.to— 3 ÉìMMimkWmTi "'"' °"''° 3 1822 01776 7286 c^^^-t UNIVERSITY OF CAUFORN A. SAN DIEGO LA JOLU. CALIFORNIA 92093 1^ SCIENCE & ENGtNttK. 0031(7 A>. >^ University of California, San Diego Please Note: This item is subject to recall. Date Due 0031(7 CI 39a (4/91) UCSD Lib. «WISB^I NON-EUCLIDEAN GEOMETRY A CRITICAL AND HISTORICAL STUDY OF ITS DEVELOPMENT BY ROBERTO BONOLA Professor in the University of Pavia AUTHORISED ENGLISH TRANSLATION WITH ADDITIONAL APPENDICES BY H. S. CARSLAW Professor in the University of Sydney, N. S.W. WITH AN INTRODUCTION BY FEDERIGO ENRIQUES Professor in the University of Bologna -«m CHICAGO THE OPEN COURT PUBLISHING COMPANY 1912 COPYRIGHT BY THE OPEN COURT PUBLISHING COMPANY CHICAGO, U. S. A. 1912 All rights resewed Printed by W. Drugulin, Leipzig, (Germany) Introduction. The translator of this little volume has done me -the honour to ask me to write a few lines of introduction. And I do this willingly, not only that I may render homage to the memory of a friend^ prematurely torn from life and from science, but also because I am convinced that the work of Roberto Bonola deserves all the interest of the studious. In it, in fact, the young mathematician will find not only a clear exposition of the principles of a theory now classical, but also a critical account of the developments which led to the foundation of the theory in question. It seems to me that this account, although concerned with a particular field only, might well serve as a model for a history of science, in respect of its accuracy and its breadth of information, and, above all, the sound philo- sophic spirit that permeates it. The various attempts of successive writers are all duly rated according to their relative importance, and are presented in such a way as to bring out the continuity of the progress of science, and the mode in which the human mind is led through the tangle of partial error to a broader and broader view of truth. This progress does not consist only in the ac- quisition of fresh knowledge, the prominent place is taken by the clearing up of ideas which it has involved; and it is remarkable with what skill the author of this treatise has elucidated the obscure concepts which have at particular periods of time presented themselves to the eyes of the investigator as obstacles, or causes of confusion. I will cite as an example his lucid analysis of the idea of there [V Introduction. being in the case of Non-Euclidean Geometry, in contrast to Euclidean Geometry, an absolute or natural measure of geometrical magnitude. The admirable simplicity of the author's treatment, the elementary character of the constructions he employs, the sense of harmony which dominates every part of this little work, are in accordance, not only with the artistic temperament and broad education of the author, but also with the lasting devotion which he bestowed on the Theory of Non-Euclidean Geometry from the very beginning of his scientific career. May his devotion stimulate others to pursue with ideals equally lofty the path of historical and philosophical criticism of the principles of science! Such efforts may be regarded as the most fitting introduction to the study of the high problems of philosophy in general, and subsequently of the theory of the understanding, in the most genuine and profound signification of the term, following the great tradition which was interrupted by the romantic movement of the nineteenth century. Bologna, October ist^ 191 1. Federigo Enriques. Translator's Preface. Bonola's Non-Euclidean Geometry is an elementary historical and critical study of the development of that subject. Based upon his article in Enriques' collection of Monographs on Questions of Elementary Geometry^, in its final form it still retains its elementary character, and only in the last chapter is a knowledge of more advanced mathematics required. Recent changes in the teaching of Elementary Geometry in England and America have made it more then ever ne- cessary that those who are engaged in the training of the teachers should be able to tell them something of the growth of that science; of the hypothesis on which it is built; more especially of that hypotheses on which rests Euclid's theory of parallels; of the long discussion to which that theory was subjected; and of the final discovery of the logical possibility of the different Non-Euclidean Geometries. These questions, and others associated with them, are treated in an elementary way in the pages of this book. In the English translation, which Professor Bonola kindly permitted me to undertake, I have introduced some changes made in the German translation.^ For permission to do so I desire to express my sincere thanks to the firm of B. G. Teubner and to Professor Liebmann. Considerable new material has also been placed in my hands by Professor Bonola, including a slightly altered discussion of part of 1 Enriques, F., Questioni riguardan/i la geometria elementare, (Bologna, Zanichelli, 1900). 2 Wissenschaft und Hypothese, IV. Band : Die nichteuklidische Geometrie. Historisch-kritische Darstellung ihrer Entwicklung. Von R, Bonola, Deutsch v. H. Liebmann. (Teubner, Leipzig, 190SJ. VI Translator's Preface. Saccheri's work, an Appendix on the Independence of Pro- jective Geometry from the Parallel Postulate, and some further Non-Euclidean Parallel Constructions. In dealing with Gauss's contribution to Non-Euclidean Geometry I have made some changes in the original on the authority of the most recent discoveries among Gauss's papers. A reference to Thibaut's 'proof, and some addit- ional footnotes have been inserted. Those for which I am responsible have been placed within square brackets. I have also added another Appendix, containing an elementary proof of the impossibility of proving the Parallel Postulate, based upon the properties of a system of circles orthogonal to a fixed circle. This method offers fewer difficulties than the others, and the discussion also establishes some of the striking theorems of the hyperbolic Geometry. It only remains for me to thank Professor Gibson of Glasgow for some valuable suggestions, to acknowledge the interest, which both the author and Professor Liebmann have taken in the progress of the translation, and to express my satisfaction that it finds a place in the same collection as Hilbert's classical Grundlagen der Geometrie. P. S. As the book is passing through the press I have received the sad news of the death of Professor Bonola. With him the Italian School of Mathematics has lost one Of its most devoted workers on the Principles of Geometry. Professor Enriques, his intimate friend, from whom I heard of Bonola's death, has kindly consented to write a short introduction to the present volume. I have to thank him, and also Professor W. H. Young, in whose hands, to avoid delay, I am leaving the matter of the translation of this introduction and its passage through the press. The University, Sydney, August 1 9 1 1 . H. S. Carslaw. Author's Preface. The material now available on the origin and develop- ment of Non-Euclidean Geometry, and the interest felt in the critical and historical exposition of the principles of the various sciences, have led me to expand the first part of my article — Sulla teoria delle parallele e sulle geometrie iioji- euclidee — which appeared sÌ5i years ago in the Questioni ri- guardanti la geometria elemefiiare, collected and arranged by Professor F. Enriques. That article, which has been completely rewritten for the German translation* of the work, was chiefly concerned with the systematic part of the subject. This book is devoted, on the other hand, to a fuller treatment of the history of parallels, and to the historical development of the geometries of Lo- fi atschewky-Boly ai and RiEMANN. In Chapter I., which goes back to the work of Euclid and the earliest commentators on the Fifth Postulate, I have given the most important arguments, by means of which the Greeks, the Arabs and the geometers of the Renaissance attempted to place the theory of parallels on a firmer foundation. In Chapter II., relying chiefly upon the work of Saccheri, Lambert and Legendre, I have tried to throw some light on the transition from the old to the new ideas, which became prevalent in the beginning of the 19th Century. In Chapters III. and IV., by the aid of the in- I Enriques, F., Fragen der Elementargeometrie. I. Teil: Die Grundlagen der Geometrie. Deutsch von H. Thieme. {1910.) II. Teil: Die geometrischen Aufgaben, ihre Losung und Losbarkeit. Deutsch von H. Fleischer. (1907.) Teubner, Leipzig. YUj Author's Preface. vestigations of GausS; Schweikart, TaurinuS; and the con- structive work of Lobatschewsky and Bolyai, I have ex- plained the principles of the first of the geometrical systems, founded upon the denial of Euclid's Fifth Hypothesis. In Chapter V., I have described synthetically the further deve- lopment of Non-Euclidean Geometry, due to the work of RiEMANN and Helmholtz on the structure of space, and to Cayley's projective interpretation of the metrical proper- ties of geometry. In the whole of the book I have endeavoured to pre- sent, the various arguments in their historical order. How- ever when such an order would have made it impossible to treat the subject simply, I have not hesitated to sacrifice it, so that I might preserve the strictly elementary character of the book. Among the numerous postulates equivalent to Euclid's Fifth Postulate, the most remarkable of which are brought together at the end of Chapter IV., there is one of a statical nature, whose experimental verification would furnish an empirical foundation of the theory of parallels. In this we have an important link between Geometry and Statics (Genocchi); and as it was impossible to find a suitable place for it in the preceding Chapters, the first of the two Notes'^ in the Appendix is devoted to it. The second Note refers to a theory no less interesting. The investigations of Gauss, Lobatschewsky and Bolyai on the theory of parallels depend upon an extension of one of the fundamental conceptions of classical geometry. But a conception can generally be extended in various directions. In this case, the ordinary idea of parallelism, founded on the hypothesis of non-intersecting straight Unes, coplanar and I In the English translation these Notes are called Appendix I. and Appendix II. Author's Preface. IX equidistant, was extended by the above-mentioned geometers, who gave up Euclid's Fifth Postulate (equidistance), and later, by Clifford, who abandoned the hypothesis that the lines should be m the same plane. No elementary treatment of Clifford's parallels is avail- able, as they have been studied first by the projective method (Clifford-Klein) and later, by the aid of Different- ial-Geometry (BiANCHi-FuBiNi). For this reason the second Note is chiefly devoted to the exposition of their simplest and neatest properties in an elementary and synthetical manner. This Note concludes with a rapid sketch of Clif- ford-Klein's problem, which is allied historically to the parallelism of Clifford. In this problem an attempt is made to characterize the geometrical structure of space, by assum- ing as a foundation the smallest possible number of postul- ates, consistent with the experimental data, and with the principle of the homogeneity of space. This is, briefly, the nature of the book. Before sub- mitting the little work to the favourable judgment of its readers, I wish most heartily to thank my respected teacher, Professor Federigo Enriques, for the valuable advice with which he has assisted me in the disposition of the material and in the critical part of the work; Professor Corrado Segre, for kindly placing at my disposal the manuscript of a course of lectures on Non-Euclidean geometry, given by him, three years ago, in the University of Turin; and my friend. Professor Giovanni Vailati, for the valuable references which he has given me on Greek geometry, and for his help in the cor- rection of the proofs. Finally my grateful thanks are due to my publisher Cesare Zanichelli, who has so readily placed my book in his collection of scientific works. Pavia, March, 1906. Roberto Bonola. Table of Contents. Chapter I. pages The Attempts to prove Euclid's Parallel Postulate. S I — 5. The Greek Geometers and the Parallel Postulate I — 9 S 6. The Arabs and the Parallel Postulate .. .. 9 — 12 S 7 — 10. The Parallel Postulate during the Renais- sance and the ly^^ Century 12 — 21 Chapter II. The Forerunners of Non-Euclidean Geometry. S II — 17. Gerolamo Saccheri (1667 — 1733) .. .. 22—44 S 18 — 22. Johann Heinrich Lambert (172S — 1777) 44—51 S 23 — 26. The French Geometers towards the End of the l8th Century 51 — 55 S 27 — 28. Adrien Marie Legendre (1752 — 1833) .. 55— 60 S 29. Wolfgang Bolyai (1775 — 1856) 60—62 § 30. Friedrich Ludwig Wachter (1792 — 1817) .. 62 — 63 § 30 (bis) Bernhard Friedrich Thib/vut (1776—1832) 63 Chapter III. The Founders of Non-Euclidean Geometry. S 31—34- Karl Friedrich Gauss (1777— 1855) .. 64 — 75 S 35. Ferdinand Karl Schweikart (1780—1859) .. 75—77 S 36 — 38. Franz Adolf Taurinus (1794—1874) .. 77 — 83 Chapter IV. The Founders of Non-Euclidean Geometry (Cont). S 39—45- Nicolai Ivanovitsch Lobatschewsky (1793—1856) , 84—96 S 46 — 55. Johann Bolyai (1S02 — 1860) 96—113 S 56—58. The Absolute Trigonometry I13— 118 § 59. Hypotheses equivalent to Euclid's Postulate .. 118—121 § 60 — 65. The Spread of Non-Euclidean Geometry 121 — 128 Chapter V. The Later Development of Non-Euclidean Geometry. S 66. Introduction 129 Table of Contents. XI pages Differential Geometry and Non-Euclidean Geometry. S 67—69. Geometry upon a Surface .. 130 — 139 § 70 — 76. Principles of Plane Geometry on the Ideas of RiEMANN 139 — 150 § 77. Principles of Riemann's Solid Geometry.. .. 151 — 152 g 78. The Work of Helmholtz and the Investigations of Lie 152 — 154 Projective Geometry and Non-Euclidean Geometry. S 79 "83. Subordination of Metrical Geometry to Projective Geometry 154 — 164 S 84 — 91. Representation of the Geometry of Lobat- SCHEWSKV-BOLYAI On the Euclidean Plane .. .. 164—175 S 92. Representation of Riemann's Elliptic Geometry in Euclidean Space 175 — 176 S 93. Foundation of Geometry upon Descriptive Pro- perties , 176 — 177 S 94. The Impossibility of proving Euclid's Postulate 177 — ^^o Appendix I. The Fundamental Principles of Statics and Euclid's Postulate. S I — 3. On the Principle of the Lever 181 — 184 S 4 — 8. On the Composition of Forces acting at a Point 184 — 192 S 9 — 10. Non-Euclidean Statics 192 — 195 S II — 12. Deduction of Plane Trigonometry from Statics 195 — 199 Appendix II. Clifford's Parallels and Surface. Sketch of Clifford- Klein's Problem. S 1—4. Clifford's Parallels 2co— 206 S 5—8. Clifford's Surface 206—211 S 9 — 11. Sketch of Clifford-Klein's Problem .. 211 — 215 Appendix III. The Non-Euclidean Parallel Construction and other Allied Constructions. S 1 — 3. The Non-Euclidean Parallel Construction .. 216—222 § 4. Construction of the Common Perpendicular to two non-intersecting Straight Lines 222-— 223 S 5. Construction of the Common Parallel to the Straight Lines which bound an Angle 223—224 XII Table of Contents. pages S 6. Construction of the Straight Line which is per- pendicular to one of the lines bounding an acute Angle and Parallel to the other 224 S 7- The Absolute and the Parallel Construction .. 224 — 226 Appendix IV. The Independence of Projective Geometry from Euclid's Postulate. S I. Statement of the Problem 227 — 228 § 2. Improper Points and the Complete Projective Plane 228—229 § 3. The Complete Projective Line 229 S 4. Combination of Elements 229 — 231 § 5. Improper Lines 231 — 233 S 6. Complete Projective Space 233 S 7. Indirect Proof of the Independence of Pro- jective Geometry from the Fifth Postulate .. 233 — 234 S 8. Beltrami's Direct Proof of this Independence 234—236 S 9. Klein's Direct Proof of this Independence .. 236— -237 Appendix V. The Impossibility of proving Euclid's Postulate. An Elementary Demonstration of this Impossibility founded upon the Properties of the System of Circles orthogonal to a Fixed Circle. § I. Introduction 238 S 2 — 7. The System of Circles passing through a Fixed Point 239 250 S 8 — 12. The System of Circles orthogonal to a Fixed Circle - 250—264 Index of Authors .. 265 Chapter I. The Attempts to prove Euclid's Parallel Postulate The Greek Geometers and the Parallel Postulate, § I. Euclid (circa 330 — 275, B. C.) calls two straight lines parallel, when they are in the same plane and being produced indefinitely in both directions, do not meet one another in either direction (Def. XXIII.).^ He proves that two straight lines are parallel, when they form with one of their transversals equal interior alternate angles, or equal corresponding angles^ or interior angles on the same side which are supplementary. To prove the converse of these propositions he makes use of the following Postulate (V.) : If a straight Ime falling on t7V0 straight lines make the ifiterior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. The Euclidean Theory of Parallels is then completed by the following theorems: Straight lines which are parallel to the same straight line are parallel to each other (Bk. I., Prop. 30). I With regard to Euclid's text, references are made to the critical edition of J. L. Heiberg (Leipzig, Teubner, 1883). [The wording of this definition (XXIIF, and of Postulate V below, are taken from Heath's translation of Heiberg's text. (Camb. Univ. Press, 1908Ì.] I 2 I. The Attempts to prove Euclid's Parallel Postulate. Through a given point one and only one straight line can be drawn which will be parallel to a given straight line (Bk. I. Prop. 31). The straight lines joining the extremities of two equal and parallel straight lines are equal and parallel (Bk. I. Prop. 33). From the last theorem it can be shown that two parallel straight lines are equidistant from each other. Among the most noteworthy consequences of the Euclidean theory are the well-known theorem on the sum of the angles of a tri- angle, and the properties of similar figures. § 2. Even the earliest commentators on Euclid's text held that Postulate V. was not sufficiently evident to be accepted without proof, and they attempted to deduce it as a consequence of other propositions. To carry out their pur- pose, they frequently substituted other definitions of parallels for the Euclidean definition, given verbally in a negative form. These alternative definitions do not appear in this form, which was believed to be a defect. Proclus (410 — 485) — in his Commefitary on the First Book of Euclid^ — hands down to us valuable informa- tion upon the first attempts made in this direction. He states, for example, that Posidonius (i^' Century, B. C.) had pro- posed to call two equidistant and coplanar straight lines par- allels. However, this definition and the Euclidean one correspond to two facts, which can appear separately, and » Wher the text of Proclus is quoted, we refer to the edi- tion of G. FriEDLEIN: Frodi Diadochi in primum Eudidis element- orum librian commeyitarii, [Leipzig, Teubner, 1873). [Compare also W. B. Frankland, The First Book of Eudid''s Elements with a Commentary based prindpally upon that of Produs Diadochus, (Camb. Univ. Press, 1905). Also Heath's Euclid, Vol. I., Introduction, Chapter IV., to which most important work reference has been made on p. l]. The Commentary of Proclus. •» Proclus (p. 177), referring to a work by Geminus (1^' Cen- tury, B. C), brings forward in this connection the examples of the hyperbola and the conchoid, and their behaviour with respect to their asymptotes, to show that there might be parallel lines in the Euclidean sense, (that is, lines which produced indefinitely do not meet), which would not be parallel in the sense of Posidonius, (that is, equidistant). Such a fact is regarded by Geminus, quoting still from Proclus, as the most paradoxical [TrapaòoHÓTaTOv] in the whole of Geometry. Before we can bring Euclid's definition into line with that of Posidonius, it is necessary to prove that if two coplanar straight lines do not meet, they are equidistant; or, that the locus of points, which are equidistant from a straight line, is a straight line. And for the proof of this proposition Euclid requires his Parallel Postulate. However Proclus (p. 364) refuses to count it among the postulates. In justification of his opinion he remarks that its converse {The sum of hvo angles of a triangle is less than two right angles), is one of the theorems proved by Euclid. (Bk. I. Prop. 17); and he thinks it impossible that a theorem whose con- verse can be proved, is not itself capable of proof. Also he utters a warning against mistaken appeals to self- evidence, and insists upon the (hypothetical) possibi- lity of straight lines which are asymptotic (p. 191 — 2). Ptolemy (2°^ Century, A. D.) — we quote again from Proclus (p. 362 — 5)— attempted to settle the question by means of the following curious piece of reasoning. 1* A I. The Attempts to prove Euclid's Parallel Postulate. Let AB, CD, be two parallel straight lines and FG a transversal (Fig. i). Let a, P be the two interior angles to the left of FG, and a, P' the two interior angles to the right. Then a + P will be either greater than, equal to, or less than a + p'. It is assumed that if any one of these cases holds for one pair of parallels (e. g. a + P ^ 2 right angles) this case will also hold for every other pair. Now FB, GD, are parallels; as are also FA and GC. Since a + P ^ 2 right angles, it follows that a' + P' ^ 2 right angles. Thus a+P + a'+P'>>4 right angles, which is obviously absurd. Hence a + P cannot be greater than 2 right angles. In the same way it can be shown that a + P cannot be less than 2 right angles. Therefore we must have a + p = 2 right angles (Proclus, p. 365). From this result Euclid's Postulate can be easily obtained. § 3. Proclus (p. 371), after a criticism of Ptolemy's reasoning, attempts to reach the same goal by another path. His demonstration rests upon the following proposition, which he assumes as evident: — The distance between two points upon two intersecting straight lines can be made as great as 7V e please, by prolonging the two lines sufficiently} From this he deduces the lemma : A straight line which meets one of two parallels must also meet the other. I For the truth of this proposition, which he assumes as self- evident, Proclus relies upon the authority of Aristotle. Cf. De Coelo I., 5. A rigorous demonstration of this very theorem was given by Saccheri in the work quoted on p. 22. Proclus (continued). C His proof of this lemma is as follows: Let AB, CD, be two parallels and £G a transversal, cutting the former in ^ (Fig. 2). — D Fig. 2. The distance of a variable point on the ray J^G from the line AB increases without limit, when the distance of that point from jF is increased indefinitely. But since the distance between the two parallels is finite, the straight line EG must necessarily meet CD. Proclus, however, introduced the hypothesis that the distance between two parallels remains finite; and from this hypothesis Euclid's Parallel Postulate can be logically de- duced. § 4. Further evidence of the discussion and research among the Greeks regarding Euclid's Postulate is given by the following paradoxical argument. Relying upon it, accord- ing to Proclus, some held that it had been shown that two straight lines, which are cut by a third, do not meet one another, even when the sum of the interior angles on the same side is less than two right angles. Let -(4C be a transversal of the two straight lines AB, CD and let E be the middle point of AC (Fig. 3). On the side of ^C on which the sum of the two internal angles is less than two right angles, take the segments AF and CG upon AB and CD each equal to AE. The two lines AB and CD cannot meet between the points AF and CG, since in any triangle each side is less than the sum of the other t\vo. A 1 ~ K E C H ( Ì L _D 6 I. The Attempts to prove Euclid's Parallel Postulate. The points F and G are then joined, and the same process is repeated, starting from the hne FG. The segments FK and GL are now taken on AB and CD, each equal to half of FG. The two lines AB, CD are not able to meet between the points 7% K and G, L. Since this operation can be repeated indefini- tely, it is inferred that the two lines AB, CD will never meet. The fallacy in this ar- gument is contained in the use of infinity, since the segments AF, FK could tend to zero, while their sum might remain finite. The author of this paradox has made use of the principle by means of which Zeno (495 — 435 B. C.) maintained that it could be proved that Achilles would never overtake the tortoise, though he were to travel with double its velocity. This is pointed out, under another form, by Proclus (p. 369 — 70), where he says that this argument proves that the point of intersection of the lines could not be reached (to determine, ópiZieiv) by this process. It does not prove that such a point does not exist.* Proclus remarks further that 'since the sum of two angles of a triangle is less than two right angles (Euclid Bk. I. Prop. 17), there exist some lines, intersected by a third, which meet on that side on which the sum of the interior Fig- 3- I [Suppose we start with a triangle ABC and bisect the base BC in D. Then on BA take the segment BE equal to BD, and on CA the segment CB' equal to CD, and join EF. Then repeat this process indefinitely. The vertex A can never be reached by this means, although it is at a finite distance.] Proclus (continued). j angles is less than two right angles. Thus if it is asserted that for every difference between this sum and two right angles the lines do not meet, it can be replied that for greater differences the lines intersect.' 'But if there exists a point of section, for certam pairs of lines, forming with a third interior angles on the same side whose sum is less than two right angles, it remains to be shown that this is the case for all the pairs of lines. Since it might be urged that there could be a certain deficiency (from two right angles) for which they (the lines) would not inter- sect, while on the other hand all the other lines, for which the deficiency was greater, would intersect.' (Proclus, p. 371.) From the sequel it will appear that the question, which Proclus here suggests, can be answered in the affirmative only in the case when the segment AC of the transversal remains unaltered, while the lines rotate about the points A and C and cause the difference from two right angles to vary. § 5. Another very old proof of the Fifth Postulate, reproduced in the Arabian Commentary of Al-Nirizi' (ptt Century), has come down to us through the Latin translation of Gherardo da Cremona* (12th Century), and is attributed to Aganis.3 The part of this commentary relating to the definitions, postulates and axioms, contains frequent references to the 1 Cf. R. O. Besthorn u. J. L. Heiberg, 'Codex Leidensis,' 399, I. Euclidis Elementa ex interpretatione Al-Hadsckdschadsch cum commentariis Al-N^ariziif (Copenhagen, F. Hegel, 1893 — 97)- 2 Cf. M. CuRTZE, *Anariin in decern libros priores eleinentoriim Euclidis Commentarii.' Ex interpretatione Gherardi Cremonensis in Codice Cracoviensi 569 servata, (Leipzig, Teubner, 1899). 3 With regard to Aganis it is right to mention that he is identified by Curtze and Heiberg with Geminus. On the other hand P. Tannery does not accept this identification. Cf. Tannery, *Z cut off the segment ZC, the same multiple of ZiV as Z£ is of ZM. In the case taken in the figure ZC = 4 ZiV. The point C thus obiaified is the point of intersection of the two straight ii?ies AB and GD. C F G To prove this it would be necessary to show that the equal segments ZN, JVS, . . ., which have been cut off one after the other firom the line ZD, have equal projections on Z£. We do not discuss this point, as we must return to it later (p. 11). In any case the reasoning is suggested directly by Aganis' figure. The distinctive feature of the preceding construction is to be noticed. It rests upon the (implicit) use of the so-called Postulate of Archimedes, which is necessary for the deter- mination of the segment J/Z, less than LZ and a submult- iple of EZ. The Arabs and the Parallel Postulate. § 6. The Arabs, succeeding the Greeks as leaders in mathematical discovery, like them also investigated the Fifth Postulate. Some, however, accepted without hesitation the ideas and demonstrations of their teachers. Among this number is Al-Nirlzi (9th Century), whose commentary on the definitions, IO I. The Attempts to prove Euclid's Parallel Postulate. postulates and axioms ot the First Book is modelled on the Introduction to the ''Elements^ of Simplicius, while his demon- stration of the Fifth Euclideari Hypothesis is that of Aganis, to which we have above referred. Others brought their own personal contribution to the argument. Nasìr-Eddìn [1201 — 1274], for example, although in his proof of the Fifth Postulate he employs the criterion used by Aganis, deserves to be mentioned for his original idea of explicitly putting in the forefront the theorem on the sum of the angles of a triangle, and for the exhaustive nature of his reasoning.' The essential part of his hypothesis is as follows: If two straight lines ;- afid s are the 07ie perpendictilar and the other oblique to the segment AB, the perpendiculars drawn frotn s upon r are less than AB on the side on 7vhich s makes an acute angle with AB, and greater on the side on which s makes an obtuse angle with AB. It follows immediately that \i AB and A'B' are two equal perpendiculars to the line BB' from the same side, the line AA' is itself perpendicular to both AB and A'B'. Further we have AA' = BE' ; and therefore the figure AA'B'B is a quadrilateral with its angles right angles and its opposite sides equal, i. e., a rectangle. From this result Nasìr-Eddìn easily deduced that the sum of the angles of a triangle is equal to two right angles. For the right-angled triangle the theorem is obvious, as it is half of a rectangle; for any triangle we obtain it by breaking up the triangle into two right-angled triangles. With this introduction, we can now explain shortly how the Arabian geometer proves the Euclidean Postulate [cf. Aganis]. I Cf. : Eiiclidis elementorum libri XII studii N'assiredini, (Rome, 1594). This work, written in Arabic, was republished in 1657 and 1801. It has not been translated into any other language. Nasìr-Eddìn's Proof. II o'c m' k' h' a Fig- 5- Let AB, CD be two rays, the one oblique and the other perpendicular to the straight Hne AC (Fig. 5). From AB cut oflf the part AH, and from ZTdraw the perpendicular HH' to AC. If the point H' falls on C, or on the opposite side of C from A^ the two rays AB and CD must intersect. If, however, H' falls between A and C, draw the line AL perpendicular to AC and equal to HH' . Then, from what we have said above, HL = AH' . In AH^io- , that one is greater which is adjacent to the shorter side, and vice versa. 1 [The Postulate of Archimedes is stated by Hilbert thus: Let Al be any point upon a straight line between the arbitrarily chosen points A and B. Take the points A2, A^, . . . so that Ai lies between A and A2, A2 between Ai and /i;„ etc.; moreover let the segments AAi, A1A2, ^2^3, ... be all equal. Then among this series of points, there always exists a ceitain point Ad, such that B lies between A and Aa-\ 2 This hypothesis is used by Saccheri in its intuitive form, viz. : a segment, which passes continuously from the length a to the length b, different from a, takes, during its variation, every length intermediate between a and b. 24 II. The Forerunners of Non-Euclidean Geometry. Let ABCD be a quadrilateral with two right angles A and B, and two equal sides AD and BC (Fig. 9). On the Euclidean hypothesis the angles Cand D are also right angles. Thus, if we assume that they are able to be both obtuse, or both acute, we implicitly deny the Fifth Postulate. Saccheri discusses these three hypotheses regarding the angles C, D. He named them: The Hypothesis of the Right Angle [<^ 6"= <^ Z> = I right angle] : The Hypothesis of the Obtuse A/igle [-^ C = <^ Z> > I right angle] : The Hypothesis of the Acute Angle [^ C= <^Z> < I right angle]. One of his first important results is the following: Accordifig as the Hypothesis of the Eight Angle, of the Obtuse Angle, or of the Acute Attgle is true i?i the two right- angled isosceles quadrilateral, we must have AB = CD, ABy- CD, or AB < CD, respectively. [Prop. 111.] In fact, on the Hypothesis of the Eight Angle, by the preceding Lemma, we have immediately AB = CD. On the Hypothesis of the Obtuse Angle, the perpendicular 00' at the middle point of the segment y^/>' divides the fundamental quadrilateral into two equal quadrilaterals, with right angles at O and O'. Since the angle D ^ angle A, then we must have AO ^ DO , by this Lemma. Thus AB > CD. On the Hypothesis of the Acute Angle these ^'^ 9- inequalities have their sense changed and we have AB < CD. Using the reductio ad absurdum argument, we obtain the converse of this theorem. [Prop. IV.] O •The Three Hypotheses. 25 If the Hypothesis of the Right Angle is true in only one 4ase, then it is true in every other case. [Prop. V.] Suppose that in the two right-angled isosceles quadrilat- eral AB CD the Hypothesis of the Eight Angle is verified. In AD and BC (Fig. lo) take the points ZTand K equi- distant from AB; join HK a.nd form the P quadrilateral ABKH. m If HK is perpendicular to AH and BK, the Hypothesis of the Right Angle is ^^j- also verified in the new quadrilateral. H If it is not, suppose that the angle AHK is acute. Then the adjacent angle ^ ^ DHK is obtuse. Thus in the quadrilateral '^' ^ ' ABKH, from the Hypothesis of the Acute Angle, it follows that AB = -^ H=- < C = I right angle. Ikit from the former, there follows the equidistance of the two straight lines DC, AB^; and thus the validity of the Hypo- thesis of the Right Angle in all the two right-angled isosceles quadrilaterals, whose altitude is equal to the line DA, is established. The same hypothesis is also true in a quadri- lateral of any height, since the line called at one time the base may later be regarded as the height. If the Hypothesis of the Obtuse Angle is true in only one case, then it is true in every other case. [Prop. VI.] Referring to the standard quadrilateral ^j9CZ> (Fig. 11), n K 1 C suppose that the angles C and D are ob- tuse. Upon AD and BC take the points H and K equidistant from AB. In the first place we note that the segment HK cannot be perpendicular to the two sides AD and BC, since in that A Oj B case the Hypothesis of the Right Angle Fig. II. would be verified in the quadrilateral ABKH, and consequently in the fundamental quadrilateral. Let us suppose that the angle AHK is acute. Then I It is true that Giordano in his argument refers to the points of the segment DC, which he shows are equidistant from the base AB of the quadrilateral. However the same argument is applicable to all the points which lie upon DC, or upon DC produced. Cf. Bonola's Note referred to on p. 15. Proof for one Quadrilateral Sufficient. 27 by the Hypothesis of the Acute Angle, HK^ AB. But as the Hypothesis of the Obtuse Angle holds in AB DC, we have AB^ CD. Therefore HK^ AB > CD. If we now move the straight line HK continuously, so that it remains perpendicular to the median 00' of the fundamental quadrilateral, the segment HK, contained between the oppo- site sides AD, BC, which in its initial position is greater than AB, will become less than AB in its final position DC. From the postulate of continuity we may then conclude that, between the initial position HK and the final position DC, there must exist an intermediate position H' K' , for which H'K' = AB. Consequently in the quadrilateral ABK'H' the Hypo- thesis of the Right Angle would hold [Prop. III.J, and therefore, by the preceding theorem, the Hypothesis of the Obtuse Angle could not be true in ABCD. The argument is also valid if the segments ^j^, BK arc greater than AD, since it is impossible that the angle AHK could be acute. Thus the Hypothesis of the Obtuse Angle holds in ABKH as well as in ABCD. Having proved the theorem for a quadrilateral whose sides are of any size, we proceed to prove it for one whose base is of any size: for example the base BK [cf Fig. 12]. Since the angles K, H, are obtuse, the perpendicular at K to KB will meet the segment AH in the point M, making the angle AMK obtuse [theorem of the ex- terior angle]. Then in ABKM we have AB > KM, by Lemma I. Cut off from AB the segment -5iV equal to MK. Then we can construct the two right angled isosceles quadrilateral BKMN, with the angle MNB obtuse, since it is an exterior angle of the triangle 28 II- The Forerunners of Non-Euclidean Geometry. ANAL It follows that the Hypothesis of the Obtuse A?igle holds in the new quadrilateral. Thus the theorem is completely demonstrated. 