PROJECTIVE &EHMER UNIVERSITY OF CALIFORNIA AT LOS ANGELES pa/iAOJJoq sew n qojijM WQJJ Ajejqn am o\ leuaiew s\m 88CI.-S6006 VINbOdHVO 'S313ONV SOT 8861.96 xog Z|. jon BujJjJBd - 9AUQ 3A3N 3Q SOS Ainiovd Auvuan IVNOIOHU AN ELEMENTARY COURSE IN SYNTHETIC PROTECTIVE GEOMETRY BY DERRICK NORMAN LEHMEB ASSOCIATE PKOFESSOK OF MATHEMATICS I M \KUS1TV UK CALIKOUMA GINN AND COMPANY BOSTON NEW YORK - CHICAGO LONDON ATLANTA DALLAS COLUMBUS SAN FRANCISCO COPYRIGHT, 1917, BY DERRICK NORMAN LEHMEK ALL BIGHTS RESERVED PRINTED IN THE UNITED STATES OF AMERICA 226.7 gbe fltfttmeum GINN AND COMPANY PRO- PRIETORS BOSTON U.S.A. A H ^V ' Engineering & notical Sciences Library PREFACE The following course is intended to give, in as simple a way as possible, the essentials of synthetic protective geometry. While, in the main, the theory is developed along the well-beaten track laid out by the great masters of the subject, it is believed that there has been a slight smoothing of the road in some places. Especially will this be observed in the chapter on Involution. The author has never felt satisfied with the usual treatment of that subject by means of circles and {inharmonic ratios. A purely projective notion ought not to be based on metrical foundations. Metrical developments should be made there, as elsewhere in the theory, by the introduction of infinitely distant elements. The author has departed from the century-old custom of writing in parallel columns each theorem and its dual. He has not found that it conduces to sharpness of vision to try to focus his eyes on two things at once. Those who prefer the usual method of procedure can, of course, develop the two sets of theorems side by side ; the author has not found this the better plan in actual teaching. As regards nomenclature, the author has followed the lead of the earlier writers in English, and has called the system of lines in a plane which all pass through a point a pencil of rays instead of a bundle of rays, as later writers seem inclined to do. For a point considered 369.-503 iv PROJECTIVE GEOMETRY as made up of all the lines and planes through it he has ventured to use the term point system, as being the natural dualization of the usual term J>/1nt!. N. LKI-IMER BEKKELKY, CALIFORNIA CONTENTS CHAPTER I ONE-TO-ONE CORRESPONDENCE SECTION PAGE 1. Definition of one-to-one correspondence 1 2. Consequences of one-to-one correspondence 2 3. Applications in mathematics 2 4. One-to-one correspondence and enumeration 3 5. Correspondence between a part and the whole 4 6. Infinitely distant point 4 7. Axial pencil ; fundamental forms 6 8. Perspective position 5 0. Projective relation 6 10. Infinity-to-one correspondence 7 11. Infinitudes of different orders 7 12. Points in a plane 8 13. Lines through a point 8 14. Planes through a point 8 15. Lines in a plane 8 16. Plane system and point system 9 17. Planes in space 9 18. Points in space 9 19. Space system 9 20. Lines in space 10 21. Correspondence between points and numbers 10 22. Elements at infinity 11 Problems 12 CHAPTER II RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE- TO-ONE CORRESPONDENCE WITH EACH OTHER 23. Seven fundamental forms 14 24. Projective properties 14 25. Desargues's theorem 15 vii viii PROJECTIVE GKOMKTKV SECTION PAGE 26. Fundamental theorem concerning two complete quadrangles 16 27. Importance of the theorem 17 28. Restatement of the theorem 18 29. Four harmonic points 18 30. Harmonic conjugates 19 31. Importance of the notion of four harmonic points .... 19 32. Protective invariance of four harmonic points 20 33. Four harmonic lines 20 34. Four harmonic planes 20 35. Summary of results 21 36. Definition of projectivity 21 37. Correspondence between harmonic conjugates 21 38. Separation of harmonic conjugates 22 39. Harmonic conjugate of the point at infinity 23 40. Projective theorems and metrical theorems. Linear con- struction 23 41. Parallels and mid-points 24 42. Division of a segment into equal parts 25 43. Numerical relations 25 44. Algebraic formula connecting four harmonic points .... 25 45. Further formulae 26 46. Anharmonic ratio 27 Problems 27 CHAPTER III COMBINATION OF TWO PROJECTIVELY RELATED FUNDAMENTAL FORMS 47. Superposed fundamental forms. Self-corresponding elements 29 48. Special case 30 49. Fundamental theorem. Postulate of continuity 31 50. Extension of theorem to pencils of rays and planes .... 32 51. Projective point-rows having a self-corresponding point . . 32 52. Point-rows in perspective position 33 53. Pencils in perspective position 33 54. Axial pencils in perspective position 33 55. Point-row of the second order 33 56. Degeneration of locus 34 CONTENTS ix SECTION PAGE 57. Pencils of rays of the second order 34 58. Degenerate case 34 69. Cone of the second order 35 Problems 35 CHAPTER IV POINT-ROWS OF THE SECOND ORDER 60. Point-row of the second order defined 37 61. Tangent line 37 62. Determination of the locus 38 63. Restatement of the problem 38 64. Solution of the fundamental problem 38 65. Different constructions for the figure 39 66. Lines joining four points of the locus to a fifth 40 67. Restatement of the theorem 40 68. Further important theorem 40 69. Pascal's theorem 40 70. Permutation of points in Pascal's theorem 41 71. Harmonic points on a point-row of the second order ... 42 72. Determination of the locus 42 73. Circles and conies as point-rows of the second order ... 43 74. Conic through five points 43 75. Tangent to a conic 44 76. Inscribed quadrangle 44 77. Inscribed triangle 45 78. Degenerate conic 46 Problems 46 CHAPTER V PENCILS OF RAYS OF THE SECOND ORDER 79. Pencil of rays of the second order defined 48 80. Tangents to a circle 48 81. Tangents to a conic 49 82. Generating point-rows lines of the system 49 83. Determination of the pencil 49 84. Brianchon's theorem . . 51 x PROJECTIVE GEOMETRY SECTION PAGE 85. Permutation of lines in Brianchon's theorem 51 86. Construction of the pencil by Brianchon's theorem .... 51 87. Point of contact of a tangent to a conic 52 88. Circumscribed quadrilateral 52 89. Circumscribed triangle 53 90. Use of Brianchon's theorem 53 91. Harmonic tangents 53 92. Projectivity and perspectivity 53 93. Degenerate case 54 94. Law of duality 54 Problems , 64 CHAPTER VI POLES AND PULARS 96. Inscribed and circumscribed quadrilaterals 66 96. Definition of the polar line of a point 56 97. Further denning properties 57 98. Definition of the pole of a line 57 99. Fundamental theorem of poles and polars 57 100. Conjugate points and lines 57 101. Construction of the polar line of a given point 58 102. Self-polar triangle 58 103. Pole and polar projectively related 58 104. Duality 59 105. Self-dual theorems 60 106. Other correspondences 60 Problems , 60 CHAPTER VII METRICAL PROPERTIES OF THE CONIC SECTIONS 107. Diameters. Center 62 108. Various theorems 62 109. Conjugate diameters 62 110. Classification of conies 63 111. Asymptotes 63 CONTENTS xi SECTION PAUK 112. Various theorems 63 113. Theorems concerning asymptotes 63 114. Asymptotes and conjugate diameters 64 115. Segments cut off on a chord by hyperbola and its asymp- totes 64 116. Application of the theorem 64 117. Triangle formed by the two asymptotes and a tangent . . 65 118. Equation of hyperbola referred to the asymptotes .... 65 119. Equation of parabola 66 120. Equation of central conies referred to conjugate diameters 68 Problems 70 CHAPTER VIII INVOLUTION 121. Fundamental theorem 71 122. Linear construction 72 123. Definition of involution of points on a line 72 124. Double-points in an involution 73 125. Desargues's theorem concerning conies through four points . 74 126. Degenerate conies of the system 74 127. Conies through four points touching a given line .... 75 128. Double correspondence 75 129. Steiner's construction 76 130. Application of Steiner's construction to double correspond- ence 77 131. Involution of points on a point-row of the second order . 78 132. Involution of rays 79 133. Double rays 80 134. Conic through a fixed point touching four lines 80 135. Double correspondence 80 136. Pencils of rays of the second order in involution .... 81 137. Theorem concerning pencils of the second order in involu- tion 81 138. Involution of rays determined by a conic 81 139. Statement of theorem 81 140. Dual of the theorem 82 Problems .... 82 xii PROJECTIVE GEOMETRY CHAPTER IX METRICAL PROPERTIES OF INVOLUTIONS SECTION PAGE 141. Introduction of infinite point ; center of involution ... 84 142. Fundamental metrical theorem . 85 143. Existence of double points 85 144. Existence of double rays 86 145. Construction of an involution by means of circles .... 86 146. Circular points 87 147. Pairs in an involution of rays which are at right angles. Circular involution 88 148. Axes of conies 88 149. Points at which the involution of rays determined by a conic is circular 89 150. Properties of such a point 90 151. Position of such a point 90 152. Discovery of the foci of the conic 91 153. The circle and the parabola . H2 154. Focal properties of conies 93 155. Case of the parabola 94 156. Parabolic reflector 94 157. Directrix. Principal axis. Vertex 94 158. Another definition of a conic 94 159. Eccentricity 95 160. Sum or difference of focal distances 95 Problems 96 CHAPTER X ON THE HISTORY OF SYNTHETIC PROJECTIVE GEOMETRY 161. Ancient results 98 162. Unifying principles 101 163. Desargues 101 164. Poles and polars 102 165. Desargues' s theorem concerning conies through four points 102 166. Extension of the theory of poles and polars to space . . 103 167. Desargues's method of describing a conic 104 168. Reception of Desargues's work 104 CONTENTS xiii SECTION PAGE 169. Conservatism in Desargues's time 105 170. Desargues's style of writing 105 171. Lack of appreciation of Desargues 107 172. Pascal and his theorem 108 173. Pascal's essay 108 174. Pascal's originality 109 175. De la Hire and his work 109 176. Descartes and his influence Ill 177. Newton and Maclaurin 112 178. Maclaurin's construction 112 179. Descriptive geometry and the second revival 113 180. Duality, homology, continuity, contingent relations ... 114 181. Poncelet and Cauchy 115 182. The work of Poncelet 116 183. The debt which analytic geometry owes to synthetic geometry 116 184. Steiner and his work 117 185. Von Staudt and his work 118 186. Recent developments 119 INDEX . 121 AN ELEMENTARY COURSE IN SYNTHETIC PROJECTIVE GEOMETRY CHAPTER I ONE-TO-ONE CORRESPONDENCE 1. Definition of one-to-one correspondence. Given any two sets of individuals, if it is possible to set up such a correspondence between the two sets that to any individual in one set corresponds one and only one individual in the other, then the two sets are said to be in one-to-one correspondence with each other. This notion, simple as it is, is of fundamental importance in all branches of science. The process of counting is nothing but a setting up of a one-to-one correspond- ence between the objects to be counted and certain words, ' one,' ' two,' ' three,' etc., in the mind. Many savage peoples have discovered no better method of counting than by setting up a one-to-one correspondence between the objects to be counted and their fingers. The scientist who busies himself with naming and classifying the objects of nature is only seating up a one-to-one correspondence between the objects and cer- tain words which serve, not as a means of counting the l 2 PROJECTIVE GEOMETRY objects, but of listing them in a convenient way. Thus he may be able to marshal and array his material in such a way as to bring to light relations that may exist between the objects themselves. Indeed, the whole notion of language springs from this idea of one-to-one correspondence. 2. Consequences of one-to-one correspondence. The most useful and interesting problem that may arise in connection with any one-to-one correspondence is to determine just what relations existing between the individuals of one assemblage may be carried over to another assemblage in one-to-one correspondence with it. It is a favorite error to assume that whatever holds for one set must also hold for the other. Magicians are apt to assign magic properties to many of the words and symbols which they are in the habit of using, and scientists are constantly confusing objective things with the subjective formulas for them. After the physicist has set up correspondences between physical facts and mathematical formulas, the " interpretation " of these formulas is his most important and difficult task. 3. In mathematics, effort is constantly being made to set up one-to-one correspondences between simple notions and more complicated ones, or between the well- explored fields of research and fields less known. Thus, by means of the mechanism employed in analytic geom- etry, algebraic theorems are made to yield geometric ones, and vice versa. In geometry we get at the proper- ties of the conic sections by means of the properties of the straight line, and cubic surfaces are studied by means of the plane. ONE-TO-ONE CORRESPONDENCE 4. One-to-one correspondence and enumeration. If a one-to-one correspondence has been set up between the objects of one set and the objects of another set, then the inference may usually be drawn that they have the same number of elements. If, however, there is an infinite number of individuals in each of the two sets, the notion of counting is necessarily ruled out. It may be possible, never- theless, to set up a one-to-one correspondence between the ele- ments of two sets even when the number is infinite. Thus, it is easy to set up such a correspondence between the points of a line an inch long and the points of a line two inches long. For let the lines (Fig. 1) be AB and A'B'. Join A A' and BB', and let these joining lines meet in >S'. For every point C on AB a point C 1 may be found on A'B' by joining C to S and noting the point C' where CS meets A'B'. Similarly, a point C may be found on AB for any point C' on A'B'. The corre- spondence is clearly one-to-one, but it would be absurd to infer from this that there were just as many points on AB as on A'B'. In fact, it would be just as reasonable to infer that there were twice as many points on A'B' as on AB. For if we bend A'B' into a circle with center at S (Fig. 2), we see that for every point C on AB there are two points on A'B'. Thus c FIG. 2 4 PROJECTIVE GEOMETRY it is seen that the notion of one-to-one correspondence is more extensive than the notion of counting, and includes the notion of counting only when applied to finite assemblages. 5. Correspondence between a part and the whole of an infinite assemblage. In the discussion of the last para- graph the remarkable fact was brought to light that it is sometimes possible to set the elements of an assem- blage into one-to-one correspondence with a part of those elements. A moment's reflection will convince one that this is never possible when there is a finite number of elements in the assemblage. Indeed, \\c may take this property as our definition of an infinite assemblage, and say that an infinite assemblage is one that may be put into one-to-one correspondence with part of itself. This has the advantage of being a positive definition, as opposed to the usual negative definition of an infinite assemblage as one that cannot be counted. 6. Infinitely distant point. We have illustrated above a simple method of setting the points of two lines into one-to-one correspondence. The same illustration will serve also to show how it is possible to set the points on a line into one-to-one correspondence with the lines through a point. Thus, for any point C on the line AB there is a line SC through S. We must assume the line AB extended indefinitely in both directions, however, if we are to have a point on it for every line through lines cutting across the other, so that the corre- spondence might be called co-to-one. Thus the assem- blage of lines cutting across two lines is of higher order than the assemblage of points on a line ; and as we have called the point-row an assemblage of the first order, the system of lines cutting across two lines ought to be called of the second order. 11. Infinitudes of different orders. Now it is easy to set up a one-to-one correspondence between the points in a plane and the system of lines cutting across two lines which lie in different planes. In fact, each line of the system of lines meets the plane in one point, and each point in the plane determines one and only one line cutting across the two given lines namely, the line of intersection of the two planes determined by the given point with each of the given lines. The assemblage 8 PKOJECTIVE GEOMETRY of points in the plane is thus of the same order as that of the lines cutting across two lines which lie in different planes, and ought therefore to be spoken of as of the second order. We express all these results as follows: 12. If the infinitude of points on a line is taken as the infinitude of the first order, then the infinitude of lines in a pencil of rays and the infinitude of planes in an axial pencil are also of the first order, while the infinitude of lines cutting across two "skew" lines, as well as the infinitude of points in a plane, are of the second order. 13. If we join each of the points of a plane to a point not in that plane, we set up a one-to-one correspondence between the points in a plane and the lines through a point in space. Thus fJtr infinitude of lines through a point in space is of the second order. 14. If to each line through a point in space we make correspond that plane at right angles to it and passing through the same point, we see that the infinitude of planes through a point in space is of the second order. 15. If to each plane through a point in space we make correspond the line in which it intersects a given plane, we see that the infinitude of lines in a plane is of the second order. This may also be seen by setting up a one-to-one correspondence between the points on a plane and the lines of that plane. Thus, take a point S not in the plane. Join any point M of the plane to S. Through S draw a plane at right angles to MS. This meets the given plane in a line m which may be taken as corresponding to the point M. Another very important ONE-TO-ONE CORRESPONDENCE 9 method of setting up a one-to-one correspondence be- tween lines and points in a plane will be given later, and many weighty consequences will be derived from it. 16. Plane system and point system. The plane, con- sidered as made up of the points and lines in it, is called a plane system and is a fundamental form of the second order. The point, considered as made up of all the lines and planes passing through it, is called a point system and is also a fundamental form of the second order. 17. If now we take three lines in space all lying in different planes, and select I points on the first, m points on the second, and n points on the third, then the total number of planes passing through one of the selected points on each line will be Imn. It is reasonable, there- fore, to symbolize the totality of planes that are deter- mined by the GO points on each of the three lines by Go 3 , and to call it an infinitude of the third order. But it is easily seen that every plane in space is included in this totality, so that the totality of planes in space is an infinitude of the third order. 18. Consider now the planes perpendicular to these three lines. Every set of three planes so drawn will determine a point in space, and, conversely, through every point in space may be drawn one and only one set of three planes at right angles to the three given lines. It follows, therefore, that the totality of points in space is an infinitude of the third order. 19. Space system. Space of three dimensions, con- sidered as made up of all its planes and points, is then a fundamental form of the third order, which we shall call a space system. 10 PROJECTIVE GEOMETRY 20. Lines in space. If we join the twofold infinity of points in one plane with the twofold infinity of points in another plane, we get a totality of lines of space which is of the fourth order of infinity. The totality of lines in space gives, then, a fundamental form of the fourth order. 