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ELEMENTS 
 
 OP 
 
 PROJECTIVE GEOMETRY 
 
 CREMONA 
 
HENRY FROWDE 
 
 Oxford University Press Warehouse 
 Amen Corner, E.C. 
 
ELEMENTS 
 
 OP 
 
 PROJECTIVE GEOMETRY 
 
 BY 
 
 LUIGI CREMONA 
 
 LL.D. EDIN., FOE. MEMB. E. S. LOND., HON. F.B.S. EDIN. 
 
 HON. MEMB. CAMB. PHIL. 80C. + 
 
 PEOFESSOB OF MATHEMATICS IN THE HNIVEESITT OF BOMB 
 
 TRANSLATED BY 
 
 CHARLES LEUDESDORF, M.A. 
 
 FELLOW OF PEMBEOKE COLLEGE, OXFOED 
 
 
 AT THE CLARENDON PRESS 
 1885 r 
 I All rights reserved ] 
 

 1T 
 
 £7 
 
 3<rfff 
 
UN 
 
 AUTHOR'S PREFACE TO THE 
 FIRST EDITION*. 
 
 Amplissima et pulcherrima scientia figurarum. At quam est inepte sortita 
 nomen Geometrise ! — Nicod. Frischlinus, Dialog. I. 
 
 Perspective methodus, qu& nee inter inventas nee inter inventu possibiles ulla 
 compendiosior esse videtur . . . — B. Pascal, Lit. ad Acad. Paris., 1654. 
 
 Da veniam scriptis, quorum non gloria nobis 
 
 Causa, sed utilitas officiumque fuit. — Ovid, ex Pont., iii. 9. 55. 
 
 This book is not intended for those whose high mission it 
 is to advance the progress of science ; they would find in it 
 nothing new, neither as regards principles, nor as regards 
 methods. The propositions are all old ; in fact, not a few of 
 them owe their origin to mathematicians of the most remote 
 antiquity. They may be traced back to Euclid (285 B.C.), to 
 Apollonius of Perga (247 B.C.), to Pappus of Alexandria (4th 
 century after Christ) ; to Desargues of Lyons (1593-1662); 
 to Pascal ( 1 623-1 662) ; to De la Hire ( 1640-17 18); to 
 Newton (1642-1 727) ; to Maclaurin (1698-1746); to J. H. 
 Lambert (1728-1777), &c. The theories and methods which 
 make of these propositions a homogeneous and harmonious 
 whole it is usual to call modern, because they have been dis- 
 covered or perfected by mathematicians of an age nearer to 
 ours, such as Carnot, Brianchon, Poncelet, Mobius, Steiner, 
 Chasles, Staudt, etc. ; whose works were published in the 
 earlier half of the present century. 
 
 Various names have been given to this subject of which we 
 are about to develop the fundamental principles. I prefer 
 
 * With the consent of the Author, only such part of the preface to the original 
 Italian edition ( t 87 2) is here reproduced as may be of interest to the English reader. 
 
 3^1<) 
 
vi author's preface to the first edition. 
 
 not to adopt that of Higher Geometry [Geometrie superieure, 
 holier e Geometrie), because that to which the title ' higher ' at 
 one time seemed appropriate, may to-day have become very 
 elementary; nor that of Modem Geometry (neuere Geometrie), 
 which in like manner expresses a merely relative idea ; and is 
 moreover open to the objection that although the methods 
 may be regarded as modern, yet the matter is to a great extent 
 old. Nor does the title Geometry of position [Geometrie der Lage) 
 as used by Staudt* seem to me a suitable one, since it 
 excludes the consideration of the metrical properties of figures. 
 I have chosen the name of Projective Geometry f, as expressing 
 the true nature of the methods, which are based essentially on 
 central projection or perspective. And one reason which has 
 determined this choice is that the great Poncelet, the chief 
 creator of the modern methods, gave to his immortal book 
 the title of Traite des proprietes projectives des figures (1822). 
 
 In developing the subject I have not followed exclusively 
 any one author, but have borrowed from all what seemed 
 useful for my purpose, that namely of writing a book which 
 should be thoroughly elementary, and accessible even to those 
 whose knowledge does not extend beyond the mere elements of 
 ordinary geometry. I might, after the manner of Staudt, 
 have taken for granted no previous notions at all ; but in that 
 case my work would have become too extensive, and would 
 no longer have been suitable for students who have read the 
 usual elements of mathematics. Yet the whole of what such 
 students have probably read is not necessary in order to 
 understand my book ; it is sufficient that they should know 
 the chief propositions relating to the circle and to similar 
 triangles. 
 
 It is, I think, desirable that theoretical instruction in 
 
 * Equivalent to the Descriptive Geometry of Caylet (SixtK memoir on quantics, 
 Phil. Trans, of the Royal Society of London, 1859; p. 90). The name Geome'trie 
 de position as used by Carnot corresponds to an idea quite different from that 
 which I wished to express in the title of my book. I leave out of consideration 
 other names, such as Geometrie segmentaire and Organische Geometrie, as referring 
 to ideas which are too limited, in my opinion. 
 
 f See Klein, Ueber die sogenannte nicht-Euklidische Geometrie (Gottinger 
 Nachrichten, Aug. 30, 1871). 
 
author's preface to the first edition. vii 
 
 geometry should have the help afforded it by the practical 
 constructing and drawing of figures. I have accordingly laid 
 more stress on descriptive properties than on metrical ones ; and 
 have followed rather the methods of the Geometrie der Lage of 
 Staudt than those of the Geometrie superieure of Chasles*. 
 It has not however been my wish entirely to exclude metrical 
 properties, for to do this would have been detrimental to 
 other practical objects of teaching f. I have therefore intro- 
 duced into the book the important notion of the anharmonic 
 ratio, which has enabled me, with the help of the few above- 
 mentioned propositions of the ordinary geometry, to establish 
 easily the most useful metrical properties, which are either 
 consequences of the projective properties, or are closely related 
 to them. 
 
 I have made use of central projection in order to establish 
 the idea of infinitely distant elements ; and, following the example 
 of Steiner and of Staudt, I have placed the law of duality 
 quite at the beginning of the book, as being a logical fact 
 which arises immediately and naturally from the possibility 
 of constructing space by taking either the point or the plane as 
 element. The enunciations and proofs which correspond to 
 one another by virtue of this law have often been placed in 
 parallel columns ; occasionally however this arrangement has 
 been departed from, in order to give to students the oppor- 
 tunity of practising themselves in deducing from a theorem 
 its correlative. Professor Eeye remarks, with justice, in the 
 preface to his book, that Geometry affords nothing so stirring 
 to a beginner, nothing so likely to stimulate him to original 
 work, as the principle of duality; and for this reason it is 
 very important to make him acquainted with it as soon 
 as possible, and to accustom him to employ it with con- 
 fidence. 
 
 The masterly treatises of Poncelet, Steiner, Chasles, and 
 
 * Cf. Reye, Geometrie der Lage (Hannover, t866; 2nd edition, 1877), p. xi. of 
 the preface. 
 
 f Cf. Zech, Die hbhere Geometrie in Hirer Anwendung auf Kegelschnitte und 
 Flachen zweit-er Ordnwng (Stuttgart, 1857), preface. 
 
V1U AUTHOR S PREFACE TO THE FIRST EDITION. 
 
 Staudt * are those to which I must acknowledge myself most 
 indebted ; not only because all who devote themselves to 
 Geometry commence with the study of these works, but also 
 because I have taken from them, besides the substance of 
 the methods, the proofs of many theorems and the solutions of 
 many problems. But along with these I have had occasion 
 also to consult the works of Apollonius, Pappus, Desargues, 
 De la Hire, Newton, Maclaurin, Lambert, Carnot, 
 Brianchon, Mobius, Bellavitis, &c. ; and the later ones of 
 Zech, Gaskin, Witzschel, Townsend, Reye, Poudra, 
 Fiedler, &c. 
 
 In order not to increase the difficulties, already very con- 
 siderable, of my undertaking, I have relieved myself from the 
 responsibility of quoting in all cases the sources from which 
 I have drawn, or the original discoverers of the various pro- 
 positions or theories. I trust then that I may be excused if 
 sometimes the source quoted is not the original onef, or if 
 occasionally the reference is found to be wanting entirely. 
 In giving references, my desire has been chiefly to call the 
 attention of the student to the names of the great geometers 
 and the titles of their works, which have become classical. 
 The association with certain great theorems of the illustrious 
 names of Euclid, Apollonius, Pappus, Desargues, Pascal, 
 Newton, Carnot, &c. will not be without advantage in assist- 
 ing the mind to retain the results themselves, and in exciting 
 that scientific curiosity which so often contributes to enlarge 
 our knowledge. 
 
 Another object which I have had in view in giving refer- 
 ences is to correct the first impressions of those to whom the 
 name Projective Geometry has a suspicious air of novelty. Such 
 
 * Poncelet, Traite des proprUUs projectives des figures (Paris, 1822). Steiner, 
 Systematische Entwickelung der Abhdngigkeit geometrischer Gestalten von einander, 
 &c. (Berlin, 1832). Chasles, Traite de Geometrie superieure (Paris, 1852); Traite 
 des sections coniques (Paris, 1865). Staudt, Geometrie der Lage (Nurnberg, 1847). 
 
 •f In quoting an author I have almost always cited such of his treatises as are 
 of considerable extent and generally known, although his discoveries may have 
 been originally announced elsewhere. For example, the researches of Chasles in 
 the theory of conies date from a period in most cases anterior to the year 1830; 
 those of Staudt began in 1831 ; &c. 
 
author's preface to the first edition. ix 
 
 persons I desire to convince that the subjects are to a great 
 extent of venerable antiquity, matured in the minds of the 
 greatest thinkers, and now reduced to that form of extreme 
 simplicity which Gergonne considered as the mark of perfection 
 in a scientific theory*. In my analysis I shall follow the 
 order in which the various subjects are arranged in the book. 
 
 The conception of elements lying at an infinite distance is due 
 to the celebrated mathematician Des argues ; who more than 
 two centuries ago explicitly considered parallel straight lines 
 as meeting in an infinitely distant point f, and parallel planes 
 as passing through the same straight line at an infinite 
 distance J. 
 
 The same idea was thrown into full light and made 
 generally known by Poncelet, who, starting from the postu- 
 lates of the Euclidian Geometry, arrived at the conclusion 
 that the points in space which lie at an infinite distance must 
 be regarded as all lying in the same plane §. 
 
 Des argues || and Newton 11 considered the asymptotes 
 of the hyperbola as tangents whose points of contact lie at an 
 infinite distance. 
 
 The name homology is due to Poncelet. Homology, with 
 reference to plane figures, is found in some of the earlier 
 treatises on perspective, for example in Lambert** or per- 
 haps even in Desargues tt> who enunciated and proved the 
 theorem concerning triangles and quadrilaterals in perspective 
 or homology. This theorem, for the particular case of two 
 triangles (Art. 1 7), is however really of much older date, as it 
 is substantially identical with a celebrated porism of Euclid 
 
 * ' On ne peut se flatter d' avoir le dernier mot d'une theorie, tant qu'on ne 
 peut pas l'expliquer en peu de paroles a u'n passant dans la rue' (cf. Chasles, 
 Apercu Mstorique, p. 115). 
 
 + (Euvres de Desargues, reunies et analysies par M. Poudra (Paris, 1864), 
 tome i. Brouillon-projet d'une atteinte aux ivinements des rencontres d'un cdne 
 avec unplan (1639), pp. 104, 105, 205. 
 
 % Loc. cit., pp. 105, 106. 
 
 § Traite des proprietes projectives des figures (Paris, 1822), Arts. 96, 580. 
 
 || Loc. dt., p. 210. 
 
 1T Philosophiae naturalis principia matliematica (1686), lib. i. prop. 27, scholium. 
 
 ** Freie Perspective, 2nd edition (Zurich, 1774). 
 
 tf Loc. cit., pp. 413-416. 
 
X AUTHOK S PREFACE TO THE FIRST EDITION. 
 
 (Art. 114), which has been handed down to us by Pappus*. 
 Homological figures in space were first studied by Pon- 
 
 CELET f. 
 
 The law of duality, as an independent principle, was enun- 
 ciated by Gergonne % ; as a consequence of the theory of 
 reciprocal polars (under the name jjrincipe de reciprocite polaire) 
 it is due to Poncelet§. 
 
 The geometric forms (range of points, flat pencil) are found, 
 the names excepted, in Desargues and the later geometers. 
 Steiner II has defined them in a more explicit manner than 
 any previous writer. 
 
 The complete quadrilateral was considered by CarnotIT; 
 the idea was extended by Steiner ** to polygons of any 
 number of sides and to figures in space. 
 
 Harmonic section was known to geometers of the most 
 remote antiquity ; the fundamental properties of it are to be 
 found for example in Apollonius ft- De la Hire \% gave the 
 construction of the fourth element of a harmonic system by 
 means of the harmonic property of the quadrilateral, i. e. by 
 help of the ruler only. 
 
 From 1832 the construction of projective forms was taught 
 by Steiner §§. 
 
 The complete theory of the anharmonic ratios is due to 
 Mobius mi, but before him Euclid, Pappus! If, Desargues****, 
 and Brianchon ttt had demonstrated the fundamental pro- 
 position of Art. 63. Desargues %%% was the author of the theory 
 
 * Chasles, Les trots livres deporismes d'Euclide, &c. (Paris, i860), p. 102. 
 
 t Loc. cit., pp. 369 sqq. 
 
 X Annates de Mathdmatiques, vol. xvi. (Montpellier, 1826), p. 209. 
 
 § Ibid., vol. viii. (Montpellier, 1818), p. 201. 
 
 || Systematische Entwichelung, pp. xiii, xiv. Collected Works, vol. i. p. 237. 
 
 f De la correlation des figures de Geome'trie (Paris, 1801), p. 122. 
 
 ** Loc. cit., pp. 72, 235; §§ 19, 55. 
 
 +t Conicorum lib. i. 34, 36, 37, 38. 
 
 XX Sectiones conicae (Parisiis, 1685), i. 20. 
 
 §§ Loc. cit, p. 91. 
 
 IHI Der barycentrische Calcul (Leipzig, 1827), chap. v. 
 
 ITIf Mathematicae Collectiones, vii. 129. 
 
 *** Loc. cit., p. 425. 
 
 ttt Mimoire sur les lignes du second ordre (Paris, 181 7), p. 7. 
 
 XX+ Loc. cit., pp. 119, 147, 171, 176. 
 
AUTHORS PREFACE TO THE FIRST EDITION. XI 
 
 of involution, of which a few particular cases were already- 
 known to the Greek geometers *, 
 
 The generation of conies by means of two projective forms 
 was set forth, forty years ago, by Steiner and by Chasles ; 
 it is based on two fundamental theorems (Arts. 149, 150) 
 from which the whole theory of these important curves can 
 be deduced. The same method of generation includes the 
 organic description of Newton f and various theorems of 
 Maclaurin. 
 
 But the projectivity of the pencils formed by joining two 
 fixed points on a conic to a variable point on the same had 
 already been proved, in other words, by Apollonius %. 
 
 When only sixteen years old (in 1640) Pascal discovered 
 his celebrated theorem of the mystic hexagram §, and in 1 806 
 Brianchon deduced the correlative theorem (Art. 153) by 
 means of the theory of pole and polar. 
 
 The properties of the quadrilateral formed by four tangents 
 to a conic and of the quadrangle formed by their points of 
 contact are to be found in the Latin appendix (De linea- 
 rum geometricarum projorietatibus generalibus tractatus) to the 
 Algebra of Maclaurin, a posthumous work (London, 1748). 
 He deduced from these properties methods for the con- 
 struction of a conic by points or by tangents in several cases 
 where five elements (points or tangents) are given. This 
 problem, in its full generality, was solved at a later date by 
 Brianchon. 
 
 The idea of considering two projective ranges of points on 
 the same conic was explicitely set forth by Bellavitis ||. 
 
 To Carnot^[ we owe a celebrated theorem (Art. 385) con- 
 cerning the segments which a conic determines on the sides of 
 
 * Pappus, Mathematical Collectiones, lib. vii. props. 37-56, 127, 128, 130-133. 
 
 + Loc. cit., lib. i. lemma xxi. 
 
 t Conicorum lib. iii. 54, 55, 56. I owe this remark to Prof. Zeuthen (1885). 
 
 § Letter of Leibnitz to M. Perier in the (Euvres de B. Pascal (Bossut's 
 edition, vol. v. p. 459). 
 
 || Saggio di geometria derivata (Nuovi Saggi dell' Accademia di Padova, vol. 
 iv, 1838), p. 270, note. 
 
 U Geometrie de position (Paris, 1803), Art. 379. 
 
xii author's preface to the first edition. 
 
 a triangle. Of this theorem also certain particular cases were 
 known long before *. 
 
 In the Freie Perspective of Lambert we meet with elegant 
 constructions for the solution of several problems of the first 
 and second degrees by means of the ruler, assuming however 
 that certain elements are given ; but the possibility of solviDg 
 all problems of the second degree by means of the ruler and a 
 fixed circle was made clear by Poncelet ; afterwards Steiner, 
 in a most valuable little book, showed the manner of practically 
 carrying this out (Arts. 238 sqq.). 
 
 The theory of pole and polar was already contained, under 
 various names, in the works already quoted of Desargues f 
 and De la Hire J ; it was perfected by Monge §, Brian- 
 chon ||, and Poncelet. The last-mentioned geometer derived 
 from it the theory of polar reciprocation, which is essentially 
 the same thing as the law of duality, called by him the ' prin- 
 cipe de reciprocite polaire.' 
 
 The principal properties of conjugate diameters were ex- 
 pounded by Apollonius in books ii and vii of his work on 
 the Conies. 
 
 And lastly, the fundamental theorems concerning foci are to 
 be found in book iii of Apollonius, in book vii of Pappus, 
 and in book viii of De la Hire. 
 
 Those who desire to acquire a more extended and detailed 
 knowledge of the progress of Geometry from its beginnings 
 until the year 1 830 (which is sufficient for what is contained 
 in this book) have only to read that classical work,Hhe Apergu 
 historique of Chasles. 
 
 * Apollonius, Conicorum lib. iii. 16-23. Desargues, loc. cit., p. 202. De 
 la Hire, loc. cit., book v. props. 10, 12. Newton, Enumeratio linearum tertii 
 ordinis (OpticJcs, London, 1704), p. 142. 
 
 f Loc. cit., pp. 164, 186, 190, sqq. 
 
 X Loc. cit., i. 21-28; ii. 23-30* 
 
 § Oeome'trie descriptive (Paris, 1 795), Art. 40. 
 
 II Journal de VEcole Poly technique, cahier xiii. (Paris, 1806). 
 
AUTHOR'S PREFACE TO THE 
 ENGLISH EDITION. 
 
 In April last year, when I was in Edinburgh on the occasion 
 of the celebration of the tercentenary festival of the University 
 there, Professor Sylvester did me the honour of saying that in 
 his opinion a translation of my book on the Elements of Projec- 
 tive Geometry might be useful to students at the English Uni- 
 versities as an introduction to the modern geometrical methods. 
 The same favourable judgement was shown to me by other 
 mathematicians, especially in Oxford, which place I visited in 
 the following month of May at the invitation of Professor Syl- 
 vester. There Professor Price proposed to me that I should 
 assist in an English translation of my book, to be carried out 
 by Mr. C. Leudesdorf, Fellow of Pembroke College, and to be 
 published by the Clarendon Press. I accepted the proposal 
 with pleasure, and for this reason. In my opinion the English 
 excel in the art of writing text-books for mathematical teach- 
 ing ; as regards the clear exposition of theories and the 
 abundance of excellent examples, carefully selected, very few 
 books exist in other countries which can compete with those 
 of Salmon and many other distinguished English authors that 
 could be named. I felt it therefore to be a great honour that 
 my book should be considered by such competent judges 
 worthy to be introduced into their colleges. 
 
 Unless I am mistaken, the preference given to my Elements 
 over the many treatises on modern geometry published on the 
 Continent is to be attributed to the circumstance that in it I 
 have striven, to the best of my ability, to imitate the English 
 
XIV AUTHOR S PREFACE TO THE ENGLISH EDITION. 
 
 models. My intention was not to produce a book of high 
 theories which should be of interest to the advanced mathe- 
 matician, but to construct an elementary text-book of modest 
 dimensions, intelligible to a student whose knowledge need not 
 extend further than the first books of Euclid. I aimed there- 
 fore at simplicity and clearness of exposition; and I was 
 careful to supply an abundance of examples of a kind suitable 
 to encourage the beginner, to make him seize the spirit of the 
 methods, and to render him capable of employing them. 
 
 My book has, I think, done some service in Italy by helping 
 to spread a knowledge of projective geometry; and I am 
 encouraged to believe that it has not been unproductive of 
 results even elsewhere, since I have had the honour of seeing 
 it translated into French and into German. 
 
 If the present edition be compared with the preceding ones, 
 it will be seen that the book has been considerably enlarged 
 and amended. All the improvements which are to be found 
 in the French and the German editions have been incor- 
 porated ; a new Chapter, on Foci, has been added ; and every* 
 Chapter has received modifications, additions, and elucidations, 
 due in part to myself, and in part to the translator. 
 
 In conclusion, I beg leave to express my thanks to the 
 eminent mathematician, the Savilian Professor of Geometry, 
 who advised this translation ; to the Delegates of the 
 Clarendon Press, who undertook its publication; and to 
 Mr. Leudesdorf, who has executed it with scrupulous 
 fidelity. 
 
 L. CREMONA. 
 Rome, May 1885. 
 
TABLE OF CONTENTS. 
 
 Peeface 
 
 PAGE 
 V 
 
 CHAPTER I. 
 Definitions (Arts. 1-7) . i 
 
 CHAPTER II. 
 
 CENTRAL PROJECTION; FIGURES IN PERSPECTIVE. 
 
 Figures in perspective (Arts. 8-11) ...... 3 
 
 Point at infinity on a straight line (12) - . . . . . 4 
 
 Line at infinity in a plane (13) -5 
 
 Triangles in perspective; theorems of Desargues (14-17) . 6 
 
 CHAPTER III. 
 
 HOMOLOGY. 
 
 Figures in homology (Arts. 18-21) 9 
 
 Locus of the centre of perspective of two figures, when one is 
 
 turned round the axis of perspective (22) . . . .12 
 
 Construction of homological figures (23) 13 
 
 Homothetic figures (23) . . . . . . . . 18 
 
 CHAPTER IV. 
 
 HOMOLOGICAL FIGURES IN SPACE. 
 
 Relief-perspective (Art. 24) 20" 
 
 Plane at infinity (26) . . . . . . . .21 
 
XVI TABLE OP CONTENTS. 
 
 CHAPTER V. 
 
 GEOMETRIC FORMS. 
 
 PAOK 
 
 The geometric prime-forms (Arts. 27-31) . . . .22 
 Their dimensions (32) 24 
 
 CHAPTEE VI. 
 
 THE PRINCIPLE OF DUALITY. 
 
 Correlative figures and propositions (Arts. 33-38) . . .26 
 CHAPTER VII. 
 
 PROJECTIVE GEOMETRIC FORMS. 
 
 First notions (Arts. 39-41) 33 
 
 Forms in perspective (42, 43) . . . . . . -35 
 
 Fundamental theorems (44,45) 36 
 
 CHAPTER VIII. 
 
 HARMONIC FORMS. 
 
 Fundamental theorem (Arts. 46, 49) 39 
 
 Harmonic forms are projective (47, 48, 50, 51) . . . -41 
 Elementary properties (52-57) ...... 44 
 
 * Constructions (58-60) 47 
 
 CHAPTER IX. 
 
 ANHARMONIC RATIOS. 
 
 Distinction between metrical and descriptive (Art. 61) . . 50 
 Rule of signs : elementary segment-relations (62) . . .50 
 Theorem of Pappus, and converse (63-66) . . . .52 
 
 Properties of harmonic forms (68-71) . . . . • 57 
 
 T J3 twenty-four anharmonic ratios of a group of four elements (72) 59 
 
 In two projective forms, corresponding groups of elements are ^ 
 
 [ equianharmonic (73) . . . . . . .62 
 
 Metrical property of two projective ranges (74) . . .62 
 
 Properties of two homological figures (75-77) . . . -63 
 
TABLE OF CONTENTS. xvii 
 
 CHAPTER X. 
 
 CONSTRUCTION OF PROJECTIVE FORMS. 
 
 PAOE 
 
 Two forms are projective if corresponding groups of elements are 
 
 equianharmonic (Art. 79) . . . . . . .66 
 
 Forms in perspective ; self-corresponding elements (80) . .67 
 Superposed projective forms (81, 82) . . . . .68 
 
 A geometric form of four elements, when harmonic (83) . . 69 
 Constructions (84-86, 88-90) . . . . . . . 70 
 
 Hexagon whose vertices lie on two straight lines ; theorem of 
 
 Pappus (87) 75 
 
 Properties of two projective figures (91-94) . . . -79 
 
 Construction of projective plane figures (95, 96) . . . 8 r 
 Any two such figures can be placed so as to be in homology (97) 84 
 
 CHAPTER XI. 
 
 PARTICULAR CASES AND EXERCISE?. 
 
 Similar ranges and pencils (Arts. 99-103) . . . .86 
 
 Equal pencils (104-108) . . . ■ 89 
 
 Metrical properties of two collinear projective ranges (109) ' . 91 
 
 Examples (110-112) 93 
 
 Porisms of Euclid and of Pappus (113, 114) . . . -95 
 Problems solved with the ruler only (115-118) . . .96 
 
 Figures in perspective ; theorems of Chasles (119, 120) . . 98 
 
 CHAPTER XII. 
 
 involution. 
 
 Definition; elementary properties (Arts. 121-124) . . . too 
 Metrical property ; double elements (125, 126) . . .102 
 Two pairs of conjugate elements determine an involution (127) 104 
 The two kinds of involutions (128) ..... 105 
 
 b 
 
XV111 TABLE OF CONTENTS. 
 
 PAGE 
 
 Another metrical property (130) . . . . . .106 
 
 Property of a quadrangle cut by a transversal (131, 132) . 107 
 
 The middle points of the diagonals of a complete quadrilateral 
 
 are collinear (133) ........ 108 
 
 Constructions (134) ........ 109 
 
 Theorems of Ceva and Menelaus (136-140) . . . .110 
 
 Particular cases (142) .113 
 
 CHAPTER XIII. 
 
 PKOJECTIVE POEMS IN KELATION TO THE CIECLE. 
 
 Circle generated by two directly equal pencils (Art. 143) . 114 
 
 Fundamental property of points on a circle (144) . . .114 
 
 \s Fundamental property of tangents to a circle (146) . . .115 
 
 Harmonic points and tangents (145, 147, 148) . . .115 
 
 CHAPTER XIV. 
 
 PROJECTIVE FORMS IN RELATION TO THE CONIC SECTIONS. 
 
 Fundamental theorems (Art. 149) 
 
 Generation of conies by means of two projective forms (150) 
 ^ Anharmonic ratio of four points or tangents of a conic (151) 
 
 Five points or five tangents determine a conic (152) 
 / Theorems of Pascal and Brianchon (153, 154) . 
 
 Theorems of Mobius (155) and Maclaurin (156) . 
 
 Properties of the parabola (157, 158) .... 
 
 CHAPTER XV. 
 constructions and exercises. 
 
 118 
 119 
 
 122 
 
 123 
 
 124 
 126 
 
 127 
 
 Properties of the hyperbola ; theorems of Apollonius (158-160) 129 
 
 Pascal's and Brianchon's theorems applied to the construction 
 
 of a conic by points or by tangents (Art. 161) . . . 131 
 
 Cases in which one or more of the elements lie at infinity (162, 
 
 163) 132 
 
TABLE OF CONTENTS. XIX 
 
 CHAPTER XVI. 
 
 DEDUCTIONS EBOM THE THEOBEMS OF PASCAL AND BEIANCHON. 
 
 PAGE 
 
 Theorem on the inscribed pentagon (Art. 164) . . -136 
 
 Application to the construction of conies (165) . . . 137 
 
 Theorem on the inscribed quadrangle (166, 168) . . .138 
 Application to the construction of conies (167, 168) . . 139 
 
 The circumscribed quadrilateral and the quadrangle formed by 
 
 the points of contact of the sides (169-173) . . . 140 
 
 Theorem on the inscribed triangle (174) . . . . 143 
 
 Application to the construction of conies (175) . . . 143 
 
 The circumscribed triangle and the triangle formed by the 
 
 points of contact of the sides (176-178) . . . 144 
 
 Theorem on the circumscribed pentagon (179) . . 145 
 
 Construction of conies subject to certain conditions (180-182) . 146 
 
 CHAPTER XVII. 
 
 DESAEGUES' THEOEEM. 
 
 Desaegues' theorem and its correlative (Art. 183) . . .148 
 Conies circumscribing the same quadrangle, or inscribed in the 
 
 same quadrilateral (184) 149 
 
 Theorems of Poncelet (186-188) 151 
 
 Deductions from Desaegues' theorem (189-194) . . . 152 
 Group of four harmonic points or tangents (195, 196) . 157 
 
 Property of the hyperbola (197) 158 
 
 Theorem 'ad quatuor tineas' quoted by Pappus (198) . .158 
 Correlative theorem (199) . 159 
 
 CHAPTER XVIII. 
 
 SELF-COEEESPONDING ELEMENTS AND DOUBLE ELEMENTS. 
 
 Projective ranges of points on a conic (Art. 200) . . 161 
 
 Projective series of tangents to a conic (201) . . . .163 
 
 Involution of points or tangents of a conic (202, 203) . . 165 
 
TABLE OF CONTENTS. 
 
 PAGl 
 
 Harmonic points and tangents (204, 205) . . . .168 
 
 Construction of the self-corresponding elements of two super- 
 posed projective forms, and of the double elements of an 
 involution (206) . . . . . . . .169 
 
 Orthogonal pair of rays of a pencil in involution (207) . .172 
 Construction for the common pair of two superposed involutions 
 
 (208,209) 173 
 
 Other constructions (210, 211) . . . . . .174 
 
 CHAPTER XIX. 
 
 PROBLEMS OF THE SECOND DEGEEE. 
 
 Construction of a conic determined by five points or tangents 
 
 (Arts. 212,213, 216, 217) 176 
 
 Particular cases (214, 215) 178 
 
 Construction of a conic determined by four points and a tangent, 
 
 or by four tangents and a point (218) . . . .180 
 
 Case of the parabola (219, 220) 181 
 
 Construction of a conic determined by three points and two tan- 
 gents, or by three tangents and two points (221) . .182 
 Construction of a polygon satisfying certain conditions (222- 
 
 225) 184 
 
 Construction of the points of intersection, and of the common 
 
 tangents, of two conies (226-230) ..... 188 
 
 Various problems (231-236) . . . . . . .190 
 
 Geometric method of false position (237) . . . 193 
 
 Solution of problems of the second degree by means of the ruler 
 
 and a fixed circle (238) . . . . . . . 194 
 
 Examples of problems solved by this method (239-249) . .194 
 
 CHAPTER XX. 
 
 POLE AND POLAR. 
 
 Definitions and elementary properties (Arts. 250-254) . .201 
 Conjugate points and lines with respect to a conic (255, 256) . 204 
 Constructions (257) 205 
 
TABLE OF CONTENTS. XXI 
 
 Self-conjugate triangle (258-262) . 
 
 Involution of conjugate points and lines (263, 264) 
 
 Complete quadrangles and quadrilaterals having the same 
 
 diagonal triangle (265-267) 
 
 Conies having a common self-conjugate triangle (268, 269) 
 Properties of conies inscribed in the same quadrilateral, or cir 
 
 cumscribing the same quadrangle (270-273) . 
 Properties of inscribed and circumscribed triangles (274, 275) 
 
 CHAPTER XXI. 
 
 THE CENTKE AND DIAMETERS OF A CONIC. 
 
 The diameter of a system of parallel chords (Arts. 276-278) 
 
 Case of the parabola (279-280) 
 
 Centre of a conic (281-283, 285) . 
 
 Conjugate diameters (284, 286-288, 290) 
 
 Case of the circle (289) .... 
 
 Theorem of Mobius (291) 
 
 Involution property of a quadrangle inscribed in a conic (292) 
 
 Ideal diameters and chords (290, 294) . 
 
 Involution of conjugate diameters : the axes (296-298) . 
 
 Various properties of conjugate diameters ; theorems of Apol- 
 
 lonius (299-315) .... 
 Conies inscribed in the same quadrilateral ; theorem of Newton 
 
 (317, 318) 
 
 Constructions (285, 290, 293, 301, 307, 311, 316, 319) 
 
 206 
 209 
 
 210 
 
 212 
 
 213 
 215 
 
 217 
 218 
 218 
 219 
 222 
 224 
 224 
 226 
 227 
 
 228 
 
 236 
 
 238 
 
 CHAPTER XXII. 
 
 POLAR RECIPROCAL FIGURES. 
 
 Polar reciprocal curves (Arts. 320, 321) 239 
 
 The polar reciprocal of a conic with regard to a conic (322, 323) 240 
 
 Polar reciprocal figures are correlative figures (324, 325) . . 241 
 Two triangles which are self-conjugate with regard to the same 
 
 conic (326) 242 
 
XX11 TABLE OF CONTENTS. 
 
 PAGK 
 
 Triangles self-conjugate with regard to one conic, and inscribed 
 
 in, or circumscribed to, another (327-329) . . -243 
 
 Two triangles circumscribed to the same conic (330, 331) . 244 
 Triangles inscribed in one conic and circumscribed to another 
 
 (332) 244 
 
 Hesse's theorem (334) 245 
 
 Reciprocal triangles with regard to a conic are homological (336) 246 
 Conic with regard to which two given triangles are reciprocal 
 
 (338) 247 
 
 Polar system (339) . . ,. • 248 
 
 CHAPTER XXIII. 
 
 FOCI. 
 
 Foci of a conic defined (Arts. 340-341) 249 
 
 The involution determined by pairs of orthogonal conjugate rays 
 
 on each of the axes (342) ...... 250 
 
 The foci are the double points of this involution (343) , . . 251 
 Focal properties of tangents and normals (344-346, 349, 360- 
 
 362) 252 
 
 The circle circumscribing the triangle formed by three tangents 
 
 to a parabola passes through the focus (347) . . -253 
 
 The directrices (348, 350, 351) 254 
 
 The latus rectum (352) 257 
 
 The focal radii (353) • 258 
 
 The eccentricity (354, 355) 259 
 
 Locus of the feet of perpendiculars from a focus on the tangents 
 
 to a conic (356-359) . 260 
 
 Constructions (363, 364, 366, 367) 264 
 
 Confocal conies (365) . . . . . . . .266 
 
 Locus of intersection of orthogonal tangents to a conic (368, 369) 268 
 Property of the director circle in relation to the self-conjugate 
 
 triangles of a conic (370-375) . ..... 270 
 
 The polar reciprocal of a circle with respect to a circle (376-379) 273 
 
TABLE OF CONTENTS. XX111 
 
 CHAPTER XXIV. 
 
 COKOLLARIES AND CONSTRUCTIONS. 
 
 Various properties and constructions connected with the hyper- 
 bola and the parabola (Arts. 380-384) 
 
 Caenot's theorem, and deductions from it (385-389, 391) 
 
 Constructions of conies (390, 392-394) .... 
 
 The rectangular hyperbola (395) 
 
 Method of determining to which kind of conic a given arc be 
 longs (396) 
 
 Constructions of conies (397-404) ..... 
 
 Trisection of an arc of a circle (406) .... 
 
 Construction of a conic, given three tangents and two points, or 
 three points and two tangents (408) 
 
 Newton's organic description of a conic (409, 410) . 
 
 Various problems and theorems (411-418, 421) 
 
 Problems of the second degree solved by means of the theory of 
 pole and polar (419) 
 
 Problems solved by the method of polar reciprocation (420) 
 
 Exercises (422) 
 
 Index 303 
 
 277 
 279 
 284 
 
 285 
 
 289 
 289 
 294 
 
 295 
 296 
 297 
 
 300 
 301 
 302 
 
UNIVEB 
 
 ELEMENTS OF PROJECTIVE GEOMETRY. 
 
 CHAPTEK I. 
 
 DEFINITIONS. 
 
 1. By a figure is meant any assemblage of points, straight 
 lines, and planes ; the straight lines and planes are all to be 
 considered as extending to infinity, without regard to the 
 limited portions of space which are enclosed by them. By 
 the word triangle, for example, is to be understood a system 
 consisting of three points and three straight lines connecting 
 these points two and two ; a tetrahedron is a system consisting 
 of four planes and the four points in which these planes inter- 
 sect three and three, &c. 
 
 In order to secure uniformity of notation, we shall always denote 
 points by the capital letters A,B ,C , ... , straight lines by the small 
 letters a , b , c , ... , planes by the Greek letters a , /3 , y , ... . Moreover, 
 AB will denote that part of the straight line joining A and B which 
 is comprised between the points A and B ; Aa will denote the plane 
 which passes through the point A and the straight line a ; aa the 
 point common to the straight line a and the plane a; a/3 the, straight 
 line formed by the intersection of the planes a, £ ; ABC the : plane of 
 the three points A , B , C ; afiy the point common to the three pJrne.s 
 a , /3 , y ; a.BC the point common to the plane a and the straight line 
 BC ; A.fiy the plane passing through the point A- and the straight 
 line /3y ; a. Be the straight line common to the plane a and the plane 
 Be ; A. fie the straight line joining the point A to the point fie, &c. 
 The notation a.BC = A / we shall use to express that the point common 
 to the plane a and the straight line BC coincides with the point A' ; 
 u — ABC will express that the straight line u contains -the points 
 A,B,C,&c. 
 
 2. To project from a fixed point 8 (the centre of projection) a 
 figure ABCD..., abed...) composed of points and straight 
 lines, is to construct the straight lines or projecting rays 
 
2 DEFINITIONS. [3-7, 
 
 8 A , 8B , SC , #Z) , ... and the planes (projecting planes) 
 8a ,Sb,Sc ,Sd , ... . We thus obtain a new figure composed of 
 straight lines and planes which all pass through the centre 8. 
 
 3. To cut by a fixed plane a (transversal plane) a figure 
 (a/3yS,... abed...) made up of planes and straight lines, is to 
 construct the straight lines or traces <ra , aft , <ry , ... and the 
 points or traces era ,<rb ,vc,... . By this means we obtain a new 
 figure composed of straight lines and points lying in the 
 plane <r. 
 
 4. To project from a fixed straight line s (the axis) a figure 
 ABC I) ... composed of points, is to construct the planes sA,sB , 
 
 sC, The figure thus obtained is composed of planes which 
 
 ^.pass through the axis s. 
 
 5. To cut by a fixed straight line s (a transversal) a figure aftyb . . . 
 composed of planes, is to construct the points sa,s(3 ,sy,... . In 
 this way a new figure is obtained^ composed of points all lying 
 on the fixed transversal s. 
 
 6. If a figure is composed of straight lines a,b;c ,... which all 
 pass through a fixed point or centre 8, it can be projected from 
 a straight line or dwmYpassing through S; the result is a figure 
 composed of planes sa,sb,sc,... . 
 
 7. If a figure is composed of straight lines a , b , c , . . . all lying 
 in a fixed plane, it may be cut by a straight line (transversal) 
 s lying in the same plane ; the figure which results is formed 
 by the points sa^b^sc,...*. 
 
 * The operations of projecting and cutting {projection and section) are the two 
 fundatafcrital oies of the' Projective Geometry. 
 
CHAPTEK II. 
 
 CENTRAL PROJECTION; FIGURES IN PERSPECTIVE. 
 
 Fig. i. 
 
 8. Consider a plane figure made up of points A,B,C ,... and 
 straight lines AB, AC,..., BC, .... Project these from a centre 
 8 not lying in the plane (a-) of the figure, and cut the rays 
 8A,SB,SC,... and the planes SAB,SAC,...,SBC ,... by a trans- 
 versal plane a (Fig. i). The traces on the plane </ of the 
 projecting rays and planes will 
 
 form a second figure, a picture 
 
 of the first. When we carry 
 
 out the two operations by which 
 
 this second figure is derived 
 
 from the first, we are said to 
 
 project from a centre (or vertex) 8 
 
 a given figure a upon -a plane of 
 
 projection </. The new figure a 
 
 is called the perspective image -or- 
 
 the central projection of the 
 
 original one. Of course, if the second figure be projected 
 
 back from the centre 8 upon the plane a, the first figure will 
 
 be formed again; i.e. the first figure is the projection of the 
 
 second from the centre 8 upon the picture-plane o\ The two 
 
 figures o- and a are said to be in perspective position, or simply 
 
 in perspective. 
 
 9. If A', B', C',... are the traces of the rays SA,SB,SC,... on 
 the plane </, we may say that to the points A,B,C ,... of the 
 first figure correspond the points A', B', C", ... of the second, 
 with the condition that two corresponding points always lie 
 on a straight line passing through 8. If the point A describe 
 a straight line a in the plane a, the ray SA will describe a 
 plane 8a ; and therefore'^' will describe a straight line a', the 
 intersection of the planes 8a and </. The straight lines a and a' , 
 
 B 2 
 
4 CENTRAL PROJECTION; FIGURES IN PERSPECTIVE. [10 
 
 in which the planes a and </ are cut by^tngrjthe same project- 
 ing plane, may thus be called corresponding lines. It follows 
 from this that to the straight lines AB,AC, . . . , BC, . . . correspond 
 the straight lines A'B\ A'C\ ..., B'C\ ..-. and that to all 
 straight lines which pass through a given point A of the plane a- 
 correspond straight lines which pass through the corresponding 
 point A' of the plane a. 
 
 10. If the point A describe a curve in the plane a, the 
 corresponding point A' will describe another curve in the 
 plane <r', which may be said to correspond to the first curve. 
 Tangents to the two curves at corresponding points are clearly 
 corresponding straight lines ; and again, the two curves are cut 
 by corresponding straight lines in corresponding points. Two 
 corresponding curves are therefore of the same degree*. 
 
 11. The two figures may equally well be generated by the 
 simultaneous motion of a pair of corresponding straight lines 
 a , a'. If a revolve about a fixed point A, then a' will always 
 pass through the corresponding point A ' . 
 
 Similarly, if a envelop a curve, then a will envelop the 
 corresponding curve. The lines a and a\ in corresponding 
 
 positions, touch the two curves at 
 corresponding points ; and again, to 
 the tangents to the first curve from 
 a point A correspond the tangents to 
 the second from the corresponding 
 point A '. Two corresponding curves 
 are therefore of the same class f- 
 
 12. Consider two straight lines 
 a and a' which correspond to one 
 aE&other in the figures a , a (Fig. 2). 
 Every ray drawn through S in 
 their plane meets them in two 
 points, say A and A\ which cor- 
 respond to one another. If the ray 
 change its position and revolve round S, the points A and A ' 
 change their positions simultaneously ; when the ray is about to 
 
 * The degree of a curve is the greatest number of points in which it can be cut 
 by any arbitrary plane. In the case of a plane curve, it is the greatest number 
 of points in which it can be cut by any straight line in the plane. 
 
 ■\ The class of a plane curve is the greatest number of tangents which can be 
 drawn to it from any arbitrary point in the plane. 
 
13] CENTRAL PROJECTION ; FIGURES IN PERSPECTIVE. 5 
 
 become parallel to &, the point A' approaches I' (the point 
 where a' is cut by the straight line drawn through 8 parallel to 
 a) and the point A moves away indefinitely. In order that the 
 property that to one point of a! corresponds one point of a 
 may hold universally, we say that the line a has a point at 
 infinity I T with which the point A coincides when A/ coincides 
 with /', viz. when the ray, turning about 8, becomes parallel 
 to a. The straight line a has only one goint at infinity, it 
 being assumed that we can draw through 8 only one ray 
 parallel to a *; 
 
 The point I\ the image of the point at infinity I, is called 
 \ the vanishing point of a'. 
 
 Similarly, the straight line a' has a point J' at infinity, 
 which corresponds to the point / where a is cut by the ray\ 
 drawn through 8 parallel to a'. 
 
 Two parallel straight lines have the same point at infinity. 
 All straight lines which are parallel to a given straight line 
 must be considered as having a common point of intersection 
 at infinity. 
 
 Two straight lines lyifig in the same plane always intersect 
 in a point (finite or infinitely distant). 
 
 J 13. If now the straight line a takes all possible positions in 
 I the plane <r, the corresponding straight line a' will always be 
 J determined by the intersection of the planes </ and 8a. As a 
 moves, the ray 81 traces out a plane it parallel to or and the 
 point /' describes the straight line ira', which we may denote 
 by i' '. This straight line i / is then such that to any point lying 
 on it corresponds a point at infinity in the plane o-, which point 
 belongs also to the plane it. 
 
 We assume that the locus of. these points at infinity in the 
 plane a is a straight line i because it may be considered as 
 the intersection of the planes it and cr. But this "locus must 
 correspond to the straight line i'm the plane a ; thus the law 
 that to every straight line in the plane v' corresponds a straight 
 line in the plane o- holds without exception. 
 
 The plane o: has only one straight line at infinity, because 
 through the point .8 only one plane parallel to o- can be drawn. 
 The straight line i\ the image of the straight line at infinity, 
 is called the vanishing line of <j . It is parallel to era. 
 
 * This is one of the fundamental hypotheses of the Euclidian Geometry. 
 
6 CENTRAL PROJECTION; FIGURES IN PERSPECTIVE. [14 
 
 In the same way, the plane </ has a straight line at infinity 
 which corresponds to the intersection of the plane o- with the 
 plane it' drawn through S parallel to </. 
 
 Two parallel planes have the same straight line at infinity 
 in common. All planes parallel to a given plane must be 
 considered as passing through a fixed straight line at infinity. 
 
 If a straight line is parallel to a plane, the straight line at 
 infinity in the plane passes through the point at infinity on 
 the line. If two straight lines are parallel, they meet in the 
 same point the straight line at infinity in their plane. 
 
 Two planes always cut one another in a straight line (finite 
 or infinitely distant). 
 
 A straight line and a plane (not containing the line) always 
 intersect in a point (finite or infinitely distant). 
 
 Three planes which do not contain the same straight line 
 have always a common point (finite or infinitely distant) . 
 / 14. Theorem. If 'two plane figures ABC ..., A' 'B'C ...,(Fig. i) 
 
 /lying in different planes a and a, are in perspective, i. e. if the rays 
 AA\ BB', CC f ,. . . meet in a point 0, then the corresponding straight 
 lines AB and A'B\ AC and A'C',..., BC and B'C\... will cut 
 one another in points lying on the same straight line, viz. the inter- 
 section of the planes of the two figures. 
 
 " It is to be shown that if M is a point lying on the 
 straight line oV, and if a straight line a, lying in the plane cr, 
 passes through M, then the corresponding straight line a' will 
 also pass through M. But this is evidently the case, since the 
 two straight lines a and a f are the intersections of the same 
 projecting plane with the two planes <r and </, and conse- 
 quently the three straight lines o-</, «, and a' meet in a point, 
 vfe that common to the three planes* The straight line 
 aa is the locus of the points which correspond to themselves 
 in the two figures. 
 
 The vanishing line i' in the plane a is parallel to the straight 
 
 r\ line o-</, since i f and the corresponding straight line i, which 
 
 lies entirely at an infinite distance in the plane it, must inter- 
 
 v sect one another on oV. Similarly, the vanishing line j of 
 
 the plane a is parallel to aa. 
 
 If each of the figures is a triangle, the theorem reads as 
 follows : — 
 
 If two triangles ABC and A'B'C\ lying respectively in the 
 
16] CENTRAL PROJECTION ; FIGURES IN PERSPECTIVE. 7 
 
 planes <r and a, are such that the straight lines AA r , BB', CC 
 meet in a point S, then the three pairs of corresponding 
 sides, BC and B'C, CA and CA', AB and A'B', intersect in 
 points lying on the straight line aa. 
 
 15. Conversely, if to the points A,B, C ,. . , and to the straight 
 lines AB,AC, ..., BC, ... of 'a 'plane figure g- correspond severally 
 the points A', B', C,.. and the straight lines A'B', A'C, . r> B'C', . . 
 of another plane figure or' * 9 in such a way that the corresponding 
 lines AB and A'B', AC and A'C,..., BC and B'C,... meet in 
 points lying on the line of intersection (&&'), of the planes .a-, and <r', 
 thqn the two figures are in perspective. 
 
 For if S be the point which is common to the three 
 planes AB . A'B^AG- . A'C\ BC . B'C, the three edges 
 A A', B$', CC of the* trihedral angliT formed by the same 
 planes will meet in S^ Similarly, the three planes AB .A'B', 
 AB . A'B', BD . B'B' meet in a point which is common to the 
 edges AA', BB', DD', and this point is again & since the two 
 straight lines AA', BB' suffice to determine it. Therefore all 
 the straight lines A A' ', BB', CC, BB' ... pass through, the 
 same point #; that is, the two given figures are in perspective, 
 and S is their centre of projection.. 
 
 If each^ of Jjfcjie figures is a triangle, we have the theorem: 
 If two triangles ABC and A' B'C', lying respectively in the 
 planes o- and cr', are such that the sides BC and B'C, CA 
 and CA', £b and A'B' intersect one another two and two 
 in points lying on the straight line o-</, then the straight lines 
 AA', BB', CC meet in a poini S. 
 
 \ 16. Theorem. If two triangles A 1 B 1 C 1 andA 2 B 2 C 2 , lying in the 
 same plane, are such that the straight lines A X A 2 , B X B 2 , C X C 2 meet 
 in the same point 0,then the three points of intersection of the sides 
 B 1 C 1 and B 2 C 2 , C V A X and C 2 A 2 , A X B X and A 2 B 2 lie on a straight 
 line. (Fig. 3.) 
 
 Through the point which is common to the straight 
 lines A X A 2 , B 1 B 2 , C 1 C 2 , draw any straight line outside the plane 
 a, and in this straight line take two points ^ and S 2 . Project 
 the** triangle A 1 B 1 C 1 from S 1 and the triangle A 2 B 2 C 2 from S 2 . 
 The points A x , A 2 , 0, S 2 , S 1 lie in the same plane ; therefore 
 S 1 A 1 and S 2 A 2 meet one another (in A suppose) ; similarly 
 S 1 B 1 and S 2 B 2 (in B suppose) and 8 1 C 1 and S 2 C 2 (in C suppose). 
 
 * The planes a and a' are to be regarded as distinct from each other. 
 
CENTRAL PROJECTION; FIGURES IN PERSPECTIVE. [17 
 
 Thus the triangle ABC is in perspective both with A 1 B 1 C 1 and 
 with A 2 B 2 C 2 . The straight lines BC i B^, B 2 C 2 intersect in 
 pairs and therefore meet in one and the same point A 6 *. 
 Similarly CA, C X A X , and A 2 C 2 meet in a point B , and AB, 
 
 B X A X , and A 2 B 2 in a point 
 C . The three points A 0i 
 B , C lie on the straight 
 line which is common to 
 the planes o- and ABC. 
 The theorem is therefore 
 proved. 
 
 17. Conversely, If two 
 triangles A 1 B 1 C 1 and A 2 B 2 C 2 , 
 lying in the same plane, are 
 sack that the sides B 1 C 1 and 
 B 2 C 2i C X A X and C 2 A 2 , A X B X 
 and A 2 B 2 cut one another in 
 three collinear points A 0i B , C , then the straight lines 
 
 pairs 
 
 •^1^2' 
 
 B X B 2 , C X C 2 , which 
 
 corresponding angular points, will 
 pass through one and the same point 0.. (Fig. 3.) 
 
 Through the straight line A B C draw anothlr plane, . 
 and project, from an arbitrary centre S v the triangle A 1 B 1 C 1 
 upon this plane. If ABC be the projection, the straight lines BC^ 
 B 1 C 1 will cut one another in the poin£ A 0i through which B 2 C 2 
 will also pass; similarly AC will pass through B and AB 
 through C . The straight lines AA 2 , BB 2 , CC 2 intersect in 
 pairs, without however all three lying in the same plane ; 
 they will therefore all meet in one point S 2 . The straight 
 lines S X S 2 and A X A 2 lie in the same plane, "since 8 X A X and S 2 A 2 
 intersect" in A :' therefore S X S 2 meets the three straight lines 
 A 1 A 2 ,B 1 B 2 , C X C 2 , i.e. A X A 2 , B 1 B 2 ,C 1 C 2 all meet in one point 0, 
 viz. that which is common to the plane or and the straight 
 line S^f. 
 
 * BC is the intersection of the planes S^C^ and S 2 B 2 C 2) which do not coin- 
 cide; so that the straight lines BC, B- l C l , and B 2 C 2 do not all three lie in one 
 plane. The three planes BC .B i C 1) BC .B 2 C 2 , and B x Ci . B 2 C 2 (or <r) intersect 
 in the same point A . 
 
 + Poncelet, Proprietes projeetives des figures (Paris, 1822), Art. 168. The 
 theorems of Arts. 11 and 12 are due to Desargues (CEuvres, ed. Poudra, vol. i. 
 P- 413). 
 
CHAPTER III. 
 
 HOMOLOGY. 
 
 18. Consider a plane a and another plane </, in which latter 
 lies any given figure made up of points and straight lines. 
 Take two points ^ and S 2 lying outside the given planes, 
 and project from each of them as centre the given figure </ on 
 to the plane o\ In this way two new figures (o^ and a 2 say) 
 will be formed, which lie in the plane o-, and which are the 
 projections of one and the same figure o-' upon one and the 
 same plane a, but from different centres of projection. Let 
 two points A i and A 2 , or two straight lines a r and a 2 , in the 
 figures a x and <r 2 be said to correspond to each other when 
 they are the images of one and the same point A' or of 
 one and the same straight line a? of the figure a. We have 
 thus two figures o^ and <r 2 lying in the same plane <r, and 
 so related that to the points A 1 , B 1 , C 1 , ... and the lines 
 A 1 B 1 , A 1 C li ... i B 1 C li ..., of the one correspond the points 
 A 2 ,B 2 ,C 2 ,..., and the lines A 2 B 2 ,A 2 C 2i . . . , B 2 C 2 , . . . , of the other. 
 Since any two corresponding straight lines of a' and <r x intersect 
 in a point lying on the straight line o-v, and again any two 
 corresponding straight lines of o-' and a 2 intersect in a point 
 lying on the same straight line ao-', it 'follows that three 
 corresponding straight lines of cr[ <r 1 , and <r 2 meet in one 
 and the same point, which is determined as the intersection 
 of the straight line of a with the straight line vo'. That is 
 to say, two corresponding straight lines of the figures o-j and 
 <r 2 always intersect on a fixed straight line, the trace of a' on a-. 
 If moreover A 1 and A 2 are a pair of corresponding points of v-l 
 and a- 2 , the rays /S^^, S 2 A 2 have a point A' in common, and 
 therefore lie in the same plane : consequently A Y A 2 and S X S 2 
 intersect in a point 0. Thus we arrive at the property that 
 every straight line, such as A±A 2 , which connects a pah 1 of 
 
10 HOMOLOGY. [19 
 
 corresponding points of the figures <t x and <r 2 , passes through 
 a fixed point 0, which is the intersection of S x 8 2 and <r. 
 From this we conclude that two figures o^ and a 2 which 
 are the projections of one and the same figure on one and 
 the same plane, but from different centres of projection, 
 possess all the properties of figures in perspective (Art. 8) 
 although they lie in the same plane. To the points and the 
 straight lines of the first correspond, each to each, the points 
 and the straight lines of the second figure ; two corresponding 
 points always lie on a ray passing through a fixed point ; 
 and two corresponding straight lines always intersect on 
 a fixed straight line s. Such figures are said to be 7wmological, 
 or in homology \ is termed the centre of homology, and s the 
 axis of homology *. They may also be said to be in plane 
 perspective ; being called the centre of perspective, and s the 
 axis of perspective. 
 
 19. Theorem. In the plane a are given two figures a x and <r 2 
 which are such that to the points A x , B x , C l9 ... and to the straight 
 lines A Y B X , A-fi^y ..., B 1 C li ... of the one correspond, each to 
 each, the points A 2 , B 2 , C 2 , ... and the straight lines A 2 B 2 , A 2 C 2i 
 ...., B 2 C 2 , ...of the other. If the points of intersection of corre- 
 sponding straight lines lie on a fixed straight line, then the straight 
 lines which join corresponding points will all pass through a fixed 
 point 0. 
 
 Let A Y and A 2 , B L and B 2 , C x and C 2 be three pairs 
 of corresponding points ; they form two triangles A X B X C Y anct 
 A 2 B 2 C 2 whose corresponding sides B 1 C 1 and B 2 C 2 , C X A X and 
 C 2 A 2 ,A X B Y and A 2 B 2 intersect in three collinear points. By 
 the theorem of Art. 17 the rays A X A 2 ,B X B 2 , C 1 C 2 will there- 
 fore meet in the same point ; but two rays A Y A 2 and B 1 B 2 
 suffice to determine this point; in whatever way then the 
 third pair of points C lt C 2 may be chosen, the ray C 1 C 2 will 
 always pass through 0. 
 
 The figures a- 11 <r 2 are therefore in homology, being the 
 centre, and s the axis, of homology. 
 
 Corollary. — It follows that if two figures lying either in the same 
 or in different planes are in perspective, and if the plane of one 
 of the figures be made to turn round the axis of perspective, 
 then corresponding straight lines A X A 2 , B y B 2 , &c, will always be 
 
 * Poncelet, ProprtiUs projective*, Arts. 297 and the following. 
 
21] HOMOLOGY. 11 
 
 concurrent ; i.e:ihe two figures will remain always in perspective. The 
 centre of perspective will of course change its position ; it will be seen 
 further on (Art. 22) that it describes a certain circle. 
 
 20. Theorem. If to the straight lines a, b, c, ... and to the 
 points ab , ac, ..., be, ..., of a figure correspond severally the 
 straight lines </, b', c\ ... and the points a'b', a f c' , ... , b f c\ . . . 
 of another coplanar figtire, so that the pairs of corresponding points 
 ab and a'b\ ac and a'c\ be and b'c\ ... are collinear with a 
 fixed point 0; then the corresponding straight lines a and a', 
 
 b and b', c and c' 3 will intersect in points which lie on a straight 
 line. 
 
 Let a and a\ b and b\ c and c' be three pairs of corre- 
 sponding straight lines; since by hypothesis the straight lines 
 which join the corresponding vertices of the triangles abc,a'b'c' 
 •all meet in a point 0, it follows (Art. 16) that the correspond- 
 ing sides a and a\ b and b\ c and c' intersect in three points 
 lying on a straight line. But two points aa\ bb', suffice to 
 determine this straight line ; it remains therefore the same if 
 instead of c and c f any other two corresponding rays are 
 considered. Two corresponding straight lines therefore always 
 intersect on a fixed straight line, which we may call s; thus 
 the given figures are in homology, being the centre, and s 
 the axis, of homology. 
 
 21. Consider two homological figures a x and ct 2 lying in 
 the plane a; let be their centre, s their axis of homology. 
 Through the point and outside the plane a draw any 
 straight line, and on this take a point S 1 , from which as 
 centre project the figure o^ upon a new plane </ drawn in any 
 way through s. In this manner we construct in the plane </ a 
 figure A'B r C r ... which is in perspective with the given one 
 o- 1 = A 1 B 1 C l .... If we consider two points A' and A 2 of the 
 figures (/ and a 2 , which are derived from one and the same 
 point A x of <r x , as corresponding to each other, then to every 
 point or straight line of a' corresponds a single point or straight 
 line of <r 2 , and vice versa ; and every pair of corresponding 
 straight lines, such, as A'B' and A 2 B 2 , intersect on a fixed 
 straight line a a' or s. Consequently (Art. 15) the figures </ 
 and <T 2 are in perspective, and the rays A'A 2 , B'B 2 , ... all 
 pass through a fixed point S 2 . Moreover every ray A f A 2 
 meets the straight line 08^ since the points A', A 2 lie on the 
 
12 
 
 HOMOLOGY. 
 
 sides S^ , OA 1 of the triangle OA^ . The rays A'A 2 ,B f B 2 , ... 
 do not all lie in the same plane, because the points A 2 ,B 2 , ... 
 lie arbitrarily in the plane <r ; the point 8 2 therefore lies on 
 the straight line OS v 
 
 From this we conclude that two homological figures may 
 be regarded, in an infinite number of ways, as the projections, 
 from two distinct points, of one and the same figure ; this 
 figure lying in a plane passing through the axis of homology, 
 and the two points being collinear with the centre of homology. 
 22. Consider two figures in perspective, lying in the planes 
 a, (/ respectively (or two figures in plane perspective in the 
 same plane a) ; let (Fig. 4) be the centre and s the axis of 
 
 perspective, and let j and 
 i' be the vanishing lines of 
 the two figures. If J and 
 F are points lying on these 
 vanishing lines, the points 
 J' and / which correspond 
 to each of them respec- 
 tively in the other figure 
 will be at infinity on the 
 rays OJ, OF respectively. 
 Further, the two corresponding straight lines //, FJ' must meet 
 in some point on s ; there are consequently an infinite number 
 of parallelograms having one vertex at 0, the opposite one on 
 s, and the other two vertices on/ and i' respectively. 
 
 Now, supposing the two figures to keep their positions in 
 their planes unaltered, let the plane <r be made to turn round 
 aa' or s. Every pair of corresponding straight lines must 
 always meet on s ; consequently the two figures will always 
 remain in perspective (Arts. 15, 19), and the point will 
 describe some curve in space. 
 
 In order to determine this curve, consider any one of the 
 above-mentioned parallelograms OJS1'. It remains always 
 a parallelogram, and the length of JS is invariable ; therefore 
 also OV is of constant length. The locus of the centre of 
 perspective is therefore a circle whose centre lies on the 
 vanishing line ** and whose plane is perpendicular to this line 
 and therefore to the axis of perspective *. 
 
 * Mobius, Barycentrische Calcid (Leipzig, 1827), § 230 (note, p. 326). 
 
23] 
 
 HOMOLOGY. 
 
 13 
 
 Kg. 5- 
 
 23. (1) Given the centre and the axis s of homology, and two 
 corresponding points A and A ' (collinear with 0) ; to construct the 
 figure homological with a given figure. 
 
 Take a second point B of the given figure (Fig. 5). To obtain the 
 corresponding point B / , we notice that the ray BB / must pass through 
 and that the straight lines AB,A'B' which correspond to one 
 another must intersect on s ; thus B' will be the point where OB 
 meets the straight line joining A' to the intersection of AB with s*. 
 In the same way we can construct any number of pairs of correspond- 
 ing points ; in order to draw the 
 straight line r' which corresponds 
 to a given straight line r, we have 
 only to find the point B / which 
 corresponds to a point B lying on 
 the line r, and to join the points 
 B' and rs. 
 
 In order to find the point I f 
 (the vanishing point) which corre- 
 sponds to the infinitely distant 
 point 7 on a given straight line (a ray 01, for example, drawn from 
 0), we repeat the construction just given for the point B'\ i.e. we join 
 another point A of the first figure to the point at infinity I on 01 
 (that is, we draw AI parallel to 01), and then join A' to the point 
 where A I meets s, and produce the joining line to cut 01 in V. 
 Then V is the required point. 
 
 All points analogous to I f (i. e. those which correspond to the points 
 at infinity in the given figure) fall on a 
 straight line V ', parallel to s; i' is the 
 vanishing line of the second figure. If, in 
 the preceding construction, we interchange 
 the points A and J/f, we shall obtain a 
 point J (a vanishing point) lying on the 
 vanishing line j of the first figure. 
 
 (2) Suppose that instead of two corre- 
 sponding points A,A / there are given (Fig. 6) 
 two corresponding straight lines a, a'. 
 These will of course intersect on 5 ; and 
 
 Fig. 6. 
 
 every ray passing through will cut them in two corresponding 
 
 * This construction shows that if B lies upon s, then B ' will coincide with B ; 
 i. e. that every point of s is its own correspondent. 
 
 f Otherwise : Draw through A' any straight line J' A', then through A and 
 the intersection of J' A' with s draw a straight line J A, and through O draw OJ' 
 parallel to A' J'. Then the intersection of OJ' and J A is the vanishing point J, 
 and a straight line j drawn through J parallel to s is the vanishing line of the 
 first figure. 
 
14 HOMOLOGY. [23 
 
 points A, A'. In order to obtain the straight line b' which corre- 
 sponds to any straight line b in the first figure, we have only to join 
 the point bs to the point of intersection of a f with the ray passing 
 tlirough and ab*. 
 
 (3) The data of the problem may also be the centre 0, the axis s, and 
 the vanishing line j of the first figure (Fig. 7). 
 In this case, if a straight line, a of the first 
 figure cuts j in J and s in P; the point 
 J' corresponding to J will be collinear with 
 J and and at an infinite distance from 0. 
 And as the straight line of corresponding to 
 a must pass both through J' and through P, 
 it is the parallel drawn through P to OJ. 
 
 To find the point A' corresponding to a 
 given point A, we must draw the straight 
 line a' which corresponds to a straight line a drawn arbitrarily 
 through A ; the intersection of a! with OA is the required point A / . 
 
 (4) Assuming a knowledge of the constructions just given, let 
 again be the centre, s the axis, of homology, and j the vanishing 
 line of the first figure. 
 
 In the first figure let a circle C be given (Figs. 8, 9, 10) ; to this 
 circle will correspond in the second figure a curve C which we can 
 construct by determining, according to the method above, the points 
 and straight lines which correspond to the points and tangents of C. 
 
 Two corresponding points will always be collinear with 0, and two 
 corresponding chords (i.e. straight lines MN , M'N', where M and M ', 
 iVand N\ are two pairs of corresponding points) will always intersect on 
 s ; as a particular case two corresponding tangents m and m' (i. e. tan- 
 gents at corresponding points if and M') will meet' in a point lying on s. 
 
 It follows clearly from this that the curve C possesses, in common 
 with the circle, the two following properties : 
 
 (1) Every straight line in its plane either cuts it in two points, or 
 is a tangent to it, or has no point in common with it. 
 
 (2) Through any point in the plane can be drawn either two 
 tangents to the curve, or only one (if the point is on the curve), 
 or none. 
 
 Since two homological figures can be considered as arising from the 
 superposition of two figures in perspective lying in different planes 
 (Art. 22), the curve C is simply the plane section of an oblique cone 
 on a circular base ; i. e. the cone which is formed by the straight lines 
 tvhich run from any point in space to all points of a circle. 
 
 * It follows from this that if a passes through O, then a' will coincide with a ; 
 i. e. every straight line passing through corresponds to itself. 
 
HOMOLOGY. 
 
 15 
 
 For this reason the curve C is called a conic section or simply a 
 conic ; thus the curve which is homological with a circle is a conic. 
 
 The points on the straight line j correspond to the points at 
 infinity in the second figure. Now the circle C may cut j in two 
 
 Fig. 8. 
 
 points J v J 2 (Fig. 8), or it may touch j in a single point J (Fig. 9), 
 or it may have no point in common with j (Fig. 10). 
 
 Fig. 9. 
 
 In the first case (Fig. 8) the curve C f will have two points <//, «//, at 
 an infinite distance, situated in the direction of the straight lines OJ lt 
 OJ v To the two straight lines which touch the circle in J x and J 2 
 will correspond two straight lines (parallel respectively to OJ x and 
 
16 
 
 HOMOLOGY. 
 
 OJ 2 ) which must be considered as tangents to the curve C" at its 
 points at infinity J/, J/. These two tangents, whose points of 
 contact lie at infinity, are called asymptotes of the curve C '; the 
 curve itself is called a hyperbola. 
 
 In the second case (Fig. 9) the curve C" has a single point J' at 
 infinity ; this must be regarded as the point of contact of the straight 
 line at infinity j', which is the tangent to C corresponding to the 
 
 Fig. 10. 
 
 tangent j at the point J of the circle. This curve C is called a para- 
 bola. 
 
 Fig. 11. 
 
 In the third case (Fig. 10) the curve has no point at infinity; it is 
 called an elli2)se. 
 
23] 
 
 HOMOLOGY. 
 
 17 
 
 In the same way it may be shown that if in the first figure a conic 
 C is given, the corresponding curve C in the second figure will be a 
 conic also. 
 
 (5). The centre of homology is a point which corresponds to itself, and 
 every ray which passes through it corresponds to itself. If then a 
 curve C pass through 0, the corresponding curve C / will also pass 
 through 0, and the two curves will have a common tangent at this 
 point. Fig. 1 1 shows the case where one of the curves is taken to be a 
 circle, and the axis of homology s and the point A corresponding 
 to the point A' of the circle are supposed to be given. 
 
 Similarly, every point on the axis of homology corresponds to 
 itself. If then a curve belonging to the first figure touch s at a 
 certain point, the corresponding curve in the second figure will touch 
 s at the same point. In Fig. 1 2 is shown a circle which is to be 
 transformed homologically by means of its tangents ; moreover it is 
 
 Fig. 12. 
 
 supposed that the axis of homology touches the circle, that the centre 
 of homology is any given point, and that the straight line a of the 
 second figure is given which corresponds to the tangent a! of the 
 circle. 
 
 (6). Two particular cases may be noticed : 
 
 (1) The axis of homology s may lie altogether at infinity ; then two 
 corresponding straight lines are always parallel, or, what amounts to 
 the same thing, two corresponding angles are always equal. In this 
 
 C 
 
18 HOMOLOGY. [23 
 
 case the two figures are said to be similar and similarly -placed, or 
 homothetic *, and the point is called the centre of similitude. 
 
 Let M x , if/ and M 2 , M/ be two pairs of corresponding points 
 of two homothetic figures, so that M 1 M/ i M 2 M 2 * meet in 0, while 
 M 1 M 2 , M{M{ are parallel. By similar triangles 
 
 OM 1 : OM( = OM % : OMf = M,M, : M{M{ t 
 
 so that the ratio OM: OM' is constant for all pairs of corresponding 
 points M and M' '. This constant ratio is called the ratio of similitude 
 of the two figures. 
 
 The tangents at two corresponding points M, M* must meet on the 
 axis of homology s, i.e. they are parallel to one another. If then the 
 tangent at M pass through 0, it must coincide with the tangent at 
 M '. It follows that if the two figures are such that common tangents 
 can be drawn to them, every common tangent passes through a centre 
 of similitude. 
 
 Take two points C, C collinear with and such that 
 
 OC OM '- . . r , 
 
 -ftft, — tttf? = ™tio of similitude. 
 OC QM 
 
 Then if CM, CM' be joined, they will evidently be parallel, and 
 CM: CM'= ratio of similitude. Therefore if M lie on a circle, centre 
 C and radius p, M' will lie on another circle whose centre is C and 
 whose radius p' is such that p : p'= ratio of similitude. In two homo- 
 thetic figures then to a circle always corresponds a circle. Further, if 
 CC be again divided at 0', so that 
 
 O'C :0'C'=OC\OC'= p:p'= ratio of similitude, 
 
 it is clear that 0' will be a second centre of similitude for the two 
 circles. It can be proved in a similar manner that any two central 
 conies (see Chap. XXI) which are homothetic, and for which a point 
 is the centre of similitude, have a second centre of similitude O f ; 
 and that 0, O f are collinear with the centres C, C f of the two conies, 
 and divide the segment CC internally and externally in the ratio of 
 similitude. If the conies have real common tangents, and C will 
 be the points of intersection of these taken in pairs — the two external 
 tangents together, and the two internal tangents together. 
 
 (2) The point 0, on the other hand, may lie at an infinite distance ; 
 then the straight lines which join pairs of corresponding points are 
 parallel to a fixed direction. In this case the figures have been termed 
 homological by affinity t, the straight line s being termed the axis of 
 
 * Homothetic figures may be regarded as sections of a pyramid or a cone made 
 by parallel planes ; s, the line of intersection of the two planes, lies at an infinite 
 distance. This is the case in Art. 8 if a and a' are parallel planes. 
 
 t Euleb, Introductio ... ii. cap. 18 ; Mobius, Baryc. Calcul, § 144 et seqq. 
 
23] HOMOLOGY. 19 
 
 affinity *. To a point at infinity corresponds in this case a point at 
 infinity, and the straight line at infinity corresponds to itself. It 
 follows from this that to an ellipse corresponds an ellipse, to a hyper- 
 bola a hyperbola, to a parabola a parabola, to a parallelogram a 
 parallelogram. 
 
 * If two figures are so related, they may be regarded as plane sections of a 
 prism or of a cylinder. This is the case in Art. 8 if the centre IS of projection is 
 infinitely distant. The projection is then called parallel projection. In the 
 particular case where the parallels SA, SB ,SC, . . . are perpendicular to the plane 
 of projection it is called orthogonal projection. 
 
 C 2 
 
CHAPTEE IV. 
 
 HOMOLOGICAL FIGURES IN SPACE. 
 
 24. Suppose now a figure to be given which is made up of points, 
 planes, and straight lines lying in any manner in space; the relief - 
 'perspective* of this is made in the following manner. A point in 
 space is taken as centre of perspective or homology ; a plane of 
 homology it is taken, every point of which is to be its own image ; 
 and in addition to these is taken a point A' which is to be the image 
 of a point A of the given figure, so that A A' passes through 0. Let 
 now B be any other point; in order to obtain its image B\ the plane 
 OAB is drawn, and we then proceed in this plane as if we had to 
 construct two homological figures, taking as the centre and the 
 intersection of the planes OAB and n as the axis of homology, and A , A' 
 as two corresponding points. The point B' will be the intersection of 
 OB with the straight line passing through A f and the point where the 
 straight line AB cuts the plane rr (Art. 23, fig. 4). Let be a third 
 point ; its image G f will be the point of intersection of OC with 
 A'D or with B'E (in n), where B and E are the points in which 
 the plane n is met by AC, BO respectively. 
 
 This method will yield, for every point of the given figure, the 
 corresponding point of the image, and two corresponding points will 
 always lie on a straight line passing through 0. Every plane a- 
 passing through cuts the two solid figures (the given one and its 
 image) in two homological figures, for which is the centre, and the 
 straight line an the axis, of homology. It follows from this that to 
 every straight line of the given figure corresponds a straight line in 
 the image, and that two corresponding straight lines lie always in a 
 plane passing through and meet each other- in a point lying on the 
 plane it. 
 
 Further : to every plane a, belonging to the given figure, and not 
 passing through 0, will correspond a plane a' in the image. For to the 
 straight lines a, b, c,... of the plane a correspond severally the straight 
 
 * This problem may present itself in the construction of bas-reliefs and of 
 theatre decorations (Poncelet, Prop. proj. 584 ; Poudra, Perspective-relief, 
 Paris, i860). 
 
26] HOMOLOGICAL FIGUKES IN SPACE. 21 
 
 lines a', b\ c', . . . ; and to the points ab, ac,...,bc,... the points a'b', a'c', 
 ...,6V,... . In other words, the straight lines a', b', </,... are such that 
 they intersect in pairs, but do not all meet in the same point; they 
 lie therefore in the same plane a'*. Two corresponding planes a, a' 
 intersect on the plane tt ; for all the points and all the straight 
 lines of this last plane correspond to themselves, and therefore the 
 straight line a'n coincides with the straight line air. 
 
 The two planes a, a' evidently contain two figures in perspective 
 (like the planes <r, a' of Arts. 12 and 14). 
 
 25. In every plane o- passing through lies a vanishing line i', 
 which is the image of the point at infinity in the same plane. The 
 vanishing lines of the planes tr v cr 2 have a common point, which is the 
 image of the point at infinity on the line o-j <r 2 . The vanishing lines 
 of all the planes o- are therefore such as to cut each other in pairs ; 
 and as they do not pass all through the same point (since the planes 
 through do not pass all through the same straight line), they must 
 lie in one and the same plane <//. 
 
 This plane <f/ 9 which may be called the vanishing 2>lane, is parallel 
 to the plane tt, since all the vanishing lines of the planes o- are 
 parallel to the same plane n. The vanishing plane <f>' is thus the 
 locus of the straight lines which correspond to the straight lines at 
 infinity in all the planes of space, and is consequently also the locus 
 of the points which correspond to the points at infinity in all the 
 straight lines of space : for the line at infinity in any plane a is the 
 same thing as the line at infinity in the plane through parallel to 
 a ; so also the point at infinity on any straight line a coincides with 
 the point at infinity on the straight line drawn through parallel 
 to a. 
 
 26. The infinitely distant points of all space are then such that 
 their images are the points of one and the same plane <j>' (the vanishing 
 plane). It is therefore natural to consider all the infinitely distant 
 points in space as lying in one and the same plane $ (the plane at 
 infinity) of which the plane <£' is the image f. 
 
 The idea of the plane at infinity being granted, the point at infinity 
 on any straight line a is simply the point a<f>, and the straight line at 
 infinity in any plane a is the straight line a(f>. Two straight lines are 
 parallel if they intersect in a point of the plane <f> ; two planes are 
 parallel if their line of intersection lies in the plane $, &c. 
 
 * Since c' cuts both a! and b' without passing through the point a'b', therefore 
 c' has two points in common with the plane a'b', and consequently lies entirely in 
 the plane a'b'. And similarly for the other straight lines. 
 
 t Poncelet, Prop. proj. 580. 
 
CHAPTER V. 
 
 GEOMETRIC FORMS. 
 
 27. A range or row of points is a figure A,B,C,... composed 
 of points lying on a straight line (which is called the base of 
 the range) ; such is, for example, the figure resulting from the 
 operations of Art. 5 or Art. 7. 
 
 An axial pencil is a figure a, /3,y, ... composed of planes all 
 passing through the same straight line (the axis of the pencil) ; 
 such is the figure resulting from the operations of Art. 4 or 
 Art. 6. 
 
 Aflat pencil is a figure a, b, <?,... composed of straight lines 
 lying all in the same plane and radiating from a given point 
 (the centre or vertex of the pencil) ; such would be the figure 
 obtained by applying the operation of Art. 2 to a range, or 
 that of Art. 3 to an axial pencil. 
 
 A sheaf {sheaf of planes, sheaf of lines) is a figure made up of 
 planes or straight lines, all of which pass through a given 
 point (the centre of the sheaf) ; like that which results from 
 the operation of Art. 2. 
 
 A plane figure [plane of points, plane of lines) is a figure which 
 consists of points or straight lines all of which lie in the same 
 plane; such is the figure resulting from the operation of 
 Art. 3. 
 
 28. The first three figures can be derived one from the other 
 by a projection or a section*. 
 
 From a range A ,B , C , ... is derived an axial pencil 
 s\A , B, C , ...) by projecting the range from an axis $ (Art. 4) ; 
 and a flat pencil ( A , B , C , . . .) by projecting it from a centre 
 
 * The series of planes sA , sB , sC , . . . ; of rays OA, OB , OC , . . . ; of points sa, 
 *&, sy, . . . ; and of straight lines aa, <r(3, ay, ... will be denoted by s (A , B , C , . . . ), 
 0(A , B , C , ...,),* (a, 0, 7, ... ,), and <r(a, j8, 7, ... ) respectively. To denote the 
 series of points A , B , C , . . . the symbols A, B, C, ... and ABC . . . will be used 
 indifferently. 
 
30] GEOMETRIC FORMS. 23 
 
 (Art. 2). From an axial pencil a , ft , y , . . . is derived a range 
 s(a, ft , y } ...) by cutting the pencil by a transversal line s 
 (Art. 5) ; and a flat pencil a (a , ft , y , ... ) by cutting it by a 
 transversal plane <r (Art. 3). From a flat pencil a , b , c , . . . is 
 derived a range a, (a , b , c , ...) by cutting it by a transversal 
 plane <r (Art. 3) ; and an axial pencil (a , b , c , . . . ) by project- 
 ing it from a centre (Art. 2). 
 
 29. In a similar manner the last two figures of Art. 27 can 
 be derived one from the other by help of one of the operations 
 of Art. 2 or Art. 3 ; in fact, if we project from a centre a 
 plane of points or lines we obtain a sheaf of lines or planes ; 
 and reciprocally, if we cut a sheaf of lines or planes by a 
 transversal plane we obtain a plane of points or lines. Two 
 plane figures in perspective (Art. 1 2) are two sections of the 
 same sheaL^ 
 
 30. The elements or constituents of the range are the points ; f 
 those of the axial pencil, the planes ; those of the flat pencil, I 
 the straight lines or rays. 
 
 In the plane figure either the points or the straight lines 
 may be regarded as the elements. If the points are considered 
 as the elements, the straight lines of the figure are so many 
 ranges ; if, on the other hand, the straight lines or rays are 
 considered as the elements, the points of the figure are the 
 centres of so many flat pencils. 
 
 The plane of points (i. e. the plane figure in which the ele- 
 ments are points) contains therefore an infinite number of 
 ranges * and the plane of lines (i. e. the plane figure in which 
 the elements are lines f) contains an infinite number of flat 
 pencils. 
 
 In the sheaf either the planes, or the straight lines or rays, 
 may be regarded as the elements. If we take the planes as 
 elements, the rays of the sheaf are the axes of so many 
 axial pencils ; if, on the other hand, the rays are considered 
 as the elements, the planes of the sheaf are so many flat 
 pencils. 
 
 The sheaf contains therefore an infinite number of axial 
 
 * One of these ranges has all its points at an infinite distance ; each of the 
 others has only one point at infinity. 
 
 t The straight line at infinity belongs to an infinite number of flat pencils, each 
 of which has its centre at infinity, and consequently all its rays parallel. 
 
24 | GEOMETRIC FORMS. [31 
 
 pencils or an infinite number of flat pencils, according as its 
 planes or its straight lines are regarded as its elements. 
 
 31. Space may also be considered as a geometrical figure, 
 whose elements are either points or planes. 
 
 Taking the points as elements^the straight lines of space 
 are so many ranges^andjthe planes of space so many planes of 
 points. If, on the other hand, the planes are considered as 
 elements, the straight lines of space are the axes of so many 
 axial pencils, and)points of space are the centres of so many 
 sheaves of planes. 
 
 Space contains therefore an infinite number of planes of 
 points * or an infinite number of sheaves of planes f , according 
 as we take the point or the plane as the element in order to 
 construct it. 
 
 32. The first three figures, viz. the range, the axial pencil, 
 and the flat pencil, which possess the property that each can 
 be derived from the other by help of one of the operations of 
 Arts. 2, 3, ... , are included together under one name, and are 
 termed the one-dimensional geometric pri?ne-forms. 
 
 The fourth and fifth figures, viz. the sheaf of planes or lines 
 and the plane of points or lines, which may in like manner be 
 derived one from the other by means of one of the operations 
 of Arts. 2, 3, ... , and which moreover possess the property of 
 including in themselves an infinite number of one-dimensional 
 prime-forms, are likewise classed together under one title, as 
 the two-dimensional geometric prime-forms. 
 
 Lastly, space, which includes in itself an infinite number of 
 two-dimensional prime-forms, is considered as constituting the 
 three-dimensional geometric prime form. 
 
 There are accordingly six geometric prime-forms ; three of 
 one dimension, two of two dimensions, and one of three di- 
 mensions J. 
 
 Note. — With reference to the use of the word dimension in the 
 preceding Article, it is clear, from what has been said in Art. 28, 
 that we are justified in considering the range, the flat pencil, and 
 the axial pencil, as of the same dimensions, since to every point in 
 
 * One of them lies entirely at infinity. 
 
 + Among these, there are an infinite number which have their centre at an in- 
 finite distance, and whose rays are consequently parallel. 
 
 J v. Staudt, Geometrie der Lage (Niirnberg, 1847), Arts. 26, 28. 
 
32] GEOMETRIC FORMS. 25 
 
 the first corresponds one ray in the second and one plane in the 
 third. The number of elements in each of these forms is infinite, 
 but it is the same in all three. 
 
 Similarly we conclude from Art. 29 that we are justified in con- 
 sidering the plane figure as of the same dimensions with the sheaf. 
 
 But the plane of points (lines) contains (Art. 30) an infinite number 
 of ranges (flat pencils) ; and each of these ranges (flat pencils) itself 
 contains an infinite number of points (rays). Thus the plane figure 
 contains a number of points (lines) which is an infinity of the second 
 order compared with the infinity of points in a range, or of rays in a 
 flat pencil ; and must therefore be considered as of two dimensions if 
 the range and flat pencil are taken to be of one dimension. 
 
 So too the sheaf of planes (or lines) contains (Art. 30) an infinite 
 number of axial pencils (or of flat pencils), and each of these itself 
 contains an infinite number of planes (or of rays). Therefore also 
 the sheaf of planes or lines must be of double the dimensions of the 
 axial pencil or the flat pencil. 
 
 Again, space, considered as made up of points, contains an infinite 
 number of planes of points, and considered as made up of planes, it 
 contains an infinite number of sheaves of planes. Space thus contains 
 an infinite number of forms of two dimensions, which latter, again, 
 contain each an infinite number of forms of one dimension. Space 
 must accordingly be regarded as of three dimensions. 
 
 We may also say that : 
 
 Forms of one dimension are those which contain a simple infinity 
 (oo) of elements; 
 
 Forms of two dimensions are those which contain a double infinity 
 (oo 2 ) of elements ; 
 
 Forms of three dimensions are those which contain a triple infinity 
 (oo 3 ) of elements. 
 
CHAPTER VI. 
 
 THE PRINCIPLE OF DUALITY*. 
 
 33. Geometry (speaking generally) studies the generation 
 and the properties of figures lying (i) in space of three dimen- 
 sions, (2) in a plane, (3) in & sheaf. » In each case, any figure 
 considered is simply an assemblage of elements ; or, what 
 amounts to the same thing* it is the aggregate of the elements 
 with which a moving or variable element coincides in its 
 successive positions. • The moving element which generates the 
 figures may be, in the first case, the point or the plane ; in the 
 second case the point or the straight line ; in the third case 
 the plane or the straight line., There are therefore always 
 two correlative or reciprocal methods by which figures may be 
 generated and their properties deduced, and it is in this 
 that geometric Duality consists. By this duality is meant the 
 co-existence of figures (and consequently of their properties 
 also) in pairs ; two such co-existirig (correlative or reciprocal) 
 figures having the same genesis and only differing from one 
 another in the nature of the generating element. 
 
 In the Geometry of space therange and the axial pencil, the 
 plane of points and the sheaf of planes, the plane of lines and 
 the sheaf of lines, are correlative forms. The flat pencil is a 
 form which is correlative to itself. 
 
 In the Geometry of the plane the range and the flat pencil 
 are correlative forms. 
 
 In the Geometry of the sheaf the axial pencil and the flat 
 pencil are correlative forms. 
 
 The Geometry of the plane and the Geometry of the sheaf, 
 considered in three-dimensional space, are correlative to each 
 other. 
 
 34. The following are examples of correlative propositions 
 
 * v. Stadpt, Gcom. der Lags, Art. 66. 
 