1/ the Hypothesis of the Acute Angle is true in only one case, then it is true in every other case. [Prop. VII.] This theorem can be easily proved by using the method of reductio ad absurdum. § 13. From the theorems of the last article Saccher: easily obtains the following important result with regard to triangles : According as the Hypothesis of the Right Angle, the Hy- pothesis of the Obtuse Angle, or the Hypothesis of the Acute Angle, is found to be true, the sum of the angles of a triangle will be respectively equal to, greater than^ or less thafi two right angles. [Prop. IX.] Let ABC [Fig. 1 3] be a triangle of which ^ is a right P angle. Complete the quadrilateral by draw- ing AD perpendicular to AB and equal to BC; and jon CD. On the Hypothesis of the Bight Afigle, the two triangles ABC and ADC are equal. Therefore -^BAC^^^DCA. It follows immediately that in the tri- angle ABC, ^A-\- ^B + <^ C= 2 right angles. On the Hypothesis of the Obtuse Angle, sinc^AB^DC, we have ^ACB^ AC, and therefore, in the same triangle, CA, is fundamental in our argument and we shall refer to it con- stantly. Two points Jt, E' are now taken on j, of which the first is situated between A and C, and the second not. Further let the perpendiculars from E^ E' to the line r meet it at E and E' . The following theorems now hold: \\{ EE^AB,\ I. I or L the angles BAC^ DCA areright angles. I E'E' = AB J \\i EE>AB,\ II. ' or ^, the angles ^^C, Z>6>y are obtuse. I E'E' AB. From FE cut off j^/= AB, and join / to A and C. Then the following equalities hold: ^ BA/= <è: EIA and ^r DCJ ^ ^ FJC. Further, by the theorem of the exterior angle [Bk. !. 16], we have ■22 n. The Forerunners of Non-Euclidean Geometry. ^ FIA + <^ FIC^ <^ FEA + <: FEC = 2 right angles. But ^BAC ^ <^E>CA-><3Z£AI+ ^DCI. Therefore ^BAC ^ ^ DC A > <: FIA + <: i^/C> 2 right angles. But, since < BAC^ <^ BCA, it follows that -^BAC^ 1 right angle. . . . Q. E. D. In the second place, suppose that E'F' <^ rCE\ ^ F'rA<_ <:J'ce', we obtain < BAE', we have the following results for the segment MN which is perpendicular to both the hnes AC and BD (Fig. 16). If / right angle, then AIN^AB. If ^ BAC== <^ nCA < / right angle, then MN< AB. Further it is easy to see that (i) If I right angle, then <^ FEM and r right angle, then <^ KHM and ^ KRF are each greater than i right angle. If ^ BAM < / right angle, then <^ KHM and <^ KRF are each less than i right angle. These results are also true, as can easily be seen, if the point F falls between M and N. In conclusion, the last three theorems, which clearly 34 II. The Forerunners of Non-Euclidean Geometry. coincide with Saccheri's theorems upon the two right-angled isosceles quadrilateral, are equivalent to the following result, proved without using Archimedes' Postulate: — R P H A M C K B N D If the truth of the Hypothesis of the Right Angle, of the Obtuse Angle, or of the Acute Angle, respectively, is known in only o?ie case, its truth is also kno2V7i in every other case. If we wish now to pass from the theorems on quad- rilaterals to the corresponding theorems on triangles, we need only refer to Saccheri's demonstration [cf. p. 28], since this part of his argument does not in any way depend upon the postulate in question. We have thus obtained the result which was to be proved. § 15. To make our exposition of Saccheri's work more concise, we take from Prop. XI. and XII. the contents of the following Lemma II: Let ABC be a tria?igle of ivhich C is a right angle: let H be the tniddle point of AB, and K the foot of the perpen- dicular fro?n H upon AC. Then we shall have AK = KC, Oil the Hypothesis of the Right Angle; AK <^ KC, on the Hypothesis of the Obtuse Angle; AK ^ KC, on the Hypothesis of the Acute Angle. On the Hypothesis of the Right Angle the result is obvious. The Projection of a Line. 35 On the Hypothesis of the Obtuse Angie, since the sum of the angles of a quadrilateral is greater than four right angles, it follows that ^AHK <,^HBC. Let HL be the perpendi- cular from H to BC (Fig. i8). Then the result just obtained, and the fact that the two triangles AHK, HBL have equal hypotenuses, give rise to the following inequality : AK<^HL. But the quadrilateral HKCL has three right angles and there- fore the angle H is obtuse {Hypothesis of the Obtuse Angle], It follows that HL < KC, and thus AK<^KC. The third part of this Lemma can be proved in the same way. It is easy to extend this Lemma as follows (Fig. 1 9) : } Fig. 19. Lemma LLI. Lf oti the one arm of an angle A equal seg- ments AA^, A-i_A^, A^A.^y . . . are taken, and AA^ , A^A^^ AjA^'. . . are their projections upon the other arm of the afigle, then the following results are true: AAi = A-iA^ = A^A^ = . . . on the Hypothesis of the Right Angle; aa,'<:a,'a,'a,'a/>a;a:>... on the Hypothesis of the Acute Angle. To save space the simple demonstration is omitted. 3* 36 II. The Forerunners of Non-Euclidean Geometry. We can now proceed to the proof of Prop. XI. and XII. of Saccheri's work, combining them in the following theorem: On the Hypothesis of the Right Angle and on the Hypo- thesis of the Obtuse Angle, a line perpendicular to a given straight line and a lifie cutting it at an acute angle intersect each other. Fig. ao. Let (Fig, 2 o) LP and AD be two straight lines of which the one is perpendicular to AP, and the other is inclined to AP at an acute angle DAP. After cutting off in succession equal segments AD, DF^, upon AD, draw the perpendiculars DB and P^M^ upon the line AP. From Lemma III. above, we have PM^ > AB, or AM^ ^ 2 AB, on the two hypotheses. Now cut off Pip2 equal to AP^, from AP^ produced, and let M^ be the foot of the perpendicular from P2 upon AP. Then we have AM2 ^ 2 AMi, and thus AM2 > 2' AB. This process can be repeated as often as we please. In this way we would obtain a point Pu upon the line AD such that its projection upon the line AP would deter- mine a segment A.if„ satisfying the relation Two Hypotheses give Postulate V. 37 AM" > 2" AB. But if n is taken sufficiently great, [by the Postulate of Archimedes'^^ we would have 2'' AB^AP, and therefore AMn > AP. Therefore the point P lies upon the side AMn of the right- angled triangle AM„ Fn- The perpendicular PL cannot intersect the other side of this triangle; therefore it cuts the hypotenuse.^ Q^. E. D. It is now possible to prove the following theorem : T?ie Fifth Postulate is true on the Hypothesis of the Right Angle at id on the Hypothesis of the Obtuse Angle [Prop. XIII.]. Let (Fig. 2i) AB, CD be two straight lines cut by the line AC. Let us suppose that ^ BAC + ^ ^CZ> < 2 right angles. Then one of the angles BAC, ACD, for example the first, will be acute. From C draw the perpen- dicular CH upon AB. In the triangle ACH, from the hypo- theses which have been made, A we shall have <^A-\r <^C + 2 right angles. 1 The Postulate of Archimedes, of which use is here made, includes implicitly the infinity of the straight line. 2 The method followed by Saccheri in proving this theorem is practically the same as that of Nasìr-Eddìn. However Nasir- Eddìn only deals with the Hypothesis of the Right Angle, as he had formerly shown that the sum of the angles of a triangle is equal to two right angles. It is right to remember that Saccheri was familiar with and had criticised the work of the Arabian Geometer. 28 II. The Forerunners of Non-Euclidean Geometry. But we have assumed that <^ BAC + <^ ACD < 2 right angles. These two results show that <^ AHC > «9C BCD. Thus the angle HCD must be acute, as ^ is a right angle. It follows from Prop. XI., XII. that the lines AB and CD intersect.^ This result allows Saccheri to conclude that the Hypo- thesis of the Obtuse Angle is false [Prop. XIV.]. In fact, on this hypothesis Euclid's Postulate holds [Prop. XIII.], and consequently, the usual theorems which are deduced from this postulate also hold. Thus the sum of the angles of the fundamental quadrilateral is equal to four right angles, so that the Hypothesis of the Eight Angle is true.^ § i6. But Saccheri wishes to prove that the Fifth Postulate is true in every case. He thus sets himself to destroy the Hypothesis of the Acute Angle. To begin with he shows that o?i this hypothesis, a straight line being given, there can be drawn a perpendicular to it and a line cutting it at an acute angle, which do not intersect each other [Prop. XVII.]. To construct these lines, let ^-5C (Fig. 22) be a triangle of which the angle C is a right angle. At B draw BD mak- ing the angle ABD equal to the angle BAC. Then, on the 1 This proof is also found in the work of Nasìr-Eddìn, which evidently inspifed the investigations of Saccheri. 2 It should be noted that in this demonstration SacCHERI makes use of the special type of argument of which we spoke in Sii. In fact, from the assumption that the Hypothesis of the Ob- tuse Angle is true, we arrive at the conclusion that the Hypothesis of the Right Angle is true. This is a characteristic form taken in such cases by the ordinary reductio ad absurdum argument. Saccheri and the Third Hypothesis. 39 Hypothesis of the Acute Atigle, the angle CBD is acute, and of the two hnes CA, BD, which do not meet [Bk. I, 27], one makes a right angle with BC. In what follows we consider only the Hypothesis of the Acute Aiigle. Let (Fig. 23) a,b be two straight lines in the same plane which do not meet. A, A2 ^ Fig. 23 From the points A^, A^^ on a draw perpendiculars A^Bt,, A.^B^ to b. The angles A^, A, of the quadrilateral thus obtained can be (i) one right, and one acute: (ii) both acute: (iii) one acute and one obtuse. In the first case, there exists already a common per- pendicular to the two lines a, b. In the second case, we can prove the existence of such a common perpendicular by using the idea of continuity [Saccheri, Prop. XXII.]. In fact, if the straight line A-, B^ is moved continuously, while kept perpendicular to b, until it reaches the position A^B^, the angle B^At_A2 starts as an acute angle and increases until it becomes an obtuse angle. There must be an intermediate position AB in which the angle BAA^ is a right angle. Then AB is the common perpendicular to the two lines a, b. In the third case, the lines a, b do not have a common 40 II. The Forerunners of Non-Euclidean Geometry. perpendicular, or, if such exists, it does not fall between B^ and B2. Evidently there will be no such perpendicular if, for all the points Ar situated upon a, and on the same side of A^, the quadrilateral ^i^.^r^;- has always an obtuse angle at Ar. With this hypothesis of the existence of two coplanar straight lines which do not intersect, and have no common perpendicular, Saccheri proves that such lines always ap- proach nearer and nearer to each other [Prop. XXIII.], and that their distance apart finally becomes smaller than any segment, taken as small as we please [Prop. XXV.]. In other words, if there are two coplanar straight Hues, which do not cut each other, and have no common perpendicular, then these lines must be asymptotic to each other." To prove that such asymptotic lines effectively exist, Saccheri proceeds as follows: — ^ Fig. 24. Among the lines of the pencil through A, coplanar with the line b, there exist lines which cut b, as, e. g., the line AB perpendicular to b; and lines which have a common 1 With this result the question raised by the Greeks, as to the possibility of asymptotic lines in the same plane, is answered in the affirmative. Cf. p. 3. 2 The statement of Saccheri's argument upon the asymptotic lines differs in this edition from that given in the Italian and German editions. The changes introduced were suggested to me by some remarks of Professor Carslaw. The Existence of Asymptotic Lines ai perpendicular with ò, as, e. g., the line AA' perpendicular to A£ [cf. Fig. 24]. If AI' cuts Ò, every other line of the pencil, which makes a smaller angle with AB than the acute angle BAjP, also cuts è. On the other hand^ if the line A Q, different from AA', has a common perpendicular with ò, every other line, which makes with AB a larger acute angle than the angle BAQ, has a common perpendicular with ò [cf § 39, case (ii).] Also it is clear that, if we take the lines of the pencil through A, from the ray AB towards the ray AA', we shall not find, among those which cut d, any line which is the last line of that set. In other words, the angles BAB, which the lines AB, cutting ^, make with AB, have an i/J'per limit, the angle BAX, such that the line AX does not cut b. Then Saccheri proves [Prop. XXX.] that, if we start with AA and proceed in the pencil through A in the direction opposite to that just taken, we shall not find any last line in the set of lines which have a common perpendicular with b\ that is to say, the angles BA Q, where A Q has a common perpendicular with b, have a lower limit, the angle BA V, such that the line ^y does not cut b and has not a com- mon perpendicular with b. It follows that A Vis a. line asymptotic to b. Further Saccheri proves that the two hnes AX and A V coincide [Prop. XXXII.]. His argument depends upon the consideration of points at infinity; and it is better to sub- stitute for it another, founded on his Prop. XXI., viz., On the Hypothesis of the Right Angle, and on that of the Acute Angle, the distance of a point on one of the lines containing an angle from the other bounding line increases indefinitely as this point moves further and further along the line. 42 II. The Forerunners of Non-Euclidean Geometry. The suggested argument is as follows: Ar^= p Fig. 25- If AX [Fig. 25] does not coincide with A Y, we can take a point P on AY, such that the perpendicular FF' from F to AX satisfies the inequality ( 1 1 FF' > AB. [Prop. XXL] On the other hand, if FQ is the perpendicular from F to b, the property of asymptotic lines [Prop. XXIII] shows that AB>FQ. But F is on the opposite side of AX from I?. Therefore PQ > FF. Combining this inequality with the preceding, we find that AB>PF. which contradicts (i). Hence AX coincides with A Y. We may sum up the preceding results in the following theorem : — A t B b Fig. 26. On the Hypothesis of the Acute Angle, there exist in the pencil of lines through A two lities p and q, asymptotic to b, one towards the right, and the other towards the left, which divide the pencil into two parts. The first of these consists of the lines which intersect b, and the second of those which have a common perpendicular ivith it.^ I In Saccheri's work tliere will be found many other inter- esting theorems before he reaches this result. Of these the Saccheri's Conclusion. 43 § 17. At this point Saccheri attempts to come to a decision, trusting to intuition and to faith in the validity of the Fifth Postulate rather than to logic. To prove that the Hypothesis of the Acute Angle is absolutely false, because it is repugnant to the 7iature of the straight line [Prop. XXXIIL] he relies upon five LemmaS;, spread over sixteen pages. In sub- stance, however, his argument amounts to the statement that if the Hypothesis of the Acute Angle were true, the lines p (Fig. 2 6) and b would have a comfnon perpendicular at their conunon point at i?iftnity, which is contrary to the nature of the straight lifie. The so-called demonstration of Saccheri is thus founded upon the extension to irifnity of certain properties which are valid for figures at a finite distance. However, Saccheri is not satisfied with his reasoning and attempts to reach the wished-for proof by adopting anew the old idea of equidistance. It is not worth while to reproduce this second treatment as it does not contain any- thing of greater value than the discussions of his prede- cessors. Stillj though it failed in its aim, Saccheri's work is of great importance. In it the most determined eftort had been made on behalf of the Fifth Postulate; and the fact that he did not succeed in discovering any contradictions among the consequences of the Hypothesis of the Acute Angle, could not help suggesting the question, whether a consistent log- ical geometrical system could not be built upon this hypo- following is noteworthy: If two straight liiies continually approach each other and their distance apart remains always greater than a given segment, then the Hypothesis of the Acute Angle is impossible. Thus it follows that, if we postulate the absence of asymptotic straight lines, we must accept the truth of the Euclidean hypo- thesis. AA TI. The Forerunners of Non-Euclidean Geometry. thesis, and the Euclidean Postulate be impossible of demon- stration.^ Johann Heinrich Lambert [1728 — 1777]- § 18. It is difficult to say what influence Saccheri's work exercised upon the geometers of the iS^li century. However, it is probable that the Swiss mathematician Lambert \vas familiar with it, ^ since in his Theorie der Par- allellitiien [1766] he quotes a dissertation by G. S. Klugel [1739 — i8i2]3, where the work of the Italian geometer is carefully analysed. Lambert's Theorie der Fara/lellmien was published after the author's death, being edited by J. Bernoulli and C. F. Hindenburg. It is divided into three parts. The first part is of a critical and philosophical nature. It deals with the two-fold question arising out of the Fifth Postulate: whether it can be proved with the aid of the preceding propositions only, or whether the help of some other hypothesis is required. The second part is devoted to 1 The publication of Saccheri's work attracted considerable attention. Mention is made of it in two Histories of Mathematics: that of J. C. Heilbronner (Leipzig, 1742) and that of Montucla (Paris, 1758). Further it is carefully examined by G. S. Klugel in his dissertation noted below (Note (3)). Nevertheless it was soon forgotten. Not till 1889 did E. Beltrami direct the attention of geometers to it again in his Note: Un precursore italiatio di Legendre e di Lobatschewsky. Rend. Ace. Lincei (4), T. V. p. 441 — 448. Thereafter Saccheri's work was translated into English by G. B. Halsted (Amer. Math. Monthly, Vol. I. 1S94 et seq.); into German, by Engel and Stackel (77/. der P. 1895); into Italian, by G. Boccardini (Milan, Hoepli, 1904). 2 Cf. SegrE: Congetture intorno alla influenza di Girolamo Saccheri sulla forrjiazione della geometria ìion euclidea. Atti Acc. Scienze di Torino, T. XXXVIIL (1903). 3 Conatiiufn praecipuorum theoriam parallelarum demonstrandi recensio, guani publico examini submitteni A. G. Kaestner et auctor respondens G. S. Kliigel, (Gòttingen, 1763). Lambert's Three Hypotheses. 45 the discussion of different attempts in which the Euclidean Postulate is reduced to very simple propositions, which however, in their turn, require to be proved. The third, and most important, part contains an investigation resembling that of Saccheri, of which we now give a short summary/ § 19. Lambert's fundamental figure is a quadrilateral with three right angles, and three hypotheses are made as to the nature of the fourth angle. The first is the Hypothesis of the Right Angle; the second, the Hypothesis of the Obtuse Angle; and the third, the Hypothesis of the Acute Angle. Also in his treatment of these hypotheses the author does not depart far from Saccheri's method. ^\vt first hypothesis leads easily to the Euclidean system. In rejecting the second hypothesis, Lambert relies upon a figure formed by two straight lines a, b, perpendicular to a third line ^^ (Fig. 27). From points £, B^, B^y.-Bn, taken in succession upon B Bj B, B„ the line b, the perpen- diculars, BAy B-,A^, B^A^, : . B„An are drawn to the hne a. He proves, in the first place, that these per- A A.^ Aj An pendiculars continually ^'^- ^7- diminish, starting from the perpendicular BA. Next, that the difterence between each and the one which succeeds it continually increases. Therefore we have BA—BnAn > n {BA—B^A^. But, if n is taken sufficiently large, the second member I Cf. Magazin fur reine und angewandte Math., 2. Stuck, p. 137 — 164. 3. Stuck, p. 325 — 358, (1786). Lambert's work was again published by Engel and Stackel {Th. der P.) p. 135 — 208, preceded by historical notes on the author. aF) li. The Forerunners of Non-Euclidean Geometry. of this inequality becomes as great as we please {Postulate of Archimedes] \ whilst the first member is always less than £A. This contradiction allows Lambert to declare that the second hypothesis is false. In examining the third hypothesis, Lambert again avails himself of the preceding figure. He proves that the perpen- diculars £A, BxA^, . . B,iAn continually increase, and that at the same time the difference between each and the one which precedes it continually increases. As this result does not lead to contradictions, like Saccheri he is compelled to carry his argument further. Then he finds, that, on the third hypothesis the sum of the angles of a triangle is less than two right angles; and going a step further than Saccheri, he discovers that the defect of a polygon, that is, the differ- ence between 2 {n — 2) right angles and the sum of its angles, is proportional to the area of the polygon. This result can be obtained more easily by observing that both the area and the defect of a polygon, which is the sum of several others, are, respectively, the sum of the areas and of the defects of the polygons of which it is composed.^ § 20. Another remarkable discovery made by Lambert has reference to the measurement of geometrical magnitudes. It consists precisely in this, that, whilst in the ordinary geo- metry only a relative meaning attaches to the choice of a 1 The Postulate of Archimedes is again used here in a form which assumes the infinity of the straight line (cf. Saccheri, Note P- 37)- 2 It is right to point out that in the Hypothesis af the Aade Angle Saccheri had already met the defect here referred to, and also noted implicitly that a quadrilateral, made up of several others, has for its defect the sum of the defects of its parts (Prop. XXV). However he did not draw any conclusion from this as to the area being proportional to the defect. Relative and Absolute Units. 47 particular unit in the measurement of lines, in the geometry founded upon the third hypothesis^ we can attach to it an absolute meaning. First of all we must explain the distinction, which is here introduced, between absolute and relative. In many questions it happens that the elements, supposed given, can be divided into two groups, so that those oi i\\Q first grotip remain fixed, right through the argument, while those of the second group may vary in a number of possible cases. When this happens, the explicit reference to the data of the first group is often omitted. All that depends upon the varying data is considered relative; all that depends upon the fixed data is absolute. For example, in the theory of the Domain, of Ration- ality, the data of the second group [the variable data] are taken as certain simple irrationalities [constituting a base\., and "ùx^ first group consists simply of unity [i], which is often passed over in silence as it is common to all domains. In speaking of a number, we say that it is rational relatively to a given base, if it belongs to the domain of rationality defined by that base. We say that it is rational absolutely, if it is proved to be rational with respect to the base i, which is common to all domains. Passing to Geometry, we observe that in every actual problem, we generally take certain figures as given and therefore the magnitudes of their parts. In addition to these variable data [of the second group\ which can be chosen in an arbitrary manner, there is always implicitly assumed the presence of the fundamental figures, straight lines, planes, pencils, etc. [fixed data or of the first group]. Thus, every construction, every measurement, every property of any figure ought to be held as relative, if it is essentially relative to the variable data. It ought, on the other hand, to be spoken of as absolute, if it is relative only to the fixed data ^8 II- The Forerunners of Non-Euclidean Geometry. [the fundamental figures], or, if, being enunciated in terms of the variable data, it only appears to depend upon them, so that it remains fixed when these vary. In this sense it is clear that in ordinary geometry the measurement of lines has necessarily a relative meaning. Indeed the existence of similar figures does not allow us in any way to individualize the size of a line in terms of funda- mental figures [straight line, pencil, etc.]. For an angle on the other hand, we can choose a method of measurement which expresses one of its absolute pro- perties. It is sufficient to take its ratio to the angle of a complete revolution, that is, to the entire pencil, this being one of the fundamental figures. We return now to Lambert and his geometry corre- sponding to the third hypothesis. He observed that with every segment we can associate a definite angle, which can easily be constructed. From this it follows that every seg- ment is brought into correspondence with the fundamental figure [the pencil]. Therefore, in the new [hypothetical] geometry, we are entitled to ascribe an absolute meaning also to the measurement of segments. To show in the simplest way how to every segment we can find a corresponding angle, and thus obtain an ab- solute numerical measurement of lines, let us imagine an equilateral triangle constructed upon every segment. We are able to associate with every segment the angle of the triangle corresponding to it and then the measure of this angle. Thus there exists a one-one correspondence between segments and the angles comprised between certain limits. But the numerical representation of segments thus ob- tained does not enjoy the distributive property which belongs to lengths. On taking the sum of two segments, we do not obtain the sum of the corresponding angles. However, a function of the angle, possessing this property, can be ob- The Absolute Unit of Length. aq tained, and we can associate with the segment, not the said angle, but this function of the angle. For every value of the angle between certain limits, such a function gives an absolute vieasure of segments. The absolute unit of length is that segment for which this function takes the value i. Now if a certain function of the angle is distributive in the sense just indicated, the product of this function and an arbitrary constant also possesses that property. It is there- fore clear that we can always choose this constant so that the absolute unit segment shall be that segment which corre- sponds to any assigned angle: e. g., 45". The possibility of constructing the absolute unit segment, given the angle, de- pends upon the solution of the following problem : To construct, on the Hypothesis of the Acute Angle, an equilateral triangle with a given defect. So far as regards the absolute m.easure of the areas of polygons, we remark that it is given at once by the defect of the polygons. We can also assign an absolute measure for polyhedrons. But with our intuition of space the absolute measure of all these geometrical magnitudes seems to us impossible. Hence if tue deny the existence of an absolute unit for segments, we can, with Lambert, reject the third hypothesis. § 21. As Lambert realized the arbitrary nature of this statement, let it not be supposed that he believed that he had in this way proved the Fifth Postulate. To obtain the wished-for proof, he proceeds with his investigation of the consequences of the third hypothesis, but he only succeeds in transforming his question into others equally difficult to answer. Other very interesting points are contained in the Theorie der Parallellinien, for example, the close resemblance 4 co II. The Forerunners of Non-Euclidean Geometry. to spherical geometry^ of the plane geometry which would hold, if the second hypothesis were valid, and the remark that spherical geometry is independent of the Parallel Postulate, Further^ referring to the third hypothesis^ he made the follow- ing acute and original observation: Froin this I should al- most conchcde that the third hypothesis tvould occur in the case of an imaginary sphere. He was perhaps brought to this way of looking at the question by the formula {A-\-B-\- C — it) r^, which expresses the area of a spherical triangle. If in this we write for the radius r, the imaginary radius K -i r we obtain r^\yi—A—B—C\; that is, the formula for the area of a plane triangle on Lambert's third hypothesise § 22. Lambert thus left the question in suspense. In- deed the fact that he did not publish his investigation allows us to conjecture that he may have discovered another way of regarding the subject. Further, ,it should be remarked that, from the general want of success of these attempts, the conviction began to be formed in the second half of the a 8th Century that it would be necessary to admit the Euclidean Postulate^ or some other equivalent postulate, without proof. In Germany, where the writings upon the question followed closely upon each other, this conviction had al- ready assumed a fairly definite form. We recognize it in A, G. Kastner,^ a well-known student of the theory of parallels, and in his pupil, G. S. Klugel, author of the 1 In fact, in Spherical Geometry the sum of the angles of a quadrilateral is greater than four right angles, etc. 2 Cf. Engel u. Stackel; Th. der P. p. 146. 3 For some information about Kastner, cf. Engel u. StAckel; Th. der P. p. 139 — 141. Klùgel's Work. e i valuable criticism of the most celebrated attempts to de- monstrate the Fifth Postulate, referred to on p. 44 [note 3]. In this work Klugel finds each of the proposed proofs insufficient and suggests the possibility of non-intersecting straight lines being divergent YMoglich ware es freilick, da^ Gerade, die sich nihct schneiden, voiieinander abweiche?i\. He adds that the apparent contradiction which this presents is not the result of a rigorous proof, nor a consequence of the definitions of straight lines and curves, but rather something derived from experience and the judgment of our senses. \Dafi so etwas widersinnig ist, wissen wir nicht infolge strenger Sc/iiusse Oder vcrmoge deutlicher Begriffe V07i der geraden und der kntmmen Linie, viebnehr durch die Erfahrung und durch das Urteil unserer Augen]. The investigations of Saccheri and Lambert tend to confirm Klugel's opinion, but they cannot be held to be a proof of the impossibihty of demonstrating the Euclidean hypothesis. Neither would a proof be reached if we proceed- ed along the way opened by these two geometers, and de- duced any number of other propositions, not contradicting the fundamendal theorems of geometry. Nevertheless that one should go forward on this path, without Saccheri's presupposition that contradictions would be found there, constitutes historically the decisive step in the discovery that Euclìd's Postulate could not be proved, and in the creation of the Non-Euclidean geometries. But from the work of Saccheri and Lambert to that of LoBATSCHEWSKY and B0LYAI, which is based upon the above idea, more than half a century had still to pass ! The French Geometers tov,;'ards the End of the i8th Century. § 23. The critical study of the theory of parallels, which had already led to results of great interest in Italy and (-2 II. The Forerunners of Non-Euclidean Geometry. Germany, also made a remarkable advance in France to- wards the end of the iSth Century and the beginning of the 19th. D'Alembert [1717 — 1783]; in one of his articles on geometry, states that 'La definition et les propriétés de la ligne droite, ainsi que des lignes parallèles sont l'écueil et pour ainsi dire le scandale des elements de Geometrie.' ^ He holds that with a good definition of the straight line both difficulties ought to be avoided. He proposes to define a parallel to a given straight line as any other coplanar straight line, which joins two points which are on the same side of and equally distant from the given line. This definition allows parallel lines to be constructed immediately. However it would still be necessary to show that these parallels are equidistant. This theorem was offered, almost as a challenge, by D'Alembert to his contemporaries. § 24. De Morgan, in his Budget of Paradoxes^, relates that Lagrange [1736 — 1813], towards the end of his life, wrote a memoir on parallels. Having presented it to the French Academy, he broke off" his reading of it with the ex- clamation: 'II faut que j'y songe encore!' and he withdrew the MSS. Further Houel states that Lagrange, in conversation with BiOT, affirmed the independence of Spherical Trigon- ometry from Euclid's Postulate.-^ In confirmation of this statement it should be added that Lagrange had made a spe- cial study of Spherical Trigonometry,'^ and that he inspired. 1 Cf. D'Alembert: Melanges de Littcrature, d'Hisioire, et de Philosophie, T. V. S II (l7S9)- Also: Encychfédie Méihodiqiie Mathématique ; T. II. p. 519, Article: Parallèles (1785). 2 A. DE Morgan: A Budget of Paiadoxes,^.\'J2,. (London, 1872). 3 Cf. J. Houel: Essai critique sur les principes fondamenlaux de la geometrie élèmentaire, p. 84, Note (Paris, G. VJLLARS, 1 883). 4 Cf. Miscellanea Taurinensia, T. II. p. 299—322 (1760 — 5i). D'Alembert, Lagrange, and Laplace. £2 if he did not write, a memoir ''Sur les principes fondamentaux de la Mecanique [1760 — i]^, in which Foncenex discussed a question of independence, analogous to that above noted for Spherical Trigonometry. In fact, Foncenex shows that the analytical law of the Composition of Forces acting at a point does not depend on the Fifth Postulate, nor upon any other which is equivalent to it.^ § 25. The principle of similarity, as a fundamental notion, had been already employed by Wallis in 1663 [cf. § 9]. It reappears at the beginning of the 19th Century, sup- ported by the authority of two famous geometers: L. N. M. Carxot [1753 — 1823] and Laplace [1749 — 1827]. In a Note [p. 481] to his Geometrie de Position [1803] Carnot affirms that the theory of parallels is allied to the principle of similarity, the evidence for which is almost on the same plane as that for equality, and that, if this idea is once admitted, it is easy to establish the said theory rigorously. Laplace [1824] observes that Newton's Law [the Law of Gravitation], by its simplicity, by its generality and by the confirmation which it finds in the phenomena of nature, must be regarded as rigorous. He then points out that one of its most remarkable properties is that, if the dimensions of all the bodies of the universe, their distances from each other, and their velocities, were to decrease proportionally, the heavenly bodies would describe curves exactly similar to those which they now describe, so that the universe, reduced step by step to the smallest imaginable space, would always present the same phenomena to its observers. These pheno- mena, he continues, are independent of the dimensions of the universe, so that the simphcity of the laws of nature only allows the observer to recognise their ratios. Referring again to this 1 Cf. Lagrange: Oeiivres, T. VIL p. 331 — 2fil- 2 Cf. Chapter VL Ca II. The Forerunners of Non-Euclidean Geometry. astronomical conception of space, he adds in a Note: 'The attempts of geometers to prove Euclid's Postulate on Parallels have been up till now futile. However no one can doubt this postulate and the theorems which Euclid deduced from it. Thus the notion of space includes a special property, self-evident, without which the properties of parallels cannot be rigorously established. The idea of a bounded region, e. g., the circle, contains nothing which depends on its absolute magnitude. But if we imagine its radius to diminish, we are brought without fail to the diminution in the same ratio of its circum- ference and the sides of all the inscribed figures. This pro- portionality appears to me a more natural postulate than that of Euclid, and it is worthy of note that it is discovered afresh in the results of the theory of universal gravitation.' ^ § 26. Along with the preceding geometers, it is right also to mention J. B. Fourier [1768 — 1830], for a discussion on the straight line which he carried on with Monge.^ To bring this discussion into line with" the investigations on parallels, we need only go back to D'Alembert's idea that the demonstration of the postulate can be connected with the definition of the straight line [cf § 23]. Fourier, who regarded the distance between two points as a prime notion^ proposed to define first the sphere; then the plane, as the locus of points equidistant from two given points;^ then the straight line, as the locus of the points equidistant from three given points. This method 1 Cf. Laplace. Oeuvres, T. VI. Livre, V. Ch. V. p. 472. 2 Cf. Seances de P Ecole ftormale: De bats, T. I. p. 28 — ^^ (1795). This discussion was reprinted in Mathésis. T. IX. p. 139 -141 (1883)- 3 This definition of the plane was given by Leibnitz about a century before. Cf. Opuscules et fragtnents incdiis, edited by L. CouTURAT, p. 554 — 5. (Paris, Alcan, 1903). Fourier and Lesfendre. 