An Elementary Course in Synthetic Projective Geometry
by Lehmer, Derrick Norman
Edition 1, (November 4, 2005)
The following course is intended to give, in as simple a way as possible, the essentials of synthetic projective 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 anharmonic 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 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 plane system. He has also rejected the term foci of an involution, and has not used the customary terms for classifying involutions—hyperbolic involution, elliptic involution and parabolic involution. He has found that all these terms are very confusing to the student, who inevitably tries to connect them in some way with the conic sections.
Enough examples have been provided to give the student a clear grasp of the theory. Many are of sufficient generality to serve as a basis for individual investigation on the part of the student. Thus, the third example at the end of the first chapter will be found to be very fruitful in interesting results. A correspondence is there indicated between lines in space and circles through a fixed point in space. If the student will trace a few of the consequences of that correspondence, and determine what configurations of circles correspond to intersecting lines, to lines in a plane, to lines of a plane pencil, to lines cutting three skew lines, etc., he will have acquired no little practice in picturing to himself figures in space.
The writer has not followed the usual practice of inserting historical notes at the foot of the page, and has tried instead, in the last chapter, to give a consecutive account of the history of pure geometry, or, at least, of as much of it as the student will be able to appreciate who has mastered the course as given in the preceding chapters. One is not apt to get a very wide view of the history of a subject by reading a hundred biographical footnotes, arranged in no sort of sequence. The writer, moreover, feels that the proper time to learn the history of a subject is after the student has some general ideas of the subject itself.
The course is not intended to furnish an illustration of how a subject may be developed, from the smallest possible number of fundamental assumptions. The author is aware of the importance of work of this sort, but he does not believe it is possible at the present time to write a book along such lines which shall be of much use for elementary students. For the purposes of this course the student should have a thorough grounding in ordinary elementary geometry so far as to include the study of the circle and of similar triangles. No solid geometry is needed beyond the little used in the proof of Desargues' theorem (25), and, except in certain metrical developments of the general theory, there will be no call for a knowledge of trigonometry or analytical geometry. Naturally the student who is equipped with these subjects as well as with the calculus will be a little more mature, and may be expected to follow the course all the more easily. The author has had no difficulty, however, in presenting it to students in the freshman class at the University of California.
The subject of synthetic projective geometry is, in the opinion of the writer, destined shortly to force its way down into the secondary schools; and if this little book helps to accelerate the movement, he will feel amply repaid for the task of working the materials into a form available for such schools as well as for the lower classes in the university.
The material for the course has been drawn from many sources. The author is chiefly indebted to the classical works of Reye, Cremona, Steiner, Poncelet, and Von Staudt. Acknowledgments and thanks are also due to Professor Walter C. Eells, of the U.S. Naval Academy at Annapolis, for his searching examination and keen criticism of the manuscript; also to Professor Herbert Ellsworth Slaught, of The University of Chicago, for his many valuable suggestions, and to Professor B. M. Woods and Dr. H. N. Wright, of the University of California, who have tried out the methods of presentation, in their own classes.