34] 
 
 THE PRINCIPLE OF DUALITY. 
 
 in the Geometry ofjy^ace. Two correlative propositions are 
 deduced one from the other by interchanging the elements 
 point and plane. 
 
 1. Two points A , B determine 
 a straight line (viz. the straight 
 line A B which passes through the 
 given points) which contains an 
 infinite number of other points. 
 
 2. A straight line a and a point 
 B (not lying on the line) deter- 
 mine a plane, viz. the plane aB 
 which connects the line with the 
 point. 
 
 3. Three points A ,B , C which 
 are not collinear determine a 
 plane, viz. the plane ABC which 
 passes through the three points. 
 
 4. Two straight lines which 
 cut one another lie in the same 
 plane. 
 
 5. Given four points A ,B ,Cf, 
 D; if : the- straight lines AB ,GD 
 meet, the four points will lie in 
 a plane, and consequently the 
 straight lines BC and AD, CA 
 and BD will also meet two and 
 two. 
 
 6 . Given any number of straight 
 lines ; if each meets all the others, 
 while the lines do not all pass 
 through a point, then they must 
 lie all in the same plane (and 
 constitute a plane of lines)*. 
 
 1. Two planes a, /3 determine a 
 straight line (viz. the straight line 
 a/3, the intersection of the given 
 planes), through which pass an in- 
 finite number of other planes. 
 
 2. A straight line a and a plane 
 /3 (not passing through the line) 
 determine a point, viz. the point 
 a/3 where the line cuts the plane. 
 
 3. Three planes a , /3 , y which do 
 not pass through the same line 
 determine a point, viz. the point 
 a/3y w r here the three planes meet 
 each other. 
 
 4. Two straight lines which lie 
 in the same plane intersect in a 
 point. 
 
 5. Given four planes a, j3, y, 8 ; 
 if the straight lines a/3 , yd meet, 
 the four planes will meet in 
 a point, and consequently the 
 straight lines /3y and a8,ya and 
 /3S, will also meet two and two. 
 
 6 . Given any number of straight 
 lines ; if each meets all the others, 
 while the lines do not all lie in 
 the same plane, then they must 
 pass all through the same point 
 (and constitute a sheaf of lines) f. 
 
 7. The following problem admits of two correlative solutions : 
 1 Given a plane a and a point A in it, to draw through A a straight 
 line lying in the plane a which shall cut a given straight line r which 
 does not lie in a and does not pass through A.' 
 
 * See note to Art. 20. 
 
 t For let a,b, c, ... be the straight lines ; as ab, ac, be are three planes distinct 
 from each other, the common point must be the intersection of the straight lines 
 a, b, c, . . . . 
 
 / 
 
28 THE PRINCIPLE OF DUALITY. [35 
 
 Join A to the point ra. Construct the line of inter- 
 
 section of the plane a with the 
 plane rA. 
 
 8. Problem. Through a given 8. Problem. In a given plane 
 
 point A to draw a straight line a, to draw a straight line to cut 
 to cut each of two given straight each of two given straight lines b 
 lines b and c (which do not lie in and c (which do not meet and do 
 the same plane and do not pass not lie in the plane a). 
 through A). 
 
 Solution. Construct the line Solution. Join the point a b to 
 of intersection of the planes Ab, the point ac. 
 Ac. 
 
 35. In the Geometry of Space, the figure correlative to a triangle 
 (system of three points) is a trihedral angle (system of three planes) ; 
 the vertex, the faces, and the edges of the latter are correlative to the 
 plane, the vertices, and the sides respectively of the triangle ; thus 
 the theorem correlative to that of Arts. 15 and 17 will be the fol- 
 lowing : 
 
 If two trihedral angles a'fty', J'tf'-J* are such that tlie edges ft'y' 
 and $"y", yV and y"a", a'fi' and a"$" lie in three planes a , /3 , y 
 which pass through the same straight line, then the straight lines 
 da", $'$", -/y" will lie in the same plane. 
 
 The proof is the same as that of Arts. 1 5 and 1 7, if the elements 
 point and plane are interchanged. If, for example, the two trihedral 
 angles have different vertices S', S" (Art. 1 5), then the points where 
 the pairs of edges intersect are the vertices of a triangle whose sides 
 are da n , fi'fr", y'y f> '; these latter straight lines lie therefore in the 
 same plane (that of the triangle). 
 
 So also the proof for the case where the two trihedral angles have 
 the same vertex S will be correlative to that for the analogous case of 
 two triangles A'B'C and A"B"C" which lie in the same plane (Art. 
 17). The theorem may also be established by projecting from a point 
 S the figure corresponding to the theorem of Art. 16. 
 
 The proof of the theorem correlative to that of Arts. 14 and 16 is 
 left as an exercise for the student. It may be enunciated as follows : 
 
 If two trihedral angles a / /3 / y / , a fr $"y" are such that tlie straight lines 
 a' a", fi'fi", y'y" lie in tlie same plane, then the pairs of edges ft'y' and 
 &"y", yV and y"a", a'$ f and d'$" determine three planes zvhich pass 
 all through the same straight line. 
 
 36. In the Geometry of the plane, two correlative proposi- 
 tions are deduced one from the other by interchanging the 
 words point and line, as in the following examples : 
 
36] 
 
 THE PRINCIPLE OF DUALITY. 
 
 29 
 
 1. Two points A,B determine 
 a straight line, viz. the line AB. 
 
 2. Four points A,B,G,D (Fig. 
 13), no three of which are col- 
 linear, form a figure called a com- 
 plete quadrangle*. The four 
 
 y> G 
 
 Fig. 13. 
 
 points are called the vertices, and 
 the six straight lines joining them 
 in pairs are called the sides of 
 the quadrangle. 
 
 Two sides which do not meet 
 in a vertex are termed opposite-^ 
 there are accordingly three pairs 
 of opposite sides, BO and AD, 
 CA and BD, AB and CD. The 
 
 1 . Two straight lines a , b de- 
 termine a point, viz. the point 
 ab. 
 
 2. Four straight lines a,b,c,d 
 (Fig. 14), no three of which are 
 concurrent, form a figure called a 
 complete quadrilateral*. The four 
 
 Fig. 14. 
 
 straight lines are called the sides 
 of the quadrilateral, and the six 
 points in which the sides cut one 
 another two and two are called 
 the vertices. 
 
 Two vertices which do not lie 
 on the same side are termed op- 
 2)0site; there are accordingly 
 three pairs of opposite vertices, be 
 and ad, ca and bd, ab and cd. 
 
 Fig. 16. 
 
 points E,F,Gin which the oppo- The straight lines e,f,g which 
 site sides intersect in pairs are join pairs of opposite vertices are 
 
 * The complete quadrangle has also been called a tetrastigm, and the complete 
 quadrilateral a tetragram. Townsend, Modern Geometry, ch. vii. 
 
30 
 
 THE PRINCIPLE OP DUALITY. 
 
 [36 
 
 termed the diagonal 2>oints ; and 
 the triangle EFG is termed the 
 diagonal triangle of the complete 
 quadrangle. The complete quad- 
 rangle includes three simple 
 quadrangles, viz. ACBD, A BCD, 
 . and ABDC (Fig. 15). 
 
 3. And so, in general : 
 A complete polygon (complete 
 n-gon, or n-point *) is a system 
 of n points or vertices, with the 
 
 — straight lines or sides 
 
 called the diagonals ; and the 
 triangle efg is termed the diagonal 
 triangle of the complete quadri- 
 lateral. The complete quadri- 
 lateral includes three simple/ 
 quadrilaterals, viz. acbd, adcb, and 
 acbd (Fig. 16). 
 
 A complete multilateral (or 
 n-side t) is a system of n straight 
 
 lines or sides, with the — ' 
 
 points or vertices in which they 
 intersect one another two and 
 two. 
 4. The theorems of Arts. 16 and 17 are correlative each to the 
 other. • ' 
 
 Theorem. If two complete 
 quadrilaterals abed, a'b'c'd' are 
 such that five pairs of vertices 
 ab and a'b' ', be and b'c', ca and 
 c / a / , ad and a'd f , bd and b'd' 
 lie upon five straight lines which 
 meet in a point S, then the re- 
 maining pair cd and c'd' will also 
 lie on a straight line through S 
 (Fig. 18). 
 
 which join them two and two. 
 
 Vz 5. Theorem. If two complete 
 ^quadrangles ABC I), A'B'C'D' 
 are such that five pairs of sides 
 AB and A'B', BC and B'C, CA 
 and C'A ',AD and A'D', BD and 
 B'D' cut one another in five 
 points lying on a straight line s, 
 then the remaining pair CD and 
 CD' will also intersect one ano- 
 ther on s (Fig. 17). - 
 
 Fig. 17. 
 
 Since the triangles ABC, 
 A' B'C' are by hypothesis in 
 
 * Or polystigrn ; Townsend, loc. < it. 
 
 Fig. 18. 
 
 Since the triangles (tri- 
 lateral) abc, a'b'c' are by 
 
 f Or polygram. 
 
37] 
 
 THE PRINCIPLE OF DUALITY. 
 
 31 
 
 perspective (Arts. 17, 18), 
 the straight lines AA' , BB', 
 CC will meet in one point 
 S. So toxuthe triangles ABD, 
 J.'B'D' are in perspective ; there- 
 fore DD' also will pass through 
 S, the point common to AA' 
 and BB' . It follows that the 
 triangles BCD, B'C'D'are also in 
 perspective : therefore CD and 
 CD' meet in a point on the 
 straight line s, which is deter- 
 mined by the point of intersec- 
 tion of BC and B'C and by that 
 of BD and B'D' *. 
 
 hypothesis in perspective (Art. 
 18), the points aa', bb', cc' 
 will lie on one straight line s. 
 So too the triangles abd, a'b'd' are 
 in perspective; therefore the point 
 dd' lies on the straight line 8 
 which passes through the points 
 aa', bb'. It follows that the 
 triangles (trilaterals) bed, b'e'd' 
 are also in perspective ; therefore 
 cd and c'd' lie on a straight line 
 through the point S, which is 
 determined by the straight lines 
 (6c) (6V) and (bd) (b'd') *. 
 
 37. In the Geometry of space the following are correlative : 
 A complete n-gon (in a plane). A complete n-flat (in a sheaf) ; 
 
 i.e. a figure made up of rt planes 
 
 (or faces) which all pass through 
 the same point (or vertex), toge- 
 
 ther with the 
 
 w(n— i) 
 
 edges in 
 
 A complete multilateral of n 
 sides, or n-side (in a plane). 
 
 which these planes intersect two 
 and two. 
 
 A complete n-edge (in a sheaf) ; 
 i.e. a figure made up of n straight 
 lines radiating from a common 
 point (or vertex), together with 
 
 the 
 
 n in— i) 
 
 planes (or faces) 
 
 which pass through these straight 
 lines taken in pairs. 
 
 Thus the following theorems are correlative, in the Geometry^of 
 space, to the two theorems above (Art. 36, No. 5), which latter 
 are themselves correlative to each other in the Geometry of the 
 plane. 
 
 If two complete four-flats in a 
 sheaf (be their vertices coincident 
 or not) apyb, a'fi'y'd' are such 
 that five pairs of corresponding 
 
 If two complete four-edges in a 
 sheaf (be their vertices coincident 
 or not) abed, a'b'c'd' are such that 
 five pairs of corresponding faces cut 
 
 * These two theorems hold good equally when the two quadrangles or quadri- 
 laterals lie in different planes ; in fact, the proofs are the same as the above, word 
 for word. 
 
32 THE PRINCIPLE OF DUALITY. [38 
 
 edges lie in five planes which one another in five straight lines 
 
 pass all through the same straight which lie all in one plane <r, then 
 
 line s, then the sixth pair of corre- the line of intersection of the 
 
 sponding edges will lie also in a sixth pair of corresponding faces 
 
 plane passing through s. will lie also in the plane <r. 
 
 The proofs of these theorems are left as an exercise to the student. 
 They only differ from those of the theorems No. 5, Art. 36 in the 
 substitution for each other of the elements point and plane ; and just 
 as theorems 5, Art. 36 follow from those of Arts. 15 and 16, so the 
 theorems enunciated above follow from those of Art. 35. When 
 the two four-flats have the same vertex 0, the theorem on the left- 
 hand side may also be established by projecting from the point 
 (Art. 2) the §gure corresponding to the right-hand theorem of 
 No. 5, Art. 36. And in this case we may by the same method 
 deduce the theorem on the right-hand side above from that on the left- 
 hand of No. 5, Art. 36. 
 
 38. In the Geometry of the sheaf, two correlative theorems are 
 derived one from the other by interchanging the elements plane and 
 straight line. Just as the Geometry of the sheaf is correlative to 
 that of the plane, with regard to three-dimensional space, so one 
 of the Geometries is derived from the other by the interchange of 
 the elements point and plane. The Geometry of the sheaf may also 
 be derived from that of the plane by the operation of projection from 
 a centre (Art. 2). 
 
 From the Geometry of the sheaf may be derived that of spherical 
 figures, by cutting the sheaf by a sphere passing through the centre 
 of the sheaf. 
 
CHAPTEE VII. 
 
 PROJECTIVE GEOMETRIC FORMS. 
 
 39. By means of projection from a centre we obtain from 
 a range a flat pencil, from a flat pencil an axial pencil, from a 
 plane of points or lines a sheaf of lines or planes. Con- 
 versely, by the operation of section by a transversal plane 
 we obtain from a flat pencil a range, from an axial pencil a 
 flat pencil, from a sheaf a plane figure. The two operations, 
 projection from a point and section by a transversal plane, 
 may accordingly be regarded as complementary to each other ; 
 and we may say that if one geometric form has been derived 
 from another by means of one of these operations, we can con- 
 versely, by means of the complementary operation, derive the 
 second form from the first. And similarly for the operations : 
 projection from an axis and section by a transversal line. 
 
 Suppose now that by means of a series of operations, each of 
 which is either a projection or a section, a form f 2 has been 
 derived from a given form^ , then another form/3 from / 2 , and 
 so on, until by n—\ such operations the form /„ has been 
 arrived at. Conversely, we may return from/„ to/ x by means 
 of another series of n— 1 operations which are complementary 
 respectively to the last, last but one, last but two, &c. of the 
 operations by which we have passed from^ to/ n . The series 
 of operations which leads from f x to /„ , and the series which 
 leads from f n to /j , may be called complementary, and the 
 operations of the one series are complementary respectively to 
 those of the other, taken in the reverse order. 
 
 In the above the geometric forms are supposed to lie in 
 space (Art. 31). If we confine ourselves to plane Geometry, the 
 complementary operations reduce to projection from a centre and 
 
34 PROJECTIVE GEOMETRIC FORMS. [40 
 
 section hy a transversal line. In the Geometry of the sheaf, 
 section by a plane and projection from an axis are comple- 
 mentary operations. 
 
 40. Two geometric prime forms of the same dimensions 
 are said to be protectively related, or simply projective, when one 
 can be derived from the other by any finite number of projec- 
 tions and sections (Arts 2, 3,... 7). 
 
 For example, let a range u be given ; project it from a 
 centre 0, thus obtaining a flat pencil ; project this flat pencil 
 from another centre 0\ by which means an axial pencil with 
 00' as axis is produced ; cut this axial pencil by a straight 
 line u 2 , thus obtaining a range of points lying on u 2 ; project 
 this range from an axis, and cut the resulting axial pencil by a 
 plane, by which means a flat pencil is produced, and so on ; then 
 any two of the one-dimensional geometric forms which have 
 been obtained in this manner are projective according to 
 definition. 
 
 When we say that a form A,B,C,1),... is projective with 
 another form A', B\ C\ D\ ... we mean that, by help of the 
 same series of operations, each of which is either a projection 
 or a section, A' is derived from A, B f from B, C from C, &c. 
 The elements A and A\ B and B\ C and £',.-• are termed 
 corresponding elements*. 
 
 For example, a plane figure is said to be projective with 
 another plane figure, when from the points A,B,C,... and from 
 the straight lines AB, AC..., BC,... of the one are derived 
 the points A',B\ G',.., and the straight lines A' B\ A' ' C' t ... 
 B'G\. . . of the other, by means of a finite number of projections 
 and sections. 
 
 In two projective plane figures, to a range in the one cor- 
 responds in the other a range which is projective with the 
 first range ; and to a flat pencil in the one figure corresponds 
 in the other a flat pencil which is projective with the first 
 pencil. 
 
 41. From what has been said above it is easy to see 
 that two geometric forms which are each projective with 
 
 * Two projective forms are termed homographic when the elements of which 
 they are constituted are of the same kind ; **. e. when the elements of both are 
 points, or lines, or planes. It will be seen later on (Art. 67) that this definition of * 
 homography is equivalent to that given by Chasles (Geometrie suptrieure, Art. 99). 
 
42] 
 
 PROJECTIVE GEOMETRIC FORMS. 
 
 35 
 
 a third are projective with one another. For if we first go 
 through the operations which lead from the first form to the 
 third, and then go through those which lead from the third to 
 the second, we shall have passed from the first form to the 
 second. 
 y/ 42. Geometric forms in perspective. 
 
 The following forms are said to be in perspective : 
 
 Fig. 20. 
 
 Two ranges (Fig. 19), if they are sections of the same fiat 
 pencil (Art. 12). 
 
 Two fiat pencils (Fig. 20), if they project, from different 
 centres, one and the same range; or if they are sections of 
 the same axial pencil. \ ^.c 
 
 [Note. — If we project a range u == ABC . . . from two different centres 
 and 0' not lying in the same plane with it, we obtain two flat 
 pencils in perspective. These pencils, again, may be regarded as 
 sections of the same axial pencil made by the transversal planes Ou, 
 Ou' '; the axial pencil namely which is composed of the planes 00' A, 
 00' B, 00' C, ..., and which has for axis the straight line 00'. This 
 is the general case of two flat pencils in perspective ; they have not the 
 same centre and they lie in different planes ; at the same time, they 
 project the same range and are sections of the same axial pencil. 
 There are two exceptional cases : (1). If we project the row u from 
 two centres and 0' lying in the same plane with u, then the 
 two resulting flat pencils lie in the same plane and are consequently 
 no longer sections of an axial pencil ; ( 2 ). If an axial pencil is cut by 
 two transversal planes which pass through a common point on the 
 axis, we obtain two flat pencils which have the same centre 0, and 
 which consequently no longer project the same range.] 
 
 Two axial pencils, if they project, from two different centres, 
 the same flat pencil. 
 
 A range and a fiat pencil, a range and an axial pencil, or aflat 
 pencil and an axial pencil, if the first is a section of the 
 second. 
 
 D % 
 
36 PROJECTIVE GEOMETRIC FORMS. [43 
 
 Two plane figures, if they are plane sections of the same 
 sheaf. 
 
 Two sheaves, if they project, from two different centres, the 
 same plane figure. 
 
 A plane figure and a sheaf, if the former is a section of the 
 latter. 
 / From the definition of Art. 40 it follows at once that two 
 (one-dimensional) forms which are in perspective are also pro- 
 tectively related ; but two projective forms are not in general 
 in perspective position. 
 / 43. Two figures in homology are merely two projective 
 plane figures superposed one upon the other, in a particular 
 position ; for by Art. 2 1 two homological figures may always 
 be regarded (and this in an infinite number of ways) as pro- 
 jections of one and the same third figure. 
 j If two projective plane figures are superposed one upon the 
 other in such a manner that the straight line connecting any 
 pair of corresponding points may pass through a fixed point ; 
 or, again, in such a manner that any pair of corresponding 
 straight lines may intersect on a fixed straight line ; then the 
 two figures are in homology (Arts. 19, 20). 
 
 In two homological figures, two corresponding ranges are in 
 perspective (and therefore of course are projectively related) ; 
 and the same is the case with regard to two corresponding 
 pencils. 
 
 44. Theorem. Two one-dimensional geometric forms, each con- 
 sisting of three elements, are always projective. 
 
 To prove this, we notice in the first place that it is 
 enough to consider the case of two ranges ABC, A'fi'C; for, 
 if one of the given forms is a pencil, fiat or axial, we may 
 substitute for it one of its sections by a transversal. ■ 
 
 (1) If the two straight lines ABC , A' B' C Y\z in different 
 planes, join AA',BB f ,CC f , and cut these straight lines 
 by a transversal **. Then the two given forms are seen 
 to be simply two sections of the axial pencil sAA' ', sBB\ 
 sCC f . 
 
 (2) If the two straight lines lie in the same plane (Fig. 21), 
 join AA', and take on this straight line any two points S, S'j 
 
 * To do this, we have only to draw through any point of AA' a straight line 
 which meets BB' and CC (Prob. 8, Art. 34). 
 
44.] 
 
 PROJECTIVE GEOMETRIC F0RM8. 
 
 ET] 
 
 37 
 
 draw SB , S'B' to cut in B", and SC , S'C to cut in C", and join 
 B"C", cutting SS' in J". Then A'B'C' may be derived from 
 
 5 u~vW 
 
 Fig. 21. 
 
 Fig. 2 2. 
 
 A5C by two projections, viz. we first project ABC from S 
 into ^"£"0";and then A"B"C" from £' into ^'£'0'. 
 
 (3) In the case where the two points A and .4' coincide (Fig. 
 2%), the two given forms are directly in perspective ; the centM. 
 of perspective is the point where BB f and CC intersect. 
 
 (4) If the two sets of points ABC^A'B'C lie on the same straight 
 line (Fig. 23), it is only necessary 
 to project one of them A'B'C on 
 to another straight line A 1 B 1 C 1 
 (from any centre 0); then let 
 any* two centres S and S 1 be 
 taken (as in Fig. 31) on AA lt 
 and let the straight line A"B"C" 
 be constructed in the manner 
 already shown in case (2). Then 
 A'B'C may be derived from 
 ABC by three projections, viz. 
 we first project ABC from S 
 into A"B"C", then A"B"C" from 
 
 S 1 into A^C^ and lastly A^B^ from into A'B'C. 
 
 (5) If A coincides with A', and B with B', we may mak e 
 use of a centre # and two transversals # x , s 2 drawn through ^4 
 in the plane SABCC. If the triad ABC be projected from # 
 upon s 1 (giving A^^), and the triad A'B'C be projected 
 from S upon s 2 (giving A 2 B 2 C 2 ) ; then the triads A X B X C X and 
 A 2 B 2 C 2 will be in perspective, because ^ coincides with A 2 (in 
 the point AA'). 
 
 In every case, then, it has been shown that the triads 
 
 Fig- 23. 
 
38 
 
 PROJECTIVE GEOMETRIC FORMS. 
 
 [45 
 
 Fig. 24. 
 
 ABC ,A'B'C can be derived from each other by a finite 
 number of projections and sections; therefore by Art. 40 
 they are projective. 
 
 As a particular case, ABC must be projective with BAC, for 
 example. In order actually to project one of these triads into 
 the other, take (Fig. 24) any two points L and N collinear 
 
 with C. Join AL ,BN, meeting 
 in K, and BL , AN, meeting in 
 M. Then BAC can be derived 
 from ABC by first projecting 
 ABC from K into LNC, and then 
 LNC from if into BAC. 
 
 In order to project ABC into 
 
 BCA, we might first project 
 
 ABC in BAC, and then BAC 
 
 into BCA. 
 
 45. Theorem. Any one-dimensional geometric form, consisting 
 
 of four elements, is projective with any of the forms derived from it 
 
 by interchanging the elements in pairs. For instance, ABCB is 
 
 projective with BADC. 
 
 Let A , B , C , B be four given points (Fig. 25), and let 
 B r> [ EFGB be aprojection of thesSfteints 
 
 from a centre M on a straighijuiine 
 BF passing through B. If AF , CM 
 meet in N, then MNGC will be a 
 projection of EFGB from centre A, 
 and BABC a projection of MNGC 
 from centre F; therefore (Arts. 
 40, 41) the form BABC is pro- 
 jective with ABCB. In a similar manner it can be shown 
 that CBAB and BCBA are projective with ABCB*. 
 
 From this it follows for example that if a flat pencil abed is 
 projective with a range ABCB, then it is projective also with 
 BABC, with CBAB, and with BCBA; i.e. if two geometric forms, 
 each consisting of four elements, are projectively related, then the 
 elements of the one can be made to correspond respectively to the 
 elements of the other in four different ways. 
 
 * Staudt, Geometric der Lage, Art. 59. 
 
 W 
 
 Fig. 25. 
 
CHAPTER VIII. 
 
 HARMONIC FORMS. 
 
 J 46. Theorem*. 
 
 Given three points A, B, C on 
 a straight line s) if a complete 
 quadrangle (KLMX) be con- 
 structed (in any plane through s) 
 in such a manner that two oppo- 
 site sides (KL , MX) meet in A y 
 two otheropposite sides (KX, ML) 
 meet in B, and the fifth side (LX) 
 passes through C, then the sixth 
 side (it J/) will cut the straight 
 line 8 in appoint D which is de- 
 termined by the three given 
 points ; i.e. it does not change its 
 position, in whatever manner the 
 arbitrary elements of the quad- 
 rangle are made to vary (Fig. 
 26). 
 
 Given in a plane three straight 
 lines a, b. c which meet in a point 
 S ; if a complete quadrilateral 
 (Jclmn) be constructed in such a 
 manner that two opposite vertices 
 (kl, mn) lie on a, two other oppo- 
 site vertices (hi , ml) lie on b, and 
 the fifth vertex (nl) lies on c, 
 then the sixth vertex (&;?i) will 
 lie on a straight line d which 
 passes through S, and which is 
 determinate ; t. e. it does not 
 change its position, in whatever 
 manner the arbitrary elements of 
 the quadrilateral are made to 
 vary (Fig. 27). 
 
 Fig. 2;. 
 
 For if a second complete For if a second complete 
 
 quadrangle (K'L'M'X') be cen- quadrilateral {k'l'm'n ') be con- 
 
 * Staudt, loc. cit., Art.. 93. 
 
40 
 
 HARMONIC FORMS. 
 
 [46 
 
 < 
 
 structed (either in the same plane, 
 or in any other plane through s), 
 which satisfies the prescribed con- 
 ditions, then the two quadrangles 
 will have five pairs of correspond- 
 ing sides which meet on the given 
 straight line ; therefore the sixth 
 pair will also meet on the same 
 line (Art. 36, No. 5, left). 
 
 From this it follows that if the 
 first quadrangle be kept fixed 
 while the second is made to vary- 
 in every possible way, the* point 
 D will remain fixed ; which 
 proves the theorem. 
 ^ The four points A BCD are 
 called harmonic, or we may say 
 that the group or the geometric 
 form constituted by these four 
 points is a harmonic one, or that 
 A BCD form a harmonic range. 
 Or again: Four points A BCD of 
 a straight line, taken in this order, 
 are called harmonic, if it is pos- 
 sible to construct a complete quad- 
 rangle such that two opposite sides 
 pass through A, two other opposite 
 sides through B, tlie fifth side 
 through C, and the sixth through D. 
 It follows from the preceding theo- 
 rem that when such a quadrangle 
 exists, t. e. when the form ABCD 
 is harmonic, it is possible to con- 
 struct an infinite number of other 
 quadrangles satisfying the same 
 conditions. It further follows that, 
 given three points ABC of a 
 range (and also the order in which 
 they are to be taken), the fourth 
 point D, which makes with them 
 a harmonic form, is determinate 
 and unique, and is found by the 
 construction of one of the quad- 
 rangles (see below, Art. 58). 
 
 structed which satisfies the pre- 
 scribed conditions, then the two 
 quadrilaterals will have five pairs 
 of corresponding vertices collinear 
 respectively with the given point ; 
 therefore the sixth pair will also 
 lie in a straight line passing 
 through the same point (Art. 36, 
 No. 5, right). 
 
 From this it follows that if the 
 first quadrilateral be kept fixed 
 while the second is made to vary 
 in every possible way, th straight 
 line t&^ill remain fixed ; which 
 proves the theorem. 
 
 The four straight lines or rays 
 abed are called harmonic, or we 
 may say that the group or the 
 geometric form constituted by 
 these four lines is a harmonic 
 one, or that abed form a harmonic 
 pencil. Or again : Four rays 
 abed of a pencil, taken in this 
 order, are called harmonic, if it is 
 possible to construct a complete 
 quadrilateral such that two oppo- 
 site vertices lie on a, two other 
 opposite vertices on b, the fifth 
 vertex on c, and the sixth on d. It 
 follows from the preceding theo- 
 rem that when such a quadri- 
 lateral exists, i.e. when the form 
 abed is harmonic, it is possible to 
 construct an infinite number of 
 other quadrilaterals satisfying the 
 same conditions. It further 
 follows that given three rays abc 
 of a pencil (and also the .order in 
 which they are to be taken), the 
 fourth ray d, which makes with 
 them a harmonic form, is deter- 
 minate and unique, and is found 
 by the construction of one of the 
 quadrilaterals (see below, Art. 58). 
 
48] HARMONIC FORMS. 41 
 
 rf47. If from any 'point S the harmonic range ABCB be projected- / 
 upon any other straight line, its projection A'B'C'B' will also be a- 1 l * 
 harmonic range (Fig. 28). 
 
 Imagine two planes drawn one through each of the straight 
 lines AB , A'B', and suppose that in the first of these planes 
 is constructed a complete quadrangle 
 of which two opposite sides meet in 
 A, two other opposite sides meet in B, 
 and a fifth side passes through C\ 
 then the sixth side will pass through 
 D (Art. 46), since by hypothesis ABCB Fi gt 2 8. 
 
 is a harmonic range. Now project 
 
 this quadrangle fr om A r point S on to the second plane ; then 
 a new quadrangle is obtained of which two opposite sides 
 meet in A f , two other opposite sides meet in B', and whose 
 fifth and sixth sides pass respectively through C f and B' ; 
 therefore A'B'C'B' is a harmonic range^^ 
 
 48. An examination of Fig. 27 will show that the harmonic 
 pencil abed is cut by any transversal whatever in a har- 
 monic range. For let 8 be the centre of the pencil and m be 
 any transversal ; in a take any point R ; join R to B by the 
 straight line k and to B by the straight line I; and join A to 
 hb or P by the straight line n. As abed is a harmonic pencil 
 and five vertices of the complete quadrilateral Hmn lie on a, b, 
 and d, the sixth vertex In or Q must lie on the fourth ray c. 
 Then from the complete quadrangle PQRS it is clear that 
 ABCB is a harmonic range. 
 
 Conversely, if the harmonic range ABCB (Fig. 27) be given, 
 and any centre whatever of projection S be taken, then the 
 four projecting rays S(A,B,C,B) will form a harmonic 
 pencil. 
 
 For draw through A any straight line to cut SB in P and 
 SC in Q, and join BQ, cutting AS in R. The quadrangle PQRS 
 is such that two opposite sides meet in A, two other opposite 
 sides in B, and the fifth side passes through C ; consequently 
 the sixth side must pass through B (Art. 46, left), since by 
 hypothesis the range ABCB is harmonic. But then we have 
 a complete quadrilateral Tclmn which has two opposite vertices 
 A and R lying on SA, two other opposite vertices B and P on 
 SB, a fifth vertex Q on SC, and the sixth B on SB; therefore 
 
42 HARMONIC FORMS. [40 
 
 (Art. 46, right) the four straight lines which project the range 
 ABCD from S are harmonic. We may therefore enunciate the 
 following proposition : 
 
 A harmonic pencil is cut by any transversal whatever in a 
 harmonic range ; and, conversely, the rays ivhich project a harmonic 
 range from any centre whatever form a harmonic pencil. 
 
 Corollary. In two homological figures, to a range of four harmonic 
 points corresponds a range of four harmonic points ; and to a pencil 
 of four harmonic rays corresponds a pencil of four harmonic rays. 
 
 49. The theorem on the right in Art. 46 is correlative to 
 that on the left in the same Article. In this latter theorem 
 all the quadrangles are supposed to lie in the same plane ; but 
 from the preceding considerations it is clear that the theorem 
 is still true and may be proved in the same manner, if the 
 quadrangles are drawn in different planes. 
 
 Considering accordingly this latter theorem (Art. 46, left) 
 as a proposition in the Geometry of space, the theorem corre- 
 lative to it will be the following : 
 
 If three planes a , /3 , y all pass through one straight line s, and if 
 a complete four -flat (see Art. 37) k\\w he cofistructed, of which two 
 opposite edges k\ , \lv lie in the plane a, two other opposite edges kv , \fx 
 lie in the plane ft, and the edgekv lies in the plane y ; then the sixth 
 edge kjjl will always lie in a fixed plane b (passing through s), which 
 does not change, in whatever manner the arbitrary elements of the 
 four-flat he made to vary. 
 
 For if we construct (taking either the same vertex or any 
 other lying on s) another complete four-flat which satisfies the 
 prescribed conditions, the two four-flats will have five pairs of 
 corresponding edges lying in planes which all pass through 
 the same straight line s; therefore (Art. 37, left) the sixth pair 
 also will lie in a plane which passes through s. The four 
 planes, a , /3 , y , 5 are termed harmonic planes ; or we may 
 say that the group or the geometric form constituted by 
 them is harmonic ; or again that they form a harmonic (axial) 
 pencil. 
 
 50. If a complete four-flat k\\xv be cut by any plane not 
 passing through the vertex of the pencil, a complete quadri- 
 lateral is obtained ; and the same transversal plane cuts the 
 planes a , j3 , y , 5 in four rays of a flat pencil of which the first 
 
51] HARMONIC FORMS. 43 
 
 two rays contain each a pair of vertices of the quadrilateral 
 while the other two pass each through one of the remaining 
 vertices. Consequently (Art. 46, right) an axial pencil of four 
 harmonic planes is cut by any transversal plane in a flat pencil 
 of four harmonic rays. 
 
 Similarly, if the harmonic axial pencil of four planes 
 a, (3, y,b is cut by any transversal line in four points A, B, C, B, 
 these form a harmonic range. For if through the transversal 
 line a plane be drawn, it will cut the planes a , /3 , y , 5 in four 
 straight lines a , b , c , d. This group of straight lines is har- 
 monic, by what has just been proved; but ABCB is a section 
 of the flat pencil a, b,c, d; consequently (Art. 48) the four 
 points A , B , C , D are harmonic. Conversely, if four points 
 forming a harmonic range be projected from an axis, or if four 
 rays forming a harmonic pencil be projected from a point, the 
 resulting axial pencil is harmonic. 
 
 51. If then we include under the title of harmonic form the 
 group of four harmonic points (the harmonic range), the group 
 of four harmonic rays (the harmonic flat pencil), and the 
 group of four harmonic planes (the harmonic axial pencil), we 
 may enunciate the theorem : 
 
 Every projection or section of a harmonic form is itself a harmonic 
 form: or, 
 
 Every form tvhich is projective with a harmonic form is itself 
 harmonic. 
 
 Conversely, two harmonic forms are ahc ays projective with one 
 another. 
 
 To prove this proposition, it is enough to consider two 
 groups each of four harmonic points ; for if one of the forms 
 were a pencil we should obtain four harmonic points on cut- 
 ting it by a transversal. Let then ABCB, A'B'C'B' be two 
 harmonic ranges, and project ABC into A'B'C in the manner 
 explained in Art. 44; the same operations (projections and 
 sections) which serve to derive A'B'C from ABC will give for 
 B a point B x \ from which it follows that the range A'B'C'B t 
 will be harmonic, since the range ABCB is harmonic. But 
 A'B'C'B' are also four harmonic points, by hypothesis ; there- 
 fore B x must coincide with B', since the three points A'B'C 
 determine uniquely the fourth point which forms with 'them a 
 harmonic range (Art. 46, left). 
 
44 HARMONIC FORMS. [52 
 
 We may add here a consequence of the definitions given in 
 Arts. 49 and 50 : 
 
 The form which is correlative to a harmonic form is itself 
 
 52. If a , b, c , d are rays of a pencil (Fig. 28), then a and b 
 are said to be separated by c and d, when a straight line pass- 
 ing through the centre of the pencil, and rotating so as to 
 come into coincidence with each of the rays in turn, cannot 
 pass from a to b without coinciding with one and only one of 
 the two other rays c and d *. The same definition applies to 
 the case of four planes of a pencil, and to that of four points of 
 a range (Fig. 26) ; only it must be granted that we may pass 
 from a point A to a point B in two different ways, either by 
 describing the finite segment AB or the infinite segment which 
 begins at A, passes through the point at infinity, and ends at B. 
 
 This definition premised, the follow- 
 
 ° ° ing property may be enunciated as at 
 
 < £ ° < once evident: Four elements of a one- 
 
 Fig. 29. dimensional geometric form [i. e. four 
 
 points of a range, four rays of a 
 pencil, &c.) can always be so divided into two pairs that 
 one pair is separated by the other, and this can be done in 
 one way only. In Fig. 26, for example, the two pairs which 
 separate one another are AB , CD ; and if A'B'G'D' is a form 
 projective with ABC I), the pair A'B' will be separated by the 
 pair CD' \ for the operations of projection and section do not 
 change the relative position of the elements. 
 
 53. Let now ABCJD (Fig. 30) be four harmonic points, i. e. four 
 points obtained by the construction of Art. 46, left. This 
 allows us to draw in an infinite number of ways a complete 
 quadrangle of which A and B are two diagonal points 
 (Art. 36, No. 2, left), while the other two opposite sides pass 
 through C and B. It is only necessary to state this con- 
 struction in order to see that the two points A and B are 
 precisely similar in their relation to the system, and that the 
 same is true with regard to C and B. It follows from this 
 that if ABCB is a harmonic range, then BACB, ABBC, BABC, 
 which are obtained by permuting the letters A and B or C 
 and B, or both at the same time, are harmonic ranges also. 
 
 * a and b, c and d, may also be termed alternate pairs of rays. 
 
55] HARMONIC FOEMS. 45 
 
 Consequently (Art. 51) the harmonic range AB CD for example 
 is projective with BACB, i. e. we can pass from one range to the 
 other by a finite number of projections and sections. In fact 
 if the range ABCB be projected from K on CQ, we obtain the 
 range LNCQ, which when projected from M on AB gives 
 BACB. 
 
 54. If A , B , C , B are four harmonic points, then A and B are 
 necessarily separated by C and B. 
 
 For if (Fig. 30) the group ABCDbe projected on the straight 
 line KM, first from the centre L and then from the centre N, 
 the projections are KMQB A c b d 
 
 and MKQB respectively. * ~" "^^T" ~T\ ~/f[ ~^^ 
 These two groups of \/ ^^tyds^'s' 
 
 points, consisting as they \ y^^^y^ 
 
 do of the same elements, mx" J/Jr y 
 
 must show the same kind vC \ I 
 
 of arrangement; therefore . Nl 
 
 the points K and M are Fig 3 , 
 
 separated by Q and B, 
 and therefore A and B are separated by C and B. 
 
 55. Let the straight lines AQ , BQ be drawn (Fig. 31), the. 
 former meeting MB in U and NB in S, while the latter meets 
 KL in T and MN in V. The complete quadrangle LTQU has 
 two opposite sides meeting in A, two other opposite sides 
 meeting in B, and a fifth side (LQ or LN) passes through C ; 
 therefore the sixth side UT will pass through B (Art. 46). In 
 like manner the sixth side V8 of the complete quadrangle 
 NYQ8 must pass through B, and the sixth sides of the com- 
 plete quadrangles KSQT , M UQ V through C. We have thus a 
 quadrangle STUV two of whose opposite sides meet in C, two 
 
46 HARMONIC FORMS. [56 
 
 other opposite sides in B, while the fifth and sixth sides pass 
 respectively through A and B. This shows that the relation 
 to which the points C and B are subject (Art. 53) is the same 
 as the relation to which the points A and B are subject ; or, 
 in other words, that the pair A , B may be interchanged 
 with the pair C , B. Accordingly, if ABCB is a harmonic 
 range, then not only the ranges BACB , ABBC , BABC } but 
 also CBAB,BCAB, CBBA,BCBA are harmonic*. s 
 
 The points A and B are termed conjugate points, as also are 
 C and B. Or either pair are said to be harmonic conjugates 
 with respect to the other. The points A and B are said to be 
 harmonically separated by the points C and B, or the points G 
 and B to be harmonically separated by A and B. We may 
 also say that the segment AB is divided harmonically by the 
 segment CB, or that the segment CB is divided harmonically 
 by AB. If two points A and B (Fig. 30) are separated har- 
 monically by the points C and B in which the straight line 
 AB is cut by two straight lines QC and QB, we may also say 
 that the segment AB is divided harmonically by the straight 
 lines QC , QB, or by the point Cand the straight line QB,&c; 
 and that the straight lines QC , QB are separated harmonically 
 by the points A, B\ &c. 
 
 Analogous properties and expressions exist in the case of 
 four harmonic rays or four harmonic planes. 
 
 [Note. — In future, whenever mention is. made of the harmonic 
 system ABCB, it is always to be understood that A and B, C and D, are 
 conjugate pairs ; it being at the same time remembered that (Art. 54) 
 A and B, C and B, are necessarily alternate pairs of points.] 
 
 ^rJ 56. The following theorem is another consequence of the 
 proposition of Art. 46, left: 
 
 hi a complete quadrilateral, each 
 diagonal is divided harmonically by 
 the other two f. 
 
 Let A and A', B and B', C 
 
 and C be the pairs of opposite 
 
 vertices of a complete quadri- 
 
 lg ' 32 " lateral (Fig. 32), and -let the 
 
 diagonal AA'he cut by the other diagonals BB' and CC in F 
 
 * Reye, Geometric der Lage (Hanover, 1866), vol. i. p. 34. 
 t Carnot, Gtomttrie de position (Paris, 1803), Art. 225. 
 
58] 
 
 HARMONIC FORMS. 
 
 47 
 
 and E respectively. Consider now the complete quadrangle 
 BB'CC '; one pair of its opposite sides meet in A, another 
 such pair in A', a fifth side passes through E, the sixth 
 through F. The points A , A' are therefore harmonically 
 separated by F and E. Similarly a consideration of the two 
 complete quadrangles CC'AA' and AA'BB' will show that 
 B , B f are harmonically separated by i^and B\ and C , C by 
 B and E. 
 
 57. In the complete quadrangle BB'CC' the diagonal points 
 are A , A', and J^;..also since the range BB'FB is harmonic, so 
 too is the pencil of four rays which project it from A (Art. 48) ; 
 therefore : 
 
 In a complete quadrangle, any two sides which meet in a diagonal 
 point^are divided harmonically hy the two other diagonal points. 
 
 This theorem is merely however the correlative (in accord- 
 ance with the principle of Duality in plane Geometry) of that 
 proved in the preceding Article. 
 
 58. The theorems of Art. 46 can be at once applied to the 
 solution, by means of the ruler only, of the following pro- 
 blems : 
 
 Given three points of a har- 
 monic range, to find the fourth. 
 
 Solution. Let A , B , C (Fig. 
 33) be the given points (lying 
 on a given straight line) and let 
 
 Given three rays of a har- 
 monic 'pencil, to construct the 
 fourth. 
 
 Solution. Let a , b , c (Fig. 
 34) be the given rays (lying in 
 one plane and passing through a 
 
 Fig. 33- 
 
 A and B be conjugate to each 
 other. Draw any two straight 
 lines through A, and a third 
 through C to cut these in L and 
 
 .Fig. 34- 
 given centre S), and let a and b 
 be conjugate to each other. 
 Through any point Q lying on c 
 draw any two straight lines to 
 
48 
 
 HARMONIC FORMS. 
 
 [59 
 
 N respectively. Join BL cutting 
 AN in M, and BN cutting AL in 
 K ; then if KM be joined it will 
 cut the given straight line in the 
 required point D, conjugate to 
 
 c*. 
 
 cut a in A and R, and b in P 
 and B, respectively. Join AB and 
 RP ; these will cut in a point D, 
 the line joining which to S is the 
 required ray d, conjugate to c. 
 
 Kg. 35- 
 
 59. In the problem of Art. 58, left, let C lie midway between A 
 and B. We can, in the solution, so arrange the arbitrary elements 
 
 that the points K and M shall- move off 
 to infinity ; to effect this we must con- 
 struct (Fig. 35) a parallelogram ALBN 
 on AB as diagonal ; then since the other 
 diagonal LN passes through C, the point 
 D will lie at infinity. 
 
 If, conversely, the points A , B , D are 
 
 given, of which the third point D lies 
 
 at infinity, we may again construct a 
 
 parallelogram ALBN on AB as diagonal ; then the fourth point C, 
 
 the conjugate of D, must be the point where LN meets the given 
 
 straight line : that is, it must be the middle point of AB. Therefore,; 
 
 If in a harmonic range ABCD 
 the point C lies midway between the 
 two conjugates A and B, then the fourth 
 point D lies at an infinite distance; 
 and conversely, if one of the points D 
 lies at infinity, its conjugate C is the 
 point midway between the two others, 
 A and B. 
 
 60. In the problem of Art. 58, 
 right,, let c be the bisector of the 
 angle between a and b. (Fig. 36). If 
 Q be taken at infinity on c, the segments AB , PR become equal to 
 one another and lie between the parallels 
 A P , BR ; consequently the ray d will be 
 perpendicular to c, i.e. given a harmonic 
 pencil of four rays, abed; if one of 
 them c bisect the angle between tJie two 
 conjugates a and b, the fourth ray d 
 will be at right angles to c. 
 Conversely : if in a harmonic pencil abed 
 (Fig- 37) two conjugate rays c , d are at right angles, then they are the 
 bisectors, internal and external, of the angle between the oilier two rays 
 a,b. 
 
 * De la Hire, Sectiones Conicae (Parisiis, 1685), lib. i, prQp. ao. 
 
 Fig. 36. 
 
 d N^X d 
 
 h/ C \B 
 
 £ V 
 
 Fig. 37. 
 
60] 
 
 HARMONIC FORMS. 
 
 49 
 
 
 For if the pencil be cut by a transversal AB drawn parallel to d, 
 the section A BCD will be a harmonic range (Art. 48) ; and as D 
 lies at infinity, C must lie midway between A and B (Art. 59) ; conse- 
 quently, if S be the centre of the pencil, A SB is an isosceles triangle 
 and SC the bisector of its vertical angle, 
 
CHAPTEK IX. 
 
 ANHARMONIC RATIOS. 
 
 J 61. Geometrical propositions divide themselves into two 
 classes. Those of the one class are either immediately con- 
 cerned with the magnitude of figures, as Euc. I. 47, or they 
 involve more or less directly the idea of quantity or measure- 
 ment, as e.g. Euc. I. 12. Such proposition : called nietrieal. 
 The other class of propositions relate merely to the position 
 of the figures with which they deal, and the idea of quantity 
 does not enter into them at all. Such propositions are called 
 descriptive. Most of the propositions in Euclid's Elements are 
 metrical, and it is not easy to find among them an example of 
 a purely descriptive theorem. Prop. 2, Book XI, may serve 
 as an instance of one. Projective Geometry on the other 
 hand, dealing with projective properties (i. e. such as are not 
 altered by projection), is chiefly concerned with descriptive 
 properties of figures^ In fact, since the magnitude of a geo- 
 metric figure is altered by projection, metrical properties are 
 as a rule not projective. But there is one important class of 
 metrical properties (anharmonic properties) which are pro- 
 jective, and the discussion of which therefore finds a place in 
 the Projective Geometry. To these we proceed; but it is 
 necessary first to establish certain fundamental notions. 
 
 62. Consider a straight line ; a point may move along it in 
 two different directions, one of which is opposite to the other. 
 Let it be agreed to call one of these the positive direction, and 
 the other the negative direction. Let A and B be two points 
 on the straight line ; and let it be further agreed to represent 
 by the expression AB the length of the segment comprised 
 between A and B, taken as a positive or as a negative number 
 of units according as the direction is positive or negative in 
 which a point must move in order to describe the segment ; 
 this point starting from A (the first letter of the expression 
 AB) and ending at B. 
 
62] ANHARMONIC RATIOS. 51 
 
 In consequence of this convention, which is termed the rule 
 
 of signs, the two expressions AB , BA are quantities which are 
 
 equal in magnitude but opposite in sign, so that BA = — AB, or 
 
 AB+BA = (1) 
 
 Now let A, B, C be three points lying on a straight line. If 
 C lies between A and B (Fig. 38 a), 
 
 (a) 
 
 c 
 
 B 
 
 B 
 
 c 
 
 A 
 
 (i) K 
 
 B 
 
 C 
 
 c 
 
 B 
 
 A 
 
 (c)° 
 
 A 
 
 B 
 
 B AC 
 
 Fig. 38. 
 
 we have AB = AC+ CB ;- 
 
 whence - CB-AC+ AB = 0, 
 
 or BC+CA + AB=:0. 
 
 Again, if iTlies between A and C (Fig. 38 b), 
 
 AC=AB + BC; 
 whence BC-AC+AB = 0, 
 
 or BC+CA + AB=0. 
 
 Lastly, if A lies between B and C (Fig. 38 c), 
 
 CB=CA + AB; 
 whence -CB+CA + AB = 0, 
 
 or BC+CA + AB= 0. 
 
 Accordingly : 
 
 If A,B, C are three collinear points, then whatever their relative 
 positions may be, the identity 
 
 BC + CA + AB=zO (2) 
 
 always holds good. 
 
 From this identity may be deduced an expression for the 
 distance between two points A and B in terms of the distances 
 
 E % 
 
52 ANHARMONIC RATIOS. [63 
 
 of these points from an origin chosen arbitrarily on the 
 straight line which joins them. 
 
 For since OA + AB + B0 = 0, 
 
 .-. AB = OB-OA;. (3) 
 
 or again, AB = AO+0B*. * 
 
 The results (l) and (2) may be extended ; they are in fact 
 particular cases of the following general proposition : 
 If A x , A 2 ,... A n be n collinear points, then 
 
 A X A 2 + A 2 A Z + ... + A n _ x A n + A n A x = 0, 
 the truth of which follows at once from (3), since the expres- 
 sion on the left hand is equal to 
 
 . . (OA 2 - OA,) + (OA 3 - OA 2 ) + . . . +(OA 1 -OA n ), ' 
 which vanishes. 
 
 Another useful result is that if A, B, C, B be four collinear 
 points, BC.AB+CA.BB + AB.QB= 0. 
 
 This again follows from (3), since the left-hand side 
 = (BCr-BB)AB+. .. + ... 
 = 0. 
 Many other relations of a similar kind between segments 
 might be proved, but they are not necessary for our purpose. 
 We will give only one more, viz. 
 
 If A, B , C , be any four collinear points, then 
 
 OA 2 . BC+ OB 2 . CA+OC 2 . AB = -BC. CA . AB. 
 For by (3) the left-hand side is equal to 
 
 ( oa 2 - oc 2 ) bc +(ob 2 - oc 2 ) ca 
 
 = ca(oa+oc)bc+cb(ob+oc)ca 
 = bc.ca(oa-ob) 
 
 = -BC.CA.AB. 
 It may be noticed that this last theorem is true even if do 
 not lie on the straight line ABC, but be any point whatever. 
 For if a perpendicular 00' be let fall on ABC, 
 OA 2 .BC+OB 2 . CA+OC 2 . AB 
 = (00 /2 + O'A 2 ) BC "+ ... + ... 
 = O'A 2 . BC + O'B 2 .CA+O'C 2 . AB 
 
 + 00' 2 (BC + CA + AB) 
 = -BC.CA.AB, 
 by what has just been proved. 
 
 63. Consider now Fig. 39, which represents the projection 
 
 * Mobius, Barycentrische Calcul, § 1. 
 
63] ANHA.KMONIC BATIOS. 53 
 
 J from a centre S of the points of a straight line a on to another 
 straight line a' '; let us examine the relation which exists 
 between the lengths of two corresponding segments AB } A'B'. 
 
 fa 
 
 I 
 
 - r 
 
 v 
 
 
 Eig. 39. Fig. 40. 
 
 From the similar triangles SAJ , A f SV 
 JA:J£::rS:FA';* 
 so from the similar triangles SB J, B'SI\ 
 JB: JS ::I'S \TB'\ 
 .-. J A J'A' = JB. TB' = JS . I'&i 
 i.e. the rectangle JA.I'A' has a constant value for all pairs 
 of corresponding points A and A '. 
 
 If the constant JS.I'S be denoted by k, we have 
 
 therefore by subtraction, 
 
 I'V-FA' = '<%-??. 
 J A .JB 
 
 But I'B'-I'A' = ^'5', and JA-JB = BA = -AB; 
 
 ••• ^'=jA-^- 
 
 If we consider four points A , B , C , D (Fig. 40) of the 
 straight line a and their four projections, A\B\C\1)\ we 
 obtain, in a similar manner, 
 
 * We suppose that in all equations involving segments the rule of signs is 
 observed. See Mobius, Baryc. Calcul, § 1 ; Townsend, Modem Geometry, 
 chapter v. 
 
• 
 
 54 ANHARMONIC RATIOS. [63 
 
 —h 
 
 B ' c ' = 7STTc' BC ' 
 
 B '"-jF!n>' BI) > 
 
 whence by division 
 
 A'C\A'B' _ AC % AB 
 ~Wc r 'B r B , ~~ BC' BB % 
 
 This last equation, which has been proved for the case of 
 
 projection from a centre S, holds also for the case where 
 
 ABC I) and A'B'C'B' are the intersections of two transversal 
 
 lines s and *' (not lying in the same plane) with four planes 
 
 a, (3, y , b which all pass through one straight line u ; in other 
 
 words, when A'B'C'B' is a projection of ABCB made from an 
 
 axis u (Art. 4). For let the four planes a , j3 , y , b be cut in 
 
 A", B", C", B" respectively by a straight line %" which meets 
 
 s and /. The straight lines AA'\ BB" y C€", BB" are [the 
 
 intersections of the planes a , (3 , y , b respectively by the plane 
 
 mt' % and therefore meet in a point S ; that namely in which 
 
 the plane ss" is cut by the axis u. So also A' A", B'B", C'C'\ 
 
 B'B" are four straight lines lying in the plane *V' and 
 
 meeting in a point 8' of the axis u (that namely in which the 
 
 plane //' is cut by the axis u.) Therefore A"B"C"B" is a 
 
 projection of ABCB from centre S and a projection A'B'C'B' 
 
 from centre S' ; so that 
 
 A"C'\ A"B" _ AC , AB _ A'C\A'B f 
 
 B"C" '' B"B" ~ BC : BB~~ B'C ' B'B' ' 
 
 _ , AC AB 
 
 Ine number -jr^ : -^r- 
 
 is called the anharmonic ratio of the four collinear points 
 A , B , C , B. The result obtained above may therefore be 
 expressed as follows : 
 
 The anharmonic ratio of four collinear points is unaltered ly any 
 projection whatever *. 
 
 * Pappus, Mathemalicae Collecliones, book vii. prop. 129 (ed. Hultsch, Berlin, 
 1877, vol.ii. p. 871). 
 
65] ANHARMONIC RATIOS. 55 
 
 Or again : 
 
 If two ranges, each of four points, are projective, they have the 
 same anharmonic ratio, or, as we may say, are equianharmonic *. 
 
 64. Dividing one by the other the expressions for A'C and 
 B'C, we have 
 
 A'C _ AC.AJ 
 
 B'C~ BC'BJ' 
 In this equation the right-hand member is the anharmonic 
 ratio of the four points A , B , C ,J ; consequently the left-hand 
 member must be the anharmonic ratio of A', B', C, J'-, thus 
 the anharmonic ratio of four points A',B',C, J', of which the last 
 lies at infinity, is merely the simple ratio A'C : B'C. 
 
 This may also be seen by observing that if A' and B' 
 remain fixed while JD' moves off to infinity on the line A'B', 
 then 
 
 A'B' 
 limiting value of wrfy = 1 5 
 
 ,. ... , . A'C A'B' A'C 1 
 
 .-. limiting value of -^, : j^, = ^/ • 
 
 Similarly, on the same supposition, - 
 
 ., ... , £ A'B' A'C B'C 
 
 limiting value of j^, : ^j^- f = j^ ; 
 
 i.e. the anharmonic ratio of the four points A',B' ,B', C, of which 
 the third lies at infinity, is equal to the simple ratio B'C : A'C. 
 
 65. From this results the solution of the following 
 Problem. — Given three collinear points A , B, C; to find a fourth 
 
 B so that the anharmonic ratio of the range ABCB may be a 
 numher A given in sign and magnitude (Fig. 41). 
 
 Solution. — Draw any transversal through C, and take on 
 it two points A', B' such that the 
 ratio CA' : CB' is equal to A :. 1^ the 
 given value of the anharmonic 
 ratio ; the two points A' and B' 
 lying on the same or on opposite 
 sides of C according as A is positive 
 or negative. Join AA ', BB ', meeting 
 
 in S; the straight line through S parallel to A'B' will cut AB 
 in the point B required f. For if B' be the point at infinity on 
 
 * Townsend, Modern Geometry, Art. 278. 
 
 •j* Chasles, Geometrie superieure (Paris, 1852), p. 10. 
 
56 ANHARMONIC RATIOS. [66 
 
 A'B', and we consider ABCB as a projection of A'B'C'B' 
 (C coincides with C) from the centre S, then the anharmonic 
 ratio of ABCB is equal to that of A'B'C'B', that is, to the 
 simple ratio A'C : B'C or A. 
 
 The above is simply the graphical solution of the equation 
 
 AC AB _ 
 
 BC : BB~ ' 
 
 AB AC % 
 or -^ji = ~Wn : A = f*, 
 
 BB BC 
 
 or in other words of the problem : 
 
 Given two points A and B } to find a point B collinear with them 
 such that the ratio of the segments AB, BB to one another may he 
 equal to a number given in sign and magnitude. 
 
 As only one such point B can be found, the proposed 
 problem admits of only one solution ; this is also clear 
 from the construction given, since only one line can be drawn 
 through 8 parallel to A'B'. Consequently there cannot be 
 two different points B and B L such that ABCB and ABCB^ 
 have the same anharmonic ratio. Or : 
 
 If the groups ABCB, ABCB 1 are eqziianharmonic, the point B x 
 must coincide with B. 
 
 66. Theorem. (Converse to that of Art. 63.) If two ranges 
 ABCB, A'B'C'B', each of four points, are equianharmonic, they afe 
 projective with one another. 
 
 For (by Art. 44) we can always pass from the triad 
 ABC to the triad A' B'C by a finite number of projections or 
 sections ; let B" be the point which these operations give as 
 corresponding to B. Then the anharmonic ratio of A'B'C'B" 
 will be equal to that of ABCB, and consequently to that of 
 A'B'C'B'-, whence it follows that B " coincides with B ', and that 
 the ranges ABCB , A'B'C'B' are projective with one another. 
 
 67. It follows then from Arts. 63 and 66 that the necessary 
 and sufficient condition that two ranges ABCB , A'B'C'B\ 
 consisting each of four points, should be projective, is the 
 equality (in sign and magnitude) of their anharmonic ratios. 
 
 The anharmonic ratio of four points ABCB is denoted by 
 the symbol (ABCB)*; accordingly the projectivity of two 
 forms ABCB and A'B'C'B' is expressed by the equation 
 (ABCB) = (A'B'C'B'). 
 
 * Mobius, Barycentrische Calcul, § 183. 
 
69] ANHARMONIC RATIOS. 57 
 
 From what has been proved it is seen that if two pencils 
 each consisting of four rays or four planes are cut by any two 
 transversals in ABCD and A'B'C'D' respectively, the equation 
 (ABCD) = (A'B'C'D') is the necessary and sufficient condition 
 that the two pencils should be projective with one another. 
 v/ The anharmonic ratio of a pencil of four rays a ,b , c ,d or 
 four planes a , /3 , y , 8 may now be defined as the constant 
 anharmonic ratio of the four points in which the four elements 
 of the pencil are cut by any transversal, and may be denoted 
 by (abed) or (a(3yb). 
 
 This done, we can enunciate the general theorem : 
 If two one-dimensional geometric forms ; consisting each of four 
 elements^ are projective, they are equianharmonic ; and if they are 
 equianharmonic, they are projective. y 
 
 68. Since two harmonic forms are always projectively 
 related (Art. 51), the preceding theorem leads to the con- 
 clusion that the anharmonic ratio of four harmonic elements 
 is a constant number. For if ABCD is a harmonic system, 
 BACD is also a harmonic system (Art. 53), and the two 
 systems ACIID and BC4-D are projectively related*; thus 
 
 (AC^D) = (BCAB), 
 AB \AB _BABB % 
 l ' 6 ' CB' CB~ PA : CB ; 
 
 , AC AB 
 
 Wh6nce BC : BB=- 1 > 
 
 t i.e. (ABCB) = -1; 
 
 therefore the anharmonic ratio of four harmonic elements is equal 
 to -If. 
 
 69. The equation (ABCD) = - 1, or 
 
 AC AD _ M 
 
 BC + BD = °> W 
 
 which expresses that the range ABCD is harmonic, may be put into 
 two other remarkable forms. 
 
 Since AD = CD-CA (Art. 62) and BD = CD-CB, the equation 
 (1) gives 
 
 CA (CD-CB) + CB (CD-CA) = 0, 
 
 UD=Hk+m> (2 > 
 
 * In Fig. 30 ACBD may be projected (from K on NC) into LCNQ ; and then 
 LCNQ may be projected (from M on AD) into BCAD. 
 t Mobius, loc. cit. p. 269. 
 
58 
 
 AXHARXOXIC RATIOS. 
 
 .70 
 
 t. e. CD is the harmonic mean between CA and CB ; a formula which 
 determines the point D when A , B , C are given. 
 
 Again, if is the middle point of the segment CD, so that we have 
 0D = CO = -0C, then 
 
 = OC-OA ; AD = OD-OA = -(0C+ Oi) ; 
 BC=OC-OB; BD = -(0C+OB). 
 Substituting these values in (1) or in 
 
 we have 
 
 OC-OA 
 
 Fig. 4*. 
 
 OC+OA 
 
 qc 
 *' o^ 
 
 or 0C*=0A.0B, (3) 
 
 t. e. half the segment CD is a mean proportional between the distances 
 of A and B from the middle point of CD. 
 
 The equation (3) shows that the segments OA and OB must have 
 the same sign, and that therefore can never lie between A and B. 
 
 If now a circle be drawn to pass 
 through A and B (Fig. 42), O will lie 
 outside the circle, and OC will be the 
 length of the tangent from to it* 
 (Euc. HL 37). The circle on v CZ) as 
 diameter will therefore cut the first 
 circle (and aU circles through *i and B) 
 orthogonally. Conversely, if two circles cut each other orthogonally, 
 they will cut any diameter of one of them in two pairs of harmonic 
 points t. 
 
 70. The same formula (3) gives 
 the solution of the following pro- 
 blem: 
 
 Given two collinear segments AB 
 and A'B'; to determine another 
 segment CD which shall divide each 
 of them harmonically (Figs. 43, 44). 
 Take any point G not lying on 
 the common base AB', and draw the circles GAB . GA'B' meeting 
 
 * If through a point any chord be drawn to cut a circle in P and Q, the 
 rectangle OF . OQ is, called the potctr of the point with regard to the circle. 
 Stkikkr, CrtUJs Jowrwdy vol. i. (Berlin, i8a6) ; Collected Works, toL i. p. aa. 
 We may then say that OC* is the povar of the point O with regard to the circle 
 in Fig. 4^. 
 
 f Poxcelet, Propr.proj. Art. 79. 
 
 Rg; 43. 
 
no 
 
 AXHARMOXIC RATIOS. 
 
 59 
 
 Fig. 44- 
 
 again in H. Join GH *, and produce it to cut the axis in 0. Then 
 from the first circle 
 
 OA.OB=OG. OH (Euc. m. 36), 
 and from the second 
 
 oa'.qb'=og.oh; * 
 
 oa.ob = oa'.ob: 
 
 is therefore the middle point of the segment required; the 
 points C and D will be the 
 intersections with the axis 
 of a circle described from 
 the centre with radius 
 equal to the length of the 
 tangent from to either of 
 the circles GAB, G' A'B'. 
 
 The problem admits of a 
 real solution when the point 
 falls outside both the 
 segments AB, A'B', and consequently outside both the circles GAB. 
 GA'B' (Figs. 43, 44). There is no real solution when the segments 
 AB, A'B' overlap (Fig. 45) ; in this case lies within both segments. 
 
 71. Let ABCD be a harmonic 
 range, and let A and B (a pair of 
 conjugates) approach indefinitely near 
 to one another and ultimately coin- 
 cide. If C lie at an infinite distance, 
 then D must coincide with A and B, 
 since it must lie midway between these 
 two points (Art. 59). If C lie at a 
 finite distance, and assume any position not coinciding with that of A 
 or B. then equation (2) of Art. 69 gives CD = CA= CB, i.e.!) coincides 
 with A and B. 
 
 Again, let A and C (two non-conjugate points) coincide, and B 
 (the conjugate of A) lie at an infinite distance. In this case A must 
 lie midway between C and D, so that D will coincide with A and C. 
 If B lie at a finite distance , and assume any position not coinciding 
 with that of A or C, then equation (1) of Art. 69 gives AD = 0, i.e. 
 the point D coincides with A and C. So that : 
 
 If of four points forming a harmonic range, any two coincide, one 
 of the other two points will also coincide with them, and the fourth 
 is indeterminate. 
 
 72. The theorem of Art. 45 leads to the following result : given 
 four elements .4 , B , C , D of a one-dimensional geometric form, the 
 
 * GH is the radical axis of the two circles, and all points on it are of equal 
 power with regard to the circles. 
 
 Fig- 45- 
 
60 ANHARMONIC RATIOS. [72 
 
 anharmonic ratios (ABCD) , (BADC) , (CDAB) , (DCBA) are all equal 
 \j to one another. 
 
 I. Four elements of such a form can be permuted in twenty-four 
 different ways, so as to form the twenty-four different groups 
 
 ABCD , BADC , CDAB , DCBA , 
 
 ABDC , BACD , DCAB , CDBA , 
 
 ACBD , CADB , BDAC , DBCA , 
 
 ACDB , CABD , DBAC , BDCA , 
 
 ADBC , DACB , BCAD , CBDA , 
 
 ADCB , DABC , CBAD , BCD A , 
 
 here arranged in six lines of four each. The four groups in each 
 
 line are projective with one another (Art. 45), and have therefore 
 
 the same anharmonic ratio . In order to determine the anharmonic 
 
 ratios of all the twenty-four groups, it is only necessary to consider 
 
 one group in each line ; for example, the six groups in the first 
 
 column. These six groups are so related to each other that when 
 
 any one of them is known the other five can be at once determined. 
 
 II. Consider the two groups ABCD and ABDC, which are derived 
 one from the other by interchanging the last two elements. Their 
 anharmonic ratios 
 
 and (ABDC) or ^ : § 
 
 are one the reciprocal of the other ; thus 
 
 (ABCD) (ABDC) =1 (1) 
 
 Similarly, (ACBD) (ACDB) = 1, (2) 
 
 and (ADBC) (ADCB) =1 (3) 
 
 III. Now if A, B, C, D are four collinear points, it has been seen 
 (Art. 62) that the identical relation 
 
 BC. AD + CA . BD + AB . CD = 
 always holds. Dividing by BC .AD, we have ^ -r jj ' Jy 
 
 ACBD AB.CD . * • ~* 
 im7AD + CB7AD= l > 
 AC AD AB AD_ 
 ° r BC '' BD + CB '' CD ~ ' 
 
 that is (Arts. 63, 67), 
 
 (ABC 'D) + (ACBD) = 1 (4) 
 
 Similarly, (ABDC) + (ADBC) = 1, . (5) 
 
 and (ACDB) + (ADCB) = 1 (6) 
 
72] ANHARMONIC RATIOS. 61 
 
 IV. If X denote the anharmonic ratio of the group ABCD, i.e. if 
 
 (ABCD) = A, 
 
 the formula (1) gives (ABDC) = -» 
 
 and (4) gives. (ACBD) = 1 - A ; 
 
 then by (2) (ACDB) = -?— , 
 
 1 — A 
 
 and by (6) (AJDCB) = 1 _ ^ = JL ; 
 
 and finally, by (3) or (5) 
 
 (ADBC) = ^1 . * 
 A 
 
 V. The six anharmonic ratios may also be expressed in terms of 
 the angle of intersection 6 of the circles described on the segments 
 AB, CD as diameters ; it being supposed that A and B are separated 
 by C and D. It will be found that 
 
 (ABCD) = ■ 
 
 -tan 2 -, 
 
 ^Z)(7) = -cot 2 |, 
 
 (ACBD) = 
 
 a 
 
 sec 2 -, 
 
 (ACDB) = cos 2 |, 
 
 (ADCB) = 
 
 SIB'-, 
 
 (ADBC) = cosec 2 |-t 
 
 VI. If in the group ABCD two points A and B coincide, then 
 AC = BC i AD = BD, and therefore 
 
 (ABCD) = (AACD) = 1. 
 But if X = 1 , the other anharmonic ratios become 
 
 (ACAD) =1-1=0, and (ACDA) = oo ; 
 \J thus when of four elements two coincide, the anharmonic ratios have 
 the values 1, 0, oo. 
 
 If (ABCD) = — 1, ». e. if the range ABCD is harmonic, the formulae 
 of (IV) give (iM) = 2 and ( ACDB ) = i . 
 
 so that when the anharmonic ratio of four points has the value 2 or 
 ^, these points, taken in another order, form a harmonic range. 
 
 VII. Conversely, the anharmonic ratio of a range ABCD, none of 
 whose points lies at infinity, cannot have any of the values 0, 1, oo, 
 without some two of its points coinciding. 
 
 AC AD 
 For if in (IV) A = 0, -^ : -^ = , and either AC or BD must 
 
 vanish ; i. e. either A coincides with C, or B with D. 
 
 * Mobius, loc. cit. p. 249. 
 
 f Casey, On Cyclides and S2>hero~quariics (Phil. Trans. 1871), p. 704. 
 
62 ANHARMONIC RATIOS. [73 
 
 If X = 1, (ACBD) = 1— X = 0, so that either A coincides with B, 
 op C with D. 
 
 And if X = oo, (ABDC) = - = 0, so that either A coincides with 
 D, or B with C. 
 
 VIII. By considering the expressions given for the six anharmonic 
 ratios in (IV) it is clear that whatever be the relative positions of the 
 points A, B , C , D , two of the ratios (and their two reciprocals) are 
 always positive and a third (and its reciprocal) negative ; and thus we 
 see that the anharmonic ratios of four points no two of which coincide 
 may have all values positive or negative except + 1, 0, or oo. 
 
 73. From the theorems of Arts. 63 and 66, which express 
 the necessary and sufficient condition that two ranges, each 
 consisting of four elements, should be protectively related, we 
 conclude that 
 
 If two geometric forms of one dimension are projective^ then any 
 two corresponding groups of four elements are equianharmonic *. 
 As a particular case, to any four harmonic elements of 
 
 the one form correspond four 
 harmonic elements of the other 
 (Art. 51). 
 
 74. Let A, A' and B,B' be any 
 two pairs of corresponding points 
 of two projective ranges (Fig. 
 46) ; let / be the point at infinity 
 belonging to the first range, and V the point corresponding 
 to it in the second range ; so let J' be the point at infinity 
 belonging to the second range, and J its correspondent in 
 the first range. By Art. 73 
 
 (AMJ) = (A'B'rJ')', 
 .% (BAJI) = {A'BTJ') (Art. 72); 
 from which, since / and J' lie at infinity, 
 
 BJ:AJ= AT : B f V (Art. 64), 
 and JA.I'A' = JB.I'B'; 
 
 i.e. the product JA.I'A' has a constant value for all pairs of 
 corresponding points f. 
 
 [This proposition has already been proved in Art. 63 for 
 the particular case of two ranges in perspective.] 
 
 * Steiner, Systematische Entwickelung . . (Berlin, 1832), p. 33, § 10; Collected 
 Works, ed. Weierstrass (Berlin, 1881), vol. i. p. 262. 
 
 f Steineu, loc. cit. p. 40, § 12. Collected Works, vol. i. p. 267. 
 
76] ANHARMONIC RATIOS. 63 
 
 ' (UNI 
 
 75. In two homological figures, four collinear points or 
 four concurrent straight lines of the one figure form a group 
 which is equianharmonic with that consisting of the points or 
 lines corresponding to them in the other figure (Art. 73). Let 
 be the centre of homology, 31 and M ' any pair of corre- 
 sponding points in the two figures, N and N' another pair of 
 corresponding points lying on the ray 03131', and X the 
 point in which this ray meets the axis of homology. Since 
 the points 031 NX , 031' N'X correspond severally to one 
 another, 
 
 (0X31N) = (OXM'N'), 
 OM ON OM' ON' 
 
 or 
 
 MX 'NX ~ M'X' N'X 9 
 031 OM' ON ON' 
 
 " MX 'M'X NX 'N'X 
 and consequently the anharmonic ratio (0XM31') is con- 
 stant for all pairs of corresponding points M and 31' taken on 
 a ray OX passing through the centre of homology. 
 
 Next let L and L' be another pair of corresponding points, 
 and Y the point in which the ray OLL' cuts the axis of 
 homology. Since the straight lines LM , L'31' must meet in 
 some point Z of the axis XY, it follows that OYLL' is a pro- 
 jection of 0X31M' from ^as centre, and therefore 
 
 (0YLL') = (0XM31'); 
 consequently the anharmonic ratio (0XM31') is constant for 
 all pairs of corresponding points in the plane. 
 
 Consider now a pair of corresponding straight lines a and 
 a', the axis of homology s, and the ray o joining the centre of 
 homology to the point aa'. The pencil osaa' is cut by 
 every straight line through in a range of four points 
 analogous to 0XM31'; consequently the anharmonic ratio 
 (osaa') is constant for all pairs of corresponding straight 
 lines a and a', and is equal to the anharmonic ratio 
 (0X31M'). 
 
 This anharmonic ratio is called the coefficient or parameter 
 of the homology. It is clear that two figures in homology 
 can be constructed when, in addition to the centre and axis, 
 we are given the parameter of the homology. 
 
 76. When the parameter of the homology is equal to — 1, 
 all ranges and pencils similar to 0X31M', osaa', are harmonic. 
 
64 ANHARMONIC RATIOS. [77 
 
 In this case the homology is called harmonic* or involutorial, 
 and two corresponding points (or lines) correspond to one 
 another doubly ; that is to say, every point (or line) has the 
 same correspondent whether it be regarded as belonging to 
 the first or the second figure. (See below, Arts. 122, 123.) 
 
 Harmonic homology presents two cases which deserve special 
 notice : ( i ) when the centre of homology is at an infinite distance, in 
 the direction perpendicular to the axis of homology; (2) when the 
 axis of homology is at an infinite distance. In the first case we have 
 what is called symmetry with respect to an axis ; the axis of homology 
 (in this case called also the axis of symmetry) bisects orthogonally 
 the straight line joining any pair of corresponding points, and bisects 
 also the angle included by any pair of corresponding straight lines. 
 The second case is called symmetry with resjyect to a centre. The 
 centre of homology (in this case called also the centre of symmetry) 
 bisects the distance between any pair of corresponding points, and 
 two corresponding straight lines are always parallel. In each of 
 these two cases the two figures are equal and similar (congruent) t ; 
 oppositely equal in the first case, and directly equal in the second. 
 
 77. Considering again the general case of two homological 
 
 figures, let a , b , m, , n be four rays of a pencil in the first 
 
 figure, and a\ b\ m\ n' the straight lines corresponding to 
 
 them in the second. Then 
 
 (mnab) = (m'n'a'b'). 
 
 Now let an arbitrary transversal be drawn to cut mnab in 
 
 MNAB, and draw the corresponding (or another) transversal 
 
 to cut m'n'a'b' in M'N'A'B' ; then 
 
 {MNAB) = (M'N'A'B'), 
 
 MA M'A' _ NA N'A' 
 
 ° r MB '' M'B' ~ NB '' N'B' ' 
 
 MA M'A' 
 Consequently, the ratio -^^ : ,, /7?/ depends only on the 
 
 straight lines ah (and a'b'), and not at all on the straight line 
 m (or m'). 
 
 The ratio MA : NA is equal to that of the distances of the 
 points M, iV'from the straight line a, which distances we may 
 denote by (M, a), (N,.a); thus 
 
 * Bellavitis, Saggio di Geometria derivata (Nuovi Saggi of the Academy of 
 Padua, vol. iv. 1838), § 50. 
 
 f Two figures are said to be congruent when the one may be superposed upon 
 the other so as exactly to coincide with it. 
 
77] ANHARMONIC RATIOS. 65 
 
 (M,a) (M', a') , 
 
 that is to say * : 
 
 In two homological figures (or, more generally, in two protectively 
 related figures) the ratio of the distances of a variable point Mfrom 
 two fixed straight lines a ,b in the first figure hears a constant ratio 
 to the analogous ratio of the distances of the corresponding point 
 M' from the corresponding straight lines a' , b f in the other 
 f,gure. 
 
 Suppose b to pass through the centre of homology ; then 
 M and M' are collinear with and V coincides with b, so that 
 
 (M, b):(M',b') = OM: OM' \ 
 and therefore 
 
 OM {M,a) , 
 
 Gwwy) = constaat - 
 
 If N and N' are another pair of corresponding points, we 
 have then 
 
 OM m (M, a) ON m (N,a) ' 
 031' : (if', a') ~ ON' '' (N', a')' 
 Now suppose the straight line a f to move away indefinitely ; 
 then a becomes the vanishing line in the first figure ; the ratio 
 
 \jqr — 7v will in the limit become equal to unity, and thus 
 
 OM /ll/r . ON '. x 
 — ,:(if,.) = _: W .) 
 
 = constant ; 
 in other words f : 
 
 In two homological figures, the ratio of the distances of any point 
 
 in the first figure from the centre of homology and from the vanishing 
 
 line respectively, bears a constant ratio to the distance of the corre* 
 
 sponding point in the second figure from the centre of homology. 
 
 * Chasles, Gdometrie supirieure, Art. 512. 
 f Chasles, Sections conigues, Art. 267. 
 
CHAPTER X. 
 
 CONSTRUCTION OF PROJECTIVE FORMS. 
 
 
 78. Let ABC and A'B'C be two triads of corresponding 
 elements of two projective forms of one dimension (Fig. 47), 
 and imagine any series of operations (of projection and section) 
 
 by which we may have 
 passed from ABC to 
 A'B'C. Then whatever 
 this series be*, it will 
 also lead from any other 
 element D of the first 
 form to the element D' 
 which corresponds to it 
 in the second. For if D 
 could give, as the result 
 of these operations, an 
 element D" different from D', then^lhe anharmonic ratios 
 {ABCD) and {A'B'C'D") would be equal; but by hypothesis 
 {ABCD) = {A'B'C'D') ; therefore {A'B'C'D') = {A'B'C'D"), 
 which is impossible unless D" coincide with D' (Art. 65). 
 
 79. Theorem (converse to that of Art. 73) : 
 
 Given two forms of one dimension ; if to the elements A ,B,C,D,... 
 of the one correspond respectively the elements A', B', C, D', ... 
 of the other in such a manner that any four elements of the first form 
 are equianharmonic with the four corresponding elements of the second, 
 then the two forms are projective. 
 
 For every series of operations (of projection or section), 
 which leads from the triad ABC to the triad A'B'C, leads at 
 the same time from the element D to another element D" such 
 that {ABCD) = {A'B'C'D"). But {ABCD) = {A'B'C'D') by 
 hypothesis ; therefore {A'B'C'D') = {A'B'C'D"), and D" must 
 
 * In Fig. 47 the series of operations is : a projection from S, a section by u", 
 a projection from S', and a section by vf. 
 
80] CONSTRUCTION OF PROJECTIVE FORMS. 67 
 
 coincide with B' (Art. 65). And since the same conclusion is 
 true for any other pair whatever of corresponding elements, it 
 follows that the two forms are projective (Art. 40). 
 
 80. From Art. 78 the following may be deduced as a par- 
 ticular case : 
 
 If among the elements of two projective forms of one dimension 
 titer e are two corresponding triads ABC and A'B'C which are in 
 perspective, then the two forms themselves are in perspective. 
 
 (1). If, for example, the forms are two ranges ABCB ... and 
 A'B'C'B' ...; then if the three straight lines AA', BB' , CC 
 meet in a point S, the other analogous lines BB',... will all 
 pass through S (Figs. 19, 40). 
 
 Suppose, as a particular case, that the points A , A' coincide 
 (Fig. 22), so that the two ranges have a pair of corresponding 
 points A and A' united in the point of intersection of their 
 bases *. The triads ABC, A'B'C are in perspective, their centre 
 of perspective being the point where BB' and CC meet; 
 accordingly : 
 \f If two projective ranges have a self-corresponding point 3 they are in 
 perspective. 
 
 Conversely it is evident that two ranges which are in per- 
 spective have always a self-corresponding point. 
 
 (2). Again, if the two forms are two flat pencils abed ... and 
 a'b'c'd'... lying in the same plane; then if the three points 
 aa', W, cc' lie on one straight line s, the analogous points dd'. . . 
 will all lie on the same straight line (Fig. 20). If the line s 
 he altogether at infinity, we have the following property : 
 
 If in two projective flat pencils, 
 
 three pairs of corresponding rays 
 
 are parallel to one another, then 
 
 every pair of corresponding rays are 
 
 ■ parallel to one another. 
 
 The hypothesis is satisfied 
 in the particular case where the Fi g 
 
 rays a and a' coincide (Fig. 48), 
 so that the two pencils have a self-corresponding ray in the 
 
 * In the case of two projective forms we shall in future employ the term 
 self-corresponding to denote an element which is such that it coincides with its 
 correspondent ; thus A or A' above maybe called a self -corresponding point of the 
 two ranges. 
 
 F 2 
 
68 
 
 CONSTRUCTION OF PROJECTIVE FORMS. 
 
 [81 
 
 straight line which joins their centres ; then s is the straight 
 line joining lib' and cc'. Accordingly: 
 
 When two projective fiat pencils {lying in the same plane) have a 
 self-corresponding ray t they are in perspective. 
 
 Conversely, two coplanar flat pencils which are in perspec- 
 tive have always a self-corresponding ray. 
 
 (3). If one" of the systems is a range ABC I) ... and the other 
 a flat pencil abed ... (Fig. 28), the hypothesis amounts to 
 assuming that the rays a , b, c pass respectively through the 
 points A,B ,C ; then we conclude that also d, ... will pass 
 through D, . . . &c. 
 
 81. Two ranges may be superposed one upon the other, so as 
 to lie upon the same straight line or base, in which case they 
 may be said to be collinear. For example, if two pencils (in 
 the same plane) S = abc ... and = a'b'c' ... (Fig. 49) are cut 
 
 by the same transversal, they will 
 determine upon it two ranges 
 ABC ... , A'B'C ... which will be 
 projectively related if the two 
 pencils are so. The question arises 
 whether there exist in this case 
 any self-corresponding points, i.e, 
 whether two corresponding points of the two ranges coincide 
 in any point of the transversal. 
 
 If, for instance, the transversal s be drawn so as to pass 
 through the points aa' and bb\ then A will coincide with A\ 
 
 and B with B' ; in this case 
 consequently there are two 
 self-corresponding points. 
 
 Again, if a range u be 
 projected (Fig. 50) from two 
 centres S and (lying in 
 the same plane with u), 
 two flat pencils abc ... and 
 a'b'c'. . . will be formed, which 
 have a pair of corresponding rays a , a' united in the line SO. 
 And if a transversal s be drawn through the point in which 
 this line cuts u, we shall obtain two projective ranges ABC ..., 
 A'B'C ... lying on a common base s, and such that they have 
 one self-corresponding point AA'. 
 
 c c 
 
 Fig. 49. 
 
 Fig. 50- 
 
83] CONSTRUCTION OF PROJECTIVE FORMS. 69 
 
 And lastly, we shall see hereafter (Art. 109) that it is possible 
 for two collinear projective ranges to be such as to have no 
 self-corresponding point. 
 
 So also two flat pencils (in the same plane) may have a 
 common centre, in which case they may be termed concentric ; 
 such pencils are formed when two different ranges are pro- 
 jected from the same centre (Fig. 51). And two axial pencils 
 may have a common axis ; such pencils 
 are formed when we project two dif- 
 ferent ranges from the same axis, or 
 the same flat pencil from different 
 centres. Again, if two sheaves are cut 
 by the same plane, two plane figures 
 are obtained; if, on the other hand, Fig. 51. 
 
 two plane figures are projected from 
 
 the same centre, two concentric sheaves are formed. In all 
 these cases the forms in question may be said to be superposed 
 one upon the other; and the investigation of their self- 
 corresponding elements, when the two forms are projectively 
 related, is of great importance. 
 
 82. Theorem. Two superposed projective {one-dimensional} 
 forms either have at most two self -corresponding elements, or else 
 every element coincides with its correspondent. 
 
 For if there could be three self-corresponding elements 
 A, B , C suppose ; then if B and B f are any other pair of cor- 
 responding points, we should have (Art. 73) (ABCB) = (ABCB'), 
 and consequently (Art. 65) B would coincide with B'. Unless 
 then the two forms are identical, they cannot have more than 
 two self-corresponding elements. \ 
 
 83. Theorem (converse to that of Art. 53). If a one-dimen~ 
 sionalform consisting of four elements A,B ,C ,B is projective with 
 a second form deduced from it by interchanging two of the elements 
 (e.g. BACB), then the form will be a harmonic one, and the two 
 interchanged elements will be conjugate to each other. 
 
 First Proof. If (ABCB) = (BACB), then (Art. 72. IV) A = £ ; 
 
 A 
 
 .-. A 2 = i, and since we cannot take A = + 1 (Art. 72. VIII) 
 
 we must have A = — 1, i.e. the form is a harmonic one. 
 
 Second Proof Suppose, for example, that A , B , C , B are four 
 
 collinear points (Fig. 52). Let K , M , Q , B be a projection of 
 
J 
 
 70 CONSTRUCTION OF PROJECTIVE FORMS. [84 
 
 these points on any straight line through B, made from an 
 arbitrary centre L. Since ABCB is projective with KMQB 
 and also (by hyp.) with BACB, the forms KMQD and BACB 
 
 are projective with one another. 
 And they have a self-corre- 
 sponding point D ; consequently 
 they are in perspective (Art. 
 80), and KB, MA, QC will meet 
 in one point A 7 . But this being 
 the case, we have a complete 
 quadrangle KLMN, of which one pair of opposite sides meet in 
 A, another such pair in B, while the fifth and sixth sides pass 
 respectively through C and B. Accordingly (Art. 46) ABCB 
 is a harmonic range. 
 
 84. Let there be given two projectively related geometric 
 forms of one dimension. Any series of operations which suf- 
 fices to derive three elements of the one from the three corre- 
 sponding elements of the other will enable us to pass from 
 the one form to the other (Art. 78); and any two given triads 
 of elements are always projective, i.e. can be derived one from 
 the other by means of a certain number of projections and 
 sections. Hence we conclude that : 
 
 Given three pairs of corresponding elements of two projective forms 
 of one dimension, any number of other pairs of corresponding elements 
 can he constructed. 
 
