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NON-EUCLIDEAN GEOMETRY r %' BELL'S MATHEMATICAL SERIES FOR SCHOOLS AND COLLEGES General Editor: William 1*. Milne, M.A., D.Sc NON-EUCLIDEAN GEOMETRY Bell's Mathematical Series FOR SCHOOLS AND COLLEGES. General Editor: WILLIAM ['. MILNE, M.A., D.Sc. FIRST LIST OF 'VOLUMES. Crown %vo. is. bd. Problem Papers in Arithmetic for Pre- paratory Schools, By T. Cooi'kk Smith, M.A. Lite Scholar of University College, Oxford; Mathe- matical Master, St. Peter's Court, BroadaniTB, Crown Bt>o. is. dd. Arithmetic. By H, Frekman, M.A. Some- time Scholar :tl Christ's College, Cambridge; Mathematical Master nt the Haberdashers' Aske's School. Crown 8TW. 2s. (id. Statics. Part I. By R. C. Fawort, M.A., B.Sc Sometime Scholar of Corpus Christi College, Cambridge ; Head of the Military and Eogfoeeruig Side, Clifton College. Crown ivo. SJ. The Elements of Non-Euclidean Geometry. By D. M. V. .Sojijikkvii.i.k, M.A., D.Sc, Lecturer in Mathematics, University of St. Andrews. Others in active preparation. LONDON: G. BELL AND SONS, LTD. THE ELEMENTS OF NON-EUCLIDEAN GEOMETRY BY D. M. Y. SOMMERV1LLE, M.A., D.Sc. LECTDRBK IK MATHEMATICS, UN1VBKSITV OK ST.^ANtlKKWS LONDON G. BELL AND SONS, LTD, 1914 PREFACE The present work is an extension and elaboration of a course of lectures on Non-Euclidean Geometry which I delivered at the Colloquium held under the auspices of the Edinburgh Mathematical Society in August, 1913. Non-euclidean geometry is now a well-recognised branch of mathematics. It is the general type of geometry of homogeneous and continuous space, of which euclidean geometry is a special form. The creation or discovery of such types has destroyed the unique character of euclidean geometry and given it a setting amongst geometrical systems. There has arisen, so to speak, a science of Com- parative Geometry. Special care has, therefore, been taken throughout this book to show the bearing of non-euclidean upon euclidean geometry ; and by exhibiting euclidean geometry as a really degenerate form — in the sense in which a pair of straight lines is a degenerate come — to explain the apparent want of symmetry and the occasional failure of the principle of duality, which only a study of non-euclidean geometry can fully elucidate. VI PREFACE There are many ways of presenting the subject. In the present work the primary exposition follows the lines of elementary geometry in deduction from chosen postulates. This was the method of Euclid, and it was also the method of the discoverers of non-euclidean geometry. Restrictions have, however, been made. It was felt that a rigorously logical treatment, with a detailed examination of all the axioms or assumptions, would both overload the boob and tend to render it dry and repulsive to the average reader. It is hoped, however, that the principles have been touched upon sufficiently to indicate the nature of the problems involved, especially in such cases where they throw light upon ordinary geometry. It is impossible thoroughly to appreciate non-euclidean geometry without a knowledge of its history. I have there- fore given in the first chapter a fairly full historical sketch of the subject up to the epoch of its discovery. Chapters II. and III. develop the principal results in hyperbolic and elliptic geometries. Chapter IV. gives the basis of an analytical treatment, the matter chosen for illustration here being, for the most part, such as was not touched on in the preceding chapters. This completes the rudiments of the subject. The next two chapters exhibit non-euclidean geometry in various lights, mathematical and philosophical, and bring up the history to a later stage. In the last three chapters some of the more interesting branches of geometry are worked out for the non-euclidean case, with a view to providing the serious student with a stimulus to pursue the subject in its higher developments. The reader will find PREFACE vu a list of text-books and references to all the existing litera- ture up to 1910 in my Biblwgraphif of Non-EucMdeav Geometry {London : Harrison, 1911). Most of the chapters are furnished with exercises for working. As no examination papers in the subject are yet available, the examples have all been specially devised, or culled from original memoirs. Many of them are theorems of too special a character to be included in the text. In preparing the treatise, the needs of the student reading privately have been kept steadily in view. Hence it is hoped that the work will prove useful to the " Scholarship Candidate " in our Secondary Schools who wishes to widen his geometrical horizon, to the Honours Student at our Universities who chooses Geometry as his special subject, and to the teacher of Geometry, in general, who desires to see in how far strict logical rigour can be made compatible with a treatment of the subject capable of comprehension by schoolboys. In acknowledging my indebtedness to previous writers on the subject, special mention should be made of Bonola s article in the collection, Qnestioni rignardanti la geamehio demmtare, editct by Enriques (Bologna, 1900 ; German translation, Leipzig, 1911); and Liemnann's N&kt- eiiljidiwhe Geometrie (Leipzig, 2nd ed., 1912). I take this opportunity of expressing my obligations to Mr. Peter Fraser, M.A., B.Sc., Lecturer in Mathematics at the University of Bristol, and Mr. E. K. Wakeford, Trinity College, Cambridge, for kindly criticising the work while in manuscript form and giving many valuable suggestions. VIII PREFACE I am also greatly indebted to Mr. W. P. Milne, M.A., D.Sc., Clifton College, Bristol, for continued assistance, by criti- cism and suggestion, all through the preparation of the book. To Dr. A. E. Taylor, Professor of Moral Philosophy at the University of St. Andrews, and Mr. C. D. Broad, B.A., Fellow of Trinity College, Cambridge, and Assistant to the Professor of Logic at the University of St. Andrews, I have also to express my thanks for reading and criticising Chapter VI. In correcting the proofs I have profited by the assistance of my wife and by the excellence of Messrs. MacLehose's printing work. D. M. y. s. The University, St. Andrews, April, 1914, CONTENTS CHAPTER I. HISTORICAL. 1 . The origins of geometry 2. Euclid 'a Elements ; axioms and postulates 3. The Parallel-postulate ; attempts to prove it • 4. Playf air's Axiom 5. Thibaut'a rotation proof ; the direction fallacy 6. Bertrand's proof by infinite areas - - - - 7. Equidistant linos " 8. First glimpses of non-ouclidoan geometry ; Snccheri 9. Lambert 10 Gauss 11. Schweikarl, Taurinus and Wachter 12. Legondre „.-•-- 13. The theory of parallels in Britain - 14. The discovery of non-euclidoan geometry ; Lobachevsky 16, Bolyai 16. The later development Examples I. * PACK 1 2 3 4 6 7 !) 11 18 14 14 in is 20 21 24 25 CHAPTER n. ELEMENTARY HYPERBOLIC GEOMETRY. 1 . Fundamental assumptions or axioms - 2. Parallel linos ; the three geometries - 3. Definition of parallel lines 4. Plane GEOMRfair. Properties of parallelism 27 20 30 31 x CONTENTS PAfiE 5, Thoorera on the transversal - - 3a 0. Theorem of the exterior angle 34 7. The parallel-angle 35 8. Theorems on the quadrilateral 36 9. Intersecting lines are divergent ----- 37 10. Parallels are asymptotic in one direction and divergent in the other - 3g 11. Parallel lines meet at infinity at a zero angle - ■ - 33 12. Non-intorscctors ; common perpendicular - - - 40 13. Classification of pairs of lines - - 41 14. Solid Geometry. Planes, dihedral angles, etc. - - 42 15. Systems of parallel lines, and lines poqjendicular to a plane --------- 43 10. Pencils and bundles of lines 45 1 7. Points at infinity ; the absolute - - - - - 46 18. Idea! points 47 1ft. Parallel lines and planes 40 20. Principle of duality ...... -DO 21. The circle ; horocyele ; equidistant-curve - - - 81 22. The sphere ; horosphero ; equidistant-surface (52 23. Circles determined by three points or three tangents - 53 24. Geometry of a bundle of lines and planes 65 25. Trigonometry. Tho circular functions 66 20. Ratio of arcs of concentric horocyclee 67 27. The parallel-angle 68 28. Two formulae for the horocyelu - - - - - St 29. Tho right-angled trianglo ; notation ; complementary angles and segments ------ 63 30. Correspondence between rectilinear and spherical triangles 03 31. Associated triangles ----- • - 65 32. Trigonometrical formulae for a right-angled triangle - 60 33. Engel-Napior rules ....... 67 34. Spherical trigonometry the same ns in euclidean apace - 00 35. Correspondence lx»twoen a right-angled triangle and a tri-reetangular quadrilateral 70 30. Napier's rules for a tri-rectangular quadrilateral - - 74 37. Formulae for any triangle ..... 74 38. Euclidean geometry holds in the infinitesimal domain - 75 30. Circumference of a circle 76 40. Sum of the angles of a triangle ; defect and area - 77 CONTENTS 41. Relation between the unite of length and area ; area of a sector and of a triangle by integration - ii. The maximum triangle ):i. Gauss' proof of tlie defect-area theorem - - - - 14. Area of a polygon • l.->. Another proof that the geometry on the horosphere is euclidean - Examples II. XI PAnK 70 81 82 83 84 84 CHAPTER III. ELLIPTIC GEOMETRY. 1. Absolute polo and polar 2. Spherical and elliptic geometries - - - - 3. The elliptic plane is a one-sided surface - - - - 4. Absolute polar system - - - 5. Projective geometry ; summary of theorems ■ 6. The absolute 7. Principle of duality - - 8. Relation betweon distance arid angle 0. Area of a triangle 10. The circle ; duality 1 1. Cylinder s rectilinear generators - 12. Common perpendiculars to two lines in space - 1 3. Para tactic linos or Clifford's parallels - - - - 14. Construction for common perpendiculars 15. Paratactio lines cut the same two generators of the absolute 111. Comparison between pnrntAxy and euclidoiiii pnrnllclism • 17. Clifford's Surface - - - - - - 18. Trigonometrical formulae ; circumference of a circle IB. The right-angled triangle .... 20. Associated triangles - • • 2 1 . Napier's Rules - 22. Spherical trigonometry the same as in euclidean space - 23. Tho trirectangnlar quadrilateral - • - Examples HI. ss 8fl 1)1 02 93 98 100 101 103 104 105 106 108 109 110 HI 112 114 no 117 no 120 121 122 xa CONTENTS CHAPTER IV. ANALYTICAL GEOMETRY. nun 1. Coordinates of a point ; Weierstrass' coordinates - - 125 2. The absolute 127 3. Equation of a straight -lino ; Weierstrass' line-coordinates 12S 4. Distance between two pointe 129 5. The absolute in elliptic geometry 130 6. Anglo between two tines • • - - . - 131 T. Distance of a point from a line ..... 132 8. Point of intersection of two lines ; imaginary points - 132 9. Line joining two points - - - - - - - 134 10. Minimal lines ........ 134 11. Concurrency and collincarity - - - - - - 135 12. The circle 13G 13. Coordinates of point dividing tho join of two points into given parts ........ 137 14. Middle point of a segment .... - 138 15. Properties of triangles ; centroid, in- and eireum-centres 130 16. Explanation of apparent exception in euelidean geometry 13!) 17. Polar triangles ; orthocontre and orthaxis - - - 141 18. Desargues' Theorem ; configurations .... 142 1!). Dcsmic system ........ 144 20. Concurrency and collincarity 145 21. Position-ratio; croBS-ratio ; projection - - - 147 Examples TV. 149 CHAPTER V. REPRESENTATIONS OF NOX-EUCLIDEAN GEOMETRY IN EUCLIDEAN SPACE. 1. The prohliin ... 2. Projective Representation 3. The absolute 4. Euclidean geometry 153 154 154 155 CONTEXTS XIII 8. 9. 10. 11. 12. 13. 1-1. 15. 16. 17. 18. 10. 20. 21. 22. 23. 24. 25. 28. 27. 23. 29. 30. 31. 32. 33. 34. 35. 36. 37. :ts. PAOI The oiroular points - - - - - - -166 Expression of angle by logarithm of cross- ratio - • ISO Projective expression for distance and angle in non- euclidean geometry - - - - - 157 Metrical geometry reduced to projective ; Cay ley-Klein - 158 Example : construction of middle points of a segment - 1 B0 Classification of geometries with projective metric - - 160 Distance in euelidean geometry 161 Geometry in which the perimeter of a triangle is constant 162 Extension to three dimensions 103 Application to proof that geometry on the horospfoero is ouclidcan - - - - - - - - 164 Geodesic Representation 165 Geometry upon a curved surface . - - - - 106 Measure of curvature - - - - - - -168 Surf acos of constant curvature ; Gauss' theorem - - 168 The Pseudosphero 1»8 The Cayley-Klein representation as a projection • - 170 Meaning of Weierstrass* coordinates - - - - 171 Conporkal Representation. Stereographic projection 172 The orthogonal circle or absolute I ' I Conformal representation Wi Point-pairs "6 Pencils of lines ; concentric eirclP3 . . - - 176 Distanco botween two points - - - - - -179 Motions ---------- 179 Reflexions - - - - - - - - -180 Complex numbers 181 Circular transformation ; conformal and nomographic - 182 Inversion ....----- 183 Types of motions - - - - - - - 185 The distance-function - - - - - - - 186 The line-element -------- 187 Simplification by taking fixed circle as a straight line - 188 Angle at which an equidistant-curve meets its axis - - 189 Extension to three dimensions - - - - - 191 XIV CONTEXTS CHAPTER VI. " SPACE-CURVATURE " AND THE PHILOSOPHICAL BEARING OF NON-ECCLIDEAN GEOMETRY. PASS 1. Four periods in the history of non-euelideau geometry - 102 2. " Curved space " 193 3. Differential geometry ; Kienuiun 194 4. Free mobility of rigid bodies ; Hebnholtz - - -105 G. Continuous groups of transformations ; Lie - • - 1fl7 6. Assumption of coordinates ...... |gg 7. Space curvature and the fourth dimension - - - 199 8. Proof of the consistency of non-ouclidean geometry - 202 9. Which is tho true geometry T 203 10. Attempts to determine the space constant by astronomical measurements 203 11. Philosophy of space - ..... 207 12. Tho inextricable entanglement of apace and matter - 209 CHAPTER VII. RADICAL AXES, HOMOTHETIC CENTRES, AND SYSTEMS OF CIRCLES. 1. Common points and tangents to two circles - - 211 2. Power of a- point with respect to a circle - - - 212 3. Power of a point with respect to an oquidistanl-eurve - 213 4. Reciprocal property - - - - - - -216 fi. Angles of intersection of two circles .... 218 0. Radical axes - - - - - - - - - 2 1 f! ", nomothetic centrea - -221 8. Radical centres and nomothetic axes - - - 221 9. Coaxal circles in elliptic geometry ----- 222 10. Uomocentric circles ....... 224 11. ('iiinpurison with nielidetm geomeay .... Jgfl 12. Linear equation of a circle - - - - - 227 CONTEXTS xv 13. Systems of circles 1 1. Correspondence between circles and planes in hyperbolic geometry ; marginal images The margined images of two planes intereect in the marginal image of the line of intersection of tho planes - Two planes intersect ol the same angle as their marginal images Systems of circles -------- Types of pencils of circles Examples VII. -------- 13. 17. IS. rAtit 22S 229 231 231 232 233 235 CHAPTER VLU. INVERSION AND ALLIED TRANSFORMATIONS 0. 7. 8. 9. in. II. ia. II IV in California) and circular transformations • A circular transformation is California] Every congruent transformation of spoco gives a circular trtiiisforinutioii of the plane - Converse .......-- The general circular transformation is compounded of a congruent transformation and a circular trans- formation which leaves unaltered all the straight lines through a fixed point - Inversions and radiations ...... Formulae for inversion Comparison with euelidcan inversion . - - - Congruent transformations ; transformation of coordinates Equations of transformation - - ... Position of a point in terms of a complex parameter Expression for congruent transformation by means of nomographic transformation of tho complex para- meter ... - Groups of motions Connection with quaternions ------ Equation of a circle in terms of complex parameters EquaJ ion of inversion Examples Vin. 230 237 238 239 240 241 244 246 247 248 248 249 250 251 252 253 XVI CONTENTS CHAPTER IX. THE CONIC. 1. Equation of the second degree 2. Classification of conies - 3. Centres and axes; foei and Uiracl rices - 4. Focal distance property - 6. Focus-tangent properties 6. Focus-directrix property 7. Geometrical proof of the focal distance property Examples XX. ' Index - NON-EUCLTDEAN GEOMETRY CHAPTER I. HISTORICAL. 1, The origins of geometry. Geometry, according to Herodotus, and the Greek deriva- tion of the word, had its origin in Egypt in the mensuration of land, and the fixing of boundaries necessitated by the repeated inundations of the Nile. It consisted at first of isolated facts of observation and rude rules for calculation, until it came under the influence of Greek thought. The honour of. having introduced the study of geometry from Egypt falls to Thales of Miletus (640-546 B.O.), one of the seven " wise men " of Greece. This marks the first step in the raising of geometry from its lowly level ; geometric ele- ments were abstracted from their material clothing, and the geometry of lines emerged. With Pytiiaoohas (about 580-500 b.c.) geometry really began to be a metrical science, and in the hands of his followers and the succeeding Platom'sts the advance in geometrical knowledge was fairly rapid. Already, also, attempts were made, by Hippocrates of Chios (about 430 B.C.) and others, to give a connected and logical presentation of the science in a series of propositions based upon a few axioms and definitions. The most famous of such attempts is, of course, that of Euclid (about 300 B.C.), and so great was his prestige that he acquired, like N. -E. G. A * HISTORICAL [1.2 Aristotle, the reputation of infallibility, a fact which latterly became a distinct bar to progress. 2. Euclid's Elements. The structure of Euclid's Elements should be familiar to every student of geometry, but owing to the multitude of texts and school editions, especially in recent years, when Euclid's order of the propositions has beeu freely departed from, Euclid's actual scheme is apt to be forgotten. We must turn to the standard text of Heiberg 1 in Greek and Latin, or its English equivalent by Sir Thomas Heath. 2 Book L, which is the only one that immediately concerns us, opens with a list of definitions of the geometrical figures, followed by a number of postulates and common notions, called also by other Greek geometers "axioms." Objection may be taken to many of the definitions, as they appeal simply to i h<* intuition. The definition of a straight line as " a line which lies evenly with the points on itself " contains no state- ment from which we can deduce any propositions. We now recog- nise that wo must start with some terms totally uudefined, and rely upon postulates to assign a more definite character to the objects. A right angle and ft square are defined before it has been shown ihnt objects corresponding to the definitions can exist. An axiom or common notion was considered by Euclid as a pro- position which is so self-evident that it needs no demonstration ; a postulate as a proposition which, though it may not be self-evident, cannot be proved by any simpler proposition. This distinction has been frequently misunderstood —to such an extent that later editors of Euclid have placed some of the postulates erroneously among the axioms, A notable instance is the parallel-postulate, No. o, which has figured for ages as Axiom 11 or 12. The common notions of Euclid are five in number, and deal exclusively with equalities and inequalities of magni- tudes. 1 12 vols., Leipzig, 1883-EM. ! 3 vols., Cambridge, 1008. 1-S] THE PARALLEL-POSTULATE The postulates are also five in number and are exclusively geometrical. The first three refer to the construction of straight lines and circles. The fourth asserts the equality of a! right angles, and the fifth is the famous Parallel- Postulate : " If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely,- meet on that side on which are the angles less than two right angles." 3. Attempts to prove the parallel-postulate. It seems impossible to suppose that Euclid ever imagined this to be self-evident, yet the history of the theory of parallels is full of reproaches against the lack of self-evidence of this " axiom." Sir Henry Savile 1 referred to it as one of the great blemishes in the beautiful body of geometry ; D'Aleinbert 1 called it " l'ecueil et le seandalc des elemens de Geometrie," The universal converse of the statement. " if two straight lines crossed by a transversal meet, they will make the interior angles on that side less than two right angles," is proved, with the help of another unexpressed assumption (that the straight line is of unlimited length), in Prop. 17 ; while the eontrapositivc, " if the interior angles on either side are not less than two right angle.* (<.-., bv Prop. 13, if they are equal to two right angles) the straight lines will not meet," is proved, again with the same assumption, in Prop. 28. Such considerations induced geometers {and others), even up to the present day, to attempt its demonstration. From the invention of printing onwards a host of parallel-postu- late demonstrators existed, rivalled only by the " circle- squarers," the " flat- earth ers," and the candidates for the 1 Praetectionea, Oxford, 1621 (p. 140), - Melange* de LiUeratwe, Amsterdam, 1759 (p. 180). HISTORICAL [1.4 Wotfskehl " Fcrmat " prize. Great ingenuity was expended, but no advance was made towards a settlement of the question, for each successive demonstrator showed the false- ness of his predecessor's reasoning, or pointed out an un- noticed assumption equivalent to the postulate which it was desired to prove. Modern research has vindicated Euclid, and justified his decision in putting this great proposition among the independent assumptions which are necessary for the development of euclidean geometry as a logical system. All this labour has not been fruitless, for it has led in modern times to a rigorous examination of the principles, not only of geometry, but of the whole of mathematics, and even logic itself, the basis of mathematics. It has had a marked effect upon philosophy, and has given us a freedom of thought which in former times would have received the award meted out to the most deadly heresies. 4. In a more restricted field the attempts of the postulate- demonstrators have given us an interesting and varied assortment of equivalents to Euclid's axiom. It would take up too much of our space to examine the numerous demon- strations, 1 but as some of the equivalent assumptions have come into school text-books, and there appears still to exist a belief that the Euclidean theory of parallels is a necessity of thought, it will be useful to notice & few of them. One of the commonest of the equivalents used for Euclid's axiom in school text- books is " Playfair's axiom " (really due to Ludlain 2 ) : " Two intersecting straight lines cannot 1 A iLseftil account of these is Riven by W. B. Franldand in his Ttieoriex of PamUdimu Cambridge. J £110. • The Rudiments of Mathematics, Cam bridge, 1785 (p. W5). L<] THE ROTATION PROOF both be parallel to the same straight line," which i3 equiva- lent to the statement, " Through a given point not more than one parallel can be drawn to a given straight line," and from this the properties of parallels follow very Fio, I, elegantly. The statement is simpler in form than Euclid's, but it is none the less an assumption. Another equivalent is : " The sum of the angles of a triangle is equal to two right angles." I do not think that fl HISTORICAL [i. 5 anyone has been bo bold as to assume this as an axiom, but there have been many attempts to establish the theory of parallels by obtaining first an intuitive proof of this statement. A very neat proof, but particularly dangerous unless it be regarded merely as an illustration, is the " Rotation Proof," due to Thibaut. 1 5, Let a ruler (Fig. 1) be placed with its edge coinciding with a side AC of a triangle, and let it be rotated succes- sively about the three vertices A, B, C, in the direction ABC, so that it comes to coincide in turn with AB. BC and CA. When it returns to its original position it must have rotated through four right angles. But this whole rotation is made up of three rotations through angles equal to the exterior angles of the triangle. The fault of this " proof" is that the three successive rotations are not equivalent at all to a single rotation through four right angles about a definite point, but are equivalent to a translation, through a distance equal to the perimeter of the triangle, along one of the sides. The construction may be performed equally well on the surface of a sphere, with a ruler bent in the form of an arc of a great circle ; and yet the sum of the exterior angles of a spherical triangle is always less than four right angles. A similar fallacy is contained in all proofs based upon the idea of direction. Take the following : AB and CD (Fig. 2) are two parallel roads which are intersected by another road BC. A traveller goes along AB. and at B turns into the road BC, altering his direction by the angle at B. At C he turns into his original direction, and therefore must haw turned back through the same angle. But this requires 1 Qrundrits dcr rtinen UathemalU; 2nd ed. Uottingcn, 1809. t. 6] THE DIRECTION FALLACY a definition of sameness of direction, and this can only be effected when the theory of parallels has been established. The difficulty is made clear when we try to see what we mean by the relative compass-bearing of two points on the earth's surface. If we travelled due west from Plymouth along a parallel of latitude, we should arrive at Newfound- land', but the direct or shortest course would start IB a " direction " WNW. and finish in the " direction " WSW. A — Fid. S Other statements from which Euclid's postulate may be deduced are "Three points are either collinear or concyclic." (\\. Bolyai. 1 ) " There is no upper limit to the area of a triangle. (Gauss. 2 ) " Similar figures exist." (Wallis. 3 ) 6 Another class of demonstrations is based upon con- siderations of infinite areas. The following is " Beetrand's Proof." * 1 Kvrzer OrunArm. 1851 (]'■ *<>)■ " Letter to W. Boiyai, Kith December, 17911. 3 Opera, Oxford, UI93 (t. ii. p. 670). «L. Bertram), Devdojipement wm^mt dc la partie iiementaire da nimhimaltgucs, Goneva, 1778 (t. ii. p. 19). HISTORICAL [,. G Let a line AX (Fig. 3), proceeding to infinity in the direc- tion of X, be divided into equal parts AB, BC, .... and let the lines AA\ BB', ... each produced to infinity, make equal angles with AX. Then the infinite strips A' ABB', B'BCC, ... can all be superposed and have equal areas, but it requires infinitely many of these strips to make up the area A' AX, contained between the lines A A' and AX, each produced to infinity. Again, let the angle A' AX be divided into equal parts A'AP, PAQ, ... . Then all these sectors can be superposed and have equal areas, but it requires only a finite number of them to make up the area A' AX. Fia. a. Hence, however small the angle A'AP may be, the area A'AP is greater than the area A' ABB', and cannot there- fore be contained within it. AP must therefore cut BB' ; and this result is easily recognised as Euclid's axiom. The fallacy here consists in applying the principle of super- position to infinite areas, as if they were finite magnitudes. If we consider (Vig. i) two infinite rectangular strips A'ABH' Bad A'PQli' with equal bases AB, PQ, and partially superposed, then the two strips are manifestly unequal, or else tiie principle of 7] INFINITE AREAS 9 superposition is at fault. Again, suppose we have two rectangular strips A' ABB', C'CDD' (Fig. o). Mark ofT equal lengths A A,. A t A t . ... along A A', each equal to CD. and equal lengths CC lt CiC t , ... along t'C, each equal to AB, and divide the strips at these points into rectangles. Then all the rectangles arc equal, and, if we O 9 O' A p ** FlO. 4. number thein consecutively, then to every rectangle in the one strip there corresponds the similarly numbered rectangle in the other strip. Hence, if the ordinary theorems of congruence and equality of areas are assumed, we must admit that the two strips arc equal in area, and that therefore the area is independent of the magnitude of A li. Such deductions are just as valid aa the de- duction of Euclid's axiom from a consideration of infinite areas. Pin. fi. 7. It will suffice to give one other example of the attempts i" huso tin* theory of parallels on intuition. Suppose that, instead of Euclid's definition of parallels as " straight lines, which, being in the same plane, and being produced indefinitely in both directions, do not meet one another in 10 HISTORICAL [1.7 either direction,'" we define them as " straight lines which an; everywhere equidistant," then the whole Euclidean theory of parallels cornea out with beautiful simplicity. In particular, the sum of the angles of any triangle ABC (Fig. 6) is proved equal to two right angles by drawing through the vertex A a parallel to the base BG. Then, if we draw per- pendiculars from A, B, C on the opposite parallel, these perpendiculars are all equal. The angle EAB = LB and the angle CAF=LG. It is scarcely necessary to point out, however, that this definition contains the whole debatable assumption. We have no warrant for assuming that a pair of straight lines can exist with the property aseribed to them in the defini- tion. To put it another way, if a perpendicular of constant length move with one extremity on a fixed tine, is the locus of its free extremity another straight line ? We shall find reason later on to doubt this. In fact, non-euclidean geometry has made it clear that the ideas of parallelism and equidistance are quite distinct. The term " parallel " Greek va/MXX^w~rttnnmg alongside) originally con- noted equidistance, but the term is used by Euclid rather in the sense " asymptotic " (Greek d-enJ/uirTwroy - non-inter- secting), and this term has come to be used in the limiting 1.81 SACCHERI II case of curves which tend to coincidence, or the limiting case between intersection and non-intersection. In non- euclidean geometry parallel straight lines are asymptotic in this sense, and equidistant straight lines in a plane do not exist. This is just one instance of two distinct ideas which are confused in euclidean geometry, but are quite distinct in non-euclidean. Other instances will present themselves. 8. First glimpses of Non-Euclidean geometry. Among the early postulate-demonstrators there stands a unique figure, that of a Jesuit, Gerolamo Saccheri (1667- 1738), contemporary and friend of Ceva. This man devised an entirely different mode of attacking the problem, in an attempt to institute a reductio ad absnrdtun. 1 At that time the favourite starting-point was the conception of parallels as equidistant 8'iaight lines, but Saccheri, like some of his predecessors, saw that it would not do to assume this in the definition. He starts with two equal perpendiculars AC and BD to a line AB. When the ends 0, D are joined, it is easily proved that the ang es at C and D are equal ; but are they right angles ? Saccheri keeps an open mind, and proposes three hypotheses : (1) The Hypothesis of the Right Angle. (2) The Hypothesis of the Obtuse Angle, (3) The Hypothesis of the Acute Angle. The object of his work is to demolish the last two hypo- theses and leave the first, the Euclidean hypothesis, supreme; 1 Ettdide* ab omni iianvj vindicate*, Milan. 1733. KngBlh trims, by Halslod. Amer. Math. MonUUg, vols. IS, 1894-08; German byStiickcl and Engrf, [)it ThMrie d*r FaralUUinicn. Leipzig, 1895. (This book by StHckel and Engcl contains a, valuable history of the theory of parallels.) L2 HISTORICAL [i. 8 but the task- turns out to be more arduous than he expected. He establishes a number of theorems, of which the most important are the following : If one of the three hypotheses is true in any one case, the same hypothesis is true in every case. On the hypothesis of the right angle, the obtuse angle, or the acute angle, the sum of the angles of a triangle is equal to, greater than, or less than two right angles. On the hypothesis of the right angle two straight lines intersect, except in the one case in which a transversal cuts them at equal angles. On the hypothesis of the obtuse angle two straight lines always intersect. On the hypothesis of the acute angle there is a whole pencil of lines through a given point which do not intersect a given straight liue, but have a common perpendicular with it, and these are sepa- rated from the pencil of lines which cut the given line by two lines which approach the given line more and more closely, and meet it at infinity. The locus of the extremity of a perpendicular of constant length which moves with its other end on a fixed line is a straight line on the first hypothesis, but on the other hypotheses it is curved ; on the hypothesis of the obtuse angle it is convex to the fixed line, and on the hypothesis of the acute angle it is concave. ^ Saccheri demolishes the hypothesis of the obtuse angle in his Theorem 14 by showing that it contradicts Euclid I. 17 (that the sum of any two angles of a triangle is less Hum two right angles) : but he requires nearly twenty more theorems before he can demolish the hypothesis of the acute angle, which he does by showing that two lines which meet in a point at infinity can be perpendicular at that t !l] LAMBERT 13 point to the same straight line. In spite of all his efforts, however, he does not seem to be quite satisfied with the validity of his proof, and he offers another proof in which he loses himself, like many another, in the quicksands of the infinitesimal. If Saccheri had had a little more imagination and been less bound down by tradition, and a firmly implanted belief that Euclid's hypothesis was the only true one, he would have anticipated by a century the discovery of the two non-euclidcan geometries which follow from his hypotheses of the obtuse and the acute angle. 9. Another investigator, J. H. Lambert (1728-1777), 1 fifty years after Saccheri, also fell just short of this dis- covery. His startiug-point is very similar to Saccheri's, and he distinguishes the same three hypotheses ; but he went further than Saccheri. He actually showed that on the hypothesis of he obtuse angle the area of a triangle is proportional to the excess of the sum of its angles over two right angles, which is the case for the geometry on the sphere, and he concluded that the hypothesis of the acute angle would be verified on a sphere of imaginary radius. He also luade the noteworthy remark that on the third hypothesis there is an absolute unit of length which would obviate the necessity of preserving a standard foot in the Archives. He dismisses the hypothesis of the obtuse angle, since it requires that two straight lines should enclose a space, but his argument against the hypothesis of the acute angle, such as the non-existence of similar figures, he characterises ' Tlimrif. dtr ParalkUinitn, 178(1. (Rcprint«l in Stiicke] and Easel, Th. tier Far., 1895.) 14 HISTORICAL fi. 10 as arguments rib amore et inmdia ducla. Thus lie arrived at no definite conclusion, and his researches were only published some years after his death. 10. About this time (1799) the genius of Gauss (1777- 1855) was being attracted to the question, and, although he published nothing on the subject except a few reviews, it is clear from his correspondence and fragments of his notes that he was deeply interested in it. He was a keen critic of the attempts made by his contemporaries to establish the theory of parallels ; and while at first lie inclined to the orthodox belief, encouraged by Kant, that Kuclidean' geometry was an example of a necessary troth, he gradually came to see that it was impossible to demon- strate it. He declares that he refrained from publishing anything because he feared the clamour of the Boeotians, or, as we should say, the Wise Men of Gotham ; indeed at this time the problem of parallel lines was greatly dis- credited, and anyone who occupied himself with it was liable to be considered as a crank. Gauss was probably the first to obtain a clear idea of the possibility of a geometry other than that of Euclid, and we owe the very name Non-Euclidean Geometry to him. 1 It is clear that about the year 1820 he was in possession of many theorems of non-euclidean geometry, and though he medi- tated publishing his researches when he had sufficient leisure to work them out in detail with his characteristic elegance, he was finally forestalled by receiving in 1832, from his friend W. Bolyai, a copy of the now famous Appendix by his son, John Bolyai. 11. Among tbc contemporaries and pupils of Gauss tliere are throe names which deserve mention. F. K. SchwkikaRt (17804859), 1 Letter to Taurinus, 8th November, 1834. HJ GAUSS 15 Professor of Law in Marburg, sent to Gauss in 1818 a pago of MR. explaining a system of geometry which tie calls " Astral Geometry," in which the sum of the angles or a triangle is always less than two right angles, and in which there is an absolute unit of length. He did not publish any account of his researches, but he induced his nephew, F. A. Taurinus (1794-1874}, to take up the question. His uncle's ideas did not appeal to him, however, but a few years later he attempted a treatment of the theory of parallels, and having received some encouragement from Gauss, he published a small book, Tkeorie der Pamlkllinim, in 1825. After its publication he came across Camerer's new edition of Euclid in Greek and 1-atin. wliiili, in an Excursus to Euclid I. 29. contains a very valuable history of the theory of parallels, and there he found that his methods had been anticipated by Saccheri and Lambert Nest year, accord- ingly, he published another work, Gfiomelriae prima elentrnia, and in the Appendix to this he works out some of tho most Important trigonometrical formulae for non-euclidean geometry by using the fundamental formulae of spherical geometry with an imaginary radius. Instead of the notation of hyperbolic functions, which was thin scarcely in use. be expresses his results in terms of logarithms and exponentials, and calls his geometry the " Logarithmic Spherical Geometry." Though Taurinus must bo regarded as an Independent rJiscovcrer of non-euclidean trigonometry, be always retained (be belief, unlike Gauss and Schweikart, that Euclidean geometry was necessarily the true one. Taurinus himself was aware, however, of the impor- tance of liia contribution to the theory of parallels, and it was a bitter disappointment to him when he found that his work attracted no attention. In disgust he burned (he remainder of the edition of his Elementa, which is now ono of the rarest of books. The third to be mentioned as having arrived at the notion of a geometry in which Euclid's postulate is denied is F. L WachteR (1792-1817), a student under Gauss. It is remarkable that he affirms that even if the postulate be denied, the geometry on a sphere becomes identical with the geometry of Euclid when the radius is indefinitely increased, though it is distinctly shown that the limiting surface is not a plane. This was one of the greatest discoveries of Lobachovsky and Bolyai. If Waehter had lived he might have bean tJ» t Saeo v ewc of rmn-flrwtfcVwi geometry, far hi- tnabjbl into the ouestion was far beyond that of the ordinary pa rail el -postulate demonstrator. 16 HISTORICAL [i. 12 12. While Gauss, Scliwetkart, Taurinus and others were working in Germany, and had arrived independently at some of the results of non-euclidean geometry, and were, in fact, just on the threshold of its discovery, in France and Britain the ideas were still at the old stage, though there was a considerable interest iu the subject, inspired chie* 1 j by A, M. Legendre (1752-1333). Legendre's researches were published in the various editions of his Elements, from 1794 to 1823, and collected in an extensive article in the Memoirs of the Paris Academy in 1833. Assuming all Euclid's definitions, axioms and postulates, except the parallel-postulate and all that follows from it, he proves some important theorems, two of which, Proposi- tions A and B, are frequently referred to in later work as Legendre's First and Second Theorems. Prop. A. The mm of (lie three angles of a rectilinear tri- angle cannot be greater than two right angles (v). (Elements, 3rd cd. 1800.) In Fig. 7, A A 1 A 2 ... A„ is a straight line, and the tri- angles AtfB^, A^BtA^ ... are all congruent, and the vertices B^ ... B u are joined by a broken hue. Suppose, if possible, that sLAoBoAt + BtAoAt+AoAiB^v. Now Z.B A A t =B J A 1 A s and sLBtAyBi+BiAyAz+AoAiB^ir. I. 12] LEGENDRE 17 Therefore LA ( fi a A t >B n A l B u and therefore A^A^B^. Let A^A, -BoB,=d; then A ll B \-B a B i +B 1 B 3 + ...+B n .,B„+BJ i ,=2A a B +nB Q B 1 =2AgB + nA A l -nd=A a A a + 2A B ~rtd, /.- A 9 A»-A£ a + BoB l + ...+B n A* + (nd-2AtB ). But, by increasing n, nil can be made to exceed the fixed length 2.4 ,>B(,: and hence A^A,^ which is the length of the straight line joining A a and A n , can be greater than the sum of the parts of the broken line which joins the same two points, which is absurd. There are several points in this proof that require careful examination. In the first place, the assumption that nd can always exceed 2.-l W by takmp n sullieiently great lies at the basis of geometrical eunlinttitjf, and is equivalent to the denial of the existence of infini- tesimals. This is generally known as the Axiom- of Archimedes. The question of continuity is fundamental in dealing with the foundations of geometry, but it would be outside ttio scope of this book to enter further into this extensive and difficult subject. Twice in this proof we have assumed the " theorem of the exterior angle " of a triangle (Euclid I. 16), first in the statement that AbA^BqBi, and second in the assumption that the straight line joining two points is the shortest path (Euclid I. 20). This is equivalent to the rejection of Saccheri's hypothesis of the obtuse angle. If this hypothesis be followed to its logical conclusion, it can be shown (see Chap. III.) that two straight lines in a plane will ulways intersect, when produced in either direction. The straight line is then re-entrant-, and there are at least two straight paths connecting any two points. The straight hne Aji u would not then of necesstfv be the shortest path from A 9 to A n . f.'K-G. B 18 HISTORICAL ft. 12 Prop. B. If (here exists a single triangle in which the mm of the angles is equal to two right angles, then m every triangle the sum of the augks must likewise be equal to two right angles. This proposition was already proved by Saccheri, along with the corresponding theorem for the case in which the sum of the angles is less than two right angles, and we need not reproduce Legendre's proof, which proceeds by con- structing successively larger and larger triangles, in each of which the sum of the angles = sr. Legendre makes an attempt to prove that the m» of the angles of a triangle is equal to two rigid angles, as follows {EUments, 12th ed. 1823) : Let A^B i C l (Fig. 8) be a triangle, in whieh A& is the greatest side and Bfi, the least. Join Ay to M lt the middle point of lift,, and produce AJt t to C 2 so that A l C 2 = Aj; j . On AyB, take AJi^KB^AyMy, and join C 2 K. Then we get a second triangle A 2 B % V 2 , in which A 2 coincides with A u and in which A 2 B„ is the greatest side and B$ t the least. Denote the angles of the triangles AyByCy, A^B.,0^ by single letters. Then a Afi 2 K m AyB x My and a C z KB, m CyMyA t Therefore LA 1 CJ(=B l! LKC S B 2 = C\, LMyAfi^B^. Therefore A t ^A 2 +B, and B^C^C^. i. 131 LECEXDBE 19 Hence Ay +By+C t =A 2 + B 2 + C\ sad area -4,^,(7, =area A 2 B 9 C 2 . By repeating this construction we get a series of triangles with the same area and angle-sum. TSowA 2 <!,Ay, A^lA^lAy, ... , A nM < %n Ay, B t <A ls B 3 <A 2 <^A lt ..., Bn+yK^-yA^ Hence the angles A n and B a both tend to zero, while the vertex C„ ultimately lies on A H B a , The sum of the angles thus reduces to the single angle C», which is ulti- mately equal to two right angles. In this proof there is a latent assumption and also a fallacy. In the first place it is tacitly assumed that the straight line is not re-entrant, for if it were re-entrant the " theorem of the exterior angle," upon which the proofs of the inequalities depend, could not be accepted, and the whole proof is invalidated. Again, if we grant the theorem of the exterior angle, B n and C„ both go to infinity, and we cannot draw any conclusions as regards the magnitude of the angle C„. I-egendre's other attempts make use of infinite areas. He makes reference to Bertrand's proof, and attempts to prove the necessity of Playfair's axiom in this way : if it be denied, then a straight line would be contained entirely within the angle formed by two rays, but this is impossible since the area enclosed by the angle is less than " half the area of the whole plane." 13. In Britain the investigations of Legendre stimulated such men as Playfaih and Leslie (Professors at Edin.- tmigh), Ivory, Perrostet Thompson, and Henry Meikle. °f these, however, none but Meikle had advanced beyond 20 HISTORICAL [1. 14 the stage of Legendre. Meikle J actually proved in detail, what had been foreshadowed fifty years before by Lambert, that if the sum of the artglea of a triangle is less than two right angles the defect is proportional to its area. He rejected the hypothesis because he would not admit the existence of a triangle with all its angles zero. He also proved independently Saccheri's general form of Legendre's second theorem. But by this time the epoch-making works of Lobachevsky and Bolyai had been published, and the discovery of a logically consistent system of geometry in which the parallel-postulate is denied proved once for all that all attempts to deduce this postulate from the other axioms are doomed to failure. It was not, however, in Germany after all that non-euclidean geometry at last saw the light, but simultaneously in remote districts of Russia and Hungary. 