THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES \yf? s^jv-faivArfo+v^-tup GEOMETRICAL RESEARCHES ON THE THEORY OF PARALLELS BY NICHOLAS LOBACHEVSKI Imperial Russian Real Councillor of State and Regular Professor of Mathematics in the University of Rasan BERLIN, 1840 Translated from the Original BY GEORGE BRUCE HALSTED A. M., Ph. D., Ex-Fellow of Princeton and Johns-Hopkins University NEW EDITION LA SALLE, ILLINOIS OPEN COURT PUBLISHING COMPANY 1914 Plonographed by John S. Swift Co., Inc., Chicago. St. Louis, New York, Cincinnati Copyright by The Open Court Publishing Co., Chicago, 1914. Mathematical Sciences Library 'Q PR 1? PA PI? o rKMAUti, Lobachevski was the first man ever to publish a non-Euclidean 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 Lobachevski 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 Lobachevski's of Geometry from Euclid's noted empirical axiom." Cayley says, "It is well known that Euclid's twelfth axiom, even in Playf air's form of it, has been considered as needing demonstration; and that Lobachevski constructed a perfectly consistent theory, where- in this axiom was assumed not to hold good, or say a system of non- Euclidean plane geometry. There is a like system of non-Euclidean solid geometry." GEORGE BRUCE HALSTED. 2407 San Marcos Street, Austin, Texas. May 1, 1891. TRANSLATOR'S INTRODUCTION. "Prove all things, hold fast that which is good," does not mean dem- onstrate everything. "Prom 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, without 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 Egypt's priests. The Papyrus of the Rhind, belonging to the British Museum, -but given to the world by the erudition of a German Egyptologist, Eisen- 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 Euclid, 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 phoeniz 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. Finally back again in its original Greek, it is published first in queenly Basel, then in stately Oxford. The latest edition in Greek is from Leipsic's learned presses. 6 THEORY OP 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, where they have, by some strange chance, Billingsly's own copy of the Arabic- Latin version of Campanus bound with the Editio Princeps in Greek and enriched with his autograph emendations. Even 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 Epic 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 Euclid'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 Europe'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. Euclid 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 un wieldly postulate or axiom: " If 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 un wieldly thing from the set of fundamental axioms. TRANSLATOR 8 INTRODUCTION. 7 Numerous and desperate were the attempts to deduce it from reason- ings about the nature of the straight line and plane angle. In the " Encyclopcedie der Wissenschaften und Kunste; Von Erech 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 Euclid's definition by pseudo-reasoning which tacitly assumes Euclid's postulate, e. g. he says p. 35: "For, 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 Euclid'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 Euclid's in form if not in essence. Of these the best is that catted 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, "Euclid and his Modern Rivals," by C. L. Dodgson, late Mathematical Lecturer of Christ Church, Oxford, the Lewis Carroll, author of Alice in Wonderland. 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 Lobachevski, 'Etudes Geom6triques sur la Theorie des Parallels' - - - a small tract of which Gauss, therein quoted, says : L'auteur a traits la matiere en main de maltre 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." 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 Hoiiel, 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 Nicolas Lobachevski (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 quick-following fellows could be called Non-Euclidean. 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 geometric non-euclidienne ne renferme en elle rien de contradictoire, quoique, 8. premiere vue, beaucoup de ses rfeul- tats aien 1'air de paradoxes. Ces contradictions apparents doivent etre regardees comme Peffet d'une illusion, due a 1'habitude que nous avons prise de bonne heure de considerer la geometric euclidienne 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. TRANSLATOR'S INTRODUCTION. V A former friend of Gauss at Goettingen was the Hungarian Wolfgang Bolyai. His principal work, published by subscription, has the follow- ing title: Tentamen Juventutem studiosam in elementa Matheseos purae, ele- meutaris ac sublimiorls, methodo intuitiva, evidentiaque huic propria, in- troducendi. Tomus Primus, 1832; Secundus, 1833. 8vo. Maros-Va- sarhelyini. In the first volume with special numbering, appeared the celebrated Appendix of his son John Bolyai with the following title : APPENDIX. 