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PRESENTED BY

Dr, Lloyd L* Dines

PEOJECTIVE GEOMETRY

OSWALD VEBLEN

PROFESSOR OP MATHEMATICS, PRINCETON UNIVERSITY
AND

JOHN WESLEY YOUNG

PROFESSOR OF MATHEMATICS, UNIVERSITY OF KANSAS

VOLUME I

GINN AND COMPANY

BOSTON * NEW YORK CHICAGO LONDON

ENTERED AT STATIONERS' HALL

fltfrenteum

GINN AND COMPANY PRO-
PRIETORS BOSTON U.S.A.

PREFACE

Geometry, which had been for centuries the most perfect example
of a deductive science, during the creative period of the nineteenth
century outgrew its old logical forms. The most recent period has
however brought a clearer understanding of the logical foundations
of mathematics and thus has made it possible for the exposition of
geometry to resume the purely deductive form. But the treatment
in the books which have hitherto appeared makes the work of lay-
ing the foundations seem so formidable as either to require for itself
a separate treatise, or to be passed over without attention to more
than the outlines. This is partly due to the fact that in giving the
complete foundation for ordinary real or complex geometry, it is
necessary to make a study of linear order and continuity, a study
which is not only extremely delicate, but whose methods are those
of the theory of functions of a real variable rather than of elemen-
tary geometry.

The present work, which is to consist of two volumes and is in-
tended to be available as a text in courses offered in American uni-
versities to upper-class and graduate students, seeks to avoid this
difficulty by deferring the study of order and continuity to the sec-
ond volume. The more elementary part of the subject rests on a
very simple set of assumptions which characterize what may be
called "general protective geometry." It will be found that the
theorems selected on this basis of logical simplicity are also elemen-
tary in the sense of being easily comprehended and often used.

Even the limited space devoted in this volume to the foundations
may seem a drawback from the pedagogical point of view of some
mathematicians. To this we can only reply that, in our opinion,
an adequate knowledge of geometry cannot be obtained without
attention to the foundations. "We believe, moreover, that the
abstract treatment is peculiarly desirable in protective geometry,
because it is through the latter that the other geometric disciplines
are most readily coordinated. Since it is more natural to derive

iii

iv PllEFAOE

the geometrical disciplines associated with the names of Euclid,
Descartes, Lobatchewsky, etc., from protective geometry than it
is to derive projective geometry from one of them, it is natural to
take the foundations of projective geometry as the foundations of
all geometry.

The deferring of linear order and continuity to the second vol-
ume has necessitated the deferring of the discussion of the metric
geometries characterized by certain subgroups of the general pro-
jective group. Such elementary applications as the metric proper-
ties of conies will therefore be found in the second volume. This
will be a disadvantage if the present volume is to be used for a
short course in which it is desired to include metric applications.
But the arrangement of the material will make it possible, when
the second volume is ready, to pass directly from Chapter VIII of
the first volume to the study of order relations (which may them-
selves be passed over without detailed discussion, if this is thought
desirable), and thence to the development of Euclidean metric
geometry. We think that much is to be gained pedagogioally as
well as scientifically by maintaining the sharp distinction between
the projective and the metric.

1 The introduction of analytic methods on a purely synthetic basis
in Chapter VI brings clearly to light the generality of the set of
assumptions used in this volume. What we call " general projective
geometry " is, analytically, the geometry associated with a general
number field. All the theorems of this volume are valid, not alone
in the ordinary real and the ordinary complex projective spaces, but
also in the ordinary rational space and in the finite spaces. The
bearing of this general theory once fully comprehended by the
student, it is hoped that he will gain a vivid conception of the
organic unity of mathematics, which recent developments of postu-
lational methods have so greatly emphasized.

The form of exposition throughout the book has been condi-
tioned by the purpose of keeping to the fore such general ideas as
group, configuration, linear dependence, the correspondence be-
tween and the logical interchangeability of analytic and synthetic
methods, etc. Between two methods of treatment we have chosen
the more conventional in all cases where a new method did nol
to have unquestionable advantages. We have tried also to

Jb'JbC.U.FAU.U V

avoid in general the introduction of new terminology. The use
of the word on in connection with duality was suggested by Pro-
fessor Frank Morley.

We have included among the exercises many theorems which in
a larger treatise would naturally have formed part of the text.
The more important and difficult of these have been accompanied
by references to other textbooks and to journals, which it is hoped
will introduce the student to the literature in a natural way. There
has been no systematic effort, however, to trace theorems to their
original sources, so that the book may be justly criticized for not
always giving due credit to geometers whose results have been
used.

Our cordial thanks are due to several of our colleagues and stu-
dents who have given us help and suggestions. Dr. H. H. Mitchell
has made all the drawings. The proof sheets have been read in whole
or in part by Professors Birkhoff, Eisenhart, and Wedderburn, of
Princeton University, and by Dr. R. L. Borger of the University
of Illinois. Finally, we desire to express to Ginn and Company our
sincere appreciation of the courtesies extended to us.

It is expected that the second volume will appear during the

coming year.

O. VEBLEN
J. W. YOUNG-

August, 1910

CONTENTS

INTRODUCTION

SECTION PAGE

1. Undefined elements and unproved propositions 1

2. Consistency, categoricalness, independence. Example of a mathematical

science 2

3. Ideal elements in geometry 7

4. Consistency of the notion of points, lines, and plane at infinity 9

5. Protective and metric geometry 12

CHAPTER I
THEOREMS OF ALIGNMENT AND THE PRINCIPLE OF DUALITY

6. The assumptions of alignment 15

7. The plane 17

8. The first assumption of extension 18

9. The three-space 20

10. The remaining assumptions of extension for a space of three dimensions . 24

11. The principle of duality 26

12. The theorems of alignment for a space of n dimensions 29

CHAPTER II
PROJECTION, SECTION, PERSPECTIVITY. ELEMENTARY CONFIGURATIONS

13. Projection, section, perspectivity 34"

14. The complete n-point, etc 36

15. Configurations 38

16. The Desargues configuration 39

17. Perspective tetrahedra 43

18. The quadrangle-quadrilateral configuration 44

19. The fundamental theorem on quadrangular sets 47

20. Additional remarks concerning the Desargues configuration 51

CHAPTER III

PROJEOTIVITIES OF THE PRIMITIVE GEOMETRIC FORMS OF ONE, TWO,
AND THREE DIMENSIONS

21. The nine primitive geometric forms 55

22. Perspectivity and projectivity 56

23. The projectivity of one-dimensional primitive forms 59

vii

viii CONTENTS

SECTION PAGE

24. General theory of correspondence. Symbolic treatment ....... 64

25. The notion of a group ..................... 06

26. Groups of correspondences. Invariant elements and figures ..... 07

27. Group properties of pro jectivi ties ................ 08

28. Protective transformations of two-dimensional forms ........ 71

29. Protective collineations of three-dimensional forms ......... 75

CHAPTER IV

HARMONIC CONSTRUCTIONS AND THE FUNDAMENTAL THEOREM OF
PROJECTIVE GEOMETRY

30. The projectivity of quadrangular sets .............. 70

31. Harmonic sets ........................ 80

32. Nets of rationality on a line .................. 84

33. Nets of rationality in the plane ................. 86

34. Nets of rationality in space ................... 80

35. The fundamental theorem of projectivity ............. 03

36. The configuration of Pappus. Mutually inscribed and circumscribed tri-

angles .......................... 08

37. Construction of projectivities on one-dimensional forms ....... 100

38. Involutions ......................... 102

39. Axis and center of homology .................. 103

40. Types of collineations in the plane ............... 106

CHAPTER V
CONIC SECTIONS

41. Definitions. Pascal's and Brianchon's theorems .......... 100

42. Tangents. Points of contact .................. 112

43. The tangents to a point conic form a line conic ........... 116

44. The polar system of a conic ................. . 120

45. Degenerate conies ...................... 120

46. Desargues's theorem on conies ................. 127

47. Pencils and ranges of conies. Order of contact .......... 128

CHAPTER VI
ALGEBRA OF POINTS AND ONJG-DIMENSIONAL COORDINATE SYSTEMS

48. Addition of points ...................... 141

49. Multiplication of points .................... 144

50. The commutative law for multiplication ............. 148

51. The inverse operations .................... 148

52. The abstract concept of a number system. Isomorphism ....... 149

53. Nonhomogeneous coordinates ................. 150

54. The analytic expression for a projectivity in a one-dimensional primitive

form ........................... 152

55. Von Staudt's algebra of throws ................. 157

CONTENTS ix

SECTION PAGE

56. The cross ratio 159

57. Coordinates in a net of rationality on a line , 1C2

58. Homogeneous coordinates on a line 163

59. Protective correspondence "between the points of two different lines . . 166

CHAPTEE VII
COORDINATE SYSTEMS IN TWO- AND THEEE- DIMENSIONAL FORMS

60. Nonhomogeneous coordinates in a plane 169

61. Simultaneous point and line coordinates 171

62. Condition that a point be on a line 172

63. Homogeneous coordinates in the plane 174

64. The line on two points. The point on two lines 180

65. Pencils of points and lines. Projectivity 181

66. The equation of a conic 185

67. Linear transformations in a plane . * 187

68. Collineations between two different planes 190

69. Nonhomogeneous coordinates in space 190

70. Homogeneous coordinates in space 194

71. Linear transformations in space 199

72. Finite spaces 201

CHAPTER VIII
PROJECTIVITIES IN ONJE- DIMENSIONAL FORMS

73. Characteristic throw and cross ratio 205

74. Projective projectivities 208

75. Groups of projectivities on a line 209

76. Projective transformations between conies 212

77. Projectivities on a conic 217

78. Involutions 221

79. Involutions associated with a given projectivity 225

80. Harmonic transformations 230

81. Scale on a conic 231

82. Parametric representation of a conic 234

CHAPTER IX
GEOMETRIC CONSTRUCTIONS. INVARIANTS

83. The degree of a geometric problem 236

84. The intersection of a given line with a given conic 240

85. Improper elements. Proposition K 2 241

86. Problems of the second degree 245

87. Invariants of linear and quadratic binary forms 251

88. Proposition K n 254

89. Taylor's theorem. Polar forms 255

X. CONTENTS

90. Invariants and covariants of binary forms 257

91. Ternary and quaternary forms and their invariants 258

92. Proof of Proposition K n 200

CHAPTER X
PROJECTIVE TRANSFORMATIONS OF TWO-DIMENSIONAL FORMS

93. Correlations between two-dimensional forms 202

94. Analytic representation of a correlation between two planes 200

95. General projective group. Representation by matrices 208

96. Double points and double lines of a collineation in a piano 271

97. Double pairs of a correlation 278

98. Fundamental conic of a polarity in a plane 282

99. Poles and polars with respect to a conic. Tangents 284

100. Various definitions of conies 285

101. Pairs of conies 287

102. Problems of the third and fourth degrees 294

CHAPTER XI
FAMILIES OF LINES

103. The regains 21)8

104. The polar system of a regulus 300

105. Projective conies 304

106. Linear dependence of lines 1)11

107. The linear congruence 312

108. The linear complex 319

109. The Pliicker line coordinates 327

110. Linear families of lines 329

111. Interpretation of line coordinates as point coordinates in S 6 331

INDEX 335

PROJECTIVE GEOMETRY

INTRODUCTION

1. Undefined elements and unproved propositions. Geometry deals
with the properties of figures in space. Every such figure is made up
of various elements (points, lines, curves, planes, surfaces, etc.), and
these elements bear certain relations to each other (a point lies on a
line, a line passes through a point, two planes intersect, etc.). The
propositions stating these properties are logically interdependent, and
it is the object of geometry to discover such propositions and to
exhibit their logical interdependence.

Some of the elements and relations, by virtue of their greater
simplicity, are chosen as fundamental, and all other elements and
relations are defined.in terms of them. Since any defined element or
relation must be defined in terms of other elements and relations,
it is necessary that one or more of the elements and one or more of
the relations between them remain entirely undefined; otherwise a
vicious circle is unavoidable. Likewise certain of the propositions
are regarded as fundamental, in the sense that all other propositions
are derivable, as logical consequences, from these fundamental ones.
But here again it is a logical necessity that one or more of the prop-
ositions remain entirely improved ; otherwise a vicious circle is again
inevitable.

The starting point of any strictly logical treatment of geometry
(and indeed of any "branch of mathematics) must then le a set of un-
defined elements and relations, and a set of unproved propositions
involving them ; and from these all other propositions (theorems) are
to le derived ly the methods of formal logic. Moreover, since we
assumed the point of view of formal (Le. symbolic) logic, the unde-
fined elements are to be regarded as mere symbols devoid of content,
except as implied by the fundamental propositions. Since it is mani-
festly absurd to speak of a proposition involving these symbols as

2 INTEODUCTION [INTROD.

self-evident, the unproved propositions referred to above must be re-
garded as mere assumptions. It is customary to refer to these funda-
mental propositions as axioms or postulates, but we prefer to retain the
term assiwiption as more expressive 'of their real logical character.

We understand the term a mathematical science to mean any set
of propositions arranged according to a sequence of logical deduction.
From the point of view developed above such a science is purely
abstract. If any concrete system of things may be regarded as sat-
isfying the fundamental assumptions, this system is a concrete ap-
plication or representation of the abstract science. The practical
importance or triviality of such a science depends simply on the
importance or triviality of its possible applications. These ideas will
be illustrated and further discussed in the next section, where it will
appear that an abstract treatment has many advantages quite apart
from that of logical rigor.

2. Consistency; categoricalness, independence. Example of a math-
ematical science. The notion of a class * of objects is fundamental
in logic and therefore in any mathematical science. The objects
which make up the class are called the elements of the class. The
notion of a class, moreover, and the relation of belonging to a class
(being included in a class, being an element of a class, etc.) are primi-
tive notions of logic, the meaning of which is not here called in
question.f

The developments of the preceding section may now be illustrated
and other important conceptions introduced by considering a simple
example of a mathematical science. To this end let S be a class, the
elements of which we will denote by A 9 B,C,... Further, let there
be certain undefined subclasses $ of S, any one of which we will call
an m-class. Concerning the elements of S and the m-classes we now
make the following

ASSUMPTIONS :

I. If A and J3 are distinct elements of S, there is at least one
m-class containing both A and B.

* Synonyms for class are set, aggregate, assemblage, totality; in German, Menge;
inPrench, ensemble.

1 Of. B. Russell, The Principles of Mathematics, Cambridge, 1003 ; and L, Cou-
turat, Les Principes des mathe'matiques, Paris, 1905.

$ A class S' is said to be a subclass of another class S, if every element of S' is
an element of S.

2] A MATHEMATICAL SCIENCE 3

II. If A and B are distinct elements of S, there is not more than
one m-class containing loth A and B.

III. Any two m-classes have at least one element of S in common.

IV. There exists at least one m-class. '

V. JEJvery m-class contains at least three elements of S.
VI. All the elements of S do not belong to the same m-class,

VII. No m-class contains more than three elements of S.

The reader will observe that in this set of assumptions we have
just two undefined terms, viz., element of S and m-class, and one
undefined relation, belonging to a class. The undefined terms, more-
over, are entirely devoid of content except such as is implied in the
assumptions.

Now the first question to ask regarding a set of assumptions is :
Are they logically consistent? In the example above, of a set of
assumptions, the reader will find that the assumptions are all true
statements, if the class S is interpreted to mean the digits 0, 1, 2, 3,
4, 5, 6 and the m-classes to mean the columns in the following table :

0123456

(1) 1234560

3456012

This interpretation is a concrete representation of our assumptions.
Every proposition derived from the assumptions must be true of this
system of triples. Hence none of the assumptions can be logically
inconsistent with the rest ; otherwise contradictory statements would
be true of this system of triples.

Thus, in general, a set of assumptions is said to le consistent if a
single concrete representation of the assumptions can le given*

Knowing our assumptions to be consistent, we may proceed to de-
rive some of the theorems of the mathematical science of which they
are the basis :

Any two distinct elements of S determine one and only one m-class
containing "both these elements (Assumptions I, II).

* It will be noted that this test for the consistency of a set of assumptions
merely shifts the difficulty from one <do main to another. It is, however, at present
the only test known. On the question as to the possibility of an absolute test of
consistency, cf. Hilbert, Grundlagen der Geometrie, 2d ed., Leipzig (1903), p. 18, and
Verhandlungen d. III. intern, math* Kongresses zu Heidelberg, Leipzig (1904),
p. 174; Padoa, L'Enseignement niathe'matique, Vol. V (1903), p. 85.

4 INTEODUCTJOK [INTBOD.

The m-class- containing the elements A and B may conveniently
be denoted by the symbol AB.

Any two m-classes have one and only one dement of S in common
(Assumptions II, III).

Tliere exist three elements of S which are not all in the same
m-class (Assumptions IV, V, VI).

In accordance with the last theorem, let A, B, C be three elements
of S not in the same m-class. By Assumption V there must be a
third element in each of the m-classes AB, BC, CA, and by Assump-
tion II these elements must be distinct from each other and from
A, By and C. Let the new elements be D, E, G, so that each of
the triples ABD, BCE, CAG belongs to the same m-class. By
Assumption III the m-classes AE and BG, which are distinct from
aH the m-classes thus far obtained, have an element of S ia common,
which, by Assumption II, is distinct from those hitherto mentioned ;
let it be denoted by F, so that each of the triples AEF and BIG
belong to the same m-class. No use has as yet been made of As-
sumption VII. We have, then, the theorem :

Any class S subject to Assumptions / VI contains at least seven
elements.

Now, making use of Assumption VII, we find that the m-classes
thus far obtained contain only the elements mentioned. The m-classes
CD and AEF have an element in common (by Assumption III)
which cannot be A or E, and must therefore (by Assumption VII)
be F. Similarly, AOG and the m-class DE have the element G in
common. The seven elements A, B, C, I), E, F, G have now been
arranged into m-classes according to the table

A B C D E F G

(!') B C D E F G A

D E F G A B G

in which the columns denote m-classes. The reader may note at once
that this table is, except for the substitution of letters for digits,
entirely equivalent to Table (1); indeed (I/) is obtained from (1) by
replacing by A, 1 by B, 2 by (7, etc. We can show, furthermore,
that S can contain no other elements than A, B, <7, D, E, F, G. For
suppose there were another element, T. Then, by Assumption III,

2] CATEGORICALKESS 5

the m-classes TA and BFG would have an element in common. This
element cannot be B> for then ABTD would belong to the same
w-class ; it cannot be F, for then AFTJE would all belong to the same
w-class; and it cannot be G, for then AGTC would all belong to the
same m-class. These three possibilities all contradict Assumption VII.
Hence the existence of T would imply the existence of four elements
in the wi-class BFG, which is likewise contrary to Assumption VII.

The properties of the class S and its m-classes may also be repre-
sented vividly by the accompanying figure (fig. 1). Here we have
represented the elements of S *by
points (or spots) in a plane, and
have joined by a line every triple
of these points which form an m-
class. It is seen that the points
may be so chosen that all but one
of these lines is a straight line.
This suggests at once a similarity
to ordinary plane geometry. Sup-
pose we interpret the elements of
S to be the points of a plane, and interpret the m-classes to be the
straight lines of the plane, and let us reread our assumptions with this
interpretation. Assumption VII is false, but all the others are true
with the exception of Assumption III, which is also true except when
the lines are parallel. How this exception can be removed we will
discuss in the next section, so that we may also regard the ordinary
plane geometry as a representation of Assumptions I- VI.

Returning to our miniature mathematical science of triples, we are
now in a position to answer another important question : To what eoo-
tent do Assumptions I VII characterize the class S and the m-classes ?
We have just seen that any class S satisfying these assumptions may
be represented by Table (I/) merely by properly labeling the ele-
ments of S. In other words, if S x and S 2 are two classes S subject
to these assumptions, every element of S x may be made to correspond *
to a unique element of S 2 , in such a way that every element of S 2
is the correspondent of a unique element of S y and that to every
w-class of Sj there corresponds an m-class of S r The two classes are

* The notion of correspondence is another primitive notion which we take over
without discussion from the general logic of classes.

6 INTBODUCTION" [INTROB.

then said to be in one-to-one reciprocal correspondence, or to be simply
isomorphic* Two classes S are then abstractly equivalent ; i.e. there
exists essentially only one class S satisfying Assumptions I- VII.
This leads to the following fundamental notion :

A set of assumptions is said to "be categorical, if there is essentially
only one system for which the assumptions are 'valid ; i.e. if any, two
such systems may le made simply isomorphic.

We have just seen that the set of Assumptions I-VII is categor-
ical. If, however, Assumption VII be omitted, the remaining set of
six assumptions is not categorical. We have already observed the
possibility of satisfying Assumptions I- VI by ordinary plane geom-
try. Since Assumption III, however, occupies as yet a doubtful posi-
tion in this interpretation, we give another, which, by virtue of its
simplicity, is peculiarly adapted to make clear the distinction between
categorical and noncategorical. The reader will find, namely, that
each of the first six assumptions is satisfied by interpreting the class S
to consist of the digits 0, 1, 2, - *, 12, arranged according to the fol-
lowing table of m-classes, every column constituting one m-class :

0123456 789 10 11 12

9 1234567891011120

^ 3 4 5 6 7 8 9 10 11 12 1 2

9 10 11 12 012 345 6 78

Henee Assumptions I-VI are not sufficient to characterize completely
the class S, for it is evident that Systems (1) and (2) cannot be made
isomorphic. On the other hand, it should be noted that all theorems
derivable from Assumptions I-VI are valid for both (1) and (2).
These two systems are two essentially different concrete representa-
tions of the same mathematical science.

This brings us to a third question regarding our assumptions : Are
they independent ? That is, can any one of them be derived as a log-
ical consequence of the others ? Table (2) is an example which shows
that Assumption VII is independent of the others, because it shows
that they can all be true of a system in which Assumption VII is
false. Again, if the class S is taken-to mean the three letters A, B, (7,

* The isomorphism of Systems (1) and (1") is clearly exhibited in fig, 1, where
each point is labeled both with a digit and with a letter. This isomorphism may,
moreover, be established in 7-64 different ways.

2,3] - IDEAL ELEMENTS 7

and the m-classes to consist of the pairs AB, JBC, CA, then it is
clear that Assumptions I, II, III, IV, VI, VII are true of this class
S, and therefore that any logical consequence of them is true with
this interpretation. Assumption V, however, is false for this class,
and cannot, therefore, be a logical consequence of the other assump-
tions. In like manner, other examples can be constructed to show
that each of the Assumptions I- VII is independent of the remain-
ing ones.

3. Ideal elements in geometry. The miniature mathematical science
which we have just been studying suggests what we must do on a
larger scale in a geometry which describes our ordinary space. We
must first choose a set of undefined elements and a set of funda-
mental assumptions. This choice is in no way prescribed a priori,
but, on the contrary, is very arbitrary. It is necessary only that the
undefined symbols be such that all other elements and relations that
occur are definable in terms of them ; and the fundamental assump-
tions must satisfy the prime requirement of logical consistency, and
be such that all other propositions are derivable from them by formal
logic. It is desirable, further, that the assumptions be independent*
and that certain sets of assumptions be categorical. There is, further,
the desideratum of utmost symmetry and generality in the whole
body of theorems. The latter means that the applicability of a theo-
rem shall be as wide as possible. This has relation to the arrange-
ment of the assumptions, and can be attained by using in the proof
of each theorem a minimum of assumptions-!

Symmetry can frequently be obtained by a judicious choice of
terminology. This is well illustrated by the concept of "points at
infinity" which is fundamental in any treatment of projective geome-
try. Let us note first the reciprocal character of the relation expressed
by the two statements :

A point lies on a line. A line passes through a point.

To exhibit clearly this reciprocal character, we agree to use the phrases
A point is on a line ; A line is on a point

'* This is obviously necessary for the precise distinction between an assumption
and a theorem.

t If the set of assumptions used in the proof of a theorem is not categorical, the
applicability of the theorem is evidently -wider than in the contrary case. Cf . exam-
ple of preceding section*

g INTRODUCTION [INTROD.

to express this relation. Let us now consider the following two
propositions :

1. Any two distinct points of 1'. Any two distinct lines of a

a plane are on one and only one plane are on one and only one

Urn* P int -

Either of these propositions is obtained from the other by simply
interchanging the words point and line. The first of these propositions
we recognize as true without exception in the ordinary Euclidean
geometry. The second, however, has an exception when the two
lines are parallel. In view of the symmetry of these two propositions
it would clearly add much to the symmetry and generality of all
propositions derivable from these two, if we could regard them both
as true without exception. This can be accomplished by attributing
to two parallel lines apoint of intersection. Such a point is not,
of course, a point in the ordinary sense ; it is to be regarded as an
ideal point, which we suppose two parallel lines to have in common.
Its introduction amounts merely to a change in the ordinary termi-
nology. Such an ideal point we call a point at infinity ; and we
suppose one such point to exist on every line.f

The use of this new term leads to a change in the statement,
though not in the meaning, of many familiar propositions, and makes
us modify the way in which we think of points, lines, etc. Two non-
parallel lines cannot have in common a point at infinity without
doing violence to propositions 1 and 1'; and since each of them has a
point at infinity, there must be at least two such points. Proposition
1, then, requires that we attach a meaning to the notion of a line on
two points at infinity. Such a line we call a line at infinity, and
think of it as consisting of all the points at infinity in a plane.
In like manner, if we do not confine ourselves to the points of a
single plane, it is found desirable to introduce the notion of a plane
through three points at infinity which are not all on the same line
at infinity. Such a plane we call a plane at infinity, and we think

* By line throughout we mean straight line.

f It should be noted that (since we are taking the point of view of Euclid) we do
not think of a line as containing more than one point at infinity ; for the supposi-
tion that a line contains two such points would imply either that two parallels can
be drawn through a given point to a given line, or that two distinct lines can have
more than one point in common.

3,4] CONSISTENCY OF IDEAL ELEMENTS 9

of it as consisting of all the points at infinity in space. Every ordi-
nary plane is supposed to contain just one line at infinity ; every sys-
tem of parallel planes in space is supposed to have a line at infinity
in common with the plane at infinity, etc.

The fact that we have difficulty in presenting to our imagination
the notions of a point at infinity on a line, the line at infinity in a
plane, and the plane at infinity in space, need not disturb us in this
connection, provided we can satisfy ourselves that the new terminol-
ogy is self-consistent and cannot lead to contradictions. The latter
condition amounts, in the treatment that follows, simply to the con-
dition that the assumptions on which we build the subsequent theory
be consistent. That they are consistent will be shown at the time
they are introduced. The use of the new terminology may, however,
be justified on the basis of ordinary analytic geometry. This we
do in the next section, the developments of which will, moreover,
be used frequently in the sequel for proving the consistency of the
assumptions there made.

4. Consistency of the notion of points, lines, and plane at infinity.
We will now reduce the question of the consistency of our new ter-
minology to that of the consistency of an algebraic system. For this
purpose we presuppose a knowledge of the elements of analytic geom-
etry of three dimensions.* In this geometry a point is equivalent
to a set of three numbers (x, y, z). The totality of all such sets of
numbers constitute the analytic space of three dimensions. If the
'numbers are all real numbers, we are dealing with the ordinary "real"
space ; if they are any complex numbers, we are dealing with the ordi-
nary " complex " space of three dimensions. The following discussion
applies to either case.

A plane is the set of all points (number triads) which satisfy a
single linear equation

ax + ly + cz + d = 0.

A Ime is the set of all points which satisfy two linear equations,

a^x + \y + CjZ + d^ 0,
0,

* Such knowledge is not presupposed elsewhere in this hook, except in the case
of consistency proofs. The elements of analytic geometry are indeed developed
from the beginning (cf. Chaps. VI, VH).

10 INTRODUCTION [INTEOD.

provided the relations

do not hold.*

Now the points (x, y, z), with the exception of (0, 0, 0), may also be
denoted by the direction cosines of the line joining the point to the
origin of coordinates and the distance of the point from the origin ;

say by f

I, m,n, -)>
dj

where d = V^+^+r, and I = -> m = --> n = ^ The origin itself

d d cL

may be denoted by (0, 0, 0, k), where i is arbitrary. Moreover, any
four numbers (x v aj a , 8 , a 4 ) (x = 0), proportional respectively to

I, m, n, - )> will serve equally well to represent the point (x, y, ),

provided we agree that (x v x 2 , x s , x 4 ) and (cx v cx^ c% s , cx 4 ) represent
the same point for all values of c different from 0. For a point
(x 9 y, z) determines

ex 7 c?/

J cmi,

where e is arbitrary (c = 0), and (x v x v x v a; 4 ) determines

/-t \ ^1 ^o ^o

(1) # = -^y=:->;s:=-3>

aj 4 a? 4 as,

provided a? 4 = 0.

We have not assigned a meaning to (x v # 2 , x s , x 4 ) when ^ 4 = 0, but

it is evident that if the point ( d, cm, en, - ) moves away from the

\ *v

origin an unlimited distance on the line whose direction cosines are
/, m, n, its coordinates approach (cl, cm, en, 0). A little consideration
will show that as a point moves on any other line with direction

* It should be noted that we are not yet, in this section, supposing anything
known regarding points, lines, etc., at infinity, but are placing ourselves on the
"basis of elementary geometry.

4] CONSISTENCY OF IDEAL ELEMENTS 11

cosines Z, m, n, so that its distance from the origin increases indefi-
nitely, its coordinates also approach (cl, cm, en, 0). Furthermore, these
values are approached, no matter in which of the two opposite direc-
tions the point moves away from the origin. We now define (x v # 3 ,
# 8 , 0) as a point at infinity or an ideal point. We have thus associ-
ated with every set of four numbers (x v x z , x s , x^ a point, ordinary
or ideal, with the exception of the set (0, 0, 0, 0), which we exclude
entirely from the discussion. The ordinary points are those for which
x is not zero, and its ordinary Cartesian coordinates are given by the
equations (1). The ideal points are those for which # 4 = 0. The num-
bers (x v # 2 , # 3 , x) we call the homogeneous coordinates of the point.
We now define a plane to be the set of all points (x v # 2 , # 8 , # 4 )
which satisfy a linear homogeneous equation :

ax^+ bx% -f cx 9 + dx^ = 0.

It is at once clear from the preceding discussion that as far as all
ordinary points are concerned, this definition is equivalent to the one
given at the beginning of this section. However, according to this
definition all the ideal points constitute a plane # 4 = 0. This plane
we call the plane at infinity. In like manner, we define a line to
consist of all points (x v x 2 , X B> 4 ) which satisfy two distinct linear
homogeneous equations :

a^+ \x 2 + c&+ d&= 0,
cx + d = 0.

Since these expressions are to be distinct, the corresponding coefficients
throughout must not be proportional. According to this definition
the points common to any plane (not the plane at infinity) and the
plane # 4 = constitute a line. Such a line we call a line at infinity,
and there is one such in every ordinary plane. Finally, the line de-
fined above by two equations contains one and only one point with
coordinates (x v x v x s , 0) ; that is, an ordinary line contains one and
only one point at infinity. It is readily seen, moreover, that with
the above definitions two parallel lines have their points at infinity
in common.

Our discussion has now led us to an analytic definition of what
may be called, for the present, an analytic projectile space of three
dimensions. It consists of :

12 INTRODUCTION [INTROD.

Points : All sets of four numbers (x v x# X B) # 4 ), except the set
(0, 0, 0, 0), where (cx v cx^ cx z) cxj is regarded as identical with
(x# # 2 , s g , # 4 ), provided c is not zero.

Planes: All sets of points satisfying one linear homogeneous
equation.

lines: All sets of points satisfying two distinct linear homoge-
neous equations.

Such a protective space cannot involve contradictions unless our
ordinary system of real or complex algebra is inconsistent. The defi-
nitions here made of points, lines, and the plane at infinity are,
however, precisely equivalent to the corresponding notions of the
preceding section. We may therefore use these notions precisely in
the same way that we consider ordinary points, lines, and planes.
Indeed, the fact that no exceptional properties attach to our ideal
elements follows at once from the symmetry of the analytic formu-
lation; the coordinate # 4 , whose vanishing gives rise to the ideal
points, occupies no exceptional position in the algebra of the homo-
geneous equations. The ideal points, then, are not to be regarded
as different from the ordinary points.

All the assumptions we shall make in our treatment of projective
geometry will be found to be satisfied by the above analytic creation,
which therefore constitutes a proof of the consistency of the assump-
tions in question. This the reader will verify later.

5. Projective and metric geometry. In projective geometry no
distinction is made between ordinary points and points at infinity,
and it is evident by a reference forward that our assumptions pro-
vide for no such distinction. We proceed to explain this a little
more fully, and will at the same time indicate in a general way
the difference between projective and the ordinary Euclidean metric
geometry.

Confining ourselves first to the plane, let m and m r be two distinct
lines, and P a point not on either of the two lines. Then the points
of m may be made to correspond to the points of m/ as follows : To
every point A on m let correspond that point A r on m f in which m f
meets the line joining A to P (fig. 2). In this way every point on
either line is assigned a unique corresponding point on the other
line. This type of correspondence is called perspective, and the points
on one line are said to be transformed into the points of the other by

4, 5]

PEOJECTIYE AND METRIC GEOMETBY

a perspective transformation with center P. If the points of a line m
be transformed into the points of a line m 1 by a perspective transfor-
mation with center P, and then the points of m 1 be transformed into the
points of a third line m ff by a perspective transformation with a new
center Q ; and if this be continued any finite number of times, ulti-
mately the points of the line m will have been brought into corre-
spondence with the points of a line m (n) , say, in such a way that every
point of m corresponds to a unique point of m (n \ A correspondence
obtained in this way is called projectile, and the points of m are said

to have been transformed into the points of m (n) by a protective
transformation.

Similarly, in three-dimensional space, if lines are drawn joining
every point of a plane figure to a fixed point P not in the plane TT
of the figure, then the points in which this totality of lines meets
another plane 7r f will form a new figure, such that to every point of
TT will correspond a unique point of TT', and to every line of TT will
correspond a unique line of TT'. We say that the figure in TT has been
transformed into the figure in TT' by & perspective transformation with
center P. If a plane figure be subjected to a succession of such per-
spective transformations with different centers, the final figure will
still be such that its points and lines correspond uniquely to the
points and lines of the original figure. Such a transformation is again
called a projectile transformation. In projective geometry two figures
that may be made to correspond to each other by means of a projec-
tive transformation are not regarded as different. In other words,

14 INTRODUCTION [INTROD.

protective geometry is concerned with those properties of figures that
are left unchanged when the figures are subjected to a projective
transformation.

It is evident that no properties that involve essentially the notion
of measurement can have any place in projective geometry as such ;*
hence the term projective, to distinguish it from the ordinary geom-
etry, which is almost exclusively concerned with properties involving
the idea of measurement. In case of a plane figure, a perspective
transformation is clearly equivalent to the change brought about in
the aspect of a figure by looking at it from a different angle, the
observer's eye being the center of the perspective transformation.
The properties of the aspect of a figure that remain unaltered when
the observer changes his position will then be properties with which
projective geometry concerns itself. For this reason von Staudt called
this science Geometric der Lage.

In regard to the points and lines at infinity, we can now see why
they cannot be treated as in any way different from the ordinary
points and lines of a figure. For, in the example given of a per-
spective transformation between lines, it is clear that to the point at
infinity on m corresponds in general an ordinary point on m ! 9 and
conversely. And in the example given of a perspective transforma-
tion between planes we see that to the line at infinity in one plane
corresponds in general an ordinary line in the other. In projective
geometry, then, there can be no distinction between the ordinary
and the ideal elements of space.

* The theorems of metric geometry may however be regarded as special cases
of projective theorems.

CHAPTER I

THEOREMS OF ALIGNMENT AND THE PRINCIPLE OF DUALITY

6. The assumptions of alignment. In the following treatment of
protective geometry we have chosen the point and the line as unde-
fined elements. We consider a class (cf. 2, p. 2) the elements of
which we call points, and certain undefined classes of points which
we call lines. Here the words point and line are to be regarded
as mere symbols devoid of all content except as implied in the as-
sumptions (presently to be made) concerning them, and which may
represent any elements for which the latter may le valid propositions.
In other words, these elements are not to be considered as having
properties in common with the points and lines of ordinary Euclidean
geometry, except in so far as such properties are formal logical conse-
quences of explicitly stated assumptions.

We shall in the future generally use the capital letters of the
alphabet, as A, B, C, P, etc., as names for points, and the small let-
ters, as a, &, c, Z, etc., as names for lines. If A and B denote the same
point, this will be expressed by the relation A = B ; if they repre-
sent distinct points, by the relation A^B. If A = B, it is sometimes
said that A coincides with B, or that A is coincident with B. The
same remarks apply to two lines, or indeed to any two elements of
the same kind.

All the relations used are defined in general logical terms, mainly
by means of the relation of 'belonging to a class and the notion of one-
to-one correspondence. In case a point is an element of one of the
classes of points which we call lines, we shall express this relation
by any one of the phrases : the point is on or lies on or is a point of
the line, or is united with the line ; the line passes through or con-
tains or is united with the point. We shall often find it convenient
to use also the phrase the line is on the point to express this relation.
Indeed, all the assumptions and theorems in this chapter will be
stated consistently in this way. The reader will quickly become ac-
customed to this " on " language, which is introduced with the purpose

16 THEOREMS OF ALIGNMENT AND DUALITY [CHAJP.I

of exhibiting in its most elegant form one of the most far-reaching
theorems of projective geometry (Theorem 11). Two lines which have
a point in common are said to intersect in or to meet in that point, or
to le on a common point. Also, if two distinct points lie on the same
line, the line is said to join the points. Points which are on the
same line are said to be collinear ; points which are not on the same
line are said to be noncollinear. Lines which are on the same point
(i.e. contain the same point) are said to be copunctal, or concurrent*
Concerning points and lines we now make the following assump-
tions :

THE ASSUMPTIONS OF ALIGNMENT, A :

A 1. If A and B are distinct points, there is at least one line on
both A and B.

A 2. If A and B are distinct points, there is not more than one
line on both A and B.

A3. If A, B, are points not all on the same line, and D and
E (D = E] are points such that B, C, D are on a line and C, A, E

are on a line, there is a point F
such that A, B, F are on a line
and also D, E, F arc on a line
(fig. 3).f

It should be noted that this set
of assumptions is satisfied by the
triple system (1), p. 3, and also
by the system of quadruples (2),

p. 6, as well as by the points and lines of ordinary Euclidean geom-
etry with the notion of " points at infinity " (cf. 3, p. 8), and by

* The object of this paragraph is simply to define the terms in. common use in
terms of the general logical notion of belonging to a class. In later portions of
this book we may omit the explicit definition of such common terms when such
definition is obvious.

f The figures are to be regarded as a concrete representation of our science, in
which the undefined " points" and "lines" of the science are represented by
points and lines of ordinary Euclidean geometry (this requires the notion of ideal
points; cf. 3, p. 8). Their function is not merely to exhibit one of the many
possible concrete representations, but also to help keep in mind the various rela-
tions in question. In using them, however, great care must be exercised not
to use any properties of such figures that are not formal logical consequences
of the assumptions ; in other words, care must be taken that all deductions are
made formally from the assumptions and theorems previously derived from the
assumptions* *

7] THE PLANE 17

the " analytic projective space " described in 4. Any one of these
representations shows that our set of Assumptions A is consistent*

The following three theorems are immediate consequences of the
first two assumptions.

THEOREM 1. Two distinct points are on one and only one line.
(Al,A2)f

The line determined by the points A, B (A = B) will often be
denoted by the symbol or name AB.

THEOREM 2. If C and D (C = D) are points on the line AB, A and
B are points on the line CD. (Al, A 2)

THEOREM 3. Two distinct lines cannot be on more than one common
point. (A 1, A 2)

Assumption A3 will be used in the derivation of the next theo-
rem. It may be noted that under Assumptions A 1, A 2 it may be
stated more conveniently as follows : If A 9 B, C^are ^points not jtll^on
the samej-ine, the line joining any point -Z> on the. line BO to _any
point JEJ (D = E) on the line CA meets the line AB in a point F.
This is the form in which this assumption" is~ generally "used in the
sequel.

7. The plane. DEFINITION. If P t Q, E are three points not on
the same line, and I is a line joining Q and R, the class S 2 of all
points such that every point of S 2 is collinear with P and some
point of I is called the plane determined by P and I.

We shall use the small letters of the Greek alphabet, a, /3, y, TT, etc.,
as names for planes. It follows at once from the definition that P and
every point of I are points of the plane determined by P and I.

THEOREM 4. If A and B are points on a plane TT, then every point
on the line AB is on TT. (A)

Proof. Let the plane TT under consideration be determined by the
point P and the line I

* In the multiplicity of the possible concrete representations is seen one of the
great advantages of the formal treatment quite aside from that of logical rigor. It
is clear that there is a great gain in generality as long as the fundamental assump-
tions are not categorical (cf. p. 6). In the present treatment our assumptions are
not made categorical until yery late.

t The symbols placed in parentheses after a theorem indicate the assumptions
needed in its proof. The symbol A will be used to denote the whole set of Assump-
tions A 1, A 2, A3.

FIG.

18 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I

1. If both A and B are on Z, or if the line AB contains P, the
theorem is immediate.

2. Suppose A is on I, B not on I, and AB does not contain P (fig. 4).
Since B is a point of w, there is a point 2?' on I collinear with B and P.

If (7 be any point on AB 9 the line
joining C on -4J5 to P on .B1?'
will have a point T in common
with AB'=l (A 3). Hence is a
point of TT.

3. Suppose neither A nor .2? is
on I and that -4J? does not con-
tain P (fig. 5). Since A and .# are
points of TT, there exist two points
J 7 and ff on Z collinear with A, P and 5, P respectively. The line join-
ing A on A'P to .# on PB 1 has a point Q in common with J?'.4' (A 3),
Hence every point of the line AB AQ
is a point of TT, by the preceding case.
This completes the proof.
If all the points of a line are points
of a plane, the line is said to be a line of
the plane, or to lie in or to le in or to
le on the plane; the plane is said to
pass through, or to contain the line,
or we may also say the plane is on the
line. Further, a point of a plane is said
to le in or to lie in the plane, and the
plane is on the point

8. The first assumption of extension. The theorems of the pre-
ceding section were stated and proved on the assumption (explicitly
stated in each case) that the necessary points and lines exist. The
assumptions of extension, M, insuring the existence of all the points
which we consider, will be given presently. The first of these, how-
ever, it is desirable to introduce at this point.
AN ASSUMPTION OF EXTENSION :
E 0. There are at least three points on every line.
This assumption is needed in the proof of the following
THEOREM 5. Any two lines on the same plane TT are on a common
point. (A, E 0)

FIG. 5

ASSUMPTION OF EXTENSION

Proof. Let the plane TT be determined by the point P and the line I,
and let a and 6 be two distinct lines of TT.

1. Suppose a coincides with I (fig. 6). If I contains P, any point

B of 6 (E 0) is colHnear with P and
some point of Z=a, which proves the
theorem when I contains P. If I does
not contain P } there exist on 5 two
points A and I? not on I (E 0), and
since they are points of TT, they are
collinear with P and two points A 1
and B ! of / respectively. The line
joining A on A'P to .# on PB 1 has a
point JS in common with A'B 1 (A 3)
ine^ theplane

PIG-. 6

i.e. I = a and & have a point in common. Hence
TT has a point in common with L
2. Let a and 5 both be distinct
from L (i) Let a contain P (fig. 7).
The line joining P to any point
B of 5 (E 0) has a point B r in com-
mon with I (Case 1 of this proof).
Also the lines a and 5 have points
,4' and R respectively in common
with I (Case 1). Now the line
A!P = a contains the points A J of
EB } and P of ^^ and hence has a point A in common with

JFia. 7

Hence

in common

= Z>.

h as a point
it]^ any line om

gA P. (ii) Let neither a nor
contain P (fig. 8). As before,
& and 6 meet Z in two points Q
and 2 respectively. Let B 1 be a
point of I distinct from Q and E
(E 0). The line PB r then meets
a and 6 in two points A and .Z?
respectively (Case 2, (i)). If
J[ = B y the theorem is proved. If A = .#, the line 6 has the point
E in common with Q^ r and the point B in common with B f A, and
hence has a point in common with AQ = a (A3).

20 THEOREMS OF ALIGNMENT AND DUALITY [CHAP, i

THEOREM 6. Tlie plane a determined ly a line I and a point P is
identical with the plane /3 determined ly a line m and a point Q,
provided m and Q are on a. (A, E 0)

Proof. Any point B of /3 is collinear with Q and a point A of m
(fig. 9). A and Q are both points of a, and hence every point of the

line AQ is a point of a (Theorem 4).
Hence every point of /3 is a point
of a. Conversely, let B be any point
of a. The line BQ meets w in a
point (Theorem 5). Hence every
point of a is also a point of /J.

COROLLARY. There is one and only
one plane determined "by three non-
collinear points, or ly a line and a
point not on the line, or ly two inter-
secting lines. (A, E 0)
The data of the corollary are all equivalent by virtue of EO. We
will denote by ABC the plane determined by the points A, B, C\
by aA the plane determined by the line a and the point A 9 etc.

THEOREM 7. Two distinct planes which are on two common points
A y B (A = 3) are on all the points of the line AB, and on no other com-
, mon points. (A, EO)

Proof. By Theorem 4 the line AS lies in each of the two planes,
which proves the first part of the proposition. Suppose C 9 not on AB,
were a point common to the two planes. Then the plane determined by
,A, B, C would be identical with each of the given planes (Theorem 6),
which contradicts the hypothesis that the planes are distinct.

COROLLARY. Two distinct planes cannot "be on more than one com-
mon line. (A, E 0)

9. jnbiejhjee-space. DEFINITION. If P, Q, E, T are four points
not in the same plane, and if TT is a plane containing Q, R, and T,
the class S g of all points such that every point of S 3 is- collinear with P
and some point of TT is called the space of three dimensions , or the
three-space determined by P and TT.

If a point belongs to a three-space or is a point of a three-space, it
is said to le in or to lie in or to le on the three-space. If all the points
of a line or plane are points of a three-space S 3 , the line or plane is said

THE THREE-SPACE

FHK 10

to lie in or to le in or to le on the S 3 . Also the three-space is said to
le on the point, line, or plane. It is clear from the definition that P and
every point of TT are points of the three-space determined by P and TT.

THEOREM 8. If A and B are distinct points on a three-space S s ,
every point on the line AB is on S 3 . (A)

Proof. Let S 3 be determined by
a plane TT and a point P.

1. If A and B are both in TT, the
theorem is an immediate conse-
quence of Theorem 4.

2. If the line AB contains P,
the theorem is obvious.

3. Suppose A is in TT, B not in
TT ; and AB does not contain P
(fig. 10). There then exists a point
S f (= A) of TT collinear with B

and P (def.). The line joining any point M on AB to P on BB f has
a point M f in common with 2^-4 (A 3). But M f is a point of w, since
it is a point of AB f . Hence jYis a point of S 3 (del).

4. Let neither A nor J5 lie in TT, and let AB not contain P (fig. 11).
The lines PA and P# meet TT in
two points A r and B r respectively.
But the line joining A on A r P to
B on P-B' has a point C in common
with B r A r . C is a point of TT, which
reduces the proof to Case 3.

It may be noted that in this
proof no use has been made of E 0.

In discussing Case 4 we have
proved incidentally, in connection
with EO and 'Theorem 4, the fol-
lowing corollary:

COROLLARY 1. If S 3 is a three-space determined ly a point P and a
plane TT, then TT and any line on S 3 but not on TT are on one and only
one common point (A, E 0)

COROLLARY 2. JEvery point on any plane determined by three non-
eollinear points on a three-space S 8 is on S s . (A)

FIG. 11

22 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I

Proof. As before, let the three-space be determined by TT and P,
and let the three noncollinear points 'be A, B, 0. Every point of the
line BC is a point of S s (Theorem 8), and every point of the plane
ABC* is collinear with A and some point of BC.

COROLLARY 3. If a three-space S 8 is determined ly a point P and
a plane TT, then TT and any plane on S 3 distinct from TT are on one
and only one common line. (A, E 0)

Proof. Any plane contains at least three lines not passing through
the same point (del, A 1). Two of these lines must meet TT in two
distinct points, which are also
points of the plane of the lines
(Cor. 1). The result then follows
from Theorem 7.

THEOREM 9. If a plane a and
a line a not on a are on the same
threes-space S gJ then a and a are
on one and only one common point.
(A,EO)

Proof. Let S 3 be determined by
the plane TT and the point P.

1. If a coincides wither, the theo-
rem reduces to Cor. 1 of Theorems.

2. If # is distinct from TT, it has

a line I in common with TT (Theorem 8, Cor. 3). Let A be any point
on a not on a (EO) (fig. 12), The plane aA, determined by A and a,
meets TT in a line m & I (Theorem 8, Cor. 3). The lines I, m have
a point B in common (Theorem 5). The line AB in aA meets a in
a point Q (Theorem 5), which is on a, since AB is on a. That a
and a have no other point in common follows from Theorem 4.

COROLLARY 1. Any two distinct planes on a three-space are on one
and only one common line. (A, E 0)

The proof is similar to that of Theorem 8, Cor. 3, and is left as an
exercise.

COROLLARY 2. Conversely, if two planes are on a common line, there
exists a three-space on loth. (A, E 0)

* The proof can evidently be so worded as not to imply Theorem 6.

FIG. 12

9] THE THREE-SPACE 23

Proof. If the planes a and ft are distinct and have a line I in
common, any point P of /3 not on I "will determine with a a three-
space containing I and P and hence containing ft (Theorem 8, Cor. 2).

COROLLARY 3. Three planes on a three-space ivhich are not on a
common line are on one and only one common point. (A, E 0)

Proof. This follows without difficult} 7 * from the theorem and Cor. 1.

Two planes are said to determine the line which they have in com-
mon, and to intersect or meet in that line. Likewise if three planes
have a point in common, they are said to intersect or meet in the point.

COROLLARY 4. If a, y3, 7 are three distinct planes on the same S 3
but not on the same line, and if a line I is on each of t'wo planes p, v
which are on the lines /3<y and ya respectively, then it is on a plane \

which is on the line aft. (A, E 0) p ,

r Q

Proof. By Cor. 3 the planes a, I ,r

j3, 7 have a point P in common, /\| 7 ~~7 7T7

so that the lines /3y, ya, a/3 all XT ^-9 -f- -i

contain P. The line I, being com- / X v / s x / /

mon to planes through @y and ya,
must pass through P, and the
lines I and a/3 therefore intersect
in P and hence determine a plane
\ (Theorem 6, Cor.).

THEOREM 10. The three-space
S 3 determined by a plane TT, and

a point P is identical with the three-space S determined 1y a plane
7r f and a point P', provided TT' and P f are on S 3 . (A, EO)

Proof. Any point A of S^ (fig. 13) is collinear with P 1 and some
point A f of TT'; but P f and A f are both points of S 3 and hence A is a
point of S 3 (Theorem 8). Hence every point of S^ is a point of S s .
Conversely, if A is any point of S 3 , the line AP f meets TT' in a point
(Theorem 9). Hence every point of S s is also a point of S.

COROLLARY. TJiere is one and only one three-space on four given
points not on the same plane, or a plane and a line not on the plane,
or two nonintersecting lines. (A, E 0)

The last part of the corollary follows from the fact that two
nonintersecting lines are equivalent to four points not in the same
plane (EO).

24 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I

It is convenient to use tlie term coplanar to describe points in the
same plane. And we shall use the term skew lines for lines that have
no point in common. Four noncoplanar points or two skew lines
are said to determine the three-space in which they lie.

10. The remaining assumptions of extension for a space of three
dimensions. In 8 we gave a first assumption of extension. We will
now add the assumptions which insure the existence of a space of
three dimensions, and will exclude from our consideration spaces of
higher dimensionality.

ASSUMPTIONS OF EXTENSION, E :

E.l. There exists at least one line.

E 2. All points are not on the same line.

E 3. All points are not on the same plane.

E 3'. If S 3 is a three-space, every point is on S 3 .

The last may be called an assumption of closure.*

The last assumption might be replaced by any one of several equiv-
alent propositions, such as for example :

Every set of five points lie on the same three-space ; or

Any two distinct planes have a line in common. (Of. Cor. 2, Theo-
rem 9)

There is no logical difficulty, moreover, in replacing the assumption
(E3 ; ) of closure given above by an assumption that all the points
are not on the same three-space, and then to define a " four-space "
in a manner entirely analogous to the definitions of the plane and
to the three-space already given. And indeed a meaning can be given
to the words point and line such that this last assumption is satisfied
as well as those that precede it (excepting E3 ; of course). We
could thus proceed step by step to define the notion of a linear
space of any number of dimensions and derive the fundamental
properties of alignment for such a space. But that is aside from our
present purpose. The derivation of these properties for a four-space
will furnish an excellent exercise, however, in the formal reasoning
here emphasized (cf. Ex. 4, p. 25). The treatment for the ^dimensional
case will be found in 12, p. 29.

* The terms extension and closure in this connection were suggested by N. J. Lennes.
It will be observed that the notation has been so chosen that Ei insures the exist-
ence of a space of i dimensions, the line and the plane, being regarded as spaces of
one and two dimensions respectively.

10] ASSUMPTIONS OF EXTENSION 25

The following corollaries of extension are readily derived from the
assumptions just made. The proofs are left as exercises.

COROLLARY 1. At least three coplanar lines are on every point.
COROLLARY 2. At least three distinct planes are on every line.
COROLLARY 3. All planes are not on the same line.
COROLLARY 4. All planes are not on the same point.
COROLLARY 5. If $ 3 is a three-space, every plane is on $ 8 .

EXERCISES

1. Prove that through a given point P not on either of two skew lines I
and I' there is one and only one line meeting both the lines Z, V.

2. Prove that any two lines, each of which meets three given skew lines,
are skew to each other.

_^ 3. Our assumptions do not as yet determine whether the number of points
on a line is finite or infinite. Assuming that the number of points on one line
is finite and equal to n + 1, prove that

i. the number of points on every line is n + 1;
ii. the number of points on every plane is n 2 + n + 1;
iii. the number of points on every three-space is n 8 + n 2 + n + 1;
*& iv. the number of lines on a three-space is (n 2 + 1) (n z + n + 1);

v. the number of lines meeting any two skew lines on a three-space is

*< vi. the number of lines on a point or on a plane is n 2 + n + 1.
- -* 4. Using the definition below, prove the following theorems of alignment for
a four-space on the basis of Assumptions A and E :

DEFINITION. If P, Q, R, S, T are five points not on the same three-space,
and S 3 is a three-space on Q, R, S, T, the class S 4 of a.11 points such that every
point of S 4 is collinear with P and some point of S 3 is called the four-space
determined by P and S 8 .

i. If A and B are distinct points on a four-space, every point on the line A B
is on the four-space. \

ii. Every line on a four-space which is not on a given three-space of the
four-space has one and only one point in common with the three-space.

iii. Every point on any plane determined by three noncollinear points on
a four-space is on the four-space.

iv. Every point on a three-space determined by four noncoplanar points
of a four-space is on the four-space.

v. Every plane of a four-space determined by a point P and a three-space
S 3 has one and only one Jjne in common with S 3 , provided the plane is not on S 3 .

vi. Every three-space on a four-space determined by a point P and a three-
space S 3 has one and only one plane in common with S 3 , provided it does
not coincide with S a .

26 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I

viL If a three-space S 3 and a plane a not on S 3 are on the same four-space,
S 3 and a have one and only one line in common.

viii. If a three-space S 3 and a line I not on S g are on the same four-space,
S 3 and I have one and only one point in common.

ix. Two planes on the same four-space hut not on the same three-space
have one and only one point in common.

x. Any two distinct three-spaces on the same four-space have one and only
one plane in common.

xi. If two three-spaces have a plane in common, they lie in the same four-space.

xii. The four-space S 4 determined l>y a three-space S 3 and a point P is
identical with the four-space determined by a three-space 83 and a point P',
provided 83 and P' are on S 4 .

5. On the assumption that a line contains n + 1 points, extend the results
of Ex. 3 to a four-space.

11. The principle of duality. It is in order to exhibit the theorem
of duality as clearly as possible that we have introduced the syrn-
metrical, if not always elegant, terminology :

A point is on a line. A line is on a point.

A point is on a plane. A plane is on a point.

A line is on a plane. A plane is on a line.

A point is on a three-space. A three-space is on a point.

A line is on a three-space. A three-space is on a line.

A plane is on a three-space. A three-space is on a plane.

The theorem in question rests on the f olio wing observation : If any
one of the preceding assumptions, theorems, or corollaries is expressed
by means of this "on" terminology and then a new proposition is
formed by simply interchanging the words point and plane, then
this new proposition will be valid, i.e. will be a logical consequence
of the Assumptions A and E. We give below, on the left, a complete
list of the assumptions thus far made, expressed in the " on " termi-
nology, and have placed on the right, opposite each, the corresponding
proposition obtained by interchanging the words point and plane
together with the reference to the place where the latter proposition
occurs in the preceding sections :

ASSUMPTIONS A 1, A 2. If u4and THEOREM 9, COR. 1. If a and /9
are distinct points, there is one are distinct planes, there is one and
and only one line on A and B. only one line on a and /3.*

* By virtue of Assumption E 8' it is not necessary to impose the condition that the
elements to be considered are in the same three-space. This observation should empha-
size, however, that the assumption of closure is essential in the theorem to he nrovad

ill]

THE PKINCIPLE OF DUALITY

ASSUMPTION A 3. If A, B, Q are
points not all on the same line, and
D and JS (D 3* E] are points such
that B, C 9 D are on a line and 9
A, E are 011 a line, then there is a
point F such that A, B, F are on a
line and also D, fi, F are on a line.

