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diff --git a/old/17001-8.txt b/old/17001-8.txt new file mode 100644 index 0000000..45dcae5 --- /dev/null +++ b/old/17001-8.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: ISO 8859-1 + + +***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 rôle 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 coördinates 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 coördinates, 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 coördinates 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 Nürnberg +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 résumé 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 "Méchanique 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, Möbius, 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 Mézières, where Monge was a draftsman and pupil, viewed the +results with distrust. Monge afterward became professor of mathematics at +Mézières 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 Traité des Propriétés +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 coördinates 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 Plücker. 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 rôle. 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 + +Möbius (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 événements des + rencontres d'un cône 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 Mathéématiques_, 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 Géométrie, 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, Géométrie Descriptive. 1800. + + 18 Poncelet, Traité des Propriétés Projectives des Figures. 1822. (See + p. 357, Vol. II, of the edition of 1866.) + + 19 Gergonne, _Annales de Mathématiques, XVI, 209. 1826._ + + 20 Steiner, Systematische Ehtwickelung der Abhängigkeit 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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363.264 326.285 cm +0 g 0 G +1 0 0 1 -363.264 -326.285 cm +BT +/F16 10.909 Tf 363.264 326.285 Td[(22)]TJ +ET +1 0 0 1 374.173 326.285 cm +0 g 0 G +1 0 0 1 -374.173 -326.285 cm +BT +/F16 10.909 Tf 109.907 312.567 Td[(37.)-250(Correspondence)-250(between)-250(harmonic)-250(conjugates)-366(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 312.567 cm +0 g 0 G +1 0 0 1 -363.264 -312.567 cm +BT +/F16 10.909 Tf 363.264 312.567 Td[(22)]TJ +ET +1 0 0 1 374.173 312.567 cm +0 g 0 G +1 0 0 1 -374.173 -312.567 cm +BT +/F16 10.909 Tf 109.907 298.848 Td[(38.)-250(Separation)-250(of)-250(harmonic)-250(conjugates)-586(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 298.848 cm +0 g 0 G +1 0 0 1 -363.264 -298.848 cm +BT +/F16 10.909 Tf 363.264 298.848 Td[(23)]TJ +ET +1 0 0 1 374.173 298.848 cm +0 g 0 G +1 0 0 1 -374.173 -298.848 cm +BT +/F16 10.909 Tf 109.907 285.13 Td[(39.)-250(Harmonic)-250(conjugate)-250(of)-250(the)-250(point)-250(at)-250(infinity)-335(.)-500(.)-500(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 285.13 cm +0 g 0 G +1 0 0 1 -363.264 -285.13 cm +BT +/F16 10.909 Tf 363.264 285.13 Td[(23)]TJ +ET +1 0 0 1 374.173 285.13 cm +0 g 0 G +1 0 0 1 -374.173 -285.13 cm +BT +/F16 10.909 Tf 109.907 271.411 Td[(40.)-247(Projective)-241(theorems)-240(and)-241(metrical)-241(theorems.)-247(Linear)]TJ 25.091 -13.549 Td[(construction)-506(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 257.862 cm +0 g 0 G +1 0 0 1 -363.264 -257.862 cm +BT +/F16 10.909 Tf 363.264 257.862 Td[(24)]TJ +ET +1 0 0 1 374.173 257.862 cm +0 g 0 G +1 0 0 1 -374.173 -257.862 cm +BT +/F16 10.909 Tf 109.907 244.144 Td[(41.)-250(Parallels)-250(and)-250(mid-points)-278(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 244.144 cm +0 g 0 G +1 0 0 1 -363.264 -244.144 cm +BT +/F16 10.909 Tf 363.264 244.144 Td[(25)]TJ +ET +1 0 0 1 374.173 244.144 cm +0 g 0 G +1 0 0 1 -374.173 -244.144 cm +BT +/F16 10.909 Tf 109.907 230.425 Td[(42.)-250(Division)-250(of)-250(segment)-250(into)-250(equal)-250(parts)-723(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 230.425 cm +0 g 0 G +1 0 0 1 -363.264 -230.425 cm +BT +/F16 10.909 Tf 363.264 230.425 Td[(26)]TJ +ET +1 0 0 1 374.173 230.425 cm +0 g 0 G +1 0 0 1 -374.173 -230.425 cm +BT +/F16 10.909 Tf 109.907 216.707 Td[(43.)-250(Numerical)-250(relations)-585(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 216.707 cm +0 g 0 G +1 0 0 1 -363.264 -216.707 cm +BT +/F16 10.909 Tf 363.264 216.707 Td[(26)]TJ +ET +1 0 0 1 374.173 216.707 cm +0 g 0 G +1 0 0 1 -374.173 -216.707 cm +BT +/F16 10.909 Tf 109.907 202.988 Td[(44.)-250(Algebraic)-250(formula)-250(connecting)-250(four)-250(harmonic)-250(points)]TJ +ET +1 0 0 1 363.264 202.988 cm +0 g 0 G +1 0 0 1 -363.264 -202.988 cm +BT +/F16 10.909 Tf 363.264 202.988 Td[(26)]TJ +ET +1 0 0 1 374.173 202.988 cm +0 g 0 G +1 0 0 1 -374.173 -202.988 cm +BT +/F16 10.909 Tf 109.907 189.27 Td[(45.)-250(Further)-250(formulae)-946(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 189.27 cm +0 g 0 G +1 0 0 1 -363.264 -189.27 cm +BT +/F16 10.909 Tf 363.264 189.27 Td[(28)]TJ +ET +1 0 0 1 374.173 189.27 cm +0 g 0 G +1 0 0 1 -374.173 -189.27 cm +BT +/F16 10.909 Tf 109.907 175.551 Td[(46.)-250(Anharmonic)-250(ratio)-668(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 175.551 cm +0 g 0 G +1 0 0 1 -363.264 -175.551 cm +BT +/F16 10.909 Tf 363.264 175.551 Td[(28)]TJ +ET +1 0 0 1 374.173 175.551 cm +0 g 0 G +1 0 0 1 -374.173 -175.551 cm +BT +/F16 10.909 Tf 109.907 161.833 Td[(PROBLEMS)-971(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 161.833 cm +0 g 0 G +1 0 0 1 -363.264 -161.833 cm +BT +/F16 10.909 Tf 363.264 161.833 Td[(29)]TJ +ET +1 0 0 1 374.173 161.833 cm +0 g 0 G +1 0 0 1 -374.173 -161.833 cm +BT +/F16 10.909 Tf 93.543 148.114 Td[(CHAPTER)-446(III)-446(-)-446(COMBINATION)-446(OF)-446(TW)-1(O)-446(PROJEC-)]TJ 15.273 -13.549 Td[(TIVELY)-250(RELATED)-250(FUNDAMENTAL)-250(FORMS)-685(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 134.565 cm +0 g 0 G +1 0 0 1 -363.264 -134.565 cm +BT +/F16 10.909 Tf 363.264 134.565 Td[(31)]TJ +ET +1 0 0 1 374.173 134.565 cm +0 g 0 G +1 0 0 1 -374.173 -134.565 cm +BT +/F16 10.909 Tf 109.907 120.847 Td[(47.)-2212(Superposed)-904(fundamental)-904(forms.)-2213(Self-)]TJ 25.091 -13.55 Td[(corresponding)-250(elements)-424(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 107.297 cm +0 g 0 G +1 0 0 1 -363.264 -107.297 cm +BT +/F16 10.909 Tf 363.264 107.297 Td[(31)]TJ +ET +1 0 0 1 374.173 107.297 cm +0 g 0 G +1 0 0 1 -374.173 -107.297 cm +BT +/F16 10.909 Tf 109.907 93.579 Td[(48.)-250(Special)-250(case)-585(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 93.579 cm +0 g 0 G +1 0 0 1 -363.264 -93.579 cm +BT +/F16 10.909 Tf 363.264 93.579 Td[(32)]TJ +ET +1 0 0 1 374.173 93.579 cm +0 g 0 G +1 0 0 1 -374.173 -93.579 cm +BT +/F16 10.909 Tf 109.907 79.86 Td[(49.)-250(Fundamental)-250(theorem.)-250(Postulate)-250(of)-250(continuity)-695(.)-500(.)-500(.)]TJ +ET +1 0 0 1 363.264 79.86 cm +0 g 0 G +1 0 0 1 -363.264 -79.86 cm +BT +/F16 10.909 Tf 363.264 79.86 Td[(33)]TJ +ET +1 0 0 1 374.173 79.86 cm +0 g 0 G +1 0 0 1 -374.173 -79.86 cm +BT +/F16 10.909 Tf 109.907 66.142 Td[(50.)-250(Extension)-250(of)-250(theorem)-250(to)-250(pencils)-250(of)-250(rays)-250(and)-250(planes)-281(.)]TJ +ET +1 0 0 1 363.264 66.142 cm +0 g 0 G +1 0 0 1 -363.264 -66.142 cm +BT +/F16 10.909 Tf 363.264 66.142 Td[(34)]TJ +ET +1 0 0 1 374.173 66.142 cm +0 g 0 G +1 0 0 1 -280.63 -27.36 cm +0 g 0 G +1 0 0 1 280.63 0 cm +0 g 0 G +endstream +endobj +53 0 obj << +/Type /Page +/Contents 54 0 R +/Resources 52 0 R +/MediaBox [0 0 419.528 595.276] +/Parent 41 0 R +>> endobj +52 0 obj << +/Font << /F16 6 0 R >> +/ProcSet [ /PDF /Text ] +>> endobj +57 0 obj << +/Length 11922 +>> +stream +1 0 0 1 46.771 548.934 cm +0 g 0 G +1 0 0 1 -46.771 -548.934 cm +BT +/F16 10.909 Tf 46.771 548.934 Td[(Contents)-21391(xi)]TJ +ET +1 0 0 1 327.401 548.934 cm +0 g 0 G +1 0 0 1 -327.401 -548.934 cm +BT +/F16 10.909 Tf 63.135 518.175 Td[(51.)-302(Projectiv)-1(e)-267(point-rows)-268(having)-267(a)-268(self-corresponding)]TJ 25.091 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Td[(59.)-250(Cone)-250(of)-250(the)-250(second)-250(order)-697(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)]TJ +ET +1 0 0 1 316.492 394.965 cm +0 g 0 G +1 0 0 1 -316.492 -394.965 cm +BT +/F16 10.909 Tf 316.492 394.965 Td[(37)]TJ +ET +1 0 0 1 327.401 394.965 cm +0 g 0 G +1 0 0 1 -327.401 -394.965 cm +BT +/F16 10.909 Tf 63.135 381.258 Td[(PROBLEMS)-971(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)-500(.)]TJ +ET +1 0 0 1 316.492 381.258 cm +0 g 0 G +1 0 0 1 -316.492 -381.258 cm +BT +/F16 10.909 Tf 316.492 381.258 Td[(38)]TJ +ET +1 0 0 1 327.401 381.258 cm +0 g 0 G +1 0 0 1 -327.401 -381.258 cm +BT +/F16 10.909 Tf 46.771 367.55 Td[(CHAPTER)-208(IV)-207(-)-208(POINT-ROWS)-208(OF)-207(THE)-208(SECOND)-208(ORDER)]TJ +ET +1 0 0 1 316.492 367.55 cm +0 g 0 G +1 0 0 1 -316.492 -367.55 cm +BT +/F16 10.909 Tf 316.492 367.55 Td[(42)]TJ +ET +1 0 0 1 327.401 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892 0 R] +>> endobj +914 0 obj << +/Type /Pages +/Count 6 +/Parent 2382 0 R +/Kids [908 0 R 928 0 R 943 0 R 961 0 R 972 0 R 984 0 R] +>> endobj +992 0 obj << +/Type /Pages +/Count 6 +/Parent 2382 0 R +/Kids [990 0 R 1010 0 R 1030 0 R 1041 0 R 1062 0 R 1070 0 R] +>> endobj +1092 0 obj << +/Type /Pages +/Count 6 +/Parent 2382 0 R +/Kids [1083 0 R 1094 0 R 1103 0 R 1112 0 R 1119 0 R 1122 0 R] +>> endobj +1127 0 obj << +/Type /Pages +/Count 6 +/Parent 2382 0 R +/Kids [1125 0 R 1138 0 R 1146 0 R 1161 0 R 1172 0 R 1185 0 R] +>> endobj +1206 0 obj << +/Type /Pages +/Count 6 +/Parent 2383 0 R +/Kids [1200 0 R 1212 0 R 1223 0 R 1241 0 R 1260 0 R 1283 0 R] +>> endobj +1293 0 obj << +/Type /Pages +/Count 6 +/Parent 2383 0 R +/Kids [1290 0 R 1295 0 R 1298 0 R 1301 0 R 1304 0 R 1316 0 R] +>> endobj +1333 0 obj << +/Type /Pages +/Count 6 +/Parent 2383 0 R +/Kids [1328 0 R 1344 0 R 1350 0 R 1364 0 R 1373 0 R 1387 0 R] +>> endobj +1400 0 obj << +/Type /Pages +/Count 6 +/Parent 2383 0 R +/Kids [1398 0 R 1407 0 R 1413 0 R 1426 0 R 1437 0 R 1450 0 R] +>> endobj +1463 0 obj << +/Type /Pages +/Count 6 +/Parent 2383 0 R +/Kids [1460 0 R 1470 0 R 1479 0 R 1488 0 R 1493 0 R 1505 0 R] +>> endobj +1525 0 obj << +/Type /Pages +/Count 6 +/Parent 2383 0 R +/Kids [1510 0 R 1527 0 R 1530 0 R 1537 0 R 1550 0 R 1564 0 R] +>> endobj +1583 0 obj << +/Type /Pages +/Count 6 +/Parent 2384 0 R +/Kids [1578 0 R 1593 0 R 1598 0 R 1614 0 R 1624 0 R 1637 0 R] +>> endobj +1651 0 obj << +/Type /Pages +/Count 6 +/Parent 2384 0 R +/Kids [1647 0 R 1657 0 R 1670 0 R 1680 0 R 1689 0 R 1703 0 R] +>> endobj +1721 0 obj << +/Type /Pages +/Count 6 +/Parent 2384 0 R +/Kids [1717 0 R 1727 0 R 1733 0 R 1741 0 R 1777 0 R 1818 0 R] +>> endobj +1902 0 obj << +/Type /Pages +/Count 6 +/Parent 2384 0 R +/Kids [1851 0 R 1904 0 R 1960 0 R 2034 0 R 2093 0 R 2167 0 R] +>> endobj +2234 0 obj << +/Type /Pages +/Count 6 +/Parent 2384 0 R +/Kids [2198 0 R 2236 0 R 2239 0 R 2243 0 R 2250 0 R 2253 0 R] +>> endobj +2265 0 obj << +/Type /Pages +/Count 6 +/Parent 2384 0 R +/Kids [2260 0 R 2271 0 R 2280 0 R 2289 0 R 2295 0 R 2305 0 R] +>> endobj +2320 0 obj << +/Type /Pages +/Count 6 +/Parent 2385 0 R +/Kids [2312 0 R 2325 0 R 2331 0 R 2338 0 R 2342 0 R 2350 0 R] +>> endobj +2359 0 obj << +/Type /Pages +/Count 2 +/Parent 2385 0 R +/Kids [2354 0 R 2361 0 R] +>> endobj +2380 0 obj << +/Type /Pages +/Count 36 +/Parent 2386 0 R +/Kids [9 0 R 41 0 R 64 0 R 119 0 R 208 0 R 279 0 R] +>> endobj +2381 0 obj << +/Type /Pages +/Count 36 +/Parent 2386 0 R +/Kids [358 0 R 443 0 R 498 0 R 577 0 R 654 0 R 719 0 R] +>> endobj +2382 0 obj << +/Type /Pages +/Count 36 +/Parent 2386 0 R +/Kids [778 0 R 878 0 R 914 0 R 992 0 R 1092 0 R 1127 0 R] +>> endobj +2383 0 obj << +/Type /Pages +/Count 36 +/Parent 2386 0 R +/Kids [1206 0 R 1293 0 R 1333 0 R 1400 0 R 1463 0 R 1525 0 R] +>> endobj +2384 0 obj << +/Type /Pages +/Count 36 +/Parent 2386 0 R +/Kids [1583 0 R 1651 0 R 1721 0 R 1902 0 R 2234 0 R 2265 0 R] +>> endobj +2385 0 obj << +/Type /Pages +/Count 8 +/Parent 2386 0 R +/Kids [2320 0 R 2359 0 R] +>> endobj +2386 0 obj << +/Type /Pages +/Count 188 +/Kids [2380 0 R 2381 0 R 2382 0 R 2383 0 R 2384 0 R 2385 0 R] +>> endobj +2387 0 obj << +/Type /Outlines +/First 22 0 R +/Last 2268 0 R +/Count 16 +>> endobj +2268 0 obj << +/Title 2269 0 R +/A 2266 0 R +/Parent 2387 0 R +/Prev 2257 0 R +>> endobj +2257 0 obj << +/Title 2258 0 R +/A 2255 0 R +/Parent 2387 0 R +/Prev 2247 0 R +/Next 2268 0 R +>> endobj +2247 0 obj << +/Title 2248 0 R +/A 2245 0 R +/Parent 2387 0 R +/Prev 1738 0 R +/Next 2257 0 R +>> endobj +1738 0 obj << +/Title 1739 0 R +/A 1736 0 R +/Parent 2387 0 R +/Prev 1497 0 R +/Next 2247 0 R +>> endobj +1724 0 obj << +/Title 1725 0 R +/A 1722 0 R +/Parent 1497 0 R +/Prev 1714 0 R +>> endobj +1714 0 obj << +/Title 1715 0 R +/A 1712 0 R +/Parent 1497 0 R +/Prev 1710 0 R +/Next 1724 0 R +>> endobj +1710 0 obj << +/Title 1711 0 R +/A 1708 0 R +/Parent 1497 0 R +/Prev 1700 0 R +/Next 1714 0 R +>> endobj +1700 0 obj << +/Title 1701 0 R +/A 1698 0 R +/Parent 1497 0 R +/Prev 1696 0 R +/Next 1710 0 R +>> endobj +1696 0 obj << +/Title 1697 0 R +/A 1694 0 R +/Parent 1497 0 R +/Prev 1686 0 R +/Next 1700 0 R +>> endobj +1686 0 obj << +/Title 1687 0 R +/A 1684 0 R +/Parent 1497 0 R +/Prev 1677 0 R +/Next 1696 0 R +>> endobj +1677 0 obj << +/Title 1678 0 R +/A 1675 0 R +/Parent 1497 0 R +/Prev 1667 0 R +/Next 1686 0 R +>> endobj +1667 0 obj << +/Title 1668 0 R +/A 1665 0 R +/Parent 1497 0 R +/Prev 1663 0 R +/Next 1677 0 R +>> endobj +1663 0 obj << +/Title 1664 0 R +/A 1661 0 R +/Parent 1497 0 R +/Prev 1654 0 R +/Next 1667 0 R +>> endobj +1654 0 obj << +/Title 1655 0 R +/A 1652 0 R +/Parent 1497 0 R +/Prev 1644 0 R +/Next 1663 0 R +>> endobj +1644 0 obj << +/Title 1645 0 R +/A 1642 0 R +/Parent 1497 0 R +/Prev 1634 0 R +/Next 1654 0 R +>> endobj +1634 0 obj << +/Title 1635 0 R +/A 1632 0 R +/Parent 1497 0 R +/Prev 1630 0 R +/Next 1644 0 R +>> endobj +1630 0 obj << +/Title 1631 0 R +/A 1628 0 R +/Parent 1497 0 R +/Prev 1621 0 R +/Next 1634 0 R +>> endobj +1621 0 obj << +/Title 1622 0 R +/A 1619 0 R +/Parent 1497 0 R +/Prev 1611 0 R +/Next 1630 0 R +>> endobj +1611 0 obj << +/Title 1612 0 R +/A 1609 0 R +/Parent 1497 0 R +/Prev 1607 0 R +/Next 1621 0 R +>> endobj +1607 0 obj << +/Title 1608 0 R +/A 1605 0 R +/Parent 1497 0 R +/Prev 1590 0 R +/Next 1611 0 R +>> endobj +1590 0 obj << +/Title 1591 0 R +/A 1588 0 R +/Parent 1497 0 R +/Prev 1586 0 R +/Next 1607 0 R +>> endobj +1586 0 obj << +/Title 1587 0 R +/A 1584 0 R +/Parent 1497 0 R +/Prev 1575 0 R +/Next 1590 0 R +>> endobj +1575 0 obj << +/Title 1576 0 R +/A 1573 0 R +/Parent 1497 0 R +/Prev 1571 0 R +/Next 1586 0 R +>> endobj +1571 0 obj << +/Title 1572 0 R +/A 1569 0 R +/Parent 1497 0 R +/Prev 1561 0 R +/Next 1575 0 R +>> endobj +1561 0 obj << +/Title 1562 0 R +/A 1559 0 R +/Parent 1497 0 R +/Prev 1557 0 R +/Next 1571 0 R +>> endobj +1557 0 obj << +/Title 1558 0 R +/A 1555 0 R +/Parent 1497 0 R +/Prev 1547 0 R +/Next 1561 0 R +>> endobj +1547 0 obj << +/Title 1548 0 R +/A 1545 0 R +/Parent 1497 0 R +/Prev 1543 0 R +/Next 1557 0 R +>> endobj +1543 0 obj << +/Title 1544 0 R +/A 1541 0 R +/Parent 1497 0 R +/Prev 1534 0 R +/Next 1547 0 R +>> endobj +1534 0 obj << +/Title 1535 0 R +/A 1532 0 R +/Parent 1497 0 R +/Prev 1501 0 R +/Next 1543 0 R +>> endobj +1501 0 obj << +/Title 1502 0 R +/A 1499 0 R +/Parent 1497 0 R +/Next 1534 0 R +>> endobj +1497 0 obj << +/Title 1498 0 R +/A 1495 0 R +/Parent 2387 0 R +/Prev 1308 0 R +/Next 1738 0 R +/First 1501 0 R +/Last 1724 0 R +/Count -26 +>> endobj +1485 0 obj << +/Title 1486 0 R +/A 1483 0 R +/Parent 1308 0 R +/Prev 1476 0 R +>> endobj +1476 0 obj << +/Title 1477 0 R +/A 1474 0 R +/Parent 1308 0 R +/Prev 1467 0 R +/Next 1485 0 R +>> endobj +1467 0 obj << +/Title 1468 0 R +/A 1465 0 R +/Parent 1308 0 R +/Prev 1457 0 R +/Next 1476 0 R +>> endobj +1457 0 obj << +/Title 1458 0 R +/A 1455 0 R +/Parent 1308 0 R +/Prev 1447 0 R +/Next 1467 0 R +>> endobj +1447 0 obj << +/Title 1448 0 R +/A 1445 0 R +/Parent 1308 0 R +/Prev 1443 0 R +/Next 1457 0 R +>> endobj +1443 0 obj << +/Title 1444 0 R +/A 1441 0 R +/Parent 1308 0 R +/Prev 1433 0 R +/Next 1447 0 R +>> endobj +1433 0 obj << +/Title 1434 0 R +/A 1431 0 R +/Parent 1308 0 R +/Prev 1423 0 R +/Next 1443 0 R +>> endobj +1423 0 obj << +/Title 1424 0 R +/A 1421 0 R +/Parent 1308 0 R +/Prev 1418 0 R +/Next 1433 0 R +>> endobj +1418 0 obj << +/Title 1419 0 R +/A 1416 0 R +/Parent 1308 0 R +/Prev 1404 0 R +/Next 1423 0 R +>> endobj +1404 0 obj << +/Title 1405 0 R +/A 1402 0 R +/Parent 1308 0 R +/Prev 1395 0 R +/Next 1418 0 R +>> endobj +1395 0 obj << +/Title 1396 0 R +/A 1393 0 R +/Parent 1308 0 R +/Prev 1384 0 R +/Next 1404 0 R +>> endobj +1384 0 obj << +/Title 1385 0 R +/A 1382 0 R +/Parent 1308 0 R +/Prev 1380 0 R +/Next 1395 0 R +>> endobj +1380 0 obj << +/Title 1381 0 R +/A 1378 0 R +/Parent 1308 0 R +/Prev 1370 0 R +/Next 1384 0 R +>> endobj +1370 0 obj << +/Title 1371 0 R +/A 1368 0 R +/Parent 1308 0 R +/Prev 1361 0 R +/Next 1380 0 R +>> endobj +1361 0 obj << +/Title 1362 0 R +/A 1359 0 R +/Parent 1308 0 R +/Prev 1356 0 R +/Next 1370 0 R +>> endobj +1356 0 obj << +/Title 1357 0 R +/A 1354 0 R +/Parent 1308 0 R +/Prev 1341 0 R +/Next 1361 0 R +>> endobj +1341 0 obj << +/Title 1342 0 R +/A 1339 0 R +/Parent 1308 0 R +/Prev 1336 0 R +/Next 1356 0 R +>> endobj +1336 0 obj << +/Title 1337 0 R +/A 1334 0 R +/Parent 1308 0 R +/Prev 1325 0 R +/Next 1341 0 R +>> endobj +1325 0 obj << +/Title 1326 0 R +/A 1323 0 R +/Parent 1308 0 R +/Prev 1321 0 R +/Next 1336 0 R +>> endobj +1321 0 obj << +/Title 1322 0 R +/A 1319 0 R +/Parent 1308 0 R +/Prev 1312 0 R +/Next 1325 0 R +>> endobj +1312 0 obj << +/Title 1313 0 R +/A 1310 0 R +/Parent 1308 0 R +/Next 1321 0 R +>> endobj +1308 0 obj << +/Title 1309 0 R +/A 1306 0 R +/Parent 2387 0 R +/Prev 1130 0 R +/Next 1497 0 R +/First 1312 0 R +/Last 1485 0 R +/Count -21 +>> endobj +1280 0 obj << +/Title 1281 0 R +/A 1278 0 R +/Parent 1130 0 R +/Prev 1276 0 R +>> endobj +1276 0 obj << +/Title 1277 0 R +/A 1274 0 R +/Parent 1130 0 R +/Prev 1272 0 R +/Next 1280 0 R +>> endobj +1272 0 obj << +/Title 1273 0 R +/A 1270 0 R +/Parent 1130 0 R +/Prev 1268 0 R +/Next 1276 0 R +>> endobj +1268 0 obj << +/Title 1269 0 R +/A 1266 0 R +/Parent 1130 0 R +/Prev 1257 0 R +/Next 1272 0 R +>> endobj +1257 0 obj << +/Title 1258 0 R +/A 1255 0 R +/Parent 1130 0 R +/Prev 1253 0 R +/Next 1268 0 R +>> endobj +1253 0 obj << +/Title 1254 0 R +/A 1251 0 R +/Parent 1130 0 R +/Prev 1249 0 R +/Next 1257 0 R +>> endobj +1249 0 obj << +/Title 1250 0 R +/A 1247 0 R +/Parent 1130 0 R +/Prev 1238 0 R +/Next 1253 0 R +>> endobj +1238 0 obj << +/Title 1239 0 R +/A 1236 0 R +/Parent 1130 0 R +/Prev 1234 0 R +/Next 1249 0 R +>> endobj +1234 0 obj << +/Title 1235 0 R +/A 1232 0 R +/Parent 1130 0 R +/Prev 1230 0 R +/Next 1238 0 R +>> endobj +1230 0 obj << +/Title 1231 0 R +/A 1228 0 R +/Parent 1130 0 R +/Prev 1220 0 R +/Next 1234 0 R +>> endobj +1220 0 obj << +/Title 1221 0 R +/A 1218 0 R +/Parent 1130 0 R +/Prev 1209 0 R +/Next 1230 0 R +>> endobj +1209 0 obj << +/Title 1210 0 R +/A 1207 0 R +/Parent 1130 0 R +/Prev 1197 0 R +/Next 1220 0 R +>> endobj +1197 0 obj << +/Title 1198 0 R +/A 1195 0 R +/Parent 1130 0 R +/Prev 1192 0 R +/Next 1209 0 R +>> endobj +1192 0 obj << +/Title 1193 0 R +/A 1190 0 R +/Parent 1130 0 R +/Prev 1182 0 R +/Next 1197 0 R +>> endobj +1182 0 obj << +/Title 1183 0 R +/A 1180 0 R +/Parent 1130 0 R +/Prev 1178 0 R +/Next 1192 0 R +>> endobj +1178 0 obj << +/Title 1179 0 R +/A 1176 0 R +/Parent 1130 0 R +/Prev 1169 0 R +/Next 1182 0 R +>> endobj +1169 0 obj << +/Title 1170 0 R +/A 1167 0 R +/Parent 1130 0 R +/Prev 1158 0 R +/Next 1178 0 R +>> endobj +1158 0 obj << +/Title 1159 0 R +/A 1156 0 R +/Parent 1130 0 R +/Prev 1153 0 R +/Next 1169 0 R +>> endobj +1153 0 obj << +/Title 1154 0 R +/A 1151 0 R +/Parent 1130 0 R +/Prev 1143 0 R +/Next 1158 0 R +>> endobj +1143 0 obj << +/Title 1144 0 R +/A 1141 0 R +/Parent 1130 0 R +/Prev 1134 0 R +/Next 1153 0 R +>> endobj +1134 0 obj << +/Title 1135 0 R +/A 1132 0 R +/Parent 1130 0 R +/Next 1143 0 R +>> endobj +1130 0 obj << +/Title 1131 0 R +/A 1128 0 R +/Parent 2387 0 R +/Prev 995 0 R +/Next 1308 0 R +/First 1134 0 R +/Last 1280 0 R +/Count -21 +>> endobj +1109 0 obj << +/Title 1110 0 R +/A 1107 0 R +/Parent 995 0 R +/Prev 1099 0 R +>> endobj +1099 0 obj << +/Title 1100 0 R +/A 1097 0 R +/Parent 995 0 R +/Prev 1079 0 R +/Next 1109 0 R +>> endobj +1079 0 obj << +/Title 1080 0 R +/A 1077 0 R +/Parent 995 0 R +/Prev 1075 0 R +/Next 1099 0 R +>> endobj +1075 0 obj << +/Title 1076 0 R +/A 1073 0 R +/Parent 995 0 R +/Prev 1059 0 R +/Next 1079 0 R +>> endobj +1059 0 obj << +/Title 1060 0 R +/A 1057 0 R +/Parent 995 0 R +/Prev 1054 0 R +/Next 1075 0 R +>> endobj +1054 0 obj << +/Title 1055 0 R +/A 1052 0 R +/Parent 995 0 R +/Prev 1050 0 R +/Next 1059 0 R +>> endobj +1050 0 obj << +/Title 1051 0 R +/A 1048 0 R +/Parent 995 0 R +/Prev 1046 0 R +/Next 1054 0 R +>> endobj +1046 0 obj << +/Title 1047 0 R +/A 1044 0 R +/Parent 995 0 R +/Prev 1038 0 R +/Next 1050 0 R +>> endobj +1038 0 obj << +/Title 1039 0 R +/A 1036 0 R +/Parent 995 0 R +/Prev 1026 0 R +/Next 1046 0 R +>> endobj +1026 0 obj << +/Title 1027 0 R +/A 1024 0 R +/Parent 995 0 R +/Prev 1022 0 R +/Next 1038 0 R +>> endobj +1022 0 obj << +/Title 1023 0 R +/A 1020 0 R +/Parent 995 0 R +/Prev 1018 0 R +/Next 1026 0 R +>> endobj +1018 0 obj << +/Title 1019 0 R +/A 1016 0 R +/Parent 995 0 R +/Prev 1007 0 R +/Next 1022 0 R +>> endobj +1007 0 obj << +/Title 1008 0 R +/A 1005 0 R +/Parent 995 0 R +/Prev 1003 0 R +/Next 1018 0 R +>> endobj +1003 0 obj << +/Title 1004 0 R +/A 1001 0 R +/Parent 995 0 R +/Prev 999 0 R +/Next 1007 0 R +>> endobj +999 0 obj << +/Title 1000 0 R +/A 997 0 R +/Parent 995 0 R +/Next 1003 0 R +>> endobj +995 0 obj << +/Title 996 0 R +/A 993 0 R +/Parent 2387 0 R +/Prev 896 0 R +/Next 1130 0 R +/First 999 0 R +/Last 1109 0 R +/Count -15 +>> endobj +981 0 obj << +/Title 982 0 R +/A 979 0 R +/Parent 896 0 R +/Prev 977 0 R +>> endobj +977 0 obj << +/Title 978 0 R +/A 975 0 R +/Parent 896 0 R +/Prev 969 0 R +/Next 981 0 R +>> endobj +969 0 obj << +/Title 970 0 R +/A 967 0 R +/Parent 896 0 R +/Prev 958 0 R +/Next 977 0 R +>> endobj +958 0 obj << +/Title 959 0 R +/A 956 0 R +/Parent 896 0 R +/Prev 954 0 R +/Next 969 0 R +>> endobj +954 0 obj << +/Title 955 0 R +/A 952 0 R +/Parent 896 0 R +/Prev 950 0 R +/Next 958 0 R +>> endobj +950 0 obj << +/Title 951 0 R +/A 948 0 R +/Parent 896 0 R +/Prev 940 0 R +/Next 954 0 R +>> endobj +940 0 obj << +/Title 941 0 R +/A 938 0 R +/Parent 896 0 R +/Prev 935 0 R +/Next 950 0 R +>> endobj +935 0 obj << +/Title 936 0 R +/A 933 0 R +/Parent 896 0 R +/Prev 925 0 R +/Next 940 0 R +>> endobj +925 0 obj << +/Title 926 0 R +/A 923 0 R +/Parent 896 0 R +/Prev 921 0 R +/Next 935 0 R +>> endobj +921 0 obj << +/Title 922 0 R +/A 919 0 R +/Parent 896 0 R +/Prev 917 0 R +/Next 925 0 R +>> endobj +917 0 obj << +/Title 918 0 R +/A 915 0 R +/Parent 896 0 R +/Prev 905 0 R +/Next 921 0 R +>> endobj +905 0 obj << +/Title 906 0 R +/A 903 0 R +/Parent 896 0 R +/Prev 900 0 R +/Next 917 0 R +>> endobj +900 0 obj << +/Title 901 0 R +/A 898 0 R +/Parent 896 0 R +/Next 905 0 R +>> endobj +896 0 obj << +/Title 897 0 R +/A 894 0 R +/Parent 2387 0 R +/Prev 752 0 R +/Next 995 0 R +/First 900 0 R +/Last 981 0 R +/Count -13 +>> endobj +865 0 obj << +/Title 866 0 R +/A 863 0 R +/Parent 752 0 R +/Prev 861 0 R +>> endobj +861 0 obj << +/Title 862 0 R +/A 859 0 R +/Parent 752 0 R +/Prev 857 0 R +/Next 865 0 R +>> endobj +857 0 obj << +/Title 858 0 R +/A 855 0 R +/Parent 752 0 R +/Prev 845 0 R +/Next 861 0 R +>> endobj +845 0 obj << +/Title 846 0 R +/A 843 0 R +/Parent 752 0 R +/Prev 841 0 R +/Next 857 0 R +>> endobj +841 0 obj << +/Title 842 0 R +/A 839 0 R +/Parent 752 0 R +/Prev 837 0 R +/Next 845 0 R +>> endobj +837 0 obj << +/Title 838 0 R +/A 835 0 R +/Parent 752 0 R +/Prev 833 0 R +/Next 841 0 R +>> endobj +833 0 obj << +/Title 834 0 R +/A 831 0 R +/Parent 752 0 R +/Prev 821 0 R +/Next 837 0 R +>> endobj +821 0 obj << +/Title 822 0 R +/A 819 0 R +/Parent 752 0 R +/Prev 816 0 R +/Next 833 0 R +>> endobj +816 0 obj << +/Title 817 0 R +/A 814 0 R +/Parent 752 0 R +/Prev 811 0 R +/Next 821 0 R +>> endobj +811 0 obj << +/Title 812 0 R +/A 809 0 R +/Parent 752 0 R +/Prev 801 0 R +/Next 816 0 R +>> endobj +801 0 obj << +/Title 802 0 R +/A 799 0 R +/Parent 752 0 R +/Prev 796 0 R +/Next 811 0 R +>> endobj +796 0 obj << +/Title 797 0 R +/A 794 0 R +/Parent 752 0 R +/Prev 785 0 R +/Next 801 0 R +>> endobj +785 0 obj << +/Title 786 0 R +/A 783 0 R +/Parent 752 0 R +/Prev 781 0 R +/Next 796 0 R +>> endobj +781 0 obj << +/Title 782 0 R +/A 779 0 R +/Parent 752 0 R +/Prev 771 0 R +/Next 785 0 R +>> endobj +771 0 obj << +/Title 772 0 R +/A 769 0 R +/Parent 752 0 R +/Prev 761 0 R +/Next 781 0 R +>> endobj +761 0 obj << +/Title 762 0 R +/A 759 0 R +/Parent 752 0 R +/Prev 756 0 R +/Next 771 0 R +>> endobj +756 0 obj << +/Title 757 0 R +/A 754 0 R +/Parent 752 0 R +/Next 761 0 R +>> endobj +752 0 obj << +/Title 753 0 R +/A 750 0 R +/Parent 2387 0 R +/Prev 586 0 R +/Next 896 0 R +/First 756 0 R +/Last 865 0 R +/Count -17 +>> endobj +740 0 obj << +/Title 741 0 R +/A 738 0 R +/Parent 586 0 R +/Prev 731 0 R +>> endobj +731 0 obj << +/Title 732 0 R +/A 729 0 R +/Parent 586 0 R +/Prev 726 0 R +/Next 740 0 R +>> endobj +726 0 obj << +/Title 727 0 R +/A 724 0 R +/Parent 586 0 R +/Prev 709 0 R +/Next 731 0 R +>> endobj +709 0 obj << +/Title 710 0 R +/A 707 0 R +/Parent 586 0 R +/Prev 702 0 R +/Next 726 0 R +>> endobj +702 0 obj << +/Title 703 0 R +/A 700 0 R +/Parent 586 0 R +/Prev 692 0 R +/Next 709 0 R +>> endobj +692 0 obj << +/Title 693 0 R +/A 690 0 R +/Parent 586 0 R +/Prev 681 0 R +/Next 702 0 R +>> endobj +681 0 obj << +/Title 682 0 R +/A 679 0 R +/Parent 586 0 R +/Prev 672 0 R +/Next 692 0 R +>> endobj +672 0 obj << +/Title 673 0 R +/A 670 0 R +/Parent 586 0 R +/Prev 668 0 R +/Next 681 0 R +>> endobj +668 0 obj << +/Title 669 0 R +/A 666 0 R 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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: American + +Character set encoding: UTF-8 +--> + +<!DOCTYPE TEI.2 SYSTEM "http://www.gutenberg.org/tei/marcello/0.4/dtd/pgtei.dtd"> + +<TEI.2 lang="en-us"> +<teiHeader> + <fileDesc> + <titleStmt> + <title>An Elementary Course in Synthetic Projective Geometry</title> + <author>Lehmer, Derrick Norman</author> + </titleStmt> + <editionStmt> + <edition n="1">Edition 1</edition> + </editionStmt> + <publicationStmt> + <publisher>Project Gutenberg</publisher> + <date value="2005-11-4">November 4, 2005</date> + <idno type="etext-no">17001</idno> + <idno type="DPid">projectID3fa113afbb55e</idno> + <availability> + <p>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 online at www.gutenberg.org/license</p> + </availability> + </publicationStmt> + + <sourceDesc> + <bibl> + <title>An Elementary Course in Synthetic Projective Geometry</title> + <author>Lehmer, Derrick Norman</author> + <imprint> + <pubPlace>Boston</pubPlace> + <publisher>Ginn and Company</publisher> + <date>1917</date> + </imprint> + </bibl> + </sourceDesc> + </fileDesc> + + <encodingDesc> + <classDecl> + <taxonomy id="lc"> + <bibl> + <title>Library of Congress Classification</title> + </bibl> + </taxonomy> + </classDecl> + </encodingDesc> + <profileDesc> + <langUsage> + <language id="en-us">United States English</language> + </langUsage> + <textClass> + <classCode scheme="lc"> + *** <!-- LoC Class (PR, PQ, ...) --> + </classCode> + <keywords> + <list> + <!-- <item></item> any keywords for PG search engine --> + </list> + </keywords> + </textClass> + </profileDesc> + <revisionDesc> + <change> + <date value="2005-11">November 2005</date> + <respStmt> + <name>Joshua Hutchinson, </name> + <name>Cornell University, </name> + <name>Online Distributed Proofreading Team</name> + </respStmt> + <item>Project Gutenberg Edition</item> + </change> + <change> + <date value="2006-6">June 2006</date> + <respStmt> + <name>Joshua Hutchinson</name> + </respStmt> + <item>Added PGHeader/PGFooter.</item> + </change> + </revisionDesc> +</teiHeader> + +<pgExtensions> + <pgStyleSheet> + figure { text-align: center; page-float: 'htb' } + .w95 { } + @media pdf { + .w95 { width: 95% } + } + </pgStyleSheet> +</pgExtensions> + +<text> +<front> + +<div> +<divGen type="pgheader" /> +</div> + +<div> +<divGen type="encodingDesc" /> +</div> + +<div rend="page-break-before: right"> +<divGen type="titlepage" /> +</div> + +<div rend="page-break-before: right"> +<index index="toc" /><index index="pdf" /> +<pb n="iii" /><anchor id="Pgiii" /> +<head>Preface</head> + +<p>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.</p> + +<p>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.</p> + +<p>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 <hi rend="font-style: italic">pencil of rays</hi> instead of a <hi rend="font-style: italic">bundle of rays</hi>, as later +writers seem inclined to do. For a point considered +<pb n="iv" /><anchor id="Pgiv" /> +as made up of all the lines and planes through it he +has ventured to use the term <hi rend="font-style: italic">point system</hi>, as being +the natural dualization of the usual term <hi rend="font-style: italic">plane system</hi>. +He has also rejected the term <hi rend="font-style: italic">foci of an involution</hi>, and +has not used the customary terms for classifying involutions—<hi rend="font-style: italic">hyperbolic +involution</hi>, <hi rend="font-style: italic">elliptic involution</hi> and +<hi rend="font-style: italic">parabolic involution</hi>. 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.</p> + +<p>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.</p> + +<p>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 +<pb n="v" /><anchor id="Pgv" /> +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.</p> + +<p>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.</p> + +<p>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.</p> + +<pb n="vi" /><anchor id="Pgvi" /> + +<p>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.</p> + +<p rend="text-align: right">D. N. LEHMER</p> + +<p><hi rend="font-variant: small-caps">Berkeley, California</hi></p> + +</div> + +<div rend="page-break-before: right"> + <index index="toc" /><index index="pdf" /> + <head>Contents</head> + <divGen type="toc" /> +</div> + +</front> + +<body> +<div rend="page-break-before: right"> +<pb n="1" /><anchor id="Pg1" /> +<index index="toc" /><index index="pdf" /> +<head>CHAPTER I - ONE-TO-ONE CORRESPONDENCE</head> +<p></p> + +<div> +<index index="toc" level1="1. Definition of one-to-one correspondence" /><index index="pdf" /> + +<head></head><p><anchor id="p1" /><hi rend="font-weight: bold">1. Definition of one-to-one correspondence.</hi> +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 <hi rend="font-style: italic">one-to-one correspondence</hi> 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 +<pb n="2" /><anchor id="Pg2" /> +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.</p></div> + +<div> +<index index="toc" level1="2. Consequences of one-to-one correspondence" /><index index="pdf" /> +<head></head><p><anchor id="p2" /><hi rend="font-weight: bold">2. Consequences of one-to-one correspondence.</hi> +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.</p></div> + +<div> +<index index="toc" level1="3. Applications in mathematics" /><index index="pdf" /> +<head></head><p><anchor id="p3" /><hi rend="font-weight: bold">3.</hi> 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.</p> + +<pb n="3" /><anchor id="Pg3" /> +</div> + +<div> +<index index="toc" level1="4. One-to-one correspondence and enumeration" /><index index="pdf" /> +<head></head> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image01.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 1</head> +<figDesc>Figure 1</figDesc> +</figure></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image02.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 2</head> +<figDesc>Figure 2</figDesc> +</figure></p> + +<p><anchor id="p4" /><hi rend="font-weight: bold">4. One-to-one correspondence and enumeration.</hi> 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 <hi rend="font-style: italic">AB</hi> and <hi rend="font-style: italic">A'B'</hi>. Join <hi rend="font-style: italic">AA'</hi> and <hi rend="font-style: italic">BB'</hi>, +and let these joining lines meet in <hi rend="font-style: italic">S</hi>. For every point <hi rend="font-style: italic">C</hi> +on <hi rend="font-style: italic">AB</hi> a point <hi rend="font-style: italic">C'</hi> may be found +on <hi rend="font-style: italic">A'B'</hi> by joining <hi rend="font-style: italic">C</hi> to <hi rend="font-style: italic">S</hi> and +noting the point <hi rend="font-style: italic">C'</hi> where <hi rend="font-style: italic">CS</hi> +meets <hi rend="font-style: italic">A'B'</hi>. Similarly, a point <hi rend="font-style: italic">C</hi> +may be found on <hi rend="font-style: italic">AB</hi> for any +point <hi rend="font-style: italic">C'</hi> on <hi rend="font-style: italic">A'B'</hi>. The correspondence +is clearly one-to-one, +but it would be absurd to infer +from this that there were just +as many points on <hi rend="font-style: italic">AB</hi> as on <hi rend="font-style: italic">A'B'</hi>. In fact, it would +be just as reasonable to infer that there were twice as +many points on <hi rend="font-style: italic">A'B'</hi> as on <hi rend="font-style: italic">AB</hi>. For if we bend <hi rend="font-style: italic">A'B'</hi> +into a circle with center at <hi rend="font-style: italic">S</hi> (Fig. 2), we see that for +every point <hi rend="font-style: italic">C</hi> on <hi rend="font-style: italic">AB</hi> there are two points on <hi rend="font-style: italic">A'B'</hi>. Thus +<pb n="4" /><anchor id="Pg4" /> +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.</p></div> + +<div> +<index index="toc" level1="5. Correspondence between a part and the whole" /><index index="pdf" /> +<head></head><p><anchor id="p5" /><hi rend="font-weight: bold">5. Correspondence between a part and the whole of an +infinite assemblage.</hi> 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.</p></div> + +<div> +<index index="toc" level1="6. Infinitely distant point" /><index index="pdf" /> +<head></head><p><anchor id="p6" /><hi rend="font-weight: bold">6. Infinitely distant point.