Return this book on or before the Latest Date stamped below. Theft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. University of Illinois Library ov 30 1967] 3, Nov 1< 7APR 2m i HOY Wey Io pew HAR2 4 19g8 DEC 20 I9f | NOV 29 Rett MAY 1 6 1079 JAN 02 1987 | JUN 13 $979 | “"" © 6 fae atin 1 c ore : % booed ‘ 4 Og aP $r2. e BEC O4 199) Mhyg on | gee os AUG 4 Kecy giny 10 184 MAR 81978 ey agg FEB 1 ORECT 1 | re al ¢. re : ot | DEC L 7 ve L161—O-1096 GEOMETRICAL RESEARCHES THE THEORY OF PARALLELS. BY NICHOLAUS LOBATSCHEWSKY, IMPERIAL RUSSIAN REAL COUNCILLOR OF STATE AND REGULAR PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF KASAN. BERLIN, 1840. TRANSLATED FROM THE ORIGINAL BY GEORGE BRUCE HALSTED, A. M., Ph. D., Ex-Fellow of Princeton College and Johns Hopkins University, Professor of Mathematics in the University of Texas. AUSTIN: PUBLISHED BY THE UNIVERSITY OF TEXAS, 1891. TRANSLATORS PREFACE. Lobatschewsky was the first man ever to publish a non-Euclidian geom- etry. Of the immortal essay now first appearing in English Gauss said, ‘‘The author has treated the matter with a master-hand and in the true geom- eter’s spirit. I think I ought to call your attention to this book, whose perusal can not fail to give you the most vivid pleasure.” Clifford says, ‘It is quite simple, merely Euclid. without the vicious assumption, but the way things come out of one another is quite lovely.” * * * «What Vesalius was to Galen, what Copernicus was to Ptolemy, that was Lobatschewsky to Euclid.” Says Sylvester, “In Quaternions the example has been given of Al- gebra released from the yoke of the commutative principle of multipli- cation—an emancipation somewhat akin to Lobatschewsky’s of Geometry from Euclid’s noted empirical axiom.” Cayley says, “It is well known that Huclid’s twelfth axiom, even in Playfair’s form of it, has been considered as needing demonstration; and that Lobatchewsky constructed a perfectly consistent theory, where- in this axiom was assumed not to hold good, or say a system of non- Euclidian plane geometry. There is a like system of non-Euclidian solid geometry.” GEORGE BRUCE HALSTED. 2407 San Marcos Street, Austin, Texas. May 1, 1891. [3] TRANSLATORS INTRODUCTION. “Prove all things, hold fast that which is good,” does not mean dem- onstrate everything. rom nothing assumed, nothing can be proved. “Geometry without axioms,” was a book which went through several editions, and still has historical value. But now a volume with such a title would, vithout opening it, be set down as simply the work of a paradoxer. The set of axioms far the most influential in the intellectual history of the world was put together in Egypt: but really it owed nothing. to the Egyptian race, drew nothing from the boasted lore of Hgypt’s priests. The Papyrus of the Rhind, belonging to the British Museum, but given to the world by the erudition of a German Egyptologist, Hisen- lohr, and a German historian of mathematics, Cantor, gives us more knowledge of the state of mathematics in ancient Egypt than all else previously accessible to the modern world. Its whole testimony con- firms with overwhelming force the position that Geometry as a science, strict and self-conscious deductive reasoning, was created by the subtle intellect of the same race whose bloom in art still overawes us in the Venus of Milo, the Apollo Belvidere, the Laocoon. In a geometry occur the most noted set of axioms, the geometry of Huclid, a pure Greek, professor at the University of Alexandria. Not only at its very birth did this typical product of the Greek genius assume sway as ruler in the pure sciences, not only does its first efflor- escence carry us through the splendid days of Theon and Hypatia, but unlike the latter, fanatics can not murder it; that dismal flood, the dark ages, can not drown it. Like the pheenix of its native Egypt, it rises with the new birth of culture. An Anglo-Saxon, Adelard of Bath, finds it clothed in Arabic vestments in the land of the Alhambra. Then clothed in Latin, it and the new-born printing press confer honor on each other. Tinally back again in its original Greek, it is published first in queenly Venice, then in stately Oxford, since then everywhere. The latest edition in Greek is just issuing from Leipsic’s learned presses. [5] 6,\- THEORY OF PARALLELS. How the first translation into our cut-and-thrust, survival-of-the-fittest English was made from the Greek and Latin by Henricus Billingsly, Lord Mayor of London, and published with a preface by John Dee the Magician, may be studied in the Library of our own Princeton College, where they have, by some strange chance, Billingsly’s own copy of the Latin version of Commandine bound with the Editio Princeps in Greek and enriched with his autograph emendations. [ven to-day in the vast system of examinations set by Cambridge, Oxford, and the British gov- ernment, no proof will be accepted which infringes Euclid’s order, a sequence founded upon his set of axioms. ; The American ideal is success. In twenty years the American maker expects to be improved upon, superseded. The Greek ideal was per- fection. The Greek Hpic and Lyric poets, the Greek sculptors, remain unmatched. The axioms of the Greek geometer remained unquestioned for twenty centuries. How and where doubt came to look toward them is of no ordinary interest, for this doubt was epoch-making in the history of mind. Among Euclid’s axioms was one differing from the others in pro- lixity, whose place fluctuates in the manuscripts, and which is not used in Kuclid’s first twenty-seven propositions. Moreover it is only then brought in to prove the inverse of one of these already demonstrated. All this suggested, at Hurope’s renaissance, not a doubt of the axiom, but the possibility of getting along without it, of deducing it from the other axioms and the twenty-seven propositions already proved. Huclid demonstrates things more axiomatic by far. He proves what every dog knows, that any two sides of a triangle are together greater than the third. Yet when he has perfectly proved that lines making with a transversal equal alternate angles are parallel, in order to prove the in- verse, that parallels cut by a transversal make equal alternate angles, he brings in the unwieldly postulate or axiom: “Tf a straight line meet two straight lines, so as to make the two in- terior angles on the same side of it taken together less, than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles which are less than two right angles.” Do you wonder that succeeding geometers wished by demonstration to push this unwieldly thing from the set of fundamental axioms. TRANSLATOR’S INTRODUCTION. if Numerous and desperate were the attempts to deduce it from reason- ings about the nature of the straight line and plane angle. In the ‘“‘Hincyclopceedie der Wissenschaften und Kunste; Von Ersch und Gru- ber;” Leipzig, 1838; under ‘Parallel,’ Sohncke says that in mathe- matics there is nothing over which so much has been spoken, written, and striven, as over the theory of parallels, and all, so far (up to his time), without reaching a definite result and decision. Some acknowledged defeat by taking a new definition of parallels, as for example the stupid one, ‘“ Parallel lines are everywhere equally dis- tant,” still given on page 33 of Schuyler’s Geometry, which that author, like many of his unfortunate prototypes, then attempts to identify with Huclid’s definition by pseudo-reasoning which tacitly assumes Euclid’s