D. N. LEHMER
Preface Contents CHAPTER I - ONE-TO-ONE CORRESPONDENCE 1. Definition of one-to-one correspondence 2. Consequences of one-to-one correspondence 3. Applications in mathematics 4. One-to-one correspondence and enumeration 5. Correspondence between a part and the whole 6. Infinitely distant point 7. Axial pencil; fundamental forms 8. Perspective position 9. Projective relation 10. Infinity-to-one correspondence 11. Infinitudes of different orders 12. Points in a plane 13. Lines through a point 14. Planes through a point 15. Lines in a plane 16. Plane system and point system 17. Planes in space 18. Points of space 19. Space system 20. Lines in space 21. Correspondence between points and numbers 22. Elements at infinity PROBLEMS CHAPTER II - RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE CORRESPONDENCE WITH EACH OTHER 23. Seven fundamental forms 24. Projective properties 25. Desargues's theorem 26. Fundamental theorem concerning two complete quadrangles 27. Importance of the theorem 28. Restatement of the theorem 29. Four harmonic points 30. Harmonic conjugates 31. Importance of the notion of four harmonic points 32. Projective invariance of four harmonic points 33. Four harmonic lines 34. Four harmonic planes 35. Summary of results 36. Definition of projectivity 37. Correspondence between harmonic conjugates 38. Separation of harmonic conjugates 39. Harmonic conjugate of the point at infinity 40. Projective theorems and metrical theorems. Linear construction 41. Parallels and mid-points 42. Division of segment into equal parts 43. Numerical relations 44. Algebraic formula connecting four harmonic points 45. Further formulae 46. Anharmonic ratio PROBLEMS CHAPTER III - COMBINATION OF TWO PROJECTIVELY RELATED FUNDAMENTAL FORMS 47. Superposed fundamental forms. Self-corresponding elements 48. Special case 49. Fundamental theorem. Postulate of continuity 50. Extension of theorem to pencils of rays and planes 51. Projective point-rows having a self-corresponding point in common 52. Point-rows in perspective position 53. Pencils in perspective position 54. Axial pencils in perspective position 55. Point-row of the second order 56. Degeneration of locus 57. Pencils of rays of the second order 58. Degenerate case 59. Cone of the second order PROBLEMS CHAPTER IV - POINT-ROWS OF THE SECOND ORDER 60. Point-row of the second order defined 61. Tangent line 62. Determination of the locus 63. Restatement of the problem 64. Solution of the fundamental problem 65. Different constructions for the figure 66. Lines joining four points of the locus to a fifth 67. Restatement of the theorem 68. Further important theorem 69. Pascal's theorem 70. Permutation of points in Pascal's theorem 71. Harmonic points on a point-row of the second order 72. Determination of the locus 73. Circles and conics as point-rows of the second order 74. Conic through five points 75. Tangent to a conic 76. Inscribed quadrangle 77. Inscribed triangle 78. Degenerate conic PROBLEMS CHAPTER V - PENCILS OF RAYS OF THE SECOND ORDER 79. Pencil of rays of the second order defined 80. Tangents to a circle 81. Tangents to a conic 82. Generating point-rows lines of the system 83. Determination of the pencil 84. Brianchon's theorem 85. Permutations of lines in Brianchon's theorem 86. Construction of the penvil by Brianchon's theorem 87. Point of contact of a tangent to a conic 88. Circumscribed quadrilateral 89. Circumscribed triangle 90. Use of Brianchon's theorem 91. Harmonic tangents 92. Projectivity and perspectivity 93. Degenerate case 94. Law of duality PROBLEMS CHAPTER VI - POLES AND POLARS 95. Inscribed and circumscribed quadrilaterals 96. Definition of the polar line of a point 97. Further defining properties 98. Definition of the pole of a line 99. Fundamental theorem of poles and polars 100. Conjugate points and lines 101. Construction of the polar line of a given point 102. Self-polar triangle 103. Pole and polar projectively related 104. Duality 105. Self-dual theorems 106. Other correspondences PROBLEMS CHAPTER VII - METRICAL PROPERTIES OF THE CONIC SECTIONS 107. Diameters. Center 108. Various theorems 109. Conjugate diameters 110. Classification of conics 111. Asymptotes 112. Various theorems 113. Theorems concerning asymptotes 114. Asymptotes and conjugate diameters 115. Segments cut off on a chord by hyperbola and its asymptotes 116. Application of the theorem 117. Triangle formed by the two asymptotes and a tangent 118. Equation of hyperbola referred to the asymptotes 119. Equation of parabola 120. Equation of central conics referred to conjugate diameters PROBLEMS CHAPTER VIII - INVOLUTION 121. Fundamental theorem 122. Linear construction 123. Definition of involution of points on a line 124. Double-points in an involution 125. Desargues's theorem concerning conics through four points 126. Degenerate conics of the system 127. Conics through four points touching a given line 128. Double correspondence 129. Steiner's construction 130. Application of Steiner's construction to double correspondence 131. Involution of points on a point-row of the second order. 