examine the axioms and the fundamental postulates of the plane 
 geometry, as laid down by Euclid in his first book, and if, in case 
 
30 
 
 of pseudospherical surfaces^ we replace the straight line by the 
 geodesic line, v:e see thai when we leave out of consideration the 
 XII postulatej concerning parallels^ all others will hold without 
 change in the pseudospherical geometry. This is, in particular, 
 the case with the principle of congruence and also ivith tlie prin- 
 ciple that a geodesic line is uniquely determined by two of its 
 points. Those propositions of plane geometry which do not 
 depend upon the parallel postulate hold, therefore, for the pseudo- 
 spherical geometry ; the others undergo such changes as to become 
 identical with the old ones, as soon as the radius It of the pseudo- 
 sphetical surface is made infinite, 
 
 " The above considerations show readily the uselessness of 
 
 ALL ATTEMPTS MADE TO PROVE THE PARALLEL POSTULATE. 
 
 If this proof could be logically deduced from the other principles, 
 
 IT WOULD HAVE TO HOLD EQUALLY FOR THE PSEUDOSPHERI- 
 CAL SURFACES IN EuCLIDIAN SPACE. 
 
 " And, in fact, if in the plane geometry we drop the Euclid- 
 ian postulate, we are led to the so-called abstract or non- 
 Euclidian geometry, whose foundations were laid down by 
 Bolyai and Lobatchevski, and which (the straight line being 
 taken as unlimited) perfectly coincides with the pseudospheri- 
 cal geometry." 
 
 Now, this objection is certainly valid, if we were bound to con- 
 sider plane figures only. Fortunately, we can have constructions 
 involving distances, or straight lines, and angles, nx)t bound to lie 
 in any particular surface at all. Tlie conception of a quadri- 
 lateral without fixed area, consisting of four fixed distances, ivhich 
 are equal by pairs ( opposite sides ), and whose angles are vari- 
 able, and equal by pairs, is an important conception in elementary 
 geometry. A triangle is fixed as soon as its sides are fixed : 
 it cannot " rack " ; its area and its angles are fixed with its 
 sides ; it is necessarily a plane figure. And if the proposition 
 is true that the area of a geodesic triangle = a const, times the 
 excess or deficiency of the sum of the angles of the triangle over 
 TT, i. e., A = db K\A -\- B -\- C—ir), where A is always posi- 
 tive and K is the radius of curvature, we have no means of 
 varying this area, when the geodesic triangle is conceived as bound 
 up with the surface, But,^ the immaterial quadrilateral,^ which 
 consists always of two triangles, is not bound up with any fixed 
 
 *See Definition 10 of my Dissertation (p. 44). 
 
31 
 
 area and is not bound to move in any particular surface^ and its 
 angles, under certain imposed conditions, being in certain deter- 
 minate relations among themselves, are still variable, — and^ by conn 
 sidering the continuous series of deformations given in Theorem 
 18,"^ we arrive at the conxilusion that there must exist a quadrilat- 
 eral with equal opposite sides, whose two angles adjacent to the 
 same side are equal to two right angles, not knowing, however, 
 whether the quadrilateral is a plane quadrilateral or not. And 
 then only, with the aid of the lemma. Theorem 19, which proves 
 that the sum of any two face-angles of a triedral angle is greater 
 than the third, we find either that such a quadHlateral must be a 
 plane quadrilateral, or that the sum of the three angles is grecder 
 than two right angles. But the second alternative, which holds 
 for surfaces of positive curvature, was proved to be impossible in 
 the case of a plane, by Theorems Hr-17; hence, only the first altet^- 
 native remains, from which it immediately follows that 
 
 THE SUM OF THE THREE ANGLES OF A RECTILINEAR TRI- 
 ANGLE IS EQUAL TO TWO RIGHT ANGLES, OR, in other words, 
 
 THE GEO]METRY OF THE PLANE IS PARABOLIC. 
 
 Now, it so happens that this proof has been formulated in such 
 a manner as to take in consideration the existence of both the 
 spherical and the pseudospherical geometries. It actually leaves 
 room for the existence of surfaces of constant curvature — such where 
 the sum of the three angles of a geodesic triangle is greater than 
 two right angles, and such where this sum is less than two right 
 angles. The first prove to be finite in extent, all their normals 
 meeting in a point at a finite distance, — as follows immedi- 
 ately from Theorems 14 and 17, since these theorems assume 
 the impossibility for two geodesies in a plane to intersect 
 in two points, and the infinite extent of the plane, (both of 
 which assumptions have been proved in Theorems 2-5 and 8) ; 
 while the second class of surfaces have their normals non-co- 
 planar, or meeting at an imaginary point. Hence, the plane 
 appears to be the limiting case between the two, being infinite 
 in extent and having its normals meeting in a point at oo. The 
 considerations of Bianchi and others, which tJiey put forward 
 against the possibility of proving the Euclidian postulate, on the 
 ground that if it could be proved for the plane, it would have to 
 hold for the pseudosphere (or for the sphere), evidently have no 
 value against such a proof where both of these cases have been con- 
 
 *For the pages corresponding to the references given on this page see 
 Table of Contents, above. 
 
32 
 
 sidered, mid where the proof first establishes the existence of a 
 kind of metrics somewhere in space, that afterwards is proved to 
 he possible only for the surface of zero curvature which is the 
 plane. Then only follows the proof of JEuclid^s postulate proper , 
 for the plane, which by its metrics is now singled out among all 
 surfaces of constant curvature, as the only one for which the sum 
 of the angles of a triangle equals two right angles. 
 
 If we examine in detail the axioms which have been made 
 use of in my Dissertation, we find that they are actually those 
 which are necessary for defining the group of continuous motions 
 in general, without, however, postulating that a point in space is 
 definable by three coordinates. Of the postulates that Helmholtz 
 enumerates in his work "Ueber die Thatsachen, die der Geo- 
 metrie zum Grunde liegen," Kon. Ges. der Wis. zu Gottingen, 
 1868, I assume only the first and the third, stated as follows : 
 
 1) Continuity in motion — defined exactly in Axiom 1, on 
 page 63 of my memoir. 
 
 2) Free mobility of rigid bodies and consequent independ- 
 ence from position of spacial figures of all kinds. 
 
 To these, whose order, of course, is reversed, I add impene- 
 trability and infinite divisibility. The number of dimensions I 
 do not assume, but deduce from the postulates. 
 
 A word still with regard to the non-Euclidian geometry. 
 What is its significance and the place it has to occupy in the 
 general science of geometry ? 
 
 I shall here, even at the risk of repetition, recapitulate and 
 supplement in a concise way what I have said with reference 
 to the two non-Euclidian groups of motions obtained by Lie 
 from the Riemann-Helmholtz axioms as modified by him. 
 I think that in this regard the position of Beltrami in his 
 '^ Saggio " and " Teoria fondamentale " is, in general, the only 
 tenable one. In our point-space of three dimensions the Lo- 
 batchevskian and Riemannian geometries for two dimensions 
 are realized respectively on the pseudospherical surfaces and on 
 the surfaces of constant positive curvature, and they can have no 
 other concrete interpretation. With special conventions, how- 
 ever, as to the meaning of ^' distance '^ and '^ angle,'' we get the 
 Poincar^ interpretation by means of what he calls the quad- 
 ratic geometries, i. e., geometries on certain quadric surfaces. 
 Professor Klein's interpretation of the generalized metrics can 
 in no way make the plane become either an elliptic or a hy- 
 perbolic plane, and if Cayley's generalized metrics can be turned 
 
33 
 
 into account for obtaining the elliptic, hyperbolic, and parabolic 
 geometries, it is only for different surfaces actually existing in 
 the Euclidian space that this interpretation can be of any 
 value. The very arbitrariness of these different kirids of metrics, 
 which depends upon the arbitrary value of the constant c in the 
 formida * (x, y)^ c log [.r, y, 0, 0'~\y where 0, 0' are the points 
 of intersection of the range xy with the fundamental quadric, 
 shows tliat the different metises must refer to a whole series of dif- 
 ferent tivo-dimensional manifolds, differing in curvature and con- 
 stituting the elements 31^ of some R^ The aggregate of all these, 
 however, will, in our case, constitute a flat manifold of three 
 dimensions, namely, the point-space of our experience, — just as 
 the aggregate of all possible plane curves of different curva- 
 ture passing through a point in a plane, constitute a plane 
 manifold of two dimensions. 
 
 As to the non-Euclidian metrics in three dimensions, I can- 
 not see any interpretation for this, unless the space to which 
 these metrics refer, be a derivative manifold contained in a 
 higher manifold of four dimensions, since again the very para- 
 meter of the curvature suggests only a particular case out of 
 an infinity of possibilities, arrived at by giving the param- 
 eter all possible values, ranging from — oo to -f oo, and the 
 aggregate of all these would then plainly make a manifold of four 
 dimensions. Here, again, Beltrami seems to have hit the truth 
 with regard to this interpretation of the geometries of Riemann 
 and Lobatchevski for space of three dimensions. Thus, at the 
 beginning of his '^ Saggio," Beltrami says : " We have sought to 
 render account to ourselves of the results to which this new 
 doctrine (the geometry of Lobatchevski) leads ; and following 
 a process which seemed to us actually to conform to the good tradi- 
 tions of scientific investigation, we have tmed to find a real basis 
 to these results. We think we have found one for the planimetric 
 par-t, but it seems to us impossible to find one for the case of three 
 dimensions.'' And then, at the end of the same work, after 
 having proved the interpretability of Lobatchevski's plani- 
 metry, in every particular, by the geometry upon pseudospheri- 
 cal surfaces, Beltrami goes on to say : " From the very nature 
 of the interpretation, we can easily foresee that there can exist no 
 analogous interpretation, equally real, for the non-Euclidian 
 stereometry. In fact, in order to obtain the interpretation 
 
 *Math. Ann., Bd. 6. 
 
34 
 
 which we have just given, it was necessary to substitute for the 
 plane a surface which cannot be reduced to a plane, that is to 
 say, whose linear element can in no way be reduced to the 
 form \/dx^ + dy^y which essentially characterizes the plane itself. 
 Consequently, if we were lacking the notion of surfaces non-appli- 
 cable to a plane, it would be impossible for us to attribute a veri- 
 table geometric meaning to the constructions which we have developed 
 up to this point. Now, the analogy leads one naturally to think 
 that if there can exist a similar interpretation for the non-Euclidian 
 stereometry, this interpretation must be deducible from the con- 
 sideration of a space whose linear element is not reducible to the 
 form Vdx^ + dy'^ + dz^, — a form which essentially characterizes 
 the Euclidian space. And since up till now, as it seems to us, 
 we have been wanting the notion of a space different from the 
 Euclidian, or, at least, such a space is beyond the domain of 
 ordinary geometry, it is reasonable to suppose that, even if the 
 analytical considerations upon which the preceding constructions 
 are based, were susceptible of being extended and carried over from 
 the region of two variables into that of three variables, nevertheless 
 the results obtained in this last ca^e could not be interpreted by the 
 ordinary geometry. This conjecture acquires a degree of prob- 
 ability bounding very closely on certainty, when one under- 
 takes to extend the preceding analysis to the case of three 
 variables. 
 
 " Putting 
 
 18 
 
 E' -" - -" -" ^ "V 
 
 + (a^ — t^ — u^)dv^ + 2uvdudv + 2vtdvdt + 2tudtdu'], 
 
 which takes the place of (1) in two dimensions,* — it is easy to 
 assure oneself that the analytic deductions obtained from 
 formula (1) subsist integrally for the new expression, and that 
 the value of ds given by this last is effectually the value of the 
 linear element of a space in which the non-Euclidian (Loba- 
 
 * The formula (1 ), here referred to, is the one which Beltrami gives at the 
 beginning of his work, for the linear element of a surface of constant negative 
 
 curvature, ds^ ■= W j-z — ^^ '- — ; this serves as 
 
 ya'' — w- — v^Y 
 the basis for his interpretation. 
 
35 
 
 tchevsJdan) stereometry finds an interpretation, as completey speak- 
 ing ANALYTICALLY, OS that given for the planimetry, 
 " But putting 
 
 t=: r cos p^j u = r sin p^ cos p^j v = r sin p^ sin p^, 
 and 
 
 Radr 
 
 we get 
 
 ds' =dp'-{-(R sinh ^ \\dpl + sin^ p^dp^, 
 
 a formula which shows that /), p^, p^ are the orthogonal curvi- 
 linear coordinates of the space considered. 
 
 " Now, M. Lam6 has proved that, taking as curvilinear co- 
 ordinates of points in space the parameters p, p^y p^, of three 
 families of orthogonal surfaces — in which case the distance be- 
 tween two infinitely near points is represented by an expression 
 of the form 
 
 ds" = H^p" + H\dp\ + Hldpl— 
 
 the three iJ^s, as functions of the /j's, must satisfy two distinct 
 systems, each consisting of three partial differential equations 
 of the types, 
 
 ^ _ _L ^ ^ J_ ^ ^2 
 
 and 
 
 dp, Ui ^j + ^p,U2 ^P2r H^ ^p > ^^ 
 
 (Le9ons sur les coordin^es curvilignes, pp. 76 and 78). 
 " In our case 
 
 H— ly H, — R sinh -^, H^ — R sinh -^ sin p, , 
 
 and for these values, the first system is evidently satisfied ; but 
 the second system is satisfied only for R= oo. Hence the 
 expression (18) cannot belong to the linear element of the ordi- 
 nary Euclidian space, and the formulce founded upon this exp^-es- 
 
36 
 
 slon cannot be constructed by means of figures given us by the or- 
 dinary geometry.'^ 
 
 And, again, in his " Teoria fondamentale degli spazii di cur- 
 vatura costante/' Beltrami says : 
 
 " Thus all conceptions of the non-Euclidian geometry find 
 a perfect correspondence in the geometry of a space of constant 
 negative curvature. It is only necessary to observe that while 
 the conceptions of the planimetry obtain a true and proper inter- 
 pretatioUy since they can be constructed upon a real surface, those, 
 on the contrary, which refer to three dimensions, are susceptible 
 only of an analytic representation, since the space in which such 
 a representation could be realized is different from that to which 
 we ordinarily give the name of space. At least, it does not seem 
 that experience could be brought into agreement with the results 
 of this more general geometry, unless we suppose R to be infi- 
 nitely great, that is, the curvature of the space to be zero. 
 This circumstance might, of course, also be due only to the 
 smallness of the triangles which we can measure, or to the 
 smallness of the region to which our observations extend them- 
 selves.'^ 
 
 In his "Teoria fondamentale,^' Beltrami shows, from a general 
 discussion of n-dimensional manifolds, that the linear element 
 in the Riemannian geometry of three dimensions may be taken 
 to be the same as the linear element upon a hypersphere in a 
 space of four dimensions. The equation of the hypersphere, 
 with center at origin, will be 
 
 fj^u' + v^JrW^^ a\ 
 and hence, 
 
 da^ = de + du^ -f dv^ + dw" 
 
 is at once the representation of the linear element upon the hyper- 
 sphere of radius a, and in a Riemannian space of curvature 1 ja^. 
 To obtain the linear element of the three-dimensional space 
 of Lobatchevski, he substitutes ds— — Eda/w, and by elimi- 
 nating 10 he gets (18). The curved Lobatchevskian space, of 
 infinite extent, is then imaged upon the interior of the sphere 
 of Euclidian space, 
 
 e-^u' + v^^ a\ — 
 
 the geodesies of that space being represented by chords of the 
 
37 
 
 sphere. Every geodesic has two distinct real points at oo, which 
 are imaged upon the representative sphere by the two ends of 
 the corresponding chord, so that the spherical surface itself cor- 
 responds to the locus of all points at oo in the Lobatchevskian 
 space. 
 
 But since we proved * that to assume a four-dimensional 
 point-space is to commit a logical error, and since Beltrami's 
 results have certainly given a conclusive analytical proof that we 
 could obtain Lobatchevski's geometry for three dimensions, if 
 we could actually construct a curved three-dimensional manifold, 
 contained in a four-dimensional plane manifold, — we may sur- 
 mise that the only way to obtain a concrete and true interpreta- 
 tion of the Lobatchevskian (as well as the Riemannian) stere- 
 ometry is to be found in Pliicker's idea that our space becomes a 
 manifold of a higher number of dimensions, when, instead of the 
 point, we take as its element a figure depending upon n para- 
 meters, making space a manifold of n dimensions. Therefore, 
 it would seem that one of the simplest ways to look for such a 
 concrete interpretation would be to start with line geometry, 
 which makes space an H^ — the next simplest after the point- 
 space, which is an R^ — and seek what in this geometry would 
 be meant by the terms : " distance,'' " angle," ^' linear element," 
 '^curvature," "parallel," "perpendicular," and other metrical 
 terms ; and see whether the results thus arrived at, — by con- 
 sidering in it the special three-dimensional manifolds (com- 
 plexes) possessed of " curvature," (since by excluding the postu- 
 late of parallels, which is now proved for a flat manifold, 
 such as our point- space must undoubtedly be, we actually ob- 
 tain a manifold of constant curvature), — whether these results 
 do agree with those obtained in the Lobatchevskian and 
 Riemannian geometries, respectively. But such an investiga- 
 tion would go far beyond the limits of the present Dissertation. 
 
 * In the introductory chapter of the Dissertation. 
 
ON THE FOUNDATIONS OF THE EUCLIDIAN 
 GEOMETEY. 
 
 Chapter I. 
 
 SPACE AND ITS DIMENSIONS. 
 
 Definition 1. — Geometry is the science which treats of spacdal 
 forms and magnitudes and their mutual relations. Dealing 
 with magnitudes, and with spacial forms only in so far as these 
 are determined by their magnitudinal relations, Geometry is a 
 branch of the general science of quantity — Mathematics, 
 
 A few introductory remarks are necessary in the way of more 
 accurately specifying the subject of geometry, which I prefer 
 to put in the form of definitions. These definitions, however, 
 will not be only nominal ; most of them prove the actual ex- 
 istence of the objects they define. 
 
 Definition 2. — Space is that in which all bodies exist. It is 
 the condition sine qua non of material objects. This truth is 
 expressed in physics by the assertion that matter , or the sub- 
 stance of which all bodies consist, has extension, or, in other 
 words, material objects occupy space. 
 
 Definition 3. — Experience teaches us that matter is also im- 
 penetrable, i. e., that every material object occupies a definite por- 
 tion of space, which is fixed by certain limits or boundaries and 
 tohich cannot at the same time be occupied by any other material 
 object. The portion of space that is for a time exclusively occupied 
 by a certain material object is called the place of that object. 
 For the sake of accurate terminology I propose to call it the 
 geometrical place. 
 
 Definition 4. — Experience further teaches us that the re- 
 sources of space with regard to its capacity of containing ma- 
 terial objects, or of affording place to the material substance, 
 are absolutely limitless. Thus, notwithstanding the impenetra- 
 bility of material substance, explained above, beside any occu- 
 pied space there is always room enough for the existence of other 
 material objects, and any vacant space is always conceived of only 
 as susceptible of being filled up with matter. Moreover, any 
 material object is conceived of as capable of being divided into 
 
 38 
 
39 
 
 any number of portions, and these again subdivided into lesser 
 ones, and so on, ad infinitum. In this respect, extended sub- 
 stance and, hence, also space, follow perfectly the nature of ah~ 
 stract quantity, ranging both ways — from the finite to the in- 
 definitely small, on the one hand, and to the indefinitely large, 
 on the other. 
 
 Definition 5. — The geometrical place of a body, being that 
 portion of space which is occupied by that body to the exclusion oj 
 any other body, lias the same spacialform and dimensions as the 
 body which fills it up. We mean by this, that whatever meas- 
 urements in regard to extension the whole body, or its several 
 parts, may have, the same are attributed to its geometrical 
 place, and whatever arrangement of extended parts makes up 
 the form of the body, the same belongs likewise to the geomet- 
 rical place, and vice versa, — so that in these regards the geomet- 
 rical place may be substituted for the body, and conversely. 
 The geometrical place alone, apart from all other physical prop- 
 erties, or, in fact, apart from the matter filling it up, is dealt 
 with in geometry, — and is regarded by this science : 
 
 1°. As a magnitude — that is, not only as something that 
 can be greater or less, the reason for which is given in Definition 
 4, but as something that can be measured, that is accurately 
 compared, with a view of an exact quantitative determination, 
 with a standard mxxgnitude of the same kind, which is arbi- 
 trarily taken as a unit, and can be repeated any number of 
 times, or divided into a certain number of equal parts, thereby 
 becoming either equal to, or greater, or smaller than the mag- 
 nitude in question ; and 
 
 2°. As a form, consisting of a definite arrangement of parts 
 according to some law, which can also be expressed by numbers. 
 
 Definition 6. — The geometrical place of a body is called a 
 solid in geometry, meaning by it, that it is mentally represented 
 as preserving a fixed form and dimensions. The geometrical 
 solid is a mere ideal abstraction and has nothing to do with phys- 
 ical solidity, from which, however, it is originally derived. 
 Geometry does not, therefore, treat it as impenetrable. It is, 
 indeed, only the impression left by a body in surrounding space 
 conceived of as capable of preserving the impression after the 
 body itself has been removed. As a magnitude or thing to be 
 measured and expressed in numbers, mthout regard to its 
 form or outer appearance, it is called volume. 
 