55 of presenting the problem of the foundations of geometry agrees with the opinions adopted at a later date by other geometers, who made a special study of the question of parallels [W. Bolyai, N. Lobatschewsky, de Tilly]. In this sense the discussion between Fourier and Monge finds a place among the earliest documents which refer to NoJi- Euclidea7i geometry} Adrien Marie Legendre [1752 — 1833I. § 27. The preceding geometers confined themselves to pointing out difficulties and to stating their opinions upon the Postulate. Legendre, on the other hand, attempted to transform it into a theorem. His "investigations, scattered among the different editions of his Elements de Geometrie [1794 — 1823], are brought together in his Reflexions sur différentes manières de démontrer la théorie des paralleles ou le [the'orème sur la somme des trois afigles du triangle. [Mém. Ac. Se, Paris, T. XIII. 1833.] In jthe most interesting of his attempts, Legendre, like Saccheri, approaches the question from the side of the sum of the angles of a triangle, which sum he wishes to prove equal to two right angles. With this end in view, at the commencement of his work he succeeds in! rejecting Saccheri's Hypothesis of the Obtuse Angle, since he estabhshes that the sum of the angles of any triangle is either less than ^Hypothesis of the Acute Angle] or equal to {Hypothesis of the Right Angle] two right angles. We reproduce a neat and simple proof which he gives of this theorem : Let n equal segments ^1^2, -^2^3, • . . ^«^«+1 be taken I To this we add that later memoirs and investigations showed that Fourier's definition also fails to build up the Eucli- dean theory of parallels, without the help of the Fifth Postulate, or some other equivalent to it. c5 II. The Forerunners of Non-Euclidean Geometry. one after the other on a straight Hne [Fig. 28]. On the same side of the Hne let n equal triangles be constructed, having for their third angular points B^B^. . . .B,f The segments Bj_B2i BiB^,... Bn—x Bni which join these vertices, are equal and can be taken as the bases of n equal triangles, B^A2B2, B, E^ B3 B^ B^^2^J^3'--- -^«-i A,iB„. The figure is completed by adding the triangle which is equal to the others. Let the angle ^i of the triangle A^B^Az be denoted by P, and the angle A2 of the consecutive triangle by a. Then p < a. In fact, if P ^ a, by comparing the two triangles A^B^Az and B1A2B2, which have two equal sides, we would deduce A,A2>B,B2. Further, since the broken line A^B^Bz . . . -^«+1 ^«-f i is greater than the segment AiA^-^i , A^Bx + n. B^B^ + ^„+i ^«+1 > n. A^A^, i. e., 2 Aj,Bi^n{AiA2 — B^B^). But if n is taken sufficiently great, this inequality con- tradicts the Postulate of Archimedes. Therefore A^A^ is not greater than B^Bz , and it follows that it is impossible that P i>> a. Thus we have P < a. From this it readily follows that the sum of the angles ot the triangle A^B^A^ is less than or equal to two right angles. This theorem is usually, but mistakenly, called Legendre's First Theorem. We say mistakenly, because Saccheri had already established this theorem almost a century earlier [cf p. 38] when he proved that the Hypothesis of the Obtuse Angle was false. Lea:endre's First Proof. 57 The theorem usually called Legendre's Second Theorem was also given by Saccheri, and in a more general form [cf. p. 29]. It is as follows: If the sum of the angles of a triangle is less than or equal to two right afigles in only one triangle, it is respectively less than or equal to two right angles in every other triangle. We do not repeat the demonstration of this theorem, as it does not differ materially from that of Saccheri. We shall rather show how Legendre proves that the sum of the three angles of a tria?igle is equal to two right angles. Suppose that in the triangle ABC [cf. Fig. 2 9] ^A ■\- i6^ 1822]; and some notes found among his papers, [1831].^ Comparing the various passages in Gauss's letters, we can fix the year 1792 as the date at which he began his 'Med- itations' . The following portion of a letter to W. Bolyai [Dec. 1 7, 1799] proves that Gauss, Hke Saccheri and Lambert before him, had attempted to prove the truth of Postulate V. by as- suming it to be false. 'As for me, I have already made some progress in my work. However the path I have chosen does not lead at all to the goal which we seek, and which you assure me you have reached.3 It seems rather to compel me to doubt the truth of geometry itself. 'It is true that I have come upon much which by most people would be held to constitute a proof: but in my eyes it proves as good as nothing. For example, if one could show that a rectilinear triangle is possible, whose area would be greater than any given area, then I would be ready to prove the whole of geometry absolutely rigorously. 'Most people would certainly let this stand as an Axiom; but I, no! It would, indeed, be possible that the area might 1 [It would be more correct to say over thirty.] 2 Cf. Gauss, Werke, Bd. VIE. p. 157—268. 3 It is to be remembered that W. Bolyai was working at this subject in Gottingen and thought he had overcome his diffi- culties. Cf. 3 29. 5 ^S III. The Founders of Non-Euclidean Geometry. always remain below a certain limit, however far apart the three angular points of the triangle were taken.' In 1804, replying to W. Bolyai on his Theoria parall- elamm, he expresses the hope that the obstacles by which their investigations had been brought to a standstill would finally leave a way of advance open.^ From all this, Stackel and Engel, who collected and verified Gauss's correspondence on this subject, come to the conclusion that the great geometer did not recognize the existence of a logically sound Non-Euclidean geometry by intuition or by a flash of genius : that, on the contrary, he had spent upon this subject many laborious hours before he had overcome the inherited prejudice against it. Did Gauss, when he began his investigations, know the writings of Saccheri and Lambert? What influence did they exert upon his work? Segre, in his Congetture^ already re- ferred to [p. 44 note 2], remarks that both Gauss and W. Bolyai, while students at Gottingen, the former from 1795 — 98, the later from 1796 — 99, were interested in the theory of parallels. It is therefore possible that, through Kastner and Seyffer, who were both deeply versed in this subject they had obtained knowledge both of the Euclides ab omni naevo vindicatus and of the Theorie der Faralleiiinien. But the dates of which we are certain, although they do not con- tradict this view, fail to confirm it absolutely. § 33. To this first period of Gauss's work, after 1 8 1 3 there follows a second. Of it we obtain some knowledge chiefly from a few letters, one written by Wachter to Gauss [18 1 6]; others [sent |by Gauss to Gerling [i8i9],jTaurinus [1824] and Schumacher [183 i]; and also from some notes found among Gauss's papers. I [It should be noticed that these efforts were still directed towards proving the truth of Euclid's postulate.] Gauss's "Meditations". ^ These documents show us that Gauss, in this second period, had overcome his doubts, and proceeded with his de- velopment of the fundamental theorems of a new geometry, which he first czWs, Anti-Euclidean [cf.WACHTER's letter quoted on p. 62]; then Astral Geometry [following Schweikart, cf. p. 76]; dina ^nsWy, Non-Euclidean [cf letter to Schumacher]. Thus he became convinced that the Non-Euclidean Geometry did not in itself involve any contradiction, though at first sight some of its results had the appearance of paradoxes [letter to Schumacher, July 12, 183 1]. However Gauss did not let any rumour of his opinions get abroad, being certain that he would be misunderstood. [He was afraid of the clamour of the Boeotiatis; letter to Bessel, Jan. 27, 1829]. Only to a few trusted friends did he reveal something of his work. When circumstances compel him to write to Taurinus [1824] on the subject, he begs him to keep silence as to the information which he imparted to him. The notes found among Gauss's papers contain two brief synopses of the new theory of parallels, and probably belong to the projected exposition of the Non-Euclidean Geo metry, with regard to which he wrote to Schumacher [on May 17, 1 831]: *In the last few weeks I have begun to put down a few of my own Meditations, which are already to some extent nearly 40 years old. These I had never put in writing, so that I have been compelled three or four times to go over the whole matter afresh in my head. Also I wished that it should not perish with me.' § 34. Gauss defines parallels as follows : ^ If the coplanar straight lines AM, BN, do not intersect each ether, while, on the other hand, every straight line through I [In this section upon Gauss's work on Parallels fuller use has been made of the material in his Collected Works (Gauss, Werke, Bd. VIII, p. 202—9)]. S* Fig. 33- 68 III. The Founders of Non-Euclidean Geometry. A between AM and AB cuts BN, then AM is said to be paral- lel to BN{^g. ii\ He supposes a straight B !—-.__ line passing through A^ to start from the position AB, and then to rotate continu- ously on the side towards ^^^ which BN is drawn, till it reaches the position AC^ in Cèjt BA produced. This line be- gins by cutting j^iVand in the end it does not cut it. Thus there can be one and only one position, separating the lines which intersect ^iVfrom those which do not intersect it. This must be "ùxt first of the lines, which do not cut BN: and thus from our definition it is the parallel AM) since there can obviously be no last line of the set of lines which intersect BN. It will be seen in what way this definition differs from Euclid's. If Euclid's Postulate is rejected, there could be dif- ferent lines through A, on the side towards which BN is drawn, which would not cut BN. These lines would all be parallels to BN according to Euclid's Definition. In Gauss's definition only the first of these is said to be parallel \.oBN. Proceeding with his argument Gauss now points out that in his definition the starting points of the lines AM and BN are assumed, though the lines are supposed to be pro- duced indefinitely in the directions of AM and BN. I. He proceeds to show that the parallelism of the line AM to the line BN is independent of the points A a?id B, pro- vided the sense in which the lines are to be produced indefinitely remain the same. It is obvious that we would obtain the same parallel AM Gauss's Theory of Parallels. 69 if we kept A fixed and took instead of B another point B' on the line BN, or on that Hne produced backwards. It remains to prove that \i AMis parallel to BJV (or the point A, it is also the parallel to BNiox any point upon AM, or upon AM produced backwards. Instead of ^ [Fig. 34] take another starting point A' upon AM. Through A\ between A'B and A'M, draw the line A'F in any direction. B|< Through Q, any point on A'F, between A' and F, draw the line AQ. Then, from the definition, A ■ AQ must cut BN, so that it is clear QF must also cut BN. Thus AA'M is the first of the lines which do not cut BN, and A'M is parallel to BN. Again take the point A' upon AM produced backwards [Fig- 35]- ^M Fig- 34. Fig. 35- Draw through A', between A'B and A'M, the line A'F in any direction. Produce A'F backwards and upon it take any point Q. Then, by the definition, QA must cut BN, for example, 70 III. The Founders of Non-Euclidean Geometry. in R. Therefore AP lies within the closed figure AARB, and must cut one of the four sides AA^ AR, RB, and BA. Obviously this must be the third side RB, and therefore AM is parallel to BN. II. The Reciprocity of the Parallelism can also be estab- lished. In other words, if AM is parallel to BN, then BN is also parallel to AM. Gauss proves this result as follows: From any point B upon BN draw BA perpendicular to AM. Through B draw any line BN' between BA and BN. At B, on the same side of AB as BN, make <^ ABC^ V2 ^N'BN There are two possible cases: Case (i), when BC cuts AM [cf. Fig. 36]. Case (ii), when BC does not cut AM [cf. Fig. 37]. Fig. 36. Case (i). Let BC cut AM in D. Take AE = AD, and join BE. Make ^BDF^^BED. Since AM is parallel to BN^ DF must cut BM^ for example, in G. From EM cut off EH equal to DG. Then, in the triangles BEH s^nà BDG, it follows that Gauss's Theory of Parallels (contd.). 71 JM. Therefore «^ EBD = ^HBG. But <^ EBD = ^N'BN. Therefore BJV and BII coincide, and BN' must cut But BN" is any line through B, between BA and BN. Therefore BN is parallel to AM. B Fig. 37- Case (ii). In this case let Z> be any arbitrary point upon AM. Then with the same argument as above, ■^ EBB = <^ GBH, But ^ ABD < < ^^C. Therefore <^ ^^Z> < ^ iV^'^iV. Therefore <^ GBH<.^N'BN. Therefore BN' must cut AM. But ^iV" is any line through B, between BA and BN. Therefore BN is parallel to AM. Thus in both cases we have proved that \iAM\5 parallel to BN, then BN is parallel to AM. "■ The next theorem proved by Gauss in this synopsis is as follows: [I Gauss's second proof of this theorem is given in the German translation. However it will be found that in it he assumes that BC cuts AM, and to prove this the argument used above is necessary.] 72 III- The Founders of Non-Euclidean Geometry. III. If the line (i) is parallel to the line (2) arid to the line (3), then (2) and (3) are parallel to each other. Case (i). Let the line (i) lie between (2) and (3) [cf. Fig. 38]. Let A and B be two points on (2) and (3), and let AB cut (i) in C. Through A let an arbitrary line AD be drawn between AB and (2). Then it must ^é: ^ cut (i), and on being pro- duced must also cut (3). Since this holds for every line such as AD, (2) is parallel to (3). Case (ii). Let the line (i) be outside both (2) and (3), and let (2) He between (i) and (3) [cf. Fig. 39]. If (2) is not parallel to (3), through any point chosen at random upon (3), a line different from (3) can be drawn which is parallel to (2). This, by Case (i), is also par- allel to (i), which is absurd. This short Note on Parall- els closes with the theorem that if tivo lities AM and BN are parallel, these lines produced backwards cannot tneet. From all this it is evident that the parallelism of Gauss xtitzxis parallelism in a given sense. Indeed his definition of parallels deals with a line drawn from A on a. definite side of the transversal AB: e. g., the ray drawn to the right, so that we might speak of AM as the parallel to BJV towards the right. The parallel from A to BJV towards the left is not necessari- ly AM. If it were, we would obtain the Euclidean hypothesis. Fig. 38. ?• 39- Corresponding Points. n The two lines, in the third theorem, which are each pa- rallel to a third line, are thus both parallels in the same sense (both left-hand, or both right-hand parallels). In a second memorandum on parallels, Gauss goes over the same ground, but adds the idea of Corresponding Points on two parallels AA , BB' . Two points A, B are said to corre- spond^ when AB makes equal internal angles with the parallels en the same side [cf. Fig. 40]. Fig. 40. Fig. 41. With regard to these Corresponding Points he states the following theorems: (i) If A, B are two correspofiding points upon tivo paral- lels, and M is the middle poitit of AB, the line MN, perpen- dicular to AB, is parallel to the two given lines, and every point on the same side of MN as A is nearer A than B. (ii) If A, B are two corresponding points upon the parallels {\) and {2), and A', B' two other correspo7iding points on the same lifies, then AA = BB', and co?iversely. (iii) If A, B, C are three points on the parallels (i), (2) and (3), such that A and B, B and C, correspond, then A and C also correspond. >jA III. The Founders of Non-Euclidean Geometry. The idea of Corresponding Points, when taken in con- nection with three Hnes of a pencil (that is, three concurrent lines [cf. Fig. 41] allows us to define the circle as the locus of the points on the lines of a pencil which correspond to a given point. But this locus can also be constructed when the lines of the pencil are parallel. In the Euclidean case the locus is a straight line : but putting aside the Euclidean hypothesis, the locus in question is a line, having many properties in common with the circle, but yet not itself a circle. Indeed if any three points are taken upon it, a circle cannot be drawn through them. This line can be regarded as the limiting case of a circle, when its radius becomes infinite. In the Non- Euclidean geometry of Lobatschewsky and Bolyai, this locus plays a most important part, and we shall meet it there under the name of the Horocycle.' This work Gauss did not need to complete, for in 1832 he received from Wolfgang Bolyai a copy of the work of his son Johann on Absolute Geometry. From letters before and after the date at which he interrupted his work, we know that Gauss had discovered in his geometry an Absolute Unit of Length [cf. Lambert and Legendre], and that a constant k appeared in his formulae, by means of which all the problems of the Non-Euclidean Geometry could be solved [letter to Taurinus, Nov. 8, 1824]. Speaking more fully of these matters in 1831 [letter to I [Lobatschewsky ; Gremkreis, Courbe-Umite or Iloricycle. BOL- YAI; Parazykl, L-lÌ7iie. It is interesting to notice that Gauss, even at this date, seems to have anticipated the importance of the Ilorocycle. The definition of Corresponding Points and the statement of their properties is evidently meant to form an introduction to the dis- cussion of the properties of this curve, to which he seems to have given the name Trope.'] The Perimeter of a Circle. 7C Schumacher], he gave the length of the circumference of a circle of radius r in the form ■nk\e^—e ^) . With regard to k, he says that, if we wish to make the new geometry agree wth the facts of experience, we must suppose k infinitely great in comparison with all known measurements. For >è ^ 00 , Gauss's expression takes the usual form for the perimeter of a circle. ' The same remark holds for the whole of Gauss's system of geometry. It contains Euclid's system, as the limiting case, when /è = 00 . ^ Ferdinand Karl Schweikart [1780 — 1859]. § 35. The investigations of the Professor of Jurispru- dence, F. K. ScHWEiKART,3 date from the same period as those of Gauss, but are independent of them. In 1807 he published Die Theorie der Parallellinien nebst dem Vorschlage ihrer Verbannung aus der Geometrie. Contrary to what one might expect from its title, this work does not contain a treatment of parallels independent of the Fifth Postulate, but one based on the idea of the parallelogram. But at a later date, Schweikart, having discovered a new order of ideas, developed a geometry independent of Euclid s hypothesis. When in Marburg in December, 1818, he handed the following memorandum to his colleague Ger- LiNG, asking him to communicate it to Gauss and obtain his opinion upon it: 1 To show this we need only use the exponential series. 2 For other investigations by Gauss, cf. Note on p. 90. 3 He studied law at Marburg and from 1796 — 98 attended the lectures on Mathematics given in that University by Professor J. K, F. Hauff, the author of various memoirs on parallels, cf. Th. der P. p. 243. n^ III. The Founders of Non-Euclidean Geometry. Memorandum. 'There are two kinds of geometry— a geometry in the strict sense — the Eudidean; and an astral geometry [astra- hsche Grofienlehre]. 'Triangles in the latter have the property that the sum of their three angles is not equal to two right angles.' 'This being assumed, we can prove rigorously: a) That the sum of the three angles of a triangle is less than two right angles; b) that the sum becomes ever less, the greater the area of the triangle; c) that the altitude of an isosceles right-angled triangle continually grows, as the sides increase, but it can never become greater than a certain length, which I call the Cofistant. Squares have, therefore, the following form [Fig. 42]. 'If this Constant were for us the Radius of the Earth, (so that every line drawn in the universe from one fixed Star to another, distant 90° from the first, would be a tangent to the surface of the earth), it would be infinitely great in comparison with the spaces which occur in daily life. 'The Euclidean geometry holds only on the assumption that the Constant is infinite. Only in this case is it true that the three angles of every triangle are equal to two right angles: and this can easily be proved, as soon as we admit that the Constant is infinite.' ^ Schweikart's Astral Geometry and Gauss's Non-Euclid- Fig. 42. Schweikart's Work. nj ean Geometry exactly correspond to the systems of Sac- CHERi and Lambert for the Hypothesis of the Acute Angle. Indeed the contents of the above memorandum can be ob- tained directly from the theorems of Saccheri, stated in Klùgel's Conatuum, and from Lambert's Theorem on the area of a triangle. Also since Schweikart in his Theorie of 1807 mentions the works of the two latter authors, the direct influence of Lambert, and, at least, the indirect influence of Saccheri upon his investigations are established.^ In March, 1 8 1 9 Gauss replied to Gerling with regard to the Astral Geometry. He compliments Schweikart, and declares his agreement with all that the sheet of paper sent to him contained. He adds that he had extended the Astral Geometry so far that he could completely solve all its pro- blems, if only Schweikart's Constant were given. In con- clusion, he gives the upper limit for the area of a triangle in the form J [log hyp (I + \2)Y ' Schweikart did not publish his investigations. Franz Adolf Taurinus [1794 — 1874]. § 36. In addition to carrying on his own investigations on parallels, Schweikart had persuaded [1820] his nephew Taurinus to devote himself to the subject, calling his atten- 1 Cf. Gauss, Werke, Bd. VIII, p. iSo— 181. 2 Cf. Segre's Congetture, cited above on p. 44. 3 The constant which appears in this formula is Schweikart's Constant C, not Gauss-'s constant /', in terms of which he expressed the length of the circumference of a circle, (cf. p. 75). The two constants are connected by the following equation: log (1+1/2)- 78 in. The Founders of Non-Euclidean Geometry. tion to the Astral Geometry, and to Gauss's favourable ver- dict upon it. Taurinus appears to have taken up the subject seriously for the first time in 1824, but with views very different from his uncle's. He was then convinced of the absolute truth of the Fifth Postulate, and always remained so, and he cherish- ed the hope of being able to prove it. FaiHng in his first at- tempts, under the influence of Gauss and Schweikart, he again began the study of the question. In 1825 he publish- ed a Theorie der Parallellinien^ containing a treatment of the subject on Non-Euclidean lines, the rejection oi the Hypothesis of the Obtuse Angle, and some investigations resembling those of Saccheri and Lambert on the Hypothesis of the Acute Angle. He found in this way Schweikart's Constant, which he called a Parameter. He thought an absolute unit of length impossible, and concluded that all the systems, corre- sponding to the infinite number of values of the parameter, ought to hold simultaneously. But this, in its turn, led to con- siderations incompatible with his conception of space, and thus Taurinus was led to reject the Hypothesis of the Acute Angle while recognising the logical compatibility of the propo- sitions which followed from it. In the next year Taurinus published his Geometriae Pri- ma Elementa [Cologne, 1826], in which he gave an improved version of his researches of 1825. This work concludes with a most important appendix, in which the author shows how a system of analytical geometry could be actually constructed on the Hypothesis of the Acute Angle. ^ With this aim Tauriuus starts from the fundamental for- mula of Spherical Trigonometry — I For the final influence of Saccheri and Lambert upon Tau- rinus, cf. SeGRE's Congetture, quoted above on p. 44. The Work of Taurinus. yg a b C.Ò.C . COS -r = COS -7 COS -r + sm ^ sm -, cos A, In it he transforms the real radius k into the imaginary radius ik. Using the notation of the hyperboHc functions, we thus have (i) cosh -T = cosh — cosh — sinh — sinh -^ cos A. This is the fundamental formula of the Logarithmic- Spherical Geometry \logarithmisch-spharischen Geometrie'\ of Taurinus. It is easy to show that in this geometry the sum of the angles of a triangle is less than 180°. For simplicity we take the case of an equilateral triangle, putting a=b=c in (i). Solving, for cos A, we obtain cosh — (i*) cos ^ = cosh— + I But sech T ^/a- Thus A is less than 60°, and the sum of the angles of the triangle is less than 180°. It is instructive to note, that, from (i*). Lt. (cos A) = Vz. a == o So that in the Hmit when a becomes zero, A is equal to 60°. Therefore, in the log. -spherical geotnetry, the sum of the angles of a triangle tends to x8o° when the sides tend to zero. We may also note that from (i*) Lt. (cos A) = V2 ; k « so that in the limit when k is infinite, A is equal to 60°. There- fore, when the constant k tends to infinity, the angles of the equilateral triangle are each equal to 60°, as in the ordinary geometry. 8o ni. The Founders of Non-Euclidean Geometry. More generally, using the exponential forms for the hy- perbolic functions, it will be seen that in the limit when k is infinite (i) becomes a^ = b"^ -^ c- — 2bc cos A, the fundamental formula of Euclidean Plane Trigonometry. § 37. The second fundamental formula of Spherical Trigonometry, cos A = — cos B cos C + sin ^ sin C cos -y > by simply interchanging the cosine with the hyperbolic cosine, gives rise to the second fundamental formula of the log.-spher- zVa/ geometry: a, (2) COS A = — cos B cos C + sin B sin C cosh -r. For A = o and C= 90°, we have (3) cosh X = •' ^" ^■^ k sin B The triangle corresponding to this formula has one angle zero and the two sides containing it are infinite and parallel [asymptotic]. [Fig. 43.] The angle B^ between the side which Fig. 43- is parallel and the side which is perpendicular to CA, is seen from (3) to be a function of a. From this onward we can call it the Angle of Parallelism for the distance a [cf. Lobat- SCHEWSKY, p. 87]. For B = 45°, the segment BC^ which is given by (3), is Schweikart's Constant [cf. p. 76]. Thus, denoting it by P, The Angle of Parallelism. gl cosh ^ = V2, from which, solving for k, we have k^ ^-_. log (I + V2) This relation connecting the two constants /' and ^ was given by Taurinus. The constant k is the same as that em- ployed by Gauss [cf. p. 75] in finding the length of the cir- cumference of a circle. § 38. Taurinus deduced other important theorems in the log.-spherical geometry by further transformations of the formulae of Spherical Trigonometry, replacing the real radius by an imaginary one. For example, that the area of a triangle is proportional to its defect [Lambert, p. 46] : that the superior limit of that area is „ , , -.,-.,, [Gauss, p. 77 ; [log(l-{-^2)]2 that the length of the circumference of a circle of radius r is 2Tr/è sinh -. [Gauss, p. 75]; that the area of a circle of radius r is 2TtZ'^ (cosh -T- — i); that the area of the surface of a sphere and its volume, are respectively ■y 47T/&^ smh^ -,, and 2TT/è3 (sinh , cosh y — — ). We shall not devote more space to the different anaiyt- 6 32 III- The Founders of Non-Euclidean Geometry. ical developments, since a fuller discussion would cast no fresh light upon the method. However we note that the results of Taurinus confirm the prophecy of Lambert on the Third Hypothesis [cf. p. 50], since the formulae of the log.-spherical geometry, interpreted analytically, give the fun- damental relations between the elements of a triangle traced upon a sphere of imaginary radius.^ To this we add that Taurinus in common with Lambert recognized that Spherical Geometry corresponds exactly to the system valid in the case of the Hypothesis of the Obtuse Angle: further that the ordinary geometry forms a hnk be- tween spherical geometry and the log.-spherical geometry. Indeed, if the radius k passes continuously from the real domain to the purely imaginary one, through infinity, we pro- ceed from the spherical system to the log.- spherical system, through the Euclidean. Although Taurinus, as we have already remarked, ex- cluded the possibility that a log.-spherical geometry could be vahd on the plane, the theoretical interest, which it offers, did not escape his notice. Calling the attention of geo- meters to his formulae, he seemed to prophecy the existence I At this stage it should be remarked that Lambert, simul- taneously with his researches on parallels, was working at the tri- gonometrical functions with an imaginary argument, whose connection with Non-Euclidean Geometry was brought to light by Taukinus. Perhaps Lambert recognised that the formulae of Spherical Trig- onometry were still real, even when the real radius was changed in a purely imaginary one. In this case his prophecy with regard to the Hypothesis of the Acute Angle (cf. p. 50) would have a firm foundation in his own work. However we have no authority for the view that he had ever actually compared his investigations on the trigonometrical functions with those on the theory of parallels. Cf. P. StAckel: Bcmerkungen sit Lamberts Theorie der Parallellinien. Biblioteca Math. p. 107 — lio. (1899). Some Conclusions by Taurinus. 83 of some concrete case in which they would find an inter- pretation. * I The important service rendered by Schweikart and Tau- rinus towards the discovery of the Non-Euclidean Geometry was recognised and made known by Engel and Stackel. In their Th. der P., they devote a whole chapter to those authors, and quote the most important passages in Taurinus' writings, besides some letters which passed between him, Gauss and Schweikart. Cf. Stackel: Franz Adolf Taurinus, Abhandl. zur Geschichte der Math., IX, p. 397 — 427 (1899). Chapter IV. The Founders of Non-Euclidean Geometry (Contd.). Nicolai Ivanovitsch Lobatschewsky [1793 — 1856],' § 39. Lobatschewsky studied mathematics at the Uni- versity of Kasan under a German J. M. C. Bartels [1769 — 1836], who was a friend and fellow countryman of Gauss. He took his degree in 18 13 and remained in the University, first as Assistant, and then as Professor. In the latter position he lectured upon mathematics in all its branches and also upon physics and astronomy. As early as 181 5 Lobatschewsky was working at paral- lels, and in a copy of his notes for his lectures [1815 — 17] several attempts at the proof of the Fifth Postulate, and some investigations resembling those of Legendre have been found. However it was only after 1823 that he had thought of the Imaginary Geometry. This may be inferred from the manuscript for his book on Elementary Geometry, where he says that we do not possess any proof of the Fifth Postulate, but that such a proof may be possible-^ 1 For historical and critical notes upon Lobatschewsky we refer once and for all to F. Engel's book: N. I. Lobàtschefskij : Zzaci geo7netrische Abhandlungen ans de?n Russischen ubersetzt tiitf Anmerktoigen und mit einer Biographic dcs Verfassers, (Leipzig, Teubner, 1899). 2 [This manuscript had been sent to St. Petersburg in 1823 to be published. However it was not printed, and it was dis- Lobatschewsky's Works. ge Between 1823 and 1825 Lobatschewsky had turned his attention to a geometry independent of Euclid's hypothe- sis. The first fruit of his new studies is the Exposition suc- cincie des principes de la geometrie avec une demonstration ri- goureuse dii théorcme des parallcles, read on 1 2 [24] Feb., 1826, to the Physical Mathematical Section of the University of Kasan. In this "Lecture", the manuscript of which has not been discovered, Lobatschewsky explains the prin- ciples of a geometry, more general than the ordinary geo- metry, where two parallels to a given line can be drawn through a point, and where the sum of the angles of a tri- angle is less than two right angles [The Hypothesis of the Acute Angle of Saccheri and Lambert]. Later, in 1829 — 30, he published a memoir On the Prin- ciples of Geometry ^'^ containing the essential parts of the preceding "Lecture", and further apphcations of the new theory in analysis. In succession appeared the Imaginary Geometry [1835],^ New Principles of Geometry, with a Com- covered in the archives of the University of Kasan. in 1898. It is clear from some other remarks in this work that he had made further advance in the subject since 1815 — 17. He was now con- vinced that all the first attempts at a proof of the Parallel Postulate were unsuccessful, and that the assumption that the angles of a triangle could depend only on the ratio of the sides and not upon their absolute lengths was unjustifiable (cf. Engel, loc.cit. p. 369 — 70).] 1 Kasan Bulletin, (1829 — 1830). Geometrical Works of Lobat- schewsky (Kasan 1883 — 18S6), Vol. I p. 1 — 67. German translation by F. Engel p. i — 66 of the work referred to on the previous page. Where the titles are given in English we refer to works pub- lished in Russian. The Geometrical Works of Lobatschewsky contain two parts; the first, the memoirs originally published in Russian; the second, those published in French or German. It will be seen below that of the works in Vol. i. several translations are now to be had. 2 The Scientific Publications of the University of Kasan (1835). Geometrical Works, Vol. I, p. 71 — 120. German translation by 86 IV. The Founders of Non-Euclidean Geometry (Contd.). plett Theory of Parallels [1835 — 38]^ the Applications of the Lnaginary Geometry to Some Integrals [1836]^, then the Geometrie Imaginaire [183 7] 3, and in 1840, a small book containing a summary of his work, Geometrische Unter' suchungen zur Theorie der Farallellinien,'^ written in German and intended by Lobatschewsky to call the attention of mathemiaticans to his researches. Finally, in 1855, a year before his death, when he was already blind, he dictated and pubHshed in Russian and French a complete exposition of his system of geometry under the title : Pangéométrie ou precis de geometrie fondée sur une theorie generale et rigoureuse des par alleles, s § 40. Non-Euclidean Geometry, just as it was conceived by Gauss and Schweikart in 1816, and studied as an ab- H. LlEBMANN, with Notes. Abhandlungen zur Geschichte der Mathe- matik, Bd. XIX, p. 3—50 (Leipzig, Teubner, 1904). 1 Scientific Publications of the University of Kasan (1835 — 38). Geom. Works. Vol. I: p. 219 — 486. German translation by F. Engel, p. 67 — 235 of his work referred to on p. 84. English translation of the Introduction by G. B. Halsted, (Austin, Texas, 1897). 2 Scientific Publications of the University of Kasan. (1836). Geom. Works, Vol. I, p. 121 — 2l8. German translation by H, LlEB- MANN; loc. cit: p. 51 — 130. 3 Crelle's Journal, Bd. XVII, p. 295—320. (1837). Geom. Works, Vol. II, p. 581—613. 4 Berlin (1840). Geo7)i. Works, Vol. II, p. 553—578. French translation by J. Houel in Mém. de Bourdeaux, T. IV. (1866), and also va. Recherches géomèiriques sur la theorie des parallèles {?a.xis, Her- mann, 1900). English translation by G. B. Halsted, (Austin, Texas, 1891). Facsimile reprint (Berlin, Mayer and Muller, 1887). 5 Collection of Memoirs by Professors of the Royal University of Kasan on the ^o*''- anniversary of its foundation. Vol. I, p. 279 — 340. (1856). Also in Geom. Works, Vol. II, p. 617— 680. In Russian, in Scientific Publications of the University of Kasan, (1855). Italian translation, by G. Battaglini, in Giornale di Mat. T. V. p. 273—336, (1867). German translation, by H. Liebmann, Ostwald's Klassiker der exakten Wissenschaften, Nr, 130 (Leipzig, 1 902). Lobatschewsky^s Theory of Parallels. 37 stract system by Taurinus in 1826, became in 1829 — 30 a recognized part of the general scientific inheritance. To describe, as shortly as possible, the method followed by LoBATSCHEWSKY in the construction of the Imaginary Geo- metry or Pangeometry, let us glance at his G eovietrische Unter- suchungeii zur Theorie der ParallellÌ7iien of 1840. In this work Lobatschewsky states, first of all, a group of theorems independent of the theory of parallels. Then he considers a pencil with vertex A, and a straight line BC^ in the plane of the pencil, but not belonging to it. Let AD be the line of the pencil which is perpendicular to BC^ and AE that perpendicular to AD. In the Euclidean system this latter line is the only line which does not intersect BC. In the geometry of Lobatschewsky there are other lines of the pencil through A which do not intersect BC. The non-inter- secting lines are separated from the intersectijig lines by the two hues h, k (see Fig. 44), which in their turn do not meet BC. [cf. Saccheri, p. 42.] These lines, which the author calls parallels, have each a definite direction of parallelistn. The line //, of the figure, is the parallel to the right: k, to the left. The angle which the perpendicular AD makes with one of the parallels is the ajtgle 0/ parallelism for the length AD. Lobatschewsky uses the symbol TT {a) to denote the angle of parallelism corresponding to the length a. In the ordinary geometry, we have TT {a) = ()o° always. In the geometry of Lobatschewsky, it is a definite function of a, tending to 90° as a tends to zero, and to zero as a increases without limit. From the definition of parallels the author then deduces their principal properties: 2,S IV. The Founders of Non-Euclidean Geometry (Contd.). That if AD is the parallel to £C for the point A, it is the parallel to BC in that direction for every point on AD [permanency]; That if AD is parallel to BC, then BC is parallel to . AD [reciprocity] : That if the lines (2) and (3) are parallel to (i), then (2) and (3) are parallel to each other [transitivity] [cf. Gauss, p. 72]; and that If AD and BC are parallel, AD is asymptotic to BC. Finally, the discussion of these questions is preceded by the theorems on the sum of the angles of a triangle, the same theorems as those already given by Legendre, and still earlier by Saccheri. There can be little doubt that Lo- BATSCHEWSKY was familiar with the work of Legendre.