postulate, e. g. he says p. 35: ‘Tor, if not parallel, they are not every- where equally distant; and since they lie in the same plane; must ap- proach when produced one way or the other; and since straight lines continue in the same direction, must continue to approach if produced farther, and if sufficiently produced, must meet.” This is nothing but HKuclid’s assumption, diseased and contaminated by the introduction of the indefinite term ‘ direction.” How much better to have followed the third Class of his predecessors who honestly assume a new axiom differing from Huclid’s in form if not in essence. Of these the best is that called Playfair’s; “Two lines which intersect can not both be parallel to the same line.” The German article mentioned is followed by a carefully prepared list of ninety-two authors on the subject. In English an account of like attempts was given by Perronet Thompson, Cambridge, 1833, and is brought up to date in the charming volume, “ Huclid and his Modern Rivals,” by C. L. Dodgson, late Mathematical Lecturer of Christ Church, Oxford. All this shows how ready the world was for the extraordinary flaming- forth of genius from different parts of the world which was at once to overturn, explain, and remake not only all this subject but as conse- quence all philosophy, all ken-lore. As was the case with the dis- covery of the Conservation of Energy, the independent irruptions of genius, whether in Russia, Hungary, Germany, or even in Canada gave everywhere the same results. At first these results were not fully understood even by the brightest 8 THEORY OF PARALLELS. intellects. Thirty years after the publication of the book he mentions, we see the brilliant Clifford writing from Trinity College, Cambridge, April 2, 1870, ““Several new ideas have come to me lately: First I have procured Lobatschewsky, ‘Etudes Geometriques sur la Theorie des Parallels’ -— -— -— asmall tract of which Gauss, therein quoted, says: L’auteura traite la matiere en main de maitre et avec le veritable esprit geometrique. Je crois devoir appeler votre attention sur ce livre, dont la lecture ne peut manquer de vous causer le plus vif plaisir.’” Then says Clifford: ‘It is quite simple, merely Euclid without the vicious assumption, but the way the things come out of one another is quite lovely.” 3 The first axiom doubted is called a ‘vicious assumption,” soon no man sees more clearly than Clifford that all are assumptions and none vicious. He had been reading the French translation by Houel, pub- lished in 1866, of a little book of 61 pages published in 1840 in Berlin under the title Geometrische Untersuchungen zur Theorie der Parallel- linien by a Russian, Nicolaus Ivanovitch Lobatschewsky (1793-1856), the first public expression of whose discoveries, however, dates back to a discourse at Kasan on February 12, 1826. Under this commonplace title who would have suspected the dis- covery of a new space in which to hold our universe and ourselves. A new kind of universal space; the idea is a hard one. To name it, all the space in which we think the world and stars live and move and have their being was ceded to Euclid as his by right of pre-emption, description, and occupancy; then the new space and its 3 quieks -following fellows could be called Non-Huclidean. Gauss in a letter to Schumacher, dated Nov. 28, 1846, mentions that as far back as 1792 he had started on this path to a new universe. Again he says: ‘La Geometrie non-EHuclidienne ne renferme en elle rien de contradictoire, quoique, a premiere vue, beaucoup de ses resul- tats aien l’air de paradoxes. Ces contradictions apparents doivent etre regardees comme l’effet d’une illusion, due a l’habitude que nous avons prise de pue heure de considerer la geometrie Huclidienne comme rigoureuse. ” But here we see in the last word the same imperfection of view as in Clifford’s letter. The perception has not yet come that though the non- Euclidean geometry is rigorous, Euclid is not one whit less so. TRANSLATORS INTRODUCTION, 9 A clearer idea here had already come to the former room-mate of _ Gauss at Goettingen, the Hungarian Wolfgang Bolyai. His principal work, published by subscription, has the following title: Tentamen Juventutem studiosam in elementa Matheseos purae, ele- mentaris ac sublimioris, methodo intuitiva, evidentique -huic propria, in- troducendi. Tomus Primus, 1831; Secundus, 1833. 8vo. Maros-Va- sarhelyini. In the first volume with special numbering, appeared the celebrated Appendix of his son Johann Bolyai with the following title: Ap., scientiam spatii absolute veram exhibens: a veritate aut falsitate Axiomatis XI Euclidei (a priori haud unquam decidenda) independen- tem. Auctore Johanne Bolyai de eadem, Geometrarum in Exercitu Caesareo Regio Austriaco Castrensium Captaneo. Maros-Vasarhely., 1832. (26 pages of text). This marvellous Appendix has been translated into French, Italian, and German. In the title of Wolfgang Bolyai’s last work, the only one he com- posed in German (88 pages of text, 1851), occurs the following: “Und da die Frage, ob zwei von der dritten geschnittene Geraden wenn die Summa der inneren Winkel nicht = 2R, sich schnerden oder nicht?, niemand auf der Erde ohne ein Axiom (wie Euclid das XI) aufzustellen, beant- worten wird; die davon unabhengige Geometrie abzusondern, und eine auf die Ja Antwort, andere auf das Nein so zu bauen, dass die Formeln der letzen auf ein Wink auch in der ersten gultig seien.” The author mentions Lobatschewsky’s Geometrische Untersuchungen Berlin, 1840, and compares it with the work of his son Johann Bolyai, ‘‘an sujet duquel il dit- ‘Quelques exemplaires de l’ouvrage publie ici ont ete envoyes a cette epoque a Vienne, a Berlin, a Gettingen. . . De Goettingen le geant mathematique, [Gauss] qui du sommet des hauteurs embrasse du meme regard les astres et la profondeur des abimes, a ecrit qu'il etait ravi de voir execute le travail qu’il avait commence pour le laisser apres lui dans ses papiers.’”’ Yet that which Bolyai and Gauss, a mathematician never surpassed in power, see that no man can ever do, our American Schuyler, in the density of his ignorance, thinks that he has easily done. In fact this first of the Non-Euclidean geometries accepts all of Eu- clid’s axioms but the last, which it flatly denies and replaces by its con- aa 10 - THEORY OF PARALLELS. tradictory, that the sum of the angles made on the same side of a trans- versal by two straight lines may be less than a straight angle without the lines meeting. EC from the point C. c Suppose in the right-angled tri- angle ACE the sum of the three angles is equal to z — 4, in the tri- angle AHF equal to z — §, then must it in triangle ACF equal z — « — f, where z and f can not be negative. Further, let theangle BAF = a, AFC = b, sois g++ 8 =a — b; now by revolving the line AF away from the perpendicular AC we can make the angle a between AF and the parallel AB as small as we choose; so E ¥ Fia. 9. also can we lessen the angle b, consequently the two angles x and 2 can have no other magnitude than 4 = 0 and § = 0. It follows that in all rectilineal triangles the sum of the three angles is either z and at the same time also the parallel angle /7 (p) = 4 x for every line p, or for all triangles this sum is < z and at the same time also /[(p)< tz The first ear serves as foundation for the ordinary geometry and plane trigonometry. The second assumption can likewise be admitted without leading to any contradiction in the results, and founds a new geometric science, to which I have given the name Jmaginary Geometry, and which I in- tend here to expound as far as the development of the equations be- tween the sides and angles of the rectilineal and spherical triangle. 