21. Correspondence between points and numbers. In the theory of analytic geometry a one-to-one corre- spondence is assumed to exist between points on a line and numbers. In order to justify this assumption a very extended definition of number must be made use of. A one-to-one correspondence is then set up be- tween points in the plane and pairs of numbers, and also between points in space an,d sets of three numbers. A single constant will serve to define the position of a point on a line ; two, a point in the plane ; three, a point in space ; etc. In the same theory a one-to-one correspondence is set up between loci in the plane and equations in two variables ; between surfaces in space and equations in three variables ; etc. The equation of a line in a plane involves two constants, either of which may take an infinite number of values. From this it follows that there is an infinity of lines in the piano which is of the second order if the infinity of points on a line is assumed to be of the first. In the same way a circle is determined by three conditions ; a sphere by four ; etc. We might then expect to be able to set up a one-to-one correspondence between circles in a plane and points, or planes in space, or between spheres and lines in space. Such, indeed, is the case, and it is often possible to infer theorems concerning spheres ONE-TO-ONE CORRESPONDENCE 11 from theorems concerning lines, and vice versa. It is possibilities such as these that give to the theory of one-to-one correspondence its great importance for the mathematician. It must not be forgotten, however, that we are considering only continuous correspondences. It is perfectly possible to set up a one-to-one correspond- ence between the points of a line and the points of a plane, or, indeed, between the points of a line and the points of a space of any finite number of dimensions, if the correspondence is not restricted to be continuous. 22. Elements at infinity. A final word is necessary in order to explain a phrase which is in constant use in the study of projective geometry. We have spoken of the "point at infinity" on a straight line a fictitious point only used to bridge over the exceptional case when we are setting up a one-to-one correspondence between the points of a line and the lines through a point. We speak of it as "a point" and not as "points," because in the geometry studied by Euclid we assume only one line through a point parallel to a given line. In the same sense we speak of all the points at infinity in a plane as lying on a line, "the line at infinity," because the straight line is the simplest locus we can imagine which has only one point in common with any line in the plane. Likewise we speak of the " plane at infinity," because that seems the most convenient way of imagining the points at infinity in space. It must not be inferred that these conceptions have any essential connection with physical facts, or that other means of picturing to ourselves the infinitely distant configura- tions are not possible. In other branches of mathematics, 12 PROJECTIVE GEOMETRY notably in the theory of functions of a complex vari- able, quite different assumptions are made and quite different conceptions of the elements at infinity are used. As we can know nothing experimentally about such things, we are at liberty to make any assumptions we please, so long as they are consistent and serve some useful purpose. PROBLEMS 1. Since there is a threefold infinity of points in space, there must be a sixfold infinity of pairs of points in space. Each pair of points determines a line. Why, then, is there not a sixfold infinity of lines in space ? 2. If there is a fourfold infinity of lines in space, why is it that there is not a fourfold infinity of planes through a point, seeing that each line in space determines a plane through that point ? 3. Show that there is a fourfold infinity of circles in space that pass through a fixed point. (Set up a one-to-one correspondence between the axes of the circles and lines in space.) 4. Find the order of infinity of all the lines of space that cut across a given line ; across two given lines ; across three given lines ; across four given lines. 5. Find the order of infinity of all the spheres in space that pass through a given point ; through two given points ; through three given points ; through four given points. 6. Find the order of infinity of all the circles on a sphere ; of all the circles on a sphere that pass through a fixed point ; through two fixed points ; through three fixed points ; of all the circles in space ; of all the circles that cut across a given line. ONE-TO-ONE CORRESPONDENCE 13 7. Find the order of infinity of all lines tangent to a sphere ; of all planes tangent to a sphere ; of lines and planes tangent to a sphere and passing through a fixed point. 8. Set up a one-to-one correspondence between the series of numbers 1, 2, 3, 4, and the series of even numbers 2, 4, 6, 8 . Are we justified in saying that there are just as many even numbers as there are numbers altogether ? 9. Is the axiom " The whole is greater than one of its parts " applicable to infinite assemblages ? 10. Make out a classified list of all the infinitudes of the first, second, third, and fourth orders mentioned in this chapter. CHAPTER II RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE- TO-ONE CORRESPONDENCE WITH EACH OTHER 23. Seven fundamental forms. In the preceding chap- ter we have called attention to seven fundamental forms : the point-row, the pencil of rays, the axial pencil, the plane system, the point system, the space system, and the system of lines in space. These fundamental forms are the material which we intend to use in building up a general theory which will be found to include ordinary geometry as a special case. We shall be concerned, not with measurement of angles and areas or line seg- ments, as in the study of Euclid, but in combining and comparing these fundamental forms and in "generating'' new forms by means of them. In problems of con- struction we shall make no use of measurement, either of angles or of segments, and except in certain special applications of the general theory we shall not find it necessary to require more of ourselves than the ability to draw the line joining two points, or to find the point of intersections of two lines, or the line of intersection of two planes, or, in general, the common elements of two fundamental forms. 24. Protective properties. Our chief interest in this chapter will be the discovery of relations between the elements of one form which hold between the 14 FUNDAMENTAL FORMS 15 corresponding elements of any other form in one-to-one correspondence with it. We have already called atten- tion to the danger of assuming that whatever relations hold between the elements of one assemblage must also hold between the corresponding elements of any assem- blage in one-to-one correspondence with it. This false assumption is the basis of the so-called "proof by analogy" so much in vogue among speculative theorists. When it appears that certain relations existing between the points of a given point-row do not necessitate the same relations between the corresponding elements of another in one-to-one correspondence with it, we should view with suspicion any application of the " proof by analogy" in realms of thought where accurate judg- ments are not so easily made. For example, if in a given point-row u three points, A, B, and C, are taken such that B is the middle point of the segment AC, it does not follow that the three points A', B', C' in a point-row perspective to u will be so related. Relations between the elements of any form which do go over unaltered to the corresponding elements of a form protectively related to it are called protective relations. Relations involving measurement of lines or of angles are not protective. 25. Desargues's theorem. We consider first the fol- lowing beautiful theorem, due to Desargues and called by his name. If two triangles, A, B, C and A', B', C', are so situated that the lines AA', BB', and CC' all meet in a point, iln-u the pairs of sides AB and A'B', BC and B'C', CA and C'A' all meet on a straight line, and conversely. 10 PROJECTIVE GEOMET K V Let the lines AA', />/>', and CC' meet in the point M (Fig. 3). Conceive of the figure as in space, so that M is the vertex of a trihedral angle of which the given triangles are plane sections. The lines AB and A'B' are in the same plane and must meet when produced, their point of intersection being clearly a point in the plane of each triangle and there- fore in the line of intersection of these two planes. Call this point P. By similar reasoning the point Q of intersection of the lines BC and B'C' must lie on this same line as well as the point R of intersection of CA and C'A'. Therefore the points P, Q, and R all lie on the same line m. If now we con- sider the figure a plane figure, the points P, Q, and JK still all lie on a straight line, which proves the theorem. The converse is established in the same manner. 26. Fundamental theorem concerning two complete quadrangles. This theorem throws into our hands the following fundamental theorem concerning two com- plete quadrangles, a complete quadrangle being defined as the figure obtained by joining any four given points by straight lines in the six possible ways. Given two complete quadrangles, X, L, M, N and K', L\ M', N', so related that XL, X'L', MN, M 1 N' all meet in a point A; Llf, L'M', NK, X' K' (ill meet in a FUNDAMENTAL FORMS 17 point C\ and LN, L'N 1 meet in a point B on the line A C ; then the lines KM and K'M' also meet in a point D on the line AC. For, by the converse of the last theorem, KK', LL', and NN' all meet in a point S (Fig. 4). Also LL', MM, and NN' meet in a point, and therefore in the same point S. Thus KK', LL', and MM' meet in a point, and so, by Desargues's theorem itself, A, S, and D are on a straight line. 27. Importance of the theorem. The importance of this theorem lies in the fact that, A, B, and C being given, an indefinite number of quadrangles A'', L', M',N' may be found such that K'L' and M'N' meet in A, K'N' and L'M' in C', with L'N' passing through B. Indeed, the lines AK' and AM' may be drawn arbitrarily through A, and any line through B may be used to determine L' and N'. By joining these two points to C the points K' and M' are determined. Then the line 18 PROJECTIVE GEOMETKY joining K' and M', found in this way, must pass through the point D already determined by the quad- rangle K, L, M, N. The three points A, B, C, given in order, serve thus to determine a fourth point D. 28. In a complete quadrangle the line joining any two points is called the opposite side to the line joining the other two points. The result of the preceding paragraph may then be stated as follows: Given three points, A, B, C, in a straight line, if a pair of opposite sides of a complete quadrangle pass through A, and another pair through C, and one of the remaining two sides goes through B, then the other of the remaining two sides will go through a fixed point which does not depend on the quadrangle employed. 29. Four harmonic points. Four points, A, B, C, D, related as in the preceding theorem are called four harmonic points. The point D is called the fourth har- monic of B with respect to A and C. Since B and D play exactly the same role in the above construction, B is also the fourth harmonic of D with respect to A and C. B and D are called harmonic conjugates with respect to A and C. We proceed to show that A and C are also harmonic conjugates with respect to B and D that is, that it is possible to find a quadrangle of which two opposite sides shall pass through B, two through D, and of the remaining pair, one through A and the other through C. Let be the intersection of KM and LN (Fig. 5). Join to A and C. The joining lines cut out on the sides of the quadrangle four points, P, Q, It, S. Consider the quadrangle P, K, Q, 0. One pair of opposite sides FUNDAMENTAL FORMS 19 FIG. 5 passes through A, one through (7, and one remaining side through Z>; therefore the other remaining side must pass through B. Similarly, RS passes through B and PS and QR pass through D. The quadrangle P, Q, R, S therefore has two opposite sides through B, two through D, and the remain- ing pair through A and C. A and C are thus harmonic conjugates with respect to B and D, We may sum up the discussion, therefore, as follows : 30. If A and C are harmonic conjugates with respect to B and D, then B and D are harmonic conjugates with respect to A and C. 31. Importance of the notion. The importance of the notion of four harmonic points lies in the fact that it is a relation which is carried over from four points in a point-row u to the four points that correspond to them in any point-row u' perspective to u. To prove this statement we construct a quadrangle -K", L, M, N such that KL and MN pass through A, KN and LM through C, LN through 7?, and KM through D. Take now any point S not in the plane of the quad- rangle and construct the planes determined by S and all the seven lines of the figure. Cut across this set of planes by another plane not passing through ,S. This plane cuts out on the set of seven planes another 20 PKOJECTIVE GEOMETRY quadrangle which determines four new harmonic points, A', B', C", D', on the lines joining S to A, B, C, D. But S may be taken as any point, since the original quad- rangle may be taken in any plane through A, B, C, D; and, further, the points A', B', C', D' are the intersection of SA, SB, SC, SD by any line. We have, then, the remarkable theorem: 32. If any point is joined to four harmonic points, and the four lines thus obtained are cut by any fifth, the four points of intersection are again harmonic. 33. Four harmonic lines. We are now able to extend the notion of harmonic elements to pencils of rays, and indeed to axial pencils. For if we define four harmonic rays as four rays which pass through a point and which pass one through each of four harmonic points, we have the theorem Four harmonic lines are cut by any transversal in four harmonic points. 34. Four harmonic planes. We also define four har- monic planes as four planes through a line which pass one through each of four harmonic points, and we may show that Four harmonic planes are cut by any plane not passing through their common line in four harmonic lines, and also by any line in four harmonic points. For let the planes a, ft, 7, S, which all pass through the line g, pass also through the four harmonic points A, B, C, D, so that a passes through A, etc. Then it is clear that any plane TT through A, B, C, D will cut out four harmonic lines from the four planes, for they are FUNDAMENTAL FORMS 21 lines through the intersection P of g with the plane TT, and they pass through the given harmonic points A, B, C, D. Any other plane cr cuts g in a point S and cuts a, /3, 7, 8 in four lines that meet TT in four points A', B', C', D' lying on PA, PB, PC, and PD respec- tively, and are thus four harmonic lines. Further, any ray cuts a, /3, 7, S in four harmonic points, since any plane through the ray gives four harmonic lines of intersection. 35. These results may be put together as follows : Criven any two assemblages of points, rays, or planes, perspectively related to each other, four harmonic elements of one must correspond to four elements of the other which are likewise harmonic. If, now, two forms are perspectively related to a thircl, any four harmonic elements of one must correspond to four harmonic elements in the other. We take this as our definition of protective correspondence, and say: 36. Definition of projectivity. Two fundamental forms are protectively related to each other when a one-to-one cor- respondence exists bettveen the elements of the two and when four harmonic elements of one correspond to four harmonic elements of the other. 37. Correspondence between harmonic conjugates. Given four harmonic points, A, B, C, D\ if we fix A and C, then B and D vary together in a way that should be thoroughly understood. To get a clear conception of their relative motion we may fix the points L and M of the quadrangle K, L, M, .2V (Fig. 6). Then, as B describes the point-row AC, the point N describes the point-row 22 PROJECTIVE GEOMETRY AM perspective to it. Projecting N again from (7, we get a point-row K on AL perspective to the point-row N and thus protective to the point-row B. Project the point-row K from M and we get a point-row D on AC again, which is projective to the point -row B. For every point B we have thus one and only one point Z), and conversely. In other words, we have set up a one- to-one correspond- ence between the points of a single point-row, which is also a projective correspondence be- p u , Q cause four har- monic points B correspond to four harmonic points D. We may note also that the correspondence is here t-liar- acterized by a feature which does not always appear in projective correspondences : namely, the same process that carries one from B to D will carry one back from r D to B again. This special property will receive further study in the chapter on Involution. 38. It is seen that as B approaches A, D also ap- proaches A. As B moves from A toward C, D moves from A in the opposite direction, passing through the point at infinity on the line AC, and returns on the other side to meet B at C again. In other words, as B traverses AC, D traverses the rest of the line from A to C through infinity. In all positions of B, except at A or (7, B and D are separated from each other by A and C. FUNDAMENTAL FORMS 23 39. Harmonic conjugate of the point at infinity. It is natural to inquire what position of B corresponds to the infinitely distant position of D. We have proved (27) that the particular quadrangle K, L, M, N employed is of no consequence. We shall therefore avail ourselves of one that lends itself most readily to the solution of the problem. We choose the point L so that the trian- gle ALC is isosceles (Fig. 7). Since D is supposed to be at infinity, the line KM is parallel to AC. There- fore the triangles KAC and MAC are equal, and the triangle ANC is also isosceles. The triangles CNL and ANL are therefore equal, and the line LB bisects the angle ALC. B is therefore the middle point of AC, and we have the theorem The harmonic conjugate of the middle point of AC is at infinity. 40. Projective theorems and metrical theorems. Linear construction. This theorem is the connecting link be- tween the general protective theorems which we have been considering so far and the metrical theorems of ordinary geometry. Up to this point we have said noth- ing about measurements, either of line segments or of angles. Desargues's theorem and the theory of harmonic elements which depends on it have nothing to do with magnitudes at all. Not until the notion of an infinitely distant point is brought in is any mention made of distances or directions. We have been able to make all of our constructions up to this point by means of the straightedge, or ungraduated ruler. A construction 24 made with such an instrument we shall call a linear construction. It requires merely that we be able to draw the line joining two points or find the point of intersection of two lines. 41. Parallels and mid-points. It might be thought that drawing a line through a given point parallel to a given line was only a special case of drawing a line joining two points. Indeed, it consists only in draw- ing a line through the given point and through the " infinitely distant point " on the given line. It must be remembered, however, that the expression " infinitely distant point" must not be taken literally. When we say that two parallel lines meet " at infinity," we really mean that they do not meet at all, and the only reason for using the expression is to avoid tedious statement of exceptions and restrictions to our theorems. We ought therefore to consider the drawing of a line par- allel to a given line as a different accomplishment from the drawing of the line joining two given points. It is a remarkable consequence of the last theorem that a parallel to a given line and the mid-point of a given segment are equivalent data. For the construction is reversible, and if we are given the middle point of a given segment, we can construct linearly a line parallel to that segment. Thus, given that B is the middle point of AC, we may draw any two lines through A, and any line through B cutting them in points N and L. Join N and L to C and get the points K and M on the two lines through A. Then KM is parallel to AC. The bisection of a given segment and the drawing of a line parallel to the segment are equivalent data when linear construction is used. FUNDAMENTAL FORMS 25 42. It is not difficult to give a linear construction for the problem to divide a given segment into n equal parts, given only a parallel to the segment. This is simple enough when n is a power of 2. For any other number, such as 29, divide any segment on the line parallel to AC into 32 equal parts, by a repetition of the process just described. Take 29 of these, and join the first to A and the last to C. Let these joining lines meet in S. Join S to all the other points. Other problems, of a similar sort, are given at the end of the chapter. 