^ But the most important part of the Imaginary Geometry is the construction of the formulae of trigonometry. To obtain these, the author introduces two new figures: the Horocycle [circle of infinite radius, cf. Gauss, p. 74], and the Horosphere ^ [the sphere of infinite radius], which in the ordinary geometry are the straight line and plane, respect- ively. Now on the Horosphere, which is made up of 00 * Horocycles, there exists a geometry analogous to the ordinary geometry, in which Horocycles take the place of straight lines. Thus Lobatschewsky obtains this first re- markable result: The Euclidean Geometry [cf. Wachter, p. 63], and., in particular, the ordinary plane trigonometry, hold upon the Hor- osphere. 1 Cf. LoBATSCHEWSKv's Criticism of I.egendre's attempt to obtain a proof of Euclid's Postulate in his Nexu Pnnciples of Geometry (Engel's translation, p. 68). 2 [Lobatschewsky uses the terms Grcnzkreis, Grenzkugel in his German work: courbe-limite, horicycle, horisphere, su7-/ace-limite in his French work.] The Horocycle snd Horocyclic Surface. 30 This remarkable property and another relating to Co- axal Horocydes [concentric circles with infinite radius] are employed by Lobatschewsky in deducing the formulae of the new Plane and Spherical Trigonometries \ The formulas of spherical trigonometry in the new system are found to be exactly the same as those of ordinary spherical trigonometry, when the elements of the triangle are measured in right- angles. § 41. It is well to note the form in which Lobatschewsky expresses these results. In the plane triangle ABC, let the sides be denoted by a^ b, c, the angles by A, B, C; and let T7 (a), TT (a), TT (c) be the angles of parallelism corresponding to the sides a, b, c. Then Lobatschewsky's fundamental formula is , . , TT /7\ TT / \ fin T\ (l>) sin TT \c) (4) cos A cos TT {b) cos TT {c) + ^^^ = 1. ^^^ ' sm IT [cij It is easy to see that this formula and that of Taurinus [(i), p. 79] can be transformed into each other. To pass from that of Taurinus to that of Lobatschew- sky, we make use of (3) of p. 80, observing that the angle B, which appears in it, is TT {a). For the converse step, it is sufficient to use one* of Lo- batschewsky's results, namely : TT (x) _ ^ (5) tan-^' = a. "^ This is the same as the equation (3) of Taurinus, under another form. The constant a which appears in (5) is indeterminate. It represents the constant ratio of the arcs cut off two Coaxal I It can be proved that the formulae of Non-Euclidean Plane Trigonometry can be obtained without the •introduction of the Ho)-ospke7e. The only result required is the relation between the arcs cut off two Horocydes by two of their axes (cf. p. 90). Cf. H. LlEBMANN, Elementare Ableitutrg der nichteuklidiscken Trigonometrie. Ber. d. kòn. Sach. Ges. d. Wiss., Math. Phys. Klasse, (1907). QO IV. The Founders of Non-Euclidean Geometry (Contd.). Horocycles by a pair of axes, when the distance between these arcs is the unit of length. [Fig- 45-] If we choose, with Lobatschew- SKY, a convenient unit, we are able to take a equal to e, the base of Natural Logarithms. If we wish, on the other hand, to bring Lo- '^ '' ' batschewsky's results into accord with the log.-spherical geometry of Taurinus, or the Non-Eu- clidean geometry of Gauss, we take Then (5) becomes x .r U(x) ~~T (5) tan^-— = ir 2 , which is the same as (6) cosh 7- = -. — TT-— ,• A sin 1 1 (x) This result at once transforms Lobatschewsky's equa- tion (4) into the equation (i) of Taurinus. It follows that: T/ie log.-spherical geometry of Taurinus is identical with the imagiftary geometry \_pa?igeometry] of Lobatschewsky. § 42. We add the most remarkable of the results which Lobatschewsky deduces from his formulae: (a) In the case of triangles whose sides are very small [infinitesimal] we can use the ordinary trigonometrical for- mulae as the formulae ol Imaginary Trigonometry, infinitesi- mals of a higher order being neglected \ I Conversely, the assumption that the Euclidean Geometry holds for the infinitesimally small can be taken as the starting point for the development of Non-Euclidean Geometry. It is one of the most interesting discoveries from the recent examination of Lobatschewsky's Trigonometry. gj (b) If for a, b, c are substituted ia^ ib, ic, the formulae of Imaginary Trigonometry are transformed into those of or- dinary Spherical Trigonometry.^ (c) If we introduce a system of coordinates in two and three dimensions similar to the ordinary Cartesian coordinates, we can find the lengths of curves, the areas of surface^- and the volumes of solids by the methods of analytical geometry. § 43. How was LoBATSCHEWSKY led to investigate the theory of parallels and to discover the Imaginary Geometry? We have already remarked that Bartels, Lobatschew- sky's teacher at Kasan, was a friend of Gauss [p. 84]. If we now add that he and Gauss were at Brunswick together dur- ing the two years which preceded his call to Kasan [1807], and that later he kept up a correspondence with Gauss, the hypothesis at once presents itself that they were not without their influence upon Lobatschewsky's work. We have also seen that before 1807 Gauss had attempted to solve the problem of parallels, and that his efforts up till that date had not borne other fruit than the hope of overcom- ing the obstacles to which his researches had led him. Thus anything that Bartels could have learned from Gauss before 1807 would be of a negative character. As regards Gauss's Gauss's MSS. that the Princeps mathematicorum had already fol- lowed this path. Cf. Gauss, Werke, Bd. VIII, p. 255—264. Both the works of Flye St. Marie, [Thhrie analytlque sur la thèorie des parallèies, (Paris, 1871)], and of KILLING [Die 7iichteuklid- ischen Raiwiformen in analytischer Behandlung, (Leipzig, 1881)], are founded upon this principle. In addition, the formulae of trigono- metry have been obtained in a simple manner by the application of the same principle, and the use of a few fundamental ideas, by M. Simon. [Cf. M. Simon, Die Trigonometrie in der absoluten Geotnetrie, Crelle's Journal, Bd. 109, p. 187 — 198 (1892)]. I This result justifies the method followed by Taurinus in the construction of his log. -spherical geometry. Q2 IV. The Founders of Non-Euclidean Geometry (Contd.). later views, it appears quite certain that Bartels had no news of them^ so that we can be sure that Lobatschewsky created his geometry quite independently of any influence from Gauss.* Other influences might be mentioned: e. g., besides Legendre, the works of Saccheri and Lambert, which the Russian geo- meter might have known, either directly or through Klugel and MoNTUCLA. But we can come to no definite decision upon this question^. In any case, the failure of the demon- strations of his predecessors, or the uselessness of his own earlier researches [1815 — 17], induced Lobatschewsky, as formerly Gauss, to believe that the difficulties which had to be overcome were due to other causes than those to which until then they had been attributed. Lobatschewsky expresses this .thought clearly in the Nau Principles of Geometry of 1825, where he says: 'The fruitlessness of the attempts made, since Euclid's time, for the space of 2000 years, aroused in me the suspicion that the truth, which it was desired to prove, was not contained in the data themselves; that to establish it the aid of experi- ment would be needed, for example, of astronomical obser- vations, as in the case of other laws of nature. When I had finally convinced myself of the justice of my conjecture and beheved that I had completely solved this difficult question, "^I wrote, in 1826, a memoir on this subject {Exposition suc- cincte des principes de la Géomctrie\.' ^ The words of Lobatschewsky afford evidence of a phil- osophical conception of space, opposed to that of Kant, which was then generally accepted. The Kantian doctrine considered space as a subjective intuition, a necessary presup- position of every experience. Lobatschewsky's doctrine was 1 Cf. the work of F. Engel, quoted on p. 84. Zweiter Teil; Lobatschefskijs Leben unci Schriftett. Cap. VI, p. 373 — 383. 2 Cf. Segre's work, quoted on p. 44. 3 Cf. p. 67 of Engel's work named above. The Pangeometry. q-ì rather allied to sensualism and the current empiricism, and compelled geometry to take its place again among the ex- perimental sciences.^ §44. It now remains to describe the relation of Lobat- scHE\vsK"S''s Paiigeo7netry to the debated question of the Eu- clidean Postulate. This discussion, as we have seen, aimed at constructing the Theory of Parallels with the help of the first 28 propositions of Euclid. So far as regards this problem, Lobatschewsky, having defined parallelism, assigns to it the distinguishing features of reciprocity and transitivity. The property of equidistance then presents itself to Lobatschewsky in its true light. Far from being indissolubly bound up with the first 28 proposit- ions of Euclid, it contains an element entirely new. The truth of this statement follows directly from the ex- istence of the Pangeometry [a logical deductive science founded upon the said 28 propositions and on the negation of the Fifth Postulate], in which parallels are not equidistatit, but are asymptotic. Further, we can be sure that the Pangeometry is a science in which the results follow logically one from the other, i. e., are free from internal contradictions. To prove this we need only consider, with Lobatschewsky, the analyt- ical form in which it can be expressed. This point is put by Lobatschewsky toward the end of his work in the following way: 'Now that we have shown, in what precedes, the way in which the lengths of curves, and the surfaces and volumes of solids can be calculated, we are able to assert that the Pan- geometry is a complete system of geometry. A single glance I Cf. The discourse on Lobatschewsky by A. Vasiliev, (Kasan, 1893). German translation by Engel in Schlomilch's Zeit- schrift, Bd. XI, p. 205 — 244 (1895). 'English translation by Halsted, (Austin, Texas, 1 895). 94 IV. The Founders of Non-Euclidean Geometry (Contd. at the equations which express the relations existing between the sides and angles of plane triangles, is sufficient to show that, setting out from them, Pangeometry becomes a branch of analysis, including and extending the analytical methods of ordinary geometry. We could begin the exposition of Pan- geometry with these equations. We could then attempt to substitute for these equations others which would express the relations between the sides and angles of every plane triangle. However, in this last case, it would be necessary to show that these new equations were in accord with the fundamental notions of geometry. The standard equations, having been deduced from these fundamental notions, must necessarily be in accord with them, and all the equations which we would substitute for them, if they cannot be deduced from the equa- tions, would lead to results contradicting these notions. Our equations are, therefore, the foundation of the most general geometry, since they do not depend on the assumption that the sum of the angles of a plane triangle is equal to two right angles.' ' § 45. To obtain fuller knowledge of the nature of the constant k contained im- plicity in Lobatschewsky's formulae, and exphcitly in those of Taurinus, we must apply the new trigonometry to some actual case. To this end Lobatschewsky used a triangle ABC, in which the side BC {a) is equal to the radius of the earth's orbit, and ^ is a fixed star, whose direction is perpendicular to BC (Fig. 46). Denote hy 2 p the maximum parallax of the star A. Then we have 1 Cf. the Italian translation of the Pangéomélrie, Giornale di Mat,, T. V. p. 334; or p. 75 of the German translation referred to on p. 86. Astronomy and Lobatschewsky's Theory. ge Therefore I /it \ I — tan/ tan -T](a)> tan (-- -/j = .^^^^ a But tan l- T\ (a) = e J [cf. p. 90], a Therefore .'^ the attempt. In this way the theory of parallels formed the favourite occupation of the young mathematician, during his . course [1817 — 22] in the Royal College for Engineers at Vienna. I For the contents of this section, cf. Lobatschewsky, On the Principles of Geometry, See p. 22 — 24 of Engel's work named on p. 84. Also Engel's remarks on p. 24S — 252 of the same work. Johann Boiyai's Earlier Work. 97 At this time Johann was an intimate friend of Carl SzAsz [1798-185 3] and the seeds of some of the ideas, which led BoLYAi to create the Absolute Science of Space, were sown in the conversations of the two eager students. It appears that to Szasz is due the distinct idea of con- sidering the parallel through £ to the line AM as the limit- ing position of a secant BC turning in a definite direction about JB; that is, the idea of consid- ering BC as parallel to AM, when BC, in the language of Szasz, de- taches itself ^springs away) from AM (Fig. 47). To this parallel Bolyai gave the name of asymptotic parallel or asymptote, [cf Saccheri]. From the conversations of the two friends were also derived the conception of the line equidistant from a straight line, and the other most important idea of the Paracycle {lÌ7ìiiting curve or horo- ry*;/.? of Lobatschewsky). Further they recognised that the proof of Axiom XI would be obtained if it could be shown that the Paracycle is a straight line. When Szasz left Vienna in the beginning of 1821 to undertake the teaching of Law at the College of Nagy-Enyed (Hungary), Johann remained to carry on his speculations alone. Up till 1820 he was filled with the idea of finding a proof of Axiom XI, following a path similar to that of Saccheri and Lambert. Indeed his correspondence with his father shows that he thought he had been successful in his aim. The recognition of the mistakes he had made was the cause of Johann's decisive step towards his future discoveries, since he realised 'that one must do no violence to nature, nor model it in conformity to any blindly formed chimsera; 7 q3 IV. The Founders of Non-Euclidean Geometry (^Contd.) that, on the other hand, one must reguard nature reasonably and naturally, as one would the truth, and be contented only with a representation of it which errs to the smallest possible extent.' Johann Bolyai, then, set himself to construct an abso- hite theory of space, following the classical methods of the Greeks: that is, keeping the deductive method^ but without deciding a priori on the truth or error of the FifthPostulate. § 47. As early as 1823 Bolyai had grasped the real nature of his problem. His later additions only concerned the material and its formal expression. At that date he had discovered the formula: a e ■^ = tan - - , connecting the angle of parallelism with the line to which it corresponds [cf. Lobatschewsky, p. 89]. This equation is the key to all Non- Euclidean Trigonometry. To illustrate the discoveries which Johann made in this period, we quote the following extract from a letter which he wrote from Temesvar to his father, on Nov. 3, 1823: 'I have now resolved to pub- lish a work on the theory of parallels, as soon as I shall have put the material in order, and my circumstances allow it. I have not yet completed this work, but the road which I have followed has made it almost certain that the goal will be attained, if that is at all possible: the goal is not yet reached, but I have made such wonderful discoveries that I have been almost overwhelmed by them, and it would be the cause of continual regret if they were lost. When you will see them, you too will recognize it. In the meantime I can say only this : / have created a ne^v universe from nothing. All that I have sent iyou till now is but a house of cards compared to the tower. I am as fully persuaded that it will bring me honour, as if I had already completed the discovery.' J. Bolyai's Theory of Parallels. gg Wolfgang expressed the wish at once to add his son's theory to the Tentatnen since 'if you have really succeeded in the question, it is right that no time be lost in making it public, for two reasons: first, because ideas pass easily from one to another, who can anticipate its publication; and se- condly, there is some truth in this, that many things have an epoch, in which they are found at the same time in several places, just as the violets appear on every side in spring. Also every scientific struggle is just a serious war, in which I cannot say when peace will arrive. Thus we ought to conquer when we are able, since the advantage is always to the first comer.' Little did Wolfgang Bolyai think that his presentiment would correspond to an actual fact (that is, to the simulta- neous discovery of Non-Euclidean Geometry by the work of Gauss, Taurinus, and Lobatschewsky). In 1825 Johann sent an abstract of his work, among others, to his father and to J.Walter von Eckwehr [1789 — 1857], his old Professor at the Military School. Also in 1829 he sent his manuscript to his father. Wolfgang was not completely satisfied with it, chiefly because he could not see why an indeterminate constant should enter into Johann's formulae. None the less father and son were agreed in pubhshing the new theory of space as an appendix to the first volume of the Tentamen: — The title of Johann Bolyai's work is as follows. Appendix scientiam spatii absolute veram exhibens: a ventate aut falsitate Axiomatis XI. Euclidei, a priori haud unquam decidenda, independentem : adjecta ad casum falsitaiis quadratura circuii geometrica.^ I A reprint — Edition de Luxe — was issued by the Hungarian Academy of Sciences, on the occasion of the first centenary of the birth of the author (Budapest, 1902). See also the English 7* lOO IV. The Founders of Non-Euclidean Geometry (Contd.). The Appendix was sent for the first time [June, 1831] to Gauss, but did not reach its destination; and a second time, in January, 1832. Seven weeks later (March 6, 1832), Gauss replied to Wolfgang thus: "If I commenced by saying that I am unable to praise this work (by Johann), you would certainly be surprised for a moment. But I cannot say otherwise. To praise it, would be to praise myself. Indeed the whole contents of the work, the path taken by your son, the results to which he is led, coincide almost entirely with my meditations, which have occupied my mind partly for the last thirty or thirty- five years. So I remained quite stupefied. So far as my own work is concerned, of which up till now 1 have put little on paper, my intention was not to let it be published during my lifetime. Indeed the majority of people have not clear ideas upon the questions of which we are speaking, and I have found very few people who could regard with any special interest what I communicated to them on this subject. To be able to take such an interest it is first of all necessary to have devoted careful thought to the real nature of what is wanted and upon this matter almost all are most uncertain. On the other hand it was my idea to write down all this later so that at least it should not perish with me. It is therefore a pleasant surprise for me that I am spared this trouble, and I am very glad that it is just the son of my old friend, who takes the precedence of me in such a remarkable manner." Wolfgang communicated this letter to his son, adding: "Gauss's answer with regard to your work is very satis- translation by Halsted, T/ie Science Absolute of Space, (Austin, Texas^ 1896). An Italian translation by G. B. Battagmni appeared in the Giornale di Mat., T. VI, p- 97 — 115 (1868). Also a French trans- lation by HOUEL, in Mém. de la Soc des Se. de Bordeaux, T. V- p. 189 — 248 (1867). Cf. also Frischauf, Absoluie Geometrie nach Johann Bolyai, (Leipzig, Teubner, 1872). Gauss's Praise of Eolyai's Work. lOI factory and redounds to the honour of our country and of our nation." Altogether different was the effect Gauss's letter pro- duced on Johann. He was both unable and unwilling to convince himself that others, earlier than and independent of him, had arrived at the No7i- Euclidean Geometry. Further he suspected that his father had communicated his discoveries to Gauss before sending him the Appendix and that the latter wished to claim for himself the priority of the discovery. And although later he had to let himself be convinced that such a suspicion was unfounded, Johann always regarded the "Prince of Geometers" with an unjustifiable aversion. * § 48. We now give a short description of the most important results contained in Johann Bolyai's work: a) The definition of parallels and their properties in- dependent of the Euchdean postulate. b) The circle and sphere of infinite radius. The geo- metry on the sphere of infinite radius is identical with ordi- nary plane geometry. c) Spherical Trigonometry is independent of Euclid's Postulate. Direct demonstration of the formulae. d) Plane Trigonometry in Non-Euclidean Geometry. Applications to the calculation of areas and volumes. e) Problems which can be solved by elementary me- thods. Squaring the circle, on the hypothesis that the Fifth Postulate is false. While LoBATSCHEWSKY has given the Imaginary Geo- metry a fuller development especially on its analytical side, I For the contents of this and the preceding article seeSxAcKEL, Die Entdeckung der 7iichteuklidischeii Geometrie durch jfohatin Bolyai. Math. u. Naturw. Berichte aus Ungarn. Bd. XVII, [1901]. Also StAckel u. Engel. Gauss, die beiden Bolyai und die nichteuklidische Geometrie. Math. Ann. Bd. XLIX, p. 149 — 167 [1897]. Bull. So. Math. (2) T. XXI, pp. 206—228 [1897]. I02 IV. The Founders of Non-Euclidean Geometry (Contd.). BoLYAi entered more fully into the question of the depen- dence or independence of the theorems of geometry upon Euclid's Postulate. Also while Lobatschewky chiefly sought to construct a system of geometry on the negation of the said postulate, Johann Bolyai brought to light the pro- positions and constructions in ordinary geometry which are independent of it. Such propositions, which he calls ab- solutely true, pertain to the absolute science of space. We could find the propositions of this science by comparing EucUd's Geometry with that of Lobatschewsky. Whatever they have in common, e. g. the formulae of Spherical Trigon- ometry, pertains to the Absolute Geometry. Johann Bolyai, however, does not follow this path. He shows directly, that is independently of the Euchdean Postulate, that his propos- itions are absolutely true. § 49. One of BoLYAi's absolute theorems, remarkable for its simplicity and neatness, is the following: The sines of the angles of a rectilinear triangle are to one another as the circumferences of the circles whose radii are equal to the opposite sides. A A ^ b B' B Fig. 48. Let ABC be a triangle in which C is a right angle, and BB' the perpendicular through B to the plane of the triangle. Draw the parallels through A and C to BB' in the same sense. Then let the Horosphere be drawn through A (eventually the plane) cutting the lines AA\ BB' and CC, respectively, in the points A, M, and N. Bolyai's Theorem. - 10^ If we denote by a\ b\ c the sides of the rectangular triangle AAIN on the Horosphere, it follows from what has been said above [cf. § 48 (b)] that sin AMN = — . But two arcs of Horocycles on the Horosphere are pro- portional to the circumferences of the circles which have these arcs for their (horocyclic) radii. If we denote by circumf. x the circumference of the circle whose (horocyclic) radius is x', we can write: A T^jT^T circumf. b' Sin AMN = -. J—,. circumf. c On the other hand, the circle traced on the Horosphere with horocyclic radius of length x\ can be regarded as the circumference of an odinary circle whose radius (rectilinear) is half of the chord of the arc 2 x' of the Horocycle. Denoting by O •^ the circumference of the circle whose (rectilinear) radius is x, and observing that the angles ABC and AMN are equal, the preceding equation taken from sin ABC = -^. From the property of the right angled triangle ABC expressed by this equation, we can deduce Bolyai's theorem enunciated above, just as from the Euclidean equation sin ABC = — c we can deduce that the sines of the angles of a triangle are proportional to the opposite sides. {Appendix § 25.] Bolyai's Theorem may be put shortly thus: (i) O^ '• O^ '• O^ = sin ^ : sin B : sin C. If we wish to discuss the geometrical systems separately we will have (i) In the case of the Euclidean Hypothesis, O-^ = 2 TTAT. I04 ^^' ^^^ Founders of Non-Euclidean Geometry (Contd.). (!') 2 11;^ sinh -r* (i ") sinh — : sinh ^ : sinh -r == sin A : sin B : sin C. Thus, substituting in (i), we have a:b'.C'. == sin ^ : sin ^ : sin C. (ii) In the case of the Non-Eudidean Hypothesis, Then substituting in (i) we have b k ' T • " /• This last relation may be called the Sine Theorem of the Bolyai-Lobatschewsky Geometry. From the formula (i) Bolyai deduces, in much the same way as the usual relations are obtained from (i), the proportionality of the sines of the angles and the opposite sides in a spherical triangle. From this it follows that Spherical Trigonometry is independent of the Euclidean Postulate {Appendix 8 26]. This fact makes the importance of Bolyai's Theorem still clearer. § 50. The following construction for a parallel through the point Z> to the straight line ^iV belongs also to the Ab- solute Geometry [Appendix % 34]. Draw the perpendiculars DB and AE to AN [Fig. 49]. fi D Fig. 49. Also the perpendicular DE to the line AE. The angle EDB of the quadrilateral ABDE, in which three angles Bolyai's Parallel Construction. IO5 are right angles, is a right angle or an acute angle, according as ED is equal to or greater than AB. With centre A describe a circle whose radius is equal to ED. It will intersect DB at a point (9, coincident with B or situated between B and D. The angle which the line AO juakes with DB is the angle of parallelisin corresponding to the segmefit BD.^ [Appendix §27.] Therefore a parallel to AN through D can be con- structed by drawing the line DM so that and applying it to the two rightangled triangles ADE and ADB, this ratio can be expressed as O AB : O ED = sin u : sin v. From this it is clear that the ratio sin k : sin v does not vary if the line AE changes its position, remaining always perpendicular to AB, while d remains fixed. In particular, if the foot of AE tends to infinity along AJV, it tends to TT {d) and z/ to a right angle. Consequently, QAB ■.QED = sm T\{d):\. On the other hand in the right-angled triangle AOB, we have the equation QAB:QiAO = sin AOB : i. This, with the preceding equation, is sufficient to establish the equality of the angles TT {d) und AOB. 2 Cf. Appendix III to this volume. I06 IV. The Founders of Non-Euclidean Geometry (Contd.). § 51. The most interesting of the Non-Euclidean con- structions given by Bolvai is that for the squaring of the circle. Without keeping strictly to Bolyai's method, we shall explain the principal features of his construction. But we first insert the converse of the construction of § 50, which is necessary for our purpose. On the Non-Euclidea7i Hypothesis to draw the segment which corresponds to a given {acute) angle of parallelism. Assuming that the theorem, that the three perpendiculars from the angular points of a triangle on the opposite sides intersect eventually, is also true in the Geometry of Bolyai- LoBATSCHEWSKY, on the line AB bounding the acute angle BAA' take a point B, such that the parallel BB' to AA through B makes an acute angle {ABB') with AB. [Fig. 50.] Fig. 50. The two rays AA , BB', and the line AB may be regarded as the three sides of a triangle of which one angular point is Coo ) common to the two parallels AA, BB'. Then the perpendiculars from A, B, to the opposite sides, meet in he point O inside the triangle, and the perpendicular from Coo to AB also passes through O. Thus, if the perpendicular OL is drawn from O to AB, the segment AL will have been found which corresponds to the angle of parallelism BAA . Bolyai's Parallel Construction (Contd.). 107 As a particular case the angle BAA' could be 45°. Then AL would be Schweikart's Constant [cf. p. 76]. We note that the problem which we have just solved could be enunciated thus: To draw a line which shall be parallel to one of the lines bounding an acute angle and perpendicular to the other. "^ § 52. We now show how the preceding result is used to construct a square equal in area to the maximum triangle. The area of a triangle being k'{M—KA — ^B—^C), the maximum triangle, i. e. that for which the three angular points are at infinity, will have for area A = k^ TT. To find the angle oi of a square whose area is k'^n, we need only remember (Lambert, p. 46) that the area of a polygon^ as well as of a triangle, is proportional to its defect. Thus we have the equation k^ 11 = k'^ (2 TT — 4 uj), from which it follows that UU = " IT 4 45^ Assuming this^ let us consider the right-angled triangle 0AM (Fig, 51), which is the eighth part of the required square. Putting OM = :)' the following relation must hold between a, b' and c, cosh -J cosh -J = cosh -j. Finally if we take b' as side, and / as hypotenuse of a right- angled triangle, the other side of this triangle, by formula (i) of p. 7 9, is determined by the equation cosh -,~ cosh -,- = cosh -r-. Then comparing these two questions, we obtain a = a. Constructing a in this way, we can immediately find the square whose area is equal to that of the maximum triangle. § 53. To construct a circle whose area shall be equal to that of this square, that is, to the area of the maximum triangle, we must transform the expression for the area of a circle of radius r 2 TT /C'^ ( cosh -, I ) , given on p. 81, by the introduction of the angle of parallelism TT( — j, corresponding to half the radius. Then we have' for the area of this circle On the other hand if the two parallels AA and BB' are drawn from the ends of the segment AB^ making equal angles with AB^ we have I Using the result tan TT {x) ^ -- xlk The Square of area it/t*. 109 <^ A AB == <: B'BA = n (^), where AB == r [Fig. 52]. Now draw ^C, perpendicular to BB\ and ^Z> perpen- dicular to AC; also put <^ CAB = a, <^ Z)^^' = z. Then we have cot TT ( ~ j cot a + I tan z = cot f TT r ' j — cc j cot a — cot Ki) It is easy to eliminate a from this last result by means of the trigonometrical formulae for the triangle ABC and so obtain 2 tan z = — ^ .a„ n (^) Substituting this in the expression found for the area of the circle, we obtain for that area IT k^ tan^ z. This formula, proved in an- D other way by Bolyai {Appendix % 43], allows us to associate a definite angle z with every circle. If 3 were equal to 45", then we would have for the area of the correspond- ing circle. Fig. 52. 2 C0t2 I Indeed, in the rightangled triangle ABC, we have cot TT ( — ) = cosh _ k TT ( "^ ) + I' we deduce, first, that cosh — From this, since cosh -^ = 2 sinh2 L. 4- i k k 2k I IO IV. The Founders of Non-Euclidean Geometry (Contd.). That is : the a7'ea of the circle, for which the angle z is 4j°, is equal to the area of the maximtim triangle, and thus to that of the square of § 52. If z = ^AAD (Fig. 51) is given, we can find r by the following construction: (i) Draw the line AC perpendicular to AD. (ii) Draw BB' parallel to AA and perpendicular to ^C7(S5i). (iii) Draw the bisector of the strip between AA and BB\ [By the theorem on the concurrency of the bisectors of the angles of a triangle with an infinite vertex.] (iv) Draw the perpendicular AB to this bisector. The segment AB bounded by AA and BB' is the required radius r. § 54. The problem of constructing a polygon equal to a circle of area tc k'^ tan^ z is, as Bolyai remarked, closely allied with the numerical value of tan z. It is resolvable for every integral value of tan- z, and for every fractional value, provided that the denominator of the fraction, re- duced to its lowest terms, is included in the form assigned by Gauss for the construction of regular polygons [Appendix §43]- The possibility of constructing a square equal to a circle leads Johann to the conclusion "habeturque aut Axi- oma XI Euclidis verutn, aut quadratura circuii geometrica; cot TT I — ) cot a = 2 cot2 TT ( — ) +1, and next that cot a — cot TT(T)=(^+tan3n(;))cotn(^). These equations allow the expression for tan z to be written down in the required form. The Quadrature of the Circle. Ill etsi hucusque indecisum manserit, quodnam ex his duobus revera locum habeat." This dilemma seemed to him at that time [1831] im- possible of solution, since he closed his work with these words: "Superesset denique (ut res omni numero absolvatur), impossibilitatem (absque suppositione aliqua) decidenda, num X (the Euclidean system) aut aliquod (et quodam) S (the Non-Euclidean system) sit, demonstrare : quod tamen occasi- oni magis idoneae reservatur." Johann, however, never published any demonstration of this kind. § 55. After 183 1 BoLYAi continued his labours at his geometry, and in particular at the following problems: 1. The connection between Spherical Trigonometry and Non-Euclidean Trigonometry. 2. Can one prove rigorously that Euclid's Axiom is not a consequence of what precedes it ? 3. The volume of a tetrahedron in Non-Euclidean geo- metry. As regards the first of these problems, beyond estab- lishing the analytical relation connecting the two trigono- metries [cf. LoBATSCHEWSKY, p. 90], BoLYAi recognized that in the Non-Euclidean hypothesis there exist three classes of Uniform Surfaces^ on which the Non-Euclidean trigono- metry, the ordinary trigonometry, and spherical trigonometry respectively hold. To the first class belong planes and hyper- spheres [surfaces equidistant from a plane]; to the second, the paraspheres [Lobatschewsky's Horospheres] ; to the third, spheres. The paraspheres are the limiting case when we pass from the hyperspherical surfaces to the spherical. This passage is shown analytically by making a I BoLYAl seems to indicate by this name the surfaces which behave as planes, with respect to displacement upon themselves. I [2 IV, The Founders of Non-Euclidean Geometry (Contd.). certain parameter, which appears in the formulae, vary con- tinuously from the real domain to the purely imaginary through infinity [cf Taurinus, p. 82]. As to the second problem, that regarding the impos- sibility of demonstrating Axiom XT, Bolyai neither succeeded in solving it, nor in forming any definite opinion upon it. For some time he believed that we could not, in any way, decide which was true, the Euclidean hypothesis or the Non-Euclidean. Like Lobatschewsky, he relied upon the analytical possibility of the new trigonometry. Then we find Johann returning again to the old ideas, and attempting a new demonstration of Axiom XI. In this attempt he applies the Non-Euclidean formulae to a system of five coplanar points. There must necessarily be some relation between the distance of these points. Owing to a mistake in his calculations Johann did not find this relation, and for some time he believed that he had proved, in this way, the false- hood of the Non-Euclidean hypothesis and the absolute truth of Axiom XI .^ However he discovered his mistake later, but he did not carry out further investigations in this direction, as the method, when applied to six or more points, would have in- volved too comphcated calculations. The third of the problems mentioned above, that re- garding the tetrahedron, is of a purely geometrical nature. BoLYAi's solutions have been recently discovered and pub- I The title of the paper which contains Johann's demon- stration is as follows: "Beweis des bis mm auf der Erde itnmer nock zwei/elkafi gewesenen, weltberuhmten ujid, ah der gesamtnten Raum- und Bewegungslehre zu Grunae dienend, auch in der That allerh'òchst7uichtigsten 11. Eudid'schen Axioms von J. Bolyai von Bolya, k. k.Geiiie-Stabs/iauptmann in Pension. Cf. StaCKEL's paper: Untcr- suchungen aiis der Absoluten Geotnetrie aus yohann Bolyais N'achlafi. Math. u. Naturw. Berichte aus Ungarn. Ed. XVIII, p. 2S0— 307 (1902). We are indebted to this paper for this section S 55- Bolyai^s Later Work. 1 1 5 lished by Stackel [cf. p. 112 note i]. Lobatschewsky had been often occupied with the same problem from 1829', and Gauss proposed it to Johann in his letter quoted on p. 100. Finally we add that J. Bolyai heard of Lobatschewsky's Geometrisc/ie Untersuchimgen in 1848: that he made them the object of critical study ^: and that he set himself to com- pose an important work on the reform of the Principles of Mathematics with the hope of prevailing over the Russian. He had planned this work at the time of the publication of the Appendix, but he never succeeded in bringing it to a conclusion.'' The Absolute Trigonometry. § 56. Although the formulae of Non-Euclidean trigono- metry contain the ordinary relations between the sides and angles of a triangle as a limiting case [cf. p. 80], yet they do not form a part of what Johann Bolyai called Absolute Geo- metry. Indeed the formulse do not apply at once to the two classes of geometry, and they were deduced on the suppos- ition of the validity of the Hypothesis of the Acute Angle. Equations directly applicable both to the Euclidean case and to the Non-Euclidean case were met by us in § 49 and they make up Bolyai's Theorem. They are tliree in number, only two of them being independent. Thus they furnish a first set of formulae of Absolute Trigonometry. » Cf. p. 53 et seq., of the work quoted on p. 84. Also Liebmann's translation, referred to in Note 2, p. 85. 