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**rJ&** iV-i- 3ft ^V* GEOMETRICAL RESEARCHES OJf THE THEORY OF PARALLELS. BY NICHOLAUS LOBATSCHEWSKY, IMPERIAL RUSSIAN REAL COUNCILLOR OP STATE AND REGULAR PROFESSOR OF MATHEMATICS IN THE UNIVERSITY Translated from OP KASAN. 1840. BERLIN, the Original by GEORGE BRUCE A. M., Ph. D., Ex-Fellow of Princeton HALSTED, College Professor of Mathematics in the AUSTIN: and Johns University Hopkins University, of Texas. FOURTH EDITION". ISSUED FOR THE COUNCIL OF THE ASSOCIATION FOR THE IMPROVEMENT GEOMETRICAL TEACHIXG-. AND DEDICATED To the President of the J. J. SYLVESTER, A. M. Cam.; F. R. S., L., and E. ; of France; Member Association, Academy Corresponding Member Institute of Sciences in Berlin, Goettingen, Na ples, Milan, St. Petersburg, etc.; LL. D., University of Dublin, and U. of E.; D C. L Oxford; Hon. Fellow of St. John's College, Cam.; Fellow of New College, Oxford; Savilian Professor of Geometry in , the University of Oxford, By his former pupil, THE Jan. 1, 1892. TRANSLATOR. TRANSLATOR'S PREFACE. Lobatschewsky was the fi>st man ever to publish a non-Euclidian geom etry. Of the immortal essay now first appearing in English Gauss said, "The author has treated the matter with a master-hand and in the true geom spirit. I perusal can not ought to call your attention to this book, whose give you the most vivid pleasure." Clifford says, "It is quite simple, merely Euclid without the vicious assumption, but the way things come out of one another is quite lovely." * * * ''What Vesalius was to Galen, what Copernicus was to Ptolemy, that was Lobatschewsky to Euclid." Says Sylvester, "In Quaternions the example has been given of Al gebra released from the yoke of the commutative principle of multipli cation an emancipation somewhat akin to Lobatschewsky's of Geometry from Euclid's noted empirical axiom." Cay ley 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, where in this axiom was assumed not to hold good, or say a system of nonEuclidian plane geometry. There is a like system of non-Euclidian solid eter's think I fail to — geometry." GEORGE BRUCE HALSTED. 2407 San Marcos Street, Austin, Texas. May 1, 1891. [81 NOTE TO THE FOURTH EDITION. The Translator has been for this essay in its for permission University He has gratified by the and by English dress, to issue an application from Japan of the Imperial of John Bolyai's use of Tokio. already Saggio be followed the edition there for the in press a translation into celebrated "Science Absolute of trami's unexpectedly great demand by di Space," Interpretazione English and is at work upon Bel della Geometria his Teoria fondamentale Non-Euclidea; degli Spazii to di Curvatura costante. He may mention as a very practical result of his Euclidean spaces, that since 1877 he has Pure as Spherics, or Two-dimensional new Elementary Synthetic Geometry. studies in regularly taught Spherics, outlined in Book IX of his Elements of own or say Plane Geometry and non- his classes Spherics, given in his TRANSLATOR'S INTRODUCTION. things, hold fast that which is good," does not mean dem From nothing assumed, nothing can be proved. onstrate everything. "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, 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 con firms with overwhelming force the position that Geometry as a science, strict and self-conscious deductive reasoning, was created by the subtle "Prove all intellect of the Venus of In same race the Milo, whose bloom in art still Apollo Belvidere, overawes us in the the Laocoon. geometry occur the most noted set of axioms, the geometry of Euclid, a pure Greek, professor at the University of Alexandria. a Not its very birth did this typical product of the Greek genius sway as ruler in the pure sciences, not only does its first efflor only at assume escence ages, us through the splendid days latter, fanatics can not murder carry unlike the can with the not drown it. new Like the birth of culture. of Theon and Hypatia, it; that dismal flood, the phcenix of its native Egypt, it rises Anglo-Saxon, Adelard of Bath, An finds it clothed in Arabic vestments in the land of the Alhambra. clothed but dark it and the new-born Then press confer honor on printing 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 Leipsic's learned presses. m Latin, [5] 6 THEORY How the first translation into PARALLELS. OF 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 and enriched with his system emendations. Princeps in Greek to-day in the vast Even autograph by Cambridge, Oxford, and the British gov proof will be accepted which infringes Euclid's order, a of examinations set ernment, no sequence founded upon his set of axioms. The American ideal is expects fection. to be In twenty years the American maker improved upon, superseded. Epic and Lyric poets, The Greek unmatched. for success. The axioms of The Greek ideal was per- the Greek sculptors, remain the Greek geometer remained unquestioned twenty centuries. How and where doubt interest, for this doubt came in the epoch-making axioms was one differing Among Euclid's lixity, whose place fluctuates in Euclid's first brought to look toward them is of was in the ordinary from the others in pro manuscripts, and which is not used twenty-seven propositions. in to prove the inverse of history no of mind. Moreover it is of these only then demonstrated. already 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 one All this " to push this unwieldly thing from the set of fundamental axioms. TRANSLATOR'S Numerous and desperate were 7 INTRODUCTION. the attempts to deduce it from about the nature of the reason line and ings straight plane angle. In the und Kunste; Von Ersch und GruEncyclopcedie der Wissenschaften 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. defeat Some acknowledged example the stupid one, tant," still given on page 33 by taking definition of parallels, as everywhere equally dis 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 for a new "Parallel lines are the indefinite term "direction." " How much better to have followed the third class of his who honestly not in assume a essence. new axiom differing Of these the best is that called Playfair's; "Two lines the same The German article mentioned is followed a which intersect can not both be parallel to predecessors from Euclid's in form if line." by carefully prepared 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, list of ninety-two authors on the Oxford. All this shows how forth of genius ready the world was for the extraordinary flaming- from different parts of the world which was overturn, explain, and remake not only all this subject but quence all philosophy, all ken-lore. As was the case at once to as conse with the dis covery of the Conservation of of genius, whether in Energy, the independent irruptions 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 PARALLELS. OF publication of the book he mentions, we see the brilliant Clifford writing from Trinity College, Cambridge, April 2, 1870, "Several new ideas have come to me lately: First I have procured Lobatschewsky, 'Etudes Geometriques sur la Theorie des Parallels' a small tract of which Gauss, therein quoted, intellects. Thirty - says: L'auteur years after the - a - traite la matiere esprit geometrique. dont la lecture ne en Je crois devoir main de maitre et appeler peut manquer de avec votre attention vous causer le plus le veritable sur ce vif livre, 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 "vicious assumption," soon no assumptions and none clearly vicious. He had been reading the French translation by Houel, pub lished in 1866, of a little book of 61 pages published in 1S40 in Berlin under the title Geometrische Untersuchungen zur Theorie der Parallellinien by a Russian, Nicolaus Ivanovitch Lobatschewsky (1793-1856), the first public expression of whose discoveries, however, dates back to a discourse at Kasan on February 12, 1S26. Under this commonplace title who would have suspected the disman