23. Sor every given angle ¢ we can find a line p such that IT (p) = «. | Let AB and AC (fig. 10) be two straight lines which at the inter- | section point A make the acute angle z; take at random on AB a point 20 THEORY OF PARALLELS. B’; from this point drop B’A’ at right angles to AC; make A’A” — AA’; erect at A” the perpendicular A”B’; and so continue until a per- N A” K F C Fig. 10. pendicular CD is attained, which no longer intersects AB. This must of necessity happen, for if in the triangle AA’B’ the sum of all three angles is equal to z — a, then in the triangle AB’ A” it equals z — 2a, in triangle AA’B” less than z — 2a (Theorem 20), and so forth, until it finally becomes negative and thereby shows the impossibility of con- structing the triangle. The perpendicular CD may be the very one nearer than which to the point A all others cut AB; at least in the passing over from those that cut to those not cutting such a perpendicular FG must exist. Draw now from the point F the line FH, which makes with FG the acute angle HIG, on that side where lies the point A. From any point H of the line FH let fall upon AC the perpendicular HK, whose pro- longation consequently must cut AB somewhere in B, and so makes a triangle AKB, into which the prolongation of the line FH enters, and therefore must meet the hypothenuse AB somewhere in M. Since the angle GFH is arbitrary and can be taken as small as we wish, therefore ¥G is parallelto AB and AF =p. (Theorems 16 and 18.) One easily sees that with the lessening of pthe angle z increases, while, for p == 0, it approaches the value 47; with the growth of p the angle a decreases, while it continually approaches zero for p =o. Since we are wholly:at liberty to choose what angle we will under- THEORY OF PARALLELS. yA stand by the symbol // (p) when the line p is expressed by a negative number, so we will assume I(p)+- 1 —p)=-, an equation which shall hold for all values of p, positive as well as neg- ative, and for p= 0. : 24. The farther parallel lines are prolonged on the side of their paral- lelism, the more they approach one another. If to the line- AB (Fig. 11) two perpendiculars AC = BE are erected and their end-points Cand HE joined by « F F a straight line, then will the quadrilat- eral CABE have two right angles at Ms A and B, but two aeute angles at C and E (Theorem 22) which are equal B to one another, as we can easily see 4 D by thinking the quadrilateral super- Fig. 11. imposed upon itself so that the line BE falls upon upon AC and AC upon BE. Halve AB and erect at the mid-point D the line DF perpendicular to AB. This line must also be perpendicular to CH, since the quadrilat- erals CADF and FDBE fit one another if we so place one on the other that the line DF remains in the same position. Hence the line CE can not be parallel to AB, but the parallel to AB for the point C, namely CG, must incline toward AB (Theorem 16) and cut from the perpendic- ular BE a part BG < CA. Since C isa random point in the line CG, it follows that CG itself nears AB the more the farther it is prolonged. 22 : THEORY OF PARALLELS. 25. Two straight lines which are parallel to a third are also parallel to one another. Wiey 42) We will first assume that the three lines AB, CD, EF (Fig. 12) lie in one plane. If two of them in order, AB and CD, are parallel to the outmost one, EF’, so are AB and CD parallel to one another. In order to prove this, let fall from any point A of the outer line AB upon the other outer line FE, the perpendicular AEH, which will cut the middle line CD in some point C (Theorem 3), at an angle DCE < }z on the side toward EF, the parallel to CD (Theorem 22). A perpendicular AG let fall upon CD from the same point, A, must fall within the opening of the acute angle ACG (Theorem 9); every other line AH from A drawn within the angle BAC must cut EF, the parallel to AB, somewhere in H, how small soever the angle BAH may be; consequently will CD in the triangle AEH cut the line AH some- where in K, since it is impossible that it should meet EF. If AH from the point A went out within the angle CAG, then must it cut the pro- . longation of CD between the points C and G in the triangle CAG. Hence follows that AB and CD are parallel (Theorems 16 and 18). Were both the outer lines AB and EF assumed parallel to the middle line CD, so would every line AK from the point A, drawn within the angle BAH, cut the line CD somewhere in the point K, how small soever the angle BAK might be. Upon the prolongation of AK take at random a point L and join it THEORY OF PARALLELS. 93 with C by the line CL, which must cut EF somewhere in M, thus mak- ing a triangle MCE. The prolongation of the line AL within the triangle MCE can cut neither AC nor CM asecond time, consequently it must meet EI some- where in H; therefore AB and EF are mutually parallel. A_G H Bree 3; Now let the parallels AB and CD (Fig. 13) lie in two planes whose intersection line is EF. From a random point E of this latter let fall a perpendicular EA upon one of the two parallels, e. g., upon AB, then from A, the foot of the perpendicular EA, let fall a new perpen- dicular AC upon the other parallel CD and join the end-points E and C of the two perpendiculars by the line EC. The angle BAC must be acute (Theorem 22), consequently a perpendicular CG from C let fall upon AB meets it in the point G upon that side of CA on which the lines AB and CD are considered as parallel. Every line EH [in the plane FEAB], however little it diverges from HF, pertains with the line EC to a plane which must cut the plane of the two parallels AB and CD along some line CH. This latter line cuts AB somewhere, and in fact in the very point H which is common to all three planes, through which necessarily also the line EH goes; conse- quently EF is parallel to AB. | In the same way we may show the parallelism of EF and CD, Therefore the hypothesis that a line EF is parallel to one of two other parallels, AB and CD, is the same as considering EF as the intersection of two planes in which two parallels, AB, CD, lie. Consequently two lines are parallel to one another if they are parallel to a third line, though the three be not co-planar. The last theorem can be thus expressed: Three planes intersect in lines which are all parallel to each other of the parallelism of two is pre-supposed, 24 THEORY OF PARALLELS. 26. Triangles standing opposite to one another on the sphere are equiva- lent in surface. ‘ By opposite triangles we here understand such as are made on both sides of the center by the intersections of the sphere with planes; in such triangles, therefore, the sides and angles are in contrary order. In the opposite triangles ABC and A’B’C’ (Fig. 14, where one of them must be looked upon as represented turned about), we have the sides AB — A’B’, BC = B’C’, CA =C’A’, and the corresponding angles B! AY Fig. 14. at the points A, B, Care likewise equal to those in the other io at the points A’, B’, C’. Through the three points A, B, C, suppose a plane passed, and upon it fromthe center of the sphere a perpendicular dropped whose pro- longations both ways cut both opposite triangles in the points D and D/ of the sphere. The distances of the first D from the points ABC, in arcs of great circles on the sphere, must be equal (Theorem 12) as well to each other as also to the distances D’A’, D’B’, D’C’, on the other triangle (Theorem 6), consequently the isosceles triangles about the points D and D’ in the two spherical triangles ABC and A’B’C’ are congruent. In order to judge of the equivalence of any two surfaces.in general, I take the following theorem as fundamental: Two surfaces are equivalent when they arise from the mating or separating of equal parts. 