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 Capitaneo. (26 pages of text). This marvellous Appendix has been translated into French, Italian, English 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, 06 zwey von der dritten geschnittene Geraden. wenn die summe der inneren Wtnkel nicht=2R, sich schneiden oder nichtf niemand auf der Erde ohne ein Axiom (wie Euclid das XI) aufzustellen, beantworten wird; die davon unabhaengige Geometric 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 seyen." The author mentions Lobachevski's Geometrische Untersuchungen, Berlin, 1840, and compares it with the work of his son John Bolyai, "au sujet duquel il dit : 'Quelques exemplaires de 1'ouvrage publiS ici ont t6 envoyfe a cette 6poque S Vienne, a Berlin, a Gcettingue. . . De EC from the point C. Suppose in the right-angled tri- angle ACE the sum of the three angles is equal to TT a, in the tri- angle AEF equal to x ft, then must it in triangle ACF equal TC a ft, where a and ft can not be negative. Further, let the angle BAF = a, AFC = b, so is a +/? = 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 also can we lessen the angle b, consequently the two angles a and ft can have no other magnitude than a = and ft = 0. It follows that in all rectilineal triangles the sum of the three angles is either rr and at the same time also the parallel angle // (p) 4 /T for every line p, or for all triangles this sum is < TT and at the same time also 77(p)< K. The first assumption 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 Imaginary 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. For every given angle a there is a line p such that fl (p) = a. Let AB and AC (Fig. 10) be two straight lines which at the inter. section point A make the acute angle a; 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* r= AA'; erect at A' the perpendicular A'B'; and so continue until a per- 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 JT a, then in the triangle AB' A* it equals n 2a, in triangle AA'B" less than TT 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 HFG, 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 A KB, 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 FG is parallel to AB and AF = p. (Theorems 16 and 18.) One easily sees that with the lessening of p the angle a, increases, while, for p = 0, it approaches the value TT; with the growth of p the angle a decreases, while it continually approaches zero for p =00 . Since we are wholly at liberty to choose what angle we will under- THEORY OF PARALLELS. 21 stand by the symbol 77 (p) when the line p is expressed by a negative number, so we will assume 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 = BD are erected and their end-points C and D joined by D a straight line, then will the quadrilat- eral CABD have two right angles at A and B, but two acute angles at C and D (Theorem 22) which are equal to one another, as we can easily see B. I by thinking the quadrilateral super- FIQ. 11. imposed upon itself so that the line BD falls upon AC and AC upon BD. Halve AB and erect at the mid-point E the line EF perpendicular to AB. This line must also be perpendicular to CE, since the quadrilat- erals CAEF and FDBE fit one another if we so place one on the other that the line EF remains in the same position. Hence the line CD 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 BD a part BG < CA. Since C is a 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 each other. FIG. 12. "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 each other. In order to prove this, let fall from any point A of the outer line AB upon the other outer line FE, the perpendicular AE, which will cut the middle line CD in some point C (Theorem 3), at an angle DCE < TT 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 BAG 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 B AE, 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. 23 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 a second time, consequently it must meet EF some- where in H; therefore AB and EF are mutually parallel. FIG. 13. 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 BAG 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 EF, 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 if 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 ides AB = A'B', BC = B'C', CA = C'A', and the corresponding angles Fio. 14. at the points A, B, C are likewise equal to those in the other triangle at th points A', B',C'. Through the three points A, B, C, suppose a plane passed, and upon it from the 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 'ess a right-angle. In the spherical triangle ABC (Fig. 15), where each side < n, 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 ABC, prolong now the other two sides through their common intersection point C until they meet the circle in A' and B'. 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 -f 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 P -}- Y = C, and since P-f-X-f-Y-|-Z==7r, therefore we have also 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 BC, and through the mid-points D and B 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 E, then will of the triangles so made BDH = AFD, and BHE = EGG (The- orems 6 and 15), whence follows that Fio. 16. the surface of the triangle ABC equals that of the quadrilateral AFGO (Theorem 26). 