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ELEMENTS

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^ELEMENTS OF
PROJECTIVE GEOMETRY

BY
GEORGE HERBERT LING

GEORGE WENTWORTH

AND

DAVID EUGENE SMITH

GINN AND COMPANY

BOSTON • NEW YORK • CHICAGO • LONDON
ATLANTA • DALLAS • COLIIMHIIS • SAN KKANCISCO

i^fi ■z./^H

COPYKIGHT, 1922, BY GINN ANM) COMPANY

ENTERED AT STATIONERS' HALL

722.7

Cbt gtfttnKum 3Prt<*

(;iNN AND COMPANY- PRO-
PRIETORS • BOSTON • U.S.A.

PREFACE

This work has been prepared for the purpose of providing
a thoroughly usable textbook in projective geometry. It is not
intended to be an elaborate scientific treatise on the subject,
unfitted to classroom use ; neither has it been prepared for
the purpose of setting forth any special method of treatment ;
it aims at presenting the leading facts of the subject clearly,
succinctly, and with the hope of furnishing to college students
an interesting approach to this very attractive and important
branch of mathematics.

There are at least three classes of students for whom a study
of the subject is unquestionably desirable ; namely, those who
expect to proceed to the domain of higher mathematics, those
who are intending to take degrees in engineering, and those
who look forward to teaching in the secondary schools.
Although the value of the subject to the second of these
classes has not as yet been duly recognized in America,
European teachers for several decades have realized its useful-
ness as a theoretical basis for some of the practical work in this
field. For the large number of students belonging to the third
class, trigonometry, analytic geometry, and projective geometry
are the three subjects essential to a fair knowledge of elemen-
tary geometry, and it is believed that the presentation given
in this book is such as greatly to aid the future teacher. There
is a healthy and growing feeling in America that teachers of
secondary mathematics need a more thorough training in the
subject matter, even at the expense of some of the theory of
education which they now have. This being the case, one of
the best fields for their study is projective geometry.

iv PREFACE

It is recognized that students of projective geometry have
usually completed an elementary course in analytic geometry
and the calculus, that they have a taste for mathematics which
leads them to elect this branch of the science, and that there-
fore there may fittingly be some departure from the elementary
methods employed in the earlier mathematical subjects. On the
other hand, for some students at least, projective geometry is a
transition stage to higher mathematics, and the subject should
tlierefore be presented with due attention to the important
and recognized principles which must always be followed in
the preparation of a usable textbook.

It is the belief of the authors that they have followed these
principles in such a way as to afford to college students a simple
but sufficient introduction to this interesting and valuable
branch of geometry. Especial attention has been given to the
proper paging of the book, to a clear presentation of the great
basal propositions, to the illustrations accompanying the text,
to the number and careful grading of the exercises, and to the
application of projective geometry to the more elementary field
of ordinary Euclidean geometry.

CONTENTS

PART I. GENERAL THEORY

CHAPTER PAOB

I. Introduction 1

11. Principle of Duality 15

III. Metric Relations. Anharmonic Ratio .... 21

IV. Harmonic Forms 31

V. Figures in Plane Homology 37

VI. Projectivities of Prime Forms 41

VII. Superposed Projective Forms 63

PART II. APPLICATIONS

VIII. Projectively Generated Figures 79

IX. Figures of the Second Order 101

X. CoNics 115

XI. CONICS AND THE ELEMENTS AT INFINITY .... 143

XII. Poles and Polars of Conics 151

XIII. Quadric Cones 167

XIV. Skew Ruled Surfaces 173

HISTORY OF PROJECTIVE GEOMETRY 181

INDEX 185

GREEK ALPHABET

The use of letters to represent both iminbers and geometric
magnitudes has become so extensive in mathematics tjiat it is
convenient for certain purposes to employ the letters of the
Greek alphabet. In projective geometry the Greek letters are
used particularly to represent planes and angles. These letters
with their names are as follows :

al[)ha

beta

gamma

omicron

delta

l>i

ei)silon

rho

/eta

0-, s

sigma

eta

tau

theta

upsilon

iota

'i>

phi

kappa

chi

lambda

psi

omega

ELEMENTS OF
PROJECTIVE GEOMETRY

PART I. GENERAL THEORY

CHAPTER I

INTEODUCTIOK

1. Orthogonal Projection. In elementary geometry the
projection of a point upon a line or upon a plane is usually
defined as the foot of the perpendicular from the point to
the line or to the plane, and the projection of a line is defined
as the line determined by the projections of all its points.
This simple projection is called orthogonal projection.

oA
I

A A\ A2

Fig. 1 Fig. 2 Fig. 3

Thus, the point A' in Fig. 1 represents the orthogonal projection
of the point A upon the line Z; the line A{A2 in Fig. 2 represents
the projection of the line A^A^ upon the line I; and the line if in
Fig. 3 represents the projection of the curve k upon the plane a.

2. Symbols. Projective geometry, like other branches
of mathematics, employs special symbols, generally using
capital letters to denote points, small letters to denote lines,
the first letters of the Greek alphabet, a, y8, 7, S, • • • , to de-
note planes, and the Greek letters (f> and 6 to denote angles.

INTRODUCTION

3. Parallel Projection. In a plane a which contains a
line p and the pomts A^, A^, ^3, • • •, ^„, if a line I is drawn
making an angle <^ with p, each of the lines through
^j, ^3, -rdg, • • •, An parallel to I makes with p the angle 0.

//Y

% An

"^'^

The points A\^ A'^, A'^, • • •, A'^, in which these lines inter-
sect p, are called the projections of A^, A^, A^, • • •, A^ upon
jt; and are said to be found by parallel projection.

In space of three dimensions a plane figure A^A^A^ may
be projected upon a plane ir by parallel projection.

4. Central Projection. If several coplanar points ^j, A^t
A^, • • •, An are joined to a point P of their plane, and if
these lines are cut by a line p, the points of intersection

AK^

-o4.".

^^'2

of the lines FA^, PA^, PA^, . . ., PA,, with the line p are
called the projections of ^j, Jg^ ^3> • • •» ^n from the center
P upon ^ and are said to be found by central projection.

In space of three dimensions a plane figure A^A^A^A^
may be projected centrally upon a plane ir.

KINDS OF PROJECTION 3

5. Projection from an Axis. Let A^, A^, A^, • • •, A,^ be

points which are not all coplanar. Their orthogonal pro-
jections upon a Ime p may be obtamed by drawing a line
from each of them perpendicular to p, or
by passing a plane through each of them
perpendicular to p. The latter method
may be generalized by requiring merely
that the planes passed through the
points shall be parallel to a fixed plane
which is not necessarily perpendicular
to p, or by requiring that the planes
passed through the points shall pass
also through a fixed line p' instead of making them parallel
to a fixed plane, p' being different from p. Then the points
are said to be projected from an axis, and this second kmd of
projection is c&Wed projection from an aris or axial projection.

The projection of points by parallel planes is the limiting case of
projection from an axis in which the axis has receded indefinitely.

6. Operations of Projection and Section. The process of
finding the projection of a plane figure upon a line or
plane consists of two parts. The first part is called the
operation of projection and consists in the construction of a
figure composed of lines, or of planes, or of both Imes and
planes, passing through the points and lines of the figure
and through the center or axis of projection. These lines
and planes are called the projectors of the points and
lines of the figure, and constitute the projector of the figure.

The second part is called the operation of section and
consists in cutting the projector of the figure by a line or
plane called the line of projection or the plane of projection.

The center or axis of projection, and the line or plane of projection,
should be so taken as not to be parts of the figure to be projected.

4 INTRODUCTION

Exercise 1. Simple Projections

1. Draw a figure showing the orthogonal projection of a
given circle upon a given plane.

The figure may be drawn freehand, and at least three cases should
be considered : (1) the circle parallel to the plane ; (2) the circle oblique
to the plane ; (3) the circle perpendicular to the plane. In (2) consider
also the case in which the circle cuts the plane.

2. Draw a figure showing the parallel, but not necessarily
orthogonal, projection of a given square upon a given plane.

3. Draw a figure showing the central projection of a given
straight line or a given plane curve upon a given line.

4. Draw a figure showing the central projection of a given
square upon a given plane.

Consider three cases, as in Ex. 1. Consider the case in which P 13
between the square and the plane as well as the case in which it is not.

5. Draw a figure in which the four vertices of a square
are projected from a given axis upon a given line.

In projecting from a center P upon a plane tt, describe the
projectors and the projections of the following, mentionin(/ all
the noteworthy special cases under each :

6. A set of points. 10. Two intersecting lines.

7. A line. 11. Two parallel lines.

8. A triangle. 12. A quadrilateral.

9. A circle. 13. A pentagon.

14. Four points and the lines joining them in pairs.

15. A tangent to a circle at any point.

In projecting from an axis p' upon (1) a plane ir and
(2) a line jo, describe the projectors and the projections of the
following, mentioning the noteworthy special cases under each :

16. A set of points. 18. A line not parallel to^'.

17. A line parallel to p'. 19. A circle.

ELEMENTS AT INFINITY 5

7. Elements at Infinity. Consideriiig central projection
only, and supposing the center P and the plane of projec-
tion TT to be given, these questions now deserve attention :

1. Does every point of a plane a have a projector?
Does it have a projection ?

Since every point of the plane a can be joined to P by a straight
line, every point of a has a projector; but since this projector may
happen to be parallel to tt, a point of a may have no projection.

2. Is every line which passes through P the projector
of some point of a ?

No ; for certain of these lines may be jiarallel to a.

3. Does every line of a liave a projector ? Does it have
a projection ?

Consider the answers to Question 1. Draw the figure.

4. Is every plane which passes through P the projector
of some line of a ?

Consider the answer to Question 2. Draw the figure.

Certain exceptional cases have been suggested in con-
nection -with these questions. Their occurrence is due to
the existence of parallel lines and parallel planes, and the
difficulty caused by them may be removed as follows :

Every straight line is assumed to have one and only
one infinitely distant point, and this point is called the
point at infinity of the line.

Every plane is treated as having one and only one
straight line situated entirely at an infinite distance, and
as having all its infinitely distant pomts situated on that
line. This line is called the line at infinity of the plane.

Space is treated as having one and only one plane situ-
ated entirely at an infinite distance, and as having all its
infinitely distant pomts and lines situated on that plane.
This plane is called the plane at infinity.

Ci\ \
P A,

6 INTRODUCTION

8. Illustrations. Let a line c rotate in a plane a about
a point C of that plane, and when it has the positions
Cyy c^"-') let it meet a fixed line p in the points A^^
^2, . . .. Then as long as c is not parallel to p it meets
p in one point and only one. As the point of intersection
becomes more and more dis-
tant, the line c becomes more
and more nearly parallel to
p. The limiting position of c
as the point of intersection
recedes infinitely is the line c'
through C parallel to p. If,
however, the rotation of c is continued ever so little beyond
c\ the intersection of c and p is found to be at a great
distance in the other direction on p, and as the rotation
proceeds farther this point of intersection comes continu-
ously back toward Ay Hence c' is said to meet jt? in a
point at infinity. If there were several infinitely distant
points on jt?, they would with C determine several lines
through C parallel to p, or several of these points would be
common to c' and jo, or one or more of these points taken
with C would fail to determine a straight Ime. Apparent
conflict with propositions of Euclidean geometry is best
avoided by the assumption that every line not situated at an
infinite distance has one and only one infinitely distant point.

Now consider all points of a plane which are infinitely
distant. In elementary geometry we find that the only
plane locus met by every line of its plane in one and only
one point is a straight line. The locus of infinitely distant
points of the plane also possesses this property. Hence this
latter locus is called the (straight) line at infinity of the
plane. Similarly, the locus of the infinitely distant points
in space is called the plane at infinity.

PROJECTORS AND PROJECTIONS 7

9. Ideas of Projector and Projection Simplified. From the
considerations set forth in §§ 7 and 8 it appears that the
introduction of the elements at infinity lias distinct advan-
tages arising out of the fact that, from the new point of
view, statements can often be more briefly and more simply
made. The greater simplicity is due to the fact that certain
cases involved in the questions cease to be exceptional.

For example, in dealing with the first question in § 7 we now
say that every point of a plane a has a projection upon the plane tt ;
for if a projector happens to be parallel to tt, it is regarded as meet-
ing TT in one point at infinity.

It is also clear that we may now say that all the lines through a
point P, and lying in a plane determined by P and a line a, consti-
tute the jirojector from the center P of all the points of a.

Similarly, we may say that all the lines and all the planes passing
through P constitute respectively the projectors from the center P
of all the points and all the lines of any plane not passing through P;
and that all the planes through any line p form the projector from
the axis p of all the points of any line not parallel to p.

Exercise 2. Projectors and Projections

1. Draw figures illustrating the four statements in the last
three paragraphs above.

2. Consider the truth of the statement that two lines in
a plane have one and only one common point. Illustrate the
statement by a figure.

3. Do every two planes in a space of three dimensions
determine a line ? Explain the statement.

4. In what case do a straight line and a plane fail to
determine within a finite distance exactly one point ?

5. In what case do a straight line and a point fail to
determine a plane ?

6. Two straight lines which determine a plane determine,
without exception, a point.

8 INTRODUCTION

10. The Ten Prime Forms. As fundamental sets of ele-
ments we use the following sets, called the ten prime forms :

One-Dimensional Forms. 1. The totality of points of a
straight luie (the base') is called a range of points, a range,
or, less frequently, a pencil of points.

The distinction between a line and the totality of its points may
be appreciated by considering jioiuts as arranged on a line like beads
on a string. Similar considerations apply to the other prime forms.

2. The totality of planes through a straight line (the
base') is called an axial pencil.

This is also called a pencil of planes or a slieaf nf planes.

3. The totality of straight lines in a plane and through
a point of the plane is called a flat pencil.

In a flat j>encil, either the point common to the lines or the plane
containing the lines may be regarded as the base of the pencil.

The terms range of points, axial pencil, and flat pencil are used for
a finite number as well as for an infinite number of elements.

Two-Dimensional Forms. 4. The totality of points in a
plane (the base) is called a plane of points.

5. The totality of planes through a point (the base)
is called a bundle of planes.

6. The totality of lines in a plane (the base) is called
a plane of lines.

7. The totality of lines through a point (the base) is
called a bundle of lines.

It is also called a sheaf of lines, but because the word sheaf is
used in conflicting senses, we shall not use it.

Three-Dimensional Forms. 8. The totality of points of
three-dimensional space.

9. The totality of planes of three-dimensional space.

Four -Dimensional Form. 10. The totality of lines of three-
dimensional space.

THE TEX PRIME FORMS 9

Exercise 3. The Ten Prime Forms

Draw a rough sketch to illustrate each of the following :

1. Range of points. 4. Plane of points.

2. Axial pencil. 5. Bundle of planes.

3. Flat pencil. 6. Plane of lines.

Examine each of the following prime forms when the base
is at infiyiity :

7. Plat pencil. 9. Bundle of lines.

8. Axial pencil. 10. Bundle of planes.

11. Find the central projection of a range of points ; of a
flat pencil ; of a plane of points ; of a plane of lines.

12. Find the plane section of a flat pencil ; of an axial
pencil ; of a bundle of planes ; of a bundle of lines.

13. Find the axial projection of a range of points and also
of a flat pencil, the axis passing through the base.

14. Find the linear section of an axial pencil and also of a
flat pencil, the line of section being in the plane.

15. Investigate the central projection of a bundle of lines ;
of the points of space ; of the lines of space.

16. Investigate the plane section of a plane of lines ; of the
planes of space ; of the lines of space.

17. Investigate the projection from an axis of a plane of
points and also of the points of space.

18. Investigate the linear sections of a bundle of planes and
also of the planes of space.

19. Apply each of the four operations to the prime forms
not already considered in connection with it.

20. Examine the results of Exs. 11-19 and in each case
determine whether to every element of the original figure there
corresponds one element and only one element of the resulting
figure, and vice versa.

INTRODUCTION

11. Classification of Prime Forms. In each of the first
three classes of the ten prime forms mentioned in § 10
the prime forms of every possible pair are connected by
a simple relation.

Consider first a range of
points A^A^A^ > > • A^ • • • on a
base j3, and consider its pro-
jector a^a^a^ •••««••• from a
pomt P exterior to /?, this pro-
jector being manifestly a flat pencil. By setting up, or
arranging, the infinitely many pairs of elements A^, a^;
^2, a^\ ^3, a^', •••; A^^ a»; • • •, we find that for every
point of the range there is a corresponding line of the
flat pencil, and vice versa ; and that if two points are
nearly coincident, so also are the corresponding lines.

Next, make a section of an axial pencil by a plane tt.
From the planes a^, a^^ a^, • • •, »„, • . • of the axial pencil
and the lines aj, ag, ag, • • ., a,^, • • . of the section of these
planes by the plane tt, infinitely
many pairs of elements a^, a^i
a^, aj; ag, a^; • • •; a,i, a„; • • • may
be set up. Evidently for every
plane of the axial pencil there
is a line of its section (the flat
pencil), and vice versa ; similarly
for the range and the axial pencil.

A similar conclusion may be
reached regarding any two prime forms of the second class,
and also regarding the two prime forms of the third
class. In each case, by the setting up of the pairs, there is
established a one-to-one correspondence between the elements
of the two forms.

This is often written as a 1 — 1 correspondence.

PERSPECTIVITY

12. Perspectivity. Certain cases of one-to-one corre-
spondence between the elements of prime forms of the same
kind should also be noted. For example, in this figure if two
transversals j9j, jt?( cut
the lines a^, a^, - • >, a,,,

• • • of a flat pencil
in the pomts A^, A[ ;

A A' • . . .' 4 A' ' . . .

the ranges A^A^- • •
and A\A',^ • • • corre-
spond in this way.

Similarly, in this figure, if two flat pencils a^a^ •••«,,

and rtj«2

a„ ' ' ' are so situated that a

J, Mj, M2»

ttn, a,

^H^ C'L\ ' ' ' intersect in
the points A^, A^, • • •,
J„, • • • of a range, such
a correspondence exists.

The correspondences
in the cases in § 11 re-
sulted from one opera-
tion of projection or one of section. In the cases just men-
tioned the correspondences resulted from one operation
of projection and one of section. All these cases and other
similar cases may be brought into one group by means of
the following definition :

If either of two prime forms can be obtained from the
other by means of one operation of projection, or one opera-
tion of section, or by means of one operation of each kind,
the two forms are said to be perspectively related, or to be
in perspective, or to be perspective.

The symbol ^ is often used for " is perspective with."
The perspective relation is called a perspectivity.

12 INTRODUCTION

Exercise 4. Perspectivity and Projection

1. If the line a' of the plane a' is the projection, from the
center P, of the line a of the plane a, then the lines a and a
intersect in a point on the line of intersection of a and a'.

2. If the angle formed by the lines a[ and al of the plane
«' is the ])rojection, from the center P, of the angle formed by
the lines a^ and a.^ of the plane a, the pairs of lines a^, a[ ;
«2, a.2 intersect in points on the line of intersection of a and a'.

3. If two triangles A^A^A^ and A'^Al^A'^ of the planes a and
«' respectively are so situated that the lines AiA[, A^A'^, and
Af^A^ pass through a common point P, the intersections of
the pairs of sides A^A^, A'lA^; A.^A^, A'^A'^; A^A^, A'^A[ are
collinear.

4. If two polygons A^A^ - • • A^ and A'lA!^ • • • Al^ of the
planes a and «' respectively are so situated that the lines
A^A[, ^2^2> • • •> ^M^« P^ss through a common point P, the
intersections of the pairs of sides A^A^, A^A'^ ; A^A^, Al^A'^; • • -;
yl„/1j, ^,',^1, and the intersections of the pairs of diagonals
'^I'^s' ^"^i^^si A^A^, .'I1-I4; • • • ; A^A^, A^Ai] • • • ; ^^.-l„j, ^yL-'^M? * ' '
are collinear.

It will be noticed that Exs. 1-4 form a related set of problems, as is
also the case with Exs. 6-8.

5. If two lines a and a' of the planes a and a' respectively
intersect, either may be regarded as the projection of the other
from any point exterior to both lines but in their common plane.

6. State and prove the converse of Ex. 2.

7. State and prove the converse of Ex. 3.

8. State and prove the converse of Ex. 4.

9. Given three points on a line a and a point A^ not on the
line, construct a triangle that shall have vlj as a vertex and
shall have each of its sides, produced if necessary, pass through
one and only one of the three given points. How many such
triangles can be constructed ?

PERSPECTIVITY AND PROJECTION 13

10. Investigate the problem similar to Ex. 9 in which two
given points A^ and A^ of the plane a are to be vertices of the
required triangle, and show how to construct the triangle when
such a triangle exists.

11. Given the points Ai and A[ of two planes a and a' which
intersect in a given line a, and given in the plane a a triangle
constructed as required in Ex. 9, use Ex. 3 to obtain a triangle
in the plane a' that shall have ^{ as a vertex and shall have
sides which, produced if necessary, shall intersect the line a
in the points in which this line is cut by the sides of the given
triangle in a. How many constructions are possible ?

12. Investigate the cases of Ex. 11 in which a second vertex
of one or of each of the triangles is also given.

13. If two triangles AiA.2A^ and A'^Al^A^ in the same plane
a are so situated that the lines A^A'^, A^A[, and A^Al^ are con-
current, the intersections oiA^A^, ^1^2 > ^2^3? ^2^3 > -^gJi, A'^A[
are collinear.

Let A^A^ and A\A'^ meet in Cj, A^A^ and A'^A'^ in C■^, and A^A^
and A'^A\ in Cg. Take a center of projection P not in tlie plane a, and
project the wliole figure upon a plane parallel to the plane PC^C^, thus
obtaining the line at infinity as the projection of CgC^. Prove that the
projection of Co is on this line.

The development of this problem and similar problems is fully
considered in Chapter V.

14. State and prove the converse of Ex. 13.

15. State and prove the proposition of plane geometry which
corresponds to Ex. 4.

16. State and prove the converse of Ex. 15.

17. Given three points A^, A^ A^ on a line a in a plane a,
and three points A[, A 2, A'^ on a line a'also in the plane a, find
three points A[', .4", Ag, not necessarily collineav, into which
both sets of three points can be projected.

18. With the same data as in Ex. 17 find three collinear
points A[', A!/, A'^' into which the first two sets of three points
mentioned can be projected.

14 INTRODUCTION

19. In Ex. 17 consider also the case in wliicli the lines a
and a! are coincident and in which the points A[, Al, A'^ are
not necessarily all distinct from the points .1,, A,^, A^.

20. Given three points on a line a, construct a quadrilateral
such that the pairs of opposite sides shall intersect in two of
the given points, and such that one of its diagonals shall pass
through the other point.

21. Assuming the construction asked for in Ex. 20, use Ex. 4
and Ex. 16 to obtain additional quadrilaterals fulfilling the
same conditions as the first. Do the other diagonals of these
quadrilaterals intersect ?

22. Given two quadrilaterals so constructed as to fulfill the
conditions of Ex. 20, the straight lines joining corresponding
vertices of these figures are concurrent.

23. If the quadrilateral constructed in Ex. 20 moves so as
continuously to fulfill the conditions stated, the other diagonal
constantly passes through a fixed point.

24. Show how to find the fixed point mentioned in Ex. 23.

25. In Ex. 20, if the third of the given points bisects
the segment joining the other two given points, determine the
position of the fixed point mentioned in Ex. 23.

26. Given five points on a line a, construct a quadrilateral
such that each of its sides and one of its diagonals, pro-
duced if necessary, shall pass through one and only one of
the given points. Obtain additional quadrilaterals fulfilling
the same conditions.

27. In Ex. 2G investigate the relation that the other diago-
nals of any two of the quadrilaterals bear to each other and to
the given line a.

28. Extend the problem in Ex. 26 to the case of the pentagon.

29. Given two quadrilaterals so constructed as to fulfill the
conditions of Ex. 26, the straight lines joining pairs of corre-
sponding vertices of these figures are concurrent.

CHAPTER II

PRINCIPLE OF DUALITY

13. Principle of Duality. It is now liighly desirable to
consider a certain important relation between pairs of
figures in space, and also between their properties. The
nature of this relation, by the use of which the difficulties
of the subject may be reduced by almost half, is explained
by the Principle of Dnalitt/, or the Principle of Reciprocity,
which may be stated as follows:

Corresponding to any figure in space which is made vp of
or generated hy points, lines, and planes there exists a second
figure which is made up of or generated hy planes, lines, and
points, such that to every point, every line, and every plane of
the first figure there corresponds respectively a plane, a line,
and a point of the second figure, and such that to every propo-
sition which relates to points, lines, and planes of the first
figure, but which does not essentially involve ideas of measure-
ment, there corresponds a similar jyroposition regarding the
planes, lines, and points of the second figure, and these two
propositions are either both true or both false.

The two figures which are related in the manner just
described, as well as the two propositions, are said to be
dual, reciprocal, or correlative.

As a simple illustration of the principle, consider the following :

Two points determine a line. Two planes determine a line.

Two lines through a point deter- Two lines in a plane determine
mine a plane. a point.

16 PRINCIPLE OF DUALITY

14. Assumption of the Principle of Duality. The validity
of the principle of duality will not be proved in this book,
although it is possible so to formulate the axioms of pro-
jective geometry that they are unchanged if everywhere the
words point and plane are interchanged, and thus to show
this validity. Nevertheless the principle will be applied
Avith great frequency in deriving properties of figures, and
in so doing either of two courses may be adopted: On
the one hand, it may be assumed that the prmciple is valid
and is capable of a proof which is, of course, entirely inde-
pendent of any results obtained by means of the principle
itself; on the other hand, the principle may be used
simply as the basis of a rule for formulating the dual of
any proposition, the rule being justified in every case by a
proof of this dual proposition.

Of these two courses the latter is not a difficult one, for
after the principle of duality has been used to derive the
enunciation of the dual proposition, it may be applied to
the various steps of the proof of the original proposition to
obtain a new set of statements which may be examined
to see if they constitute a proof of the dual. In each case it
will be found that a proof is secured. The plan has the
further advantage that it avoids the feeling of dissatisfac-
tion and uncertainty attendant upon making a very general
and far-reaching but apparently unjustified assumption,
besides which the repeated application of the principle
leads to that confidence in its validity which comes from
increasing experimental evidence.

For this reason the second course has been adopted in
this book ; and whenever the dual of a proposition is derived
by applying the principle of duality, either the proof of
the dual is derived by the same means or such derivation
is left to serve as an exercise for the student.

DERIVATION OF DUAL PROPOSITIONS 17

15. Derivation of Dual Propositions. If a figure or a
proposition in the geometry of space is given, the first
considerations which enter into the derivation of the dual
figures or propositions are the facts that the point and the
plane are dual elements, and that every geometric figure
may be obtained by using either the point or the plane
as the primary generating element. Although neither the
point nor the plane has a superior claim over the other
to be considered as the primary generating element, it fre-
quently happens that the statement of a proposition is so
framed as to imply that one or the other of these elements
has been so used. For this reason the derivation of the dual
of any proposition generally requires more than the mere
interchange of the words point and plane. On the other
hand, it is true, almost without exception, that propositions
which have duals may be so stated that the latter may be
found from the former by such interchange.

Skill in the derivation of dual figures and propositions
is quickly gained, and the examples given in § 16 will
assist the beginner in acquiring this skill. In general it
may be said that the student will find it advantageous to
consider, in every proposition he meets, the proposition
which results from the method of treatment mentioned
above, and then to consider whether the proof of the
derived proposition can be obtained from the original
proof in the same way.

In plane geometry the pomt and the line are dual ele-
ments, and any figure may be regarded as having been gen-
erated by either of these elements. In a general way, duals
in the plane are derived by interchanging these elements.

The principle of duality for threefold space, applied to
plane geometry, yields the geometry of the bundle of lines
and planes, the line and the plane being dual elements.

PKINCIPLE OF DUALITY

16. Examples of Duality,
the more simple examples of

1. Point A.

2. Line a.

3. Two points determine a line.

4. Two lines which determine

a plane also determine a
point.

5. Three points in general de-

termine a plane.

6. Several points which lie in a

plane.

7. Several lines which lie in a

plane.

8. A plane triangle; that is,

three points and the three
lines determined by them
in pairs.

9. A plane poh/gon.

10. A range of points.

11. K flat pencil.

12. A plane of points and a plane

of lines.

13. Four points in a plane and

the six lines joining them
in pairs ; a coni])lete quad-
rangle, or four-point.

14. Four lines in a plane and the

six points determined by
the various pairs of these
lines ; a comi)lete quadri-
lateral, or four-line.

15. Four points in space and the

lines and planes deter-
mined by them.

The followmg are a few of
duality :

1'. Plane a.

2'. Line a'.

3'. Two planes determine a line.

4'. Two litres which determine

a point also determine a

j)lane.

5'. Three planes in general de-
termine a point.

6'. Several planes which pass
through a point.

T. Several lines which pass
through a point.

8'. A trihedral angle; that is,
three planes and the three
lines determined by them
in pairs.

9'. A polyhedral angle.

10'. An axial pencil.

11'. A flat pencil.

12'. A bundle of planes and a
bundle of lines.

13'. Four planes through a point
and the six lines of inter-
section in pairs; a com-
plete four-flat.

14'. Four lines through a point
and the six planes deter-
mined by the various pairs
of these lines ; a complete
four-edge.

15'. Four planes in space and
the lines and points deter-
mined by them.

EXAMPLES OF DUALITY

16. Given three collinear points

A, B, C, find iovLT points P^,
P^, P3, P^ such that the
• lines P^P^ and P^P^ shall
meet in A, the lines Pg^a
and P4P1 shall meet in B,
and the lineP^P^ shall pass
through C.

17. // (he line a' of the plane a'

is the projection from the
center P of a line a of
the plane a, the lines a and
a' intersect in a point of
the line of intersection of a
and a.

Proof. The lines a and a' lie in
the plane determined by P and
a, and hence they determine a
point. Since their jwint of in-
tersection lies in the jilane a
and also in the plane a', it
lies in the line determined by
a and a'.

16'. Given three coaxial planes a,
/3, y, find four planes tTj,
iTo, TTj, TT^ such that the
lines TTjTTg and Tr^TTi shall
lie in a, the lines tt^tt^
and TT^TTi shall lie in /?,
and the line tt^tt^ shall lie
in y.

17'. If the line a' through the point
A' is determined by A' and
the point of intersection of
the plane it tcitk the line a
through the point A, the lines
a and a' lie in a plane which
passes through the line A A',

Proof. The lines a and a' pass
through the point determined
by 7r and a, and hence they
determine a plane. Since their
jilane passes through the point
^1 and also through the point
A', it passes through the line
determined by A andyl'.

17. Figures in a Plane and Figures in a Bundle. The

Principle of Duality may also be stated for the geometry
of figures in a plane, and likewise for the geometry of
figures in a bundle. In the first case the point and the line
are dual elements, and in the second case the line and
the plane. Simple modifications of the statement of the
principle in §13 yield the statements for the two cases.

Pairs of the examples of § 16 may be used to illustrate this fact.
For example, the following pairs are duals in the plane : 1, 2 ; 6, 7 ;
8, 8; 9,9; 10, 11; 13, 14.

The following pairs are duals in the bundle : 1', 2' ; 6', 7' ; 8', 8' ;
9', 9'; 10', 11'; 13', 14'.

Many other examples of duality will be found as we proceed.

20 PRINCIPLE OF DUALITY

Exercise 5. Principle of Duality

1. By means of the Principle of Duality obtain for three-
fold space the statement and also the proof of the dual of Ex. 3,
I>age 12.

2. Similarly, find the space dual of Ex. 4, page 12.

3. Derive and prove the space dual of Ex. 13, page 13.

4. Obtain for plane geometry the statement and proof of
the dual of Ex. 13, page 13.

5. Derive the space dual of Ex. 4.

6. Verify that in the geometry of the bundle the results
of Exs. 3 and 5 are dual.

7. If a proposition in plane geometry or in the geom-
etry of the bundle has a dual, but is not self-dual in that
geometry, then in the geometry of threefold space it belongs
to a set of four propositions each of which is dual with two
of the others.