27. A three-sided solid angle equals the half sum of the surface angles less a right-angle. In the spherical triangle ABC (Fig. 15), where each side < z, desig. nate the angles by A, B, C; prolong the side AB so that a whole circle ABA’B’A is produced; this divides the sphere into two equal parts. THEORY OF PARALLELS. 25 In that half in which is the triangle ABO, prolong now the other two sides through their common intersection point C until wey meet the circle in A’ and B’. C. Fia. 15. In this way the hemisphere is divided into four triangles, ABC, ACB’, B’/CA’, A’CB, whose size may be designated by P, X, Y, Z. It is evi. dent that here P+ X=—B, P+ Z=— A. The size of the spherical triangle Y equals that of the opposite triangle ABC’, having a side AB in common with the triangle P, and whose third angle C’ lies at the end-point of the diameter of the sphere which goes from C through the center D of the sphere (Theorem 26). Hence it follows that I aa Y =O, and since P + X + Y + Z=—7z, therefore we have also | P(A) B+ C— x). We may attain to the same conclusion in another way, based solely upon the theorem about the equivalence of surfaces given above. (Theo-* rem 26.) | In the spherical triangle ABC (Fig. 16), halve the sides AB and BQ, and through the mid-points D and E draw a great circle; upon this let fall from A, B, C the perpendiculars AF’, BH, and CG. If the perpendic- » ular from B falls at H between D and H, then will of the triangles so made BDH = AFD, and BHE — EGC (The. 4 orems 6 and 15), whence follows that Fic. 16, the surface of the triangle ABC equals that of the quadrilateral AFGC (Theorem 26). 26 THEORY OF PARALLELS. If the point H coincides with the middle point E of the side BC (Fig. B 17), only two equal right-angled triangles, ADF and BDE, are made, by whose interchange the r R equivalence of the surfaces of the triangle ABC and the quadrilateral AFEC is established. If, finally, the point H falls outside the triangle A ° ABC (Fig. 18), the perpendicular CG goes, in Fig. 17. consequence, through the triangle, and so we go over from the triangle ABC to the quadrilateral AFGC by adding the B A 0 Ita. 18. triangle FAD — DBH, and then taking away the triangle CGE = EBH. Supposing in the spherical quadrilateral AF'GO a great circle passed through the points A and G, as also through F and OC, then will their arcs between AG and FC equal one another (Theorem 15), consequently also the triangles FAC and ACG be congruent (Theorem 15), and the angle FAC equal the angle ACG. ‘ Hence follows, that in all the preceding cases, the sum oot all three angles of the spherical triangle equals the sum of the two equal angles in the quadrilateral which are not the right angles. Therefore we can, for every spherical triangle, in which the sum of the three angles is 8, find a quadrilateral with equivalent surface, in which are two right angles and two equal perpéndicular sides, and where the two other angles are each 48. THEORY OF PARALLELS. ay Let now ABCD (Fig. 19) be the spherical quadrilateral, where the sides AB— DC are perpendicular to BC, and the angles A and D each 48. P A D Fig. 19. Prolong the sides AD and BC until they cut one another in E, and further beyond E, make DE = EF and let fall upon the prolongation of BC the perpendicular FG. Bisect the whole arc BG and join the mid-point H by great-circle-arcs with A and F. The triangles EFG and DCE are congruent (Theorem 15), so FG = DE AB. The triangles ABH and HGF are likewise congruent, since they are right angled and have equal perpendicular sides, consequently AH and AF pertain to one circle, the are AHF =z, ADEF likewise — 7, the angle HAD = HFE— 48S — BAH=— 48S — HFG = 48 — HFE—EFG —=45S—HAD—7z-+45; consequently, angle HFE=4(S—z); or what is the same, this equals the size of the lune AHFDA, which again is equal to the quadrilateral ABCD, as we easily see if we pass over from the one to the other by first adding the triangle EFG and then BAH and thereupon taking away the triangles equal to them DCE and HFG. Therefore $(S—z) is the size of the quadrilateral ABCD and at the same time also that of the spherical triangle in which the sum of the three angles is equal to S. 28 THEORY OF PARALLELS. 28. If three planes cut each other in parallel lines, then the swm of the three surface angles equals two right angles. Let AA’, BB’ CC’ (Fig. 20) be three parallels made by the inter- section of planes (Theorem 25). Take upon them at random three Fig. 20. points A, B, C, and suppose through these a plane passed, which con- sequently will cut the planes of the parallels along the straight lines AB, AC, and BC. Further, pass through the line AC and any point D on the BB’, another plane, whose intersection with the two planes of the parallels AA’ and BB’, CC’ and BB’ produces the two lines AD and DO, and whose inclination to the third plane of the parallels AA’ and CC’ we will designate by w. The angles between the three planes in which the parallels lie will be designated by X, Y, Z, respectively at the lines AA’, BB’, CC’; finally call the linear angles BDC = a, ADC = b, ADB=c. About A as center suppose a sphere described, upon which the inter- sections of the straight lines AC, AD AA’ with it determine a spherical triangle, with the sides p, q, and r. Call its size z Opposite the side q lies the angle w, opposite r lies X, and consequently opposite p lies the angle 7-+24—w—X, (Theorem 27). ~ In like manner CA, CD, CC’ cut a sphere about the center OC, and determine a triangle of size 8, with the sides p’, q’, r’, and the angles, w opposite q’, Z opposite r’, and consequently z-++-23—w—Z opposite p’. - Finally is determined by the intersection of a sphere about D with the lines DA, DB, DC, a spherical triangle, whose sides are 1, m, n, and the angles opposite them w+Z—2§, w+X—2a, and Y. Consequently its size 6 = 4 (X+Y +-Z—7)—a—B-+ W. Decreasing w lessens also the size of the triangles g and #, so that . a+ f—w can be made smaller than any given number. oad THEORY OF PARALLELS, 29 In the triangle can likewise the sides 1 and m be lessened even to vanishing (Theorem 21), consequently the triangle d can be placed with one of its sides | or m upon a great circle of the sphere as often as you choose without thereby filling up the half of the sphere, hence @ van- ishes together with w; whence follows that necessarily we must have | X+Y+2Z4—=rn7 29. In a rectilineal triangle, the perpendiculars erected at the mid-potnts of the sides either do not meet, or they all three cut each other in one point. Having pre-supposed in the triangle ABO (Fig. 21), that the two per- pendiculars ED and DF, which are erected upon the sides AB and BC at their mid points E and F, intersect in the point D, then draw within the angles of the triangle the lines DA, DB, DC. In the congruent triangles ADE and BDE (Theorem 10), we have AD= BD, thus follows also that. BD— CD; the triangle ADC is hence isosceles, consequently the perpendicular dropped from the vertex D upon the base AC falls upon G the mid point of the base. The proof remains unchanged also in the case when the intersection point D of the two perpen- diculars ED and FD falls in the line AC itself, or 7 £ falls without the triangle. Fie. 21. In case we therefore pre-suppose that two of those perpendiculars do not intersect, then also the third can not meet with them. 30. The perpendiculars which are erected upon the sides of a rectilineal triangle at their mid-points, must all three be parallal to each other, so soon as the parallelism uf two of them %s pre-supposed, In the triangle ABC (Fig. 22) let the lines DH, FG, HK, be erected perpendicular upon the sides at their mid- 3 points D, F, H. We will in the first place assume that the two perpendiculars DE and FG are parallel, cutting the line AB in L and M, and that the perpendicular HK lies between them. Within the angle BLE draw from the point L, at random, a straight line LG, which must cut FG somewhere in G, how small soever the angle of deviation GLE: may be. (Theorem 16). 