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diff --git a/old/17001.txt b/old/17001.txt new file mode 100644 index 0000000..f953b62 --- /dev/null +++ b/old/17001.txt @@ -0,0 +1,5185 @@ +The Project Gutenberg EBook of An Elementary Course in Synthetic +Projective Geometry by Lehmer, Derrick Norman + + + +This eBook is for the use of anyone anywhere at no cost and with almost no +restrictions whatsoever. You may copy it, give it away or re-use it under +the terms of the Project Gutenberg License included with this eBook or +online at http://www.gutenberg.org/license + + + +Title: An Elementary Course in Synthetic Projective Geometry + +Author: Lehmer, Derrick Norman + +Release Date: November 4, 2005 [Ebook #17001] + +Language: English + +Character set encoding: US-ASCII + + +***START OF THE PROJECT GUTENBERG EBOOK AN ELEMENTARY COURSE IN SYNTHETIC PROJECTIVE GEOMETRY*** + + + + + +An Elementary Course in Synthetic Projective Geometry + + +by Lehmer, Derrick Norman + + + + +Edition 1, (November 4, 2005) + + + + + +PREFACE + + +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 + +BERKELEY, CALIFORNIA + + + + + +CONTENTS + + +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. + + + + + [Figure 1] + + FIG. 1 + + + [Figure 2] + + FIG. 2 + + +*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. + + + + +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. + +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._ + + [Figure 3] + + FIG. 3 + + +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__._ + + [Figure 4] + + FIG. 4 + + +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_. + + [Figure 5] + + FIG. 5 + + +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 {~GREEK SMALL LETTER ALPHA~}, {~GREEK SMALL LETTER BETA~}, {~GREEK SMALL LETTER GAMMA~}, {~GREEK SMALL LETTER DELTA~}, which all pass through the line _g_, pass +also through the four harmonic points _A_, _B_, _C_, _D_, so that {~GREEK SMALL LETTER ALPHA~} passes +through _A_, etc. Then it is clear that any plane {~GREEK SMALL LETTER PI~} 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 {~GREEK SMALL LETTER PI~}, and they pass +through the given harmonic points _A_, _B_, _C_, _D_. Any other plane {~GREEK SMALL LETTER SIGMA~} +cuts _g_ in a point _S_ and cuts {~GREEK SMALL LETTER ALPHA~}, {~GREEK SMALL LETTER BETA~}, {~GREEK SMALL LETTER GAMMA~}, {~GREEK SMALL LETTER DELTA~} in four lines that meet {~GREEK SMALL LETTER PI~} 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 {~GREEK SMALL LETTER ALPHA~}, {~GREEK SMALL LETTER BETA~}, +{~GREEK SMALL LETTER GAMMA~}, {~GREEK SMALL LETTER DELTA~} 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._ + + [Figure 6] + + FIG. 6 + + + + +*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._ + + [Figure 7] + + FIG. 7 + + + + +*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. + + [Figure 8] + + FIG. 8 + + + + +*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'_, + + [formula] + +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,_ + +or + + _(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 + + [formula] + +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. + + + + +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 _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 + + + + + [Figure 9] + + FIG. 9 + + +*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'_. + + [Figure 10] + + FIG. 10 + + + + +*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._ + + [Figure 11] + + FIG. 11 + + +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_. + + + + +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? + +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__._ + + [Figure 12] + + FIG. 12 + + +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_, +_A__1_, _A__2_, and _A__3_, 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_, _L__1_ etc., on the fixed ray _SD_. These, +in turn, project through the fixed point _M_ into four harmonic points, +_N_, _N__1_ etc., on the fixed line _DS'_. These last four harmonic points +give four harmonic rays _CA_, _CA__1_, _CA__2_, _CA__3_. 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._ + + [Figure 13] + + FIG. 13 + + + + +*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, {~GREEK SMALL LETTER SIGMA~}, is +said to be _perspectively related_ to the pencil _S_ when every ray on _S_ +goes through the point on {~GREEK SMALL LETTER SIGMA~} 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. + + [Figure 14] + + FIG. 14 + + + + +*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. + + [Figure 15] + + FIG. 15 + + + + +*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._ + + [Figure 16] + + FIG. 16 + + + [Figure 17] + + FIG. 17 + + +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._ + + [Figure 18] + + FIG. 18 + + + + +*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._ + + + + +PROBLEMS + + +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. + + [Figure 19] + + FIG. 19 + + + + +*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, _P__1_, _P__2_, _P__3_, _P__4_, 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 _P__1_, _P__2_, +_P__3_, _P__4_ 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__._ + + [Figure 20] + + FIG. 20 + + +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, _a__1_, _a__2_, _a__3_, _a__4_, 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 _a__1_, _a__2_, _a__3_, +_a__4_ 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._ + + [Figure 21] + + FIG. 21 + + + + +*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. + + [Figure 22] + + FIG. 22 + + + + +*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_. + + [Figure 23] + + FIG. 23 + + + + +*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. + + [Figure 24] + + FIG. 24 + + + + +*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. + + [Figure 25] + + FIG. 25 + + + + +*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. + + + + +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. + +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 +_a'_, _b'_, _c'_ are also drawn parallel to each other. Show why the lines +(_ab'_, _a'b_), (_bc'_, _b'c_), (_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 intersect 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 lines joining two opposite points of +contact are four lines which meet in a point._ + + [Figure 26] + + FIG. 26 + + + + +*96. Definition of the polar line of a point.* Consider the quadrilateral +_K_, _L_, _M_, _N_ inscribed in the conic (Fig. 26). It determines the +four harmonic points _A_, _B_, _C_, _D_ which project from _N_ in to the +four harmonic points _M_, _B_, _K_, _O_. Now the tangents at _K_ and _M_ +meet in _P_, a point on the line _AB_. The line _AB_ is thus determined +entirely by the point _O_. For if we draw any line through it, meeting the +conic in _K_ and _M_, and construct the harmonic conjugate _B_ of _O_ 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 _LON_ may be any line whatever through _O_; and since _D_, +_L_, _O_, _N_ are four harmonic points, we may describe the line _AB_ as +the locus of points which are harmonic conjugates of _O_ with respect to +the two points where any line through _O_ meets the curve. + + + + +*97.* Furthermore, since the tangents at _L_ and _N_ meet on this same +line, it appears as the locus of intersections of pairs of tangents drawn +at the extremities of chords through _O_. + + + + +*98.* This important line, which is completely determined by the point +_O_, is called the _polar_ of _O_ with respect to the conic; and the point +_O_ is called the _pole_ of the line with respect to the conic. + + + + +*99.