 We proceed to illustrate this by two examples, taking 
 (i) two ranges and (s) two flat pencils; the forms being 
 in each case supposed to lie in one plane. 
 
 Given (Fig. 53) three pairs of Given (Fig. 54) three pairs 
 corresponding points A and A' , of corresponding rays a and a', 
 B and B',C and C / ', of the pro- b and b',c and c', of the projec- 
 jective ranges u and u' \ to con- tive pencils U and U' '; to con- 
 struct these ranges. struct these pencils. 
 
 We proceed as in Art. 44. On Through the point of infer- 
 tile straight line which joins any section of any two of the corre- 
 two of the corresponding points, sponding rays, say a and a f , 
 say A and A f , take two arbitrary draw two arbitrary transversals 
 points S and S'. Join SB , S'B' s and s'. Join the points sb and 
 cutting one another in B", and sfb' by the straight line b" , and 
 SC , S'C cutting one another in the points sc and sV by the 
 C"; join B /f C /f , and let it cut straight linec"; and let a" be the 
 
84] 
 
 CONSTRUCTION OF PROJECTIVE FORMS. 
 
 71 
 
 A A' in A". The operations which straight line joining the points 
 enable us to pass from ABC to b"o" and aaf. The operations 
 
 Fig- 53- 
 
 *V 54- 
 
 A'B'C are : i. a projection from 
 S; 2. a section by u" (the line on 
 which lie the points.4", B", C") ; 
 3. a projection from S' ; 4. a 
 section by u. The same opera- 
 tions lead from any other given 
 point D on u to the correspond- 
 ing point D' on u', so that the 
 rays SD and S'D' must intersect 
 in a point D" of the fixed straight 
 line u". 
 
 In this manner a range 
 u"=A"B"C"D"... 
 is obtained which is in perspec- 
 tive both with u and with vf. 
 
 which enable us to pass from abc 
 to a'b V are : 1 . a section by s ; 
 2. a projection from the point 
 U" where a", b", c" meet; 3. a 
 section by s' ; 4. a projection 
 from U'. The same operations 
 lead from any other given ray d 
 of the pencil U to the correspond- 
 ing ray df of the pencil TJ'\ so that 
 the points sd and ttdf must lie on 
 a straight line d" which passes 
 through the fixed point U". 
 In this manner a pencil 
 
 is obtained which is in perspec- 
 tive both with U and with U'. 
 
 In the preceding construction (left), D is any arbitrary point on w. 
 If D be taken to be the point at infinity on u, then (Fig. 53) SD 
 will be parallel to u; in order therefore to find the point on vf 
 which corresponds to the point at infinity on u, draw SI" parallel to 
 
72 
 
 CONSTRUCTION OF PROJECTIVE FORMS. 
 
 [85 
 
 u to cut u" in I" ; then join S'l", which will cut u' in the required 
 point I'. Similarly, if the ray through S' parallel to u cuts u" in 
 J", and SJ" be joined, this will cut u in J, the point on u which 
 corresponds to the point at infinity on u'. 
 
 If D be taken at P, the point 
 where u and u" meet, then B" 
 also coincides with P, and the 
 point P' on w' corresponding to 
 the point P on ^ is found as the 
 intersection of S'P with u'. 
 
 Similarly, if Q' be the point of 
 intersection of u' and u", the 
 point on u corresponding to Q' 
 on v! is Q, where SQ' cuts w. 
 
 85. The only condition to 
 which the centres S and S' are 
 subject is that they are to lie 
 upon the straight line which joins 
 a pair of corresponding points ; 
 in other respects their position is 
 arbitrary. We may then for in- 
 stance take S at A' and S' at A 
 ( Fi g- 55). Then the ray S'P co- 
 incides with u, and P' is accord- 
 ingly the point of intersection of 
 u and vf. So too the ray SQ' 
 coincides with u, and Q also lies 
 at the point uu'. 
 
 If then we take the points A' 
 and A as the centres S and S' 
 respectively, the straight line u" 
 will cut the bases u and u re- 
 spectively in P and Q\ the points 
 which correspond to the point 
 uu' regarded in the first instance 
 as the point P / of the line u' and 
 in the second instance as the 
 point Q of the line u. 
 
 Now in the construction of 
 the preceding Art., the straight 
 line u" was found as the locus of 
 the points of intersection of pairs 
 
 In the preceding construction 
 (right), d is any arbitrary ray 
 passing through U. If it be taken 
 to be p, the line joining U to U" 9 
 then the corresponding ray p' of 
 the pencil U f is the line joining 
 the point TJ f to the point s'p. 
 
 Similarly, if (f be the ray 
 U'U" of the pencil TJ\ the ray q 
 corresponding to it in the pencil 
 U is that which joins the points 
 U and sq'. 
 
 The only condition to which 
 the transversals s and $' are sub- 
 ject is that they are to pass 
 through the point of intersection 
 of a pair of corresponding rays ; 
 in other respects their position is 
 arbitrary. We may then for in- 
 stance take a' for s and a for s' 
 (Fig. 56). Then the point /p 
 coincides with U, and yf is ac- 
 cordingly the straight line UU'. 
 So too the point sq' coincides 
 with U\ and q also must be the 
 straight line UU'. 
 
 If then we take the rays a' 
 and a as the transversals s and 
 s' respectively, the point U" will 
 be the intersection of the rays p 
 and <f which correspond to the 
 straight line UU', regarded in 
 the first instance as the ray p' of 
 the pencil U', and in the second 
 instance as the ray q of the 
 pencil U. 
 
 Now in the construction of the 
 preceding Art., the point U" was 
 found as the centre of perspective 
 of the ranges in perspective 
 
85] 
 
 CONSTRUCTION OF PROJECTIVE FORMS. 
 
 73 
 
 of corresponding rays of the 
 pencils in perspective 
 S(ABCD..) and S'{A'B'C'D'..). 
 The straight line u" obtained by 
 the construction of the present Art. 
 is in like manner the locus of the 
 points of intersection of pairs of 
 corresponding rays of the pencils 
 A'(ABCD..) and A(A'B'C'D'..) i 
 i.e. the locus of the points in 
 which the pairs of lines A'B and 
 AB', A'C and AC, A'D and 
 AD', ... intersect. 
 
 
 s (abed . . .) and / (a'b'c'd' ' . . .). 
 
 The point U" obtained by the 
 construction of the present Art. 
 is in like manner the centre of 
 perspective of the ranges 
 
 a' (abed . . .) and a (a'b'c'd' '. . .), 
 i.e. the point in which the lines 
 joining the pairs of corresponding 
 points a'b and ab', ac and ae\ 
 a'd and ad', ... meet. 
 
 .\\ 
 
 Fig. 55- 
 
 Fig. 56. 
 
 If in place of A' and A any 
 other pair of points* B' and B, or 
 C and C, ... be taken as centres 
 of the auxiliary pencils S and S f , 
 the straight line u" must still 
 cut the two bases u and u in the 
 points P and Q' ; i.e. the straight 
 line u" remains the same. 
 
 If iheh ABC ... MN ... and 
 A'B'C ... M'N' ... are two pro- 
 jective ranges (in the same plane), 
 every pair of straight lines such 
 as MN f and M'N intersect in 
 points lying on a fixed straight 
 line. This straight line passes 
 through those points which cor- 
 
 If in place of a and a any 
 other pair of rays b' and b, or c 
 and c, ... be taken as transversals, 
 the point U" must still be the 
 intersection of p and q ; i. e. the 
 point U" remains the same. 
 
 If then abc ... mn ... and 
 a / b , c / ... mV ... are two projec- 
 tive pencils (in the same plane) 
 every straight line which joins a 
 pair of points such as mn' and 
 m'n passes through a fixed point. 
 This point is the intersection of 
 those rays which correspond in 
 
74 
 
 CONSTRUCTION OF PROJECTIVE FORMS. 
 
 [86 
 
 respond in each range to the 
 point of intersection of their bases 
 when regarded as a point of the 
 other range. 
 
 86. If the two ranges u and u' 
 are in perspective (Fig. 57) the 
 points P and Q' will coincide 
 with the point in which the 
 bases u and vf meet ; and since 
 the straight line which is the 
 locus of the points (AB', A'B), 
 (AC, A'C), (AD', A'B), ... and 
 the straight line which is the 
 locus of the points (BA f , B'A), 
 (BC, B'C), (BD', B'D), ... have 
 two points in common, viz. and 
 (AB', A' B), these straight lines 
 must coincide altogether. This 
 being so, AA'BB' is a com- 
 plete quadrangle, whose diagonal 
 points are 0,S (the point where 
 AA',BB', ... meet), and M (the 
 point of intersection of AB' and 
 A'B); consequently (Art. 57) the 
 straight lines u and u' are har- 
 monic conjugates with regard to 
 the straight lines u" and OS. If 
 therefore two transversals u and 
 u' cut a flat pencil (a, b, c, . . . ) in the 
 V oints(A,A>),(B,By(C,C') ..., 
 then the points of intersection of 
 the pairs of straight lines AB ' and 
 A'B , AC and A'C , BC and 
 B'C, ... lie on one and the same 
 straight line u", which passes 
 through the point uu' ; and the 
 straight line joining uu' to the 
 centre of the pencil is the har- 
 monic conjugate of u" with re- 
 spect to u and u'. 
 
 From this follows the solution 
 of the problem : 
 
 To draw the straight line con- 
 necting a given point M-with the 
 
 each pencil to the straight line 
 joining the centres of the pencils 
 when regarded as a ray of the 
 other pencil. 
 
 If the two pencils U and U' 
 are in perspective (Fig. 59) the 
 rays p and q' will coincide with 
 the straight line UU'; and since 
 through the point of intersection 
 of the rays (ab' , a'b), (acf , a'c), 
 (ad' , a'd), ... and through the 
 point of intersection of the rays 
 (ba' , b'a\ (be' , b'c), (bd' , I'd), ... 
 pass two different straight lines, 
 viz. UU' and (ab', a'b), these 
 points must coincide. This being 
 so, aa'bb' is a complete quadri- 
 lateral, whose diagonals are UU', 
 s (the straight line on which 
 aa' , bb', . . . intersect), and m (the 
 straight line which joins ab r and 
 a 7 6); consequently (Art. 56) the 
 points U and U' are harmonic 
 conjugates with regard to U" and 
 the point in which s meets UU'. 
 If therefore a range be projected 
 from two points U and U' by the 
 rays (a , a'), (b , b'), (c , c') . . ., then 
 the straight lines which join the 
 pairs of points (ab' , a'b), (ac , a'c), 
 (be , b'c), ... meet in one and the 
 same point U", which lies on the 
 line UU' ; and the point where 
 the straight line UU' cuts the base 
 of the range is the harmonic con- 
 jugate of U" with respect to U 
 and U'. 
 
 From this follows the solution 
 of the problem : 
 
 To construct the point where a 
 given straight line m would be in- 
 
87] 
 
 CONSTRUCTION OP PROJECTIVE FORMS. 
 
 75 
 
 inaccessible point of intersection of 
 two given straight lines u and u'. 
 
 tersected by a straight line ( U U') 
 which cannot be drawn, but which 
 is determined by its passing 
 through two given points U and 
 U'. 
 
 Through M (Figs. 57 and 58) 
 draw two straight lines to cut u 
 in A and B, and u r in B' and A' 
 
 On m (Fig. 59) take two points, 
 and join them to U by the 
 straight lines a and b, and to U' 
 
 u" M N W 
 
 A // 
 
 / XV 
 
 
 W B' C * A' W 
 
 Fig. 58. 
 
 respectively ; join AA',BB' meet- 
 ing in S. Through S draw any 
 straight line to cut u in C and 
 u' in C", and join B£' , B'C, 
 intersecting in N. The" straight 
 line joining M and N will be the 
 line u" required. 
 
 Fig. 59- 
 
 by the straight lines V and a' ; 
 let s be the straight line joining 
 the points of intersection of a , a' 
 and b , b f . On s take any other 
 point and join it to U, U' by the 
 straight lines c , <f respectively. 
 The straight line n which joins 
 the points be' and b'c will cut m 
 in the point U" required. 
 
 If the straight lines u and u are parallel to one another (Fig. 58) 
 the preceding construction gives the solution of the problem : given 
 / two parallel straight lines, to draw through a given point a straight 
 line parallel to them, making use of the ruler only. 
 
 87. If in the theorem of the If in the theorem of the pre- 
 
 preceding article the flat pencil ceding article the range consist 
 
76 
 
 CONSTRUCTION OF PROJECTIVE FORMS. 
 
 [88 
 
 consist of only three rays, the 
 theorem may be enunciated as 
 follows, with reference to the 
 three pairs of points A A', BB f , 
 CC: 
 
 If a hexagon (six - point) 
 AB'CA'BC (Fig. 60) has its ver- 
 tices of odd order (1st, 3rd, and 5th) 
 
 Fig. 60. 
 
 on one straight line u, and its ver- 
 tices of even order (2nd, 4th, and 
 6th) on another straight line u', 
 then the three pairs of opposite 
 sides (AB' and AB* , B'G and 
 BC, CA' and C'A) meet in three 
 points lying on one and the same 
 straight line u"*. 
 
 of only three points, the theorem 
 may be enunciated as follows 
 with reference to the three pairs 
 of rays aa', bb', cc f : 
 
 If a hexagon (six-side) ab'ca'bc' 
 (Fig. 61) be such that its sides of 
 odd order (1st, 3rd, and 5th) 
 
 Fig. 61. 
 
 meet in one point U, and its sides 
 of even order (2nd, 4th, and 
 6th) meet in another point U', 
 then the three straight lines 
 which join the pairs of opposite 
 vertices (ab f and a% b'c and be', 
 ca! and c'a) pass through one and 
 the same point U". 
 
 Fig. 62. Fig. 63. 
 
 88. Returning to the con- Returning to the construction 
 
 struction of Art. 84 (left), let the of Art. 84 (right), let the straight 
 
 * Pappos, loc. cit. Book vii. prop. 139. 
 
90] 
 
 CONSTRUCTION OF PROJECTIVE FOEMS. 
 
 77 
 
 centre S be taken at the point 
 where AA ' meets BB', and the 
 centre >S" at the point where A A' 
 meets CC (Fig. 62). Then since 
 SB , S'B' meet in B', and SC, 
 S'C in C, therefore B'C is the 
 straight line u". Consequently 
 any other pair of corresponding 
 points D and D' are constructed 
 by observing that the straight 
 lines SD, S'D' must m/et on B'C. 
 
 From a consideration of the 
 figure SS'CDD'B, which is a 
 >n, we derive the theorem : 
 
 line joining the points aa', cc' be 
 taken as the transversal s, and 
 that joining the points aa', W 
 as the transversal / (Fig. 63). 
 Then since the line joining the 
 points sb , s'b' is b, and the line 
 joining the points sc , sV is c', 
 therefore be' is the point U". 
 Consequently any other pair of 
 corresponding rays d and d' are 
 constructed by observing that the 
 points sd , s'd' must be collinear 
 with be'. 
 
 From a consideration of the 
 figure ss'cdd'b, which is a hexa- 
 gon (six - side) we derive the 
 theorem : 
 
 In a hexagon, of which two ver- 
 tices are the centres of two pro- 
 jective pencils, and the four others 
 are the points of intersection of 
 four pairs of corresponding rays, 
 the three points in which the pairs 
 of op>posite sides meet one another 
 are collinear. 
 
 If the three points aa', bb', ccf 
 in Art. 84 (right) lay on the 
 same straight line s (if, for ex- 
 ample, a and a' coincided), then 
 the two pencils would be in per- 
 spective ; we should therefore 
 only have to connect the two 
 centres of the pencils with every 
 point of s in order to obtain any 
 number of pairs of corresponding 
 rays (Fig. 20). 
 
 90. If the two ranges u and u' (Art. 84, left) are superposed one 
 upon the other, i.e. if the six given points AA'BB'CC lie on the 
 same straight line (Fig. 64), we first project u from an arbitrary 
 centre S' on an arbitrary straight line u x , and then proceed to make 
 the construction for the case of the ranges u~{ABC...) and 
 «i== (A 1 B X C 1 ...), i.e. to construct with regard to the pairs of points 
 (AA X ), (BBJ, (CCd in the way shown in Art. 84. A pair of corre- 
 sponding points D and D x of the ranges u and u x having been found, 
 
 In a hexagon, of which two 
 sides are segments of the bases of 
 two projective ranges, and the four 
 others are the straight lines con- 
 necting four pairs of correspond- 
 ing points, the straight lines which 
 join the three pairs of opposite 
 vertices are concurrent. 
 
 89. If in the problem of Art. 
 84 (left) the three straight lines 
 AA', BB', CC passed through the 
 same point S (if, for example, A 
 and A' coincided), then the two 
 ranges would be in perspective ; 
 we should therefore only have to 
 draw rays through JS in order to 
 obtain any number of pairs of cor- 
 responding points (Fig. 1 9). 
 
78 
 
 CONSTRUCTION OF PROJECTIVE FORMS. 
 
 [90 
 
 the ray S'D X determines upon vf the point D' which corresponds 
 toi>. 
 
 The construction is simpler in the case where two corresponding 
 
 points A and A' coincide (Fig. 65). 
 When this is so, if u x be drawn 
 through .4, the range u x will be in per- 
 spective with u ; thus, after having 
 projected u' upon u x from an arbi- 
 trary centre JS' 9 if S be the point 
 where BB' and CG X meet, it is 
 only necessary further to project u 
 from S upon u x , and then u x from 
 S' upon u'. 
 
 The two collinear ranges u and 
 u' have in general two self-corre- 
 sponding points; one at A A', and 
 a second at the point of inter- 
 Fig- 6 4- section of their common base line 
 with the straight line SS'. 
 
 If then SS' passes through the point uu x , the two ranges u and u' 
 have only one self-corresponding point. If it were desired to con- 
 struct upon a given straight line two collinear ranges having 
 A and A' for a pair of corresponding points, and a single self-corre- 
 sponding point at M (Fig. 66), we should proceed as follows. Take 
 
 any point S / , and draw any straight line u x through M ; project A / from 
 S' on u x ', join the point A x so found to A, and let A A x meet S'M in S. 
 Then to find the point on uf which corresponds to any point B on u, 
 project B from S into B x , and then B x from S' into B' ; this last is 
 the point required. 
 
 If the two pencils U , U' (Art. 84, right) are concentric, i.e. if the 
 six rays aa'bb'cc' pass all through one point, we first cut a'b'c' by 
 a transversal and then project the points of intersection from an 
 arbitrary centre U x . If a x b x c x are the projecting rays, we have then 
 
03] CONSTRUCTION OF PROJECTIVE FORMS. 79 
 
 only to consider the non-concentric pencils U and U l =(a 1 b 1 c 1 ). Or 
 we may cut abc by a transversal in the points ABC, and a'b'c' by 
 another transversal in A 'B'C, and then proceed with the two ranges 
 ABC ..., A'B'C ... in the manner explained above. 
 
 The figures corresponding to these constructions are not given; 
 the student is left to draw them for himself. He will see that in 
 these cases also the constructions admit of considerable simplification 
 if, among the given rays, there be one which is self-corresponding ; if, 
 for example, a and a' coalesce and form a single ray, &c. 
 
 91. Consider two projective (nomographic) plane figures tt and 
 7/ ; as has already been seen (Art. 40), any two corresponding straight 
 lines are the bases of two projective ranges, and any two correspond- 
 ing points are the centres of two projective pencils. 
 
 If the two figures have three self-corresponding points lying in a 
 straight line, this straight line s will correspond to itself; for it will 
 contain two projective ranges which have three self-corresponding 
 points, and every point of the straight line s will therefore (Art. 82) 
 be a self-corresponding point. Consequently every pair of corresponding 
 straight lines of n and i/ will meet in some point on s, and therefore 
 the two figures are in perspective (or in homology in the case where 
 they are coplanar). 
 
 92. If two projective plane figures which are coplanar have three 
 self-corresponding rays all meeting in a point 0, this point will be 
 the centre of two corresponding (and therefore projective) pencils 
 which have three self-corresponding rays ; therefore (Art 82) every 
 ray through will be a self-corresponding one. Hence it follows 
 that every pair of corresponding points will be collinear with ; 
 therefore the two figures are in homology. 
 
 93. If two projective plane figures which are coplanar have four 
 self-corresponding points A , B , C , D, no three of which are collinear, 
 then will every point coincide with its correspondent. 
 
 For the straight lines AB , AC , AD , BC , BD , CD are all self- 
 corresponding ; therefore the points of intersection of AB and CD, 
 AC and BD, BC and AD, i. e. the diagonal points of the quadrangle 
 A BCD, are all self-corresponding. Since the three points A , B, and 
 (AB) (CD) are self-corresponding, every point on the straight line 
 AB coincides with its correspondent ; and the same may be proved true 
 for the other five sides of the quadrangle. If now a straight line be 
 drawn arbitrarily in the plane, there will be six points on it which 
 are self-corresponding, those namely in which it is cut by the six 
 sides of the quadrangle ; and therefore every point on the straight 
 line is a self-corresponding one ; which proves the proposition. 
 
 In a similar manner it may be shown that if two coplanar pro- 
 jective figures have four self-corresponding straight lines a, b, c, d, 
 
80 CONSTRUCTION OF PROJECTIVE FORMS. [94 
 
 forming a complete quadrilateral (i.e. such that no three of them are con- 
 current), then every straight line will coincide with its correspondent. 
 
 94. Theorem. Two plane quadrangles ABCD , A'B'C'D' are 
 always projective. 
 
 (1). Suppose the two quadrangles to lie in different planes it , i/. 
 Join A A ', and on it take an arbitrary point S (different from A'), and 
 through A draw an arbitrary plane it" (distinct from it) ; then from 
 S as centre project A', B', C, D' upon it" and let A", B", C", D" 
 be their respective projections (A" therefore coinciding with A). 
 
 In the plane it join AB , CD, and let them meet in E ; so too in 
 the plane w" join A"B" , C"D", and let these meet in E". The 
 straight lines ABE , A"B"E" lie in one plane since they meet each 
 other in the point A == A" ; therefore BB" and EE" will meet one 
 another in some point S v 
 
 Now let a new plane it'" (distinct from 77) be drawn through the 
 straight line ABE, and let the points A", B" , C", D", E" be pro- 
 jected from B x as centre upon it'". Let A'", B'", C", D"' , E'" 
 be their respective projections, where A"', B'" , E'" are collinear and 
 coincide with A , B ,E respectively, and C", D'", E'" are collinear 
 also, since their correspondents C", D", E" are collinear. The straight 
 lines CDE , C"'D'"E'" lie in one plane since they meet each other 
 in the point E~E"'; therefore CC" and DD'" will meet one 
 another in some point S 2 . If now the points A f ", B'" , C" , D f " be 
 projected from S 2 as centre upon the plane it, their projections will 
 evidently be A , B , C , D. 
 
 The quadrangle ABCD may therefore be derived from the quad- 
 rangle A'B'C'D' by first projecting the latter from JS as centre upon 
 the plane it", then projecting the new quadrangle so formed in the 
 plane it" from S x upon it'", and lastly projecting the quadrangle so 
 formed in the plane it"' from >^ 2 upon it ; that is to say, by means 
 of three projections and three sections *. 
 
 (2). The case of two quadrangles lying in the same plane reduces 
 to the preceding one, if we begin by projecting one of the quadrangles 
 upon another plane. 
 
 (3). If the two quadrangles (lying in different planes) have a pair 
 of their vertices coincident, say D and D f , then two projections will 
 suffice to enable us to pass from the one to the other; or, what 
 amounts to the same thing, a third quadrangle can be constructed 
 which is in perspective with each of the given ones ABCD, 
 A'B'C'D'. 
 
 For let there be drawn through D two straight lines s and s', one 
 in each of the planes ; let s cut the sides of the triangle ABC in 
 
 * Gbassmann, Die stereometrischen Gleichungen dritten Grades und die dadurch 
 erzeugten Oberfiachen ; Crelle's Journal, vol. 49. § 4 (Berlin, 1855). 
 
96] CONSTRUCTION OF PROJECTIVE FORMS. 81 
 
 L,M , iV respectively, and let s' cut the sides of the triangle A'B'C'm 
 L',M',N' respectively. Then in the plane ss' the straight lines LL', 
 MM', NN' will form a triangle which is in perspective at once with 
 ABC and with A'B'C. 
 
 (4). If the quadrangles (still supposed to lie in different planes) 
 have two pairs of their vertices C = C, D == D' coincident, then if 
 the straight lines AA' , BB' meet one another the quadrangles will be 
 directly in perspective, the point of intersection of AA / and BB' 
 being the centre of projection ; so that we can pass at once from the 
 one quadrangle to the other by one projection from 0. If A A ' , BB' 
 are not in the same plane, so that they do not meet one another, then 
 through CD let an arbitrary plane n" be drawn, and in it let the 
 straight line be drawn which meets AB and A'B'. If in this straight 
 line two arbitrary points A", B" be taken, then A"B"C"D" will be 
 a quadrangle which is in perspective at once with ABCD and with 
 A'B'C'D'. 
 
 95. From the theorem just proved it follows that two projective 
 plane figures n and 7/ can be constructed when we are given two 
 corresponding quadrangles ABCD, A'B'C'D' ; for the operations 
 (projections and sections) which serve to derive A'B'C'D' from 
 ABCD will lead from any point or straight line whatever of v to the 
 corresponding point or straight line of 7/ ; and vice versa. 
 
 Or, again, it may be supposed that two corresponding quadrilaterals 
 are given. For if in these two corresponding pairs of opposite ver- 
 tices be taken, we have thus two corresponding quadrangles ; and the 
 operations (projections and sections) which enable us to derive 
 one of these quadrangles from the other will also derive the one 
 quadrilateral from the other. 
 
 96. Two plane figures may also be made projective in another 
 manner ; leaving out of consideration the relative position of the 
 planes in which .they lie, we may operate on each of the figures 
 separately *. Suppose that we are given, as corresponding to one 
 another, two complete quadrilaterals abed, a'b'c'd'. We begin by 
 constructing, on each pair of corresponding sides, such as a and a', 
 the projective ranges which are determined by the three pairs of 
 corresponding points ab and a'b', ac and a'c', ad and a'd'. Thi§ 
 done, to every point of any of the four straight lines a,b,c,d will 
 correspond a determinate point of the corresponding line in the 
 other figure. 
 
 (1). Now let in the first figure a transversal m be drawn to cut 
 a, b, c, d in A , B , C , D respectively ; then the points^', B', C, D' 
 which correspond to these in the second figure will in like manner lie 
 on a straight line m'. 
 
 * Staudt, Geom. der Lage, Art. 130. 
 G 
 
82 CONSTRUCTION OP PROJECTIVE PORMS. [90 
 
 For, considering the triangle abc, cut by the transversals d and m, 
 the product of the three anharmonic ratios 
 
 a {bcdm) , b (cadm) , c (abdm) 
 is equal to + i (Art. 140); but these anharmonic ratios are equal 
 respectively to the following : 
 
 a'(b'c'd').A', b'{c'a'd').B' , c'{a'b' 'd').C ', 
 so that the product of these last three is also equal to + i. And 
 therefore, since the points a'd', b / d / , c'd' are collinear, the points 
 A', B', C are also collinear (Art. 140). 
 
 By considering in the same manner the triangle abd, cut by the 
 transversals c and m, it can be shown that A', B', D' are collinear ; 
 it follows then that the four points A' ', B', C, D' all lie on the 
 same straight line m', the correspondent of m. 
 
 This proof holds good also when m passes through one of the 
 vertices of the quadrilateral abed ; if for example m pass through 
 cd, the anharmonic ratios c (abdm), d(abcm) will each be equal to + i ; 
 the reasoning, however, remains unaltered. 
 
 Thus every pair of corresponding vertices of the quadrilaterals 
 abed , a'b'c'd' (for example cd and c'd') become the centres of two 
 projective pencils, in which to c , d, (cd)(ab) correspond c', d', (c'd')ia'b') 
 respectively, and to any ray cutting a , b in two points P , Q cor- 
 responds a ray cutting a', b' in the two corresponding points P',Q'. 
 
 (2). The two ranges ABCD , A'B'C'D' in which the sides of the 
 quadrilaterals abed, a'b'c'd' are respectively cut by two corresponding 
 straight lines m, m' are projective. 
 
 For, considering the triangle bem, cut by the transversals a and d, 
 the product of the anharmonic ratios of the three ranges 
 be , B , ba , bd 
 G , cb , ca , cd 
 B , C , A , D 
 is equal to + i. And considering in like manner in the other plane 
 the triangle b'c'm! \ cut by the transversals a' and d', the product of 
 the anharmonic ratios of the three ranges 
 
 b'c\ B\ b'a', b'd' 
 
 B' , C , A* , D' 
 is also equal to + i. But the range in which b is cut by the pencil 
 cmad is equianharmonic with the range in which b' is cut by the 
 pencil c'm'a'd' ; i.e. the ranges 
 
 be j B , ba , bd 
 
 6V, B\ 6V, b'd' 
 are equianharmonic ; and for a similar reason the ranges 
 
90] CONSTRUCTION OF PROJECTIVE FORMS. 83 
 
 C , cb , ca , ed 
 C, e'b\ c'a' s c'd' 
 
 are equianharmonic. Therefore the ranges 
 B , C , A , D 
 B', Q\ A', D' 
 
 will be equianharmonic and therefore projective ; whence it follows 
 that the projective ranges m and m' are determined by means of 
 the pairs of corresponding points lying on a and a, b and b', 
 c and c'. 
 
 (3). If the straight line m turn round a fixed point M, then m' 
 also will revolve round a fixed point. 
 
 For by hypothesis the points A and B, in which m cuts a and b, 
 describe two ranges in perspective whose self-corresponding point is 
 ab. Similarly the points A', B' describe two ranges, which, being 
 respectively projective with the ranges on a , 6, are projective with 
 one another ; and which are further seen to be in perspective, 
 since they have a self-corresponding point a'b'. Consequently the 
 straight line m' will always pass through a fixed point M', the 
 correspondent of M ; and will therefore trace out a pencil. The 
 pencils generated by m and m are projective, since the ranges 
 are projective in which they are cut by a pair of corresponding 
 sides of the quadrilaterals, e.g. by a and a'. To the rays of the 
 pencil M which pass respectively through the vertices ah , ac , ad, 
 be , bd , cd of the quadrilateral abed correspond the rays of the pencil 
 M f which pass respectively through the vertices a'V, a'cf, a'd' , 6V, 
 b'd', c'd' of the quadrilateral a / b / c / d / . 
 
 This reasoning holds good also when the point M, round which 
 m turns, lies upon one of the sides of the quadrilateral, on c for 
 example ; because we still obtain two ranges in perspective upon two 
 of the other sides. - Since c is now a ray of the pencil M , c' will be 
 the corresponding ray of the pencil M'\ that is to say, M f will lie on 
 c'. If M be taken at one of the vertices, as ed, then M f will coincide 
 with c'd', &c. 
 
 (4). Now suppose the pencil Mto be cut by a transversal n, and the 
 pencil M ' to be cut by the corresponding straight line n'. "While the 
 point mn describes the range n, the corresponding point m'n' will 
 describe the range n' ; and these two ranges will be projective since 
 they are sections of two projective pencils. When the point mn falls 
 on one of the sides of the quadrilateral abed, the point mn' will fall 
 on the corresponding side of the quadrilateral a'b'c'd' ; therefore the 
 two projective ranges are the same as those which it has already 
 been shown may be obtained by starting from the pairs of correspond- 
 ing points on a and a', b and b', c and e' . 
 
 G 2 
 
84 CONSTRUCTION OF PROJECTIVE FORMS. [97 
 
 In this manner the two planes become related to one another in 
 such a way that there corresponds uniquely to every point in the one a 
 point in the other, to every straight line a straight line, to every 
 range a projective range, to every pencil a projective pencil. The 
 two figures thus obtained are the same as those which can be obtained, 
 as explained above (Art. 95) by means of successive projections and 
 sections, so arranged as to lead from the quadrilateral abed to the 
 quadrilateral a'b'c'd'. For the two figures 7/ derived from it by 
 means of these two processes have four self-corresponding straight 
 lines a', b', c', d f forming a quadrilateral, and therefore (Art. 93) 
 every element (point or straight line) of the one must coincide 
 with the corresponding element in the other ; i.e. the two figures 
 must be identical. 
 
 97. Theorem. Any two projective plane figures {the straight lines 
 at infinity in which are not corresponding lines) can be superposed 
 one upon the other so as to become homological. 
 
 Let i , j f be the vanishing lines of the two figures — i.e. the 
 straight lines in each which correspond respectively to the straight 
 line at infinity in the other. In the first place let one of the figures 
 be superposed upon the other in such a manner that i and j' may be 
 parallel to one another. Since to any point M on i corresponds a 
 point at infinity in the second figure, to the pencil of straight lines 
 in the first figure which meet in M corresponds in the second figure 
 a pencil of parallel rays. Through M draw the straight line m 
 parallel to these rays ; then m will be parallel to its correspondent m'. 
 Similarly let a second point N be taken on i and through N let the 
 straight line n be drawn which is parallel to its correspondent n' ; 
 let m and n meet in S, and rnf and n' in JS'. If through S a straight 
 line I be drawn parallel to *, its correspondent V will pass through S' 
 and will also be parallel to i, since the point at infinity on i corre- 
 sponds to itself. The corresponding pencils S and S' are therefore such 
 that three rays I , m , n of the one are severally parallel to the three 
 corresponding rays V, m', n' of the other ; and consequently (see 
 below, Art. 104) the two pencils are equal. Now let one of the planes 
 be made to slide upon the other, without rotation, until S' comes 
 into coincidence with JS; then the two pencils will become concentric ; 
 and since they are equal, every ray of the one will coincide with the 
 ray corresponding to it in the other. This being the case, every 
 pair of corresponding points will be collinear with S, and the two 
 figures will be homological, S being the centre of homology. 
 
 98. Suppose that in a plane n is given a quadrangle A BCD, and 
 in a second plane it' a quadrilateral a'b'c'd'. By means of construc- 
 tions analogous to those explained in Arts. 94-96, the points and 
 straight lines of the one plane can be put into unique correspondence 
 
98] CONSTRUCTION OF PROJECTIVE FORMS. 85 
 
 with those of the other, so that to any range in the first plane cor- 
 responds in the second plane a pencil projective with the said range, 
 and to any pencil in the first plane corresponds in the second plane a 
 range projective with the said pencil. Two plane figures related to 
 one another in this manner are called correlative or reciprocal. 
 
CHAPTEK XI. 
 
 PARTICULAR CASES AND EXERCISES. 
 
 99. Two ranges are said to be similar, when to the points 
 A,B,C, D,... of the one correspond the points^', B', C',D', ... 
 of the other, in such a way that the ratio of any two corre- 
 sponding segments AB and A'B', AC and A'C , ... is a con- 
 stant. 
 
 If this constant is unity, the ranges are said to be equal. 
 Two similar ranges are projective, every anharmonic ratio 
 such as (ABC I)) being equal to the corresponding ratio 
 
 (A'B'C'B'). For suppose the 
 bases of the two ranges to lie 
 in the same plane (Fig. 6j) 
 and let their point of inter- 
 section be denoted by P f when 
 considered as a point be- 
 longing to u' and by Q when 
 Fig 6 7> considered as a point belong- 
 
 ing to u. Let A, A' be any 
 pair of corresponding points ; P that point of u which corre- 
 sponds to P\ and Q' that point of u' which corresponds to Q. 
 Draw A A" parallel to u', and A' A" parallel to u. - 
 
 The triangles PQQ', PA A" have the angles at Q and A 
 equal and the sides about these equal angles proportionals, 
 since by hypothesis 
 
 PQ PA PA_ 
 P'Q' ~~ P'A' ~ AA" ' 
 
 Therefore the triangles are similar, and the angles QPQ' and 
 APA" are equal; and consequently the points P,Q[, A" are 
 collinear. If then the range ABC ... be projected upon PQ', 
 by straight lines drawn parallel to n f , we shall obtain the 
 range A" B" C" ... ; and from this last, by projecting it upon 
 
102] PARTICULAR CASES AND EXERCISES. 87 
 
 u' by straight lines drawn parallel to u, the range A' B' C ... 
 may be derived. 
 
 If PQ = P' Q', i.e. if the straight line PQ' makes equal 
 angles with the bases of the given ranges, the ranges are 
 equal. 
 
 To the point at infinity of u corresponds the point at infinity 
 of u\ 
 
 100. Conversely, if the points at infinity I and V of two 
 projective ranges u and u' correspond to each other, the ranges 
 will he similar. For if (Fig. 67) u be projected from /', and u f 
 from /(as in Art. 85, left), two pencils of parallel rays will be 
 formed, corresponding pairs of which intersect upon a fixed 
 straight line u". The segments A" B" of »" will be propor- 
 tional to the segments AB of u and also to the segments A' B' 
 of u' ; consequently the segments AB of u will be proportional 
 to the segments A'B' of nf. 
 
 Otherwise: if AA\ BB\ CC are three pairs of corre- 
 sponding points, and / , V the points at infinity, we have (by 
 Art. 73) 
 
 (ABCI) = A'B' CI') ; 
 
 or (by Art. 64), since /and /' are infinitely distant, 
 AC A'C 
 BC ~ B'C' 
 an equation which shows that corresponding segments are 
 proportional to one another. 
 
 Examples. If a flat pencil whose centre lies at a finite distance 
 be cut by two parallel straight lines, two similar ranges of points will 
 be obtained. 
 
 Any two sections of a flat pencil composed of parallel rays are 
 similar ranges. 
 
 In these two examples the ranges are not only projective, but also 
 in perspective : in the first case the self-corresponding point lies at 
 infinity ; in the second case it lies (in general) at a finite distance. 
 
 101. Two flat pencils, whose centres lie at infinity, are pro- 
 jective and are called similar, when a section of the one is 
 similar to a section of the other. When this is the case any 
 other two sections of the pencils will also be similar to one 
 another. 
 
 102. From the equality of the anharmonic ratios we con- 
 clude that two equal ranges are projective (Art. 79), and that 
 
88 PARTICULAR CASES AND EXERCISES. [103 
 
 conversely two projective ranges are equal (Art. 73), when the 
 corresponding segments which are bounded by the points of 
 two corresponding triads ABC and A'B'C are equal ; i.e. when 
 A'B' = AB, A'C'=AC, (and consequently B'C'=BC). 
 
 Examples. If a flat pencil consisting of parallel rays be cut by 
 two transversals which are equally inclined to the direction of the 
 rays, two directly equal ranges of points will be obtained *. 
 
 If a flat pencil of non-parallel rays be cut by two transversals 
 which are parallel to one another,. and equidistant from the centre of 
 the pencil, two oppositely equal ranges will be obtained *. 
 
 103. Two similar ranges lying on the same base, and which have one 
 self- corresponding point N at infinity, have also a second such point 
 M, which is in general at a finite distance. If A A ', BB' are two 
 pairs of corresponding points, 
 
 MA : MA' = AB : A'B' = a constant. 
 To find M therefore it is only necessary to divide the segment A A' 
 into two parts MA , MA' which bear to one another a given ratio. 
 
 This ratio MA : MA' is equal (Art. 64) to the anharmonic ratio 
 (AA'MN). If its value is — i, the points AA'MN are harmonic 
 (Art. 68), i.e. M is the middle point of AA' , and similarly also that of 
 every other corresponding segment BB',... ; in other words, the two 
 ranges, which in this case are oppositely equal, are composed of pairs 
 of points which lie on opposite sides of a fixed point M, and at equal 
 distances from it. 
 
 But if the constant ratio is equal to + i , i. e. if MA and MA ' are 
 equal in sign and magnitude, the point M will lie at infinity. For 
 since (AA'MN)=i, .-. (NMA'A)=i (Art. 45); consequently the 
 points M and N coincide. 
 
 It follows also from the construction of Art. 90 (Fig. 66) that two 
 
 ranges on the same base, which Jiave 
 a single self-corres2)onding point lying 
 at infinity, are directly equal. 
 
 For if in Fig. 66 the point M 
 
 move off to infinity, the straight lines 
 
 SS' and A X B X become parallel to the 
 
 a"; Bi given straight line u or u' on which 
 
 Yig, 68. t'he ranges lie (Fig. 68), and as 
 
 the triangles SA 1 B 1 and S'A 1 B l lie 
 
 upon the same base and between the same parallels, the segments 
 
 * Imagine a moving point P to trace out a range ABC... and its correspondent 
 P' to trace out simultaneously the equal range A'B'C'.... Then if P and P' 
 move in the same direction, the two ranges are said to be directly equal ; if P 
 and P' move in opposite directions, the ranges are said to be oppositely equal. 
 
106] PARTICULAR CASES AND EXERCISES. 89 
 
 which they intercept upon any parallel to the base are equal ; thus 
 JB=A / B / , or two corresponding segments are equal', consequently 
 AA'=BB', i.e. the segment bounded by a pair of corresponding points 
 is of constant length. We may therefore suppose the two ranges to 
 have been generated by a segment given in sign and magnitude, 
 which moves along a given straight line ; the one extremity A of 
 the segment describes the one range, and the other extremity A / 
 describes the other range. 
 
 Conversely it is evident that if a segment A A', given in sign and 
 magnitude, slide along a given straight line, its extremities A and A' 
 will describe two directly equal (and consequently projective) ranges, 
 which have a single self-corresponding point, lying at an infinite 
 distance. 
 
 104. Two flat pencils are said to be equal when to the 
 elements of the one correspond the elements of the other in 
 such a way that the angle included between any two rays 
 of the first pencil is equal in sign and magnitude to the angle 
 included between the two corresponding rays of the second. 
 
 It is evident that two such pencils can always be cut by 
 two transversals in such a way that the resulting ranges are 
 equal ; but two equal ranges are always projective ; therefore 
 also two equal flat pencils are always projective. 
 
 Conversely, two projective flat pencils abed... and a'b'c'd' ... will 
 be equal if three rays abc of the one make with each other angles 
 which are equal respectively to those which the three corresponding 
 rays make with each other. 
 
 This theorem may be proved by cutting the two pencils 
 by two transversals in such a way that the sections ABC 
 and A'B'C of the groups of rays abc smd^a'b'c' may be equal. 
 The projective ranges so formed will be equal (Art. 102); con- 
 sequently also the other corresponding angles ad and a'd\ ... of 
 the given pencils must be equal to one another. 
 
 105. Since two equal forms (ranges or flat pencils) are 
 always projective with one another, it follows that if a range 
 or a flat pencil be placed in a different position in space, 
 without altering the relative position of its elements, the form 
 in its new position will be projective with regard to the same 
 form in its original position. 
 
 106. Consider two equal pencils abed... and a'b'c'd'... in the 
 same plane or in parallel planes ; and suppose a ray of the 
 one pencil to revolve about the centre and to describe the 
 
90 PARTICULAR CASES AND EXERCISES. [107 
 
 pencil ; then the corresponding ray of the other pencil will 
 describe that other pencil, by revolving about its centre. 
 This revolution may take place in the same direction as 
 that of the first ray, or it may be in the opposite direction ; 
 in the first case the pencils are said to be directly equals and in 
 the second case to be oppositely equal to one another. 
 
 In the first case the angles aa', W ', cc',... are evidently all 
 equal, in sign as well as in magnitude ; consequently a pair 
 of corresponding rays are either always parallel or never 
 parallel. 
 
 In the second case two corresponding angles are equal in 
 magnitude, but of opposite signs. If then one of the pencils 
 be shifted parallel to itself until its centre coincides with that 
 of the other pencil, the two pencils, now concentric, will still 
 be projective (Art. 105) and will evidently have a pair of 
 corresponding rays united in each of the bisectors (internal 
 and external) of the angle included between two correspond- 
 ing rays a and a'. It follows that these rays are also the 
 bisectors of the angle included between any other pair of 
 corresponding rays. If the first pencil be now replaced in its 
 original position, so that the two pencils are no longer con- 
 centric, we see that there are in each pencil two rays, each of 
 which is parallel to its correspondent in the other pencil ; and these 
 two rays are at right angles to each other, since they are parallel 
 to the bisectors of the angle between any pair of correspond- 
 ing rays. 
 
 107. If two flat pencils abed... and a'b'c'd' ... are projective, and 
 if the angles aa' , W ', cc' included by three pairs of corresponding rays 
 are equal in magnitude and of the same sign, then the angle dd' 
 included by any other pair of corresponding rays will have the same 
 
 For if we shift the first pencil parallel to itself until it 
 becomes concentric with the second, and then turn it about the 
 common centre through the angle aa', the rays a,b,c will coin- 
 cide with the rays a', b', c' respectively. The two pencils, which 
 are still projective (Art. 105), have then three self-correspond- 
 ing rays; consequently (Art. 82) every other ray will coincide 
 with its correspondent. If now the first pencil be moved back 
 into its original position, the angle dd' will be equal to aa'. 
 108. As the angles aa', bb' , cc', ... of two directly equal 
 
109] PARTICULAR CASES AND EXERCISES. 91 
 
 pencils are equal to one another, such pencils, when concentric 
 and lying in the same plane, may be generated by the rotation 
 of a constant angle aa! round its vertex 0, supposed fixed ; the 
 one arm a traces out the one pencil, while the other arm a' 
 traces out the other pencil. 
 
 Conversely, if an angle of constant magnitude turn round 
 its vertex, its arms will trace out two (directly) equal and 
 therefore projective pencils. Evidently these pencils have no 
 self-corresponding rays. 
 
 A transversal cutting these two pencils determines on 
 itself two collinear ranges having no self-corresponding points. 
 
 What has been said in Arts. 104-108 with respect to two pencils 
 in a plane might be repeated without any alteration for the case of 
 two axial pencils in space. 
 
 109. (1). Let ABC ..., A'B'C ' ... be two projective ranges lying 
 upon the same base, and let them, by means of the pencils abc ..., 
 a'b'c' '..., be projected from different points U ,U'. Let i,j' be those 
 rays passing through U, U' respectively, which are parallel to the 
 given base, and let i' ', j be the rays corresponding to them. The 
 points I', J in which these last two rays cut the given base will then 
 be those points which correspond to the point at infinity (I or J') of 
 the base, according as that point is regarded as belonging to the 
 range ABC ... or to the range A'B'C' ... 
 
 The fact that the two corresponding groups of points are pro- 
 tectively related gives an equation between the anharmonic ratios, 
 from which we deduce (as in Art. 74) 
 
 JA.rA'=JB.I'B'= a constant; . . . . (1) 
 
 i.e. the product J A . FA' is constant for every pair of points A, A'. 
 
 Let be the middle point of the segment JI', and 0' the point 
 corresponding to regarded as a point belonging to the first range. 
 
 Since the equation (1) holds for every pair of corresponding points, 
 and therefore also for and 0', we have 
 
 jA.rA'=jo.ro\ (2) 
 
 or (OA-OJ) (pA'-0I') + 0J(00'-0r) = o ; 
 
 or since 01' = — OJ, 
 
 0A.0A'-0r(0A-0A'+00') = o (3) 
 
 Let us now enquire whether there are in this case any self- 
 corresponding points. If such a point exist, let it be denoted by E ; 
 then replacing both A and A' in (3) by E, we have 
 
 0E 2 =0r.00' (4) 
 
 We conclude that when 01', 00' is positive, i.e. when does not 
 
92 PARTICULAR CASES AND EXERCISES. [109 
 
 lie between I' and 0' , there are two self-corresponding points E and 
 F, lying at equal distances on opposite sides of 0, and dividing the 
 segment I'O' harmonically (Art. 69). 
 
 When lies between I' and 0' , there are no such points. 
 When 0' coincides with 0, there is only one such point, viz. the 
 point itself. 
 
 (2). Imagine each of the given ranges to be generated by a point 
 moving always in one direction*. If the one. range is described in 
 the order ABC, the other range will be described in the order A'B'C; 
 this order may be the same as the first, or may be opposite to it. 
 
 If the order of ABC is opposite to that of A'B'C, the same will be 
 the case with regard to the order of IJA and that of I' J' A ', and again 
 with regard to the finite segment J A and the infinite segment J 'A'; 
 i.e. the finite segments J A and I 'A' have the same sign. In con- 
 sequence therefore of equation (2), JO and I'O' have the same sign ; 
 so that does not fall between I' and / (Fig. 6ga) ; there are there- 
 fore two self-corresponding points. And these will lie outside the 
 finite segment JI', since OE is a mean proportional between 01' and 
 00'. 
 
 If the order of ABC is the same as that of A 'B'C, we arrive in a 
 
 similar manner at the con- 
 clusion that J A and I'A\ 
 and again JO and I'O', have 
 opposite signs. In this case 
 Fig. 69. then,self-correspondingpoints 
 
 exist if does not lie be- 
 tween I' and 0'; that is, if 0' lies between and I' (Fig. 696). And 
 these will lie within the segment JI', since OE is a mean proportional 
 between 01' and 00'. 
 
 (3). Suppose that there are two self-corresponding points E and 
 F (Fig. 70) ; draw through E any straight line, on which take two 
 
 points S, S'; and project one of 
 the ranges from S and the 
 other from S'. The two pencils 
 which result are in perspective, 
 since they have a self-corre- 
 sponding ray SES'; accordingly 
 the corresponding rays SA and 
 S'A', SB and S'B', ...SF and 
 Fig. 70. ( S'F' will intersect in points 
 
 lying on a straight line u' 
 which passes through F. 
 
 Let E" be the point where this straight line u" meets SS'. Then 
 * Steiner, loc. cit. p. 61. § 16, II. Collected Works, vol. i. p. 280. 
 
 « 
 
 J 
 
 
 
 
 
 u 
 
 u' 
 u 
 
 J 
 
 
 
 
 0' 
 
 u 
 
 tt' 
 
110] PARTICULAR CASES AND EXERCISES. 93 
 
 EFAA' and EFBB' are the projections of EE"SS' from the centres 
 A" and B" respectively ; therefore EFAA f and EFBB' are projective 
 with one another ; thus the anharmonic ratio of the system consisting 
 of any two corresponding points together with the two self-corre- 
 sponding points is constant. 
 
 In other words : two projective forms which are superposed one upon 
 the other, and which have two self-corresponding elements, are composed 
 of pairs of elements which give with two fixed ones a constant anhar- 
 monic ratio *. 
 
 (4). Next suppose that there are no self-corresponding points ; so 
 that lies between 0' and I' (Fig. 71). Draw from a straight line 
 U at right angles to the given base and make U the geometric 
 mean between I'O and 00'; thus I'UO' will be a right angle. 
 
 Again, draw through Uthe straight line IUJ' parallel to the given 
 base; then the angle IUI' will be equal to JUJ', and the angle 
 0U0' will be equal to 01' U and 
 
 therefore to IUI'. Thus in the il____^_ L 
 
 two projective pencils which pro- •yn^> 
 
 ject the two given ranges from U, ' Sy | \ \X 
 
 the angles ITJV, JUJ', OUO' % f 2 ^ J-^ ^ — £ 
 
 included by three pairs of cor- 
 responding rays are all equal ; lg ' * ' 
 consequently (Art. 107) the angles 
 
 A UA', BUB', ... are also all' equal to them and to one another, and 
 are all measured in the same direction t. 
 
 Thus : two collinear ranges which have no self- cor responding points 
 can always be regarded as generated by the intersection of their base 
 line with the arms of an angle of constant magnitude which revolves, 
 ahvays in the same direction, about its vertex. 
 
 110. We have seen (Art. 84) the general solution of the problem : 
 Given three pairs of corresponding elements of two projective one- 
 dimensional forms, to construct any desired number of pairs ; or, in 
 other words, to construct the element of the one form which corre- 
 sponds to a given element of the other. The solution of the following 
 particular cases is left as an exercise to the student : 
 
 1. Suppose the two forms to be two ranges u and u' which lie on 
 different bases ; and let the given pairs of elements be 
 (a) P and P', Q and Q% A and A' \ 
 
 * The above construction gives the solution of the problem: Given two pairs 
 A , A and B , B f of corresponding points, and one of the self-corresponding points 
 E, to find the other self-corresponding point. 
 
 t Chasles, loc. cit. p. 119. 
 
 t P , P / , Q, Q', IfJft J, J' have the same meaning as in Art. 84 ; A ,B , ... are 
 any given points. 
 
94 
 
 PARTICULAR CASES AND EXERCISES. 
 
 [Ill 
 
 G>) 
 
 P and P', 
 
 .4 and .4', 
 
 P and B' 
 
 (c) 
 
 / and /', 
 
 »/ and J", 
 
 P and P' 
 
 (d) 
 
 / and /', 
 
 »7 and </', 
 
 4 and A' 
 
 (e) 
 
 / and /', 
 
 P and P', 
 
 Q and #' 
 
 m 
 
 I and /', 
 
 P and P', 
 
 A and ^1 / - 
 
 (g) 
 
 / and /', 
 
 A and -4', 
 
 P and B\ 
 
 2. Solve problems (d) and (g), supposing the ranges to be collinear. 
 
 3. Solve the problems correlative to (a) and (b) when the two given 
 forms are two non- concentric pencils. 
 
 4. Suppose one of the pencils to have its centre at infinity. 
 
 5. Suppose both the pencils to have their centres at infinity. 
 
 111. He may also prove for himself the following proposition : 
 
 If the three vertices A,A',A" of a variable triangle slide respectively 
 on three fixed straight lines u, u', u" ivhich meet in a point, while two 
 of its sides A' A", A" A turn respectively round two fixed points and 
 0', then will also the third side AA f always pass through a fixed point 
 0", collinear with and 0'. 
 
 It is only necessary to show that the points A>A', A" in moving 
 describe three ranges which are two and two in perspective. Or the 
 theorem of Art. 16 may be applied to two positions of the variable 
 triangle. 
 
 This proposition proved, the following corollary may be at once 
 deduced : 
 
 If the four vertices A,A f ,A'^A ffr of a variable quadrangle slide re- 
 spectively upon four fixed straight lines 
 which all pass through the same point 0, 
 while three of its sides AA',A'A", A"A "' 
 turn respectively round three fixed points 
 C', B"' t B f , then will the fourth side 
 A"' A and the diagonals A A", A' A"' 
 pass respectively through three other fixed 
 points C", C", B", which are deter- 
 mined by the three former ones. The six 
 fixed points ^are the vertices of a complete 
 quadrilateral, i.e. they lie three by three 
 on four straight lines (Fig. 7 2). 
 
 In a similar manner may be deduced the analogous corollary 
 relating to a polygon of n vertices. 
 
 112. Theokem. If a triangle 1 2 3 circumscribes another triangle 
 U 1 U 2 U 3 , there exist an infinite number of triangles each of ivhich is 
 circumscribed about the former and inscribed in the latter (Fig. 73). 
 
 The two pencils 
 
 0,(^,^,0;...) and O z (U lt U„U z ...) 
 
 Fig. 72. 
 
114] PARTICULAR CASES AND EXERCISES. 95 
 
 obtained by projecting the range U 2 U 3 ... from 2 and from 3 , are 
 evidently in perspective. Similarly the pencils 
 
 O x (U v U 2 , U 9 ... ) and 3 (tr u TJ % , U 3 ... ) 
 
 obtained by projecting the range U x U 3 ... from O x and from 3 , are 
 in perspective. Therefore the pencils 
 
 O x (U x ,U 2 ,U 3 ...) and 2 (U X , U 2 ,U 3 ...) 
 
 are projective (Art. 41); but the rays O x U 3 and 2 U 3 coincide; 
 therefore (Art. 62) the pencils are in perspective, and their corre- 
 sponding rays intersect in pairs on U X U 2 . 
 There are then three pencils O x , 2 , 3 , 
 which are two and two in perspective ; 
 corresponding rays of the first and 
 second, second and third, third and first, 
 intersecting in pairs on the straight lines 
 U X U 2 , U 2 U 3 , U 3 U X respectively. This 
 shows that every triad of corresponding 
 rays will form a triangle which is cir- 
 cumscribed about the triangle O x 2 3 , 
 and inscribed in the triangle U X U 2 U*. Fig. 73. 
 
 113. Theorem. A variable straight 
 
 line turning about a fixed point U cuts two fixed straight lines u and 
 u' in A and A ' respectively ; if S , S' are two fixed points collinear 
 with uu f , and SA. , S'A' be joined, the locus of their point of intersection 
 M will be a straight line t. 
 
 1 To prove this, we observe that the points A and A' trace out 
 two ranges in perspective with one another, and that consequently the 
 pencils generated by the moving rays SA , S'A ' are in perspective 
 (Arts. 41, 80). 
 
 The demonstration of the correlative theorem is proposed as an 
 exercise to the student. 
 
 114. Theorem. U,S,S / are three collinear points ; a transversal 
 turning about U cuts two fixed straight lines u and u' in A and A' 
 respectively ; if SA , S / A / be' joined, their point of intersection M will 
 describe a straight line passing through the point uu'\ K 
 
 The proof is analogous 'to that of the preceding theorem. 
 The proposition just stated may also be enunciated as follows : 
 If the three sides of a variable triangle A A 'M turn respectively about 
 three fixed collinear points U, S , #', while two of its vertices A , A / 
 
 * Steiner, loc. cit. p. 85. § 23, II. Collected Works, vol. i. p. 297. 
 t Pappus, loc. cit., book VII. props. 123, 139, 141, 143. Chasles, loc. cit. 
 pp. 241, 242. 
 
 X Chasles, loc. cit. p. 242. 
 
96 
 
 PARTICULAR CASES AND EXERCISES. 
 
 [115 
 
 Fig. 74. 
 
 slide respectively upon two fixed straight lines u , u' , then will the 
 third vertex M also describe a straight line *. 
 
 In a like manner may be demonstrated the more general theorem : 
 If a polygon of n sides displaces itself in such a manner that each of 
 its sides passes through one of n fixed collinear j>oints, while n—i 
 of its vertices slide each on one of n — i fixed straight lines, then will 
 also the remaining vertex, and the point of intersection of any two 
 non-consecutive sides, describe straight lines t. 
 
 The correlative proposition is indicated in Art. 85. 
 115. Problem. Given a parallelogram ABC D and a point P in its 
 plane, to draw through P a parallel to a given straight line EF also 
 lying in the plane, making use of the ruler only. 
 
 First Solution. — Let E and F (Fig. 74) be the points where the 
 
 given straight line is cut by AB and 
 AD respectively. On AC take any 
 point K\ join EX, meeting CD in 
 G, and FK, meeting BC in H. 
 
 The triangles AEF , CGH are 
 homological (Art. 18), since AC , EG, 
 FH meet in the same point K ; and 
 the axis of homology is the straight 
 line at infinity, since the sides 
 AE , AF of the first triangle are parallel respectively to the cor- 
 responding sides CG, CH of the second. Therefore also the remaining 
 sides EF and GH are parallel to one another £. 
 
 The problem is thus reduced to one already solved (Art. 86), viz. 
 given two parallel straight lines EF and GH, to draw through a 
 given point P a parallel to them. 
 
 Second Solution §. — Produce (Fig. 75) the sides AB, BC, CD, DA 
 q P and a diagonal AC of the 
 
 given parallelogram to meet 
 the given straight line EF in 
 E, F, G, H, I respectively, and 
 join EP, GP. Through I draw 
 any straight line cutting EP in 
 A' and GP in C, and join HA', 
 FC / ; if these meet in Q, then 
 will PQ be the required straight 
 line. 
 For if B' denote the point where EP cuts FQ, and D' the point 
 
 * This is one of Euclid's porisms. See Pappus, loc. cit., preface to book VII. 
 
 f This is one of the porisms of Pappus; loc. cit., preface to book VII. 
 
 % Poncelet, Propriitis projectives, Art. 198. 
 
 § Lambert, Freie Perspective (Zurich, 1774), vol. ii. p. 169. 
 
 Pig- 75. 
 
117] PARTICULAR CASES AND EXERCISES. 97 
 
 where GP cuts IIQ, the parallelograms ABCD and A'B'C'D' are 
 homological, EF being the axis of homology. The point P corre- 
 sponds to the point of intersection of AB and CD, and the point Q 
 to that of BC and AD ; therefore PQ corresponds to the line at in- 
 finity in the first figure ; accordingly it is the vanishing line of the 
 second figure, and consequently PQ is parallel to EF (Art. 18). 
 
 116. Pkoblem. Given a circle and its centre; to draw a perpen- 
 dicular to' a given straight line, making use of the ruler only. 
 
 Draw two diameters AC , BD of the circle (Fig. 76) ; the figure 
 ABCD is then a rectangle. Accordingly, if any point K be taken on 
 the circumference, then by means of the last 
 proposition (Art. 1 1 5) a parallel KL can be 
 drawn to the given straight line EF. If 
 the point L where this parallel again meets 
 the circumference be joined to the other 
 extremity M of the diameter through K, 
 then evidently LM will be perpendicular 
 to KL, and therefore also to the given 
 straight line. N pj g# ^5. 
 
 117. Problem. Given a segment AC and 
 
 its point of bisection B, to divide BC into n equal parts, making use 
 of the ruler only. 
 
 Construct a quadrilateral JJLDN (Fig. 77) of which one pair of 
 opposite sides DL , NU meet in A, the other pair LTJ , DN in C, and 
 of which one diagonal DU passes through B ; the other diagonal LN 
 will be parallel to AC (Art. 59), and will be bisected in M by DU. 
 
 Fig. 77. 
 
 Now construct a second quadrilateral VMEO which satisfies the 
 same conditions as the first, and which moreover has M for an 
 extremity and N for middle point of that diagonal which is parallel 
 to AC. To do this it is only necessary to join AM and BN, meeting 
 in E, and to join CE ; this last will cut LN produced in a point 
 
 H 
 
98 PARTICULAR CASES AND EXERCISES. [118 
 
 sucli that NO=.MN—LM. Now construct a third quadrilateral 
 analogous to the first two, and which has N for an extremity and 
 for middle point of that diagonal which is parallel to AC. If P is 
 the other extremity of this diagonal, then OP = NO=MN=LM. 
 Proceed in a similar manner, until the number of the equal segments 
 LM ,MN ,N0, OP, ... is equal to n. 
 
 If PQ is the segment last obtained, join LB, meeting QC in Z) 
 the straight lines which join Z to the points M , N ,0 ,P, ... will 
 divide BC into n equal parts *. 
 
 118. The following problems, to be solved by aid of the ruler only, 
 are left as exercises to the student : 
 
 Given two parallel straight lines AB and u ; to bisect the seg- 
 ment AB (Art. 59). 
 
 Given a segment AB and its point of bisection C ; to draw through 
 a given point a parallel to AB (Art. 59). 
 
 Given a circle and its centre ; to bisect a given angle (Art. 60). 
 
 Given two adjacent equal angles AOC, COB; to draw a straight 
 line through at right angles to OC (Art. 60). 
 
 119. Theorem. If two triangles ABC , A' B'C, lying in different 
 planes a , </, are in perspective, and if the plane of one of them be made 
 to turn round <ra', then the point in which the rays AA / , BB', CC 
 meet will change its position, and will describe a circle lying in a 
 plane perpendicular to the line oVf. 
 
 Let D,E,F (Fig. 78) be the points of the straight line oV 
 in which the pairs of corresponding sides BC and B'C, CA and CA',AB 
 and A 'B' meet respectively (Art. 18). First consider the planes of the 
 
 triangles to have any given definite posi- 
 tion, and let be the centre of projection 
 for that position. 'Through draw 
 OG , OH, OK parallel respectively to the 
 sides of the triangle A'B'C ; as these 
 parallels lie in the same plane (parallel 
 to (/) they will ifteet the plane <r in three 
 points G , H , K of the line ttv. 
 Fig. 78. Now suppose the plane </ together 
 
 with the triangle A' B'C to turn round 
 the line <r</. The range BCDG is in perspective with the range 
 B'CDG' (where G' denotes the point at infinity on B'C) ; there- 
 fore the anharmonic ratio {BCDG) is equal to the anharmonic ratio 
 {B'CDG'), i.e. to the simple ratio B'D : CD (Art. 64), which is 
 
 * These and other problems, to be solved by aid of the ruler only, will be found 
 in the work of Lambert quoted above. 
 
 t Chasles, loc. cit., Arts. 368, 369. A proposition equivalent to this has already 
 been proved by a different method in Art. 22. 
 
120] PARTICULAR CASES AND EXERCISES. 99 
 
 constant. Since then B, C, D are fixed points, G must also be a 
 fixed and invariable point (Art. 65). From the similar triangles 
 OBG , B BD 
 
 OG :B'D::BG: BD, 
 B'D . BG 
 
 0G = 
 
 BD 
 
 i. e. OG is constant. The point therefore moves on a sphere whose 
 centre is G and whose radius is the constant value just found for OG. 
 
 In a similar manner it may be shown that moves upon each of 
 two other spheres having their centres at H and K respectively. 
 
 Since then the point must lie simultaneously on* several spheres, 
 its locus must be a circle, whose plane is perpendicular to the line 
 of centres of the spheres, and whose centre lies upon this same line. 
 
 This line GHK is the line of intersection of the planes n and a- 
 and is consequently parallel to <rc/ (since it and </ are parallel planes) ; 
 it is the vanishing line of the figure <r, regarded as the perspective 
 image of the figure </ (Art. 13). 
 
 120. Theoeem. Two concentric projective pencils lying in the same 
 plane, which have no self-corresponding rays, may be regarded as the 
 perspective image of two directly equal pencils *. 
 
 Let be the common centre of the two pencils. Cut them by a 
 transversal s, thus forming two collinear projective ranges ABC... 
 and A 'B f C' .. . which have no self-corresponding points. Draw through 
 s any plane </ ; we can determine in this plane (Art. 109) a ppint U 
 such that the segments AA ', BB' , CO ', ... subtend at it a constant 
 angle ; thus if the two ranges be projected from U as centre, two 
 directly equal pencils will be obtained. Now let the eye be placed at 
 any point of the straight -line OTJ, and let the given pencils be pro- 
 jected from this point as centre on to the plane </. In this way two 
 new pencils will be formed ; and these are precisely the two directly 
 equal pencils mentioned in the enunciation. 
 
 * Ohasles, loc. cit., Art. 180. 
 
 H % 
 
CHAPTEK XII. 
 
 INVOLUTION. 
 
 121. Considee two projective flat pencils (Fig. 79) having a 
 common centre ; let them be cut in corresponding points by 
 the transversals u and u', thus giving two projective ranges 
 ABC ... and A'B'C ... ; and let u" be 
 the straight line on which the pairs 
 of lines AB' and A'B, ... (Art. 85, left) 
 intersect. Through draw any ray 
 (not a self-corresponding ray) ; it will 
 cut u and u' in two non-corresponding 
 •p. ' points A and B' and will meet u" in 
 
 a point of the line A'B. To the ray OA 
 of the first pencil corresponds accordingly the ray OA' of the 
 second, and to the ray OB' of the second pencil corresponds 
 the ray OB of the first. In other words, to the ray OA or OB' 
 correspond two different rays OA' , OB according as the first 
 ray is regarded as belonging to the first pencil or to the 
 second. For the line A'B must cut AB' on u", and cannot 
 pass through so long as this point does not lie on u". We 
 see then that 
 
 In two superposed projective forms* [of one dimension) there 
 correspond, in general, to any given element two different elements, 
 according as the given element is regarded as one belonging to the 
 ■first or to the second form. 
 
 We say in general, because in what precedes it has been 
 assumed that does not lie upon to". 
 
 * We say two forms, because the reasoning which we have made use of in the 
 case of two concentric flat pencils may equally well be applied in the case of two 
 collinear ranges, and of two axial pencils having a common axis. The same result 
 may be arrived at by cutting the two flat pencils by a transversal, and by pro- 
 jecting them from a point lying outside their plane. 
 
123] * INVOLUTION. 101 
 
 122. But in the case where lies upon u" (Fig. 8o), if a 
 ray be drawn through to cut u and u' in A and B' respec- 
 tively, then will also A'B pass through \ in other words, to 
 the ray OA or OB' corresponds 
 the same ray OA' or OB. 
 This property may be expressed 
 by saying that the two rays 
 correspond doubly to one another ; 
 or we may say that the two rays 
 are conjugate to one another. 
 
 Now suppose, reciprocally, 
 that two concentric projective Fig. 8o. 
 
 flat pencils have a pair of rays 
 
 which correspond doubly to one another. Cut the pencils 
 by two transversals u and u\ and let A and B' denote the 
 points where these transversals intersect one of the given 
 rays ; then A' and B will denote the poir>te where they 
 intersect the other given ray. The straight lire u'\ the 
 locus of the points of intersection of the "-pairs- of 3mea such 
 as MN', M'N, formed by joining crosswise any two pairs of 
 corresponding points of the ranges u , u\ (Art. 85), will pass 
 through 0, since the lines AB' , A'B meet in that point. If 
 now there be drawn through any other ray, cutting the 
 transversals say in C and B', then will C'B also pass through 
 0, i.e. the rays OCB' and OBC also correspond doubly to 
 each other. We conclude that 
 
 When two superposed projective forms of one dimension are such 
 that any one element has the same correspondent, to whichever 
 form it be regarded as belonging, then every element possesses this 
 
 123. This particular case of two superposed projective forms 
 of one dimension is called Involution*. We speak of an 
 involution of points, of rays, or of planes, according as the v 
 elements are points of a range, rays of a flat pencil, or planes 
 of an axial pencil. 
 
 In an involution, then, the elements are conjugate to one 
 another in pairs; i.e. each element has its conjugate. To I 
 whichever of the two forms a given element be considered to « 
 
 * Desargues, Brouillon projet d'une atteinte aux evenements des rencontres oVun 
 cdne avec unplan (Paris, 1639) : edition Poudba (Paris, 1864), vol. i. p. 119. 
 
102 INVOLUTION. [124 
 
 belong, the element which corresponds to it is the same, viz. 
 its conjugate. It follows from this that it is not necessary to 
 regard the two forms as distinct, but that an involution may he 
 considered as a set of elements which are conjugate to one another in 
 pairs. 
 
 When AA',BB', CC',... are said to form an involution, it is 
 to be understood that A and ^',i?andi?', Cand C\... are pairs 
 of conjugate elements ; moreover, any element and its con- 
 jugate may be interchanged, so that AA' BB' CC ... and 
 A' A B'B CC. . . are projective forms. 
 
 124. Since an involution is only a particular case of two 
 superposed projective forms, every section and every projection of 
 an involution gives another involution *. 
 
 Two conjugate elements of the given involution give rise to 
 two conjugate elements of the new involution. It fellows 
 (Art. 18) that^the figure homological with an involution is 
 also an\ involution. 
 
 . 125, When two collinear projective ranges form an involu- 
 tion, tMeiie; corresponds to each point (and consequently also to 
 the point at infinity i" or /') a single point (1' or J) ; i.e. the 
 two vanishing points coincide in a si ngle point. Let this point, 
 the conjugate of the point at infinity, be denoted by 0. The 
 equation (1) of Art. 109 then becomes 
 
 OA . OA' = constant. 
 In other words, an involution of points consists of pairs of 
 points A, A' which possess the property that the rectangle 
 contained by their distances from a fixed point 0, lying on 
 the base, is constant f- This point is called the centre of the 
 involution. 
 
 The self-corresponding elements of two forms in involution 
 ! are called the double elements of the involution. In the case of 
 1 the involution of points AA', BB',... we have 
 
 OA.OA'=OB.OB'=,..= constant. 
 If this constant is positive, i. e. if does not lie between two 
 conjugate points, there are two double points E&ndF, such 
 that 
 
 OE 2 = OF 2 = OA .OA' = OB .OB' = ... ; 
 
 * Desargues, he. cit. p. 147. 
 t Ibid. pp. 112, 119. 
 
126] INVOLUTION. 103 
 
 therefore lies midway between E and F, and the segment 
 FF divides harmonically each of the segments AA',JBF', ... 
 (Art. 69. [3]). Accordingly: 
 
 If an involution has two double elements, these separate har 
 monically any pair of conjugate elements; or: An involution is mack 
 up of pairs of elements which are harmonically conjugate tvith regard 
 to two fixed elements. 
 
 If, on the other hand, the constant is negative, i.e.\i falls 
 between two conjugate points, there are no double points. In 
 this case there are two conjugate points situated at equal 
 distances from and on opposite sides of it, such that 
 OE= - OF', and 
 
 OF 2 = OF' 2 = -OE.OF'=-OA. OA'. 
 
 If the constant is zero, there is only one double point ; 
 but in this case there is no involution properly so called. 
 For since the rectangle OA. OA' vanishes, one out of every 
 pair of conjugate points must coincide with 0. 
 
 126. The proposition that if an involution has two double 
 elements, these separate harmonically any pair of conjugate 
 elements, may also be proved thus : 
 
 Let F a,nd F be the double elements, A sand A' any pair of 
 conjugate elements ; since the systems FFAA', EFA'A are pro- 
 jective, therefore (Art. 83) each of them is harmonic. 
 
 The following is a third proof. 
 
 Consider EAA f ... and FA' A... as two projective ranges, and 
 project them respectively from two points SandS' collinear 
 with F (Fig. 8i). The projecting pencils S(FAA'...) and. 
 S'(EA'A...) are in perspective (since 
 they have a self-corresponding ray 
 in SS'F) ; therefore the straight line 
 which joins the point of intersection 
 of SA.tmd S'A' to that of SA' and 
 S'A will contain the points of inter- 
 section of all pairs of corresponding Fig. 8i. 
 rays, and will consequently meet 
 
 the common base of the two ranges at the second double 
 point F. But from the figure we see that we have now a 
 complete quadrilateral, one diagonal of which, AA', is cut by 
 the other two in F and F; consequently (Art. 56) FFAA' is a 
 harmonic range. 
 
104 
 
 INVOLUTION. 
 
 [127 
 
 The proposition itself is a particular case of that proved in 
 Art. 109 (3). From this we conclude that the pairs of elements 
 (points of a range, rays or planes of a pencil) which, with two 
 fixed elements, give a constant anharmonic ratio, form two 
 superposed projective forms, which become an involution in the 
 case where the anharmonic ratio has the value — i (Art. 68). 
 
 127. An involution is determined ly two 'pairs of conjugate 
 
 For let A , A 'and B,B'be the given pairs. If any element C 
 be taken, its conjugate is determinate, and can be found as in 
 Art. 84, by constructing so that the form A'AB'C' shall be 
 projective with AA'BC. We then say that the six elements 
 AA\BB\CC T are in involution; i. e. they are three pairs of an 
 involution. 
 
 Suppose that the involution with which we have to deal is 
 an involution of points. Take any point G (Fig. 82) outside 
 the base, and describe circles round GAA f 'and GBB'; if His 
 the second point in which these circles meet, join GH, and let 
 it cut the base in 0. Since GHAA' lie on a circle, 
 
 G. OH=OA.OA'; 
 and since GHBB' lie on a circle, 
 
 OG.OH=OB.OB'; 
 /. OA.OA'=OB.OB'. 
 is therefore the centre of the involution determined by the 
 
 pairs of points A , A 'and B ,B'. If any other circle be drawn 
 
 through 6r and ZT, and cut the base in CandC", we have 
 
 OG.OE= OC.OC'\ 
 
 :. OC . OC' = OA . OA' = OB .OB', 
 
 and C, C are therefore a pair of conjugate points of the invo- 
 
 In other words, the circle which passes through two 
 
 lution. 
 
128] 
 
 INVOLUTION. 
 
 105 
 
 conjugate points C, C or D ,1)' and through one of the points 
 G , H always passes through the other. Accordingly : 
 
 The pairs of conjugate points of the involution are the points of 
 intersection of the base with a series of circles passing through the 
 points G and H. 
 
 128. From what jxrecedes itis^evident that if the involution 
 has double points , t he s e wil l be the points of contact ofjhe 
 base wit h the two^ircleswhich ca n be draw n to pass through 
 G and i/ _and to touch the ha,sft. It has already been seen 
 (Art. 125) that these points are harmonically conjugate with 
 regard to A and A\ and also 
 with regard to B and B'. Con- 
 sequently (Art. 70) the involution 
 has double points when one of the 
 pairs AA f , BB / lies entirely within 
 or entirely without the other \ i.e. 
 when the segments AA r and BB' 
 do not overlap (Fig. 82)-; and the 
 involution has 'no double points 
 when one pair is alternate to the other, 
 [^ and BB' overlap (Fjg. 83) *, 
 
 In the first case, the involution (as already seen) consists of 
 an infinite number of pairs of points which are harmonically 
 conjugate with regard to a pair of fixed points. 
 
 In the second case, on the other hand, the involution is 
 traced out on the base by 
 the arms of a right angle 
 which revolves about its 
 vertex. For since (Fig. 
 84) the segments A A' and 
 BB' overlap, the circles 
 described onAA' and BB f 
 respectively as diameters Fi g> 84. 
 
 will intersect in two points 
 
 G and H which lie symmetrically with regard to the base ; 
 GH being perpendicular to the base, which bisects it at } 
 the centre of the involution. It follows that 
 
 Fig. 83. 
 e. when the segments AA r 
 
 * An involution of the kind hich has double points is often called a hyperbolic 
 involution ; one of the kind which has no double points being called an elliptic 
 involution. 
 
106 INVOLUTION. [129 
 
 OG 2 = 0H 2 = AO .OA' = BO .OB', 
 and that all other circles passing through G and H and 
 cutting the base in the other pairs CC\ BB\... of the involution 
 will have their centres also on the base, and will have CC\ 
 DD\... as diameters. If then we project any of the segments 
 AA\ BB\ CC\... from G (or H ) as centre, we shall obtain in 
 each case a right angle AGA', BGB', CGC\ ... (or AEA\ 
 BEB\ CHC ,...). 
 
 We conclude that when an involution of points AA ', BB',... 
 has no double points, i. e. when the rectangle OA.OA f is equal 
 to a negative constant — & 2 , each of the segments AA\BB\... 
 subtends a right angle at every point on the circumference of 
 a circle of radius k, whose centre is at and whose plane is 
 perpendicular to the base of the involution. 
 
 This last proposition is a particular case of that of Art. 109 (4). 
 If then an angle of constant magnitude revolve in its plane about its 
 vertex, its arms will determine on a fixed transversal two projective 
 ranges, which are in involution in the case where the angle is a right 
 angle. 
 
 129. Consider an involution of parallel rays ; these meet in a point 
 at infinity, and the straight line at infinity is a ray of the involution. 
 The ray conjugate to it contains the centre of the involution of points 
 which would be obtained by cutting the pencil by-any transversal ; it 
 may therefore be called the central ray of the given involution. If, 
 reciprocally/ we project an involution of points by means of parallel 
 rays, these rays will form a new involution, whose central ray passes 
 through the centre of the given involution. 
 
 When one involution is derived from another involution by fteans 
 of projections or sections (Art. 124), the double elements of the first 
 always give rise to the double elements ofj;he setqpd. 
 
 130. Since in an involution any groupof elements is projective with 
 the group of conjugate elements, it follows that if any four points of 
 the involution be taken, theirmnharmonic ratio will be equal to that 
 of their four conjugates. In the involution A A ', BB' , CC',. . . the 
 groups of points ABA'C and A 'B'AC, for example, will be projective ; 
 therefore 4 
 
 AA' AC^_A^A A'C 
 
 BA ,: BC '~ B / A : B 7 C > 
 whence 
 
 ^ AB\ BC. CA' + A'B.B'C. C'A = o. 
 
 Conversely, if this relation hold among the segments determined by 
 sixcollinear points AA'BB'CC^ these will be three conjugate pairs of 
 
131] 
 
 INVOLUTION. 
 
 107 
 
 an involution. For the given relation shows that the anharmonic 
 ratios (ABA'C) and (A'B'AC) are equal to one another; the groups 
 ABA'C and A'B'AC are therefore projective. But A and A' corre- 
 spond doubly to each other ; therefore (Art. 122) AA / ', BB / \ GC are 
 three conjugate pairs of an involution. 
 
 181. Theoeem. The three pairs 
 of opposite sides of a complete 
 quadrangle are cut by any trans- 
 versal in three pairs of conjugate 
 points of an involution *. 
 
 Let QRST (Fig. 85) be a 
 complete quadrangle, of which the 
 pairs of opposite sides RT and 
 QS y ST and QR, QT and RS are 
 cut by any transversal in A and 
 A', B and B\ C and C respec- 
 
 Fig. 85. 
 
 tively. If P is the point of 
 intersection of QS and RT, then 
 ATPR is a projection of ACA'B* 
 from Q as centre, and ATPR is 
 also a projection of ABA'C from 
 S as centre ; therefore the- group 
 4C4'£'isprojective vnth.ABA'C\ 
 and therefore (Art. 45) with 
 A' CAB. And since A and A' 
 correspond doubly to one another 
 in the projective groups ACA'B' 
 
 * DES ARGUES, 
 
 Correlative Theorem. The 
 straight lines which connect any 
 point with the three pairs of oppo- 
 site vertices of a complete quadri- 
 lateral are three pairs of conjugate 
 rays of an involution. 
 
 Let qrst (Fig. 86) be a com- 
 plete quadrilateral, of which the 
 pairs of opposite vertices rt and qs } 
 st and qr, qt and rs are projected 
 from any centre by the rays a and 
 a', b and b', c and c' respectively. 
 
 Fig. 86. 
 
 Let p be the straight line which 
 joins the points qs and rt. The 
 pencils atpr and aca'b' are in per- 
 spective (their corresponding rays 
 intersect in pairs on q) ; similarly 
 atpr and aba'c' are in perspec- 
 tive (their corresponding rays 
 intersect in pairs on s). The 
 pencil atpr is therefore of course 
 projective with each of the 
 pencils aca'b' and aba'c* ', and 
 
 loc cit. p. 171. 
 
108 INVOLUTION. [132 
 
 and A' CAB, it follows (Art. 122) therefore aca'b' is projective with 
 that AA',BB', CC are three con- aba'c or (Art. 45) with aYab. 
 jugate pairs of an involution. And since a and a? correspond 
 
 doubly to one another in the 
 
 pencils aca'b' and a' cab, it follows 
 
 (Art. 122) that aa', bb', cc' are 
 
 three pairs of conjugate rays of 
 
 an involution. 
 
 The theorem just proved may The theorem just proved may 
 
 also be stated in the following also be stated in the following 
 
 form : form : 
 
 If a complete quadrangle move If a complete quadrilateral 
 
 in such a way that jive of its sides move in such a way that Jive of its 
 pass each through one of five fixed vertices slide each on one of five 
 collinear points, then its sixth fixed concurrent straight lines, then 
 side will also /;ass through a fixed its sixth vertex will also move on a 
 point collinear with the other five, fixed straight line, concurrent with 
 and forming an involution with the other five, and forming an in- 
 them. volution with them. 
 
 132. By combining the preceding theorem (left) with that of Art. 
 130, we see that 
 
 If a transversal be cut by the three pairs of opposite sides of a com- 
 plete quadrangle in A and A', B and B' , C and C respectively, these 
 determine upon it segments which are connected by the relation 
 
 AB\ BC. CA' + A'B.B'C. C'A = o *. 
 
 133. In the theorem of Art. 131 (right) let U and V, Pand 7\ 
 W and W denote respectively the opposite vertices rt and qs, st and 
 qr, qt and rs of the quadrilateral qrst, and let AA ', BB f , CC denote 
 respectively the points of intersection of the rays aa' , bb' ', cc' with an 
 
 * arbitrary transversal. With the help of Art. 124 the following 
 proposition may be enunciated : 
 
 If the three pairs UTJ', VV, WW of opposite vertices of a complete 
 quadrilateral be projected from any centre upon any straight line, the 
 six points AA', BB', CC so obtained will form an involution. 
 
 Suppose now, as a particular case of this, that the centre of pro- 
 jection G is taken at one of the two points of inter section of the circles 
 described on UU', W respectively as diameters. Then AG A' and 
 BGB' are right angles, and therefore also (Art. 128) CGC is a right 
 angle ; therefore the circle on WW as diameter will also pass through 
 G. Hence the three circles which have for diameters the three 
 diagonals of a complete quadrilateral pass all through the same two 
 
 * Pappus, loc. cit., book VII. prop. 130. 
 
135] 
 
 INVOLUTION. 
 
 109 
 
 points; that is, they have the same radical axis. The centres of 
 these circles lie in a straight line ; hence 
 
 The middle points of the three diagonals of a complete quadrilateral 
 are collinear *. 
 
 The proposition of Art. 131 
 (right) leads immediately to the 
 
 Construction for the sixth ray 
 c' of an involution of which jive 
 rays a, a', b,b',c are given. 
 
 For take on c (Fig. 86) an arbi- 
 trary point, through which draw 
 wnicn taKe any two points v ana any two straight lines q and t, 
 T, and join AT, BT, A'Q, B'Q ; and join the point ta to qb' , and 
 
 the point tb to qa ; if the joining 
 lines be called r , s respectively, 
 then the straight line connecting 
 the centre of the pencil with the 
 yS point rs is the required ray c'. 
 
 If, in the preceding problem (left), the point C lies at infinity, its 
 conjugate is the centre of the involution. In order then to find 
 the centre of an involution of i hich two 
 pairs AA', BB f of conjugate points are 
 given, we construct (Fig. 87) a complete 
 quadrangle QSTR of which one pair of 
 opposite sides pass respectively through A 
 and A ', another such pair through B and B f , 
 and which has a fifth side parallel to the 
 base ; the sixth side will then pass through 
 the centre 0. 
 
 134. The proposition of Art. 
 
 131 (left) leads immediately to the 
 
 Construction for the sixth point 
 
 C r of an involution of which jive 
 
 points A,A',B,B', C are given. 
 
 For draw through C (Fig. 85) 
 an arbitrary straight line, on 
 [hich take any two points Q and 
 \ and join AT, BT, A'Q, B'Q; 
 if AT, B'Q meet in R, and BT, 
 A'Q in S, the straight line RS 
 will cut the base of the involu- 
 tion in the required point C 
 
 Fig. 87. 
 
 The sixth point C which, together with 
 five given points AA'BB'C, forms an involution, is completely deter- 
 mined by the construction ; there is only one point C f which possesses 
 the property on which the construction depends (Art. 127). This 
 may be otherwise seen by regarding C f as given by the equation 
 {AA f BC) = (A'AB'C) between anharmonic ratios; for it is known 
 (Art. 65) that there is only one point C f which satisfies this equation. 
 
 135. The theorem converse to that of Art. 131 is the fol- 
 lowing : 
 
 If a transversal cut the sides of a triangle RSQ (Fig. 85) in 
 three points A', B', C r which, when taken together with three other 
 points A , B , C lying on the same transversal, form three conjugate 
 
 * Chasles, loc, cit., Arts. 344, 345. Gauss, Collected Works, vol. iv. p. 391. 
 