40 
 
 Postulate 1. — The geometrical solid or body may be mentally 
 imagined as moving about in fixed space, or changing its position 
 with respect to other bodies, whether physical or geometricaly 
 without distortion or change of form. The solid is said to pos- 
 sess geometrical rigidity, meaning by it that the disposition of the 
 parts with reject to each other is fixed and unchangeable, or, that 
 there is no internal motion. This idea of geometry is derived 
 from the fact that space is conceived of as affording a mere passive 
 capacity of being filled up with matter, all changes of form being 
 r^erred to the active principle of the material substance py^oper, 
 i. e., to physical causes alone (of. Definition 4). It is also based 
 upon the undoubted fact of universal experience that, in so 
 far as can be ascertained by observation and experiment 
 (measurements — astronomical, physical, and geodesic), no real, 
 or physical, ligid body, moving about in space, has ever been 
 known to undergo any alteration in form or dimensions on account 
 only of change of position in space, without regard to physical 
 causes which, in most cases, have been found quite adequate to 
 account for such alterations. And even if the contrary were 
 the truth in the case of real bodies, the Euclidian geometry 
 would still have nothing to do with such alterations, as it con- 
 siders only ideal rigidity, where change of form or dimensions 
 as depending upon position, is purposely eliminated for the 
 sake of simplicity, and may be left to other branches of the 
 mathematical sciences to consider (Kinematics, for instance, 
 may very properly consider such questions as a special kind of 
 liaisons — constraint — depending upon any number of para- 
 meters, those of position included). But the Euclidian geom- 
 etry considers only the simplest case, even if it were only an 
 idealized abstraction. 
 
 Definition 7. — A body is said to be equal to another geomet- 
 rically, when their geometrical places can be made to fill each 
 other without remainder of any parts of the one, not filled by 
 corresponding parts of the other. When the geometrical places 
 thus fill each other, we say that the geometrical bodies coincide, 
 — coincidence, as thus defined, being a proof of equality, in- 
 variably resorted to in geometry. 
 
 When the coincidence can take place only with the change 
 of form resulting from a mere rearrangement of parts, these 
 last preserving separately their respective magnitudes and forms, 
 the bodies are said to be of equal volume, though not equal in 
 form. 
 
41 
 
 Definition 8. — One body is said to be r/reater than another, 
 when some of the parts of the one can be made to coincide 
 with all the parts of the other, while there still remain some 
 parts of the first, having no corresponding parts of the second 
 to coincide with. The other body is then called the less. 
 
 Scholium. — Having firmly established the empirical and 
 rational basis of the notions contained in the previous defini- 
 tions, it may not be amiss to give a more compact and abstract 
 form to the logical process by which they are obtained, which is 
 free from all cavils on the part of those who think that spacial 
 forms and magnitudes may, for all we know, be certain functions 
 of absolute position, which we shall never be able to ascertain or 
 disprove. Starting with the notions of space, matter or ex- 
 tended substance in general, position, and change of position or 
 motion, and with the abstract notion of quantity, we may as- 
 sume, for the sake of abstraction, the existence of a hypothet- 
 ical impenetrable material substance, infinitely divisible, — i. e., 
 possessed of the following properties : — 
 
 1 . Impenetrability. — Every determiiiate portion or quantity of 
 this substance occupies or fills up a corresponding portion of space 
 which cannot at the same time be occupied by any other portion of 
 the same substance. Any tioo portions of space thus filled up by 
 the same quantity of the substance at different times , are said to be 
 equal in capacity y and any two portions of the substance tohich 
 can fill up the same portion of space at different times, or dif- 
 ferent portions of space of equal capacity at the same time, are said 
 to be of the same bulk. So that to each bulk, which measures 
 the quantity of the hypothetical substance, there is a correspond- 
 ing capacity of the space which is occupied by it at any moment, 
 to the exclusion of any other portion of the same substance ; 
 to a greater portion of the substance, there corresponds a greater 
 capacity of the space occupied by it, to a double or multiple 
 bulk, a double or equimultiple capacity of the space, and to 
 any part of a given bulk, a corresponding part of the capacity 
 of the space occupied. The generic term for bulk or capacity 
 alike is volume, so that the quantity of the substance (bulk) and 
 the spac£ filled up by it, to the exclusion of any more of the 
 same substance (capacity), are said to be equal in volume. 
 
 2. Infinite divisibility. — If a portion of the substance is di- 
 vided into n portions, such that they can fill up spaces of equal 
 capacity each, that portion is said to be divided into n equal parts. 
 
42 
 
 The property of infinite divisibility now becomes perfectly com- 
 prehensible, and is possessed, according to hypothesis, both by 
 the hypothetical substance, and by the spa^e giving position to this 
 substance. 
 
 3. Form ; rigidity, or plasticity. — If two equal poiiions of 
 the substance {of equal bulk or volume) are made to fill up succes- 
 sively the same fixed portion of space (not merely spaces of equal 
 capacity), then in these two portions of the substance we observe not 
 only equality in bulk of the whole, but also of corresponding parts, 
 filling up in the two cases the same corresponding parts of space ; 
 that is, the two portions of the substance, while each fills up 
 in tmm the space considered, have a similar arrangement of parts 
 that are equal in bulk in the two cases, no mutter in what man- 
 ner the division is made, and however small the parts considered. 
 The two equal portions of the substance, in their successive positions, 
 are, therefore, said to be equal not only in volume, but also in 
 form — equal form thus meaning an equal arrangement of equal 
 parts. When a portion of the substance leaves a certain posi- 
 tion, passing into another position, it may change its form (i. e., 
 the arrangement of parts may change, so that an arrange- 
 ment denoted by a, b, c, • - - k, may now have to be repre- 
 sented by • • • e " • k .../... g ... a • • • h - --, where a, 
 ^j ^j ^j ff 9) hf h ^tc., denote unequal portions). Moreover, 
 even if as a whole it does not change its position, that is, 
 if some one part of it, at least, preserves its old position, the 
 form of the whole may still change, and actually does change, 
 whenever the remaining parts change their relative positions to this 
 fixed part and to each other ; and, provided the space filled up 
 by the whole is still continuous, that is, it is still filled up com- 
 pactly and represents one concrete whole, without interruptions 
 of vacant or unoccupied portions intervening, we say, the space 
 occupied by the whole has not changed its volume, but has 
 changed its form, and so has the substance filling it up. The 
 distinction between two portions of space of equal volume and 
 two portions of both equal volume and equal form is now clear 
 and unambiguous. In fact, we have seen that, when the sub- ' 
 stance as a whole does not leave its original position, i. e., when 
 at least one of its portions preserves its position, a change in form 
 is possible only in virtue of the change of position of the remain- 
 ing parts with respect to the stationary part and with respect to 
 one another, — in other words, change of form is caused by mo- 
 
43 
 
 tion of parts of the whole with respect to one another, that is, 
 by internal motion. The same is, therefore, true with respect 
 to a substance which has left its original position entirely, fill- 
 ing now up a portion of space which has not the smallest part 
 in common with the original position. If there has been inter- 
 nal motion or a rearrangement of unequal parts besides, then 
 the substance has also changed its form ; if there has been no 
 internal motion, the original arrangement of parts having been 
 preserved, the substance has only changed its position, but not 
 its form. If a substance resists a change of form, as just de- 
 fined, whether the whole is at rest or in motion, we may say 
 that the parts are held fixed to one another, and we call 
 this state of the substance the rigid state ; the whole substance 
 is then said to form a solid or a rigid body. The portions of 
 space which represent any two successive positions of a solid 
 in motion, are said to be equal to each other, in volume and 
 form, just as two solids that can be made to fill up successively 
 the same space, are themselves said to be equal in volume and 
 form. When two equal solids are made successively to fill up 
 the same space, then, by abstracting from time, that is, disre- 
 garding the fact that the filling up can take place only at dif- 
 ferent moments, we simply say that the two solids are made to 
 fill up the same space, or, they are made to coincide with each 
 other — coincidence being a test of equality in volume and form. 
 If, on the contrary, the substance does not resist a rearrange- 
 ment of parts, these parts are not held rigidly to one another, 
 and change of form is possible without change of volume. 
 Such a portion of the substance is said to possess plasticity. 
 
 We see now that these notions, though having a firm empir- 
 ical basis, are not absolutely dependent upon the actual con- 
 dition of things. The hypothetical substance, of absolute impene- 
 trability, need not actually exist, but a^ an abstraction, agreeing, 
 in general, with our expet^ence, it may serve as a starting point 
 for the only possible science of measurement of extension ; since the 
 notions based on its assumption are clear and unequivocal, and 
 absolutely necessary to make the investigation of the laws of 
 spacial forms and magnitudes possible. Moreover, we must 
 agree to class all actual phenomena, in so far as they conform 
 to the laws deducible from these notions and from those that 
 follow in this introduction, as geometrical phenomena, that is, 
 such as depend upon the essence of extension only ; and, in so 
 
44 
 
 far as they deviate from these laws, they must be explained by 
 physical causes, and any attempt to confuse these two (as very 
 able geometers, like Clifford and others, have done) would 
 only tend to raise a dust of endless discussion, which would 
 never permit us to see the real foundations of geometry. 
 
 Definition 9.-^ A rigid physical body is said to be sur- 
 rounded by vacant space on all sides, when it can be moved in 
 all directions : forwards and backwards, to the right or to the 
 left, and so on, in all intermediate directions.* JVhen some 
 other body is posited beyond the vacant space, in any part of it, 
 the two bodies are said to be at a distance from each other, that 
 admits, either of the position of some third body between them, 
 or of the motion of one towards the other. In the latter case, 
 the bodies are said to approach each other, the distance between 
 them becoming less and less, until it vanishes altogether, ad- 
 mitting of no further approach towards each other, the bodies 
 then being in contact. These ideas of distance and contact are 
 transferred upon geometrical solids, or the geometrical places of 
 the bodies corresponding to the positions of the physical bodies 
 just mentioned. 
 
 Definition 10. — When two rigid physical bodies are brought 
 into close contact with each other, so that no further motion of 
 one toward the other is possible, they are said to have reached 
 the limits or boundaries of each other, and if these limits are to 
 some extent continuous, — i. e., when they touch each other in 
 many parts, the touchings being uninterrupted by intermediate 
 vacant space, which happens when the bodies fit each other, — the 
 limits are then called surfaces. A physical surface is, accord- 
 ingly, the continuous boundary where the rigidity of a body 
 just begins, or where the physical property of impenetrability 
 just begins to act. If the body is surrounded by vacant space, 
 the surface of the body is the boundary separating the impene- 
 trable matter of the body from the capacious space. But it is 
 neither the one nor the other, since the smallest part of the 
 body has some of its parts removed from the boundary by the 
 interposed rigidity of other parts of itself. No other rigid body 
 could possibly have access to those concealed parts without 
 overcoming rigidity, and, therefore, no part of the body, how- 
 
 * The word direction is used here in the common acceptance of its meaning, 
 viz: some course, but it is really vague. The scientific meaning of the 
 word will be given in another place in this work. 
 
45 
 
 ever small, can belong to its surface, of which the essential 
 characteristic is its being in contact with some other body, w, 
 ivith vacant space and, hence, capable of being brought into 
 contact with some other body. Surface, therefore, has no mag- 
 nitude of the same kind as a body ; in other words, it has no 
 bulk or volume, and it can never amount to any part of volume. 
 But, as is shown in the following definition, it nevertheless has 
 magnitude and form of its own ; in other words, it is a thing 
 that can be measured and expressed in numbers — these numbers 
 being in determinate relations to those expressing the volume 
 of the body bounded by the surface. One of the tasks of 
 geometry is, in fact, the discovery and determination of these 
 relations. The idea of surface is also transferred from physical 
 bodies to geometrical solids, and the geometrical surface may be 
 said to represent the geometrical place ofaphydcal surface. Geom- 
 etry regards it as a separate entity, capable of existing by 
 itself and moving about in space, or changing its position with 
 regard to other bodies and surfaces, whether physical or geo- 
 metrical, without distortion or change of form. 
 
 Definition 11. — Since a rigid body, immersed in unoccupied 
 space, or in any plastic material substance, displaces a portion 
 of the material, or occupies a portion of the void, to the exclu- 
 sion of other matter, equivalent to its own volume, and since 
 this rigid body exposes only its surface — the interior parts not 
 coming into play at all in the act of this displacement (the in- 
 terior might as well be imagined hollow or devoid of matter in 
 this connection), — it is evident that, in general, surface ought to 
 be a function of volume, increasing with the increase of the 
 last ; that is, to a large volume there must in general correspond 
 a large surface, although the converse is not a necessary conse- 
 quence. At any rate, it is quite inconceivable how a rigid im- 
 penetrable body could take up space to the exclusion of other 
 material substance, which, on account of its capability of mo- 
 tion or change of position, can be prevented from occupying the 
 same space as the body consider ed, only by the boundaries of the 
 last J — were it not that these boundaries are in themselves an extended 
 magnitude, standing in some functional relation to volume and 
 form. Accordingly, surface must have portions, all of which 
 may be exposed to vacant space, or in contact with surfaces of 
 other bodies, or some exposed and the others covered by cor- 
 responding surfaces of other bodies. In this last case, the ex- 
 
46 
 
 posed surfaces can again be brought into contact with surfaces of 
 other bodies. Two surfaces will then be equal, if they can lie 
 upon each other and mutually cover all their parts ; and one is 
 greater than the other, when a part of the first can cover the 
 whole of the second, while another part of the first will remain 
 exposed, or covered by a third surface. 
 
 Scholium. Any part of a body may be regarded as a sepa- 
 rate body (in the geometrical sense of the term) from the re- 
 maining part, since each can be imagined to move about in 
 space independently of the other, and without distortion or 
 change of magnitude ; and, while the two constitute the parts of 
 the same solid, the limit common to both is a surface of some 
 definite shape and magnitude. 
 
 Corollary. Surfaces coincide with one another when the 
 bodies limited by them coincide and, conversely, bodies limited 
 by surfaces that can be made to coincide with one another, 
 must themselves be capable of coincidence, since when the sur- 
 faces are brought into actual coincidence, none of the bodies 
 can help being everywhere within and nowhere beyond their 
 coinciding limits. 
 
 Definition 12. — Both experience and reasoning lead us to 
 the conclusion that, while volume, or unspecified space — space in 
 all possible directions , wherever motion is possible — is homogeneous, 
 surface, or that which limits a body, may be of very different 
 kinds, having almost nothing in common, except that an in- 
 definitely small, or infinitesimal, part of any surface may be 
 imagined to move towards another infinitesimal part of the 
 same surface by a continuous infinity of paths in the surface 
 itself, as will be shown later. But this common property is not 
 sufficient to make surface a magnitude, always definable with 
 mathematical precision, and capable of being expressed in a 
 voluntarily chosen unit. The indefinite size of the small part 
 that is capable of congruence would make the computation of 
 areas with mathematical precision impossible, unless there be ways 
 of reducing these to surfaces capable of coincidence in finite por- 
 tions. Volume, as a magnitude, which, in fact, is only the 
 capacity of space to contain matter of a constant ideal impene- 
 trability, is everywhere the same. Any part of volume is 
 capable of coincidence with any other corresponding part of 
 volume ; this coincidence is given directly in the fundamental 
 idea of motion together with the idea of rigidity of the moving 
 
47 
 
 bodies. Volume, therefore, or space unspecified, is homo- 
 geneous ; whereas surface, as having an infinite variety of forms, 
 is not so. And while, for instance, a smaller body, being posited 
 within, or surrounded by, a larger one, invariably occupies a 
 part of the volume of the larger, — the limits of the two unequal 
 bodies may be, and, in fact very frequently are, incapable of co- 
 incidence in any of their finite parts. In order to make surface 
 a mathematical magnitude (i. e., definable with precision), there 
 must be found at least one homogeneous surface, of which any 
 finite part is capable of coincidence with any other correspond- 
 ing part of the same, and, after taking such a surface as stand- 
 ard, there must be found rules how to reduce other surfaces, with 
 any desirable degree of precision, to this standard. Such homo- 
 geneous surfaces, the essential characteristic of which is that 
 any part of them can be imagined to slide upon the whole, re- 
 maining always in coincidence in all its parts with correspond- 
 ing portions of the whole, do really exist ; and their existence 
 is likewise a matter both of experience and of mathematical de- 
 duction. Any of these homogeneous surfaces might be taken 
 as a standard of measurement ; but one, as affording the greatest 
 advantages for computation, and being capable of indefinite ex- 
 tension, is accepted as the standard, and all others are always 
 reduced to this single standard. (It is needless to remark that 
 a difference of choice of the standard surface would, like dif- 
 ferent systems of numeration, lead only to different methods of 
 computation, but not to different results.) 
 
 To sum up : — surfaces are multiform and are, therefore, seldom 
 capable of coincidence in their finite portions. The nature of 
 measurement, however, requires a homogeneous standard, to 
 which all magnitudes of the same kind that are to be meas- 
 ured, can easily be reduced. Surfaces answering this descrip- 
 tion of homogeneity and, hence, capable of serving as a standard 
 of measurement of area and form, actually exist — of different 
 species and infinite in number — the essential characteristic of 
 all of which is the capability of any portion of such a surface 
 to slide along the whole, remaining always in coincidence with 
 different corresponding portions of the whole. The simplest 
 of these is chosen as the norm ; it will be shown that it pos- 
 sesses the additional properties of being indefinitely extended, 
 beyond any arbitrary limit, and of its side towards the interior 
 of the body which is limited by it, fitting upon the opposite 
 
48 
 
 exposed side, so that the two can be made to coincide 
 (plane). 
 
 Definition 13. — If two bodies are in partial contact of their 
 surfaces with each other, the boundary separating the part of 
 surface in contact from the part not in contact, in either of the 
 bodies, is the limit of either the covered or the exposed portion 
 of the surface. For simplicity, let us imagine the surfaces to 
 be homogeneous. Any finite portion of the uncovered surface 
 can be imagined to be in contact with the surface of some third 
 body (see Def. 11), whose form at a finite distance is immaterial, 
 and which moves upon the rest of the uncovered surface, along 
 any path in it, until it reaches the limit of the covered surface, 
 where it is checked in its motion by the rigidity of the surface 
 of the covering body, and can only move so, that, while a por- 
 tion of its surface touches the covered, another portion 
 touches the covering, body. Motion along the boundary 
 separating the covered from the uncovered surface, must 
 still be possible for an indefinitely small portion of a finite 
 body, whose contact with the two bodies, in the exposed por- 
 tions of their respective surfaces, blends along the boundary 
 for the infinitesimal element ; for, as this finite body can be 
 conceived to move along the two, remaining always in partial 
 contact with each, the infinitesimal element, touching the two 
 simultaneously, must, of necessity, find a region of motion 
 along their common boundary. This boundary will, therefore, 
 have parts of its own, viz., the specializations of position of 
 the infinitesimal touching element considered ; but these parts 
 will not be of the same kind as the parts of a surface ; neither 
 can the whole be a part of surface. In fact, it cannot be a 
 part of the covered surface, since it must likewise belong to the 
 uncovered surface ; but, however small, a portion of the cov- 
 ered surface, if not infinitesimal, will always have some of its 
 parts removed from the exposed region by intervening parts of 
 the covered region. Similarly, it cannot be a part of the ex- 
 posed surface. But we have proved that it must have parts of 
 its own. The boundary between two portions of surface is, 
 therefore, a new magnitude ; it is a line, and its parts are dif- 
 ferent from those of a solid or a surface, but like them expres- 
 sible in numbers having some determinate relation to the num- 
 bers expressing the magnitudes of volume and surface. (The 
 same reasoning holds also in the case of non-homogeneous sur- 
 
49 
 
 faces, provided a certain amount of plasticity of form is al- 
 lowed to the touching parts of the moving body). 
 
 Scholium. — The analysis of the last Definition can be made 
 more concrete by the following r6sum6, which also puts its re- 
 sult in a somewhat different light : — 
 
 When three bodies touch one another in their surfaces and 
 they also fit one another, so that no vacant space is left between 
 the parts of the touchings, the same part of surface can belong 
 only to two of the touching bodies at once ; while the boundary 
 separating the surface belonging to any pair simultaneously, 
 from the surface belonging to either of the pair and the re- 
 maining third body, belongs to all three bodies simultaneously 
 and, hence, is not a surface, but a line. It is the continuous 
 boundary of rigidity of three bodies that have come into con- 
 tact with one another, and any indefinitely small part of it can 
 be conceived to pass to the position of any other of the same 
 only by two different courses, in case the boundary is com- 
 pleted and the line returns into itself, and only by one course, 
 if the boundary is not completed. 
 
 Corollary. — From this follows immediately the statement 
 made in Definition 12, that an infinitesimal part of a surface can 
 move towards another infinitesimal part of the same surface, 
 by a continuous infinity of different paths in the very surface. 
 
 The idea of a line is also transferred from a physical sur- 
 face upon a geometrical one, and the geometrical line may 
 be said to represent the geometrical place of a physical line. 
 Geometry regards it as a separate entity, which can be conceived 
 to move about in space, or upon a suitable surface, without 
 change of form or magnitude. 
 
 Definition 14. — Any body may be regarded as divided into 
 two definite parts, having one part of their surfaces — namely, 
 that created by the section — in mutual contact, while the surface 
 of the original whole body is now also cut into two parts, each 
 belonging to one of the two bodies now taking the place of the 
 original one. The boundary separating the common surface 
 from the distinct parts will, according to Definition 13, be a line 
 on the surface of the original body, — thus being, from one side 
 at least, exposed to space. Any part of this line may be brought 
 into contact with some other body. When the whole line is in 
 contact with other bodies, leaving no part of it adjacent to va- 
 cant space, it can be regarded as covered by another, the 
 
50 
 
 duplicate of the first in form and magnitude, which is traced 
 upon the surfaces of the touching bodies, — and no body, be- 
 sides, can be in contact with the line. The line and its du- 
 plicate are then said to coincide — coincidence in this case also 
 being a proof of equality in both form and magnitude. If we 
 form, in a similar way, the duplicate of only a portion of the 
 line, this duplicate will evidently be less in magnitude than the 
 whole line. 
 
