EME 3flS 
 

NON-EUCLTDEAN 
 GEOMETRY 
 
 BY 
 
 HENRY PARKER MANNING, Pii.D. 
 
 ASSISTANT PROFESSOR OF PURE MATHEMATICS 
 IN HUoWN UNIVERSITY 
 
 BOSTON, U.S.A. 
 GINN & COMPANY, PUBLISHERS 
 
 <L be .atbrncTui 
 1901 
 

 , 
 
 ' 
 
 COPYRIGHT, 1901, BY 
 HENRY PARKER MANNING 
 
 ALL BIGHTS RESERVED 
 
CONTEXTS 
 
 IN i i;c>i>r<-Tlox 
 
 ( 1IA1TKR I 
 PANGEOMETKY 
 
 I. PROPOSITION- M 1-1 SIMM. M.I <s nn; I'IMSCM-M: or 
 
 Si i-i.iM'oM i II.N . . . . . . . . . '*> 
 
 II. I'uoro-i i ION- \\iin ii MM. TIM i. i'i: 1{ i .. i i; i< 1 1 c I-"HMIM- . 
 
 III. THE Tiiuti: ll\ r ..... BKf ... . . 11 
 
 CHAPTEB II 
 
 rilK IIVI'KKI'.OLK (.Ko.MKTKV 
 
 I. l'\i: vi I I i 1-iM - ......... ^1 
 
 II. BoUKDART-CuRVEi UTD SUBVACM, \M K-.'i 1 1 -i- i v M -( 'i i; \ i - 
 
 \\1. Si 1M V I-.S ......... J ; i 
 
 111. TRIOOKOMEI IM. \i K..KMI i i . . . . . 5:> 
 
 CHAPTEK 111 
 
 TIII-: ELLIPTIC ii-:oMi-;THy . . . <;-> 
 
 CHA1TKII IV 
 ANALYTIC NON-EUCLIDKAN ( .KnM KTH V 
 
 I. HYPERBOLIC ANALYTIC GEOMETRY ..... 09 
 
 II. ELLIPTIC ANALYTIC GEOMETRY ...... *<> 
 
 III. ELLIPTIC SOLID ANALYTIC GEOMETRY ..... s <> 
 
 Hl-H'IMi VI. NoTK .......... ( .>1 
 
PREFACE 
 
 NON-EUCLIDEAN Geometry is now recognized as an impor- 
 tant branch of Mathematics. Those who teach Geometry 
 should have some knowledge of this subject, and all who 
 are interested in Mathematics will find much to stimulate 
 them and much for them to enjoy in the novel results and 
 views that it presents. 
 
 This book is an attempt to give a simple and direct 
 account of the Non-Euclidean Geometry, and one which 
 presupposes but little knowledge of Mathematics. The first 
 three chapters assume a knowledge of only Plane and Solid 
 Geometry and Trigonometry, and the entire book can be read 
 by one who has taken the mathematical courses commonly given 
 in our colleges. 
 
 No special claim to originality can be made for what is 
 published here. The propositions have long been estab- 
 lished, and in various ways. Some of the proofs may be 
 new, but others, as already given by writers on this subject, 
 could not be improved. These have come to me chiefly 
 through the translations of Professor George Bruce Halsted 
 of the University of Texas. 
 
 I am particularly indebted to my friend, Arnold B. Chace, 
 Sc.D., of Valley Falls, K. I., with whom I have studied and 
 discussed the subject. 
 
 HENRY P. MANNING. 
 
 PROVIDED i . .January, 1901. 
 
 iii 
 
 888489 
 
NON-EUCLIDEAN GEOMETRY 
 
 INTRODUCTION 
 
 THE axioms of Geometry were formerly regarded as laws 
 of thought which an intelligent mind could neither deny nor 
 investigate. Not only were the axioms to which we have 
 been accustomed found to agree with our experience, but 
 it was believed that we could not reason on the supposition 
 that any of them are not true. It has been shown, however, 
 that it is possible to take a set of axioms, wholly or in part 
 contradicting those of Euclid, and build up a Geometry as 
 consistent as his. 
 
 We shall give the two most important Non-Euclidean 
 Geometries.* In these the axioms and definitions are taken 
 as in Euclid, with the exception of those relating to parallel 
 lines. Omitting the axiom on parallels,! we are led to three 
 hypotheses ; one of these establishes the Geometry of Euclid, 
 while each of the other two gives us a series of propositions 
 both interesting and useful. Indeed, as long as we can exam- 
 ine but a limited portion of the universe, it is not possible to 
 prove that the system of Euclid is true, rather than one of 
 the two Non-Euclidean Geometries which we are about to 
 describe. 
 
 We shall adopt an arrangement which enables us to prove 
 first the propositions common to the three Geometries, then 
 to produce a series of propositions and the trigonometrical 
 formulae for each of the two Geometries which differ from 
 
 * See Historical Note, p. '.:]. f See p. 91. 
 
 1 
 
2 NON-EUCLIDEAN GEOMETRY 
 
 ,'thkfc of Euclid,; and by analytical methods to derive some of 
 their most striking properties. 
 
 )/IWfc{cfo.:u<>t propose; to investigate directly the foundations 
 of Geometry, nor even to point out all of the assumptions 
 which have been made, consciously or unconsciously, in this 
 study. Leaving undisturbed that which these Geometries 
 have in common, we are free to fix our attention upon their 
 differences. By a concrete exposition it may be possible to 
 learn more of the nature of Geometry than from abstract 
 theory alone. 
 
 Thus we shall employ most of the terms of Geometry with- 
 out repeating the definitions given in our text-books, and 
 assumt that the figures defined by these terms exist. In 
 particular we assume : 
 
 I. The existence of straight lines determined by any two 
 points, and that the shortest path between two points is a 
 straight line. 
 
 II. The existence of planes determined by any three points 
 not in a straight line, and that a straight line joining any two 
 points of a plane lies wholly in the plane. 
 
 III. That geometrical figures can he. mored <ihot iritJtottt 
 changing their shape or size. 
 
 IV. That a point moving along a line from one position to 
 another passes through every point of the line between, and 
 that a geometrical magnitude, for example, an angle, or the 
 length of a portion of a line, varying from one value to another, 
 passes through all intermediate values. 
 
 In some of the propositions the proof will be omitted or 
 only the method of proof suggested, where the details can be 
 supplied from our common text- books. 
 
CHAPTER I 
 PANGEOMETRY 
 
 I. PROPOSITIONS DEPENDING ONLY ON THE PRINCIPLE 
 OF SUPERPOSITION 
 
 1.' Theorem. If one straight line meets another, the sum of 
 r/t, nljacent angles formed is equal to two right angles. 
 
 2. Theorem. If two straight lines intersect, the vertical 
 titi<//t'8 are equal. 
 
 3. Theory Two triangles are equal if they have a */</> 
 <iml two adjacent <//////**. <>/' tiro */,/, .v <nt</ tin' fn,-/n<lrtl <//////, 
 
 / ne equal, respectively, to the corresponding parts of (//< 
 other. 
 
 4. Theorem. In an isosceles triati'/lc t/n- ///////*'* opposite th<- 
 r<///<tl sides are equal. 
 
 Bisect the angle at the vertex and use (3). 
 
 5. Theorem. The perpendiculars erected at the middle 
 fi'iiids of the sides of a triangle ///< >/ /// a point if tir<> / 
 them meet, <tn</ tin* [>oint is the centre of a circle f/xtf <-,m 
 
 i-nHijh tin- (///- /-, rfi'-.'s of the. triangle. 
 c 
 
 Proof. Suppose Eu and FO meet at O. The triangles A TO 
 and /;/'/> are equal by (3). Also, A En and <'EO are equal. 
 
 3 
 
4 NON-EUCLIDEAN GEOMETRY 
 
 Hence, CO and BO are equal, being each equal to AO. Th. 
 triangle B CO is, therefore, isosceles, and OD if drawn bisect- 
 ing the angle BOC will be perpendicular to BC at its middle 
 point. 
 
 6, Theorem. In a circle the radius bisecting an angle at 
 the centre is perpendicular to the chord which subtends the 
 angle and bisects this chord. 
 
 7, Theorem. Angles at the centre of a circle are propor- 
 tional to the intercepted arcs and may be measured by them. 
 
 8, Theorem. From any point without a line a perpendicu- 
 lar to the line can be drawn. P 
 
 Proof. Let P' be the position which P would 
 take if the plane were revolved about AB into A 
 coincidence with itself. The straight line PP 1 
 is then perpendicular to AB. p , 
 
 9, Theorem. If oblique lines drawn from a point in a per- 
 pendicular to a line cut off equal distances from the foot of 
 the perpendicular, they are equal and make equal angles 
 with the line and with the perpendicular. 
 
 10, Theorem. If two lines cut a third at the same angle, 
 
 'F 
 
 L 
 
 /( 
 
 'E 
 
 that is, so that corresponding angles are equal, a line can be 
 drawn that is perpendicular to both. 
 
PROPOSITIONS PROVED BY SUPERPOSITION 
 
 
 Proof. Let the angles FMB and MND be equal, and through 
 H, the middle point of MN, draw LK perpendicular to CD ; 
 then LK will also be perpendicular to AB. For the two 
 triangles LMH and KNH are equal by (3). 
 
 11. Theorem. If two equal lines in a plane are erected per- 
 pendicular to a given line, the line joining their extremities 
 makes equal angles with them and is bisected at right angles 
 by a third perpendicular erected midway between them. 
 
 I) 
 
 ^UC 
 
 (Let 
 
 
 II 
 
 B 
 
 Let AC and BD be perpendicular to AB, and suppose AC 
 and BD equal. The angles at C and D made with a line join- 
 ing these two points are equal, and the perpendicular HK 
 erected at the middle point of AB is perpendicular to CD at 
 its middle point. 
 
 Proved by superposition. 
 
 12. Theorem. Criven as in the last proposition tivo perpen- 
 diculars and a third perpendicular erected midway between 
 them; any line cutting this third perpendicular at right 
 angles, if it cuts the first two at all, will cut off equal 
 lengths on them and make equal angles with them. 
 
 Proved by superposition. 
 
 Corollary. The last two propositions hold true if the angles 
 at A and B are equal acute or equal obtuse angles, HK being 
 
6 NON-EUCLIDEAN GEOMETRY 
 
 perpendicular to AB at its middle point. If AC = BD, the 
 angles at C and D are equal, and HK is perpendicular to 
 CD at its middle point or, if CD is perpendicular to HK 
 
 C K D 
 
 A II B 
 
 at any point, K, and intersects A C and BD, it will cut off equal 
 distances on these two lines and make equal angles ivith them. 
 
 II. PROPOSITIONS WHICH ARE TRUE FOR RESTRICTED 
 FIGURES 
 
 The following propositions are true at least for figures 
 whose lines do not exceed a certain length. That is, if there 
 is any exception, it is in a case where we cannot apply the 
 theorem or some step of the proof on account of the length of 
 some of the lines. For convenience we shall use the word 
 restricted in this sense and say that a theorem is true for 
 restricted figures or in any restricted portion of the plane. 
 
 1. Theorem. The exterior anglv of a triangle is greater 
 than either opposite interior angle (Euclid, I, 16). 
 
PROPOSITIONS TRUE FOR RESTRICTED FIGURES 7 
 
 Proof. Draw AD from .4 to the middle point of the oppo- 
 site side and produce it to E, making DE = AD. The two 
 triangles A DC and EBD are equal, and the angle 7*727), being 
 greater than the angle EBD. is greater than f '. 
 
 Corollary. At li'uxt ttro out/It's <>f <i trio n<il>' n re 
 
 2. Theorem. //' /// '//>///''* "/ <i ///<//<///* tin- ///////. ///>- 
 an "{ii'il <ind th>' trian<jh is isosc*-/* *. 
 
 c 
 
 Proof. The perpendicular erected at the middle point of the 
 base divides the triangle into two figures which may be made 
 to coincide and are equal. This perpendicular, therefore, 
 passes through the vertex, and the two sides opposite the 
 equal angles of the triangle are equal. 
 
 3. Theorem. In a triangle ?///// inn-i/im/ <ttif/les the side 
 oppotsitt tli,' greater of tw<> tti//<'8 in greater than the Me 
 opposite the smaller; and conversely, if the sides of a triangle 
 are unequal the opposite angles art unequal, <//// the greater 
 angle lies oppo*/'f, ///, ; ir>'<it, / Me. 
 
 4. Theorem. If two triangles have two sides of one equal, 
 respectively, to two sides of the other, but the included angle 
 of the first greater than the included angle of the second, the 
 third side of the first is greater than the third Me of the 
 second ; and conversely, if two triangles have two sides of 
 
8 
 
 NON-EUCLIDEAN GEOMETRY 
 
 - 
 
 one equal, respectively, to two sides of the other, but the third 
 side of the first greater than the third side of the second, the 
 angle opposite the third side of the first is. greater than the 
 angle opposite the third side of the second. 
 
 5, Theorem. The sum of two lines drawn from any point 
 to the extremities of a straight line is greater than the sum of 
 two lines similarly drawn but included by them. 
 
 6. Theorem. Through any point one perpendicular only 
 can be drawn to a straight line. 
 
 D\ 
 
 Proof. Let P' be the position which P would take if the 
 plane were revolved about AB into coincidence with itself. 
 If we could have two perpendiculars, PC and PD, from P to 
 AB, then CP 1 and DP' would be continuations of these lines 
 and we should have two different straight lines joining P and 
 P' } which is impossible. 
 
 Corollary. Two right triangles are equal when the hypothe- 
 nuse and an acute angle of one are equal, respectively, to the 
 hypothenuse and an acute angle of the other. 
 
 7. Theorem. The perpendicular is the shortest line that can 
 be drawn from a point to a straight line. 
 
 Corollary. In a right triangle the hypothenuse is greater 
 than either of the two sides about the right angle. 
 
PROPOSITIONS TRUE FOR RESTRICTED FIGURES 
 
 8, Theorem. If oblique lines drawn from a point in a per- 
 pendicular to a line cut off unequal distances from the foot of 
 *he perpendicular, they are unequal, and the more remote 
 is the greater ; and conversely, if two oblique lines drawn 
 from a point in a perpendicular are unequal, the greater 
 cuts off a greater distance from the foot of the perpendicular. 
 
 9. Theorem. If a perpendicular is erected at the middle 
 point of a straight line, any point not in the perpendicular is 
 nearer that extremity of the line which is on the same side of 
 the perpendicular. 
 
 Corollary. Two points equidistant from the extremities of a 
 strait/lit /hie determine a, perpendicular to the line at its middle 
 point. 
 
 10. Theorem. Two triangles are equal when they have three 
 sides of one equal, respectively, to three sides of the other. 
 
 11, Theorem. If two /////* /// a plane erected perpendicular 
 to a third are unequal, the line joining their extremities 
 makes unequal angles with them, the greater angle with the 
 shorter perpendicular. 
 
 Proof. Suppose AC> BD. Produce ED, making BE AC. 
 Then EEC = ACE. But BDC > EEC, by (1), and ACD is a, 
 part of A CE. Therefore, all the more BDC > A CD. 
 
10 
 
 NON-EUCLIDEAN GEOMETRY 
 
 12. Theorem. If the two angles at C and D are equal, the 
 perpendiculars are equal, and if the angles are unequal, the 
 perpendiculars are unequal, and the longer perpendicular 
 makes the smaller angle. 
 