110 INVOLUTION. [136 
 
 pairs of an involution, then the three straight lines RA , SB , QCmeet 
 in the same point. 
 
 To prove it, let RA , SB meet in T, and let TQ meet the 
 transversal in C v Applying the theorem of Art. 131 (left) to 
 the quadrangle QRST, we have 
 
 (AA'BCJ = (A'AB'C). 
 But by hypothesis 
 
 (AA'BC) = (A'AB'C); 
 .-. (AA'BC 1 ) = (AA'BC); 
 consequently (Art. 54) C 1 coincides with C, i.e. QC passes 
 through T. 
 
 The correlative theorem is : 
 
 If a point S he joined to the vertices of a triangle rsa (Fig. 86) by 
 three rays a', V, c' which, when taken together with three other rays 
 a,b ,c passing also through S } form three conjugate pairs of an 
 involution, then the points ra , qb , sc lie on the same straight line t. 
 
 136. Take again the figure of the complete quadrangle 
 QRST whose three pairs of opposite sides are cut by a trans- 
 versal in A and A', B and B', C and C. Let (Fig. 88) SQ and 
 RT meet in R', QR and ST in S', RS and QT in Q'. 
 
 Fig. 88. 
 
 Consider the triangle RSQ ; on each of its sides we have a 
 group of four points, viz. 
 
 SQR'A', QRS'B', RSQ'C. 
 
 The projections of these from T on the transversal are 
 
 BCAA', CABB', ABCC'. 
 
 The product of the anharmonic ratios of these last thiee 
 groups is 
 
 ( BA ■ BA\ ,CB CB\ f AC § AC\ 
 { CA : CA') \AB '' AB') ^BC '' BC'J ' 
 
138] INVOLUTION. Ill 
 
 CA'.AB'.BC 
 0r BA'. CB'. AC ' 
 
 which (Art. 130) is equal to — i. Therefore : 
 
 If any transversal meet the sides of a triangle, and if moreover from 
 as centre each vertex he projected upon the side opposite to it, 
 is of four points this obtained on each of the sides of the triangle 
 will he such that the product of their anharmonic ratios is equal to — i. 
 Conversely, if three pairs of points R'A', S'B', Q'C be taken, 
 one on each of the sides of a triangle RSQ, such that the product of 
 the anharmo?iicratios(SQR'A'),(QRS'B'), (RSQ' C) is equal to —1 ; 
 then, if the straight lines RR', SS', QQ' are concurrent, the points 
 A', B', C will be collinear; and conversely, if the points A\B\C 
 are collinear, the straight lines RR f , SS', QQ' will be concurrent. 
 
 137. Suppose now the transversal to lie altogether at 
 infinity; then the anharmonic ratios (SQR'A'), QRS'B'), and 
 (RSQ'C) become (Art. 64) respectively equal to SR' : QR', 
 QS'iRS', and RQ'\ SQ' ; so that the preceding proposition re- 
 duces to the following * : 
 
 If the straight lines connecting the three vertices of a triangle 
 RSQ with any given point T meet the respectively opposite sides 
 in R f , S', Q', the segments which they determine on the sides will be 
 connected by the relation 
 
 SR/.QS'.RQ' _ _ 
 QR'.RS'.SQ' ~ I; 
 and conversely : 
 
 If on the sides SQ , QR , RS respectively of a triangle RSQ 
 points R\ S', Q f be taken such that the above relation holds, then 
 will the straight lines RR f , SS', QQ f meet in one point T. 
 
 138. Kepeating this last theorem for two points T' and T", 
 we obtain the following : 
 
 If the two sets of three straight lines which connect the vertices of 
 a triangle RSQ with any two given points T f and T ff meet the 
 respectively opposite sides in R', S', Q f and R'\ S", Q", then will the 
 product of the anharmonic ratios (SQR'R"), (QRS'S"), and 
 {RSQ' Q") be equal to + i. 
 [For each of the expressions 
 
 SR'.QS'.RQ' SR". QS" . RQ" 
 QR'.RS'.SQ' ' QR".RS".SQ" 
 
 * Ceva's theorem. See his book, Be lineis rectis se invicem secantibus statica 
 construct™ (Mediolani, 1678), i. 2. Cf. M6BIUS, Baryc. Calc. § 198. 
 
112 INVOLUTION. [139 
 
 is equal to — i ; and the required result follows on dividing 
 one of them by the other.] 
 
 139. Considering again the triangle QBS (Fig. 88), and 
 taking the transversal to be entirely arbitrary, let ST , QT be 
 taken so as to be parallel to QB , BS respectively. Then the 
 figure QBST becomes a parallelogram; the points S' and Q' 
 pass to infinity, and B' (being the point of intersection of the 
 diagonals QS , BT) becomes the middle point of SQ. Conse- 
 quently (Art. 64) the anharmonic ratios {SQB'A'), {QBS'B'), 
 {BSQ 'C') become equal respectively to - ( QA ' : SA '), {BB' :QB'), 
 and (SC : RC). Thus*: 
 
 If a transversal cut the sides of a triangle RSQ in A', B', C 
 respectively, it determines upon them segments which are connected 
 by the relation 
 
 QA'.BB'.SC _ 
 SA'.QB'.BC'~ I; 
 and conversely: 
 
 If on the sides SQ , QB , BS respectively of a triangle 
 A', B\ C be taken such that the above relation holds, then 
 these three points be collinear. 
 
 140. Repeating the last theorem of the preceding Article for 
 two transversals, we obtain the following : 
 
 If the sides of a triangle BSQ are cut by two transversals in 
 A', B\ C and in A'\ B", C" respectively, the product of the 
 anharmonic ratios (SQA'A"), (QBB'B'% and (BSC'C") will be 
 equal to + 1. 
 
 [For each of the expressions 
 
 QA'. BB'. 8C QA"; RB" . SC" 
 SA'.QB'.BC ' SA". QB". BC" 
 
 is equal to i ; dividing one by the other, the required result 
 follows.] 
 
 Reciprocally, if on the sides of a triangle BSQ three pairs of 
 points A' A", B'B", C'C" be taken such that the product of the 
 anharmonic ratios {SQA'A"), (QBB'B"), {BSC'C") may be equal to 
 + i ; then, if the points A', B' , C are collinear, the points A", B", C" 
 will also be collinear, and if the lines BA' , SB', QC are concurrent , 
 the lines BA", SB", QC" will also be concurrent. 
 
 141. It has been shown (Art. 122) that if two projective ranges 
 * Theorem of Menelaus ; Sphaerica, iii. I. Cf. Mobius, loc. cit. 
 
142] INVOLUTION. 113 
 
 (ABC.) and (A'B'C '...), lying in the same plane, are projected from 
 the point of intersection of a pair of lines such as AB' and A'B, AC 
 and A'C,...ov BC / and B'C., the projecting rays form an involution. 
 The theorems correlative to this are as follows : 
 
 Given two projective", but not concentric, flat pencils (abc.) and 
 (a'b'c' ...) lying in the same plane ; if they be cut by the straight line 
 which joins a pair of points such as ab' and afb, ac' and a'c,... or he' 
 and b'c., the points so obtained form an involution. 
 
 Given two projective axial pencils (a/3y...) and (a'Ay* ...) whose 
 axes meet one another ; if they be cut by the plane which is deter- 
 mined by passing through a pair of lines such as a/3' and a'/3, ay and 
 ay,... or fry and /3'y..., the rays so obtained form an involution. 
 
 Given two projective flat pencils (abc.) and (a'b'c' ...) which are 
 concentric, but lie in different planes ; if they be projected from the 
 point of intersection of a pair of planes such as ab' and a'b, ac and 
 a'c,.. or be' and Kg..., the projecting planes form an involution. 
 
 142. Particular Cases. All points of a straight line which lie in 
 pairs at equal distances on opposite sides of a fixed point on the line, 
 form an involution, since every pair is divided harmonically by the 
 fixed point and the point at infinity. 
 
 Conversely, if the point at infinity is one of the double points of an 
 involution of points, then the other double point bisects the distance 
 between any point and its conjugate. If in such an involution the 
 segments A A' ', BB f formed by any two pairs of conjugate points have 
 a common middle point, then will this point bisect also the segment 
 CC formed by any other pair of conjugates. 
 
 All rectilineal angles which have a common vertex, lie in the same 
 plane, and have the same fixed straight line as a bisector, form an in- 
 volution, since the arms of every angle are harmonically conjugate 
 with regard to the common bisector and the ray perpendicular to it 
 through the common vertex. 
 
 Conversely, if the double rays of a pencil in involution include a 
 right angle, then any ray and its conjugate make equal angles with 
 either of the double rays. If in such an involution the angles included 
 by two pairs of conjugate rays aa' and bb f have common bisectors, 
 these will be the bisectors also of the angle included by any other pair 
 of conjugate rays cc' '. 
 
 All dihedral angles which have a common edge and which have the 
 same fixed plane as a bisector, form an involution ; for the faces of 
 every angle are harmonically conjugate with regard to the fixed plane 
 and the plane drawn perpendicular to it through the common edge. 
 
 Conversely, if the double planes of an axial pencil in involution are 
 at right angles to one another, then any plane and its conjugate make 
 equal angles with either of the double planes. 
 
 I 
 
CHAPTEE XIII. 
 
 PROJECTIVE FORMS IN RELATION TO THE CIRCLE. 
 
 *. 
 
 143. Consider (Fig. 89) two directly equal pencils abed... 
 and a'b'c'd'... in a plane, having their centres at and 0' 
 respectively. The angle contained by a pair of corresponding 
 rays aa', bb\ cc', ... is constant (Art. 106) ; the locus of the inter- 
 section of pairs of corresponding rays 
 is therefore (Euc. III. 21) a circle 
 passing • through and 0'. The 
 tangent to this circle at makes 
 with 00' an angle equal to any of 
 the angles OAO\ OBO', 0C0\ &c. ; 
 but this is just the angle which O'O 
 considered as a ray of the second 
 pencil should make with the ray 
 
 Yi g . 89. corresponding to it in the first pencil ; 
 
 therefore to O'O or q' considered as 
 a ray of the second pencil corresponds in the first pencil the 
 tangent a to the circle at 0. 
 
 Imagine the circumference of the circle to be described by a 
 moving point A ; the rays AO , AO' or a, a' will trace out the 
 two pencils. As A approaches 0, the ray AO' will approach 
 00' or q' and the ray AO will approach q\ and in the limit 
 when A is indefinitely near to 0, the ray AO will coincide with 
 q or the tangent at 0. This agrees with the definition of the 
 tangent at 0, as the straight line which joins two indefinitely 
 near points of the circumference. 
 
 Similarly, to the ray 00' or j? considered as belonging to the 
 first pencil corresponds the ray p of the second pencil, the 
 tangent to the circle at 0'. 
 
 144. Conversely, if any number of points A , B, C, D, . . . on a 
 circle be joined to two points and 0' lying on the same 
 
146] PROJECTIVE FORMS IN RELATION TO THE CIRCLE. 115 
 
 circle, the pencils (A, B, C, £,...) and 0' (A, B, C, D,...)so 
 formed will be directly equal, since the angle AOB is equal to 
 AO'B, AOC to AO'C,... BOC to BO'C, &c. But two equal 
 pencils are always projective with one another (Art. 104). If 
 then the points A, B, C, ... remain fixed, while the centre of 
 the pencil moves and assumes different positions on the cir- 
 cumference of the circle, the pencils so formed are all equal to 
 one another, and consequently all projective with one another. 
 The tangent at is by definition the straight line which joins 
 to the point indefinitely near to it on the circle. It follows 
 that in the projective pencils (A,B, C, ...)and 0' (A, B, C,...), 
 the ray of the first which corresponds to the ray O'O of the 
 second is the tangent at 0. 
 
 145. It has been seen (Art. 73) that in two projective forms 
 four harmonic elements of the one correspond to four harmonic 
 elements of the other. If then the four rays & (A, B, C, D) 
 form a harmonic pencil, the same is the case with regard to 
 the four rays 0' (A, B, C, D), whatever be the position of the 
 point 0' on the circle. By taking 0' indefinitely near to A, 
 we see that the pencil composed of the tangent at A and 
 the chords AB, AC, AD will also be harmonic; so again the 
 pencil composed of the chord BA, the tangent at B, and the 
 chords BC, BD will be harmonic, &c. 
 
 When this is the case, the four points A,B,C,I)ofthe circle 
 are said to he harmonic *. 
 
 146. The tangents to a circle 
 determine upon any pair of fixed 
 tangents ttvo ranges which are 
 projective with one another. 
 
 Let M (Fig. 90) be the 
 centre of the circle, PQ and 
 P'Q' a pair of fixed tangents, 
 and A A' a variable tangent. 
 The part AA ' of the variable 
 tangent intercepted between 
 the fixed tangents subtends 
 
 a constant angle at 31; for if Q, P', T are the points of 
 contact of the tangents respectively, 
 
 Fig. 90. 
 
 * Steiner, loc. cit., p. 
 
 57. § 43. Collected Works, vol. i. p. 345. 
 I 2 
 
116 PROJECTIVE FORMS IN RELATION TO THE CIRCLE. [147 
 
 angle AMA' = AMT + TMA' 
 
 = \QMT+\TMP' 
 = hQMP'*. 
 
 Accordingly, as the tangent AA' moves, the rays MA, MA' 
 will generate two projective pencils (Art. 108), and the points 
 A, A' will trace out two projective ranges. 
 
 Since the angle AM A' is equal to the half of QMP f , it is 
 equal to either of the angles QMQ', PMP f (denoting by P and 
 Q ' the same point, according as it is regarded as belonging to 
 the first or to the second tangent). Consequently Q and Q', 
 P and P' are pairs of corresponding points of the two pro- 
 jective ranges; i.e. the points of contact of the two fixed 
 tangents correspond respectively to the point of intersection of 
 the tangents. 
 
 Imagine the circle to be generated, as an envelope, by the 
 motion of the variable tangent ; the points A, A' will trace out 
 the two projective ranges. As the variable tangent approaches 
 the position PQ, the point A f approaches Q' , and A ap- 
 proaches the point which corresponds to Q', viz. Q ; and in the 
 limit when the variable tangent is indefinitely near to PQ, the 
 point A will be indefinitely near to Q or the point of contact 
 of the tangent PQ. The point of contact of a tangent must 
 therefore be regarded as the point of intersection of the 
 tangent with an indefinitely near tangent. 
 
 147. The preceding proposition shows that four tangents 
 a, b, o, d to a circle are cut by a fifth in four points A, B, C, D 
 whose anharmonic ratio is constant whatever be the position 
 of the fifth tangent. 
 
 This tangent may be taken indefinitely near to one of the 
 four fixed tangents, to a for example ; in this case A will be 
 the point of contact of a, and B,C ,D the points of intersection 
 ab, ac, ad respectively. 
 
 As a particular case, if a , b, c, d meet the tangent PQ in four 
 harmonic points, they will meet every tangent in four har- 
 monic points. The group constituted by the point of contact 
 of a and the points of intersection ab, ac, ad will also be har- 
 monic. In this case, the four tangents a, b, c, d are said to be 
 harmonic j\ 
 
 * Poncelet, Propr. proj., Art. 462. 
 
 t Steiner, loc. cit., p. 157. § 43. Collected Works, vol. i. p. 345. 
 
148] PROJECTIVE FORMS IN RELATION TO THE CIRCLE. 117 
 
 148. The range determined upon any given tangent to a circle by 
 any number of fixed tangents is projective with the pencil formed ly 
 joining their points of contact to any arbitrary point on the circle. 
 
 Let A, B, C,...X (Fig. 91) be points on the circle, and 
 a, b, c, ...cc the tangents at these points respectively. If the 
 points A f y B\ C", ... in 
 which the tangent x is cut 
 by the tangents a,b,c,... 
 be joined to the centre of 
 the circle, the joining lines 
 will be perpendicular re- 
 spectively to the chords 
 
 XA,XB, XC,... and will 
 
 Fig. 91. 
 
 therefore (Art. 108) form 
 
 a pencil equal to the pencil X (A , B, C, ...). The range A'B'C ..., 
 
 is therefore projective with the pencil X (A, B, C,...). 
 
 Corollary. If four points on a circle are harmonic , then the 
 tangents also at these points are harmonic ; and conversely. 
 
 For if, in what precedes, X (ABCIJ) is a harmonic pencil, 
 A'B'C'B' will be a harmonic range; and conversely. 
 
CHAPTER XIV. 
 
 PROJECTIVE FORMS IN RELATION TO THE CONIC SECTIONS. 
 
 149. Let the figures be constructed which are homological 
 with those of Arts. 144, 146, 148. To the points and tangents 
 of the circle will correspond the points and tangents of a conic 
 section (Art. 23). A tangent to a conic is therefore a straight 
 line which meets the curve in two points which are inde- 
 finitely near to one another; a point on the curve is the 
 point of intersection of two tangents which are indefinitely 
 near to one another. To two equal and therefore projective 
 pencils will correspond two projective pencils, and to two 
 projective ranges will correspond two projective ranges ; for 
 two pencils or ranges which correspond to one another in two 
 homological figures are in perspective. We deduce therefore 
 the following propositions : 
 
 (l). If any numher of points A, B, 0, D, ... on a conic are joined 
 to two fixed points and 0' lying on the same conic (Fig. 92), the 
 
 (A,B, C, If,...) and 
 
 0' (A, £,0,1),...) so formed 
 
 are projective with one another. 
 
 To the ray 00' of the first 
 
 pencil corresponds the tangent at 
 
 0', and to the ray O'O of the 
 
 » second pencil corresponds the 
 
 tangent at 0. 
 
 (2). Any number of tangents a,b,c,d, ...to a conic determine on a 
 
 pair of fixed tangents and 0' (Fig. 93) two projective ranges. To 
 
 the point 00' or Q of the first range corresponds the point of contact 
 
 Q of 0', and to the same point o'o or I" of the second range corre- 
 
 Fig. 92. 
 
 sponds the point of contact P of 0*. 
 
 * Steiner, he. cit., p. 139. § 38. Collected Works, vol. i. pp. 332, 333. 
 
150] PROJECTIVE FORMS IN RELATION TO CONIC SECTIONS. 119 
 
 (3). The range which a variable tangent to a conic determines upon 
 a fixed tangent is projective with the pencil formed by joining the 
 
 S 
 
 •93- 
 
 point of contact of the variable tangent to any fixed point of the 
 conic. (Fig. 94.) 
 
 150. We proceed now to the theorems converse to those of 
 Art. 149. The proofs here 
 given are due to M. Ed. Dewulf. 
 
 I. If two {non-concentric) pencils 
 lying in the same plane are pro- 
 jective with one another {but not 
 in perspective), the locus of the 
 points of intersection of pairs of 
 corresponding rays is a conic 
 passing through the centres of the 
 two pencils ; and the tangents to 
 the locus at these points are the rays which correspond in the two 
 pencils respectively to the straight line which joins the two centres. 
 
 Let and A (Fig. 95) be the respective centres of the two 
 pencils, and let 0M 1 and AM 1 , 0M 2 and AM 2 , 03I 3 and A M z ,... 
 be pairs of corresponding rays. The locus of the points M x } 
 M 2 ,3I 3i ... will pass through 0, since this point is the inter- 
 section of the ray A of the pencil A with the corresponding 
 ray of the pencil 0. Similarly A will be a point on the 
 locus. 
 
 Fig. 94. 
 
120 
 
 PROJECTIVE FORMS IN RELATION 
 
 [150 
 
 Let o be that ray of the pencil which corresponds to the 
 ray AO of the pencil A. Describe a circle touching o at 0, 
 
 Fig- 95- 
 
 and let this circle cut OA in A\ and 0M 1} 0M 2 , 0M 3 , ... in the 
 points 1//, J/"/, Jf 8 ',... respectively. 
 
 The pencils (Jf/ Jf/ M 3 f ... ) and A' (M{ M 2 ' M 3 f ...) are 
 directly equal to one another; and since by hypothesis the 
 pencil (if/ M 2 if/ ... ) or (M x M 2 M 3 ... ) is projective 
 with the pencil A (M 1 M 2 M 3 ...), therefore the pencils 
 A' (1// M 2 ' M z f ... ) and A (M x M 2 M 3 ... ) are projective. 
 But they are in perspective, since the ray A'O in the one 
 corresponds to the ray AO in the other (Artr 80) ; therefore 
 pairs of corresponding rays will intersect in points S^, S 2i 
 # 3 , ... lying on a straight line s. In order, then, to find that 
 point of the locus which lies on any given ray m of the pencil 
 A, it is only necessary to produce m to meet s in S, to join SA' 
 cutting the circle in M\ and to join 0M f ; this last line will 
 cut m in the required point M. But this construction is pre- 
 cisely the same as that employed in Art. 23 (Fig. n) in order 
 to draw the curve homological with a circle, having given the 
 axis s and centre of homology, and a pair of corresponding 
 points A and A'. The locus of the points M is therefore a 
 conic section. 
 
 II. If two (non-cottinear) ranges lying in the same plane are pro- 
 jective with one another {but not in perspective), the envelope of the 
 straight lines joining pairs of corresponding points is a conic, i.e. the 
 straight lines all touch a conic. This conic touches the bases of the 
 
150] 
 
 TO THE CONIC SECTIONS. 
 
 121 
 
 two ranges at the points which correspond in these respectively to the 
 point of intersection of their bases. 
 
 Let s and / (Fig. 96) be the bases of the two ranges, and 
 let A and A\ B and B\C and C\ , .. be pairs of corresponding 
 
 Fig. 96. 
 
 points. The curve enveloped by the straight lines AA\ BB\ 
 CC, ... will touch s, since this is the straight line joining the 
 point ss r or S' of the second range with the corresponding 
 point S of the first. Similarly, / will be a tangent to the 
 envelope. 
 
 Describe a circle touching s at S. and draw to it tangents 
 
 a , 
 
 9" from the points A,B,C, ... S' respectively. The 
 
 tangents a'\ b'\ c", . . . will determine on /' a range which is 
 projective with s and therefore also with /. But the point S' 
 corresponds to itself in the two ranges t? and /'; these are 
 therefore in perspective (Art. 80), and the straight lines A!' A\ 
 B" B\ C" C, ... will meet in one point 0. In order then to 
 draw a tangent to the envelope from any given point M lying 
 on the line s, it is only necessary to draw from M a tangent m 
 to the circle, meeting 9" in M ", and to join OM " ; this last line 
 will cut / in that point M f of the range *' which corresponds 
 to the point M of the range s, and MM' will be the required 
 
122 PROJECTIVE FORMS IN RELATION [151 
 
 tangent to the envelope. But this construction is precisely 
 the same as that made use of in Art. 23 (Fig. 12) in order to 
 draw the curve homological with a circle, taking a given tan- 
 gent to the circle as axis of homology, any given point as 
 centre of homology, and /, /' as a pair of corresponding 
 straight lines. The envelope of the lines MM' is therefore a 
 conic section. 
 
 The theorems (I) and (II) of the present Article are correlative 
 (Art. 33), since the figure formed by the points of intersection 
 of corresponding rays of two projective pencils is correlative 
 to that formed by the straight lines joining corresponding 
 points of two projective ranges. Thus in two figures which are 
 correlative to one another [according to the law of duality in a 
 plane), to points lying on a conic in one correspond tangents to a 
 conic in the other. 
 
 151. Having regard to Arts. 73 and 79, the propositions of 
 Arts. 149, 150 may be enunciated as follows : 
 
 The anharmonic ratio of the four straight lines which connect 
 four fixed points on a conic with a variable point on the same is 
 constant. 
 
 The a?iharmonic ratio of the four points in which four fixed tan- 
 gents to a conic are cut by a variable tangent to the same is 
 constant *. 
 
 The anharmonic ratio of four joints A, B, C, D lying on a conic is 
 the anharmonic ratio of the pencil (A, B , C , D) formed by joining 
 them to any point on the conic. The anharmonic ratio of four 
 tangents a, b, c, d to a conic is that of the four points (a, b, c, d), 
 where is an arbitrary tangent to the conic. 
 
 If this anharmonic ratio is equal to — 1, the group of four points 
 or tangents is termed harmonic. 
 
 The anharmonic ratio of four tangents to a conic is equal to that 
 of their points of contact \. 
 
 Consequently the tangents at four harmonic points are harmonic, 
 and vice versa. 
 
 The locus of a point such that the rays joining it to four given 
 points ABC I) form a pencil having a given anharmonic ratio is a 
 conic passi?ig through the given points. 
 
 * Steiner, he. cit., p. 156. § 43. Collected Works, vol. i. p. 344. 
 t Chasles, Geometrie Supeneure, Art. 663. 
 
152] 
 
 TO THE CONIC SECTIONS. 
 
 123 
 
 The tangent to the locus at one of these points, at A for 
 example, is the straight line which forms with AB, AC, AD a 
 pencil whose anharmonic ratio is equal to the given one. 
 
 The curve enveloped by a straight line which is cut by four give?i 
 straight lines in four points whose anharmonic ratio is given is a 
 conic touching the given straight lines. 
 
 The point of contact of one, of these straight lines, a for 
 example, forms with the points ah, ac, ad a range whose anhar- 
 monic ratio is equal to the given one *. 
 
 152. Through five given ])oints 
 0, 0' ,A,B ,C in a plane (Fig. 92), 
 no three of which lie in a straight 
 line, a conic can be described. 
 For we have only to construct the 
 two projective pencils which have 
 their centres at two of the given 
 points, and 0' for example, and 
 in which three pairs of corre- 
 sponding rays OA and O'A, OB 
 and O'B, OC and O'C intersect 
 n the three other points. Any 
 other pair OB and O'D of corre- 
 sponding rays will give a new 
 point D of the curve. 
 
 To construct the tangent at any 
 one of the given points, at for 
 example, we have only to deter- 
 mine that ray of the pencil 
 which corresponds to the ray O'O 
 of the pencil '. 
 
 Through five given points only 
 one conic can be drawn ; for if 
 there could be two such, they 
 would have an infinite number of 
 other points in common (the 
 intersections of all the pairs of 
 corresponding rays of the projec- 
 tive pencils) ; which is impossible. 
 
 Given five straight lines 
 o, o', a, b, c in a plane (Fig. 93), 
 no three of which meet in a point, 
 a conic can be described to touch 
 them. For we have only to con- 
 struct the two projective ranges 
 which are determined upon two of 
 the given lines, o and 0' for ex- 
 ample, by the three others a, b,c, 
 and of which three pairs of cor- 
 responding points oa and o'a, ob 
 and o'b, oc and o'c are given. 
 The straight lin« d which joins 
 any other pair of corresponding 
 points of the two ranges will be a 
 new tangent to the curve. 
 
 To construct the point of con- 
 tact of any one of the given 
 straight lines, that of for ex- 
 ample, we have only to determine 
 that point of the range which 
 corresponds to the point o'o of the 
 range o f . 
 
 Only one conic can be drawn to 
 touch five given straight lines ; 
 for if there could be two such, 
 they would have an infinite num- 
 ber of common tangents (all the 
 straight lines which join pairs of 
 corresponding points of the pro- 
 jective ranges) ; which is im- 
 
 * Steiner, loc. cit. f pp. 156, 157, §43. Collected Works, vol. i. pp. 344, 345. 
 
124 
 
 PROJECTIVE FORMS IN RELATION 
 
 [153 
 
 From this we see also that : 
 
 Through four given points can 
 be drawn an infinite number of 
 conies ; and two such conies have 
 no common points beyond these 
 four. 
 
 There can be drawn an infinite 
 number of conies to touch four 
 given straight lines ; and two such 
 conies have no common tangents 
 beyond these four. 
 
 153. The theorems of Art. 88 may now be enunciated in the 
 following manner : 
 
 If a hexagon is circumscribed to 
 a conic (Figs. 97 and 61), tlie 
 
 If a hexagon is inscribed in a 
 conic (Figs. 98 and 6o) 5 the three 
 
 Fig. 97. 
 
 straight lines which join the three 
 pairs of opposite vertices are con- 
 current. 
 
 This is known as Bkianchon's 
 theorem *. 
 
 Fig. 98. 
 
 pairs of opposite sides intersect 
 one another in three collinear 
 points. 
 
 This is known as Pascal's 
 theorem f. ' 
 
 154. Pascal's theorem has reference to six points of a conic, 
 Brianchon's theorem to six tangents ; these six points or tan- 
 gents may be chosen arbitrarily from among all the points on 
 the curve and all the tangents to it. Now a conic is deter- 
 mined by five points or five tangents; in other words, five 
 points or five tangents may be chosen at will from among all 
 the points or lines of the plane, but as soon as these five 
 
 * This theorem was published for the first time by Brianchon in 1806, and 
 afterwards repeated in his Memoir e sur les lignes da second ordre (Paris, 181 7: 
 P- 34)- 
 
 f This theorem was given in Pascal's E.<sai sur les Coniques, a small work of 
 six pages 8vo., published in 1640, when its author was only sixteen years old. 
 It was republished in the CEuvres de Pascal (The Hague, 1779), and again more 
 recently by H. Weissenborn, in the preface to his book Die Projection in da' 
 Ebene (Berlin, 1862). 
 
154] TO THE CONIC SECTIONS. 125 
 
 elements have been fixed, the conic is determined. Pascal's 
 theorem then expresses the condition which six points on a 
 plane must satisfy if they lie on a conic;. and Brianchon's 
 theorem expresses similarly the condition which six straight 
 lines lying in a plane must satisfy if they are all tangents to 
 a conic. And the condition in each case is both necessary 
 and sufficient. 
 
 That it is necessary is seen from the theorems themselves. 
 For six points on a conic, taken in any order, may be re- 
 garded* as the vertices of an inscribed hexagon*; but since 
 Pascal's theorem is true for every inscribed hexagon, the three 
 pairs of opposite sides must meet in three collinear points in 
 whatever order the six points be taken. 
 
 The condition is also sufficient. For suppose (Fig. 98) that 
 the hexagon AB'CA'BC, formed by taking the six points in a 
 certain order, possessesthe property that the pairs of opposite 
 sides BC'smd B'C, CA' and C'A, AB' and A '^intersect in three 
 collinear" points P, Q, B. Through the five points AB'CA'B 
 one conic (and one only) can be drawn ; if X be the point 
 where this conic cuts AG' again, then AB'CA'BX is an in- 
 scribed hexagon, and its pairs of opposite sides B'C and BXj 
 XA (or C'A) and CA', A' Band AB' will meet in three collinear 
 points. But the second and third of these points are Q and 
 B ; therefore BX must meet B'C at the point of intersection of 
 B'C and QB, i. e. at P. Both BC and BX thus pass through 
 P, and they must therefore coincide. Since then the point X 
 lies not only on AC but also on BC, it must coincide with . 
 the point C itself. 
 
 The condition is therefore sufficient ; and it has already 
 been shown to be necessary. 
 
 . By taking the six points in all the different orders possible, 
 sixty f simple hexagons can be made. From the reasoning 
 above, it follows that if any one of these hexagons possesses 
 the property that its three pairs of opposite sides intersect in 
 three collinear points, the six points will lie on a conic, and 
 consequently all the other hexagons will possess the same 
 
 * It is perhaps hardly necessary to remind the reader that the hexagons to 
 which Pascal's and Brianchon's theorems refer are not hexagons in Euclid's sense 
 — i. e. they are not necessarily convex (non-reentrant) figures. 
 
 + In general, a complete ?i-gon includes in itself | (n — 1) (n — 2) . . . 1 simple 
 
126 PROJECTIVE FORMS IN RELATION [155 
 
 property*. By analogous considerations having reference to 
 Brianchon's theorem, "properties correlative to those just estab- 
 lished may be shown to be true of a system of six straight 
 lines t. 
 
 155. Consider the two triangles which are formed, one by 
 the first, third, and fifth sides, the other by the second, fourth, 
 and sixth sides, of the inscribed hexagon AB'C A'BC (Fig. 98). 
 Let BC and B'C, CA'&nd C'A , AB' and A'B be taken as corre- 
 sponding sides of the triangles. By Pascal's theorem these 
 sides intersect in pairs in three collinear points ; and there- 
 fore (Art. 17) the two triangles are homological. Pascal's 
 theorem may therefore be enunciated as follows : 
 
 If two triangles are in homology \ the points of intersection of the sides 
 of the one with the non-corresponding sides of the other lie on a conic. 
 
 Similarly, in a circumscribed hexagon atica'bc' (Fig. 97) let 
 the vertices of even order and those of odd order respectively 
 be regarded as the angular points of two triangles, and let 
 he' and b'c, ca f and e'a, ab' and a'b be taken to be corresponding 
 vertices. By Brianchon's theorem these vertices lie two and 
 two on three straight lines which meet in a point ; therefore 
 (Art. 16) the two triangles are homological. Brianchon's 
 theorem may therefore be enunciated as follows : 
 
 If tivo triangles are in homology ', the straight lines joining the 
 angular points of the one to the non-corresponding angular points 
 of the other all touch a conic. 
 
 The two theorems maybe included under the one enunciation: 
 
 If two triangles are in homology, the points of intersection of the 
 sides of the one with the non-corresponding sides of the other lie on 
 a conic, and the straight lines joining the angular points of the one 
 to the non-corresponding angular points of the other all touch another 
 conic %. 
 
 156. Beturning to Fig. 98, let the points A, B', C, A', B he 
 regarded as fixed, and C as variable ; Pascal's theorem may 
 then be presented in the following form : 
 
 If a triangle C'BQ move in such a way that its sides PQ, QC, 
 C'P twrn round three fixed points R, A, B respectively, while two 
 
 * Steiner, loc. cit., p. 311. § 60. No. 54. Collected Works, vol. i. p. 450. 
 
 t A system of six points on a conic thus determines sixty different lines such as 
 PQR in Fig. 98, or Pascal lines as they have been called. So too a system of six 
 tangents to a conic determines sixty different Briuitchon points. 
 
 J Mobius, loc. cit., Art. 278. 
 
157] TO THE CONIC SECTIONS. 127 
 
 of its vertices P, Q slide along two fixed straight lines CB', CA f 
 respectively ', then the remaining vertex C f will describe a conic which 
 passes through the following five joints, viz. the two given points A 
 and B, the point of intersection C of the given straight lines, the 
 point of intersection B' of the straight lines Alt and CB', and the 
 point of intersection A' of the straight lines BR and CA'*. 
 
 So also Brianchon's theorem may be expressed in the 
 following form : 
 
 If a triangle c'pq (Fig. 99) move in such a way that its vertices 
 pq, qc', c'p slide along three fixed straight lines r, a, b respectively, 
 while two of its sides p , q turn round two fixed 
 points cb' , ca' respectively, then the remaining 
 side c' will envelope a conic which touches the j / 
 
 following five straight lines, viz. the two given a/7/\ 
 straight lines a and b, the straight line c which j 
 
 joins the fixed points, the straight line V which I p / j- //(^ 
 
 joins the points ar and cb', and the straight K/ \ r / \ c 
 
 line a' which joins the points br and cd '. ~&P^^_ jy 
 
 157. (1). If in the theorems of Art. 152 *' y^ 
 
 v ' • /a- 
 
 (right) one of the tangents is supposed Fi 
 
 to lie at infinity, the conic becomes a 
 
 parabola (Art. 23)/ Thus a parabola is determined by four tangents, 
 
 or (Art. 152, right) only one parabola can be drawn to touch four given 
 
 straight lines; and no two parallel tangents can be drawn to a parabola. 
 
 (2). If the same supposition is made in theorem (2) of Art. 
 149, it is seen that the points at infinity on the two tangents 
 and 0' are corresponding points of the projective ranges 
 determined on these tangents ; for the straight line which 
 joins them is a tangent to the curve. It follows (Art. 100) 
 that 
 
 The tangents to a parabola meet two fixed tangents to the same in 
 points forming two similar ranges ; or 
 
 Two fixed tangents to a parabola are cut proportionally by the 
 other tangents f . 
 
 (3). Let A and A', B and B f , C and C, ... be the points in 
 
 * This theorem was given by Maclaurin, in 1721 ; cf. Phil. Tram, of the Royal 
 Society of London for 1735, and Chasles, Apercu historique sur Corigine et le 
 developperaent des me'thodes en Geometric (Brussels, 1837 > second edition, Paris, 
 1875). If B lies at infinity, the theorem becomes identical with lemma 20. 
 book i. of Newton's Principia. 
 
 t Apollonii Pekgaei Conicorum lib. iii. 41. 
 
128 PROJECTIVE FORMS IN RELATION [158 
 
 which the various tangents to the parabola meet the two 
 fixed tangents (Fig. ioo), and let P and Q' be the respective 
 points of contact of the latter. The point of intersection of 
 
 Fig. ioo. 
 
 the two fixed tangents will be denoted by Q or P' according 
 as it is regarded as a point of the first or of the second tan- 
 gent. We have then 
 
 AB AC_ BG_ AP_ AQ^ PQ^ 
 
 TB f ~ A'C' B'C ~ A'P' ~ A'Q' " ■" ~ P'Q f ' 
 
 (4). Conversely, given two straight lines in a plane, on which lie 
 two similar ranges (which are not in perspective), the straight lines 
 connecting pairs of corresponding points will envelope a parabola which 
 touches the given straight lines at t/ie points which correspond in 
 the two ranges respectively to their point of intersection. 
 
 For the points at infinity on the given straight lines being 
 corresponding points (Art. 99), the straight line which joins 
 them will be a tangent to the envelope ; thus the envelope is 
 a conic (Art. 150 (II)) which has the line at infinity for a 
 tangent, i.e. it is a parabola. 
 
 158. In theorem I of Art. 150 (Fig. 95) suppose that the 
 point A lies at infinity, or, in other words, that the pencil A 
 consists of parallel rays. To the straight line OA, considered as 
 a ray a' of the pencil (viz. that ray which is parallel to the 
 rays of the other pencil), corresponds that ray a of the pencil 
 A which is the tangent at the point A. This ray a may be at 
 a finite, or it may be at an infinite distance. 
 
 In the first case (Fig. 101) the straight line at infinity is 
 a ray/ of the pencil A, and to it corresponds in the pencil 
 a ray/' different from a' and consequently not passing through 
 
159] 
 
 TO THE CONIC SECTIONS. 
 
 129 
 
 A ; the conic will therefore be a hyperbola (Art. 23) having 
 A(z=zaa!) &ndjj' for its points at infinity; the straight line a 
 is one asymptote and/ 7 is parallel to the other. 
 
 Fig. ioi. 
 
 Fig-. 102. 
 
 In the second case (Fig. 102) the line at infinity is the 
 tangent at A to the conic, which is therefore a parabola. 
 
 159. If in this same theorem of Art. 150 the points A and 
 are supposed both to lie at infinity (Fig. 103), the two pro- 
 jective pencils will each consist of parallel rays ; and since 
 the conic which these pencils 
 generate must pass through A and 
 it is a hyperbola (Art. 23). The 
 asymptotes of the hyperbola are 
 the tangents to the curve at its 
 infinitely distant points*; they 
 will therefore be the rays a and </ 
 of the first and second pencil which 
 correspond to the straight line at 
 infinity considered as a ray of the 
 second and first pencil respectively. 
 
 By the general theorem of Art. 149, the asymptotes of a 
 hyperbola are cut by the other tangents in points forming 
 two projective ranges, in which the points of contact (which 
 are in this case at infinity) correspond respectively to the point 
 of intersection Q of the asymptotes. The equation of Arts. 74 
 
 and 109 (l), viz. 
 
 JM . V M ' = constant 
 
 becomes therefore in this case 
 
 QM. QM' = constant, 
 
 Fig. 103. 
 
 Desargues, loc. cit., p. 210. Newton, Principia, lib. i. prop. 27, Scholium. 
 
 K 
 
130 PEOJECTIVE FORMS IN THE CONIC SECTIONS. [160 
 
 M and M' being the points of intersection of any tangent with 
 the asymptotes. We conclude therefore that 
 
 The segments which are determined by any tangent to a hyperbola 
 on the two asymptotes (measured from the point of intersection of 
 the asymptotes), are such that the rectangle contained by them is 
 constant. 
 
 This may be stated in a different form as follows : 
 
 The triangle formed by any tangent to a hyperbola and the 
 asymptotes has a constant area *. 
 
 160. Again, let the theorem of Art. 149 be applied to the 
 
 case of two fixed parallel tangents which are cut by a variable 
 
 tangent in M and M '. In the projective ranges thus generated 
 
 the points which correspond respectively to the infinitely 
 
 distant point of intersection of the two fixed tangents are 
 
 their points of contact ; if these be denoted by / and I\ we 
 
 have by Art. 74 the equation 
 
 JM. VM f - constant. 
 
 Therefore, the segments which a variable tangent to a conic cuts of 
 
 from two fixed parallel tangents {measured from the points of contact 
 
 of these latter) are such that the rectangle contained by them is 
 
 t- 
 
 * Apollonius, loc. cit., iii. 43. 
 t Ibid. iii. 42. 
 
CHAPTEE XV. 
 
 CONSTRUCTIONS AND EXERCISES. 
 
 161. By help of Pascal's and Brianchon's theorems may be 
 solved the following problems : 
 
 Given five tangents a, V, c , a', Given five points A , B', C , A', 
 
 b, to a conic, to draw from any B on a conic, to find the j>oint of 
 
 given i)oint H^ lying on one> of intersection of the curve with any 
 
 these tangents a, another tangent given straight liner drawn through' 
 
 to the curve (Fig. 104). one of these points A (Fig. 105). 
 
 Fig. 104. 
 
 It c / be the required tangent, 
 ^ab'ca'bc' is a hexagon to which 
 Brianchon's theorem applies. Let 
 r be the diagonal connecting one 
 pair ay and a f b of opposite ver- 
 tices, and let q be the diagonal 
 connecting another such pair ca' 
 and c / a (where ca is the given 
 pointZT) ; then the diagonal which 
 connects the remaining pair hd 
 and b'c must pass through the 
 point qr . If then p be the straight 
 
 Fig. 105. 
 
 If C be the required point, 
 AB'CA'BC is a hexagon to 
 which Pascal's theorem applies. 
 Let R be the point of intersection 
 of one pair ^LS' and A'B of oppo- 
 site sides, ancTlet ♦ be the point 
 of intersection of another such 
 pair CA' and r\ then QR must 
 pass through the point of inter- 
 section of the remaining pair 
 BC and B'C. If then PB be 
 joined, .it will cut the given 
 
 K % 
 
132 
 
 CONSTRUCTIONS AND EXERCISES. 
 
 [162 
 
 line joining the points qr and Vc, 
 the straight line which joins <pb 
 to the given point H is the re- 
 quired tangent. 
 
 By assuming different positions 
 for the point H, all lying on one 
 of the given tangents, and repeat- 
 ing in each case the above con- 
 struction, any desired number of 
 tangents to the conic may be 
 drawn. 
 
 Brianchon's theorem therefore 
 serves to construct, by means of its 
 tangents, the conic which is deter- 
 mined by five given tangents*. 
 
 straight line r in the required 
 point C f . 
 
 By assuming different positions 
 for the given straight line r, all 
 passing through one of the given 
 points on the conic, and repeating 
 in each case the above construc- 
 tion, any desired number of points 
 on the conic may be found. 
 
 Pascal's theorem therefore serves 
 to construct, by means of its 
 points, the conic which is deter- 
 mined by five given points f. 
 
 162. Particular eases of the problem of Art. 161 (right). 
 
 I. Suppose the point B to lie at infinity ; the problem then 
 becomes the following : 
 
 Given four points A , B\ C , A' on a hyperlola and the direction 
 of one asymptote, to find the second point of intersection C of the 
 curve with a given straight line r drawn through A (Fig. 106). 
 
 Solution. This is deduced from that of 
 the general problem by taking the point 
 B to lie at infinity in the given direction. 
 We draw through A' a straight line m in 
 this direction ; if then AB/ meets m in R, 
 and A'C meets r in Q, we join QR meeting 
 B'C in P, and draw through P a parallel 
 to.m; this parallel will cut r in the re- 
 quired point C. 
 
 II. Suppose the point A to lie at in- 
 finity; the problem is then : 
 
 Given four points B', C , A\ B on a hyper- 
 Lola and the direction of one asymptote-, to find the point of inter- 
 section of the curve with a given straight line r drawn parallel to 
 this asymptote (Fig. 107). 
 
 Solution. Draw* through -Z?'a straight line parallel to the given 
 direction. If this line meet A'B in R, and if A'C meet / in 
 
 * Brianchon, loc. cit., p. 38 ; Poncelet, loc. cit., Art. 209. 
 t Newton, Principia, prop. 22 ; Maclaurin, De linearum geometricamm pro- 
 prietatibus generations (London, 1 748), § 44. 
 
 Fig. 106. 
 
162] 
 
 CONSTRUCTIONS AND EXERCISES. 
 
 133 
 
 Q, join QR cutting B'C in P. Then if BP be joined, it will 
 cut r in the required point C 
 
 III. Suppose the two points .4' and B both to lie at'infinity. 
 The problem then becomes : 
 
 Given three points A , B\ C on a hyperbola and the directions of 
 loth asymptotes, to find the second point of intersection of the curve ■ 
 with a given straight line r drawn through A (Fig. 108). 
 
 directions of 
 
 Fig. 108. 
 
 Solution. Through the point Q, where the given straight 
 line r meets a straight line drawn through C parallel to the 
 direction of the first asymptote, draw a parallel to AB' . Let 
 P be the point where this parallel cuts B'C \ then a parallel 
 through P to the second asymptote will cut r in the required 
 point C \y^ 
 
 . IV. If the two points A and B' both lie at infinity, the 
 problem is : 
 
 Given three points C , A', B of a 
 loth asymptotes, to find the point of 
 intersection of the curve with a given 
 straight line r drawn parallel to one of 
 the asymptotes (Fig. 109). 
 
 Solution. Through Q, the point of 
 intersection of r and CA\ draw a 
 parallel to A'B ; let P be the point 
 where this parallel meets the straight 
 line drawn through C parallel to the 
 other asymptote. Then if BP be 
 joined, it will cut r in the required point C. 
 
 V. If, lastly, the points B', C , A\ B are finite and the 
 straight line AC lies at infinity, the problem becomes the 
 following : 
 
 Fig. 109. 
 
 4L 
 
134 
 
 CONSTRUCTIONS AND EXERCISES. 
 
 [163 
 
 Fig. no. 
 
 Given four points B', C \A\B of a hyperbola and the direction 
 of one asymptote, to find the direction of the other asymptote 
 (Fig. no). 
 
 Solution. Through the point B, in which A'B meets the 
 
 straight line drawn through 
 B f in the given direction, 
 draw a parallel to CA' \ let 
 P be the point where this 
 parallel cuts B'C. Then if 
 BP be joined, it will be 
 parallel to the required di- 
 rection. 
 
 It will be a useful exercise 
 for the student to deduce the constructions for these particular 
 cases from the general construction ; in order to do this it is 
 only necessary to remember that to join a finite point to a 
 point lying at infinity in a given direction we merely draw 
 through the former point a parallel to the given direction. 
 163. Particular cases of the problem of Art. 161 (left). 
 
 I. Suppose the point ac' to lie at infinity ; then the problem 
 becomes the following : 
 
 Given five tangents a , b\ c , a\ b to a conic \ to draw the tangent 
 ivhich is parallel to one of them, to a, for example (Fig. in). 
 Solution. Draw through the point a'c a straight line a 
 
 parallel to a; join ab' and a'b 
 by the straight line r, and join 
 the points qr and Vc by the 
 straight line^?. Then if through 
 the point pb a parallel be drawn 
 to a, it will be the required 
 tangent. 
 
 From a given point in the 
 Fig. ni> plane of a conic two tangents 
 
 at most can be drawn to the 
 curve (Art. 23) ; so that from a point lying on a given tangent 
 only one other tangent can be drawn. If then the conic is a 
 parabola, it cannot have a pair of parallel tangents. (This 
 has already been seen in Art. 157 (l).) 
 
 II. Suppose the straight line b to lie at infinity; the 
 problem is then: 
 
163] 
 
 CONSTRUCTIONS AND EXERCISES. 
 
 135 
 
 Fig. 112. 
 
 Given four tangents a,b',c,a' to a parabola, to draw from a given 
 point H lying on one of them, a, another tangent to the curve (Fig. 
 1 1 %). 
 
 ■ Solution. Through the point 
 ah' draw the straight line r 
 parallel to a! '; join the points 
 H and a'c by the straight line 
 q, and the points qr and Vc by 
 the straight line p. The straight 
 line drawn through H parallel 
 to p will be the required tan- 
 gent. 
 
 III. If the straight line a 
 lies at infinity, we have the problem : 
 
 Given four tangents b', c,a',b to a parabola, to draw the tangent 
 which is parallel to a given straight line (Fig. 113). 
 
 Solution. Through a'b draw 
 .the straight line r parallel to 
 V, and through a'c draw the 
 straight line q parallel to the 
 given direction ; join the 
 points qr, b'c by the straight 
 line p. The straight line 
 through pb parallel to the. 
 given direction is the tangent 
 required. 
 
 IV. If in problem II the point H assume different positions 
 on a, or if in III the given straight line assume different 
 directions, we arrive at the solution of the problem : 
 
 To construct by means of its tangents the parabola which is deter- 
 mined by four given tangents, u^ 
 
 Fig. 113 
 
CHAPTEE XVI. 
 
 DEDUCTIONS FROM THE THEOREMS OF PASCAL 
 AND BRIANCHON. 
 
 164. We have already given some propositions and con- 
 structions (Arts. 161-163) which follow immediately from the 
 theorems of Pascal and Brianchon, by supposing some of 
 the elements to pass to infinity. Other corollaries may be 
 deduced by assuming two of the six points or six tangents to 
 approach indefinitely near to one another *. 
 
 If AB'C A' ' BC are six points on a conic, Pascal's theorem 
 asserts that the pencils A(A'B'CC) and B(A'B'CC% for 
 example, are projective with one another. To the ray AB of 
 the first pencil corresponds in the second the tangent at B, so 
 that we may say that the group of four lines 
 
 AA\ AB\ AC , AB 
 is projective with the group 
 
 BA\ BB\ BC, tangent at B. 
 But this amounts evidently to saying that the point C\ which 
 was at first taken to have any arbitrary position on the curve, 
 
 has come to be indefinitely near to 
 the point B. Instead then of the 
 inscribed hexagon we have now the 
 figure made up of the inscribed 
 pentagon AB f CA r B and the tan- 
 gent h at the vertex B (Fig. 
 114); and Pascal's theorem be- 
 comes the following : 
 114. . If a pentagon is inscribed in a conic, 
 
 the points of i?itersection R , Q of two 
 pairs of non-consecutive sides (AB' and A'B, AB and CA f ), and the 
 
 * Caenot, loc. cit., pp. 455, 456. 
 
165] THEOREMS OF PASCAL AND BRIANCHON j DEDUCTIONS. 137 
 
 point P where the fifth side (B'C) meets the tangent aU the opposite 
 vertex, are collinear. { ^* 
 
 This corollary may also be deduced from the construction (Art. 84, 
 right) for two projective pencils. Three pairs of corresponding rays 
 are here given, viz. AA' and BA', AC'und BC, AB' and BB' . We 
 cut the two pencils by the transversals CA', CB' respectively; if R 
 be the point of intersection of A'B and AB', then any pair of corre- 
 sponding rays of the two pencils must cut the transversals CA', CB' 
 respectively in two points which are collinear with R. In order 
 then to obtain that ray of the second pencil which corresponds to 
 AB, viz. the tangent at B, we join R to the point of intersection Q of 
 CA' and AB, and join QR meeting CB' in P; then BP is the 
 required ray b. But this construction agrees exactly with the corol- 
 lary enunciated above. 
 
 165. By help of this corollary the two following problems can be 
 solved : 
 
 (1). Given five points A , B' , C , A',B of a conic, to draw the tangent 
 at one of them B (Fig. 114). 
 
 Solution. Join Q, the point of intersection of AB and CA', to R, 
 the point of intersection of AB' and A'B ; if P is the point where 
 QR meets B'C, then BP will be~the required tangent *. 
 
 Particular cases. 
 
 Given four points of a hyperbola and the direction of one asymptote, 
 to draw the tangent at one of the given points. (This is obtained 
 by taking one of the points A , B', C , A' to lie at infinity.) 
 
 Given four points of a hyperbola and the direction of one asymptote, 
 to draw that asymptote. (1> at infinity.) 
 
 Given three points of a hyperbola and the directions of both 
 asymptotes, to draw the tangent at one of the given points. (Two of 
 the four points A, B', C , A' at infinity.) 
 
 Given three points of a hyperbola and the directions of both 
 asymptotes, to draw one of the asymptotes. (B and one of the other 
 points at infinity.) - ^ u. 
 
 (2). Given four points A, B , A', C of a conic and the tangent at 
 one of them B, to construct the conic by points ; for example, to find 
 the point of the curve which lies on a given straight line r drawn 
 through A (Fig. 114). 
 
 Solution. Let R be the point where A'B meets r, and Q the 
 point where AB meets CA' ; and let QR cut the given tangent in P. 
 The point B' where CPcuts the given straight line r will be the one 
 required. 
 
 By supposing one or more of the elements of the figure to lie at 
 
 * Maclaurin, loc. cit., § 40. 
 
 11 
 
138 
 
 DEDUCTIONS FKOM THE THEOREMS 
 
 [166 
 
 infinity, e. g. one of the points A , A', C ; or two of these points ; or 
 the point A and the line r ; or the point B ; or the point B and one 
 of the other points ; or the point B and the given tangent ; we obtain 
 the following particular cases : 
 
 To construct by points a hyperbola, having given 
 
 three points of the curve, the tangent at one of these points, and 
 the direction of one asymptote ; 
 or : two points, the tangent at one of them, and the directions of 
 
 both asymptotes ; 
 or : three points and an asymptote ; 
 
 or : two points, one asymptote, and the direction of the other 
 asymptote. 
 Given three points of a hyperbola, the tangent at one of them, 
 and the direction of an asymptote, to find the direction of the other 
 asymptote. 
 
 To construct by points a parabola, having given three points of 
 the curve (lying at a finite distance) and the direction of the point at 
 infinity on it. 
 
 166. Returning to the hexagon AB'CA'BC inscribed in a 
 conic, let not only C be taken in- 
 definitely near to B, but also C 
 indefinitely near to B f . The figure 
 will then be that of an inscribed 
 quadrangle AB'A'B together with the 
 tangents at B and B f (Fig. 115), and 
 Pascal's theorem becomes the follow- 
 ing: 
 
 If a quadrangle is inscribed in a 
 conic, the points of intersection of the 
 two pairs of opposite sides, and the point 
 of intersection of the tangents at a pair 
 of opposite vertices, are three collinear 
 
 This property coincides with one already obtained elsewhere (Art. 
 85, right). For considering the projective pencils of which BA and B'A, 
 BA ' and B'A ', . . . are corresponding rays, it is seen that the straight 
 line which joins the point of intersection Q of BA and B'A' to the 
 point of intersection R of B'A and BA' must pass through the point 
 of intersection P of the rays which correspond in the two pencils 
 respectively to the straight line joining their centres B and B\ 
 
 167. By help of the foregoing corollary the following problems can 
 be solved : 
 
167] • OF PASCAL AND BRIANCHON. 139 
 
 (1). Given four points A ,B' ,A',B of a conic and the tangent BP 
 at one of them B, to draw the tangent at another of the points B / 
 
 (Fig. 115). 
 
 Solution. Let AB and A'B' meet in Q, and AB' and A f B in R ; 
 and let QR meet the given tangent in P. Then B'P will be the 
 required tangent *. 
 
 By supposing one of the given points, or the given tangent, to 
 lie at infinity, the solutions of the following particular cases are 
 obtained : 
 
 To draw the tangent at a given point of a hyperbola, having given 
 in addition two other points on the curve, the tangent at one of them, 
 and the direction of one asymptote ; or, one other point, the tangent 
 at' this, and the directions of both asymptotes ; or, one other point, 
 one asymptote, and the direction of the other asymptote. 
 
 To draw the asymptote of a hyperbola when its direction is known, 
 having given in addition three points on the curve and the tangent at 
 one of them ; or, two points on the curve, the tangent at one of them, 
 and the direction of the second asymptote ; or, two points on the 
 curve and the second asymptote. 
 
 To draw the tangent at a given point of a parabola, having given 
 two other finite points on the curve, and the direction of the point at 
 infinity on it. 
 
 (2). To construct a conic by points, having given three points A ,B, B f 
 on the curve and the tangents BP , B'P at two of them ; i. e. to 
 determine, for example, the point A' in which an arbitrary straight 
 line r drawn through B is cut by the conic (Fig. 116). 
 
 Solution. Join the point of intersection P of the given tangents 
 to the point R where r cuts AB ' ; and let K 
 
 PR cut AB in Q. If B 'Q be joined, it will 1 
 
 cut r in the required point A f . 
 
 By supposing one of the points A , B , B' 
 or one of the lines BP , B'P, r to lie at 
 infinity, we shall obtain the solutions of 
 the following particular cases : 
 
 To construct by points a hyperbola, 
 having given two points on the curve, the 
 tangents at these, and the direction of one • 
 asymptote ; or, one point on the curve, 
 the tangent there, one asymptote and the 
 direction of the second asymptote ; or, one point on the curve and 
 both asymptotes. 
 
 To construct by points a parabola^ having given two points on the 
 
 * Maclaurin, he. cit., § 38. 
 
 A* 
 
140 
 
 DEDUCTIONS FROM THE THEOREMS 
 
 [168 
 
 curve, the tangent at one of them, and the direction of the point at 
 infinity on the curve. 
 
 168. The tangents at the other vertices A and A f of the 
 quadrangle ABA'B' (Fig. 1 16) will also intersect on the straight 
 line joining the points (AH, A'B') and (AB', A f B). Hence the 
 theorem of Art. 166 may be enunciated in the following, its 
 complete form : 
 
 If a quadrangle is inscribed in a conic, the points of intersection 
 of the two pairs of opposite sides, and the points of intersection of 
 the tangents at the two pairs of opposite vertices, are four collinear 
 points. 
 
 If two opposite vertices of the quadrangle be taken to lie at 
 infinity, this becomes the following : 
 
 If on a chord of a hyperbola, as diagonal, a parallelogram be 
 constructed so as to have its sides parallel to the asymptotes, the 
 other diagonal will pass through the point of intersection of the 
 asymptotes. 
 
 169. Theorem. The complete quadrilateral formed by four 
 to a conic, and the complete quadrangle formed by their four 
 
 of contact, have the same diagonal triangle. 
 In the last two figures write C , JD , B , G in pladfe of 
 
 A f ,B',H, Q respectively. In the inscribed quadrangle ABCD 
 (Fig. 117) the point of intersection of the tangents at A and C, 
 
170] OF PASCAL AND BRIANCHON. 141 
 
 that of the tangents at B and D, the point of intersection of the 
 sides AD, BC, and that of the sides AB , CD all lie on one straight 
 line EG. If the same points A,B,C,D are taken in a different 
 order, two other inscribed quadrangles ACDB and ACBD are 
 obtained, to each of which the theorem of Art. 168 may be 
 applied. Taking the quadrangle ACDB, it is seen that the 
 point of intersection of the tangents at A and D, that of the 
 tangents at C and B, the point of intersection of the sides 
 AB , CD, and that of the sides AC , BD all lie on one straight 
 line FG. So too the quadrangle ACBD gives four points 
 lying on one straight line EF; viz. the points of intersection 
 of the tangents at A and B, of the tangents at C and D, of the 
 sides AD , CB, and of the sides AC , BD*. 
 
 The three straight lines EG , GF , FE thus obtained are the 
 sides of the diagonal triangle EFG (Art. 36, [2] ) of the complete 
 quadrangle whose vertices are the points A,B,C,D; and 
 since the same straight lines contain also the points in which 
 intersect two and two the tangents a,b, c, d at these points, 
 they are also the diagonals of the complete quadrilateral 
 formed by these four tangents. The theorem is therefore 
 proved. 
 
 170. In the complete quadrilateral abed the diagonal /, 
 whose extremities are the points ac , bd, cuts the other two 
 diagonals g and e in E and G respectively ; these two points 
 are therefore harmonically conjugate with regard to ac and bd 
 (Art. 56). The correlative theorem is : The two opposite sides 
 of the complete quadrangle ABCD which meet in i^are har- 
 monically conjugate with regard to the straight lines which 
 connect F with the two other diagonal points E and G (Art. 57). 
 Summing up the preceding, we may enunciate the following 
 proposition (Fig. 117): 
 
 If at the vertices of a {simple) quadrangle ABCD, inscribed in a 
 conic, tangents a,b,c, d be drawn, so as to form a (simple) quadrila- 
 teral circumscribed to the conic, then this quadrilateral possesses the 
 following properties with regard to the quadrangle: (i)the diagonals 
 of the two pass through one point (F) and form a harmonic pencil ; 
 (2) the points of intersection of the pairs of opposite sides of the two 
 lie on one straight line (EG) and form a harmonic range ; (3) the 
 
 * Maclauiun, loc. cit., § 50 ; Carnot, loc. cit., pp. 453, 454. 
 
142 
 
 DEDUCTIONS FROM THE THEOREMS 
 
 [171 
 
 diagonals of the quadrilateral pass through the points of intersection 
 of the pairs of opposite sides of the quadrangle *, 
 
 171. By help of the theorem of Art. 169, when we are given four 
 tangents a ,b ,c ,d to a conic and the point of contact A of one of 
 them, we can at once find the points of contact of the three others ; 
 and when we are given four points A , B, G, D on a conic and the 
 tangent a at one of them, we can draw the tangents at the three 
 other points t. 
 
 Solution. Draw the diagonal 
 triangle EFG of the complete 
 quadrilateral abed ; then AG, 
 AF, AE will cut b, c, d respec- 
 tively in the required points of 
 contact B,G, D. 
 
 Draw the diagonal triangle 
 EFG of the complete quadrangle 
 ABCD; then the straight lines 
 joining ag,af,ae to B, G, D re- 
 spectively will be the required 
 tangents. 
 
 172. The theorem of Art. 169 may be enunciated with re? 
 gard to the (simple) quadrilateral formed by the four straight 
 lines a, b, c,d; it then takes the following form, under which 
 it is seen to be already included in the theorem of Art. 170 J : 
 
 In a quadrilateral circumscribed to a conic, the straight lines 
 which join the points of contact of the pairs of opposite sides pass 
 through the point of intersection of the diagonals (Fig. 118). 
 
 This property coincides with one already proved with regard 
 to two projective ranges (Art. 85, left). For 
 consider the projective ranges on a and c as 
 bases, in which ab and cb, ad and cd, ... are 
 corresponding points ; the straight lines which 
 connect the pairs of points ab and cd, cb and 
 ad respectively, must intersect on the straight 
 line which connects the points corresponding 
 in the two ranges respectively to ac ; but this 
 is the straight line joining the points of contact 
 of a and c. 
 
 If the conic is a hyperbola, and we consider 
 the quadrilateral which is formed by the asymp- 
 totes and any pair of tangents, the foregoing 
 theorem expresses that the diagonals of such a quadrilateral are 
 parallel to the chord which joins the points of contact of the two 
 tangents §. 
 
 * Chasles, Sections coniques, Art. 121. 
 t Maclaurin, loc. cit., §§ 38, 39. 
 % Newton, loc. cit., Cor. ii. to lemma xxiv. 
 § Apollonius, loc. cit, iii. 44. 
 
 Fig. 11 
 
175] OF PASCAL AND BEIANCHON. 143 
 
 173. The theorem of Art. 172 gives the solution of the problem : 
 To construct a conic by tangents, having given three tangents a,b,c 
 
 and the points of contact A and G of two of them; to draw, for 
 example, through a given point H lying on a a second tangent to the 
 curve (Fig. 118), 
 
 Solution. Join the point ab to the point of intersection of AG and 
 H(bc) ; the joining line will meet c in a point which when joined to 
 H gives the required tangent d. 
 
 If one of the points A , G or one of the given tangents be supposed 
 to lie at infinity, the solution of the following particular cases is 
 obtained : 
 
 To construct by tangents a hyperbola, having given one asymptote, 
 two tangents to the curve, and the point of contact of one of them ; 
 or, both asymptotes and one tangent. 
 
 To construct by tangents a parabola, having given the point at 
 infinity on the curve, two tangents, and the point of contact of one of 
 them ; or, two tangents and the points of contact of both. 
 
 Given four tangents to a conic and the point of contact of one of 
 them, to find the points of contact of the others. 
 
 174. If in Pascal's theorem the points A,'B', C' be taken to lie 
 indefinitely near to A , B , C re- * 
 spectively, the figure becomes \vy^n 
 
 that of an inscribed triangle lew''/ 
 
 ABC together with the tangents ^^S^^rK 
 
 at its vert As (Mg. 119); and /^^M\ 
 
 the theorem reduces to the / ^^y~^^\ 
 
 * 11 • / /^-<k 
 
 following: / / ^^s£ 
 
 In a triangle r inscribed in a conic, 1 . / >• 
 
 the tangents at the vertices meet the V__^^ 
 
 respectively opposite sides in three Fi S- ll 9- 
 
 175. This gives the solution of the problem : 
 
 Given three points A, B,G of a conic and the tangents at two of 
 them A and B, to draw the tangent at the third point G (Fig. 119). 
 
 Solution. Let P, Q be the points where the given tangents at 
 A , B cut BG, CA respectively ; if PQ cut AB in B, then GR is the 
 tangent required. 
 
 The following are particular cases : 
 
 Given two points on a hyperbola, the tangents at these points, 
 and the direction of one asymptote, to construct the asymptote 
 itself. 
 
 Given one asymptote of a hyperbola, one point on the curve, the 
 
144 DEDUCTIONS FROM THE THEOREMS [176 
 
 tangent at this point, and the direction of the second asymptote, to 
 construct this second asymptote. 
 
 Given both asymptotes of a hyperbola and one point on the curve, 
 to draw the tangent at this point. 
 
 (From the solution of this problem, it follows that the segment 
 determined on any tangent by the asymptotes is bisected at the point 
 of contact). 
 
 Given two points on a parabola, the direction of the point at 
 infinity on the curve, and the tangent at one of the given points, to 
 draw the tangent at the other given point. 
 
 176. The inscribed triangle ABC and the triangle BEF 
 formed by the tangents (Fig. 119) possess the property that 
 their respective sides BC and EF, CA and FD, AB and BE in- 
 tersect in pairs in three collinear points. The triangles are 
 therefore homological, and consequently (Art. 18) the straight 
 lines AB ,BF, CF which connect their respective vertices pass 
 through one point 0. Thus we have the proposition : 
 
 In a triangle circumscribed to a conic, the straight lines which join 
 the vertices to the points of contact of the respectively opposite sides 
 are concurrent. 
 
 177. By help of this proposition the following problem can be 
 solved : 
 
 Given three tangents to a conic and the points of contact of two of 
 them, to determine the jtoint of contact of the third. 
 
 Solution. Let DEF (Fig. 119) be the triangle formed by the 
 three tangents, and let A,B be the points of contact of EF, FB re- 
 spectively. If AB and BE intersect in 0, then FQ will cut the 
 tangent BE in the required point of contact C. 
 
 Particular cases. 
 
 Given one asymptote of a hyperbola, two tangents, and the point 
 of contact of one of them, to determine the point of contact of the 
 other. 
 
 Given both asymptotes of a hyperbola, and one tangent, to deter- 
 mine the point of contact of the latter. 
 
 Given two tangents to a parabola and their points of contact, to 
 determine the direction of the point at infinity on the curve. 
 
 Given two tangents to a parabola, the point of contact of one of 
 them, and the direction of the point at infinity on the curve, to deter- 
 mine the point of contact of the other given tangent. 
 
 178. As a particular case of the theorem of Art. 176, consider a 
 parabola and the circumscribing triangle formed by the tangents at 
 any two points A, B, and the straight line at infinity, which is also 
 
179] 
 
 OF PASCAL AND BRIANCHON. 
 
 145 
 
 Fig. 1 20. 
 
 a tangent. If the tangents at A and B meet in C (Fig. 120), the 
 straight line joining C to the middle point D of the chord AB will be 
 parallel to the direction in 
 which lies the point at infinity 
 on the curve. 
 
 Again, if any point M be 
 taken on AB, and parallels 
 MP , MQ be drawn to BC , AC 
 respectively to meet AC , BC 
 in P, Q ; and if MP be drawn 
 parallel to DC to meet PQ in 
 R ; then PQ will be a tangent 
 to the parabola, and R its point of contact. 
 
 179. Just as from Pascal's theorem a series of special 
 theorems have been derived, relating to the inscribed pen- 
 tagon, quadrangle, and triangle, so also from Brianchon's 
 theorem can be deduced a series of correlative theorems re- 
 lating to the circumscribed pentagon, quadrilateral, and 
 triangle. 
 
 Suppose e.g. that two of the six tangents a,b , ,c,a / ,b,c / which 
 form the circumscribed hexagon, (Art. 153, left), b and d for 
 example, lie indefinitely near to one another. Since a tangent 
 intersects a tangent indefinitely 
 near to it in its point of contact 
 (Arts. 146, 149), the hexagon will 
 be replaced by the figure made up 
 of the circumscribed pentagon 
 aVca'b together with the point of 
 contact of the side b (Fig. 121). 
 Brianchon's theorem will then become the following : 
 
 If a pentagon is circumscribed to a conic, the two diagonals which 
 connect any two pairs of opposite vertices, and the straight line join- 
 ing the fifth vertex to the point of contact of the opposite side, meet 
 in the same point. 
 
 This theorem expresses a property of projective ranges which has 
 already (Art. 85, left) been noticed. 
 
 For consider the two projective ranges determined by the other 
 tangents on a and b as bases. Three pairs of corresponding points 
 are given, viz. those determined by a', b', and c. Project the first 
 range from the point caf and the second from cb' ; this gives two 
 pencils in perspective of which corresponding, pairs of rays intersect 
 
 L 
 
 Fig. 121. 
 
146 DEDUCTIONS FROM THE THEOREMS [180 
 
 on the straight line r which joins the points ah', ha' . In order 
 then to obtain that point of the second range which corresponds to 
 the point ah of the first, viz. the point of contact of the tangent h, we 
 draw the straight line q which joins the points ca' and ah, and then 
 the straight line p which joins cb f and qr ; then ph is the point 
 required. But this construction agrees exactly with the theorem in 
 question. 
 
 180. By means of the property of the circumscribed pentagon 
 just established the following problems can be solved : 
 
 (1). Given Jive tangents to a conic, to determine the point of contact 
 of any one of them *. 
 
 Particular case. Given four tangents to a parabola, to determine 
 their points of contact, and also the direction of the point at infinity 
 on the curve. 
 
 (2). To construct hy tangents a conic, having given four tangents 
 and the point of contact of one of them. 
 
 Particular cases. 
 
 ►To construct by tangents a hyperbola of which three tangents and 
 r one asymptote are given. 
 
 To construct by tangents a parabola, having given three tangents 
 and the direction of the point at infinity on the curve ; or three 
 tangents and the point of contact of one of them. 
 
 181. The corollaries of Brianchon's theorem which relate to the 
 circumscribed quadrilateral and triangle have already been given 
 (they are the propositions of Arts. 172 and 176) ; they are correlative 
 to the theorems of Arts. 166 and 174, just as those of Arts. 164 and 
 179 are correlative to one another. 
 
 It will be a very useful exercise for the student to solve for himself 
 the problems enunciated in the present chapter : the constructions all 
 depend upon two fundamental ones, correlative to one another, and 
 following immediately from Pascal's and Brianchon's theorems. 
 
 182. The corollaries to the theorems of Pascal and Brianchon show 
 that just as a conic is uniquely determined by five points or five 
 tangents, so al^o it is uniquely determined by four points and the 
 tangent at one of them, by four tangents and the point of contact of 
 one of them, by three points and the tangents at two of them, or 
 by three tangents and the points of contact of two of them. It 
 follows that 
 
 (1). An infinite number of conies can be drawn to pass through 
 three given points and to touch a given straight line at one of these 
 points ; or to pass through two given points and to touch at them 
 two given straight lines ; but no two of these conies can have another 
 point in common. 
 
 * Maclaubin, loc. cit, § 41 . 
 
182] OF PASCAL AND BRIANCHON. 147 
 
 (2). An infinite number of conies can be drawn to touch a given 
 straight line at a given point, and to touch two other given straight 
 lines ; or to touch two given straight lines at two given points ; but 
 no two of these conies can have another tangent in common. 
 
 If then two conies touch a given straight line at the same point 
 (i.e. if the conies touch one another at this point), they cannot have 
 in addition more than two common tangents or two common points ; 
 and if two conies touch two given straight lines at two given points 
 {i.e. if two conies touch one another at two points) they cannot have 
 any other common point or tangent. 
 
 Thus if two conies touch a straight line a at a point A , this point 
 is equivalent to two points of intersection, and the straight line a is 
 equivalent to two common tangents. ^ 
 
 L 2 
 
CHAPTEK XVII. 
 
 DESARGUES THEOREM. 
 
 183. Theorem. Any transversal 
 whatever meets a conic and the op- 
 posite sides of an inscribed quad- 
 rangle in three conjugate pairs of 
 points of an involution. 
 
 This is known as Des argues' 
 theorem *. 
 
 Let QEST (Fig. 122) be a 
 quadrangle inscribed in a conic, 
 
 Correlative Theorem. The 
 tangents from an arbitrary point to 
 a conic and the straight lines which 
 join the same point to the opposite 
 vertices of any circumscribed quad- 
 rilateral form three conjugate pairs 
 of rays of an involution. 
 
 Let qrst (Fig. 123) be a quad- 
 rilateral circumscribed about 8 
 
 Fig. 122. 
 
 and let s be any transversal cut- 
 ting the conic in P and P', and 
 the sides QT, ES, QE, TS of the 
 
 x Fig. 123. 
 
 any point S let 
 ' be drawn to the 
 
 conic ; from 
 
 tangents p, p 
 
 conic, and let the straight lines 
 
 * Desargues, loc. cit., pp. 171, 176. 
 
185] 
 
 DESARGUES THEOREM, 
 
 149 
 
 quadrangle in A , A ', B, B' re- 
 spectively. 
 
 The two pencils which join 
 the points P, R, P', T of the 
 conic to Q and S respectively are 
 projective with one another (Art. 
 149), and the same is therefore 
 true of the groups of points in 
 which these pencils are cut by 
 the transversal. That is, the 
 group of points PBP' A is pro- 
 jective with the group PA'P'B', 
 and therefore (Art. 45) with 
 P'B'PA'\ consequently (Art. 
 123) the three pairs of points 
 
 PP', AA\ BB f 
 are in involution. 
 
 184. This theorem, like that 
 of Pascal (Art. 153, right), enables 
 us to construct by points a conic 
 of which five points P,Q,R,S,T 
 are given. For if (Fig. 122) an 
 arbitrary transversal s be drawn 
 through P, cutting QT, RS, QR, 
 TS in A , A', B, B' respectively ; 
 
 *and if (as in Art. 134) the point 
 P' be found, conjugate to P in 
 the involution determined by the 
 pairs of points A, A' andi?, B' \ 
 then will P' be another point on 
 the conic to be constructed. 
 
 185. The pair of points C, C 
 in which the transversal cuts the 
 diagonals QS and RT of the 
 inscribed quadrangle belong also 
 (Art. 131, left) to the involution 
 determined by the points A , A f 
 and B, B'. 
 
 Moreover, since the points 
 A, A' and B, B f suffice to deter- 
 mine the involution, the points 
 
 a, a',b,b' be drawn which join 
 S to the vertices qt, rs, qr, ts of 
 the quadrilateral respectively. 
 
 The two groups of points in 
 which q and s are cut by the 
 tangents p, f M p' , t are pro- 
 jective with one another (Art. 
 149), and the same is therefore 
 true of the pencils formed by 
 joining these points to S. That 
 is, the group of rays pbp'a is 
 projective with the group^a / jp / 6', 
 and therefore (Art. 45) with 
 2>'b'pa' ; consequently (Art. 123) 
 the three pairs of rays 
 pp', aa', bb' 
 are in involution. 
 
 This theorem, like that of 
 Brianchon (Art. 153, left), en- 
 ables us to construct by tangents 
 a conic of which five tangents 
 p, q, r, s, t are given. For if 
 (Fig. 123) an arbitrary point S 
 be taken on p, and this point be 
 joined to the points qt, rs, qr, ts 
 respectively by the rays a,a',b,b'\ 
 and if (Art. 134) the ray p' be 
 constructed, conjugate to p in the 
 involution determined by the pairs 
 of rays a, a' and b,V ' ; then will 
 p f be another tangent to the conic 
 to be constructed. 
 
 The pair of rays c, c' which 
 connect S with the points of 
 intersection qs and rt of the 
 opposite sides of the circum- 
 scribed quadrilateral belong also 
 (Art. 131, right) to the involu- 
 tion determined by the rays a, a' 
 and b , V. 
 
 Moreover, since the rays a, a' 
 and b, b' suffice to determine the 
 involution, the rays p, p' are a 
 
150 
 
 DESARGUES THEOREM. 
 
 [186 
 
 P, P f are a conjugate pair of 
 this involution for every conic, 
 whatever be its nature, which 
 circumscribes the quadrangle 
 QEST. 
 
 Thus: 
 
 Any transversal meets the conies 
 circumscribed about a given quad- 
 rangle in pairs of points forming 
 an involution. 
 
 If the involution has double 
 points, each of these is equivalent 
 to two points of intersection P 
 and P f lying indefinitely near to 
 one another ; and will therefore 
 be the point of contact of the 
 transversal with some conic cir- 
 cumscribing the quadrangle. 
 
 There are therefore either two 
 conies which pass through four 
 given points Q, P, S, T and 
 touch a given straight line s 
 (not passing through any of the 
 given points), or there is no 
 conic which satisfies these con- 
 ditions. 
 
 186. If, from among the six 
 points AA', BB', PP' of an 
 involution, five are given, the 
 sixth is determined (Art. 1 34). If 
 ,( ! then in Fig. 122 it is supposed 
 
 that the conic is given, and that 
 the quadrangle varies in such a 
 way that the points A, A', B 
 remain fixed, then also the 
 point B' will remain invariable ; 
 consequently : 
 
 If a variable quadrangle move 
 in such a way as to remain 
 always inscribed in a given conic, 
 while three of its sides turn each 
 round one of three fixed collinear 
 points, then the fourth side will 
 turn round a fourth fixed point, 
 
 conjugate pair of this involution 
 for every conic, whatever be its 
 nature, which is inscribed in the 
 quadrilateral qrst. 
 
 Thus : 
 
 The pairs of tangents drawn 
 from any point to the conies 
 inscribed in a given quadrilateral 
 form an involution. 
 
 If the involution has double 
 rays, each of these is equivalent 
 to two tangents p and p' lying 
 indefinitely near to one another ; 
 and will therefore be the tangent 
 at S to some conic inscribed in 
 the quadrilateral. 
 
 There are therefore either two 
 conies which touch four given 
 straight lines q, r, s, t and pass 
 through a given point S (not 
 lying on any of the given lines), 
 or there is no conic which satis- 
 fies these conditions. 
 
 If, from among the six rays 
 aa', bb f , pp r of an involution, 
 five are given, the sixth is deter- 
 mined (Art. 134). If then in 
 Fig. 123 it is supposed that the 
 conic is given, and that the 
 quadrilateral varies in such a 
 way that the rays a, a', b remain 
 fixed, then also the ray b f will 
 remain invariable ; consequently: 
 
 If a variable quadrilateral move 
 in such a way as to remain always 
 circumscribed to a given conic, 
 while three of its vertices slide 
 each along one of three fixed con- 
 current straight lines, then the 
 fourth vertex will slide along a 
 
187] 
 
 DESARGUES THEOREM. 
 
 151 
 
 Fig. 124. 
 
 collinear with the three given fourth fixed straight line, concur- 
 ones. rent with the three given ones. 
 
 187. The theorem of the preceding Art. (left) may be ex- 
 tended to the case of any inscribed polygon having an even 
 number of sides. Suppose such a polygon to have 2 n sides, 
 and to move in such a way that in—i of these pass respec- 
 tively through as many fixed points all lying on a straight 
 line s (Fig. 124). Draw the 
 diagonals connecting the 
 first of its vertices with the 
 4 «S 6 th , 8 th , ... 2 (n - i) th 
 vertex, thus dividing the 
 polygon into n — 1 simple 
 quadrangles. In the first 
 of these quadrangles the first 
 three sides (which are the 
 first three sides of the polygon) pass respectively through 
 three fixed points on s ; therefore also the fourth side (which is 
 the first diagonal of the polygon) will pass through a fixed point 
 on s. In the second quadrangle the first three sides (the first 
 diagonal and the fourth and fifth side of the polygon) pass re- 
 spectively through three fixed points on * ; therefore the 
 fourth side (the second diagonal of the polygon) will pass 
 through a fixed point on s. Continuing in the same manner, 
 we arrive at the last quadrangle and find that the fourth side 
 of this (i.e. the 2n th side of the polygon) passes through a 
 fixed point on s. We may therefore enunciate the general 
 theorem : 
 
 If a variable polygon of an even number of sides move in such a 
 way as to remain aUvays inscribed in a given conic, while all its sides 
 but one pass respectively through as many fixed points lying on a 
 straight line, then the last side also will pass through a fixed point 
 collinear with the others *, 
 
 If tangents can be drawn to the conic from the fixed point 
 
 round which the last side turns, and if each of these tangents 
 
 is considered as a position of the last side, the two vertices 
 
 • which lie on this side will coincide and the polygon will have 
 
 only 2 n — 1 vertices. The point of contact of each of the two 
 
 * Poncelet, loc. cit, Art. 5T3. 
 
152 
 
 DESARGUES THEOREM. 
 
 [188 
 
 remain 
 
 tangents will therefore be one position of one of the vertices 
 of a polygon of zn—i sides inscribed in the conic so that its 
 sides pass respectively through the zn — i given collinear 
 points. 
 
 188. The solution of the correlative theorem is left as an 
 exercise to the student ; the enunciation is as follows : 
 
 If a variable polygon of an even number (zn) of sides moves so as to 
 to a given conic, while all its vertices 
 but one slide along as many fixed 
 straight lines radiating from a centre, 
 then the last vertex also loill slide 
 a fixed straight line passing 
 the same centre (Fig. 
 
 125). 
 
 If the straight line on which 
 this last vertex slides cut the 
 conic in two points, and if the 
 tangents at these be drawn, each 
 of them will be one position of 
 a side of a polygon of zn — i 
 sides circumscribed about the 
 conic so that its vertices lie each 
 on one of the z n — i given con- 
 
 Fig. 125. 
 
 current straight lines. 
 
 189. If in Fig. 122 it be sup- 
 posed that the points S and T lie 
 indefinitely near to one another on 
 the conic, or in other words that 
 ST is the tangent at S, then the 
 quadrangle QRST reduces to the 
 inscribed triangle QRS and the 
 tangent at S (Fig. 126), so that 
 Desargues' theorem becomes the 
 following : 
 
 If a triangle QRS is inscribed 
 in a conic, and if a transversal s 
 meet two of its sides in A and A', 
 the third side and the tangent at 
 the opposite vertex in B and B', 
 and the conic itself in P and P', 
 
 If in Fig. 123 the tangents 
 s and t be supposed to lie indefi- 
 nitely near to one another, so that 
 st becomes the point of contact of 
 the tangent s, then the quadri- 
 lateral qrst reduces to the circum- 
 scribed triangle qrs and the point 
 of contact of s (Fig. 127), so that 
 the theorem correlative to that of 
 Desargues becomes the following : 
 
 If a triangle qrs is circum- 
 scribed about a conic, and if from 
 any point S there be drawn the 
 straight lines a, a' to two of its 
 vertices, the straight lines b,b' to 
 the third vertex and tfie point of 
 
191] 
 
 DESARGUES THEOREM. 
 
 153 
 
 these three pairs of points are in 
 
 involution. 
 
 190. This theorem gives a 
 solution of the problem : Given Jive 
 
 Fig. 126. 
 
 points P,P',Q,B ,S on a conic, 
 to dravj the tangent at any one of 
 them S. 
 
 Forif^ 5 ^ / 5J B(Fig.i26) are the 
 points in which the straight line 
 PP' cuts the straight lines QS, SB, 
 BQ respectively, we construct (as 
 in Art. 1 34) the point B ' conjugate 
 to B in the involution determined 
 by the two pairs of points A , A' 
 and P, P'; then B'S will be the 
 required tangent. 
 |0 -191. If in Fig. 126 it be now 
 supposed in addition that the 
 points Q and B also lie inde- 
 finitely near to one another on 
 the conic, i. e. that QB is the 
 tangent at Q, then the inscribed 
 quadrangle QBST is replaced by 
 the two tangents at Q and S and 
 their chord of contact QS counted 
 twice (Fig. 128). 
 
 Since, the straight lines QT, 
 BS now coincide, A and A' will 
 
 contact of the opposite side, and 
 the tangents p, p f to the conic, 
 then these three jyairs of rays are 
 in involution. 
 
 This theorem gives a solution of 
 the problem : Given five tangents 
 
 Fig. 127. 
 p, p',q, r, 8 to a conic, to find the 
 point of contact of any one of 
 them s. 
 
 For if a,a',b (Fig. 1 2 7) are the 
 rays joining the point pp' to the 
 points qs,sr,rq respectively, we 
 construct (as in Art. 134) the ray 
 b' conjugate to b in the involu-. 
 tion determined by the two pairs 
 of rays a, a f and p ,//; then b's 
 will be the required point of con- 
 tact. 
 
 If in Fig. 127 it be now sup- 
 posed in addition that the tan- 
 gents q and r lie indefinitely near 
 to one another, i.e. that qr is the 
 point of contact of the tangent q, 
 then the circumscribed quadri- 
 lateral qrst is replaced by the 
 points of contact of the tangents 
 q and s and the point of intersec- 
 tion qs of these tangents counted 
 twice (Fig. 129). 
 
 Since the points qt, rs now co- 
 incide in a single point qs, the 
 
154 
 
 DESABGUES THEOREM. 
 
 [191 
 
 also coincide in one point, which 
 is consequently one of the double 
 points of the involution deter- 
 mined by the pairs of conjugate 
 
 rays a and a' will also coincide 
 in a single ray a, which is conse- 
 quently one of the double rays of 
 the involution determined by the 
 
 Fig. 128. 
 
 points P, P' and B, B'. 
 case, then, Desargues' 
 becomes the following : 
 
 In this 
 theorem 
 
 If a transversal cut two tan- 
 gents to a conic in B and B\ tlieir 
 chord of contact in A, and the 
 conic itself in P and P\ then the 
 point A is a double point of the 
 involution determined by the pairs 
 of points P, P' and B, B'. 
 
