[t :i.J eaeeeeeeceaeeeeeaeeeeeeeecaeereaceeeeeaE!Ji!C!P'.i!eeEE1212eaee222aceaeee-aea BULLETIN OF THE UNIVERSITY OF TEXAS. HALS TED'S LOBATSCHEWSKY'S GEOMETRY. !ee•e.ea@pepee•·eee.nnreee eea eee eae eee eee eee cee eee 222 222 222eee 222 ea22eeeee GEOMETRICAL RESEARCHES ON THE THEORY OF PARALLELS. BY NICHOLAUS LOBATSCHEWSKY, lllnBU.L RUSSIAN REAL COUNCILLOR OF STATE AND REGULAR PROFESSOR OP KATBEKATICS IN THE UNIVERSITY OF KASAN. BERLIN, 1840. TBANSLATBD FRO)[ THE ORIGINAL BY GEORGE BBUOE 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 l:NIVERSITY OF TBXA.8. 1891. TRANSLATOR'S PREFACE. Lobatschewsky was the fil'.'Bt man ever to publish a non-Euclidian geom­etry. 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 geom­eter.'s spirit. I think I ought to call your attention to this book, whose perusal can noi fail to give you the most vivid pleMure." 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 Al­gebra released from the yoke of the commutative principle of multipli­cation-an emancipation somewhat akin to Lobatschewsky's of Geometry from Euclid's noted empirical axiom." Cayley says, "It is well known that Euclid's twelfth axiom, even in Playfair's form of it, has been considered as needing. demonstration; and that Lobatchewsky constructed a perfectly consistent theory, where­in this axiom was assumed not to hold good, or say a system of non­Euclidian plane geometry. There is a. like system of non-Euclidia.n solid geometry." GEORGE BRUCE HALSTED. 2407 San Marcos Street, Austin, Tuas. May I, 1891. [8) TRANSLATOR'S INTRODUCTION. 11 Prove all things, hold fast that which is good," does not mean dem­onstrate everything. From 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 pa.radoxer. 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 pl'iests. The Papyrus of the Rhind, bE.longing to the British Museum, but given to the world by the erudition of a G13rman Egyptologist, Eisen­lohr, 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 con­firms 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 efflor­escence 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 phamix of its native Egypt. it rises with the new birth of culture. An Anglo Saxon, Adela.rd 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 Venice, then in stately Oxford, since then everywhere. The latest edition in Greek is just issuing from Leipeic's learned presses. (5] 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. 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. faction. 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 ordinsry 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 unwieldly 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 unwieldly thing from the set of fundamental axioms. Numerous and desperate were the attempts to deduce it from reMOn· ings about the nature of the straight line and plane angle. In the "Encyclopoodie der Wissenschaften und Kunste; Von Ersch und Gru­ber;" Leipzig, 1838; under "Parallel," Sohncke says that in mathe­matics 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 acknowledgeQ 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 every­where equally distant; and since they lie in the same plane; must· ap­proach 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 hil.ve 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 cafied Piayfa.ir's; " Two lines which intersect e&n 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, 11 by C. L. Dodgson, late Mathematical Lecturer of Christ Church, Oxford. All this shows how ready the world was for the extra.ordinary flaming­forth 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 dis­covery of the Conservation of Energy, the independent irruptions of genius, whether in RuSBia, Hungary, Germany, or even in Canada gave everywhere the same results. At first these results were not fully understood even by the brightest THEORY OF PARALLELS. intellects. Thirty yea.rs after the publication of the book he mentions, we see the brilliant Clifford writing from Trinity College, Cambridge, April 2, 1870, "Severa.I new ideas ba.ve come to me lately: First I have procured Loba.tschewsky, 'Etudes Geometriques sur la. Theorie des Parallels' ---a. small tract of which Gauss, therein quoted, says: L'a.uteur a 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." The first axiom doubted is called a "vicious assumption," soon no man sees more clearly than Clifford that all are assumptions and none v1c10us. He had been reading the French translation by Houcl, pub· lished in 1866, of a little book of 61 pages published in 1840 in Bc_lin under the title Geometrische Untersuchungen zur Theorie dcr Parallel· linien by a Russian, Nicolaus Ivanovitch Lobatschewsky (1793-1856), the first public expression of whose discoveries, however, dates back to a discourse at Kasa.non 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 Geometrie non-Euclidienne ne renferme en elle rien de contradictoire, quoique, a premiere vue, beaucoup de ses resul· tats aien l'air de paradoxes. Oes contradictions apparents doivent etre regardees comme l'e:ffet d'une illusion, due a l'habitude que nous avons prise de bonne heure de considerer la geometrie 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. A clearer idea here had already come to the former room-mate of Gaues at Goottingen, the Hungarian Wolfgang Bolyai. His principal work, published by subscription, has the following title: Tentamen Juventutem etudiosam in elementa Matheeeoe purae, ele­mentarie ac eublimioris, methodo intuitiva, evidentique huic propria, in. troducendi. Tomue Primus, 1831; Secundue, 1833. 