43. Numerical relations. Since three points, given in order, are sufficient to determine a fourth, as explained above, it ought to be possible to reproduce the process numerically in view of the one-to-one correspondence which exists between points on a line and numbers; a correspondence which, to be sure, we have not estab- lished here, but which is discussed in any treatise on the theory of point sets. We proceed to discover what relation between four numbers corresponds to the harmonic relation between four points. 44. Let A, B, C, D be four harmonic points (Fig. 8), and let SA, SB, SC, SD be four harmonic lines. Assume a line drawn through B parallel y IG . to SD, meeting SA in A' and SC in C'. Then A,' B, 6", and the infinitely distant point on A'C' are four harmonic points, and therefore B is the middle point of the segment A'C'. Then, since 26 PROJECT1VE GEOMETRY the triangle DAS is similar to the triangle BAA', we may write the proportion AB:AD=BA':SD. Also, from the similar triangles DSC and BCC', we have CD:CB = SD:BC'. From these two proportions we have, remembering that AD- CB the minus sign being given to the ratio on account of the fact that A and C are always separated from B and D, so that one or three of the segments AB, CD, AD, CB must be negative. 45. Writing the last equation in the form and using the fundamental relation connecting three points on a line, which holds for all positions of the three points if account be taken of the sign of the segments, the last proportion may be written ( CA - BA) : AB = - ( CA - DA) : AD, or (AB-AC):AB = (AC-ADy.AD; so that AB, AC, and AD are three quantities in har- monic progression, since the difference between the first and second is to the first as the difference between the second and third is to the third. Also, from this last proportion comes the familiar relation which is convenient for the computation of the distance AD when AB and AC are given numerically. FUNDAMENTAL FORMS 27 46. Anharmonic ratio. The corresponding relations between the trigonometric functions of the angles deter- mined by four harmonic lines are not difficult to obtain, but as we shall not need them in building up the theory of projective geometry, we will not discuss them here. Students who have a slight acquaintance with trigonometry may read in a later chapter ( 161) a development of the theory of a more general relation, called the anharmonic ratio, or cross ratio, which connects any four points on a line. PROBLEMS 1. Draw through a given point a line which shall pass through the inaccessible point of intersection of two given lines. The following construction may be made to depend upon Desargues's theorem : Through the given point P draw any two rays cutting the two lines in the points AB' and A'B, A, B, lying on one of the given lines and A', B', on the other. Join A A' and BB', and find their point of intersec- tion S. Through 5 draw any other ray, cutting the given lines in CC'. Join BC' and B'C, and obtain their point of intersection Q. PQ is the desired line. Justify this construction. 2. To draw through a given point P a line which shall meet two given lines in points A and B, equally distant from P. Justify the following construction : Join P to the point S of intersection of the two given lines. Construct the fourth harmonic of PS with respect to the two given lines. Draw through P a line parallel to this line. This is the required line. 3. Given a parallelogram in the same plane with a given segment A C, to construct linearly the middle point of A C. 28 PROJECTIVE GEOMETRY 4. Given four harmonic lines, of which one pair are at right angles to each other, show that the other pair make equal angles with them. This is a theorem of which frequent use will be made. 5. Given the middle point of a line segment, to draw a line parallel to the segment and passing through a given point. 6. A line is drawn cutting the sides of a triangle ABC in the points A', B', C', the point A' lying on the side BC, etc. The harmonic conjugate of A' with respect to B and C is then constructed and called A". Similarly, B" and C" are constructed. Show that A " B' C" lie on a straight line. Find other sets of three points on a line in the figure. Find also sets of three lines through a point. CHAPTER III COMBINATION OF TWO PROJECTIVELY RELATED FUNDAMENTAL FORMS 47. Superposed fundamental forms. Self-corresponding elements. We have seen (37) that two projective point-rows may be superposed upon the same straight line. This happens, for example, when two pencils which are projective to each other are cut across by a straight line. It is also possible for two projective pencils to have the same center. This happens, for example, when two projective point-rows are projected to the same point. Similarly, two projective axial pen- cils may have the same axis. We examine now the possibility of two forms related in this way, having an element or elements that correspond to themselves. We have seen, indeed, that if B and D are harmonic conjugates with respect to A and (7, then the point- row described by B is projective to the point-row de- scribed by D, and that A and C are self-corresponding points. Consider more generally the case of two pencils perspective to each other with axis of perspectivity u 1 (Fig. 9). Cut across them by a line u. We get thus two projective point-rows superposed on the same line u, and a moment's reflection serves to show that the point N of intersection u and u' corresponds to itself in the two point-rows. Also, the point M, where u 29 30 PKOJECTIVE GEOMETRY intersects the line joining the centers of the two pen- cils, is seen to correspond to itself. It is thus possible for two projective point- rows, superposed upon the same line, to have two self-corresponding points. Clearly M and N may fall together if the line joining the centers of the pencils happens to pass through the point of in- tersection of the lines u , / FIG. 9 and u . f 48. We may also give an illustration of a case where two superposed projective point-rows have no self-corresponding points at all. Thus we may take two lines revolving about a fixed point S and always making the same angle a with each other (Fig. 10). They will cut out on any line u in the plane two point- rows which are easily seen to be projective. For, given any four rays SP which are harmonic, the four corresponding rays SP' must also be harmonic, since they make the same angles with each other. Four harmonic points P corre- spond, therefore, to four harmonic points P'. It is clear, however, that no point P can coincide with its corre- sponding point P', for in that case the lines PS and Fir,. 10 TWO FUNDAMENTAL FORMS 31 P'S would coincide, which is impossible if the angle between them is to be constant. 49. Fundamental theorem. Postulate of continuity. We have thus shown that two projective point-rows, superposed one on the other, may have two points, one point, or no point at all corresponding to themselves. We proceed to show that If two projective point-rows, superposed upon the same straight line, have more than two self-corresponding points, they must have an infinite number, and every point corre- sponds to itself ; that is, the two point-rows are not essentially distinct. If three points, A, B, and C, are self-corresponding, then the harmonic conjugate D of B with respect to A and C must also correspond to itself. For four harmonic points must always correspond to four harmonic points. In the same way the harmonic conjugate of D with respect to B and C must correspond to itself. Combining new points with old in this way, we may obtain as many self-corresponding points as we wish. We show further that every point on the line is the limiting point of a finite or infinite sequence of self-corresponding points. Thus, let a point P lie between A and B. Construct now Z>, the fourth harmonic of C with respect to A and B. D may coincide with P, in which case the sequence is closed ; otherwise P lies in the stretch AD or in the stretch DB. If it lies in the stretch DB, construct the fourth harmonic of C with respect to D and B. This point D' may coincide with P, in which case, as before, the sequence is closed. If P lies in the stretch DD', we construct the fourth harmonic of C with respect 32 PROJECTIVE GEOMETRY to -D-O 7 , etc. In each step the region in which P lies is diminished, and the process may be continued until two self-corresponding points are obtained on either side of P, and at distances from it arbitrarily small. We now assume, explicitly, the fundamental postulate that the correspondence is continuous, that is, that the distance between two points in one point-row may be made arbitrarily small by sufficiently diminishing the distance between the corresponding points in the other. Suppose. now that P is not a self-corresponding point, but cor- responds to a point P 1 at a fixed distance d from P. As noted above, we can find self-corresponding points arbitrarily close to P, and it appears, then, that we can take a point D as close to P as we wish, and yet the distance between the correspondiug points D' and P' approaches d as a limit, and not zero, which contradicts the postulate of continuity. 50. It follows also that two projective pencils which have the same center may have no more than two self- corresponding rays, unless the pencils are identical. For if we cut across them by a line, we obtain two projec- ,tive point-rows superposed on the same straight line, which may have no more than two self-corresponding points. The same considerations apply to two projective axial pencils which have the same axis. 51. Projective point-rows having a self -corresponding point in common. Consider now two projective point- rows lying on different lines in the same plane. Their common point may or may not be a self-corresponding point. If the two point-rows are perspectively related, then their common point is evidently a self -corresponding TWO FUNDAMENTAL FORMS 33 point. The converse is also true, and we have the very important theorem : 52. If in two projective point-rows the point of inter- section corresponds to itself, then the point-rows are in perspective position. Let the two point-rows be u and u' (Fig. 11). Let A and A', B and B', be corresponding points, and let also the point M of intersection of u and u' correspond to itself. Let A A' and BB' meet in the point S. Take S as the center of two pencils, one perspective to u and the other perspective to u'. In these two pencils SA coincides with its cor- responding ray SA 1 , SB with its corresponding ray S"B', and SM / ^ with its corresponding ray SM'. The two pencils are thus identical, by the preceding theorem, and any ray SD must coincide with its cor- responding ray SD'. Corresponding points of u and u', therefore, all lie on lines through the point S. 53. An entirely similar discussion shows that If in two projective pencils the line joining their cen- ters is a self-corresponding ray, then the two pencils are perspectively related. 54. A similar theorem may be stated for two axial pencils of which the axes intersect. Very frequent use will be made of these fundamental theorems. 55. Point-row of the second order. The question nat- urally arises, What is the locus of points of intersec- tion of corresponding rays of two projective pencils 34 which are not in perspective position ? This locus, which will be discussed in detail in subsequent chapters, is easily seen to have at most two points in common with any line in the plane, and on account of this fundamental property will be called a point-row of the second order. For any line u in the plane of the two pencils will be cut by them in two protective point- rows which have at most two self-corresponding points. Such a self-corresponding point is clearly a point of intersection of corresponding rays of the two pencils. 56. This locus degenerates in the case of two per- spective pencils to a pair of straight lines, one of which is the axis of perspectivity and the other the common ray, any point of which may be considered as the point of intersection of corresponding rays of the two pencils. 57. Pencils of rays of the second order. Similar inves- tigations may be made concerning the system of lines joining corresponding points of two projective point- rows. If we project the point-rows to any point in the plane, we obtain two projective pencils having the sanu- center. At most two pairs of self-corresponding rays may present themselves. Such a ray is clearly a line joining two corresponding points in the two point-rows. The result may be stated as follows : TJie system of rays joining corresponding points in two projective point-rows has at most two rays in common with any pencil in the plane. For that reason the system of rays is called a pencil of rays of the second order. 58. In the case of two perspective point-rows this system of rays degenerates into two pencils of rays of the first order, one of which has its center at the center 35 of perspectivity of the two point-rows, and the other at the intersection of the two point-rows, any ray through which may be considered as joining two corresponding points of the two point-rows. 59. Cone of the second order. The corresponding theorems in space may easily be obtained by joining the points and lines considered in the plane theorems to a point S in space. Two projective pencils give rise to two projective axial pencils with axes intersecting. Corresponding planes meet in lines which all pass through S and through the points on a point-row of the second order generated by the two pencils of rays. They are thus generating lines of a cone of the second order, or quadric cone, so called because every plane in space not passing through S cuts it in a point-row of the second order, and every line also cuts it in at most two points. If, again, we project two point-rows to a point S in space, we obtain two pencils of rays with a common center but lying in different planes. Corre- sponding lines of these pencils determine planes which are the projections to S of the lines which join the cor- responding points of the two point-rows. At most two such planes may pass through any ray through S. It is called a pencil of planes of the second order. PROBLEMS 1. A man A moves along a straight road u, and another man B moves along the same road and walks so as always to keep sight of A in a small mirror M at the side of the road. How many times will they come together, A moving always in the same direction along the road ? 36 PROJECTIVE GEOMETRY 2. How many times would the two men in the first prob- lem see each other in two mirrors M and N as they walk along the road as before ? (The planes of the two mirrors are not necessarily parallel to .) 3. As A moves along u, trace the path of B so that the two men may always see each other in the two mirrors. 4. Two boys walk along two paths u and u', each holding a string which they keep stretched tightly between them. They both move at constant but different rates of speed, letting out the string or drawing it in as they walk. How many times will the line of the string pass over any given point in the plane of the paths ? 5. Trace the lines of the string when the two boys move at the same rate of speed in the two paths but do not start at the same time from the point where the two paths intersect. 6. A ship is sailing on a straight course and keeps a gun trained on a point on the shore. Show that a line at right angles to the direction of the gun at its muzzle will pass through any point in the plane twice or not at all. (Con- sider the point-row at infinity cut out by a line through the point on the shore at right angles to the direction of .the gun.) 7. Two lines u and u' revolve about two points U and U' respectively in the same plane. They go in the same direc- tion and at the same rate of speed, but one has an angle a the start of the other. Show that they generate a point-row of the second order. 8. Discuss the question given in the last problem when the two lines revolve in opposite directions. Can you recognize the locus ? CHAPTER IV POINT-ROWS OF THE SECOND ORDER 60. Point-row of the second order defined. We have seen that two fundamental forms in one-to-one corre- spondence may sometimes generate a form of higher order. Thus, two point-rows ( 55) generate a system of rays of the second order, and two pencils of rays ( 57), a system of points of the second order. As a system of points is more familiar to most students of geometry than a system of lines, we study first the point-row of the second order. 61. Tangent line. We have shown in the last chapter ( 55) that the locus of intersection of corresponding rays of two projective pencils is a point-row of the second order ; that is, it has at most two points in com- mon with any line in the plane. It is clear, first of all, that the centers of the pencils are points of the locus ; for to the line SS r , considered as a ray of $, must correspond some ray of S' which meets it in S f . S', and by the same argument S, is then a point where corresponding rays meet. Any ray through S will meet it in one point besides S, namely, the point P where it meets its corresponding ray. Now, by choosing the ray through S sufficiently close to the ray SS f , the point P may be made to approach arbitrarily close to S', and the ray S'P may be made to differ in position from the 37 369303 38 PROJECTIVE GEOMETRY tangent line at S' by as little as we please. We have, then, the important theorem The ray at S' which corresponds to the common ray SS' is tangent to the locus at S'. In the same manner the tangent at S may be constructed. 62. Determination of the locus. We now show that it is possible to assign arbitrarily the position of three points, A, B, and C, on the locus (besides the points S and S'*) ; but, these three points being chosen, the locus is completely determined. 63. This statement is equivalent to the following: Griven three pairs of corresponding rays in two projectwe pencils, it is possible to find a ray of one which corre- sponds to any ray of the other. 