30 THEORY OF PARALLELS. Since in the triangle LGM the perpendicular HK can not meet with MG (Theorem 29), therefore it must cut LG somewhere in P, whence follows, that HK is parallel to DH (Theorem 16), and to MG (Theorems 18 and 25). Put the side BC= 2a, AC= 2b, AB= 2c, and designate the an- gles opposite these sides by A, B, C, then we have in the case just considered A = IT(b)—/I(c), B= [I (a)—I1(¢), C= //(a)+ //(b), as one may easily show with help of the lines AA’, BB’, CC’, which are drawn from the points A, B, OC, parallel to the perpendicular HK and consequently to both the other perpendiculars DE and FG (Theo- rems 23 and 25). Let now the two perpendiculars HK and FG be parallel, then can the third DE not cut them (Theorem 29), hence is it either parallel to them, or it cuts AA’. The last assumption is not other than that the angle O> II (a)-+IT(b.) If we lessen this angle, so that it becomes equal to //(a)-+//(b), while we in that way give the line AC the new position CQ, (Fig. 23), and designate the size of the third side BQ by 2c’, then must the angle CBQ at the point B, which is increased, in accordance with what is proved above, be equal to I(a)—H(e")> H(a)—H(e), whence follows c’ >c (Theorem 23). A B Fig. 23. In the triangle ACQ are, however, the angles at A and Q equal, hence in the triangle ABQ must the angle at Q be greater than that at the point A, consequently is AB>BQ, (Theorem 9); that is c>c’, 31. We call boundary line (oricycle) that curve lying in a plane for which all perpendiculars erected at the mid-points of chords are parallel to each other. THEORY OF PARALLELS. 31 In conformity with this definition we can represent the generation of a boundary line, if we draw to a given line AB (Fig. 24) from a given Wie. 24, point A in it, making different angles CAB = //(a), chords AC = 2a; the end C of such a chord will le on the boundary line, whose points we can thus gradually determine. The perpendicular DE erected upon the chord AC at its mid-point D will be parallel to the line AB, which we will call the Awis of the bound- ary line. In like manner will also each perpendicular FG erected at the mid-point of any chord AH, be parallel to AB, consequently must this peculiarity also pertain to every perpendicular KL in general which is erected at the mid-point K of any chord CH, between whatever points C and H of the boundary line this may be drawn (Theorem 30). Such perpendiculars must therefore likewise, without distinction from AB, be called Awes of the boundary line. 32. A circle with continually increasing radius merges into the boundary line. Given AB (Fig. 25) a chord of the boundary line; draw from the end-points A and B of the chord two axes AC and BF, which consequently will make with the chord two equal angles BAC — ABH —@ (Theorem 31). Upon one of these axes AC, take any- where the point E as center of a circle, and draw the arc AF from the initial point A of the axis AC to its intersection point F with the other axis BF. The radius of the circle, FE, corresponding to the point F will make on the one side with the chord AF an angle AFH —§, and on the 32 THEORY OF PARALLELS. other side with the axis BF, the angle EFD—y,. It follows that the angle between the two chords BAF — g—3<§+7—a (Theorem 22); whence follows, a—B<4y. Since now however the angle y approaches the limit zero, ag well in consequence of a moving of the center E in the direction AC, when F remains unchanged, (Theorem 21), as also in consequence of an ap- proach of F to B on the axis BF, when the center E remains in its position (Theorem 22), so it follows, that with such a lessening of the angle y, also the angle a—, or the mutual inclination of the two chords AB and AF’, and hence also the distance of the point B on the bound- ary line from. the point F on the circle, tends to vanish. | Consequently one may also call the boundary-line a circle with i- Jinitely great radius. 33. Let AA’ — BB’= « (Figure 26), be two lines parallel toward the side from A to A’, which parallels serve 8 as axes for the two boundary arcs (arcs on i iy two boundary lines) AB==s, A’B’==s’, then is 7 S86 4% A A where € is independent of the arcs s, s’ and of Fie. 26. the straight line x, the distance of the arc s’ from s. In order to prove this, assume that the ratio of the are s to s’ is equal to the ratio of the two whole numbers n and m. Between the two axes AA’, BB’ draw yet a third axis OO’, which so cuts off from the arc AB a part AC=—¢ and from the arc A’B’ on the same side, a part A’C’=?’. Assume the ratio of ¢ to s equal to that of the whole numbers p and g, so that: Divide now s by axes into ng equal parts, then will there be mg such parts on s/ and np on t. 7 However there correspond to these equal parts on s and ¢ likewise equal parts on s’ and 7’, consequently we have y/ s! Hence also wherever the two arcs ¢ and t’ may be taken between the two axes AA’ and BB’, the ratio of ¢ to ¢’ remains always the same, as THEORY OF PARALLELS. 83 long as the distance « between them remains the same. If we there- fore for x1, put s= es’, then we must have for every x Sia SO ch Since e€ is an unknown number only subjected to the condition e>1, and further the linear unit for x may be taken at will, therefore we may, for the simplification of reckoning, so choose it that by e is to be un- derstood the base of Napierian logarithms. We may here remark, that s’—0 for x= wo, hence not only does the distance between two parallels decrease (Theorem 24), but with the prolongation of the parallels toward the side of the parallelism this at last wholly vanishes. Parallel lines have therefore the character of asymptotes. 34. Boundary surface (orisphere) we call that surface which arises from the revolution of the boundary line about one of its axes, which, together with all other axes of the boundary-line, will be also an axis of the boundary-surface. A chord is inclined at equal angles to such axes drawn through its end- points, wheresoever these two end-points may be taken on the boundary-surface. Let A, B, C, (Fig. 27), be three points on the boundary-surface; Citar , Via. 27. AA’, the axis of revolution, BB’ and CC’ two other axes, hence AB and AC chords to which the axes are inclined at equal angles A/AB —=B/BA, A’AC —C’CA (Theorem 31.) 