2 Cf. P. Stackel und J. KurschA'k: Johann Bolyals Be- nierkungen iiber JV. Lobaischefskijs Geofneh-iscke Untermchungen zur Theorie der Parallellinicn, Math. u. Naturw. Berichte aus Ungarn, Bd. XVIII, p. 250—279 (1902). 3 Cf. P. StAckel: Johann Bolyais Raumlehre, Math. u. Naturw. Berichte aus Ungarn, Bd. XIX (1903). 8 jj_^ IV. The Founders of Non-Euclidean Geometry (Contd.). Other formulae of Absolute Trigonometry were given in 1870 by the Belgian geometer, De Tilly, in ins Etudes de Mécaniqiie Abstraite. ^ The formulae given by De Tilly refer to rectilinear tri- angles, and were deduced by means of kinematical con- siderationS; requiring only those properties of a bounded region of a plane area, which are independent of the value of the sum of the angles of a triangle. In addition to the function 0-'*^> which we have already met in Bolyai's formulae, -those of De Tilly contain another function Ex defined in the following way: Let r be a straight line and / the equidistant curve, distant x from r. Since the arcs of / are proportional to their projections on r, it is clear that the ratio between a (recti- fied) arc of / and its projection does not depend on the length of the arc, but only on its distance x from r. De Tilly's function Ex is the function which expresses this ratio. On this understanding, the Formulae of Absolute Trigon- ometry for the right-angled triangle ABC 2^0. as follows: (i) \C)a = Qjc sin A [0'^ = O^sin^' (2) fcos A = Ea. sin B [cos B = Eb. sin A (3) Ec = Ea. Eb. The set (i) is equivalent to Bolyai's Theorem for the Right- Angled Triangle. All the formulae of Absolute Trigono- metry could be derived by suitable com- bination of these three sets. In particular, for the right-angled triangle, we obtain the following equation: — I Mémoires couronnés et autres Mémoires, Acad, royale de Belgique. T. XXI (1870). Cf. also the work by the same author: Essai sur les principes Jmtdamentaux dc la p-rométrie et de la Mccanique, Mém. de la Soc. des Se. de Bordeaux. T. Ili (cah. I) (1878). The Absolute Trigonometry. II c O^a {Ea + Eb. Ec) + Q'-^- ^Eb + Ec. Ea) = O'^ (^^ + Ea. Eb). This can be regarded as equivalent to the Theorem of Pythagoras in the Absolute Geometry.^ § 57. Let us now see how we can deduce the results of the Euclidean and Non-Euclidean geometries from the equations of the preceding article. Euclidean Case. The Equidistant Curve (/) is a straight line [that is, Ex = 1], and the perimeters of circles are proportional to their radii. Thus the equations (i) become (i') {a = c sin A \b = c sin B. The equations (2) give (2') cos A = sin B, cos B = sin A. Therefore A A^ B = 90°. Finally the equation (3) reduces to an identity. The equations (i') and (2') include the whole of ordin- ary trigonometry. Non- Euclidean Case. Combining the equations (i) and (2) we obtain E^a—l E2b—\ If we now apply the first of equations (2) to a right- angled triangle whose vertex A goes oft" to infinity, so that the angle A tends to zero, we shall have Lt cos A = Lt {Ea. sin B). But Ea is independent of A; also the angle B, in the limit, becomes the angle of parallelism corresponding to a, i. e. n {a). ^ Cf. R. BoNOLV, La trigonometria assoluta secondo Giovann: Bolyai. Rend. Istituto Lombardo (2). T. XXXVIII (1905). 8* Il6 IV. The Founders of Non-Euclidean Geometry (Contd.). Therefore we have sin n (a) A similar result holds for Eb. Substituting these in equation (5) we obtain cot2 TT {a) cot2 TT {b) ' from which cot IT {a) cot jr (^} This result, with the expression for Ex, allows us at once to obtain from the equations (i), (2), (3), the formulae of the Trigonometry of Bolyai-Lobatschewsky: fcot TT {a) = cot TT {c) sin A ^^ ' jcot n {b) = cot n {c) sin B, , „^ fsin A = cos B sin TT {b) I 2 ) -I Ì sin j9 = cos A sin TT («;), (3") sin TT {c) = sin TT {a) sin TT {b). These relations bet\veen the elements of every right- angled triangle were given in this form by Lobatschewsky/ If we wish to introduce direct functions of the sides, instead of the angles of parallelism TT (a), TT {b) and TT (^), it is sufficient to remember [p. 90] that tan —^ = e "'*. We can thus express the circular functions of TT {x) in terms of the hyperbolic functions of x. In this way the pre- ceding equations are replaced by the following relations: (i"'_) sinh -7- = sinh -j sin A k k b J sinh -r = sinh -7- sin B, I Cf. e. g., The Geometrische Untersuchungen of LOBATSCHEWSKY referred to on p. 86. Absolute Trigonometry and Spherical Trigonometry. 117 (2"') COS A = sin B cosh -r cos B = sin A cosh ^r. and (•?'") cosh -,- = cosh -y- cos /i cosh -7-. § 58. The following remark upon Absolute Trigono- metry is most important: // we regard the elements in its formulae as elements of a spherical triangle, we obtain a system of equations which hold also for Spherical Triangles. This property of Absolute Trigonometry is due to the fact, already noticed on p. 114, that it was obtained by the aid of equations which hold only for a limited region of the plane. Further these do not depend on the hypothesis of the angles of a triangle, so that they are valid also on the sphere. If it is desired to obtain the result directly, it is only necessary to note the following facts: — (i) In Spherical Trigonometry the circumferences of circles are proportional to the sines of their (spherical) radii, so that the first formula for right-angled spherical triangles sin ). Relative to this fundamental set, which we shall denote by \a, B, C, D\, the following hypotheses are mutually equivalent, and equivalent also to that stated by Euclid in his Fifth Postulate: a) The internal angles, which two parallels make with a transversal on the same side, are supplementary [Ptolemy]. b) Two parallel straight lines are equidistant. c) If a straight line intersects one of two parallels, it also intersects the other (Proclus); 120 IV. The Founders of Non-Euclidean Geometry (Contd.). or, Two straight lines, which are parallel to a third, are parallel to each other; or again. Through a point outside a straight line there can be drawn one and only one parallel to that line. d) A triangle being given, another triangle can be con- structed similar to the given one and of any size whatever. [Wallis.] e) Through three points, not lying on a straight line, a sphere can always be drawn. [W. Bolyai.] f) Through a point between the Hnes bounding an angle a straight hne can always be drawn which will intersect these two lines. [Lorenz.] a) If the straight line r is perpendicular to the trans- versal AB and the straight line s cuts it at an acute angle? the perpendiculars from the points of s upon r are less than AB^ on the side in which AB makes an acute angle with s. [Nasìr-Eddìn.] P) The locus of the points which are equidistant from a straight line is a straight line. f ) The sum of the angles of a triangle is equal to two right angles. [Saccherl] Now let us suppose that we diminish the fundamental set of hypotheses, cutii?ig oiit the Archimedean Hypothesis. Then the propositions (a), (b), (c), (d), (e) and (f) are mutually equivalent, and also equivalent to the Fifth Postu- late of Euclid, with respect to the fundamental set |^, ^, C]. With regard to the propositions (a), (P), (t), while they are mutually equivalent with respect to the set \A, B, C| no one of them is equivalent to the Euclidean Postulate. This result brings out clearly the importance of the Postulate of Archi- medes. It is given in the memoir of Dehn' [19°°] to which I Cf. Note on p. 30. Hypotheses Equivalent to Euclid's Postulate. i2I reference has already been made. In that memoir it is sho^vn that the hypothesis (f) on the sum of the angles of a triangle is compatible not only with the ordinary elementary geo- metry, but also with a new geometry— necessarily Non-Archi- medean—where the Fifth Postulate does not hold, and in which an infinite number of lines pass through a point and do not intersect a given straight line. To this geometry the author gave the name of Semi-Euclidean Geometry. The Spread of Non-Euclidean Geometry. § 60. The works of Lobatschewsky and Bolyai did not receive on their publication the welcome which so many centuries of slow and continual preparation seemed to promise. However this ought not to surprise us. The history of scientific discovery teaches that every radical change in its separate departments does not suddenly alter the con- victions and the presuppositions upon which investigators and teachers have for a considerable time based the present- ation of their subjects. In our case the acceptance of the Non-Euclidean Geo- metry was delayed by special reasons, such as the difficulty of mastering Lobatschewsky's works, written as they were in Russian, the fact that the names of the two discoverers were new to the scientific world, and the Kantian conception of space which was then in the ascendant. Lobatschewsky's French and German writings helped to drive away the darkness in which the new theories were hidden in the first years; more than all availed the constant and indefatigable labors of certain geometers, whose names are now associated with the spread and triumph of Non- Euclidean Geometry. We would mention particularly: C. L. Gerling [1788— 1864], R. Baltzer [1818— 1887] and Fr. Schmidt [1827 — 1901], in Germany; J. Hoùel [1823 — 122 IV. The Founders of Non-Euclidean Geometry (Contd.). 1886], G. Battaglini [1826 — 1894], E. Beltrami [1835— 1900], and A. Forti, in France and Italy. § 61. From 181 6 Gerling kept up a correspondence upon parallels with Gauss % and in 181 9 he sent him Schweikart's memorandum on Astra lgeo?/ietrie [cf. p. 75]. Also he had heard from Gauss himself [1832], and in terms which could not help exciting his natural curiosity, of a kleine Schrift on Non-Euclidean Geometry written by a young Austrian officer, son of W. Bolyai.* The bibliograph- ical notes he received later from Gauss [1844] on the works of Lobatschewsky andBoLYAi^ induced Gerling to procure for himself the Geomdrischeii Uiitersuchtingen and the Appen- dix, and thus to rescue them from the oblivion into which they seemed plunged. § 62. The correspondence between Gauss and Schu- macher, published between i860 and iSós,"^ the numerous references to the works of LoBATSCHEWSKy and Bolyai, and the attempts of Legendre to introduce even into the elemen- tary text books a rigorous treatment of the theory of pa- rallels, led Baltzer, in the second edition of his Elemmte der 1 Cf. Gauss, Werke, Bd. VIII, p. 167—169. 2 Cf. Gauss's letter to Gerling (Gauss, Werke, Bd. VIII, p. 220). In this note Gauss says with reference to the contents of the Appendix: "worin ich alle meine eigenen Ideen taid Resultate wlederfiiìde mit g7-ofier Eleganz entwickelt." And of the author of the work : „Ich halte diesm jiingC7i Geonietei' v. Bolyai fib- eni Genie erster Grafie". 3 Cf. Gauss, IVerke, Bd. VIII, p. 234—238. 4 Briefwechsel ziuischen C, F. Gauss 7cnd H. C. Schuinacher, Bd. II, p. 268—431 Bd. V, p. 246 (Altona, 1860—1863). As to Gauss's opinions at this time, see also, Sartorius von Walters' HaUSEN, Gatifi zutn Geddr/Unis, p. 8o— 8l (Leipzig, 1S56). Cf. GAUSS, Werke, Bd. VIII, p. 267—268. The Spread of Non-Euclidean Geometry. 1 23 Mathemafik {1^6'j), to substitute, for the Euclidean definition of parallels one derived from the new conception of space. Following LoBATSCHEWSKY he placed the equation A-\-B + C = 180°, which characterises the Euclidean triangle, among the experimental results. To justify this innovation, Baltzer did not fail to insert a brief reference to the possi- bility of a more general geometry than the ordinary one, founded on the hypothesis of two parallels. He also gave suitable prominence to the names of its founders.^ At the same time he called the attention of Houel, whose interest in the question of elementary geometry was well known to scientific men, ^ to the Non-Euclidean geometry, and re- quested him to translate the Geometrischen Untersiichungen and the Appendix into French. § 63. The French translation of this little book by LoBATSCHEWSKY appeared in 1866 and was accompanied by some extracts from the correspondence between Gauss and Schumacher.^ That the views of Lobatschewsky, Bolyai, and Gauss were thus brought together was extremely fortunate, since the name of Gauss and his approval of the discoveries of the two geometers, then obscure and unknown, 1 Cf. Baltzer, Elemente der Mathematik, Bd. 11 (5. Auflage) p. 12 — 14 (Leipzig, 1878). Also T. 4, p. 5 — 7, of Cremona's trans- lation of that work (Genoa, 1867). 2 In 1863 HoiJEL had published his wellknown Essai d'une exposition 7-ationelIe des principes fondametitmcx de la Geometrie èli- 7!ientaire. Archiv d. Math. u. Physik, Bd. XL (1863). 3 Ména, de la Soc. des Sci. de Bordeaux, T. IV, p. 88 — 120 (1S66). This short work was also published separately under the title Etudes géométriques sur la théorie des parallèles par N. I. LoBAT- SCHEWSKY, Conseiller d'État de l'Empire de Russie et Professeur à rUniversité de Kasan: traduit de l'allemand par J. Houel, suivie d'un Extrait de la correspondance de Gauss et de Schumacher, (Paris, G. VU.LARS, 1866). 124 ^^' T^ti^ Founders of Non-Euclidean Geometry (Contd.). helped to bring credit and consideration to the new doctrines in the most efficacious and certain manner. The French translation of the Appendix appeared in 1867.' It was preceded by a Notice sur la vie et les travaux des deux viathématiciens hotigrois W. et J. Bolyai de Bolya, written by the architect Fr. Schmidt at the invitation of HoiJEL,^ and was supplemented by some remarks by W. Bol- yai, taken from Vol. I of the Tentameli and from a short analysis, also by Wolfgang, of the Principles of Arithmetic and Geometry.3 In the same year [1867] Schmidt's discoveries regard- ing the BoLYAis were published in the Archiv d. Math. u. Phys. Also in the following year A. Forti, who had already written a critical and historical memoir on Lobatschewsky,'* 1 Mém. de la Soc. des Se. de Bordeaux, T. V, p. 189 — 248. This short work was also published separately unter the title: La Science absolute de l' espace, indèpendante de la vérité on fausseti de l'Axiome XI d'Euclide {que l'on ne pourra jamais établiì- a priori); suivie de la quadrature géometrique du cercle, dans le cas de la fausseté de l'Axiome XI, par Jean Bolyai, Capitaine au Corps du genie dans l'armée autrichienne; Précède d'iene notice sur la vie it les travaux de W. et de J. Bolyai, par M. Fr. Schmidt, (Paris, G. VlLLARS, 1868). 2 Cf. P. StAckel, Franz Schmidt, Jahresber. d. Deutschen Math. Ver., Bd. XI, p. 141 — 146 (1902). 3 This little book of \V. BoLYAl's is usually referred to shortly by the first words of the title Kicrzer Grtmdriss. It was pub- lished at Maros-Visàrhely in 1851. ■* Intorno alla geometria itnmaginaria o non euclidiana. Consid- erazioni storico-critiche. Rivista Bolognese di scienze, lettere, arti e scuole, T. Il, p. 171 — 184 (1867). It was published separately as a pamphlet of 16 pages (Bologna, Fava e Garagnani, 1867). The same article, with some additions and the title, Studii geo- metrici sulla teorica delle parallele di N. J. Lobatschewky, appeared in the politicai journal La Provincia di Pisa, Anno III, Nr. 25, 27, Hoùel and Schmidt. 1 25 made the name and the works of the two now celebrated Hungarian geometers known to the Italians/ To the credit of Hoùel there should also be mentioned his interest in the manuscripts of Johann Bolyai, then [1867] preserved, in terms of Wolfgang's will, in the library of the Reformed College of Maros-Vàsàrhely. By the help of Prince B. Boncampagni [182 i — 1894], who in his turn in- terested the Hungarian Minister of Education, Baron Eòtvòs, he succeeded in having them placed [1869] in the Hungarian Academy of Science at Budapest.^ In this way they became more accessible and were the subject of painstaking and careful research, first by Schmidt and recently by Stackel. In addition Houel did not fail in his efforts, on every available opportunity, to secure a lasting triumph for the Non- Euclidean Geometry. If we simply mention his Essai cri- tique sur les principes fondameìiteaux de la geometrie:'^ his ar- ticle, Sur l' impossibilité de démontrer par tene construction plane le postulatum d'Euclide; "* the Notices sur la vie et les iravaux de N. J. Lobatschewsky; 5 and finally his translations of various writings upon Non-Euclidean Geometry into French,^ 29, 30 (1867); and part of it was reprinted under the original title (Pisa, Nistri, 1867). * Cf. Iniorito alia vita ed agli sa-itti di Wolfgang e Giovanni Bolyai di Bolya, rnatemalici ungheresi. Boll, di Bibliografia e di Storia delle Scienze Mat. e Fisiche. T. I, p. 277—299 (1869), Many historical and bibliographical notes were added to this article of Forti's by B. Boncompagni. 2 Cf. Stackel's article on Franz Schmidt referred to above. 3 I. Ed., G. ViLLARS, Paris, 1867; 2 Ed., 1883 (cf. Note 3 p. 52). 4 Giornale di Mat. T. VII p. 84— 89; Nouvelles Annales (2) T. IX, p. 93-96. 5 Bull. des. Sc. Math. T. I, p. 66—71, 324—328, 384—388 (1870). In addition to the translations mentioned in the text, Hoùel 126 IV. The Founders of Non-Euclidean Geometry (Contd.). it will e understood how fervent an apostle this science had found in the famous French mathematician. Hoùel's labours must have urged J. Frischauf to per- form the service for Germany which the former had rendered to France. His book — Absolute Geometrie nach J. Bolyai — (1872)" is simply a free translation of Johann's Appendix, to which were added the opinions of W. Bolyai on the Found- ations of Geometry. A new and revised edition of Frisch- auf's work was brought out in 1876^. In that work reference is made to the writings of Lobatschewsky and the memoirs of other authors who about that time had taken up this study from a more advanced point of view. This volume remained for many years the only book in which these new doctrines upon space were brought together and compared. § 64. With equal conviction and earnestness Giuseppe Battaglini introduced the new geometrical speculations into Italy and there spread them abroad. From 1867 the Gior- nale di Matematica, of which he was both founder and editor, became the recognized organ of Non-Euclidean Geometry. Battaglini's first memoir — Sulla geometria immaginaria di Lobatschewsky^— y^z.% written to establish directly the prin- ciple which forms the foundation of the general theory of parallels and the trigonometry of Lobatschewsky. It was translated a paper by Battaglini (cf. note 3), two by Beltrami (cf. note 2 p. 127 and p. 147); one, by Rif.mann (cf. note p. 138). and one by Helmholtz (cf. note p. 152). 1 (xii ■\- 96 pages) (Teubner, Leipzig). 2 Eletnente der Ahsoluteii Geometrie, (vi -|- 142 pages) (Teubner, Leipzig). 3 Giornale di Mat. T. V, p. 217 — 231 (1S67). Rend. Ace. Science Fis. e Matem. Napoli, T. VI, p. 157 — 173 (1867). French translation, by HoUEL, Nouvelles Annales (2) T. VII, p. 209—21, 2Ó5— 277 (i8óS). Battaglini and Beltrami. 127 followed, a few pages later, by the Italian translation of the Pangéométrie'^; and this, in its turn, in 1868, by the translation of the Appendix. At the same time, in the sixth volume of the Giornale di Matematica, appeared E. Beltrami's famous paper, Saggio di ititerpretazione della geometria non euclidea. ^ This threw an unexpected light on the question then being debated regard- ing the fundamental principles of geometry, and the concep- tions of Gauss and Lobatschewsky.-^ Glancing through the subsequent volumes of the Giorn- ale di Matematica we frequently come upon papers upon Non-Euclidean Geometry. There are two by Beltrami [1872] connected with the above— named Saggio; several by Batt- aglini [1874 — 78] and by d'OviDio [1875 — 77]? which treat some questions in the new geometry by the projective me- thods discovered by Cayley; Houel's paper [1870] on the impossibility of demonstrating Euclid's Postulate; and others by Cassani [1873 — 81], Gunther [1876], De Zolt [1877], Frattini [1878], Ricordi [1880], etc. § 65. The work of spreading abroad the knowledge of the new geometry, begun and energetically carried forward by the aforesaid geometers, received a powerful impulse from another set of publications, which appeared about this time [1868—72]. These regarded the problem of the foundations of geometry in a more general and less elementary way than that which had been adopted in the investigations of Gauss, 1 This was also published separately as a small book, entitled, Pangeometria sunto di geometria fondata sopra una teoria generate e rigorosa delle parallele (Naples, 1867; 2a Ed. 1874). 2 It was translated into French by Houel in the Ann. Sc. de l'École Normale Sup., T. VI, p. 251—288 (1869). 3 Cf. Commemorazione di E. Beltrami by L. CREMONA: Giornale di Mat. T. XXXVIII, p. 362 (1900). Also the Nachruf by E. Pascal, Math. Ann. Bd. LVII, p. 65—107 (1903). 128 IV. The Founders of Non-Euclidean Geometry (Contd.). LoBATSCHEWSKY, and BoLYAi. In Chapter V. we shall shortly describe these new methods and developments, which are asso- ciated with the names of some of the most eminent mathe- maticians and philosophers of the present time. Here it is sufficient to remark that the old question of parallels, from which all interest seemed to have been taken by the in- vestigations of Legendre forty years earlier, once again and under a completely new aspect attracted the attention of geo- meters and philosophers, and became the centre of an extremely wide field of labour. Some of these efforts were simply directed toward rendering the works of the founders of Non-Euclidean geometry more accessible to the general mathematical public. Others were prompted by the hope of extending the results, the content, and the meaning of the new doctrines, and at the same time contributing to the pro- gress of certain special branches of Higher Mathematics,^ I Cf. e. g., É. Picard, La Science Moderne et son état actual, p. 75 (Paris, Flammarion, 1905). Chapter V. The Later Development of Non-Euclidean Geometry. § 66. To describe the further progress of Non-Eudidean Geometry in the direction of Differential Geometry and Pro- jective Geometry, we must leave the field of Elementary Mathe- matics and speak of some of the branches of Higher Mathe- matics, such as the Differential Geometry of Manifolds, the Theory of Continuous Transformation Groups^ Pure Projec- tive Geometry (the system of Staudt) and the Metrical Geometries which are subordinate to it. As it is not consistent with the plan of this work to refer, even shortly, to these more advanced questions, we shall confine ourselves to those matters without which the reader could not understand the motive spirit of the new researches, nor be led to that other geometrical system, due to Riemann, which has been alto- gether excluded from the previous investigations, as they assume that the straight line is of infinite length. This system is known by the name of its discoverer and corresponds to the Hypothesis of the Obtuse A?igle of Sac- CHERi and Lambert.^ ^ The reader, who wishes a complete discussion of the sub- ject of this chapter, should consult Klein's Vorlesungen uber die iiickteuklidische Geometrie, (Gòttingen, 1903); and BlANCHl's Lezioni sulla Geometria differenziale, 2 Ed. T. I, Cap. XI — XIV (Pisa, Spoerri, 1903). German translation by Lukat, i^t Ed. (Leipzig, 1899). Also The Elements of Non-Eicclidean Geometry by T. L. CoOLlDGE which has recently (1909) been published by the Oxford University Press. 9 I 20 V. The Later Development of Non-Euclidean Geometry. Differential Geometry and Non-Euclidean Geometry. The Geometry upon a Surface. § 67. What follows will be more easily understood if we start with a few observations: A surface being given, let us see how far we can establish a geometry upon it analogous to that on the plane. Through two points A and B on the surface there will generally pass one definite line belonging to the surface, namely, the shortest distance on the surface between the two points. This line is called the geodesic joining the two points. In the case of the sphere, the geodesic joining two points, not the extremities of a diameter, is an arc of the great circle through the two points. Now if we wish to compare the geometry upon a surface with the geometry on a plane, it seems natural to make the geodesies, which measure the distances on the one surface, correspond to the straight lines of the other. It is also natural to consider two figures traced upon the surface as {geodetical- ly) equal, when there is a point to point correspondence be- tween them, such that the geodesic distances between corre- sponding points are equal. We obtain a representation of this conception of equality, if we assume that the surface is made of z. flexible and itiex-, tensible sheet. Then by a movement of the surface, which does not remain rigid, but is bent as described above, those figures upon it, which we have called equal, are to be superposed the one upon the other. Let us take, for example, a piece of a cylindrical surface. By simple bending, without stretching, folding, or tearing, this can be applied to a plane area. It is clear that in this case two figures ought to be called equal on the surface, which coincide with equal areas on the plane, though of course two such figures are not in general equal in space. Differential Geometry and Non-Euclidean Geometry. j^j Returning now to any surface whatsoever, the system of conventions, suggested above, leads to a geometry on the sur- face, which we propose to consider ahvays for suitably bounded regions {^Normal Regions]. Two surfaces which are applicable the one to the other, by bending without stretching, will have the same geometry. Thus, for example, upon any cylindrical surface whatsoever, we will have a geometry similar to that on any plane surface, and, in general, upon any developable surface. The geometry on the sphere affords an example of a geometry on a surface essentially different from that on the plane, since it is impossible to apply a portion of the sphere to the plane. However there is an important analogy be- tween the geometry on the plane and the geometry on the sphere. This analogy has its foundation in the fact that the sphere can be freely moved upon itself, so that propositions in every way analogous to the postulates of congruence on the plane hold for equal figures on the sphere. Let us try to generalize this example. In order that a suitably bounded surface, by bending but without stretching, can be moved upon itself in the same way as a plane, a cer- tain number \K\ invariant with respect to this bending, must have a constant value at all points of the surface. This number was introduced by Gauss and called the Curvature.'^ [In English books it is usually called Gauss's Curvature or tlie Measure of Curvature.] I Remembering that the curvature at any poir t of a plane curve is the reciprocal of the radius of the osculating circle for that point, we shall now show that the curvature at a point M of the surface can be defined. Having drawn the normal n to the surface at M, we consider the pencil of planes through n, and the corre- sponding pencil of curves formed by their intersections with the surface. In this pencil of (plane) curves, there are two, orthogonal to each other, whose curvatures, as defined above, are maximum and minimum. The product of their curvatures is Gauss's Curva- ture of the Surface at M. This Curvature has one most marked 9* I 32 ^ • The Later Development of Non-Euclidean Geometry. Surfaces of Constant Curvature can be actually con- structed. The three cases K^O, A'>6>, K<^0, have to be distinguished. For K^= 6>, we have the developable surfaces [applic- able to the plane]. For K^ O, we have the surfaces applicable to a sphere of radius i : "j/ A', and the sphere can be taken as a model for these surfaces. For K<^ O, we have the surfaces applicable to the Pseudosphere, which can be taken as a model for the surfaces of constant negative curvature. Pseudosphere. Fig. 54- The Pseudosphere is a surface of revolution. The equat- ion of its meridian curve (the tractrix ^) referred to the axis characteristic. It is unchanged for every bending of the surface which does not involve stretching. Thus, if two surfaces are applicable to each other in the sense of the text, they ought to have the same Gaussian Curvature at corresponding points [Gauss], This result, the converse of which was proved by Minding to hold for vSurfaces of Constant Curvature, shows that surfaces, freely movable upon themselves, are characterised by constancy of curvature. ^ The tractrix is the curve in which the distance from the Surfaces of Constant Curvature. 133 of rotation z, and to a suitably chosen axis of ;c perpendicular to z, is kJ^y k^—x2 (i) z = k\og '^ I —Vk'-x% where k is connected with the Curvature K by the equation To the pseudosphere generated by (i) can be applied any portion of the surface of constant curvature — ,-. Surface of Constant Negative Curvature.^ Fig. 56. point of contact of a tangent to the point where it cuts its asymptote is constant. I Fig. 56 is reproduced from a photograph ef a surface con- structed by Beltrami. The actual model belongs to the collection of models in the Mathematical Institute of the University of Pavia. I 34 V. The Later Development of Non-Euclidean Geometry. § 68. There is an analogy between the geometry on a surface of constant curvature and that of a portion of a plane, both taken within suitable boundaries. We can make this analogy clear by tratislatiug the fundamental definitions and properties of the one into those of the other. This is indicat- ed shortly by the positions which the corresponding terms occupy in the following table: (a) Surface. (a) Portion of the plane. (b) Point. (b) Point. (c) Geodesic. (c) Straight line. (d) Arc of Geodesic. (d) Rectilinear Segment. (e) Linear properties of the (e) Postulates of Order for Geodesic. points on a Straight Line. (f) A Geodesic is determined (f) A Straight Line is deter- by two points. mined by two points. (g) Fundamental properties (g) Postulates of Congruence of the equality of Geode- for Rectilinear Segments sic Arcs and Angles. and Angles. (h) If two Geodesic triangles (h) If two Rectilinear triang- have their two sides and les have their two sides the contained angles e- and the contained angles qual, then the remaining equal, then the remaining sides and angles are equal. sides and angles are equal. It follows that we can retain as common to the geome- try of the said surfaces all those properties concerning bound- ed regions on a plane, which in the Euclidean system are independent of the Parallel Postulate, when no use is made of the complete plaiic [e. g., of the infinity of the straight line] in their demonstration. We must now proceed to compare the propositions for a bounded region of the plane, depending on the Euclidean hypothesis, with those which correspond to them in the geo- metry on the surface of constant curvature. We have, e. g., the proposition that the sum of the angles of a triangle is Geometry on a Surface of Constant Curvature. i^c equal to two right angles. The corresponding property does not generally hold for the surface. Indeed Gauss showed that upon a surface whose curva- ture K is constant or varies from point to point, the surface integral over the whole surface of a geodesic triangle ABC, is eqtial io the excess of its three angles over two right angles. ' i. e. \[ KdS =A-VB+ C— IT. ABC Let us apply this formula to the surfaces of constant curvature, distinguishing the three possible cases — Case 1. K=^0. In this case we have UxdS = O; that is ^ + ^ + C=tx. ABC Thus the sum of the angles of a geodesic triangle on sur- faces of zero curvature is equal to two 7'ight angles. Case II. ^=i> ^• In this case we have ABC ABC But {^dS = area of the triangle ABC = A. ^^=A-\- B-\- C—-K. From this equation it follows that ^ + ^ + C> TT, and that L=k^ {A^ B ■\- C— tt). 1 Cf. BlANCHi's work referred to above; Chapter VI. I ?5 V. The Later Development of Non-Euclidean Geometry. That is: a) The sum of the angles of a geodesic triangle on sur- faces of constant positive curvature is greater than two right afigles. b) The area of a geodesic triangle is proportional to the excess of the sum of its angles over two right angles. X--T,<0 Case III. In this case we have ABC ABC where we again denote the area of the triangle ABC hy A. Then we have From this it follows that A-ir B + CTr K^ — k2 Pseudosphere ^A-\-^B-^^C), (c-) [cf. § 70] in every normal region. 2) The postulates of congruence in the whole of the plane. Thus we would have the geometrical systems of Euclid, of LOBATSCHEWSKY-BoLVAi. and of RiEMANN {f/i£ elliptic type), which we have met above, where two straight lines have only one common point : and a second Riemann's system Kthe spherical type), where two straight lines have always two common points. § 75- We cannot be quiie certain what idea Riemann had formed of his complete plane, whether he had thought of it as the elliptic p'laiu, or the spherical plane, or had recognized the possibility of both. This uncertainty is due to the fact that in his memoir he deals with Differential Geometry and devotes only a lew words to the complete forms. Further, those who continued his labours in this direc- tion, among them Beltrami, always considered Riemann's geometry in connection with the sphere. They were thus led to hold that on the complete Riemann's plane, as on the sphere (owing to the existeurc of the opposite ends of a diameter), the postulate that a straight line is determined by two points had exceptions," and that the only form of the plane compatible with the Hypothesis of the Obtuse Angle would be the spherical plane. Cf. for example, tlic sliort reference to the geometry of space of constant positive curvature with which Beltrami concludes his memoir: Teoria fondamt'iifale lit'^^li spazii di atrvatura costante, Ann. di Mat. (2). T. 11, p. 354 — 355 (1868); or the French trans- lation of this memoir by J. llouKi., Ann. So. d. I'Ecole Norm. Sup. T. VI, p. 347-377. 10* 148 V. The Later Development of Non-Euclidean Geometry. The fundamental characteristics of the elliptic plane were given by A. Cayley [1821 — 1895] in 1859, but the connection between these properties and Non-Euclidean geometry was first pointed out by Klein in 187 i. To Klein is also due the clear distinction between the two geometries of RiEMANN, and the representation of the elliptic geometry by the geometry of the sheaf [cf S 7i]- To make the difference between the spherical and elliptic geometries clearer, let us fix our attention on two classes of surfaces presented to us in ordinary space: the surface with two faces {two-sided) and the surface with one face {one-sided). Examples of two-sided surfaces are afforded by the ordinary plane, the surfaces of the second order (conicoidal, cylindrical, and spherical), and in general all the surfaces enclosing solids. On these it is possible to distinguish two faces. An example of a one-sided surface is given by the Leaf of MÒBIUS [MoBiussche Blatt], which can be easily constructed as follows: Cut a rectangular strip AB CD. In- stead of joining the opposite sides AB and CD and thus obtaining a cylindrical surface, let these sides be joined after one of them, e. g., CD, has been rotated through two right angles about its middle point. Then what was the upper face of the rectangle, in the neighbourhood of CD, is now succeeded by the lower face of the original rectangle. Thus on Mobius' Leaf the distinction between the tivo faces becomes impossible. If we wish to distinguish the one-sided surface from the wo-sided by a characteristic, depending only on the intrinsic properties of the surface, we may proceed thus: — We fix a point on the surface, and a direction of rotation about it Then we let the point describe a closed path upon the sur- face, which does not leave the surface; for a two-sided sur- A One-Sided Surface. 149 face the point returns to its initial position and the final direction of rotation coincides with the initial one; for a one- sided surface, [as can be easily verified on the Leaf of Mobius, when the path coincides with the diametral line] there exist closed paths for which the final direction of rotation is oppos- ite to the initial direction. Coming back to the two Riemann's planes, we can now easily state in what their essential difi"erence consists : the spher- ical plaiie has the character of the two-sided surface, and the elliptic plane that of the one- sided surface. The property of the elliptic plane here ^he Leaf of Mobius. enunciated, as well as all its other propert- '^" ^^' ies, finds a concrete interpretation in the sheaf of lines. In fact, if one of the lines of the sheaf is turned about the vertex through half a revolution, the two rotations which have this line for axis are interchanged. Another property of the eUiptic plane, allied to the preceding, is this : The elliptic plane, unlike the Euclidean plane and the other Non-Euclidean planes, is not divided by its lines into two parts. We can state this property other- wise: If two points A and A' are given upon the plane, and an arbitrary straight hne, we can pass from A to A' by a path which does not leave the plane and does not cut the line.