 Or, differently stated : 
 If a variable conic 2?ass through 
 two given points P and P' and 
 touch two given straight lines, the 
 chord which joins the points of 
 contact of these two straight lines 
 will always pass through a fixed 
 point on PP'. 
 
 If the tangents QU, SU vary 
 at the same time with the conic, 
 while the points P,P',B,B' re- 
 main fixed, the chord of contact 
 
 Fig. 129. 
 
 pairs of conjugate rays p, p f and 
 b , b' . The theorem correlative to 
 that of Desargues then becomes 
 the following : 
 
 If a given point 8 be joined to 
 two points on a conic by the 
 straight lines b, b', and to the 
 point of intersection of the tan- 
 gents at these points by the straight 
 line a; and if from the same 
 point S tliere be drawn the two tan- 
 gents p, p' to the conic ; then a is 
 a double ray of the involution de- 
 termined by the pairs of raysp,p' 
 and b, V ' . 
 
 Or, differently stated : 
 
 If a variable conic touch two 
 given straight lines p and p' and 
 pass through two given points, the 
 tangents at these two points will 
 always intersect on a straight line 
 passing through pp/. . 
 
 If the points of contact of q and 
 s vary at the same time with the 
 conic, while the straight lines 
 p,p', b, b f remain fixed, the point 
 
193] 
 
 DESARGUES THEOREM. 
 
 155 
 
 QS must still always pass through 
 one or other of the double points 
 of the involution determined by 
 the pairs of points P, P ' and B,B'. 
 If then four collinear points P, P ', 
 B, B' are given and any conic is 
 drawn through P and P\ and 
 then the pairs of tangents from 
 B and B' to this conic ; then if 
 each tangent from B is taken to- 
 gether with each tangent from 
 B\ four chords of contact will be 
 obtained, which intersect one 
 another two and two in the double 
 points of the involution determined 
 byP, />' and £ ,"£'*, 
 
 192. From the theorem of the 
 last Article (left) is derived a 
 solution of the problem : Given 
 four points P,P f >Q,S on a conic 
 and the tangent at one of them Q, 
 to draw the tangent at any other 
 of the given points S (Fig. 128). 
 
 For if A , B are the points in 
 which PP' cuts QS and the given 
 tangent respectively, and we con- 
 struct the point B' conjugate to 
 B in the involution determined 
 by the pair of points P, P' and 
 the double point A ; then the 
 straight line SB' will be the tan- 
 gent required. 
 
 of intersection qs must still always 
 lie on one or other of the double 
 rays of the involution determined 
 by the pairs of rays p,y/ and 6, b'. 
 If then four concurrent straight 
 lines p , p\ b , b f are given and any 
 conic is drawn touching p and 7/, 
 and then the two pairs of tan- 
 gents to this conic at the points 
 where it is cut by b and &'; 
 then if the tangents at the two 
 points on b are combined with 
 the tangents at the two points on 
 &', each with each, four points of 
 intersection will be obtained, 
 which lie two and two on the 
 double rays of the involution de- 
 termined by p, p f and b, b' '. 
 
 From the theorem of the last 
 Article (right) is derived a solu- 
 tion of the problem : Given four 
 tangents %), p' , q, s to a conic and 
 the point of contact of one of them 
 q, to determine the point of contact 
 of any other of the given tangents 
 s (Fig. 129). 
 
 For if a , b are the rays which 
 connect pp r with qs and with the 
 given point of contact respec- 
 tively, and we construct the ray 
 b' conjugate to b in the involu- 
 tion determined by the pair of 
 rays p , p' and the double ray a ; 
 then sb f will be the required 
 point of contact. 
 
 193. Consider again the theorem of Art. 191 ; and suppose that 
 the conic is a hyperbola, and that its asymptotes are the tangents 
 given (Fig. 1 30). The chord of contact QS lies in this case entirely 
 at infinity; so that the involution (PP\ BB', ...) has one double 
 point at infinity, and therefore (Arts. 59, 125) the other double point 
 
 * Brianchon, loc. eft., pp. 20, 21, 
 
156 
 
 DESARGUES THEOREM. 
 
 [194 
 
 is the common point of bisection of the segments PP[, BB', ... We 
 conclude that : 
 
 If a hyperbola and its asymptotes be cut by a transversal, the seg- 
 ments intercepted by the curve and by the asymptotes respectively have 
 the same middle point. 
 
 Fig. 130. 
 From this it follows that 
 
 PB = B'P' and PB' = BP' *, 
 
 which gives a rule for the construction of a hyperbola when the two 
 asymptotes and a point on the curve are given t. 
 
 194. Consider once more the Consider once more the theorem 
 
 theorem of Art. 191 (left), and of Art. 191 (right), and suppose 
 
 suppose now that the points P 
 and P' are indefinitely near to one 
 another, i. e. let the transversal 
 be a tangent to the conic (Fig. 
 131). Its point of contact P will 
 
 now that the tangents p and p' lie 
 indefinitely near to one another, 
 i.e. let the point S lie on the 
 conic itself (Fig. 132). The tan- 
 gent to the conic at S will be the 
 
 a 
 
 Fig. 131. 
 
 be the second double point of the 
 involution determined by the pair 
 of points B,B f and the double 
 points ; consequently (Art. 125) 
 P and A are harmonic conjugates 
 
 Fig. 132. 
 
 second double ray of the involu- 
 tion determined by the pair of 
 rays b , V and the double ray a ; 
 consequently (Art. 125) p and a 
 are harmonic conjugates with re- 
 
 * Apollonius, loe. cit., ii. 8, 16. 
 t Ibid., ii. 4. 
 
196] 
 
 DES ARGUES THEOREM. 
 
 157 
 
 with regard to B and B' ; and 
 we conclude that : 
 
 In a triangle UBB' circum- 
 scribed to a conic, any side BB f 
 is divided harmonically by its 
 point of contact P and the point 
 where it meets the chord QS joining 
 the points of contact of the other 
 two sides. 
 
 195. From A a second tangent 
 can be drawn to the conic ; let its 
 point of contact be 0. Since the 
 four points P,A,B,B', which have 
 been shown to be harmonic, are 
 respectively the point of contact 
 of the tangent AB, and the three 
 points where this tangent cuts 
 three other tangents OA, QB, SB' 
 respectively, it follows that the 
 tangents AB, OA, QB, SB' will 
 be cut by every other tangent in 
 four harmonic points (Art. 149); 
 i.e. they are four harmonic tan- 
 gents (Art. 151). And since the 
 chord of contact QS of the con- 
 jugate tangents QB, SB' passes 
 through A the point of intersec- 
 tion of the tangents at P and 0, 
 we have the theorem : 
 
 If the chord of contact of one 
 pair of tangents to a conic pass 
 through the point of intersection of 
 another pair of tangents, then each 
 pair is harmonically conjugate 
 with regard to the other. 
 And conversely : 
 
 If four tangents to a conic are 
 harmonic, the chord of contact of 
 each pair of conjugate tangents 
 passes through the point of inter- 
 section of the other pair. 
 
 gard to b and b'; and we con- 
 clude that : 
 
 In a triangle ubb' inscribed in 
 a conic, any two sides b and b' 
 are harmonic conjugates with re- 
 gard to the tangent p at the vertex 
 in which they meet and the straight 
 line joining this vertex to the point 
 of intersection of the tangents q 
 and s at the other two vertices. 
 
 The straight line a cuts the 
 conic in a second point ; let the 
 tangent at this be o. Since the 
 four rays p ,a,b ,b' , which have 
 been shown to be harmonic, are 
 respectively the tangent at S, and 
 the straight lines which join S to 
 three other points on the conic 
 (the points of contact of o , q, and 
 s) it follows that the straight 
 lines connecting these four points 
 with any other point on the conic 
 will form a harmonic pencil (Art. 
 149); i.e. the four points are 
 harmonic (Art. 151). And since 
 the point of intersection of the 
 tangents q and s lies on the chord 
 of contact of the tangents p and o, 
 we have the theorem : 
 
 If the point of intersection of 
 the tangents at one pair of points 
 on a conic lie on the chord join- 
 ing another such pair of points, 
 then each pair is harmonically 
 conjugate with regard to the other. 
 And conversely : 
 
 If four points on a conic are 
 harmonic, the point of intersection 
 of the tangents at each pair of con- 
 jugate points lies on the chord 
 joining the other pair. 
 
 196. These two correlative propositions can be combined into one 
 
158 DESAKGUES' THEOREM. [197 
 
 by virtue of the property already established (Arts. 148, 149) that 
 the tangents at four harmonic points on a conic are themselves har- 
 monic, and conversely. We may then enunciate as follows : 
 
 If a pair of tangents to a conic meet in a point lying on the chord 
 of contact of another ptair^ then also the second pair will meet in a 
 point lying on the chord of contact of the first ; and the four tangents 
 (and likewise their points of contact) will form a harmonic system*. 
 
 Thus in Fig. 131 QS passes through A , the point of intersection of 
 PA and OA, and similarly OP passes through £7, the point of inter- 
 section of QB and SB'; and the pencil U (QSPA) is harmonic, and 
 likewise the pencil A (OPQ U). 
 
 In Fig. 132 the point qs lies on a, the chord of contact of and j), 
 and similarly the point op lies on the straight line u which joins the 
 points of contact of q and $ ; and the range u (qsap) is harmonic, and 
 the range a (o2)qu) also. 
 
 197. Example. Suppose the conic to be a hyperbola (Fig. 133). 
 Its asymptotes are a pair of tangents whose 
 chord of contact QS is the straight line at 
 infinity ; consequently the chord joining the 
 points of contact of a pair of parallel tangents 
 will pass through the point of intersection U 
 of the asymptotes ; and conversely, if through 
 U a transversal be drawn, the tangents at the 
 points P and 0, where it cuts the curve, will 
 
 Fig. 133. De parallel. The point U will lie midway 
 
 between P and 0, since in general UVPO 
 (Fig. 131) is a harmonic range, and in this case V lies at in- 
 finity. 
 
 Any tangent to the curve cuts the asymptotes in two points B and 
 B' which are harmonically conjugate with regard to the point of con- 
 tact P and the point where the tangent meets the chord of contact of 
 the asymptotes ; but this last lies at infinity ; therefore P is the 
 middle point of BB\ Thus 
 
 The part of a tangent to a hyperbola which is intercepted between 
 the asymptotes is bisected at its point of contact t. 
 
 This proposition is a particular case of that of Art. 193. 
 
 198. Theoeem J. If a quadrangle is inscribed in a conic, the 
 rectangle contained by the distances of any i^oint on the curve from 
 
 * De la Hire, loc. cit., book i. prop. 30. Steiner, loc. cit., p. 1 59, § 43 ; Collected 
 Works, vol. i. p. 346. 
 
 + Apollonius, loc. cit., ii. 319. 
 
 t To this Chasles has given the name of Pappus' theorem, since it corresponds 
 to the celebrated 'problema ad quatuor linens' of this ancient geometer. Cf. 
 Apercu historique, pp. 37, 338. 
 
199] DESARGUES' THEOREM. 159 
 
 one pair of opposite sides is to the rectangle contained by its distances 
 from the other pair in a constant ratio. 
 
 In Fig. 122, the pairs of points P and P', A and A', B and B' 
 being, by Desargues' theorem, in involution, the anharmonic ratios 
 (PP'AB) and (P'PA'B') are equal to one another, or 
 PA PB_ _ P\A/ P'B' 
 P'A : P'B ~ PA '' PB' 
 PB' PA' 
 
 P'B'' P'A' 
 
 But PA : P'A is equal to the ratio of the distances (measured in 
 any the same direction) of the points P and P' from the straight line 
 QT, and the other ratios in the foregoing equation may be interpreted 
 similarly ; we have therefore 
 
 (A±.{B)__(&).M 
 
 (A)'-{B)' {B'f(A')" 
 
 (B).(B')-(By.(B'Y' 
 where (A), (A'), (B), (B') denote the distances of the point P from 
 the sides QT, R/S, QR, ST respectively of the inscribed quadrangle 
 QRST, and (A)', (A')', (B)', (B')' denote similarly the distances of 
 the point P' from these sides respectively. (These distances may be 
 measured either perpendicularly or obliquely, so long as they are all 
 measured parallel to one another.) The ratio 
 
 (A) (A') 
 
 (B)(B') 
 is therefore constant for all points P on the conic ; which proves the 
 theorem. 
 
 199. Theoeem. If a quadrilateral is circumscribed about a conic, 
 the rectangle contained by the distances of one pair of opposite vertices 
 from any tangent is to the rectangle contained by the distances of the 
 other pair from the same tangent in a constant ratio *. 
 
 In Fig. 123 let the vertices qr, qt, st, sr of the circumscribed 
 quadrilateral qrst be denoted by R, T, T x , R x respectively ; let the 
 points where the tangents p, p' meet the side q be called P, P' 
 respectively t, and let the points where these same tangents meet the 
 side s be called P x , P x ' respectively. Since by the theorem corre- 
 lative to that of Desargues, the pairs of rays p and p', a and a', 
 b and b', are in involution, the anharmonic ratios (bapp') and 
 (b'a'p'p) are equal to one another. Hence by theorem (2) of 
 Art. 149, 
 
 * Chasles, Sections coniques, Art. 26. 
 f P' is not shown in the figure. 
 
160 DESARGUES' THEOREM. [199 
 
 (RTPP') = (T&PfPJ 
 
 = (2?^^/) by Art. 45; 
 
 #P &£* _ R^P X R X P( 
 
 •*• Tp • Tp , - T ^ ' T ^, » 
 
 RP.T.P. RP'.T.P/ 
 Whence TP^P^ TP'.RSf 
 
 But RP : TP is equal to the ratio of the distances (measured in 
 any the same direction) of the points R and T from the straight line 
 p ; so also T 1 P 1 : R X P X is the ratio of the distances of the points T x 
 and R x from the same straight line p. The foregoing equation 
 therefore expresses that the ratio 
 
 RP.T 1 P 1 
 
 TP.R.P, 
 
 is constant for every tangent p to the conic ; which proves the 
 theorem. 
 
CHAPTEK XVIII. 
 
 SELF-CORRESPONDING ELEMENTS AND DOUBLE ELEMENTS. 
 
 200. Consider two projective flat pencils, concentric or non- 
 concentric. Through their common centre or through their 
 two centres and 0' draw a conic or a circle, and let this 
 cut the rays of the first pencil in A,B,C, ... and those of the 
 second in A', B\ C", ... . Project these two series of points 
 from two new points 1 , 0/ (or from the same point) lying 
 on the conic ; the two projecting pencils Y [ABC ... ) and 
 1 / (A / B / C / ... ) are by Art. 149 projective with the two given 
 pencils (ABC ... ) and Cf (A' B' C ... ) respectively ; and are 
 therefore projective with one another. 
 
 The two series of points ABC ... and A'B'C ... are said to form 
 two projective ranges on the conic *. 
 
 I. Now project these two ranges (Fig. 134) from two of then- 
 corresponding points, say from A' and A. The projecting 
 pencils 
 
 A'(A,£,C,...) and A(A\B\C\...) 
 
 will be projective with one another ; and since they have the 
 
 self-corresponding ray AA ', they are A 
 
 in perspective. Corresponding pairs n/f~/^^\ 
 
 of rays will therefore (Art. 80) inter- ^n^V„: A^>\ 
 
 sect on a fixed straight line, so that ^j ^^n T^., 4 \ r 
 
 AB' and A'JB, AC and A'C, AD' and \AT p?T?k^ 
 
 A' I) . . . , will meet on one straight line s. ^I<C_j^ s 
 
 If any point betaken on s, the straight Fi 
 
 lines joining it to A and A' will cut 
 
 the conic again in another pair of corresponding points of 
 
 the ranges ABCD... and A'B'C'D' ... . 
 
 * Bellavitis, Saggio di Geometria derivata (Nuovi Saggi dell' Accademia di 
 Padova, vol. iv. 1838, p. 270, note). 
 
 M 
 
162 
 
 SELF-CORRESPONDING ELEMENTS 
 
 [200 
 
 If instead of A' and A any other pair of corresponding 
 points had been taken as centres of projection, say B' and B, 
 the same straight line s would have been arrived at. For 
 since AB'CA'BC is a hexagon inscribed in a conic, it follows 
 by Pascal's theorem that the point of intersection of B f C and 
 BC must lie on the straight line which joins the point of 
 intersection of A'B and AB' to that of A'C and AC (Art 153, 
 right). 
 
 II. Any point M in which the conic and the straight line s 
 intersect is a self-corresponding point of the two ranges 
 ABC ... and A'B'C ... . For if M, M' be corresponding points 
 
 Fig. 136. 
 
 of the two ranges, it has been seen that A'M; AM' must inter- 
 sect on s ; if then M lie on s, 31' must coincide with M; i. e. a 
 pair of corresponding points of the two ranges are united 
 at M. 
 
 The two ranges will therefore have two self -corresponding points, 
 or only one, or none at all, according as 
 the straight line s cuts the conic in two 
 points (Fig. 135), touches it (Fig. 136), or 
 does not cut it (Fig. 137). 
 
 III. From what precedes it is clear 
 that two projective ranges of points on a 
 conic are determined by three pairs of 
 corresponding points A and A' , B and B' , 
 C and C For in order to find other 
 pairs of corresponding points, and the 
 self-corresponding points (when such 
 exist), we have only to construct the 
 straight line s which passes through the points of intersection 
 of the three pairs of opposite sides of the hexagon AB'CA'BC' 
 (Figs. 98, 134, 135). The self-corresponding points will then 
 
 Fig. 137. 
 
201] AND DOUBLE ELEMENTS. 163 
 
 be the points where s cuts the conic, and any number of pairs 
 of corresponding points can be constructed by help of the 
 property that any pair D and B' are such that the lines A'B 
 and AB' (or B'B and BB', or CD and CD') intersect on s * 
 
 201. Instead of projective ranges of points on a conic we may- 
 consider projective series of tangents to the same. Let o, o' be two 
 projective ranges of points (either collinear or lying on different straight 
 lines as bases). Describe a conic to touch o and o', and draw to this 
 conic, from each pair of corresponding points A and A', B and B\ 
 C and C', ... the tangents a and </, b and b', c and c', ... . If now 
 these two series of tangents are cut by two other tangents o 1 and o/, 
 two new ranges of points will be obtained, which are projective with 
 the given ranges respectively (Art. 149), and are therefore projective 
 with one another. 
 
 Tivo series of tangents to a conic are said to be projective with one 
 anotJier when they are cut by any other tangent to the curve in two 
 projective ranges. 
 
 I. Suppose the first series of tangents to be cut by the tangent a', 
 and the second by the tangent a. The two projective ranges so 
 formed are in perspective, since they have the self-corresponding 
 point aa' ; the straight lines which join the pairs of corresponding 
 points a'b and ab', a'c and ac', . . . will therefore pass through one 
 point S. This point does not change if another pair of tangents 
 b f and b are taken as transversals ; for by Brianchon's theorem the 
 straight lines which join the three pairs of opposite vertices a'b and 
 ab' } a'c and ac', b'c and be' of the circumscribed hexagon ab'c a'bc' 
 must meet in a point (Art. 153, left). 
 
 II. If the point S is such that tangents can be drawn from it to 
 the conic, each of them will be a self- corresponding line of the two 
 projective series of tangents abc ... and a'b'c' .... 
 
 [The proof of this is analogous to that of the corresponding property 
 of two projective ranges of points on a conic (Art. 200, II). ] 
 
 III. Two projective series of tangents to a conic are determined 
 by three pairs of corresponding lines a and a', b and b', c and c'. 
 For in order to find other pairs of corresponding lines, and the self- 
 corresponding lines (when such exist), we have only to construct the 
 point of intersection S of the diagonals which join two and two the 
 opposite vertices of the circumscribed hexagon ab'c a'bc'. The self- 
 corresponding lines will be the tangents from S to the conic, and any 
 pair of corresponding lines d and d' may be constructed by means of 
 the property that the points a'd and ad' (or b'd and bd", or c'd and 
 cd', . . .) are collinear with S. 
 
 * Steiner, loc. cit., p. 174, § 46, Hi.; Collected Works, vol. i. p. 357. 
 M % 
 
164 SELF-CORRESPONDING ELEMENTS [201 
 
 IV. A range of points A , B, C, ... oh a conic and a series of tangents 
 a, b, c, ... to the same are said to be projective with one another, 
 when the pencil formed by joining A, B, C, ... to any point on the 
 conic is projective with the range determined by a, b, c, ... on any 
 tangent to the conic. 
 
 A range of points A, B, C, ... on a conic, or a series of tangents 
 a, b, c, ... to the same, is said to be projective with a range of points 
 on a straight line, or a pencil (flat or axial), when this last -mentioned 
 range or pencil is projective with the pencil formed by joining 
 ABC ... to any point on the conic or with the range determined by 
 a, b, c, ... on any tangent to the conic. 
 
 V. These definitions premised, we may now include under the 
 title of one-dimensional geometric form not only the range of 
 collinear points, the flat pencil, and the axial pencil, but also 
 the range of points on a conic and the series of tangents to a 
 conic * ; and with regard to these we may enunciate the general 
 theorem : Two one-dimensional forms which are each projective 
 with a third (also of one dimension) are projective with one another 
 (cf. Art. 41). 
 
 VI. From these definitions it follows also that theorem (3) of Art. 
 149 may be enunciated in the following manner : 
 
 Any series of tangents to a conic is projective with the range formed 
 by their points of contact. 
 
 VII. Let .4 , B, G, ... and A', B\ C, ... be two projective ranges of 
 points on a conic, and let a, 6, c, ... and a', b', c', ... be the tangents 
 at these points. The series of tangents a, b, c, ... and d ', b', c', ... 
 are projective with the series of points of contact A, B, C, ... and 
 A', B f , C", ... respectively, and are therefore projective with one 
 another. Let s be the straight line on which the pairs of straight lines 
 such as AB' and A % AC' and A'C y BC and B'C ... intersect ; and 
 let S be the point in which meet the straight lines joining pairs of 
 points such as ah' and a'b , ac' and a'c, be' and b'c, ... . If s cuts the 
 conic in two points M and N, these must be the self- corresponding 
 points of the ranges ABC ... and A'B'C ... ; the tangents m and n 
 at M and N respectively must therefore be the self-corresponding 
 lines of the projective series dbc ... and a'b'c' ...; consequently the 
 straight lines m and n will meet in S. 
 
 VIII. From the foregoing it follows that for the consideration of a 
 
 * The introduction of these new one- dimensional forms enables us now to add 
 to the operations previously made use of (section by a transversal straight line 
 and projection by straight lines radiating from a point) two others, viz. section of 
 a flat pencil by a conic passing through the centre of the pencil, and projection of 
 a range of collinear points by means of the tangents to a conic which touches the 
 base of the range. 
 
203] AND DOUBLE ELEMENTS. 165 
 
 series of tangents can always be substituted that of their points of 
 contact, and vice versa. 
 
 202. Instead of considering any two projective pencils as 
 in Art. 200, take an involution of straight lines radiating 
 from a point 0. Suppose these to be cut by a conic passing 
 through in the pairs of points A and A', B and B', C and 
 C, ... , and let these points be joined to any other point 1 on 
 the conic. Since by hypothesis (Arts. 122, 123) the pencils 
 0(AA'BC ...) and (A'AB'C ... ) are projective with one 
 another, the pencils X (AA'BC ... ) and O x (A'AB'C ... ) are 
 so too (Art. 149); and therefore the rays issuing from O x 
 form an involution also. In this case we say that the two pro- 
 jective ranges of points ABC ... and A'B'C ... on the conic form 
 an involution ; or that there is on the conic an involution formed by 
 the pairs of conjugate points AA', BB', CC, ... *. 
 
 I. Similarly, if there is given an involution of points on a straight 
 line o and if from the pairs of conjugate points there be drawn 
 tangents a and a', b and b', c and c', ... to a conic touching o, these 
 will be cut by any other tangent to the conic in an involution of 
 points ; in this case we say that aa r , bb' } cc', ... form an involution of 
 tangents to the conic (cf. Art. 201). 
 
 II. If several pairs of tangents aa', bb', cc', ... to a conic form an 
 involution, their points of contact A A', BB', CC, ... form an involu- 
 tion also, and conversely (Art. 201, VI). 
 
 203. Of the six points A, B' , C, A', B , C'ona conic con- 
 sidered in Art. 200, let C lie indefinitely near to A, and C in- 
 definitely near to A'. The projective ranges (ABC...) or 
 (ABA' ... ) and (A'B'C..) or (A' B f A...) will then form an 
 involution (A A ', BB', ...) and the inscribed hexagon is replaced 
 by the figure made up of the inscribed quadrangle AB'A'B&nd 
 the tangents at the opposite vertices A and ^'(Figs. 115, 138). 
 We conclude that 
 
 An involution of points on a conic is determined by two pairs 
 AA\ BB'. 
 
 I. In order to find other pairs of conjugate points, it is only 
 necessary to construct the straight line s which joins the point 
 of intersection of AB' and A'Bto that of AB and A'B'; i.e. to 
 
 * Staudt, Beitrciye zur Geometric der Lage (Niirnberg, 1856-57-60), Arts. 70 
 •qq. 
 
166 
 
 SELF-CORRESPONDING ELEMENTS 
 
 [203 
 
 draw the straight line joining the points of intersection of the 
 pairs of opposite sides of the inscribed quadrangle AB'A'B. 
 
 The points where s cuts the conic 
 are the double points. Pairs of 
 conjugate points will be constructed 
 by remembering that any pair € 
 and C are such that the straight 
 lines^C and A' C (or AC and^'(7, 
 or BC and B'C, or B'C and BC) 
 intersect on *. 
 
 II. The tangents at a pair of 
 conjugate points, such as A and A', 
 B and B', ... likewise intersect on 
 the straight line s (Art. 166). 
 
 III. Since the pairs of sides BC 
 and B'C, CA and C'A\ AB and 
 A'B' of the triangles ABC, A' B'C 
 intersect in three points lying on 
 
 a straight line *, the triangles are homological (Art. 1 7) *, and 
 the straight lines AA', BB', CC will meet in one point 8. But 
 A A ' and BB' suffice to determine this point; accordingly : 
 
 Any pair of conjugate points of the involution are collinear with a 
 fixed point 8 -, or 
 
 Every straight line drawn through S to cut the conic determines 
 on it a pair of conjugate points of the involution. 
 
 IV. It has been seen that if s cuts the conic in two points 
 M and JV, these are the double points of the involution. The 
 tangents at M and N will therefore meet in 8. 
 
 Fig. 138. 
 
 V. Conversely, the pairs of points in which a conic is cut by 
 the rays of a pencil whose centre 8 does not lie on the curve form 
 an involution. 
 
 For if A and A\ i? and B' are the points of intersection 
 of the curve with two of the rays, these two pairs A A' 
 and BB' determine an involution such that the straight 
 line joining any pair of corresponding points always passes 
 through a fixed point, viz. 8. If the involution has double 
 points, these are the intersections of the conic with the 
 
 * The triangles A'BC and AB'C', AB'C and A'BC, ABC and A' B'C are 
 likewise homological in pairs. 
 
203] 
 
 AND DOUBLE ELEMENTS. 
 
 167 
 
 straight line s which joins the point of intersection of AB and 
 A'B' to that of AB f and A'B. 
 
 VI. If from different points of a straight line s pairs of tangents 
 a and a', b and b', c and c', ... be drawn to the conic, these form an 
 involution. For if A and A', B and B', C and C, ... are the points of 
 contact of the tangents a and a', b and b', c and c', ... respectively, and 
 S is the point of intersection of the chords A A' and BB', then in the 
 involution determined by the pairs A , A f and B , J5' the straight line 
 joining any other pair of conjugate points will pass through S. The 
 point C and its conjugate lie therefore on a straight line passing 
 through S, and the tangents at these points must meet on the 
 straight line joining the points aa' and bb' , i.e. on s; the conjugate 
 of G is therefore C. This shows that A and A', B and B', C and C 
 form a range of points in involution, and that consequently a and a', 
 b and b\ c and c' form a series of tangents in involution. 
 
 VII. If M and N are the double points of an involution 
 AA', BB', CC',... of points on a conic, it has been seen that 
 AB , .4'i?', MN, are three concurrent straight lines (the same is 
 the case with regard to AB' , A'B , MN). In consequence then of 
 theorem V, above, we conclude that : 
 
 If A A' and BB f are two pairs of conjugate elements of an involu- 
 tion, and MN the double elements, then MN, AB, and A'B' (and 
 similarly MN , AB' , and A'B) are three pairs of conjugate elements 
 of another involution. 
 
 VIII. The straig-hfc line s cuts the conic (see below, Art. 
 254) when the point S lies outside the conic (Fig. 138), that is, 
 
 Fig. 139. 
 when the arcs A A' and BB' do not overlap one another ; when 
 these arcs overlap, the point S lies within the conic and the 
 straight line s does not cut the latter (Fig. 139). We therefore 
 
168 SELF-CORRESPONDING ELEMENTS [204 
 
 arrive again at the property already proved in Art. 128, viz. 
 that 
 
 An involution has two double elements when any two pairs of 
 conjugate elements are such that they do not overlap ; and it has no 
 double elements when they are such that they do overlap. 
 
 In no case can an involution, properly so called, have only 
 one double element. For if s were a tangent to the conic, S 
 would be its point of contact, and of every pair of conjugate 
 points one would coincide with S (cf. Art. 125). 
 
 204. If (MNAB...) and (MNA'B'...) are two projective 
 ranges of points on a conic, M and N will be the self-corre- 
 sponding points, and the straight line MN will pass through 
 the point of intersection of AB' and 
 A'B (Art. 200). Now let B' be sup- 
 posed to lie indefinitely near to A 
 and similarly B to A\ so that the 
 straight lines AB' and A'B become 
 in the limit the tangents at A and 
 A! respectively (Fig. 140). Since now 
 3INAA' and MNA'A are groups of 
 corresponding points of two projective 
 Fig. 140. ranges, the two pencils mnaa! and 
 
 mna'a formed by joining them to any 
 point on the conic will be projective ; and therefore mnaa! 
 is a harmonic pencil (Art. 83). We thus arrive again at the 
 second theorem of Art. 195 (right) ; viz. 
 
 If four points M^N^AyA' on a conic are harmonic, the tangents at 
 one pair of conjugate points, say A and A', intersect on the chord 
 MN joining the other pair ; 
 
 and its correlative (Art. 195, left), 
 
 If four tangents to a conic are harmonic , the point of intersection 
 of one pair of conjugates lies on the chord of contact of the other 
 pair. 
 
 From the former of these it follows that if through the 
 point of intersection S of the tangents at M and N straight 
 lines be drawn cutting the conic in A and A\B and B', C and 
 C\ ... respectively, any of these pairs of points will be har- 
 monically conjugate with regard to if and N. The tangents 
 at A and A', B and B\ C and 6", ... will therefore intersect in 
 pairs on the straight line MN. 
 
206] 
 
 AND DOUBLE ELEMENTS. 
 
 169 
 
 Fig. 141. 
 
 In other words : 
 
 If from any point there he drawn to a conic two tangents and a 
 secant, the two points of contact and 
 the two points of intersection form a 
 harmonic system. 
 
 The points (AA'), (BB'), (CC), ... 
 form an involution of which M and 
 N are the double points (Art. 203, 
 III, IV). We therefore arrive again 
 at the property of an involution 
 that if it has two double elements these are separated harmoni- 
 cally by any pair of conjugate elements (Art. 125). 
 
 205. Suppose now that the conic is a circle (Fig. 141). From the 
 similar triangles SAM , SMA', 
 
 AM : MA' : : SM : SA', 
 
 and from the similar triangles SAN, SNA' 
 
 AN:NA'::SN:SA'; 
 AM A'M . . axn 
 
 * ' AN = ~FN ' (smCe SM = Sir >' 
 or AM.A'N = AN.A'M. 
 
 But by Ptolemy's theorem (Euc. vi. D), 
 
 AA'. MN = AM. A'N+AN. A'M. 
 If then M , N , A , A ' are four harmonic points on a circle, 
 \ AA'. MN = AM . A'N = AN . A'M. 
 
 206. The properties established in Art. 200 and the following 
 Articles lead at once to the solution of the important problem : 
 
 To construct the self- corresponding elements of two superposed pro- 
 jective forms, and the double elements of an involution. $t5/ 
 
 I. Let two concentric projective pencil's be given, which are deter- "IX 
 mined by three pairs of corresponding rays 
 (Fig. 142); it is required to construct their 
 self-corresponding rays. 
 
 Through the common centre describe 
 any circle, cutting the three given pairs of 
 rays in A and A', B and B', C and C re- 
 spectively. Let AB', A'B meet in E, and 
 AC, A'C in Q ; if the straight line QR cut 
 the circle in two points M and N, then 
 OM, ON will be the required self-corre- 
 sponding rays. 
 
 Fig. 142. 
 
170 
 
 SELF-CORltESPONDING ELEMENTS 
 
 [206 
 
 II. Let A and A', B and B\C and C (Fig. 143) be three pairs 
 
 of corresponding points of two collinear ranges; it is 
 construct the self-corresponding points. 
 
 to 
 
 
 fr 
 
 Fig. 143. 
 
 Describe any circle touching the common base of the two ranges, 
 and to this circle draw from the given points the tangents a and a 7 , 
 b and b', cand c' . Let r be the straight line which joins the points* 
 ab\ a'b, and q that which joins the points ac\ a'c. If the point qr 
 lies outside the circle and from it the tangents m and n be drawn to 
 the circle, then the points om , on in which these meet the base will 
 be the required self-corresponding points of the two ranges. 
 
 Fig. 144. 
 Otherwise (Fig. 144): 
 Draw any circle whatever in the plane and take on it any point 
 
- 
 
 UNTVEKi 
 
 206] 
 
 AND DOUBLE ELEMENTS. 
 
 171 
 
 0. From project the given points upon the circumference of the 
 circle, and let A[ and A x ', B x and B x ', C x and C x be the projec- 
 tions of A and A', B and B', C and C respectively. Join A X B X , A X B X 
 meeting in R, and A X C X , A X 'C X 
 meeting in Q (or B X C X , B X 'C X meet- 
 ing in P). If the straight line PQR 
 cut the circle in two points M x , N x , 
 and these be projected from the- 
 point back upon the given base o, 
 then their projections M , N will 
 be the required self-corresponding 
 points of the given ranges *. 
 
 III. In (I) let the two pencils 
 be in involution (Fig. 145), and let 
 it be required to find the double rays. 
 
 Two pairs of conjugate rays . suffice now to determine the pen- 
 cils. Draw through the centre any circle cutting the given 
 rays in A and A\ B and B' respectively. Let AB', A'B meet in 
 R, and AB , A'B' in Q; if the straight line QR cut the circle in 
 two points M and N, then OM , ON will be the required double rays 
 of the involution., 
 
 IV. Let A and A\B and B' be two given pairs of conjugates of an 
 
 Fig- 145- 
 
 involution of points on a straight line ; it is required to find the 
 double points (Fig. 146). 
 
 Draw any circle in the plane and take on it any point 0. From 
 project the given points upon the circumference of the circle, and 
 let A x and A/, B X and B( be the projections of A and A', B and B' 
 respectively. Let A x B x ' i A l 'B 1 meet in R, and A X B X , A(B X ' in Q. If 
 QR cuts the circle in M x , N lt and these points be projected from 
 back upon the given straight line, then their projections M , N will 
 be the required double points. 
 
 * Steiner, loc. cit.y pp. 68 and 174, §§17 and 46; Collected Works, vol. i. 
 PP- 285, 356. 
 
172 
 
 SELF-CORRESPONDING ELEMENTS 
 
 [207 
 
 , Otherwise : 
 
 Describe a circle touching the base AB... (Fig. 147), and draw to 
 this circle from the points A and A', B and B', the tangents a and a', 
 
 Kg. 147. 
 
 b and b\ respectively. Let r be the straight line which joins the 
 points ab', a'b, and q that which joins the points ab, afb'. If the point 
 qr lies outside the circle, the tangents m and n from this point to the 
 circle will cut the base line of the involution in the required double 
 points. 
 
 207. Theorem. A pencil in involution is either such that every 
 ray is at right angles to its conjugate, or else it contains one and 
 only one pair of conjugate rays including a right angle. 
 
 Consider again Art. 206, III ; if the point of intersection S 
 of the straight lines AA\BB\ ... is the centre of the circle 
 (Fig. 148) then AA\ BB', ... are all diameters, and therefore 
 
 Fig. 148. 
 
 Fig. 149. 
 
 each ray OA, OB, ... will be at right angles to its conjugate 
 0A\ 0B\ ... In this case then the involution is formed by a 
 series of right angles which have their common vertex at 0. 
 
208] 
 
 AND DOUBLE ELEMENTS. 
 
 173 
 
 But if 8 is not the centre of the circle (Fig. 149), draw 
 ;he diameter through it ; if C and C are the extremities of 
 this diameter, the rays OC, OC will include a right angle. 
 But these will be the only pair of conjugate rays which 
 possess this property, since through S only one diameter can 
 be drawn. 
 
 208. This proposition is only a particular case of the 
 following one : 
 
 Too superposed involutions (or such as are contained in the same 
 one-dimensional form) have always a pair of conjugate eleynents in 
 common, except in the case where the involutions have double 
 elements and the double elements of the one overlap those of the other. 
 
 Take two involutions of rays having a common centre 0, 
 and let a circle drawn through cut the pairs of con- 
 jugate rays of the first involution in the pairs of points 
 (AA',BB\...) and those of the second in (GG',HH' ',...)• Let 
 8 be the point of intersection of AA',BB' f .,. and T that of 
 GG\HTL\.... If the straight line ST cut the circle in two 
 points i?and E', these will be a conjugate pair of each involu- 
 tion, since they are collinear with 8 and with T also. Let us 
 now examine in what cases ST will cut the circle. 
 
 * h 
 
 Fig. 150. 
 
 Fig. 151. 
 
 In the first place, it will certainly do so if one at least of the 
 points S, T lies within the circle (Art. 203, VIII), i.e. if one at 
 least of the involutions has no double elements (Figs. 150, 151). 
 
 Secondly, if both the points 8, T lie outside the circle, i. e. if 
 both the involutions have double elements, then the straight 
 line ST may or may not cut the circle. If OMflN are the 
 double elements of the first involution, 0JJ y 0V those of the 
 second, the rays OE, OW must be harmonically conjugate both 
 with regard to 0M,0N and with regard to 0U,0F; but (Art. 
 70) in order that there should exist a pair of elements which 
 
174 
 
 SELF-CORRESPONDING ELEMENTS 
 
 [209 
 
 are at the same time harmonically conjugate with regard to 
 each of the two pairs OM , Oi\^and 0U,0V, it is necessary and 
 sufficient that these two pairs should not overlap. If then 
 these pairs do not overlap, ST will cut the circle (Fig. 152) ; 
 
 Fig. 152 
 
 whereas if they do overlap, ST will not cut the circle (Fig. 
 153). The two involutions have therefore a common pair of 
 conjugate elements in all cases except this last, viz. when they 
 both have double elements and these overlap. 
 
 [In Figs. 150, 151 and 152, are shown cases of two involutions 
 having a common pair of conjugate elements E and E' ; Fig. 153 
 on the other hand illustrates the case where no such pair exists.] 
 
 209. The preceding problem, viz. that of determining the common 
 pair of conjugate elements of two involutions superposed one upon 
 the other, depends upon the following, viz. to determine (in a range, 
 in a pencil, or on a conic) a pair of elements which are harmonically 
 conjugate with regard to each of two given pairs. This problem has 
 already been solved, for the case of a range, in Art. 70 ; the following 
 is another solution : 
 
 Suppose that we have to deal with a range of points lying on a 
 straight line. Take any circle and a point on it, and project the 
 given points from upon the circumference ; let M, N and U, V be 
 their projections (Fig. 152). Let the tangents at M and N to the 
 circle meet in S, and the tangents at U and V in T. If the pair MN 
 does not overlap the pair UV, then ST will cut the circle in two 
 points E and US', which when projected back from.0 upon the given 
 straight line will give the points required. 
 
 210. The double points of the involution determined by the pairs 
 A 9 A' and B, B / are the common pair of conjugate elements of two 
 other involutions ; one of these is determined by the pairs A , B 
 
21lj AND DOUBLE ELEMENTS. 175 
 
 and A', B', the other by the pairs A, B' and A', B (Art. 203, 
 VII.) 
 
 From this follows a construction for the double points of an 
 involution of collinear joints which is determined by the pairs A, A' 
 and B, B'. Take any point G outside the base of the involution 
 and describe the circles GAB, GA'B' ; they will meet in another 
 point, say in H. Similarly let K be the second point of intersection 
 of the circles GAB', GA f B. Every circle passing through G and H 
 meets the base in a pair of conjugate points of the involution AB, A'B' 
 (Art. 127) ; so too every circle passing through G and K gives a pair 
 of conjugate points of the involution AB', A'B. If then the circle 
 GHK be described and it meet the base, the two points of intersection 
 will be the double elements of the involution AA', BB' *. 
 
 211. It follows from the foregoing that the determination of the 
 self-corresponding points of two projective ranges ABC ... and 
 A'B'C ... on a conic (and consequently of the self-corresponding 
 points of any two superposed projective forms) reduces to the con- 
 struction of the straight line s on which intersect the pairs of 
 straight lines AB' and A'B, AG' and A'G, BC' and B'C, .... Simi- 
 larly the determination of the double points of an involution AA', 
 BB', ... depends on the construction of the straight line s on which 
 intersect the pairs of straight lines AB and A'B', AB' and A'B, ... 
 or the pairs of tangents at A and A', B and B', ... . 
 
 Conversely, if any straight line s (which does not touch the conic) 
 is given, an involution of points on the conic is thereby determined ; 
 for it is only necessary to draw, from different points of s, pairs of 
 tangents to the conic, and the points of contact will be pairs of 
 conjugate points of an involution. 
 
 But, on the other hand, in order that two projective ranges of 
 points ABC ... and A'B'C ... may be determined, there must be 
 given, in addition to the straight line s, a pair of conjugate points A 
 and A' also ; then the straight lines joining A and A' to any point 
 on s will cut the conic in a pair of corresponding points B' and B. 
 
 Two projective ranges of points determine an involution ; for they 
 determine the straight line s, which determines the involution. If 
 the two ranges have two self-corresponding points, there will also be 
 the double points of the involution. 
 
 * Chasles, Giomilrie, suptrieure, Art. 263. 
 
CHAPTEK XIX. 
 
 PROBLEMS OP THE SECOND DEGREE. 
 
 212. Problem. Given five points 
 0, 0\ A, B, C on a conic, to 
 determine the points of intersection 
 of the curve with a given straight 
 line s. 
 
 Solution. Join any two of 
 the points 0, 0' to each of the 
 others A, B, C (Fig. 154); the 
 
 Problem. Given five tangents 
 0, o', a, b,c to a conic, to draw 
 a pair of tangents to the curve 
 from a given point S. 
 
 Consider the points where two 
 of the tangents 0, o' are met by 
 the others a, b, c (Fig. 155) ; the 
 
 *%■ i54 
 
 pencils (A, B, C, ... ) and 
 O f (A,B,C,...) will be projective, 
 and will cut the transversal s 
 in points forming two collinear 
 projective ranges. 
 
 A point M which corresponds 
 to itself in these two ranges will 
 
 ranges (a, b, c, ... ) and 
 o / (a, b, c, ... ) will be projective, 
 and if projected from JS as centre 
 will give two concentric projec- 
 tive pencils. 
 
 Any ray m which corresponds 
 to itself in these two pencils will 
 
213] 
 
 PROBLEMS OF THE SECOND DEGREE. 
 
 177 
 
 also be a point on the conic, since 
 a pair of corresponding rays of 
 the two pencils must meet in M. . 
 The points of intersection of the 
 conic with the straight line s are, 
 therefore found as the self-corre- 
 sponding points of the two colli-, 
 near ranges which are determined 
 on s by the three pairs of corre- 
 sponding rays OA and O'A, OB 
 and O'B, OG and O'C. There 
 may be two such self-correspond- 
 ing points, or only one, or none 
 at all ; consequently the straight 
 line s may cut the conic in two 
 points, or it may touch it, or it may 
 not meet it at all. The construction 
 of the self-corresponding points 
 themselves may be effected by 
 either of the methods explained 
 in Art. 206, II. 
 
 213. In a similar manner the 
 problem may be solved if there 
 be given four points , 0', A, B 
 on a conic and the tangent o at 
 one of them ; or three points 
 0, 0', A and the tangents o and 
 o' at two of them and 0'. In 
 the first case the two pencils are 
 determined by the three pairs of 
 rays o and O'O, OA and O'A, 
 OB and O'B ; and in the second 
 case by the three pairs o and 
 O'O, 00 f and o', OA and O'A. 
 
 If however there be given five 
 tangents, or four tangents and 
 the point of contact of one of 
 them, or three tangents and the 
 points of contact of two of 
 them, we may begin by first con- 
 structing such of the points of 
 contact of the tangents as are not 
 
 also be a tangent to the conic, 
 since a pair of corresponding 
 points of the two ranges o and o' 
 must lie on m. The tangents 
 from S to the conic are therefore 
 found as the self-corresponding 
 rays of the two concentric pencils 
 which are determined by the rays 
 joining JS to the three pairs of 
 corresponding points oa and o'a, 
 ob and o'b, oc and o'c. There 
 may be two such self-correspond- 
 ing rays, or only one, or none at 
 all ; consequently there can either 
 be drawn from the point £ two 
 tangents to the conic, or JS is a 
 point on the conic, or else from S 
 no tangent at all can be drawn. 
 The construction of the self-cor- 
 responding rays themselves may 
 be effected by the method ex- 
 plained in Art. 206, I. 
 
 In a similar manner the pro- 
 blem may be solved if there be 
 given four tangents o, o', a, b to 
 a conic and the point of contact 
 of one of them o ; or three tan- 
 gents o, o', a and the points of 
 contact and 0' of two of them 
 o and o'. In the former case the 
 three pairs of points which deter- 
 mine the two ranges are and 
 o'o , oa and o'a , ob and o'b ; 
 in the latter case they are 
 and o'o, oo' and 0', oa and 
 o'a. 
 
 If however there be given five 
 points on the conic, or four points 
 and the tangent at one of them, 
 or three points and the tangents 
 at two of them, we may begin by 
 first constructing such of the tan- 
 gents at the points as are not 
 already given (Arts. 165, 171, 
 
 N 
 
178 
 
 PROBLEMS OF THE SECOND DEGREE. 
 
 [214 
 
 already given (Arts. 180, 171, 
 177); the problem will then reduce 
 to one of the cases given above. 
 
 175); the problem will then re- 
 duce to one of the cases given 
 above. 
 
 214. In the construction given in Art. 212 (left) suppose that the 
 conic is a hyperbola and that the given 
 straight line s is one of the asymptotes (Fig. 
 156). The collinear projective ranges de- 
 termined on s by the pencils (A, B, C, ... ) 
 and 0' (A, B, C, ... ) will have in this case 
 one self-corresponding point, and this (being 
 the point of contact of the hyperbola and 
 the asymptote) will lie at an infinite dis- 
 tance. But in two collinear ranges whose 
 self-corresponding points coincide in a single 
 one at infinity, the segment intercepted between any pair of corre- 
 sponding points is of constant length (Art. 103). We therefore 
 conclude that 
 
 If from two fixed points and / on a hy2>erbola there be drawn 
 two rays to cut one another on the curve, the segment PP' which these 
 intercept on either of the asymptotes is of constant length *. 
 
 Fig. 156. 
 
 Fig. 157- 
 
 215. If in Art. 212 (left) the straight line s be taken to lie at 
 infinity, the problem becomes the following : 
 
 Given five ptoints 0, 0', A, B, C on a conic, to determine the joints 
 at infinity on it (Fig. 157). 
 
 * Brianchon, loc. cit., p. 36. 
 
216] PROBLEMS OF THE SECOND DEGREE. 179 
 
 Consider again the projective pencils (A , B , C , . . . ) and 
 0' (A, B, C, ... ), which determine on the straight line at infinity s two 
 collinear ranges whose self-corresponding points are the required 
 points at infinity on the conic. Since each of these self-corresponding 
 points must lie not only at the intersection of a pair of corresponding 
 rays of the two pencils but also on the line at infinity s, the corre- 
 sponding rays which meet in such a point must be parallel to one 
 another ; the problem therefore reduces to the determination of the 
 pairs of corresponding rays of the two pencils which are parallel to 
 one another. 
 
 In order then to solve the problem we draw through the parallels 
 OA', OB', OC to O'A, O'B, O'C respectively, and then tonstruct 
 (Art. 206, I) the self-corresponding rays of the two concentric 
 pencils which are determined by the three corresponding pairs OA 
 and OA', OB and OB', OC and OC. If there are two self-corre- 
 sponding rays OM and ON, the conic determined by the five given 
 points is a hyperbola whose points at infinity lie in the directions 
 OM, ON ; i.e. whose asymptotes are parallel to OM and ON 
 respectively. 
 
 If there is only one self-corresponding ray OM, the conic deter- 
 mined by the five given points is a parabola whose point at infinity 
 lies in the direction OM. 
 
 If there is no self-corresponding ray, the conic determined by the 
 five given points is an ellipse, since it does not cut the straight line 
 at infinity. 
 
 If in the first case (Fig. 157) it is desired to construct the asymp- 
 totes themselves of the hyperbola, we consider this latter as determined 
 by the two points at infinity and three other points, say A, B, and 
 C ; in other words, we regard the hyperbola as generated by the two 
 projective pencils, one of which consists of rays all parallel to OM, 
 and the other of rays all parallel to ON, and which are such that one 
 pair of corresponding rays meet in A, a second pair in B, and a 
 third pair in C. The rays which correspond in the two pencils 
 respectively to the straight line at infinity (the line joining the 
 centres of the pencils) will be the asymptotes required. 
 
 Let then a, b, c (Fig. 157) be the rays parallel to OM which pass 
 through A, B, C respectively, and let a', b', c' be the rays parallel 
 to ON which pass through the same points respectively. Join the 
 points ab' and a'b and the points bo' and b'c, and let K be the point 
 of intersection of the joining lines ; the straight lines drawn through 
 K parallel to OM and ON will be the .required asymptotes. 
 
 216. Problem. Given Jive points A, B,C, D, E on a conic, to 
 draw the tangents from a given point S to the conic. 
 
 This problem also can be made to depend on that of Art. 212 
 
 N % 
 
180 
 
 PROBLEMS OP THE SECOND DEGREE. 
 
 [217 
 
 Fig. 158. 
 
 (left), by making use of the properties of the involution (Art. 203) 
 obtained by cutting the conic by transversals drawn through S. 
 
 Join SA,SB (Fig. 158); these 
 straight lines will cut the conic 
 again in two new points A' and B', 
 which can be determined (making 
 use of the ruler only, and without 
 drawing the curve) by means of 
 Pascal's theorem (Art. 161, right). 
 (In the figure the points A' and 
 B / have been constructed by means 
 of the hexagons ADCBEA' and 
 BECADB' respectively). Now 
 let the point of intersection of A B 
 and A'B' be joined to that of AB' 
 and A'B ; the joining line s will pass through the points of contact of 
 the tangents from S (Art. 203). The problem therefore reduces to 
 that of determining the points of intersection of the conic and the 
 straight line s (Art. 212, left). 
 
 217. The problem, To find the points of intersection of a given 
 
 straight line s and a conic which is 
 determined by Jive given tangents, 
 may similarly be made to depend 
 on that of Art. 212 (right), by 
 making a construction (Fig. 159) 
 analogous to the foregoing one. 
 
 And the problem, To draw 
 through a given i^oint a straight 
 line which shall divide a given 
 triangle into two parts having to 
 one another a given ratio, may be 
 solved by reducing it to the follow- 
 ing construction : To draw from 
 the given point a tangent to a 
 hyperbola of which the asymptotes and a tangent are known. 
 These are left as exercises to the student. 
 
 Fig. 159- 
 
 218. Problem. To construct 
 a conic which shall pass through 
 four given points Q , R , £ , T, 
 and shall touch a given straight 
 line s which does not pass through 
 any of the given p>oints. 
 
 Solution. Let A « A' , B , B' 
 
 To construct a conic which shall 
 touch four given straight lines 
 q, r , s, t , and shall p>ass through 
 a given point S which does not lie 
 on any of the given lines. 
 
 Let a , a' , b , b' be the rays 
 
219] 
 
 PROBLEMS OF THE SECOND DEGREE. 
 
 181 
 
 be the points where the sides joining the point Sto the vertices 
 
 QT , RS , QR , ST respectively qt , rs , qr , st respectively of the 
 
 of the quadrangle QRST cut quadrilateral qrst (Fig. 161). 
 
 the straight line s (Fig. 160). Construct the double rays (if 
 
 Fig. 1 60. 
 
 Fig. 161. 
 
 Construct the double points (if such exist) of the involution de- 
 such exist) of the involution de- termined by the pairs of rays 
 
 a and a', b and &'. 
 
 If there are two double rays 
 m and n, each of them will be 
 (Art. 185, right) a tangent at 
 S to some conic inscribed in 
 the quadrilateral qrst. Each of 
 the conies qrstm , qrstn therefore 
 gives a solution of the problem; 
 and these conies can be con- 
 structed by tangents by help of 
 Brianchon's theorem (Art. 161, 
 left). 
 
 If however there are no double 
 rays, there is no conic which 
 satisfies the conditions of the 
 problem. 
 
 219. If in the foregoing Art. (left) the straight line s be taken to 
 lie at infinity, the problem becomes the following : 
 
 To construct a -parabola which shall ^;ass through four given points 
 
 ft, '*»*.>. 
 
 To solve it, take any point (Fig. 162), and through it draw 
 the rays a , a' , b , b' parallel respectively to the straight lines 
 QT , RS , QR , ST ; and construct the double rays (if such exist) 
 of the involution determined by the pairs of rays a and a' } b and b'. 
 
 termined by the pairs of points 
 A and A', B and B'. 
 
 If there are two double points 
 M and N, each of them will be 
 (Art. 185, left) the point of con- 
 tact with s of some conic cir- 
 cumscribed about the quadrangle 
 QRST. Each of the conies 
 QRSTM, QRSTN therefore gives 
 a solution of the problem ; and 
 these conies can be constructed 
 by points by help of Pascal's 
 theorem (Art. 161, right). 
 
 If however there are no double 
 points, there is no conic which 
 satisfies the conditions of the 
 problem. 
 
182 
 
 PROBLEMS OF THE SECOND DEGREE. 
 
 [220 
 
 Each of these double rays will determine the direction in which lies 
 
 the point at infinity on a parabola passing through the four given 
 points ; the problem therefore reduces to 
 the last problem of Art. 165. If however 
 the involution has no double rays, no 
 parabola can be found which satisfies the 
 conditions of the problem. 
 
 Through four given points therefore can 
 be drawn either two parabolas or none ; 
 in the first case the other conies which 
 pass through the given points are ellipses 
 and hyperbolas ; in the second case they 
 are all hyperbolas. The first case occurs 
 when each of the four points lies outside 
 the triangle formed by the other three 
 
 (i. e. when the quadrangle formed by the four points is non-reentrant) ; 
 
 the second case when one of the four points lies within the triangle 
 
 formed by the other three (i.e. when the quadrangle formed by the 
 
 four points is reentrant). 
 
 220. If in Art. 218 (right) one of the straight lines q ,r , s ,t lies 
 
 at infinity, the problem becomes the following : 
 
 To construct a parabola which shall touch three given straight lines 
 
 and shall ptass through a given point. 
 
 221. Pkoblem. To construct 
 a conic which shall pass through 
 three given points P , P\ P" and 
 shall touch two given straight lines 
 q and s, neither of which p>asses 
 through any of the given points. 
 
 Solution. This depends on the 
 theorem of Art. 191 (left). Join 
 PP\ and consider it as a trans- 
 versal which cuts the conic in 
 P and P', and the pair of tan- 
 gents q and s in the two points 
 B and B' (Fig. 163). If A and A x 
 are the double points of the in- 
 volution determined by the two 
 pairs of points P and P\B and B', 
 the chord of contact of the conic 
 and the tangents q and s must 
 pass through one of these points, 
 by the theorem quoted above. 
 
 To construct a conic which 
 shall touch three given straight 
 lines p , p', p" an ^ shall pass 
 through two given points Q and S, 
 neither of which lies on any of the 
 given straight lines. 
 
 The solution depends on the 
 theorem of Art. 191 (right). Con- 
 sider pp f as a point from which 
 the tangents p and p' have been 
 drawn to the conic, and the rays 
 b and b f to the two points Q 
 and S (Fig. 164). If a and a x are 
 the double rays of the involution 
 determined by the two pairs of 
 rays p and p', b and b\ the point 
 of intersection of the tangents at 
 Q and S to the conic must lie on 
 one of these rays, by the theorem 
 quoted above. Repeat the same 
 
221] 
 
 PROBLEMS OF THE SECOND DEGREE. 
 
 183 
 
 Repeat the same reasoning for 
 the case of the transversal PP", 
 which cuts q and s in D and D" \ 
 
 reasoning for the case of the point 
 pp'\ from which are drawn the 
 rays d and d" to the points 
 
 Fig. 163. 
 
 if C and C x are the double points 
 of the involution determined by 
 the two pairs of points P and P" ', 
 D and D'\ the chord of contact 
 must similarly pass through 
 C or C x . The problem admits 
 therefore of four solutions ; viz. 
 when the two involutions 
 (PP'iBB') and {PP" , DD") 
 both have double points, there 
 are four conies which satisfy the 
 given conditions. If the double 
 points are A , A x and C , C : 
 respectively, the chords of con- 
 tact of the four conies and the 
 tangents q and s are AC, A X C , 
 AC X , and A 1 G 1 . Of each of these 
 conies five points are known, viz. 
 P , P / i P", and the two points 
 of intersection of AC (or of A X C ', 
 or AC li or A x C xi as the case may 
 be) with q and s ; they can ac- 
 cordingly be constructed by points 
 by means of Pascal's theorem 
 (Art. 161, right). 
 
 Fig. 164. 
 
 Q and S; if c and c x are the 
 double rays of the involution de- 
 termined by the two pairs of rays 
 2) and ;;", d and d", the point of 
 intersection of the tangents must 
 similarly lie on c or c v The 
 problem admits therefore of four 
 solutions ; viz. when the two in- 
 volutions (ppf, bb') and (pp", dd") 
 both have double rays, there are 
 four conies which satisfy the given 
 conditions. If the double rays are 
 a , a x and c , c x respectively, the 
 points of intersection of the 
 tangents at Q and S to the four 
 conies are ac , a x c , ac 3 , and a x c x . 
 Of each of these conies five 
 tangents are known, viz. p ,p',p", 
 and the two straight lines which 
 join ac (or a 2 c, or ac 1} or a y c x , as 
 the case may be) to Q and S ; 
 they can accordingly be con- 
 structed by tangents by means of 
 Brianchon's theorem (Art. 161, 
 left).. 
 
184 
 
 PROBLEMS OF THE SECOND DEGREE. 
 
 222. Pboblem. To construct a polygon whose vertices shall lie on 
 given straight lines (each on each), and whose sides shall pass through 
 given points (each through each *). 
 
 Solution. For the sake of simplicity suppose that it is required 
 to construct a quadrilateral, whose vertices 1,2,3,4 shall lie 
 respectively on four given straight lines s x , s 2 , s 3 , s 4 , and whose 
 sides 12 , 23 , 34 , 41 shall pass respectively through four given points 
 
 #12 5 ^23 5 #34 J #41 ( Fi g* l6 5)« 
 
 The method and reasoning will be the 
 
 Fig. 165. 
 
 same as for a polygon of any number of sides. Take any points 
 A li B li C lf ... on«j and project them from S 12 as centre upon s 2 ; and 
 let A 2 , B 2 5 ^2 »••• be their projections. Project A 2 , B 2 , C 2 , ... from 
 S 23 as centre upon s 3i and let A 3 , B 3 , C 3i ... be their projections. 
 Project A 3 .B 3 ,C 3 ,... from S u as centre upon s 4 , and let A 4 , B i , C 4 ,. . . 
 be their projections. Finally project A 4 , 2? 4 , 4 , ... from # 4J as centre 
 upon s 15 and let A , 5 , C , ... be their projections. 
 
 The points S 12 , S n , S 3i , £ 41 are the centres of four projectively 
 related pencils ; for the first and second are in perspective (since 
 their pairs of corresponding rays A X A 2 , BJB 2 , ... and A 2 A 3 , B 2 B 3 , ... 
 intersect on s 2 ), the second and third are in perspective (pairs of 
 corresponding rays intersect on s 3 ), and similarly the third and fourth 
 are in perspective (pairs of corresponding rays intersect on s 4 ). Con- 
 sequently (Art. 150) pairs of corresponding rays of the first and 
 fourth pencils (such as A X A 2 and A 4 A) will intersect on a conic ; or 
 in other words the locus of the first vertex of the variable quadri- 
 lateral whose second, third, and fourth vertices (A 2 , A 3 , A 4 ) slide 
 respectively on three given straight lines (s 2 , s 3 , s 4 ) and whose sides 
 (A X A 2 , A 2 A 3 , A S A 4 , A 4 A) pass respectively through four given points, 
 
 * Poncelet, he. cit., p. 345. 
 
223] PROBLEMS OF THE SECOND DEGREE. 185 
 
 is a conic *. This conic passes through the points S 12 , aS"^ , the 
 centres of the pencils which generate it ; in order therefore to deter- 
 mine it, three other points on it must be known ; the intersections 
 of the three pairs of corresponding rays A X A 2 and A±A , B 1 B 2 and BJB , 
 C X C 2 and Cfi will suffice. It is then only necessary further to con- 
 struct (Art. 212) the points of intersection if and N of the straight 
 line s x with the conic determined by these five points ; either M or N 
 can then be taken as the first vertex of the required quadrilateral. 
 
 This construction may be looked at from another point of view. The 
 broken lines A X A 2 A 3 A A A , B X B 2 B 3 BJ$ , and C x C 2 C z Cfi may be regarded 
 as the results of so many attempts made to construct the required quad- 
 rilateral ; these attempts however give polygons which are not closed, 
 for A does not in general coincide with A lt nor B with JB lt nor 
 C with G x . These attempts and all other conceivable ones which might 
 similarly be made, but which it is not necessary to perform, give on 
 the straight line s x two ranges A t B 1 C x ... and ABC.) one being 
 traced out by the first vertex and the other by the last vertex of the 
 open polygon. These ranges are projective with one another, since 
 the second has been derived from the first by means of projections 
 from /S^ , S 2Z , S 34 , S A1 as centres, and sections by the transversals 
 8 2 ,s 3 , s 4 , s x . Each of the self-corresponding points therefore of the 
 two ranges will give a solution of the problem ; for, if the first vertex 
 of the polygon be taken there, the last vertex will also fall on the 
 same point, and the polygon will be closed. 
 
 In the following examples also the method remains the same 
 whatever be the number of sides of Cz M 
 
 the polygon which it is required to k M^-ff7''^k\ — "^/f ' 
 
 construct. A'^vS&Z >K' 'V 
 
 223. Peoblem. To inscribe in a / ^^S{^^^>J^/ 
 given t conic a polygon whose sides /'Z^i^ / f\ 
 
 pass respectively through given points. ^xi/l \ \V\ / / / 
 
 Solution. Suppose that it is re- <w j \ \\k' / / 
 
 quired to inscribe in the conic a c ^£-A/_YAf 
 
 . . . B iB — n 
 
 triangle whose sides pass respectively 
 
 through three given points S xi S 2 , S 3 
 
 (Fig. 1 6 6). Let us make three trials. Take then any three points 
 
 A , B , C on the conic ; join them to S x and let the joining lines 
 
 cut the conic again in A x , B x , C x ; join these points to S 2 and let 
 
 * This theorem, viz. that 'if a simple polygon move in such a way that its 
 sides pass respectively through given points and all its vertices except one slide 
 respectively along given straight lines, then the remaining vertex will describe a 
 conic,' is due to Maclaubin (Phil. Trans., London, 1735). Cf. Chasles, Aper$u 
 historique, p. 150. 
 
 f i, e. either completely traced or determined by five given points. 
 
186 
 
 PROBLEMS OF THE SECOND DEGREE. 
 
 [224 
 
 the joining lines cut the conic again in A 2 , B 2 , C 2 ; finally join 
 these points to S 3 and let the joining lines cut the conic again 
 in A', B\ C. Since the point finally arrived at, A' or B' or C", 
 does not in general coincide with the corresponding starting-point 
 A or B or C, we shall have, instead of an inscribed triangle as re- 
 quired by the problem, three polygons AA X A 2 A' , BB 1 B 2 B / , CC 1 C 2 C / 
 which are not closed. But since, by a series of projections from 
 #, , S 2 , S 3 in succession as centres, we have passed from the 
 range A, B, C , ... to the range A x , B 1 , C 1 , ... , from this last to 
 i 2 , 5 2 , C 2 ,..., and from this to A', B\ C",.--? it follows that the 
 range of points A , B , C , ... with which we started is projective with 
 the range of points A ', B',C, ..., with which we ended (Arts. 200, 201, 
 203). The problem would be solved if one of the points in the latter 
 range coincided with its correspondent in the former. If then the 
 two projective ranges ABC... and A'B'C... have self-corresponding 
 points, each of these may be taken as the first vertex of a triangle 
 which satisfies the given conditions. We have therefore only to 
 determine (Art. 200, II) the straight line on which intersect the three 
 pairs of opposite sides of the inscribed hexagon AB f CA'BC\ and to con- 
 struct (Art. 212) the points of intersection M and N of this straight 
 line with the conic ; each of them will give a solution of the problem *. 
 224. By a similar method may be solved the correlative problem : 
 
 To circumscribe about a given 
 conic (i. e. one which is either 
 completely drawn or determined 
 by Jive tangents) a polygon whose 
 vertices lie respectively on given 
 straight lines. 
 
 Suppose that it is required to 
 circumscribe about the conic a 
 triangle whose vertices lie re- 
 spectively on the straight lines 
 s 1 ,s 2 ,s 3 (Fig. 167). Take any 
 point A on the conic and draw 
 the tangent a at it ; from the 
 point where this tangent cuts 
 s 1 draw another tangent a x (let 
 its point of contact be A^) ; from 
 the point where a x cuts s 2 draw 
 a third tangent a 2 (let its point of contact be A 2 ) ; finally, from 
 the point where a 2 cuts s 3 draw the tangent a", and let its point 
 of contact be A'. The problem would be solved if the point A f 
 
 Fig. 167. 
 
 * PONCELET, IOC. dt., p. 352. 
 
225] PROBLEMS OF THE SECOND DEGREE. 187 
 
 coincided with A , i. e. if the tangents a! and a coincided with one 
 another. Suppose that other similar trials have been made, taking 
 other arbitrary points B ,C , ... on the conic to begin with ; then we 
 shall arrive in succession at the ranges of points A , B , C , ... , 
 
 ^uAj^j-j^i^i^ an( * ^ ', B\ C", ... , which are all 
 
 protectively related to one another. For the first range is projective 
 with the second (Art. 203), since the tangents at A and A lt B and B x , 
 C and C 1 , ... always intersect on s x ; and for similar reasons the 
 second and third, and the third and fourth, are projective with one 
 another; consequently (Art. 201) the same is true of the fourth and 
 the first. Since the problem would be solved if A f coincided with A, 
 or B f with B , ... , each of the self-corresponding points of the pro- 
 jective ranges ABO... and A'B'C' ... may be taken as the point of 
 contact of the first side of a triangle which satisfies the given con- 
 ditions. "We have therefore only to make three trials (Art. 200), 
 i. e. to take any three points A , B , C on the conic and to derive 
 from them the corresponding points A', B\ C ; and then to con- 
 struct the points of intersection of the conic with the straight line 
 which joins the points of intersection of the three pairs of opposite 
 sides (the Pascal line) of the inscribed hexagon AB'CA'BC*. 
 
 225. The particular case of the problem of Art. 223 in which the 
 given points S lt S 2t ... lie all upon one straight line s must be con- 
 sidered separately. If the number of sides of the required polygon 
 is even, the theorem of Art. 187 maybe applied; in this case the 
 problem has either no solution at all, or it has an infinite number of 
 solutions. Suppose it required, for example, to inscribe in the conic 
 an octagon of which the first seven sides pass respectively through 
 the points S l9 S 2t ... S 7i then by the theorem just quoted the last side 
 will pass through a fixed point S on s : this point S is not arbitrary, 
 but its position is determined by those of the points S lf S 3 , ...S 7 . 
 If then the last of the given points S 8 coincides with S, there are an 
 infinite number of octagons which satisfy the given conditions. But 
 if S s does not coincide with S, there is no solution. 
 
 If the number of sides of the required polygon is odd, the problem 
 becomes determinate. Suppose it is required to inscribe in the conic a 
 heptagon (Fig. 124) whose sides pass respectively through the given 
 collinear points S x , S 2i S 3 , ... S 7 . By the theorem of Art. 187 there 
 exist an infinite number of octagons whose first seven sides pass through 
 seven given collinear points and whose eighth side passes through 
 a fixed point S collinear with the others. If among these octagons 
 there is one such that its eighth side touches the conic, the problem 
 will be solved ; for this octagon, having two of its vertices indefinitely 
 
 * PONCELET, IOC. Clt., p. 354. 
 
188 
 
 PROBLEMS OF THE SECOND DEGREE. 
 
 [226 
 
 near to one another, will reduce to an inscribed heptagon, whose 
 Bides pass respectively through seven given points. If then tan- 
 gents can be drawn from the point S to the conic, the point of 
 contact of each of them will give a solution (Art. 187). According 
 therefore to the position of the point S with reference to the conic, 
 there will be two solutions, or only one, or none. 
 
 In Fig. 126 is shown the case of this problem where the polygon 
 to be inscribed is a triangle *. 
 
 The solution of the correlative problem, to circumscribe about a 
 given conic a polygon whose vertices lie respectively on given rays of a 
 pencil^ is left as an exercise to the student. This problem also is 
 either indeterminate or impossible if the polygon is one of an even 
 number of sides ; it is determinate and of the second degree if the 
 polygon is one having an odd number of sides (Figs. 125, 127). 
 
 226. Lemma. If two. conies cut one another in the points 
 / A f B , 9 C\ and if from 
 
 A and B respectively two 
 straight lines AFF\ BGG' be 
 drawn cutting the first conic 
 in F and G, and the second 
 in F / and G', then the chords 
 FG , F'G' will intersect in 
 a point H lying on the chord 
 OC (Fig. 168). 
 
 The transversal CG' cuts 
 the first conic and the oppo- 
 site sides of the inscribed 
 quadrangle ABGF in six 
 points of an involution (Art. 
 183, left) ; and the same is true with regard to the second conic and the 
 inscribed quadrangle ABG'F'. But the two involutions must coin- 
 cide (Art. 127), since they have two pairs of conjugate points in 
 common, viz. the points C, C in which the transversal cuts both the 
 conies, and the points in which it cuts the pair of opposite sides 
 AFF\ BGG', which belong to both quadrangles. The involutions 
 will therefore have every pair of conjugate points in common, and 
 therefore the transversal CC will meet FG and F f G f in the same 
 point H, the conjugate of the point in which it meets ABf. 
 
 227. The preceding lemma, which is merely a corollary of Desargues' 
 theorem, leads at once to the solution of the two following problems, 
 one of which is of the first, and the other of the second degree. 
 
 * Pappus, loc. cit, book vii. prop. 117. 
 
 + This may also be proved very simply by applying Pascal's theorem to each of 
 the hexagons AFGBCC, AF'G'BCC in turn. 
 
 Fig. 168. 
 
228] PROBLEMS OF THE SECOND DEGREE. 189 
 
 I. Problem. Given three of the points of intersection A,B,C of 
 two conies, and in addition two other points D, E of the first, and two 
 other points F, G of the second, to determine the fourth point of inter- 
 section of the two conies (Fig. 168). 
 
 Take two of the given points of intersection A and B, and join 
 AF, BG. These straight lines will cut the first conic again in points 
 F', G' respectively which can be determined by the method of Art. 
 161 (right). Join FG, F'G', and let them meet in H. By the fore- 
 going lemma H will lie on the chord joining the other two points of 
 intersection of the conies. This chord will therefore be HC, and 
 it remains only to determine the point C f where HC cuts either of 
 the conies ; C" will be the required fourth point of intersection of the 
 conies. 
 
 II. Problem. Given two of the points of intersection A, B, of two 
 conies, and in addition the three points D , E , N of the first and the three 
 points F,G,M of the second, to determine the other two points of inter- 
 section of the conies (Fig. 168). 
 
 Join AFaxidi BG, and let them meet the first conic again in F' ', G' 
 respectively ; join FG, F'G', and let them meet in H. The point H 
 will lie on the chord joining the two required points. Again, join 
 AM, and let it meet the first conic again in M'; join GM , G'M', and 
 let these meet in K ; then the point K also will lie on the same 
 chord. The required points therefore lie on HK, and the problem 
 reduces to the determination (Art. 212) of the points of intersection 
 C > C of the conies with HK * 
 
 228. The solution just given of problem II holds good equally when 
 the points A and B lie indefinitely near to one another, i.e. when the 
 two conies touch a given straight line at the same given point. 
 
 In this case two conies are given which touch one another at a 
 point A, and the straight line HK is constructed which joins their 
 remaining points of intersection C and C'. If HK passes through 
 A, one of the points C or C' must coincide with A, since a conic 
 cannot cut a straight line in three points. When this is the case, 
 three of the four points of intersection of the conies lie indefinitely 
 near to one another, and may be said to coincide in the point A ; 
 and the conies are said to osculate at the point A. The construction 
 gives a point H of the chord which joins A to the fourth point of 
 intersection C of the conies. It may happen that this chord coin- 
 cides with the tangent at A ; in this case A represents four coincident 
 points of intersection of the two conies (or rather, four such points 
 lying indefinitely near to one another). 
 
 * Gaskin, The geometrical construction of a conic section , &c. (Cambridge, 1852), 
 pp. 26, 40. 
 
190 PROBLEMS OF THE SECOND DEGREE. [229 
 
 229. Let now the lemma of Art. 226 be applied to the case of a 
 conic and a circle touching it at a point A. At A draw the normal 
 to the conic (the perpendicular to the tangent at A), and let it cut 
 the conic again in F and the circle again in F'. On AF as diameter 
 describe a circle ; this circle, which touches the conic at A and cuts 
 it at F, will cut it again at another point G such that AG F is a right 
 angle. Join AG and let G f be the point where it cuts the first 
 circle. Join FG,F f G'\ by the lemma they will intersect on the 
 chord UK; but they are parallel to one another, since AG'F' also is 
 a right angle. Thus for any circle whatever which touches the conic 
 at A> the chord of intersection HK with the conic has a constant 
 direction, viz. that parallel to FG. 
 
 If HK passes through A, the conic and the circle osculate at this 
 point. If then a parallel through A to FG cut the conic again in 
 C, the circle which touches the conic at A and cuts it at C will be 
 the osculating circle (circle of curvature) at A *. 
 
 [In the particular case where A is a vertex (Art. 297) of the conic, F 
 will be the other vertex, FG the tangent at F, AC the tangent at A, 
 and C will coincide with A . It is seen then that the osculating circle 
 at a vertex of a conic has not only three but four indefinitely near 
 points in common with the conic] 
 
 Conversely, the conic can be constructed which passes through 
 three given points A, P, Q and has a given circle for its osculating 
 circle at one of these points A. 
 
 For join AP, AQ, and let them cut the given circle in P f , Q' re- 
 spectively; and join PQ, P'Q f , meeting in U. If AU be joined and cut 
 the circle again in C, the required conic will pass through C. It is 
 therefore determined by the four points A ,P,Q , G and the tangent 
 at A (which is the same as the tangent to the circle there). 
 
 230. The proposition correlative to the lemma of Art. 226 may be 
 enunciated as follows : 
 
 If a and b are a pair of common tangents to two conies, and if from 
 two points taken on a and b respectively the tangents f,gbe drawn to 
 the first conic and the tangents f , g' to the second, then the points fg and 
 fg r will be collinear with the point of intersection of the second pair of 
 common tangents to the conies. 
 
 This proposition enables us to solve the problems which are corre- 
 lative to I and II of Art. 227 ; viz. given three (or two) of the com- 
 mon tangents to two conies, and in addition two (or three) tangents 
 to the first and two (or three) tangents to the second, to determine 
 the remaining common tangent (or the two remaining common tan- 
 gents) to the conies. 
 
 231. Pkoblem. Given eleven points A, B,C,D,E; A 1 ,B 1 ,C v D 1 ,F i ; P; 
 
 * Poncelet, loc. cit., Arts. 334-337. 
 
234] PROBLEMS OF THE SECOND DEGREE. 191 
 
 to construct by 2>oints the conic which passes through P and through the 
 four points of intersection of the two conies which are determined by the 
 points A ,B ,0 ,D , E and A 1 ,B 1 ,0 1 ,D 1 , E x respectively. The conies 
 are supposed not to be traced, nor are their points of intersection given *. 
 Solution. Draw through P any transversal, and construct (Art. 
 212, left) the points M and M' in which it cuts the conic ABC BE 
 and the points N and N f in which it cuts the conic A v B x O l D 1 E v 
 Since these two conies and the required one all pass through the same 
 four points, Desargues' theorem may be applied to them. If therefore 
 (Art. 134, left) the point P r be constructed, conjugate to P in the 
 involution determined by the pairs of points M and M ', N and N', 
 this point P' will lie on the required conic. By causing the trans- 
 versal to turn about the point P, other points on the required conic 
 may be obtained. 
 
 232. Pkoblem. Given ten points A,B,C,D,E; A 1 ,B 1 ,C l ,D 1 ,E 1 
 and a straight line s ; to construct a conic which shall touch s and 
 shall pass through the four points of intersection of the two conies 
 which are determined by the points A,B,0 ,D,E and A 1 ,B 1 ,0 1 ,D 1 ,E 1 
 respectively. The conies are supposed not to be traced, nor are their 
 points of intersection given. 
 
 Solution. Construct (Art. 212) the points of intersection M and M' 
 of s with the conic ABODE, and the points of intersection N and 
 N' of s with the conic A l B x C 1 D 1 E 1 , and then (Art. 134) the double 
 points of the involution determined by the two pairs of points 
 M and M' ', N and N'. If P is one of these double points, it will be 
 the point of contact (Art. 185) of s with a conic drawn through the 
 four points of intersection of the conies ABODE and A 1 B 1 C 1 D 1 E l 
 to touch s. The problem thus reduces to that of the preceding 
 Article. 
 
 233. The correlative constructions give the solutions of the corre- 
 lative problems : viz. to construct a conic which passes through a given 
 point (or which touches a given straight line), and which is inscribed 
 in the quadrilateral formed by the four common tangents to two 
 conies ; the conies being supposed each to be determined by five given 
 tangents, but not to be completely traced ; and their four common 
 tangents being supposed not to be given. 
 
 234. Pkoblem. Through a given point S to draw a straight line 
 which shall be cut by four given straight lines a ,b ,c ,d in four 2>oints 
 having a given anharmonic ratio. 
 
 Solution. It has been seen (Art. 151) that the straight lines 
 which are cut by four given straight lines in four points having a 
 given anharmonic ratio are all tangents to one and the same conic 
 
 * Poncelet, loc. cit., Art. 389. 
 
192 PROBLEMS OF THE SECOND DEGREE. [.235 
 
 touching the given straight lines ; and that if A , B , C are the points 
 where d cuts a ,b ,c respectively, and D is the point of contact of d, 
 the anharmonic ratio (ABCD) is equal to that of the four points in 
 which the straight lines a ,b , c ,d are cut by any other tangent to 
 the conic. Accordingly, if on the straight line d that point D be 
 constructed (Art. 65) which gives with the points 
 
 ad(=A),bd(=B),cd(=C) 
 
 an anharmonic ratio {ABCD) equal to the given one, and if then the 
 straight lines be constructed (Art. 213, right) which pass through S 
 and touch the conic determined by the four tangents a,b ,c,d and 
 the point of contact D of d, each of these straight lines will give a 
 solution of the proposed problem. 
 
 If one of the straight lines a , b , e , d lie at infinity, the problem 
 becomes the following : 
 
 Given three straight lines a ,b , c and a point S, to draw through S 
 a straight line such that the segment intercepted on it between a and b 
 may be to that intercepted on it between a and c in a given ratio. 
 
 To solve this, construct on the straight line a that point A which 
 is so related to the points ab (=B) and ac (==C) that the ratio 
 AB : AC has the given value ; and draw from S the tangents to the 
 parabola which is determined by the tangents a ,b , c and the point 
 of contact A of a. 
 
 The correlative construction gives the solution of the following 
 problem : On a given straight line s to find a point such that the 
 rays joining it to four given points A , B ,C ,D form a pencil having 
 a given anharmonic ratio. 
 
 235. Problem. Given two projective ranges of points lying on the 
 straight lines u , u' respectively ; to find two corresponding segments 
 MP, M'P' such that the angles MOP , M / / P / which they subtend at 
 two fixed points , r respectively may be given in sign and mag- 
 nitude. 
 
 Solution. Take on u' two points A' and D' such that the angle 
 A'O'D' may be equal to the second of the given angles ; let A and D 
 be the points on u which correspond respectively to A' and D' t and 
 let A x be a point on u such that the angle A x OD is equal to the first 
 of the given angles. The problem would evidently be solved if OA x 
 coincided with OA, since in this case the angles AOD and A'O'D' 
 would be equal to the given angles respectively. If the rays 
 O'A', OA , 0'D',OD , OA x be made to vary simultaneously, they will 
 trace out pencils which are projectively related. For those traced 
 out by O'A' and O'D' respectively are projective, and similarly 
 those traced out by OA x and OD respectively, since the angles A'O'D' 
 and A x OD are constant (Art. 108); and the pencils traced out by 
 
237] PROBLEMS OF THE SECOND DEGREE. 193 
 
 OA and O'A' respectively, and by OD , 0' ' D' respectively, are pro- 
 jective since the given ranges on u and u' are so. Consequently the 
 pencils generated by OA and OA x respectively are projective, and 
 their self-corresponding rays give the solutions of the problem. If 
 three trials be made of a similar kind to the foregoing one, three 
 pairs of corresponding rays OA and OA xi OB and OB x , OG and 0G X 
 will be obtained ; let the self-corresponding rays of the concentric 
 projective pencils determined by these three pairs be constructed 
 (Art. 206, 1). If one of these self-corresponding rays meets u in M, 
 and if the point P be taken on u such that the angle MOP is equal to 
 the first of the given ones, and if then on u' the points M', P f be found 
 which correspond to M,P respectively, the angle M'O'P' will be 
 equal to the second of the given angles, and the problem will be solved. 
 
 236. Problem. Given two projective ranges of points A ,B ,G , ... 
 and A' ,B f ,G f ... lying on the straight lines u and u' respectively, to 
 
 find two corresponding segments which shall be equal, in sign and 
 magnitude, to two given segments. 
 
 Solution. Take on u' a segment A'])' equal to the second of the 
 given ones, and let AD be the segment on u which corresponds to 
 A'D'. Take on u the point A x such that A X D is equal to the first of 
 the given segments ; then the problem would be solved if A x coincided 
 with A. If the points A , A', D', D ,A X be made to vary simulta- 
 neously, the ranges traced out by A and A' respectively will be pro- 
 jective with one another, as also those traced out by D and !)' respectively 
 (by reason of the projective relation existing between ABG... and 
 A'B'C' ...} ; and the ranges traced out by A and D respectively, and 
 similarly those traced out by A' and D' respectively, will be projective 
 with one another, since they are generated by segments of constant 
 length sliding along straight lines (Art. 103). Consequently also the 
 ranges traced out by A and A x are protectively related, and their self- 
 corresponding points give the solutions of the problem. It is there- 
 fore only necessary to obtain three pairs of corresponding points 
 A and A',B and B', G and C', by making three trials, and then to 
 construct the self- corresponding points of the ranges determined by 
 these three pairs (Art. 206, II). 
 
 237. The student cannot have failed to remark that the method 
 employed in the solution of the preceding problems has been in all 
 cases substantially the same. This method is general, uniform, and 
 direct ; and it may be applied in a more or less simple manner to all 
 problems of the second degree, i.e. to all questions which when treated 
 algebraically would depend on a quadratic equation. It consists in 
 making three trials, which give three pairs of corresponding elements 
 of two superposed projective forms ; the self-corresponding elements 
 of these systems give the solutions of the problem. This method is 
 
 
 
194 PROBLEMS OF THE SECOND DEGREE. [238 
 
 precisely analogous to that known in Arithmetic as the ' rule of false 
 position/ and it has on that account been termed a geometric method 
 of false position*. 
 
 238. Problems of the second degree (and those which are reducible 
 to such) are solved, like all those occurring in elementary Geometry, 
 by means of the ruler and compasses only, that is to say by means of 
 the intersections of straight lines and circles t. But again, the solu- 
 tion of any such problem can be made to depend on the determination 
 of the self-corresponding elements of two superposed projective forms, 
 which determination depends (Art. 206) on the construction of 
 the self -corresponding points of two projective ranges lying on a 
 circle whose position and size is entirely arbitrary. It follows 
 that a single circle, described once for all, will enable us to solve all 
 problems of the second degree which can be proposed with reference 
 to any given elements lying in one plane (the plane in which the 
 circle is drawn) J. This circle once described, any such problem will 
 reduce to that of constructing three pairs of points of the two pro- 
 jective systems whose self-corresponding elements give the solution of 
 the problem. This done, we proceed to transfer to the circumference 
 of the circle, by means of projections and sections, these three pairs of 
 points. This will give three pairs of points on the circle ; taking these 
 as the pairs of opposite vertices of an inscribed hexagon, we have only 
 further to draw the straight line which joins the points of intersection 
 of the three pairs of opposite sides (the Pascal line) of this hexagon. 
 It is hardly necessary to remark that instead of the solution of such 
 a problem being made to depend on the 
 common elements of two superposed pro- 
 jective forms, it may always be reduced to 
 the determination of the double elements 
 of an involution (Art. 211). 
 
 The following Articles (239 to 249) 
 *& contain examples of problems solved by 
 
 Fig. 169. means of the method just explained. 
 
 239. Pkoblem. Given (Fig. 169) two 
 projective ranges of points lying on the straight lines u and u respectively, 
 and two other projective ranges of points lying on the straight lines v 
 
 * Chasles, Geom. sup., p. 212. 
 
 + A problem is said to be of the first degree when it can be solved with help of 
 the ruler only, i. e. by the intersections of straight lines. See Lambert, loc. cit., 
 p. 161 ; Brianchon, loc. cit, p. 6; Poncelet, loc. cit., p. 76. 
 