sees a than Clifford that all more are 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 of by right description, and occupancy; then the new space and its fellows could be called Non-Euclidean. Gauss in as a far back Again letter to as Schumacher, dated Nov. 28, 1S46, mentions that 1792 he had started he says: pre-emption, quick-following this path to "LaGeometrie non-Euclidienne ne on a new universe. renferme en elle rien de contradictoire, quoique, a premiere vue, beaucoup de ses resultats aien l'air de paradoxes. Ces contradictions apparents doivent etre regardees comme l'effet d'une illusion, due a l'habitude que nous avons prise de bonne heure de considerer la geometrie Euclidienne comme rigoureuse." But here we see Clifford's letter. in the last word the same imperfection of view The perception has not yet come that though the Euclidean geometry is rigorous, Euclid is not one whit less so. as in non- . TRANSLATOR'S 9 INTRODUCTION. A clearer idea here had already come to the former Hungarian Wolfgang Bolyai. Gcettingen, work, published by subscription, has the following title: Gauss at the room-mate His of principal Tentamen Juventutem studiosam in elementa Matheseos purae, eleac sublimioris, methodo intuitiva, evidentique huic propria, in- mentaris Tomus troducendi. Primus, 1831; Secundus, 1833. 8vo. Maros-Va- sarhelyini. In the first volume with of his special numbering, appeared the Bolyai with the following title: absolute Axiomatis XI Euclidei (a priori tem. Auctore Johanne Caesareo 1832. Regio (26 celebrated Johann Appendix Ap., scientiam spatii son veram Bolyai exhibens: haud unquam de a veritate aut falsitate decidenda) independen- eadem, Geometrarum in Exercitu Captaneo. Maros-Vasarhely., Austriaco Castrensium pages of This marvellous text). Appendix has been translated into French, Italian, and German. In the title of Wolfgang Bolyai's last work, the only one he com posed in German (88 pages of text, 1851), occurs the following: Und da die Frage, ob zwei von der driiten geschnittene Geraden wenn die " Summa der inneren \Yinkelnicht=-2R, worten wird; die davon eine auf die Ja Antwort, sich schneiden oder Euclid das auf der Erde ohne ein Axiom (wie unabhasngige nichtf, niemand 51) aufzustellen, Geometrie andere auf das Nein beant- abzusondern, und so zu bauen, dass Formeln der letzen auf ein Wink auch die in der ersten gultig seien." Lobatschewsky 's Geometris^he Untersuchungen Berlin, 1840, and compares it with the work of his son Johann Bolyai, "an sujet duquel il dit'Quelques exemplaires de l'ouvrage publie ici De ont ete envoyes a cette epoque a Vienne, a Berlin, a Gcettingen. du sommet le des hauteurs Goettingen geant mathematique, [Gauss] qui embrasse du meme regard les astres et la profondeur des abimes, a ecrit qu'il etait ravi de voir execute le travail qu'il avait commence pour le laisser apres lui dans ses papiers.' Yet that which Bolyai and Gauss, a mathematician never surpassed in power, see that no man can ever do, our American Schuyler, in the density of his ignorance, thinks that he has easily done. In fact this first of the Non-Euclidean geometries accepts all of Eu clid's axioms but the last, which it flatly denies and replaces by its con- The author mentions . " 10 THEORY OF PARALLELS. tradictory, that the sum of the angles made on the same side of a trans versal by two straight lines may be less than a straight angle without the lines meeting. 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 son geometry I find certain imperfections which I hold to be the rea why this science, apart from transition into analytics, can as yet make As no advance from that state in which it has belonging to these imperfections, come to I consider the us from Euclid. obscurity in the fundamental concepts of the geometrical magnitudes and in the. manner and method of representing the measuring of these magnitudes, and finally the momentous gap in the theory forts of mathematicians have been so of parallels, to fill which all ef far in vain. theory Legendre's endeavors hav9 done nothing, since he only rigid way to turn into a side path and take theorems which he illogically strove to exhibit as in auxiliary refuge axioms. first My essay on the foundations of geometry I pub necessary lished in the Kasan 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 "t?elehrten Schriften dxr XJniversiUet 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 country men from following such a subject, which since Legendre had lost its \nterest. Yet I am of the opinion that the Theory of Parallels should For this was forced to leave the " not lose its claim to the attention of geometers, and therefore I aim to investigations, remarking beforehand that give of to the Legendre, all other imperfections for ex opinion contrary a definition of the straight line show themselves foreign here ample, and without any real influence on the theory of parallels. In order not to fatigue my reader with the multitude of those theo rems whose proofs present no difficulties, I prefix here only those of which a knowledge is necessary for what follows. A straight line fits upon itself in all its positions. By this I mean 1. that during the revolution of the surface containing it the straight line does not change its place if it goes through two unmoving points m the surface: (i. e., if we turn the surface containing it about two points of here the substance of my — — the line, the line does not move.) Hi] 12 THEORY PARALLELS. straight lines can not intersect in straight line sufficiently produced 2. Two 3. A beyond OF bounds, and all in such way cuts a two points. both ways must go out bounded plain into two parts. perpendicular to a third never intersect, how straight 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.) Vertical angles, where the sides of one are productions of the 6. This holds of plane rectilineal angles sides of the other, are equal. as also of plane surface angles: (i. e., dihedral angles.) themselves, among Two straight lines can not intersect, if a third cuts them at the 7. same angle. In a rectilineal triangle equal sides lie opposite equal angles, and 8. inversely. In a rectilineal triangle, a greater side lies opposite a greater 9. angle. In a right-angled triangle the hypothenuse is greater than either of the other sides, and the two angles adjacent to it are 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. A straight line which stands at right angles upon two other 11. 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 4. Two lines two. a sphere with a plane is a circle. right angles to the intersection of two per pendicular 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 sides and the included angle equal, or a side and the adjacent two angles equal. From here follow the other theorems with their explanations and proofs. 12. The intersection of 13. A straight line at THEORY All 16. lines which in straight with reference to a into two classes into — The be boundary called parallel and culling lines of the one a plane go out from line in the given straight IS PARALLELS. OF same a plane, point can, be divided nol-cidting. and the other class of those lines will to the given line. point A (Fig. 1) let fall upon the line BC the perpendicular AD, to which again draw the perpendicular AE. From the In the right angle EAD either will all straight lines which go out from the line DC, as for example AF, point A meet the or some of them, perpendicular AE, will not meet the In the uncertainty whether the per line DC. pendicular AE is the only line which does not meet DC, we will assume it may be possible that there are still other lines, for example AG, like the which do not cut Fig. 1. prolonged. In pass the not-cutting lines, as AG, ing over from the cutting lines, as we must come upon a line AH, parallel to DC, a boundary line, upon one side of which all lines AG are such as 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 p. designate by fl (p) for AD If // (p) is a right angle, so will the prolongation AE' of the perpen dicular 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 itself 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 DC, how far they AF, to soever may be = intersect the line BC. If J! (p) < >V ~» make a AD, making the same parallel to the prolonga assumption we must also then upon the other side of angle DAK = H (p) will lie also a line AK, tion DB of the line DC, so that under this distinction of sides in parallelism. 