^ This fact is 'translated' by an obvious property of the sheaf, which it would be superfluous to mention. § 76. The interpretation of the spherical plane by the sheaf of rays (straight lines starting from the vertex) is ana- logous to that given above for the elliptic plane. The trans- I A surface which completely possesses the properties of the elliptic plane was constructed by W. Boy. [Gott. Berichte, p. 20 —23 (1900); Math. Ann. Bd. LVII, p. 151 — 184 (1903)]. ICQ V. The Later Developmeru of Non-Euclidean Geometry. lation of the properties of this plane into the properties of the sheaf of rays is effected ])y the use of a 'dictionary' similar to that of § 71, in which the word J>oÌ7ìf is found opposite the word rav. The comparison of the sheaf of rays with the sheaf of lines affords a useful means of making clear the connections, and revealing the differences, whic-h are to be found in the two geometries of Rikmann. We can consider two sheaves, with the same vertex, the one of lines, the other of rays. Tt is clear that to every line of the first correspond two ra)s of the second; that every figure of the first is formed by two symmetrical figures of the second; and that, with certain restrictions, the metrical pro- perties of the two forms are the same. Thus if we agree to regard the two opposite rays of tlie sheaf of rays as forming one element only, the sheaf of rays and the sheaf of lines are identical. The same considerations :ipply to the two Riemann's planes. To every point of the elliptic plane correspond two distinct and opposite points of the spherical plane; to two lines of the first, which pass through that point, corres- pond two lines of the second, which have two points in common; etc. The elliptic plane, when compared with the spherical plane, ought to be regarded as a tfoubh' plane. With regard to the elliptic j)laue and the spherical plane, it is right to remark th:it tlii- formulae of absolute tri- gonometry, given in § 56, can he applied to them in every suitably bounded region. This follows from the fact, al- ready noted in S 58, that the formulae of absolute trigonom- etry hold on the sphere, 'and the geometry of the sphere, so far as regards normal regions, coincides with that of these two planes. Riemann's Solid Geometry. jci Principles of Riemann's Solid Geometry. § 77. Returning now to solid geometry, we start from the philosophical foundation that the postulates, although we grant them, by hypothesis, an actual meaning, express truths of experience, which can be verified only in a bounded region. We also assume, that on the foundation of these postul- ates points in space are represented by three coordinates. On such an (analytical) representation, every line is given by three equations in a single variable: and we must now proceed to determine a function j, of the parameter t, which shall express the length of an arc of the curve. On the strength of the distributive property, by which the length of an arc is equal to the sum of the lengths of the parts into which we imagine it to be divided, such a function will be fully determined when we know the element of distance (ds) between two infinitely near points, whose coordinates are jCi + dXi , x, + dx2 , X, + dxy RiEMANN starts with very general hypotheses, which are satisfied most simply by assuming that ds', the square of the element of distance, is a quadratic expression in- volving the differentials of the variables, which always re- mains positive: ds' == Zfly- dxi dxj , where the coefficients aij are functions oi x^, x^, Xy Then, admitting the principle of superposition of figures, it can be shown that the fimction a;j must be such that, with the choice of a suitable system of coordinates, ds'= ^ ^ I-U— (jri2+;.j, = -J log -^ 1 7 -^ -- . ^^xy ^^xy ^^xx ^^yy Introducing the inverse circular and hyperboUc functions, l6o V- The Later Uevelopment of Non-Euclidean Geometry. (2) D^y == ik COS ~^ . I D,y =/ècosh-^ ^ "'^ - (3) xD^y = ik Sin -^ ,- Jil- — - I The constant k, which appears in these formulae, is connected with Riemann's Curvature K by the equation Similar considerations lead to the projective interpret- ation of the conception of angle. The atigle between hoo lines is proportional to the logarithm of the anharmo7iic ratio of the pencil which they fortn with the tajigents from their point of intersection to the absolute. If we wish the measure of the complete pencil to be 2 TT, as in the ordinary measurement, we must take the fraction x : z i as the constant multiplier. Then to express ' analytically the angle between two lines u (ui, u^-, u^), V (z/i, z^2j 2^3)} we put Y«„ = Z bij Ui uj . If bij is the CO factor of the element aij in the dis- criminant of ^xxi the tangential equation of the absolute is given by and the angle between the two lines by the following formulae: — ^nv + Ku/ 2 U/ uT" (l) ^U,V^--j\Qg^ 'nil ' uii ' vv Formulie for the Angle. i6i (2) -< ?/, 7' ^= COS ~^ — ,- r I «j< I 21V <^ 2^, e- = ^ cosh -^ ^"^L _ ^ z/, Z' = sin -» ^"'^ ^^'^ ^"" (3') z/, z/ sinh ' ■ UU ' VV Similar expressions hold for the distance between two points and the angle between two planes, in the geometry of space. We need only suppose that ^xx = O, ¥„„ = O, represent the equations (in point and tangential coordinates) of the absolute of space, instead of the absolute of the plane. According as Q^x = C> is the equation of a real quadric, without generating lines, or of an imaginary quadric, the formulae will refer to the geometry of Lobatschewky-Bolyai, or that of RiEMANN.' § 82. The preceding formulae, concerning the angles between two lines or planes, contain those of ordinary geometry as a special case. Indeed if, for simplicity, we take the case of the plane, and the system of orthogonal axes, the tangential equation of the Euclidean absolute {f^e circular points, § 79) is The formula (2'j, when we insert becomes I For a full discussion of the subject of this and the pre- ceding sections, see Clebsch-Lindemann, Vorlesungen ilber Geometrie, Bd. II. Th. I, p. 461 — et seq. (Leipzig, 1891). 11 1 62 V. The Later Development of Non-Euclidean Geometry. ') so that this equation can be written COS {u, V) = /j 4 -+- m^ 7)12. the ordinary expression for the angle between the two lines (/i ;//i) and (4 m^. For the distance between two points the argument does not proceed so simply, when the absolute degenerates into the circular points. Indeed the points J/, N, where the line XY intersects the absolute, coincide in the point at infinity on this line, and the formula (i) gives in every case: D^y = 4 log {M^N^XY) = A log I = o. However, by a simple artifice we can obtain the ordinary formula for the distance as the limiting case of formula (3). To do this more easily, let us suppose the equations of the absolute (not degenerate), in point and line coor- dinates, reduced to the form : Q.i^. == ^Xi'^ + ^X^^ + X~^ = O, Then^ putting equation (3) of the preceding section gives Euclid's Geometry as a Limiting Case. 1 53 D,y = ik sin-^ /eA. Let e be infinitesimal. Omitting terms of a higher order, we can substitute K € A for sin~' K e A in this formula If we now choose k^ infinitely large, so that the product ik Y^ remains finite and equal to unity for every value of e, the said formula becomes Let € now tend to the limit zero. The tangential equation of the absolute becomes «i^ + «2^ = O; and the conic degenerates into two imaginary conjugate points on the line u^ = o. The formula for the distance, on putting takes the form which is the ordinary Euclidean formula. We have thus ob- tained the required result. We note that to obtain the special Euclidean case from the general formula for the distance, we must let k^ tend to infinity. Since Riemann's curvature is given by - — tj , this affords a confirmation of the fact that Riemann's curvature is zero in Euclidean space. § 83. The properties of plane figures with respect to a conic, and those of space with respect to a quadric, together constitute projective metrical geometry. This was first studied by Cayley,'' apart from its connection with the Non-Euclid- I Sixth Memoir upon Quantics. Phil. Trans. Vol. CXLIX, p. 6 1 -90 (1859). Also Collected Works, Vol. II, p. 561 — 592. 104 ^' ^^^ Later Development of Non-Euclidean Geometry. ean geometries. These last relations were discovered and explained some years later by F. Klein. ^ To Klein is also due a widely used nomenclature for the projective metrical geometries. He gives the name hyper- bolic geometry to Cayley's geometry, when the absolute is real and not degenerate: elliptic geoinetry, to that in which the absolute is imaginary and not degenerate: parabolic geometry, to the limiting case of these two. Thus, in the remaining articles, we can use this nomenclature to describe the three geometrical systems of Lobatschewsky-Bolyai, of RiEMANN (elliptic type), and of Euclid. Representation of the Geometry of Lobatschewsky- Bolyai on the Euclidean Plane. § 84. To the projective interpretation of the Non- Euclidean measurements, of which we have just spoken, may be added an interesting representation which can be given of the Hyperbolic Geometry on the Euclidean plane. To ob- tain it, we take on the plane a real, not degenerate, conic : e. g. a circle. Then we make the following definitions, relative to this circle : Plane == region of points within the circle. Point = point inside the circle. Straight line = chord of the circle. We can now easily verify that the postulate that a straight line is determined by two points, and the postulates regarding the properties of straight lines and angles, can be expressed as relations, which are always valid, when the above interpretations are given to these terms. But in the further development of this geometry we add I Cf. Uber die sogcnannte Nichi-Euklidische Geometrie. Math. Aim. Bd. IV, p. 573-625 (1871). Representation on the Euclidean Plane. 1 65 to these the postulates of congruence, contained in the following principle of displacement. If we are given two points A, A' on the plane, and the straight lines a, a , respectively passing through them, there are four methods of superposing the plane on itself, so that A and a coincide respectively with A and a. More precisely: ofie method of superposition is defined by taking as corre- sponding to each other, one ray of a and one ray of a , one section of the plane bounded by a and one section bounded by a. Two of these displacements are dirt'ct co?igruèncès and two converse congruences. With the preceding interpretations of the entities, point, Un; and plane, the principle here expressed is translated into the following proposition: If a conic {i\ g., a circle) is given in a platie, and two internal points A, A' are taken, as also two chords a, a', re- spectively passing through them, there are four projective trans- formations of the plafie, which change into itself the space within the conic, and which make A and a correspond respect- ively to A' and a . To fix one of them^ it is sufficient to make sure that a given extremity of a corresponds to a given extremity of a , and that to one section of the plane bounded by a, cor- responds a definite section of the plane bounded by ci . Of these four transformations, two determine on the conic a projective correspondence in the same sense, and two a pro- jective correspondence i?i the opposite sense. § 85. We shall prove this proposition, taking for sim- plicity two distinct conies T, t', in the same plane or other- wise. Let M, N be the extremities of the chord a [cf Fig. 5 8]. Also M\ N' those of a [cf Fig. 59]. 1 66 V. The Later Development of Non-Euclidean Geometry. Let F, P' be the poles of a^ a with respect to the two conies. On this understanding, the Hne PA intersects the conic T in two real and distinct points i?, S: also the line P' A intersects the conic t' in two real and distinct points A projective transformation which changes t into t', the line a into a, and the point A into A, will make the point P correspond to P\ and the hne PA to the line P' A. Fig. 59- Thus this transformation determines a projective cor- respondence between the points of the two conies, in which the pair of points M', N' corresponds to the pair of points M, N: and the pair of points R' , S' to R, S. Vice versa, a projective transformation between the two conies, which enjoys this property, is associated with a pro- jective transformation of the two planes, such as is here de- scribed.' But if we consider the two conies t, t', we see that to I For this proof, and the theorems of Projective Geometry upon which it is founded, see Chapter X, p. 251 — 253 of the work of Enriques referred to on p. 156. Projective Transformations. 167 the points of the range MNRS on T may be made to cor- respond the points of any one of the following ranges on t': M'N'R'S' n'm's'e: m'n's'jr: N'M'R'S'. In this way we prove the existence of the four project- ive transformations of which we have spoken in the propos- ition just enunciated. If we suppose that the two conies coincide, we do not need to change the preceding argument in any way. We add, how- p ever, that of the four transformations only one makes the segment AM correspond to the segment A'M\ if at the same time the shaded parts of the figure cor- respond to each other. Further the two transformations defined by the ranges / MNRS \ M'lYR'S' determine projections in the same sense, while the other two, defined by the ranges : f MNRS \ / MNRS \ \ MN'S'R' ) \ N'M'R'S' ) determine projections in the opposite setise. \ / MNRS \ )' ), \ N'M'S'K ) § 86. With these remarks, we now return to complete the definitions of S 84, relative to a circle given on the plane. Flane = region of points within the circle. l68 V. The Later Development of Non-Euclidean Geometry. Point = point within the circle. Straight Line = chord of the circle. Displacements == projective transformations of the plane which change the space within the circle into itself. Semi-Revolutions = homographic transformations of the circle. Congn/ent Figures = figures which can be transformed the one into the other by means of the projective trans- formations named above. The preceding arguments permit us to affirm at once that all the propositions of elementary plane geometry, asso- ciated with the concepts straight line, angle and congruence, can be readily translated into proj)erties relative to the system of points inside the circle, which we denote by {S). In particular let us see what corresponds in {S) to two per- pendicular lines in the ordinary plane. To this end we note that if r, s are two perpendicular lines, a semi-revolution of the plane about j will superpose r upon itself, exchanging, however, the two rays in which it is divided by s. According to the above definitions, a semi-revolution in {S) is a homographic transformation, which has for axis a chord s of the circle and for centre the pole of the chord. The lines which are unchanged in this transformation, in ad- dition to s, are the lines passing through its centre. Thus in the system (S) we must call two lines perpendicular, when they are conjugate with respect to the fundamcjital circle. We could easily verify in {S) all the propositions on perpendicular lines. In particular, that if we draw the (imag- inary) tangents to the fundamental circle from the common point of two conjugate chords in (^), these tangents form a harmonic pencil with the perpendicular lines [cf. p. 155].' I This representation of the Non-Euclidean plane has been The Distance between two Points. i5q § 87. Let US now see how the distance between two points can be expressed in this conventional measurement, which is being taken for the interior of the circle. To this end we introduce a system of orthogonal coord- inates {x, y), with origin at the centre of the circle. The distance between two points A {x, j), B {x , y) in the plane with which we are dealing cannot be represen- ted by the usual formula Y{x~xy\{y-y)\ since it is not invariant for the projective transformations which we have called displacements. The distance must be a function of the coordinates, invariant for the said transforma- tions, which for points on the straight line possesses the dis- tributive property given by the formula dist. {AE) = dist. {AC) -f dist. {CB). Now the anharmonic ratio of the four points A^ B, M, N, where M, N are the extremities of the chord AB, is a relation between the coordinates {x, y), {x\ y') of AB, remaining invariant for all projective transformations which leave the ~_ fundamental circle fixed. The most general ex- pression, possessing this invariant property, will be an arbi- trary function of this anharmonic ratio. If we remember that the said function must be distrib- utive in the sense above indicated, we must assume that, except for a multiplier, it is equal to the logarithm of the anharmonic ratio, (ABM^T) = ^^: -^^,- We shall thus have distance (AB) = ^ log {ABMN). employed by Grossmann in carrying out a number of the con- structions of Non-Euclidean Geometry. Cf. Appendix, III, p. 225. 170 V- The Later Development of Non-Euclidean Geometry. In a similar way we proceed to find the proper ex- pression for the angle between two straight lines. In this case we must notice that if we wish the right angle to be ex- it pressed by — , we must take as constant multiplier of the logarithm the factor 1:22. Then we shall have for the angle between a and b, ^^>^=^ 2/ ^^^ iabmn), where m, n are the conjugate imaginary tangents from the vertex of the angle to the circle, and {a b m n) is the an- harmonic ratio of the four lines a, b, in and «, expressed analytically by sin ia ni) sin {a n) sin [d m) ' sin (i n) % 88. A glance at what was said above on the sub- ordination of the metrical to the projective geometry (S 81) will show clearly that the preceding formulge^ regarding the distance and angle, agree with those which we would have in the Non-Euclidean plane, if the absolute were a circle. This would be sufficient to suggest that the geometry of the system (6') gives a concrete representation of the geometry of LoBATSCHEWSKY-BoLVAi. However, as we wish to discuss this point more fully, let us see how the definition and pro- perty of parallels are translated in \S). Let r (z^i, U2, u^) and r {i\, V2, v^ be two difterent chords of the fundamental circle. Let the circle be referred to an orthogonal Cartesian set of axes, with the centre for origin, and let us take the radius as unit of length. Then we have x^ -Vy^ — 1=0, u^-\-v^ — 1=0, for the point and line equation of the circle. The Angle between two Lines. 171 Making these equations homogeneous, we obtain Xj,^ + x,' — x^' = O, Ui'+U2' — //3^ = O. The angle ^r, r between the two straight hnes r and r can be calculated by means of the formula (3') of § 81, if we put ^uu '-'■ u,' + u. u. 3 ' We thus obtain sin <5C r, r V (U1V3 ViUiY — (2^22^3 — "v-^ò^ — iu-^-i^ V-^U^^' But the lines r, r are given by Xt_H^-\- X-Jl^-V X.jLl.^ = O, XtJ)-!^ + .^22^2 + -^3^3 ^= O ; and they meet in the point, x^ = u^v^, — U^^2 , X^ == U^^x—-U{U^, X^ == U1V2 — 2/2 Z'l- Thus the preceding expression for this angle takes the form / \ • V ' ' \Xx Xx X^, ) (4) sm <^r,r= ^ — " . l/(2 a relation which agrees with that given by TaurinuS; Lo- BATSCHEWSKY and BoLYAi for the angle of parallehsm [cf. p. 90]. § 90. We proceed, finally, to see how the distance be- tween two neighbouring points {the element of distance) is expressed in the system (vS), so that we may be able to compare this representation of the hyperbolic geometry with that given by Beltrami [cf. g 69]. Let {x^y)^ (x + dx,y + dy) be two neighbouring points. Their distance ds is calculated by means of (2) of § 81 if we substitute : Qxz = x'+y^— I, Qj,y = (x + dxy + iy + dyY— i, ^xy = X {x + dx) -^-y (y + dy) — i . Since the angle is small, we may substitute the sine for the angle, and we have _ ,2 (dx^ + dy»){l—x2 —y2) 4- [xdx +ydyy (X2 +-y2 _ I) ((;, ^ dxY + fy + dyY - I)) Thus, omitting terms higher than the second order, we have ^^. _ ^2 {dx^ 4- dy2) (I — x2 —y2) + ^xdx +ydyy (l X2 —y2)2 or (c) ds"- = k" (^ —y^) ^-^^ -^ixydxdy^ {l—x2)dy2 ^ -^ {l—X2—y2)2 Now we recall that Beltrami, in i868, interpreted the geometry of Lobatschewsky-Bolvai by that on the surfaces of constant negative curvature. The study of the geometry on such surfaces depends upon the use of a system of coord- inates on the surface, and the law according to which the element of distance {ds) is measured. The choice of a suitable I HA R. The Later Development of Non-Euclidean Geometry. system (?/, v) enabled Beltrami to put the square of ds in this form: (I — v^') dit^ -f- zitvdudri -\-(\ — n^) dv^ k , (I «2 — 2/2)2 where the constant k^ is the reciprocal, with its sign changed, of the curvature of the surface.' In studying the properties of these surfaces and in mak- ing a comparison between them and the metrical results of the geometry of Lobatschewsky-Bolyai, Beltrami in his classical memoir, quoted on p. 138, employed the following artifice: He represented the points of the surface on an aux- iliary plane, such that the point {u, z') of the surface corre- sponded to the point on the plane whose Cartesian coord- inates (x,}>) were {u, v). The points on the surface were then represented by points inside the circle x^ +y^ — I = O; the points at infinity on the surface by points on the cir- cumference of the circle: its geodesies by chords: parallel geodesies by chords meeting in a point on the circumference of the said circle. Then the expression for {dsY took the same form as that given in (5), which states the form to be used for the element of distance in the system {S). It follows that, by his representation of the surfaces of constant negative curvature on a plane, Beltrami was led to one of the projective metrical geometries of Cayley, and precisely to the metrical geometry relative to a funda- mental circle, given above in §§ 80, 81. I Risoluzione del problema di riportare i punti di una superficie sop>ra un piano in modo che le linee geodetiche vengano rappresentate da linee rette. Ann. di Mat. T. VII, p. 185 — 204 (1866). Also Opere Matematiche. T. I, p. 262 — 280 (Milan, 1902). Beltrami's Geometry and Projective Geometry. jyc § gi. The representation of plane hyperbolical geo- metry on theEudidean plane is capable of being extended to the case of solid geometry. To represent the solid geometry of LoBATSCHEWSKY-BoLYAi in Ordinary space we need only adopt the following definitions for the latter: Space = Region of points inside a sphere. I^owt = Point inside the sphere. Straight Line = Chord of the sphere. Plane = Points of a plane of section which are inside the sphere. Displacements = Projective transformations of space, which change the region of the points inside the sphere into itself, etc. With this 'Dictionary' the propositions of hyperbolic solid geometry can be translated into corresponding proper- ties of the Euclidean space, relative to the system of points inside the sphere.' Representation of Riemann's Elliptic Geometry in Euclidean Space. § 92. So far as regards plane geometry, we have already remarked [pp. 142 — 3] that the geometry of the ordinary sheaf of lines gives a concrete interpretation of the elliptical system of Riemann. Therefore, if we cut the sheaf by an ordinary plane, completed by the line at infinity, we obtain a representation on the Euclidean plane of the said Rie- mann's plane. I Beltrami considers the interpretation of Non-Euclidean Solid Geometry, and, in general, of the geometries of manifolds of higher order in space of constant curvature, in his memoir: Teoria fondamentale degli spazii di curvatura costante. Ann. di Mat. (2), T, II, p. 232—255 (1868). Opere Mat. T. I, p. 406—429 (Milan, 1902). 176 V. The Later Development of Non-Euclidean Geometry. If we wish a representation of the elliptic space in the Euclidean space, we need only assume in this a single-valued polarity, to which corresponds an imaginary quadric, not degenerate. We must then take, with respect to this quadric, a system of definitions analogous to those indicated above in the hyperbolic case. We do not pursue this point further, as it offers no fresh difficulty. However we remark that in this representation all the points of the Euclidean space, including the points on the plane at infinity, would have a one-one correspondence with the points of Rietnann' s space. Foundation of Geometry upon Descriptive Properties. § 93. The principles explained in the preceding sections lead to a new order of ideas in which the descriptive propert- ies appear as the first foundations of geometry, instead of congruence and displacement, of which Riemann and Helm- HOLTZ availed themselves. We note that, if we do not wish to introduce at the beginning any hypothesis on the inter- section of coplanar straight lines, we must start from a suitable system of postulates, valid in a boic7ided region of space, and that we must complete the initial region later by means oi improper points, lines and planes [cf. p. 157].^ When projective geometry has been developed, the metrical properties can be introduced into space, by adding to the initial postulates those referring to displacement or I For such developments, cf. Klein, Ioc. cit. p. 158: Pasch, Vorlesungen iiber neuere Geometrie, (Leipzig, l882)j SCHUR, Uber die Einfichrting der sogenannten idea! en Elemenie in die projective Geometrie, Math. Ann. Bd. XXXIX, p. 113 — 124 (1891): Bonola, Suila intro- duzione degli elementi improprii in geometria proiettiva. Giornale di Mat. T. XXXVIII, p. 105— 116 (1900). Foundation of Geometry upon Descriptive Properties. 177 congruence. By so doing we find that a certain polarity of space, allied to the metrical conceptions, becomes trans- formed into itself by all displacements. Then it is shown that the fundamental quadric of this polarity can only be: a) A real, non-ruled quadric; b) An imaginary quadric (with real equation); c) A degetiérate quadric. Thus the three geometrical systems, which Riemann and Helmholtz reached from the conception of the element of distance, are to be found also in this way.* The Impossibility of proving Euclid's Postulate. § 94. Before we bring to a close this historical treat- ment of our subject it seems advisable to say a few words on the impossibility of demonstrating Euclid's Postulate. The very fact that the innumerable attempts made to obtain a proof did not lead to the wished-for result, would suggest the thought that its demonstration is impossible. In- deed our geometrical instinct seems to afford us evidence that a proposition, seemingly so simple, if it is provable, ought to be proved by an argument of equal simplicity. But such considerations cannot be held to afford a proof of the impossibility in question. If we put Euclid's Postulate aside, following the devel- opments of Gauss, Lobatschewsky and Bolyai, we can construct a geometrical system in which no contradictions are met. This seems to prove the logical possibility of the Non-Euclidean hypothesis, and that Euclid's Postulate is independent of. the first principles of geometry and therefore cannot be demonstrated. However the fact that contradictions I For the proof of this result see BONOLA, Determinazione per via geometrica dei ire tipi de spazio; iperbolico, parabolico, ellittico. Rend. Gire. Mat. Palermo, T. XV, p. 56—65 (1901). 12 J 73 V. The Later Development of Non-Euclidean Geometry. have not been met is not sufficient to prove this; we must be certain that, proceeding on the same Hnes, such con- tradictions could never be met. This conviction can be gained with absolute certainty from the consideration of the formulae of Non-Euclidean geometry. If we take the system of all the sets of three numbers (x, y, z), and agree to con- sider each set as an analytical point, we can define the distance between two such analytical points by the formulae of the said Non-Euclidean Trigonometry. In this way we construct an analytical system, which offers a conventional interpretation of the Non-Euclidean geometry, and thus demonstrates its logical possibility. In this sense the formulae of the Non- Euclidean Trigon- ometry of Lobatscheiusky-Bolyai give the proof of the independ- ence of Euclid's Postulate from the first principles of geometry (regarding the straight line, the plane and congruence). We can seek a geometrical proof of the said independ- ence, on the lines of the later developments of which we have given an account. For this it is necessary to start from the principle that the conceptions, derived from our intu- ition, independently of the correspondence which they find in the external world, are a priori logically possible; and that thus the Euclidean geometry is logically possible and every set of deductions founded upon it. But the interpretation which the Non-Euclidean plane hyperbolic geometry finds in the geometry on the surfaces of constant negative curvature, offers, up to a certain point, a first proof of the im.possibility of demonstrating the Eu- clidean postulate. To put the matter in more exact terms: by this means it is established that the said postulate cannot be demonstrated on the foundation of the first principles of geometry, held valid in a bounded region of the plane. In fact, every contradiction, which would arise from the other postulate, would be translated into a contradiction Euclid's Postulate cannot be Proved. 1/9 in the geometry on the surfaces of constant negative curv- ature. However, since the comparison between the hyperbolic plane and the surfaces of constant negative curvature, exists, as we have seen, only for bounded regions^ we have not thus excluded the possibility that the Euclidean postulate might be proved for the complete plane. To remove this uncertainty, it would be necessary to refer to the abstract manifold of constant curvature, since no concrete surface exists in ordinary space, in which the ^(?w- //] aura le point d'appui [E] dans le miheu et devra, par consequent, étre charge également aux bouts [C, D\. done la charge que supporte le point d'appui du levier; qui fait la base du triangle, et qui est charge, à ses ivickelung, (3. Aufl., Leipzig, 15^97); English translation by T. J. Mc- CoRMACK (Open Court Publishing Co. Chicago, 1902). Also, for the different hypotheses from which the proof of the principle of the lever, can be obtained, see P. Duhem, Les origines de la stati- qiie, (Paris, 1905), especially Appendix C, Sur les divers axiomes d'ou se peut déduire la ihcorie du levier. Statical Hypothesis equivalent to Postulate V. i8^ deux extrémités de poids égaux, sera égale au poids double du sommet et, par consequent, égale à la somme des deux poids.* ^ § 3. Lagrange's argument contains implicitly some hypotheses of a statical nature, regarding symmetry, addition of constraints,^ etc.; and, in addition, it involves a geometrical property of the Euclidean triangle. But if we wish to omit the latter, a course which for certain reasons seems natural, the preceding conclusions will be modified. Indeed, though we may still assume that the triangle ABD is in equilibrium about the point E^ where the lines MN and CD intersect, we cannot assert that E is the middle point of CD, as this would be equivalent to assuming Euclid's Postulate. Consequently, we cannot assert that the single weight 2 P, applied at C, can be substituted for the two weights at A and B, since, if such a change could take place, a lever would be in equilibrium, with equal weights at its ends, about a point which cannot be its middle point. Vice versa, if we assume, with Archimedes, that two equal weights at the end can be replaced by a double weight at the middle point of the lever, then we can easily deduce that E is the middle point of CD, and from this it will follow that ABD is a Euclidean triangle. Hence we have established the equivalence of Euclid's Fifth Postulate and the said hypothesis of Archimedes. Such equivalence is, of course, relative to the system of hypotheses which comprises, on the one hand, the above-named statical hypotheses, and, on the other, the ordinary geometrical hypotheses. 1 Oeuvres de Lagrange, T. XI, p. 4 — 5. 2 For an analysis of \!as. physical principles on which ordinary statics is founded, cf. F. Enriques, Problemi della Scienza. Cap. V. (Bologna, 1906). German translation, (Leipzig, 1910). 184 Appendix I. The Fundamental Principles of Statics etc. With the modern notation, we can speak of forces, of the composition of forces, oi resultants, m'ìXtz.à. oi weights, levers, etc. Then the hypothesis referred to takes the following form: The resultant of two equal forces in the same plane, applied at right angles to the extremities of a straight line and towards the same side of it, is a single force at the middle point of the line, of double the intensity of the given forces. From what we have said above, if this law for the com- position of forces were true, it would follow that the ord- inary theory of parallels holds in space. On the Composition *of Forces Acting at a Point. § 4. The other fundamental principle of statics, the law of the Parallelogram of Forces, from the usual geom- etrical interpretation which it receives, is closely connected with the Euclidean nature of space. However, if we examine the essential part of this principle, namely, the analytical expression for the resultant R of two equal forces P, acting at a point, it is easy to show that it exists independently of any hypothesis on parallels. This can be made clear by deducing the formula R = a/' cos a, where 2 a is the angle formed by the two concurrent forces from the following principles: i) Two or more forces, acting at the same point, have a definite resultant. 2) The resultant of two equal and opposite forces is zero. 3) The resultant of two or more forces, acting at a point, along the same straight line, is a force through the same point, equal to the sum of tlie given forces, and along the same line. Composition of Concurrent Forces. I8: 4) The resultant of two equal forces, acting at the same point, is directed along the line bisecting the angle between the two forces. 5) The magnitude of the resultant is a continuous funct- ion of the magnitude of the components. Let us see briefly how we establish our theorem. The value i? of the resultant of two forces of equal magnitude /*, enclosing the angle 2 a, is a function of P and a only. Thus we can Avrite i?= 2/(P,a). A first application of the principles named above shows that R is proportional to P, and this result is independent of any hypothesis on parallels [cf note i, p. 195]. Thus the preceding equation can be written more simply as R == 2P/{0.). We now proceed to find the form of/" (a). § 5. Let us calculate /(a) for some particular value of the angle. (I) Let a = 45°- At the point O at which act 1 p Q the two forces Ft,, P2, of equal magnitude P, let us imagine two equal and opposite forces applied, perpendicular to R and of magni- tude — (cf. Fig. 64). At the same time let us imag- ine R decomposed into two others, directed along R and of magni- tude R We can then regard each force F as the resultant of two forces at right angles, of magnitude — . J 86 Appendix I. The Fundamental Principles of Statics etc. We thus have Z' = 2 . ^ ./(45°). On the other hand^ R being the resultant of i^i and Pa, we have R== 2 /y(45°)- From these two equations we obtain /(45°) = \ V^' (II) Again let a = 60°. In this case apply a.t O a. force R' equal and opposite to R (cf. Fig. 65). The system of the two forces R and of R' is in equilibrium. Thus by symmetry, R' = P. Therefore, R = F. But, on the other hand, i?= 2 /y(6o"). Therefore/ (60") = y. (Ill) Again let a = 36°. At O let the five forces P^, P^-.Pc^, of magnitude P^ be Special Cases. 1 87 applied, such that each of them forms with the next an angle of 72° (cf. Fig. 66). This system is in equilibrium. For the resultant R of P2 and P^, we have R= 2/y(36°). For the resultant i?' of /'i and P^ , we have R' = 2Pf{U'). On the other hand, R has the same direction as Pc^ ; that is, a direction opposite to that of R. Therefore 2 /yCsó") = 2 i'/(72°) + P. (i) Therefore 2/(36°) = 2/(72°) + i. If, instead, we take the resultants of P^ and P^ , and of P^ and P^, we obtain two forces of magnitude 2 P/ (36°), containing an angle of 144°. Taking the resultant of these two, we obtain a new force R" of magnitude 4 ^7(36°)/ (7 2°). Now R", by the symmetry of the figure, has the same line of action as P^ , but acts in the opposite direction. Thus, since equihbrium must exist, i^=4/'/(36°)/(72°). (2) Therefore i = 4/ (36°)/ (72°). From the two equations (i) and (2) we obtain /(36o)_ltV:5_/(;.o)^-f^^, 4 4 on solving for/ (36°) and/ (7 2°). § 6. By arguments similar to those used in the pre- ceding section we could deduce other values for / (a). However, if we restrict ourselves only to those just found, 1 88 Appendix I. The Fundamental Principles of Statics etc. and compare them with the corresponding values of cos a, we obtain the following table: cos 0° = 1 /(0°) = I cos 36° cos 45^ I + Vs' 4 2 cos 60° = — /(36°) = /(45°) = /(6o°) = 4 COS 72° = -I + /5 /(72°) + Vi /(90-) = o. This table suggests the identity of the two functions y(a) and cos a. For fuller p confirmation of this fact, we determine the functional equation which _/ (a) satis- fy 2 fies (cf. Fig. 67). To this end let us con- sider four forces F^, P2, F.