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 α, β, γ, δ, which all pass through the line _g_, pass
also through the four harmonic points _A_, _B_, _C_, _D_, so that α passes
through _A_, etc. Then it is clear that any plane π through _A_, _B_, _C_,
_D_ will cut out four harmonic lines from the four planes, for they are
lines through the intersection _P_ of _g_ with the plane π, and they pass
through the given harmonic points _A_, _B_, _C_, _D_. Any other plane σ
cuts _g_ in a point _S_ and cuts α, β, γ, δ in four lines that meet π in
four points _A’_, _B’_, _C’_, _D’_ lying on _PA_, _PB_, _PC_, and _PD_
respectively, and are thus four harmonic hues. Further, any ray cuts α, β,
γ, δ in four harmonic points, since any plane through the ray gives four
harmonic lines of intersection.

*35.*  These results may be put together as follows:

_Given any two assemblages of points, rays, or planes, perspectively
related to each other, four harmonic elements of one must correspond to
four elements of the other which are likewise harmonic._

If, now, two forms are perspectively related to a third, any four harmonic
elements of one must correspond to four harmonic elements in the other. We
take this as our definition of projective correspondence, and say:

*36. Definition of projectivity.*  _Two fundamental forms are protectively
related to each other when a one-to-one correspondence exists between the
elements of the two and when four harmonic elements of one correspond to
four harmonic elements of the other._

                                [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, σ, is
said to be _perspectively related_ to the pencil _S_ when every ray on _S_
goes through the point on σ which corresponds to it.

*72. Determination of the locus.*  It is now clear that five points,
arbitrarily chosen in the plane, are sufficient to determine a point-row
of the second order through them. Two of the points may be taken as
centers of two projective pencils, and the three others will determine
three pairs of corresponding rays of the pencils, and therefore all pairs.
If four points of the locus are given, together with the tangent at one of
them, the locus is likewise completely determined. For if the point at
which the tangent is given be taken as the center _S_ of one pencil, and
any other of the points for _S’_, then, besides the two pairs of
corresponding rays determined by the remaining two points, we have one
more pair, consisting of the tangent at _S_ and the ray _SS’_. Similarly,
the curve is determined by three points and the tangents at two of them.

*73. Circles and conics as point-rows of the second order.* It is not
difficult to see that a circle is a point-row of the second order. Indeed,
take any point _S_ on the circle and draw four harmonic rays through it.
They will cut the circle in four points, which will project to any other
point of the curve in four harmonic rays; for, by the theorem concerning
the angles inscribed in a circle, the angles involved in the second set of
four lines are the same as those in the first set. If, moreover, we
project the figure to any point in space, we shall get a cone, standing on
a circular base, generated by two projective axial pencils which are the
projections of the pencils at _S_ and _S’_. Cut across, now, by any plane,
and we get a conic section which is thus exhibited as the locus of
intersection of two projective pencils. It thus appears that a conic
section is a point-row of the second order. It will later appear that a
point-row of the second order is a conic section. In the future,
therefore, we shall refer to a point-row of the second order as a conic.

                               [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__ α_, where α 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 _κ_
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 _α_, _β_, _γ_, _δ_, ... and _α’_, _β’_, _γ’_, _δ’_, ... (Fig. 36).
Take now _γ_ as the center of a pencil projecting the points _α’_, _β’_,
_δ’_, ..., and take _γ’_ as the center of a pencil projecting the points
_α_, _β_, _δ_, .... 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 _(γα’, γ’α)_ to
_(γβ’, γ’β)_. 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 α and _π’_ coincide, as also do the points
_α’π_ and _γρ’_. By the above construction the line _γ’ρ_ must meet _γρ’_
on the line joining _(γα’, γ’α)_ with _(γπ’, γ’π)_. But these four points
form a quadrangle inscribed in the conic, and we know by § 95 that the
tangents at the opposite vertices _γ_ and _γ’_ meet on the line _v_. The
line _γ’ρ_ 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,