64. We proceed, then, to the solution of the funda- mental PROBLEM : G-iven three pairs of rays, aa', bb', and cc', of two protective pencils, S and S', to find the ray d' of S' which corresponds to any ray d of S. Call A the intersection of aa', B the intersection of bb', and C the intersection of cc' (Fig. 12). Join AB by the line u, and AC by the line u'. Consider u as a point- row perspective to S, and u' as a point-row perspective to S'. u and u' are projectively related to each other, since S and S' are, by hypothesis, so related. But their point of intersection A is a self-corresponding point, since a and a' were supposed to be corresponding rays. It fol- lows (52) that u and u' are in perspective position, and that lines through corresponding points all pass POINT-ROWS OF THE SECOND ORDER 39 through a point 3/, the center of perspectivity, the position of which will be determined by any two such lines. But the intersection of c with u and the intersec- tion of , S, and S' may thus be considered as chosen arbitrarily on the locus, and the following remarkable theorem follows at once. POINT-ROWS OF THE SECOND ORDER 41 Griven six points, 1, 2, 3, 4, 5, 6, on the point-row of the second order, if we call L the intersection of 12 with 45, M the intersection of 23 with 56, N the intersection of 34 with 61, then L, M, and N are on a straight line. 70. To get the notation to correspond to the figure, we may take (Fig. 13) A = 1, B = 2, S' = 3, D = 4, S = 5, and (7=6. If we make A = 1, C= 2, S= 3, D = 4, S'= 5, and B = Q, the points L and N are interchanged, but the line is left unchanged. It is clear that one point may be named arbitrarily and the other five named in 5 ! = 120 different ways, but since, as we have seen, two different assignments of names give the same line, it follows that there cannot be more than 60 differ- ent lines LMN obtained in this way from a given set of six points. As a matter of fact, the number obtained in this way is in general 60. The above theorem, which is of cardinal importance in the theory of the point-row of the second order, is due to Pascal and was discovered by him at the age of sixteen. It is, no doubt, the most important contribution to the theory of these loci since FIG. 13 42 PEOJECTIVE GEOMETRY the days of Apollonius. If the six points be called the vertices of a hexagon inscribed in the curve, then the sides 12 and 45 may be appropriately called a pair of opposite sides. Pascal's theorem, then, may be stated as follows : TJie three pairs of opposite sides of a hexagon inscribed in a point-row of the second order meet in three points on a line. 71. Harmonic points on a point-row of the second order. Before proceeding to develop the consequences of this theorem, we note another result of the utmost impor- tance for the higher developments of pure geometry, which follows from the fact that if four points on the locus project to a fifth in four harmonic rays, they will project to any point of the locus in four harmonic rays. It is natural to speak of four such points as four har- monic points on the locus, and to use this notion to define projective correspondence between point-rows of the second order, or between a point-row of the second order and any fundamental form of the first order. Thus, in particular, the point-row of the second order, cr, is said to be perspectively related to the pencil S when every ray on S goes through the point on a- which corresponds to it. 72. Determination of the locus. It is now clear that five points, arbitrarily chosen in the plane, are sufficient to determine a point-row of the second order through them. Two of the points may be taken as centers of two projective pencils, and the three others will deter- mine three pairs of corresponding rays of the pencils, and therefore all pairs. If four points of the locus are POINT-ROWS OF THE SECOND ORDER 43 given, together with the tangent at one of them, the locus is likewise completely determined. For if the point at which the tangent is given be taken as the center S of one pencil, and any other of the points for $', then, besides the two pairs of corresponding rays determined by the remaining two points, we have one more pair, consisting of the tangent at S and the ray SS f . Simi- larly, the curve is determined by three points and the tangents at two of them. 73. Circles and conies as point-rows of the second order. It is not difficult to see that a circle is a point-row of the second order. Indeed, take any point S on the circle and draw four harmonic rays through it. They will cut the circle in four points, which will project to any other point of the curve in four harmonic rays ; for, by the theorem concerning the angles inscribed in a circle, the angles involved in the second set of four lines are the same as those in the first set. If, moreover, we pro- ject the figure to any point in space, we shall get a cone, standing on a circular base, generated by two projective axial pencils which are the projections of the pencils at S and S'. Cut across, now, by any plane, and we get a conic section which is thus exhibited as the locus of intersection of two projective pencils. It thus appears that a conic section is a point-row of the second order. It will later appear that a point-row of the second order is a conic section. In the future, therefore, we shall refer to a point-row of the second order as a conic. 74. Conic through five points. Pascal's theorem fur- nishes an elegant solution of the problem of drawing a conic through five given points. To construct a sixth 44 FIG. 14 point on the conic, draw through the point numbered 1 an arbitrary line (Fig. 14), and let the desired point 6 be the second point of intersection of this line with the conic. The point L = 12 45 is obtainable at once ; also the point ^=34 61. But L and N determine Pascal's line, and the in- tersection of 23 with 56 must be on this line. Intersect, then, the line LN with 23 and obtain the point M. Join M to 5 and intersect with 61 for the desired point 6. 75. Tangent to a conic. If two points of Pascal's hex- agon approach coincidence, then the line joining them approaches as a limiting position the tangent line at that point. Pascal's theorem thus affords a ready method of drawing the tangent line to a conic at a given point. If the conic is de- termined by the points 1, 2, 3, 4, 5 (Fig. 15), and it is desired to draw the tangent at the point 1, we may call that point 1, 6. The points L and M are obtained as usual, and the intersection of 34 with LM gives N. Join N to the point 1 for the desired tangent at that point. 76. Inscribed quadrangle. Two pairs of vertices may coalesce, giving an inscribed quadrangle. Pascal's theo- rem gives for this case the very important theorem Two pairs of opposite sides of any quadrangle inscribed in a conic meet on a straight line, upon which line also intersect the two pairs of tangents at the opposite vertices. POINT-BOWS OF THE SECOND ORDER 45 For let the vertices be J, B, C, and D, and call the vertex A the point 1, 6 ; 7?, the point 2 ; (7, the point 3, 4 ; and Z>, the point 5 (Fig. 16). Pascal's theorem then indicates that L = AB-CD, M=AD-BC, and N, which is the inter- section of the tangents at A and (7, are all on a straight line u. But p IG< if we were to call A the point 2, B the point 6, 1, C the point 5, and D the point 4, 3, then the intersection P of the tangents at B and Z> are also on this same line u. Thus L, M, N, and P are four points on a straight line. The consequences of this theorem are so numerous and important that we shall devote a separate chapter to them. 77. Inscribed triangle. Finally, thjee of the vertices of the hex- agon may coalesce, giving a trian- gle inscribed in a conic. Pascal's theorem then reads as follows (Fig. 17) for this case: The three tangents at the vertices of a triangle inscribed in a conic meet the opposite sides in three points on a straight line. FIG. 17 46 PROJECTIVE GEOMETRY 78. Degenerate conic. If we apply Pascal's theorem to a degenerate conic made up of a pair of straight lines, we get the following theo- rem (Fig. 18) : If three points, A, B, C, are chosen on one line, and three points, A', B', C 1 , are chosen on IIG. 18 another, then the three points L=AB'-A'B, N = BC'-B'C, M=CA'-C'A are all on a straight line. PROBLEMS 1. In Fig. 12, select different lines u and u' and find for each pair the center of perspectivity M. 2. Given four points, A, B, C, D, in the plane, construct a fifth point P such that the lines PA, PB, PC, PD shall be four harmonic lines. Suggestion. Draw a line a through the point A such that the four lines a, AB, AC, AD are harmonic. Construct now a conic through A, B, C, and D having a for a tangent at A. 3. Where are all the points P, as determined in the preceding question, to be found ? 4. Select any five points in the plane and draw the tan- gent to the conic through them at each of the five points. 5. Given four points on the conic, and the tangent at one of them, to construct the conic. (" To construct the conic " means here to construct as many other points as may be desired.) POINT-ROWS OF THE SECOND ORDER 47 6. Given three points on the conic, and the tangent at two of them, to construct the conic. 7. Given five points, two of which are at infinity in different directions, to construct the conic. (In this, and in the following examples, the student is supposed to be able to draw a line parallel to a given line.) 8. Given four points on a conic (two of which are at in- finity and two in the finite part of the plane), together with the tangent at one of the finite points, to construct the conic. 9. The tangents to a curve at its infinitely distant points are called its asymptotes if they pass through a finite part of the plane. Given the asymptotes and a finite point of a conic, to construct the conic. 10. Given an asymptote and three finite points on the conic, to determine the conic. 11. Given four points, one of which is at infinity, and given also that the line at infinity is a tangent line) to construct the conic. CHAPTER V PENCILS OF RAYS OF THE SECOND ORDER 79. Pencil of rays of the second order defined. If the corresponding points of two projective point-rows be joined by straight lines, a system of lines is obtained which is called a pencil of rays of the second order. This name arises from the fact, easily shown ( 57), that at most two 1m es of the system may pass through any arbitrary point in the plane. For if through any point there should pass three lines of the system, then this point might be taken as the center of two projective pencils, one projecting one point-row and the other pro- jecting the other. Since, now, these pencils have three rays of one coincident with the corresponding rays of the other, the two are identical and the two point-rows are in perspective position, which was not supposed. 80. Tangents to a circle. To get a clear notion of this system of lines, we may first show that the tangents to a circle form a system of this kind. For take any two tangents, u and u', to a circle, and let A and B be the points of contact (Fig. 19). Let now t be any third tangent with point of contact at C and meeting u and u' in P and P' respectively. Join A, B, P, P', and C to 0, the center of the circle. Tangents from any point to a circle are equal, and therefore the triangles POA and POC are equal, as also are the triangles P'OB 48 PENCILS OF THE SECOND ORDER 49 and P'OC. Therefore the angle POP' is constant, being equal to half the constant angle AOC+COB. This being true, if we take any four harmonic points, P, P 2 , P%, P, on the line u, they will project to in four harmonic lines, and the tangents to the circle from these four points will meet u' in four har- monic points, PJ, P, f , P s ', P, be- cause the lines from these points to inclose the same angles as the lines from the points P v P 2 , P & , PI on u. The point-row 011 u is therefore protective to the point-row on u'.' Thus the tangents to a circle are seen to join corresponding points on two projective point-rows, and so, according to the definition, form a pencil of rays of the second order. 81. Tangents to a conic. If now this figure be pro- jected to a point outside the plane of the circle, and any section of the resulting cone be made by a plane, we can easily see that the system of rays tangent to any conic section is a pencil of rays of the second order. The converse is also true, as we shall see later, and a pencil of rays of the second order is also a set of lines tangent to a conic section. 82. The point-rows u and u' are, themselves, lines of the system, for to the common point of the two point- rows, considered as a point of u, must correspond some point of u', and the line joining these two corresponding points is clearly u' itself. Similarly for the line u. 83. Determination of the pencil. We now show that it is ^)ossible to assign arbitrarily three lines, a, b, and c, of 50 PROJECTIVE GEOMETRY the system (besides the lines u and u'~) ; but if these three lines are chosen, the system is completely determined. This statement is equivalent to the following: Given three pairs of corresponding points in two pro- jective point-rows, it is possible to find a point in owe which corresponds to any point of the other. We proceed, then, to the solution of the fundamental PROBLEM. Given three pairs of points, A A', BB', and CC', of two protective point-rows u and u', to find the point D 1 of u' which corresponds to any given point D of u. On the line , joining A and A', take two points, S and S', as centers of pencils perspective to u and u' respectively (Fig. 20). The figure will be much simplified if we take S on BB' and S' on CC'. SA and S'A' are corresponding rays of S and S', and the two pencils are therefore in perspective position. It is not difficult to see that the axis of perspectivity m is the line joining B' and C. Given any point D on w, to find the correspond- ing point D' on u' we proceed as follows: Join D to S and note where the joining line meets m. Join this point to S'. This last line meets u' in the desired point D'. We have now in this figure six lines of the system, a, b, c, d, u, and u'. Fix now the position of u, u', b, c, and d, and take four lines of the system, a^ a a , a g , a 4 , which meet b in four harmonic points. These points project to FIG. 20 PENCILS OF THE SECOND ORDER 51 Z>, giving four harmonic points on m. These again project to i>', giving four harmonic points on c. It is thus clear that the rays a^ 2 , 3 , 4 cut out two protective point- rows on any two lines of the system. Thus u and u' are not special rays, and any two rays of the system will serve as the point-rows to generate the system of lines. 84. Brianchon's theorem. From the figure also appears a fundamental theorem due to Brianchon : If 1 , 2, 3, 4, 5, 6 are any six rays of a pencil of the second order, then the lines Z = (12, 45), m = (23, 56), n =(34, 61) all pass through a point. 85. To make the notation fit the figure (Fig. 21), make = 1, ?>=2, w'=3, d 4:, ?^ = 5, c = 6; or, interchanging two of the lines, a = l, /> = 6. Thus, by dif- ferent namings of the lines, it appears that not more than 60 dif- ferent Brianchon points are possible. If we call 12 and 45 oppo- site vertices of a cir- cumscribed hexagon, then Brianchon's theorem may be stated as follows: The three lines joining the three pairs of opposite vertices of a hexagon circumscribed about a conic meet in a point. 86. Construction of the pencil by Brianchon's theorem. Brianchon's theorem furnishes a ready method of deter- mining a sixth line of the pencil of rays of the second FIG. 21 52 PROJECTIVE GEOMETRY Fio. 22 order when five are given. Thus, select a point in line 1 and suppose that line 6 is to pass through it. Then I = (12, 45), n = (34, 61), and the line m = (23, 56) must pass through (/, n). Then (23, In) meets 5 in a point of the required sixth line. 87. Point of contact of a tangent to a conic. If the line 2 approach as a limiting position the line 1, then the intersec- tion (1, 2) approaches as a limiting position the point of contact of 1 with the conic. This suggests an easy way to con- struct the point of contact of any tangent with the conic. Thus (Fig. 22), given the lines 1, 2, 3, 4, 5 to construct the point of contact of 1=6. Draw I = (12, 45), m =(23, 56); then (34, Ini) meets 1 in the required point of contact T. 88. Circumscribed quadrilat- eral. If two pairs of lines in Brianchon's hexagon coalesce, we have a theorem concern- ing a quadrilateral circum- scribed about a conic. It is easily found to be (Fig. 23) The four lines joining the two opposite pairs of vertices and the two opposite points of contact of a quadrilateral circumscribed about a conic all meet in a point. The consequences of this theorem will be deduced later. PENCILS OF THE SECOND ORDER 53 89. Circumscribed triangle. The hexagon may further degenerate into a triangle, giving the theorem (Fig. 24) The lines joining the vertices to the points of contact of the opposite sides of a triangle circumscribed about a conic all meet in a point. 90. Brianchon's theorem may also be used to solve the follow- ing problems: A R B / FIG. 24 Given four tangents and the point of contact on any one of them, to construct other tangents to a conic. Given three tangents and the points of contact of any two of them, to construct other tangents to a conic. 91. Harmonic tangents. We have seen that a variable tangent cuts out on any two fixed tangents projective point-rows. It follows that if four tangents cut a fifth in four harmonic points, they must cut every tangent in four harmonic points. It is possible, therefore, to make the following definition : Four tangents to a conic are said to be harmonic when they meet every other tangent in four harmonic points. 92. Projectivity and perspectivity. This definition sug- gests the possibility of defining a projective correspond- ence between the elements of a pencil of rays of the second order and the elements of any form heretofore discussed. In particular, the points on a tangent are said to be perspectively related to the tangents of a conic when each point lies on the tangent which corresponds to it. These notions are of importance in the higher developments of the subject. 54 PROJECTIVE GEOMETRY 93. Brian chon's theorem may also be applied to a degenerate conic made up of two points and the lines through them. Thus(Fig. 25), If a, J, c are three lines through a point S, and a', b', c' are three lines through an- other point S', then the line* l = (aV, a'6), m = (bJ, Vc), and n = (ra', c'a~) all meet in a point. 94. Law of duality. The observant student will not have failed to note the re- markable similarity between the theorems of this chap- ter and those of the preceding. He will have noted that points have replaced lines and lines have replaced points ; that points on a curve have been replaced by tangents to a curve ; that pencils have been replaced by point-rows, and that a conic considered as made up of a succession of points has been replaced by a conic considered as generated by a moving tangent line. The theory upon which this wonderful law of duality is based will be developed in the next chapter. FIG. 25 PROBLEMS 1. Given four lines in the plane, to construct another which shall meet them in four harmonic points. 2. Where are all such lines found? 3. Given any five lines in the plane, construct on each the point of contact with the conic tangent to them all. PENCILS OF THE SECOND ORDER 55 4. Given four lines and the point of contact on one, to construct the conic. (" To construct the conic " means here to draw as many other tangents as may be desired.) 5. Given three lines and the point of contact on two of them, to construct the conic. 6. Given four lines and the line at infinity, to construct the conic. 7. Given three lines and the line at infinity, together with the point of contact at infinity, to construct the conic. 8. Given three lines, two of which are asymptotes, to construct the conic. 9. Given five tangents to a conic, to draw a tangent which shall be parallel to any one of them. 10. The lines a, b, c are drawn parallel to each other. The lines ', b', c' are also drawn parallel to each other. Show why the lines (&', a7>), (be 1 , b'cj, (ca' } c'a) meet in a point. (In problems 6 to 10 inclusive, parallel lines are to be drawn.) CHAPTER VI POLES AND POLARS 95. Inscribed and circumscribed quadrilaterals. The following theorems have been noted as special cases of Pascal's and Brianchon's theorems: If a quadrilateral be inscribed in a conic, two pairs of opposite sides and the tangents at opposite vertices infcr- sect in four points, all of which lie on a straight line. If a quadrilateral be circumscribed about a conic, the lines joining two pairs of opposite vertices and the HUM joining two opposite points of contact are four lines which meet in a point. 96. Definition of the polar line of a point. Consider the quadrilateral K, L, J/, N inscribed in the conic (Fig. 26). It determines the four harmonic points A, B, G, D which pro- ject from N into the four har- monic points M, Ji, K, 0. Now the tangents at K and M meet in P, a point on the line AB. The line AB is thus determined entirely by 56 D FIG. 26 POLES AND POLAKS 57 the point 0. For if we draw any line through it, meeting the conic in K and M, and construct the harmonic conjugate B of with respect to K and M, and also the two tangents at K and M which meet in the point P, then BP is the line in question. It thus appears that the line L ON may be any line whatever through O ; and since D, L, 0, N are four harmonic points, we may describe the line AB as the locus of points which are harmonic conjugates of with respect to the two points where any line through meets the curve. 