* If a point _B_ is on the polar of _O_, then it is harmonically +conjugate to _O_ with respect to the two intersections _K_ and _M_ of the +line _BC_ 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. Similarly, 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. + + [Figure 27] + + FIG. 27 + + + + +*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 intersection 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, each vertex is the pole of the +opposite side. For _B_, _M_, _O_, _K_ are four harmonic points, and they +project to _C_ in four harmonic rays. The line _CO_, therefore, meets the +line _AMN_ in a point on the polar of _A_, being separated from _A_ +harmonically by the points _M_ and _N_. Similarly, the 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 _C_, and therefore _O_ 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 with respect to a +conic revolves about a fixed point and describes a pencil projective 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 projective 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 _OC_, which is the polar of _A_. + + + + +*104. Duality.* We have, then, in the pole and polar relation a device +for setting up a one-to-one correspondence between the points and lines of +the plane--a correspondence 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 the measurement of angles a _metrical_ theorem, +and any other kind a _projective_ theorem, we may put the case as follows: + +_Any projective theorem involves another theorem, dual to it, obtainable +by interchanging everywhere the words 'point' and 'line.'_ + + + + +*105. Self-dual theorems.* The theorems of this chapter 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 theorem on the circumscribed +quadrilateral derived from Brianchon's, which is itself clearly the dual +of Pascal's theorem, and in fact was first discovered by dualization of +Pascal's. + + + + +*106.* It should not be inferred from the above discussion that +one-to-one correspondences may not be devised that will control certain of +the so-called metrical relations. 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 conics generate a point-row of the second order. + +3. Given five points, draw the polar of a point with respect to the conic +passing through them, without drawing the conic itself. + +4. Given five lines, draw the polar of a point with respect 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 conic by a pair of protective 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. +Entering 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 is, 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 defined 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 a circumscribed parallelogram are +conjugate._ + +All these theorems are easy exercises for the student. + + + + +*110. Classification of conics.* Conics are classified according to their +relation to the infinitely distant line. If a conic has two points in +common with the line at infinity, it is called a _hyperbola_; if it has no +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 therefore all its +diameters are parallel,_ for the pole of a tangent 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._ + + [Figure 28] + + FIG. 28 + + + + +*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_ and _B_ (Fig. 28). If, then, _O_ is the intersection of +the asymptotes,--and therefore the center of the curve,-- then the triangle +_OAB_ is circumscribed about the curve. By the theorem just quoted, the +line through _A_ parallel 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 diameters, 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 + +_The segments cut off on any chord between the hyperbola and its +asymptotes are equal._ + + + + +*116.* This theorem furnishes a ready means of constructing the hyperbola +by points when a point on the curve and the two asymptotes are given. + + [Figure 29] + + FIG. 29 + + + + +*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 opposite vertices, and also _A_ and _C_, then _AC_ +and _BD_ are parallel, and parallel to _PQ_, the line joining the points +of contact 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 _O_, and the other meets _OA_ at a distance +_x_ from _O_. Then, since _P_ is the middle point of _AB_, _x_ is one half +of _OA_ and _y_ is one half of _OB_. The area of the parallelogram whose +adjacent sides are _x_ and _y_ is one half the area of the triangle _AOB_, +and therefore, by the preceding paragraph, is constant. This area is equal +to _xy . __sin__ {~GREEK SMALL LETTER ALPHA~}_, where {~GREEK SMALL LETTER ALPHA~} is the constant angle between the asymptotes. +It follows that the product _xy_ is constant, and since _x_ and _y_ are +the oblique cooerdinates of the point _P_, the asymptotes being the axes of +reference, we have + +_The equation of the hyperbola, referred to the asymptotes as axes, is +__xy =__ constant._ + +This identifies the curve with the hyperbola as defined and discussed in +works on analytic geometry. + + + + + [Figure 30] + + FIG. 30 + + +*119. Equation of parabola.* We have defined the parabola as a conic which +is tangent to the line at infinity (§ 110). Draw now two tangents to the +curve (Fig. 30), meeting in _A_, the points of contact being _B_ and _C_. +These two tangents, together with the line at infinity, form a triangle +circumscribed 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 +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 particular, _AD_ passes through _P_, the point of contact of the +tangent drawn parallel to _BC_. + +Draw now another tangent, meeting _AB_ in _B'_ and _AC_ in _C'_. Then +these three, with the line at infinity, make a circumscribed +quadrilateral. But, by Brianchon's theorem 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 _BAC_ we have, for all positions of the +tangent line _B'C_, + + _B'D' : BB' = AC : AB,_ + +or, since _B'D' = AC'_, + + _AC': BB' = AC:AB =_ constant. + +If another tangent meet _AB_ in _B"_ and _AC_ 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 by a 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(x', y')_, then, from the similar +triangles _BMD'_ and we have + + _y : y' = BD' : D'C = BB' : AB'._ + +Also + + _y : y' = B'D' : C'C = AC' : C'C._ + +If now a line is drawn through _A_ parallel to a diameter, meeting the +axis of ordinates in _K_, we have + + _AK : OQ' = AC' : CC' = y : y',_ + +and + + _OM : AK = BB' : AB' = y : y',_ + +and, by multiplication, + + _OM : OQ' = y__2__ : y'__2__,_ + +or + + _x : x' = y__2__ : y'__2__;_ + +whence + +_The abscissas of two points on a parabola are to each other as the +squares of the corresponding cooerdinates, 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 conics 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 _S_ 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._ + +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__2__._ + + [Figure 31] + + FIG. 31 + + +Since _AD_ and _BC_ are parallel tangents, _PQ_ is a diameter 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 _N_, because _AP_, _BQ_, +and _AB_ form a triangle circumscribed to the conic. Let _NS_ meet _PQ_ in +_M_. Then, from the properties of the circumscribed triangle (§ 89), _M_, +_N_, _S_, and the point at infinity on _NS_ are four harmonic points, and +therefore _N_ is the middle point of _MS_. If the cooerdinates 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,_ + +and from the similar triangles _PQA_ and _MQN_, + + _AP : PQ = MN : MQ,_ + +whence, multiplying, we have + + _+-b__2__/4 a__2__ = y__2__/4 (a + x)(a - x),_ + +where + + [formula] + +or, simplifying, + + [formula] + +which is the equation of an ellipse when _b__2_ has a positive sign, and +of a hyperbola when _b__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 direction of the +diameters. + +5. A line _u'_ is drawn through the pole _U_ of a line _u_ and at right +angles to _u_. The line _u_ revolves about a point _P_. Show that the line +_u'_ is tangent to a parabola. (The lines _u_ and _u'_ are called normal +conjugates.) + +6. Given a circle and its center _O_, to draw a line through a given point +_P_ parallel to a given line _q_. Prove the following 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 + + + + + [Figure 32] + + FIG. 32 + + +*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 theorem that results 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: + +_Given 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'__ 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 meet in a point, and thus, from the same +theorem, applied to the triangles _KLM_ and _K'L'M'_, the point _C'_ is on +the same line with _A_ and _B'_. 