8. If a proposition is self-dual in plane geometry, then the
four propositions mentioned in Ex. 7 reduce to two.

9. Can the four propositions of Ex. 7 ever reduce to one
proposition ? Discuss in full.

10. Give an example of a self-dual figure in threefold space.

11. State a simple self-dual proposition regarding the figure
mentioned in the answer to Ex. 10.

12. Are all propositions regarding the self-dual figure of
Ex. 10 themselves self-dual ? Discuss in full.

13. Given three planes passing through a line a of a bundle
whose base is A, construct in this bundle a four-edge such that
pairs of opposite edges lie in two of the given planes and one
of its diagonal lines lies in the remaining plane.

Compare this example with Ex. 20, page 14. Notice that when a prop-
osition is harder to prove than its dual the proof of the latter may be
used to suggest that of the former.

CHAPTER III

METRIC RELATIONS. ANHARMONIC RATIO

18. Metric and Descriptive Properties. Properties of geo-
metric figures are of two sorts : (1) metric, that is, those
which relate to the measurement of geometric magnitudes ;
(2) descriptive, that is, those which are not metric. Nearly
all the propositions of ordinary elementary geometry deal
^vith metric properties, while, speaking generally, those of
projective geometry deal with descriptive properties.' In
fact, it is possible to exclude almost entirely from projec-
tive geometry the consideration of the metric properties of
figures. On the other hand, even when the object in view
is the study of the descriptive properties of figures, it
frequently happens that brevity is secured by the use of
metric considerations. For this reason the metric proper-
ties of figures will be used freely whenever the nature of
the work is such as to make this course advisable.

If the student will consider the work which has thus far been done
in this book he will see that no statement has been made that depends
in any way upon measurement. The lines projected may be of any
desired length, the angles may have any desired measure, and the
closed figures may have any desired area. It is therefore evident that
the work thus far has not been metric in any way.

In this chapter, on the other hand, we proceed to establish certain
important properties which will prove to be of great service to us in
subsequent work. IMoreover, as we proceed, it will appear that by
virtue of the propositions proved in §§26 and 27 the study of these
properties is appropriate in connection with the descriptive properties
of figures.

22 METRIC llELATIONS

19. Relations of Line Segments. In measuring distances
along a straight line attention is given to direction as well
as to length. One direction along a line is selected as posi-
tive, the opposite one being negative. The direction of a
line segment is called its sense and is indicated by the
order of the end letters, JJi denoting the segment of a line
thought of as extendhig from A to B and BA the segment
of a line thought of as extending from B to A. Evidently,
therefore, we have

AB = -BA,

or AB+BA = 0.

Having adopted this convention with respect to signs,
many identical relations can be proved. For example, A,
B, Cy • • 'y J, K being coUmear pomts in any order :

O O O— O O 0 I O 0 o—

AD^B CK ' E J

1. AB+BC+CD+..- + JK + KA = (i.

2. AB.CD+BC'AD + CA • BD = 0.

3. BC ' AD^+CA . in? + AB . ClP' + BC - CA . AB = 0.

In one method of proving these relations we employ
as origin any point 0 on the given line. Then for any
segment AB we have

AB = OB-OA.

This substitution and others of a similar nature being
made in any identity of this sort, the truth of the identity
becomes apparent.

The proof may be made algebraic if the measures of the
segments OAj OB, • • . are denoted by the letters a, 5, • • •.
Moreover, having an identical relation among real alge-
braic numbers, we may deduce a corresponding relation
among line segments.

LINE SEGMENTS AND ANGLES 23

20. Relations of Angles. In measuring angles attention
is given to tlie direction of rotation as well as to the magni-
tude of the angles. Rotation is called positive if it proceeds
in the direction opposite to that taken by the hands of a
clock ; otherwise it is called negative. The direction of
rotation is called the sense of the angle and is mdicated by
the order of the letters which denote its arms, the angle
formed by the rotation of a line from the position of the
line a to that of the line b being called the angle ab.

The ambiguity which may be felt to attach to this
method of representing angles may easily be removed in
the following way : Take as the standard line any line o
through the intersection of the lines a and b, and let it be
agreed that the angle between a and b
shall be understood to mean that angle
formed by the lines a and b which does
not contain the line o, and that the angle
oa formed by o and a shall mean the
angle included between a specified one
of the halves of the infinite line o which proceed from the
point common to a and b and that half of a which is first
reached by a positive rotation from o.

Then the algebraic identities by means of which the rela-
tions between line segments were proved, as well as all
other algebraic identities, are capable of interpretations with
respect to angles. In the above case ab — ob — oa, and by
means of such identities the relations may be verified.

The dihedral angles formed by pairs of planes of an
axial pencil may be treated in a similar fashion, a standard
plane (o being used. In this case the angle between two
planes a and ^ will be denoted by a^ if the planes a, /3,
and o) have the same general positions as the lines a, 5,
and 0 in the figure shown above.

24 ANHARMONIC RATIO

21. Anharmonic Ratio. The most useful metric element
in projective geometry is called an anharmonic ratio. It is
related to a range of four points, a flat pencil of four lines,
and an axial pencil of four planes, as follows :

1. The anharmonic ratio (^ABCD^ of four collinear points
A, B, C, D, is defined as .^ . ^

'bc'Iw'

2. TJie anharmonic ratio (ahcd^ of four concurrent and
coplanar line segments a, J, c, d is defined as

sin ac ^ sin ad
sin he sin hd

3. The anharmonic ratio (a^yB^ of four coaxial planes a, ^,

7, 8 is defined as • • s

'' '' sm ay ^ sm ad

sin /Sy * sin /38

An anharmonic ratio is also called a cross ratio or a double ratio.
The anharmonic ratio (ABCD) is easily remembered by writing

A A

— — : — — and then writing C in both terms of the first fraction and

D in both terms of the second fraction.

The above definition of anharmonic ratio, though not universal,
has the approval of the leading authorities of the present time.

22. CoKOLLAKY. If A, B, C, D are collinear points^ then:

1. (^ABCD^ is negative when and only when the segment
AB contains either C or Z), hut not hoth.

2. (^ABCU) approaches AC/BG as a limit as D recedes
indefinitely in either direction.

Exercise 6. Anharmonic Ratios

1. If (ABCD^) = (ABCD^), D^ and D^ are coincident.

2. Consider Ex. 1 and § 22 for the anharmonic ratio (nhed).

3. Consider Ex, 1 and § 22 for the anharmonic ratio (n^yB).

RATIOS OF FOUR POINTS 25

23. Twenty-four Anharmonic Ratios. Corresponding to
the order A, B, C, D of four collinear points, there has been
defined the anharmonic ratio (^ABCD^. There are, however,
twenty-four possible orders for these points, that is, the 4 !
permutations of the four letters ; and therefore there are
twenty -four anharmonic ratios for the four points, as follows:

QABCD), {ABDC), QACBD), (ACDB), (ADBC), (ABCB'),
(BACD}, (BABC), (BCAD), (BCDA}, (^BBAC), {BDCA\
(CABD), (^CADB), (CBAB), (CBDA^, {CDAB^, {CBBA^,
(BABC), (^DACB), (BBAC), (^DBCA), (DCAB), (DCBA).
But by definition (§ 21)

^ -^ AC AD AC'BD

while (^ABI>C) = ^:^=^^^,

^ ^ BI) BC AC'BD

and so • {BACD^ = {ABDC).

In like manner it may be shown that the last eighteen
ratios fall into six sets of three each, all those in any set
being equal to one of the first six anharmonic ratios.

Exercise 7. Anharmonic Ratios
Given the three collinear points A, B, C, proceed as follows :

1. Find the collinear point D such that {ABCD)=^ 7.

2. Find the collinear point D such that (ABCD) = — 7.

3. Find the collinear point D such that (ABCD)= k.

4. Prove that (ABCD) = (BADC) = (CDAB) = (DCBA).

5. Determine the several pernmtations of the four elements
A, B, C, D which leave the value of (ABCD) unchanged.

6. Which of the twenty -four ratios are equal to (A BBC) ?

26 ANHARMONIC RATIO

24. Relations of the First Six Ratios. The first six
anharmonic ratios given in § 23 are also connected by sim-
ple relations. If (^ABCn)=x, we have the following:

1. (ABCD)=x.

rAnnns ^^ ^^ AC-BD
lor ^ABCD) = — :— = j^-^ = x,

^ ^ BD BC AC'BD X

3. (^ACBD^ = \-x.

For (§ 19, 2) AB ■ CD + BC • AD + CA ■ BD^ 0;

AB CD , CA-BD , AC AD ,
^•'^"•^^ AD'CB^^-ADTcB^^-BC-Bb^^-'''

Therefore (J CBD) = \-x.
1

4. (ACDB} =

For (yl CDB) = — -— , since we have simply interchanged the

'^ ^ (ACBD) ^-^ °

last two letters, as in 1 and 2 above. Hence the result follows from 3.

5. (ADBC} = ^^'

For (^ CBD) = \ — X, by 3, where we merely interchange the
second and third letters. Hence, by similar reasoning,

{A DBC) = 1 - (^ BDC) = 1-1 = ini .

6. (ADCB)=^--

x — 1

For we found from 1 and 2 that the transposition of the third and
fourth letters gave the reciprocal of the original anharmonic ratio,

1 X

and so from 5 we have (ADCB) = = •

^ ^ (A DBC) x-1

RELATIONS OF THE RATIOS

25. Equality of the Six Expressions. We may now de-
termine the values of x for which any pair of the six

1 ^ 1 x—\ , X ,

expressions x^-t \ — x^ , , and are equal,

X 1 — X X x — 1

and therefore we may determine the values of these six
expressions which correspond to the values of x so
found. The results may be put in tabular form as follows :

1-x

1-X

x-1

-1

l+V-3

2
l-V-3

l-V-3

2
l+V-3

-1

l+V-3

-1

l-V-3

l_V-3

l+V-3

l-V-3

l + V-3

-1

It will be noticed that these values of the six distinct
ratios of the 4 ! anharmonic ratios may be classified into
three groups as follows :

1. Those in which the values of x are imaginary, the
values of all the functions being also imaginary.

2. Those in which the values of x are 1, 0, or oo, the
values of the functions being also 1, 0, or oo.

3. Those in which the values of x are —1, ^, or 2, the
values of the functions being also —1, ^, or 2.

The first group is not concerned with the anharmonic ratio of
four real collinear points, and the second does not correspond to the
anharmonic ratio of four real points which are distinct. The third
group is the only important one for our present purpose, and this
will be considered in Chapter IV.

ANHARMONIC RATIO

Theorem. Prime Forms Related by projections
AND Sections

26. If two prime forms of the first class are so related, that
either may he obtained from the other by a finite number of
projections and sections, the anharmonic ratio of any four
elements of one is equal to the anharmonic ratio of the corre-
sponding four elements of the other.

Proof. I. Let either form be obtainable from the other
by means of one operation of projection or one of section.

1. Range ABCD and fiat pencil abed.

From P, the base of the flat pencil, draw PQ perpendicular
to jP, the base of the range ABCD. Then, equating pairs
of expressions for double the
areas of the triangles A CP, BCPy
ADP, BDP, we have

PA'PCs,\nac = PQ.AC,

PB. PC sin be =:PQ.BC,

PA . PD sin ad^PQ' AD,

PB . PD sin bd = PQ' BD.

A C PA sin ad
liC' PB

TT P^ sin ac
Hence •

and
whence

PB sin be

sin ac sin ad

sin bd
AC AD,
BC' BD'

bd'

sin be sin bd

(abed) = (ABCD).
2. Mange ABCD and axial pencil a^yS.

From a point P in the base of the axial pencil a^yS project
the range ABCD, obtaining as projector the flat pencil abed.

Then (ABCD) = (abcd). But in Case 3 it will be shown
that (abcd)=(a^yh). Hence (ABCD)= (a^yh).

EELATIONS OF THE RATIOS

3. Flat pencil ahcd and axial pencil a^y8.

Through a point ^ in the base of the axial pencil a/378
pass a plane perpendicular to this base, cutting the planes
a, y8, 7, S in the lines aQ, 6q, Cq, d^ and the lines a, b, c, d in
the points A, B, C, D. From the
definition of the angles between the
planes it follows that

But (aQhQCf^dQ) = {ABCn)

= (abcd^.
Hence {a^yS ) = (ahcd^ .

II. Let either form be obtainable from the other by
means of several operations of projection or of section,
or of both.

Let two prime forms /^ and /„ + i be obtainable, either
from the other, by means of n operations, and let the prime
forms which are successively produced beginning with f^
t»e /2,/3, • • ',fu,fn + v Also let e^, ej,, 4', ej," and e^+i, 4 + r
4+p ^i+i be corresponding sets of four elements of two
consecutive prime forms /;fc,/;t+i.

Then, by Part I, (e^el^e^'e^") = (e^ + 14 + 14'+ 14'+ 1)-

This relation is true for all values of k from 1 to n.

Hence (^i^iWW") = (^n+i««+i«n+i««+i)»
and the truth of the theorem is established.

27. Corollary. If two prime forms of the first class are
perspective, the anharmonic ratio of any four elements of
one form is equal to that of the four corresponding elements
of the other form.

30 ANHARMONIC RATIO

Exercise 8. Relations of the Ratios

1. Show how the table on page 27 is obtained, verifying
each result.

2. Any two letters in the anharmonic ratio (A BCD) may
be interchanged without affecting the value of the ratio, pro-
vided the other two letters are also interchanged.

T^7. . 1 ^ 1 x — 1 X

In the 8ix expressions x, — » 1— x, ' »

X 1 — X X x — 1

make successively the following substitutions for x and note the
recurrence of the original forms of the expressions:

x' — 1

3. x'. 5. 1-x'. 7. ^-r^-

4. -:• 6. :: :• 8.

x' 1-x' x'-l

9. If the point 0 and three nonconcurrent lines a, b, c are
in one plane, draw a line through O which shall cut a, b, c in
points A, B, C such that (OABC)= k, any given number.

10. Solve the dual of Ex. 9 in plane geometry.

11. Solve the space dual of Ex. 9.

12. If A^, A^, /Jj, B^, Cj, Cj, Z)j, Dg, X, Y are collinear points,
and if (A^A^XY) = {B^B^XY) = (C^C^XY) = (D^D.^XY) = -1, it
follows that (A^B^C^D^) = (A^B^C^D^).

13. li A^, A^, ■ • ', A„,X, Y are w + 2 collinear points, then
(A^A^XY)(A^A^XY) • • • (A,,_,A^XY)(A^A,XY)=1.

14. If ylj, A^, A^, X, Y are five coplanar points, and if
A^(A^A^XY) denotes the anharmonic ratio of the four lines
from A^ to A^, A^, X, Y, it follows that the product of the
anharmonic ratios A^(A^A^XY), A^(A^A^XY), A^(A^A.^XY) is 1.

15. Generalize the result found in Ex. 14.

16. If /I J, A^ A^, A^, X, Fare concyclic points, it follows that
the anharmonic ratios A'(Jj/l2^l8^^)and Y(A^A^A^A^) are equal.

CHAPTER IV
HARMONIC FORMS

28. Harmonic Range. When four collinear points A, B,
C, D are so situated that {ABCD) = —1, the four points
are said to constitute a harmonic range.

In the same way we may define a harmonic flat pencil and
a harmonic axial pencil.

Any one of these three forms is spoken of as a harmonic
form., and it follows (§ 26} that every form derived from a
harmonic form by a finite number of projections and sec-
tions is a harmonic form. If three elements of a harmonic
form are given, it is evident that the fourth element, called
the fourth harmonic to the three, is uniquely determined.

Moreover, since the anharmonic ratio is here negative,
from the above definition the elements of the first pair,
say A and B., separate those of the second pair, say C and D.

Ti. + • • /^P^m 1 ACBD , .AC AD

That IS, since (ABCD) = — 1, -• = — 1, and = ,

'^ ^ 'AD-BC BC BD

so that the two ratios have opposite signs. Therefore, either Cor D

divides A B internally and the other divides it externally.

The elements of either of these pairs, A and B, or C
and D, are said to be conjugates or harmonic conjugates
with respect to the other pair. They are also said to be
harmonically separated by the elements of the other pair.

Since from the definition (§ 21) the anharmonic ratios (ABCD)
and (CDAB) are equal, it follovv^s that the relation betw^een the pairs
of elements A, B and C, D is symmetric with respect to them.

32 HARMONIC FORMS

Exercise 9. Harmonic Ranges

1. Given three collinear points A, B, C, with C bisecting
AB, determine the fourth harmonic D.

2. Consider Ex. 1 when C is the point at infinity.

3. Consider Ex. 1 when A, B, C are any collinear points.

4. Given three concurrent lines a, h, c, with c bisecting the
angle ab, determine the fourth harmonic d.

Compare Ex. 4 with Ex. 1.

5. Consider Ex. 4 when a, h, c are any concurrent lines.
Compare Ex. 5 with Ex. 3.

6. If {ABCD) = — 1, the four points A, B, C, D may, by a
finite number of projections and sections, be projected into the
positions A, B, D, C.

7. If A BCD is a harmonic range, the line segments AC,
AB, and AD are connected by the proportion AC:AD =
AC — AB:AB — AD.

8. If AB^Cjy^ and AB^C^D^ are harmonic ranges on differ-
ent bases, the lines B^B^, ^i^a? -^1^2 ^^® concurrent, and the
lines B^B^, ^1^2' ^-Pi ^^'^ ^^^ concurrent.

9. If AJi^C^D^ and Aji^CJ)^ are harmonic ranges, and if
^j/lj, i^i^a, CjCg are concurrent at 0, then D^^ also passes
through 0.

10. If A, B, C, D, 0, P are points on a circle and are
so placed that the pencil O(ABCD) is harmonic, the pencil
P (A BCD') is also harmonic.

11. If ABCD is a harmonic range, and if O is the midpoint
of CD, then ()C^ = OA ■ OB.

12. Use the result of Ex. 11 to find a pair of points which
shall be harmonic conjugates with respect to two given pairs
of collinear points A^, B^; A^, B^.

13. Given four coplanar lines, draw when possible a line
which shall cut them in a harmonic range.

COMPLETE QUADRANGLE 33

29. Complete Quadrangle. The figure formed by four
points in a plane, no three of which are colhnear (asP, Q, E, S
in the figure below), and the six lines determined by them
is called a complete four-point or complete quadrangle.

Any two of the six lines of a complete quadrangle which
do not intersect in one of the original four points are called
opposite sides. The intersections of opposite sides are called
diagonal points, and they are the vertices of the diagonal
triangle of the complete quadrangle.

Theorem, harmonic Property of a Quadrangle

30. If four collinear points A, B, C, D are so situated that
two opposite sides of a complete quadrangle pass through A,
two opposite sides piass th'ough B, and the two remaining sides
pass through C and D respectively, then (^ABCD) — — 1.

Let P, Q, R, S be the vertices of the complete quadrangle,
and let PQ, BS pass through A ; PS, QR through B ; PR
through C; and QS through D.

Then {ABCD^ = iSQOD) = {BACD^ = (JWcm '

Hence (ABCI>y = l,

and {ABCI)} = -1.

(A BCD) cannot be equal to + 1, since no two points are coinci-
dent, as would then be the case.

HARMONIC FORMS

31. Complete Quadrilateral. The figure formed by four
lines in a plane, no three of which are concurrent (as
JO, q, r, 8 below), and the six points determined by them is
called a complete four-side or complete quadrilateral.

Any two of the six pomts of a complete quadrilateral
which do not both lie on one of the original four lines
are called opposite vertices. The lines determined by pairs
of opposite vertices are called
diagonal lines, and they deter-
mine the diagonal triangle.

The student should compare this fig-
ure with that of the complete quadrangle
in § 30, and should notice also the duality
suggested by §§29 and 31, the dual ele-
ments being the point and line.

Theorem. Harmonic Property of a Quadrilateral

32. Tf four concurrent lines a, b, c, d are so situated that
two opposite vertices of a complete quadrilateral are on a, two
opposite vertices on b, and the two remaining vertices on c and
d respectively, then (abcd^ = — 1.

Let p, q, r, s in the figure above be the sides of the
complete quadrilateral, and let p and q, and also r and s,
intersect on a ; p and «, and also q and r, intersect on b ;
p and r intersect on c ; and s and q intersect on d.

(abed) = (sqod) = (bacd) =

Then
Hence
and

(abcdy = 1,
(abed) = — 1.

(abed)

Why cannot (abed) = -f- 1 ? Students should compare this proof,
step by step, with that of § 30.

QUADRANGLES AND QUADRILATERALS 35

Exercise 10. Quadrangles and Quadrilaterals

1. State and prove the converse of § 30.

2. State and prove the converse of § 32.

3. Two vertices of the diagonal triangle of a complete
quadrangle are harmonically separated by the points in which
the line determined by them is cut by the remaining pair of
opposite sides of the quadrangle.

4. By interchanging certain elements, it is possible to
derive § 32 from § 30 and Ex. 2 above from Ex. 1. Derive a
proposition in this way from Ex. 3 and investigate its truth.

5. From § 30 derive a theorem respecting the complete
four-flat and prove it.

In the geometry of space the figure dual to the complete (luadraugle
is called the complete four-fiat, and, similarly, the complete four-edge iu
Ex. 6 is dual to the complete quadrilateral.

6. As in Ex. 5, from § 32 derive a theorem respecting the
complete four-edge and prove it.

7. The six points, other than. the diagonal points, in which
the diagonal lines meet the sides of a complete quadrangle
lie in sets of three on each of four lines.

8. From the result in Ex. 7 prove the existence of a com-
plete quadrilateral which has the same diagonal triangle as any
complete quadrangle.

9. Prove the plane duals of Exs. 7 and 8.

10. In this figure QS is parallel to AB. Show
that PC is a median and is divided harmonically.

Consider D, the intersection of QS and AB, to have
moved to infinity.

11. In the complete quadrangle shown in § 30 show that
AQ ■ PS - BC = - AC ■ BS . PQ.

12. As in Ex. 11, show that A Q- PS . BD = AD- BS • PQ.

13. Using § 30, prove Ex. 12, page 30.

36 HARMONIC FOKMS

33. Descriptive Definitions of Harmonic Forms. The har-
monic forms might originally have been defined in a purely
descriptive fashion based upon the facts just developed.
Thus, a harmonic range might have been defined as a set
of four collinear points so situated that through each of
the first two points there pass two opposite sides of a com-
plete quadrangle, and through each of the other two points
there passes one of the remaming sides of the quadrangle.
Similar definitions might have been given for the har-
monic flat pencil and the harmonic axial pencil. These
are the definitions which are usually adopted when it is
desired to avoid as far as may be possible the use of con-
siderations based upon measurement.

Exercise 11. Harmonic Forms

1. Given three collinear points A, B, C, construct the fourth
harmonic D from the descriptive definition of § 33.

In the constructions on this page use only an ungraduated ruler.

2. Given three concurrent lines a, b, c, construct the fourth
harmonic d from the descriptive definition of § 33.

3. Given a line segment J 5 and an indefinite line parallel
to A B, bisect ,1 B.

4. Given a line segment AB, its midpoint C, and any point
0 not in the line of .4/i, through O draw a line parallel to AB.

5. Given two intersecting lines and the bisector of one of
the angles formed by them, construct the bisector of the
supplementary angle formed by the lines.

6. Given a line segment AB divided at C in the ratio ??i : n,
construct a point D that divides the segment AB externally
in the same ratio.

7. Dualize for space the descriptive definitions of a harmonic
range and a harmonic flat jxjncil.

CHAPTER V

FIGURES IN PLANE HOMOLOGY

34. Homologic Plane Figures. Further interesting appli-
cations of the anharmonic ratio and illustrations of its
significance occur in homologic plane figures.

Given two figures in a plane, if to every point of one
figure there corresponds a pomt of the other, if to every
line of one there corre-
sponds a line of the other,
if the lines joining corre-
sponding points of the
two figures are concur-
rent, and if the inter-
sections of corresponding
lines are collinear, the
two figures are said to be
homologic, or in (^plane)
homology.

The point in wliich all lines joining corresponding points
are concurrent is called the center of homology, the line
which contains all intersections of corresponding lines is
called the axis of homology.

In the above illustration the two given figures are the triangles
ABC, A'B'C The corresponding points indicated are the three
pairs of vertices, but any number of other pairs of points may be
chosen. The corresponding lines are AB and A'B', BC and B'C,
CA and C'A'. The center of homology is O, and the axis of
homology is o.

38 FIGURES IN PLANE HOMOLOGY

Theorem. Figures in homology

35. If Uvo figures are in plane homology^ it follows that :

1. All sets of four collinear points consistitig of the center of
homology, a point on the axis of homology, and two correspond-
ing points of the figure have a common anharmonic ratio.

2. All sets of four concurrent lines consisting of a line
through the center of homology, the axis of homology, arid two
corresponding lines of the figure have a common anharmonic
ratio.

3. These two common anliar77ionic ratios are equal.

Proof. Let A^, A^, A^ and A[, A^, A^ be corresponding sets
of three points of two homologic figures; and let OA^A[,
OA^A'^, OA^A^ intersect o in 0^, 0^, Og respectively ; also
let A^A^, A[A'^ meet in i|; A^A^, A^A^ meet in i^ ; and
A^A^, A'^A[ meet in I^. Then it is evident that
(00,A,A[} = iO(\A^A!,~) = (00,A,A^}

CONSTANT OF HOMOLOGY 39

36. Constant of Homology. The common value of the
two anharmonic ratios found in the theorem of § 35 is called
the constant of homology for the two figures.

Given a center 0 and an axis of homology o, we can
construct a figure homologic with any given figure in such
a way that to any point A of the given figure there shall
correspond any selected point A' on the line OA, or that to
any line a of the given figure there shall correspond any
selected line a' concurrent with o and a. For the point A'
selected there is a value of the constant of homology.
Conversely, for any assigned value of the constant of
homology, the center and axis being given, one and only
one point A' corresponds to any given point ^ of a given
figure. The fact is that tvhen the center, axis, and constant
of homology/ are given, one and only one figure homologic with
a given figure can he constructed.

Examples of such constructions are given on page 40.

There are several notable special cases of the homologic
relation. One such case is that of harmonic homology in
which the constant of homology is — 1.

Another case is that in which the axis of homology is
the line at infinity. Then all pairs of corresponding lines
are parallel, and the figures are similar. In this case the
constant of homology (^OO-^A^A'^ becomes OA^/OA[, which
may be shown to be the ratio of similitude. If in addition
the constant is —1, the center 0 is a center of symmetry
for the figure composed of the two homologic figures.

A third case is that in which the center of homology is
a point at infinity. Then the constant is O^A[/0-^^Ay If
also the homology is harmonic, the axis of homology is
an axis of symmetry for the figure composed of the two
homologic figures.

40 FIGURES IN PLANE HOMOLOGY

Exercise 12. Figures in Homology

1. Given a center of homology O, an axis of homology o,
and any triangle A^A^A^, construct, homologic with A^A^A^, a
triangle that has a vertex ^^ at a given position on OA^.

As to the possibility of this construction, see Ex. 13, page 13.

Given a triangle A^^A^A^^ the center of homology 0, the axis
of homology o, and the following constants of homology^ con-
struct the figures homologic with A^A^A^ :

2. 3. 3. -3. 4. -1. 5. 1.

6. In each of Exs. 2-5, if A[A!^A'^ is the result of the con-
struction, use the same center, axis, and constant of homology
to construct the figure homologic with A[A!^Al.

7. If there are three coplanar figures f, /„ yj, and if for
a given center and a given axis of homology two of them,
/j and /jj, are homologic with the constant Cg, and if f^ and f^
are homologic with the constant c^, then f^ and f^ are homo-
logic with the constant of homology c^ equal to c^ • c^.

8. Consider carefully Exs. 2-7 for the case in which O is
a point at infinity and the case in which a is the line at infinity.

9. For a given center, axis, and constant of homology con-
struct the line corresponding to the line at infinity.

This line is called tlie vanishing line.

10. In Exs. 2-5 determine the vanishing line.

11. Under what conditions is the vanishing line at infinity?

Given a circle, the axis of homology o, and the center of
Jiomology 0, construct the figure homologic with the circle
under each of the following conditions :

12. The vanishing line does not meet the circle.

13. The vanishing line is a secant of the circle.

14. The vanishing line is a tangent to the circle.

CHAPTER VI
PROJECTIVITIES OF PEIME FORMS

37. Projective One-Dimensional Prime Forms. Whenever
there exists between the elements of two one-dimensional
prime forms a one-to-one correspondence such that, by
means of a finite number of operations of projection and
section, it is possible to pass simultaneously from all the
elements of one prime form to the corresponding elements
of the other, the two prime forms are said to be projectively
related or to be projective.

The correspondence existing between two projective one-
dimensional prime forms is called a projectivity.

The symbol j- is frequently used for " is projective with."

Theorem. Projective Prime Forms

38. Prime forms which are projective with the same prime
form are projective with each other.

Proof. If Wj operations yield a form f^ from a form /j,
and if n^ operations yield /g from/g, then n^ + n^ operations
yield /g from f^

It is not the purpose of this book to discuss the projectivities
of prime forms other than one-dimensional ones, but it may be
stated that between the prime forms of higher dimensions there
exist relations which have the same general character as those
just defined, and that these relations are also called projectivities.
A complete study of projective geometry would include the consider-
ation of these higher projectivities and of many important geometric
propositions relating to them.

42 PROJECTIVITIES OF PRIME FORMS

Theorem. Projectivity of Triads

39. Between tivo one-dimensional prime forms, each of
which consists of three elements in a specified order, there
exists a projectiinty.

Proof. If either of the prime forms is not a range, it is
possible by operations of projection and section to obtain
from it a range which is projective with it. Hence it is
necessary to prove the proposition only for the case in
which both prime forms are ranges of three points.

This theorem is what was formerly called a lemma, a proposition
inserted merely for the purpose of leading up to a fundamental theo-
rem ; in this case, the one given in § 40. The proof involves the
consideration of the three cases below.

1. The ranges may he coplanar and upon different bases.

Let the ranges be ^j^jCj on the base jt>j and ^2^2 ^2 ^^
the beise p^, and let both ranges be in tlie same plane.

Draw the line through A^ and A^, and on it take any
points I^ and J^, not coincident with A^ and Ac^ respectively.

Draw ^5j, ^Cj, J^B^, J^C^; and let P^B^, J^B^ intersect
at B, and let J^C^, I^C^ intersect at C.

Through B and C draw the line p, cutting A-^A^ at A.

Then range A^B^C^ — range ^BC — range A^B^C^.
Hence range A^B^C^ — range A^B^C^. § 38

PROJECTIVITY OF TRIADS 43

2. The ranges may he coplanar and upon the same base.
Let the ranges A^B^C^ and ^2-^2^2 ^^ ^^ ^^^ same base p.

~-^ — ■ 0 '■ a p

^^A, ^2 B, cr^

0 ^ CI n ' 0 ^ P

B2 C2 ^2

It is here assumed that the points A^ B^ C^ may be the
same except for order as the points A^^ B^, Cj, or may be
partly or wholly distinct from these points.

In any case from a center P, exterior to the base p,
project A^B^C^ upon a new base p'. Then apply Case 1
to show that the range so obtained on the base p' is pro-
jective with the range A^B^Cy It then follows that the
range ^2^2 ^2 ^^ projective with the range A^B^Cy

3. TJie ranges may not he coplanar.

Let the bases p-^^ and p^ of the ranges A-^B-^C^ and ^2-^2 ^2
not be in any one plane.

Join any point Oj of the base p^ to any point 0^ of the
base jt>2 by the line p. Select three points A^ B, C on p.

Then, by Case 1, it follows that

range ^2^2 ^2 a ^^^^^ ABC^
and range A^B^C-^ — range ABC.