8vo. Maros-Va­earhelyini. In the first volume with special numbering, appeared the celebrated Appendix of his son Johann Bolyai with the following title: Ap., ecientiam epatii absolute veram exhibens: a veritate aut falsitate Axioms.tis XI Euclidei (a priori ha.ud unqua.m decidenda) independen· tern. Auctore Johanne Bolyai de eadem, Geometrarum in Exercitu Ca.esareo Regio Austria.co Castrensium Ca.ptaneo. Ma.ros-Vasarhely., 1832. (26 pa.ges of text). This ma.rvellous Appendix has been tra.nsla.ted into French, lta.lian, and Gerina.n. In the title of W olfga.ng Bolyai's last work, the only one he com­posed in Germa.n (88 pages of text, 1851), occurs the following : " U nd da. die Frage. ob zwei von der dritten geschnittene Geraden wenn die Summa der inneren Winkel nicht = 2 R, sich schne-iden oder n-icht?, niema.nd auf der Erde ohne ein Axiom (wie Euclid da.e XI) a.ufzustellen, beant. worten wird; die davon unabha!ngige 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 eeien." The author mentions Lobatschewsky's Geometrische U ntersuchungen I Berlin, 1840, a.nd compares it with the work of his eon Johann Bolyai, "an eujet duquel il dit· 'Quelques exempla.ires de l'ouvra.ge publie ici ont ete envoyes a cette epoque a Vienne, a Berlin, a Grettingen. . . De Goettingen le geant mathema.tique, [Gauss] qui du sommet des hauteurs embrasse du meme regs.rd les a.stres et la profondeur des abimes, a ecrit qu'il etait ravi de voir execute le travail qu'il a.vait commence pour le la.ieser apres lui da.ns see pa.piers.'" Yet that which Bolyai and Gauss, a mathema.tician never surpassed fo 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 repla.ces by its con· tradictory, that the sum of the angles ma.de on the same Bide of a trans­versal by two straight lines may be less than a straight angle without the lines meeting. A perfectly consistent and elegant geometry then follows, in which the sum of the angles of a triangle is always less than a straight angle, and not every triangle has its vertices concyclic. THEORY OF PARALLELS. In geometry I find certain imperfections which I hold to be the rea­son why this science, apart from transition into analytics, can as yet make no advance from that state in which it has come to us from Euclid. As belonging to these imperfections, I consider the obscurity in the fundamental concepts of the geometrical magnitudes and in th(I manner and method of representing the measuring of these magnitudes, and finally the momentous gap in the theory of para.llele, to fill which all ef. forts of mathematicians have been so far in vain. For this theory Legendre's endeavors ha.v9 done nothing, since he was forced to leave the only rigid way to turu into a side pa.th and take refuge in auxiliary theorems which he illogically stroYe to exhibit as necessary axioms. My first essa.y on the foundations of geometry I pub. lished in the Kasa.n Messenger for the year 1829. In the hope of having satisfied all requirements, I undertook hereupon a treatment of the whole of this science, and published my work in separate parts in the "Ge. khrten &hriften der Universitret Kasan" for the years 1836, 1837, 1838, under the title "New Elements of Geometry, with a complete Theory of Parallels." The extent of this work perhaps hindered my country­men from following such a subject, which since Legendre had lost its interest. Yet I am of the opinion that the Theory of Parallels should not lose its claim to the attention of geometers, and therefore I aim to give here the substance of my investigations, remarking beforehand that contrary to the opinion of Legendre, all other imperfections-for ex­ample, the definition of a straight line-show themselves foreign here and without any real influence on the theory of parallels. In order not to fatigue my reader with the multitude of those theo­rems whose proofs present no difficulties, I prefix here only those of which a knowledge is necessary for what follows. 1. A straight line fits upon itself in all its positions. By this I mean that during the revolution of the surface containing it the straight line does not change its place if it goes through two unmoving points in the surface: (i. e., if we turn the surface containing it about two points of the line, the line does not move.) [11] 2. Two straight lines can not intersect in two points. 3. A straight line sufficiently produced both ways must go out beyond all bounds, and in such way cuts a bounded plain.into two parts. 4. Two straight lines perpendicular to a third never intersect, how far soever they be produced. 5. A straight line always cuts another in going from one side of it over to the other side: (i. e., one straight line must cut another if it has points on both sides of it.) 6. Vertical angles, where the sides of one are productions of the sides of the other, are equal. This holds of plane rectilineal angles among themselves, as also of plane surface angles: (i.e., dihedral angles.) 7. Two straight lines can not intersect, if a third cuts them at the same angle. 8. In a rectilineal triangle equal sides lie opposite equal angles, and inversely. 9. In a rectilineal triangle, a greater side lies opposite a greater angle. In a right-angled triangle the hypothenuse is greater than either of the other sides, and the two angles adjacent to it aro acute. 10. Rectilineal triangles are congruent if they have a side and two angles equal, or two sides and the included angle equal, or two sides and the angle opposite the greater equal, or three sides equal. 