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 +given, then the point __C'__ corresponding to any given point __C__ is +uniquely determined._ + +The construction of this sixth point is easily accomplished. 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'_ and _CC'_. + + + + +*123. Definition of involution of points on a line.* + +_Three pairs of points on a line are said to be in involution 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._ + + [Figure 33] + + FIG. 33 + + + + +*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 together, 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. + + [Figure 34] + + FIG. 34 + + + + +*125. Desargues's theorem concerning conics through four points.* Let +_DD'_ be any pair of points in the involution determined as above, and +consider the conic passing through the five points _K_, _L_, _M_, _N_, +_D_. We shall use Pascal's theorem to show that this conic also passes +through _D'_. The point _D'_ is determined as follows: Fix _L_ and _M_ as +before (Fig. 34) and join _D_ to _L_, giving on _MN_ the point _N'_. Join +_N'_ to _B_, giving on _LK_ the point _K'_. Then _MK'_ determines the +point _D'_ on the line _AA'_, given by the complete quadrangle _K'_, _L_, +_M_, _N'_. Consider the following six points, numbering them in order: _D += 1_, _D' = 2_, _M = 3_, _N = 4_, _K = 5_, and _L = 6_. We have the +following intersections: _B = (12-45)_, _K' = (23-56)_, _N' = (34-61)_; +and since by construction _B_, _N_, and _K'_ are on a straight line, it +follows 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 conics 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 points belong also to the same +involution with the points cut out by the system of conics, as indeed we +might infer from the fact that the three pairs of opposite sides of the +quadrangle may be considered as degenerate conics of the system. + + + + +*127. Conics through four points touching a given line.* It is further +evident that the involution determined on a line by the system of conics +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 conics or none may be drawn +tangent to any given line._ + + [Figure 35] + + FIG. 35 + + + + +*128. Double correspondence.* We have seen that corresponding 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, however, 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-rows, which are not, however, in involution unless the +angle between the lines is a right angles. For a point _P_ may correspond +to a point _P'_, which in turn will correspond to some other point 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. + + [Figure 36] + + FIG. 36 + + + + +*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 +protective point-rows, to 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'_, _C'_, _D'_, ... be superposed on +the line _u_. Project them both to a point _S_ and pass any conic _{~GREEK SMALL LETTER KAPPA~}_ +through _S_. We thus obtain two projective pencils, _a_, _b_, _c_, _d_, +... and _a'_, _b'_, _c'_, _d'_, ... at _S_, which meet the conic in the +points _{~GREEK SMALL LETTER ALPHA~}_, _{~GREEK SMALL LETTER BETA~}_, _{~GREEK SMALL LETTER GAMMA~}_, _{~GREEK SMALL LETTER DELTA~}_, ... and _{~GREEK SMALL LETTER ALPHA~}'_, _{~GREEK SMALL LETTER BETA~}'_, _{~GREEK SMALL LETTER GAMMA~}'_, _{~GREEK SMALL LETTER DELTA~}'_, ... (Fig. 36). +Take now _{~GREEK SMALL LETTER GAMMA~}_ as the center of a pencil projecting the points _{~GREEK SMALL LETTER ALPHA~}'_, _{~GREEK SMALL LETTER BETA~}'_, +_{~GREEK SMALL LETTER DELTA~}'_, ..., and take _{~GREEK SMALL LETTER GAMMA~}'_ as the center of a pencil projecting the points +_{~GREEK SMALL LETTER ALPHA~}_, _{~GREEK SMALL LETTER BETA~}_, _{~GREEK SMALL LETTER DELTA~}_, .... These two pencils are projective to each other, and +since they have a self-correspondin ray in common, they are in perspective +position and corresponding rays meet on the line joining _({~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER ALPHA~}', {~GREEK SMALL LETTER GAMMA~}'{~GREEK SMALL LETTER ALPHA~})_ to +_({~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER BETA~}', {~GREEK SMALL LETTER GAMMA~}'{~GREEK SMALL LETTER BETA~})_. 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 correspondence: _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 coincide (Fig. 37). Take now any point _C_, which we +will also call _R'_. We must show that the corresponding point _C'_ must +also coincide with the point _B_. Join all the points to _S_, as before, +and it appears that the points {~GREEK SMALL LETTER ALPHA~} and _{~GREEK SMALL LETTER PI~}'_ coincide, as also do the points +_{~GREEK SMALL LETTER ALPHA~}'{~GREEK SMALL LETTER PI~}_ and _{~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER RHO~}'_. By the above construction the line _{~GREEK SMALL LETTER GAMMA~}'{~GREEK SMALL LETTER RHO~}_ must meet _{~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER RHO~}'_ +on the line joining _({~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER ALPHA~}', {~GREEK SMALL LETTER GAMMA~}'{~GREEK SMALL LETTER ALPHA~})_ with _({~GREEK SMALL LETTER GAMMA~}{~GREEK SMALL LETTER PI~}', {~GREEK SMALL LETTER GAMMA~}'{~GREEK SMALL LETTER PI~})_. But these four points +form a quadrangle inscribed in the conic, and we know by § 95 that the +tangents at the opposite vertices _{~GREEK SMALL LETTER GAMMA~}_ and _{~GREEK SMALL LETTER GAMMA~}'_ meet on the line _v_. The +line _{~GREEK SMALL LETTER GAMMA~}'{~GREEK SMALL LETTER RHO~}_ is thus a tangent to the conic, and _C'_ and _R_ are the same +point. That two projective point-rows superposed 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 quadrangle which +has two pairs of opposite sides going through two pairs of points. The +details we leave to the student. + + [Figure 37] + + FIG. 37 + + + [Figure 38] + + FIG. 38 + + + + +*131. Involution of points on a point-row of the second order.* It is +important to note also, in Steiner's construction, 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 __ points of two +projective point-rows superposed on the conic, and when they correspond to +each other doubly._ 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 +perspective position, and their axis of perspectivity _u_ (Fig. 38) is the +line which joins the point _(AB', A'B)_ to the point _(AC', 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 drawn 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'__._ + + + + +*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 conics +through four points we have the dual: + +_The tangents from a fixed point to a system of conics tangent to four +fixed lines form a pencil of rays in involution._ + + + + +*134.