14 THEORY All remaining lines or their OF PARALLELS. prolongations within the two right angles lie within the if intersect, they parallels; they pertain on the other hand to the non-intersecting AG, if they lie upon the other sides of the \ ,-r parallels AH and AK, in the opening of the two angles EAH J 77 (p), E'AK // (p), between the parallels and EE'-the per pendicular to AD. Upon the other side of the perpendicular EE' will in like manner the prolongations AH' and AK' of the parallels AH and AK likewise be parallel to BC ; the remaining lines pertain, if in the angle K'AH', to the intersecting, but if in the angles K'AE, H'AE' to the non-intersecting. In accordance with this, for the assumption 77 (p) \ ~- the lines can be only intersecting or parallel ; but if we assume that 77 (p) < \ rr, then turned toward BC angle HAK = 2 pertain 77 (p) to those that between the = = — r- — = we must allow two in addition ing and we parallels, one on the one and one on the other side; distinguish the remaining lines into non-intersect must intersecting. For both assumptions it line becomes serves as where lies the intersecting parallel, so that DC, how the cuts the mark of parallelism that the for the smallest deviation toward the side small soever if AH is angle parallel to HAF may be. DC, every line AF THEORY 17. A. straight OF 15 PARALLELS. line maintains the characteristic of parallelism a,t all its points. Given AB (Fig. 2) parallel to CD, to which latter AC is perpendic Fio. 2. ular. points taken perpendicular. We will consider two its the at random on the line AB and production beyond point E lie on that side of the perpendicular on which AB is looked upon as parallel 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 Let the (by Theorem 1 6) must cut CD somewhere in G. Thus we get a triangle into which the line EF goes; now since this latter, from the con struction, can not cut AC, and can not cut AG or EK a second time ACG, (Theorem 2), therefore it must meet CD somewhere at H Now let E' be a point on the production (Theorem 3). of AB and E'K' perpendic 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 AF, whose production will cut CD in G (Theorem 16.) Thus we get a triangle AGC, into which goes the production of the line E'F'; since now this line can not cut AE a second time, and also can not cut AG, since the angle BAG BE'G', (Theorem 7), therefore ular to the = 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 parallel. 16 THEORY 18. Two lines Let AC be if we CE a OF PARALLELS. always mutually parallel. perpendicular on CD, to are which AB is parallel draw from C the line making any acute angle ECD with CD, and let fall perpendicular AF upon CE, we obtain a rightangled triangle ACF, in which AC, being the hypothenuse, from A the is greater than the side AF (Theorem 9.) Make AG = AF, and slide Fig. 3. AG, when AB and FE wiJ figure take the position AK and GH, such that the angle BAK FAC, con sequently 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 lino AB from the point A. Hence it follows that CE will always intersect AB, how small soever nay be the angle ECD, consequently CD is parallel to AB (Theorem 1 6.) 19. In a rectilineal triangle the sum of the three angles can not be greats, han two right angles. Suppose in the triangle ABC (Fig. 4) the sum of the three angles is iqual to tz -\- a; then choose in case )f the inequality of the sides the imallest BC, halve it in D, draw :rom A through D the line AD md make the prolongation of it, DE, equal to AD, then join the Fig. 4. joint E to the point C by the line EC. In the congruent triangles ADB and straight CDE, the angle ABD DEC (Theorems 6 and 10); whence follows DOE, and BAD ihat also in the triangle ACE the sum of the three angles must be equal » 71 + a; but also the smallest angle BAC (Theorem 9) of the triangle ABC in passing over into the new triangle ACE has been cut up into ,he two parts EAC and AEC. Continuing this process, continually EFAB until AF coincides with the = = = THEORY 17 PARALLELS. OF 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 \a; since now, however, the third angle can not be greater than r., so must a be either null or negative. 20. If in any rectilineal triangle the sum of the three angles is equal to - two right angles, so is this also the If in the rectilineal = , C, be angles, the three angles A B upon the angle opposite side AC perpendicular p. This will cut the tri- A angle into two right-angled triangles, in each of which the sum of three angles must also be i»C the either bo greater ti, i G' n, since it can and in their combination not less than not in n- right-angled triangle with the perpendicular sides p a quadrilateral whose opposite sides are equal and whose adjacent sides p and q are at right angles (Fig. 6.) By repetition of this quadrilateral we can make another with sides np and q, and finally a quadrilateral ABCD with sides at right angles So we obtain than a and q, and from this to each other, such that AB = np, AD any whole numbers. = mq, DC Such = np, BC = mq, where quadrilateral is divided by the diagonal congruent right-angled triangles, BAD and rz. BCD, in each of which the sum of the three angles The numbers n and m can be taken sufficiently great for the rightangled triangle ABC (Fig. 7) whose perpendicular sides AB np, BC another given (right-angled) triangle BDE mq, to enclose within itself m and n are a DB into two = = = as soon as tho 2 right-angles — car. fit each other. / - - triangle. (Fig. 5) the sum of Let fall from the vertex of acute. the third every other for ABC then must at least two of its — and triangle case i 18 THEORY Drawing the line successive two have DC, a we OF obtain side in PARALLELS. right-angled triangles of which every common. triangle ABC is formed by the union of the two triangles ACD and DCB, in neither of which can the sum of the angles be greater than tt; consequently it must be equal to 7-., in order that the sum in the compound triangle may be equal to -. The In the triangle BDC consists of the two triangles DEC DBE, consequently must in DBE the sum of the three angles be equal to 77, and in general this must be true for every triangle, since each can be cut into two right-angled triangles. From this it follows that only two hypotheses are allowable : Either is the sum of the three angles in all rectilineal triangles equal to 77, or same way the and this sum 21. is in all less than From make with Let fall a n. given point we can always draw a straight line that shall given straight line an angle as small as we choose. from the given point A (Fig. 8) upon the given line BC the a D E C Fig. 8. perpendicular AB ; take upon BC at random AD; make DE AD, and draw AE. = the point D ■ draw the line THEORY OF 19 PARALLELS. In the right-angled triangle ABD let the angle ADB a; then must triangle ADE the angle AED be either |a or less (Theo and 20). Continuing thus we finally attain to such an angle, = in the isosceles 8 rems AEB, as 22. is less than any If perpendiculars to the same straight line are parallel to each sum of the three angles in a rectilineal triangle is equal to two then the other, given angle. two right angles. Let the lines AB and CD pendicular (Fig. 9) be parallel to each other and per to AC. Draw from A the lines AE and AF to the which are points taken on E and F, the line CD at any distances FC > EC from the point C. Suppose in the right-angled tri angle ACE the sum of the three angles angle AEF equal to tt /9, then must it — — ft, where Further, a and R can not be let the angle BAF = is in a, in the equal to ACF triangle equal n - — tri — a negative. 