Therefore range ^2^2 ^2 a i"^"g6 A^B^Cy
From these results the existence of a projectivity follows.

44 PROJECTIVITIES OF PRIME FORMS

Theorem. Fundamental Theorem of Prime Forms

40. Betiveen two one-dimensional prime forms there exists
one and only one projectivity in which three elements of one
form, in a specified order, correspond to three elements of the
other form, also in a specified order.

Proof. Let us first consider three special cases.

1. Suppose that the two prime forms are a fiat pencil and
the range obtained hy cutting the pencil by a line.

Let the lines of the flat pencil be a, 6, <?, • • ., Z, . • • and
let these lines be cut by a line p in the points A,B,C,''',
L, ' ' '. Then these prime forms are perspective in such a
way that a. A', b, B; c, C; ' • •; I, L; ' ' • are pairs of cor-
responding elements.

Assume that, if possible, a second projectivity exists in
which the first three of these pairs of elements correspond,
but in which I corresponds to 3f and not to L.

Then, from the perspectivity,

(abcl) = (ABCL);
and, from the second projectivity,

(abcl) = (ABCM}.
Then (ABCL^^^ABCM),

which is impossible unless L = M.

Hence the second projectivity cannot exist, and the only
projectivity existing between the forms is the perspectivity.

FUNDAMENTAL THEOREM

2. Suppose that the two prime forms are an axial pencil
and the range obtained hy cutting the pencil hy a line.

Fig. 1

Fig. 2

Let the planes of the axial pencil be a, /8, 7, • • • , X, • • •
(Fig. 1), and let p cut the planes in A, B, C, • • -, L, • • -.

A proof similar to that on page 44 should be given by the student.

3. Suppose that the two prime forms are ranges.

Let A^, B^, Cj (Fig. 2) be three points of the first range,
and let them correspond respectively to the points A^, B^, C^
of the second. A projectivity exists between these sets of
three points, and this projectivity can be extended to include
the whole of both ranges. Let ij correspond to L^.

If possible let there be a second projectivity connecting
the ranges, in which ij corresponds to M^, a point other
than Z-g. Then, from the two projectivities,

(A,B,C,L,') = (A^B^C,^L^')

and (^1^1 qZi) = (A^B^C^M^-).

Therefore (A^B^C^L^') = (A^B^C^M^),
which is impossible unless L^ = M^.

Hence in this case two projectivities cannot exist.

46 PROJECTIVITIES OF PRIME FORMS

4. Consider now the general case.

Let the two forms be /, /', and let the three pairs of
corresponding elements be 1, 1' ; 2, 2' ; 3, 3'.

/ — r'^""c"'^D"^

A' B' C D' E'.

If / is a range, let the elements 1, 2, 3, 4, 5, . • • be A,

B, C, D, E, • ' •; but if / is not a range, let the elements
1, 2, 3, 4, 5, . • • be cut by a line in the points A, B, C,
D, U, ' • '. Similarly, if /' is a range, let the elements 1',
2', 3', 4', 5', . . . be A', B', C\ D', E',...; but if/' is not a
range, let the elements 1', 2', 3', 4', 5', • • . be cut by a line
in the points A', B', C\ D\E' ....

Between the ranges ABCD • • • and A'B'C'D' . . . there is
just one projectivity in which 4, A'-, B, B' ; C, C are cor-
responding elements. Let D, D' be corresponding elements.

Then form 1234 ...- range ^J5Ci) •• •

- range A'B'C'D' ... - form 1'2'3'4', ....
Hence form 1234 ... - form 1'2'3'4' • . ..

Suppose that, if possible, between / and /' there is a
second projectivity in which 1, 1'; 2, 2'; 3, 3' are pairs of
corresponding elements and 4, 5' are corresponding elements.

Then range ABCD...- form 1234 .. .,

and form 1'2'3'5' ... - range A'B'C'E' • . .;

whence range ABCD • • • -r- range A'B'C'E' ....

Accordingly there would be a second projectivity between
the ranges ABC ... and A'B'C . . ., in which A, A'; By B';

C, C are pairs of corresponding points. This, however, is
impossible. Hence the theorem is true in all cases.

fundame:ntal theorem 47

41. Corollary. There is one projectivity and only one
vrojectivity between one-dimensional forms on the same base
which makes three distinct elements of a one-dimensional form
correspond each to itself. This projectivity makes every element
of the form correspond to itself.

Exercise 13. Projectivities of Triads

1. li A, B^, B^, Cj, Cj ^i'6 fiv6 distinct points on a line, find
a set of projections and sections, minimum in number, which
connects the triads AB^C^ and ^52^2-

2. Examine Ex. 1 for the case in which B^ coincides with B^.

3. li A, B, C are three points on a line, find the set of
projections and sections, minimum in number, which connects
these points with themselves in any selected order.

Consider Ex. 3 for each of the six possible orders of the points.

4. If A^, A^, B^, -Bjj, Cj, Cj are six distinct collinear points,
find a set of projections and sections, minimum in number,
which connects the triads A^B^C^ and A^^C^.

5. If .4, B^, B^, Cj, Cg are five points, no four of which are
coplanar, find a set of projections and sections, minimum in
number, which connects the triads ABf!^ and AB^C^.

This is a special case of a more general problem.

6. In how many ways can a projectivity between the triads
specified in Ex. 5 be established ?

7. If A^, A^, 5j, B^, Cj, Cj are six points in space, no four
of which are coplanar and no three of which are collinear,
the triads A^B^C^ and Afi^C^ are projective. Specify a set of
projections and sections which constitutes such a projectivity.

8. Investigate the possibility of establishing a projectivity
between the triads specified in Ex. 5 such that any fourth
point Z>j in the plane AB^C^ shall correspond to any fourth
point Dj in the plane AB,^C^.

48 PROJECTIVITIES OF PRIME FORMS

Theorem. Metric condition for a projectivity

42. If between the elements of two one-dimensional prime
forms there exists a one-to-one correspondence such that the
anharmonic ratio of every set of four elements of one prime
form is equal to that of the set of four corresponding elements
of the other prime fonm^ the correspondence is a projectivity.

Ai B, C, D,

Az Bz Cg D2

Proof. It is sufficient to consider the case of two ranges
because if either of the given prime forms is not a range,
it is possible by operations of projection and section to
obtain from it a range which is projective with it.

From § 39 it follows that a projectivity exists between
three points Ay, J5j, C^ of the first range and the three
corresponding points A^, B^, C^ of the second range. Then
if a fourth pair of corresponding points B^, D^ on the ranges
are found, it follows by hypothesis that

(:a,b^c^d^) = Ca^b^c^d^').

Also let range A-^Bj^C^D^^ — range A,^B^C^D'^.

Then (^2^2<^2^2) = (^1^1 CiA)' § 26

and so (A^B^C^d;,) = (A^B^C^D^).

Accordingly Z)^ coincides with D^ ; and therefore

range ^2-^2 ^2^2 a ^^^^g^ A^i^i^r

Thus, any fourth point -Z>j of the first range has the same
corresponding point D^ in the given one-to-one correspond-
ence as it has in the projectivity between the ranges which is
determined by the triads of points A-^B^C^^ and ^3^2 ^2*
Hence the correspondence is this projectivity.

METRIC COXDITIOX FOIl A PROJECTIVITY 49

If it is given that two one-dimensional prime forms are
projective, it should be noted that it is necessary only that
tliree elements of one of them and the corresponding ele-
ments of the other be specified in order that by the con-
struction of § 39 the whole projectivity may be established ;
and this construction furnishes a standard method of
establishing a projectivity.

Associated with the general results of §§ 39-42, several which are
special in their nature and application are considered in §§ 43-46.

Exercise 14. Projectivity of Prime Forms

1. Given three points A^, B^, C^ of a range and the corre-
sponding points A^, B^, C^ of a second range projective with
the first, the bases being different, construct the point D^ of
the second range corresponding to a fourth point D^ of the first.

2. Consider Ex. 1 when the point D^ is at infinity.

3. Consider Ex. 1 when the ranges are coplanar and D^
is their intersection.

4. Obtain simplified constructions for Exs. 1 and 2 when
A^ and A^ coincide at the intersection of the ranges.

5. Consider Ex. 1 when the bases are coincident.

6. Consider Ex. 5 when A^, B^; A^, B^; C\, D^ are all pairs
of coincident points.

7. Prove the theorem suggested by the result of Ex. 6.

8. If two ranges are projective, to every harmonic range in
one of them there corresponds a harmonic range in the other.

9. Assuming that if four points A, B, C, D are properly
divided into pairs a common pair of harmonic conjugates with
respect to these pairs exists, prove the converse of Ex. 8.

10. Investigate the question of dividing a set of four col-
linear points into pairs so that a common pair of harmonic
conjugates may exist.

PROJECTIVITIES OF PRIME FORMS

Theorem. Projective Ranges in Perspective

43. Tivo projective ranges whose bases are not coplanar
are perspective,

Proof. Let A^B^C^- > > L^' - • on the base p^ and ^2^2 ^2
• • • ij • • • 0^^ the base p^ be projective ranges in which
Ay, A^ ; 5|, B^\ Cj, Cg ; • • • are corresponding elements,
their bases not being coplanar.

On the line A.^A^ take a point A^ and let C be the point
in which the line C^C^ intersects the plane determined by
the point A and the line B^B^. Let the line AC intersect
the line B^B^ in the point B.

Denote by or, y8, 7, . • •, \, • • • the planes determined by the
line ^J?Cand the points Jp 5j, Cj, • • •, ij, • • • respectively.

The axial pencil ay87 • • • X • • • is perspective with the
range A^B^C^ . . . i^ . . . and cuts the line jt?2 i^i a range pro-
jective with the range A^B^^ Cj • • • X^ • . • .

Moreover, in the projectivity thus established the points
A^y B^ Cg correspond to the points ^j, Bj, C^ respectively.
This projectivity is therefore the same as the one that was
originally assumed to exist between the ranges (§ 40).

Hence from the range A^B^C^ • • • on the base p^ it is
possible to pass to the range A^B^C^ • • . on the base p^
by one operation of projection from the axis ABC and
one operation of section by the line p^ Hence tlie pro-
jectivity existing between the ranges is a perspectivity.

PROJECTIVE RANGES
Theorem. Condition for Perspective Ranges

44. Two projective ranges on different bases in a plane
IT are perspective if and only if the point common to them is
self-corresponding in the projectivity.

:?''

'ca

(■

Fig. 1

Fig. 2

Proof. First, let the ranges A^B^C^ • • • (Fig. 1) on the
base jOj and .42^2^2 • • • on the base p^ be projective in such
a way that X, the intersection of the two ranges, is self-
corresponding in the projectivity.

Let the lines A^A^ and B^B,^ intersect at P, and draw PX.

The projectivity is completely determined by the corre-
spondence of Ay, By X with A^, B^, X, but this correspond-
ence is established by the projection of A^ B^ X from the
center P and the section of the resulting flat pencil by
the line p^. Hence the ranges are perspective.

Next, let the ranges be perspective (Fig. 2).

Let the intersection of p^ and p^, regarded as a point of
the first range, be denoted by X^ Since its correspond-
ing point would be found by cutting the projector of X^
from a center P in the plane tt, or from an axis p not in
this plane, as the case may be, by the line p^, it is the
intersection Xy That is, X^ is self-corresponding.

PROJECTIVITIES OF PRIME FORMS

Theorem. Ranges Perspective with a Third range

45. Two projective (but not perspective^ rallies on different
bases in a plane are both perspective with the same range
on any line that does not pass through the intersection of the
bases of the given ranges.

Proof. The construction used in Case 1 of § 39 estab-
lishes the existence of a range which is perspective with
each of the two given coplanar ranges. It is now necesgary
to show that the points P^ and P^ can be so selected that
the line -45 C shall be any line which does not pass through
the intersection of p^ and p^.

Let p be any line not concurrent with p^ and p^. Let
the intersection of p^ and p be i>j, and that of p^ and p
be E^. Suppose that the point D^ corresponding to Z)j, and
the point E^ corresponding to E^, have been found by the
method of Case 1 of § 39, or by any other suitable method.
The problem now is to choose points P^ and P^ such that
the line ABC shall coincide with the line p or D^E^

and let this line meet A^A^ in j^. Draw
and let this line meet A. A., in B. Join P to B.,

Draw Dj^j,

^i^v

■ , and let these lines meet DiE^ in B, C,

PERSPECTIVE RANGES 53

Then range ABCD^E^ . • . = range A^B^C^D^E^ ....
Hence range ABCD^E^ ' ' ' a ^^g^ ^^2^2^2^2 * * '•

But the triad AD^E^ is perspective with the triad A^D^E^y
and these triads determine completely the nature of the
projectivity between the range ABCD^E^ • . . and the range

^2^2^2^2^2 ' • '•

Therefore the two ranges A^B^C^D^E^ ... on jOj and
A^B^C^D^E^ . . . on jt?2 ^-^e perspective with a range on
the given line p.

Exercise 15. Projective and Perspective Forms

1. State and prove the dual of § 43.

State and prove the duals of each of the following :

2. § 44, in the plane. 5. § 45, in the plane.

3. § 44, in space. 6. § 45, in space.

4. Ex. 3, in the bundle. 7. Ex. 6, in the bundle.

Solve the duals of each of the following :

8. Ex. 1, page 49, in space.

9. Ex. 3, page 49, in the plane.

10. Ex. 3, page 49, in space.

11. Ex. 1, page 49, in the plane, considering the case in
which the ranges are coplanar.

12. If three ranges in a plane have concurrent bases and if
two of the ranges are perspective with the third, these two
ranges are perspective with each other.

13. Three fixed lines Pi,p^,p^ radiate from A^, one of three
fixed collinear points A^, A^, A^. A point P^ moves on^^, and
the lines A^P^, A^P^ cut ^j^j 2\ respectively in P^, P^. Show that
P^P^ has a fixed point.

54 PROJECTIVITIES OF PRIME FORMS

46. Special Case. In the construction described in § 39
the only hmitation upon the position of the points ^ and 7^
was that ^ should not be at ^^ and that J^ should not be at
A^' The necessity for this limitation is due to the complete
failure of the construction which would result from the
coincidence of the lines ij^i, ^^i, ^C^ We shall now
consider a special case which is noteworthy because it
puts in evidence a line that has important relations to all
pairs of corresponding points of the ranges and furnishes
a basis for several simple constructions.

Let ^ be taken at A^, and ^ at Ay Then the line p is
determined by the intersection of I\B^(^A^B^^ and I^B^
(^A^B^ and the intersection of I^C^^^A^C^) and ^C^i^A^^C^}'
It contains likewise the intersections of A^D^ ^1^2 > ^2-^1'
A^E^ ; and so on. It can now be shown that this line p
can be located independently of the placing of ^ and ^
on the particular line A^A^.

If the ranges A^B^C^ ' • • and A^B^C^ • • • are projective
but not perspective, the point of intersection of p^ and p^
will not be self-corresponding. Regarding this point as
belonging to the range A^B^C^ • • -, call it Xj. In the other
range there will correspond to X^ a point Xj. Regarding
the intersection as a point of range A^B^C^ • • ., call it Y^.
To it will correspond a point Y^ of the first range. Then
^jXj and ^2^1 intersect on the line p. But their inter-
section is Xj.

PERSPECTIVE RANGES 55-

Similarly, Y^, the intersection of A^Y^ and A^Y^, is on the
line jt). Accordingly the line p is the line determined by the
points of the first and second ranges which correspond to
the point of intersection of the ranges regarded as a pomt
of the second and first ranges respectively. It follows that
the line p is determined by the projectivity between the
ranges and bears the same relation to any two correspond-
mg points of the ranges that it does to A^, A^. Accordingly
not only do A^B^, A^B^; A-^C^^ A^C^; A^D^, A^D^\ •••
intersect on p, but so do B^ C^i B^ C^ ; B^D^, ^2^1 5 * * * 5
CjDg, C^D^; C^E^, C^E^', .••; and so on.

If the ranges A^B^C^ — and A^B^C^ • • • are perspective,
their common point X is self -corresponding. Let p be the
line joining X to the intersection of A^B^ and A^B^

We then see that the flat pencils A^(^A^B^C^ • • • X* • •)
and A^^A^Bj^C^ • • • A'. • •) are projective and have a com-
mon line Aj^A^. Hence these pencils are perspective, and
the axis of perspective is p. Hence, as in the other case,
Aj^B^, A^B^ ; A-^ Cg, A^C^ ; A^D^^ A^D^ ; > "; B^C^, B^C^; • . •;
(7jZ>2, C^D^ all intersect on p.

Exercise 16. Perspective Forms

1. If a simple reentrant hexagon has its first, third, and
fifth vertices on one straight line of a plane tt and has its
second, fourth, and sixth vertices on a
second straight line of that plane, the
intersections of the first and fourth,
the second and fifth, and the third and
sixth sides are collinear.

2. State and prove the proposition dual in a plane to Ex. 1.

3. State and prove the proposition dual in space to Ex. 1.

4. State and prove the proposition dual in the bundle to Ex. 3

56 PROJECTIVITIES OF PRIME FORMS

Problem. Line to an Inaccessible Point

47. To draw a line which shall pass through the ifiacces-
sible intersection of two given straight lines and also through
a given point 0 of their plane.

Solution. To draw a line which shall satisfy the first
condition, let the given lines be p^ and p^. Select a point
P in the plane of the lines p^ and jp^, but not on either
line, and through P pass three lines cutting p^ in ^j, B^, Cj

and ^2 ^ ^2' -^2' ^2-

Since the ranges Ylji5jCj and A^B^C^ are perspective,
the pairs of lines A^B^, A^B^; A^C^, ^2^v ^\^i-> ^2^1 ii^ter-
sect on a line p which passes through the intersection of
the lines p^ and p^ (§ 46).

Now to satisfy the second condition we must so choose
the point P that the line p will pass through 0.

Through 0 draw two lines and let them intersect p^ in
A^ and B^ and intersect p^ in A^ and B^. Take P to be
the intersection of A^A^ and B^B^^,. Through P draw an-
other line cutting p^ in C^ and p^ in C^. Then the line p
which is determined by the intersections of two of the three
pairs of lines A^B^, A^B^; A^C^^ A^C^; B^C„, B^C^ satisfies
the two given conditions.

PROBLEMS OF CONSTRUCTION 67

Problem. Line Parallel to a Given Liwe

48. Given two parallel lines and a point in their plane,
to draw through the point a line parallel to the given lines.

The solution is left for the student, who should write out the
proof after the method used in § 47.

The case in § 49 is somewhat different.

PROBLEM. Line Parallel to a Given Line

49. Gfiven in a plane a point 0, a line p^ not passing
through 0, and a parallelogram none of whose sides is known
to be parallel to p^, to draw through 0 a line parallel to p^

Pi Ai

Solution. This problem may be reduced to the preced-
ing one as follows:

Let the adjacent sides a^ and Jj of the parallelogram
meet p^ in A^ and B^^ respectively. Let C be any point on
the diagonal I\P^, and let the lines CAj^ and CB^ meet a^
and ^2 ill ^2 ^^^ -^2 respectively.

Then the triangles J^A^B^ and F^A^B^ are homologic, and
since the intersections of -Pi^i, ^^2-^2 ^^^ ^^ ^i^v -^2^2 ^^^
at infinity, the axis of homology is at infinity.

Hence A^B^ is parallel to p^.

We now have two parallel lines A^B^ and p^, and by
means of § 48 we can draw through the given point 0 a
line parallel to the given hue p^.

58 PROJECTIVITIES OF PRIME FORMS

Theorem. Similar Ranges

50. If in ttoo projective ranges the points at infinity are
corresponding points^ the ratios of all pairs of corresponding
segments are the same, and coiiversely.

Proof. Let the points at infinity of two ranges be J^
and J^, and suppose that the points t/j, A^, J?^, Cj, • • • of
tlie first range correspond respectively to the points J^,
A^i B^, C2, • ' ' of the second range.

Then (A^B^C^J^) = (A^iVi).

or A^C^-.B^C^^A.C.iB.C,. §22

Hence ^g Cg : ^ 1 C^ = ^2 ^"2 ^ ^1 <^i-

Similarly, it can be shown that

A^B^: A^Bi = A^C^: A,C, = A^D^: A,D,= • . .

= B^C^: B^C\ = B^I)^: B,B,= . . . = r,

where r is independent of the pair of corresponding seg-
ments involved.

Conversely, suppose that all the pairs of corresponding
segments have the same ratio.

Let A^, A^; B^ B^; C^, C^\ Jj, K^ be pairs of correspond-
ing points, t/j being the point at infinity of the first range.

Then (^1^1 Vi) = {A^B^C^K^),

-^1^1 ^2^2

. AK^

-^1^1 -^2^2

B^K^

But

A,C,

: ^2^2 =^1^1 '

B^C^.

Therefore

A^iq :

B^K^=\,

and hence K<^ must be at infinity and should be called J^.
Accordingly J^ and J^, the points at infinity of the two
ranges, are corresponding points.

SIMILAR AND CONGRUENT FORMS 59

51. Similar Ranges. Two projective ranges whose points
at infinity are corresponding points are said to be similar.

An example of similar ranges is furnished by sections of a flat
pencil by parallel lines, or by sections of a flat pencil of parallel rays
by any two lines.

If similar ranges are on the same base, the point at
infinity is a self-corresponding point. If the ratio of corre-
sponding segments (§ 50) is 1:1, there is no other such
point, but in any other case a second point exists.

52. Congruent Ranges. Similar ranges in which the ratio
of corresponding segments is unity are said to be congruent.

If two similar ranges have parallel bases, their common point at
infinity is self-corresponding and the ranges are perspective.

53. Similar and Congruent Pencils. The terms similar
and congruent are applied to certain special cases of flat
pencils and axial pencils, the mere mention of this fact
being sufficient for the present treatment.

Two projective flat pencils whose bases are at infinity
are said to be similar if linear sections of these pencils
are similar ranges.

In this and in similar cases the student should draw the figure.

Two projective flat pencils whose bases are in the finite
part of the plane are said to be equal or congruent if every
pair of lines of one pencil contains an angle equal to the
angle contained by the corresponding pair of lines of the
other pencil.

Similar and congruent axial pencils are defined in the same way.

Flat pencils and axial pencils are said to be proper or
improper according as their bases are -or are not in the
finite part of space.

60 PROJECTIVITIES OF PRIME FORMS

Exercise 17. Projectivities

1. There exist infinitely many sets of projections and sec-
tions which connect a prime form with itself.

2. There exist infinitely many sets of projections and sec-
tions which connect two projective prime forms.

3. A plane quadrangle is projective with a properly chosen
parallelogram.

4. A plane quadrangle is projective with a properly chosen
square.

5. A plane quadrangle is projective with every square.

6. Every two plane quadrangles are projective. Specify
a set of operations that connects the quadrangles.

7. A triangle and any point in its plane, but not in its
perimeter, are projective with any equilateral triangle and
its center.

8. Solve the duals of Ex. 6.

9. Two ranges on the same base are projective if they are
so related that every pair of corresponding points is a harmonic
conjugate with respect to a given pair of points on the base.

Apply Ex. 12 on page 30.

10. If Jj and K^ are the points at infinity of two projective
ranges A^B^C^ • • • and AJi^C^ - • •) J^ and K^ being the points
corresponding to them, then A^^K^ • A^J^ = B^K^ ■ B^J^ = • • ■.

11. If the ranges in Ex. 10 are on the same base, and if 0^
is the midpoint of ./^A'^ and if O^ is the point corresponding to 0^,
then 0/, . O^A^ - 0/, • O/, - O^K^ • O^A^ + O^K^ • Ofi^ = 0.

12. If A^ and A^ in Ex. 11 coincide with A, it follows that

13. If P^ is a fixed point on the side PJ^^ of a given triangle
PjPjPg, and if a moving line cuts P^P^ in P^, P^P^ in Q, and
PJ'^ in R, P^Q and P^R meeting in P, then P^P cuts P^P^ in
a fixed point.

PROJECTIVITIES 61

14. Generalize Ex. 12, page 53.

15. Given A^B^C^ . . •, A^^^ • • -, and A^B^C^ • • •, three pro-
jective but not perspective ranges whose bases p^, p^, p^ are
coplanar and concurrent, prove that, if any triad of corre-
sponding points is collinear, there exists a range ABC • • •
which is perspective with each of the three given ranges.

16. If Pj, P^, Pg in Ex. 15 are the centers from which the
three ranges on p^, p^, p^ respectively are projected into the
range ABC ■ • >, each of the sides of the triangle P^P^P^ cuts
a pair of the given ranges in points which correspond in one
of the projectivities.

17. If in Ex. 16 P moves along the base p of the range
ABC • ■ •, the lines P^P, P^P and P^P trace on any fourth line
of the plane three projective ranges which have one point that
is self-corresponding for all three projectivities.

18. If Xj, X^ ; Fj, F2 ; A^, A^axe. pairs of corresponding points
of two coplanar projective ranges, and if (§ 46) X^ and Y^ coin-
cide at the intersection of the bases, determine the position
of B^ which corresponds to any fourth point B^ of the first
range, basing the solution on § 46.

19. Solve the plane dual of Ex. 18.

20. Solve the space dual of Ex. 18.

In solving the remainder of the problems in this exercise, use only the
ungraduated ruler. These exercises are chiefly due to Steiner, one of the
founders of the science of projective geometry.

21. Given a segment AB extended to twice its length, divide
it into any number of equal parts.

22. In an indefinite straight line given a segment AB
divided at C in the ratio of two given whole numbers, draw
through any given point a line parallel to the given line.

23. Given two parallel lines ^ and^', and given on^ a seg-
ment AB, from any given point on p lay off a segment of p
equal to any given multiple of AB.

24. Divide AB va. Ex. 23 in the ratio of two given integers.

62 PROJECTIVITIES OF PEIME FORMS

25. Given a parallelogram, divide any segment in its plane
in the ratio of two given integers.

26. In a plane, given three parallel lines which cut a fourth
line in the ratio of two given integers, draw through a given
point a line parallel to a given line.

27. If two parallel line segments which have to each other
the ratio of two given integers are given, draw a line parallel
to a given third line.

28. Given two parallel lines and a segment divided in the
ratio of two given integers, draw through a given point a line
parallel to a given fourth line.

29. Given two nonparallel segments, each divided in the
ratio of two given integers, draw a line parallel to a given line.

30. Given a square, a point, and a line, all in one plane,
draw through the point a line perpendicular to the given line.

31. Given a square and a right angle, both in the same
plane, bisect the angle.

32. Given a square and an angle, both in the same plane,
construct any multiple of the angle.

In each of the following problems, in addition to the data mentioned,
one circle fully drawn and its center are assumed to be given.

33. Through a given point draw aline parallel to a given
line.

34. Through a given point draw a line perpendicular to a
given line.

35. Through a given point draw a line which shall make
with a given line an angle equal to a given angle.

36. From a given point draw a line parallel and equal to
a given line.

37. Determine the intersections, if any, of a given line and
a circle of given center and radius.

38. Determine the intersections, if any, of two circles of
given centers and given radii.

CHAPTER YII

SUPERPOSED PROJECTIVE FORMS

54. Superposed Projective Forms. Hitherto only inci-
dental reference has been made to the existence of pro-
jective forms on the same base, but the general results
obtained apply to them except when the contrary is in-
dicated. It is proposed now to consider some special
properties of one-dimensional projective prime forms on
the same base. Such forms are called superposed forms.

In the first place, the existence of superposed projective
forms is established if, when two projective ranges are on
different bases, as A^B^C^ • • • on the base p^ and A^B^C^ • • •
on the base p^, the second range is projected on the base p^
from a center P so taken as not to be on the line A^A^.
The result is a range A[B[C[ • • • on the base p^ which is pro-
jective but not identical with Jj^jCj • • •, since A^ and A\
are not coincident.

In the second place, if each of two superposed projective
forms is operated upon by section or projection, the result-
ing forms are superposed and projective.

64 SUPERPOSED PROJECTIVE FORMS

Theorem, existence of superposed Projective Forms

55. There exist pairs of projective prime forms which
have common bases and which have not all their corre-
sponding elements coincident.

The proof of this theorem is evident from § 54.

Theorem. Invariance under Projection

56. If from two superposed projective prime forms two
other superposed prime forms are obtained by projection or
by section^ the latter forms are projective.

The proof of this theorem is evident from § 54.

57. Self-corresponding, or Double, Elements. It has been
shown (§ 39) that three points A^^ B^, C^ of a Ime are
projective with any three points of the same line, even
though some or all of the two sets of three are the same ;
and it follows from § 40 that the correspondence of the
three pairs of points establishes for all points of the line a
projectivity in which some of the points may coincide with
their corresponding points. It is evident that similar con-
siderations apply to flat pencils and to axial pencils.

Elements of superposed projective prime forms that
coincide with those to which they correspond are called self-
corresponding elements, or double elements ; and the deter-
mination of the number of such elements is a problem
of importance. Since from superposed flat pencils or
axial pencils we may by section obtain superposed ranges
whose self-corresponding points are situated on the self-
corresponding elements of the pencils, the discussion of this
question is substantially the same for all one-dimensional
prime forms, and hence will be limited to the case of
superposed ranges.

SELF-CORRESPONDING ELEMENTS 65

Theorem. Self-corresponding Elements

58. The number of self -corresponding elements of two dis-
tinct superposed projective one-dimensional prime forms is
two, one, or none, and all three of these cases occur.

Proof. The proof consists of the following four parts :

1. There may he two self -corresponding elements.

If A^, -Bj, Cj, C^ are four points on a line p, the triads
Jji?jCj and A^B^C.^ determine a projectivity between dis-
tinct ranges on p with A^ and B^, but not with either Cj
or Cg, as self-corresponding points.

2. There may he just one self-corresponding element.

On a line p take four points A, B^, B^, Cj, and through
A pass two lines a and p'. On a take any point J^. Let
the line I^B^ cut p' in B', the line B'B^ cut a in ^, the line
I^Ci cut p' in C", and the line ijC cut p in C^.

The triads AB^C^ and AB^C^ determine a projectivity
between distinct ranges on p in which the point A is self-
corresponding. Moreover, each of these ranges on p is
perspective with the range AB'C' • • • on j9' and hence, if
?7j and U^ are corresponding points of the range on p^, it
follows that I{ U^ and I^ U-^ must intersect on p'. This would
not happen if fTj and U^ were coincident, except at A.
Therefore A is the only self -corresponding point.

6(> SUPERPOSED PROJECTIVE FORMS

3. There may he no self-corresponding element.

Taking a range A^B^C^ • -- on 2i base p, let its projector
from a point P, not in the base, be the pencil a^h^c^ • ■.
Tlu-ough P clraw the lines a^, b^, c^, - - -, making any fixed
angle, say 30°, with ftp b^, c^, - • - respectively, and let these
lines meet the base p in A^, B^, Cj, • • •.

Then range A^B^C^ '"a P^^^il a^b^e^^ . • .

- pencil a^b^c^ • . •

- range ^2^2 ^2 ' ' -

Since the ranges A^B^C^ - • • and ^2-^2^2 * * ' ^^^^® "^ ^^^'
responding element, this part of the theorem is proved.

4. There cannot be three self-corresponding elements.

It follows from § 41 that if three elements are self-
correspondmg, all elements are self-corresponding, and the
forms are coincident. Hence if the forms are distinct, there
cannot be three self-corresponding elements.

In selecting the triads of elements which determine the projec-
tivity between superposed forms we may include self-corresponding
elements. Thus, for ranges the projectivity is determined if the two
self-corresponding points X, Y and a pair of corresponding points
Ay A^ are given, for then the triads of corresponding points are
A'F^i and XYA^. Also, if one self-corresponding point X and two
pairs of corresponding points A^ A^; B^, B^ are given, the deter-
mination is complete. Here the triads are XA^B^ &nAXA^B^. In each
of these cases simple constructions serve to determine additional
pairs of corresponding points.

In this connection the student may review the solutions of Exs. 1
and 2, page 47, which furnish constructions for these cases.