ASSUMPTION EO. There are at
least three points on every line.

ASSUMPTION El. There exists
at least one line.

ASSUMPTION E 2. All points are
not on the same line.

ASSUMPTION E 3. All points are
not on the same plane.

ASSUMPTION E3'. If S, is a

three-space, every point is on S 3 .

THEOBEM 9, COR. 4. If a, ft, <y
are planes not all on the same line,
and ^ and v(p*pv) are planes such
that , % //, are on a line and % a, v
are on a line, then there is a plane X
such that a, /3, X are on a line and
also /, v, X are on a line.

COR. 2, p. 25. There are at
least three planes on every line.

ASSUMPTION E 1. , There exists
at least one line.

COR. 3, p. 25. All planes are
not on the same line.

COB. 4, P- 25. All planes are
not on the same point.

COE. 5, p. 25. If S s is a three-

space, every plane is on S s .

In all these propositions it is to be noted that a line is a class
of points whose properties are determined by the assumptions, while
a plane is a class of points specified by a definition. This definition
in the "on" language is given below on the left, together with a*
definition obtained from it by the interchange of point and plane.
Two statements in this relation to one another are referred to as
(space) duals of one another.

If P, Q, JR are points not on
the same line, and I is a line on
Q and It, the class S 2 of all
points such that every point of
S 2 is on a line with P and some

If X, JJL, v are planes not on the
same line, and I is a line on" ^
and v > the class B 2 of all planes
such that every plane of B 2 is on
a line with X and some plane on
I is called the "bundle determined
by X and I.

point on I is called the plane
determined by P and I

Now it is evident that, since X, /A, v and I all pass through a point 0>
the bundle determined by X and I is simply the class of all planes on-
the point 0. In like manner, it is evident that the dual of the defini-
tion of a three-space is simply a definition of the class of all planes on
a three-space. Moreover, dual to the class of all planes on a line we
have the class of all points on a line, ie. the line itself, and conversely.

28 THEOBEMS OF ALIGNMENT AND DUALITY [CHAP. I

With the aid of these observations we are now ready to establish
the so-called principle of duality :

THEOEEM 11. THE J^J^E^OF^DUALITY^FOR A SPACE OF THREE
DIMENSIONS. Any proposition deducible from Assumptions A and E
concerning points, lines, and planes of a three-space remains valid, if
stated in the "ow-^Jcraj^fo^j^A^ the words "point" and "plane^
are intercJianqed. (A, E)

Proof. Any proposition deducible from Assumptions A and E is
obtained from the assumptions given above on the left by a certain
sequence of formal logical inferences. Clearly the same sequence of
logical inferences may be applied to the corresponding propositions
given above on the right. They will, of course, refer to the class of
all planes on a line when the original argument refers to the class of
all points on a line, i.e. to a line, and to a bundle of planes when the
original argument refers to a plane. The steps of the original argu-
ment lead to a conclusion necessarily stated in terms of some or all
of the twelve types of " on " statements enumerated at the beginning
of this section. The derived argument leads in the same way to a
conclusion which, whenever the original states that a point P is on a
line l y says that a plane TT' is one of the class of planes on a line V,
i.e. that TT' is on V\ or which, whenever the original argument states
that a plane TT is on a point P, says that a bundle of planes on a
point P f contains a plane -TT', Le. that P r is on TT'. Applying similar
considerations to each of the twelve types of " on " statements in
succession, we see that to each statement in the conclusion arrived
at by the original argument corresponds a statement arrived at by
the derived argument in which the words point and plane in the
original statement have been simply interchanged.

Any proposition obtained in accordance with the principle of dual-
ity just proved is called the space dual of the original proposition.
The point and plane are said to be dual elements ; the line is self-
duaL We may derive from the above similar theorems on duality ^in
a plane and at a point. For, consider a plane TT and a point P not on
TT, together with all the lines joining P with every point of TT. Then
to every point of TT will correspond a line through P, and to every
line of TT will correspond a plane through P. Hence every proposi-
tion concerning the points and lines of TT is also valid for the corre-
sponding lines and planes through P. The space dual of the latter

ii, 12] SPACE OF N DIMENSIONS 29

proposition is a new proposition concerning lines and points on a
plane, which could have been obtained directly by interchanging
the words point and line in the original proposition, supposing the
latter to be expressed in the "on" language. This gives

THEOREM 12. THE THEOREM OF DUALITY IN A PLANE. Any prop-
osition deducible from Assumptions A and E concerning the points
and lines of a plan&remains valid, if stated in the "on" terminology,
when the words "point" and "line" are interchanged. (A, E)

The space dual of this theorem then gives

THEOREM 13. THE THEOREM OF DUALITY AT A POINT. Any prop-
osition deducible from Assumptions A and E concerning the planes^

^alid y if stated in the "on" termi-

nology, when the words "plane" and " line" are interchanged. (A, E)

The principle of duality was first stated explicitly by Gergonne (1826), but
was led up to by the writings of Poncelet and others during the first quarter
of the nineteenth century. It should be noted that this principle was for
several years after its publication the subject of much discussion and often
acrimonious dispute, and the treatment of this principle in many standard
texts is far from convincing. The method of formal inference from explicitly
stated assumptions makes the theorems appear almost self-evident. This may
well be regarded as one of the important advantages of this method.

It is highly desirable that the reader gain proficiency in forming the duals
of given propositions. It is therefore suggested as an exercise that he state
the duals of each of the theorems and corollaries in this chapter. He should
in this case state both the original and the dual proposition in the ordinary
terminology in order to gain facility in dualizing propositions without first
stating them in the often cumbersome "on" language. It is also desirable
tnat he dualize several of the proofs by writing out in order the duals of each
proposition used in the proofs in question.

EXERCISE

Prove the theorem of duality for a space of four dimensions : Any propo-
sition derivable from the assumptions of alignment and extension and closure
for a space of four dimensions concerning points, lines, planes, and three-
spaces remains valid when stated in the " on " terminology, if the words
point and three-space and the words line and plane be interchanged. '

* 12. The theorems of alignment for a space of n dimensions. We
have already called attention to the fact that Assumption E3',-
whereby we limited ourselves to the consideration of a space of only

* This section may be omitted on a first reading.

30 THEOEEMS OF ALIGNMENT AND DUALITY [CHAP. I

three dimensions, is entirely arbitrary. This section is devoted to the
discussion of the theorems of alignment, i.e. theorems derivable from
Assumptions A and E 0, for a space of any number of dimensions.
In this section, then, we make use of Assumptions A and E only.

DEFINITION. If ^, P 19 -Z?, , P n are n + 1 points not on the same
(n l)-space, and S n _ l is an ( l)-space on J%, P 2> - , P n) the class
S n of all points such that every point of S n is on a line with J% and
some point of S n _ 1 is called the n-space determined by J^ and S B-1 .

As a three-space has already been defined, this definition clearly
determines the meaning of "w-space" for every positive integral value
of n. We shall use S n as a symbol for an %-space, calling a plane a
2-space, a line a 1 -space, and a point a 0-space, when this is convenient.
S is then a symbol for a point.

DEFINITION. An S r is on an S t and an S^ is on an S r (r<t) 9 pro-
vided that every point of S r is a point of S t .

DEFINITION. Jo points are said to be independent, if there is no S^g
which contains them all.

Corresponding to the theorems of J 6-9 we shall now establish
the propositions contained in the following Theorems S n l, S n 2,
S ra 3. As these propositions have all been proved for the case n = 3,
it is sufficient to prove them on the hypothesis that they have already
been proved for the cases n = 3, 4, - -, n 1 ; i.e. we assume that the
propositions contained in Theorem S n _ 1 l, a, &, c, d, e, /have been
proved, and derive Theorem S n l, #,,/ from them. By the prin-
ciple of mathematical induction this establishes the theorem for any n.

THEOREM S n l. Let the n-space S n "be defined ly the point R and the

a. There is an n-space on any n + 1 independent points.
1). Any line on two points of S n has one point in common with R n _ a ,
and is on S n .

c. Any S r (r < n) on r + 1 independent points of S n is on S n .

d. Any S r (r < n) on r + 1 independent points of S n has an S rnl in
common with R n _ lf provided all r + 1 points are not on R n _ r

e. Any line I on two points of S n has one point in common with
any S n _ I on S m .

/ -Sf .T and T n _ a (T not on T n _^ are any point and any
(n l.)-space respectively of the n-space determined ly R and R w-1 ,
the latter n-space is the same as that determined ly T and T w _ r

12] SPACE OF JV DIMENSIONS 31

Proof, a. Let the n + 1 independent points be JJ, P v - - . , P n . Then
the points JJ, P 2 , -, JJ are independent; for, otherwise, there would
exist an S n _ 2 containing them all (definition), and this S n _ 2 with J?
would determine an S a-1 containing all the points JJ, j, - , %, con-
trary to the hypothesis that they are independent. Hence, by Theorem
S n _ 1 l a, there is an S^ on the points JJ, JJ, , Jg; and this S H ^
with JJ determines an w-space which is on the points JJ, JJ, J?, ,..

&. If the line I is on R or R n _ 1? the proposition is evident from the
definition of S n . If I is not on R or R n _ 1} let A and J5 be the given
points of I which are on S n . The lines R ^ and R Q B then meet R^
in two points A! and B J respectively. The line I then meets the two
lines J3'R , R ^'; and hence, by Assumption A3, it must meet the
line A'B* in a point P which is on R n-1 by Theorem S K _ 1 15. To
show that every point of I is on S n , consider the points A, A r , P. Any
line joining an arbitrary point Q of I to R , meets the two lines PA
and A A 1 , and hence, by Assumption A3, meets the third line A 1 P.
But every point of AlP is on R n _ l (Theorem S n-1 l 6), and hence Q
is, by definition, a point of S n .

c. This may be proved by induction with respect to r. For r 1 it
reduces to Theorem S B 1 J. If the proposition is true for r = k 1,' all
the points of an S k on k + 1 independent points of S n are, by definition
and Theorem S^l/, on lines joining one of these points to the points
of the S k _ l determined by the remaining k points. But under the
hypothesis of the induction this S k _ 1 is on S n , and hence, by Theorem
S n l 6, all points of S k are on S n .

d. Let r+1 independent" points of S tt be J, J?, - -, P r and let j?J be
not on R n _ r Each of the lines JJ P k (k = 1, , r) has a point Q k in
common with R n _ l (by S B 1 J). The points Q 19 Q z , - - -, Q ? . are inde-
pendent; for if not, they would all be on the same S ? ._ 2 , which,
together with J^, would determine an S,^ containing all the points
PI (by S ri-1 l 6). Hence, by S r-1 l a, there is an S r _ : on j, Q*,- ,Q r
which, by c, is on both S r and S n .

e. We will suppose, first, that one of the given points is R . Let
the other be A. By definition I then meets R u-1 in a point A', and, by
5,^15, in only one such point. If R is on S n _ 1} no proof is required
for this case. Suppose, then, that R is not on S n _ 1 , and let C be any
point of S n . r The line R (7 meets R^ in a point C r (by definition).
By d, S n _ 1 has in common with R tt _! an (n 2)-space, S n _ 2 , and, by

32 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I

Theorem S W _ 1 1^, this has in common with the line A f C f at least
one point D f . All points of the line D f C are then on S a-1 , by S n-1 l b.
Now the line I meets the two lines C'D f and CC r ; hence it meets the
line CD ] (Assumption A3), and has at least one point on S n _ r

We will now suppose, secondly, that both of the given points are
distinct from R . Let them be denoted by A and J3, and suppose that
R is not on S tt _ r By the case just considered, the lines R Q A and
R Q B meet S n-1 in two points A! and B 1 respectively. The line Z, which
meets R ^' and R Q B r must then meet A'B 1 in a point which, by
Theorem S n-1 lJ, is on S B-1 .

Suppose, finally, that R is on S tt-1 , still under the hypothesis that Z
is not on R . By d, S n _ l meets R n-1 in an (n 2)-space Q rt _ 2 , and
the plane R Z meets R B-1 in a line I 1 . By Theorem S n _ l ie 9 V and
Q n _ 2 have in common at least one point P. Now the lines Z and R P
are on the plane R Z, and hence have in common a point Q (by Theorem
S a l e = Theorem 5). By S W-1 1 b the point Q is common to S n _ 1 and Z.

/. Let the %-space determined by T and T n-1 be denoted by T w .
Any point of T n is on a line joining T with some point of T n _ r
Hence, by &, every point of T n is on S n . Let P be any point of S n
distinct from T . The line T P meets T n-1 in a point, by e. Hence
every point of S B is a point of T B .

GOROLLAEY. On n + 1 independent points there is one and but one S n .

This is a consequence of Theorem S n l a and S n l/. The formal
proof is left as an exercise.

THEOEEM S n 2. An S r and an S k having in common an S p , but
not an S p+1 , are on a common S r+ ^._ p and are not both on the so/me
s > if n<r + kp.

Proof. If Jc~p, S k is on S r . If Jc >p, let JJ be a point on S k not
on S 9 . Then 1J and S r determine an S r+1 , and P^ and S p an S |)+l>
such that S p+l is contained in S r+1 and S t . If Is > p + 1, let P% be a
point of S k not on S p+1 . Then P z and S r+1 determine an S r+2 , while
J> and S p+1 determine an S^ +2 , which is on S r+2 and S r This process
can be continued until there results an S^, containing all the points
of S k . By Theorem S n l, Cor., we have i=sJcp. At this stage in the
process we obtain an S r+ j.^ which contains both S r and S k .

The argument just made shows that 1J, JJ, , P k _ p , together with
any set Q v Q z> . . ., Q r+1 , of r + 1 independent points of S r , constitute

12] SPACE OF N DIMENSIONS S3

a set of r + k p + 1 independent points, each of which is either in
S, or S A . If S r and S k were both on an S n , where n< r + & p, these
could not be independent.

THEOKEM S W 3. ^w S r and an S k contained in an S tt ar &0#i 00, the
same S,. +jt _ w .

Proof. If there were less than r + k n+1 independent points
common to S, and S^ say r + kn points, they would, by Theorem S n 2,
determine an S q , where q = r + k (r + k n 1) = n -f 1.

Theorems S n 2 and S M 3 can be remembered and applied very easily
by means of a diagram in which S M is represented by n + 1 points.
Thus, if n = 3, we have a set of four points. That any two S 's have
an S x in common corresponds to the fact that any two sets of three
must have at least two points in common. In the general case a set
of r + 1 points and a set of k + 1 selected from the same set of n + 1
have in common at least r + k n + 1 points, and this corresponds
to the last theorem. This diagram is what our assumptions would
describe directly, if Assumption E were replaced by the assumption:

JSvery line contains two and only tivo points.

If one wishes to confine one's attention to the geometry in a space
of a given number of dimensions, Assumptions E 2, E 3, and E 3' may
be replaced by the following :

En. Not all points are on the same S XJ if k < n.

En'. If S is an S n , all points are on S.

Eor every S n there is a principle of duality analogous to that which,
we have discussed for n = 3. In S n the duality is between S k and S n _ t _ l
(counting a point as an.$ ), for all &'s from to n 1. If n is odd,
there is a self-dual space in S n ; if n is even, S ff contains no self-dual
space.

EXERCISES

1. State and prove the theorems of duality in S 5 ; in S tt .

2. If m + 1 is the number of points on a line, how many S A /s are there in
anS B ?

* 3. State the assumptions of extension by which to replace Assumption En
and En' for spaces of an infinite number of dimensions. Make use of the
transfmite numbers.

* Exercises marked * are of a more advanced or difficult character.

CHAPTER II

PROJECTION, SECTION, PERSPECTIVITY. ELEMENTARY
CONFIGURATIONS

13. Projection, section, perspectivity. The point, line, and plane
are the simple elements of space * ; we have seen in the preceding
chapter that the relation expressed by the word on is a reciprocal
relation that may exist between any two of these simple elements.
In the sequel we shall have little occasion to return to the notion of
a line as being a class of points, or to the definition of a plane ; but
shall regard these elements simply as .entities for which the relation
" on " has been denned. The theorems of the preceding chapter are to
be regarded as expressing the fundamental properties of this relation.!
We proceed now to the study of certain sets of these elements, and
begin with a series of definitions.

DEFINITION. A figure is any set of points, lines, and planes in space.
A plane figure is any set of points and lines on the same plane. A
point figure is any set of planes and lines on the same point.

It should be observed that the notion of a point figure is the space
dual of the notion of a plane figure. In the future we shall fre-
quently place dual definitions and theorems side by side. By virtue
of the principle of duality it will be necessary to give the proof of
only one of two dual theorems.

DEFINITION. Given a figure F DEFINITION. Given a figure F
and a point P\ every point of F and a plane TT; every plane of F
distinct from P determines with distinct from TT determines with
P a line, and every line of F not TT a line, and every line of F not
on P determines with P a plane; on TT determines with TT a point;
the set of these lines and planes the set of these lines and points
through P is called the projection on TT is called the section $ of F

* The word space is used in place of the three-space in which are all the elements
considered.

t We shall not in future, however, confine ourselves to the " on " terminology,
but shall also use the more common expressions.

| A section by a plane is often called a plane section.

13] PEOJECTION, SECTION", PEESPECTIYITY 35

of F from P, The individual lines by TT. The individual lines and

and planes of the projection are points of the section are also

also called the projectors of the called the traces of the respective

respective points and lines of F. planes and lines of F.

If F is a plane figure and the point P is in the plane of the figure, the
definition of the projection of F from P has the following plane dual :

DEFINITION. Given a plane figure F and a line I in the plane of F;
the set of points in which the lines of F distinct from I meet I is
called the section of F by Z. The line I is called a transversal, and
the points are called the traces of the respective lines of F.

As examples of these definitions we mention the following: The
projection of three mutually intersecting nonconcurrent lines from a
point P not in the plane of the lines consists of three planes through P;
the lines of intersection of these planes are part of the projection only
if the points of intersection of the lines are thought of as part of the
projected figure. The section of a set of planes all on the same line
by a plane not on this line consists of a set of concurrent lines, the
traces of the planes. The section of this set of concurrent lines in a
plane by a line in the plane not on their common point consists of
a set of points on the transversal, the points being the traces of the
respective lines.

DEFINITION. Two figures F 1? F 2 are said to be in (1, 1) correspond-
ence or to correspond in a one-to-one reciprocal way, if every element
of F.,^ corresponds (cf. footnote, p. 5) to a unique element of F 2 in such
a way that every element of F 2 is the correspondent of a unique ele-
ment of F r A figure is in (1, 1) correspondence with itself, if every
element of the figure corresponds to a unique element of the same
figure in such a way that every element of the figure is the corre-
spondent of a unique element. Two elements that are associated in
this way are said to be corresponding or homologous elements.

A correspondence of fundamental importance is described in the
following definitions:

DEFINITION. If any two homol- DEFINITION. If any two homol-
ogous elements of two corre- ogous elements of two corre-
sponding figures have the same spending figures have the same
projector from a fixed point 0, trace in a fixed plane o>, such
such that all the projectors are that all the traces of either

36 PROJECTION", SECTION, PEESPECTIVITY [CHAP. II

distinct, the figures are said to figure are distinct, the figures are
be perspective from 0. The point said to be perspective from co,
is called the center of perspec- The plane w is called the plane
tivity. of perspectivity.

DEFINITION. If any two homologous lines in two corresponding
figures in the same plane have the same trace on a line I, such
that all the traces of either figure are distinct, the figures are said
to be perspective from 1. The line I is called the axis of perspectivity.

Additional definitions of perspective 'figures will be given in the
next chapter (p. 56). These are sufficient for our present purpose.

DEFINITION. To project a figure in a plane a from a point onto a
plane a r , distinct from a, is to form the section by cc r of the projection
of the given figure from 0. To project a set of points of a line I from
a point onto a line V 9 distinct from I but in the same plane with I
and 0, is to form the section by l f of the projection of the set of points
from 0.

Clearly in either case the two figures are perspective from 0, pro-
vided is not on either of the planes a, a 1 or the lines I, l r .

EXERCISE

What is the dual of the process described in the last definition ?

The notions of projection and section and perspectivity are fun-
damental in all that follows.* They will be made use of almost
immediately in deriving one of the most important theorems of pro-
jective geometry. We proceed first, however, to define an important
class of figures.

14. The complete n-point, etc. DEFINITION. A complete n-point in
space or a complete space n-point is the figure formed by n points, no
four of which lie in the same plane, together with the n(n 1)/2
lines joining every pair of the points and the n(n l)(n 2)/6 planes
joining every set of three of the points. The points, lines, and planes
of this figure are called the vertices, edges, and faces respectively of
the complete 7i-point

* The use of these notions in deriving geometrical theorems goes back to early
times. Thus, e.g., B. Pascal (1623-1662) made use of them in deriving the theorem
on a hexagon inscribed in a conic which bears his name. The systematic treatment
of these notions is due to Ponceletj cl his Traits des proprie'te's proiectives des
figures, Paris, 1822.

14] JV-POINT, iV-PLANE, JV-LIXE 37

The simplest complete %-point in space is the complete space
four-point. It consists of four vertices, six edges, and four faces,
and is called a tetrahedron. It is a self-dual figure.

EXERCISE

Define the complete n-plane in space by dualizing the last definition. The
planes, lines, and points of the complete n-plane are also called the faces,
edges, and vertices of the n-plane.

DEFINITION. A complete n-point in a plane or a complete plane
n-point is the figure formed by n points of a plane, no three of
which are collinear, together with the n(n 1)/2 lines joining every
pair of the points. The points are called the vertices and the lines
are called the sides of the n-point. The plane dual of a complete
plane ^-point is called a complete plane n-line. It has n sides and
n(n V)/2 vertices. The simplest complete plane n-point consists of
three vertices and three sides and is called a triangle.

DEFINITION. A simple space nyoint is a set of n points J^, J^, P v ,P n
taken in a certain order, in which no four consecutive points are
coplanar, together with the n lines J?J, JJJJ, , PJE( joining suc-
cessive points and the n planes J%J%Z, -, J^^ determined by
successive lines. The points, lines, and planes are called the vertices,
edges, and faces respectively of the figure. The space dual of a simple
space 7i-point is a simple space n-plane.

DEFINITION. A simple plane n-point is a set of n points P^, P%, P%, - JJ
of a plane taken in a certain order in which no three consecutive points
are collinear, together with the n lines JJ.2J, P^, -, P n l^ joining suc-
cessive points. The points and lines are called the vertices and sides
respectively of the figure. The plane dual of a simple plane n-point is
called a simple plane n-line.

Evidently the simple space n-point and the simple space n-plane are
identical figures, as likewise the simple plane n-point and the simple
plane n-line. Two sides of a simple n-line which meet in one of its
vertices are adjacent. Two vertices are adjacent if in the dual relation.
Two vertices of a simple n-point J%P% - P n (n even) are opposite if, in
the order J^ P n , as many vertices follow one and precede the other
as precede the one and follow the other. If n is odd, a vertex and a
side are opposite if, in the order .TJJfJ - P n , as many vertices follow the
side and precede the vertex as follow the vertex and precede the side.

PROJECTION, SECTION, PEESPECTIVITY [CHAP.II

The space duals of the complete plane %-point and the complete plane
%-line are the complete n-plane on a point and the complete n-line on a
point respectively. They are the projections from a point, of the plane
^-line and the plane ^-point respectively.

15. Configurations. The figures defined in the preceding section
are examples of a more general class of figures of which we will now
give a general definition.

DEFINITION. A figure is called a configuration, if it consists of a
finite number of points, lines, and planes, with the property that each
point is on the same number & 12 of lines and also on the same num-
ber a ia of planes ; each line is on the same number & 21 of points and the
same number # 23 of planes ; and each plane is on the same number a 81
of points and the same number a 32 of lines.

A configuration may conveniently be described by a square matrix :

1 point

2 line

3 plane

point

line

plane

31 32 33

In this notation, if we call a point an element of the first kind, a
line an element of the second kind, and a plane one of the third kind,
the number a (i = j) gives the number of elements of the /th kind
on every element of the ith kind. The numbers a iv a 22t a ss give the
total number of points, lines, and planes respectively. Such a square
matrix is called the symbol of the configuration.

A tetrahedron, for example, is a figure consisting of four points,
six lines, and four planes ; on every line of the figure are two points
of the figure, on every plane are three points, through every point
pass three lines and also three planes, every plane contains three lines,
and through every line pass two planes. A tetrahedron is therefore
a configuration of the symbol

COlSTFIG-UBATIOlSrS

The symmetry shown in this symbol is due to the fact that the figure
in question is self-dual. A triangle evidently has the symbol

Since all the numbers referring to planes are of no importance in
case of a plane figure, they are omitted from the symbol for a plane
configuration.

In general, a complete plane ?i-point is of the symbol

n
2

w-1

and a complete space w-point of the symbol

2 Jn(n-l) w-2

3 3 t n (n-l)(a-2)

Further examples of configurations are figs. 14 and 15, regarded as
plane figures.

EXERCISE

Prove that the numbers in a configuration symbol must satisfy the condition

(* / = 1, 2, 8)

16. The Desargues configuration. A very important configuration
is obtained by taking the plane section of a complete space five-point.
The five-point is clearly a configuration with the symbol

and it is clear that the section by a plane not on any of the vertices
is a configuration whose symbol may be obtained from the one just
given by removing the first column and the first row. This is due
to the fact that every line of the space figure gives rise to a point in

PBO JECTION, SECTION, PEESPECTIYITY [CHAP, n

the plane, and every plane gives rise to a line. The configuration in
the plane has then the symbol

We proceed to study in detail the properties of the configuration just
obtained. It is known as the configuration of Desargues.

We may consider the vertices of the complete space five-point as con-
sisting of the vertices of a triangle A, B 9 C and of two points O l} 2

Fio. 14

not coplanar with any two vertices of the triangle (fig. 14). The sec-
tion by a plane a not passing through any of the vertices will then
consist of the following :

A triangle A^B^C^ the projection of the triangle ABC from : on a.

A triangle J 2 2? 2 <7 2 , the projection of the triangle ABC from 2 on a.

The trace of the line 0^0 y

The traces A zi B^ <7 3 of the lines BC, CA, AB respectively.

The trace of the plane ABC, which contains the points A z , B^ C y

The traces of the three planes AO^O^ BO^O^ COft^, which contain
respectively the triples of points OA^A^ OB^B^ OC^C y

The configuration may then be considered (in ten ways) as consist-
ing of two triangles A^B^ and A^B^C V perspective from a point and

THEOREM OF DESAEGUES 41

having homologous sides meeting in three collinear points J 3 , B B , <7 8 .
These considerations lead to the following fundamental theorem: .

THEOREM 1. THE THEOREM OF DESARGUES.* If two triangles in the
same plane are 'perspective "from a point, the three pairs of homologous
sides meet in collinear points; i.e. the triangles are perspective from
a line. '(A, E)

Proof. Let the two triangles be A^B^ and A 2 J3 Z C 2 (fig. 14), the
lines A^A^ B^^ ^aA meeting in the point 0. Let B^A V B^A 2 inter-
sect in the point <? 3 ; A^C V A 2 C Q in B^\ B^C V B Z C Z in A z . It is required
to prove that A z , B B , <? s are coUinear, Consider any line through
which is not in the plane of the triangles, and denote by O v 3 any
two distinct points on this line other than 0. Since the lines A 2 Z
and A^O^ lie in the plane (A^A^ 1 2 ), they intersect in a point A.
Similarly, B^O^ and J? 2 2 intersect in a point B, and likewise C^ and
C 2 2 in a point C. Thus ABCO^O^ together with the lines and planes
determined by them, form a complete five-point in space of which the
perspective triangles form a part of a plane section. The theorem
is proved by completing the plane section. Since AB lies in a plane
with AJ& V and also in a plane with A Z B 2 , the lines AJ& V A Z B 2) and
AB meet in 9 . So also A 1 C V A Z C Z) and AC meet in B B ; and B t C v
B Z C 2 , and BC meet in A y Since A z) B z , C s lie in the plane ABC and
also in the plane of the triangles A^B^C^ and A^B^C^ they are collinear.

THEOREM l f . If two triangles in the same plane are perspective
from a line, the lines joining pairs of homologous vertices are, con-
current; i.e. the triangles are perspective from a point. (A, E)

This, the converse of Theorem 1, is also its plane dual, and hence
requires no further proof.

COROLLARY. If two triangles not in the same plane are perspective
from a point, the pairs of homologous sides intersect in collinear
points; and conversely. (A, E)

A more symmetrical and for many purposes more convenient nota-
tion for the Desargues configuration may be obtained as follows:
Let the vertices of the space five-point be denoted by ^, J^, P t) P^ J
(fig. 15). The trace of the line P^ in the plane section is then '
naturally denoted by ^ in general, the trace of the line JJJ^ by 7?.
(i, j = 1, 2, 3, 4, 5, i = /). Likewise the trace of the plane j?J^5 ma 7
* G-irard Desargues, 1593-1662.

42 PKQJECTICH8", SECTION, PER8PECTIVITY [CHAP, n

be denoted by l vt (i, j, k = 1, 2, 3, 4, 5). This notation makes it pos-
sible to te]l at a glance which lines and points are united. Clearly a
point is on a line of the configuration if and only if the suffixes of
the point are both among the suffixes of the line. Also the third
point on the line joining JJ and J^ is the point JJ, ; two points are
on the same line if and only if they have a suffix in common, etc.

EXERCISES

1. Prove Theorem V without making use of the principle of duality.

2. If two complete w-points in different planes are perspective from a point,
the pairs of homologous sides intersect in collinear points. "What is the dual
theorem ? What is the corresponding theorem concerning any two plane figures
in. different planes ?

3. State and prove the converse of the theorems in Ex. 2.

4. If two complete ra-points in the same plane correspond in such a way
that homologous sides intersect in points of a straight line, the lines joining
homologous vertices are concurrent ; i.e. the two w-points are perspective from
a point. Dualize.

5. What is the figure formed by two complete n-points in the same plane
when they are perspective from a point? Consider particularly the cases n = 4 and
n = 5. Show that the figure corresponding to the general case is a plane section
of a complete space (n + 2)-point. Give the configuration symbol and dualize.

6. If three triangles are perspective from the same point, the three axes of
perspectivity of the three pairs of triangles are concurrent ; and conversely.
Dualize, and compare the configuration of the dual theorem with the case n = 4
of Ex. 5 (cf. fig. 15, regarded as a plane figure).

17]

PEESPECTIVE TETEAHEDEA

17. Perspective tetrahedra. As an application of tlie corollary of
the last theorem we may now derive a theorem in space analogous to
the theorem of Desargues in the plane.

THEOREM 2. If two tetrahedra are perspective from a point, tlie six
p airs of homologous edges intersect in coplanar points, and the four
pairs of homologous faces intersect in coplanar lines ; i.e. the tetra-
are perspective from a plane. (A, E)

:*'

FIG. 16

Proof. Let the two tetrahedra be -Zy^J-ZJ and JJ'JJ'JJ'JEJ', and let
the lines JJJJ', J^', jZ^', JJ-Z^ meet in the center of perspectivity 0.
Two homologous edges P % P 3 and J?'J' then clearly intersect ; call the
point of intersection P v . The points ^ 2 , J? s , JJ S lie on the same line,
since the triangles JJJSJJ and -ZJ'Ia'JJ' are perspective from (The-
orem 1, Cor.). By similar reasoning applied to the other pairs of
perspective triangles we find that the following triples of points are
collinear:

The first two triples have the point P^ in common, and hence
determine a plane;, each of the other two triples has a point in

44 PKOJECTION, SECTION, PEESPECTIYITY [CHAP.II

common with each, of the first two. Hence all the points P %1 lie in
the same plane. The lines of the four triples just given are the lines
of intersection of the pairs of homologous faces of the tetrahedra.
The theorem is therefore proved.

THEOREM 2'. If two tetrahedra are perspective from a plane, the
lines joining pairs of homologous vertices are concurrent, as likewise
the planes determined ly pairs of homologous edges ; i.e. the tetrahedra
are perspective from a point (A, E)

This is the space dual and the converse of Theorem 2.

EXERCISE

"Write the symbols for the configurations of the last two theorems.

18. The quadrangle-quadrilateral configuration.

DEFINITION. A complete plane DEFINITION. A complete plane
four-point is called a complete four-line is called a complete
quadrangle. It consists of four quadrilateral It consists of four
vertices and six sides. Two sides sides and six vertices. Two ver-
not on the same vertex are called tices not on the same side are
opposite. The intersection of two called opposite. The line joining
opposite sides is called a diag- two opposite vertices is called a
onal point. If the three diagonal diagonal line. If the three diag-
points are not collinear, the tri- onal lines are not concurrent, the
angle formed by them is called triangle formed by them is called
the diagonal triangle of the the diagonal triangle of the
quadrangle.* quadrilateral*

The assumptions A and E on which all our reasoning is based do
not suffice to prove that there are more than three points on any line.
In fact, they are all satisfied by the triple system (1), p. 3 (cf. fig. 17).
In a case like this the diagonal points of a complete quadrangle, ,ajce
^collinear.and.the di^onSTlmes of sT^qm^ete quadrilateral concur-
rent^ as may readily be verified. Two perspective triangles cannot
exist in such a plane, and hence the Desargues theorem becomes

* In general, the intersection of two sides of a complete plane w-point which do
not have a vertex in common is called a diagonal poM of the ra-point, and the line
joining two vertices of a complete plane n-line which do not lie on the same side
is called a diagonal line of the 7i-line. A complete plane n~point (n-line) then has
n(n - 1) (n - 2) (n -3)/8 diagonal points (lines). Diagonal points and lines are
sometimes called false vertices and false sides respectively.

ifl ASSUMPTION H

trivial. Later on we shall add an assumption* which excludes all
such cases as this, and, in fact, provides for the existence of an in-
finite number of points on a line. A part of what is contained in
this assumption is the following:

ASSUMPTION H . Tlie diagonal
points of a complete quadrangle
are noncollinear.

Many of the important theorems
of geometry, however, require the
existence of no more than a finite
number of points. "We shall there-
fore proceed without the use of w P IG 17

further assumptions than A and E,

understanding that in order to give our theorems meaning there must
~be postulated, the existence of the points specified in their hypotheses.
In most cases the existence of a sufficient number of points is
insured by Assumption H , and the reader who is taking up the
subject for the first time may well take it as having been added
to A and E. It is to be used in the solution of problems.

We return now to a further study of the Desargues configuration.
A complete space five-point may evidently be regarded (in five ways)
as a tetrahedron and a complete four-line at a point A plane section
of a four-line is a quadrangle and the plane section of a tetrahedron
is a quadrilateral. It follows that (in five ways) the Desargues con-
figuration may be regarded as a quadrangle and a quadrilateral.
Moreover, it is clear that the six sides of the quadrangle pass through
the six vertices of the quadrilateral. In the notation described on
page 41 one such quadrangle is JJ 2 , ^ 3 , JJ 4 , JJ 5 and the corresponding
quadrilateral is Z 234 , Z 235 , ? 2 45> Z 845 .

The question now naturally arises as to placing the figures thus ob-
tained in special relations. As an application of the theorem of De-
sargues we will show how to construct f a quadrilateral which has the
same diagonal triangle as a given quadrangle. "We will assume in our
discussion that the diagonal points of any quadrangle form a triangle.

* Merely saying that there are more than three points on a line does not insure
that the diagonal points of a quadrangle are noncollinear. Cases where the diagonal
points are collinear occur whenever the number of points on a line is 2 + !

t To construct a figure is to determine its elements in terms of certain given
elements.

46 PROJECTION, SECTION", PEKSPECTIVITY [CHAP, n

Let PV P^ p *> % be the vertices of the given complete quadrangle,
and let D 12 , Z> 13 , D u be the vertices of the diagonal triangle, D 12 being
on the side Pf^ > 1B on the side '%P B , and D 14 on the side JJJJ (fig. 18).
We observe first that the diagonal triangle is perspective with each of the
four triangles formed ly a set of three of the vertices of the quadrangle,
the center of perspectivity being in each case the fourth vertex. This
gives rise to four axes of perspectivity (Theorem 1), one corresponding
to each vertex of the quadrangle.* These four lines clearly form the
sides of a complete quadrilateral whose diagonal triangle is Z> 12 , D 13 , Z> 14 .

It may readily be verified, by selecting two perspective triangles,
that the figure just formed is, indeed, a Deja^^ This

special case of the Desargues configuration is called the qruadrangle-
qiiadrilateral configuration.^

EXERCISES

1. If p is the polar of P with regard to the triangle ABC, then P is the
pole of p with regard to the same triangle ; that is, P is obtained from p by
a construction dual to that used in deriving p from P. From this theorem it
follows that the relation between the quadrangle and quadrilateral in this

* The line thus uniquely associated with a vertex is called the polar of the point
with respect to the triangle formed by the remaiDing three vertices. The plane dual
process leads to a point associated with any line. This point is called the pole of the
line with respect to the triangle.

t A further discussion of this configuration and its generalizations will be found
in the thesis of H. F. McNeish. Some of the results in this paper are indicated in
the exercises.

18,19] QUADBAjSTGULAK SETS 47

configuration is mutual ; that is, if either is given, the other is determined.
For a reason ^rhich will be evident later, either is called a covariant of the
other.

2. Show that the configuration consisting of two perspective tetrahedra,
their center and plane of perspectivity, and the projectors and traces may be
regarded in six ways as consisting of a complete 5-point P 12 , P 13 , P u , P 15 , P 16
and a complete 5-plane ir S456 , *- 3456 , 7r 233G , 7r 2S46 , 7r 234 -, the notation being
analogous to that used on page 41 for the Desargues configuration. Show-
that the vertices of the 5-plane are on the faces of the 5-point.

3. If P 15 P 2 , P g , P 4 , P 5 , are vertices of a complete space 5-point, the ten
points Dy, in which an edge p {j meets a face P l P l P m (i,/, fc, I, m all distinct),
are called diagonal points. The tetrahedra P 2 P 8 P 4 P 5 and -Z^-^is^w^iB are per-
spective with P x as center. Their plane of perspectivity, v 19 is called the polar
of P x with regard to the four vertices. In like manner, the points P 2 , P 3 , P 4 , P 5
determine their polar planes 7r 2 , 7r 3 , 7r 4 , ir e . Prove that the 5-point and the polar
5-plane form the configuration of two perspective tetrahedra ; that the plane
section of the 5-point by any of the five planes is a quadrangle-quadrilateral
configuration ; and that the dual of the above construction applied to the 5-plane
determines the original 5-point.

4. If P is the pole of IT with regard to the tetrahedron A^A 2 A Z A^ then is v
the polar of P with regard to the same tetrahedron ?

19. The fundamental theorem on quadrangular sets.

THEOREM 3. If two complete quadrangles %%J%% and P^PjPjPl
correspond P^ to P^, 1% to P^ etc. in suc7i a way that five of the
pairs of homologous sides intersect in points of a line I, then the sixth
pair of homologous sides will intersect in a point of I. (A, E)

This theorem holds whether the quadrangles are in the same or
in different planes.

Proof. Suppose, first, that none of the vertices or sides of one of
the quadrangles coincide with any vertex or side of the other. Let
%Pv %J%, -ZJ-Z2, JJJJ, P^PI be the five sides which, by hypothesis,
meet their homologous sides P^Pj, P^Pj, PJPJ, PJPJ, P^Pj in points
of I (fig. 19). We must show that P Z P^ and IgP meet in a point
of I. The triangles ^P^ and P^P* are, by hypothesis, perspec-
tive from l\ as also the triangles P^P and P^P^P^. Each pair is
therefore (Theorem I/) jpersgective from appoint, and this point is in
each case the intersection^^ of the lines JJZJ 7 and P^Ef. Hence the
triangles P^P^ and ^^^ are perspective from and their pairs
of homologous sides intersect in the points of a line, which is evi-
dently I, since it contains two points of L But P^ and $P are

PBOJECTIOST, SECTION, PEESPECTIYITY [CHAP.H

two homologous sides of these last two triangles. Hence they inter-
sect in a point of the line I.

If a vertex or side of one quadrangle coincides with a vertex or
side of the other, the proof is made by considering a third quadrangle*
whose vertices and sides are distinct from those of both of the others,
and which has five of its sides passing through the five given points

of intersection of homologous sides of the two given quadrangles. By
the argument above, its sixth side will meet the sixth side respectively
of each of the two given quadrangles in the same point of I. This
completes the proof of the theorem.

1. It should be noted that the theorem is still valid if the line I con-
tains one or more of the diagonal points of the quadrangles. The case in which
I contains two diagonal points is of particular importance and will be discussed
in Chap. IV, 31.

NOTE 2. It is of importance to note in how far the quadrangle P^P^P^P^
is determined when the quadrangle P i P 2 P 8 P 4 and the line I are given. It may
be readily verified that in such a case it is possible to choose any point P{ to
correspond to any one of the vertices P t , P 2 , P 8 , P 4 , say P x ; and that if m is
any line of the plane IP{ (not passing through Pf ) which meets one of the sides,
say a, of P^PJP^P^ (not passing through P x ) in a point of I, then m may be
chosen as the side homologous to a. But then the remainder of the figure is
uniquely determined.

* This evidently exists whenever the theorem is not trivially obvious.

19 ] QUADBAKG-ULAK SETS 49

THEOREM 3'. If two complete quadrilaterals a^a^ and a(a^a(a[
correspond a l to a[, 3 to a! 2) etc. in such a way that five of the lines
joining homologous vertices pass through a point P, the line joining the
sixth pair of homologous vertices will also pass through P. (A, E)

This is the plane dual of Theorem 3 regarded as a plane theorem.

DEFINITION. A set of points in which the sides of a complete quad-
rangle meet a line I is called a quadrangular set of points.

Any three sides of a quadrangle either, form a triangle or meet in
a vertex ; in the former case they are saifl. to form a triangle triple,
in the latter > point triple of lines. In a quadrangtilar set of points
on a line I any three points in which the lines of a triangle triple meet I
is called a triangle triple of points in the set ; three points in which
the lines of a point triple meet I are called a point triple of points.
A quadrangular set of points will be denoted by

Q(ABC, DBF),

where ABC is a point triple and DEF is a triangle triple, and
where A and D, B and E, and and F are respectively the inter-
sections with the line of the set of the pairs of opposite sides of
the quadrangle.

The notion of a quadrangular set is of great importance in much
that follows. It should be noted again in this connection that one
or two * of the pairs A, D or B, U or (7, F may consist of coincident
points ; this occurs when the line of the set passes through one or
two of the diagonal points.f

We have just seen (Theorem 3) that if we have a quadrangular
set of points obtained from a given quadrangle, there exist other
quadrangles that give rise to the same quadrangular set In the
quadrangles mentioned in Theorem 3 there corresponded to every
triangle triple of one a triangle triple of the other.

DEFINITION. When two quadrangles giving rise to the same
quadrangular set are so related with reference to the set that to a
triangle triple of one corresponds a triangle triple of the other, the

* All three may consist of coincident points in a space in which the diagonal points
of a complete quadrangle are collinear.

t It should he kept in mind that similar remarks and a similar definition may he
made to the effect that the lines joining the vertices of a quadrilateral to a point P
form a quadrangular set of lines, etc. (cf. 30, Chap. IV).

PKOJECTION, SECTION, PEBSPECTIVITY [CHAP, n

quadrangles are said to be similarly placed (fig. 20); if a point triple
of one corresponds to a triangle triple of the other, they are said to
be oppositely placed (fig. 21).

It will be shown later (Chap. IV) that quadrangles oppositely
placed with respect to a quadrangular set are indeed possible.

FIG. 20

FIG. 21

With the notation for quadrangular sets defined above, the last
theorem leads to the following

COROLLARY. If all "but one of the points of a quadrangular set Q (ABC,
DBF) are given, the remaining one is uniquely determined. (A, E)

For two quadrangles giving rise to the same quadrangular set
with the same notation must be similarly placed, and must hence
be in correspondence as described in -the theorem.

19,20] DESAEGUES CONFIGURATION 51

The quadrangular set which is the section by a 1-space of a complete 4-point
in a 2-space, the Desargues configuration which is the section by a 2-space of
a complete 5-point in a 3-space, the configuration of two perspective tetra-
hedra which maybe considered as the section by a 3-space of a complete 6-point
in a 4-space are all special cases of the section by an n-space of a complete
(n + 3)-point in an (n + l)-space. The theorems which we have developed for
the three cases here considered are not wholly parallel. The reader will find
it an entertaining and far from trivial exercise to develop the analogy in full.

EXERCISES

1. A necessary and sufficient condition that three lines containing the ver-
tices of a triangle shall be concurrent is that their intersections P, Q, It with
a line I form, with intersections E, F, G of corresponding sides of the triangle
with I, a quadrangular set Q(PQR, EFG).'

2. If on a given transversal line two quadrangles determine the same quad-
rangular set and are similarly placed, their diagonal triangles are perspective
from the center of perspectivity of the two quadrangles.

3. The polars of a point P on a line I with regard to all triangles which
meet I in three fixed points pass through a common point P' on L

4. In a plane TT let there be given a quadrilateral a 19 2 , a s , a 4 and a point
not on any of these lines. Let A 19 A 2 , A%, A 4 be any tetrahedron whose four
faces pass through the lines a v 2 , a 3 , a respectively. The polar planes of
with respect to all such tetrahedra pass through the same line of TT.

20. Additional remarks concerning the Desargues configuration.

The ten edges of a complete space five-point may be regarded (in
six ways) as the edges of two simple space five-points. Two such
five-points are, for example, JJjZyjJJJj and P^P^P^. Corresponding
thereto, the Desargues configuration may be regarded in six ways
as a pair of simple plane pentagons (five-points). In our previous
notation the two corresponding to the two simple space five-points
just given are ^J^^^^i an< i -^is-^B^a^L^Jr Every vertex* of each
of these pentagons is on a side of the other.

Every point, P^ for instance, has associated with it a unique line
of the configuration, viz. 1 M5 in the example given, whose notation
does not contain the suffixes occurring in the notation of the point.
The line may be called the polar of the point in the configuration,
and the point the pole of the line. It is then readily seen that the
polar of any point is the axis of perspectivity of two triangles
whose center of perspectivity is the point. In case we regard the
configuration as consisting of a complete quadrangle and complete

52 PEOJBCTIOK, SECTION, PERSPECTIVITY [CHAP, n

quadrilateral, it is found that a pole and polar are homologous vertex
and side of the quadrilateral and quadrangle. If we consider the
configuration as consisting of two simple pentagons, a pole and polar
are a vertex and its opposite side, e.g. ^ and -? 4 ^ 5 .

The Desargues configuration is one of a class of configurations
having similar properties. These configurations have been studied
by a number of writers.* Some of the theorems contained in these
memoirs appear in the exercises below.

EXERCISES

In discussing these exercises the existence should be assumed of a sufficient number
of points on each line so that the figures in question do not degenerate. In some cases
it may also be assumed that the diagonal points of a complete quadrangle are not
cottinear. Without these assumptions our theorems are true, indeed, but trivial.

1. What is the peculiarity of the Desargues configuration obtained as the
section of a complete space five-point by a plane which contains the point of
intersection of an edge of the five-point with the face not containing this edge ?
also by a plane containing two or three such points ?

2. Given a simple pentagon in a plane, construct another pentagon in the
same plane, whose vertices lie on the sides of the first and whose sides con-
tain the vertices of the first (cf. p. 51). Is the second uniquely determined
when the first and one side of the second are given?

S, 3. If two sets of three points A, B, C and A', B' } C' on two coplanar lines
Z and V respectively are so related that the lines A A', BB', CC f are concurrent,
then the points of intersection of the pairs of lines AB f and BA', BC' and CB',
CA' and A C' are collinear with the point ZZ'. The line thus determined is called
the polar of the point (A A', BB') with respect to I and Z'. Dualize.

4. Using the theorem of Ex. 3, give a construction for a line joining any
given point in the plane of two lines Z, V to the point of intersection of Z, V
without making use of the latter point.

5. Using the definition in Ex. 3, show that if the point P' is on the polar p
of a point P with respect to two lines Z, Z', then the point P is on the polar p'
of P' with respect to Z, V.

6. If the vertices A 19 A^ -4 3 , A of a simple plane quadrangle are respec-
tively on the sides a z , a 3 , a s , a 4 of a simple plane quadrilateral, and if the inter-
section of the pair of opposite sides A^A^ A%A is on the line joining the pair
of opposite points a^, <z 2 a 8 , the remaining pair of opposite sides of the quad-
rangle will meet on the line joining the remaining pair of opposite vertices of
the quadrilateral. Dualize.

* A. Cayley, Collected Works, Vol. I (1846), p. 317. G. Veronese, Mathema-
tische Annalen, Vol. XIX (1882). Further references will be found in a paper by
W. B. Carver, Transactions of the American Mathematical Society, Vol. VI (1905),
p. 534.

20] EXEECISES 53

7. If two complete plane n-points A^ A 2 , - , A n and A{, A^ , A' n are
so related that the side A^A^ and the remaining 2 (n 2) sides passing through
^ and ^L 2 meet the corresponding sides of the other rc-point in points of a line Z,
the remaining pairs of homologous sides of the two n -points meet on I and the
two n-points are perspective from a point. Dualize.

8. If five sides of a complete quadrangle A^A^A^A^ pass through five
vertices of a complete quadrilateral a 1 a a a 8 a 4 in such a way that A^A^ is on
<2 3 a 4 , A 2 A S on a 4 a 1? etc., then the sixth side of the quadrangle passes through
the sixth vertex of the quadrilateral. Dualize.

9. If on each of three concurrent lines a, &, c two points are given, A lt A 3
on a; jB 1? B 2 on &; C v C 2 on c, there can be formed four pairs of triangles
AfBjC L (i,j, k = 1, 2) and the pairs of corresponding sides meet in six points
which are the vertices of a complete quadrilateral (Veronese, Atti dei Lincei,
1876-1877, p. 649).

10. With nine points situated in sets of three on three concurrent lines
are formed 36 sets of three perspective triangles. For each set of three dis-
tinct triangles the axes of perspectivity meet in a point; and the 36 points
thus obtained from the 36 sets of triangles lie in sets of four on 27 lines,

o/> q

giving a configuration (Veronese, loc. cit.).

11. A plane section of a 6-point in space can be considered as 3 triangles
perspective in pairs from 3 collinear points with corresponding sides meeting
in 3 collinear points.

12. A plane section of a 6-point in space can be considered as 2 perspective
complete quadrangles with corresponding sides meeting in the vertices of a
complete quadrilateral.

13. A plane section of an n-point in space gives the configuration *

which may be considered (in n C n _ k ways) as a set of (n ) fc-points perspective

in pairs from n _x,C a points, which form a configuration
the points of intersection of corresponding sides form a coi

_*A n fc 2

& " n-*^

an<

ifiguration

/A fc-2
3 k C s

14. A plane section of a 7-point in space can be considered (in 120 ways)
as composed of three simple heptagons (7-points) cyclically circumscribing
each other.

15. A plane section of an 11-point in space can be considered (in 19 ways)
as composed of four 11-points cyclically circumscribing each other.

16. A plane section of an n-point in space for n prime can be considered

(in [n 2 ways) as n ~ simple n-points cyclically circumscribing each other.

* The symbol n C r is used to denote the number of combinations of n things
taken r at a time.

54 PROJECTION, SECTION, PERSPECTIVES [CHAP, n

17. A plane section of a 6-point in space gives (in six ways) a 5-point whose
sides pass through the points of a configuration

18. A plane section of an n -point in space gives a complete (n 1) -point
whose sides pass through the points of a configuration

* 19. The n-space section of an ??2-point (m z. n + 2) in an (n + l)-space can be
considered in the w-space as (m &) ^-points (in m (? m _ x ways) perspective in pairs
from the vertices of the ?i-space section of one (m I')-point ; the r-spaces of
the /j-point figures meet in (r 1) -spaces (r = l,2, ,w 1) which form the
rc-space section of a -point.

* 20. The figure of two perspective (n + l)-points in an w-space separates
(in n + 3 ways) into two dual figures, respectively an (n + 2)-point circum-
scribing the figure of (n + 2) (n l)-spaces.

* 21. The section by a 3-space of an n-point in 4-space is a configuration

M c,

71-2

-2<?2

M C 3

n-3

The plane section of this configuration is

22. Let there be three points on each of two concurrent lines l v / 2 . The
nine lines joining points of one set of three to points of the other determine
six triangles whose vertices are not on / t or l z . The point of intersection of ^
and Z 2 has the same polar with regard to all six of these triangles.

23. If two triangles are perspective, then are perspective also the two
triangles whose vertices are points of intersection of each side of the given
triangles with a line joining a fixed point of the axis of perspectivity to the
opposite vertex.

*24. Show that the configuration of the two perspective tetrahedra of
Theorem 2 can be obtained as the section by a 3-space of a complete 6-point
in a 4-space.

*25. If two 5-points in a 4-space are perspective from a point, the corre-
sponding edges meet in the vertices, the corresponding plane faces meet in the
lines, and the corresponding 3-space faces in the planes of a complete 5-plane
in a 3-space.

* 26. If two (n -f l)-points in an ra-space are perspective from a point,
their corresponding r-spaces meet in (r 1) -spaces which lie in the same
(n l)-space (r=rl, 2 , 1) and form a complete configuration of
(n + 1) (n 2) -spaces in (n 1) -space.

CHAPTER III

PROJECTIVITIES OF THE PRIMITIVE GEOMETRIC FORMS OF
ONE, TWO, AND THREE DIMENSIONS

21. The nine primitive geometric forms.

DEFINITION. A. pencil of points DEFINITION. A. pencil of planes
or a range is the figure formed by or an axial pencil * is the figure
the set of all points on the same formed by the set of all planes on
line. The line is called the axis the same line. The line is called
of the pencil. the axis of the pencil.

As indicated, the pencil of points is the space dual of the pencil
of planes. ^/

DEFINITION. A pencil of lines or a flat pencil is the figure formed
by the set of all lines which are at once on the same point and the
same plane ; the point is called the vertex or center of the pencil

The pencil of lines is clearly self-dual in space, while it is the
plane dual of the pencil of points. The pencil of points, the pencil
of lines, and the pencil of planes are called the primitive geometric
forms of the first grade or of one dimension.

DEFINITION. The following are known as the primitive geometric
forms of the second grade or of two dimensions :

The set of all points on a plane The set of all planes on a point
is called a plane of points. The is called a bundle of planes. The
set of all lines on a plane is called set of all lines on a point is called
a plane of lines: The plane is a bundle of lines . The point is
called the "base of the two forms, called the center of the bundles.
The figure composed of a plane The figure composed of a bundle
of points and a plane of lines of lines and a bundle of planes
with the same base is called a with the same center is called
planar field. simply a "bundle.

DEFINITION. The set of all planes in space and the set of all points'
in space are called the primitive geometric forms of the third grade
or of three dimensions.

* The pencil of planes is also called "by some writers a sheaf.
55

56 PRIMITIVE GEOMETRIC FORMS [CHAP, in

There are then, all told, nine primitive geometric forms in a space
of three dimensions.*

22. Perspectivity and projectivity. In Chap. II, 13, we gave a
definition of perspectivity. This definition we will now apply to the
case of two primitive forms and will complete it where needed. We
note first that, according to the definition referred to, two pencils of
points in the same plane are perspective provided every two homol-
ogous points of the pencils are on a line of a flat pencil, for they
then have the same projection from a point. Two ^planes of points
(lines) are perspective, if every two homologous elements are on a
line (plane) of a bundle of lines (planes). Two pencils of lines in the
same plane are perspective, if every two homologous lines intersect
in a point of the same pencil of points. Two pencils of planes are
perspective, if every two homologous planes are on a point of a pencil
of points (they then have the same section by a line). Two bundles of
lines (planes) are perspective, if every two homologous lines (planes)
are on a point (line) of a plane of points (lines) (they then have the
same section by a plane), etc. Our previous definition does not, how-
ever, cover all possible eases. In the first place, it does not allow for
the possibility of two forms of different kinds being perspective, such
as a pencil of points and a pencil of lines, a plane of points and a
bundle of lines, etc. This lack of completeness is removed for the
case of one-dimensional forms by the following definition. It should
be clearly noted that it is in complete agreement with the previous
definition of perspectivity ; as far as one-dimensional forms are con-
cerned it is wider in its application.

DEFINITION. Two one-dimensional primitive forms of different kinds,
not having a common axis, are perspective, if and only if they corre-
spond in such a (1, 1) way that each element of one is on its homol-
ogous element in the other ; two one-dimensional primitive forms of
the same kind are perspective, if and only if every two homologous
elements are on an element of a third one-dimensional form not
having an axis in common with one of the given forms. If the third
form is a pencil of lines with vertex P, the perspectivity is said to be

* Some writers enumerate only six, by defining the set of all points and lines on
a plane as a single form, and by regarding the set of all planes and lines at a point
and the set of all points and planes in space each as a single form. We have fol-
lowed the usage of Enriques, Vorlesungen iiber Projektive Geometric.

22] PEESPECTIYITY 57

central with center P; if the third form is a pencil of points or a pencil
of planes with axis Z, the perspectivity is said to be axial with axis Z.

As examples of this definition we mention the following: Two
pencils of points on skew lines are perspective, if every two homol-
ogous elements are on a plane of a pencil of planes ; two pencils of
lines in different planes are perspective, if every two homologous
lines are on a point of a pencil of points or a plane of a pencil of
planes (either of the latter conditions is a consequence of the other) ;
two pencils of planes are perspective, if every two homologous planes
are on a point of a pencil of points or a line of a pencil of lines (in
the latter case the axes of the pencils of planes are coplanar). A pen-
cil of points and a pencil of lines are perspective, if every point is on
its homologous line, etc.