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 can have no lessen the angle b, consequently the two angles a and R 0 and R 0. magnitude than a It follows that in all rectilineal triangles the sum of the three angles is either tt and at the same time also the parallel angle 77 (p) \ n for or for all triangles this sum is < tt and at the same time line p, every also 77 (p) < i -. 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 we can find a line p such that II (p) aother = = = = Let AB and AC section point (Fig. 10) be two A make the acute angle straight a; take lines which at the inter. at random on AB a point 20 OF THEORY PARALLELS. B'; from this point drop B'A' at right angles AA'; erect at A" the perpendicular A"B"; and A.' A" to so AC; make A'A" continue until r K a - - per- c Fig. 10. pendicular CD is attained, which no longer intersects AB. This must of necessity happen, for if in the triangle AA'B' the sum of all three 2 a, a, then in the triangle AB' A" it equals 77 angles is equal to 77 until 2a (Theorem 20), and so forth, in triangle AA'B" less than 77 — — — finally becomes negative structing the triangle. it The point perpendicular and thereby shows the CD may be the very A all others cut AB ; at least in the cut to those not such impossibility one nearer of con than which to the passing perpendicular FG must over from those that exist. cutting 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 AKB, into which the prolongation of the line FH enters, and therefore must meet the hypothenuse AB somewhere in M. Since the is we wish, therefore GFH and can small as be taken as angle arbitrary FG is parallel to AB and AF p. (Theorems 1 6 and 18.) One easily sees that with the lessening of p the angle a increases, while, for p 0, it approaches the value -J--; 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 underDraw now a from the = = . THEORY stand the by number, symbol TI (p) so we will OF 21 PARALLELS. when the line p is expressed by a negative assume 77(p)+77(-P)=;r, equation which ative, and for p an = 24. lelism, The the shall hold for all values of p, 0. farther parallel lines are prolonged they approach one another. positive as on the side well of as neg their paral more If to the line AB (Fig. 1 BE are erected perpendiculars AC by c F a straight line, then will the quadrilat eral CABE have two right angles at A and B, but two acute angles at C and E (Theorem 22) which are equal °to one another, as we can easily see A> ^IG- *!■ by thinking the quadrilateral superimposed upon itself so that the line BE falls upon upon AC and AC 1) two = and their end -points C and E joined upon BE. Halve AB and erect at the mid-point D the line DF perpendicular to perpendicular to CE, since the quadrilat erals CADF and FDBE fit one another if we so place one on the other Hence the line CE can that the line DF remains in the same position. not be parallel to AB, but the parallel to AB for the point C, namely CG, must incline toward AB (Theorem 1 6) and cut from the perpendic AB. This line must also be ular BE a part BG < CA. Since C is nears AB the a random more point in the line the farther it is CG, prolonged. it follows that CG itself 22 THEORY 25. one Two straight lines which OF are PARALLELS. parallel to a third are also parallel another. Fig. 12. We will first one plane. assume that the three lines If two of them in order, AB and CD AB, CD, EF (Fig. 12) lie in AB and CD, are parallel another. to the In order parallel to one point A of the outer line AB upon the other outer line FE, the perpendicular AE, which will cut the middle on the line CD in some point C (Theorem 3), at an angle DCE < £ side toward EF, the parallel to CD (Theorem 22). A perpendicular AG let fall upon CD from the same point, A, must fall within the opening of the acute angle ACG (Theorem 9); every other line AH from A drawn within the angle BAC must cut EF, the parallel to AB, somewhere in H, how small soever the angle BAH may be; consequently will CD in the triangle AEH cut the line AH some where in K, since it is impossible that it should meet EF. If AH from the point A went out within the angle CAG, then must it cut the pro longation of CD between the points C and G in the triangle CAG. Hence follows that AB and CD are parallel (Theorems 16 and 18). Were both the outer lines AB and EF assumed parallel to the middle line CD, so would every line AK from the point A, drawn within the angle 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 outmost one, to prove this, EF, so are let fall from any - THEORY with C ing a by the line CL, triangle MCE. The prolongation neither AC where in nor CM OF which must cut EF somewhere in of the line AL within the a second H; therefore time, consequently AB and EF are Fig, Now let the parallels 23 PARALLELS. From EF. fall EA upon MCE mak can it must meet EF cut some mutually parallel. 13. AB and CD intersection line is triangle M, thus (Fig. 1 3) random lie in two E of planes whose this latter let point 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 The angle BAC must be of the two perpendiculars by the line EC. 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 piarallel to each other if the parallelism of two is presupposed. a perpendicular one a of the two 24 THEORY OF PARALLELS. Triangles standing opposite to one another on the sphere are equiva lent in surface. on both By opposite triangles we here understand such as are made sides of the center by the intersections of the sphere with planes; in such triangles, therefore, the sides and angles are in contrary order. In the opposite triangles ABC and A'B'C (Fig. 14, where one of them must be looked upon as represented turned about), we have the sides AB A'B', BC B'C, CA CA', and the corresponding angles 26. = = = Fig. 14. points A, B, C are likewise equal to those in the other triangle at the 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' The distances of the first D from the points ABC, in of the sphere. 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. at the 27. A three-sided solid angle equals the half sum of the surface angles right-angle. In the spherical triangle ABC (Fig. 1 5), where each side < -, 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. less a THEORY In that half in which is the sides through their common OF 25 PARALLELS. triangle ABC, prolong now the other two intersection point C until they meet the circle in A' and B'. Fig. 15. In this way the hemisphere is divided into four triangles, B'CA', A'CB, whose size may be designated dent that here P + X A. B, P + Z = The size of the ABC, ACB', by P, X, Y, Z. It is evi. = spherical triangle Y equals that of the opposite triangle triangle P, and whose lies at the end-point of the diameter of the sphere which through the center D of the sphere (Theorem 26). Hence ABC, having third angle C a side AB in with the common goes from C it follows that P -j- Y= C, and since P-|-K-|-Y-f-Z P = i(A + B = ;T, therefore we have also + C-;r). 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), and through the mid-points D and E draw a great circle; upon this let fall from A, B, C the perpendiculars AF, BH, and CG. If the perpendic- p halve the sides AB and BC, B ular from B falls at H between D and E, then will of the triangles BDH = orems 6 AFD, and and BHE 15), the surface of the (Theorem 26). = so EGC made (The- whence follows that triangle ABC equals that Fig. 16. of the quadrilateral AFGO 26 THEORY If the point OF PARALLELS. H coincides with the middle B y/\ ~~^~~~~-^ point E of the side BC (Fig. 1 7), only equal right-angled triangles, ADF and BDE, are made, by whose interchange the of the triangle ABC of the two surface.s equivalence and the quadrilateral AFEC is established. / I If, finally, the point H falls outside the triangle -~^ ABC in Y~ (Fig. 18), the perpendicular CG goes, so we go and the Fig. 17. triangle, consequence, through the over from the triangle ABC to the quadrilateral AFGC by adding / Fig. 18. EBH. DBH, and then taking away the triangle CGE triangle FAD in the a AFGC circle spherical quadrilateral Supposing passed great 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 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 4-S. = = THEORY Let each ABCD now sides AB = DC OF 27 PARALLELS. (Fig. 19) be the spherical quadrilateral, perpendicular to BC, and the angles are where the A and D £S. Fig. 19. Prolong the sides AD beyond E, make further and BC until DE = they another in E, and prolongation BG and join the cut one EF and let fall upon the perpendicular FG. Bisect the whole arc mid-point by great-circle-arcs with A and F. The triangles EFG and DCE are congruent (Theorem 15), of BC the H DC = The so FG = AB. 