</hi> 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 <hi rend="font-style: italic">C</hi> on the line <hi rend="font-style: italic">AB</hi> +there is a line <hi rend="font-style: italic">SC</hi> through <hi rend="font-style: italic">S</hi>. We must assume the line +<hi rend="font-style: italic">AB</hi> extended indefinitely in both directions, however, if +we are to have a point on it for every line through <hi rend="font-style: italic">S</hi>; +and even with this extension there is one line through +<hi rend="font-style: italic">S</hi>, according to Euclid's postulate, which does not meet +the line <hi rend="font-style: italic">AB</hi> and which therefore has no point on +<hi rend="font-style: italic">AB</hi> to correspond to it. In order to smooth out this +<pb n="5" /><anchor id="Pg5" />discrepancy we are accustomed to assume the existence +of an <hi rend="font-style: italic">infinitely distant</hi> point on the line <hi rend="font-style: italic">AB</hi> and to assign +this point as the corresponding point of the exceptional +line of <hi rend="font-style: italic">S</hi>. 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 +<hi rend="font-style: italic">point-row</hi>, and the totality of lines through a point a +<hi rend="font-style: italic">pencil of rays</hi>, we say that the point-row and the pencil +related as above are in <hi rend="font-style: italic">perspective position</hi>, or that they +are <hi rend="font-style: italic">perspectively related</hi>.</p></div> + +<div> +<index index="toc" level1="7. Axial pencil; fundamental forms" /><index index="pdf" /> +<head></head><p><anchor id="p7" /><hi rend="font-weight: bold">7. Axial pencil; fundamental forms.</hi> 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 +<hi rend="font-style: italic">axial pencil</hi>, and the three assemblages—the point-row, +the pencil of rays, and the axial pencil—are called +<hi rend="font-style: italic">fundamental forms</hi>. 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.</p></div> + +<div> +<index index="toc" level1="8. Perspective position" /><index index="pdf" /> +<head></head><p><anchor id="p8" /><hi rend="font-weight: bold">8. Perspective position.</hi> 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 +<pb n="6" /><anchor id="Pg6" /> +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.</p></div> + +<div> +<index index="toc" level1="9. Projective relation" /><index index="pdf" /> +<head></head><p><anchor id="p9" /><hi rend="font-weight: bold">9. Projective relation.</hi> 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, <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">S</hi>, then a point-row <hi rend="font-style: italic">a</hi> perspective +to <hi rend="font-style: italic">A</hi> will be in one-to-one correspondence with +a point-row <hi rend="font-style: italic">b</hi> perspective to <hi rend="font-style: italic">B</hi>, but corresponding points +will not, in general, lie on lines which all pass through +a point. Two such point-rows are said to be <hi rend="font-style: italic">projectively +related</hi>, 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 <hi rend="font-style: italic">continuous</hi>; 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 +<pb n="7" /><anchor id="Pg7" /> +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."</p></div> + +<div> +<index index="toc" level1="10. Infinity-to-one correspondence" /><index index="pdf" /> +<head></head><p><anchor id="p10" /><hi rend="font-weight: bold">10. Infinity-to-one correspondence.</hi> 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 <hi rend="font-style: italic">m</hi> points on one and <hi rend="font-style: italic">n</hi> points on the other. The +number of lines joining the <hi rend="font-style: italic">m</hi> points of one to the +<hi rend="font-style: italic">n</hi> points jof the other is clearly <hi rend="font-style: italic">mn</hi>. 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]<hi rend="vertical-align: super">2</hi>. 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.</p></div> + +<div> +<index index="toc" level1="11. Infinitudes of different orders" /><index index="pdf" /> +<head></head><p><anchor id="p11" /><hi rend="font-weight: bold">11. Infinitudes of different orders.</hi> 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 +<pb n="8" /><anchor id="Pg8" /> +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:</p></div> + +<div> +<index index="toc" level1="12. Points in a plane" /><index index="pdf" /> +<head></head><p><anchor id="p12" /><hi rend="font-weight: bold">12.</hi> 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.</p></div> + +<div> +<index index="toc" level1="13. Lines through a point" /><index index="pdf" /> +<head></head><p><anchor id="p13" /><hi rend="font-weight: bold">13.</hi> 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. <hi rend="font-style: italic">Thus the infinitude of lines through a +point in space is of the second order.</hi></p></div> + +<div> +<index index="toc" level1="14. Planes through a point" /><index index="pdf" /> +<head></head><p><anchor id="p14" /><hi rend="font-weight: bold">14.</hi> 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 <hi rend="font-style: italic">the infinitude of +planes through a point in space is of the second order.</hi></p></div> + +<div> +<index index="toc" level1="15. Lines in a plane" /><index index="pdf" /> +<head></head><p><anchor id="p15" /><hi rend="font-weight: bold">15.</hi> If to each plane through a point in space we +make correspond the line in which it intersects a given +plane, we see that <hi rend="font-style: italic">the infinitude of lines in a plane is of +the second order.</hi> 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 <hi rend="font-style: italic">S</hi> +not in the plane. Join any point <hi rend="font-style: italic">M</hi> of the plane to <hi rend="font-style: italic">S</hi>. +Through <hi rend="font-style: italic">S</hi> draw a plane at right angles to <hi rend="font-style: italic">MS</hi>. This +meets the given plane in a line <hi rend="font-style: italic">m</hi> which may be taken as +corresponding to the point <hi rend="font-style: italic">M</hi>. Another very important +<pb n="9" /><anchor id="Pg9" /> +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.</p></div> + +<div> +<index index="toc" level1="16. Plane system and point system" /><index index="pdf" /> +<head></head><p><anchor id="p16" /><hi rend="font-weight: bold">16. Plane system and point system.</hi> The plane, considered +as made up of the points and lines in it, is called +a <hi rend="font-style: italic">plane system</hi> 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 <hi rend="font-style: italic">point system</hi> +and is also a fundamental form of the second order.</p></div> + +<div> +<index index="toc" level1="17. Planes in space" /><index index="pdf" /> +<head></head><p><anchor id="p17" /><hi rend="font-weight: bold">17.</hi> If now we take three lines in space all lying in +different planes, and select <hi rend="font-style: italic">l</hi> points on the first, <hi rend="font-style: italic">m</hi> points +on the second, and <hi rend="font-style: italic">n</hi> points on the third, then the total +number of planes passing through one of the selected +points on each line will be <hi rend="font-style: italic">lmn</hi>. 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]<hi rend="vertical-align: super">3</hi>, and to call it an infinitude of the <hi rend="font-style: italic">third</hi> order. But +it is easily seen that every plane in space is included in +this totality, so that <hi rend="font-style: italic">the totality of planes in space is an +infinitude of the third order.</hi></p></div> + +<div> +<index index="toc" level1="18. Points of space" /><index index="pdf" /> +<head></head><p><anchor id="p18" /><hi rend="font-weight: bold">18.</hi> 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 <hi rend="font-style: italic">the totality of points +in space is an infinitude of the third order.</hi></p></div> + +<div> +<index index="toc" level1="19. Space system" /><index index="pdf" /> +<head></head><p><anchor id="p19" /><hi rend="font-weight: bold">19. Space system.</hi> Space of three dimensions, considered +as made up of all its planes and points, is then +a fundamental form of the <hi rend="font-style: italic">third</hi> order, which we shall +call a <hi rend="font-style: italic">space system.</hi></p> + +<pb n="10" /><anchor id="Pg10" /></div> + +<div> +<index index="toc" level1="20. Lines in space" /><index index="pdf" /> +<head></head><p><anchor id="p20" /><hi rend="font-weight: bold">20. Lines in space.</hi> 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. <hi rend="font-style: italic">The +totality of lines in space gives, then, a fundamental form +of the fourth order.</hi></p></div> + +<div> +<index index="toc" level1="21. Correspondence between points and numbers" /><index index="pdf" /> +<head></head><p><anchor id="p21" /><hi rend="font-weight: bold">21. Correspondence between points and numbers.</hi> 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 +<pb n="11" /><anchor id="Pg11" /> +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 <hi rend="font-style: italic">continuous</hi> 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.</p></div> + +<div> +<index index="toc" level1="22. Elements at infinity" /><index index="pdf" /> +<head></head><p><anchor id="p22" /><hi rend="font-weight: bold">22. Elements at infinity.</hi> 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, +<pb n="12" /><anchor id="Pg12" /> +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.</p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>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?</p> + +<p>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?</p> + +<p>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.)</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<pb n="13" /><anchor id="Pg13" /> + +<p>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.</p> + +<p>8. Set up a one-to-one correspondence between the series +of numbers <hi rend="font-style: italic">1</hi>, <hi rend="font-style: italic">2</hi>, <hi rend="font-style: italic">3</hi>, <hi rend="font-style: italic">4</hi>, ... and the series of even numbers +<hi rend="font-style: italic">2</hi>, <hi rend="font-style: italic">4</hi>, <hi rend="font-style: italic">6</hi>, <hi rend="font-style: italic">8</hi> .... Are we justified in saying that there are just +as many even numbers as there are numbers altogether?</p> + +<p>9. Is the axiom "The whole is greater than one of its +parts" applicable to infinite assemblages?</p> + +<p>10. Make out a classified list of all the infinitudes of the +first, second, third, and fourth orders mentioned in this +chapter.</p> + +</div> +</div> + +<pb n="14" /><anchor id="Pg14" /> +<div rend="page-break-before: always"> +<index index="toc" /><index index="pdf" /> +<head>CHAPTER II - RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE +CORRESPONDENCE WITH EACH OTHER</head> + +<div> +<index index="toc" level1="23. Seven fundamental forms" /><index index="pdf" /> +<head></head><p><anchor id="p23" /><hi rend="font-weight: bold">23. Seven fundamental forms.</hi> 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.</p></div> + +<div> +<index index="toc" level1="24. Projective properties" /><index index="pdf" /> +<head></head><p><anchor id="p24" /><hi rend="font-weight: bold">24. Projective properties.</hi> Our chief interest in this +chapter will be the discovery of relations between +the elements of one form which hold between the +<pb n="15" /><anchor id="Pg15" /> +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 <hi rend="font-style: italic">u</hi> three points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">C</hi>, are taken +such that <hi rend="font-style: italic">B</hi> is the middle point of the segment <hi rend="font-style: italic">AC</hi>, +it does not follow that the three points <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi> +in a point-row perspective to <hi rend="font-style: italic">u</hi> 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 <hi rend="font-style: italic">projective +relations.</hi> Relations involving measurement of lines or +of angles are not projective.</p></div> + +<div> +<index index="toc" level1="25. Desargues's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p25" /><hi rend="font-weight: bold">25. Desargues's theorem.</hi> We consider first the following +beautiful theorem, due to Desargues and called +by his name.</p> + +<p><hi rend="font-style: italic">If two triangles, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi> and <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, are so situated +that the lines <hi rend="font-style: italic">AA'</hi>, <hi rend="font-style: italic">BB'</hi>, and <hi rend="font-style: italic">CC'</hi> all meet in a point, then +the pairs of sides <hi rend="font-style: italic">AB</hi> and <hi rend="font-style: italic">A'B'</hi>, <hi rend="font-style: italic">BC</hi> and <hi rend="font-style: italic">B'C'</hi>, <hi rend="font-style: italic">CA</hi> and +<hi rend="font-style: italic">C'A'</hi> all meet on a straight line, and conversely.</hi></p> + +<pb n="16" /><anchor id="Pg16" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image03.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 3</head> +<figDesc>Figure 3</figDesc> +</figure></p> + +<p>Let the lines <hi rend="font-style: italic">AA'</hi>, <hi rend="font-style: italic">BB'</hi>, and <hi rend="font-style: italic">CC'</hi> meet in the point <hi rend="font-style: italic">M</hi> +(Fig. 3). Conceive of the figure as in space, so that +<hi rend="font-style: italic">M</hi> is the vertex of a trihedral angle of which the given +triangles are plane sections. The lines <hi rend="font-style: italic">AB</hi> and <hi rend="font-style: italic">A'B'</hi> 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 <hi rend="font-style: italic">P</hi>. By similar +reasoning the point +<hi rend="font-style: italic">Q</hi> of intersection of +the lines <hi rend="font-style: italic">BC</hi> and +<hi rend="font-style: italic">B'C'</hi> must lie on this same line as well as the point <hi rend="font-style: italic">R</hi> +of intersection of <hi rend="font-style: italic">CA</hi> and <hi rend="font-style: italic">C'A'</hi>. Therefore the points +<hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">Q</hi>, and <hi rend="font-style: italic">R</hi> all lie on the same line <hi rend="font-style: italic">m</hi>. If now we consider +the figure a plane figure, the points <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">Q</hi>, and <hi rend="font-style: italic">R</hi> +still all lie on a straight line, which proves the theorem. +The converse is established in the same manner.</p></div> + +<div> +<index index="toc" level1="26. Fundamental theorem concerning two complete +quadrangles" /><index index="pdf" /> +<head></head><p><anchor id="p26" /><hi rend="font-weight: bold">26. Fundamental theorem concerning two complete +quadrangles.</hi> This theorem throws into our hands the +following fundamental theorem concerning two complete +quadrangles, a <hi rend="font-style: italic">complete quadrangle</hi> being defined +as the figure obtained by joining any four given points +by straight lines in the six possible ways.</p> + +<p><hi rend="font-style: italic">Given two complete quadrangles, <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi> and +<hi rend="font-style: italic">K'</hi>, <hi rend="font-style: italic">L'</hi>, <hi rend="font-style: italic">M'</hi>, <hi rend="font-style: italic">N'</hi>, so related that <hi rend="font-style: italic">KL</hi>, <hi rend="font-style: italic">K'L'</hi>, <hi rend="font-style: italic">MN</hi>, <hi rend="font-style: italic">M'N'</hi> all +meet in a point <hi rend="font-style: italic">A</hi>; <hi rend="font-style: italic">LM</hi>, <hi rend="font-style: italic">L'M'</hi>, <hi rend="font-style: italic">NK</hi>, <hi rend="font-style: italic">N'K'</hi> all meet in a +<pb n="17" /><anchor id="Pg17" /> +point <hi rend="font-style: italic">Q</hi>; and <hi rend="font-style: italic">LN</hi>, <hi rend="font-style: italic">L'N'</hi> meet in a point <hi rend="font-style: italic">B</hi> on the line +<hi rend="font-style: italic">AC</hi>; then the lines <hi rend="font-style: italic">KM</hi> and <hi rend="font-style: italic">K'M'</hi> also meet in a point <hi rend="font-style: italic">D</hi> +on the line <hi rend="font-style: italic">AC</hi>.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image04.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 4</head> +<figDesc>Figure 4</figDesc> +</figure></p> + +<p>For, by the converse of the last theorem, <hi rend="font-style: italic">KK'</hi>, <hi rend="font-style: italic">LL'</hi>, +and <hi rend="font-style: italic">NN'</hi> all meet in a point <hi rend="font-style: italic">S</hi> (Fig. 4). Also <hi rend="font-style: italic">LL'</hi>, <hi rend="font-style: italic">MM'</hi>, +and <hi rend="font-style: italic">NN'</hi> meet in a point, and therefore in the same +point <hi rend="font-style: italic">S</hi>. Thus <hi rend="font-style: italic">KK'</hi>, <hi rend="font-style: italic">LL'</hi>, and <hi rend="font-style: italic">MM'</hi> meet in a point, +and so, by Desargues's theorem itself, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">D</hi> are +on a straight line.</p></div> + +<div> +<index index="toc" level1="27. Importance of the theorem" /><index index="pdf" /> +<head></head><p><anchor id="p27" /><hi rend="font-weight: bold">27. Importance of the theorem.</hi> The importance of +this theorem lies in the fact that, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">C</hi> being +given, an indefinite number of quadrangles <hi rend="font-style: italic">K'</hi>, <hi rend="font-style: italic">L'</hi>, <hi rend="font-style: italic">M'</hi>, <hi rend="font-style: italic">N'</hi> +my be found such that <hi rend="font-style: italic">K'L'</hi> and <hi rend="font-style: italic">M'N'</hi> meet in <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">K'N'</hi> +and <hi rend="font-style: italic">L'M'</hi> in <hi rend="font-style: italic">C</hi>, with <hi rend="font-style: italic">L'N'</hi> passing through <hi rend="font-style: italic">B</hi>. Indeed, +the lines <hi rend="font-style: italic">AK'</hi> and <hi rend="font-style: italic">AM'</hi> may be drawn arbitrarily +through <hi rend="font-style: italic">A</hi>, and any line through <hi rend="font-style: italic">B</hi> may be used to +determine <hi rend="font-style: italic">L'</hi> and <hi rend="font-style: italic">N'</hi>. By joining these two points to +<hi rend="font-style: italic">C</hi> the points <hi rend="font-style: italic">K'</hi> and <hi rend="font-style: italic">M'</hi> are determined. Then the line +<pb n="18" /><anchor id="Pg18" /> +joining <hi rend="font-style: italic">K'</hi> and <hi rend="font-style: italic">M'</hi>, found in this way, must pass +through the point <hi rend="font-style: italic">D</hi> already determined by the quadrangle +<hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi>. <hi rend="font-style: italic">The three points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, given in +order, serve thus to determine a fourth point <hi rend="font-style: italic">D</hi>.</hi></p></div> + +<div> +<index index="toc" level1="28. Restatement of the theorem" /><index index="pdf" /> +<head></head><p><anchor id="p28" /><hi rend="font-weight: bold">28.</hi> In a complete quadrangle the line joining any +two points is called the <hi rend="font-style: italic">opposite side</hi> to the line joining +the other two points. The result of the preceding +paragraph may then be stated as follows:</p> + +<p>Given three points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, in a straight line, if a +pair of opposite sides of a complete quadrangle pass +through <hi rend="font-style: italic">A</hi>, and another pair through <hi rend="font-style: italic">C</hi>, and one of the +remaining two sides goes through <hi rend="font-style: italic">B</hi>, then the other of +the remaining two sides will go through a fixed point +which does not depend on the quadrangle employed.</p></div> + +<div> +<index index="toc" level1="29. Four harmonic points" /><index index="pdf" /> +<head></head><p><anchor id="p29" /><hi rend="font-weight: bold">29. Four harmonic points.</hi> Four points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>, +related as in the preceding theorem are called <hi rend="font-style: italic">four +harmonic points</hi>. The point <hi rend="font-style: italic">D</hi> is called the <hi rend="font-style: italic">fourth harmonic +of <hi rend="font-style: italic">B</hi> with respect to <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi></hi>. Since <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> play +exactly the same rôle in the above construction, <hi rend="font-style: italic"><hi rend="font-style: italic">B</hi> is +also the fourth harmonic of <hi rend="font-style: italic">D</hi> with respect to <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi></hi>. +<hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> are called <hi rend="font-style: italic">harmonic conjugates with respect to +<hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi></hi>. We proceed to show that <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi> are also +harmonic conjugates with respect to <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi>—that is, +that it is possible to find a quadrangle of which two +opposite sides shall pass through <hi rend="font-style: italic">B</hi>, two through <hi rend="font-style: italic">D</hi>, +and of the remaining pair, one through <hi rend="font-style: italic">A</hi> and the other +through <hi rend="font-style: italic">C</hi>.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image05.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 5</head> +<figDesc>Figure 5</figDesc> +</figure></p> + +<p>Let <hi rend="font-style: italic">O</hi> be the intersection of <hi rend="font-style: italic">KM</hi> and <hi rend="font-style: italic">LN</hi> (Fig. 5). +Join <hi rend="font-style: italic">O</hi> to <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>. The joining lines cut out on the +sides of the quadrangle four points, <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">Q</hi>, <hi rend="font-style: italic">R</hi>, <hi rend="font-style: italic">S</hi>. Consider +the quadrangle <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">Q</hi>, <hi rend="font-style: italic">O</hi>. One pair of opposite sides +<pb n="19" /><anchor id="Pg19" /> +passes through <hi rend="font-style: italic">A</hi>, one through <hi rend="font-style: italic">C</hi>, and one remaining side +through <hi rend="font-style: italic">D</hi>; therefore the other remaining side must +pass through <hi rend="font-style: italic">B</hi>. Similarly, <hi rend="font-style: italic">RS</hi> passes through <hi rend="font-style: italic">B</hi> and +<hi rend="font-style: italic">PS</hi> and <hi rend="font-style: italic">QR</hi> pass +through <hi rend="font-style: italic">D</hi>. The +quadrangle <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">Q</hi>, +<hi rend="font-style: italic">R</hi>, <hi rend="font-style: italic">S</hi> therefore +has two opposite +sides through <hi rend="font-style: italic">B</hi>, +two through <hi rend="font-style: italic">D</hi>, +and the remaining +pair through +<hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>. <hi rend="font-style: italic">A</hi> and +<hi rend="font-style: italic">C</hi> are thus harmonic conjugates with respect to <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi>. +We may sum up the discussion, therefore, as follows:</p></div> + +<div> +<index index="toc" level1="30. Harmonic conjugates" /><index index="pdf" /> +<head></head><p><anchor id="p30" /><hi rend="font-weight: bold">30.</hi> If <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi> are harmonic conjugates with respect +to <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi>, then <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> are harmonic conjugates with +respect to <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>.</p></div> + +<div> +<index index="toc" level1="31. Importance of the notion of four harmonic points" /><index index="pdf" /> +<head></head><p><anchor id="p31" /><hi rend="font-weight: bold">31. Importance of the notion.</hi> 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 <hi rend="font-style: italic">u</hi> to the four points that correspond to +them in any point-row <hi rend="font-style: italic">u'</hi> perspective to <hi rend="font-style: italic">u</hi>.</p> + +<p>To prove this statement we construct a quadrangle +<hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi> such that <hi rend="font-style: italic">KL</hi> and <hi rend="font-style: italic">MN</hi> pass through <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">KN</hi> +and <hi rend="font-style: italic">LM</hi> through <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">LN</hi> through <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">KM</hi> through <hi rend="font-style: italic">D</hi>. +Take now any point <hi rend="font-style: italic">S</hi> not in the plane of the quadrangle +and construct the planes determined by <hi rend="font-style: italic">S</hi> and +all the seven lines of the figure. Cut across this set of +planes by another plane not passing through <hi rend="font-style: italic">S</hi>. This +plane cuts out on the set of seven planes another +<pb n="20" /><anchor id="Pg20" /> +quadrangle which determines four new harmonic points, +<hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, <hi rend="font-style: italic">D'</hi>, on the lines joining <hi rend="font-style: italic">S</hi> to <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>. But +<hi rend="font-style: italic">S</hi> may be taken as any point, since the original quadrangle +may be taken in any plane through <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>; +and, further, the points <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, <hi rend="font-style: italic">D'</hi> are the intersection +of <hi rend="font-style: italic">SA</hi>, <hi rend="font-style: italic">SB</hi>, <hi rend="font-style: italic">SC</hi>, <hi rend="font-style: italic">SD</hi> by any line. We have, then, the +remarkable theorem:</p></div> + +<div> +<index index="toc" level1="32. Projective invariance of four harmonic points" /><index index="pdf" /> +<head></head><p><anchor id="p32" /><hi rend="font-weight: bold">32.</hi> <hi rend="font-style: italic">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.</hi></p></div> + +<div> +<index index="toc" level1="33. Four harmonic lines" /><index index="pdf" /> +<head></head><p><anchor id="p33" /><hi rend="font-weight: bold">33. Four harmonic lines.</hi> We are now able to extend +the notion of harmonic elements to pencils of rays, and +indeed to axial pencils. For if we define <hi rend="font-style: italic">four harmonic +rays</hi> as four rays which pass through a point and which +pass one through each of four harmonic points, we have +the theorem</p> + +<p><hi rend="font-style: italic">Four harmonic lines are cut by any transversal in four +harmonic points.</hi></p></div> + +<div> +<index index="toc" level1="34. Four harmonic planes" /><index index="pdf" /> +<head></head><p><anchor id="p34" /><hi rend="font-weight: bold">34. Four harmonic planes.</hi> We also define <hi rend="font-style: italic">four harmonic +planes</hi> as four planes through a line which pass +one through each of four harmonic points, and we may +show that</p> + +<p><hi rend="font-style: italic">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.</hi></p> + +<p>For let the planes α, β, γ, δ, which all pass through +the line <hi rend="font-style: italic">g</hi>, pass also through the four harmonic points +<hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>, so that α passes through <hi rend="font-style: italic">A</hi>, etc. Then it is +clear that any plane Ï€ through <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> will cut out +four harmonic lines from the four planes, for they are +<pb n="21" /><anchor id="Pg21" /> +lines through the intersection <hi rend="font-style: italic">P</hi> of <hi rend="font-style: italic">g</hi> with the plane +Ï€, and they pass through the given harmonic points +<hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>. Any other plane σ cuts <hi rend="font-style: italic">g</hi> in a point <hi rend="font-style: italic">S</hi> and +cuts α, β, γ, δ in four lines that meet Ï€ +in four points <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, <hi rend="font-style: italic">D'</hi> lying on <hi rend="font-style: italic">PA</hi>, <hi rend="font-style: italic">PB</hi>, <hi rend="font-style: italic">PC</hi>, and <hi rend="font-style: italic">PD</hi> respectively, +and are thus four harmonic hues. Further, any +ray cuts α, β, γ, δ in four harmonic points, since any +plane through the ray gives four harmonic lines of +intersection.</p></div> + +<div> +<index index="toc" level1="35. Summary of results" /><index index="pdf" /> +<head></head><p><anchor id="p35" /><hi rend="font-weight: bold">35.</hi> These results may be put together as follows:</p> + +<p><hi rend="font-style: italic">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.</hi></p> + +<p>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:</p></div> + +<div> +<index index="toc" level1="36. Definition of projectivity" /><index index="pdf" /> +<head></head><p><anchor id="p36" /><hi rend="font-weight: bold">36. Definition of projectivity.</hi> <hi rend="font-style: italic">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.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image06.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 6</head> +<figDesc>Figure 6</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="37. Correspondence between harmonic conjugates" /><index index="pdf" /> +<head></head><p><anchor id="p37" /><hi rend="font-weight: bold">37. Correspondence between harmonic conjugates.</hi> Given +four harmonic points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>; if we fix <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>, +then <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> vary together in a way that should be +thoroughly understood. To get a clear conception of +their relative motion we may fix the points <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">M</hi> of +the quadrangle <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi> (Fig. 6). Then, as <hi rend="font-style: italic">B</hi> describes +the point-row <hi rend="font-style: italic">AC</hi>, the point <hi rend="font-style: italic">N</hi> describes the point-row +<pb n="22" /><anchor id="Pg22" /> +<hi rend="font-style: italic">AM</hi> perspective to it. Projecting <hi rend="font-style: italic">N</hi> again from <hi rend="font-style: italic">C</hi>, we +get a point-row <hi rend="font-style: italic">K</hi> on <hi rend="font-style: italic">AL</hi> perspective to the point-row +<hi rend="font-style: italic">N</hi> and thus projective to the point-row <hi rend="font-style: italic">B</hi>. Project the +point-row <hi rend="font-style: italic">K</hi> from <hi rend="font-style: italic">M</hi> and we get a point-row <hi rend="font-style: italic">D</hi> on +<hi rend="font-style: italic">AC</hi> again, which is projective to the point-row <hi rend="font-style: italic">B</hi>. For +every point <hi rend="font-style: italic">B</hi> we have thus one and only one point +<hi rend="font-style: italic">D</hi>, 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 <hi rend="font-style: italic">B</hi> correspond to four harmonic points <hi rend="font-style: italic">D</hi>. +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 <hi rend="font-style: italic">B</hi> to <hi rend="font-style: italic">D</hi> will carry one back from +<hi rend="font-style: italic">D</hi> to <hi rend="font-style: italic">B</hi> again. This special property will receive further +study in the chapter on Involution.</p></div> + +<div> +<index index="toc" level1="38. Separation of harmonic conjugates" /><index index="pdf" /> +<head></head><p><anchor id="p38" /><hi rend="font-weight: bold">38.</hi> It is seen that as <hi rend="font-style: italic">B</hi> approaches <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">D</hi> also approaches +<hi rend="font-style: italic">A</hi>. As <hi rend="font-style: italic">B</hi> moves from <hi rend="font-style: italic">A</hi> toward <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> moves +from <hi rend="font-style: italic">A</hi> in the opposite direction, passing through the +point at infinity on the line <hi rend="font-style: italic">AC</hi>, and returns on the +other side to meet <hi rend="font-style: italic">B</hi> at <hi rend="font-style: italic">C</hi> again. In other words, as <hi rend="font-style: italic">B</hi> +traverses <hi rend="font-style: italic">AC</hi>, <hi rend="font-style: italic">D</hi> traverses the rest of the line from <hi rend="font-style: italic">A</hi> to +<hi rend="font-style: italic">C</hi> through infinity. In all positions of <hi rend="font-style: italic">B</hi>, except at <hi rend="font-style: italic">A</hi> or +<hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> are separated from each other by <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>.</p> + +<pb n="23" /><anchor id="Pg23" /></div> + +<div> +<index index="toc" level1="39. Harmonic conjugate of the point at infinity" /><index index="pdf" /> +<head></head><p><anchor id="p39" /><hi rend="font-weight: bold">39. Harmonic conjugate of the point at infinity.</hi> It is +natural to inquire what position of <hi rend="font-style: italic">B</hi> corresponds to the +infinitely distant position of <hi rend="font-style: italic">D</hi>. We have proved (§ 27) +that the particular quadrangle <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi> 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 <hi rend="font-style: italic">L</hi> so that the triangle +<hi rend="font-style: italic">ALC</hi> is isosceles (Fig. 7). Since +<hi rend="font-style: italic">D</hi> is supposed to be at infinity, the +line <hi rend="font-style: italic">KM</hi> is parallel to <hi rend="font-style: italic">AC</hi>. Therefore +the triangles <hi rend="font-style: italic">KAC</hi> and <hi rend="font-style: italic">MAC</hi> +are equal, and the triangle <hi rend="font-style: italic">ANC</hi> is also isosceles. The +triangles <hi rend="font-style: italic">CNL</hi> and <hi rend="font-style: italic">ANL</hi> are therefore equal, and the line +<hi rend="font-style: italic">LB</hi> bisects the angle <hi rend="font-style: italic">ALC</hi>. <hi rend="font-style: italic">B</hi> is therefore the middle +point of <hi rend="font-style: italic">AC</hi>, and we have the theorem</p> + +<p><hi rend="font-style: italic">The harmonic conjugate of the middle point of <hi rend="font-style: italic">AC</hi> is at +infinity.