* If a conic of the system should go through the fixed point, it is +clear that the two tangents would coincide and indicate a double ray of +the involution. The theorem, therefore, follows: + +_Two conics 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 two pairs +of rays correspond to each other doubly. From this it is clear that we +might have deemed 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 +corresponding 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 analogous 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. + + + + +*136. Pencils of rays of the second order in involution.* We may also +extend the notion of involution to pencils 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 pencil 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 involution, 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, corresponding 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 __ being conjugate with +respect to the conic. The double points, if any exist, are the points +where the line meets the conic._ + + + + +*140.* The dual theorem reads: _A conic determines at every point in the +plane an involution of rays, corresponding rays being conjugate with +respect to the conic. The double rays, if any exist, are the tangents from +the point 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 conics or none may be drawn to pass +through points of it and also to be tangent to the four lines. + +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 remain 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. + +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 _AA'_, _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 _P'_, and +if it be tangent to two given lines, the chord of contact of these two +tangents will always pass through a fixed point on _PP'_. + +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 9 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 + + + + + [Figure 39] + + FIG. 39 + + +*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 +the _center_ of the involution. Since corresponding points in the +involution have been shown to be harmonic conjugates 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 _B'_, thus determining on _AK_ +and _AN_ the points _L_ and _N_. _LN_ meets _AA'_ in the center _O_ of the +involution. + + + + +*142. Fundamental metrical theorem.* From the figure we see that the +triangles _OLB'_ and _PLM_ are similar, _P_ being the intersection of KM +and LN. Also the triangles _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 fundamental 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 corresponding 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 root 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 +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 _OA_. A +similar discussion may be made for the case where _A_ and _A'_ lie on +opposite sides of _O_. The results may be stated as follows, without any +reference to the center: + +_Given two pairs of points in an involution of points, if the points of +one pair are separated from each other by the points of the other pair, +then the 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 has two 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. + + [Figure 40] + + FIG. 40 + + + + +*145. Construction of an involution by means of circles.* The equation +just derived, _OA . OA' = OB . 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 circle 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 __ 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'_. +Further, the line _GG'_ meets _AA'_ in the center of the involution. To +find the double points, if they exist, we draw a tangent from _O_ 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 +_O_. This simple and effective 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 theory of analytic geometry +indicates that the circle is a special conic section that happens to pass +through two particular imaginary points on the line at infinity, called +the _circular points_ and usually denoted by _I_ 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 conics through four points will cut any line in the plane +in a point-row in involution. + + [Figure 41] + + FIG. 41 + + + + +*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'_. This +circle cuts out two points _M_ and _M'_ in the involution. The rays _GM_ +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 involution 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 conics.* At the close of the last chapter (§ 140) we gave +the theorem: _A conic determines at every point in its plane an involution +of rays, corresponding rays __ being conjugate with respect to the conic. +The double rays, if any exist, are the tangents from the point to the +conic._ In particular, taking the point as the center of the conic, we +find that conjugate diameters form a system of rays in involution, of +which the asymptotes, if there are any, are the double rays. Also, +conjugate diameters are harmonic conjugates with respect to the +asymptotes. By the theorem of the last paragraph, there are two conjugate +diameters which are at right angles to each other. These are called axes. +In the case of the parabola, where the center is at infinity, and on the +curve, there are, properly speaking, no conjugate diameters. While the +line at infinity might be considered as conjugate to all the other +diameters, it is not possible to assign to it any particular direction, +and so it cannot be used for the purpose of defining an axis of a +parabola. There is one diameter, however, which is at right angles to its +conjugate system of chords, and this one is called the _axis_ of the +parabola. The circle also furnishes an exception in that every diameter is +an axis. The involution in this case is circular, every ray being at right +angles to its conjugate ray at the center. + + + + +*149. Points at which the involution determined by a conic is circular.* +It is an important problem to discover whether for any conic other than +the circle it is possible to find any point in the plane where the +involution determined as above by the conic is circular. We shall proceed +to the curious problem of proving the existence of such points and of +determining their number and situation. We shall then develop the +important properties of such points. + + + + +*150.* It is clear, in the first place, that such a point cannot be on +the outside of the conic, else the involution would have double rays and +such rays would have to be at right angles to themselves. In the second +place, if two such points exist, the line joining them must be a diameter +and, indeed, an axis. For if _F_ and _F'_ were two such points, then, +since the conjugate ray at _F_ to the line _FF'_ must be at right angles +to it, and also since the conjugate ray at _F'_ to the line _FF'_ must be +at right angles to it, the pole of _FF'_ must be at infinity in a +direction at right angles to _FF'_. The line _FF'_ is then a diameter, and +since it is at right angles to its conjugate diameter, it must be an axis. +From this it follows also that the points we are seeking must all lie on +one of the two axes, else we should have a diameter which does not go +through the intersection of all axes--the center of the conic. At least one +axis, therefore, must be free from any such points. + + [Figure 42] + + FIG. 42 + + + + +*151.