^, P^ of magnitude P, acting at one point, forming with each other the following angles -^ p,p, = <: p^^p, = 2 p ^P,P,=- 2(a-P) -^ P,P, == 2 (a + P). We shall determine the resultant P of these four forces in two different ways. Taking Pt_ with P^ , and P^ with F^ we obtain two forces i?i and i?j, of magnitude The General Case. i8q inclined at an angle 2 p. Taking the resultant of Rt_ and R2, we have a force li, such that i? = 4-/y(a)/(P). On the other hand, taking /'i with /'_,, and F^ with F^^, we obtain two resultants, both along the direction of R, and of magnitudes 2Ff{a + ^\2Ff{a-^), respectively. These two forces have a resultant equal to their sum, and thus F = 2^/(a + p) + 2i'/(a— p). Comparing the two values of i?, we find that (i) 2/(a)/(P) =/(a + p) +/(a-P) is the functional equation required. If we now remember that cos (a + P) + cos (a — P) = 2 cos a cos P, and take account of the identity between f (a) and cos a in the preceding table for certain values of a, and the hy- pothesis that f (a) is continuous, without further argument we can write / (a) = cos a. It follows that F = 2 F cos a. The validity of this formula of the Euclidean space is thus also established for the Non-Euclidean spaces. § 7. The law of composition of two equal concurrent forces leads to the solution of the general problem of the resultant, since we can assign, without any further hypothesis, the components of a force F along two rectangular axes through its point of application O. J go Appendix I. The Fundamental Principles of Statics etc. Let the two perpendicular lines be taken as the axes of X and y, and let i? make the angles a, P with them Through O draw the line which makes an angle a with Ox and an angle P with Oy. Imagine two equal and oppos- ite forces Pi and Pz to act along this line at O, their mag- nitude being — . Also imagine the force 7? replaced by the two equal forces P, of magni- tude — , actmg m the same direction as P. Then the system P^, P^, P, Pha.s R for resultant. But Pi and P, taken together, have a resultant X = P cos a along Ox: and P2 and P, taken together, have a resultant Y= Rcoi p along Oy. These two forces are the components of P along the two perpendicular lines. As to their magnitudes, they are identical with what we would obtain in the ordinary theory founded upon the principle of the Parallelogram of Forces. However, the lines OX and O V, which represent the com- ponents upon the axes, are not necessmily the projections of R, as in the Euclidean case. Indeed we can easily see that, if these lines were the orthogonal projections of R upon the axes, the Euclidean Hypothesis would hold in the plane. § 8. The functional method applied in S 6 to the composition of two equal forces acting at a point, is derived from D. DE FoNCENEx [1734 — 1799]- r>y a method ana- Rectangular Components of a Fece. Iqi logous to that which led us to the equation for / (a) (= y), FoNCENEX arrived at the differential equation' P + ^y=^ o. From this, on integrating and taking account of the initial conditions of the problem, he obtained the known expression for/ (a). However the application of the principles of the In- finitesimal Calculus, requires the continuity and differentiabil- ity of/ (a), conditions, which, as Foncenex remarks, involve the (physical) nature of the problem. But as he wishes to go 'jusqu'aux difficultes les moins fondees', he avails himself of the Calculus of Finite Differences, and of a Difference Equation, which allows him to obtain / (a) for all values of a which are commensurable with it. The case a incom- mensurable is treated 'par une méthode famiUère aux Géo- mètres et frequente surtout le écrits des Anciens'; that is, by the Method of Exhaustion.^ All Foncenex' argument, and therefore that given in 1 We could obtain this equation from (l) p. 189 as follows: Put p = a'a and suppose that /(a) can be expanded by Taylor's Series for every value of a. Then we have 2/(a) (/ (o) + '/a /' (o) + 'l^ f" (o) . . .^ = 2/(a) + 2 ^-/" {«) + .. Equating the coefficients of do^ and putting y = /(a) and k'i = — /" (o), we have d2y — il 4- ^2^ = o. da' 2 Cf. Foncenex : Si/r les prindpes /ondameittatix de la Mecan- ique. Misc. Taurinensia. T. II, p. 305 — 315 (1760 — 1761). His argument is repeated and explained by A. GENOCCm in his paper: Sur un Mémoire de Daviet de Foncenex et sur les geometries non- euclidiennes. Torino, Memorie (2), T. XXIX, p. 366 — 371 (1877). IQ2 Appendix I. The Fundamental Principles of Statics etc. § 6, is independent of Euclid's Postulate. However, it should be remarked that Foncenex' aim was not to make the law of composition of concurrent forces independent of the theory of parallels, but rather to prove the law itself. Probably he held, as other geometers [D. Bernouilli, D'Alembert], that it was a truth independent of any ex- perimental foundation. Non-Euclidean Statics. § 9. Having thus shown that the analytical law for the composition of concurrent forces does not depend on Euclid's Fifth Postulate, we proceed to deduce the law accord- ing to which forces perpendicular to a line will be composed. Let A, A be the points of application of two lorces Pi, P2 of equal magnitude P (cf Fig. 69). Let C be the middle point of AA, and B a point on the perpendicular BC to AA. Joining AB and AB, and putting <^ BAC = a, <^ ABC = p, it is clear that the force P^ can be regarded as a component of a force T-s,, acting at A and along BA. The magnitude of this force is given by P sin a r= -. — Equal Forces perpendicular to a Line. ig^ The other component Q^, at right angles to P^, is given by Q = T'cos a = /'cot a. Repeating this process with the force F2 , we obtain the following system of coplanar forces : (i) System F^, F^- (2) System/',, P,, Q,, Q,. (3) System 7;, T^. If we assume that we can move the point of application of a force along its line of action, it is clear that the first two systems are equivalent, and because (2) is equivalent to (3), we can substitute for the two forces jPi , P2, the two forces 7; and 7;. The latter, being moved along their lines of action to B, can be composed into one force P = 2rcosp = 2/'^- ^ sin a This, in its turn, can be moved to C, its direction per- pendicular to A A remaining unchanged. This result, which is obviously independent of Euclid's Postulate, can be applied to the three systems of geometry: Euclid^s Geometry. In the triangle ABC we have cos P = sin a. Therefore R= 2 P. Geometry of Lobatschewsky-Bolyai. In the triangle ABC, if we denote the side AA by 2 b, we have cos p ^ ^ / \ -. == cosh -r (p. II 7). sin a k ^^ '^ Thus Ò i? ==• 2 jP cosh , 1^ 1 94 Appendix I. The Fundamental Principles of Statics etc. Riemann's Geometry. In the same triansjle we have Therefore cos 6 h -. = COS -r- • sin a /C R = 2 P COS — Conclusioti. It is only in EucUdean space that the resultant of two equal forces, perpendicular to the same line, is equal to the sum of the two given forces. In the Non-Euclidean spaces the resultant depends, in the manner indicated above, on the distance between the points at which the two forces are applied.^ § IO. The case of two unequal forces P^ Q, per- pendicular to the same straight line, is treated in a similar manner. In the Euclidean Geometry we obtain the known results; R^ P -V (2, R _ P _ Q p-\- 1 q P In the Geometry of Lobatschewsky-Bolyai the problem of the resultant leads to the following equations: R = P cosh y + <2 cosh y, R _ P Q sinh T sinh -r sinh -7- k K K Then, by the usual substitution of the circular functions for the hyperboHc, we obtain the corresponding result for Riemann's Geometry: I For a fuller treatment of Non-Euclidean Statics, the reader is referred to the following authors: J. M. de Tilly, Etudes de Mécafiique abstraiie, Mém. couronnés et autres mém., T. XXI (1870). J. Andrade, La Statique et les Géo??iétries de Lobatscheivsky , d'Euclide, et de Riemann. Appendix (II) of the work quoted on p. 181. Unequal Forces. igc 7? == Z' COS y + (2 COS -|-, R P Q ■ p-\- 9 • <] ■ P sm — T— sin -T- sm -r k k k In these formulce /, q, denote the distances of the points of application of P and Q from that of R. These results can be summed up in a single formula, valid for Absolute Geometry; R = F.EP+ Q. Eq, R _ -P __ Q 07/ +7) ~0(^)~Ò(?)' To obtain these results directly, it is sufficient to use the formulas of Absolute Trigonometry, instead of the Euclidean or Non-Euclidean, in the argument of which a sketch has just been given. Deduction of Plane Trigonometry from Statics. § II. Let us see, in conclusion, how it is possible to treat the converse (\\XQsi\on: given the law of composition of forces, to deduce the fundamental equations of trigonometry. To this end we note that the magnitude of the resultant R of two equal forces' F, perpendicular to a line A A' of length 2 b, will in general be a function of P and b. Denoting this function by cp {P, b), we have or more simply^ if = cp (P, b), R = P(?{b). I The proportionality of R and P follows from the laiu of association on which the composition of forces depends. In fact, let us imagine each of the forces P, acting at A and A', to be Iq6 Appendix I. The Fundamental Principles of Statics etc. On the other hand in § 9 (p. 193), we were brought to the following expression for J^: sm a Eliminating i? and J^, between these, we have /7\ cos p op (Ò) = - -*- • ^ ^ ^ sin a Thus if the analytical expression for (p (/;) is known, this formula will supply a relation between the sides and angles of a right-angled triangle. To determine qp (fi), it is necessary to establish the corresponding functional equation. With this view, let us apply perpendicularly to the line AA', the four equal forces J^j, F^, P^,, P^,, in such a way that the points of application of jP^ and F^, F^ and jP,, are distant 2 {a-\-b) and 2 {b — a), respectively (cf. Fig. 70). We can determine the resultant R of these four forces in two different ways: (i) Taking F,_ with F2, and F^^ with F^, we obtain two forces Ri, R^ of magnitude: F^{ay, replaced by n equal forces, applied at A and A'. Combining these, we would have for R the expression y? = « cp (^, b\. Comparing this result with the equation given in the text, we have Similarly we have cp (kP, b)-^k(^{P, b), for every rational value of /c; and the formula may be extendeii to irrational values. Then putting P= i and k = P v^t obtain 9 [P, Ò) = P(^ (6). Q. E. D. Deduction of Trigonometry from Statics. 197 and taking R^, R2 together, we obtain R = F(^ {a) qp {ù). (ii) Taking F^ with F^ , we obtain a force of magnitude : F(p{è + a), and taking F2 with F^, we obtain another of magnitude: F(p(^ — a). Taking these two together we have, finally, R = F(i>(ò + a) + Fcp(ù—a). A r-a- b~a R. p. p. R- R Fig. 70. From the two expressions for R we obtain the functional equation which qp (^) satisfies, namely, (2) cp(ò) (p{a) = cp(l> + a) + cp (i> — a). This equation, if we put cp {ò) = 2 fib), is identical with that met in § 6 (p. 189), in treating the composition of concurrent forces. The method followed in finding (2) is due to D'Alem- bert.^ However, if we suppose a and b equal to each other, and if we note that qp io) = 2, the equation reduces to (3) [9(^)]' = qp (2:r) + 2. This last equation was obtained previously by Foncenex, in connection with the equilibrium of the lever.^ 1 Opuscules mathématiqiies, T. VI, p. 371 (1779). 2 Cf. p. 319—322 of the work by FOxNCENEX, referred to above. Iq8 Appendix I. The Fundamental Principles of Statics etc. § 12. The statical problem of the composition of forces is thus reduced to the integration of a functional equation. FoNCENEX, who was the first to treat it in this way^, thought that the only solution of (3), was cp (x) = const. If this were so, the constant would be 2, as is easily verified. Later Laplace and D'Alembert integrated (3), obtaining cp (x) = e <^ + e ^ . where <: is a constant, or any function which takes the same value when x is changed to 2 x/ The solution of Laplace and D'Alembert, applied to the statical problem of the preceding section, leads to the case in which c- is a function of x. Further, since we cannot admit values of c such a.sa+i ù, where a, i> are both different from zero, we have three possible cases, according as c is real, a pure imaginary, or infinite.^ Corresponding to these 1 We have stated above (p. 53), when speaking of FoNCENEX' memoir, that, if it v?as not the vv'ork of Lagrange, it was certainly inspired by him. This opinion, accepted by Genocchi and other geometers, dates from Delambre. The distinguished biographer of Lagrange puts the matter in the following words: "// (Za- gi-aiigé) fournissait à Fonceiiex la parile analyllque de ses mémoires en ltd laissajtl le soin de développer les raisonnements sur lesqueh portaiettl ses formules. En effet, on remarque drja dans ces mémoires (of Foncenex) cede marche purement analitique, qui depuis a fait le caractère des grandes productions de Lagrange. II avail trouvè tt?ie nouvelle théorie dii levier". Notices sur la voie et les ouvrages de M. le Comic Lagrange. Mém. Inst, de France, classe Math, et Physique, T. Xm, p. XXXV (1 8 1 2). 2 Cf. D'Alembert: Sur les principes de la Mécaniqtce : Mém. de l'Ac. des Sciences de Paris (1769). — Laplace: Recherches sur l'intrgraiion des equations diffirentiellcs : Mém. Ac. sciences de Paris (savants étrangers) T. VII (1733). Oeuvres de Laplace, T. Vili, p. 106 — 7. 3 We can obtain this result directly by integrating the equa- The Three Geometries, 199 three cases, we have three possible laws for the composition of forces, and consequently three distinct types of equations connecting the sides and angles of a triangle. These results are brought together in the following table, where k denotes a real positive number. Value of c Form of q) (^) Trigonometri- cal equations Nature of plane c = k X X ek'^-e T_2cosh^ k b cos p cosh-; . — „ X' sin a hyperbolic c = ik i X ix ,k -\-e k = 2 cos — ' k b cos p elliptic c = 00 X X e^+e «' = 2 cos 6 1 =-. — - sm a parabolic Conclusion: The law for the composition of forces per- pendicular to a straight line, leads, in a certain sense, to the relations which hold between the sides and angles of a triangle, and thus to the geometrical properties of the plane and of space. This fact was completely established by A. Genocchi [181 7 — 1889] in two most important papers', to which the reader is referred for full historical and bibliographical notes upon this question. tion (2), or, what amounts to the same thing, equation (l) of S 6. Cf., for this, the elementary method employed by Cauchy for finding the function satisfying (i). Oeuvres de Cauchy , (sér. 2). T. ni, p. 106— 113. I One of them is the Memoir referred to on p. 19 1. The other, which dates from 1869, is entitled: Dei primi principii della meccanica e della geometria in relazione al postulato d'Euclide. Annali della Società italiana delle Scienze (3). T. II, p. 153 — 189. Appendix II. Clifford's Parallels and Surface. Sketch of Clifford-Klein's Problem. Clifford's Parallels. § I. Euclid's Parallels are straight lines possessing the following properties: a) They are coplanar. b) They have no common points. c) They are equidistant. If we give up the condition (c) and adopt the views of Gauss, Lobatschewsky and Bolyai, we obtain a first ex- tension of the notion of parallelism. But the parallels which correspond to it have very few properties in common with the ordinary parallels. This is due to the fact that the most beautiful properties we meet in studying the latter depend principally on the condition (c). For this reason we are led to seek such an extension of the notion of parallelism, that, so far as possible, the new parallels shall still possess the characteristics, which, in Euclidean geometry, depend on their equidistance. Thus, following W. K. Clifford [1845 — 1879], we give up the property of coplanariiy, in the definition of parallels, and retain the other two. The new definition of parallels will be as follows: Two straight lines, iti the same or in different planes, are called parallel, when the points of the one are equidistant from the points of the other. Clifford's Parallels. 201 § 2. Two cases, then, present tlieip.selves, according as these parallels lie, or do not lie, in the same plane. The case in which the equidistant straight lines are coplanar is quickly exhausted, since the discussion in the earher part of this book [§ 8] allows us to state that the corresponding space is the ordinary Euclidean. We shall, therefore, suppose that the two equidistant straight lines r and s T are not in the same plane, and that the perpendiculars drawn from r to J are equal. Obvi- s , A R ously these lines will also be per- , , Fig. 71. pendicular to r. Let AA , BB be two such perpendiculars (Fig. 71). The skew quad- rilateral ABB' A , which is thus obtained, has its four angles and two opposite sides equal. It is easy to see that the other two opposite sides AB, AB' are equal, and that the interior alternate angles, which each diagonal — e. g. AB' — makes with the two parallels, are equal. This follows from the congruence of the two right-angled triangles AAB' and ABB'. If now we examine the solid angle at A, from a theorem valid in all the three geometrical. systems, we can write -^ AAB = i right angle. This inequality, taken along with the fact that the angles AB' A and B' AB are equal, can be written thus: <^ AAB' 4- <^ AB' A > i right angle. Stated in this way, we see that the sum of the acute angles in the right-angled triangle AA B' is greater than a right angle. Thus in the said triangle the Hypothesis of the Obtuse Angle is verified, and consequently parallels ?iot iti the same plane can exist only in the space of Riemann. 202 Appendix II. Clifford's Parallels and Surface. § 3. Now to prove that in the elliptic space of Riemann there actually do exist pairs of straight lines, not in the same plane and equidistant, let us consider an arbitrary straight line r and the infinite number of planes perpendicular to it. These planes all pass through another line r, the polar of r in the absolute polarity of the elliptic space. Any line whatever, joining a point of r with a point of/, is perpend- icular both to r and to /, and has a constant length, equal to half the length of a straight line. From this it follows that r, r are two equidistant straight lines^ not in the same plane. But two such equidistants represent a very particular case, since all the points of r have the same distance not only from /, but from all the points of r. r r / H A M / W K B Fig. 72. To establish the existence of straight lines in which the last peculiarity does not exist, we consider again two lines r and /, one of which is the polar of the other (Fig. 72). Upon these let the equal segments AB^ AB' be taken, each less than half the length of a straight line. Joining A with A^ and B with B' , we obtain two straight lines ^, b, not polar the one to the other, and both perpendicular to the lines r, r . It can easily be proved that a, b are equidistant. To show this, take a segment AH upon AA; then on the The Polars as Parallels. 203 supplementary line ^ to AHA, take the segment ^i^ equal to AH. If the poinfs H and M are joined respectively with £^ and B, we obtain two right-angled triangles A£H, ABM, which, in consequence of our construction, are congruent. We thus have the equality HB' = B3f. Now if H and B are joined, and the two triangles HBB' and HBM zx^ compared, we see immediately that they are equal. They have the side HB common, the sides HB' and MB equal, by the preceding result, and finally BB' and HM are also equal, each being half of a straight line. This means, in other words, that the various points of the straight line a are equidistant from the line b. Now since the argument can be repeated, starting from the line b and dropping the perpendiculars to a, we conclude that the line HK^ in addition to being perpendicular to b, is also perpend- icular to a. We remark, further, that from the equality of the various segments AB, HK, A B\ . . . the equality of the re- spective supplementary segments is deduced, so that the two lines a, b, can be regarded as equidistant the one from the other, in two different ways. If then it happened that the line AB were equal to its supplement, we would have the ex ceptional case, which we noted previously, where a, b are the polars of each other, and consequently all the points of a are equidistant from the different points of b. § 4. The non-planar parallels of elliptic space were discovered by Clifford in 1873.^ Their most remarkable properties are as follows: 1 The two different segments, determined by two points on a straight line, are called supplementary. 2 Preliminary Sketch of Biquaternions. Proc. Lond. Math. Soc. Vol. IV. p. 381— 395(1873). Clifford's Mathematical Papers, p. 181—200. 204 Appendix II. Clifford's Parallels and Surface. fi) If a siraigJit line meets two parallels, it makes with the»! equal eorrespo?iding angles, equal interior alternate angles, etc. (ii) If in a skew quadrilateral the opposite sides are equal and the adjacent angles supplemcjitary, then the opposite sides are parallel. Such a quadrilateral can therefore be called a ske:a parallelogram . The first of these two theorems can be immediately verified; the second can be proved by a similar argument to that employed in § 3. (iii) If two straight lines are equal and parallel, ajid their extremities are suitably joined, we obtain a skezv paral- lelogram. This result, which can be looked upon, in a certain sense, as the converse of (ii), can also be readily established. (iv) Through a?iy point (AI) in space, which does not lie on the polar of a straight line (r), two parallels can be drawn to that line. Indeed, let the perpendicular MN be drawn from M to r, and let N' be the point in which the polar of MN meets r (Fig. 73). From this polar cut off the two segments N' M' , N'AI", equal to NM, and join the points M', M" to M. The two lines /, r", thus ob- tained, are the required par- allels. If M lay on the polar of r, then MN would be equal to half the straight line; the two points M' , M" would coincide: and the two parallels /, r" would also coincide. Fig- 73- Properties of Clifford's Parallels. 205 The angle between the t.vo parallels /, r" can be measured by the segment MM", which the two arms of the angle intercept on the polar of its vertex. In this way we can say that half of the angle between r and r", that is, the angle 0/ parallelism, is equal to the distance of parallelism. To distinguish the two parallels /, r", let us consider a helicoidal movement of space, with MN for axis, in which the pencil of planes perpendicular to MJV, and the axis J/' J/ ' of that pencil, obviously remain fixed. Such a movement can be considered as the resultant of a translation along MJV, accompanied by a rotation about the same axis: or by two translations, one along MN, the other along M'M". If the two translations are of equal amount, we obtain a space vector. Vectors can be right-handed or left-handed. Thus, referr- ing to the two parallels /, r", it is clear that one of them will be superposed upon r by a right-handed vector of magnitude AfJV, while the other will be superposed on r by a left-handed vector of the same magnitude. Of the two lines r, r", one could be called the right-handed parallel and the other the left-handed parallel to r. (v) Two right-handed {or left-handed) parallels to a straight line are I'ight-handed {or left-handed) parallels to each other. Let b, c be two right-hand- ed parallels to a. From the two points A, A of a, distant from each other half the length of a straight Hne, draw the perpendiculars AB, AB' on b, and the perpendiculars AC, AC on c (cf. Fig. 74). The lines AB', AC are the polars of AB and AC. Therefore ^ BAC = <^B'AC. B /^/ B" A A' Fig. 74- 206 Appendix II. Clifford's Parallels and Surface. Further^ by the properties of parallels AB = AB\ AC^AC. Therefore the triangles ABC, A JS C are equal Thus it follows that BC = B'C. Again, since BB' = AA = CC\ the skew quadrilateral BBC' C has its opposite sides equal. But to establish the parallelism of b, c, we must also prove that the adjacent angles of the said quadrilateral are supplementary (cf ii). For this we compare the two solid angles B {AB' C) and B' (AB"C'). In these the following relations hold: ^ABB' = -^ AB'B" = I right angle ^ ABC = <^ AB'C. Further, the two dihedral angles, which have BA and B'A' for their edges, are each equal to a right angle, dimin- ished (or increased) by the dihedral angle whose normal section is the angle ABB'. Therefore the said two solid angles are equal. From this the equality of the two angles B' BC, B'B'C follows. Hence we can prove that the angles B, B' of the quadri- lateral BB' C C are supplementary, and then (on drawing the diagonals of the quadrilateral, etc.) that the angle B is supplementary to C, and C supplementary to C, etc. Thus b and c are parallel. From the figure it is clear that the parallelism between b and c is right-handed, if that is the nature of the parallelism between the said lines and tlie line a. Clifford's Surface. § 5. From the preceding argument it follows that all the lifies which meet three right-handed parallels are left-handed parallels to each other. Clifford's Surface. 207 Indeed, if ABC is a transversal cutting the three lines a, b, c, and if three equal segments AA\ BB\ CC are taken on these lines in the same direction," the points A'B'C lie on a line parallel to ABC. The psjallelism between ABC and A'B'C is thus left-handed. From this we deduce that three parallels a, b, c, define a ruled surface of the second order (Clifford's Surface). On this surface the lines cutting a, b, c form one system of generators {g^: the second system of generators {gd) is formed by the infinite number of lines, which, like a, ^, c, meet {gs). Clifford's Surface possesses the following charact- eristic properties: a) Two generators of the same system are parallel to each other. b) Two generators of opposite systems cut each other at a constant atigle. § 6. We proceed to show that Clifford's Surface has t7vo distinct axes of rcvolutiofi. To prove this, from any point M draw the parallels d (right-hand- ed), s (left-handed), to a line r, and denote by Ò the distance MN of each parallel from r (cf. Fig. 75). Keeping d fixed, let s rotate about r, and let /, /', /" , positions which s takes in this rotation Fig. 75- . be the successive I It is clear that if a direction is fixed for one line, it is then fixed for every line parallel to the first. 208 Appendix II. Clifford's Parallels and Surface. It is clear that s, s', s" . . . are all left-handed parallels to r and that all intersect the line d. Thus s in its rotation about r generates a Clifford's Surface. Vice versa, if d and j- are two generators of a Clifford's Surface, which pass through a point M of the surface, and 2 Ò the angle between them, we can raise the perpendicular to the plane sd at M and upon it cut off the lines AIL = MiV = Ò. Let Z> and ^ be the points where the polar of ZiV meets the lines d and s, respectively, and let i^be the middle point ofZ'^= 2Ò. Then the lines HL and HIV are parallel, both to s and d. Of the two lines HZ and HIV choose that which is a right-handed parallel to d and a left-handed parallel to s, say the line HIV. Then the given Clifford's Surface can be generated by the revolution of s or d about HIV. In this way it is proved that every Clifford's Surface possesses one axis of rotation and that every point on the surface is equidistant from it. The existence of another axis of rotation follows im- mediately, if we remember that all the points of space, equi- distant from HN., are also equidistant from the line which is the polar of HN. This line will, therefore, be the second axis of rotation of the Clifford's Surface. § 7. The equidistance of the points of Clifford's Surface from each axis of rotation leads to another most remarkable property of the surfaces. In fact, every plane passing through an axis r intersects it in a line equidistant from the a.xis. The points of this line, being also equally distant from the point {O) in which the plane of section meets The Axes of Clifford's Surface. 209 the other axis of the surface, lie on a circle, whose centre (O) is the pole of /■ with respect to the said line. Therefore the meridians and the parallels of the surface are circles. The surface can thus be generated by making a circle rotate about the polar of its cetitre, or by making a circle move so that its centre describes a straight line, while its plane is maintained constantly perpendicular to it (Bianchi).' This last method of generating the surface, common also to the Euclidean cylinder, brings out the analogy be- tween Clifford's Surface and the ordinary circular cyhnder This analogy could be carried further, by considering the properties of the hehcoidal paths of the points of the surface, when the space is submitted to a screwing motion about either of the axes of the surface. § 8. Finally, we shall show that the geometry on Clif- ford's Surface, understood in the sense explained in §§ 67, 68, is identical with Euclidean geometry. To prove this, let us determine the law according to which the element of distance between two points on the surface is measured. Let u, V, be respectively a parallel and a meridian through a point O on the surface, and M any arbitrary point upon it. Let the meridian and parallel through M cut off the arcs OP, OQ from u and v. The lengths u, ?> of these arcs will be the coordinates of Q Jlf. The analogy between the system of coordinates here adopted and the Cartesian orthogonal system is evident (cf. Fig. 76). Fig. 75. I Sulla siipeificie a curvatiaa nulla in geometria ellittica. Ann. di Mat. (2) XXIV, p. 107 (1896). Also Lezioni di Geometria Differ- enziale. 2a Ed., Voi. I, p. 454 (Pisa, 1902). 14 2 IO Appendix II. Clifford's Parallels and Surface. Let M' be a point whose distance from M is infini- tesimal. If {u, v) are the coordinates of J/, we can take {u + du, V + dv) for those of M' . Now consider the infinitesimal triangle MM' N., whose third vertex N is the point in which the parallel through AI intersects the meridian through M' . It is clear that the angle MNM' is a right angle, and that the sides MN, NM' are equal to du^ dv. On the other hand, this triangle can be regarded as rectilinear (as it lies on the tangent plane at M). So that, from the properties of infinitesimal plane triangles, its hypo- tenuse and its sides, by the Theorem of Pythagoras, are con- nected by the relation ds^ = du^ -^ dv^. But this expression for ds* is characteristic of ordinary geometry, so that we can immediately deduce that the pro- perties of the Euclidean plane hold i?i every normal region on a Clifford's Surface. An important application of this result leads to the evaluation of the area of this surface. Indeed, if we break it up into such congruent infinitesimal parallelograms by means of its generators, the area of one of these will be given by the ordinary expression dx dy sin 9, where dx, dy are the lengths of the sides and is the con- stant angle between them (the angle between two generators). The area of the surface is therefore E dx dy sin = sin 9 2 dx • 2 dy. But both the sums 2 dx, 2 dy represent the length / of a straight line. Therefore the area A of Clifford's Surface takes the very simple form. The Area of Clifford's Surface. 211 A = /^ sin e, which is identical v/ith the expression for the area of a EucHdean parallelogram (Clifford).' Sketch of Clifford-Klein's Problem. § 9. Clifford's ideas, explained in the preceding sections, led Klein to a new statement of the fundamental problem of geometry. In giving a short sketch of Klein's views, let us refer to the results of § 68 regarding the possibility of interpret- ing plane geometry by that on the surfaces of constant curvature. The contrast between the properties of the Eu- chdean and Non-Euclidean planes and those of the said surfaces was there restricted to suitably bounded regions. In extending the comparison to the unbounded regions, we are met, in general, by differences; in some cases due to the presence of singular points on the surfaces (e. g., vertex of a cone); in others, to the different connectivities of the surfaces. Leaving aside the singular points, let us take the cir- cular cylinder as an example of a surface of constant curv- ature, everywhere regular, but possessed of a connectivity different from that of the Euclidean plane. The difference between the geometry of the plane and that of the cylinder, both understood in the complete sense, has been already noticed on p. 140, where it was observed that the postulate of congruence between two arbitrary straight lines ceases to be true on the cylinder. Nevertheless there are numerous properties common to the two geometries, I Preliminary Sketch, cf. p. 203 above. The properties of this surface were referred to only very briefly by Clifford in 1873. They are developed more fully by Klein in his memoir: Zur nichl- euklidischen Geometrie, Math. Ann. Bd. XXXVII, p. 544—572 (1890). 14* 212 Appendix II. Clifford's Parallels and Surface. which have their origin in the double characteristic, that both the plane and the cylinder have the same curvature, and that they are both regular. These properties can be summarized thus: i) The geometry of a?iy normal region of the cylinder is identical with that of any normal region of the plane. 2) The geometry of any normal region whatsoever of the cylinder, fixed with respect to an arbitrary point upon it, is identical with the geometry of any normal region what- soever of the plane. The importance of the comparison between the ge- ometry of the plane and that of a surface, founded on the properties (i) and (2), arises from the following consid- erations : A geometry of the plane, based upon experimental criteria, depends on two distinct groups of hypotheses. The first group expresses the validity of certain facts, directly observed in a region accessible to experiment {postulates of the normal region); the second group extends to inaccessible regions some properties of the initial region {postulates of extension). The postulates of extension could demand, e. g., that the properties of the accessible region should be valid in the entire plane. We would then be brought to the two forms, the parabolic and the hyperbolic plane. If, on the other hand, the said postulates demanded the extension of these pro- perties, with the exception of that which attributes to the straight line the character of an open line, we ought to take account of the elliptic plane as well as the two planes mentioned. But the preceding discussion on the regular surfaces of constant curvature suggests a more general method of enun- ciating the postulates of extension. We might, indeed, simply demand that the properties of the initial region should hold in the neighbourhood of every point of the plane. In this Clifford-Klein's Problem. 21 3 case, the class of possible forms of planes receives con- siderable additions. We could, e. g., conceive a form with zero curvature, of double connectivity, and able to be com- pletely represented on the cyhnder of Euclidean space. The object of Clifford-Klein' s problem is the determination of all the two dimensional manifolds of constant curvature, which are everyiohere regular. § 10. Is it possible to realise, with suitable regular surfaces of constant curvature, in the Euclidean space, all the for7tis of Clifford-Klein ? The answer is in the negative, as the following example clearly shows. The only regular developable surface of the Euclidean space, whose geometry is not identical with that of the plane, is the cylinder with closed cross-section. On the other hand, Clifford's Surface in the elliptic space is a regular surface of zero curvature, which is essentially different from the plane and cylinder. However with suitable conventions we can represent Clifford's Surface even in ordinary space. Let us return again to the cylinder. If we wish to un- fold the cylinder, we must first render it simply connected by a cut along a generator {g); then, by bending without stretching, it can be spread out on the plane, covering a strip between two parallels igxigz)- There is a one-one correspondence between the points of the cylinder and those of the strip. The only exception is afforded by the points of the generator (^), to each of which correspond two points, situated the one on^i, the other on g2. However, if it is agreed to regard these two points as idefitical, that is, as a single point, then the correspondence becomes one-one without exception, and the geometry of the strip is completely identical with thai of the cylinder. 214 Appendix II. Clifford's Parallels and Surface. A representation analogous to the above can also be adopted for Clifford's Surface. First the surface is made simply connected by two cuts along the intersecting gener- ators {g, g). In this way a skew parallelogram is obtained in the elliptic space. Its sides have each the length of a straight line, and its angles G and 9' [O + 0'= 2 right angles] are the angles between g and g. This being done, we take a rhombus in the Eu- clidean plane, whose sides are the length of the straight line in the elliptic plane, and whose angles are 0, 6'. On this rhombus Clifford's Surface can be represented congruenti}' (developed). The correspondence between the points of the surface and those of the rhombus is a one-one correspond- ence, with the exception of the points of^ and^', to each of which correspond two points, situated on the opposite sides of the rhombus. However, if we agree to regard these points as identical, two by two, then the correspondence becomes one-one without exception, and the geometry of the rhombus is completely identical 7oith that of Clifford's Surface.'