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image07.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 7</head> +<figDesc>Figure 7</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="40. Projective theorems and metrical theorems. Linear +construction" /><index index="pdf" /> +<head></head><p><anchor id="p40" /><hi rend="font-weight: bold">40. Projective theorems and metrical theorems. Linear +construction.</hi> 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 +<pb n="24" /><anchor id="Pg24" /> +made with such an instrument we shall call a <hi rend="font-style: italic">linear</hi> +construction. It requires merely that we be able to +draw the line joining two points or find the point of +intersection of two lines.</p></div> + +<div> +<index index="toc" level1="41. Parallels and mid-points" /><index index="pdf" /> +<head></head><p><anchor id="p41" /><hi rend="font-weight: bold">41. Parallels and mid-points.</hi> 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 <hi rend="font-style: italic">linearly</hi> a line parallel to +that segment. Thus, given that <hi rend="font-style: italic">B</hi> is the middle point of +<hi rend="font-style: italic">AC</hi>, we may draw any two lines through <hi rend="font-style: italic">A</hi>, and any line +through <hi rend="font-style: italic">B</hi> cutting them in points <hi rend="font-style: italic">N</hi> and <hi rend="font-style: italic">L</hi>. Join <hi rend="font-style: italic">N</hi> and +<hi rend="font-style: italic">L</hi> to <hi rend="font-style: italic">C</hi> and get the points <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi> on the two lines +through <hi rend="font-style: italic">A</hi>. Then <hi rend="font-style: italic">KM</hi> is parallel to <hi rend="font-style: italic">AC</hi>. <hi rend="font-style: italic">The bisection of +a given segment and the drawing of a line parallel to the +segment are equivalent data when linear construction is used.</hi></p> + +<pb n="25" /><anchor id="Pg25" /></div> + +<div> +<index index="toc" level1="42. Division of segment into equal parts" /><index index="pdf" /> +<head></head><p><anchor id="p42" /><hi rend="font-weight: bold">42.</hi> It is not difficult to give a linear construction +for the problem to divide a given segment into <hi rend="font-style: italic">n</hi> equal +parts, given only a parallel to the segment. This is +simple enough when <hi rend="font-style: italic">n</hi> is a power of <hi rend="font-style: italic">2</hi>. For any other +number, such as <hi rend="font-style: italic">29</hi>, divide any segment on the line +parallel to <hi rend="font-style: italic">AC</hi> into <hi rend="font-style: italic">32</hi> equal parts, by a repetition of +the process just described. Take <hi rend="font-style: italic">29</hi> of these, and join +the first to <hi rend="font-style: italic">A</hi> and the last to <hi rend="font-style: italic">C</hi>. Let these joining lines +meet in <hi rend="font-style: italic">S</hi>. Join <hi rend="font-style: italic">S</hi> to all the other points. Other +problems, of a similar sort, are given at the end of +the chapter.</p></div> + +<div> +<index index="toc" level1="43. Numerical relations" /><index index="pdf" /> +<head></head><p><anchor id="p43" /><hi rend="font-weight: bold">43. Numerical relations.</hi> 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.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image08.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 8</head> +<figDesc>Figure 8</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="44. Algebraic formula connecting four harmonic points" /><index index="pdf" /> +<head></head><p><anchor id="p44" /><hi rend="font-weight: bold">44.</hi> Let <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> be four +harmonic points (Fig. 8), and +let <hi rend="font-style: italic">SA</hi>, <hi rend="font-style: italic">SB</hi>, <hi rend="font-style: italic">SC</hi>, <hi rend="font-style: italic">SD</hi> be four +harmonic lines. Assume a +line drawn through <hi rend="font-style: italic">B</hi> parallel +to <hi rend="font-style: italic">SD</hi>, meeting <hi rend="font-style: italic">SA</hi> in <hi rend="font-style: italic">A'</hi> and +<hi rend="font-style: italic">SC</hi> in <hi rend="font-style: italic">C'</hi>. Then <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, and the infinitely distant +point on <hi rend="font-style: italic">A'C'</hi> are four harmonic points, and therefore +<hi rend="font-style: italic">B</hi> is the middle point of the segment <hi rend="font-style: italic">A'C'</hi>. Then, since +<pb n="26" /><anchor id="Pg26" /> +the triangle <hi rend="font-style: italic">DAS</hi> is similar to the triangle <hi rend="font-style: italic">BAA'</hi>, we +may write the proportion</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AB : AD = BA' : SD.</hi> +</p> + +<p>Also, from the similar triangles <hi rend="font-style: italic">DSC</hi> and <hi rend="font-style: italic">BCC'</hi>, we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">CD : CB = SD : B'C.</hi> +</p> + +<p>From these two proportions we have, remembering that +<hi rend="font-style: italic">BA' = BC'</hi>,</p> + +<p rend="text-align: center"> +<formula notation="tex">\[ +\frac{AB \cdot CD}{AD \cdot CB} = -1, +\]</formula> +</p> + +<p>the minus sign being given to the ratio on account of the +fact that <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi> are always separated from <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi>, +so that one or three of the segments <hi rend="font-style: italic">AB</hi>, <hi rend="font-style: italic">CD</hi>, <hi rend="font-style: italic">AD</hi>, <hi rend="font-style: italic">CB</hi> +must be negative.</p></div> + +<div> +<index index="toc" level1="45. Further formulae" /><index index="pdf" /> +<head></head><p><anchor id="p45" /><hi rend="font-weight: bold">45.</hi> Writing the last equation in the form</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">CB : AB = -CD : AD,</hi> +</p> + +<p>and using the fundamental relation connecting three +points on a line,</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">PR + RQ = PQ,</hi> +</p> + +<p>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</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">(CB - BA) : AB = -(CA - DA) : AD,</hi> +</p> + +<p>or</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">(AB - AC) : AB = (AC - AD) : AD;</hi> +</p> + +<p>so that <hi rend="font-style: italic">AB</hi>, <hi rend="font-style: italic">AC</hi>, and <hi rend="font-style: italic">AD</hi> 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</p> + +<p rend="text-align: center"> +<formula notation="tex">\[ +\frac{2}{AC} = \frac{1}{AB} + \frac{1}{AD}, +\]</formula> +</p> + +<p>which is convenient for the computation of the distance +<hi rend="font-style: italic">AD</hi> when <hi rend="font-style: italic">AB</hi> and <hi rend="font-style: italic">AC</hi> are given numerically.</p> + +<pb n="27" /><anchor id="Pg27" /></div> + +<div> +<index index="toc" level1="46. Anharmonic ratio" /><index index="pdf" /> +<head></head><p><anchor id="p46" /><hi rend="font-weight: bold">46. Anharmonic ratio.</hi> 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 <hi rend="font-style: italic">anharmonic ratio</hi>, or <hi rend="font-style: italic">cross ratio</hi>, which connects +any four points on a line.</p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p><hi rend="font-weight: bold">1</hi>. 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 <hi rend="font-style: italic">P</hi> draw +any two rays cutting the two lines in the points <hi rend="font-style: italic">AB'</hi> and +<hi rend="font-style: italic">A'B</hi>, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, lying on one of the given lines and <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, on the +other. Join <hi rend="font-style: italic">AA'</hi> and <hi rend="font-style: italic">BB'</hi>, and find their point of intersection +<hi rend="font-style: italic">S</hi>. Through <hi rend="font-style: italic">S</hi> draw any other ray, cutting the given +lines in <hi rend="font-style: italic">CC'</hi>. Join <hi rend="font-style: italic">BC'</hi> and <hi rend="font-style: italic">B'C</hi>, and obtain their point +of intersection <hi rend="font-style: italic">Q</hi>. <hi rend="font-style: italic">PQ</hi> is the desired line. Justify this +construction.</p> + +<p><hi rend="font-weight: bold">2.</hi> To draw through a given point <hi rend="font-style: italic">P</hi> a line which shall +meet two given lines in points <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi>, equally distant from +<hi rend="font-style: italic">P</hi>. Justify the following construction: Join <hi rend="font-style: italic">P</hi> to the point +<hi rend="font-style: italic">S</hi> of intersection of the two given lines. Construct the +fourth harmonic of <hi rend="font-style: italic">PS</hi> with respect to the two given lines. +Draw through <hi rend="font-style: italic">P</hi> a line parallel to this line. This is the +required line.</p> + +<p><hi rend="font-weight: bold">3.</hi> Given a parallelogram in the same plane with a given +segment <hi rend="font-style: italic">AC</hi>, to construct linearly the middle point of <hi rend="font-style: italic">AC</hi>.</p> + +<pb n="28" /><anchor id="Pg28" /> + +<p><hi rend="font-weight: bold">4.</hi> 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.</p> + +<p><hi rend="font-weight: bold">5.</hi> Given the middle point of a line segment, to draw a +line parallel to the segment and passing through a given +point.</p> + +<p><hi rend="font-weight: bold">6.</hi> A line is drawn cutting the sides of a triangle <hi rend="font-style: italic">ABC</hi> in +the points <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi> the point <hi rend="font-style: italic">A'</hi> lying on the side <hi rend="font-style: italic">BC</hi>, etc. +The harmonic conjugate of <hi rend="font-style: italic">A'</hi> with respect to <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">C</hi> is +then constructed and called <hi rend="font-style: italic">A"</hi>. Similarly, <hi rend="font-style: italic">B"</hi> and <hi rend="font-style: italic">C"</hi> are +constructed. Show that <hi rend="font-style: italic">A"B"C"</hi> 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.</p> +</div> +</div> + +<div rend="page-break-before: always"> +<pb n="29" /><anchor id="Pg29" /> +<index index="toc" /><index index="pdf" /> +<head>CHAPTER III - COMBINATION OF TWO PROJECTIVELY RELATED +FUNDAMENTAL FORMS</head> + +<div> +<index index="toc" level1="47. Superposed fundamental forms. Self-corresponding +elements" /><index index="pdf" /> +<head></head> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image09.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 9</head> +<figDesc>Figure 9</figDesc> +</figure></p> + +<p><anchor id="p47" /><hi rend="font-weight: bold">47. Superposed fundamental forms. Self-corresponding +elements.</hi> 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 <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> are harmonic +conjugates with respect to <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>, then the point-row +described by <hi rend="font-style: italic">B</hi> is projective to the point-row described +by <hi rend="font-style: italic">D</hi>, and that <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi> are self-corresponding +points. Consider more generally the case of two pencils +perspective to each other with axis of perspectivity <hi rend="font-style: italic">u'</hi> +(Fig. 9). Cut across them by a line <hi rend="font-style: italic">u</hi>. We get thus +two projective point-rows superposed on the same line +<hi rend="font-style: italic">u</hi>, and a moment's reflection serves to show that the +point <hi rend="font-style: italic">N</hi> of intersection <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> corresponds to itself +in the two point-rows. Also, the point <hi rend="font-style: italic">M</hi>, where <hi rend="font-style: italic">u</hi> +<pb n="30" /><anchor id="Pg30" /> +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 <hi rend="font-style: italic">M</hi> and <hi rend="font-style: italic">N</hi> may +fall together if the line +joining the centers of the +pencils happens to pass +through the point of intersection +of the lines <hi rend="font-style: italic">u</hi> +and <hi rend="font-style: italic">u'</hi>.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image10.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 10</head> +<figDesc>Figure 10</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="48. Special case" /><index index="pdf" /> +<head></head><p><anchor id="p48" /><hi rend="font-weight: bold">48.</hi> 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 <hi rend="font-style: italic">S</hi> and always making the +same angle a with each other +(Fig. 10). They will cut out on +any line <hi rend="font-style: italic">u</hi> in the plane two point-rows +which are easily seen to be +projective. For, given any four +rays <hi rend="font-style: italic">SP</hi> which are harmonic, the +four corresponding rays <hi rend="font-style: italic">SP'</hi> must +also be harmonic, since they make +the same angles with each other. +Four harmonic points <hi rend="font-style: italic">P</hi> correspond, +therefore, to four harmonic points <hi rend="font-style: italic">P'</hi>. It is clear, +however, that no point <hi rend="font-style: italic">P</hi> can coincide with its corresponding +point <hi rend="font-style: italic">P'</hi>, for in that case the lines <hi rend="font-style: italic">PS</hi> and +<pb n="31" /><anchor id="Pg31" /> +<hi rend="font-style: italic">P'S</hi> would coincide, which is impossible if the angle +between them is to be constant.</p></div> + +<div> +<index index="toc" level1="49. Fundamental theorem. Postulate of continuity" /><index index="pdf" /> +<head></head><p><anchor id="p49" /><hi rend="font-weight: bold">49. Fundamental theorem. Postulate of continuity.</hi> +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</p> + +<p><hi rend="font-style: italic">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.</hi></p> + +<p>If three points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">C</hi>, are self-corresponding, +then the harmonic conjugate <hi rend="font-style: italic">D</hi> of <hi rend="font-style: italic">B</hi> with respect to <hi rend="font-style: italic">A</hi> +and <hi rend="font-style: italic">C</hi> must also correspond to itself. For four harmonic +points must always correspond to four harmonic points. +In the same way the harmonic conjugate of <hi rend="font-style: italic">D</hi> with +respect to <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">C</hi> 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 <hi rend="font-style: italic">P</hi> lie between <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi>. Construct +now <hi rend="font-style: italic">D</hi>, the fourth harmonic of <hi rend="font-style: italic">C</hi> with respect to <hi rend="font-style: italic">A</hi> and +<hi rend="font-style: italic">B</hi>. <hi rend="font-style: italic">D</hi> may coincide with <hi rend="font-style: italic">P</hi>, in which case the sequence +is closed; otherwise <hi rend="font-style: italic">P</hi> lies in the stretch <hi rend="font-style: italic">AD</hi> or in the +stretch <hi rend="font-style: italic">DB</hi>. If it lies in the stretch <hi rend="font-style: italic">DB</hi>, construct the +fourth harmonic of <hi rend="font-style: italic">C</hi> with respect to <hi rend="font-style: italic">D</hi> and <hi rend="font-style: italic">B</hi>. This +point <hi rend="font-style: italic">D'</hi> may coincide with <hi rend="font-style: italic">P</hi>, in which case, as before, +the sequence is closed. If <hi rend="font-style: italic">P</hi> lies in the stretch <hi rend="font-style: italic">DD'</hi>, +we construct the fourth harmonic of <hi rend="font-style: italic">C</hi> with respect +<pb n="32" /><anchor id="Pg32" /> +to <hi rend="font-style: italic">DD'</hi>, etc. In each step the region in which <hi rend="font-style: italic">P</hi> lies is +diminished, and the process may be continued until two +self-corresponding points are obtained on either side of +<hi rend="font-style: italic">P</hi>, and at distances from it arbitrarily small.</p> + +<p>We now assume, explicitly, the fundamental postulate +that the correspondence is <hi rend="font-style: italic">continuous</hi>, that is, that <hi rend="font-style: italic">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.</hi> Suppose +now that <hi rend="font-style: italic">P</hi> is not a self-corresponding point, but corresponds +to a point <hi rend="font-style: italic">P'</hi> at a fixed distance <hi rend="font-style: italic">d</hi> from <hi rend="font-style: italic">P</hi>. +As noted above, we can find self-corresponding points +arbitrarily close to <hi rend="font-style: italic">P</hi>, and it appears, then, that we can +take a point <hi rend="font-style: italic">D</hi> as close to <hi rend="font-style: italic">P</hi> as we wish, and yet the +distance between the corresponding points <hi rend="font-style: italic">D'</hi> and <hi rend="font-style: italic">P'</hi> +approaches <hi rend="font-style: italic">d</hi> as a limit, and not zero, which contradicts +the postulate of continuity.</p></div> + +<div> +<index index="toc" level1="50. Extension of theorem to pencils of rays and planes" /><index index="pdf" /> +<head></head><p><anchor id="p50" /><hi rend="font-weight: bold">50.</hi> 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.</p></div> + +<div> +<index index="toc" level1="51. Projective point-rows having a self-corresponding +point in common" /><index index="pdf" /> +<head></head><p><anchor id="p51" /><hi rend="font-weight: bold">51. Projective point-rows having a self-corresponding +point in common.</hi> 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 +<pb n="33" /><anchor id="Pg33" /> +point. The converse is also true, and we have the very +important theorem:</p></div> + +<div> +<index index="toc" level1="52. Point-rows in perspective position" /><index index="pdf" /> +<head></head><p><anchor id="p52" /><hi rend="font-weight: bold">52.</hi> <hi rend="font-style: italic">If in two protective point-rows, the point of intersection +corresponds to itself, then the point-rows are in +perspective position.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image11.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 11</head> +<figDesc>Figure 11</figDesc> +</figure></p> + +<p>Let the two point-rows be <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> (Fig. 11). Let +<hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">B'</hi>, be corresponding points, and let +also the point <hi rend="font-style: italic">M</hi> of intersection of <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> correspond +to itself. Let <hi rend="font-style: italic">AA'</hi> and <hi rend="font-style: italic">BB'</hi> meet in the point <hi rend="font-style: italic">S</hi>. Take +<hi rend="font-style: italic">S</hi> as the center of two pencils, +one perspective to <hi rend="font-style: italic">u</hi> and the other +perspective to <hi rend="font-style: italic">u'</hi>. In these two +pencils <hi rend="font-style: italic">SA</hi> coincides with its corresponding +ray <hi rend="font-style: italic">SA'</hi>, <hi rend="font-style: italic">SB</hi> with its +corresponding ray <hi rend="font-style: italic">SB'</hi>, and <hi rend="font-style: italic">SM</hi> +with its corresponding ray <hi rend="font-style: italic">SM'</hi>. +The two pencils are thus identical, by the preceding +theorem, and any ray <hi rend="font-style: italic">SD</hi> must coincide with its corresponding +ray <hi rend="font-style: italic">SD'</hi>. Corresponding points of <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>, +therefore, all lie on lines through the point <hi rend="font-style: italic">S</hi>.</p></div> + +<div> +<index index="toc" level1="53. Pencils in perspective position" /><index index="pdf" /> +<head></head><p><anchor id="p53" /><hi rend="font-weight: bold">53.</hi> An entirely similar discussion shows that</p> + +<p><hi rend="font-style: italic">If in two projective pencils the line joining their centers +is a self-corresponding ray, then the two pencils are +perspectively related.</hi></p></div> + +<div> +<index index="toc" level1="54. Axial pencils in perspective position" /><index index="pdf" /> +<head></head><p><anchor id="p54" /><hi rend="font-weight: bold">54.</hi> 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.</p></div> + +<div> +<index index="toc" level1="55. Point-row of the second order" /><index index="pdf" /> +<head></head><p><anchor id="p55" /><hi rend="font-weight: bold">55. Point-row of the second order.</hi> The question naturally +arises, What is the locus of points of intersection +of corresponding rays of two projective pencils +<pb n="34" /><anchor id="Pg34" /> +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 <hi rend="font-style: italic">point-row of the +second order</hi>. For any line <hi rend="font-style: italic">u</hi> 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.</p></div> + +<div> +<index index="toc" level1="56. Degeneration of locus" /><index index="pdf" /> +<head></head><p><anchor id="p56" /><hi rend="font-weight: bold">56.</hi> 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.</p></div> + +<div> +<index index="toc" level1="57. Pencils of rays of the second order" /><index index="pdf" /> +<head></head><p><anchor id="p57" /><hi rend="font-weight: bold">57. Pencils of rays of the second order.</hi> 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: <hi rend="font-style: italic">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.</hi> For that reason the system of rays is called <hi rend="font-style: italic">a +pencil of rays of the second order.</hi></p></div> + +<div> +<index index="toc" level1="58. Degenerate case" /><index index="pdf" /> +<head></head><p><anchor id="p58" /><hi rend="font-weight: bold">58.</hi> 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 +<pb n="35" /><anchor id="Pg35" /> +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.</p></div> + +<div> +<index index="toc" level1="59. Cone of the second order" /><index index="pdf" /> +<head></head><p><anchor id="p59" /><hi rend="font-weight: bold">59. Cone of the second order.</hi> The corresponding +theorems in space may easily be obtained by joining +the points and lines considered in the plane theorems +to a point <hi rend="font-style: italic">S</hi> in space. Two projective pencils give rise +to two projective axial pencils with axes intersecting. +Corresponding planes meet in lines which all pass +through <hi rend="font-style: italic">S</hi> 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 <hi rend="font-style: italic">cone of the second +order</hi>, or <hi rend="font-style: italic">quadric cone</hi>, so called because every plane in +space not passing through <hi rend="font-style: italic">S</hi> 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 <hi rend="font-style: italic">S</hi> 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 <hi rend="font-style: italic">S</hi> of the lines which join the corresponding +points of the two point-rows. At most two +such planes may pass through any ray through <hi rend="font-style: italic">S</hi>. It +is called <hi rend="font-style: italic">a pencil of planes of the second order</hi>.</p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p><hi rend="font-weight: bold">1. </hi> A man <hi rend="font-style: italic">A</hi> moves along a straight road <hi rend="font-style: italic">u</hi>, and another +man <hi rend="font-style: italic">B</hi> moves along the same road and walks so as always +to keep sight of <hi rend="font-style: italic">A</hi> in a small mirror <hi rend="font-style: italic">M</hi> at the side of the +road. How many times will they come together, <hi rend="font-style: italic">A</hi> moving +always in the same direction along the road?</p> + +<pb n="36" /><anchor id="Pg36" /> + +<p>2. How many times would the two men in the first problem +see each other in two mirrors <hi rend="font-style: italic">M</hi> and <hi rend="font-style: italic">N</hi> as they walk +along the road as before? (The planes of the two mirrors +are not necessarily parallel to <hi rend="font-style: italic">u</hi>.)</p> + +<p>3. As A moves along <hi rend="font-style: italic">u</hi>, trace the path of B so that the +two men may always see each other in the two mirrors.</p> + +<p>4. Two boys walk along two paths <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> 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?</p> + +<p>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.</p> + +<p>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.)</p> + +<p>7. Two lines <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> revolve about two points <hi rend="font-style: italic">U</hi> and <hi rend="font-style: italic">U'</hi> +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.</p> + +<p>8. Discuss the question given in the last problem when +the two lines revolve in opposite directions. Can you +recognize the locus?</p> +</div> +</div> + +<div rend="page-break-before: always"> +<pb n="37" /><anchor id="Pg37" /> +<index index="toc" /><index index="pdf" /> +<head>CHAPTER IV - POINT-ROWS OF THE SECOND ORDER</head> + +<div> +<index index="toc" level1="60. Point-row of the second order defined" /><index index="pdf" /> +<head></head><p><anchor id="p60" /><hi rend="font-weight: bold">60. Point-row of the second order defined.</hi> 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.</p></div> + +<div> +<index index="toc" level1="61. Tangent line" /><index index="pdf" /> +<head></head><p><anchor id="p61" /><hi rend="font-weight: bold">61. Tangent line.</hi> 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 <hi rend="font-style: italic">SS'</hi>, considered as a ray of <hi rend="font-style: italic">S</hi>, must +correspond some ray of <hi rend="font-style: italic">S'</hi> which meets it in <hi rend="font-style: italic">S'</hi>. <hi rend="font-style: italic">S'</hi>, +and by the same argument <hi rend="font-style: italic">S</hi>, is then a point where +corresponding rays meet. Any ray through <hi rend="font-style: italic">S</hi> will meet +it in one point besides <hi rend="font-style: italic">S</hi>, namely, the point <hi rend="font-style: italic">P</hi> where +it meets its corresponding ray. Now, by choosing the +ray through <hi rend="font-style: italic">S</hi> sufficiently close to the ray <hi rend="font-style: italic">SS'</hi>, the point +<hi rend="font-style: italic">P</hi> may be made to approach arbitrarily close to <hi rend="font-style: italic">S'</hi>, and +the ray <hi rend="font-style: italic">S'P</hi> may be made to differ in position from the +<pb n="38" /><anchor id="Pg38" /> +tangent line at <hi rend="font-style: italic">S'</hi> by as little as we please. We have, +then, the important theorem</p> + +<p><hi rend="font-style: italic">The ray at <hi rend="font-style: italic">S'</hi> which corresponds to the common ray <hi rend="font-style: italic">SS'</hi> +is tangent to the locus at <hi rend="font-style: italic">S'</hi>.</hi></p> + +<p>In the same manner the tangent at <hi rend="font-style: italic">S</hi> may be +constructed.</p></div> + +<div> +<index index="toc" level1="62. Determination of the locus" /><index index="pdf" /> +<head></head><p><anchor id="p62" /><hi rend="font-weight: bold">62. Determination of the locus.</hi> We now show that +<hi rend="font-style: italic">it is possible to assign arbitrarily the position of three +points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">C</hi>, on the locus (besides the points <hi rend="font-style: italic">S</hi> +and <hi rend="font-style: italic">S'</hi>); but, these three points being chosen, the locus is +completely determined.</hi></p></div> + +<div> +<index index="toc" level1="63. Restatement of the problem" /><index index="pdf" /> +<head></head><p><anchor id="p63" /><hi rend="font-weight: bold">63.</hi> This statement is equivalent to the following:</p> + +<p><hi rend="font-style: italic">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.</hi></p></div> + +<div> +<index index="toc" level1="64. Solution of the fundamental problem" /><index index="pdf" /> +<head></head><p><anchor id="p64" /><hi rend="font-weight: bold">64.</hi> We proceed, then, to the solution of the fundamental</p> + +<p><hi rend="font-variant: small-caps">Problem</hi>: <hi rend="font-style: italic">Given three pairs of rays, <hi rend="font-style: italic">aa'</hi>, <hi rend="font-style: italic">bb'</hi>, and <hi rend="font-style: italic">cc'</hi>, +of two protective pencils, <hi rend="font-style: italic">S</hi> and <hi rend="font-style: italic">S'</hi>, to find the ray <hi rend="font-style: italic">d'</hi> of <hi rend="font-style: italic">S'</hi> +which corresponds to any ray <hi rend="font-style: italic">d</hi> of <hi rend="font-style: italic">S</hi>.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image12.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 12</head> +<figDesc>Figure 12</figDesc> +</figure></p> + +<p>Call <hi rend="font-style: italic">A</hi> the intersection of <hi rend="font-style: italic">aa'</hi>, <hi rend="font-style: italic">B</hi> the intersection of <hi rend="font-style: italic">bb'</hi>, +and <hi rend="font-style: italic">C</hi> the intersection of <hi rend="font-style: italic">cc'</hi> (Fig. 12). Join <hi rend="font-style: italic">AB</hi> by the +line <hi rend="font-style: italic">u</hi>, and <hi rend="font-style: italic">AC</hi> by the line <hi rend="font-style: italic">u'</hi>. Consider <hi rend="font-style: italic">u</hi> as a point-row +perspective to <hi rend="font-style: italic">S</hi>, and <hi rend="font-style: italic">u'</hi> as a point-row perspective +to <hi rend="font-style: italic">S'</hi>. <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> are projectively related to each other, +since <hi rend="font-style: italic">S</hi> and <hi rend="font-style: italic">S'</hi> are, by hypothesis, so related. But their +point of intersection <hi rend="font-style: italic">A</hi> is a self-corresponding point, since +<hi rend="font-style: italic">a</hi> and <hi rend="font-style: italic">a'</hi> were supposed to be corresponding rays. It follows +(§ 52) that <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> are in perspective position, +and that lines through corresponding points all pass +<pb n="39" /><anchor id="Pg39" /> +through a point <hi rend="font-style: italic">M</hi>, the center of perspectivity, the +position of which will be determined by any two such +lines. But the intersection of <hi rend="font-style: italic">a</hi> with <hi rend="font-style: italic">u</hi> and the intersection +of <hi rend="font-style: italic">c'</hi> with <hi rend="font-style: italic">u'</hi> are corresponding points on <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>, +and the line joining them is clearly <hi rend="font-style: italic">c</hi> itself. Similarly, +<hi rend="font-style: italic">b'</hi> joins two corresponding points on <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>, and so the +center <hi rend="font-style: italic">M</hi> of perspectivity of <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> is the intersection +of <hi rend="font-style: italic">c</hi> and <hi rend="font-style: italic">b'</hi>. To find <hi rend="font-style: italic">d'</hi> in <hi rend="font-style: italic">S'</hi> corresponding to a given +line <hi rend="font-style: italic">d</hi> of <hi rend="font-style: italic">S</hi> we note the point <hi rend="font-style: italic">L</hi> where <hi rend="font-style: italic">d</hi> meets <hi rend="font-style: italic">u</hi>. Join +<hi rend="font-style: italic">L</hi> to <hi rend="font-style: italic">M</hi> and get the point <hi rend="font-style: italic">N</hi> where this line meets <hi rend="font-style: italic">u'</hi>. +<hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">N</hi> are corresponding points on <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>, and <hi rend="font-style: italic">d'</hi> +must therefore pass through <hi rend="font-style: italic">N</hi>. The intersection <hi rend="font-style: italic">P</hi> of +<hi rend="font-style: italic">d</hi> and <hi rend="font-style: italic">d'</hi> is thus another point on the locus. In the same +manner any number of other points may be obtained.</p></div> + +<div> +<index index="toc" level1="65. Different constructions for the figure" /><index index="pdf" /> +<head></head><p><anchor id="p65" /><hi rend="font-weight: bold">65.</hi> The lines <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> might have been drawn in +any direction through <hi rend="font-style: italic">A</hi> (avoiding, of course, the line +<hi rend="font-style: italic">a</hi> for <hi rend="font-style: italic">u</hi> and the line <hi rend="font-style: italic">a'</hi> for <hi rend="font-style: italic">u'</hi>), and the center of perspectivity +<hi rend="font-style: italic">M</hi> would be easily obtainable; but the above +construction furnishes a simple and instructive figure. +An equally simple one is obtained by taking <hi rend="font-style: italic">a'</hi> for <hi rend="font-style: italic">u</hi> +and <hi rend="font-style: italic">a</hi> for <hi rend="font-style: italic">u'</hi>.</p> + +<pb n="40" /><anchor id="Pg40" /></div> + +<div> +<index index="toc" level1="66. Lines joining four points of the locus to a fifth" /><index index="pdf" /> +<head></head><p><anchor id="p66" /><hi rend="font-weight: bold">66. Lines joining four points of the locus to a fifth.