59. Classes of Projectivities. The projectivity between
superposed forms is said to be hyperbolic when there are
two self-corresponding elements, parabolic when there is
one, and elliptic when there is none.

HYPERBOLIC PROJECTIVITY 67

Theorem. Anharmonic Ratio

60. If tivo superposed projective ranges have two self-
corresponding points, the anharmonic ratio of these two points
with any pair of corresponding points is independent of the
choice of the latter.

Proof. Let X, Y be self-corresponding points of two
superposed projective ranges, and let A-^, A^; B^, B^ be
any two pairs of corresponding points.

Then {XYA^B{) = (^XYA^B^^.

mi p XAh XB, XAn XBq
Therefore : = : ,

and hence

YA^ YB^ YA^ YB^
XA^ ^ XA^ _ XB^ XB^

YA^ YA^ YB^ YB^
Therefore (XYA^A^} = (XYB^B^).

Exercise 18. Superposed Ranges

Given a range XYA^B^ • • • , find two points A^, B^ of a
range on the same base, projective with the given range, such
that X, Y shall he self-corresponding points and the ratio
(^XYA^A^ shall he as follows:

1. 4. 2. -4. 3. -1. 4. L

6. Consider Exs. 1 and 3 when Fis the point at infinity.

In Exs. 1-5 observe the situation of pairs of corresponding points with
respect to the self -corresponding points.

6. Construct two superposed ranges in which the point at
infinity shall be the only self-corresponding point.

7. Given a self-corresponding point and two pairs of corre-
sponding points, construct the other self-corresponding point
of two superposed projective ranges.

68 SUPERPOSED PROJECTIVE FORMS

Theorem, congruence of projective Ranges

61. Two superposed projective ranges that have the point at
infinity as their only self-corresponding point are congruent.

Proof. Let A^B^C^ • • • and A^B^C^ • • . be two projective
ranges on the same base jo, the only self-corresponding
point of the ranges being the point J at infinity. Let X^
and Aj be corresponding points not at infinity.

Then (^A^B^X^J^ = {A^B^X^J^ ;
whence ^ = ^. . §22

Therefore ^1^ = ±&.-I=d^ -\ = ^&,

i?jAj B^X^ B^X^ B^X^

A,B,_B,X,

^2^2 ^2^2

Suppose now .that, if possible, X^ is taken so that
B^X^: B^X^ = A^B^: A^B^.

Then Mi = JJa^,

B^X-^ B^X^

and hence X^ and Xj must coincide.

But this yields a self-corresponding point distinct from
t/, which is contrary to the hypothesis.

Hence the choice of X^ in the finite part of the line, as
assumed above, must be impossible. But it is possible
unless A^B^ : A^B^^l.

A B
Hence —^-^ = 1, or A^B^ = A^B^

A^B^

It follows then that any two corresponding segments
are equal and that the ranges are congruent.
An interesting alternative proof is given in § 62.

CONGRUENCE OF PROJECTIVE RANGES 69

62. Alternative Proof. Let jo' be a line parallel to p,
and hence intersecting/* at J. Let the range A^B^ — J" be
projected from any center P^ upon the base p\ the resulting
range being A[B[ - • - J.

Because the latter range is
perspective with A^B^ • ' > J,
the point J at infinity must
be self-corresponding (§ 44).
Therefore the range A^B^ ... J"
is projective with the range
A[B'^ ... J" in such a way that the point J at infinity is
self-corresponding, and hence these ranges are perspective.

The center of perspective of these ranges must be on
A[A^. Let it be !{. Then B^ must be the intersection of
B[P^ and p. Draw P^P^.

It will be proved that I^ is parallel to p and p'.

If P^P^ is not parallel to p and p', let it meet p m X^
and p' in X[. Then

range A^B^ . . . Xj • • • J^ range A[B[ . • • X{ • • • J"

= range A^B^ ... X^ ... J".

Hence the given ranges have two self-corresponding
points Xj and J, which is contrary to the hypothesis, and
so I^I^ is parallel to p.

From similar triangles it follows that

A^B,^P^A,_P,A^_A^B^
A[B[ P,A[ P,A[ A'X
and hence that ^i A — ^2^2*

Similarly, any two corresponding segments are equal.
Hence the ranges are congruent.

Compare the above figure with the one in § 58.

70 SUPERPOSED PROJECTIVE FORMS

Theorem. Angle subtended by Corresponding Points

63. Given two superposed projective ranges having no
self-corresponding point, it is possible to find a point at
which all pairs of corresponding points in the range subtend
equal angles.

Proof. Let A^B^C^ • • • and ^2^2 ^2 • • • be projective
ranges on a base jt), and let Jj and K^ be the point at
infinity of p. Let K^ and J^ be the points of the two
ranges which correspond to the point at infinity.

Bisect -fiTjJj in Oj, and let 0^ of the second range corre-
spond to Oj of the first range.

Then, J^ and K^ being the point at infinity and A^, B^
being any points of the first range, we have

(A B J K^ = ^^'^^ • ^^^1 = 1 • ^i^i = A-^i,
^ ^ ^ ^ ^^ B^J^' B^K^ ' B^K^ A^K^

and (^2^2'^2^2) = ^; §22

-t>2'^2

and since {A^B^J^K^^ = (A^B^J^K^), we have

AiK^ . A^J^ = B^K^ . ^2^2 = . . . = OyK^ • O^J^.
Hence

(Oi^i - Oi^i) (Oi^2 - ^1^2) - ^1^1 • (C>i^2 - 0^0^ = 0,
and Oi^i . OjJg - O^A^ • O^J^ - O^K^ . O^A^

-f Oj^i . Oi^2 - ^\^\ ' O1J2 + O^K^ '0^0^ = 0.
If now A^ and A^ should coincide at A, a self-corre-
sponding point, then, since O^J^ = — O^K^, 0^ being the
midpoint of J^Ki, it would follow that

a;A^ = -o,K, . O1O2 = 0,J^ . 0,0^,

Then OiJ^ and 0^0^ would agree in sign; that is, Oj
would not lie between J^ and 0^.

CONSTANT ANGLE 71

On the other hand, if 0^ does not lie between 0^ and
Jg, a point A, related to 0^, 0^, J^ as above, may be found
and will be self-corresponding. Hence, when there is no self-
corresponding point, Oj must be within the segment O^J^.

Let -STj, Jj' ^v ^2 ^ indicated on the base. Erect at
Oj the perpendicular O^F meeting at P the semicircle
whose diameter is O^J^.

Through the point F draw a line parallel to -p.

O2 O,

Then angle O^FJ^ = 90°, and angles K<^PK^ (acute),
J^FJ^, and O^FO^ are equal. But the triads K^J^^O^ and
K^J^O^ determine the projectivity of the ranges.

Let -4j, A^ be any pair of corresponding points.

The projectors from F of the "ranges K^O^J^A^ and
^jOjJj^i are pencils which by the rotation of the second
through the common value 6 of the three angles K^FK^,
J^FJy, O^FO^ could be made to have three common lines
while the projectivity would not be destroyed.

Then all the corresponding lines would be made to
coincide ; and by the rotation of FA^ through the angle d
it would be brought into the position FA^

Hence angle A^FA^ = 0, and the theorem follows.

In Case 3 of § 58 the existence of superposed projective ranges
having no self-corresponding point was established by means of an
example. In this example the two ranges could have been generated
simultaneously by the intersection of their base with the arms of an
angle of constant size rotating about its vertex. It has just been
established that every pair of such superposed projective ranges can
be generated in this way.

72 SUPERPOSED PROJECTIVE FORMS

Theorem. Involution of elements

64. If the projectivity between two superposed projective
one-dimensional forms is such that when any one element is
taken as belonging to the first form that element has the same
corresponding element as it has when it is taken as belonging
to the second form^ then every element has this property.

Proof. We shall consider the theorem for ranges only,
the proof being similar for other forms.

Between two triads A^B^C^ and A^B^C^ that lie on a
base p there exists a projectivity which determines two
superposed projective ranges on this bsise. In connec-
tion with this projectivity every point of the line p may
be given two names. Thus, a point might be L^ and R^.
Moreover, the original triads A^B^C^ and A^B^C^ may have
some points in common.

Suppose now that B^ is taken to be the same as A^^ and
B^ the same as A^. Let a point Z>j be taken to be the
same as C^. The theorem is then proved if we can show
that i>2 coincides with Cj.

From the conditions of the case

iA,A^C,C^^ = (A,B,C,D,) = (A^B^C^D^) = (A^A.C^D^) ;
whence A^C^ : A^C^ = A^D^ : A^D^,

and D^ coincides with Cj.

65. Involution. When two superposed prime forms are
connected by a projectivity such as that described in the
above theorem, they are said to form an involution.

Elements of an involution which correspond to each
other are said to be conjugate.

The projectivity is also called an involution.

66. Corollary. Every projection and every section of
an involution of elements yields an involution.

INVOLUTION OF ELEMENTS 73

Theorem. Anharmonic Ratios in Hyperbolic Involution

67. In a hyperbolic involution the anharmonic ratio of the
two self -corresponding elements and any pair of correspond-
ing elements is — 1.

Proof. If X, Y are the self-corresponding points and
^1, A^ are corresponding points of a point involution, then

Hence (^XYA^A^y^l, and (^XYA^A^^ = -1.

(X YA^A^) cannot be equal to 1, for X,Y,A.^,A^ are distinct points.

The classification of projectivities into hyperbolic, parabolic, and
elliptic applies to the involutions, and it is easy to prove that invo-
lutions of all these classes exist. Hence §§ 58-61 apply in the case
of involutions. In particular, the value of the anharmonic ratio
mentioned in § 60 has been determined in the above theorem.

The converse of this theorem is easily proved; and hence,
when any two elements of a range or pencil are given, it becomes
easy to determine any desired number of pairs of corresponding
elements of an involution of which the two given elements shall
be self-corresponding. A similar remark may be made regarding
the more general case of § 60.

68. Corollary 1. In a hyperbolic point involution the
point at infinity, if not self-corresponding, corresponds to a
point midway between the self -corresponding points.

69. Corollary 2. In a hyperbolic point involution, if
the point at infinity is a self-corresponding point, the other
self -corresponding point bisects the line joining every pair of
corresponding points.

70. Corollary 3. iw a hyperbolic involution of lines
(or planes^, the line (or plane^ which bisects the angle between
the self-corresponding lines (or planes^ has for its correspond-
ing line (or plane^ the one which is perpendicular to it.

74 SUPERPOSED PROJECTIVE FORMS

71. Involution Determined. An involution is determined
whenever enough is known to establish two determining
triads of the projectivity. Consequently the following
data are sufficient for this purpose:

1. Two pairs of corresponding elements.

2. One self-corresponding element and a pair of corre-
sponding elements.

3. Two self-corresponding elements.

Considering the argument for ranges only, the other
cases admitting of similar treatment, we see that the fol-
lowing triads determine projectivities :

In Case 1 the triads A^A^B^, A^A^B^, where A^, A^ ; B^, B^
are corresponding pairs of points.

In Case 2 the triads XA^A^, XA^A^, where X is a self-
corresponding point and -4j, A^ are a pair of correspond-
ing points.

In Case 3 the triads XYA^, XYA^, where X, Y are self-
corresponding points and A^, A^ are a pair of harmonic
conjugates with respect to them.

72. Center of Involution. Let the pairs of points J, 0 ;
-(4 J, A^ ; -Bj, B^ (J being the point at infinity) be corre-
sponding.

Then (^^ JS^ 0J) = {A^B^J 0) ;

whence —^ = — ^ ,

B^O A^O

and OAi • OA^ = OB^ . OB^ = ....

Then in a point involution the point corresponding to
the point at infinity is such that the product of its dis-
tances from any pair of corresponding points is independent
of the choice of the pair. This point is called the center
of the im^olution.

POINT INVOLUTION 75

73. Two Cases of Point Involution. A further examina-
tion of the relations just found suggests two possible cases :

1. The product OA^ • OA^ may be negative.

2. The product OA^ • OA^ may be positive.

In Case 1 no self-corresponding point X can exist ; for
if it did we should have OX^ negative, which is impossible
for real values of OX. The involution is therefore elliptic.

Also, from the relation OA^^ • OA^ = OB^ • OB^, if each
product is negative, it follows that the point 0 separates
every pair of corresponding points. Moreover, if OA^ is
longer than OB^, it is evident that OA^ is shorter than OB^,
and so any two pairs of corresponding points, as A^, A^;
i?j, i?2' mutually separate each other.

In Case 2 by similar reasoning we establish that the
involution is hyperbolic with the self-corresponding points
equidistant from 0, that any two corresponding points lie
in the same direction from O, and that no two pairs of
corresponding points mutually separate each other.

These metric properties are sometimes used to define elliptic and
hyperbolic point involutions.

Exercise 19. Point Involutions

1. Choosing two pairs of corresponding points that will
determine an elliptic involution, find by construction a third
pair of corresponding points and also find the center.

2. Solve Ex. 1 for a hyperbolic involution.

3. Find a pair of corresponding points of an involution of
which A' (a given point) and / (the point at infinity) are the
self-corresponding points.

4. Given one self-corresponding point of an involution and
a pair of corresponding points, find by construction the other
self-corresponding point.

76 SUPERPOSED PROJECTIVE FORMS

Theorem. Line Involution

74. Every involution of lines has one pair of correspond-
ing lines at right angles ; if it has two pairs at right angles^
all its pairs are at right angles.

Proof. Let aj, a^ ; ftj, hc^ be pairs of corresponding lines
of an elliptic involution on a base P. If a^, a^ or 5j, h^
are at right angles, the first part needs no proof.

If neither of these pairs of lines is at right angles, cut
the four lines by any line p in the points ^j, A^^ B^, B^.

Describe the circles PA^A^ and PB^B^ meeting again
in Q. heiPQcnt p in O, and let the perpendicular bisector
of PQ cut p in V.

Describe the circle whose center is V and whose radius
is VPt or VQ, and let it cut p in Cj and C^,

Then OC^ . OC^ = OP'OQ

= OA^ . 0^2 = ^^1 • OB^.

Hence the lines c^, or PC^, and c^, or PC^, belong to the
original involution. Furthermore, they are at right angles,
because C^PC^ is a semicircle.

For the case of a hyperbolic involution see § 70.

LINE INVOLUTION 77

Again, if two pairs of corresponding lines are at right
angles, these pairs separate each other, and the involution
is elliptic.

Suppose now that the lines a^, a^ and also the lines 6j, h^
are at right angles. Then the segments PA^A^ and PB^B^
are semicircles, and their common chord PQ is at right
angles to p at 0. Now if c^, c^-, any other pair of corre-
sponding lines of the involution, cut p at Cj, Cg, then

OCi . OC2 = OP . OQ,

and the segment C^PC^ is a semicircle.

Accordingly, <?j, c^, any two corresponding lines of the
involution, are at right angles.

Exercise 20. Review

1. On a given base p, if each of two ranges is projective
with a third range in such a way that X and Y are self-
corresponding points for both projectivities, then these two
ranges are projective with each other, and X and Y are self-
corresponding points in the projectivity connecting them.

2. In Ex. 1, if the anharmonic ratios (§ 60) associated with
the two projectivities are r^ and r^, find the anharmonic ratio
associated with the third projectivity.

3. Interpret the results of Exs. 1 and 2 when the first two
projectivities are involutions.

4. A projectivity between superposed forms is determined
if two self-corresponding elements and the anharmonic ratio
of these elements and two corresponding elements are given.

5. Given two self-corresponding points X and Y of two
superposed projective i-anges, find a third range on the same
base between which and each of the other two ranges there
exists a projectivity for which X and Y are self-corresponding
points. Consider the number of solutions.

78 SUPERPOSED PROJECTIVE FORMS

6. A point involution is completely determined by its
center and one self-corresponding point.

7. A point involution is completely determined by its center
and a pair of corresponding points.

8. The circles which pass through two given fixed points
determine, on any line which cuts them, corresponding points
of an involution.

9. If .1, B, C, D are fixed coplanar but noncollinear points
and a point 0 moves in their plane in such a way that the
anharmonic ratio of the i^encil 0{ABCD) is a given constant k,
the lines OC and OD meet the line AB in corresponding points
of two superposed projective ranges.

10. Consider Ex. 9 for the case in which ^ = — 1.

11. In Ex. 9 the path of O intersects the line AB in A and B
and in no other points, and hence it passes through all four
of the points A, B, C, D.

12. Every line through one of the points A, B, C, Z) in Ex. 9
cuts the path of 0 in one and only one additional point.

13. In Ex. 12 find the other point in which the path of O is
cut by any given line through A.

14. In Ex. 9, assuming that as a point moving along a curve
approaches a fixed point the secant through the two points ap-
I)roaches the tangent to the curve at the fixed point, construct
the tangent to the path of O at the point A.

15. If ylj, A^, A^ are noncollinear points, a^, a^, a^ noncon-
current lines, P^ a point on a^, P^A^ and a^ intersect in P^,
P^A^ and Og in Pg, and P^A^ and a^ in P[, then, as P^ and P[
move along a^, they trace two superposed projective ranges.

16. In Ex. 15 under what circumstances (if any) do the
lines Pj^i, P^A^, and P^A^ form a triangle with its vertices
on the three given lines ?

17. What modifications of the data in Ex. 15 render it cer-
tain that in all positions the lines P^A^, P^^A,^, and P^A^ form a
triangle with its vertices on the same three given lines ?

PAKT 11. APPLICATIONS
CHAPTER VIII

PKOJECTIVELY GENERATED FIGURES

75. Statement of the General Problem. Application of
the properties of prime forms will now be made to the
study of a problem which is connected with a somewhat
wide range of topics in geometry. This problem, to the
discussion of which the remainder of the book is devoted,
may be stated in these words:

To determine the character of all geometric configurations
whose generating elements are determined hy corresponding
elements of two projective one-dimensional prime forms.

The problem divides naturally into a number of cases
according as the two projective one-dimensional prime
forms are any one of the following pairs:

1. Two ranges, considered in §§ 84 and 85.

2. Two flat pencils, considered in §§ 86-89.

3. Two axial pencils, considered in §§ 90 and 95.

4. A range and a flat pencil, considered in § 96.

5. A range and an axial pencil, considered in § 97.

6. A flat pencil and an axial pencil, considered in § 98.

76. Locus of a Point. If a point moves in space subject
to a given law, the figure consisting of all the points with
which the moving point may coincide, and of no others,
is called the locus of the point.

80 PROJECTIVELY GENERATED FIGURES

77. Envelope of a Plane. If a plane moves in space
subject to a given law, the figure which is tangent to all
the planes with which the moving plane may coincide, and
to no others, is called the envelope of the plane.

78. Envelope of a Line. If a line moves in a plane sub-
ject to a given law, the figure which is tangent to all the
lines with which the moving line may coincide, and to no
others, is called the (^plane} envelope of the line.

79. Generation of a Figure by a Line. If a line moves
in space subject to a given law, the figure consisting of all
the lines with which the moving line may coincide, and of
no others, is said to be generated by the moving line.

80. Order of a Figure. The greatest number of points
of a figure that lie on a line which is not entirely in the
figure is called the order of the figure.

Thus a circle may be met by a line in two, one, or no points.
Consequently the order of the circle is two. Similarly, the order of a
straight line is one. The order of a plane is also one, while that of
a sphere is two.

81. Class of a Figure in Space. The greatest number of
tangent planes of a figure in space which pass tlirough a
line that does not have all the planes through it tangent
to the figure is called the class of the figure.

Thus a sphere may have tangent to it two, one, or no planes which
pass through a straight line, and hence the sphere is of class two.

82. Class of a Figure in a Plane. The greatest number
of tangent lines which can be drawn to a plane figure from
any point in its plane is called the class of the figure.

Thus, of the lines in a plane which pass through a given point
two, one, or none may be tangent to a g^ven circle, and hence the
circle is of class two.

DEFINITIONS 81

83. Dual Elements. From the point of view of the Prin-
ciple of Duality the following are corresponding elements :

1. In geometry of three dimensions, a point on a figure
and a plane tangent to a figure.

It can also be shown that a line on a figure is self-dual.

2. In geometry of the plane, a point on a figure and a line
tangent to a figure.

3. In geometry of the bundle, a line on a figure and biplane
tangent to a figure.

Exercise 21. Preliminary Definitions

1. What is the envelope of a system of tangents to a given
circle ? What is the dual of a circumscribed polygon ?

2. What is the order and what is the class of the projector
of a circle from a point not in its plane ?

The student may consult the chapters on higher plane curves in texts
on elementary analytic geometry, such as the one in this series, and deter-
mine the orders and the classes of the curves considered there.

3. In Ex. 2 what is the order and what is the class of any
plane section of the figure ?

4. Find the surface generated by a line so moving as to be
constantly parallel to and at a given distance from a given line,
and state the order and the class of this surface.

5. In Ex. 4 consider the various plane sections of the sur-
face, stating the order and the class of each.

6. Find the locus in space of a point which so moves as to
be constantly at a given distance from the nearest point of a
given line segment, stating the order and the class of the locus.

7. Find the order and the class of the plane sections of the
figure obtained in Ex. 6.

8. Find the envelope of a plane which so moves as to be
constantly at a given distance from the nearest point of a
given line segment, stating the order and the class of the locus.

82 PROJECTIVELY GENERATED FIGURES

Theorem. Ranges with a common Element

84. The envelope of the line which so moves as always to
contain corresponding points of tivo coplanar projective ranges
is of the second class, unless the ranges are superposed and
without a self-corresponding point, in which case the envelope
is the common base. If the ranges are perspective, the enve-
lope consists of tivo points, one of tvhich is common to the
ranges. If the ranges are not perspective and not superposed,
the envelope is tangent to the base of each range at that point
of it which corresponds to the point common to the ranges
when this common point is regarded as belonging to the other
range.

Proof. The two projective ranges referred to in § 75
may or may not have one common element, and in this
theorem we consider two ranges having such an element.
In this case the ranges must evidently be coplanar.

The element determmed by two corresponding points of
the ranges is a straight line, and hence in this case the
problem is that of determining the envelope of a line
which so moves as always to contain two corresponding
points of two coplanar projective ranges.

These ranges may be (1) superposed ; (2) not superposed
but perspective ; (3) neither superposed nor perspective.

1. Let the ranges be superposed.

Then all pairs of corresponding but not self-corresponding
points determine the common base of the ranges ; and the
lines through any self-corresponding point are infinitely
many. The only figure that has all these lines and no others
as tangents consists of the one or two self -corresponding
points or, if there is no self-corresponding point, consists
of the common base of the ranges.

RANGES WITH A COMMON ELEMENT 83

2. Let the ranges he not superposed hut perspective.

Then all lines determined by distinct corresponding
points pass through the center of perspective, and every
line through the point common to the ranges joins that
point to its corresponding point, that is, to itself. Hence
the envelope consists of two points ; namely, the center of
perspective and the point common to the ranges.

3. Let the ranges he neither superposed nor perspective.
To the common point Xj, or Y^, there correspond in the

ranges two points X^ and Fj. Since the base p^ of the first
range joins Y^ to F^, the enve- p

lope is tangent to p^ ; similarly,
it is tangent to p^.

Consider now Fj and Fj, two
corresponding points nearly co-
incident with Xj and Xj respec-
tively. The line v joining them
is nearly coincident with p^. If,
now, Fj approaches coincidence
with Xj, V and Fj approach p^
and Xg. But if a moving tan-
gent to a curve approaches a
fixed tangent as a limiting line, the intersection of these
two approaches the point of contact of the fixed tangent
as a limiting pomt. Hence in this case Xg is the point of
contact of p^ with the envelope. Similarly, Fj is the point
of contact of py

Finally, the class of the envelope is two ; for two tan-
gents to the envelope, namely, p^ and A^A^, pass through
a point Ay If through any point 0 there should pass three
tangents to the envelope, 0 would be a center of perspec-
tive for the ranges ; but the ranges, are not perspective.

84 PROJECTIVELY GENERATED FIGURES

Theorem, ranges with no common element

85. The lines which join corresponding points of two pro-
jective ranges that have no common point are the intersections
of corresponding planes of two projective axial pencils which
have no common plane.

Proof. If two projective ranges have no common ele-
ment, their bases cannot meet, and therefore the ranges
cannot be coplanar.

Let A^B^C^ • • • and A^B^C^ • • • be projective ranges on
the bases pi and p^ which are not coplanar.

Let a^, y8j, 7^, • • • be the planes determined by the line pi
and the points A^, B^, Cg, • • • respectively, and let a^, jS^, 72? • • •
be the planes determined by the line p^ and the points
^1, i?j, Cj, • • • respectively. Then we have

axial pencil cc^^ff^ ' * ' a ^^"S^ ^2 ^2 ^2 * * *
-range ^1^1 Cj...
- axial pencil ^2^272 * * -

Also the line A^A^ is the intersection of the planes a^
and a^. Moreover, if the axial pencils had a common
plane, the ranges along their axes would both be in this
plane. But this is contrary to hypothesis.

Hence the proof is complete.

As a consequence of this theorem the discussion of the figure
which is generated by these two projective ranges may be deferred
until we consider the figure which is generated by projective axial
pencils that have no common plane (§ 95).

KANGES AND PENCILS 85

Theorem, coplanar Flat Pencils

86. The locus of the point which so moves as always to be
common to tivo corresponding lines of two coplanar projective
flat pencils is of the second order, unless the pencils are super-
posed and without a self-corresponding line, in which case the
locus is the common vertex. If the pencils are perspective, the
locus consists of two straight lines, one of which is common to
the pencils. If the pencils are not superposed, the locus con-
tains the base of each pencil and at each of these points is
tangent to the line of the corresponding pencil which corre-
sponds to the common line of the pencils when this common
line is regarded as belonging to the other pencil.

Proof. Two flat pencils may or may not have a common
base. In the former case the common base may be a plane
contaming both pencils or a point which is the vertex of
both pencils. We shall now deal with the first of these
subcases, the second being discussed in § 87.

Essentially, then, the theorem involves the problem of
finding the locus of the intersection of two coplanar pro-
jective flat pencils. It is evident that these pencils may
be (1) superposed ; (2) not superposed but perspective ;
(3) neither superposed nor perspective.

1. Let the pencils be superposed.

In this case the flat pencils have in common not only
their planes but also their vertices. The points common to
the corresponding lines include in any case the common
vertex and also include all points of any self -corresponding
lines of the pencil. Hence in this case the locus is the one
or two self-corresponding lines of the pencils or, in case
there is no self-corresponding line, the common point of all
the lines of the pencil.

86 PROJECTIVELY GENERATED FIGURES

2. Let the pencils he not superposed but perspective.

In this case the pencils have a self-corresponding line,
all the points of which are in the locus. In addition the
intersections of pairs of correspondmg lines are in the
locus. But these intersections are on a straight line, and
accordingly the locus is a pair of straight lines.

3. Let the pencils he neither superposed nor perspective.

Since the common line is not self-corresponding, let it
be called x^ and ^j* ^^ the two pencils it has corresponding
to it the lines ^j and x^.

The base J^ of the first pencil, being the intersection of
the lines t/^ and i/^, is on the locus. Similarly, the base F^
of the second pencil is on the locus.

Consider now a line v^ that is nearly coincident with a;j,
and the corresponding line v^ that is nearly coincident with
Zj. The point Fof the locus determined by Vj and v^ is nearly
coincident with i^. If, now, the point V approaches F^ along
the locus, the line Vg, or VJ^, approaches x^. But the limit-
ing position of a secant Vl^ as V approaches ^ is the tan-
gent to the locus at i^. Hence the locus is tangent to the
line x^ at I^. Similarly, it follows that the line y^ is tangent
to the locus at J^.

RANGES AND PENCILS 87

Finally, the order of the locus is two; for any line of
the first pencil, as a^, meets the locus at J^ and at A, the
intersection of a^ and its corresponding line a^. Moreover,
if any line o cuts the locus in three points, the triads of
lines of the pencils which would intersect in these points
would be perspective, and so would the complete pencils.
But the pencils are not perspective. The theorem is,
therefore, completely proved.

Exercise 22. Ranges and Pencils

1. Two fixed lines AO^B and CO^D are each perpendicular
to 0^0^, and a moving line cuts them in P^ and P^ so that the
ratio O^P^ : O^P^ is a constant. Find the envelope of the moving
line. Consider the case in which O^P^ and O^P^ are equal.

Compare the ranges traced by P^ and Pg.

2. Two fixed lines intersect at right angles at O, and a
moving line cuts them at equal distances from O. Find the
envelope of the moving line.

3. Examine Ex. 1, substituting the condition that O^P^ and
O^P^ maintain a constant difference.

4. Examine Ex. 2, substituting the condition that the dis-
tances from 0 maintain a constant difference.

5. Examine Ex. 2, substituting the condition that one dis-
tance exceeds a given multiple of the other by a fixed amount.

6. Two lines revolve at the same angular velocity in opposite
senses about the fixed points 0^ and 0^ respectively. Initially
they make angles of 90° and 45° respectively with the line 0^0.^.
Find the locus of their intersection.

7. Consider Ex. 6, substituting the condition that initially
the lines coincide.

8. Examine Exs. 6 and 7 on the assumption that the lines
revolve in the same sense.

88 PROJECTIVELY GENERATED FIGURES

THEOREM. Flat Pencils with a common Vertex
87. TJie envelope of the plane, which so moves as always to
contain corresponding lines of two projective fiat pencils that
are not coplanar hut have a common vertex, is of the second
class. If the flat pencils are perspective, the envelope consists
of ttco straight lines, one of tvhich is common to the pencils.
If the pencils are not perspective, the envelope is a surface
tangent to the plane of each fiat pencil along the line which
corresponds to the common line of the pencil when this cotnmon
line is regarded as belonging to the other flat pencil. All the
planes and the mrface generated pass, through the common
vertex of the pencil.

Proof. Since each pair of corresponding lines of the
given flat pencils determines both a point and a plane,
we are concerned with the problem of finding the aggre-
gate of elements, either points or planes, determined by
corresponding lines of two projective flat pencils which
have a common vertex. The first of these cases is trivial.

It may here be assumed that the flat pencils are not coplanar, as
the case of coplanar pencils has just been discussed.

From the point of view of loci the points determined by
corresponding lines either will be the common vertex alone
or, if the common line happens to be self-corresponding,
will be this common line itself.

FLAT PENCILS 89

On the other hand, any two corresponding lines deter-
mine a plane which passes through the common vertex.
If we cut the whole configuration by a plane that does not
pass through the common vertex, we obtain two pro-
jective coplanar ranges and the lines joining corresponding
points. This section of the envelope is then one of the
figures described in § 84, Consequently the envelope
sought is the projector, from the common base of the two
flat pencils, of one of these figures. If the flat pencils are
perspective, this projector consists of two straight lines
through the common vertex. If the flat pencils are not
perspective, the projector is a conic surface and is tangent
to the plane of each flat pencil. This is evident from the
fact that the plane of either pencil is determined by the
common line of the pencils and x^ that one of its own lines
which corresponds to the common line. The plane of this
pencil has the line x in common with the envelope.

Through the common line of the pencils there pass the
two planes of the pencils, and these are tangent to the
envelope. If tlu-ough any line there should pass more than
two tangent planes, the given pencils would be perspective.

88. Flat Pencils having no Common Base. The discus-
sion of the case of projective flat pencils that have a com-
mon line but are not coplanar and do not have a common
vertex can be given quickly. No intersection of corre-
sponding lines can occur except on the common line. The
common line may be self-corresponding, in which case any
point on it is common to corresponding lines, and any
plane through it contains corresponding lines. If the
common line is not self-corresponding, there is in each
pencil a line corresponding to it. With these lines it deter-
mines two points, the vertices of the pencils, and two planes,
the planes of the pencils.

90 PROJECTIVELY GENERATED FIGURES

89. Projective Flat Pencils having no Common Element.
Figures generated by means of projective flat pencils hav-
ing no common line are so simple as not to demand special
study. They will be noticed in passing, but no formal
theorem regarding them need be stated.