* Let now _P_ be a point on one of the axes (Fig. 42), and draw any +ray through it, such as _q_. As _q_ revolves about _P_, its pole _Q_ moves +along a line at right angles to the axis on which _P_ lies, describing a +point-row _p_ projective to the pencil of rays _q_. The point at infinity +in a direction at right angles to _q_ also describes a point-row +projective to _q_. The line joining corresponding points of these two +point-rows is always a conjugate line to _q_ and at right angles to _q_, +or, as we may call it, a _conjugate normal_ to _q_. These conjugate +normals to _q_, joining as they do corresponding points in two projective +point-rows, form a pencil of rays of the second order. But since the point +at infinity on the point-row _Q_ corresponds to the point at infinity in a +direction at right angles to _q_, these point-rows are in perspective +position and the normal conjugates of all the lines through _P_ meet in a +point. This point lies on the same axis with _P_, as is seen by taking _q_ +at right angles to the axis on which _P_ lies. The center of this pencil +may be called _P'_, and thus we have paired the point _P_ with the point +_P'_. By moving the point _P_ along the axis, and by keeping the ray _q_ +parallel to a fixed direction, we may see that the point-row _P_ and the +point-row _P'_ are projective. Also the correspondence is double, and by +starting from the point _P'_ we arrive at the point _P_. Therefore the +point-rows _P_ and _P'_ are in involution, and if only the involution has +double points, we shall have found in them the points we are seeking. For +it is clear that the rays through _P_ and the corresponding rays through +_P'_ are conjugate normals; and if _P_ and _P'_ coincide, we shall have a +point where all rays are at right angles to their conjugates. We shall now +show that the involution thus obtained on one of the two axes must have +double points. + + [Figure 43] + + FIG. 43 + + + + +*152. Discovery of the foci of the conic.* We know that on one axis no +such points as we are seeking can lie (§ 150). The involution of points +_PP'_ on this axis can therefore have no double points. Nevertheless, let +_PP'_ and _RR'_ be two pairs of corresponding points on this axis (Fig. +43). Then we know that _P_ and _P'_ are separated from each other by _R_ +and _R'_ (§ 143). Draw a circle on _PP'_ as a diameter, and one on _RR'_ +as a diameter. These must intersect in two points, _F_ and _F'_, and since +the center of the conic is the center of the involution _PP'_, _RR'_, as +is easily seen, it follows that _F_ and _F'_ are on the other axis of the +conic. Moreover, _FR_ and _FR'_ are conjugate normal rays, since _RFR'_ is +inscribed in a semicircle, and the two rays go one through _R_ and the +other through _R'_. The involution of points _PP'_, _RR'_ therefore +projects to the two points _F_ and _F'_ in two pencils of rays in +involution which have for corresponding rays conjugate normals to the +conic. We may, then, say: + +_There are two and only two points of the plane where the involution +determined by the conic is circular. These two points lie on one of the +axes, at equal distances from the center, on the inside of the conic. +These points are called the foci of the conic._ + + + + +*153. The circle and the parabola.* The above discussion applies only to +the central conics, apart from the circle. In the circle the two foci fall +together at the center. In the case of the parabola, that part of the +investigation which proves the existence of two foci on one of the axes +will not hold, as we have but one axis. It is seen, however, that as _P_ +moves to infinity, carrying the line _q_ with it, _q_ becomes the line at +infinity, which for the parabola is a tangent line. Its pole _Q_ is thus +at infinity and also the point _P'_, so that _P_ and _P'_ fall together at +infinity, and therefore one focus of the parabola is at infinity. There +must therefore be another, so that + +_A parabola has one and only one focus in the finite part of the plane._ + + [Figure 44] + + FIG. 44 + + + + +*154. Focal properties of conics.* We proceed to develop 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 involution 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 + +_The lines joining a point on the conic to the foci make equal angles with +the tangent._ + +It follows that rays from a source of light at one focus are reflected by +an ellipse to the other. + + + + +*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 + + [Figure 45] + + FIG. 45 + + +_A diameter makes the same angle with the tangent at its extremity as that +tangent does with the line from its point of contact to the focus (Fig. +45)._ + + + + +*156.* This last theorem is the basis for the construction 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 _directrix_. The axis which contains +the foci is called the _principal axis_, and the intersection of the axis +with the curve is called the _vertex_ of the curve. The directrix is at +right angles to the principal axis. In a parabola the vertex is equally +distant from the focus and the directrix, these three points and the point +at infinity on the axis being four harmonic points. In the ellipse the +vertex is nearer to the focus than it is to the directrix, for the same +reason, and in the hyperbola it is farther from the focus than it is from +the directrix. + + [Figure 46] + + FIG. 46 + + + + +*158. Another definition of a conic.* Let _P_ be any point on the +directrix through which a line is drawn meeting the conic in the points +_A_ and _B_ (Fig. 46). Let the tangents at _A_ and _B_ meet in _T_, and +call the focus _F_. Then _TF_ and _PF_ are conjugate lines, and as they +pass through a focus they must be at right angles to each other. Let _TF_ +meet _AB_ in _C_. Then _P_, _A_, _C_, _B_ 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 : FM = FB : BN_. Dropping perpendiculars _AA_ and _BB'_ upon the +directrix, this becomes _FA : AA' = FB : BB'_. We have thus the property +often taken as the definition of a conic: + +_The ratio of the distances from a point on the conic to the focus and the +directrix is constant._ + + [Figure 47] + + FIG. 47 + + + + +*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 parabola. This ratio +is called the _eccentricity_. + + [Figure 48] + + FIG. 48 + + + + +*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 symmetrical with +respect to both. If the distances from 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'_; _(r +- r') : (d +- d')_. In the ellipse _(d + d')_ is constant, being +the distance between the directrices. In the hyperbola this distance is +_(d - d')_. It follows (Fig. 48) that + +_In the ellipse the sum of the focal distances of any point on the curve +is constant, and in the hyperbola the difference between the focal +distances is constant._ + + + + +PROBLEMS + + +1. Construct the axis of a parabola, given four tangents. + +2. Given two conjugate lines at right angles to each other, and let them +meet the axis which has no foci on it in the points _A_ and _B_. The +circle on _AB_ as diameter will pass through the foci of the conic. + +3. Given the axes of a conic in position, and also a tangent with its +point of contact, to construct the foci and determine the length of the +axes. + +4. Given the tangent at the vertex of a parabola, and two other tangents, +to find the focus. + +5. The locus of the center of a circle touching two given circles is a +conic with the centers of the given circles for its foci. + +6. Given the axis of a parabola and a tangent, with its point of contact, +to find the focus. + +7. The locus of the center of a circle which touches a given line and a +given circle consists of two parabolas. + +8. Let _F_ and _F'_ be the foci of an ellipse, and _P_ any point on it. +Produce _PF_ to _G_, making _PG_ equal to _PF'_. Find the locus of _G_. + +9. If the points _G_ of a circle be folded over upon a point _F_, the +creases will all be tangent to a conic. If _F_ is within the circle, the +conic will be an ellipse; if _F_ is without the circle, the conic will be +a hyperbola. + +10. If the points _G_ in the last example be taken on a straight line, the +locus is a parabola. + +11. Find the foci and the length of the principal axis of the conics in +problems 9 and 10. + +12. In problem 10 a correspondence is set up between straight lines and +parabolas. As there is a fourfold infinity of parabolas in the plane, and +only a twofold infinity of straight lines, there must be some restriction +on the parabolas obtained by this method. Find and explain this +restriction. + +13. State and explain the similar problem for problem 9. + +14. The last four problems are a study of the consequences of the +following transformation: A point _O_ is fixed in the plane. Then to any +point _P_ is made to correspond the line _p_ at right angles to _OP_ and +bisecting it. In this correspondence, what happens to _p_ when _P_ moves +along a straight line? What corresponds to the theorem that two lines have +only one point in common? What to the theorem that the angle sum of a +triangle is two right angles? Etc. + + + + + +CHAPTER X - ON THE HISTORY OF SYNTHETIC PROJECTIVE GEOMETRY + + + + +*161. Ancient results.* The theory of synthetic projective geometry as we +have built it up in this course is less than a century old. This is not to +say that many of the theorems and principles involved were not discovered +much earlier, but isolated theorems do not make a theory, any more than a +pile of bricks makes a building. The materials for our building have been +contributed by many different workmen from the days of Euclid down to the +present time. Thus, the notion of four harmonic points was familiar to the +ancients, who considered it from the metrical point of view as the +division of a line internally and externally in the same ratio(1) the +involution of six points cut out by any transversal which intersects the +sides of a complete quadrilateral as studied by Pappus(2); 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 doubtless familiar with the theorem, _Two lines determine a +point, and two points determine a line_, but they had no glimpse of the +wonderful law of duality, of which this theorem is a simple example. The +principle of projection, by which many properties of the conic sections +may be inferred from corresponding properties of the circle which forms +the base of the cone from which they are cut--a principle so natural to +modern mathematicians--seems not to have occurred to the Greeks. The +ellipse, the hyperbola, and the parabola were to them entirely different +curves, to be treated separately with methods appropriate to each. Thus +the focus of the ellipse was discovered some five hundred years before the +focus of the parabola! It was not till 1522 that Verner(3) of Nuernberg +undertook to demonstrate the properties of the conic sections by means of +the circle. + + + + +*162. Unifying principles.* In the early years of the seventeenth +century--that wonderful epoch in the history of the world which produced a +Galileo, a Kepler, a Tycho Brahe, a Descartes, a Desargues, a Pascal, a +Cavalieri, a Wallis, a Fermat, a Huygens, a Bacon, a Napier, and a goodly +array of lesser lights, to say nothing of a Rembrandt or of a +Shakespeare--there began to appear certain unifying principles connecting +the great mass of material dug out by the ancients. Thus, in 1604 the +great astronomer Kepler(4) introduced the notion that parallel lines +should be considered as meeting at an infinite distance, and that a +parabola is at once the limiting case of an ellipse and of a hyperbola. He +also attributes to the parabola a "blind focus" (_caecus focus_) at +infinity on the axis. + + + + +*163. Desargues.* In 1639 Desargues,(5) an architect of Lyons, published +a little treatise on the conic sections, in which appears the theorem upon +which we have founded the theory of four harmonic points (§ 25). +Desargues, however, does not make use of it for that purpose. Four +harmonic points are for him a special case of six points in involution +when two of the three pairs coincide giving double points. His development +of the theory of involution is also different from the purely geometric +one which we have adopted, and is based on the theorem (§ 142) that the +product of the distances of two conjugate points from the center is +constant. He also proves the projective character of an involution of +points by showing that when six lines pass through a point and through six +points in involution, then any transversal must meet them in six points +which are also in involution. + + + + +*164. Poles and polars.* In this little treatise is also contained the +theory of poles and polars. The polar line is called a _traversal_.(6) The +harmonic properties of poles and polars are given, but Desargues seems not +to have arrived at the metrical properties which result when the infinite +elements of the plane are introduced. Thus he says, "When the _traversal_ +is at an infinite distance, all is unimaginable." + + + + +*165. Desargues's theorem concerning conics through four points.* We find +in this little book the beautiful theorem concerning a quadrilateral +inscribed in a conic section, which is given by his name in § 138. The +theorem is not given in terms of a system of conics through four points, +for Desargues had no conception of any such system. He states the theorem, +in effect, as follows: _Given a simple quadrilateral inscribed in a conic +section, every transversal meets the conic and the four sides of the +quadrilateral in six points which are in involution._ + + + + +*166. Extension of the theory of poles and polars to space.* As an +illustration of his remarkable powers of generalization, we may note that +Desargues extended the notion of poles and polars to space of three +dimensions for the sphere and for certain other surfaces of the second +degree. This is a matter which has not been touched on in this book, but +the notion is not difficult to grasp. If we draw through any point _P_ in +space a line to cut a sphere in two points, _A_ and _S_, and then +construct the fourth harmonic of _P_ with respect to _A_ and _B_, the +locus of this fourth harmonic, for various lines through _P_, is a plane +called the _polar plane_ of _P_ with respect to the sphere. With this +definition and theorem one can easily find dual relations between points +and planes in space analogous to those between points and lines in a +plane. Desargues closes his discussion of this matter with the remark, +"Similar properties may be found for those other solids which are related +to the sphere in the same way that the conic section is to the circle." It +should not be inferred from this remark, however, that he was acquainted +with all the different varieties of surfaces of the second order. The +ancients were well acquainted with the surfaces obtained by revolving an +ellipse or a parabola about an axis. Even the hyperboloid of two sheets, +obtained by revolving the hyperbola about its major axis, was known to +them, but probably not the hyperboloid of one sheet, which results from +revolving a hyperbola about the other axis. All the other solids of the +second degree were probably unknown until their discovery by Euler.(7) + + + + +*167.* Desargues had no conception of the conic section of the locus of +intersection of corresponding rays of two projective pencils of rays. He +seems to have tried to describe the curve by means of a pair 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 violent abuse and held up to +ridicule and scorn! "Incredible 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 reviewers 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 friendship 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. + + + + +*169. Conservatism in Desargues's time.