</hi> +Suppose that the points <hi rend="font-style: italic">S</hi>, <hi rend="font-style: italic">S'</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, and <hi rend="font-style: italic">D</hi> are fixed, +and that four points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">A<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">A<hi rend="vertical-align: sub">2</hi></hi>, and <hi rend="font-style: italic">A<hi rend="vertical-align: sub">3</hi></hi>, are taken on the +locus at the intersection with it of any four harmonic +rays through <hi rend="font-style: italic">B</hi>. These four harmonic rays give four +harmonic points, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">L<hi rend="vertical-align: sub">1</hi></hi> etc., on the fixed ray <hi rend="font-style: italic">SD</hi>. These, +in turn, project through the fixed point <hi rend="font-style: italic">M</hi> into four +harmonic points, <hi rend="font-style: italic">N</hi>, <hi rend="font-style: italic">N<hi rend="vertical-align: sub">1</hi></hi> etc., on the fixed line <hi rend="font-style: italic">DS'</hi>. +These last four harmonic points give four harmonic +rays <hi rend="font-style: italic">CA</hi>, <hi rend="font-style: italic">CA<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">CA<hi rend="vertical-align: sub">2</hi></hi>, <hi rend="font-style: italic">CA<hi rend="vertical-align: sub">3</hi></hi>. Therefore the four points <hi rend="font-style: italic">A</hi> +which project to <hi rend="font-style: italic">B</hi> in four harmonic rays also project +to <hi rend="font-style: italic">C</hi> in four harmonic rays. But <hi rend="font-style: italic">C</hi> may be any +point on the locus, and so we have the very important +theorem,</p> + +<p><hi rend="font-style: italic">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.</hi></p></div> + +<div> +<index index="toc" level1="67. Restatement of the theorem" /><index index="pdf" /> +<head></head><p><anchor id="p67" /><hi rend="font-weight: bold">67.</hi> The theorem may also be stated thus:</p> + +<p><hi rend="font-style: italic">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.</hi></p></div> + +<div> +<index index="toc" level1="68. Further important theorem" /><index index="pdf" /> +<head></head><p><anchor id="p68" /><hi rend="font-weight: bold">68.</hi> A further theorem of prime importance also +follows:</p> + +<p><hi rend="font-style: italic">Any two points on the locus may be taken as the centers +of two projective pencils which will generate the locus.</hi></p></div> + +<div> +<index index="toc" level1="69. Pascal's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p69" /><hi rend="font-weight: bold">69. Pascal's theorem.</hi> The points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>, <hi rend="font-style: italic">S</hi>, and +<hi rend="font-style: italic">S'</hi> may thus be considered as chosen arbitrarily on the +locus, and the following remarkable theorem follows +at once.</p> + +<pb n="41" /><anchor id="Pg41" /> + +<p><hi rend="font-style: italic">Given six points, 1, 2, 3, 4, 5, 6, on the point-row of +the second order, if we call</hi></p> + +<p rend="text-align: center"><hi rend="font-style: italic">L the intersection of 12 with 45,</hi></p> +<p rend="text-align: center"><hi rend="font-style: italic">M the intersection of 23 with 56,</hi></p> +<p rend="text-align: center"><hi rend="font-style: italic">N the intersection of 34 with 61,</hi></p> + +<p><hi rend="font-style: italic">then <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, and <hi rend="font-style: italic">N</hi> are on a straight line.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image13.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 13</head> +<figDesc>Figure 13</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="70. Permutation of points in Pascal's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p70" /><hi rend="font-weight: bold">70.</hi> To get the notation to correspond to the figure, we +may take (Fig. 13) <hi rend="font-style: italic">A = 1</hi>, <hi rend="font-style: italic">B = 2</hi>, <hi rend="font-style: italic">S' = 3</hi>, <hi rend="font-style: italic">D = 4</hi>, <hi rend="font-style: italic">S = 5</hi>, and +<hi rend="font-style: italic">C = 6</hi>. If we make <hi rend="font-style: italic">A = 1</hi>, <hi rend="font-style: italic">C=2</hi>, <hi rend="font-style: italic">S=3</hi>, <hi rend="font-style: italic">D = 4</hi>, <hi rend="font-style: italic">S'=5</hi>, and. +<hi rend="font-style: italic">B = 6</hi>, the points <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">N</hi> are interchanged, but the line +is left unchanged. +It is clear that one +point may be named +arbitrarily and the +other five named in +<hi rend="font-style: italic">5! = 120</hi> 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 <hi rend="font-style: italic">LMN</hi> 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 <hi rend="font-style: italic">60</hi>. 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 +<pb n="42" /><anchor id="Pg42" /> +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:</p> + +<p><hi rend="font-style: italic">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.</hi></p></div> + +<div> +<index index="toc" level1="71. Harmonic points on a point-row of the second order" /><index index="pdf" /> +<head></head><p><anchor id="p71" /><hi rend="font-weight: bold">71. Harmonic points on a point-row of the second order.</hi> +Before proceeding to develop the consequences of this +theorem, we note another result of the utmost importance +for the higher developments of pure geometry, +which follows from the fact that if four points on the +locus project to a fifth in four harmonic rays, they will +project to any point of the locus in four harmonic rays. +It is natural to speak of four such points as four harmonic +points on the locus, and to use this notion to +define projective correspondence between point-rows of +the second order, or between a point-row of the second +order and any fundamental form of the first order. +Thus, in particular, the point-row of the second order, +σ, is said to be <hi rend="font-style: italic">perspectively related</hi> to the pencil <hi rend="font-style: italic">S</hi> when +every ray on <hi rend="font-style: italic">S</hi> goes through the point on σ which +corresponds to it.</p></div> + +<div> +<index index="toc" level1="72. Determination of the locus" /><index index="pdf" /> +<head></head><p><anchor id="p72" /><hi rend="font-weight: bold">72. Determination of the locus.</hi> 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 +<pb n="43" /><anchor id="Pg43" /> +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 <hi rend="font-style: italic">S</hi> +of one pencil, and any other of the points for <hi rend="font-style: italic">S'</hi>, then, +besides the two pairs of corresponding rays determined +by the remaining two points, we have one more pair, +consisting of the tangent at <hi rend="font-style: italic">S</hi> and the ray <hi rend="font-style: italic">SS'</hi>. Similarly, +the curve is determined by three points and the +tangents at two of them.</p></div> + +<div> +<index index="toc" level1="73. Circles and conics as point-rows of the second order" /><index index="pdf" /> +<head></head><p><anchor id="p73" /><hi rend="font-weight: bold">73. Circles and conics as point-rows of the second order.</hi> +It is not difficult to see that a circle is a point-row of +the second order. Indeed, take any point <hi rend="font-style: italic">S</hi> 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 <hi rend="font-style: italic">S</hi> and <hi rend="font-style: italic">S'</hi>. 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.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image14.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 14</head> +<figDesc>Figure 14</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="74. Conic through five points" /><index index="pdf" /> +<head></head><p><anchor id="p74" /><hi rend="font-weight: bold">74. Conic through five points.</hi> Pascal's theorem furnishes +an elegant solution of the problem of drawing a +conic through five given points. To construct a sixth +<pb n="44" /><anchor id="Pg44" /> +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 +<hi rend="font-style: italic">L = 12-45</hi> is obtainable at once; also +the point <hi rend="font-style: italic">N = 34-61</hi>. But <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">N</hi> +determine Pascal's line, and the intersection +of 23 with 56 must be on +this line. Intersect, then, the line <hi rend="font-style: italic">LN</hi> +with 23 and obtain the point <hi rend="font-style: italic">M</hi>. Join +<hi rend="font-style: italic">M</hi> to 5 and intersect with 61 for the desired point 6.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image15.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 15</head> +<figDesc>Figure 15</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="75. Tangent to a conic" /><index index="pdf" /> +<head></head><p><anchor id="p75" /><hi rend="font-weight: bold">75. Tangent to a conic.</hi> 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 +<hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">M</hi> are obtained as usual, +and the intersection of 34 with <hi rend="font-style: italic">LM</hi> +gives <hi rend="font-style: italic">N</hi>. Join <hi rend="font-style: italic">N</hi> to the point 1 for +the desired tangent at that point.</p></div> + +<div> +<index index="toc" level1="76. Inscribed quadrangle" /><index index="pdf" /> +<head></head><p><anchor id="p76" /><hi rend="font-weight: bold">76. Inscribed quadrangle.</hi> Two pairs of vertices may +coalesce, giving an inscribed quadrangle. Pascal's theorem +gives for this case the very important theorem</p> + +<p><hi rend="font-style: italic">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.</hi></p> + +<pb n="45" /><anchor id="Pg45" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image16.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 16</head> +<figDesc>Figure 16</figDesc> +</figure></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image17.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 17</head> +<figDesc>Figure 17</figDesc> +</figure></p> + +<p>For let the vertices be <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, and <hi rend="font-style: italic">D</hi>, and call the +vertex <hi rend="font-style: italic">A</hi> the point 1, 6; <hi rend="font-style: italic">B</hi>, the point 2; <hi rend="font-style: italic">C</hi>, the point +3, 4; and <hi rend="font-style: italic">D</hi>, the point 5 (Fig. 16). Pascal's theorem then +indicates that +<hi rend="font-style: italic">L = AB-CD</hi>, +<hi rend="font-style: italic">M = AD-BC</hi>, +and <hi rend="font-style: italic">N</hi>, which +is the intersection +of the +tangents at <hi rend="font-style: italic">A</hi> +and <hi rend="font-style: italic">C</hi>, are all +on a straight +line <hi rend="font-style: italic">u</hi>. But +if we were to +call <hi rend="font-style: italic">A</hi> the point 2, <hi rend="font-style: italic">B</hi> the point 6, 1, <hi rend="font-style: italic">C</hi> the point 5, and +<hi rend="font-style: italic">D</hi> the point 4, 3, then the intersection <hi rend="font-style: italic">P</hi> of the tangents +at <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> are also on this same +line <hi rend="font-style: italic">u</hi>. Thus <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi>, and <hi rend="font-style: italic">P</hi> 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.</p></div> + +<div> +<index index="toc" level1="77. Inscribed triangle" /><index index="pdf" /> +<head></head><p><anchor id="p77" /><hi rend="font-weight: bold">77. Inscribed triangle.</hi> 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:</p> + +<p><hi rend="font-style: italic">The three tangents at the vertices of a triangle inscribed +in a conic meet the opposite sides in three points on a +straight line.</hi></p> + +<pb n="46" /><anchor id="Pg46" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image18.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 18</head> +<figDesc>Figure 18</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="78. Degenerate conic" /><index index="pdf" /> +<head></head><p><anchor id="p78" /><hi rend="font-weight: bold">78. Degenerate conic.</hi> 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):</p> + +<p><hi rend="font-style: italic">If three points, +<hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, are +chosen on one +line, and three +points, <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, +<hi rend="font-style: italic">C'</hi>, are chosen on +another, then the +three points <hi rend="font-style: italic">L = AB'-A'B</hi>, <hi rend="font-style: italic">M = BC'-B'C</hi>, <hi rend="font-style: italic">N = CA'-C'A</hi> +are all on a straight line.</hi></p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>1. In Fig. 12, select different lines <hi rend="font-style: italic">u</hi> and trace the locus +of the center of perspectivity <hi rend="font-style: italic">M</hi> of the lines <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>.</p> + +<p>2. Given four points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>, in the plane, construct +a fifth point <hi rend="font-style: italic">P</hi> such that the lines <hi rend="font-style: italic">PA</hi>, <hi rend="font-style: italic">PB</hi>, <hi rend="font-style: italic">PC</hi>, <hi rend="font-style: italic">PD</hi> shall be +four harmonic lines.</p> + +<p><hi rend="font-style: italic">Suggestion.</hi> Draw a line <hi rend="font-style: italic">a</hi> through the point <hi rend="font-style: italic">A</hi> such that the four +lines <hi rend="font-style: italic">a</hi>, <hi rend="font-style: italic">AB</hi>, <hi rend="font-style: italic">AC</hi>, <hi rend="font-style: italic">AD</hi> are harmonic. Construct now a conic through +<hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, and <hi rend="font-style: italic">D</hi> having <hi rend="font-style: italic">a</hi> for a tangent at <hi rend="font-style: italic">A</hi>.</p> + +<p>3. Where are all the points <hi rend="font-style: italic">P</hi>, as determined in the +preceding question, to be found?</p> + +<p>4. Select any five points in the plane and draw the tangent +to the conic through them at each of the five points.</p> + +<p>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.)</p> + +<pb n="47" /><anchor id="Pg47" /> + +<p>6. Given three points on the conic, and the tangent at +two of them, to construct the conic.</p> + +<p>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.)</p> + +<p>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.</p> + +<p>9. The tangents to a curve at its infinitely distant points +are called its <hi rend="font-style: italic">asymptotes</hi> if they pass through a finite part +of the plane. Given the asymptotes and a finite point of a +conic, to construct the conic.</p> + +<p>10. Given an asymptote and three finite points on the +conic, to determine the conic.</p> + +<p>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.</p> +</div> +</div> + +<div rend="page-break-before: always"> +<index index="toc" /><index index="pdf" /> +<pb n="48" /><anchor id="Pg48" /> +<head>CHAPTER V - PENCILS OF RAYS OF THE SECOND ORDER</head> + +<div> +<index index="toc" level1="79. Pencil of rays of the second order defined" /><index index="pdf" /> +<head></head><p><anchor id="p79" /><hi rend="font-weight: bold">79. Pencil of rays of the second order defined.</hi> 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.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image19.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 19</head> +<figDesc>Figure 19</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="80. Tangents to a circle" /><index index="pdf" /> +<head></head><p><anchor id="p80" /><hi rend="font-weight: bold">80. Tangents to a circle.</hi> 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, <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>, to a circle, and let <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi> +be the points of contact (Fig. 19). Let now <hi rend="font-style: italic">t</hi> be any +third tangent with point of contact at <hi rend="font-style: italic">C</hi> and meeting <hi rend="font-style: italic">u</hi> +and <hi rend="font-style: italic">u'</hi> in <hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">P'</hi> respectively. Join <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">P'</hi>, and +<hi rend="font-style: italic">C</hi> to <hi rend="font-style: italic">O</hi>, the center of the circle. Tangents from any +point to a circle are equal, and therefore the triangles +<hi rend="font-style: italic">POA</hi> and <hi rend="font-style: italic">POC</hi> are equal, as also are the triangles <hi rend="font-style: italic">P'OB</hi> +<pb n="49" /><anchor id="Pg49" /> +and <hi rend="font-style: italic">P'OC</hi>. Therefore the angle <hi rend="font-style: italic">POP'</hi> is constant, being +equal to half the constant angle <hi rend="font-style: italic">AOC + COB</hi>. This +being true, if we take any four harmonic points, <hi rend="font-style: italic">P<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">P<hi rend="vertical-align: sub">2</hi></hi>, +<hi rend="font-style: italic">P<hi rend="vertical-align: sub">3</hi></hi>, <hi rend="font-style: italic">P<hi rend="vertical-align: sub">4</hi></hi>, on the line <hi rend="font-style: italic">u</hi>, they will project to <hi rend="font-style: italic">O</hi> in four +harmonic lines, and the tangents +to the circle from these four +points will meet <hi rend="font-style: italic">u'</hi> in four harmonic +points, <hi rend="font-style: italic">P'<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">P'<hi rend="vertical-align: sub">2</hi></hi>, <hi rend="font-style: italic">P'<hi rend="vertical-align: sub">3</hi></hi>, <hi rend="font-style: italic">P'<hi rend="vertical-align: sub">4</hi></hi>, because +the lines from these points +to <hi rend="font-style: italic">O</hi> inclose the same angles as +the lines from the points <hi rend="font-style: italic">P<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">P<hi rend="vertical-align: sub">2</hi></hi>, +<hi rend="font-style: italic">P<hi rend="vertical-align: sub">3</hi></hi>, <hi rend="font-style: italic">P<hi rend="vertical-align: sub">4</hi></hi> on <hi rend="font-style: italic">u</hi>. The point-row on <hi rend="font-style: italic">u</hi> is therefore projective +to the point-row on <hi rend="font-style: italic">u'</hi>. 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.</p></div> + +<div> +<index index="toc" level1="81. Tangents to a conic" /><index index="pdf" /> +<head></head><p><anchor id="p81" /><hi rend="font-weight: bold">81. Tangents to a conic.</hi> 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.</p></div> + +<div> +<index index="toc" level1="82. Generating point-rows lines of the system" /><index index="pdf" /> +<head></head><p><anchor id="p82" /><hi rend="font-weight: bold">82.</hi> The point-rows <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> are, themselves, lines of +the system, for to the common point of the two point-rows, +considered as a point of <hi rend="font-style: italic">u</hi>, must correspond some +point of <hi rend="font-style: italic">u'</hi>, and the line joining these two corresponding +points is clearly <hi rend="font-style: italic">u'</hi> itself. Similarly for the line <hi rend="font-style: italic">u</hi>.</p></div> + +<div> +<index index="toc" level1="83. Determination of the pencil" /><index index="pdf" /> +<head></head><p><anchor id="p83" /><hi rend="font-weight: bold">83. Determination of the pencil.</hi> We now show that +<hi rend="font-style: italic">it is possible to assign arbitrarily three lines, <hi rend="font-style: italic">a</hi>, <hi rend="font-style: italic">b</hi>, and <hi rend="font-style: italic">c</hi>, of +<pb n="50" /><anchor id="Pg50" /> +the system (besides the lines <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>); but if these three +lines are chosen, the system is completely determined.</hi></p> + +<p>This statement is equivalent to the following:</p> + +<p><hi rend="font-style: italic">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.</hi></p> + +<p>We proceed, then, to the solution of the fundamental</p> + +<p><hi rend="font-variant: small-caps">Problem.</hi> <hi rend="font-style: italic">Given three pairs of points, <hi rend="font-style: italic">AA'</hi>, <hi rend="font-style: italic">BB'</hi>, and +<hi rend="font-style: italic">CC'</hi>, of two projective point-rows <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi>, to find the point +<hi rend="font-style: italic">D'</hi> of <hi rend="font-style: italic">u'</hi> which corresponds to any given point <hi rend="font-style: italic">D</hi> of <hi rend="font-style: italic">u</hi>.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image20.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 20</head> +<figDesc>Figure 20</figDesc> +</figure></p> + +<p>On the line <hi rend="font-style: italic">a</hi>, joining <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>, take two points, <hi rend="font-style: italic">S</hi> +and <hi rend="font-style: italic">S'</hi>, as centers of pencils perspective to <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> +respectively (Fig. 20). The figure +will be much simplified if we take +<hi rend="font-style: italic">S</hi> on <hi rend="font-style: italic">BB'</hi> and <hi rend="font-style: italic">S'</hi> on <hi rend="font-style: italic">CC'</hi>. <hi rend="font-style: italic">SA</hi> and +<hi rend="font-style: italic">S'A'</hi> are corresponding rays of <hi rend="font-style: italic">S</hi> +and <hi rend="font-style: italic">S'</hi>, and the two pencils are +therefore in perspective position. +It is not difficult to see that the +axis of perspectivity <hi rend="font-style: italic">m</hi> is the line +joining <hi rend="font-style: italic">B'</hi> and <hi rend="font-style: italic">C</hi>. Given any point +<hi rend="font-style: italic">D</hi> on <hi rend="font-style: italic">u</hi>, to find the corresponding +point <hi rend="font-style: italic">D'</hi> on <hi rend="font-style: italic">u'</hi> we proceed as +follows: Join <hi rend="font-style: italic">D</hi> to <hi rend="font-style: italic">S</hi> and note +where the joining line meets <hi rend="font-style: italic">m</hi>. Join this point to <hi rend="font-style: italic">S'</hi>. +This last line meets <hi rend="font-style: italic">u'</hi> in the desired point <hi rend="font-style: italic">D'</hi>.</p> + +<p>We have now in this figure six lines of the system, +<hi rend="font-style: italic">a</hi>, <hi rend="font-style: italic">b</hi>, <hi rend="font-style: italic">c</hi>, <hi rend="font-style: italic">d</hi>, <hi rend="font-style: italic">u</hi>, and <hi rend="font-style: italic">u'</hi>. Fix now the position of <hi rend="font-style: italic">u</hi>, <hi rend="font-style: italic">u'</hi>, <hi rend="font-style: italic">b</hi>, <hi rend="font-style: italic">c</hi>, and +<hi rend="font-style: italic">d</hi>, and take four lines of the system, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">2</hi></hi>, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">3</hi></hi>, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">4</hi></hi>, which +meet <hi rend="font-style: italic">b</hi> in four harmonic points. These points project to +<pb n="51" /><anchor id="Pg51" /> +<hi rend="font-style: italic">D</hi>, giving four harmonic points on <hi rend="font-style: italic">m</hi>. These again project +to <hi rend="font-style: italic">D'</hi>, giving four harmonic points on <hi rend="font-style: italic">c</hi>. It is thus clear +that the rays <hi rend="font-style: italic">a<hi rend="vertical-align: sub">1</hi></hi>, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">2</hi></hi>, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">3</hi></hi>, <hi rend="font-style: italic">a<hi rend="vertical-align: sub">4</hi></hi> cut out two projective point-rows +on any two lines of the system. Thus <hi rend="font-style: italic">u</hi> and <hi rend="font-style: italic">u'</hi> are +not special rays, and any two rays of the system will +serve as the point-rows to generate the system of lines.</p></div> + +<div> +<index index="toc" level1="84. Brianchon's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p84" /><hi rend="font-weight: bold">84. Brianchon's theorem.</hi> From the figure also appears +a fundamental theorem due to Brianchon:</p> + +<p><hi rend="font-style: italic">If <hi rend="font-style: italic">1</hi>, <hi rend="font-style: italic">2</hi>, <hi rend="font-style: italic">3</hi>, <hi rend="font-style: italic">4</hi>, <hi rend="font-style: italic">5</hi>, <hi rend="font-style: italic">6</hi> are any six rays of a pencil of the +second order, then the lines <hi rend="font-style: italic">l = (12, 45)</hi>, <hi rend="font-style: italic">m = (23, 56)</hi>, +<hi rend="font-style: italic">n = (34, 61)</hi> all pass through a point.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image21.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 21</head> +<figDesc>Figure 21</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="85. Permutations of lines in Brianchon's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p85" /><hi rend="font-weight: bold">85.</hi> To make the notation fit the figure (Fig. 21), make +<hi rend="font-style: italic">a=1</hi>, <hi rend="font-style: italic">b = 2</hi>, <hi rend="font-style: italic">u' = 3</hi>, <hi rend="font-style: italic">d = 4</hi>, <hi rend="font-style: italic">u = 5</hi>, <hi rend="font-style: italic">c = 6</hi>; or, interchanging +two of the lines, <hi rend="font-style: italic">a = 1</hi>, +<hi rend="font-style: italic">c = 2</hi>, <hi rend="font-style: italic">u = 3</hi>, <hi rend="font-style: italic">d = 4</hi>, <hi rend="font-style: italic">u' = 5</hi>, +<hi rend="font-style: italic">b = 6</hi>. Thus, by different +namings of the +lines, it appears that +not more than 60 different +<hi rend="font-style: italic">Brianchon points</hi> +are possible. If we +call 12 and 45 opposite +vertices of a circumscribed +hexagon, +then Brianchon's theorem may be stated as follows:</p> + +<p><hi rend="font-style: italic">The three lines joining the three pairs of opposite vertices +of a hexagon circumscribed about a conic meet in a point.</hi></p></div> + +<div> +<index index="toc" level1="86. Construction of the penvil by Brianchon's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p86" /><hi rend="font-weight: bold">86. Construction of the pencil by Brianchon's theorem.</hi> +Brianchon's theorem furnishes a ready method of determining +a sixth line of the pencil of rays of the second +<pb n="52" /><anchor id="Pg52" /> +order when five are given. Thus, select a point in line +1 and suppose that line 6 is to pass through it. Then +<hi rend="font-style: italic">l = (12, 45)</hi>, <hi rend="font-style: italic">n = (34, 61)</hi>, and the line <hi rend="font-style: italic">m = (23, 56)</hi> must +pass through <hi rend="font-style: italic">(l, n)</hi>. Then <hi rend="font-style: italic">(23, ln)</hi> meets 5 in a point of +the required sixth line.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image22.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 22</head> +<figDesc>Figure 22</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="87. Point of contact +of a tangent to a conic" /><index index="pdf" /> +<head></head><p><anchor id="p87" /><hi rend="font-weight: bold">87. Point of contact +of a tangent to a conic.</hi> +If the line 2 approach +as a limiting position the +line 1, then the intersection +<hi rend="font-style: italic">(1, 2)</hi> 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 <hi rend="font-style: italic">1=6</hi>. +Draw <hi rend="font-style: italic">l = (12,45)</hi>, <hi rend="font-style: italic">m =(23,56)</hi>; +then <hi rend="font-style: italic">(34, lm)</hi> meets 1 in the +required point of contact <hi rend="font-style: italic">T</hi>.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image23.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 23</head> +<figDesc>Figure 23</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="88. Circumscribed quadrilateral" /><index index="pdf" /> +<head></head><p><anchor id="p88" /><hi rend="font-weight: bold">88. Circumscribed quadrilateral.</hi> +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)</p> + +<p><hi rend="font-style: italic">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.</hi> The +consequences of this theorem will be deduced later.</p> + +<pb n="53" /><anchor id="Pg53" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image24.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 24</head> +<figDesc>Figure 24</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="89. Circumscribed triangle" /><index index="pdf" /> +<head></head><p><anchor id="p89" /><hi rend="font-weight: bold">89. Circumscribed triangle.</hi> The hexagon may further +degenerate into a triangle, giving the theorem (Fig. 24) +<hi rend="font-style: italic">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.</hi></p></div> + +<div> +<index index="toc" level1="90. Use of Brianchon's theorem" /><index index="pdf" /> +<head></head><p><anchor id="p90" /><hi rend="font-weight: bold">90.</hi> Brianchon's theorem may +also be used to solve the following +problems:</p> + +<p><hi rend="font-style: italic">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.</hi></p></div> + +<div> +<index index="toc" level1="91. Harmonic tangents" /><index index="pdf" /> +<head></head><p><anchor id="p91" /><hi rend="font-weight: bold">91. Harmonic tangents.</hi> 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:</p> + +<p><hi rend="font-style: italic">Four tangents to a conic are said to be harmonic when +they meet every other tangent in four harmonic points.</hi></p></div> + +<div> +<index index="toc" level1="92. Projectivity and perspectivity" /><index index="pdf" /> +<head></head><p><anchor id="p92" /><hi rend="font-weight: bold">92. Projectivity and perspectivity.</hi> 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 <hi rend="font-style: italic">perspectively related</hi> 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.</p> + +<pb n="54" /><anchor id="Pg54" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image25.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 25</head> +<figDesc>Figure 25</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="93. Degenerate case" /><index index="pdf" /> +<head></head><p><anchor id="p93" /><hi rend="font-weight: bold">93.</hi> Brianchon's theorem may also be applied to a +degenerate conic made up of two points and the lines +through them. Thus(Fig. 25),</p> + +<p><hi rend="font-style: italic">If <hi rend="font-style: italic">a</hi>, <hi rend="font-style: italic">b</hi>, <hi rend="font-style: italic">c</hi> are three lines +through a point <hi rend="font-style: italic">S</hi>, and <hi rend="font-style: italic">a'</hi>, <hi rend="font-style: italic">b'</hi>, +<hi rend="font-style: italic">c'</hi> are three lines through another +point <hi rend="font-style: italic">S'</hi>, then the lines +<hi rend="font-style: italic">l = (ab', a'b)</hi>, <hi rend="font-style: italic">m = (bc', b'c)</hi>, +and <hi rend="font-style: italic">n = (ca', c'a)</hi> all meet in +a point.</hi></p></div> + +<div> +<index index="toc" level1="94. Law of duality" /><index index="pdf" /> +<head></head><p><anchor id="p94" /><hi rend="font-weight: bold">94. Law of duality.</hi> 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 <hi rend="font-style: italic">law of duality</hi> is based +will be developed in the next chapter.</p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>1. Given four lines in the plane, to construct another +which shall meet them in four harmonic points.</p> + +<p>2. Where are all such lines found?</p> + +<p>3. Given any five lines in the plane, construct on each +the point of contact with the conic tangent to them all.</p> + +<pb n="55" /><anchor id="Pg55" /> + +<p>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.)</p> + +<p>5. Given three lines and the point of contact on two of +them, to construct the conic.</p> + +<p>6. Given four lines and the line at infinity, to construct +the conic.</p> + +<p>7. Given three lines and the line at infinity, together +with the point of contact at infinity, to construct the conic.</p> + +<p>8. Given three lines, two of which are asymptotes, to +construct the conic.</p> + +<p>9. Given five tangents to a conic, to draw a tangent +which shall be parallel to any one of them.</p> + +<p>10. The lines <hi rend="font-style: italic">a</hi>, <hi rend="font-style: italic">b</hi>, <hi rend="font-style: italic">c</hi> are drawn parallel to each other. +The lines <hi rend="font-style: italic">a'</hi>, <hi rend="font-style: italic">b'</hi>, <hi rend="font-style: italic">c'</hi> are also drawn parallel to each other. +Show why the lines (<hi rend="font-style: italic">ab'</hi>, <hi rend="font-style: italic">a'b</hi>), (<hi rend="font-style: italic">bc'</hi>, <hi rend="font-style: italic">b'c</hi>), (<hi rend="font-style: italic">ca'</hi>, <hi rend="font-style: italic">c'a</hi>) meet in a +point. (In problems 6 to 10 inclusive, parallel lines are to +be drawn.)</p> +</div> +</div> + +<div rend="page-break-before: always"> +<index index="toc" /><index index="pdf" /> +<pb n="56" /><anchor id="Pg56" /> +<head>CHAPTER VI - POLES AND POLARS</head> + +<div> +<index index="toc" level1="95. Inscribed and circumscribed quadrilaterals" /><index index="pdf" /> +<head></head><p><anchor id="p95" /><hi rend="font-weight: bold">95. Inscribed and circumscribed quadrilaterals.</hi> The +following theorems have been noted as special cases of +Pascal's and Brianchon's theorems:</p> + +<p><hi rend="font-style: italic">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.</hi></p> + +<p><hi rend="font-style: italic">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.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image26.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 26</head> +<figDesc>Figure 26</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="96. Definition of the polar line of a point" /><index index="pdf" /> +<head></head><p><anchor id="p96" /><hi rend="font-weight: bold">96. Definition of the polar line of a point.