26 THEORY OP PARALLELS. If the point H coincides with the middle point E of the side BC (Fig. B IV), only two equal right-angled triangles, ADF and BDE, are made, by whose interchange the equivalence of the surfaces of the triangle ABC and the quadrilateral AFEC is established. If, finally, the point H falls outside the triangle 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 triangle FAD = DBH, and then taking away the triangle CGE = EBEL Supposing in the spherical quadrilateral AFGC a great circle passed through the points A and G, as also through F and C, 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 of 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 perpendicular sides, and where the two other angles are each S. THEORY OF PARALLELS. .27 Let now ABCD (Fig. 19) be the spherical quadrilateral, where the aides AB ~ DC are perpendicular to BC, and the angles A and D each S. Fio. 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 = DC = 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 arc AHF = TT, ADEF likewise = n, the angle HAD = HFE == S BAH = S HFG = |S HFE EFG |S HAD-7r+S; consequently, angle HFE = (S ;r); 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 TT) 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 sum 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 DC, 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 A A', 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 A A' with it determine a spherical triangle, with the sides p, q, and r. Call its size a. Opposite the side q lies the angle w, opposite r lies X, and consequently opposite p lies the angle ;r-|-2a w X, (Theorem 27). In like manner CA, CD, CC' cut a sphere about the center C, and determine a triangle of size ft, with the sides p', q', r', and the angles, w opposite q', Z opposite r', and consequently ;r-{-2/9 wZ 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-\-Z2ft, w-\-"K 2#, and Y. Consequently its size /7(a)+/7(b.) If we lessen this angle, so that it becomes equal to 77 (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 /7(a) /7(c')>/7(a) //(c), whence follows c' >c (Theorem 23). A 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 H! FIG. 24. point A in it, making different angles CAB= /7(a), chords AC = 2a; the end C of such a chord will lie 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 Axis 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 Axes 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 BAG = ABF = a (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 K w A of the axis AC to its intersection point FIG. 25. 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 AFE=^9, and on the 32 THEORY OF PARALLELS. other side with the axis BF, the angle EFD = f. It follows that the angle between the two chords BAF = a fii, 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'= for = GO , 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 (prisphere) 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; FIG. 27. A A', 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', CO', drawn through 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 A A', BB', must be parallel to the three axes AA', BB', CO', (Theorems 23 and 25); just such a perpen- dicular BE' upon the chord AC in the plane of the parallels A A', 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 AA' and BB' lie, and the plane of the triangle ABC be designated by 77 (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 A B 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 congruent right-angled triangles AFD and DFB, consequently we have FA = FB. Erect now at F the line FF' perpendicular to the plane of the tri- angle ABC. Since the angle D'DF = /7(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 EE', FF' upon EF 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 FE, (Theorem 1 1). The triangles AEF and FEC are congruent, since they are right-angled and have the sides about the right angles equal, hence is AF = FC = FB. 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 CG is at right angles to FG, and hence at the same time also to GG', so consequently is the angle C'CG = B'BG, (Theorem 23). 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 TT, (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. a/, 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 Let now ABC (Fig. 28) be a rectilineal right-angled triangle, where the hypothenuse AB = c, the other sides AC = b, BC = a, and the FIG. 28. angles opposite them are 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 CO' parallel toAA'. The planes in which these three parallels lie make with each other the angles: /7(a) at AA', a right angle at CC' (Theorems 11 and 13), consequently //(') 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 = 77(c), Ten = /I (ft), mJc = /7(a) and the opposite angle? are /7(b), //('), $x. Therefore we must, with the existence of a rectilineal triangle whoa.. sides are a, b, c and the opposite angles U (a), IJ(ft) fa also admit th-s existence of a spherical triangle (Fig. 29) with the sides 77 (c), /7(a) and the opposite angles 77(b), //(')> fa 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', ft, and the oppsite angles /7(b'), /7(c), fa Hence we may pass over from a, b, c, a, ft, to b, a, c, ft, a, and also to a, a', ft, b', c. 