132. Involution of rays 133. Double rays 134. Conic through a fixed point touching four lines 135. Double correspondence 136. Pencils of rays of the second order in involution 137. Theorem concerning pencils of the second order in involution 138. Involution of rays determined by a conic 139. Statement of theorem 140. Dual of the theorem PROBLEMS CHAPTER IX - METRICAL PROPERTIES OF INVOLUTIONS 141. Introduction of infinite point; center of involution 142. Fundamental metrical theorem 143. Existence of double points 144. Existence of double rays 145. Construction of an involution by means of circles 146. Circular points 147. Pairs in an involution of rays which are at right angles. Circular involution 148. Axes of conics 149. Points at which the involution determined by a conic is circular 150. Properties of such a point 151. Position of such a point 152. Discovery of the foci of the conic 153. The circle and the parabola 154. Focal properties of conics 155. Case of the parabola 156. Parabolic reflector 157. Directrix. Principal axis. Vertex 158. Another definition of a conic 159. Eccentricity 160. Sum or difference of focal distances PROBLEMS CHAPTER X - ON THE HISTORY OF SYNTHETIC PROJECTIVE GEOMETRY 161. Ancient results 162. Unifying principles 163. Desargues 164. Poles and polars 165. Desargues's theorem concerning conics through four points 166. Extension of the theory of poles and polars to space 167. Desargues's method of describing a conic 168. Reception of Desargues's work 169. Conservatism in Desargues's time 170. Desargues's style of writing 171. Lack of appreciation of Desargues 172. Pascal and his theorem 173. Pascal's essay 174. Pascal's originality 175. De la Hire and his work 176. Descartes and his influence 177. Newton and Maclaurin 178. Maclaurin's construction 179. Descriptive geometry and the second revival 180. Duality, homology, continuity, contingent relations 181. Poncelet and Cauchy 182. The work of Poncelet 183. The debt which analytic geometry owes to synthetic geometry 184. Steiner and his work 185. Von Staudt and his work 186. Recent developments INDEX
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 correspondence 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 setting up a one-to-one correspondence between the objects and certain words which serve, not as a means of counting the 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 geometry, algebraic theorems are made to yield geometric ones, and vice versa. In geometry we get at the properties of the conic sections by means of the properties of the straight line, and cubic surfaces are studied by means of the plane.
*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, nevertheless, to set up a one-to-one correspondence between the elements 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 AA' and BB', and let these joining lines meet in S. For every point C on AB a point C' 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 correspondence 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 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 paragraph the remarkable fact was brought to light that it is sometimes possible to set the elements of an assemblage 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, we 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 S; and even with this extension there is one line through S, according to Euclid's postulate, which does not meet the line AB and which therefore has no point on AB to correspond to it. In order to smooth out this discrepancy we are accustomed to assume the existence of an infinitely distant point on the line AB and to assign this point as the corresponding point of the exceptional line of S. With this understanding, then, we may say that we have set the lines through a point and the points on a line into one-to-one correspondence. This correspondence is of such fundamental importance in the study of projective geometry that a special name is given to it. Calling the totality of points on a line a point-row, and the totality of lines through a point a pencil of rays, we say that the point-row and the pencil related as above are in perspective position, or that they are perspectively related.
*7. Axial pencil; fundamental forms.* A similar correspondence may be set up between the points on a line and the planes through another line which does not meet the first. Such a system of planes is called an axial pencil, and the three assemblages—the point-row, the pencil of rays, and the axial pencil—are called fundamental forms. The fact that they may all be set into one-to-one correspondence with each other is expressed by saying that they are of the same order. It is usual also to speak of them as of the first order. We shall see presently that there are other assemblages which cannot be put into this sort of one-to-one correspondence with the points on a line, and that they will very reasonably be said to be of a higher order.
*8. Perspective position.* We have said that a point-row and a pencil of rays are in perspective position if each ray of the pencil goes through the point of the point-row which corresponds to it. Two pencils of rays are also said to be in perspective position if corresponding rays meet on a straight line which is called the axis of perspectivity. Also, two point-rows are said to be in perspective position if corresponding points lie on straight lines through a point which is called the center of perspectivity. A point-row and an axial pencil are in perspective position if each plane of the pencil goes through the point on the point-row which corresponds to it, and an axial pencil and a pencil of rays are in perspective position if each ray lies in the plane which corresponds to it; and, finally, two axial pencils are perspectively related if corresponding planes meet in a plane.