 X Poncelet, loc. cit., p. 187; Steiner, Die geometrischen Constructionen aus- 
 gefiihrt mittelst der geraden Linie und eines festen Kreises (Berlin, 1833), p. 67 ; 
 Collected Works, vol. i. pp. 461-522 ; Staudt, Geometric der Lage (Niirnberg, 
 !847)> § 23- 
 
241] PROBLEMS OF THE SECOND DEGREE. 195 
 
 and v respectively ; it is required to draw through a given point two 
 straight lines 8 and s', which shall cut u and u' in a pair of corre- 
 sponding points and also v and v in a pair of corresponding p>oints. 
 
 Through draw any straight line cutting u f , i/in A f ,P' respectively; 
 let A be the point on u which corresponds to A', and let P be the 
 point on v which corresponds to P' '. The problem would be solved if 
 the straight lines OA and OP coincided with one another. If these 
 straight lines be made to change their positions simultaneously, they 
 will trace out two concentric projective pencils (determined by three 
 trials of a similar kind to the one just made) ; and the self-corre- 
 sponding rays of these pencils will give the solutions of the problem. 
 
 240. In the preceding problem the straight lines u and u f might 
 be taken to coincide, and similarly v and v'. If all four straight 
 lines coincided with one another, the problem would become the 
 following : 
 
 Given two projective ranges u, u' and two other projective ranges 
 v, v all lying on one straight line, to find a pair of points which shall 
 correspond to one another when regarded as points of the ranges u, u 
 respectively, and likewise when regarded as points of the ranges v, v* 
 respectively. 
 
 241. Peoblem. Between twio given straight lines u and u x to place 
 a segment such that it shall subtend given angles at two given points 
 and S (Fig. 170). 
 
 Fig. 170. 
 
 Draw any ray SA to meet u in A; draw SA X to meet u x in A 1 so 
 that ASA X may be equal to the second of the given angles ; join 0A lt 
 and draw 0A r to meet u in A' so that A f OA x may be equal to the 
 first of the given angles. Then the problem would be solved if OA 
 coincided with 0A\ Three trials of a similar kind to the one just made 
 will give three pairs of corresponding rays (OA and 0A ( \ OB and 0B / i 
 OC and 0C) of the two projective pencils which would be traced out 
 by causing OA and OA' to change their positions simultaneously; 
 the self-corresponding rays OM and ON of these pencils will give the 
 solutions (MM x and NN^) of the problem. 
 
 O % 
 
196 PROBLEMS OF THE SECOND DEGREE. [242 
 
 242. Pkoblem. Given two projective ranges u and u; if a pair of 
 corresponding points A and A' of these ranges be taken, it is required 
 to find another pair of corresponding points M and M' such that the 
 ratio of the length of the segment AM to that of the segment A'M' may 
 be equal to a given number X. 
 
 Let A and A',B and B', C and C be three pairs of corresponding 
 points of the two ranges. On u take two new points B", C" such that 
 AB"=\.A'B' said AC"=\.A'C. The points A , B", C" determine 
 a range which is similar (Art. 99) to the range A', B',C, ... and 
 therefore projective with A , B , C , ... . The collinear ranges 
 A , B", C" , . . . and A ,B ,C , ... have already one self-corresponding 
 . point in A ; their other self-corresponding point M (Art. 90) will 
 give the solution of the problem, since AM=AM"=\.A'M'. This 
 problem is therefore of the first degree. 
 
 243. Problem. Given two collinear projective ranges ABC ... and 
 A'B 'C . . . , to find a pair of corresponding points M and M' such 
 that the segment MM f shall be bisected at a given point 0. 
 
 Take three points A?', B", C" such that is the middle point of each 
 of the segments AA",BB", CC" '; the points A",B", C" determine 
 a range which is equal to the range ABC ... , and therefore projective 
 with the range A'B'C .... Construct the self- corresponding points of 
 the collinear projective ranges A'B'C ... and A"B"C" ... ; if M' or 
 M ff is one of them, then MM ' will have its middle point at 0, and 
 will be a segment such as is required. 
 
 244. Pkoblem. Given a straight line and two points E ,F on it ; 
 to determine on the straight line two points M and M' such that the 
 segment MM' may be equal in length to a given segment, and the 
 anharmonic ratio (EFMM / ) equal to a given number. 
 
 Take on the given straight line any three points A,B, C ', then 
 find on it three points A', B f ,C such that the anharmonic ratios 
 {EFAA') , (EFBB') , (EFCC) may each be equal to the given number; 
 and again three points A", B", C" such that the segments AA", BB h ', 
 CC" may each be equal in length to the given segment. The ranges 
 ABC ... and A'B'C ... will be projectively related (Arts. 79, 109), and 
 the same will be the case with regard to the ranges ABC ... and 
 A"B"G" . . .(Art. 103); therefore A'B'C ... and A"B"C" ... will be 
 projective with one another. If these ranges have self-corresponding 
 points, and if M ' or M" is one of them, the segment MM' and the 
 anharmonic ratio (EFMM') will have the given values, and the 
 problem is solved. 
 
 245. Problem. To inscribe in a given triangle PQR.a rectangle 
 of given area (Fig. 171). 
 
 Suppose MSTU to be the rectangle required ; if MS' be drawn 
 parallel to PR, a parallelogram MSPS' will be formed which is equal 
 
245] 
 
 PROBLEMS OF THE SECOND DEGREE. 
 
 197 
 
 Fig. 171. 
 
 in area to the rectangle ; so that for the given problem may be 
 substituted the following equivalent one : 
 
 To find on the base QB of a given triangle PQB a point M such that 
 if MS , MS' be drawn parallel 
 to the sides PQ , PR to meet 
 PB, PQ in S, S' respectively, 
 the rectangle contained by PS 
 and PS' shall be equal to a 
 given square k*. 
 
 Take any point A on QB, 
 draw AD parallel to PQ to 
 meet PB in D, and take on 
 PQ a point D' such that 
 the rectangle contained by 
 PD and PD' maybe equal to k* ; then draw D'A' parallel to PB to 
 meet QB in A '. If the points A and A' coincided with one another, 
 the problem would be solved. 
 
 Now let the points A , D , D', A' be made to vary simultaneously ; 
 they will trace out ranges which are all projective with one another. 
 For since D is the projection of A made from the point at infinity on 
 PQ, and A' the projection of D' made from the point at infinity on 
 PB , the first and second ranges are in perspective, and the third and 
 fourth likewise. But the second and third ranges are projective with 
 one another, since the relation PD . PD'=k 2 shows (Art. 74) that the 
 points D and D', in moving simultaneously, describe two projective 
 ranges such that the point P , regarded as belonging to either range, 
 corresponds to the point at infinity regarded as belonging to the other*. 
 
 Three similar trials give three pairs of points similar to A and 
 A'; if the self-corresponding points of the ranges determined by these 
 pairs be constructed, they will give the solutions of the problem. 
 
 Instead of taking the point A quite arbitrarily in the three trials, 
 any particular positions may be chosen for it, and by this means the 
 construction may often be simplified. This remark applies to all the 
 problems which we have discussed. With regard to the present one, 
 it is clear that if A be taken at infinity, its projection D will also lie 
 at infinity; consequently D' will coincide with P, and therefore A' 
 with B. Again, if A be taken coincident with Q, its projection D 
 will coincide with P , and consequently D ', and therefore also A', will 
 pass off to infinity. We have thus two trials, neither of which requires 
 
 * If the two ranges be called u and vf, and the construction of Art. 85 (left) 
 be referred back to, it will be seen that the auxiliary range u" lies in this case 
 entirely at infinity. If then a pair of corresponding points D and D' have been 
 found, and we wish to find the point E' which corresponds to any other point E 
 of PR (=w), we have only to join D'E, and to draw BE' parallel to D'E to 
 meet PQ (=«,') in £'. 
 
198 PROBLEMS OF THE SECOND DEGREE. [240 
 
 any construction ; the pairs which result from them are composed 
 respectively of the point at infinity and R , and of Q and the point at 
 infinity. If the pair given by the third trial be called B ,B' } and if 
 A , A f stand for any pair whatever, we have (Art. 74) 
 
 QA.BA'szQB.MB', 
 
 and therefore, if M is a self-corresponding point, 
 QM.RM=QB.RB\ 
 
 from which the self- corresponding points could be found. But it is 
 better in all cases to go back to the general construction of Art. 206. 
 In this case the three pairs of conjugate points of the two ranges 
 which are given are : B and B' ; the point at infinity and R; Q and 
 the point at infinity. Let then any circle be taken, and a point on 
 its circumference ; from draw the straight lines OB , OB', OR , OQ, 
 and a parallel to QR, and let these cut thecircle again inB 1 , 2?/, R v Q ti 
 and i" respectively*. Join the point of intersection of B x R 1 and B(I 
 with that of B X I and B{Q X ; if the joining line cut the circle in 
 two points M x and N % , the straight lines which join these to will 
 meet QR in the self-corresponding points M and N, and these give 
 the solutions of the problem. 
 
 246. Problem. To construct a polygon, whose sides shall pass 
 respectively through given points, and all whose vertices except one shall 
 lie respectively on given straight lines ; and which shall be such that 
 the angle included by the sides which meet in the last vertex is equal to 
 a given angle. 
 
 Suppose, for example, that it is required to construct a triangle 
 LMN (Fig. 172) whose sides MN ,NL ,LM shall pass through the 
 
 given points , V, U respectively, 
 and whose vertices M , N shall lie 
 on the given straight lines u , v re- 
 spectively; and which shall be such 
 that the angle MLN is equal to a 
 given angle. 
 
 Through draw any straight 
 line to cut u in A and v in B; 
 join BV, and through U draw the 
 Fig. 172. straight line TJX making with BV 
 
 an angle equal to the given one. 
 Let TJX meet u in A' \ the problem would be solved if the point 
 A f coincided with A. If the rays OA, UA' be made to vary 
 simultaneously, they will determine on u two projective ranges ; the 
 solutions of the problem will be found by constructing the self-corre- 
 sponding points of these ranges. 
 
 * Of these points only I is marked in the figure. 
 
248] 
 
 PROBLEMS OF THE SECOND DEGREE. 
 
 199 
 
 247. The following problem is included in the foregoing one : 
 
 A ray of light emanating from a given point is reflected from n 
 given straight lines in succession ; to determine the original direction 
 which the ray must have, in order that this may make with its direction 
 after the last reflexion a given angle. 
 
 Let u x , u 2 , ... u n be the given straight lines (Fig. 173). If the 
 ray OA x strike u x at A x , then by the 
 law of reflexion the incident and re- 
 flected rays will make equal angles 
 with u x ; but the incident ray passes 
 through the fixed point ; therefore the 
 reflected ray will always pass through 
 the point O x which is symmetrical to 
 with regard to Wj*. So again, if 
 the ray after one reflexion strikes u 2 at 
 A 2 , it will be reflected according to 
 
 the same law; consequently the ray after two reflexions will pass 
 through a fixed point 2 which is symmetrical to x with regard to 
 u 2 ; and so on. The paths of the ray before reflexion, and after one, 
 two, ... n reflexions form therefore a polygon OA x A 2 A 3 ... , whose n+ 1 
 sides pass respectively through n + 1 fixed points , O x , 2 , ... O n , 
 and which is such that n of its vertices lie respectively on n given 
 straight lines u x ,u 2 , ... u n ; while the angle included by the sides 
 which meet in the last vertex is to be equal to a given angle. Thus 
 the problem reduces, as was stated, to that of Art. 246. 
 
 248. Pkoblem. To construct a polygon whose vertices shall lie 
 respectively on given straight lines, and whose sides shall subtend given 
 angles at given points respectively. 
 
 Suppose it required to construct a triangle whose vertices 1,2,3 
 shall lie on the given straight lines u x , u 2 , u 3 respectively, and whose 
 sides 23 , 31, 12 shall subtend at 
 the given points S x , S 2 , S 3 respec- 
 tively the angles a> x , o> 2 , o> 3 which 
 are given in sign and magnitude 
 (Fig. 174). On u x take any point 
 A ; join AS Z , and make the angle 
 ASJB equal to a> 3 ; let S 3 B cut u 2 
 in B. Join BS X ; make the angle 
 BS X C equal to a> x , and let S X C cut 
 u 3 in C. Join CS 2 ; make the 
 angle CS 2 A ' equal to © 2 , and let 
 S^' cut u x in A'. The problem would be solved if S 2 A' coincided 
 
 Fig. 174. 
 
 * i.e. a, point 0^ such that 00 t is bisected at right angles by u x . 
 
200 PROBLEMS OF THE SECOND DEGREE. [240 
 
 with S 2 A. If S 2 A be made to turn about S 2 , the other rays 
 S 3 A , S S B , S t B , Sfi, S 2 C, and S 2 A' will change their positions simul- 
 taneously, and will trace out pencils which are all protectively 
 related. For the ranges traced out by S d A and S a B respectively 
 will be projective (Art. 108) since the angle AS 3 B is constant; the 
 ranges traced out by S Z B and S t B respectively are projective since 
 they are in perspective ; and so on. The solutions of the problem will 
 therefore be given by the self-corresponding rays of the concentric 
 projective pencils which are generated by S 2 A and S 2 A f respectively. 
 
 In the same manner is solved the more general problem in which 
 the straight lines joining S li S ii ... to the vertices of the polygon are 
 no longer to include given angles, but are to be such that together 
 with pairs of given straight lines meeting in S t , S 2 , . . . respectively 
 they form at each of these points a pencil of four rays having a given 
 anharmonic ratio. If at each of the points the pencil is to be 
 harmonic, and the given straight lines such as to include a right 
 angle, the problem can be enunciated as follows (Art. 60) : 
 
 To construct a polygon whose vertices shall lie respectively on given 
 straight lines, and whose sides shall subtend at given points angles 
 whose bisectors are given, 
 
 249. The same method gives the solution of the problem : 
 
 To construct a polygon whose sides shall pass respectively through 
 given jwints, and which shall be such that the pairs of adjacent sides 
 divide given segments respectively in given anharmonic ratios*. 
 
 Particular cases of this problem may be obtained by supposing that 
 each pair of adjacent sides is to intercept on a given straight line a 
 segment given in magnitude and direction ; or a segment which is 
 divided by a given point into two parts having a given ratio to one 
 another t. 
 
 * That is to say, two adjacent sides are to cut a given straight line, on which 
 are two given points A, JB, in two other points C, D such that the anharmonic 
 ratio (ABCD) may be equal to a given number. 
 
 f Chasles, Geom. sup., pp. 219-223; andTowNSEND, Modern Geometry (Dublin, 
 1865), vol. ii. pp. 257-275. 
 
CHAPTEE XX. 
 
 POLE AND POLAR. 
 
 250. Let any point 8 be taken in the plane of a conic 
 (Fig. 175), and through it let any number of transversals be 
 drawn to cut the conic in pairs of points A and A\B and B\ 
 C and C\ ... . The tangents a and a' , b and b' } c and c' at these 
 points will, by Arts. 203, 204, intersect in pairs on a fixed 
 straight line s, on which lie also the points of contact of the tan- 
 gents from S to the conic (when the 
 position of 8 is such that tangents 
 can be drawn). Further, the pairs of 
 chords AB'*nd-A'B,AC'm&A' <?,... 
 BC and B'C,... AB and A! B\ 
 AC and A'C\ ... BC and B'C\ ... 
 will intersect on s. Another pro- 
 perty of the straight line s may 
 be noticed. In the complete quad- 
 rangle AA!BB\ each of the straight 
 lines A A! and BB' is divided har- 
 monically by the diagonal point 8 
 and the point where it is cut by 
 the straight line s which joins the 
 other diagonal points (Art. 57); 
 consequently A and A' (and simi- 
 larly B and B n , C and C\ ...) are harmonic conjugates with 
 regard to 8 and the point where A A ' (or BB\ CC\...) is cut 
 by *. 
 
 The straight line s determined in this manner by the point 
 S is called the polar of 8 with respect to the conic ; and, re- 
 ciprocally, the point S is said to be the pole of the straight 
 line s. 
 
 The polar of a given point 8 is therefore at the same time: (l) the 
 
 Fig- 175. 
 
202 
 
 POLE AND POLAR. 
 
 [251 
 
 locus of the points of intersection of tangents to the conic at the pairs 
 of points where it is cut by any transversal through 8 ; (2) the locus 
 of the points of intersection of pairs of opposite sides of quadrangles 
 inscribed in the conic such that their diagonals meet in S; (3) the 
 locus of points taken on any transversal through 8 such that they are 
 harmonically conjugate to 8 with regard to the pair of points in 
 which the transversal is cut by the conic ; (4) the chord of contact of 
 the tangents from 8 to the conic, when 8 has such a 'position 
 that it is possible to draw these * f. 
 
 251. Reciprocally, any given straight line s determines a 
 point S, of which it is the polar. For let A and B (Fig. 176) be 
 any two points on the conic ; the tangents a and b at these points 
 will cut s in two points from which can be drawn two other tan- 
 gents a' and b' to the conic. Let A f and B' be the points of 
 contact of these, and let AA',BB' meet in 8; then the polar of 
 8 will pass through the points aa' and bb\ and must therefore 
 coincide with s. 
 
 If then from any point on s a pair of tangents can be drawn to the 
 conic, their chord of contact will pass through 8. 
 
 Fig. 176 
 
 252. The complete quadrangle AA'BB' and the complete 
 quadrilateral aa'bb' (Fig. 176) have the same diagonal 
 
 * Apollonius, loc. cit, lib. vii. 37; Desargues, loc. ci£.,pp. 164 sqq. ; De la 
 Hire, loc. cit., books i. and ii. 
 
 + (4) follows from (3) by what has been proved in Art. 71. 
 
254] 
 
 POLE AND POLAR. 
 
 203 
 
 triangle (Art. 169). The vertices of this triangle are S, the point 
 of intersection F pf AB and A!B\ and the point of intersection 
 E of AB r and A'B ; its sides are s, the straight line/ joining the 
 points ab and a f b\ and the straight line e joining the points ab f 
 and a'b. Thus if from any two points taken on the straight line s 
 pairs of tangents a and a\ b and b' be drawn to the conic , the 
 diagonals of the quadrilateral aba'b' will pass through S. 
 
 253. The straight lines a, a', b, b' (Fig. 177) form a quadri- 
 lateral circumscribed about the conic, one of whose diagonals is 
 s, and whose other two diagonals 
 meet in S. Thus if from any point 
 on s a pair of tangents be drawn to 
 the conic, they will be harmonically 
 conjugate with regard to s and the 
 straight line joining the point to 8 
 (Art. 56). 
 
 254. If then a conic is given, 
 every point in its plane has its 
 polar and every straight line has its 
 pole*. The given conic, with 
 reference to which the pole and 
 polar are considered, may be 
 called the auxiliary conic. 
 
 I. If a point in the plane of a conic is such that from it 
 two tangents can be drawn to the curve, it is said to lie 
 outside the conic, or to be an external point ; if it is such that 
 no tangent can be drawn, it is said to lie inside the conic, or 
 to be an internal point. If then the pole lies outside the 
 conic (Art. 203, VIII) the polar cuts the curve, and it cuts 
 it at the points of contact of the tangents from the pole to the 
 conic t« 
 
 If the pole lies inside the curve, the polar does not cut the 
 conic. 
 
 II. If a point on the conic itself be taken as pole and a 
 transversal be made to revolve round this point, one of its 
 points of intersection with the conic will always coincide with 
 the pole itself. Since then the polar is the locus of the points 
 where the tangents at these points of intersection meet, and 
 
 * Desargues, loc. cit, p. 190. 
 t See also Art. 250, (4). 
 
204 POLE AND POLAR. [255 
 
 in this case one of the tangents is fixed, it follows that the 
 polar of a point on the conic is the tangent at this point ; or 
 that if the pole is a point on the conic, the polar is the tangent at 
 this point. 
 
 . III. Keciprocally, if every point of the polar lies outside the 
 conic, the pole lies inside the conic ; if the 'polar cuts the 
 conic, the pole is the point where the tangents at the two 
 points of intersection meet ; and if the polar touches the conic, 
 the pole is its point of contact. 
 
 255. If two points are such that the first lies on the polar 
 of the second, then will also the "second lie on the polar of the first. 
 
 Consider Fig. 176; let E be taken as pole and let F 
 be a point lying on the polar of E. If the straight line EF 
 cuts the conic, it will cut it in two points which are harmoni- 
 cally conjugate with regard to E and F (Art. 250 [3] ) ; 
 consequently one of the points E , F will lie inside and 
 the other outside the conic, and by Art. 250 (3) again, if i^be 
 taken as pole, E will be a point on its polar. 
 
 If the straight line EF does not cut the conic, the chord of 
 contact of the tangents from E will pass through F, since this 
 chord is the polar of E \ and therefore by Art. 250 (1) E will 
 lie on the polar of F. 
 
 The above proposition may also be expressed in the follow- 
 ing manner : 
 
 If a straight line f pass through the pole of another straight line 
 e, then will also e pass through the pole off. 
 
 For let E, F be the poles of e, f respectively ; since by 
 hypothesis E lies on the polar of F, therefore F will lie on 
 the polar of E\ that is to say, e will pass through F, the pole 
 of/. 
 
 Two points such as E and F, which possess the property 
 that each lies on the polar of the other, are termed conjugate or 4 
 reciprocal points with respect to the conic. And two straight 
 lines such as e and/, each of which passes through the pole of 
 the other, are termed conjugate or reciprocal lines with respect 
 to the conic. 
 
 The foregoing proposition may then be enunciated as 
 follows : 
 
 If two points are conjugate to one another with respect to a conic, 
 their polars also are conjugate to one another, and conversely. 
 
 / 
 
257] 
 
 POLE AND POLAR. 
 
 205 
 
 256. The same proposition can be put into yet another 
 form, viz. 
 
 Ever?/ point on the polar of a given point E has for its polar a 
 straight line passing through E. 
 
 Every straight line passing through the pole of a given straight 
 line e has for its pole a point lying on e *. 
 
 In other words, if a variable pole F be supposed to describe 
 a given straight line e, the polar of F will always pass through 
 a fixed point E, the pole of the given line ; and conversely, if 
 a straight line f revolve round a fixed point E, the pole of f 
 will describe a straight line e, the polar of the given point E. 
 
 Or again : the pole of a given straight line e is the centre of the 
 pencil formed by the polars of all points on e ; and the polar of a 
 given point E is the locus of the poles of all straight lines passing 
 through Ef. 
 
 257. Problem. Given apoint 
 S, to construct its polar with, 
 respect to a given conic. 
 
 I. Let the conic be determined 
 by five points A,B,C,D,E 
 (Fig. 178). 
 
 Given a straight line s, to con- 
 struct its pole with respect to a 
 given conic. 
 
 I. Let the conic be determined 
 by five tangents a , b , c , d , e (Fig, 
 i?9> 
 
 i Fig. 178. 
 
 Join SA,SB, and find the 
 points A',B' where these cut the 
 conic again respectively (Art. 161, 
 right). The straight line s which 
 joins the point of intersection of 
 AB' and A'B to that of AB and 
 
 Fig. 179. 
 
 From the points sq, , sb draw 
 the second tangents a' ,b f respec- 
 tively to the conic (Art. 161, 
 left). The point S in which the 
 diagonals of the quadrangle aba'b' 
 intersect one another will be 
 
 * Desargues, loc. cit, p. 191. 
 f Poncelet, loc. cit., Art. 195. 
 
206 
 
 POLE AND POLAR. 
 
 [258 
 
 A'B' will be the polar of the the pole of the given straight 
 
 given point (Art. 250 [2] ). line. 
 
 II. Let the conic be determined II. Let the conic be deter- 
 by five tangents a,6,c,d,e mined by five points A, B,C,D,E 
 (Fig. 180). ( ri g- lSl )- 
 
 Fig. 180. 
 
 Through S draw two trans- 
 versals u and v, and construct 
 their poles U and V (as on the 
 right hand side above) ; U V will 
 be the polar of JS (Art. 256). To 
 simplify matters the transversal 
 u may be drawn through the 
 point ab ; if then the second 
 tangent c' be drawn to the conic 
 (Art. 161) from the point uc, U 
 will be the point of intersection 
 of the diagonals of the quadri- 
 lateral acbc'. So too if the 
 transversal v be drawn through 
 the point ac for example, and the 
 second tangent b' be drawn to the 
 conic from the point vb, then V 
 will be the point of intersection 
 of the diagonals of the quadri- 
 lateral abcb'. 
 
 .. 
 
 Fig. 181. 
 
 On s take two points U and 
 V, and construct their polars u 
 and v (as on the left hand side 
 above) ; the point uv will be the 
 pole of s (Art. 256). To simplify 
 matters the point £7 may be taken 
 on the straight line AB ; if then 
 UC be joined, and the second 
 point C in which it meets the 
 conic be constructed, u will be the 
 straight line joining the points of 
 intersection of the pairs of oppo- 
 site sides of the quadrangle 
 ACBC. So too if V be taken on 
 the straight line AC for example, 
 and VB be joined, and its second 
 point of intersection B' with the 
 conic be constructed, then v will 
 be the straight line joining the 
 points of intersection of the pairs 
 of opposite sides of the quadrangle 
 ABCB'. 
 
 258, 
 
 Let E and F (Fig. 182) be a pair of conjugate points 
 
259] 
 
 POLE AND POLAB. 
 
 207 
 
 and let G be the pole of EF; then G will be conjugate both to 
 E and to F, so that the three points E, F, G are conjugate to one 
 another two and two. Every side therefore of the triangle 
 EFG is the pole of the opposite vertex, and the three sides are 
 conjugate lines two and two. 
 
 A triangle such as EFG, in which each vertex is the pole 
 of the opposite side with regard to a given conic is called a 
 self-conjugate or self-polar triangle with regard to the conic. 
 
 259. To construct a triangle self-conjugate with regard to a given 
 conic. 
 
 One vertex E (Fig. 182) maybe taken arbitrarily; construct its 
 
 Fig. 182. 
 
 polar, take on this polar any point F, and construct the polar of F. 
 This last will pass through E, since E and F are conjugate points ; 
 if G be the point where it cuts the polar of E, then E and G , 
 F and G, will be pairs of conjugate points ; and therefore EFG is a 
 self- conjugate triangle. 
 
 In other words : take any point E and draw through it any two 
 transversals to cut the conic in A and D, B and C respectively ; join 
 AC, BD, meeting in F, and AB , CD meeting in G ; then EFG is a 
 self- conjugate triangle. 
 
 Or again, one side e may be taken arbitrarily, and its pole E con- 
 structed ; if through E any straight line / be drawn, and its pole 
 
208 POLE AND POLAK. [260 
 
 (which will lie on e) be constructed and joined to the pole of e by the 
 straight line g, then efg will be a triangle such as is required ; for the 
 straight lines e ,/, g are conjugate two and two. 
 
 Thus, after having taken the side e arbitrarily, we may proceed as 
 follows : take two points on e and from them draw pairs of tangents 
 a and d, b and c, to the conic ; join the points ac , bd by the straight 
 line/, and the points ab,cd by the straight line g ; then will efg be 
 a self-conjugate triangle. 
 
 260. From what has been said above the following property 
 is evident : 
 
 The diagonal points of the complete quadrangle formed by any 
 four points on a conic are the vertices of a triangle which is self- 
 conjugate with regard to the conic. And the diagonals of the 
 complete quadrilateral formed by any four tangents to a conic 
 are the sides of a triangle which is self-conjugate with regard to the 
 conic *. 
 
 Or, in other words : 
 
 The triangle whose vertices are the diagonal points of a complete 
 quadrangle is self-conjugate with regard to any conic circumscribing 
 the quadrangle. And the triangle whose sides are the diagonals of a 
 complete quadrilateral is self-conjugate with regard to any conic 
 inscribed in the quadrilateral. 
 
 261. From the properties of the circumscribed quadrilateral and 
 the inscribed quadrangle (Arts. 166 to 172) it follows moreover 
 that : 
 
 If EFG (Fig. 182) is a triangle self-conjugate with regard to 
 a given conic, and ABC is a triangle inscribed in the conic, 
 such that two of its sides CA, AB pass through two of the 
 vertices F, G respectively of the other triangle, then will the re- 
 maining side BC pass through the remaining vertex E, and every 
 side of the inscribed triangle will be divided harmonically 'by the 
 corresponding vertex of the self- conjugate triangle and the side which 
 joins the other two vertices of it. 
 
 The three straight lines EA,FB , GC meet in one point D on the 
 conic ; the two triangles are therefore in perspective, and the three 
 pairs of corresponding sides FG and BC, GE and CA, EF and AB, 
 will meet in three collinear points. 
 
 Hence it follows that a self-conjugate triangle EFG and a point A 
 of a conic determine an inscribed quadrangle ABCD, whose diagonal 
 
 * Desargues, he. cit., p. 186. 
 
263] POLE AND POLAR. 209 
 
 triangle is EFG. The points B ,C ,D are those in which the 
 straight lines AG, AF, AE cut the conic again. 
 
 The enunciation of the correlative property is left to the student *. 
 
 262. Of the three vertices of the triangle EFG, one always 
 lies inside the conic, and the two others outside it. For if E 
 is an internal point, its polar does not cut the conic, and con- 
 sequently F and G are both external to the conic. If, on the 
 other hand, E is an external point, its polar cuts the conic, and 
 F and G are harmonic conjugates with regard to the two 
 points of intersection ; of the two points F and G therefore, 
 one must be internal and the other external to the conic. 
 
 From this property and that of Art. 254, 1, we conclude that 
 of the three sides of any self-conjugate triangle, two always 
 cut the curve, and the third does not. 
 
 263. (1). On every straight line there are an infinite number of 
 pairs of points which are conjugate to one another with respect to a 
 given conic, and these form an involution f. 
 
 (2). Through every point pass an infinite number of pairs of 
 straight lines which are conjugate to one another with respect to a 
 given conic, and these form an involution f. 
 
 (3). If a point describes a range, its polar with respect to a given 
 conic will trace out a pencil which is projective with the given range. 
 And, conversely, if a straight line describes a pencil, its pole with 
 respect to a given conic will trace out a range which is projective with 
 the given pencil %. 
 
 To prove these theorems, consider Fig. 183, and suppose 
 in it the conic and the three points A,B ,G to be given. 
 Let the point C be supposed to move along the conic. 
 Then the rays AC, EC will trace out two pencils which are 
 projective with one another (Art. 149 [l]) ; and therefore the 
 ranges in which these pencils cut the polar of G will be pro- 
 jective also ; that is to say, the conjugate points i^and E will 
 describe two collinear projective ranges. In these ranges the 
 points F and E correspond to one another doubly, since the 
 polar of E passes through F, and the polar of F passes 
 through E ; consequently the ranges in question are in in- 
 volution. 
 
 From what has been said it follows also that the pairs of 
 
 * Poncelet, loe. cit., p. 104. + Desargues, loc. cit., pp. 192, 193. 
 
 X Mobius, Baryc. Cole, § 290. 
 
 P 
 
210 
 
 POLE AND POLAR. 
 
 [264 
 
 conjugate lines GF, GE in like manner form an involution, 
 and that the range of poles E,F,... is projective with the 
 pencil of polars GF, GE,.... 
 
 264. If the straight line incuts the conic, the two points of 
 
 Fig. 183. 
 
 intersection are the double points of the involution formed by 
 the pairs of conjugate poles. The centre of the involution lies 
 on the diameter which passes through the pole G of the given 
 straight line (Art. 290). 
 
 If the point G is external to the conic, the tangents from G 
 to the conic are the double rays of the involution formed by 
 the pairs of conjugate polars. 
 
 Consequently (Art. 125): 
 
 A chord of a conic is harmonically divided by any pair of points 
 lying on it which are conjugate with respect to the conic ; and 
 
 The pair of tangents drawn from any point to a conic are har- 
 monic conjugates with respect to any pair of straight lines meeting in 
 the given point which are conjugate with respect to the conic. 
 
 If the point G lies at infinity, the pairs of conjugate straight 
 lines form an involution of parallel rays, the central ray of 
 which is a diameter of the conic (Arts. 129, 276). 
 
 265. Theorem. If tivo complete quadrangles have the same 
 diagonal points, their eight vertices lie either four and four on two 
 straight lines or else they all lie on a conic. 
 
265 
 
 POLE AND POLAR. 
 
 211 
 
 Fig. 184. 
 
 Let ABCD and A'B 'CD' (Fig. 184) be two quadrangles 
 
 having the same diagonal points E ,F,G; so that 
 BC,AD, B'C, A'D' all meet in E, 
 CA , BB , C'A' , B'D' „ „ F, 
 AB, CD, A'B', CD' „ „ G. 
 (1). In the first place let the eight vertices be such that some 
 
 three of them are collinear. Suppose 
 
 for example that A ' lies on AB. Since 
 
 AB and A'B' meet in G, therefore 
 
 B' also must lie on AB; and since 
 
 the straight lines GE , GF are har- 
 monically conjugate with regard both 
 
 to AB, CD and to A'B' ,CD', and 
 
 AB coincides with A'B', therefore also 
 
 CB coincides with CD'. Thus the 
 
 four points C,D,C,D' are collinear, 
 
 and the eight points A,B,C,D, A',B',C,D' lie four and four 
 
 on two straight lines. 
 
 (2). But if this case be excluded, *. e. if no three of the eight 
 
 vertices He in a straight line, 
 
 then a conic can be drawn 
 
 through any five of them. Let 
 
 a conic be drawn through 
 
 A,B,C,D,A' (Fig. 185); then 
 
 shall B',C,D' lie on the 
 
 same conic. For since E,F,G 
 
 are the diagonal points of 
 
 the inscribed quadrangle 
 
 ABCD, G is the pole of EF, 
 
 and therefore G and the 
 
 point where its polar EF meets the transversal GB'A' are 
 
 harmonically conjugate with regard to the points where this 
 
 transversal cuts the conic. But one of these last points is 
 
 A', therefore the other is B' '; for since E,F,G are also the 
 
 diagonal points of the quadrangle A'B' CI/, the points A' 
 
 and B' are harmonically conjugate with regard to G and the 
 
 point where EF cuts A'B'. In a similar manner it can be 
 
 shown that C and D' also lie on the same conic. The eight 
 
 vertices A,B,C ,D,A',B',C,D' therefore lie on a conic, and 
 
 the proposition is proved. 
 
 p % 
 
 Fig. 185. 
 
212 POLE AND POLAK. [266 
 
 Since the straight lines AB and A'B' meet in G, therefore 
 A A' and BB', as also AB' and A'B, will meet on EF, the polar 
 of G. This property gives the means of constructing the 
 point B f when the points A, B,C ,B,A' are given. The point 
 C will then be found as the point of intersection of A'F and 
 B'E, and the point B' as that of B'F, A'F, and C'G. 
 
 266. Suppose now that two conies are given which are 
 inscribed in the same quadrilateral. Let the four common 
 tangents which form this quadrilateral be a, b, c, d, and let their 
 points of contact with the conies heA,B,C,B and A', B',C',B f 
 respectively. By the theorem of Art. 169, the triangle formed 
 by the diagonals of the circumscribed quadrilateral abed has for 
 its vertices the diagonal points of the inscribed quadrangle 
 ABCB and also those of the inscribed quadrangle A'B'C'B'; 
 thus ABCB and A'B'C'B' have the same diagonal points. 
 Accordingly, by the theorem of Art. 265, the eight points 
 A,B, C , B , A',B', C',B' lie either four and four on two straight 
 lines, or they lie all on a conic. 
 
 267. By writing, as usual, line for point, and point for line, 
 the propositions correlative to those of Arts. 265 and 266 can 
 be proved, viz. 
 
 If two complete quadrilaterals have the same three diagonals, their 
 eight sides either pass four and four through two points, or else they 
 all touch a conic. 
 
 If two conies intersect in four points, the eight tangents to them 
 at these points either pass four and four through two points, or they 
 all touch a conic *. 
 
 268. If there be given the diagonal points E, F, G and one 
 vertex A of a quadrangle ABCB, the quadrangle is completely 
 determined, and can be constructed. For B is that point on 
 AE which is harmonically conjugate to A with respect to E 
 and the point where FG cuts AE ; so C is that point on AF 
 which is harmonically conjugate to A with respect to jF'and 
 the point where GE cuts AF; and B is that point on AG 
 which is harmonically conjugate to A with respect to G and 
 the point where incuts AG. 
 
 But if there be given the diagonal points E, F, G of a 
 quadrangle ABCB and the conic with respect to which EFG 
 is a self-conjugate triangle, the quadrangle is not completely 
 * Staudt, loc. cit., p. 293. 
 
270] 
 
 POLE AND POLAR. 
 
 213 
 
 determined. For we may take arbitrarily on the conic a point 
 A as one vertex of the quadrangle ABCB ; then the other 
 vertices B, C, B are the second points of intersection of the 
 conic with the straight lines AG, AF, AE respectively. Hence 
 it follows that : 
 
 All conies with respect to which a given triangle EFG is self- 
 conjugate, and which pass through a fixed point A, pass also through 
 three other fixed points B ,C ,D. 
 
 269. Peoblem. To construct a conic passing through two given 
 points A and A', and with respect to which a given triangle EFG 
 shall be self -conjugate. 
 
 Solution. Construct, in the manner just shown, the three points 
 B, C, D which form with A a complete quadrangle having E , F, and G 
 for its diagonal points. Five points A, A', B , C, D on the conic are 
 then known, and by means of Pascal's theorem any number of other 
 points on it may be found. Or we may construct the three points 
 B', C / ,D / which form with A' a complete quadrangle having E, F, and 
 G for its diagonal points ; the eight points A, B,C, D , A', B' ', C', D f 
 will then all lie on the conic required. 
 
 270. Consider again the problem (Art. 218) of describing a conic 
 to touch four given straight lines a,b,c,d and to pass through a given 
 point S (Fig. 186). The diagonals 
 
 of the quadrilateral abed form a 
 triangle EFG which is self- conju- 
 gate with regard to the conic; 
 consequently, if the three points 
 P , Q , R be constructed which 
 together with S form a quadrangle 
 having 2? , F, and G for its diagonal 
 points, the three points so con- 
 structed will lie also on the 
 required conic. Now it may 
 happen that there is no conic 
 which satisfies the problem, or 
 again there may be two conies which satisfy it (Art. 218, right); 
 in the second case, since the construction for the points P, Q, R 
 is linear, the two conies will both pass through these points. Thus : 
 
 If two conies inscribed in the same quadrilateral abed %)ass through 
 the same point S, they will intersect in three other points P, Q, R ; and 
 the triangle formed by the diagonals of tlie circumscribed quadrilateral 
 abed will coincide with that formed by the diagonal points of the 
 inscribed quadrangle PQRS. 
 
 In order to find a construction for the points P, Q, R, consider 
 
 Fig. 1 86. 
 
214 POLE AND POLAR. [271 
 
 the point P for example which lies on ES (Fig. 186). It is seen 
 that the segment SP must be divided harmonically by E and its 
 polar FG (Art. 250) ; but the diagonal (ab) (cd) which passes through 
 E is also divided harmonically, at E and F. We have therefore 
 two harmonic ranges, which are of course projective (Art. 51) and 
 which are in perspective since they have a self-corresponding point at 
 E ; therefore the straight lines P (ab), S (cd), and FG, which join the 
 other pairs of corresponding points, will meet in a point (Art. 80). 
 We must therefore join S to one extremity of one of the diagonals 
 passing through E, for example to the point cd, and take the point 
 where the joining line meets FG. This point, when joined to the 
 other extremity ab of the diagonal, will give a straight line which 
 will meet ES in the required point P*. 
 
 271. The propositions and constructions correlative to those of 
 the last three Articles, and which will form useful exercises for the 
 student, are the following : 
 
 All conies with respect to which a given triangle is self-conjugate, 
 and which touch a fixed straight line, touch three other fixed straight lines. 
 
 To construct a conic to touch two given straight lines, and with 
 respect to which a given triangle shall be self-conjugate. 
 
 If two conies circumscribing the same quadrangle have a common 
 tangent, they have three other common tangents. 
 
 To construct the three remaining common tangents to two conies 
 which pass through four given points and touch a given straight line 
 (Art. 218, left). 
 
 Fig. 187. 
 
 272. Let ABCD (Fig. 187) be a complete quadrilateral whose 
 diagonal points are E, F, and G. Let also 
 
 L and P be the points where FG meets AD and BG respectively. 
 J/" and Q „ „ GE „ ED and CA 
 
 -tfandP „ „ EF „ GD and AB 
 
 The six points so obtained are the vertices of a complete quad- 
 rilateral. For the triangle EFG is in perspective with each of the 
 
 * Bbianchon, loc. cit., p. 45 ; Maclaurin, De tin. Georn., § 43. 
 
274] POLE AND POLAR. 215 
 
 triangles ABC, DCB, CD A, BAD, the centres of perspective being 
 D, A, B, C respectively ; whence it follows that the four triads of 
 points PQR, PMN, LQN, and LMR lie on four straight lines (the 
 axes of perspective). 
 
 These four axes form a quadrilateral whose diagonals LP, MQ, NR 
 form the triangle EFG. Accordingly, a conic inscribed in the 
 quadrangle ABCD and passing through L will pass also through 
 N , P, and R (Art. 270) ; similarly a conic can be inscribed in 
 the quadrangle ABDC to pass through R, M, N, and Q', and 
 a conic can be inscribed in the quadrangle ACBD to pass through 
 Q, P, M, and L. 
 
 It will be seen that for each of these conies the four tangents 
 shown in the figure (the four sides of the complete quadrangle ABCD) 
 are harmonic, and that the same will therefore be the case with 
 regard to their points of contact (Arts. 148, 204). For take one of 
 the sides of the quadrangle, for example AB ; a consideration of the 
 complete quadrangle CDEF shows that this side is harmonically 
 divided in R and G. Now the points A, B, G are the points of 
 intersection of the tangent AB with the other three tangents, and R 
 is the point of contact of AB ; therefore the four tangents are cut by 
 any other tangent to the conic in four harmonic points *. 
 
 273. If ABCD is a parallelogram, the points E, G, M, Q pass off 
 to infinity, and LNPR also becomes a parallelogram. Of the three 
 conies considered above the first will in this case be an ellipse which 
 touches the sides of the parallelogram ABCD at their middle points ; 
 the second a hyperbola which touches the sides AB and CD at their 
 middle points and has AC and BD for asymptotes ; and the third a 
 hyperbola having the same asymptotes and touching the sides AD 
 and BC at their middle points. 
 
 274. From that corollary to Brianchon's theorem which has 
 reference to a quadrilateral circumscribed about a conic 
 (Art. 172) we have already, in Art. 173, deduced a method for 
 the construction of tangents to a conic when we are given 
 three tangents a , b , c and the points of contact B , C of two 
 of them (Fig. 183). We take any point E on BC and join it 
 to the points ab,ac by the straight lines g, f, respectively; if 
 the point in which g meets c be joined to that in which /meets 
 b, the joining line d will be a tangent to the conic. 
 
 The four tangents a , b , , d form a complete quadrilateral 
 two of whose diagonals # = (ah) (cd) and/= (ac) (bd) intersect 
 
 * Steiner, loc. cit., p. 160, § 43, 4; Collected Works, vol. i. p. 347; Staudt, 
 Beitrage zur Geometrie der Laye, Art. 329. 
 
216 POLE AND POLAK. [275 
 
 in E\ therefore also (Art. 172) the chords of contact AD and 
 BC of the tangents a and d, b and c respectively will intersect 
 in E. The straight lines joining E to the points ab and ac, 
 being two of the diagonals of the quadrilateral abed, are con- 
 jugate lines with respect td the conic ; consequently : 
 
 If a triangle abe is circumscribed about a conic, the straight lines 
 which join two of its vertices ab and ac to any point E on the polar 
 of the third vertex be are conjugate to one another with respect to the 
 conic. 
 
 And conversely : 
 
 If two straight lines (c and b) touch a conic, any two conjugate 
 straight lines (fandg) drawn from any point (E) on their chord of 
 contact will cut the two given tangents in points such that the 
 straight line (a) joining them touches the conic. 
 
 275. Let us now investigate the correlative property. Sup- 
 pose three points A , B , C on a conic to be given, and the 
 tangents b , c at two of these points (Fig. 183). If a straight 
 line e drawn arbitrarily through the point be cut AB in G and 
 AC in F; then if GC and FB be joined they will intersect in 
 a point D lying on the conic. 
 
 The four points A,B,B,C form a complete quadrangle 
 two of whose diagonal points lie on e\ therefore (Art. 166) 
 
 the point be and the point of 
 intersection of the tangents at 
 A and B will lie on e. The 
 points G and F, being two of the 
 diagonal points of the quadrila- 
 teral ABCB, are conjugate with 
 respect to the conic; consequently 
 Fi l8g If a triangle ABC (Fig. 188) is 
 
 inscribed in a conic, the points F 
 and G in which two of the sides are cut by any straight line drawn 
 through the pole S of the third side are conjugate to one another 
 with respect to the conic. 
 And conversely : 
 
 If two given points (B , C) on a conic be joined to two conjugate 
 points (G ,F) which are collinear with the pole (S) of the chord (BC) 
 joining the given points, then the joining lines will intersect in a 
 point (A) lying on the conic. 
 
CHAPTEE XXL 
 
 THE CENTRE AND DIAMETERS OF A CONIC. 
 
 Fig. 189. 
 
 276. Let an infinitely distant point be taken as pole, 
 and through it let a transversal be drawn (Fig. 1 89) to cut the 
 conic in two points A and 
 A'. The segment A A' will 
 be harmonically divided by 
 the pole and the point where 
 it is cut by the polar (Art. 
 250); this point will there- 
 fore be the middle point of 
 AA' (Art. 59). That is to 
 say: 
 
 If any number of parallel chords of a conic be drawn, the locus of 
 their middle points is a straight line ; and this straight line is the 
 polar of the point at infinity in which the chords intersect *. 
 
 277. This straight line is termed the diameter of the chords 
 which it bisects. If the diameter meets the conic in two 
 points, these will be the points of contact of the tangents 
 drawn to the conic from the pole, i. e. of those tangents which 
 are parallel to the bisected chords. If the tangents at the ex- 
 tremities A and A r of one of these chords be drawn, they will 
 meet in a point on the diameter. If A A ' and BB r are two of 
 the bisected chords, the straight lines AB and A f B\ AB / and 
 A'B will intersect in pairs on the diameter (Art. 250). 
 
 If, conversely, from a point on the diameter can be drawn 
 a pair of tangents a and a' to the conic, their chord of contact 
 A A' will be bisected by the diameter; and if through the 
 same point there be drawn the straight line which is harmoni- 
 cally conjugate to the diameter with respect to the two 
 
 * Apollonius, Conic., lib. i. 46, 47, 48 ; lib. ii. 5, 6, 7, 28-31, 34-37. 
 
218 THE CENTRE AND DIAMETERS OP A CONIC. [278 
 
 tangents, this straight line will be parallel to the bisected 
 chords. If from two points on the diameter there be drawn 
 two pairs of tangents a and a\ b and b', the straight line join- 
 ing the points ab and a'U and that joining the points ab K and 
 a'b will both be parallel to the bisected chords (Art. 252). 
 
 278. To each point at infinity, that is, to each pencil of 
 parallel rays, corresponds a diameter. The diameters all pass 
 through one point ; for they are the polars of points lying on 
 one straight line, viz. the straight line at infinity; the point 
 in which the diameters intersect is the pole of the straight 
 line at infinity (Art. 256). 
 
 279. Since every parabola is touched by the straight line at 
 infinity, and the point of contact is the pole of this straight 
 line (Art. 254, II), it follows (Art. 278) that all diameters of a 
 parabola are parallel to one another (they all pass through the 
 point at infinity on the curve) ; and conversely, every straight 
 line which cuts a parabola at infinity is a diameter of the curve. 
 
 280. If 8 is any point from which a pair of tangents a and 
 a' can be drawn to the conic (Fig. 189), the chord of contact 
 AA', the polar of 8, will be bisected at B by the diameter 
 which passes through 8 ; for 8 and the point at infinity on 
 AA' are conjugate points with respect to the conic. If the 
 diameter cuts the curve in M and M' ? the tangents at these 
 points are parallel to AA', and MM' is divided harmonically 
 by the pole 8 and the polar AA' (Art. 250). 
 
 If then the conic is a parabola (Fig. 190) the point 31' moves 
 off to infinity, and therefore M is the 
 middle point of the segment SR ;' thus 
 
 The straight line which joins the middle 
 point of a chord of a parabola to the pole 
 of the chord is bisected by the curve *. 
 281. When the conic is not a parabola, 
 Fi x the straight line at infinity is no longer 
 
 a tangent to the curve, and consequently 
 the pole of this straight line, or the point of intersection of the 
 diameters, is a point lying at a finite distance. Since any two 
 points on the conic which are collinear with the pole are 
 separated harmonically by the pole and the polar (Art. 250), the 
 pole will lie midway between the two points on the curve 
 
 * Apollonius, loc. cit., lib. i. 35. 
 

 284] THE CENTRE AND DIAMETERS OF A CONIC. 
 
 when the polar lies at infinity. Every chord of the conic 
 therefore which passes through the pole of the straight line at 
 infinity is bisected at this point. 
 
 On account of this property the pole of the straight line at 
 infinity or the point in which all the diameters intersect is 
 called the centre of the conic. 
 
 282. Applying the properties of poles and polars in general 
 (Arts. 250—253) to the case of the 
 centre and the straight line at in- 
 finity, it is seen (Fig. 191) that : 
 
 If A and A' are any pair of points 
 on the conic collinear with the 
 centre, the tangents at A and A'~ 
 are parallel. rig -g 
 
 If A and A',B and B' are any two 
 pairs of points on the conic which are collinear with the 
 centre, the pairs of chords AB and A'B', AB' and A' B are 
 parallel, so that the figure ABA'B' is a parallelogram. 
 
 If a and V are any pair of parallel tangents, their chord 
 of contact passes through the centre, as also does the straight 
 line lying midway between a and a' and parallel to both. 
 
 If a and a', b and V are any two pairs of parallel tangents, the 
 straight line joining the points ab and a'b' and that joining the 
 points ab' and a'b both pass through the centre ; in other 
 words, if aba'b' is a parallelogram circumscribed to the conic, 
 its diagonals intersect in the centre. 
 
 283. If the conic is a hyperbola, the straight line at in- 
 finity cuts the curve ; consequently the centre is a point 
 exterior to the curve (Art. 254, I) in which intersect the tan- 
 gents at the infinitely distant points, i.e. the asymptotes 
 (Fig. 197). 
 
 If the conic is an ellipse, the straight line at infinity does 
 not cut the curve ; consequently the centre is a point inside 
 the curve (Figs. 191, 192). 
 
 284. Two diameters of a central conic (ellipse or hyper- 
 bola*) are termed conjugate when they are conjugate straight 
 
 * In the case of the parabola there are no pairs of conjugate diameters ; for 
 since the centre lies at infinity, the diameter drawn parallel to the chords which 
 are bisected by a given diameter must coincide always with the straight line at 
 infinity. 
 
220 THE CENTRE AND DIAMETERS OP A CONIC. [285 
 
 lines with respect to the conic, i. e. when each passes through 
 the pole of the other (Art. 255). 
 
 Since the pole of a diameter is the point at infinity on any 
 
 of the chords which the diameter 
 
 bisects, it follows that the diameter 
 
 V conjugate to a given diameter b is 
 
 parallel to the chords bisected by b ; 
 
 conversely, V bisects the chords which 
 
 are parallel to b*. 
 
 Fig. 192. * Any two conjugate diameters form 
 
 with the straight line at infinity 
 
 a self-conjugate triangle (Art. 258), of which one vertex is 
 
 the centre of the conic and the other two are at infinity. 
 
 Since in a self-conjugate triangle two of the sides cut the 
 conic and the third side does not (Art. 262), and since the 
 straight line at infinity cuts a hyperbola but does not cut an 
 ellipse, it follows that of every two conjugate diameters of a 
 hyperbola one only cuts the curve, while an ellipse is cut by 
 all its diameters. 
 
 285. Pkoblbm. Given jive points A ,B ,C ,D ,E on a conic, to 
 determine its centre. 
 
 Solution. We have only to repeat the construction given in Art. 
 257 5 II (right), assuming the straight line s to lie in this case at 
 infinity. Draw through G a parallel to AB, and determine the point C 
 in which this parallel meets the conic again ; draw also through B a 
 parallel to AC, and determine the point B f in which this parallel 
 meets the conic again. The straight line u which joins the points of 
 intersection of the pairs of opposite sides of the quadrangle ACBG\ 
 and the straight line v which joins the points of intersection of the 
 pairs of opposite sides of the quadrangle ABGB' , will meet in the 
 required point 0, which is the pole of the straight line at infinity 
 and therefore the centre of the conic t. 
 
 The straight lines u and v are the diameters conjugate respec- 
 tively to AB and AG', if through there be drawn the straight 
 lines v! , v' parallel to AB , AG respectively, then u and u', v and v' 
 will be two pairs of conjugate diameters. 
 
 If the conic is determined by five tangents, its centre may be 
 found by a method which will be explained further on (Art. 319). 
 
 * Apollonius, loc. cit., lib. ii. 20. 
 
 t If w and v should be parallel, the conic is a parabola, whose diameters are 
 parallel to u and v. 
 
288] THE CENTRE AND DIAMETERS OF A CONIC. 221 
 
 286. Four tangents to a conic form a complete quadrilateral 
 whose diagonals are the sides of a self-conjugate triangle (Art. 
 260). Suppose the four tangents to be parallel in pairs (Fig. 
 191); then one diagonal will pass to infinity, and con- 
 sequently the other two will be conjugate diameters (Art. 
 284); thus: 
 
 The diagonals of any parallelogram circumscribed to a conic are 
 conjugate diameters. 
 
 The points of contact of the four tangents form a complete 
 quadrangle whose diagonal points are the vertices of the self- 
 conjugate triangle (Arts. 169, 260). In the case where the 
 four tangents are parallel in pairs one of these diagonal points 
 is the centre of the conic, and the other two lie at infinity. 
 That is to say, the six sides of the quadrangle are the sides 
 and diagonals of an inscribed parallelogram ; its sides are 
 parallel in pairs to the diagonals of the circumscribed paral- 
 lelogram, and its diagonals intersect in the centre of the 
 conic. 
 
 287. Conversely, let ABA!B' (Fig. 191) be any inscribed 
 parallelogram, and consider it as a complete quadrangle. 
 Since its three diagonal points must be the vertices of a self- 
 conjugate triangle, one of them, will be the centre of the conic, 
 and the other two will be the points at infinity on two conju- 
 gate diameters ; thus: 
 
 In any parallelogram inscribed in a conic, the sides are parallel to 
 two conjugate diameters and the diagonals intersect in the centre. 
 
 Or again : 
 
 The chords which join a variable point A on a conic to the ex- 
 tremities B and B f of a fixed diameter are always parallel to two 
 conjugate diameters. 
 
 288. The following conclusions can be drawn at once, from 
 Art. 286. 
 
 Any two parallel tangents (a and a') are cut by any pair of 
 conjugate diameters in two pairs of points, the straight lines 
 connecting which give two other parallel tangents (b and b'). 
 
 If from the extremities (A and A') of any diameter straight 
 lines be drawn parallel to any two conjugate diameters, they 
 will meet in two points on the curve, and the chord joining 
 these will be a diameter. 
 
 Given any two parallel tangents a and a' whose points of 
 
222 THE CENTRE AND DIAMETERS OF A CONIC. [289 
 
 contact are A and A' respectively, and any third tangent b ; if 
 from A a parallel be drawn to the diameter passing through 
 a'b this parallel will meet the tangent b at its point of con- 
 tact B. 
 
 Given any two parallel tangents a and a' whose points of 
 contact are A and A' respectively, and another point B on the 
 conic ; the tangent at B will meet the tangent a in a point 
 lying on that diameter which is parallel to A'B, and it will 
 meet the tangent a' in a point lying on that diameter which is 
 parallel to AB. 
 
 289. Suppose now that the conic is a circle (Fig. 193), i.e. 
 the locus of the vertex of a right angle 
 AMB whose arms AM and BM turn round 
 fixed points A and B respectively. These 
 arms in moving generate two equal and 
 consequently projective pencils ; therefore 
 the tangent at A will be the ray of the 
 first pencil which corresponds to the ray 
 Flg ' m ' BA of the second (Art. 143). The tangent 
 
 at A must therefore make a right angle with BA ; and simi- 
 larly the tangent at B will be perpendicular to AB. The 
 tangents at A and B are therefore parallel, and consequently 
 AB is a diameter, and the middle point of AB is the centre 
 of the circle (Art. 282). 
 
 I. Since AB is a diameter, the straight lines AM and BM 
 will be parallel to a pair of conjugate diameters, whatever be 
 the position of M (Art. 287) ; therefore : 
 
 Every pair of conjugate diameters of a circle are at right angles to 
 one another. 
 
 II. Since the diagonals of any parallelogram circumscribed 
 about the circle are conjugate diameters, they will inter- 
 sect at right angles ; thus any parallelogram which circumscribes a 
 circle must be a rhombus. 
 
 III. In a rhombus, the distance between one pair of opposite 
 sides is equal to the distance between the other pair ; thus by 
 allowing one pair of opposite sides of the circumscribed rhom- 
 bus to vary while the other pair remain fixed, we see that the 
 distance between two parallel tangents is constant. This 
 distance is the length of the straight line joining the points of 
 contact of the tangents, for this straight line, which is a 
 
290] THE CENTKE AND DIAMETERS OF A CONIC. 223 
 
 diameter, cuts at right angles the conjugate diameter and the 
 tangents parallel to it ; therefore all diameters of a circle are 
 equal in length. 
 
 IV. The diagonals of any inscribed parallelogram are 
 diameters ; but all diameters are equal in length ; therefore 
 any parallelogram inscribed in a circle must be a rectangle. 
 
 290. Returning to the general case where the conic is any 
 whatever (Fig. 189), let s be any straight line and 8 its pole. 
 All chords parallel to s will be bisected by the diameter 
 passing through 8 ; for since 8 and the point at infinity on s 
 are conjugate points with respect to the conic, the polar of the 
 second point will pass through the first. We may also say that : 
 
 If a diameter pass through a fixed point, the conjugate diameter 
 will be parallel to the polar of this point. 
 
 I. If the diameter passing through 8 cuts the conic in two 
 points M and M', then MM' is divided harmonically by the 
 pole 8 and the polar s*\ thus if is the middle point of MM', 
 that is, the centre of the conic, and R the point where MM' is 
 cut by the polar s, we have (Art. 69) 
 
 OS.OR= OM 2 . 
 
 II. From this follows a construction for the semi-diameter 
 conjugate to a chord AA' of a conic, having given the extremities 
 A and A' of the chord and three other points on the conic. 
 We determine (Art. 285) the centre 0, and join it to the 
 middle point R of AA' ; we then construct the tangent at 
 A and take its point of intersection S with OR. If now a 
 point M be taken on OR such that OM is the mean propor- 
 tional between OR and OS, then OM will be the required 
 semi-diameter. 
 
 If lie between R and S, so that OR and OS have opposite 
 signs, the diameter OR will not cut the conic ; but in this case 
 also the length OM, the mean proportional between OR and 
 OS, is called the magnitude of the semidiameter conjugate to the 
 chord AA'. 
 
 An analogous definition can be given for the case of any 
 straight line (Art. 294). 
 
 III. If the conic is a circle, the perpendicularity of the 
 conjugate diameters in this case gives the theorem : 
 
 * Apollonius, loc. cit„ i. 34, 36 ; ii. 29, 30. 
 
224 THE CENTRE AND DIAMETERS OF A CONIC. [291 
 
 The polar of any point with respect to a circle is perpendicular to 
 the diameter which passes through the pole. 
 
 291. From this last property can be derived a second de- 
 monstration of the very important theorem of Art. 263 (3), viz. 
 
 The range formed by any number of collinear points, and the 
 pencil formed by their polars with respect to any given conic, are two 
 projective forms. 
 
 Consider as poles the points A,B ,C ,... lying on a straight 
 line, s (Fig. 194); the diameters (A,B,C ,...) obtained by 
 
 joining them to the centre of 
 the conic will form a pencil 
 which is in perspective with the 
 
 range A,B ,C , Another pencil 
 
 will be formed by the polars 
 a,b,c,... of the points A, B, C , ... 
 since these polars all pass through 
 Fi a point S (Art. 256), the pole of 
 
 s. If now the conic is a circle, 
 then by the property proved in Art. 290, III, the straight lines 
 0(A,B,C,...) are perpendicular respectively to a, b,c , . . . ; and 
 the two pencils are in this case equal. The range of poles 
 A,B,C,... is therefore projective with the pencil of polars 
 a , b , c , . . . with regard to a circle. 
 
 This result may now be extended and shown to hold not 
 only for a circle but for any conic. For any given conic may 
 be regarded as the projection of a circle (Arts. 149, 150). In 
 the projection, to harmonic forms correspond harmonic forms 
 (Art. 51); consequently to a point and its polar with regard 
 to the conic will correspond a point and its polar with regard 
 to the circle, and to a range of poles and the pencil formed 
 by their polars with regard to the conic will correspond a 
 range of poles and the pencil formed by their polars with 
 regard to the circle. But it has been seen that this range and 
 pencil are projective in the case of the circle ; therefore the 
 same is true with regard to the range and pencil in the case 
 of the conic, and the theorem is proved. 
 
 292. Theoeem. A quadrangle is inscribed in a conic, and a point 
 is taken on the straight line which joins the points of intersection of the 
 pairs of opposite sides. If from, this point be drawn tlie straight lines 
 connecting it with the two pairs of opposite vertices, and also a pair of 
 
292] 
 
 THE CENTRE AND DIAMETERS OP A CONIC. 
 
 225 
 
 Fig. 195. 
 
 tangents to the conic , these straight lines will be three conjugate pairs of 
 an involution. 
 
 Let ABCD be a simple quadrangle inscribed in a conic (Fig. 
 195); let the diagonals AC , BD meet in F, and the pairs of opposite 
 sides BC ,AD and AB ,CD in E 
 and G respectively ; the points 
 E,F,G will then be conjugate two 
 and two with respect to the conic 
 (Art. 259). Take any point / on 
 EG and join it to the vertices of 
 the quadrangle, and draw also the 
 tangents IP, IQ to the conic. The 
 two tangents are harmonically 
 separated by IE, IF (Art. 264), 
 since these are conjugate straight 
 lines, F being the pole of IE. But 
 the rays IE , IF are harmonically 
 conjugate also with regard to I A , IC ; for the diagonal A C of the 
 complete quadrilateral formed by AB , BC , CD, and DA is divided 
 harmonically by the other two diagonals BD and EG, and the two 
 pairs of rays in question are formed by joining / to the four 
 harmonic points on AC. For a similar reason the rays IE , IF 
 are harmonically conjugate with regard to 
 IB , ID. The pair of tangents, the rays 
 I A , IC, and the rays IB , ID are therefore 
 three conjugate pairs of an involution, of 
 which IE, IF are the double rays (Art. 125). 
 
 I. By virtue of the theorem correlative 
 to that of Desargues (Art. 183, right), a 
 conic can be inscribed in the quadrilateral 
 ABCD so as to touch the straight lines 
 IF and IQ. 
 
 II. The theorem correlative to the one 
 proved above may be thus enunciated : 
 
 If a simple quadrilateral ABCD (Fig. 
 196) is circumscribed about a conic, and if 
 through the point of intersection of its diagonals any transversal be 
 drawn, this will cut the conic and the pairs of opposite sides AB 
 and CD, BC and AD, in three pairs of conjugate points of an invo- 
 lution. 
 
 III. By virtue of Desargues' theorem (Art. 183, left), a conic can 
 be described to pass through the four vertices of the quadrilateral and 
 through the two points where the conic is cut by the transversal *. 
 
 * Chasles, Sections coniyues, Arts. 122, 126. 
 Q 
 
 Fig. 196. 
 
226 THE CENTRE AND DIAMETERS OF A CONIC. [293 
 
 293. The theory of conjugate points with regard to a conic gives 
 a solution of the problem : 
 
 To construct the points of intersection of a given straight line s with 
 a conic which is determined by five points or by five tangents. 
 
 Take on s any two points XI and V, construct their polars u and v 
 (Art. 257), and let U' and V be the points where these meet s. If 
 the involution determined by the two pairs of reciprocal points U 
 and TJ\ V and V, has two double points M and N, these will be the 
 required points of intersection of the conic with s. If V and V' 
 should coincide, the conic touches s at the point in which they 
 coincide. If the involution has no double points, the conic does 
 not cut s *. 
 
 By a correlative method may be solved the problem : to draw from 
 a given point S a pair of tangents to a conic which is determined by 
 five tangents or by five points. 
 
 294'. Let A and A' be a pair of points lying on a straight line s 
 which are conjugate with respect to the conic, and let be the point 
 where s meets the diameter passing through its pole S (the diameter 
 bisecting chords parallel to s). Then will be the centre of the 
 involution formed on s by the pairs of conjugate points such as A 
 and A% and therefore (Art. 125) 
 
 OA . OA f = constant. 
 If s cuts the conic in two points M and if, these will be the double 
 points of the involution, and 
 
 OA.OA'=OM 2 =01V 2 . 
 If 8 does not cut the conic, the constant value of OA . OA' will be 
 negative (Art. 125); in this case there exists a pair H and H' of 
 conjugate points of the involution, or of conjugate points with regard 
 to the conic, such that lies midway between them, and 
 OA . OA' = OH . OH' =-OH*=- OH' 2 . 
 
 The segment HH' has been called an ideal chord f of the conic, 
 just as MJ¥ in the first case is a real chord. Accepting this defini- 
 tion we may say that a diameter contains the middle points of all 
 chords, real and ideal, which are parallel to the conjugate diameter. 
 
 When two conies are said to have a real common chord MN, it is 
 meant that they both pass through the points M and N. When two 
 conies are said to have an ideal common chord HH', this signifies 
 that II and H ' are conjugate points with regard to both conies, and 
 that the diameters of the two conies which pass through the 
 respective poles of HH' both pass through the middle point of 
 IIII'. 
 
 * Staudt, Geomelrie der Lage, Art. 305. 
 
 t PONCELET, IOC. Clt., p. 29. 
 
297] THE CENTRE AND DIAMETERS OF A CONIC. 227 
 
 295. A pencil of rays in involution has in general (Art. 207) 
 one pair of conjugate rays which include a right angle. 
 Therefore 
 
 Through a given point can always be drawn one pair of straight 
 lines which are conjugate with respect to a given conic and which 
 include a right angle ; and these are the internal and external bisec- 
 tors of the angle made with one another by the tangents drawn from 
 the given point, when this is exterior to the conic. 
 
 296. In Art. 263 (Fig. 183) let the point G be taken to co- 
 incide with the centre of the conic (hyperbola or ellipse) ; 
 two conjugate lines such as GF, GF will then become conju- 
 gate diameters, and we see that the pairs of conjugate diameters 
 of a conic form an involution. If the conic is a hyperbola, the 
 asymptotes are the double rays of the involution (Arts. 264, 
 283); thus any two conjugate diameters of a hyperbola are har- 
 monically conjugate with regard to the asymptotes*. If the conic 
 is an ellipse, the involution has no double rays. 
 
 Consider two pairs of conjugate elements of an involution ; 
 the one pair either overlaps or does not overlap the other, and 
 according as the first or the second is the case, the involution 
 has not, or it has, double points (Art. 128) ; thus: 
 
 Of any two pairs of conjugate diameters of an ellipse, the one 
 aa' is always separated by the other W (Fig. 192) ; 
 
 Of any two pairs of conjugate diameters of a hyperbola, the one 
 aa' is never separated by the other bb f (Fig. 197). 
 
 297. The involution of conjugate 
 diameters will have one pair of con- 
 jugate diameters including a right 
 angle (Art. 295). If there were a 
 second such pair, every diameter 
 would be perpendicular to its con- 
 jugate (Art. 207), and in that case 
 the angle subtended at any point F] * 
 on the curve by a fixed diameter 
 would be a right angle (Art. 287), and consequently the conic 
 would be a circle. Every conic therefore which is not a para- 
 bola or a circle has a single pair of conjugate diameters which 
 are at right angles to one another. These two diameters 
 a and a' are called the axes of the conic (Figs. 192, 197). In the 
 
 * De la Hike, loc. cit., book ii. prop. 13, Cor. 4. 
 Q2 
 
228 THE CENTRE AND DIAMETERS OF A CONIC. [298 
 
 hyperbola (Fig. 197) the axes are the bisectors of the angle 
 between the asymptotes m and n (Arts. 296, 60). 
 
 In the ellipse both axes cut the curve (Art. 284); the 
 greater (a f ) is called the major, the smaller (a) the minor axis. 
 In the hyperbola only one of the axes cuts the curve ; this one 
 (a') is called the transverse axis, the other (a) the conjugate axis. 
 The points in which the conic is cut by the axis a' in either 
 case are called the vertices. 
 
 Regarding an axis as a diameter which bisects all chords 
 perpendicular to itself, it is seen that the parabola also has 
 an axis. For since all chords at right angles to the common 
 direction of the diameters are parallel to one another, their 
 middle points lie on one straight line, which is the axis a of 
 the parabola (Fig. 190). The parabola has one vertex at 
 infinity ; the other, the finite point in which the axis a cuts the 
 curve, is generally called the vertex of the parabola. 
 
 298. Since each of the orthogonal conjugate diameters of a 
 central conic (ellipse or hyperbola) bisects all chords perpen- 
 dicular to itself, it follows that the conic is symmetrical with re- 
 spect to each of the diameters in question (Art. 76). The ellipse 
 and the hyperbola have therefore each two axes of symmetry ; 
 the parabola, on the other hand, has only one such axis. 
 
 The ellipse and hyperbola are also symmetrical with respect 
 to a point ; the centre of symmetry being in each case the 
 pole of the straight line at infinity. 
 
 In general, given a conic, a point 8, and s the polar of S with 
 
 / , respect to the conic; if S be 
 
 r*zz™-~/ft~~-~zsdx, taken as centre and s as axis of 
 
 / m^^tOv^/ harmonic homology (Art. 76), the 
 
 x rJ^§^p^™%7 conic is homological with itself 
 
 / yS L 299. In the theorem of Art. 
 
 I/' 275 suppose the inscribed triangle 
 
 ^ Fig I98 to be AA X M (Fig. 198); that is, 
 
 let two of its vertices A and A x 
 be collinear with the centre of the conic, which is taken 
 to be an ellipse or hyperbola. The pole of the side AA X will 
 be the point at infinity common to the chords bisected by 
 the diameter AA X , and the theorem will become the following : 
 
 * See also Art. 396, below. 
 
302] THE CENTRE AND DIAMETERS OF A CONIC. 229 
 
 The straight lines which join two conjugate points P and P' to the 
 extremities A and A x of that diameter whose conjugate is parallel to 
 FT?' intersect on the conic. 
 
 300. The pairs of conjugate points taken, similarly to P 
 and P\ on the diameter conjugate to AA X form an involution 
 (Art. 263) whose centre is the centre of the conic. If this 
 involution has two double points B and B lt these lie on the 
 curve, which is therefore an ellipse. If the involution has no 
 double points, the conic is a hyperbola (Art. 284) ; in this 
 case two points B and B 1 can be found which are conjugate 
 in the involution and consequently conjugate with respect to 
 the conic, and which lie at equal distances on opposite sides 
 of (Art. 125). In both cases the length of the diameter 
 conjugate to AA ± is interpreted as being the segment BB 1 
 (Arts. 290, 294). 
 
 In the ellipse we have (Art. 294) 
 
 OP . OF = constant = OB 2 = OB •, 
 and in the hyperbola 
 
 OP. 0P'= constant = OS. 0# 1= - 0B 2 = - OB 2 . 
 
 301. The foregoing theorem enables us to solve the problem : 
 
 To construct by points a conic, having given a fair of conjugate 
 diameters AA X and BB X in magnitude and fwsition. 
 
 Fig. 199. 
 
 In the case of the ellipse (Fig. 198) the four points A,A l ,B,B 1 
 all lie on the curve ; in the case of the hyperbola (Fig. 199) let AA X 
 be that one of the two given diameters which meets the conic. 
 
 Construct on the diameter BB X several pairs of conjugate points 
 P and P f of the involution determined by having as centre 
 and B and B x in the first case as double points, in the second case 
 as conjugate points. The straight lines AP and A V P' (as also A X P 
 and AP') will intersect on the curve. 
 
 302. The straight lines OX, OX' drawn parallel to AP,A Y P f 
 respectively are a pair of conjugate diameters (Art. 287). The 
 
230 THE CENTRE AND DIAMETERS OF A CONIC. [303 
 
 pairs of conjugate diameters form an involution (Art. 296) ; 
 consequently the pairs of points analogous to X, X' (in which 
 the diameters cut the tangent at A) also form an involution, 
 the centre of which is A, since OA and the diameter OB 
 parallel to AX are a pair of conjugate diameters. If the conic 
 is a hyperbola, the involution of conjugate diameters has two 
 double rays, which are the asymptotes ; therefore the points 
 K and K x , in which AX meets the asymptotes, are the double 
 points of the involution XX\ ...*. 
 
 303. Since OP AX is a parallelogram, AX= — OP; and from 
 the similar and equal triangles 0P'A t and AX'O, AX'= OP'f. 
 But OP. 0P'= ± OB 2 (Art. 125) ; therefore AX.AX f = + 0B 2 ; 
 or 
 
 The rectangle contained by the segments intercepted on a fixed tangent 
 to a conic between its point of contact and the points where it is cut 
 by any two conjugate diameters is equal to the square ( + OB 2 ) on the 
 semi-diameter drawn parallel to the tangent. 
 
 304. We have seen (Art. 302) that in the case of the hyper- 
 bola K and K x are the double points of the involution of which 
 A is the centre and X,X' a pair of conjugate points ; thus 
 
 AX.AX'=AK 2 =0B 2 . 
 Therefore AK= OB, and OAKB is a parallelogram. Accord- 
 ingly: 
 
 If a parallelogram be described so as to have a pair of conjugate 
 semi-diameters of a hyperbola as adjacent sides, one of its diagonals 
 will coincide with an asymptote %. 
 
 Further, the other diagonal AB is parallel to the second asymptote. 
 For consider the harmonic pencil (Art. 296) formed by the two 
 asymptotes and the two conjugate diameters OA , OB. The 
 four points in which this pencil cuts AB will be harmonic ; 
 but one of the asymptotes OK meets AB in its middle point, 
 therefore the other will meet it at infinity (Art. 59). 
 , 305. Let X 1 be the point where the diameter OX meets the 
 tangent at A v Since OX' and 0X X are a pair of conjugate 
 lines which meet in a point on the chord of contact AA X of 
 
 * In Fig. 199 only one of the points jK, K x is shown. 
 
 + In order to account for the signs, it need only be observed that in the case of 
 the ellipse OP and OP' are similar, but AX and AX' opposite to one another 
 in direction ; while in the case of the hyperbola OP and OP' are opposite, but 
 AX and AX' similar as regards direction. 
 
 X Apollonius, loc. cit., book ii. 1. 
 
309] THE CENTRE AND DIAMETERS OF A CONIC. 231 
 
 the tangents AX and A X X X , the straight line X / X 1 (Art. 274) 
 ■will be a tangent to the conic. 
 
 The point of contact of this tangent is M, the point of inter- 
 section of AP and A X P' (Art. 299). 
 
 306. It is seen moreover that X / X 1 is one diagonal of the 
 parallelogram formed by the tangents at A and A 1 and the 
 parallels to AA X drawn through P and P / '; this may also be 
 proved in the following manner. All points of a diameter 
 have for their polars straight lines which are parallel to the 
 conjugate diameter (Art. 284) ; if then through the conjugate 
 points P and P' parallels be drawn to AA X , the first will be 
 the polar of P'and the second the polar of P; consequently 
 these parallels are conjugate lines. If now the theorem of 
 Art. 274 be applied to these conjugate lines and the two tan- 
 gents at A and A 1 , we obtain the following proposition : 
 
 If a parallelogram is such that one pair of its opposite sides are 
 tangents to a conic, and the other pair are straight lines, conjugate 
 with regard to the conic and drawn parallel to the chord of contact of 
 the two tangents, then its diagonals also will be tangents to the conic. 
 
 307. This gives the following solution of the problem : 
 
 To construct a conic by tangents, having given a pair of conjugate 
 diameters AA X and BB X in magnitude and direction. 
 
 Suppose BB X to be that diameter which meets the conic in the case 
 where the latter is a hyperbola. On BB X determine a pair of con- 
 jugate points P and P / of the involution which has the centre of 
 the conic as centre and the points B , B x either as double points or 
 as conjugate points, according as the conic to be drawn is an ellipse 
 or a hyperbola. Draw through A and A 1 parallels to BB X , and 
 through P and P / parallels to AA X ; the diagonals of the parallelo- 
 gram so obtained will be tangents to the required conic. 
 
 308. The segments AX and A X X^ are equal in magnitude and 
 opposite in sign ; and it has been seen that AX.AX'= + OB 2 ; 
 therefore AX'. A.X^ ± OB 2 ; or 
 
 The rectangle contained by the segments intercepted upon two 
 parallel fixed tangents between their points of contact and the points 
 where they are cut by a variable tangent (X'Xj) is equal to the 
 square (+ OB 2 ) on the semi-diameter parallel to the fixed tangents*. 
 
 309. Since the straight line OB is parallel to AX and A Y X X 
 and half-way between them, the segments determined by AM 
 
 * See Art. 160. 
 
232 THE CENTRE AND DIAMETERS OF A CONIC. [310 
 
 and A X M respectively on A^X X and AX (measured from A L and 
 A respectively) are double of OP and 0P' : , but by the 
 theorem of Art. 300 the rectangle OP. OP' is constant ; thus 
 
 The straight lines connecting the extremities of a given diameter 
 with any point on the conic meet the tangents at these extremities in 
 two points such that the rectangle contained by the segments of the 
 tangents intercepted between these points and the points of contact is 
 
 310. Since X is (Art. 288) the point of intersection of the 
 tangent at A and the tangent parallel to X'X Y , the proposition 
 of Art. 303 may also be expressed as follows : 
 
 The rectangle contained by the segments (AX, AX') determined by 
 two variable parallel tangents upon any -fixed tangent is equal to the 
 square (+ OB 2 ) on the semi-diameter parallel to the fixed tangent. 
 
 311. From the theorems of Arts. 299, 300 is derived the solution 
 of the following problem : 
 
 Given the two extremities A and A t of a diameter of a conic, a third 
 point M on the conic, and tlie direction of the diameter conjugate to 
 AA J , to determine the length of the latter diameter (Fig. 199). 
 
 Through 0, the middle point of AA X , draw the diameter whose 
 direction is given ; let it be cut by AM and A X M in P and P f respec- 
 tively, and take OB the mean proportional between OP and OP'; 
 then OB will be the half of the length required. 
 
 312. The proposition of Art. 303 gives a construction for pairs of 
 
 conjugate diameters, and in par- 
 ticular for the axes, of an ellipse of 
 which two conjugate semi-diameters 
 OA and OB are given in magnitude 
 and direction (Fig. 200). 
 
 Through A draw a parallel to 
 OB ; this will be the tangent at A, 
 and will be cut by any two conju- 
 D gate diameters in two points X and 
 
 X' such that 
 AX.AX'=-OB 2 . 
 If now there be taken on the normal at A two segments AC and 
 AD each equal to OB, every circle passing through C and D will cut 
 this tangent in two points X and X' which possess the property ex- 
 pressed by the above equation ; these points are therefore such that 
 the straight lines joining them to the centre will give the direc- 
 tions of a pair of conjugate diameters. If the circle be drawn 
 
 * Apollonius, loc. cit., lib. iii. 53. 
 
 I ° 
 
 
 \x' 
 
 I / 
 
 ^s. \ / 
 
 w\ 
 
 \^^ j 
 
 
 
 X 
 
 \ X 
 
 } 
 
 / 
 
314] THE CENTRE AND DIAMETERS OF A CONIC. 233 
 
 through the angle XOX' becomes a right angle, and consequently 
 OX , OX' will be the directions of the axes. 
 
 Since the circular arcs CX', X f B are equal, the angles COX', X'OD 
 are equal ; consequently OX' , OX are the internal and external 
 bisectors of the angle which 00 , OD make with one another. In 
 order then to construct the semi-axes OP , OQ in magnitude, let fall 
 perpendiculars AX X , AX/ on OX, OX' respectively. Then XandX 15 
 X f and X( are pairs of conjugate points; therefore OP will be the 
 geometric mean between OX and 0X X , and OQ the geometric mean 
 between OX' and OX/ *. 
 
 313. Through the extremities A and A! (Fig. 201) of two 
 conjugate semi-diameters OA and OA! of a conic draw any two 
 parallel chords AB and A'B'. To find 
 the points B and B' we have only to 
 join the poles of these chords; this 
 will give the diameter OX' which passes 
 through their middle points. 
 
 Let OX be the diameter conjugate 
 to OX', i.e. that diameter which is Fi 20I 
 
 parallel to the chords AB, A'B'. The 
 
 pencils O(XX'AB) and 0(X'XA'B') are each harmonic (Art. 59), 
 and are therefore projective with one another ; consequently 
 the pairs of rays 0(XX', AA', BB') are in involution (Art. 123). 
 But the two pairs (XX', AA') determine the involution of 
 conjugate diameters (Arts. 1 27, 296); therefore also OB and OB' 
 are conjugate diameters. Thus 
 
 If through the extremities A and A! of two conjugate semi-diameters 
 parallel chords AB , A'B' be drawn, the points B and B' will be the 
 extremities of two other conjugate semi-dia?neters. 
 
 Two diameters AA and BB determine four chords AB 
 which form a parallelogram (Arts. 260, 287). The diameters 
 conjugate respectively to them form in the same way another 
 parallelogram, which has its sides parallel to those of the first ; 
 that is, every chord AB is parallel to two chords A'B', and not 
 parallel to two other chords A'B'. 
 
 314. Let E,Kbe the points where AB is cut by OA', OB' 
 respectively. The diameter OX' which bisects A'B' will also 
 bisect UK ; therefore AB and IIK have the same middle point ; 
 thus AH=KB and AK=HB. The triangles OAK and OBH 
 
 * Chasles, Aperg>j, historique, pp. 45, 362; Sections coniques, Art. 205. 
 
234 THE CENTRE AND DIAMETERS OP A CONIC. [315 
 
 are therefore equal in area (Euc. I. 37), as also AKB' and 
 BRA', and therefore also OAB' and OA!B are equal. Accord- 
 ingly : 
 
 The parallelogram described on two semi-diameters (OA , OB') as 
 adjacent sides is equal in area to the parallelogram described similarly 
 on the two conjugate semi-diameters. 
 
 In the same way the triangles OAB and OA'B' can be 
 proved equal. 
 
 The triangles AHA', BKB' are equal for the same reason ; 
 and OAR, OBK &re equal, and therefore also OAA' and OBB'. 
 Therefore 
 
 The parallelogram described on a pair of conjugate semi-diameters 
 as adjacent sides is of constant area *. 
 
 315. Let M and N be the middle points of the non-parallel 
 chords AB and A'B'. Since AB and A'B' are parallel to a 
 pair of conjugate diameters (Art. 287) and since ON is the 
 diameter conjugate to the chord A'B', therefore ON will be 
 parallel to AB; so also OM will be parallel to A'B'. The 
 angles OMA and ON A! are therefore equal or supplementary ; 
 and since the triangles OMA and ON A' are equal in area (being 
 halves of the equal triangles OAB and OA'B'), we have (Euc. 
 
 VL 15), 
 
 OM.AM=±ON.NA'f. 
 
 Now project (Fig. 202) the points A,M,B,A', N, B' from 
 the point at infinity on OB as centre 
 upon the straight line B'B'. The 
 ratio of the parallel segments AM 
 and ON, OM and NA' is equal to 
 that of their projections ; we con- 
 clude therefore from the equality 
 just proved that the rectangle 
 contained by the projections of 
 OM and AM is equal to that 
 Fi S- 202 - contained by the projections of 
 
 ON and NA'. As the projecting 
 rays are parallel to OB, the projections of OM and MA are 
 
 * Apollonius, loc. cit., lib. vii. 31, 32. 
 
 t The signs + and — caused by the relative direction of the segments 
 OM, NA' and ON, AM correspond respectively to the case of the ellipse (Fig. 
 201) and to that of the hyperbola (Fig. 202). 
 
316] THE CENTRE AND DIAMETERS OP A CONIC. 235 
 
 each equal to half the projection of BA or of OA. Since N 
 is the middle point of A'B' } the projection of ON will be equal 
 to half the sum of the projections of OA' and OB', and the 
 projection of NA' will be equal to half the projection of B'A', 
 that is, to half the difference between the projections of OA 
 and OB'. We have therefore 
 
 (proj. OA) 2 = + proj. (OA' + OB') 
 xproj. {OB'-OA'), 
 or (proj. OA') 2 ± (proj. OA) 2 = (proj. OB') 2 . 
 
 In the same manner, by projecting the same points on OB 
 by means of rays parallel to OB' (Fig. 203), we should 
 obtain 
 
 . (proj. OA) 2 ± (proj. OA') 2 = (proj. OB) 2 . 
 
 This proves the following proposition : 
 
 If any pair of conjugate diameters are projected upon a fixed 
 diameter by means of parallels to the 
 diameter conjugate to this last, then 
 the sum (in the ellipse) or difference 
 (in the hyperbola) of the squares on 
 the projections is equal to the square 
 on the fixed diameter. 
 
 By the Pythagorean theorem 
 (Euc. I. 47) the sum of the 
 squares on the orthogonal pro- 
 jections of a segment on two Fi 
 straight lines at right angles to 
 
 one another is equal to the square on the segment itself. If 
 then a pair of conjugate diameters are projected orthogonally 
 on one of the axes of a conic and the squares on the pro- 
 jections of each diameter on the two axes are added together, 
 the following proposition will be obtained : 
 
 The sum, (for the ellipse) or difference (for the hyperbola) of the 
 squares on any pair of conjugate diameters is constant, and is equal 
 to the sum or the difference of the squares on the axes *. 
 
 316. If five points on a conic are given, then by the method 
 explained in Art. 285 the centre and two pairs of conjugate 
 diameters u and u', v and v f can be constructed. If these pairs 
 overlap one another, the conic is an ellipse ; in the contrary case it 
 
 * Apollonius, loc. cit., lib. vii. 12, 13, 22, 25. 
 
236 THE CENTRE AND DIAMETERS OF A CONIC. [317 
 
 is a hyperbola (Art. 296). If in this second case the double rays of 
 the involution determined by the two pairs u and ?/, v and v' be 
 constructed, they will be the asymptotes of the hyperbola. 
 
 If in either case the orthogonal pair of conjugate rays of the in- 
 volution be constructed, they will be the axes of the conic. 
 