^ § II. These representations of the cylinder and of Clifford's Surface show us how, for the case of zero curva- ture, the investigation of Clifford-Klein's forms can be reduced to the determination of suitable Euclidean polygons, eventually degenerating into strips, whose sides are two by two transformable, one into the other, by suitable movements of the plane, their angles being together equal to four right- angles (Klein).* Then it is only necessary to regard the points of these sides as identical, two by two, to have a representation of the required forms on the ordinary plane. I Cf. Clifford loc. cit. Also Klei.n's memoir referred to on p. 2X1. * Cf. the memoir just named. Clifford-Klein's Problem. 215 It is possible to present, in a similar way, the investi- gation of Clifford-Klein's forms for positive or negative values of the curvature, and the extension of this problem to space.' I A systematic treatment of Clifford-Klein's problem is to be found in Killing's Eiiifilhrung in die Gnindlagen der Geometrie. Bd. I, p. 271 — 349 (Paderborn, 1893). Appendix III. The Non-Euclidean Parallel Construction and other Allied Constructions. § I. The Non-Euclidean Parallel Construction depends upon the correspondence between the right-angled triangle and the quadrilateral with three right angles. Indeed, when this correspondence is known, a number of different con- structions are immediately at our disposal.* To express this correspondence we introduce the following notation: In the right-angled triangle, as usual, a, b are the sides: c is the hypotenuse: X is the angle opposite a and fi that opposite b. Further the angles of parallelism for a, b are denoted by a and p: and the lines which have X, ]x. for angles of parallelism are denoted by /, tn. Also two lines, for which the corresponding angles of parallelism are com- plementary, are distinguished by accents, e. g.: n {d) = I - n(^), n(/') = ^ - ^ (^^- Then with this notation: To every right-angled triangle {a, b, c, X, \x) there corresponds a quadrilateral with three right-angles^ whose fourth angle (acute) is P, a7id whose sides are c, m\ a, /, taken in order from the corner at which the angle is p. The converse of this theorem is also true. I Cf. p. 256 of Engel's work referred to on p. Correspondence between Quadrilateral and Triangle. 217 The following is one of the constructions, which can be derived from this theorem, for drawing the parallel through A to the line BC (cf. Fig. 77). Let AB be the perpendicular from A to BC. At A draw the line perpendicular to AB, and from any point C in BC draw the perpendicular CD ^ 3 to this line. With centre A and rad- ius BC (equal to c) describe a circle cutting CD in E. Now we have ^ EAD = M, and therefore •^ BAE = -^ — ^ = n (;//). But the sides of the quadrilateral are c, m', a, /, taken in order from C. Therefore A£ is parallel to BC. If a proof of this construction is required without using the trigonometrical forms, one might attempt to show direct- ly that the line AE produced, (simply owing to the equality of BC and A£), does not cut BC produced, and that the two have not a common perpendicular. If this were the case, they would be parallel. Such a proof has not yet been found. Again, we might prove the truth of the construction using the theorem, that in a prism of triangular section the sum of the three dihedral angles is equal to two right angles': so that for a prism with n angles the sum is (2 n—4) right angles. This proof is given in § 2 below. . ^ Cf. LoBATSCHEWSKY (Engel's translation) p. 172. 2l8 Appendix III. The Non-Euclidean Parallel Construction. Finally, the correspondence stated in the above theorem — only part of which is required for the Parallel Construction of Fig. 78 ■ — can be verified without the use of the geo- metry of the Non-Euclidean space. This proof is given in S 3- § 2. Direct proof of the Parallel Construction by fneans of a Prism. Q Fig. 78. Let ABCD be a plane quadrilateral in which the angles at Z>, Ay B are right angles. Let the angle at C be denoted by p, AD by a, DC by /, CB by c, and BA by m. At A draw the perpendicular ^Q to the plane of the quadrilateral. Through B, C, and Z? draw ^Q, CQ and Z>S2 parallel to A^. Also through A draw AQ parallel to BC, cutting CD in E {ED = b^, and let the plane through A^ and AE cut CZPQ in EQ.. From the definition, we have ^EAD n (?//) Further the plane ^lAB is at right angles to a, and the plane Q.DA at right angles to /, since ^A and AB are per- pendicular to a, while QZ> and a; are perpendicular to /. Direct Proof of the Parallel Construction. 2I9 IT Also <^ AB9. = <^ OAB = -^ — ^ In the prism Q {ABCD) the faces which meet in Q^, ^.B, QD are perpendicular. Also the four dihedral angles make up four right angles. It follows that the faces of the prism C (DBQ), which meet along CQ, are perpendicular. Also it is clear that in £ (DQA) the faces which meet in £A are perpendicular, while the dihedral angle for the edge CD is the same as for £D (thus equal to a). We shall now prove the equality of the other dihedral angles in these prisms C {DBQ.) and E (DQA) — those con- tained by the faces which meet in CB and AE. In the first prism this angle is equal to the angle be- tween the planes ABCD and CBQ. It is thus equal to |U, i. e. it is equal to <^ ABQ.. In the second prism, the angle between the planes meeting in EQ belongs also to the prism Q {ADE). In this the angle at Q.D is a right-angle, and that at QA is equal IT to H- Thus the third angle is equal to |li. Therefore the prisms C {DB9.) and E (DQJ) are congruent Therefore ^ BCQ = ^ QEA, and the lines which have these angles of parallelism are also equal. Thus c = BC and ^i = AE are equal, which was to be proved. Further it follows that ^ DEA = <^ DCQ; i. e.the angle Xj, opposite the side a of the triangle, is given by X^ = TT (/) = X. Finally ^ DCB == ^ DEQ; i.e. P = 17 (d,), or Ù, = a. 220 Appendix III. The Non-Euclidean Parallel Construction. Thus the correspondence between the triangle and the quadrilateral is proved.^ § 3. Proof of the Correspondence by Plane Geometry. In the right-angled triangle ABC produce the hypo- tenuse AB to D, where the perpendicular at D is parallel to C^(cf. Fig. 79). Fig- 79- Then with the above notation BD = m. Draw through A the parallel to Z>0 and CBQ. Then ^ CAQ = p = n {b), and it is also equal to X + j9; and let the ray b' = B' O be drawn, the image of the line b in a (cf. Fig. 82). Then, using the preceding con- struction^ let the line BiBj,' be drawn parallel to the two lines which con- tain the angle BOB . This line, from the symmetry of the figure with respect to a, is perpendicular to OA. It follows that BiB\ is parallel to one of the lines which contain the angle AOB and perpendicular to the other. ^.? 7. The constructions given above depend upon metrical considerations. However it is also possible to make use of the fact that to the metrical definitions of perpend- B' Fig. Sz. Projective Constructions. 225 icularity and parallelism a projective meaning can be given (§ 79), and that projective geometry is independent of the parallel postulate (§ 80). Working on these lines, what will be the construction for the parallels through a point A to a. given line? Let the points /'i, 1*2, P^ ^^^ P^i ^2', P^ be given on g so that the points P^ , P^ •, P^, are all on the same side of Pi, Pi, P^, and p,p,' = p,p; = p,p;. Join AP-i, ÀP2, APt^ and denote these Hnes by s^, s^, and Sy Similarly let AP^', AP^', AP.' be denoted by Si', $2 and J3'. Then the three pairs of rays through A^ determ- ine a projective transformation of the pencil is) into itself, the double elements of which are obviously the two parallels which we require. These double elements can be constructed by the methods of projective geometry.^ The absolute is then determined by five points: i. e., by five pairs of parallels; and so all further problems of metrical geometry are reduced to those of projective geometry. If we represent (cf. § 84) the Lobatschewsky-Bolyai Geometry (e. g., for the Euclidean plane) so that the image of the absolute is a given conic (not reaching infinity), then it has been shown by Grossmann^ that most of the problems for the Non-Euchdean plane can be very beautifully and easily solved by this 'translation'. However we must not forget that this simplicity disappears, if we would pass from the 'translation' back to the 'original text'. 1 Cf. for example, Enriques, Geometria proiettiva, (referred to on p. 156) S 73- 2 Gross.mann, Die fiiiidamentalen Konstriiklioneti der nicht- eiiklidiscken Geometrie, Programm der Thurgauischen Kantonschule, (Frauenfeld, 1904). 15 226 Appendix III. The Non-Euclidean Parallel Construction. In the Non-Euclidean plane the absolute is inaccessible, and its points are only given by the intersection of pencils of parallels. The points Outside of the absolute, while they are accessible in the 'translation', cannot be reached in the 'text' itself. In this case they are pencils Of straight lines, which do not meet in a point, but go through the (ideal) pole of a certain line with respect to the absolute. If, then, we would actually carry out the constructions, difficulties will often arise, such as those we meet in the translation of a foreign language, when we must often sub- stitute for a single adjective a phrase of some length. Appendix IV. The Independence of Projective Geometry from Euclid's Postulate. § I. Statement of the Frobietn. In the following pages we shall examine more carefully a question to which only passing reference was made in the text (cf. § 80), namely, the validity ofProjective Geometry in Non-Euclidean Space, since this question is closely related to the demonstration of the independence of that geometry from the Fifth Postulate. In elliptic space (cf § 80) we may assume that the usual projective properties of figures are true, since the postulates of projective geometry are fully verified. Indeed the absence of parallels, or, what amounts to the same thing, the fact that two coplanar lines always intersect, makes the foundation of projectivity in elliptic space simpler than in Eu- clidean space, which, as is well known, must be first com- pleted by the points at infinity. However in hyperbolic space the matter is more com- plicated. Here it is not sufficient to account for the absence of the point common to two parallel lines, an exception which destroys the validity of the projective postulate: — two coplanar lines have a coinmon point. We must also remove the Other exception — the existence of coplanar lines which do not cut each other, and are not parallel {the non-inter- secting lines). The method, which we shall employ, is the same as that used in dealing with the Euclidean case. We introduce fictitious points^ regarded as belonging to two co- planar lines which do not meet. IS* 228 ApP- I^' ^^^ Indcpend. of Proj. Geo. from Euclid's Post. In the following paragraphs, keeping for simplicity to two dimensions only, we show how these fictitious points can be introduced on the hyperbolic plane, and how they enable us to establish the postulates of projective geometry without exception. Naturally no distinction is now made be- tween \kit proper poitits, that is, the Ordinary points, and the fictitious points, thus introduced. § 2. Improper Points and the Complete Projective Plane. We start with the pencil of lines, that is, the aggregate of the lines of a plane passing through a point. We note that through any point of the plane, which is not the vertex of the pencil^ there passes one, and only one, line of the pencil. On the hyperbolic plane, in addition to the pencil, there exist two other systems of lines which enjoy this property, namely; — (i) the set of parallels to a line iti one direction', (ii) the set of perpendiculars to a line. If we extend the meaning of the term, pencil of lines, we shall be able to include under it the two systems of lines above mentioned. In that case it is clear that t7vo arbi- trary lines of a plane will determine a pencil, to 7a hie h they belong. If the two lines are concurrent , the pencil is formed by the set of lines passing through their common point; if they are parallel, by the set of parallels to both, in the same direction; finally, if they are nofi-ifitersectifig, by all the lines which are orthogonal to their common perpendicular. In the first type of pencil (Ù^e proper pencil), there exists a point common to all its lines, the vertex of the pencil; in the two other types (the improper pencils), this point is lacking. IVe shall now introduce, by convention, a fictitious entity, called an improper point, and regard it as pertainitig to all the lines of the pencil. With this convention, every pencil has a vertex, The Complete Line and Plane. 22Q which will be a proper point, or an improper point, accord- ing to the different cases. The hyperbolic plane, regarded as the aggregate of all its points, proper and improper, will be called the complete projective plane. § 3. The Complete Projective Line. The improper points are of two kinds. They may be the vertices of pen- cils of parallels, or the vertices of pencils of non-intersecting lines. The points of the first species are obtained in the same way, and have the same use, as the points at infinity common to two Euclidean parallels. For this reason we shall call them points at infinity on the hyperbolic plane, when it is necessary to distinguish them from the others. The points of the second species will be called ideal points. It will be noticed that, while every line has only one point at infinity on the Euclidean plane, it has tivo points at infinity on the hyperbolic plane, there being two distinct directions of parallelism for each line. Also that, while the line on the Euclidean plane, with its point at infinity, is closed, the hyperbolic line, regarded as the aggregate of its proper points, and of its two points at infinity, is open. The hyperbolic line is closed by associating with it all the ideal points, which are common to it and to all the lines on the plane which do not intersect it. From this point of view we regard the line as made up of two segments., whose common extremities are the two points at infinity of the line. Of these segments, one contains, in addition to its ends, all the proper points of the line; the other all its improper points. The line, regarded as the aggregate of its points, proper and improper, will be called the complete projective line. § 4. Combination of Elements. We assume for the concrete representation of a point of the complete projective plane: — 2'ZO App. IV. The Independ. of Proj. Geo. from Euclid's. Post. (i) its physical image, if it is a proper point; (ii) a line which passes through it, and the relative direction of the line, if it is a point at infinity; (iii) the common perpendicular to all the lines passing through it, if it is an ideal point. We shall denote a proper point by an ordinary capital letter; an improper point by a Greek capital; and to this we shall add, for an ideal point, the letter which will stand for the representative line of that point. Thus a point at infinity will be denoted, e. g., by Q, while the ideal point, through which all lines perpendicular to the line o pass, will be denoted by Qo- On this understanding, if we make no distinction be- tween proper points and improper points, not only can we affirm the unconditional validity of the projective postulate: two arbitrary lines have a common point: but we can also construct this point, understanding by this construction the process of obtaining its concrete representation. In fact, if the lines meet, in the ordinary sense of the term, or are parallel, the point can be at once obtained. If they are non-inter- secting, it is sufficient to draw their common perpendicular, according to the rule obtained in Appendix III S 4- On the other hand, we are not able to say that the second postulate of projective gtovaeXry—tivo points determine a line — and the corresponding constructions, are valid un- conditionally. In fact no line passes through the ideal point Qo and through the point at infinity Q on the line tent AB as AB OA. OB We shall now show that the Ideal length of an Ideal Segment is unaltered by inversivi with regard to any circle of the system. Fig. So. Let OD be any circle of the system and let C be its centre (Fig. 87). Then inversion changes an Ideal Line into an Ideal Line. Let the Ideal Segment AB invert into the Ideal Segment A'B'. These two Ideal Lines intersect at the point D, where the circle of inversion^meets AB. Then the Ideal Length of AD AD 1 A'D the Ideal Length of ^'^ ~ OA. ODj OA' . OD __ AD OA ~' 'ad ' OA' But from the triangles CAD, CAD and OAC, OA'C, we find Ideal Displacements. 245 AD CA CD CA CO AO A^O' Thus the Ideal Length oi AD = the Ideal Length oiA'D. Similarly we find BD and B'D have the same Ideal Length, and therefore AB and A'B' have the same Ideal Length. Fig. 87. § 6. Ideal Displacements. The length of a segment must be unaltered by dis- placement. This leads us to consider the definition of Ideal Displacement. Any displacement may be produced by re- peated applications of reflection; that is, by taking the image of the figure in a line (or in a plane, in the case of solid geometry). For example, to translate the segment AB (cf. Fig. 88) into another position on the same straight line, we 246 Appendix V. Impossibility of proving Euclid's Postulate. may reflect the figure, first about a line perpendicular to and bisecting BB' , and then another reflection about the middle point of AB would bring the ends into their former positions relative to each other. Also to move the segment AB into A B B' — I A' Fig. the position AB' (cf. Fig. 89) we can first take the image ot AB in the line bisecting the angle between AB and AB\ and then translate the segment along AB' to its final position. We proceed to show that inversion about any circle of the system is equivalent to reflection of the Ideal Points and Lines in the Ideal Line which coincides with the circle of Ì7iversion. Let C (Fig. 90) be the centre of any circle of the system, and let A be the inverse of any point A with regard to this circle. Then the circleO AA' is orthogonal to the circle of inversion. In other words, such inversion changes any point A into a point A on the Ideal Line perpendicular to the circle of in- version. Also the Ideal Line AA is 'bisected' by that circle at M, since the Ideal Segment AM inverts into the segment AM, and Ideal Lengths are unaltered by such inversion. Again let AB be any Ideal Segment, and by inversion Fig. 89. Ideal Reflection. 247 with regard to any circle of the system let it take up the position AS (Fig. 8 7). We have seen that the Ideal Length of the segment is unaltered: and it is clear that the two segments, when produced, meet on the circle of inversion, and make equal angles with it. Also the Ideal Lines A A Fig. 90. and BB' are perpendicular to, and 'bisected' by, the Ideal Line with which the circle of inversion coincides. Such an inversion is, therefore, the same as reflection, and translation will occur as a special case of the above, when the circle of inversion is orthogonal to the given Ideal Line. We thus define Ideal Reflection m an Ideal Line as in- version with this line as the circle of inversion. It is unnecessary to say more about Ideal Displace- fne?its than that they will be the result of Ideal Reflection. With these definitions it is now possible to 'translate' every proposition in the ordinary plane geometry into a 248 Appendix V. Impossibility of proving Euclid's Postulate. corresponding proposition in this Ideal Geometry. We have only to use the words Ideal Points, Lines, Parallels, etc., instead of the ordinary points, lines, parallels, etc. The argument employed in proving a theorem, or the con- struction used in solving a problem, will be applicable, word for word, in the one geometry as well as in the other, for the elements involved satisfy the same laws. This is the 'dictionary' method so frequently adopted in the previous pages of this book. § 7. Extension to Solid ^ Geometry. The System of Spheres passing through a fixed point. These methods may be extended to solid geometry. In this case the inversion of the system of points, lines, and planes gives rise to the system of points, circles intersecting in the centre of inversion, and spheres also intersecting in that point. The geometry of this system of spheres could be derived from that of the system of points^ lines and planes, by interpreting each proposition in terms of the inverse figures. For our purpose it is better to regard it as derived from the former by the invention of the terms: Ideal Point, Ideal Line, Ideal Plane, Ideal Length and Ideal Dis- placement. The Ideal Point is the same as the ordinary point, but the point O is excluded from the domain of Ideal Points. The Ideal line through two Ideal Points is the circle of the system which passes through these two points. The Ideal Flafie through three Ideal Points, not on an Ideal Line, is the sphere of the system which passes through these three points. Thus the plane geometry, discussed in the preceding articles, is a special case of this plane geometry. Ideal Parallel Lines are defined as before. The line through A parallel to ^C is the circle of the system, lying Extension to Solid Geometry. 249 on the sphere through O, Ay B, and C, which touches the circle given by the Ideal Line .BC at O and passes through A. It is clear that an Ideal Line is determined by two points, as a straight line is determined by two points. An Ideal Plane is determined by three points, not on an Ideal Line, as an ordinary plane is determined by three points, not on a straight line. If two points of an Ideal Line lie on an Ideal Plane, all the points of the line do so : just as if two points of a straight line lie on a plane, all its points do so. The intersection of two Ideal Planes is an Ideal Line; just as the intersection of two ordinary planes is a straight line. The measurement of angles in the two spaces is the same. For the measurement of length we adopt the same de- finition of Ideal Length as in the case of two dimensions. The Ideal Length of an Ideal Segment is the length of the rectilinear segment to which it corresponds. To these defi- nitions it only remains to add that of Ideal Displacement. As in the two dimensional case, this is reached by means of Ideal Reflection : and it can easily be shown that if the system of Ideal Poitits, Lines and Planes is inverted with regard to one of its spheres, the result is equivalent to a reflection of the system in this Ideal Platie. This Ideal Geometry is identical with the ordinary Euclidean Geometry. Its elements satisfy the same laws: every proposition vaUd in the one is also valid in the other: and from the results of Euclidean Geometry those of the Ideal Geometry can be inferred. In the articles that follow we shall establish an Ideal Geometry whose elements satisfy the axioms upon which the Non-Euclidean Geometry of Bolyai-Lobatschewsky is based. The points, lines and planes of this geometry will be figures of the Euclidean Geometry, and from the known properties of these figures, we could state what the corresponding the- orems of this Non-Euclidean Geometry would be. Also from 2 co Appendix V. Impossibility of proving Euclid's Postulate. some of its constructions, the Non-Euclidean constructions could be obtained. This process would be the converse of that referred to in dealing with the Ideal Geometry of the preceding articles; since, in that case, we obtained the the- orems of the Ideal Geometry from the corresponding Eu- clidean theorems. The Geometry of the System of Circles Orthogonal to a Fixed Circle. § 8. Ideal Poiiiis, Ideal lines and Ideal Parallels. In the Ideal Geometry discussed in the previous articles, the Ideal Point was the same as the ordinary point, and the Ideal Lines and Planes had so far the characteristics of straight lines and planes that they were lines and surfaces respectively. Geometries can be constructed in which the Ideal Points, Lines and Planes are quite rem.oved from ordinary points, lines, and planes: so that the Ideal Points no longer have the characteristic of having no parts: and the Ideal Lines no longer boast only length, etc. What is required in each geometry is that the entities concerned satisfy the axioms which form the foundations of geometry. If they satisfy the axioms of Euclidean Geometry, the argu- ments, which lead to the theorems of that geometry, will give corresponding theorems in the Ideal Geometry: and if they satisfy the axioms of any of the Non-Euclidean Geom- etries, the arguments^ which lead to theorems in that Non- Euclidean Geometry, will lead equally to theorems in the corresponding Ideal Geometry. We proceed to discuss the geometry of the system of circles orthogonal to a fixed circle. Let the fundamental circle be of radius k and centre O. Let A, A" be any two inverse points, A being inside the circle. Every such pair of points {A, A'), is an Ideal Point {A) of the Ideal Geometry with which we shall 71010 deal. Circles orthogonal to a fixed Circle. 251 If two such pairs of points are given — that is, two Ideal Points (A, B), (Fig. 92) — these determine a circle which is orthogonal to the fundamental circle. Every such circle is a?i Ideal Line of this Ideal Geometry. Fig. 91. Hence any two different Ideal Points determine an Ideal Line. In the case of the system of circles passing through a fixed point O, this point O was excluded from the domain of the Ideal Points. In this system of circles all orthogonal to the fundamental circle, the coincident pairs of points lying on the circumference of that circle are excluded from the domain of the Ideal Points. We define the angle between two Ideal Lines as the angle between the circles which coincide with these lines. We have now to consider in what way it will be proper to define Parallel Ideal Lines. Let AàI be the Ideal Line through A, perpendicular to the Ideal Line BC; in other words, the circle of the system passing through A', A'\ and orthogonal to the circle through £', B", C and C" (cf Fig. 92). 2^2 Appendix V. Impossibilty of proving Euclid's Postulate. Imagine AM to rotate about A so that those Ideal Lines through A cut the Ideal Line BC at a gradually decreasing angle. The circles through A which touch the given Fig. 92. circle £C at the points [/, V, where it meets the fundamental circle, are Ideal Lines of the system. They separate the lines of the pencil of Ideal Lines through A, which cut the Ideal Line -BC, from those which do not cut that line. All the lines in the angle q), shaded in the figure, do not cut the line £C; all those in the angle ^), not shaded, do cut this line. This property is exactly what is assumed in the Parallel Postulate upon which the Non-Euclidean Geometry of BoLYAi-LoBATSCHEWSKY is based. We therefore are led to define Parallel Ideal Lines in this Plane Ideal Geometry as follows: TAe Ideal Lines through an Ideal Point parallel to a given Ideal Line are the two circles of the syston passing Some Theorems in this Geometry. 253 through the given pointy which touch the circle with 7vhich the given line coincides at the poi?tts where it meets the fundam- ental circle. Thus we have in this Ideal Geometry two parallels through a point to a given line: a right-handed parallel, and a left-handed parallel: and these separate the lines of the pencil which intersect the given line from those which do not intersect it. Some Theorems of this Non-Euclidean Geometry. § 9. At this stage we can say that any of the theorems of the BoLYAi-LoBATSCHEwsKY Non-EucHdean Geometry, in- volving angle properties only, will hold in this Ideal Geo- metry and vice versa. Those involving lengths we cannot yet discuss, as we have not yet defined Ideal Lengths. For example, it is obvious that there are triangles in which all the angles are zero (cf. Fig. 93). The sides of such triangles are parallel in pairs. Thus the sum of the angles of an Ideal Triangle is certainly not always equal to two right angles. We can prove that this sum is always less than two right angles by a simple application of inversion, as follows: Let Ci, C2, C3 be three circles of the system, forming an Ideal Triangle. Invert these circles from the point of intersection / of C^ and C2 , which hes inside the fundament- al circle. Then d and C2 become two straight lines d' and C2' through /. Also the fundamental circle C inverts into a circle C cutting Ci and C2 at right-angles, so that its centre is /. Again, the circle C. inverts into a circle C3', cutting C at right-angles. Hence its centre lies outside C. We thus obtain a 'triangle', in which the sum of the angles is less than two right-angles, and since these angles are equal to the angles of the Ideal Triangle, this result holds also for the Ideal Triangle. 2^4 Appendix V. Impossibility of proviug Euclid's Postulate. Finally, it can be shown that there is always one, and only one, circle of the system cutting two non-intersecting circles of the system at right-angles. In other words, two Fig. 93- non-intersecting Ideal Lines have a common perpendicular. All these results must be true in the Hyperbolic Geometry. § IO. Ideal Lengths and Ideal Displacements. Before we can proceed to the discussion of the metrical properties of this geometry, we must define the Ideal Length of an Ideal Segment. It is clear that this must be such that it will be unaltered, if we take the points A\ B", as defining the segment AB, instead of the points A\ B'. It must make the complete line infinite in length. It must satisfy the distri- butive law 'distance' AB = 'distance' AC -\- 'distance' CB, Ideal Lengths. 255 if C is any other point on the segment AB^ and it must also remain \in.di\i&xQàhy Ideal Displacement. We defaie the Ideal length of a?iy segment AB as 'V_ I B'^ \ 77/ WI/J where U, V are the points where the Ideal Line AB meets the fundamental circle (cf. Fig. 91). lot \a'i Fig. 94. This expression obviously involves the Anharmonic Ratio of the points UABV. It will be seen that this de- finition satisfies the first three of the conditions named above. It remains for us to examine what must represent dis- placement in this Ideal Geometry. Let us consider what is the effect of inversion with regard to a circle of the system upon the system of Ideal Points and Lines. Let A A" be any Ideal Point A (cf. Fig. 94). Let the 2CS Appendix V. Impossibility of proving Euclid's Postulate. circle of inversion meet the fundamental circle in C, and let D be its centre. Let A', A" invert into B', B" . Since the circle A A' C touches the circle of inversion at C, its inverse also touches that circle at C. But a circle passes through A ^ A", B' and B'\ and the radical axes of the three circles AA'C, B'B"C, AA'B'B" are concurrent. Hence B' B" passes through 6>, and OB' . OB" = 0C\ Therefore inversion with regard to any circle of the system changes an Ideal Point into an Ideal Point. But it is clear that the circle AA'B'B" is orthogonal to the fundamental circle, and also to the circle of inversion. Thus the Ideal Line joining the Ideal Point A and the Ideal Point B, into which it is changed by this inversion, is perpendicular to the Ideal Line coincidiiig with the circle of itwersion. We shall now prove that it is 'bisected' by that Ideal Line. Let the circle through AB meet the circle of inversion at M, and the fundamental circle in U and V. It is clear that U and V are inverse points with regard to the circle of inversion [cf. Fig. 95]. Then we have: B'V _ CV 'AU~'CA"> A'V CV ^^^ B'U ~ CB' ' Thus A'V B'V CV2 CV2 A'U B'U CA'.CB' CM 2 Therefore A'V I M'V M'V I B'V /M'V\ 2 A'U M'U M'U\ B'U Ideal Reflection. 257 Hence the Ideal Length of AM is equal to the Ideal Length of MB. Thus we have the following result: Inversion with regard to a circle of the system changes any Ideal Point A ifito an Ideal Point B, such that the Ideal Line AB is perpendicular to, and ''bisected' by, the Ideal LÌ7ie coinciding with the circle of inversion. Fig. 95 In other words, inversion with regard to such a circle causes any Ideal Point A to take the position of its image in the corresponding Ideal Line. We proceed to examine what effect such inversion has upon an Ideal Line. Since a circle^ orthogonal to the fundamental circle, 17 2 e 8 Appendix V. Impossibility of proving Euclid's Postulate. inverts into a circle also orthogonal to the fundamental circle, any Ideal Line AB inverts into another Ideal Line ab, pass- ing through the point M, where AB meets the circle of in- version (cf. Fig. 96). Also the points U, V invert into the Fig. 96. points ti and v on the fundamental circle; and the lines AB and ab are equally inclined to the circle of inversion. It is easy to show that the Ideal Lengths of AM and BM are equal to those of aM a.nd ^ J/ respectively, and it follows that the Ideal Length of the segment AB is unaltered by this inversion. Also we have seen that Aa and Bb are perpendicular to, and 'bisected' by, the Ideal Line coinciding vnth this circle. // follows from these results that inversion with regard to any circle of the system has the same effect upon an Ideal Segment as reflection in the corresponding Ideal Line. We are thus agaifi able to defi7ie Ideal Reflection in any Ideal Line as the inversion of the system of Ideal Points and Ideal Displacement. 259 Lines 7mt/i regard to the circle which ^ciacides with this Ideal Line. It is unnecessary to define Ideal Displacements., as any displacement can be obtained by a series of reflections and any Ideal Displacement by a series of Ideal Reflections. We notice that the definition of the Ideal Length of any Segment fixes the Ideal Unit of Length. We may take this on one of the diameters of the fundamental circle, since these lines are also Ideal Lines of the system. Let it be the segment OP (Fig. 97). Fig. 97- Then we must have /0V\ PV\ l°g \ou\pu) i. e. 1 PU log py = I. Therefore PU PF ~ ^' and the point P divides the diameter in the ratio e: 1. The Unit Segment is thus fixed for any position in the 17=" 200 Appendix V. Impossibility of proving Euclid's Postulate. domain of the Ideal Points, since the segment OP can be 'moved' so that one of its ends coincides with any given Ideal Point. A different expression for the Ideal Length would simply mean an alteration in the unit, and taking logarithms to any other base than §4, 88, 92, 122, 128, 139, 144- Leibnitz, G. W. F. (1646—1716). 54- Lie, S. (1842 — 1899). 152—4, 179. Liebmann, H. 86, 89, 113, 145, 180, 220. Lindemann, F. 161. Lobatschewsky, N. J. (1793 — 1856). 44, 5L 55> 63, 65, 74 80, 84 — 99, 101—3, 104—6; 111—3, 116, 121 — 8, 137, 141 145, H7, 152, 154, 157—8 161, 164, 170, 173—5, 177—8 193 — 4, 217, 220, 222, 225 238, 249, 252—3, 264. Lorenz, J. F. (1738—1807). 58, 120. Lukat, M. 129. Liitkemeyer, G. 145 — 6. Mc Cormack, T. J. 182. Mach, E. 181. Minding, F. (1806—1885). 132, ^37- Mobius, A. F. (1790—1868). 148 — 9- Monge, G. (1746—1818). 54—5. Montucla, J. E. (1725 — 1799) 44. 92. Nasìr-Eddìn, (1201 — 1274). 10, 12—3, 16, 37—8, 120. Newton, L (1642—1727). 53. 01bers,H.W.M.(i758— 1840). 65. Oliver, (ist Half of the 17* Cent- ury). 17. j Ovidio, (d') E. 127. Paciolo, Luca (circa 1445 — 1514)- 17- Pascal, E. 127, 139. Pasch, M. 176. Picard, C. É. 128. Poincaré, H. 154, 180. Poncelet, J. V. (1788—1867). 155, 236. Posidonius, (ist Century B.C.I 2, 3, 8, 14. Proclus, (410— 485^ 2—7, 12—3, 18—20, 119. Ptolemy, (87—165). 3—4, 119. Riccardi, P. (182S— 1898). 17. Ricordi, E. 127. Riemann, B. (1826—1866). 126, 129, 138—9, 141—3,145 — 154, 157—8, 160—1, 163—4, 175 — 7, 179—180, 194, 201—2. Saccheri, G. (1667—1733). 4, 22—4, 26, 28—30, 34, 36—46, 51, 55-7, 65—6, 78, 85, 87 —8, 97, 120, 129, 139, 141, 144- Sartorius v. Waltershausen, W. (1809—1876). 122. Saville, H. (1549—1622). 17. Schmidt, F. (1826 — 1901). 121, 124—5. Schumacher, H. K. (1780—1850). 65-7, 75, 122—3, 152. Schur, F. H. 176. Schweikart, F. K. (1780—1859). 67, 75 -78, 80, 83,86, 107, 122. 268 Index of Authors. Segre, C. 44, 66, 77— S, 92. Seyffer, K. F. (1762— 1S22). 60, 66. Simon, H. 91. Simplicius, (6'h Century). 8, 10. Sintsoff, D. 139. Stackel, P. 16, 44—5, 50, 60 — 1, 63, 66, 82 — 3, lOT, 112—3, 124-5. Staudt.G. C.v. (1798—1867). 129, 154, 233, 236. Szasz, C. (1798—1853). 97. Tannery, P. (1S43— 1904). 7, 20. Tacquet, A. (1612 — 1660). 17. Tartaglia, N. (1500—1557). 17. Taurinus, F. A. (1794—1874). 65—6, 74, 77-9' Si— 3, 87, 89-91, 94, 99, 112, 137, 173. Thibaut, B. F. (1775— 1S32). 63. Tilly (de), J. M. 55, 114, 194. Townsend, E. J. 236. Vailati, G. 18, 22. Valerio Luca (? 1522 — 1618). 17. Vasiliev, A. 93, Wachter, F. L. (1792—1817). 62 —3, 66, 88. Wallis, J. (1616 — 1703). 12, 15 —7, 29, 53' 120. Weber, H. 180. Wellstein, J. 180, 23S. Zamberti, B. (ist Half of the l6th Century). 17. Zeno, (495— 435V 6. Zolt, A. (de) 127. The Open Court Mathematical Series A Brief History of Mathematics. By the late Dr. Karl Fink, Tubingen, Germany. 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