 Corollary. — Lines coincide with one another, when the sur- 
 faces limited by them coincide, since none of the lines limiting 
 these surfaces can be within or mthout the limited surfaces. 
 The converse, however, is not necessarily true (except in the 
 case of the plane, as will be shown later). 
 
 Scholium. — Lines, like surfaces, are multiform; but there 
 exist also homogeneous lines, of which any part is capable of coin- 
 cidence with any corresponding part of the same. One of such 
 homogeneous lines, capable of indefinite extension and uniquely 
 determined by any two of its elements, is accepted as the stand- 
 ard of line-measurement (straight line). 
 
 Definition 15. — When the surfaces of two bodies are in par- 
 tial contact, the bounding line being, in its turn, brought into 
 partial contact with a corresponding line upon the surface of a 
 third body, or, in other words, a part of the line being covered, — 
 any indefinitely small part of the remaining uncovered line may 
 again be covered with a corresponding infinitesimal line upon 
 the surface of still another body ; then the parts of the last body, 
 immediately adjacent to the line in question, may be imagined to 
 move along the uncovered part of the line, only by two opposite 
 paths, until they reach the limit of the covered part of the line, 
 where further motion is checked altogether by the rigidity of 
 the surface of the body which has effected the first partial 
 covering of the line. This limit, therefore, separating the 
 covered part of the line from the uncovered, has no parts at all, 
 since an infinitesimal element, coming up to it, finds no exten- 
 sion to move upon. (There would be no advantage in this case 
 in starting with a finite part of the line moving upon the un- 
 covered part, by allowing plasticity of form in case of non- 
 homogeneity, since — a line being a region of motion, only for an 
 infinitesimal part of a body, touching two surfaces at once — 
 the motion upon it must be a filing in, or successional motion, 
 all along the line ; that is, the motion of a row of individual 
 
51 
 
 members, where there is a unicursal succession of each mem- 
 ber into the place of the one immediately preceding, going on 
 either indefinitely, or returning to the original starting place, 
 — the members being the infinitesimal portions of the line, and 
 all of them belonging to the same series.) The proof that the 
 limit we have found is a part neither of the covered, nor of 
 the uncovered, portion of the line, is perfectly similar to the 
 proofs given for the limits of a solid and a surface, and need 
 not be repeated again. The boundary of a line is no magnitude. 
 It is called a point in geometry, which regards only its position 
 or geometrical place. We say its motion generates a line, 
 meaning by this that a line represents a field of motion for it, 
 or, otherwise, the path of a moving point. It is regarded in 
 geometry as a separate entity, which can move about in space 
 independently. It is neither homogeneous nor heterogeneous, 
 since it has no parts. The geometrical places of two points 
 always coincide with each other, as soon as they are brought 
 into the same position in space, within a body, upon a surface, 
 or a line. It is, therefore, regarded as the element of space. 
 
 Corollary. — When two lines coincide, their ends, or the points 
 limiting them, coincide also, i, e., these ends have the same 
 positions, two by two. 
 
 To sum up what has been stated in the foregoing definitions : — 
 
 The boundary separating impenetrable substance from capa- 
 cious space, or the region where no motion is possible, from 
 the region where motion is wholly unimpeded, is a surface, and 
 admits only of motion in contact along finite regions, with the 
 condition of plasticity of the touching surface of the moving 
 body for the case of non-homogeneity. 
 
 The boundary separating uncovered surface from covered 
 surface, or the region where motion in contact is possible, from 
 the region where motion in contact for finite bodies is also im- 
 possible, is a line, and belonging neither to the first nor to the 
 second, it admits motion in contact for an indefinitely small 
 portion of a body. 
 
 The limit separating an uncovered portion of a line from a 
 covered portion, or the region where motion in contact is pos- 
 sible for an infinitesimal portion of a body, from the region of 
 absolute exclusion of motion, is a point, which is, thus, the 
 position of an infinitely small portion of a body at rest ; it has, 
 therefore, no dimensions and only position. 
 
62 
 
 Space, in its totality, being the repository of extended sub- 
 stance which is capable of motion (change of position) and 
 endowed with the properties of impenetrability, rigidity, and 
 infinite divisibility, limiting and bounding vacant space in cer- 
 tain definite ways, — gives rise to three difiPerent kinds of spacial 
 magnitude, so connected that one is the limit of the other and 
 is limited by the third. The point is the result of limitless 
 divisibility of any of the three kinds of extended spacial mag- 
 nitude. 
 
 Space is, therefore, a tridimensional manifoldness, only be- 
 cause of its three chief attributes, giving three different kinds 
 of specializations of position, limiting each other and so con- 
 nected that there is always a certain determinate relation be- 
 tween the units of one kind of space and the units of the other 
 kinds. It was shown in Definition 11, that the rigid surface 
 by means of which the property of impenetrability makes itself 
 effective, must be some function of the volume ; reasons were 
 also given why a surface should be a field of motion for finite re- 
 gions in contact, and a line, a field of motion for infinitesimal 
 regions of contact ; from which will follow at once, that a line 
 is a differential element of a surface, and a point, an element of 
 a line. Limitless space, in its totality, would be a one-dimen- 
 sional magnitude, ranging from zero to infinity, in terms of 
 volume alone, — were it not for the invariable relations between 
 the units of volume and those of the two subcategories of speci- 
 fied space, resulting from the fact that they always limit one 
 another. In order, therefore, to suppose that space has more 
 than three dimensions, we must conceive that a point has di- 
 mensions, since we began our analysis from free unlimited space, 
 and found it, in itself, without considering its limits, to be one- 
 dimensional ; and only in relation to its limits does it become 
 tridimensional, since its first two limits (but not the third) are 
 also magnitudes and bear certain fixed relations to unspecified 
 space as a magnitude. Since u, du, d^u^ where u represents a 
 piece of unspecified space, i. e, volume, are all variables, but 
 not so d^Uj it follows that, if u is a function of .r, it must be of 
 the third order : lo = ax^ may do for the simplest representation 
 of such a function, where dx will represent the differential of 
 a line ; dx may, of course, be a homogeneous function of the first 
 degree, of a number of differentials dx^, dx^, dx^, • • ■ dx^ ; but 
 then there must be ti — 3 linear relations between the dx^a, re- 
 
53 
 
 ducing the number of independent ones to three. It is absurd, 
 by reasoning in a reversed order, to infer by a kind of induc- 
 tion, that, just as a point in moving generates a line, a line, 
 in moving out of its regions, generates a surface, and a sur- 
 face generates a body, so tridimensional space, in moving 
 out of itself, will produce a new kind of space. In the first 
 'place, we would have to prove in general, that if our reasoning 
 holds for n, it will hold for n + 1, as we always do in mathe- 
 matics in such a kind of induction. Riemann^s construction of an 
 (n -f- l)-fold variability, out of an n-fold one and a variability 
 of one dimension, is based on the assumption that the n-fold 
 variability passes over into another one, entirely differe'iit, in a 
 determinate way, so that each point of the first passes over into 
 a definite point of the other, which is not at the same time a ptolnt of 
 the first, — an assumption that must he proved in each particular 
 case. In our case, this assumption is actually equivalent to as- 
 suming, that space can move out of space, which is absurd by the 
 very definition of space, viz., space is that which gives place to 
 material objects, whether at rest or in motion ; so that wherever 
 motion is at all possible for a tri-dimensional piece of space, there 
 is space again. In the second place, even if we admit the possi- 
 bility of space moving out of itself into some other region, we have 
 not admitted any new property of space which might be the ob- 
 ject of measurement, since we tacitly assume that the new region 
 is not space, so that space would remain again tridimensional. 
 And even if we should admit a new unknown property of 
 space, we would still have to prove that tridimensional space 
 is its limit ; and that its units bear a fixed mathematical rela- 
 tion to the units of space we have already considered. Indeed, 
 it is not sufficient for a phenomenon to have a certain number of 
 properties, in order to consider that phenomenon of as many 
 dimensions as the number of its properties. The properties 
 must be the limits of each other, and their units must stand in 
 a certain invariable mathematical relation to one another. 
 
 The following is an analytical deduction of the number of 
 dimensions of space considered as a point-manifold, which 
 was written up and added, as a supplement to the introductory 
 chapter of the main body of the Dissertation, two years after 
 the latter had been completed. The purpose of this analysis 
 is to prove the a-priori necessity of three dimensions, when the 
 point, as usually defined in elementary geometry, is taken as 
 
54 
 
 the element of space. It will follow that to make space a four- 
 dimensional manifold, without changing its element to some 
 other geometrical entity, will involve a contradiction in terms. 
 The discussion is divided into 13 paragraphs. — 
 
 1) Let the whole original manifold he S, which we suppose to 
 be continuous J i. e., any two different positions in it can be 
 reached from each other only through an unbroken series of other 
 positions, all in S, whose number is infinite. 
 
 2) Let an invariable piece of it — 11, endowed with impen- 
 etrability, rigidity, and infinite divisibility, be imagined as 
 capable, as a whole, of changing its position in S. The invari- 
 ability of U is characterized by the fact that the mutual dispo- 
 sition, or arrangement, and the relation, of the parts of f/, ob- 
 tained in a determinate way by any arbitrary subdivision, is to 
 remain unaltered with respect to one another and with respect 
 to the whole of U, considered as an entire manifold in itself. 
 We say that internal motion, of parts of U within C7, is ex- 
 cluded, and any two parts that have been separated from each 
 other by certain continuous series of other parts, in one posi- 
 tion of U within >S^, will remain so in any other position. We 
 thus arrive at the notion of Bulk or Volume. Further, this 
 notion is made more precise by postulating, that, when such a 
 rigid piece has once occupied a definite portion of vacant or 
 unoccupied 8, to the exclusion of any other impenetrable 
 piece, it will always be brought into coincidence with that 
 portion again, as soon as a finite part of it, no matter how 
 small, will be brought into coincidence with the corresponding 
 part of 8 it has occupied originally ; any other small portion 
 of U will then also have to occupy its original position within 
 8. (According to this postulate, it is perfectly indifferent 
 whether Z7is conceived to move within 8, or U is conceived 
 as stationary and 8 as changing its relation with respect to 
 Uf in giving it position in different portions of itself — all 
 these portions being, of necessity, equal in bulk or volume to 
 one another and to U.) If we should measure only bulk or 
 volume, we would get only one dimension. The property of 
 impenetrability of the movable pieces, however, leads us to 
 distinguish a new category of manifold, subordinate to the cate- 
 gory >S' and contained in it, in the following way : — 
 
 3) Suppose U fixed, and another piece F, of same nature, 
 moving up to it, reaches the boundary of the latter. It will 
 
65 
 
 then be prevented from occupying the same place as U by the 
 impenetrability of the two, exhibited in their boundaries, so 
 that a certain kind of motion of V is checked when the latter 
 comes into contact with U. Here we have the notion of the 
 boundary of U separating it from vacant Sy or from other pieces 
 V of same nature as itself. 
 
 4) This first derivative boundary of U, which we may denote 
 by Z7', is neither a portion of Z7, nor of V that has come into 
 contact with Uj nor of the vacant 8, in which U is posited ; it 
 may, or may not, have portions of its own. Example : — the 
 limits of a period of time, no matter how great, have no parts, 
 since there is no possibility for an invariable piece of time to 
 change its position with respect to other invariable pieces, and 
 move up to them to come in contact with their boundaries. In 
 our case, however, U' must have parts of its own, as will 
 become evident from the following considerations : 
 
 5) By infinite divisibility of a rigid piece F, we may arrive 
 at the notion of a plastic substance, each infinitesimal portion 
 of which occupies a corresponding infinitesimal portion of the 
 manifold S, and the whole retaining only the property of im- 
 penetrability, having lost, however, rigidity. It will repre- 
 sent, in fact, a plastic substance, the smallest portions of which 
 are easily capable of separation and change of position with 
 respect to one another and with respect to the whole of their 
 aggregate. The rigid piece U considered, if immersed in this 
 plastic substance, will displace a portion of it equivalent to the 
 bulk of the portion immersed. To a smaller portion of U im- 
 mersed will correspond a smaller bulk of the plastic material 
 displaced, and to a greater portion immersed will correspond a 
 greater bulk displaced. Now this displacement is necessarily 
 effected only by the boundary of the immersed portion of Uj 
 and by that part of it alone which has, before immersion, been 
 exposed to vacant /S', as distinguished from the remaining part 
 which — separating the immersed portion of U from the non- 
 immersed — could have no effect in the act of displacement 
 considered. It follows, therefore, that to a greater bulk dis- 
 placed — and hence to a greater portion of U immersed — cor- 
 responds a greater portion of the boundary U' y which we may 
 call C7.' (i = immersed), as distinguished from the remaining 
 portion of U'y which, as belonging to the exposed portion of 
 U only, we may call Z7J (e = exposed). By changing con- 
 
56 
 
 tinuously the portion of U immersed, from an infinitesimal 
 bulk to the whole bulk of U, we arrive at the notion of a con- 
 tinuously increasing boundary 1[J[, from an infinitesimal to the 
 whole of U' , and a correspondingly decreasing Z7J, from the 
 whole of U' to an infinitesimal of f/J, and then to zero. 
 
 6) The limit separating the two portions of V (i. e.j U[ from 
 CT), at each and every stage of the process, is evidently a new 
 kind of boundary — : TJ", — and the infinitesimal portion of 
 U\ between two very near Z7"'s corresponding to two very 
 
 nearly equal bulks of TJ immersed, in the continuous process 
 described above, may be denoted by dU' , meaning an infini- 
 tesimal of TJ' y and the aggregate of all these, ^dUlj may be 
 taken to equal the whole of U' belonging to the whole of U, 
 just as ?7= ^dUj^. (In the synthetic discussion of the dimen- 
 sions, above (Definition 12), allusion was made to the geomet- 
 rically indispensable notion of a homogeneous Z7', which might 
 serve as a standard of measurement for different C/''s, and 
 later the actual existence and construction of such a U' will 
 be rigorously proved.) A homogeneous U' may be considered 
 a field of motion for finite portions of C7', covered by corre- 
 sponding portions of V belonging to some moving rigid F, 
 which, in the process of motion, touches U in the variable por- 
 tions of U' considered. U' in itself, without the piece U 
 which is bounded by it, has an independent existence, only as an 
 abstraxition (like, for instance, force without matter). In fact, 
 we can speak of it as moving about, either in 8 or in its own 
 region, only in vii^tue of corresponding motions of U (or V) to 
 which U' (or V) belongs. 
 
 7) Corollary. — It follows also, that U', as a boundary be- 
 tween U and unoccupied adjoining portions of S, or between 
 U and F, must be considered as having the property of im- 
 penetrability. In fact, U' is conceived of as that which pre- 
 vents any portion of F, no matter how small, from penetrating 
 into the region occupied by any portion of Z7, and vice versa. 
 But, as a region of motion for portions of itself i. e., as a mani- 
 fold in itself it cannot possess the property of impenetrability, and 
 must, in fact, be of the same nature with respect to portions of V 
 or U' moving in the whole, as vacant S is with respect to impene- 
 trable U or F. This is a reason why the boundary between 
 U^ and Cr can be obtained only through the medium of an 
 auxiliary F, which may also be considered rigid, and portions 
 
57 
 
 of whose boundary — F', very near F'( = U^y c = covered), 
 will then serve as a check to a third rigid piece Wy of same 
 nature as U and V, a finite portion of whose boundary — W^ 
 is conceived as covering corresponding finite portions of UJ 
 (exposed with respect to V only) and moving in the manifold 
 Z7', until a finite portion of W^ comes into coincidence with a 
 corresponding portion of F', which thus becomes V^^ = W^^. 
 
 8) It is important to observe that, when a piece of the original 
 manifold F comes into coincidence with another piece of the 
 same kind U", in general only finite portions of their boundaries 
 coincide and thus become IT = FJ ; the remaining portions of 
 their boundaries combine in forming a new combined boundary 
 of the piece {U-j- F), taken as a whole, — this combined 
 boundary being now represented by (^7+ F)'= C/"^' -f- Fj'.* 
 It is, therefore, evident that, when TF, considered in No. 7, 
 moves up to F, the two portions of its boundary — W^^ and 
 W^^ — become now combined into the boundary separating W 
 from the combined piece {U-i- F), m., 
 
 (wi + w:j={u+v):j 
 
 and, in general, for the same reason as above, there will yet re- 
 main a finite portion of ( Z7 + F)' exposed which we may denote 
 hyiU+V):. 
 
 9) We see now that U'\ — originally obtained as the limit 
 separating U! from C7', when F was considered as the plastic 
 substance in which a portion of U was immersed, and then 
 identified with the boundary separating U^{— FJ) from C/', and 
 also from F', in case Fis considered rigid and a portion of its 
 boundary FJ covering an equal portion of C/', — by means of 
 the process considered in No. 7 becomes broken up into two por- 
 tions : one lying in the region of(U+ F)^ exclusively, and sepa- 
 rating U^^ from Fj" , or each of these from the same correspond- 
 ing portion of U^(= F^'), and the other lying exclusively in 
 the region of (C/-h F)f^ = (TF,; + TF/J, and separating U;^ 
 from V' . Let the first be called U'' and the second U'J . 
 
 10) By introducing another piece T, which we make to play 
 the r6le of W for the breaking up of U^l into C/^J^ ^^ and 
 
 * This combined boundary is in no way different in character from U', 
 being like it continuous, and having finite parts represented by A ( C7"+ T)' 
 
 -=AC7^ + AFe'. 
 
58 
 
 U" c , and then still another piece X for the breaking 
 up of U'' ^^ into two pieces, and so on, we see that U" consists 
 of as many parts as we please. Moreover, by making W move 
 lip to T, so that ( W'^^ -f M7J, — conceived as changing its posi- 
 tion continuously and as changing its form and magnitude if 
 need be,* — shall always remain during this process in coinci- 
 dence with an equivalent variable ( C/ + F)^^, we shall convince 
 ourselves that an infinitesimal d{W^^-\- TF^'J, in the neigh- 
 borhood of U", remaining always in coincidence with a variable 
 d{TJ-\-V)'^^, Avill displace itself and find a region of motion 
 along the element d{U -\- V)' taken in the neighborhood of U" 
 and conceived, in toto, as a locus for the different positions of the 
 moving infinitesimal portion of the first derivative boundary. 
 U" itself, therefore, as a whole, will prove to be a locus in quo 
 for U'J^ — Urn d{U + V)l^. When W comes up to T, it is there 
 
 checked, — and we arrive at the conception of U'", separating the 
 region of motion for U^'^ from the region where such a motion 
 is impossible. It is, of course, the same boundary as that 
 which separated U^'^^^t fronn L^^^ ^^, obtained at the beginning of 
 this paragraph. 
 
 11) In summing up, we see that, while S is a region of mo- 
 tion for pieces like C/, F, etc., the first derivative boundary 
 U' is the limit of d U, and that, not being a portion of U, it has 
 portions of its own, being a region of motion for corresponding 
 'finite portions of the boundaries of a movable covering piece 
 F. IJ" is the limit of dU' as dU' = 0, and, not being a part 
 of Z7', on account of its sej^arating the region of motion in con- 
 tact for a finite piece W from the region where such a motion 
 is impossible, it still has portions of its own. For, although 
 W cannot find upon it a region of motion, even for a finite 
 portion of its boundary, it can find upon it a region of motion 
 for the limit of an infinitesimal portion of its boundary, namely 
 for lim dW (= W"). For, as it was shown, TFcan be made 
 
 
 
 to move so, that, while two finite portions of its boundary, dis- 
 tinct but contiguous in W", move each upon a corresponding 
 portion of U' and F' respectively, an infinitesimal of its boun- 
 dary d W, contiguous to TF", and taken as near W" as we please, 
 
 * This was previously shown to be possible, by supposing the first deriva- 
 tive boundary to be homogeneous, for simplicity, or by allowing sufficient 
 plasticity to W\ 
 
69 
 
 on either side of it, will move and remain in coincidence with 
 corresponding variable infinitesimal portions of both U' and 
 V , In passing as near to the limit as we please, we come to 
 the conception of portions of the limit W" itself, moving in the 
 whole of it as in a locus in quo. 
 
 The third derivative boundary U" separates the region of 
 motion for the limit of an indefinitely small portion of the 
 boundary of the first order from the region where motion is 
 impossible even for the limit of an infinitesimal portion of such 
 a boundary. An infinitesimal portion of U", very near U'", on 
 both sides of it, namely dU'\ may be considered an element 
 of U"j so that we have ^d U" = U", 
 
 12) And now, if we agree that any manifold derived from 8 
 in the manner indicated, will have to be capable of giving place 
 to impenetrable substance, or to its boundary which is directly 
 connected with the substance, we find that the last derived 
 boundary, TJ"\ which does not, even near its limit, afford a 
 region of motion to an infinitesimal of the first derivative 
 boundary d U' of the impenetrable substance, has only position 
 but no dimensions. So that we have three categories of space, 
 corresponding to three properties of bodies — rigidity, im- 
 penetrability, and infinite divisibility, — each limiting the pre- 
 ceding and limited by the following. 
 