14. The discovery of Non-Euclidean geometry. Nikolai Ivanovich Lobachevsky (1793-1856), Professor of Mathematics at Kazan, was interested in the theory of parallels from at least 1815. Lecture notes of the period 1815-17 are extant, in which Lobachevsky attempts in various ways to establish the Euclidean theory. He proves Legendre's two propositions, and employs also the ideas of direction and infinite areas. In 1823 he prepared a treatise on geometry for use in the University, but it obtained so unfavourable a report that it was not printed. The MB. remained buried in the University Archives until it was discovered and printed in 1909. In this book he states that " a rigorous proof of the postulate of Euclid has 1 Edinburgh New Philos. Joum., 38 (1844 J, p. 313. i 18] LOBACHEVSKY AKD BOLYAI 2\ not hitherto been discovered ; those which have been given may be called explanations, and do not deserve to be considered as mathematical proofs in the full sense." Just three years afterwards, he read to the physical and Hiullicmatical section of the University of Kazan a paper entitled " Exposition succinte des principes de la geometric avec une demonstration rigoureuse du theoreme des paralleles." In this paper, the manuscript of which has unfortunately been lost, Lobachevsky explains the prin- ciples of his " Imaginary Geometry," which is more general than Euclid's, and in which two parallels can be drawn to a given line through a given point, and in which the sum of the angles of a triangle is always less than two right angles. In the course of a busy life Lobachevsky wrote some half dozen extensive memoirs expounding the new geometry. The first of these were in Russian, and therefore inaccessible. In 1840 he tried to reach a wider circle with a small book in German entitled Geonietrische Untersuchunffen zur Theorie der Paiallctlinien, and just before his death he wrote a summary of bis researches under the title " Pangeometry," which he put into French and contributed to the memorial volume published at the jubilee of his own University. 1 IB. Bolyai Janos (John) (1802-1860) was the son of Holv.m Farkas (Wolfgang) (1775-1856), a fellow-student and friend of Gauss at Gottingen, The father was early interested in the theory of parallels, and without doubt discussed the subject with Gauss while at Gottingen. The professor of mathematics at that time. A. G. Kaestner. had ' An Eniilisli translation of tho Geawrtiixtlit I'Memm-huni/fii was PuMiahcd by Hoisted (Austin, Texas. ISiUJ. An extensive Life ot Uroaohoveky was published, together with Oerman translations of two °i the Russian papers, by Engel (Leipzig, 1SS18). 22 HISTORICAL [i, 18 himself attacked the problem, and with hi help G. S. KlLigel, one of his pupils, compiled in 17G3 the earliest history of the theory of parallels. In 1804 Wolfgang Bolyai, just after his appointment as professor of mathematics in Maros-Vasarhely, sent to Gauss a " Theory of Parallels," the elaboration of his Gottingen studies. In this he gives a demonstration very similar to that of Meikle and some of Perronet Thompson's, in which he tries to prove that a series of equal segments placed end to end at equal angles, like the sides of a regular polygon, must make a complete circuit. Though Gauss clearly revealed the fallacy. Botyai persevered and sent Gauss, in 1808, a further elaboration of his proof. To this Gauss did not reply, and Bolyai, wearied with his ineffectual endeavours to solve the riddle of parallel lines, took refuge in poetry and composed dramas. During the next twenty years, amid various interruptions, he put together his system of mathematics, and at length, in 1832-3. published in two volumes au elementary treatise l on mathematical discipline which contains all his ideas with regard to the first principles of geometry. Meanwhile John Bolyai, while a student at the Royal College for Engineers at Vienna, had been giving serious attention to the theory of parallels, in spite of his father's solemn adjuration to let the loathsome subject alone. At first, like his predecessors) he attempted to find a proof for the parallel-postulate, but gradually, as he foeussed his attention more and more upon the results which would follow from a denial of the axiom, there developed in his mind the idea of a general or " Absolute Geometry " which 1 Tentamm jumntitkm /thidimam in elementa malkeseos . , . iniro- ihimidi, Muro.-i.\"ttsarhdy. IS32-3. ,. I.V BOLYAI 23 would contain ordinary or euclidean geometry as a special or limiting case. Already, in 1823, lie had worked out the main ideas of the non-euclidean geometry, and in a letter of 3rd November he announces to his father his intention of publishing a work on the theory of parallels, " for," he says, " I have made such wonderful discoveries that I am myself lost in astonishment, and it would be an irre- parable loss if they remained unknown. When you read them, dear Father, you too will acknowledge it. I cannot say more now except that out of nothing I have created a new and another world. All that I have sent you hitherto is as a house of cards compared to a tower." Wolfgang advised his son, if his researches had really reached the desired goal, to get them published as soon as possible, for new ideas are apt to leak out, and further, it often happens that a new discovery springs up spontaneously in mam- places at once. " like the violets in springtime." Bolyai's presentment was truer than he suspected, for at this very moment Lobaehevsky at Kazan, Gauss at Gotiinjren, Taurinus at Cologne, were all on the verge of this great discovery It was not, however, till 1832 that at length the work was published. It appeared in Vol. I. of his father's Tenfamen, under the title "Appendix, scientiam absolute veram exliibens." W. Bolyai wrote one other book, 1 in German, in which he refers to the subject, but the son, although he continued to work at his theory of space, published nothing further. Lobaehevsky 's Geome.fr inrhe Untersuckungen- came to his 1 ffwwr Omndrist einfet Versuelis, Maros-Vasarhely, lS.Tl. J. Bolyai's '■ Appendix " has been translated into French, Italian, Herman, Kn^Mali and Magyar; English by Halstcd (Austin, Texas, 18i)l}. A complete life of the Bolyai. with Herman translations of the " Appendix," parts of the Teiitam'en, etc., has been published by EUUckcl {Leipzig. HI13), a* ■ companion book to Engel's Lo6o(«/w/»K/. 24 HISTORICAL (I. IS knowledge in 1848, and this spurred hini on to complete the great work on " Rauinlehre," which he had already planned at the time of the publication of his " Appendix," but he left this in large part as a rafts indigestaque moles, and he never realised his hope of triumphing over his great Russian rival. On the other hand, Lobachevsky never seems to have heard of Bolyai, though both were directly or indirectly in communication with Gauss. Much has been written on the relationship of these three discoverers, but it is now generally recognised that John Boiyai and Lobachevsky each arrived at their ideas independently of Gauss and of each other ; and, since they possessed the convictions and the courage to publish them which Gauss lacked, to them alone is due the honour of the discovery, 16. The succeeding history of non-euclidean geometrv will be passed over here very briefly. 1 The ideas inaugu- rated by Lobachevsky and Bolyai did not for many years attain any wide recognition, and it was only after Baltzer had called attention to them in 1867, and at his request Hoiiel had published French translations of the epoch- making works, that the subject of non-euclidean geometry began to be seriously studied. It is remarkable that while Saccheri and Lambert both considered the two hypotheses, it never occurred to Lobachevsky or Bolyai or their predecessors, Gauss, 'Some of the later history will be Riven in Chap. VI. The best history <>t the subject is !{. Brmnla : I M gcometria tum-fiieiidfa : etpo- «iionr Htonm-rrthra ,1,1 «,to m-ilvppo (Holoimn. l!HH>) ; K na li.ili traiH i- tion (based on (he German translation by Liebmann. Leipzig 190S) by H. S. Cm-slaw (Chicago, 1<H2). A full' classified HbBoonipEy is lo ito found in fvmttiiervillc s BMiixjrnphyofwm-t.uclid, ■,,/. inrhidino the theory nfparallvb, the fumdalimw of gtomftn/ ,,»d */wrre ,,f n <Km«iuJoZ (London, iSJl I). Ex. i. RtEMAXN AXD KLEIN 25 Schweikart, Taurinus and Waehter, to admit the hypo- thesis that the sum of the angles of a triangle may be greater than two right angles. This involves the conception of a straight line as being unbounded but yet of finite length. Somewhere " at the back of beyond " the two ends of the. line meet and close it. We owe this conception first to Bernhard Riemanh (1826-1866) in his Dissertation of 1854 J (published only in 1866 after the authors death), but in his Spherical Geometry two straight lines intersect twice like two great circles on a sphere. The conception of a geometry in which the straight line is finite, and is, without exception, uniquely determined by two distinct points, is due to Felix Kletk. 2 Klein attached the now usual nomenclature to the three geometries ; the geometry of Lobachevsky he called Hyperbolic, that of Rieniann Elliptic, and that of Euclid Parabolic. EXAMPLES I. 1. If the angle in a semicircle is constant, prove that it is a right angle. 2. A !i is a fixed line and P ft variable point such thai the angle APH is constant. Show that the tangents at A and B to the curve locus of P are equally inclined to AB. 3. If every chord in the locus of Question 2 has the property that it subtends a constant angle at points on the curve, prove that the sum of the angles of a, triangle must be equal to two right angles. Examine the fallacies in the following proofs of Euclid's axiom : 4. If the side c and the angles A and B of a triangle arc given the triangle is determined, and therefore the angle 0-J\A, B, c). But aince this equation must be homogeneous, it cannot contain the 1 " fiber die Hvpoilirsen, welcho tier Geometric xn Grande liegen" ; English translation by W, K. Clifford, Nature. S (1873). : " CberdiesngenannteNicht-EiiklidischeGeoraetrie," Math. Anaalen, 4 US71), 8(1873). 26 HISTORICAL [Ex. . aide c. Hence C'-JIA, B). Let ABC be ii right-angled triangle, and draw the (JcrpcurlitTuJar CD on tlio hypotenuse. Then the two triangles ABC, ACD have two angles equal, and therefore tho third angle ACD—B. Similarly BCD=A. Therefore A + B+C =2 right angles. (A. M. Legendre, 1794.) 5. Let OB and NA be both perpendicular to ON, then OB does not meet NA. Ijet 00, making a finite angle GOB, be tho last line through which meets NA. Then A F .d can be produced beyond its point of intersection with OG to if, ami OK still meets NA. Hence 00 is not tho last, and therefore all lines through within the angle NOB must meet NA. (J. D. Gergonnc, 1812.) 8. One altitude AD of an equilateral triangle ABC divides it into two right-angled triangles, in which one acute unfile is double the other. If the three altitudes meet in 0, each of the triangles .-I OK, He, has one angle equal to half the angle of the equiliiti-ru! hinngle ; hence tho angle OAE-\AOE Heme the sum of the angles of the iriangle ABO is equal to half (lie .sum of the angles at 0, i.e. equal to two right angles. (J. K. F. Hauff, 1819.) 7. A A' ±AB and ABB' is acute. From D, any point on J3.B', is drawn the perpendicular BE to AB. O is any point on A A', and BC cuts Di£ in F. O is the middle point of EF, and BO meets AG in H. An isosceles triangle is drawn with base EF and sides equal to ED, making the base angles-ra. Rotate the plane of the ligure about AB through the angle «, Denote the point* in their new positions by suffixes. Then DyO±EF, and B1I is the pro- jection of WW,', // is t herefoiv the projection of a point on both WW,' and AA t % and these lines therefore meet (K. Th. E. Gronau, 1902.) CHAPTER n. ELEMENTARY HYPERBOLIC GEOMETRY. 1. Fundamental assumptions. In establishing any system of geometry we must start by naming certain objects which we cannot define in terms of anything more elementary, and make certain assump- tions, from which by the laws of logic we can develop a consistent system. These assumptions are the axioms of the science." The axioms of geometry have been class) tied by Hilbert ' under five groups : 1. Axioms of convection, or classification, connecting point, line and plane. Axioms of order, explaining the idea of" between." .-I.i/hm'n of i inij/mnHre. Axiom of prmillete. 5. Axioms of continuity. Without entering into these in more detail, 2 we shall assume as deductions from them, the theorems relating to the comparison and addition of segments and angles. The method of superposition can be used as a /nfou rfe parler. Strictly speaking, a geometrical figure is inoapablo of being moved ; * ' T). Hilbert, Qnuuttagen. der Gmmetrie, Leipzig, 1800, 4th ed, 1013 ; English translation by Townsend. Chicago, 15)02. "The reader who wishes In rtody r In- dsvakpSMBi of non-niclidean geometry from u set of axioms may refer to .1. L. Coolidge, Element* of Nttn.ftuclidean Geometry. Oxford, 1900 'Cf. Chap. VI. §4. 2. :;, I. 28 ELEMENTARY HYPERBOLIC GEOMETRY [». t linos are not drawn, nor arc figures constructed. It is only the act of the mind which fixes the attention on certain: geometrical figures wliioli already exist, developing them out, so to speak, like the picture on a photographic plate. And when we speak or applying one figure to another by superposition, all (hut we mean is that a comparison is made between the two figures and certain results deduced by the axiom of congruence. VVhen a geometrical figure, e.o. a line or a point, is spoken of as moving, we are really trans' ferring our attention to a succession of lines or points in different positions. The measurement of angles is independent of the theory of parallels. Vertically opposite angles are equal ; the sum of the four angles made by two intersecting lines is an absolute constant, and one quarter of this is a right angle. An absolute unit of angle, therefore, exists. A " flat- angle," which is equal to two right angles, is generally denoted by the symbol tt. Through a given point only one perpendicular can be drawn to a given straight line, the usual construction for this being always possible. The question of the numerical value of it, or, what is the same thing, of the unit of angle, need not concern us until we come to consider trigonometrical formulae (see g 39). VVe mav, however, state at once tnat when r is treated as a number it has just the value which wo are already accustomed to assign to it, viz, 3 J to a rough approximation, or. accurately, 4 times the sum of the infinite series - 3+Z-T+-- But it is necessary to warn the reader that ir does not stand for the ratio of the circumference of a circle to its diameter for in non-euclidean geometry this ratio is not constant s and the" radian, or unit angle, in terms of which a flat-ande is represented by the number tt, does not admit of the familiar construction by means of a circle. We shall assume, as deductions from the axioms of congruence, the congruence-theorems for triangles (Euc. I. 4, 8, 26), and those on the base-angles of an isosceles triangle (Euc. I. 5, G), which imply the symmetry of the plane. The theorems relating to inequalities among the II. 2] FUN DAM EXTAL ASSUMPTIONS 2!1 sides and angles of a triangle (Euc. I. 16-20) are true within a restricted region. In particular, the "theorem of the exterior angle," upon which the others depend, is proved in § 6 to be true without exception in hyperbolic geometry. An important axiom of order which must be explicitly mentioned is Pasch's Axiom. 1 If a straight line cuts one side of a triangle and- does not pass through a vertex, it teill also cut one of (he other sides (" side " being understood to mean the segment subtended by the opposite interior angle of the triangle). A large part of geometry can be constructed without the axioms of continuity, 2 but we shall in general assume continuity. The watershed, so to speak, between the euclidean and the non-euclidean geometries which we are about to develop, is the axiom of parallels. 2. Parallel lines. Consider (Fig. 9) a straight line I and a point not on the line. Let ON be ± I, and take any point P on I The line OP cuts I in P. As the point P moves along / away from N there are two possibilities to consider : (1) P may return to its starting point after having traversed a finite distance. This is the hypothesis of Elliptic Geometry. ('2) P may continue moving, and the distance NP tend to infinity. This hypothesis is true in ordinary geometry. The ray OP then tends to a definite limiting 3 position OL, 1 M. Punch, VortemtHgm iibcr neuere Qeomttrit, Leipzig, 1882 ; 2nd cd. 'Of. 0. B. Hetrted, Rational Geometry, New Vtirk, 1904. :> This assumes continuity. Wc might dispense with this assumption by assuming a definite fine Oh which separates the interscctors of 8 A from the non-intersectors. 30 ELEMENTARY HYPERBOLIC GEOMETRY fn. 3 and OL is said to be parallel to NA. If P moves along / in the opposite sense, OP will tend to another limiting position, OM, and OM || NB. In Euclidean Geometry, the two rays OL and OM form one line, and the angles NOL and NOM are right angles. The hypothesis of Hyperbolic Geometry is that the rays OL, OM are distinct, so that Playfair's axiom is contradicted. 3. In this chapter we shall develop the fundamental theorems of Hyperbolic Geometry. Definition of Parallel Lines. AA' is said to be parallel to BB in the sense thus indicated when (') A A' and BB' lie in the same plane, (2) AA' does not meet BB', both being produced indefinitely, and (3) every ray drawn through A within the angle BAA' meets the ray BB'. Through any point two parallels OL and OM can be drawn to a given line AB, so that OL || NA and OM || NB. The angles NOL and NOM are, by symmetry, equal, and this angle depends only on the length <if the perpendicular ON =p. It is called the angle of parallelism or the parallel- angle, and is denoted by U{p). There are two distinct senses of parallelism. tl. 4) P.YRALLEL LIXES 31 The two parallel lines separate all the lines through into two classes, those which intersect AB and those which are non-intersectors of AB. Properties of Plane Figure3, Parallelism, etc. 4. Parallel lines possess in common with cuclidean parallels the following properties : ( 1 ) The properly of parallelism is maintained, in the same sense, throughout the ichole lenglh of (he Urn. (Property of IraiifmissibilUg.} Let AA' [| BB', and let P be any point in A A '. We have FIO. 10. to prove that within the angle BPA' every ray through /' cuts BB'. and no other ray through P cuts BB'. There are two cases to be considered, according as P is on the side of A in the direction of parallelism or not. In the first case draw any line PQ through P within the angle BPA', and on it take a point K. Then the line AK must cut BB' in some point L, and BP in M. Hence PQ, which cannot cut. ag;tiu cither .1//, or BM, must cut the third side BL of the triangle BML (Pasch's Axiom). But PA' does not cut BB' ; therefore PA' || BB'. In the second case it is only necessary to take K on QP produced backwards. 32 ELEMENT ARY HYPERBOLIC GEOMETRY [m 4 (2) Parallelism, is reciprocal, i.e. if AA' || BB', then BB'\\AA' (Fig. 11). The bisectors of the angles BAA', ABB' meet in a point M, since each meets the other parallel, Draw perpendi- culars MP, MQ, ME from M on AA', BB', AB. By a comparison of the triangles these perpendiculars are equal. Draw MM' bisecting the angle PMQ. Then, if PQ is Joined, PQ1MM', and makes equal angles with AA' and BB'. The lines AA', BB' are therefore symmetrical PIO. 11. with respect to MM', and the reciprocity is therefore established. P, Q arc called corresponding points on the two parallels. (3} Parallelism is transitive, i.e. if A A' II BB' and BB' || CC, then 44' || CC. There are two cases to be considered. [a) Let BB' lie between ^^' and CC (Fig. 12). We may suppose ABC to be collinear. Within the angle OAA' draw any line AP. Since 44' || BB', AP cuts SB' in a point Q. Then, since QB' \\ CC, PQ produced within the angle CQB' must cut CC. Also A A' itself does not cut CC ; therefore AA' || CC". (ft) In the same figure let A A' and 2?S' be || CC. Then It- 5] PARALLELISM 33 any line within the angle CAA' must cut CC, and therefore BB'. Also 44' itself cannot cut BB', for then we would have two intersecting straight fines A A' and BB' both parallel to CC in the given sense. Therefore A A' |j R6'. Parallels in hyperbolic geometry are, however, sharply distinguished from euclidean parallels by the following property : The distance between two parallels diminishes in the direction of parallelism and tends to zero ; in the other direction the distance increases and tends to infinity. PlQ. 12. Before we can prove this we shall require several pre- liminary theorems. 5. If a transversal meets two lines making the sum of the i hi u nor angles on the same side equal to ttca right angles, the two lines cannot meet and are not parallel. Let PQ be a transversa] cutting the two lines AA' Bad BB' m p m & q (FJg . \s), a nd making the sum of the angles APQ + PQB equal to two right angles. Then , si nee the smn of the angles PQB+B'QP = W , therefore the alternate angles APQ and B'QP are equal. . *Wt PQ at M and draw MKlAA' and ML1.BB' 34 ELEMENTARY HYPERBOLIC GEOMETRY \n. 6 Then the triangles MKP and MLQ are congruent and LKMF=LLMQ, Therefore KML is o straight line, perpendicular to both AA' and BB'. By symmetry, if A A? and #S' meet on one side, they will also meet on the other. This is only possible in elliptic geometry. Also if A A' and BB' are parallel in one sense, they will be parallel also in the opposite sense, which is only true in euclidean geometry. Hence, in hyperbolic geometry they neither intersect nor are parallel. Fro. is. It follows that if a tramversal meets two parallel Urns it am Irs the sum of the interior angles on tJie side of parallelism less //torn two right angles. 6. An exterior angle of a triangle is greater than either of the interior opposite angles. Let ABC (Fig. 14) be a triangle with BC produced to D. Then if the exterior angle AG1) is not greater than the interior angle ABC it will be either equal to it or leas. Suppose first that tACD = ABC, then £ACB+ABC=t and BA, CA cannot meet (except in elliptic geometry). Second, if LACD<ABC._ draw BA' making LA'BC = AGD, ,i. 7] THEOREM OF TTIE EXTEKIOE ANGLE 35 Then BA' lies within the angle ABC and must meet AC, while the sum of the angles A'BC \ A'CB = ir. But this is impossible (except in eiUptic geometry). FlO. 14, Hence the " theorem of the exterior angle " is true, except possibly in elliptic geometry. 7. The parallel angle II (p) diminishes as tlte distance p increases. Let AA' and BB' be || MM' (Fig. 15), and ABM ±MM' ; and suppose AM >BM. Fm. is. Then £ MAA' +ABB'< it. (§ 5, Cor.) Bl « LMBB'+ABB'=tt. Therefore Z,MAA'< MBB'. to avoid further prolixity we shall assume, or leave as sxerdges to the reader, the theorems that II (p) is uniquely 36 ELEMENTARY HYPERBOLIC GEOMETRY [n. 8 defined for any value of p, and that there is a unique value of p corresponding to any acute angle as parallel-angle. Further II (p) is a continuous function of p. As y>-»-a>, II (p) -+0, and as p -+0, II (p) -* 1 [. The analytical expression for TI (p) will be found later in § 27. The range of p may be extended into the negative region. If we suppose the point A to move to the other side of the line MM', the angle MAA' will become obtuse, and we have, in fact, n(-y)+n(p)=Tr. 8. (a) Let ABNM be a quadrilateral with right angles at the adjacent vertices M, N, and let MA=NB. If we bisect MN perpendicularly by PQ we see from symmetry that the angles M AH and NBA are equal. Draw AA' and BB' \\ MN. Then, since MA =NB, the angles MAA' and NBB' are equal. But Z A' AB ■ + B'BA < w : therefore L B'BC >A'AB. Hence LMAB<NBC, and the angles MAB and NBA are acute. Thus, hyperbolic geometry implies Saccheri's Hypothesis of the Acute Angle, It follows, by considering the quadrilateral AMQP, that □. 0] SACCHERFS QUADRILATERAL if a quadrilateral has three right angles tlie fourth angle tmtst be acule. (6) // AM, BN are perpendiculars to MN and AM>BN, then the angle MAB<NBA. Via. 17. CatoSMA'=NB. Then LNBA >NBA' - MA'B > MAB, from the theorem of the exterior angle. Conversely, if /-MAB<NBA, then MA>NB. {Proof by r&luctio ad absurdum, using (a) and (&).) 9. The distance between two intersecting lines increases vrithoitt limit: Take two points P, P' on OA such that OP'>0P, and drop perpendiculars PM> P'M' on ON. Then the angle M'P'O and MPO are both acute. 38 ELEMENTARY HYPERBOLIC GEOMETRY [n. 10 Therefore L M'P'P < MPF, and hence M'P' > MP. Take any kiigth 'V. Let O.V h<- the distance correspond- ing to the parallel-angle AOJ, and draw NN' ION. Then NN' || 04. Take A 7 // >G, and draw a line BK making the acute angle N'HK. Then ///T, which lies within the angle OBN', must meet OA in some point E. Draw KL±0N. Then, since the angle tftftf is obtuse, LLKB<NHK\ therefore LK>NB>G. Hence the perpendicular PM can exceed any 'ength. 10. (a) The distance between two -parallels diminishes in the direction of parallelism and tends to zero. Let AA' || MM', and let AM. BN be two. perpendiculars dropped on MM' from points on AA'. such that B lies <m the side of A in the direction of parallelism. Then the angles MAA' and NBA' are both acute; therefore /LMAB<NBA, therefore NB<MA {§8(6), converse). Choose any length e, however small, and make MP<e. Draw PBLMA. If PX || MAT', ZMPJC is acute ; there- fore PB lies within the angle APX and will meet A A' in some point B, since PA' ,| AA'. Make ^tf/JP' =NBP, BP' ^BP, and draw P'M'±NM'. Then #P' neither meets nor is parallel to AM/', and BA' must lie within the angle M'BP', and therefore meets M'P' n . nl DISTANCE BETW KEN PARALLELS 3d in some point 4'. Then AT 'A' < M'P' < e . Hence the distance between the parallels diminishes indefinitely. Parallel lines are therefore asymptotic, and not equi- distant as in euclidean geometry. (/j) In the direction opposite to that of pamlklim the iisiawe between two parallels increases without limit. We have, in Fig. 19, AM>BN. Draw AL\\M'M (Fig. 20). From P, any point on A' A, draw PN±M'M, via. so. cutting AL in fi, and draw PK± AL. Then PN >PR>PK, and PK, the distance of P from Ah, can exceed any length. Hence PN can exceed any length. 11. Two parallel lines can therefore be regarded, as meeting 'it >,ilir,it>f. and, pother, the angle of intersection must be considered as being equal to zero. Fio. 21. Let AA'\\BB', and choose any small angle e. Draw 40 ELEMENTARY HYPERBOLIC GEOMETRY fn 12 AP making /_PAA'<e. Then AP cuts BB' in some point P. Make PQ=PA, and join AQ. Then £AQP = PAQ<PAA'<c. Hence as BQ^oa , AQ tends to the position AA', and /.AQB^O. 12. Non-intersectors. If two lines are both perpendicular to a third, they cannot meet and are not parallel ; and conversely, if two lines are non-intersecting and not parallel, they will have a common perpendicular. Let AB* and LX be any two lines (Fig. 22). From any point A on the one line draw a perpendicular AL to the Flo. 22. other. Then if AL is not perpendicular to both linea it makes an acute angle with AB' at one side, say the angle LAB' is acute. If BM is another perpendicular on the side of AL next the acute angle, and such that the angle MBB' is also acute, then MB< LA. The distance between the two lines thus diminishes in this direction, but unless the lines intersect or are parallel, it cannot diminish inde- finitely. Draw MM' \\ BB\ and let the perpendicular C'N' to LX from any point C on BB' meet MM' in Q. Theu n. 13] NON I XTI-: US 1KTUKS 41 0'N'>QN'. But QN'->oo ; therefore CW->«. Thus the distance between the two given lines AB' and LX at first diminishes and finally tends to infinity. It must, therefore, have a minimum value, and, if UV is that mini- mum distance, UV must be perpendicular to both lines. Hilherfs 1 construction for the common perpendicular. Take A, B, any two points on the one line (Fig. 22), and draw perpendiculars AL, BM on the other. If AL = BM, the common perpendicular is found by bisecting LM perpendicularly. Suppose AL>BM. Make LP = MB, and the angle LPP' = MBB'. Draw LL' || PP' and MM' || BB'. Then, from the congruence of the figures XLPP' and XMBB', the angles XLL' and XMM' are equal. Hence LL' is not parallel to MM', and therefore is not parallel to BB' ; nor docs it meet MM', therefore it must cut BB', There- fore, since PP' || LL', it must meet BB' in some point C. Make BC m PC. Draw the perpendiculars ON, C'N' to LX. Then, comparing MBC'N' and LPCN, we find CN-CN\ and the common perpendicular is found by bisecting NN' perpendicularly by UV. 13. If we make the common perpendicular to two non- intersecting lines zero, the two lines will coincide, but if the common perpendicular at the same time goes off to infinity the two lines may become parallel. Two straight lines maj' therefore be (1) Intersecting, and have a real angle of intersection, but no common perpendicular. (2) Non-intersecting, and have a real shortest distance or common perpendicular, but no real angle, 1 Gmndlatfen dtr (hointlH",, 2nd ed. (1803), Appendix III. § 1. 42 ELEMENTARY HYPERBOLIC GEOMETRY [n. W (3) Parallel, with a zero angle and zero shortest distance or common perpendicular at infinity. Before the principles of non-euclidean geometry became known, lines were Bomotimes classified as convergent, divergent and equi- distant. In fact, from the assumption that two straight Uaec oanoof first converge and then diverge wil (toot intersecting, Robert Simson (1756) was enabled to prove Euclid's postulate. In non-euclidean geometry equidistant straight lines cannot exist. Intersectors are convergent or divergent in the same sense as in euciktean geo- metry; parallels are convergent and asymptotic in one direction and divergent in the other ; non-intersectors are ultimately divergent in both directions. Planes, Dihedral Angles, etc, 14. If two planes have a point in common they have a line in common. 1 The diiiednd angle between two planes is measured in the usual way by the angle between two intersecting lines, one in each, plane, perpendicular to the line of intersection. If this angle is a right angle the planes are perpendicular. The usual proof in euclidean geometry that the dihedral anglo measured in this way is independent of the point chosen on the line of intersection involves pamllsk and another proof is required. Take P, I", any two points on the line of intersection of two planes a, fi (Fig. 23). Draw PA -P'A'±PP' in (be plane a, und PB=P'B'±PP' in the plane ji. Join PA' and P'A intersecting in U, and PIT and P'B intersecting in F. Then, by comparing the triangles PAP' and P'A'P, we find PA'^P'A Bad LPAV =P'A'U. Hence PU =P'U. Similarly PB'=P'B and i'F = P'F. Hence, by comparing triangles PUV, P'VV, we find lUPV = VP'V. Then, by oomparing triangles PA'B' and P'AR, we find AB=A'B'. Lastly, hy comparing triangles APB and A' P'B', w© obtain I.APB =A'P'B'. For the following theorems the usual proofs are valid. 1 This is mi wmmplion, explicitly excluding space of four or more dimensions. 151 SOLID GEOMETRY 43 If a straight line p is perpendicular to each of two inter- secting lines a, b at their point of intersection 0, it is perpendicular to every line through in the plane ab, and is said to be perpendicular to the plane ab. Every plane through p is perpendicular to the plane ab. The line of intersection of two planes which are both perpendicular to a plane a is perpendicular to a. Two parade! lines tie in the same plane (by definition). Two lines a, b, which are both perpendicular to a plane y, lie in a plane, lor if a, h cut y in A, B, then the planes aB and bA are both perpendi- cular to y, and therefore coincide. Three planes which have a point in common intersect in paira in three concurrent- straight lines. Three lines which intersect in pairs are either concurrent or coplanar. 15. {u) If two lities AA' and CC are both parallel to " third line BB', then AA' j| CC (Fig. 24). (The case in which all three lines lie in the same plane rw 44 ELEMENTARY HYPERBOLIC GEOMETRY fn. 15 haa already been proved in §4.) Take three fixed points A, B, C on the three lines, and any other point P on BB'. Fro. St. Join PA, PC. As P moves along BB', the plane PAC rotates about ,10. In tlie limit, AP and CP become parallel to BB', and coincide respectively with AA' and CC ; therefore AA' and CC lie in the same plane. Again, if CP is fixed while the plane PAC revolves, PA tends to PB' and the plane CPA to CPB'. CA, the line Fia. 2fi. of intersection of the planes CPA and V'CAA', therefore tends to CC, and CC \\ AA'. This result may be stated also in the form : two planes 101 LINES IX SPACE 45 whk'h pffss respectively through tieo parallel lines intersect in a line parallel to the two given lines. (b) If three planes a, 0, y intersect m lines a, b, c, such that a and b are neillier parallel nor intersect, then a, b, c are all perpendicular to the same plane (Fig. 25). Let AB be the common perpendicular to a, b. Then the plane through Ala passes through B and is X the plain; ab, and therefore ± b. Let this plane cut c in C. Then the planes ac and be are both perpendicular to the plane ABC, and therefore a, 6, c are all perpendicular to this plane. 16. Pencils and bundles of lines. A system of coplanar lines through a point is called a pencil of lines with vertex 0. The whole system of lines and planes through in space is called a bundle of lines and planes. FlO. S6. If a system of lines is such that each is parallel in the same sense to a given line, they are all parallel in pairs (§ 15 (a) ), and form a pencil or bundle of parallel lines, or a parallel handle. This is completely determined by one line with a given direction. 46 ELEMENTARY HYPERBOLIC GEOMETRY [a, n Denote a bundle of lines with vertex by 0, and a bundle of lines parallel to fin a given sense by Q. Then these two bundles uniquely determine a line Oil, which passes through and is parallel to Tin the given sense. Two parallel bundles Q, ff uniquely determine a line mi', which is parallel to both I and V, The line QIV may be constructed thus : Take any point A and determine Ail and Ail' (pig. 26). Bisect the angle HAQ', and take the distance AN corresponding to the parallel-angle kQAQ'. The line through N ±AN in the plane QAQ' is i\AQ and to Ail', So also we can prove that any three bundles, ordinary or parallel, uniquely determine a plane, for each pair determines a straight line, and the three straight lines thus determined are coplanar. 17, Points at infinity. To an ordinary bundle corresponds a point 0, but to a parallel bundle there is only a direction corresponding. We shall extend the class of points by introducing a class nl fictitious points called points at infinity. These points function in exactly the same way as ordinary or, as we shall nail them, actual points, and determine lines and planes with each other or with actual points. On every line there are two points at infinity, and the assemblage of points at infinity in a plane is a curve of the second degree or conic,' since it is met by any line in two points. In three dimensions the assemblage is a surface of the second degree or quadric. This figure, the assem- blage of all the points at infinity, is called the Absolute. ' The definition of a eonie which m shall use is " a plane curve which is cu by any straight fine in iu plane in two potato." W- UwwptaSSfan of the case of " imaginary " intersection see Chap III § 5 a ^ mmma n. 18] POINTS AT INFINITY 47 When two points at infinity approach coincidence, the line determined by them becomes a tangent to the absolute. As Q, Q' approach, the angle QAQ' in Fig. 26 tends to zero and AN-><x» . The line QiY therefore goes off to infinity. Such a line is called a line at mfimiy. Similarly we obtain dunes at infinity, which are tangent planes to the Absolute. In euclidean geometry there ia just one parallel through a given point to a given lino in a plane, and the two points at Infinity upon a line coincide. The assemblage of points at infinity in a plane then reduces to a double line, the line at infinity, which is a degenerate case of a eonio. There ia in this caw; only one real line at infinity ; but any line whose equation in rectangular coordinates is of the form x±iy+c-0 is at an infinite distance from the origin, since I ;• _o, tmd the assemblage of these lines consists of two imaginary pencils. The equation of the line at infinity is asOr + 0y+ I =0, and the equations of the two pencils are <u + Aa=0, u>' + Aa=0, where w, b>' —x + iy. The absolute in euclidean geometry thus consists, as a locus of points, of the line at infinity it taken twice, and, as an envelope of lines of two imaginary pencils <u+Aa = 0, «u' + Aa- 0. with their verti.es on the line at infinity. These two imaginary vertices are the points of intersection of the point-circle um>' ■ iP + lf -0 with the line at infinity Since the equation of any eirele can be written uu'+na=0, where u--0 represents a straight line, we see that every circle passe.* through the two points (wiu'=0, a = 0), and for this reason these twu imaginary points are called the circular poitO*. In euclidean geometry of three dimensions the absolute consists, aa a locus of poinK of the plane at infinity taken twice, and, as an envelope of planes, of all the planes through tangents to an imaginary eirele. the intersection of the point-sphere 3? + >f + zr=0 with the plane at infinity. The whole of metrical geometry is determined by the form of the Ai -'"lute ; this will be more fully treated in Chap. V. 18. Ideal points. // a system of lines is such that any kto are co-planar, while tlic>i do not all lie in the same plane and are neither parallel nor intersect, then they are all perpendicular to a fixed plane. 48 ELEMENTARY HYPERBOLIC GEOMETRY [,.. is If a, b are any two of the lines they determine a plane T which is perpendicular to both. If c is any third line' which does not lie in the plane ab, it is the intersection of two planes m and be, which are both 1 T , and therefore o is X »(gl5(6)). We shall call this system, which is completely determined by two of the lines, or by a certain plane v, a bundle of lines with an ideal vertex 0. The plane * is called the axis of the bundle. All those lines of the system which lie m a plane are perpendicular to a straight line I, the inter- section of their plane with the fixed plane, and form a pencil of lines with ideal vertex 0. The line I is called the axis of the pencil. The ideal paints thus introduced behave exactly like actual points. They can be regarded as lying outside the absolute, and are therefore ultra-spatial or ultra-infinite points. Two ideal points may determine a real or actual line Considering only points in a plane, the two ideal points are determined by two lines a, a'. If a, a' are non-intersecting the common perpendicular to these lines belongs to both pencils, and ,s therefore the line determined by the two ideal points. If « || «', the line 00' is a line at infinity • if a cute «', 00' is an ideal tine, which contains only ideal points. An ideal line lies entirely outside the absolute Similarly, m three-dimensional hyperbolic geometry we lm-i! xl-nl planes, It is left to the reader to show now that any two points actual, at infinity or ideal, always determine uniquely a line actual, at infinity or ideal ; and that any three points actual, at infinity or ideal, always determine uniquely a plane; actual, at infinity or ideal. it. is] IDEAL POINTS 49 These relations, in two dimensions, can be pictured more dearly if we draw a conic to represent the absolute, or assemblage of points at infinity (Fig. 26 bis). Actual points are then represented by points in the interior of the conic, ideal points by points outside the conic. Lines which intersect on the conic represent parallel lines, those Pointi at >* T Fro. 2» bis. winch intersect outside the conic represent non-inter- sectors. If Oil and 012' are the tangents to the conic from an ideal point 0, all the lines of the pencil with vertex are perpendicular to ilQ'. For the present this may be used as a mere graphical representation. Its full meaning can only be understood in the light of projective geometry. (See Chap. III. §§ 5, 6, and Chap. V. §§ 1-14.) 19. Extension to three dimensions. If the point of intersection of a line with a plane is at infinity, the line is said to be parallel to the plane. If the point of intersection is ideal there is a unique line and a unique plane perpendicular to both the given line and the ^IViill |>l;uie. x. ■ K. a. D 50 ELEMENTARY HYPERBOLIC GEOMETRY [a 20 Two planes intersect in a line. If this line is at infinity the planes are said to be parallel : if it is ideal the two planes are non-intersecting and there is a unique line perpendicular to both. All planes parallel to a given lino in a given sense pass through the same point at infinity and intersect in paira in a parallel bundle of lines. All planes perpendicular to a given plane pass through the same ideal point and intersect in pairs in a bundle "of hues with ideal vertex. The following theorem is of great importance : Through a line which is parallel to a plane passes just one plane which is parallel to the given plane. Let the line I out the plane « in the point at infinity 0. Through D passes just one line at infinity », and this" line determines with I a unique plane, which is parallel to a. The actual construction may be obtained thus : Take any point A on I and draw AN J_ «, Through A draw AB J. the plane IN. Then Bl is the plane required. Through a line which meets a plane a in an ideal point pass two planes parallel to the plane «, for two tangents can be drawn from to the section of the absolute made by the plane a. 20. Principle of duality. There is a correspondence between points and lines in a plane, and between points and planes in space, which gives rise to a sort of duality. To an actual plane a corre- sponds uniquely an ideal point A, all the lines and planes through which are perpendicular to the plane « ; and to an actual point A corresponds an ideal plane «, which is perpendicular to all the lines and planes through the n. 211 THE CIRCLE 51 point A. Let B be any point on a ; then the plane R which corresponds to B must pass through A, since every plane perpendicular to « passes through A. If the plane it is at infinity the corresponding point A is its point of contact with the absolute. The points and planes are therefore poles and polars with regard to the absolute. This reci- procity will appear again in elliptic geometry, where the elements are all real. The Circle and the Sphere. 21. The circle. In a plane the locus of a point which is at a constant distance from a fixed point is a circle. The fixed point is the centre, and the constant distance the radius. A circle cuts aU its radii at right angles. This follows in the limit when we consider a chord PQ, which forms an isosceles triangle with the two radii CP, CQ. That is, a circle is the orthogonal trajectory of a pencil of lines with a real vertex. Let the vertex go to infinity ; then the lines of the pencil become parallel, and the circle takes a limiting form, which is not, as in ordinary geometry, a straight line, but is a Uniform curve. This curve, a circle with infinite radius, is called a horocycle; it is the orthogonal trajectory of a pencil of parallel lines. The parallel lines, normal to the horocycle, are called its radii. AH horocycles are superposable. To obtain the orthogonal trajectory of u pencil of lines with ideal vertex we proceed thus : Let AA' be the axis of the pencil (Fig. 27), and draw perpendiculars to A A'. Cut off equal distances MP, NQ, ... along these perpendiculars. Then the locus of P is again a uniform curve, which is not, as in ordinary geometry, a 52 ELEMENTARY HYPERBOLIC GEOMETRY fn.22 straight line; and the curve cuts all the perpendiculars to AA' at right angles. It is therefore the orthogonal trajectory required. From the property that the curve is e.|iiiclistaiit from the straight line A A' it is called an equi- -7- FIO. 27. distant-curve. The complete curve consists of two branches, symmetrical about the axis, and also symmetrical about any line (radius) which is perpendicular to the axis. As the axis tends to infinity, the perpendiculars tend to become parallel, and the equidistant-curve becomes a horocycle. We can thus pass continuously from an iii|iiirlistatit-curve to a circle. When the axis goes to infinity the centre also appears at infinity ; then the axis becomes ideal and the centre becomes real. There are therefore three sorts of circles : (1) Proper circles, with real centre and ideal axis. (2) Horocffcks, with centre and axis at infinity. (3) Equidistant-curves, with ideal centre and real axis. A straight line, or rather two coincident lines, is the limiting case of an equidistant-curve when the distance vanishes. 22. The sphere. In space of three dimensions the locus of a point which is equidistant from a fixed point is a sphere. It is the ortho- gonal trajectory of a bundle of fines with a real vertex. When the centre is at infinity the surface is called a horo- a. 23] THE CIRCLE AND THE SPHERE 53 sphere, and when the centre is ideal the surface is an dji'idistanl-surface to a plane as axis. A plane section of a sphere is always a circle ; the greatest section, or the section of least curvature, is a diametral section passing through the centre, that is, a great circle on the sphere. A plane section of a horosphere is a circle, except when the section is normal to the surface, i.c. passes through a normal, in which case the section i3 a horocycle. A section of an equidistant surface by a plane which does not cut the axial plane is a circle ; if the plane cuts the axial plane the section is an equidistant-curve with the line of intersection as axis ; if the plane is parallel to the axial plane the section is a horocycle. 