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 tt, the BAH HFE HFG HFE-EFG +S HAD |S £S angle £S— HAD— tt-HS; consequently, angle HFE |(S— tt); or what —- = = - = -- = = - - = = is the same, this equals the size of the lune quadrilateral ABCD, AHFDA, if which again is from pass easily 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 in which the sum of the same time also that of the spherical triangle three angles is equal to S. equal the to the one to the other — as we see we over 28 THEORY OF PARALLELS. If three planes cut each other in parallel lines, then the sum of the three surface angles equals two right angles. Let A A', BB' CC (Fig. 20) be three parallels made by the inter section of planes (Theorem 25). Take upon them at random three 28. Fig. 20. B, C, and suppose through these a plane passed, which con will cut the planes of the parallels along the straight lines points A, sequently 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 AA', BB', CC; c. finally call the linear angles BDC a, ADC b, ADB About A as center suppose a sphere described, upon which the inter sections of the straight lines AC, AD AA' with it determine a spherical triangle, with the sides p, q, and r. Call its size a. Opposite the side q lies the angle w, opposite r lies X, and consequently opposite p lies the angle 7T-\-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, v: opposite q', Z opposite r', and consequently 7T-{-'2ft— w— Z opposite p'. Finally is determined by the intersection of a sphere about D with the lines DA, DB, DC, a spherical triangle, whose sides are 1, m, n, and the angles opposite them w-\-7i "2ft, w-^-'K la, and Y. Consequently = — = = — its size d=i (X+Y+Z—-)— a—ft+w. Decreasing w lessens also the size of the triangles a u-\-ft w can be made smaller than any given number. — and ft, so that THEORY OF 29 PARALLELS. In the triangle o can likewise the sides 1 and m be lessened even to vanishing (Theorem 21), consequently the triangle d can be placed with one of its sides 1 or m upon a great circle of the sphere as often as you choose without thereby filling up the half of the sphere, hence 3 van ishes together with w; whence follows that necessarily we must have X+Y+Z=-;r 29. In a rectilineal triangle, the perpendiculars erected at the mid-points of the sides either do not meet, or they all three cut each other in one point. 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 triangle ADC is hence isosceles, consequently the perpendicular dropped from the vertex D upon the base AC falls upon G the mid point of the base. The proof remains unchanged also in the case when the intersection point D of the two perpen ° diculars ED and FD falls in the line AC itself, or7 A falls without the Fl°- 2L triangle. therefore pre-suppose that two of those perpendiculars do intersect, then also the third can not meet with them. The perpenelicidars which are erected upon the sides of a rectilineal 30. In case we not mid-points, must all three be parallal to each other, so soon is presupposed. as the parallelism of two of them In the triangle ABC (Fig. 22) let the lines DE, FG, HK, be erected triangle at their upon the sides at their mid We will in the first place H. perpendicular points D, F, assume that the two perpendiculars DE and the line AB in L FG parallel, cutting and M, and that the perpendicular HK lies between them. Within the angle BLE draw are from the point L, LG, which how small at random, a straight line must cut FG somewhere in soever the angle G, Fig. 22. of deviation GLE may be. (Theorem 16). 30 OF THEORY PARALLELS. with triangle LGM the perpendicular HK can not meet whence in P, MG (Theorem 29), therefore it must cut LG somewhere follows, that HK is parallel to DE (Theorem 16), and toMG (Theorems 18 and 25). Since in the the 2c, and designate 2b, AB case just the in have then we these sides by A, B, C, gles opposite Put the side BC 2a, AC = an- = = considered A= 77(b)—77(c), B= 77 (a)—77(c), C as one are may easily consequently rems 23 and Let now to both the other perpendiculars DE and FG (Theo 25). the two perpendiculars the third DE not cut them them, 77(a)+77(b), help of the lines AA', BB', CC, which points A, B, C, parallel to the perpendicular HK drawn from the and = show with parallel, then can it either parallel to HK and FG be (Theorem 29), hence is it cuts AA'. or The last assumption is not other than that the angle C>77(a)+77(b.) If while we lessen this we angle, give in that way so that it becomes the line AC the new equal to 77(a) -(-77(b), position CQ, (Fig. 23), 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 and whence follows c' >c 77(a)-77(c')> 77(a)—77(c), (Theorem 23). Fig. 23. In the triangle ACQ are, however, the angles at A and Q equal, 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 hence in the which all perpendiculars each other. erected at the mid-points of chords are parallel to THEORY In a conformity boundary line, OF with this definition if we draw to a 31 PARALLELS. we can represent the generation of line AB given (Fig. 24) from a given Fig. 24. point A in it, making different angles CAB the end C of such chord will lie a on the = 77(a), chords AC boundary line, whose = 2a; points gradually determine. 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 In like manner will also each perpendicular FG erected at the ary line. 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, we can thus The be called Axis of the boundary line. continually increasing radius A circle toith 32. merges into the boundary line. Given AB end-points (Fig. 25) a chord of the A and B of the chord two / / BF, which consequently will make with the chord two equal angles AC and BAC = Upon ABF one where the -- ■- a point / Ifcfe^ V\. (Theorem 31). of these E as axes AC, ty/ take any- center of a draw from the boundary line; axes circle, and draw the arc AF from the initial point A of the axis AC to its intersection point vJ \ jf— -^ — c Fig. 25. F with the other axis BF. The radius of the on the one circle, FE, corresponding side with the chord AF an to the angle point AFE = ft, F will make and on the 32 PARALLELS. OF THEORY BF, the angle EFD other side with the axis between the two chords BAF angle whence Since follows, now = y. It follows that the ft<ft+y— a (Theorem 22); = a— ft<iy- — a however the angle r approaches the limit zero, as well m when r moving of the center E in the direction AC, remains unchanged, (Theorem 21), as also in consequence of an ap in its proach of F to B on the axis BF, when the center E remains the of position (Theorem 22), so it follows, that with such a lessening chords angle y, also the angle .