*9. Projective relation.* It is easy to imagine a more general correspondence between the points of two point-rows than the one just described. If we take two perspective pencils, A and S, then a point-row a perspective to A will be in one-to-one correspondence with a point-row b perspective to B, but corresponding points will not, in general, lie on lines which all pass through a point. Two such point-rows are said to be projectively related, or simply projective to each other. Similarly, two pencils of rays, or of planes, are projectively related to each other if they are perspective to two perspective point-rows. This idea will be generalized later on. It is important to note that between the elements of two projective fundamental forms there is a one-to-one correspondence, and also that this correspondence is in general continuous; that is, by taking two elements of one form sufficiently close to each other, the two corresponding elements in the other form may be made to approach each other arbitrarily close. In the case of point-rows this continuity is subject to exception in the neighborhood of the point "at infinity."
*10. Infinity-to-one correspondence.* It might be inferred that any infinite assemblage could be put into one-to-one correspondence with any other. Such is not the case, however, if the correspondence is to be continuous, between the points on a line and the points on a plane. Consider two lines which lie in different planes, and take m points on one and n points on the other. The number of lines joining the m points of one to the n points jof the other is clearly mn. If we symbolize the totality of points on a line by [infinity], then a reasonable symbol for the totality of lines drawn to cut two lines would be [infinity]2. Clearly, for every point on one line there are [infinity] lines cutting across the other, so that the correspondence might be called [infinity]-to-one. Thus the assemblage 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 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 the 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 method of setting up a one-to-one correspondence between 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, considered 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 l 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 lmn. It is reasonable, therefore, to symbolize the totality of planes that are determined by the [infinity] points on each of the three lines by [infinity]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, considered 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.
*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 correspondence 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 between points in the plane and pairs of numbers, and also between points in space and 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 plane 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 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 correspondence 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 configurations are not possible. In other branches of mathematics, notably in the theory of functions of a complex variable, 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.
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.
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 chapter 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 segments 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 construction 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. Projective properties.* Our chief interest in this chapter will be the discovery of relations between the elements of one form which hold between the corresponding elements of any other form in one-to-one correspondence with it. We have already called attention to the danger of assuming that whatever relations hold between the elements of one assemblage must also hold between the corresponding elements of any assemblage 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 judgments 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 projectively related to it are called projective relations. Relations involving measurement of lines or of angles are not projective.
*25. Desargues's theorem.* We consider first the following 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, then the pairs of sides AB and A'B', BC and B'C', CA and C'A' all meet on a straight line, and conversely.
Let the lines AA', BB', 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 therefore 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 consider the figure a plane figure, the points P, Q, and R 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 complete 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, _K_, _L_, _M_, _N_ and _K'_, _L'_, _M'_, _N'_, so related that _KL_, _K'L'_, _MN_, _M'N'_ all meet in a point _A_; _LM_, _L'M'_, _NK_, _N'K'_ all meet in a _ point _Q_; and _LN_, _L'N'_ meet in a point _B_ on the line _AC_; 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, B, 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 K', L', M', N' my 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 joining K' and M', found in this way, must pass through the point D already determined by the quadrangle 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 harmonic 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 O be the intersection of KM and LN (Fig. 5). Join O to A and C. The joining lines cut out on the sides of the quadrangle four points, P, Q, R, S. Consider the quadrangle P, K, Q, O. One pair of opposite sides passes through A, one through C, and one remaining side through D; 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 remaining 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 B, and KM through D. Take now any point S not in the plane of the quadrangle 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 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 quadrangle 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 harmonic 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 α, β, γ, δ, which all pass through the line g, pass also through the four harmonic points A, B, C, D, so that α passes through A, etc. Then it is clear that any plane π through A, B, C, D will cut out four harmonic lines from the four planes, for they are lines through the intersection P of g with the plane π, and they pass through the given harmonic points A, B, C, D. Any other plane σ cuts g in a point S and cuts α, β, γ, δ in four lines that meet π in four points A', B', C', D' lying on PA, PB, PC, and PD respectively, and are thus four harmonic hues. Further, any ray cuts α, β, γ, δ 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:
Given 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 third, any four harmonic elements of one must correspond to four harmonic elements in the other. We take this as our definition of projective correspondence, and say:
*36. Definition of projectivity.* Two fundamental forms are protectively related to each other when a one-to-one correspondence exists between 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, N (Fig. 6). Then, as B describes the point-row AC, the point N describes the point-row AM perspective to it. Projecting N again from C, we get a point-row K on AL perspective to the point-row N and thus projective 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 D, and conversely. In other words, we have set up a one-to-one correspondence between the points of a single point-row, which is also a projective correspondence because four harmonic points B correspond to four harmonic points D. We may note also that the correspondence is here characterized 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 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 approaches 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 C, B and D are separated from each other by A and C.