23. Circles determined by three points or three tangents. Lot A, B, C be three given points I to ''"id the centre of a, circle passing through A, B, C. Bisect the joins of the three points perpendicularly. If two of tho perpendiculars meet, all three will he concurrent in the centre required. 54 ELEMENTARY HYPERBOLIC GEOMETRY [n. Suppose the perpendicular bisectors A' A - ' and LL' to AB and BC are non- intersecting Let N'L' be their common iwpendtoafc* Leu this line cut the perpendicular at M in if', and draw iho popendicobn A A', BB', CC. Then, since NN> bisects AB perpendicularly and is J. A'B'. AA'=BB\ Similarly BB' =CC. In the quadrilaterals AA'M'M, CC'M'M, AA'=CC, AM=MG Z ,' e ^"!f„ at . A i\f amI °' m ^ ht ' * llerafore « t«e quadri- lateral «rjrjf be folded over i/jf, C will coincide with J, and •moo only one perpendicular can he drawn from A upon 11/' «v wl 1 cornc.de with ^ and the angles at W am right, Henee ii , MM , NN are all perpendicular to A'C, and ABC he on an equidistant-curve with axis A'B'C, Suppose ' AW'HiZ,'; then VJ/' must be parallel to both. For, ,1 «!J« ' Cn by th ° first M " the fchreo lines »» concurrent ■ and fUW » a non-mtersector to LL', then by the second case Li/ and A A are nou-mtcrsecttog. Therefore J/Jf'||£i/ ABC then lie on a horocyete. ' ' In addition to the circle, equidistant-curve or horocycle, which ean he drawn through ABC in this way, there exist thr^ cqui- chstant-ourves auch that one of the points lies on one branch while the other two he on the other branch. Bisect AB, AC at if and ,V Join MN and draw the perpendiculars A A', BB', CC" to MN (See Fig. 50, p. 77.) Then AA'^BB'^CC, and an equlranT curve with u. i/A passes through £, C and ,1. A triangle has therefore four eiroumeireles, at least three of which are cquitlisUmt- curvee. There cannot be more than one real circumcentro. This point, which wo may call the cireumcentre, is the point of concur- rence of the perpendicular bisectors of the sides, and may be real at inlinity. or ideal. J ^ _ If L is the middle point of BC, the perpendicular from h on MN is also ± BC, since it bisects the quadrilateral BB'C'C. Hence the altitude. , of the triangle LMN are concurrent in the cireumcentre of the triangle ABC. A triangle therefore possesses a unique arttocnto, real at infinity, or ideal. If the orthocentro is Meal there is a real orthaxu, which is perpendicular to the three altitudes. The construction for the circles touching the sides of a triangle is, as usual, obtained by bisecting the angles. Three of the circles may be equidistant-curves or horocyclea. 24] GEOMETRY OF A BUNDLE 55 24. Geometry of a bundle of lines and planes. In plane geometry we have points, lines, distances and angles ; in a bundle of lines and planes through a point we have lines, planes, plane angles and dihedral angles Let us change the language to make it resemble the language of plane geometry. Tn translating from one language to another we require a dictionary. The following will suffice : " Point " - - Line through 0. " Line " - - - Plane through 0. " Distance " between Angle between two lines two ,: points " - through 0. " Angle " between Dihedral angle between two " lines " - two planes through 0. " Parallel lines " - Parallel planes. Then two " points " determine a " line " and two " lines " determine a " point." " Parallel lines " only exist when bat infinity or ideal. When is at infinity, through a given " point " there passes just one " tine " " parallel " to a given " line " (§ 19), and when is ideal, two " parallels " can be drawn through a given " point " to a given " line." There are therefore three kinds of geometry of a bundle according as the vertex ia actual, at infinity or ideal, and these are exactly of the same form as elliptic, parabolic {i.e. eticlidean) and hyperbolic plane geometry. If a sphere be drawn with centre cutting the lines and planes f the bundle, we can get a further correspondence. When is an actual point we have a proper sphere. We nave then the following dictionary : " Point " - Pair of antipodal points on sphere. " Line " - - Great circle. 56 ELEMENTARY HYPERBOLIC GEOMETRY [i,.a 6 "Distance" - " Angle " between " lines " Length of are. Angle between great circles. Hence the geometry on a proper sphere, where great circles represent lines, and pairs of antipodal points repre- sent points is the same as elliptic geometry. (See farther Chapter III.) When is ideal, the sphere becomes an equidistant- surface, and its geometry is hyperbolic ; when is at infinity it becomes a horosphere, and ite geometry is cuchdean : '< point " in each case being represented by a point, and "lines" by normal sections, which are also shortest lines or geodesies on the surface. We have here the important and remarkable theorem that the geometry on (lie liorosphere is eudidean. Trigonometrical Formulae. 25. We shall now proceed to investigate the metrical relations of figures, leading up to the trigonometric;,! formulae for a triangle. The starting point is found in a relation connecting the arcs of concentric horocycles and this leads to the expression for the angle of parallelism The great theorem which enables us to introduce the circular functions, sines and cosines, etc., of an angle is that the geometry of shortest lines (horocycles) traced on a horosphem ra the same as plane ouclidean geometry. Let- A, B, C be three points on a horosphere with centre Q The planes ABQ, etc., cut the surface in horocycles, and we have a triangle ABC formed of shortest lines or geodesies which , are arcs of horocycles. The angles of this triangle are the angles between the tangents to the arcs or the n. »] TRIGONOMETRY r,7 dihedral angles between the planes QAB, 9.BC, Q.CA. If the angle at C is a right angle, then the ratios of the arcs are lie AB AC =sinJ, 2b=cosj4, etc. Hie circular functions could bo introduced independently of the horosphere by defining them as analytical functions of the angle 0, vte. i 0j ff, 3! %m6 = 0-^ + ", -■ 61 coa »= 1 — ijT + j i - *•• i the unit of angle being such that the measure of a fiat-angle is 7T -3-14160... . Wo may call this "circular measure." Then it could be shown that if ABC "m » rectilinear triangle, right-angled at V, the ratios BC/AB and AC/AB tend to the limits sin A and cos A as BC, AC and AB aU tend to zero, while the angle A is fixed (Cf. Chap. HL § 18, footnote). 26. Ratio of arcs of concentric horocycles. Let A SI, B t Q be two parallel lines, and let them be cut by horocycles A t B lt A n B^ /i 3 B 3 with centre at infinity O. A Then the ratio of the arcs A& : A t B, depends only on the distance A x A t =x. (See Ex. 26 and 26.) AJ! 2 A x B t U. *£-JWi then ffi-M — lfcr*>** Therefore f(x + y) =/(«) - f(y). 68 ELEarENTARY HYPERBOLIC GEOMETRY |.,.» This Is the fundamental law of indices, and the function is therefore the exponential function : f(x) »e» c being an absolute constant greater than unity. f( x ) is a pure ratio, and must be independent of the' arbitrary umt of length which is selected ; therefore bg/(«) or x logfi must be a pure ratio. Hence log c must be the reciprocal of a length. We shall put log, c = 1 ft ; then . , . /(*)=«* where k is an absolute linear constant and e is the base of the natural logarithms, k is called the space-constant ■ its actual value in numbers of course depends upon the* arbitrary unit of length which is selected, but it forms itself a natural unit of length, and it is often convenient to make its value unity. This is one of the most remarkable facts m non-euclidean geometry, that there is an absolute umt for length as well as for angle. It can be proved (see §39) that k M th length of the are of a horocyde which is sucli thai the tangent at one extremity is parallel bo the radius through the other extremity. 27. The parallel-angle. We can apply this now to find the value of the parallel- angle II {,,) m terms of p. This is the simplest case of the determination of the relations between the sides and angles of a triangle. The triangle in this case has two sides infinite, one angle right and another angle zero. Let AP H Bl\ and ABLBP (Kg. 30}. Erect a perpen- d 1C ular at A to the plane of ABY. Draw the parallels BQ and FU Draw the horosphere with centre at infinity Q and passing through A, and let it cut Bll in B' and TO in C. Let BB' = y, and the a res B'C, C'A ,AB'bea,b c l! 87] THE PARALLEL-ANGLE m Since BV±A£ the plane IIBV .L plane QAB, and since the angles which the arcs AB', B'C, C'A make with the lines QA, ilB, Off are all right, LC'AB'=VAB=T\{p), LAB'C = \, FIB. 30. and since geometry on the horosphere is euclidean, LAC'B'^-ll{p). ft f* Hence sin II (p) - , and cos ll(p)=L\ ff therefore tan. , .n(p) = , ■* ' o + c, The arc of the horocyelc b is a standard length, viz. the length of the arc which is such that the tangent at one extremity is parallel to the radius at the other extremity. In Fig. 30 BK is such an arc, and =6. Hence b .(1) = e*. (2) 60 ELEMENTARY HYPERBOLIC GEOMETRY, ,, „ I ^ ^ the PARALLEL-ANGLE 61 Now fold the plane QAV about QA until it lies in the plane ilAB (Fig. 31). Draw TB"±BQ. Then, if we draw FIO. 31. the horocyclic arc VCT with ceritref2i thisar{J = 6 ■o/r -p ; therefore £'#" = j, _ y _ Therefore ^t_ c _/-r v ,« Hence, multiplying (2) and (3) and using (1), we have This relation may be put into other forms, e.g. cot 11 (p)=sinh£; other equivalent forms can be read off from the accom- panymg figure (Fig. 32), treating the figure as a euclidean triangle. We shall effect a simplification by taking in the following paragraphs (§§ 28-37) the constant k as the unit of length. Utpi Flo. 32. The formulae may be restored in their general form by dividing by it every letter which represents a length. 28. Two formulae for the horocycle. Let AB=s be an arc of a horocyclc with centre £1, and let S be the length of an arc of a horocycle such that the tangent at one end is perpendicular to the radius at the other end (Kg. 33). Let s<S. Extend BA to M so that BM=8; then AM=S-s. Take A, on the radius through A so that the perpendicular at A. to A X A is parallel to MM* Then the arc A& -S. Let BQ l cut Afil in D. Then DA^DB-t s*y. 1** DA - %i. Then we have, comparing the arcs A^ and AM , S-s^Se-'-' 1 W Extend AB to N so that BN «& Take 4, on the radius through A so that the perpendicular at A % to AA % is parallel to MJf. Then the arc A 2 M^S. And since BH || DA t , and 4 S S2 2 |! DB, Z)4, =-!>»-=*. Then g+*«Sfi»-». ■ < 2 ) Adding these two equations, we get or e" = i( e ' +e '') =cosl1 ' ^ 62 ELEMENTARY HYPERBOLIC GEOMETRY t „J Substituting in (1), wc get V e'+e- t J- A ^r^ St&ahL - (a) Then • -W-Stanh/.eoBh/.Sainln „/ B) These two formulae (a) and (b) give the tangent and ordmate at the extremity of an arc of a horo^Ctiz t * is the arc AB of a horocycle, t the length of the talent AT .ntercepted between the point of contact A an i TZ s=-Stanh(=£sinb^. ,,.30] THE RIGHT-ANGLED TRIANGLE 63 29. The right-angled triangle; complementary angles and segments. Let ABC be a right-angled triangle with right angle at C. Denote the sidea by a, b, the hypotenuse by c, the angles opposite the sides by A, /jl. Let a = II (a), etc. Then we have five segments and five angles connected by the relations a=II(a), $ = IL(b), y = II(c), X=H(0, M=U(m). Fta. Si. Let a' denote hw-a] then we have the complementary segments and angles «' = ll(a'), £'=11(0'), etc. We have to deal with the circular functions of the angles, and the complementary angles are of course connected by the relations sin a - cos a, tan «' = cot a, etc. We have also to deal with the hyperbolic functions of the segments, and we have the relations sinli a = cot H {«) = cot a = tan a = tan 17 (o') = cosech a ' , cosh a = cosee II (a) = cosec n = sec a = sec II (a') = coth a', 30. Correspondence between rectilinear and spheri- cal triangles. Draw AQ ± the plane of the triangle (Fig. 35), and draw Bil and CQ [| /IQ. 64 ELEMENTARY HYPERBOLIC GEOMETRY [„. m Then BQ || 0ft, and BC ± plane 4CQ ; therefore BC±C9 the P«a*eUIM 1 plane ^C, and the angle between the planes QfiCand ABV =11(1). AW since the planes 9AB Fin. S5. OK7 00* intersect in parallel lines, the sum of the angles of mtaaeetioii-T; therefore the angle between theplaL 9ABmd9BC = ^-\. Now draw a sphere with centre B, and we get a rieat- aagted spherical triangle with hypotenuse f Jll(a) skies M andy = n( C ), measured by the angles which they sub- tend at the centre, and opposite angles X'«?-X and .31] ASSOCIATED TRIANGLES = 11(6), i.e. to the rectilinear triangle (c, cA, b/x) corre- sponds the spherical triangle (a, ft\\ y B). 31. Associated triangles. To the spherical triangle (a, /u X', y B) we get four other associated triangles by drawing the polars of the two vertices (cf. Chapter III, § 20). This gives a star pentagon (Fig. 36) whose outer angles are all right angles. The five associated right-angled triangles have the parts indicated in the figure. The inner simple pentagon has the measure of each side equal to that of the opposite exterior angle. If we write down in cyclic order the parts X', ft', «, y', B as they occur on the sides of the simple pentagon, the parts of the five associated spherical triangles can be written down by cyclic permutation of these letters, thus : I. a , ft X', y B. 2. /, a ft, jS'X'. 3. B , y o , X ft . 4. X', £'y\ ft a . 5. fi, X B , a' y'. N.-E. O, 66 ELEMENTARY HYPERBOLIC GEOMETRY [a. 32 „.33l Corresponding to these we get five associated rectilinear triangles : NAPIER'S RULES 67 1. e, 2.6', 3. I , 4. m, 5. a', X, c' b a, V y, b p. f «' m* y a 0' c' X Hence, if we establish a relationship between the sides and angles of one triangle, we can obtain fonr other relation- ships by applying the same result to the associated triangles, or by a cyclic permutation of the letters (I'm'ac'b) (Ima'cb') (xy« v 73)(x MU v/3')- 32. Trigonometrical formulae for a right-angled triangle. Produce the hypotenuse AB to D so that the per- pendicular at D to AD is parallel to AC, Then AD=l, BD-l-c. Draw the horocyclic arcs with centre fi passing through D and B. Then (by § 28 (a), (b) and (8).) g 1+ s s =S tanh?, s 2 =S tanh (l~e), s, =se _ " =S sinh a/cosh (J - c). Therefore tanh I =tanh (I - c) + sinh a cosh (I - c), sinh a cosh I - sinh I cosh {I - c) - cosh I sinh (I - c) = sinh c. sinha = sinhc sinX. From the associated triangles we get sinh c' =sinh b' sin fx ; therefore sinh b =sinhc sin w . sinh b = sinh I emu sinh/' =siuh»isiny sinhm' =sinho' sin ,6' Prom (3), (4) and (5) we get (1), (4) aud (5) (1), (2) and (3) (1), (2) and (5) (2), (3) and (4) sinh b = tanh « cot X. cosh c = cot X cot ju. sinh a = tanh 6 cot.u. cosh c = cosh a cosh b. cos X - tanh b coth c. cosX =cosha sin/*, cos/* = cosh b sin\. cos ft. = tanh a coth c. ( 10) (1) (2) (3) (4) (5) (6) (7) (8) (9) 33. Engel-Napier rules, These ten formulae, which connect all the five parte of the triangle in set* of three, are of exactly the same form as the formulae of spherical trigonometry, with hyperbolic Fro. 37. fin, 38. 68 ELEMENTARY HYPERBOLIC GEOMETRY [it 33 functions of the aides instead of circular functions, and can be written down by Napier's rules.' If we write the five parts b, X, e, ft, a in cyclic order as they occur in the triangle (Pig. 38), then sine (middle part) = product of cosines of opposite parts | A ] = product of tangents of adjacent parts it being understood that the circular functions of the angles' and the hyperbolic functions of the sides, are taken and each function of x is the " complement " of the correspond- in.-; function of s, i.e. cosh i =sinh c, tan X = cot X, etc. | Xotb.— A lias the same meaning as X', but c is not the same as c'.] Tim rale may be put in another form, which fa more homogeneous. II. Hi NAPIER'S RULES i><) Fro. 99. tS !" th0 fo ™« la * i» t«nn« of the segments a', *, c, - V. The formulae become : t ,,»•»» . coah c =sinh I sinh m =eoth n' eotli 6' £l rffl£ *?r h t° Sl '° mc lF was kid h - v Lo'^evskv, XZ Found.. - r 7 *£""£?' o!m P- * H ma! «* use of the dianriims in tinflM Kl? • ?2£ J?-" n, r ifieti forraa of «N*rt Rales W ™«oN 8h od L kngel ■■> on edition of Lobachevsky's , VeM » JPowdWftw* p s« * and four otber pairs obtained by tbe cyclic permutation (hna'cb') or {a'fcinb'}. If we write the five parts a', I, c, m, b' in cyclic order (Fig. 39), then cosh (middle part) = product of hyp. sines of adjacent parts ( B ) = product of hyp. cotangents of opposite parts. It is easily verified that this rule holds, with circular functions instead of hyperbolic functions, for a spherical triangle in euclidean space with hypotenuse e, aides I' and m' and opposite angles a' and 6'. 34. Expressing the formulae in terms of «', X, y, m. /3'. we get, since cosh x = eosec fr", sinh x = cot £, coth a; = sec £, where x stands for any one of the letters a, b, c, I, m, and £ for the corresponding Greek letter a, j3, y, X, ft, sin y = tan X tan ft = cos a' cos /3', Fig. 40. with four other pairs obtained by the cyclic permutation («'XyM|8'). They may be read off from Fig. 40 by applying Rule (b) with circular functions. But these are the formulae for a right-angled spherical triangle with hypotenuse y', sides «' and £' and opposite angles ft' and X' ; or one with hypotenuse a, sides ft and y L ( 70 ELEMENTARY HYPERBOLIC GEOMETRY [n. 35 n. 88] TRI-RECTANGULAR QUADRILATERAL 71 and opposite angles X' and £ But this is just the spherical triangle which we found to correspond to the rectilinear triangle (§ 30), Hence (he formulae for a spherical triangle in kyperbohe space are exactly the same as those for a. spherical triangle in eudUean sjxice, when we take as the measure of a side the angle which it subtends at the centre, and as the measure of on angle the dihedral angle between the planes passing through the sides and the centre (Cf Chap. III. § 21.) It may be noted that the letters M ', a , y\ P, V in Fig. 40 are the same, and in the same order, as those on the sides of the simple pentagon in Kg, 36. 35, Correspondence between a right-angled triangle and a tn-rectangular quadrilateral. c£ua!%£ ° A ' md mXBA and " CA (Fig " 4J) " Tl,en A D ■* a-li=Il{c+l).... -( ij P-X=Yl(c+m) (j/j Similarly Draw Bfl II AC, and DQ±BA and | AC (Fig. 42). Then AD ~l, ££8=", and a+/*=II(e-I) <*' Similarly + A=n (C - m ) « \_o- -\C FlO. 42. Wote.— If l>c, then ir-(cn-/i) = II(c-J); ifl=c «+/*=II(0)=-; which are both contained in (2) if we understand that Draw PQ1C4 and || A*, and MJ-iKJ and H iL4 (Fig. 43). o. KlO. JL f 72 ELEMENTARY HYPERBOLIC GEOMETRY [., 35 Then AD = *, BB = m , MOa = U lm - a) , DCi^U^l), md n{m-a] + Tl(l + b)^. (3) Similarly mm+a t + uil -(,)=- _ ( g,. In the txi-rectangular quadrilateral with angle & and eide8 ffl order n{OT+a)+n(f-6)=!T. u c^ J Re. 44. *. «, v, ta, draw SB fl ZW; and X ^5 (Fig. 44). Then fi £ = u% for ii (!t ) + n ( «')= J ^fl^ntn ^fl=H(* + «',, and n(«,)+n((+ W 'j = ^. ,. fI Similarly IJ(f)+n<» + ir> ft .... .' ' .Draw AH || CD, and £fl^fl and B 67) (Fig. 45). Then «^ Mil = !!(«,), B/lii -r IT<< -«'), and JI(w)+e=IT{(-«') m si,nilarI y n«) * e=ij( W -yj (lr> /a 1 '™- 2^/1 MfcJPiS' and ™ P%iT«- r^-u , and d /UP=/, fl=I7(/), and we have H(©'-t»)+n(w+/)«=|. (I11) Similarly n (y '-u) + n(r ( /) = 5. (m<) 11.35] TRI-RECTANGULAR QUADRILATERAL 73 Now the quadrilateral is determined by l and «. L*t (=c and a=»i'. Then II(e + i») = 0-n{w)=/3- A, from (I) and (1'). n(c-H») = 0+n(u>)=j3+A ( from (I') and (2')- Therefore 0=fi and ri(w>)=A ortc=J and/=6. A Then, comparing (HI) and (3), we have Jl{m-v)= z-Ii(f+b) = Tl{m-a)i therefore v=a. 74 ELEMENTARY HYPERBOLIC GEOMETRY [u. 36 Hence, to a right-angled triangle {c, a A, b ft) there corresponds a tri-rectangular quadrilateral with angle and sides, starting from the angle, c, m' t a, I By reversing the order of tho sides, we get the quadrilateral (/?, I am' a), to which corresponds a triangle (I, m'y, b a'), or {I, b a', m'y). Fig. 47. Flo, 47. If we take a', I, c, m, b' as quantities determining the parts a, A, c, ft, b of the triangle, then we get a triangle corresponding to the quantities b', a', I, c, m, and similarly, by cyclic perm illation, we get five associated triangles. This forms an independent proof of the result deduced in ;<3I from spherical triangles. 36. Wo can deduce from this correspondence that the relations between the parts of a tri-rectangular quadrilateral can be written down by rules exactly analogous to Napier's rules. If the angle is O andthe sidesja order are o, m, I, b, write down in a circle tho parts C, a, m, I, b. Then sine (middle part) = product of cosines of opposite parts = product of tangents of adjacent parts, wiili the same understanding as in the ease of the triangle (§ 33 (a)). If wo write the parts in the cyclic order C, I, a, b, m, we get rules analogous to (b) at the end of § 33, viz. : eos (middle part) = product of sines of adjacent parts =product of cotangents of opposite parts. 37. The formulae for a general triangle can be obtained from those for a right-angled triangle by dividing the triangle into two right-angled trianglea (Pig. 48). Thus, sinh p =sinh a sin B = sinh b sin A. Hence sinh a = sinh ft _ sin lw sin A sin B ™ sin C a. 381 THE GENERAL TRIANGLE 75 Again, cosh c - cosh c, cosh c s + sinh c, sinh r a , cos C - cos C'j cos C. - sin C\ sin C« . cosh a ■ cosh c, cosh p. cosh 6 - cosh c 2 cosh p, sinh c t - sinh a sin C, , sinh c a = sinh b sin C t , cos (7, = coth a tanb p, cos C 2 =coth b tanh p. c Also F10. 48. Therefore cosh c =cosh a cosh b sech 3 ?) +sinh a sinh & sin C x sin C' a = cosh (t cosh b sech V + sinh a sinh 6 x (coth « coth b tanh 2 j) - cos C) = cosh a cosh b - sinh « sinh b cos C. Similarly - cos C =cos A cos B - sin A sin B cosh c. It is needless to write down other formulae, which may be obtained from the corresponding formulae of spherical trigonometry by putting cosh for cos and tsinh for sin, when operating upon the sides, leaving the functions of the angles unaltered. 38. The formulae of hyperbolic trigonometry become those of euclidean plane trigonometry when the constant . 76 ELEMENTARY HYPERBOLIC GEOMETRY j „. aa To a first approximation The formula cosh I =cosh l coah I ~ -«nh ^sinh * cos becomes 1 +1* J\ + I ?Vi + U ^ « & „ or <^=« !! +6 a -2a6co3C. This shows that when we are dealing with a small region M. small in comparison with k, the geometry is sensibly the same as that of Euclid. 39. Circumference of a circle. Let <fe be the length of the are PQ of a, circle of ^^ wU h aubtends an angle rf£? at the centre. Then ainJl 2j = ainh^8inl^ I Honce the length of the whole circumference is faft sm |, * Here, for the fiat time, we require to consider the actual value of r, for the formula to, ^ 1, which is here warned, fa true only nRTSKirSS th0 tangent at p - *- «- •» «-* * = £#sinh r , 1/ ^ sink • - sinh - sin ft ft k ' sinli = tanh 7 cot ft „.4i>] CIRCUMFERENCES OF A CIRCLE 77 Let the centre go ta infinity, so that the circle becomes a horo- cvole. Then r-+ oo , 6~*0, and tan/' Comparing these with the formulae in 6; 28, we find S -h. P Jbsiiih? = *.^--»-* and fctanh 7 = .s. —£—-*«. ho * !"!•• r.'. 40. Sum of the angles and area of a triangle. Join MN 7 the middle points of AB, AC, and construct the equidistant-curve with MN as axis, which passes through FIO. 50. B, C and A. Then the perpendiculars AA', BB', CC to MN ate all equal, and <LB'BM =MAA', LC'CN =NAA\ 78 ELEMENTARY HYPERBOLIC GEOMETRY [ir. 40 Denote by ABE the angle which AB makes with the tangent to the equidistant-curve at B ; the angle B'BE is a right angle. Then ABAC + ABE + ACE =B'BM + MBE + C'CN + NCE = r . Hence the sum of the angles of the triangle ABC = - - 2CBE. The difference r-{A+B + C)n called the defect of the triangle. Again, the area of the triangle ABC - BMNC + MA A' +NAA' =B'BCC. Hence all triangles with base BC and vertex on the other branch of the equidistant-curve which passes through B, C and A have the same area and the same angle-sum or defect. Now, if we are given any two triangles, we can transform one of them into another of the aame area and defect, and having one of its sides equal to one of the sides of the other triangle. Let ABC, DEF be, the two triangles, and let DF be the greatest of the six sides. Construct an equidistant-curve passing through B, C and A. With centre C and radius equal to DF, draw a circle cutting the branch of the equi- distant-curve on which A ties in A'. Then the triangle A'BC has the same area and defect as the triangle ABC, and has the side A'C equal to DF. Again, if the perpendicular bisector of the base BC of a triangle ABC meets the other branch of the equidistant- curve in A', the isosceles triangle A'BC has the same area and defect as the triangle ABC. Hence, if two triangles have the same area they can be transformed into the same isosceles triangle, and have therefore the same defect, and conversely. n. 41] AREA AND ANGLE-SUM 79 Now, let a triangle ABC with area A and defect S be divided into two triangles ABD, ADC with areas ^ and A 2 and defects Si and S t ■ Then S^tt-BAD-B-ADB, & % = Tr-DAC-C-ADC. Therefore &i + & t =%*--A-B-C -■* = ■* -A-B-C = &, and A! + A a = A. If Ai = A a , then S t =S, and A=2A 15 5 = 2^. Hence the defect is proportional to the area, or ± = \{t-A-B-C). The value of this constant \ depends upon the unite of angle and area which are employed ; but when these have been chosen it is given absolutely. 41, Relation between the units of length and area. In euclidean geometry the units of length and area are immediately connected by taking as the unit of area the area of a square whose side is the unit of length. In fact the relationship is so obvious that there is constant con- fusion, though we are not always aware of it, between the area of a rectangle and the product of two numbers. Thus modern treatment has tended to confuse the theorems of the second book of Euclid, which are purely geometrical theorems relating to areas of squares and rectangles, with algebraic theorems relating to "squares" and products of numbers. The expression " product of two lines " has no meaning until we frame a suitable definition consistent with the rest of the subj ect-mattcr. The area of a rectangle is not equal to the product of its sides, but the number of units of area in the area of a rectangle is equal to the product of the numbers of units of length in its sides. 80 ELEMENTARY HYPERBOLIC GEOMETRY [ir. 41 n.42] AREAS 81 It would take us too far out of our way to examine completely the notion of area. We shall simply take advantage of the fact, that when we are dealing with a very small region of the plane we can apply euclidean geometry. Thus, while there exists no such thing as a euclidean square in non-euclidean geometry, if we take a regular quadri- lateral 1 with all its sides very small we may take as its area the square of the number of units of length in its sides ; or, more accurately, the units of length and area are so adjusted that the ratio of the area of a regular quadri- lateral to the square of the number of units of length in its side tends to the limit unity as the sides are indefinitely diminished. I*t us apply tliis to find the area of a sector of a circle POQ, the angle POQ --- $ being very small. Fig. si. Produce OP, OQ to P' and Q\ lMOP = OQ = r PP'-QO'- d* Then * ~ w ~ ar area of PQQ'P> = dr .PQ = k$drwDh -. k Hence the area of the sector = Jf tf/coah - - 1) = 2k i $mnh t —, al- and the area of the whole circle is 4irjt» sinh* -1 w ^A regular polygon k one which has all ita sides equal and all ita angles We can apply this now to find the area of a triangle by another method. It is sufficient to take a triangle ABC with a right angle at C. Divide it into small sectors by lines drawn through A. Then the area ifl given by fW4fcoah|-l\ Express c in terms of A and the constant 6, write tanh ^=(, and put cosM =y, and we get, after some reductions, u c ia ~ d y ~ d,J cosh , 4A = ■ - — _ = — ... , - ' The integral of this term, from y= 1 to y =cosM, is 1 ,2y-(l+C) 1 BOS' '(l- 2 cosh* .sinMJ 1-0 2 = | cos-' (1 - 2 coa?/f ) = fcir " 2B). Hence the area of the triangle ■^B-4)**(*-f^-Jl) 42. It appears then that, when the angles are measured in " circular " measure, the constant \ =F, and the formula for the area of a triangle becomes /± = k*(ir-A-B-C). As the area of a triangle increases the sum of the angles diminishes, but, so long as the vertices are real, the angles are positive quantities ; the area cannot therefore exceed v&. This is therefore the maximum limit to the area of a triangle whcu its angles all tend to zero. A triangle of maximum area has all its vertices at infinity and its sides are parallel in pairs. n.-b. a. P 82 ELEMENTARY HYPERBOLIC GEOMETRY [n. 48 43. On account of its neatness, wo add the proof that Causa gave of the formula for the area of a triangle, in a letter' to W. Botyai acknowledging the receipt of the " Appendix." Gauss starts by assuming that the area enclosed by a straight line and two tinea through a [joint parallel to it is finite, and a certain function /(jt-.£) or the angle between the two parallels ; and further, that the area of a triangle whoso vertices are alt at iiiliiiily is a certain finite quantity t Then wo have, from Fig, 52, /(jr-jp)+/(a>)=f. i'lti, 62. Again, from Pig. 53, f(<f>) +/( $ +/( 5 r - <p - $ =(, Flo. 53. Whence /('/') = A& where k is a constant, and therefore (=Air. 1 Cth March, 1832 (Gauss' Werke, viii. 221). n.441 GAUSS* PROOF Of ANGLE-SUM 83 Now, by producing the sides of any triangle with angles ", /J, y, and drawing parallels, we have ( l-'ig. Si) Therefore Fltt. 54. *=/(«)+/(#+/(y)+A- A = A(7T-U-/3-7). 44. Area of a polygon. The area of a polygon can be found by breaking it up into triangles. By joining one vertex to each of the others, we divide an R-gon into n-2 triangles. The sum of the angles of the rt-gon is equal to the sum of the angles of &£ n-2 triangles. Let Aj, A s ,... be the areas, and S lt S 2 , ... the defects of these triangles ; then, if S is the sum of the angles and A the area of the polygon, A = £A -EW =h*(n~2 .tt-S). If S' is the sum of the exterior angles, flf +#«-**; therefore which is independent of n. 84 ELEMENTARY HYPERBOLIC GEOMETRY [a 45 45. Wo add here another proof of the result that tlte geometry of ionoydea on the horosphere w the same as the geometry of Mraiijht lines on the eudidean plane. Let the three parallel lines iu apace Ail, J5fi, Cfl be cut by a Ex. iu] EXAMPLES FI9. 65. horosphere with centre fl in A, B, 0, and make AA'=BB'=CC, so that A'B'C lie again on a horosphere with centre il. [See Ex. 8.) Let the dihedral angles between the planes BCil, CAB, ABU be a, B, y, and let the angles of the rectilinear triangle A'B'C be <*', 8', y', and its area A. Then, &S ,1.1' increase, (be angles ilA'B', RA'C, etc., all tend to right angles ; hence a', B', y" tend to the values a, B, y. Also A -0. NowA-P(r-«'-j3'-/); hence a + Bi 7=3-, i.e. when tone planes intersect in pail* in three parallel lines the sum of the dihedral angles ia equal to two right angles. Heme 1 In- sum of the atigli>« nf a geodesic triangle on the horosphere ia equal to two right angles. EXAMPLES n. 1. Prove that the four axes of the circuineircles of a triangle form a complete quadrilateral whose diagonal triangle is the given triangle ; ami state the reciprocal theorem. 2. Tf a simple quadrilateral is inscribed in a circle, horoeycle or one branch of an equidistant-curve, prove that the sum of one pair of opposite angles is equal to the sum of the other pair of opposite angles. Show that this holds also for a crossed quadrilateral if the angles are measured always in the same sense, and for a quadri- lateral whose vertices are distributed between the two branches of an equidistant-curve if the angles on opposite branches are reckoned of opposite sign. 3 If a simple quadrilateral is circumscribed about a circle, prove that the sum of one pair of opiwsite sides is equal to the sum of the other pair of opposite aides. Examine the case of a crossed quadri- lateral circumscribed about a circle, equidistant-curve or horoeycle. 4. If a is the chord of an arc a of a horoeycle, prove that o = 2fc sinh $a/k. 5. If 6 is the angle which the chord of a horoeycle makes with the tangent at either end, and a is the arc, prove that a = 2k tan ft 6 If * is the angle which the tangent at one extremity of an arc a of a horoeycle makes with the radius through the other extremity, prove that a -k cos ft 7. Prove that the arc of an equidistant- curve of distance a, corresponding to a segment* on its axis, is x eosh ajk. 8 HA B are corresponding points on the parallels A A', BB', and A, C are corresponding points on the parallels AA',CC\ prove that B, C are corresponding points on the parallels BS , CO . 9 Prove the following construction for the parallel from to AM. Draw ON±NA. Take any point A on AM, draw OB J.OA and ABJ.OB. With centre and radius equal to AM, draw a circle cutting AB in P. Then OP |l AM. 10. Prove that the radius of the inscribed circle of a triangle of maximum area is U'foff.3. 11. In a quadrilateral of maximum area, if 2a, 26 are the lengths of the common perpondiculars of opposite sides, prove that sinh -sinh 7 = 1. k k 12. A regular quadrilateral is symmetrically inscribed in a regular maximum quadrilateral ; prove that each of its angles is cos «fc 13. If the three escribed circles of a triangle are all horocyctcs, prove that each side of the triangle is cosh""?, and that the radius of the inscribed circle is tanh"M. «»" «» radiua o£ the crcumcircle is tanh _1 J (fc being unity). 86 ELEMENTARY HYPERBOLIC GEOMETRY [Ex. u. 14. In euclidcan geometry prove that any convex quadrilateral can by repetition of itself bo made to cover the whole plane without overlapping. 15. In hyperbolic geometry prove that any convex polygon with an even number of sides can by repetition of itself be made to cover the whole plane without overlapping, provided the sum of its angles is equal to or a submultiple of four right, singles. Show that the same is possible if the number of sides is odd, provided the sum of the angles is equal to or a submultiple of two right angles. 16. If a is the side and a the angle of a regular «-gon, prove that _ v .a a cos -= sm ^ cosh — . n 2 21: 17. If r is the radius of the inscribed circle, if that of the circum- scribed circle of a regular n-gon with side a and angle a, prove that sin li ■ a cot - tanh ~, and cosh = - = cot - cot -. fr n. 2k k n 2 18. A regular network is formed of regular a-gons, p at each point. Show that the area of each polygon is l?ir{2nlp - n 4 2), 18. A semiregular network is formed of triangles and hexagons with the same length of side, three of each being a t each point. Prove that the length of the side is 2k cosh" 1 Vi(4+V3). 20. A -wiiii regular network consists of regular polygons all with the sa-rae length of side. At each vertex there arc p, %-gons, Pt »a-gon s . p, n a -gons, etc If each %-gon has area A,, prove that Kx. n.i EXAMPLES 87 ^*[<>**)-%} 21. If a ring of « equal circles can be placed round an equal circle, each one touching the central circle and two adjacent ones, prove that the radius of each eirele is given by 2 cosh r sin - = 1 k n 22. Prove that the area included between an arc of an cquidistant- eurve of distance a, its axis, and two ordinates at distance z is 23. AA' ;; BR' and they make equal angles with AB, ACX. BB' awl AD LAG. If the angle A'AD=z, prove that the area of the circle whose radius is AB is equal to ?rA? tan's. (J. Bolyai.) 24. Prove that the volume of a sphere of radius r is / 2r r\ 25. These parallel lines AQ, BB, CQ are cut by two horocycles with centre SI in A, B, C and A', B', C. Prove that the arcs AB:BC=A'B' BC. 23. AtB r , A t B„ AtB s arc arcs of concentric horocycles as in Fig. 29, and AiA t =A,A^ Prove that j4,B, : A ,B 1 =j4, J B t : A t B t , Hence show that the ratio A t Bi : A t B t depends only on the length 27. Prove that the sides of a pentagon whose angles are all right angles are connected by the relations cosh (middle side) = product of hyp. cots, of adjacent sides = product of hyp. sines of opposite sides. 28. A simple spherical pentagon, each of whoso vertices is the pole of the opposite side, is projected from the centre upon any plane. IYove that the projection is a pentagon whose altitudes are con- current ; and that the product of the hyp. tangents of the segments into which each altitude is divided is the same. fat) sinh J> CHAPTER III. ELLIPTIC GEOMETRY. 1. The hypothesis of elliptic geometry is that the straight line, instead of being of infinite length, is closed and of finite length. Two straight lines in the same plane will always meet, even when they are both perpendicular to a third straight. line. Let I, m, n, be three straight lines drawn perpendicular to another straight line a at the points L, M, N. Let m, n meet in A ; n, I in B ; and l t m in 0, "When LB is produced it will meet a again either in L or in some other point. Let V be the first point in which it again meets a. Then, from isosceles triangles, we have BL = BN =BL' CL=CM=CU. Hence B and C are both the middle m .2l ABSOLUTE POLE AXD POLAR 89 point of the segment LBV, and must therefore coincide. In the same way A, B and all coincide. Hence all the perpendiculars to a given line a on one side of it meet in a point A, and A is equidistant from all points on the line. The point A is called the absolute pok of the line a, and a is called the absolute pohr of A. If P is any point on a, the distance AP is called a quadrant, and A is said to be orthogonal to P, or A and P are called absolute conjugate points. 2. The perpendiculars drawn in the other sense will similarly meet in a point A'. The question arises : are A and A' distinct points ? On the hypothesis that A and A' are distinct points, two straight lines have two points in common. It could be proved that in this case any two straight lines would intersect in a pair of points distant from one another two quadrants. A consistent system of geometry results, which is exactly like the geometry on a sphere, straight lines being represented by great circles, and is therefore called Spheri- cal Geometry. The two points of intersection of two lines :iig called antipodal points. Two points determine a line uniquely except when they are antipodal points ; a pair of antipodal points determine a whole pencil of lines. On the hypothesis that A and A' are one and the same point, two straight lines always cut in just one point, and two distinct points uniquely determine a line. This gives again a consistent system of geometry, which is called Elliptic Geometry. 1 1 Somotimea both of these systems are called Elliptic geometry, and ttey are distinguished as the Antipodal or Double form aud (ho Polar or Single form. We shall, however, keep the term Elliptic geometry for the' latter form. 90 ELLIPTIC GEOMETRY fiu. a rn. 31 GEOMETRY OF A BUNDLE 01 While spherical geometry admits more readily of being realised by means of tbe sphere, elliptic geometry is by far the more symmetrical, and our attention will be confined entirely to this type. Elliptic geometry has also the advan- tage that it more nearly resembles euclidean geometry, since in euclidean geometry all the perpendiculars to a straight line in a plane have to be regarded as passing through one point (at infinity). Another mode of representation of these two geometries exists, which exhibits them both with equal clearness. Consider a bundle of straight lines and planes through a point 0. If we call a straight line of the bundle a " point," and a plane of the bundle a " line," we have the following theorems with their translations. (Cf. Chap. II. § 24.) Two lines through Two "points" uniquely uniquely determine a - plaue determine a " line." through 0. Two planes through intersect always in a single line through 0. All the planes through perpendicular to a given plane a through pass through a fixed line a through Two " lines " intersect always in a single " point." All the " lines " perpendi- cular to a given " line " a pass through a fixed " point" A, which is orthogonal to every " point " lying in a. 0, which is orthogonal to every line through lyiug in a. Hence elliptic geometry can be represented by the geometry of a bundle of lines and planes. In the same way spherical geometry can be represented by the geometry of a bundle of rays (or half-lines) and half-planes. Two rays which together form one and the same straight line represent a pair of antipodal points. In elliptic geometry all straight lines are of the same finite length 2q, equal to two quadrants. If we extend these considerations to three dimensions, all tbe perpendiculars to a plane « pass through a point A, the absolute pole of a, and the locus of points a quadrant distant from a point A is a plane a, the absolute polar of A. 3. The plane in elliptic geometry, or, as we may call it, the elliptic, plane, differs in an important particular from the euclidean ot hyperbolic plane. It is not divided by a straight line into two distinct regions. Imagine a set of three rectangular lines Oxyz with Oif on the°line AM and Oz always cutting the fixed line AP. Fig. 57. As moves along AM it will return to A, but now Oz is turned downwards and Ox points to the left instead of to the right. The point z has thus moved in the plane PAM and come to the other side of the line AM without actually crossing it. A concrete illustration of this peculiarity is afforded by what i» called Menus' sheet, which consists of a band of paper half twisted and with its ends joined. A line traced 92 ELLIPTIC GEOMETRY [in.41 m . 5 ] ABSOLUTE POLAR SYSTEM 93 along the centre of the band will return to its starting point, but on the opposite surface of the sheet. The two The elliptic Fra, 68. sides of the sheet are continuously connected, plane is therefore a one-sided surface. If we carry out the same procedure for tlio euclideau plane, we shall obtain exactly similar results, with the exception that a point passes through infinity in going from one side of the Line to the other. This ia well illustrated by the ease of a curve which runs along an asymptote. Ordinarily the curve lies on opposite sides of the asymp- tote at the two ends, and thus appears to cross the asymptote. When it does actually cross the asymptote at infinity it has a point of inflexion there and lies on the same side of the asymptote at each end. 4. Absolute polar system. To every point in space corresponds a plane, and vice versa, which are absolute pole and polar. If the polar of a point A passes through B, the polar of B will pass through A, because the distance AR is a quadrant. Let A, B be two points on a line I ; the potars of A and B intersect in a line I'. Let A' and B' be any two points on l' ■ then the polar of A' passes through both A and B. Hence the polars of all points on the line V pass through the Hue I If P is any point on I, its polar will pass through A' and B'. Therefore the polars of all points on the line I pass through the line I'. To every line I, therefore, there corresopnds a line I , the absolute polar of I, such that the polar of any point on I passes through V and vice versa. All points of l' are a quadrant distant from all points of I, and every line which meets both I and I' cuts them at. right angles. If we confine ourselves to a plane, to every point m the plane corresponds a line in the plane and vice versa. These relations arc exactly the same as those that we get in ordinary geometry by taking poles and polars with regard to a conic in a plane, or a surface of the second degree in space. The points on the conic or quadnc surface have the property that they lie on their polars ; the polar is a tangent to the conic or quadric and the pole is the point of contact. 5 Projective geometry. These relations of polarity with regard to a conic belong to pure projective geometry, and have nothing whatever to do with actual measurement, distances or angles. All the theorems of projective geometry can be at once transferred to non-cuclidean geometry, for, so long as we are not dealing with actual metrical relations, non-euchdean geometry is in no way whatever distinguished from euclideau. Pure projective geometry tabes no notice of points at infinity, for infinity bare implies infinite distance, and is therefore irrelevant to the subject. It has there- fore nothing to do with parallel lines. 94 ELLIPTIC GEOMETRY fin. 5 m. si PROJECTIVE GEOMETRY 95 l.'siinriinwtclv most Ba^irii best-hooka on projective geometry start by assuming euclidean metrical geometry. A harmonic range is defined in terms of tlic ratios of seg- ments, and a conic is obtained as the " projection " of a circle. This treatment unnecessarily limits the generality of projective geometry, and attaches a quite unmerited importance to euclidean metric. Tho use of analytical geometry might [» thought to supply a means for a general treatment, for tho algebraic relations brtwoa) numbers which express the relations of projective geometry are just theorems of arithmetic, and these may be applied to any subject matter which cou be subjected to numerical treatment, whether that subject matter is euclidean or non-euclidean geometry. But ihc difficulty in applying this procedure is that the subject matter must first be prepared for numerical treatment. This mean* either postponing the introduction of projective geometry until metrical geometry, with a system of coordinates, has been established. 