97. Furthermore, since the tangents at L and JVmeet on this same line, it appears as the locus of intersections of pairs of tangents drawn at the extremities of chords through 0. 98. This important line, which is completely deter- mined by the point 0, is called the polar of with respect to the conic ; and the point is called the pole of the line with respect to the conic. 99. If a point B is on the polar of 0, then it is har- monically conjugate to with respect to the two inter- sections K and J/ of the line BO with the conic. But for the same reason O is on the polar of B. We have, then, the fundamental theorem If one point lies on the polar of a second, then the second lies on the polar of the first. 100. Conjugate points and lines. Such a pair of points are said to be conjugate with respect to the conic. Simi- larly, lines are said to be conjugate to each other with respect to the conic if one, and consequently each, passes through the pole of the other. 58 PEOJECTIVE GEOMETRY 101. Construction of the polar line of a given point. Given a point P, if it is within the conic (that is, if no tangents may be drawn from P to the conic), we may construct its polar line by drawing through it any two chords and joining the two points of inter- section of the two pairs of tangents at their extremities. If the point P is outside the conic, we may draw the two tangents and construct the chord of contact (Fig. 27). 102. Self-polar triangle. In Fig. 26 it is not difficult to see that AOC is a self-polar triangle, that is, c;u -h vertex is the pole of the opposite side. For B, M, 0, K are four harmonic points, and they project to C in four harmonic rays. The line CO, therefore, meets the line . / .]/.V~ in a point on the polar of A, being separated from A harmonically by the points M and N. Similarly, 1 lie- line CO meets KL in a point on the polar of A, and therefore CO is the polar of A. Similarly, OA is the polar of (7, and therefore is the pole of AC. 103. Pole and polar projectively related. Another very important theorem comes directly from Fig. 26. As a point A moves along a straight line its polar ?'//// respect to a conic revolves about a fixed point and describes a pencil protective to the point-row described by A. For, fix the points L and N and let the point A move along the line AQ\ then the point-row A is project ivc to the pencil LK, and since K moves along the conic, the pencil LK is projective to the pencil NK, which in turn is projective to the point-row C', which, finally, is projective to the pencil 0(7, which is the polar of A. POLES AND POLAKS 59 104. Duality. We have, then, in the pole and polar relation a device for setting up a one-to-one correspond- ence between the points and lines of the plane a cor- respondence which may be called projective, because to four harmonic points or lines correspond always four harmonic lines or points. To every figure made up of points and lines will correspond a figure made up of lines and points. To a point-row of the second order, which is a conic considered as a point-locus, corresponds a pencil of rays of the second order, which is a conic considered as a line-locus. The name ' duality ' is used to describe this sort of correspondence. It is important to note that the dual relation is subject to the same exceptions as the one-to-one correspondence is, and must not be appealed to in cases where the one-to-one correspondence breaks down. We have seen that there is in Euclidean geometry one and only one ray in a pencil which has no point in a point-row perspective to it for a corresponding point ; namely, the line parallel to the line of the point-row. Any theorem, therefore, that involves explicitly the point at infinity is not to be translated into a theorem concerning lines. Further, in the pencil the angle between two lines has nothing to correspond to it in a point-row perspective to the pencil. Any theorem, therefore, that mentions angles is not translatable into another theorem by means of the law of duality. Now we have seen that the notion of the infinitely distant point on a line involves the notion of dividing a segment into any number of equal parts in other words, of measuring. If, therefore, we call any theorem that has to do with the line at infinity or with 60 PEOJECTIVE GEOMETRY the measurement of angles a metrical theorem, and any other kind a protective theorem, we may put the case as follows: Any protective theorem involves another theorem, dual to it, obtainable by interchanging everywhere the words ^ and 105. Self -dual theorems. The theorems of this chap- ter will be found, upon examination, to be self-dual; that is, no new theorem results from applying the process indicated in the preceding paragraph. It is therefore useless to look for new results from the theo- rem on the circumscribed quadrilateral derived from Brianchon's, which is itself clearly the dual o Pascal's theorem, and in fact was first discovered by dualization of Pascal's. 106. It should not be inferred from the above discus- sion that one-to-one correspondences may not be devised that will control certain of the so-called metrical rela- tions. A very important one may be easily found that leaves angles unaltered. The relation called similarity leaves ratios between corresponding segments unaltered. The above statements apply only to the particular one- to-one correspondence considered. PROBLEMS 1. Given a quadrilateral, construct the quadrangle polar to it with respect to a given conic. 2. A point moves along a straight line. Show that its polar lines with respect to two given conies generate a point-row of the second order. POLES AND POLARS 61 3. Given five points, draw the polar of a point with re- spect to the conic passing through them, without drawing the conic itself. 4. Given five lines, draw the polar of a point with re- spect to the conic tangent to them, without drawing the conic itself. 5. Dualize problems 3 and 4. 6. Given four points on the conic, and the tangent at one of them, draw the polar of a given point without drawing the conic. Dualize. 7. A point moves on a conic. Show that its polar line with respect to another conic describes a pencil of rays of the second order. Suggestion. Replace the given 0011101)7 a pair of projective pencils. 8. Show that the poles of the tangents of one conic with respect to another lie on a conic. 9. The polar of a point A with respect to one conic is a, and the pole of a with respect to another conic is A '. Show that as A travels along a line, A ' also travels along another line. In general, if A describes a curve of degree n, show that A ' describes another curve of the same degree n. (The degree of a curve is the greatest number of points that it may have in common with any line in the plane.) CHAPTER VII METRICAL PROPERTIES OF THE CONIC SECTIONS 107. Diameters. Center. After what has been said in the last chapter one would naturally expect to get at the metrical properties of the conic sections by the introduction of the infinite elements in the plane. En- tering into the theory of poles and polars with these elements, we have the following definitions : The polar line of an infinitely distant point is called a diameter, and the pole of the infinitely distant line is called the center, of the conic. 108. From the harmonic properties of poles and polars, The center bisects all chords through it ( 39). Every diameter passes through the center. All chords through the same point at infinity (that /.*, each of a set of parallel chords) are bisected by the diameter which is the polar of that infinitely distant point. 109. Conjugate diameters. We have already denned conjugate lines as lines which pass each through the pole of the other ( 100). Any diameter bisects all chords parallel to its conjugate. The tangents at the extremities of any diameter are parallel, and parallel to the conjugate diameter. Diameters parallel to the sides of an inscribed paral- lelogram are conjugate. All these theorems are easy exercises for the student. 62 METRICAL PROPERTIES 63 110. Classification of conies. Conies are classified ac- cording to their relation to the infinitely distant line. If a conic has t\vo points in common with the line at infinity, it is called a hyperbola ; if it has 110 point in common with the infinitely distant line, it is called an ellipse ; if it is tangent to the line at infinity, it is called a parabola. 111. In a hyperbola the center is outside the curve ( 101), since the two tangents to the curve at the points where it meets the line at infinity determine by their intersection the center. As previously noted, these two tangents are called the asymptotes of the curve. The ellipse and the parabola have no asymptotes. 112. The center of the parabola is at infinity, and there- fore all its diameters are parallel, for the pole of a tan- gent line is the point of contact. The locus of the middle points of a series of parallel chords in a parabola is a diameter, and the direction of the line of centers is the same for all series of parallel chords. The center of an ellipse is within the curve. 113. Theorems concerning asymptotes. We derived as a consequence of the theorem of Brianchon ( 89) the proposition that if a triangle be circumscribed about a conic, the lines joining the vertices to the points of contact of the opposite sides all meet in a point. Take, now, for two of the tangents the asymptotes of a hyperbola, and let any third tangent cut them in A iind B (Fig. 28). If, then, is the intersection of the asymptotes, and therefore the center of the curve, 64 PROJECTIVE GEOMETRY FIG. 28 then the triangle OAB is circumscribed about the curve. By the theorem just quoted, the line through A par- allel to OB, the line through B parallel to OA, and the line OP through the point of contact of the tangent AB all meet in a point C. But OACB is a parallelogram, and PA = PB. Therefore The asymptotes cut off on each tangent a segment which is bisected by the point of contact. 114. If we draw a line OQ parallel to AB, then OP and OQ are conjugate diam- eters, since OQ is parallel to the tangent at the point where OP meets the curve. Then, since A, P, B, and the point at infinity on AB are four harmonic points, we have the theorem Conjugate diameters of the hyperbola are harmonic conjugates with respect to the asymptotes. 115. The chord A"B", parallel to the diameter OQ, is bisected at P' by the conjugate diameter OP. If the chord A"B" meet the asymptotes in A', B', then A', P', B', and the point at infinity are four harmonic points, and therefore P' is the middle point of A'B'. Therefore A'A"=B'B" and we have the theorem TJie segments cut off on any chord between the hyperbola and its asymptotes are equal. 116. This theorem furnishes a ready means of con- structing the hyperbola by points when a point on the curve and the two asymptotes are given. METRICAL PROPERTIES 65 117. For the circumscribed quadrilateral, Brianchon's theorem gave ( 88) The lines joining opposite vertices and the lines joining opposite points of contact are four lines meeting in a point. Take now for two of the tangents the asymptotes, and let AB and CD be any other two (Fig. 29). If B and D are op- posite vertices, and also A and (7, then A C and BD are par- allel, and parallel to PQ, the line joining the points of con- tact of AB and CD, for these are three of the four lines of the theorem just quoted. The fourth is the line at infinity which joins the point of contact of the asymptotes. It is thus seen that the triangles ABC and ADC are equivalent, and therefore the triangles AOB and COD are also. The tangent AB may be fixed, and the tangent CD chosen arbitrarily ; therefore The triangle formed by any tangent to the hyperbola and the two asymptotes is of constant area. 118. Equation of hyperbola referred to the asymptotes. Draw through the point of contact P of the tangent AB two lines, one parallel to one asymptote and the other parallel to the other. One of these lines meets OB at a distance y from 0, and the other meets OA at a distance x from 0. Then, since P is the middle point 66 PROJECTIYE GEOMETRY of AB, x is one half of OA and y is one half of OB. The area of the parallelogram whose adjacent sides arc x and y is one half the area of the triangle J"/>, and therefore, by the preceding paragraph, is constant. This area is equal to xy sin a, where a is the constant angle between the asymptotes. It follows that the product ./// is constant, and since x and y are the oblique coordi- nates of the point P, the asymptotes being the axes of reference, we have The equation of the hyperbola, referral t<> the asymptotes as axes, is xy = constant. This identifies the curve with the hyperbola as de- fined and discussed in works on analytic geometry. 119. Equation of parabola. We have defined the parabola as a conic which is tangent to the line at infinity ( 110). Draw now two tan- gents to the curve (Fig. 30), meeting in A, the points of con- tact being B and C. These two tangents, together with the line at infinity, form a triangle circum- Fie. 30 scribed about the conic. Draw through B a parallel to AC, and through C a parallel to AB. If these meet in D, then AD is a METRICAL PROPERTIES 67 diameter. Let AD meet the curve in P, and the chord BC in Q. P is then the middle point of AQ. Also, Q is the middle point of the chord BC, and therefore the diameter AD bisects all chords parallel to BC. In par- ticular, AD passes through P, the point of contact of the tangent drawn parallel to BC. Draw now another tangent, meeting AB in B' and A C in C'. Then these three, with the line at infinity, make a circumscribed quadrilateral. But, by Brianchon's the- orem applied to a quadrilateral ( 88), it appears that a parallel to AC through B', a parallel to AB through C", and the line BC meet in a point D'. Also, from the similar triangles BB'D' and B A C we have, for all positions of the tangent line B' C', ff T} , . BB , = AC . ABj or, since B'D' = AC', AC' : BB' = AC:AB = constant. If another tangent meet AB in B" and A C in C", we have AC' : BB' = AC" : BB", and by subtraction we get C'C":B'B" = constant; whence The segments cut off on any two tangents to a parabola l>i/ <( variable tangent are proportional. If now we take the tangent B'C' as axis of ordinates, and the diameter through the point of contact O as axis of abscissas, calling the coordinates of B (x, y) and of C (V, y), then, from the similar triangles BMD' and CQ'D', we have y : y' = BD' : D'C = BB' : AB'. Also y : y' = B'D' :C'C=AC': C'C. 68 PROJECTIVE GEOMETRY If now a line is drawn through A parallel to a diameter, meeting the axis of ordinates in X, we have AK : OQ' = AC' : CC' = y:y', and OM : AK = BB' : AB' = y:y', and, by multiplication, or x : x 1 = y* : y' * ; whence The abscissas of two points on a parabola are to each other as the squares of the corresponding coordinates, a diameter and the tangent to the curve at the extremity of the diameter being the axes of reference. The last equation may be written y 2 = 2px, where 2p stands for y'' 2 :x'. The parabola is thus identified with the curve of the same name studied in treatises on analytic geometry. 120. Equation of central conies referred to conjugate diameters. Consider now a central conic, that is, one which is not a parabola and the center of which is therefore at a finite distance. Draw any four tangents to it, two of which are parallel (Fig. 31). Let the parallel tangents meet one of the other tangents in A and B and the other in C and D, and let P and Q be the points of contact of the parallel tangents R and 8 of the others. Then AC, BD, PQ, and RS all meet in a point W ( 88). From the figure, PW: WQ = AP:QC = PD: BQ, or AP BQ = PD QC. METRICAL PROPERTIES 69 If now DC is a fixed tangent and AB a variable one, we have from this equation AP BQ = constant. This constant will be positive or negative according as PA and BQ are measured in the same or in opposite directions. Accordingly we write AP-BQ = b\ Since AD and BC are parallel tangents, PQ is a diam- eter and the conjugate diameter is parallel to AD. The middle point of PQ is the center of the conic. We take now for the axis of abscissas the diameter PQ, and the conjugate diameter for the axis of ordinates. Join A to Q and B to P and draw a line through S parallel to the axis of ordinates. These three lines all meet in a point JV, because AP, BQ, and AB form a triangle circumscribed to the conic. Let NS meet I'Q in J/. Then, from the properties of the circum- scribed triangle ( 89), M, N, S, and the point at infinity on ^S are four harmonic points, and therefore N is the middle point of MS. If the coordinates of S are (x, y), so that OM is x and MS is y, then MN= y/2. Now from the similar triangles PMN and PQB we have BQ:PQ = NM: PM, P IG 70 PROJECTIVE GEOMETRY and from the similar triangles PQA and MQN, AP:PQ = MN:MQ, whence, multiplying, we have 6 2 /4 a 2 = y>/4 (a + x)(a - *), PQ where a = ~ , or, simplifying, x z /a 2 + y 2 / i' 2 = 1, which is the equation of an ellipse when b 2 has a posi- tive sign, and of a hyperbola when 6 2 has a negative sign. We have thus identified point-rows of the second order with the curves given by equations of the second degree. PROBLEMS 1. Draw a chord of a given conic which shall be bisected by a given point P. 2. Show that all chords of a given conic that are bisected by a given chord are tangent to a parabola. 3. Construct a parabola, given two tangents with their points of contact. 4. Construct a parabola, given three points and the direc- tion of the diameters. 5. A line u' is drawn through the pole U of a line u and at right angles to it. The line u revolves about a point P. Show that the line u 1 is tangent to a parabola. (The lines u and u' are called normal conjugates.) 6. Given a conic and its center O, to draw a line through a given point P parallel to a given line q. Prove the fol- lowing construction : Let p be the polar of P, Q the pole of q, and A the intersection of p with OQ. The polar of A is the desired line. CHAPTER VIII INVOLUTION 121. Fundamental theorem. The important theorem concerning two complete quadrangles ( 26), upon which the theory of four harmonic points was based, can easily be extended to the case where the four lines KL, K'L', MN, M'N' do not all meet in the same point A, and the more general theo- rem that re- sults may also be made the basis of a theory no less important, which has to do with six points on a line. The theorem is as follows : Q-iven two complete quadrangles, K, L, M, N and K', L', M', N', so related that KL and K'L' meet in A, MN and M'N' in A', KN and K'N' in B, LM and L'M 1 in B', LN and L'N' in C, and KM and K'M' in C', then, if A, A', B, B', and C are in a straight line, the point C' also lies on that straight line. The theorem follows from Desargues's theorem (Fig. 32). It is seen that KK', LL', MM', NN' all 71 FIG. 32 72 PROJECTIVE GEOMETRY meet in a point, and thus, from the same theorem, ap- plied to the triangles KLM and K'L'M', the point C' is on the same line with A and 11 '. As in the simpler case, it is seen that there is an indefinite number of quadrangles which may be drawn, two sides of which go through A and A', two through B and B', and one through C. The sixth side must then go through C'. Therefore, 122. Two pairs of points, A, A' and B, B', being yt'i'rii, then the point C' corresponding to any given point C is uniquely determined. The construction of this sixth point is easily accom- plished. Draw through A and A' any two lines, and cut across them by any line through C in the points L and N. Join N to B and L to B', thus determining the points K and M on the two lines through A and A'. The line KM determines the desired point C'. Manifestly, starting from C', we come in this way always to the same point C. The particular quadrangle employed is of no consequence. Moreover, since one pair of opposite sides in a complete quadrangle is not distinguishable in any way from any other, the same set of six points will be obtained by starting from the pairs AA ! and CC', or from the pairs BB 1 and CC'. . 