 13. Theorem. If two lines are perpendicular to a third, 
 points on either equidistant from the third are equidistant 
 from the other. 
 
 C \ ,K D 
 
 H . 
 
 Proof. Let AB and CD be perpendicular to HK, and on CD 
 take any two points, C and D, equidistant from K', then C 
 and D will be equidistant from AB. For by superposition we 
 can make D fall on C, and then DB will coincide with CA 
 
 by (6). 
 
 The following propositions of Solid Geometry depend di- 
 rectly on the preceding and hold true at least for any 
 restricted portion of space. 
 
 14. Theorem. If a line is perpendicular to two intersecting 
 lines at their intersection, it is perpendicular to all lines of 
 their plane passing through this point. 
 
 15. Theorem. "If two planes are perpendicular, a line drawn 
 in one perpendicular to their intersection is perpendicular to 
 the other, and a line drawn through any point of one perpen- 
 dicular to the other lies entirely in the first, 
 
THE THREE HYPOTHESES 
 
 11 
 
 16. Theorem. If a line is perpendicular to a plane, any 
 plane through that line is perpendicular to the plane. 
 
 17. Theorem. If a plane is perpendicular to each of two 
 meeting planes, it is perpendicular to their intersection. 
 
 in. THE THREE HYPOTHESES 
 
 The angles at the extremities of two equal perpendiculars 
 
 tlier right angles, (icntr anul.'s, or obtus<> angles, at least 
 
 for restricted figures. We shall distinguish the three cases 
 
 .king of them as the hypothesis of the right angle, the 
 
 liesis of the acute angle, and the hypothesis of the obtuse 
 
 . respectively. 
 
 Theorem. Tin 1 line joining the extremities of two equal 
 ,'ti<li<'iitars is, at least for any restricted portion of the 
 . equal to, greater fJmii* or less than the line joining 
 feet in the three hypotheses, respectively. 
 
 C K D 
 
 A H B 
 
 Proof. Let AC and BD be the two equal perpendiculars and 
 
 \ a third perpendicular erected at the middle point of AB. 
 
 HA and KC are perpendicular to HK, and KC is equal 
 
 to, greater than, op less than HA, according as the angle at C 
 
 .s equal to, less than, or greater than the angle at A (II, 12). 
 
 Hence, CD, the double of KC, is equal to, greater than, or less 
 
 AB in the three hypotheses, respectively. 
 
12 
 
 NON-EUCLIDEAN GEOMETRY 
 
 Conversely, if CD is given equal to, greater than, or less 
 than AB, there is established for this figure the first, second, 
 or third hypothesis, respectively. 
 
 Corollary. If a quadrilateral has three right angles, the sides 
 adjacent to the fourth angle are equal to, greater than, or less 
 than the sides opposite them, according as the fourth angle is 
 right, acute, or obtuse. 
 
 2. Theorem. If the hypothesis of a right angle is true in a 
 single case in any restricted portion of the plane, it holds 
 true in every case and throughout the entire plane. 
 
 C' D' 
 
 Proof. We^ have now a rectangle ; that is, a quadrilateral 
 with four right angles. By the corollary to the last propo- 
 sition, its opposite sides are equal. Equal rectangles can be 
 placed together so as to form a rectangle whose sides shall be 
 any given multiples of the corresponding sides of the given 
 rectangle. 
 
 Now let A'B' be any given line and A'C' and B'D' two equal 
 lines perpendicular to A'B' at its extremities. Divide A'C 1 , if 
 necessary, into a number of equal parts so that one of these 
 parts shall be less than AC, and on AC and BD lay off AM 
 and BN equal to one of these parts, and draw MN. ABNM 
 is a rectangle ; for otherwise MN would be greater than or 
 
THE THREE HYPOTHESES 
 
 13 
 
 less than A B and CD, and the angles at M and N would all be 
 acute angles or all obtuse angles, which is impossible, since 
 their sum is exactly four right angles. Again, divide A'B' 
 into a sufficient number of equal parts, lay off one of these parts 
 on .1 H and on MN, and form the rectangle APQM. Rectangles 
 equal to this can be placed together so as exactly to cover the 
 figure A 'B'D'C 1 , which must therefore itself be a rectangle. 
 
 3. Theorem. If the hypothesis of the acute angle or the 
 hypothesis of the obtuse angle holds true in a single case 
 wit bin a restricted portion of the plane, the same hypothesis 
 hold* true for every case within any such portion of the plane. 
 
 C K D 
 
 Proof. Let CD move along A C and BD, always cutting off 
 equal distances on these two lines ; or, again, let A C and BD 
 move along on the line AB towards HK or away from HK, 
 always remaining perpendicular to AB and their feet always 
 at equal distances from H. The angles at C and D vary 
 continuously and must therefore remain acute or obtuse, as 
 the case may be, or at some point become right angles. There 
 would then be established the hypothesis of the right angle, 
 and the hypothesis of the acute angle or of the obtuse angle 
 could not exist even in the single case supposed. 
 
 The angles at C and D could not become zero nor 180 in a 
 restricted portion of the plane ; for then the three lines A C, 
 CD, and BD would be one and the same straight line. 
 
14 
 
 NON-EUCLIDEAN GEOMETRY 
 
 4. Theorem. The sum of the angles of a triangle, at least 
 in any restricted portion of the plane, is equal to, less than, 
 or greater than two right angles, in the three hypotheses, 
 
 respectively. 
 
 A 
 
 FIG. 1. 
 
 D C 
 
 FIG. 2. 
 
 Proof. Given any right triangle, ABD (Fig. 1), with right 
 angle at B, draw AC perpendicular to AB and equal to BD. 
 In the triangles ADC and DAB, AC = BD and' AD is common, 
 but DC is equal to, greater than, or less than AB in the three 
 hypotheses, respectively. Therefore, DAC is equal to, greater 
 than, or less than ADB in the three hypotheses, respectively 
 (II, 4). Adding BAD to both of these angles, we have ADB 
 -{BAD equal to, less than, or greater than the right angle 
 BAG. 
 
 Now at least two angles of any restricted triangle are acute. 
 The perpendicular, therefore, from the vertex of the third 
 angle upon its opposite side will meet this side within the 
 triangle and divide the triangle into two right triangles. 
 Therefore, in any restricted triangle the sum of the angles 
 is equal to, less than, or greater than two right angles in the 
 three hypotheses, respectively. 
 
 We will call the amount by which the angle-sum of a tri- 
 angle exceeds two right angles its excess. The excess of a 
 polygon of n sides is the amount by which the sum of its 
 angles exceeds n 2 times two right angles. 
 
 It will not change the excess if we count as additional 
 vertices any number of points on the sides, adding to the sum 
 of the angles two right angles for each of these points. 
 
THE EXCESS OF POLYGONS 15 
 
 5. Theorem. The excess of a polygon is equal to the sum of 
 the excesses of any system of triangles into which it may be 
 divided. 
 
 Proof. If we divide a polygon into two polygons by a straight 
 or broken line, we may assume that the two points where it 
 meets the boundary are vertices. If the dividing line is a 
 broken line, broken at p points, the total sum of the angles of 
 the two polygons so formed will be equal to the sum of the 
 angles of the original polygon plus four right angles for each 
 of these p points, and the sides of the two polygons will be 
 the sides of the original polygon, together with the p + 1 
 parts into which the dividing line is separated by the p points, 
 each part counted twice. 
 
 Let N be the sum of the angles of the original polygon, and 
 n the number of its sides. Let S' and n', S" and n" have the 
 same meanings for the two polygons into which it is divided. 
 Then we have, writing R for right angle, 
 
 .S"+ S" = .S' + 4y;/,\ 
 
 and n' + n" = n + 2 (p + 1). 
 
 Therefore, S' - 2 (n' -2)R + S" - 2 (n" - 2) R 
 
 = S + pR -2(n + 2p-2)R 
 = S - 2 (n - 2) R. 
 
 Any system of triangles into which a polygon may be divided 
 is produced by a sufficient number of repetitions of the above 
 process. Always the excess of the polygon is equal to the 
 sum of the excesses of the parts into which it is divided. 
 
16 NON-EUCLIDEAN GEOMETRY 
 
 We may extend the notion of excess and apply it to any 
 combination of different portions of the plane bounded com- 
 pletely by straight lines. 
 
 Instead of considering the sum of the angles of a polygon, 
 we may take the sum of the exterior angles. The amount by 
 which this sum falls short of four right angles equals the 
 excess of the polygon. We may speak of it as the deficiency 
 of the exterior angles. 
 
 The sum of the exterior angles is the amount by which we 
 turn in going completely around the figure, turning at each 
 vertex from one side to the next. If we are considering a 
 combination of two or more polygons, we must traverse the 
 entire boundary and so as always to have the area considered 
 on one side, say on the left. 
 
 6. Theorem. The excess of polygons is always zero, always 
 negative, or always positive. 
 
 Proof. We know that this theorem is true of restricted tri- 
 angles, but any finite polygon may be divided into a finite 
 number of such triangles, and by the last theorem the excess 
 of the polygon is equal to the sum of the excesses of the 
 triangles. 
 
 When the excess is negative, we may call it deficiency, or 
 speak of the excess of the exterior angles. 
 
 Corollary. The excess of a polygon is numerically greater 
 than the excess of any part which may be cut off from it by 
 straight lines, except in the first hypothesis, when it is zero. 
 
AREA OF TRIANGLES 
 
 17 
 
 The following theorems apply to the second and third 
 hypotheses. 
 
 7. Theorem. By diminishing the sides of a triangle, or even 
 one side while the other two remain less than some fixed 
 length, we can diminish its area indefinitely, and the sum 
 of its angles will approach two right angles as limit. 
 
 C D 
 
 A B 
 
 Proof. Lt-t ABDC be a quadrilateral with three right angles, 
 A, B, and ('. A perpendicular moving along AB will con- 
 stantly increase or decrease ; for if it could increase a part 
 of the way and decrease a part of the way there would be 
 different positions where the perpendiculars have the same 
 length ; a perpendicular midway between them would be per- 
 pendicular to CD also, and we should have a rectangle. 
 
 Divide AB into equal parts, and draw perpendiculars 
 through the points of division. The quadrilateral is divided 
 into n smaller quadrilaterals, which can be applied one to 
 another, having a side and two adjacent right angles the same 
 in all. Beginning at the end where the perpendicular is the 
 shortest, each quadrilateral can be placed entirely within the 
 
 next. Therefore, the first has its area less than - th of 
 
 n 
 
 the area of the original quadrilateral, and its deficiency or 
 excess less than - th of the deficiency or excess of the whole. 
 
 Now any triangle whose sides are all less than A C or BD, and 
 one of whose sides is less than one of the subdivisions of AB, 
 
18 NON-EUCLIDEAN GEOMETRY 
 
 can be placed entirely within this smallest quadrilateral. Such 
 a triangle has its area and its deficiency or excess less than 
 
 - th of the area and of the deficiency or excess of the original 
 n 
 
 quadrilateral. 
 
 Thus, a triangle has its area and deficiency or excess less 
 than any assigned area and deficiency or excess, however 
 small, if at least one side is taken sufficiently small, the 
 other two sides not being indefinitely large. 
 
 8. Theorem. Two triangles having the same deficiency or 
 excess have the same area. 
 
 Proof. Let A OB and A 'OB 1 have the same deficiency or excess 
 and an angle of one equal to an angle of the other. If we place 
 them together so that the equal angles coincide, the triangles 
 will coincide and be entirely equal, or there will be a quad- 
 rilateral common to the two, and, besides this, two smaller 
 triangles having an angle the same in both and the same 
 deficiency or excess. Putting these together, we find again 
 a quadrilateral common to both and a third pair of triangles 
 having an angle the same in both and the same deficiency or 
 excess. We may continue this process indefinitely, unless we 
 come to a pair of triangles which coincide ; for at^rtrtime can 
 one triangle of a pair be contained entirely within the other, 
 since they have the same deficiency or excess. 
 
AREA OF TRIANGLES 19 
 
 Let so denote the sum of the sides opposite the equal angles 
 of the first two triangles, sa the sum of the adjacent sides, and 
 s 'a that portion of the adjacent sides counted twice, which is 
 common to the two triangles when they are placed together. 
 Writing o' and a' for the second pair of triangles, o" and a" 
 for the third pair, etc., we have 
 
 sa = s'd -\- so', so = sa', 
 
 sa ' = s'a' + so", so' = sa", 
 
 .sV = s'a" + W. etc. = .sV". etc. 
 
 .-. <S v, =,' a +.vV +s'a lv H ---- . 
 
 Therefore, the expressions N'//. xV. xV, diminish indefi- 
 nitely. Each of these is made up of a side counted twice 
 from one and a side counted twice from the other of a pair of 
 triangles. Thus, if we carry the process sufficiently far, the 
 remaining triangles can be made to have at least one side as 
 small as we please, while all the sides diminish and are less, 
 for example, than the longest of the sides of the original 
 triangles. Therefore, the areas of the remaining triangles 
 diminish indefinitely, and^as the difference of the areas 
 remains the same for each pair of triangles, this difference 
 must be zero. The triangles of each pair and, in particular, 
 the first two triangles have the same area. 
 
 Let A BC and DEF have the same deficiency or excess, and 
 suppose AC<DF. Produce AC to C", making AC' = DF. 
 
20 
 
 NON-EUCLIDEAN GEOMETRY 
 
 Then there is some point, B ! , on AB between A and B such 
 that AB'C' has the same deficiency or excess and the same 
 area as ABC. Place AB'C' upon DEF so that AC' will coin- 
 cide with DP, and let DE'F be the position which it takes. 
 If the triangles do not coincide, the vertex of each opposite 
 the common side DF lies outside of the other. The two tri- 
 angles have in common a triangle, say D OF, and besides this 
 there remain of the two triangles two smaller triangles which 
 have one angle the same in both and the same deficiency 
 or excess. These two triangles, and therefore the original 
 triangles, have the same area. 
 
 9. Theorem. The areas of any two triangles are propor- 
 tional to their deficiencies or excesses. 
 
 Proof. A triangle may be divided into n smaller triangles 
 having equal deficiencies or excesses and equal areas by lines 
 drawn from one vertex to points of the opposite side. Each of 
 
 these triangles has for its deficiency or excess - th of the defi- 
 
 1 
 ciency or excess of the original triangle, and for its area - th 
 
 of the area of the original triangle. 
 
 When the deficiencies or excesses of two triangles are com- 
 mensurable, say in the ratio m : n, we can divide them into 
 m and n smaller triangles, respectively, all having the same 
 deficiency or excess and the same area. The areas of the 
 given triangles will therefore be in the same ratio, in : n. 
 
THE SINE AND COSINE 21 
 
 When the deficiencies or excesses of two triangles, A and B, 
 are not commensurable, we may divide one triangle, A, as 
 above, into any number of equivalent parts, and take parts 
 equivalent to one of these as many times as possible from the 
 
 other, leaving a remainder which has a deficiency or excess 
 less than the deficiency or excess of one of these parts. The 
 portion taken from the second triangle forms a triangle, B'. 
 A and B' have their areas proportional to their deficiencies or 
 excesses, these being commensurable. Now increase indefi- 
 nitely the number of parts into which A is divided. These 
 parts will diminish indefinitely, and the remainder when we 
 take B 1 from B will diminish indefinitely. The deficiency or 
 excess and the area of B' will approach those of B, and the 
 triangles .1 and B have their areas and their deficiencies or 
 excesses proportional. 
 