 These are : — 
 
 8, a region of motion for pieces of the impenetrable substance 
 itself — Uy F, etc., and their boundaries, both finite and infini- 
 tesimal ; 
 
 U\ the first derivative boundary of impenetrable substance, 
 a region of motion in contact of finite pieces like Z7, F, i. e., a 
 region of motion for a finite portion of the boundary of a piece 
 of impenetrable substance, and, lastly, 
 
 Z7", the second derivative boundary, a region of motion for 
 the limit of an infinitesimal portion of the first detnvoiive boun- 
 dary dU\ 
 
 The boundary, U'", between two portions of this last region 
 of motion, because of its limiting the region of motion for an 
 infinitesimal portion of the boundary belonging to impenetrable 
 substance, from the region of no motion even for an infinitesi- 
 mal portion of the boundary, must be of zero dimensions, but 
 still capable of having definite position, being the primary irre- 
 ducible element of space. It follows, therefore, since each of 
 
60 
 
 the boundaries is capable, near the corresponding limit, of 
 being considered an infinitesimal element of the category which 
 is bounded by it, the last category in order derived, namely 
 U"y has one dimension ; the one preceding, two dimensions ; and 
 the original one U, or a piece of vacant S as measured by C7, 
 three dimensions. 
 
 13) * By the very process of the deduction of the three 
 qualitatively different categories of space (regions of different 
 kinds of motion, which, however, are the boundaries of each 
 other, in a series), we have arrived at the notion of a manifold 
 of two dimensions, objectively not independent from the main 
 category, but, none the less, having a true abstract reality, and 
 which, by its very nature, as a manifold in itself and not 
 as a boundary, is devoid of the property of impenetrability 
 (see No. 7). In this derived manifold, therefore, boundaries 
 of portions can be established only by means of a piece of the 
 higher manifold, having the property of impenetrability in the 
 region of the dimensions of the derived manifold considered, 
 and, for this very reason, suggesting a dimension over and above 
 those of the derived manifold. It is no wonder, then, that the 
 general reasoning, applicable to the original manifold, is not 
 applicable to the derived manifold. The figure given as an 
 objection, instead of disproving the reasoning, is only another 
 proof of its validity. Mark, that in order to attribute impene- 
 trability to the limits of the circle, you must postulate it to 
 be infinitely thin, and V an infinitely thin film, — which, of 
 course, is equivalent to postulating a third dimension, but in a 
 very disguised form. 
 
 Now, this infinite thinness is already capable of being increased 
 indefinitely. In other words, the assumption of impenetrability, 
 by the reasoning employed above, would involve a third dimen- 
 sion, outside of the given manifold of two dimensions, leaving 
 this last unchanged. A reference to Nos. 7, 8, 9, 10 will fully 
 justify this assertion. 
 
 And still otherwise, — perhaps this way of looking at the 
 thing may be more satisfactory : — To such intelligent beings 
 
 * 13) is a reply to an objection raised during a conference -when this was 
 presented : Why by the same general reasoning we are unable to prove a 
 third dimension even in the domain of an admittedly two dimensional mani- 
 fold, as is, for instance, the first derivative manifold. The nature of the ob- 
 jection will be understood by the reader from the reply, which alone is given 
 here explicitly. 
 
61 
 
 as would have no sense of the third dimension, if such beings 
 were at all possible, the speculation about a third dimension 
 would not only involve no logical contradiction, but, on the con- 
 trary, would be a perfectly logical and necessary generalization. 
 For, they would have to postulate some dimension — not directly 
 given in experience — as a medium for the continuous passage 
 of W to Ty remaining always in contact with both U and F, 
 which, even in two dimensions are not essentially separated, 
 since they are both on one and the same side of U' and V, 
 But the speculation about a fourth dimension for such beings as 
 
 a 
 
 
 
 
 S 
 
 
 
 
 .§ 
 
 
 T 
 
 
 
 
 
 
 'O 
 
 
 
 
 *6 
 
 ^ 
 
 "^ 
 
 V 
 
 O ^ 
 
 y' 
 
 
 \ 
 
 -^ "fi / 
 
 f 
 
 
 1 \ 
 
 o S / 
 
 
 
 j \ 
 
 *» « / 
 
 
 
 \ 
 
 M / 
 
 
 
 \ 
 
 (U o 
 
 
 
 \ 
 
 
 
 
 \ 
 
 1.1 
 
 S o \ 
 
 u 
 
 V" 
 
 i i 
 
 ^■s \ 
 
 Ci OB 
 
 \ 
 
 
 i M 
 
 / 
 
 •^ bo 
 
 V 
 
 ^^ 
 
 / 
 
 t^-^ 
 
 c^ ■ 
 
 f^ 
 
 
 k5^1 
 
 
 w 
 
 
 
 
 
 .2 
 
 
 
 
 Fig. a. 
 
 have already risen to the empirical verification of the abstract 
 deduction of three dimensions, would certainly involve a logi- 
 cal inconsistency. For, starting with the manifold S as de- 
 fined above, in its most general aspect, without boundaries at 
 all, and, for all we know, having n dimensions, where n may 
 be any entire positive number, we were led, by a simple analysis 
 of its definition, to three different manifolds, containing each 
 other in series, and the third derivative boundary which is 
 the boundary of the lowest category in this series, was found 
 to be something that can have, at most, position, but no dimen- 
 sions. If preferred, you may say that we have proved that the 
 maximum number of dimensions of space as defined by the 
 properties of 8, is three, and that, having logically arrived at 
 
62 
 
 the maximum, we find it in perfect agreement with our intuitive 
 experience, which, of course, also served us as a starting basis 
 in defining the manifold S, at the beginning of the present dis- 
 cussion. 
 
 I deem it necessary to repeat at the conclusion that I ac- 
 knowledge the fruitfulness of the idea of making space a mani- 
 fold of a higher number of dimensions, by dropping the property 
 of impenetrability in the physical sense, and assuming a figure 
 depending on n parameters, as the element of space. 
 
Chaptek II. 
 
 THE SPHERE, THE CIRCLE, THE STRAIGHT LINE, THE ANGLE, THE 
 TRIANGLE, THE PLANE, ETC. 
 
 Definition I. — A pair of fixed points in space, or any two 
 points in rigid connection which can he made to coincide with these, 
 are said to he at an invayiahle distance from each other. 
 
 Corollary I. — If a pair of points, A and B, in rigid con- 
 nection, are capahle of coincidence, one hy one, with another pair, 
 C and D, likewise in rigid connection between themselves, the two 
 pairs loill he at equal distances, each point from its pair, i. e., 
 distance of B from A = distance of C from D or of D from C. 
 
 Corollary II. — Two pairs of fixed points in space. A, B, and 
 C, D, hoth of which are capahle of coincidence successively with 
 the same freely movahle pair in rigid connection, E and F, are re- 
 spectively at equal distances. 
 
 For shortness we shall call a pair of points in rigid con- 
 nection, free to move as a whole, simply a rigid pair of points, 
 and will denote them thus (AB). 
 
 Axiom 1. — If any surface or line can be made to coincide 
 with another surface or line, it can do so only by passing from 
 one position to the other by a continuous path, consisting of an 
 infinity of such positions, every position in which is a surface or 
 line, respectively congruent with the moving one. 
 
 Lemma 1. — From the principle of rigidity and the defini- 
 tions of body, surface, line, and point, it follows that a body is 
 absolutely fixed in space when, and only when, a finite portion 
 of the surface limiting it, or in any way rigidly fixed in it, is 
 fixed in space. 
 
 For, whenever a body is fixed in space, the whole of its sur- 
 face, limiting it from all space around, and hence every finite 
 portion of this surface, is fixed in space ; and whenever the 
 body is moved, the whole of its surface, and hence every finite 
 portion of it, no matter how small, changes its position. We 
 cannot say, however, the same of a point in the surface, since 
 a point, having no magnitude, but only position, does not limit 
 the surface to which it belongs and which may be conceived to 
 
 63 
 
64 
 
 change its position as a whole, and hence to interchange the 
 positions of all congruent lines in it that are drawn from a 
 common point limiting them all and remaining fixed. Hence, 
 a body and, therefore, also any surface or line, rigidly fixed in 
 it, may be conceived to move when only one point in the body, 
 the surface, or the line, is held fixed in space. Such a motion 
 of the body is called rotation, revolution, or turning, about the 
 fixed point. 
 
 Theorem 1. — If (AB) denote a rigid pair of points, of which 
 A is fixed and (B) is made to assume all possible positions 
 compatible with the rigidity of the pair and the fixity of A, 
 then (B) will describe a homogeneous surface called a spherical 
 suifcvce, limiting a body called a sphere, A is called the center of 
 the sphere J or of the spherical surface, and the invariable distance 
 AB, from the center to any point in the surface, is called the 
 radius of the sphere. 
 
 In fact, the moving point (B) can pass from any fixed point 
 B with which it initially coincided to any other point J5', J5", 
 and so on, by a continuous infinity of paths crossing and re- 
 crossing one another in all conceivable ways, provided all these 
 points are at the same distance from A sls B (preceding Lemma 
 and Axiom 1). Let (B) pass from B to B' by a continuous 
 path of some determinate rigid form, BB'; while doing so, any 
 rigid line connecting A and (jB) will describe a portion of a 
 surface. This surface, in its turn, conceived as a rigid form, 
 can (according to Lemma) be imagined so to move while A is 
 fixed, that every one of its points describe a line not already con- 
 tained in the original position of the surface itself, — thus describ- 
 ing a body. The line (BB'), conceived rigid and always limit- 
 ing the moving surface {ABB'), will then be dragged along with 
 it in its motion and will describe a surface, since each of its 
 points describes a line not coincident with the original position 
 of the moving line ; in other words, each point of the line 
 (BB') passes to another point not already contained in the 
 original position of the line. Moreover, since the surface 
 {ABB') can be conceived to sweep through all that portion of 
 space around A whose aggregate of points is characterized by 
 the property that their distances from A are correspondingly 
 equal to those of the aggregate of points contained in {ABB') 
 in its original position, it follows that the surface described by 
 the line {BB') will contain the whole aggregate of points which 
 
65 
 
 are at the same distances from A as (B) in its original posi- 
 tion. It is also evident from the moae the body described 
 by the surface (ABB') is generated, that it is a continuous 
 body, i. e.y the aggregate of points composing it is a continuous 
 aggregate, which allows to pass from any point in the body to 
 any other through any third point belonging to the body, by a 
 continuous path lying wholly in the body (^. e., every point of 
 which belongs to the body). The surface, therefore, is also con- 
 tinuous, in the same sense, — namely, that we can pass by con- 
 tinuous motion from any point in it to any other, through any 
 third, by a path every point of which is in the surface. More- 
 over, the moving point will during such a motion remain at the 
 given distance from JL, hence, on the surface of an imagined 
 fixed sphere ; and if the moving point is conceived to be fixed in 
 a sphere, of same center A, but which is dragged along with (B) 
 in its motion, we see that every portion of a spherical surface 
 is congruent with every other of same limits ; that is to say, 
 the spherical surface is a homogeneous surface. 
 
 Corollary I. — Since the spherical surface separates a continu- 
 ous portion of space from all other space, that is, all points 
 that can be reached by one continuous path wholly contained 
 in the body, from all points of space that can not be reached 
 by a continuous motion from the center, unless the boundary 
 of the body is crossed by a path of which a portion, at least, 
 does not belong to the body, — it follows that this surface is 
 also a closed surface. 
 
 Definition II. — All points in the sphere (body limited by the 
 spherical surface) generated by the given radius AB^ from the 
 given center A, which can be reached by continuous motion from 
 the center, without crossing the surface at all, or after crossing and 
 recrossing it an even number of times, are said to be within the 
 surface or inside the sphere, and the distances of all these points 
 from the center (excluding those of the surface itself) are said 
 to be smaller than the distance AB. All points that can be 
 reached by continuous motion from the center, only after crossing 
 the surface once, or crossing and recrossing it an odd number of 
 times, are said to be without the surface or outside the sphere, and 
 their distances from A are said to be greater than the distance AB, 
 
 Corollary II. — When a sphere is conceived to move within 
 the boundaries of a fixed spherical surface which is always in 
 coincidence with the surface of the moving sphere, its center 
 
(or centers, if there can be more than one), remaining at a con- 
 stant distance (or constant distances) from each and every one 
 of the same fixed aggregate of points belonging to the same 
 surface, will remain fixed in position. But since, when the 
 sphere moves around one fixed center, from which it is con- 
 ceived to be generated, every other point in it, at a distance 
 from the center, moves upon the surface of a sphere (Theorem 
 1), it follows that, given a spherical surface as a whole, its center 
 is uniquely determined. A complete spherical surface is, there- 
 fore, said to he the locus of all points equidistant from a unique 
 fixed pointy called the center. 
 
 Corollary III. — If two spheres coincide in any finite portion 
 of their surfaces, they coincide throughout, and, hence, have the 
 same center and radius. 
 
 For, by holding the finite common portion of each spherical 
 surface fixed, each of the surfaces remains fixed (Lemma 1). 
 But if either of the spheres is conceived as moving within its 
 fixed boundaries, around its fixed centre, every point at a dis- 
 tance from the fixed centre, moves upon a corresponding sphere, 
 and passes into the position of every possible point on the last, 
 and of no other point. Hence, there is only one center com- 
 mon to both, since the portion of each movable spherical sur- 
 face, which is initially in coincidence with the corresponding 
 common portion of the two fixed surfaces, can, without leaving 
 either of the spherical surfaces, pass over into any congruent 
 portion of either. 
 
 It follows from what has preceded, that a distance is a geo- 
 metrical magnitude. No two points at different distances from 
 a given point, can be connected with the given point by the 
 same line, which might serve as a path of motion from one end- 
 point of each distance to the other. If we take, to fix ideas, 
 as the connection between A and B, a line of some determinate 
 shape, lying wholly within the sphere described by (AB), sl 
 concentric sphere, described by a distance AC less than AB, 
 will cut every line {AB) into two parts, one of which will lie 
 within the smaller sphere, and the other, outside. Thus we 
 see that to a smaller distance corresponds a smaller path of 
 motion from one end of the distance to the other. This sug- 
 gests the idea of representing the distance-magnitude by a line. 
 But, then, we must find a line such that to a fixed distance, 
 from one fixed point on it to another, should correspond a 
 
67 
 
 unique position of the line, and to a smaller portion of the line 
 should also correspond a smaller distance, — which is not al- 
 ways the case with any line. Is there such a path between 
 A and jB? I say, that if we eliminate all paths between A 
 and B which can assume different positions while A and B 
 remain fixed, — since their distance is unaltered, — there must 
 still be left a path between them which is unique for this fixed 
 position of the ends, and which will also satisfy the other re- 
 quirement, — as I shall proceed to prove with the utmost rigor 
 in the following two propositions and their corollaries. 
 
 Measuremefnt of Distances From a Fixed Origin ; Addition and 
 Subtraction of Given Distances, 
 
 Theorem 2. — Let S represent a sphere described by radius 
 a from origin 0. OA = a, as soon as A is on its surface. De- 
 scribe from A a sphere S' with a radius 6, where b <^a. Then 
 is outside 8' (by Defin. 2), and, hence, some portions of S are 
 outside S'; for if the whole of 8 were inside 8'j the center of 8, 
 which is separated from its own outside by some portions of 8y 
 would be inside 8' a fortiori. Similarly, some portions of 8' 
 are outside 8; otherwise A would have to be within the surface 
 of 8f and not on it. But some portions of 8 are also inside 8'; for 
 otherwise it would be impossible to reach by continuous motion 
 from A any point belonging to 8, before crossing the surface of 
 8^ once, — which is absurd, since A itself is on the surface of 8, 
 and hence belongs to >S'; a portion of 8 is, therefore, inside 8\ 
 Also some portions of 8' are inside 8 ; for if 8' were wholly 
 outside 8y it could not contain in its interior any point belong- 
 ing to 8 — contrary to what has just been proved to be the case. 
 Hence, the spheres penetrate each other, — having one portion 
 of space in common, limited by the portions of their surfaces 
 which they have inside each other, and two other portions of 
 space, enclosed each between the interior side of the one and 
 the exterior of the other. The two surfaces must intersect 
 along a line every point of which is at equal distances from 
 and from A, respectively. Hence, a rigid connection of any 
 point C on this line, with both centers and A, will be cap- 
 able of displacing itself in such a manner that, while and A 
 remain fixed, C shall take all different positions on the line. 
 Describe now from a sphere 8" concentric with 8y with a 
 radius equal to the distance from of a point lying in the in- 
 
terior of both 8 and S'. Its surface will pass through this 
 point, hence will cut this space ^into two portions, of which 
 only one will lie inside S". This new sphere will still have a 
 portion of its surface inside 8' and the other portion outside, 
 since it encloses centre 0, which lies outside 8'. Describing 
 again a concentric sphere from 0, with a radius whose end-point 
 passes now through a point lying in the interior of 8' and /S"', 
 we shall still more reduce the portion of volume enclosed in the 
 interior of /S" and /S"', to that only which is left inside this new 
 sphere 8'" and 8\ (The portion of the surface of 8' exposed is 
 continually increasing during this process of variation of >S', 8'\ 
 /S"" • • • etc., since some portions of it enclosed by a greater sphere 
 described from are exterior to any of the smaller spheres.) Evi- 
 dently, by continuing this process far enough and in a suitable 
 manner, the variable portion of volume can be made less than 
 any assignable small volume. This will happen when the two 
 spheres, the constant one 8', described from A, and the variable 
 8^'"\ described from 0, touch each other in one or several points, 
 or in one or several lines ; for they cannot touch in any portion 
 of surface without their coinciding in every part (Cor. 3 to 
 Theorem 1). Now, they cannot touch in several distinct lines 
 or distinct points; for any line of fixed form, connecting and 
 A with a point of the contact, could be displaced so, that, while 
 and A remain fixed, the point on the line which has been 
 in coincidence with the touching point originally taken, shall 
 coincide with any other point in the common touching parts, 
 which must be at the same distances from and from Ay re- 
 spectively. Hence, since by the principle of continuous congru- 
 ence in motion (Axiom 1), the figure described by the displaced 
 line is a continuous one, — the figure described by a determinate 
 point in it is also a continuous one. It must, therefore, be 
 either one line, — any portion of which can, by a continuous 
 motion along itself, pass without deformation into any other of 
 same limits, — or one point. 
 
 Scholium. — By continually decreasing the radius of the 
 variable sphere, and dropping the condition of its having to 
 pass through a point within 8 ', we shall, at last, arrive to a 
 series of spheres which have all their volume outside 8', and 
 have not one point in common with the constant sphere. It is evi- 
 dent, therefore, that in this process of passing continuously from 
 spheres having some portions inside 8', to such whose volume 
 
69 
 
 rt^'o;^ o-^^^. 
 
 and surface are wholly exterior to /S", — so that there is an ap- 
 preciable distance from each and every point belonging to the 
 region of these exterior spheres, and each and every point be- 
 longing to the region of S'j — we must on the way encounter 
 a sphere, such that any sphere described by a radius greater 
 than its own, has some portions of volume inside S', and any 
 sphere described by a smaller radius, has every point of its re- 
 gion at some appreciable distance from the region of S'. And 
 in crossing from the series of greater to the series of smaller 
 spheres, we obviously can choose two, one from either series, 
 whose radii differ as little as we please from each other and 
 from the radius of the bounding sphere. Evidently the 
 minimum sphere of the first series and the maximum of the 
 second series, both tend towards the same limit — the bound- 
 ing sphere, which, we say, touches the constant sphere. This 
 bounding sphere must have at least one point in common with 
 the constant sphere, and may have even a continuous line of 
 contact, until otherwise shown, — which is done in what follows. 
 Similar remarks hold in the next 
 supposition of the present demon- 
 stration, viz.j w^hen we have to 
 add a distance to a given one, in- 
 stead of subtracting (see p. 70). 
 Now, the contact cannot be a 
 line. For this would imply that 
 the surface of the sphere >S'^"^, 
 lying wholly outside S' according 
 to hypothesis, is divided by this 
 line into two parts : one,* which 
 together with the exposed por- 
 tion of >S" that has been continu- 
 ally increasing during the process 
 of variation of S^''\ makes up a 
 closed surface, separating the in- 
 terior of both spheres from the 
 whole of the exterior space ; and 
 the second part, which, as derived 
 from the portion of the variable 
 surface that has remained inside the combined closed surface of 
 both spheres during all the process of variation, must still be- 
 
 FlG. 1, a 
 
 Fig. la and lb on p. 70 illustrate this in two ways. 
 
70 
 
 loDg to the interior and, hence, must be enclosed between the 
 outside of 8 ' and the interior side of the part of surface of 8^'^'^ 
 previously considered. Some volume must, therefore, be enclosed 
 between the outside of S' and the outside of the portion of sur- 
 face of /S^*^"^, last considered. Describing now a sphere from 
 with a radius of some point in this portion of volume, which, 
 consequently, is greater than the radius of the touching sphere 
 /S'^"^, we see that the new spherical surface must enclose every point 
 on 8^"^^ and, hence, also the line of contact. It must, therefore, 
 enter into the interior of 8' along a line lying on the covered 
 portion of 8' , and again come out from there by another line, 
 lying on the exposed portion of same, — which is possible. 
 Hence, the contact must be in a single point. 
 