* It is not necessary 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 Pascal 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, authority counted for +less at this time in Paris than it did in Italy, and the tragedy enacted +in Rome when Galileo was forced to deny his inmost convictions at the +bidding 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 Desargues 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.(8) "You may have +two designs, 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 properties of the conic sections which they do not +already know; and the other is to write for the curious unlearned, 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 +painting, architecture, etc., shall become common and easy to 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 certain 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 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 survived. Curiously enough, this is the one +term singled out for the sharpest criticism and ridicule by his reviewer, +De Beaugrand.(9) That Descartes knew the character 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(10) to Mersenne, "I can hardly form an idea of what he may +have written concerning conics." 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! + + + + +*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,"(11) 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 intersection of __MK__ and __SK__; __V__ +the intersection of __MV__ and __SV__; __A__ the intersection of __MA__ +and __SA__ (__A__ is the intersection of __SV__ and __MK__), and __{~GREEK SMALL LETTER MU~}__ the +intersection of __MV__ and __SK__; and if through two of the four points +__A__, __K__, __{~GREEK SMALL LETTER MU~}__, __V__, which are not in the same straight line with +__M__ and __S__, such as __K__ and __V__, we pass the circumference of a +circle cutting the lines __MV__, __MP__, __SV__, __SK__ in the points +__O__, __P__, __Q__, __N__; I say that the lines __MS__, __NO__, __PQ__ +are of the same order."_ (By "lines of the same order" Pascal means lines +which meet in the same point or are parallel.) By projecting the figure +thus described upon another plane he is able to state his theorem for the +case where the circle is replaced by any conic section. + + + + +*173.* It must be understood that the "Essay" was only a resume of a more +extended treatise on conics which, owing partly to Pascal's extreme youth, +partly to the difficulty of publishing scientific works in those days, and +also to his later morbid interest in religious matters, was never +published. Leibniz(12) 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 out, not fully realizing, perhaps, how much +was his own and how much he owed to his teacher. Writing in 1679, he +says,(13) "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 interesting, and +will serve as an exercise for the student in synthetic projective +geometry. It is as follows: _Draw two parallel lines, __a__ and __b__, and +select a point __P__ in their plane. Through any point __M__ of the plane +draw a line meeting __a__ in __A__ and __b__ in __B__. Draw a line through +__B__ parallel to __AP__, and let it meet __MP__ in the point __M'__. It +may be shown that the point __M'__ thus obtained does not depend at all on +the particular ray __MAB__ used in determining 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 correspondence 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_ +transform 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 _b_, any two lines which may meet in a finite part +of the plane. 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 synthetic projective +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 attention 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 experience 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 analysis 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 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,(14) "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 Newton 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 projective geometry. Thus Newton's "organic +method"(15) of generating conic sections is closely related to the method +which 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, __O__ 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 intersection, __A__, of the other 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.(16) 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 __ around three fixed points, __R__, +__A__, __B__, respectively, while two of its vertices, __P__, __Q__, slide +along two 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, Moebius, Abel, and many +others. The renaissance may be said to date from the invention by +Monge(17) of the theory of _descriptive geometry_. Descriptive geometry is +concerned with the representation 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 perpendicular to the two +planes of projection. These two planes are then made to coincide by +revolving the horizontal into the vertical about their common line. Such +is the method of descriptive geometry which in the hands of Monge acquired +wonderful generality and elegance. Problems concerning fortifications were +worked so quickly by this method that the commandant at the military +school at Mezieres, where Monge was a draftsman and pupil, viewed the +results with distrust. Monge afterward became professor of mathematics at +Mezieres and gathered around him a group of students destined to have a +share in the advancement of pure geometry. Among these were Hachette, +Brianchon, Dupin, Chasles, Poncelet, and many others. + + + + +*180. Duality, homology, continuity, contingent relations.* Analytic +geometry had left little to do in the way of discovery of new material, +and the mathematical 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, homology, 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 geometry +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 contingent +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 elements 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." + + + + +*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 projective +properties of figures(18) Cauchy says, "In his preliminary 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 principle is, properly speaking, +only a strong induction, which cannot be indiscriminately applied to all +sorts of questions in geometry, nor even in analysis. The reasons 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 working 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 cooerdinates as well as +with three, but the existence of a space of four dimensions to correspond +to it does not therefore follow. When the geometer speaks of the two real +or imaginary intersections 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 +homology_, which is, in effect, the collineation 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 transformations 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,(19) 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 synthetic geometry.