 Corollary. The areas of two polygons are to each oth< , at 
 tli fir deficiencies or excesses. 
 
 10. Theorem. Given a right triangle with a fixed angle ; 
 if the sides of the triangle diminish indefinitely, the ratio of 
 the opposite side to the hypothenuse and the ratio of the 
 adjacent side to the hypothenuse approach as limits the sine 
 and cosine of this angle. 
 
 Proof. Lay off on the hypothenuse any number of equal 
 lengths. Through the points of division A l} A t , draw per- 
 pendiculars AtC-H A^Ci, - to the base, and to these lines 
 
22 NON-EUCLIDEAN GEOMETRY 
 
 produced draw perpendiculars A 2 D 1} A B D Z , - each from the 
 next point of division of the hypothenuse. 
 
 The triangles OA 1 C 1 and A^A^D^ are equal (II, 6, Cor.). 
 
 therefore. and 7^ 
 
 OA Z ^ OAi OA 2 ^ OA l 
 
 the upper sign being for the second hypothesis and the lower 
 sign for the third hypothesis. 
 
 A 
 
 JJo 
 
 #1 
 
 A* 
 
 Pi< 
 
 A C r !" j ^ Cj 2 r 
 
 Assume 
 
 OA,._ > OA 
 
 OA l 
 
 PC, 
 OAi 
 
 and 
 
 Since OA r _ l = (r 1) OA^ 
 
 and also = (? 1) A r _ 1 A r) 
 
 (~\C* f)f^ C* A C* A 
 
 , . , . . l/Vx.1 H^ l/V/ r i , OjYlii O-. lJ"lj 1 
 
 the inequalities - ^ and - 
 
 OA^OA r _^ OA l ' OA r _i 
 
 applied to the angle at A r _ l become 
 
 A A~ lD A~ 1 % r OA ^ and ~A L A$~OA = *' 
 
 A r _ l A. r U^i r _ l Yl r _j/l r USi r _ l 
 
 The first of these two inequalities may be written 
 
 A D 4/4 
 
 A r 1-ky 1 > " d r l^ 1 )- 
 
THE SINE AND COSINE 
 Add 1 to both members, 
 
 -OAT*^*^ 
 
 But 
 
 OA r < 0^ r _! 
 
 Again, C r _ l C r ^ D r _ l A r . 
 
 Hence, from the second inequality above, we have 
 
 or 
 
 Add 1 to both members, 
 
 <>r r ^ <>A 
 
 ' 1 OC 
 
 The ratios ^ and being less than 1, and always increas- 
 
 ing or always decreasing when the hypothenuse decreases, 
 approach definite limits. These limits are continuous func- 
 tions of .1 ; if we vary the angle of any right triangle contin- 
 uously, keeping the hypothenuse some fixed length, the other 
 two sides will vary continuously, and the limits of their ratios 
 to the hypothenuse must, therefore, vary continuously. 
 
 < 'ailing the limits for the moment sA and cA, we may extend 
 their definition, as in Trigonometry, to any angles, and prove 
 that all the formulae of the sine and cosine hold for these 
 functions. Then for certain angles, 30, 45, 60, we can prove 
 
24 
 
 NON-EUCLIDEAN GEOMETRY 
 
 that they have the same values as the sine and cosine, and 
 their values for all other angles as determined from their 
 values for these angles will be the same as the corresponding 
 values of the sine and cosine. 
 
 C 
 
 Draw a perpendicular, CF, from the right angle C to the 
 hypothenuse AB. The angle FOB is not equal to A, but the 
 difference, being proportional to the difference of areas of 
 the two triangles ABC and FBC, diminishes indefinitely when 
 the sides of the triangles diminish. From the relation 
 
 ^_ AC_ FB -BC _ 
 ~AC ~AB + ~BC ~AB ~ ' 
 
 we have, by passing to the limit, 
 
 (cA) 2 + (sA)' 2 = 1. 
 
 Let x and y be any two acute angles, and draw the figures 
 used to prove the formulae for the sine and cosine of the sum 
 of two angles. 
 
 The angles x and y remaining fixed, we can imagine all of 
 the lines to decrease indefinitely, and the functions sx, ex, sy, 
 etc., are the limits of certain ratios of these lines. 
 
 CA_ _ CE_ OB EA_ BA_ 
 OA OB OA BA OA' 
 
 PC _OD OB CD BA 
 ~OA~~OE'O~A~'BA~OA 
 oc \ 
 
 in the second figure ) . 
 
 (JA J 
 
THE SINE AND COSINE 
 
 25 
 
 The angles at M are equal in the two triangles EMB and 
 CMO, and we may write 
 
 CM ME 4- 8 ME CM 4 8 
 
 where 8 has the limit zero. 
 
 v 
 
 M 
 
 The angle EAB, or x', is not the same as x, but differs from 
 ./ only by an amount which is proportional to the difference of 
 the areas of the triangles OMC and MAB, and which, there- 
 fore, diminishes indefinitely. Thus, the limits of sx' and ex' 
 a iv .sv and ex. 
 
 Finally, as the two triangles A('\ and Ji/>\ have the angle 
 \ in ((iiinion. we may write 
 
 />.V CW 4-8' CJV - DN + 8' 
 
 AN BN 
 
 /;.v ~ .i.v 
 where the limit of 8' is zero. 
 
 Now at the limits our identities become 
 s (x + y) = sx - cy + ex sy, 
 c(x H- y) = ex cy sx sij. 
 
26 NON-EUCLIDEAN GEOMETRY 
 
 By induction, these formulae are proved true for any angles. 
 Other formulae sufficient for calculating the values of these 
 functions from their values for 30, 45, and 60 are obtained 
 from these two by algebraic processes. 
 
 If the sides of an isosceles right triangle diminish indefi- 
 nitely, the angle does not remain fixed but approaches 45, 
 and the ratios of the two sides to the hypothenuse approach 
 as limits s45 and c45. Therefore, these latter are equal, 
 and since the sum of their squares is '1, the value of each is 
 
 =9 the same as the value of the sine and cosine of 45. 
 V2 
 
 Again, bisect an equilateral triangle and form a triangle in 
 which the hypothenuse is twice one of the sides. When the 
 sides diminish, preserving this relation, the angles approach 
 30 and 60. Therefore, the functions, s and c, of these angles 
 have values which are the same as the corresponding values of 
 the sine and cosine of the same angles. 
 
 Corollary. When any plane triangle diminishes indefinitely, 
 the relations of the sides and angles approach those of the sides 
 and angles of plane triangles in the ordinary geometry and 
 trigonometry with which we are familia r. 
 
 11. Theorem. Spherical geometry is the same in the three 
 hypotheses, and the formulae of spherical trigonometry are 
 exactly those of the ordinary spherical trigonometry. 
 
 Proof. On a sphere, arcs of great circles are proportional to 
 the angles which they subtend at the centre, and angles on a 
 sphere are the same as the diedral angles formed by the planes 
 of the great circles which are the sides of the angles. Their 
 relations are established by drawing certain plane triangles 
 which may be made as small as we please, and therefore may 
 be assumed to be like the plane triangles in the hypothesis 
 of a right angle. These relations are, therefore, those of the 
 ordinary Spherical Trigonometry. 
 
THE THREE GEOMETRIES 27 
 
 The three hypotheses give rise to three systems of Geometry, 
 which are called the Parabolic, the Hyperbolic, and the Elliptic 
 Geometries. They are also called the Geometries of Euclid, of 
 Lobachevsky, and of Kiemann. The following considerations 
 exhibit some of their chief characteristics. 
 
 C D D' D' 
 
 Given PC perpendicular to a line, CF; on the latter we take 
 CD = PC, 
 DD' = Pit. 
 . D'D" = PD', etc. 
 
 Now if PC is sufficiently short (restricted), it is shorter 
 than any other line from /' to the line C'F; for any line as 
 short as PC or shorter would be included in a restricted por- 
 tion of the plane about the point P, for which the perpendicu- 
 lar is the shortest distance from the point to the line. 
 
 Therefore, PD > PC, .'. CD' > 2 < 'It. 
 
 PD' > PC, etc. ; ( 'It" > 3 ( 'D, etc. 
 
 Again, in the three hypotheses, respectively, 
 , and 
 
 DPD' < \ CPD, CD'P < $ CDP, 
 
 D'PD" < $DPD', etc., CD"P < \ CD'P, etc. 
 
 At P we have a series of angles. In the first hypothesis 
 there is an infinite number of these angles, and the series 
 forms a geometrical progression of ratio -J, whose value is 
 
28 NON-EUCLIDEAN GEOMETRY 
 
 exactly In the second hypothesis there is also an infinite 
 
 number of these angles, and the terms of the series are less 
 than the terms of the geometrical progression. The value of 
 
 the series is. therefore, less than In the third hypothesis 
 
 we have a series whose terms are greater than those of the 
 geometrical progression, and, therefore, whether the series is 
 
 convergent or divergent, we can get more than by taking a 
 
 sufficient number of terms. In other words, we can get a right 
 angle or more than a right angle at P by repeating this process 
 a certain finite number of times. 
 
 The angles at D, D', D", are exactly equal to the terms 
 of the series of angles at P. In the first two hypotheses they 
 approach zero as a limit. 
 
 The distances CD, CD', CD", - increase each time by more 
 than a definite quantity, CD-, therefore, if we repeat the 
 process an unlimited number of times, these distances will 
 increase beyond all limit. Thus, in the first and second 
 hypotheses we prove that a straight line must be of infinite 
 length. 
 
 In the hypothesis of the obtuse angle the line perpendicular 
 to PC at the point P will intersect CF in a point at a certain 
 finite distance from C, one of the D's, or some point between. 
 On the other side of PC this same perpendicular will intersect 
 FC produced at the same distance. But we have assumed that 
 two different straight lines cannot intersect in two points ; 
 therefore, for us the third hypothesis cannot be true unless 
 the straight line is of finite length returning into itself, and 
 these two points are one and the same point, its distance from 
 C in either direction being one-half the entire length of the 
 line. In this way, however, we can build up a consistent 
 Geometry on the third hypothesis, and this Geometry it is 
 which is called the Elliptic Geometry. 
 
THE THREE GEOMETRIES 
 
 29 
 
 The constructions would have been the same, and very 
 nearly all the statements would have been the same, if we had 
 taken CD any arbitrary length on CF. 
 
 The restriction which we have placed upon some of 
 the propositions of this chapter is necessary in the third 
 hypothesis. 
 
 Thus, in the proof that the exterior angle of a triangle is 
 greater than the opposite interior angle, the line AD drawn 
 through the vertex A to the middle point 7) of the opposite 
 
 side was produced so as to make AE = 2AD. If AD were 
 greater than half the entire length of the straight line deter- 
 mined by A and D, this would bring the point E past the point 
 A } and the angle CBE, which is equal to the angle C, instead 
 of being a part of the exterior angle CBF, becomes greater 
 than this exterior angle. 
 
 Again, if two angles of a triangle are equal and the side 
 between them is just an entire straight line, it does not follow 
 necessarily that the opposite sides are equal. It may be said, 
 
30 NON-EUCLIDEAN GEOMETRY 
 
 however, that the opposite sides form one continuous line, and, 
 therefore, this figure is not strictly a triangle, but a figure 
 
 C 
 
 somewhat like a lune. The points A and B are the same 
 point, and the angles A and B are vertical angles. 
 
 Finally, though we assume that the shortest path between 
 two points is a straight line, it is not always true that a 
 straight line drawn between two points is the shortest path 
 between them. We can pass from one point to another in 
 two ways on a straight line ; namely, over each of the two 
 parts into which the two points divide the line determined by 
 them. One of these parts will usually be shorter than the 
 other, and the longer part will be longer than some paths 
 along broken lines or curved lines. 
 
 When, however, the straight line is of infinite length, that 
 is, in the hypothesis of the right angle and in the hypothesis 
 of the acute angle, all the propositions of this chapter hold 
 without restriction. 
 
 The Euclidean Geometry is familiar to all. We will now 
 make a detailed study of the Geometry of Lobachevsky, and 
 then take up in the same way the Elliptic Geometry.. 
 
CHAPTER II 
 THE HYPERBOLIC GEOMETRY 
 
 \V K have now the hypothesis of the acute angle. Two lines 
 in a plane perpendicular to a third diverge on either side of 
 their common perpendicular. The sum of the angles of a 
 triangle is less than two right angles, and the propositions 
 of the last chapter hold without restriction. 
 
 I. PARALLEL LINES 
 
 From any point, P, draw a perpendicular, PC, to a given 
 lint'. I/;, and let PD be any other line from P meeting CB 
 in /A If D move off indefinitely on CB, the line PD will 
 approarh a limiting position /'/'. 
 
 P 
 
 PE is said to be parallel to CB at P. PE makes with PC 
 an angle, CPE, which is called the angle of parallelism for 
 tho perpendicular distance PC, It is less than a right angle 
 by an amount which is the limit of the deficiency of the tri- 
 angle PCD. On the other side of PC we can find another 
 line parallel to CA and making with PC the same angle of 
 parallelism. We say that PE is parallel to AB towards that 
 part which is on the same side of PC with PE. Thus, at any 
 
 31 
 
32 NON-EUCLIDEAN GEOMETRY 
 
 point there are two parallels to a line, but only one towards 
 one part of the line. Lines through P which make with PC 
 an angle greater than the angle of parallelism and less than 
 its supplement do not meet AB at all. We write II (p) to 
 denote the angle of parallelism for a perpendicular distance, p. 
 
 1. Theorem. A straight line maintains its parallelism at 
 all points. 
 
 A -^ v 
 
 B 
 
 H D 
 
 Let AB be parallel to CD at E and let F be any other point 
 of AB on either side of E, to prove that AB is parallel to CD 
 at F. 
 
 Proof. To H, on CD, draw EH and FH. If H move off 
 indefinitely on CD, these two lines will approach positions of 
 parallelism with CD. But the limiting position of EH is the 
 line AB passing through F, and if the limiting position of FH 
 were some other line, FK, F would be the limiting position of 
 If) the intersection of EH and FH. 
 
 2. Theorem. If one line is parallel to another, the second 
 is parallel to the first. 
 
 Given AB parallel to CD, to prove that CD is parallel to AB. 
 
 Proof. Draw AC perpendicular to CD. The angle CAB 
 will be acute; therefore, the perpendicular CE from C to AB 
 must fall on that side of A towards which the line AB is 
 parallel to CD (Chap. I, II, 1). The angle ECD is then acute 
 and less than CEB, which is a right angle. That is, we have 
 CAB < A CD, and CEB > ECD. 
 
PARALLEL LINES 
 
 33 
 
 If the line CE revolve about the point C to the position of 
 ' 'A. the angle at E will decrease to the angle A, and the angle 
 at C will increase to a right angle. There will be some posi- 
 tion, say CF, where these two angles become equal ; that is, 
 
 CFB = FCD. 
 
 FE 
 
 Draw MN perpendicular to CF at its middle point and 
 revolve the figure about MX as an axis. CD will fall upon 
 the original position of AB, and AB will fall upon the original 
 position of CD. Therefore, CD is parallel to A B. 
 
 Corollary. FB and CD are both parallel to MN. 
 