It is of great importance to note that our definitions of perspective
primitive forms are dual throughout; ie. that if two forms are per-
spective, the dual figure will consist of perspective forms. Hence any
theorem proved concerning perspectivities can at once be dualized ; in
particular, any theorem concerning the perspectivity of two forms of
the same kind is true of any other two forms of the same kind.

We use the notation [P] to denote a class of elements of any kind
and denote individuals of the class by P alone or with an index or
subscript. Thus two ranges of points may be denoted by [P] and [Q].
To indicate a perspective correspondence between them we write

The same symbol, ^, is also used to indicate a perspectivity between
any two one-dimensional forms. If the two forms' are of the same
kind, it implies that there exists a third form such that every pair
of homologous elements of the first two forms is on an element of
the third form. The third form may also be exhibited in the notation
by placing a symbol representing the third form immediately over
the sign of perspectivity, ^.
Thus the symbols

denote that the range [P] is perspective by means of the center A with
the range [Q], that each Q is on a line r of the flat pencil [>], and
that the pencil [r] is perspective by; the axis a with the flat pencil [*].

58 PEIMITIVE GEOMETRIC FOEMS [CHAP, in

A class of elements containing a finite number of elements can
be indicated by the symbols for the several elements. When this
notation is used, the symbol of perspectivity indicates that elements
appearing in corresponding places in the two sequences of symbols

are homologous. Thus

123 4: = AJ3C D

implies that 1 and A, 2 and B, 3 and <7, 4 and D are homologous.

DEFINITION.* Two one-dimensional primitive forms [cr] and [V] (of
the same or different kinds) are said to be projective, provided there
exists a sequence of forms [r], |V] ? , [r (n) ] such that

[<r] == [r] = [r'] = -== [T ( >] = [>'].
. LJ A LJ A L J A A L J A L J

The correspondence thus established between [cr] and [V] is called
a protective correspondence or projectivity , or also a protective trans-
formation. Any element cr is said to be projected into its homologous
element cr' by the sequence of perspectivities.

Thus a projectivity is the resultant of a sequence of perspectivities.
It is evident that [cr] and JV ] may be the same form, in which case
the projectivity effects a permutation of the elements of the form.
For example, it is proved later in this chapter that any four points
A, B, C, D of a line can be projected into B, A } D, C respectively.

A projectivity establishes a one-to-one correspondence between the
elements of two one-dimensional forms, which correspondence we may
consider abstractly without direct reference to the sequence of perspec-
tivities by which it is defined. Such a correspondence we denote by

Projectivities we will, in general, denote by letters of the Greek
alphabet, such as TT. If a projectivity TT makes an element a- of a
form homologous with an element a 1 of another or the same form,
we will sometimes denote this by the relation ^(^)== .';,. Jn this
case we may say the* projectivity transforms cr into <r'. Here the
symbol TT( ) is used as a functional symbol "f acfiEg orTthe variable J
cr, which represents any one of the elements of a given form.

* This is Poncelet's definition of a projectivity.
t Just like F(x), sin(a;), log(fc), etc.

J The definition of variable is " a symbol x which represents any one of a class
of elements [cc]." It is in this sense that we speak of " a variable point."

23] PEOJECTIYITY 59

23. The projectivity of one-dimensional primitive forms. The

projectivity of one-dimensional primitive forms will be discussed
with reference to the projectivity of pencils of points. The corre-
sponding properties for the other one-dimensional primitive forms
will then follow immediately by the theorems of duality (Theorems
11-13, Chap. I).

THEOREM 1. If A, B } C are three points of a line I and A 1 , B r , C f
three points of another line V 9 then A can "be projected into A r , B into
B f , and C into C 1 ly means of two centers of perspectivity. (The lines
may be in the same or in different planes.) (A, E)

Proof. If the points in any one of the pairs AA\ BE 1 , or CC' are
coincident, one center is sufficient, viz., the intersection of the lines
determined by the other
two pairs. If each of these
pairs consists of distinct
points, let 8 be any point
of the line AA r , distinct
from A and A 1 (fig. 22).
From 8 project A, B, C
on any line V r distinct
from I and V 9 but con-
taining A f and a point
of I If B", C" are the
points of Z" correspond-
ing to B, C respectively,

the point of intersection S 1 of the lines B f B rf and C f O fr is the second
center of perspectivity. This argument holds without modification,
if one of the points A 9 B, C coincides with one of the points A r , B ! , O r
other than its corresponding point.

COROLLARY 1. IfA,B,C and A!, B ! 9 C 1 are on the same line, three
centers of perspectivity are sufficient to project A, B, C into A 1 , B f , C f
respectively. (A, E)

COROLLARY 2. Any three distinct elements of a one-dimensional
primitive form are protective with any three distinct elements of
another or the same one-dimensional primitive form. (A, E)

For, when the two forms are of the same kind, the result is ob-
tained from the theorem and the first corollary directly from the

60 PRIMITIVE GEOMETEIO FOKMS [CHAP.III

theorems of duality (Theorems 1113, Chap. I). If they are of differ-
ent kinds, a projection or section is sufficient to reduce them to the
same kind.

THEOREM 2. The projectivity ABCD -j^B ADC holds for any four
distinct $>oints A, B, C, D of a line. (A, E)

Proof. From a point S, not on the line I = AB, project ABCD into
AB r C f D f on a line V through A and distinct from I (fig. 23). From D
project AB'C f D f on the line SB. The last four points will then project
into BADC by means of the center C f . In fig. 23 we have

S D C r

ABCD~=AB r C f D r = BB f C"S = BADC.

A A A

It is to be noted that a geometrical order of the points ABCD has no bearing
on the theorem. In fact, the notion of such order has not yet been introduced

into our geometry and, indeed, cannot
be introduced on the basis of the
present assumptions alone. The theo-
rem merely states that the correspond-
ence obtained ly interchanging any two
of four collinear points and also inter-
changing the remaining two is projectwe.
The notion of order is, however, im-
plied in our notation of projectivity
and perspectivity. Thus, for example,
we introduce the following definition :

DEFINITION. Two ordered pairs of elements of any one-dimensional
form are called a throw;"]! the pairs are AB, CD, this is denoted by
T(AB, CD). Two throws are said to be equal, provided they are
projective ; in symbols, T(AB, CD) = T (A f B f , C'D f ), provided we have

The last theorem then states the equality of throws :
J(AB, CD) = T(BA, DC) = T(CD, AB) = T(DC,

The results of the last two theorems may be stated in the follow-
ing form :

THEOREM 1'. If 1 , %, 3 are elements of any one-dimensional prim-
itive form, there exist projective transformations 'which will effect any
one of the six permutations of these three elements.

PEOJECTIYITY

THEOREM 2'. If 1, 2, 3, 4 are any four distinct elements of a one-
dimensional primitive form, there exist ^rojectixe transformations
which will transform 1234 into an y ne f ^ ie following permuta-
tions of itself: 1234, 143, 3412, 4321.

A projective transformation has been defined as the resultant of any
sequence of perspectivities. We proceed now to the proof of a chain
of theorems, "which lead to the fundamental result that any projective
transformation between two distinct one-dimensional primitive forms
of the same kind can be obtained as the resultant of two perspectivities.

THEOREM 3. If [P], [P'], [P' f ] are pencils of points on three distinct

S S 1

concurrent lines I, V, l n respectively, sucli that [P] = [P f ] and [P r ] =

S"
"], then likewise [P] = [P"], and the three centers of perspectivity

S, S f S" are collinear. (A, E)

TIG-. 24

Proof. Let be the common point of the lines 1 9 V, l n . 1S.P V P^P Z
are three points of [P], and ^J^'J?' and JJ /7 J5"JJ f/ the corresponding
points of [P'], [P n ] (fig. 24), it is clear that the triangles P^P^,
P^PI, P^P^ !I are perspective from 0* By Desa-jg^e^Jbh^eorgm
(Theorem 1, Chap. II) homologous sides of ^any pair, of these three
triangles meet in ^nll jnaar ? ppiTrf^ Thp finn elusion of the theorem then

follows readily from the hypotheses.

* If the points in each of these sets of three are collinear, the theorem is obvious
and the three centers of perspectivity coincide.

PRIMITIVE GEOMETEIC FOEMS

[CHAP. Hi

COROLLARY. If n concurrent lines l v l v Z 8 ,

n-l,n

(A,E)

, l n are connected by
if I

perspectivities L ^_, A L - 2J A L - aj A
are distinct lines, then we have [J^] == |

Proof. This follows almost immediately from the theorem, except
when it happens that a set of four successive lines of the set Z 1 Z 2 Z S - l n
are such that the first and third coincide and likewise the second and
fourth. That this case forms no exception to the corollary may be
shown as follows : Consider the perspectivities connecting the pencils
of points on the lines l v Z 2 , Z 3 , Z 4 on the hypothesis that Z 1 = Z 8 , Z a = Z 4
(fig. 25.) Let l v l z meet in <9, and let the line 3^S M meet Z x in A v

and l z in -4 2 ; let A 3 = A l and J^ 4 be the corresponding points of Z 3 and
Z 4 respectively. Further, let JS V B Z9 B and C v C z> <7 8 , (7 4 be any
other two sequences of corresponding points in the perspectivities.
Let S A be determined as the intersection of the lines A^A.^ and B^B^
The two quadrangles S 12 S^B 2 C 2 and $ 41 34 J? 4 <7 4 have five pairs of
homologous sides meeting l t =l s in the points OA^B^B^Cy /Hence
the side S^C^ meets ^ in C^ (Theorem 3, Chap. II).

THEOREM 4. J/ K] K] [^] ^ re pencils of points on distinct

cr Sf

Kwe^ Z y Z 2 , Z respectively, such that K] = [^]==K], awd ^ [P'] is
the pencil of points on any line V containing the intersection of l v I
and also a point of 1 2 , "but not containing $ 2 , then there exists a point

S f S
v such that [JJ] = [P ! ] = [J>]. (A, E)

23] PEOJEGTIYITY 63

Proof. Clearly we have

But by tlie preceding theorem and the conditions on V we have

#
[-ZJ] = [P'], -where S a ' is a point of ^^ Hence we have

K] = [^'] = K]-

This theorem leads readily to the next theorem, which is the result
toward which we have been working. We prove first the following
lemmas :

LEMMA 1. Any axial perspectivity between the points of two skeiv
lines is equivalent to (and may le replaced ly) two central perspectives.
(A,E)

For let [P], [P'] be the pencils of points on the skew lines. Then
if S and S r are any two points on the axis s of the axial perspectivity,
the pencils of lines $[P], S r [P f ] * are so related that pairs of homol-
ogous lines intersect in points of the line common to the planes of the
two pencils [P] and S f [P f ], since each pair of homologous lines lie,
by hypothesis, in a plane of the axial pencil s[P]~s[P f ].

LEMMA 2. Any projectivity "between pencils of points may be defined
"by a sequence of central perspectivities.

For any noncentral perspectivities occurring in the sequence defining
a projectivity may, in consequence of Lemma 1, be replaced by sequences
of central perspectivities.

THEOREM 5. If two pencils of points [P] and [P ; ] on distinct lines
are projective, there exists a pencil of points [Q] and two points S, S r

S S f
such that we haw [P] = [Q] = [P']. (A, E)

Proof. By hypothesis and the two preceding lemmas we have a
sequence of perspectivities

& & S S S n

* Given a class of elements [P]; the symbol S[P] is used to denote the class
of elements SP determined by a given element S and any element of [P]. Hence,
if [P] is a pencil of points and 8 a point not in [P], S [P] is a pencil of lines with
center 8 ; if s is a line not on any P, s [P] is a pencil of planes with axis s.

64 PEIMITIVE GEOMETEIC POEMS [CHAP, in

We assume the number of these perspectivities to be greater than two,
since otherwise the theorem is proved. By applying the corollary of
Theorem 3, when necessary, this sequence of perspectivities may be
so modified that no three successive axes are concurrent. We may
also assume that no two of the axes Z, Z p Z 2 , Z s , , V of the pencils
[P], [J], [JJ], [JJ], - [P'J are coincident; for Theorem 4 may evidently
be used to replace any l t (= J t ) by a line ZJf (= Z t ). Now let l[ be the
line joining the points ZZ t and Z 2 Z 8 , and let us suppose that it does not
contain the center 2 (fig. 26). If then [2J 7 ] is the pencil of points
on l[ t we may (by Theorem 4) replace the given sequence of per-

orf or or cr

spectivities by [P] == [P^] = [P 2 ] = [P s ] = and this sequence

may in turn be replaced by

t sy si s,

(Theorem 3). If $ 2 is on the line
joining ll : and Z 2 Z 3 , we may replace
Z x by any line I" through the inter-
l section of Z X Z 2 which meets Z and
y IG t 26 does nofc contain the point 3 l (The-

orem 4). The line joining Z 2 Z 3 to

ZZ[' does not contain the point 3% which replaces 3 S . For, since ^ 2 is
on the line joining Z 3 Z 2 to ll v the points Z 3 Z 2 and ZZ X are homologous
points of the pencils [P 8 ] and [P] ; and if 3% were on the line join-
ing Z 3 Z 2 to ll[ r , the point Z 3 Z 2 would also be homologous to ll[ f . We
may then proceed as before. By repeated application of this process
we can reduce the number of perspectivities one by one, until finally
we obtain the pencil of points [Q] and the perspectivities

S S r

' As a consequence we have the important theorem :

THEOEEM 6. Any two projective pencils of points on skew lines are
axially perspective. (A, E)

Proof. The axis of the perspectivity is the line 33 r of the last
theorem.

24. General theory of correspondence. Symbolic treatment. In
preparation for a more detailed study of projective (and other) corre-
spondences, we will now develop certain general ideas applicable to

CORRESPONDENCE 65

all one-to-one reciprocal correspondences as defined in Chap. II, 13,
p. 35, and show in particular how these ideas may be conveniently
represented in symbolic form.* As previously indicated (p. 58), we
will represent such correspondences in general by the letters of the
Greek alphabet, as A, B, F, . The totality of elements affected
by the correspondences under consideration forms a system which we
may denote by S. If, as a result of replacing every element of a system
$! by the element homologous to it in a correspondence A, the sys-
tem Sj is transformed into a system S 2 , we express this by the relation
A(S 1 ) = S a . In particular, the element homologous with a given ele-
ment P is represented by A (P).

I. If two correspondences A, B are applied successively to a sys-
tem Sj, so that we have A (S t ) = S 2 and B (S 2 ) = S 3 , the single corre-
spondence F which transforms S x into S 3 is called the resultant or
product of A by B; in symbols S s = B (S 2 ) = B (A(S 1 )) = BA (SJ, or,
more briefly, B A = F. Similarly, for a succession of more than two
correspondences.

II. Two successions of correspondences A m A m ^i - A x and
BgB^-L - B! have the same resultant, or their products are equal,
provided they transform S into the same S'; in symbols, from the

A W A W _ 1 - - A,(S) = B A.! - - - B,(S)
follows A^A^ - . - A x = B ff B^. . . B r

III. The correspondence which makes every element of the sys-
tem correspond to itself is called the identical correspondence or simply
the identity, and is denoted by the symbol 1. It is then readily seen
that for any correspondence A we have the relations

IV. If a correspondence A transforms a system S x into S 2 , the corre-
spondence which transforms S 3 into S x is called the inverse of A and is
represented by A" 1 ; i.e. if we have A (S x ) = S 2 , then also A"" 1 (S 2 ) = S r
The inverse of the inverse of A is then clearly A, and we evidently
have also the relations

AA- I =AT I A=I.

* In this section we have followed to a considerable extent the treatment given
"by H. Wiener, Berichte der K. sachsischen Gesellschaft der Wissenschaften, Leipzig.
Vol. XLH (1890), pp. 249-252.

66 PEIMITIYE GEOMETBIC FOBMS [CHAP. Ill

Conversely, if A, A' are two correspondences such that we have
AA' = 1, then A' is the inverse of A. Evidently the identity is its
own inverse.

V. The product of three correspondences A, B, F always satisfies
the relation (FB) A = F (BA) (the associative law). For from the
relations A(S 1 )=S 2 , B(S 2 )=S 3 , F(S 3 )=S 4 foUows at once BA(S 1 )=S 3 ,
whence F(BA) (S~) =S 4 ;" and also FB (S 2 ) = S 4 , and hence (FB) A (SJ
= S 4 , which proves the relation in question. More generally, in any
product of correspondences any set of successive correspondences may
be inclosed in parentheses (provided their order be left unchanged),
or any pair of parentheses may be removed; in other words, in a
product of correspondences any set of successive correspondences may
be replaced by their resultant, or any correspondence may be replaced
by a succession of which the given correspondence is the resultant.

VI. In particular, we may conclude from the above that the inverse
of the product M - - BA is A" 1 B"" 1 M" 1 , since we evidently have
the relation M - - - BAA^B" 1 . . . M"^! (cf. IV).

VII. Further, it is easy to show that from two relations A = B and
F = A follows AF = BA and FA = AB. In particular, the relation
A = B may also be written AB~ x = 1, B~ x A = 1, B A~ l = 1, or A~ *B = 1.

VIII. Two correspondences A and B are said to be commutative
if they satisfy the relation BA = AB.

IX. If a correspondence A is repeated n times, the resultant is writ-
ten AA A = A n . A correspondence A is said to be of period n, if n
is the smallest positive integer for which the relation A n = 1 is satisfied.
When no such integer exists, the correspondence has no period ; when
it does exist, the correspondence is said to be periodic or cyclic.

X. The case n = 2 is of particular importance. A correspondence
of period two is called involutoric or reflexive.

25. The notion of a group. At this point it seems desirable to
introduce the notion of a group of correspondences, which is funda-
mental in any system of geometry. We will give the general abstract
definition of a group as follows : *

DEFINITION. A class G of elements, which we denote by a, &,
c, -, is said to form a group with respect to an operation or law of

* We have used here substantially the definition of a group given by L. E. Dickson,
Definitions of a Group and a Pield by Independent Postulates, Transactions of the
American Mathematical Society, Vol. VI (1905), p. 199.

25,26] GROUPS 67

combination o, acting on pairs of elements of G, provided the fol-
lowing postulates are satisfied :

G 1, For every pair of (equal or distinct) elements a, b of G, the
result a o & of acting with the operation o on the pair in the order
given * is a uniquely determined element of G.

G 2. The relation (a o &) o c = a o (b o c) holds for any three (equal or
distinct] elements a, 1), c of G.

G3. There occurs in G an element i, such that the relation aoi = a
holds for ever?/ clement a of G.

G4. For every element a in G there exists an element a 1 satisfying
the relation a o a'= i.

From the above set of postulates follow, as theorems, the following :

The relations a o a r = i and a o i = a im%)ly respectively the relations
a'o a i and ioa~ a.

An element i of G is called an identity element, and an element a 1
satisfying the relation a o a 1 = i is called an inrerse element of a.

There is only one identity element in G.

For every element a of & there is only one inverse.

We omit the proofs of these theorems.

DEFINITION. A group which satisfies further the following postulate
is said to be commutative (or dbelian) :

G 5. TJie relation a o 5 = I o a is satisfied for every pair of de-
ments a, b in G.

26. Groups of correspondences. Invariant elements and figures.

The developments of the last two sections lead now immediately
to the theorem:

A set of correspondences forms a group provided the set contains
the inverse of any correspondence in the set and provided the resultant
of any two correspondences is in the set

Here the law of combination o of the preceding section is simply
the formation of the resultant of two successive correspondences.

DEFINITION. If a correspondence A transforms every element of a
given figure F into an element of the same figure, the figure F is said
to be invariant under A, or to be left invariant by A. In particular,

* I.e. ao 5 and & o a are not necessarily identical. The operation o simply defines
a correspondence, whereby to every pair of elements a, 6 in G in a given order corre-
sponds a unique element j this element is denoted by a o &.

68 PEIMITIVE G-EOMETEIC FOEMS [CHAP, in

an element which is transformed into itself by A is said to be an
invariant element of A; the latter is also sometimes called a double
element or a fixed element (point, line, plane, etc.).

We now call attention to the following general principle :
The set of all correspondences in a group G which learn a given
figure invariant forms a group.

This follows at once from the fact that if each of two corre-
spondences of G leaves the figure invariant, their product and their
inverses will likewise leave it invariant ; and these are all in G, since,
by hypothesis, G is a group. It may happen, of course, that a group
defined in this way consists of the identity only.

These notions are illustrated in the following section :
27. Group properties of projectivities. From the definition of a pro-
jectivity between one-dimensional forms follows at once

THEOREM 7. The inverse of any projectivity and the resultant of
any two projectivities are projectivities.

On the other hand, we notice that the resultant of two perspec-
tivities is not, in general, a perspectivity ; if, however, two perspec-
tivities connect three concurrent lines, as in Theorem 3, their resultant
is a perspectivity. A perspectivity is its own inverse, and is therefore
reflexive. As an example of the general principle of 26, we have
the important result :

THEOREM 8. The set of all projectivities leaving a given pencil of
points invariant form a group.

If the number of points in such a pencil is unlimited, this group con-
tains an unlimited number of projectivities. It is called the general
projective group on the line. Likewise, the set of all projectivities on a
line leaving the figure formed by three distinct points invariant forms a
subgroup of the general group on the line. If we assume that each per-
mutation (cf. Theorem I/) of the three points gives rise to only a single
projectivity (the proof of which requires an additional assumption),
this subgroup consists of six projectivities (including, of course, the
identity). Again, the set of all projectivities on a line leaving each of two
given distinct points invariant forms a subgroup of the general group.

We will close this section with two examples illustrative of the
principles now under discussion, in which the projectivities in ques-
tion are given by explicit constructions.

27] G-BOUP OF PEOJECTIVITIES 69

EXAMPLE 1. A group of projectivities leaving eacli of two given
points invariant. Let M 9 N be two distinct points on a line I, and
let m, n be any two lines through 2I 9 N respectively and coplanar
with I (fig. 27). On m let there be an arbitrary given point S. If ^
is any other point on m distinct from J/ ; the points 8, S l together
with the line n define a projectivity TT^ on I as follows : The point
TTj (A) = A r homologous to any point A of I is obtained by the two

S 8 l
perspectivities [A] = [^i]~[-4/], where [A^ is the pencil of points

on n. Every point jS % then, distinct from J/, defines a unique pro-
jectivity 7r 4 ; we are to show that the set of all these projectivities TT,
forms a group. We show first that the product
of any two TT V 7r 2 is a uniquely determined pro-
jectivity 7T 3 of the set (fig. 27).
In the figure, A! = TT^ (A)
and A n = Tr z (A f ) have been

M B B' B" A A' A" N

constructed. The point S s giving A n directly from A by a similar con-
struction is then uniquely determined as the intersection of the lines
A n A v m. Let B be any other point of I distinct from M 9 N 9 and let
J5'= ir^(S) and J?*= ^(B 1 ) be constructed ; we must show that we have
B"=z*jr z (B). We recognize the quadrangular set Q(If3'A r , NA"B n ) as
defined by the quadrangle SS 2 B Z A Q . But of this quadrangular set all
points except B n are also obtained from the quadrangle S^B^A^
whence the line S^ determines the point B n (Theorem 3, Chap. II).
It is necessary further to show that the inverse of any projectivity in
the set is in the set. For this purpose we need simply determine $ 2
as the intersection of the line AA Z with m and repeat the former argu-
ment. This is left as an exercise. Finally, the identity is in the set,
since it is TT V when S^S.

70 PRIMITIVE GEOMETRIC FORMS [CHAP.III

It is to be noted that in this example the points Jf and N are
double points of each projectivity in the group; and also that if P, P r
and Q, Q r are any two pairs of homologous points of a projectivity
we have Q (MPQ, NQ'P ! ). Moreover, it is clear that any projectivity,
of the group is uniquely determined by a pair of homologous elements,
and that there exists a projectivity which
will transform any point A of I into any
other point B of I, provided only that
A and B are distinct from
M and N. By virtue of
the latter property the
group is said
to be transitive.

A A' A/ A"

FIG. 28

EXAMPLE 2. Commutative projectivities. Let M be a point of a
line Z, and let m, m f be any two lines through H distinct from I, but
in the same plane with I (fig. 28.) Let 8 be a given point of m } and
let a projectivity TT I be defined by another point S l of m which deter-

*? a

mines the perspectivities [A] = [-4 X ] == [A r ] 9 where [^ x ] is the pencil

of points on m f . Any two projectivities defined in this way ly points S l
are commutative. Let 7r 2 be another such projectivity, and construct
the points A r =^(A) 9 A' f ='jr 2 (A r ) 9 and A[ = Tr z (A). The quadrangle
JSS^A^A^ gives Q(MAA r , ITA ff A[)- and the quadrangular set determined
on I by the quadrangle SS^A^ has the first five points of the former
in the same positions in the symbols. Hence we have vr^A^) = A rr , and
therefore TT^ = TT^.

EXERCISES

1. Show that the set of all projectivities TT^ of Example 2 above forms a
group, which is then a commutative group.

2. Show that the projectivity wj of Example 1 above is identical with the
projectivity obtained by choosing any other two points of m as centers of
perspectivity, provided only that the two projectivities have one homologous

27,28] TWO-DBmNSIOjSTAL PEOJEOTIYITIES 71

pair (distinct from M or 2V) in common. Investigate the general question as
to how far the construction may be modified so as still to preserve the propo-
sition that the projectivities are determined by the double points J/, -V and
one pair of homologous elements.

3. Discuss the same general question for the projectivities of Example 2.

4. Apply the method of Example 2 to the projectivities of Example 1.
"Why does it fail to show that any two of the latter are commutative ? State
the space and plane duals of the two examples.

5. ABCD is a tetrahedron and a, ^8, y, S the faces not containing A ,B,C,D
respectively, and I is any line not meeting an edge. The planes (I A , IB, 1C, ID)
are projective with the points (la, 1/3, ly, /S).

6. On each of the ten sides of a complete 5-point in a plane there are three
diagonal points and two vertices. Write down the projectivities among these
ten sets of five points each.

28. Projective transformations of two-dimensional forms.

DEFINITION. A projective transformation between the elements of
two two-dimensional or two three-dimensional forms is any one-to-
one reciprocal correspondence between the elements of the two forms,
such that to every one-dimensional form of one there corresponds
a projective one-dimensional form of the other.

DEFINITION. A collineation is any (1, 1) correspondence between

-"^j^a^iEWM*--* J \ / JT

two two-dimensionaior two three-dimensional forms in which to every
element of one of the forms corresponds an element of the same kind
in the other form, and in which to every one-dimensional form of one
corresponds a one-dimensional form of the other. A. projective, colline-
ation is one in which this correspondence is projective. Unless other-
wise specified, the term collineation will, in the future, always denote
a projective collineation.*

In the present chapter we shall confine ourselves to the discus-
sion of some of the fundamental properties of collineations. In this
section we discuss the collineations between two-dimensional forms,
and shall take the plane (planar field) as typical; the corresponding
theorems for the other two-dimensional forms will then follow from
duality.

The simplest correspondence between the elements of two distinct
pla r nes TT, ir 1 is a perspective correspondence, whereby any two homol-
ogous elements are on the same element of a bundle whose center
is on neither of the planes TT, TT'. The simplest collineation in a plane,

* In how far a collineation must be projective will appear later.

72 PRIMITIVE GEOMETEIC FOEMS [CHAP.III

ie. which transforms every element of a plane into an element of the
same plane, is the following :

DEFINITION. A perspectwe^ collineation in a plane is a protective
collineation leaving invariant every point on a given line o and every,
line on a given point 0. The line o and the point are called the
axis and center respectively of the perspective collineation. If the
center and axis are not united, the collineation is called a planar
homology; if they are united, a planar elation.

A. perspective collineation in a plane TT may be constructed as
follows : Let any line o and any point of TT "be chosen as axis and
center respectively, and let T^ be any plane through o distinct from TT.
Let O v 2 be any two points collinear with and in neither of the
planes TT, 7r r The perspective collineation is then obtained by the

O l 0*

two perspectivities [P] === [PJ = [P'], where P is any point of TT and

.ZJ, P r are points of TT I and TT respectively. Every point of the line o
and every line through the point clearly remain fixed by the trans-
formation, so that the conditions of the definition are satisfied, if
only the transformation is projeetive. But it is readily seen that
every pencil of points is transformed by this process into a perspec-
tive pencil of points, the center of perspectivity being the point 0\
and every pencil of lines is transformed into a perspective pencil, the
axis of perspectivity being o. The above discussion applies whether

or not the point is on the line o,

THEOREM 9. A perspective col-
lineation in a plane is uniquely
defined if the center, axis, and any
two homologous points (not on the
axis or center) are given, with the
single restriction that the homol-
ogous points must le collinear
with 0. (A, E)

Proof. Let 0, o be the center and axis respectively (fig. 29). It is
clear from the definition that any two homologous points must be
collinear with 0, since every line through. is invariant ; similarly
(dually) any two homologous lines must be concurrent with o. Let
A, A 1 be the given pair of homologous points collinear with 0. The

28] TWO-DIMEXSIO-X T AL PEOJECTITITIES 73

point B f homologous to any point B of the plane is then determined.
We may assume 3 to be distinct from 0, A, A 1 and not to be on o.
B ! is on the line OB, and if the line AB meets o in C, then, since C
is invariant by definition, the line AB = AC is transformed into A r C.
B f is then determined as the intersection of the lines OB and A'C.
This applies unless B is on the line A A'} in this case we determine
as above a pair of homologous points not on AA' 9 and then use the
two points thus determined to construct B 1 . This shows that there
can be no more than one perspective collineation in the plane with
the given elements.

To show that there is one we may proceed as follows : Let ^ be
any plane through o distinct from TT, the plane of the perspectivity,
and let O t be any point on neither of the planes TT, TT I . If the line A0 1
meets ^ in A v the line A ? A^ meets 00^ in a point 2 . The perspec-
tive collineation determined by the two centers of perspectivity O v
and the plane ^ then has 0, o as center and axis respectively and A 9 A r
as a pair of homologous points.

COROLLARY 1. A perspective collineation in a plane transforms every
one-dimensional form into a perspective one-dimensional form. (A, E)

COROLLARY 2. A perspective collineation with center and axis o
transforms any triangle none of whose vertices or sides are on o *or
into a perspective triangle, the center of perspectivity of the triangles
being the center of the collineation and the asois of perspectivity being
the axis of the collineation. (A, E)

COROLLARY 3. The only planar collineations (whether required to
be protective or not] which leave invariant the points of a line o and
the lines through a point are homologies if is not on o, and
elations if is on o. ,(A, E)

Proof. This will be evident on observing that in the first paragraph
of the proof of the theorem no use is made of the hypothesis that the
collineation is protective.

COROLLARY 4. If H is a perspective collineation such that H (0) = 0,
H(o) = o, E(A) A r 9 H(B) = B f where A, A r , B, B J are collinear with
a point K of o, then we have Q(OAB, KB'A 1 ). (A, E)

Proof. If C is any point not on AJ! and H(<7) = C", the lines AC
and A r C r meet in a point L of o, and BO and B f C r meet in a point M
of o\ and the required quadrangle is CC'LM (cf. fig. 32, p. 77).

PRIMITIVE GEOMETRIC FORMS [CHAP, m

THEOREM 10. Any complete quadrangle of a plane can le trans-
formed into any complete quadrangle of the same or a different plane
ly a projectile collineation which, if the quadrangles are in the same
plane, is the resultant of a finite numler of perspective collineations.
(A,E)

Proof. Let the quadrangles be in the same plane and let their ver-
tices be A, B, 0, D and A', B\ C", D J respectively. We show first that
there exists a collineation leaving any three vertices, say A', B 1 , C f , of

, yl / / v

>\/Vi \

' k /

/ /

Pio. 80

the quadrangle A f B ! C f D r invariant and transforming into the fourth,
D f , any other point Z> 8 not on a side of the triangle A r $ r C r (Gg- 30). Let
D be the intersection of ^D 3 , ffD 1 and consider the homology with
center A 1 and ajds JS f C f transforming D 3 into D. Next consider the
homology with center B 1 and axis C f A r transforming ^into D f . Both
these homologies exist by Theorem 9. The resultant of these two
homologies is a eollineation leaving fixed A 1 , B r , C 1 and transforming
Z> 3 into D f . (It should be noticed that one or both of the homologies
may be the identity.)

Let O t be any point on the line containing A and A r and let o be
any line not passing through A or A*. By Theorem 9 there exists a

28,29] THEBB-DIMEirSIOSrAL PBOJECTIVITIES 75

perspective collineation ^ transforming A to A* and having O l and t
as center and axis. Let B v C v D^ be points such that

In like manner, let o z be any line through A f not containing B^ or
B r and let 2 be any point on the line B^B'. Let 7r 3 be the perspec-
tive collineation with axis 2 , center 2 , and transforming B : to B*.
Let <7 = 7 and D= w). Here"

Now let 3 be any point on the line C 2 6 f/ and let 7r 8 be the per-
spective collineation which has A f B r =o s for axis, 3 for center, and
transforms <7 2 to <7 ; . The existence of 7r g follows from Theorem 9 as
soon as we observe that C f is not on the line A f B f , by hypothesis,
and 6 2 is not on A ! B ! \ because if so, C^ would be on A r B and there-
fore C would' be on AB. Let 7r 3 (D 2 ) = D 3 . It follows that

The point D 3 cannot be on a side of the triangle A f B f C f because
then D 2 would be on a side t of A ! B r C 2 , and hence D^ on a side of
A r B : G v and, finally, D on a side of -ABC. Hence, by the first para-
graph of this proof, there exists a projectivity 7r 4 such that

n-t(A r B r C'Z> 9 ) = A'B'C'D*.

The resultant tr^ir^ir^rr^ of these four collineations clearly transforms
A, B, C, D into A f 9 B r , C f , D ! respectively. If the quadrangles are in
different planes, we need only add a perspective transformation between
the two planes.

COROLLAEY. There exist projectile collineations in a plane which
will effect any one of the possible % permutations of the vertices of
a complete quadrangle in the plane. (A, E)

29. Projective collineations of three-dimensional forms. Protective
collineations in a three-dimensional form have been defined at the
beginning of 28.

DEFINITION. A projective collineation in space which leaves inva-
riant every point of a plane a> and every plane on a point is called a
perspective collineation. The plane a> is called the plane ofperspectivity;
the point is called the center. If is on a>, the collineation is said
to be an elation in space ; otherwise, a homology in space.

76 PEIMITIYE GBOMETEIC FOKMS [CHAP.III

THEOREM 11. I/O is any point and a any plane, there exists one
and only one perspective collineation in space having 0, co for center
and plane of per spectivity respectively, which transforms any point A
(distinct from and not on co) into any other point A! (distinct from
and not on co) collinear with AO. (A, E)

Proof. We show first that there cannot be more than one per-
spective collineation satisfying the conditions of the theorem, by
showing that the point B f homologous to any point B is uniquely

determined by the given conditions. We may assume B not on a
and distinct from and A. Suppose first that B is not on the line
AO (fig. 31). Since BO is an invariant line, B 1 is on B0\ and if
the line AB meets co in L, the line AB = AL is transformed into
the line A ! L. Hence B r is determined as the intersection of BO
and A r L. There remains the case where B is on AO and distinct
from A and (fig. 32). Let 0, C 1 be any pair of homologous points
not on AO, and let AC and BC meet to in L and M respectively.
The line MB = M is transformed into M C r 9 and the point B J is then
determined as the intersection of the lines BO and MC f . That this
point is independent of the choice of the pair C, C J now follows
from the fact that the quadrangle MLCC 1 gives the quadrangular
set Q(KAA f , OB r B), where K is the point in which AO meets co
(K may coincide with without affecting the argument). The point
B* is then uniquely determined by the five points 0, E, A, A 1 , B.
The correspondence defined by the construction in the paragraph
above has been proved to be one-to-one throughout. On the line AO
it is projective because of the perspectivities (fig. 32)

29] THBEE-DIMENSIO]S T AL PEOJECTIYITIES 77

C '

On OB, any other line through 0, it is projeetive because of the per-
spectivities (fig. 31) , ^

w=w=

That any pencil of points not through is transformed into a
perspective pencil, the center of perspectivity being 0, is now easily
seen and is left as an exercise for the reader. From this it follows

PIG. 32

that any one-dimensional form is transformed into a projeetive form,
so that the correspondence which has been constructed satisfies the
definition of a projeetive collineation.

THEOREM 12. Any complete five-point in space can le transformed
into any other complete Jive-point in space ly a projeetive collineation
which is the resultant of a finite number of perspective collineations. (A,E)

Proof. Let the five-points be ABODE and AWC'D'E* respectively.
We will show first that there exists a collineation leaving A f f C f D f
invariant and transforming into JS f any point 2S Q not coplanar with
three of the points A f J3 r C f Z> f . Consider a homology having A f B ! C l as
plane of perspeetivity and D 1 as center. Any such homology trans-
forms J into a point on the line JS Q I> f . Similarly, a homology with
plane A f f l> ! and center C r transforms Iff into a point on the line HfC r .
If E Q D ! and S r O f intersect in a point JS V the resultant of two homol-
ogies of the kind described, of which the first transforms H into JS^
and the second transforms JE^ into Jff, leaves A f ffC f Z> f invariant and
transforms JH Q into If. If the lines E^D* and JS f C r are skew, there
is a line through B 1 meeting the lines E^D 1 and E f C r respectively

78 PEIMITIVE GEOMETEIC POEMS [CHAP, in

in two points E^ and JS 2 . The resultant of the three homologies, of
which the first has the plane A*'C f and center D f and transforms
JSo to JE V of which the second has the plane A f C f D f and center B 1
and transforms E l to JE Z , and of which the third has the plane A f 'D r
and center C r and transforms J 2 to E ! , is a collineation leaving A J 3 f C 1 D t
invariant and transforming E^ to E 1 . The remainder of the proof is
now entirely analogous to the proof of Theorem 10. The details are
left as an exercise.

COROLLARY. There exist projectile collineations which, will effect
any one of the possible 120 permutations of the vertices of a complete
five-point in space. (A, E)

' EXERCISES

1. Prove the existence of perspective collineations in a plane without
making use of any points outside the plane.

2. Discuss the figure formed by two triangles which are homologous
under an elation. How is this special form of the Desargues configuration
obtained as a section of a complete five-point in space?

3. Given an elation in a plane with center and axis o and two homol-
ogous pairs A , A' and B, B' on any line through 0, show that we always
have Q(OAA', OB'B).

4. What permutations of the vertices of a complete quadrangle leave a
given diagonal point invariant ? every diagonal point ?

5. Write down the permutations of the six sides of a complete quadrangle
brought about by all possible permutations of the vertices.

6. The set of all homologies (elations) in a plane with the same center
and axis form a group.

7. Prove that two elations in a plane having a common axis and center
are commutative. Will this method apply to prove that two homologies with
common axis and center are commutative?

8. Prove that two elations in a plane having a common axis are commu-
tative. Dualize. Prove the corresponding theorem in space.

9. Prove that the resultant of two elations having a common axis is an
elation. Dualize. Prove the corresponding theorem in space. What groups
of elations are defined by these theorems ?

10. Discuss the effect of a perspective collineation of space on : (1) a pencil
of lines ; (2) any plane ; (3) any bundle of lines j (4) a tetrahedron ; (5) a
complete five-point in space.

11. The set of all collineations in space (in a plane) form a group.

12. The set of all projective collineations in space (in a plane) form a group.

13. Show that under certain conditions the configuration of two perspective
tetrahedra is left invariant by 120 collineations (cf. Ex. 3, p. 47).

CHAPTER IV

HARMOITCC CONSTRUCTIONS AM) THE FUNDAMENTAL THEOREM
OF PROTECTIVE GEOMETRY

30. The projectivity of quadrangular sets. We return now to a
more detailed discussion of the notion of quadrangular sets introduced
at the end of Chap. II. We there defined a quadrangular set of points
as the section by a transversal of the sides of a complete quadrangle ;
the plane dual of this figure we call a quadrangular set of lines;*
it consists of the projection of the vertices of a complete quadrilateral
from a point which is in the plane of the quadrilateral, but not on
any of its sides ; the space dual of a quadrangular set of points we
call a quadrangular set of planes; it is the figure formed by the
projection from a point of the
figure of a quadrangular set
of lines. We may now prove
the following im-
portant theorem :

THEOREM 1.
TJie section by a
transversal of a
quadrangular
set of lines is a
quadrangular
set of points.
(A,E)

Proof. By Theorem 3', Chap. II, p. 49, and the dual of Note 2, on
p. 48, we may take the transversal I to be one of the sides of a com-
plete quadrilateral the projection of whose vertices from a point P
forms the set of lines in question (fig. 33). Let the remaining three
sides of such a quadrilateral be a, &, <?. Let the points Jo> ca, and al

* It would be more natural at this stage to call such a set a quadrilateral set of
lines; the next theorem, however, justifies the term we have chosen, which has the
advantage of uniformity.

FIG. 33

THE FUNDAMENTAL THEOEEM

[CHAP. IV

be denoted by A, B, and C respectively. The sides of the quadrangle
PABCra&et I in the same points as the lines of the quadrangular set
of lines.

COROLLARY. A set of collinear points which is projective with a
quadrangular set is a quadrangular set. (A, E)

THEOKEM I/. The projection from a point of a quadrangular set of
points is a quadrangular set of lines. (A, E)

This is the plane dual of the preceding ; the space dual is :

THEOREM I/ 7 . The section ly a plane of a quadrangular set of planes
is a quadrangular set of lines. (A, E)

COROLLARY. If a set of elements of a primitive one-dimensional form
is protective with a quadrangular set, it is itself a quadrangular set.
(A,E)

31. Harmonic sets. DEFINITION. A quadrangular set Q (123, 124)
is called a harmonic set and is denoted by H (12, 34). The elements
3, 4 are called harmonic conjugates with respect to the elements 1, 2 ;
and 3 (or 4) is called the harmonic conjugate of 4 (or 3) with respect
to 1 and 2.

From this definition we see that in a harmonic set of points
\\(AC, BD), the points A and C are diagonal points of a complete

FIG. 34

quadrangle^ while the points B and D are the intersections of the
remaining two opposite sides of the quadrangle with the line AC
(fig. 34). Likewise, in a harmonic set of lines H (ac, Id), the lines a
and c are two diagonal lines of a complete quadrilateral, while the

31] HAEMO]SriC SETS 81

lines 5 and d are the lines joining the remaining pair of opposite
vertices of the quadrilateral to the point of intersection ac of the
lines a and c (fig. 35). A harmonic set of planes is the space dual
of a harmonic set of points, and is therefore the projection from a
point of a harmonic set of lines.

In case the diagonal points of a complete quadrangle are collinear, any
three points of a line form a harmonic set and any point is its own harmonic
conjugate with regard to any two points collinear with it. Theorems on har-
monic sets are therefore trivial in those spaces for which Assumption H is
not true. We shall therefore base our reasoning, in this and the following
two sections, on Assumption H Q J though most of the theorems are obviously
true also in case H is false. This is why some of the theorems are labeled as
dependent on Assumptions A and E, whereas the proofs given involve H Q also.

The corollary of Theorem 3, Chap. II, when applied to harmonic
sets yields the following:

THEOREM 2. The harmbnic conjugate of an element with respect to
two oilier elements of a one-dimensional primitive form is a unique
element of the form. (A, E)

Theorem 1 applied to the special case of harmonic sets gives

THEOREM 3. Any section or projection of a harmonic set is a
harmonic set. (A, E)

COROLLARY. If a set of four elements of any one-dimensional prim-
itive form is projectile with a harmonic set, it is itself a harmonic set.
(A,E)

THEOREM 4. If 1 and 2 are harmonic conjugates with respect to
3 and 4, 3 and 4 are harmonic conjugates with respect to 1 and 2.
(A, E, H )

Proof. By Theorem 2, Chap. Ill, there exists a projectivity

1234^-3412.

But by hypothesis we have H(34, 12). Hence by the corollary of
Theorem 3 we have H (12, 34).

By virtue of this theorem the pairs 1, 2 and 3, 4 in the expression
H (12, 34) play the same r81e and may be interchanged,*

*The corresponding theorem for the more general expression Q(12S, 456)
cannot be derived without the use of an additional assumption (cf . Theorem 24,
Chap. IV).

82 THE FUNDAMENTAL THEOBEM [CHAP.IV

THEOREMS. Given two harmonic sets H(12, 34) and H(1'2', 3'4'),
there exists a projectivity such that 1234^ 1'2'3'4'. (A, E)

Proof. Any projectivity 123 ^ 1'2'3' (Theorem 1, Chap. Ill) must
transform 4 into 4' by "virtue of Theorem 3, Cor., and the fact that
the harmonic conjugate of 3 with respect to 1 and 2 is unique (Theo-
rem 2). This is the converse of Theorem 3, Cor.

COROLLABY 1. If H (12, 34) and H (12', 3'4') are two harmonic sets
of different one-dimensional forms having the element 1 in common,
we ham 1234 = 12'3'4'. (A, E)

For under the hypotheses of the corollary the projectivity 123-^ 1'2'3'
of the preceding proof may be replaced by the perspectivity 123 == 12'3'.

COEOLLAEY 2. If H (12, 3 4) is a harmonic set, there exists a projec-
tivity 1234^-1243. (A,E)

This follows directly from the last theorem and the evident fact
that if H (12, 34) we have also H (12, 43). The converse of this
corollary is likewise valid ; the proof, however, is given later in this
chapter (ef. Theorem .27, Cor. 5).

We see as a result of the last corollary and Theorem 2, Chap. Ill,
that if we have H(12, 34), there exist projectivities which will trans-
form 1234 into any one of the eight permutations

1234, 1243, 2134, 2143, 3412, 3421, 4312, 4321.*

In other words, if we have H(12, 34), we have likewise H(12, 43),
H(21, 34), H(21, 43), H(34, 12), H(34, 21), H(43, 12), H(43, 21).

THEOREM 6. The two sides of a complete quadrangle which meet in
a diagonal point are harmonic conjtyates with respect to the two sides
of the diagonal triangle which meet in this point. (A, E)

Proof. The four sides of the complete quadrangle which do not
pass through the diagonal point in question form a quadrilateral
which defines the set of four lines mentioned as harmonic in the
way indicated (fig. 36).

It is sometimes convenient to speak of a pair of elements of a
form as harmonic with a pair of elements of a form of different ^
kind. For example, we may say that two points are harmonic with
two lines in a plane with the points, if the points determine two

* These transformations form the so-called eigM-group.

31]

HABM02SIC SETS

lines through the intersection of the given lines which are harmonic
with the latter; or, what is the same thing, if the line joining the
points meets the lines in two points
harmonic with the given points.
With this understanding we may
restate the last theorem as follows:
The sides of a complete quadrangle
which meet in a diagonal point are
harmonic with the other two diago-
nal points. In like manner, we may
say that two points are harmonic
with two planes, if the line joining
the points meets the planes in a
pair of points harmonic with the
given points ; and a pair of lines is
harmonic with a pair of planes, if FIG. 35

they intersect on the intersection

of the two planes, and if they determine with this intersection two
planes harmonic with the given planes.

EXERCISES

1. Prove Theorem 4 directly from a figure without using Theorem 2,
Chap. III.

2. Prove Theorem 5, Cor. 2, directly from a figure.

3. Through a given point in a plane construct a line which passes through
the point of intersection of two given lines in the plane, without making use

_cjfjhe latter point.

4. A line meets the sides of a triangle ABC in the points A I9 B v C 1? and
the harmonic conjugates A%, B 2 , C 3 of these points with respect to the two
vertices on the same side are determined, so that we have H(^jB, C^C*),
H(BC,A l A 2 )^nd.H(CA,B l B^ Show that A 19 B v C 2 ; ^C^A^C^A^
are collinear; that AA^ BB 2 , CC 2 are concurrent; and that AA 2 , BB V CC X ;
AA V BB^ CCtf AA^BBv CC 2 are also concurrent.

5. If each of two sides AB, EC of a triangle ABC meets a pair of opposite
edges of a tetrahedron in two points which are harmonic conjugates with
respect to .4, B and B, C respectively, the third side CA will meet the third
pair of opposite edges in two points which are harmonic conjugates with
respect to C, A.

6. A, B, C, D are the vertices of a quadrangle the sides of which meet a
given transversal I in the six points P 1? P 2 , P s , P 4 , P 5 , P 6 ; the harmonic conju-
gate of each of these points with respect to the two corresponding vertices of the

84 THE FUOTDAMElSrTAL THEOEEM [CHAP. IV

quadrangle is constructed and these six points are denoted by P{, P%, P%, P,
P$, PQ respectively. The three lines joining the pairs of the latter points
which lie on opposite sides of the quadrangle meet in a point P, which is the
harmonic conjugate of each of the points in -which these three lines meet I
with respect to the pairs of points P' denning the lines.

7. Denning the polar line of a point with respect to a pair of lines as the
harmonic conjugate line of the point with regard to the pair of lines, prove
that the three polar lines of a point as to the pairs of lines of a triangle form
a triangle (called the cogredient triangle) perspective to the given triangle.

8. Show that the polar line denned in Ex. 7 is the same as the polar line
denned in Ex. 8, p. 52.

9. Show that any line through a point and meeting two intersecting
lines Z, V meets the polar of with respect to I, I' in a point which is the
harmonic conjugate of with respect to the points in which the line through
meets Z, V.

10. The axis of perspectivity of a triangle and its cogredient triangle is the
polar line (cf . p. 4C) of the triangle as to the given pointt

11. If two triangles are perspective, the two polar lines of a point on their
axis of perspectivity meet on the axis of perspectivity.

12. If the lines joining corresponding vertices of two n-lines meet in a point,
the points of intersection of corresponding sides meet on a line.

13. (Generalization of Exs. 7, 10.) The n polar lines of a point P as to the n
(n l)-lines of an n-line in a plane form an ra-line (the cogredient w-line)
whose sides meet the corresponding sides of the given n-line in the points of
a line p. The line p is called the polar of P as to the ra-line.*

14. (Generalization of Ex. 11.) If two si-lines are perspective, the two
polar lines of a point on their axis of perspectivity meet on this axis.

15. Obtain the plane duals of the last- two problems. Generalize them to
three- and 7i-dimensional space. These theorems are fundamental for the con-
struction of polars of algebraic curves and surfaces of the n-th degree*

32. Kets of rationality on a line, DEFINITION. A point P of a line
is said to be "harmonically related to three given distinct points A, B t G
of the line, provided P is one of a sequence of points A, B, C, IT 1} j6T 2 , _H" 3 ,
- of the line, finite in number, such that ^ is the harmonic conju-
gate of one of the points A, B, C with respect to the other two, and
such that every other point H t is harmonic with three of the set A, B, C,
H v HI, - - 3 j5" _ r The class of all points harmonically related to three
distinct points A, B, C on a line is called the one-dimensional net of
rationality defined by A, B, (7; it is denoted by R(A0). A net of
rationality on a line is also called a linear net

* This is a definition "by induction of the polar line of a point with respect to an
n-line.

32] NETS OF BATIOXALITY 85

THEOREM 7. If A, 3, C, D and A 1 , B f , C f , I) 1 are respectively points
of two lines stick that ABCD-^A r B f C r Z> f , and if D is harmonically
related to A, 3, C, then D r is harmonically related to A r ,B r , C f . (A, E)

This follows directly from the fact that the projectivity of the theo-
rem makes the set of points Hj which defines D as harmonically related
to A,B, C pro jective with a set of points H( such that every harmonic set
of points of the sequence A 9 B, C, H v H^ - -, D is homologous with a
harmonic set of the sequence^ 7 , B f , C f , JBJ, H 9 ^ '-,D f (Theorem 3, Cor.).

COROLLARY. If a class of points on a line is projectile with a net
of rationality on a line, it is itself a net of rationality.

THEOREM 8. If K, L, JJ are three distinct points of R (ABC), A, B, C
are points of R (KLM). (A, E)

Proof. From the projectivity ABCKj^ BAKC follows, by Theorem?,
that C is a point of R (ABIC). Hence all points harmonically related
to A, B, C are, by definition, harmonically related to A, B, K. Since K
is, by hypothesis, in the net R (ABC), the definition also requires that
all points of R(ABK] shall be points of R(ABC). Hence the nets
R(ABC) and R(ABK) are identical; and so R(ABC) = R(ABK)

COROLLARY. A net of rationality -on a line is determined ly any
distinct three of its points.

THEOREM 9. If all but one of the six (or five, or four) points of a
quadrangular set are points of the same net of rationality R, this
one point is also a point of R. (A, E)

Proof. Let the sides of the quadrangle PQRS (fig. 37) meet the
line I as indicated in the points A, A^ B,B^ C } C v so that B = B^\
and suppose that the first five of these are points of a net of rationality

R =:

We must prove that G : is a point of R. Let the pair of lines ES and
PQ meet in B f . We then have

S M

BCB^A = BQB f P = BAjBiCr

Since A is in. R(BCBJ, it follows from this projectivity, in view of
Theorem 7, that C l is in R (BA&) == R.

DEFINITION. A point P of a line is said to be guadrangularly
related to three gwen distinct points A, B, C of the line, provided

THE FUNDAMENTAL THEOEEM

[CHAP. IV

P is one of a sequence of points A, B, C, H v H^ H# of the line,
finite in number, such that H^ is the harmonic conjugate of one of
the points A, B, with respect to the other two, and such that every
other point H t is one of a quadrangular set of which the other five
belong to the set A, B, C, & v H v - - -, ^_ r

COROLLAET. The class of all points Quadrangularly related to three
distinct collinear points A, B, C is R(ABC). (A, E)

From the last corollary it is plain that R (ABO) consists of all points that
can be constructed from A , .5, C by means of points and lines alone ; that is
to say, all points -whose existence can be inferred from Assumptions A, E, H .
The existence or nonexistence of further points on the line ABC is unde-
termined as yet. The analogous class of points in a plane is the system of all
points constructive, by means of points and lines, out of four points A,B,C, D,
no three of which are collinear. This class of points is studied by an indirect
method in the next section.

33. Nets of rationality in the plane. DEFINITION. A point is said
to be rationally related to two noncollinear nets of rationality R v R 2
having a point in common, provided it is the intersection of two lines
each of which joins a point of R l to a distinct point of R 2 . A line is
said to be rationally related to R and R 2 , provided it joins two points
that are rationally related to them. The set of all points and lines
rationally related to R v R 2 is called the net of rationality in a plane
(or of two dimensions) determined by R v R 2 ; it is also called the
planar net defined by R x> R 2 .

From this definition it follows directly that all the points of R x
and R 2 are points of the planar net defined by R v R 2 .

133] NETS OF EATIONALITY 87

THEOKEM 10. Any line of the planar net R 2 defined "by R v R 2 meets

Proof. We prove first that if a line of the planar net R 2 meets R v
it meets R 2 . Suppose a line I meets R x in A^ it then contains a second
point P of R 2 . By definition, through P pass two lines, each of which
joins a point of R 1 to a distinct point of R 2 . If I is one of these lines,
the proposition is proved ; if these lines are distinct from Z, let them
meet R x and R 2 respectively in the points B v B 2 and JJ, J% (fig. 38).
If is the common point of R v R 2 , we then have

where -4 2 is the point in which I meets the line of R 2 . Hence A* is a
point of R 2 (Theorem 7).

Now let I be any line of the net R 3 , and let P, Q be two points
of the net and on I (def.). If one of these points is a point of R x or
R 2 , the theorem is proved by the case just considered. If not, two
lines, each joining a point of R l to a distinct point of R s , pass through
P; let them meet R l in A v B It and R 2 in A z , B^ respectively (fig. 38).
Let the lines QA^ and QB l meet R 2 in A[ and B^ respectively (first case).

Then if I meets the lines of R l and R 2 in Jj* and P% respectively, the
quadrangle PQA I B 1 gives rise to the quadrangular set Q(I^A^B Z9
OB^) of which five points are points of R 2 ; hence P z is a point of R 2
(Theorem 9). J? is then a point of R x by the first case of this proof.

THEOREM 11. The intersection of any two lines of a planar net is
a point of the planar net (A, E)

THE FUNDAMENTAL THEOEEM

[CHAP. IV

Proof. This follows directly from the definition and the last theo-
rem, except when one of the lines passes through 0, the point common
to the two linear nets R 1? R 2 defining the planar net. In the latter
case let the two lines of the planar net be l v 1 2 and suppose 1 2 passes
through 0, while ^ meets R 1? R 2 in A v A z respectively (fig. 39). If the
point of intersection P of IJ 2 were not a point of the planar net, 1 2

would, by definition,
contain a point Q of
the planar net, dis-
tinct from and P.
The lines QA l and
QA a would meet R 2
and R 1 in two points
2? 2 and 2^ respec-
tively. The point <7 2
in which the line
PB^ met the line of
R 2 would then be the
harmonic conjugate

FIG. 39

of JB Q with respect to and A 2 (through the quadrangle

<7 2 would therefore be a point of

R 2 , and hence

P would be a

point of the planar net, being the intersection of the lines A^A 2
and B^dy

THEOREM 12. TJie points of a planar net R 2 on a line of the planar
net form a linear net. (A, E)

Proof. Let the planar net be defined by the linear nets R x , R 2 and
let I be any line of the planar net. Let P be any point of the planar
net not on I or R l or R 2 . The lines joining P to the points of R 2 on I
meet R x and R 2 by Theorems 10 and 11. Hence P is the center of
a perspectivity which makes the points of R 2 on I perspective with
points of R 1 or R 2 . Hence the points of I belonging to the planar net
form a linear net. (Theorem 7, Cor.)