 ^rt jon o/ j'^ ^ . We say that the radius of /S'*^"\ 
 
 described from and touching 
 8' , is = (a — 6), and the distance 
 from to any point in /S'*^'*^ and, 
 in particular, to the point of 
 contact, is (a — h); the distance 
 of this last point from A being 
 equal to 6, the distance from 
 to A={a — h)-\-h = a — in 
 perfect agreement with original 
 hypothesis. 
 
 If, instead of diminishing the 
 radius of the sphere /S, we should 
 continuously increase it, we 
 would, by a reasoning perfectly 
 similar to the preceding, come 
 to the conclusion that the space 
 between the exterior portion of 
 the surface of 8^""^ and the inte- 
 rior of 8' y would continuously 
 diminish (and likewise the exposed portion of the surface of 8') 
 and could be made less than any assignable volume, — when 
 the two surfaces would have to touch, either along a single 
 line, or in a single point. Now, it cannot be along a line ; 
 for, then, — since /S^**^ in the limit will have to enclose in its 
 interior both the portion of 8' which has been interior to 8 at 
 the outset and which has kept on increasing during the pro- 
 cess of variation of /S*^"^, and also the remaining portion of 
 
 Fig. 1, 6 
 
71 
 
 aS" which has been exterior to >S^"^ and has been decreasing 
 during the process, — there must be, in the limit, a portion 
 of volume, contained between the portion of the surface of S' 
 considered last, on the outside of it, and that portion of the 
 surface of /S'^"^ which is now separated from the first por- 
 tion of S' by the surface of the other portion of S' . The two 
 portions of surface, enclosing this volume and belonging to 
 the two spheres respectively, are separated by the line of con- 
 tact. Now, if we describe a sphere from by the distance 
 from it to any point within this space, this sphere will lie 
 wholly within 8^""^ and must be one of the spheres which have 
 cut S' before reaching the limit. It must, therefore, first emerge 
 from that space ; but as it cannot cut 8^""^ it must cut ;S" in two 
 lines, one on each of the portions of its surface, separated from 
 each other by the line of contact of /S" and 8^''\ — which is im- 
 possible. Hence, the contact 8' and 8^""^ must be in a single 
 point. 
 
 We then say that the radius of the sphere /S^"^ described from 
 0, enclosing 8' and touching it in one point, is equal to a + 6. 
 The distance from to any point in >S'^"^, and, in particular, to 
 the point of contact, is = (a -f 6) ; the distance of this last point 
 from A being equal to b, the distance from to ^ is (a -j- ^) — 
 6 = a, in perfect agreement with the original hypothesis. 
 
 Suppose now we have to add to OA a distance AB greater 
 than OA. We describe then from A a sphere 8', with radius A B ; 
 it will enclose and, hence, at least a portion 8 ; it may, how- 
 ever, enclose it all. Describe now a sphere 8^ from 0, with radius 
 equal to AB ; it will enclose A, and hence 8^ and 8' must have 
 common portions ; but neither can enclose the whole of the 
 other, since, as one cannot be greater than the other, they would 
 have to coincide, which is impossible — the centres being distinct. 
 Hence, they will intersect as previously. Increasing now the 
 radius of the sphere described from 0, we can prove, as in the 
 last case, that some >S'*^''^ will come to touch 8' in one point B, 
 and enclose it all, so that OB will be equal to OA + ABy as 
 previously. Also, by diminishing the radius of 8^ continuously, 
 we shall get another point of contact, whose distance from 
 we shall call OA — AB = a — b <,0. Such a point of con- 
 tact as this last can never be obtained from the above rules of 
 addition and subtraction by adding any positive distance, either 
 
72 
 
 greater or less than OAj nor by subtracting from OA any posi- 
 tive distance less than OA. Hence, points corresponding to 
 negative distances from 0, where OA is positive, are distinct 
 not only from one another, but also from all points mark- 
 ing positive distances from under same hypothesis. We see 
 now, that we can add and subtract distances as abstract 
 quantities. 
 
 Scholium. — The rules for addition and subtraction given 
 above are based upon the fact that every two rigid pairs of points 
 connected at one end give rise to an infinity of distances 
 between the free ends, having a perfectly determinate higher 
 and lower limit (maximum and minimum), which we have de- 
 fined, respectively, as the sum and the difference of the two 
 given distances represented by the rigid pairs themselves. An 
 additional reason for singling out the maximum and minimum 
 distances from the host of all the aggregate, is found in the 
 further fact that, when the ends of one of the two connected 
 rigid pairs are fixed in position, the position of the remaining 
 free end is not fixed for any one of the derived distances, with the 
 exception of these two limiting distances. It is also evident, 
 that one of the essential conditions which the addition and sub- 
 traction of measurable quantities must satisfy, namely, that the 
 sum increase with the increase of each of the terms, and that any 
 two quantities differing in value should always give a determinate 
 difference (see Weber, "Traits D'Alg^bre Sup^rieure," 1. 1, p. 9), 
 is perfectly satisfied by our rules for all the three cases considered 
 in the theorem. In order, however, that these rules be perfectly 
 consistent, and should lead to no contradictions in their appli- 
 cation, it is further necessary to prove that, in the first place, 
 the operation of addition obeys the associative and commutative 
 laws of addition of abstract quantities, and, in the second place, 
 the rule for subtraction is actually the inverse of the rule for 
 addition. This can be done by the aid of the following two 
 lemmas : 
 
 Lemma 1, Theorem 3. — If the point B has been so de- 
 termined by our rule of addition, from the fixed points and 
 A, that OB = OA -f AB, then, taking B as origin and a mov- 
 able rigid pair (AO') congruent with (AO), the fixed point 
 Oj determined so that B0^= BA -\- AO' coincides with the 
 original starting point 0. 
 
73 
 
 Demonstration. — Let 0, Ay B be the triplet of points of the 
 original operation ; S'^, S'[, aS"J, the corresponding spheres. 
 Then, by construction, if C is any point on S'[ not coincident 
 with Bj we have OB '> OC. The spheres S^ and 8'^ have same 
 centre and pass through A and J5, respectively, and the 
 sphere 8 'I, with center Aj is wholly interior to 8^, with the 
 exception of point J5, at which the two spheres touch. Let 
 now the sphere (-S'y), conceived rigid, rotate about the fixed 
 point A ; every point (P) belonging to the rotating sphere, 
 will remain upon a sphere described about J. as a center, 
 with radius (AP). Hence, the center of the moving sphere, 
 (0), will move upon the surface of 8^ described from A with 
 radius (A 0) ; and, since (B) is the only point of the surface of 
 (/S'q) which has, at the beginning of the motion, been at a dis- 
 tance (AB) from A, this point (^B) will be the only point in the 
 moving surface (8'^) 
 which will, during all the 
 rotation, remain upon the 
 surface of 8'^. All other 
 points in the moving sur- 
 face (^o), having been at 
 the beginning of the rota- 
 tion outside iS"/, will re- 
 main outside during any 
 moment of the rotation. 
 Hence, the moving sphere 
 (81) will, during all the 
 rotation, remain in inte- 
 rior contact with the sta- 
 tionary sphere 8'^, touch- 
 ing it at any moment in 
 some point jB', which 
 marks the corresponding 
 position in space, at that 
 moment, of the moving point (B) rigidly fixed in the moving sur- 
 face of {8q). Let 0' be the corresponding position, at the moment 
 considered, of the centre (0). It is now evident that the rotation 
 of (81) can be so arranged that its centre (0) describe the whole 
 of the spherical surface 8 1. (It is sufficient for this purpose, that 
 the radius (AO) describing aS'J, be rigidly fixed in the sphere 
 (8^), which moves along with the radius about A fixed.) Simi- 
 
 FlG. 2. 
 
74 
 
 larly, (S'^) can rotate so, that (B) describe the whole of the 
 surface S'^. Our original construction of the triplet of points 
 0, A J and Bj will, in either case, at any moment be presented 
 by a congruent triplet 0', Aj B'; hence, O'B' = O'A + AB'* 
 We conclude, therefore, that : — 
 
 First. To any point 0' on the surface of S^ corresponds one, 
 and only one, point B' on the concentric surface S'^, such that 
 0^S^= WA + 35"= OB, a nd, at the sam^ time, WJ = '0A 
 and AB' = AB, and, further, 0' B' is the maximum distance 
 between the ends of the connected rigid pairs {0' A) and 
 {AB'). 
 
 Second. To any point B' on the surface 8 'I corresponds some 
 point 0' on the surface /S'J, such that our original construction, 
 repeated from 0' and A^ instead of and Aj will lead us ex- 
 actly to the point B' . We cannot say, however, until proven 
 so, that to B' corresponds only one point 0' . For, although 
 we know that there is only one point B' on aS"/, corresponding 
 to a fixed point 0' on 8\, such that O'B' > 0' C — C being any 
 other point on /S"/, not coincident with B', we cannot affirm 
 
 that, conversely, B' 0' is 
 also the maximum dis- 
 tance from fixed B' to 
 any point on the surface 
 Sly or, in other words, 
 that BW> B'D, where 
 D is any point on S \ not 
 coincident with 0' ; and 
 if it is not the maximum 
 distance, then we know 
 from the preceding, that 
 a spherical surface de- 
 scribed from B' with ra- 
 dius = B'O'y will cut the 
 surface /S J in a closed line, 
 every point of which is 
 exactly at the same dis- 
 Ym. 2. tances from B' and from 
 
 J. as 0', and can, therefore, replace 0' in our construction. 
 Let us now start with the fixed points B and Ay as in the 
 theorem, and let the movable rigid pair {AO'), congruent with 
 
75 
 
 (AO), be put in the position AO^y where BO^ — BA + ^O^, 
 in the sense of the definition of addition given in Theorem 2. 
 I say, that Oj, which is by construction unique on S\, and which 
 satisfies the inequality B0^'> BO' , where 0' is any other point 
 on S\ not coincident with Oi, cannot be any other point than 
 the original starting point 0. For, let it be some other point 
 D 'j D \Q unique on 8\, and BD^ BO. In this new con- 
 struction, B being the starting point and D, the final point in 
 the operation, we know that, when (J5) moves upon S j, its cor- 
 responding unique point (D) moves upon S\, (It is hardly 
 necessary to explain that the sphere rotating about A fixed in 
 the new construction, and corresponding to Sl'in the old one, 
 is {S^l) touching internally in Z) the stationary sphere /SJ, and 
 supposed, at the beginning of the motion, to have its center at B.) 
 We know, further, that to every point (D) on S \, there is at 
 least one point {B) (and maybe an infinity of such points) on 
 /Sj, such that the construction repeated from the point (^), or 
 from any one of such points, and from A, will lead exactly to 
 {D). Let now (D) move up to ; none of its corresponding 
 points {B) can, for this new position of (D), coincide with B, 
 since, by construction, BD^ BO. Hence, the point corre- 
 sponding to in the new construction, must be some other point 
 C on S'[, — and we have CO = BD > BO — which is in con- 
 tradiction with the result obtained from the first construction, 
 namely, OB '>0C. Hence 0^ marking that position of the end of 
 the rigid pair {AO') in the new construction, which corresponds 
 to the maximum distance BO' from B to any point on S\y 
 cannot coincide with any point ou S\ except 0. We conclude 
 hence, that if OS = OA -f AB, then W) = ~BA -j-~AO — in 
 the sense of our rule of addition, — and if the distances are 
 measured from the same origin 0, then 6 + a = a -f 6. 
 
 Q. E. D. 
 
 Lemma 2, Theorem 4. — If 0^ = OA -\- AB, then a 
 sphere described from B with radius BA, will touch externally 
 the sphere described from with radius OA. In other words, 
 if from the rule of addition OA — OB — AB^ then OA is the 
 minimum distance between the extreme ends, of the fixed rigid 
 pair {OB) and the movable rigid pair {BA) jointed to the first 
 at B ; or, otherwise still, the rule for subtraction is an actual 
 inverse of the rule for addition. 
 
76 
 
 Demonstration. — Let the theorem not be true. Then a 
 construction of a triplet of points, 0, A, and B, is possible, like 
 that in the figure, where the sphere S 3 — touching internally 
 at B the sphere S'^ of center 0, and having its center A on 
 the spherical surface S^j likewise described from — has a 
 radius AB such, that the sphere 8'^ described from B with 
 this radius, cuts aS'^ in a closed line AA'. Then, we can describe 
 from B as center another sphere S'^f such as will touch exter- 
 nally the sphere aS'^ in the point C, and, consequently, whose 
 radius BC<^BA, Hence, the sphere S^^ described from C 
 with radius CD = ABy having B and some space in the 
 
 neighborhood, in its inte- 
 rior, cuts S'q or encloses it 
 wholly in its interior. By 
 conceiving now the spher- 
 ical shell contained between 
 the concentric surfaces S^ 
 and SI, to be rigid and to 
 rotate about the stationary 
 sphere /S'^, dragging along 
 (8^) rigidly fixed to this 
 shell, ((7) will come to A^ 
 and 8^^ will now coincide 
 with a congruent sphere 8'[ 
 described from A with ra- 
 dius CD — AB and cut- 
 ting S'q in a closed line or 
 enclosing it wholly in its 
 interior ; but this is contrary to hypothesis, according to which 
 the sphere described from A with radius AB, is 8^, tangent 
 internally to 8^. Our theorem, therefore, can never be untrue, 
 but must be true without exception. Hence, the rule for sub- 
 traction is a real consequence of the rule for addition. 
 
 Q. E. D. 
 Scholium. — Now we can prove that the operation of addi- 
 tion obeys the associative and commutative laws in their full 
 generality. We have, however, to remark : — first, that from 
 the very sense of the rule for addition follows the equality : 
 a 4- 6 + c = (a + 6) -f c, and, in general, any number of terms 
 following the first, written separately with signs +, may be 
 
 Fig. 3. 
 
OF THE * 
 
 UNIVERSITY 
 
 77 
 
 incorporated in a parenthesis enclosing the first, and hence 
 such a parenthesis may be dropped ; and second, that in any 
 parenthesis, by Lemma 1, we may interchange the order of the 
 first two terms, if they are both positive. 
 
 Theorem 5. — The operation of addition obeys the associ- 
 ative and commutative laws, so that any number of terms, be- 
 ginning from, and ending with, any term we please, may be en- 
 closed in a parenthesis, and the result added to the preceding 
 terms, — and any term may be transposed forwards and back- 
 wards, through any number of other terms. 
 
 Demonstration. — We have by lemma 2 : 
 
 6 + c — c — 6 = & — 6 = 0, 
 and also 6 + c — (6 + c) = (6 -f c) — (6 -f c) = ; 
 
 .*. — (6 + c) = — c — 6. 
 Therefore, a+6 + c — (6-fc) = a + 5 + c — c — 6 
 
 ... a -h 6 + c - (6 + c) + (6 + c) = a + (6 + c), 
 
 or a + 6 -f c = a + (6 + c), 
 
 and still otherwise, 
 
 (a + 6) + c = a + (6 + c). 
 
 Put now 5 = c? -f 6 + • • • , 
 
 c = (/ + ^) + • • • + (g + • • • + r) 4- • • • 4- < ; 
 then we get 
 
 a + (cZ + e +...) + [(/+ ^) + ... + (5 + ... + r) + ..- + q 
 
 = [a+((^+e+.'. •)] + Kf+gH' • '+{{q+' • •+0+- • -+0] 
 
78 
 
 = («+^) + (e+- • •+/+^) + (- • • + ?+• • •+r+- • - + 
 
 =etc. 
 
 Suppose, now, we have the sum, 
 
 and we wish to transpose 6 + c over the terms 
 We write then, 
 
 = a+ [(6 + c) + (cZ+6+/)] 
 
 = a + [(d^ + e +/) + (6 + c)] = a + (d + e +/+ 6 + c) 
 
 We add then the remaining terms to both sides of the equation, 
 and get the desired transposition. Q. E. D. 
 
 Scholium. — The process of obtaining any integral multiple 
 of a given distance, and also, of obtaining the ratio (commensur- 
 able or incommensurable) of two given distances, ought no 
 longer to detain us, and we shall only remark that this process 
 is a direct consequence of the rules for addition and subtraction 
 and of the postulate of continuity, which says that between 
 any two positions of any geometrical entity (a point included) 
 there is always an infinity of other such positions, all in space, 
 and that at least one infinite series of such positions must be 
 passed through, to reach either of the two given positions from 
 the other. By the associative law of addition and its extension 
 to subtraction in considering this operation as the addition of neg- 
 ative terms, we can get a given series of points, either by adding 
 and subtracting separate terms, each to, or from the preceding 
 sum, or by adding to the same given term a series of distances 
 equivalent to one, two, three • • •, n • • •, of the added and subtracted 
 terms taken in any order we please. This last process, in as- 
 suming that the term to be added (in a positive and a negative 
 sense) increases continuously, leads us to the important concep- 
 tion of a homogeneous line, completely determined by two 
 
79 
 
 points in it, which serves as the basis of all line-measurements, 
 and is fully described in the following definitions and succeed- 
 ing corollaries. 
 
 Definition III. — By conceiving the sphere S' described from 
 A (in Theorem 2), to vary continuously beginning from one 
 whose radius is indefinitely small, and, passing all imaginable 
 distances, to become indefinitely large, we shall get all possible 
 distances OB, positive and negative, whose general expression 
 is {a -f x). To each such distance from corresponds, by con- 
 struction, one, and only one, point, and the aggregate of all these 
 points will, evidently, form a continuous line, which, satisfying 
 fully the conditions necessary for a line suitable to represent 
 the distance-magnitude, may be called the distance-line or the 
 straight line. 
 
80 
 
 Corollary I. — To every distance on the line, as measured 
 from 0, there will correspond a point 5 at a corresponding 
 distance from A, such that no other point in space can have 
 the same distances from and A, respectively. 
 
 For, if we describe two spheres from and^^Jl with the cor- 
 responding distances as radii, these two spheres will have 
 only one point in common, namely, the point on the distance 
 line ; but if there were another point in space at the same dis- 
 tances from and J., this last point would also be on both 
 these spheres, which is contrary to construction. 
 
 Corollary II. — If J. and B^ and A' and B' , are two pairs of 
 points, at equal distances from each other singly, then the 
 straight lines constructed from each pair as from and A or- 
 iginally, will coincide with each other along the whole of their 
 extent as soon as the congruent pairs {AB) and (^A'B') are 
 made to coincide. This follows immediately from the fact that 
 the distance-line is unique when constructed from two points 
 of fixed position. 
 
 Corollary III. — Every point not on the distance-line between 
 A and B or its prolongation, is such that we can find a con- 
 tinuous series of points of which this is one, which have the 
 same distances from A and B, respectively, as the given one. 
 
 For, since the given point is not on the distance-line con- 
 structed from A and B, it follows that the two spheres through 
 the point, described from A and B as centers, will not touch in 
 this point, but cut along a line which is the locus of points 
 having the same distances from A and B as the given one ; 
 hence our corollary. 
 
 Corollary IV. — If we imagine a rigid body to be placed so, 
 that two of its points coincide with A and B considered pre- 
 viously, and to be fixed there, then all the points of the body 
 fall in either of the following two categories : — 
 
 (1) A continuous series of them coincide with corresponding 
 points on the distance-line through A and B, which is fixed in 
 space as long as A and B are fixed ; 
 
 (2) The remaining points coincide, each with a point in space 
 whose distances from A and B^ respectively, it has in common 
 with every point on the intersection of the two spheres described 
 from A and from B through this point. 
 
 It is evident, therefore, that any point in the body, of the 
 second category, can be displaced along the line of intersec- 
 
81 
 
 tion of the two spheres, on which it originally fell, while (A) 
 and (B) are still fixed in A and By letting all other points in 
 the body take care of themselves, or, rather, the rigidity of the 
 body — which consists in preserving the relative distances of the 
 points of the body — take care of their individual positions dur- 
 ing this displacement. We find then that every other point of 
 the second category will have, in all its displacements, to re- 
 main on a corresponding line of intersection of two spheres 
 from A and B. Every point in the first category, however, 
 will have to remain stationary, since there is no other point in 
 space having the same distances from A and By besides the one 
 with which it coincided originally. These conditions are seen 
 also to be perfectly compatible with the relative distances of 
 the individual points of the body from one another, as soon 
 as a continuous series of spheres, described from A and B as 
 centers, is imagined, together with their mutual intersections ; 
 for, as the points of the body move along these lines of intersec- 
 tion, these lines, together with their spheres, can be conceived 
 to glide upon themselves, never changing their form, since a 
 sphere is a homogeneous surface ; hence, the mutual distances 
 of the moving points are defined alike throughout their motion. 
 Such a motion of a rigid body, about a stationary straight line 
 (axis), we call rotation^ and the body moving with such a mo- 
 tion, is said to rotate or to revolve about the axis ABj or, simply, 
 about the two fixed points A and B. 
 
 Corollary V. — Since in the rotation of the solid, considered 
 in the preceding corollary, any two points in the solid, (C) and 
 (i)), which lie on the axis {AB)y will remain in coincidence 
 with a pair of congruent points C and I) fixed upon the dis- 
 tance-line AB, and since no other points in the solid, besides 
 those lying on the axis, remain fixed, — it follows that if a solid 
 is moved so, that a given pair of points in it, (C) and (D), re- 
 main fixed in space, then all the points in the moving solid fall 
 in two categories : — such as remain on a fixed axis determined 
 by C and D and constructed from any two 'points in itj A and 
 By and such as move constantly upon corresponding intersec- 
 tions of two systems of spheres of centers A and By respectively. 
 