 F 
 
 Proof. FB and CD are symmetrically situated with respect 
 to MN, and cannot intersect MN since they do not intersect 
 each other. Draw FH to H, on CD, intersecting MN in K. 
 If H move off indefinitely on CD, FH will approach the posi- 
 tion of FB as a limit. Now K cannot move off indefinitely 
 before H does, for FK < FH. But again, when H moves off 
 indefinitely, K cannot approach some limiting position at a 
 
34 NON-EUCLIDEAN GEOMETRY 
 
 finite distance on MN; for FB, and therefore CD, would then 
 intersect MN and each other at this point. Therefore, H and 
 K must move off together, and the limiting position of FH 
 must be at the same time parallel to CD and MN. 
 
 In the same way we can prove that any line lying in a 
 plane between two parallels must intersect one of them or be 
 parallel to both. 
 
 3. Theorem. Two lines parallel to a third towards the same 
 part of the third are parallel to each other. 
 
 First, when they are all in the same plane. 
 
 Let AB and EF be parallel to CD, to prove that they are 
 parallel to each other. 
 
 Proof. Suppose AB lies between the other two. To H, any 
 point on CD, draw AH and EH, and let K be the point where 
 EH intersects AB. As H moves off indefinitely on CD, AH 
 and EH approach as limiting positions AB and EF. Now A' 
 cannot move off indefinitely before H does, for EK < EH. 
 But again, when H moves off indefinitely, K cannot approach 
 some limiting position at a finite distance on AB ; for this point 
 would be the intersection of AB and EF, and the limiting 
 position of H, whereas H moves off indefinitely on CD. There- 
 fore, H and K must move off together, and the limiting posi- 
 tion of EH must be at the same time parallel to CD and AB. 
 
PARALLEL LINES 
 
 35 
 
 If AB, lying between the other two, is given parallel to CD 
 and EF, EF must be parallel to CD', for a line through E 
 parallel to CD would be parallel to AB, and only one line can 
 be drawn through E parallel to A B towards the same part. 
 
 i> 
 
 Second, when the lines are not all in the same plane. 
 
 Let AB and CD be two parallel lines and let E be any point 
 not in their plane. 
 
 Proof. ToHonCDdTSiw AHandEH. As H moves off indefi- 
 nitely, AH approaches the position of AB, and the plane E I // 
 the position of the plane EAB. Therefore, the limiting posi- 
 tion of EH is the intersection of the planes ECD and E. 1 /;. 
 The intersection of these planes is, then, parallel to CD, and 
 in the same way we prove that it is parallel to AB. 
 
 Now, if EF is given as parallel to one of these two lines 
 towards the part towards which they are parallel, it must be 
 the intersection of the two planes determined by them and 
 the point E, and therefore parallel to the other line also. 
 
 4. Theorem. Parallel lines continually approach each other. 
 
 Let AB and CD be parallel, and from A and B, any points 
 on AB, drop perpendiculars AC and BD to CD. Supposing 
 that B lies beyond A in the direction of parallelism, we are 
 to prove that BD < AC. 
 
 Proof. At H, the middle point of CD, erect a perpendicular 
 meeting AB in K. The angle BKH is an acute angle, and the 
 
36 
 
 NON-EUCLIDEAN GEOMETRY 
 
 angle AKH is an obtuse angle. Therefore, a perpendicular to 
 HK at K must meet CA in some point, E, between C and A 
 
 D 
 
 C H 
 
 and DB produced in some point, F, beyond B. 
 (Chap. I, I, 12) ; therefore, DB < CA. 
 
 But DF = CE 
 
 Corollary. If AB and CD are parallel and AC makes equal 
 angles with them (like FC in 2 above), then EF, cutting off 
 equal distances on these two lines, AE = CF, on the side towards 
 which they are parallel, will be shorter than A C. 
 
 M 
 
 H 
 
 Proof. MN, perpendicular to AC at its middle point, is 
 parallel to AB and bisects EF, the figure being symmetrical 
 with respect to MN. EH, the half of EF, is less than AM, 
 and therefore EF is less than A C. 
 
 5. Theorem. As the perpendicular distance varies, starting 
 from zero and increasing indefinitely, the angle of parallelism 
 decreases from a right angle to zero. 
 
 Proof. In the first place the angle of parallelism, which is 
 acute as long as the perpendicular distance is positive, will be 
 
THE ANGLE OF PARALLELISM 
 
 37 
 
 made to differ from a right angle by less than any assigned 
 value if we take a perpendicular distance sufficiently small. 
 
 For, ADE being any given angle as near a right angle as 
 we please, we can take a point, Z, on DE and draw LR perpen- 
 dicular to DA at R. The angle RDL must increase to become 
 the angle of parallelism for the perpendicular distance RD. 
 
 Now let p be the length of a given perpendicular PM, and 
 let a be the amount by which its angle of parallelism differs 
 
 7T 
 
 from ; that is, say 
 
 I'M, being perpendicular i<> M.\, and // any point on MN, the 
 angle MPH approaches as a limit the angle of parallelism, 
 II ( p), when H moves off indefinitely on MN. The line PH 
 meets the line MX as long as MPH < n (y>), and by taking 
 MTH sufficiently near II (p), but less, we can make the angle 
 MIIP as small as we please (see p. 27). 
 
 In figure on page 38, let A C be perpendicular to AB, D being 
 any point on A C and DE parallel to AB. Draw DK beyond DE, 
 making .with DE an angle, EDK = U (p), and make DK = p. 
 TF, perpendicular to DK at K, will be parallel to DE and AB. 
 
38 
 
 NON-EUCLIDEAN GEOMETRY 
 
 By placing PMN of the last figure upon DKT, we see that 
 DC will meet KT in a point, G if 
 
 KDC < H O), 
 
 that is, if ADE>2a. 
 
 Then in the right triangle DKG, 
 
 Starting from the point G, we can repeat this construction, 
 and each time we subtract from the angle of parallelism an 
 amount greater than a. We can continue this process until 
 the angle of parallelism becomes equal to or less than 2 a. 
 
 If the point D move along AC, DE remaining constantly 
 parallel to AB, the angle at D will constantly diminish, and 
 by letting D move sufficiently far on A C we can reach a point 
 where this angle becomes equal to or less than 2 a. 
 
 Suppose D is at the point where the angle of parallelism is 
 just 2 a. Then, if we draw DK and TF as before, KT will be 
 
PERPENDICULARS IX A TRIANGLE 
 
 39 
 
 parallel to DC. All the parallels to AB lying between AB 
 and this position of TF meet A C, and as the parallel moves 
 towards this position of TF, the angle of parallelism at D 
 approaches zero, and the point D moves off indefinitely. 
 
 For an obtuse angle we may take^> negative, and we have 
 
 6. Theorem. The perpendiculars erected at the middle 
 points of the sides of a triangle are all parallel if two of 
 tin in are parallel. 
 
 Let .1, B, and C be the vertices of the triangle, and D, E, 
 and F, respectively, the middle points of the opposite sides. 
 Suppose the perpendiculars at D and E are given parallel, to 
 prove that the perpendicular at F is parallel to them. 
 
40 ! NON-EUCLIDEAN GEOMETRY 
 
 Proof. Draw CM through C parallel to the two given par- 
 allel perpendiculars. CM forms with the two sides at C angles 
 
 of parallelism II '.( - J and II f - J , of which the angle at C is 
 
 the sum or difference according as C lies between the given 
 perpendiculars or on the same side of both. By properly 
 diminishing these angles at C, keeping the lengths of CA 
 and CB unchanged, we can make the perpendiculars at their 
 middle points D and E intersect CM, and therefore each other, 
 
 at any distance from C greater than and greater than - 
 
 2i 2i 
 
 Let A'fi'C" be the triangle so formed, the point where the 
 two given perpendiculars meet, and C'M' the line through 0. 
 In the triangle A'B'C', the three perpendiculars meet at the 
 point (Chap. I, I, 5). Now we can let move off on C'M', 
 the construction remaining the same. That is, we let the 
 lines C'A' and C'B' rotate about C" without changing their 
 lengths, in such a manner that the three perpendiculars D'O, 
 E'O, and F'O shall always pass through 0. As moves off 
 
 indefinitely, the angles at C' approach n ( - J and II f - J as 
 
 limits, and the three perpendiculars approach positions of 
 parallelism with C'M 1 and with each other. But the triangle 
 A'B'C' approaches as a limit a triangle which is equal to ABC, 
 having two sides and the included angle equal, respectively, to 
 the corresponding parts of the latter. Therefore, in ABC the 
 three perpendiculars are all parallel. 
 
 7. Theorem. Lines which do not intersect and are not 
 parallel have one and only one common perpendicular. 
 
 Proof. Let AB and CD be the two lines, and from A, any 
 point of AB, drop AC perpendicular to CD. If AC is not 
 itself the common perpendicular, one of the angles which it 
 makes with A B will be acute. Let this angle be on the side 
 
PERPENDICULARS IN A TRIANGLE 
 
 1 
 
 towards AB, so that BAC < ~ Draw AE parallel to CD 
 
 6 
 
 on this same side of A C. The angle EA C is less than BA C, 
 since AB is not parallel to CD and does not intersect it. Let 
 A H be any line drawn in the angle EA C, intersecting CD at 
 //. If H, starting from the position of C, move off indefinitely 
 
 OMB 7) 
 
 on the line CZ>, the angle BA H will decrease from the magni- 
 tude of the angle BAC to the angle BA E. The angle AHC 
 will decrease indefinitely from the magnitude of the angle at 
 (\ which is a right angle and greater than BAC. There will 
 be some position for which BAH = AHC. In this position 
 the line NM through the middle point of AH perpendicular 
 to one of the two given lines will be perpendicular to the 
 other, as proved in Chap. I, I, 10. 
 
 If there were two common perpendiculars we should have a 
 rectangle, which is impossible in the Hyperbolic Geometry. 
 
 8. Theorem. If the perpendiculars erected at the middle 
 
 points of the sides of a triangle do not meet and are not 
 llel, they are all perpendicular to a certain line. 
 
NON-EUCLIDEAN GEOMETRY 
 
 Proof. We can draw a line, AB, that will be perpendicular 
 to two of these lines, and the perpendiculars from the three 
 vertices of the triangle upon this line will be equal, by Chap. I, 
 II, 13. A perpendicular to AB erected midway between any 
 two of these three is perpendicular to the corresponding side 
 of the triangle at its middle point (Chap. 1, 1, 11). Thus, all 
 three of the perpendiculars erected at the middle points of the 
 sides of the triangle are perpendicular to AB. 
 
 A line is parallel to a plane if it is parallel to its projection 
 on the plane. 
 
 9. Theorem. A line may be drawn perpendicular to a plane 
 and parallel to any line not in the plane. 
 
 ^B 
 R, D A 
 
 M 
 X 
 
 x- 
 
 N 
 
 Proof. Let AB be the given line and MN the plane. If AB 
 meets the plane MN at a point, A, we take on its projection a 
 length, AC, such that the angle at A equals II (A C). Then 
 CD, perpendicular to the plane at C, will be parallel to AB. 
 In the same way, on the other side of the plane a perpendic- 
 ular can be drawn parallel to BA produced. 
 
 If AB does not meet MN, then at least in one direction it 
 diverges from MN. Through H, any point of the projection 
 of AB on the plane, we can draw a line, HK, parallel to AB 
 towards that part of AB which diverges from MN, and then 
 draw CD parallel to this line and perpendicular to the plane. 
 
 Unless AB is parallel to MN it will meet the plane at some 
 point, or the plane and line will have a common perpendicular, 
 and the line will diverge from the plane in both directions. 
 
H( >UNDARY-CURVES 43 
 
 In the latter case there are two perpendiculars that are parallel 
 to the line, one parallel towards each part of the line. 
 
 Two perpendiculars cannot be parallel towards the same 
 part of a line ; for then they would be parallel to each other, 
 and two lines cannot be perpendicular to a plain 1 and parallel 
 to each other. 
 
 II. liolNDAKY-CURVES AND SURFACES, AND EQUI- 
 DISTANT-CURVES AND SURFACES 
 
 Having given the line AB y at its extremity, A, we take any 
 arbitrary angle and produce the side AC so that the perpen- 
 dicular erected at its middle point shall be parallel to A/:. 
 The locus of the point C is a curve which is called oricycle, or 
 boundary-curve. Mi is its axis. 
 
 From their definition it follows that all boundary-curves 
 are equal, and the boundary -curve is symmetrical with respect 
 to its axis ; if revolved through two right angles about its 
 axis, it will coincide with itself. 
 
 1. Theorem. Any line parallel to the axis of a boundary- 
 curve may be taken for axis. 
 
 Let AB be the axis and rn any liin- parallel to Ml, to 
 prove that CD may ! taken as axis. 
 
44 
 
 NON-EUCLIDEAN GEOMETRY 
 
 Proof. Draw A C ; also to E, any other point on the curve, 
 draw AE and CE. The perpendiculars erected at the middle 
 points of A C and of AE are parallel to AB and CD and to each 
 
 D 
 
 other. Therefore, the perpendicular erected at the middle 
 point of CE, the third side of the triangle A CE, is parallel to 
 them and to CD. CD then may be taken as axis. 
 
 Corollary. The boundary-curve may be slid along on itself 
 without altering its shape ; that is, it has a constant curvature. 
 
 2. Theorem. Two boundary -curves having a common set of 
 axes cut off the same distance on each of the axes, and the 
 ratio of corresponding arcs depends only on this distance. 
 
 Proof. Take any two axes and a third axis bisecting the 
 arc which the first two intercept on one of the two boundary- 
 
BOUNDARY-CURVES 
 
 45 
 
 curves. By revolving the figure about this axis we show that 
 the curves cut off equal distances on the two axes. 
 
 Let A A ', BB', and CC' be any three axes of the two boundary- 
 curves AB and A'B'; let their common length be x and let 
 them intercept arcs s and t on AB, s' and t' on .4'^'. 
 
 When s = t, s' = t\ and, in general, 
 
 s .<?' 
 
 as we prove, first when s and t are commensurable, and then 
 by the method of limits when they are incommensurable. 
 
 The ratio , is, therefore, a constant for the given value of x. 
 s 
 
 Write 
 
 From three boundary-curves having the same set of axes, 
 
 we find /(* + y)=/(s)/(y). 
 
 This property is characteristic of the exponential function 
 whose general form is f(x) = e** Therefore, - = e ax , the 
 value of a depending on the unit of measure (see below p. 76). 
 
 * Putting y = x, 2 x, (n 1) x in succession, we find 
 
 f(nx) = [/(z)] 
 for positive integer values of n, x being any positive quantity. 
 
 Now 
 
 and this is the rth power of the sth root of the first member of the 
 equation 
 
46 
 
 NON-EUCLIDEAN GEOMETRY 
 
 3. Theorem. The area enclosed by two boundary -curves 
 having the same axes and by two of their common axes is 
 proportional to the difference of the intercepted arcs. 
 
 Proof. Let s and s' be the lengths of the intercepted arcs, 
 and I the distance measured on an axis between them. Let t, 
 t', and k be the' corresponding quantities for a second figure 
 constructed in the same way. 
 
 If the corresponding lines in the two figures are all equal, 
 the areas are equal, for they can be made to coincide. If 
 only k = /, the areas are to each other as corresponding arcs, 
 say as s' : t', proved first when the arcs are commensurable, and 
 then by the method of limits when they are incommensurable. 
 