COROLLARY. The planar net R* defined ly two linear nets R v R 2 is
identical with the planar net R| defined "by two linear nets R 8 , R 4 , pro-
vided R s , R 4 are linear nets in R^. (A, E)

For every point of R 2 is a point of R| by the above theorem, and
every point of R^ is a point of Rf by Theorem 10.

33,34] NETS OF KATIONALITY 89

EXERCISE

If A, B 9 C, D are the vertices of a complete quadrangle, there is one and
only one planar net of rationality containing them ; and a point P belongs to
this net if and only if P is one of a sequence of points ABCDD^D^ , finite
in number, such that D l is the intersection of two sides of the original quad-
rangle and such that every other point D t is the intersection of two lines join-
ing pairs of points of the set ABCDD^ A-i-

34. Nets of rationality in space. DEFINITION. A point is said to
be rationally related to two planar nets R 2 , R| in different planes but
having a linear net in common, provided it is the intersection of two
lines each, of which joins a point of R 2 to a distinct point of R 2 2 .
A line is said to be rationally related to R 2 , R|, if it joins two, a plane
if it joins three, points which are rationally related to them. The set
of all points, lines, and planes rationally related to R 2 , R| is called the
net of rationality in space (or of three dimensions) determined by
RI > R% 5 ft is a l so called the spatial net defined by R 2 , R 2 2 .

Theorems analogous to those derived for planar nets may now be
derived for nets of rationality in space. We note first that every point
of R 2 and of R * is a point of the spatial net R 3 defined by R x 2 , R * (the
definition applies equally well to the points of the linear net common
to R 2 , R|) ; and that no other points of the planes of these planar nets
are points of R 3 . The proofs of the fundamental theorems of align-
ment, etc., for spatial nets can, for the most part, be readily reduced
to theorems concerning planar nets. We note first :

LEMMA. Any line joining a point A^ of R? to a distinct point P of
R 3 meets R 2 . (A, E)

Proof. By hypothesis, through P pass two lines, each of which
joins a point of R 2 to a distinct point of R 2 2 . We may assume these
lines distinct from the line PA V since otherwise the lemma is proved.
Let the two lines through P meet R a 2 , R| in B v B 2 and C v C z respec-
tively (fig. 40). If A v B v C l are not collinear, the planes PA^ and
PA^ meet R 2 in the lines A^B^ and A^ respectively, which meet
the linear net common to R 2 , R 2 in two points S, T respectively
(Theorems 11, 12). The same planes meet the plane of R 2 2 in the lines
SB 2 and TC^ respectively, which are lines of R 2 , since S 9 T are points
of R 2 2 . These lines meet in a point A 2 of R 2 2 (Theorem 11), which
is evidently the point in which the line PA meets the plane of R 2 2 .
If A v B v C^ are collinear, let J 2 be the intersection of PA^ with the

THE FUNDAMENTAL THEOEEM

[CHAP. IV

plane of R 2 2 , and S the intersection of A l B l with the linear net
common to R? and R 2 2 . Since A 1 is in R (SBfiJ, the perspectivity

SC 1 B l A = SC^Ao implies that A 2 is in R(SB 2 C 2 ) and hence in R 3 2 .

m P

Fio. 40

THEOREM 13. Any line of the spatial net R 3 defined by R*, R| meets
R*andR*. (A, E)

FIG. 41

Proof. By definition the given line I contains two points A and B
of the net R 8 (fig. 41). If A or B is on Rf or R|, the theorem reduces
to the lemma. If not, let ^ be a point of R.?, and A z and B 2 the points
in which, by the lemma, %A and P^B meet R 2 2 ; also let JJ' be any

84] STETS OF RATIONALITY 91

point of R x 2 not in the plane P^AB, and let P^A and P^B meet R 3 2 in A
and .#2". The lines J 2 2 and -.4^| meet in a point of R 2 2 (Theorem 11),
and this point is the point of intersection of I with the plane of R|.
The argument is now reduced to the case considered in the lemma.

THEOREM 14 The points of a spatial net lying on a line of the
spatial net form a linear net. (A, E)

Proof. Let I be the given line, R 2 and R 2 2 the planar nets defining
the spatial net R 3 , and L : and L 2 the points in which (Theorem 13)
I meets Rf and R 2 2 (L t and L 2 may coincide). Let A^ be any point of
R 2 not on I or on R 2 , and S the point in which A^L^ meets the linear
net common to R 2 and R 2 2 (fig. 42). If L^ and L 2 are distinct, the lines

PIG. 42 FIG. 43

SL t and SL 2 meet R 2 and R 2 2 in linear nets (Theorem 12); and, by
Theorem 13, a line joining any point P of R 3 on I to A l meets each
of these linear nets. Hence all points of R 3 on I are in the planar
net determined by these two linear nets. Moreover, by the definition
of R 8 , all the points of the projection from A of the linear net on SL 2
upon I are points of R 3 . Hence the points of R 3 on I are a linear net.
If x = ,= S, then, by definition, there is on I a point A of R*, and
the line AA^ meets R| in a point A z (fig. 43). The lines SA^ and SA 2
meet R? and R| in linear nets R x and R 2 by Theorem 12. If JB X is
any point of R x other than Jt p the line .A^ meets R 2 2 in a point J? 2 by
Theorem 13. By Theorem 12 all points of I in the planar net deter-
mined by Rj and R 2 form a linear net, and they obviously belong to R 3 .
Moreover, any point of R 3 on I, when joined to A v meets R 2 2 by Theo-
rem 13, and hence belongs to the planar net determined by R x and R 2 .
Hence, in this case also, the points of R 8 on I constitute a linear net.

92 THE FUNDAMENTAL THEOEEM [CHAP, iv

THEOREM 15. The points and lines of a spatial net R 3 which lie on
a plane a of the net form a planar net. (A, E)

Proof. By definition a contains three noncollinear points A, B, C of
R 8 , and the three lines AB, BC, CA meet the planar nets R* and R 2 2 ,
which determine R 3 , in points of two linear nets R x and R 2 , consisting
entirely of points of R 3 . These linear nets, if distinct, determine a
planar net R 2 in a, which, by Theorem 10, consists entirely of points
and lines of R 3 . Moreover, any line joining a point of R 3 in a to A
or B or C must, by Theorem 13, meet R x and R 2 and hence be in R\
Hence all points and lines of R 3 on a are points and lines of R 2 . This
completes the proof except in case R t = R a , which case is left as an
exercise.

COROLLARY 1. A net of rationality in space is a space satisfying
Assumptions A and E, if "line" le interpreted as "linear net" and
"plane" as "planar net." (A, E)

For all assumptions A and E, except A 3, are evidently satisfied ;
and A 3 is satisfied because there is a planar net of points through
any three points of a spatial net R 3 , and any two linear nets of this
planar net have a point in common.

This corollary establishes at once all the theorems of alignment in
a net of rationality in space, which are proved in Chap. I, as also the
principle of duality. "We conclude then, for example, that two planes
of a spatial net meet in a line of the net, and that three planes of a
spatial net meet in a point of the net (if they do not meet in a line),
etc. Moreover, we have at once the following corollary :

COROLLARY 2. A spatial net is determined ly any two of its planar

nets. (A, E)

EXERCISES

1. If A,B, C,Z>, E are the vertices of a complete space five-point, there is
one and only one net of rationality containing them all. A point P belongs to this
net if and only if P is one of a sequence of points ABCDEI^I^ , finite 'in
number, such that I I is the point of intersection of three faces of the original
five-point and every other point J t - is the intersection of three distinct planes
through triples of points of the set ABCDEI^ - - - 1^^.

2. Show that a planar net is determined if three noncollinear points and a
line not passing through any of these points is given.

3. Under what condition is a planar net determined by a linear net and two
points not in this net ? Show that two distinct planar nets in the same plane
can have at most a linear net and one other point in common.

34, 35] . THE FUNDAMENTAL THEOREM: 93

4. Show that a set of points in a plane -which is protective with L, planar
net is a planar net.

5. A line joining a point P of a planar net to any point not in the net, but
on a line of the net not containing P : has no other point than P in common
with the net.

6. Two points and two lines in the same plane do not in general belong to
the same planar net.

7. Discuss the determination of spatial nets by points and planes, similarly
to Exs. 2, 3, and 6.

8. Any class of points projective with a spatial net is itself a spatial net.

9. If a perspective collineation (homology or elation) in a plane with
center A and axis I leaves a net of rationality in the plane invariant, the
net contains A and /.

10. Prove the corresponding proposition for a net of rationality in space
invariant tinder a perspective transformation.

11. Show that two linear nets on skew lines always belong to some spatial
net ; in fact, that the number of spatial nets containing two given linear
nets on skew lines is the same as the number of linear nets through two given
points.

12. Three mutually skew lines and three distinct points on one of them
determine one and only one spatial net in which they lie.

13. Give further examples of the determination of spatial nets by lines,

35. The fundamental theorem of projectivity. It has been shown
(Chap. Ill) that any three distinct elements of a one-dimensional
form may be made to correspond to any three distinct points of a
line by a projective transformation. Likewise any four elements of
a two-dimensional form, no three of which belong to the same one-
dimensional form, may be made to correspond to the vertices of a
complete planar quadrangle by a projective transformation ; and any
five elements of a three-dimensional form, no four of which belong
to the same two-dimensional form, may be made "to correspond to
the five vertices of a complete spatial five-point by a projeetive
transformation.

These transformations are of the utmost importance. Indeed, it is
the principal object of projective geometry to discover those prop-
erties of figures which remain invariant when the figures are sub-
jected to projective transformations. The question now naturally
arises, Is it possible to transform any four elements of a one-
dimensional form into any four elements of another one-dimensional
form ? This question must be answered in the negative, since a har-
monic set must always correspond to a harmonic set. The question

94 THE FUNDAMENTAL THEOEEM [CHAP.IV

tlien arises whether or not a projective correspondence between one-
dimensional forms is completely determined when three pairs of
homologous elements are given. A partial answer to this funda-
mental question is given in the next theorem.

LEMMA 1. If a projectivity leaves three distinct points of a line fixed,
it leaves fixed every point of the linear net defined ~by these points.

This follows at once from the fact that if three points are left
invariant by a projectivity, the harmonic conjugate of any one of
these points with respect to the other two must also be left inva-
riant by the projectivity (Theorems 2 and 3, Cor.). The projectivity
in question must therefore leave invariant every point harmonically
related to the three given points.

THEOREM 16. THE FUNDAMENTAL THEOREM OF PROJECTIVITY FOR A
NET OF RATIONALITY ON A LINE. If A, B, C, D are distinct points of
a linear net of rationality, and A r , J3 f , C f are any three distinct points
of another or the same linear net, then for any proj'ectivities giving
ABCD-^AWC'D* and ABCD^A r r O r D[ 9 we have D J =D[. (A, E)

Proof. If TT, TT X are respectively the two pro jectivities of the theorem,
the projectivity Tr^" 1 leaves A'JB f C f fixed and transforms D f into D[.
Since D f is harmonically related to A f , B r , C r (Theorem 7), the theorem
follows from the lemma.

This theorem gives the answer to the question proposed in its
relation to the transformation of the points of a linear net. The
corresponding proposition for all the points of a line, ie. the prop-
osition obtained from the last theorem by replacing " linear net " by
"line," cannot be proved without the use of one or more additional
assumptions (cf. 50, Chap. VI). We have seen that it is equiva-
lent to the proposition: If a projectivity leaves three points of a
line invariant, it leaves every point of the line invariant. Later, by
means of a discussion of order and continuity (terms as yet unde-
fined), we shall prove this proposition. This discussion of order
and continuity is, however, somewhat tedious and more difficult
than the rest of our subject ; and, besides, the theorem in question
is true in spaces,* where order and continuity do not exist. It has

* Different, of course, from ordinary space; " rational spaces*' (cf. p. 98 and
the next footnote) are examples in which continuity does not exist; u finite spaces,"
of which examples are given in the introduction ( 2), are spaces in which neither
order nor continuity exists.

35] THE FUNDAMENTAL THEOREM 95

therefore seemed desirable to give some of the results of this
theorem before giving its proof in terms of order and continuity.
To this end we introduce here the following provisional assumption
of projectivity, which will later be proved a consequence of the order
and continuity assumptions which will replace it. This provisional
assumption may take any one of several forms. "We choose the fol-
lowing as leading most directly to the desired theorem :

AN ASSUMPTION OF PROJECTIVITY :

P. If a projectivity leaves each of three distinct points of a line-
invariant, it leaves every point of the line invariant*

We should note first that the plane and space duals of this assump-
tion are immediate consequences of the assumption. The principle of
duality, therefore, is still valid after our set of assumptions has been
enlarged by the addition of Assumption P.

"We now have :

THEOEEM 17. THE FUNDAMENTAL THEOREM OF PROJECTTTE GEOM-
ETRY, f If 1, @, 3,4- are any four elements of a one-dimensional primitive
form, and l f , % ? , 3 l are any three elements of another or the same one-
dimensional primitive form, then for any projectivities giving 134~^
1W&4' and 1^34^1 r ^3 f 4[ } we have 4'=4[. (A, E, P)

Proof. The proof is the same under the principle of duality as that
of Theorem 16, Assumption P replacing the previous lemma.

This theorem may also be stated as follows :

A projectivity 'between one-dimensional primitive, forms is uniquely
determined when three pairs of homologous elements are given. (A, E, P)

COROLLARY. If tico pencils of points on different lines are projectile
and have a self-corresponding point, they are perspective. (A, E, P)

* We have seen in the lemma of the preceding theorem that the projectivity
described in this assumption leaves invariant every point of the net of rationality
defined by the three given points. The assumption simply states that if all the points
of a linear net remain invariant under a protective transformation, then all the points
of the line containing this net must also remain invariant. It will be shown later
that in the ordinary geometry the points of a linear net of rationality on a line corre-
spond to the points of the line whose coordinates, when represented analytically, are
rational numbers. This consideration should make the last assumption almost, if
not quite, as intuitionally acceptable as the previous Assumptions A and E.

t On this theorem and related questions there is an extensive literature to which
references can be found in the Encyklopadie articles on Protective Geometry and
Foundations of Geometry. It is associated with the names of von Staudt, Klein,
Zeuthen, Liiroth, Darboux, F. Schur, Pieri, Wiener, Hilbert. Cf . also 50, Chap. VI.

96 THE FUNDAMENTAL THEOREM [CHAP.IV

Proof. For if is the self-corresponding point, and AA f and BB'
are any two pairs of homologous points distinct from 0, the perspec-
tivity whose center is the intersection of the lines A A', BB 1 is a
projectivity between the two lines which has the three pairs of
homologous points 00, AA f , BB f , which must be the projectivity of
the corollary by virtue of the last theorem.

The corresponding theorems for two- and' three-dimensional forms
are now readily derived. We note first, as a lemma, the propositions
in a plane and in space corresponding to Assumption P.

LEMMA 2. A projectile transformation which leaves invariant each

* j. * four > j. ,c a plane three , . , , 7 , ^
of a set of - points of * no of WIMMI oelonq to me same
J J five * J space four J y

line ^ . . . . _, - the plane. , .

, leaves invariant every point of (A. K P)

plane J * J space. ^ '

Proof. If A, B, C, D are four points of a plane no three of which
are collinear, a projective transformation leaving each of them in va-
riant 'must also leave the intersection of the lines AB, CD invariant.
By Assumption P it then leaves every point of each of the lines AB,
CD invariant. Any line of the plane which meets the lines AB and
CD in two distinct points is therefore invariant, as well as the inter-
section of any two such lines. But any point of the plane may be
determined as the intersection of two such lines. The proof for the
case of a projective transformation leaving invariant five points no
four of which are in the same plane is entirely similar. The existence
of perspective collineations shows that the condition that no three
(four) of the points shall be on the same line (plane) is essential.

THEOREM 18. A projective collineation* "between two planes (or
within a single plane) is uniquely determined when four pairs of
homologous points are given, provided no three of either set of four
points are collinear. (A, E, P)

Proof. Suppose there were two collineations TT, TT X having the given
pairs of homologous points. The collineation irjr~ l is then, by the
lemma, the identical collineation in one of the planes. This gives at
once TTj = TT, contrary to the hypothesis.

* We confine the statement to the case of the collineation for the sake of sim-
plicity of enunciation. Projective transformations which are not collineations will
be discussed in detail later, at which time attention will be called explicitly to the
fundamental theorem.

35] THE FUSTDAMEXTAL THEOBEM 97

By precisely similar reasoning we have :

THEOREM 19. A protective collineation in space is uniquely deter-
mined when five pairs of liomoloyous points are given, provided no
four of either set of five points are iii the same plane. (A, E, P)

The fundamental theorem deserves its name not only because so
large a part of projective geometry is logically connected with it, but
also because it is used explicitly in so many arguments. It is indeed
possible to announce a genial course of procedure that appears in
the solution of most " linear " problems, i.e. problems which depend on
constructions involving points, lines, and planes only. If it is desired
to prove that certain three lines l v 1 2 , / 3 pass through a point, find two
other lines m v m^ such that the four points mj v mj^ mJ ZJ m^n^ may
be shown to be projective with the four points mj v mj^, mj,^ mjn^
respectively. Then, since in this projectivity the point m^n^ is self-
corresponding, the three lines l v l, 1 B joining corresponding points
are concurrent (Theorem 17, Cor.). The dual of this method appears
when three points are to be shown collinear. This method may be
called the principle of projectivity, and takes its place beside the
principle of duality as one of the most powerful instruments of pro-
jective geometry. The theorems of the next section may be regarded
as illustrations of this principle. They are all propositions from which
the principle of projectivity could be derived, ie. they are propositions
which might be chosen to replace Assumption P.

We have already said that ordinary real (or complex) space is a
space in which Assumption P is valid. Any such space we call a
properly projective space. It will appear in Chap. VI that there
exist spaces in which this assumption is not valid. Such a space,
ie. a space satisfying Assumptions A and E but not P, we will call
an improperly projective space.

From Theorem 15, Cor. 1 and Lemma 1, we then have

THEOREM 20. A net of rationality in space is a properly projective
space. (A, E)

It should here be noted that if we added to our list of Assump-
tions A and E another assumption of closure, to the effect that all
points of space belong to the same net of rationality, we should
obtain a space in which all our previous theorems are valid, in-
cluding the fundamental theorem (without using Assumption P).

THE FUNDAMENTAL THEOREM

[CHAP. IV

Such a space may be called a rational space. In general, it is clear
that any complete five-point in any properly or improperly protective
space determines a subspace which is rational and therefore properly
projective.

36. The configuration of Pappus. Mutually inscribed and circum-
scribed triangles.

THEOREM 21. If A, B, C are any three distinct points of a line I,
and A\ B\ C f any three distinct points of another line I' meeting I,
the three points of intersection of the pairs of lines AB r and A!B> BC 1
and B'C, CA 1 and C f A
are collinear. (A, E, P)

Proof. Let the three points of intersection referred to in the theorem
be denoted by C r ", A", B" respectively (fig. 44). Let the line "C"
meet the line B'C in a point D (to be proved identical with A");
also let B"C" meet V in A v the line A'B meet AC r in B v the line AB'
meet A r C in B[. We then have the following perspectivities :

By the principle of projectivity then, since in the projectivity thus
established C n is self-corresponding, we conclude that the three lines
A^A'i B n B v DB meet in the point '. Hence D is identical with A n ,
and A", B ff , C" are collinear.

It should be noted that the figure of the last theorem is a con-
figuration of the symbol

COXFIGUEATICXtf OF PAPPUS 99

It is known as the concur crtiajL. of Pappus* It should also be noted
that this configuration may be considered as a simple plane hexagon
(six-point) inscribed in two intersecting lines. If the sides of such a
hexagon be denoted in order by 1, 2, 3, 4, 5, 6, and if we call the sides
1 and 4 opposite, likewise the sides 2 and 5, and the sides 3 and 6 (cf.
Chap. II, 14), the last theorem may be stated in the following form :
COROLLARY. If a simple hexagon le inscribed in two intersecting lines,
the three pairs of opposite sides will intersect in collinear points.^

Finally, we may note that the nine points of the configuration of
Pappus may be arranged in sets of three, the sets forming three
triangles, 1, 2, 3, such
that 2 is inscribed in
1, 3 in 2, and 1 in 3.
This observation leads
to another theorem con-
nected with the Pappus
configuration.

THEOREM 22. If
A^B^C^ be a triangle

inscribed in a triangle p iG 45

A^B! V there exists a

certain set of triangles each of which is inscribed in the former and
circumscribed about the latter. (A, E, P)

Proof. Let \a\ be the pencil of lines with center A^ [&] the pencil
with center B^ and [c] the pencil with center C^ (fig. 45). Consider the

perspectivities [a] 2 [&] == [c]. In, the projectivity thus estab-
lished between [a] and [c] the line A l C l is self-corresponding; the
pencils of lines [a], [c] are therefore perspective (Theorem 17, Cor.
(dual)). Moreover, the axis of this perspectivity is C 2 A^ for the lines
Aft* and C^ are clearly homologous, as also the lines A^A Z and O^A^.
Any three homologous lines of the perspective pencils [a], [&], [c] then
form a triangle which is circumscribed about A 1 B 1 C 1 and inscribed
in A^B^C^

* Pappus, of Alexandria, lived about 340 A.D. A special case of this theorem may
be proved without the use of the fundamental theorem (cf. Ex. 3, p. 52).

t In this form it is a special case of Pascal's theorem on conic sections
(cf. Theorem 3, Chap. V).

100

THE FUNDAMENTAL THEOREM

[CHAP. IV

EXERCISES

1. Given a triangle ABC and two distinct points A', Z>'; determine a point C"
sucli that the lines A A', BB', CC'are concurrent, and also the lines A I?, BC', CA'
are concurrent, i.e. such that the two triangles are perspective from two dif-
ferent points. The two triangles are there said to he doubly perspective.

2. If two triangles ABC and A'B'C' are doubly perspective in such a way
that the vertices A, B, C are homologous with A', B', C' respectively in one
perspectivity and with J3', C', A' respectively in the other, they will also be per-
spective from a third point in such a way that A, B, C are homologous respec-
tively with C', A', B'^ i.e. they will be triply perspective.

3. Show that if A", B", C" are the centers of perspectivity for the triangles
in Ex. 2, the three triangles ABC, A'B'C', A"B"C" are so related that any two
are triply perspective, the centers of perspectivity being in each case the vertices
of the remaining triangle. The nine vertices of the three triangles form the
points of one configuration of Pappus, and the nine sides form the lines of
another configuration of Pappus.

37. Construction of projectivities on one-dimensional forms.
THEOREM 23. A necessary and sufficient condition for the projectivity
on a line MNAB -^ MNA'B f ( 31 * JV) is Q(3IAB, NB'A'). (A, E, P)

Proof. Let n be any line on* N not passing through A (fig. 46). Let O l
be any point not on n or on MA, and let A l and B t be the intersections
respectively of O t A and O^B with n. Let 2 be the intersection

and B'B V

Then

By Theorem 17 the projectivity so determined on the line AM is the
same as

The only possible double points of the projectivity are N and the
intersection of AN with O a 2 . Hence 0^ passes through M 9 and
Q(MAB, NB f A f ) is determined by the quadrangle O^A^

37] ONE-DIMENSIONAL PEOJECTIYITIES 101

Conversely, if Q(1LLB, XJB'A') we have a quadrangle O
and hence

and by this construction 31 is self-corresponding, so that

If iu the above construction we have J/"= JV, we obtain a projec-
tivity with the single double point J/= JV.

DEFINITION. A projectivity on a one-dimensional primitive form
with a single double element is called parabolic. If the double ele-
ment is J/, and AA f 9 BB* are any two homologous pairs, the pro-
jectivity is completely determined and is conveniently represented
by MMAB ^ MMA'B'.

COROLLARY. A necessary and sufficient condition for a parabolic
projectivity MHAB-^MMA'B* is Q(3IA, MB'A f ). (A, E, P)

THEOREM 24 If we hare

Q(ABC, A'3 f C f ),
we have also Q(A f B'C r , ABC).

Proof. By the theorem above,

Q(A3C, A'B'C 1 )

implies AA r BC^ AA'C'B',

which is the inverse of A r AB r C f -^ A A OB 9

which, by the theorem above, implies

Q(A'B f C' 9 ABC).

The notation Q (ABC, A f B f C l ) implies that A, B, C are the traces of a
point triple of sides of the quadrangle determining the quadrangular set.
The theorem just proved states the existence of another quadrangle
for which A\ B r , C 1 are a point triple, and consequently A, B, C are a
triangle triple. This theorem therefore establishes the existence of
oppositely placed quadrangles, as stated in 19, p. 50. This result
can also be propounded as follows:

THEOREM 25. If two quadrangles I^P 2 P Z P^ and Q^^Q^Q^ are so related
J? to Q v J% to Q y etc. that five of the sides l$I$(i,j =1,2,8,4;
i =/) meet the five sides of the second which are opposite to Q t Qj in points
of a line I, the remaining sides of the two quadrangles meet on L (A,E,P)

102 THE FUNDAMENTAL THEOEE3M [CHAP. IV

Proof. The sides of the first quadrangle meet I in a quadrangular
set Q (P 12 P 13 P^ P*Pu&) ; hence Q (P 2 PA P*PM* But, by hypoth-
esis, five of the sides of the second quadrangle pass through these
points as follows : Q^ through J^, Q^Q 9 through P^ 9 Q& through P^>
Q z Qt through 2J a , QQ 2 through P is , Q 2 Q Z through P u . As five of these
conditions are satisfied, by Theorem 3, Chap. II, they must all be
satisfied.

EXERCISES

1. Given one double point of a projectivity on a line and two pairs of
homologous points, construct the other double point.

2. If a, b, c are three nonconcurrent lines and A', B', C' are three collinear
points, give a construction for a triangle whose vertices A. , B, C are respectively
on the given lines and whose sides BC, CJL, AB pass respectively through the
given points. What happens when the three lines a , I, c are concurrent ? Dualize.

38. Involutions, DEFINITION. If a projectivity in a one-dimensional
form is of period two, it is called an involution. Any pair of homol-
ogous points of an involution is called a conjugate pair of the involution
or a pair of conjugates.

It is clear that if an involution transforms a point A into a point A r ,
then it also transforms A r into A] this is expressed by the phrase that
the points A, A 1 correspond to each other doully. The effect of an invo-
lution is then simply a pairing of the elements of a one-dimensional
form such that each element of a pair corresponds to the other ele-
ment of the pair. This justifies the expression "a conjugate pair"
applied to an involution.

THEOREM 26. If for a single, point A of a line which is not a double
point of a projectivity TT on the. line we have the relations 7r(A)t=A !
and 7r(A') = A, the projectivity is an involution. (A, E, P)

Proof. For suppose P is any other point on the line (not a double
point of ?r), and suppose 7r(P) = P f . There then exists a projectivity
giving AA'PP 1 ^ A'AP'P

(Theorem 2, Chap. III). By Theorem 17 this projectivity is TT, since
it has the three pairs of homologous points A, A*\ A r , A\ P, P r . But
in this projectivity P r is transformed into P. Thus every pair of
homologous points corresponds doubly.

COROLLARY. An involution is completely determined when two pairs
of conjugate points are given. (A, E, P)

INVOLUTIONS 103

THEOREM 27. A necessary and sufficient condition that three pairs
of points A, A r ; B, B' ; C, C 1 be conjugate pairs of an involution is
Q(ABC,A f B'C f ). (A, E, P)
Proof. By hypothesis we have

AA'BC-^A'AB'C'.

By Theorem 2, Chap. Ill, we also have
A'AS'C'-fiA
which, with the first projectivity, gives

A necessary and sufficient condition that the latter projectivity hold
is Q(ABC, AWC r ) (Theorem 23).

COROLLARY 1. If an involution has doulle points, they are harmonic
conjugates with respect to every pair of the involution. (A, E, P)

For the hypothesis A~A f ,B B J gives at once H(AB 9 CC f ) as the
condition of the theorem.

COROLLARY 2. An involution is completely determined when two
double points are given, or when one double point and one pair of
conjugates are given. (A, E, P)

COROLLARY 3. If J/, N are distinct doulle points of a projectivity
on a line, and A,A r ; B,B' are any two pairs of homologous elements,
the pairs M, N; A, B J ; A 1 , B are conjugate pairs of an involution.*
(A, E, P)

COROLLARY 4 If an involution has one double element, it has another
distinct from the first. (A, E, P)

COROLLARY 5. The projectivity ABGD-j^ABDC "between four dis-
tinct points of a line implies the relation H (AB, CD). (A, E, P)

Eor the projectivity is an involution (Theorem 26) of which A, B
are double points. The result then follows from Cor. 1.
39. Axis and center of tiomology.

THEOREM 28. If \A] and [B] THEOREM 28'. If [I] and \m\
are any two projectile pencils are any two projectile pencils of
of points in the same plane on lines in the same plane on distinct

* This relation is sometimes expressed by saying, s l The pairs of points are in
involution." From what precedes it is clear that any two pairs of elements of a
one-dimensional form are in involution, but in general three pairs are not.

104

THE FUNDAMENTAL THEOEEM

[CHAP. IV

distinct lines l v 1 2> there exists a
line I siich that if A v B I and A^ B z
are any two pairs of homologous
points of the two pencils, the lines
A^ and A^ intersect on I
(A,E,P)

DEFINITION. The line I is called
the axis of homology of the two
pencils of points.

points S v S Q9 there exists a point S
such that if a v \ and a 2> & 2 are
any two pairs of homologous lines
of the two pencils, the points a^
and a 2 6 1 are collinear with S.
(A, E, P)

DEFINITION. The point S is
called the center of homology of
the pencils of lines.

Proof. The two theorems being plane duals of each other, we may
confine ourselves to the proof of the theorem on the left. From the
projectivity [S] ^ [A] follows A^B^B^A] (fig. 47). But in this pro-
jectivity the line A^ is self-corresponding, so that (Theorem 17, Cor.)

the two pencils are perspective. Hence pairs of corresponding lines
meet on a line I ; e.g. the lines A^ s and B^ z meet on I as well as
A^B Z and B^Ay To prove our theorem it remains only to show that
B 2 A S and A 2 B^ also meet on L But the latter follows at once from
Theorem 21, since the figure before us is the configuration of Pappus.
COROLLARY. If [A], [B] are not COROLLARY. If [I], [m] are not
perspective) the axis of homology is perspective, the center of homology

the line joining the points homol-
ogous with the point l^ regarded
first as a point of l^ and then as
a point of l y

is the point of intersection of the
lines homologous with the line S^
regarded first as a line of [I] and
then as a line of [m].

For in the perspectivity J^B] = B l [A'] the line ^ corresponds to

) > and hence the point l^ corresponds to K t in the projectivity
[#] 7^ [A], Similarly, ll z corresponds to l^ y

39] OEKTEE AXD AXIS OF HOROLOGY 105

EXERCISES

1. There is one and only one projectivity of a one-dimensional form leaving
invariant one and only one element 0, and transforming a given other element
-4 to an element B.

2. Two protective ranges on skew lines are always perspective.

3. Prove Cor. 5, Theorem 27, without using the notion of involution.

4. If MNAB-^MyA'B'* then MXAA'- J/JV.RB'.

5. If P is any point of the axis of homology of two projective ranges
M A"^' tliel1 tlie P r J ectiv % ^[-4] 77 **[#] is an involution. Dualize.

6. Call the faces of one tetrahedron a ls 03, a 3 , a 4 and the opposite vertices
*i 1? -dU, -4 3 , J 4 respectively, and similarly the faces and vertices of another tetra-
hedron ft, ft, ft, ft and B 19 B s , B+ If 4i, J a , .4,, ^ lie on ft, ft, ft, ft
respectively, and B^ lies on a x , B on a 2 , -5 3 on a 3 < then ^ lies on a 4 . Thus
each of the two tetrahedra related in this fashion is both inscribed and cir-
cumscribed to the other.

7. Prove the theorem of Desargues (Chap. II) by the principle of pro-
jectivity.

8. Given a triangle ABC and a point A\ show how to construct two points
B', C' such that the triangles ABC and A'B'C' are perspective from four
different centers.

9. If two triangles A^B^ and A^B 2 C Z are perspective, the three points

(4 A, A^) = C,, (AiC v A&) = B 9 , (B X C 3 , J5 2 CO = A v

if not collinear, form a triangle perspective with the first two, and the three
centers of perspectivity are collinear.

* 10. (a) If it is a projectivity in a pencil of points [J.] on a line a with inva-
riant points A v AV and if [], \M ] are the pencils of points on two lines /, w
through A^AQ respectively, show by the methods of Chap. Ill that there exist
three points S 19 S 3 , S s such that we have

where v (A ) = A' ; that S v S z , -4 2 are collinear ; and that ^ 2 , g , A ^ are collinear.
(&) Using the fundamental theorem, show that there exists on the line ^A 2
a point S such that we have

(c) Show that (5) could be used as an assumption of projectivity instead of
Assumption P ; i.e. P could be replaced by : If TT is a projectivity with fixed
points ^t 1? A%, giving 7r(-4) = A' in a pencil of points 4], and [] is a pencil
of points on a line I through A 19 there exist two points S v S 2 such that

S l S,

106

THE FUNDAMENTAL THEOREM

[CHAP. IV

* 11. Show that Assumption P could be replaced by the corollary of
Theorem 17.

* 12. Show that Assumption P could be replaced by the following : If we
have a projectivity in a pencil of points defined by the perspecthities

[-X] = [] = [A"],

and [JIT] is the pencil of points on the line 5^, there exist on the base of []
two points Si, S such that we have also

=[*']

40. Types of collineations in the plane. We have seen in the
proof of Theorem 10, Chap. Ill, that if O^O, is any triangle, there
exists a collineation II leaving O v 2 , and O s invariant, and trans-
forming any point not on a side of the triangle into any other such

III

FIG. 48

point. By Theorem 18 there is only one such collineation II. By the
same theorem it is clear that II is fully determined by the projec-
tivity it determines on two of the sides of the invariant triangle, say
2 3 and 1 S . Hence, if H x is a homology with center O l and axis
2 O st which determines the same projectivity as II on the line O^,
and if H 2 is a homology with center 3 and axis 1 8 , which deter-
mines the same projectivity as II on the line 8 , then it is evident

that

<! TYPES OF COLLIXEATIOXS 107

It is also evideut that no point not a vertex of the invariant triangle
can be fixed unless II reduces to a homology or to the identity. Such
a transformation II when it is not a hornology is said to be of Type I 9
and is denoted by Diagram / (fig. 48).

EXERCISE

Prove that two honiologies with, the same center and axis are commutative,
and hence that two projectivities of Type I with the same invariant figure are
commutative.

Consider the figure of two points 1? 2 and two lines o v 2 , such
that 1 and 2 are on o v and o l and 2 are on O r A collineation II
which is the product of a homology H, leaving 3 and o. 2 invariant,
and an elation E, leaving O l and o l invariant, evidently leaves this
figure invariant and also leaves invariant no other point or line- If A
and B are two points not on the lines of the invariant figure, and we
require that TL(4\=sB

this fixes the transformation (with two distinct double lines) among
the lines at O v and the parabolic transformation among the lines at 2 ,
and thus determines II completely. Clearly if II is not to reduce to a
homology or an elation, the line AB must not pass through O l or 2 .
Such a transformation II, when it does not reduce to a homology or
an elation or the identity, is said to be of Type II and is denoted by
Diagram II (fig. 48).

EXERCISE

Two projective collineations of Type IT, having the same invariant figure,
are commutative.

DEFINITION The figure of a point and a line o on is called a
lineal element Oo.

A collineation having a lineal element as invariant figure must effect
a parabolic transformation both on the points of the line and on the
lines through the point. Suppose Aa and Sb are any two lineal ele-
ments whose points are not on o or collinear with 0, and whose lines
are not on or concurrent with o. Let E x be an elation with center
and axis OA 9 which transforms the point (oa) to the point (ol). Let E 2
be an elation of center (AB, o) and axis o, which transforms A to B.
Then II = E 2 E X has evidently no other invariant elements than and o
and transforms Aa to BL

108 THE FUNDAMENTAL THEOEEM [CHAP. IV

Suppose that aaother projeetivity II' would transfer Aa to Bl with
Oo as only invariant elements. The transformation II 7 would evidently
have the same effect on the lines of and points of o as II. Hence
II'II- 1 would he the identity or an elation. But as ITrT 1 ^) = . it
would be the identity. Hence II is the only projectivity which trans-
forms Aa to b with Oo as only invariant.

A transformation having as invariant figure a lineal element and no
other invariant point or line is said to be of Type III, and is denoted
by Diagram /// (fig. 48).

A homology is said to be of Type /Fand is denoted by Diagram IV.

An elation is said to be of Type V and is denoted by Diagram F.

It will be shown later that any collineation can be regarded as be-
longing to one of these five types. The results so far obtained may be
summarized as follows :

THEOREM 29. A protective collineation with given invariant figure F,
if of Type I or II will transform any point P not on a line of F into
any other such point not on a line joining P to a point of F; if of
Type III will transform any lineal element Pp such that p is not on
a point, or P on a line, of F into any other such element Qq; if of
Type IV or V, will transform any point P into any other point on the
line joining P to the center of the collineation.

The r61e of Assumption P is well illustrated by this theorem. In case of
each of the first three types the existence of the required collineation was proved
by means of Assumptions A and E, together with the existence of a sufficient
number of points to effect the construction. But its uniqueness was established
only by means of Assumption P. In case of Types IV and F, both existence
and uniqueness follow from Assumptions A and E.

EXERCISES

1. State the dual of Theorem 29.

2. If the number of points on a line is^? + 1, the number of collineations
with a given invariant figure is as follows :

Type/, O-2)O~3).
Type//, o- 2) (>-_!),
Type m,p(p~ I)*.
Type/F, p~2.
TypeF, p-1.

In accordance with the results of this exercise, when the number of points
on a line is infinite it is said that there are oo 2 transformations of Type / or //;
oo 3 of Type ///; and oo 1 of Types /Fand F.

CHAPTER V*

CONIC SECTIONS

41. Definitions. Pascal's and Brianchon's theorems.

DEFINITION. The set of all points of intersection of homologous
lines of two protective, nonperspective flat pencils which are on the
same plane but not on the same point is called a point conic (fig. 49).
The plane dual of a point conic is called a line conic (fig. 50). The
space dual of a point conic is called a cone of planes; the space dual

. 49

FIG. 50

of a line conic is called a cone of lines. The point through which
pass all the lines (or planes) of a cone of lines (or planes) is called
the vertex of the cone. The point conic, line conic, cone of planes,
and cone of lines are called one-dimensional forms of the second degree.^

The following theorem is an immediate consequence of this defi-
nition.

THEOREM 1. The section of a cone of lines ly a plane not 6n the
vertex of the cone is a point conic. The section of a cone of planes by
a plane not on the vertex is a line conic.

Now let A^ and B l be the centers of two flat pencils defining a
point conic. They are themselves, evidently, points of the conic, for the
line A^BI regarded as a line of the pencil on A 1 corresponds to some
other line through B^ (since the pencils are, by hypothesis, protective

* All the developments of this chapter are on the basis of Assumptions A, E, F,
and H .

t A fifth one-dimensional form a self-dual form of lines in space called the
regulus will be defined in Chap. XL This definition of the first four one-dimen-
sional forms of the second degree is due to Jacob Steiner (1796-1863). Attention
will be called to other methods of definition in the sequel.

109

110

CONIC SECTIONS

[CHAP. V

but not perspective), and the intersection of these homologous lines
is B r The conic is clearly determined by any other three of its
points, say A 2 , B z , <7 2 , because the projectivity of the pencils is then

determined by

(AJ*&) 7: B l (AJ}&)

(Theorem 17, Chap. IV).

Let us now see how to determine a sixth point of the conic on a
line through one of the given points, say on a line Z through B y If the
line I is met by the lines A^, A^C^ B^A^ B^G Z in the points S> T, U, A

FIG. 51

-j^ UB^A. The other

respectively (fig. 5 1), we have, by hypothesis, S
double point of this projectivity, which we will call C v is given by the
quadrangular set Q^ST, C^AU) (Theorem 23, Chap. IV). A quad-
rangle which determines it may be obtained as follows : Let the lines
A^B^ and A^BZ meet in a point (7, and the lines AC and A^C^ in a
point B ; then the required quadrangle is A^A^CB^ and C l is determined
as the intersection of AJB with L

C l Trill coincide with 2 , if an d Q ^J *f B is on A 2 B^ (fig. 52). This means
that A C, A^Cty and A^B 2 are concurrent in B. In other words, A must be the
point of intersection of B X C 2 with the line joining C = (A^B^ (A^) and
B = (JljCy^o&j)* an< i I must be the line joining B z anil A . This gives, then,
a construction for a line which meets a given conic in only one point.

The result of the preceding discussion may be summarized as
follows : TJie four points A z , B 2 , <7 2 , C l are points of a point conic

HI] PASCAL'S THEOREM 111

determined ly two projectile pencils on A^ and B v if and only if the
three points C' = (.4^,) (A&), B = (A^C.) (A 2 C^, A = (B^) (B.,C\) are
collinear. The three points in question are clearly the intersections
of pairs of opposite sides of the simple hexagon A^B^A^B^C y

Since A v B v C^ may be interchanged with -.4.,, J5 2 , C 2 respectively
in the above statement, it follows that A v Z? 1? C v C n are points of a
conic determined by protective pencils on A 2 and B 2 . Thus, if C^ is
any point of the first conic, it is also a point of the second conic,
and vice versa. Hence we have established the following theorem :

THEOREM 2. STEIXER'S THEOREM. If A and B are any two given
points of a conic, and P is a variable point of this conic, ice have

In view of this theorem the six points in the discussion may be
regarded as any six points of a conic, and hence we have

THEOREM 3. PASCAL'S THEOREM.* The necessary and sufficient con-
dition that siQTjwwsno tJiree of u'hicJi are collinear, be points of
the same conic is that the three pairs of opposite sides of a simple
hexagon of ivhicli they are vertices shall meet in collinear points.^

The plane dual of this theorem is ^

THEOREM 3'. jteiAXCHoy's THEOREM. Tlie necessary and sufficient
condition that six lines, no three of ichich are concurrent, be lines of
a line conic is that the lines joining the three pairs of opposite vertices
of any simple hexagon of whicli the given lines are sides, shall be
concurrent.^

As corollaries of these theorems we have

COROLLARY 1. A line in the plane of a point conic cannot have more
than two points in common with the conic.

COROLLARY 1'. A point in the plum of a line conic cannot be on
more than two lines of the conic.

* Theorem 3 was proved by B. Pascal in 1640 when only sixteen years of age.
He proved it first for the circle and then obtained it for any conic by projection
and feection. This is one of the earliest applications of this method. Theorem 8'
was first given by C. J. Brianchon in 1806 (Journal de 1'ficole Polytechnique,
Vol. VI, p. 301).

t The line thus determined by the intersections of the pairs of opposite sides of
any simple hexagon whose vertices are points of a point conic is called the Pascal
line of the hexagon. The dual construction gives rise to the Brianchon poM of a
hexagon whose sides belong to a line conic.

112 COOTC SECTIONS [OHAP.V

Also as immediate corollaries of these theorems we have
THEOREM 4. There is one and only one point conic containing Jim

given points of a plane no three of which are collinear.

THEOREM -4'. There is one and only one line conic containing Jive

given lines of a plane no three of which are concurrent.

EXERCISES

1. What are the space duals of the above theorems?

2. Prove Brianehon's theorem without making use of the principle of
duality.

3. A necessary and sufficient condition that six points, no three of "which
are collinear, be points of a point conic, is that they be the points of inter-
section (&'), (fa'), (ca'), (a') ( c&/ )> ( a O f tne si des #j &> c an< i a/ ? &'> c/ f two
perspective triangles, in which a and a', "b and &', c and c' are homologous.

42. Tangents. Points of contact. DEFINITION. "A line p in the
plane of a point conic which meets the point conic in one and only
one point P is called a tangent to the point conic at P. A point P in
the plane of a line conic through which passes one and only one line
p of the line conic is called a point of contact of the line conic on'jp.

THEOREM 5. Through any point of a point conic there is one and
only one tangent to the point conic.

Proof. If PQ is the given point of the point conic and P is any
other point of the point conic, while P is a variable point of this
conic, we have, by Theorem 2,

Any line through P Q meets its homologous line of the pencil on 7^ in
a point distinct from P, except when its homologous line is P^.
Since a projectivity is a one-to-one correspondence, there is only one
line on -ZJ which has I(P^ as its homologous line.

THEOREM 5'. 0% any line of a line conic there is one and only one
point of contact of the line conic.

This is the plane dual of the preceding theorem.

EXERCISE
Give the space duals of the preceding definitions and theorems.

Returning now to the construction in the preceding section for the
points of a point conic containing five given points, we recall that

42] TA2s T GE]N T TS 113

the point of intersection Ci of a line I through B was determined by
the quadrangular set Q(J3 Z ST, C^U). The points and O l can,
by the preceding theorem, coincide on one and only one of the lines
through B z * For this particular line Z, A becomes the intersection

. 52

of the tangent at B z with I, and the collinearity of the points A, B 9 C
may be stated as follows :

THEOREM 6. If the vertices of a simple plane five-point are points
of a point conic, the tangent to the point conic at one of the vertices
meets the opposite, side in a point collinear with the points of inter-
section of the other two pairs of nonadjacent sides.

This theorem, by its derivation, is a degenerate case of Pascal's
theorem. It may also be regarded as a degenerate case in its state-
ment, if the tangent be thought of as taking the place of one side
of the simple hexagon.

It should be clearly understood that the theorem has been obtained by
specializing the figure of Theorem 3, and not by a continuity argument.
The latter would be clearly impossible, since our assumptions do not require
the conic to contain more than a finite number of points.

Theorem 6 may be applied to the construction of a tangent to
a point conic at any one of five given points P^ P%, P%, P^ jg of the
point conic (fig. 53). By this theorem the tangent p l at JJ must be

* As explained in the fine print on page 110, this occurs when I passes through
the point of intersection of BiOz with the line joining C (A^Bz) (A%Bi) and

114

CONIC SECTIONS

[CHAP. V

such that the points 2\(P^) = -i (%%) (%**) - B > and (%P 3 ) (P B P^ = C
are collinear. But B and C are determined by ^, P%, ^, -^, J^, and
hence p l is the line joining ^ to the intersection of the lines BC
and P & %.

FIG. 53 '

In like manner, if P v P%, P^, P, and jp 1 are given, to construct the
point P$ on any line I through P of a point conic containing P^ P Z) P SJ P
and of which p l is the tangent at P 19 we need only determine the points
A =1\(P Z P^ B = I (JJJ?), and C = (AB) (P 2 P 3 ) ; then P^C meets f in ^
(fig. 53).

FIG. 54

In case I is the tangent p 4 at P^ P 5 coincides with ^ and the fol-
lowing points are collinear (fig. 54) :

42] TAXGEXTS 115

Hence we have the following theorem :

THEOREM 7. If the vertices P 13 P 2 , %, P of a simple quadrangle are
points of a point conic, the tangent at % and the side P S P, the tangent
at P and the side JJJEJ, and the pair of sides JJJ* and RE. meet in three
collinear points.

If %, , P B , P 5 and the tangent p l at P are given, the construction
determined by Theorem -3 for a point P of the point conic on a line /
through P 3 is as follows (fig. 53): Determine C-=(P 1 P 5 )(P 2 P B ), A = pj,
and B = (A C)(P^ ; then P,B meets I in %.

In case I is the tangent at P z , P coincides with P 3 and we have the
result that C = (P 1 J)(P 2 P B ), A^p^p,, B = (P 1 K)(P 5 P) are collinear
points, which gives

PIG. 5o

THEOREM 8. If the vertices of a complete quadrangle are points of
a point conic 9 the tangents at a pair of vertices meet in a poi%t of the
line joining the diagonal points of the quadrangle which are not on
the side joining the two vertices (fig. 55).

The last two theorems lead to the construction for a point conic
of which there are given three points and the tangents at two of
them. Eeverting to the notation of Theorem 7 (fig. 54), let the given
points be P, P P z and the given tangents be p^ p r Let I be any line
through P y If PZ is the other point in which I meets the point conic,
the points A^p^(P^ 9 B^p^(P^, and C =(%%)(%%) are collinear.
Hence, if C^l(P^ and B=p(AC), then F 2 is the intersection of I
with B%.

In case I is the tangent p z at JJ, the points JEJ and P B coincide, and
the points

116 CONIC SECTIONS [CHAP.V

are collinear. Hence the two triangles P^P^ and p^sPt are per-
spective, and we obtain as a last specialization of Pascal's theorem
(fig. 56)

THEOREM 9. A trianyle whose vertices are points of a point conic
is perspective with the triangle formed "by the tangents at these points,
the tangent at any vertex "being ho7nologous with the side of the first
triangle which does not contain this vertex.

COROLLARY. If I(, J^, P are three points of a point conic, the lines
P^PV P$P are harmonic with the tangent at P z and the line joining P%
to the intersection of the tangents at jfj and P.

Proof. This follows from the definition of a harmonic set of lines,
on considering the quadrilateral P^A, AB> BP^ PJ( (fig. 56).

FIG. 56

43. The tangents to a point conic form a line conic. If P v P^ P 8 , P

are points of a point conic and p v p Z9 p s , p are the tangents to the
conic at these points respectively, then (by Theorem 8) the line join-
ing the diagonal points (JJ JJ) (P B I%) and (JfJJIJ) (J%I) contains the inter-
section of the tangents p v p s and also the intersection of p 2> p^. This
line is a diagonal line not only of the quadrangle Jf^J^, but also of
the quadrilateral PiPzPsP^ Theorem 8 may therefore be stated in
the form:

THEOREM 10. Tlu complete quadrangle formed ly four points of
a point conic and the complete quadrilateral of the tangents at these
points have the same diagonal triangle.

Looked at from a slightly different point of view, Theorem 8 gives
also

THEOREM 11. The tangents to a point conic form a line conic.

43] TAXGEXTS 117

Proof. Let JJ, P 2 , P 3 be any three fixed points on a conic, and let P
be a variable point of this conic. Let p v p*> Pv P be respectively the
tangents at these points (fig. 57). By the corollary of Theorem 28,
Chap. IV, JJJE} is the axis of homology of the projectivity between the
pencils of points on p 1 and p 2 defined by

But by Theorem 10, if Q=(P 1 P 2 ) (P Z P), the points pj> zi PiP& and Q are
collinear. For the same reason the points p z p z , pp v Q are collinear.
It follows, by Theorem 28, Chap. IV, that the homolog of the variable

PIG. 57

point Pip is p 2 p> i.e. p is the line joining pairs of homologous points
on the two lines p v p^ so that the totality of the lines p satisfies the
definition of a line conic.

COROLLARY. The center of homology of the projectivity P l [P] -^ P% [P]
determined "by the points P of a point conic containing P^ 1^ is the
intersection of the tangents at 1^ P^. The axis of honwlogy of the
projectimty p l [jp] -^ p% [p] determined by the lines p of a line conic
containing the lines p lt p 2 is the line joining the points of contact

f Pv P*

THEOREM 12. If J^ is a fixed and P a variable point of a point

conic, and p v p are the tangents at these two points respectively, then

118 CONIC SECTIONS [CHAP.V

Proof. Using the notation of the proof of Theorem 11 (fig. 57),
we have

where Q is always on JJJJ. But we also have

and, by Theorem 11,

Combining these projectivities, we have

The plane dual of Theorem 11 states that the points of contact of
a line conic form a point conic. In view of these two theorems and
their space duals we now make the following

DEFINITION. A conic section or a conic is the figure formed by a
point conic and its tangents. A cone is the figure formed by a cone
of lines and its tangent planes.

The figure formed by a line conic and its points of contact is then
likewise a conic as defined above ; i.e. a conic (and also a cone) is a
self-dual figure.

The duals of Pascal's theorem and its special cases now give us a
set of theorems of the same consequence for point conies as for line
conies. We content ourselves with restating Brianchon's theorem
(Theorem 3') from this point of view.

BRIANCHON'S THEOREM. If the sides of a simple hexagon are tan-
gents to a conic, the lines joining opposite vertices are concurrent;
and conversely.

It follows from the preceding discussion that in forming the plane
duals of theorems concerning conies, the word conic is left unchanged,
while the words point (of a conic) andjJa^$wL(of a conic) are inter-
changed. "We shall also, in the future, make use of the phraseTa conic
passes through a point P, and P is on the conic, when P is a point
of a conic, etc.

DEFINITION. If the points of a plane figure are on a conic, the figure
is said to be inscribed in the conic; if the lines of a plane figure
are tangent to a conic, the figure is said" to be circumscribed about
the conic.

TA]S T GE^TS 119

EXERCISES

1. State the plane and space duals of the special cases of Pascal's theorem.

2. Construct a conic, given (1) five tangents, (2) four tangents and the
point of contact of one of them, (3) three tangents and the points of contact
of two of them.

3. ABX is a triangle whose vertices are on a conic, and a, I. x are the tan-
gents at A, B, X respectively. If .4, L are given points and X is variable,
determine the locus of (1) the center of perspectivity of the triangles ABX
and abx ; (2) the axis of perspectivity.

JS 4. X, Y, Z are the vertices o a variable triangle, such that X, Y are always
on two given lines a, I respectively, while the sides XY, ZX, ZY always pass
through three given points P, A, B respectively. Show that the locus of the
point Z is a point conic containing A,B,D = (ah), J/ = (AP)l, and JV~ = (BP)a
(Maclaurin's theorem). Dualize. (The plane dual of this theorem is known
as the theorem of Braiken ridge.)

5. If a simple plane fl-point varies in such avray that its sides always pass
through n given points, while n 1 of its vertices are always on n 1 given
lines, the nth vertex describes a conic (Poncelet).

6. If the vertices of two triangles are on a conic, the six sides of these two
triangles are tangents of a second conic ; and conversely.' Corresponding to
every point of the first conic there exists a triangle having this point as a
vertex, whose other two vertices are also on the first conic and whose sides
are tangents to the second conic. Dualize.

7. If two triangles in the same plane are perspective, the points in \\hich
the sides of one triangle meet the nonhomologous sides of the other are on
the same conic ; and the lines joining the vertices of one triangle to the non-
homologous vertices of the other are tangents to another conic.

8. If A , B, C, D be the vertices of a complete quadrangle, whose sides
AB, AC, AD, BC, BD, CD are cut by a line in the points P, Q, R, S, T, V
respectively, and if E, F^ff, K, L, JJf are respectively the harmonic conjugates
of these points with respect to the pairs of vertices of the quadrangle so that
we have H (AB, PE), H (A C, QF), etc., then the six points E, F, G t A", , AT
are on a conic which also passes through the diagonal points of the quadrangle
(Holgate, Annals of Mathematics, Ser. 1, Vol. Til (1893), p. 73).

9. If a plane a cut the six edges of a tetrahedron in six distinct points,
and the harmonic conjugates of each of these points with respect to the two
vertices of the tetrahedron that lie on the same edge are determined, then the
lines joining the latter six points to any point of the plane a are on a cone,
on which are also the lines through and meeting a pair of opposite edges of the
tetrahedron (Holgate, Annals of Mathematics, Ser. 1, Tol. VII (1893), p. 73).

10. Given four points of a conic and the tangent at one of them, construct
the tangents at the other three points. Dualize.

11. A, A', B, B' are the, vertices of a quadrangle, and m, n are two lines
in the plane of thja quadrangle which meet on AA'. M Is a variable point

120

CONIC SECTIONS

[CHAP. V

OIL m, the lines BM, B'M meet n in the points N, N' respectively ; the lines
ANj A'N' meet in a point P. Show that the locus of the lines PJ\1 is a line
conic, -which contains the lines m, p = P(n, BB'), and also the lines A A', BB',
A'B'y AB (Amodeo, Lezioni di Geometria Projettiva, Naples (1905), p. 331).

12. Use the result of Ex. 11 to give a construction of a line conic deter-
mined by five given lines, and show that by means of this construction it is
possible to obtain two lines of the conic at the same time (Amodeo, loc. cit.).

13. If a, 5, c are the sides of a triangle whose vertices are on a conic, and
m, m' are two lines meeting on the conic which meet a, &, c in the points A, B, C
and A', B', C' respectively, and which meet the conic again in N 9 N' respec-
tively, we have ABCNj^A'B'C'N' (cf. Ex. 6).

14. If A, B, C, D are points on a conic and a, 5, c, d are the tangents to
the conic at these points, the four diagonals of the simple quadrangle ABCD
and the simple quadrilateral abed are concurrent.

44. The polar system of a conic.

THEOREM 13. If P is a point in THEOREM 13'. If pis a line in the
the plane of a conic, lut not on the plane of a conic, but not tangent to

conic, the points of intersection of
the tangents to the conic at all the
pairs of points which are collinear
with P are on a line, which also con*-
tains the harmonic conjugates of P
with respect to these pairs of points.

the conic, the lines joining the points
of contact of pairs of tangents to the
conic which meet on p pass through
a point P, through which pass also
the harmonic conjugates ofp with
respect to these pairs of tangents.

. 58

Proof. Let P v P% and P^, P be two pairs of points on the conic which
are collinear with P, and let p v p z be the tangents to the conic at JJ, P 2
respectively (fig. 58). If 2> v D 2 are the points (%%)(%%) an

] POLAR SYSTEM 121

respectively, the line D^D^ passes through the intersection Q of jt\ 3 $
(Theorem 8). Moreover, the point P r in which Z^D* meets 2J2* is the
harmonic conjugate of P with respect to P 13 R (Theorem 6, Chap. IT).
This shows that the line D^D* QP ! is completely determined by the
pair of points P^ P,. Hence the same line QP' is obtained by replacing
P 3> P by any other pair of points on the conic collinear with P, and
distinct from P v P,. This proves Theorem 13. Theorem 13' is the
plane dual of Theorem 13.