*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 triangle ALC is isosceles (Fig. 7). Since D is supposed to be at infinity, the line KM is parallel to AC. Therefore 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 between 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 nothing 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 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 drawing 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 parallel 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.
*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 established 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 to SD, meeting SA in A' and SC in C'. Then A', B', C', 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 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 : B'C.
From these two proportions we have, remembering that BA' = BC',
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
CB : AB = -CD : AD,
and using the fundamental relation connecting three points on a line,
PR + RQ = PQ,
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
(CB - BA) : AB = -(CA - DA) : AD,
(AB - AC) : AB = (AC - AD) : AD;
so that AB, AC, and AD are three quantities in hamonic 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.
*46. Anharmonic ratio.* The corresponding relations between the trigonometric functions of the angles determined 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.
*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 AA' and BB', and find their point of intersection S. Through S 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 AC, to construct linearly the middle point of AC.
*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 pencils 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 C, then the point-row described by B is projective to the point-row described 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' (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 intersects the line joining the centers of the two pencils, 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 intersection of the lines u and u'.
*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 correspond, therefore, to four harmonic points P'. It is clear, however, that no point P can coincide with its corresponding point P', for in that case the lines PS and 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 corresponds 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 D, 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 to DD', 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 corresponds to a point P' 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 corresponding 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 projective 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 point. The converse is also true, and we have the very important theorem:
*52.* If in two protective point-rows, the point of intersection 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 AA' 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 corresponding ray SA', SB with its corresponding ray SB', 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 corresponding 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 centers 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 naturally arises, What is the locus of points of intersection of corresponding rays of two projective pencils 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 projective 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 perspective 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 investigations 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 same 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: The system of rays joining corresponding points in two protective 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 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. Corresponding lines of these pencils determine planes which are the projections to S of the lines which join the corresponding 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.
*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?
2. How many times would the two men in the first problem 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 u.)
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. (Consider 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 direction 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 correspondence 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 common 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', considered as a ray of S, must correspond some ray of S' which meets it in S'. 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', 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 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:
Given three pairs of corresponding rays in two projective pencils, it is possible to find a ray of one which corresponds to any ray of the other.
*64.* We proceed, then, to the solution of the fundamental
PROBLEM: Given 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 follows ( 52) that u and u' are in perspective position, and that lines through corresponding points all pass through a point M, the center of perspectivity, the position of which will be determined by any two such lines. But the intersection of a with u and the intersection of c' with u' are corresponding points on u and u', and the line joining them is clearly c itself. Similarly, b' joins two corresponding points on u and u', and so the center M of perspectivity of u and u' is the intersection of c and b'. To find d' in S' corresponding to a given line d of S we note the point L where d meets u. Join L to M and get the point N where this line meets u'. L and N are corresponding points on u and u', and d' must therefore pass through N. The intersection P of d and d' is thus another point on the locus. In the same manner any number of other points may be obtained.
*65.* The lines u and u' might have been drawn in any direction through A (avoiding, of course, the line a for u and the line a' for u'), and the center of perspectivity M would be easily obtainable; but the above construction furnishes a simple and instructive figure. An equally simple one is obtained by taking a' for u and a for u'.
*66. Lines joining four points of the locus to a fifth.* Suppose that the points S, S', B, C, and D are fixed, and that four points, A, A1, A2, and A3, are taken on the locus at the intersection with it of any four harmonic rays through B. These four harmonic rays give four harmonic points, L, L1 etc., on the fixed ray SD. These, in turn, project through the fixed point M into four harmonic points, N, N1 etc., on the fixed line DS'. These last four harmonic points give four harmonic rays CA, CA1, CA2, CA3. Therefore the four points A which project to B in four harmonic rays also project to C in four harmonic rays. But C may be any point on the locus, and so we have the very important theorem,
Four points which are on the locus, and which project to a fifth point of the locus in four harmonic rays, project to any point of the locus in four harmonic rays.
*67.* The theorem may also be stated thus:
The locus of points from which, four given points are seen along four harmonic rays is a point-row of the second order through them.