1 which is just the fault we wish to avoid, or the establishment of a system of projective coordinates independent of distance. In cither case we have to assume much more than is really necessary. For convenience of reference we shall give a summary nf the theorems of projective geometry which we shall require, assuming that proofs of these are available which do not involve metrical geometry. (Reference may be made to Reye, Geometry of Position, Part I., translated by Holgate, New York, 1898, or Veblen and Young, Projective Geometry, Vol. L, Boston, 1910.) If two ranges of points arc made to correspond in such a way that to every point P on the one range corresponds uniquely a point P' on the other, and vice verm, the ranges are said to be homograph r c Notation. {P)-k{P'\. The simplest way of obtaining a range which is homo- 1 Of. Chap. IV. §21. graphic with a given range is as follows. Take any point 0, not on the axis of the range : join to the points of the range, and cut these rays by any transversal. The range on this transversal is called the projection of the first range and is nomographic with it. In this special position, in which the lines joining pairs of corresponding points are concurrent, the ranges arc said to be in perspective, with centre 0. Notation. {P}*o{P'} or {P}a(P'}. It can be proved that two homographic ranges can always be connected by a finite number of projections, and in fact this number can in general be reduced to two. It can be proved that (ABCD)-x(BADC)-x(CDAB)-{I)CBA), but in general the four points are projective in no other order. Properties which are unaltered by projection are called projective properties. Thus, points which are collinear. or lines which are concurrent, retain these properties after projection. A harmonic range is projected into a harmonic range. We cannot define a harmonic range in terms of the ratios of segments, because a segment is not projective. We define a harmonic range thus ; Let X, Y be two given points on a line, and P a third point. (See Fig. 85, Chap. IV.) Through P draw any line PST t and on it take any two poiuts 8, T. Join S and T to X and F ; let SX cut TY in F, and SY cut TX in V. Join UY, and let it cut XY in Q. Q is called the harmonic conjugate of P with regard to X and Y. This construction can be proved to be unique ; P, Q are distinct, and are separated by and 96 ELLIPTIC GEOMETRY pit. 5 separate X and Y. If (XY, PQ) is a harmonic range (XY, PQ)-x(XY, QP). If we start with three points on a line, we can derive an indefinite number of other points by the above quadrilateral construction, and in fact we can find a new point lying between any two given points. AH the points derived in this way form a net of rationality. They do not give all the points on the line. To secure this we would require an assumption of continuity. If three points A, B, C of one line are projected on to three points A', B', C" of another line, the correspondence between all the points of the two ranges is determined. This is the fundamental, theorem of projective geometry. Two homographic ranges can exist on the same line. If three points A, B, C are self corresponding, it follows by the fundamental theorem that all the points are self- corresponding. Hence two homographic ranges oa the same line cannot have more than two self-corresponding points. That it is possible in certain cases to have two self- corresponding points is shown in Fig. 59. I is the given line, l t an intermediate line on which a range of points {P} is projected from centre 5, , and £„ is a second centre of projection from which the projected range {P,} is projected on to I. In this way P' corresponds to P. Let Z, cut I in Y, and let SjSg cut Z in X. Then X and Y are self-corresponding points. If Z, passes through X, the two self-corresponding points will coincide. If \P) and {P'\ are two homographic ranges on the same line, such that to P corresponds P\ in general to P' will correspond another point P". If P" coincides with P, the »] INVOLUTIONS 97 points of the line are connected in pairs and are said to form an involution. If D x and D., are the double or self- corresponding points of an involution, and X, X' axe a pair of corresponding points, (D^JCX^^iD^JC'X), so that (DjZ>aAX) is a harmonic range. If two real self-corre- sponding points do not exist, we introduce by definition conjugate pairs of " imaginary " points, much in the same way as ideal points were introduced into hyperbolic geometry. When the double points are real the involution is said to be hyperbolic, and when they arc imaginary it is said to be elliptic. If the double points coincide, the conjugate of any point P coincides with D, and the involution is said to be parabolic. If ( p} and I p' | are two homographic pencils with different vertices, the locus of the points of intersection of corre- sponding lines is a curve with the property that any line cuts it in two points, real, coincident or imaginary. A line / N.-E. G. 98 ELLIPTIC GEOMETRY | in. <■. in. ol THE ABSOLUTE cuts the two pencils in homograpiiic ranges, and the self- corresponding points of these ranges are points on the locus. FlO. 60. This curve is called a point-conic ; it is the general curve of the second degree, characterised by the property that any line cuts it in two points. Similarly the envelope of the lines joining pairs of corresponding points on two homograpiiic ranges is a curve of the second class, or line-conic, characterised by the property that from any pomt two tangents can be drawn to it. It can be proved that a point-conic is also a line-conic, and vice versa. The term conic can then be applied to either. 6. The absolute. Let us return now to the absolute polar system in a plane. We shall prove the theorem : In every polar system in a plane which has Ike reciprocal properly (km " if the polar of a point A passes through B, (he polar of B passes through A," tliere is afmtl conic, the locus of points or ike envelope of lines which an inchlent with their pohirs. Consider a line I, The polar of a point P on I cuts I in ;i point P' t and the polar of P' passes through P. Hence the points of I are connected in pairs and form an involution whose double points arc incident with their polars. Every line therefore cuts the locus in two points, and the locus ia a point-conic. Similarly the envelope is a line-conic. If I cuts the locus in P and Q, the polars of P and Q are lines of the One-conic. Further, the polar of P does not cut the locus in any second point, since the polar of any point upon it passes through P ; hence the polar of P is a tangent to the point-conic, and the point- and line-couics form one and the same conic. Similarly, in three dimensions a polar system determines a surface of the second degree or quadric surface. Applying this theorem to the absolute polar system, we mid a conic in a plane or a quadric surface in space which is given absolutely. But as a real point cannot lie on its polar, since it is at the fixed distance of a quadrant from any point of it, this conic or quadric can have no real points. This imaginary conic, or in space the imaginary quadric surface, is called the Absolute. Let P, P' and Q. Q' be two pairs of conjugate points on a line g, so that PP' = QQ' = a quadrant. Therefore Fro. 61. PQ ~ P'Q'. Let g cut the absolute in X and Y ; then P, P' and Q, Q' are harmonic conjugates with regard to X and Y. Let Q coincide with X ; then Q' will also coincide with X, and the equation PQ^P'Q' becomes PX=FX = PX-PP'. 100 ELLIPTIC GEOMETRY [in. 7 Therefore 11- PP'/PX, Therefore PX must be infinite. Every point on the absolute is therefore at an infinite distance from any real point, and the absolute is, like the real conic in hyperbolic geometry, the locus of points at infinity. 7. Principle of duality. The polar system with regard to the absolute conic establishes the principle of duality, In euclidean geometry t he principle of duality holds so long as we are dealing with, purely descriptive properties, i.e. it holds in projective geometry, which is independent of any hypothesis regarding parallel lines, but it has only a very limited range in metrical geometry, and is often applied more as a principle of analogy than as a scientific principle with a logical foundation. Thus, four circles can be drawn to touch three given lines, but only one circle can be drawn to pass through three points. A circle is the locus of a point which is always at a fixed distance from a given point, but we cannot con- sider it also as the envelope of a line which makes a constant ainilc with a fixed straight line. In hyperbolic geometry, when we consider equidistant curves as circles, we find it true that four circles are deter- mined by three points ; and if we introduce freely points at infinity and ideal points, we can make the principle of duality fit fairly well. In elliptic geometry, however, the principle of duality has its widest field of validity, and extends to the whole of metrical geometry. The reason for this is found in t lie nature of the absolute and the measure of distance and angle. In a pencil of lines with vertex there are always two absolute hues, the tangents from to the absolute, in. 8] PRINCIPLE OF DUALITY 101 and in all three geometries these two lines are conjugate imaginaries. They form the double lines of the elliptic involution of pairs of conjugate or rectangular lines through 0. In a range of points on a line I there are similarly two absolute points, the points of intersection of I with the absolute. They form the double points of the involution of pairs of conjugate points with regard to the absolute. But in hyperbolic, elliptic and euclidean geometry this involution is respectively hyperbolic, elliptic and parabolic. Thus it is only in elliptic geometry that the involution on a line is of the same nature as that in a pencil. 8. As a consequence of this, in elliptic geometry the distance between heo points is proportional to the ajujle between their absolute polars. Consider two lines OP, OQ. Let P', Q' be the poles of OP and OQ. Then P'Q' is the polar of 0. PP' = QQ'=q. Now distances measured along PQ are proportional to the angles at 0. Therefore ^ = / - />0 ^ a „d P'Q' = PQ. Therefore the distance d between the poles of the lines is 102 ELLIPTIC GEOMETRY [m.a connected with the angle a between the lines by the relation a= —a, 7T and if the unit of distance is such that ? = T, then d = a. Here we must observe that two points have two distances, viz. d and 2q - d ; two lines have two angles, q and «■ - «. In the above relation we have made the smaller distance correspond to the smaller angle. Consider, however, a triangle ABC, in which we shall suppose each of the sides < q, and each of the angles < ^. 33 FlO. 6S. The absolute polar figure is another triangle A'B'C, in which B'C is the polar of A, etc. Let AB, AC meet B'C in M and N, Then and B'C IT IT III. fl AREA OF A TRIANGLE 103 To the angle j4, <\, corresponds therefore the segment To a segment d corresponds an angle '(2q-d), and to an angle a corresponds a segment — (it - «). if the segment d>d', the corresponding angle a<a'. Tn applying the principle of duality, therefore, we must interchange point and line, segment and angle, greater and less. 9. Area of a triangle. Two lines enclose an area proportional to the angle between them, = 2#vi, say, where k is a linear constant, MB fit and the area of the whole plane is 2A 2 tt. In Fig. 64 the areas enclosed by the angles of the triangle are shaded, and these areas cover the area of the triangle three times, and the rest of the plane only once. We have, therefore, 2k a -{A +B+C) -2ftV +2A, whence A=k*(A+B + C -x). 104 ELLIPTIC GEOMETRY firt. 10 The area of a triangle is therefore proportional to the excess of the sum of its angles over two right angles. If A u B u C l are the exterior angles, A=&(2tt-A 1 -Bi-C 1 ). The absolute polar of the triangle ABC is a triangle A'B'V with sides a', b', c' = ?$■ {-w - A), etc. Hence A=k*l(4q-a'-b'-c% at or the perimeter of a triangle falls sftort of Aq by m amount ■proportional to lite area of (lie polar triangle. These results hold also for the sum of the exterior angles and the perimeter of any simple polygon. 10. The circle. A circle is the locus of points equidistant from a fixed point, the centre, and by the principle of duality it is also i in I-..-., the envelope of lines which make a constant angle with a fixed line, the axis. in ill THE CIRCLE AND THE SPHERE 105 Let C be the centre and c the polar of C. Let P be any point on the circle, and draw the tangent PT. Then CMXTM and also J. TP. Therefore T is the pole of CP. PM =q-r=-u; therefore u is a constant angle = | - 1 r. Further, since PM = q -f, the circle is an equidistant-curve with c as axis. Just as in hyperbolic geometry, the circle or equidistant-curve lies symmetrically on both sides of the axis, but the two branches are continuously connected. In elliptic geometry, therefore, equidistant-curves arc proper circles. When the radius of a circle is a quadrant the circle becomes a double straight line, the axis taken twice. 11. In three dimensions the surface equidistant from a plane is a proper sphere. A remarkable surface exists which is equidistant from a line. With this property it resembles a cylinder in ordinary space. A section by a plane perpendicular to the axis is a circle. A section by a plane through the axis is an equidistant-curve to the axis, but this is also a circle, and the surface can be generated by revolving a circle about its axis. It thus also resembles an anchor ring (Fig. 66). ^ But sections perpendicular to the axis do not cut it in pairs of circles, but only in single circles, and so it also resembles a hyperboloid of' one sheet. Every point is at a distance d from the axis I, and is therefore at a distance q - d from the line P, the absolute conjugate of L The surface has there- fore two conjugate axes, and can be generated by the revolution of a circle about either of these. It is therefore a double surface of revolution. It is a surface of the second degree, since a straight line cuts it in two points. 106 ELLIPTIC GEOMETRY frrr. 12 From its resemblance to a hyperboloid, the existence of rectilinear generators is suggested. If it does possess rectilinear generators, these lines must be everywhere Fig. SO.' equidistant from either axis. We shall therefore investigate tlie existence of such lines, and return in § 17 to a description of this surface {Cliffords surface). 12. Common perpendicular to two lines in space. Consider two lines a, b not in the same plane. Let a' and b' be their absolute polars. Any line which cute both a and a' is perpendicular to both; hence any line which ,i! TU t P io . turc of Clifford"* ™rfacc will bo best understood after readme Limp, V. In the conformal representation of nnn-cuelidcan Konnu-iry in euciidean space, planes and spheres ore all represented by spheres straight lines and circles by circles. Clifford's surface is represented by an anchor-ring, one axis Wing represented by the axis of the ring, the other axis being represented by a line at infinity. The circular sections of the surface by plane* through an axis (which arc lines of oun are represented by the meridians and parallels of the anchor-ring (which arc also lines of curvature). The rectilinear generators are represented by the l.itariLTiir oircular sections of the ring. The two systems of these lnst-mentioncd eireles arc depicted in the figure. Thev inierseet at a- constant angle. m. 121 LINES IN SPACE 107 meets the four lines a, b, a\ b' cuts them all at right angles. Now, three of these lines a, 6, a' determine a ruled surface of the second degree, and the fourth line b' cuts this surface in two points P, Q. The two generators p, q of the opposite system through P and Q are common transversals of the four lines a, a', b, ¥, and therefore cut all four at right iWgks, The two lines a, b have therefore two common perpendiculars. The two common perpendiculars p, q are absolute polars, For. since O, a' and 6, b' cut p and q at right angles, they also cut the polars p' and q', but they have only two common transversals; therefore p' must coincide with q, and q' with p. Of the two common perpendiculars one is a minimum and the other a maximum perpendicular from one line on PlO. 67. the other. Take any point P t on a and draw P,Q 2 1&, Q^P.La, and so on. Then P t Q»>Q s Pt>P*Q a > - > s0 that" the perpendiculars form a decreasing sequence which must tend to a finite limit A&. So if we continue the sequence in the other direction, drawing PA ±o, Q,P„ _!.&,..., we have an increasing sequence which tends to the other common perpendicular A 2 B Z . 108 ELLIPTIC GEOMETRY \m. 13 13. Paratactic lines. If A l B 1 = A f B 2 , all the intermediate perpendiculars must also be equal ; the two lines have then an infinity of common perpendiculars, and therefore the four lines a, b, a', b' all belong to the same regulus of a ruled surface of the second degree. The two lines are equidistant, though not coplanar ; they are analogous to parallel lines in ordinary geometry and possesses many of their properties. They were dis- covered by W. K. Clifford, and have therefore been called Clifford's parallels. A more distinctive name, suggested by Study, is paratactic lines. Through any point two lines can be drawn paratactic to a given straight line, one right-handed and the other left- handed. Each is obtained from the original line by screwing it along the perpendicular NO either right-handedly or left- handedly. The angle through which it has to be turned is pro- portional to the distance through which it has to be moved. ■ In the plane ONM draw OMlON, cutting the given line in M, the pole of ON in this plane. Draw MP the P FIO. 6& polar of ON, which is therefore perpendicular to NM, and along it cut off MP - MP' =d. Then OP and OP' are the two lines through paratactic to NM. Also d - 2(f Q. (ii. 141 PARATAXY 109 14. The above construction for a common perpendicular to two skew lines can only be carried out in elliptic geo- metry, for in hyperbolic geometry the polar of a real point is ideal, and in euelidean geometry it is at infinity. Con- sider a svstem of pairs of planes at right angles to each other Hi.. 69. drawn through the line a. This forms an elliptic involution, the double elements of which are the imaginary planes through a which touch the absolute. These planes are cut by the line 6 in a range of points forming an elliptic involu- tion. Although the double points of this involution are imaginary, the centres B lt B % of the segments determined by the double points are always real. These form a pair of elements of the involution a quadrant apart. In elliptic geometry there are two real centres, in euelidean geometry one is at infinity, and in hyperbolic geometry one is ideal. The perpendiculars to b at B lt B s are the two common perpendiculars. Since B, and B z are conjugate points and A&l B^^ A,B t is the polar of B t in the plane Ajb ; therefore A , B 8 _L AyB v But the plane aB t A. the plane aB s ; therefore AJlt is 1 the plane aB«; therefore A^Xa. Similarly A^B^La. The points A lt A z are the centres of a similar elliptic involution on a. no ELLIPTIC GEOMETRY |rn, fa 15. Two paratactic lines cut the same two generators, of the nam system of the absolute. Let three common transversals fc, l t , l 3 cut the four lines a, a', b, V in A,, A,' r B., B.', and the absolute in Xi, Y, (*-l, 2, 3). Then, since Aj, A x ' and B lt B,' are FIO. 70. harmonic conjugates with regard to X,, Y u (A^', B l B 1 i ) is an involution with double points X„ Y\. Also, by a fundamental property of a ruled surface of the second degree, i4Ai'B l B l ')-x(A s A t 'B£ i ')-xiA 9 di'B i B 3 '). Therefore (A 1 A 1 'B i B l 'X,Y l )7,(A 2 A i 'B i B.,'X 2 Y s ) ^{A 3 A 3 'B 3 B 3 'X,Y A ). Therefore X x X t X 3 and Y s F,y s are two generators x, y of the same regulus as A,A S A 3 , etc. ; they are also generators of the same regulus of the absolute, since they cut it in more ni. IB] PAEATAXY 111 than two points. Hence all the common perpendiculars to two paratactic lines cut the same two generators of the absolute. Let a cut the absolute in a lt u 2 . Through each of these points passes one generator (g t and g t ) of the absolute of the opposite system to a;, y, and therefore cutting x, y in &&» nm- Si and St mu8t als0 Delon 8 to the same Te ^ m as l u l t , k, since they cut the surface in more than two points. Therefore they cut a\ b and b' also. Hence a and b cut the same two generators of the absolute, q.e.d, Conversely, if a and b cut two generators g z , g 2 of the absolute in ai 8 lt u*#z, let A,, h t be the two generators of the other system through a l ,u 2 , then the polar of a is the intersection of the planes (gjij, {^J, and therefore cuts both g, and (jr 2 . Hence gr, and g % are common transversals of a, a\ b, 6\ " But, by § 12, if a, a', b, b' do not all belong to the same regulus, they have only two common trans- versals, which are absolute polars. Now g t and g t are not absolute polars (each being its own polar), hence a, a\ b, b' belong to the same regulus, and have an infinity of common transversals. Therefore a,, b are paratactic. The two seta of generators of the absolute may be called right-handed and left-handed. Two lines which cut the same two left (right) generators of the absolute are called left (right) paratactic lines. We see, therefore, that all the common transversals of two right paratactic lines are left paratactic lines. Further, if a and b are both right (left) paratactic to c, then a is right (left) paratactic to b ; for a, b, c all cut the same two generators of the absolute. 16. Paratactic lines have many of the properties of ordinary euclidean parallels. In particular they have the 112 ELLIPTIC GEOMETRY tin. 17 characteristic property of being equidistant. They are not, however, coptanar. We shall use the symbol fl for right parataxy, and U for left parataxy. If AB n CD and if AC and BD are both 1 CD, they are also ± AB; AC=BD and Li BD, and AB=CD. Also AD cuts both pairs of lines at equal angles. The figure ABDC is a skew rectangle ; its opposite sides are equal and paratactic. If ABUCD and = CD, then joining AC, BD and AD, LADC = LDAB, and we find two congruent triangles ACD and DBA ; therefore AC=BD and LACD=LDBA. Conversely, if AB = CD and LABD = LDCA, or if CAB + ACD = 2 right angles, then AB is paratactic to CD. Hence, if ^3 = and n CD, it follows that AC = and U BD. ABDC is analogous to a parallelogram. Real parataxy can only exist in elliptic space. For if ABDC is a skew rectangle, the lines AB, AC, AD are not co-planar. Therefore ACAD + BAD > LCAB, i.e. > a right angle. But LBAD = LADQ; therefore L_ CAD + ADC +ACD>2 right angles ; therefore the geometry is elliptic, 17. Clifford's surface. If «0 c and 6 P c, so that a U b, the common transversals of a, b, c are all LI , and form one regulus of a ruled surface of the second degree ; the lines of the other regulus are all 11 a, b and c. If a' is a generator of the opposite system to a, b, c, then any line which cuts a and is u a' cuts 6 and c. The surface is therefore generated by a line which cuts a 17] CLIFFORD'S SURFACE 113 fixed line and is paratactic to another fixed line. By § 16 it cuts the fixed line at a constant angle, W. Let OP be the fixed line, which is cut by the variable line OP' (Fig. 68). Draw OJVLthe plane POP' and =d=2$ql-!r. Draw OM bisecting the angle POP' {=2$), and draw NMLON in the plane MON. NM is then para- tactic to both OP and OP' , and ON is supposed to be drawn in the direction such that NM (1 OP and U OP'. Any other tine which cuts OP and is U OP', i.e. any generator of the left-handed system, is also U NM. and is at the same distance d from NM. Hence NM is an axis of revolution of the surface, and similarly the polar of NM is also an axis of revolution. This surface is, therefore, just the surface of revolution of a circle about its axis which we considered in §11. In fact, through any point P of this surface there pass two lines paratactic to the axis, and since these lines arc equi- distant from the axis, they lie entirely in the surface. This surface, which is called Clifford's Surface, is therefore a ruled surface. All the generators of one set are n to the axis I, and ail the generators of the other set are U I. Two generators of opposite systems cut at a constant angle IT - d. From the figure in § 15, it appears that Clifford's surface cuts the absolute in two generators of each system. Suppose the surface is cut along two generators. The whole surface is covered with a network of lines inter- secting at a fixed angle W, and can be conformly repre- sented upon a euclidean rhombus with this angle. The geometry on this surface is therefore exactly the same as that upon a finite portion of the euclidean plane bounded N.-E. Q. H 114 ELLIPTIC GEOMETRY | in. 13 by a rhombus whose opposite sides are to be regarded as coincident. As an immediate consequence, the area of the surface is found to be 4tp . sin 20, since the side of the rhombus = 2j. We have therefore the remarkable result Fia. 7i. that both in hyperbolic and in elliptic space there exist surfaces {viz. horospberes and Clifford's surfaces respec- tively) upon which euchdean geometry holds. Circumference of a 18. Trigonometrical formulae, circle. In investigating the trigonometrical formulae we shall use a method which might equally well have been employed in hyperbolic geometry. The starting point is the assump- tion that euclidean geometry holds in the infinitesimal domain. 1 1 The truth of this assumption in indicated by the fact that when the sides (if ;i t rtuneje tend to iiero, the Hum of the angles tend* In tin- value jr. The steps of the proof are as follows. Let A BC be a triangle with right angle H 0. We have to prove (1) that the nitio AC: AB tends to a limit. This limit is a function of the angle A, say f(A). We liuvo to prove (2) that/(j-IJ is continuous, and (3) that its vnluc is cos A. The last step is best obtained bv the formation of a functional equation fi» + 4>) -> f{0~<t>)=2fW ■/{•/>)■ See Coolidge, Chap. IV. m. 18] CIRCUMFERENCE OF A CIRCLE 115 Let AOB be a smaU angle a, OA =OB=r, AA'=BB' =oV, AB = a, A'B' = a + da. The angle OAB is nearly = £ , Let OAB^OBA^-0, OA'B =OB'A' = ~-(6+d6). altfa Flo. 72, Draw BM, making the angle ABM - ABO - ^ - $. Then, neglecting higher inlinitesimals, we have A'M = AB; therefore MB' =da, and BM=dr. Therefore da = 2dr sin A MBB' m 2dr sin $, da or dr =20. Again, the area ABB' A' =adr, and the sum of its exterior angles , * therefore adr= -2&?dQ, (§9) or i.e. M dr~ k dr*' d?a a _ The solution of this is a=Csm(^ + <f>). Differentiating, -*- = 26 = v cos U + <f>) • 116 ELLIPTIC GEOMETRY [ra, 19 When r=0, «=0 and 2(^-#J =■*--«, therefore 20= a, so that = C sin q4, a = , cos <p ; whence 0=0 and C=ka. Hence we have, finally, j ■ 7 a = tea sin v. Since a and a are small, we can take a as the arc of a circle of radius r. The whole circumference of a circle is therefore 2irk sin j- When r = |7rjt, the circumference is 2-&, which is twice the length of a complete line, and therefore q = l-xk. 19. Trigonometrical formulae for a right-angled triangle. Keep one part, say b, fixed. Let BAB' =dA, BB' =da, Fio. 73. AB'C=B+dB, N&^dc. Then cfc=ffacosB, •0) h sin y rfj4 = NB = rfff sin S (2) • in. 201 THE EIGHT- ANGLED TRTANGLE 117 The area of BAB' is obtained in two ways, (1) by inte- grating f, . c , . , a A ac = 1 fcshi Jo = £ 3 <L4(l -cos jj ; (2) in terms of the angular excess = kHdA + 7r-B + B+dB-Tr) =&{dA +dB). e (3) Equating these, dB= -cos, dA Eliminate (fa and dA between these three equations, and we get kdBb&a C ,= -ks\n ,dA = - da sin B = - dc tan B, giving a differential equation in B and c. The integral of this is c sin j sin B =f(b), since b is the only constant part. Putting B = ™, c=6, and we find /(6) = sin g ft f* Hence we have sin , =sin 'sin B. 20. Associated triangles. Id order to obtain the other relations between the sides and angles, we shall establish a sequence of associated triangles which form the basis for Napier's rules in spherical trigonometry. This sequence has already been referred to, and a similar sequence was found in hyperbolic geometry. 118 ELLIPTIC GEOMETRY fin. 20 We shall first introduce the following notation. Let u=a/k, a' = 2 -«; then the angles «, £, y correspond to the sides a, b, c of the triangle. Draw the absolute polars of the vertices A and B. These form, with the sides produced of the given triangle, a star Fro. 74. pentagon. Mark on each side the angle which corresponds to it : and we get the figure (Fig. 74). Each of the five outer angles is a right angle. Each vertex of the simple pentagon is the pole of the opposite side. We obtain then five associated right-angled triangles. If we write down the five quantities A, «', y, /3', B which correspond to the parts of the first triangle A, «, y, 8, B t the corresponding quantities in the same order for the second triangle are «'. y, B', B, A, but these are the same as the five quantities corresponding to the first triangle permuted cyclically ; and they are represented in proper order by the sides of the simple pentagon. Ui.21] NAPIER'S RULES 119 21. Napier's rules. Now we have proved for the first triangle that .6 . c . n sin ,,=sih .sniff. Writing this in terms of A, «', y, &\ B, we have cos j8' = sin y sin B, and since this equation can be applied to each of the five triangles, and therefore transformed by cyclic permutation, we can state a general rule as follows : Write the five angles A, «', y, fi', B in order on the sides of a simple pentagon. Then, calling any one part the middle part and the other two pairs the adjacent parts and the opposite parts, we have cos (middle part) = product of sines of adjacent parts, (a) Taking in succession y, A, B as middle parts, we get cos y =sin a sin B', cos A= sin a sin S, cosB = sin/3'sin^. Hence cos y =cot A cot B, i..e. cos (middle part) = product of cotangents of opposite parts (b) There is a relation of one of these forms between any three parts of the triangle. For convenience we write down the ten relations in terms of a, b,c, A, B. cos t = cos ,-' cos j - cot A cot B, k k K Bin t = sin v sin A = tan y cot ff , a . n . c b cos A = cos t sin ff - cot % tan t . and two other pairs formed by interchanging a, 6 and A, B. 120 ELLIPTIC GEOMETRY im. 22 These are exactly the same as the relations which exist between the parts of a spherical triangle. Tlte trigonometry of the elliptic plane is tlierefore exactly the same as ordinary s /i//i'r >'>;,/ lrifjo7ioinetry If we write the parts «, B', y', A\ 8 in the order in which they occur in the triangle, we get the more familiar rules of Napier : sine (middle part) = product of cosines of opposite parts = product of tangents of adjacent parts. 22. Id elliptic space the formulae for spherical trigonometry are die same as in euclidean space, when we take as the measure of a side of a spherical triangle the angle which it subtends at the centre, and as the measure of an angle the dihedral angle between the planes passing through the sides and the centre. Fro. Let be the centre of the sphere, and let 0A, OB, OC cut the polar plane of in A', B', C. Then we get a hi. as] SPHERICAL TRIGONOMETRY 121 rectilinear triangle A'B'O with sides a', 6', c'. The angles which the radii 0A', etc., make with the sides are right angles ; hence A' = the dihedral angle between the planes OAB and OAC, is. A' =A. Also a' =ka. Hence the rela- tions between a. j8, y, A, B, C are the same as those between ^i >iri A\ R, C, which are the same as those of ordinary spherical trigonometry. The measurement of angle, plane or dihedral, is the same in all three kinds of space, and spherical trigonometry involves only angular measurement. This explains why spherical trigonometry is the same in all three geome- tries. 23. The trirectangular quadrilateral. As in hyperbolic geometry, there is a correspondence between a right-angled triangle and a trirectangular quadrilateral In fact Fro. 70. wo see that, by producing two opposite sides of the quadrilateral to meet, we get, corresponding to the trirectangular quadrilateral CamBi, a right-angled triangle with hypotenuse ?,jrfc-&=6, sides a and {irk -m- m. and tho opposite angles I and r - C. 122 ELLIPTIC GEOMETRY [Ex. in. If we write the parts IT r + O. / b a IT nt. ir 2 Tl " F 2~ fc' I in oyclio order, then we have tbe rules : sine (middle part) = product of cosines of opposite parts -product of tangents of adjacent parts. EXAMPLES m. 1. Prove that the bisectors of the vertical angle of a triangle divide the base into segments whose nines are in the ratio of tbe sines of the sides. 2. Prove that the arc of an equidistant- curve of distance a, corre- sponding to a segment x 011 its axis, is x cos «//,'. 3. Prove that the area of a circle of radius r is iirfcsm 2 — . 2k 4. Prove that the area included between an arc of an equidistant- ourve of distance o, its axis, and two ordinate's m distance x, is ■ . a kxmn -. 5. Prove that the area of tke whole plane is 2wl.*, and the volume of the whole of spaco is ir^fc 5 . 8. Prove that the volume of a sphere of radius r is .,/2r . 2A (In the following examples it is unity.) 7. If R is i In- radius of the cironmsphere of a regular tetrahedrou whose side is a, show that sin in - \/| sin H, 8. If 2d is the distance between opposite edges of a cube of edge 2a, 8ft 1 be distance between opposite faces, and ft the radius of the circumaphere, prove that sin x d - 2 tan s ffl, sin'ft = sin'a/oos 2o, sin ! fl = 3sin*a, 9. A semiregukr network is formed of triangles and quadrilaterals, two of each at each node. Prove that this can only exist in elliptic spaco, and that the length of the aide ia .'. -. Kx. in.] EXAMPLES 12^ 10. In elliptic geometry show that there can exist six equal circles, each touching each of the others, and of radius given by 2eoBrsin- = l; three equal circles each having double contact with the other two, and of radius % ; and (with overlapping) four 4 1 circles each touching the other three, and of radius cos- 1 — , 11. Prove that five spheres, each of radius g, can be placed each touching the other four ; eight spheres, each of radius -, and each having double contact with four others ; and four spheres of radius - each having double contact with the other throe, 4 * 12. For a regular polyhedron : a = length of edge. a = angle subtended by edge at centre, 8 - angle of each polygon. S = dihedral angle between faces, w = number of sides of each face. p = number of edges at each vertex. R = radius of circumscribed sphere. r = radios of inscribed sphere. p = radius of sphere touching the edges. R a = radius of circum circle of each face. r = radius of incirclc of each face. Prove the relations : 7T . ir a cos ■ sin cos i, n p 2 ir . 8 a cos - = sm - cos -, n I £ a sin " = sin R sin £, sin p = tan cot -, a V sin - =sin fl„sin -, sin r a = tan ; oot -, cos R = cos r cos /?„, cosp=cosr coar,,, sin r„= tan root-, sin r - sin p sin 3. n .it / / ,0 ,t\ ,^ 2 co a y^-cos.-j. 124 ELLIPTIC GEOMETRY I Ex. hi. 13. For a regular tetrahedron prove that cos 8=cosa/(l +2cos«>. hexahedron „ co9S=(cosa-l}/2 coso „ octahedron „ cosS= -i/(l + 2cosa>. is •■ dodeoahedron prove that f08 6={2coso-(l+V5)}/{4cosa + (l-^5)}. For a regular icosijiedroti prove that cos 8= {( 1 - V5) cos a - { ] + Vo) }/2(l + 2 coao). 14. Prove that elliptic space can be filled twice over by 5 regular tetrahedra of side cwT't - }), with 3 at each edge and 4 at each vertex. 15. Prove that elliptic space can be filled in tJio following ways : (1) 4 cubes, of edge |, 3 at oaoh edge and 4 at each vortex. (2) 8 tefcrahedra, of edge I, 4 at each edge and 8 at each vertex. (3) 12 octahedra, of edge ~ , 3 at each edge and 6 at each vertex. (4) 60 dodecahedrn, of edge cos" 1 — S z H t 3 at each edge and 4 at each vertex. 8 (5) 300 tetrahodra, of edge cos \ 5 at each edge and 20 at each vertex. CHAPTER TV. ANALYTICAL GEOMETRY. 1. Coordinates. We shall assume elliptic geometry as the standard case, and construct a system of coordinates. The formulae can be adapted immediately to hyperbolic geometry by changing the sign of fc 8 . Take two rectangular axes Ox, Ojf. Let P be any point, and draw the perpendiculars PM=u and PN =v. Let 0P=r, xOP = 6. M 71 FTC. 77. r, $ are the polar coordinates of the point, it, v might be taken as rectangular coordinates, but we shall find it more convenient to take certain functions of these. We have . u , r a sin ■,•= sin , cosy, sin , =sin . sinO. 126 ANALYTICAL GEOMETRY [IV. I For any point on OP, therefore, sin V = sin "tan B. This is the equation of OP in terms of the coordinates u and v. Consider any line. Draw the perpendicular ON=p, and let xON =«. -p and a are always real, and completely fio, ?a determine the line. If P is any point on the line with coordinates u, «, 7) <f tan ^ cot , =cos (8 - a). Therefore tan |cos| = sin|cos«+ain %in rt . This equation is linear and homogeneous in ■ u , v r m l' sm r C0S F We shall effect a great simplification, therefore, if we take as coordinates certain multiples of these functions. The equation of a straight line being now of the first degree, the degree of any homogeneous equation in these coordi- nates gives the number of points in which a straight line B] COORDINATES 127 meets the curve, i.e. the degree of the equation is the same as the degree of the curve. In order that the coordinates of a real point may be real numbers, both in elliptic and in hyperbolic geometry, we shall define the coordinates as follows : a; = « sin -r = « sm v cos tf, v y = k sin j_ = k sin , sin B, r 2=COSt. These are called Wekrstmss' point-coordinates. The three homogeneous coordinates are connected by a fixed relationship. We have i.e. :c* + ^ + ifcV=A a . As any equation in x, y, z may be made homogeneous by using this identical relation, we need only, in general, use the ratios of the coordinates. 2. The absolute. In hyperbolic geometry, putting ik instead of k, we find the coordinates a;=Asiuli . , j/=Asinh v, z=cosh r* and x, y, z are connected by the relationship x* + y*-k*z i =-k t . 128 ANALYTICAL GEOMETRY Iiv. 3 tf / is infinite, x, y, z are all infinite, but they have definite limiting ratios. Let a, 8, y be the actual values, *, y, z the ratios, so that a = \x, 8 =Xy, y = \z, and \-h» . ' Then a* + 8 a -k*y*=-li?; therefore X»+tf-kH*=-^ = 0. Hence the ra&os of the coordinates of a point at infinity satisfy the equation x*+y*-]fr?=0. This is the equation of the absolute, which is therefore a curve of the second degree or a conic. In hyperbolic geometry it is a real curve ; in elliptic geometry the equation is x* +y s +A?« s =0, which represents an imaginary conic. 3. Normal form of the equation of a straight line. Line-coordinates. We found the equation of a straight line in terms of the perpendicular p and the angle «, which this perpendicular makes with the z-axis, in the form x cos a +y sin a = kz tan ?. which may be written The ratios ( : n : f determine the line, and can be taken as its line-coordinates. It is convenient to take certain multiples of these as actual homogeneous coordinates, viz. £=cosa cost* F/=Sin a cos?' k C- -k&ln i ft .v. 4] THE STRAIGHT LINE which are connected by the identical relation These are called Weierstrass' line-coordinates. In hyperbolic geometry £ = cos a cosh*?. »j=sina cosh?. f=-&sinh^. and the identical relation is If p-»-co, £, i), f all ->oo. Let the actual values be a, 8, y, and let a =\£ 8=\r}, y = Xf ; then Hence the coordinates of a fine at infinity satisfy the equation it^ + *V - f* = 0, A homogeneous equation in line-coordinates £, q, f repre- sents an envelope of lines. This equation represents an envelope of class 2, {.e. a conic. This is the same conic as we had before and represents the absolute, since it expresses the condition that the line (£ v, f) should be a tangent to x* +y z - khP =0. 4. Distance between two points. Let P{x y y, z) and P'(x', y', z') be the two points, PP' =d. Then, if the polar coordinates are (r, 6) and (/, $'), d t t' . t . t' cos t ■ cos r cos -, + sin y sm -r cos (9 - &'} , xx 1 yi/ or, in terms of the ratios of the coordinates, jRc'+yy' + k?zz' d cos , = k j& +&+&**/** +#*+'!&* N.-E. O. 1*1 ANALYTICAL GEOMETRY [tv. fl It is convenient to introduce here a brief notation. If (x, y, z), (as*, y', z') are the coordinates of two points, we shall define xx 1 +yy' +l^zzf = (xx') t and we shall speak of the points {x) and {if). Then the distance between the points (x) and (a;') is g' ven b y d (xx'\ COS , = i ■! , k J(xx)J(x'x') 5 In elliptic geometry the distance-function is periodic. d Suppose d = hvk ; then cos , =0, and xx" + y>/ + k % zz' =0, i.e. all points on this hne are at the distance £tt& or a quadrant from (x\ y 1 , z'). This is therefore the equation of the absolute polar of (x', y\ z'). It is the polar with respect to the conic a?+^ + itV=0. This is therefore the equation of the absolute. d Suppose d = irifc ; then cos t = -1, and, with actual values of the coordinates, xx' + yy" +k?zz' ■ -k-. but x*+y 2 +kH 2 = **, and x'*+y'* +kH'*=k*; therefore, multiplying the first equation by 2 and adding to the others, (x + x'f + (y + y'f + k*{z + z'f =0, which requires that x' = - x, y' = - y, z' = -z. In spherical geometry these would represent antipodal points. In elliptic geometry antipodal points coincide, LV. o] DISTANCE AND ANGLE 131 and therefore in every case, if two points have their co- ordinates in the same ratios, they must coincide. 6. Angle between two lines. From the figure (Fig. 79) we have sin v 1 =sin -, sin 0,, cos t =cot t tan j", sin ^ = sin [sin^j, coB/3 t =coti tan y, no. 7u. cos </>i = sin $ 1 cos ~J, 0, + a = ir - <f>, cos 2 = sin /S,; cos jf, ft + fi z = « 3 - n t , cos (0! + 2 ) =sin /3, sin 0„ cos V} cos 7 , 2 - coscc 2 , sin ¥ sin ,"■ k k k cos (/?, + S ) = cot 2 T tan y" tan y - sin 8i si n /3 2 ^CGSfas-a,). ,n 132 ANALYTICAL GEOMETRY jiv.7 Therefore -f . p, . p s cos ^ =coaec 8 T sm V sin v fC A' tC + (cos «j - «! - cot* v tan y ten y ) cos y 1 cos y 2 = sin 41 sin ', a + cos jJ cos ^f cos (a a - n^) = l^ + i^t + mnz, or, in terms of the ratios, COS (/> J Pfr + &** + & n'^? + *V + & If (££) =0 is the line-equation of the absolute. 9 </&&M&£,) 7. Distance of a point from a line. If tl is the distance of a point from a line, kirk-d is the distance of the point from the pole of the line. Let the coordinates of the point be (x, y, z) and of the line (f, >;, f). The pole of the line is (&*f , khi, 0. Therefore &+ny+& m £* + mi 1 cj ■ + £ a + W Jg* + >? + &I& J(asc) V( I . d sin . = -== 8. Point of intersection of two lines (&, i; lt £,), The coordinates of the point of intersection are pro- portional to Uf s - nsfi » &&-£*&» &J»-&m)- iv. 8] INTERSECTION OF LINES 133 Tf the actual values are a, ft, y, so that « =X#, etc., then +# s (£'J2-£tfl) 2 ] = A !! ifc a [l-COS 3 f]; therefore X = cosec ^, where is the angle between the lines. I f the coordinates of the point of intersection satisfy the equation K a + ii/ 2 + A-V=0, i.e. if the lines intersect on the absolute, X is infinite and tj> is zero. The two lines in this case are parallel. If the ratios of the coordinates make x 2 + tf + k i z t <0, X is imaginary. The two lines have then no real point of intersection, and the angle is imaginary, The lines may be said to intersect outside the absolute. (These two cases can, of course, only happen in hyperbolic geometry.) In the latter case the two lines have a common perpendi- cular. Let gx + iftf + £z =0 be perpendicular to both ; then Ut+vh + &/** =0- a* + w> +&VA 2 =0 ; therefore f : n : i=tt^2-ldi '■ 6& " £•£ : ^(&9»-&Ji) 5 but this line is just the polar of their point of intersection. The length p of the common perpendicular is equal to k<f>, and we have a ^ COS <j> = COS I = i ,f 2 + W 2 + U ^ 2 . The actual Weierstrass coordinates of an ideal point are therefore purely imaginary numbers of the form (ix, ly, iz), and their ratios are real. If we let the coordinates {z, //. s) be any complex numbers, we get points bolonging to the whole " complex domain." This 134 ANALYTICAL GEOM ETR V [iv. includes (1) real actual points, for which the ratios x : // : s are real and i- i if - && has the same sign as L~ (2) real ideal points, for which the ratios x:y :i are real, and (** + j*-i-JPstyJP is negative, (8) imaginary points, for which at least one of the ratios x;y:B is imaginary. The line joining a pair of conjugate imaginary points is a real lino, actual, at infinity or ideal. The distance between a pair of conjugate imaginary points is real only if their join is ideal. 9. Line joining two points. Similarly the line-coordinates of the line joining two points (xj, y, , Zj), (x s , y t , z t ) are proportional to #,2 2 -y.fr, ZjOSg-ZjSj, Xjifi - sr-ji/, . The actual values of the line- coordinates are found by multiplying by the factor cosec ., where d is the distance between the two points. If the ratios of the line-coordinates satisfy the equation £* + «* + £*//?=(), the line is at infinity, and the distance d is zero. If the ratios make g* + ir + ^/i 2 <0, the line is wholly ideal, and the distance d is imaginary. 10. Minimal lines, When the join of two points is a tangent to the absolute. the distance between the two points is zero. For this reason the tangents to the absolute are called int'itininl lines. In euclidean geometry the distance between two points {x it #,), <*,.*) is zero if {Xl - Xs? + { y,. Vtr=0t Li.it ,,.. //,= d ){*i-Xi}, i.e. if the join of the two points passes through one of the circular points (Chap, II. §17). The line at infinity itself passes through both of the circular points, ant! it is the only real line which passes through them. The distance between two [joints at infinity should thus be BMO. But again, any point on the line at infinity is in- ly, li] CONCUBBENCY AND COLLINEARITY 135 finitely distant from any other point. Hence the distance between two points, both of which are at infinity, becomes indeterminate. In relation to the rest of the plane wc must consider such distances m infinite, and the geometry of points at infinity becomes quite unmanageable. The geometry upon the line at infinity by itself, however, is really elliptic, since the absolute upon this line consists of a pair of imaginary points ; the " distance " between two points at infinity could then be represented hy the angle which they subtend at any finite point. 11. Concurrency and collinearity. The condition that the lines (£, 17, , &)> etc., be con- current is £1 >h & £2 >h u £3 'la & =0. The condition that the points (a^, y t , «,), etc., be collinear is a% tfx % ,=0. aj £ y 2 z 2 #3 Vs z 'i These conditions arc, of course, the same as those in ordinary analytical geometry, with homogeneous co- ordinates. Since the equation of a straight line is homogeneous and of the first degree in the coordinates, all theorems of ordinary geometry which do not involve the actual values of the coordinates, or the distance-formulae, will be true also in non-euclidean geometry. These theorems are those of projective geometry. The difference between euclidean and non-euclidean geometry only appears in the form of the identical relation which connects the point and line coordinates, i.e. in the form of the absolute. 