DEFINITION. The line thus asso- DEFINITION. The point thus

ciated with any point P in the associated with any line p in the

plane of a conic, but not on the plane of a conic, but not tangent

conic, is called the polar of P to the conic, is called thereof p

with respect to the conic. If P with respect to the conic. If p is

is a point on the conic, the polar a tangent to the conic, the pole is

is defined as the tangent at P. defined as the point of contact of P.

THEOKEM 14. TJie line joining THEOREM 14'. The point of

two diagonal points of any com- intersection of two diagonal lines

plete quadrangle whose vertices of any complete quadrilateral

are points of a conic is the polar wJiose sides are tangent to a conic

of the other diagonal point with is the pole of the other diagonal

respect to the conic. line with respect to the conic.

Proof. Theorem 14 follows immediately from the proof of Theo-
rem 13. Theorem 14' is the plane dual of Theorem 14

THEOREM 15. The polar of a THEOREM 15'. Tlie pole of a

point P with respect to a conic line p with respect to a conic is

passes through the points of con- on the tangents to the conic at the

tact of the tangents to the conic poi/its in which p meets the conic,

through P, if such tangents exist, if such paints exist

Proof. Let P l be the point of contact of a tangent through. P, and
let P%> PS be any pair of distinct points of the conic collinear with P.
The line through JJ and the intersection of the tangents at jg, P%
meets the line J%P% in the harmonic conjugate of P with, respect to
j?fjf (Theorem 9, Cor.). But the line thus determined is the polar of P
(Theorem 13). This proves Theorem 15. Theorem 15' is its plane dual

THEOREM 16. If p is the polar of a point P with respect to a conic,
P is the pole of p with respect to the same conic.

122 COXIC SECTIONS [CHAP.V

If P is not on the conic, this follows at once by comparing Theo-
rem 13 with Theorem 13'. If P is on the conic, it follows immediately
from the definition.

THEOREM 17. If the polar of a point P passes through a point Q,
the polar of Q passes through P.

Proof. If P or Q are on the conic, the theorem is equivalent to
Theorem 15. If neither P nor Q is on the conic, let PP^ be a line

meeting the conic in two points, P^ P 2 . If one of the lines P^Q, P 2 Q
is a tangent to the conic, the other is also a tangent (Theorem 13);
the line P^ = PP is then the polar of Q, which proves the theorem
under this hypothesis. If, on the other hand, the lines I$Q,P 2 Q meet
the conic again in the points P& P respectively (fig. 59), the point
(IJJFJ) (P Z P^ is on the polar of Q (Theorem 14). By Theorems 13 and 14
the polar of (J?-E) (P Z P^) contains the intersection of the tangents at
P v R and the point Q. By hypothesis, however, and Theorem 13, the
polar of P contains these points also. Hence we have (P^ (P%P^) = P,
which proves the theorem.

COROLLAEY 1. If two vertices of a triangle are the yoles of their
opposite sides with respect to a conic, the third vertex is the pole of
its opposite side.

DEFINITION. Any point on the polar of a point P is said to be
conjugate to P with regard to the conic; and any line on the pole

POLAE SYSTEM 123

of a line p is said to be conjugate to p with regard to the conic.
The figure obtained from a given figure in the plane of a conic by
constructing the polar of every point and the pole of every line of
the given figure with regard to the conic is called the polar or polar
reciprocal of the given figure with regard to the conic.* A triangle,
of which each vertex is the pole of the opposite side, is said to be
self -polar or self-conjugate with regard to the conic.

COROLLARY 2. TJie diagonal triangle of a complete quadrangle ichose
vertices are on a conic, or' of a complete quadrilateral whose sides are
tangent to a conic, is self -polar with regard to the co/iic; and, conversely,
every self -polar triangle is the diagonal triangle of a complete quad-
rangle whose points are on the conic, and of a complete quadrilateral
whose sides are tangent to the conic. Corresponding to a given self-polar
triangle, one vertex or side of such a quadrangle or quadrilateral may
be chosen arbitrarily on the conic.

Theorem 17 may also be stated as follows : If P is a variable point
on a line q, its polar p is a variable line through the pole Q of q. In the
special case where q is a tangent to the conic, we have already seen
(Theorem 12) that we have

[^AM-

If Q is not on q, let A (fig. 60) be a fixed point on the conic, a the
tangent at A, JT the point (distinct from A, if AP is not tangent) in
which AP meets the conic, and x the tangent at X. "We then have, by
Theorem 12,

By Theorem 13, (ax) is on p, and hence p = Q (ax). Hence we have

[*]*[*]

If P ! is the point pq, this gives

But since the polar of P ! also passes through P, this projectivity is
an involution. The result of this discussion may then be stated as
follows :

* It was by considering the polar reciprocal of Pascal's theorem that Brianchon
derived the theorem named after him. This method was fully developed "by Poncelet
and Gergonne in the early part of the last century in connection with the principle
of duality.

124 C03STC SECTIONS [CHAP.Y

THEOREM 18. On any line not a tangent to a given conic the pairs
of conjugate points are pairs of an involution. If the line meets the
conic in two points, these points are the doulle points of the involution.

COROLLARY. As a point P varies over a pencil of points, its polar
with respect to any conic varies over a projectile pencil of lines.

. DEFINITION. The pairing of the points and lines of a plane brought
about by associating \vith every point its polar and with every line its
pole with respect to a given conic in the plane is called a polar system.

EXERCISES

1. If in a polar system two points are conjugate to a third point A , the
line joining them is the polar of A.

2. State the duals of the last two theorems.

?*' 3. If a and Z> are two nonconjugate lines in a polar system, every point A
of a has a conjugate point B on 5. The pencils of points [A] and B] are
protective ; they are perspective if and only if a and I intersect on the conic
of the polar system.

< 4. Let A be a point and 5 a line not the polar of A with respect to a given
conic, but in the plane of the conic. If on any line I through A we determine
that point P which is conjugate with the point $, the locus of P is a conic
passing through A and the pole B of 5, unless the line AB is tangent to the

] POLAB SYSTEM 125

conic, In which case the locus of P is a line. If AB is not tangent to the conic,
the locus of P also passes through the points in which b meets the given conic
(if such points exist), a-nd also through the points of contact of the tangents to
the given conic through A (if such tangents exist). Dualize (Reye-Holgate,
Geometry of Position, p. 106).

5. If the vertices of a triangle are on a given conic, any line conjugate to
one side meets the other two sides in a pair of conjugate points. Conversely,
a line meeting two sides of the triangle in conjugate points passes through
the pole of the side (von Staudt).

6. If two lines conjugate with respect to a conic meet the conic in two
pairs of points, these pairs are projected from any point on the conic by a
harmonic set of lines, and the tangents at these pairs of points meet any
tangent in a harmonic set of points.

7. ^Vith a given point not on a given conic as center and the polar of this
point as axis, the conic is transformed into itself by a homology of period two.

8. The Pascal line of any simple hexagon whose vertices are on a conic is
the polar with respect to the conic of the Brianchon point of the simple hexagon
whose sides are the tangents to the conic at the vertices of the first hexagon.

9. If the line joining two points A, B, conjugate with respect to a conic,
meets the conic in two points, these two points are harmonic with A 9 B.

10. If in a plane there are given two conies Cf and C|, and the polars oi
all the points of Cf with respect to C| are determined, these polars are the
tangents of a third conic.

11. If the tangents to a given conic meet a second conic in pairs of points,
the tangents at these pairs of points meet on a third conic.

12. Given five points of a conic (or four points and the tangent through
one of them, or any one of the other conditions determining a conic), sho^w
how to construct the polar of a given point with respect to the conic,

13. If two pairs of opposite sides of a complete quadrangle are pairs ot
conjugate lines with respect to a conic, the third pair of opposite sides are
conjugate -with respect to the conic (Hesse).

14. If each of two triangles in a plane is the polar of the other with respect
to a conic, they are perspective, and the axis of perspectivity is the polar of the
center of perspectivity (Chasles).

15. Two triangles that are self -polar with respect to the same conic have
their six vertices on a second conic and their six sides tangent to a third
conic (Steiner).

16. Regarding the Desargues configuration as composed of a quadrangle
and a quadrilateral mutually inscribed (cf. 18, Chap. II), show that the
diagonal triangle of the quadrangle is perspective with the diagonal triangle
of the quadrilateral.

17. Let A , B be any two conjugate points with respect to a conic, and let
the lin^s AM, BM joining them to an arbitrary point of the conic meet the
latter again in the points C, D respectively. The lines AD, BC will then meet
on the conic, and the lines CD and AB are conjugate. Dualize.

126 CONIC SECTIONS [CHAP.V

45. Degenerate conies. For a variety of reasons it is desirable to
regard two coplanar lines or one line (thought of as two coincident
lines) as degenerate cases of a point conic; and dually to regard
two points or one point (thought of as two coincident points) as
degenerate cases of a line conic. This conception makes it possible
to leave out the restriction as to the plane of section in Theorem 1.
For the section of a cone of lines by a plane through the vertex of
the cone consists evidently of two (distinct or coincident) lines, i.e.
of a degenerate point conic ; and the section of a cone of planes by
a plane through the vertex of the cone is the figure formed by some
or all the lines of a flat pencil, i.e. a degenerate line conic.

EXERCISE

Dualize in all possible ways the degenerate and nondegenerate cases of
Theorem 1.

Historically, the first definition of a conic section was given by the ancient
Greek geometers (e.g. Mensechmus, about 350 B.C.), who defined them as the
plane sections of a "right circular cone." In a later chapter we will show
that in the " geometry of reals " any nondegenerate point conic is protectively
equivalent to a circle, and thus that for the ordinary geometry the modern
projective definition given in 41 is equivalent to the old definition. We are
here using one of the modern definitions because it can be applied before devel-
oping the Euclidean metric geometry.

Degenerate conies would be included in our definition (p. 109), if
we had not imposed the restriction on the generating projective
pencils that they be nonperspective ; for the locus of the point of
intersection of pairs of homologous lines in two perspective flat
pencils in the same plane consists of the axis of perspectivity and
the line joining the centers of the pencils.

It will be seen, as we progress, that many theorems regarding non-
degenerate conies apply also when the conies are degenerate. For
example, Pascal's theorem (Theorem 3) becomes, for the case of a
degenerate conic consisting of two distinct lines, the theorem of
Pappus already proved as Theorem 21, Chap. IV (cf. in particular the
corollary). The polar of a point with regard to a degenerate conic
consisting of two lines is the harmonic conjugate of the point with
respect to the two lines (cf. the definition, p. 84, Ex. 7). Hence the
polar system of a degenerate conic of two lines (and dually of two
points) determines an involution at a point (on a line).

45,40] THEOREM OF DBS ARGUES 127

EXERCISES

1. State Brianchon's theorem (Theorem 3') for the case of a degenerate
line conic consisting of two points.

2. Examine all the theorems of the preceding sections -with reference to
their behavior when the conic in question becomes degenerate.

46. Desargues's theorem on conies.

THEOREM 19. If the vertices of a complete quadrangle are oil a conic
which meets a line in two points, the latter are a pair in the invo-
lution determined on the line ly the pairs of o^iosite sides of the
quad/ 'anyk >*

Proof. Reverting to the proof of Theorem 2 (fig. 51), let the line
meet the conic in the points B v C and let the vertices of the quad-
rangle be A v A*, B v C 2 . This quadrangle determines on the line an
involution in which S 9 A and T, U are conjugate pairs. But in the
proof of Theorem 2 we saw that the quadrangle A^A^BC determines
Q^ST, C^U). Hence the two quadrangles determine the same
involution on the line, and therefore B 2 , C l are a pair of the involution
determined by the quadrangle A^B^Cy

Since the quadrangles A^AJS^ and A^BC determine the same
involution on the line when the latter is a tangent to the conic, we
have as a special case of the above theorem :

COROLLARY. If the vertices of a complete quadrangle are on a conic,
the pairs of opposite sides meet the tangent at any other point in pairs
of an involution of which the point of contact of the tanyent is a double
point.

The Desargues theorem leads to a slightly different form of statement for
the construction of a conic through five given points : On any line through
one of the points the complete quadrangle of the other four determine an
involution ; the conjugate in this involution of the given point on the line
is a sixth point on the conic.

As the Desargues theorem is related to the theorem of Pascal, so
are certain degenerate cases of the Desargues theorem related to the
degenerate cases of the theorem of Pascal (Theorems 6, 7, 8, 9). Thus
in fig. 53 we see (by Theorem 6) that the quadrangle BCRP^ deter-
mines on the line P^ an involution in which the points P^ P of the
conic are one pair, while the points determined by p v P^ and those

* First given by Desargues in 1639 ; cf. OEuvres, Paris, Vol. I (1864), p. 188.

128 CONIC SECTIONS [CHAP.V

determined by % P^ P^ are two other pairs. This gives the following
special case of the theorem of Desargues :

THEOBEM 20. If the vertices of a triangle are on a conic, and a line I
meets the conic in two points, the latter are a pair of the involution
determined on I by the pair of points in which two sides of the triangle
meet Z, and the pair in which the third side and the tangent at the
opposite vertex meet I In case I is a tangent to the conic, the point of
contact is a double point of this involution.

In terms of this theorem we may state the construction of a conic through
four points and tangent to a line through one of them as follows : On any line
through one of the points which is not on the tangent an involution is deter-
mined in which the tangent and the line passing through the other two points
determine one pair, and the lines joining the point of contact to the other two
points determine another pair. The conjugate of the given point on the line
in this involution is a point of the conic.

A further degenerate case is derived either from Theorem 7 or
Theorem 8. In fig. 54 (Theorem 7) let I be the line P 2 P 3 . The quad-
rangle ABP^ determines on I an involution in which JJ, P B are one
pair, in which the tangents at P l} P determine another pair, and in
which the line P^ determines a double point. Hence we have

THEOREM 21. If a line I meets a conic in two points and J%, P are
any other two points on the conic, the points in which I meets the conic
are a pair of an involution through a double point of which passes the
line P^P and through a pair of conjugate points of which pass the
tangents at P lt P^ If I is tangent to the conic, the point of contact is
the second double point of this involution.

The construction of the conic corresponding to this theorem may be stated
as follows : Given two tangents and their points of contact and one other point
of the conic. On any line I through the latter point is determined an involution
of which one double point is the intersection with I of the line joining the two
points of contact, and of which one pair is the pair of intersections with I of
the two tangents. The conjugate in this involution of the given point of the
conic on Us a point of the conic.

EXERCISE

State the duals of the theorems in this section.

47. Pencils and ranges of conies. Order of contact. The theorems
of the last section and their plane duals determine the properties of
certain systems of conies which we now proceed to discuss briefly.

47]

PENCILS AXD RANGES

129

DEFINITION. The set of all conies
through the vertices of a complete
quadrangle is called a pencil of
conies of Type I (fig. 61).

Theorem 19 and its plane dual
THEOREM 22. Any line (not
tlirougli a vertex of the deter-
mining quadrangle) is met ly the
conies of a pencil of Type I in the
pairs of an involution*

DEFINITION. The set of all conks
tangent to the sides of a complete
quadrilateral is called a range of
conies of Type I (fig. 62).
give at once :

THEOREM 22'. The tangents
tlirougli any point (not on a side
of the determining quadrilateral)
to the conies of a range of Type I
are the pairs of an involution.

FIG. 63

GOBOLLABY. Through a gen-
eral^ point in the plane there is
one and only one, and tangent to
a general line there are two or no
conies of a given pencil of Type I.

EIG. 64

COBOLLABY. Tangent to a gen-
eral line in the plane there is one
and only one, and through a gen-
eral point there are two or no
conies of a gimn range of Type I.

* This form of Desargues's theorem is due to Ch. Sturm, Annales de Mathe*ma-
tiques, Vol. XVII (1826), p. 180.

t The vertices of the quadrangle are regarded as exceptional points.

130

CONIC SECTIONS

[CHA.P. V

DEFINITION. The set of all conies
through the vertices of a triangle
and tangent to a fixed line through
one vertex is called a pencil of
conies of Type II (fig. 63).

DEFINITION. The set of all conies
tangent to the sides of a triangle
and passing through a fixed point
on one side is called a range of
conies of Type II (fig. 64).

Theorem 20 and its plane dual then give at once :

THEOREM 23. Any line in the
plane of a pencil of conies of
Type II (which does not pass
through a vertex of the determin-
ing triangle) is met by the conies
of the pencil in the pairs of an
involution*

COROLLARY. Through a general
point in the plane there is one and
only one conic of the pencil; and
tangent to a general line in the
plane there are two or no conies
of the pencil.

THEOREM 23'. The tangents
through any point in the plane
of a range of conies of Type II
(which is not on a side of the
determining triangle) to the conies
of the range are the pairs of an
involution.

COROLLARY. Tangent to a gen-
eral line in the plane there is one
and only one conic of the range;
and through a general point in
the plane there are two or no
conies of the range.

DEFINITION. The set of all conies through two given points and
tangent to two given lines through these points respectively is called

a pencil or range of conies of Type
IV* (fig. 65).

Theorem 21 now gives at once:

THEOREM 24. Any line in the plane
of a pencil of conies of Type IV (which
does not pass through either of the
points common to all the conies of
the pencil) is met ly the conies of the
pencil in the pairs of an involution.
Through any point in the plane (not
on either of the lines that are tangent
to all the conies of the pencil) the
tangents to the conies of the pencil are the pairs of an involution. Tlie
line joining the two points common to all the conies of the pencil meets

* The classification of pencils and ranges of conies into types corresponds to the
classification of the corresponding plane collineations (cf. Exs. 2, 4, 7, below).

FIG. Co

47] PENCILS AND RANGES 131

any line in a double point of the involution determined on that line.
And {lie point of intersection of the common tangents is joined to any
point ly a double line of the involution determined at that point

COROLLAEY. Throitgli any general point or tanyent to any general
line in the plane there is one and only one conic of the pencil.

EXERCISES

1. What are the degenerate conies of a pencil or range of Type 7? The
diagonal triangle of the fundamental quadrangle (quadrilateral) of the pencil
(range) is the only triangle which is self -polar with respect to two conies of
the pencil (range).

2. Let A' 2 and B 2 be any two conies of a pencil of Type /, and let P be any
point in the plane of the pencil. If p is the polar of P with respect to .-I 2 , and
P' is the pole of p with respect to B' 2 , the correspondence thus established
between [P] and [P'] is a projective collineation of Type J, whose invariant
triangle is the diagonal triangle of the fundamental quadrangle. The set of
all projective collineations thus determined by a pencil of conies of Type I
form a group. Dualize.

3. What are the degenerate conies of a pencil or range of Type //?

4. Let a pencil of conies of Type // be determined by a triangle ABC and
a tangent a through A. Further, let ' be the harmonic conjugate of a with
respect to AB and AC, and let A' be the intersection of a and BC. Then
A, a and A', a' are pole and polar with respect to every conic of the pencil ; and
no pair of conies of the pencil have the same polars with regard to any other
points than A and A'. Dualize, and show that all the collineations determined
as in Ex. 2 are in this case of Type //.

5. What are the degenerate conies of a pencil or range of Type JT?

6. Show that any point on the line joining the two points common to all
the conies of a pencil of Type IV has the same polar with respect to all the
conies of the pencil, and that these all pass through the point of intersection
of the two common, tangents.

7. Show that the collineations determined by a pencil of Type IV by the
method of Ex. 2 are all homologies (i.e. of Type IV).

* The pencils and ranges of conies thus far considered have in com-
mon the properties (1) that the pencil (range) is completely defined
as soon as two conies of the pencil (range) are given ; (2) the conies
of the pencil (range) determine an involution on any line (point) in
the plane (with the exception of the lines (points) on the determining
points (lines) of the pencil (range)). Three other systems of conies may
be defined which likewise have these properties. These new systems

* The remainder of this section may be omitted on a first reading.

132

COtflC SECTIONS

[CHAP. V

may be regarded as degenerate cases of the pencils and ranges already
defined. Their existence is established by the theorems given below,
which, together with their corollaries, may be regarded as degenerate
cases of the theorem of Desargues. "We shall need the following

LEMMA. Any conic is transformed lij a protective collineation in
the plane of the conic into a conic sucli that the tangents at homologous
points are homologous.

Proof. This follows almost directly from the definition of a conic.
Two projective flat pencils are transformed by a projective collineation
into two projective flat pencils. The intersections of pairs of homologous
lines of one pencil are therefore transformed into the intersections
of the corresponding pairs of homologous lines of the transformed
pencils. If any line meets the first conic in a point P, the transformed
line will meet the transformed conic in the point homologous with P.
Therefore a tangent at a point of the first conic must be transformed
into the tangent at the corresponding point of the second conic.

THEOREM 25. If a line p Q is a tangent to a conic A 2 at a 'point P OJ
and Q is any point of A 2 , then through any point on the plane of A 2

but not on A 2 or p ,
there is one and only
one conic B* through
PS and Q, tangent to
Pv and such that there
is no point except P^
which has the same
polar with regard to
loth A* and B\

Proof. If P J is any point of the plane not on p Q or 'A 2 , let P be
the second point in which P Q P ! meets A* (fig. 66). There is one and
only one elation with center P Q and axis P Q Q changing P into P ;
(Theorem 9, Chap. III). This elation (by the lemma above) changes
A* into another conic B* through the points P^ and Q and tangent
to jp . The lines through J^ are unchanged by the elation, whereas
their poles (on p ) are subjected to a parabolic projectivity. Hence
no point on p Q (distinct from P Q ) has the same polar with regard to A*
as with regard to B*. Since A 2 is transformed into B* by an elation,
the two conies can have no other points in common than P Q and Q.

47]

PEXCILS AXB EAXGES

133

That there is only one eonic B' 2 through P r satisfying the con-
ditions of the theorem is to be seen as follows : Let QP meet p in 3,
and QP 1 meet p in S r (fig. 66;. The points S and S r must have the
same polar with regard to A 2 and any conic J5 2 , since this polar
must be the harmonic conjugate of p^ with regard to P Q Q and P Q P.
Let p be the tangent to A 2 at P and p f be the tangent to If 2 at P f ,
and let p and p r meet p Q in I and 2 17 respectively. The points

FIG. 67

T and T J have the same polar, namely P Q P, with regard to A 2 and
any conic B*. By the conditions of the theorem the projectivity

must be parabolic. Hence, by Theorem 23, Cor.,, Chap. IV,

Q(P ST,P Q T f S r ).

Hence p and p f must meet on P^Q in a point B so as to form the quad-
rangle EQPP 1 . This determines the elements P Q , Q, P f > p Q , p f of B*>
and hence there is only one possible conic J5 2 .

COROLLARY 1. The conies A 2 and 3 s can have no other points in
common than P and Q.

COROLLARY 2. Any line I not on P^ or Q which meets A 2 and B*
meets them in pairs of an involution in which the points of intersection
of I with P Q Q <md p Q are conjugate.

Proof. Let I meet Jf in N and N v J5 2 in L and L v J%Q in M, and
j? in M 1 (fig. 67). Let K and ^ be the points of A 2 which are trans-
formed by the elation into L and L^ respectively. By the definition of
an elation K and S t are colHnear with Jf, while K is on the line JJ
and J^ on LJ%. Let KN^ meet p in jK, and j^JJ meet JCS^ in &

134

COXIC SECTIONS

[CHAP. V

Then, since JN", E, X v K are on the conic to which. p Q is tangent at ^, we
have, by Theorem 6, applied to the degenerate hexagon P^K^EN^N,
that S 9 L v and E are collinear. Hence the complete quadrilateral
Sit, JEYy JOTp I has pairs of opposite vertices on P Q 3I and P Q M V P Q N
and J^Y r JJ and P L r Hence

ox . The set of all conies
through a point Q and tangent to
a line p^ at a point -Z^ and such
that no point of p except % has
the same polar with regard to two
conies of the set, is called a pencil
of conies of Type III (tig. 68).

DEFINITION. The set of all conies
tangent to a line q and tangent to
a line p Q at a point P Q , and such
that no line on P except p has
the same pole with regard to two
conies of the set, is called a range
of conies of Type III (fig. 69).

FIG. 08

FIG. 09

Two conies of such a pencil (range) are said to have contact of the
second order, or to osculate, at JJ.

Corollary 2 of Theorem 25 now gives at once:

THEOREM 26. Any line in the
plane of n peticil of conies of
Type III, which is not on either of
the common points of the pencil, is
met hy the conies of the pencil in the
pa i/'s of a n iii i*oht t ion. TJi ro ugh
any point iti the plane except the
ctnnitwn points there is one and
only one conic of th e pencil; and
tanyent to any line not through
either of the common points there
are ttro or no w flics of the pencil.

THEOREM 26'. Tlirougli any point
in tlie plane of a range of conies of
Type III, which is not on either of
the common tangents of the range,
the tangents to the conies of the pen-
cil are the pairs of an involution.
Tangent to any line in the plane ex-
cept the common tangents there is
one and only one conic of the range;
and through any point not on either
of the common tangents there are
two or no conies of the range.

* This argument has implicitly proved that three pairs of points of a conic, as
Nii P Q> such that the lines joining them meet in a point M, are projected
from any point of the conic by a quadrangular set of lines (Theorem 16, Chap. VUI).

47] PENCILS AND BAXGES 135

The pencil is determined by The range is determined by

the two common points, the com- the two common tangents, the

mon tangent, and one conic of the common point, and one conic of

pencil. the range.

EXERCISES

1. What are the degenerate conies of this pencil and range?

2. Show that the collineation obtained by making correspond to any point P
the point P' which has the same polar p with regard to one given conic of the
pencil (range) that P has with regard to another given conic of the pencil (range)
is of Type 1/7.

THEOREM 27. If a line p^ is tangent to a conic A 2 at a point P Qt
there is one and only one conic tangent to p$ at P^ and passing
through any other point P 1 of the plane of A 2 not on p Q or A 2
which determines for every point of p Q the same polar line as does A~.

Proof. Let P be the second point in which P^P* meets A 2 (fig. 70).
There is one and only one elation of which 2J is center and p Q axis,
changing P to P 1 . This elation changes A 2 into a conic B* through

P r , and is such that if q is any tangent to A 2 at a point Q, then q is
transformed to a tangent $ f of B* passing through gjp , and Q is trans-
formed into the point of contact Q r of q r , collinear with Q and JJ.
Hence there is one conic of the required type through P f .

That there is only one is evident, because if I is any line through JP',
any conic B* must pass through the fourth harmonic of P J with regard
to lp Q and the polar of lp a as to A* (Theorem 13). By considering two
lines I we thus determine enough points to fix B\

COBOLLAEY 1. By duality there is one and only one conic B z tangent
to any line not passing through P y

136 CONIC SECTIONS [CHAF.V

COROLLARY 2. Any line I not on P^ which meets A* and B* meets
them in pairs of an involution one doitble point of which is lp Q) and
the other the point of I conjugate to lp Q with respect to A 2 . A dual
statement holds for any point L not on p Q .

COROLLARY 3. The conies A 2 and B' 2 can have no other point in
commoti thati J?J cnid no other tangent iti common than p Q .

Proof. If they had one other point P in common, they would have
in common the conjugate of P in the involution determined on any
line through P according to Corollary 2.

DEFINITION. The set of all conies tangent to a given line p Q at a
given point P , and such that each point on p has the same polar
with regard to all conies of the set, is called a pencil or range of
conies of Type V. Two conies of such a pencil are said to have
contact of the third order, or to hyperosculate at JJ.

Theorem 27 and its first two corollaries now give at once:

THEOREM 28. Any line I not on the common point of a pencil of
Type V is met ly the conies of the pencil in pairs of an involution
one double point of which is the intersection of I with the common
tangent. Through any point L not on the common tangent the pairs
of tangents to the conies of the pencil form an involution one double
line of which is the line joining L to the common point. There is one
conic of the set through each point of the plane not on the common
tangent, and one conic tangent to each line not on the common point.

The pencil or range is determined by the common point, the common
tangent, and one conic of the set

EXERCISES

1. What are the degenerate conies of a pencil of Type F?

2. Show that the collineation obtained by making correspond to any
point P the point Q which has the same pole p with regard to one conic of
a pencil of Type V that P has with regard to another conic of the pencil is
an elation.

3. The lines polar to a point A with regard to all the conies of a pencil
of any of the five types pass through a point A'. The points A and A' are
double points of the involution determined by the pencil on the line A A'.
Construct A'. Dualize. Derive a theorem on the complete quadrangle as a
special case of this one.

4. Construct the polar line of a point A with regard to a conic C 2 being
given four points of C* and a conjugate of A with regard to <7 2 .

47] PENCILS AXD BADGES 137

5. Given an involution I on a line Z, a pair of points A and A' on I not
conjugate in I, and any other point B on L construct a point B' such that .4
and A' and B and 5' are pairs of an involution I' whose double points are a
pair in I. The involution I' may also be described as one which is commu-
tative with I, or such that the product of I and T is an involution.

6. There is one and only one conic through three points and having a
given point P and line p as pole and polar.

7. The conies through three points and having a given pair of points as
conjugate points form a pencil of conies.

MISCELLANEOUS EXERCISES

1. If and o are pole and polar with regard to a conic, and *1 and B are

two points of the conic collinear with 6>, then the conic is generated by the

two pencils A [P] and B [P'] where P and P' are paired in the involution

on o of conjugates with regard to the conic.

/ 2. Given a complete five-point ABCDE. The locus of all points X such

that X (BCDE) T A (BCDE)

is a conic. A

V 3. Given two projective nonperspective pencils, [p] and [<?]. Every line I

upon which the projectivity Z[^] -r-l^q] is involutorie passes through a fixed
point 0. The point is the pole of the line joining the centers of the pencils
with respect to the conic generated by them.

^ 4. If two complete quadrangles have the same diagonal points, their eight
vertices lie on a conic (Cremona, Projective Geometry (Oxford, 1885), Chap.XX).

5. If two conies intersect in four points, the eight tangents to them at
these points are on the same line conic. Dualize and extend to the cases
where the conies are in pencils of Types II-V.

6. All conies with respect to which a given triangle is self -conjugate, and
which pass through a fixed point, also pass through three other fixed points.
Dualize.

7. Construct a conic through two given points and with a given self-
conjugate triangle. Dualize.

8. If the sides of a triangle are tangent to a conic, the lines joining two
of its vertices to any point conjugate with regard to the conic to the third
vertex are conjugate with regard to the conic. Dualize.

9. If two points P and Q on a conic are joined to two Conjugate points P f 9 Q?
on a line conjugate to PQ, then PP' and QQf meet on the conic.

10. If a simple quadrilateral is circumscribed to a conic, and if I is any
transversal through the intersection of its diagonals, / will meet the conic and
the pairs of opposite sides in conjugate pairs of an involution. Dualize.

11. Given a conic and three fixed collinear points A, B, C. There is a fourth
point D on the line AB such that if three sides of a simple quadrangle in-
scribed in the conic pass through A, B, and C respectively, the fourth passes,
through D (Cremona, Chap. XVII).

138 CONIC SECTIONS [CHAP.V

12. If a variable simple n-line ( even) is inscribed in a conic in such a way
that n 1 of its sides pass through n 1 fixed collinear points, then the other
side passes through another fixed point of the same line. Dualize this theorem.

13. If two conies intersect in two points A. B (or are tangent at a point A)
and two lines through A and B respectively (or through the point of contact
.4) meet the conies again in 0, (/ and L, Z/, then the lines OL and O'L' meet
on the line joining the remaining points of intersection (if existent) of the
two conies.

14. If a conic C 2 passes through the vertices of a triangle which is self-
polar with respect to another conic K 2 , there is a triangle inscribed in C 2 and
self -polar with regard to J 2 , and having one vertex at any point of C 2 . The
lines which cut C 3 and K* in two pairs of points which are harmonically con-
jugate to one another constitute a line conic C|, which is the polar reciprocal
of C 2 with regard to K* (Cremona, Chap. XXII).

15- If a variable triangle is such that two of its sides pass respectively
through two fixed points 0' and lying on a given conic, and the vertices oppo-
site them lie respectively on two fixed lines u and u', while the third vertex
lies always on the given conic, then the third side touches a fixed conic, which
touches the lines u and u'. Dualize (Cremona, Chap. XXII).

16. If P is a variable point on a conic containing A, B, C, and Ms a vari-
able line through P such that all throws T (PA, PB: PC, /) are projective,
then all lines I meet in a point of the conic (Schroter, Journal fur die reine und
angewandte Mathematik, Tol. LXII, p. 222).

17. Given a fixed conic and a fixed line, and three fixed points A, B, C on
the conic, let P be a variable point on the conic and let PA, PB, PC meet
the fixed line in A', B'* C'. If is a fixed point of the plane and (OA', PB'} = K
and (j&TC") = /, then K describes a conic and I a pencil of lines whose center is
on the conic described by K (Schroter, loc. cit.).

18. Two triangles ABC and PQR are perspective in four ways. Show that
if ABC and the point P are fixed and Q, R are variable, the locus of each of
the latter points is a conic (cf. Ex. 8, p. 105, and Schroter, Mathematische
Annalen, Vol. II (1870), p. 553).

19. Given six points on a conic. By taking these in all possible orders
60 different simple hexagons inscribed in. the conic are obtained. Each of
these simple hexagons gives rise to a Pascal line. The figure thus associated
with any six points of a conic is called the hexagramma mysticum.* Prove the
following properties of the hexagramma niysticum :

i. The Pascal lines of the three hexagons PiPoPgP^Pg, P^P^P Z P^P 5 P^
and PiPgPsP^PgP^ are concurrent. The point thus associated with such a set
of three hexagons is called a Steiner point.
ii. There are in all 20 Steiner points.

* On the Pascal hexagram cf . Steiner-Schroter, Vorlesungen tiber Synthetische
Geometrie, Vol. II, 28 ; Salmon, Conic Sections in the Notes ; Christine Ladd,
American Journal of Mathematics, Vol. IE (1879), p. 1.

47] EXERCISES 139

iii. From a given simple hexagon five others are obtained by permuting
in all possible ways a set of three vertices no two of which are adjacent. The
Pascal lines of these six hexagons pass through two Steiner points, which are
called conjugate Steiner points. The 20 Steiner points fall into ten pairs of
conjugates.

iv. The 20 Steiner points lie by fours on 15 lines called Steiner lines.
v. AVhat is the symbol of the configuration composed of the 20 Steiner
points and the 15 Steiner lines ?

20. Discuss the problem corresponding to that of Ex. 19 for all the special
cases of Pascal's theorem.

21. State the duals of the last two exercises.

22. If in a plane there are given two conies, any point A has a polar with
respect to each of them. If these polars intersect in A', the points A , A' are
conjugate with respect to both conies. The polars of A / likewise meet in A.
In this way every point in the plane is paired with a uniqxte other point. By
the dual process every line in the plane is paired with a unique line to which
it is conjugate with respect to both conies. Show that in this correspondence
the points of a line correspond in general to the points of a conic. All such
conies which correspond to lines of the plane have in common a set of at most
three points. The polars of every such common point coincide, so that to each
of them is made to correspond all the points of a line. They form the excep-
tional elements of the correspondence. Dualize (Reye-Holgate, p. 110).*

23. If in the last exercise the two given conies pass through the vertices of
the same quadrangle, the diagonal points of this quadrangle are the " common
points " mentioned in the preceding exercise (Reye-Holgate, p. 110).

24. Given a cone of lines with vertex and a line u through 0. Then a
one-to-one correspondence may be established among the lines through by
associating with every such line a its conjugate a' with respect to the cone
lying in the plane au. If, then, a describes a plane ?r, of will describe a cone of
lines passing through u and through the polar line of ?r, and which has in
common with the given cone any lines common to it and to the given cone
and the polar plane of u (Reye-Holgate, p. 111).*

25. Two conies are determined by the two sets of five points A, B, C,D,E
and A,B,C 9 H, K. Construct the fourth point of intersection of the two conies
(Castelnuovo, Lezioni di Geometria, p. 391).

26. Apply the result of the preceding Exercise to construct the point P such
that the set of lines P(A, JB, C, D, E) joining P to the vertices of any given
complete plane five-point be projective with any given set of five points on a
line (Castelnuovo, loc. cit.).

27. Given any plane quadrilateral, construct a line which meets the sides
of the quadrilateral in a set of four points projective with any given set of
four collinear points.

* The correspondences defined in Exs. 22 and 24 are examples of so-called
quadratic correspondences.

140 COKIC SECTIONS [CHAP.V

28. Two sets of five points .1, B, C, Z>, E and .4, B, H, K 9 L determine
two conies which, intersect again in two points A", I". Construct the line XT
and show that the points JY", F are the double points of a certain involution
(Castelnuovo, loc. cit.).

29. If three conies pass through two given points A, B and the three pairs
of conies cut again in three pairs of points, show that the three lines joining
these pairs of points are concurrent (Castelnuovo, loc. cit.).

30. Prove the converse of the second theorem of Desargues : The conies
passing through three fixed points and meeting a given line in the pairs of
an involution pass through a fourth fixed point. This theorem may be used
to construct a conic, given three of its points and a pair of points conjugate
with respect to the conic. Dualize (Castelnuovo, loc. cit.).

31. The poles of a line with respect to all the conies of a pencil of conies
of Type 1 are on a conic which passes through the diagonal points of the
quadrangle defining the pencil. This conic cuts the given line in the points
in which the latter is tangent to conies of the pencil. Dualize.

32. Let p be the polar of a point P with regard to a triangle ABC. If P
varies on a conic which passes through ^i , B, C* then p passes through a fixed
point Q (Cayley, Collected Works, Vol. I, p. 301).

33. If two conies are inscribed in a triangle, the six points of contact are
on a third conic.

CHAPTER VI

ALGEBRA OF POINTS AND OUTE -DIMENSIONAL COORDINATE

SYSTEMS

48, Addition of points. That analytic methods may be introduced
into geometry on a strictly protective basis was first shown by von
Staudt.* The point algebra on a line which is defined in this chapter
without the use of any further assumptions than A, E, P is essentially
equivalent to von Staudt's algebra of throws (p. 60), a brief account
of which will be found in 55. The original method of von Staudt
has, however, been considerably clarified and simplified by modern
researches on the foundations of geometry.f All the definitions and
theorems of this chapter before Theorem 6 are independent of As-
sumption P. Indeed, if desired, this part of the chapter may be read
before taking up Chap. IV.

Given a line I, and on I three distinct (arbitrary) fixed points which
for convenience and suggestiveness we denote by P , P^ P n , we define
two one-valued operations $ on pairs of points of I with reference to
the fundamental points P Q) JJ, P^. The fundamental points are said
to determine a scale on I.

DEFINITION. In any plane through I let L and IL be any two lines
through P^, and let Z be any line through 1% meeting / and II in
points A and A f respectively (fig. 71). Let P x and P y be any two points
of I, and let the lines P X A and P y A* meet II and Z in the points X
and Y respectively. The point P x+y , in which the line XT meets Z, is
called the sum of the points P x and P v (in symbols J^ + ^ = -^ +y ) in

* K G. C. von Staudt (1798-1867), BeitrSge zur Geometric der Lage, Heft 2 (1857),
pp. 166 et seq. This "book is concerned also with the related question of the inter-
pretation of imaginary elements in geometry.

t Of., for example, G. Hessenberg, Ueber einen Geometrischen Calcul, Acta
Mathematiea, Vol. XXIX, p. 1.

J By a one-valued operation o on a pair of points A, B is meant any process
whereby with every pair A^ B is associated a point O, which is unique provided
the order of A, B is given; in symbols AoBC. Here "order" has no geo-
metrical significance, but implies merely the formal difference of AoB and Bo A.
If AoB BoA, the operation is commutative; if (AoB)oC=Ao(BoC), the opera-
tion is associative.

141

142

ALGEBEA OF POINTS

[CHAP. VI

the scale ^, P v P n . The operation of obtaining the sum of two points
is called addition*

THEOREM 1. If P x and P y are distinct from P Q and P ee) Q(P a ,P x P Q9
y ) is a necessary and sufficient condition for the equality

This follows immediately from the definition, AXA!Y being a
quadrangle which determines the given quadrangular set.

COROLLARY 1. If P x is any point of I, we have P x +P Q = P + P x =iI x>

This is also an immediate consequence of the definition.

COROLLARY 2. The operation of addition is one-valued for every
pair of points P& P y of I, except for the pair P mi Jf^. (A, E)

This follows from the theo-
rem above and the corollary of

* The historical origin of this con-
struction will he evident on inspection
of the attached figure. This is the
figure which results, if we choose for
& the "line at infinity" in the plane
in the sense of ordinary Euclidean
geometry (cf. p. 8). The construction
is clearly equivalent to a translation
of the vector P Q P y along the line Z,
which brings its initial point into coincidence with the terminal point of the vector
PoP x , which is the ordinary construction for the sum of two vectors on a line.

48] ADDITION 143

Theorem 3, Chap. II, in case P x and P y are distinct from JfJ and P um
If one of the points P x , P y coincides with P Q or P 9 , it follows from
Corollary 1.

COROLLARY 3. The operation of addition is associative; i.e.

for any three points P^ P v , P s for which the above expressions are
defined. (A, E)

Proof (fig. 73). Let J^-f J^ be determined as in the definition by-
means of three lines ?, 7, 7 and the line XF. Let the line ^Y be
denoted by Z ', and by means of l*JL 9 7 ' construct the point (J + P y ) 4- ^,

. 73

which is determined by the line XZ, say. If now the point P y -}~P z
be constructed by means of the lines l m , l, 7 ; , and then the point
P x + (P y + %) be constructed by means of the lines l m , l! l^ it will be
seen that the latter point is determined by the same line XZ.

COROLLARY 4 The operation of addition is commutative; i.e.

for every pair of points P x , P y for which the operation is defined. (A, E)

Proof. By reference to the complete quadrangle AXA ! Y (fig. 71)
there appears the quadrangular set Q(P m P y P , 1J, jyj +y ), which by the
theorem implies that P y +I^^P x ^ jr But, by definition, P x +P y = P x + y .
Hence

144 ALGEBBA. OF PODTTS [CHAP.TI

THEOREM 2. Any three points P x , 1^, P a (P a = R) satisfy the relation
P ppp p p p p

J -<x> J - J - J -i - L 'o- t - c ' t

i.e. the correspondence established by making each point P x of I corre-
spond to P x r = P x +P ai where P a (= P*) is any fixed point of I, is protective.
(A, E)

Proof. The definition of addition (fig. 71) gives this projectivity as
the result of two perspectivities:*

The set of all projectivities determined by all possible choices of P a in the
formula Pi-P^P^ is the group described in Example 2, p. 70. The sum of
two points P a and P b might indeed have been defined as the point into which
P b is transformed when P is transformed into P a by a projectivity of this
group. The associative law for addition would thus appear as a special case
of the associative law which holds for the composition of correspondences in
general ; and the commutative law for addition would be a consequence of the
commutativity of this particular group.

FIG. 74

49. MtdtipUcation of points. DEFIXITIOX. In any plane through I
let ? , l v k be any three lines through P Q9 P v P^ respectively, and let ^
meet 4" and I* in points A and B respectively (fig. 74). Let P x) P y be any
two points of 1 9 and let the lines P X A and P y B meet 1 9 and Z in the points
X and T respectively. The point 2^ in which the line XY meets I is

* To make fig. 71 correspond to the notation of this theorem, P must be
identified with P a .

49] MULTIPLICATION 145

called the product ofP x by P y (in symbols P x -P y =P^)m the scale P^ P^ P x
on /. The operation of obtaining the product of two points is called
multiplication* Each of the points P x , P u is called a factor ' of the
product P x - P r

THEOREM 3. If P x and P v are any tv:o i^ints of I distinct from
P Q , P l} P X) Q(P Q P x P ly P*P y P xlf ) is necessary and sufficient for the equality
P X -^P^. (A,E)

This follows at once from the definition, AXJBY being the defining
quadrangle.

COROLLARY 1. For any point P, (=F P^) on I ice Jtrn-e the relations
P^P = P X P, = P X ; P^P^P^P^P,; P X P^P^P^P^ (A,E)

This follows at once from the definition.

COROLLARY 2. TJie operation of multiplication is one-valued for
every pair of points P x , P y of I, except P^ - P^ and P^ P^. (A, E)

This follows from Corollary 1, if one of the points P 3) P y coincides
with %, P v or P n . Otherwise, it follows from the corollary, p. 50, in
connection with the above theorem.

* The origin of this construction may also "be seen in a simple construction of
metric Euclidean geometry, which results from the construction of the definition
"by letting the line Z he the "line at infinity" (cf. p. 8). In the attached figure
which gives this metric construction we have readily, from similar triangles, the
proportions: j^ = y^ = JiP , ^

PO JPy PO y Po-Pary

which, on taking the segment P Pi= 1, gives the desired result PoPy= PoP

146

ALGEBBA OF POINTS

[CHAP. VI

COBOLLARY 3. The operation of multiplication is associative; i.e. we
ha re (^ - P y ) P z = P A - (P y P z ) for every three points P x , JJ, P z for which
these products are defined. (A, E)

Proof (fig. 76). The proof is entirely analogous to the proof for
the associative law for addition. Let the point P x P y be constructed

FIG. 70

as in the definition by means of three fundamental lines 1 Q) 1 13
the point P xy being determined by the line XT. Denote the line .
by l[, and construct the point P^ P z = (P x P y ) *P S) using the lines Z , l[, /
as fundamental. Further, let the point P y P s = ^ be constructed by
means of the lines Z , V v ?, and then let P x *I> yz = P x * ( 1J) be con-
structed by means of Z , I v I*. It is then seen that the points P x P us
and P^ JP are determined by the same line.

By analogy with Theorem 1, Cor. 4, we should now prove that mul-
tiplication is also commutative. It will, however, appear presently
that the commutativity of multiplication cannot be proved without
the use of Assumption P (or its equivalent). It must indeed be clearly
noted at this point that the definition of multiplication requires the
first factor P x in a product to form with JjJ and P l a point triple of
the quadrangular set on I (cf. p. 49); the construction of the product
is therefore not independent of the order of the factors. Moreover,
the fact that in Theorem 3, Chap. II, the quadrangles giving the points
of the set are similarly placed, was essential in the proof of that

49] MULTIPLICATION 147

theorem. "We cannot therefore use this theorem to prove the com-
mutative law for multiplication as in the case of addition.

An important theorem analogous to Theorem 2 is, however, inde-
pendent of Assumption P. It is as follows :

THEOREM 4 If the relation P x -P >f = P xtJ holds Ittvecn any three
'points P^ P v , P jy on I distinct from f^ we Jut re P^P^P^ ^P^P^
and also .SJJJfJJJ P^P^P^; i.e. the correspondence established "by
makintj each point P x of I correspond to P^P x *P a (or to P x '~P a -Pj},
where P a is a ny fixed point of I distitict from zero, is projective. (A, E)

Proof. The definition of multiplication gives the first of the above
projectivities as the result of two perspectivities (fig. 76) :

The second one is obtained similarly. In fig. 76 we have

The set of all projectivities determined by all choices of P a in the for-
mula Px = P x -P a is the group described in Example 1, p. 69. The proper-
ties of multiplication may be regarded as properties of that group in the same
svay that the properties of addition arise from the group described in Example
2, p. 70. In particular, this furnishes a second proof of the associative law
for multiplication.

THEOREM 5. Multiplication is distributive with respect to addition;
i.e. if P x) P y , P z are any three points on I (for which the operations
"below are defined), we have

P z .(P x +P y )=P s .P I +P,.P y , a nd(Z+P l ).P z =P I .P s + P v .P,. (A,E)
Proof. Place

P x +P y = P x + y , P Z -P^P^ P s -P y = P, y , and P z -P^ = P^+ yy
By Theorem 4 we then have

5 -? ^l-VV^c -i- y "A * ^0 -? *zx *zy2z(x + y)'

But by Theorem 1 we also have (&%%> %>%%+) Hence, by
Theorem 1, Cor., Chap. IV, we have Q^J^JJ, P^P^^ which,
by Theorem 1, implies -^ B -r-^=-^ (a:+ y ) . The relation

is proved similarly.

148 ALQEBEA OF POINTS [CHAP.VI

50. Tlie commutative law for multiplication. With the aid of
Assumption P we will now derive finally the commutative law for
multiplication :

THEOREM 6. TJie operation of multiplication is commutative; i.e.
we hare P x JJ = P y - P for every pair of points P x , P y of I for which
these two products are defined. (A., E, P)

Proof. Let us place as "before P x - P y = P xy , and J - P x = P yc . Then, by
the first relation of Theorem 4, and interchanging the points P^ P^
we have

and from the second relation of the same theorem we have

By Theorem 17, Chap. IV, this requires P^ P^.

In view of the fact already noted, that the fundamental theorem
of protective geometry (Theorem 17, Chap. IT) is equivalent to
Assumption P, the proof just given shows further:

THEOKEM 7. Assumption P is necessary and sufficient for the com-
mutative law for multiplication* (A, E)

51. The inverse operations. DEFINITION. Given two points P a , J%
on Z, the operation determining a point JP satisfying the relation
P^ + P x =iI is called subtraction; in symbols IP a =zP x . The point
P x is called the difference of J? from P a . Subtraction is the inverse of
addition.

The construction for addition may readily be reversed to give a con-
struction for subtraction. The preceding theorems on addition then give :

THEOEEM 8. Subtraction is a one-valued operation for every pair
of points P a > P b on I, except the pair P^ j. (A, E)

COROLLAEY. We have in particular P a P a P Q for every point
P a (^PJonl (A, E)

* The existence of algebras in which multiplication is not commutative is then
sufficient to establish the fact that Assumption P is independent of the previous
Assumptions A and E. For in order to construct a system (cf. p. 6) which satisfies
Assumptions A and E without satisfying Assumption P, we need only construct an
analytic geometry of three dimensions (as described in a later chapter) and use as a
basis a noncommutative number system, e.g. the system of quaternions. That the
fundamental theorem of projective geometry is equivalent to the commutative
law for multiplication was first established by Hilbert, who, in his Foundations of
Geometry, showed that the commutative law is equivalent to the theorem of Pappus
(Theorem 21, Chap. IV). The latter is easily seen to be equivalent to the funda-
mental theorem.

60,31,32] ABSTRACT XU^IBEE SYSTEM 149

DEFINITION. Given two points P a , P b on / ; the point P x determined by
the relation J^-^ = ^ is called the quotient of J% by P a (also the rutio
of P b to P a ) ; in symbols JIJ/JJ = Jf, or J : P a = JJ. The operation deter-
mining j/2 is called diinsio/i; it is the inverse of multiplication.*

The construction for multiplication may also be reversed to give a
construction for division. The preceding theorems on multiplication
then give readily :

THEOREM 9. Division is a one-valued operation for every pair of
points P a , P b on I except the pairs P Q , P Q and .%, -%. (A, E)

COKOLLAHY. We hare in particular PJ% = P^ ./_ = J?, ^/J = ,
etc., for every poi lit P a o>i I distinct from JjJ and 1^. (A, E)

Addition, subtraction, multiplication, and division are known as
the four rational operations.

52. The abstract concept of a number system. Isomorphism. The
relation of the foregoing discussion of the algebra of points on a line
to the foundations of analysis must now -be briefly considered. "With
the aid of the notion of a group (cf. Chap. Ill, p. 66), the general con-
cept of a number system is described simply as follows :

DEFINITION. A set N of elements is said to form a number system,
provided two distinct operations, which we will denote by and
respectively, exist and operate on pairs of elements of N under the
following conditions:

1. The set N forms a group with respect to e.

2. The set N forms a group with respect to o, except that if i + is
the identity element of N with respect to e, no inverse with respect
to o esists for i + .f If a is any element of N, a i+ = ^0 a = i+.

3. Any three elements a, &, c of N satisfy the relations a (b c)
= (5)(ac) and (l@c)ea=(bQa)e(cd).

The elements of a number system are called numbers; the two oper-
ations and are called addition and mult implication respectively.
If a number system forms commutative groups with respect to both
addition and multiplication, the numbers are said to form

* What we have defined is more precisely right-handed division. The left-handed
quotient is defined similarly as the point P x determined by the relation P x P a P&,
In a commutative algebra they are of course equivalent.

t The identity element i+ in a number system is usually denoted by (zero).

$ The class of all ordinary rational numbers forms a field; also the class of real
numbers; and the class of all integers reduced modulo p (p a prime), etc.

150 ALGEBRA OF POINTS [CHAP.YI

On the basis of this definition may be developed all the theory
relating to the rational operations i.e. addition, multiplication, sub-
traction, and division in a number system. The ordinary algebra
of the rational operations applying to the set of ordinary rational or
ordinary real or complex numbers is a special case of such a theory.
TJie whole terminology of tins algebra, in so far as it is definable in
terms of the four rational operations, will in the future be assumed
as defined. We shall not, therefore, stop to define such terms as
reciprocal of a number, exponent, equation, satisfy, solution, root, etc.
The element of a number system represented by a letter as a will be
spoken of as the value of a. A letter which represents any one of a
set of numbers is called a variable; variables will usually be denoted
by the last letters of the alphabet.

Before applying the general definition above to our algebra of
points on a line, it is desirable to introduce the notion of the
abstract equivalence or isomorphism between two number systems.
ox^ If two number systems are such that a one-to-one

reciprocal correspondence exists between the numbers of the two
systems, such that to the sum of any two numbers of one system
there corresponds the sum of the two corresponding numbers of the
other system; and that to the product of any two numbers of one
there corresponds the product of the corresponding numbers of the
other, the two systems are said to be abstractly equivalent or (simply)

When two number systems are isomorphic, if any series of oper-
ations is performed on numbers of one system and the same series
of operations is performed on the corresponding numbers of the
other, the resulting numbers will correspond.

53. Noiihomogeneous coordinates. By comparing the corollaries
of Theorem 1 with the definition of group (p. 66), it is at once
seen that the set of points of a line on which a scale has been estab-
lished, forms a group with respect to addition, provided the point jg
be excluded from the set. In this group J?J is the identity element,
and the existence of an inverse for every element follows from
Theorem 8. In the same way it is seen that the set of points on
a line on which a scale has been established, and from which the

* For the general idea of the isomorphism between groups, see Burnside's Theory
of Groups, p. 22.

53] COOEDIXATES 151

point .g has been excluded, forms a group with respect to multipli-
cation, except that no inverse with respect to multiplication exists
for j?J; P^ is the identity element in this group, and Theorem 9 insures
the existence of an inverse for every point except J. These con-
siderations show that the first two conditions in the definition of a
number system are satisfied by the points of a line, if the operations 9
and o are identified with addition and multiplication as defined in.
48 and 49. The third condition in the definition of a number
system is also satisfied in view of Theorem 5. Finally, in view of
Theorem 1, Cor. 4, and Theorem 6, this number system of points on
a line is commutative with respect to both addition and multipli-
cation. This gives then:

THEOREM 10. TJie set of all points on o, line on which a scale has
Tjeeii established, and from -ich ieli the point J2 is excluded, forms a field
with res'pect to the operations of addition and multiplication preciously
defined. (A, E, P)

This provides a new way of regarding a point, viz.,, that of regarding
a point as a nwnber of a number system. This conception of a point
will apply to any point of a line except the one chosen as J. It is
desirable, however, both on account of the presence of such an excep-
tional point and also for other reasons, to keep the notion of point
distinct from the notion of number, at least nominally. This we do
by introducing a field of numbers a, I, c,- ,l,k,- , %,?/,%, which
is isomorphie witli the field of points on a line. The numbers of the
number field may, as we have seen, be the points of the line, or they
may be mere symbols which combine according to the conditions
specified in the definition of a number system ; or they may be ele-
ments defined in some way in terms of points, lines, etc.*

In any number system the identity element with respect to addi-
tion is called zero and denoted by 0, and the identity element with
respect to multiplication is called one or unity, and is denoted by 1.
We shall, moreover, denote the numbers 1+1, 1+1+1, - - -, a, -
by the usual symbols 2, 3, - - -, a, -.f In the isomorphism of our
system of numbers with the set of points on a line, the point JJ must
correspond to 0, the point JJ to the number 1 ; and, in general, to every

* See, for example, 55, on von Staudt's algebra of throws, where the numbers
are thought of as sets of four points.

t Cf., however, in this connection 57 below.

152 ALQEBEA OF PODTTS [CHAP. VI

point will correspond a number (except to j), and to every number
of the field will correspond a point. In this way every point of the
line (except .) is labeled by a number. This number is called the
(nonhomogeneous) coordinate of the point, to which it corresponds.
This enables us to express relations between points by means of
equations between their coordinates. The coordinates of points, or
the points themselves when we think of them as numbers of a
number system, we will denote by the small letters of the alphabet
(or by numerals), and we shall frequently use the phrase "the point a?"
in place of the longer phrase " the point whose coordinate is x" It
should be noted that this representation of the points of a line by
numbers of a number system -is not in any way dependent on the
commutativity of multiplication; i.e. it holds in the general geom-
etries for which Assumption P is not assumed.

Before leaving the present discussion it seems desirable to point
out that the algebra of points on a line is merely representative,
under the principle of duality, of the algebra of the elements of any
one-dimensional primitive form. Thus three lines Z , l v ! of a flat
pencil determine a scale in the pencil of lines; and three planes
# , a v a:,, of an axial pencil determine a scale in this pencil of planes ;
to each corresponds the same algebra.

54. The analytic expression for a projectivity in a one-dimensional
primitive form. Let a scale be established on a line I by choosing
three arbitrary points for P Q9 J%, , ; and let the resulting field of points
on a line be made isomorphic with a field of numbers 0, 1, a, - , so
that JJ corresponds to 0, ^ to 1, and, in general, P a to a. For the
exceptional point -g, let us introduce a special symbol co with excep-
tional properties, which will be assigned to it as the need arises.
It should be noted here, however, that this new symbol oo does not
represent a number of a field as defined on p. 149.