-;. ft, or the mutual inclination of the two AB and AF, and hence also the distance of the point B on the bound ary line from the point F on the circle, tends to vanish. Consequently one may also call the boundary-line a circle with in finitely great radius. consequence of a — Let AA' 33. = B3' = be two lines (Figure 26). x the side from A to A', which parallels serve boundary arcs (arcs on boundary lines) AB=s, A'B'=.s', then is as axes two toward >|L for the two s' where parallel = se — x is Fig. 26. independent of the arcs s, s' and of straight line x, the distance of the arc s' from s. In order to prove this, assume that the ratio of the arc s to s' is equal to the ratio of the two whole numbers n and m. Between the two axes A A', BB' draw yet a third axis CC, which e the so cuts off from the the same side, a arc AB part A'C part AC a = t' . that of the whole numbers p and q, = — Divide on now s by s' and np However there equal parts on axes on into nq arc t to * A'B' on equal to that- so p s', I= m parts from the Assume the ratio of n s= t and — s. q equal parts, then will there be too such t. correspond to these equal parts l', consequently we have on s and t likewise s' and f t Hence also wherever the two two axes AA' and BB', s arcs t the ratio of and V may be taken between the t to V remains always the same as THEORY long as fore for the distance x= i, put x s= OF between them remains the e.s', then s' Since e is an 33 PARALLELS. we = unknown number and further the linear unit for x must If same. have for every we there x se~ x. only subjected may be taken at to the condition e>i, therefore may, will, we for the simplification of reckoning, so choose it that by e is to be un derstood the base of Napierian logarithms. We may here remark, that s'= 0 for a;= oo 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. 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 endpoints, 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; 34. Fig. 27. AA' the axis of and AC chords — B'BA, A AC revolution, to which the = CCA BB' and CC two other axes, hence AB axes are inclined at (Theorem 31.) equal angles A'AB 34 THEORY OF PARALLELS. end-points of the third chord BC, are likewise parallel and lie in one plane, (Theorem 25). A perpendicular DD' erected at the mid-point D of the chord AB and in the plane of the two parallels AA', BB', must be parallel to the three axes AA', BB', CC, (Theorems 23 and 25); just such a perpendicular EE' upon the chord AC in the plane of the parallels AA', CC will be parallel to the three axes AA', BB', CC, and the perpendicular DD'. Let now the angle between the plane in which the parallels AA' and BB' lie, and the plane of the triangle ABC be designated by 77(a), If a is positive, then erect where a may be positive, negative or null. FD a within the triangle ABC, and in its plane, perpendicular upon the chord AB at its mid-point D. obe drawn outside the Were a a negative number, then must FD when of the chord AB; a=0, the point F triangle on the other side Two axes BB', CC, drawn througn the = = coincides with D. In all cases consequently Erect angle now arise two we congruent right-angled triangles AFD and DFB, have FA = FB. at F the line FF' perpendicular to the plane of the tri ABC. angle D'DF= 77(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 Since the AF A perpendicular fall upon the base = FC = FB. from the vertex F of the isosceles its triangle BFC let mid-point G; BC, goes through plane passed 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 CCG B'BG, (Theorem 23). through = this a THEORY OF 35 PARALLELS. Hence follows, that for the boundary-surface each of the be considered as Principal-plane boundary axes may axis of revolution. we will call each plane passed through an axis of the surface. Accordingly every Principal-plane cuts boundary line, while for another position tersection is a the boundary-surface in the cutting plane this in of the 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 Con ary surface by the intersections with the three principal planes. of the and sides angles pertains to sequently the same interdependence the boundary-triangles, that is proved in the ordinary geometry for the rectilineal triangle. In what follows, we will designate the size of a line by a letter 35. with an accent x', in order to indicate that this has a rela. line, which is represented by the same letter which relation is given by the equation added, e. g. tion to that of another without accent x, n(x) + n(x') Let the now ABC hypothenuse (Fig. 28) AB = be a rectilineal = i-. right-angled triangle, c, the other sides AC = Fig. 28. angles opposite them are BAC=77(4ABC = 77(/3). b, BC = where a, and the 36 THEORY At the point triangle ABC, PARALLELS. OF A erect the line AA' at and from the points right angles to the plane B and C draw BB' and CC of the parallel to AA'. planes in which these three parallels lie make with each other and 13), the angles: 77(,y.) at AA', a right angle at CC (Theorems 11 at BB' consequently H(a') (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 run 77(c), kn 77 (ft), mk 77(a) and the opposite angles The = = = are 77(b), /7(«'), **■ Therefore sides are a, we b, existence of 77(a) c must, with the existence of a rectilineal triangle whose opposite angles 77 (a), 17 (ft) £-, also admit the> and the spherical triangle (Fig. 29) with opposite angles 77(b), 77(«'), \~. a and the the sides 77(c), 77 (ft), 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, «', ft, and the oppsite angles 77 (b'), //(C), iTT. Hence a', ft, b', we may pass over from a, b, c, a, ft, to b, a, c, ft, a, and also to a, 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 {, B"A r, and the angles opposite them 77(a), I7(o'), I-tt, and where consequently (Theo rem 34): = — p r sin 77(a), q = r cos Now break the connection of the three = 11(a). principal-planes along the line BB' and turn them out from each other so that they with all the lines lying in them come to lie in one plane, where consequently the arcs p q, r will unite to a single arc of a boundary-line, which goes through the , THEORY point one A and has AA' for side will lie, the OF 37 PARALLELS. axis, in such a manner that q and p, the side b of the arcs (Fig. 30) triangle, on the which is Fig. 30. perpendicular to A A' at A, the axis CC going from the end of b par through C" the union point of p and q, the side a per point C, and from the end-point of a the axis pendicular BB' parallel to AA' which goes through the end-point B" of the arc p. AA' at On the other side of AA' will lie, the side c perpendicular to and the to BB' axis the AA', through going parallel the point A, and allel to AA' and to end-point CC at the B" of the arc r remote from the end The size of the line CC" express by In like If point we depends upon b, point of b. dependence which we will CC"=/(b). manner we will have BB" describe, taking CC ' as =/ (c). axis, axis C to its intersection D with the /». CD by t, then is BD BB"= BD+DB" boundary line from the BB' and designate the arc a new = BD-f-CC", consequently y(c) =/(*)+ /(b)Moreover, we perceive, that (Theorem 32) t^pem =zr sin 77(a) e^K ABC (Fig. 28) were If the perpendicular to the plane of the triangle lines c and r remain would the then erected at B instead of at the point A, the to t and q, straight lines a the same, the arcs q and t would change = 38 THEORY and b into b and a, and the PARALLELS. OF angle 77(a) into 77 (ft), consequently we would have q=r sin whence follows by substituting cos and if we change a and 77 ft sin (a) II(ft) e'<», the value of q, = sin 7/ (ft) e/«, into b' and c, 77(b) = sin 77(c) e«a>; further, by multiplication with e-^b) sin 77 e-«b> = sin 7/ (c) e-«c> (b) Hence follows also sin 77 e-«»>= sin (a) 77(b) e-«b>. however, the straight lines a and b are independent of another, and moreover, for b=0, /(b)=0, 77(b)—-^-tt, so we have Since now, one for every straight line a e-/(»)=sin 77(a). Therefore, sin 77 (c) sin II (ft) Hence we obtain besides = = by sin sin cos 77(a) sin 77(b), 11(a) sin 77 (a). mutation of the letters 77(a) cos 77Q9) sin 77(b), 7/(b) cos 77(c) cos 77(a), cos 77(a) cos 77(c) cos 77 (ft). If we designate in the right-angled spherical triangle (Fig. 29) the sides 77(c), 77 (ft), 77(a), with the opposite angles 77(b), 77 (a'), by the = cos = = letters a, b, c, A, B, then the obtained equations take on the form of those which we know as proved in spherical trigonometry for the right- angled triangle, namely, sin a=sin c sin sin b=sin c sin cos cos cos from which A, B, A=cos a sin B, B=cos b, sin A, c=cos a, cos b; equations pass angles in general. Hence