136 ANALYTICAL GEOMETRY [iv. 12 12. The circle. A circle is the locus of points equidistant from a. fixed point. Let {x 1 , jfe «,) be the centre and r the radius ; then the equation of the circle is or, when rationalised, (aacftxiXj) cos 2 £ = (xar,) a . This equation is of the second degree, and from its form we see that it represents a conic touching the absolute (xx) =0 at the points where it is cut by the line (xx l ) =0. (axe,) =0 ia the polar of the centre, and is therefore equidistant from the circle, i.e. it is the axis of the circle. Hence A circle is a conk having double contact with the absolute ; ite axis is the common- chord and its centre is the pole of the common chord. The equidistant-curve. Let (£, i hl f,) be the coordinates trf the axis, and d the constant distance; then the equation of the curve is s \ a d = & + *& + & or J (xx) ($i) sin^ = {£sr + .,# + &f. This again represents a conic having double contact with the absolute, the common chord being the axis. The pole of the axis is equidistant from the curve, and so the equi- distant-curve is a circle. In elliptic geometry both centre and axis are real, in hyperbolic geometry the centre atone is real for a proper circle, and the axis alone is real for an equidistant-curve. iv. 13] THE CIRCLE 137 The horocycle. In hyperbolic geometry, the equation of the absolute being x* + if - k*z 2 = 0, the equation of a horocycle is of the form 2? + y & - kH z = \(ax + by + cz)% where *+*-£ 13. Coordinates of a point dividing the join of two points into given parts. If (x 1 , Jf t ,?j), (aij, 3/g, z 2 ) are any two points, the coordinates of any point on the line joining them are (Kx^+fixt, Ay, + /*?/«. Az,+,u2 2 ), for if ax + by+cz = is the equation of the line, so that it is satisfied by the coordinates of the two given points, it will he satisfied also by the coordinates of any point with coordinates of this form. Similarly, if we consider these as the line-coordinates of two lines, the coordinates of any line through their point of intersection are of this form. In fact the line X(«jx + 6,y + Cjz) + tt(a& + b$ + c#) =0, whose coordinates are (A<ri + /ia*, ...)• passes through the intersection of the two given lines a 1 x + b i y+e t z = Q and a& + o a y + CjZ = 0. To find the coordinates of a point dividing the join of two points whose actual coordinates are (34 , y t , z,) and (a! 2 , y 2 , z 2 ) into two parts r, and r s , where r l +r i -r. Let(\Xj+ux s ,...) be the actual coordinates of the re- quired point. Then g, (Xx! + nx t )+ y, (Ay, + /xy a ) + *fej (As, + pxj - k* cos J ; therefore A + /* cos ,- % = cos } • 138 ANALYTICAL GEOMETRY Similarly Xcos , +fi = cos r\ k whence A sin, = sin-- and Msin|=sin^ and the actual coordinates are - r. at, sin ^ + x s ain k [rv. 14 sm I If (^i. fh, %). etc., are only the ratios of the coordinates, we must first find their actual values by dividing by the factor (zx)/k. If the line is divided externally into two parts r, and r t , we have only to observe the proper signs off, r t and ■/■„. 14, Middle point of a segment. In particular, if r, -r, we get the ratios of the coordinates of the middle point of the segment (a:, +x it g, +y t , 3, + Zg ) t or, if at,, etc., are only proportional to the coordinates, the ratios of the coordinates of the middle point are % t ¥» the actual values being obtained by dividing by 2 cos X. The join of too points has a second middle point with coordinates -J!JL=- ,% n : . : . . the actual values being obtained by dividing by 2ain^. In elliptic geometry these points are both real and a quadrant apart ; ' in 1 In spherical geometry the tmo middle points of a segment are antipodal, and are not (as in elliptic geometry) harmonic: conjaentcs with respect to the given points. rv. 16] CEXTROTDS 139 hyperbolic geometry the factor 2 sin gr, becomes 2* sinn ^ and the coordinates of the second middle point are all imaginary. 15. Properties of triangles. Centroid, in- and circum- centres. Through each vertex of a triangle (a;,), (sj s ), (a%) pass two medium, and the medians are concurrent in sets of three in four cenlroids, denoted, in the notation of §4, by JU±- *)■ Vtatjai) V(ij*s) V(!KjXs; J The same combination of signs is taken for all three co- ordinates, and theTe are four different combinations of signs, one corresponding to each of the centroids. Similarly, the middle points of the aides are collinear in sets of three in four lines, the axes of the circumscribed circles. The bisectors of the angles are concurrent in sets of three in four points, the centres of the inscribed circles ; and their points of intersection with the opposite sides are collinear in sets of three in four lines. 16. Explanation of apparent exception in euclidean geometry. In euclidean geometry four cycles can be drawn to touch the sides of a triangle, but apparently only one can be circumscribed. Of the four circumcireles of a triangle in hyperbolic geometry, three are equidistant-curves. In euclidean geometry the equidistant-curve through B, C and A reduces to the line EC and the line through A \\ BC. (Ci. Chap. II. f 23.) 140 ANALYTICAL GEOMETRY flv. 16 The conception of a pair of parallel straight lines as forming a circle iu euclidean geometry is consistent with the definition of a circle as a conic having double contact with the absolute, for the absolute in this case is a pair of coincident straight lines, and this is cut by a pair of parallel lines in two pairs of coincident points. A single straight line is not, of course, a tangent to the absolute, though it cuts it in two coincident points : this case is just the same as that of a line which passes through a double point on a curve, but which is not considered as being a tangent. But when we have a pair of parallel lines cutting the absolute S> in four points all coincident, we can regard O as being a tangent to the curve consisting of this pair of lines. Fig. 80 represents the case approximately when the absolute is still a proper conic and the pair of straight lines is also a proper conic, having double contact with the absolute. Fia. so. The axis of the circle consisting of a pair of parallel lines is the line lying midway between them ; the absolute pole of this (a point at infinity) is the centre. When the 171 ORTBOCENTRE 141 axis passes through the centre, i.e. when it coincides with the line at infinity, the circle becomes a horocycle, which is thus represented in euclidean geometry by a straight line together with the line at infinity. Two equidistant-enrves, with parallel axes, have the same centre at infinity. In hyperbolic geometry two equidistant-curves, with parallel axes intersecting at infinity at 0, have their centres on the tangent at 0, and therefore at a zero distance apart though not coincident. 17. Polar triangles. Orthoeentre and orthaxis. If A, B, C is a triangle and A', B', C the absolute poles of the sides o, 6, c, then the sides a', b', c' of the second triangle are the absolute polars of the vertices A, B, C of the given triangle. Two such triangles are called polar triangles. Tf the coordinates of A, B, are (a^, ft, Zj), etc., the equations of their polars are (2X1) =0, etc. The point-coordinates of the vertices A', B\ C are &&-?&' *x**-*A. (Siya-Ssffi)/* 2 . etc - The equation of AA', which joins (x,,ft, »i) to the point of intersection of (axe 9 ) - and {xx 3 ) =0, is (xXi) (x&) - {xx 3 ) (x^) =0. Writing down two other equations by a cyclic permuta- tion of the suffixes, we get the equations of BB' and CC, and the sum of these vanishes identically. Hence AA', BB', CC are concurrent. AA'LBC and B'C ; hence the point of concurrence is the common ortkomUre of the triangles ABC, A' B'C. The absolute poles of A A'. BB', CC, i.e. the points on the sides of the triangles distant a quadrant from the 142 ANALYTICAL GEOMETRY [iv. 18 opposite vertices, will be collmear in a line called the ortkaxis, o, which is the absolute polar of the ortho- centre. The two triangles ABC, A' BO' are in perspective with centre and axis o. 18. Desargues' theorem. Configurations. The last result is a particular case of Desargues' theorem for perspective triangles, which, since it expresses a pro- jective property, is true in non-enclideau geometry, and can be proved (using space of three dimensions) in a purely projective manner. In the figure for Desargues' theorem (Fig. 81) we have two triangles with their corresponding vertices lying on three concurrent lines, and their corresponding sides inter- secting in three collinear points. There are thus 10 points and 10 lines : through each point pass 3 lines, and on each r*. 181 CONFIGURATIONS 143 line lie 3 points. A figure of points and lines with this property, that through every point pass the same number of lines and on every line lie the same number of points, is called a configuration. If p m denotes the number of lines through a point, p i0 the number of points on a line, p 0Q the whole number of points, and p u the whole number of lines, the configuration may be denoted by the symbol 7>ao Poi 3>io Pn Desargues' configuration is represented by and is reciprocal. A convenient notation for the points is by pairs of the numbers from 1 to 5. The three points which lie on one line are denoted by the combinations with the same three numbers. The configuration formed by the six middle points 8f«, M & , etc., of the sides of a triangle ABV and the four points of concurrency G , <?„ G%, G 3 of the medians is a Desargues configuration of a special kind (Fig. 82). The points G form a complete quadrangle, and the points H are the vertices of a complete quadrilateral, both having ABC as diagonal triangle. This is called, therefore, the quadrangle- quadrilateral configuration. Each vertex M ra of the quadri- lateral lies on a side G r G. of the quadrangle. Similarly, the eix bisectors of the angles and the four lines of collinearity of the points in which they meet the sides of the triangle form the same configuration. 144 ANALYTICAL GEOMETRY I FIO. 83, 19. Desmic system. In three dimensions we have similar interesting con- figurations. If fa), (x t ), (x 3 ), (x t ) are four points in space, v(*a) A*&i) Az&z) -J(z& 4 y represent ^ the eight centroids of the four points. Each centroid is on a line joining one of the points to the cen- troid of the other three. If the four given points be denoted by A lt A t , A s , A t . and the other points corresponding to the different com- binations of signs be represented as follows : it. 20] DESMIC SYSTEM 145 + + + + B u + + - - B 2 , + - + - B 3 , + - - + B t , + c„ + - + + O a , + + - + c a , + + + - c t , then the join of any B with any C passes through an A, e.g. B a - C t gives A 3 . So the 12 points lie in sets of 3 on 10 lines. They form three tetrahedra, any two of which are in perspective in four different ways, the centres of perspective being the vertices of the third tetrahedron. Corresponding planes of two perspective tetrahedra inter- sect in four lines which are coplanar, and these planes are the faces of the third tetrahedron. A system of tetrahedra of this kind is called a desmic system. In a similar way it may be proved that the centres or axial planes of the 8 circum- or in-scribed spheres form with the given tetrahedron a desmic system. A simple example of a desmic system in ordinary space is afforded by the corners of a cube, its centre and the points of concurrency (at infinity) of its edges. 20. Concurrency and collinearity. In euclidean geometry we have the two useful theorems of Menelaus and Ceva as tests for collinearity and con- currency. Theorems corresponding to these hold also in non- euclidean geometry. I. If a transversal meets the sides of a triangle ABC in XYZ, and a, j8, y are the angles of intersection, taken positively, we have (Fig. 83) sinBX_ siny sinC'Ysinq sin^Z = _sin£ sin BZ~ ~ sirTa ' sin CX ~sm8' sin A Y sin y ' N.-e. a. K 146 ANALYTICAL GEOMETRY [rv. 20 the positive directions on the sides being in the cyclic order ABC. Hence sin BX si n CY sin AZ sin CX sin AY sin HZ A = +1. Fio. 83. II. If three concurrent lines tlirough the vertices meet the opposite sides of a triangle ABC in XYZ, and a, ft, y are the angles between the lines (Fig, 84), siuiJA'siiiy , sin C<£ _ sinj3 sin OB "sin X sin 00 "sin X' sin BX _ sin OB sin y sin CX sin OC ' sin ft' singy _ sin 00 sin a sin>4F~ sinOZ siny" sin AZ sin 0,4 s in ft sin BZ ~ sin OB sin a ' therefore Similarly and Therefore sin .a* sinC'F sin^Z --]. sin OX' sin AY smBZ' Conversely, the points X, Y, Z are col linear, or AX, BY, CZ are concurrent, according as siiiBX sinCY sin AZ _ .sin OX ' sin At ' ^mBZ " + l ot ~ L w.sil CONCURRENCY AND OOLLINEARITY 147 This condition may be put in another form. Since siaBX sin AB sin BAX ... .... , . • . - ,__. = -. ... ■ . i-nri the condition reduces to sin CX sin AC sin CMa nn.K4£ sin Cay sin ACZ sin 321 ' sin ABY ' sin BCZ~ ~ ' in which form it is the same as the condition in cuclidean geometrv. From this it follows at once that if AX, BY, CZ are three concurrent lines through 0, their isogonal conjugates with respect to the sides of the triangle are concurrent in the iBogonal conjugate of 0. 21. Position-ratio. Cross-ratio. If X, Y, P are colli near, the ratio . ., „ is called the position-ratio of P with respect to X and Y, and the double ta tio EJL_- _^?ML_* is called the cross-ratio of the range sin YP sin YQ ( ,Y Y, PQ). Similar definitions can be given for pencils of rays, and the whole theory of cross-ratio can be developed on the same lines as in ordinary geometry. 148 ANALYTICAL GEOMETRY fiv. 21 Thus, the cross-ratio of a pencil is equal to that of any transversal, and cross-ratios are unaltered by projection. T Further, it can be shown that (ABCD)=(BADC) - (CDAB)=(DCBA), (ABCD) . {ABDC) - 1 , (ABCD) + (ACBD) - 1. The harmonic property of the complete quadrilateral follows. For (Fig. 85), (XYPQ)t? t (UYMQ), and also^FtfMQ). Therefore ( VV, MQ) = ( VU, MQ) = - 1. If A (a^ , y t , s,) and B(x 2 , y 2 , z s ) are two fixed points, and P a variable point with coordinates fo + \x t , y t + \y« , s, + As a ), then, if AP=r t , PB = r 2 , AB = r, we found A=sin -/ sin , 2 = the position -ratio of P with respect to A and B. If Q is the point corresponding to the parameter p, the cross-ratio (AB, PQ)=-. The cross- Ex. IV.] CROSS-RATIO 149 ratio of the two pairs of points corresponding to the para- meters A, A' and n, m' is These results are the same as in euclidean geometry. EXAMPLES IV. 1. Prove that the actual Weierstrass line -coordinates of the absolute polar of {%, y, z) are (x/k, y/k, kz), and the actual point- coordinates of the absolute pole of {£, »;, f ) are (k£, to), (/k). 2. If the distance between the points (-r 1( y u Zj), (*,, y„ z.) vanishes, prove that their join touches the absolute, 3. If to+ir,, ifi + iy t , «! + «i) * rc the actual Weierstrass coordi- nates of a point (£,, y, . etc., being real numbers), prove that (z, , y, , Zt) and (a:,, y,, n t ) are conjugate with regard to the absolute, 4. If («i + «Xj,...) (a, i t'<ju....)are the actual Weierstrass coordinates of two points at a real distance (a-, , ?/, , etc., being real numbers), prove that, for all values of X, (*, + Au, , ...) and {x,+ Aa,,..,) are conjugate with regard to the absolute. 5. If ds is the element of arc of a curve and dx, dy, «fe the differen- tials of the Weieratrass coordinates, prove that di?=dx t +d?f ! +k?dz l . li" r, ff are the polar coordinates, prove that (&*=(?»* -t-fr* sin* f/fl*. 6. ABCD i» a skew quadrilateral, PQIiS are points on the four sides AB, BC, CD, DA. Prove thai if sin AP sin BQ sin Oil sin Z>S=sin BP sin CQ sin DR sin A3, the four points PQKS lie in one plane. 7. 1, 2, 3, 4 are the vertices of a tetrahedron. A plane cuts each of the six edges. If the edge 12 is out at A, and the ratio mnlAfwi2A is denoted by (12), prove that (12)(23}(34)(41) = 1. Conversely, ir {12)(23)(:U){ 11}- 1, prove that the points 12, 23, 34, 41 (i.e. the corresponding points on these edges) are coplanar. 8. If (12)(23)(34)(41)-I=(12)(24){43)(31) = (13K32){24)(41), prove that either (i) the sets of points 12, 23, 31, etc., are collinear, or (ii) the lines (12, 34), (13, 24), (14, 23) are concurrent. 150 ANALYTICAL CEOMETRY [Ex. iy. 9. Four circles (ouch in succession, each raw touching two others (the number of external contacts being even) ; show that the four points of contact lie on a circle, and that the four tangents at the points of contact touch a circle. 10. Four spheres touch in succession, each one touching two others (the number of external contacts being even) ; show that the Four points of contact lie on a circle, and that the Four tangent planes at the points of contact touch a sphere. .Show Further that, whatever the nature of the contacts, the four tangent planes pass through one point. 11. Five spheres touch in succession, each one touching two others (the number of externa) contacts being even) ; show that the live points of contact lie on a sphere, and that the five tangent planes at the points of contact touch a sphere. (Edue. Times (n.s.), xi p. 57.) 12. D, E, F are the feet of the pcr]>endiculars from a point on the sides of the triangle ABC. Prove that cos BD cos CE cob A F = cos CD cos A E cos BF. 13. A BC is a given triangle, anil I is any line. P, Q, R are the feet of the perpendiculars fr .1 . B, C on I. PP J ±BC, QQ'LCA, RR'IAB. Prove that PP', QQ\ BR' meet in a point {the arik&pole oil). 14. Prove that the locus of a point such that the ratio of the cosines of its distances from two fixed points is const-ant is a straight line. 15. If L, M, N ; £n, All, A\ ; etc., are the points of contact of the in- and e-scribed circles of the triangle ABC with the sides a, b, e. and 2s=a + b + e, prove the relations: A M, ■ AJS t = BN t = BL t = CL 3 =CM 3 =s, AM=AN = BN s =BLs=CL t = OM l =&-a, etc. 16. Establish the reciprocal relations to those in Question 15 for the circumcircles. 17. Prove that the envelope of a line which makes with two fixed lines a triangle of constant perimeter is a circle. Prove also that the envelope is a circle if the excess of the sum of two sides over the third side is constant. What is the reciprocal theorem ? Ex, rv. EXAMPLES 151 (In the following questions, 18-22, the geometry is hyperbolic. The formulae are analogous to well-known formulae in spherical trigonometry.) 18. If /.■«. kb. kc are the sides, and A, B, C the angles of a triangle, prove that A _ /sinh j) sinh (s - a) gm A _ /sinh (« - b) sinh (s -ej 2 t sinh b sinh c 2 \ sinh b sinh c 19. If r, r,, r t , r, are the radii of the in- and e-scribed circles of a triangle ABC, prove that tanh r sinh /s = tanh r, sinh {# tt} = tanh r, sinh (a - b) =tanhr, sinh (s-c) =v'sinh (s -a) sinh (a - b) sinh (a - c). 20. Prove that tanh r, tanh r* tanhrj =\/sinh (a - a) sinh (a-b) sinh (» - c), 21. If R is the radius of the circumcircle of the triangle ABC, prove that . a , b c sinh a audi 6 sinh o 2 cosh -cosh _ cosh -tanh /? = — — r = . - = - ; „. 2 2 2 sin A am B sin If D u D t , D 3 are the distances of the circumscribed equi- distant-curves, prove that . a , , b . , c - _ sinh a 2 cosh- sinh -sinh-oolh D,- , ■•> ete . 2 2 2 sin A 22. Prove that eothif+tanh /J, -i fanh /) 3 I tanh JJ a ='2 cosh# sin A, 'sinh a, col h R I tanh O, - tanh /J, - tanh l) 3 -2 cosh (n - a) sin i4/sinh a, etc 23. Prove that, in the desmic configuration in § 1 9, the following sets of points are coplanar: AiA.BtB^Ct, AtAJBiBJSiCt. and those obtained from these by cyclic permutation of ABC or of 234. Deduce that the configuration has the symbol 12 4 6 3 16 S 6 4 12 152 ANALYTICAL GEOMETRY [Ex. 24. If ono pair of altitudes of a tetrahedron ABCl) intersect, prove that the other pair will also intersect; and if one altitude intersects two others, all fonr arc concurrent. If these conditions are satisfied, prove that cos AB cosCO^cos AC cos BD=coa AD ma BO. 25. Prove that there is a circle which touches the in- and the e-scribed circles of a triangle. [In spherical geometry this is Hart's circle, and corresponds to the nine-point circle in ordinary geometry. See M'Clelland and Preston's Spherical Trigonometry, Chap. VI Art. 88.] ^ 26. Prove that there is a circle which touches the four circum- ctrcles of a triangle, [In euclidean geometry the circumscribed equidistant-curves are three pairs of parallel lines and form a triangle A'B'C, of which A, B, C are the middle points of the sides, The circumcircle of ABC m the nine-point circle of A'B'C. and touches the inscribed circle of A 'B'C. That is. the last-named circle touches the four " circumcireles ■ of the triangle A BC.J CHAPTER V REPRESENTATIONS OF NON-EUCLIDEAN GEOMETRY IN EUCLIDEAN SPACE. 1. The problem of Representation is one that faces us whenever we try to realise the figures of non-euclidean geometry. There already exists in the mind, whether intuitively or as the result of experience, a more or less clear idea of euclidean geometry. This geometry has from time immemorial been applied to the space in which we live; and now, when it is pointed out to us that there are other conceivable systems of geometry, each as self-consistent as Euclid's, it is a matter of the greatest difficulty to conjure up a picture of space endowed with non-cucltdunn properties. The image of euclidean space constantly presents itself and suggests as the easiest solution of the difficulty a representation of non-euclidean geometry by the figures of euclidean geometry. Thus, upon a sheet of paper, which is for us the rough model of a euclidean plane, we draw figures to represent the entities of non-euclidean geometry. Sometimes we represent the non-euclidean straight lines by straight lines and sometimes by curves, according as the idea of straightness or that of shape happens to be uppermost in the mind. But we must never forget that the figures that we are constructing are only repre- sentations, and that the non-euclidean straight line is 154 CONCRETE REPRESENTATIONS [v. 2 every bit as straight as its euclidean counterpart. The problem of representing non-euclidean geometry cm the euclidean plane is exactly analogous to that of map- projection. Projective Representation. 2. The fact that a straight line can be represented by an equation of the first degree enables us to represent non- euclidean straight lines by straight lines on the euclidean plane. Distances and angles will not, however, be truly J represented, and we must find the functions of the euclidean distances and angles which give the actual distances and angles of non-euclidean geometry. 3. The absolute is represented by a conic. In hyperbolic geometry this conic is real, in elliptic geometry it is wholly imaginary, but in every case the polar of a real point is a real line. The conic always has a real equation. In the case in which the absolute is a real conic, we could if we like, represent it by a circle, but except in special cases this does not give any gain in simplicity. Two lines whose point of intersection is on the absolute are parallel ; two lines whose point of intersection lies outside the absolute are non-intersectors. The points outside the absolute have to be regarded as ultra-infinite, and are called ideal points. They are distinguished from other imaginary points by the fact that, while their actual coordinates are all imaginary, the ratios of their coordinates are real. In the present representation they are repre- 1 In the sense of map-projections ; i.e. angles which are equal in the euclidean representation, when measured hy euclidean standards, do not in general represent oipin] angles in the non-euclidean geometry, but, again in the sense of map- project ions, figures are distorted. •■1 THE ABSOLUTE 155 sented by real points ; other imaginary points are repre- sented by imaginary points. (Cf. Chap. IV. § 8.) A real line has two points at infinity, and part of the line lies in the ideal region. A line which touches the absolute Idtrtt point! Fro. SB. has one point at infinity, and all the rest of the line is ideal. A line which lies outside the absolute is wholly ideal. Through any point two parallels can be drawn to a given line, viz. the points joining the given point to the two points at infinity on the given line. A triangle which has its three vertices on the absolute has a constant area. In elliptic geometry the absolute is imaginary, and there are no ideal points. 4. Euclidean geometry. Euclidean geometry is a limiting case, where the space- constant k-><x> . The coordinates of a point become Hie usual rectangular coordinates x and y with z = l. The equation of the absolute becomes in point-coordi nates z = 0, and in line-coordinates £« + »j a =0, ie. the absolute degenerates as a locus to a straight line counted twice— the straight line at infinity, and as an envelope to two imaginary pencils of lines, i + i», = and £-iij=0, whose 156 CONCRETE REPRESENTATIONS fv.s vertices lie on the line at infinity since the line-coordinates of their join are f = 0, ,=0, f«£ and its equation is therefore z=0. The equations of the lines of these ima- ginary penci Is are of the forms x + iy + at = 0,x-iy + cz=Q. The formula for the distance between two points, ja£+0£+* s a*'__^ r cos 7= ._ ■Jx 2 + -f + Ktyyfx^+y'* + jfcVi' becomes 1 -\ • jj.^ + yy' + k *) ■ £ . (l _ J . ^/ ) V 2 JF" / ~( ii ^y | 6- 1 ■ ^ + ^. +i ' a tii s \ or -i- 1 (^T + to-yT r 2 = (j;-^) 3 + (y-y')2. 5. The circular points. The equation of a circle becomes of the general form x* +y* +z(ax+by + cz)=Q, and this represents a conic passing through the points of intersection of the line s=0 with the pair of imaginary lines x + iy =0 and x - iy = 0, i.e. every circle passes tliiou-h the vertices of the imaginary pencils. For this-; reason these two points are called the circular points. This pro- perty of the circle is the equivalent of the property that it has double contact with the absolute. (Chap. IV. § Hi.) 6. Now, in ordinary geometry the angle between two lines can be expressed in terms of the two lines joining their point of intersection to the circular points. 1 BteE uffsssT " mtC 8Ur b a "" ;oriC deS foyc,rs '" AW Ann - J/ ""' ■ v. 71 ANGLE AND DISTANCE FUNCTIONS 157 Let the equations of the two lines w, «' through the origin be y=x tan 6, y =x tan 0', and denote the two lines joining to the circular points by m, w \ their equa- tions are y = ix, y^ -kk. The cross-ratio of the pencil (ttu',<m') is tan (J -i^ tan 0' -i tan 6 + i ' tan 0' +i Now i-t&ad a cos 0- sin cosfl+isinfl i + tan& icosfl+ainrJ cos 0-* sin W Therefore (« u', «»') = e s '<* ' *\ and =c 2,s . = 0' - Q = | log (u»'. mw'). i.e. fte aflfffe between two tines is a certain multiple of the logarithm of ilw cross-ratio of the pencil formed by the too lines and the lines joining their point of intersection to the circular points. 7. Now let us return to the case where the absolute is a real conic a£ + y* - Bz z = 0. Consider two points P(x,y,z), P'{x', y', z'). The point (x + \x r , y+\t/, s+Xz') lies on their join. If this point is on the absolute, (x + \x') 2 + (y+\yy-&(z + te'?=0, i.e. X* (x' a + y'* - A-V s ) + 2X (xx 1 + yy' - 1M) + (x % + f- -&*z*)=0. Let X,, A j be the roots of this quadratic. The line PP' cuts the absolute in the two points X, Y, corresponding to these parameters, and the cross-ratio of the range {PF,XY)=h Let {PF)=d = k<j>, and X s + y* - £*z 2 = r*, x' a + y' 1 - AV 2 = r' 2 ; 158 CONCRETE REPRESENTATIONS [v, 8 then the quadratic for \ becomes A V 3 + 2\rr' cos ih + r 3 = ; whence X lf X 2 = ( - cos oi:t J - sin 3 0)r/r' = - e +f *r/r'. Therefore XJK^e-*'* and £ = A* log (W, XT). Therefore d = Hi log (PP', 1 V ) , i.e. f/i« distance between two points is a certain multiple of the logarithm of tJte cross- ratio of live range formed by the two points and the two points in which their join cuts the absolute. In a similar way it can be shown that the angle between two straight lines is a certain multiple of the logarithm of tJte cross-ratio of the pencil formed by the two lines and the two tangents from their point of intersection to the absolute. If the unit angle is such that the auglu between two lines which are conjugate with regard to the absolute is |tt, then <f> = \i\og{pp', xg). 8. By this representation the whole of metrical geometry is reduced to projective geometry, for cross-ratios are unaltered by projection. Any projective transformation which leaves the absolute unaltered will therefore leave distances and angles unaltered. Such transformations are called congruent transformations and form the most general motions of rigid bodies. This projective metric is associated with the name of Cayley. 1 who invented the term Absolute, lie was the 1 "A sixth memoir upon quantics," London Phil. Trans. R. Sor... 149 (1859). Cayley wrote <i number of papers dealing specially with non- euolidean geometry, but although he imisl he regarded us une of the epoch-makers, he never quite arrived at a just appreciation of the science, lit liis mind nou-cuclidean geometry scarcely attained to an indejieiidcnt existence, hot was always either the geometry upon a certain claas of curved surfaces, like spherical geometry, or a mode of representation of certain projective rotations in euolidean geometry. 9] PROJECTIVE METRIC first to develop the theory of the absolute, though only as a geometrical representation of the algebra of qualities. Klein ' introduced the logarithmic expressions and showed the connection between Cayley's theory and Lobaehevsky s geometry. 2 9. As an example of a projective solution of a metrical problem, let us find the middle points of a, segment PQ. Let PQ cat the absolute in X, Y, and let 31,, M s he the double points of the involu- tion {PQ, XY). Then (X YPM,)7:{ YXQMJ7=:(X YM k Qh therefore tli9t.(/*jl/,)=dist(Jf 1 Q), jtf„ ju% are therefore the middb points of the segment (PQ). Since M u M t are harmonic conjugates with respect to X, Y and also with respect to P, Q, the construction is therefore as follows. 1 " Uber die sogenannte Nicht-Euklidischo Geometrie," Math Ami., 4(1871), 6(1873). 1 Since the definition of the cross-ratio of a range is the same in non* euolidean geometry, the logarithmic expressions for distance and angle hold not only in the eucb'dean representation of the geometry, but also in Hie actual non-ouclidean geometry itself. ICO CONCRETE REPRESENTATIONS [v. 10 Join 0, the pole of PQ, to P and Q, cutting the absolute in A A', BB\ AB\ A'B intersect in M, and AB, A'fi' in M 3 . For by this con- struction OM l M s is a self-conjugate triangle and M,, M 3 are harmonic conjugates with respect to A', Y, and also with respect to P, Q. 10. Classification of geometries with projective metric. Having arrived at the result that metrical plane geometry is projective geometry in relation to an absolute conic, distances and angles being determined by the projective expressions dist. (PQ) = K log (X Y, PQ), angle (pq) = h log (xy, pq), we may reverse the process, and define distances and angles by these expressions. We thus get a general system of geometry which will include euclidean, hyperbolic and elliptic geometries as special cases. The nature of the geometry will be determined when the absolute conic is fixed, and the values of the constants K and k have been determined. Generally speaking, the values of these constants depend only on the units of distance and angle which are selected, but there is an essential distinction according as the constants are real, imaginary or infinite. There is no distinction, for example, between the cases corresponding to different real values of K. This simply corresponds to a different choice of the arbitrary unit of length ; just as in angular measurement the constant k may be chosen so that the measure of a right angle may be - or 180 or any other number. As each of the constants may conceivably be real, infinite or imaginary, there are nine species of plane geometry. The points of the absohite are at an infinite distance v. llj THE EUCLIDEAN CASE 161 from all other points, and the rangente to the absolute make an infinite angle with all other lines. If the measure of angle is to be the same as in ordinary geometry, the tangents from a real point to the absolute must be imaginary ; the cross-ratio (pq, xy) will be imagi- nary, and k must be purely imaginary. When p, q are conjugate with regard to the absolute they are at right angles, and if the unit angle is such that the angle in this case is ^ , then k = \i. Then there are three cases according as the absolute is a real proper conic (hyperbolic geometry, K real), an ima- ginary conic (elliptic geometry. K imaginary), or degenerate to two coincident lines and two imaginary points (parabolic or euclidean geometry. K infinite). 11. In the last case there is a difficulty in determin- ing the distance. Since X. Y coincide, the cross-ratio (PQ, XY) is zero and K must be infinite; but the dis- tance becomes now indeterminate. Suppose PY=PX + e, where e is small. Then (PQ^y^fxVA^Q^i^A)' 1 QXi x+6 \QX"Pxr neglecting squares and higher powers of e. and (PQ) = K\og(PQ,XY) = K*Q x -±>). Let K—~kx> and «— >0 in such a way that Ke- >a finite limit A. Then (pQ)-*px-qx- S\-E. Q. 162 CONCRETE R E PRESENTATIONS [v. 12 Now, to fix X we must choose a point E such tbat (PE) =1, the unit distance. PX.EX PQ XB.XQ , yp ™. PE ' P~X7QX = PE • PQ-\ At ' B ^ h Then (PQ) = If we measure distances from P = as origin, <GQ)=(AD, EQ) = (0«>, £1)=^^, which agrees with the expression in euclideau geometry, since ;=1, and 01 = 1. oo 1 This case differs in one marked respect from the case of elliptic geometry. In that system there is a natural unit of length, which may be token as the length of the complete straight line — the period, in fact, of linear measurement ; just as in ordinary angular measurement there is a natural unit of angle, the complete revolution. In euclideau geometry, however, the unit of length has to be chosen conventionally, the natural unit having become infinite. 12. The other geometries, in which the measure of angle is either hyperbolic or parabolic, are of a somewhat bizarre nature. For example, if the absolute degenerates to two imaginary lines a, <i)', and two coincident points ii, the case is just the reciprocal of the cuclidean case; Sinoar measurement in elliptic, K being imaginary, and angular measurement is parabolic, k being infinite. In this geometry the straight lino is of finite length =!rA'i". If the positive direction along any one line is defined, the positive directions along alt other lines in a plane are determined, for this is determined by the sense of rotation about tiie point Ii. The sides of a triangle are defined as the segments which subtend the opposite angles which do not contain ii, just as in ouclidean geometry the angles of a triangle are defined as the angles which arc subtended by the opposite segments which do not cut the line at infinity. Thus the sides of the triangle A BC in (he figure ( Fig, 88) are represented by the heavy lines. If the positive direction on each v. 13] RECIPROCAL OF EUCLIDEAN CASE 103 line is then defined as the direction corresponding to clockwise rotation about il. then a + ti-t e=the length of the complete line, i.e. the pciimeier of a triangk is constant and = 5rJsT»» Fio, SB. 13. Extension to three dimensions. In three dimensions the absolute is a quadric surface. If the measures of angle between lines and between planes are to be elliptic, the tangent planes through an actual line must be imaginary, and the tangents through an actual point in an actual plane must be imaginary. (1) Let the quadric be real. If the quadric has real generators, i.e. if it is a ruled quadric, every plane cuts it in a real conic, for it cuts all the generators in real points. Actual points must lie within the section and actual lines must cut the surface. But the tangent planes passing through a line which cuts a ruled quadric are real, and so the measure of dihedral angles would be hyperbolic. The quadric cannot therefore be ruled. Through a hue which does not cut a non-ruled quadric 164 CONCRETE REPRESENTATIONS [v. 1 4 two real tangent planes pass; hence actual lines, and there- fore planes, must cut the surface, and actual points are within the surface, " This gives hyperbolic geometry. The absolute could be represented by a real sphere. All points outside the sphere are ideal points. (2) Let the quadric be imaginary. The measure of distance is also elliptic, and the geometry is elUftic. (3) Let the quadric degenerate. If the quadric degene- rates to a cone, necessarily with real vertex, the measure of dihedral angle must be parabolic. If the quadric degenerates to two planes, unless the planes coincide they will have a real line of intersection and the measure of plane angle must be parabolic. Hence the quadric must degenerate to two coincident planes. A quadric which reduces, as a locus, to two coincident planes, reduces as an envelope to a conic lying in this plane. If the measure of angle is elliptic this conic must be ima- ginary. This is the case of euclidean geometry. The absolute consists of an imaginary conic in the plane at infinity. Any quadric which passes through this conic is cut by every piano in a conic which passes through the two absolute, or circular, points in this plane, i.e. every plane section is a circle, and the quadric is a sphere. The imaginary conic itself must be regarded as a circle since it is the plane section of a sphere. This is the imaginary circle at infinity. 14. Other three-dimensional geometries can bo constructed in which the measure of plane or dihedral angle is hyperbolic or para- bolic, but they are not of much interest, as they resemble ordinary geometry too slightly. One application of these bwarro geometries may bo given. It is obvious that in euclidean space the geometry on the plane at infinity is elliptic, since the absolute consists of the imaginary circle in this i.->| CKODKSIC RKi'lIKSKNTATluX 165 plane, and it follows, as we have already seen, that the geometry of complete straight lines through a point is elliptic, the geometry of rays, or of points on a sphere, being of course the spherical or antipodal variety. Now consider three-dimensional hyperbolic space. A tangent plane to the absolute rata the absolute in a degenerate conic con- sisting of two imaginary straight lines and two coincident points fl ; hence the geometry on such a plane is the reciprocal of euclidean, i.f. the measure of distance is elliptic, while angular measurement is parabolic. Now, the polar of a |>oiiit (or line) on this plane is a plane (or line) passing through H Hence, by this second reciproca- tion, we find that the geometry of a bundle of parallel lines and planes is euclidean, and if we cut the system by the surface (horosphere) which cuts each line and plane orthogonally, wo find that the geometry on the horosphere ts euclidean. Geodesic Representation. 15. It has been seen that the trigonometrical formulae of elliptic geometry with constant k are exactly the same as those of spherical trigonometry on a sphere of radius k, and therefore elliptic geometry can be truly represented on a sphere, straight lines being represented by great circles, and antipodal points being regarded as identical. Within a limited region of the sphere which contains no pair of antipodal points, the geometry is exactly the same as elliptic geometry. We do not require, as in Cayley's representation, to obtain a distance- or angle-function ; distances and angles are represented by the actual distances and angles on the sphere. The corresponding representation for hyperbolic geometry appears at first sight to be imaginary, since hyperbolic geometry is the same as the geometry upon a sphere of purely imaginary radius. It is possible, however, to obtain a real representation of this kind, though confined to a limited portion of the hyperbolic plane. 166 CONCRETE REPRESENTATIONS [v. 10 16. Geometry upon a curved surface. We must first understand what we mean by the geometry upon a surface which is not, like the sphere, uniform. The straight line joining two given points has the property that the distance measured along it is less than that measured along any other Hue joining the same two points. This is the property which we shall retain upon the surface. A curve lying on a surface and having this minimum property is called a geodesic. The geodesies of a sphere are all great circles. Now, if a surface can be bent in any way, without stretching, creasing or tearing, geodesies will remain geodesies, lengths of lines and magnitudes of angles will remain unaltered, and the geometry on the surface remains precisely the same. Two surfaces which can be transformed into one another in this way are called ap)>li- cable surfaces. If, for example, a plane is bent into the form of a cylinder, the geometry, at least of a limited region, will be unaltered. The same holds for any surface which can be laid flat or developed on the plane. The sphere is a surface which cannot be developed on the plane, and it possesses a geometry of its own, A complete Sphere cannot in fact be bent- at all without cither si i-ptching or kinking, but a limited portion of it can be bent into different shapes without altering the character of the geometry. 17. Measure of curvature. Now the sphere and the plane possess this property in common, that congruent figures, e.g. triangles with equal corresponding sides and angles, can be constructed in any positions on the surface, or, to use the language of kinc- 17] MEASURE OF CURVATURE 167 matics, a rigid figure is freely movable on the surface. It follows that the surface is applicable to itself at all its points. This property is expressed analytically by saying that there is a certain quantity, called the measure of curvature, which is the same at all points of the surface and is not altered by bending. To see what this invariant quantity is, consider any plane section of a surface passing through a tangent hue OT at ; the section is a curve having this line as tangent at 0. The more obliquely the plane cuts the surface the sharper is the curvature of the section, until, when the plane touches the surface at 0, the section is just a point.' The section of least curvature occurs when the plane is per- pendicular to the tangent plane, or passes through the normal to the surface. Again, if we revolve the cutting plane about the normal, the curvature of the section will vary continuously and have a maximum and a minimum value. These occur for sections at right angles, and are called the principal curva- tures of the surface at 0. The curvature of a curve at a point being defined as the reciprocal of the radius of the circle of closest fit to the curve at 0, the product of the principal u uiv a fa tt es at 0, denoted by M, is called the measure of curvature of the surface at 0. If the two curva- tures are in the same sense M is positive, if in opposite senses M is negative ; if one is zero, as in the case of a cylinder or any developable surface, M is zero. For a sphere of radius it, M is the same at all points and = 1/ft*. 1 This holds for a convex surface like a sphere. In the general case the section of a surf.vr l.v a tangent plane is a curve which has a node ai Hie pnim <>r .outset, with real or imaginary tangents. In the cn»c ot a surface of the second degree the section consists of two straight lines, real or imaginary. In the text we arc considering tho case of a node with imaginary tanjsenU, which apjicars just as a point. 168 CONCRETE REPRESENTATIONS [v. 18 18. Surfaces of constant curvature. C4auss, who founded the differential geometry of surfaces, as well as being almost the discoverer of non-euclidean geometry, discovered the beautiful theorem * that wlmi a surface is bent in any way without stretching or kin/ting, the measure of curvature at every point remains vnaUered. It follows, then, that the only surfaces upon which, free mobility is possible are those which are applicable, upon themselves in all positions, and therefore for which M has the same value at all points. There are three kinds of surfaces of constant curvature, (1) those of constant positive curvature, of which the sphere is a type ; (2) those of constant negative curvature, saddle- backed at all points tike a " diabolo " ; (3) those of zero curvature, the plane and all developables. 19. The pseudosphere. Fortunately, we do not require to take an imaginary sphere as the type of surfaces of constant negative curva- ture. There are different forms of such surfaces, even of revolution, but the simplest is the surface called the Pseudosphere, which is formed by revolving a tractrix about its asymptote. The tractrix is connected with the simpler curve, the catenary, which is the form in which a uniform chain hangs under gravity. The equation of the catenary referred to the axes Ox, Oy is y=Acosh| It has the property that the distance of the foot of the ordinate N from the tangent at Q is coustant and equal to k, while QP= the im£ 1 2™**' Disquisaio,m S^nmUts circa superficies cumu, Oiittingcn, v. in | THE PSEUDOSPHERE \m arc AQ. It follows then that if a string AQ is unwound from the curve, its extremity will describe a curve AP with the property that the length of the tangent PN is constant and equal to h This curve is the Tractrix. Ox is an asymptote. Now, if the tractrix is revolved about the asymptote we get a surface of revolution whose principal sections at P are the meridian section in which the tractrix lies, and a section through the normal PT at right angles to the plane of the curve. The radii of curvature of these sections are respectively PQ and PT, and we have PQ . PT = PN 2 = t?, but since the curvatures are in opposite senses, the measure of curvature = - 1/k 1 . The pseudosphere, therefore, gives a real surface upon which hyperbolic geometry is verified— within a limited region. The surface does not, of course, represent the whole of the hyperbolic plane, for it has only a single point 170 CONCRETE REPRESENTATIONS fv.M at infinity. The meridian curves are geodesies passing through this point at infinity, and therefore represent a system of parallel lines. So the surface only corresponds Mb. bo, to a port-ion of the hyperbolic plane bounded by two lines and an arc of a horocvcle. 