We may now derive the analytic relation between the coordinates of
the points on I, which expresses a projective correspondence between
these points. Let x be the coordinate of any point of I. We have seen
that if the point whose coordinate is x is made to correspond to either
of the points

(I) 3/=v + a, (a 3= oo)

or (II) #'=a#, (a = 0)

54]

LINEAR FRACTIONAL TRANSFORMATION

153

where a is the coordinate of any given point on /, each of the result-
ing correspondences is projective (Theorem 2 and Theorem 4). It is
readily seen, moreover, that if x is made to correspond to

(in) *'=*,

&,*

the resulting correspondence is likewise projective. For we clearly
have the following construction for the point l/,c (fig. 77): With the
same notation as before for the construction of the product of two

FIG. 77

numbers, let the line xA meet 7. in X. If T is determined as the
intersection of IX with 7 , the line BY determines on Z a point x 1 ,
such that text =1, by definition. "We now have

The three projectivities (I), (II), and (III) are of fundamental
importance, as the next theorem will show. It is therefore desirable
to consider their properties briefly ; we will thus be led to define the
behavior of the exceptional symbol oo with respect to the operations
of addition, subtraction, multiplication, and division.

The projectivity a/= a? + a, from its definition, leaves the point %>,
which we associated with co, invariant. We therefore place oo 4- a = oo
for all values of a (a = oo). This projectivity, moreover, can have no
other invariant point unless it leaves every point invariant ; for the
equation x = x + a gives at once a = 0, if x 3= oo. Further, by prop-
erly choosing a, any point x can be made to correspond to any point a/;

154 ALGEBKA OF POINTS [CHAP. YI

but when one such pair of homologous points is assigned in addition
to the double point cc, the projectivity is completely determined.
The resultant or product of any two projectivities % f =x + a and
#'= + & is clearly a/= x + (a -f &). Two such projectivities are
therefore commutative.

The pro jeetivity d = ax, from its definition, leaves the points and cc
invariant, and by the fundamental theorem (Theorem 17, Chap. IV)
cannot leave any other point invariant without reducing to the iden-
tical projectivity. As another property of the symbol co we have
therefore cc = a cc (a =r= 0). Here, also, by properly choosing a, any
point x can be made to correspond to any point a/, biit then the pro-
jectivity is completely determined. The fundamental theorem in this
case shows, moreover, that any projectivity with the double points 0, co
can be represented by this equation. The product of two projectivities
x r = a:c and xf = 6.r is clearly ic r = (at) x, so that any two projectivities of
this type are also commutative (Theorem 6).

Finally, the projectivity x f =l/x, by its definition, makes the
point co correspond to and the point to co. "We are therefore led
to assign to the symbol co the following further properties : 1/co = 0,
and 1/0 = x. This projectivity leaves 1 and 1 (defined as 1)
invariant. Moreover, it is an involution because the resultant of two
applications of this projectivity is clearly the identity; ie. if the
projectivity is denoted by TT, it satisfies the relation 7r 2 = 1.

THEOREM 11. Any projectivity o/i a line is the product of projec-
timties of the three types (I), (JJ), and (III), and may be expressed
in the form

n\ rf

(1) '

Conversely, every equation of this form represents a projectimty, if
ad-lc^ 0. (A, E, P)

Proof. We will prove the latter part of the theorem first. If we
suppose first that c 3= 0, we may write the equation of the given

transformation in the form

, __ ad

(2) ^.2 + 4.

c cx + d

This shows first that the determinant ad "be must be different from
; otherwise the second term on the right of (2) would vanish, which

54] LIXEAK FEACTIOXAL TEAXSFORMATIOX 155

would make every x correspond to the same point tt/c, while a pro-
jectivity is a one-to-one correspondence. Equation (1), moreover,
shows at once that the correspondence established by it is the result-
ant of the five :

,7 1 /7 ''"^V / "

x l ex, x n = #! + it, x s = > x = ( o -- .f.j, d x -T ---

<>. 2 \ c f c

If c = 0, and ad =?= 0, this argument is readily modified to show that
the transformation of the theorem is the resultant of projectivities of
the types (I) and (II). Since the resultant of any series of projectiv-
ities is a projectivity, this proves the last part of the theorem.

It remains to show that every projectivity can indeed be repre-

sented bv an equation jsf = - - - To do this simplv, it is desirable
- " ex + a

to determine first what point is made to correspond to the point oc by
this projectivity. If we follow the course of this point through the
five projectivities into which we have just resolved this transforma-
tion, it is seen that the first two leave it invariant, the third trans-
forms it into 0, the fourth leaves invariant, and the fifth transforms
it into a/c] the point oo is then transformed by (1) into the point
a/c. This leads us to attribute a further property to the symbol cc,

viz., , 7

a ,

- , when x = cc.

According to the fundamental theorem (Theorem 17, Chap. IV), a pro-
jectivity is completely determined when any three pairs of homolo-
gous points are assigned. Suppose that in a given projectivity the
points 0, 1, oo are transformed into the points p, q> r respectively.
Then the transformation

clearly transforms into p, 1 into gr, and, by virtue of the relation
just developed for oo, it also transforms oo into r. It is, moreover, of the
form of (1). The determinant adlc is in this case (j p)(r q}(r~p)>
which is clearly different from zero, if p, q, r are all distinct. This
transformation is therefore the given projectivity.

COEOLLAEY 1. The projectivity sJ = a/x(a, = 0, or oo) transforms
"into oo and oo into 0. (A, E, P)

156 ALGEBKA OF POINTS [CH^.TI

For it is the resultant of the two projectivities, x l = l/x and
x f = ax v of which the first interchanges and oo, while the second
leaves them both invariant. We are therefore led to define the symbols
a/Q and a/cc as equal to co and respectively, when a is neither
no r oo.

COROLLARY 2. Any projectivity leaving the point co invariant may
1e expressed in the form x f = ax + L (A, E, P)

COROLLARY 3. A/iy projectivity may le expressed analytically "by
the lilinear equation c%3/+ dx f ax~ 6 = 0; and conversely, any
~bilinec.tr equation defines a protective correspondence "between its two
variables. (A, E, P)

COROLLARY 4. If a projectixity leaves any points invariant, the
coordinates of these doulle points satisfy the quadratic equation
car + (da)x-1> = Q. (A, E, P)

DEFINITION. A system of mn numbers arranged in a rectangular
array of w rows and n columns is called a matrix. If m = n, it is
called a square matrix of order n*

The coefficients \ , ) of the protective transformation (1) form a

\c aj * J v '

square matrix of the second order, which may be conveniently used to
denote the transformation. Two matrices ( a , j and ( a f ) repre-

sent the same transformation, if and only if a : a f == b : &'= c : e r = d : d f .
The product of two projectivities

, , . ax + 1 , , , v a f x f + V
' and z" =*7r.(x f )= . . 7 .

is given by the equation

x _ (aa 1 + cV) x + la 1 + db f
-

This leads at once to the rule for the multiplication of matrices,
which is similar to that for determinants.

DEFINITION. The product of two matrices is defined by the equation

(a 1 V\ (a l\ _ (aa! + cV la} + db f \
\c f d f )\c d) \ac f +cd f I

* For a development of the principal properties of matrices, cf . B6cher, Intro-
duction to Higher Algebra, pp. 20 ff.

54,53] THROWS 157

This gives, in connection with the result just derived,
THEOREM 12. The product of two projectmties

is represented ly the product of their matrices; in sytnloh,

COROLLAEY 1. The determinant of the 'product of tv:o 2)rojectivitie$
is equal to the product of their determinants. (A, E,.P)

COROLLARY 2. The inverse of the projectimty 7r~i a \ is given
d ~ l where A, B, C, D are the cofactors

of a, I, c, d respectively in- the determinant , . (A, E, P)

This follows at once from Corollary 3 of the last theorem by inter-
changing x 9 x 1 . We may also verify the relation by forming the

product flrV= [ T 7 * V which transformation is equiva-
\ aa oc/

lent to (~: !j ]. The latter is called the identical matrix.

/ 7% \

COROLLARY 3. Any involution is represented ly ( a ), that is

\c -"" a I

ly x f = 2i , with the condition that a?+t>c^ 0. (A, E, P)
J ex a

55. Von Staudt's algebra of throws. "We wDl now consider the
number system of points on a line from a slightly different point of
view. On p. 60 we defined a throw as consisting of two ordered
pairs of points on a line ; and defined two throws as equal when they
are protective. The class of all throws which are protective (ie. equal)
to a given throw constitutes a class which we shall call a mark.
Every throw determines one and only one mark, but each mark
determines a whole class of throws.

According to the fundamental theorem (Theorem 17, Chap. IV), if
three elements A 9 B, C of a throw and their pkces in the symbol
T(A3, CD) are given, the throw is completely determined by the
mark to which it belongs. A given mark can be denoted by the
symbol of any onfe of the (projective) throws which define it. We
shall also denote marks by the small letters of the alphabet. And so,
since the equa]ity sign (=) indicates that the two symbols between

158 ALGEBEA OF POINTS [CHAP.YI

which it stands denote the same thing, we may write T (AB, CD) =
a = l t if a, I, T(AB, CD) are notations for the same mark. Thus
T (AB, CD) = T (BA 9 DC) = T (CD, AB) = T(DC, BA) are all symbols
denoting the same mark (Theorem 2, Chap. III).

According to the original definition of a throw the four elements
which compose it must be distinct. The term is now to be extended
to include the following sets of two ordered pairs, where A, B, C are
distinct. The set of all throws of the type 7(AB, CA) is called a
mark and denoted by <x> ; the set of all throws of the type T(AB, CB)
is called a mark and is denoted by ; the set of all throws of the type
T(AB, CC) is a mark and is denoted by 1. It is readily seen that
if %, %> j are any three points of a line, there exists for every point
P of the line a unique throw T (_? JJ, JJ, P) of the line ; and con-
versely, for every mark there is a unique point P. The mark co, by
what precedes, corresponds to the point R> ; the mark to JJ ; and
the mark 1 to J?.

DEFINITION. Let T(AB, CDJ be a throw of the mark a, and let
T(AB, <7D 2 ) be a throw of the mark &; then, if Z> 3 is determined by
iB, ADJ>^ 9 the mark c of the throw T(AB 9 CD S ) is called the
of the marks a and &, and is denoted by a + b; in symbols,
a -f- J = c. Also, the point D[ determined by Q (AD^C y BDJD^ deter-
mines a mark with the symbol T(AB, CD f B ) = c r (say), which is called
the product of the marks a> and 6 ; in symbols, ab = c 7 . As to the
marks and 1, to which these two definitions do not apply, we define
further : #-f-0 = + # = , ^-0 = 0-^ = 0, and a - 1 = 1 a = a.

Since any three distinct points A } B, C may be projected into a fixed
triple j, ^ JJ, it follows that the operation of adding or multiplying
marks may be performed on their representative throws of the form
T(%>PQ> JiP)- By reference to Theorems 1 and 3 it is then clear that
the class of all marks on a line (except oo) forms a number system, with
respect to the operations of addition and multiplication just defined,
which is isomorphic with the number system of points previously
developed.

This is, in brief, the method used by von Staudt to introduce ana-
lytic methods into geometry on a purely geometric basis.* "We have

* Cf . reference on p. 141. Von Staudt used the notion of an involution on a line
in defining addition and multiplication ; the definition in terms of quadrangular sets
is, however, essentially the same as his by virtue of Theorem 27, Chap. IV.

5o,oG] CROSS RATIO 159

given it here partly on account of its historical importance; partly
because it gives a concrete example of a number system isoinurpliic
with the points of a line * ; and partly because it gives a natural
introduction to the fundamental concept of the cross ratio uf four
points. This we proceed to derive in the next section.

56. Ttie cross ratio. We have seen in the preceding section that
it is possible to associate a number with every throw of four points
on a line. By duality all the developments of this section apply also
to the other one-dimensional primitive forms, i.e. the pencil of lines
and the pencil of planes. TTith every throw of four elements of any
one-dimensional primitive form there may be associated a definite
number, which must be the same for every throw protective with the
first, and is therefore an invariant under any protective transforma-
tion, i.e. a property of the throw that is not changed when the throw
is replaced by any protective throw. This number is called the cross
ratio of the throw. It is also called the doulle ratio or the anhar-
monic ratio. The reason for these names will appear presently.

In general, four given points give rise to six different cross ratios.
For the 24 possible permutations of the letters in the symbol
T(AB, CD} fall into sets of four which, by virtue of Theorem 2,
Chap. Ill, have the same cross ratios. In the array below, the per-
mutations in any line are protective with each other, two permuta-
tions of different lines being in general not projective :

AB, CD

BA, DC

DC, BA

CD, AB

AB, DC

BA, CD

CD, BA

DC, AB

AC, BD

CA, DB

DB, CA

BD, AC

AC, DB

CA, BD

BD, CA

DB, AC

AD, BC

DA, CB

.CB, DA

BC, AD

AD, CB

DA, BC

BC, DA

CB, AD

If, however, the four points form a harmonic set H (AB, CD), the
throws T(AB, CD) and T(AB, DC) are projective (Theorem 5,
Cor. 2, Chap. IV). In this case the permutations in the first two rows
of the array just given are all projective and hence have the same cross
ratio. The four elements of a harmonic set, therefore, give rise to only
three crostTratios. The values of these cross ratios are readily seen

* Cf. 53. Here, with every point of a line on which a scale has been estab-
lished, is associated a mark which Js the coordinate of the point.

160 ALQEBEA OF POINTS [CHAP.TI

to be 1, J-, 2 respectively, for the constructions of our number
system give* at once H (gJJ, JJP 1 ) 9 H (JJ, 1J1J), and H ( jfj, JJJ>).

We now proceed to develop an analytic expression for the cross
ratio "Be, (# r r a , # 3 : 4 ) of any four points on a line (or, in general, of any
four elements of any one-dimensional primitive form) whose coordi-
nates in a given scale are given. It seems desirable to precede this
derivation by an explicit definition of this cross ratio, which is inde-
pendent of von Staudt's algebra of throws.

DEFINITION. The cross ratio I&.fax^ x 9 x^ of elements x v # a , # 8 , x
of any one-dimensional form is, if x 19 # a , x z are distinct, the coordi-
nate X of the element of the form into which # 4 is transformed by
the projectivity which transforms x v x s , x s into oo, 0, 1 respectively ;
i.e. the number, X, defined by the projectivity o^oy^-^coOlX. If
two of the elements x v # 2 , # 8 coincide and x is distinct from all of
them, we define E (x^, x 3 x) as that one of B (x 2 x 19 x^), B (# 8 # 4 ,
jTjA'j,), Ii (j: 4 r 8 , ayj, for which the first three elements are distinct.

THEOREM 13. TJie cross ratio B (x^x^ x z x^ of the four dements
whose coordinates are respectively x v % 2 , x s> x is given ~by the relation

\ I "~~"

(A, E, P)

Proof, The transformation

is evidently a projectivity, since it is reducible to the form of a
linear fractional transformation, viz.,

in which the determinant (a^ c s ) (# 2 ^ S )(cc 3 ^ is not zero, pro-
vided the points x v % Q> x s are distinct. This projectivity transforms
x v x 2 , X B into oo, 0, 1 respectively. By definition, therefore, this pro-
jectivity transforms x into the point whose coordinate is the cross
ratio in question, Le. into the expression given in the theorem. If
x v x 2 , X B are not all distinct, replace the symbol 1$ (ayj a , x z x^) by one
of its equal cross ratios B? (x z x v x^), etc. ; one of these must have
the first three elements of the symbol distinct, since in a cross ratio
of four points at least three must be distinct (def.).

56] OEOSS BATIO 161

COROLLARY 1. We have in particular

B Uy>: 2 , ,0,'j) = cc, B (.<.y\, V*.' 2 ) ^ an $ ^ ( i W W = 1
-z/^, c 2 , # 3 are fl/iy rtra? distitid dements of the form. (A, E)

COROLLARY 2. T&e ?*0s$ /Y^/O ^/ hanno/iic set H(ay: 2 , a.y 4 ) Z6 5
B (^yj 2 , iy; 4 ) == 1, for we have H (ac 0, 1 1 ). (A, E s P)

"COROLLARY 3. J/ R (x*^ ^v^) = X, rt^ other five cross ratios of the
throws composed of the fovr dements x v # 2 , .r s , A' 4 /?

5 (' Vs ;<: A) = ' E ( V ! * A v a) ^ ~- '

,,, 4fl
(A, E, P)

The proof is left as an exercise.

COROLLARY 4. If x v # 2 , C 3 , x form a harmonic set H (j/^,

(A, E, P)

The proof is left as an exercise.

COROLLARY 5. If a, I, c are any three distinct elements of a one-
dimensional primitive form, and a 1 , b', d are any three other distinct
elements of the same form, then the correspondence established ~by the
relation E (ab 9 ex) = B (aV, c ! x r ) is projective. (A, E, P)

Proof. Analytically this relation gives

a c b x__a r c f Vx*
a x 1) c a f x f U c f

which, when expanded, evidently leads to a bilinear equation in
the variables x, x f , which defines a projective correspondence by
Theorem 11, Cor. 3.
That the cross ratio

is invariant under any projective transformation may also be verified directly
by observing that each of the three types (I), (II), (III) of projectivities on
pp. 152, 153 leaves it invariant. That every projectivity leaves it invariant
then follows from Theorem 11.

162 ALGEBRA OF POINTS [CHAP.YI

57. Coordinates in a net of rationality on a line. We now con-
sider the numbers associated with the points of a net of rationality
on a line. The connection between the developments of this chapter
and the notion of a linear net of rationality is contained in the
following theorem :

THEOREM 14. The coordinates of the points of the net of rationality
R(JJJ.B)/0m a number system, or field, which consists of all numbers
each of tchich can be obtained by a finite number of rational algebraic
operations o/i and 1, and only these. (A, E)

Proof. By Theorem 14, Chap. IV, the linear net is a line of the
rational space constituted by the points of a three-dimensional net of
rationality. By Theorem 20, Chap. IV, this three-dimensional net is
a properly protective space. Hence, by Theorem 10 of the present
chapter, the numbers associated with R(Oloo) form a field.

All numbers obtainable from and 1 by the operations of addi-
tion, subtraction, multiplication, and division are in R(Olco), because
(Theorem 9, Chap. IV) whenever x and y are in R (Oloo) the quadran-
gular sets determining x + y, xy, % y, x/y have five out of six
elements in R(Oloc). On the other hand, every number of R(Oloo)
can be obtained by a finite number of these operations. This follows
from the fact that the harmonic conjugate of any point a in R(Oloo)
with respect to two others, 5, c, can be obtained by a finite number
of rational operations on a, I, c. This fact is a consequence of Theo-
rem 13, Cor. 2, which shows that x is connected with a, b, c by the

relation

(x b) (a c) -f (x c) (a b) = 0.

Solving this equation for x, we have

2 be ab ac

a number * which is clearly the result of a finite number of rational
operations on a, 1 9 c. This completes the proof of the theorem. We
have here the reason for the term net of rationality.

It is well to recall at this point that our assumptions are not yet sufficient
to identify the numbers associated with a net of rationality with the system
of all ordinary rational numbers. We need only recall the example of the
miniature geometry described in the Introduction, 2, which contained only

* The expression for x cannot be indeterminate unless & = c.

57, os] HOMOGENEOUS COORDINATES 163

three points on a line. If in that triple-system geometry we perform the con-
struction for the number 1 + 1 on any line in which we have assigned the
numbers 0, 1, to the three points of the Hue in any way, it will be found
that this construction yields the point 0. This, is due to the fact previously
noted that in that geometry the diagonal points of a complete quadrangle
are collinear. In every geometry to which Assumptions A, E, P apply we

may construct the points 1 + 1, 1 + 1 + 1 thus forming a sequence of

points which, with the usual notation for these sums, we may denote by 0, 1,
2, 3, 4, .--. Two possibilities then present themselves: either the points,
thus obtained are all distinct, in ^liich case the net R(Olcc) contains all the
ordinary rational numbers ; or some point of this sequence coincides with one
of the preceding points of the sequence, in which case the number of points
in a net of rationality is finite. We shall consider this situation in detail in
a later chapter, and will then add further assumptions. Here it should be
emphasized that our results hitherto, and all subsequent results depending only
on Assumptions A, E, P, are valid not only in the ordinary real or complex
geometries, but in a much more general class of spaces, which are character-
ized merely by the fact that the coordinates of the points on a line are the
numbers of a field, finite or infinite.

58. Homogeneous coordinates on a line. The exceptional character
of the point J2, as the coordinate of which we introduced a symbol
co with exceptional properties, often proves troublesome, and is, more-
over, contrary to the spirit of protective geometry in which the points
of a line are all equivalent ; indeed, the choice of the point & was
entirely arbitrary. It is exceptional only in its relation to the opera-
tions of addition, multiplication, etc., which we have defined in terms
of it. In this section we will describe another method of denoting
points on a line by numbers, whereby it is not necessary to use any
exceptional symbol.

As before, let a scale be established on a line by choosing any three
points to be the points P^ J?, R ; and let each point of the line be
denoted by its (nonhomogeneous) coordinate in a number system
isomorphic with the points of the line. We will now associate with
every point a pair of numbers (x v # 2 ) of this system in a given order,
such that if x is the (nonhomogeneous) coordinate of any point dis-
tinct from Jg,, the pair (x v x^f associated with the point is satisfies the
relation x = xj%y. TVith the point R we associate any pair of the
form (&, 0), where k is any number (&=0) of the number system
isomorphic with the line. To every jwint of ilic line corresponds a pair
of numbers, and to every pair of numbers in the field, except fJie pair

164 ALGEBPtA OF POINTS [CHAP.TI

(0, 0), corresponds a unique poiiit of the line. These two numbers are
called homogeneous coordinates of the point with which they are
associated, and the pair of numbers is said to represent the point.
This representation of points on a line by pairs of numbers is not
unique, since only the ratio of the two coordinates is determined;
Le. the pairs (x r r 2 ) and (mx v mx 2 ) represent the same point for all
values of m different from 0. The point P Q is characterized by the
fact that ^ = ; the point H by the fact that # 2 = ; and the point
% by the fact that x\ x y

THEOREM 15. Li homogeneous coordinates a projectwity on a line is
represented by a linear homogeneous transformation in two variables,

(1) p*l~**i + lp, (orf-fc^O)

V pZJ = 0^+. d%,

where p is an arbitrary factor of proportionality. (A, E, P)
Proof. By division, this clearly leads to the transformation

cx + d

provided x^ and # 2 are both different from 0. If # 3 = 0, the trans-
formation (1) gives the point (a?/, a?/) = (a t c) ; i.e. the point H =
(1, 0) is transformed by (1) into the point whose nonhomogeneous
coordinate is a/c. And if # 2 '=0, we have in (1) (x p % 2 ) = (d, c);
i.e. (1) transforms the point whose nonhomogeneous coordinate is
rf/e into the point R. By reference to Theorem 11 the validity
of the theorem is therefore established.

As before, the matrix ( a , ) of the coefficients may conveniently

be used to represent the projectivity. The double points of the pro-
jectivity, if existent, are obtained in homogeneous coordinates as
follows : The coordinates of a double point (& v # 2 ) must satisfy the
equations px ^ = aXl + bz z ,

px z =cx : +dXy
These equations are compatible only if the determinant of the system

vanishes. This leads to the equation

c d *~~ p

58]

HOMOGENEOUS COORDINATES

165

for the determination of the factor of proportionality p. This equa-
tion is called the characteristic equation of the matrix representing
the projeetivity. Every value of p satisfying this equation then leads
to a double point when substituted in one of the equations (3) ; viz.,
if p 1 be a solution of the characteristic equation, the point

( t /: r :c z ) = (- 5, a - pj = (d - p v - c)
is a double point."*

In homogeneous coordinates the cross ratio TkiAB, CD) of four
points A = (a v 2 ), B = (l v & 2 ), C = (c v <g, D = (d v rf 2 ) is given by

(ad) (bd)

where the expressions (ac) 9 etc., are used as abbreviations for Q-fia l9
etc. This statement is readily verified by writing down the above
ratio in terms of the nonhomogeneous coordinates of the four points.
We will close this section by giving to the two homogeneous coor-
dinates of a point on a line an explicit geometrical significance. In
view of the fact that the coordinates of such a point are not uniquely
determined, a factor of proportionality being entirely arbitrary, there
may be many such interpretations. On account of the existence of
this arbitrary factor, we may impose a further condition on the coor-
dinates (x v # 2 ) of a point, in addition to the defining relation xjx^x,
where x is the nonhomogeneous coordinate of the point in question,
We choose the relation x^+ # 2 = 1. If this relation is satisfied,

= B( 10, oo a?),

= B( loo, 00).

Thus homogeneous coordinates subject to the condition o^ 4- # 2 = 1
can be defined by choosing three points A 9 B, C arbitrarily, and letting
x l = & (AB, CX} and a? 2 = B (AC, BX). The ordinary homogeneous
coordinates would then be defined as any two numbers proportional
to these two cross ratios.

* This point is indeterminate only if 5 = c = and a = d. The projectiyity is
then the identity.

1 -1

#! X 2

! SS 2

I -I

1 -1

166 ALGEBRA OE POINTS [CHAP. VI

59. Projective correspondence between the points of two different
lines. Hitherto we have confined ourselves, in the development of
analytic methods, to the points of a single line, or, under duality, to
the elements of a single one-dimensional primitive form. Suppose
now that we have two lines I and m with a scale on each, and let
the nonhomogeneous coordinate of any point of I be represented by
x, and that of any point of m by y. The question then arises as to
how a projective correspondence between the point x and the point y
may be expressed analytically. It is necessary, first of all, to give a
meaning to the equation y = x. In other words : What is meant by say-
ing that two points $ on /, and y on m have the same coordinate ?
The coordinate a? is a number of a field and corresponds to the point
of which it is the coordinate in an isomorphism of this field with the
field of points on the line 1. "We may think of this same field of
numbers as isomorphic with the field of points on the line m. In
bringing about this isomorphism nothing has been specified except
that the fundamental points J, Q . determining the scale on m
must correspond to the numbers 0, 1 and the symbol oo respectively.
If the correspondence between the points of the line and the numbers
of the field were entirely determined by the respective correspond-
ences of the points ^, JJ, R> just mentioned, then we should know
precisely what points on the two lines I and m have the same coor-
dinates. It is not true of all fields, however, that this correspondence
is uniquely determined when the points corresponding to 0, 1, oo are
assigned.* It is necessary, therefore, to specify more definitely how
the isomorphism between the points of m and the numbers of the
field is brought about. One way to bring it about is to make use of
the projectivity which carries the fundamental points 0, 1, oo of I
into the fundamental points 0, 1, cc of m, and to assign the coordinate
& of any point A of I to that point of m into which A is transformed
by this projectivity. In this projectivity pairs of homologous points
will then have the same coordinates. That the field of points and the
field of numbers are indeed made isomorphic by this process follows
directly from Theorems 1 and 3 in connection with Theorem 1, Cor.,
Chap. IV. TFe may now readily prove the following theorem :

* This is shown by the fact that the field of all ordinary complex numbers can
be isomorphic with itself not only by making each number correspond to itself, but
also by making each number a + ib> correspond to its conjugate a i&.

59] EXERCISES 167

THEOREM 16. Any projecthe correspondence between the points [:e]
and [y] of two distinct lines may le represented analytically by tlu
relation y x 'by properly choosing the coordinates on the two lines.
If the coordinates on the two lines are so related that tltc relation
y = x represents a projectile correspondence, then any projective cor-
respondence between the points of the tico lines is given by a relation

CM + T> ,11

y = _ , f cu ( fa -fc Q\

C& + d

(A, E, P)

Proof. The first part of the theorem follows at once from the pre-
ceding discussion, since any projectivity is determined by three pairs
of homologous points, and any three points of either line may be
chosen for the fundamental points. In fact, we may represent any
projectivity between the points of the two lines by the relation y = x,
by choosing the fundamental points on I arbitrarily; the fundamental
points on m are then uniquely determined. To prove the second part
of the theorem, let TT be any given projective transformation of the
points of the line I into those of m, and let TT O be the projectivity
y sc, regarded as a transformation from m to /. The resultant
TTJIT = ^ is a projectivity on ?, and may therefore be represented by
se/ = (ax + 1) /(ex + d). Since TT = ir~ V a , this gives readily the result
that TT may be represented by the relation given in the theorem.

EXERCISES

1. Give constructions for subtraction and division in the algebra of points
on a line.

2. Give constructions for the sum and the product of two lines of a pencil
of lines in which a scale has been established.

3. Develop the point algebra on a line by using the properties expressed in
Theorems 2 and 4 as the definitions of addition and multiplication respec-
tively. Is it necessary to use Assumption P from the beginning?

4. Using Cor. 3 of Theorem 9, Chap. Ill, show that addition and multi-
plication may be defined as follows : As before, choose three points P , P v
P on a line Z as fundamental points, and let any line through Poo be labeled
2,0. Then the sum of two numbers P x and P y is the point P x+ y into which P v
is transformed by the elation with axis I* and center P m which transforms
P into P.,.; and the product P^-Pj, is the point P^ into which P y is trans-
formed by the homology with axis Z w and center P which transforms P l into
P.,,. Develop the point algebra on this basis without using Assumption P,
except in the proof of the commutativity of multiplication.

168 ALGEBRA OF POIXTS [CHAP.TI

5. If the relation ax = ly holds between four points a, 5, x, y of a line,
show that we have Q(0fca, X#JT). Is Assumption P necessary for this result?

6. Prove l>v direct computation that the expression -^ ^ : -8 % i s

fc *>* 7* 7* - **

*1 -^ *g ^4

unchanged in value when the four points x v x 2 , # s? o? 4 are subjected to any-
linear fractional transformation u/ =

cx + d

7. Prove that the transformations

X' = X, X' = ~, X' = 1-X, X'=-i-r, *' = r^7' *' = ^

A I A A "" JL A

form a group. What are the periods of the various transformations of this
group? (Cf. Theorem 13, Cor. 3.)

8. If A, B, C, P v P 2 , , P n are any n -f 3 points of a line, show that
every cross ratio of any four of these points can be expressed rationally in
terms of the n cross ratios A, = B (AB< CP ), i = 1, 2, , n. When n = 1
this reduces to Theorem 13, Cor. 3. Discuss in detail the case n = 2.

9. If B: (a^o, x^x 4 ) = X, show that

1-X = 1 X

^-^"^-^2 *3-*l'

The relation of Cor. 3 of Theorem 13 is a special case of this relation.

10. Show that if R (AB, CD) = E (AB< DC), the points form a harmonic

set H (-43, CD).

11. If the cross ratio E (.15, CD) = X satisfies the equation X 2 -X + 1 = 0,

then B (.IB, CD) = B (4 C 5 D) = B (AD, JJC) = X,

and B (AB.DC) = B (-4 C, BD) = B (4D, CB) = - X 2 .

12. If A, B, T, F, Z are any five points on a line, show that

B (-4B, AT) B (4B, rZ) B (-4B, ZZ) = 1.

13. State the corollaries of Theorem 11 in homogeneous coordinates.

14. By direct computation show that the two methods of determining the
double points of a projectivity described in 54 and 58 are equivalent.

15. If Q(lBC,ATZ),then

B (AX, FC) + B (BY, ZA) -f B (CZ 9 ZB) = 1.

16. If M v Jf 2 , J/ 3 are any three points in the plane of a line I but not on
L the cross ratios of the lines ?, -PJ/p PM^ PM Z are different for any two
points P on L

17. If A , B are any two fixed points on a line Z, and X, F are two variable
points such that B (AB, XY) is constant, the set [Z j is projective with the
set [F].

CHAPTER VII

COORDINATE SYSTEMS IN TWO- AND THREE-DIMENSIONAL*

FORMS

60. Nonhomogeneotts coordinates in a plane. In order to repre-
sent the points and lines of a plane analytically we proceed as follows :
Choose any two distinct lines of the plane, which we will call the
axes of coordinates, and determine on each a scale (48) arbitrarily,
except that the point of intersection of the lines shall be the
0-point on each scale (fig. 78). This point we call the origin. Denote
the fundamental
points on one of
the lines, which
we call the x-axis }
by O a , I,,., oOp ; and
on the other line,
which we will call
the y-axis, by O y ,
l y , oo y . Let the

line cc^cc, be de- ^ Tg

noted by Z.

Now let P be any point in the plane not on Z. Let the lines Pcc y
and Poo,,, meet the #-axis and the y-axis in points whose nonhornoge-
neous coordinates are a and ~b respectively, in the scales just estab-
lished. The two numbers a, "b uniquely determine and are uniquely
determined by the point P. Thus every point in the plane not on l m
is represented by a pair of numbers ; and, conversely, every pair of
numbers of which one belongs to the scale on the as-axis and the
other to the scale on the y-axis determines a point in the plane (the
pair of symbols co^ oo y being excluded). The exceptional character
of the points on l m mil be removed presently ( 63) by considera-
tions similar to those used to remove the exceptional character of

*A11 the developments of this chapter are on the basis of Assumptions
A,E,P.

169

170 COORDINATE SYSTEMS [CHAP, vn

the point cc in the case of the analytic treatment of the points of a
line ( 58). The two numbers just described, determining the point
P, are called the nonhomogencous coordinates of P with reference to

FIG. 79

the two scales on the x- and the ^-axes. The point P is then repre-
sented analytically by the symbol ( , 6). The number a is called the
^-coordinate or the abscissa of the point, and is always written first
in the symbol representing the point; the number & is called the
^/-coordinate or the ordinate of the point, and is always written last
in this symbol.

The plane dual of the process just described leads to the corre-
sponding analytic representation of a line in the plane. For this pur-
pose, choose any two distinct points in the plane, which we will call
the centers of coordinates ; and in each of the pencils of lines with
these centers determine a scale arbitrarily, except that the line o join-
ing the two points shall be the 0-line in each scale. This line we call
the origin. Denote the fundamental lines on one of the points, which
we will call the u-center, by O tt , l a , o> tt ; and on the other point, which
we will call the v-center, by O p , l r , cc p . Let the point of intersection
of the lines oo w , oo r be denoted by & (fig. 79).

Sow let I be any line in the plane not on R. Let the points lco v
and /oc tt be on the lines of the ^-center and the ^-center, whose non-
homogeneous coordinates are m and n respectively in the scales just
established. The two numbers m, n uniquely determine and are
uniquely determined by the line L Thus every line in the plane not
on is represented by a pair of numbers ; and, conversely, every pair
of numbers of which one belongs to the scale on the ^-center and the
other to the scale on the ^-center determines a line in the plane (the
pair of symbols OO M , oo p being excluded). The exceptional character

60, 01]

COORDINATES IS A PLANE

171

of the lines on j will also be removed presently. The two numbers
just described, determining the line /, are called the nonhomogcueous
coordinates of I with reference to the two scales on the it- and
^-centers. The line I is then represented analytically by the symbol
\m, %]. The number m is called the u-co'drdinate of the line, and is
always written first in the symbol just given ; the number 11 is called
the ^-coordinate of the line, and is always written second in this
symbol. A variable point of the plane will frequently be represented
by the symbol (x 9 y) ; a variable line by the symbol [u, r]. The coor-
dinates of a point referred to two axes are called point coordinates ;
the coordinates of a line referred to two centers are called line coor-
dinates. The line Z M and the point R are called the singular line and
the singular point respectively.

61* Simultaneous point and line coordinates. In developing further
our analytic methods we must agree upon a convenient relation
between the axes and centers of the point and line coordinates respec-
tively. Let us consider any triangle in the plane, say with vertices

0, Z7, V. let the lines OU and OF be the y- and 0-axes respectively,
and in establishing the scales on these axes let the points 7, V be
the points cc y , co a respectively (fig. 80), "Further, let the points J7, 7
be the ^-center and the ^-center respectively, and in establishing the

172 COORDINATE SYSTEMS [CHAP, vn

scales on these centers let the lines UO, TO be the lines co u , cc v
respectively. The scales are now established except for the choice of
the 1 points or lines in each scale. Let us choose arbitrarily a point
I,, on the #-axis and a point l y on the y-axis (distinct, of course, from
the points 0, U, F). The scales on the axes now being determined,
we determine the scales on the centers as follows : Let the line on
V and the point l x on the a-axis be the line l u ; and let the line
on F and the point l v on the y-axis be the line l p . All the scales
are now fixed. Let T be the projectivity (59, Chap. VI) between
the points of the i'-axis and the lines of the w-center in which points
and lines correspond when their x- and z^-coordinates respectively
are the same. If TT' is the perspeetivity in which every line on the
^-center corresponds to the point in which it meets the #-axis, the
product TT'TT transforms the <?>axis into itself and interchanges and
GO.,, and I, and l x . Hence TrV is the involution a/ = 1/#. Hence
it follows that the line on U whose coordinate is u is on the point of
the x-axis whose coordinate is 1/u; and the point on the x-axis
whose coordinate is x is oil the line of the u-center whose coordinate
is I/a*. This is the relation between the scales on the #-axis and
the 2^-center.

Similar considerations with reference to the y-axis and the -^-center
lead to the corresponding result in this case : The line on V 'whose coor-
dinate is v is on that point of the y-axis whose coordinate is 1/v;
and the point of the y-axis whose coordinate is y is on that line of the
v-ceiiter whose coordinate is 1/y.

62. Condition that a point be on a line. Suppose that, referred to
a system of point-and-line coordinates described above, a point P has
coordinates (a, V) and a line / has coordinates \m, n]. The condition
that P be on I is now readily obtainable. Let us suppose, first, that
none of the coordinates a, I, m, n are zero. We may proceed in either
one of two dual ways. Adopting one of these, we know from the
results of the preceding section that the line [m, n] meets the #-axis
in a point whose ^-coordinate is 1/m, and meets the 7/-axis in a
point whose ^-coordinate is 1/n (fig. 81). Also, by definition, the
line joining P = (a, 1) to V meets the #-axis in a point whose ^-coor-
dinate is a ; and the line joining P to Z7 meets the y-axis in a point
whose y-coorclinate is &. If P is on I, we clearly have the following
perspeetivity :

62]

COOEDINATES IN A PLANE

173

=
A

(1)
Hence we have

which, when expanded (Theorem 13, Chap. VI), gives for the desired
condition

(3) ma + nl + 1 = 0.

This condition has been shown to be necessary. It is also sufficient,
for, if it is satisfied, relation (2) must hold, and hence would follow
(Theorem 13, Cor. 5, Chap. TI)

-- Oacc x j- -- Oac A
m A n v

But since this projectivity has the self-corresponding element 0, it
is a perspectivity which leads to relation (1). But this implies that
P is on L

FIG. 81

If now & = (&= 0), we have at once 5 = l/n ; and if & = (a^ 0),
we have likewise a = 1/m for the condition that P be on But
each of these relations is equivalent to (3) when a = and J =
respectively. The combination a = 0, 5 = gives the origin which
is never on a line [m, ?i] where m^O^n. It follows in the same
way directly from the definition that relation (3) gives the desired
condition^ if we have either m = or n = 0. The condition (3) is
then valid for all cases, and we have

174 COOEDIXATE SYSTEMS [CHAP.VH

THEOREM 1. The necessary and sufficient condition that a point
P = (a, 6) le on a line I = [?n, n] is that the relation ma + ^5 + 1 =
be satisfied.

DEFINITION. The equation DEFINITION. The equation
which is satisfied by the eoordi- which is satisfied by the coordi-
nates of all the points on a given nates of all the lines on a given
line and no others is called the point and no others is called the
point equation of the line. line equation of the point.

COEOLLARY 1. The point equa- COROLLAEY 1'. The line equa-
tion of the line [m, n] is tion of the point (a, V) is
m + ny + 1 = 0. au + bv + 1 = 0.

EXERCISE
Derive the condition of Theorem 1 by dualizing the proof given.

63. Homogeneous coordinates in the plane. In the analytic repre-
sentation of points and lines developed in the preceding sections the
points on the line Z7F=0 and the lines on the point were left
unconsidered. To remove the exceptional character of these points
and lines, we may recall that in the case of a similar problem in the
analytic representation of the elements of a one-dimensional form we
found it convenient to replace the nonhomogeneous coordinate x of
a point on a line by a pair of numbers x v # 2 whose ratio xjx^ was
equal to x(x = oo), and such that x z = when x = oo.

A similar system of homogeneous coordinates can be established for
the plane. Denote the vertices 0, U 9 V of any triangle, which we will
call the triangle of reference, by the " coordinates " (0, 0, 1), (0, 1, 0),
(1, 0, 0) respectively, and an arbitrary point T, not on a side of the
triangle of reference, by (1, 1, 1). The complete quadrangle OUVT
is called the frame of reference * of the system of coordinates to be
established. The three lines UT, VT, OT meet the other sides of the
triangle of reference in points which we denote by l x = (1, 0, 1),
1 F =(0, 1, 1), 1,='(1, 1, 0) respectively (fig. 82).

We will now show how it is possible to denote every point in the
plane by a set of coordinates (x v # 2 , # 8 ). Observe first that we have
thus far determined three points on each of the sides of the triangle

* Frame of reference is a general term that may be applied to the fundamental
elements of any coordinate system.

63] HOMOGENEOUS COORDINATES 175

of reference, viz.: (0, 0, 1), (0, 1, 1), (0, 1, 0) on OU; (0, 0, 1), (1, 0, 1),
(1, 0, 0) on OF; and (0, 1, 0), (1, 1, 0), (1, 0, 0) on CT. The coordi-
nates which we have assigned to these points are all of the form
(x v # 2 , # 3 ). The three points on OU are characterized by the fact that
x l = 0. Fixing attention on the remaining coordinates, we choose the
points (0, 0, 1), (0, 1, 1), (67 X 0} as the fundamental points (0, I)/
(1, 1), (1, 0) of a system of homogeneous coordinates on the line OU.
If in this system a point has coordinates (/, 7/1), we denote it in our
planar system by (0, 1, m). In like manner, to the points of the other
two sides of the triangle of reference may be assigned coordinates of
the form (Is, 0, m) and (k, I, 0) respectively. We have thus assigned
coordinates of the form (x v x^ r s ) to aH the points of the sides of the
triangle of reference. Moreover, the coordinates of every point on
these sides satisfy one of the three relations 2^= 0, ^ 2 = 0, # 3 = 0.

Now let P be any point in the plane not on a side of the triangle
of reference. P is uniquely determined if the coordinates of its pro-
jections from any two of the vertices of the triangle of reference on
the opposite sides are known. Let its projections from U and V on
the sides OY and OU be (k, 0, n) and (0, 1', n r ) respectively. Since
under the hypothesis none of the numbers k> %, l f , n f is zero, it is
clearly possible to choose three numbers (x v % 2 , x z ) such that x l : x 3
= &:?&, and # a : x z = V : n f . We may then denote P by the coordinates
(x v x# x s ). To make this system of coordinates effective, however,
we must show that the same set of three numbers (x v x & x & ) can be
obtained by projecting P on any other pair of sides of the triangle
of reference. In other words, we must show that the projection of
P = (x^ x# x s ) from on the line TJY is the point (x v x 2 , 0). Since
this is clearly true of the point jP = (l, 1, 1), we assume P distinct
from 51 Since the numbers # 1} # 2 , % 3 are all different from 0, let us
place x 1 : x z = a?, and # 2 : # 3 = y, so that x and y are the nonhomoge-
neous coordinates of (x v 0, # 3 ) and (0, # 2 , x s ) respectively in the scales
on 07 and OU defined by = O x , l a , F= oo r and = O y , l y , U= ay
Finally, let OP meet U Y in the point whose nonhomogeneous coor-
dinate in the scale defined by ?7= O z , l z , F= co z is %} and let OP
meet the line 1 X U in A. We now have

* * * yf * z - J y^y y >

176 COOBDIKATE SYSTEMS [CHAP.VH

where C is the point in which TA meets OU. This projectivity
between the lines UV and OU transforms O s into co y , cc s into O y , and
1 3 into l y . It follows that C has the coordinate 1/2 in the scale on
Or. We have also

which gives

= B

Substituting x = ^ : # 3 , and ?/ = # 3 : <? 3 , this gives the desired relation
z = x t : </.'
follows :

z = x t : </.' 2 . The results of this discussion may be summarized as

FIG. 82

THEOREM 2. DEFINITION. If P is any point not on a side of the
triangle of reference OUV, there exist three numbers x v x z > x s ( a ^ Dif-
ferent from 0) such that the projections of P from the vertices 0.17.
T on the opposite sides have coordinates (x v x^ 0), (x v 0, # 8 ), (0, # 2 , # 8 )
respectively. TJiese three numbers are called the homogeneous coordf-
"nafes^qflP, and P is denoted ly (x v x^ x^ Any set of three numbers
(not all equarto'Q) determine uniquely a point whose (homogeneous)
coordinates they are.

The truth of the last sentence in the above theorem follows from
the fact that, if one of the coordinates is 0, they determine uniquely
a point on one of the sides of the triangle of reference ; whereas, if
none is equal to 0, the lines joining U to (x v 0, # 8 ) and V to (0, # 2 ,j*! 8 )
meet in a point whose coordinates by the reasoning above are (x v x y xj).

63] HOMOGENEOUS COOEDDsATES ITT

COKOLLAKY. The coordinates (x v 2 2 , z s ) and (l\c v k% 2 , hc 2 ) tktcmiiiu
the same point, ifk is not 0.

Homogeneous line coordinates arise by dualizing the above discus-
sion in the plane. Thus we choose any quadrilateral in the plane as
frame of reference, denoting the sides by [1, 0, 0], [0, 1, 0], [0, 0, 1],
[1, 1, 1] respectively. The points of intersection with [1, 1, 1] of the
lines [1, 0, 0], [0, 1, 0], [0, 0, 1] are joined to the vertices of the tri-
angle of reference opposite to [1, 0, 0], [0, 1, 0], [0, 0, 1] respectively
by lines that are denoted by [0, 1, 1], [1, 0, 1], [1, 1, 0]. The three
lines [1, 0, 0], [1, 1, 0], [0, 1, 0] are then taken as the fundamental
lines [1, 0], [1, 1], [0, 1] of a homogeneous system of coordinates in
a flat penciL If in this system a line is denoted by [u v wj, it is
denoted in the planar system by [w 1? 2 , 0]. In like manner, to the
lines on the other vertices are assigned coordinates of the forms
[0, ^ 2 , J and [u v 0, J respectively. As the plane dual of the
theorem and definition above we then have at once

THEOREM 2 f . DEFINITION. If 1 is any line not on a, vertex of the
triangle of reference, there exist three numbers u v u 09 u s all different
from zero, such that the traces of I on the three sides of the triangle of
reference are projected from the respective opposite, vertices by the lines
\u v u z) 0], [u v 0, ttj, [0, 2 , ?/]. FJiese three numbers are called the
homogeneous coordinates of /, and I is denoted ly [u 19 u^ w s ]. Any
set of three numbers (not all zero) determine uniquely a line whose
coordinates they are.

Homogeneous point and line coordinates may be put into such
a relation that the condition that a point (x^ % 9 or s ) be on a line
[!, a , tt s ] is that the relation u^+ ^ 2 # 3 -f u 3 x s = be satisfied. We
have seen that if (x v x v x 9 ) is a point not on a side of the triangle of
reference, and we place x = xjx z> and y = # 2 /# 3 , the numbers (x, y)
are the nonhomogeneous coordinates of the point (x v x s , a? 3 ) referred
to OF as the re-axis and to OU as the y-asis of a system of nonho-
mogeneous coordinates in which the point T=(l, 1, 1) is the point
(1, 1) (0, U, 7 being used in the same significance as in the proof of
Theorem 2). By duality, if [u v u# -M 3 ] is any line not on any vertex
of the triangle of reference, and we place u = uju^ and v == uju y
the numbers [^, v] are the nonhomogeneous coordinates of the line
[u v u& J referred to two of the vertices of the triangle of reference

178

COORDINATE SYSTEMS

[CHAP. YII

as CT-eenter and F-center respectively, and in which the line [1, 1, 1]
is the line [1, 1]. If, now, we superpose these two systems of nonhom-
ogeneous coordinates in the way described in the preceding section,
the condition that the point (x, y) be on the line [u 9 v] is that the
relation uz + vy + l = Q be satisfied (Theorem 1). It is now easy to
recognize the resulting relation between the systems of homogeneous
coordinates with which we started. Clearly the point (0, 1, 0) = U is
the *7-center, (1, 0, 0) = F is the F-center, and (0, 0, 1) = is the third

FIG. 83

vertex of the triangle of reference in the homogeneous system of line
coordinates. Also the line whose points satisfy the relation # x = is
the line [1, 0, 0], the line for which # s = is the line [0, 1, 0], and
the line for which # 3 = is the line [0, 0, 1]. Finally, the line
[1, 1] = [1, 1, 1], whose equation in nonhomogeneous coordinates is
+ y + 1 = 0, meets the line- x^ = in the point (0, 1, 1), and the
line # 2 = in the point ( 1, 0, 1). The two coordinate systems are
then completely determined (fig. 83).

It now follows at once from the result of the preceding section
that the condition that (x v x z , # 3 ) be on the line [u v u^ u z ] is

ufli 4-

0, if none of the coordinates x lt # 2 , # 8 , u v

G3] HOMOGENEOUS COORDINATES 179

is zero. To see that the same condition holds also when one (or more)
of the coordinates is zero, we note first that the points (0, 1, 1),
( 1, 0, 1), and ( 1, 1, 0) are collinear. They are, in fact (fig. 83), on
the axis of perspectivity of the two perspective triangles OTFand
la-lylc, the center of perspectivity being J, It is now clear that

the line [1, 0, 0] passes through the point (0, 1, 0;,
the line [0, 1, 0] passes through the point (1, 0, 0),
the line [1, 1, 0] passes through the point ( 1, 1, 0).

There is thus an involution between the points (x v ;<? 2 , 0) of the line
ac 9 = and the traces (j?/, %.!, 0) of the lines with the same coordinates,
and this involution is given by the equations

In other words, the line \u v u, 0] passes through the point ( u z) it v 0).
Any other point of this line (except (0, 0, 1)) has, by definition, the
coordinates ( ^ 3 , u v x s ). Hence all points (x^ # 2 , # 3 ) of the line
[u v u z , 0] satisfy the relation 7^ 4- ?M*2 + Vs 0- The same argu-
ment applied when any one of the other coordinates is zero estab-
lishes this condition for all cases. A system of point and a system
of line coordinates, when placed in the relation described above, will
be said to form a system of homogeneous point and line coordinates in
the plane. The result obtained may then be stated as follows :

THEOREM 3. In a system of homogeneous point and line coordinates
in a plane the necessary and sufficient condition that a point (% v # 2 , .T 3 )
le on a line [u v 2 , ^ 3 ] is that the relation u^ + 11^ 4- u^ = le
satisfied.

COROLLAEY. The equation of a line through the origin of a system
of nonhomogeneous coordinates is of the form mx 4- ny = 0.

EXERCISES

1. The line [1, 1, 1] is the polar of the point (1, 1, 1) with regard to the
triangle of reference (cf. p. 46).

2. The same point is represented by (a l9 a 2 , a s ) and (& r 5 2 , 5 8 ) if and only

if the two-rowed determinants of the matrix (J 1 ^ 2 ^ 3 ) are all zero.

V*l &2 V

3. Describe nonhomogeneous and homogeneous systems of line and plane
coordinates in a bundle by dualizing in space the preceding discussion. In
such a bundle what is the condition that a line be on a plane ?

180

COORDINATE SYSTEMS

[CHAP. Til

64. The line on two points. The point on two lines. Given two
points, A = (a v a z , a,) and B = (b v & 2 , & s ), the question now arises as
to what are the coordinates of the line joining them; and the dual
of this problem, namely, given two lines, m = [m v m 2 , m s ] and n =
[n v n n , '7& 8 ], to find the coordinates of the point of intersection of the
two lines.

THEOREM 4. The equation of THEOREM 4'. The equation of
the line joining the points (a^a^a^) the point of intersection of the
and (b v 5 2> 6 3 ) is lines [m x ,m 2 ,m 8 ] and [n v n^n s ] is

= 0.

Proof. "When these determinants are expanded, we get

3 =0,

respectively. The one above is the equation of a line, the one below
the equation of a point. Moreover, the determinants above both
evidently vanish when the variable coordinates are replaced by the
coordinates of the given elements. The expanded form just given
leads at once to the following:

COROLLAET V. The coordinates
of the point of intersection of the
lines [m v m 2 , m s ], [n v n 2 , w 8 ] are

COROLLAEY 1. The coordinates
of the line joining the points
(a v 2 , a,), (b v 6 2 , b s ) are

There also follows immediately from this theorem:
COROLLARY 2. The condition COROLLARY 2'. The condition
that three points A, B, C be col- that three lines m, n, p be con-
linear is current is

= 0.

-0.

EXAMPLE. Let us verify the theorem of Desargues (Theorem 1, Chap. II)
analytically. Choose one of the two perspective triangles as triangle of refer-
ence, say A' = (0, 0, 1), J5 7 = (0, 1, 0), C" = (1, 0, 0), and let the center of per-
spectivity be P = (1, 1, 1). If the other triangle is ABC, we may place

64,65] PEOJECTITE PENCILS 181

A = (1, 1, rt), B = (1, ft, 1), C = (c, 1, 1) : for the equation of the line PA'
is x l j: 2 = ; and since J. is, by hypothesis, on this line, its first two coordi-
nates must be equal, and may therefore be assumed equal to 1 ; the third
coordinate is arbitrary. Similarly for the other point*. Now, from the above
theorems and their corollaries we readily obtain in succession the following :

The coordinates of the line A'& are [1, 0, 0].

The coordinates of the line AB are [1 ah, a 1, & 1].

Hence the coordinates of their intersection C" are

C"'=(0. l -&,,i-l).
Similarly, we find the coordinates of the intersection A" of the lines B'C\ BC

tobe J"=(l-e,6-l,0):

and, finally, the coordinates of the intersection B" of the lines C'A'* CA to be

3"= (e- 1,0,1 -a).
The points A", B'\ C" are readily seen to satisfy the condition for collinearity.

EXERCISES

1. Work through the dual of the example just given, choosing the sides of
one of the triangles and the axis of perspectivity as the fundamental lines of
the system of coordinates. Show that the work may be made identical, step
for step, with that above, except for the interpretation of the symbols.

2. Show that the system of coordinates may be so chosen that a quadrangle-
quadrilateral configuration is represented by all the sets of coordinates that
can be formed from the numbers and 1. Dualize.

3. Derive the equation of the polar line of any point with regard to the
triangle of reference. Dualize.

65. Pencils of points and lines.. Projectivity. A convenient ana-
lytic representation of the points of a pencil of points or the lines of
a pencil of lines is given by the following dual theorems :

THEOREM 5. Any point of a THEOREM 5'. Any line of a
pencil of points may le repre- pencil of lines may "be represented
sented ly ly

P = (X 2 a 1 + \l v X 2 a 3 + \l z , p = [|* a m 1

where A = (a x & 2 , a s ) and B = where m = [m v m 2 , m g ] and n =
(&!> 5 2 , 6 S ) are any two distinct [n v n, raj are any two distinct
points of the pencil. lines of the pencil.

Proof. We may confine ourselves to the proof of the theorem on
the left. By Theorem 4, Cor. 2, any point (x v x^ # s ) of the pencil of
points on the line AB satisfies the relation

182 COORDINATE SYSTEMS [CHAP.VII

\ x i x * x s\

(1) fl x s ,=0.

A &s 5 a!
We may then determine three numbers p, X 2 ', X/, such that we have

(2) p* 1 = \'a 1 +Xft- (* = 1,2,3)

The number p cannot be under the hypothesis, for then we should
have from (2) the proportion x : a a : 3 = 5^ S 3 : 5 8 , which would imply
that the points A and B coincide. We may therefore divide by p.
Denoting the ratios \ f /p and \[/p by X 2 and \, we see that every
point of the pencil may be represented in the manner specified.
Conversely, every point whose coordinates are of the form specified
clearly satisfies relation (1) and is therefore a point of the pencil.

The points A and B in the above representation are called the "base
points of this so-called parametric representation of the elements of
a pencil of points. Evidently any two distinct points may be chosen
as base points in such a representation. The ratio \/\ is called the
parameter of the point it determines. It is here written in homoge-
neous form, which gives the point A for the value \=0 and the
point B for the value X 2 = 0. In many cases, however, it is more
convenient to write this parameter in nonhomogeneous form,

P = (a, + 7J> V a 2 + \h, a, + X6 3 ),

which is obtained from the preceding by dividing by X 2 and replacing
\/\ by ^- I n this representation the point B corresponds to the
value X = oc. "We may also speak of any point of the pencil under
this representation as the point X t : X 2 or the point X when it corre-
sponds to the value X X /X 2 = X of the parameter. Similar remarks and
the corresponding terminology apply, of course, to the parametric
representation of the lines of a flat pencil. It is sometimes convenient,
moreover, to adopt the notation A -f- XJ? to denote any point "of the
pencil whose base points are A, B or to denote the pencil itself ; also,
to use the notation m, + pn to denote the pencil of lines or any line
of this pencil whose base lines are m, n.

In order to derive an analytic representation of a projectivity
between two one-dimensional primitive forms in the plane, we seek
first the condition that the point X of a pencil of points A + \B be
on the line p of a pencil of lines m + pn. By Theorem 3 the condition
that the point X be on the line /t is the relation

65] PKOJECTIYE PENCILS 183

When expanded this relation gives

t=3 i=3

l a l = 0.