123. Definition of involution of points on a line. TJiree pairs of points on a line are said to be in htrolii- tion if through each pair may be drawn a pair of opposite sides of a complete quadrangle. If two pairs are fixed and one of the third pair describes the line, then the other also describes the line, and the points of the line are said to be paired in the involution determined by the two fixed pairs. INVOLUTION 73 124. Double-points in an involution. The points C and C" describe projective point-rows, as may be seen by fixing the points L and M. The self-corresponding points, of which there are two or none, are called the double-points in the involution. It is not difficult to see that the double- points in the involution are harmonic conjugates with respect to corresponding points in the involution. For, fixing as before the points L and M, let the intersection of the lines CL and C'M be P (Fig. 33). The locus of P is a conic which goes through the double-points, because the point-rows C and C' are projective, and therefore so are the pencils LC and MC' which generate the locus of P. Also, when C and C' fall to- gether, the point P coincides with them. Further, the tangents at L and M to this conic described by P are the lines LB and MB. For in the pencil at L the ray LM common to the two pencils which generate the conic is the ray LB' and corresponds to the ray MB of M, which is therefore the tangent line to the conic at M. Similarly for the tangent LB at L. LM is therefore the polar of B with respect to this conic, and B and B' are therefore harmonic conjugates with respect to the double-points. The same discussion applies to any other pair of corresponding points in the involution. Ft' \ FIG. 33 74 PKOJECTIVE GEOMETRY 125. Desargues's theorem concerning conies through four points. Let DD' be any pair of points in the in- volution determined as above, and consider the conic passing through the five points A', 7>, J/, JV, I). We shall use Pascal's theorem to show that this conic also passes through D'. The point D' is determined as fol- lows : Fix L and M as before (Fig. 34) and join D to L, giving on MN the point N'. Join N 1 to B, giving on LK the point K'. Then MK' de- termines the point D' on the line A A', given by the complete quad- rangle K'j L, M, N'. Consider the following six points, numbering them in order: Z>=1, D' = 2, M = 3, JV~=4, 7f= 5, and L = Q. We have the following intersections: #=(12-45), K'= (23-56), N'= (34-61); and since by construction B, N\ and K' are on a straight line, it fol- lows from the converse of Pascal's theorem, which is easily established, that the six points are on a conic. We have, then, the beautiful theorem due to Desargues : The system of conies through four points meets any line in the plane in pairs of points in involution. 126. It appears also that the six points in involution determined by the quadrangle through the four fixed Fiu. INVOLUTION 75 points belong also to the same involution with the points cut out by the system of conies, as indeed we might infer from the fact that the three pairs of oppo- site sides of the quadrangle may be considered as degenerate conies of the system. 127. Conies through four points touching a given line. It is further evident that the involution determined on a line by the system of conies will have a double-point where a conic of the system is tangent to the line. We may therefore infer the theorem Through four fixed points in the plane two conies or none may be drawn tangent to any given line. 128. Double correspondence. We have seen that cor- responding points in an involution form two projective point-rows superposed on the same straight line. Two projective point-rows superposed on the same straight line are, how- ever, not necessarily in involution, as a simple example will show. Take two lines, a and a', which both revolve about a fixed point S and which always make the same angle with each other (Fig. 35). These lines cut out on any line in the plane which does not pass through S two projective point- FlG g5 rows, which are not, however, in involution unless the angle between the lines is a right angle. For a point P may correspond to a point P', which in turn will correspond to some other point 76 PROJECTIVE GEOMETRY than P. The peculiarity of point-rows in involution is that any point will correspond to the same point, in whichever point-row it is considered as belonging. In this case, if a point P corresponds to a point P', then the point P' corresponds back again to the point P. The points P and P' are then said to correspond doubly. This notion is worthy of further study. 129. Steiner's construction. It will be observed that the solution of the fundamental problem given in 83, Given three pairs of points of two projective point-rows, to / G / B A u FIG. 36 construct other pairs, cannot be carried out if the two point-rows lie on the same straight line. Of course the method may be easily altered to cover that case also, but it is worth while to give another solution of the problem, due to Steiner, which will also give further information regarding the theory of involution, and which may, indeed, be used as a foundation for that theory. Let the two point-rows A, B, C, D, > and A', B 1 , C', D', be superposed on the line u. Project them both to a point S and pass any conic K through S. We thus obtain two projective pencils, a, b, c,d, and INVOLUTION 77 a', b', c', d', at S, which meet the conic in the points a, & 7, B, and a', ', 7', 8', ... (Fig. 36). Take now 7 ;is the center of a pencil projecting the points a', /3', 8', , and take 7' as the center of a pencil projecting the points a, y9, 8, . These two pencils are projective to each other, and since they have a self-corresponding ray in common, they are in perspective position and corresponding rays meet on the line joining (70;', 7'*) to (7/3', 7'/3). The correspondence between points in the two point-rows on u is now easily traced. 130. Application of Steiner's construction to double correspondence. Steiner's construction throws into our hands an important theorem concerning double corre- spondence : If two projective point-rows, superposed on the same line, have one pair of points which correspond to each other doubly, then all pairs correspond to each other doubly, and the line is paired in involution. To make this appear, let us call the point A on u by two names, A and P', according as it is thought of as belonging to the one or to the other of the two point- rows. If this point is one of a pair which correspond to each other doubly, then the points A' and P must coin- cide (Fig. 37). Take now any point C, which we will also call E'. We must show that the corresponding point C' must also coincide with the point ft. Join all the points to S, as before, and it appears that the points a and TT' coincide, as also do the points O-'TT and 7/3'. By the above construction the line y'p must meet 7/0' on the line joining (70;', 7'*) with (777', 7V). But these four points form a quadrangle inscribed in the conic, and we know by 95 that the tangents at the opposite 78 PROJECTIVE GEOMETRY vertices 7 and 7' meet on the line v. The line y'p is thus a tangent to the conic, and C 1 and R are the same point. That two projective point-rows super- posed on the same line are also in involution when one pair, and therefore all pairs, correspond doubly may be shown by taking S at one vertex of a complete FIG. 37 quadrangle which has two pairs of opposite 1 sides going through two pairs of points. The details we leave to the student. 131. Involution of points on a point-row of the second order. It is important to note also, in Steiner's con- struction, that we have obtained two point-rows of the second order superposed on the same conic, and have paired the points of one with the points of the other in such a way that the correspondence is double. We may then extend the notion of involution to point-rows of the second order and say that the points of a conic are paired in involution when they are corresponding INVOLUTION 79 points of two protective point-rows superposed on the conic, inul t/'/n'ii tlti'i/ correspond to each other doithly. With this definition we may prove the theorem : The lines joining corresponding points of a point-row of the second order in involution all pass through a fixed point U, and the line joining any two points A, B meets the line joining the two corresponding points A', B' in the points of a line u, which is the polar of U with respect to the conic. For take A and A' as the centers of two pencils, the first perspective to the point-row A', B', C' and the second perspective to the point-row A, B, C. Then, since the common ray of the two pencils corresponds to itself, they are in perspec- tive position, and their axis of perspectivity u (Fig. 38) is the line which joins the point (AB 1 , A'B^) to the point (AC 1 , A'C). It is then immediately clear, from the theory of poles and polars, that BB' and CC' pass through the pole U of the line u. 132. Involution of rays. The whole theory thus far developed may be dualized, and a theory of lines in involution may be built up, starting with the complete quadrilateral. Thus, The three pairs of rays which may be draivn from a point through the three pairs of opposite vertices of a complete quadrilateral are said to be in involution. If the pairs aa' and bb' are fixed, and the line c describes a pencil, the corresponding line c' also describes a pencil, and the rays of the pencil are said to be paired in the involution determined by aa' and bb'. 80 PEOJECTIVE GEOMETRY 133. Double rays. The self-corresponding rays, of which there are two or none, are called double rays of the involution. Corresponding rays of the involution are harmonic conjugates with respect to the double rays. To the theorem of Desargues ( 125) which has to do with the system of conies through four points we have the dual: The tangents from a fixed point to a system of conies tan- gent to four fixed lines form a pencil of rays in inrnlutim. 134. If a conic of the system should go through the fixed point, it is clear that the two tangents would co- incide and indicate a double ray of the involution. The theorem, therefore, follows: Two conies or none may be drawn through a fixed point to be tangent to four fixed lines. 135. Double correspondence. It further appears that two projective pencils of rays which have the same center are in involution if one pair of rays correspond to each other doubly. From this it is clear that \\v might have defined six rays in involution as six rays which pass through a point and also through six points in involution. While this would have been entirely in accord with the treatment which was given the corre- sponding problem in the theory of harmonic points and lines, it is more satisfactory, from an aesthetic point of view, to build the theory of lines in involution on its own base. The student can show, by methods entirely analo- gous to those used in the second chapter, that involution is a projective property ; that is, six rays in involution are cut by any transversal in six points in involution. INVOLUTION 81 136. Pencils of rays of the second order in involution. We may also extend the notion of involution to pen- cils of rays of the second order. Thus, the tangents to a conic are in involution when they are corresponding rays of two protective pencils of the second order superposed upon the same conic, and when they correspond to each other doubly. We have then the theorem : 137. The intersections of corresponding rays of a pen- cil of the second order in involution are all on a straight line u, and the intersection of any two tangents ab, when joined to the intersection of the corresponding tangents a'b', gives a line which passes through a fixed point U, the pole of the line u with respect to the conic. 138. Involution of rays determined by a conic. We have seen in the theory of poles and polars ( 103) that if a point P moves along a line m, then the polar of P revolves about a point. This pencil cuts out on m another point-row P', projective also to P. Since the polar of P passes through P', the polar of P' also passes through P, so that the correspondence between P and P' is double. The two point-rows are therefore in invo- lution, and the double points, if any exist, are the points where the line m meets the conic. A similar involution of rays may be found at any point in the plane, corre- sponding rays passing each through the pole of the other. We have called such points and rays conjugate with respect to the conic ( 100). We may then state the following important theorem : 139. A conic determines on every line in its plane an involution of points, corresponding points in the involution 82 being conjugate with respect to the conic. The double point*, if any exist, are the points where the line meets lli>- '>ni<- A7r/-//////. * >it every point in the plane an involution of rays, corri'xj>i>ml- ing rays being conjugate with respect to the conic. The double rays, if any exist, are tin- t/>tt to the conic. PROBLEMS 1. Two lines are drawn through a point on a conic so as always to make right angles with each other. Show that the lines joining the points where they meet the conic again all pass through a fixed point. 2. Two lines are drawn through a fixed point on a conic so as always to make equal angles with the tangent at that point. Show that the lines joining the two points where the lines meet the conic again all pass through a fixed point. 3. Four lines divide the plane into a certain number of regions. Determine for each region whether two conies or none may be drawn to pass through points of it and also to be tangent to the four lines. (See 144.) 4. If a variable quadrangle move in such a way as always to remain inscribed in a fixed conic, while three of its sides turn each around one of three fixed collinear points, then the fourth will also turn around a fourth fixed point collinear with the other three. 5. State and prove the dual of problem 4. 6. Extend problem 4 as follows : If a variable polygon of an even number of sides move in such a way as always to ivmain inscribed in a fixed conic, while all its sides but one pass through as many fixed collinear points, then the last side will also pass through a fixed point collinear with the others. INVOLUTION 83 7. If a triangle QRS be 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 C and C", then A A', BB', CC' are three pairs of points in an involution. 8. Use the last exercise to solve the problem : Given five points, Q, R, S, C, C', on a conic, to draw the tangent at any one of them. 9. State and prove the dual of problem 7 and use it to prove the dual of problem 8. 10. If a transversal cut two tangents to a conic in B and B', their 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 BB' and PP'. 11. State and prove the dual of problem 10. 12. If a variable conic pass through two given points, P and /'', and if it be tangent to two given lines, the chord of contact of these two tangents will always pass through one of two fixed points on PP 1 . 13. Use the last theorem to solve the problem: Given four points, P, P', Q, S, on a conic, and the tangent at one of them, Q, to draw the tangent at any one of the other points, S. 14. Apply the theorem of problem 10 to the case of a hyperbola where the two tangents are the asymptotes. Show in this way that if a hyperbola and its asymptotes be cut by a transversal, the segments intercepted by the curve .and by the asymptotes respectively have the same middle point. 15. In a triangle circumscribed about a conic, any side is divided harmonically by its point of contact and the point where it meets the chord joining the points of contact of the other two sides. CHAPTER IX METRICAL PROPERTIES OF INVOLUTIONS 141. Introduction of infinite point; center of involution. We connect the projective theory of involution with the metrical, as usual, by the introduction of the elements at infinity. In an involution of points on a line the point which corresponds to the infinitely distant point is called FIG. 39 the center of the involution. Since corresponding points in the involution have been shown to be harmonic con- jugates with respect to the double points, the center is midway between the double points when they exist. To construct 'the center (Fig. 39) we draw as usual through A and A' any two rays and cut them by a line parallel to AA' in the points K and M. Join these points to B and /?', thus determining on AK and A'N the points L and N, LN meets AA' in the center of the involution. 84 METRICAL PROPERTIES 85 142. Fundamental metrical theorem. From the figure we see that the triangles OLE' and PLM are similar, P being the intersection of KM and LN. Also the tri- angles KPN and BON are similar. We thus have OB:PK=ON:PN and OB':PM=OL:PL-, whence OB - OB' : PK PM= ON OL : PN PL. In the same way, from the similar triangles OAL and PKL, and also OA'N and PMN, we obtain OA OA': PK PM= ON-OL:PN- PL, and this, with the preceding, gives at once the funda- mental theorem, which is sometimes taken also as the definition of involution : OA OA'=OB OB' = constant, or, in words, The product of the distances from the center to two cor- responding points in an involution of points is constant. 143. Existence of double points. Clearly, according as the constant is positive or negative the involution will or will not have double points. The constant is the square of the distance from the center to the double points. If A and A' lie both on the same side of the center, the product OA OA' is positive ; and if they lie on opposite sides, it is negative. Take the case where they both lie on the same side of the center, and take also the pair of corresponding points BB'. Then, since OA OA' = OB OB', it cannot happen that B and B' are separated from each other by A and A'. This is evident enough if the points are on opposite sides of the center. If the pairs are on the same side of the 86 PROJECTI YE GEOMETRY center, and B lies between A and A', so that OB is greater, say, than OA, but less than OA', then, by the equation OA OA' = OB OB\ we must have OB' also less than OA' and greater than <>.[. A similar discus- sion may be made for the case where .4 and A' lie on opposite sides of 0. The results may be stated as follows, without any reference to the center: Criven two pairs of points in <,/,//;,, ,,f j,,,intx^ if the points of one pair are separated from each other by the points of the other pair, then thu involution has no double points. If the points of one pair are not separated from each other by the points of the other pair^ then the involution lias tivo double points. 144. An entirely similar criterion decides whether an involution of rays has or has not double rays, or whether an involution of planes has or has not double planes. 145. Construction of an involution by means of circles. The equation just derived, OA OA' = OB <>!>', indicates another simple way in which points of an involution of points may be constructed. Through A and A' draw any circle, and draw also any cir- cle through B and B' to cut the first in the two points G and G' (Fig. 40). Then any circle through G and G' will meet the line in pairs of points in the involution determined by AA' and BB'. For if such a circle meets the line in the points CC', then, by the theorem in the geometry of the circle which says that if any chord is FIG. 40 METRICAL PROPERTIES 87 drawn through a fixed point within a circle, the product of its segments is constant in whatever direction the chord is drawn, and if a secant line be drawn from a fixed point without a circle, the product of the secant and its external segment is constant in whatever direction the secant line is drawn, we have OC OC'=OG OG' = constant. So that for all such points OA OA' = OB OB'=OC OC'. Fur- ther, the line GG' meets A A' in the center of the invo- lution. To find the double points, if they exist, we draw a tangent from to any of the circles through GG'. Let T be the point of contact. Then lay off on the line OA a line OF equal to OT. Then, since by the above theorem of elementary geometry OA OA' '= OT 2 = OF 2 , we have one double point F. The other is at an equal distance on the other side of 0. This simple and effec- tive method of constructing an involution of points is often taken as the basis for the theory of involution. In projective geometry, however, the circle, which is not a figure that remains unaltered by projection, and is essentially a metrical notion, ought not to be used to build up the purely projective part of the theory. 