Since pencils which lie in the same plane or in differ-
ent planes that intersect in a line of the pencils have a
common element, it follows that the pencils under consider-
ation lie in planes whose intersection does not pass through
the vertex of either pencil. Upon this line the pencils
determine ranges which may be either identical or distinct.

If the ranges are identical, their common base is the
locus of the intersections of corresponding lines of the pen-
cils. Likewise each pair of corresponding lines determines
a plane through the line joining the bases of the flat pencils ;
and the envelope of these planes is this line. Hence the
figure determined is either the line determined by the ver-
tices of the pencils or the line determined by the planes
of the pencils, according to the point of view.

If the superposed ranges mentioned above are not iden-
tical, the only pairs of corresponding lines of the flat pen-
cils which determine elements are those which meet in the
self-corresponding points of the superposed ranges. These
determine two, one, or no points on the line common to the
planes of the pencils, or two, one, or no planes through
the line joining the vertices. Hence, from the point of view
of loci, the figure generated by means of the projective flat
pencils consists of two, one, or no points ; and from the
point of view of envelopes, the figure consists of two, one,
or no planes. Though neither of the configurations obtained
has any special interest for us, it is clear that they conform
in a general way to the type of figures which we obtain
in the other cases.

FLAT AND AXIAL PENCILS 91

Theorem. Axial Pencils with a common plane

90. The surface generated by the line which so moves as
always to be contained in corresponding planes of two axial
pencils that have a common plane is of the second order
unless the axial pencils are superposed and are without self-
corresponding planes, in ivhich case the surface degenerates
into the common axis of the pencils. If the axial pencils are
perspective, the surface consists of two planes, one of which
is common to the pencils. If the pencils are neither perspective
nor superposed, the surface contains the axis of each pencil
and along each of these axes is tangent to the plane which
corresponds to the common plane of the pencil when this com-
mon plane is regarded as belonging to the other pencil. The
generating line continually passes through the intersections of
the axes of the pencils.

Proof. Two projective axial pencils may have or may
not have a common element. In this theorem we con-
sider only the former case. Evidently the axes of the two
pencils are coplanar. Then the pencils may be (1) super-
posed ; (2) not superposed but perspective ; (3) neither
superposed nor perspective.

1. Let the axial pencils be superposed.

Then all pairs of corresponding planes intersect in the
axis, but any line in a self-corresponding plane may be
regarded as common to two coincident corresponding
planes. The surface generated by the lines common to
corresponding planes consists, therefore, of one or two
self-corresponding planes if there are such planes or, if
there are no such planes, this surface degenerates into
a line, the common axis of the pencil. This last case is
of minor importance only.

92 PROJECTIVELY GENERATED FIGURES

2. Let the axial pencils he not superposed hut perspective.
The axes of the pencils intersect in a point 0 which

lies in every plane of both pencils. Then the Ime deter-
mined by any pair of corresponding planes passes through
this point. Moreover, if a plane is passed so as not to con-
tain this point, it cuts the axial pencils in coplanar per-
spective flat pencils, and the locus of the intersections of
corresponding lines of these pencils is two straight lines,
one of which is the line through the bases of the flat pencils
(§ 86). Consequently the surface generated by the inter-
sections of corresponding planes of the axial pencils is the
projector of the two straight lines from the point 0 ; that
is, the surface is two planes, one of which is the common
plane of the axial pencil.

3. Let the axial pencils he neither superposed nor perspective.

P2\,Pl

As in the previous case, the axes jOj and p^ intersect in
a pomt 0 through which passes every line determined by
corresponding planes of the axial pencils. A plane which
does not pass through 0 cuts the axial pencils in coplanar
projective but not perspective flat pencils whose bases are
-Pj and Pj, the locus of whose intersections is described by
§ 86. The surface generated by the lines common to cor-
responding planes of the axial pencils is the projector from
the point 0 of the locus just mentioned.

RULED SURFACES

91. Regulus. Any three straight lines, no two of which
are coplanar, are met by infinitely many straight lines
which, taken as an aggregate, are said to form a regulus.

The three given Imes are called the directrices of the
regulus, and the lines which meet the directrices are called
the generators of the regulus.

Kegulus

QuADRic Surface

Skew Quadric Ruled Surfaces

92. Quadric Surface. The aggregate of the points of the
lines of a regulus constitute a surface called a quadric
surface, and the generators of the regulus are also called
the generators of this surface.

There are quadric surfaces which are not constituted in this way.

93. Ruled Surface. A surface generated by the move-
ment of a straight line is called a ruled surface.

94. Skew Ruled Surface. A ruled surface in which no
two consecutive generators intersect is called a shew ruled
surface.

For a full discussion of skew ruled surfaces see § 187.

94 PROJECTIVELY GENERATED FIGURES

Theorem. Axial pencils with No common element

95. The lines of intersection of two projective axial pencils
which have no common element generate a skew ruled surface
of the second order in which lie the bases of the pencil. Every
section of this surface by a plane through a generating line
is a pair of straight lines. All other plane sections are curves
of the second order.

Proof. The proof may be divided into five parts :

1. The ruled surface is skew.

Each generator intersects each axis. If two generators
intersect, they determine a plane which contains two points
and therefore all points of each axis. This plane must
then be a common element of the pencils, which is con-
trary to hypothesis. Hence no two generators intersect,
and so the surface is skew.

2. The bases of the two axial pencils lie in the surface.

Through any point A of the base of either axial pencil
there pass all the planes of that pencil and one plane a
of the other. Hence A is on the line determined by the
plane a and its corresponding plane. Hence the surface
contains every point of the base of either pencil.

3. Every section of the surface by a plane which does not
pass through a generating lirie is a curve of the second order.

Any plane tt which does not contain a generating line
cuts the axial pencils in two projective flat pencils. If
these flat pencils were perspective, their common line would
be self-corresponding and the plane tt would cut two planes
of the axial pencils in their common line, that is, in a gen-
erating line; and this is contrary to hypothesis. Then
(§ 86) the section is a curve of the second order.

AXIAL PENCILS 96

4. Every section of the surface hy a plane through a gen-
erating line is a pair of straight lines, one of which is the
generating line and the other of which meets every one of the
generating lines mentioned above.

Let the surface be generated by the projective axial
pencils «^i/Sj7j . • • and oc^fi^'^^ ....

Any plane that is an element of one of these axial
pencils intersects the surface in two straight lines, of
which one line is the base of its pencil, and the other
line is the line of intersection of this plane with the plane
that corresponds to it in the other pencil.

Now let a plane tt that does not belong to either axial
pencil be passed through a generating line a which is deter-
mined by the planes a^ and a^ of the axial pencils.

This plane cuts the axial pencils in projective flat pencils
ah^c^ . . . and ah^c^ .... It cuts the generating lines and
hence the surface generated in the locus determined by
these flat pencils. The latter have a self-corresponding
line a, and accordingly they are perspective.

Consequently the locus determined by these flat pencils
consists of two lines ; namely, a and the line in which lie
the points of intersection of the lines Jj, h^ ; c^, c^; . . ..

The line containing the points of intersection of the
pairs of lines 5j, b^; c^, c^; - ' • intersects every one of the
generating lines determined by the axial pencils. For if
we consider the line determined by the corresponding
planes /Sj, ySj, we find that it passes through the intersec-
tion of the lines b^, b^. Similarly, it may be shown that
it meets the line determined by any other pair of corre-
spondmg planes except a^, a^. The line and a, the inter-
section of a^, a^, are in the same plane tt, and hence the
statement is established.

90 PROJECTIVELY GENEKATED FIGURES

5. Tlie surface itself is of the second order.

Let any line p which is not a generating line intersect
the surface hi a point P. Pass a plane tt through the line jt?.
The section of the surface is a curve of the second order,
and consequently the line p cannot meet it in more than
two points. Furthermore, it cannot meet the surface in
points not in this curve. Hence p cannot meet the sur-
face in more than two points. That many lines actually
meet the surface in two points follows from Case 4. Hence
the surface is of the second order.

96. A Range and a Flat Pencil. If a range and a flat
pencil are projective, they may or may not be coplanar.

If the range and pencil are coplanar, each point of the
range taken with the correspondmg line of the flat pencil
determines the plane containmg both the range and the flat
pencil ; and this plane is the figure sought.

If the range and pencil are not coplanar, let the range
be Jj/ijCj • • . on the base p^, and the flat pencil a^b^c^ • • •
on the base ij. The plane determined by A^ and «j is the
same as the plane determined by I^A^ and a^ Hence this
case yields the same result as that of two noncoplanar
projective flat pencils which have a common base ^ (§ 87).

97. A Range and an Axial Pencil. The case of a range
and an axial pencil may also be considered briefly.

A point and a plane have not been considered as deter-
mining a third element. If, however, the point is in the
plane, they may be said to determine either the point or
the plane ; and since all, two, one, or none of the points
of a range might lie on the corresponding planes of an
axial pencil projective with the range, the configuration
sought in the problem might be regarded as consisting of
pouits or planes as indicated. The case is not important.

SUMMARY 97

98. A Flat Pencil and an Axial Pencil. The case of a
flat pencil and an axial pencil demands but little attention.
There are two subcases, a somewhat trivial one in which
the base of the flat pencil is on the base of the axial pencil,
and another subcase which has been dealt with from a
different point of view.

A little consideration shows that the first subcase may-
be regarded as yielding all, two, one, or none of the
elements of the flat pencil, for these elements contain all
points common to pairs of corresponding elements of the
flat pencil and the axial pencil.

In the second subcase the lines of the flat pencil and
the corresponding planes of the axial pencil determine a
locus of points. But the plane of the flat pencil cuts the
axial pencil in a second flat pencil projective with the first ;
and the locus in question is also determined by the inter-
sections of corresponding lines of this second flat pencil
and the given flat pencil. Hence the locus is the same as
that described in § 86.

99. Summary of Results. From the discussion in this
chapter there have come to notice the following figures,
exclusive of certain trivial ones:

1. Certain plane curves of the second class.

2. Certain plane curves of the second order.

3. Certain conical surfaces of the second class.

4. Certain conical surfaces of the second order.

5. Certain ruled surfaces of the second order in space.

The study of these curves and surfaces will be under-
taken with a view to exhibiting the symmetry among them
and to establishing their more striking properties, as well
as with a view to showing the power of methods which
are based upon the principles that have been set forth.

98 PKOJECTIVELY GENERATED FIGURES

Exercise 23. Review

1. If two points of a circle are each joined to four other
points of the circle, the anharmonic ratios of the two pencils so
formed are equal.

2. If a point moving on a circle is constantly joined to two
fixed points on the circle, the flat pencils generated by the two
joining lines are projective.

3. If a variable tangent to a circle meets two fixed tangents,
the ranges traced by the intersections are projective.

s/ 4. A line so moves as to cut the sides BC, CD of a square
A BCD in points A', Y such that the angle XAY is constant.
Find the nature of the envelope of the moving line.

5. A line so moves as always to be at a constant distance
from a fixed point. Find the nature of the envelope of the line.

6. A wire fence consists of a number of horizontal strands of
wire at equal intervals, crossed by a number of vertical strands
also at equal intervals between each pair of posts. One of two
posts is pushed into an oblique position. What sort of surface
passes through all the wires between the two posts ?

,^ 7. Two lighthouses are in a north-and-south line. The lamps
revolve at the same uniform rate in the same angular sense,
and each lamp throws two shafts of light in opposite directions.
If the lamps are so adjusted that when one light shines north
and south the other shines northeast and southwest, find the
nature of the locus of the spot illuminated simultaneously by
both lights. Has this locus any infinitely distant points ?

8. Consider Ex. 7 when thg lamps are so adjusted that
periodically the shafts of light coincide.

9. Consider Exs. 7 and 8 when the two lamps revolve in
opposite senses.

10. In Ex. 7, if one light rotates twice as quickly as the
other, do the rays generate projective flat pencils?

REVIEW EXERCISES 99

11. A number of lamps, ea«h of which throws two shafts of
light in opposite directions and rotates at a uniform rate, are
to be placed so that their rays shall all continually converge
upon an object which moves along a circle. If all the lamps
have the same angular rate, specify a possible arrangement.

12. In Ex. 11 specify an arrangement and adjustment in
which all the lamps do not have the same angular rate.

13. Each of the circles Cj, c^, • • •, Cn is tangent to the circle
next preceding and to the one next following but to no others
of the set, and P^, P^, - • -, P^ are variable points on the respec-
tive circles such that each line PkPk+i contains the point of
contact of the circles Ck, c^.+i. If 0^ is any point on c^, and 0„
is any point on c„, find the nature of the locus of the intersec-
tion of O^Pj and 0„P„, the figure being plane.

14. Consider Ex. 13 with the omission of the restriction
that no circle shall be tangent to any other except the one
next preceding and the one next following.

15. Each of the circles c^, c^, • • •, c^ is tangent to the circle
next preceding and to the one next following but to no others
of the set, and t^^, t,^, ■ ■ •, t^ are variable tangents to the respec-
tive circles such that the point ^^•^^•+l is on the common tangent
of c/t, C/. + 1. If o is any fixed tangent to c^, and o„ is any fixed
tangent to c„, find the nature of the envelope of the line join-
ing the intersection of Oj, t^ and that of o„, t^^.

16. Each of two circles in one plane is divided into ten equal
arcs. For each circle the tangent at one point of division and
the secants through that point and the other points of division
are drawn. If these two sets of lines are produced indefinitely,
their 100 points of iatersection lie in sets of ten upon 10 curves
of the second order.

< 17. At the same moment two trains leave a junction on
straight diverging lines and travel at uniform rates. If two
passengers, one on the rear platform of each train, watch each
other, find the envelope of their line of sight.

100 PROJECTIVELY GENERATED FIGURES

18. Solve Ex. 17 with the modification that the trains leave
nearly but not quite at the same time.

19. Each of the right circular cones C^, C^, • • •, C^ is tangent
along a straight line to the one following it. The variable lines
p , p^ • • -ijhi on the respective cones are such that each plane
PkPkJt\ contains the line of contact of the cones C^t, Cjt+i- If
o,, o„ are any fixed lines lying on Cj, C„ respectively, find the sur-
face generated by the intersection of the planes o^j)^, o«i>u-

20. As a train is running along a straight track at a uniform
rate, an automobile moves, also at a uniform rate, down a hill
along a straight road which passes beneath the railroad. Find
the figure generated by the line joining two fixed points, one
on the train and the other on the automobile.

21. Initially a plane cuts two fixed intersecting planes per-
pendicularly in the lines o^ and o^. As it moves it cuts these
planes in the lines p^,p^ which cut o^ and o^ at their intersection
and always make equal angles with them. Find its envelope.

Examine the flat pencils traced by Pj and Pg-

22. If A BCD is a regular tetrahedron, and a line so moves
as always to intercept on AB and CD equal distances from
.1 and C, find the surface generated by the line.

23. Consider Ex. 22 if the line continually divides AB and
CD proportionally.

24. A sloping telephone wire and an electric-light pole cast
upon the side of a house shadows which intersect. If the wind
causes the source of light to swing in a straight line, find the
path traced by the intersection of the shadows.

25. Consider Ex. 24 if the source of light swings in a circle
which intersects the wire and the pole.

26. Two fixed lines o^ and o^ intersect and pierce a plane w
at the points O^ and O^. Two planes tt^ and tt^ revolve about o^
and Oj respectively so that their intersections with a> describe
equal angles in the same time. Find the nature of the envelope
of the intersections of tTj and tt .

CHAPTER IX

FIGURES OF THE SECOND ORDER

100. Purpose of the Discussion. The results obtained in
the preceding chapter may be given a more general as well
as a more compact and symmetric form. It will be noted
that these results relate to three types of figures, namely,
figures in a plane ; figures in a bundle, or conical figures ;
and figures in space. These tliree types of figures will be
considered separately.

101. Plane Figures. It has already been shown that the
figure obtamed as the envelope of the line joining corre-
sponding points of coplanar projective ranges is a curve
of the second class, and the one obtained as the locus of
the intersections of corresponding lines of projective flat
pencils is a curve of the second order. Whether all curves
of the second class and all curves of the second order may
be generated in this way, whether the ranges and flat pen-
cils which give rise to the curves in question have special
situations relative to those curves, and what relation, if any,
exists between curves of the second class and those of the
second order, are questions whose answers will exhibit clearly
the importance and generality of the results obtained. These
questions will now be discussed, but for the sake of brevity
the treatment will be limited, the parts of the argument
which are omitted being indicated. The student should
not lose sight of the omission, and he should later seek
to complete the argument which answers the questions.

101

102 FIGUKES OF THE SECOND ORDER

102. Generalization of Results. By a line of reasoning
which will not be given here the following can be proved :

Every eurve of the second class is the envelope of the lines
joining corresponding points of two coplanar projective ranges
ivhose bases are tangent to the curve.

The application of the Principle of Duality for the plane
to this result immediately leads to the further result :

Every curve of the second order is the locus of the inter-
sections of corresponding lines of two coplanar projective flat
pencils whose bases are on the curve.

Accordingly it is clear that the developments in the
preceding chapter relate to all curves of the second class
and to all of the second order.

Consider the second question suggested in § 101. We
have already seen that the bases of the projective ranges
by means of which the curves of the second class were
obtained are tangent to these curves at certain of their
points. It will be shown (§ 106) that any two tangents
to a curve of the second class may be taken as the bases
of projective ranges such that the given curve is the
envelope of the lines joining corresponding points of these
ranges. Correspondingly, it may be shown that any two
pomts on a curve of the second order may be taken as
bases of projective flat pencils such that the given curve
is the locus of the intersections of corresponding lines of
these pencils ; and because to most students the idea of the
locus of points is more familiar than that of an envelope,
the latter proposition will be established first.

The proofs of these statements regarding curves of the
second order depend upon an auxiliary proposition which
will now be stated and proved. Tliis theorem will later
(§ 120) be generalized.

GENERALIZATION OF RESULTS 103

Theorem, auxiliary Proposition

103. If ^, ^, P^, ^ are four points on a curve of the sec-
ond order wliich is the locus of the intersections of correspond-
ing lines of the two coplanar projective fiat pencils whose bases
are ^ and P^, then the pairs of lines I^P^, P^P^ ; ^^, P^P^ ;
^^, PqP^ determine collinear points.

Proof. By hypothesis the points on the curve in ques-
tion determine projective flat pencils whose bases are I{
and P^. These flat pencils may be denoted by -?^(^^-^-^)
and P,CP,P,P,P,).

Cut these pencils by the lines .^^ and ^^ respectively,
and let the resulting ranges be Q^^^IQq and P^I^R^R^.

These ranges are not only projective but also have a
self-corresponding element P^. Hence they are perspective,
and the lines I^Q^ (or P^P^^^ P^R^ (or P^P^-, and Q^R^ pass
through 0, the center of perspective.

Therefore the points Q^, R^ are collinear with the inter-
section of I^P^ and P^P^. But ^g is the intersection of P^P^
and i^^, and R^ is the intersection of j^ig and ^i^.

Hence the proposition is proved.

By means of the above theorem the desired proposition, known
as Steiuer's theorem, may now be proved.

104 FIGURES OF THE SECOND ORDER

Theorem. SiEmER's Theorem

104. Every curve of the second order is the locus of the
intersections of coplanar projective flat pencils whose bases
are any two points of the curve.

Proof. Let -^, ^ be the bases of the projective flat
pencils which generate the curve, and let -^, I^, I^ be any
three fixed pohits on the curve.

Let -^ be any point moving along the curve and occu-
pying successive positions, as j^, i^', • • •.

It will be shown that the flat pencils generated by the
moving lines I^I^ and im are projective.

For each position of J^ the pairs of lines I^, I^;
^^, ^^ ; I^I^y P^Il determine as collinear the fixed point
0 and the points Q^, R^ (§ 103).

As II moves, Q^ and R^ move along the fixed lines I[Pq
and P^Pq and trace ranges on them.

Then flat pencil ^(^^' • • •) 7^ range R^R^ . . .(on ^^)

= range Q^Ql-- "(on P^P^)
= flat pencil ^(^^'...).

Accordingly the curve is the locus of the intersections
of corresponding lines of the projective flat pencils whose
bases are I^ and ^, any two points of the curve.

STEINER'S THEOREM 105

Theorem. Auxiliary proposition

105. If <2» '3, ^4, ^6 ^^^ /•'^^ tangents to a curve of the
second class which is the envelope of the lines joining cor-
responding points of two coplanar projective ranges whose
bases are ^ and t^, then the pairs of points t^t^, t^t^ ; t^t^, t^t^ ;
^3^4, fg^j determine concurrent lines.

This theorem is the dual of the theorem of § 103 and leads to the
dual of Steiner's theorem. It will later (§ 121) be generalized into
a highly important proposition. The proof is left for the student

Theorem. Dual of Steiner's Theorem

106. Every curve of the second class is the envelope of the
lines joining corresponding points of the coplanar projective
ranges whose bases are any two tangents to the curve.

It is particularly important that by means of the Principle of
Duality, or otherwise, the student should follow out in detail the
proof of this proposition, as well as the proof of the proposition which
immediately precedes it (§ 105). There is not much difficulty in
obtaining the steps of the proof as duals of the corresponding steps
of the proof in § 104, but the figure and the verification of the various
steps of the argument in connection with this figure require close
attention on the part of the student.

107. Relations between Curves of the Second Order and
Curves of the Second Class. The two sorts of plane curves
which have been obtained can now be shown to be iden-
tical. In other words, it can be shown that every curve
of the second order is of the second class, and conversely.
Only one of these proofs will be given, since the other can
be derived from it by means of the Principle of Duality.
In this proof use will be made of a limitmg case of the prop-
osition in §103, in which two of the four arbitrarily chosen
points on the curve have moved up to coincidence with
the bases of the pencils, and this will first be established.

106 FIGURES OF THE SECOND ORDER

Theorem. Inscribed Quadrangle

108. If I^y Jl are two points on a curve of the second
order which is generated hy two coplanar projective fiat pen-
cils whose bases are i^ and ^, the tangents at J^ and ^, the
tangents at i^ aiid i^, and the pairs of opposite sides I^I^,
im; J^I^, -^i^ of the inscribed quadrangle imi^I^ inter-
sect in collinear points.

Proof. In the figure of § 103 let ^ and ^ move along
the curve so as to approach ^ and i^ as limiting points.

Then I^ and I^ approach the tangents to the curve at
J^ and ^ respectively, and 0 approaches the intersection of
the tangents at ^ and ^ as a limiting point.

The hexagon I^I^I^P^P^Il approaches the quadrangle
im^Il, together with the tangents at ^ and I^.

During the motion of the points and lines the intersec-
tions of the pairs of lines -^^, ^^; ^^, ^^; ^^, i^i^
remain colUnear, and the limiting positions which they
approach are collinear.

Since I{, P^ are not special points on the curve (§ 104),
it follows that the tangents at J3, Pq meet on the same line.

Then the theorem of § 103 takes the form of this theorem.

This proposition will be used as auxiliary to the proof of the
identity of curves of the second order with those of the second class.

STEINER'S THEOREM 107

Exercise 24. Steiner's Theorem and Related Theorems

1. In the theorem of § 103, if P^, P^, P^, P^ are fixed points,
and if P„ and P. so "move that the intersection R otPP.PP

o D 6 2 8' 5 6

moves on a fixed line through the intersection of P^P^, P^P ,
then the intersection Q^ of P^P^, P^P^ moves on the same line,
and Qg, R^ trace on this line two superposed projective ranges.

2. In Ex. 1 find the self-corresponding points of the super-
posed projective ranges.

3. Prove the proposition which is related to the theorem of
§ 108 as Ex. 1 is related to the theorem of § 103.

4. Solve the problem which is related to Ex. 3 as Ex. 2 is
related to Ex. 1.

5. If fg, fg are two tangents to a curve of the second class
which is the envelope of the lines joining corresponding
points of two coplanar projective ranges whose bases are t^, t^,
the points of contact of t^, t^, the points of contact of t^, t^,
and the pairs of opposite vertices t^t^, t^t^ ; t^t^, t^t^ of the
circumscribed quadrilateral t/^t^t^ determine concurrent lines.

6. A variable hexagon ^i^2^3^4^5^6 ii^scribed in a curve
of the second order so moves that P,P„, P.P.: P„P„, P^P.
always intersect in fixed points O and R respectively. Find
the locus of the intersection of P^P^, P^P^

7. Prove the duals of Exs. 1, 3, and 6 for the plane.

8. Solve the dual of Ex. 2 for the plane.

9. By an argument independent of that given in this chap-
ter prove Steiner's theorem for the special case of the circle.

10. By an argument independent of that referred to in § 106
prove the theorem stated there for the special case of the circle.

11. Pj, Pg, Pg, P^, P5 are five fixed coplanar points no three
of which are in a straight line; find the locus of a point
P which so moves that the intersections of the pairs of lines
P^P^, P^P^ ; P^P^, P^P ; P^P^, PP^ are constantly collinear.

108 FIGURES OF THE SECOND ORDER

Theorem. Identity op Curves

109. Every plane curve of the second order is of the
second class, and conversely.

Proof. When a curve of the second order is given, the
pencils of lines drawn from two of its points to all of
its points are projective. From any three pairs of corre-
sponding lines all additional pairs can be obtained by the
method of § 39. In particular, since the tangent at either
of the two points corresponds to the line joining the two
points, it can be drawn by the same method; and there-
fore, when the whole curve is given, the tangents at as
many points as may be desired can be drawn.

Let a curve of the second order be given, and select
on it any three points ^, ^, ^. Draw J^, ^^, i^^ and
the tangents MI^N, NI^L, and LI^M. The projectivity is
determined by the triads I^M, ij^, J^ and ^i^, j^i, I^I^.

Consider the curve as the locus of a moving point P.
It will be shown that as P moves along the curve the
tangent at P so moves as to meet the tangents at i^ and ^
in corresponding points X and Y of two projective ranges.

IDENTITY OF CURVES 109

Let 0, K be the intersections of the sides -^j^, I^P and
^P, I^ of the quadrangle PI{I^I^ inscribed in the given
curve. For this inscribed quadrangle, M is the intersection
of tangents at opposite vertices and 0, K are intersections
of pairs of opposite sides, and hence it follows from § 108
that the points M, 0, K are collinear. By means of a like
reasonmg the points 0, K, Y are proven collinear. Hence
the points M, 0, Y are collinear.

Similarly, the pomts L, 0, X may be proved collinear.

Since ^, -^, ^ are fixed points, the tangents at these
points are also fixed. As the point P moves, so do the
lines P^P, P^P, P^P, XY, l^O, LO, MO, and the points
0, X, Y. The Imes LO, MO generate perspective flat pen-
cils with bases at L and M, and the points X and Y trace
on the lines MI^N and LI^N ranges perspective with these
pencils and hence projective with each other. Therefore,
as P moves, the tangent at P so moves as always to meet
the tangents at J^ and ^ in correspondmg points of two
projective ranges.

The given curve is the envelope of the tangent at P
and, by § 104, is of the second class.

The converse of this proposition being also its dual for
the plane, its proof is also the dual of the above argu-
ment. The theorem is therefore established.

110. Conic. A plane curve which is of the second order
and second class can be shown to be a curve ordinarily
called a conic section or a conic.

The proof will not be given, but, independently of its other uses,
the word conic will be employed to designate any curve of the
second order. The use of properties of conies not deduced from the
definition here given will be avoided.

A conic section, or conic, is a curve of the second order
and second class.

110 FIGURES OF THE SECOND ORDER

Theorem, order and class of Surfaces (m the Bundle)

111. Every surface (in the bundle) that is of the second
order is of the second class, and conversely.

Proof. A comparison of the problems whose discussion
led to §§84 and 90 shows that these theorems deal with
figures and state results that, for threefold space, are dual.
Similarly, the theorems of §§86 and 87 are dual. It was
indicated, but not proved, that § 84 is true for all plane
curves of the second class and that § 87 is true for all
plane curves of the second order. Correspondingly, §§90
and 87 can be shown to apply to all surfaces (in the
bundle) that are of the second class and all surfaces (in
the bundle) that are of the second order. Moreover, the iden-
tity of the curves of the second order with curves of the
second class having been established in § 109, the identity of
these surfaces of the second order with those of the second
class may be established by reasoning dual to that of
§ 109. This involves the derivation of an auxiliary theorem
dual for space to that of § 103 and one that is dual to the
limiting case of the latter as worked out in § 108. The
student will find that the principal difficulty is connected
with the drawing of appropriate figures for these cases.

In this way the classes of figures numbered 3 and 4 in § 99 are
shown to be identical.

112. Quadric. Since a surface (in the bundle) of the
second order and second class may be thought of as gener-
ated by the motion of a line that always passes through
a fixed point, it is said to be a conic surface or a cone ;
and on account of its order and class it is called a
quadric conic surface.

A quadric conic surface or quadric cone is a surface (in
the bundle) of the second order and second class.

ORDER AND CLASS OF SURFACES 111

Theorem. Skew Ruled Surfaces

113. Every ruled surface (not in the plane or in the bundle')
of the second order is of the second cla^s^ and conversely.

Proof. In §§85 and 95 certain ruled surfaces of the
second order that exist in threefold space but not in the
plane or in the bundle were found to be generated by means
of both projective ranges and projective axial pencils.
It can further be shown that all ruled surfaces of the
second order can be so generated. Moreover, a complete
discussion of these surfaces from the point of view of
both methods of generation would have led to results
similar to those obtained for the conies and for the quadric
cones. Among other things it would have appeared that
the planes which pass through a point P not on the sur-
face, and are tangent to the surface, would generate a
quadric cone, and that of these planes not more than two,
but in some cases two, would pass through a line that
contains F. Hence these surfaces are of the second class.

The converse may also be proved.

The discussion indicates that the class of figures numbered 5 in
§ 99 is self-dual in threefold space.

114. Summary. We have now shown that the config-
urations whose generating elements are determined by
corresponding elements of two projective one-dimensional
prime forms are as follows:

1. All plane curves of the second order and second class
(conies).

2. All conic surfaces of the second order and second cla^s
(quadric cones').

3. All ruled mrfaces of the second order and second class
not in the plane or in the bundle.

112 FIGURES OF THE SECOND ORDER

Exercise 25. Review

1. State and prove the dual for space of § 103.

2. State and prove the dual for the bundle of Ex. 1.

3. Compare the dual for space of the theorem in § 105
with the result of Ex. 2.

4. Give the dual for the plane of the statement and proof
of Ex. 11, page 107.

5. Give the dual for space of the statement and proof
of the theorem in § 108.

6. Compare the space dual of the result of Ex. 5, page 107,
with the dual for the bundle of the result of Ex. 5, above.

7. Derive the dual for space of Ex. 1, page 107.

8. Establish three projectivities between flat pencils which
shall lead to the generation of conies having respectively two,
one, and no points of intersection with any given straight line.

In this connection consider § 68.

9. Consider Ex. 8 for the case in which the given straight
line is the line at infinity of a given plane.

Tlie student will observe that the solution of this problem establishes
the existence of conies with two, one, and no points at infinity.

10. Establish three projectivities between ranges which
shall lead to the development of conies having respectively
two, one, and no tangents whose points of contact are on
any straight line.

11. Consider Ex. 10 for the case in which the given straight
line is the line at infinity of a given plane.

12. Prove the dual for space of Ex. 6 on page 107.

13. Solve the dual for the plane of Ex. 8.

14. Consider Ex. 13 for the case in which the given point
is at infinity in a given direction.

15. Solve the dual for space of Ex. 8.

REVIEW EXERCISES 113

16. Five concurrent lines, no three of which are in any one
plane, all lie on one conic surface of the second order.

17. Prove the proposition regarding five parallel lines which
corresponds to Ex. 16.

18. Establish three projectivities between axial pencils
which shall lead to the generation of conic surfaces having
their vertices at infinity and having as right sections curves
with two, one, and no points at infinity respectively.