1 20. The Cay ley- Klein representation as a projection. Through the medium of the geodesic representation we can now get a geometrical interpretation of the Cayley- Klein representation. If we project a sphere centrally, great circles are projected into straight lines, since their planes pass through the centre of projection. Hence the Cayley- Klein representation of elliptic geometry can be regarded as the central or gnomonic projection of the geometry on a sphere. The equations of transformation are easily found. 'The intrinsic equation of the Iraetrix is a=fctog ensce ifi. and since y = isiu^ we have <j—ke ■■*. The ratio of the corresponding urcs of two horocycles (sections _|_ to tin axis) is therefore «"-"*, which agrees with the expression wc have already found (Chap. II. § 28). v. 21] CENTRAL PROJECTION 171 Let the plane of projection be chosen for convenience as the tangent plane at 0, and take rectangular axes, Oz through the centre of the sphere 8, and Ox, Oy in the plane no. «i. of projection. Let the coordinates of P, any point on the sphere, be (x, y, z) and the coordinates of its projection F be(/,/,0). Tbm#+tf+(* -if-*, af_,/ OF OS k x~y~ON OS-NP h-z ~J®-x*-tf~ * 21. Meaning of Weierstrass' coordinates. Let it, v, w be the angles which SP makes with the planes yz, zx and xy ; u, v, w can he regarded as coordinates on the sphere, and z=&sini&, y=h.&mv, z-k = k$mw. Then k sin u, jfcsinw and sin ware Weierstrass' coordi- 172 CONCRETE REPRESENTATIONS \v. 22 nates, denoted by X, Y, Z, and connected by the relation 3P+r*+l*E*-JP, In terms of the spatial coordinates ofP, X=x, Y=y,kZ=z-k. We see thus that Weierstrass 1 coordinates are pro- portional to the sines of the distances of P from the sides of a self-polar triangle, and are therefore analogous to trilinear coordinates. Dually, the line-coordinates of a line are defined as pro- portional to the sines of the distances of the line from the vertices of the fundamental triangle. We may also define the point-coordinates as proportional to the cosines of the distances from the vertices, and the line-coordinates as proportional to the eosiues of the angles which the line makes with the sides of the triangle. In the three- dimensional representation the point-coordinates are the direction-sines of the point referred to rectangular axes. Conformal Representation. 22. Stereographic projection. There is another very useful projection of a sphere, the stereographic projection. In this case the centre of pro- jection is taken on the surface. Let S be the centre of projection, and the centre of the sphere of radius k. Take the plane of projection perpendi- cular to SC, and at distance SO=d. Choose rectangular axes with OS as axis of z. Let the coordinates of P, any point on the sphere, be (x, y, z), and the coordinates of its projection P' be (x', y', 0). Then, since SPA is a right angle - SOP', SP . SP' = SA . SO = 2H. The formulae of transformation are : x' = y' ^OP' ^SP' _ d m x^+y'*+d* x y ON SP 4-z x* + if + (z-df a83 * v. 221 STEREOGRAPHIC PROJECTION 173 If the plane of projection is chosen to pass through C, and the xy plane of P is the tangent plane at S,d = k, and the formulae become : x' _ y' I 2fc* = x'^+ y' & + # x~y~z 85*+/ + ^ 2& £ FIO. 62. A plane ax + by + cz + d=Q becomes 2fc* (atf +by'+ek)+ d{a;' B + y Ji + k*) = 0, which represents a circle. Hence all circles on the are represented by circles. Consider two planes h+my + nz=nh l'x+m'y+n'z = n'k through the centre, and cutting the sphere in great 174 CONCRETE REPRESENTATIONS [v. 23 circles. The angle between the great circles is equal to the angle between the planes, and is given by cos = W + mm' + nn'. The projections of the great circles are the circles 2A- 2 (h + my + nk) - nk (s* + y 2 + &*) = 0, The angle <f> at which the circles cut is given by therefore cos = H' + mm' + nn\ i.e. the projections cut at the same angle as the great circles. Stereographic projection is three-dimensional inversion, for SP.SP' = const., circles are changed into circles, and angles are unaltered. A small figure on the sphere will therefore be projected into a similarly shaped small figure on the plane, with corresponding angles all equal. For this reason the representation is called nmformul. 23. The orthogonal circle or absolute. A circle in the projection, which represents a straight line in the non-euclidean geometry, has for its equation * & + f) ~2k(h + my) - n& = 0, and this cuts at right angles the fixed circle This circle, imaginary in elliptic geometry, real in hyper- bolic, is the projection of the absolute, which cuts all straight lines at right angles. v. 25] CONFORMAL REPRESENTATION 175 24. Conforms! representation. We shall now consider generally the problem of the conformal representation of the non-euclidean plane upon the euclidean plane, straight lines being represented by circles. We shall set aside stereographic projection entirely, as this assumes the geodesic representation on a sphere, and treat the problem purely as a problem in correspondence. It may be shown directly that the circles which represent straight lines all cut a fixed circle orthogonally. For, if the circles x* + 1/* + 2#x h Ify + c = represent straight lines, they must have the property of being determined uniquely by two points. Hence the three coefficients g, /, c must be connected by a fixed equation of the first degree, say -'/.'/' +2#' = c + c', but this is just the condition that the circle should cut orthogonally the fixed circle x i +y t +2g'z + 2fy + e'=0. In elliptic geometry this circle is imaginary, in hyperbolic geometry it is real. If the fixed circle reduces to a point (the transition between a real and an imaginary circle), all the circles which represent straight lines pass through this fixed point. Now, if we invert the system with this point as centre, the circles become straight lines. Hence the straight lines of euclidean geometry can be represented by a system of circles passing through a fixed point. 25. Point-pairs. This representation has a fault which we must try to correct. In non-euclidean geometry, hyperbolic or elliptic, 17(1 CONCRETE REPRESENTATIONS |v. 28 two straight lines intersect in only one point ; but the circles which represent tlieni intersect in a pair of points. In the representation of hyperbolic geometry the fixed circle is real, and two orthogonal circles may intersect in real, imaginary or coincident points, according as the straight lines which they represent are interseetors, non- intorsectors or parallel. The pair of points which corre- spond to a single point are inverses with respect to the fixed circle. We must therefore consider pairs of points which are inverses with respect to the fixed circle as forming just one point. In the representation of spherical geometry, as distinct from elliptic geometry, the points of a pair will be con- sidered as distinct and constituting a pair of antipodal points. In the representation of eucMean geometry, one of the points of a pair is always the fixed point itself. 26. Pencils of lines. Concentric circles. To a pencil of lines through a point P corresponds a pencil of circles through the two points P u P t which correspond to P. The radical axis of this system is itself a circle of the system, and is in no way distinguished from any other circle of the system. The representation of straight lines by circles is not necessarily a conformal one, nor is it by any means the only possible conformal representation. If the representation is conformal we can show that when straight lines are represented by circles, circles also are represented by circles. 1 For a system of concentric circles cut all the lines of a pencil with vertex P at right angles. They will therefore be 1 For the converse of this theorem, see Ch»j>. VIII. 8 2. PEN-OILS OF LINES 177 Fig. 03. n.-b. o 178 CONCRETE REPRESENTATIONS \v. 26 represented by a system of curves catting orthogonally a system of coaxal circles ; but. this is also a system of coaxal circles, and has P lt P 3 as limiting points (Fig. 93). A circle is represented actually by a pair of circles ; these are inverses with respect to the fixed circle, and are coaxal with the fixed circle. The two limiting points form its centre. Corresponding to a pencil of lines through an ideal point F, i.e. a system of lines cutting a fixed line I at right angles, we have a system of circles cutting at right angles the fixed circle and the circle /' which represents the fixed line (Fig. 94). But this is a coaxal system whose radical axis is the common chord of the fixed circle and the circle I' ; its limiting points are the common points Q x and Q* of V and the fixed circle. The circles with centre P are equi- distant-curves, and are represented by a system of circles passing through Q„ Q 2 , Corresponding to a pencil of lines through a point at 9) .' / FlO. 00. 28] CONCENTRIC CIRCLES 179 infinity P, i.e. a system of parallel lines, we have a system of circles cutting the fixed circle orthogonally at a fixed point on it (Fig. 95). The horocycles, circles with centre P, are represented by circles touching the fixed circle at thia point. 27. The distance between two points. If ABC is a circle which represents a circle in non- euclidean geometry, and is the point which represents its centre, the radii are represented by arcs of circles through cutting the given circle and the fixed circle orthogonally. The arcs OA, OB. OC, .,. represent equal distances in non- euclidean geometry. We require then to find what function of the positions of the points and A represents the distance between their corresponding points. 28. motions. In order to investigate this function we shall make use of the idea of motion. By a motion or displacement in the general sense is meant not a change in position of a single point or of any bounded figure, but a displacement of the ISO CONCRETE REPRESENTATIONS [v. 29 whole space, or, if we are dealing only with two dimensions, of the whole plane. A motion is a transformation which changes each point P uniquely into another point P' in such a way that distances and angles are unchanged. It follows that straight lines remain straight lines, and the displacement is a particular case of a collineation (the general one-one point- transformation which changes straight lines into straight lines). Further, it will change circles into circles, and the fixed circle must remain fixed as a whole. We require therefore to find what is the sort of transforma- tion of the euclidean plane which will change circles into circles and leave a fixed circle unaltered. 29. Reflexions. The process of inversion with respect to a circle at once suggests itself, since this transformation leaves angles unaltered and changes rircles into circle*. KuHher, since rq. or. the fundamental circle must be unaltered as a whole, the circle of inversion must cut it orthogonally. Let us then v. :w) MOTIONS AST) REFLEXIONS 181 consider inversion in a circle which represents a straight line. In euclidean geometry, when the circle of inversion becomes a straight line, inversion reduces to reflexion in this line. Now any motion or displacement in euclidean geometry can be reduced to a pair of reflexions in two suitably chosen lines. If AB is displaced to A'B' (Fig. 97), first take MO the perpendicular bisector of A A' ; the reflexion of AB in MO is A'B X . Then take M'O the perpendicular bisector of Bfl, which passes through A', and the reflexion of A'B Y is A'B', Since OB = OB 1 = OB', lies also on the perpen- dicular bisector of BB', and is in fact the centre of rotation for the given displacement. 30. Complex numbers. We shall find now what is the most general transformation which changes circles into circles and the fundamental circle into itself. The equation of any circle is a* -m/ 2 + 2gx + 2fy+c = 0. The procedure is greatly simplified by the introduction of complex numbers and the use of Argand's diagram. Let z^x+iy, p=g+if, and write the conjugate complex numbers z=x-iy, p-g-tf- Then the equation of the circle becomes zz + pz+pz+c=0, (1) a lineo-linear expression in z, z. Its centre is z= -p 7 and the square of its radius is pp - c. 182 CONCRETE REPRESENTATIONS [v. 31 31. Circular transformation, conformal and homo- graphic. Now it is proved in the theory of functions that any transformation of the form *-/</), z=f(z') is conformal, leaving angles unchanged. A real trans- formation of this form which leaves the form of the equation (1) unaltered, i,e. which changes circles into circles, is one in which z, z' both occur only to the first power, 1 or az' +8 -_aZ+J3 where «, 8, y, S are any complex numbers such that uS=j=8y. By this transformation any complex number z is trans- formed into a complex number z', and the point [x, y) corresponding to z is transformed into the point (x', i/') corresponding to z'. If z,, z a , 23,34 are any four complex numbers which are transformed into z,', z 2 ', 2*,', a/, and if we define the cross- ratio (zjZsj, SjZj) Zi -Z t " z« - z t then (z,z„ , Zgz 4 ) = (%'«,', 23'?/). The transformation is therefore said to be tomographic. Let % be the modulus, and 13 the amplitude of the complex number z, -23, so that z t -z 3 =r n e m <=, then the cross- ratio (2,2,, z&,) has modulus -S /[** and amplitude 1 Xba only other type of real tranaformation having this proporty is Mint which diffore only from thi* one by interchnngine; z and i. Bat tln.» oniy differs from the former by a. reflexion in the z-nxis, z— z\ ' v.MJ HOMOCRA1MIH' TRANSFORMATIONS 183 The cross-ratio is real only when its amplitude is a multiple of it, is. when the points corresponding to the four numbers z,, z t , Zs, z t are eoncyclic, and its value is then PiPa ' *A (Fig. 98). Flo. (IS. The transformation has to satisfy the further condition that it transforms the fundamental circle into itself. It can be proved that if the fundamental circle is a? +?/* + /£=(), or 22 + A"=0, the general form of the transformation is tit - Kj$ ~ frf+a " K the. fundamental circle is Jf =--0. or 2 = s, it can be proved that the general transformation is az' +b ~cz' ±d' where <*, 6, c, d are real and ad+bc. 32. Inversion. Consider now the equations of transformation of inversion in a circle cutting the fundamental circle orthogonally. s = - z = 184 CONCRETE REPRESENTATIONS \v. :t-2 The equation of a circle cutting zz + K = orthogonally ia zz+pz+pz-K = 0. Let G( -p) be the centre, P(z) and P'(z'} a pair of inverse points. Fra. as. Let the complex numbers represented by CP and CP' be u, u'. Then z=~p + u, z'=-p+ u '. Also, since u, «' have the same amplitude, and the product of their moduli is equal to the square of the radius of the circle of inversion, uu'=pp + K. Therefore (z + p) (z' +p)-pp-g^Q i or &'+pz+pz'~K=Q, ._ -pi' + K z'+P ' %.e. v. 331 IXVKltSIuNS AXD MOTION'S 185 A second inversion in the circle zz + qz +qz -K-0 gives (K + ]xj)z"-K{p-q) (p-q)2r + (K + pg)- This will not hold when the circle of inversion is a straight Hue — (j>. Here inversion becomes reflexion, and we have z=re if >, s'=rc (:! *- e) , ?-l**-*»; therefore z = z'e-'* , This combined with an inversion gives -pz" + K .... r +p Then, Let tf> - I - ^, /? =6* ; then e 2 '+ - -«-»* « -|. if p,8=«, the transformation becomes iisf-K jS % ~ f-izT+n ■ Hence in either case the transformation is of this form. Hence the general <l is placement of a plane figure is eqaivalent to a pair of inversions in two circles which cut the fundamental circle orthogonally. 33. Types of motions. In the general displacement there are always two points which are unaltered, for if z' =z we have the quadratic equation 0z z + {a-a)z + K/3=O. If we substitute z = - Kf%, the equation becomes %' a + (« < -«)z'+^/9 = 0; therefore z' is also a root. The two points are therefore inverses with regard to the fundamental circle. This point-pair corresponds to the centre of rotation in the general displacement. In hyperbolic geometry there are 186 CONCRETE REPRESENTATIONS [v. 34 three distinct types of displacement according as the centre of rotation ia real, ideal, or at infinity. The first case is similar to ordinary rotation ; the second case is motion of translation along a fixed line, and points not on this line describe equidistant-curves ; in the third case all points describe arcs of horocycles. 34. The distance-function. We have now to find the expression for the distance between two points P, Q, i.e. the function of their co- ordinates or complex numbers (zj, z 2 ). which remains invariant during a motion. The two points determine uniquely a circle cutting the fundamental circle orthogonally hi A', F, This circle represents the straight line jo inin g PQ, and X, Y represent the points at infinity on this line. If the motion is one of translation along this line, the straight line as well as the fundamental circle are unaltered, and X, Y arc fixed points. Let x, y be the complex numbers corresponding to X, Y; then the cross-ratio (z^, xy) Temains constant. If we suppose, therefore, that for points on this line the distance (PQ) is a function of this cross-ratio, wc can write (PQ)=f(h> z 2 ). If P> 6- R are three points on the line, corresponding to the numbers %, % %, this function has to satisfy the relation (PQ) + (QR) = (PR), or Ah,^)+/(s2,^)=f(h-^)- This is a functional equation by which the form of the function is determined. Consider z as a parameter deter- mining the position of a point, and differentiate with regard to ?j. Then, since v«i 2. m Zi _ y z ^_ x pY g X , v. 35] we have THE DISTANCE-FUNCTION 187 , QY 3 fPX\ », . , RY 3 fPX\ 1 k- ^ qx §£ \py) =f {Zl ' h) rx % \py)' Hence f'(z u z 2 )_QX RYfPX RY\_(PX QY\JW a ,W) J (z, ,z s )~QY' RX " VP Y ' RXJ -\PY' QXJ (z jZz , xy)' i.e. (%, s 2 )/'(^, z 2 )=( z i, ts)f'(*i, %)= const. =^. Integrating, we find and substituting in the functional equation we find C-0. Hence (PQ) = M log ( 2l z 3J xg) = M log (py ■ §£) =M l0g {PQ - XY) > (PQ, X Y) being the cross-ratio of the four points P, Q, X, Y on the circle, i.e. the cross-ratio of the pencil Q(PQ, XY), where is any point on the circle. In hyperbolic geometry K= -&, and the fundamental circle is real. The distance between two conjugate points is kivk, and the cross- ratio (PQ, XY)= -1. Then {PQ)=^l- Therefore & = h. 35. The line-element. If, returning to the stereograph ic projection, we take the formulae in § 22, we can find an expression for the line-element (Is. We have, r, y, z heing the coordinates of a point on the sphere, Expressing this in terms of x' and >/, we get 4JftP (<&'* + (ft/'") rf« 3 =- (x's+y'+d 1 ? 188 CONCRETE REPRESENTATIONS [t.3D In particular, if rf=3£,so that the plane of projection k the tangent plane at A (Fig. 92), we get where n — [/$ft 36. There is a gain in simplicity when the fundamental circle is taken as a straight line, say the axis of x. Then straight lines are represented by circles with their centra? on the axis of x. Pairs of points equidistant from the axis «&•**" B V f .A \ ■r ;G Pis. 100. of x represent the same point, and we may avoid dealing with pairs of points by considering ouly those points above v. 37] AN APPARENT PARADOX 189 the a:-axis. A proper circle is represented by a circle lying entirely above the ar-axis ; a horoeyele by a circle touching the x-axis ; an equidistant-curve by the upper part of a circle cutting the z-axis together with the reflexion of the part which lies below the axis. Through three points A, B, pass four circles. If A', B', C are the reflexions of A, B, C, the four circles are represented by ABC, A'BC, AB'G, ABC. The last three are certainly equidistant-curves ; the first may be a proper circle, a horoeyele or an equidistant- curve. 37. Angle at which an equidistant-curve meets its axis. Fig, 100 shows that the two branches of an equidistant- curve oat at inlinity at a Unite angle, a fact that is not apparent in the Cay ley - Klein representation. Let APBQA (Fig. 101) be the equidistant- curve, AMB its axis, represented by the circle on A B as diameter, 190 CONCRETE REPRESENT ATIOKS (V. 37 and let C be the centre of the circle APB. Draw CX ± A U meeting the two branches of the equidistant- curve and its axis in P, Q, M. Let PAQ = 2a; then CAX=a, tan a=^-. Let </ be the dia- XA tan ce of the equidistant-curve from its axis. The hue PX being ± AB represents a straight line; it cute ,1 S in X, and the second point at infinity on the line is represented by Y at infinity. Henee *«ft \og(PM, XY)=klog~, also d = k]og¥£, ,!/ A QX Now CX={{PX-QX); therefore PX-QX l/'' - J \ d 2 V We can get a geometrical meaning for this result. Draw PLJ.PN and PE\\NE (Fig. 102). Then the equidistant-curve and the parallel and the axis all meet at infinity at E. Fro. 102. The angle LPE is then = a. Consider a chord PQ of the equidistant-curve : like a circle, the curve cute the chord at equal angles. Keeping P fixed, let Q go to infinity. PQ becomes parallel to NX, and makes a zero angle with it ; hence the angle between the curve and the axis is equal to the angle LPE. The explanation of the apparent contradiction shown in the Caytey-Klein representation, where the two branches of the eani- distnnt-curve form one continuous curve, lies in the fact that the angle between two lines becomes indeterminate when their point of intersection is on the absolute and at the same time one of the lines touches the absolute. If the first alone happens the angle is zero, if the second the angle is infinite. v.sa] EXTENSION TO THREE DIMENSIONS 191 38. Extension to three dimensions. The conforinal representation of non-euclidean geometry can be extended to three dimensions, planes being repre- sented by spheres cutting a fundamental sphere orthogon- ally. A propel sphere is represented by a sphere which does not cut the fundamental sphere, a horosphere by a sphere touching the fundamental sphere, and an equidistant- snx&Qfi by a sphere cutting the fundamental sphere. A horocycle is represented by a circle touching the funda- mental sphere. The horocycles which lie on a horosphere all pass through the same point on the sphere, viz. the point of contact. This is exactly similar to the system of circles on a plane representing tin* straight lines of euclidean geometry, and thus we have another verification that the geometry on the horosphere is euclidean, This suggests that the three geometries can be repre- sented on the plane of any one of them by systems of circles cutting a fixed circle orthogonally. CHAPTER VI. "SPACE CURVATURE" AND THE PHILOSOPHICAL BEARINU OF XON-EUCLIDEAN GEOMETRY. 1. Pour periods in the history of non-euclidean geometry. The projective and the geodesic representations of non- euclidean geometry have an important bearing on the history of the subject, for it was through these that Cayley and Rtcmaiiu arrived independently at non-euclidean geometry. Klein has divided the history of non-euclidean geometry into three periods. The first period, which contains Gauss, Lobachevsky and Bolyai, is characterised by the syn- thetic method, and applies the methods of elementary geometry. The second period is related to the geodesic representation, and employs the methods of differential geometry. It begins with Hihmanx's classical dissertation, and includes also the work of Helmholtz, Lie and Beltrami on the formula for the line-element. The Surd period is related to the projective representation, and applies the principles of pure projective geometry. It begins with Cayley, whose ideas were developed and put into relationship with non-euclidean geometry by Ki.kis\ To these a fourth period, has now to be added, which is connected with no representation, but is concerned with the vi. 2] CURVED SPACE" 193 logical grounding of geometry upon sets of axioms. It is inaugurated by Pasch, though we must go back to von Stactxt for the true beginnings. This period contains Hilbert and an Italian school represented by Peano and Pieri • in America its chief representative is Veblen. It has led to the severe logical examination of the founda- tions of mathematics represented by the Principia Mathe- matica of Russell and Whitehead. 2. "Curved space." If we attempt to extend the geodesic representation of non-euclidean geometry to space of three dimensions, we find ourselves at a loss, for the representation of plane geometry already requires tliree dimensions. It is quite a legitimate mathematical conception, however, to extend space to four dimensions. A limited portion of elliptic space of three dimensions could be represented on a portion of a " hypersphcre " in space of four dimensions, or the whole of elliptic space of three dimensions could be repre- sented completely on a hyperspbere, with the understanding that a point in elliptic space is represented by a pair of antipodal points on the hypersphere. A hypersphere is a locus of constant curvature, just as a sphere is a surface of constant curvature. Analog)' with the geometry of surfaces leads to the conception of the curvature of a three-dimensional locus in space of four dimensions, and just as the curvature of a surface can be determined at any point by intrinsic considerations, such as by measuring the angles of a geodesic triangle, so by similar measurements in the three-dimensional locus we could, without going outside that locus, obtain a notion of its curvature. n.-e. a. N 194 PHILOSOPHICAL [vi. 3 3, Application of differential geometry. This was the path traversed by Riemann iu his cele- brated Dissertation. Space, he teaches us, is an example of a " manifold " of three dimensions, distinguished from other manifolds by nature of its homogeneity and the possibility of measurement. Space is unbounded, but not necessarily infinite. Thereby he expresses the possibility that the straight line may be of finite length, though without end— a conception that was absent from the minds of any of his predecessors. The position of a point P can be determined by three numbers or coordinates, .r, y, z ; and if x + dx, y +dy t z + dz are the values of the coordinates for a neighbouring point Q, then the length of the small element of length PQ, =ds, must be expressed in terms of the increments dx, dy, dz. If the increments are all increased in the same ratio, ds will be increased in the same ratio, and if all the increments are changed in sign the value of ds will be unaltered. Hence ds must be an even root, square, fourth, etc., of a positive homogeneous function of dx, dy, dz of the second, fourth, etc., degree. The simplest hypothesis is that & 2 is a homogeneous function of dx, dy, dz of the second degree, or by proper choice of coordinates d$*=& homogeneous linear expression in dx-, dy-, dz*. For example, with rectangular coordinates in ordinary space, ds* = dx*+dy z +dz % . By taking the analogy of Gauss' formulae for the curva- ture of a surface, Ricmann defines a certain function of the differentials as the measure of curvature of the manifold. In order that congruence of figures may be possible, it is necessary that the measure of curvature should be every- where the same ; but it may be positive or zero. (Riemann had no conception of Lobachevsky's geometry, for which VI. 4| RIEMAXX AND HELMHOLTZ IflG the measure of curvature is negative.) He gives without proof the following expression for the line-element. If a denotes the measure of curvature, then ds = ^fa*/{\+$a2x*). (Cf. Chap. V. § 35.) If k is what has already been called the space-constant, « ■ I//.: 2 . 4, Free mobility of rigid bodies. About the same time that Niemann's Dissertation was being published, Hermann von Helmholtz (1821-189-1) was conducting very similar investigations from the point of view of the general intuition of space, being incited thereto by his interest in the physiological problem of the localisation of objects in the field of vision. Helmholtz ' starts Irani the idea of congruence, and, by assuming certain principles such as Hint of free mobility of rigid bodies, and nmnodromy, i.e. that a body returns unchanged to its original position after rotation about an axis, he proves — what is arbitrary in Riemanu's investiga- tion—that the square of the line-element is a homogeneous function of Hie second decree in the differentials. That the form of the function which expresses the distance between two points is limited by the possibility of the existence of congruent figures in different positions is shown as follows. Suppose we have five points in space, A, B, C, D, E. The position of each point is determined by three coordinates, and connecting each pair of points there is a certain expression involving the coordinates, which corresponds to the distance between the two points. Let 1 " Uober die Tbatsachen, die d(>r Geometric zuni (jmncle beget),'* Q Winger KachricJiten, 1868. An abstract of this paper "' ;,y published in 18b*G. 196 PHILOSOPHICAL fvi. 4 us try to construct a figure A'D'C'D'E' with exactly the same distances between pairs of corresponding points as the figure ABODE. A' may be taken arbitrarily. Than B' must lie on a certain surface, since its coordinates are connected by one equation. C" has to satisfy two condi- tions, and therefore lies on some curve, and then D' is completely determined by its distances from A', B' and C". Similarly E' will be completely determined by its distances from A', B' and C, but we cannot now guarantee that the distance D'E' will be equal to DE. The distance- function is thus limited by one condition. And with more than five points a still greater number of conditions must be satisfied. 1 It is customary to speak, as Helmholtss does, of the transformation of a figure into another congruent figure as a dwpfammenl of a single rigid figure from one position to another. This language often enables us to abbreviate our statements. Titus, employing this language, we may argue fur the general case as Follows. If there are u points, the figure lias 3» degrees of freedom, and there are £«(«- 1) equations connecting the distances of pairo cif points. But a rigid body has only 6 degrees of freedom j therefore the number of equations determining the distance- function is !«(jT.-l)-3M + 6=£{M-3Hn-'i>. But it is necessary to avoid here a dangerous confusion. Points in space are fixed objects and cannot be conceived as altering their positions. When we speak of a motion of a rigid figure we are thinking of material bodies. The assumption which llelmholtz makes, which is expressed by the phrase, the " free mobility of rigid bodies," is thus 1 This method was employed by J. M. do Tilly, Brnxdlts, Mem, Acad, Rmj. (8vo collection), 47 (1893), to find the expression for the distance- function without using iniinitesuimls. vi. 5] HELMIIOLTZ AND LIE 107 simply an assumption that there is such a thing as absolute space. While, psychologically, the idea of congruence may be based on the idea of rigid bodies, if it were really dependent upon the actual existence of rigid bodies it would have a very insecure foundation. Not only are the most solid bodies within out experience elastic and deformable, but modern researches in physics have given a high degree of probability to the conception that all bodies suffer a change in their dimensions when they are in motion relative to the aether. As all bodies, including our measuring rods, suffer equally in this distortion, however, we can never be conscious of it. 5. Continuous groups of transformations. Helmholtz's researches, though of great importance in the history of the foundations of geometry, lacked the thoroughness which we would have expected had the author been a mathematician by profession. The whole question was considered over again from a severely mathematical point of view by Sophus Lie 1 (1842-1899), who reduced the idea of motions to trans- formations between systems of coordinates, and congruence to invariance under such transformations. The underlying idea is that of a group of transformations. Suppose we have a set of operations R, S } T, ... such that (1) the operation R followed by the operation S is again an operation (denoted by the product RS) of the set, and (2) (RS)T =R(ST), then the set of operations is said to form a group. The operation, if it exists, which leaves the operand 1 S. lie, Theorie der TrajuiforiinUitnuigruppen, vol. iii. {Leipzig, 18fKJ), Abt V. Kap. 20-24 ; and " Obcrdio Grundlagen der Geometric," Leipager Berkhtc, 42 (1890). PHILOSOPHICAL [vr. 5 unaltered, is called the identical transformation, and is denoted by I. Tims, if R, S, T are the operations of rotation about a fixed point through J, 2 and 3 right angles, the operations 1, R, 8, T form a group, and this is a sub-group of the group consisting of the 8 operations of rotation through every multiple of j- The transformations which Lie considers are infinitesimal transformations, and the groups are continuous groups, auch as the group of nil the rotations about a fixed point. All tbe transformations which change points into points, straight lines into straight lines, and planes into planes form a continuous group which is called the general pro- jective group. The assumption from which Lie starts in his geometrical investigation is the " axiom of free mobility in the infini- tesimal " : " If, at least within a certain region, a point P and a line- element through P are fixed, continuous motion is still possible, but if, in addition, a plane-clement through P is fixed, no motion is possible." Starting then with the group of projective transforma- tions, he determines the character of the transformations so that this assumption may be verified, and he proves that they form a group which leaves unaltered eii her a non-ruled surface of the second degree (real or imaginary ellipsoid, hyperboloid of two sheets or elliptic paraboloid), or a plane and an imaginary conic lying on this plane. This invariant figure is just the Absolute. The motions of space, therefore, form a sub-group of the general projective group of point- transformations which leave the Absolute invariant. And vi. 7] COORDINATES 199 so, without Helmholtz's axiom of monodromy, but using a definite assumption of free mobility. Lie establishes that the only possible types of metrical geometry are the three m which the absolute is a real non-ruled quadric (hyperbolic, geometry), an imaginary quadric (elliptic geometry), and a plane with an imaginary conic (euclidean geometry). 6, Assumption of coordinates. There are several points on which the investigations of Riemann, Helmholtz and Lie admit of criticism. The outstanding difficulty which strikes one at once lies in the use of coordinates. How can we define the coordinates of a point before we have fixed the idea of congruence ? This question has been settled by an appeal to the famous procedure of von Statidt (1798-1867), the founder of projective geometry. He has shown 1 how, by means of repeated application of the quadrilateral-construction for a harmonic range (see Chap. HI. § 5), numbers may be assigned to all the points of a line. This, and other questions involved, have now been solved by the modern procedure of Paseh, Hilbert and the Italian school repre- sented by Pieri. This procedure, which marks a return to the classical method of Euclid, consists in developing geometry as a purely logical system deduced from an appropriately chosen system of axioms or assumptions. 7. Space-curvature and the fourth dimension. A misunderstanding, which is especially common among philosophers, has grown around Riemanivs use of the term " curvature." Helmholtz, whose philosophical 1 C K. Cli. v. Standi, Qmmeirizdtr Lage.. Xiirnterg, 1847, and Beitriige zurQwmvtriederLage, Niiraberg, 1850-57-60. 200 PHILOSOPHICAL fvi. 7 writings 1 are much better known than his mathematical researches, has unfortunately contributed largely to this error. The use of the terra " space- curvature " has led to the idea that non-euclidean geometry of three dimensions necessarily implies space of four dimensions, for curvature of space has no meaning except to relation to a fourth dimension. But when we assert that space has only three dimensions, we thereby deny that space has four dimensions. The geometry of this space of three dimensions, whether it is euclidean or non-euclidean, follows logically from certain assumed premises, one of which will certainly be equivalent to the statement that space has not more than three dimensions (cf. Chap. n. § 14, footnote). The origin of the fallacy lies in the failure to recognise that the geometry on a curved surface is nothing but a representa- tion of the non-euclidean geometry. This is brought out still more clearly by the fact that, as non-euclidean geometry, elliptic or hyperbolic, can be represented on certain curved surfaces in euclidean space, the converse is also true, that euclidean geometry can be represented t>n certain curved surfaces in elliptic or hyper- bolic space ; and, of course, we do not consider the euclidean plane as being a curved surface. While, therefore, the conception of non-euclidean space of three dimensions in no way implies necessarily space- curvature or a fourth dimension, it is still an interesting speculation to suppose that we exist really in a space of four dimensions, but with our experience confined to a certain curved locus in this space, just as Helmholtz's "two-dimensional beings" were confined to the surface 1 EL v. Hclmlinliz. -'The origin and meaning of geometrical axioms," Mind, 1 (1.S7C), 3 (1878) s also in Papular Scientific Lactam (London, 18S1), vol. ii. vi. 7] THE FOURTH DIMENSION 201 of a sphere in space of three dimensions, and acquired in this way the idea that their geometry is non-euclidean. W. K. Clifford ' has gone further than this and imagined that the phenomena of electricity, etc., might be explained by periodic variations in the curvature of space. But we cannot now say that this three-dimensional universe in which we have our experience is space in the old sense, for space, as distinct from matter, consists of a changeless set of terms in changeless relations. There are two alternatives. We must either conceive that space is really of four dimen- sions and our universe is an extended sheet of matter existing in this space, the aether % if we like ; and then, just as a plane surface is to our three-dimensional intelli- gence a pure abstraction, so our whole universe will become an ideal abstraction existing only in a mind that perceives space of four dimensions— an argument which has been brought to the support of Bishop Berkeley ! 3 Or, we must resist our innate tendencies to separate out space and bodies as distinct entities, and attempt to build up a monistic theory of the physical world in terms of a single set of entities, material points, conceived as altering their relations with time.* In either case it is not space that is altering its qualities, but matter which is changing its form or relations with time. 1 The Common Swwe of tin timet Sciences (London, 1 883). chap. iv. § tU. 1 Cf. W. W. Rouse Bull, " A hypothesis relating to the nature of the ether and gravity," Messenger of Math., 21 (ISitl). s See C H. Hinton, Scientific Romance, First Series, p. 31 (London, 188S). For other four-dtoettriona] theories of physical phenomena see Hinton. The Fourth Dimension (l-oudon. ISKM). l ('f A N. Whitehead. "On mathematical concepts of the material world,'" hit. Trans., A 205 (1900). 202 PHILOSOPHICAL [vi . 8 8. Proof of the consistency of non-euclidean geometry. The characteristic feature of the second period in the history of non-euclidean geometry is brought out for the first time by Beltrami' (1835-1900), who showed that Lobachevsky's geometry is represented upon a surface of constant curvature. This is historically the first euclidean representation of non-euclidean geometry, and is of import- ance in providing a proof of the consistency of the non- euclidean systems. While the development of hyperbolic geometry in the hands of Lobachevsky and Bolyai led to no apparent internal contradiction, a doubt remained that inconsistencies might yet be discovered if the investigations were pushed far enough. This doubt was removed by Beltrami's concrete representation by means of the pseudo- sphere, which reduced the consistency of non-euclidean geometry to depend upon that of euclidean geometry, which everyone admits to be self-consistent. Any concrete representation of non-euclidean geometry in euclidean space can be applied with the same object. In fact, the Cayley representation is more suitable for this purpose, since it affords an equally good representation of three-dimensional geometry. The advantage of Beltrami's representation is that, distances and angles are truly repre- sented, and the arbitrariness which may perhaps be felt in the logarithmic expressions for distances and angles is eliminated. At the present time no absolute test of consistency is v'f' ^o lm l; f m '°. d J '""rpn*"'"™* ^"a gcowttria non-eurii.hn. .Naples imn, JS.-ltrnuu n!»o showetl Mint, since the equation of a geodesic in * :...!-.«■ coordinates ia liomr, tin- surface eon t>o represented on a plane, geodesies being represented by straight lines, and real points bcin K r "l ,r ''~';"" 11 ,V " l " , "' h "-' lu:1,m;l fi.vn! Hrckr. il. , -A , , be < ran! aiUon from the geodesic to the projective representation of On lev vi. 10 1 TEST OF CONSISTENCY 203 known to exist, and the only test which we can apply is to construct a concrete representation by means of a body of propositions whose consistency is universally granted. In the case of non-euclidean geometry the test which has just been applied suffices to prove the impossibility of demonstrating Euclid's postulate. For, if Euclid's postu- late could be mathematically or logically proved, this would establish an inconsistency in the non-euclidean systems ; but any such inconsistency would appear again in the concrete representation. The mathematical truth of the euclidean and the non-euclidean geometries is equally si rong. 9. Which is the true geometry ? There being no <i priori means of deciding from the mathematical or logical side which of the three forms of geometry does in actual fact represent the true relations of things, three questions arise : (1) Can the question of the true geometry be decided a posterior*, or experimentally ? (2) Can it be decided on philosophical grounds '? (3) Is it, after all, a proper question to ask, one to which an answer can be expected '. 10. Attempts to determine the space-constant by astronomical measurements. Let us consider what form of experiment we can contrive to determine, if possible, the geometrical character of space. Essentially it must consist in the measurements of distances and angles, the sort of triangulation which is employed to determine the figure of the earth, but on a prodigiously larger scale. If we could measure the angles 204 PHILOSOPHICAL fvt. 10 of some very large triangle, the difference between their sum and two right angles migM give us the necessary data for determining the value of the space-constant. We do not say that such an experiment will give us the necessary data, for, as we shall see presently, the whole argument is destroyed by a vicious circle {§ 12) ; but let us assume, for the sake of illustrating the argument, that the experiment can be made, and see to what conclusions it leads. The largest triangles, whose vertices are all accessible and whose angles we can measure directly, are far too small to allow of any discrepancy being observed. We must turn to astronomy to provide us with triangles of a suitable size. The largest triangles, of which two vertices are accessible, are those determined by a star and the observer in two different positions. Let S be a star and E t , E % two positions of the earth at opposite ends of a diameter of its orbit, the sun ; and Fig. 103. let CSLE t E». The angle E t SC, subtended by the earth's radius R at the star, is called the poraflms of the star : blowing this angle and applying euclidean geometry, we can find the-star's distance. There are two methods of determining the angle EfiC. The first, or direct method, is to measure the angle SEfi by vi. 10] ASTRONOMICAL MEASUREMENTS 205 the transit circle. Then the parallax is, by assumption of euclidean geometry, the angle ?