This is a bilinear equation whose coefficients depend only on the coor-
dinates of the base points and base lines of the two pencils and not
on the individual points for which the condition is sought. Placing

this equation becomes 6>X + Dp. -4X B = 0,

which may also be written*
*

The result may be stated as follows: Any perspective relation "between
two one-dimensional primitive forms of different "kinds is obtained by
establishing a protective correspondence between the parameters of the
two forms. Since any protective correspondence between two one-
dimensional primitive forms is obtained as the resultant of a sequence
of such perspectivities, and since the resultant of any two linear frac-
tional transformations of type (1) is a transformation of the same
type, we have the following theorem :

THEOREM 6. Any projectile correspondence "between two one-dimen-
sional primitive forms in the plane is obtained "by establishing a
protective relation

between the parameters p> X of the two forms.

In particular we have

COROLLARY 1. Any projectimty in a one-dimensional primitive
form in the plane is given by a relation of the form

where X is the parameter of the form,

^ pi does not vanish because the correspondence between
A ana /i is (1,1).

184 COOBDISrATE SYSTEMS [CHAP.TII

COROLLARY 2. If X 1? X 2 , X 3 , \ are the parameters of four elements
A v A*, A 3 , A of a one-dimensional primitive form,) the cross ratio
B (A^A^ A S A 4 ) is given ly

A projectivity between two different one-dimensional forms may
be represented in a particularly simple form by a judicious choice of
the base elements of the parametric representation. To fix ideas, let
us take the case of two projective pencils of points. Choose any two
distinct points A 9 B of the first pencil to be the base points, and let
the homologous points of the second pencil be base points of the
latter. Then to the values X = and X = oo of the first pencil must
correspond the values p = and \L = oo respectively of the second.
In this case the relation of Theorem 6, however, assumes the form
ju, = &X. Hence, since the same argument applies to any distinct
forms, we have

COEOLLARY 3. If tico distinct projective one-dimensional primitive
forms in the plane are represented parametrically so that the base
elements form two homologous pairs, the projeetivity is represented by
a relation of the form p = k\ between the parameters p, X of the two
forms.

This relation may be still further simplified. Taking again the case
discussed above of two projective pencils of points, we have seen that,
in general, to the point (a^ + l v a 2 + J a , a s + J 8 ), ie. to X = 1, corre-
sponds the point (0/ + &J/, tf 2 '-f-&& 2 ', #/+&&), ie. the point /-& = .
Since the point !?'=(&/, J/, 6 g ') is also represented by the set of coordi-
nates (&/, &/, &/), it follows that if we choose the latter values for the
coordinates of the base point B 1 , to the value X = 1 will correspond
the value fi = 1, and hence we have always p. = X. In other words,
we have

COROLLARY 4. If two distinct one-dimensional forms are protective,
the lase elements may le so chosen that the parameters of any two
homologous elements are equal.

Before closing this section it seems desirable to call attention
explicitly to the forms of the equation of any line of a pencil and of
the equation of any point of a pencil which is implied by Theorem 5 ;
and Theorem 5 respectively. If we place m = m^ + w 2 # 2 H- m B x s and

65, 66] EQUATION OF A COXIC 185

71 = 71^+ nx 2 + }2 3 c>: 3 , it follows from the first theorem mentioned
that the equation of any line of the pencil whose center is the inter-
section of the lines m = 0, n = is given by an equation of the form
ni + fin = 0. Similarly, the equation of any point of the line joining
A. = a l u l + 2 3 + 3 3 = and B = J^ + 6 a w a -j- Zu? s = is of the
form A +\B = 0.

66. The equation of a conic. The results of 65 lead readily to
the equation of a conic. By this is meant an equation in point (line)
coordinates which is satisfied by all the points (lines) of a conic, and
by no others. To derive this equation, let A 9 B Le two distinct points
on a conic, and let

tn = m^i^-r wzA' a -t- ?/u,c = 0,

(1) d = w^ + M tf v 2 + /v' 3 = 0,

be the equations of the tangent at A, the tangent at J?, and the line
A3 respectively. The conic is then generated as a point locus by
two protective pencils of lines at A and B, in which m, p at A are
homologous with jp, n at B respectively. This projectivity between
the pencils

7 m + \2 } - >

^ ; p + IAH Q

is given (Theorem 6, Cor. 3) by a relation

(3) /A = &X

between the parameters /*, X of the two pencils. To obtain the equa-
tion which is satisfied by all the points of intersection of pairs of
homologous lines of these pencils, and by no others, we need simply
eliminate ^, X between the last three relations. The result of this
elimination is

(4) /-#wztt=0,

which is the equation required. By multiplying the coordinates of
one of the lines by a constant we may make k = 1.

Conversely, it is obvious that the points which satisfy any equation
of type (4) are the points of intersection of homologous lines in the
pencils (2), provided that /z, = k\. If m, n, p are fixed, the condition
that the conic (4) shall pass through a point (a v a v a z ) is a linear
equation in Je. Hence we have

186

COOEDIKATE SYSTEMS

[CHAP. VII

THEOEEM 7. If m = 0, n = 0,
p = are the equations of two
distinct tangents of a conic and
the line joining (heir points of con-
tact -respectively, the point equa-
tion of the conic is of the form

p~ kmn = 0.

The coefficient "k is determined by
any third point on the conic. Con-
versely, the points which satisfy
an equation of the above form
constitute a conic of which m =
and n = are tangents at points
on p = 0.

COROLLAEY. By properly choos-
ing the triangle of reference, the
point equation of any conic may
te put in the form

THBOEEM 7.' If A = 0, B = 0,
C = are the equations of two
distinct points of a conic and the
intersection of the tangents at these
points respectively, the line equa-
tion of the conic is of the form

where r^= 0, # 3 = are two tan-
gents, and # 2 = is the line join-
ing their points of contact.

The coefficient k is determined ly
any third line of the conic. Con-
versely, the lines which satisfy an
equation of the above form consti-
tute a conic of which A = and
B = are points of contact of the
tangents through C = 0.

COEOLLAEY. By properly choos-
ing the triangle of reference, the
line equation of any conic may
be put in the form

w 2 2 2^3 = 0,

where ^= 0, % 3 = are two points,
and u 2 = is the intersection of
the tangents at these points.

It is clear that if we choose the point (1, 1, 1) on the conic, we have
k = 1. Supposing the choice to have been thus made, we inquire
regarding the condition that a line [u v u z , u^ be tangent to the conic

This condition is equivalent to the condition that the line whose

equation is

^a^-f- ^ 2 # 2 + U B % & =

shall have one and only one point in common with the conic. Elimi-
nating # 8 between this equation and that of the conic, the points
common to the line and the conic are determined by the equation

UjSB? + Ufl^ + 1(, 3 %j = 0.

The roots of this equation are equal, if and only if we have

u% 4 u^ = 0.

tK5,C7] LIXEAE TEAXSFOP^IATIOXS 1ST

Since this is the line equation of all tangents to the conic, and since
it is of the form given in Theorem 7', Cor., above, we have here a new
proof of the fact that the tangents to a point conic form a li/ic conic
(cf. Theorem 11, Chap. V).

When the linear expressions for M, n, p are substituted in the equa-
tion j> 2 ~kmn = of any conic, there results, when multiplied out, a
homogeneous equation of the second degree in x v # 2 , tt i s , which may
be written in the form

(1) a n x{ + a 22 ,<7 + a^c* + 2 a 12 ^ 2 + 2 a^c^ + 2 a^x^ = 0.

We have seen that the equation of every conic is of this form. We
have not shown that every equation of this form represents a conic
(see 85, Chap. IX).

EXERCISE
Shcrw that the conic

degenerates into (distinct or coincident) straight lines, if and only if we have

Dualize. (A, E, P, H )

67. Linear transformations in a plane. We inquire now concern-
ing the geometric properties of a linear transformation

(1)

Such a transformation transforms any point (x v # 2 , # 3 ) of the plane
into a unique point (#/, # 2 ; , x) of the plane. Eeciprocally, to every
point x 1 will correspond a unique point x, provided the determinant
of the transformation

^31 ^82 a X

is not 0. For we may then solve equations (1) for the ratios x^ix^n
in terms of a?/: x : x% as follows :

(2) />',

p'x,

188 COORDINATE SYSTEMS [CHAP.VII

here the coefficients A tJ are the cofactors of the elements a t) respec-
tively in the determinant A.

Further, equations (1) transform every line in the plane into a
unique line. In fact, the points x satisfying the equation

UjJL\ + M 2 # 3 + tt a # 8 =

are, by reference to equations (2), transformed into points x 1 satisfy-
ing the equation

(A u ii 1 + A^IS + A^ x[ 4- (A^ + A^u^ + A^u t ) ^

+ (A^U! + A^u s + A 5Q u s ) 3 ' = 0,

which is the equation of a line. If the coordinates of this new line be
denoted by [/, u^ u\ 9 we clearly have the following relations between
the coordinates [K V u^ w s ] of any line and the coordinates [/, u^, u\
of the line into which it is transformed by (1):

(3)

We have seen thus far that (1) represents a collineation in the plane
in point coordinates. The equations (3) represent the same collineation
in line coordinates.

It is readily seen, finally, that this collineation is protective. For
this purpose it is only necessary to show that it transforms any
pencil of lines into a projective pencil of lines. But it is clear that if
m = and n are the equations of any two lines, and if (1) trans-
forms them respectively into the lines whose equations are m 1 =
and ?2/=0, any line ??i + Xft=0 is transformed into m f +\n f Q,
and the correspondence thus established between the lines of the
pencils has been shown to be projective (Theorem 6).

Having shown that every transformation (1) represents a projective
collraeation, we will now show conversely that every projective
collineation in a plane may be represented by equations of the form
(1). To this end we recall that every such collineation is completely
determined as soon as the homologous elements of any complete
quadrangle are assigned (Theorem 18, Chap. IV). If we can show
that likewise there is one and only one transformation of the form
(1) changing a given quadrangle into a given quadrangle, it will
follow that, since the linear transformation is a projective collineation,
it is the given projective collineation.

67] LINEAR TRANSFORiEATIOXS 189

Given any protective collineation in a plane, let the fundamental
points (0, 0, 1), (0, 1, 0), (1, 0, 0), and (1, 1, 1) of the plane (which
form a quadrangle) be transformed respectively into the points
A - (a v a v aj, B = (b v 6 2 , J s ), C = (c v c z , ), and I) = (d v tl, tl), form-
ing a quadrangle. Suppose, now, we seek to determine the coefficients
of a transformation (1) so as to effect the correspondences just indi-
cated. Clearly, if (0, 0, 1) is to be transformed into (a v a, ,), we

must have .

a ls =\a v r/ 23 = X 2 , a 3G =X s ,

X being an arbitrary factor of proportionality, the value (=?= 0) of which
we may choose at pleasure. Similarly, we obtain

Since, by hypothesis, the three points A, B, C are not collinear, it
follows from these equations and the condition of Theorem 4, Cor. 2,
that the determinant A of a transformation determined in this way
is not 0. Substituting the values thus obtained in (1), it is seen that
if the point (1, 1, 1) is to be transformed into (d v d^ d s ), the following
relations must hold :

/w7 1 =<r

pel* = c

Placing p = 1 and solving this system of equations for v 9 p, \ we
obtain the coefficients a tj of the transformation. This solution is
unique, since the determinant of the system is not zero. Moreover,
none of the values X, p, v will be ; for the supposition that v = 0,
for example, would imply the vanishing of the determinant

d l \

which in turn would imply that the three points D, B, A are collinear,
contrary to the hypothesis that the four points A, B> C 9 D form a
complete quadrangle.

Collecting the results of this section, we have

THEOREM 8. Any protective colKneatwn in the plane may be repre-
sented in point coordinates Tyy equations of form (1) or in line coordi-
nates ly equations of form (3), and in each case the determinant of

190 COOBDINATE SYSTEMS [CHAP.VII

the transformation is different from ; conversely, any transforma-
tion of one of these forms in which the determinant is different from
represents a protective collineation in the plane.

COROLLARY 1. In nonhomogeneous point coordinates the equations of
a projectile collineation are

COROLLARY 2. If the singular line of the system of nonhomogeneous
point coordinates is transformed into itself, these equations can 5e
written x ! =

a* 6,

=0.

68. Collineations between two different planes. The analytic form
of a collineation between two different planes is now readily derived.
Let the two planes be a and /3, and let a system of coordinates be
established in each, the point coordinates in a being (x v % 2 , # 3 ) and
the point coordinates in y8 being (y v y z , y s ). Further, let the isomor-
phism between the number systems in the two planes be established
in such a way that the correspondence established by the equations

Vi** *v y =a fc y.^afe

is protective. It then follows, by an argument (cf. 59, p. 166),
which need not be repeated here, that any collineation between the
two planes may be obtained as the resultant of a projectivity in the
plane a, which transforms a point X, say, into a point X\ and the pro-
jectivity r = X f between the two planes. The analytic form of any
protective collineation letween the two planes is therefore :

with the determinant A of the coefficients 'different from 0. And, con-
versely, every such transformation in which A = represents a projec-
tive collineation letween the two planes.

69. Nonhomogeneous coordinates in space. Point coordinates in
space are introduced in a way entirely analogous to that used for the
introduction of point coordinates in the plane. Choose a tetrahedron
of reference OUVW and label the vertices = 0^= O y = 0,,, Z7= oo a ,

COORDINATES 1ST SPACE

191

F= oo y , JT= ex), (fig. 84) ; and on the lines O^oo,, O^oo,, O s co 2 , called
respectively the x-axis, the y-axis, the 2-axis, establish three scales by
choosing the points 1 X9 l y , 1 2 . The planes Ooo a co y , Ooo x co g9 Ott y ao a are
called the xy-plane, xz-plane, yz-plane respectively. The point is
called the origin. If P is any point not on the plane co x oo y oo z> which
is called the singular plane of the coordinate system, the plane
P 00^003 meets the #-axis in a point whose nonhomogeneons coordinate
in the scale (0^, 1^, co x ) we call a. Similarly, let the plane
meet the 2/-axis in a point
whose nonhomogeneous
coordinate in the scale
(0,, l y) co y ) is I ; and let
the plane Poo x co y meet the
z-axis in a point whose
nonhomogeneous coordi-
nate in the scale (O a , 1 Z9 <x> z )
is c. The numbers a, &, c
are then the nonhomo-
geneous #-, y- 9 and z-coor-
dinates of the point P.
Conversely, any three
numbers a 9 1 9 c determine
three points A 9 B 9 C on
the x- 9 y-> and #-axes respectively, and the three planes AaOyOo^ B<x> x <x> a ,
Cco x <x> y meet in a point P whose coordinates are a 9 6, c. Thus every
point not on the singular plane of the coordinate system determines
and is determined by three coordinates. The point P is then repre-
sented by the symbol (a 9 &, c).

The dual process gives rise to the coordinates of a plane. Point
and plane coordinates may then be put into a convenient relation, as
was done in the case of point and line coordinates in the plane, thus
giving rise to a system of simultaneous point and plane coordinates
in space. We will describe the system of plane coordinates with
reference to this relation. Given the system of nonhomogeneous point
coordinates described above, establish in each of the pencils of planes
on the lines VW 9 UW, UV a scale by choosing the plane UVW as
the zero plane M = O r = O w in each of the scales, and letting the planes
VW 9 UW 9 UV be the planes oo tt , oo , oo^ respectively. In the ^scale

FIG. 84

192 COORDINATE SYSTEMS [CHAP.VH

let that plane through. 7W be the plane l u , which meets the aj-axis
in the point l a . Similarly, let the plane l v meet the ?/-axis in the
point l y ; and let the plane l w meet the z-axis in the point !.
The swscale, inscale, and w-scale being now completely determined,
any plane TT not on the point (which is called the singular point
of this system of plane coordinates) meets the #-, y-, and #-axes in
three points L, M> N which determine in the u-, v-, and ^-scales planes
whose coordinates, let us say, are l } m, n. These three numbers are
called the nonhomogeneous plane coordinates of TT. They completely
determine and are completely determined by the plane TT. The plane
TT is then denoted by the symbol [I, m, n].

In this system of coordinates it is now readily seen that the con-
dition that the point (a, 5, c) be on the plane \l 9 m,ri\ is that the relation
la + mb + nc + l = Q le satisfied. It follows readily, as in the planar
case, that the plane |7, m> n] meets the #-, y-, and #-axes in points
whose coordinates on these axes are 1/, 1/m, and 1/n respec-
tively.* In deriving the above condition we will suppose that the
plane TT = |7, m 9 n] does not contain two of the points U, F, IF, leav-
ing the other case as an exercise for the reader. Suppose, then, that
U= co x and F= ao y are not on TT. By projecting the ys-plane with
U as center upon the plane TT, and then projecting TT with F as center
on the ##-plane, we obtain the following perspectivities :

where (x, y, z) represents any point on ?r. The product of these two
perspectivities is a projectivity between the y^-plane and the ##-plane,
by which the singular line of the former is transformed into the sin-
gular line of the latter. Denoting the ^-coordinate of points in the
2/2-plane by z f , this projectivity is represented (according to Theorem
8, Cor. 2, and 68) by relations of the form

We proceed to determine the coefficients a v l v c v The point of
intersection of TT with the y-axis is (0, 1/m, 0), and is clearly

* This statement remains valid even if one or two of the numbers Z, m, n are
zero (they cannot all be zero unless the plane in question is the singular plane
which we exclude from consideration), provided the negative reciprocal of be
denoted by the symbol o>.

09] COOBDINATES DT SPACE 193

transformed by the projectivity in question into the point (0, 0, 0).
Hence (1) gives

CI= -,IT

The point of intersection of TT with the z-axis is, if w=0, (0, 0, 1/n)
and is transformed into itself. Hence (1) gives

n m

7 M

or 5. = ---

If n = 0, we have at once \ = 0.

Finally, the point of intersection of TT with the #-axis is ( 1/7, 0, 0),
and the transform of the point (0, 0, 0). Hence we have

or a, = ---

1 m

Hence (1) becomes y = -- x -- z -- >

m m m

a relation which must be satisfied by the coordinates (x, y, z) of any
point on TT. This relation is equivalent to

Ix + my 4- nz -j- 1 = 0.
Hence (a, 6, c) is on [7, m,n], if

(2) Za + m& + nc + 1 = 0.

Conversely, if (2) is satisfied by a point (a, 6, c), the point (0, 5, e) = P
is transformed by the projectivity above into (#, 0, c) = $, and hence
the lines JP 27 and Q V which meet in (, &, c) meet on TT.

DEFINITION. An equation which DEFINITION. An equation which
is satisfied by all the points (x, y, ) is satisfied by all the planes [u,v,w]
of a plane and by no other points on a point and by no other planes
is called the point equation of the is called the plane equation of the
plane. point.

The result of the preceding discussion may then be stated as follows :
THEOREM 9. The point equation THEOREM 9'. The plane equation
of the plane [I, m, n] is of the point (a, I, c) is

= Q. au + bv + cw + 1 = 0.

194 COORDINATE SYSTEMS [OHAP.VII

70. Homogeneous coordinates in space. Assign to the vertices 0, U,
V, W of any tetrahedron of reference the symbols (0, 0, 0, 1), (1, 0, 0, 0),
(0, 1, 0, 0), (0, 0, 1, 0) respectively, and assign to any fifth point T
not on a face of this tetrahedron the symbol (1, 1, 1, 1). The five
points 0, U 9 V, W, T are called the frame of reference of the system
of homogeneous coordinates now to be described. The four lines join-
ing T to the points 0, U, V, W meet the opposite faces in four points,
which we denote respectively by (1, 1, 1, 0), (0, 1, 1, 1), (1, 0, 1, 1),
(1, 1, 0, 1). The planar four-point (0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0),
(0, 1, 1, 1) we regard as the frame of reference (0, 0, 1), (0, 1, TO),
(1, 0, 0), (1, 1, 1) of a system of homogeneous coordinates in the plane.
To any point in this plane we assign the coordinates (0, # 2 , # 3 , # 4 ), if
its coordinates in the planar system just indicated are (# 2 , # 3 , # 4 ). In
like manner, to the points of the other three faces of the tetrahedron of
reference we assign coordinates of the forms (x v 0, # 3 , # 4 ), (x v x 2t 0, # 4 ),
and (x v x z> x s , 0). The coordinates of the points in the faces opposite
the vertices (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) satisfy respec-
tively the equations x^ = 0, # 2 = 0, x z = 0, x = 0.

To the points of each edge of the tetrahedron of reference a notation
has been assigned corresponding to each of the two faces which meet
in the edge. Consider, for example, the line of intersection of the
planes x l =Q and # 2 = 0. Eegarding this edge as a line of x t =^ 0, the
coordinate system on the edge has as its fundamental points (0, 0, 1, 0),
(0, 0, 0, 1), (0, 0, 1, 1). The first two of these are vertices of the tetra-
hedron of reference, and the third is the trace of the line joining
(0, 1, 0, 0) to (0, 1, 1, 1). On the other hand, regarding this edge as a
line of # 2 =0, the coordinate system has the vertices (0, 0, 1, 0) and
(0, 0, 0, 1) as two fundamental points, and has as (0, 0, 1, 1) the trace
of the line joining (1, 0, 0, 0) to (1, 0, 1, 1). But by construction the
plane (0, 1, 0, 0)(1, 0, 0, 0)(1, 1, 1, 1) contains both (0, 1, 1, 1) and
(1, 0, 1, 1), so that the two determinations of (0, 0, 1, 1) are identical.
Hence the symbols denoting points in the two planes x l = and
# 2 =0 are identical along their line of intersection. A similar result
holds for the other edges of the tetrahedron of reference.

THEOREM 10, DEFINITION. If P is any point not on a face of the
tetrahedron of reference, there exist four numbers x v # 2 , # 3 , x^ all
different from zero, suck that the projections of P from the four vertices
(1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) respectively upon their

70] COORDINATES IN SPACE 195

opposite faces are (0, <e a , x v xj, (x v 0, X B) a?J, (^ re s , 0, # 4 ), (x v x 2 , x z , 0).
l^ese four numbers are called the homogeneous coordinates of P and
P is denoted ly (x v x 2 , X B , a?J. Any ordered set of four members, not
all zero, determine uniquely a point in space whose coordinates they are.

Proof. The line joining P to (1, 0, 0, 0) meets the opposite face in
a point (0, x z , x 2 , # 4 ), which is not an edge of the tetrahedron of refer-
ence, and such therefore that none of the numbers # 2 , X B , x 4 is zero.
Likewise the line joining P to (0, 1, 0, 0) meets the opposite face in
a point (as/, 0, # 8 ', # 4 ), such that none of the numbers x[, # 8 ', x is zero.
But the plane P(l, 0, 0, 0) (0, 1, 0, 0) meets x l = in the line joining
(0, 1, 0, 0) to (0, x 2 , # 3 , # 4 ), and meets x 2 = in the line joining
(1, 0, 0, 0) to (as/, 0, # 3 ', xl). By the analytic methods already devel-
oped for the plane, the first of these lines meets the edge common
to x 1 = and x 2 = in the point (0, 0, a? 8 , # 4 ), and the second meets
it in the point (0, 0, #/, xl). But the points (0, 0, a? 8 , # 4 ) and
(0, 0, # 8 ', x^) are identical, and hence, by the preceding paragraph, we
have # s /# 4 = %l/%l* Hence, if we place x l = x^xjxl, the point
(as/, 0, x, xl) is identical with (x v 0, x 9 , x). The line joining P to
(0, 0, 1, 0) meets the face # 8 = in a point (x? 9 x, 0, x^. By the
same reasoning as that above it follows that we have x/xl r = xjx
and %2/xl 1 ttj%# so that the point (as/', x^ 0, x^) is identical with
(x v # 2 , 0, a? 4 ). Finally, the line joining P to (0, 0, 0, 1) meets the face
# 4 = in a point which a like argument shows to be (x v x 2 , X B , 0).

Conversely, if the coordinates (x v # 2 , a? 3 , # 4 ) are given, and one of
them is zero, they determine a point on a face of the tetrahedron
of reference. If none of them is zero, the lines joining (1, 0, 0, 0)
to (0, x z9 x 39 # 4 ) and (0, 1, 0, 0) to (x v 0, # 8 , # 4 ) are in the plane
(1, 0, 0, 0) (0, 1, 0, 0) (0, 0, x s) x 4 ), and hence meet in a point which,
by the reasoning above, has the coordinates (x v x z , # 8 , # 4 ).

COEOLLAEY. The notations (x^x^x^x^) and (kx v Jcx 2 , Jcx B , kx^)
denote the same point for any value of 7c not equal to zero.

Homogeneous plane coordinates in space arise by the dual of the
above process. The four faces of a tetrahedron of reference are denoted
respectively by [1, 0, 0, 0], t [0, 1, 0, 0], [0, 0, 1, 0], and [0, 0, 0, 1].
These, together with any plane [1, 1, 1, 1] not on a vertex of the
tetrahedron, form the frame of reference. The four lines of inter-
section of the plane [1, 1, 1, 1] with the other four planes in the order

196 COOKDINATE SYSTEMS [CHAP.VII

above are projected from the opposite vertices by planes which are
denoted by [0,1,1,1], [1,0,1,1], [1,1,0,1], [1,1,1,0] respectively.
The four planes [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1], and [0, 1, 1, 1] form,
if the first in each of these symbols is suppressed, the frame of
reference of a system of homogeneous coordinates in a bundle (the
space dual of such a system in a plane). The center of this bundle
is the vertex of the tetrahedron of reference opposite to [1, 0, 0, 0].
To any plane on this point is assigned the notation [0, u v u# ^J, if
its coordinates in the bundle are [u z , u s> wj. In like manner, to the
planes on the other vertices are assigned coordinates of the forms
[u v 0, u s , u], [u v u v 0, ttj, [u v u 2 , u 3 , 0]. The space dual of the last
theorem then gives :

THEOKEM 10'. DEFINITION. Ifir is any plane not on a vertex of the
tetrahedron of reference, there exist four numbers u v u^ u 9) u *> a ^ differ-
ent from zero, such that the traces of IT on the four faces [1, 0, 0, 0],
[0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1] respectively are projected from the
opposite vertices "by the planes [0, u^, u %9 wj, [u v 0, w s , wj, [u v u z) 0, % 4 ],
\u v u z ,u 2 , 0], These four members are called the homogeneous coordinates
0/7T, and TT is denoted ly [u v u z , u s , u\. Any ordered set of four num-
lers, not all zero, determine uniquely a plane whose coordinates they are.

By placing these systems of point and plane coordinates in a proper
relation we may now readily derive the necessary and sufficient con-
dition that a point (x v a? a , x s> a? 4 ) be on a plane \u v u%, u &) u^\. This
condition will turn out to be

We note first that in a system of point coordinates as described above
the sis points (- 1, 1, 0, 0), (- 1, 0, 1, 0), (- 1, 0, 0, 1), (0, - 1, 1, 0),
(0, 0, 1, 1), (0, 1, 0, 1) are coplanar, each being the harmonic con-
jugate, with respect to two vertices of the tetrahedron of reference, of
the point into which (1, 1, 1, 1) is projected by the line joining the
other two vertices. The plane containing these is, in fact, the polar
of (1, 1, 1, 1) with respect to the tetrahedron of reference (cf. Ez. 3,
p. 47). Now choose

as the plane [1, 0, 0, 0] the plane ^= 0,
as the plane [0, 1, 0, 0] the plane # 2 = 0,
as the plane [0, 0, 1, 0] the plane x s = 0,
as the plane [0, 0, 0, 1] the plane x 4 = 0,

70] COORDINATES IN SPACE 197

as the plane [1, 1, 1, 1] the plane containing the points ( 1, 1, 0, 0),
(-1,0,1,0), (-1,0,0,1)-

With this choice of coordinates the planes [1, 0, 0, 0], [0, 1, 0, 0],
[0, 0, 1, 0], and [1, 1, 1, 0] through the vertex F 4 , say, whose point
coordinates are (0, 0, 0, 1), meet the opposite face x = in lines
whose equations in that plane are

Hence the first three coordinates of any plane [u v ^ 2 , u v 0] on F 4
are the line coordinates of its trace on # 4 = 0, in a system so chosen
that the point (SG V x 2 , # 3 ) is on the line [u v u S9 u s ] if and only if the
relation u^ + U Q X & + u s x s = is satisfied. Hence a point (x v x z , x v 0)
lies on a plane [u l9 u 2 , u^ 0] if and only if we have ufa + UyfB 9 +
^ 3 # s = 0. But any point (sc v # 2 , # 8 , # 4 ) on the plane [u v u^ u^ 0] has,
by definition, its first three coordinates identical with the first three
coordinates of some point oa the trace of this plane with the plane
x = 0. Hence any point (x 19 x 2 , x 3 , x 4 ) on [u v u 2 , u s , 0] satisfies the
condition ufa + u^x z + u z x s + ufa*= 0. Applying this reasoning to
each of the four vertices of the tetrahedron of reference and dualizing,
we find that if one coordinate of \u v u 2 , ^ 3 , %J is zero, the necessary
and sufficient condition that this plane contain a point (x v % 2 , # 8 , x 4 )
is that the relation

U^ + U& + Ufa + Ufa =

le satisfied; and if one coordinate of (x v x z , X B , x 4 ) is zero, the neces-
sary and sufficient condition that this point be on the plane [^ v u &9 u^u^\
is likewise that tJie relation just given "be satisfied.

Confining our attention now to points and planes no coordinate of
which is zero, let xjx^x 9 xjx^y^ a? 8 /# 4 = 2, and let uju^% 9
w a /w 4 =0, u z f%^w. Since x 9 y, z are the ratios of homogeneous
coordinates on the lines # 2 = # 3 = 0, x^ = x 3 = 0, and x = x% = respec-
tively, they satisfy the definition of nonhomogeneous coordinates
given in 69. And since the homogeneous coordinates have been
so chosen that the plane (u v % z , u^ u^ meets the line x z = x z = in
the point ( u 9 0, 0, u^ = ( 1/u, 0, 0, 1), it follows that u, v } w are
nonhomogeneous plane coordinates so chosen that a point (x, y, z) 9
none of whose coordinates is zero, is on a plane [%, v> w\ none of
whose coordinates is zero, if and only if we have (Theorem 9)

ux + *oy -I- ^^ + 1 = ;

198

COORDINATE SYSTEMS

[CHAP. VII

that is, if and only if we have

U& 4- Ufa + u s x 3 + Ufa = 0.

This completes for all cases the proof of

THEOREM 11. The necessary and sufficient condition that a point
(Xy x 2i X B) # 4 ) le on a plane [u v u^ u s> u^ is that the relation

Ufa + Ufa + Ufa + Ufa =
be satisfied.

By methods analogous to those employed in 64 and 65 we may
now derive the results of Exs. 1-8 below.

EXERCISES

1. The equation of the plane through the three points A = (a v a 2 , # 3 , a 4 ),
B = (1 19 b z , 3 , Z> 4 ), C = (c x , c 2 , c 3 , c 4 ) is

#1 2T 5.' a

5 1 & 2 5 3 & 4

= 0.

Dualize.

2. The necessary and sufficient condition that four points -4, B, C 7 D be
coplanar is the vanishing of the determinant

b l 5 3 b z 5 4

3. The necessary and sufficient condition that three points A, B, C be
collinear is the vanishing of the three-rowed determinants of the matrix

4. Any point of a pencil of points containing A and -B may be represented by

P =

5. Any plane of a pencil of planes containing m = [%, m 2 , w 8 , wi 4 ] and
n = [n lf n 2 , TZ S , n 4 ] may be represented by

6. Any projectivity between two one-dimensional primitive forms (of points
or planes) in space is expressed by a relation between their parameters X, p,
of the form

If the base elements of the pencil are homologous, this relation reduces to

70,71] LINEAE TRAITSFOBMATION 1 199

7. If \ v X 21 \ 3 , \ 4 are the parameters of four points or planes of a pencil,
tlieir cross ratio is

8. Any point (plane) of a plane of points (bundle of planes) containing
the noncollinear points A, B, C (planes a, /?, y) may be represented by

P = (A^ + X^ + Ag^, A.ja 2 4- Xj&j -f X 3 c 2 , X^g + X 2 Z> 8 -f A 8 <? 8 , A.^ + X^ -f \ 3 c 4 ).

9. Derive the equation of the polar plane of any point with regard to the
tetrahedron of reference.

10. Derive the equation of a cone.

*11. Derive nonhomogeneous and homogeneous systems of coordinates in
a space of four dimensions.

71. Linear transformations in space. The properties of a linear
transformation in space

are similar to those found in 68 for the linear transformations in a
plane. If the determinant of the transformation

is different from zero, the transformation (1) will have a unique in-
verse, viz.:

< ;

where the coefi&cients ^t y . are the cof actors of the elements a y . respec-
tively in the determinant A.

The transformation is evidently a collineation, as it transforms the

^ G %!#! + ^ 2 # 2 + w 8 # s + i6 4 a; 4 =

into the plane

2 4- A BB u s -h

200 COOKDINATE SYSTEMS [CHAP, vn

Hence the collineation (1) produces on the planes of space the trans-
formation

( ' a-iol

aru[ = A^Ut 4-

To show that the transformation (1) is projective consider any
pencil of planes

In accordance with (2) this pencil is transformed into a pencil of the
form

(a^x 1 + a fa + ax B + a 4 \) + X (6^ + 1 fa + I fa + 1^ = 0,

and these two pencils of planes are projective (Ex. 6, p. 198).

Finally, as in 67, we see that there is one and only one trans-
formation (1) changing the points (0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0),
(1, 0, 0, 0), and (1, 1, 1, 1) into the vertices of an arbitrary complete
five-point in space. Since this transformation is a projective collinea-
tion, and since there is only one projective collineation transforming
one five-point into another (Theorem 19, Chap. IV), it follows that
every projective collineation in space may be represented by a linear
transformation of the form (1). This gives

THEOREM 12. Any projective collineation of space may le repre-
sented in point coordinates ly equations of the form (1), or in plane
coordinates by equations of the form (3). In each case the determinant
of the transformation is different from zero. Conversely, any trans-
formation of this form in which the determinant is different from zero
represents a protective collineation of space.

COEOLLAET 1. In nonhomogeneous point coordinates a projective
collineation is represented ly the linear fractional equations

in which the determinant A is different from zero.

71,72] FINITE SPACES 201

COROLLARY 2. If the singular plane of the nonhomogeneous system
is transformed into itself, these equations reduce to
x f = a

72. Finite spaces. It will be of interest at this point to emphasize
again the generality of the theory which we are developing. Since
all the developments of this chapter are on the basis of Assumptions
A, E, and P only, and since these assumptions imply nothing regard-
ing the number system of points on a line, except that it be commu-
tative, it follows that we may assume the points of a line, or, indeed,
the elements of any one-dimensional form, to be in one-to-one recip-
rocal correspondence with the elements of any commutative number
system. We may, moreover, study our geometry entirely by analytic
methods. From this point of view, any point in a plane is simply a
set of three numbers (x v # 2 , # 3 ), it being understood that the sets
(x v x 2> X B ) and (Jcx v Jcx 2 , Jcx s ) are equivalent for all values of A in the
number system, provided k is different from 0. Any line in the plane
is the set of all these points which satisfy any equation of the form
2^ + u z x 2 + u 8 x B = 0, the set of all lines being obtained by giving
the coefficients (coordinates) [u v u^ u$] all possible values in the
number system (except [0, 0, 0]), with the obvious agreement that
[u^u^Uz] and [ku 19 ku s , JcUi] represent the same line (^=^0). By
letting the number system consist of all ordinary rational numbers,
or all ordinary real numbers, or all ordinary complex numbers, we
obtain respectively the analytic form of ordinary rational, or real, or
complex projective geometry in the plane. All of our theory thus
far applies equally to each of these geometries as well as to the
geometry obtained by choosing as our number system any field
whatever (any ordinary algebraic field, for example).

In particular, we may also choose a finite field, i.e. one which con-
tains only a finite number of elements. The simplest of these are
the modular fields, the modulus being any prime number^.* If we

* A modular field with modulus p is obtained as follows : Two integers n, n f
(positive, negative, or zero) are said to be congruent modulo p, written n=n', mod.p,
if the difference n n' is divisible by p. Every integer is then congruent to one
and only one of the numbers 0, 1, 2, , p 1. These numbers are taken as the
elements of our field, and any number obtained from these ly addition, subtraction*

202 COORDINATE SYSTEMS [CHAP.VH

consider, for example, the case p = 2, our number system contains
only the elements and 1. There are then seven points, which we
will label A, B, C, D, E, F, G, as follows : A = (0, 0, 1), B = (0, 1, 0),
<? = (!, 0, 0), D = (0, 1, 1), J0 = (l, 1, 0), ^ = (1, 1, 1), ff (1, 0, 1).
The reader will readily verify that these seven points are arranged
in lines according to the table

A B C D F G
B C D E F G A
D E F G A B C,

each column constituting a line. Tor example, the line a? 1 = clearly
consists of the points (0, 0, 1) = A, (0, 1, 0) = B, and (0, 1, 1) = D, these
"being the only points whose first coordinate is 0. We have labeled
the points of this finite plane in such a way as to exhibit clearly its
abstract identity with the system of triples used for illustrative pur-
poses in the Introduction, 2.*

EXERCISES

1. Verify analytically that two sides of a complete quadrangle containing a
diagonal point are harmonic with the other two diagonal points.

2. Show analytically that if two projective pencils of lines in a plane have
a self -corresponding line, they are perspective. (This is equivalent to Assump-
tion P.)

3. Show that the lines whose equations are x 1 + \x 2 = 0, x% 4- pjc 8 = 0, and
#s +vx t = are concurrent if X/w = 1 ; and that they meet the opposite
sides of the triangle of reference respectively in collinear points, if X/xv = 1.

4. Find the equations of the lines joining (c v c 2 , c 8 ) to the four points
(1, 1, 1), and determine the cross ratios of the pencil.

and multiplication, if not equal to one of these elements, is replaced by the element
to which it is congruent. The modular field with modulus 5, for example, consists of
the elements 0, 1, 2, 3, 4, and we have as examples of addition, subtraction, and
multiplication 14-3 = 4, 2 + 3 = (since 5 = 0, mod. 5), 1 - 4 = 2, 2 . 3 = 1, etc,
Furthermore, if a, 5 are any two elements of this field (a^ 0), there is a unique
element x determined by the congruence ax = 6, mod. p; this element is defined
as the quotient b/a (For the proof of this proposition the reader may refer to any
standard text on the theory of numbers.) In the example discussed we have, for
example, 4/3 = 3.

* For references and a further discussion of finite projective geometries see a
paper by O. Veblen and W. H. Bussey, Finite Projective Geometries, Transactions
of the American Mathematical Society, Vol. VII (1906), pp. 241-259. Also a sub-
sequent paper by 0. Veblen, Collineations in a Finite Projective Geometry, Trans-
actions of the American Mathematical Society, Vol. VIII (1907), pp. 266-268.

72] EXERCISES 203

5. Show that the throw of lines determined on (c l9 c 2 , c s ) by the four
points (1, 1, 1) is projective with (equal to) the throw of lines determined
on (b 19 & 3 , & 3 ) by the points (a l9 a 2 , 3 ), if the following relations hold:

a 3 = 0,

ajajb? + a^bj + a^ajb} = 0,

and that the sis cross ratios are a 3 /a 3 , a 9 /a 19 a^/a^ , a s /a, - aja^
2 /i (C. A. Scott, Mod. Anal. Geom., p. 50).

6. Write the equations of transformation for the five types of planar col-
lineations described in 40, Chap. IV, choosing points of the triangle of
reference as fixed points.

7. Generalize Ex. 6 to space.

8. Show that the set of values of the parameter X of the pencil of lines
m + \n = is isomorphic with the scale determined in this pencil by the lines
for which the fundamental lines are respectively the lines X = 0, 1, oo

9. Show directly from the discussion of 61 that the points whose non-
homogeneous coordinates x 9 y satisfy the equation y = x are on the line joining
the origin to the point (1, 1).

10. There is then established on this line a scale whose fundamental points
are respectively the origin, the point (!?!) and the point in which the line meets
the line fc . The lines joining any point P in the plane to the points oo y , 003.
meet the line y = x in two points whose coordinates in the scale just determined
are the nonhomogeneous coordinates of P, so that any point in the plane
(not on loo) is represented by a pair of points on the line y = x. Hence, show
that in general the points (a;, y) of any line in the plane determine on the
line y = x a projectivity with a double point on Z ; and hence that the equa-
tion of any such line is of the form y = ax -f- b. What lines are exceptions to
this proposition ?

11. Discuss the modular plane geometry in which the modulus is^? = 3 ;
and by properly* labeling the points show that it is abstractly identical with
the system of quadruples exhibited as System (2) on p. 6.

12. Show in general that the modular projective plane with modulus p
contains p* + p + 1 points and the same number of lines ; and that there are
p 4- 1 points (lines) on every line (point).

13. The diagonal points of a complete quadrangle in a modular plane pro-
jective geometry are collinear if and only if p = 2.

14 * Show that the points and lines of a modular plane all belong to the
same net of rationality. Such a plane is then properly projective without the
use of Assumption P.

15. Show how to construct a modular three-space. If the modulus is 2,
show that its points may be labeled 0, 1, . . . , 14 in such a way that the
planes are the sets of seven obtained by cyclic permutation from the set
1 4 6 11 12 13 (i.e. 1 2 5 7 12 13 14, etc.), and that the lines are ob-
tained from the lines 014, 1 2 8, 5 10 by cyclic permutations. (For a

204 GOOEDINATE SYSTEMS [CHAP.VII

study of this space, see G. M, Conwell, Annals of Mathematics, Vol. 11
(1910), p. 60.)

16. Show that the ten diagonal points of a complete five-point in space
(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0), (1, 1, 1, 1) are given by the
remaining sets of coordinates in which occur only the digits and 1.

17. Show that the ten diagonal points in Ex. 16 determine in all 45 planes,
of which each of a set of 25 contains four diagonal points, while each of the
remaining 20 contains only three diagonal points. Through any diagonal
point pass 16 of these planes. The diagonal lines, i.e. lines joining two
diagonal points, are of two kinds : through each of the diagonal lines of the
first kind pass five diagonal planes ; through each line of the second kind pass
four diagonal planes.

18. Show how the results of Ex. 17 are modified in a modular space with
modulus 2 ; with modulus 3. Show that in the modular space with modulus
5 the results of Ex. 17 hold without modification.

* 19. Derive homogeneous and nonhomogeneous coordinate systems for
a space of n dimensions, and establish the formulas for an n-dimensional
projective collineation.

CHAPTER VIII

PROJECTTVITIES IN ONE-DIMENSIONAL FORMS*

73. Characteristic throw and cross ratio.

THEOREM 1. If M 9 N are double points of a projectivity on a line,
and AA f , BB f are any two pairs of homologous points (i.e. if
MNAB -^ MNA'ff), then MNAA -^ MNBB 1 .

Proof. Let S 9 S f be any two distinct points on a line through
M (fig. 85), and let the lines SA and S f A r meet in A" 9 and SB and

S'B' meet in B". The points A", B", Nsxe then collinear (Theorem 23,
Chap. IV). If the line A n B" meets SS' in a point Q> we have

A" B"

MNAA f = MQ8S 1 = MNBB f .

A A

This proves the theorem, which may also be stated as follows :

The throws consisting of the pair of double points in a given order
and any pair of homologous points are all equal.

DEFINITION. The throw "T(MN 9 AA 1 ), consisting of the double points
and a pair of homologous points of a projectivity, is called the charac-
teristic throw of the projectivity ; and the cross ratio of this throw
is called the characteristic cross ratio of the projectivity. |

* All the developments of this chapter are on the basis of Assumptions A, E, P, H .

t Since the double points enter symmetrically, the throws T (MN^ AA 7 ) and
T(JO(f, AA") may be used equally well for the characteristic throw. The corre-
sponding cross ratios R (MN, A A') and B (NM, AA*) are reciprocals of each other
(cf. Theorem 13, Cor. 3, Chap. VI).

205

206 ONE-DIMENSIONAL PROJECTIVITIES [CHAP.VIII

COROLLARY 1. A projectivity on a line with two given distinct
double points is uniquely determined ly its characteristic throw or
cross ratio.

COROLLARY 2. The characteristic cross ratio of any involution with
double points is 1.

This follows directly from Theorem 27, Cor. 1, Chap. IV, and
Theorem 13, Cor. 2, Chap. VI.

If m, n are nonhomogeneous coordinates of the double points, and
Jc is the characteristic cross ratio of a projectivity on a line, we have

x t m f x m_ ,
x 1 n ' x n

for every pair of homologous points x, x f . This is the analytic expres-
sion of the above theorem, and leads at once to the following analytic
expression for a projectivity on a line with two distinct double points
m, n :

COROLLARY 3. Any projectivity on a line with two distinct double
points m, n may be represented ~by the equation

x'n x n
x r , x being any pair of homologous points.

For when cleared of fractions this is a bilinear equation in x j , x
which obviously has m, n as roots. Moreover, since any projectivity
with two given distinct double points is uniquely determined by one
additional pair of homologous elements, it follows that any projec-
tivity of the kind described can be so represented, in view of the fact
that one such pair of homologous points will always determine the
multiplier k These considerations offer an analytic proof of Theo-
rem 1, for the case when the double points Jf, 2f are distinct.

It is to be noted, however, that the proof of Theorem 1 applies
equally well when the points M, N coincide, and leads to the follow-
ing theorem :

THEOREM 2. If in a paralolic projectivity with double point M the
points A A 1 and BB f are two pairs of homologous points , the parabolic
projectivity with double point M which puts A into B also puts A 1
into B'.

COROLLARY. The characteristic cross ratio of any parabolic projcc-
timty is unity.

73] CHARACTERISTIC THROW 207

The characteristic cross ratio together with the double point is
therefore not sufficient to characterize a parabolic projectivity com-
pletely. Also, the analytic form for a projectivity with double points
m, n, obtained above, breaks down when m = n. We may, however,
readily derive a characteristic property of parabolic projectivities,
from which will follow an analytic form for these projectivities.

THEOREM 3. If a parabolic projectivity with doiible point M trans-
forms a point A into A! and A f into A fl , the pair of points A, A rf is
harmonic with the pair A f M; i.e. we have H (MA r , AA ff ).

Proof. By Theorem 23, Chap. IV, we have Q(MAJL*, MA" A').

Analytically, if the coordinates of M 9 A, A r , A rf are m, %, x r , x n
respectively, we have, by Theorem 13, Cor. 4, Chap. VI,

2 = 1 1

x ffit x " in x - yn
This gives

x'm x m x n m x'm

which shows that if each member of this equation be placed equal to
t, the relation

(1)

' m x

is satisfied by every pair of homologous points of the sequence obtained
by applying the projectivity successively to the points A, A r , A",
It is, however, readily seen that this relation is satisfied by every pair
of homologous points on the line. Tor relation (1), when cleared of
fractions, clearly gives a bilinear form in x f and x, and is therefore a
projectivity; and this projectivity clearly has only the one double
point m. It therefore represents a parabolic projectivity with the
double point m y and must represent the projectivity in question, since
the relation is satisfied by the coordinates of the pair of homologous
points A, A\ which are sufficient with the double point to determine
the projectivity.

We have then :

COROLLABY 1. Any parabolic projectivity with a double point, M,
may be represented by the relation (1).

DEFINITION. The number t is called the characteristic constant of
the projectivity (1).

208 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP. VIII

COROLLARY 2. Conversely, if a projectivity with a double point
M transforms a point A into A', and A* into A !r , such that we have
H (MA 1 , -4-4"), the projectwity is parabolic.

Proof. The double point M and the two pairs of homologous
points AA r , A'A n are sufficient to determine the projectivity uniquely;
and there is a parabolic projectivity satisfying the given conditions.

74. Projective projectivities. Let TT be a projectivity on a line I,
and let TT I be a projectivity transforming the points of I into the
points of another or the same line V. The projectivity Ti^Tnrf 1 is then
a projectivity on V. For Tr^ 1 transforms any point of V into a point
of Z, TT transforms this point into another point of I, which in turn is
transformed into a point of V by ir v Thus, to every point of V is made
to correspond a unique point of l f , and this correspondence is projec-
tive, since it is the product of projective correspondences. Clearly,
also, the projectivity ir^ transforms any pair of homologous points of
TT into a pair of homologous points of w^w-Trf 1 .

DEFINITION. The projectivity Tj^Trrrf 1 is called the transform of V
"by TT X ; two projectivities are said to be projectile or conjugate if one
is a transform of the other by a projectivity.

The question now arises as to the conditions under which two pro-
jectivities are projective or conjugate. A necessary condition is evi-
dent. If one of two conjugate projectivities has two distinct double
points, the other must likewise have two distinct double points; if
one has no double points, the other likewise can have no double points ;
and if one is parabolic, the other must be parabolic. The further
conditions are readily derivable in the case of two projectivities with
distinct double points and in the case of two parabolic projectivities.
They are stated in the two following theorems :

THEOREM 4. Two projectivities each of which has two distinct double
points are conjugate if and only if their characteristic throws are equal.

Proof. The condition is necessary. For if TT, TT' are two conjugate
projectivities, any projectivity T^ transforming TT into TT' transforms
the double points M, N of TT into the double points M', N f of TT', and
also transforms any pair of homologous points A, A l of TT into a pair
of homologous points A 1 , A[ of ir 1 ; i.e.

Tr^MNAA^ = M'N'A'Af.
But this states that their characteristic throws are equal.

74, 75] GKOUPS ON A LINE 209

The condition is also sufficient ; for if it is satisfied, the projec-
tivity TT^ defined by

clearly transforms TT into TT'.

COROLLARY. Any two involutions with doiible points. are conjugate.
THEOREM 5. Any two parabolic projectivities are conjugate.
Proof. Let the two parabolic projectivities be defined by
ir(MMA) = MMA V and 7r r (M r M'A r ) = Jf'Jf^'.
Then the projectivity ^ defined by

ir^MAAJ = M'A'Af

clearly transforms TT into TT'.

Since the characteristic cross ratio of any parabolic projectivity is
unity, the condition of Theorem 4 may also be regarded as holding
for parabolic projectivities.

75. Groups of projectivities on a line. DEFINITION. Two groups G
and G ; of projectivities on a line are said to be conjugate if there
exists a projectivity ir^ which transforms every projectivity of G into a
projectivity of G', and conversely. We may then write 7r 1 G'7r~ ;l =G / ;
and G' is said to be the transform of G ly nr.

We have already seen (Theorem 8, Chap. Ill) that the set of all
projectivities on a line form a group, which is called the general pro-
jective group on the line. The following are important subgroups :

1. TJie set of all projectivities leaving a given point of the line
invariant.

Any two groups of this type are conjugate. For any projectivity
transforming the invariant point of one group into the invariant point
of the other clearly transforms every projectivity of the one into
some projectivity of the other. Analytically, if we choose % = oo as
the invariant point of the group, the group consists of all projectivities

of the form

x r = ase + 'b.

2. The set of all projectivities leaving two given distinct points
invariant

* Any two groups of this type are conjugate. Tor any projectivity
transforming the two invariant points of the one into the invariant
points of the other clearly transforms every projectivity of the one

210 ONE-DIMENSIONAL PEOJECTIYITIES [CHAP.VIII

into a projectivity of tlie other. Analytically, if x v x 2 are the two
invariant points, the group consists of all projectivities of the form

AU( ____

x f ~

The product of two such projectivities with multipliers k and Id is
clearly given by

x f x z

This shows that any two projectivities of this group are commuta-
tive. This result gives

THEOREM 6. Any two projectivities which have two double points
in common are commutative.

This theorem is equivalent to the commutative law for multiplication.
If the double points are the points and oo, the group consists of all projec-
tivities of the form x f = ax.

3. The set of all parabolic projectivities with a common double point
In order to show that this set of projectivities is a group, it is only
necessary to show that the product of two parabolic projectivities
with the same double point is parabolic. This follows readily from
the analytic representation. The set of projectivities above described
consists of all transformations of the form

^ l

where x^ is the common double point (Theorem 3, Cor. 1). If

___

/VM .. fy> f)i __ /y

ifj " " IAJ, tfj ^^ tAj*

are two projectivities of this set, the product of the first by the second
is given by j j

-. =: ' ~r & T" ^o,

a/ x l x %i x

which is clearly a projectivity of the set. It shows, moreover, that
any two projectivities of this group are commutative. Whence

THEOREM 7. Any two parabolic projectivities on a line with the
same double point are commutative.

This theorem is independent of Assumption P, although this assumption*
is implied in the proof we have given. The theorem has already been proved
without this assumption in Example 2, p. 70.

75] GBOUPS ON A LINE 211

Any two groups of this type are conjugate. For every projectivity
transforming the double point of one group into the double point of
the other transforms the one group into the other, since the projec-
tive transform of a parabolic projectivity is parabolic.

DEFINITION. Two subgroups of a group G are said to be conjugate
under G if there exists a transformation of G which transforms one
of the subgroups into the other. A subgroup of G is said to be self-
conjugate or invariant under G if it is transformed into itself by
every transformation of G; i.e. if every transformation in G trans-
forms any transformation of the subgroup into another (or the same)
transformation of the subgroup.

We have seen that any two groups of any one of the three types
are conjugate subgroups of the general projective group on the line.
We may now give an example of a self-conjugate subgroup.

Tlie set of all parabolic projectivities in a group of Type 1 alove is
a self-conjugate subgroup of this group. It is clearly a subgroup, since
it is a group of Type 3. And it is self-conjugate, since any conjugate
of a parabolic projectivity is parabolic, and since every projectivity of
the group leaves the common double point invariant.

EXERCISES

1. Write the eqtiations of all the projective transformations which permute
among themselves (a) the points (0, 1), (1,0), (1, 1) ; (&) the points (0, 1),
(1,0), (1,1), (a,&); (c) the points (0,1), (1,0), (1,1), (-1,1). What
are the equations of the self-conjugate subgroup of the group of transforma-
tions (a)?

2. If a projectivity x r (ax + I}/ (ex -f d) having two distinct double ele-
ments be written in the form of Cor. 3, Theorem 1, show that

and that

ad be

3. If a parabolic projectivity x f = (ax + b)/(cx + d) be written in the form
of Theorem 3, Cor. 1, show that x^ = (a d)/2 c, and t = 2 c/(a + ?).

4. Show that a projectivity with distinct double points x v x z and charac-
teristic cross ratio k can be written in the form

or 1
a?! 1 1
ar k 1

212 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP.VIII

5. Show that the parabolic projectivity of Theorem 3, Cor. 1, may be
written in the form

i ^i

1
1
1

x
1

1 i

1
1

6. If by means of a suitably chosen transformation of a group any of the
elements transformed may be transformed into any other element, the group
is said to be transitive. If by a suitably chosen transformation of a group any
set of n distinct elements may be transformed into any other set of n distinct
elements, and if this is not true for all sets of n + 1 distinct elements, the
group is said to be n-ply transitive. Show that the general projective group on
a line is triply transitive, and that of the subgroups listed in 75 the first
is doubly transitive and the other two are simply transitive.

7. If two projecti vities on a line, each having two distinct double points,
have one double point in common, the characteristic cross ratio of their prod-
uct is equal to the product of their characteristic cross ratios.

76. Projective transformations between conies. "We have consid-
ered hitherto projectivities between one-dimensional forms of the
first degree only. We shall now see how projectivities exist also be-
tween one-dimensional forms of the second degree, and also between
a one-dimensional form of the first and one of the second degree.
Many familiar theorems will hereby appear in a new light.

As typical for the one-dimensional forms of the second degree we
choose the conic. The corresponding theorems for the cone then
follow by the principle of duality.

Let TTj be a projective collineation between two planes a, a v and
let C z be any conic in a. Any two projective pencils of lines in a
are then transformed by ^ into two projective pencils of lines in a v
such that any two homologous lines of the pencils in a are trans-
formed into a pair of homologous lines in o^; for if TT be the projec-
tivity between the pencils in a', T^TTTrf 1 will be a projectivity between
the pencils in a (cf. 74). Two projective pencils of lines generating
the conic <7 2 thus correspond to two pencils of lines in a generating
a conic Cf. The transformation ir^ then transforms every point of C 2
into a unique point of Cf. Similarly, it is seen that ir^ transforms
every tangent of (7 2 into a unique tangent of Cf.

DEFINITION. Two conies are said to be protective if to every point of
one corresponds a poinjb of the other, and to every tangent of one

76]

TEAJNTSFOEMATION OF COJSTICS

213

corresponds a tangent of the other, in such a way that this correspond-
ence may be brought about by a projective collineation between the
planes of the conies. The projective collineation is then said to
generate the projectivity between the conies.

Two conies in different planes are projective, for example, if one is the pro-
jection of the other from a point on neither of the two planes. If the second
of these is projected back on the plane of the first from a new center, we
obtain two conies in the same plane that are projective. We will see presently
that two projective conies may also coincide, in which case we obtain a pro-
jectivity on a conic.

THEOREM 8. Two conies that are projectile with a third are
projective.

Proof. This is an immediate consequence of the definition and the
fact that the resultant of two collineations is a collineation.

We proceed now to prove the fundamental theorem for projec-
tivities between two conies.

THEOREM 9. 'A projectivity "between two conies is uniquely deter-
mined if three distinct points (or tangents) of one are made to corre-
spond to three distinct points (or tangents] of the other.