19. Establish between two given ranges which are not in
the same plane a projectivity such that if the surface gener-
ated is cut by any given plane in the finite part of space, the
section shall be two straight lines.

20. In a bundle tTj, tt^, tt^, tt^, tt^ are five fixed planes, no
three of which are coaxial. Find the envelope of a plane ir
which moves so that the planes determined by the intersec-
tions of TT^, TT^ and TT^, TTg, of TT^, TTg and TTg, TT, and of TTg, TT^

and TT, TT^ are constantly coaxial.

21. Consider Ex. 19 for the case in which the given plane
is at infinity.

22. Given a plane and two projective axial pencils which
have no common element, establish between other pencils a
projectivity which shall lead to the generation of a surface
that shall be the projector from a given center of the intersec-
tion of the given plane and the surface generated by the given
axial pencils.

23. Derive the dual of Ex. 22 for space.

24. Given in a plane three nonconcurrent bases p^, p^, p^
passing through the points A^,A^,A^ respectively, specify three
projectivities which connect ranges on the bases p^,p^'i P^tPs'^
p^,p^ respectively, and which are such that the three conies
that they determine shall coincide and also be tangent to the
three lines p^, p,^, p^ at A^, A^, A^ respectively.

25. Examine Ex. 24 for the case in which the three points
A^, A^, A^ are collinear.

114 FIGURES OF THE SECOND ORDER

26. In Ex. 24 select such lines p^, p^, p^ and such points
A^, A.^, A^ that the conic generated shall be a circle.

27. Given in a bundle three non-coaxial planes tTj, tt^, tt,
passing through the lines a^, a^, Og respectively, specify three
projectivities which connect flat pencils in the planes tTj, tt^;
TTjj, TTg; TTg, TT^ respectively, and which are such that the three
conic surfaces they determine shall coincide and shall be
tangent to the planes tt^, tt^, tTj along the lines a^, a.^, a^.

28. Solve the dual of Ex. 27 for the bundle.

29. Consider Ex. 19 for the case in which the section by the
plane at infinity is to be two straight lines.

30. Establish such a projectivity between two given axial
pencils, not in the same bundle, that if the surface generated
is cut by any given plane in the finite part of space, the section
shall be circular.

31. Find two axial pencils, not in the same bundle, between
which such a projectivity may be established that the corre-
sponding surface generated shall be cut by a given plane in a
given circle of that plane.

32. Given two projective axial pencils, not in a bundle, pass
a plane which shall cut the surface generated by them in two
straight lines.

33. Given three bases p^, p^, p^ in space, no two of which
intersect, specify three projectivities between ranges on the
bases p^, p^\ p.^, p^\ p^, j)^ respectively, such that the three skew
ruled quadric surfaces determined by them shall coincide.

34. Solve the dual for space of Ex. 33.

35. Develop completely the proof of the theorem corre-
sponding to the theorem of § 109 for the case of figures of
the second order in the bundle.

36. For the case of figures of the second order in three-
fold space describe accurately the figure for the theorem
corresponding to that of § 109, and outline the proof.

CHAPTER X
CONICS

115. Detennination of Conies by Certain Conditions. Some
of the more important properties of the curves and sur-
faces to which attention has been drawn in the preceding
chapters will now be deduced. The conies will be dealt
with much more fully than the other figures because of
their more frequent application and also because, after
their properties have been set forth, the corresponding
properties of the quadric cone may be obtained by means
of the Principle of Duality.

Chapters X-XII are devoted to the conies.

In Chapter XIII there is given a discussion of quadric
cones, this being confined to a few topics in addition to those
suggested by the developments obtamed for the conies.
Notwithstanding this limitation, students should give due
attention to these properties of quadric cones.

In Chapter XIV will be found a brief introduction to
the study of the properties of skew quadric ruled surfaces.
A thorough study of these figures may well be deferred
until the student has an opportunity to approach the sub-
ject from the point of view of analytic geometry also, when
a comparison of the analytic and synthetic treatments will
heighten the interest.

The first body of facts to be established relates to sets
of data which completely determine conies. It constitutes
the important theorem stated in § 116, which consists of
six simple propositions.

115

116 CONICS

Theorem. A Conic Determined

116. In a plane there is one and only one conic which has
one of the following properties :

1. It passes through five given points, no four of which are
collinear.

2. It passes through four given points, no three of which
are collitiear, and at any one of these points is tangent to a
given line which passes through this point hut not through
any other of the given points.

3. It passes through three given points, not collinear, and
at each of two of these points is tangent to a given line which
passes through this point but not through any other of the
three given points.

4. It passes through tivo given points, and at each of these
is tangent to a given line which parses through that point hut
not through the other given point, and in addition is tangent
to a third given line which is not concurrent with the other
two given lines.

5. It parses through a given point and is tangent at that
point to a given line through the point, and is tangent to
each of three other given lines so situated that of four given
lines no three are concurrent.

6. It is tangent to five given lines, no four of which are
concurrent.

Without doubt the student will generally use in his work an
abridgment of this statement. The longer statement given above
may be regarded as an interpretation of the shorter one in § 117,
making clear the meaning of the determination of a figure by means
of certain data. The student will find that very frequently in geom-
etry this abridged form of statement is used in the sense expressed
more fully by the other one. Occasional expansions of shorter
statements into the corresponding longer ones are well worth the
attention of the student.

A CONIC DETERMINED 117

Theorem. Alternative Statement of § 116

117. A conic is determined by any one of the following
sets of elements that are associated with it:

1. Five of its points.

2. Four of its points and the tangent at one of these points.

3. TJiree of its points and the tangents at two of these points.

4. TJiree of its tangents and the points of contact on two of
these tangents.

5. Four of its tangents and the point of contact on one
of these tangents.

6. Five of its tangents.

Proof. We shall deal with the cases in the above order.
1. A conic is determined by five of its points.

Let i^, i^, ^, i^, i^ be five points of a plane, no four of
them being coUinear. Join jP and i^ to i^, ^, P^.

The triads of lines P^P^, P^P^, P^P^ and ^i^, ^i^, P^P^
determine a projectivity between the flat pencils whose
bases are P^, ^, and hence they determine a conic through
the five points. Any conic through these points could be
generated from the projectivity determined by the same
triads (§ 104) and would be the same as the one mentioned.

If three of the five points are situated on a line I, the other two
points should be taken as the bases of the pencils. In this case
the conic consists of the line I and the line through the bases.

118 CONICS

2. A conic is determined hy four of its points and the
tangent to it at one of these points.

Let ^, J^, I^, II be four points of a plane, no three of
them being collinear, and in the plane let t^ be any line
which passes through i^ but which does not pass through
any other of the four given points.

Draw from 7J the Imes I^, I^I^, and draw from J^ the
lines P^P,, P^P^, P^P,.

Consider ^ and ^ as bases of flat pencils. The lines t^,
J^P^ ; ^^, I^P^ ; J^^, ^^ being taken as corresponding, one
and only one projectivity is thereby established between
the flat pencils. This projectivity determines one conic
passing through the points ij, ^, ^, P^ and having the
line t^ as its tangent at I{.

No other conic can fulfill these conditions, since in that
case the conic would also be generated from the projectivity
just mentioned, and hence this conic would coincide with
the first one.

Therefore the second statement is proved.

If three of the four points other than the one at which the tangent
is given are on a line I, the conic consists of the given tangent and
the line I. If Pj and two only of the other points are collinear, no
conic is determined.

A CONIC DETERMINED • 119

3. A conic is determined hy three of its points and the
tangents to it at tivo of these points.

Let ^, ^, P^ be three noncollinear points of a plane,
and in the plane let t^ and t^ be lines which pass through
j^ and -^ respectively but through no other of the three
given points. Draw the lines i^^, ^^, and I^P^.

The triads t^, I^, I^P^ and -^^, t^, I^P^ determine a pro-
jectivity between the pencils whose bases are I[ and ^,
and this projectivity determines a conic passing through
^, ^, P^ and tangent at ^ and ^ to t^ and t^ respectively.

As in the other cases, it may be shown that there is
only one such conic. Hence the third statement is proved.

If the three points are on a line I and if one of the given tangents
is I, the conic consists of I and the other tangent. If neither of the
tangents coincides with I or both tangents coincide with I, the conic
may be thought of as the line I taken twice.

4. A conic is determined hy three of its tangents and the
points of contact of two of these tangents.

Since Nos. 3 and 4 are dual in the plane, the proof of
No. 4 follows at once.

5. A conic is determined by four of its tangents and the
point of contact of one of these tangents.

Since Nos. 2 and 5 are dual in the plane, the proof of
No. 5 follows at once.

6. A conic is determined hy five of its tangents.

Since Nos. 1 and 6 are dual in the plane, the proof of
No. 6 follows at once.

120 . CONICS

118. Construction of Conies. The statements of § 117
establish the existence of conies that fulfill certain condi-
tions, and suggest but do not solve the problem of con-
structing the conies under these various conditions. The
solution of this problem can be based upon the notion
of projectivity involved in § 117, but it can also be based
upon two very celebrated theorems which will be considered
on page 121. After these theorems have been proved, the
problems of the construction of conies will be treated from
both points of view.

Before considering these two theorems, however, it will
be found necessary to make some extension of the common
notion of a hexagon with which the student is familiar from
elementary geometry.

119. Hexagon. If any six coplanar points are taken in
a given order, the figure formed by the lines through all
pairs of successive points, as well as through the first and
last points, is called a hexagon.

As in the ordinary case of the hexagon, the first and

fourth, the second and fifth, and the third and sixth sides

are called opposite sides.

Thus, in the above figures the pairs of opposite sides are P-yP^,
p p . p p p p . p p p p

In each of the above figures the diagonals from P^ are PiP^,
P,P^, and P^P,.

A similar generalization applies to each of the other polygons. It
thus ajjpears that opix)site sides of a quadrilateral may intersect and
that a diagonal may lie wholly outside a polygon.

THEOREMS OF PASCAL AND BEIANCHON 121
Theorem. Pascal's Theorem

120. If a hexagon is inscribed in a conic, the three
intersections of the three pairs of opposite sides are collinear.

Proof. This is the theorem of § 103 with the restriction
upon i^, ^ removed by Steiner's theorem (§ 104).

Unless a cross hexagon is taken, the figure is usually very large.
The proposition is due to Blaise Pascal (1623-1662).

Theorem. Brianchon's Theorem

121. If a hexagon is circumscribed about a conic, the three
lines joining the three pairs of opposite vertices are concurrent.

Proof. This is simply a generalization of § 105.

The student should write out the proof of this theorem.

The proposition is due to Charles Julien Brianchou (1785-1864).

122 CONICS

122. Pascal Line. The line containing the points of inter-
section of the three pairs of opposite sides of a hexagon
in a conic is called the Pascal line of the hexagon.

123. Brianchon Point. The point of concurrence of the
three lines joining the opposite vertices of a hexagon about
a conic is called the Brianchon point of the hexagon.

124. Converses of the Theorems of Pascal and Brianchon.
The converses of the theorems of Pascal and Brianchon
can be established as in the exercise below, and each of
them may then be given a different interpretation. Thus,
if six coplanar points are chosen and joined to form a hexa-
gon, a conic passes through any five of them. Does it pass
through the sixth point ? It does if and only if the three
points of intersection of the pairs of opposite sides are
collinear. Hence Pascal's theorem and its converse imply
the necessary and sufficient conditions for the passing of
a conic through six given coplanar points. Brian chon's
theorem can be interpreted in a corresponding fashion.

Exercise 26. Theorems of Pascal and Brianchon

1. State and prove the converse of Pascal's theorem.

2. If two pairs of opposite sides of a hexagon inscribed in
a conic are parallel, the other two opposite sides are parallel.

3. A hexagon is to be inscribed in a conic in such a way that

a given line shall be its Pascal line. Determine the maximum
number of sides of the hexagon that may be given, and solve
the problem.

4. Solve Ex. 3 for the ease when the given line is at infinity.

5. State and prove the converse of Brianchon's theorem,

6. Circumscribe a hexagon about a given conic in such a way
that a given point shall be its Brianchon point, as many of the
vertices of the hexagon as possible being given in advance.

THEOREMS OF PASCAL AND BRIANCHON 123

125. Limiting Cases of the Theorems of Pascal and Brian-
chon. There are several limiting cases of the theorems
of Pascal and Brianchon which have useful applications
and which require mention at this point. They arise out of
approach to coincidence of vertices of an inscribed hexagon
of a conic and also of sides of a circumscribed hexagon.

Let im^P^P^P^ be a hexagon mscribed in a conic. If ^
approaches I{ along the conic, the line I^ approaches the
tangent t^ at the point ^, and the hexagon approaches the
figure composed of the pentagon P^I^P^P^I^ and the tangent t^
to the conic at I^. The pairs of opposite sides are ^j, P^P ;
I[P^, I^Pq ; -^J^, J^I^. These pairs determine collinear points.

Similarly, ^ may approach 7^ and either j^ approach ^
or J^ approach ^, yielding an inscribed quadrilateral and
tangents to the conic at two of the vertices of the quad-
rilateral. A third case is that in which ^ approaches ^,
^ approaches ^, and ^ approaches I^.

In each case the propriety of extending Pascal's theorem, and
others, to limiting cases in which two distinct elements are allowed
to become coincident is left for the student's consideration.

In the case of a circumscribed hexagon, if one side
approaches coincidence with a second, their point of in-
tersection approaches a limiting position at the point
of contact of the second side. There arise out of the
approach of sides to coincidence a number of limiting
cases of Brianchon's theorem which can be worked out
and which will be needed from time to time.

Other limiting cases of these propositions are those in
which the points or lines of the figures are not all in the
finite part of the plane. For example, one or two vertices
of the Pascal hexagon and one side of the Brianchon
hexagon may be at infinity. Coincident elements and
infinitely distant elements may be present in one hexagon.

124 CONICS

Problem, conic through Five Points

126. Griven five points in a plane, no four of them being
collinear, construct the conic which is determined by them.

Solution. This problem admits of two simple solutions.

1. Method based on a projectivity.

Let the given points be I{, J^, ^, ^, ^, Then in any
chosen direction from any pomt, as J^, there can be found
another point of the conic which is not collinear with two
of the others. Let the chosen direction be along the line jOj,
and let the point to be found be called P. Draw i^^,
F,P,, F,P,, F^P^, P^P,, P^P,. The triads of lines P,P„ P,P„ P,P,
and P^J^, J^Jl, I^P^ determine the projectivity between two
flat pencils which generate the required conic.

The point P is the intersection of p^ and its correspond-
ing line of the pencil whose base is ^ ; and it may be found
by the method used in § 39, Case 1. Draw this line and
produce it to meet j9j, thus determining P.

By varying the position of the line p^ any number of
additional points of the conic may be found.

Evidently it is not feasible to obtain all the points of the conic
by this method, nor is the method convenient in practice. In this
respect it is similar to the method of plotting in analytic geometry.

CONIC THROUGH FIVE POINTS 125

2. Method based on PascaVs theorem.

Let the given points be ^, ^, ^, ^, ^. As before, an-
other point P can be found on a chosen line p^ that passes
tlirough any one of the points, as J^.

The given points i^, ^, ^, ^, ^ and the point P, which
is to be found, are the vertices of a hexagon inscribed in
the conic determined by the five given points.

Then ^^, ^^; ^^, P^P; P^P^, PJ^ intersect on the Pascal
line of this hexagon. Of these six lines, I^P is not given
and PI^ is the given line py The Pascal line is determined
by the intersection of i^^, ^^ and that of ^^, p^

Draw the Pascal line and let it meet P^I^ in Q^. Draw
()2^. This line Q^P, must coincide with the line J^P and
must intersect the line p^ in the required point P.

Since every line through ij determines a point on the
conic, it is possible to locate any number of points.

This method, like the first one, bears a certain resemblance to
the method of plotting in analytic geometry. From the point of
view of convenience it is decidedly superior to the first method.

The student will observe that, since the conic is of the
second order, the line p^ cuts it in one and only one point
other than ^, and also that in either solution of the prob-
lem the use of the ruler alone is sufficient.

126 CONICS

Problem. Four Pornxs and a Tangent

127. G-iven four points in a plane, no three of them col-
linear, and a line passing through one and only one of these
points, construct the conic which passes through the given
points and at one of them is tangent to the given line.

Solution. As was the case in § 126, there are two simple
methods of construction. In each method any number of
additional points of the conic may be found by determining
where the conic would be cut by lines which pass through
one of the points.

1. Method based on a projectivity.

Let the given points be J^, I^, P^, P^, and let the given
line be t^ passing through the point -^. Draw any line p^
through the point !{. Join i^ to each of the points P^, P^, -^,
and join P^ to each of the points P^, I^.

The triads ^^, ^^, P^P^ and t^, P^P^, P^P^ determine a
projectivity between the flat pencils whose bases are I{, J^,
and the required conic is the locus of the intersections of
corresponding lines of the projective pencils.

The line through J^ which corresponds to p^ of the pencil
whose base is J^ can be determined by the method used in
§ 39, Case 1, and P, the intersection of this line with jOj, is
the point required.

By varying the position of p^ any number of points of
the conic may be found.

FOUR POINTS AND A TANGENT

2. Method based on PascaVs theorem.

127

Let P be the point on p^ that is to be found. It is
determined if the direction of P.P can be determined.

The pentagon I^P^Pr^P is inscribed in the required
conic, and the line t^ is tangent to the conic at J^.

The intersections of ^^, ^^; t^, P^P ; and ^i^, PP^
(or jt?j) are on the Pascal line.

Produce P^I^ and P^Pr^ to meet at Q^, and produce P^P^
and p^ to meet in ^3. Draw the Pascal line. Let t^ meet
this line in Q^ ; join ^ and Q^.

Then the lines I^Q^ ^^nd i^P are coincident and the
intersection of P,Q^ and jPj is the required point P.

Problem. Three points and Two Tangents

128. Given in a plane three noncolUnear points and two
lines, each of which passes through one and only one of the
given points, construct the conic which passes through the three
given points and at each of two of them is tangent to the
given line through that point.

The solution is left for the student. It should be effected by two
methods, as in the two preceding theorems. As in the other prob-
lems the second method is to be preferred for practical reasons.

An appreciation of the superior convenience of the second method
is best secured by making the actual construction necessary for find-
ing by the first method the line through P^ which corresponds to p^
of the first pencil.

128 CONICS

Problem. Three Tangents and Two Points

129. Given three nonconcurrent lines in a plane, and on
each of two of these lines a point which is not on any other
of the three, construct the conic which is tangent to each of
the given lines and has each of the given points as the point
of contact of the given line on which it lies.

Of what problem is this the dual? The solution is left for
the student.

Problem. Four Tangents and One Point

130. Given four lines in a plane, no three of them con-
current, and a point on otie but not on two of them, construct
the conic which is tangent to each of these lines and has the
given point as the poitit of contact of the given line on which
it lies.

Of what problem is this the dual? The solution is left for
the student.

Problem. Five Tangents

131. Given five lines in a plane, no four of them con-
current, construct the conic which is tangent to each.

Of what problem is this the dual? The solution is left for
the student.

Problem, constructing a Tangent

132. Given five or more points of a conic, construct the
tangent to the conic at any one of these points.

The solution is left for the student.

Problem. Finding a Point of contact

133. Given five or more tangents to a conic, determine by
construction the poifit of contact of any one of these tangents.

The solution is left for the student.

PROBLEMS OF CONSTRUCTION 129

Exercise 27. Problems of Construction

1. If two projective flat pencils generate a circle, they are
congruent.

2. Using the result in Ex. 1, find any number of additional
points of a circle when three of its points are given.

3. Find any number of points of a circle when two of its
points and the tangent at one of them are given.

4. Solve the problem in § 126 when one of the five points
is at infinity in a given direction.

5. Solve the problem in § 126 when two of the points are
at infinity in given directions.

6. Solve the problem in § 127 when the given line is at
infinity.

7. Solve the problem in § 127 when one of the four, points
is at infinity in a given direction.

8. Solve the problem in § 128 when one of the points is
at infinity in a given direction and the tangent at that point
is given to be the line at infinity.

9. Solve the problem in § 129 when the two given points
are at infinity.

10. Solve the problem in § 131 when one of the five given
lines is the line at infinity.

11. Solve the problem in § 132 when the point at which the
tangent is to be constructed is at infinity in a given direction.

12. Solve the problem in § 133 when the given tangent
whose point of contact is to be found is the line at infinity.

y~ 13. If a parallelogram is inscribed in a conic, the tangents
to the conic at the vertices form a parallelogram circumscribed
about the conic.

•^14. If P^, P^, Pg, P^, P5 are fixed points and P moves on the
conic determined by them, find the envelope of the Pascal line
of the hexagon P^I^P^P^PrP-

130 CONICS

Theorem, involution on complete Quadrangle

134. If a straight line cuts all the sides of a complete
quadrangle but does not pass through any vertex, it cuts the
three pairs of opposite sides of the quadrangle in conjugate
points of an involution.

Proof. Let a straight line p cut the pairs of opposite
sides of the complete quadrangle whose vertices are I{, I^,
^, i^ in A, A'; B, B'; C, C; and let il/, iV, 0 be the diagonal
points of the quadrangle.

Then range ^^^'C^ flat pencil J^(MF^A'F^^

-^ range MI^A'Pj^
-^ flat pencil JJ (MP^A'F^)
■^ range AC'A'B'.

But range ^5^'C^ range A'CAB. § 23

Hence range AC'A'B' -^ range A'CAB.

Accordingly, A, A'; B, B'; C, C are conjugate points of
an involution on p. § 65

This theorem is auxiliary to, and is in fact a special case of, an
interesting and important theorem which was first established by
the French geometer Girard Desargues (1593-1662).

Desargues's theorem offers another line of apjiroach to some of
the preceding constructions and to other similar problems. In
particular, on page 133, it is applied to the solution of § 126.

DESARGUES'S THEOREM
Theorem. Desargues's Theorem

131

135. If a complete quadrangle is inscribed in a conic, and
if a straight line cuts the conic in ttvo points distinct from
each other and from the vertices of the quadrangle, these two
points form a conjugate pair of the involution of points on
the line, which is determined hy the intersections of the line
with the pairs of opposite sides of the quadrangle.

Proof. Let the complete quadrangle P^I^I^Il be inscribed
in a conic, and let a line p which does not pass through
any of the four vertices cut the conic in P, P' and the
pairs of opposite sides m. A, A'; B, B' ; C, C'.

Then range P^P'^- flat pencil i^(P^P'^)

-flat pencil ^ (P^P'i^) - range PA'P'B'.
But range P^'P'^' -^ range P'B'PA'. § 23

Therefore range PPP'^ - range P'P'PJ'.

Hence P, P'; A, A'; B, B' are conjugate points of an
involution on p determined by the pairs A, A' ; B, B'.

The involution formed by the intersections of p with the
pairs of opposite sides of the quadrangle I^I^P^P^ is also
determined by the pairs of points A, A'; B, B'.

Accordingly, P, P' are conjugate points of the involution
of points determined by the intersections of the line p with
the pairs of opposite sides of the quadrangle P^P^P^P^.

132 CONICS

136. Restatement of Desargues's Theorem. It should be
noted that many conies pass through the four points ij, i^,
i^, i^ and that to each of such conies Desargues's theorem
apphes. Moreover, it should be remembered that the pairs
of lines 7^-^, ^/^; I^F^, I^P^ are degenerate conies through
the four pomts. Hence Desargues's theorem is capable of
restatement as follows :

The infinitely many conies, including pairs of lines, wJiich
pass through four given coplanar points, no three of which are
collinear, determine on any line which intersects them (hut does
not pass through any one of the points^ infinitely many pairs
of points of an involution.

137. Corollary. If the involution determined by the
conies is hyperbolic, two of the conies which pass through the
four points touch the straight line; if it is elliptic, no conic
through the four points is tangent to the straight line.

Exercise 28. Application of Desargues's Theorem

1. What sort of involution is determined upon a side of the
diagonal triangle of the quadrangle mentioned in the theorem
of § 135 ?

2. Test the validity of the proof of Desargues's theorem when
it "is applied to a line through one of the given points, say P^.

Given four points in a plane, no three of which are col-
linear, show how to draw a straight line subject to each of the
following conditions :

3. There shall be two conies passing through the four points
and tangent to the line.

4. There shall be one conic passing through the four points
and tangent to the line.

5. There shall be no conic such as described in Ex. 4.

DESAKGUES'S THEOREM 133

Problem. Conic through Five Points

138. G-iven five points, no four of which are coUinear, con-
struct the conic ivhich is determined hy them.

Solution. Let us consider two cases.

1. No three of the five given jyoinis are coUinear.

Let the five points be i^, ^, i*, i^, 7^. Draw JJi^, ^i^,
i^i^, P^J^, F^F^, J^I^, and any transversal p^ through F^.

It is now required to find the point F in which this hue
again cuts the conic determined by the five given points.

The points Ap A^, B^ B^, in which p.^ is cut by the hues
J^/^, i^i^, F^F^, B^I^, determine an invohition in which ij
and the required point F are corresponding points.

Hence the point F can be determined from the pro-
jectivity between the ranges in the involution. For in-
stance, the three known points A^ B^ A^ and the required
point F are projective with the four known points A^, B^,
Jp Fy Various special devices for finding F based upon
the method of § 39, Cases 1 and 2, can be found.

By varying the position of the line p^ any number of
points on the conic can be found.

2. Three of the five given points are coUinear.

In this case the required conic is a pair of straight lines,
one the hue through the three points and the other the line
through the other two pomts.

134 CONICS

Problem. Position of Self-corresponding Elements

139. Given two superposed projective one-dimensional prime
fortnSy construct the position of the self-corresponding elements.

Solution. If the superposed prime forms are not ranges,
it is possible by operations of projection and of section to
obtain from them two superposed projective ranges. Hence
it is necessary to solve the problem only for the case in
which the prime forms are ranges.

Let A^B^C^ and A^B^C^ be two triads of corresponding
points of superposed projective ranges on a base p.

Describe any circle coplanar with the line p. Join any
point F of the circle to each of the six given points, and let
these lines cut the circle again in A[, B[, Cj and A^-, B^-, Cg.
Join A[ to A'^, B'^, C^, and A'^ to B[, C{.

Then flat pencil A'^ (A[B[C[ - • •)

- flat pencil P(A[B[C[ • . .) § 53
-range A^B^C^ . --

-^ range ^2^2 ^2 * * *

- flat pencil P (A'^B'^C;^ . . .)

- flat pencil A[ {A'^B'^C^ • • •)•

But the flat pencils A'^ iA[B[C[ - • •), A iAK^^'i ' * 0 ^ave
a self -corresponding element, and hence are perspective.

CONSTRUCTIONS OF THE SECOND ORDER 135

Let X' be a point, if there be any, in which p\ the axis of
perspectivity, cuts the circle, and let PX' meet p at X.

In the four flat pencils previousl)'- mentioned the corre-
sponding lines are A\X', PX', A'^X', PX', and hence in the
superposed pencils whose bases are at P, PX' is a self-
corresponding line. Therefore X is a self-corresponding
point of the ranges on p.

Conversely, it is true that, corresponding to each self-
corresponding point of the ranges on the line p, there is an
intersection of the line p' with the circle.

Hence, to find the self -corresponding points on p we join
P to the intersections of p' and the circle, and produce
these lines to intersect p. There may be no, one, or two
intersections with p, and each of these intersections is a
self-corresponding point.

140. Constructions of the Second Order. All constructions
made before § 139 were effected wholly by the use of
straight lines, and at every stage the results were uniquely
determinate ; that is, all the problems had one and only
one solution. If the solutions of the problems analogous
to these constructions are effected by the methods of
analytic geometry, it is found that only equations of the
first degree are used. For this reason these and similar
problems are said to be of the first order.

Beginning with § 141, attention will be given to prob-
lems whose solutions by the method of analytic geometry
would involve the use of at least one equation of the second
degree, as in § 139. Correspondingly, each construction will
require the use (at least once) of a curve of tlie second
order, and for simplicity the circle will be taken.

The problem in § 139 furnishes a basis for others, and hence has
been deferred from its most natural jilace, which was in connection
with the treatment of superposed projective forms in Chapter VII.

136 CONICS

Problem, intersections of a Line and a Conic

141. Given any of the sets of elements mentioned in §117
as determining a conic, construct the intersections (if there are
any) of the conic with a given straight line p in its plane.

Solution. If the set of elements is not five points, by
means of §§ 127-133 find five points P^, Pg, i^, i^, P^ on the
conic. Join any two of the points, as ij and j^, to F^, P^, i^,
and let these lines meet the line p in P^, P^, P^ and i^", i^", Jp".

These triads determine superposed projective ranges on jj.
By the method of § 139 find the self -corresponding points
(if there are any) of these ranges.

Since these self-corresponding pomts are common to
corresponding lines of the projective flat pencils whose
bases are ij and 1^, they are on the conic. Moreover, they
are the only points of p which are on the conic. There
may, therefore, be two, one, or no intersections.

Problem. Tangents from a Point

142. Given any of the sets of elements mentioned in § 117
as determining a conic, construct the tangents (if there are any)
to the conic from a given point P in its plane.

Solution. If the set of given elements is not five tangents
to the conic, by means of §§ 127-133 find five tangents
^1' ^2' ^3' ^4' ^5 ^® ^'^® conic. The tangents ^3, t^, t^ cut t^, t^
in triads of points which, being joined to P, determine a
projectivity between flat pencils whose base is P.

Find the self-corresponding lines of these pencils. Any
such line passes through P and also joins corresponding
points of the projective ranges on t^, t^ which serve to
generate the conic. Hence the line is tangent to the conic.
The number of these lines is two, one, or none.

PROBLEMS OF CONSTRUCTION
Problem. Four Points and a Tangent

137

143. Construct a conic which shall pass through four given
points, no three of which are collinear, and shall be tangent to
a given Jine that does not contain any of the points.

Solution. Let the given points be ij, i^, i^, i^, and let
the given line be ty Let t^ cut the lines J^i^, F^P^ in A^ A^
and cut the lines P^P^, i^-?^ in By, B^.

Find the self-corresponding points of the involution on t^
which is determined by these pairs of pomts. Through the
four points and any self-corresponding point i^ construct
a conic (§ 138). This conic is tangent to fj at i^. For if
it cuts t^ in a second point i^, then the point i^ is not
self-corresponding, and this is contrary to fact.

Hence, for every self -corresponding point of the involu-
tion on t^ one conic can be constructed.

There may be no, one, or two self-corresponding points
(§ 139). Hence no, one, or two conies may be constructed.

Theorem. Four Points and a Tangent

144. The number of conies which pass through four given
points, no three of ivhich are collinear, and are tangent to
a given straight line which does not pass through any of the
points, is none, 'one, or two.

The proof is left for the student.

138 CONICS

Problem. Four Tangents and a Point

145. Construct a conic ivhich shall be tangent to each
of four given straight lines, no three of which are concur-
rent, and which shall pass through a given point exterior to
the lines.

This problem is the dual of § 143 and may be solved as such.
The solution is left for the student. Likewise, a theorem dual to
§ 144 results from the proof of the construction.

146. Special Case of Desargues's Theorem. To complete
a set of constructions which include §§ 126, 129-133, 143,
and 145, two others are necessary, and these are given in
§§ 148 and 149. In order to solve these two problems special
cases of Desargues's theorem (and its dual) may be used.

Instead of the four distinct pomts of the conic con-
sidered in Desargues's theorem, let the first and second
points move up to coincidence, and also let the third and
fourth points move up to coincidence. Then the line join-
ing the first and second points and that joining the third
and fourth points become tangents to the conic. Also the
lines joining the first and third points, the second and
fourth points, the first and fourth points, and the second
and third points move into coincidence upon the chord of
contact of the two tangents mentioned.

Theorem. Special Form of Desargues's Theorem

147. Two straight lines and the conies which are tangent
to them at two given points intersect a given line that does
not pass through either of these points in pairs of points of an
involution, one of the self-corresponding points of which is the
intersection of the given line with the chord of contact.