 When I and k are commensurable, suppose 
 
 in n 
 
 Thus, assuming that f(x) is a continuous function of x, we have proved 
 that for all real positive values of x and n 
 
 f(nx) = [/(x)], 
 and if we put x for n and 1 for x, we have 
 
 We will write /(I) = e a ; then 
 
 /(x) = e*. 
 
EQUIDISTANT-CURVES 47 
 
 We can draw a series of boundary-curves at distances equal to 
 a on the axes and divide the areas into m and n parts, respec- 
 tively. If f is the ratio of arcs corresponding to the distance 
 a, these parts will be proportional to the quantities 
 
 f i t \ r fi't . . /v i 
 
 v , f r j i i) v r 
 
 The two areas are then to each other in the ratio 
 
 But .vV" = s and t'i = f, 
 
 so that this is the same as the ratio 
 
 s s' : t t'. 
 
 When I and k are incommensurable, we proceed as in other 
 similar demonstrations. 
 
 This theorem is analogous to the one which we have proved 
 about polygons : the area is proportional to the amount of 
 rotation in excess of four right angles in going around the 
 figure, for the rate of rotation in going along a boundary- 
 curve is constant. 
 
 The locus of points at a given distance from a straight line 
 is a curve which may be called an equidistant-curve. The 
 perpendiculars from the different points of this curve upon 
 the base line are equal and may be called axes of the curve. 
 
 An equidistant-curve fits upon itself when revolved through 
 two right angles about one of its axes or when slid along upon 
 itself. It has a constant curvature. 
 
48 
 
 NON-EUCLIDEAN GEOMETRY 
 
 It can be proved, exactly as in the case of two boundary- 
 curves having the same set of axes, that arcs on an equidis- 
 tant-curve are proportional to the segments cut off by the 
 axes at their extremities on the base line or on any other 
 equidistant-curve having the same set of axes. 
 
 4. Theorem. The boundary-curve is a limiting curve between 
 the circle and the equidistant- curve ; it may be regarded as a 
 circle with infinitely large radius, or as an equidistant-curve 
 whose base line is infinitely distant. 
 
 F C 
 
 H 
 
 Proof. Take a line of given length, AB = 2 a 'say, making 
 an angle, A, with a fixed line, A C. Construct another angle at 
 B equal to the angle A, and draw a perpendicular to AB at its 
 middle point, D. 
 
 If the angle at A is sufficiently small, we have an isosceles 
 triangle with AB for base, and its vertex at a point, F, on A C. 
 With F as centre, we can draw a circle through the points A 
 and B. Now let the angle at A gradually increase, the rest 
 of the figure varying so as to keep the construction. F will 
 move off indefinitely, and when A = II (a) the three lines AF, 
 BF, and DF will become parallel, and B will become a point 
 on the boundary-curve AB', which has AC for axis. 
 
 On the other hand, if the angle at A were taken acute, but 
 greater than II (a), we should have three lines, AE, BH, and 
 
BOUNDARY-CURVES AS LIMIT CURVES 49 
 
 DF, perpendicular to a line, EH, the base line of an equidis- 
 tant-curve through the points A and B. Now let the angle A 
 gradually decrease, the rest of the figure varying so as to pre- 
 serve the construction. The quadrilateral ADFE, having three 
 right angles and the fourth angle A decreasing, must increase 
 in area. We get this same movement if we think of AD and 
 DF remaining fixed in the plane while AE revolves about .1, 
 making the angle A decrease. Thus the only way in which 
 the area of the quadrilateral can increase is for EH to move 
 off along on AC and become more and more remote from A. 
 When A becomes equal to II (a), BH and DF become parallel 
 to AC. and />' falls on the boundary -curve AB'. 
 
 Calling the radius of a circle axis, we find that circles, 
 boundary-curves, and equidistant-curves have many properties 
 in common : 
 
 The perpendicular erected at the middle point of any chord 
 is an axis. In particular, a tangent is perpendicular to the axis 
 drawn from its point of contact. These are curves cutting 
 at right angles a system of lines through a point, a system 
 of parallel lines, and the perpendiculars to a given line, 
 respectively. 
 
 Two of these curves having the same set of axes cut off 
 equal lengths on all these axes, and the ratio of corresponding 
 arcs on two such curves is a constant depending only on the 
 way in which they divide the axes. 
 
 Three points determine one of these curves ; that is, through 
 any three points not in a straight line we can. draw a curve 
 which shall be either a circle, a boundary-curve, or an equi- 
 distant-curve, and through any three points only one such 
 curve can be drawn. Any triangle may be inscribed in one 
 and only one of these curves. 
 
 Each of these curves can be moved on itself or revolved about 
 any axis through 180 into coincidence with itself. 
 
50 
 
 NON-EUCLIDEAN GEOMETRY 
 
 A boundary-surface or orisphere is a surface generated by 
 the revolution of a boundary-curve about one of its axes. 
 
 5, Theorem. Any line parallel to the axis of a boundary- 
 surface may be regarded as axis. 
 
 A 
 
 K 
 
 D' 
 
 Let A A 1 be the axis, meeting the surface at .4, and BB' a 
 line parallel to the axis through any other point, B, of the 
 surface; to prove that BB' may be regarded as axis. 
 
 Proof. Let C be a third point on the surface. Draw CC" 
 through C, and through D, E, and F, the middle points of the 
 sides of the plane triangle ABC, draw DD', EE', and FF' all 
 parallel to A A'. Finally, let OO' be parallel to these lines and 
 perpendicular to the plane ABC. The projecting planes of 
 the other parallels all pass through 00' (see I, 9). 
 
 Since A A' is axis to the surface, EE' and FF' are perpen- 
 dicular to AC and AB } respectively. Draw FK perpendicular 
 to the plane ABC at F. It will lie in the projecting plane 
 OFF'. AB, being perpendicular to FF' and to FK, is perpen- 
 
BOUNDARY-SURFACES 51 
 
 dicular to this plane, OFF', and therefore to OF. In the same 
 way we prove that AC is perpendicular to OE. Therefore, 
 EC is perpendicular to OD (Chap. I, I, 5). But OD is the 
 intersection of the plane ABC with the plane ODD'. Hence, 
 IK' is perpendicular to this plane and to DD' (Chap. I, II, 15). 
 
 DD' being parallel to BB' lies in the plane determined by 
 BB' and BC, and in this plane only one perpendicular can be 
 drawn to BC at its middle point. Therefore, if we pass any 
 plane through BB' and from B draw a chord to any other 
 point, C, o its intersection with the surface, the perpendicular 
 in this plane to BC, erected at the middle point of BC, will 
 be parallel to BB'. This proves that the section is a boundary- 
 curve, having BB' for axis, and that -the surface can be gener- 
 ated by the revolution of such a boundary-curve around BB'. 
 
 Therefore, BB' may be regarded as axis of the surface. 
 
 A plane passed through an axis of a boundary-surface is 
 called a principal plane. Every principal plane cuts the sur- 
 face in a boundary -curve. Any other plane cuts the surface in 
 a circle ; for the surface may be regarded as a surface of revo- 
 lution having for axis of revolution that axis which is perpen- 
 dicular to the plane. This perpendicular may be called the axis 
 of the circle, and the point where it meets the surface, the pole 
 of the circle. The pole of a circle on a boundary -surf ace is 
 at the same distance from all the points of the circle, distance 
 being measured along boundary -lines on the surface. 
 
 Any two boundary -surfaces can be made to coincide, and a 
 boundary-surface can be moved upon itself, any point to the 
 position of any other point, and any boundary -curve through 
 the first point to the position of any boundary -curve through 
 the second point. We may say that a boundary-surface has 
 a constant curvature, the same for all these surfaces. Figures 
 on a boundary-surface can be moved about or put upon any 
 other boundary-surface without altering their shape or size. 
 
52 NON-EUCLIDEAN GEOMETRY 
 
 We can develop a Geometry on the boundary-surface. By 
 line we mean the boundary-curve in which the surface is cut 
 by a principal plane. The angle between two lines is the 
 same as the diedral angle between the two principal planes 
 which cut out the lines on the surface. 
 
 6. Theorem. Geometry on the boundary -surf ace is the same 
 as the ordinary Euclidean Plane G-eometry. 
 
 Proof. On two boundary-surfaces with the same system of 
 parallel lines for axes corresponding triangles are similar ; that 
 is, corresponding angles are equal, having the same measures 
 as the diedral angles which cut them out, and corresponding 
 lines are proportional by (2). But we can place these figures 
 on the same surface ; therefore, on one boundary-surface we 
 can have similar triangles. Thus, we can diminish the sides 
 of a triangle without altering their ratios or the angles. We 
 can do this indefinitely ; for the ratio of corresponding lines 
 on the two surfaces, being expressed by the function e ax of the 
 distance between them, can be made as large as we please by 
 taking x sufficiently large. If we assume that figures on the 
 boundary-surface become more and more like plane figures 
 when we diminish indefinitely their size, it follows that a 
 triangle on this surface approaches more and more the form 
 of an infinitesimal plane triangle, for which the sum of the 
 angles is two right angles, and the angles and sides have 
 the same relations as in the Euclidean Plane Geometry. All 
 the formulae of Plane Trigonometry with which we are familiar 
 hold, then, for triangles on the boundary-surface. 
 
 On the boundary -surface we have the "hypothesis of the 
 right angle." Eectangles can be formed, and the area of a 
 rectangle is proportional to the product of its base and alti- 
 tude, while the area of a triangle is half of the area of a 
 rectangle having the same base and altitude. 
 
TRIGONOMETRICAL FORMULAE 
 
 53 
 
 An equidistant-surface is a surface generated by the revo- 
 lution of an equidistant-curve about one of its axes. It is 
 the locus of points at a given perpendicular distance from a 
 plane. Any perpendicular to the plane may be regarded as 
 an axis, and the surface is a surface cutting at right angles a 
 system of lines perpendicular to the plane. The surface has 
 a constant curvature, fitting upon itself in any position. 
 
 III. TRIGONOMETRICAL FORMULAE 
 
 1. Let ABC be a plane right triangle. Erect A A' perpen- 
 dicular to its plane and draw BB' and CC' parallel to A A'. 
 Draw a boundary-surface through A, having these lines for 
 axes and forming the boundary-surface triangle AB"C". Also 
 construct the spherical triangle about the point B. 
 
 c6-n(a) 
 
 The angle A is the same in the plane triangle and in the 
 boundary -surf ace triangle. The planes through A A' are per- 
 pendicular to ABC. Hence, the spherical triangle has a right 
 angle at the vertex which lies on c, and EC being perpendic- 
 ular to CA is perpendicular to the plane of CC' and A A 1 . 
 Therefore, the plane BCC' is perpendicular to the plane ACC'j 
 
54 NON-EUCLIDEAN GEOMETRY 
 
 and the diedral whose edge is EC has for plane angle the 
 angle ACC' = H(b). Since the boundary-surface triangle is 
 right-angled at C", the angle B", or what is the same thing, the 
 diedral whose edge is BB', is the complement of the angle A. 
 
 In the spherical triangle the side opposite the right angle 
 is II (a), the two sides about the right angle are II (c) and B, 
 and the opposite angles are II () and 90 A. 
 
 Applying to these quantities the trigonometrical formulae 
 for spherical right triangles, we get at once the relations that 
 connect the sides and angles of plane right triangles. 
 
 Produce to quadrants the two sides about the angle whose 
 value is the complement of A. We form in this way a spher- 
 ical right triangle in which the side opposite the right angle 
 is the complement of II (c), the two sides about the right angle 
 are the complements of II (a) and n (&), and their opposite 
 angles are the complements of B and A. From this triangle 
 we deduce the following rule for passing from the formulae of 
 spherical right triangles to those of plane triangles : 
 
 Interchange the two angles (or the two sides) and everywhere 
 use the complementary function, taking the corresponding 
 angle of parallelism for the sides. 
 
 The formulae for spherical right triangles are 
 
 sin a sin b 
 
 sin A = . sin B = - 
 
 sin c sin c 
 
 tan b tan a 
 
 cos A = cos B 
 
 tan c tan c 
 
 tan a tan b 
 
 tan A = - tan B = 
 
 sin b sm a 
 
 cos B cos A 
 
 sm A = sin B = 
 
 cos b , . cos a 
 
 cos c = cos a cos b. 
 cos c = cot A cot B. 
 
TRIGONOMETRICAL FORMULAE 
 
 55 
 
 From these, by the rule given on the previous page, we 
 derive the following formulae for plane right triangles : 
 
 cosB = 
 
 cos n (a) 
 
 sin B = 
 
 cot n ( 
 
 cot II (a) 
 cot B = - 
 
 cos II (b) 
 
 sin A 
 cosB = 
 
 cos A = 
 
 sin A = 
 
 cot A = 
 
 cos A = 
 
 cos n (b) 
 
 cos n (c) 
 
 cot II (a) 
 
 sin n (b) sin n (a) 
 
 sin II (c) = sin II (a) sin II (b). 
 sin II (c) = tan A tan B.* 
 
 We can obtain the formulae for oblique plane triangles by 
 dropping a perpendicular from one vertex upon the opposite 
 side, thus forming two right triangles. 
 
 2. Take the relation 
 
 sin II (a) = 
 
 sin B 
 cos A 
 
 Let p, q, and r be the sides of the triangle AB"C" of our 
 last demonstration and p', y', and r the corresponding sides 
 
 * We can arrange the parts of a right triangle so as to apply Napier's 
 rules ; namely, the arrangement would be 
 
 co-b 
 
56 
 
 NON-EUCLIDEAN GEOMETRY 
 
 of the triangle formed in the same way on a boundary-surface 
 tangent to the plane ABC at B. 
 
 g 
 COS A = * 
 
 Now <7 and q' are corresponding arcs 
 on two boundary -curves which have the 
 same set of parallel lines as axes, and 
 their distance apart, x, is the distance 
 from a boundary-curve of the extremity 
 of a tangent of arbitrary length, a. Thus, 
 we have for corresponding arcs 
 
 s' 
 
 - = sin n (a). 
 
 3, To MN, a given straight line, erect a perpendicular at a 
 point, O, and on this perpendicular lay off OA = y below MN, 
 and OB and BP each equal to x above MN 9 x and y being any 
 arbitrary lengths. At P draw PR perpendicular to OP and 
 
THE ANGLE OF PARALLELISM 
 
 57 
 
 extending towards the left, and through B draw EF making 
 with OP an angle II (z), and therefore parallel on one side to 
 ON and on the other side to PR. Finally, draw AK and AH, 
 the two parallels to EF through A. 
 
 At the point .4 we have four angles of parallelism : 
 CAK=CAH = U(AC), 
 
 Therefore, U(y) = U (A C) + BA C, 
 
 and HO + 2z) = n(.4C)- BAG. 
 