spherical trigonometry is t we can over not to those for all dependent spherical tri upon whether in a THEORY rectilineal angles 36. ABC or triangle the sum OF 39 PARALLELS. of the three angles is equal to two right not. We will (Fig. 31), now consider anew in which the sides the are right-angled rectilineal triangle a, b, c, and the opposite angles H(a),n(ft),iTT. Prolong the hypothenuse c through the point B, and make BD=y3; at the point D erect upon BD the perpendicu lar DD', which consequently will be parallel to BB' the prolongation of the side a beyond the point B. Parallel to DD' from the point A draw AA', which is at the same time also parallel to CB', (Theorem 25), therefore is the angle , A'AD=77(c+/?), A' AC 77 (b), consequently U(h)=n(a)+ii(c+ft). = Fig gL If from B enuse we lay off ft c, then at the end on the hypoth point D, (Fig. triangle erect upon AB perpendicular DD', and from the point A parallel to DD' draw AA', so will BC with its prolongation CC be the third parallel; then is, angle CAA'=77 (b), T>AA' U(c—ft), consequently 77(c— ft) II(a)-{- 77(b). The last equation is then also still valid, when c=^9, or c<ft. If c=ft (Fig. 33), then the perpendicu- 32), within the the — — frIG. 32. ular A A' erected upon AB at the point A 40 is THEORY parallel we to the side PARALLELS. OF BC=a, with its prolongation, CC, consequently 77(a)+77(b)=^, whilst also 77(c—$=|-, (Theorem 23). c<;3, then the end of ft falls beyond the point A at D (Fig. 34) have If upon the prolongation lar DD' erected upon of the AD, Here the AB. perpendicu parallel to it from A, will likewise be parallel to the side BC=a, with its prolongation CC. Here we have the angle DAA' 77 (ft c), consequently 77(a)+ 77(b) =-_77(y3-c)=7/(c-/3), (Theorem 23). The combination of the two equations hypothenuse and the line AA' B = — found gives, 277(b)=77(c-/3)+/7(c+;?), 277(«)=/7(c-/?)-77(c+,3), whence follows cos 77(b) cos [ jll(c—ft) +j 77(c+/?)] cos 77(a) "cos [ ^n(c—ft)—^ 77(c+/?)j here the Substituting rem value, (Theo 35) cos Fig. 34. 77(b) =cos77(c), 77(a) [tan £77 (c)]2=tan \U (c—ft) tan £77 (c+ft). ft is an arbitrary number, as the angle 77 (;3) ■ cos we have Since here at the one THEORY side of OF 41 PARALLELS. may be chosen at will between the limits 0 and between the limits 0 and co , so we may deduce c quently ft consecutively /3=c, 2c, 3c, &c, that for every positive \tz, conse by taking number n, [tan£ 77(c)] n=tan£77(«c). If consider we the ratio of two lines n as x and c, and assume that cot£77(c)=ec, then find for every line we tan£77(x)=e— tive, where x in may be any arbitrary 77(x)=0 for x=co e since general, whether it be positive or nega - number, which is greater than unity, . Since the unit by which the lines measured is arbitrary, Napierian Logarithms. are the base of the so we may also understand by 37. Of the equations found above in Theorem 35 it is sufficient to e following, know the two 77(c)=. sin U (a) sin /7(b) (a)=sin 77(b) cos IJ(ft), sin sin77 applying the latter to both the sides and b about the a right angle, in order .from the combination to deduce the remaining two of Theorem without ambiguity of the algebraic sign, since here all angles are 35, acute. In similar a manner we tan (1.) (2.) We will (Fig. 35) now cos consider ^^ p. / n. 1 d ^ for which triangle whose sides are a, b, c, B, C. the side the ■pio. 35. triangles, rectilineal If A and B are acute angles, then the perpendicular p from the vertex of the angle C falls within the triangle and cuts c ■** a equations 77(c)=sin 77(a) tan 77(a), 77(a)=cos 77(c) cos 77 (ft). 0PP0site angles A, and tlie / ^v y attain the two we c into two A and parts, x on the side of the side of the angle angle B. Thus arise two right-angled obtain, by application of equation (1), tan tan c — 77(a)=sin B tan 77(p), 77(b)=sin A tan 77 (p), x on 42 PARALLELS. OF THEORY equations remain unchanged also when one of the angles, right angle (Fig. 36) or ani obtuse angle (Fig. 37). which is a Fig. 36. Therefore For tion a we g. B, Fig. 37. have universally (3.) sin A tan triangle e. with acute for every triangle 77(a)=sin B tan 77(b). angles A, B, (Fig. 35) we have also (Equa 2), 77(x)=cos A cos 77(b), 77(c x)=cos B cos 77(a) which equations also relate to triangles, in which one of the angles A or B is a right angle or an obtuse angle. As example, for B=£- (Fig. 36) we must take x=c, the first equa cos cos — tion then goes over into that which we have found above the other, however, is self-sufficing. For of the we have (Theorem 23), If A is x Equation 77(x c)=cos (- B) cos 77(a); cos 77(x— c)= cos 77(c— x) also cos (tt— B)= cos B. a — — — and right or an — obtuse angle, then must c— x and x be put for in order to carry back this case upon the preceding. In order to eliminate x from both equations, we notice that and rem 2, B>i,T (Pig- 37) the first equation remains unchanged, instead second, however, we must write correspondingly cos but as c— x, 36) [tan+/7(c—x)]s l+[tan|77(c— x)]2 1— cos77(c — x)= " i_|_e2i : — 2c £/7(c)] 2 [cot £/7(x)] s l-t-[tan£77(c)p[cot£77(x)]2 cos //(c) cos7/(x) 1— [tan — 1 — cos 77(c)cos77(x) (Theo THEORY If we substitute here the OF 43 PARALLELS. for expression cos 77(x), cos77(c — x), we ob tain cos cos77(c)= //(a) cos B-f cos 77(b) cos A l-|-cos77(a) cos//(b) cosA cosB whence follows cos and 77(a) cosB= cos /7(c)-cosA cos/7(b) 1 cos A cos — 77(b) cos 77(c) finally [sin 77 (c) ] 2 =[ 1 —cos B cos 77 (c) cos 77 (a) ] [ 1 —cos A cos 77 (b) cos 77 (c) ] In the same must we way also have (*■) [sin 77 (a) ] 2 =[ 1 —cos C cos 77 (a) cos 77 (b) J [ 1 —cos B cos 77 (c) cos 77 (a) ] [sin 77(b) ]2 =[1— cos A cos 77 (b) cos 77(c) ] [1—cos C cos 77(a) cos 77(b) ] From these three equations we find [sin //(b)!* ["sin/7 (cYl2 h 77 [sin (a)]3 ,,, Hence follows without ,* If ambiguity a s (5.) K ' tion we v,, cos A cos of n/M sign, sin 77(c) 77(b) y—t ^1— 1 sm 77(a) sin 77(c) corresponding to equa tt, x cos 7/(c) A 77(b) v ;-r w substitute here the value of sin (3.) sin then we 77(c)= tan . sin 0 77(a) cos 77(c) obtain cos cos but r =[1— cosAcos77(b)cos//(c)1-. L \ I \ )} 77(C): by substituting this expression for cot A sin C sin (a ";\ *• 77(a) sin C sin A sin 77 (b)-|-cos A sin C cos77 cos cos 17(a) cos C= 1 j-fr-r 77 (b) In the meantime the cos — B sm-R-sin -AJ 77(a) equation (3) comes cosA — cos . 0 sm 11 (a). w equation (6) gives by changing ^-7-=cotB 77(b) (a) cos 77(b) ; equation (4), 77 (b)-l-cos C= By elimination of sin 77(b) with help of the cos in 77(c) sinC sin 77 (a) 4- cos C. the letters, 44 From the last two equations follows, cos sin B sin C _, „ (7.) ' v PARALLELS. OF THEORY cosC= A+cosB ' — — : . „. 77 (a) sm and equations for the interdependence of the sides a, b, c, therefore will be, opposite angles A, B, C, in the rectilineal triangle All four the [Equations (3), (5), (6), (7).] / sin A tan sin B tan 77 (b), 77 (a) = 77 (b) cos sin i cos (8.) v ; A cos 77 (c) + 77(b) 8in sin 77(c) -= 77(a)" cos 77(b) J \cotAsinCsin77(b) + cosC— cos 17 v (a) COS A -f- B COS COS C b, 77(a) 77(a) cos 77 (a) manner equations . = = = a, 1 £a" — a, also for the other sides b and 8 pass for such over b sin A = a a2 =b2 a sin cos Of these (a) are cot The sin B sin C : sin 77 c, of the sin and in like ' very small, we may content our triangle approximate determinations. (Theorem 36.) If the sides a, selves with the 1, sin -|- c2 (A -f- C) A -4- cos triangles c. into the following: B, — = 2 be cos A, b sin A, (B + C) = 0. equations the first two are assumed in the ordinary geom two lead, with the help of the first, to the conclusion etry; the last A -f- B +C = -. 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. Therefore the Hence there is no means, other than astronomical observations, to use THEORY OF 45 PARALLELS. 