The proof is left for the student.

PROBLEMS OF CONSTRUCTION 139

Problem. Three Points and Two Tangents

148. Construct a conic which shall pass through each of
three given noncollinear points and he tangent to each of two
given lines that do not pass through any of the points.

Solution. Let the points be i^, ^, i^, and let the lines
be ^1, ^2- Let t^, t^ cut the line P^P^ in A^, A^ and the line
P^P^ in i?j, i>2. We shall first find the points of contact of
<i, <2 with the conic.

Find the self-corresponding points (if there are any) of
the involutions determined by i^, P^ and Jj, A^ and by
^, P^ and B^, B^ respectively. Let the line through JT/^,
one of the first of these, and M^, any one of the second, cut
^2 in P^ and t^ in P^. Pass a conic through i^, and tangent
to t^ and ^2 at P^ and P^ respectively (§ 128).

Since the point corresponding to P^ in the involution
on A^A^ is completely determined by Mj^ and the pair of
points Jj, ^2' it follows that this conic must pass through
i^ (§ 147), for the involution determined on P^I^ by conies
tangent at ^ and P^ to ^g ^^^ h ^^ ^^^^ determined by
a self-corresponding point and the pair of points A^, A^.
Similarly, this conic can be shown to pass through i^.

Hence, for every possible pair of points ^, i^ one conic
may be constructed.

There may be no, one, two, or four pairs of points, as
i^, i^ (§ 144), and for each a conic may be constructed.

140 CONICS

Theorem. Three Points and Two Tangents

149. The number of conies which pass through three given
noncollinear points and which are tangent to two given lines
that do not contain any of the points is none, one, two, or four,
as the case may he.

The student should write out the proof, which is essehtially that
of § 148.

PROBLEM. Three Tangents and Two Points

150. Construct a conic which shall he tangent to three given
nonconcurrent lines and shall pass through two given points
which are exterior to the lines.

The student should write out the solution, wliich is simply the
dual of that of § 148.

Exercise 29. Review

* 1. If the sides of an angle of constant size rotating about
a fixed vertex intersect respectively two fixed lines, the line
joining these intersections envelops a conic. '■(r-_ i>i^*lz/*i

2. Two vertices of a variable triangle move along two fixed
lines, and the three sides respectively pass through three fixed
collinear points. Eind the locus of the third vertex.

3. Consider Ex. 2 for the case in which the three fixed
points are not collinear.

4. If two triangles are in plane homology, the intersections
of the sides of one triangle with the noncorresponding sides
of the other lie on a conic.

5. State Pascal's theorem for the case in which the first
and second, the third and fourth, and the fifth and sixth
vertices have become coincident.

6. The complete quadrilateral formed by four tangents to
a conic, and the complete quadrangle formed by their four
points of contact, have the same diagonal triangle.

EEVIEW EXERCISES 141

7. If a variable quadrangle P^P^P^P^ inscribed in a conic
has as fixed points P^, P^, and the intersection of P^P^, P^P^, the
other vertices of its diagonal triangle move along the same
fixed straight line.

8. If Pg, Pg are fixed points on a given conic, and if P is
a moving point, as P moves along the conic the Pascal line
of the hexagon, consisting of the triangle PPgPg and the
tangents to the conic at the points P^, P^, Pg, envelops a conic.

9. If Pj, Pg, P^ are fixed vertices of a complete quadri-
lateral whose fourth vertex P moves along a given conic
through Pj, Pg, P_j, all the vertices of the diagonal triangle trace
straight lines and all the sides pass through fixed points.

10. If the points P^ and P^ trace superposed projective
ranges on the base AB of a fixed triangle ABC, if Pj is a fixed
point not on any side of the triangle, if P^P^ meets A C in P^,
and if P^Pg meets BC in Pg, find the locus of P, the inter-
section of AP, and JSP..

5 4

11. In Ex. 10 find the envelope of P^P^

12. State Desargues's theorem for the case in which a pair
of the four given coplanar points become coincident.

13. State Desargues's theorem for the case in which two
pairs of the given coplanar points become coincident.

14. Three sides, AB, AD, CD respectively, of a variable
quadrangle inscribed in a given conic pass through three given
points of a line. Find the envelope of BC.

15. Extend Ex. 14 to the case of a simple inscribed polygon
having 2 n sides.

16. From the data of § 127 construct, by means of § 147,
tangents at additional points of the conic.

17. Prove the dual of § 149, namely, that the number of
conies which can be constructed under the conditions of § 150
is none, one, two, or four.

18. Solve the dual of § 139 for the plane.

142 CONICS

19. If the lines 2^1 and p^ are drawn through the vertices P^
and Pj respectively of a given quadrangle, the conies which
pass through the vertices of the quadrangle determine perspec-
tive ranges on p^ and j-j^.

20. If the lines p^ and p^ are drawn through the vertex P^
of a given quadrangle, the system of conies which pass through
the vertices of the quadrangle determine projective ranges
on p^ and p^.

21. State and prove the dual of Ex. 19 for the plane.

22. Construct a conic which shall pass through two given
points Pj and P^, shall be tangent to a given line t^ at the
point Pg, and shall be tangent to a second given line t^.

Apply Ex. 12 for the line t^. Find the self-corresponding points of
the involution.

23. Construct a conic which shall be tangent to a given
line t^ at the point P^, to t^ at P^, and to t^.

24. Consider the problem of §141 for the case in which the
given line is the line at infinity.

25. Consider the problem of §143 for the case in which the
given line is the line at infinity.

26. Consider the problem of § 148 for the case in which one
of the given lines is the line at infinity.

27. Solve the dual of Ex. 22 for the plane.

28. Construct a triangle which shall be inscribed in a given
triangle and have its sides pass through three given points.

Observe that if a triangle has two vertices, as required, but not the
third, the sides through the latter cut a side of the given triangle in
corresponding points of superposed projective ranges.

29. Construct a triangle which shall be inscribed in a given
conic and have its sides pass through three given points.

30. If a conic can be described through the six vertices of
two given triangles, another conic can be described which shall
be tangent to the six sides of the two given triangles.

CHAPTER XI

CONICS AND THE ELEMENTS AT INFINITY

151. Classification of Conies. In the discussion of conies
in the preceding chapter no classification was made, nor
was any account taken of the fact that on certain occasions
the term straight line may mean " straight line at infinity "
and the term point may mean " point at infinity." These
considerations can be associated very advantageously.

Ellipse Pakabola Hyperbola

In projective geometry, conies are classified by means of
their relations to t^Q line at infinity. This line, like any
other, may intersect a conic in no, one, or two points, and
hence conies are divided into three classes as follows :

1. Ellipses, or conies that do not intersect the line at
infinity.

2. Parabolas, or conies that intersect the line at infinity
in one point (or are tangent to the line at infinity).

3. Hyperbolas, or conies that intersect the line at infinity
in two distinct points.

While these conies are familiar to the student from his work in
analytic geometry, the study of conies will now be considered from
a different point of view.

143

144 CONICS AND THE ELEMENTS AT INFINITY

152. Elements at Infinity. In the interpretation of the
results already obtamed, in so far as the ellipse is con-
cerned, it will be seen that the expressions point on the curve
and tangent to the curve always mean a point and a line in
the finite part of the plane.

On the other hand, in connection with the parabola, one
and only one tangent, and one and only one point of the
curve (the point of contact of that tangent), may be taken
to be at infinity.

In the case of the hyperbola there are two points on the
curve which are at infinity, but the line at infinity is not
a tangent. At each of the infinitely distant points of the
curve there is, however, a tangent which has no infinitely
distant point except its point of contact.

It follows that in the cases of the parabola and hyperbola
the interpretations of the theorems of Pascal and Brianchon
and of similar theorems obtained by the methods of pro-
jective geometry vary according as all or only part of the
elements are assumed to be in the finite part of the plane.

In the light of the procedure indicated, the results which have
been obtained are capable of restatements which vary for the three
types of conies, but which have a great interest, because they bring
these results into clearer relation to those obtained by the methods
of analytic geometry.

153. Asymptote. A line, not the line at infinity, which
is tangent to a conic at an infinitely distant point is called
an asymptote.

In this figure a is an asymptote.

Every hyperbola has, then, two asymp-
totes, and the other conies have none,
though sometimes the parabola is said
to have the line at infinity as an asymptote. This latter form of
statement is convenient when geometry is treated algebraically,
but it will not be adopted in this text.

ELEMENTS AT INFmiTY 145

154. Special Interpretations. As indicated in § 152, each

of the results that have been derived for the conies should
be examined for interpretations based upon the rela-
tions of the elements at infinity to the three types of
conies. The great variety of results that can be obtained
prevents a systematic and detailed reexamination in this
place of all the theorems and constructions that have
been derived. A few of these will be obtained, but for the
most part their derivation must be left to the student, a
work which will prove both interesting and profitable.

Of the elements (points and lines) which determine a
conic not more than two points and not more than one
line may be at infinity, except in the limiting case of
coincident points or coincident tangents. The existence
of one infinitely distant point on a conic determines that
the curve is not an ellipse, and the existence of two such
points determines that the curve is a hyperbola. Similarly,
the tangency of the line at infinity to the curve determines
it to be a parabola.

On the other hand, when all the given determining
elements are in the finite part of the plane, the conic may
prove to be of any one of the three types. The determi-
nation of the character of the conic of which certain ele-
ments are given is a particularly interesting case. It is the
problem of § 141 as modified in Ex. 24, page 142.

In view of what is said above, we shall now restate
the important theorem of § 116.

In each case the student should draw the figure and satisfy him-
self that the statement is correct and that it is a special case of one
of the corresponding statements in §§ 116 and 117. He should also
supplement the results here set forth by the others which can be
obtained if a thorough examination of the theorem is made for its
various interpretations.

146 CONICS AND THE ELEMENTS AT INFINITY

Theorem, a conic Determined

155. 1. There is one and only one conic (^parabola or
hyperbola) zvhich passes through four points in the finite
part of a plane and one infinitely distant point in a speci-
fied direction.

This follows from the theorem stated in § 116, No. 1.

2. Tliere is one and only one hyperbola which passes through
three points in the finite part of a plane and has given direc-
tions for its asymptotes.

This follows from the theorem stated in §116, No. 1.

3. There is one and only one parabola which passes through
three noncollinear points in the finite part of a plane and has
its infinitely distant point in a given direction.

This follows from the theorem stated in § 116, No. 2.

^ 4. There is one and only one hyperbola which passes through
any point in the finite part of the plane and has two given
straight lines as asymptotes.

This follows from the theorem stated in § 116, No. 3.

5. There is one and only one hyperbola which has two given
lines as asymptotes and is tangent to a third liiie which is not
parallel to either of the others.

This follows from the theorem stated in § 116, No. 4.

6. There is one and only one parabola ivhich is tangent to
each of three nonconcurrent lines lying in the finite part of the
plane and has its infinitely distant point in a given direction.

This follows from the theorem stated in § 116, No. 5.

4- 7. There is one and only one parabola which is tangent to
any four lines of a plane, no three of which are concurrent and
no two of which are parallel.

This follows from the theorem stated in § 116, No. 6.

PASCAL'S THEOREM 147

Theorem. Special Interpretation of Pascal's Theorem

156. The chords from a point on a hyperbola to each of
Uvo other points on the hyperbola intersect the lines through
these two points parallel to one of the asymptotes, the inter-
sections being collinear with the intersection of the tangents at
the two points.

The proof of this theorem is included in the proof given in § 157.

Theorem. Further interpretation of Pascal's Theorem

xl57. The chords from a point on a parabola to each of
two other points on the parabola intersect the lines from these
tivo points to the infinitely distant point of the curve, the inter-
sections being collinear with the intersection of the tangents
at the two points.

Proof. These two theorems are closely related, being
obtained by applying to such conies the case of Pascal's
theorem in which two pairs of vertices coincide. If a
conic is known to have one point at infinity, it may be
either a hyperbola or a parabola.

Consider a hexagon inscribed in a conic in such a way
that the first and second vertices coincide, the fourth and
fifth vertices coincide, and the sixth vertex is at infinity.
We may also assume that the curve is a hyperbola or that
it is a parabola. If it is a hyperbola the sides of the
hexagon which intersect at the infinitely distant point are
parallel to the same asymptote. In either case one pair
of opposite sides is a pair of tangents.

The two theorems considered above are merely state-
ments of Pascal's theorem for the two cases described,
the terms used being appropriate in connection with these
two kinds of conies.

148 CONICS AND THE ELEMENTS AT INFINITY

Theorem. Special interpretation of Brianchon's Theorem

158. Cfiven five tangents to a parahola, the line parallel
to the first tangent and concurrent with the third and fourth
tangents cuts the line parallel to the fifth tangent and concur-
rent with the second and third tangents on the line joining the
intersection of the first and second tangents to that joining the
fourth and fifth.

Proof. In Brianchon's theorem (§ 121), simply let one
of the tangents be the line at infinity, and the proof
follows at once.

Pascal's theorem and Brianchon's theorem have a large number
of special interpretations. Of these we have space for only the three
given in §§ 156-158. They have been selected not because of
their intrinsic importance but because they indicate the method
of procedure.

159. Special Constructions. On account of the elements
at infinity the problems which were considered in Chap-
ter X may also be given special statements for certain
cases. Thus a point at infinity may be specified by its
direction ; and since a hyperbola is determined by means
of any three of its points in the finite part of the plane
and its two points at infinity, it is determined by the three
points mentioned and the directions of the two points at
infinity. These latter directions are also the directions of
the asymptotes.

In actual constructions certain special situations arise.
Thus, drawing a line to a given infinitely distant point is
the same as drawing a line parallel to a given line. To
effect this with the ungraduated ruler it is necessary to
have additional data, as in the exercises on pages 99 and
100. Problems involving considerations of this sort will
be considered in §§ 160-162.

PROBLEMS OF CONSTRUCTION 149

Problem, construction of the Hyperbola

160. Given in a plane three noncoUinear points and tivo
pairs of parallel lines, each pair having a direction different
from those of the lines joining the three poitits and also
different from that of the other pair, construct a hyperbola
through the three given points and having asymptotes parallel
to the two given pairs of lines.

The student should write out the solution, making appropriate
modifications of the methods employed in §§ 126 and 138.

Problem. Determination of a Conic

161. Given a set of elements sufficient for the determination
of a conic, determine the nature of the conic and the directions
of its infinitely distant points (if there are any^.

Among the constructions of the second order the construction in
§ 141 deserves attention in this connection, and this problem is one
of its special forms. If elements sufficient for the determination of
the conic are given, the finding of the intersections of the conic
with the line at infinity includes determining whether the conic is
an ellipse, a parabola, or a hyperbola.

As in the original case, if five points of the conic are not given
they may be found by construction. Let them be P^, P^, P^, P^, Py
The triads of lines P^P^, PJ\, P^P^ and P^P^, P^P^, P^P^ deter-
mine the projectivity by means of which any number of additional
points of the conic may be found.

A difficulty now arises in following the original construction,
because the line p is at infinity. The triads of points of the super-
posed projective ranges on this line that are determined by the triads
of the flat pencil are now not available from the point of view of
construction by the ruler. If, however, the possibility of drawing
lines parallel to all given lines is assumed, the resulting diflSculty
disappears. For the purpose of drawing the necessary parallels, the
compasses must be used more freely than in the construction in
§ 141. With this difference the construction follows as before, and
the student should write out the solution in full.

160 CONICS AND THE ELEMENTS AT INFINITY

Problem. Construction of the Parabola

162. Given four points in a plane, no three of them col-
inear, construct the parabola which passes through these points.

By § 144 the number of such parabolas is none, one, or two.
The solution of this problem, which is based on that of § 143, is
left for the student.

Exercise 30. Elements at Infinity

As suggested on pages 144-148, investigate with respect to
the elements at infinity the following cases already considered :

§120.

§128.

Page 142, Ex. 19.

§121.

§149.

10.

Page 142, Ex. 22.

§126.

Page 141, Ex. 10.

11.

Page 142, Ex. 28.

§127.

§147.

12.

Page 142, Ex. 29.

In Exs. 1-12 practice in special interpretation, not the finding of im-
portant results, is the object.

13. Using tlie compasses only once arid the ruler, find five
points in the finite part of the plane which shall determine an
ellipse other than a circle.

14. Consider Ex. 13 for the case of a parabola.

15. Consider Ex. 13 for the case of a hyperbola.

16. In the finite part of the plane find four points through
which no parabola passes.

17. In the finite part of the plane find four points through
which two parabolas pass.

18. Find four points through which both a circle and a
parabola pass.

19. On a straight line p passing through a given point Pj,
find a point P such that through P^, P and two other given
points P^y Pg there pass (1) two parabolas ; (2) one parabola ;
(3) no parabola. In each case indicate all possible positions of P.

CHAPTER XII

POLES AND POLARS OF COXICS

163. Polar of a Point. In § 165 it will be proved that
if three lines, concurrent at a point 0, cut a conic in
A^, A^ ; B^, B^ ; Cj, Cg, the harmonic conjugates Ay B^ C
of 0 with respect to A^, A^;
J?j, B^; Cj, Cj are collinear.
Since OC^C^ may be any line
through 0, it follows that all
harmonic conjugates of 0 with
respect to the pairs of points
in which Imes through 0 cut
the conic are on the line de-
termined by A and B, two of
these conjugates.

The line thus determined by two harmonic conjugates
is called the polar of the point 0 with respect to the conic.

164. Pole of a Line. Suppose that there are given a conic
and a line o. If from a point of the Ime o two tangents
are drawn to the conic, then
ky the harmonic conjugate of
o with respect to the two tan-
gents, may be constructed.
This line will be spoken of as
a harmonic conjugate of o with
respect to the conic, or simply
as a harmonic conjugate of o. The point of intersection of
two harmonic conjugates of o will be called the pole of o.

151

152

POLES AND POLARS OF CONICS

Theorem. Triangles m Homology

165. If through a point 0 three lines are drawn cutting
a conic in the pairs of points A^, A^ ; B^, B^ ; Cj, Cg, the
triangles AyB^C-^ «wt? -42^2^2 ^^^ *^^ harmonic homology.

Proof. The triangles A^B^ Cj and A^B^C^ are in homology,
as is indicated in the note under Ex. 13, page 13.

The axis of homology passes through X^, Xj, A'g, the
intersections of B^C^, B^C^\ C^Ay, C^A^\ A^B^, A^B^.

The Pascal line of the hexagon A^C^C^A^B^B^ contains
the points L, 0, M, which are the intersections of the oppo-
site sides ^jCj, A^B^; C^C^, B^B^; C^A^, A^By Moreover,
the Pascal line cuts the line AjAg in a point N.

Also range LMON = range A^A^ OA

— range OAA^A^
But (^LM0N) = -1.

Therefore (OAA^A^^ = 1 ^ (OAA^A^) = - 1.

Hence the constant of homology is —1.

§23

§30
§24

POLES AND POLARS 153

Exercise 31. Poles and Polars
Prove the theorem of § 165 for the following triangles :
1. A^B^C^, A^B^C^. 2. A^B^C^, A^B^C^. 3. A^B^C^, A^B^C^.

4. In the figure of §165 A^B^, A^B^; A^C^, A^C^,; Bf^
B^C^ also intersect on the line X^X^X^.

5. Draw a large and accurate figure consisting of the figure
of § 165 and the additional lines which would be introduced
in proving Exs. 1, 2, and 3.

It is suggested that the sets of lines introduced on account of Exs. 1-3
be given distinctive colors.

6. State and prove the dual of § 165 for the plane.

7. Construct carefully the figure which is the dual of that
required in Ex. 5.

8. If two triangles are homologic and the constant of
homology is —1, the six vertices are on a conic.

9. If two triangles are homologic and the constant of
homology is — 1, the six sides are tangent to a conic.

10. By means of § 165 find a figure harmonically homo-
logic with any polygon inscribed in a conic.

11. Inscril)e in a conic a polygon which shall be harmoni-
cally self-homologic.

12. Use § 165 to obtain a line that bisects a given set of
parallel chords of a conic.

13. Given a circle or a carefully drawn ellipse, parabola, or
hyperbola, show experimentally that the polar of a given
point O is the same line whatever pair of three given lines
through O is used in constructing it.

14. Construct the polar o of a given point O of a conic, and
then find the intersection of the polars of two points on o.

15. Construct the polar of a vertex of the diagonal triangle
of a complete quadrangle inscribed in a conic.

154 POLES AND POLARS OF CONICS

166. Properties of a Polar. The polar of a point with
respect to a conic has the following important properties:

1. The polar of a point 0 with respect to a conic con-
tains all harmonic conjugates of 0 with respect to the conic.

In the preceding discussion the polar, though deter-
mined by A and B only, contains C, no matter what is the
direction of OC^C^. Accordingly, any two of the harmonic
conjugates of O determine a line through all of them.

In each of these cases the student should draw the figure and be
certain that the suggested proof is clearly followed.

2. The polar of a point O with respect to a conic contains
the other intersections of opposite sides of any inscribed com-
plete quadrangle of which 0 is a diagonal point.

In the discussion of § 165, A^B^A^B^ is any inscribed
quadrangle of which O is a diagonal point; and the other
intersections of pairs of opposite sides of this quadrangle
are situated on the axis of homology, which coincides with
the polar of 0.

3. The polar of a point 0 with respect to a conic contains
the intersections of pairs of tangents to the conic at the points
in which any line through 0 cuts the conic.

Let the line OB^B^ (§165) approach coincidence with
the line OA^A^. Then B^A^ approaches the tangent at Jj*
and B^A^ approaches the tangent at A^\ and at all stages
these lines intersect on the polar of 0.

4. The polar of a point 0 with respect to a conic contains \
the points of contact of the tangents (if there are any^ from
O to the conic.

If the line OA^A^ rotates about 0 and approaches the
position of tangency to the conic, the points A^^ A, A^
approach coincidence at the point of contact.

PROPERTIES OF POLES AND POLARS 155

167. Properties of a Pole. Applying the Principle of
Duality to the statements of § 166, we have the following ;

1. The pole of a line o with respect to a conic is on all
harmonic conjugates of o with respect to the conic.

The student should draw the figure in each of these cases and
should write out the duals of the proofs suggested in § 166.

2. The pole of a line o with respect to a conic is on the
other lines joining opposite vertices of any circumscribed com-
plete quadrilateral of which o is a diagonal line,

3. The pole of a line o vrith respect to a conic is on the
chord of contact (^produced if necessary^ of the tangents from
any point of o to the conic.

4. The pole of a line o with respect to a conic is the in-
tersection of the tangents to the conic at the points (if there
are any^ in which the line o cuts the conic.

A comparison of the properties of pole and polar as stated in
§§ 166 and 167 leads to various interesting conclusions. A few of
these are stated in §§ 168-171.

Exercise 32. Construction of Poles and Polars

1. Give a construction based upon § 166, 2, for the polar of
a given point with respect to a given conic.

2. Construct the tangents to a conic from a given point O.

3. Find the pole of a given line with respect to a conic,

4. At a given point on a conic draw a tangent to the conic.

5. For any conic construct the polar o of a given point 0,
and then find the pole of the line o.

The figure should be drawn very carefully.

6. With respect to a given conic find the polar o^ of a given
point Oj, the polar o^ of a given point O^ on o^, and the polar o^
of the intersection of o^ and o^.

156 POLES AND POLARS OF CONICS

Theorem. Relation of Pole and Polar

168. If a 'point 0 is the pole of the line o, the line o is the
polar of the point 0.

Proof. Let 0 be the pole of the line o, and let Op 0^
be pomts on o. Let the chords of contact of the tangents
to the curve from Oj and O^ cut o in Og and 0^. These
chords pass through 0 (§ 167, 3). Draw 0^0 and O^O.

Since the lines 0^0, 0^0 are conjugates of the line o
(§ 167, 1), the points Og, O^ are conjugates of 0, and
therefore o is the polar of 0 (§ 164).

169. Inside and Outside of a Conic. If a point 0 moves
up to a position on a conic, its polar o becomes a tangent ;
but if O is not on the conic, either no tangent or two tan-
gents pass through it. According as 0 is on two tangents
or on no tangent, it is said to be outside or inside the curve.

If 0 is outside the curve, its polar cuts the conic in the
two points of contact of the tangents from 0 to the conic.

Suppose O is inside the curve ; then, smce the polar meets
all tangents to the conic, infinitely many of its points are
outside the curve. Moreover, the polar does not meet the
curve ; for if it did, tangents could be drawn from 0 to the
intersections. The proof of the theorem that every point
of the polar, that is, every harmonic conjugate of 0, is
outside the conic is, however, somewhat complicated and
will be omitted from this book, the fact being assumed.

RELATIONS OF POLES AND POLARS 157

Theorem, points on a polar

170. If the 'point O^ is on the polar of the point Og, then
Oj is on the polar of the point Oj.

Proof. Of the points Oj and 0^, at least one must be
outside the conic. For if Oj and 0^ are both inside tlie
conic, then (§ 169) every point of o^, including Oj, is out-
side the conic, which is contrary to the hypothesis made.

Let O2 be outside the conic. Then § 166, 3 and 4, yields
the desired conclusion.

Let O2 be inside the conic. Then Oj, being on Og, is out-
side the conic. Let Oj cut the conic in the points A, B.
It will cut the line Oj 0^ in 0<^ ; for otherwise the tangents
of which O^A is the chord of contact would not meet on
©2, as they must by § 166, 3. Hence O3 is on 0^.

171. Corollary. If a point 0^ traces out a range whose
base is o^, its polar Oj traces out a fat pencil whose base
is Og, the pole of o^.

The relation between the range traced by the point O^ and the
flat pencil described by o^ is stated in the theorem of § 177.

172. Conjugate Points and Conjugate Lines. Two points
so situated that each is on the polar of the other are said
to be conjugate^ and two lines so situated that each contains
the pole of the other are said to be conjugate.

Accordingly, harmonic conjugates are sjiecial cases of conjugates.

158 POLES a:n^d polars of conics

173. Self-Polar, or Self-Conjugate, Triangle. If any point
Oj, not on a conic, is taken, its polar Oj may be found. On
this line let any point 0^, not on the conic, be taken, and
let its polar o^ be found. Then o^ passes through Oj. Let
the intersection of Oj and o^ be Og.

The polar Og of Og is the line 0^0^, and Og is conjugate
to both Oj and Oj. The triangle O^O^O^ is such that each
side is the polar of the op-
posite vertex. Every such
triangle is said to be self-
polar, or self-conjugate, with
respect to the conic. Evi-
dently there is an infi-
nite number of triangles
which are self-polar with respect to a given conic.

No self -polar triangle has two vertices inside the conic ;
for if one vertex is inside the curve, its polar in which the
other two vertices lie is entirely outside the curve.

If we should attempt to construct a self-polar triangle
all of whose vertices are outside the conic, we might
choose a point Oj outside the conic and draw its polar Oj.
Let A, B be the two points in which this line would cut
the conic. Then the other vertices 0^, Og would be on Oj
and would be separated by A, B. If we should take 0^
outside the conic, it would remain to determine whether 0^
would be inside or outside the conic; that is, whether
tangents could be drawn from Og to the conic. The con-
siderations adduced thus far would not enable us to give
a sufficiently brief but complete discussion of this question,
but an application of principles of continuity, which have
'not been developed in this book, would enable us to go
farther and to establish the proposition that every self-
polar triangle has one and only one vertex within the conic

SELF-POLAR TRLA.NGLE

159

Theorem. Diagonal Triangle

174. The diagonal triangle of a complete quadrangle
inscribed in a conic is self -polar ; and, conversely, a self-polar
triangle is the diagonal triangle of an inscribed complete
quadrangle.

Proof. Let O^O^O^ (Fig- 1) be the diagonal triangle of a
complete quadrangle A^A^B^B^ inscribed in a given conic.
From § 166, 2, it follows that O^O^O^ is a self-polar triangle.

Conversely, let OjOgOg (Fig. 2) be a self-polar triangle
with respect to a given conic.

From A^j any point on the conic, draw A^O^, A^O^, and
let them meet the conic again in A^, B^. Draw O^B^,
O^B^, O^A^, O^A^, and let O^A^, O^A^ meet in B^. Also
let Og^j, OjOg meet in K, let O^A^, ^3^1 meet in L, and
let OjJj, OgOg meet in M.

Then (^A^B^KO^^ = -l = {A^A^O^Ly

Hence r&n^e A^B^KO^^TdmgQ A^A^O^L,
and the points A^, B^, Oj are collinear.

Again, since range A^B^MO^ = harmonic range A^A^O^L
and M is on the polar of Oj, then B^ is on the conic.
Hence the quadrangle A^A^B^B^ is inscribed.

160 POLES AKD POLARS OF CONICS

Theorem. Ranges and their Conjugates

175. On a line the range of points and the range of their
conjugates with respect to a conic constitute an involution.

Proof. Let 0^ be the pole of a line o^ with respect to
a given conic. Through 0^ draw lines cutting the conic
in ^1, J J ; and also in J3^, B^ ; Bl, B^ ; B[', B^'; .... Then
OjOjOg is the diagonal triangle of Aj^A^B^B^; 0^0{0^ of
A^A^B'^B[ ; • . .. The pairs of points 0^, 0^\ 0[, 0'^', -- -
are conjugate, and, associating with A^A.^ all lines through
Oj, we obtain all pairs of conjugates on o^.

But range 0^0[0i; . . . OgO^O^' • . •

= pencil A,(iO,0[0!^ . • • O^O'^O'^ • . •)
^ pencil ACOjOi'Oi" . . • 0,0',0'^ . . .)
^ pencil A^^B^B'^B'^ . . . B^B[B!< . . .)
-^ pencil A^ {B^B'^B'^ • • • B^B[B'^ • . •)
grange 030^0^'...0i0{0i"....

ELEMENTS AND THEIR CONJUGATES 161

Theorem, pencils and their Conjugates

176. The pencil of lines through a point and the pencil of
their conjugates with respect to a conic constitute an involution.

The proof of this dual of § 175 is left for the student.

The properties of poles and polars furnish one basis for the estab-
lishment of the validity of the Principle of Duality for figures in
a plane, and Poncelet practically used them in this way.

Theorem, point Describing a Range

177. If a point describes a range, the polar of the point
with respect to a conic describes aflat pencil which is projective
with that range.

Proof. Let O be the pole of the range, and let the point
take the positions Oj, Og, Og, O^, • • .. Then its polar
always passes through O. The polars of Oj, Og, Og, O4 • • •
intersect 0, the base of the range, in 0[, 0^, O3, 0^, • • •, the
conjugates of Oj, O^, Og, O4, • • • respectively.

Then range O^O^O^O^ . • • ^ range 0[0'^0^0[ ... § 175

= pencil OjOjOgO^ ....

Hence range O^O^O^O^- ' --r pencil o-^o^o^o^ ....

162 POLES AND POLARS OF CONICS

178. Duality in Plane Figures. It is now possible to indi-
cate a line of argument by which the Principle of Duality
may be established for plane geometry. In the plane of a
given figure, composed of, or generated by, points and lines,
take any conic and construct the polar of every point and
the pole of every line. Then a new figure is obtained in
which there is a point for every line and a line for every
point of the first figure, and in which to the intersection of
any two lines of the first figure there corresponds the line
determined by the poles of these two lines in the second.
To any locus of points in the first figure there corresponds
an envelope of lines in the second. Hence it is evident that
a duality in figures exists.

179. Duality in Properties of Figures. Likewise, if any
nonmetric proposition is true for some or all of the points
and lines of the first figure in § 178, it follows that this figure
cannot be constructed by choosing arbitrarily in the plane the
sets of points and lines which constitute it, but that, certain
points and lines being selected, the choice of the remaining
ones is restricted. For the proposition, by its assertion of a
relation, or of relations, existing among the points and the
lines of a given figure, is a denial of the possibility of choos-
ing all of them arbitrarily. Hence the second figure is not
merely a set of lines and points, each chosen arbitrarily in the
plane. In fact, there exists a certain limitation upon the
choice of the lines and points of the second figure, and
the statement of this limitation constitutes the proposition
correlative to the one regarding the first figure. Hence
there is a duality in properties of figures.