 The direction of the axes can be found without first constructing 
 the centre and two pairs of conjugate diameters*. Let A, B, C, F, G 
 be the five given points (Fig. 168) ; describe a circle round three of 
 them ABC, and construct (Art. 227, I) the fourth point of intersection 
 C" of this circle with the conic determined by the five given points. 
 Any transversal will cut the two curves and the two pairs of opposite 
 sides of the common inscribed quadrangle ABCC in pairs of points 
 forming an involution (Art. 183). The double points P and Q (if 
 such exist) of this involution will be conjugate with regard to each 
 of the curves (Arts. 125, 263); i.e. they will be the pair common 
 (Art. 208) to the two involutions which are formed on the transversal 
 by the pairs of points conjugate with regard to the circle and by the 
 pairs of points conjugate with regard to the conic (Art. 263). Suppose 
 that the straight line at infinity is taken as the transversal. As this 
 straight line does not meet the circle, one at least of these two 
 involutions will have no double points, and consequently (Art. 208) 
 the points P and Q do really exist. Since these points are infinitely 
 distant and are conjugate with regard to both curves they will be 
 (Arts. 276, 284) the poles of two conjugate diameters of the circle 
 and also of two conjugate diameters of the conic ; but conjugate 
 diameters of the circle are perpendicular to one another (Art. 289) ; 
 therefore P and Q are the poles of the axes of the conic. Further, 
 the segment PQ is harmonically divided by either pair of opposite 
 sides of the quadrangle ABCC ; consequently P and Q are the 
 points at infinity on the bisectors of the angles included by the 
 pairs of opposite sides (Art. 60). In order then to find the required 
 directions of the axes, we have only to draw the bisectors t of the 
 angle included by a pair of opposite sides of the quadrangle ABCC, 
 for example by AB and CC (Fig. 168). 
 
 317. Let qrst (Fig. 161) be a complete quadrilateral, and S 
 any point. It has already been seen (Art. 185, right) that the 
 pairs of rays a and a', b and b', which join S to two pairs of 
 opposite vertices, belong to an involution of which the tangents 
 drawn from S to any conic inscribed in the quadrilateral are a 
 pair of conjugate rays. Suppose the involution to have two 
 double rays m and?*; they will be harmonically conjugate 
 
 * Poncelet, loc. cit., Art. 394. 
 t See also the note to Art. 387. 
 
318] THE CENTRE AND DIAMETEES OF A CONIC ■ 237 
 
 with regard to such a pair of tangents (Art. 125), and will 
 consequently be conjugate lines with respect to the conic. 
 But (Art. 218, right) m and n are the tangents at 8 to the 
 two conies which can be inscribed in the quadrilateral qrst 
 so as to pass through S. Therefore 
 
 If tioo conies which are inscribed in a given quadrilateral pass 
 through a given point, their tangents at this point are conjugate lines 
 with respect to any conic inscribed in the quadrilateral. 
 
 Instead of taking an arbitrary point S, let m be supposed 
 given. If this straight line does not pass through any of the 
 vertices of the quadrilateral, there will be one conic, and only 
 one, which touches the five straight lines m,q,r,s,t (Art. 
 152). Let S be the point where this conic touches m; there 
 will be a second conic which is inscribed in the quadrilateral 
 and which passes through S ; let the tangent to this at S be n. 
 The straight lines m and n will then be conjugate to one 
 another with respect to all conies inscribed in the quadrilateral ; 
 and therefore (Art. 255), 
 
 The poles of any straight line m with respect to all conies inscribed 
 hi the same quadrilateral lie on another straight line n. 
 
 Moreover, since the straight lines m and n are the double 
 rays of the involution of which the rays drawn from S to two 
 opposite vertices are a conjugate pair, therefore m and n 
 divide harmonically each diagonal of the quadrilateral. 
 
 318. I. The correlative propositions to those of Art. 317 are 
 the following : 
 
 If a straight line touches two conies which circumscribe the same 
 quadrangle, the two points of contact are conjugate to one another 
 with respect to all conies circumscribing the quadrangle. 
 
 The polars of any given point M with respect to all the conies 
 circumscribing the same quadrangle meet in a fixed point N. The 
 segment MN is divided harmonically at the two points where it is cut 
 by any pair of opposite sides of the complete quadrangle. 
 
 II. Suppose in the second theorem of 
 Art. 317 that the straight line m lies at 
 infinity ; then the poles of m will be the 
 centres of the conies (Art. 281), and n will 
 
 bisect each of the diagonals of the quadri- 
 
 to u Fig. 204. 
 
 lateral (Art. 59); therefore: 
 
 The centres of all conies inscribed in the same quadrilateral lie 
 
238 
 
 THE CENTRE AND DIAMETERS OF A CONIC. 
 
 [319 
 
 on the straight line (Fig. 204) which passes through the middle 
 points of the diagonals of tlhe quadrilateral*. 
 
 III. Suppose similarly in theorem I of the present Article 
 that the point 31 lies at infinity ; the polars of M will become 
 the diameters conjugate to those which have M as their 
 common point at infinity ; thus : 
 
 In any conic circumscribing a given quadrangle, the diameter which 
 is conjugate to one drawn in a given fixed direction will pass through 
 a fixed point. 
 
 319. Newton's theorem (Art. 318, II) gives a simple method for 
 
 finding the centre of a conic deter- 
 mined by five tangents a,b, c, d, e 
 (Fig. 205). The four tangents 
 a,b,c,d form a quadrilateral; 
 join the middle points of its 
 diagonals. Let the same be done 
 with regard to the quadrilateral 
 abce ; the two straight lines thus 
 obtained will meet in the required 
 centre 0. 
 
 The five tangents, taken four 
 and four together, form five quad- 
 rilaterals ; the five straight lines 
 which join the middle points of the 
 diagonals of each of the quadri- 
 laterals will therefore all meet in 
 the centre of the conic inscribed 
 
 Fig. 205. 
 
 in the pentagon abcde. 
 
 The same theorem enables us to find the direction of the diameters 
 of a parabola which is determined by four tangents a, b,c, d. For 
 each point on the straight line joining the middle points of the 
 diagonals of the quadrilateral abed is the pole of the straight line at 
 infinity with regard to some conic inscribed in the quadrilateral 
 (Art. 318, II); therefore the point at infinity on the line will be 
 the pole with regard to the inscribed parabola (Arts. 254 III, and 23). 
 The straight line therefore which joins the middle points of the 
 diagonals is itself a diameter of the parabola (Fig. 204). 
 
 * Newton, Principia, book i. lemma 25. Cor. 3. 
 
CHAPTEE XXII. 
 
 POLAR RECIPROCAL FIGURES. 
 
 320. An auxiliary conic K being given, it has been seen 
 (Art. 256) that if a variable pole describes a fixed straight 
 line its polar turns round a fixed point, and reciprocally, that 
 if a straight line considered as polar turns round a fixed point, 
 its pole describes a fixed straight line. 
 
 Consider now as polars all the tangents of a given curve C, 
 or in other words suppose the polar to move, and to envelope 
 the given curve. Its pole will describe another curve, which 
 may be denoted by C'. Thus the points of C' are the poles of 
 the tangents of C. 
 
 But it is also true that, reciprocally, the points of C are 
 the poles of the tangents of C'. For let 
 M' and N' be two points on C' (Fig. 206) • 
 their polars m and n will be two tangents 
 to C and the point mn where they meet 
 will be the pole of the chord M'N' (Art. 
 256). Now suppose the point N'to approach 
 M' indefinitely; the chord M'N' will ap- 
 proach more and more nearly to the position 
 of the tangent at M' to the curve C'; the Fi 6 
 
 straight line n will at the same time ap- 
 proach more and more nearly to coincidence with m, and the 
 point mn will tend more and more to the point where m 
 touches C. In the limit, when the distance M'N' becomes 
 indefinitely small, the tangent to C' at M' will become the 
 polar of the point of contact of m with C. Just then as 
 the tangents of C are the polars of the points of C', so also 
 are the tangents of C' the polars of the points of C ; if a 
 straight line m touches the curve C at M, the pole M ' of m 
 
240 POLAR RECIPROCAL FIGURES. [321 
 
 is a point of the curve C ' and the polar m ' of 31 is a tangent 
 to the curve C' at M f . 
 
 Two curves C and C' such that each is the locus of the 
 poles of the tangents of the other, and at the same time also 
 the envelope of the polars of the points of the other, are said 
 to be polar reciprocals * one of the other with respect to the 
 auxiliary conic K. 
 
 321. An arbitrary straight line r meets one of the reciprocal 
 curves in n points say ; the polars of these points are n tan- 
 gents to the other curve all passing through the pole R f of r. 
 To the second curve therefore can be drawn from any given 
 point R f the same number of tangents as the first curve has 
 points of intersection with the straight line r, the polar of R' \ 
 and vice versa. In other words, the degree and class of a curve 
 are equal to the class and degree respectively of its polar reciprocal 
 tvith respect to a conic. 
 
 322. Now suppose the curve C to be a conic, and a , b two 
 tangents to it; they will be cut by all the other tangents 
 c,d,e,... in corresponding points of two projective ranges 
 (Art. 149). In other words, C may be regarded as the curve 
 enveloped by the straight lines c,d ,e , ... which connect the 
 pairs of corresponding points of two projective ranges lying 
 on a and b respectively (Art. 150). 
 
 The curve C' will pass through the poles A\ B\C\D\W,... 
 of the tangents a, b, c, d, e, ... of C. The straight lines 
 A'(C' S D',I!',...) will be the polars of the points a(c,d,e,...) 
 and will form a pencil projective with the range of poles 
 lying on the straight line a (Art. 291); so too the straight lines 
 i?'(C", i)', J5", ...) will be the polars of the points b(c ,d ,e,...) 
 and will form a pencil projective with the range of poles lying 
 on I. But the ranges a(c,d,e,...) and b(c ,d,e,...) are 
 projective; therefore also the pencils A\C\B\E\...) and 
 £'(C D',E', ...) are projective. Consequently C' is the locus 
 of the points of intersection of corresponding rays of two 
 projective pencils ; that is (Art. 150) a conic. Accordingly : 
 
 The polar reciprocal of a conic with respect to another conic is a 
 conic f. 
 
 323. When an auxiliary conic K is given and another conic 
 
 * PONCELET, IOC. tit., Art. 232. 
 
 t Ibid., Art. 231. 
 
324] POLAR RECIPROCAL FIGURES. 241 
 
 C whose polar reciprocal C ' is to be determined, the question 
 arises whether C' is an ellipse, a hyperbola, or a parabola. The 
 straight line at infinity is the polar of the centre of K ; there- 
 fore the points at infinity on C ' correspond to the tangents of 
 C which pass through 0. It follows that the conic C ' will be an 
 ellipse or a hyperbola according as the point is interior or exterior 
 to the conic C, and O f will be a parabola when lies upon C. 
 
 If A is the pole of a straight line a with respect to C, and a' 
 the polar of A and A! the pole of a with respect to K, then 
 will A' be the pole of a' with respect to C', since to four poles 
 forming a harmonic range correspond four polars forming a 
 harmonic pencil (Art. 291) and vice versa. Therefore the 
 centre M' of C' will be the pole with respect to K of the 
 straight line m which is the polar of with respect to C. To 
 two conjugate diameters of C' will correspond two points of 
 m which are conjugate with respect to C, &c. 
 
 324. Let there be given in the plane of the auxiliary conic 
 a -figure (Art. 1) or complex of any kind composed of points, 
 straight lines, and curves ; and let the polar of every point, the 
 pole of every line, and the polar reciprocal of every curve, be 
 constructed. In this way a new figure will be obtained ; the 
 two figures are said to be polar reciprocals one of the other, since 
 each of them contains the poles of the straight lines of the 
 other, the polars of its points, and the curves which are the 
 polar reciprocals of its curves. To the method whereby the 
 second figure has been derived from the first the name of polar 
 reciprocation is given. 
 
 Two figures which are polar reciprocals one of the other are 
 correlative figures in accordance with the law of duality in plane 
 Geometry (Art. 33) ; for to every point of the one corresponds 
 a straight line of the other, and to every range in the one corre- 
 sponds a pencil in the other. They lie moreover in the same 
 plane ; their positions in this plane are determinate, but may 
 be interchanged, since every point in the one figure and the 
 corresponding straight line in the other are connected by the 
 relation that they are pole and polar with respect to a fixed 
 conic. Thus two polar reciprocal figures are correlative figures 
 which are coplanar, and which have a special relation to one 
 another with respect to their positions in the plane in which 
 they lie. On the other hand, if two figures are merely 
 
242 POLAB RECIPROCAL FIGURES. [325 
 
 correlative in accordance with the law of duality, there is no 
 relation of any kind between them as regards their position *. 
 
 325. If one of the reciprocal figures contains a range (of 
 poles) the other contains a pencil (of polars), and these two 
 corresponding forms are projective (Art. 291). If then the 
 points of the range are in involution, the rays of the corre- 
 sponding pencil will also be in involution, and to the double 
 points of the first involution will correspond the double rays 
 of the second (Art. 124). If there is a conic in one of the 
 figures there will also be one in the other figure (Art. 322); to 
 the points of the first conic will correspond the tangents of 
 the second, and to the tangents of the first will correspond the 
 points of the second; to an inscribed polygon in the first 
 figure will correspond a circumscribed polygon in the second 
 (Art. 320). If the first figure exhibits the proof of a theorem 
 or the solution of a problem, the second will show the proof of 
 the correlative theorem or the solution of the correlative 
 problem ; that namely which is obtained by interchanging the 
 elements ' point ' and ' line.' 
 
 326. Theoeem. If two triangles are both self-conjugate with 
 regard to a given conic, their six vertices lie on a conic, and their six 
 sides touch another conic f. 
 
 Let ABC and DEF be two triangles (Fig. 207) each of 
 which is self-conjugate (Art. 258) with regard 
 to a given conic K. Let DE and DF 
 cut BC in B x and C x respectively, and let AB 
 and AC cut EF in E x and F x respectively. The 
 point B is the pole of CA, and C is the pole of 
 AB ; B x is the pole of the straight line joining 
 the poles of BC and DE, i.e. of AF; and C x is 
 the pole of the straight line joining the poles of 
 BC and DF, i. e. of AE. The range of poles BCB X C X is therefore (Art. 
 291) projective with the pencil of polars A(CBFE), and therefore 
 with the range of points F X E X FE in which this pencil is cut by the 
 transversal EF. Thus 
 
 (BCB X C X ) = (F X E X FE) 
 
 = (E X F X EF) by Art. 45, 
 which shows that the two ranges in which the straight lines BC and 
 ^respectively are cut by AB, CA, DE, FD are protectively related. 
 
 * Steiner, loc. cit., p. vii of the preface; Collected Works, vol. i. p. 234. 
 t Steiner, loc. cit., p. 308, § 60, Ex. 46 ; Collected Works, vol. i. p. 448 ; 
 Chasles, Sections coniqnes, Art. 215. 
 
329] POLAR RECIPROCAL FIGURES. 243 
 
 These six straight lines therefore, the six sides of the given triangles, 
 all touch a conic C (Art. 150, II). 
 
 The poles of these six sides are the six vertices of the triangles ; 
 these vertices therefore all lie on another conic C' which is the polar 
 reciprocal of C with regard to the conic K *. 
 
 327. The proposition of the preceding Article may also be expressed 
 as follows : Given two triangles which are self-conjugate with regard 
 to the same conic K ; if a conic C touch five of the six sides it will 
 touch the sixth side also, and if a conic pass through five of the six 
 vertices it will pass through the sixth vertex also. 
 
 It follows that if a conic C touch the sides of a triangle abc which is 
 self-conjugate with regard to another conic K , there are an infinite 
 number of other triangles which are self-conjugate with regard to the 
 second conic and which circumscribe the first. 
 
 For let d be any tangent to C ; from D, its pole with regard to K, 
 draw a tangent e to C, and let / be the polar with regard to K of the 
 point de ; then the triangle def will be self- conjugate with regard to 
 K (Art. 259). But C touches five sides a, b, c, d, e of two triangles 
 which are both self-conjugate with respect to K; therefore it must 
 also touch the sixth side / ; which proves the proposition. 
 
 328. If the point D is such that from it a pair of tangents e' and 
 f can be drawn to K, the four straight lines e,f, ef, f will form a 
 harmonic pencil (Art. 264), since e and / are conjugate straight lines 
 with respect to the conic K ; consequently the straight lines e' and f f 
 are conjugate to one another with respect to C. 
 
 The locus of D is the conic C ' which is the polar reciprocal of C 
 with regard to K ; therefore : 
 
 If a conic C is inscribed in a triangle which is self-conjugate with 
 respect to another conic K, the locus of a point such that the j)airs of 
 tangents drawn from it to the conies C and K form a harmonic pencil 
 is a third conic C' which is the polar reciprocal of C with respect to K. 
 
 329. Correlatively : If a conic C' circumscribes a triangle which 
 is self-conjugate with respect to another conic K, there are an infinite 
 number of other triangles which are inscribed in C and are self -con- 
 jugate with respect to K ; and the straight lines which cut C f and "&. in 
 two pairs of joints which are harmonically conjugate to one another all 
 touch a third conic C which is the polar reciprocal of C f with regard 
 to K. 
 
 * "We may show independently that the six vertices lie on a conic as follows. 
 It has been seen that the pencil of polars A {CBFE) is projective with the range 
 of poles BCB^Px ; it is therefore projective with the pencil D {BCB X CC) formed by 
 joining these to the point D. Therefore 
 
 A {CBFE) = D (BCB&) = D {BCEF) 
 = D{CBFE) by Art. 45, 
 which shows (Art. 150, I) that A, B, 0, D, E, F lie on a conic. 
 
 R % 
 
244 POLAR RECIPROCAL FIGURES. [330 
 
 330. Theorem. If two triangles circumscribe the same conic, 
 their six vertices lie on another conic. 
 
 Let OQ'R' and O'PS be two triangles each circum- 
 scribing a given conic C (Fig. 208). The two tangents PS 
 
 and Q'R' are cut by the four 
 other tangents O'P, OQ', OP', O'S 
 in two groups of corresponding 
 points PQPS and P'Q'R'S' of two 
 projective ranges u and u' (Art. 
 149); consequently the. pencils 
 O(PQPS) and O'(P'Q'P'S') 
 formed by connecting these points 
 with and 0' respectively are 
 projective. Therefore the points P, Q',B',S, in which their 
 pairs of corresponding ray s intersect, lie on a conic C' (Art. 150,1) 
 passing through the centres and 0' '; which proves the theorem. 
 
 331. The theorem correlative and converse to the foregoing 
 one is the following : 
 
 If two triangles are inscribed in the same conic, their six sides 
 touch another conic*. 
 
 This may be proved by considering the triangles OQ'R' 
 and O'PS as both inscribed in the conic C', and by reasoning 
 in a manner exactly analogous, but correlative, to that above. 
 
 332. It follows at once that: 
 
 If two triangles circumscribe If two triangles are inscribed 
 
 the same conic, the conic which in the same conic, the conic which 
 
 passes through five of their ver- touches five of their sides touches 
 
 tices passes through the sixth the sixth side also, 
 vertex also. 
 
 Or: 
 
 If two conies are such that a triangle can be inscribed in the one 
 so as to circumscribe the other, then there exist an infinite number of 
 other triangles which possess the same property f. 
 
 333. There are in the figure (Fig. 208) four projective 
 forms : the two ranges u and u f , which determine the tangents to 
 the conic C, and the two pencils and ', which determine the 
 points of C'; the pencil is in perspective with the range u 
 
 * Brianchon, loc. cit., p. 35; Steiner, he. cit, p. 173, § 46, II; Collected 
 Works, vol. i. p. 356. 
 
 f PONCELET, loc. Cit., Art. 565. 
 
334 
 
 POLAR RECIPROCAL FIGURES. 
 
 245 
 
 and the pencil O f is in perspective with the range u'. If then 
 any tangent to C cut the bases u and u' of the two ranges in 
 A and A' respectively, the rays OA and O'A' will meet in a 
 point M lying onC'; and, conversely, if any point M on C' be 
 joined to the centres and 0', the joining lines will cut u and 
 u' respectively in two points A and A! such that the straight 
 line joining them is a tangent to C. Therefore : 
 
 If a variable triangle AA'M is 
 such that two of its sides pass 
 respectively through two fixed 
 points O f and lying on a given 
 conic, and the vertices opposite to 
 them lie respectively on two fixed 
 straight lines u and u f , while the 
 third vertex lies always on the 
 given conic, then the third side 
 will touch a fixed conic which 
 touches the straight lines u and u' '. 
 
 If a variable triangle AA'M is 
 such that two of its vertices lie 
 respectively on two fixed tangents 
 u and u f to a given conic, and 
 the sides opposite to them pass 
 respectively through two fixed 
 points O f and 0, while the third 
 side always touches the given 
 conic, then the third vertex will lie 
 on a fixed conic which passes 
 through the points and 0' '. 
 
 334. Theorem. If the extremities of each of two diagonals of a 
 complete quadrilateral are conjugate points with respect to a given 
 conic, the extremities of the third diagonal also will be conjugate points 
 with respect to the same conic *. 
 
 Let ABXY (Fig. 209) be a complete quadrilateral such 
 that A is conjugate to X, and B to Y, with respect to a given 
 conic K (not shown in 
 the figure). Let the sides 
 AB , XT meet in C, and 
 the sides AY,BX in Z; 
 then shall C and Z be 
 conjugate points with 
 respect to the conic K. 
 
 Suppose the polars of 
 the points A,B,C (with 
 respect to K ) to cut the 
 
 straight line ABC in A', B',C r respectively. The three 
 pairs of conjugate points A and A!, B and B f , C and C are 
 in involution; consequently, considering XYZ as a triangle 
 cut by a transversal A!B'C\ it follows by Art. 135 that the 
 
 * Hesse, De octo punctis intersectionis trium superficiei'um secundi ordinis 
 (Dissertatio pro venia legendi, Regiomonti, 1 840), p. 1 7. 
 
246 POLAR RECIPROCAL FIGURES. [335 
 
 straight lines XA', YB', ZC meet in one point Q. Since 
 evidently X^'is the polar of A and YB' the polar of B with 
 respect to K, their point of intersection Q is the pole of AB. 
 Since then C is a point on AB and is conjugate to C, its f>olar 
 will be QC ; but QC passes through Z; therefore Cand i^are 
 conjugate points, which was to be proved. 
 
 335. The proof of the following, the correlative theorem, is 
 left as an exercise to the student : 
 
 If two pairs of opposite sides of a complete quadrangle are conju- 
 gate lines with respect to a conic, the two remaining sides also are 
 conjugate lines with respect to the same conic. 
 
 In order to obtain such a complete quadrangle, it is only 
 necessary to take the polar reciprocal of the quadrilateral con- 
 sidered in Hesse's theorem, i. e. the figure which is formed by 
 the polars of the six points A and X, B and Y, C and Z. 
 
 336. The following proposition is a corollary to that of 
 Art. 334: 
 
 Two triangles which are reciprocal with respect to a conic are in 
 *. 
 Let ABC (Fig. 210) be any triangle; the polars of its 
 vertices with respect to a given 
 conic form another triangle A!B'C f 
 reciprocal to the first, that is, such 
 that the sides of the first triangle 
 are also the polars of the vertices of 
 the second. Let the sides CA and C'A' 
 meet in E, and the sides AB and 
 Fig. 210. A'B' in F. 
 
 The points B and E are conjugate 
 with respect to the conic, since E lies on C'A\ the polar of B ; 
 similarly C and .Fare conjugate points. Thus in the quadri- 
 lateral formed by BC, CA, AB, and EF, two pairs of opposite 
 vertices B and E, C,and F are conjugate; therefore the 
 third pair are conjugate also, viz. A and the point B where 
 BC meets EF. The polar B'C of A therefore passes through 
 B\ thus BC and B'C meet in a point I) lying on EF. 
 Since then the pairs of opposite sides of the two triangles meet 
 one another in three collinear points, the triangles are in 
 homology, and the straight lines AA', BB', CC which join 
 
 * Chasles, loc. cit., Art. 135, 
 
339] 
 
 POLAR RECIPROCAL FIGURES. 247 
 
 the pairs of vertices meet (Art. 1 7) in a point 0, the pole of 
 the straight line DEF. 
 
 337. By combining this theorem with that of Art. 155 the 
 following property may be enunciated : 
 
 If two triangles are reciprocals with respect to a given conic K, 
 the six points in which the sides of the one intersect the non- 
 corresponding* sides of the other lie on a conic C, and the six straight 
 lines which connect the vertices of the one with the non-corresponding 
 vertices of the other touch another conic C r , the polar reciprocal of 
 C with respect to K (Art. 322); these straight lines are in fact 
 the polars with regard to K of the six points just mentioned. 
 
 If one of the triangles A'B'C is inscribed in the other 
 ABC, the three conies C, C', and K coincide in one which is 
 circumscribed about the former triangle and inscribed in the 
 latter (Arts. 174, 176). •» 
 
 338. Problem. Given two triangles ABC, A'B'C which are in 
 homology ; to construct (when it exists) the conic with regard to which 
 they are reciprocal. 
 
 Take one of the sides, BG for example ; the points in which it is 
 cut by G'A' a,n&A'B' are conjugate to the points .Sand G respectively, 
 and these two pairs of conjugate points determine an involution 
 (Art. 263), the double points of which (if they exist) are the points 
 where BG is cut by the conic in question. In order then to find the 
 points in which this conic cuts BG, it is only necessary to construct 
 these double points. In this way the points in which the sides of 
 the triangles meet the conic can be found, and the latter is determined. 
 Since A' and B are the poles of BG and G'A', these points and that 
 in which C'A' meets BG will be the vertices of a self-conjugate 
 triangle (Art. 258). If then, in finding the points of intersection of 
 the conic and the straight lines BG and C'A' in the manner just 
 explained, it should hajypen that the two involutions found have 
 neither of them double points, the conclusion is that no conic exists 
 such as is required ; for if it did exist, it must be cut by two of the 
 side^ of the self-conjugate triangle (Art. 262). 
 
 339. The centre of homology of the given triangles (Fig. 210) is 
 the pole of the axis of homology DBF; and the projective corre- 
 spondence (Art. 291) between the points (poles) lying on the axis 
 and the straight lines (polars) radiating from the centre of homology 
 is determined by the three pairs of corresponding elements D and 
 
 * Two sides BG and WC of the triangles may be termed corresponding, when 
 each lies opposite to the pole of the other. And two vertices A and A' may be 
 termed corresponding, when each lies opposite to the polar of the other. 
 
248 POLAR RECIPROCAL FIGURES. - [339 
 
 A A', E and BB' , F and CC. Consequently it is possible to construct 
 with the ruler only (Art. 84) the polar of any other point on the 
 axis, and the pole of any other ray passing through the centre 0. 
 
 What has just been said with regard to the point and the axis 
 of homology may also be said with regard to any vertex of one of 
 the triangles and its polar (the corresponding side of the other 
 triangle). For if e. g. the vertex A' and the side BG be considered, 
 the projective correspondence between the straight lines radiating 
 from A' and the points lying on BC is determined by the three pairs 
 of corresponding elements A'B' and C, A'C and B, A f O and D. 
 
 This being premised, it will be seen that the polar of any point P 
 and the pole of any straight line p can be constructed with the help 
 of the ruler only. For suppose P to be given ; it has been shown 
 that the pole^s of the straight lines PO, PA, PB, PC, PA', ... can 
 be constructed, and these all lie on a straight line X which is the 
 required polar of P. So again i f th e straight line p is given, the 
 polars of the points in which it meeVtfC, CA, ... can be constructed, 
 and will meet in a point which is the pole of p. 
 
 It will be noticed that all these determinations of poles and polars 
 are linear (i. e. of the first degree) and independent of the construction 
 (Art. 338) of the auxiliary conic, which is of the second degree, 
 since it depends on finding the double elements of an involution. 
 The construction of the poles and polars is therefore always possible, 
 even when the auxiliary conic does not exist. In other words : the 
 two given triangles in homology determine between the points and 
 the straight lines of the plane a reciprocal correspondence such that 
 to every point corresponds a straight line and to every straight line 
 a point, to the rays of a pencil the points of a range projective with 
 the pencil, and vice versa. Any point and the straight line corre- 
 sponding to it may be called pole and polar, and this assemblage of 
 poles and polars, which possesses all the properties of that determined 
 by an auxiliary conic (Art. 254), may be called a polar system. 
 
 Two triangles in homology accordingly determine* a polar system. 
 If an auxiliary conic exists, it is the locus of the points which lie 
 on the polars respectively corresponding to them, and it is at the 
 same time the envelope of the straight lines which pass through the 
 poles respectively corresponding to them. If no auxiliary conic 
 exists, there is no point which lies on its own polar *. 
 
 * Staudt, loc. cit, Art. 241. 
 
CHAPTEE XXIII. 
 
 FOCI *, 
 
 340. It has been seen (Art. 263) that the pairs of straight 
 lines passing through a given point 8 and conjugate to one 
 another with respect to a given conic form an involution. Let 
 a plane figure be given, containing a conic C ; and let the figure 
 homological with it be constru^lfcd, taking 8 as centre of homo- 
 logy ; let C ' be the conic corresponding to C in the new figure. 
 Since in two homological figures a harmonic pencil corre- 
 sponds to a harmonic pencil, any pair of straight lines through 
 8 which are conjugate with respect to C will be conjugate 
 also with respect to C'. The polars of S with respect to the 
 two conies will be corresponding straight lines ; if then the 
 polar of S with respect to C be taken as the vanishing line 
 in the first figure, the polar of 8 with respect to C' will 
 lie at infinity; i. e. the point 8 will be the centre of the 
 conic C'. 
 
 In this case therefore any two straight lines through 8 
 which are conjugate with respect to C will be a pair of conju- 
 gate diameters of C'. If 8 is external to C, the double rays of 
 the involution formed by the conjugate lines through 8 are the 
 tangents from S to C, and therefore the asymptotes of C', 
 which is in this case a hyperbola. If S is internal to C, 
 the involution has no double rays, and therefore C' is an 
 ellipse. 
 
 We conclude then that to every point S in the plane of a given 
 conic C corresponds a conic G r homological with C and having its 
 centre at S; which conic C' is a hyperbola or an ellipse according as 
 8 is external or internal to the given conic C. 
 
 * Steineb, Vorlesungenuber synthetische Geometrie (ed. Schroter), II ter Abschnitt, 
 § 35 ; Zech, Hohere Geometrie (Stuttgart, 1857), § 7 > R EYE > Geometrie der Lage 
 (2nd ed., Hannover, 1877), Vortrag 13. 
 
250 FOCI. [341 
 
 341. For certain positions of the point S the conic C ' will 
 be a circle. When S has one of these positions it is called a 
 focus* of the conic C. Since all pairs of conjugate diameters 
 of a circle cut one another orthogonally the involution at S of 
 conjugate lines with respect to C will in this case consist 
 entirely of orthogonal pairs. 
 
 If C is a circle, its centre is a focus ; for every pair of 
 conjugate lines which meet in 0, i.e. 
 every pair of conjugate diameters of C, 
 cut orthogonally. And a circle C has no 
 other focus but its centre 0. For let any 
 points be taken (Fig. 211) distinct from 
 and a straight line SQ be drawn not 
 passing through 0; and let P be the 
 pole of $Q. Then since PO must be 
 perpendicular to SQ, the conjugate lines 
 SP, SQ cannot be orthogonal, and there- 
 fore 8 cannot be a focus of C. 
 
 The foci of a conic C which is not a 
 circle are of necessity internal points ; 
 this follows from what has been said above (Art. 340). 
 Further, they lie on the axes. For if F is a focus and 
 the centre of the conic, the pole of the diameter FO will lie 
 on the perpendicular drawn through F to FO; therefore 
 FO is perpendicular to its conjugate diameter, i. e. FO is an 
 axis of the conic. 
 
 Again, the straight line connecting two foci F and F x is an 
 axis. For if straight lines perpendicular to FF X be drawn 
 through F and F 1 these will both be conjugate to FF X , and 
 their point of intersection will therefore be the pole of FF X ; 
 but this point lies at infinity ; therefore FF t is an axis. 
 
 342. Let a point P be taken arbitrarily on an axis a of a 
 conic ; through P draw a straight line r, and from It, the pole 
 of r, draw the straight line / perpendicular to r ; let P' be the 
 point where / meets the axis. The straight lines passing 
 through P and those passing through P' and conjugate to 
 them respectively form two projective pencils ; for the second 
 pencil is composed of rays which project from P f the range 
 
 * Db la Hire, Sectiones conicae (Parisiis, 1685), lib. viii. prop. 23 ; Poncelet, 
 Proprtitis projectives, Art. 457 et seqq. 
 
343] FOCI, 251 
 
 formed by the poles of the rays of the first pencil, which range 
 is (Art. 291) projective with the first pencil itself. The two 
 pencils in question have three pairs of corresponding rays 
 which are mutually perpendicular ; for if A be the point at 
 infinity which is the pole of the axis a, the rays PA , PP\ r of 
 the first pencil correspond to the rays P'P,P'A, / of the 
 second, and the three former rays are severally perpendicular 
 to the three latter. The two pencils therefore by the inter- 
 section of corresponding rays generate a circle of which 
 PP' is a diameter ; and therefore every pair of corresponding 
 rays of the two pencils P and P f intersect at right angles. 
 Thus: 
 
 To every point P lying on an axis of the conic corresponds a point 
 P f on the same axis such thai any two conjugate straight lines which 
 pass one through P and the other through P r are perpendicular to 
 one another. 
 
 ■flie pairs of points analogous to P, P f form an involution. 
 For let the ray r move parallel to itself ; the corresponding 
 rays r' (which are all perpendicular to r) will all be parallel to 
 each other. The pencil of parallels r is projective (Art. 291) 
 with the range which the poles R of the rays r determine upon 
 the diameter conjugate to that drawn parallel to r\ and the 
 pencil of parallels r ' is in perspective with this same range. 
 Therefore the pencils r,r / are projective, and consequently the 
 points P, P f in which a pair of corresponding rays ;•, r f of the 
 pencils cut the axis a trace out two projective ranges. To the 
 straight line at infinity regarded as a ray r corresponds 
 in the second pencil the diameter parallel to the rays r' '; and 
 similarly, to the line at infinity regarded as a ray r' corre- 
 sponds in the first pencil the diameter parallel to the rays r. 
 Therefore the point at infinity on the axis has the same corre- 
 spondent whether it be regarded as a point P or as a point P': 
 viz. the centre of the conic. We conclude that the pairs of 
 points P \P' constitute an involution of which the centre is the centre 
 of the conic, 
 
 343. If the involution formed by the points P, P r on the 
 axis a has double points, each of them will be a focus of the 
 conic, since every straight line through such a double point 
 will be conjugate to the perpendicular drawn to it through 
 the point itself. 
 
252 FOCI. [344 
 
 If the involution has no double points, each of the two 
 points (Art. 128) at which the pairs PP' subtend a right angle 
 will be a focus of the conic. For every pair of mutually 
 perpendicular straight lines which meet in such a point will 
 pass through two points P, P', and will therefore be conju- 
 gate lines with respect to the conic. 
 
 From this it follows that one at least of the two axes of a 
 conic contains two foci. Further, a conic has only two 
 foci ; for every straight line which joins two foci is an 
 axis (Art. 341), and no conic (except it be a circle) has more 
 than two axes. 
 
 Consequently a central conic {ellipse or hyperbola) has two foci, 
 which are the double points of the involution PP f on an axis and are 
 also the points at which the pairs of points PP'ofthe involution on 
 the other axis subtend a right angle. 
 
 The axis which contains the foci may on this account be 
 called the focal axis. Since the foci are internal points, it is 
 seen that in the hyperbola the focal axis is that one which 
 cuts the curve (the transverse axis). 
 
 Since the centre of the conic is the centre of the 
 involution PP', it bisects the distance between the two 
 foci. 
 
 From what has been said it follows that two perpendicular 
 straight lines which are conjugate with respect to a conic meet the focal 
 axis in two points which are harmonically conjugate with respect to 
 the foci ; and they determine upon the other axis a segment which 
 subtends a right angle at either focus. 
 
 344. The normal at any point on a curve is the perpen- 
 dicular at this point to the tangent. Since the tangent and 
 normal at any point on a conic are conjugate lines at right 
 angles, they meet the focal axis in a pair of points harmoni- 
 cally conjugate with respect to the foci ; and they determine 
 on the other axis a segment which subtends a right angle at 
 either focus (Art. 343). Accordingly: 
 
 If a circle be draivn to pass through the two foci and through any 
 point on the conic, it will have the two points in which the non-focal 
 axis is cut by the tangent and normal at that point as extremities of 
 a diameter. 
 
 And again (Art. 60): 
 
 The tangent and normal at any point on a conic are the bisectors 
 
347] FOCI. 253 
 
 of the angle made with one another ly the rays which join that point 
 to the foci*. 
 
 These rays are called the focal radii of the given poinlw 
 
 345. A pair of conjugate lines which intersect at ight 
 angles in a point 8 external to the conic are harmonically 
 conjugate with respect to the tangents from 8 to the conic 
 (Art. 264) as well as with respect to the rays joining 8 to the 
 foci (Art. 343); therefore: 
 
 The angle between two tangents and that included ly the straight 
 lines which join the point of intersection of the tangents to the foci 
 have the same bisectors \. 
 
 346. In the parabola, the point at infinity on the axis, re- 
 garded as a point P, coincides with its correspondent P'; for 
 the straight line at infinity, being a tangent to the conic at 
 the said point P, passes through iti~own pole. 
 
 Accordingly one of the double points of the involution 
 determinftrbn the axis by the pairs, of conjugate orthogonal 
 rays, i.e. one of the foci, is at infinity. The other double point 
 lies at a finite distance, and is generally spoken of as the focus 
 of the parabola ; 
 
 Since in the case of the parabola one focus is at infinity, 
 the theorems proved" above (Arts. 343-345) become the follow- 
 ing: 
 
 Two conftigate orthogonal rays, and in particular the tangent and 
 normal at any point on the parabola, meet the axis in tioo points 
 which are equidistant from the focus. 
 
 The tangent and normal at a point on a parabola are the bisectors 
 of the angle which the focal radius of the point makes with the 
 diameter passing through the point %. 
 
 The straight line which connects the focus with the point of inter- 
 section of two tangents to a parabola makes with 'either of the 
 tange?its the same angle that the axis makes with the other tangent. 
 
 347. From the last of these may be immediately deduced 
 the following theorem : 
 
 The circle circumscribing a tria?igle formed by three tangents to a 
 parabola passes through the focus. 
 
 Let P QB (Fig. 2 1 z) be a triangle formed by three tangents 
 
 * Apollonius, loc. cit.y iii. 48. 
 
 f Ibid., iii. 46. 
 
 % De la Hire, loc. cif., lib. viii. prop. 2. 
 
254 
 
 FOCI. 
 
 [348 
 
 Fig. 212. 
 
 to a parabola, and let F be the focus. Considering the 
 tangents which meet in P, the angle FPQ is equal to that 
 
 made by PR with the axis ; and 
 considering the tangents which 
 meet in P, the angle FPQ is 
 equal to that made by PP with 
 the axis. Hence the angles 
 FPQ , FPQ are equal, and there- 
 fore P, Q , P , F lie on the same 
 circle. 
 
 Corollary. The locus of the 
 
 foci of all parabolas which touch the 
 
 three sides of a given triangle is the 
 
 circumscribing circle of the triangle. 
 
 This corollary gives the construction for the focus of a 
 
 parabola which touches four given straight lines. And since 
 
 only one such parabola can be drawn (Art. 157), we conclude 
 
 that: 
 
 Given four straight lines, the circles circumscribing the four 
 triangles which can be formed by taking the lines three and three 
 together all pass through the same point. 
 
 348. The polar of a focus is called a directrix. 
 
 The two directrices are straight lines perpendicular to the 
 
 transverse axis and external to the conic, since t^e foci lie 
 
 on the transverse axis and are internal to the conic (Art. 343). 
 
 In the case of the parabola, the straight line at infinity 
 
 is one directrix ; the other 
 lies at a finite distance, 
 and is generally spoken 
 of as the directrix of the 
 parabola. 
 
 If F be a focus, and if 
 
 the tangent at any point 
 
 X on a conic cut the 
 
 corresponding directrix in 
 
 Y, this point I will be the 
 
 pole of the focal radius 
 
 FX. Therefore FX, FY are conjugate lines with respect to 
 
 the conic, and since they meet in a focus, they will be at 
 
 right angles: consequently: 
 
 Fig. 213. 
 
349] FOCI. 255 
 
 The part of a tangent to a conic intercepted between its point of 
 contact and a directrix subtends a right angle at the corresponding 
 focus. 
 
 349. Let the tangent and normal at any point M on a 
 conic meet the focal axis in P ,P f respectively, and let them 
 meet the other axis in Q , Q' respectively (Fig. 213). From M 
 let perpendiculars MP", MQ" be drawn to the axes. 
 From the similar triangles OPQ , Q"MQ 
 
 OP:OQ=Q"M:Q"Q,\ 
 and from the right-angled triangle Q'MQ 
 
 Q"M:Q"Q = Q'Q":Q"M; 
 .-. OP:OQ = Q'Q":Q"M 
 = Q'Q":OP", 
 or OP.OP"=OQ.Q'Q" 
 
 = OQ(Q'0+OQ"), 
 
 so that OP.OP"-OQ.OQ"=OQ.Q'0 (l) 
 
 But P and P" are a pair of conjugate points, since MP" is 
 the polar of P; similarly Q and Q" are conjugate points. 
 Therefore (Art. 294) 
 
 OP . OP" = OA 2 and OQ . OQ" = ± OP 2 , 
 where OA , OP are the lengths of the semiaxes, and the double 
 sign refers to the two cases of the ellipse and the hyperbola. 
 Again, the points Q , Q' subtend a right angle at either of the 
 two foci F,F' (Art. 343) so that 
 
 OQ . Q'O = OF 2 . 
 Substituting, (l) becomes 
 
 OF 2 = OA 2 + OP 2 . 
 This shows that in the ellipse OA > OP ; so that the focal 
 axis is the axis major. 
 
 Eeferring now to Figs. 214 and 215, 
 FA = FO+OA, 
 FA'= FO + OA' =FO-OA; 
 .: FA.FA' = F0 2 -OA 2 
 = + OP 2 . 
 If F> be the point in which a directrix cuts the focal axis, 
 the vertices A and A f of the conic will be harmonically conju- 
 gate with respect to F and the point B where the polar of F 
 cuts AA' (Art. 264); therefore, since bisects AA f , 
 OA 2 = OF.OD. 
 
256 
 
 FOCI. 
 
 [350 
 
 The parabola has one vertex at infinity; consequently 
 the other lies midway between the focus and the directrix 
 (Fig. 218). 
 
 350. If a focus F of a conic C be taken as centre of homo- 
 logy, and a conic C' be constructed homological with C and 
 
 Fig. 214. Fig. 215. 
 
 having its centre at F, it has been seen (Arts. 340, 341) that 
 C' is a circle. But by what has been proved in Art. 77, if 
 if and M' are a pair of corresponding points of C and C', 
 
 FM 
 
 : MP = constant, 
 
 FM' 
 
 or 
 
 FM 
 MP 
 
 FM' x constant, 
 
 where MP (Figs. 214, 215) is the distance of M from the 
 vanishing line, that is from the polar of F, i. e. the correspond- 
 ing directrix. Now FM' is constant, because C' is a circle ; 
 therefore 
 
 The distance of any point on a conic from a focus bears a constant 
 ratio to its distance from the corresponding directrix. 
 
 Moreover, this ratio is the same for the two foci. For let 
 (Figs. 214, 215) be the centre of the conic, F, F' the foci, A , A' 
 the vertices lying on the focal axis, D,l)' the points in which 
 this axis is cut by the directrices ; then (Art. 294) 
 
 OA 2 = OA' 2 = 0F.0D = OF'. OB'. 
 But OF' = - OF, so that A'D' = - AD and F'A'= - FA, 
 and therefore FA:AD= F'A' : A'D', 
 
 which shows that the ratio is the same for F and for F'. 
 
 In the case of the parabola the ratio in question is unity, 
 
352] FOCI. 257 
 
 because (Art. 349) the vertex of a parabola is equally distant 
 from the focus and the directrix. Therefore 
 
 The distance of any point on a parabola from the focus is equal to 
 its distance from the directrix. 
 
 351. Conversely, the locus of a point M which is such that its 
 distance from a fixed point F bears a constant ratio e to its distance 
 
 from a fixed straight line d is a conic of which F is a focus and d the 
 corresponding directrix *. 
 
 For let MP (Figs. 214, 215) be drawn perpendicular to d; 
 then by hypothesis 
 
 FM _ 
 MP ~ e * 
 
 Let now the figure be constructed which is homological 
 with the locus of M ; F being taken as centre of homology, 
 and d as vanishing line. If M' be the point corresponding to 
 M s then (Art. 77) 
 
 FM 
 
 -j^f-. : MP = constant. 
 
 FM' 
 
 These two equations show that FM f is constant ; thus the 
 locus of M ' is a circle, centre F. The locus of M is there- 
 fore a conic (Art. 23) having one focus at F (Art. 341). And 
 since the straight line at infinity is the polar of F with 
 respect to the circle, the straight line d is the polar of F with 
 respect to the conic ; i.e. it is the directrix corresponding to F. 
 
 352. The length of a chord of a conic drawn through a focus 
 perpendicular to the focal axis is called the latus rectum or 
 the parameter of the conic. 
 
 Let MFM' (Fig. 216) be a chord of a conic drawn through a 
 focus F, and let iV be the point where it cuts the corresponding 
 directrix. Let LFL f be the latus rectum drawn through F. 
 Then since the directrix is the polar of the focus, N and F 
 are harmonic conjugates with regard to M and M f . There- 
 fore 
 
 211 
 
 + 
 
 NF~ NM^ NM' 
 and if perpendiculars MK , FD , M'K' be let fall on the 
 directrix, 
 
 2 
 
 7 + 
 
 FD ~ M'K' T MK 
 
 * Pappus, Math. Collect., lib. vii. prop. 238. 
 
258 
 
 But by Art. 350 
 
 FOCI. 
 
 [353 
 
 'K'\ FD : MK = M'F: FL : FM: 
 " FL" M f F^ FM' 
 
 that is to say : 
 
 In any conic, half the latus 
 the segments of any focal chord. 
 
 is a harmonic mean between 
 
 - 
 
 
 "X 
 
 
 
 X 
 
 
 ■M 
 
 
 Tu 
 
 
 
 0' 
 
 
 
 
 \ 
 
 
 
 
 JP 
 
 
 
 K' 
 
 -7^ 
 
 
 
 
 N 
 
 ID 
 
 
 
 
 
 3 
 
 L' 
 
 
 A.' 
 
 Fig. 2 1 6. 
 
 Corollary. If 31, M f be taken at A', A respectively, 
 
 FL = i (~AF + FA') 
 
 - i _^1_ 
 ~ 2 AF.FA' 
 
 OA 
 
 = Tm ( b y Art - 349 )> 
 
 so that FL= + — — , 
 
 — 0^4 
 
 which gives the length of the semi-latus rectum in terms of 
 the semi-axes. 
 
 In the parabola r—j = o , so that FL = 2 FA. 
 
 353. Theorem. In the ellipse the sum, and in the hyperbola the 
 difference, of the focal radii of any point on the curve is constant*. 
 Let M be any point on a central conic (Figs. 314, 215) whose 
 
 * Apolloxius, he. cit., iii. 51, 52. 
 
356] FOCI. 
 
 foci are F, F' and directrices d, d'; and let (if, d) &c. denote 
 as usual the distance of M from d, &c. By Art. 351 
 FM F'M 
 
 (M,d)~(M,d')- € ' 
 FM±F'M 
 •*' (M,d)±(M i d / )~ € ' 
 But (Fig. 214) in the ellipse (Jf, d) + (M t d'), and (Fig. 215) 
 in the hyperbola (M, d) — (M, d') is equal to the distance DJ)' 
 between the two directrices ; therefore 
 
 FM±F'M=*.JDD', 
 which proves the proposition. 
 
 Conversely : The locus of a point the sum (difference) of whose- 
 distances from two fixed points is constant is an ellipse (a hyperbola) 
 ofzvhich the given points are the foci. 
 
 354. If in the proposition of the last Article the point M be 
 taken at a vertex A, 
 
 e.DD'=FA±F'A 
 
 = zOA 
 
 = AA' ) 
 so that the length of the focal axis is the constant value of the sum 
 or difference of the focal radii. It is seen also that the constant 
 e is equal to the ratio of the length of the focal axis to the 
 distance between the directrices. 
 
 355. Since by Art. 294 
 
 OA 2 = OF.OD, 
 or AA /2 = FF'.DI>', 
 
 _ AA' _ FF 
 A € -DD'-AA' ; 
 so that the constant e is equal to the ratio of the distance 
 between the foci to the length of the focal axis. Now in the 
 ellipse FF' <AA\ in the hyperbola FF' >AA\ in the parabola 
 FF'=AA'= 00, in the circle FF'=o. Therefore the conic is 
 an ellipse, a hyperbola, a parabola, or a circle, according as 
 e<i, e>i, e = i, or e = o. This constant e is called the eccen- 
 tricity of the conic. 
 
 356. Theoeem. The locus of the feet of perpendiculars let fall 
 from a focus upon the tangents to an ellipse or hyperbola is the 
 
 circle described on the focal axis as diameter *. 
 
 * Apollonius, loc. tit., iii. 49, 50. 
 
 S 2 
 
260 
 
 FOCI. 
 
 [357 
 
 Take the case of the ellipse (Fig. 217). If F, F' are the foci, 
 and M is any point on the curve, join F'M and produce it to G 
 making MG equal to MF. Then F'G will (Art. 354) be equal 
 
 to AA ' whatever be the posi- 
 tion- of M ; thus the locus of G 
 is a circle, centre F' and radius 
 equal to AA'. 
 
 If FG be joined, it will cut 
 the tangent at M perpendicu- 
 larly, since this tangent (Art. 
 344) bisects the angle F3IG; 
 and the point U where the two 
 lines intersect will be the mid- 
 dle point of FG because FMG 
 is an isosceles triangle. There- 
 fore OU is parallel to F'G and equal to \F'G, that is, to OA ; 
 i. e. the locus of U is the circle on AA ' as diameter. 
 
 A similar proof holds good for the hyperbola, except that from the 
 greater of the two MF, MF' must be cut off a, part MG equal to the less. 
 
 357. If FU, FU' (Fig. 217) are the perpendiculars let fall 
 from a focus F on a pair of parallel tangents, U,F,U' will 
 evidently be collinear. And since U and U' both lie on the 
 circle described oh A A ' as diameter, 
 
 FU . FU' = FA . FA' 
 
 = + OW (Art. 349), 
 according as the conic is an ellipse or a hyperbola. 
 
 Thus the product of the distances of a pair of parallel tangents 
 from a focus is constant. 
 
 Since the perpendicular let fall from the other focus F' on 
 the tangent at M is equal to FU', it follows that 
 
 The product of the distances of any tangent to an ellipse (hyper- 
 bola) from the two foci is constant, and equal to the square of half 
 the minor (conjugate) axis. 
 
 Conversely : The envelope of a straight line which moves in such a 
 way that the ptroduct of its distances from two fixed points is constant 
 is a conic ; an ellipse if the value of the constant is positive, a hyperbola 
 if it is negative. 
 
 358. Let F (Fig. 218) be the focus of a parabola, A the 
 vertex, M any point on the curve, N the point of intersection 
 of the tangents at M and A. If NF' be drawn to the infinitely 
 
360] 
 
 FOCI. 
 
 26 L 
 
 distant focus I' (i. e. if NF' be drawn parallel to the axis), the 
 angles ANF f , FNM will be equal (Art. 346). But ANF' is a 
 right angle ,'theref ore FNM 
 is a right angle also. Thus 
 
 The foot of the perpen- 
 dicular let fall from the focus 
 of a parabola on any tangent 
 lies on the tangent at the 
 vertex. 
 
 Coeollaby. Since any 
 point on the circumscribing 
 circle of a triangle may be 
 regarded (Art. 347) as the focus of a parabola inscribed in the 
 triangle, it follows at once from the theorem just proved that if from 
 any jwint on the circumscribing circle of a triangle perpendiculars be 
 let fall on the three sides, their feet will be collinear *. 
 
 359. The theorem of Art. 356 may be put into the following form: 
 If a right angle move in its plane in such a way that its vertex 
 
 describes a fixed circle, while one of its arms passes always through a 
 fixed point, the envelope of its other arm will be a conic concentric with 
 the given circle, and having one focus at the fixed point. The conic is 
 an ellipse or a hyperbola according as the given point lies within or 
 without the given circle t. 
 
 So too the corresponding theorem (Art. 358) for the parabola may 
 be expressed in a similar form as follows : 
 
 If a right angle move in its plane in such a way that its vertex 
 describes a fixed straight line, while 
 one of its arms passes always through 
 a fixed point, the other arm will en- 
 velope a parabola having the fixed 
 point for focus and the fixed straight 
 line for tangent at its vertex. 
 
 360. I. Let the tangents at 
 the vertices of a central conic 
 be cut in P, P r by the tangent 
 at any point M (Fig. 219). The 
 three tangents form a triangle 
 circumscribed about, the conic, 
 two of the vertices of which Fi s- 2I 9- 
 
 are P and P r , the third (at infinity) being the pole of the 
 
 * For other proofs of this see Art. 416. 
 
 f Maclauein, Geometria Organica, pars II a . prop. xi. 
 
262 FOCI. [361 
 
 axis AA'. Therefore (Art. 274) the straight lines drawn from 
 P and P' to any point on the axis will be conjugate to one 
 another with respect to the conic. Thus, in particular, the 
 straight lines joining P and P' to a focus will be conjugate 
 to one another; but conjugate lines which meet in a focus 
 are mutually perpendicular (Art. 343); consequently the circle 
 on PP' as diameter will cut the axis A A' at the foci*. 
 
 II. Let the tangent PMP' cut the axis A A' at N; then J^is 
 the harmonic conjugate of if with respect to P, P f (Art. 194). 
 Consider now the complete quadrilateral formed by the 
 lines FP,F'P,FP\F'P'. Two of its diagonals are FF' and 
 PP'; the third diagonal must then cut FF' and PP' in points 
 which are harmonically conjugate to N with regard to F, F' 
 and P , P' respectively. It must therefore be the normal at 
 M to the conic t- 
 
 361. Let TM , TN (Fig. 220) be a pair of tangents to a conic, 
 M and N their points of contact, 
 F a focus, d the corresponding 
 directrix. If the chord MN cut d 
 in P, this point is the pole of TF; 
 therefore TFP is a right angle (Art. 
 343) J. 
 
 But MN is divided harmonically 
 by FT and its pole P ; thus 
 F(MNTP) is a harmonic pencil, 
 and consequently FT,FP are the 
 bisectors of the angle MFN. Accordingly : 
 
 One of the bisectors of the angle which a chord of a conic subtends 
 at a focus passes through the pole of the chord. The other bisector 
 meets the chord at its point of intersection with the directrix corre- 
 sponding to the focus. 
 
 . Or the same thing may be stated in a different manner, thus : 
 The straight line which joins a focus to the point of intersection of 
 a pair of tangents to a conic makes equal (or supplementary) angles 
 with the focal radii of their points of contact §. 
 
 * Apollonius, loc. tit., iii. 45. Desargues, (Euvr'es, i. pp. 209, 210. 
 t Apollonius, loc. tit, iii. 47. 
 
 X If the points M and N are taken indefinitely near to one another, this reduces 
 to the theorem already proved in Art. 348. 
 § De la Hire, loc. tit., lib. viii. prop. 24. 
 
 Fig. 220. 
 
362] 
 
 FOCI. 
 
 263 
 
 362. Let the tangents TM , TN be cut by any third tangent 
 in M' ' , N' respectively (Figs. 221, 222); let L be the point of 
 contact of this third tangent. The following relations will 
 hold among the angles of the figures : 
 
 N'FL = NFN' = I NFL, 
 LFM' = M'FM = I LFM, 
 whence by addition 
 
 N'FL + LFM' = $ (NFL + LFM), 
 or N'FM' = \ NFM = NFT = TFM* 
 
 Let now the tangents TM, TN be fixed, while the tangent 
 M'N' is supposed to vary. By what has just been proved, 
 the angle subtended at the focus by the part M'N' of the 
 
 Fig. 221, 
 
 Fig. 222. 
 
 variable tangent intercepted between the two fixed ones is 
 constant. As the variable tangent moves, the points M ' , N' 
 describe two projective ranges (Art. 149), and the arms 
 FM ', FN' of the constant angle M'FN' trace out two con- 
 centric and directly equal pencils (Art. 108). Accordingly: 
 
 * In this reasoning it is supposed that FM', FN', FT are all internal bisectors ; 
 4. e. that either the conic is an ellipse or a parabola, or that if it is a hyperbola, 
 the three tangents all touch the same branch (Fig. 221). If on the contrary two 
 of the tangents, for example TM and TN, touch one branch and the third M'N' 
 the other branch (Fig. 222), then FM' and FN' will be external bisectors. In 
 that case, 
 
 N'FL = ± NFL- - 
 
 LFM' = I LFM + - 
 2 a 
 
 (the angles being measured all in the same direction) ; 
 
 .-. N'FM' = \ NFM, just as in the case above. 
 
264 
 
 FOCI. 
 
 [363 
 
 The ranges which a variable tangent to a conic 
 fixed tangents are projected from either focus by 
 ' pencils. 
 
 on two 
 of two 
 
 This theorem clearly holds good for the cases of the parabola and 
 its infinitely distant focus, and the circle and its centre. For the 
 parabola it becomes the following : 
 
 Two fixed tangents to a parabola intercept on any variable tangent 
 to the same a segment whose projection on a line perpendicular to the 
 axis is of constant length. 
 
 The general theorem may also be put into the following form : 
 
 One vertex F of a variable triangle M'FN' is fixed, and the angle 
 
 M'FN f is constant, while the other vertices M', N' move respectively on 
 
 fixed straight lines TM, TN. The envelope of the side M'N' is a 
 
 conic of which F is a focus, and which touches the given lines TM, TN. 
 
 363. The problem, Given the two foci F, F' of a conic and a 
 
 tangent t, to construct the conic, is 
 determinate, and admits of a single 
 solution, as follows. 
 
 Join FF' (Figs. 223, 224) and let 
 it cut t in P; take P' the harmonic 
 conjugate of P with respect to F and 
 F'. If a straight line P'M be drawn 
 perpendicular to t, it will be the 
 normal corresponding to the tangent 
 t (Art. 344), i.e. if will be the point 
 of contact of t. Draw MP" perpendicular to FF' ; it will be the 
 polar of P, and P , P" will be conjugate points with respect to the 
 
 Fig. 223. 
 
 Fig. 224. 
 
 conic. If then FF' be bisected at 0, and on FF' there be taken 
 two points A, A' such that OA* = OA" 1 = OP. OP", A and A' will 
 
364] 
 
 FOCI. 
 
 265 
 
 be the vertices of the conic. The conic is therefore completely deter- 
 mined ; for three points on it are known (M, A , A') and the tangents 
 at these three points (t and the straight lines AC, A'C drawn 
 through A , A' at right angles to AA'). 
 
 An easy method of constructing the conic by tangents is to describe 
 any circle through i^and F', cutting AC, A / C / in H and K, II' and K f 
 respectively (Fig. 224). Then if the chords UK', H'K be drawn 
 which intersect crosswise in the centre of the circle (which lies on 
 the non-focal axis), these will be tangents to the conic (Art. 360). 
 Every circle through F and F' which cuts AC and A'C thus deter- 
 mines two tangents to the conic. 
 
 The conic is an ellipse or a hyperbola according as t cuts the 
 segment FF r externally or internally. 
 
 The conic is a parabola when F f is at infinity (Fig. 225). In this 
 case produce the axis FF to P' making FP' equal to PF, and draw 
 P'M perpendicular to t ; then M will be the point of contact of the 
 given tangent t. Draw MP" perpendicular to the axis ; then P and 
 P" will be conjugate points with 
 respect to the parabola. And since 
 the involution of conjugate points 
 on the axis has one double point 
 at infinity, the middle point A 
 of PP" will be the other double 
 point, i.e. the vertex of the parabola. 
 The parabola is therefore com- 
 pletely determined, since two points 
 on it are known (M and A), and 
 the tangents at these points (t and 
 the straight line drawn through A 
 at right angles to the axis). 
 
 364. On the other hand, the problem, To construct the conic which 
 has its foci at two given points F, F f and which passes through a 
 given point M, which is also a determinate one, admits of two solutions. 
 For if the locus of a point be sought the sum of whose distances from 
 F and F' is equal to the constant value FM+F'M, an ellipse is 
 arrived at ; but if the locus of a point be sought the difference of whose 
 distances from i^and i^'is equal to FM^FM', a hyperbola is found. 
 
 This may also be seen from the theorem of Art. 344, which shows 
 that if the straight lines t, t' be drawn bisecting the angle FMF' 
 (Fig. 223) each of these lines will be a tangent at M to a conic which 
 satisfies the problem, the other line being the corresponding normal 
 to this conic. The finite segment FF' is cut or not by the tangents 
 according as the conic is a hyperbola or an ellipse. There will 
 consequently be two conies which have F, F' for foci and which pass 
 
 Fig. 225. 
 
266 FOCI. [365 
 
 through M\ a hyperbola having for tangent at M that bisector t r 
 which cuts the segment FF f , and for normal the other bisector t ; 
 and an ellipse having t for tangent at M and t f for normal. 
 
 These two conies, having the same foci, are concentric and have 
 their axes parallel. They will cut one another in three other points 
 besides M ; and their four points of intersection will form a rectangle 
 inscribed in the circle of centre and radius OM ; in other words, 
 the three other points will be symmetrical to M with respect to the 
 two axes and the centre. This is evident from the fact that a conic 
 is symmetrical with respect to each of its axes. 
 
 365. Through every point M in the plane then pass two conies, 
 an ellipse and a hyperbola, having their foci at F and F'. In other 
 words, the system of confocal conies having their foci at F and F' is 
 composed of an infinity of ellipses and an infinity of hyperbolas ; and 
 through every point in the plane pass one ellipse and one hyperbola, 
 which cut one another there orthogonally and intersect in three other 
 points. 
 
 Two conies of the system which are of the same kind (both 
 ellipses or both hyperbolas) clearly do not intersect at all. 
 
 Two conies of the system however which are of opposite kinds 
 (one an ellipse, the other a hyperbola) always intersect in four points, 
 and cut one another orthogonally at each of them. This may be 
 seen by observing that the vertices of the hyperbola are points lying 
 within the segment FF', and therefore within the ellipse. On the 
 other hand, there must be points on the hyperbola which lie outside 
 the ellipse ; for the latter is a closed curve which has all its points 
 at a finite distance, while the former extends in two directions to 
 infinity. The hyperbola therefore, in passing from the inside to the 
 outside of the ellipse, must necessarily cut it. 
 
 No two conies of the system can have a common tangent ; because 
 (Art. 363) only one conic can be drawn to have its foci at given 
 points and to touch a given straight line. 
 
 Any straight line in the plane will touch a determinate conic of 
 the system, and will be normal, at the same point, to another 
 conic of the system, belonging to the opposite kind. The first of 
 these conies is a hyperbola or an ellipse according as the given 
 straight line does or does not cut the finite segment FF' . 
 
 366. If first point F' lies at infinity, the problem of Art. 364 
 becomes the following : Given the axis of a parabola, the focus F, and 
 a point M on the curve, to construct the parabola. 
 
 Just as in Art. 364, there are two solutions (Fig. 226). The 
 tangents at M to the two parabolas which satisfy the problem are 
 the bisectors of the angle made by MF with the diameter passing 
 through If; therefore the parabolas cut orthogonally at M and 
 
367] 
 
 FOCI. 
 
 267 
 
 consequently intersect at another point, symmetrical to M with 
 respect to the axis. The parabolas cannot intersect in any other 
 finite point, since they touch one another at infinity *. 
 
 The tangents to the two parabolas at M cut the axis in two points 
 P, P' which lie at equal distances 
 on opposite sides of F; and if P" is 
 the foot of the perpendicular let fall 
 rom M on the axis, the vertices A , A' 
 of the parabolas are the middle points 
 
 respec- 
 
 Fig. 226. 
 
 of the segments PP", P'P' 
 tively. 
 
 Suppose A and P " to fall on the same 
 side of F. Then since P'P" < P'P, 
 and P'A' is the half of P'P", and 
 P'F the half of P'P, therefore. 
 P'A'<P'F; i.e. A and A' fall on 
 opposite sides of F. It follows that in 
 the system composed of the infinity of 
 parabolas which have a common axis 
 and focus, two parabolas intersect (orthogonally and in two points) 
 or do not intersect, according as their vertices lie on opposite sides 
 or on the same side of the common focus. 
 
 Since F, A, A' are the middle points of PP', PP", P'P" respec- 
 tively, we have the relations 
 
 FP + FP'=o, 
 2FA = FP + FP", 
 2FA'=FP' + FP", 
 whence the following are easily deduced : 
 FP"=FA +FA',f 
 FP = FA-FA'=A'A, 
 FP' = FA' -FA = AA'. 
 
 These last relations enable us at once to find the points P, P', P" 
 when A and A' are known. The point M (and the symmetrical point 
 in which the parabolas intersect again) can then be constructed by 
 observing that FM is equal to FP or FP'. 
 
 367. It has been seen that a conic is determined when the two 
 foci and a tangent are given. It can also be shown that a conic 
 is determined when one focus and three tangents are given', this follows 
 
 * That is to say, if the figure be constructed which is homological with that 
 formed by the two parabolas, it will consist of two conies touching one another 
 at a point situated on the vanishing line of the new figure, and intersecting in 
 two other points. 
 
 t Hence the middle point of A A' is also the middle point of FP". 
 
268 FOOL [368 
 
 at once from the proposition at the end of Art. 362. For let LMN 
 (Fig. 227) be the triangle formed by the three given tangents, and F 
 the given focus. Then the conic is seen to be the envelope of the 
 
 base M'N' of a variable triangle 
 M'FN', which is such that the 
 vertex F is fixed, the angle 
 M'FN' is always equal to the 
 constant angle MFN, and the 
 vertices M ', N' move on the fixed 
 straight lines LM , LN respec- 
 tively. 
 
 In order to determine the 
 other focus F', we make use of 
 the theorem of Art. 345. At 
 the point M make the angle 
 LMF' equal to FMN ; and at 
 ■p. 227 " the point N make the angle 
 
 LNF' equal to FNM (all these 
 angles being measured in the same direction) ; then the point of 
 intersection of MF', NF' will be the second focus F'. 
 
 The investigation of the circumstances under which the conic is an 
 ellipse, a hyperbola, or a parabola, is left as an exercise to the student. 
 The following are the results : 
 
 (1) The conic is an ellipse if F lies within the triangle LMN) or 
 if F lies without the circle circumscribing LMN and within one of 
 the (infinite) spaces bounded by one of the sides of the triangle and 
 the other two produced : 
 
 (2) a hyperbola if F lies inside the circle but outside the triangle ; 
 or if it lies within one of the (infinite) F-shaped spaces which have 
 one of the angular points of the triangle LMN for vertex and are 
 bounded by the sides meeting in that angular point, both produced 
 backwards : 
 
 (3) a parabola if F lies on the circle circumscribing the triangle 
 LMN, as we have seen already (Art. 347) *. 
 
 368. Let TM , TN (Fig. 228) be a pair of tangents to an 
 ellipse or hyperbola which intersect at right angles. If per- 
 pendiculars FU,F'U' and FF, F'V be let fall upon them 
 respectively from the foci F and F', then evidently TU= FF 
 and TU'= V'F'. But by Art. 357 we have FF. F'F'= ± OB 2 ; 
 therefore TU . TU'= ± OB 2 . But since U and V both lie on 
 
 * Steineb, Developpement (Tune serie de thdoremes relatifs aux sections coniqucs 
 (Annates de Gergonne, t. xix. 1828, p. 47) ; Collected Works, vol. i. p. 198. 
 
369] 
 
 FOCI. 
 
 269 
 
 the circle described upon the focal axis AA' as diameter (Art. 
 356), the rectangle TV . TV' is the power of the point T with 
 respect to this circle, and is equal to OT 2 — A 2 . Thus 
 
 OT 2 = OA 2 ± OB 2 = constant, 
 so that we have the following theorem * : 
 
 The locus of the point of intersection of two tangents to an ellipse 
 or a hyperbola which cut at right angles is a concentric circle. 
 
 This circle is called the director circle of the conic f. 
 
 In the ellipse OT 2 = OA 2 + OB 2 , so that the director circle circum- 
 scribes the rectangle formed by the tangents at the extremities of 
 the major and minor axes. In the hyperbola OT 2 ■= OA 2 —OB 2 , so 
 that pairs of mutually perpendicular tangents exist only if OA > OB. 
 If OA = OB, i.e. if the hyperbola is equilateral (Art. 395), the di- 
 rector circle reduces simply to the centre ; that is, the asymptotes 
 are the only pair of tangents which cut at right angles. If OA < OB, 
 the director circle has no real existence ; the hyperbola has no pair 
 of mutually perpendicular tangents. 
 
 1/ 
 
 Fig. 228. 
 
 Fig. 229. 
 
 369. Consider now the case of the parabola (Fig. 229). Let 
 F be the focus, A the vertex, TH and TK a pair of mutually 
 perpendicular tangents. If these meet the tangent at the 
 vertex in i^Tand irrespectively, the angles FHT , FKT will be 
 right angles (Art. 358), so that the figure THFK is a rectangle. 
 Therefore TH=KF; and since the triangles TEH, FAK are 
 evidently similar, TF=AF. The locus of the point T is 
 
 * De la Hire, loc. cit., lib. viii. props. 27, 28. 
 + Gaskin, The geometrical construction of a conic section, 
 1852), chap. iii. prop. 10 et seqq. 
 
 (Cambridge, 
 
270 
 
 FOCI. 
 
 [370 
 
 therefore a straight line parallel to HK, and lying at the same 
 distance from HK (on the opposite side) that F does. That is 
 to say : 
 
 The locus of the point of intersection of two tangents to a para- 
 bola which cut at right angles is the directrix*. 
 
 Since the director circle of a conic is concentric with the latter, it 
 must in the case of the parabola have an infinitely great radius. In 
 other words, it must break up into the line at infinity and a finite 
 straight line. And we have just seen that this finite straight line is 
 the directrix. 
 
 370. The director circle possesses a property in relation to 
 the self-conjugate triangles of the 
 conic which we will now proceed 
 to investigate. Let XYZ (Fig. 230) 
 be a triangle which is self-conjugate 
 with respect to a conic whose centre 
 is 0. Join OX and let it cut YZ 
 in X' and the conic in A'. Draw 
 OB* parallel to YZ-, let it cut XY 
 in L and the conic in B'\ and draw 
 ZL' parallel to OX to meet OB' in L'. 
 Then A' and OB ' are evidently 
 conjugate semi-diameters; alsoXand 
 X', L and L' are pairs of conjugate 
 points with respect to the conic. Therefore 
 
 OX. OX' = + OA' 2 , and 01 .OL' = ± 03'* t 
 where the positive or the negative signs are to be taken 
 according as the semidiameters OA' , 0B f are real or ideal 
 (Art. 294). 
 
 Thus for the ellipse 
 
 OX . 0X F +OL.OL'= OA' 2 + OB' 2 
 = OA 2 + OB 2 , 
 and for the hyperbola 
 
 OX . 0X'+ OL . OL' = ±(0A /2 - OB' 2 ) 
 = ±(OA 2 -OB 2 ), 
 so that in both cases (Art. 368) 
 
 OX.OX'+OL.OL'^OT 2 , (1) 
 
 where OT is the radius of the director circle. 
 
 * De la Hire, he. cit., lib. viii. prop. 26. 
 
 Fig. 230. 
 
371] FOCI. 271 
 
 Now let a circle be described round the triangle XYZ, and 
 let U be the point where it cuts OX again ; then 
 X'Y.X'Z=X'X.X'U; 
 
 .-. ru = ^-rz 
 ~ox xz ' 
 
 (from the similar triangles OLX, X'YX) 
 
 01 or 
 = ox 0L ■ 
 
 Therefore equation (l) gives 
 
 OT 2 = OX.OX'+OX.X'U 
 = OX.OU, 
 
 that is to say : The centre of a conic has with respect to the circum- 
 scribing circle of any triangle self-conjugate to the conic a constant 
 power, which is equal to the square of the radius of the director 
 circle. 
 
 Or in other words : 
 
 The circle circumscribing any triangle which is self-conjugate with 
 regard to a conic is cut orthogonally by the director circle *, 
 
 The following particular cases of this theorem are of interest : 
 
 I. The centre of the circle circumscribing any triangle which is self- 
 conjugate with respect to a parabola lies on the directrix. 
 
 II. The circle circumscribing any triangle which is self-conjugate 
 with respect to an equilateral hyperbola passes through the centre of 
 the conic. 
 
 371. Consider a quadrilateral circumscribed about a conic. Since 
 each of its diagonals is cut harmonically by the other two, the circle 
 described on any one of the diagonals as diameter is cut orthogonally 
 by the circle which circumscribes the diagonal triangle (Art. 69). 
 But the diagonal triangle is self-conjugate with respect to the conic 
 (Art. 260), and therefore its circumscribing circle cuts orthogonally 
 the director circle (Art. 370). Consequently the director circle and 
 the three circles described on the diagonals as diameters all cut 
 orthogonally the circle circumscribing the diagonal triangle. Now 
 by Newton's theorem (Art. 318) the centres of the four first-named 
 circles are collinear; and circles whose centres are collinear and 
 which all cut the same circle orthogonally have a common radical 
 axis. Therefore : 
 
 The director circle of a conic, and the three circles described on 
 
 * Gaskin, he. cit., p. 33. 
 
272 FOCI. [372 
 
 the diagonals of any circumscribed quadrilateral as diameters, are 
 coaxial. 
 
 In the parabola the director circle reduces to the directrix and 
 the straight line at infinity; in this case then the above theorem 
 becomes the following : 
 
 If a quadrilateral is circumscribed about a imrabola, the three 
 circles described on the diagonals of the quadi-ilateral as diameters have 
 the directrix for their common radical axis. 
 
 372. If in the theorem of Art. 371 the quadrilateral be supposed 
 to be given, and the conic to vary, we arrive at the following 
 theorem : 
 
 The director circles of all the conies inscribed in a given quadri- 
 lateral form a coaxial system, to which belong the three circles having 
 as diameters the diagonals of the quadrilateral. 
 
 There is one circle of such a system which breaks up into two 
 straight lines : that namely which degenerates into the radical axis 
 and the straight line at infinity. Now the director circle breaks up 
 into two straight lines — viz. the directrix and the line at infinity — 
 in the case of a parabola (Art. 369). Therefore the common radical 
 axis of the system of coaxial director circles is the directrix of the 
 parabola which can be inscribed in the quadrilateral. 
 
 If the circles of the system do not intersect, there are two of them 
 which degenerate into point-circles (the limiting points). Now the 
 director circle degenerates into a point in the case of the equilateral 
 hyperbola (Art. 368). Therefore when the circles do not cut one 
 another, the two limiting points of the system are the centres of 
 the two equilateral hyperbolas which can in this case be inscribed 
 in the quadrilateral. If the circles do intersect, the system has no 
 real limiting points ; and in this case no equilateral hyperbola can be 
 inscribed in the quadrilateral. 
 
 The circles which cut orthogonally the circles of a coaxial system 
 form another coaxial system ; if the first system has real limit- 
 ing points, the second system has not, and vice versa. In order 
 then to inscribe an equilateral hyperbola in a given quadrilateral, 
 it is only necessary to describe circles on two of the diagonals of the 
 quadrilateral as diameters, and then to draw two circles cutting the 
 former two orthogonally. When the problem is possible, these two 
 orthogonal circles will intersect ; and their two points of intersection 
 are the centres of the two equilateral hyperbolas which satisfy the 
 conditions of the problem. 
 
 373. If five points are taken on a conic, five quadrangles may be 
 formed by taking these points four and four together ; and the 
 diagonal triangles of these five quadrangles are each of them self- 
 conjugate with respect to the conic. If the circumscribing circles of 
 
376] FOCI. 273 
 
 these five diagonal triangles be drawn, they will give, when taken 
 together in pairs, ten radical axes. These ten radical axes will all 
 meet in the same point, viz. the centre of the conic. 
 
 374. Consider again a quadrilateral circumscribing a conic ; let 
 P and P\ Q and Q\ R and IV be its three pairs of opposite vertices. 
 If these be joined to any arbitrary point S, and if moreover from this 
 point S the tangents t , t f are drawn to the conic, it is known by the 
 theorem correlative to that of Desargues (Art. 183, right) that t and t', 
 SP and SP', SQ and SQ', SR and SR' are in involution. Now let 
 one of the sides of the quadrilateral (say P'Q'R') be taken to 
 be the straight line at infinity, so that the inscribed conic is a 
 parabola ; and let 8 be taken at the orthocentre (centre of perpen- 
 diculars) of the triangle PQR formed by the other three sides of 
 the quadrilateral. Then each of the three pairs of rays SP and SP', 
 SQ and SQ', SR and SR' cut orthogonally; therefore the same will 
 be the case with the fourth pair t and t'. But tangents to a 
 parabola which cut orthogonally intersect on the directrix (Art. 369) ; 
 therefore : 
 
 The orthocentre of any triangle circumscribing a parabola lies on 
 the directrix. 
 
 375. If in the theorem of the last Article the triangle be supposed 
 to be fixed, and the parabola to vary, we obtain the theorem : 
 
 The directrices of all parabolas inscribed in a given triangle meet in 
 the same point, viz. the orthocentre of the triangle. 
 
 Given a quadrilateral, one parabola (and only one) can always be 
 inscribed in it. By taking the sides of the quadrilateral three and 
 three together, four triangles are obtained ; and the four ortho- 
 centres of these triangles must all lie on the directrix of the parabola. 
 It follows that 
 
 Given four straight lines, the orthocentres of the four triangles 
 formed by taking them three and three together are collinear. 
 
 376. Let C be any given conic, and let C r be its polar 
 reciprocal with respect to an auxiliary conic K. The particular 
 case in which K is a circle whose centre coincides with a focus 
 F of the conic C is of great interest ; we shall now proceed to 
 consider it. 
 
 If r, / be any two straight lines which are conjugate with 
 respect to C, and if R , R' be their poles with respect to K, it 
 is known (Art. 323) that R , R r will be conjugate points with 
 respect to C '. Consider now two such lines r , / which pass 
 through F; they will be at right angles since every pair of 
 conjugate lines through a focus cut one another orthogonally. 
 
 T 
 
274 FOCI. [377 
 
 They will therefore be perpendicular diameters of the circle K, 
 and their poles B , R' with respect to K will be the points at 
 infinity on /, r respectively. These points are conjugate 
 with respect to C', and the straight lines joining them to the 
 centre of this conic are therefore a pair of conjugate diameters 
 of C'; consequently two conjugate diameters of C' are always 
 mutually perpendicular. This proves that C' is a circle; i.e. 
 the polar reciprocal of a conic, with respect to a circle which has 
 its centre at one of the foci, is a circle. 
 
 By taking the steps of the above reasoning in the opposite 
 order, the converse proposition may be proved, viz. 
 
 The polar reciprocal of a circle with respect to an auxiliary 
 circle is a conic having one focus at the centre of the auxiliary 
 circle. 
 
 As in Art. 323, it is seen that the conic is an ellipse, a 
 hyperbola, or a parabola, according as the centre of the 
 auxiliary circle lies within, without, or upon the other 
 circle. 
 
 377. If d be the directrix of the conic C corresponding to 
 the focus F, and if its pole be taken with respect to the circle 
 K, this point will evidently be the centre of the circle C' 
 (Art. 323). 
 
 The radius of the circle C' may also easily be found. For 
 in Fig. i\6 let two points X, X' be taken in the latus rectum 
 LFL' such that 
 
 FX.FL = FX'.FL'=k 2 , 
 
 where k denotes the radius of the circle K ; and let straight 
 lines be drawn through X and X' perpendicular to XFX'. These 
 straight lines are evidently parallel tangents of the circle C', 
 and the distance XX' between them is therefore equal in length 
 to the diameter of C'. But 
 
 \X , X=FX=^ i 
 FL 
 
 so that the radius of the circle C' is equal to -^ • 
 
 The eccentricity e of the conic C may be expressed in a 
 simple manner in terms of quantities depending upon the 
 two circles K and C'. For if 0' be the centre and p the 
 
379] FOCI. 275 
 
 radius of the latter circle, it has been seen that the directrix 
 is the polar of 0' with respect to K ; therefore (Fig. 216) 
 
 FD.FO'=k 2 . 
 But it has just been proved that 
 
 FL . p = k 2 ; 
 
 FL FO' 
 
 therefore (Art. 351), € = ^tft = 
 
 x tit p 
 
 378. The proposition of Art. 376 may be proved in a 
 different manner, so as to lead at once to the position and size 
 of the circle C'. 
 
 Take any point if on the (central) conic C (Fig. 217); from 
 the focus F draw FU perpendicular to the tangent at 31, and on 
 FU take a point Z such that FZ . FU= k 2 , k being as before 
 the radius of the circle K. Then the locus of Z is the polar 
 reciprocal of C with respect to K. 
 
 Now it is known (Arts. 356, 357) that CHies on the circle 
 on A A ' as diameter, and that if UF cut this circle again at U' 
 
 FU.FU'= + OB 2 . 
 Therefore FZ :FU' = k 2 : + OB 2 ; 
 
 which proves (Art. 23 [6]) that the locus of Z is a circle whose 
 centre 0' lies on FO and divides it so that FO f :FO = k 2 : OB 2 , 
 
 OA 
 
 and whose radius p is equal to k 2 . jp^ , that is, (Art. 352 Cor.) 
 
 to ~ And again, since OF. 0D= OA 2 and FD = FO+ 02, 
 
 (Figs. 214, 215), 
 
 .-. F£.FO=OF 2 -OA 2 = + OB 2 =Jc 2 ~, 
 
 by what has just been proved. 
 
 .-. FO'.FD = Jc 2 ; 
 i.e. 0' is the pole of the directrix d with respect to K. 
 
 In the particular case where h = OB, p = OA ; that is to say : 
 The polar reci])rocal of an ellipse (hyperbola) with respect to a circle 
 having its centre at a focus and its radius equal to half the minor 
 {conjugate) axis is the circle described on the major {transverse) axis as 
 diameter. 
 
 379. In the case where C is a parabola, let M be any point 
 on the curve (Fig. 218); let fall FN perpendicular to the tangent 
 at 11, and take on FN a point Z such that FZ. FN=& 2 . Then, 
 
 T 2 
 
276 FOCI. [379 
 
 as before, the locus of Z will be the polar reciprocal of C with 
 respect to K. Draw ZQ perpendicular to ZF to cut the axis 
 of the parabola in Q. 
 
 Then a circle will evidently go round QANZ, so that 
 FA.FQ = FN.FZ=k 2 ; 
 therefore Q is a fixed point, and the locus of Z is the circle on 
 QF as diameter. If 0' be the centre, p the radius of this 
 circle, 
 
 In the particular case where k is equal to half the latus rectum, 
 that is, to 2 FA , we have p = k ; that is to say : 
 
 The polar reciprocal of a parabola with respect to a circle having its 
 centre at the focus and its radius equal to half the latus rectum is a 
 circle of the same radius, having its centre at the point of intersection of 
 the axis with the directrix. 
 
CHAPTEE XXIV. 
 
 COROLLARIES AND CONSTRUCTIONS. 
 
 380. In the theorem of Art. 275 suppose the vertices B and C of 
 the inscribed triangle ABC (Fig. 188) to be the points at infinity on 
 a hyperbola ; then S will be the centre of the curve, and the theorem 
 will become the following : 
 
 If from any point ion a hyperbola parallels be drawn to the 
 asymptotes, they will meet any given diameter in two points #and G 
 which are conjugate to one another with regard to the curve. Or : 
 
 If through two points lying on a diameter of a hyperbola, which 
 are conjugate to one another with regard to the curve, parallels be drawn 
 to the asymptotes, they will intersect on the curve. 
 
 From this follows a method for the construction of a hyperbola by 
 points, having given the asymptotes and a point M on the curve. 
 
 On the straight line SM, which joins if to the point of inter- 
 section S of the asymptotes, take two conjugate points of the in- 
 volution determined by having S for centre and M for a double 
 point. These points will be conjugate to one another with respect 
 to the conic (Art. 263) ; if then parallels to the asymptotes be 
 drawn through them, the two vertices of the parallelogram so formed 
 will be points on the hyperbola which is to be constructed. 
 
 381. Let similarly the theorem of Art. 274 be applied to the 
 hyperbola, taking the sides b and c of the circumscribed triangle abc 
 to be the asymptotes ; it will then become the following : 
 
 If through the points where the asymptotes are cut by any tangent 
 to a hyperbola any two parallel straight lines be drawn, these will 
 be conjugate to one another with respect to the conic. Or : 
 
 Two parallel straight lines which are conjugate to one another with 
 respect to a hyperbola cut the asymptotes in points, the straight lines 
 joining which are tangents to the curve. 
 
 From this we deduce a method for the construction, by means of its 
 tangents, of a hyperbola, having given the asymptotes b and c and one 
 tangent m. 
 
 Draw parallel to m two conjugate rays of the involution (Art. 129) 
 determined by having m for a double ray and the parallel diameter 
 for central ray. The two straight lines so drawn will be conjugate 
 to one another with respect to the conic ; if then the points where 
 they cut the asymptotes be joined to one another, we shall have two 
 tangents to the curve. 
 
278 COROLLARIES AND CONSTRUCTIONS. [382 
 
 382. Let B and C be any two points on a parabola, and A the 
 point where the curve is cut by the diameter which bisects the chord 
 BC. Let F and G be two points lying on this diameter which are 
 conjugate with respect to the parabola, i.e. two points equidistant 
 from A (Art. 142) ; by the theorem of Art. 275, BF and CG, and 
 likewise BG and CF, will meet on the curve. 
 
 This enables us to construct by points a parabola which circum- 
 scribes a given triangle ABC and has the straight line joining A to 
 the middle 'point of BG as a diameter. 
 
 Or we may proceed according to the following method : 
 On BC take two points H and H / which shall be conjugate to one 
 another with regard to the parabola, i.e. any two points dividing BC 
 harmonically. Since H and H' are collinear with the pole of the 
 diameter passing through A, therefore by the theorem of Art. 275, a 
 point on the parabola will be found by constructing the point of 
 intersection of AH with the diameter passing through W , and another 
 will be found as the point where AW meets the diameter passing 
 through H. 
 