 For, rotating upon the fixed axis constructed from, and 
 therefore determined by, C and D, no other points in the solid 
 besides those lying on the axis can remain fixed in space. Any 
 two points [A) and {B) in the axis, however, remaining fixed, the 
 
82 
 
 solid rotates also about the distance-line constructed from A and 
 By and this must coincide with that determined by C and D. 
 
 Corollary YI. — It follows,* that there is no difference in the 
 form of the distance-line when constructed from any pair of 
 points in space, and that any two such lines will coincide with 
 each other throughout the whole of their extent as soon as two 
 congruent points in both are made to coincide. 
 
 Corollary YII. — Any portion of a straight line will, by the 
 preceding corollary, coincide with any other portion of the 
 same straight line, as soon as their limits coincide. In other 
 words, a straight line is homogeneous, i. e., any portion can 
 move upon the whole without deformation. 
 
 Corollary VIII. — If we move up the segment (AB) of a 
 straight line upon itself a distance AA'y so that the position of 
 (AB) at the end of the motion will be A' BB' , then the whole 
 line AA' BB' is also a straight line. 
 
 A a! B B' 
 
 Fig. 5. 
 
 In fact, AA' B and A' BB' , separately considered, being, re- 
 spectively, the original and final position of the same segment 
 )AB), are two segments of a straight line, having the portion 
 A' B in common. Hence, by Corollary YI, they must coin- 
 cide, each with a corresponding segment upon the unique dis- 
 tance-line determined by A' B, i. e., AA' BB' is likewise a seg- 
 ment of a straight line. 
 
 It follows, that in this way we can prolong a segment of a 
 straight line indefinitely far, solely by shoving it along itself 
 and its successive prolongations. 
 
 Corollary IX. — A straight line cannot have more than one 
 branch on each side of a point belonging to it. It cannot, for 
 instance, have the branch ^^ on one side of the point B, and 
 J5(7and BD on the other side of it. 
 
 Fig. 6. 
 
 *This corollary follows also very readily from the associative law of ad- 
 dition of distances. 
 
83 
 
 For, otherwise, by revolving a solid containing both branch 
 ABD and branch ABC, about ABD fixed, BC would be 
 displaced — which is impossible if ABC is a straight line. 
 
 Another proof is obtained thus : — If the line ABD is con- 
 structed from ABy then the points between B and A and be- 
 tween B and D are obtained by continuously increasing the 
 distance x — respectively to be subtracted from and added to 
 AB — from zero to infinity. But, in doing so, the radius of 
 the corresponding sphere 8' described from B as center, on 
 whose surface the corresponding points lie, increases continu- 
 ously, i. e.j these points recede more and more from 
 the center By both ways, and can never come back to 
 it without crossing the intermediate spherical surface, 
 or, which is the same thing, without retracing their 
 steps backwards. But as this is not permissible in 
 the continuous description of the distance-line, we 
 can never come back to B, Hence, it is impossible 
 that the additional branch BChe ever described. 
 
 Theorem 6. — A straight line ABy issuing from a 
 fixed point A and prolonged indefinitely on the other 
 side of By will not return to A again, after any num- 
 ber of prolongations, each equal to ABy however large 
 that number. 
 
 For, if this were possible, then taking just half of 
 the whole extent, whose end let be (7, we would have 
 two distinct straight lines between A and (7, namely, 
 ABC and CB'Ay since CB' A is supposed to be differ- 
 ent from ABCy as the point B is not supposed to re- 
 trace its steps backwards beginning from (7. 
 
 Another proof is exactly like the second proof of 
 Corollary IX to Theorem 5, where the impossibility 
 of returning back to a point in the distance-line is Fia. 7. 
 deduced from the fact that, as we continually move 
 away from it both ways, all the points passed in either sense, 
 become separated from those not yet reached, by a series of 
 closed spherical surfaces, which, by the definition of an increas- 
 ing distance, can never be crossed backwards. 
 
 Corollary. — From this theorem and from the preceding one 
 it follows, that two distinct straight lines can have no more than 
 one point in common, like AOB and COD. As soon as, 
 besides 0, another point E in CODy is made to coincide with 
 
84 
 
 E' in A OB, the two lines must coincide up to the ends of 
 the smaller segments of either, on both sides of 0, and their 
 prolongations must coincide as far as we please to take them. 
 
 Fig. 8. 
 
 Definition lY. — A pair of straight lines having only one 
 point in common are said to diverge and form an angle at the 
 point Oj which is the vertex of the angle, and the straight lines 
 themselves are the sides of the angle. Such a pair of straight 
 lines is called a crossing pair of straight lines. 
 
 Scholium. — It is evident, since an angle has four segments, 
 having six combinations in pairs, if we leave out the two pairs 
 belonging to the same straight line each, we get only four — the 
 number of angles as defined above. (We shall learn later an 
 extension of the definition, which will give an indefinite num- 
 ber of angles.) 
 
 We shall consider at present one angle, made only by two 
 segments of distinct straight lines issuing only on one side of 
 the vertex 0, like A OB. We imagine this to be a rigid figure, 
 
 in which any two points 
 k and I, each on one side, 
 preserve always a con- 
 stant distance between 
 each other. 
 
 The vertex and one 
 
 side of any other angle 
 
 A' 0' B' can, evidently, 
 
 be applied to the vertex 
 
 Fia. 9. and either side of A OB. 
 
 If, at the same time, a 
 
 point in the remaining side 0' B' can he made to coincide with a 
 
 point equidistant from on OB — the corresponding free side of 
 
 the angle AOB — without breaking the rigidity of either of 
 
86 
 
 the figures as explained above, the whole side O'B' will coin- 
 cide with OB, and the two angles will he said to he equal; if on 
 the contrary y this is not possible, the angles are unequal. 
 
 Let us see whether there is a way of measuring angles. 
 
 Theorem 7. — If two straight lines, ^OCand BOB, inter- 
 sect so that they form two adjacent angles A OB and BOG (such 
 as have one side in common) equal to each other, all four 
 angles are equal. 
 
 Demonstration. — For, put an exactly equal movable figure, 
 which we denote by small letters upon the given one, so that 
 ZaOh of the movable figure shall coincide with BOC of the 
 
 c b 
 
 d A 
 
 c 
 D 
 
 Fig. 10. 
 
 original figure ; then Oc, the prolongation of aO, will coincide 
 with OD, the prolongation of BO, and Od, the prolongation 
 of 60, will coincide with OA, the prolongation of CO. 
 
 Hence ZBOC= ZaOh= Z.hOG= ICOD = /.cOd^^ ZDOA 
 = ZdOa= Z A OB ; that is, all four angles are equal. 
 
 Q. E. D. 
 
 Definition V. — Each of the four equal angles formed hy the 
 two intersecting lines, is called a right angle, and the lines are said 
 to he perpendicular to each other. 
 
 Theorem 8. — Let the straight lines AOA' and BOB' inter- 
 sect each other at right angles. Let one of these be fixed, and the 
 other turn round in space, — A OA ' , say, taking up all possible 
 
86 
 
 positions compatible with the rigidity of the figure. Then OA 
 will generate a surface, which is indefinite in extent if OA is 
 indefinite in extent, and which is capable of revolving upon 
 itself around without deformation, i. e., any portion of it 
 inclosed between any two positions of OA^ like OA^ and OA^, 
 will be capable of coincidence with any other portion inclosed 
 between two other positions of OA making an angle equal to 
 AfiA^\ moreover, if BOB' is turned over about the fixed 
 point 0, so that the segment OB' come to coincide with the 
 original position of OB, and OB with that of OB' , the whole 
 surface will, without deformation, turn about and come into 
 coincidence with the trace of its original position, upside down, 
 as soon as BOB' comes into coincidence with B' OB, 
 
 *' 
 
 
 
 
 
 B* 
 
 Fig. 
 
 11. 
 
 ./ 
 
 Demonstration. — The first half of the proposition follows 
 immediately from the principle of continuity of congruence in 
 motion (Axiom 1) and from the property of a straight line, 
 combined with the fact that in the rotation of a rigid body any 
 point of the second category moves upon a closed line — the 
 intersection of two spheres described from any two points in 
 the fixed axis as centers and passing through the point in 
 question (Corollary TV. to Definition 3)* The second half 
 
 * If the two points on the axis are taken equidistant from 0, the plane 
 represents the system of concentric circles which served for Lobatchevski as 
 a definition of a plane in his work (see " Urkundenznr Geschichte der Nicht- 
 Eukl. Geom.," Engel, 1898, pp. 93-109). 
 
87 
 
 follows from the equality of the angles A OB and A OB'. For 
 we can imagine from the start, that a duplicate of the figure 
 BOAOB' has been brought into coincidence with B' OAOB ; 
 so that BOA and the duplicate of B' OA generate the same sur- 
 face on one side, as the duplicate of BOA and B' OA itself y on 
 the other side. Hence the theorem. 
 
 Definition YI. — The surface is called a plane, OA — the 
 generator, and, in any of its fixed positions, a half-ray, which 
 together with its prolongation on the other side of composes 
 a complete ray or element. The point is the origin ; the axis 
 B' OB is said to be normal to the plane AOAy 
 
 Corollary I. — It follows, that we can revolve the plane 
 about any element or ray passing through 0, until its upper 
 side comes into coincidence with the original lower side, and 
 
 J^ 
 
 
 
 
 
 ^ 
 
 Fig. 
 
 11. 
 
 y" 
 
 / 
 
 ^^4 
 
 moe. versa — the ray itself remaining fixed ; since, in this mo- 
 tion, the ray will always coincide with itself while the nm-mal 
 is displaced and revolves around into coincidence with its own 
 reversed position. 
 
 Corollary II. — It follows also, that a plane divides all space 
 into two equal portions, which become coincident with each 
 other as soon as the normal coincides with its reversed position. 
 
 Corollary III. — The plane described by the segment OA, 
 using the perpendicular OB as an axis, is identically the same 
 as that described by the prolongation of OA, viz., OA' ; also 
 this plane is unique as long as BOB' is fixed. 
 
88 
 
 Corollary TV. — Given the plane A OA^ whose normal is 
 OB, and a straight line OC not lying in this plane, then 
 Z.BOC is not a right angle. 
 
 For, joining by a straight line, h and a, a point on the normal 
 and some point in the plane, not 0, and then revolving {BhOci) 
 around Bh fixed, ihd) must somewhere in its motion intercept, 
 in some point c, the straight line OC, which is supposed to re- 
 main stationary until this occurs ; since, after ( Oa) comes 
 round back to its original position, (60a) will have described 
 a closed surface, separating a finite portion of space from all 
 other space. This surface will consist, partly of the portion of 
 the plane described by the segment {Oa) of the element OA^ 
 in which a is situated, and partly of the lateral surface de- 
 scribed by (ah), every point of which will be at some appreci- 
 able distance from 0. Hence, a sphere described by the 
 smallest distance from of any point in the lateral surface, 
 
 Fig. 12. 
 
 will enclose an appreciable portion of OC; while a sphere de- 
 scribed by a distance greater than that of the point in the lateral 
 surface farthest from (only the finite segment (ah) is con- 
 sidered), will enclose in its interior a portion of OC having 
 some points exterior to the closed surface. Hence, a variable 
 point on OC, in moving continuously from a position on the 
 interior portion towards a position on the exterior portion, 
 must cross the closed surface. But as C can meet neither an 
 element of the plane A Oa, nor the normal OB, in any other 
 
89 
 
 point than 0, it must pierce the lateral surface described hj 
 (ba) in some point c. Thus, the generator of the lateral sur- 
 face (ba) in the position of bca intercepts OC in the point c. 
 Let then the element (OA^)) starting from its new position, de- 
 scribe its own plane anew, dragging along OC in its motion. 
 Then, this last will describe a surface which will lie wholly on 
 that side of the plane A OA^ where the half of the normal OB 
 is situated. Reversing now the plane, the surface generated 
 by OC will lie wholly on the side where OB' was originally, 
 and will be separated from its original position by double the 
 space between its original position and the plane A OAy There- 
 fore, the surface generated by 0(7 in its rotation around the 
 axis BOB', is not a plane (Corollary II), and hence Z.BOC is 
 not a right angle. 
 
 Corollary V. — It follows, that all planes coincide with one 
 another as soon as their normals and origins coincide. 
 
 Theorem 9. — All right angles are equal. 
 
 Demonstration. — If the vertex of any right angle is made 
 to coincide with the origin of our plane, and one of its sides, 
 with the normal OB of the plane, the other side must coincide 
 with some one of the elements of the plane, say OA^, by pre- 
 ceding corollary. Hence, all right angles can be made to coin- 
 cide, or are equal. Q. E. D. 
 
 The preceding theorems about the properties of a plane and 
 of right angles will suffice to render more concrete our notions 
 about angles in general as geometrical magnitudes. First, we 
 observe that we have found a natural unit for measuring angles, 
 namely, the right angle ; and, secondly, we shall soon see how 
 the plane will afford us a means of comparing all possible 
 angles with our standard unit, the right angle. The following 
 preliminary remarks are necessary. 
 
 Scholium I. — Any pair of crossing lines will be capable of 
 being applied to the plane so, that the vertex shall lie on the 
 origin, and the lines themselves shall lie wholly in the plane. 
 
 In fact, one of the two lines can always be brought into co- 
 incidence with any ray in the plane so, that the vertex fall on 
 the origin. If, now, the other line falls upon another ray, the 
 proposition is proved for the case under consideration. If not, 
 this other line will have pierced the plane (which is admissible 
 during the process of applying ; we may imagine, if necessary, 
 
90 
 
 a portion of the plane removed during the application and then 
 restored back to its original position ; see also definition of 
 rigidity), and will have only one point in common with the 
 plane, viz., the origin, like OB or 00 in Fig. 12. If, now, 
 we revolve the plane about the fixed ray with which the first 
 side is in coincidence, it will sweep through all space during a 
 revolution that brings the normal into its reversed position. 
 Hence, some time during this revolution, it will have to inter- 
 cept another point on the line which has not been in it origi- 
 nally and which is supposed to have remained fixed in position ; 
 but just as this interception occurs, this line will have two 
 points in common with some ray in the plane, and will, hence, 
 coincide with it along the whole of its extent — i. e., both lines 
 of the crossing pair will lie in the plane. Hence, all pairs of 
 crossing lines can be made to coincide, each with a pair of 
 crossing rays in the plane. In other w^ords, all possible angles 
 are to be found among the different angles between the different 
 rays of any plane. 
 
 Scholium II. — If we apply any two perpendicular lines of 
 indefinite extent to a corresponding pair of rays in the plane, 
 we see that the whole extent of the plane can, from any initial 
 position of a ray, OA say, be divided into four congruent parts, 
 each of which will be enclosed by a right angle. The end of 
 any fixed distance from 0, measured along a ray, will generate 
 in its motion around a homogeneous curve every point of 
 which is equidistant from 0. The curve is called the circum- 
 ference of a circle, and is its center. A portion of the plane 
 enclosed between any two rays measured one way from center 
 to circumference and termed radii is called a sector ^ that is, a 
 part of the whole portion of the plane enclosed by the circumfer- 
 ence ; such a sector is congruent with any other sector enclosed 
 between two radii making the same angle with each other. It 
 is now evident that, since, whenever the angle between two 
 fixed radii is equal to the angle between two other fixed radii, 
 both the sectors of the circle and the segments of the circumfer- 
 encCy or arcs, enclosed by the respective pairs of radii, are equal 
 each to each — the angles can be measured either by the cor- 
 responding sectors or by the arcs enclosed between their sides, 
 so that a greater angle corresponds to a greater sector or arc, and 
 a multiple or part of an angle corresponds to the same multiple 
 or part of the corresponding sector or arc. Since every possible 
 
91 
 
 position of two intersecting straight lines, with respect to each 
 other, has been proved to find its analogue in some position of 
 two intersecting rays of a plane, it is sufficient to investigate 
 these last. Now, we can conceive the whole aggregate of dis- 
 placements possible for a ray in the plane (Fig. 13) with re- 
 spect to a fixed ray A 'a' OaA, to be bound up with the corre- 
 sponding displacements of a circle rigidly fixed to the moving 
 ray and revolving with it about the center, so that the circum- 
 ference moves in its own trace. Then we see that, if (Oc), one 
 segment of the moving ray, makes in any of its positions an 
 angle AOq with OJ., measured by the arc — passed over by the 
 
 moving point in the circumference — from a to c, then the pro- 
 longation of (Oc) must have been displaced just as much on the 
 other side of OA' — the prolongation of OA — and must form 
 an equal angle A' OC — as measured from the fixed segment 
 OA' or, along the circumference of the circle, from a' to c', in 
 the same sense as ac. For, any fixed point in the moving cir- 
 cumference must have been displaced an equal arc with any 
 other. So we see that the two half-rays of the same ray lie on 
 opposite sides of the two half-rays of any other ray not coinci- 
 dent with it — since any fixed ray can be taken as the initial ray 
 — and the angle made by the prolongations of any two half-rays 
 will be exactly equal to the angle made by the half-rays them- 
 selves. Also, that any point in the moving ray will be dis- 
 
92 
 
 placed a quarter, a half, three quarters, and a whole circumfer- 
 ence, when the ray is displaced one, two, three, or four, right 
 angles ; further, that the half-ray will fall upon its own prolon- 
 gation — for a displacement of two right angles, and will come 
 to its original position — for a displacement of four right angles. 
 It follows, then, that a half-ray, in any of its positions, makes 
 with the two half-rays of another ray two angles, one of which 
 is just as much less than a right angle, as the other is greater, — 
 
 their sum being equal to two right angles {acute and ohtvLse 
 angles). 
 
 The following theorem, concerning angles in space, becomes 
 at once evident by the application of the angles to the plane. 
 
 Theorem 10. — Two adjacent* angles whose two non-com- 
 mon sides form the opposite prolongations from the vertex of 
 the same straight line, are together equal to two right angles ; 
 the two non-adjacent angles, made by two straight lines and 
 their prolongations, respectively, are equal ; and all the four 
 angles taken together are equal to four right angles. 
 
 Definition VII. — A figure bounded by three straight lines 
 intersecting, two by two, in three points and making three 
 angles, each less than two right angles, is called a triangle. 
 We must not, at the beginning, consider the triangle as contain- 
 ing any surface that may be limited by the sides of the triangle. 
 
 * Adjacent angles are defined as usual. 
 
93 
 
 Scholium. — It is evident from the way we have constructed 
 the straight line, that any three points at fixed distances from 
 each other are capable of congruence with any other triplet of 
 points of the same fixed distances ; since, if in any such a triad 
 we revolve one of the points around the straight line connect- 
 ing the remaining pair, we get the locus of all the points which 
 are at the same two distances from the fixed pair as the given 
 third one, and no point outside this locus can have the same 
 distances from the fixed pair, — as shown in Theorem 2-5 and 
 corollaries. Hence, when the corresponding points of the two 
 equidistant pairs in the two triads are brought into coincidence, 
 the third ones, in each triad, will be capable of coincidence. 
 
 Corollary. — Since as soon as the ends of two equal distances 
 coincide the straight lines representing these distances coincide, 
 it follows that any two triangles whose sides are respectively 
 equal to each other are congruent. 
 
 Theorem 11. — Two triangles are equal when two sides and 
 the included angle of one are respectively equal to two sides 
 and the included angle of the second. 
 
 The demonstration given by Euclid in his Elements for 
 this theorem, holds here word for word, and need not be 
 repeated. 
 
 Corollary. — In every isosceles triangle the angles at the base 
 (opposite the equal sides) are equal. For, its duplicate can be 
 applied to it so, that the equal sides be interchanged by turning 
 over ; the base and, hence, the angles interchanged, will still 
 coincide with the corresponding ones in the original triangle. 
 
 These two theorems concerning congruent angles and tri- 
 angles are sufficient to deduce a most fundamental property of 
 the plane — namely, that the origin may be transposed to any 
 point in the plane, and hence, any straight line having only 
 two points in common with the plane, anywhere, will lie wholly 
 in the plane ; hence, the plane itself is capable of translation or 
 rotation upon itself without deformation. 
 
 Theorem 12. — A straight line having two points in common 
 with a plane, will lie wholly in that plane. 
 
 Demonstration. — If the two points are upon the same ray, 
 the straight line coincides with the ray, i. e., lies in the plane. 
 Suppose, however, that the two points lie upon different rays ; 
 we then can prove that any other point of the straight line lies 
 upon some other ray of the plane. 
 
94 
 
 Let BOB' , where OB = OB' ^ be the normal of the plane 
 AOC, and let the straight line X'X cut the rays OA and OC 
 in the points A and C, respectively ; we have to prove that any 
 point jD of the line X' X lies upon a ray OD. Connecting OD, 
 we join B and B' y respectively, with Ay C, and I), 
 
 K~~~-~~~^. 
 
 Fig. 14. 
 
 Then ABOC^AB'OC and ABOA = AB'OA, by last 
 proposition. 
 
 .'.BC=:B'C,BA = B'Ay whence AABC= AAB'Cy by 
 corollary to Definition VII. 
 
 Whence, Z.BAD= AB'AJD; 
 
 . • . ABAD = AB'ADy and ^D = J5'D ; 
 
 whence, again, ABOD = AB' OD, 
 
 . • . ABOD = right angle by Theorem 10. 
 
 OD is, therefore, a ray. Now, since this is true for any point 
 D in the line X'X, the whole of its extent lies in the plane. 
 
 Q. E. D. 
 
 Definition YIII. — A plane can now be re-defined as the sur- 
 face in which every straight line lies wholly if it passes through 
 two of its points. 
 