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 y, B'A r ? and the angles opposite them //(), //('), fa and where consequently (Theo- rem 34): p = r sin IJ(a), q = r cos 77(). 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, g, r will unite to a single arc of a boundary-line, which goes through the THEORY OF PARALLELS. 37 point A and has A A' for axis, in such a manner that (Fig. '30) on the one side will lie, the arcs q and p, the side b of the triangle, which is Fro. 30. perpendicular to AA' at A, the axis CC' going from the end of b par- allel to A A' and through C' the union point of p and q, the side a per- pendicular to CC' at the point C, and from the end-point of a the axis BB' parallel to A A' 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' =/(b). In like manner we will have BB' =/(c). If we describe, taking CC ' as axis, a new boundary line from the point C to its intersection D with the axis BB' and designate the arc CD by t, then is ED = f (a). BB'=: BD-f-DB' = BD-fCC', consequently Moreover, we perceive, that (Theorem 33) 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 and r remain the same, the arcs q and t would change to t and q, the straight lines a So THEORY OP PARALLELS. and b into b and a, and the angle 77 (a) into 77(^9), consequently we would have q= rsn whence follows by substituting the value of q, cos 77 (a) = sin 77 (/9) e*>, and if we change a and ft into b' and c, sin77(b)^sin77(c)e a >; further, by multiplication with e-^ b) sin II (b) e/< b >= sin 77 (c) e*e> Hence follows also sin II (a) e-K> sin 77 (b) e^). Since now, however, the straight lines a and b are independent of one another, and moreover, for b=0, /(b)=0, 77(b)=r^r, so we have for every straight line a e-/<>=sin77(a). Therefore, sin 77 (c) = sin 77 (a) sin 77 (b), sin 77 (/9) = cos 77 (a) sin 77 (a). Hence we obtain besides by mutation of the letters sin 77 (a) = cos 77 Q3) sin 77 (b), cos 77 (b) = cos 77 (c) cos 77 (a), cos II (a) = cos 77(c) cos 77(). If we designate in the right-angled spherical triangle (Fig. 29) the sides 77(c), 77 (/3), 77 (a), with the opposite angles 77 (b), 77 (?r (Fig. 37) the first equation remains unchanged, instead of the second, however, we must write correspondingly cos 77(x c)=cos (TT B) cos 77(a); but we have cos /7(x c) = cos 77(c x) (Theorem 23), and also cos (x 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) l-[tanj/7(c-x)] ~l+[tan|/7(c x)]? cos/7(c 1 [tan |/7(c)]8[cot j/7( : l+[tan/7(c)]*[co COB /7(c) cos/7(x) 1 cos /7(c)cos/7(x) THEORY OP PARALLELS. 43 If we substitute here the expression for cos 77(x), cos/7(c *), we ob- tain j,. , _ COB 77(a) cos B-j-cos77(b) cos A "A C >= i^coe//^) coe/7(b) cosA cosB whence follows cos /7(a) cosB^ <** %)~<**A cos77(b) 1 cosA cos/7(b) cos 77(c) and finally [sin/7(c)]=[l cosBcos/7(c)cos/7(a)][l -cosAcos77(b)cos/7(c)] In the same way we must also have w [sin 77(a) ] =[1 -cos C cos 77 (a) cos 77 (b) ] [1 cos B cos 77 (c) cos 77(a) ] [sin 77(b) ] 8 =[1 -cos A cos /7 (b) cos /7(c) ] [1 cos C coe /7(a) cos 77(b)] From these three equations we find [sin77(b)][sin77(c)]* ., ., [siny/(a)] -=[! cosAcos/7(b)cos/7(c)]. Hence follows without ambiguity of sign, IT /i.x rr, -^ . sin/7(b)sin/7Yc) (5.) cos A cos // (b) cos //(c) H -- . v ' . = 1. sin /7 (a) If we substitute here the value of sin 77 (c) corresponding to equa- tion (3.) sin JJ(c)=^- tan H (a) cos JJ (c) then we obtain cos Hfc) =. cos /7(a) sinC sin A sin /7 (b)-j-cos A sin C cos 77 (a) cos 77 (b); but by substituting this expression for cos 77 (c) in equation (4), (6.) cot A sin C sin 77 (b)+cos 0= cos77(a) By elimination of sin 77(b) with help of the equation (3) comes cos 77 (a) cos A . ^cosC^l -- r -R-sinCsin77(a). cos 77 (b) sin B In the meantime the equation (6) gives by changing the letters, cos 77(a) cos 77 (b) ==cotB sin C sin 77(a)-J-cosC. 44 THEORY OF PARALLELS. From the last two equations follows, . _ _ sin B sin C (7.) cos A4-C08 B cos C : . . sm77(a) All four equations for the interdependence of the sides a, b, c, and the opposite angles A, B, C, in the rectilineal triangle will therefore be, [Equations (3), (5), (6), (7).] sin A tan 77 (a) = sin B tan 77 (b), sin 77 (b) sin 77 (c) cos A cos 77 (b) cos 77 (c) -\ cot A sin C sin 77 (b) -f- cos C = cos A -f- cos B cos C = 77(a) cos 77 (b) cos 77 (a) ' sin B sin C sin 77 (a) If the sides a, b, c, of the triangle are very small, we may content our* selves with the approximate determinations. (Theorem 36.) cot 77 (a) = a, sin 77 (a) = 1 |a cos 77 (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: b sin A = a sin B, a 8 =b* + c 8 2bc cos A, a sin ( A -j- C) = b sin A, cos A -f 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 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 OP PARALLELS. 46 for judging of the exactitude which pertains to the calculations of the ordinary geometry. This exactitude is very far-reaching, as I 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 hundredth 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