34 THEORY OF PARALLELS. Two axes BB’, CC’, drawn througn the end-points of the third chord BC, are likewise parallel and lie in one plane, (Theorem 25). A perpendicular DD/ erected at the mid-point D of the chord AB and in-the plane of the two parallels AA’, BB’, must be parallel to the three axes AA’, BB’, CC’, (Theorems 23 and 25); just such a perpen- dicular EK’ upon the chord AC in the plane of the parallels AA’, CC’ will be parallel to the three axes AA’, BB’, CC’, and the perpendicular DD’. Let now the angle between the plane in which the parallels A.A’ and BB’ lie, and the plane of the triangle ABC be designated by // (a), where a may be positive, negative or null. If a@ is positive, then erect FD —a within the triangle ABC, and in its plane, perpendicular upon the chord AB at its mid-point D. Were a a negative number, then must FD = a be drawn outside the triangle on the other side of the chord AB; when a—0, the point F coincides with D. In all cases arise two Agee right-angled triangles AFD and DFR, consequently we have FA = FB. Hrect now at I the line Fi’ perpendicular to the plane of the tri. angle ABC. Since the angle D/DF = //(a), and DF —a, so FF’ is parallel to DD’ and the line EE’, with which also it lies in one plane perpendicu- lar to the plane of the triangle ABC.. Suppose now in the plane of the parallels EK’, FF’ upon HF the per- pendicular EK erected, then will this be also at right angles to the plane of the triangle ABC, (Theorem 13), and to the line AE lying in this plane, (Theorem 11); and consequently must AE, which is perpendicu- lar to EK and EE’, be also at the same time perpendicular to FH, (Theorem 11). The triangles AEF and FEC are congruent, since they are right-angled and have the sides about the right angles equal, hence is Ane ECs i A perpendicular from the vertex F of the isosceles triangle BFC let fall upon the base BC, goes through its mid-point G; a plane passed through this perpendicular FG and the line FF’ must be perpendicular to the plane of the triangle ABC, and cuts the plane of the parallels BB’, CC’, along the line GG’, which is likewise parallel to BB’ and CC’, (Theorem 25); since now OG is at right angles to FG, and hence at the same time also to GG’, so consequently is ee angle C’OG = B’/BG, (Theorem 23). a - THEORY OF PARALLELS. 35 Hence follows, that for the boundary-surface each of the axes may be considered as axis of revolution. Principal-plane we will call each plane passed through an axis of the boundary surface. Accordingly every Principal-plane cuts the boundary-surface in the boundary line, while for another position of the cutting plane this in- tersection is a circle. Three principal planes which mutually cut each other, make with each other angles whose sum is z, (Theorem 28). These angles we will consider as angles in the boundary-triangle whose sides are arcs of the boundary-line, which are made on the bound: ary surface by the intersections with the three principal planes. Con- sequently the same interdependence of the angles and sides pertains to the boundary-triangles, that is proved in the ordinary geometry for the rectilineal triangle. 35. In what follows, we will designate the size of a line by a letter ‘with an accent added, e. g. x’, in order to indicate that this has a rela, tion to that of another line, which is represented by the same letter without accent x, which relation is given by the equation IT (2) + I(x!) = $7. Let now ABC (Fig. 28) be a rectilineal right-angled triangle, where the hypothenuse AB =e, the other sides AC=—b, BC =a, and the Fig. 28. angles opposite them are BAC = /[I(a), ABC = /T(§). 36 THEORY OF PARALLELS. At the point A erect the line AA’ at right angles to the plane of the triangle ABC, and from the points B and C draw BB’ and CC’ parallel to AA’, The planes in which these three parallels lie make with each other the angles: //(z) at AA’, a right angle at CC’ (Theorems 11 and 13), consequently //(z’) at BB’ (Theorem 28). The intersections of the lines BA, BC, BB’ with a sphere described about the point B as center, determine a spherical triangle mnk, in which the sides are mn = [](c),. kn= I1(8), mk— [](a) and the opposite angles are [I(b), IM(u!), 42. Therefore we must, with the existence of a rectilineal triangle whose sides are a, b, c and the opposite angles // (z), //() 47, also admit the existence of a spherical triangle (Fig. 29) with the sides //(c), /7({), [](a) and the opposite angles //(b), //(a’), $z- Tc p) Fig. 29. Of these two triangles, however, also inversely the existence of the spherical triangle necessitates anew that of a rectilineal, which in con- sequence, also can have the sides a, a’, 8, and the oppsite angles /](b’), Ic), 3x. Hence we may pass over from a, b, ¢, 2, [, to b, a, c, 8, a, and also toa, hs san ht yp ae | Suppose through the point A (Fig. 28) with AA’ as axis, a bound- ary-surface passed, which cuts the two other axes BB’, CC’ , in B” and C”, and whose intersections with the planes the parallels form a bound- ary-triangle, whose sides are B’C” =p, C”A=gq, B”A=7, and the angles opposite them //(2), //(a’), 4x, and where consequently (Theo- rem 34): p—rsin /[](a), g=rcos [](a): Now break the connection of the three principal-planes along the line BB’, and turn them out from each other so that they with all the lines lying in them come to lie in one plane, where consequently the arcs p, q, r will unite to a single arc of a boundary-line, which goes through the | THEORY OF PARALLELS. yl point A and has AA’ for axis, in such a manner that (Fig. 30) on the one side will lie, the arcs g and p, the side b of the triangle, which is Fic. 30. perpendicular to AA’ at A, the axis CC’ going from the end of b par- allel to AA’ and through C’” the union point of p and g, the side a per- pendicular to CC’ at the point C, and from the end-point of a the axis BB’ parallel to AA’ which goes through the end-point B” of the arc p. On the other side of AA’ will lie, the side c perpendicular to AA’ at the point A, and the axis BB’ parallel to AA’, and going through the end-point B” of the arc r remote from the end point of b. The size of the line CC” depends upon b, which dependence we will express by CC” — f(b). In like manner we will have BB” — / (c). If we describe, taking CC’ as axis, a new boundary line from the point CO to its intersection D with the axis BB’ and designate the arc CD by ¢, then is BD = (a). BB’ — BD+ DB’ — BD-+CC’, consequently #(c)=F(a)-+ £(0). Moreover, we perceive, that (Theorem 32) | t—pet) —r sin [](a) ef), If the perpendicular to the plane of the triangle ABC (Fig. 28) were erected at B instead of at the point A, then would the lines c andr remain the same, the arcs g and ¢ would change to ¢ and gq, the straight lines a 38 THEORY OF PARALLELS. and b into b and a, and the angle //(z) into //(f), consequently we would have q=rsin //(f) e%™, whence follows by substituting the value of 4g, cos /} (z) = sin /T(f) e%™, and if we change «and f into b’ and ¢, sin //(b) =sin //(c)e™; further, by multiplication with e/) sin /7 (b) ef) = sin [J (c) es Hence follows also sin // (a) e/) — sin /7(b) ef), Since now, however, the straight lines a and b are independent of one another, and moreover, for b—0, f(b)=—0, //(b)—47, so we have for every straight line a e—f@) —sin /] (a). Therefore, sin // (c) sin // (a) sin /7(b), sin []() —cos // (a) sin // (a). Hence we obtain besides by mutation of the letters sin // (a) —cos //() sin /7(b), cos [](b) = cos [/ (c) cos J] (2), cos [] (a) cos /](c) cos /7({). If we designate in the right-angled spherical triangle (Fig. 29) the sides /J(c), //(), [7 (a), with the opposite angles //(b), //(a’), by the letters a, b, c, A, B, then the obtained equations take on the form of those which we know as proved in spherical trigonometry for the right- angled triangle, namely, 3 sin a=sin c sin A, sin b=sin ¢ sin B, cos A=cos asin B, cos B=cos b, sin A, COs C=Cos a, cos b; from which equations we can pass over to those for all spherical tri- angles in general. Hence spherical trigonometry is not dependent upon whether in a THEORY OF PARALLELS. 