 Corollary. — It immediately follows, that any angle can be 
 made to lie anywhere in the plane ; and so also a triangle, — for 
 one side of the triangle will lie wholly in the plane as soon as 
 
95 
 
 its two ends lie in the plane ; now, if also the third vertex is 
 brought into coincidence with some point in the plane, by re- 
 volving the triangle around the side held fixed in the plane, 
 the other two sides will likewise come wholly into the plane. 
 
 Scholium. — When a movable half-ray in a plane describes 
 a positive (continuously increasing) angle, and in doing so it 
 slides upon a fixed straight line termed transversal and inter- 
 secting its initial position in a point other than the vertex, the 
 segments which it cuts off upon this transversal — measured 
 from the fixed point of intersection towards the corresponding 
 positions of the variable point of intersection with the moving 
 half-ray — will continuously increase as long as the moving 
 half-ray meets the transversal ; that is, until a segment is 
 reached which exceeds in length any arbitrarily given finite seg- 
 ment, no matter how great. For, any point of the transversal, 
 that is at a finite distance greater than the given length from 
 the origin of the segments, can readily be joined with the ver- 
 tex of the variable angle, and will therefore form the limit of 
 one of the segments greater than the given one. 
 
 Definition IX. — The normal is said to be perpendicular to 
 the plane, because it is perpendicular to every straight line 
 passing through it and lying wholly in the plane. 
 
 Theorem 13. — A straight line perpendicular to any two 
 intersecting straight lines at their point of intersection, is per- 
 pendicular to the plane in which these straight lines are situ- 
 ated. The straight lines need not be the rays of the original 
 construction of the plane, but any two straight lines intersect- 
 ing these and, hence, 
 
 ^ M 
 
 \ 
 \ 
 \ 
 \ 
 
 lying in their plane. L — 
 
 The proof is word for 
 word that given by Eu- 
 clid in Book XI, prop- 
 osition lY, of his Ele- 
 ments. 
 
 Corollary I. — An im- 
 mediate consequence is 
 that at every point of a ^' 
 
 plane we can erect a per- Fia. 15. 
 
 pendicular to the plane. 
 
 For, if K is such a point, not the origin of the original 
 construction, and LKM and KN are any two perpendicular 
 
 \' 
 
96 
 
 straight lines through K in the same plane, then, fixing the 
 line LKM and revolving {NK) about it the amount of one 
 quadrant, into the position of iV^' -ST, this last is now perpen- 
 dicular to KM and KN) hence, it is perpendicular to the plane 
 in which they are situated. In other words, KN' may now be 
 regarded as a normal, and all straight lines in the given plane 
 through Kj as the rays. 
 
 Corollary II. — Since any two planes coincide as soon as their 
 origins and normals coincide, it follows that a plane will coin- 
 cide with itself when the origin is displaced, in any manner 
 whatever, to any other point in the same plane, provided the 
 normal at the origin is made to coincide with the normal at the 
 point to which the origin is shifted ; also, that the origin or, 
 indeed, any point in a plane, treated as such, may displace itself 
 continuously, describing any curve in the plane — the whole 
 plane remaining unaltered in shape or position ; and, further, 
 that the plane may be turned upside down, shifted upon itself 
 in any manner whatever, without altering its position or shape 
 as a whole (Leibnitz). 
 
 At this stage of our investigation our elementary figures, viz., 
 the angle, the triangle, and the circle, can be made more concrete. 
 Since each of these figures can be made to lie wholly in a plane, 
 we may suppose their boundaries to limit corresponding por- 
 tions of a plane. This is the reason of their being called plane 
 figures, in contradistinction to those which cannot be made 
 to lie wholly in a plane. Thus, a network of straight lines 
 and circles may be conceived to cover these plane figures, just 
 like the plane itself ; crossing and recrossing one another in all 
 possible ways, since every point in a plane may be conceived as 
 an origin of rays and angles and as a center for the circumfer- 
 ences of the circles described by the fixed points in these rays. 
 Geometry invariably makes use of this conception of the plane, 
 in the way of d^jiat : " Describe a circle, from such and such a 
 point as centre, and with such and such a distance as radius,^' 
 etc., — just as it does with respect to homogeneous space in 
 general. 
 
CHAPTER III 
 
 THE QUADRILATERAL AND THE IMMATERIAL QUADRILATERAL. 
 PARALLEL STRAIGHT LINES 
 
 The following propositions of the first book of Euclid's Ele- 
 ments can now be proved very rigorously, either by Euclid's 
 demonstrations, or in a more elegant way — like the one used by 
 Legendre and his followers, — since none of the hypotheses, 
 whether tacit or explicit, which Euclid assumes in the shape of 
 postulates, definitions, and axioms, in these demonstrations, are 
 now wanting a solid basis. I omit the demonstrations, refer- 
 ring the reader to Euclid or Legendre, whose treatment of these 
 propositions is admitted to be rigorous and faultless, once you 
 grant the postulates and axioms upon which their proofs rest, 
 and which have now been established with the utmost rigor. 
 The propositions referred to are : 
 
 IX*, X, XI, XII, XIV, the converse of XY, XVI, XVII, 
 (for the last two I prefer to give my own proofs, which will 
 throw some light on the nature of parallels); then XVIII, 
 XIX,t XXI, XXII, XXIII, XXIV, XXV, XXVI. Now 
 we come to Euclid's treatment of parallels, which has been 
 acknowledged to be the weakest spot in the Euclidian geometry. 
 I propose to treat this subject in an entirely difierent manner, 
 using again, as in the rest of this treatise, the kinematic method, 
 which is much more powerful than the static method adopted 
 by Euclid, and which, I flatter myself with the belief, will es- 
 tablish on a foundation firmer than ever before, the most im- 
 
 * The propositions left out have been proved by ns explicitly or implicitly, 
 excepting VII, which is unnecessary, and VI, which is a consequence of XXVI, 
 case 1, since the duplicate of a triangle with two equal angles can be turned 
 over and applied to the original triangle, with which it will coincide. 
 
 fThe Twentieth of Euclid and also the more general proposition that a 
 straight line is the shortest path between two points (not the shortest dis- 
 tance, since there is only one distance between two fixed points) can be 
 proved immediately by the consideration, — that any point of the second cate- 
 gory with respect to two fixed points, lies upon the intersection of two 
 spheres described from the points with radii whose sum is greater than the 
 sum of the radii of the two spheres in contact described from the same 
 fixed points, whose point of contact is a point of the first category, on the 
 straight line between the fixed points, by the very construction. 
 
 97 
 
98 
 
 portant truths of similarity and proportion, on which rests all 
 the grand superstructure of our actual mathematics, both pure 
 and applied. 
 
 I proceed to prove XVI and XYII of Euclid's first book. 
 
 Theorem 14. — The sum of any two interior angles of 
 a triangle is less than two right angles. 
 
 a + b<2rt Z's, 
 
 Demonstration. — If a + 6 ^ 2 rt. Z's, then, because /.ABE 
 -f 6 = 2 rt. Z's (Theorem 10), it follows a + 6 ^Z.ABE-\- h, 
 and a ^ /.ABE. For a similar reason h ^ /.DAB. Now, apply- 
 ing the lower part of the diagram, 
 namely DABEy to the upper ABC 
 so, that A shall fall upon B and B 
 upon Ay and the angle a, being 
 ^ /EBAy shall coincide with or in- 
 close the latter, and b shall coincide 
 with or inclose /DAB, — AD will 
 lie, either on the side BC, or within 
 the angle 6, and BE, either on A (7, or 
 within the angle a. In either case AD 
 and BE will intersect — namely, at C, 
 the vertex of the triangle, or in some 
 point within the triangle; and when 
 returned to their original positions, they 
 — while making the prolongations of 
 CA and CB, respectively — will in- 
 tersect at some other point C, below 
 AB. Or, in other words, the two lines CAD and CBE will 
 intersect in two points Cand C, — which is impossible (The- 
 orems 2, 5 and, corollaries). Therefore, it is impossible that 
 in the triangle ABC, a + 6 ^ 2 rt. Z's. Q. E. D. 
 
 Corollary I. — If in the formula a + 6 ^ 2 rt. Z's, we sepa- 
 rate the case of equality, namely, a + 6 = 2 rt. Z's, we shall have 
 a=/ ABE and b = / DAB ; and the two straight lines mak- 
 ing such angles with the secant^ cannot meet either below or 
 above the secant, — for, in either case, they would have to 
 meet simultaneously also on the other side of the secant, which 
 is impossible. Therefore, if two straight lines intersected by 
 a third one make with it two interior angles on the same side 
 
 Fig. 16. 
 
99 
 
 of the secant, equal to two right angles, they cannot meet, even 
 if produced indefinitely both ways. 
 
 Corollary II. — Any exterior angle of a triangle is greater 
 than either of the interior and opposite angles. 
 
 Because any of the interior angles with its adjacent exterior 
 angle equals two right angles (Theorem 10) ; while with eithe of 
 the opposite interior angles it is less than two right angles ; hence, 
 each of these last ones is less than the exterior opposite to it. 
 
 Corollary III. — If one of the interior angles of a triangle 
 is a right or an obtuse angle, either one of the remaining two is 
 less than a right angle ; therefore, in any triangle there can be 
 no more than one right or obtuse angle, and not less than two 
 acute angles. 
 
 Corollary IV. — A perpendicular is the shortest line from a 
 point to a straight line (Proposition XYIII of Euclid's Ele- 
 ments). A perpendicular is, therefore, assumed to represent 
 the distance from a point to a straight line. 
 
 Definition X. — A quadrilateral is a figure bounded by four 
 sides containing four interior angles. A rectilinear quadrilat- 
 eral is one bounded by the segments of four straight lines — 
 of fixed length each — between four points which are the ver- 
 tices of the quadrilateral, — each vertex being connected only 
 with two adjacent ones. A plane quadrilateral is one that can 
 be placed wholly in a plane. We shall have occasion to use 
 quadrilaterals of fixed sides only, but not of fixed angles. Of 
 course, such are, so to speak, non-material ones, i. e., bounding 
 no fixed plane area, and in such, the relative distances of the 
 four vertices are fixed only for the four (out of all six pos- 
 sible) pairs, constituting the ends of the four sides respec- 
 tively. While the three distances, AB, BCy CA, must 
 necessarily fix all distance-relations between three points — a 
 distance being a relation between one pair (number of com- 
 binations of three different things taken two at a time), only 
 six distances will be sufficient to fix uniquely the distance- 
 relations between four points (number of combinations of four 
 different things taken two at a time) ; hence, four distances are 
 insufficient in the case of a quadrilateral. If, however, the 
 quadrilateral is restricted to lie in a plane, we have an addi- 
 tional relation, — and only one additional distance, like one diago- 
 nal (the distance between either pair of the opposite vertices), 
 will be sufficient to determine the complete form of the quadri- 
 
100 
 
 lateral. This property of a quadrilateral is expressed by say- 
 ing that a quadrilateral can " rack." We shall call such a quad- 
 rilateral with variable angles an immaterial quadrilateral. 
 
 Theorem 15. — In any rec- 
 tilinear quadrilateral, whether 
 plane or not, whose opposite 
 sides are equal, the opposite an- 
 gles are equal. 
 
 Demonstration. — Let AB =: 
 DC, AD = BC. Then join- 
 ing A C and DB, we get 
 
 Fig. 17. 
 
 aABC= a CJDA 
 
 and 
 
 A ABB = A CDB ; 
 
 . • . ZABC= Z CDA, ZBAD = ZDCB. 
 
 Q. E. D. 
 
 Corollary I.— It follows, that ZABD = ZBDC and ZADB 
 = ZCBD; similarly ZCAB== ZACD and ZACB = ZCAD, 
 — that is, the angles made by the same diagonal with the two 
 opposite sides, are equal. 
 
 Corollary II. — It also follows, that in a plane quadrilateral, 
 with equal opposite sides, two non-opposite sides and the 
 angle enclosed by them are suf- 
 ficient to determine the whole 
 quadrilateral. 
 
 Theorem 16. — Two plane 
 quadrilaterals ABCD and 
 A' B' CD' , having three sides and 
 two included angles in the one 
 equal respectively to the corres- 
 ponding three sides and included 
 angles in the other, are equal to 
 each other. 
 
 Demonstration. — If 
 
 DA = D'A\ 
 AB = A'B', 
 
 Fig. 18. 
 
 and also, 
 
 BC=^B'C', 
 
101 
 
 Osim'fi) 
 
 0, 
 
 Of 
 
 X' 
 
 ZDAB = ZD'A'B\ 
 
 ZABO= ZA'B'C, 
 
 then, by superposition, we see that the two quadrilaterals coin- 
 cide with each other. Q. E. D. 
 
 Theorem 17. — In a plane quadrilateral, two of whose op- 
 posite sides are equal and in 
 which the interior angles made 
 by these with one of the re- 
 maining sides, are supplemen- A«(m+/) 
 tary, this last side cannot be 
 greater than the side opposite 
 to it. 
 
 Let OA ^ O^A^, Z AOO^ 
 + / 00,A^ = 2 rt. Z 's ; 
 
 then 00^ > AA^. 
 
 For, let 00^>AA^, Then 
 since OA = O^A^ and Z AOO^ 
 := Z Afifi^j the supplement of 
 Z OOyA^y we can apply the 
 lower side of the quadrilateral 
 to its upper side O^A^j and we 
 get ^,^,< 0,0, = 00,. 
 
 Now, suppose 00^ — AA^=l, 
 some length ; then, every time 
 is transposed a distance = 00^ 
 along the straight line XX', A 
 is transposed a distance = 00^ 
 — L Let OA = ml -{- ?„ where 
 m is a positive integer, and l^ 
 is either zero or a length less 
 than I. Then, applying this 
 quadrilateral to itself, along the 
 same straight line XX , 2(m Fia. 19. 
 
 + 1) times, we get, — 
 
 <^<^2U-f 1)= 2 (m + 1) 00, = 2 (m + 1) ^^, + 2 (m -h 1) ^ 
 >2(m+l) AA,-\-2ml-{.2l, = AA,A,>>>A,^^^,^+ OA -\- 
 
 ^2U+l) ^2(m+l)> 
 
102 
 
 that is, the straight line between and 0^^^^ ^^ is greater than 
 the broken line between these same two points — which is 
 absurd * (Euclid XX, Book 1). Hence, 00^ > AA^. 
 
 Q. E. D. 
 
 Corollary. — The sum of the three angles of a triangle can- 
 not be greater than two right angles. 
 
 For, if ^50 is such a triangle, then apply- 
 ing to it an equal triangle BCD, where BD 
 = ^(7 and CD = AB, we get ZDBC+ ZCBA 
 + ZBAC=^ ZC+ Z B+ ZA>2rt.Z's; 
 therefore, if Z D'BC+ ZB+ ZA = 2 rt. 
 Z's, and D'B=DB, we have ZD'BC< 
 ZDBC, whence CD' < CD (Euclid I, XV) ; 
 we have also, D'B = CA and ZD'BA-{- 
 ZBAC= 2 rt. Z's, ^. e., D'BACis just such 
 B a plane quadrilateral as has been discussed in 
 the theorem, and CD' <AB, — which is im- 
 possible. Hence, the sum of the three angles 
 of a triangle cannot be greater than two right 
 angles. Q. E. D. 
 
 FIG. 20. Theorem 18. — An immaterial quadrilateral 
 
 whose opposite sides are equal, can so " rack " or change its 
 angles that, while one of them is passing through all possible 
 magnitudes, the middle points of one pair of opposite sides 
 shall always remain at a distance from each other, equal to 
 each of the other pair of opposite sides. 
 
 Demonstration. — Let ABCD be a quadrilateral whose oppo- 
 site sides are equal, and E and _F, the mid-points of AD and 
 BC, respectively. 
 
 We observe first, that, ABCD being an immaterial quadrilat- 
 eral, the only restriction imposed upon the relative position 
 of its sides is that resulting from the equality of either pair 
 of its opposite sides. In any deformation or " racking " of the 
 quadrilateral, the sides need not stay in one plane, if this is re- 
 quired by the conditions of the relative motion to which they 
 are subjected, so that any pair may be in a different plane from 
 the remaining pair. (B), for instance, is free to move in some 
 path or other, situated in a spherical surface around (C), this 
 
 * See note, p. 116. 
 
103 
 
 last moving in a sphere about (Z)), and this one about (A) — 
 provided the four distances do not change during these motions. 
 Initially the distance EF is not given ; it is, however, capable 
 of determination. At least in one of the infinite number of 
 relative positions which can be assumed by the points (J.), (B), 
 (C), and (D), as just defined, the distance jEJi^will be equal to 
 each of the sides (AB)j (CD) ; this will, evidently, be the case 
 
 when all four points (B), (C) (J.), (D) will fall upon the same 
 straight line, say, upon B' , C, A' , I)', respectively, where B' C 
 = 2BF=2CF = A'I)'=2AE=:2EB, and A'B'=:AB^ 
 DC^B'C. 
 
 For ZB = ZD — always (Theorem 15); hence, when the 
 first becomes zero, the second must also become zero ; that is, 
 when (BC) falls upon (AB), (AD) falls upon (DC); hence, 
 (A'B') will be in the same straight line with (B' C) when and 
 only when (A' D) i^ in the same straight line with (DC), 
 where the primed letters denote some particular position of the 
 sides ; in other words, B' will be in the same straight line with 
 A' C y and D' will be in the same straight line with these. 
 Hence, F and E are the middle points of B' C and A' D' , 
 respectively. We have, then, 
 
 EF= EA' + A'B'- FB' = ^A'D' + A'B' - ^C'B' 
 
 = A'B' ==AB= CD, 
 since A'D'/2 = C'B'/2, 
 
 In this position, then, let (EF) be hinged to the two ends of 
 a straight line of fixed length, or let E and F become fixed, so 
 that (BC) singly may revolve around i^in a sphere, and (AD) 
 singly, around E in a sphere. I say that the quadrilateral may 
 be brought out of coincidence with a single straight line and 
 may assume all values for any one of its angles, provided the 
 equality of opposite angles is preserved, not only in the whole 
 
104 
 
 quadrilateral, but also in the partial quadrilaterals. For, if 
 not, the reason of this must be sought only in the mutual 
 geometrical relations of the distances, which are the only parts 
 of our immaterial quadrilateral given, including now the addi- 
 tional distance jEJi^ between the mid-points of (AB) and (JBC) ; 
 that is to say, these distances must be incompatible with an 
 angle b other than zero, so that an attempt to change this value 
 would involve a contradiction in theory and, hence, a break in 
 the distance-relations in practice. Calling the interior angles 
 of the two partial quadrilaterals as in the figure, we get six 
 independent relations between the angles, which, on the suppo- 
 sition that the motion sought is possible, must hold for all 
 values of the angle 6, the value 6 = included. We have 
 also, in addition, two independent relations, derived from the 
 supposition that (BFC) and (AED) are to remain straight 
 lines in the whole course of the motion. We have thus at first 
 appearance relations enough, and just enough, to determine each 
 angle, if such a determination is at all involved in the mutual 
 relations of the angles and distances. Accordingly, if the mo- 
 tion postulated were impossible, these relations would have to 
 lead us to a contradiction for any value of b other than 6 = 0. 
 Now, instead of leading to such a contradiction, the first six 
 relations give us, as an immediate consequence, a new relation 
 verified by the remaining two, — showing that our supposition 
 does not involve any contradiction, and that such a motion can 
 be realized. The relations referred to are : 
 
 ^ = S,II. 
 
 1) a = ai 
 
 4)y5=6' 
 
 2)7 = c 
 
 . • . a = 7, 1 ; 5) 3 = cZ 
 
 3) a=c 
 
 6)b==d. 
 
 r 7) a + S = 2 rt. Z's 
 Besides \ 
 
 8) /3 + 7 = 2 rt. Z^s. 
 
 Combining I and II, we get, 
 
 
 a+8 = y3+7,ni,— 
 
 which agrees with (7) and (8). Of course, I and II can be 
 realized without the relations (7) and (8), when (BFC) and 
 (AED) are not restricted to remain straight lines. But the 
 
105 
 
 argument is thereby not invalidated — that the motion presup- 
 posed in the theorem cannot be impossible under the only con- 
 ditions given. It may, of course, be impossible under certain 
 
 Fig. 21. 
 
 additional conditions, like the restriction of the lines to slide on 
 certain surfaces ; but such a restriction is excluded in the given 
 conditions, where only abstract distances are given. The motion 
 itself may involve certain positions as possible and others as 
 impossible, — which circumstance we shall presently proceed to 
 investigate only in so far as will be necessary for our main 
 purpose, which is the establishment of the theory of parallels. 
 Remark. — Such a motion would also be possible, even in 
 the case of spherical arcs of great circles, provided these arcs 
 are at liberty to move out of the surface to which they belong 
 initially, — that is, provided an additional restriction is not im- 
 posed upon them that every point of all five must remain, dur- 
 ing the motion, at a constant distance from the center of the 
 sphere to which they belong when the spherical angle h is 
 zero. In other words, in order that such a motion be realized 
 for arcs of great circles of the same sphere, the planes deter- 
 mined by the arcs in motion can all have a common point of 
 intersection only when the spherical angle of two of these linked 
 arcs is zero or tt — i. e., when they coincide with one and the 
 same great circle ; for all other values of the angles these 
 planes cannot intersect in the same point, — in other words, the 
 arcs must leave the common surface to which they originally 
 belong, as can easily be proved when we come to a maturer 
 stage of the science of geometry, to which the treatment of 
 spherical geometry properly belongs. A similar remark holds 
 with respect to geodesies upon a pseudosphere. It seems, that 
 proceeding from these considerations, it ought to be possible to 
 prove that, whenever the normals to a surface-element of a ho- 
 mogeneous surface meet in a point, or, what is the same thing, 
 whenever the surface has constant positive curvature, there must 
 
106 
 
 be an excess of the sum of the three angles of a triangle over 
 two right angles ; and whenever the normals lie in different 
 planes, or the homogeneous surface has constant negative cur- 
 vature, there is a deficiency of the sum of the three angles of 
 a triangle from two right angles. 
 