-SEfi. The second method, that of Bessel, is to compare the position of the star <S with those of neighbouring stars which, from their faintness and other considerations, are believed to be much farther away than S. Considering S' as at infinity, and again assuming euclidean geometry, Eft' || OS and the parallactic angle E t SC =S'E 1 S. But on the hypothesis that geometry is hyperbolic, these two methods will give different results, and the angle. SEfi t-S'^iS is in fact not equal to ^, but is the parallel- angle corresponding to the distance R. Let 2$ be the small difference (^-SE t c) -S'E V S ; then 2e=J-n(ff). Also e » : = tan \\\{R) =tan(j - f)) -(1 -tan 0)/(l +tan $) ; therefore fl/fr=log,{(l +tan 0)1(1 -tan 6)} =2 tan B, approx. ■ Now we have records of the determination of the star « Centauri by both methods. An early measurement by the direct method yielded the value I'M", while Bessel's method gives the value Q-76" ±0-01". Taking 20 therefore equal to 0-38", we have tan = 92 x 10" 8 , and kffi = 530000 approx. The direct method is not susceptible of very great accuracy, and the value P14* for the parallax is probably much too large, but even from these data, if we admit the soundness of our argument, we should be 206 PHILOSOPHICAL fvi, 10 warranted in stating that the space-constant must be at least half a million times the radius of the earth's orbit. The data, so far as they go, seem to favour the hypothesis of hyperbolic geometry rather than that of elliptic, since the calculation leads to a real value, for k. Thp hypothesis of elliptic geometry, however, leads to the result that a star would be visible in opposite directions unless there is some absorption of light in space, 1 If we assume that the light from a. .star which is at a distance of h-n-k (i.e. half the total length of the straight line in elliptic space) is so diminished by absorption that the star becomes invisible, then the parallax of the farthest, visible stars, measured by the direct method, would, as on the euclidean hypothesis without absorption, be zero. And if the light is totally absorbed in a distance of say \irk, the case would be similar to that on the hyperbolic hypothesis, or on the assumption of absorption in euclidean space. Thus, if we admit the hypothesis of absorption of light in free space, it becomes impossible to draw any definite conclusion as to the nature of actual space, except perhaps that the space-constant is very large. The diivci appeal to experiment therefore leads only to the conclusion that the space-constant, if not infinite, must be very large compared with any of (he usual units of length, and is very large in comparison with the. distances which we have ordinarily to deal wiHi. These experiments do not contradict euclidean geometry, but they only verify it within the limits of experimental error. No amount of 1 A complication, however, arista owing to tin- Unite rate nf propagation of light. The two images «l tho star Been in opposite directions will represent, the star tit clinVrettt times, and in "pnenil therefore in different positions, so that, even if then- Merc no adsorption of light, the appearance Of in., sky would nol necessarily be symmetrical. (C'f. W. IJ. Franklund. Math. Gazette, July 1013.) I 'I] INTUITION 207 experimental evidence of this kind can ever prove that the geometry of 3pace is strictly euclidean, for ( here will always be a margin of error. On the other band, so far as we have gone, it remains conceivable that further refinements in our instruments and more accurate information regarding the laws of absorption of light might enable us to establish an vpjicr limit to the value, of the space-constant, and thus demonstrate that the. geometry of actual space is non- euchdean. 11, Philosophy of space. This way of regarding experience as the source of our spatial ideas is in striking contrast to Kant's attitude towards space, which is expressed by his dicta : that space is not an empirical concept derived from external experience, but a framework already exist ing in the mind without which no external phenomena would be possible. 1 From the new point of view, geometry applied to actual space has become an experimental science, or a branch of applied mathe- matics. We are not forced to accept ite axioms, but shall onlv do so when we find them convenient and in sufficiently close agreement with the facts of experience. Since Kant's time the intuitive has become discredited. We now know that there are things which formerly appeared to be intui ti ve which are in fact false. Thus, it was formerly believed that every continuous function possessed a differential co- efficient ; the proposition appeared, indeed, to be intuitive. But Weierstrass gave an example which showed that the belief was false. In the extreme empiricist view the parallel-postulate has to be ranked with the law of gravita- tion as a law of observation, which is verified within the limits of experimental error. 1 1. Kant, Critiqm of Pure Reason, ohap, i. 208 PHILOSOPHICAL |>I. 11 As regards the second question, therefore, the powers of philosophy have been narrowly circumscribed by the stricture kid upon intuition. Obviously the fact that a coherent mental picture can be formed of euclidean space does not constitute a proof that this is the form of actual space, since the same thing applies to the non-euclideaii systems. But the philosopher may say he has an intuition of euclidean space. What does this mean ? Has he an intuition that the sum of the angles of a triangle is equal to two right angles ? Does he perceive intuitively that two straight lines which are both perpendicular to a third remain equidistant ! What intuitions or beliefs would the philosopher have had if he had been deprived of powers of locomotion and the sense of touch, and been provided with only one eye ? He would believe, because his eye told him bo, that two railway lines converge to a point, that objects change their shapes when they are moved about : and he would perhaps demonstrate that the sum of the angles of a triangle is greater than two right angles. His intuitions are merely beliefs, and perhaps not even true ones. We have really to distinguish between different kinds of apace. The space of experience is brought to our knowledge through the senses principally of sight and touch, and is a composite of two spaces, " visual space " and " tactual space." Pure visual space, which is the limited field of our imaginary one-eyed sessile philosopher, is a crude elliptic two-dimensional space ; l the three-dimensional form of tactual space is conditioned probably in part by the semi-circular canals of the ear. From this composite 1 Cf, Thomas Rcid. ,4n Inquiry into the Human Mind. Edinburgh, 1704, chap. vi. " On Seeing," § " Of the geometry of visibles." vi. 12] VISUAL AND TACTUAL SPACE 20!) space, which is far front being the beautiful mathematical continuum which we have arrived at after generations of thought, we get by abstraction a conceptual space which is conditioned only by the laws of logic, but to which we find it convenient to ascribe the particular form which we call euclidean space, for the reason that this is the simplest of the logically possible forms which correspond with sufficient closeness to the space of experience. Whether there is, besides these, an intuitional space, we shall leave to philo- sophy to settle if it can. We may, perhaps, leave Kant in possession of an a priori space as the framework of his external intuitions, but this space is amorphous, and only experience can lead us to a conception of its geometrical properties. 12. The inextricable entanglement of space and matter. A further point--and this is the " vicious circle : ' of which we spoke above— arises in connection with the astronomical attempts to determine the nature of space. These experi- ments are based upon the received laws of astronomy and optics, which are themselves based upon the euclidean assumption. It might well happen, then, that a discre- pancy observed in the sum of the angles of a triangle could admit of an explanation by some modification of these laws or that even the absence of any such discrepancy might still be compatible with the assumptions of non-euclidean geometry. "' All measurement involves both physical and geometrical assuniptiun.s and ihi> two things, apace and matter, ore not gtVflB separately, but analysed out of a common experience. Subject to the general condition (bat space is to be changeless and matter to move about in space, we can explain the same observed results N.-K. O. O 210 PHILOSOPHICAL \vi. 12 in many different ways by making compensatory chanpr* in the qualities that we assign to space and the qualities we assign to matter. Hence it seems theoretically impossible to decide by any cxporiment what are the qualities of one of them in distinction from the other." ' It was on such grounds that Poincare.* maintained the essential impropriety of the question, " Which is the true geometry ? " In Ms view it is merely a matter of con- venience. Facts are and always will be most simply described on the euclidean hypothesis, but they can still be described on the non-euclidean hypothesis, with suitable modifications of the physical laws. To ask which is the true geometry is then just as unmeaning as to ask whether the old or the metric system is the true one. The con- clusion thus arrived at by Poincare is quite akin to the modern doctrine in physics expressed by the Principle of Relativity. Just as, according to this doctrine, it is impossible by any means to obtain a knowledge of absolute motion, so, according to Poincare. it is beyond our power to obtain a knowledge of absolute space. 1 Mr. C, I). Broad, with whom I have discussed thif chapter, has put this point of view so well that I quote his words. 5 H. Poincare, 1st science el rhypatklse (Pans, !ft03), chap. v. ; Etagfci translation by W. .). Grcenstreet, London, !U€5. CHAPTER Vn. RADICAL AXES, HOMOTrTRTTC CENTRES AND SYSTEMS OF CIRCLES. 1. Common points and tangents to two circles. Two circles intersect in four points and have four common tangents. Various cases arise according as these points uih.I lines are coincident or imaghttfj in pain, In hyperbolic geometry two equidistant-curves whose axes intersect have their common points and tangents all real. A proper circle which cuts both branches of ao equidistant-enrve has four real common tangents with it. If it cuts only one branch, two of the common points and two of the common tangents are imaginary. Two proper circles cannot have more than two of their common points real ; their common tangents are then two real and two imaginary. If two proper circles do not intersect, their common tangents are all real or all imaginary. The case of four real common points and four imaginary common tangents cannot occur in hyperbolic geometry ; two real and two imaginary common points can only occur along with two real and two imaginary common tangents. In elliptic geometry, if two circles intersect in two real and two imaginary points, they have two real and two imaginary common tangents. If each lies entirely outside the other, their common points are all imaginary and their 212 SYSTEMS OF CIRCLES fvn. 2 common tangents are all real. The absolute polars of two such circles have four real common points and their common tangents all imaginary. If one lies entirely within the other, their common points and tangents are all imaginary. The case of four real common points and Four real common tangents cannot occur in elliptic geometry, 2. The power of a point with respect to a circle. Let. be a fixed point in the plane of a proper circle with centre C and radius a. Through draw any secant cutting Fio. 104. the circle in P, Q. Draw CN 1.0PQ. Let OC=d and COP = B, OP=r, OQ=r', so that Otf = l(r'4-r),Ptf = Hr'-r). Now 1 cmd = co&CN cos i(r' +r), cos a = cos CN cos £ (/ - r). _. . cosd cosi(/+r) 1 - tan \r tan lr' . Therefore CM a = C03 I ^j = i+tau £r tan jf ' therefore tan \r tan £/ = const, = tan it(d +a) ton i(rf - a). 1 Elliptic geometry is token as the standard case, and the space-con- stant it w taken as the unit of length. vn, 3] POWER OF A POINT 2LJ In hyperbolic geometry tan is replaced by i tanh. This product may be called the power of the point with respect to the circle. It is positive if is outside, negative if is inside the circle. In the former case, if I is the length of the tangent from to the circle, the power of is equal to tan*Jfc 3. Power of a point with respect to an equidistant- curve. (1) Let the secant cut one branch of the curve in P, Q, i.e. in hyperbolic geometry the secant does not cut the o M PlO. 105. axis of the curve, in elliptic geometry neither of the finite segments OP, OQ cuts the axis. Let M be the middle point of PQ, and draw MN JL the axis ; then MN is also 1 PQ, Draw OH 1 the axis. Let OH^d, MN-x,0P=-r, OQ=r>, so that OJtf=- 4 {/+*), PJlf = A (/-/). Then, from the trirectangular quadrilaterals OHNM, PKNM, sin d = cos A (/ +r) sin x, sin a =cos A, (/ - r) sin a; ; sin d __ cos A (/ + r) 1 -tan A.rtanA/ . sin r(.~cos !(/ -r) I +tan Artan \r' ' therefore tan \r tan lr' = const. =tan A {a -rf)/tan | (d +o). therefore 214 SYSTEMS OF CIRCLES fvn. 3 (2) Let the secant cut both branches of the curve, i.e. the point of intersection A with the axis is real and one of the segments OP, OQ cuts the axis. no. ioa. Let OAN = 6, ON =rf, OP =r, OQ =r', so that 0A = k{r'+r), PA = !.<r'-r). sintf = sin ,1 (r' + *■ ) sin f), sina=sin .](/ -r) sin 6. sind = sinj ( / +r ) = ta n lr' -r tan \r sia a = sin £(/ - r) ~ tan \r' - tan \r tanjr_ _tani(d-n) i t — const. — z * » i — taiUr tani{d + a) Then Therefore "M: Therefore Note. Figs. 105 and 106 have been drawn for the ease of hyperbolic geometry. In elliptic geometry the equidistant- curve is convex towards the axis. In Fig. 105. in this case, either OR < PK or lies between P and 0. If is the same point in the VII. A POWER 01-' A POINT 215 two figures, the values of tan {r tan i/ and tanArtau*/, respec- tively for the secant which cuts and the secant which does not cut t be axis, are equal. Hence, if a variable line tJirough a fixed point cuts « circle in P, Q and its axis in A, either the ratio or the product of the tangents of half the segments OP, OQ is constant, according as (1) one, or (2) both or neither of the segments contains the point A. If OT is a tangent- to the curve, the constant is equal to tatPlOT, and is catted the power of the point with respect to the circle. Tho two cases are simply explained in elliptic geometry. Let the clotted circle A A' represent the axis of the circle, which is no. 107, represented in the diagram by a pair of circles. The secant cuts the two circles in P, P' ; Q,Q'; and the axis in A, A'. These pairs of course represent single points. AA' = PP'=QQ' = ir; therefore OCi'=v-OQ. tan hOP Therefore tan \OP tan \0Q = tan \ OP cot \OQ' = ^rgjg> 216 ■SYSTEMS OK ('.] lU'LKS F\n. 4 4. Reciprocally, if P is a variable point on a fixed line PN, p Fia, 108. and the tangents FT, PT from P to a fixed circle make angles 0, & frith PN, we have in Fig. 108, sind=sin /-sin \($' + 6), sin a = sin r sin \(ff - Q) t sin d _ 3i n|(e'+fl) _ tan iff + tan W whence sin a sin !, [ff - $) tan h $' - tan 1 ' vu. 4j POWKK OF A LINK 217 This result is true also in euclidean geometry, the constant reducing to {d - a)/(d +<?). For an equidistant-curve, let the line cut the axis in N at an angle « ( Kg. 109). Then. 8 and ff being taken positively, TPL<*TPLml(6+&% NPL = \(8-ff), cos a - cos x sin 1 (0 - W\ cos a = cos x sin J, (# -i- 8') ; whence, as before. . — ^a, ia constant, tan An If the angles 8. $' are measured in the same sense, then for & we must put - -8', and we have tan if) tan W = const. If, the angles 8, 8' being measured in the same sense, both or neither of them contains the line joining / J to the centre, then we have (Fig. 110) LPC = i(t)' +*■ +8) =| + l(ff + 0), NPC = ^-!,(ff+e), TPC=wpr=!A-r-ff+Q)-*-W -fl- Fiu. 110. 218 Then SYSTEMS OF CIRCLES |yn. 8 sin d =sin r cos A ($' + $), sin a = sin r cos A (f)' - 0), and tan IB ten ifl' =const. Hence, if from a mriable paint on a fixed line I the tangents to a circle are p, q, and (fie line to lite centre is a, either Ike ratio or the product of the tangents of half the angles Ity), ity) is constant, according as (I) one, or (2) both or neither of the angles contains the line a. 5. Angles of intersection of two circles. Since two circles may intersect in four points, there are four angles of intersection to consider. It is easy to show geometrically that if two circles have only two real points of intersection, the two angles of intersection are equal. Suppose a circle cuts an equidistant-curve in four points, P, P' on one branch, Q, Q 1 on the other branch. Then, no. in. drawing PM, P'M' _L the axis and joining P, P' to C, the centre of the circle, L CPP' = CP'P, Z MPP' - M'P'P i therefore /_ MPC = M'P'C, and the angles of intersection at P, P' are equal, and simi- larly the angles of intersection at Q, Q' are equal. vn. 0] INTERSECTION OP CIRCLES 219 But LCPM + CQN =2CPQ, since LMPQ=NQP ; therefore OPM and CQN are not in genera! equal. If <1>M=CQN, then each =CPQ=CQP; M and N then coincide, and lies on the axis of the equidistant-curve. Similarly, it may be shown that if two equidistant-curves intersect in four- points, the angles at the points of inter- section which are on different branches are equal, but all four angles cannot be equal unless the axes are at right angles. When two of the angles of intersection are right, the circles are said to cut orthogonally, All four angles cannot be right, for then the centre C of the one circle must lie on the axis of the other, and if CT, CT' are the tangents to the second circle, CT is a radius of the first circle. But CT is a quadrant ; therefore the first circle must reduce to two coincident straight lines. 6. Radical axes. Let P, P', Q, Q' be the points of intersection of two circles, with axes «=0 and /3=0. Then, if 5=0 is the equation of the absolute, the equations of the circles can be written S - <t 2 - 0, 8 - j3* = 0. The equation [S -«*) -{S- ) S 1! )=0 represents a conic passing through their common points, but this breaks up into the two straight lines a±fi=Q, and these represent a pair of common chords which pass through the point of intersection of the axes. Tliey form with a and /3 a har- monic pencil. If y is the polar of the intersection of the axes, i.e. the line of centres, the other pairs of common chords pass through «y and j3y. If we take any point on one of the first pair of common 220 SYSTEMS OF ni:''l.KN j \ n, 'i chords, say PP', the power of with respect to either circle is tan |OP/tan \()I y . These two lines are therefore the locus of points from which equal tangents can be drawn to the two circles. But if we take, a point on PQ, the power with respect to one circle is the product, and with respect to the other circle the ratio of tan WP and tan \0Q, and this chord does not possess the property of equal tangents. Fro. n± Hence, of the three pairs of common chords of two circles, one paii* pass through the intersection of the axes and are harmonically separated by them, and possess the property that the tangents from any point on either to the two circles are equal. These two lines are called the radical axes of the two circles. 1 YTI. 8] RADICAL AXES 221 7. Homothetic centres. Rcciproeallv, two circles have four common tangents, which intersect in three pairs of points. One pair lie on the line joining the centres, and are harmonically separated by them, the other pairs he on the lines joining the centres to the pole of the line of centres. The first pair possess the property that any line drawn through one of them cuts the two circles a t eq ual a ngles. These two poi nts are called the homothetic centres of the two circles. 8. Radical centres and homothetic axes. The three pairs of radical axes of thee circles taken in pairs pass thmwjhfonr points, the radical centres of the three circles. Let ABC be the triangle formed by the axes «, 6, c of the three circles ; a pair of radical axes eii« 2 , $$%, y&z passes through each of these points. Fu:. IIS. If one radical axis y, of the circles A, B, and one radical axis a of the circles A, C intersect in P, then the tangents from P to the three circles are all equal. Therefore P lies SYSTEMS OF CIRCLES [vn. :; on a radicnl axis «, of the circles B, C. Wc have then «i. B t , ji concurrent in P. Let y t cut Ul in S, and join BS. Then, since (ab, y,yj is harmonic, B(AC, SP) is harmonic , therefore BS is ft , i.e. a% ,B l ,y 1 pass through 5. Similarly «2' A, Vz are concurrent in Q, and « 2 , B 3 , y^ in /?. The quadrangle PQRS has ^£6' as harmonic triangle. Reciprocally, the three -pairs of homolketie centres of three circles taken in pairs lie in sets of three on four limes, the nomothetic axes of the three circles. They form a complete quadrilateral, whose harmonic triangle is the triangle formed by the centres of the circles. 9, Coaxal circles in elliptic geometry. The locus of the centre of a circle which passes through two fixed points D u D z on a line I consists of the two perpendicular bisectm-s OL, O'L of the segments D,/^ and 0»A (Kg- H4). All the circles through /),, Z>, therefore fall into two groups; any two circles belonging to the same group have I as a radical axis. Each group is there- fore called a system of coaxal circles with common points />,. D q . When the centre is at 0, the circle is a minimum, and it increases up to a maximum, which is just the straight line / itself, when the centre is at L. Let C ls C\ on OL = V be the centres of two circles of the one system, and take two points R tl A, on I Draw the tangents KJJ U KJU* A' 2 F„ K s f s to the circles C„ C % . Then K,0 1 =K l V t and I.F,-*^,. Hence the points V lie on a circle with centre A"„ and V on a circle with centre K t . Also, since K t U t is a tangent, to the circle 6 T , and a radius of the circle K u C,*/, is a tangent to the circle E t ; and since O t U l m(J i y i , C\ lies on a radical axis of the circles K„ K t . Hence the circles K have V as a vn. I>] COAXAL CIRCLES 223 radical axis. We get then a system of coaxal circles K associated with the system C, and every circle of the one system cuts orthogonally every circle of the other system. As KD, diminishes the circle tends to vanish. Z),, I) s are called the Hmiliitg points of the A' system. If K lies *. \\y. _— — — i»&-j (' 1 t / 3 y fig. 114. in the segment D t OD zt the circle is imaginary. As K approaches 0', the circle becomes the straight line ¥. The A' system is a non-intersecting system, i.e. it has imaginary common points. The system has imaginary limiting points. If the segment D 1 D i vanishes so that the common points 224 SYSTEMS OF CIRCLES [vH. 10 D u 7> 2 coincide at 0. the circles C all touch the line I at 0, and the circles K all touch the line (' at 0. If the segment D } D 2 becomes », so that the common points coincide at 0'. the circles all reduce to straight lines passing through 0', while the circles K become con- centric circles with centre 0'. 10. Homocentric circles. The locus of the centre of a circle which touches two fixed lines d u & % through a point L consists of the two bisectors o, o' of the angles between rf,, d s (Fig. 115). All the circles touching t^, d» therefore fall into two groups ; any two circles belonging to the same group have L as a hoinothetic centre. Each group is therefore called a system of homocentric circles with common tangents rf, , d%. When the centre is at L, the circle is a minimum and reduces to the point L itself; as the centre moves along o', the circle increases up to a maximum when the centre is at 0, the pole of o. Let fli, c 2 through It, the intersection of o and /, be the axes of two circles of the one system. Take two lines k, , fc, through L, and let « n , « 2 , v, , v z be the tangents to the circles C,, C a at their points of intersection with jfc,. k„. Then the angles (£,*',) = (/f- t K 2 ) and {k i v i ) = (k i r i ). Hence the lines u are tangents to a circle with axis Jc u and v are tangents to a circle with axis k 2 . Also, since (i^) lies on the circle ( , and on the axis of the circle A',, and since the angle (<?,«,) = (<V'i). fi passes through a hoinothetic centre of the circles A',, A* 2 . Hence the two circles K have // as a hoinothetic centre. We get then a system of homocentric circles K associated with the system C, and every circle of the one system is tangentially distant a quadrant from every circle to. io| HOMOCENTRIC CIRCLES of the other system. As k approaches d, the circle K becomes the straight line rf, . d t , d 2 are called the limiting lines of the K system. If h lies outside the angle djd.^, Fig. 115. the eircle becomes imaginary. As k approaches ©', the circle reduces to the point L". The A' system has imaginary common tangents; the C system has imaginary limiting lines. If the angle d^^ vanishes, so that the common tangents N.-E. Q. F 220 SYSTEMS OF CIRCLES tvn. 11 d t , t? a coincide with o', the circles V all reduce to points on o\ while the circles K become concentric circles with axis a". If the angle d t d 2 becomes w, so that the common tangents d t! f/ a coincide with o, the circles all touch o at L, and the circles K all touch o at //. 11, In euclidean and hyperbolic geometry this duality does not hold, since in euclidean geometry the envelope of a system of lines cutting a fixed line at a constant angle is a point at infinity, and in hyperbolic geometry it is an ideal circle. In hyperbolic geometry, as K goes to infinity the circle becomes a horocyelc. Between the horocyele and the straight line lies a system of branches of equidistant- curves. The other branches complicate the figure as they intersect the other circles of the system. The same thing, of course, occurs in the other coaxal system passing through In euclidean geometry a system of coaxal circles is a linear system, i.e. through a given point only one circle of the system passes. In non-euclidcan geometry, through three given points four circles pass, -i.e. four circles can be drawn through any point P to pass through two fixed points X, Y. Denote these circles by PX F, / J 'A' Y, etc. ; then of the four circles, PXY, P'XY have their centres on the one per- pendicular bisector of XY, and belong to the one coaxal system, while PX'Y, PXY' belong to the other. Hence, through a given point there pass only two circles of a given coaxal system. In euclidean geometry a system of coaxal circles is equivalent to a system of conies through four points ; in non-euciidean geometry it is equivalent to a system of conies through two points and having vtr. I2| LINEAR SYSTEMS double contact with a fixed conic. 1 The reciprocal system, i.e. a system of circles touching two lines, is equivalent in non-euclidcan geometry, to a system of conies touching two lines and having doable contact with a fixed conic ; and in euclidean geometry to a system of conies passing through two points and touching two huos. Hence the complexity of the latter system compared with a system of coaxal circles. The analytical treatment of systems of coaxal circles in non- euclidean geometry can he reduced to the consideration of linear systems in the following way, 12. Linear equation of a circle. If {x, y, z) are the actual Weierstrass coordinates, the equation of a circle, with centre (.r, , #, , s,) and radius r, is **« + Wi + ^^i - ** coa 7 • Let n=Aa",, lj=ky u c = A£%, rf= - AJPeos-, so that k W -h jW + e 1 = A'A-^a:, 1 + y» + L% 3 ) = XH-* =jj>, say. Then the equation reduces to ax + by + cz+tl II. The non-homogeneous linear equation, willi real cnefheicnts, in actual Weierstrasis coordinates, therefore represents a circle with centre (a, b, c/A*), axis (& > by I «=0, and radius r, such that on . -the positive value of -, k p where f- = IrtC- \ W+A In elliptic geometry fr 5 is positive and p 2 is always positive. The centre is always real, and the radius is real if iP~k*a s +&!£+<?. If d=0 the circle becomes a straight line, and if d=p it reduces to a [joint. In hyperbolic geometry, changing the sign of A*, we have r d jj»=cS-A% a -i 5 i 2 , aud cosh ,=tho positive value of -. K p 1 It is therefore exactly equivalent to a svstem of circles in eticlidenn geometry having double contact with n fixed conic. The limiting points are represented by the foci of the conic. 228 SYSTEMS OF CIRCLES fvn. 13 The centre is real, ideal or at inlinity, according as jf>, = or <0. The curve is therefore A real circle if tP>& -IftP-tW^-Q, reducing to a point if An imaginary circle if <?■ - Ifia* - Ifi&xP^O. An equidistant-curve if c 1 <i(- a (!i 2 -! -Ir), reducing to a straight line if<J=0. A ho recycle if ^ = jt s (a' + 6»). The two equations ax + bi/ + cz+d=Q represent the same circle. In dliptio geometry this is verified, since (a:, y,z) and { -x, -y, 2) represent the same point. In hyperbolic geometry, for a proper circle or a horocyclc only one of these equations can be satisfied, since z must be positive ; for an equidistant-curve the two equations represent the two branches. The points of intersection of two circles «,a; + 6, y + cfi ±d l —0 and a& + b&+c&+d r =0 are found by solving these equations simul- taneously with the equation x i +y I +&& = li*. These give four sets of values of &, y*, #, and therefore four points of intersection. 13. Systems of circles. If S, =0 and S,-0 are equations of circles in this form, 5, + AS--0 represents a circle, and for all values of A represents a pencil of circles passing through two fixed points. If <l t =d t , the circle S, -S,=0 of the system reduces to a straight line, and if d t +d a =0 the circle S,+8 t =0 is another straight line. These are the radical axes of tho two circles. S i + AjS, + /kiS»=0 represents a linear two-parameter system or 1 iiiudlc of circles. Ha circle of the system passes through the point V, Sj' + AS.'+iiiS.'^O, and we have (SfiJ - 5,'5 3 ) + A [5*8,' - 5,'S a } = 0, which represents a linear one parameter system or pencil of circles. Hcneo all circles of a bundle which pass through one fixed point form a coaxal system and pass through another fixed point. If rf, + Ad- + /wtf, =0, we get a pencil of straight lines If the vertex of this pencil is real, choose it as origin ; then the linear system can be reduced to the form x + ky+piz 1 e)=0 Then one radical axis of every pair of circles of the system passes through the origin. i.e. the circles have a common radical centre at the origin. If tangents are drawn from this point to the circles of the system they are all equal, and hence all circles of a bundle eut orthogonally a TO. Hi MARGINAL IMAGES 229 fixed circle. In hyperbolic geometry the orthogonal circle may 1k> a proper circle, real or imaginary, an equidistant -curve or a horo- cyclo. If tho orthogonal circle reduces to a point, all the circles pass through this point. All these results admit, in the case of elliptic geometry, of a simple interpretation by means of the central projection of the sphere. To a plane corresponds a circle, to an axial pencil of planes corresponds a pencil of circles, and to a bundle of planes through a fixed point corresponds a bundle of circles, the orthogonal circle of which corresponds to the polar plane of with respect to the sphere. This representation fails in hyperbolic geometry, since the sphere becomes imaginary, but there is a correspondence in tween the circles of the hyperbolic plane and the planes of hyperbolic space. 14, Correspondence between circles and planes in hyperbolic geometry. Marginal images, 1 Consider a fixed plane F and a plane E. From any point P on E drop a perpendicular PQ on F. The assem- Pra. 110. ■This theory is due, analytically, in F. Hnusdorff, " Anftlytische Beitrage zur niehtcuklidieeheii (Jcumutric," tteiptiger BtrkMa, 61 (189!)}, p. 177, and geometrically to H. Liebmann, " Synthetische Ableitung der Kreisi'erwnndtachnflen in tier LnbaUoiicfslrijschen Geometric," Mpzitjer Berichu, 54 ( I !X>2}. p. 250. Cf. also Liebmann, Niehlettktidische Qeomttrie, 2nd ed., Leipzig, 1912, p, 63. 230 SYSTEMS OF CIRCLES , [vn. II blage of all points Q lie within a curve called the marginal linage of the plane E on the plane F. (1) Let E and F he non-intersci 'tin». and have a common perpendicular AB (Fig. 116). Through A in the plane B draw any line AP, and in the pbae P4JJ, which cuts F in 5Q, draw <?PJ_£<3 and ||/(P. Then <? lies on the marginal image of E. If AB =p and BQ =p\ then sinh p sinh j/ = 1. Hence // is constant, and the marginal image is a circle with centre B and radius p' given by sinh p sinh p' = 1. (2) Let E cut J at an angle <, in the line MN (Kg. 117). Draw a plane 1 MA*, cutting E in MP and F in MQ. Kin. 117. Draw QP 1 MQ and || M P. Then „ - IT (MQ). Hence MQ is constant, and the locus of Q is an equidistant-curve with axis MN and distance a such that !!(«)=«. (3) Let E be parallel to F. Then the line MN goes to infinity, and the equidistant-curve becomes a homcyclc. Hence there is a (1, 1) correspondence between the circles in a plane and -the planes in space. vn. 181 MARGINAL IMAGES 231 15. When the planes E, E' intersect in a straigfd line I, their marginal images intersect in. two points which form the ■marginal image of the line l. Let the planes E, E' cut the absolute in conies C, C, and let be the absolute pole of the plane F, Then the marginal images /, I' of B, E' are the projections of C, C on the plane F with ceutre of projection 0. The conies C, C intersect only in two points P, Q, the points of inter- section of I with the absolute. The cones OC, OC cut the absolute each in a second conic C,, C,'. Now C, C cut in P, Q ; l\ . t 7 "it is P, , Q, ; G, Ci „ R, S ', C, C\ „ Ri, Si, and the points P, P, «ive the same projection on F, and so also do the other pairs. Hence the marginal images cut in four points, two of which form the marginal image of the line of intersection I, 16. The angle between two planes is equal to the angk of intersection of (heir marginal images. Let Ei, E t be two planes cutting in TT 1 . Let P,Qj be i'lii. 118. 232 SYSTEMS OF CIRCLES |vn. 17 the common perpendicular of E x and F, P t Q 2 that of g, and P. The plane P&Qft X P cute TT in £. Draw PA'i JV*' in the, planes ^.fi.nrr, and SS' ±F and 1 1 TT'. The planes P,ft.S fl, and P 2 <^S = P, are ± F. The marginal images r u r s of ff t , £ 2 are circles with centres Q,, Q s and intersecting in 5. We have then four planes E u B a , R 2 , R, whoso lines of intersection 7T', P a /> 2 ', SS', jyy are parallel. Therefore z (g,g J + (g^jj + (R 2 R t) + (JW = 2 _ But ftft) -£-(*£■) an d pyy. f -(^ Therefore 0M-fV£ 17. Systems of circles. A pencil of planes through a line I is represented on F by a system of circles through two fixed points, the marginal image of I. The planes perpendicular to I form a pencil of planes with ideal axis f , the absolute polar of I These are represented on F by a system of circles through two imaginary fixed points, the marginal image of I', and every circle of the first system cuts orthogonally every circle of the second system. These form therefore conjugate systems of coaxal circles. A bundle of planes thxongh a point P is represented by a system of circles any two of which intersect in a pair of points which are the marginal image of a line through P. If is the foot of the perpendicular from P on F, the two points of each pair lie on a line through and are equi- distant from 0. O is the radieal centre of the system, and all the circles cut orthogonally the circle which is the marginal image of the polar plane of P. vu. IE PENCILS OP CIRCLES 233 18. Types of pencils of circles. (I) Lot the axis I of the pencil of planes he a non-mtersector of the piano F, and let ffl be the common perpendicular of I and F. Lot. .-l f B be the marginal image of I ; O is the middle point of AB. Then the marginal images of the plant's through I are circles through the real pointeil, B. One of these is the line A B ; then, as the plane 8 is tilted, we get branches of equidistant-curves, then a horaeycJe, and lastly circles, onding with the state on AB as diameter, the circle of least diameter. (I') In the conjugate system the planes E arc all perpendicular to a Used line I, and the axis V is ideal. The marginal images are first a straight line through 0±AB, then a series of equidisfunt- curves with axes 1 A B and increasing distances, then a horocyele, and lastly a series of circles with diminishing radii tending to the limiting point A ; and a similar series tending to the other limiting point B. (2) Let the axis ( cut F in 0. The marginal images are first the straight line AB. then a scries of equidistant-curvrs with axes through 0, one branch passing- through each of the points A, B, ending with the equidistaut-eurve of greatest distance whose axis J.AB. (2a) lf l±F, A and B coincide, and the morginal images are concurrent straight lines through O. (26) If Hies in F. A and B are at infinity in opposite senses. Tho marginal images are equidistant- curves with common axis AB. (2') The conjugate system to (2) is similar to (1'), hut instead of starting with a straight line we ha%'e first an equidistant-curve with a minimum distance (2'a) When l±F the limiting points coincide and the marginal images become concentric circles (2'b) When ( lies in F the limiting points are at infinity La opposite senses, and the marginal images are straight lines 1. AB. (3) Let 1 1! F, then one of the points, B say, is at infinity. The marginal images are equidistant-curves through A with axes parallel to AB. one being the straight line AB and one the horocyele ± AB. (3') In the conjugate system the limiting point B is at infinity. We have in <>l oqutilisiant-eurvea with increasing distances. 234 SYSTEMS OF CIRCLES [vn, 18 then a horoeycle, and lastly a series of circles with diminishing radii ending with the limiting point A. (4) Let I be at infinity with P as point at infinity, and BOppose P is not on F. The planes E are all parallel The marginal image of I is a point A, i.e. A, B coincide. A is the orthogonal projec- tion of P on F. There ia a real plane through l±F cutting F in a line *. Wo have then, as the marginal images of the planes E, first the straight lino *, then a series of equidistant curves, then a horoeycle, and finally a series of circles, all touching I at A, which is both a limiting point and a common point. ( ■(') The conjugate system is of exactly the same form, since the absolute polar of the line I at infinity touching the absolute at / J is also a line touching the absolute at P. The marginal images all touch a line ± t, A being the point of contact. (5) In (4) let P be on F. so that A coincides with P at infinity. The parallel planes E make a constant angle a with if. We have then, as marginal images, a series of equidistant-curves with axes parallel to the direction through A, and constant distance a. such that il(o)=o. (5a) If the conjugate axis t lies in F, a= £jt, and the equidistant- curves reduce to a system of parallel straight lines. {5b) It I lies in F , u=0, and the marginal images are a system or concentric horoeycles. {6') The conjugate system to (5) is a system of exactly the same form with the angle a'=- - a. (5'a) is the same as (06) and (a'b) the same as {;V«), Note— In f>} we appear to have a pencil of circles with coincident common points, but we must consider this actually as a pencil with one real common point, and an ideal common point which is the ravened o «ri*4 rasped to fee absolute, Smttarfy En $ m should regard the two branches of the equidistant-curves separately, and regard the whole system as consisting of two pencils, each with one actual and one ideal common point. (Cf, IJx. VIII. 19. 20.) This is rendered clearer if. in finding the marginal images.' we confine our attention to the parts of the planes and lines which lie on one side of the plane F. Ex. vi i.J EXAMPLES EXAMPLES Vn. 1. Prove that if the common tangents to two circles are all real, the distances between the points of coated are equal in pairs, and that all four distances will be equal only if the axes of the circles an orthogonal. 2. Prove that the second radical axis of two circles which pass through A, I! pa**"* through the middle point of one of the segments All. HA. 3. Prove thatx i ky-kz=0 represents, for parameter A, a pencil of lines parallel to the positive axis of x. 4. In elliptic geometry, prove that the circles a,*-i hjij + c 1 z±d, = *> and ap + etc. - will cut orthogonally i f IP (<t s a t + bfa) + e,c, ± d^d. a 0. 5. If a bundle of circles contains a pencil of lines parallel to the positive axis of x, show thai the equation of the bundle can bo written in the form {x+plt)+ ky-rti(x+p)=0. 6. If a bundle of eircles contains a pencil of lines perpendicular to the axis of x. show that the equation of the bundle can be written 7 Prove that tho orthogonal circle of the bundle of circles x-i Ay <- /*(z -4 c)=0 is es=L 8. Prove thiit every circle of the system {x- pfc)+Xp+ft(* -p)=Q cuts orthogonally the horooycle p{x~kz)--k. 9. If the orthogonal circle of the bundle of circles aj+Aj/+fi(2 + c)=0 is imaginary, prove that every circle of the system passes through the ends of a diameter of the fixed circle z +e-0. 10. Prove that tho locus of the centres of point-circles of the bundle z + A(0 + 6)i-/is=O is the equidistant-curve by- +K 11. If sinhpsinh p'=l. prove that ll(,,)t-n(ji.') = ^, and p'«lag0Oth|. 12. Given a circle, equidistant- curve or horocyclc in a plane F, show how to construct the plane A* of which it is the marginal image on the plane f '. CHAPTER Vm. INVERSION - AND ALLIED TRANSFORMATIONS.' 1. In euclidcan geometry, Inversion, or the transformation by reciprocal radii, is a transformation which changes any point P into a point P', and P' into P ; the line PP' passes through a fixed point 0. the centre of inversion, and the segments OP, OP" are connected by the relation OP .OP = constant. This transformation lias the properties that it changes circles into circles and transforms angles unaltered in magnitude. It is a special case of a conformal trans- formation which preserves angles, and of the more special type of conformal transformation, the rimekr transforniutiim which changes circles into circles. We shall consider in this chapter the circular trans- formations in the non-euclidcan plane, and first we shall prove the following theorem. 2. Any print-transformation, which changes circles into circles is conformal. Two circles which intersect at equal angles at A, B are transformed into two circles which intersect at equal angles at A', B', i.e. certain pairs of equal angles are transformed into pairs of equal angles. We shall show that this holds for all pairs of equal angles. 1 See the references to Hiuistloril and Liobiwuin in chap, vii. § JU. vm. 81 CIRCULAR TRANSFORMATIONS 237 Let the lines o„ 6, through S, and a it 6 a through S. z make equal angles in tke same sense, Let flj, « 2 meet in 0, FIG. no. Draw and let S,S S make equal angles with a 1 and %. 6 through >? 3 making u t b t -6h», then ba i =aj} > . Two circles A £ can be drawn with their centres on the bisector of the angle at 0, passing through 8, and S 3 and having a,, a 2 and fc, , b as tangents ; and similarly a circle C can he drawn with its centre on the perpendicular bisector of Sj£, and having 6, b 3 as tangents. The equal angles 0,6, and ba s are transformed into equal A A angles, and the equal angles ba t and aJ> a are also trans- farmed into equal angles, is. a pair of equal angles in any position are transformed into a pair of equal angles. Hence two adjacent right angles are transformed into two adjacent right angles, half a right angle is transformed into half a right angle, and so on. Hence angles are unchanged. 3. Consider two planes J, , F*. All the planes in space are represented on each of these by circles, and we have a 1 238 INVERSION AND TRANSFORMATIONS [vm. 4 correspondence between the circles of F t and the circles of F* through the medium of the planes of space. Then, if F a is made to coincide with F u we have established a corre- spondence between the circles of F^ itself, i.e. we have effected a transformation of F t , changing circles into circles. Instead of supposing the plane F t to move, we may suppose F to be a fixed plane and let the whole of space move rigidly, with the exception of F. To a circle C corresponds a plane E. This plane is moved to E' and gives another circle C. To a pencil of circles corresponds an axial pencil of planes, and this gives again a pencil of circles. To a bundle of circles with common radical centre corresponds a bundle of planes through a point P ; P is moved to P', and we get another bundle of circles with common radical centre 0'. Hence this effects a transformation of the plane F, changing a point into a point, and a circle into a circle. It does not change a straight line into a straight line, but in general into a ctrola The motion of space which has just been considered is a kind of congruent transformation, i.e. it does not alter distances or angles. Rut a congruent transformation considered more generally may reverse the order of objects, changing, for example, a right-hand ^Iovi> into a left-hand glove. Such a transformation is produced by a reflexion in a plane. A motion is equivalent to two reflexions. We may extend the above result, therefore, and say : Every congruent transformation of sjmce gives rise to a circular transformation of a plane. 4. Conversely : Every paint -transformation of the plane which duinges circles into circles can be represented by a congruent transfornuttion of space. vm.5] CIRCULAR TRANSFORMATIONS 239 To a circle C corresponds a plane E, and to the corre- sponding circle C" corresponds a plane E'. Hence a plane is transformed iuto a plane, and the angle between two planes is equal to the angle between the corresponding planes. Further, a pencil of circles is transformed into a pencil of circles (since the transformation is a point-trans- formation) ; hence a straight line, the axis of a pencil of planes, is transformed into a straight line. Also a bundle of circles is tmnsformed into a similar system ; hence a point, the vertex of a bundle of planes, is transformed into a point. The transformation of space therefore changes points, lines and planes iuto points, lines and planes, and leaves angles unaltered, ie. it is a congruent trans- formation. 1 6. The general circular transformation which we have been considering is more general than inversion, for in- version leaves unaltered a point 0, the centre of inversion, and also all straight lines through 0. In general a system of lines through a point is trans- formed iuto a pencil of circles. In a pencil of circles through two points A, B there is always one straight line, the straight line AB ; and if a pencil of circles contains two straight lines it must consist entirely of straight lines; for the planes corresponding to the two lines are both perpendicular to F, and any plane through their line of intersection is also perpendicular to F. Now a pencil of lines through a point A is transformed into a pencil of circles through A', B' . Hence one line of the pencil is transformed into the straight line A' ft. Hence J Tn ouclidran space these conditions would specify only a similar transformation. In non-euelidean geometry, when the angles of a triangle are given, its aides are also determined. 