39 rectilineal triangle the sum of the three angles is equal to two right angles or not. 86. We will now consider anew the right-angled rectilineal triangle ABC (Fig. 31), in which the sides are a, b, c, and the opposite angles I(x), I1(8), ¥7- Prolong the hypothenuse c through the point B, and make BD=§; at the point D erect upon BD the perpendicu- lar DD’, which consequently will be parallel to BB’, the prolongation of the side a beyond the point B. Parallel to DD’ from the point A draw AA’, which is at the same time also parallel to CB’, (Theorem 25), therefore is the angle A/AD=II (e+) A’AC= J](b), consequently IT(b)= II (a) +I (c+) B If from B we lay off 8 on the hypoth- enuse c, then at the end point D, (Fig. 32), within the triangle erect upon AB the perpendicular DD’, and from the point A parallel to DD’ draw AA’, so will BC with its prolongation CC’ be the third parallel; then is, angle CAA’=// (b), DAA’= /T (c—8), consequently //(c— 6)=/T(a)+/1(b). The last equation is then also still valid, when c=, or c A — B A C B r Fia. 36. Fie. 37. Therefore we have universally for every triangle (3.) sin A tan //(a)=sin B tan //(b). For a triangle with acute angles A, B, (Fig. 35) we have also (Equa- tion 2), cos /](x)=cos A cos //(b), cos //(c—x)=cos B cos //(a) | which equations also relate to triangles, in which one of the angles A or B is a right angle or an obtuse angle. As example, for B=4z (Fig. 36) we must take x=c, the first equa- tion then goes over into that which we have found above as Equation 2, the other, however, is self-sufficing. For B>4z (Fig. 37) the first equation remains unchanged, instead of the second, however, we must write correspondingly cos [](x—c)=cos (z—B) cos //(a); but we have cos //(x—c)=—cos /[](c—x) (Theorem 23), and also cos (t—B)=—cos B. If A is a right or an obtuse angle, then must c—x and x be put for x and c—x, in order to carry back this case upon the preceding. In order to eliminate x from both equations, we notice that (Theo- rem 36) 1—[tan} {,\c—x) ]? 1-+[tan 4//(c—x) |? 1—e2x— 2¢ =P sseouas 1—[tan 4//(c)]?[cot £//(x)]? cos [1(c)—cos/1(x) ~~ 1—cos II(c)cos [1(x) cos /](c—x)= THEORY OF PARALLELS. 43 If we substitute here the expression for cos //(x), cos //(c—x), we ob- tain cos /I(a) cos B+-cos/I(b) cos A cos [](¢)— 1+¢0s //(a) cos //(b) cosA cosB whence follows cos [I(c)—cos cos /(b) B= 3 [Tc cos [](a) cos 1—cosA cos/1(b) cos [/(c) and finally [sin /7 (c) ]2 =[1—cos Boos /](c) cos /] (a) ][1—cos A cos /] (b) cos JT (c) ] In the same way we must also have (4. [sin // (a) ]? =[1—cos C cos /] (2) cos /] (b) ][1—cos B cos /] (c) cos /] (a) | [sin IT(b) ]? =[1—cos A cos /] (b) cos /7(c) ][1—cos C cos /7 (a) cos /7(b) | From these three equations we find [sin /7(b)]? [sin (c)]? [sin 11 (a) Hence follows without ambiguity of sign, =[1—cosA cos //(b) cos II(c)] 2 : : sin // (b)sin // (c (5.) cos A cos IT(b) cos /1(c) $a ia If we substitute here the value of sin //(c) corresponding to equa- tion (3.) : sin A. he sin C tan // (a) cos [7 (c) then we obtain cos // (a) sin C sin A sin //(b)+-cos A sin C cos// (a) cos /7(b); but by substituting this expression for cos //(c) in equation (4), cos IT (c) —= (6.) cot A sinC sin /] Micee cae By elimination of sin //(b) with help of the equation (3) comes cos [T (a) 7; cosA oe TE) cos C=1 — ——sysinC sin II (a). In the meantiine the equation (6) gives by changing the letters, cos IT (a) cos II(b) —cot B sinC sin // (a)-+-cosC. 44 THEORY OF PARALLELS. From the last two equations follows, sin B sinC sin [/ (a) All four equations for the interdependence of the sides a, b, c, and (7.) cos A-+-cos B cosC— the opposite angles A, B, C, in the rectilineal triangle will therefore be, [ Equations (3), (5), (6), (7).] /sin A tan //(a) = sin B tan // (b), sin /] (b) sin //(c) cos A cos //(b) cos /[(c) + — sin [7 (a) =a (8.) eee A sin C sin /] (b) + cos C =e sin BsinC cos A + cos Bcos C = mT If the sides a, b, c, of the triangle are very small, we may content our- selves with the approximate determinations. (Theorem 36.) | cot /] (a) = a, sin // (a) = 1 — $a? cos /] (a) = a, and in like manner also for the other sides b and c. The equations 8 pass over for such triangles into the following: bsin A = asin B, a2 =b2 + c? — 2becosA, asin (A -++ C) = bsin A, cos A. + cos(B + C) = 0. ) Of these equations the first two are assumed in the ordinary geom- etry; the last two lead, with the help of the first, to the conclusion A+B+C=rnz. Therefore the imaginary geometry passes over into the ordinary, when we suppose that the sides of a rectilineal triangle are very small. I have, in the scientific bulletins of the University of Kasan, pub. lished certain researches in regard to the measurement of curved lines, of plane figures, of the surfaces and the volumes of solids, as well as in relation to the application of imaginary geometry to analysis. The equations (8) attain for themselves already a sufficient foundation for considering the assumption of imaginary geometry as possible. Hence there is no means, other than astronomical observations, to use THEORY OF PARALLELS. 45 _ for judging of the exactitude which pertains to the calculations of the ordinary geometry. : This exactitude is very far-reaching, as | have shown in one of my investigations, so that, for example, in triangles whose sides are attain- able for our measurement, the sum of the three angles is not indeed dif- ferent from two right angles by the hundreath part of a second. In addition, it is worthy of notice that the four equations (8) of plane geometry pass over into the equations¢for spherical triangles, if we put a,/— 1, b ,/— 1, c,/— 1, instead of the sides a, b, c; with this change, however, we must also put . 1 sin // (a) Babee (5) cos [7 (a) = (4/—1) tana, gtd eres ae 1), . and similarly also for the sides b and c. In this manner we pass over from equations (8) to the following: sin A sin b = sin Bsina, cosa = cos b cose + sin b sinc cos A, cot A sin C + cosC cos b = sin b cota, cos A = cosa sin B sin C — cos B cosC. TRANSLATOR’S APPENDIX. ELLIPTIC GEOMETRY. Gauss himself never published aught upon this fascinating subject, Geometry Non-Euclidean; but when the most extraordinary pupil of his long teaching life came to read his inaugural dissertation before the Philosophical Faculty of the University of Goettingen, from the three themes submitted it was the choice of Gauss which fixed upon the one “Ueber die Hypothesen welche der Geometrie zu Grunde liegen.” Gauss was then recognized as the most powerful mathematician in the world. I wonder if he saw that here his pupil was already beyond him, when in his sixth sentence Riemann says, ‘therefore space is only a special case of a three-fold extensive magnitude,” and continues: “From this, however, it follows of necessity, that the propositions of geometry can not be deduced from general magnitude ideas, but that those peculiarities through which space distinguishes itself from other thinkable threefold extended magnitudes can only be gotten from ex- perience. Hence arises the problem, to