11. A straight line which stands at right angles upon two other straight lines not in one plane with it is perpendicular to all straight lines drawn through the common intersection point in the plane of those two. 12. The intersection of a sphere with a plane is a circle. 13. A straight line at right angles to the intersection of two ·per­pendicular planes, and in one, is perpendicular to the other. 14. In a spherical triangle equal sides lie opposite equal angles, and inversely. 15. Spherical triangles are congruent (or symmetrical) if they have two sides and the included angle equal, or a side and the adjacent angles equal. From here follow the other theorems wit.h 'their explanations and proofs. THEORY OF PARALLELS. 13 16. All straight lines which in a plane go out from a point can, with reference to a given straight line in the same plane, be divided into two cla.sses-into cutting and not-cutting. The boundary lines of the one and the other class of those lines will be called parallel to the given line. From the point A (Fig. I) let fall upon the line BC the perpendicular AD, to which a.gain draw the perpendicular AE. In the right angle EAD either will all straight lines which go out from the point A meet the »'-------'!!'------1» line DC, as for example AF, or some of them, like the perpendicular AE, will not meet the line DC. In the uncertainty whether the per· pendicular AE is the only line which does not meet DC, we will assume it may be possible that •'. B there are still other lines, for example AG, Fm. I. which do not cut DC, how far soever they may be prolonged. In pass­ing over from the cutting lines, a.s AF, to the not-cutting lines, as AG, we must. come upon a line AH, parallel to DC, a boundary line, upon one side of which all lines AG are such a.s do not meet the line DC, while upon the other side every straight line AF cuts the line DC. The angle HAD between the parallel HA and the perpendicular AD is called the parallel angle (angle of parallelism), which we will here designate by fl (P) for AD = p. If ll(p) is a right angle, so will the prolongation AE' of the perpen­dicular AE likewise be parallel to the prolongation DB of the line DC, in addition to which we remark that in regard to the four right angles, which are made at the point A by the perpendiculars AE and AD, and their prolongations AE' and AD', every straight line which goes out from the point A, either itseif or at least its prolongation, lies in one of the two right angles which are turned toward BC, so that except the parallel EE' all others, if they are sufficiently produced both ways, must intersect the line BC. If fl (p) < t ;:, then upon the other side of AD, making the same angle DAK = fl (P) will lie also a line AK, parallel to the prolonga­tion DB of the line DC, so that under this assumption we must also make a distinction of sides in parallelism. All remaining lines or their prolongations within the two rip:ht angles turned toward BC pertain to those that intersect, if they lie within the angle RAK = 2 ll (p) between the parallels; they pertam on the other hand to the non-intersecting AG, if they lie upon the other sides of the parallels AH and AK, in the opening of the two angles EAR = i tr -ll(p), E'AK= t;: -ll (p), between the parallels and EE' the per· pendicular to AD. Upon the other side of the perpendicular EE' will m like manner the prolongations AH' and AK' of the parallels AH and AK likewise be parallel to BC; the remaining linP.s pertain, if in the angle K'AH', to tho intersecting, but if in the angles K'AE, H'AE' to the non-intersecting. In accordance with this, for the assumption n(p) = t rr.. the lines can be only intersecting or parallel; but if we assume that ll(p) < t:r, then we must allow two parallels, one on the one and one on the oLher side; in addition we must distinguish the remaining lines into non-intersect­ing and intersecting. For both assumptions it serves as the mark of parallelism that the line becomes intersecting for the smallest deviation toward the side where lies the parallel, so that if AH is parallel to DC, every line AF cuts DC, how small soever the angle RAF may be. THEORY OJi' PARALLELS. 17. A straight line maintains the characteristic of parallelism at all 1"ts -points. Given AB (Fig. 2) parallel to CD, to which latter AC is perpendic FIG. 2. ular. We will consider two points taken at random on the line AB and its production beyond the perpendicular. Let the point Elie on that side of the perpendicular on which AB is looked upon as p11orallel to CD. Let fall from the point E a perpendicular EK on CD and so draw EF that it falls within the angle BEK. Connect the points A and F by a straight line, whose production then (by Theorem 16) must cut CD somewhere in G. Thus we get a triangle ACG, into which the line EF goes; now since this latter, from the con­struction, can not cut AC, and can not cut AG or EK a second time (Theorem 2), therefore it must meet CD somewhere at H (Theorem 3). Now let E' be a point on the production of AB and E'K' perpendic­ular to the production of the line CD; draw the line E'F'' making so small an angle AE'F' that it cuts AC somewhere in F'; making the same angle with AB, draw also from A the line Ali,, whose production will cut CD in G (Theorem 16.) Thus we get a triangle AGO, into which goes the production of the line E'F'; since now this line can not cut A.E a second time, and also can not cut ..l\.G, since the angle BAG= BE'G', (Theorem 7), therefore must it meet CD somewhere in G'. Therefore from whatever points E and E' the lines EF and E'F' go out, and however little they may diverge from the line AB, yet will they always cut CD, to which AB is pa.ra.l.leL 18. Two lines are always mutually parallel. Let AC be a perpendicular on CD, to which AB is parallel if we draw from C the line A _._.