146. It ought to be mentioned that the theoiy of analytic geometry indicates that the circle is a special conic section that happens to pass through two partic- ular imaginary points on the line at infinity, called the circular points and usually denoted by / and J. The above method of obtaining a point-row in involution is, then, nothing but a special case of the general theorem of the last chapter ( 125), which asserted that a system of conies through four points will cut any line in the plane in a point-row in involution. 88 PROJECTIVE GEOMETRY 147. Pairs in an involution of rays which are at right angles. Circular involution. In an involution of rays there is no one ray which may be distinguished from all the others as the point at infinity is distinguished from all other points on a line. There is one pair of rays, however, which does differ from all the others in that for this particular pair the angle is a right angle. This is most easily shown by using the construction that employs circles, as indicated above. The centers of all the circles through G and G' lie on the perpendicular bisector of the line GG'. Let this line meet the line AA' in the point C (Fig. 41), and draw the circle with center C which goes through G and G'. ^ ^TT This circle cuts out two points M and M' in the involution. The rays G M and GM ' are clearly at right angles, being inscribed in a semicircle. If, therefore, the involution of points is projected to G, we have found two corresponding rays which are at right angles to each other. Given now any invo- lution of rays with center G, we may cut across it by a straight line and proceed to find the two points M and M' . Clearly there will be only one such pair unless the perpendicular bisector of GG' coincides with the line AA'. In this case every ray is at right angles to its corresponding ray, and the involution is called circular. 148. Axes of conies. At the close of the last chapter ( 140) we gave the theorem : A conic determines at every point in ite plane an involution of yv///.s, corresponding METRICAL PROPERTIES 89 being conjugate with respect to the conic. The double rays, if any exixt, are the tangents from the point to the <-rt of the plane. 154. Focal properties of conies. We proceed to de- velop some theorems which will exhibit the importance of these points in the theory of the conic section. Draw a tangent to the conic, and also the normal at the point of contact P. These two lines are clearly conjugate normals. The two points T and N, therefore, where they meet the axis which contains the foci, are corresponding points in the invo- lution considered above, and are therefore harmonic conjugates with respect to the foci (Fig. 44); and if we join them to the point P, we shall obtain four harmonic lines. But two of them are at right angles to each other, and so the others make equal angles with them (Problem 4, Chapter II). Therefore Tlie lines joining a point on the conic to the foci make equal anyles with the ttini/ent. It follows that rays from a source of light at one focus are reflected by an ellipse to the other. 94 PROJECTIVE GEOMETRY 155. In the case of the parabola, where one of the foci must be considered to be at infinity in the direction of the diameter, we have A diameter makes the same angle with the tangent at it* extremity as that tangent does with the line from its point of contact to the focus (Fig. 45). FIG. 45* 156. This last theorem is the basis for the construc- tion of the parabolic reflector. A ray of light from the focus is reflected from such a reflector in a direction parallel to the axis of the reflector. 157. Directrix. Principal axis. Vertex. The polar of the focus with respect to the conic is called the <-t,-i.i: The axis which contains the foci is called the ju-in>-i/> meet in T, and call the focus F. Then TF and PF are conjugate lines, and as tlu-v pass through a focus they must be at right angles to each other. Let METRICAL PROPERTIES 95 TF meet AB in 0. Then P, A, C', />' are four harmonic points. Project these four points parallel to TF upon the directrix, and we then get the four harmonic points P, M, Q, N. Since, now, TFP is a right angle, the angles MFQ and NFQ are equal, as well as the angles AFC and BFC. Therefore the triangles MAF and NFB are similar, and FA : AM= FB : BN. Dropping perpendiculars AA' and BB' upon the directrix, this be- comes FA : A A' = FB : BB'. We have thus the property often of a conic: FIG. 40 taken as the definition The ratio of the distances from a point on the conic to the focus and the directrix is constant. 159. Eccentricity. By taking the point at the vertex of the conic, we note that this ratio is less than unity for the ellipse, greater than unity for the hyperbola, and equal to unity for the pa- rabola. This ratio is called the eccentricity. 160. Sum or difference of focal distances. The ellipse and the hyperbola have two foci and two directrices. The eccentricity, of course, is the same for one focus as for the other, since the curve is sym- metrical with respect to both. If the distances from 96 PROJECTIVE GEOMETRY Fi<;. is a point on a conic to the two foci are r and r', and the distances from the same point to the corresponding directrices are d and d' (Fig. 47), we have r:d = r': d'=(rr l ) : (<2, on a line, and join them to any point S not on that line. Then the triangles ASB, CSD, ASD, CSB, having all the same altitude, are to each other as their bases. Also, since the area of any triangle is one half the product of any two of its sides by the sine of the angle included between them, we have AB x CD_AS x BS sin ASB xCSxDS sinCSD_sin ASBx sinCSD AD x CB~AS x DS sin ASD xCSxBS sin CSB~sin ASD x sin CSB' Now the fraction on the right would be unchanged if instead of the points A, B, C, D we should take any other four points A', B', C', I? lying on any other line cutting across SA, SB, SC, SD. In other 98 SYNTHETIC PROJECTIVE GEOMETRY 99 the involution of six points cut out by any transversal which intersects the sides of a complete quadrilateral words, the fraction on the left is unaltered in value if the points A, B, C, D are replaced by any other four points perspective to them. Again, the fraction on the left is unchanged if some other point were taken instead of S. In other words, the fraction on the right is unaltered if we replace the four lines SA, SB, SC, SD by any other four lines perspective to them. The fraction on the left is called the anhar- imnii'- ratio of the four points A, B, C, D ; the fraction on the right is called the anharmonic ratio of the four lines SA, SB, SC, SD. The anharmonic ratio of four points is sometimes written (ABCD), so that ADx CB If we take the points in different order, the value of the anharmonic ratio will not necessarily remain the same. The twenty-four different ways of writing them will, however, give not more than six different values for the anharmonic ratio, for by writing out the fractions which define them we can find that (ABCD) = (BADC) = (CDAB) = (DCBA). If we write (ABCD) = a, it is not difficult to show that the six values are a; I/a; I- a; 1/(1 - a) ; (a-l)/a; a/(a-l). The proof of this we leave to the student. If A, B, C, D are four harmonic points (see Fig. 6, p. 22), and a quad- rilateral KLMX is constructed such that KL and MN pass through A, KN and LM through C, LN through B, and KM through D, then, projecting A, B, C, D from L upon KM, we have (ABCD) = (KOMD), where is the intersection of KM with LN. But, projecting again the points K, 0, M, D from N back upon the line AB, we have (KOMD) = (CBAD). From this we have (ABCD) = (CBAD), or a I/a ; whence a = lora = 1. But it is easy to see that a = 1 implies that two of the four points coincide. For four harmonic points, therefore, the six values of the anharmonic ratio reduce to three, namely, 2, , and 1. Incidentally we see that if an interchange of any two points in an anharmonic ratio does not change its value, then the four points are harmonic. Many theorems of project! ve geometry are succinctly stated in terms of anharmonic ratios. Thus, the anharmonic ratio of any four 100 PROJECTIVE GEOM KT 1 { Y was studied by Pappus* ; but these notions were not made the foundation for any general theory. Taken by themselves, they are of small consequence ; it is their relation to other theorems and sets of theorems that gives them their importance. The ancients were doubt- less familiar with the theorem, Two lines dett-rmiin- projective pencils of rays. He seems to have tried to describe the curve by means of a parr of compasses, moving one leg back and forth along a straight line instead of holding it fixed as in drawing a circle. He does not attempt to define the law of the movement necessary to obtain a conic by this means. 168. Reception of Desargues's work. Strange to say, Desargues's immortal work was heaped with the most vio- lent abuse and held up to ridicule and scorn ! " Incredi- ble errors ! Enormous mistakes and falsities ! Really it is impossible for anyone who is familiar with the science concerning which he wishes to retail his thoughts, to keep from laughing ! " Such were the comments of re- viewers and critics. Nor were his detractors altogether ignorant and uninstructed men. In spite of the devotion of his pupils and in spite of the admiration and friend- ship of men like Descartes, Fermat, Mersenne, and Roberval, his book disappeared so completely that two centuries after the date of its publication, when the French geometer Chasles wrote his history of geometry, there was no means of estimating the value of the work done by Desargues. Six years later, however, in 1845, Chasles found a manuscript copy of the " Bruillon- project," made by Desargues's pupil, De la Hire. * Euler, Introductio in analysin intinitorum, Appendix, cap. V. 1748. SYNTHETIC PROJECTIVE GEOMETRY 105 169. Conservatism in Desargues's time. It is not neces- sary to suppose that this effacement of Desargues's work for two centuries was due to the savage attacks of his critics. All this was in accordance with the fashion of the time, and no man escaped bitter denunciation who attempted to improve on the methods of the ancients. Those were days when men refused to believe that a heavy body falls at the same rate as a lighter one, even when Galileo made them see it with their own eyes at the foot of the tower of Pisa. Could they not turn to the exact page and line of Aristotle which declared that the heavier body must fall the faster ! " I have read Aristotle's writings from end to end, many times," wrote a Jesuit provincial to the mathematician and astronomer, Christoph Scheiner, at Ingolstadt, whose telescope seemed to reveal certain mysterious spots on the sun, " and I can assure you I have nowhere found anything similar to what you describe. Go, my son, and tranquilize yourself ; be assured that what you take for spots on the sun are the faults of your glasses, or of your eyes." The dead hand of Aristotle barred the advance in every department of research. Physicians would have nothing to do with Harvey's discoveries about the circulation of the blood. "Nature is accused of tolerating a vacuum ! " exclaimed a priest when Pas- cal began his experiments on the Puy-de-Dome to show that the column of mercury in a glass tube varied in height with the pressure of the atmosphere. 170. Desargues's style of writing. Nevertheless, author- ity counted for less at this time in Paris than it did in Italy, and the tragedy enacted in Rome when Galileo 106 PROJECTIVE GEOMETKY was forced to deny his inmost convictions at the bid- ding of a brutal Inquisition could not have been staged in France. Moreover, in the little company of scientists of which Desargues was a member the utmost liberty of thought and expression was maintained. One very good reason for the disappearance of the work of De- sargues is to be found in his style of writing. He failed to heed the very good advice given him in a letter from his warm admirer Descartes.* " You may have two de- signs, both very good and very laudable, but which do not require the same method of procedure : The one is to write for the learned, and show them some new prop- erties of the conic sections which they do not already know; and the other is to write for the curious .un- learned, and to do it so that this matter which until now has been understood by only a very few, and which is nevertheless very useful for perspective, for paint- ing, architecture, etc., shall become common and easy t<> all who wish to study them in your book. If you have the first idea, then it seems to me that it is necessary to avoid using new terms ; for the learned are already accustomed to using those of Apollonius, and will not readily change them for others, though better, and thus yours will serve only to render your demonstrations more difficult, and to turn away your readers from your book. If you have the second plan in mind, it is cer- tain that your terms, which are French, and conceived with spirit and grace, will be better received by persons not preoccupied with those of the ancients. . . . But, if you have that intention, you should make of it a great * CEuvrcs de Desargues, t. II, 132. SYNTHETIC PKOJECTIVE GEOMETRY 107 volume ; explain it all so fully and so distinctly that those gentlemen who cannot study without yawning ; who cannot distress their imaginations enough to grasp a proposition in geometry, nor turn the leaves of a book to look at the letters in a figure, shall find nothing in your discourse more difficult to understand than the description of an enchanted palace in a fairy story." The point of these remarks is apparent when we note that Desargues introduced some seventy new terms in his little book, of which only one, involution, has sur- vived. Curiously enough, this is the one term singled out for the sharpest criticism and ridicule by his re- viewer, De Beaugrand.* That Descartes knew the char- acter of Desargues's audience better than he did is also evidenced by the fact that De Beaugrand exhausted his patience in reading the first ten pages of the book. 171. Lack of appreciation of Desargues. Desargues's methods, entirely different from the analytic methods just then being developed by Descartes and Fermat, seem to have been little understood. " Between you and me/' wrote Descartes f to Mersenne, " I can hardly form an idea of what he may have written concerning conies." Desargues seems to have boasted that he owed nothing to any man, and that all his results had come from his own mind. His favorite pupil, De la Hire, did not realize the extraordinary simplicity and generality of his work. It is a remarkable fact that the only one of all his associates to understand and appreciate the methods of Desargues should be a lad of sixteen years! * CEuvres de Desargues, t. II, 370. t CEuvres de Descartes, t. II, 499. 108 PROJECTIVE GEOMETRY 172. Pascal and his theorem. One does not have to believe all the marvelous stories of Pascal's admiring sisters to credit him with wonderful precocity. We have the fact that in 1640, when he was sixteen years old, he published a little placard, or poster, entitled "Essay pour les conique," * in which his great theorem appears for the first time. His manner of putting it may be a little puzzling to one who has only seen it in the form given in this book, and it may be worth while for the student to compare the two methods of stating it. It is given as follows : "If in the plane of M, S, Q we draw through M the two lines MK and MV, and through the point S the two lines SK and SV, and let K be the inter- section of MK and SK ; V the intersection of MV '2. SYNTHETIC PROJECTIYE GEOMETRY 109 days, and also to his later morbid interest in religious matters, was never published. Leibniz * examined a copy of the complete work, and has reported that the great theorem on the mystic hexagram was made the basis of the whole theory, and that Pascal had deduced some four hundred corollaries from it. This would indicate that here was a man able to take the unconnected materials of projective geometry and shape them into some such symmetrical edifice as we have to-day. Unfortunately for science, Pascal's early death prevented the further development of the subject at his hands. 174. In the " Essay " Pascal gives full credit to Desargues, saying of one of the other propositions, " We prove this property also, the original discoverer of which is M. Desargues, of Lyons, one of the greatest minds of this age . . . and I wish to acknowledge that I owe to him the little which I have discovered." This acknowledgment led Descartes to believe that Pascal's theorem should also be credited to Desargues. But in the scientific club which the young Pascal attended in company with his father, who was also a scientist of some reputation, the theorem went by the name of ' la Pascalia,' and Descartes's remarks do not seem to have been taken seriously, which indeed is not to be wondered at, seeing that he was in the habit of giving scant credit to the work of other scientific investigators than himself. 175. De la Hire and his work. De la Hire added little to the development of the subject, but he did put into print much of what Desargues had already worked * Chasles, Histoire de la G6omtrie. 70. 110 PROJECTIVE GEOMETRY out, not fully realizing, perhaps, how much was his own and how much he owed to his teacher. Writing in 1679, he says,* " I have just read for the first time M. Desargues's little treatise, and have made a copy of it in order to have a more perfect knowledge of it.'' It was this copy that saved the work of his master from oblivion. De la Hire should be credited, among other things, with the invention of a method by which figures in the plane may be transformed into others of the same order. His method is extremely interest- ing, and will serve as an exercise for the student in synthetic projective geometry. It is as follows: Draw two parallel lines, a and ft, and select a point P in tlu-ir plane. Through any point M of the plane draw a line meeting a in A and b in B. Draw a lin*- fJtr<>ti1< f< /- mining it, so that we have set up a one-to-one correspondence between the points M and M' in the plane. The student may show that as M describes a point-row, M' describes a point-row projective to it As M describes a conic, M' describes another conic. This sort of correspon- dence is called a collineation. It will be found that the points on the line b transform into themselves, as does also the single point P. Points on the line a trans- form into points on the line at infinity. The student should remove the metrical features of the construction and take, instead of two parallel lines a and ft, any two lines which may meet in a finite part of the plane. * CEuvres de Desargues, t. I, 231. SYNTHETIC PROJECTIVE GEOMETRY 111 The collineation is a special one in that the general one has an invariant triangle instead of an invariant point and line. 176. Descartes and his influence. The history of syn- thetic protective geometry has little to do with the work of the great philosopher Descartes, except in an indirect way. The method of algebraic analysis invented by him, and the differential and integral calculus which developed from it, attracted all the interest of the mathematical world for nearly two centuries after Desargues, and synthetic geometry received scant atten- tion during the rest of the seventeenth century and for the greater part of the eighteenth century. It is difficult for moderns to conceive of the richness and variety of the problems which confronted the first workers in the calculus. To come into the possession of a method which would solve almost automatically problems which had baffled the keenest minds of antiquity ; to be able to derive in a few moments results which an Archimedes had toiled long and patiently to reach or a Galileo had determined experimentally ; such was the happy expe- rience of mathematicians for a century and a half after Descartes, and it is not to be wondered at that along with this enthusiastic pursuit of new theorems in anal- ysis should come a species of contempt for the methods of the ancients, so that in his preface to his " Mechanique Analytique," published in 1788, Lagrange boasts, " One will find no figures in this work." But at the close of the eighteenth century the field opened up to research by the invention of the calculus began to appear so thoroughly explored that new methods and new objects 112 PRO.TECTIVE GEOMETRY of investigation began to attract attention. Lagrange himself, in his later years, turned in weariness from analysis and mechanics, and applied himself to chemistry, physics, and philosophical speculations. " This state of mind," says Darboux,* " we find almost always at certain moments in the lives of the greatest scholars." At any rate, after lying fallow for almost two centuries, the field of pure geometry was attacked with almost religious enthusiasm. 