</hi> Consider +the quadrilateral <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi> inscribed in the conic +(Fig. 26). It +determines the +four harmonic +points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, +<hi rend="font-style: italic">D</hi> which project +from <hi rend="font-style: italic">N</hi> in to +the four harmonic +points <hi rend="font-style: italic">M</hi>, +<hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">O</hi>. Now +the tangents at <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi> meet in <hi rend="font-style: italic">P</hi>, a point on the +line <hi rend="font-style: italic">AB</hi>. The line <hi rend="font-style: italic">AB</hi> is thus determined entirely by +<pb n="57" /><anchor id="Pg57" /> +the point <hi rend="font-style: italic">O</hi>. For if we draw any line through it, meeting +the conic in <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi>, and construct the harmonic +conjugate <hi rend="font-style: italic">B</hi> of <hi rend="font-style: italic">O</hi> with respect to <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi>, and also +the two tangents at <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi> which meet in the point +<hi rend="font-style: italic">P</hi>, then <hi rend="font-style: italic">BP</hi> is the line in question. It thus appears +that the line <hi rend="font-style: italic">LON</hi> may be any line whatever through <hi rend="font-style: italic">O</hi>; +and since <hi rend="font-style: italic">D</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">O</hi>, <hi rend="font-style: italic">N</hi> are four harmonic points, we +may describe the line <hi rend="font-style: italic">AB</hi> as the locus of points which +are harmonic conjugates of <hi rend="font-style: italic">O</hi> with respect to the two +points where any line through <hi rend="font-style: italic">O</hi> meets the curve.</p></div> + +<div> +<index index="toc" level1="97. Further defining properties" /><index index="pdf" /> +<head></head><p><anchor id="p97" /><hi rend="font-weight: bold">97.</hi> Furthermore, since the tangents at <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">N</hi> meet +on this same line, it appears as the locus of intersections +of pairs of tangents drawn at the extremities of chords +through <hi rend="font-style: italic">O</hi>.</p></div> + +<div> +<index index="toc" level1="98. Definition of the pole of a line" /><index index="pdf" /> +<head></head><p><anchor id="p98" /><hi rend="font-weight: bold">98.</hi> This important line, which is completely determined +by the point <hi rend="font-style: italic">O</hi>, is called the <hi rend="font-style: italic">polar</hi> of <hi rend="font-style: italic">O</hi> with +respect to the conic; and the point <hi rend="font-style: italic">O</hi> is called the <hi rend="font-style: italic">pole</hi> +of the line with respect to the conic.</p></div> + +<div> +<index index="toc" level1="99. Fundamental theorem of poles and polars" /><index index="pdf" /> +<head></head><p><anchor id="p99" /><hi rend="font-weight: bold">99.</hi> If a point <hi rend="font-style: italic">B</hi> is on the polar of <hi rend="font-style: italic">O</hi>, then it is harmonically +conjugate to <hi rend="font-style: italic">O</hi> with respect to the two intersections +<hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi> of the line <hi rend="font-style: italic">BC</hi> with the conic. But +for the same reason <hi rend="font-style: italic">O</hi> is on the polar of <hi rend="font-style: italic">B</hi>. We have, +then, the fundamental theorem</p> + +<p><hi rend="font-style: italic">If one point lies on the polar of a second, then the +second lies on the polar of the first.</hi></p></div> + +<div> +<index index="toc" level1="100. Conjugate points and lines" /><index index="pdf" /> +<head></head><p><anchor id="p100" /><hi rend="font-weight: bold">100. Conjugate points and lines.</hi> Such a pair of points +are said to be <hi rend="font-style: italic">conjugate</hi> with respect to the conic. Similarly, +lines are said to be <hi rend="font-style: italic">conjugate</hi> to each other with +respect to the conic if one, and consequently each, +passes through the pole of the other.</p> + +<pb n="58" /><anchor id="Pg58" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image27.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 27</head> +<figDesc>Figure 27</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="101. Construction of the polar line of a given point" /><index index="pdf" /> +<head></head><p><anchor id="p101" /><hi rend="font-weight: bold">101. Construction of the polar line of a given point.</hi> +Given a point <hi rend="font-style: italic">P</hi>, if it is within the conic (that is, if no +tangents may be drawn from <hi rend="font-style: italic">P</hi> 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 <hi rend="font-style: italic">P</hi> is +outside the conic, we may draw the two tangents and +construct the chord of contact (Fig. 27).</p></div> + +<div> +<index index="toc" level1="102. Self-polar triangle" /><index index="pdf" /> +<head></head><p><anchor id="p102" /><hi rend="font-weight: bold">102. Self-polar triangle.</hi> In Fig. 26 it is not difficult +to see that <hi rend="font-style: italic">AOC</hi> is a <hi rend="font-style: italic">self-polar</hi> triangle, that is, each +vertex is the pole of the opposite side. For <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">O</hi>, <hi rend="font-style: italic">K</hi> +are four harmonic points, and they project to <hi rend="font-style: italic">C</hi> in four +harmonic rays. The line <hi rend="font-style: italic">CO</hi>, therefore, meets the line +<hi rend="font-style: italic">AMN</hi> in a point on the polar of <hi rend="font-style: italic">A</hi>, being separated from +<hi rend="font-style: italic">A</hi> harmonically by the points <hi rend="font-style: italic">M</hi> and <hi rend="font-style: italic">N</hi>. Similarly, the +line <hi rend="font-style: italic">CO</hi> meets <hi rend="font-style: italic">KL</hi> in a point on the polar of <hi rend="font-style: italic">A</hi>, and +therefore <hi rend="font-style: italic">CO</hi> is the polar of <hi rend="font-style: italic">A</hi>. Similarly, <hi rend="font-style: italic">OA</hi> is the +polar of <hi rend="font-style: italic">C</hi>, and therefore <hi rend="font-style: italic">O</hi> is the pole of <hi rend="font-style: italic">AC</hi>.</p></div> + +<div> +<index index="toc" level1="103. Pole and polar projectively related" /><index index="pdf" /> +<head></head><p><anchor id="p103" /><hi rend="font-weight: bold">103. Pole and polar projectively related.</hi> Another very +important theorem comes directly from Fig. 26.</p> + +<p><hi rend="font-style: italic">As a point <hi rend="font-style: italic">A</hi> 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 <hi rend="font-style: italic">A</hi>.</hi></p> + +<p>For, fix the points <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">N</hi> and let the point <hi rend="font-style: italic">A</hi> move +along the line <hi rend="font-style: italic">AQ</hi>; then the point-row <hi rend="font-style: italic">A</hi> is projective +to the pencil <hi rend="font-style: italic">LK</hi>, and since <hi rend="font-style: italic">K</hi> moves along the conic, +the pencil <hi rend="font-style: italic">LK</hi> is projective to the pencil <hi rend="font-style: italic">NK</hi>, which in +turn is projective to the point-row <hi rend="font-style: italic">C</hi>, which, finally, is +projective to the pencil <hi rend="font-style: italic">OC</hi>, which is the polar of <hi rend="font-style: italic">A</hi>.</p> + +<pb n="59" /><anchor id="Pg59" /></div> + +<div> +<index index="toc" level1="104. Duality" /><index index="pdf" /> +<head></head><p><anchor id="p104" /><hi rend="font-weight: bold">104. Duality.</hi> 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 <hi rend="font-style: italic">measuring</hi>. If, therefore, we call any +theorem that has to do with the line at infinity or with +<pb n="60" /><anchor id="Pg60" /> +the measurement of angles a <hi rend="font-style: italic">metrical</hi> theorem, and any +other kind a <hi rend="font-style: italic">projective</hi> theorem, we may put the case +as follows:</p> + +<p><hi rend="font-style: italic">Any projective theorem involves another theorem, dual to +it, obtainable by interchanging everywhere the words 'point' +and 'line.'</hi></p></div> + +<div> +<index index="toc" level1="105. Self-dual theorems" /><index index="pdf" /> +<head></head><p><anchor id="p105" /><hi rend="font-weight: bold">105. Self-dual theorems.</hi> The theorems of this chapter +will be found, upon examination, to be <hi rend="font-style: italic">self-dual</hi>; +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.</p></div> + +<div> +<index index="toc" level1="106. Other correspondences" /><index index="pdf" /> +<head></head><p><anchor id="p106" /><hi rend="font-weight: bold">106.</hi> 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 <hi rend="font-style: italic">similarity</hi> +leaves ratios between corresponding segments unaltered. +The above statements apply only to the particular one-to-one +correspondence considered.</p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>1. Given a quadrilateral, construct the quadrangle polar +to it with respect to a given conic.</p> + +<p>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.</p> + +<pb n="61" /><anchor id="Pg61" /> + +<p>3. Given five points, draw the polar of a point with respect +to the conic passing through them, without drawing +the conic itself.</p> + +<p>4. Given five lines, draw the polar of a point with respect +to the conic tangent to them, without drawing the +conic itself.</p> + +<p>5. Dualize problems 3 and 4.</p> + +<p>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.</p> + +<p>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.</p> + +<p><hi rend="font-style: italic">Suggestion.</hi> Replace the given conic by a pair of protective pencils.</p> + +<p>8. Show that the poles of the tangents of one conic with +respect to another lie on a conic.</p> + +<p>9. The polar of a point <hi rend="font-style: italic">A</hi> with respect to one conic is <hi rend="font-style: italic">a</hi>, +and the pole of <hi rend="font-style: italic">a</hi> with respect to another conic is <hi rend="font-style: italic">A'</hi>. Show +that as <hi rend="font-style: italic">A</hi> travels along a line, <hi rend="font-style: italic">A'</hi> also travels along another +line. In general, if <hi rend="font-style: italic">A</hi> describes a curve of degree <hi rend="font-style: italic">n</hi>, show +that <hi rend="font-style: italic">A'</hi> describes another curve of the same degree <hi rend="font-style: italic">n</hi>. (The +degree of a curve is the greatest number of points that it +may have in common with any line in the plane.)</p> +</div> +</div> + +<div rend="page-break-before: always"> +<index index="toc" /><index index="pdf" /> +<pb n="62" /><anchor id="Pg62" /> + +<head>CHAPTER VII - METRICAL PROPERTIES OF THE CONIC SECTIONS</head> + +<div> +<index index="toc" level1="107. Diameters. Center" /><index index="pdf" /> +<head></head><p><anchor id="p107" /><hi rend="font-weight: bold">107. Diameters. Center.</hi> 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:</p> + +<p>The polar line of an infinitely distant point is called +a <hi rend="font-style: italic">diameter</hi>, and the pole of the infinitely distant line is +called the <hi rend="font-style: italic">center</hi>, of the conic.</p></div> + +<div> +<index index="toc" level1="108. Various theorems" /><index index="pdf" /> +<head></head><p><anchor id="p108" /><hi rend="font-weight: bold">108.</hi> From the harmonic properties of poles and polars,</p> + +<p><hi rend="font-style: italic">The center bisects all chords through it (§ 39).</hi></p> + +<p><hi rend="font-style: italic">Every diameter passes through the center.</hi></p> + +<p><hi rend="font-style: italic">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.</hi></p></div> + +<div> +<index index="toc" level1="109. Conjugate diameters" /><index index="pdf" /> +<head></head><p><anchor id="p109" /><hi rend="font-weight: bold">109. Conjugate diameters.</hi> We have already defined +conjugate lines as lines which pass each through the +pole of the other (§ 100).</p> + +<p><hi rend="font-style: italic">Any diameter bisects all chords parallel to its conjugate.</hi></p> + +<p><hi rend="font-style: italic">The tangents at the extremities of any diameter are +parallel, and parallel to the conjugate diameter.</hi></p> + +<p><hi rend="font-style: italic">Diameters parallel to the sides of a circumscribed parallelogram +are conjugate.</hi></p> + +<p>All these theorems are easy exercises for the student.</p> + +<pb n="63" /><anchor id="Pg63" /></div> + +<div> +<index index="toc" level1="110. Classification of conics" /><index index="pdf" /> +<head></head><p><anchor id="p110" /><hi rend="font-weight: bold">110. Classification of conics.</hi> 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 <hi rend="font-style: italic">hyperbola</hi>; if it has no point in +common with the infinitely distant line, it is called an +<hi rend="font-style: italic">ellipse</hi>; if it is tangent to the line at infinity, it is called +a <hi rend="font-style: italic">parabola</hi>.</p></div> + +<div> +<index index="toc" level1="111. Asymptotes" /><index index="pdf" /> +<head></head><p><anchor id="p111" /><hi rend="font-weight: bold">111.</hi> <hi rend="font-style: italic">In a hyperbola the center is outside the curve</hi> +(§ 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 <hi rend="font-style: italic">asymptotes</hi> of the curve. The +ellipse and the parabola have no asymptotes.</p></div> + +<div> +<index index="toc" level1="112. Various theorems" /><index index="pdf" /> +<head></head><p><anchor id="p112" /><hi rend="font-weight: bold">112.</hi> <hi rend="font-style: italic">The center of the parabola is at infinity, and therefore +all its diameters are parallel,</hi> for the pole of a tangent +line is the point of contact.</p> + +<p><hi rend="font-style: italic">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.</hi></p> + +<p><hi rend="font-style: italic">The center of an ellipse is within the curve.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image28.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 28</head> +<figDesc>Figure 28</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="113. Theorems concerning asymptotes" /><index index="pdf" /> +<head></head><p><anchor id="p113" /><hi rend="font-weight: bold">113. Theorems concerning asymptotes.</hi> 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 <hi rend="font-style: italic">A</hi> +and <hi rend="font-style: italic">B</hi> (Fig. 28). If, then, <hi rend="font-style: italic">O</hi> is the intersection of the +asymptotes,—and therefore the center of the curve,— +<pb n="64" /><anchor id="Pg64" /> +then the triangle <hi rend="font-style: italic">OAB</hi> is circumscribed about the curve. +By the theorem just quoted, the line through <hi rend="font-style: italic">A</hi> parallel +to <hi rend="font-style: italic">OB</hi>, the line through <hi rend="font-style: italic">B</hi> parallel to <hi rend="font-style: italic">OA</hi>, and the +line <hi rend="font-style: italic">OP</hi> through the point of +contact of the tangent <hi rend="font-style: italic">AB</hi> +all meet in a point <hi rend="font-style: italic">C</hi>. But +<hi rend="font-style: italic">OACB</hi> is a parallelogram, and +<hi rend="font-style: italic">PA = PB</hi>. Therefore</p> + +<p><hi rend="font-style: italic">The asymptotes cut off on +each tangent a segment which is +bisected by the point of contact.</hi></p></div> + +<div> +<index index="toc" level1="114. Asymptotes and conjugate diameters" /><index index="pdf" /> +<head></head><p><anchor id="p114" /><hi rend="font-weight: bold">114.</hi> If we draw a line <hi rend="font-style: italic">OQ</hi> +parallel to <hi rend="font-style: italic">AB</hi>, then <hi rend="font-style: italic">OP</hi> and <hi rend="font-style: italic">OQ</hi> are conjugate diameters, +since <hi rend="font-style: italic">OQ</hi> is parallel to the tangent at the point +where <hi rend="font-style: italic">OP</hi> meets the curve. Then, since <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">B</hi>, and +the point at infinity on <hi rend="font-style: italic">AB</hi> are four harmonic points, +we have the theorem</p> + +<p><hi rend="font-style: italic">Conjugate diameters of the hyperbola are harmonic +conjugates with respect to the asymptotes.</hi></p></div> + +<div> +<index index="toc" level1="115. Segments cut off on a chord by hyperbola and its asymptotes" /><index index="pdf" /> +<head></head><p><anchor id="p115" /><hi rend="font-weight: bold">115.</hi> The chord <hi rend="font-style: italic">A"B"</hi>, parallel to the diameter <hi rend="font-style: italic">OQ</hi>, is +bisected at <hi rend="font-style: italic">P'</hi> by the conjugate diameter <hi rend="font-style: italic">OP</hi>. If the +chord <hi rend="font-style: italic">A"B"</hi> meet the asymptotes in <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, then <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">P'</hi>, <hi rend="font-style: italic">B'</hi>, +and the point at infinity are four harmonic points, and +therefore <hi rend="font-style: italic">P'</hi> is the middle point of <hi rend="font-style: italic">A'B'</hi>. Therefore +<hi rend="font-style: italic">A'A" = B'B"</hi> and we have the theorem</p> + +<p><hi rend="font-style: italic">The segments cut off on any chord between the hyperbola +and its asymptotes are equal.</hi></p></div> + +<div> +<index index="toc" level1="116. Application of the theorem" /><index index="pdf" /> +<head></head><p><anchor id="p116" /><hi rend="font-weight: bold">116.</hi> This theorem furnishes a ready means of constructing +the hyperbola by points when a point on the +curve and the two asymptotes are given.</p> + +<pb n="65" /><anchor id="Pg65" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image29.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 29</head> +<figDesc>Figure 29</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="117. Triangle formed by the two asymptotes and a tangent" /><index index="pdf" /> +<head></head><p><anchor id="p117" /><hi rend="font-weight: bold">117.</hi> For the circumscribed quadrilateral, Brianchon's +theorem gave (§ 88) <hi rend="font-style: italic">The lines joining opposite vertices +and the lines joining opposite points of contact are four +lines meeting in a point.</hi> Take now for two of the +tangents the asymptotes, and let <hi rend="font-style: italic">AB</hi> and <hi rend="font-style: italic">CD</hi> be any +other two (Fig. 29). +If <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">D</hi> are opposite +vertices, and +also <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">C</hi>, then +<hi rend="font-style: italic">AC</hi> and <hi rend="font-style: italic">BD</hi> are parallel, +and parallel to +<hi rend="font-style: italic">PQ</hi>, the line joining +the points of contact +of <hi rend="font-style: italic">AB</hi> and <hi rend="font-style: italic">CD</hi>, +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 <hi rend="font-style: italic">ABC</hi> and <hi rend="font-style: italic">ADC</hi> are +equivalent, and therefore the triangles <hi rend="font-style: italic">AOB</hi> and <hi rend="font-style: italic">COD</hi> +are also. The tangent AB may be fixed, and the tangent +<hi rend="font-style: italic">CD</hi> chosen arbitrarily; therefore</p> + +<p><hi rend="font-style: italic">The triangle formed by any tangent to the hyperbola +and the two asymptotes is of constant area.</hi></p></div> + +<div> +<index index="toc" level1="118. Equation of hyperbola referred to the asymptotes" /><index index="pdf" /> +<head></head><p><anchor id="p118" /><hi rend="font-weight: bold">118. Equation of hyperbola referred to the asymptotes.</hi> +Draw through the point of contact <hi rend="font-style: italic">P</hi> of the tangent +<hi rend="font-style: italic">AB</hi> two lines, one parallel to one asymptote and the +other parallel to the other. One of these lines meets +<hi rend="font-style: italic">OB</hi> at a distance <hi rend="font-style: italic">y</hi> from <hi rend="font-style: italic">O</hi>, and the other meets <hi rend="font-style: italic">OA</hi> at +a distance <hi rend="font-style: italic">x</hi> from <hi rend="font-style: italic">O</hi>. Then, since <hi rend="font-style: italic">P</hi> is the middle point +<pb n="66" /><anchor id="Pg66" /> +of <hi rend="font-style: italic">AB</hi>, <hi rend="font-style: italic">x</hi> is one half of <hi rend="font-style: italic">OA</hi> and <hi rend="font-style: italic">y</hi> is one half of <hi rend="font-style: italic">OB</hi>. +The area of the parallelogram whose adjacent sides are +<hi rend="font-style: italic">x</hi> and <hi rend="font-style: italic">y</hi> is one half the area of the triangle <hi rend="font-style: italic">AOB</hi>, and +therefore, by the preceding paragraph, is constant. This +area is equal to <hi rend="font-style: italic">xy · <hi rend="font-style: normal">sin</hi> α</hi>, where α is the constant angle +between the asymptotes. It follows that the product <hi rend="font-style: italic">xy</hi> +is constant, and since <hi rend="font-style: italic">x</hi> and <hi rend="font-style: italic">y</hi> are the oblique coördinates +of the point <hi rend="font-style: italic">P</hi>, the asymptotes being the axes +of reference, we have</p> + +<p><hi rend="font-style: italic">The equation of the hyperbola, referred to the asymptotes +as axes, is <hi rend="font-style: italic">xy =</hi> constant.</hi></p> + +<p>This identifies the curve with the hyperbola as defined +and discussed in works on analytic geometry.</p></div> + +<div> +<index index="toc" level1="119. Equation of parabola" /><index index="pdf" /> +<head></head> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image30.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 30</head> +<figDesc>Figure 30</figDesc> +</figure></p> + +<p><anchor id="p119" /><hi rend="font-weight: bold">119. Equation of +parabola.</hi> 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 +<hi rend="font-style: italic">A</hi>, the points of contact +being <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">C</hi>. +These two tangents, +together with the +line at infinity, form +a triangle circumscribed +about the +conic. Draw through <hi rend="font-style: italic">B</hi> a parallel to <hi rend="font-style: italic">AC</hi>, and through +<hi rend="font-style: italic">C</hi> a parallel to <hi rend="font-style: italic">AB</hi>. If these meet in <hi rend="font-style: italic">D</hi>, then <hi rend="font-style: italic">AD</hi> is a +<pb n="67" /><anchor id="Pg67" /> +diameter. Let <hi rend="font-style: italic">AD</hi> meet the curve in <hi rend="font-style: italic">P</hi>, and the chord +<hi rend="font-style: italic">BC</hi> in <hi rend="font-style: italic">Q</hi>. <hi rend="font-style: italic">P</hi> is then the middle point of <hi rend="font-style: italic">AQ</hi>. Also, <hi rend="font-style: italic">Q</hi> +is the middle point of the chord <hi rend="font-style: italic">BC</hi>, and therefore the +diameter <hi rend="font-style: italic">AD</hi> bisects all chords parallel to <hi rend="font-style: italic">BC</hi>. In particular, +<hi rend="font-style: italic">AD</hi> passes through <hi rend="font-style: italic">P</hi>, the point of contact of +the tangent drawn parallel to <hi rend="font-style: italic">BC</hi>.</p> + +<p>Draw now another tangent, meeting <hi rend="font-style: italic">AB</hi> in <hi rend="font-style: italic">B'</hi> and <hi rend="font-style: italic">AC</hi> +in <hi rend="font-style: italic">C'</hi>. 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 <hi rend="font-style: italic">AC</hi> through <hi rend="font-style: italic">B'</hi>, a parallel to <hi rend="font-style: italic">AB</hi> through <hi rend="font-style: italic">C'</hi>, +and the line <hi rend="font-style: italic">BC</hi> meet in a point <hi rend="font-style: italic">D'</hi>. Also, from the similar +triangles <hi rend="font-style: italic">BB'D'</hi> and <hi rend="font-style: italic">BAC</hi> we have, for all positions of the +tangent line <hi rend="font-style: italic">B'C</hi>,</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">B'D' : BB' = AC : AB,</hi> +</p> + +<p>or, since <hi rend="font-style: italic">B'D' = AC'</hi>,</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AC': BB' = AC:AB =</hi> constant. +</p> + +<p>If another tangent meet <hi rend="font-style: italic">AB</hi> in <hi rend="font-style: italic">B"</hi> and <hi rend="font-style: italic">AC</hi> in <hi rend="font-style: italic">C"</hi>, we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic"> +AC' : BB' = AC" : BB", +</hi></p> + +<p>and by subtraction we get</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">C'C" : B'B" =</hi> constant; +</p> + +<p>whence</p> + +<p><hi rend="font-style: italic">The segments cut off on any two tangents to a parabola +by a variable tangent are proportional.</hi></p> + +<p>If now we take the tangent <hi rend="font-style: italic">B'C'</hi> as axis of ordinates, +and the diameter through the point of contact <hi rend="font-style: italic">O</hi> as axis +of abscissas, calling the coordinates of <hi rend="font-style: italic">B(x, y)</hi> and of +<hi rend="font-style: italic">C(x', y')</hi>, then, from the similar triangles <hi rend="font-style: italic">BMD'</hi> and +we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">y : y' = BD' : D'C = BB' : AB'.</hi> +</p> + +<p>Also</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">y : y' = B'D' : C'C = AC' : C'C.</hi> +</p> + +<pb n="68" /><anchor id="Pg68" /> + +<p>If now a line is drawn through <hi rend="font-style: italic">A</hi> parallel to a diameter, +meeting the axis of ordinates in <hi rend="font-style: italic">K</hi>, we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AK : OQ' = AC' : CC' = y : y',</hi> +</p> + +<p>and</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OM : AK = BB' : AB' = y : y',</hi> +</p> + +<p>and, by multiplication,</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OM : OQ' = y<hi rend="vertical-align: super">2</hi> : y'<hi rend="vertical-align: super">2</hi>,</hi> +</p> + +<p>or</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">x : x' = y<hi rend="vertical-align: super">2</hi> : y'<hi rend="vertical-align: super">2</hi>;</hi> +</p> + +<p>whence</p> + +<p><hi rend="font-style: italic">The abscissas of two points on a parabola are to each +other as the squares of the corresponding coördinates, a +diameter and the tangent to the curve at the extremity of +the diameter being the axes of reference.</hi></p> + +<p>The last equation may be written</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">y<hi rend="vertical-align: super">2</hi> = 2px,</hi> +</p> + +<p>where <hi rend="font-style: italic">2p</hi> stands for <hi rend="font-style: italic">y'<hi rend="vertical-align: super">2</hi> : x'</hi>.</p> + +<p>The parabola is thus identified with the curve of the +same name studied in treatises on analytic geometry.</p></div> + +<div> +<index index="toc" level1="120. Equation of central conics referred to conjugate +diameters" /><index index="pdf" /> +<head></head><p><anchor id="p120" /><hi rend="font-weight: bold">120. Equation of central conics referred to conjugate +diameters.</hi> Consider now a <hi rend="font-style: italic">central conic</hi>, 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 <hi rend="font-style: italic">A</hi> +and <hi rend="font-style: italic">B</hi> and the other in <hi rend="font-style: italic">C</hi> and <hi rend="font-style: italic">D</hi>, and let <hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">Q</hi> be +the points of contact of the parallel tangents <hi rend="font-style: italic">R</hi> and <hi rend="font-style: italic">S</hi> +of the others. Then <hi rend="font-style: italic">AC</hi>, <hi rend="font-style: italic">BD</hi>, <hi rend="font-style: italic">PQ</hi>, and <hi rend="font-style: italic">RS</hi> all meet in +a point <hi rend="font-style: italic">W</hi> (§ 88). From the figure,</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">PW : WQ = AP : QC = PD : BQ,</hi> +</p> + +<p>or</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AP · BQ = PD · QC.</hi> +</p> + +<pb n="69" /><anchor id="Pg69" /> + +<p>If now <hi rend="font-style: italic">DC</hi> is a fixed tangent and <hi rend="font-style: italic">AB</hi> a variable one, +we have from this equation</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AP · BQ = <hi rend="font-style: normal">constant.</hi></hi> +</p> + +<p>This constant will be positive or negative according as +<hi rend="font-style: italic">PA</hi> and <hi rend="font-style: italic">BQ</hi> are measured in the same or in opposite +directions. Accordingly we write</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AP · BQ = ± b<hi rend="vertical-align: super">2</hi>.</hi> +</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image31.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 31</head> +<figDesc>Figure 31</figDesc> +</figure></p> + +<p>Since <hi rend="font-style: italic">AD</hi> and <hi rend="font-style: italic">BC</hi> are parallel tangents, <hi rend="font-style: italic">PQ</hi> is a diameter +and the conjugate diameter is parallel to <hi rend="font-style: italic">AD</hi>. The +middle point of <hi rend="font-style: italic">PQ</hi> is the +center of the conic. We take +now for the axis of abscissas +the diameter <hi rend="font-style: italic">PQ</hi>, and the +conjugate diameter for the +axis of ordinates. Join <hi rend="font-style: italic">A</hi> to +<hi rend="font-style: italic">Q</hi> and <hi rend="font-style: italic">B</hi> to <hi rend="font-style: italic">P</hi> and draw a +line through <hi rend="font-style: italic">S</hi> parallel to +the axis of ordinates. These +three lines all meet in a point +<hi rend="font-style: italic">N</hi>, because <hi rend="font-style: italic">AP</hi>, <hi rend="font-style: italic">BQ</hi>, and <hi rend="font-style: italic">AB</hi> +form a triangle circumscribed +to the conic. Let <hi rend="font-style: italic">NS</hi> meet +<hi rend="font-style: italic">PQ</hi> in <hi rend="font-style: italic">M</hi>. Then, from the properties of the circumscribed +triangle (§ 89), <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi>, <hi rend="font-style: italic">S</hi>, and the point at infinity +on <hi rend="font-style: italic">NS</hi> are four harmonic points, and therefore <hi rend="font-style: italic">N</hi> is the +middle point of <hi rend="font-style: italic">MS</hi>. If the coördinates of <hi rend="font-style: italic">S</hi> are <hi rend="font-style: italic">(x, y)</hi>, +so that <hi rend="font-style: italic">OM</hi> is <hi rend="font-style: italic">x</hi> and <hi rend="font-style: italic">MS</hi> is <hi rend="font-style: italic">y</hi>, then <hi rend="font-style: italic">MN = y/2</hi>. Now +from the similar triangles <hi rend="font-style: italic">PMN</hi> and <hi rend="font-style: italic">PQB</hi> we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">BQ : PQ = NM : PM,</hi> +</p> + +<pb n="70" /><anchor id="Pg70" /> + +<p>and from the similar triangles <hi rend="font-style: italic">PQA</hi> and <hi rend="font-style: italic">MQN</hi>,</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">AP : PQ = MN : MQ,</hi> +</p> + +<p>whence, multiplying, we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">±b<hi rend="vertical-align: super">2</hi>/4 a<hi rend="vertical-align: super">2</hi> = y<hi rend="vertical-align: super">2</hi>/4 (a + x)(a - x),</hi> +</p> + +<p>where</p> + +<p rend="text-align: center"> +<formula notation="tex"> +\[ +a=\frac{PQ}{2}, +\] +</formula> +</p> + +<p>or, simplifying,</p> + +<p rend="text-align: center"> +<formula notation="tex"> +\[ +x^2/a^2 + y^2/\pm b^2 = 1, +\] +</formula> +</p> + +<p>which is the equation of an ellipse when <hi rend="font-style: italic">b<hi rend="vertical-align: super">2</hi></hi> has a positive +sign, and of a hyperbola when <hi rend="font-style: italic">b<hi rend="vertical-align: super">2</hi></hi> has a negative +sign. We have thus identified point-rows of the second +order with the curves given by equations of the second +degree.</p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>1. Draw a chord of a given conic which shall be bisected +by a given point <hi rend="font-style: italic">P</hi>.</p> + +<p>2. Show that all chords of a given conic that are bisected +by a given chord are tangent to a parabola.</p> + +<p>3. Construct a parabola, given two tangents with their +points of contact.</p> + +<p>4. Construct a parabola, given three points and the direction +of the diameters.</p> + +<p>5. A line <hi rend="font-style: italic">u'</hi> is drawn through the pole <hi rend="font-style: italic">U</hi> of a line <hi rend="font-style: italic">u</hi> and +at right angles to <hi rend="font-style: italic">u</hi>. The line <hi rend="font-style: italic">u</hi> revolves about a point <hi rend="font-style: italic">P</hi>. +Show that the line <hi rend="font-style: italic">u'</hi> is tangent to a parabola. (The lines <hi rend="font-style: italic">u</hi> +and <hi rend="font-style: italic">u'</hi> are called normal conjugates.)</p> + +<p>6. Given a circle and its center <hi rend="font-style: italic">O</hi>, to draw a line through +a given point <hi rend="font-style: italic">P</hi> parallel to a given line <hi rend="font-style: italic">q</hi>. Prove the following +construction: Let <hi rend="font-style: italic">p</hi> be the polar of <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">Q</hi> the pole of +<hi rend="font-style: italic">q</hi>, and <hi rend="font-style: italic">A</hi> the intersection of <hi rend="font-style: italic">p</hi> with <hi rend="font-style: italic">OQ</hi>. The polar of <hi rend="font-style: italic">A</hi> is +the desired line.</p> +</div> +</div> + +<div rend="page-break-before: always"> +<pb n="71" /><anchor id="Pg71" /> +<index index="toc" /><index index="pdf" /> +<head>CHAPTER VIII - INVOLUTION</head> + +<div> +<index index="toc" level1="121. Fundamental theorem" /><index index="pdf" /> +<head></head> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image32.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 32</head> +<figDesc>Figure 32</figDesc> +</figure></p> + +<p><anchor id="p121" /><hi rend="font-weight: bold">121. Fundamental theorem.</hi> 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 +<hi rend="font-style: italic">KL</hi>, <hi rend="font-style: italic">K'L'</hi>, <hi rend="font-style: italic">MN</hi>, +<hi rend="font-style: italic">M'N'</hi> do not +all meet in the +same point <hi rend="font-style: italic">A</hi>, +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:</p> + +<p><hi rend="font-style: italic">Given two complete quadrangles, <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi> and +<hi rend="font-style: italic">K'</hi>, <hi rend="font-style: italic">L'</hi>, <hi rend="font-style: italic">M'</hi>, <hi rend="font-style: italic">N'</hi>, so related that <hi rend="font-style: italic">KL</hi> and <hi rend="font-style: italic">K'L'</hi> meet in <hi rend="font-style: italic">A</hi>, +<hi rend="font-style: italic">MN</hi> and <hi rend="font-style: italic">M'N'</hi> in <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">KN</hi> and <hi rend="font-style: italic">K'N'</hi> in <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">LM</hi> and <hi rend="font-style: italic">L'M'</hi> +in <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">LN</hi> and <hi rend="font-style: italic">L'N'</hi> in <hi rend="font-style: italic">C</hi>, and <hi rend="font-style: italic">KM</hi> and <hi rend="font-style: italic">K'M'</hi> in <hi rend="font-style: italic">C'</hi>, then, +if <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">B'</hi>, and <hi rend="font-style: italic">C</hi> are in a straight line, the point <hi rend="font-style: italic">C'</hi> +also lies on that straight line.