___ CE making any acute angle ECD with CD, and let fall from A the perpendicular AF upon CE, we obtain a right. angled triangle ACF, in which AC, being the hypothenuse, is greater than the side AF <11-----------.......,----R ~-----------......----u (Theorem 9.) Fm. 3. Make AG= AF, and slide the figure EFAll until AF coincides with AG, when AB and FE will take the position AK and GH, such that the angle BAK = F AC, con. eequently AK must cut the line DC somewhere in K (Theorem 16), thus forming a triangle AKC, on one side of which the perpendicular GH intersects the line AK in L (Theorem 3), and thus determines the dis. tance AL of the intersection point of the lines AB and CE on the line AB from the point A. Hence it follows that CE will always intersect AB, how small soever may be the angle ECD, consequently CD is parallel to AB (Theorem 16.) 19. In a rectilineal triangle the sum of the three angles can not be greater than two right angles. Suppose in the. triangle ABC (Fig. 4) the sum of the three angles is equal to ;r +a; then choose in case E of the inequality of the sides the smallest BC, halve it in D, draw from A through D the line AD and make the prolongation of it, DE, equal to AD, then join the A point E to the point C by the straight line EC. In the congruent triangles ADB and CDE, the angle ABD =DOE, and BAD= DEC (Theorems 6 and 10); whence follows that also in the triangle ACE the sum of the three angles must be equal to;:+ a; but also the smallest angle BAO (Theorem !l) of the triangle ABC in passing over into the new triangle ACE has been cut up into the two parts EAC and AEC. Continuing this process, continually THEORY OF PARALLELS. halving the side opposite the smallest angle, we must finally attain to a triangle in which the sum of the three angles is :-: +a, but wherein are two angles, each of which in absolute magnitude is less than ta; since now, however, the third angle can not be greater than rr, so must a be either null or negative. 20. If in any rectilineal triangle the sum of the tl1ree angles is equal to two right angles, so is tliis also the case for every other triangle. If in the rectilineal triangle ABC (Fig. 5) the sum of the three angles = :-:, then must at least two of its angles, A 11 and C, be acute. Let fall from the vertex of ~ the third angle Il upon the opposite side AC the perpendicular p. This will cut the tri-A~ EC from the point C. Suppose in the right-angled tri. angle ACE the sum of the three angles is equal to ;r -a., in the tri. arigle AEF equal to ;r -(3, then must it in triangle ACF equal 1T -a -(3, where a and (3 can not be negative. Further, let the angle BAF =a, AFC= b, so is a.+ f3 =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 u. and (3 can have no other magnitude than a = 0 and p= 0. It follows that in all rectilineal triangles the sum of the three angles is either rr and at the same time also the para1lel angle fl (P) = t rr for every line p, or for all triangles this sum is < ;r and at the same time also fl (p) < i ;r. 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 be­tween the sides and angles of the rectilineal and spherical triangle. 23. For every given angle a. we c11n find a line p such that fl (P) = a. Let All and AC (Fig. I 0) be two straight lines which at the inter. section point A make the acute angle .:t; take at random on AB a point THEORY OF P A.RA.LLELS. 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· FIG. IO. 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 ;: -a, then in the triangle AB' A" it equals n -2a, in triangle AA"B" less than ;: -2a (Theorem 20), and so forth, until it finally becomes negative and thereby shows the impossibility of con­structing 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 li'ne FR, which makes with FG the acute angle HFG, on that side where lies the point A. From any point Hof the line FR let fall upon AC the perpendicular HK, whose pro­longation consequently must cut AB somewhere in B, and so makes a triangle AKB, into which the prolongation of the line FR 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 ·t.:; with the growth of p the angle a decreases, while it continually approaches zero for p =oo . Since we are wholly at liberty to choose what angle we will under. THEORY OF P.A.RALLELB. stand by the symbol n (p) when the line pis expressed by a negative number, so we will &sSume ll(p)+fl( -p)=;., an equation which shall hold for all values of p, positive as well as neg­ative, and for p = 0. 24. The farther parallel lines are prolonged on the side of their paral­lelism, the more they approach one another. If to the line AB (Fig. 11) two perpendiculars AC = BE are erected and their end-points C and E joined by c F F. a straight line, then will the quadrilat-1---r-1 eral CABE have two right angles at 0 A and B, but two a.cute angles at C and E (Theorem 22) which a.re equal to one another, as we can easily see A 11 B. by thinking the quadrilateral super- Fm. 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 CE, since the quadrilat­erals 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 is a random point in the line CG, it follows that CG itself nears AB the more the farther it is prolonged. 25. 'I'wo straight lines which are parallel to a third are al.so parallel to one another. 