 383. In the theorem of Art. 274 suppose the tangent c to lie at 
 infinity ; then we see that 
 
 If a and b are two tangents to a parabola, and if from any point 
 on the diameter passing through the point of contact of a there be 
 drawn two straight lines, one passing through the point ab and the other 
 parallel to b, these wiH be conjugate to one another with regard to the 
 parabola. 
 
 This enables us to construct by tangents a parabola, having given 
 two tangents a and t, the point of contact A of one of them a. and the 
 direction of the diameters. 
 
 Draw the diameter through A and let it meet t in ; the second 
 tangent t' from will be the straight line which is harmonically 
 conjugate to t with respect to the diameter OA (the polar of the point 
 at infinity on a) and the parallel through to a. If now two straight 
 lines /* and h' be drawn through which shall be conjugate to one 
 another with regard to the parabola, i.e. two straight lines which are 
 harmonic conjugates with regard to t and t f , the parallel to $ drawn 
 from the point ha and the parallel to h drawn from the point h r a 
 will both be tangents to the required parabola. 
 
 384. If in the theorem of Art. 274 the straight line a be supposed 
 to lie at infinity, and b and c to be two tangents to a parabola, we 
 obtain the following : 
 
 The parallels drawn to two tangents to a parabola, from any point 
 on their chord of contact, are conjugate lines with regard to tJie conic. 
 
 By another application of the same theorem we deduce a result 
 already proved in Art. 178, viz. that 
 
385] 
 
 COROLLARIES AND CONSTRUCTIONS. 
 
 279 
 
 If, from a point on the chord of contact of a pair of tangents b and c 
 to a parabola, two straight lines h and h' be drawn parallel to b and c 
 respectively, the straight line joining the points he and h'b will be a 
 tangent to the curve *. 
 
 From this may be deduced a construction for the tangents to a 
 parabola determined by two tangents and their points of contact. 
 
 385. Theorem. If a conic cut the sides BC , CA , AB of a 
 triangle ABC in the points B and B', E and E', F and F' re- 
 spectively, then will 
 
 BB.BB' CE.CE' AF.AF' _ 
 
 cb.cb'' ae.aw"bf7bF~ y r\ 
 
 This celebrated theorem is due to Carnot f. 
 Consider the sides of the triangle ABC (Fig. 231) as 
 
 Fig. 231. 
 cut by the transversals BE and B' E' in the points B and B', 
 i?and E', G and G'; by the theorem of Menelaus (Art. 139) 
 BBCJ^AG_ 
 
 CB' AE'BG ~ T ' ( ' 
 
 BB' CE' AG' 
 
 and 
 
 = 1. 
 
 CB' AE' BG' 
 Again, BEE'B' is a quadrangle inscribed in the conic, and by 
 Desargues' theorem (Art. 183) the transversal^^ meets the 
 opposite sides and the conic in three pairs of points in involu- 
 tion; therefore (Art. 130) the anharmonic ratios (ABFG) and 
 (BAF'G') are equal; thus (Art. 45) (ABFG) = (ABG'F'), or 
 (ABFG): (ABG'F')=i, which gives 
 
 AF.AF' AG. AG' 
 
 .... (4) 
 
 BF.BF'' BG. 
 
 De la Hire, loc. cit., lib. iii. prop. 21. 
 
 = I. 
 
 BG' 
 
 + G&omitrie de position, p. 437. 
 
280 COROLLARIES AND CONSTRUCTIONS. [386 
 
 Multiplying together (2), (3), and (4), we obtain the relation 
 stated in the enunciation*. 
 
 386. Conversely, if on the sides BC \CA,AB respectively of 
 a triangle ABC there he taken three pairs of points B and B\ 
 B and B', F and F' such that the segments determined by them and 
 the vertices of the triangle satisfy the relation (1) of Art. 385, 
 these six 'points lie on a conic. 
 
 For let the conic be drawn which passes through the five 
 points B,B', E, B\ F, and let F" be the point where it cuts 
 AB again. By Carnot's theorem a relation holds which differs 
 only from (l) in that it has F" in the place of F'. This 
 relation, combined with (1), gives 
 
 AF':BF'=AF":BF", 
 whence (ABF'F") = i ; 
 
 and therefore (Art. 72, VII) F" coincides with F f . 
 
 * Carnot's theorem, being evidently true for the circle (since in this case 
 BD . BD' = CD . CD'> &c), may be proved without making use of involution 
 properties as follows : 
 
 Let I , J, K be the points at infinity on BO , CA , AB respectively, and sup- 
 pose Fig. 231 to have been derived by projecting from any vertex on any plane a 
 triangle A X B X C X whose sides are cut by a circle in D x and D x ', E x and E x ', F x and F' 
 respectively. Let J, , /„ K x be the points on the sides B X C X , C X A X , A V B 1 which 
 project into I, J, Irrespectively; they will of course be collinear. Then 
 BD 
 
 So 
 
 
 CD 
 
 m (BCDI) (Art. 64) 
 
 = (B l C l DJ 1 ) (Art. 63) 
 
 B X D X B X I X 
 ~~C 1 D 1 ''C 1 I l ' 
 
 BD 
 
 BD' 
 
 CD' 
 
 .BD' 
 
 B X D X ' 
 
 ~ C X D{ 
 
 B X D X 
 
 B 1 I 1 
 ■ B X D X ' 
 
 B X I X 2 
 
 CD, 
 
 CD' 
 
 ~ C X D X . 
 
 CJ? 
 " BJ*' 
 
 C x D x ' 
 
 (Euc, 
 
 ' c x i x * 
 
 i". 35, 36.) 
 
 CE 
 AE 
 AF 
 
 .CE' 
 
 .AE' 
 .AF' 
 
 A X J X * 
 B X K* 
 
 
 
 Similarly, 
 
 and BF.BF'-AKr 
 
 Multiplying these three equations together, and remembering that by the 
 theorem of Menelaus the product on the right-hand side is equal to unity, we 
 have the result required. 
 
 Carnot's theorem is true not only for a triangle but for a polygon of any num- 
 ber of sides ; the proof just given can clearly be extended so as to show this, the 
 theorem of Menelaus being capable of extension to the case of a polygon. 
 
 Menelaus' theorem is included in that of Carnot. It is what the latter reduces 
 to when the conic degenerates into two straight lines of which one lies at infinity. 
 
387] COROLLARIES AND CONSTRUCTIONS. 281 
 
 387. If the point A pass off to infinity (Fig. 232) the ratios 
 AF: AE and AF' : AE / become in the limit each equal to unity, 
 and the equation (1) of Art. 385 accordingly reduces to 
 BD . BD' CF.CF' _ 
 
 CD.CD'' BF.BF'" T * • W 
 
 Draw parallel to BC a straight line to cut CFE' in Q and 
 the conic in P and P'\ the preceding equation, applied to the 
 triangle whose vertices are C, Q, and the point at infinity 
 where PP' and BC meet, gives 
 
 QE.QE' CD . CD' _ 
 CE.CE'' QP.QP'~ *' 
 
 Fig. 232. 
 
 Multiplying together these last two equations, we obtain 
 BD.BD' _ QP.QP\ 
 BF.BF f ~ QE.QE'' 
 
 that is to say : 
 
 If through any point Q there be drawn in given directions two 
 transversals to cut a conic in P,P' and E,E' respectively, then the 
 rectangles QP . QP' and QE . QE' are to one another in a constant 
 ratio* f. 
 
 * Apollonius, loc. city lib. iii. 16-23. Desargues, loc. cit., p. 202. De la 
 Hibe, loc. cit., bk. v. props. 10, 12. 
 
 f From this follows at once the result already proved in a different manner in 
 Art. 316, viz. that if a conic is cut by a circle, the chords of intersection make 
 equal angles with the axes. 
 
 For let P , P', E, E' be the points of intersection of a circle with the conic ; 
 then (Euc. iii. 35) QP . QP' = QE . QE'. But if MOM', NCN' be the diameters 
 of the conic parallel respectively to QPP' and QEE', we have, by the theorem 
 in the text, 
 
 QP.QP'-.QE. QE' = CM. CM': CN.CN' 
 = CM* : CN*. 
 Therefore CM = CN, and consequently CM and CN (and therefore also QPP' 
 and QEE') make equal angles with the axes. 
 
282 COROLLARIES AND CONSTRUCTIONS. [388 
 
 388. Suppose in equation (5) of Art. 387 that the conic is a 
 hyperbola and that in place of BC is taken an asymptote RK 
 of the curve; then the ratio RB .RB' : KB .KB' becomes 
 equal to unity, and therefore 
 
 rf.hf'Lke.ke', 
 
 that is to say : 
 
 If through any point R [or R') lying on an asymptote there be 
 drawn, parallel to a given straight line, a transversal to cut a hyper- 
 bola in two points F and F' (B and B'), then the rectangle 
 HF.RF' (R'B . R'B') contained by the intercepts will be constant. 
 
 If the diameter parallel to the given direction R'B meets 
 the curve, then if S and 8' are the points where it meets it, 
 and if is the centre, 
 
 R'B . R'B' = 08.08' = - OS 2 . 
 
 If the diameter OT parallel to the given direction RF does 
 not meet the curve, a tangent can be drawn which shall be 
 parallel to it. The square on the portion of this tangent 
 intercepted between its point of contact and the asymptote 
 will be equal to the rectangle RF. RF' by the theorem now 
 under consideration; but this portion is (Art. 303) equal to 
 the parallel semidiameter OT; therefore RF. RF'= OT 2 , or : 
 
 If a transversal, cut a hyperbola in F and F' (in B and B') 
 and an asymptote in H (in R'), the rectangle HF.RF' (R'B .R'B') 
 is equal to + the square on the parallel semidiameter OT (08); the 
 positive or negative sign being taken according as the curve has or 
 has not tangents parallel to the transversal. 
 
 389. If the transversal cuts the other asymptote in L 
 (ml'), then by Art. 193 
 
 HF'=FL or H'B'=BL', 
 
 and consequently 
 
 FH. FL = - OT 2 or BR'. BL' = OS 2 ; 
 
 therefore : 
 
 If a transversal drawn from any point F (B) on a hyperbola 
 cut the asymptotes in H and L (in H' and L'), the rectangle 
 FH . FL (BR'. BL') is equal to + the square on the parallel 
 semidiameter ; the negative or positive sign being taken according as 
 the curve has or has not tangents parallel to the transversal. 
 
391] COROLLARIES AND CONSTRUCTIONS. 283 
 
 390. From the proposition of the last Article may be deduced a 
 construction for the axes of a hyperbola, having given a pair of conjugate 
 semidiameters OF and OT in magnitude and direction (Fig. 233). 
 
 We first construct the asymptotes. Of the two given semidiameters, 
 let OF be the one which cuts the curve. 
 Draw through F a parallel to OT', this 
 will be the tangent at F. Take on this 
 parallel FP and FQ each equal to OT ; 
 then OP and OQ will be the asymptotes 
 (Art. 304). In order now to obtain the 
 directions of the axes, we have only to 
 find the bisectors of the angle included 
 by the asymptotes, or, in other words, the 
 two perpendicular rays OX, Y which 
 are conjugate to one another in the in- 
 volution of which OP and OQ are the double rays (Arts. 296, 297). 
 
 To determine the lengths of the axes, draw through F a parallel 
 to OX, and let it cut the asymptotes in B and B'\ and on OX take 
 OS the mean proportional between FB and FB'. Then will OS be 
 the length of the semiaxis in the direction OX; and OX will or will 
 not cut the curve according as the segments FB , FB f have or have 
 not the same direction. Again, construct the parallelogram of which 
 OS is one side, which has an adjacent side along OY, and one 
 diagonal along an asymptote ; its side OR will be the length of the 
 semiaxis in the direction OY (Art. 304). 
 
 391. In the plane of a triangle ABC take any two points and 
 0'; if OA , OB , 00 meet the respectively opposite sides BO, CA, AB 
 of the triangle in D, E, F, Ceva's theorem (Art. 137) gives 
 
 BD CE AF_ _ 
 
 cd'ae'bf~ l ' 
 
 Similarly, if (/A , O'B , O'C meet the respectively opposite sides in 
 D',E',F', then 
 
 BD' CE' AF' _ 
 CD'' AE''BF' ~ *' 
 
 If these equations be multiplied together, equation (1) of Art. 385 
 is obtained ; therefore : 
 
 If from any two points the vertices of a triangle are projected upon 
 the respectively opposite sides, the six points so obtained lie on a conic. 
 
 For example, the middle points of the sides of a triangle and the 
 feet of the perpendiculars from the vertices on the opposite sides 
 are six points on a conic *. 
 
 * This conic is a circle (the nine-point circle). See Steiner, Annales de Mathe- 
 matiques (Montpellier, 1828), vol. xix. p. 42 ; or his Collected Works, vol. i. p. 195. 
 
284 
 
 COROLLARIES AND CONSTRUCTIONS. 
 
 392. Pkoblem. To construct a conic which shall jmss through three 
 given points A , B , G, and with regard to which the pairs of corre- 
 sponding points of an involution lying on a given straight line u shall 
 he conjugate points. 
 
 Let AB and AC (Fig. 234) be joined, and let them meet u in D 
 and E. Let the points corresponding in the involution to D and E 
 respectively be D' and E'\ let D" be the harmonic conjugate of D 
 
 Fig. 234. 
 with respect to A and B, and let E" be the harmonic conjugate of 
 E with respect to A and C. Thus D will be conjugate (with respect 
 to the required conic) both to D' and to D", and therefore D'D" 
 will be the polar of D s . So too E'E" vti\\ be the polar of E. 
 
 Join BE , CD, and let them cut E'E" and D'D" in E and D 
 respectively ; then E will be conjugate to E and D to D. If then 
 two points B', C be found such that the ranges BB'EE and 
 CC'DD are harmonic, they will both belong to the required conic. 
 
 In the figure, F and F', G and G' are the pairs of points which 
 determine on u the involution of conjugate points. 
 
 393. Pkoblem. To construct a 
 conic which shall pass through four 
 given points Q , R , S , T and shall 
 divide harmonically a given seg- 
 ment MN (Fig. 235). 
 
 Let the pairs of ^opposite sides 
 of the quadrangle QRST meet the 
 straight line MN in A and A', 
 B and B'. If the required conic 
 cuts MN, the two points of inter- 
 section will be a pair of the invo- 
 lution determined by A and A', 
 B and B' (Art. 183). If then the 
 involution of which M and N are 
 the double points and the involution 
 determined by the pairs of points A and A', B and B' have a pair 
 P and P' in common, the required conic will pass through each of 
 the points P and P' (Arts. 125, 208). 
 
395] COROLLARIES AND CONSTRUCTIONS. 285 
 
 In order to construct these points, describe any circle (Art. 208) 
 and from any point on it project the points A , A ', B , B', M , N 
 upon the circumference, and let A x , A x , B 1 , B\ , M x , N x be their 
 respective projections. If the chords A X A X and B X B( meet in V, and 
 the tangents at M x and N x meet in U, all straight lines passing 
 through U determine on the circumference, and consequently (by 
 projection from 0) on the straight line MN, pairs of conjugate points 
 of the first involution, and the same is true, with regard to the 
 second involution, of straight lines passing through V. If the straight 
 line UV meets the circle in two points P x and P x , let these be joined 
 to 0; the joining lines will cut MN in the required points P and P f . 
 
 Let W be the pole of UV with respect to the circle. Every 
 straight line passing through W and cutting the circle determines 
 on it two points which are harmonically conjugate with regard to 
 P x and P x ; and these points, when projected from on MN, will 
 give two points which are harmonically conjugate with regard to 
 P and P', and which are therefore conjugate to one another with 
 respect to the required conic. If then UV does not cut the circle, so 
 that the points P and P f cannot be constructed, draw through W 
 two straight lines cutting the circle, and project the points of inter- 
 section from the centre upon the straight line MN ; this will give 
 two pairs of points which will determine the involution on MN of 
 conjugate points with respect to the conic. The problem therefore 
 reduces to that treated of in the preceding Article. 
 
 394. Problem. To construct a conic which shall pass through 
 four given points Q , R , S , T, and through two conjugate points 
 [which are not given) of a known involution lying on a straight line u. 
 
 This problem is similar to the preceding one ; since it amounts 
 to constructing the pair of conjugate points common to the given 
 involution and to that determined on u by the pairs of opposite 
 sides of the quadrangle QRST (Art. 183). 
 
 Such a common pair will always exist when the given involution 
 has no double points ; and the two points composing it will both lie 
 on the required conic. If the given involution has two double points 
 M and N, the present problem becomes identical with that of 
 Art. 393. 
 
 The problem clearly admits of only one solution, and the same is 
 the case with regard to those of the two preceding Articles. 
 
 395. Consider a hyperbola whose asymptotes are perpen- 
 dicular to one another, and to which, on this account, is given 
 the name of rectangular hyperbola (Fig. 236). Since the 
 asymptotes are harmonically conjugate with regard to 
 any pair of conjugate diameters (Art. 296), they will in 
 
286 COROLLARIES AND CONSTRUCTIONS. [395 
 
 this case be the bisectors of the angle included between 
 any such pair (Art. 60). But the parallelogram described 
 on two conjugate semidiameters as adjacent sides has its 
 diagonals parallel to the asymptotes (Art. 304); in this case 
 therefore every such parallelogram is a rhombus ; that is, every 
 
 diameter is equal in length to its 
 conjugate. On account of this 
 property the rectangular hyper- 
 bola is also called equilateral*. 
 
 I. Since the chords joining the 
 
 extremities P and P' of any 
 
 diameter to any point M on the 
 
 curve are parallel to a pair of 
 
 conjugate diameters (Art. 287), 
 
 the angles made by PM and 
 
 Fig. 236. P'M with either asymptote are 
 
 equal in magnitude and of 
 
 opposite sign. If the points P and P' remain fixed, while M 
 
 moves along the curve, the rays PM and P'M trace out two 
 
 pencils which are oppositely equal to one another (Art. 106). 
 
 II. Conversely, the locus of the points of intersection of pairs 
 of corresponding rays of two oppositely equal pencils is an equilateral 
 hyperhola. 
 
 For, in the first place, the locus is a conic, since the two 
 pencils are projective (Art. 1 50). Further, the two pencils have 
 each a pair of rays which include a right angle, and which 
 are parallel respectively to the corresponding rays of the other 
 pencil (Art. 106); the conic has thus two points at infinity 
 lying in directions at right angles to one another, and is there- 
 fore an equilateral hyperbola. It will be seen moreover that the 
 centres P and P f of the two pencils are the extremities of a 
 diameter. For the tangent p at P is the ray corresponding to 
 P'P regarded as a ray p 'of the second pencil, and the tangent 
 q' at P' is the ray corresponding to PP' regarded as a ray q 
 of the first pencil (Art. 150); but the angles^ andj/^' must 
 be equal and opposite; therefore, since p' and q coincide, 
 p and q' must be parallel to one another. 
 
 III. The angular points of a triangle ABC and its ortho- 
 centre (centre of perpendiculars) B are the vertices of a 
 
 * Apollonius, loc. cit., vii. 21. De la Hire, loc. cif., book v. prop. 13. 
 
395] COROLLARIES AND CONSTRUCTIONS. 287 
 
 complete quadrangle in which each side is perpendicular to 
 the one opposite to it, and whose six sides determine on the 
 straight line at infinity three pairs of points subtending each 
 a right angle at any arbitrary point 8. The three pairs of rays 
 formed by joining these points to 8 belong therefore to an in- 
 volution in which every ray is perpendicular to its conjugate 
 (Arts. 131 left, 124, 207). 
 
 But this involution of rays projects from S the involution 
 of points which, in accordance with Desargues' theorem, is 
 determined on the straight line at infinity by the pairs of 
 opposite sides of the quadrangle and by the conies (hyper- 
 bolas*) circumscribed about it. The pairs of conjugate rays 
 therefore of the first involution give the directions of the 
 asymptotes of these conies ; thus : 
 
 If a conic pass through the angular points of a triangle and 
 through the orthocentre, it must be an equilateral hyperbola-^. 
 
 IV. Conversely, if an equilateral hyperbola be drawn to 
 pass through the vertices A,B ,C of a triangle, it will pass 
 also through the orthocentre D. For imagine another hyper- 
 bola which is determined (Art. 162, I) by the four points 
 A, B , C , D and by one of the points at infinity on the given 
 hyperbola. This new hyperbola will be an equilateral one by 
 the foregoing theorem, and will consequently pass through the 
 second point at infinity on the given curve ; and since the 
 two hyperbolas thus have five points in common (A,B,C, and 
 two at infinity) they must be identical; which proves the 
 proposition. Therefore : 
 
 If a triangle be inscribed in an equilateral hyperbola, its ortho- 
 centre is a point on the curve. 
 
 V. If the point B approach indefinitely near to A, i. e. if BAC 
 becomes a right angle, we have the following proposition : 
 
 If EFG (Fig. 236) is a triangle, right-angled at B, which is 
 
 * No ellipse or parabola can be circumscribed about the quadrangle here con- 
 sidered (Art. 219). 
 
 f This may be deduced directly from Pascal's theorem. For let a conic be 
 drawn through A,B,C,D, and let I x and I 2 be the points where it meets the 
 line at infinity. Since ABCD^Iiis a hexagon inscribed in a conic, the inter- 
 sections of AB and DI lt of BC and I X I 2 , and of CD and I 2 A, are three collinear 
 points. Therefore the straight line joining the point in which D/j meets AB 
 to that in which AI 2 meets CD must be parallel to BC. Thus AI 2 must be at 
 right angles to Dl x , and as these lines are parallel to the asymptotes of the conic 
 the latter is a rectangular hyperbola. 
 
288 COROLLARIES AND CONSTRUCTIONS. . [396 
 
 inscribed in an equilateral hyperbola, the tangent at E is perpendicular 
 to the hypotenuse FG. 
 
 VI. Through four given points Q,R,S,T can be drawn only 
 one equilateral hyperbola (Art. 394). The orthocentre of each 
 of the triangles QRS , BST ', STQ , QRT lies on the curve * 
 
 VII. Given four tangents to an equilateral hyperbola, to construct 
 the curve. 
 
 Since the diagonal triangle of the quadrilateral formed by the 
 four tangents is self-conjugate with respect to the hyperbola, 
 the centre of the latter will lie on the circle circumscribing 
 this triangle (Art. 370, II). But the centre of the hyperbola 
 lies also on the straight line which joins the middle points of 
 the diagonals of the quadrilateral (Art. 318, II). Either of the 
 points of intersection of this straight line with the circle will 
 therefore give the centre of an equilateral hyperbola satisfying 
 the problem ; there are therefore two solutions. For another 
 method of solution see Art. 372. 
 
 VIII. The polar reciprocal of any conic with respect to a circle K 
 having its centre on the director circle is an equilateral hyperbola. 
 
 For since the tangents to the conic from the centre of the 
 circle K are mutually perpendicular, the conic which is the 
 polar reciprocal of the given one must cut the straight line at 
 infinity in two points subtending a right angle at 0. That is 
 to say, it must be an equilateral hyperbola. 
 
 396. Suppose given a conic, a point JS, and its polar s ; and let a 
 straight line passing through JS cut the conic in A and A'. Let the 
 figure he constructed which is homological with the given conic, 
 JS being taken as centre of homology, s as axis of homology, and A , A' 
 as a pair of corresponding points. Then every other point B r which 
 corresponds to a point B on the conic will lie on the conic itself. 
 For if AB meets the axis s in P, then B', the point of intersection of 
 SB and A'P, is likewise a point on the conic (Art. 250). The curve 
 homological with the given conic will therefore be the conic itself. 
 Any two corresponding points (or straight lines) are separated har- 
 monically by JS and s ; this is, in fact, the case of harmonic homology 
 (Arts. 76, 298). 
 
 To the straight line at infinity will therefore correspond the 
 
 * These theorems are due to Brianchon and Poncelet ; they were enunciated 
 in a memoir published in vol. xi. of the Annales de Mathe'maUques (Montpellier, 
 1 821), and were given again in vol. ii. (p. 504) of Poncelet's Applications 
 a" Analyse et de G6ometrie (Paris, 1864). 
 
397] 
 
 COROLLARIES AND CONSTRUCTIONS. 
 
 289 
 
 ib) 
 
 ls: 
 
 straight line j which is parallel to s and which lies midway between 
 S and s ; and the points in which j meets the conic will correspond 
 to the points at infinity on the same conic. 
 
 From this may be derived a very simple method of determining 
 whether a given arc of a conic, however small, belongs to an ellipse, a 
 parabola, or a hyperbola. 
 
 Draw a chord s joining any two points in the arc ; construct its 
 pole S, and draw a straight line 
 j parallel to s and equidistant 
 from S and s. If j does not cut 
 the arc, the latter is part of an 
 ellipse (Fig. 237 a). If j touches 
 the arc at a point J, the arc belongs 
 to a parabola of which SJ is a 
 diameter (Fig. 237 b). If, finally, 
 j cuts the are in two points J x , J 2 
 (Fig. 237c), the arc will be part 
 of a hyperbola whose asymptotes 
 are parallel to 8J X and SJ 2 *. 
 
 397. Pkoblem. Given a tangent 
 to a conic, its point of contact, and 
 the position (but not the magnitude) of a pair of conjugate diameters; 
 to construct the conic (Fig. 238). 
 
 Suppose the point of intersection of the given diameters, and 
 P and Q the points in which they are cut by the given tangent. 
 Through the point of contact M of this tangent draw parallels to 
 OQ , OP to meet OP, OQ in P' and Q' respectively. Since the 
 polar of M (the tangent) passes through P, the polar of P will pass 
 
 (C) 
 
 / 
 
 Fig. 237. 
 
 Fig. 238. 
 
 through M ; and since the polar of P is parallel to OQ, it must be 
 MP') therefore P and P' are conjugate points. 
 
 If now points A and A' be taken on OP such that OA and OA' may 
 each be equal to the mean proportional between OP and OP' } then 
 AA f will be equal in length to the diameter in the direction OP 
 (Art. 290). In the same way the length of the other diameter BB' 
 will be found by making OB and OB f each equal to the mean pro- 
 portional between OQ and OQ'. 
 
 * Poncelet, loc. cit., Arts. 225, 226. 
 
 u 
 
290 
 
 COROLLARIES AND CONSTRUCTIONS. 
 
 If the points P and P' fall on the same side of 0, the involution 
 of conjugate points has a pair of double points A and A' (Art. 128); 
 that is to say, the diameter OP meets the curve. If, on the other 
 hand, P and P' lie on opposite sides of 0, the involution has no 
 double points, and the diameter OP does not meet the curve. In 
 this case A and 4' are two conjugate points lying at equal distances 
 from 0. The figure shows two cases : that of the ellipse (a) and that 
 of the hyperbola (6). 
 
 398. Pboblem. Given a point M on a conic and the positions of 
 two pairs of conjugate diameters a and </, b and b\ to construct the 
 conic. 
 
 I. First solution (Fig. 239). Through M draw chords parallel to 
 each diameter, and such that their middle points lie on the respec- 
 tively conjugate diameters. The other extremities A^A f 1 B i B f of 
 
 Fig. 239. 
 
 the four chords so drawn will be four points all of which lie on the 
 required conic. 
 
 II. Second solution (Fig. 240). Denoting the diameter MOM' by 
 c, if the ray c' be constructed which is conjugate to c in the in- 
 volution determined by the pairs of rays a and V, b and b', then c' 
 will be the diameter conjugate to c (Art. 296). Through M draw 
 MP parallel to a, and through M' draw M'P' parallel to a'; these 
 parallels will intersect on the conic (Art. 288) ; let them cut c' in P 
 and P' respectively. These last two points are conjugate with re- 
 spect to the conic (Art. 299); thus if on c' two other points be found 
 which correspond to one another in the involution determined by the 
 pair P, P f and the central point 0, then MQ and M'Q' will intersect 
 on the conic. If then on c' two points N and N f be taken such that 
 the distance of either of them from is a mean proportional be- 
 tween OP and 0P\ they will be the extremities of the diameter c' 
 (Art. 290). 
 
 III. Third solution. Through the extremities M and M f of the 
 diameter which passes through the given point draw parallels to a 
 and a' '; they will meet in a point A lying on the conic. Through 
 the same points draw parallels to b and b' ; these will meet in another 
 point B also lying on the conic (Art. 288). Produce AO to A', 
 
401] 
 
 COROLLARIES AND CONSTRUCTIONS. 
 
 291 
 
 Fig. 241. 
 
 making OA' equal to AO; and similarly BO to B', making OB' equal 
 to BO; then will A' and B' be points also lying on the required 
 conic (Art. 281). 
 
 399. Problem. Given in position two pairs of conjugate diameters 
 a and a', b and V of a eonic, and a tangent f, to construct the conic. 
 
 I. First solution (Fig. 241). Let 
 be the point of intersection of the 
 given diameters, that is, the centre 
 of the conic. Draw parallel to t and 
 at a distance from equal to that 
 at which t lies, a straight line f\ this 
 will be the tangent parallel to t. Let 
 the points of intersection of t and if 
 with a and a' be joined ; this will give two other parallel tangents 
 u and u f (Art. 288). Another pair of parallel tangents v and 1/ will 
 be obtained by joining the points where t and € meet b and b'. 
 
 II. Second solution. The conjugate diameters a and a', b and &', 
 will meet t in two pairs of points A and A', B and B f which deter- 
 mine an involution whose centre is the point of contact of t (Art. 302). 
 The problem therefore reduces to one already solved (Art 397). If 
 the involution has double points, the straight lines joining these points 
 to will be the asymptotes. 
 
 400. Problem. Given two points M and N on a conic and the 
 position of a pair of conjugate diameters a and a', to construct the 
 conic (Fig. 242). 
 
 Let M' and N' be the other extremities of the diameters passing 
 through M and N. Through M and W draw MH, M'H parallel to 
 a and a' respectively ; similarly, through N and N' draw NK , N'K 
 parallel to a and a' respectively. The points H and K will lie on 
 the required conic. 
 
 Fig. 242. Fig. 243. 
 
 401. Peoblem. Given two tangents m and n to a conic, and the 
 position of a pair of conjugate diameters a and a', to construct the 
 conic (Fig. 243). 
 
 Draw the straight lines m' and n' parallel respectively to m and n, 
 and at distances from the centre equal respectively to those at 
 which m and n lie ; then mf will be the tangent parallel to m, and n' 
 
 u % 
 
292 COROLLARIES AND CONSTRUCTIONS. [402 
 
 the tangent parallel to n. Join the points where m and mf meet a and 
 a' by the straight lines t and t', and the points where n and n' meet 
 a and a' by the straight lines u and vf. The four straight lines 
 t , tf, u , v! will all be tangents to the required conic (Art. 288). 
 
 402. Problem. Given jive points on a conic, to construct a 'pair of 
 conjugate diameters which shall make with one another a given angle *. 
 
 Construct first a diameter AA / of the conic (Art. 285); and on it 
 describe a segment of a circle containing an angle equal to the given 
 one. Find the points in which the circle of which this segment is a 
 part cuts the conic again (Art. 227) ; if M is one of these points, AM 
 and A'M will be parallel to a pair of conjugate diameters. Since 
 then AM A' is equal to the given angle, the problem will be solved by 
 drawing the diameters parallel to AM and A'M. 
 
 If the segment described is a semicircle, this construction gives 
 the axes. 
 
 403. Problem. To construct a conic with respect to which a given 
 triangle EFG shall be self-conjugate, and a given point P shall be tJie 
 pole of a given straight line pf. 
 
 Let p meet FG in A. The polar of A will pass through E the 
 pole of FG, and through P the pole of p, and will therefore be 
 EP. Similarly FP , GP will be the polars of the points B , C in 
 which p is cut by GE , EF respectively. Let A' be the point in 
 which FG intersects EP; then F and G, A and A', are two pairs 
 of conjugate points with respect to the conic, and if the involution 
 which they determine has a pair of double points L and L' , these points 
 will lie on the required conic (Art. 264). The same construction may 
 be repeated in the case of the other two sides of the triangle EFG. 
 
 If the point P lies within the triangle EFG, the points A',B' C 
 lie upon the sides FG , GE , EF respectively (not produced %). The 
 straight line p may cut two of the sides of the triangle, or it may lie 
 entirely outside the triangle. In the first case the involutions lying 
 on the two sides of the triangle which are cut by p are both of the 
 non-overlapping (hyperbolic) kind, and therefore each possesses double 
 points (Art. 128); these give four points of the required curve, and 
 the problem reduces to that of describing a conic passing through four 
 given points and with respect to which two other given points are 
 conjugates (Art. 393). In the second case, on the other hand, the 
 pairs of conjugate points on each of the sides of the triangle EFG 
 overlap, and the involutions have no double points (Art. 128); in 
 
 * Dk la Hire, loc. cit., book ii. prop. 38. 
 
 t Staudt, Geometrie der Lage, Art. 237. 
 
 % We shall say that a point A' lies on the side FG of the triangle, when it lies 
 between F and G ; and that a straight line cuts the side FG, when its point of 
 intersection with FG lies between F and G. 
 
405] 
 
 C0K0LLAR1ES AND CONSTRUCTIOl 
 
 Of 
 
 >RN\ 
 
 this case the conic does not cut any of the sides of the self- 
 conjugate triangle ; therefore (Art. 262) it does not exist. 
 
 If the point P lies outside the triangle, one only of the three 
 points A', B', C lies on the corresponding side; the two others lie 
 on the respective sides produced. If these two other sides are cut by 
 p , none of the involutions possesses double points, and the conic does 
 not exist. If on the other hand p cuts the first side, or if p lies 
 entirely outside the triangle, the conic exists, and may be constructed 
 as above. 
 
 In all cases, whether the conic has a real existence or not, the polar 
 system (Art. 339) exists. It is determined by the self-conjugate 
 triangle EFG, the point P, and the straight line p. To construct 
 this system is a problem of the first degree, while the construction of 
 the conic is a problem of the second degree. 
 
 404. Pkoblem. Given a pentagon ABODE, to describe a conic with 
 regard to which each vertex shall be the pole of the opposite side *. 
 
 Let F be the point of intersection of AB and CD . If the conic 
 K be constructed (Art. 403) with regard to which ADF is a self- 
 conjugate triangle and E the pole of BC, then the points B and C in 
 which BC is cut by AF and DF respectively will be the poles of ED 
 and EA, the straight lines which join E to the points D and A 
 respectively. Every vertex of the pentagon will therefore be the 
 pole of the opposite side ; that is, K will be the conic required. 
 
 If the conic C be constructed which passes through the five vertices 
 of the pentagon, and also the conic C ' which touches the five sides 
 of the pentagon (Art. 152), these two conies will be polar reciprocals 
 one of the other with respect to K (Art. 322). 
 
 405. Pkoblem. Given Jive points A , B , C , D , E (no three of 
 which are collinear), to determine 
 a point M such that the pencil 
 MiABCDE) shall be projective with 
 a given pencil abede (Fig. 244). 
 
 Through D draw two straight 
 lines DD', DE' such that the pencil 
 D (ABCD'E') is projective with 
 abede (Art. 84, right). Construct 
 the point E' in which DE' meets 
 the conic which passes through the 
 four points ABCD and touches DD' at D (Art. 165) ; then construct 
 the point M in which the same conic meets EE'. M will be the 
 point required. For since M , A , B , C , D , E' lie on the same conic, 
 the pencil M(ABCDE') is projective with the pencil D (ABCD'E'), 
 
 Fig. 244. 
 
 * Staudt, loc. cit., Arts. 238, 258. 
 
294 COROLLARIES AND CONSTRUCTIONS. [406 
 
 which by construction is projective with the given pencil abcde. 
 Since then ME' and ME are the same ray, the problem is solved. 
 
 As an exercise may be solved the correlative problem, viz. 
 
 Given Jive straight lines a , b , c , d , e, no three of which are con- 
 current, to draw a straight line m to meet them in Jive points forming 
 a range projective with a given range ABCDE *. 
 
 406. Peoblem. To trisect a given arc AB of a circled. 
 
 On the given arc take (Fig. 245) a point N, and from B measure 
 in the opposite direction to AN an arc BN' equal to twice the arc 
 AN. If BT be the tangent at B, and if be the centre of the circle 
 
 rig- 245. 
 
 of which the arc AB is a part, the angles AON and TBN f are equal 
 and opposite. If N and N' vary their positions simultaneously, the 
 rays ON and BN' will describe two oppositely equal pencils, and the 
 locus of their point of intersection M will therefore (Art. 395, II) be 
 an equilateral hyperbola passing through and B. The asymptotes 
 of this hyperbola are parallel to the bisectors of the angle made by 
 AO and i^with one another ; for these straight lines are correspond- 
 ing rays (being the positions of the variable rays ON and BN' for 
 which the arcs AN and BN' are each zero). The centre of the 
 hyperbola is the middle point of the straight line OB which joins the 
 centres of the two pencils. 
 
 The hyperbola having been constructed by help of Pascal's theorem, 
 the point P will have been found in which it cuts the arc AB. Two 
 corresponding points N and N' coalesce in this point ; therefore 
 the arc AP is half of the arc PB, and P is that point of trisection 
 of the arc AB which is the nearer to A. 
 
 The hyperbola meets the circle in two other points E and Q. The 
 point R is one of the points of trisection of the arc which together 
 
 * Staudt, loc. cit., Art. 263. 
 
 t Staudt, Beitrdge, Art. 432. Chasles, Sections coniques, Art. 37. 
 
408] COROLLARIES AND CONSTRUCTIONS. 295 
 
 with AB makes up a semicircle ; and the point Q is one of the points 
 of tri section of the arc which together with AB makes up the cir- 
 cumference of the circle. 
 
 407. It has been seen (Art. 191) that if P', P", Q\ Q" (Fig. 246) 
 are four given collinear points, and if any conic be described to pass 
 through P r and P", and then a tangent be drawn to this conic from 
 Q' and another from Q", the chord joining the points of contact 
 of these tangents passes through one of the double points M ', N' of 
 the involution which is determined by the two pairs of points 
 P f and P", Q' and Q". The two tangents which can be drawn from 
 Q', combined with the two from Q", give four such chords of contact, 
 of which two pass through M* and two through N'. From this may 
 
 Fig. 246. 
 
 be deduced a construction for the double points of the involution 
 P'P", Q'Q", or, what is the same thing (Art. 125), for the two 
 points M', N' which divide each of the two given segments P'P" 
 and Q'Q" harmonically. 
 
 Describe any circle to pass through P f and P", and draw to it from 
 Q' the tangents f and u', and from Q" the tangents f and u". The 
 chord of contact of the tangents t' and t fr and that of the tangents u r 
 and u" will cut the straight line P'P" in the two required points 
 M' and N'. 
 
 408. This construction has been applied by Brianchon * to the 
 solution of the two problems considered in Art. 221, viz. 
 
 I. To construct a conic of which two points P', P y/ and three 
 tangents q , q', q" are given. 
 
 Join P'P", and let it cut the three given tangents in Q , #', Q" 
 respectively (Fig. 246). Describe any circle through P', P" and 
 draw to it tangents from Q, Q', Q /f . ' The chords which join the 
 points of contact of the tangents from Q" to the points of contact of 
 the tangents from Q meet P'P" in two points M and N\ and simi- 
 larly the tangents from Q" combined with those from- Q' determine 
 two points M ' and N'. 
 
 The chord of contact of the tangents q\ q" to the required conic 
 will therefore pass through one of the points M , N, and that of the 
 
 * Brianchon, loc. cit., pp. 47, 51. 
 
296 COROLLARIES AND CONSTRUCTIONS. [409 
 
 tangents q f , q" will pass through one of the points M', N\ The 
 four combinations MM ', MN\ NM', NN' give the four solutions of 
 the problem. 
 
 The problem therefore reduces to the following : To describe a conic 
 which shall touch three given straight lines q , q', q" in such a way that 
 the chords of contact of the two pairs of tangents q , q" and q\ q" shall 
 pass respectively through two given points M and M' . Let QQ'Q" (Fig. 
 
 247) denote the triangle formed by the 
 three given tangents, and let A , A', A" 
 be the points of contact to be deter- 
 mined. By a corollary to Desargues' 
 theorem (Art. 194), the side q = Q'Q" 
 is divided harmonically at the point of 
 contact A and at the point where it is 
 cut by the chord A' A". If these four 
 harmonic points be projected on MQ" 
 from A" as centre, it follows that the segment RQ" intercepted on 
 MQ" between q" and qf is divided harmonically by M and the 
 chord A' A". 
 
 Let then MQ" be joined ; it will cut q" in some point R \ and 
 let the point V be determined which is harmonically conjugate to 
 M with regard to R and Q". In order to do this, draw through M 
 any straight line to cut q" and q' in S and T respectively ; join 
 SQ" and TR, meeting in TJ\ and join QU, meeting RQ" in V. 
 Join VM'\ it will meet q f and q" in A' and A" ; and finally if 
 MA" be joined, it will cut Q'Q" in A. 
 
 II. To construct a conic qf which three points P , P\ P" and two 
 tangents q , qf are given. 
 
 Join PP', and let it meet q and q' in Q and Q' respectively; join 
 PP", and let it meet q and q / in R and R' respectively. Describe 
 a circle round PP'P'\ and to it draw tangents from Q and Q'; the 
 chords of contact will meet PP' in two points M and N. Similarly 
 draw the tangents from R and R / ; the chords of contact will meet 
 PP" in two other points M / and N'. Then each of the straight lines 
 MN\ NN\ M'N ', MM' will meet the tangents q and q' in two of 
 the points of contact of these two tangents with a conic circumscribing 
 the triangle PP'P". 
 
 This construction differs from that given in Art. 221 (left) only in 
 the method of finding the double points M and N, M f and N'. 
 
 409. Theorem. If two angles AOS and AO'S of given magnitude 
 turn about their respective vertices and 0' in such a way that the 
 point of intersection S qf one pair of arms lies always on a fixed straight 
 line u, the point of intersection of the other pair of arms will describe 
 a conic (Fig. 248). 
 
411] 
 
 COROLLARIES AND CONSTRUCTIONS. 
 
 297 
 
 The proof follows at once from the property that the pencils 
 traced out by the variable rays OA and 
 OS, OS and O'S , O'^and O'A are 
 projective two and two (Arts. 42, 
 108), and that consequently the 
 pencils traced out by OA and O'A 
 are projective. This theorem is due 
 to Newton, and was given by him 
 under the title of The Organic De- 
 scription of a conic *. 
 
 410. The following, which depend 
 on the foregoing theorem, may serve 
 as exercises to the student : — 
 
 1. Deduce a construction for a 
 
 conic passing through five given points , 0', A , B , C. 
 
 2. Given these five points, determine the magnitude of the angles 
 AOS , AO'S and the position of the straight line u in order that the 
 conic generated may pass through the five given points. 
 
 3. On the straight line 00 ' which joins the vertices of the two 
 given angles a segment of a circle is described containing an angle 
 equal to the difference between four right angles and the sum of the 
 given angles. Show that according as the circle of which this seg- 
 ment is a part cuts, does not cut, or touches the straight line u, so 
 the conic generated will be a hyperbola, an ellipse, or a parabola. 
 
 ' 4. Determine the asymptotes of the conic, supposing it to be a 
 hyperbola ; or its axis, in the case where it is a parabola. 
 
 5. When is the conic (a) a circle, (b) an equilateral hyperbola, 
 (c) a pair of straight lines ? 
 
 6. Examine the cases in which the two given angles are directly 
 equal, or oppositely equal, or supple- 
 mentary t. 
 
 411. Theokem. If a variable triangle 
 AM A' move in such a way that its sides turn 
 severally round three given points , ' , S 
 (Fig. 249) while two of its vertices A , A' slide 
 along two fixed straight lines u , vf respec- 
 tively, the locus of the third vertex M is a 
 conic passing through the following five points, 
 viz. , 0' ', uu', and the intersections B and C of u and u' with O'S 
 and OS respectively %. 
 
 * Principia, lib. i. lemma xxi; Enumeratio linearum tertii ordinis (OpUcks, 
 1704), p. 158, § xxxi. 
 f Maclaurin, Geometria Organica (London, 1720), sect. i. prop. 2. 
 X See Art. 156. 
 
COROLLARIES AND CONSTRUCTIONS. 
 
 [412 
 
 412. 
 
 this). 
 
 413. 
 
 Theoeem. (The theorem of Art. 411 is a particular case of 
 If a variable polygon move in such a way that its n sides turn 
 severally round n fixed points O x , 0% , . . . O n 
 (Fig. 250) while n — 1 of its vertices slide 
 respectively along n — 1 fixed straight lines 
 u x ,u 2 , ... u n _ 1 , then the last vertex will 
 describe a conic; and the locus of the point 
 of intersection of any pair of non-adjacent 
 sides will also be a conic *. 
 
 The proof of this theorem and its cor- 
 ■Fig. 250. relative is left to the student t. 
 
 Theoeem. From two given points A and A' tangents AB , AG 
 and A'B',A'C are drawn to a conic; then will the four points oj 
 contact B , C , B',C', and the two given points A , A' all lie on a conic 
 (Fig. 251 J). 
 
 Let A'C, A'B f meet BC in D and E respectively; these points 
 
 will evidently be the poles of AC, AB' 
 respectively. The pencil A(BCB'C) 
 is projective with the range of poles 
 BCED (Art. 291), and therefore with 
 the pencil A'(BCED) or A'(BCB;C)>, 
 which proves the theorem. 
 
 414. Theoeem (correlative to that 
 of Art. 413). From two given points 
 A and A' tangents AB, AC and A'B', 
 A'C are drawn to a conic; then will 
 the four tangents and the two chords 
 of contact all touch a conic J. 
 
 For (Fig. 251) the range of points BC 
 (AB, AC, A'B', A'C) or BCED is 
 projective with the pencil A (BC B'C) formed by their polars; but 
 this pencil is projective with the range B'C(AB,AC,A'B', A'C); 
 therefore the six lines AB, AC, A'B', A'C, BC, B'C all touch a 
 conic. 
 
 415. Theoeem. On each diagonal of a complete quadrilateral is 
 taken a pair of points dividing it harmonically ; if of these six points 
 three (one from each diagonal) lie in a straight line, the other three will 
 also lie in a straight line. 
 
 Coeollaey. The middle points of the three diagonals of a complete 
 quadrilateral are collinear. 
 
 * This theorem is due to Maclaurin and Braikenridge (Phil. Trans., 
 London, 1735). 
 
 t Poncblbt, loc. cit., Art. 502. 
 
 X Chasles, Sections coniques, Arts. 213, 214. 
 
418] COBOLLABIES AND CONSTRUCTIONS. 299 
 
 416. Theorem. If from any point on the circle circumscribing 
 a triangle ABC straight lines OA' , OB', OC be inflected to meet the 
 sides BC, CA, AB in A', B' , C respectively, and to make with them 
 equal angles (both as regards sign c » 
 
 and magnitude) ; then the three /^^"""^n" 
 
 points A',B',C will be collinear /^\\\^s^^L f 
 
 (Fig. 252). (// y\\ \r*- 
 
 Through draw OA", OB", OC" c *U1 \ \\ /\ 
 
 parallel to BC, CA, AB respec- Xi^^^-W- ^s. 
 
 tively; then it is easily seen that b\^^^J^a "ft 
 
 the angles AOA", BOB", COC" Fig> 2g2 
 
 have the same bisectors. The same 
 
 will therefore be true with regard to the angles AOA', BOB', COC; 
 
 consequently (Art. 142) the arms of these last three angles will 
 
 form an involution, and therefore (Art. 135) the points A', B', C will 
 
 be collinear * t. 
 
 417. Theorem. If from the vertices of a triangle circumscribed 
 about a circle straight lines be inflected to meet any tangent to the 
 circle, so that the angles they subtend at the centre may be equal 
 (in sign and magnitude), then the three straight lines will meet in a 
 point J. 
 
 The proof is similar to that of the theorem in the preceding 
 Article. 
 
 418. Problems. (1). Given three collinear segments AA\ BB', CC; 
 to find a point at which they all subtend equal angles (Art. 109). 
 
 In what case can these angles be right angles % (See Art. 
 128). 
 
 (2). Given two projective ranges lying on the same straight line ; 
 to find a point which is harmonically conjugate to a given point on 
 the line, with respect to the two self-corresponding points of the two 
 ranges (which last two points are not given) §. 
 
 (3). Given two pairs of points lying on a straight line ; to deter- 
 mine on the line a fifth point such that the rectangle contained by its 
 distances from the points of the first pair shall be to that contained 
 
 * Chasles, loc. cit, Art. 386. 
 
 f Otherwise : Since the triangles BOC ', COB' are similar, 
 BC : CB' m OB : OC. 
 So also CA' : AC = OC : OA, 
 
 an d AB'\BA' = OA:OB; 
 
 whence by multiplication, paying attention to the signs of the segments, 
 
 BC .CA'.AB' = - CA . B'C . A'B, 
 which shows (Art. 139) that A', B' , C are collinear. 
 
 % Chasles, loc. cit, Art. 387. 
 
 § Chasles, Giom. sup., Art. 269. 
 
300 COROLLAKIES AND CONSTKUCTIONS. [419 
 
 by its distances from the points of the second pair in a given 
 ratio *. 
 
 (4). Through a given point to draw a transversal which shall cut 
 off from two given straight lines two segments (measured from a 
 fixed point on each line) which shall have a given ratio to one 
 another ; or, the rectangle contained by which shall be equal to a 
 given one f. 
 
 419. It will be a useful exercise for the student to apply the 
 theory of pole and polar to the solution of problems of the first 
 and second degree, supposing given a ruler, and a fixed circle and 
 its centre. We give some examples of problems treated in this 
 manner : 
 
 I. To draw through a given point P a straight Une parallel to a 
 given straight line q. 
 
 The pole Q of q and the polar p of P (with respect to the given 
 circle) must be found ; if A be the point where p is cut by the 
 straight line OQ joining Q to the centre of the circle, then the polar 
 a of A will be the straight line required. 
 
 II. To draw from a given point P a perpendicular to a given straight 
 line q. 
 
 Draw through P a straight line parallel to OQ ; it will be the 
 perpendicular required. 
 
 III. To bisect a given segment AB. 
 
 Let a and b be the polars of A and B respectively, and c that 
 diameter of the given circle which passes through ab ; if d be the 
 harmonic conjugate of c with respect to a and b, the pole of d will be 
 the middle point of AB. 
 
 IV. To bisect a given arc MN of a circle. 
 
 Construct the pole S of the chord MN ; the diameter passing 
 through S will cut MN in the middle point of the latter. 
 
 V. To bisect a given angle. 
 
 If from a point on the circle parallels be drawn to the arms of the 
 given angle, the problem reduces to the preceding one. 
 
 VI. Given a segment AC ; to produce it to B so that AB may be 
 double of AC. 
 
 Let 1 a and c be the polars of A and C respectively, d the diameter 
 of the given circle which passes through ac, and b the ray which 
 makes the pencil abed harmonic ; the pole of b will be the required 
 point B. 
 
 * This is the problem ' de sectione determinata ' of Apollonius. See Chasles, 
 G6om. sup., Art. 281. 
 
 t These are the problems ' de sectione rationis ' and ' de sectione spatii ' of 
 Apollonius. See Chasles, Giom. sup., Arts. 296, 298. 
 
420] COROLLARIES AND CONSTRUCTIONS. 301 
 
 VII. To construct the circle whose centre is at a given point U and 
 whose radius is equal to a given straight line UA. 
 
 Produce AU to B, making UB equal to AU (by VI), and draw 
 perpendiculars at A and B to AB (by II). Bisect the right angles at 
 A and B (by V) ; and let the bisecting lines meet in C and D. We 
 have then only to construct the conic of which AB and CD are a pair 
 of conjugate diameters (Art. 301). 
 
 420. The following problems* depend for their solution on the 
 theorem of Art. 376. 
 
 I. Given three points A, B, C on a conic and one focus F, to 
 construct the conic. 
 
 With centre F and any radius describe a circle K, and let the 
 polars of A, B, C with respect to this circle be a, b, c respectively. 
 Describe a circle touching a, b, c and take its polar reciprocal with 
 respect to K; this will be the conic required. 
 
 Since there can be drawn four circles touching a, b, c (the inscribed 
 circle of the triangle abc and the three escribed circles), there are 
 four conies which satisfy the problem. 
 
 II. Given two points A, B on a conic, one tangent t, and a focus F> 
 to construct the conic. 
 
 Describe a circle K as in the last problem, and let a, b be the 
 polars of A, B, and I 1 the pole of t, with respect to K. Draw a circle 
 to pass through T and to touch a and b ; the polar reciprocal of this 
 circle with respect to K will be the conic required. 
 
 Since four circles can be drawn to pass through a given point and 
 touch two given straight lines, this problem also admits of four 
 solutions. 
 
 III. Given one point A on a conic, two tangents b, c, and a focus F, 
 to construct the conic. 
 
 Describe a circle K as in the last two problems ; let a be the polar 
 of A, and let B, C be the poles of 6, c respectively with regard to this 
 circle. Draw a circle to pass through B and C and to touch a ; its 
 polar reciprocal with respect to K will be the conic required. 
 
 Since two circles can be described through two given points to 
 touch a given straight line, this problem admits of two solutions. 
 
 IV. Given three tangents a, b, c to a conic and one focus F, to 
 construct the conic. 
 
 Describe a circle K as in the last three problems, and let A, B, C 
 be the poles of a, b, c respectively with regard to this circle. Draw 
 the circle through A, B, C and take its polar reciprocal with respect 
 to K ; this will be the conic required. 
 
 This problem clearly admits of only one solution. 
 
 * Solutions of these problems were given by De la Hire (see Chasles, Aperfu 
 historique, p. 125), and by Newton (Principia, lib. i. props. 19, 20, 21). 
 
302 COKOLLAKIES AND CONSTBUCTIONS. 
 
 421. Problem. Given the axes of a conic in position (not in 
 magnitude) and a pair of conjugate straight lines which cut one anotlter 
 orthogonally, to construct the foci. 
 
 If be the centre of the conic, and P, P / and Q, Q f the points in 
 which the two conjugate lines respectively cut the axes, then of the 
 two products OP . OP f and OQ . OQ', one will be positive and the 
 other negative. This determines which of the two given axes is the 
 one containing the foci. If now a circle be circumscribed about 
 the triangle formed by the two given conjugate lines and the non- 
 focal axis, it will cut the focal axis at the foci (Art. 343). 
 
 422. The following are left as exercises to the student. 
 
 1. Given the axes of a conic in position, and also a tangent and 
 \j its point of contact, construct the foci, and determine the lengths of 
 
 the axes (Art. 344). 
 
 2. Given the focal axis of a conic, the vertices, and one tangent, 
 construct the foci (Art. 360). 
 
 (V^ 3. Given the tangent at the vertex of a parabola, and two other 
 
 tangents, find the focus (Art. 358). 
 ^^—-4. Given the axis of a parabola, and a tangent and its point of 
 contact, find the focus (Art. 346). 
 
 5. Given the axis and the focus of a parabola, and one tangent, 
 construct the parabola by tangents (Arts. 346, 349, 358). 
 
 6. The locus of the pole of a given straight line r with respect to 
 any conic having its foci at two given points is a straight line «*' 
 perpendicular to r. The two lines r, / are harmonically separated 
 by the two foci. 
 
 7. The locus of the centre of a circle touching two given circles 
 consists of two conies having the centres of the given circles for foci. 
 
 ^^ 8. The locus of a point whose distance from a given straight line is 
 equal to its tangential distance from a given circle consists of two 
 parabolas. 
 
 9. In a central conic any focal chord is proportional to the square 
 of the parallel diameter. 
 
 10. In a parabola, twice the distance of any focal chord from its 
 pole is a mean proportional between the chord and the parameter. 
 
INDEX. 
 
 Affinity, pp. 18, 19. 
 Angle of constant magnitude turning 
 round its vertex traces out two 
 directly equal pencils, 91. 
 bisection of an, 300. 
 trisection of an, 294. 
 Angles, two, of given magnitude ; gene- 
 ration of a conic by means of, 297. 
 Anharmonic ratio defined, 54, 57. 
 unaltered by projection, 54. 
 of a harmonic form is — 1, 57. 
 cannot have the values + I, o, or 00, 
 
 62. 
 of four points or tangents of a conic, 
 122. 
 Anharmonic ratios, the six, 60, 61. 
 Apollonius, x, xi, xii. 
 
 on the parabola, 127, 218. 
 
 on the hyperbola, 130, 142, 156, 
 
 158, 286. 
 on the diameters of a conic, 217, 223, 
 
 230, 232, 234, 235. 
 on focal properties of a conic, 253, 
 
 258, 259, 262. 
 section-problems, 300. 
 Arc of a conic, determination of kind 
 of conic to which it belongs, 289. 
 of a circle, trisection of, 294. 
 of a circle, bisection of, 300. 
 Asymptotes, tangents at infinity, 1 6, 1 2 9 . 
 meet in the centre of the conic, 219. 
 determination of the, given five 
 points on the conic, 178, 179. 
 Auxiliary conic, 203, 239, 240. 
 
 circle of a conic, 260. 
 Axes of a conic defined, 227, 228. 
 case of the parabola, 228. 
 focal and non-focal, 252. 
 bisectors of the angle between its 
 chords of intersection with any 
 circle, 236, 281. 
 Axes of a conic, construction of the, 
 given a pair of conjugate diameters, 
 232, 283. 
 given five points, 236, 292. 
 Axis of perspective or homology, 10. 
 of affinity, 18. 
 of symmetry, 64. 
 
 Bellavitis, xi, 64, 161. 
 Bisection of a given segment or angle 
 by means of the ruler only, 300. 
 
 Brianchon, x, xi, xii, 124, 125. 
 Brianchon's theorem, xi, 124. 
 points, the sixty, 125. 
 
 Carnot's theorem, xi, 279, 280. 
 Centre of projection, 1, 3. 
 
 of perspective or homology, 10, 12, 
 
 98. 
 of similitude, 18. 
 of symmetry, 64. 
 of an involution, 102. 
 Centre of a conic, the pole of the line at 
 
 infinity, 218. 
 bisects all chords, 219. 
 the point of intersection of the 
 
 asymptotes, 219. 
 when external and when internal to 
 
 the conic, 219. 
 locus of, given four tangents, 237. 
 construction of the, given five points, 
 
 220. 
 construction of the, given five tan- 
 gents, 238. 
 Ceva, theorem of six segments, ill. 
 Chasles, xi, xii. 
 
 on homography, 34. 
 
 method of generating conies, 127. 
 
 correlative to the theorem ■ ad qua- 
 
 tuor lineas,' 159. 
 on the geometric method of false 
 
 position, 194. 
 solutions of problems of the second 
 
 degree, 200. 
 Circle, curve homological with a, 14, 15. 
 generated by the intersection of two 
 
 directly equal pencils, 114. 
 harmonic points and tangents of a, 
 
 115, 116. 
 fundamental projective properties of 
 
 points and tangents of a, 115. 
 of curvature at a point on a conic, 
 
 190. 
 cutting a conic ; the chords of inter- 
 section make equal angles with 
 
 the axes, 236, 281. 
 circumscribing triangle formed by 
 
 three tangents to a parabola, 253. 
 auxiliary, of a conic, 260. 
 Class of a curve, 4. 
 
 is equal to the degree of its polar 
 
 reciprocal with regard to a conic, 
 
 240. 
 
304 
 
 INDEX. 
 
 
 Coefficient of homology, 63. 
 Collinear projective ranges, 68. 
 
 their self- corresponding points, 78, 
 
 9 1 , 9 2 . 93- 
 
 construction for these, 1 70. 
 Complementary operations, 33. 
 Concentric pencils, 69. 
 
 construction for their self-correspond- 
 ing rays, 169. 
 Cone, sections of the, 14, 18. 
 Confocal conies, 266. 
 Congruent figures, 64. 
 Conic, homological with a circle, 15, 16. 
 
 generated by two projective pencils, 
 119. 
 
 generated as an envelope from two 
 projective ranges, 120. 
 
 determined by five points or five 
 tangents, 123. 
 
 fundamental projective property of 
 points and tangents, 118. 
 
 projective ranges of points and series 
 of tangents of a, 161. 
 
 homological with itself, 228, 288. 
 
 polar reciprocal of a, 240. 
 
 homological with a given conic, and 
 having its centre at a given point, 
 249. 
 
 confocal with a given conic, and 
 passing through a given point, 266. 
 Conic, construction of a, having given 
 
 five points or tangents, 131, 149, 176, 
 179, 180, 297. 
 
 four points and the tangent at one 
 of them, 137, 177. 
 
 three points and the tangents at two 
 of them, 139, 177. 
 
 three tangents and the points of con- 
 tact of two of them, 143, 177. 
 
 four tangents and the point of con- 
 tact of one of them, 146, 177. 
 
 four points and a tangent, 180. 
 
 four tangents and a point, 180. 
 
 three points and two tangents, 182, 
 296. 
 
 three tangents and two points, 182, 
 295. 
 
 the asymptotes and. one point or 
 tangent, 277. 
 
 the two foci and one tangent, 264. 
 
 the two foci and one point, 265. 
 
 one focus and three tangents, 268, 
 301. 
 
 one focus and three points, 301. 
 ne focus, two points and a tangent, 
 3d. 
 
 one focus, two tangents and a point, 
 301. 
 
 a pair of conjugate diameters, 229, 
 231. 
 
 a pair of conjugate diameters in posi- 
 tion, and two points or tangents, 
 291, 
 
 a pair of conjugate diameters in posi- 
 tion, and a tangent and its point 
 of contact, 289. 
 
 two pairs of conjugate diameters in 
 position, and one point or tan- 
 gent, 290, 291. 
 
 two reciprocal triangles, 247. v 
 
 a self- conjugate triangle, and appoint 
 and its polar, 292. 
 
 a self-conjugate pentagon, 293. 
 
 three points and the osculating circle 
 at one of them, 190. 
 Conic, construction of a, homological 
 with itself, 228, 288. 
 
 passing through three points and 
 determining a known involution 
 on a given line, 284. 
 
 passing through four points and di- 
 viding a given segment harmoni- 
 cally, 284. 
 
 passing through four points and 
 through a pair of conjugate points 
 of a given involution, 285. 
 Conies, osculating, 189. 
 
 having a common self-conjugate 
 triangle, 213, 214. 
 
 circumscribing the same quadrangle, 
 150, 214, 237. 
 
 inscribed in the same quadrilateral, 
 150, 213, 214, 237. 
 Conjugate axis of a hyperbola, 228. 
 Conjugate diameters, defined, 219. 
 
 of a circle cut orthogonally, 222. 
 
 form an involution, 227. 
 
 parallelogram described on a pair as 
 adjacent sides is of constant area, 
 
 234- 
 sum or difference of squares is con- 
 stant, 235. 
 construction of, given two pairs, 232. 
 construction of, given five points on 
 
 the conic, 236. ' 
 including a given angle, construction 
 
 of, 292. 
 Conjugate lines meeting in a point, one 
 
 orthogonal pair can be drawn, 
 
 227. 
 orthogonal, the involution determined 
 
 by them on an axis of the conic, 
 
 251. 
 orthogonal, with respect to a para- 
 bola, 253. 
 Conjugate points and lines with regard 
 to a conic, 204. 
 involution-properties of, 209. 
 Conjugates, harmonic, 46. 
 
 in an involution, 101. 
 Construction of a figure homological 
 with a given one, 13. 
 for the fourth element of a harmonic 
 
 form, 47. 
 for the fourth point of a range whose 
 anharmonic ratio is given, 55, 
 
INDEX. 
 
 305 
 
 / 
 
 of pairs of corresponding elements of 
 two projective forms, when three 
 are given, 70. 
 
 for the self-corresponding elements 
 of two superposed projective forms, 
 169. 
 
 for the sixth element of an involu- 
 tion, 109. 
 
 of pairs of elements of an involution, 
 given two, 104. 
 
 for the centre of an involution, 109. 
 
 for the double elements of an in- 
 volution, 169, 175, 295. 
 
 for the common pair of two super- 
 posed involutions, 1 73. 
 
 for the pole of a line or polar of a 
 point, 2*05, 206. 
 
 of a triangle self-conjugate to a conic, 
 207. 
 
 of the centre and axes of a conic, 
 220, 236, 238, 283, 292. 
 
 of conjugate diameters, 232, 236, 292. 
 
 for diameters of a parabola, having 
 given four tangents, 238. 
 
 for the focus of a parabola, given 
 four tangents, 254. 
 
 for the foci of a conic, given the axes 
 and a pair of orthogonal conjugate 
 lines, 302. 
 Copolar and coaxial triangles, 7, 8. 
 Correlative figures, 26, 85, 241. 
 Curvature, circle of, 190. 
 
 Degree of a curve, 4. 
 
 is equal to the class of its polar reci- 
 procal with respect to a conic, 240. 
 De la Hire, x, xii. 
 
 Desargues, ix, x, xii, 101, 102, 107, 148. 
 Desargues' theorem, 148. 
 Descriptive, the term, as distinguished 
 
 from metrical, 50. 
 Diagonal triangle, of a quadrangle or 
 quadrilateral, 30. 
 common to the complete quadri- 
 lateral formed by four tangents to 
 a conic, and the complete quad- 
 rangle formed by their points of 
 contact, 140. 
 Diagonals of a complete quadrilateral, 
 each is cut harmonically by the 
 other two, 46. 
 their middle points are collinear, 
 
 109, 299. 
 form a triangle self-conjugate to any 
 conic inscribed in the quadri- 
 lateral, 208. 
 if the extremities of two are conju- 
 gate points with regard to a conic, 
 those of the third are so too, 245. 
 Diameters of a conic defined, 217. 
 of a parabola, 218. 
 conjugate, 219. 
 ideal, 223. 
 
 of a parabola, construction for, given 
 four tangents, 238. 
 Dimension of a geometric form, 25. 
 Directly equal ranges, defined, 88. 
 generated by the motion of a seg- 
 ment of constant length, 89. 
 Directly equal pencils, defined, 90. 
 two, the projection of two concentric 
 
 projective pencils, 89. 
 two, generate a circle by their inter- 
 section, 114. 
 subtended at a focus of a conic by 
 the points in which a variable 
 tangent cuts two fixed ones, 264. 
 Director circle, defined, 269. 
 
 the locus of the intersection of or- 
 thogonal tangents, 269. 
 cuts orthogonally the circumscribing 
 circle of any self-conjugate tri- 
 angle, 270. 
 Directrix, defined, 254., 
 
 property of focus and, 256. 
 Directrix of a parabola, the locus of 
 the intersection of orthogonal tan- 
 gents, 270. 
 the locus of the centre of the cir- 
 cumscribing circle of a self-con- 
 jugate triangle, 271. 
 the locus of the orthocentre of a 
 circumscribing triangle, 273. 
 Division of a given bisected segment 
 into n equal parts, by means of 
 the ruler only, 97. 
 Double elements of an involution, 
 they separate harmonically 
 
 'of conjugates, 103, 
 construction for the, i6Qj_£9&. 
 Duality, the principle of, 26-32. 
 
 tion, ifllX 
 any pair J 
 
 Eccentricity, 259. 
 
 of the polar reciprocal of a circle 
 with respect to another circle, 274. 
 Ellipse, 16. 
 
 its centre an internal point, 219. 
 is cut by all its diameters, 220. 
 is symmetrical in figure, 228. 
 Envelope of connectors of correspond- 
 ing points 'of two projective 
 is a conic, 120. 
 if the ranges are similar, it is a para- 
 bola, 128. 
 of a straight line the product of 
 whose distances from two given 
 points is constant, 260. 
 Equal ranges and pencils, 86-90. 
 Equianharmonic forms and figures are 
 projective, and vice versa, 54, 56, 
 62,66. 
 Equilateral hyperbola, why so called 
 286. 
 triangles self-conjugate with regard 
 
 to a, 271. 
 inscribed in a quadrilateral, 272. 
 
306 
 
 INDEX. 
 
 circumscribing a triangle passes 
 through the orthocentre, 287. 
 
 is the polar reciprocal of a conic with 
 regard to a point on the director 
 circle, 288. 
 
 construction of, given four tangents, 
 272, 288. 
 Euclid, porisms of, x, 96. 
 External and internal points with re- 
 gard to a conic, 203. 
 
 False position, geometrical method of, 
 
 194. 
 Focal axis of a conic, 252. 
 
 radii of a point on a conic, 253. 
 
 radii, their sum or difference is con- 
 stant, 258. 
 Foci, denned, 250. 
 
 are points such that conjugate lines 
 meeting in them cut orthogonally, 
 250. 
 
 are internal points lying on an axis, 
 250. 
 
 are the double points of the involu- 
 tion determined on an axis by pairs 
 of orthogonal conjugate lines, 251 . 
 
 of a parabola, one at infinity, 253. 
 
 of parabolas inscribed in a given 
 triangle, locus of, 254. 
 
 properties of, with regard to tangent 
 and normal, 259-264. 
 
 reciprocation with respect to the, 
 2 74, 275. 
 
 construction of, under various con- 
 ditions, 302. 
 Focus of a parabola, 253. 
 
 inscribed in a given triangle, locus 
 of, 254. 
 
 reciprocal of the curve with regard 
 to, 275. 
 Forms, geometric, defined, 22, 164. 
 
 elements of, 23, 164. 
 
 prime, of one, two, three dimensions, 
 24. 
 
 dual generation of, 23, 24, 26. 
 
 projective, 34-38. 
 
 'larmonic, 39-49. 
 
 projective, when in perspective, 67. 
 projective, superposed, 68, 69. 
 
 Gaskin, 189, 269, 271. 
 Gergonne, x. 
 
 Harmonic forms defined, 39, 40. 
 forms are projective, 41, 43. 
 pairs of points necessarily alternate, 45. 
 conjugates, 46. 
 point or ray, construction for the 
 
 fourth, 47. 
 forms, metrical relations, 57, 58. 
 homology, 64, 228, 288. 
 points and tangents of a circle, 115, 
 
 116, 169. 
 and of a conic, 122, 157, 168. 
 
 Hesse, theorem, relating to the ex- 
 tremities of the diagonals of a 
 complete quadrilateral, 245. 
 Hexagon, inscribed in a line-pair, 76. 
 
 circumscribed to a point-pair, 76. 
 
 inscribed in a conic, 124. 
 
 circumscribed to a conic, 124. 
 
 complete, contains sixty simple hexa- 
 gons, 125. 
 Homographic, the term, 34. 
 
 figures, construction of, 81. 
 
 figures may be placed in homology, 
 84. 
 Homological figures, construction of, 
 13-20. 
 
 metrical relations between, 63-65. 
 Homology, defined, 9, 10. • 
 
 in space, 20. 
 
 plane of, 20. 
 
 coefficient or parameter of, 63. 
 
 harmonic, 64, 228, 288. 
 Homothetic figures, 18. 
 Hyperbola, tangent - properties of a, 
 129, 130. 
 
 and asymptotes cut by a transversal, 
 156, 282. 
 
 tangent cut off by the asymptotes 
 is bisected at the point of contact, 
 158.. 
 
 centre is an external point, 219. 
 
 is cut by one only of every pair of 
 conjugate diameters, 220. 
 
 is symmetrical in figure, 228. 
 
 properties of the asymptotes and 
 conjugate points and lines, 277. 
 
 equilateral, 285. 
 
 Ideal diameters and chords, 223, 226. 
 Infinity, points and line at, 5. 
 
 line at, a tangent to the parabola, 16. 
 plane at, 21. 
 Internal and external points with re- 
 gard to a conic, 203. 
 Intersection of a conic with a straight 
 line; constructions, 176, 177, 180, 
 226. 
 , of two conies; constructions, 189. 
 /Involution, defined, 101. 
 
 the two kinds, elliptic and hyper- 
 bolic, 105, 168. 
 construction for the sixth element of 
 
 an, ipg. 
 determined by two pairs of conju- 
 gates, 104, 165. 
 of points or tangents of a conic, 165.- 
 construction for the double elements 
 
 of an, 169, 295. 
 formed by cutting a conic by a pencil, 
 
 166. 
 of conjugate points or lines with 
 ; regard to a conic, 209. 
 of conjugate diameters of a conic, 
 227. 
 
INDEX. 
 
 307 
 
 Involution-properties of the complete 
 quadrangle and quadrilateral, 107. 
 
 of a conic and an inscribed or circum- 
 scribed quadrangle, 148, 225. 
 
 of a conic and an inscribed or cir- 
 cumscribed triangle, 152, 157. 
 
 of a conic, two tangents, and their 
 chord of contact, 154. 
 
 of conjugate points and lines with 
 regard to a conic, 209. 
 
 Lambert, ix, xi, 96-98. 
 
 Latus rectum, 257, 258. 
 
 Locus of the centre of perspective of 
 
 two figures when one is turned 
 
 round the axis of perspective, 1 2,98. 
 of the intersection of corresponding 
 
 rays of two projective pencils is a 
 
 conic, 119. 
 ad quatuor lineas, 158. 
 of middle points of parallel chords 
 
 of a conic, 217. 
 of poles of a straight line with regard 
 
 to conies inscribed in a quadri- 
 lateral, 237. 
 of the centre of a conic, given four 
 
 tangents, 237. 
 of foot of perpendicular from the 
 
 focus of a conic on a tangent, 260. 
 of the intersection of orthogonal 
 
 tangents to a conic, 269. 
 
 Maclaurin, xi, 127, 141, 185, 297^298^ 
 
 Major and minor axes of an ellipse, 228. 
 
 Menelaus, theorem on triangle cut by 
 a transversal, 112, 280. 
 
 Metrical, the term, distinguished from 
 descriptive, 50. 
 
 Mobius, theorem on figures in per- 
 spective, 12. 
 on anharmonic ratio, x, 56, 61. 
 
 Monge, xii. 
 
 Newton, locus of centre of a conic in- 
 scribed in a quadrilateral, 238. 
 organic description of a conic, xi, 
 297. 
 Nine-point circle, 283. 
 Normal, 252. 
 
 Oppositely equal pencils, 90. 
 
 they generate an equilateral hyper- 
 bola by their intersection, 286. 
 Oppositely equal ranges, 88. 
 Organic description of a conic, 297. 
 Orthocentre of a triangle circumscribing 
 , a parabola lies on the directrix, 273. 
 of a triangle inscribed in an equi- 
 lateral hyperbola lies on the curve, 
 287. 
 Orthogonal projection, 19. 
 
 pair of rays in a pencil in involution, 
 172. 
 
 pair of conjugate diameters of a 
 
 conic, 227. 
 conjugate lines with respect to a 
 conic, 251, 252. 
 Osculating conies, 189. 
 circle of a conic, 190. 
 
 Pappus, x, xii. 
 
 on a hexagon inscribed in a line- 
 pair, 7$,- 
 
 porismsof, 95, 96. 
 
 fundamental property of the an- 
 harmonic ratios, 54. 
 
 problem 'ad quatuor lineas,' 158. 
 
 on the focus and directrix property 
 of a conic, 257. 
 Parabola, touches the line at infinity, 
 16. 
 
 is determined by four points or tan- 
 gents, 127. 
 
 two fixed tangents are cut propor- 
 tionally by the other tangents, 
 127. 
 
 generated as an envelope from two 
 similar ranges, 128. 
 
 diameters of a, 218. 
 
 construction of the diameters, having 
 given four tangents, 238. 
 
 focal properties of the, 253, 254. 
 
 focus and directrix property, 257. 
 
 self-conjugate triangle, property of, 
 271. 
 
 inscribed in a triangle, its directrix 
 passes through the orthocentre, 
 273- 
 Parabola, construction of a, given four 
 points, 181. 
 
 given four tangents, 135. 
 
 given three tangents and a point, 
 182. 
 
 under various conditions, 138, 139, 
 143, 146. 
 
 given the axis, the focus, and one 
 point, 266. 
 
 given two tangents, the point of con- 
 tact of one of them, and the 
 direction of the axis, 278. 
 
 given two tangents and their points 
 of contact, 279. 
 Parallel lines meet at infinity, 5. 
 
 projection, 19. 
 
 lines, construction of, with the ruler 
 only, 96, 300. 
 Parallelogram, inscribed in or circum- 
 scribed about a conic, 219, 221. 
 
 described on a pair of conjugate semi- 
 diameters of a conic is of constant 
 area, 234. 
 Parameter of homology, 63. 
 Pascal's theorem, xi,_j_24. 
 
 lines, the sixty, 125. 
 Pencil, flat, defined, 22. 
 
 axial, 22. 
 
 X 2 
 
308 
 
 INDEX. 
 
 harmonic, 40, 42. 
 in involution, 101. 
 in involution, orthogonal pair of 
 
 rays of a, 172. 
 cut by a conic in pairs of points 
 
 forming an involution, 166. 
 Pentagon, inscribed in a conic, 136. 
 circumscribed to a conic, 145. 
 self- conjugate with regard to a conic, 
 
 293- 
 Perpendiculars, centre of, see Ortho- 
 centre. 
 
 from a focus on tangents to a conic, 
 the locus of their feet a circle, 259. 
 
 from the foci of a conic on a tangent, 
 their product constant, 260. 
 
 from any point of the circumscribing 
 circle of a triangle to the sides, 
 their feet collinear, 261, 299. 
 
 construction of, with the ruler only, 
 
 97, 3°°- 
 Perspective, figures in, 3. 
 triangles in, 7, 8, 246. 
 forms in, 35. 
 plane, 10. 
 relief, 20. 
 Plane of points or lines, 22. 
 Planes, harmonic, 42. 
 
 involution of, 101. 
 Points, harmonic, on a straight line, 40. 
 harmonic, on a circle, 116. 
 harmonic, on a conic, 122, 157. 
 projective ranges of, on a conic, 161. 
 Polar reciprocal curves and figures, 
 240, 241. 
 of a conic with respect to a conic 
 
 is a conic, 240. 
 of a circle with respect to a circle, 
 
 274. 
 of a conic with respect to a focus, 
 
 274* 275. 
 of a conic with respect to a point on 
 the director circle, 288. 
 Polar system, defined, 248. 
 
 determined by two triangles in per- 
 spective, 248. 
 determined by a self-conjugate tri- 
 angle and a point and its polar, 293. 
 Pole and polar, defined, 201, 202. 
 reciprocal property of, 204. 
 theory of, applied to the solution 
 
 of problems, 300. 
 construction of, 205, 206, 248. 
 Poles, range of, projective with the 
 pencil formed by their polars, 209, 
 224. 
 of a straight line with regard to all 
 conies inscribed in the same quad- 
 lateral lie on a fixed straight line, 
 
 Polygon, inscribed in a conic, whose 
 sides pass through fixed points, 
 151, 185/187. 
 
 circumscribed to a conic, whose ver- 
 tices slide on fixed lines, 152, 186. 
 
 whose sides pass through fixed points 
 and whose vertices lie on fixed 
 lines, 184. 
 Poncelet, ix, x, xii. 
 
 on variable polygons inscribed in or 
 circumscribed to a conic, 151, 184- 
 187. 
 
 on ideal chords, 2 26. 
 
 on polar reciprocal figures, 240. 
 
 on triangles inscribed in one conic 
 and circumscribed about another, 
 
 . 2 44- 
 Porisms, of Euclid and Pappus, 95, 96. 
 
 of in- and circumscribed triangle, 94, 
 244. 
 
 of the inscribed and self-conjugate 
 triangle, 243. 
 
 of the circumscribed and self-con- 
 jugate triangle, 243. 
 Power of a point with respect to a 
 
 circle, 58. 
 Prime-forms, the six, 24. 
 Problems, solved with ruler only, 96-98. 
 
 of the second degree, 1 76-200. 
 
 solved by means of the ruler and a 
 fixed circle, 194, 300. 
 
 solved by polar reciprocation, 301. 
 Projection, operation of, 2, 22, 164. 
 
 central, 3. . 
 
 orthogonal, 19. 
 
 parallel, 19. 
 
 of a triad of elements into any other 
 given triad, 36. 
 
 of a quadrangle into any given quad- 
 rangle, 80. 
 
 of a plane figure into another plane 
 figure, 81. 
 Projective forms and figures, 34. 
 
 forms, when in perspective, 67 . 
 
 forms, when harmonic, 69. 
 
 ranges, metrical relations of, 62. 
 
 forms, construction of, 70-74. 
 
 figures, construction of, 81-84. 
 
 plane figures can be put into homo- 
 logy 84. 
 
 properties of points and tangents of 
 a circle, 11 4-1 17. 
 
 properties of points and tangents of 
 a conic, 11 8-1 30. 
 Projectivity of any two forms ABC 
 &nd A' B'C f , 36. 
 
 of two forms ABCD and BABC, 38. 
 
 of harmonic forms, 41,-43. 
 
 of the anharmonic ratio, 54."^" 
 
 of any two plane quadrangles, 80. 
 
 of a range of poles and the pencil 
 formed by their polars, 209, 2 24. 
 
 Quadrangle, complete, defined, 29. 
 two plane quadrangles always pro- 
 jective, 80. 
 
INDEX. 
 
 309 
 
 harmonic properties, 39, 47. 
 
 involution properties, 107, 225. 
 
 inscribed in a conic, 138, 140, 208, 
 225. 
 
 if two pairs of opposite sides are con- 
 jugate lines with regard to a conic, 
 the third pair is so too, 246. 
 Quadrangles having the same diagonal 
 points ; their eight vertices lie on 
 a conic or a line-pair, 210. 
 Quadrilateral, complete, defined, 29. 
 
 harmonic properties, 39, 46. 
 
 involution properties, 107, 225. 
 
 middle points of diagonals are col- 
 linear, 109, 299. 
 
 circumscribed to a conic, 142, 208, 
 225, 272. 
 
 locus of centres of inscribed conies, 
 
 237. 
 theorem of Hesse relating to the ex- 
 tremities of the three diagonals, 
 
 245. 
 Quadrilaterals having the same di- 
 agonals ; their eight sides touch a 
 conic or a point-pair, 212. 
 
 Range, defined, 22. 
 
 harmonic, 40. 
 Ranges, projective, on a conic, 161. 
 Ratio, of similitude, 18. 
 
 harmonic, 57. 
 
 anharmonic, 54-62. 
 Reciprocal figures, 85. 
 
 points and lines with regard to a 
 conic, 204. 
 
 triangles, two, are in perspective, 246. 
 Reciprocation, polar, 241. 
 
 with respect to a circley 274, 275. 
 
 applied to solution of problems, 301. 
 Rectangular hyperbola, see Equilateral. 
 Ruler only, problems solved with, 96- 
 
 98. 
 Ruler and fixed circle, problems solved 
 by help of the, 194, 300. 
 
 Section, operation of, 2, 22, 164. 
 of a cone, 14, 18. 
 of a cylinder, 19. 
 Segment, dividing 'two given ones har- 
 monically, 58, 103, 295. 
 of constant magnitude sliding along 
 a line generates two directly equal 
 ranges, 89. 
 bisected, its division into n equal 
 parts by aid of the ruler only, 97. 
 Segments of a straight line, metrical 
 
 relations between, 51, 52. 
 Self-conjugate pentagon with regard +0 
 
 a conic, 293. 
 Self-conjugate triangle, 207-209. 
 
 circumscribing circle of a, its pro- 
 perties, 271. 
 
 Self-conjugate triangles with regard to 
 
 a conic, two; properties of, 242. 
 Self-corresponding elements, defined, 67. 
 of two superposed projective forms, 
 
 68, 69, 78, 91-93. 
 general construction for these, 169. 
 of two coplanar projective figures, 79. 
 of two projective ranges on or series 
 of tangents to a conic, 162, 163. 
 Sheaf, defined, 22. 
 Signs, rule of, 51. 
 
 Similar ranges and pencils, 86, 87, 
 128. 
 and similarly placed figures, 18. 
 Staudt, vi, vii. 
 
 on the geometric prime-forms, 24. 
 on the principle of duality, 26. 
 on harmonic forms, 39. 
 on the construction of two projective 
 
 figures, 81. 
 on the polar system, 248. 
 on an involution of points on a conic, 
 165. 
 Steiner, vii, 3, xii 
 
 on the sixty Pascal lines and Brian- 
 
 chon points, 125. 
 on the solution of problems of the 
 second degree by means of a ruler 
 and a fixed circle, 194. 
 Superposed geometric forms, 68, 69. 
 construction of their self-correspond- 
 ing elements, 169. 
 plane figures, if projective, cannot 
 have more than three self-corre- 
 sponding elements, 79. 
 Supplemental chords, 221. 
 Symmetry, a special case of homology, 
 64. 
 
 Tangents, harmonic, of a circle, 11 6, 
 
 in- 
 harmonic, of a conic, 168. 
 
 to a conic, series of projective, 163, 
 164. 
 
 orthogonal, to a conic, 269. 
 
 to a conic from a given point ; con- 
 structions, 176, 177, 179, 226. 
 
 common, to two conies; construc- 
 tions, 190. 
 Tetragram and Tetrastigm, 29. 
 Townsend, 200. 
 
 Transversal, cut by the sides of a tri- 
 angle, 112. 
 
 cutting a quadrangle or a quadri- 
 lateral, 107, 108. 
 
 cutting a conic and an inscribed 
 quadrangle, 150. 
 
 drawn through a point to cut a 
 conic; property of the product of 
 the segments, 281. 
 
 cutting a hyperbola and its asymp- 
 totes, 156, 282. 
 Transverse axis of a hyperbola, 228. 
 
310 
 
 INDEX. 
 
 Triangle, inscribed in one triangle and 
 circumscribed about another, 94. 
 
 inscribed in a conic, 143, 216. 
 
 circumscribed to a conic, 144, 216. 
 
 inscribed or circumscribed, involu- 
 tion-properties, 152, 157. 
 
 self-conjugate with regard to a conic, 
 207, 270. 
 
 circumscribed to a parabola, 253, 
 
 273. 
 
 self-conjugate with regard to a para- 
 bola, 271. 
 
 self-conjugate with regard to an 
 equilateral hyperbola, 271. 
 
 cut by a conic, Carnot's theorem, 
 279. 
 
 inscribed in an equilateral hyper- 
 bola, 287. 
 
 Triangles, two, self-conjugate with re- 
 gard to a conic ; properties of, 242. 
 
 inscribed in one conic and self-con- 
 jugate to another, 243. 
 
 circumscribed to one conic and self- 
 conjugate to another, 243. 
 
 inscribed in one conic and circum- 
 scribed to another, 244. 
 
 reciprocal, are in perspective, 246. 
 
 formed by two pairs of tangents to a 
 conic and their chords of contact, 
 298. 
 Trisection of an arc of a circle, 294. 
 
 Vanishing points and lines, 5. 
 
 plane, 21. 
 Vertex of a conic, 228, 256. 
 
 circle of curvature at a, 19 D. 
 
 THE END 
 
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