</hi></p> + +<p>The theorem follows from Desargues's theorem +(Fig. 32). It is seen that <hi rend="font-style: italic">KK'</hi>, <hi rend="font-style: italic">LL'</hi>, <hi rend="font-style: italic">MM'</hi>, <hi rend="font-style: italic">NN'</hi> all +<pb n="72" /><anchor id="Pg72" /> +meet in a point, and thus, from the same theorem, applied +to the triangles <hi rend="font-style: italic">KLM</hi> and <hi rend="font-style: italic">K'L'M'</hi>, the point <hi rend="font-style: italic">C'</hi> is on +the same line with <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B'</hi>. 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 +<hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>, two through <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">B'</hi>, and one through <hi rend="font-style: italic">C</hi>. +The sixth side must then go through <hi rend="font-style: italic">C'</hi>. Therefore,</p></div> + +<div> +<index index="toc" level1="122. Linear construction" /><index index="pdf" /> +<head></head><p><anchor id="p122" /><hi rend="font-weight: bold">122.</hi> <hi rend="font-style: italic">Two pairs of points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">A'</hi> and <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">B'</hi>, being given, +then the point <hi rend="font-style: italic">C'</hi> corresponding to any given point <hi rend="font-style: italic">C</hi> is +uniquely determined.</hi></p> + +<p>The construction of this sixth point is easily accomplished. +Draw through <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi> any two lines, and +cut across them by any line through <hi rend="font-style: italic">C</hi> in the points +<hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">N</hi>. Join <hi rend="font-style: italic">N</hi> to <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">L</hi> to <hi rend="font-style: italic">B'</hi>, thus determining +the points <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi> on the two lines through <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>, +The line <hi rend="font-style: italic">KM</hi> determines the desired point <hi rend="font-style: italic">C'</hi>. Manifestly, +starting from <hi rend="font-style: italic">C'</hi>, we come in this way always to the +same point <hi rend="font-style: italic">C</hi>. 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 <hi rend="font-style: italic">AA'</hi> and +<hi rend="font-style: italic">CC'</hi>, or from the pairs <hi rend="font-style: italic">BB'</hi> and <hi rend="font-style: italic">CC'</hi>.</p></div> + +<div> +<index index="toc" level1="123. Definition of involution of points on a line" /><index index="pdf" /> +<head></head><p><anchor id="p123" /><hi rend="font-weight: bold">123. Definition of involution of points on a line.</hi></p> + +<p><hi rend="font-style: italic">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.</hi></p> + +<pb n="73" /><anchor id="Pg73" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image33.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 33</head> +<figDesc>Figure 33</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="124. Double-points in an involution" /><index index="pdf" /> +<head></head><p><anchor id="p124" /><hi rend="font-weight: bold">124. Double-points in an involution.</hi> The points <hi rend="font-style: italic">C</hi> and +<hi rend="font-style: italic">C'</hi> describe projective point-rows, as may be seen by fixing +the points <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">M</hi>. The self-corresponding points, of +which there are two or none, are called the <hi rend="font-style: italic">double-points</hi> 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 <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">M</hi>, let the intersection +of the lines <hi rend="font-style: italic">CL</hi> and <hi rend="font-style: italic">C'M</hi> be <hi rend="font-style: italic">P</hi> (Fig. 33). The locus of <hi rend="font-style: italic">P</hi> is +a conic which goes through the double-points, because the +point-rows <hi rend="font-style: italic">C</hi> and +<hi rend="font-style: italic">C'</hi> are projective, +and therefore so +are the pencils +<hi rend="font-style: italic">LC</hi> and <hi rend="font-style: italic">MC'</hi> +which generate +the locus of <hi rend="font-style: italic">P</hi>. +Also, when <hi rend="font-style: italic">C</hi> +and <hi rend="font-style: italic">C'</hi> fall together, +the point +<hi rend="font-style: italic">P</hi> coincides with +them. Further, the tangents at <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">M</hi> to this conic +described by <hi rend="font-style: italic">P</hi> are the lines <hi rend="font-style: italic">LB</hi> and <hi rend="font-style: italic">MB</hi>. For in the +pencil at <hi rend="font-style: italic">L</hi> the ray <hi rend="font-style: italic">LM</hi> common to the two pencils which +generate the conic is the ray <hi rend="font-style: italic">LB'</hi> and corresponds to the +ray <hi rend="font-style: italic">MB</hi> of <hi rend="font-style: italic">M</hi>, which is therefore the tangent line to the +conic at <hi rend="font-style: italic">M</hi>. Similarly for the tangent <hi rend="font-style: italic">LB</hi> at <hi rend="font-style: italic">L</hi>. <hi rend="font-style: italic">LM</hi> is +therefore the polar of <hi rend="font-style: italic">B</hi> with respect to this conic, and +<hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">B'</hi> are therefore harmonic conjugates with respect +to the double-points. The same discussion applies to any +other pair of corresponding points in the involution.</p> + +<pb n="74" /><anchor id="Pg74" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image34.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 34</head> +<figDesc>Figure 34</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="125. Desargues's theorem concerning conics through +four points" /><index index="pdf" /> +<head></head><p><anchor id="p125" /><hi rend="font-weight: bold">125. Desargues's theorem concerning conics through +four points.</hi> Let <hi rend="font-style: italic">DD'</hi> be any pair of points in the involution +determined as above, and consider the conic +passing through the five points <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N</hi>, <hi rend="font-style: italic">D</hi>. We +shall use Pascal's theorem to show that this conic also +passes through <hi rend="font-style: italic">D'</hi>. The point <hi rend="font-style: italic">D'</hi> is determined as follows: +Fix <hi rend="font-style: italic">L</hi> and <hi rend="font-style: italic">M</hi> as before (Fig. 34) and join <hi rend="font-style: italic">D</hi> to <hi rend="font-style: italic">L</hi>, +giving on <hi rend="font-style: italic">MN</hi> +the point <hi rend="font-style: italic">N'</hi>. +Join <hi rend="font-style: italic">N'</hi> to <hi rend="font-style: italic">B</hi>, +giving on <hi rend="font-style: italic">LK</hi> +the point <hi rend="font-style: italic">K'</hi>. +Then <hi rend="font-style: italic">MK'</hi> determines +the +point <hi rend="font-style: italic">D'</hi> on +the line <hi rend="font-style: italic">AA'</hi>, +given by the +complete quadrangle +<hi rend="font-style: italic">K'</hi>, <hi rend="font-style: italic">L</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">N'</hi>. Consider the following six points, +numbering them in order: <hi rend="font-style: italic">D = 1</hi>, <hi rend="font-style: italic">D' = 2</hi>, <hi rend="font-style: italic">M = 3</hi>, <hi rend="font-style: italic">N = 4</hi>, +<hi rend="font-style: italic">K = 5</hi>, and <hi rend="font-style: italic">L = 6</hi>. We have the following intersections: +<hi rend="font-style: italic">B = (12-45)</hi>, <hi rend="font-style: italic">K' = (23-56)</hi>, <hi rend="font-style: italic">N' = (34-61)</hi>; and since by +construction <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">N</hi>, and <hi rend="font-style: italic">K'</hi> 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:</p> + +<p><hi rend="font-style: italic">The system of conics through four points meets any line +in the plane in pairs of points in involution.</hi></p></div> + +<div> +<index index="toc" level1="126. Degenerate conics of the system" /><index index="pdf" /> +<head></head><p><anchor id="p126" /><hi rend="font-weight: bold">126.</hi> It appears also that the six points in involution +determined by the quadrangle through the four fixed +<pb n="75" /><anchor id="Pg75" /> +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.</p></div> + +<div> +<index index="toc" level1="127. Conics through four points touching a given line" /><index index="pdf" /> +<head></head><p><anchor id="p127" /><hi rend="font-weight: bold">127. Conics through four points touching a given line.</hi> +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</p> + +<p><hi rend="font-style: italic">Through four fixed points in the plane two conics or +none may be drawn tangent to any given line.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image35.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 35</head> +<figDesc>Figure 35</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="128. Double correspondence" /><index index="pdf" /> +<head></head><p><anchor id="p128" /><hi rend="font-weight: bold">128. Double correspondence.</hi> 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, <hi rend="font-style: italic">a</hi> and <hi rend="font-style: italic">a'</hi>, which +both revolve about a fixed point <hi rend="font-style: italic">S</hi> +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 <hi rend="font-style: italic">S</hi> two projective point-rows, +which are not, however, in +involution unless the angle between the lines is a right +angles. For a point <hi rend="font-style: italic">P</hi> may correspond to a point <hi rend="font-style: italic">P'</hi>, +which in turn will correspond to some other point +<pb n="76" /><anchor id="Pg76" /> +than <hi rend="font-style: italic">P</hi>. 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 <hi rend="font-style: italic">P</hi> corresponds to a point <hi rend="font-style: italic">P'</hi>, then +the point <hi rend="font-style: italic">P'</hi> corresponds back again to the point <hi rend="font-style: italic">P</hi>. +The points <hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">P'</hi> are then said to <hi rend="font-style: italic">correspond doubly</hi>. +This notion is worthy of further study.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image36.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 36</head> +<figDesc>Figure 36</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="129. Steiner's construction" /><index index="pdf" /> +<head></head><p><anchor id="p129" /><hi rend="font-weight: bold">129. Steiner's construction.</hi> It will be observed that +the solution of the fundamental problem given in § 83, +<hi rend="font-style: italic">Given three pairs of points of two protective point-rows, to +construct other pairs</hi>, 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 <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>, ... and <hi rend="font-style: italic">A'</hi>, +<hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, <hi rend="font-style: italic">D'</hi>, ... be superposed on the line <hi rend="font-style: italic">u</hi>. Project +them both to a point <hi rend="font-style: italic">S</hi> and pass any conic <hi rend="font-style: italic">κ</hi> through <hi rend="font-style: italic">S</hi>. +We thus obtain two projective pencils, <hi rend="font-style: italic">a</hi>, <hi rend="font-style: italic">b</hi>, <hi rend="font-style: italic">c</hi>, <hi rend="font-style: italic">d</hi>, ... and +<pb n="77" /><anchor id="Pg77" /> +<hi rend="font-style: italic">a'</hi>, <hi rend="font-style: italic">b'</hi>, <hi rend="font-style: italic">c'</hi>, <hi rend="font-style: italic">d'</hi>, ... at <hi rend="font-style: italic">S</hi>, which meet the conic in the points +<hi rend="font-style: italic">α</hi>, <hi rend="font-style: italic">β</hi>, <hi rend="font-style: italic">γ</hi>, <hi rend="font-style: italic">δ</hi>, ... and +<hi rend="font-style: italic">α'</hi>, <hi rend="font-style: italic">β'</hi>, <hi rend="font-style: italic">γ'</hi>, <hi rend="font-style: italic">δ'</hi>, ... (Fig. 36). Take now +<hi rend="font-style: italic">γ</hi> as the center of a pencil projecting the points <hi rend="font-style: italic">α'</hi>, <hi rend="font-style: italic">β'</hi>, +<hi rend="font-style: italic">δ'</hi>, ..., and take <hi rend="font-style: italic">γ'</hi> as the center of a pencil projecting +the points <hi rend="font-style: italic">α</hi>, <hi rend="font-style: italic">β</hi>, <hi rend="font-style: italic">δ</hi>, .... 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 <hi rend="font-style: italic">(γα', γ'α)</hi> +to <hi rend="font-style: italic">(γβ', γ'β)</hi>. The correspondence between points in +the two point-rows on <hi rend="font-style: italic">u</hi> is now easily traced.</p></div> + +<div> +<index index="toc" level1="130. Application of Steiner's construction to double +correspondence" /><index index="pdf" /> +<head></head><p><anchor id="p130" /><hi rend="font-weight: bold">130. Application of Steiner's construction to double +correspondence.</hi> Steiner's construction throws into our +hands an important theorem concerning double correspondence: +<hi rend="font-style: italic">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.</hi> To +make this appear, let us call the point <hi rend="font-style: italic">A</hi> on <hi rend="font-style: italic">u</hi> by two +names, <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">P'</hi>, 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 <hi rend="font-style: italic">A'</hi> and <hi rend="font-style: italic">P</hi> must coincide +(Fig. 37). Take now any point <hi rend="font-style: italic">C</hi>, which we will +also call <hi rend="font-style: italic">R'</hi>. We must show that the corresponding +point <hi rend="font-style: italic">C'</hi> must also coincide with the point <hi rend="font-style: italic">B</hi>. Join all +the points to <hi rend="font-style: italic">S</hi>, as before, and it appears that the points +α and <hi rend="font-style: italic">Ï€'</hi> coincide, as also do the points <hi rend="font-style: italic">α'Ï€</hi> and <hi rend="font-style: italic">γÏ'</hi>. +By the above construction the line <hi rend="font-style: italic">γ'Ï</hi> must meet <hi rend="font-style: italic">γÏ'</hi> +on the line joining <hi rend="font-style: italic">(γα', γ'α)</hi> with +<hi rend="font-style: italic">(γπ', γ'Ï€)</hi>. But these +four points form a quadrangle inscribed in the conic, +and we know by § 95 that the tangents at the opposite +<pb n="78" /><anchor id="Pg78" /> +vertices <hi rend="font-style: italic">γ</hi> and <hi rend="font-style: italic">γ'</hi> meet on the line <hi rend="font-style: italic">v</hi>. The line <hi rend="font-style: italic">γ'Ï</hi> +is thus a tangent to the conic, and <hi rend="font-style: italic">C'</hi> and <hi rend="font-style: italic">R</hi> 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 <hi rend="font-style: italic">S</hi> 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.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image37.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 37</head> +<figDesc>Figure 37</figDesc> +</figure></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image38.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 38</head> +<figDesc>Figure 38</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="131. Involution of points on a point-row of the second order." /><index index="pdf" /> +<head></head><p><anchor id="p131" /><hi rend="font-weight: bold">131. Involution of points on a point-row of the second +order.</hi> 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 <hi rend="font-style: italic">the points of a conic +are paired in involution when they are corresponding +<pb n="79" /><anchor id="Pg79" /> +points of two projective point-rows superposed on the conic, +and when they correspond to each other doubly.</hi> With this +definition we may prove the theorem: <hi rend="font-style: italic">The lines joining +corresponding points of a point-row of the second order in +involution all pass through a fixed point <hi rend="font-style: italic">U</hi>, and the line +joining any two points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi> meets the line joining the +two corresponding points <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi> in the +points of a line <hi rend="font-style: italic">u</hi>, which is the polar +of <hi rend="font-style: italic">U</hi> with respect to the conic.</hi> For +take <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi> as the centers of two +pencils, the first perspective to the +point-row <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi> and the second +perspective to the point-row <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>. +Then, since the common ray of the +two pencils corresponds to itself, they are in perspective +position, and their axis of perspectivity <hi rend="font-style: italic">u</hi> (Fig. 38) +is the line which joins the point <hi rend="font-style: italic">(AB', A'B)</hi> to the +point <hi rend="font-style: italic">(AC', A'C)</hi>. It is then immediately clear, from +the theory of poles and polars, that <hi rend="font-style: italic">BB'</hi> and <hi rend="font-style: italic">CC'</hi> pass +through the pole <hi rend="font-style: italic">U</hi> of the line <hi rend="font-style: italic">u</hi>.</p></div> + +<div> +<index index="toc" level1="132. Involution of rays" /><index index="pdf" /> +<head></head><p><anchor id="p132" /><hi rend="font-weight: bold">132. Involution of rays.</hi> 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,</p> + +<p><hi rend="font-style: italic">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 <hi rend="font-style: italic">aa'</hi> and <hi rend="font-style: italic">bb'</hi> are fixed, and the line <hi rend="font-style: italic">c</hi> describes a pencil, +the corresponding line <hi rend="font-style: italic">c'</hi> also describes a pencil, and the +rays of the pencil are said to be paired in the involution +determined by <hi rend="font-style: italic">aa'</hi> and <hi rend="font-style: italic">bb'</hi>.</hi></p> + +<pb n="80" /><anchor id="Pg80" /></div> + +<div> +<index index="toc" level1="133. Double rays" /><index index="pdf" /> +<head></head><p><anchor id="p133" /><hi rend="font-weight: bold">133. Double rays.</hi> The self-corresponding rays, of +which there are two or none, are called <hi rend="font-style: italic">double rays</hi> 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:</p> + +<p><hi rend="font-style: italic">The tangents from a fixed point to a system of conics tangent +to four fixed lines form a pencil of rays in involution.</hi></p></div> + +<div> +<index index="toc" level1="134. Conic through a fixed point touching four lines" /><index index="pdf" /> +<head></head><p><anchor id="p134" /><hi rend="font-weight: bold">134.</hi> 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:</p> + +<p><hi rend="font-style: italic">Two conics or none may be drawn through a fixed point +to be tangent to four fixed lines.</hi></p></div> + +<div> +<index index="toc" level1="135. Double correspondence" /><index index="pdf" /> +<head></head><p><anchor id="p135" /><hi rend="font-weight: bold">135. Double correspondence.</hi> 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.</p> + +<pb n="81" /><anchor id="Pg81" /></div> + +<div> +<index index="toc" level1="136. Pencils of rays of the second order in involution" /><index index="pdf" /> +<head></head><p><anchor id="p136" /><hi rend="font-weight: bold">136. Pencils of rays of the second order in involution.</hi> +We may also extend the notion of involution to pencils +of rays of the second order. Thus, <hi rend="font-style: italic">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.</hi> We have then the theorem:</p></div> + +<div> +<index index="toc" level1="137. Theorem concerning pencils of the second +order in involution" /><index index="pdf" /> +<head></head><p><anchor id="p137" /><hi rend="font-weight: bold">137.</hi> <hi rend="font-style: italic">The intersections of corresponding rays of a pencil +of the second order in involution are all on a straight +line <hi rend="font-style: italic">u</hi>, and the intersection of any two tangents <hi rend="font-style: italic">ab</hi>, when +joined to the intersection of the corresponding tangents <hi rend="font-style: italic">a'b'</hi>, +gives a line which passes through a fixed point <hi rend="font-style: italic">U</hi>, the pole +of the line <hi rend="font-style: italic">u</hi> with respect to the conic.</hi></p></div> + +<div> +<index index="toc" level1="138. Involution of rays determined by a conic" /><index index="pdf" /> +<head></head><p><anchor id="p138" /><hi rend="font-weight: bold">138. Involution of rays determined by a conic.</hi> We +have seen in the theory of poles and polars (§ 103) +that if a point <hi rend="font-style: italic">P</hi> moves along a line <hi rend="font-style: italic">m</hi>, then the polar +of <hi rend="font-style: italic">P</hi> revolves about a point. This pencil cuts out on <hi rend="font-style: italic">m</hi> +another point-row <hi rend="font-style: italic">P'</hi>, projective also to <hi rend="font-style: italic">P</hi>. Since the +polar of <hi rend="font-style: italic">P</hi> passes through <hi rend="font-style: italic">P'</hi>, the polar of <hi rend="font-style: italic">P'</hi> also passes +through <hi rend="font-style: italic">P</hi>, so that the correspondence between <hi rend="font-style: italic">P</hi> and +<hi rend="font-style: italic">P'</hi> is double. The two point-rows are therefore in involution, +and the double points, if any exist, are the points +where the line <hi rend="font-style: italic">m</hi> 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 <hi rend="font-style: italic">conjugate</hi> with +respect to the conic (§ 100). We may then state the +following important theorem:</p></div> + +<div> +<index index="toc" level1="139. Statement of theorem" /><index index="pdf" /> +<head></head><p><anchor id="p139" /><hi rend="font-weight: bold">139.</hi> <hi rend="font-style: italic">A conic determines on every line in its plane an +involution of points, corresponding points in the involution +<pb n="82" /><anchor id="Pg82" /> +being conjugate with respect to the conic. The double points, +if any exist, are the points where the line meets the conic.</hi></p></div> + +<div> +<index index="toc" level1="140. Dual of the theorem" /><index index="pdf" /> +<head></head><p><anchor id="p140" /><hi rend="font-weight: bold">140.</hi> The dual theorem reads: <hi rend="font-style: italic">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.</hi></p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>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.</p> + +<p>5. State and prove the dual of problem 4.</p> + +<p>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.</p> + +<pb n="83" /><anchor id="Pg83" /> + +<p>7. If a triangle <hi rend="font-style: italic">QRS</hi> be inscribed in a conic, and if a +transversal <hi rend="font-style: italic">s</hi> meet two of its sides in <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>, the third +side and the tangent at the opposite vertex in <hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">B'</hi>, and +the conic itself in <hi rend="font-style: italic">C</hi> and <hi rend="font-style: italic">C'</hi>, then <hi rend="font-style: italic">AA'</hi>, <hi rend="font-style: italic">BB'</hi>, <hi rend="font-style: italic">CC'</hi> are three +pairs of points in an involution.</p> + +<p>8. Use the last exercise to solve the problem: Given five +points, <hi rend="font-style: italic">Q</hi>, <hi rend="font-style: italic">R</hi>, <hi rend="font-style: italic">S</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">C'</hi>, on a conic, to draw the tangent at any +one of them.</p> + +<p>9. State and prove the dual of problem 7 and use it to +prove the dual of problem 8.</p> + +<p>10. If a transversal cut two tangents to a conic in <hi rend="font-style: italic">B</hi> and +<hi rend="font-style: italic">B'</hi>, their chord of contact in <hi rend="font-style: italic">A</hi>, and the conic itself in <hi rend="font-style: italic">P</hi> +and <hi rend="font-style: italic">P'</hi>, then the point <hi rend="font-style: italic">A</hi> is a double point of the involution +determined by <hi rend="font-style: italic">BB'</hi> and <hi rend="font-style: italic">PP'</hi>.</p> + +<p>11. State and prove the dual of problem 10.</p> + +<p>12. If a variable conic pass through two given points, +<hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">P'</hi>, 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 <hi rend="font-style: italic">PP'</hi>.</p> + +<p>13. Use the last theorem to solve the problem: Given +four points, <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">P'</hi>, <hi rend="font-style: italic">Q</hi>, <hi rend="font-style: italic">S</hi>, on a conic, and the tangent at one of +them, <hi rend="font-style: italic">Q</hi>, to draw the tangent at any one of the other points, <hi rend="font-style: italic">S</hi>.</p> + +<p>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.</p> + +<p>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.</p> +</div> +</div> + + +<div rend="page-break-before: always"> +<pb n="84" /><anchor id="Pg84" /> +<index index="toc" /><index index="pdf" /> +<head>CHAPTER IX - METRICAL PROPERTIES OF INVOLUTIONS</head> + +<div> +<index index="toc" level1="141. Introduction of infinite point; center of involution" /><index index="pdf" /> +<head></head> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image39.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 39</head> +<figDesc>Figure 39</figDesc> +</figure></p> + +<p><anchor id="p141" /><hi rend="font-weight: bold">141. Introduction of infinite point; center of involution.</hi> +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 <hi rend="font-style: italic">center</hi> 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 +<hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi> any two rays and cut them by a line parallel +to <hi rend="font-style: italic">AA'</hi> in the points <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">M</hi>. Join these points to +<hi rend="font-style: italic">B</hi> and <hi rend="font-style: italic">B'</hi>, thus determining on <hi rend="font-style: italic">AK</hi> and <hi rend="font-style: italic">AN</hi> the points <hi rend="font-style: italic">L</hi> +and <hi rend="font-style: italic">N</hi>. <hi rend="font-style: italic">LN</hi> meets <hi rend="font-style: italic">AA'</hi> in the center <hi rend="font-style: italic">O</hi> of the involution.</p> + +<pb n="85" /><anchor id="Pg85" /></div> + +<div> +<index index="toc" level1="142. Fundamental metrical theorem" /><index index="pdf" /> +<head></head><p><anchor id="p142" /><hi rend="font-weight: bold">142. Fundamental metrical theorem.</hi> From the figure +we see that the triangles <hi rend="font-style: italic">OLB'</hi> and <hi rend="font-style: italic">PLM</hi> are similar, <hi rend="font-style: italic">P</hi> +being the intersection of KM and LN. Also the triangles +<hi rend="font-style: italic">KPN</hi> and <hi rend="font-style: italic">BON</hi> are similar. We thus have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OB : PK = ON : PN</hi> +</p> + +<p>and</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OB' : PM = OL : PL;</hi> +</p> + +<p>whence</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OB · OB' : PK · PM = ON · OL : PN · PL.</hi> +</p> + +<p>In the same way, from the similar triangles <hi rend="font-style: italic">OAL</hi> and +<hi rend="font-style: italic">PKL</hi>, and also <hi rend="font-style: italic">OA'N</hi> and <hi rend="font-style: italic">PMN</hi>, we obtain</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OA · OA' : PK · PM = ON · OL : PN · PL,</hi> +</p> + +<p>and this, with the preceding, gives at once the fundamental +theorem, which is sometimes taken also as the +definition of involution:</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">OA · OA' = OB · OB' = <hi rend="font-style: normal">constant</hi>,</hi> +</p> + +<p>or, in words,</p> + +<p><hi rend="font-style: italic">The product of the distances from the center to two corresponding +points in an involution of points is constant.</hi></p></div> + +<div> +<index index="toc" level1="143. Existence of double points" /><index index="pdf" /> +<head></head><p><anchor id="p143" /><hi rend="font-weight: bold">143. Existence of double points.</hi> 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 <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi> lie both on the same side +of the center, the product <hi rend="font-style: italic">OA · OA'</hi> 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 <hi rend="font-style: italic">BB'</hi>. Then, +since <hi rend="font-style: italic">OA · OA' = OB · OB'</hi>, it cannot happen that <hi rend="font-style: italic">B</hi> and +<hi rend="font-style: italic">B'</hi> are separated from each other by <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>. This is +evident enough if the points are on opposite sides of +the center. If the pairs are on the same side of the +<pb n="86" /><anchor id="Pg86" /> +center, and <hi rend="font-style: italic">B</hi> lies between <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi>, so that <hi rend="font-style: italic">OB</hi> is +greater, say, than <hi rend="font-style: italic">OA</hi>, but less than <hi rend="font-style: italic">OA'</hi>, then, by the +equation <hi rend="font-style: italic">OA · OA' = OB · OB'</hi>, we must have <hi rend="font-style: italic">OB'</hi> also +less than <hi rend="font-style: italic">OA'</hi> and greater than <hi rend="font-style: italic">OA</hi>. A similar discussion +may be made for the case where <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">A'</hi> lie on +opposite sides of <hi rend="font-style: italic">O</hi>. The results may be stated as +follows, without any reference to the center:</p> + +<p><hi rend="font-style: italic">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.</hi></p></div> + +<div> +<index index="toc" level1="144. Existence of double rays" /><index index="pdf" /> +<head></head><p><anchor id="p144" /><hi rend="font-weight: bold">144.</hi> 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.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image40.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 40</head> +<figDesc>Figure 40</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="145. Construction of an involution by means of circles" /><index index="pdf" /> +<head></head><p><anchor id="p145" /><hi rend="font-weight: bold">145. Construction of +an involution by means of circles.</hi> The equation just derived, <hi +rend="font-style: italic">OA · OA' = OB · OB'</hi>, indicates another +simple way in which points of an involution of points may be +constructed. Through <hi rend="font-style: italic">A</hi> and <hi +rend="font-style: italic">A'</hi> draw any circle, and draw also any +circle through <hi rend="font-style: italic">B</hi> and <hi +rend="font-style: italic">B'</hi> to cut the first in the two points <hi +rend="font-style: italic">G</hi> and <hi rend="font-style: +italic">G'</hi> (Fig. 40). Then any circle through <hi rend="font-style: +italic">G</hi> and <hi rend="font-style: italic">G'</hi> will meet the +line in pairs of points in the involution determined by <hi +rend="font-style: italic">AA'</hi> and <hi rend="font-style: +italic">BB'</hi>. For if such a circle meets the line in the points <hi +rend="font-style: italic">CC'</hi>, then, by the theorem in the geometry +of the circle which says that <hi rend="font-style: italic">if any chord +is +<pb n="87" /><anchor id="Pg87" /> +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</hi>, we have <hi rend="font-style: italic">OC · +OC' = OG · OG' =</hi> constant. So that for all such points +<hi rend="font-style: italic">OA · OA' = OB · OB' = OC · +OC'</hi>. Further, the line <hi rend="font-style: italic">GG'</hi> +meets <hi rend="font-style: italic">AA'</hi> in the center of the +involution. To find the double points, if they exist, we draw a tangent +from <hi rend="font-style: italic">O</hi> to any of the circles through +<hi rend="font-style: italic">GG'</hi>. Let <hi rend="font-style: +italic">T</hi> be the point of contact. Then lay off on the line <hi +rend="font-style: italic">OA</hi> a line <hi rend="font-style: +italic">OF</hi> equal to <hi rend="font-style: italic">OT</hi>. Then, +since by the above theorem of elementary geometry +<hi rend="font-style: italic">OA · OA' = OT<hi rend="vertical-align: super">2</hi> = OF<hi rend="vertical-align: super">2</hi></hi>, we have one double +point <hi rend="font-style: italic">F</hi>. The other is at an equal +distance on the other side of <hi rend="font-style: italic">O</hi>. 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.</p></div> + +<div> +<index index="toc" level1="146. Circular points" /><index index="pdf" /> +<head></head><p><anchor id="p146" /><hi rend="font-weight: bold">146.</hi> 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 +<hi rend="font-style: italic">circular points</hi> and usually denoted by <hi rend="font-style: italic">I</hi> and <hi rend="font-style: italic">J</hi>. 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.</p> + +<pb n="88" /><anchor id="Pg88" /> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image41.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 41</head> +<figDesc>Figure 41</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="147. Pairs in an involution of rays which are at right +angles. Circular involution" /><index index="pdf" /> +<head></head><p><anchor id="p147" /><hi rend="font-weight: bold">147. Pairs in an involution of rays which are at right +angles. Circular involution.