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 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 AE, which will cut the middle line CD in some point C (Theorem 3), at an angle DCE < t rr 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 E°F, 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 some­where in K, since it is impossible that it should meet EF. IfAH from the point A went out within the angle CAG, then must it cut the pro­longation 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 BAE, 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 P .ARALLELS. with C by the line CL, which must cut EF somewhere in M, thus ma.k. ing a triangle MCE. The prolongation of the line AL within the triangle MOE can cut neither A(! nor CM a second time, consequently it must meet EF some­where 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 perpen­dicular AC upon the other parallel CD and join the end-points E and C of the two perpendiculars by the line EC. The angle BAO 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 cute AB somewhere, and in fact in the very point H which is common to all three planes, through which necessarily also the lino EH goes; conse. quently E]' 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 expreseed: Three planes intersect in line.~ which are all parallel to each other if the parallelism of two is pre-supposed. THEORY OF P.A.RALLELB. 26. Triangles standing opposite to one another on the sphere are equiva· lent in surface. By opposite triangles we here understand suc.h 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 ABO and A'B'O' (Fig. 14, where one of them must be looked upon as represented turned about), wo have the sides AB = A'B', BO= B'O', OA=O'A', and the corresponding angles B' Fm. 14. at the points A, B, 0 are likewise equal to those in the other triangle at the points A', B', 0'. Through the three points A, B, 0, 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 ABO, in a.res 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 trial!gle (Theorem 6), consequently the isosceles triangles about the points D and D' in the two spherical triangles ABC and A'B'O' 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 equi'valent 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 k.~s a right-angle. In the spherical triangle ABO (Fig. 15), where each side <-;r, desig. na.te the angles by A, B, 0; prolong the side AB so that a whole circle ABA'B'A is produced; this divides the sphere into two equal parts. THEORY OF P.A.RALLBLB. 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'. C' FIG. 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 P + Y = C, and since P + X +Y + Z = rr, therefore we have also P=t(A+B+C-rr). 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 .II E draw a great circle; upon this let fall from A, B, C the perpendiculars AF, BH, and CG. If the perpendic-:r _ _.;;t--~--~ ular from B falls at H between D and E, then will of the triangles so ma'1e BDH = AFD, and BHE =EGO (The. orems 6 and 15), whence follows that Fm. 16. the surface of the triangle ABC equals that of the quadrilateral AFGC (Theorem 26). 17), only two equal right.angled triangles, ADF and BDE, are made, by whose interchange the equivalence of the surfaces of the triangle ABO Jr and the quadrila_teral 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 eo we go over from the triangle ABC to the quadrilateral AFGC by adding the B FIG. 18. triangle FAD= DBH, and then taking away the triangle CGE = EBH. 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 F AC and ACG be congruent (Theorem 15), and the angle F AC 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 S, 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 tS. 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 iS· 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 a.re BG and join the mid·point H by great-circle-arcs with A and F. The triangles EFG and DCE a.re congruent (Theorem 15), so FG = DC=AB. The triangles ABH and HGF are likewise congruent, since they a.re right angled and have equal perpendicular sides, consequently AH and AF pertain to one circle, the a.re AHF = rr, ADEF likewise = rr, the angle HAD= HFE= tS -BAH= tS -HFG = tS -HFE-EFG =tS-HAD-rr+tS; consequently, angle HFE = t(S-rr}; or what is the same, this equals the si~e of the lune AHFDA, which a.gain is equal to the quadrilateral ABCD, as we easily see if we p&BB over from the one to the other by first adding the triangle EFG and then BAH and thereupon ta.king away the triangles equal to them DCE and HFG. Therefore t (S-rr) 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. If three planes cut each other in paralkl lines, then the sum of the three s-urface angles equaUi two right angles. Let AA', BB' CC' (Fig. 20) be three pa.ra.llels made by the inter. section of planes (Theorem 25). Take upon them at random three FIG. 20. points A, B, C, a.nd suppose through these a. plane passed, which con· sequently will cut the planes of the parallels a.long 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' a.nd 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. spher~ described, upon which the inter­sections of the straight lines AC, AD AA' 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 conseqµently opposite p lies the angle 7r+2a-w-X, (Theorem 27). In like manner CA, CD, CC' cut a. sphere about the center C, and determine a. triangle of size f3, with the sides p', q', r', and the angles, w opposite q', Z opposite r', and consequently 7r+2f3-w-Z opposite p'. Finally is determined hy the intersection of a. sphere about D with the lines DA, DB, DC, a spherical triangle, whose sides a.re 1, m, n, and the angles opposite them w+Z-2f3, w+X-2a, and Y. Consequently its size ct=t(X+Y+Z-rr)-a-f3+w. Decreasing w lessens also the size of the triangles a and f3, so that a.