 Analytically, the proof of the possibility of such a motion 
 as defined in the theorem, is obtained also by showing that, if 
 we add to the above eight relations an arbitrary relation h— 6, 
 the determinant of the nine linear equations in nine variables — 
 when the equations are made homogeneous — vanishes identi- 
 cally, proving that h can have any value. The determinant in 
 question is — if a, a, b, jS, c, 7, d, 3, 1 is the order in which 
 the variables are written — as follows : 
 
 1-10000000 
 
 1-1 
 
 
 
 
 
 10 
 
 
 1-10 
 1-10 
 10 
 
 
 1 
 
 
 
 
 
 1 -TT 
 
 
 1 
 
 
 
 - 
 
 = 0. 
 
 an 
 
 0-^ 
 
 Q. E. D. 
 
 Theorem 19. Lemma. — Given 
 angle and a straight line through its ver- 
 tex, not in the plane of the angle, the sum of 
 the angles made by this line with the sides 
 of the angle, is greater than the angle. 
 
 The theorem needs proof only for the 
 supposition that Z COA and Z COB, 
 taken singly, are each less than Z A OB. 
 Connect any two points, a and 6, on the 
 sides of the given angle AOB, by a 
 straight line ab ; this last will be in the 
 plane of the angle (Theorem 12), and 
 hence OC, the line through outside the 
 
107 
 
 plane, cannot meet ah. Pass now a Im^Od in the plane A OB, 
 at an angle aOd equal to Z ^0(7 and cutting ah in some point 
 d — which it must do, since it cannot cut either Oh or Oa 
 again (Theorem 6 and corollary), and because the straight line 
 from will pierce a sphere inclosing the whole triangle ahO 
 (Theorems 2-5 and corollaries). Lay off on OCa segment Oc 
 equal to Od, and join ca, ch ; then ca = da (Theorem 11), and 
 in the triangle ach^ ac -^ ch'^ ad + dh (Euclid I, XX *) ; hence, 
 ZcOh > ZdOh (Euclid I, XXY). . •. ZaOc + ZcOh> 
 ZaOd+ Z dOh, or ZAOC + I COB> ZAOB. 
 
 Q. E. D. 
 
 Theorem 20. — In the motion considered in Theorem 18, the 
 four sides of each quadrilateral will, at any instant, be in one plane. 
 
 For from I, a = 7, or from II, /3 == 8, combined with (7), 
 a + 3 = 2 rt. Z's, it follows that 7 -f- 8 = 2 rt. Z's. Now, join- 
 ing FD, we find that if CF is always in the plane of A DFEy 
 then CD J FB^ FA, and AB are in the same plane, and the theo- 
 rem is conceded. But if CF is sometimes out of the plane 
 
 ^ , ,_ 
 
 n 
 
 Id — -..>,..^^ 
 
 cj 
 
 f^^ 
 
 ' — ~-— ~^<J/ 
 
 f 
 
 gt/f 
 
 I 
 
 ^JB 
 
 Fig. 23. 
 
 I) FE, then Z CFF determines a plane different from plane DFE 
 — that is, BF is not in the plane of Z CFE ; hence, by preced- 
 ing lemma, Z CFD + Z DFF> Z CFE, or Z CFD + Z FDC 
 > 3 (Theorem 15) ; and since 0=7, we get c -\- Z CFD + 
 Z FDC> 7 4- 5, or Z nCF+ Z CFD-j- Z FDC> 2rt. Z^s— 
 which is impossible (cor. to Theorem 17). Hence, CF is always 
 in the plane DFE, and the theorem is proved as before. 
 
 Q. E. D. 
 
 Corollary I. — It follows, that any immaterial quadrilateral 
 vnth equal opposite sides, can move in a plane so, that one of its 
 angles assume any value we please, and that the distarwe hetween 
 the middle points of one pair of opposite sides constantly remain 
 equal to each of the remaining pair of opposite sides. 
 
 * See also note, p. 97. 
 
108 
 
 Corollary II. — It is evident that the sum of the interior 
 angles and, hence, also of the exterior angles, adjacent to the 
 same side in any of the positions of such a moving quadri- 
 lateral, is equal to two right angles, since 2 rt. Z's = a -f- /^ = 
 y-^8 = d-\-c=a-\-h — b-\-c, etc. 
 
 Theorem 21. — The sum of the interior angles adjacent to 
 the same side of any plane quadrilateral with equal opposite 
 sides, is equal to two right angles. 
 
 For, an immaterial quadrilateral having sides of equal 
 length with the given one, can assume such a position in a 
 plane, that the angle enclosed between any two of its non-oppo- 
 site sides, be equal to the corresponding angle in the given 
 quadrilateral. And as the sum of the interior angles adjacent 
 to the same side of this immaterial quadrilateral in the particu- 
 lar position, is equal to two right angles, it follows by Corollary 
 II to Theorem 15, that the sum of the interior angles adjacent 
 to the same side in our given quadrilateral, is likewise equal to 
 two right angles. Q. E. D. 
 
 Corollary I. — The sum of the three interior angles of any 
 triangle is equal to two right angles. 
 
 For, if in the triangle ABC we produce AB to E, Z CBE> 
 C; hence, making Z CBD = ZBCA^ BD falls within angle 
 CBE (Theorem 14, Cor. II). Taking BD =^ AC, and joining 
 
 CD, we have, A CBD = a BCA, 
 and CD = AB (Theorem 11), 
 hence, we have a plane quadri- 
 lateral with equal opposite sides 
 therefore, Z CAB -h Z ABD = 
 2 rt. Z's, or ZA + ZB + ZG 
 = 2 rt. Z's. 
 
 Corollary II. — The exterior 
 angle of a triangle is equal to the two interior and opposite 
 angles. 
 
 Corollary III. — The four angles in any plane quadrilateral 
 equal 4 rt. Z^s, since it can be divided into two triangles having 
 their six angles coincident with the four given ones. 
 
 Theorem 22. — Two straight lines perpendicular to a third 
 one in the same plane with it, — 
 
 1) Will both be perpendicular to any other perpendicular 
 drawn from any point in the one to the other ; 
 
109 
 
 2) Will both have the same inclination towards any secant 
 to both ; 
 
 3) Will be everywhere equidistant from each other, i, e., any 
 perpendicular from one to the other will be of the same length 
 with any other. 
 
 Demonstration. — 1) Let J.j5, CD be both perpendicular to 
 the same straight line KL in the same plane ; and draw any 
 other perpendicular UN, from any other point 31 in AB to 
 CD ; and join LM. The sums of the interior angles of each of 
 the two triangles are equal to two right angles singly, and to- 
 
 FiG. 25. 
 
 gether they are equal to four right angles. But the six angles 
 of both triangles make up together the four angles of the quad- 
 rilateral KLMN. Of these last, the angles K, L, and N are 
 each a right angle (by construction) ; therefore, Z 31 also is 
 a right angle, and hence any perpendicular to CD, from any 
 point in AB, will also be perpendicular to AB. For the same 
 reason, any perpendicular to ABj from any point in CD, will 
 also be perpendicular to CD. 
 
 2) In the same diagram, if P3fL is a secant toAB and CD, 
 which are both perpendicular to a third line in the same plane, 
 then draw from L a perpendicular LK to AB, and from Jf, a per- 
 pendicular 3fN to CD. The first will also be perpendicular to 
 CD J and the second, to AB (section 1 of our proposition). Now, 
 in the triangle L3fN we have Z L3IN -{- Z NL3I = 2 rt. Z's — 
 Z N= rt. Z (corollary to Theorem 21); besides, /.L3IN-\- 
 Z LiOr= 31= rt. Z. Hence, Z L3^1N -h /_ NL3f=: Z L3fN-{- 
 Z L3IK; Z NL3I= Z L3IK, or Z PLD = Z P3IB (Theorem 
 10). That is, both AB and CD have the same inclination 
 o wards any common secant. 
 
110 
 
 3) If KL, MN are any two perpendiculars to both AB and 
 CDy which are situated in the same plane, and ML joins the 
 opposite angles M and X, then Z NLM = Z LMK ; Z LMN 
 = Z MLK (section 2 of our proposition), and ML is common 
 to both triangles ; therefore, A L3IN= A MLK, and MN= 
 KLj — and as the same reasoning applies equally to any other 
 two perpendiculars to both, all the points of either line AB ov 
 CDy are at equal distances from the other. 
 
 Definition XI. — Two straight lines perpendicular to a third 
 in the same plane, as having the same inclination towards any 
 common secant and being everywhere equidistant from each 
 other, are said to be parallel to each other, meaning — beside 
 each other, or going in the same direction and being everywhere 
 at the same distance from each other. 
 
 Corollary I. — Any perpendicular to one of a pair of parallel 
 lines, in the same plane with both, if produced indefinitely 
 must meet the other at right angles. 
 
 For, if from the point of intersection with the first a perpen- 
 dicular be drawn to the other, it must also be perpendicular to 
 the first (section 1) and, therefore, coincide with the perpen- 
 dicular to the same, previously drawn. 
 
 Corollary II. — From any given point without a given 
 straight line only one parallel can be drawn, namely, that line 
 which is at right angles to the perpendicular from the point to 
 the given line. 
 
 Corollary III. — Any two points at the same distance from 
 a given straight line, in the same plane with, and on the same 
 side of it, determine another straight line parallel to the first. 
 For, joining these points and drawing the perpendiculars to the 
 given straight line, we get a plane quadrilateral which is con- 
 gruent with the duplicate turned over, so that the equal perpen- 
 diculars become interchanged in position ; hence, the angles op- 
 posite the given right angles are equal, and as their sum equals 
 two right angles (Theorem 19, corollary III), each is equal to 
 one right angle ; that is, the line connecting the points, and the 
 given line, are both perpendicular to the same straight line in 
 the same plane with these. Hence, they are parallel. 
 
 Corollary IV. Any two straight lines lying in the same 
 plane and having same inclinations towards a common secant 
 are parallel ; for if a parallel to one of the given lines is con- 
 structed through the intersection of the other with the secant, 
 the corollary becomes evident (section 2). 
 
Ill 
 
 Theorem 23. — Not more than two points of equal distances 
 from a given straight line, can be situated in another straight 
 line which is in a different plane from that passing through the 
 given line and one of these points. 
 
 Demonstration. — Let AB^ CE be the two straight lines, of 
 which EC is not in the plane BA C passing through the line 
 AB and the point (7; and let the distances of the points C 
 and E from AB (i, e., the perpendiculars EB and CA drawn 
 from these points to AB) be equal. Then no other point on 
 EC can be of the same distance from AB. 
 
 For, produce the plane BAG indefinitely beyond BG, and 
 draw initGDLAGy.'.W AB (Definition XI), and BD 1 AB. 
 BD will meet CD, and BI) = AG= BE (Theorem 22, Cor. 
 I and Definition XI). Join AD, BG, EA. The right-angled 
 triangles BAE^ ABB, and j5J.(7are equal (Theorem 11) ; there- 
 fore, EA = J)A=^ GB, and Z EAB = Z DAB = Z GBA. 
 Also, since EA is not in the plane GAB, Z GAE -f Z EAB 
 > Z GAB (Theorem 19), or Z GAE + Z EAB > Z GAD 
 -^ZDAB; hence, ZGAE> Z GAD. Now the triangle 
 EAG must have Z EGA < rt. Z. For, if we imagine it re- 
 moved from its original position and applied to a DA G in 
 such a way that EA shall coin- 
 cide with its equal DA — the 
 other side, inclosing the greater 
 angle EAG, will fall without y 
 
 the smaller angle DAG and / 
 
 will take the position AG', I y >>^ j _-^b 
 
 as in the diagram ; and join- c'^ 
 ing GG ', we obtain an isosceles ^ p^^ 26 ^ 
 
 triangle AGG' , of which the 
 
 angles (7. and C, at the base, are equal, and each less than 
 a right angle (Cor. to Theorem 11 and Cor. Ill to Theorem 
 14). Then, DG and GG' form an angle DGG' < 2 rt. Z's, 
 having its opening towards A ; that is, the prolongation oi G' G 
 is separated from A and from any point on AD by the half-ray 
 CD. Therefore the half-ray G' (7, including its prolongation, in 
 passing continuously, not through A, to the position C' D contain- 
 ing one point of the segment AD, must first pass through some 
 point on the prolongation o^ AD before reaching its final posi- 
 tion. Hence Z A G' D is less than some angle which is less than 
 Z ^(7'C(Scholium to Theorem 12); or ZAG'D<ZAG'G< 
 
112 
 
 UV yiB 
 
 Fig. 26. 
 
 rt. Z. Now, any point on the prolongation of EC to the 
 left, must also be without the plane DC A, since one part of 
 EC cannot be without, and the other within, the plane DC A 
 (Theorem 12) ; and any straight line connecting C and an- 
 other point of the same distance from AB, without the plane 
 DC Ay makes with CA an angle < rt. Z and, consequently, 
 cannot be the prolongation of EC, since such a prolongation 
 forms with CA an angle > rt. Z (Theorem 10). Therefore, 
 no point of the same distance from AB 2^ C, can be on the 
 prolongation of EC to the left. For a similar reason, no point 
 of the same distance from AB as the two given points, can be 
 
 on the prolongation of CE to 
 the right, because CE is not 
 in the plane AEB passing 
 through AB and the point E, 
 Neither can a point of an 
 ^E equal distance from AB be 
 situated on CE, between C 
 and E, since then, either E 
 or C would be on the prolong- 
 ation of a straight line connecting two points of equal dis- 
 tances from another straight line — one of the points connected 
 being in a different plane from that passing through the other 
 point and the other straight line, — which has just been demon- 
 strated to be impossible. Therefore, no other point besides E 
 and (7, on the same straight line with them, or on its prolonga- 
 tions, is possible, of the same distance from ^j5 as ^ and C. 
 
 Q. E. D. 
 
 Corollary I. — Hence, any straight line in space which con- 
 nects three points at equal distances from another straight line, 
 must lie wholly in the plane passing through the other line 
 and one of its own points, — and, therefore, is parallel to the 
 other ; and all that is said in Theorem 22, with reference to a 
 parallel straight line, is also true of any straight line in space 
 having, at least, three points at equal distances from a given 
 straight line. 
 
 Corollary II. — It also follows, that if two equal lines inter- 
 sect two others in four points, and three of these lines are at 
 right angles to one another, all four are in the same plane, and, 
 therefore, are parallel, each pair singly. 
 
 Corollary III. — A parallel can also be defined as the locus 
 
113 
 
 o/ all points equidistant from a straight line, and coUinear vnth 
 two given points. 
 
 The definition of alternate interior and exterior angles of a 
 line intersecting two others in the same plane, is as usually- 
 given. If the definition of parallels is taken provisionally 
 in the sense in which we have defined them, we see that we 
 have proved propositions XXVII, XXVIII, XXIX of Eu- 
 clid, also XXXII and some others, from which a number of 
 corollaries can be drawn, regarding the conditions of paral- 
 lelism of straight lines in space, which we will not give 
 here. We are now in a position to prove propositions XXX, 
 XXXIII, and XXXIV of Euclid, using his proofs word for 
 word. This will now enable us to deduce Euclid's famous 
 Eleventh Axiom, and thereby extend the definition of paral- 
 lelism to any two lines situated in the same plane and not meet- 
 ing each other at a finite distance. 
 
 Theorem 24.— 
 Two straight lines 
 in the same plane, 
 of which one is 
 perpendicular to a 
 third one, and the 
 other makes an 
 acute angle with 
 it, if sufficiently 
 produced on that 
 side of the secant 
 where the acute 
 angle is situated, 
 must meet each 
 other somewhere. 
 
 Let 5 C be per- 
 pendicular to AB, 
 and AE make an 
 
 acute angle with ^^ 
 
 AB at A, then 
 
 AE and BC, if 
 
 produced towards the opening of the acute angle, will meet in 
 
 some point/. 
 
 Demonstration. Take some point a on AE, draw a perpen- 
 dicular from it to A B (Euclid, prop. XI). This perpendicular 
 
 I m n pBq 
 
 Fig. 27. 
 
114 
 
 will cut AB in some point I between A and JB, and not on its 
 prolongation AR ; otherwise ZBAE, being according to sup- 
 position an acute angle, would be greater than a right angle 
 (Theorem 14, Cor. II) — which is absurd. Let now Al be con. 
 tained in AB m times (m being a whole number), or m times 
 with some remainder less than AL Take upon AEy beginning 
 from A and proceeding towards -£J, m + 1 parts equal to Aa, 
 namely Aa, ab, 6c . • •, and let / be the end of the last part. 
 
 A Im n pBq 
 
 Fig. 27. 
 
 Draw now perpendiculars aa^, hh^, cc^- • -ff^ to AD which is made 
 perpendicular to AB (Eucl., prop. XI) ; they will also be 
 parallel to AB and to one another (Defin. XI). Draw also 
 6m, en J • • -/g, parallel to BC, AD, al, — which will all meet at 
 right angles all the perpendiculars to AD (Cor. 1 to Theorem 22). 
 Let the vertices of these right angles be g, A, ^, • • •/. All these 
 perpendiculars will represent two sets of parallels. The triangles 
 Aaly ahg, ---efk will be equal to one another because they have 
 one side and two adjacent angles equal respectively, in all 
 of them — those adjacent angles being alternate angles (Theorem 
 
115 
 
 22, section 2, and Euclid, prop. XXYI). Therefore Al = 
 ag — hh- " ek, and therefore also — according to Euclid, proposi- 
 tion XXXIII — Al = lm = mn • . • = pq. For the same reason 
 a^a = Aly bj) = Am, c^c = An • • 'f^f== Aq. But Aq, contain- 
 ing m -f 1 times Al is greater than AB ; hence f^f>AB, In 
 other words, the distance of f from AD is greater than the dis- 
 tance between the parallels BC, AD, which is everywhere the 
 same and equal to AB ; hence, y must lie beyond the space in- 
 closed between both the parallels — and since it is a point on 
 AE, AE must intersect ^Cin some point of/' below/. 
 
 Q. E. D. 
 
 Corollary I. — Two straight lines in the same plane, which 
 cannot meet how far soever produced both ways, are parallel 
 to each other. For, any perpendicular to one of them, drawn 
 from any point in the other, cannot make an oblique angle with 
 the latter — otherwise they would meet on the side of the acute 
 angle ; it must, therefore, be at right angles to both — or both 
 the lines must be parallel (Theorem 22, Defin. XI). 
 
 Q. E. D. 
 
 Corollary II. — Any straight line intersecting one of a pair 
 of parallel lines, and in one plane with the other, if produced 
 sufficiently far, must meet the latter. 
 
 For, being in one plane with both, were it not to meet the 
 second somewhere, it would be parallel to it (preceding Corol- 
 lary) and hence also to the first (Euclid, prop. XXX) ; but it is 
 not, since it intersects the first. Therefore it must likewise 
 intersect the other, somewhere at a finite distance. Q. E. D. 
 
 With this, the theory of parallels, as well as that of the elements 
 of geometrical measurement — distance, straight line, plane, 
 angle, circle, etc., — is firmly established. We see, then, that 
 
 THE FOUNDATIONS OF THE EUCLIDIAN GEOMETRY rest OU a 
 
 much firmer basis than mere arbitrary assumptions verified by 
 experience to a very great degree of approximation. They 
 are, rather, implanted in the very nature of our logic, being to 
 a great degree what the Kantists call a priori, and are empirical 
 only in so much as all our conceptions of quantity, form, and 
 motion depend upon experience. 
 
AUTOBIOGEAPHY. 
 
 The author of this DissertatioD, Israel Euclid Rabinovitch, 
 was born in the month of June of the year 1861, in the town 
 of Berditchev, Kiev Government, Russia. He obtained his 
 elementary education in an old-fashioned rabbinical school, and 
 afterwards prepared himself by self-instruction for entrance 
 into a Russian University, by having completed a course of a 
 Russian classical gymnasium. Having, however, found it diffi- 
 cult to obtain admission there, he left in 1887 for the United 
 States, with the express purpose of eventually pursuing an 
 academic course of studies in one of the American universities. 
 He entered the University of Pennsylvania in 1891 and, hold- 
 ing a scholarship there, studied the elements of mathematics, 
 including some engineering courses, under Professors Crawley, 
 Fisher, Barker, and Spangler, during 1891-1892 and part of 
 1892-1893, when he left on account of illness. During the 
 academic year of 1894-1895 he attended special courses of 
 mathematics and German in Harvard University. In 1896 
 he entered the Johns Hopkins University, as a student in ad- 
 vanced standing, and has since been pursuing courses in mathe- 
 matics under Professors Morley, Craig, Chessin, and Hulburt, 
 and Dr. Cohen, and in physics under Professor Ames and Dr. 
 Bliss, and in philosophy under Professor Griffin. He was made 
 a candidate for the degree of Doctor of Philosophy in 1899, 
 having selected Mathematics as his principal subject, and 
 Physics and History of Philosophy as first and second subordi- 
 nates, and was appointed Fellow in Mathematics in June, 1900. 
 
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