*68.* A further theorem of prime importance also follows:
Any two points on the locus may be taken as the centers of two projective pencils which will generate the locus.
*69. Pascal's theorem.* The points A, B, C, D, S, and S' may thus be considered as chosen arbitrarily on the locus, and the following remarkable theorem follows at once.
Given 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 C = 6. If we make A = 1, C=2, S=3, D = 4, S'=5, and. B = 6, 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 different 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 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:
The 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 importance 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 harmonic 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, σ, is said to be perspectively related to the pencil S when every ray on S goes through the point on σ 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 determine three pairs of corresponding rays of the pencils, and therefore all pairs. If four points of the locus are 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 S', 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'. Similarly, the curve is determined by three points and the tangents at two of them.
*73. Circles and conics 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 project 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 furnishes an elegant solution of the problem of drawing a conic through five given points. To construct a sixth 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 N = 34-61. But L and N determine Pascal's line, and the intersection 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 hexagon 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 determined 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 theorem 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.
For let the vertices be A, B, C, and D, and call the vertex A the point 1, 6; B, the point 2; C, the point 3, 4; and D, the point 5 (Fig. 16). Pascal's theorem then indicates that L = AB-CD, M = AD-BC, and N, which is the intersection of the tangents at A and C, are all on a straight line u. But 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 D 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, three of the vertices of the hexagon may coalesce, giving a triangle 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.
*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 theorem (Fig. 18):
If three points, A, B, C, are chosen on one line, and three points, A', B', C', are chosen on another, then the three points L = AB'-A'B, M = BC'-B'C, N = CA'-C'A are all on a straight line.
1. In Fig. 12, select different lines u and trace the locus of the center of perspectivity M of the lines u and u'.
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 tangent 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.)
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 infinity 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 lines 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 projecting 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 O, 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 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, P1, P2, P3, P4, on the line u, they will project to O in four harmonic lines, and the tangents to the circle from these four points will meet u' in four harmonic points, P'1, P'2, P'3, P'4, because the lines from these points to O inclose the same angles as the lines from the points P1, P2, P3, P4 on u. The point-row on u is therefore projective 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 projected 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 possible to assign arbitrarily three lines, _a_, _b_, and _c_, of _ 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 projective point-rows, it is possible to find a point in one which corresponds to any point of the other.
We proceed, then, to the solution of the fundamental
PROBLEM. Given three pairs of points, AA', BB', and CC', of two projective point-rows u and u', to find the point D' of u' which corresponds to any given point D of u.
On the line a, 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 u, to find the corresponding 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, a1, a2, a3, a4, which meet b in four harmonic points. These points project to D, giving four harmonic points on m. These again project to D', giving four harmonic points on c. It is thus clear that the rays a1, a2, a3, a4 cut out two projective 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 l = (12, 45), m = (23, 56), n = (34, 61) all pass through a point.
*85.* To make the notation fit the figure (Fig. 21), make a=1, b = 2, u' = 3, d = 4, u = 5, c = 6; or, interchanging two of the lines, a = 1, c = 2, u = 3, d = 4, u' = 5, b = 6. Thus, by different namings of the lines, it appears that not more than 60 different Brianchon points are possible. If we call 12 and 45 opposite vertices of a circumscribed 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 determining a sixth line of the pencil of rays of the second order when five are given. Thus, select a point in line 1 and suppose that line 6 is to pass through it. Then l = (12, 45), n = (34, 61), and the line m = (23, 56) must pass through (l, n). Then (23, ln) 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 intersection (1, 2) approaches as a limiting position the point of contact of 1 with the conic. This suggests an easy way to construct 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 l = (12,45), m =(23,56); then (34, lm) meets 1 in the required point of contact T.
*88. Circumscribed quadrilateral.* If two pairs of lines in Brianchon's hexagon coalesce, we have a theorem concerning a quadrilateral circumscribed 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.
*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 following problems:
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 suggests the possibility of defining a projective correspondence 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.
*93.* Brianchon'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, b, c are three lines through a point S, and a', b', c' are three lines through another point S', then the lines l = (ab', a'b), m = (bc', b'c), and n = (ca', c'a) all meet in a point.
*94. Law of duality.* The observant student will not have failed to note the remarkable similarity between the theorems of this chapter 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.
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.
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.)