177. Newton and Maclaurin. But in hastening on to the epoch of Poncelet and Steiner we should not omit to mention the work of Xewton and Maclaurin. Although their results were obtained by analysis for the most part, nevertheless they have given us theorems which fall naturally into the domain of synthetic pro- jective geometry. Thus Newton's " organic method " f of generating conic sections is closely related to the method w r hich we have made use of in Chapter III. It is as follows: If two angles, AOS and AO'S, of given magnitudes turn about their respective vertices, and O', in such a way that the point of intersection, S, of one pair of arms always lies on a straight line, the point of inter- section, A, of the oilier pair of arms will describe a conic. The proof of this is left to the student. 178. Another method of generating a conic is due to Maclaurin 4 The construction, which we also leave for the student to justify, is as follows : If a triangle C'PQ move in such a way that its sides, PQ, QC', and C'P, turn * See Ball, History of Mathematics, French edition, t. II, 23.3. f Newton, Principia, lib. i, lemrna XXI. J Maclaurin, Philosophical Transactions of the Royal Society of London, 1735. SYNTHETIC PROJECTIVE GEOMETRY 113 around three fixed points, R, A, B, respectively, while two of its vertices, P, Q, slide along ttvo fixed lines, CB' and CA', respectively, then the remaining vertex will describe a conic. 179. Descriptive geometry and the second revival. The second revival of pure geometry was again to take place at a time of great intellectual activity. The period at the close of the eighteenth and the beginning of the nineteenth century is adorned with a glorious list of mighty names, among which are Gauss, Lagrange, Legendre, Laplace, Monge, Carnot, Poncelet, Cauchy, Fourier, Steiner, Von Staudt, Mobius, Abel, and many others. The renaissance may be said to date from the in- vention by Monge * of the theory of descriptive geometry. Descriptive geometry is concerned with the representa- tion of figures in space of three dimensions by means of space of two dimensions. The method commonly used consists in projecting the space figure on two planes (a vertical and a horizontal plane being most convenient), the projections being made most simply for metrical purposes from infinity in directions perpen- dicular to the two planes of projection. These two planes are then made to coincide by revolving the hori- zontal into the vertical about their common line. Such is the method of descriptive geometry which in the hands of Monge acquired wonderful generality and ele- gance. Problems concerning fortifications were worked so quickly by this method that the commandant at the military school at Mezieres, where Monge was a drafts- man and pupil, viewed the results with distrust. Monge afterward became professor of mathematics at Mezieres * Monge, Ge"omtrie Descriptive. 1800. 114 PROJECTIVE GEOMETRY and gathered around him a group of students destined to have a share in the advancement of pure geometry. Among these were Hachette, Brianchou, Dupin, I' hash's, Poncelet, and many others. 180. Duality, homology, continuity, contingent rela- tions. Analytic geometry had left little to do in the way of discovery of new material, and the mathemati- cal world was ready for the construction of the edifice. The activities of the group of men that followed Monge were directed toward this end, and we now begin to hear of the great unifying notions of duality, homol- ogy, continuity, contingent relations, and the like. The devotees of pure geometry were beginning to feel the need of a basis for their science which should be at once as general and as rigorous as that of the analysts. Their dream was the building up of a system of geom- etry which should be independent of analysis. Monge, and after him Poncelet, spent much thought on the so- called " principle of continuity," afterwards discussed by Chasles under the name of the "principle of con- tingent relations." To get a clear idea of this principle, -consider a theorem in geometry in the proof of which certain auxiliary elements are employed. These ele- ments do not appear in the statement of the theorem, and the theorem might possibly be proved without them. In drawing the figure for the proof of the theorem, however, some of these elements may not appear, or, as the analyst would say, they become imaginary. " No matter," says the principle of contingent relations, "the theorem is true, and the proof is valid whether the elements used in the proof are real or imaginary." SYNTHETIC PROJECTIVE GEOMETRY 115 181. Poncelet and Cauchy. The efforts of Poncelet to compel the acceptance of this principle independent of analysis resulted in a bitter and perhaps fruitless controversy between him and the great analyst Cauchy. In his review of Poncelet's great work on the projec- tive properties of figures * Cauchy says, " In his pre- liminary discourse the author insists once more on the necessity of admitting into geometry what he calls the ' principle of continuity.' We have already discussed that principle . . . and we have found that that prin- ciple is, properly speaking, only a strong induction, which cannot be indiscriminately applied to all sorts of questions in geometry, nor even in analysis. The rea- sons which we have given as the basis of our opinion are not affected by the considerations which the author has developed in his Traite des Proprietes Projectives des Figures." Although this principle is constantly made use of at the present day in all sorts of investigations, careful geometricians are in agreement 'with Cauchy in this matter, and use it only as a convenient work- ing tool for purposes of exploration. The one-to-one correspondence between geometric forms and algebraic analysis is subject to many and important exceptions. The field of analysis is much more general than the field of geometry, and while there may be a clear notion in analysis to correspond to every notion in geometry, the opposite is not true. Thus, in analysis we can deal with four coordinates as well as with three, but the existence of a space of four dimensions * Poncelet, Traite" des Proprie'te's Projectives des Figures. 1822. (St-e p. 357, Vol. II, of the edition of 1866.) 116 PROJECTIVE GEOMETRY to correspond to it does not therefore follow. When the geometer speaks of the two real or imaginary inter- sections of a straight line with a conic, he is really speaking the language of algebra. Apart from the algebra involved, it is the height of absurdity to try to distinguish between the two points in which a line fails to meet a conic! 182. The work of Poncelet. But Poncelet's right to the title " The Father of Modern Geometry " does not stand or fall with the principle of contingent relations. In spite of the fact that he considered this principle the most important of all his discoveries, his reputation rests on more solid foundations. He was the first to study figures in homoloyy, which is, in effect, the colline- ation described in 175, where corresponding points lie on straight lines through a fixed point. He was the first to give, by means of the theory of poles and polars, a transformation by which an element is transformed into another of a different sort. Point-to-point trans- formations will sometimes generalize a theorem, but the transformation discovered by Poncelet may throw a theorem into one of an entirely different aspect. The principle of duality, first stated in definite form by Gergonne,* the editor of the mathematical journal in which Poncelet published his researches, was based by Poncelet on his theory of poles and polars. He also put into definite form the notions of the infinitely distant elements in space as all lying on a plane at infinity. 183. The debt which analytic geometry owes to syn- thetic geometry. The reaction of pure geometry on * Gergonne, Annales de MatMmatiques, XVI, 209. 1820. SYNTHETIC PROJECTIVE GEOMETRY 117 analytic geometry is clearly seen in the development of the notion of the class of a curve, which is the number of tangents that may be drawn from a point in a plane to a given curve lying in that plane. If a point moves along a conic, it is easy to show and the student is recommended to furnish the .proof that the polar line with respect to a conic remains tangent to another conic. This may be expressed by the statement that the conic is of the second order and also of the second class. It might be thought that if a point moved along a cubic curve, its polar line with respect to a conic would remain tangent to another cubic curve. This is not the case, however, and the investigations of Poncelet and others to determine the class of a given curve were afterward completed by Pliicker. The notion of geo- metrical transformation led also to the very important developments in the theory of invariants, which, geo- metrically, are the elements and configurations which are not affected by the transformation. The anharmonic ratio of four points is such an invariant, since it remains unaltered under all projective transformations. 184. Steiner and his work. In the work of Poncelet and his contemporaries, Chasles, Brianchon, Hachette, Dupin, Gergonne, and others, the anharmonic ratio en- joyed a fundamental role. It is made also the basis of the great work of Steiner,* who was the first to treat of the conic, not as the projection of a circle, but as the locus of intersection of corresponding rays of two pro- jective pencils. Steiner not only related to each other, * Steiner, Systematische Entwickelung der Abhangigkeit geome- trischer Gestalten von einander. 1832. 118 PEOJECTIVE GEOMETKY in one-to-one correspondence, point-rows and pencils and all the other fundamental forms, but he set into correspondence even curves and surfaces of higher de- grees. This new and fertile concept ion gave him an easy and direct route into the most abstract and diffi- cult regions of pure gepmetry. Much of his work was given without any indication of the methods by which he had arrived at it, and many of his results have only recently been verified. 185. Von Staudt and his work. To complete the the- ory of geometry as we have it to-day it only remained to free it from its dependence on the semimetrical basis of the anharmonic ratio. This work was accomplished by Von Staudt,* who applied himself to the restatement of the theory of geometry in a form independent of analytic and metrical notions. The method which has been used in Chapter II to develop the notion of four harmonic points by means of the complete quadrilateral is due to Von Staudt. His work is characterized by a most remarkable generality, in that he is able to discuss real and imaginary forms with equal ease. Thus he assumes a one-to-one correspondence between the points and lines of a plane, and defines a conic as the locus of points which He on their corresponding lines, and a pencil of rays of the second order as the system of lines which pass through their corresponding points. The point-row and pencil of the second order may be real or imaginary, but his theorems still apply. An illustra- tion of a correspondence of this sort, where the conic is imaginary, is given in 15 of the first chapter. In * Von Staudt, Geometric der Lage. 1847. SYNTHETIC PROJECTIVE GEOMETRY 119 defining conjugate imaginary points on a line, Von Staudt made use of an involution of points having no double points. His methods, while elegant and power- ful, are hardly adapted to an elementary course, but Reye* and others have done much toward simplifying his presentation. 186. Recent developments. It would be only confus- ing to the student to attempt to trace here the later developments of the science of protective geometry. It is concerned for the most part with curves and surfaces of a higher degree than the second. Purely synthetic methods have been used with marked success in the study of the straight line in space. The struggle be- tween analysis and pure geometry has long since come to an end. Each has its distinct advantages, and the mathematician who cultivates one at the expense of the other will never attain the results that he would attain if both methods were equally ready to his hand. Pure geometry has to its credit some of the finest discov- eries in mathematics, and need not apologize for having been born. The day of its usefulness has not passed with the invention of abridged notation and of short methods in analysis. While we may be certain that any geometrical problem may always be stated in analytic form, it does not follow that that statement will be simple or easily interpreted. For many mathematicians the geometric intuitions are weak, and for such the method will have little attraction. On the other hand, there will always be those for whom the subject will have a peculiar glamor who will follow with delight * Reye, Geometrie der Lage. Translated by Holgate, 1897. 120 PROJECTIVE GEOMETRY the curious and unexpected relations between the forms of space. There is a corresponding pleasure, doubtless, for the analyst in tracing the marvelous connections between the various fields in which he wanders, and it is as absurd to shut one's eyes to the beauties in one as it is to ignore those in the other. "Let us cultivate geometry, then," says Darboux,* "without wishing in all points to equal it to its rival. Besides, if we were tempted to neglect it, it would not be long in finding in the applications of mathematics, as once it has al- ready done, the means of renewing its life and of developing itself anew. It is like the Giant Antseus, who renewed his strength by touching the earth." * Ball, loc. cit. p. 261. INDEX (The numbers refer to the paragraphs) Abel (1802-1829), 179 Analogy, 24 Analytic geometry, 21, 118, 119, 120, 146, 176, 180 Anharmonic ratio,46,161, 184,185 Apollonius (second half of third century B.C.), 70 Archimedes (287-212 B.C.), 176 Aristotle (384-322 B.C.), 169 Asymptotes, 111, 113, 114, 115, 116, 117, 118, 148 Axes of a conic, 148 Axial pencil, 7, 8, 23, 50, 54 Axis of perspectivity, 8, 47 Bacon (1561-1626), 162 Bisection, 41, 109 Brianchon (1785-1864), 84, 85, 86, 88, 89, 90, 95, 105, 113, 174, 184 Calculus, 176 Carnot (1796-1832), 179 Cauchy (1789-1857), 179, 181 Cavalieri (1598-1647), 162 Center of a conic, 107, 112, 148 Center of involution, 141, 142 Center of perspectivity, 8 Central conic, 120 Chasles (1793-1880), 168, 179, 180, 184 Circle, 21, 73, 80, 145, 146, 147 Circular involution, 147, 149, 150, 151 Circular points, 146 Class of a curve, 183 Classification of conies, 110 Collineation, 175 Concentric pencils, 50 Cone of the second order, 59 Conic, 73, 81 Conjugate diameters, 114, 148 Conjugate normal, 151 Conjugate points and lines, 100, 109, 138, 139, 140 Constants in an equation, 21 Contingent relations, 180, 181 Continuity, 180, 181 Continuous correspondence, 9, 10, 21,49 Corresponding elements, 64 Counting, 1, 4 Cross ratio, 46 Darboux, 176, 186 De Beaugrand, 170 Degenerate pencil of rays of the second order, 58, 93 Degenerate point-row of the second order, 56, 78 De la Hire (1640-1718), 168, 171, 175 Desargues (1593-1662), 25, 26, 40, 121, 125, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 174, 175 Descartes (1596-1650), 162, 170, 171, 174, 176 Descriptive geometry, 179 Diameter, 107 Directrix, 157, 158, 159, 160 Double correspondence, 128, 130 Double points of an involution, 124 Double rays of an involution, 133, 134 Duality, 94, 104, 161, 180, 182 Dupin (1784-1873), 174, 184 Eccentricity of conic, 159 Ellipse, 110, 111, 162 121 122 PKOJECTIVE GEOMETRY Equation of conic, 118, 119, 120 Euclid (ca. 300 B.C.), 6, 22, 104 Euler (1707-1783), 166 Fermat (1601-1665), 162, 171 Foci of a conic, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162 Fourier (1768-1830), 179 Fourth harmonic, 29 Fundamental form, 7, 16, 23, 36, 47, 60, 184 Galileo (1564-1642), 162, 169, 170, 176 Gauss (1777-1855), 179 Gergonne (1771-1859), 182, 184 Greek geometry, 161 Hachette (1769-1834), 179, 184 Harmonic conjugates, 29, 30, 39 Harmonic elements, 36, 49, 91, 163, 185 Harmonic lines, 33, 34, 35, 66, 67 Harmonic planes, 34, 35 Harmonic points, 29, 31, 32, 33, 34, 35, 36, 43, 71, 161 Harmonic tangents to a conic, 91,92 Harvey (1578-1657), 169 Homology, 180, 182 Huygens (1629-1695), 162 Hyperbola, 110,111,113, 114, 115, 116, 117, 118, 162 Imaginary elements, 146, 180, 181, 182, 185 Infinitely distant elements, 6, 9, 22, 39,' 40, 41, 104, 107, 110 Infinity, 4, 5, 10, 12, 13. 14, 15, 17, 18, 19, 20, 21, 22, 41 Involution, 37, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 161, 163, 170 Kepler (1571-1630), 162 ranse (1730-1813). 176, 179 Laplace (1749-1827), 179 Legendre (1752-1833), 179 Leibniz (1646-1716), 173 Linear construction, 40, 41, 42 Maclaurin (1698-1746), 177, 178 Measurements, 23, 40, 41, 104 Mersenne (1588-1648), 168. 171 Metrical theorems, 40, 104, 106, 107, 141 Middle point, 39, 41 Mobius (1790-1868), 179 Monge (1746-1818), 179, 180 Napier (1550-1617), 162 Newton (1642-1727), 177 Numbers, 4, 21, 43 Numerical computations, 43, 44, 46 Ollr-to-ono Correspondence. l.L'. 3, 4, 5, 6, 7, 9, 10, 11, 24. ::J. 37, 43, 60, 104, 106, 184 Opposite sides of a hexagon. 70 Opposite sides of a quadrilatcr.il, 28, 29 Order of a form, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21 Pappus (fourth century A.D.), 161 Parabola, 110, 111, 112, 119, 162 Parallel lines, 39, 41, 162 Pascal (1623-1662), 69, 70, 74. 76, 76, 77, 78, 95, 105, 125, 162, 169, 171, 172, 173 Pencil of planes of the second order, 59 Pencil of rays, 6, 7, 8, 23 ; of the second order, 57, 60, 79, 81 Perspective position, 6, 8, 35, 37, 51, 53, 71 Plane system, 16, 23 Planes on space, 17 Point of contact, 87, 88, 89, 90 Point system, 16, 23 Point-row, 6, 7, 8, 9, 23 ; of the second order, 55, 60, 61, 66, 67, 72 Points in space, 18 Pole and polar, 98, 99, 100, 101, 138, 104, 166 123 Poncelet (1788-1807), 177, 171), 180, 181, 182, 183, 184 Principal axis of a conic, 157 Projection, 161 Projective axial pencils, 59 Projective correspondence, 9, 35, 36, 37, 47, 71, 92, 104 Projective pencils, 53, 64, 68 Projective point-rows, 51, 79 Projective properties, 24 Projective theorems, 40, 104 Quadrangle, 26, 27, 28, 29 Quadric cone, 59 Quadrilateral, 88, 95, 96 Roberval (1602-1675), 168 Ruler construction, 40 Scheiner, 169 Self-corresponding elements, 47, 48, 49, 50, 51 Self -dual, 105 Self-polar triangle, 102 Separation of elements in involu- tion, 148 Separation of harmonic conju- gates, 38 Sequence of points, 49 Sign of segment, 44, 45 Similarity, 106 Skew lines, 12 Space system, 19, 23 Sphere, 21 Steiner (1796-1863), 129, 130, 131, 177, 179, 184 Steiner's construction, 129, 130, 131 Superposed point- rows, 47, 48, 49 Surfaces of the second degree, 166 System of lines in space, 20, 23 Systems of conies, 125 Tangent line, 61, 80, 81, 87, 88, 89, 90, 91, 92 Tycho Brahe (1546-1601), 162 Verner, 161 Vertex of conic, 157, 159 Von Staudt (1798-1867), 179, 185 Wallis (1616-1703), 162 University of California SOUTHERN REGIONAL LIBRARY FACILITY 405 Hilgard Avenue, Los Angeles, CA 90024-1388 Return this material to the library from which it was borrowed. 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