 Now in the right triangle A !',< 
 
 cos n (y + a-) = 
 
 _ cos n (A C) 
 
 cosBAC 
 
 or 
 
 1 cos II (y 4- x) _ cos BA C cos II (A C) 
 1 4- cos n (y 4- x) ~~ cos^^C 4- cos II (4 C) 
 
 cos [n (^ C) + 5-4 C] cos 
 
58 NON-EUCLIDEAN GEOMETRY 
 
 whence, 
 
 x) = tan $U (y) tan Jn(y + 
 
 tan -J- II (aj) is then a function of #, say f(x) 9 satisfying the 
 condition 
 
 
 and putting successively in this equation y + x, y -\-2x, etc., 
 for y, we may add 
 
 no;) 
 
 
 II (0) = and tan -J- II (0) = 1 ; therefore, putting y = in 
 
 L 
 
 the first and last of all these fractions, we have 
 
 or 
 
 This equation is characteristic of the exponential function.* 
 II (x) being an acute angle, tan II (a;) < 1 ; therefore, we may 
 write /(I) = e~ a> , so that /(a;) = e~ a>x . a' depends on the unit 
 of measure ; we will take the unit so that a' = 1. Finally, 
 since II ( x) = TT II (x), 
 
 tan H (- x) = cot n (a:) = [tan ^ II (x)]- ^ 
 That is, for all real values of x 
 
 tan -i- II (ic) = e"*, 
 
 * See footnote, p. 45. 
 
THE ANGLE OF PARALLELISM 59 
 
 1 COS II (x) 
 
 or : . ^ / = cos ix + i sm ix* 
 
 sm II (x) 
 
 * i stands for V 1. The best way to get the relations between the 
 exponential and trigonometrical functions is by their developments in 
 series : X 2 ~ n 
 
 cos* = l-| + |-. .. + (-!> 
 sinx = x-2L + ^-, + ( 
 
 These series are convergent for all values of x. 
 Putting ix for x, we have 
 
 i.e. , e* cos x -j- i sin x. 
 
 Also e~ IX = cos x i sin x. 
 
 .-. cosx = $(e f * 
 
 Again, putting ix for x, we have 
 
 e x = cos ix i sin ix, 
 c~ x = cos ix + i sin ix ; 
 and cos ix = (e* + e--'), 
 
 sin ix = (e 1 e- x ). 
 
 2i l 
 
 cos ix = 1 + ^-, + ?-, -f 
 
 For real values of x, cos ix and are real and positive, and vary 
 from 1 to co as x varies from to co. 
 
 In the equation cos 2 ix + sin 2 ix = 1, the fii'st term is real and positive 
 for real values of x, the second term is real and negative ; therefore, sinix 
 is in absolute value less than cos ix, and tan ix is in absolute value less 
 than 1. tan ix varies in absolute value from to 1 as x varies from to w. 
 
60 NON-EUCLIDEAN GEOMETRY 
 
 Changing the sign of x, we have 
 
 1 + cos H (35) 
 
 : . v = cos ix i sin ix, 
 
 sin II (x) 
 
 and, adding and subtracting, 
 
 1 
 
 _, . = cos ix, 
 sin n (a) 
 
 cot II (x) = i sin zee. 
 
 The nature of the angle of parallelism is, therefore, expressed 
 by the equations 
 
 sin H(x) = > 
 
 v ' cos tx 
 
 tan n (x) = . > 
 sin ix 
 
 taxiix 
 
 cosllfo) = : 
 i 
 
 4, Substituting in the formulae of plane right triangles, we 
 find that they reduce to those of spherical right triangles with 
 ia, ib, and ic for a, b, and c, respectively. The formulae 'of 
 oblique triangles are obtained from those of right triangles 
 in the same way as on the sphere, and thus all the formulae 
 of Plane Trigonometry are obtained from those of Spherical 
 Trigonometry simply by making this change. 
 
 As fundamental formulae for oblique triangles we write 
 
 sin A _ sinJS _ sin C 
 sin ia sin ib sin ic 
 
 cos ia = cos ib cos ic + sin ib sin ic cos A, 
 cos A = cos B cos C + sin B sin C cos ia. 
 In the notation of the Il-f unction, these are 
 
 sin A tan n (a) = sin B tan n (b) = sin C tan n (c), 
 
INFINITESIMAL TRIANGLES 61 
 
 sinn (ft) sin n( C ) = _ cos cog cog ^ 
 
 sm n (a) 
 
 , sin B sin C 
 
 COS yl = COS B COS C H : . . 
 
 sin II (a) 
 
 5. Since for very small values of x we have approximately 
 
 sin ix = ixj 
 
 x* 
 cos ix = 1 4- > 
 
 tan ix = to-, 
 our formulae for infinitesimal triangles reduce to 
 
 sin A _ sin _ sin C 
 a b t c 
 
 a 2 = b* + c* 2bc cos A, 
 cos .4 = cos (B + C). 
 
 6. Triangles on an equidistant-surface are similar to their 
 projections on the base plane ; that is, they have the same 
 angles and their sides are . proportional. Thus the formulae 
 of Plane Trigonometry hold for any equidistant-surface if 
 with the letters representing the sides we put, besides i, a 
 constant factor depending on the distance of the surface from 
 the plane. 
 
CHAPTER III 
 
 THE ELLIPTIC GEOMETRY 
 
 IN the hypothesis of the obtuse angle a straight line is 
 of finite length and returns into itself. This length is the 
 same for all lines, since any two lines can be made to coin- 
 cide. Two straight lines always intersect, and two lines 
 perpendicular to a third intersect at a point whose distance 
 from the third on either line is half the entire length of a 
 straight line. 
 
 1. A straight line does not divide the plane. Starting from 
 the point of intersection of two lines and passing along one of 
 them a certain finite distance, we come to the intersection 
 point again without having crossed the other line. Thus, we 
 can pass from one side of the line to the other without having 
 crossed it. 
 
 There is one point through which pass all the perpendiculars 
 to a given line. It is called the pole of that line, and the line 
 is its polar. Its distance from the line is half the entire 
 length of a straight line, and the line is the locus of points 
 at this distance from its pole. Therefore, if the pole of one 
 
 62 
 
POLES AND POLARS 63 
 
 line lies on another, the pole of the second lies on the first, and 
 the intersection of two lines is the pole of the line joining 
 their poles. 
 
 The locus of points at a given distance from a given line is 
 a circle having its centre at the pole of the line. The straight 
 line is a limiting form of a circle when the radius becomes 
 equal to half the entire length of a line. 
 
 We can draw three lines, each perpendicular to the other 
 two, forming a trirectangular triangle. Tt is also a self -polar 
 triangle ; each vertex is the pole of the opposite side. 
 
 2, All the perpendiculars to a plane in space meet at a 
 point which is the pole of the plane. It is the centre of 
 a system of spheres of which the plane is a limiting form 
 when the radius becomes equal to half the entire length of a 
 straight line. 
 
 Figures on a plane can be projected from similar figures on 
 any sphere which has the pole of the plane for centre. That 
 is, they have equal angles and corresponding sides in a con- 
 stant ratio that depends only on the radius of the sphere. 
 Two corresponding angles are equal, because they are the same 
 as the diedral angles formed by the two planes through the 
 centre of the sphere which cut the sphere and the plane in 
 the sides of the angles. Corresponding lines are proportional ; 
 for if two arcs on the sphere are equal, their projections on the 
 plane are equal ; and that, in general, two arcs have the same 
 ratio as their projections on the plane is proved, first when 
 they are commensurable, and by the method of limits when 
 they are incommensurable. 
 
 Geometry on a plane is, therefore, like Spherical Geometry, 
 but the plane corresponds to only half a sphere, just as the 
 diameters of a sphere correspond to the points of half the 
 surface. Indeed, the points and straight lines of a plane 
 correspond exactly to the lines and planes through a point, 
 
64 NON-EUCLIDEAN GEOMETRY 
 
 but we can realize the correspondence better that compares 
 the plane with the surface of a sphere. If we can imagine 
 that the points on the boundary of a hemisphere at opposite 
 extremities of diameters are coincident, the hemisphere will 
 correspond to the elliptic plane. There is no particular line 
 of the plane that plays the part of boundary. All lines of 
 the plane are alike ; the plane is unbounded, but not infinite 
 in extent. 
 
 The entire straight line corresponds to a semicircle. We 
 will take such a unit for measuring length that the entire 
 length of a line shall be TT ; the formulae of Spherical Trigo- 
 nometry will then apply without change to our plane. Dis- 
 tances on a line will then have the same measure as the angles 
 which they subtend at the pole of the line, and the angle 
 between two lines will be equal to the distance between their 
 poles. The distance from any point to its polar, half the 
 entire length of a straight line, may then be called a quadrant. 
 
 We can form a self -polar tetraedron by taking three mutually 
 perpendicular planes and the plane which has their intersec- 
 tion for pole. The vertices of this tetraedron are the poles of 
 the opposite faces. At each vertex is a trirectangular triedral, 
 and each face is a trirectangular triangle. 
 
 3. Theorem. All the planes perpendicular to a fixed line 
 intersect in another fixed line, catted its polar or conjugate. 
 The relation is reciprocal, and all the points of either line 
 are at a quadrant's distance from all the points of the other. 
 
 Proof. Let the two planes perpendicular to the line A B at 
 H and K intersect in CD. Pass a plane through AB and R, 
 any point of CD. This plane will intersect the two given 
 planes in HR and KR. HR and KR are perpendicular to AB ; 
 therefore, R is at a quadrant's distance from H and K. R is 
 then the pole of AB in the plane determined by AB and R, 
 
POLAR LINES 
 
 65 
 
 and is at a quadrant's distance from every point of AB. But 
 R is any point of CD ; therefore, any point of either line is at 
 a quadrant's distance from each point of the other line, and a 
 point which is at a quadrant's distance from one line lies in 
 
 the other line. Again, any point, //, of AB, being at a quad- 
 rant's distance from all the points of CD, is the pole of CD in 
 the plane determined by it and CD. Thus, HR and KR are 
 both perpendicular to CD, and the plane determined by AB 
 and R is perpendicular to CD. 
 
 The opposite edges of a self-polar tetraedron are polar lines. 
 
 All the lines which intersect a given line at right angles 
 intersect its polar at right angles. Therefore, the distances 
 of any point from two polar lines are measured on the same 
 straight line and are together equal to a quadrant. Two 
 points which are equidistant from one line are equidistant 
 from its polar. 
 
 The locus of points which are at a given distance from a 
 fixed line is a surface of revolution having both this line and 
 its polar as axes. We may call it a surface of double revolu- 
 tion. The parallel circles about one axis are meridian curves 
 for the other axis. If a solid body, or, we may say, all space, 
 move along a straight line without rotating about it, it will 
 rotate about the conjugate line as an axis without sliding 
 
66 
 
 NON-EUCLIDEAN GEOMETRY 
 
 along it. A motion along a straight line combined with a 
 rotation about it is called a screw motion. A screw motion 
 may then be described as a rotation about each of two con- 
 jugate lines or as a sliding along each of two conjugate lines. 
 
 4. Theorem. In the elliptic geometry there are lines not 
 in the same plane which have an infinite number of common 
 perpendiculars and are everywhere equidistant. 
 
 Given any two lines in the same plane and their common per- 
 pendicular. If we go out on these lines in either direction from 
 the perpendicular, they approach each other. Now revolve 
 one of them about this perpendicular so that they are no longer 
 in the same plane. After a certain amount of rotation the lines 
 will have an infinite number of common perpendiculars and be 
 equidistant throughout their entire length. 
 
 Proof. Let p be the length of the common perpendicular 
 AC, and take points B and D on the two lines on the same 
 side of this perpendicular at a distance, a. 
 
 BD<p, but if CD revolve about AC, BD will become longer 
 than p by the time CD is revolved through a right angle ; for 
 BCD will then be a right triangle, with BD for hypothenuse 
 and BC, the hypothenuse of the triangle ABC, for one of 
 its sides, so that we shall have BD>BC and BO AC. 
 
PARALLEL LINES 67 
 
 Suppose, when CD has revolved through an angle, 6, BD 
 becomes equal to^? and takes the position BD'. The triangles 
 ABC and D'BC are equal, having corresponding sides equal. 
 Therefore, BD' is perpendicular to CD'. BD' is also perpen- 
 dicular to BA ; for if we take the diedral A-BC-D' and place 
 it upon itself so that the positions of B and C shall be inter- 
 changed, A will fall on the position of D', and D' on the 
 position of .4, and the angle D'BA must equal the angle ACD'. 
 Therefore, Hl>' as well as CA is a common perpendicular to 
 the lines AB and CD'. 
 
 Now at the point C we have a triedral whose three edges are 
 CB, CD, and CD'. Moreover, the diedral along the edge CD 
 is a right diedral ; therefore, the three face angles of the 
 triedral satisfy the same relations as do the three sides of a 
 spherical right triangle ; namely, 
 
 cos BCD' = cos BCD cos DC It'. 
 But /;r /) = - ACR and BCD' = ABC. 
 
 Hence, this relation may be written 
 
 c-oaABC = sin A < 'B cos B. 
 Again, in the right triangle ABC 
 
 cos ABC 
 
 . ' . COS B = COS y>, 
 7T 
 
 or, since B and p are less than ? 
 
 9= p. 
 
 The angle 0, therefore, does not depend upon a. If we take 
 any two lines in a plane and turn one about their common 
 perpendicular through an angle equal in measure to the length 
 
68 NON-EUCLIDEAN GEOMETRY 
 
 of that perpendicular, the two lines will then be everywhere 
 equidistant. 
 
 As we have no parallel lines in the ordinary sense in this 
 Geometry, the name parallel has been applied to lines of this 
 kind. They have many properties of the parallel lines of 
 Euclidean Geometry. 
 
 Through any point two lines can be drawn parallel to a 
 given line. These are of two kinds, sometimes distinguished 
 as right-wound and left-wound. They lie entirely on a surface 
 of double revolution, having the given line as axis. The sur- 
 face is, therefore, a ruled surface and has on it two sets of 
 rectilinear generators like the hyperboloid of one sheet. 
 
CHAPTER IV 
 ANALYTIC NON-EUCLIDEAN GEOMETRY 
 
 WE shall use the ordinary polar coordinates, p and 0, and for 
 the rectangular coordinates, x and y, of a point, we shall use 
 the intercepts on the axes made by perpendiculars through the 
 point to the axes. The formulae depend upon the trigonomet- 
 rical relations, and in our two Geometries differ only in the 
 use of the imaginary factor i with lengths of lines. 
 
 I. HYPERBOLIC ANALYTIC GEOMETRY 
 
 
 1. The relations between polar and rectangular coordinates : 
 The angles at the origin which the radius vector makes with 
 the axes are complementary. From the two right triangles 
 
 we have 
 
 tan ix = cos tan ip, 
 
 tan iy = sin 6 tan ip. 
 Therefore, tan 2 ip = tan 2 ix + tan 2 iy, 
 
 tan iy 
 
 tan ix 
 
 69 
 
 tan 6 = 
 
70 
 
 NON-EUCLIDEAN GEOMETRY 
 
 xy 
 
 2. The distance, 8, between two points : 
 
 cos i& = cos ip cos ip' + sin ip sin ip' cos (0' 6). 
 
 8 and one of the points being fixed, this may be regarded as 
 the polar equation of a circle. 
 
 \ 
 
 3, The equation of a line : 
 
 Let p be the length of the perpendicular from the origin 
 upon the line, and a the angle which the perpendicular makes 
 
HYPERBOLIC ANALYTIC GEOMETRY 
 
 71 
 
 with the axis of x. From the right triangle formed with this 
 perpendicular and p we have 
 
 tan ip cos (0 a) = tan ip. 
 