+f3-w can be made smaller than any given number. THEORY OF PARALLELS. 29 In the triangle (j can likewise the sides l and m be lessened even to vanishing (Theorem 21), consequently the triangle (j can be placed with one of its sides l or m upon a great circle of the sphere as often as you choose without thereby filling up the half of the sphere, hence b van­ishes together with w; whence follows that necessarily we must have X+Y+Z = ;; 29. In a rectilineal triangle, the perpendi'culars erected at the mi'd-points of t\e sides either do not meet, or they all three cut each other in one po·int. Having pre-supposed in the triangle ABC (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 B 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 ""-----'!----"'0 diculars ED and FD falls in the line AC itself, or .., Fm. 21. falls without. the triangle. 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 mi'd-points, must all three be parallal to each other, so soon as the parallelism of two of them is pre-supposed. In the triangle ABC (Fig. 22) let the lines DE, FG, HK, be erected perpenll(a)+ll(b.) If we lessen this angle, so that it becomes equal to ll(a) +ll(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 fl(a)-fl(c')> ll(a)-ll(c), whence follows c' >c (Theorem 23). A Fm. 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 (oricyde) 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. 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 FIG. 24. point A in it, ma.king different angles CAB= //(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 a.t 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 ca.ch perpendicular FG ereeted a.t the mid.point of a.ny chord AH, be pa.ra.llel to AB, consequently must this peculiarity also pertain to every perpendicular KL in genera.I which is erected a.t the mid-point K of a.ny chord CH, between whatever points C a.nd H of the boundary line thia 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 'Tllerges into the boundary lim. Given AB (Fig. 25) a. chord of the boundary line; draw from the end-points A and B of the chord two axes AC a.nd BF, which consequently wi!l make with the chord two equal angles BAO = ABF =a (Theorem 31 ). Upon one of these axes AC, take a.ny· D where the point E as center of a circle, and draw the a.re AF from the initial point A of the a.xis AC to its intersection point F with the other a.xis BF. The radius of the circle, FE, corresponding to the point F will make on the one side with the chord AF a.n angle AFE = f3, a.nd on the THEORY OF P.A.RALLELS. other side with the axis BF, the angle EFD = r· It follows that the angle between the two chords BAF = a-f31, 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 un­derstood the base of Nap1erian logarithms. W c may here remark, thaL s1 = 0 for x = oo , hence not only does the distance between two parallels decrease (Theorem 24), but with the prolongation of th41 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 end­poims, wluiresoever these two end.points may be taken on the boundary-surface. Let A, B, C, (Fig. 27), be three points on the boundary.surface; , ' ' B Tl -J4 K r.' c FIG. 27. AA', the axis of revolution, DB' 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.) THEORY OF P.A.R.ALLELS. 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 AA1, BB', must be parallel to the three axes AA', BB', CC', (Theorems 23 and 25); just such a perpen· dicular EE' upon the chord AC in the plane of the parallels AA', CO' 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 A BC be designated by ll(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=O, the point F coincides with D. In all cases arise two congruent right-angled triangles AFD and DFB, consequently we havo FA = FB. Erect .aow at F the line FF' perpendicular to the plane of the tri­angle ABC. Since the angle D'DF = ll(a), and DF= a, so FF' is parallel to DD' and the line EE', with which also it lies in one plane perpendicu­lar to the plane of the triangle ABC. Suppose now in the plane of the parallels EE', Fli'' 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 perpendicu­lar to EK and EE' , be also at the same time perpendicular to FE, (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 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 Fli'' 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 P .A.RALLELS. Hence follows, that for the boundary-surface each of the axes may be eoJJ.siuered 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 in­tersection is a circle. Three principal planes wb1cll mutually cut each other, make with each other angles whose sum is ;r, (Theorem 28). These angles we will consider a.s angles in the boundary-triangle whose sides are arcs of the boundary-line, which a.re ma.de on the bound ary surface by the intersections with the three principal planes. Con­aequently 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 ea.me letter without accent x, which relation is given by the equation n(x) + fl(x') = -f;r. 