* The +reaction of pure geometry on 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 Pluecker. The notion of geometrical +transformation led also to the very important developments in the theory +of invariants, which, geometrically, 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 enjoyed a fundamental role. It is made also the basis +of the great work of Steiner,(20) 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 projective pencils. Steiner not only related to +each other, 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 degrees. This new and fertile conception gave him +an easy and direct route into the most abstract and difficult regions of +pure geometry. 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 theory 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,(21) 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 lie 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 illustration of a correspondence of this sort, +where the conic is imaginary, is given in § 15 of the first chapter. In +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 powerful, are hardly adapted to an elementary course, but Reye(22) and +others have done much toward simplifying his presentation. + + + + +*186. Recent developments.* It would be only confusing 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 between 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 discoveries 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 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,(23) "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 +already done, the means of renewing its life and of developing itself +anew. It is like the Giant Antaeus, who renewed, his strength by touching +the earth." + + + + + +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 conics, 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 + +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, 86, 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 + +Lagrange (1736-1813), 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 + +Moebius (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 + +One-to-one correspondence, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 24, 36, 87, 43, +60, 104, 106, 184 + +Opposite sides of a hexagon, 70 + +Opposite sides of a quadrilateral, 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, 75, 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, 164, 166 + +Poncelet (1788-1867), 177, 179, 180, 181, 182, 183, 184 + +Principal axis of a conic, 157 + +Projection, 161 + +Protective 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 involution, 148 + +Separation of harmonic conjugates, 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 conics, 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 + + + + + + +FOOTNOTES + + + 1 The more general notion of _anharmonic ratio_, which includes the + harmonic ratio as a special case, was also known to the ancients. + While we have not found it necessary to make use of the anharmonic + ratio in building up our theory, it is so frequently met with in + treatises on geometry that some account of it should be given. + + Consider any four points, _A_, _B_, _C_, _D_, on a line, and join + them to any point _S_ not on that line. Then the triangles _ASB_, + _GSD_, _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 + + [formula] + + 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'_, _D'_ lying on any other line cutting across _SA_, _SB_, + _SC_, _SD_. In other 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 _anharmonic 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 + + [formula] + + 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 + + [formula] + + 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 quadrilateral _KLMN_ 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 _O_ is the intersection of + _KM_ with _LN_. But, projecting again the points _K_, _O_, _M_, _D_ + from _N_ back upon the line _AB_, we have _(KOMD) = (CBAD)_. From + this we have + + _(ABCD) = (CBAD),_ + + or + + [formula] + + whence _a = 0_ or _a = 2_. But it is easy to see that _a = 0_ + 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, [formula], 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. + + [Figure 49] + + FIG. 49 + + + Many theorems of projective geometry are succinctly stated in terms + of anharmonic ratios. Thus, the _anharmonic ratio of any four + elements of a form is equal to the anharmonic ratio of the + corresponding four elements in any form projectively related to it. + The anharmonic ratio of the lines joining any four fixed points on a + conic to a variable fifthpoint on the conic is constant. The locus + of points from which four points in a plane are seen along four rays + of constant anharmonic ratio is a conic through the four points._ We + leave these theorems for the student, who may also justify the + following solution of the problem: _Given three points and a certain + anharmonic ratio, to find a fourth point which shall have with the + given three the given anharmonic ratio._ Let _A_, _B_, _D_ be the + three given points (Fig. 49). On any convenient line through _A_ + take two points _B'_ and _D'_ such that _AB'/AD'_ is equal to the + given anharmonic ratio. Join _BB'_ and _DD'_ and let the two lines + meet in _S_. Draw through _S_ a parallel to _AB'_. This line will + meet _AB_ in the required point _C_. + + 2 Pappus, Mathematicae Collectiones, vii, 129. + + 3 J. Verneri, Libellus super vigintiduobus elementis conicis, etc. + 1522. + + 4 Kepler, Ad Vitellionem paralipomena quibus astronomiae pars optica + traditur. 1604. + + 5 Desargues, Bruillon-project d'une atteinte aux evenements des + rencontres d'un cone avec un plan. 1639. Edited and analyzed by + Poudra, 1864. + + 6 The term 'pole' was first introduced, in the sense in which we have + used it, in 1810, by a French mathematician named Servois (Gergonne, + _Annales des Matheematiques_, I, 337), and the corresponding term + 'polar' by the editor, Gergonne, of this same journal three years + later. + + 7 Euler, Introductio in analysin infinitorum, Appendix, cap. V. 1748. + + 8 OEuvres de Desargues, t. II, 132. + + 9 OEuvres de Desargues, t. II, 370. + + 10 OEuvres de Descartes, t. II, 499. + + 11 OEuvres de Pascal, par Brunsehvig et Boutroux, t. I, 252. + + 12 Chasles, Histoire de la Geometrie, 70. + + 13 OEuvres de Desargues, t. I, 231. + + 14 See Ball, History of Mathematics, French edition, t. II, 233. + + 15 Newton, Principia, lib. i, lemma XXI. + + 16 Maclaurin, Philosophical Transactions of the Royal Society of + London, 1735. + + 17 Monge, Geometrie Descriptive. 1800. + + 18 Poncelet, Traite des Proprietes Projectives des Figures. 1822. (See + p. 357, Vol. II, of the edition of 1866.) + + 19 Gergonne, _Annales de Mathematiques, XVI, 209. 1826._ + + 20 Steiner, Systematische Ehtwickelung der Abhaengigkeit geometrischer + Gestalten von einander. 1832. + + 21 Von Staudt, Geometrie der Lage. 1847. + + 22 Reye, Geometrie der Lage. Translated by Holgate, 1897. + + 23 Ball, loc. cit. p. 261. + + + + +***END OF THE PROJECT GUTENBERG EBOOK AN ELEMENTARY COURSE IN SYNTHETIC PROJECTIVE GEOMETRY*** + + + + +CREDITS + + +November 2005 + + Project Gutenberg Edition + Joshua Hutchinson, Cornell University, Online Distributed + Proofreading Team + +June 2006 + + Added PGHeader/PGFooter. + Joshua Hutchinson + + +A WORD FROM PROJECT GUTENBERG + + +This file should be named 17001-0.txt or 17001-0.zip. + +This and all associated files of various formats will be found in: + + + http://www.gutenberg.org/dirs/1/7/0/0/17001/ + + +Updated editions will replace the previous one -- the old editions will be +renamed. + +Creating the works from public domain print editions means that no one +owns a United States copyright in these works, so the Foundation (and +you!) can copy and distribute it in the United States without permission +and without paying copyright royalties. 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