 This is the polar equation of the line. We get the equation 
 in x and y by expanding and substituting ; namely, 
 
 cos a tan ix + sin a tan iy = tan ip. 
 The equation a tan ix + I tan iy = i 
 represents a line for which 
 
 tan a i> 
 
 Now, for real values of p, tan 2 ip < 1 (see footnote, p. 59). 
 The line is therefore real if a and b are real, and if 
 
 \ 
 
 \ 
 
 4. The distance, 8, of a point from a line : 
 
 Let the radius vector to the point intersect the line at A, 
 and let p l be the radius vector to A. We have two right 
 
72 
 
 NON-EUCLIDEAN GEOMETRY 
 
 triangles with equal angles at A, and from the expressions 
 for the sines of these angles we get the equation 
 
 sn 
 
 sin i(p pi) 
 
 This equation holds for all points, xy, of the plane, 8 being 
 negative when the point is on the same side of the line as the 
 origin, and zero when the point is on the line. 
 
 Now, 
 
 . sintp . . . . 
 
 sin i& = r- sin ip sin ip cos ip. 
 
 tanip 
 
 = //I ' 
 
 cos (0 a) 
 
 sin ip = sin ip cos ip cos (0 a), 
 
 tan ip 1 
 and sin i& = cos ip cos ip [tan ip cos (0 a) tan ip~\. 
 
 $ being fixed, this may be regarded as the polar equation of 
 an equidistant-curve. 
 
 \ 
 
 \ 
 
 5. The angle between two lines : 
 
 <f> being the angle which a line makes with the radius vector 
 at any point, we have 
 
THE ANGLE BETWEEN TWO LINES 73 
 
 cos < = cos ip sin (0 a), 
 . _ sin ip 
 sint'p 
 
 For two lines intersecting at this point, 
 
 sin ii sin ip 
 
 sm sin <>, = 
 
 Bin* ip 
 
 sin /?! sin 
 
 = sm *P! sm ip 2 -t 
 
 Now, from the equation of the line 
 
 sin ip l 
 
 * = cos ii cos (0 i), 
 
 tantp 
 
 sin ip a 
 
 = cos ip z cos (0 or 2 ) ; 
 
 tan ip 
 
 so that sin fa sin < 2 = sin ip l sin ///., 
 
 + cos ipi cos tp 2 cos (0 a i) cos (0 or 2 ). 
 
 Again, cos ^ cos fa = cos i^ cos ip 9 sin (0 <*i) sin (0 or 2 ). 
 Adding these equations, we have 
 
 cos (< 2 <i) == s i n '(PI sin tp a + cos tpi cos tp 2 cos (o- 2 a^). 
 Two lines are perpendicular if 
 
 cos (a z a^ + tan ip l tan ip 2 = 0. 
 The lines a tan to; + ^ tan iy = t, 
 
 a' tan ix H- 6' tan iy = i 
 are perpendicular if aa' + bb' = 1. 
 
 6. The equation of a circle in x and y : 
 
 sin ip cos = cos ip tan taj, 
 sin tp sin = cos ip tan iy ; 
 1 1 
 
 also, cos ip = 
 
 4- tan 2 ip VI + tan 2 ix + tan 2 i?/ 
 
74 
 
 NON-EUCLIDEAN GEOMETRY 
 
 The equation of a circle may, therefore, be written 
 (1 4- tan 2 ix + tan 2 iy) (1 4- tan 2 ix' 4- tan 2 iy') cos 2 i8 
 
 = (1 4- tan ix tan tic' + tan iy tan ty') 2 . 
 
 7. The equation of a boundary-curve : 
 
 Let the axis of the boundary-curve which passes through 
 the origin make an angle, a, with the axis of x 9 and let the 
 point where the boundary -curve cuts this axis be at a distance, 
 k, from the origin, positive if the origin is on the convex side 
 of the curve, negative if the origin is on the concave side of 
 the curve. The boundary -curve is the limiting position of a 
 circle whose centre, on this axis, moves off indefinitely. 
 
 p' being the radius vector to the centre, the radius of the 
 circle is p' k, and its equation may be written 
 
 cos i(p' k) = cos ip cos ip' + sin ip sin ip' cos (6 ), 
 or, expanding and dividing by cos ip' } 
 
 cos ik + tan ip' sin ik = cos ip 4- sin ip tan ip 1 cos (0 a). 
 
THE EQUATION OF A BOUNDARY-CURVE 
 
 75 
 
 Now, let p' increase indefinitely, tan ip 1 tends to the limit i, 
 so that the limit of the first member of the equation is 
 
 cos ik + i sin ik, or e~ k , 
 and the polar equation of the curve is 
 
 e~ k = cos ip [1 4- / tan ip cos (0 nr)] ; 
 or, in x y coordinates, 
 
 (1 + tan 2 ix 4- tan 2 iy) e~ zk 
 
 = (1 4 i cos a tan i.r 4 i sin a tan ty) 2 . 
 
 Let & be negative and equal, say, to b, and let a = ; 
 also, let a be the ordinate of the point A where the curve 
 cuts the axis of y. 
 
 Substituting in the equation, we find 
 
 = cos a. 
 
 Through A draw a line parallel to the axis of x, and, there- 
 fore, making an angle, II (a), with the axis of y. If we draw 
 a boundary-curve through the origin having the same set of 
 parallel lines for axes, so that the two boundary-curves cut 
 
76 NON-EUCLIDEAN GEOMETRY 
 
 off a distance, b, on these axes, we know that the ratio of 
 corresponding arcs is 
 
 s' 1 
 
 - = sinll(a) = -> (See p. 56.) 
 
 s v ' cosia 
 
 therefore, - = e~ b . (See p. 45.) 
 
 8. The equation of an equidistant-curve : 
 
 The polar equation of (4) reduced to an equation in x and y 
 takes the form 
 
 (1 + tan 2 ix + tan 2 iy) sin 2 iS 
 
 = cos 2 ip (cos a tan ix -\- sin a tan iy tan ip) 2 . 
 
 9. Comparison of the three equations : 
 The equation 
 
 (1 + tan 2 ix + tan 2 iy) c 2 = (i a tan ix b tan iyf 
 
 represents a circle, a boundary-curve, or an equidistant-curve, 
 according as a 2 + b 2 < 1, =1, > 1, respectively. 
 
 c 
 
 la 
 
 C 
 
 10. Differential formulae : 
 
 Suppose we have an isosceles triangle in which the angle A 
 at the vertex diminishes indefinitely. In the formula 
 
 sin A __ sinC 
 sin ia sin ic 
 
 we may put for sin A, sin ia, sin C ; 
 
 A, ia, 1, 
 
 respectively. Therefore, 
 (I.) 
 
DIFFERENTIAL FORMULA 77 
 
 Corollary. In a circle of radius r, the ratio of any arc to the 
 angle subtended at the centre is sin ir. 
 
 Again, in the right triangle ABC, let the hypothenuse c 
 revolve about the vertex A. Differentiating the equation 
 
 cos B 
 
 sm.4 = -> 
 
 cos tl> 
 
 where b is constant, we have 
 
 siuBdB 
 
 cos Ad A = - 
 
 COS ll> 
 
 cos^l 
 
 But sm B = > 
 
 cosia 
 
 . ' . dB = cos ia cos ib dA , 
 or (II.) dB = cos ic dA. 
 
 Now, using polar coordinates, we have an infinitesimal right 
 triangle whose hypothenuse, ds, makes an angle, say <, with 
 the radius vector (see figure on page 78). The two sides about 
 
 the right angle are dp and r-^ dO ; 
 
 therefore, ds* = dp* sin' 2 ip dO*, 
 
 sin ip dO 
 
 tan</> = T-^-T-' 
 t dp 
 
78 
 
 NON-EUCLIDEAN GEOMETRY 
 
 For two arcs cutting at right angles, let d' denote differ- 
 entiation along the second arc : 
 
 sin ip dO _ I d^p_ 
 
 i dp 
 
 sinip d'O 
 
 or 
 
 dp d'p 
 
 == 
 
 sin 2 ip. 
 
 11. Area: 
 It equals 
 
 We will consider only the case where the origin is within 
 the area to be computed and where each radius vector meets 
 the bounding curve once, and only once. 
 
 Integrating with respect to p, from p = 0, we have 
 
 or 
 
 X2JT 
 (cos ip 1) d&, 
 . 
 
 /27T 
 
 I cos ip dO - 2 TT. 
 
 * The unit of area being so chosen that the area of an infinitesimal 
 rectangle may be expressed as the product of its base and altitude. 
 
AREA 
 
 79 
 
 Suppose P and P' are two " consecutive " points on the 
 curve, PM and P'M' the tangents at these points, and < the 
 
 angle which the tangent makes with the radius vector. The 
 angle MP'M' indicates the amount of turning or rotation at 
 these points as we go around the curve. 
 Now, by (IT.), 
 
 M/>'M' = </< + cos ipdO. 
 
 In going around the curve, < may vary but finally returns 
 to its original value. That is, for our curve 
 
 = 0, 
 
 COB <pdO. 
 
 and the amount of rotation is 
 
 s: 
 
 Hence, the area is equal to the excess over four right angles 
 in the amount of rotation as we go around the curve. This 
 theorem can be extended to any finite area. 
 
 12. A modified system of coordinates : 
 
 Our equations take simple forms if we write lit for tan ix, 
 w for tan ///, lr for tan ip, and so on for all lengths of lines. 
 
80 
 
 NON-EUCLIDEAN GEOMETRY 
 
 Thus, we have u 2 4- v 2 = r 2 .* 
 
 The equation of a line is 
 
 au + bv = 1, 
 and the equation 
 
 (1 - u 2 - v 2 ) c 2 = (l- au - bv) 2 
 
 represents a circle, a boundary-curve, or an equidistant-curve, 
 according as a 2 + b 2 < 1, = 1, > 1, respectively. 
 
 II. ELLIPTIC ANALYTIC GEOMETRY 
 
 The Elliptic Analytic Geometry may be developed just as 
 we have developed the Hyperbolic Analytic Geometry, and the 
 formulae are the same with the omission of the factor i. But 
 these formulae are also very easily obtained from the relation 
 of line and pole, and we shall produce them in this way. 
 
 The formulae of Elliptic Plane Analytic Geometry may be 
 applied to a sphere in any of our three Geometries. 
 
 1. The relations between polar and rectangular coordinates : 
 
 tan x = cos tan p, tan y = sin 6 tan p j 
 
 * If we draw a quadrilateral with three 
 right angles and the diagonal to the acute 
 angle, and use a, 6, and c in the same way that 
 w, v, and r are used above, the five parts 
 lettered in the figure have the relations of a 
 
 right triangle in the Euclidean Geometry ; e.g., 
 
 = , etc. 
 
ELLIPTIC ANALYTIC GEOMETRY 
 
 therefore, tan 2 p = tan 2 x 4 tan' 2 y, 
 
 81 
 
 xy 
 
 2. The distance, 8, between two points : 
 
 cos 8 = cos p cos p' 4 sin p sin p 1 cos (0' 0). 
 This may be regarded as the polar equation of a circle of 
 radius 8, p' and 6' being the polar coordinates of the centre. 
 Now, sin p cos = cos p tan #, 
 
 sin p sin = cos p tan y ; 
 1 1 
 
 also, 
 
 COSp = 
 
 1 + tan 2 p VT+ tan 2 x 4 tan 2 y 
 The equation of a circle in rectangular coordinates may, there- 
 fore, be written 
 
 (1 + tan 2 x 4 tan 2 y) (1 + tan 2 x' + tan 2 y') cos 2 8 
 
 = (14- tan x tan sc' 4- tan y tan y') 2 . 
 
 * The line which has the origin for pole forms with the coordinate axes 
 a trirectangular triangle, and z, y, and 6 may be regarded as representing 
 the directions of the given point from its three vertices. 
 
 On a sphere, if we take as origin the pole of the equator, p and are 
 colatitude and longitude, x and y, one with its sign changed, are the 
 " bearings " of the point from two points 90 apart on the equator. 
 
82 
 
 NON-EUCLIDEAN GEOMETRY 
 
 3, The equation of a line : 
 
 When 8 = > the circle becomes a straight line. For this 
 we have, therefore, the equation 
 
 tan x tan x' + tan y tan ij + 1 = 0. 
 x'y' is the pole of the line. 
 From the equation 
 
 tan p cos (6 a) = tan_/?, 
 or cos a tan x + sin a tan y = 
 
 we find 
 
 cos a 
 
 tan x' = 9 
 
 tan y = 
 
 sn 
 
 as can be shown geometrically, the polar coordinates of this 
 
 point being TT 
 
 P T TT oc. 
 
 The equation a tan x + b tan y + 1 = 
 represents a real line for any real values of a and b. 
 
 4. The distance, 8, of a point from a straight line : 
 
 This is the complement of the distance between the point 
 and the pole of the* 1 line ; it is expressed by the equation 
 
DIFFERENTIAL FOBMULJE 
 
 83 
 
 sin 8 = cos p sinp + sin p eosp cos (0 a) 
 = cos p cos jt? [tan p cos (0 a) tan 7;]. 
 
 5. The angle, <}>, between two lines : 
 
 This is equal to the distance between their poles ; therefore, 
 
 cos <j> = sin j9 sinp' 4- cos p cos p 1 cos (a 1 ). 
 The two lines a tan x + I tan y -f 1 = 0, 
 a' tan a; + ft'tany + 1 = 
 are perpendicular if aa' -f bb 1 4- 1 = 0. 
 
 6, Differential formulae : 
 The formula 
 
 sn 
 
 sn 
 
 sin a sin c 
 
 becomes, when A diminishes indefinitely, 
 (I.) a = sine- A. 
 
 Corollary. In a circle of radius r, the ratio of any arc to the 
 *> (bt ended at the centre is sin r. 
 
 From the right triangle ABC, if If remain fixed, we get, by 
 differentiating the equation 
 
 cos B 
 
 sin ^4 = 
 
 cosb 
 
84 NON-EUCLIDEAX GEOMETRY 
 
 (II.) dB = - cose dA. 
 
 Thus, we have for differential formulae in polar coordinates 
 
 tan < = sin p > * 
 dp 
 
 * If <f> is constant, as in the logarithmic spiral of Euclidean Geometry, 
 we can integrate this equation ; namely, 
 
 tan = dd. 
 sinp 
 
 .-. tan log tan - = + c, 
 
 Writing <?' for etan<^ this is tan - = c' e t&n< t>. 
 
 2 
 
 On the sphere this is the curve called the loxodrome. 
 
A MODIFIED SYSTEM OF COORDINATES 85 
 
 and for two arcs cutting at right angles 
 dp d'p 
 
 9*i m -*** 
 
 The formula for area is * 
 
 J J sin p dp <16. 
 
 We integrate first with respect to p, and if the area contains 
 the origin and each radius vector meets the curve once, and 
 only once, our expression becomes 
 
 /27T 
 
 2-7T- I cospde. 
 
 */0 
 
 The entire rotation in going around the curve is found as 
 on page 79, and is 
 
 X2ff 
 cos p dO. 
 _ 
 
 Thus the area is equal to the amount by which this rotation 
 is less than four right angles. 
 
 For example, the area of a circle of radius p is 2?r (1 cosp), 
 and the amount of turning in going around it is 2?r cos p. The 
 area of the entire plane is 2 TT. 
 