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 •' Fw. 28. angles opposite them are BAC= fl(.1.), ABC= fl{ft)­ 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: fl(:1.) at AA', a right angle at CC' (Theorems 11 and Ill), consequently fl(a') at BB' (Theorem 28). The intersections of the lines BA, BC, BB' with a sphere described abcmt the point B as center, determine a spherical triangle mnk, in which the sides are mn = fl(c ), kn= fl ((3), mk = fl(a) and the opposite angles are fl(b), fl(:/), t ;r. Therefore we must, with the existence of a rectilineal triangle whose sides are a, b, c and the opposite angles fl (a), fl(/3) f;r, also admit the existence of a spherical triangle (Fig. 29) with the sides ll(c), ll(ft), ll(a) and the opposite angles fl(b), fl(a'), fl!'· Gm~ Fm. 29. Of these two triangles, however, also inversely i;he existence of the spheri"..al triangle necessitates anew that of a rectilineal, which in con­sequence, also can have the sides a, a', f3, and the oppsite angles ll(b'), ll(c), f;r. Hence we may pass over from a, b, c, a, f3,,to b, a, c, f3, a, and also to a, a.', /3, b', c. Suppose through the point A (Fig. 26) wit,h AA' as axis, a bound­ary-surface passed, which cuts the two other axes BB', CC', in B" and c•, and whose intersections with the planes the parallels form a bound­ary-triangle, whose sides are B"C" = p, C"A = q, B"A= r, and the angles opposite them fl(a), fl(a'). f:rr. and where consequently (Theo­rem 34): p= r sin fl(a), q = rcos fl(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 point A and ha.a AA1 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 1:1 s' Fm. 30. perpendicular to AA' at A, the axis CC' going from the end of b par­allel to AA' 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 a.xis BB' parallel to AA' which goes through the end-point B" of the a.rep. On the other side of AA' will lie, the side c perpendicular to A A' 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" = f (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 BD = f (a). BB1 = BD+DB"= BD+CC0 , consequently j (c) =/(a)+ I (b). Moreover, we perceive, that (Theorem 32) t pef(b) =r sin fl(a.) ef(bJ . 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 and b into b and a, and the angle n(a) into n(/3). consequently we would have q= r sin n(/3) 0J<•1, whence follows by substituting the value of q, cos fl (:.t) =sin ll(/3) eJ<•>, and if we change a and pinto b' and c, sin [/ (b) = sin n(c) 0 f(•>; further, by multiplication with 9/(b) sin n (b) 9 / (b) =sin n(c) 9/(c) Renee follows also sin n (a) 9/(a) = sin n(b) e/(b). Since now, however, the straight lines a and b are independent of one another, and moreover, for b=O, /(b)= O, ll(b)=!rr, so we have for evEiry straight line a. e-/(a) = sin n(a.). Therefore, sin n(c) = sin n(a.) sin n(b), sin 11(/3) = cos n(a) sin n(a.). Hence we obtain besides by mutation of the letters sin ll(a) = cos ll(/3) sin//(b), cos ll(b) = cos n(c) co_s ll(a), cos ll(a) = cos ll(c) cos ll(p). If we designate in the right.angled spherical triangle · (Fig. 29) ~he sides ll(c), fl(p), ll(a), with the opposite angles ll(b), ll(a.'), by the letters a, b, c, A, B, then the obtained eqnations take on the form of those which we know as proved in spherical trigonometry for the right· angled triangle, namely, sin &=sin c sin A, sin b=sin c sin B, cos A=cos a sin 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 P.A.RALLELS. 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 n(a), n(ft), trr. Prolong the hypothenuse c through the point B, and make BD=(1; at the point D erect upon BD the perpendicu· l&r 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', J> (Theorem 25), therefore is the angle A'AD=ll(c+f1), A'AC= fl(b), consequently ll(b)=fl(a,)+ll (c+f1). A b c FIG. 31. If from B we lay off (1 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 00' be the third parallel; then is, angle CAA'=ll (b), DAA '= ll(c-(3), consequently ll(c­/3) = fl(a)+ fl (b). The last equation is then also still valid, when c=f3, or c! c' FIG. 33. we have ll(.'l}+ll(b)=trr, whilst also ll(c-{i)=trr, (Theorem 23). If cf;r (Fig. 37) the first equation remains unchanged, instead of the second, however, we must write correspondingly cos U(x-c)=cos (rr-B) cos ll(a); but we have cos ll(x-c)=-cos ll(c-x) ('l'heorem 23), and also cos (;r-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 npon the preceding. In order to eliminate x from both equations, we notice that (Theo· rem 36) 1-[tan-t ll\.,;-x) ]2 cosfl(c-x)= l -t-[tanf//(c:::...x) ]2 J-e2x-2c l+e21-2c l-[tantll(c)]2[cottfl(x)]• -1+[tantfl(c)]2[cottll(x)]& cos fl(c)-cosll(x) -1-cos ll(c)cosfl(x) THEORY OF PARALLELS. 43 If we substitute here the expression for cos fl(x), coafl(c-x), we ob· ta.in cosll(c)= COB fl(a) COB B+cosfl(b) COB A i+cosfl(a) cosfl(b) cosA cosB whence follows cos ll(a) cosB= cos ll(c)-cosA coslf(b) 1-cosAcosfl(b) cos //(c) and finally [sin fl(c) )1 =(1-cosBcos fl(c) cos fl(a)] (1-cos A cos fl(b) C98 //(c)] In the same way we must also have (4.) [sin ll(a) ]' =[l-cos Ccos n(e.) COB II(b) J[l-cosBcos /7 (c)cos II (a)] (sin ll(b) )1 =(1-cosAcos fl (b)cos ll(c)] [1-cosC cos fl(a)cos ll(b)] From these three equations we find [sin fl(b)]s [sinll(c)]s ----. -(1-cosAcos fl(b)cosll(c)]s. [ sm 11 (a)]2 Hence follows without ambiguity of sign, rr sinfl(b)sinll(c) (5.) cosAcosll(b)cos11(c)+ . fl ) 1. sm (a If we substitute here the value of sin ll(c) corresponding to equa· tion (3.) sin A sin ll(c)= -.