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 put a^/ l,bi/- l.cy'- l, instead of change, however, we must also put we — sin and similarly In this 77(a) = the sides a, cos 77(a) tan 77(a) —- = (a), (\/— l) =-. sm a tan a, ——— (y — I), c. pass over from equations sin A sin b = sin B sin a, manner we =: cos cot A sin C cos A = b cos c -f- cos a c; with this cos also for the sides b and cos a b, cos -|- sin b C cos b sin B sin C sin = — (8) to the c cos A, sin b cot a, cos B cos C. following: TRANSLATOR'S APPENDIX. ELLIPTIC GEOMETRY. Gauss himself published aught upon this fascinating subject, Geometry extraordinary pupil of his long teaching life came to read his inaugural dissertation before the Philosophical Faculty of the University of Goettingen, from the three never Non-Euclidean ; but when the most themes submitted it " the choice of Gauss which fixed upon the one Hypothesen welche der Geometrie zu Grunde liegen." Ueber die Gauss was then was recognized I wonder if he world. saw when in his sixth sentence special "From of case a powerful mathematician in the pupil was already beyond him, Riemann says, "therefore space is only a as the most that here his three-fold extensive this, however, it follows of and magnitude," necessity, that the continues: propositions of general magnitude-ideas, but that those peculiarities through which space distinguishes itself from other thinkable threefold extended magnitudes can only be gotten from ex geometry can perience. not be deduced from Hence arises the problem, to find the simplest facts from a which the metrical relations of space are determinable 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 These facts are, as all facts, not necessary, ent aim made fundamental. 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." but Riemann extends the idea of curvature to spaces of three and dimensions. on the The curvature of the freely figures plane is constant and it can move sphere is constant and without deformation. zero, and on it The curvature of the two-dimentional [47] more positive, and The curvature of figures slide without stretching. space of Lobatschewsky and 48 Bolyai completes ures can move to the sphere being constant and negative, and in it fig stretching or squeezing. As thus corresponding without it is called the It would formation. or we pseudo-sphere. live, the supposed stretch when fact that we space. suppose If you think so, take to we can move Riemann, be a without de special case of At We presume its curvature null. interfere to not does squeeze us space our move, is But is it not absurd to thing? it into us we then, according space of constant curvature. once PARALLELS. the group, In the space in which a OF THEORY envisaged as a peculiar property of our speak of space as interfering with any a knife and a raw potato, and try to cut solid. seven-edged on in this astonishing discourse comes the epoch-making idea, that though space be unbounded, it is not therefore infinitely great. "In the extension of space-constructions to the im Riemann says: measurably great, tne unbounded is to be distinguished from the in finite; the first pertains to the relations of extension, the latter to the a Father size-relations. our space is an unbounded three-fold extensive manifoldness, is hypothesis, which is applied in each apprehension of the outer world, according to which, in each moment, the domain of actual perception is filled out, and the possible places of a sought object constructed, and The unboundedwhich in these applications is continually confirmed. ness of space possesses therefore a greater empirical certainty than any From this however the Infinity in no way follows. outer experience. Rather would space, if one presumes bodies independent of place, that is ascribes to it a constant curvature, necessarily be finite so soon as this curvature had ever so small a positive value. One would, by extend the of the in a ing beginnings geodesies lying surface-element, obtain an unbounded surface with constant positive curvature, therefore a sur "That an face which in a take the form of Here ception we homaloidal three-fold extensive manifoldness would a sphere, and so is finite." have for the first time in human that universal space may yet be thought finite. the marvelous per only 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. Assume that a TRANSLATOR'S 49 APPENDIX. Two intersecting complete straight lines enclose a plane figure, a digon. digons are congruent if their angles are equal. All complete straight lines are of the same length, I. In a given plane all the per pendiculars to given straight line intersect in a single point, whose distance from the straight line is \l. Inversely, the locus of all the points at a distance -J7, on straight lines passing through a given point and lying in given plane, is a straight line perpendicular to all the Two » a, radiating lines. The total volume of the universe is I3/-The angle by The ence of the sum an excess greater the angles of a plane triangle is greater than a straight proportional to its area. area of the triangle, the greater the excess or differ angle sum the Royal Astronomer for Ireland: Says "It is necessary to to which mers are from of the have we have found tt- large triangles, and the largest triangles triangles which the astrono The measuring. largest available triangles measure access means are, of course, the of those which have the diameter of the earth's orbit It is fixed star at the vertex. of annual base and a of the stars precisely the investiga mighty tri * * * angles had the sum of its three angles equal to two right angles. "Astronomers have sometimes been puzzled by obtaining a negative vestigations parallax as a very curious circumstance that the in a are tions which would be necessary to test whether parallax indeed as always might result tion. * * generally or inquiries oi were really curved then a negative parallax space observations which possessed mathematical perfec arisen from the from * of these No doubt this has the result of their labors. this nature, but if one errors It must remain large enough triangles the sum which an open are inevitable in question angles of the three whether if we had would still be two right angles." nothing within our experience possibility that the space in which we * * * find ourselves may be curved in the manner here supposed. "The subjective impossibility of conceiving of the relation of the most distant points in such a space does not render its existence in credible. In fact our difficulty is not unlike that which must have been felt by the first man to whom the idea of the sphericity of the earth Says Prof. Newcomb: which will justify » "There is denial of the 50 was THEORY suggested OF PARALLELS. in conceiving how by he could return to the point traveling in a constant direction from which he started without during his of gravity." any sensible change in the direction In accordance with Professor Cayley's Sixth Memoir upon Quantics: journey feeling " points is equal to C times the logarithm of the joining the two points is divieled by the funda The distance between two cross ratio in which the line mental quadric." projective expression for distance, and Laguerre's for an angle were in 1871 generalized by Felix Klein in his article Ueber die sogenannte Nicht-Euklidische Geometrie, and in 1872 (Math. Ann., Vol. 6) This equivalence of projective metrics with non-Euclidean geometry, space being of constant negative or positive curvature ac cording as the fundamental surface is real and not rectilineal or is im aginary. We have avoided mentioning space of four or more dimensions, wishing to preserve throughout the synthetic standpoint. For a bibliography of hyper-space and non-Euclidean geometry see articles by George Bruce Plalsted in the American Journal of Mathe he showed the matics, Vol. I., pp. 261-276, 3S4, 385; We notice that Clark regular courses fessors, and we in non-Euclidian presume, without authority on Vol. II, and Cornell geometry by pp. 65-70. University are giving their most eminent Pro looking, that the same is true of Har Hopkins University, with Prof. Newcomb an origi this far-reaching subject. vard and the Johns nal University ■*&y •c*. V /* % ^ j©&£^ fc *4 * :ar >>^''."4 V I