 7. A modified system of coordinates : 
 
 Writing u for tan x, v for tan //, r for tan p, etc., we have 
 
 u* + v 2 = r 2 .f 
 
 The equation of a line then becomes 
 au + b + l= 0, 
 and the equation of a circle 
 
 (1 + w 2 + ?; 2 ) c 2 = (1 + au + bv) 2 . 
 
 * The unit of area being properly chosen. 
 t The footnote on page 80 applies here also. 
 
86 
 
 NON-EUCLIDEAN GEOMETRY 
 
 III. ELLIPTIC SOLID ANALYTIC GEOMETRY 
 
 We will develop far enough to get the equation of the 
 surface of double revolution. 
 
 / 
 
 
 T 
 
 A 
 
 A 
 
 O x 
 
 
 / 
 
 1, Coordinates, lines, and planes : 
 
 Draw three planes through the point perpendicular to the 
 axes. For coordinates x, y, z, we take the intercepts which 
 these planes make on the axes. 
 
 The lines of intersection of these three planes are perpen- 
 dicular to the coordinate planes (Chap. I, II, 16 and 17) ; in 
 fact, all the face angles in the figure are right angles except 
 those at P and the three angles BA'C, CB'A, and AC'B, which 
 are obtuse angles. 
 
 Let p be the radius vector to the point P, and a, ft, and y 
 the three angles which it makes with the three axes. Then 
 
 cos 2 a -h cos 2 ft + cos 2 y = 1, 
 
 cos a = 
 
 tana? 
 
 tan/o 
 tan- x -f tan 2 y -f tan 2 z = tan 2 p. 
 
 etc. ; 
 
ELLIPTIC SOLID ANALYTIC GEOMETRY 87 
 
 For the angle between two lines intersecting at the origin 
 cos 6 = cos a cos a' -f cos ft cos ft' 4- cos y cos y'. 
 
 The angle subtended at the origin by the two points xyz 
 and x'y'z' is given by the equation 
 
 tan x tan x 1 -f- tan ?/ tan ?/' -f- tan z tan ' 
 cos0 = - -* - 
 
 tanp tan/a 
 
 For the distance between two points 
 
 cos 8 = cos p cos p' 4- sin p sin p' cos 0. 
 
 7T 
 
 This gives us the equation of a sphere, and for 8 = the 
 
 A 
 
 equation of a plane. The latter in rectangular coordinates is 
 tan x tan x' + tan y tan // 4- tan z tan 2' -f 1 = 0. 
 
 Let ^> be the length of the perpendicular from the origin 
 upon the plane, and a, ft, y the angles which this perpendicular 
 makes with the axes. Then we have for its pole 
 
 COS a 
 
 tana-' = tan p' cos a = > etc. : 
 tan jo 
 
 hence, the equation of the plane may be written 
 
 cos a tan x 4- cos ft tan // 4- cos y tan z tan p. 
 
 2. The surface of double revolution : 
 
 Take one of its axes for the axis of 2, suppose k the distance 
 of the surface from this axis, and let 6 denote the angle which 
 the plane through the point P and the axis of z makes with 
 the plane of xz. We may call z and latitude and longitude. 
 
 Produce OA and CB. They will meet at a distance, > from 
 
 Ju 
 
 the axis of z in a point, 0', on the other axis of the surface, and 
 there form an angle that is equal in measure to z. 
 
88 
 
 NON-EUCLIDEAN GEOMETRY 
 
 From the right triangle O'AB 
 
 But 
 and 
 
 Therefore, 
 or 
 
 Similarly, 
 
 tan O'A 
 
 cos z = 
 
 tan O'B 
 
 tan O'A = cot x, 
 tan O'B = cot CB = 
 
 cotk 
 
 cosz = 
 
 tana? 
 
 tan y 
 
 cos 6 
 tan k cos 6 
 
 tana? 
 tan k cos 6 
 
 COS 
 
 tan k sin 
 
 cosz 
 
 Squaring and adding, we have for the equation of the surface 
 tan 2 x + tan 2 y = tan 2 k sec 2 . 
 
 J3 
 
 For the length of the chord joining two points on the sur- 
 face, we have 
 
 cos 8 = cos p cos p' (1 -f- tan x tan x' + tan y tan y 1 -f- tan 2 tan '). 
 Now, tan 2 p = tan 2 & sec 2 + tan 2 z ; 
 
 therefore, sec 2 p sec 2 & sec 2 *, 
 
 or cos p = cos k cos *. 
 
THE SURFACE OF DOUBLE REVOLUTION 89 
 
 That is, in terms of 2, z', 6, and 0', we have 
 
 cos 8 = cos 2 k cos (' )+ sin 2 k cos (0' 0). 
 From this we can get an expression for ds, the differential 
 element of length on the surface : 
 
 cos ds = cos 2 k cos dz + sin 2 k cos dO, 
 
 ds 2 
 or, since cos ds 1 > etc., 
 
 a 
 
 ds* = cos 2 k dz* + sin 2 k dO*. 
 
 z and 6 are proportional to the distances measured along 
 the two systems of circles. These circles cut at right angles, 
 and may be used to give us a system of rectangular coordinates 
 on the surface. The actual lengths along these two systems of 
 circles are Osink and 2 cos A: (see Cor. p. 83). If, therefore, 
 we write 
 
 a = sin k, ft = z cos k, 
 
 we shall have a rectangular system on the surface where the 
 coordinates are the distances measured along these two systems 
 of circles which cut at right angles. 
 The formula now becomes 
 
 ds 2 = da 2 + dp. 
 
 An equation of the first degree in a and ft represents a curve 
 which enjoys on this surface all the properties of the straight 
 line in the plane of the Euclidean Geometry. Through any 
 two points one, and only one, such line can be drawn, because 
 two sets of coordinates are just sufficient to determine the 
 coefficients of an equation of the first degree. The shortest 
 distance between two points on the surface is measured on 
 such a line. For, the distance between two points on a path 
 represented by an equation in a and ft is the same as the dis- 
 tance between the corresponding points and on the correspond- 
 ing path in a Euclidean plane in which we take a and ft for 
 rectangular coordinates. It must, therefore, be the shortest 
 
90 NON-EUCLIDEAN GEOMETRY 
 
 when the path is represented by an equation of the first degree 
 in a and ft. Such a line on a surface is called a geodesic line, 
 or, so far as the surface is concerned, a straight line. The 
 distance between any two points measured on one of these 
 lines is expressed by the formula 
 
 Triangles formed of these lines have all the properties of 
 plane triangles in the Euclidean Geometry : the sum of the 
 angles is TT, etc. In fact this surface has the same relation 
 to elliptic space that the boundary-surface has to hyperbolic 
 space. 
 
 The normal form of the equation of a line is 
 
 a. cos to -f- ft sin to = p. 
 
 The rectilinear generators of the surface make a constant 
 angle, k, with all the circles drawn around the axis which 
 is polar to the axis of z. These generators are then "straight, 
 lines " on the surface, and their equation takes the form 
 
 a cos k ft sin k = p. 
 
HISTORICAL NOTE 
 
 THE history of Non-Euclidean Geometry has been so well 
 and so often written that we will give only a brief outline. 
 
 There is one axiom of Euclid that is somewhat complicated 
 in its expression and does not seem to be, like the rest, a 
 simple elementary fact. It is this : * 
 
 If two lines are cut by a third, and the sum of the interior 
 angles on the same side of the cutting line is less than two 
 right angles, the lines will meet on that side when sufficiently 
 produced. 
 
 Attempts were made by many mathematicians, notably by 
 Legendre, to give a proof of this proposition ; that is, to show 
 that it is a necessary consequence of the simpler axioms pre- 
 ceding it. Legendre proved that the sum of the angles of a 
 triangle can never exceed two right angles, and that if there 
 is a single triangle in which this sum is equal to two right 
 angles, the same is true of all triangles. This was, of course, 
 on the supposition that a line is of infinite length. He could 
 not, however, prove that there exists a triangle the sum of 
 whose angles is two right angles, t 
 
 At last some mathematicians began to believe that this state- 
 ment was not capable of proof, that an equally consistent 
 
 * See article on the axioms of F.uclid by Paul Tannery, Bulletin des 
 Sciences MatMmatiques, 1884. 
 
 t See, for example, the twelfth edition of his Elements de Geometrie, 
 Livre I, Proposition XIX, and Note II. See also a statement by Klein in 
 an article on the Non-Euclidean Geometry in the second volume of the 
 first series of the Bulletin des Sciences Mathematiques. 
 
 91 
 
92 NON-EUCLIDEAN GEOMETRY 
 
 Geometry could be built up if we suppose it not always true, 
 and, finally, that all of the postulates of Euclid were only 
 hypotheses which our experience had led us to accept as 
 true, but which could be replaced by contrary statements in 
 the development of a logical Geometry. 
 
 The beginnings of this theory have sometimes been ascribed 
 to Gauss, biit it is known now that a paper was written by 
 Lambert,* in 1766, in which he maintains that the parallel 
 axiom needs proof, and gives some of the characteristics of 
 Geometries in which this axiom does not hold. Even as long 
 ago as 1733 a book was published by an Italian, Saccheri, in 
 which he gives a complete system of Non-Euclidean Geometry, 
 and then saves himself and his book by asserting dogmatically 
 that these other hypotheses are false. It is his method of 
 treatment that has been taken as the basis of the first chapter 
 of this book.t 
 
 Gauss was seeking to prove the axiom of parallels for many 
 years, and he may have discovered some of the theorems which 
 are consequences of the denial of this axiom, but he never 
 published anything on the subject. 
 
 Lobachevsky, in Russia, and Johann Bolyai, in Hungary, 
 first asserted and proved that the axiom of parallels is not 
 necessarily true. They were entirely independent of each 
 other in their work, and each is entitled to the full credit of 
 this discovery. Their results were published about 1830. 
 
 It was a long time before these discoveries attracted much 
 notice. Meanwhile, other lines of investigation were carried 
 on which were afterwards to throw much light on our subject, 
 not, indeed, as explanations, but by their striking analogies. 
 
 Thus, within a year or two of each other, in the same 
 journal (Crelle) appeared an article by Lobachevsky giving 
 
 * See American Mathematical Monthly, July-August, 1895. 
 t The translation of Saccheri by Halsted has been appearing in the 
 American Mathematical Monthly. 
 
HISTORICAL NOTE 98 
 
 the results of his investigations, and a memoir by Minding on 
 surfaces on which he found that the formulae of Spherical 
 Trigonometry hold if we put ia for a, etc. Yet these two 
 papers had been published thirty years before their connection 
 was noticed (by Beltrami). 
 
 Again, Cayley, in 1859, in the Philosophical Transactions, 
 published his Sixth Memoir on Quantics, in which he developed 
 a projective theory of measurement and showed how metrical 
 properties can be treated as projective by considering the 
 anharmonic relations of any figures with a certain special 
 figure that he called the absolute. In 1872 Klein took up 
 this theory and showed that it gave a perfect image of the 
 Non-Euclidean Geometry. 
 
 It has also been shown that we can get our Non-Euclidean 
 Geometries if we think of a unit of measure varying according 
 to a certain law as it moves about in a plane or in space. 
 
 The older workers in these fields discovered only the 
 Geometry in which the hypothesis of the acute angle is 
 assumed. It did not occur to them to investigate the assump- 
 tion that a line is of finite length. The Elliptic Geometry 
 was left to be discovered by Eiemann, who, in 1854, took up a 
 study of the foundations of Geometry. He studied it from 
 a very different point of view, an abstract algebraic point of 
 view, considering not our space and geometrical figures, except 
 by way of illustration, but a system of variables. He investi- 
 gated the question, What is the nature of a function of these 
 variables which can be called element of length or distance ? 
 and found that in the simplest cases it must be the square 
 root of a quadratic function of the differentials of the varia- 
 bles whose coefficients may themselves be functions of the 
 variables. By taking different forms of the quadratic expres- 
 sions we get an infinite number of these different kinds of 
 Geometry, but in most of them we lose the axiom that bodies 
 may be moved about without changing their size or shape. 
 
94 NON-EUCLIDEAN GEOMETRY 
 
 Two more names should be included in this sketch, Helm- 
 holtz and Clifford. These did much to make the subject 
 popular by articles in scientific journals. To Clifford we owe 
 the theory of parallels in elliptic space, as explained on page 68. 
 He showed that we can have in this Geometry a finite surface 
 on which the Euclidean Geometry holds true.* 
 
 The chief lesson of Non-Euclidean Geometry is that the 
 axioms of Geometry are only deductions from our experience, 
 like the theories of physical science. For the mathematician, 
 they are hypotheses whose truth or falsity does not concern 
 him, but only the philosopher. He may take them in any form 
 he pleases and on them build his Geometry, and the Geome- 
 tries so obtained have their applications in other branches of 
 mathematics. 
 
 The "axiom," so far as this word is applied to these geo- 
 metrical propositions, is not " self-evident, 7 ' and is not neces- 
 sarily true. If a certain statement can be proved, that is. if 
 it is a necessary consequence of axioms already adopted, then 
 it should not be called an axiom. When two or more mutually 
 contradictory statements are equally consistent with all the 
 axioms that have already been accepted, then we are at liberty 
 to take either of them, and the statement which we choose 
 
 * Some of the more interesting accounts of Non-Euclidean Geometry 
 are: Encyclopedia Britannica, article "Measurement/* by Sir Robert Ball. 
 Recue Generate des Sciences, 1891, " Les Geometries Non-Euclidean.*' by 
 Poincare\ Bulletin of the American Mathematical Society, May and June, 
 1900, " Lobachevsky's Geometry," by Frederick S. Woods. Mathema- 
 tiscfie Annalen, Bd. xlix, p. 149, 1897, and Bulletin des Sciences Mathe- 
 matiques, 1897, "Letters of Gauss and Bolyai'*; particularly interesting 
 is one letter in which Gauss gives a formula for the area of a triangle on 
 the hypothesis that we can draw three mutually parallel lines enclosing a 
 finite area always the same. The last two articles refer to the publica- 
 tions of Professors Engel and Stackel, which give in German a full history 
 of the theory of parallels and the writings and lives of Lobachevsky and 
 Bolyai See also the translations by Prof. George Bruce Halsted of 
 Lobachevsky and Bolyai and of an address by Professor Vasiliev. 
 
HISTORICAL *OTE 95 
 
 becomes for our Geometry an axiom. Our Geometry is a study 
 of the consequences of this axiom. 
 
 The assumptions which distinguish the three kinds of Geom- 
 etry that we have been studying may be expressed in different 
 forms. We may say that one or two or no parallels can be 
 drawn through a point ; or, that the sum of the angles of a 
 triangle is equal to, less than, or greater than two right angles ; 
 or, that a straight line has two real points, one real point, or 
 no real point at infinity ; or, that in a plane we can have 
 similar figures or we cannot have similar figures, and a straight 
 line is of finite or infinite length, etc. But any of these forms 
 determines the nature of the Geometry, and the others are 
 deducible from it. 
 
ADVERTISEMENTS 
 
PLANE AND SOLID 
 
 Analytic Geometry 
 
 By FREDERICK H. BAILEY, A.M. (Harvard), and FREDERICK 
 
 S. WOODS, Ph.D. (Gottingen), Assistant Professors 
 
 of Mathematics in Massachusetts Institute 
 
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 book is intended for students beginning the study 
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WENTWORTH'S GEOMETRY 
 
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