-tan f/(a)cos fl(c) smC then we obtain fl _ COB fl(a)sinC cos (c) -sin A sin /l(b)+cos A sin C cosf/ (a) cos ll(b); but by substituting-this expression for cos fl(c) in equation (4), . . cos fl(b) (6.) cot A sm C sm fl (b)+cosC=cos /l(a) By elimination of sin ll(b) with help of the equation (3) comes cos fl(a) cos A . . II ----COB c=I = ~Bsm c S!Il (a). cos II(b) sm ln the meanti1110 the equation (6) gives by changing the letters, cos II(a) -cot B sin C sin II (a)+cos C. cos /J(b) ­ From the last two equations follows, sin B sin C (7.) cosA+cosB cosC --sinfl(a) All four equations for the interdependenco 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).1 sin A tan Jl (a)= sin B tan IT (b), sin n(b) sin n(c) ~ cosAcosll(b)cosfl(c)+ -sin /J(a) --= 1, (8.) ( cos fl(b) cot A sin C sin fl (b) + cosC -cos /l (a) , sin BsinC cosA+cosBcosC= . fl ) . Sill (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 fl(a) =a, sin fl(a) = 1 -ta2 cosfl(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.2 =b2 + c2 -2bc cos A, a sin (A+ C) = b sin 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=rr. 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 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 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 ll(a) =--­ . cos (a), COB ll(a) = ('\/'-1) tan&, l tan n (a)=---­ sin a('\/'-l), 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 B sin a, COB &= COB b COB c +sin b sin c cos A, cot A sin 0 +cos 0 cos b = sin b cot a, cos A = cos a sin B sin 0 -cos B cos C. TRANSLATOR'S APPENDIX. ELLIPTIC GEOMETRY. Gauss himself never published aught upon this fascinating subject, Geometry Non-Euclidean; but when the most extra.ordinary pupil of his long teaching life came to read his inaul!'ural dissertation before the Philosophical Faculty of the University of Goettingen, from the three themes submitted it was tho choice of Gauss which fixed upon the one "Ueber die Hypothesen welche der Geometr1e zu Grundo licgcn." Gauas 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 contmues: "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 a.rises 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 Euclid 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] Bolyai completes the group, being constant and negative, a.nd 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, de­formation. It would then, according to Riemann, be a special case of a space of constant curvature vVe 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 tho 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 in­finite; the first pertains to the relations of extension, the latter to the size-relations. "That our space is an uv.bounded three.fold extensive manifoldness, is an hypothesis, which is applied in each apprehension of tho 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 unbounded­ness 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 ono 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. 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, l. In a given plane all the per· pendiculars to a given straight line intersect in a single point, whose distance from tho straight lino is ·F Inversely, the locus of all the points at a distance tl 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 za /;r. The sum of the angles of n. piano trianglo is g"reater than a straight angle by an excess proportional to its area. The greater the area of the triangle, tho greater the excess or differ· ence of the angle sum from ;r. Says tho Royal Astronomer for Ireland: "It is necessary to measure large triangles, and the largest triangles to whfoh we have access are, of course, tho triangles which tho astrono. mers have found means of measuring. The largest available triangles are those which have tho diameter of the earth's orbit as a base and a fixed star at tho vertex. It is a very curious circumstance that tho in. vestigations of annual parallax of tho stars arc precisely the investiga­tions which would be necessary to test whether ono of these mighty tri­angles had the sum of its three angles equal to two right angles. * * * "Astronomers have sometimes been puzzled by obtaining a negative parallax n.s 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 negatirn parallax might result from observations which possessed mathematical perfec­tion. ,,., * * It must remain an open question whether if we had huge enough triangles tho sum of tho 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 tho spar.e in which we find ourselves may be curved in the manner hero 1rnpposed. * * * "The subjective impossibility of conceiving of the relation of the most distant points in such a space docs not render its existence in. credible. In fact our difficulty is not unlike that which must 11avo been felt by the first man to whom the idea of the sphericity of the earth 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 i"s equal to c times the logarithm of the cross ratio in which the l£ne joining the two points is divided by the funda· mental quadri'c." This projective expression for distance, and Laguerre's for au angle were in 18'71 generalized by Felix Klein in his article Ueber die soge· nannte Nicht-Euklidische Geometrie, and in I 872 (Math. Ann., Vol. 6) he showed the equivalence of projective metrics with non-Euclidean geometry, space being of constant negative or po&itive curvature ac­cording as the fundamental surface is real and not rectilineal or is im. agina:ry. 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· ma.tics, Vol. I., pp. 261-276, 384, 385; Vol. II., pp. 65-'70. We notice that Clark University and Cornell University are giving regular courses in non-Euclidian geometry by their most eminent Pro­fessors, and we presume, without looking, that the same is true of Har­vard and the Johns Hopkins University, with Prof. Newcomb an origi­nal authority on this far-reaching subject.