find the simplest facts from which the metrical relations of space are determinable—a problem which from the nature of the'thing is not fully determinate; for there may be obtained several systems of simple facts which suffice to deter- mine the metrics of space; that of Huclid as weightiest is for the pres- ent aim made fundamental. These facts are, as all facts, not necessary, but only of empirical certainty; they are hypotheses. Therefore one can investigate their probability, which, within the limits of observation, of course is very great, and after this judge of the allowability of their extension beyond the bounds of observation, as well on the side of the immeasurably great as on the side of the immeasurably small.” Riemann extends the idea of curvature to spaces of three and more dimensions. The curvature of the sphere is constant and positive, and on it figures can freely move without deformation. The curvature of the plane is constant and zero, and on it figures slide without stretching. The curvature of the two-dimentional space of Lobatschewsky and [47] 48 THEORY OF PARALLELS, Bolyai completes the group, being constant and negative, and in it fig- ures can move without stretching or squeezing. As thus corresponding to the sphere it is called the pseudo-sphere. In the space in which we live, we suppose we can move without de- formation. It would then, according to Riemann, be a special case of a space of constant curvature We presume its curvature null. At once the supposed fact that our space does not interfere to squeeze us or stretch us when we move, is envisaged as a peculiar property of our space. But is it not absurd to speak of space as interfering with any- thing? If you think so, take a knife and a raw potato, and try to cut it into a seven-edged solid. Father on in this astonishing discourse comes the epoch-making idea, that though space be unbounded, it is not therefore infinitely great. Riemann says: ‘‘In the extension of space-constructions to the im- measurably great, the unbounded is to be distinguished from the in- finite; the first pertains to the relations of extension, the latter to the size-relations. “That our space is an unbounded three-fold extensive manifoldness, is an hypothesis, which is applied in each apprehension of the outer world, according to which, in each moment, the domain of actual perception is filled out, and the possible places of a sought object constructed, and which in these applications is continually confirmed. The unbounded- ness of space possesses therefore a greater empirical certainty than any outer experience. From this however the Infinity in no way follows. Rather would space, if one presumes bodies independent of place, that is ascribes to it a constant curvature, necessarily be finite so soon as this curvature had ever so small a positive value. One would, by extend- ing the beginnings of the geodesics lying in a surface-element, obtain an unbounded surface with constant positive curvature, therefore a sur- face which in a homaloidal three-fold extensive manifoldness would take the form of a sphere, and so is finite.” Here we have for the first time in human thought the marvelous per- ception that universal space may yet be only finite. ; Assume that a straight line is uniquely determined by two points, but take the contradictory of the axiom that a straight line is of infinite size; then the straight line returns into itself, but two having inter- sected get back to that intersection point without ever again meeting. TRANSLATOR’S APPENDIX. 49 Two intersecting complete straight lines enclose a plane figure, a digon. Two digons are congruent if their angles are equal. All complete straight lines are of the same length, 7. In a given plane all the per- pendiculars to a given straight line intersect in a single point, whose distance from the straight line is 47. Inversely, the locus of all the points at a distance 4/ on straight lines passing through a given point and lying in a given plane, is a straight line perpendicular to all the radiating lines. The total volume of the universe is /3/z. The sum of the angles of a plane triangle is greater than a straight angle by an excess proportional to its area. The greater the area of the triangle, the greater the excess or differ- ence of the angle sum from z. Says the Royal Astronomer for Ireland: “It is necessary to measure large triangles, and the largest triangles to which we have access are, of course, the triangles which the astrono- mers have found means of measuring. The largest available triangles are those which have the diameter of the earth’s orbit as a base and a _ fixed star at the vertex. It is a very curious circumstance that the in- vestigations of annual parallax of the stars are precisely the investiga- tions which would be necessary to test whether ono of these mighty tri- angles had the sum Of its three angles equal to two rightangles. * * * “ Astronomers have sometimes been puzzled by obtaining a negative parallax as the result of their labors. No doubt this has generally or indeed always arisen from the errors which are inevitable in inquiries of this nature, but if space were really curved then a negative parallax might result from observations which possessed mathematical perfec- tion. * * * It must remain an open question whether if we had large cnough triangles the sum of the three angles would still be two right angles.” Says Prof. Newcomb: ‘There is nothing within our experience which will justify a denial of the possibility that the space in which we find ourselves may be curved in the manner here supposed. * * * “The subjective impossibility of conceiving of the relation of the most distant points in such a space does not render its existence in- credible. In fact our difficulty is not unlike that which must have been felt by the first man to whom the idea of the sphericity of the earth 5O THEORY OF PARALLELS. was suggested in conceiving how by traveling in a constant direction he could return to the point from which he started without during his journey feeling any sensible change in the direction of gravity.” In accordance with Professor Cayley’s Sixth Memoir upon Quantics: “The distance between two points is equal io c times the logarithm of the cross ratio in which the line joining the two points"is divided by the funda- mental quadric.”’ This projective expression for distance, and Laguerre’s for an angle were in 1871 generalized by Felix Klein in his article Ueber die soge- nannte Nicht-EKuklidische Geometric, and in 1872 (Math. Ann., Vol. 6) he showed the equivalence of projective metrics with non-Euclidean geometry, space being of constant negative or positive curvature ac- cording as the fundamental surface is real and not rectilineal or is im- aginary. | We have avoided mentioning space of four or more dimensions, wishing to preserve throughout the synthetic standpoint. For a bibliography of hyper-space and non-Euclidean geometry see articles by George Bruce Halsted in the American Journal of Mathe- matics, Vol. L, pp. 261-276, 384, 385; Vol. IL., pp. 65-70. We notice that Clark University and Cornell University are giving regular courses in non-Euclidian geometry by their most eminent Pro- fessors, and we presume, without looking, that the same is true of Har- vard and the Johns Hopkins University, with Prof. Newcomb an origi- nal authority on this far-reaching subject. ~~ wf ms Siu k \ 7 ; . q a Toppan, +e ee in Pat eS oe ee a. A