</hi> 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 <hi rend="font-style: italic">G</hi> and <hi rend="font-style: italic">G'</hi> lie on the perpendicular +bisector of the line <hi rend="font-style: italic">GG'</hi>. Let +this line meet the line <hi rend="font-style: italic">AA'</hi> +in the point <hi rend="font-style: italic">C</hi> (Fig. 41), and +draw the circle with center <hi rend="font-style: italic">C</hi> +which goes through <hi rend="font-style: italic">G</hi> and <hi rend="font-style: italic">G'</hi>. +This circle cuts out two points +<hi rend="font-style: italic">M</hi> and <hi rend="font-style: italic">M'</hi> in the involution. The rays <hi rend="font-style: italic">GM</hi> and <hi rend="font-style: italic">GM'</hi> are +clearly at right angles, being inscribed in a semicircle. +If, therefore, the involution of points is projected to +<hi rend="font-style: italic">G</hi>, we have found two corresponding rays which are +at right angles to each other. Given now any involution +of rays with center <hi rend="font-style: italic">G</hi>, we may cut across it +by a straight line and proceed to find the two points +<hi rend="font-style: italic">M</hi> and <hi rend="font-style: italic">M'</hi>. Clearly there will be only one such pair +unless the perpendicular bisector of <hi rend="font-style: italic">GG'</hi> coincides with +the line <hi rend="font-style: italic">AA'</hi>. In this case every ray is at right angles +to its corresponding ray, and the involution is called +<hi rend="font-style: italic">circular</hi>.</p></div> + +<div> +<index index="toc" level1="148. Axes of conics" /><index index="pdf" /> +<head></head><p><anchor id="p148" /><hi rend="font-weight: bold">148. Axes of conics.</hi> At the close of the last chapter +(§ 140) we gave the theorem: <hi rend="font-style: italic">A conic determines at every +point in its plane an involution of rays, corresponding rays +<pb n="89" /><anchor id="Pg89" /> +being conjugate with respect to the conic. The double rays, +if any exist, are the tangents from the point to the conic.</hi> +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 <hi rend="font-style: italic">axis</hi> 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.</p></div> + +<div> +<index index="toc" level1="149. Points at which the involution determined by +a conic is circular" /><index index="pdf" /> +<head></head><p><anchor id="p149" /><hi rend="font-weight: bold">149. Points at which the involution determined by +a conic is circular.</hi> 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.</p> + +<pb n="90" /><anchor id="Pg90" /></div> + +<div> +<index index="toc" level1="150. Properties of such a point" /><index index="pdf" /> +<head></head><p><anchor id="p150" /><hi rend="font-weight: bold">150.</hi> 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 <hi rend="font-style: italic">F</hi> +and <hi rend="font-style: italic">F'</hi> were two such points, then, since the conjugate +ray at <hi rend="font-style: italic">F</hi> to the line <hi rend="font-style: italic">FF'</hi> must be at right angles to it, +and also since the conjugate ray at <hi rend="font-style: italic">F'</hi> to the line <hi rend="font-style: italic">FF'</hi> +must be at right angles to it, the pole of <hi rend="font-style: italic">FF'</hi> must +be at infinity in a direction at right angles to <hi rend="font-style: italic">FF'</hi>. +The line <hi rend="font-style: italic">FF'</hi> 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.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image42.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 42</head> +<figDesc>Figure 42</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="151. Position of such a point" /><index index="pdf" /> +<head></head><p><anchor id="p151" /><hi rend="font-weight: bold">151.</hi> Let now <hi rend="font-style: italic">P</hi> be a point on one of the axes (Fig. 42), +and draw any ray through it, such as <hi rend="font-style: italic">q</hi>. As <hi rend="font-style: italic">q</hi> revolves +about <hi rend="font-style: italic">P</hi>, its pole <hi rend="font-style: italic">Q</hi> moves along a line at right angles +to the axis on which <hi rend="font-style: italic">P</hi> lies, describing a point-row <hi rend="font-style: italic">p</hi> +projective to the pencil of rays <hi rend="font-style: italic">q</hi>. The point at infinity +in a direction at right angles to <hi rend="font-style: italic">q</hi> also describes a point-row +projective to <hi rend="font-style: italic">q</hi>. The line joining corresponding +points of these two point-rows is always a conjugate +line to <hi rend="font-style: italic">q</hi> and at right angles to <hi rend="font-style: italic">q</hi>, or, as we may call it, +a <hi rend="font-style: italic">conjugate normal</hi> to <hi rend="font-style: italic">q</hi>. These conjugate normals to <hi rend="font-style: italic">q</hi>, +joining as they do corresponding points in two projective +point-rows, form a pencil of rays of the second +<pb n="91" /><anchor id="Pg91" /> +order. But since the point at infinity on the point-row +<hi rend="font-style: italic">Q</hi> corresponds to the point at infinity in a direction +at right angles to <hi rend="font-style: italic">q</hi>, these point-rows are in perspective +position and the normal conjugates of all the lines +through <hi rend="font-style: italic">P</hi> meet in a point. This point lies on the +same axis with <hi rend="font-style: italic">P</hi>, as is seen by taking <hi rend="font-style: italic">q</hi> at right angles +to the axis on which <hi rend="font-style: italic">P</hi> lies. The center of this pencil +may be called <hi rend="font-style: italic">P'</hi>, and thus we have paired the point <hi rend="font-style: italic">P</hi> +with the point <hi rend="font-style: italic">P'</hi>. By moving the point <hi rend="font-style: italic">P</hi> along the +axis, and by keeping the +ray <hi rend="font-style: italic">q</hi> parallel to a fixed +direction, we may see that +the point-row <hi rend="font-style: italic">P</hi> and the +point-row <hi rend="font-style: italic">P'</hi> are projective. +Also the correspondence is +double, and by starting +from the point <hi rend="font-style: italic">P'</hi> we arrive +at the point <hi rend="font-style: italic">P</hi>. Therefore +the point-rows <hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">P'</hi> 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 <hi rend="font-style: italic">P</hi> and the corresponding rays through +<hi rend="font-style: italic">P'</hi> are conjugate normals; and if <hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">P'</hi> 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.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image43.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 43</head> +<figDesc>Figure 43</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="152. Discovery of the foci of the conic" /><index index="pdf" /> +<head></head><p><anchor id="p152" /><hi rend="font-weight: bold">152. Discovery of the foci of the conic.</hi> We know +that on one axis no such points as we are seeking can +lie (§ 150). The involution of points <hi rend="font-style: italic">PP'</hi> on this axis +<pb n="92" /><anchor id="Pg92" /> +can therefore have no double points. Nevertheless, let +<hi rend="font-style: italic">PP'</hi> and <hi rend="font-style: italic">RR'</hi> be two pairs of corresponding points on +this axis (Fig. 43). Then we know that <hi rend="font-style: italic">P</hi> and <hi rend="font-style: italic">P'</hi> are +separated from each other by <hi rend="font-style: italic">R</hi> and <hi rend="font-style: italic">R'</hi> (§ 143). Draw +a circle on <hi rend="font-style: italic">PP'</hi> as a diameter, and one on <hi rend="font-style: italic">RR'</hi> as a +diameter. These must intersect in +two points, <hi rend="font-style: italic">F</hi> and <hi rend="font-style: italic">F'</hi>, and since the +center of the conic is the center +of the involution <hi rend="font-style: italic">PP'</hi>, <hi rend="font-style: italic">RR'</hi>, as is +easily seen, it follows that <hi rend="font-style: italic">F</hi> and <hi rend="font-style: italic">F'</hi> +are on the other axis of the conic. +Moreover, <hi rend="font-style: italic">FR</hi> and <hi rend="font-style: italic">FR'</hi> are conjugate +normal rays, since <hi rend="font-style: italic">RFR'</hi> is +inscribed in a semicircle, and the +two rays go one through <hi rend="font-style: italic">R</hi> and the other through <hi rend="font-style: italic">R'</hi>. +The involution of points <hi rend="font-style: italic">PP'</hi>, <hi rend="font-style: italic">RR'</hi> therefore projects +to the two points <hi rend="font-style: italic">F</hi> and <hi rend="font-style: italic">F'</hi> in two pencils of rays in +involution which have for corresponding rays conjugate +normals to the conic. We may, then, say:</p> + +<p><hi rend="font-style: italic">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.</hi></p></div> + +<div> +<index index="toc" level1="153. The circle and the parabola" /><index index="pdf" /> +<head></head><p><anchor id="p153" /><hi rend="font-weight: bold">153. The circle and the parabola.</hi> 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 +<pb n="93" /><anchor id="Pg93" /> +axis. It is seen, however, that as <hi rend="font-style: italic">P</hi> moves to infinity, +carrying the line <hi rend="font-style: italic">q</hi> with it, <hi rend="font-style: italic">q</hi> becomes the line at infinity, +which for the parabola is a tangent line. Its pole +<hi rend="font-style: italic">Q</hi> is thus at infinity and also the point <hi rend="font-style: italic">P'</hi>, so that <hi rend="font-style: italic">P</hi> +and <hi rend="font-style: italic">P'</hi> fall together at infinity, and therefore one focus +of the parabola is at infinity. There must therefore be +another, so that</p> + +<p><hi rend="font-style: italic">A parabola has one and only one focus in the finite +part of the plane.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image44.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 44</head> +<figDesc>Figure 44</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="154. Focal properties of conics" /><index index="pdf" /> +<head></head><p><anchor id="p154" /><hi rend="font-weight: bold">154. Focal properties of conics.</hi> 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 <hi rend="font-style: italic">P</hi>. These +two lines are clearly conjugate +normals. The two points <hi rend="font-style: italic">T</hi> and +<hi rend="font-style: italic">N</hi>, 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 <hi rend="font-style: italic">P</hi>, 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</p> + +<p><hi rend="font-style: italic">The lines joining a point on the conic to the foci make +equal angles with the tangent.</hi></p> + +<p>It follows that rays from a source of light at one +focus are reflected by an ellipse to the other.</p> + +<pb n="94" /><anchor id="Pg94" /></div> + +<div> +<index index="toc" level1="155. Case of the parabola" /><index index="pdf" /> +<head></head><p><anchor id="p155" /><hi rend="font-weight: bold">155.</hi> 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</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image45.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 45</head> +<figDesc>Figure 45</figDesc> +</figure></p> + +<p><hi rend="font-style: italic">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).</hi></p></div> + +<div> +<index index="toc" level1="156. Parabolic reflector" /><index index="pdf" /> +<head></head><p><anchor id="p156" /><hi rend="font-weight: bold">156.</hi> 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.</p></div> + +<div> +<index index="toc" level1="157. Directrix. Principal axis. Vertex" /><index index="pdf" /> +<head></head><p><anchor id="p157" /><hi rend="font-weight: bold">157. Directrix. Principal axis. Vertex.</hi> The polar of +the focus with respect to the conic is called the <hi rend="font-style: italic">directrix</hi>. +The axis which contains the foci is called the <hi rend="font-style: italic">principal +axis</hi>, and the intersection of the axis with the curve is +called the <hi rend="font-style: italic">vertex</hi> 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.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image46.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 46</head> +<figDesc>Figure 46</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="158. Another definition of a conic" /><index index="pdf" /> +<head></head><p><anchor id="p158" /><hi rend="font-weight: bold">158. Another definition of a conic.</hi> Let <hi rend="font-style: italic">P</hi> be any point +on the directrix through which a line is drawn meeting +the conic in the points <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi> (Fig. 46). Let the tangents +at <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi> meet in <hi rend="font-style: italic">T</hi>, and call the focus <hi rend="font-style: italic">F</hi>. Then +<hi rend="font-style: italic">TF</hi> and <hi rend="font-style: italic">PF</hi> are conjugate lines, and as they pass through +a focus they must be at right angles to each other. Let +<pb n="95" /><anchor id="Pg95" /> +<hi rend="font-style: italic">TF</hi> meet <hi rend="font-style: italic">AB</hi> in <hi rend="font-style: italic">C</hi>. Then <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">B</hi> are four harmonic +points. Project these four points parallel to <hi rend="font-style: italic">TF</hi> upon +the directrix, and we then get +the four harmonic points <hi rend="font-style: italic">P</hi>, +<hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">Q</hi>, <hi rend="font-style: italic">N</hi>. Since, now, <hi rend="font-style: italic">TFP</hi> is +a right angle, the angles <hi rend="font-style: italic">MFQ</hi> +and <hi rend="font-style: italic">NFQ</hi> are equal, as well +as the angles <hi rend="font-style: italic">AFC</hi> and <hi rend="font-style: italic">BFC</hi>. +Therefore the triangles <hi rend="font-style: italic">MAF</hi> +and <hi rend="font-style: italic">NFB</hi> are similar, and +<hi rend="font-style: italic">FA : FM = FB : BN</hi>. Dropping +perpendiculars <hi rend="font-style: italic">AA</hi> and <hi rend="font-style: italic">BB'</hi> +upon the directrix, this becomes +<hi rend="font-style: italic">FA : AA' = FB : BB'</hi>. We +have thus the property often taken as the definition +of a conic:</p> + +<p><hi rend="font-style: italic">The ratio of the distances from a point on the conic to +the focus and the directrix is constant.</hi></p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image47.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 47</head> +<figDesc>Figure 47</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="159. Eccentricity" /><index index="pdf" /> +<head></head><p><anchor id="p159" /><hi rend="font-weight: bold">159. Eccentricity.</hi> 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 +<hi rend="font-style: italic">eccentricity</hi>.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image48.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 48</head> +<figDesc>Figure 48</figDesc> +</figure></p></div> + +<div> +<index index="toc" level1="160. Sum or difference of focal +distances" /><index index="pdf" /> +<head></head><p><anchor id="p160" /><hi rend="font-weight: bold">160. Sum or difference of focal +distances.</hi> 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 +<pb n="96" /><anchor id="Pg96" /> +a point on a conic to the two foci are <hi rend="font-style: italic">r</hi> and <hi rend="font-style: italic">r'</hi>, and +the distances from the same point to the corresponding +directrices are <hi rend="font-style: italic">d</hi> and <hi rend="font-style: italic">d'</hi> +(Fig. 47), we have <hi rend="font-style: italic">r : d = r' : d'</hi>; +<hi rend="font-style: italic">(r ± r') : (d ± d')</hi>. In the +ellipse <hi rend="font-style: italic">(d + d')</hi> is constant, +being the distance between +the directrices. In the hyperbola +this distance is <hi rend="font-style: italic">(d - d')</hi>. +It follows (Fig. 48) that</p> + +<p><hi rend="font-style: italic">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.</hi></p> +</div> + +<div> +<index index="toc" /><index index="pdf" /> +<head>PROBLEMS</head> + +<p>1. Construct the axis of a parabola, given four tangents.</p> + +<p>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 <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi>. The circle on <hi rend="font-style: italic">AB</hi> as diameter will +pass through the foci of the conic.</p> + +<p>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.</p> + +<p>4. Given the tangent at the vertex of a parabola, and +two other tangents, to find the focus.</p> + +<p>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.</p> + +<p>6. Given the axis of a parabola and a tangent, with its +point of contact, to find the focus.</p> + +<pb n="97" /><anchor id="Pg97" /> + +<p>7. The locus of the center of a circle which touches a +given line and a given circle consists of two parabolas.</p> + +<p>8. Let <hi rend="font-style: italic">F</hi> and <hi rend="font-style: italic">F'</hi> be the foci of an ellipse, and <hi rend="font-style: italic">P</hi> any +point on it. Produce <hi rend="font-style: italic">PF</hi> to <hi rend="font-style: italic">G</hi>, making <hi rend="font-style: italic">PG</hi> equal to <hi rend="font-style: italic">PF'</hi>. +Find the locus of <hi rend="font-style: italic">G</hi>.</p> + +<p>9. If the points <hi rend="font-style: italic">G</hi> of a circle be folded over upon a +point <hi rend="font-style: italic">F</hi>, the creases will all be tangent to a conic. If <hi rend="font-style: italic">F</hi> is +within the circle, the conic will be an ellipse; if <hi rend="font-style: italic">F</hi> is without +the circle, the conic will be a hyperbola.</p> + +<p>10. If the points <hi rend="font-style: italic">G</hi> in the last example be taken on a +straight line, the locus is a parabola.</p> + +<p>11. Find the foci and the length of the principal axis of +the conics in problems 9 and 10.</p> + +<p>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.</p> + +<p>13. State and explain the similar problem for problem 9.</p> + +<p>14. The last four problems are a study of the consequences +of the following transformation: A point <hi rend="font-style: italic">O</hi> is fixed +in the plane. Then to any point <hi rend="font-style: italic">P</hi> is made to correspond +the line <hi rend="font-style: italic">p</hi> at right angles to <hi rend="font-style: italic">OP</hi> and bisecting it. In this +correspondence, what happens to <hi rend="font-style: italic">p</hi> when <hi rend="font-style: italic">P</hi> 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.</p> + +</div> +</div> + +<div rend="page-break-before: always"> +<index index="toc" /><index index="pdf" /> +<pb n="98" /><anchor id="Pg98" /> +<head>CHAPTER X - ON THE HISTORY OF SYNTHETIC PROJECTIVE GEOMETRY</head> + +<div> +<index index="toc" level1="161. Ancient results" /><index index="pdf" /> +<head></head><p><anchor id="p161" /><hi rend="font-weight: bold">161. Ancient results.</hi> 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<note place="foot"><p>The +more general notion of <hi rend="font-style: italic">anharmonic ratio</hi>, 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.</p> + +<p>Consider any four points, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>, on a line, and join them to +any point <hi rend="font-style: italic">S</hi> not on that line. Then the triangles <hi rend="font-style: italic">ASB</hi>, <hi rend="font-style: italic">GSD</hi>, <hi rend="font-style: italic">ASD</hi>, +<hi rend="font-style: italic">CSB</hi>, 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</p> + +<p rend="text-align: center"><formula notation="tex">\[ +\frac{AB \times CD}{AD \times CB} = \frac{AS \times BS \sin ASB \times CS +\times DS \sin CSD}{AS \times DS \sin ASD \times CS \times BS \sin CSB} = +\frac{\sin ASB \times \sin CSD}{\sin ASD \times \sin CSB} +\]</formula></p> + +<p>Now the fraction on the right would be unchanged if instead of the +points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> we should take any other four points <hi rend="font-style: italic">A'</hi>, <hi rend="font-style: italic">B'</hi>, <hi rend="font-style: italic">C'</hi>, <hi rend="font-style: italic">D'</hi> +lying on any other line cutting across <hi rend="font-style: italic">SA</hi>, <hi rend="font-style: italic">SB</hi>, <hi rend="font-style: italic">SC</hi>, <hi rend="font-style: italic">SD</hi>. In other +words, <hi rend="font-style: italic">the fraction on the left is unaltered in value if the points +<hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> are replaced by any other four points perspective to them.</hi> +Again, the fraction on the left is unchanged if some other point were +taken instead of <hi rend="font-style: italic">S</hi>. In other words, <hi rend="font-style: italic">the fraction on the right is +unaltered if we replace the four lines <hi rend="font-style: italic">SA</hi>, <hi rend="font-style: italic">SB</hi>, <hi rend="font-style: italic">SC</hi>, <hi rend="font-style: italic">SD</hi> by any other four +lines perspective to them.</hi> The fraction on the left is called the <hi rend="font-style: italic">anharmonic +ratio</hi> of the four points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi>; the fraction on the right +is called the <hi rend="font-style: italic">anharmonic ratio</hi> of the four lines <hi rend="font-style: italic">SA</hi>, <hi rend="font-style: italic">SB</hi>, <hi rend="font-style: italic">SC</hi>, <hi rend="font-style: italic">SD</hi>. The +anharmonic ratio of four points is sometimes written (<hi rend="font-style: italic">ABCD</hi>), so that</p> + +<p rend="text-align: center"> +<formula notation="tex"> +\[ +\frac{AB \times CD}{AD \times CB} = (ABCD). +\] +</formula> +</p> + +<p>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 <hi rend="font-style: italic">(ABCD) = (BADC) = (CDAB) = (DCBA)</hi>. +If we write <hi rend="font-style: italic">(ABCD) = a</hi>, it is not difficult to show that +the six values are</p> + +<p rend="text-align: center"> +<formula notation="tex"> +\[ +a; 1/a; 1-a; 1/(1-a); (a-1)/a; a/(a-1). +\] +</formula> +</p> + +<p>The proof of this we leave to the student.</p> + +<p>If <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> are four harmonic points (see Fig. 6, p. *22), and a quadrilateral +<hi rend="font-style: italic">KLMN</hi> is constructed such that <hi rend="font-style: italic">KL</hi> and <hi rend="font-style: italic">MN</hi> pass through +<hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">KN</hi> and <hi rend="font-style: italic">LM</hi> through <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">LN</hi> through <hi rend="font-style: italic">B</hi>, and <hi rend="font-style: italic">KM</hi> through <hi rend="font-style: italic">D</hi>, then, +projecting <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">C</hi>, <hi rend="font-style: italic">D</hi> from <hi rend="font-style: italic">L</hi> upon <hi rend="font-style: italic">KM</hi>, we have <hi rend="font-style: italic">(ABCD) = (KOMD)</hi>, +where <hi rend="font-style: italic">O</hi> is the intersection of <hi rend="font-style: italic">KM</hi> with <hi rend="font-style: italic">LN</hi>. But, projecting again +the points <hi rend="font-style: italic">K</hi>, <hi rend="font-style: italic">O</hi>, <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">D</hi> from <hi rend="font-style: italic">N</hi> back upon the line <hi rend="font-style: italic">AB</hi>, we have +<hi rend="font-style: italic">(KOMD) = (CBAD)</hi>. From this we have</p> + +<p rend="text-align: center"> +<hi rend="font-style: italic">(ABCD) = (CBAD),</hi> +</p> + +<p>or</p> + +<p rend="text-align: center"> +<formula notation="tex"> +\[ +a=a/(a-1); +\] +</formula> +</p> + +<p>whence <hi rend="font-style: italic">a = 0</hi> or <hi +rend="font-style: italic">a = 2</hi>. But it is easy to see that <hi +rend="font-style: italic">a = 0</hi> 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 +notation="tex">$\frac{1}{2}$</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.</p> + +<p rend="text-align: center"> +<figure rend="w95" url="images/image49.png"> +<head><hi rend="font-variant: small-caps">Fig.</hi> 49</head> +<figDesc>Figure 49</figDesc> +</figure></p> + +<p>Many theorems of projective geometry are succinctly stated in +terms of anharmonic ratios. Thus, the <hi rend="font-style: italic">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.</hi> We leave these +theorems for the student, who may +also justify the following solution of +the problem: <hi rend="font-style: italic">Given three points and +a certain anharmonic ratio, to find a +fourth point which shall have with the +given three the given anharmonic ratio.</hi> +Let <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, <hi rend="font-style: italic">D</hi> be the three given points +(Fig. 49). On any convenient line +through <hi rend="font-style: italic">A</hi> take two points <hi rend="font-style: italic">B'</hi> and <hi rend="font-style: italic">D'</hi> +such that <hi rend="font-style: italic">AB'/AD'</hi> is equal to the +given anharmonic ratio. Join <hi rend="font-style: italic">BB'</hi> and <hi rend="font-style: italic">DD'</hi> and let the two lines +meet in <hi rend="font-style: italic">S</hi>. Draw through <hi rend="font-style: italic">S</hi> a parallel to <hi rend="font-style: italic">AB'</hi>. This line will meet +<hi rend="font-style: italic">AB</hi> in the required point <hi rend="font-style: italic">C</hi>.</p></note> +the involution of six points cut out by any transversal +which intersects the sides of a complete quadrilateral +<pb n="100" /><anchor id="Pg100" /> +as studied by Pappus<note place="foot"><p> +Pappus, Mathematicae Collectiones, vii, 129.</p></note>; +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, <hi rend="font-style: italic">Two lines determine a +point, and two points determine a line</hi>, 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 +<pb n="101" /><anchor id="Pg101" /> +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<note place="foot"><p>J. Verneri, Libellus super vigintiduobus elementis conicis, etc. 1522.</p></note> +of Nürnberg undertook to demonstrate +the properties of the conic sections by means of +the circle.</p></div> + +<div> +<index index="toc" level1="162. Unifying principles" /><index index="pdf" /> +<head></head><p><anchor id="p162" /><hi rend="font-weight: bold">162. Unifying principles.</hi> 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<note place="foot"><p>Kepler, Ad Vitellionem paralipomena quibus astronomiae pars +optica traditur. 1604.</p></note> 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" +(<hi rend="font-style: italic">caecus focus</hi>) at infinity on the axis.</p></div> + +<div> +<index index="toc" level1="163. Desargues" /><index index="pdf" /> +<head></head><p><anchor id="p163" /><hi rend="font-weight: bold">163. Desargues.</hi> In 1639 Desargues,<note place="foot"><p>Desargues, Bruillon-project d'une atteinte aux événements des +rencontres d'un cône avec un plan. 1639. Edited and analyzed by +Poudra, 1864.</p></note> 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). +<pb n="102" /><anchor id="Pg102" /> +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.</p></div> + +<div> +<index index="toc" level1="164. Poles and polars" /><index index="pdf" /> +<head></head><p><anchor id="p164" /><hi rend="font-weight: bold">164. Poles and polars.</hi> In this little treatise is also +contained the theory of poles and polars. The polar +line is called a <hi rend="font-style: italic">traversal</hi>.<note place="foot"><p>The term 'pole' was first introduced, in the sense in which we +have used it, in 1810, by a French mathematician named Servois +(Gergonne, <hi rend="font-style: italic">Annales des Mathéématiques</hi>, I, 337), and the corresponding +term 'polar' by the editor, Gergonne, of this same journal three years +later.</p></note> 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 <hi rend="font-style: italic">traversal</hi> is at an infinite +distance, all is unimaginable."</p></div> + +<div> +<index index="toc" level1="165. Desargues's theorem concerning conics through +four points" /><index index="pdf" /> +<head></head><p><anchor id="p165" /><hi rend="font-weight: bold">165. Desargues's theorem concerning conics through +four points.</hi> 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 +<pb n="103" /><anchor id="Pg103" /> +any such system. He states the theorem, in effect, as +follows: <hi rend="font-style: italic">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.</hi></p></div> + +<div> +<index index="toc" level1="166. Extension of the theory of poles and polars to +space" /><index index="pdf" /> +<head></head><p><anchor id="p166" /><hi rend="font-weight: bold">166. Extension of the theory of poles and polars to +space.</hi> 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 <hi rend="font-style: italic">P</hi> in +space a line to cut a sphere in two points, <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">S</hi>, and +then construct the fourth harmonic of <hi rend="font-style: italic">P</hi> with respect to +<hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">B</hi>, the locus of this fourth harmonic, for various +lines through <hi rend="font-style: italic">P</hi>, is a plane called the <hi rend="font-style: italic">polar plane</hi> of <hi rend="font-style: italic">P</hi> +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 +<pb n="104" /><anchor id="Pg104" /> +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.<note place="foot"><p> +Euler, Introductio in analysin infinitorum, Appendix, cap. V. +1748.</p></note></p></div> + +<div> +<index index="toc" level1="167. Desargues's method of describing a conic" /><index index="pdf" /> +<head></head><p><anchor id="p167" /><hi rend="font-weight: bold">167.</hi> 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.</p></div> + +<div> +<index index="toc" level1="168. Reception of Desargues's work" /><index index="pdf" /> +<head></head><p><anchor id="p168" /><hi rend="font-weight: bold">168. Reception of Desargues's work.</hi> 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.</p> + +<pb n="105" /><anchor id="Pg105" /></div> + +<div> +<index index="toc" level1="169. Conservatism in Desargues's time" /><index index="pdf" /> +<head></head><p><anchor id="p169" /><hi rend="font-weight: bold">169. Conservatism in Desargues's time.</hi> 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.</p></div> + +<div> +<index index="toc" level1="170. Desargues's style of writing" /><index index="pdf" /> +<head></head><p><anchor id="p170" /><hi rend="font-weight: bold">170. Desargues's style of writing.</hi> Nevertheless, authority +counted for less at this time in Paris than it did in +Italy, and the tragedy enacted in Rome when Galileo +<pb n="106" /><anchor id="Pg106" /> +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.<note place="foot"><p> +Å’uvres de Desargues, t. II, 132.</p></note> "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 +<pb n="107" /><anchor id="Pg107" /> +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, <hi rend="font-style: italic">involution</hi>, has survived. +Curiously enough, this is the one term singled +out for the sharpest criticism and ridicule by his reviewer, +De Beaugrand.<note place="foot"><p> +Å’uvres de Desargues, t. II, 370.</p></note> 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.</p></div> + +<div> +<index index="toc" level1="171. Lack of appreciation of Desargues" /><index index="pdf" /> +<head></head><p><anchor id="p171" /><hi rend="font-weight: bold">171. Lack of appreciation of Desargues.</hi> 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<note place="foot"><p> +Å’uvres de Descartes, t. II, 499.</p></note> 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!</p> + +<pb n="108" /><anchor id="Pg108" /></div> + +<div> +<index index="toc" level1="172. Pascal and his theorem" /><index index="pdf" /> +<head></head><p><anchor id="p172" /><hi rend="font-weight: bold">172. Pascal and his theorem.</hi> 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,"<note place="foot"><p> +Å’uvres de Pascal, par Brunsehvig et Boutroux, t. I, 252. +</p></note> 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: <hi rend="font-style: italic">"If in the plane of <hi rend="font-style: italic">M</hi>, <hi rend="font-style: italic">S</hi>, <hi rend="font-style: italic">Q</hi> we draw +through <hi rend="font-style: italic">M</hi> the two lines <hi rend="font-style: italic">MK</hi> and <hi rend="font-style: italic">MV</hi>, and through the +point <hi rend="font-style: italic">S</hi> the two lines <hi rend="font-style: italic">SK</hi> and <hi rend="font-style: italic">SV</hi>, and let <hi rend="font-style: italic">K</hi> be the intersection +of <hi rend="font-style: italic">MK</hi> and <hi rend="font-style: italic">SK</hi>; <hi rend="font-style: italic">V</hi> the intersection of <hi rend="font-style: italic">MV</hi> and +<hi rend="font-style: italic">SV</hi>; <hi rend="font-style: italic">A</hi> the intersection of <hi rend="font-style: italic">MA</hi> and <hi rend="font-style: italic">SA</hi> (<hi rend="font-style: italic">A</hi> is the intersection +of <hi rend="font-style: italic">SV</hi> and <hi rend="font-style: italic">MK</hi>), and <hi rend="font-style: italic">μ</hi> the intersection of <hi rend="font-style: italic">MV</hi> +and <hi rend="font-style: italic">SK</hi>; and if through two of the four points <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">K</hi>, +<hi rend="font-style: italic">μ</hi>, <hi rend="font-style: italic">V</hi>, which are not in the same straight line with <hi rend="font-style: italic">M</hi> and +<hi rend="font-style: italic">S</hi>, such as <hi rend="font-style: italic">K</hi> and <hi rend="font-style: italic">V</hi>, we pass the circumference of a circle +cutting the lines <hi rend="font-style: italic">MV</hi>, <hi rend="font-style: italic">MP</hi>, <hi rend="font-style: italic">SV</hi>, <hi rend="font-style: italic">SK</hi> in the points <hi rend="font-style: italic">O</hi>, <hi rend="font-style: italic">P</hi>, +<hi rend="font-style: italic">Q</hi>, <hi rend="font-style: italic">N</hi>; I say that the lines <hi rend="font-style: italic">MS</hi>, <hi rend="font-style: italic">NO</hi>, <hi rend="font-style: italic">PQ</hi> are of the same +order."