240 INVERSION AND TRANSFORMATIONS [vm. 8 tlirounh any point A there is one straight line g which is transformed into a straight line g'. Let h be another line which is transformed into a straight line h'. Then the pencil igh), which consists entirely of straight lines, is trans- formed into a pencil (g'h% which consists entirely of straight lines. lfg t h intersect in 0, then^'. k' intersect in 0\ and FIO. 120, the corresponding angles at and 0' are equal. Let lie moved into coincidence with 0' and g with g' . Then either h and h' coincide or can be brought into coincidence by flapping the whole plane over about g', i.e. by a reflexion i n //. Then all the other lines of the first pencil will coincide with their correspondents, since angles are unaltered. Hence, Uie gemral circular transformation is compotmded of a congruent transformation of tlie plane and a circular transformatio, t which leaves unaltered all the straight lines through a fixed point. 6. Of this simpler form of circular transformation, which keeps one point fixed, there are three types, according as the fixed point is real, ideal or at infinity. These are called the hyperbolic, elliptic and parabolic types. And further, there are two forms of each, according as vm.7] INVERSIONS AND RADIATIONS 241 the corresponding congruent transformation of space is a reflexion or a motion. In the first case points are connected in pairs, since the relation between a point and its image is symmetrical. If P is transformed into P', then by the same transformation P' is transformed into P. A repetition of the transforma- tion will reproduce the slahts quo. The transformation is therefore periodic with period 2. or, as it is called, involu- tory. This form of transformation is called an inversion. In the other case, by repeated transformation the trans- formed points on a fixed line always go in the same direction. This form is called a radiation. 7. We shall now determine the metrical relations which * define inversion. (1) Hyperbolic Inversion, with real centre 0. Draw a line 0D=d perpendicular to the plane F, and through I) .A' draw a plane K±OD. We shall obtain a hyperbolic inversion by a reflexion of space in the plane K. Take any point P in F and draw PAA.F, and QA±0D anil H PA. Let Q' be the reflexion of Q, so that DQ' =QD, and in the plane POQ draw Q'A' LOD, and P'A'LF n,-e. a. H i, 242 INVERSION AND TRANSFORMATIONS [vm. 7 and || Q'A'. Then P' is the point which corresponds to P. Construct the point C which corresponds to D. Let OC = c,QD=DQ' =y, OP =z, OP' -as'. Then ainhe srrnW = l = sinh as sinh (d +?/)= sinh as' sinh (d-y); therefore sinha:=cosech{rf+;y) and cosha; = coth(<£+y). tanh ^x = coth x - cosech x = cosh (d + y ) - sinh (d + y) = e - W+rt, Similarly tanh Jx'=e- {d -5'> and tanh §c = e-''. Hence tauh Jx tanh J35' =e~- <( =tanlvHc. This is the formula for inversion in a circle of radius c. (2) Elliptic Inversion, with ideal centre 0. The fixed lines are all perpendicular to a fixed line I. Draw a plane K through I making an angle u with F, and take this as the plane of reflexion. Then in Fig. 122, where AOD = DO A' = $, PA ±0P and || OA, P'A' ±0P' and || OA', OP -z, OP' =x>, OC=c, we have u(x)=u+e, u(x')= <t -e. vm. 71 FORMULAE FOR INVERSION 243 Therefore 1 1 {x) + IT (as') = 2a = 2T7 (c). This is the formula for inversion in an equidistant-curve of distance c. If a = I, this gives x' = -x, a reflexion in a straight line. If a = j, we have II (x) + II (x') «», or sinh x sinli x' = 1 , a form of transformation which was frequently used by Lobacbevsky in establishing the trigonometrical formulae. (3) Parabolic Inversion, with centre £2 at infinity. The corresponding congruent transformation of space consists of a reflexion in a plane E [| F. In Fig. 123 XD is the trace of the fixed plane K, C the marginal image of D ; VA is the trace of a plane || F, V'A' FIO. 123. the trace of the reflexion of VA in K t and P, P' the marginal images of A and A'. Draw the horocyclic arcs PU t CV, P'U'. Let CP=x, CP'=x', x being positive and x' negative. Then PV =0X =P'V =i = l, since each is the arc of a 244 INVERSION AND TRANSFORMATIONS fm s horocycle having the tangent at one end parallel to the radius at the other end. XV=XV, CY=PU.e*, CV'~P'U'.<*; also CF + CT" = 2£X Therefore e r + e* = 2. This is the formula for inversion in a horocycle. 8. There is one property in which non-euclidean inversion differs from euclidean. In euclidean inversion the inverse P' of a point P with respect to a circle of radius OA is the Fig. 12*. point of intersection of the radius OP with the polar of P. This does not hold in non-euclidean geometry. If P'T is a tangent to the circle, and OP=r, 0P' = /, we have cos TOP' = coth OP' tanh a = tanh OP coth a. Hence tanh r tanh / = tanh%, whereas the distances of the inverse points are connected by the relation ^ y ^ ^ = ^^ In euclidean geometry these both reduce to the same, m > = a a . The transformation which is called in euclidean geometiy " quadric inversion," and which is obtained by the above vnt.9] CONGRUENT TRANSFORMATIONS 245 construction with the circle replaced by any conic, is, there- fore, in non-euclidean geometry, not a generalization of inversion. 9. Congruent transformations. Transformation of coordinates. The equations which determine a congruent transforma- tion of the plane are given at once by the equations for transformation of coordinates. Let the rectangular axes Ox, Oy be moved into the position O'x', O'y', still remaining rectangular. Let the Fro. 125. coordinates of 0' be (a , b e , c ), and the equations of O'x', O'y', a^ + i^y +0^=0. Then, the geometry being hyperbolic, -{D t 346 INVERSION AND TRANSFORMATIONS [vm. 8 the new axes are rectangular and pass Also, since through 0', a,a +6,6 +BiC =0, (2) whence Oq : b : c : 1 =6jC s - 6^ : c,a a - c^ : afi a - a^ : R. To determine the factor R, we have - k*R? = (V* - & A) 2 + («v»i - c^o,) 1 - ^{fflj&j - a A) a = -#>(V + 6,2 - Cl «/JP) («*• + 6," - Cl »/P) + &{<i i a 2 + b l b t -c 1 c 2 fk z y = -&. Therefore R = ±l. There are two cases in the transformation, according as the new axes follow the same order as the old, or are reversed. The second case is obtained from the first by interchanging x and y. If the axes are supposed to be fixed while the whole plane moves, we call these two cases respectively a motion and a reflexion of the plane. A reflexion may be produced by flapping the plane over about a line in it. If the axes are unchanged we have a 8 =0, 6 t = l, 0,-1, &i=0, e = l, an d therefore aj> t - a t b l = \. If the axes are interchanged a 2 = l, 6 a =0, ^=0, 6, = 1, c fl = l, and therefore Oi& a -«.&= -I. Hence for a motion R= +1, for a reflexion R = - 1. We have then b ] e z -b 2 c 1 =Ra , Cia 2 -c 8 a 1 =E6 , &iCo + Vi/** = -R<t 2 , a l c +<t l p t /k s =Rb,,, (ijb 2 -a 3 b 1 =Rc e , fesC + &0C2/P = P% , Ojc + fcoCa/A 2 = - Bfr, , a 2 b ~a i fi t =Rc 1 . vu>. 10] CONGRUENT TRANSFORMATIONS 247 10. Let P be any point, whose coordinates referred to the old and the new axes are (x, y t z) and (a;', y', z'). Then, expressing the distances of P from 0'x\ O'y' and O', we have x'=fcsinhM'P/fc, y 1 =k sinh N'Pfk, z* = cosh O'P/k; hence a;' = e^a; + bji/ + <\z, \ t/=a t x + b i y+c 2 z, V ... (a) - JfcV =aoa; + &(,!/ - fc%s.J where the nine coefficients are connected by the six relations (l)and(2). Further, since x'* +r/ i -kh' s = -h 2 , we have a* + a a a - «,»/# = 1 , Vi + b&t + 6«Po = °> V + 6 a 2 - 6 a /fc* = 1 , 0,0! + cja s + c^ = 0, Multiplying the equations (a) respectively by %, a v - ajk* and adding, we get a; = a 1 a:'+<* s / +«(£'. 1 Similarly ?/ =&!»' + &$' +W- \ •• ( A ') -A*z=c,!b'h-Cj1/' -IPc&', J from which we see that the coordinates of 0/, Ox', referred to the new axes are (a u «,, a a ), (6 t , b it b a ), (c,, c 2 , c^. (a') is the inverse transformation to (a). Both can be represented by the scheme a; y ikz x' 0] Oj Cjlik y' a t b 2 cjik ikz' a-oM bfjik c which may be read either horizontally or vertically. i 248 INVERSION AND TRANSFORMATIONS [v.n. n The determinant of the substitution = + 1 f or a motion, - 1 for a reflexion. 11. The transformation admits of a very simple repre- sentation. Since x i +y i -kh i = -&, we can write {x + iy) {% - iy) = k*(z - 1) (z + 1 ). Let Then and x = x+iy = k(z- 1) c- j8- ?^_ _ t(g-l) ft(s + I) s-iy' A(z + 1)~ s+iy ' 1-XX 1-XX 1-XX The coordinates of a point are then expressed in terms of the complex parameter X, just as in Argand's diagram. 12. Let X' be the parameter of P referred to the new axes. We have to express X' in terms of X. x' + iy' = (a-, + iajx + (6, + ib J y + (e, + w t ) z, W + I ) = (p - a & - b$ + k*c£)lh Multiplying these by I -XX, we get N =k{a l + ta-^(X+\) + ik(b t +?o g )(X -X) + («i+m 2 )(1+XX) =&X[(% + ia z ) -{(6, +?6 2 )] +fcX[(a 1 +«,) +*<«, +{& a )] + (i 4-XA)to +«;»). D-i(l -XX) -a (X +X) -i6 (X -X) +fe (l +XX). Now («.„ - t6 )[(a, + w 2 ) + i(fc, + $,)] = - c oC] + Re! + i( - Co c a + JfcJ - - ( Co _ #) (c, + ic^, and K+*o)l(«i+Wt)-ifo+#J]= -(Co + fljfcj+tc,). vm. 13J THE COMPLEX PARAMETER 249 Let X be the parameter of 0', so that Then, for a motion, K= +1, and iVX = (c, + M! £ )(X -X)(l -XoX), Z) = i(c + l)(l-XX ){l-XoX). Now c^+c^=P(c Q z -l); therefore c x + ic 2 = - As/c * - 1 e*, where # = tt + tan - ' J and a a + 6 8 = f(c 2 -1); therefore X (c + 1) = Je a ~lef i , where i/r = tan - 1 -*. "0 Therefore A T - A(c + 1)(X - X )(l - XtX)e iW " ♦>• Hence \' = ^ = N X-X <L «.«»->» ^e "# 1-XX Let e**-*-itfS, and X a = -)3, X 5= -,3; then mXh-j8 X' = £X+5' i.e. the general transformation of coordinates, or the general motion in the plane, can be represented by a type of homo- graphic transformation of a complex parameter. 13. If S denotes the operation wbich changes X into X' by tbe above equation, then the product of two such operations S x aud S A leads to yi « a X +p 3 "^aX'+aj _ u 2 {ai\ +A) ^-ffgt&X+ci) = aX +0 where a = ajct s + /8i/3g, /3 = u iA + ui j3* ; \ 250 INVERSION AND TRANSFORMATIONS [m U therefore $ } S 2 is an operation of the same form. The operations 8 have therefore the property that the product of any two of them is again an operation S. Further, it can be proved that (ftS a )S 3 =ft(S 2 ft,), and the operations are associative, A set of operations satisfy- ing these two conditions is called a group. This group "is called the group of non-eiwlidean motions. The nomographic transformation which represents a motion is a particular case of the general homographic transformation y _ «X+j fl >X + <T where «, ft y, § are any complex numbers. This trans- formation belongs to a more general group, the group of hmtiogmphic tmnsformutions, and the group of motions is a sub-group of this larger group. 14. In elliptic geometry a motion is represented by the transformation aK-B \' =■= — ". 8\+~a If S is the product of two operations ft, S t , we have a = am t - ft ft,, 8 = a ,ft + ^8,. Put_a=d + ia, 8=b-tc, where a, b, o, d are real and i =•%/ - 1 ; then we have o= «irf 2 +6 1 c 2 -Cj6 2 + (i 1 a a , 6= -a 1 c 3 +6 1 rf 2 + c 1 <? a -i-d 1 6 2 , C= ffii&s-ftjfflg + c^ + ^Cj, Now these relations are exactly the same as those which we obtain from the equation ™+fy + ck + d = (a 1 i+b t j + c 1 k+d l )(a s i + b 2 j+c z k+dJ, where i? = j a _ $■ _ _ j jk=i= - kj, U =j =-ik, ij - & = _#, vm. is] GROUTS AND QUATERNIONS 251 Here ai + bj+ck + d (=q) is a quaternion. Hence the rule for compounding operations of the group \'={a\-|8)/(8\+a) is exactly the same as that for quaternions. The meaning of this can be explained as follows. The operation q( )q l performed upon a vector ( ) has the effect of a rotation about a definite line. The product of two such operations h(Si< kf l }H~ l -Ubl )?rV lta ?( )?"'■ where q = q->qi, and is therefore another operation of the same form. These operations form the group of rotations about a fixed point, or the group of motions on the sphere. 15. If we take polar coordinates (r, 0), •Y T t x +vy =& sinh , (cos + isin 0) =& sinh j_e i9 , z+l = cosh t + 1 =2 cosh* sr- Therefore A = tanh 9 , e . ,r 2JW sinh ,- = -. & 1-XX XX = tanh 3 ^, >/xXe» = X, cosh £ =^xx' The equation of a straight line aa; + &iy + C2=0 becomes, when expressed in terms of X, ai(\+X)+»&*(X-X) + c(l+\A)-0, i.c e(AA + l)+A(a+i&)X-i-&(a-*)A=0, which is of the form XX-dA-aX+l=0. If the line passes through the origin, c=0, and the equation reduces to 5X+aX=0. i 232 INVERSION AND TRANSFORMATIONS [m 10 The equation of a circle, cosh ^ = coah ^ cosh t - sinh f sinh t cos (0 - a), with centre (c, a) and radius a, becomes Xx(cosh ^ +coah |) - sinh r(Xe"*' + Ac" 1 ) + cosh j- cosh "=0. In general, therefore, the equation XX -SiX-aX +6=0 represents a circle (equidistant-curve or horocycle) which reduces to a straight line when 6 = 1. 16. The general homographic transformation of A leaves the form of the equation of a circle unaltered, and is there- fore a circular transformation. The transformation of inversion is included in this. Inversion is characterised by connecting points in pairs. The parameters A, A' of a pair of inverse points must therefore be connected by a lineo-Iinear equation of one of the forms y _aX+8 yX+6' y_a\+8 yX+V ■(I) y\+S' yX+S' 4* The first form characterises motions, the second re- flexions, when y = 8 and <5 = 5. Inversion belongs to the second form and is a symmetrical transformation, i.e. X' is expressed in terms of X by exactly the same equation as that which expresses A in terms of A'. If y=/=0, we can take X' = »*+# X+S Ex.vm.] INVOLUTORY TRANSFORMATION 253 Then X =-<?"— ® and A = -$x'+8 X' -a Hence S = -u and 8 = 8, i.e. 8 is a real number = -6. The transformation for inversion is therefore of the form ., aX-b X — <i •(I) If the points X, A' coincide, so that A' =X, AA -aX -«X +6=0 (r) which is the equation of the circle of inversion. If X, A' are a pair of corresponding points, equation (i) gives A'X-aX'-aX+6=G. If y =0, we can take X' =«A +8. Proceeding as before we find aa = l and 8= -a/3. The transformation then reduces to the form oX'+aA = 6. In this case the circle of inversion is the inverse with regard to the absolute of a circle which passes through the origin. These results should be compared with the corresponding formulae for euclidean geometry in Chapter V. §§ 31, 32. EXAMPLES VEX 1. In elliptic geometry, show that the general transformation of coordinates is expressed by A'=} ; ^k+ta-E f ™ °' are real. ~{c-id)k+{a-ib}' 2. Prove that the general homographic transformation X'= ■ . , changes circles into circles. ak * li 3. Show that the transformations A'=—r— -. form a group. yA + o i 254 INVERSION AND TRANSFORMATIONS [Ex. vm. 4. Show that the general reflexion nf the plane in hyperbolic geometry is represented by \' = ~-!-. tfA, + a. 5. Show that the reflexions of (lie plane tlo not form a group, but that the product of two reflexions is a motion. 6. Show that the operations of the group A'=^— ^ leave un- Ex. vm.] EXAMPLES 255 altered the equation A A = 1 /3A + a 7. Show that the equation »/=0 is unaltered by the operations of the group A'= - — -, where a, b, c, d are real. CA +« 8. Show (lint the equation x=0 is unaltered by the ojjerations of the group A'= — — where a, b, c, d are real. tCA + tt 9. If the points A,, A,, A 3 are collincar, prove that 1 + A,a, A, A, =0. 1 + AjAj Aj Aj 1 + A..A, A, A, 10. Verify that if A, A^cosh |+eo8h *\ - sinh |{ A,e - '»+ A>«) + cosh C - - cosh "=0, the points A,, A, are collinear with the point whose polar coordi- nates are (c, a). 11. Prove that the formula for a hyperbolic radiation, correspond- ing to a translation in space through distance rf, is sinh x'=e a sinh x. 12. Prove that the transformation tanb rtanh /= const, changes a straight hue into a curve of the second degree. 13. Prove that the inverse of the absolute in a circle of radius c is a circle of radius equal to k log cosh c/k. (This circle is called the vanishing circle ; cf. vanishing plane in the theory of perspective.) 14. Prove that the inverse of a straight line is a circle cutting the vanishing circle orthogonally. 15. Prove that the inverse of a horocyele is a circle touching the vanishing circle. 16. Prove that any circle which cuts the circle of inversion ortho- gonally is unaltered by inversion. 17. Prove that the inverse of a system of parallel lines is a system of circles all tombing at the sumo point. 18. Prove that a horosphere which cuts the vanishing sphere orthogonally is inverted into a plane touching the vanishing sphere ; and that a horocyele traced on the horosphere is inverted into a circle lying in this plane and passing through the point of contact.' Hence deduce that the geometry on the horosphere is euclidean. 19. Show that the equations x +}iy= U («- 1) and x+fa/ = kt{s + \), where ( -tanh a/it, represent two pencils of branches of equidistant - curves, the first passing through the origin, the second through the point on the axis of x at distance 2a from the origin. Prove that the inverses of these systems with respect to a circle with centre and radius 2a are respectively x+fiy -kt 3 (z+ 1) and t(x + /ty)=k{s - 1 ). 20. Prove that the inverse of the pencil of straight lines x + p.y = ktz, where f - tanh a/k, wit h respect to a circle with centre and radius a, is the pencil of circles 2p*(»+/»gfl -kt [(p*+ l)z+(p*- 1)1, where p at tanh fafk. Show that the common points of this pencil are on the axis of k at distances from the origin equal to a and 6, where tanh \bjk =p*. 21. Prove that inversion with regard to the absolute is represented by X'A=1. .Show that this transformation leaves every straight line unaltered, and changes the circle a*+%+« + rf=0 into ax+by+cz-d^O, i.e. interchanges the two branches of an equi- distant-curve. 22. 1'rove that two successive inversions in the two branches of mi equidistant-curve of distance k sinh" l l, followed by a reflexion in its axis, are equivalent to an inversion in the absolute. i CHAPTER IX. THE CONIC. 1. A conic is a curve of the second degree, i.e. one which is cut by any straight line in two points. Since the equation of a straight line in Weicrstrass' coordinates is homogeneous and of the first degree, the equation of the conic will be a homogeneous equation of the second degree. In Cayley's representation a conic will be represented by a conic. This is the chief beauty of Cayley's representation, that the degree of a curve is kept unaltered. The projective properties of a conic are the same as in ordinary geometry, and it is only in metrical properties that there is any distinction. Since metrical geometry is reduced to projective geometry in relation to the absolute conic, the metrical geometry of a conic in non-euclidean space reduces to the projective geometry of a pair of conies. The metrical properties are those which are not altered by any projective transformation which transforms the absolute into itself The metrical geometry of a conic therefore reduces to a study of the invariants and co- variants of a pair of conies. We shall confine ourselves here to au enumeration of the different types of conies, and a few theorems relating to the focal properties of the central conies which bear the closest resemblance to those in ordinary geometry. is. -J CLASSIFICATION OF CONICS 257 2. Classification of conies. Tn euclidean geometry, leaving out degenerate forms, there are three species of conies, according as they cut the line at infinity in real, coincident or imaginary points. These are the hyperbola, the parabola and the ellipse. Also, as a special case of the ellipse, we have the circle, whose imaginary intersections with the line at infinity are the two circular points. In non-euclidean geometry conies are classified similarly with reference to their intersections with the absolute. Two conies cut in four points, and reciprocally they have four common tangents. The points and lines which a conic has in common with the absolute are called the absolute pout* and tangents. These elements may be all real, or imaginary or coincident in pairs. When two absolute points are coincident, two absolute tangents are also coincident. When two points are real and two ima- ginary, the same is true for the tangents. When the points are all real, the tangents may be all real or all imaginary. When the points are all imaginary, the conic must be within the absolute (for we need not notice a conic which is wholly ideal), and the tangents are all imaginary. Conies are therefore oteflrittod im> follows : (J) Absolute points and tangents all real. team hypabdht, with two real branches concave towards a point between thorn. (2) Absolute points nil real, absolute tangents sill imaginary. liiiptrM'i. with two real branches, resembling an ordinary hyperbola. (3) Absolute points and tangent* all imaginary. fillips, n rinsed curve. (4) Absolute points and tangents two real and two imaginary. Semi-hifperhofti. with one real branch. N.-E. O, R 258 THE C0X1C ix. 3 I (5) Absolute points and tangents imi coincident and two real. Concave hyperbolic parabola, two real branches touching the absolute at the same point. (6) Absolute points two coincident and two real, absolute tangents two coincident and two imaginary. Convex hyperbolic parabola, one real branch and an ideal branch touching the absolute. (7) Absolute points and tangents two coincident and two imaginary. Elliptic parabola, resembling nu ordinary parabola. (8) Absolute points and tangents three coincident and one real. (habiting parabola, one real branch osculating the absolute at one end. (9) Absolute points and tangents, two pairs of each real and coincident. EquidiKtanl-mrve. (10) Absolute points and tangents all imaginary and coincident in pairs. Proper circle. (11) Absolute points and tangents all coincident. Iloroeijck. In elliptic geometry the absolute points and tangents are all imaginary, and we have only ellipses and proper circles. 3. The four absolute points form a complete quadrangle. The diagonal points forai a triangle C\C Z C' S which is self- con j ugate with regard to the conic and the absolute. Every chord through any of these points is bisected at the point. The points C,C' 2 C 3 are therefore centres of the conic, and their joins are the axes. The four absolute tangents form a complete quadrilateral. Its diagonal triangle is formed by the three axes. In euciidean geometry the foci of a conic are the intersection* of the tangents from the circular points. Those are the absolute tangents, and we call therefore the three pairs of •*■« CENTRES, AXES, FOCI, ETC. intersections of the absolute tangents the/oe* of the conic. Similarly the three pairs of joints of the absolute points are called focal lines. The polars of the foci with regard to the conic are called tlttYctrux's. Tv. m pass fhruujih each centre and are per- pendicular to the opposite axis. Fill. The poles of the focal lines with regard to the conic are called director point.?. Two lie on each axis. In euciidean geometry the focal lines degenerate in two pairs to the linn at infinity. The third pair become the asymptotes. Four of the director points coincide with the centre, and the other pair coincide with the points at infinity on the conic. In euciidean geometry the a.si/mptntcs are the tangents to the conic at the points where it cuts the absolute ; but in non-euclidean geometry the lines which N.E. O. Illi 200 THE CONIC r tx. I ix. 15] FOf'M, PROPERTIES 261 i most closely resemble the euelidean asymptotes are the taDgents to the conic from a centre, and arc therefore six in number. 4. By taking the triangle formed by the centres as triangle of reference, the equations of the absolute and the conic can be taken in the form . , . or in line- coordinates a b e The coordinates of the common (mints are given by a? : >?: z'—b-o-.c-aia-b, and the coordinates of the common tangents £»: */» : f==a(6- e) : b{c-a) : c(a-b). The focus F t is the intersection of two absolute tangents ■Ja (6 -c)x+\/b(c -a)y - <Jc[a -b}z =0, Ja (b - c)x - Jb (e - a)y+ s /c(a^b)z=0 ; therefore its coordinates are 0, \/c{a b), ,/i(c-a). F L ' is the intersection of the other pair of absolute tangents, and Its coordinates are „ , — r ,- Ifi/. d' arc tho distances of a point P (x, y, z\ from F lr F,', y s/e(a - b) + z -Jb (c - a) coa if- T - d -i y^( g -b)+zjc{c-a) + sin Hence ~»^ i ^- (^-^-^-^Hftla-W-^-tf I &+tf+z*)a(c-b\ _o+b (a-b)tj t -(c~ a)z* e +6 - c-6* o(a?T!/'+#)^ = c~& , i.e. e#/*er //w SUin or the difference of Ike distances of any point on a conic from a pair of foci is constant. Reciprocally, either the sum or (he difference of the angles which, any tangent to a conic makes with a pair of focal lines is constant. A tangent makes a triangle with a pair of focal lines. In the case in which the sum of the interior angles is con- stant the sum of the angles of the triangle is constant, and hence the area is constant. This result may be compared with the property of a hyperbola in euelidean geometry, a tangent to which makes with the asymptotes a triangle of constant area, 6. The conic, the absolute, and a pair of focal lines form three conies passing through the same four points. Any lino is cat bv these three conies in involution. Let the line cut the conic in P, Q. the absolute in X, Y, and the local lines in M, N. Then (XY, PQ, MN) is an involution. Ijet G, G' be the middle points of PQ. so that (XY, PG)~(XY, GQ) and (XY, PG')7:(XY t G'Q). Then G, G' are the double points of the involution, and (AT. MQ)*(YX, NG)-*(XY, GN); therefore G, G" are also the middle points of MN, i.e. the segments determined by the points of intersection of any line I with a conic and the points in which I cuts a pair of focal faies have the same two middle pcmits. Reciprocally, the tangents from any point P to a conic and the lines joining P to a pair of foci have the same two bisectors. If P lies on the conic, the tangent and normal to the conic at P are the bisectors ofPF, PF'. I THE CONIC [nc.a 6. Take a focus F with coon) iiuites The equation of the corresponding directrix is y-Jb(a -b)+zJc[c-a)-0. Let d bo the distance of any point P on the conic from the directrix and r its distance from the focus; then sin.tf- smr = Therefore Va? + sf~+3?jfb~- c)(a -b-c) ' yjb(a - b) +z*/c(c - a) ■J-j^ + if-i&Jtiib-c) siiir_ la mid \ ■b-e i.e. the ratio of the sines of the dtstamxs of a point- on a conic from a focus and the c»firs/it»ttlin<f directrix is constant. Reciprocally, the ratio of the sine of th& angle which a tangent to a conic makes with a focal line to the sine of Us distance from die corresponding director point w constant. 7. It is interesting to obtain a geometrical proof of the focal distance property. 1 Lot Si be the absolute and any conic Inning four real common tangents with 12. Let two jmirs of the common iuiigen(« intersect in the pair of foci F, F'. Let P be any point on C. .loin PF and PF', cutting 12 in X, 7 and A", Y'. Then we have to prove that dist, (/\F)±dist. (W)=const„ or, in terms of cross-ratios, log (PF, XY) ± log {PF', A'}")=conat., i.e. either the product or the quotient of the cross- ratios is constant. Let A' A"', YY' cut FF' in A and /J. A" J' ami A" J" cut FF' in A' and B'. Then (PF, XY)~K x -( F'F, A A') and (FJ", ZTITCtCW, A'BpZ{ F'F, BA'}. 1 For part of thin proof I am indebted to Dr. VV. P. Milne. a. 71 FOCAL DISTANCE PROPERTY 2611 Therefore (PF, XY)HPF\ X'Y')={F'F, AA')MF'F, BA')=(F'F,AB). Similarly {PF, XY) . (PF', X'Y')=(FF\ A'B'). Wo have therefore to prove one of these cross-ratios constant. Four eonics through the points X, X', Y, Y' are Si ; X Y, A" i ; XX', YY'; X Y', X'Y. Let 12 cut FF' in V, V. We have then an Fie. 1S7. involution determined by (UV, FF% and this contains also the pairs A, B and A', B". If therefore {FF', AB) is a given cross-ratio, A, B must be fixed points. • Now, supposing that A, B are fixed points, the point P is con- structed thus : Si is a fixed conic and F, F' two fixed points. FF cuts Si in fixed point* U, V, and .4, B are a pair of fixed points m the involution determined by (FF', UV). Through F anv line u is drawn cutting Si in X, Y. XA cuts 11 again in X', and'wc get the line X'F'=n' corresponding to u. P is 264 THE CONIC fix. 7 the point of intersection of u, ■*'. If F'X' cuts again in Y% then iU"r% m B ' tU ° point ^responding to A m the involution {ft , UV). Since « cuts in two points, there are two lines u' correspond in K to u, and similarly there are two lines u corresponding to u' The rays « «' are therefore connected by a (2, 2) correspondence. The locus of P is therefore a curve of the fourth dc^ve. liuf v.lun /(coincides with H", so also do both the corresponding lines V,and MM Bfw: (h-^n-rr ill,- |.»:iis eonUins ilu, | m ,. W uvirr It therefore consists of this line doubled and a conic. Also, if u is a tangent to the two lines «' coincide, and P is a double point on «, Therefore m is a tangent to the locus of P Hence the conic which is the locus of P touches the four taneents drawn from F. F' to fl. »s » Further, if P is taken on fi, X and X' coincide with V>; hence the tiiii-I'THs I,, fi „ | tt-s points of teittBBOtiOOa ui:h MBS. l!ir..iM> Cither .4 or >S. re 4, .ff arc therefore the fixed points in which the tangents to 11 at its intersections with C cut FF', and therefore (PF, XY)+(PF, X'Y') is constant. Tho fooi f, F' are real only when the absolute tangents are nil aginary. In the case of the conver hyperbola the order of tho points P. F X, Y and P, f, X'. Y' is the same, and the difference of the focal dtstanr&s is constant (Fig. 128). Pw, X2S. Ex.de.] FOCAL DISTANCE PROPERTY 265 In the case of the ellipse the points P, F, X, Y and P, F', Y', X' have the same order, and the mm of the focal distances is constant (Fig. 129). EXAMPLES IX. 1, If tho equation of the absolute is aP + tf+HW^O, prove that the coordinates of the three pairs of foci of the conic x t la+f/b + l^z'/c=(i are («, rf^g /:?), (**^J, 0. ^} where £».=&- e, f3=c-a, y—a-b. 2. In hyperbolic geometry, where the otpiation of the absolute is ^+2^-^=0, show that the equation x*/a +,</*/& -iV/e=0 represents (1) an imaginary conic if a>0, 6>0, e<0, (2) a real ellipse if a, b, c are positive and c docs not lie between a and 6, (3) an ideal ellipse if «<0, 6>e>0 or 6<0. «>c>0, (4) a concave hyperbola if a, b, c are all positive and e lies between a and b, (5) a convex hyperbola if «<0, e>6>0 or 6<0, c>«>0. •>m THK OONIC [Ex. ix. Ex, jx.J EXAMPLES 267 i a. In elliptic geometry, prove that an ellipse, real or imaginary, has always one pair of real and two pairs of imaginary foci. 4. In hyperbolic geometry, prove that the three pairs of Foci are (1) one real and two imaginary for a real or imaginary ellipse or a convex hyperbola, (2) all ideal for an ideal ellipse or a concave hyperbola. 5. A,B are tixed points and APB is a right angle ; show that tho locus of P is an ellipse. If AB=2a, prove that the real foci are on AR at a distance from 0, the middle point of AB, such that Unhz/£=tanh s «/fc, or a5=H*logcosh2a/J(:. Hence prove the following construction for the iboi • Draw OR making the angle AOR=Tl(a) and cutting the circle on AB as diameter in R. If. Then F, F are the feet of the perpendiculars on AB from R. K. 6. A, B are fixed points and P is a variable point, such that I he angle APR is constant ; prove that the loans of P is a curve of the fourth degree, 7. A, B are fixed points and j° is a variable point, such that cosh ^i. cosh — A. K is constant ; prove (hat the locus of P is an ellipse. 8. Prove that the locus of a point, such that the ratio of the sines of its distances from two fixed points is constant, is a conic. 9. A, B are fixed points and P is a variable point, such that the sum or the difference of tho angles ARP, BAP is constant; prove that in each case the locus of P is a conic passing through A and B. 10. A variable lino cuts off on two fixed axes intercepts whose sum or difference is constant ; prove that in each case the envelope of tho line is a conic touching the axes. 11. Prove that the product, of the sines of the distances from a pair of foci to a tangent is constant State the reciprocal theorem. 12. Prove that the locus of points from which tangents to a central conic are at right angles is a conic meeting the given conic where it meets its directrices. State the reciprocal theorem. 13. Prove that the locus of a point which makes with two given points a trianplc, whose perimeter is constant, is a conic with the two given points as foci. Show that the locus is also a conic if the excess of the sum of two sides over the third side is constant. 14. Prove that the envelope of a line which makes with two given lines' a triangle of constant area is a conic. Show that the envelope is also a conic if the excess of the sum of two angles of tho triangle over the third is constant. 15. Prove that -^r + r^t + — y =« represents, for all values 0+ A 0+ A C+ A of A., & system of confocal conies. 16. Show that in the conformal representation, in which straight lines are represented by circles, a conic is represented by a qnnrtic curve having two nodes at the circular points, i.e. a bicircular quartic. INDEX Tho numbore rcfrr to tin? tsiifls, except itwso iirccwtetl by Ex., which nta to ',.■ Bxampiu; n suriiiites fuomote. Absolute, the, 40, 08, 1 54, JOS. ks equation, 127. 129, 174. in Euo. (Jeom,. 47, 155, 104. Absolute (leometry, 22. polar system. H, space, 107, 210. Absolute unit, of angle, 88. of longlh, 13, IS, 5S. Mil Absorption of ligbt, 309. Actual points, 40. Author, 197, 201. Al.ESlBKKT, d', 3. AhiLudes of a triangle. 54, 141. of a, t.tialii'dii.ii, Ex. iv. 24. A n eh or- ring, 106 n. Angle, 28, 121 nhranlt 42. flat, 28. formula, 131. logarithmic expression, 157. of parallelism, 30, 35, 58. right. 28. in a semi ci rcli-. Ex, i I ; ix. 5. Anglo-smn of a triangle, 5. 10, 12, IB, 10, 18,21. and area, 13, 30, 77. 82, 104. Antipodal points, 55, 80, 130, 178. Ap p heebie surfaces , Wo. Archimedes, axiom of, 17. Area, of circle. 80 ; Ex. ii. 23 j iii, 3. of equidistant-curve, Ex. ii. 22 ; iii. 4. of plane, 10 ; Ex. iii. 5. of polygon, 83, 104. of triangle, 13, 20, 77-70, 81, 82, 103-104. Area, of triangle, maximum, 7,81. infinite, 7-0', 19, 20. unit of, 79. Argand's diagram, 181,248. AlIISTOTLE, 2. Astral geometry, 15. Astronomy, 203-207. Asymptote.'!, 259. Asymptotic linos, 10, 30, 42. Axioms, 2, 27. of Archimedes, 17. of Pnscii, 20. Axis of a circle, 52, 104, 136. of a conic, 258. of a pencil of lines, 48. radical, 219,228. Halt., W. W. Rouse, 201 u. IUi.tzer, H. B., 24. 15ki.tha.mi, E., 202. Berkeley, Q., 201. Bkuthakd, L., 7. Hieircular (piartic, Ex. ix. Iff. Bisectors of angles, Ex. iii. 1 ; 139. Bolyai.J., 14, 15,21-24. Bot.yai, W., 7, 14, 21-24. li.'Not.A, K., 24 n. I Broab, C. D., 210n. Bundle of circles, 228. of lines, 45, 55, DO. Camerer, J. W-, 15. Carsiaw.H. S., 24 n. Cayley, A., 158, 102. i Centre of circle, 51, 104. of conic. 2ss. nomothetic, 221. 270 INDEX INDEX 271 '% Centra, radical, 221, Controid, 139. Ceva's theorem, 145. Cirolo, 51, 104. circumference, 7(4, 1 14. equal ton, 1 3ft, 227. 808, in relation to Absolute, 130, 258. in Hue. (3enm., 47, 140. of infinite radius, 51 [see oho !! "ii n-.\ (■]<■). through three, points, 53, 189. Circle at infinity, 104. Circular functions, 57, I14ji. measure of angle, 57, 81. points, 47, 156. transformations, 181 ; Chap. viii. Cireumcentre of triunglo, 64. Circumeireles of triangle, 53, 189 ; Ex. ii. I, 13 ; iv. 10, 21, 22. of regular polygon, Ex. ii. 17. Circumscribed quadrilateral, Ex. ii. 3. sphere, Ex. iii. 7, 12, Clifford, W. K., 25 «, 201. Clifford's parallels. 108. surface, 10(1. 112. Coaxal oircles, 222, 232. Collinaarity, 135, 145 ; Ex. viii. fl. Collinoation, 180. CftmpWW hearings, 7. Complementary scgmonts, 03. Complex numbers, 181, 248. Concurrency, 135, 145 ; Ex. iv. 12. Configurations, 143. Confocal conies, Ex. ix. IB. Conforrual representation, 172- 191. i i wisformation, 182: Chap, viii, Congruenee, 28, 194-197. ■ if infinite areas, 8. Congruent transformation, 158, 238, 245. Conies, 40 n, 08 ; Chap. ix. ; Ex. viii. 12. Conjugate coaxal circles, 232. harmonic, 05, 148. isogonal, 147. points, 80. Conaifltency of N.-E, G., 202. Continuity, 17, 29, 90. (.'(Divergent lines, 42. Cotit.tinii-:, .1 I. ., Z~ ... Coordinates, 125, 199. homogeneous, 135, lino, 128, 172. polar, 125. tri linear, 172, Weieretrasa', 127, 129, 171, 227, Corresponding points, 32 ; Ex. ii. Cross-ratio, 147, Cube, Ex, iii. 8. Curvature, measure of, 160. of space, 193, 199. surfaces of constant, 168, Cyclic quadrilateral, Ex. ii. 2. Cylinder, 105. Defect of triangle, 20, 78. Definitions, Euclid's, 2. I )■■;.'. derate eases in Eue. Geom., 11,47, 75, 82, 139-141. 166- 150, 161-162, 217, 220, 244, 259, 201. Desargues' theorem, 142. Deamio system, 144 ; Ex. iv. 23. Dovolopabte surfaces, 160. Dihedral angles, 42, Dimensions of space, 208. Direction fallacy, 0, 20. Director points, 259. Directrix, 259, Displacement, 179, 190. Distance, absolute unit of, 13, 15, in fine. Ili-nm., 75, IfiH. Mil. formula, 120. 132, 158, 186. Divergent lines, 12. Duality. 69, I nil. 226. Egypt, 1, Element of length, 187, 104 ; Ex, iv. 5. Ellipse, 267, Elliptic geometry, 25, 20, 55 ; Chap. iii. ; 200, 208 («M also Spherical geometry), inversion or radiation, 240. involution, 97. Empiricism, 207. EnsHXs F., 1 1 it, 13 n, 21 n, 08 n. Engol-Napi'i' nth-*, 07. Envelopes, 104, 120; Ex. iv. 17; ix. 10, 14. Equidistance, 10, 42. Equidistant- curve, 12, 52, 105, 258. equation, 130, 228. length of arc, Ex. ii. 7. iii. 2. Eimidislant-surfaco, 53, 105. Escribed circle*, Ex. ii. 13. ICCl'I-lU, 1, 2. Euclidean geometry, 30, 47, 76, 711. 00,92, 134, 139-141, 155- 157, 101, 176, 220, 259, 201. Kvcc-s ni a trinnule. Hit Exterior angle, theorem, 17, 19, 29, 31-35. Focal distance properly, 200, 202. Focal lines, 25!). Foci, 259 ; Ex. ix. 1, 3, 4, 5 K. .ens-directrix property, 202. Wam dimensions of space, 42 ?t, 193, 199. FRANKi-ASn, W. B.. 4 n, 200 >i- Froe. mobility, 107, 108, 195-100. Fundamental theorem of projec- tive geometry, '.Iii Gauss, C. F,, on area of triangle, 7, 82-83. on curved surfaces, 108. on parallels, 14, 22, 24. pentagram, 08 n. C-dili-sir.,, 166 Geometry, Absolute, 22. Analytical, Chap, iv. Astral, 15. Bizarre, 162, 104. I H I'ferontial, 194. Elliptic, Chap. iii. Euclidean, g.v. Hyperbolic, Chap. ii. Imaginary, 21. in the infinitesimal, 70, 1 14. Log. -spherical, 15. Non-Euclidean, 14, 20. of a bundle. 55, W. on Clifford's surface, 1 13. (leomotry on curved surface, 160. on equidistant-surface, 50. on horosphore, 15, 50, 84, 165 ; Ex. vih. 18. on imaginary sphere, 13, 15, 105. on plane at infinity, 135, 105. on sphere, 50, 105, origins, 1. Parabolic, 25. Projective, 93-98. Spherical, 25, 89, 130. 138 ». with hyperbolic or parabolic measure of angle. 102, 104, with projective metric. 100. QaaaaKKB, ■' l> . Bx- ■ ■ *». Greek geometry. 1. GllKENSTBKET, W. ■!., 210 «, Chonai:. K '(' !■'. . Ex. i. 7. Groups, 197, 250. Halsted, G, B., 1 1 ii, 21 n. 23 n. Harmonic range, 95. Hart's circle. Ex iv. 25. Haitst, J. K. F., Ex i. 0. Havsdorfk. F, 229 ». HK\TIt. 'I'. L. 2. Ih'iHiiKii, J- L., 2. Hnst,MH(M.T2, H. von, 105- 1 '.IT. I'.l'.l Heboootus, 1. Hii.bert, D, 27, 41, 11&. HlNTON, C. H.. 201 II. IlllTDnltATF-S, I. I [0 HiATK, T. F„ 94. lloiiuHviitric circles, 221 Homojjmphic transformation, 182, 240. Homography, 94. Hornochetic centres and axes, 221. Horocyele, 51, 258, equation, 137, 228. leu-lb .if arc, 57 ; E\ n I Horosphero, 52. geometry on, 16, 56, 84, 160 1 Ex. viii. 18. Ui.Tki.. -I .21 Hyperbolas, 257. Hyperbolic functions, 15, 03. geometry, Chap. ii. ; 25, 30. 272 INDEX 1KDEX 273 I Hyperbolic inversion or radia- Won, 240. involution, 97. Ideal elements, 47, 154. Imaginary points, 97, 133, 134. Indujinablea, 2, 27, Inequalities, 28. Infinite us. unbounded, 194. Iiilimt,- areas. 7-9. |ii. 2u. Infinitesimal domaio, 7ii, I 1 4, transformation, 198. Infinity, points at, 4fi. Inscribed circles of n triangle, 54 139; Ex. ii. 10, 13; iv. 15, 19, .'11 of a regular polygon, Ex. ii. 17. Inscribed quadrilateral. En. ii. 2 Intersection of lines, 132. of circles, 211, 228. angle of, of circles, 218. Iniiiition, 2, 207. Inversion, 241, 252 in Eire, tieotn . ISO, 183. 244. quadrie, 244; Ex. viii. 12. Involut '.17. Involulory transformation, 241. Isogonol conjugates, 147. Isosceles triangle, 28. Ivory, J., 19. Kak.st.vkb, A. O., 21. Kant, I., 1 4, 207. Ki.kin, IV. 2.7. 159, 192. Kluokj,, G. S„ 22. Laoukkkk, E., )i5ii ,1. Lamukrt, J. H., 13-14, 15, 20. Lbhbn-dke, A. M., 10-18,20; Ex. i. 4. Lehi.ii-:, J., 19. Lit:, S., 197. I.ikiimann. H., 24 n, 229 n. Limiting points, 223. tines, ii:'.". Lin e-c oordinates, 128, Linear systems of eii-ctes. 220. Line-element. 187, 194 : Ex t>, B. LoBAfHEVSKY, N. I., 15 "II - ■> ] 23,24, ISSn, 243. Loci, Ex. iv, 14; i.\. 5-fl, 12, IS, Logarithmic expression for dis- tanbe anil angto, 157. LogariUmiie-spherieal geometry, 15 Lublam, V\'., 4, Manifold, 194. Marginal images, 229. Maximum triangle, 7, SI ; Ex ii 19. quadrilateral, Ex. ii. II, 12. Medians. 139. Meiklk, H., 19, 22. Menelaus" theorem, 145. Middle point of segment. 138, 159. Milne, \V. I'., 202 ft. Minimal lines, 134 ; Ex. iv. 2. Mo bins' sheet, 91. Monodromy, 195. Motions. 28, 17!), IS5, 190, 248. Naj?ier, J., 08 n. rnli*, 08, 74, 119, 122. Net of rationality, 9li. Networks, Ex. ii. 11, IT), 18-20; iii. 9. Nine-point circle, Ex. iv. 25, 20. -Non-euclidcan geometry, 14, 20. Non-intorsectors, 12, 31, -10. Normal to conic, 201. One-sided sru i,ei<. u2. Order, 27. Orfhaxis, 54, 141. Otthoecntre, 54, III. Orthogonal points, 89. circles, 219 ; Ex. vii. 4. trajectory, 51, Onhopolo, Ex. iv, 1.!. .. 9g 75 UAp). 90, 38, 59; Ex. vii. II. Parabolas, 258. Parebolic geometry. 25. inversion of radiation, 249. involution, 97. Parallax, 204. Parallel angle, 30, 35, 5H. lines, 29, 30 ; Ex. ii. ; 133. planes, 50. postulate, Chap, i. ; 27, 203. Farotaxy, 103. PaBOH, M., 193. axiom, 29. Peano, G„ 193. Pencil of circles, 228, 233. of hoes, 45, 170. Pentagmmmo. mirifieum, 68, 08 n, 118. Perimeter of triangle, 104, 103. Perpendicular to a line, 28. to plane, 43. to eoplannr lines, 12, 40. ro two lines in apace, 100, 109. Perspective triangles, 142. PerspcctiviiY, 95 PrERi, ML, 193. Planes, 42. Pinion ists, 1. Playfair, J., 19. axiom, 4, 30. POINCAKK, H„ 210. Polar system, 92, 98. triangles, 102, 141. Pole and polar, 51, 89, 130 ; Ex. tv. 1. Polygon, area, 83, 104. regular, 80 m ; Ex. ii. 19-20. Polyhedrn, Ex. iii 7, 8, 12-15. Position-ratio, 147. Postulates, 2, 3, 27. Power of a point, 212. of 11 line, 210 Projection, 95, 148. gnomon io, 170. stereographic, 172. !Yejer-ii\-e geometry. 93-98, 135. group, 198. metric, 158. Pseudoaphere, 198. Pythagoras, 1. Quadrangle-quadrilateral con- figuration. 143. Quadrant, 89, 110. Quadrilateral, circumscribed, Ex. ». 3. complete, '48. inscribed, Ex. ii. 2. of maximum area, Ex. ii 11. 1 - tr-irectangukr, 70, 121. Quadrilateral -ei instruction, 96, 199. Quaternions, 251. Radian, 28. Radiation and inversion, 241 ; Ex. viii, 11, Radical axes, 219, 228. centres, 221. Radius of circle, 51. Kaiitmality, net of, 00. Reciprocity of parallelism, 32, Rectangle, 9. 79, 112. Reflexions. 238, 240. in Krrc, Geom., 180. Reid, T., 208 n. Relativity-principle, 197, 210. Reye, T., 94. Ribuakn, B., 26, 194. Rigid figure, 107, 195. Rotation proof of parallel -post u hue, 6. Rotations, 185. Russell, B., 193. Saccheri, G.. 11-13, 15, 18,24. Swit.E, H., 3. Sc-hweikaiit, F. K., 14. Sections of sphere, cic., 63. Segments, congruence of, 27. Sell-corresponding elements, 00. Semi-circular eonals of the ear, 20H. Semi-hyperbola, 257. Similar figure*, 7, 13, 1 runsformation, 239 n. Simson, R., 42. Sommbbviixe, D. M. Y., 24«, Space-constant, 58, construction for, 77. physical measurement of, 203. Space-fillings, Ex. ii. 14, 15, 18- 21 ; iii: 14. 15. Sphere, 62, 105. .if infinite radius, 15 <«er Horo- spliere). volume, Ex. ii, 24 ; iii. 6. Spherical geometry, 25, 89, 130, 138 » . triangles, Q3. trigonometry, 09, 70, 120, 274 INDEX Square, 2. St.ukbl, P., Jin, 13 », 23 n. Staudt, G. K. CIi. von, 103. IBB. Stereogruphie projection, 172. Straight lino, Euclid's dot,, 2. as shortest path, 17. ro-eutrant, 17. 19, 11)1. Superposition, 8, 27. Synimolry of space, 28. Systems of circles, 228, 232. Tactual space, 208. Tangents to two circles, 21 1. to a conic, 2U1. Tauki.vus, F. A., Iff. IVlriilH'tlniii, li.\. iv. 7, 8. dl Miiio, I 1 1 radius of eircunmpherc, Ex, iii. 7. with concurrent altitudes. F.\ iv. 24. Tiialks, I. Tii i iiaut, B. F., 6. TH.I.IMI-WIV. T IV'rronnt, 19, 22 Tilly, .1. 11. de, ]im«. Townsbno, E. J., 27 re. Transformations, circular, 181 ; CI lap. viii. conformal, 182; Chap, viii. congruent, 138, 238. 245. hom "graphic, 182, 24B. infinitesimal, IDS. of coordinates, 245. <>r inversion, ISO, is:!. 21 1. 2H2 Transitivity of paralleli-tn, 32. Transmiwibuity of parallelism, i Trims vo real theorem, 33. Triangle, angle-sum, 5, 10, 12, 13, 15, Hi, 18, 2«, 21. 77. 104. area, 13, 20, 77-79, 81, 82, 103- 104. associated, 05. 74, 117. exterior angle, If, 111, 2B, 34- 35. of maximum area, 7, 81. periinotcr, 104, 163. perspective, 142. points connected with, 139. rectilinear und spitoriciil, n:i. right-angled, (13, 1 Hi. side, 2D. trigonometry of. lili. 1 Mi. Tiiiingulation, 203. 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I). 6 volfi, Bucolics — Gears ii;s * -Eneid, 1*3— ^nuid, 4.6— .Entiitl, 7^— .Entdd. iu-r.-. XeilOpJlOn'S Au:ilinsis. TranslalKtl by the Rev. J, s, Watkjn, M.A. With Map. j voE. Books 1, and II. -Books III., IV., and V.— Books VI. and VII. llellcniis. Books I. and II. Trans- lated by the Rev. II. n a LB, M.A. Far ether Translations from tht Classics, ?tt the Catalogue ef Bokiis Libraries, ■which -mill he fbrwkrdtd en application MATHEMATICS FuU Catalogttc of Naihtmathal Books post fret en npptktslhn Cambridge Mathematical Series Public School Arithmetic. By W, M. Baker, M.A,, imd A* A. BoUB»«, M.A, js, Hd. Or with Aiiihwers, 4J. 6*/* THe Student's Aritbmeiiie. By w M« Uakf-R, M.A., and A. A* Bouhnb, M.A. With or without AflBsratlt, 2J. oV. New School Arithmetic. ByC ^rnui.h- wmy* M.A., and f. E. Romnstnt. M.A, W r jih Of wittujut Answers. 4J. eV/^ In Two Parts. 7t. fyt. ftflwtfe* Key W Pari 1 1., St. fu/. net. New school Examples in a wrpartittf volume, }*. 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