180. Polar Reciprocal or Polar Dual. A figure obtained
from a given figure by the method explained in § 178 is
called the polar reciprocal or polar dual of the given figure.

CENTER AND DIAMETERS OF A CONIC 163

181. Center and Diameters of a Conic. As in some pre-
ceding cases, useful metric relations are obtained by con-
sideration of the elements at infinity. Thus, the line at
infinity has a pole, and from the property of the harmonic
range this pole is seen to bisect every chord of a conic
which passes through it. Because of this symmetry of the
curve with respect to the pole of the line at infinity, this
point is called the center of the conic. If the conic is
a parabola, the center is also the point of contact of the
parabola with the line at infinity; and since this point is
at infinity and the notion of symmetry loses its usual force,
the parabola is generally said not to have a center. In the
case of the other conies the center is not at mfinity, the
center of the ellipse being inside the curve and that of
the hyperbola being outside. In the case of the latter it
is the intersection of two tangents to the curve whose
points of contact are at infinity. These tangents are, of
course, the asymptotes.

Again, every point on the line at infinity has a polar
which passes through the center. The polar of a point at
infinity is called a diameter of a conic. In the case of a
parabola, since all the points at infinity are on the line at
infinity, the diameters intersect in a point at infinity and
hence are parallel. Also, any point on the line at infinity
being chosen, all lines through that point are parallel ; and
if any one of these lines meets the conic, the harmonic con-
jugate of the chosen point at infinity that is situated on
that line bisects the segment of it which is intercepted by
the curve. Hence, every diameter bisects each of the set of
parallel chords of the conic which passes through its pole.
Likewise, the tangents at the points in which a diameter
cuts the conic pass through the pole of that diameter ; that
is, they are parallel to the chords bisected by the diameter.

164 POLES AND POLARS OF CONICS

182. Conjugate Diameters and Principal Axes. If two
diameters are conjugate lines with respect to a conic, each
diameter passes through the pole of the other, which is
the point at infinity, and so each is parallel to the chords
bisected by the other. Such diameters are called conjugate
diameters. According to § 176 they constitute an involu-
tion, and since, in general, only one pair of corresponding
lines of an involution is at right angles, in general only
one pair of conjugate diameters is at right angles. These
two diameters are called the principal diameters and form
the principal axes. Moreover, it follows from § 74 that if
there are more pairs of conjugate diameters which are at
right angles, all pairs have this property. In this case it
can be shown that the conic is a circle.

When the involution of diameters is hyperbolic, the
self -corresponding elements are the asymptotes ; and these
separate harmonically every pair of conjugate diameters.

Two conjugate diameters and the line at infinity consti-
tute a seK-polar triangle, and of such a triangle just two
sides cut the conic. Hence both of two conjugate diam-
eters of an ellipse meet the curve, but only one of any
two conjugate diameters cuts the hyperbola.

Exercise 33. Review

1. Find the locus of the harmonic conjugates of a given
point with respect to a given pair of straight lines.

2. Find the point which is the harmonic conjugate of a
given point with respect to each of two given pairs of
straight lines.

3. Find the point which has an infinite number of harmonic
conjugates with respect to each of two given pairs of straight
lines, and find the locus of these conjugates.

REVIEW EXERCISES 165

4. Find three points each of which is conjugate to the
other two with respect to a pair of opposite sides of a given
complete quadrangle.

5. Each of the three points found in Ex. 4 is conjugate to
the other two with respect to any conic which passes through
the four vertices of the quadrangle.

6. Given any five points, no four of which are collinear,
construct, with the ruler only, the polar of a given point with
respect to that conic.

7. Solve the dual of Ex. 6.

8. Construct a self-polar triangle for a given conic, using
the ruler only.

9. Construct the common self-polar triangle for all conies
which pass through four given points.

10. If all parts of a figure which consists of a conic and
a self-polar triangle are erased except the triangle and two
points of the curve, reconstruct the figure.

11. Through three given points construct a conic which has
a given point and a given line as pole and polar.

12. Solve the dual of Ex. 11.

13. Through two given points construct a conic which is
tangent to a given line and has a given point and a given line
as pole and polar.

14. Through four given points construct a conic which has
a given pair of points as conjugates.

15. Through four given points and through some pair (not
specified) of points of a given involution on a straight line
construct a conic.

16. Given four points P^, P^, P^, P^ and two fixed lines j)^,
p^ passing through P^ and P^ respectively, find the envelope of
the line joining the other intersections of these two lines with
a variable conic through the four points.

166 POLES AND POLARS OF CON^ICS

17. Given the two points common to two conies and three
other points of each, find all the intersections of the conies.

18. Given five points of a conic, find any diameter and the
diameter conjugate thereto.

19. Given five points on a conic, construct the center, the
axis, and the asymptotes of the conic.

The student should consult § 70 and § 74.

20. If a conic has more than one pair of conjugate diam-
eters which are at right angles, the conic is a circle.

21. Given six points on a conic and the tangents at these
points, the Pascal line of the inscribed hexagon is the polar
of the Brianchon point of the circumscribed hexagon.

22. If the pole of the line PJ^^ with respect to one conic
which passes through the points P^, P^, P^, P^ coincides with
the pole of P^P^ with respect to a second conic through these
points, the pole of -P^Pg with respect to the second conic coin-
cides with the pole of ^8^4 with respect to the first.

23. Construct a conic which has a given triangle as a self-
polar triangle and a given point and a given line as pole
and polar.

24. Construct a conic which has a given point as center
and a given self-polar triangle.

25. Construct a conic which has a given pair of lines as
conjugate diameters and a given point and a given line as pole
and polar.

26. Construct a conic that has each side of a given pentagon
as polar of the vertex opposite to it.

27. In the construction of Ex. 26 find the polar reciprocal of
the conic determined by the vertices of the pentagon.

28. If a moving point traces a given conic, find the envelope
of the polar of the point with respect to another given conic.

29. The lines joining the vertices of a triangle to the corre-
sponding vertices of the triangle polar to it are concurrent.

CHAPTER XIII
QUADRIC CONES

183. Properties of Quadric Cones. A large number of the
properties of quadric cones can be derived as duals of
those of the conic. Thus, it is evident that a quadric
cone is determined by its relation to certain sets of five
planes and lines. Moreover, the theorems of Pascal,
Brianchon, and Desargues, and their limiting cases, have
duals which relate to the tangent planes and generating
lines of the quadric cones.

In the problems of construction of the first order the
possibility of drawing lines through pairs of points and of
finding pomts as the intersections of lines was assumed.
For the corresponding problems of this chapter there
should be assumed the possibility of drawing lines common
to pairs of planes and of constructing planes determined
by pairs of lines.

In the problems of the second order the constructibility
of at least one conic, ordinarily a circle, was assumed. At
this point the corresponding assumption is that of the con-
structibility of at least one quadric cone. On the basis
of these assumptions the problems analogous to those of
Chapter X can be adequately treated.

Similarly, the theory of polar planes and lines of quad-
ric cones follows from that of poles and polars of conies,
and can be made to furnish an evidence of the truth of
the Principle of Duality for the bundle.. The development
of the subject along these lines is left as an exercise.

167

168 QUADRIC CONES

Theorem. Sections of Quadric Cones

184. A quadric cone the vertex of which is not infinitely
distant may he so cut by a plane as to yield an ellipse, a
parabola, or a hyperbola.

Fig. 1

Proof. Every quadric cone can be generated by means
of the lines of intersection of corresponding planes of
projective axial pencils that belong in the same bundle,
and every plane section of a quadric cone is a conic.

Suppose now that a quadric cone (Fig. 1) is generated
from the intersections a^, a^, a^, ' - • of pairs of correspond-
ing planes a^, a^ ; a^, a^; a^, a^; • • - of two axial pencils
whose bases are the lines p and p' ; and let the base of the
bundle be P, a point not at mfinity.

First, to secure a plane section which is a hyperbola, let
TT be a plane not through P but parallel to a^ and a^.

This plane cuts the cone in a conic, and since the plane
cuts aj and a^ at infinity, the conic has two distinct points
at infinity. The conic is not a pair of straight lines, since
its projector from P is not a pair of planes.

Hence the conic is a hyperbola.

SECTIONS OF QUADRIC CONES

169

Next, to obtain a section which is a parabola, let a[ be the
plane tangent to the cone (Fig. 2) along the generator a^.

This plane contains the whole of the line a^ but meets
any other generator, as a^, in only one point, namely, P.

Cut the cone by a plane a^ parallel to a[. This plane
cuts any generator other than a^, as a^^ at a finite distance
from P, and it cuts aj at infinity.

Hence the section of the cone has one and only one
point at mfinity, and the conic is a parabola.

Fig. 2

Fig. 3

Finally, to obtain a section which is an ellipse, let ttj be
a plane (Fig. 3), through P but not coincident with any
plane of either axial pencil, and let ttj be a plane not
through P but parallel to tt^ at a finite distance from it.

The intersection of any pair of corresponding planes, as
a and a', since it meets ir^ at P, cannot be parallel to 7r2.

Hence the intersection of ttj and the cone has no point
at infinity, and the conic is an ellipse.

If the vertex of the cone is at infinity, the generating lines are
all parallel. It will be seen (§ 186) that the surface must contain no,
one, or two infinitely distant generators. In these cases the sections
made by planes not parallel to the generators will be all ellipses, all
parabolas, or all hyperbolas respectively. Hence the theorem fails.

170 QUADRIC CONES

185. Axes of a Quadric Cone. Let o be a line through
the vertex of a quadric cone, and let a> be the polar
plane of o with respect to the cone. In co there is one
line, and there may be many lines, perpendicular to o.
Through o and a line o' in co perpendicular to o pass a
plane tt, and let I and I' be the hnes in which the plane tt
cuts the cone. Then the lines o and o' are conjugates with
respect to I and I', and, being perpendicular to each other,
they bisect the angles formed by I and I'. Hence, through
any line o there is one plane which cuts the cone in lines
that form an angle of which a is the bisector.

If and only if the line o is perpendicular to its polar
plane co, all planes through o cut the cone in lines that
make an angle of which o is the bisector. In this case the
line 0 is an axis of symmetry with respect to the cone.

Manifestly every axis of symmetry is perpendicular to
its polar plane. Also, a line o' parallel to o, an axis of
symmetry, cuts a cone in two points, and the segment
joining these points is bisected by the polar plane, since
the axis of symmetry, the line joining the vertex to the in-
tersection of the plane co with o', and the lines I and V are
harmonic. Hence the polar plane of an axis of symmetry
is a plane of symmetry.

It can be shown that every quadric cone has one axis
Oj of symmetry and a plane co^ of symmetry which is per-
pendicular to it. The plane Wj cuts the cone in a conic
that has two principal axes which we may call o^ and Og.
Of the lines Oj, o^, o^ each pair is conjugate to the third
line and determines the polar plane of this line. Moreover,
each of these polar planes is perpendicular to its polar line,
and hence to the other two polar planes. There are, there-
fore, three axes of symmetry, perpendicular eaoli to each,
and three planes of symmetry, perpendicular each to each.

CYLINDERS

171

186. Cylinders. Hitherto it has been assumed that the
axes of the generating axial pencils intersect in the finite
part of space. If, however, the axes of the generating axial
pencils are parallel, then all the generating lines are parallel
to them, and the vertex of the surface is at infinity. In
this case the surfaces generated are called cylinders.

Hyperbolic Cylinder Parabolic Cylinder Elliptic Cylinder

Cylinders are classified with reference to their relation
to the plane at infinity. The plane at infinity may cut
the cylinder in two, one, or no straight lines. In these
cases a section perpendicular to the generating lines of
the cylinder is a hyperbola, a parabola, or an ellipse
respectively; and the cylinder is said to be hyperholic^
parabolic^ or elliptic, as the case may be.

In the case of a cylinder the plane at infinity and cer-
tain of its lines are in the bundle to which the cylinder
belongs. The plane at infinity has a polar line which is
an axis of symmetry and is called the axis of the cylinder.
It can be shown that certain planes through this axis and
all planes perpendicular to it are planes of symmetry. In
the case of the parabohc cylinder the polar line of the
plane at infinity is at infinity and lies in the cylinder.

172 QUADRIC CONES

Exercise 34. Quadric Cones

1. Prove the theorem regarding quadric cones which corre-
sponds to Steiner's theorem regarding conies.

2. Every conic surface of the second class is of the second
order. Prove also the converse.

3. The hexahedral angle whose faces are determined by
the six pairs of alternate edges of another hexahedral angle
which is inscribed in a quadric cone has its faces tangent to a
quadric cone.

4. Inscribe in a quadric cone a trihedral angle whose three
edges shall be in three given planes.

5. If a variable simple four-flat so moves as always to be
circumscribed about a given quadric cone, while three of its
edges move each in one of three fixed coaxial planes, then
the fourth edge moves on a fourth fixed plane coaxial with
the three given cones.

6. State the properties of polar lines and planes of quadric
cones corresponding to those of poles and polars of conies which
are given in §§ 166 and 167.

7. Find the points of intersection of a given straight line
with a quadric cone of which five determining elements are
also given.

8. In a bundle a^, fi^, y^ and a,^, /S^, y^ are two sets of fixed
coaxial planes. Two planes tt^ and tt^ so move that the lines
determined by tt^ and a^, ir^ and a^; tt^ and )8j, tt^ and /3.^',
IT and yj, tt^ and y.^ lie in three coaxial planes. Find the surface
generated by the line common to tt^ and tt.^.

9. In a bundle the edges of a trihedral angle, whose planes
are a, /8, y and which is self-polar with respect to a given quadric
cone, determine with any line o the planes a', ^', y'. If the
polar plane of o is w, the pairs of lines determined by w with
a and a', by w with ^ and /3', and by w with y and y' form a
pencil in involution.

CHAPTER XIV

SKEW RULED SURFACES

187. Skew Ruled Surfaces. The third set of figures
which were projectively generated was found to consist of
the ruled surfaces of the second order that are not conic.
Before discussing these we shall consider the classification
of ruled surfaces in general.

One classification of ruled surfaces is based upon the
law governing the motion of the generating line. At any
instant the motion of this line may be a revolution about
one of its o^vn points or it may be a displacement by
virtue of which the line immediately ceases to intersect
its present position. In the former case it is sometimes
said that every pair of consecutive generators intersect,
and m the latter case it is said that no two consecutive
generators intersect. Surfaces generated in the first way
are called developable surfaces, and those generated in the
second way are called skeiv surfaces. Cones and cylinders
are examples of developable surfaces, but they are of a
special type, inasmuch as each of their generators inter-
sects every other one. Likewise, skew ruled surfaces of the
second order and second class are special in character, for
no generator intersects any other of the set, even though
they be not consecutive. Generators usually cut other
generators of the set if the latter are not consecutive.

In this chapter only a few specially interesting facts
regarding the surfaces of the second order and second
class will be established.

173

174

SKEW KULED SUKEACES

Theorem, second Set of Generating Lines

188. Every skew ruled surface generated hy the intersec-
tions of corresponding planes of two projective axial pencils
has also a second set of generating lines whose relation to the
surface is similar to that of the first set. Each member of
either set of generators intersects no others of its own set, but
intersects every one of the other set. Through every point of
the surface there pass two generators, one of each set.

Proof. Every plane through any generatmg line a of
the surface cuts the surface along the line a and also
along a second line a-^, and nowhere else.

Moreover, the line a^ cuts each of the generating lines
that have been noted. The infinitely many planes through
a cut the surface in infinitely many lines a^, a^, a^, • • •,
each of which cuts every one of the generators ; and
every point of the surface lies on one and only one of
the new lines.

No two of these lines intersect; for if they did, all the
generators would lie in the plane determined by them.

SECOND SET OF GENERATING LINES 175

Consider now the two sets of points B^^ B^, ^3? • • •
and Cj, Cg, Cg, • • • in which the lines a^, ag, a^, - • • intersect
two other generators b and c.

These sets of points are the intersections of the gener-
ators b and c with the planes of the axial pencil whose
base is a and whose planes pass through the lines a^, a^,
flg, • . • ; and consequently they constitute perspective (but
not coplanar) ranges.

Then the axial pencil whose base is b and whose planes
pass through Cj, Cj, Cg, • • • and the axial pencil whose
base is c and whose planes pass through B^, B^j -Sg, • • •,
being perspective respectively with the ranges C^C^C^' ' •
and B^B^B^ • • •, are projective with each other.

Corresponding planes of these axial pencils intersect in
the lines a^, a^., ag, • • •, which are therefore generating
lines of a skew quadric ruled surface.

The latter skew ruled surface must coincide with the
original one, since the lines a^, a^, ^g, • • • contain all the
points of the original ruled surface, and no others. Hence
these lines must constitute a second set of generators for
that ruled surface.

189. Corollary. The lines of either set of generators
determine projective ranges on any ttco lines of the other set.

190. Conjugate Reguli. Two reguli which are related as
are the two in § 188 are called conjugate reguli.

The theorem of § 188 may be restated as a corollary to
this definition as shown below.

191 . Corollary. Every skew ruled surface of the second
order carries two conjugate reguli. Each line of either regulus
intersects no lines of its own regulus, but intersects each line
of the other regulus. Through every point of the surface there
pass two lines, one from each regulus.

176 SKEW KULED SURFACES

Theorem. Determination by Generators

192. Given three straight lines no two of which are coplatiar,
there exists one and only one skew quadric ruled surface of which
each of these lines is a generator.

Proof. Let a, b, c be three straight hues no two of which
are coplanar.

Through each point of a one line and only one can be
.drawn to meet all three of the lines.

Let p^, j»2, and p^ be three lines which meet a, b, c.

Then p^ together with the three lines a, b, c, and p^
together with the same lines, determine triads of planes of
axial pencils whose bases are p^ and p^. These triads
determine a projectivity between the pencils, and this
projectivity determines one skew quadric ruled surface of
which a, b, c are generators.

Any two corresponding planes of the projective axial
pencils intersect in a line that meets p^ and p^. This line
also meets p^; for if p^ meets a, b, c in Jg, B^, Cg respec-
tively, the two axial pencils above mentioned are each per-
spective with the same range on jOg, the perspectivities being
determined by the correspondence of A^, B^, Cg to the planes
p^a, p^b, p^e in the one case, and the planes p,^a^ p^b, p.^c in
the other. It follows that corresponding planes cut p^ in
the same point, and hence their line of intersection cuts py

Accordingly, the surface is uniquely determined by the
three generators a, b, c.

CLASSIFICATION 177

imiimniiipi^^

Hyperbolic Paraboloid, the conjugate reguli being formed
by rods. Certain sections are parabolas, other sections
are hyperbolas. The curvature of the surface is not
secured by the pressure of one set of rods upon the other

Hyperboloid of One Sheet, the conjugate reguli being
formed by straight rods. The surface in the neighbor-
hood of its center of symmetry is shown. Horizontal
sections are ellipses, vertical sections are hyperbolas

gjjjjjpjjjM^^

178 SKEW RULED SURFACES

193. Skew Quadric Ruled Surfaces Classified. These sur-
faces are classified according to the nature of their sections
by the plane at infinity. As has been shown, any plane
section of one of these surfaces is a conic and degenerates
into two straight lines if the cutting plane contains a
generator. The surfaces are, therefore, of two sorts:

1. Hyperbolic paraboloids, or those whose intersections
with the plane at infinity are pairs of generators.

2. Hyperboloids of one sheet, or those whose intersections
with the plane at infinity are nondegenerate conies.

It may be noted that if a hyperbolic paraboloid is regarded
as generated by the joining lines of corresponding points of
two projective ranges, the points at infinity of the ranges
are found to be corresponding points. Hence in this case
(and in this case only) the ranges are similar. Accord-
ingly, if corresponding points of two similar (but not
coplanar) ranges are connected by threads, a good model
of a hyperbolic paraboloid may be constructed.

To exhibit both sets of generating lines it is better to use a
quadrilateral A BCD hinged at two opposite vertices, as B and D, so
that the triangles ABD,
CBD can be adjusted to
lie in different planes.
Congruent ranges can be
taken on A B and CD and
also on BC and DA. Cor-
responding points can be
joined by strings, and in
this manner an excellent
model can be constructed
with very little trouble.

Directions for constructing a string model of the hyperboloid of
one sheet are not so easily given. The existence of such a surface is
evident, since it is generated by the lines joining corresponding
points of projective ranges which are not coplanar and not similar.

CLASSIFICATION

OmmiiJliiiii^^

179

Hyperbolic Paraboloid, the conjugate reguli being formed

by straight rods. The surface near the vertex, or

saddle point, is shown. Certain sections through the

vertex are parabolas, others are hyperbolas

Hyperbolic Paraboloid, the conjugate reguli being formed

by strings, ff every string in either set is cut, the strings

in the other set retain their positions

180 SKEW RULED SURFACES

Exercise 35. Skew Ruled Surfaces

1. A regulus is determined by two nonintersecting lines
and three noncollinear points, no two of which are coplanar
with any of the lines.

2. If a regulus contains a line at infinity, the conjugate
regulus also contains a line at infinity.

3. Determine three pairs of quadric ruled surfaces which
have in common two given noncoplanar lines and also respec-
tively no, one, and two generators of the other set.

4. Determine a quadric ruled surface which contains two
given noncoplanar lines and a given point exterior to them.
How many such surfaces are there ? Find any additional lines
which are common to such surfaces.

5. If four generators of a regulus cut one generator of
the conjugate regulus in a harmonic range, they cut every
generator of the conjugate regulus in a harmonic range.

Four generators of a regulus which have the property mentioned in
Ex. 6 are called harmonic generators.

6. Given any three lines in space, no two of which are
coplanar, find a fourth line which, with the three given lines,
constitutes a set of harmonic generators of a regulus.

7. If a line so moves as constantly to intersect each of two
noncoplanar lines and also to remain parallel to a given plane,
the line generates a hyperbolic paraboloid.

8. If a range and a flat pencil which do not lie in the same
plane or in parallel planes are projective, and if from each
point of the range a line is drawn parallel to the correspond-
ing line of the flat pencil, these parallel lines all lie on a
hyperbolic paraboloid.

9. The locus of the harmonic conjugates of any point with
respect to a ruled surface is a plane.

10. The lines (or planes) of any bundle which are tangent
to a quadric ruled surface generate a quadric cone.

HISTORY OF PROJECTIVE GEOMETRY

The history of geometry may be divided roughly into
four periods: (1) The synthetic geometry of the Greeks,
including not merely the geometry of Euclid but the
work on conies by Apollonius and the less formal contri-
butions of many other writers; (2) the birth of analytic
geometry, in which the synthetic geometry of Desargues,
Kepler, Roberval, and other writers of the seventeenth
century merged into the coordinate geometry of Descartes
and Fermat ; (3) the application of the calculus to geom-
etry,— a period extending from about 1650 to 1800, and
including the names of Cavalieri, Newton, Leibniz, the Ber-
noullis, L'HSpital, Clairaut, Euler, Lagrange, and D'Alem-
bert, each one, especially after Cavalieri, being primarily
an analyst rather than a geometer ; (4) the renaissance
of pure geometry, beginning with the nineteenth century
and characterized by the descriptive geometry of Monge,
the projective geometry of Poncelet, the modern synthetic
geometry of Steiner and Von Staudt, the modern analytic
geometry of Pliicker, the non-Euclidean hypotheses of
Lobachevsky, Bolyai, and Riemann, and the laying of the
logical foundations of geometry, — a period of remarkable
richness in the development of all phases of the science.

It is in this fourth period that projective geometry has
had its development, even if its origin is more remote.
The origin of any branch of science can always be traced
far back in human history, and this fact is patent in the
case of tliis phase of geometry. The idea of the projection

181

182 • HISTORY

of a line upon a plane is very old. It is involved in the
treatment of the intersection of certam surfaces, due to
Archytas, in the fifth century B.C., and appears in various
later works by Greek writers. Similarly, the invariant prop-
erty of the anharmonic ratio was essentially recognized
both by Menelaus in the first century a.d. and by Pappus
in the third century. The notion of infinity was also famil-
iar to several Greek geometers, so that various concepts
that enter into the study of projective geometry were com-
mon property long before the science was really founded.

One of the first important steps to be taken in modern
times, in the development of this form of geometry, was
due to Desargues, a French architect. In a work on conic
sections, published in 1639, Desargues set forth the founda-
tion of the theory of four harmonic points, not as done
today, but based on the fact that the product of the dis-
tances of two conjugate pohits from the center is con-
stant. He also treated of the theory of poles and polars,
although not using these terms. In 1640 Pascal, then only
a youth of sixteen, published a brief essay on conies setting
forth the well-known theorem that bears his name.

The descriptive geometry of Monge is a kind of pro-
jective geometry, although it is not what we ordinarily
mean by this term. He was a French geometer of the
period of the Revolution, and had been in possession of
his theory for over thirty years before the publication of
his "Geometrie Descriptive " (1795). It is true that certain
of the features of this work can be traced back to De-
sargues, Taylor, Lambert, and Frezier, but it was Monge
who worked out the theory as a science. Inspired by the
general activity of the period, but following rather in the
steps of Desargues and Pascal, Carnot treated chiefly of
the metric relations of figures. In particular he investigated

HISTORY 183

these relations as connected with the theory of transver-
sals, — a theory whose fundamental property of a four-
rayed pencil goes back to Pappus, and which, though
revived by Desargues, was set forth for the first time in
its general form by Carnot in his " Geometric de Posi-
tion " (1803), and supplemented in his " Theorie des
Transversales " (1806). In these works Carnot introduced
negative magnitudes, the general quadrilateral, the general
quadrangle, and numerous other similar features of value
to the elementary geometry of today.

Projective geometry had its origin somewhat later than
the period of Monge and Carnot. Newton had discovered
that all curves of the third order can be derived by central
projection from five fundamental types. But in spite of this
the theory attracted so little attention for over a century
that its origin is generally ascribed to Poncelet. A pris-
oner in the Russian campaign, confined at Saratoff on the
Volga (1812-1814), " prive," as he says, " de toute espece
de livres et de secours, surtout distrait par les malheurs
de ma patrie et les miens propres," Poncelet still had the
vigor of spirit and the leisure to conceive the great work,
" Traite des Propri6t^s Projectives des Figures," which he
published in 1822. In this work was first made promi-
nent the power of central projection in demonstration and
the power of the principle of continuity in research. His
leading idea was the study of projective properties, and
as a foundation principle he introduced the anharmonic
ratio, — a concept, however, which dates back to Menelaus
and Pappus, and which Desargues had also used. Mobius,
following Poncelet, made much use of the anharmonic
ratio in his " Barycentrische Calcul" (1827), but he gave
it the name Doppelschnitt-l'erhaltniss (^ratio hisectionalis),
a term now in common use under Steiner's abbreviated

184 HISTORY

form Doppelverhdltniss. The name anharmonie ratio or
anharmonie function (rapport anharmonique, or fonetion
anharmonique) is due to Chasles, and cross-ratio was sug-
gested by Clifford. The anharmonie point-and-line prop-
erties of conies have been elaborated by Brianchon, Chasles,
Steiner, Dupin, Hachette, Gergonne, and Von Staudt. To
Poncelet is due the theory of figures homologiques, the per-
spective axis and perspective center (called by Chasles
the axis and center of homology), an extension of Carnot's
theory of transversals, and the cordes ideales of conies, which
Pliicker applied to curves of all orders. Poncelet also
discovered what Salmon has called " the circular points at
infinity," thus completing and establishing the first great
principle of modem geometry, — the principle of continuity.
Brianchon (1806), tlirough his application of Desargues's
theory of polars, completed the foundation ijvhich Monge
had begun for Poncelet's theory of reciprocal polars (1829).

Steiner (1832) gave the first complete discussion of the
projective relations between rows, pencils, etc., and laid the
foundation for the subsequent development of pure geom-
etry. He practically closed the theory of conic sections, of
the corresponding figures in three-dimensional space, and
of surfaces of the second order. With him opens the period
of special study of curves and surfaces of higher order. His
treatment of duality and his application of the theory of pro-
jective pencils to the generation of conies are masterpieces.

Cremona began his publications in 1862. His elementary
work on projective geometry (1875) is familiar to English
readers in Leudesdorf s translation. The recent contribu-
tions have naturally been of an advanced character, seek-
ing to lay more strictly logical foundations for the science,
and in this line the American work by Professors Veblen
and Young is noteworthy.

INDEX

PAGE

Anharmonic ratio . . 21, 24, 183

Asymptote 144

Axial pencil 8, 96

Axial projection 3

Axis, of a cylinder .... 171

of homology 37

of projection 3

of symmetry 170

Base 8

Brianchon point 122

Brianchon theorem . . 121, 148
Bundle 8

Center, of a conic 163

of homology 37

of involution 74

of projection 2

Central projection ..... 2

Class of a figure 80

Classification, of conies . . . 143

of prime forms .... 10

of projectivities .... 60

Congruent elements .... 72

Congruent pencils 59

Congruent ranges . . . . 59, 68

Conic 109, 115, 143, 156

Conjugate reguli 175

Conjugates ... 31, 72, 157, 164
Constant of homology ... 39
Constructions of second order 135
Correlative propositions . . 15

Cylinder 171

185

PAOB

Desargues's theorem .... 131
Descriptive properties ... 21
Developable surface .... 173

Diagonal lines 34

Diagonal points 33

Diagonal triangle . . . . 33, 34

Diameter 163

Directrix 93

Double elements 64

Duality 15, 81, 162

Elements at infinity ... 5, 144

Ellipse 143

Elliptic cylinder 171

Elliptic projectivity .... 66
Envelope 80

Figures of second order . . 101
Flat pencil . . 8, 59, 89, 90, 96, 97

Four-point 33

Four-side 34

Fundamental theorem ... 44

Generation of a figure ... 80
Generator 93

Harmonic conjugates . .31, 151

Harmonic form 31, 36

Harmonic homology .... 39

Harmonic pencil 31

Harmonic range 31

Hexagon 120

Homology 37

186

INDEX

PAGE

Hyperbola 143

Hyperbolic cylinder .... 171

Hyperbolic involution ... 73

Hyperbolic paraboloid ... 178

Hyperbolic projectivity ... 66

Hyperboloids 178

Infinity 5, 144

Involution 72, 74

Line at infinity 5

Line involution 76

Locus 79

Metric properties 21

One-dimensional forms ... 8

One-to-one correspondence . 10

Opposite sides 33, 120

Opposite vertices 34

Order, of a construction . . 135

of a figure 80

Orthogonal projection ... 1

Parabola 143

Parabolic cylinder 171

Parabolic projectivity ... 66

Parallel projection 2

Pascal line 122

Pascal's theorem . . . 121, 147

Pencil 8

Perspectivity 11

Plane, at infinity 5

of points 8

of symmetry 1 70

Plane figures 101

Point at infinity 5

Point involution 75

Polar 151, 154

Polar reciprocal 162

PAQB

Pole 151, 166

Prime forms 8

Principal axes 164

Principal diameter .... 164
Principle of duality . . 15, 162

Projection i, 7

Projectivity 41, 66

Projector 3, 7

Quadrangle 33

Quadric 110

Quadric cone 167

Quadric surface ... 93, 110, 178
Quadrilateral 34

Kange of points 8, 96

Reciprocity 16

Regulus 93, 176

Relation, of angles .... 23
of anharmonic ratios . . 26

of segments 22

Ruled surface 93

Section 3

Self-conjugate triangle . . . 158
Self-corresponding elements . 64
Self -polar triangle . . . . . 158

Sense 22,23

Sheaf 8

Similar figures 39

Similar ranges 69

Similitude, ratio of ... . 39
Skew surface . . 93, 111, 173, 178

Steiner's theorem 104

Superposed forms 63

Symbols 1, H, 41

Ten prime forms 8

Three-dimensional forms . . 8
Two-dimensional forms ... 8

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