</hi> (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.</p></div> + +<div> +<index index="toc" level1="173. Pascal's essay" /><index index="pdf" /> +<head></head><p><anchor id="p173" /><hi rend="font-weight: bold">173.</hi> It must be understood that the "Essay" was +only a résumé 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 +<pb n="109" /><anchor id="Pg109" /> +days, and also to his later morbid interest in religious +matters, was never published. Leibniz<note place="foot"><p> +Chasles, Histoire de la Géométrie, 70.</p></note> 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.</p></div> + +<div> +<index index="toc" level1="174. Pascal's originality" /><index index="pdf" /> +<head></head><p><anchor id="p174" /><hi rend="font-weight: bold">174.</hi> 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.</p></div> + +<div> +<index index="toc" level1="175. De la Hire and his work" /><index index="pdf" /> +<head></head><p><anchor id="p175" /><hi rend="font-weight: bold">175. De la Hire and his work.</hi> De la Hire added +little to the development of the subject, but he did put +into print much of what Desargues had already worked +<pb n="110" /><anchor id="Pg110" /> +out, not fully realizing, perhaps, how much was his +own and how much he owed to his teacher. Writing in +1679, he says,<note place="foot"><p> +Å’uvres de Desargues, t. I, 231.</p></note> "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: <hi rend="font-style: italic">Draw +two parallel lines, <hi rend="font-style: italic">a</hi> and <hi rend="font-style: italic">b</hi>, and select a point <hi rend="font-style: italic">P</hi> in their +plane. Through any point <hi rend="font-style: italic">M</hi> of the plane draw a line +meeting <hi rend="font-style: italic">a</hi> in <hi rend="font-style: italic">A</hi> and <hi rend="font-style: italic">b</hi> in <hi rend="font-style: italic">B</hi>. Draw a line through <hi rend="font-style: italic">B</hi> +parallel to <hi rend="font-style: italic">AP</hi>, and let it meet <hi rend="font-style: italic">MP</hi> in the point <hi rend="font-style: italic">M'</hi>. It +may be shown that the point <hi rend="font-style: italic">M'</hi> thus obtained does not +depend at all on the particular ray <hi rend="font-style: italic">MAB</hi> used in determining +it, so that we have set up a one-to-one correspondence +between the points <hi rend="font-style: italic">M</hi> and <hi rend="font-style: italic">M'</hi> in the plane.</hi> The student +may show that as <hi rend="font-style: italic">M</hi> describes a point-row, <hi rend="font-style: italic">M'</hi> describes +a point-row projective to it. As <hi rend="font-style: italic">M</hi> describes a conic, +<hi rend="font-style: italic">M'</hi> describes another conic. This sort of correspondence +is called a <hi rend="font-style: italic">collineation</hi>. It will be found that the +points on the line <hi rend="font-style: italic">b</hi> transform into themselves, as does +also the single point <hi rend="font-style: italic">P</hi>. Points on the line <hi rend="font-style: italic">a</hi> 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 <hi rend="font-style: italic">a</hi> and <hi rend="font-style: italic">b</hi>, any +two lines which may meet in a finite part of the plane. +<pb n="111" /><anchor id="Pg111" /> +The collineation is a special one in that the general +one has an invariant triangle instead of an invariant +point and line.</p></div> + +<div> +<index index="toc" level1="176. Descartes and his influence" /><index index="pdf" /> +<head></head><p><anchor id="p176" /><hi rend="font-weight: bold">176. Descartes and his influence.</hi> 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 "Méchanique +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 +<pb n="112" /><anchor id="Pg112" /> +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,<note place="foot"><p> +See Ball, History of Mathematics, French edition, t. II, 233. +</p></note> "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.</p></div> + +<div> +<index index="toc" level1="177. Newton and Maclaurin" /><index index="pdf" /> +<head></head><p><anchor id="p177" /><hi rend="font-weight: bold">177. Newton and Maclaurin.</hi> 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"<note place="foot"><p> +Newton, Principia, lib. i, lemma XXI.</p></note> +of generating conic sections is closely related to the +method which we have made use of in Chapter III. +It is as follows: <hi rend="font-style: italic">If two angles, <hi rend="font-style: italic">AOS</hi> and <hi rend="font-style: italic">AO'S</hi>, of given +magnitudes turn about their respective vertices, <hi rend="font-style: italic">O</hi> and <hi rend="font-style: italic">O'</hi>, +in such a way that the point of intersection, <hi rend="font-style: italic">S</hi>, of one pair +of arms always lies on a straight line, the point of intersection, +<hi rend="font-style: italic">A</hi>, of the other pair of arms will describe a conic.</hi> +The proof of this is left to the student.</p></div> + +<div> +<index index="toc" level1="178. Maclaurin's construction" /><index index="pdf" /> +<head></head><p><anchor id="p178" /><hi rend="font-weight: bold">178.</hi> Another method of generating a conic is due to +Maclaurin.<note place="foot"><p> +Maclaurin, Philosophical Transactions of the Royal Society of +London, 1735.</p></note> The construction, which we also leave for +the student to justify, is as follows: <hi rend="font-style: italic">If a triangle <hi rend="font-style: italic">C'PQ</hi> +move in such a way that its sides, <hi rend="font-style: italic">PQ</hi>, <hi rend="font-style: italic">QC'</hi>, and <hi rend="font-style: italic">C'P</hi>, turn +<pb n="113" /><anchor id="Pg113" /> +around three fixed points, <hi rend="font-style: italic">R</hi>, <hi rend="font-style: italic">A</hi>, <hi rend="font-style: italic">B</hi>, respectively, while two of +its vertices, <hi rend="font-style: italic">P</hi>, <hi rend="font-style: italic">Q</hi>, slide along two fixed lines, <hi rend="font-style: italic">CB'</hi> and <hi rend="font-style: italic">CA'</hi>, +respectively, then the remaining vertex will describe a conic.</hi></p></div> + +<div> +<index index="toc" level1="179. Descriptive geometry and the second revival" /><index index="pdf" /> +<head></head><p><anchor id="p179" /><hi rend="font-weight: bold">179. Descriptive geometry and the second revival.</hi> +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, Möbius, Abel, and many +others. The renaissance may be said to date from the invention +by Monge<note place="foot"><p> +Monge, Géométrie Descriptive. 1800.</p></note> of the theory of <hi rend="font-style: italic">descriptive geometry</hi>. +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 Mézières, where Monge was a draftsman +and pupil, viewed the results with distrust. Monge +afterward became professor of mathematics at Mézières +<pb n="114" /><anchor id="Pg114" /> +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.</p></div> + +<div> +<index index="toc" level1="180. Duality, homology, continuity, contingent relations" /><index index="pdf" /> +<head></head><p><anchor id="p180" /><hi rend="font-weight: bold">180. Duality, homology, continuity, contingent relations.</hi> +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."</p> + +<pb n="115" /><anchor id="Pg115" /></div> + +<div> +<index index="toc" level1="181. Poncelet and Cauchy" /><index index="pdf" /> +<head></head><p><anchor id="p181" /><hi rend="font-weight: bold">181. Poncelet and Cauchy.</hi> 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<note place="foot"><p> +Poncelet, Traité des Propriétés Projectives des Figures. 1822. +(See p. 357, Vol. II, of the edition of 1866.)</p></note> +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 Traité des Propriétés 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 coördinates as well as with +three, but the existence of a space of four dimensions +<pb n="116" /><anchor id="Pg116" /> +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. <hi rend="font-style: italic">Apart from the +algebra involved</hi>, it is the height of absurdity to try to +distinguish between the two points in which a line +<hi rend="font-style: italic">fails to meet a conic!</hi></p></div> + +<div> +<index index="toc" level1="182. The work of Poncelet" /><index index="pdf" /> +<head></head><p><anchor id="p182" /><hi rend="font-weight: bold">182. The work of Poncelet.</hi> 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 <hi rend="font-style: italic">in homology</hi>, 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,<note place="foot"><p> +Gergonne, <hi rend="font-style: italic">Annales de Mathématiques, XVI, 209. 1826.</hi></p></note> +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.</p></div> + +<div> +<index index="toc" level1="183. The debt which analytic geometry owes to synthetic +geometry" /><index index="pdf" /> +<head></head><p><anchor id="p183" /><hi rend="font-weight: bold">183. The debt which analytic geometry owes to synthetic +geometry.</hi> The reaction of pure geometry on +<pb n="117" /><anchor id="Pg117" /> +analytic geometry is clearly seen in the development of +the notion of the <hi rend="font-style: italic">class</hi> 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 Plücker. 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.</p></div> + +<div> +<index index="toc" level1="184. Steiner and his work" /><index index="pdf" /> +<head></head><p><anchor id="p184" /><hi rend="font-weight: bold">184. Steiner and his work.</hi> In the work of Poncelet +and his contemporaries, Chasles, Brianchon, Hachette, +Dupin, Gergonne, and others, the anharmonic ratio enjoyed +a fundamental rôle. It is made also the basis of +the great work of Steiner,<note place="foot"><p> +Steiner, Systematische Ehtwickelung der Abhängigkeit geometrischer +Gestalten von einander. 1832.</p></note> +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, +<pb n="118" /><anchor id="Pg118" /> +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.</p></div> + +<div> +<index index="toc" level1="185. Von Staudt and his work" /><index index="pdf" /> +<head></head><p><anchor id="p185" /><hi rend="font-weight: bold">185. Von Staudt and his work.</hi> 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,<note place="foot"><p> +Von Staudt, Geometrie der Lage. 1847.</p></note> +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 +<pb n="119" /><anchor id="Pg119" /> +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<note place="foot"><p> +Reye, Geometrie der Lage. Translated by Holgate, 1897.</p></note> +and others have done much toward simplifying +his presentation.</p></div> + +<div> +<index index="toc" level1="186. Recent developments" /><index index="pdf" /> +<head></head><p><anchor id="p186" /><hi rend="font-weight: bold">186. Recent developments.</hi> 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 +<pb n="120" /><anchor id="Pg120" /> +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,<note place="foot"><p> +Ball, loc. cit. p. 261.</p></note> +"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."</p> +</div> +</div> + + +<div rend="page-break-before: always"> +<index index="toc" /><index index="pdf" /> +<head>INDEX</head> + +<p rend="text-align: center">(The numbers refer to the paragraphs)</p> + +<p rend="text-indent: 0">Abel (1802-1829), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Analogy, <ref target="p24">24</ref><lb /></p> + +<p rend="text-indent: 0">Analytic geometry, <ref target="p21">21</ref>, <ref target="p118">118</ref>, <ref target="p119">119</ref>, +120, <ref target="p146">146</ref>, <ref target="p176">176</ref>, <ref target="p180">180</ref><lb /></p> + +<p rend="text-indent: 0">Anharmonic ratio, <ref target="p46">46</ref>, <ref target="p161">161</ref>, <ref target="p184">184</ref>, <ref target="p185">185</ref><lb /></p> + +<p rend="text-indent: 0">Apollonius (second half of third +century B.C.), <ref target="p70">70</ref><lb /></p> + +<p rend="text-indent: 0">Archimedes (287-212 B.C.), <ref target="p176">176</ref><lb /></p> + +<p rend="text-indent: 0">Aristotle (384-322 B.C.), <ref target="p169">169</ref><lb /></p> + +<p rend="text-indent: 0">Asymptotes, <ref target="p111">111</ref>, <ref target="p113">113</ref>, <ref target="p114">114</ref>, <ref target="p115">115</ref>, +116, <ref target="p117">117</ref>, <ref target="p118">118</ref>, <ref target="p148">148</ref><lb /></p> + +<p rend="text-indent: 0">Axes of a conic, <ref target="p148">148</ref><lb /></p> + +<p rend="text-indent: 0">Axial pencil, <ref target="p7">7</ref>, <ref target="p8">8</ref>, <ref target="p23">23</ref>, <ref target="p50">50</ref>, <ref target="p54">54</ref><lb /></p> + +<p rend="text-indent: 0">Axis of perspectivity, <ref target="p8">8</ref>, <ref target="p47">47</ref><lb /></p> + +<p rend="text-indent: 0">Bacon (1561-1626), <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Bisection, <ref target="p41">41</ref>, <ref target="p109">109</ref><lb /></p> + +<p rend="text-indent: 0">Brianchon (1785-1864), <ref target="p84">84</ref>, <ref target="p85">85</ref>, <ref target="p86">86</ref>, +88, <ref target="p89">89</ref>, <ref target="p90">90</ref>, <ref target="p95">95</ref>, <ref target="p105">105</ref>, <ref target="p113">113</ref>, <ref target="p174">174</ref>, <ref target="p184">184</ref> <lb /></p> + +<p rend="text-indent: 0">Calculus, <ref target="p176">176</ref><lb /></p> + +<p rend="text-indent: 0">Carnot (1796-1832), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Cauchy (1789-1857), <ref target="p179">179</ref>, <ref target="p181">181</ref><lb /></p> + +<p rend="text-indent: 0">Cavalieri (1598-1647), <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Center of a conic, <ref target="p107">107</ref>, <ref target="p112">112</ref>, <ref target="p148">148</ref><lb /></p> + +<p rend="text-indent: 0">Center of involution, <ref target="p141">141</ref>, <ref target="p142">142</ref><lb /></p> + +<p rend="text-indent: 0">Center of perspectivity, <ref target="p8">8</ref><lb /></p> + +<p rend="text-indent: 0">Central conic, <ref target="p120">120</ref><lb /></p> + +<p rend="text-indent: 0">Chasles (1793-1880), <ref target="p168">168</ref>, <ref target="p179">179</ref>, <ref target="p180">180</ref>, +184<lb /></p> + +<p rend="text-indent: 0">Circle, <ref target="p21">21</ref>, <ref target="p73">73</ref>, <ref target="p80">80</ref>, <ref target="p145">145</ref>, <ref target="p146">146</ref>, <ref target="p147">147</ref><lb /></p> + +<p rend="text-indent: 0">Circular involution, <ref target="p147">147</ref>, <ref target="p149">149</ref>, <ref target="p150">150</ref>, +151<lb /></p> + +<p rend="text-indent: 0">Circular points, <ref target="p146">146</ref><lb /></p> + +<p rend="text-indent: 0">Class of a curve, <ref target="p183">183</ref><lb /></p> + +<p rend="text-indent: 0">Classification of conics, <ref target="p110">110</ref><lb /></p> + +<p rend="text-indent: 0">Collineation, <ref target="p175">175</ref><lb /></p> + +<p rend="text-indent: 0">Concentric pencils, <ref target="p50">50</ref><lb /></p> + +<p rend="text-indent: 0">Cone of the second order, <ref target="p59">59</ref><lb /></p> + +<p rend="text-indent: 0">Conic, <ref target="p73">73</ref>, <ref target="p81">81</ref><lb /></p> + +<p rend="text-indent: 0">Conjugate diameters, <ref target="p114">114</ref>, <ref target="p148">148</ref><lb /></p> + +<p rend="text-indent: 0">Conjugate normal, <ref target="p151">151</ref><lb /></p> + +<p rend="text-indent: 0">Conjugate points and lines, <ref target="p100">100</ref>, +109, <ref target="p138">138</ref>, <ref target="p139">139</ref>, <ref target="p140">140</ref><lb /></p> + +<p rend="text-indent: 0">Constants in an equation, <ref target="p21">21</ref><lb /></p> + +<p rend="text-indent: 0">Contingent relations, <ref target="p180">180</ref>, <ref target="p181">181</ref><lb /></p> + +<p rend="text-indent: 0">Continuity, <ref target="p180">180</ref>, <ref target="p181">181</ref><lb /></p> + +<p rend="text-indent: 0">Continuous correspondence, <ref target="p9">9</ref>, <ref target="p10">10</ref>, +21, <ref target="p49">49</ref><lb /></p> + +<p rend="text-indent: 0">Corresponding elements, <ref target="p64">64</ref><lb /></p> + +<p rend="text-indent: 0">Counting, <ref target="p1">1</ref>, <ref target="p4">4</ref><lb /></p> + +<p rend="text-indent: 0">Cross ratio, <ref target="p46">46</ref><lb /></p> + +<p rend="text-indent: 0">Darboux, <ref target="p176">176</ref>, <ref target="p186">186</ref><lb /></p> + +<p rend="text-indent: 0">De Beaugrand, <ref target="p170">170</ref><lb /></p> + +<p rend="text-indent: 0">Degenerate pencil of rays of the +second order, <ref target="p58">58</ref>, <ref target="p93">93</ref><lb /></p> + +<p rend="text-indent: 0">Degenerate point-row of the +second order, <ref target="p56">56</ref>, <ref target="p78">78</ref><lb /></p> + +<p rend="text-indent: 0">De la Hire (1640-1718), <ref target="p168">168</ref>, <ref target="p171">171</ref>, +175<lb /></p> + +<p rend="text-indent: 0">Desargues (1593-1662), <ref target="p25">25</ref>, <ref target="p26">26</ref>, <ref target="p40">40</ref>, +121, <ref target="p125">125</ref>, <ref target="p162">162</ref>, <ref target="p163">163</ref>, <ref target="p164">164</ref>, <ref target="p165">165</ref>, <ref target="p166">166</ref>, +167, <ref target="p168">168</ref>, <ref target="p169">169</ref>, <ref target="p170">170</ref>, <ref target="p171">171</ref>, <ref target="p174">174</ref>, <ref target="p175">175</ref><lb /></p> + +<p rend="text-indent: 0">Descartes (1596-1650), <ref target="p162">162</ref>, <ref target="p170">170</ref>, +171, <ref target="p174">174</ref>, <ref target="p176">176</ref><lb /></p> + +<p rend="text-indent: 0">Descriptive geometry, <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Diameter, <ref target="p107">107</ref><lb /></p> + +<p rend="text-indent: 0">Directrix, <ref target="p157">157</ref>, <ref target="p158">158</ref>, <ref target="p159">159</ref>, <ref target="p160">160</ref><lb /></p> + +<p rend="text-indent: 0">Double correspondence, <ref target="p128">128</ref>, <ref target="p130">130</ref><lb /></p> + +<p rend="text-indent: 0">Double points of an involution, <ref target="p124">124</ref><lb /></p> + +<p rend="text-indent: 0">Double rays of an involution, <ref target="p133">133</ref>, +134<lb /></p> + +<p rend="text-indent: 0">Duality, <ref target="p94">94</ref>, <ref target="p104">104</ref>, <ref target="p161">161</ref>, <ref target="p180">180</ref>, <ref target="p182">182</ref><lb /></p> + +<p rend="text-indent: 0">Dupin (1784-1873), <ref target="p174">174</ref>, <ref target="p184">184</ref> <lb /></p> + +<p rend="text-indent: 0">Eccentricity of conic, <ref target="p159">159</ref><lb /></p> + +<p rend="text-indent: 0">Ellipse, <ref target="p110">110</ref>, <ref target="p111">111</ref>, <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Equation of conic, <ref target="p118">118</ref>, <ref target="p119">119</ref>, <ref target="p120">120</ref><lb /></p> + +<p rend="text-indent: 0">Euclid (ca. 300 B.C.), <ref target="p6">6</ref>, <ref target="p22">22</ref>, <ref target="p104">104</ref><lb /></p> + +<p rend="text-indent: 0">Euler (1707-1783), <ref target="p166">166</ref> <lb /></p> + +<p rend="text-indent: 0">Fermat (1601-1665), <ref target="p162">162</ref>, <ref target="p171">171</ref><lb /></p> + +<p rend="text-indent: 0">Foci of a conic, <ref target="p152">152</ref>, <ref target="p153">153</ref>, <ref target="p154">154</ref>, <ref target="p155">155</ref>, +156, <ref target="p157">157</ref>, <ref target="p158">158</ref>, <ref target="p159">159</ref>, <ref target="p160">160</ref>, <ref target="p161">161</ref>, <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Fourier (1768-1830), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Fourth harmonic, <ref target="p29">29</ref><lb /></p> + +<p rend="text-indent: 0">Fundamental form, <ref target="p7">7</ref>, <ref target="p16">16</ref>, <ref target="p23">23</ref>, <ref target="p36">36</ref>, +47, <ref target="p60">60</ref>, <ref target="p184">184</ref> <lb /></p> + +<p rend="text-indent: 0">Galileo (1564-1642), <ref target="p162">162</ref>, <ref target="p169">169</ref>, <ref target="p170">170</ref>, +176<lb /></p> + +<p rend="text-indent: 0">Gauss (1777-1855), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Gergonne (1771-1859), <ref target="p182">182</ref>, <ref target="p184">184</ref><lb /></p> + +<p rend="text-indent: 0">Greek geometry, <ref target="p161">161</ref> <lb /></p> + +<p rend="text-indent: 0">Hachette (1769-1834), <ref target="p179">179</ref>, <ref target="p184">184</ref><lb /></p> + +<p rend="text-indent: 0">Harmonic conjugates, <ref target="p29">29</ref>, <ref target="p30">30</ref>, <ref target="p39">39</ref><lb /></p> + +<p rend="text-indent: 0">Harmonic elements, <ref target="p86">86</ref>, <ref target="p49">49</ref>, <ref target="p91">91</ref>, +163, <ref target="p185">185</ref><lb /></p> + +<p rend="text-indent: 0">Harmonic lines, <ref target="p33">33</ref>, <ref target="p34">34</ref>, <ref target="p35">35</ref>, <ref target="p66">66</ref>, <ref target="p67">67</ref><lb /></p> + +<p rend="text-indent: 0">Harmonic planes, <ref target="p34">34</ref>, <ref target="p35">35</ref><lb /></p> + +<p rend="text-indent: 0">Harmonic points, <ref target="p29">29</ref>, <ref target="p31">31</ref>, <ref target="p32">32</ref>, <ref target="p33">33</ref>, +34, <ref target="p35">35</ref>, <ref target="p36">36</ref>, <ref target="p43">43</ref>, <ref target="p71">71</ref>, <ref target="p161">161</ref><lb /></p> + +<p rend="text-indent: 0">Harmonic tangents to a conic, +91, <ref target="p92">92</ref><lb /></p> + +<p rend="text-indent: 0">Harvey (1578-1657), <ref target="p169">169</ref><lb /></p> + +<p rend="text-indent: 0">Homology, <ref target="p180">180</ref>, <ref target="p182">182</ref><lb /></p> + +<p rend="text-indent: 0">Huygens (1629-1695), <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Hyperbola, <ref target="p110">110</ref>, <ref target="p111">111</ref>, <ref target="p113">113</ref>, <ref target="p114">114</ref>, <ref target="p115">115</ref>, +116, <ref target="p117">117</ref>, <ref target="p118">118</ref>, <ref target="p162">162</ref> <lb /></p> + +<p rend="text-indent: 0">Imaginary elements, <ref target="p146">146</ref>, <ref target="p180">180</ref>, <ref target="p181">181</ref>, +182, <ref target="p185">185</ref><lb /></p> + +<p rend="text-indent: 0">Infinitely distant elements, <ref target="p6">6</ref>, <ref target="p9">9</ref>, +22, <ref target="p39">39</ref>, <ref target="p40">40</ref>, <ref target="p41">41</ref>, <ref target="p104">104</ref>, <ref target="p107">107</ref>, <ref target="p110">110</ref><lb /></p> + +<p rend="text-indent: 0">Infinity, <ref target="p4">4</ref>, <ref target="p5">5</ref>, <ref target="p10">10</ref>, <ref target="p12">12</ref>, <ref target="p13">13</ref>, <ref target="p14">14</ref>, <ref target="p15">15</ref>, +17, <ref target="p18">18</ref>, <ref target="p19">19</ref>, <ref target="p20">20</ref>, <ref target="p21">21</ref>, <ref target="p22">22</ref>, <ref target="p41">41</ref><lb /></p> + +<p rend="text-indent: 0">Involution, <ref target="p37">37</ref>, <ref target="p123">123</ref>, <ref target="p124">124</ref>, <ref target="p125">125</ref>, <ref target="p126">126</ref>, +127, <ref target="p128">128</ref>, <ref target="p129">129</ref>, <ref target="p130">130</ref>, <ref target="p131">131</ref>, <ref target="p132">132</ref>, <ref target="p133">133</ref>, +134, <ref target="p135">135</ref>, <ref target="p136">136</ref>, <ref target="p137">137</ref>, <ref target="p138">138</ref>, <ref target="p139">139</ref>, <ref target="p140">140</ref>, +161, <ref target="p163">163</ref>, <ref target="p170">170</ref> <lb /></p> + +<p rend="text-indent: 0">Kepler (1571-1630), <ref target="p162">162</ref> <lb /></p> + +<p rend="text-indent: 0">Lagrange (1736-1813), <ref target="p176">176</ref>, <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Laplace (1749-1827), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Legendre (1752-1833), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Leibniz (1646-1716), <ref target="p173">173</ref><lb /></p> + +<p rend="text-indent: 0">Linear construction, <ref target="p40">40</ref>, <ref target="p41">41</ref>, <ref target="p42">42</ref> <lb /></p> + +<p rend="text-indent: 0">Maclaurin (1698-1746), <ref target="p177">177</ref>, <ref target="p178">178</ref><lb /></p> + +<p rend="text-indent: 0">Measurements, <ref target="p23">23</ref>, <ref target="p40">40</ref>, <ref target="p41">41</ref>, <ref target="p104">104</ref><lb /></p> + +<p rend="text-indent: 0">Mersenne (1588-1648), <ref target="p168">168</ref>, <ref target="p171">171</ref><lb /></p> + +<p rend="text-indent: 0">Metrical theorems, <ref target="p40">40</ref>, <ref target="p104">104</ref>, <ref target="p106">106</ref>, +107, <ref target="p141">141</ref><lb /></p> + +<p rend="text-indent: 0">Middle point, <ref target="p39">39</ref>, <ref target="p41">41</ref><lb /></p> + +<p rend="text-indent: 0">Möbius (1790-1868), <ref target="p179">179</ref><lb /></p> + +<p rend="text-indent: 0">Monge (1746-1818), <ref target="p179">179</ref>, <ref target="p180">180</ref> <lb /></p> + +<p rend="text-indent: 0">Napier (1550-1617), <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Newton (1642-1727), <ref target="p177">177</ref><lb /></p> + +<p rend="text-indent: 0">Numbers, <ref target="p4">4</ref>, <ref target="p21">21</ref>, <ref target="p43">43</ref><lb /></p> + +<p rend="text-indent: 0">Numerical computations, <ref target="p43">43</ref>, <ref target="p44">44</ref>, +46 <lb /></p> + +<p rend="text-indent: 0">One-to-one correspondence, <ref target="p1">1</ref>, <ref target="p2">2</ref>, +3, <ref target="p4">4</ref>, <ref target="p5">5</ref>, <ref target="p6">6</ref>, <ref target="p7">7</ref>, <ref target="p9">9</ref>, <ref target="p10">10</ref>, <ref target="p11">11</ref>, <ref target="p24">24</ref>, <ref target="p36">36</ref>, +87, <ref target="p43">43</ref>, <ref target="p60">60</ref>, <ref target="p104">104</ref>, <ref target="p106">106</ref>, <ref target="p184">184</ref><lb /></p> + +<p rend="text-indent: 0">Opposite sides of a hexagon, <ref target="p70">70</ref><lb /></p> + +<p rend="text-indent: 0">Opposite sides of a quadrilateral, +28, <ref target="p29">29</ref><lb /></p> + +<p rend="text-indent: 0">Order of a form, <ref target="p7">7</ref>, <ref target="p10">10</ref>, <ref target="p11">11</ref>, <ref target="p12">12</ref>, <ref target="p13">13</ref>, +14, <ref target="p15">15</ref>, <ref target="p16">16</ref>, <ref target="p17">17</ref>, <ref target="p18">18</ref>, <ref target="p19">19</ref>, <ref target="p20">20</ref>, <ref target="p21">21</ref> <lb /></p> + +<p rend="text-indent: 0">Pappus (fourth century A.D.), +161<lb /></p> + +<p rend="text-indent: 0">Parabola, <ref target="p110">110</ref>, <ref target="p111">111</ref>, <ref target="p112">112</ref>, <ref target="p119">119</ref>, <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Parallel lines, <ref target="p39">39</ref>, <ref target="p41">41</ref>, <ref target="p162">162</ref><lb /></p> + +<p rend="text-indent: 0">Pascal (1623-1662), <ref target="p69">69</ref>, <ref target="p70">70</ref>, <ref target="p74">74</ref>, <ref target="p75">75</ref>, +76, <ref target="p77">77</ref>, <ref target="p78">78</ref>, <ref target="p95">95</ref>, <ref target="p105">105</ref>, <ref target="p125">125</ref>, <ref target="p162">162</ref>, +169, <ref target="p171">171</ref>, <ref target="p172">172</ref>, <ref target="p173">173</ref><lb /></p> + +<p rend="text-indent: 0">Pencil of planes of the second +order, <ref target="p59">59</ref><lb /></p> + +<p rend="text-indent: 0">Pencil of rays, <ref target="p6">6</ref>, <ref target="p7">7</ref>, <ref target="p8">8</ref>, <ref target="p23">23</ref>; + of the second order, <ref target="p57">57</ref>, <ref target="p60">60</ref>, <ref target="p79">79</ref>, <ref target="p81">81</ref><lb /></p> + +<p rend="text-indent: 0">Perspective position, <ref target="p6">6</ref>, <ref target="p8">8</ref>, <ref target="p35">35</ref>, <ref target="p37">37</ref>, +51, <ref target="p53">53</ref>, <ref target="p71">71</ref><lb /></p> + +<p rend="text-indent: 0">Plane system, <ref target="p16">16</ref>, <ref target="p23">23</ref><lb /></p> + +<p rend="text-indent: 0">Planes on space, <ref target="p17">17</ref><lb /></p> + +<p rend="text-indent: 0">Point of contact, <ref target="p87">87</ref>, <ref target="p88">88</ref>, <ref target="p89">89</ref>, <ref target="p90">90</ref><lb /></p> + +<p rend="text-indent: 0">Point system, <ref target="p16">16</ref>, <ref target="p23">23</ref><lb /></p> + +<p rend="text-indent: 0">Point-row, <ref target="p6">6</ref>, <ref target="p7">7</ref>, <ref target="p8">8</ref>, <ref target="p9">9</ref>, <ref target="p23">23</ref>; + of the second order, <ref target="p55">55</ref>, <ref target="p60">60</ref>, <ref target="p61">61</ref>, <ref target="p66">66</ref>, + <ref target="p67">67</ref>, <ref target="p72">72</ref><lb /></p> + +<p rend="text-indent: 0">Points in space, <ref target="p18">18</ref><lb /></p> + +<p rend="text-indent: 0">Pole and polar, <ref target="p98">98</ref>, <ref target="p99">99</ref>, <ref target="p100">100</ref>, <ref target="p101">101</ref>, +138, <ref target="p164">164</ref>, <ref target="p166">166</ref><lb /></p> + +<p rend="text-indent: 0">Poncelet (1788-1867), <ref target="p177">177</ref>, <ref target="p179">179</ref>, +180, <ref target="p181">181</ref>, <ref target="p182">182</ref>, <ref target="p183">183</ref>, <ref target="p184">184</ref><lb /></p> + +<p rend="text-indent: 0">Principal axis of a conic, <ref target="p157">157</ref><lb /></p> + +<p rend="text-indent: 0">Projection, <ref target="p161">161</ref><lb /></p> + +<p rend="text-indent: 0">Protective axial pencils, <ref target="p59">59</ref><lb /></p> + +<p rend="text-indent: 0">Projective correspondence, <ref target="p9">9</ref>, <ref target="p35">35</ref>, +36, <ref target="p37">37</ref>, <ref target="p47">47</ref>, <ref target="p71">71</ref>, <ref target="p92">92</ref>, <ref target="p104">104</ref><lb /></p> + +<p rend="text-indent: 0">Projective pencils, <ref target="p53">53</ref>, <ref target="p64">64</ref>, <ref target="p68">68</ref><lb /></p> + +<p rend="text-indent: 0">Projective point-rows, <ref target="p51">51</ref>, <ref target="p79">79</ref><lb /></p> + +<p rend="text-indent: 0">Projective properties, <ref target="p24">24</ref><lb /></p> + +<p rend="text-indent: 0">Projective theorems, <ref target="p40">40</ref>, <ref target="p104">104</ref> <lb /></p> + +<p rend="text-indent: 0">Quadrangle, <ref target="p26">26</ref>, <ref target="p27">27</ref>, <ref target="p28">28</ref>, <ref target="p29">29</ref><lb /></p> + +<p rend="text-indent: 0">Quadric cone, <ref target="p59">59</ref><lb /></p> + +<p rend="text-indent: 0">Quadrilateral, <ref target="p88">88</ref>, <ref target="p95">95</ref>, <ref target="p96">96</ref> <lb /></p> + +<p rend="text-indent: 0">Roberval (1602-1675), <ref target="p168">168</ref><lb /></p> + +<p rend="text-indent: 0">Ruler construction, <ref target="p40">40</ref> <lb /></p> + +<p rend="text-indent: 0">Scheiner, <ref target="p169">169</ref><lb /></p> + +<p rend="text-indent: 0">Self-corresponding elements, <ref target="p47">47</ref>, +48, <ref target="p49">49</ref>, <ref target="p50">50</ref>, <ref target="p51">51</ref><lb /></p> + +<p rend="text-indent: 0">Self-dual, <ref target="p105">105</ref><lb /></p> + +<p rend="text-indent: 0">Self-polar triangle, <ref target="p102">102</ref><lb /></p> + +<p rend="text-indent: 0">Separation of elements in involution, +148<lb /></p> + +<p rend="text-indent: 0">Separation of harmonic conjugates, +38<lb /></p> + +<p rend="text-indent: 0">Sequence of points, <ref target="p49">49</ref><lb /></p> + +<p rend="text-indent: 0">Sign of segment, <ref target="p44">44</ref>, <ref target="p45">45</ref><lb /></p> + +<p rend="text-indent: 0">Similarity, <ref target="p106">106</ref><lb /></p> + +<p rend="text-indent: 0">Skew lines, <ref target="p12">12</ref><lb /></p> + +<p rend="text-indent: 0">Space system, <ref target="p19">19</ref>, <ref target="p23">23</ref><lb /></p> + +<p rend="text-indent: 0">Sphere, <ref target="p21">21</ref><lb /></p> + +<p rend="text-indent: 0">Steiner (1796-1863), <ref target="p129">129</ref>, <ref target="p130">130</ref>, <ref target="p131">131</ref>, +177, <ref target="p179">179</ref>, <ref target="p184">184</ref><lb /></p> + +<p rend="text-indent: 0">Steiner's construction, <ref target="p129">129</ref>, <ref target="p130">130</ref>, +131<lb /></p> + +<p rend="text-indent: 0">Superposed point-rows, <ref target="p47">47</ref>, <ref target="p48">48</ref>, <ref target="p49">49</ref><lb /></p> + +<p rend="text-indent: 0">Surfaces of the second degree, <ref target="p166">166</ref><lb /></p> + +<p rend="text-indent: 0">System of lines in space, <ref target="p20">20</ref>, <ref target="p23">23</ref><lb /></p> + +<p rend="text-indent: 0">Systems of conics, <ref target="p125">125</ref> <lb /></p> + +<p rend="text-indent: 0">Tangent line, <ref target="p61">61</ref>, <ref target="p80">80</ref>, <ref target="p81">81</ref>, <ref target="p87">87</ref>, <ref target="p88">88</ref>, +89, <ref target="p90">90</ref>, <ref target="p91">91</ref>, <ref target="p92">92</ref><lb /></p> + +<p rend="text-indent: 0">Tycho Brahe (1546-1601), <ref target="p162">162</ref> <lb /></p> + +<p rend="text-indent: 0">Verner, <ref target="p161">161</ref><lb /></p> + +<p rend="text-indent: 0">Vertex of conic, <ref target="p157">157</ref>, <ref target="p159">159</ref><lb /></p> + +<p rend="text-indent: 0">Von Staudt (1798-1867), <ref target="p179">179</ref>, <ref target="p185">185</ref> <lb /></p> + +<p rend="text-indent: 0">Wallis (1616-1703), <ref target="p162">162</ref><lb /></p> + +</div> + +</body> + +<back rend="page-break-before: right"> +<div> +<pgIf output="pdf"> + <then> + <div> + <divGen type="pgfooter" rend="page-break-before: right" /> + </div> + </then> + <else> + <div> + <head>Footnotes</head> + <divGen type="footnotes" /> + </div> + + <div> + <divGen type="pgfooter" rend="page-break-before: right" /> + </div> + </else> +</pgIf> +</div> +</back> + + </text> +</TEI.2> + +<!-- +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 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b/old/17001-tei/images/image49.png 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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