FIG. 86

Proof. Let <7 2 , <7 2 be the two conies (fig. 86), and let A, B, be
three points of (7 2 , and A f , B r , C r the corresponding points of Cf. Let
P and P f be the poles of AB and A'Jff with respect to <7 2 and <7 X
respectively. If now the collineation TT is defined by the relation
7r(ABCP)=zA f B r C'P f (Theorem 18, Chap. IV), it is clear that the
conic (7 2 is transformed by TT into a conic through the points A', B f , C r 9
with tangents A'P* and B f P f . This conic is uniquely determined by
these specifications, however, and is therefore identical with CJ. The
collineation TT then transforms C 2 into Cf in such a way that the
points A, B, C are transformed into A 1 , B f , C f respectively. Moreover,

214 ONE-DIMENSIONAL PEOJECTIYITIES [CHAP.VIII

suppose 7r f were a second collineation transforming C 2 into C* in the
way specified. Then 7r'"V would be a collineation leaving A, B, C, P
invariant ; i.e. TT = TT'.

The argument applies equally well if A f B'C f are on the conic O 2 ,
i.e. when the two conies 2 , C? coincide. In this case the projectivity
is on the conic C. This gives

COROLLARY 1. A projectivity on a conic is uniquely determined when
three pairs of homologous elements (points or tangents] are given.

Also from the proof of the theorem follows

COROLLARY 2. A collineation in a plane which transforms three
distinct points of a conic into three distinct points of the same conic and
which transforms the pole of the line joining two of the first three
points into the pole of the line joining the two corresponding points
transforms the conic into itself.

The two following theorems establish the connection between pro-
jectivities between two conies and projectivities between one-dimen-
sional forms of the first degree.

THEOREM 10. If A and B 1 are THEOREM 10'. If a and V are
any two points of two projective any two tangents of two protective
conies C* and Of respectively, the conies C z and C* respectively, the
pencils of lines with centers at A pencils of points on a and V are
and B ! are protective if every pair protective if every pair of homol-
of homologous lines of these pencils ogous points on these lines is on
pass through a pair of homologous a pair of homologous tangents of
points on the two conies respectively, the conies respectively.

Proof. It will suffice to prove the theorem on the left. Let A' be
the point of (7f homologous with A. The collineation which generates
the projectivity between the conies then makes the pencils of lines at
A and A f protective, in such a way that every pair of homologous
lines contains a pair of homologous points of the two conies. The pen-
cil of lines at B f is projective with that at A 1 if they correspond in
such a way that pairs of homologous lines intersect on C* (Theorem
2, Chap. V). This establishes a projective correspondence between
the pencils at A and B r in which any two homologous lines pass
through two homologous points of the conies and proves the theorem.

It should be noted that in this projectivity the tangent to C 2 at A
corresponds to the line of the pencil at B' passing through A f .

70]

TBAJtfSFOKMATION OF COLICS

215

COROLLARY. Conversely, if two
conies correspond in such a way
that every pair of homologous tan-
gents is on a pair of homologous
points of two projective pencils of
points whose axes are tangents of
the conies, they are protective.

COROLLARY. Conversely, if two
conies correspond in such a way
that every pair of homologoits
points is on a pair of homologous
lines of two protective pencils of
lines whose centers are on the
conics 9 they are projective.

Proof. This follows from the fact that the projectivity between the
pencils of lines is uniquely determined by three pairs of homologous
lines. A projectivity between the conies is also determined by the
three pairs of points (Theorem 9), in which three pairs of homolo-
gous 4 lines of the pencils meet the conies. But by what precedes
and the theorem above, this projectivity is the same as that described
in the corollary on the left. The corollary on the right may be proved
similarly. If the two conies are in the same plane, it is simply the
plane dual of the one on the left.

By means of these two theorems the construction of a projectivity
between two conies is reduced to the construction of a projectivity
between two primitive one-dimensional forms.

It is now in the spirit of our previous definitions to adopt the
following :

DEFINITION. A Hue conic and
a pencil of points whose axis is
a line of the conic are said to be
perspective if they correspond in
such a way that every line of the
conic passes through the homolo-
gous point of the pencil of points.
A line conic and a pencil of lines
are said to be perspective if every
two homologous lines meet in a
point of a pencil of points whose

DEFINITION. A point conic and
a pencil of lines whose center is a
point of the conic are said to be
perspective if they correspond in
such a way that every point of
the conic is on the homologous
line of the pencil. A point conic
and a pencil of points are said to
be perspective if every two homol-
ogous points are on the same line
of a pencil of lines whose center
is a point of the conic.

axis is a line of the conic.

The reader will now readily verify that with this extended use of
the term perspective, any sequence of perspectivities leads to a pro-
jectivity. For example, two pencils of lines perspective with the same
point conic are projective by Theorem 2, Chap. V; two point conies

216 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP.VIII

perspective with the same pencil of lines or with the same pencil of
points are projective by Theorem 10, Cor., etc.

Another illustration of this extension of the notion of perspectivity
leads readily to the following important theorem :

THEOEEM 11. Two conies which are not in the same plane and have
a common tangent at a point A are sections of one and the same cone.

Proof. If the two conies (7 2 , C* (fig. 87) are made to correspond
in such a way that every tangent x of one is associated with that

tangent #'of the other
which meets x in a
point of the common
tangent a of the conies,
they are projective.
For the tangents of
the conies are then

87 perspective with the

same pencil of points

(cf. Theorem 10', Cor.). Every pair of homologous tangents of the two
conies determines a plane. If we consider the point of intersection
of three of these planes, say, those determined by the pairs of tangents
W, erf, dd f , and project the conic C* on the plane of <7 2 from 0, there
results a conic in the plane of C*. This conic has the lines 6, c, d for
tangents and is tangent to a at A ; it therefore coincides with C 2
(Theorem 6', Chap. V). The two conies (7 2 , C? then have the same
projection from 0, which proves the theorem.*

EXERCISES

1. State the theorems concerning cones dual to the theorems of the preced-
ing sections.

2. By dualizing the definitions of the last article, define what is meant by
the perspectivity between cones and the primitive one-dimensional forms.

3. If two projective conies have three self-corresponding points, they are
perspective with a common pencil of lines.

4. If two projective conies have four self -corresponding elements, they
coincide.

5. State the space duals of the last two propositions.

*It will be seen later that this theorem leads to the proposition that any conic
may be obtained as the projection of a circle tangent to it in a different plane.

76, 77]

PEOJECTIVITIES OX A CONIC

217

6. If a pencil of lines and a conic in the plane of the pencil are projective,
but not perspective, not more than three lines of the pencil pass through their
homologous points on the conic. (Hint. Consider the points of intersection of
the given conic with the conic generated by the given pencil and a pencil of
lines perspective with the given conic.) Dualize.

7. The homologous lines of a line conic and a projective pencil of lines in
the same plane intersect in points of a " curve of the third order " such that
any line of the plane has at most three points in. common with it. (This fol-
lows readily from the last exercise.)

8. The homologous elements of a cone of lines and a projective pencil of
planes meet in a " space curve of the third order" such that any plane has
at most three points in common with it.

9. Dualize the last two propositions.

77. Projectivities on a conic. We have seen that two protective'
conies may coincide (Theorems 8-10), in which case we obtain a
projective correspondence among the points or the tangents of the

conic. The construction of the projectivity in this case is very
simple, and leads to many important results. It results from the
following theorems":

THEOREM 12. If A, A f are any THEOREM 12'. If a, a f are any

two distinct homologous points of two distinct homologous tangents

a projectivity on a conic, and B,3 ! ; of a projectivity on a conic, and

C, C r ; etc., are any other pairs of I, V; c, c f ; etc., are any other pairs

218 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP, vin

homologous points, the lines A r B of homologous tangents, the points
and AB', A f C and AC 1 , etc., meet a f b and aV, a f c and ac f , etc., are
in points of the same line; and collinear with the same point;
this line is independent of the pair and this point is independent of
AAl chosen. the pair aa 1 chosen.

Proof. The pencils of lines A* (ABC- - ) and A(A'B f C f < - ) are pro-
jective (Theorem 10), and since they have a self-corresponding line
AA 1 , they are perspective, and the pairs of homologous lines of these
two pencils therefore meet in the points of a line (fig. 88). This
proves the first part of the theorem on the left. That the line thus
determined is independent of the homologous pair AA! chosen then
follows at once from the fact this line is the Pascal line of the simple
hexagon AB'CA'BC 1 , so that the lines B J G and BC f and all other
analogously formed pairs of lines meet on it. The theorem on the
right follows by duality.

DEFINITION. The line and the point determined by the above dual theo-
rems are called the axis and the center of the projectivity respectively.

COROLLARY 1. A (nonidentical) COROLLARY 1'. A (nonidenti-

projectivity on a conic is wiiquely cal) projectivity on a conic is

determined when the axis of ho- wniquely determined when the

mology and one pair of distinct center and one pair of distinct

homologous points are given. homologous tangents are given.

These corollaries follow directly from the construction of the pro-
jectivity arising from the above theorem. This construction is as
follows : Given the axis o and a pair of distinct homologous points
AA f , to get the point P 1 homologous with any point P on the conic ;
join P to A! ; the point P f is then on the line joining A 'to the point
of intersection of A'P with o. Or, given the center and a pair of
distinct homologous tangents aa f , to construct the tangent p f homolo-
gous with any tangent p ; the line joining the point a r p to the center
meets a in a point of p r .

COROLLARY 2. Every double COROLLARY 2'. Every doulle

point of a projectivity on a conic line of a projectivity on a conic

is on the axis of the projectivity ; contains the center of the projec-

and, conversely, every point com- tivity; and, conversely, every tan-

mon to the axis and the conic is gent of a conic through the center

a double point. is a double line of the projectivity.

77]

PEOJECTIYITIES ON A CONIC

219

COROLLARY 3. A projectimty COROLLARY 3'. A projectimty

among the points on a conic is among the tangents to a conic is

parabolic if and only if the axis parabolic if and only if the center

is tangent to the conic. . is a point of the conic.

THEOREM 13. A projectimty among the points of a conic determines
a projectimty of the tangents in which the tangents at pairs of homol-
ogous points are homologous.

Proof. This follows at once from the fact that the collineation in
the plane of the conic which generates the projectivity transforms
the tangent at any point of the conic into the tangent at the homol-
ogous point, and hence also generates a projectivity between the
tangents.

THEOREM 14. The center of a projectivity of tangents on a conic
and the axis of the corresponding projectimty of points are pole and
polar with respect to the conic.

FIG. 89

Proof. Let AA ! , BB ! t CC ! (fig. 89) he three pairs of homologous
points (AA 1 being distinct), and let A'B and AB ! , A'C and AC f , meet
in points E and S respectively, which determine the axis of the pro-
jectivity of points. Now the polar of E with respect to the conic is
determined by the intersections of the pairs of tangents at A f , B and
A) B 1 respectively; and the polar of S is ' determined by the pairs of
tangents at A 1 , C and A, C r respectively (Theorem 13, Chap. V). The
pole of the axis JZS is then determined as the intersection of these

220 ' ONE-DIMENSIONAL PEOJEOTIVITIES [CHAP.VIII

two polars (Theorem 17, Chap. V), But by definition these two polars
also determine the center of the projectivity of tangents.

This theorem is obvious if the projectivity has double elements ; the proof
given, however, applies to all cases.

The collineation generating the projectivity on the conic transforms
the conic into itself and clearly leaves the center and axis invariant.
The set of all collineations in the plane leaving the conic invariant
form a group (cf. p. 67). In determining a transformation of this
group, any point or any line of the plane may be chosen arbitrarily
as a double point or a double line of the collineation ; and any two
points or lines of the conic may be chosen as a homologous pair of
the collineation. The collineation is then, however, uniquely deter-
mined. In fact, we have already seen that the projectivity on the
conic is uniquely determined by its center and axis and one pair of
homologous elements (Theorem 12, Cor. 1); and the theorem just
proved shows that if the center of the projectivity is given, the axis
is uniquely determined, and conversely.

COROLLARY 1. A plane protective collineation which leaves a non-
degenerate conic in its plane invariant is of Type I if it lias two
doutle points on the conic, tmless it is of period tivo, in which case it
is of Type IV; and is of Type III if the corresponding projectivity
on the conic is parabolic.

COROLLARY 2. An elation or a collineation of Type II transforms
every nondegenerate conic of its plane into a different conic.

COROLLARY 3. A plane $rojective collineation which leaves a conic
in its plane invariant and has no double point on the conic has one
and only one double point in the plane.

THEOREM 15. The group of protective collineations in a plane leav-
ing a nondegenerate conic invariant is simply isomorphic* with the
general protective group on a line.

Proof. Let A be any point of the invariant conic. Any projectivity
on the conic then gives rise to a projectivity in the flat pencil at A in
which two lines are homologous if they meet the conic in a pair of
homologous points. And, conversely, any projectivity in the flat

* Two groups are said to be simply isomorpMc if it is possible to establish a (1,1)
correspondence between the elements of the two groups such that to the product of
any two elements of one of the groups corresponds the product of the two corre-
sponding elements of the other.

77,78] INVOLUTIONS 221

pencil at A gives rise to a projectivity on the conic. The group of all
projectivities on a conic is therefore simply isomorphic with the group
of all projectivities in a fiat pencil, since it is clear that in the corre-
spondence described between the projectivities in the flat pencil and
on the conic, the products of corresponding pairs of projectivities will
be corresponding projectivities. Hence the group of plane collineations
leaving the conic invariant is simply isomorphic with the general pro-
jective group in a flat pencil and hence with the general projective
group on a line.

78. Involutions. An involution was defined (p. 102) as any projec-
tivity in a one-dimensional form which is of period two, ie. by the
relation I 2 = 1 (I = 1), where I represents an involution. This relation
is clearly equivalent to the other, I = I~ 1 (l= 1), so that any projec-
tivity (not the identity) in a one-dimensional form, which is identical
with its inverse, is an involution. It will be recalled that since an in-
volution makes every pair of homologous elements correspond doubly,
i.e. A to A! and A f to A, an involution may also be considered as a
pairing of the elements of a one-dimensional form ; any such pair is
then called a conjugate, pair of the involution. We propose now to
consider this important class of projectivities more in detail. To this
end it seems desirable to collect the fundamental properties of invo-
lutions which have been obtained in previous chapters. They are as
follows :

1. If the relation 7r z (A) A Jwlds for a single element A (not a
double element of TT) of a one-dimensional form, the projectivity TT is
an involution, and the relation holds for every element of the form
(Theorem 26, Chap. IV).

2. An involution is uniquely determined when two pairs of conjib-
gate elements are given (Theorem 26, Cor., Chap. IV).

3. The opposite pairs of any quadrangular set are three pairs of
an involution (Theorem 27, Chap. IV).

4. If M, N are distinct double elements of any projectivity in a
one-dimensional form and A, A r and B, B l are any two pairs of
homologous elements of the projectivity, the pairs of elements MN 9 AB J
A f B are three pairs of an involution (Theorem 27, Cor. 3, Chap. IV),

5. If M, N are double elements of an involution, they are distinct,
and every conjugate pair of the involution is harmonic with M, N
(Theorem 27, Cor. 1, Chap. IV).

222

ONE-DIMENSIONAL PEOJECTIVITIES [CHAP, vm

6. An involution is uniquely determined, if two double elements are
given, or if one double element and another conjugate pair are given.
(This follows directly from the preceding.)

7. An involution is represented analytically ly a bilinear form
cscx r a (JG + x') & = 0, or ly the transformation

ex a

(Theorem 12, Cor. 3, Chap. YI).

8. An involution with double elements m, n may be represented ly
the transformation

(Theorem 1, (Tors. 2, 3, Chap. VIII).

We recall, finally, the Second Theorem of Desargues and its various
modifications ( 46, Chap. V), which need not be repeated at this
place. It has been seen in the preceding sections that any projec-
tivity in a one-dimensional primitive form may he transformed into a
projectivity on a conic. We shall find that the construction of an in-
volution on a conic is especially simple, and may be used to advantage
in deriving further properties of involutions. Under duality we may

confine our consideration

*\ to the case of an involu-

' tion of points on a conic.
y THEOREM 16. The lines
joining the conjugate points
of an involution on a conic
all pass through the center
of the involution.

Proof. Let A, A' (fig. 90)
be any conjugate pair (A
not a double point) of an
involution of points on a
conic C 2 . The line A A! is then an invariant line of the collineation gener-
ating the involution. Every line joining a pair of distinct conjugate
points of the involution is therefore invariant, and the generating
collineation must be a perspective collineation, since any collineation
leaving four lines invariant is either perspective or the identity

. 90

78] INVOLUTIONS 223

(Theorem 9, Cor. 3, Chap. III). It remains only to show that the
center of this perspective collineation is the center of the involution.
Let B y B f (B not a double point) be any other conjugate pair of the
involution, distinct from A, A 1 . Then the lines AB f and A'B inter-
sect on the axis of the involution. But since B t B r correspond to each
other doubly, it follows that the lines AB and A f B f also intersect
on the axis. This axis then joins two of the diagonal points of the
quadrangle AA ! BB r . The center of the perspective collineation is
determined as the intersection of the lines AA r and BB ! , i.e. it is
the third diagonal point of the quadrangle AA'BB*. The center of
the collineation is therefore the pole of the axis of the involution
(Theorem 14, Chap. V) and is therefore (Theorem 14, above) the center
of the involution.

Since this center of the involution is clearly not on the conic, the
generating collineation of any involution of the conic is a homology,
whose center and axis o are pole and polar with respect to the conic.
A homology of period two is sometimes called a harmonic homol-
ogy, since it transforms any point P of the plane into its harmonic
conjugate with respect to and the point in which OP meets
the axis. It is also called a protective reflection or a point-line reflec-
tion. Clearly this is the only kind of homology that can leave a conic
invariant.

The construction of the pairs of an involution on a conic is now
very simple. If two conjugate pairs A 9 A* and B, B f are given, the lines
AA 1 and BB J determine the center of the involution. The conjugate
of any other point C on the conic is then determined as the intersec-
tion with the conic of the line joining C to the center. If the involu-
tion has double points, the tangents at these points pass through the
center of the involution ; and, conversely, if tangents can be drawn to
the conic from the center of the involution, the points of contact of
these tangents are double points of the involution.

The great importance of involutions is in part due to the following
theorem :

THEOREM 17. Any projectimty in a one-dimensional form may le
obtained, as the product of two involutions.

Proof. Let II be the projectivity in question, and let A be any
point of the one-dimensional form which is not a double point.

224 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP.VIII

Further, let U(A) = A! and ft (A 1 ) = A". Then, if I, is the involution
of which A 1 is a double point and of which AA n is a conjugate pair
(Prop. 6, p. 222), we have

so that in the projectivity I^II the pair AA f corresponds to itself
doubly. I^II is therefore an involution (Prop. 1, p. 221). If it be
denoted by I 2 , we have 1^11 = 1^ or TL^I^I^ which was to be
proved.

This proof gives at once :

COROLLARY 1. Any projectivity II may le represented as the prod-
uct of two involutions, II = I 1 -I 2 , either of which (but not lotli) has
an arbitrary point (not a double point of II) for a double point.

Proof. We have seen above that the involution ^ may have an
arbitrary point (A 1 ) for a double point. If in the above argument we
let I 2 be the involution of which A f is a double point and AA rf is a
conjugate pair, we have II I & (A'A' r ) = A r! A r ; whence II I 2 is an invo-
lution, say I r We then have II = 1^ I 2 , in which I 2 has the arbitrary
point A f for a double point.

The argument given above for the proof of the theorem applies
without change when A = A ff , i.e. when the projectivity II is an in-
volution. This leads readily to the following important theorem :

COROLLARY 2. If A A! is a conjugate pair of an involution I, the
involution of which A, A! are double points transforms I into itself,
and the two involutions are commutative.

Proof. The proof of Theorem 17 gives at once I I^Lj, where \
is determined as the involution of which A,A f are double points. We
have then I x - 1 = I 2 , from which follows, by taking the inverse of both
sides of the equality, I ^ = I^ 1 = I 2 , or l l - 1 = ! I a , or I x I I x = I.

As an immediate corollary of the preceding we have

COROLLARY 3. The product of two involutions with double points
A, A f and B, B ! respectively transforms into itself the involution in
which A A f and B B' are two conjugate pairs.

Involutions related as are the two in Cor. 2 above are worthy of
special attention.

DEFINITION. Two involutions are said to be harmonic if their
product is an involution.

78,79] INVOLUTIONS 225

THEOREM 18. Two harmonic involutions are commutative.

Proof. If I v I 2 are harmonic, we have, by definition, I x - 1 2 = I 3 , where
I 3 is an involution. This gives at once the relations 1^ I 2 - 1 3 = 1 and

COROLLARY. Conversely, if two distinct involutions are commutative,
they are harmonic.

For from the relation Ij-I^Ig-^ follows (I 1 I 2 ) 2 = 1; i.e. I^I^
is an involution, since I^I^l.

DEFINITION. The set of involutions harmonic with a given involu-
tion is called a pencil of involutions.

It follows then from Theorem 17, Cor. 2, that the set of all involu-
tions in which two given elements form a conjugate pair is a pencil.
Thus the double points of the involutions of such a pencil are the
pairs of an involution.

79. Involutions associated with a given projectivity* In deriving
further theorems on involutions we shall find it desirable to suppose
the projectivities in question to be on a conic.

THEOREM 19. If a projectivity on a conic is represented as the product
of two involutions, the axis of the projectivity is the line joining the
centers of the two involutions.

Proof. Let the given projec-
tivity be II = I 2 - I x ; I v I 2 being
two involutions. Let O v 2 be
the centers of I x , I 2 respectively
(fig. 91), and let "A and B be
any two points on the conic
which are not double points of
either of the involutions \ or I 2
and which are not a conjugate
pair of I t or I 3 . If, then, we
have II (AB) = A'B 1 , we have, by
hypothesis, \(AB) = A^B^ and

) = A f B'- ? AV B 1 being uniquely determined points of the conic,

such that the lines AA V BB^ intersect in 1 and the lines A^AI 9
intersect in 2 . The Pascal line of the hexagon AA^BB^ff then
passes through O v O z and the intersection of the lines AB 1 and A'B.
But the latter point is a point on the axis of II. This proves the theorem.

226 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP, vni

COROLLARY. A projectivity on a conic is the product of two involu-
tions, the center of one of which may le any arbitrary point (not a
double point) on the axis of the projectivity ; the center of the other
is then uniquely determined.

Proof. Let the projectivity II be determined by its axis I and any
pair of homologous points A, A 1 (fig. 91). Let 1 be any point on the
axis not a double point of II, and let I x be the involution of which
O x is the center. If, then, l l (A) = A v the center 2 of the involution
I such that II = I 3 - I x is clearly determined as the intersection of the
line A^A! with the axis. For by the theorem the product / 2 J a is a
projectivity having I for an axis, and it has the points A, A f as a homol-
ogous pair. This shows that the center of the first involution may
be any point on the axis (not a double point). The modification of
this argument in order to show that the center of the second involu-
tion may be chosen arbitrarily (instead of the center of the first) is
obvious.

THEOREM 20. There is one and only one involution commutative
with a given nonparabolic noninvolutoric projectivity. If the projec-
tivity is represented on a conic, the center of this involution is the
center of the projectivity.

Proof. Let the given nonparabolic projectivity II be on a conic,
and let I be any involution commutative with H ; Le. such that we
have II I = I- II. This is equivalent to II I II"" 1 = I. That is to say,
I is transformed into itself by II. Hence the center of I is transformed
into itself by the collineation generating II. But by hypothesis the
only invariant points of this collineation are its center and the points
(if existent) in which its axis meets the conic. Since the center of I
cannot be on the conic, it must coincide with the center of II. More-
over, if the center of I is the same as the center of II, I is trans-
formed into itself by the collineation generating II, II -LIT" * = I.
Hence H I = I H. Hence I is the one and only involution commu-
tative with II,

COROLLARY 1. There is no involution commutative with a parabolic
projectivity.

DEFINITION. The involution commutative with a given nonpara-
bolic noninvolutoric projectivity is called the involution belonging to
the given projectivity. An involution belongs to itself.

79] INVOLUTION'S 227

COROLLARY 2. If a nonparabolic projectivity has double points, the
involution belonging to the projectivity has the same double points.

For if the axis of the projectivity meets the conic in two points,
the tangents to the conic at these points meet in the pole of the axis.

It is to be noted that the involution I belonging to a given projec-
tivity II transforms II into itself, and is transformed into itself by II.
Indeed, from the relation II I = I II follow at once the relations
I.II-I = II and DM'II~' 1 = L Conversely, from the equation

iM-n- 1 follows n.i = Mi.

THEOREM 21. The necessary and sufficient condition that two invo-
lutions on a conic be harmonic is that their centers be conjugate with
respect to the conic.

Proof. The condi- 3 \^^-^ A

tion is sufficient. For
let I 1? I 2 be two invo-
lutions on the conic
whose centers O v 2
respectively are con-
jugate with respect
to the conic (fig. 92).
Let A be any point
of the conic not a
double point of either involution, and let I 1 (A)^A 1 and I Z (AJ=A'.
If, then, l l (A f ) = A[, the center O l is a diagonal point of the quadrangle
AA^A!A[, and the center 2 is on the side A^A 1 . Since, by hypothesis,
2 is conjugate to O l with respect to the conic, it must be the diago-
nal point on AjA f , i.e. it must be collinear with AA[. We have then
l z >I^(AA J )A f A, i.e. the projectivity l^\ is an involution I 8 . The
center 3 of the involution I 3 is then the pole of the line 1 3 with
respect to the conic (Theorem 19). The triangle 1 a 8 is therefore
self-polar with respect to the conic. It follows readily also that the
condition is necessary. Tor the relation I^Ig Ig leads at once to
the relation 1^ = 1^1,. If O v 3 , 3 are the centers respectively of
the involutions I v I 2 , I 3 , the former of these two relations shows
(Theorem 19) that B is the pole of the line 0^; while the latter
shows that 3 is the pole of the line 0^0 y The triangle 0^0. 2 9 is
therefore self-polar.

228

ONE-DIMENSIONAL PEOJECTIVITIES [CHAP, vm

COROLLARY 1. Given any two involutions, there exists a third invo-
lution which is harmonic with each of the given involutions.

For if we take tlie two involutions on a conic, the involution whose
center is the pole with respect to the conic of the line joining the
centers of the given involutions clearly satisfies the condition of the
theorem for each of the latter.

COROLLARY 2. Three involutions each of which is harmonic to the
other two constitute, together with the identity, a group.

COROLLARY 3. The centers of all involutions in a pencil of involu-
tions are collinear.

THEOREM 22. The set of all projectivities to which belongs the same
involution I forms a commutative group.

Proof. If II, IIj are two projectivities to each of which belongs the
involution I, we have the relations I -II -I = 13 and I-II 1 -I=:II 1 ,
from which follows I-II~ 1 I = n~ 1 and, by multiplication, the rela-
, which shows that the set
forms a group. To show
that any two projectivities
of this group are commu-
tative, we need only sup-
pose the projectivities
given on a conic. Let A
be any point on this
conic, and let II (A) = A !
and TL l (A r ) = A, so that
^ 1 / . Since the

Hani I.II.M.IVI=]

A A,

same involution I belongs,

by hypothesis, both to II and 11^ these two projectivities have the
same axis ; let it be the line I (fig. 93). The point IL^ (A) = A 1 is now
readily determined (Theorem 12) as the intersection with the conic of
the line joining A f to the intersection of the line AA[ with the axis I.
In like manner, H(A^) is determined as tie intersection with the
conic of the line joining A to the intersection of the line A^A r with
the axis I Hence H (AJ = A[, and hence II U^A) = A[.

It is noteworthy that when the common axis of the projectivities
of this group meets the conic in two points, which are then common
double points of all the projectivities of the group, the group is the

79] INVOLUTIONS 229

same as the one listed as Type 2, p. 209. If, however, our geometry
admits of a line in the plane of a conic but not meeting the conic, the
argument just given proves the existence of a commutative group
none of the projectivities of which have a double point.

THEOREM 23. Two involutions have a conjugate pair (or a double
point) in common if and only if the product of the two invohctions
has two double points (or is parabolic).

Proof. This follows at once if the involutions are taken on a conic.
For a common conjugate pair (or double point) must be on the line
joining the centers of the two involutions. This line must then meet
the conic in two points (or be tangent to it) in order that the involu-
tions may have a conjugate pair (or a double point) in common.

EXERCISES

1 . Dualize all the theorems and corollaries of the last two sections.

2. The product of two involutions on a conic is parabolic if and only if the
line joining the centers of the involutions is tangent to the conic. Dualize.

3. Any involution of a pencil is uniquely determined when one of its con-
jugate pairs is given.

4. Let II be a noninvolutoric projectivity, and let I be the involution be-
longing to II; further, let TL(AA') = A' A", A being any point on which the
projectivity operates which is not a double point, and let 1(4') = A. Show,
by taking the projectivity on a conic, that the points A'A are harmonic
with the points A A".

5. Derive the theorem of Ex. <4 directly as a corollary of Prop. 4, p. 221,
assuming that the projectivity n has two distinct double points.

6. From the theorem of Ex. 4 show how to construct the involution be-
longing to a projectivity n on a line without making use of any double points
the projectivity may have.

7. A projectivity is uniquely determined if the involution belonging to it
and one pair of homologous points are given.

8. The product of two involutions I 1? I 2 is a projectivity to which belongs
the involution which is harmonic with each of the involutions I x , I 2 .

9. Conversely, every projectivity to which a given involution I belongs can
be obtained as the product of two involutions harmonic with I.

10. Show that any two projectivities n i9 II 3 may be obtained as the
product of involutions in the form H l = I'I 19 II 3 = I 3 -I; and hence that the
product of the two projectivities is given by IVI^ = I 2 -I r

11. Show that a projectivity H = I-Ii may also be written IE = I 2 -I, I 2
being a uniquely determined involution ; and that in this case the two invo-
lutions Ij, I 2 are distinct unless n is involutoric.

230 ONE-DIMENSIONAL PEOJECTIVITIES [CHAP.VIII

12. Show that if I x , I 2 , I 3 are three involutions of the same pencil, the
relation (I^^-Ig) 2 ^ 1 must hold.

13. If aa', W, cc' are the coordinates of three pairs of points in involution,

u J.T. 4. '- & &'- c c'-a -

show that r = 1.

a 7 c 6 a c' b

80. Harmonic transformations. The definition of harmonic involu-
tions in the section above is a special case of a more general notion
which can he defined for (1, 1) transformations of any kind whatever.

DEFINITION. Two distinct transformations A and B are said to be
harmonic if they satisfy the relation (AB" 1 ) 2 = 1 or the equivalent
relation (BA- 1 )** 1, provided that AB" 1 ^ 1.

A number of theorems which are easy consequences of this defini-
tion when taken in conjunction with the two preceding sections are
stated in the following exercises. (Of. C. Segre, Note sur les homo-
graphies binaires et leur faisceaux, Journal fur die reine und ange-
wandte Mathematik, Vol. 100 (1887), pp. 317-330, and H. Wiener,
Ueber die aus zwei Spiegelungen zusammengesetzten Verwandt-
schaften, Berichte d. K. sachsischen Gesellschaft der Wissenschaften,
Leipag, Vol. 43 (1891), pp. 644-673.)

EXERCISES

1. If A and B are two distinct involutoric transformations, they are har-
monic to their product AB.

2. If three involutoric transformations A, B, T satisfy the relations
(ABF) 2 = 1 , ABF 5* 1 , they are all three harmonic to the transformation AB.

3. If a transformation S is the product of two involutoric transformations
A, B (i.e. 2 = AB) and T is an involutoric transformation harmonic to 21, then
we have (ABr) 2 =l.

4. If A, J3, C, A', B', C' are six points of a line, the involutions A, B, T,
such that T(AA') ~B'B, A(') = CC, *&(CC?)=A'A, are all harmonic to
the same projectivity. Show that if the six points are taken on a conic, this
proposition is equivalent to Pascal's theorem (Theorem 3, Chap. V).

5. The set of involutions of a one-dimensional form which are harmonic
to a given nonparabolic projectivity form, a pencil. Hence, if an involution
with double points is harmonic to a projectivity with two double points, the
two pairs of double points form a harmonic set.

6. Let be a fixed point of a line I, and let C be called the mid-point of a
pair of points A, J3 } provided that C is the harmonic conjugate of with
respect to A and B. If A, B, C, A', B', C'are any six points of I distinct
from 0, and AB' have the same mid-point as A'B, and BC' have the same
mid-point as B'C> then CA' will have the same mid-point as C?A*

80,81] SCALE ON A CONIC 231

7. Any two involutions of the same one-dimensional form determine a
pencil of involutions. Given two involutions A, B and a point M, show how
to construct the other double point of that involution of the pencil of which
one double point is M.

8. The involutions of conjugate points on a line I with regard to the conies
of any pencil of conies in a plane with I form a pencil of involutions.

9. If two nonparabolic projectivities are commutative, the involutions
belonging to them coincide, unless both projectivities are involutions, in which
case the involutions may be harmonic.

10. If [II] is the set of projectivities to which belongs an involution I and
A and B are two given points, then we have [II (.4)] -^ [II (By].

11. A conic through two of the four common points of a pencil of conies
of Type J meets the conies of the pencil in pairs of an involution. Extend
this theorem to the other types of pencils of conies. Dualize.

12. The pairs of second points of intersection of the opposite sides of a
complete quadrangle with a conic circumscribed to its diagonal triangle are in
involution (Sturm, Die Lehre von den Geometrischen Verwandtschaften,
Vol. I, p. U9).

81. Scale on a conic. The notions of a point algebra and a scale
which we have developed hitherto only for the elements of one-
dimensional primitive forms may also be studied to advantage on a
conic. The constructions for the sum and the product of two points
(numbers) on a conic is remarkably simple. As in the case on the
line, let 0, 1, co be any three arbitrary distinct points on a conic C\
Regarding these as the fundamental points of our scale on the conic,
the sum and the product of any two points x, y on the conic (which
are distinct from co) are denned as follows :

DEFINITION. The conjugate of in the involution on the conic
having oo for a double point and x, y for a conjugate pair is called
the sum of the two points x 9 y and is denoted by x + y (fig. 94, left).
The conjugate of 1 in the involution determined on the conic by the
conjugate pairs 0, oo and x, y is called the product of the points x,
y and is denoted by x - y (fig. 94, right).

It will be noted that under Assumption P this definition is entirely
equivalent to the definitions of the sum and product of two points on
a line, previously given (Chap. VI). To construct the point x + y on
the conic (fig. 94), we need only determine the center of the involution
in question as the intersection of the tangent at oo with the line joining
the points x, y. The point x + y is then determined as the intersection
with the conic of the line joining the center to the point 0. Similarly,

232

ONE-DIMENSIONAL PKOJECTIYITIES [CHAP. VIII

to obtain the product of the points x, y we determine the center of the
involution as the intersection of the lines Oo> and xy. The point x - y
is then the intersection with the conic of the line joining this center to

FIG. 94

the point 1. The inverse operations (subtraction and division) lead to
equally simple constructions. Since the scale thus defined is obviously
projective with the scale on a line, it is not necessary to derive again
the fundamental properties of addition and subtraction, multiplication
and division. It is clear from this consideration that the points of a
conic form a field with reference to the operations just defined. This
fact will be found of use in the analytic treatment of conies.

At this point we will make use of it to discuss the existence of the
square root of a number in the field of points. It is clear from the

FIG. 95

preceding discussion that if a number x satisfies the equation x* = a,
the tangent to the conic at the point x must pass through the inter-
section of the lines Ooo and 1 a (fig. 95). A number a will therefore
have a square root in the field if and only if a tangent can le drawn to

81]

SCALE ON A CONIC

233

the conic from the intersection of the lines Oco and 1 a; and, conversely,
if the number a has a sqiiare root in the field, a tangent can be drawn
to the conic from this point of intersection. It follows at once that if
a number a has a square root x, it also has another which is obtained
by drawing the second tangent to the conic from the point of inter-
section of the lines Oco and la. Since this tangent meets the conic
in a point which is the harmonic conjugate of x with respect to Oco,
it follows that this second square root is x. It follows also from
this construction that the point 1 has the two square roots 1 and 1
in any field in which 1 and 1 are distinct, i.e. whenever R Q is satisfied.

We may use these considerations to derive the following theorem,
which will be used later.

THEOREM 2-4. If AA r , BB* are any two distinct pairs of an involu-
tion, there exists one and only one pair CC f distinct from BB 1 such
that the cross ratios
%(AA r , BB>)
B (AA r , CC r )
equal.

Proof. Let the
involution be taken
on a conic, and let
the pairs AA! and
BB f be represented
by the points Oco
and la respectively (fig. 96). Let xx 1 be any other pair of the invo-
lution. We then have, clearly from the above, xx* = a. Further, the
cross ratios in question give

U (Oco, 1 a) = - , B (Oco, 005') = -..
x a x'

These are equal, if and only if x j = ax, or if xx f = ax*. But this implies
the relation a = ax 2 , and since we have a = 0, this gives x* = 1. The
only pair of the involution satisfying the conditions of the theorem
is therefore the pair (7(7' = 1, a.

EXERCISES

1. Show that an involution which has two harmonic conjugate pairs has
double points if and only if 1 has a square root in the field.

2. Show that any involution may be represented by the equation afx a.

_ .,.
FIG. 90

234

ONE-DIMENSIONAL PEOJECTIVITIES [CHAP. VIII

3. The equation of Ex. 13, p. 230, is the condition that the lines joining
the three pairs of points aa' ? W/, cc' on a conic are concurrent.

4. Show that if the involution x'x = a has a conjugate pair W such that
the cross ratio R(0o>, W) has the value X, the number a\ has a square root
in the field.

82. Parametric representation of a conic. Let a scale be established
on a conic <7 2 by choosing three distinct points of the conic as the
fundamental points, say, 0=0, M= oo, A= 1. Then let us establish
a system of nonhomogeneous point coordinates in the plane of the

conic as follows : Let

- N the line OH be the x-

axis, with as origin
and M as <x> x (fig. 97).
Let the tangents at
and M to the conic
meet in a point N 9 and
let the tangent ON be
the y-axis, with -2V" as
oo y . Finally, let the
point A be the point
(1, 1), so that the line
AN meets the #-axis
in the point for which
# = 1, and AM meets
the ^-axis in the point for which y = 1. Now let P = X be any point
on the conic. The coordinates (a?, y) of P are determined by the
intersections of the lines PN and PM with the #-axis and the y-axis
respectively. We have at once the relation

FIG. 97

since the points 0, oo, 1, X on the conic are perspective from M with
points 0, oo, 1, y on the y-axis. To determine x in terms of X, we note,
first, that from the constructions given, any line through N meets the
conic (if at all) in two points whose sum in the scale is 0. In par-
ticular, the points 1, 1 on the conic are collinear with N and the
point 1 on the #-axis, and the points X, X on the conic are collinear
with N and the point x on the re-axis. Since the latter point is also
on the line joining and co on the conic, the construction for multi-
plication on the conic shows that any line through, the point x on

82] PAEAMETEIC BEPEESENTATION 235

the #-axis meets the conic (if at all) in two points whose product is
constant, and hence equal to X 2 . The line joining the point x on the
#-axis to the point 1 on the conic therefore meets the conic again in
the point X 2 . But now we have 0, co, 1, X 2 on the conic perspective
from the point 1 on the conic with the points 0, co, 1, $ on the
#-axis. This gives the relation

x = X 2 .

We may now readily express these relations in homogeneous form.
If the triangle OMN is taken as triangle of reference, ON being
x^ = 0, 031 being os a = 0, and the point A being the point (1, 1, 1),
we pass from the nonhomogeneous to the homogeneous by simply
placing x = xjx v y = xjx y The points of the conic C' 2 may then
le represented ly the relations

(1) a^ia^ifljBssX^X,:!.

This agrees with our preceding results, since the elimination of \
between these equations gives at once

a: '-^053=0,

which we have previously obtained as the equation of the conic.

It is to be noted that the point M on the conic, which corresponds
to the value X = oo, is exceptional in this equation. This exceptional
character is readily removed by writing the parameter X homogene-
ously XsaX^Xy Equations (1) then readily give

THEOREM 25. A conic may le represented analytically ly the equa-
tions x^ : X 2 : x s = \f : \\ : X 2 2 .

This is called a parametric representation of a conic.

EXERCISES

1. Show that the equation" of the line joining two points X 19 X 2 on the conic
(1) above is x l (X x -f X 2 ) # 2 + \iX 2 x s = ; and that the equation of the tan-
gent to the conic at a point \i is x l 2 A^g + X 2 # 3 = 0. Dualize.

2. Show that any collineation leaving the conic (1) invariant is of the form

(Hint. Use the parametric representation of the conic and let the projectivity
generated on the conic by the collineation be Xf = a\i + /JXg, X^ = y\i + 8X 2 .)

CHAPTER IX

GEOMETRIC CONSTRUCTIONS. INVARIANTS

83. The degree of a geometric problem. The specification of a line
by two of its points may be regarded as a geometric operation* The
plane dual of this operation is the specification of a point by two
lines. In space we have hitherto made use of the following geometric
operations : the specification of a line by two planes (this is the
space dual of the first operation mentioned above) ; the specification
of a plane by two intersecting lines (the space dual of the second
operation above) ; the specification of a plane by three of its points
or by a point and a line ; the specification of a point by three planes
or by a plane and a line. These operations are known as linear
operations or operations of the first degree, and the elements deter-
mined by them from a set of given elements are said to be obtained
by linear constrictions, or by constrictions of the first degree. The
reason for this terminology is found in the corresponding analytic
formulations. Indeed, it is at once clear that each of the two linear
operations in a plane corresponds analytically to the solution of a
pair of linear equations ; and the linear operations in space clearly
correspond to the solution of systems of three equations, each of the
first degree. Any problem which can be solved by a finite sequence
of linear constructions is said to be a linear prollem or a problem
of the first* degree. Any such problem has, if determinate, one and
only one solution.

In the usual representation of the ordinary real projective geometry in a
plane by means of points and lines drawn, let us say, with a pencil on a sheet
of paper, the linear constructions .are evidently those that can be carried
out by the use of a straightedge alone. There is no familiar mechanical

* An operation on one or more elements is defined as a correspondence whereby
to the set of given elements corresponds an element of some sort (cf. 48). If the
latter element is uniquely defined by the set of given elements (in general, the order
of the given elements is an essential factor of this determination), the operation is
said to be one-valued. The operation referred to in the text is then a one-valued
operation defined for any two distinct points and associating with any such pair
(the order of the points is in this case immaterial) a new element, viz. a line.

236

8B] DEGEEE OF A GEOMETBIC PEOBLEM 237

device for drawing lines and planes in space. But a picture (which is the
section by a plane of a projection from a point) of the lines and points of
intersection of linearly constructed planes may be constructed with a straight-
edge (cf. the definition of a plane).

As examples of linear problems we mention : (a) the determination
of the point homologous with a given point in a projectivity on a
line of which three pairs of homologous points are given; (6) the
determination of the sixth point of a quadrangular set of which five
points are given; (c) the determination of the second double point
of a projectivity on a line of which one double point and two pairs of
homologous points are given (this is equivalent to (&) ) ; (d) the deter-
mination of the second point of intersection of a line with a conic, one
point of intersection and four other points of the conic being given, etc.

The analytic relations existing between geometric elements offer
a convenient means of classifying geometric problems.* Confining
ourselves, for the sake of brevity, to problems in a plane, a geometric
problem consists in constructing certain points, lines, etc., which bear
given relations to a certain set of points, lines, etc., which are sup-
posed given in advance. In fact, we may suppose that the elements
sought are points only ; for if a line is to be determined, it is sufficient
to determine two points of this line ; or if a conic is sought, it is suffi-
cient to determine five points of this conic, etc. Similar considera-
tions may also be applied to the given elements of the problem,
to the effect that we may assume these given elements all to be
points. This merely involves replacing any given elements that are
not points by certain sets of points having the property of uniquely
determining these elements. Confining our discussion to problems in
which this is possible, any geometric problem may be reduced to
one or more problems of the following form : Given in a plane a
certain finite number of points, to construct a point which shall bear
to the given points certain given relations.

In the analytic formulation of such a problem the given points
are supposed to be determined by their coordinates (homogeneous or
nonhomogeneous), referred to a certain frame of reference. The ver-
tices of this frame of reference are either points contained among the
given points, or some or all of them are additional points which we

* The remainder of this section follows closely the discussion given in Castel-
nuovo, Lezioni di geometria, Rome-Milan, Vol. I (1904), pp. 467 fl.

238 G-EOMETEIC CONSTEUCTIONS [CHAP.IX

suppose added to the given points. The set of all given points then
gives rise to a certain set of coordinates, which we will denote by
1, a, &, c, ,* and which are supposed known. These numbers to-
gether with all numbers obtainable from them by a finite number of
rational operations constitute a set of numbers,

which we will call the domain, of rationality defined ly the data.^ In
addition to the coordinates of the known points (which, for the sake
of simplicity, we will suppose given in nonhomogeneous form), the
coordinates (x, y) of the point sought must be considered. The con-
ditions of the problem then lead to certain analytic relations which
these coordinates x, y and a, 6, c - - must satisfy. Eliminating one
of the variables, say #, we obtain two equations,

/i()-o, /.fayHO,

the first containing x but not y ; the second, in general, containing
both x and y. The problem is thus replaced by two problems : the
first depending on the solution of f (x) = to determine the abscissa
of the unknown point ; the second to determine the ordinate, assum-
ing the abscissa to be known.

In view of this fact we may confine ourselves to the discussion of
problems depending on a single equation with one unknown. Such
problems may be classified according to the equation to which they
give rise. A problem is said to be algebraic if the equation on which
its solution depends is algebraic, i.e. if this equation can be put in
the form

(1) ar+a l ar~ l +ajf-*+> + a n = 0,

in which the coefficients a l9 & 2 , - , a n are numbers of the domain of
rationality defined by the data. Any problem which is not algebraic
is said to be transcendental Algebraic problems (which alone will
be considered) may in turn be classified according to the degree n of

* In case homogeneous coSrdinates are used, a, &, c, . denote the mutual ratios
of the coordinates of the given elements.

t A moment's consideration will show that the points whose coordinates are
numbers of this domain are the points obtainable from the data by linear construc-
tions. Geometrically, any domain of rationality on a line may be defined as any
class of points on a line which is closed under harmonic constructions ; i.e. such
that if A, 2J, C are any three points of the class, the harmonic conjugate of A with
respect to 3 and C is a point of the class,

83] DEGREE OF A GEOMETEIC PEOBLEM 239

the equation on which their solutions depend. We have thus problems
of the first degree (already referred to), depending merely on the solution
of an equation of the first degree; problems of the second degree,
depending on the solution of an equation of the second degree, etc.

Account must however be taken of the fact that equation (1)
may be reducible within the domain K ; in other words, that the left
member of this equation may be the product of two or more poly-
nomials whose coefficients are numbers of K. In fact, let us suppose,
for example, that this equation may be written in the form

where j> 19 fa are two polynomials of the kind indicated, and of degrees
n^ and n 2 respectively (n t + n z = ri). Equation (1) is then equivalent
to the two equations

<W*) = o> &(*) = o.

Then either it happens that one of these two equations, e.g. the first,
furnishes all the solutions of the given problem, in which case fa being
assumed irreducible in K, the problem is not of degree n, but of degree
n^ < n ; or, both equations furnish solutions of the problem, in which
case fa also being assumed irreducible in K, the problem reduces to
two problems, one of degree ^ and one of degree n y In speaking of
a problem of the nth degree we will therefore always assume that
the associated equation of degree n is irreducible in the domain of
rationality defined by the data. Moreover, we have tacitly assumed
throughout this discussion that equation (1) has a root ; we shall see
presently that this assumption can always be satisfied by the intro-
duction, if necessary, of so-called improper elements. It is important
to note, however, since our Assumptions A, E, P do not in any way
limit the field of numbers to which the coordinates of all elements
of our space belong, and since equations of degree greater than one
do not always have a root in a given field when the coefficients of
the equation belong to this field, there exist spaces in which problems
of degree higher than the first may have no solutions. Thus in the
ordinary real projective geometry a problem of the second degree
will have a (real) solution only if the quadratic equation on which
it depends has a (real) root.

The example of a problem of the second degree given in the next
section will serve to illustrate the general discussion given above.

240 GEOMETRIC CONSTRUCTIONS [CHAP. DC

84. The intersection of a given line with a given conic. Given a
conic defined, let us say, by three points A, B, and the tangents at
A and B ; to find the points of intersection of a given line with this
conic. Using nonhomogeneous coordinates and choosing as #-axis one
of the given tangents to the conic, as y-axis the line joining the points
A and J5, and as the point (1, 1) the point C, the equation of the conic
may be assumed to be of the form

The equation of the given line may then be assumed to be of the form

The domain of rationality defined by the data is in this case

K -[!,*, 2].

The elimination of y between the two equations above then leads to
the equation

(1) # 2 j;# 2= 0.

This equation is not in general reducible in the domain K. The
problem of determining the points of intersection of an arbitrary line
in a plane with a given conic in this plane is then a problem of the
second degree. If equation (1) has a root in the field of the geometry, it
is clear that this root gives rise to a solution of the problem proposed ;
if this equation has no root in the field, the problem has no solution.
If, on the other hand, one point of intersection of the line with the
conic is given, so that one root of equation (1), say a? ~ r, is known,
the domain given by the data is

and in this domain. (1) is reducible ; in fact, it is equivalent to the

equation

(x r + p) (a? r) = 0.

The problem of finding the remaining point of intersection then
depends merely on the solution of the linear equation

x T + 2> = ;

* There is no loss in generality in assuming this form ; for if in the choice of
coordinates the equation of the given line were of the form x = c, we should merely
have to choose the other tangent as se-axis to bring the problem into the form here
assumed.

84,85] PKOPOSITION K 2 241

that is, the problem is of the first degree, as already noted among
the examples of linear problems.

It is important to note that equation (1) is the most general form
of equation of the second degree. It follows that every problem of the
second degree in a plane can ~be reduced to the construction of the points
of intersection of an arbitrary line with a particular conic. We
shall return to this later ( 86).

85. Improper elements. Proposition K 2 . We have called attention
frequently to the fact that the nature of the field of points on a line
is not completely determined by Assumptions A, E, P, under which
we are working. We have seen in particular that this field may be
finite or infinite. The example of an analytic space discussed in the
Introduction shows that the theory thus far developed applies equally
well whether we assume the field of points on a line to consist of all
the ordinary rational numbers, or of all the ordinary real numbers,
or of all the ordinary complex numbers. According to which of these
cases we assume, our space may be said to be the ordinary rational
space, or the ordinary real space, or the ordinary complex space.
Now, in the latter we know that every number has a square root.
Moreover, each of the former spaces (the rational and the real) are
clearly contained in the complex space as subspaces. Suppose now
that our space S is one in which not every number has a square
root. In such a case it is often convenient to be able to think of our
space S as forming a subspace in a more extensive space S', in which
some or all of these numbers do have square roots.

We have seen that the ordinary rational and ordinary real spaces
are such that they may be regarded as subspaces of a more exten-
sive space in the number system associated with which the square
root of any number always exists. In fact, they may be regarded as
subspaces of the ordinary complex space which has this property.
For a general field it is easy to prove that if a v a z , -, a n are any
finite set of elements of a field F, there exists a field F, containing
all the elements of F, such that each of the elements a v & 2 , , a n is a
square in P. This is, of course, less general than the theorem that
a field P exists in which every element of F is a square, but it is
sufficiently general for many geometric purposes. In the presence of
Assumptions A, E, P, H it is equivalent (cf. 54) to the following
statement :

242 GEOMETRIC CONSTKUCTIONS [CHAP.IX

PROPOSITION K 2 . If any finite number of involutions arc given in
a space S satisfying Assumptions A, E, P, there exists a space S' of
which S is a subspace,* such that all the given involutions have
double points in S'.

A proof of this theorem will be found at the end of the chapter.
The proposition is, from the analytic point of view, that the domain
of rationality determined by a quadratic problem may be extended so
as to include solutions of that problem. The space S' may be called
an extended space. The elements of S may be called proper elements,
and those of S' which are not in S may be called improper. A projec-
tive transformation which changes every proper element into a proper
element is likewise a proper transformation; one which transforms
proper elements into improper elements, on the other hand, is called
an improper transformation. Taking Proposition K 2 for the present as
an assumption like A, E, P, and H , and noting that it is consistent
with these other assumptions because they are all satisfied by the ordi-
nary complex space, we proceed to derive some of its consequences.

THEOREM 1. A proper one-dimensional projectivity without proper
double elements may always be regarded in an extended space as
having two improper double elements. (A, E, P, H , K 2 )f

Proof. Suppose the projectivity given on a conic. If the involu-
tion which belongs to this projectivity had two proper double points,
they would be the intersections of the axis of the projectivity with
the conic, and hence the given projectivity would have proper double
points. Let S 7 be the extended space in which (K 2 ) the involution
has double points. There are then two points of S' in which the
axis of the projectivity meets the conic, and these are, by Theorem 20,
Chap. VIII, the double points of the given projectivity.

COROLLARY 1. If a line does not meet a conic in proper points, it
may be regarded in an extended space as meeting it in two improper
points. (A, E, P, H , K 2 )

COROLLARY 2. Every quadratic equation with proper coefficients has
two roots which, if distinct, are both proper or both improper. (A, E,
P, H , K 2 )

* We use the word subspa.ce to mean any space, every point of which is a point
of the space of which it is a subspace. With this understanding the subspace may
be identical with the space of which it is a subspace. The ordinary complex space
then satisfies Proposition K 2 . t Of. Ex., p. 261.

85] PBOPOSITION K 2 243

For the double points of any projectivity satisfy an equation of
the form car + (d a) x I = (Theorem 11, Cor. 4, Chap. VI), and
any quadratic equation may be put into this form.

THEOREM 2. Any two involutions in the same one-dimensional form
have a conjugate pair in common, which may be proper or improper.
(A, E, P, H , K 2 )

This follows at once from the preceding and Theorem 23, Chap. VIII.

COROLLARY. In any involution there exists a conjugate pair, proper
or impro%)er, which is harmonic with any given conjugate pair. (A,
E, P, H , K 2 )

For the involution which has the given pair for double elements
has (by the theorem) a pair, proper or improper, in common with the
given involution. The latter pair satisfies the condition of the theorem
(Theorem 27, Cor. 1, Chap. IV).

We have seen earlier (Theorem 4, Cor., Chap. VIII) that any two
involutions with double points are conjugate. Under Proposition K 2
we may remove the restriction and say that any two involutions are
conjugate in an extended space dependent on the two involutions. If
the involutions are on coplanar lines, we have the following :

THEOREM 3. Two involutions on distinct lines in the same plane
are perspective (the center of perspectivity leing proper or improper),
provided the point of intersection of the lines is a double point for
loth or for neither of the involutions. (A, E, P, K 3 )

Proof. If the point of intersection of the two lines be a double
point of each of the involutions, let Q and E be an arbitrary pair
of one involution and Q 1 and R 1 an arbitrary pair of the other involu-
tion. The point of intersection of the lines QQ r and RE J is then a
center of a perspectivity which transforms elements which determine
the first involution into elements which determine the second. If
the point is a double point of neither of the two involutions, let
M be a double point of one and M r of the other (these double points
are proper or else exist in an extended space S r which exists by
Proposition K 2 ). Also let .Fand N* be the conjugates of in the two
involutions. Then by the same argument as before, the point of
intersection of the lines MM f , NN f may be taken as the center of
the perspectivity.

244 GEOMETRIC CONSTRUCTIONS [CHAP, ix

It was proved in 66, Chap. VII, that the equation of any point
conic is of the form

(1) a n x* + a,^* + a^&l + 2 a u ^ a + 2 a^x^ + 2 a 2S x 2 x s = ;

but it was not shown that every equation of this form represents a
conic. The line x l = contains the point (0, x v x 5 ) satisfying (1),
provided the ratio x 2 : x s satisfies the quadratic equation

a^p* + 2 a#B 9 x 9 + a^x* = 0.

Similarly, the lines # 2 = and x z = contain points of the locus
denned by (1), provided two other quadratic equations are satisfied.
By Proposition K 2 there exists an extended space in which these
three quadratic equations are solvable. Hence (1) is satisfied by the
coordinates of at least two distinct points P, Q (proper or improper).*

A linear transformation

(2)

evidently transforms the points satisfying (1) into points satisfying
another equation of the second degree. If, then, (2) is so chosen as
to transform P and Q into the points (0, 0, 1) and (0, 1, 0) respec-
tively, (1) will be transformed into an equation which is satisfied by
the latter pair of points, and which is therefore of the form

(3) ax* + CjXfo + c z x : x s + c 8 o^ 2 = 0.

If c l = 0, the points satisfying (3) lie on the two lines
x l = 0, ax 1 + c 2 # 8 + c s # 2 = ;

and hence (1) is satisfied by the points on the lines into which these
lines are transformed by the inverse of (2). If c x 3* 0, the trans-
formation

^i = <

(4) a^-

* Proposition K 2 has been used merely to establish the existence of points satis-
fying (1). In case there are proper points satisfying (1), the whole argument can be
made without K2.

85,86] PEOBLEMS OF THE SECOND DEGREE 245

transforms the points (x v x y a? 8 ) satisfying (3) into points (x[ 9 xj, x)
satisfying

(5) L - *-& x* + ( 6l xi + c,x>) x> = 0.

But (5) is in the form which was proved in Theorem 1, Chap. VII,
to be the equation of a conic. As the points which satisfy (5) are
transformed by the inverse of the product of the collineations (2) and
(4) into points which satisfy (1), we see that in all cases (1) repre-
sents a point conic (proper or improper, degenerate or nondegenerate).

This gives rise to the two following dual theorems :

THEOREM 4. Every equation of the form

a^x* + a^%* + a^xl + 2 a^x^ + 2 a^x z 4- 2 a 25 x 2 x B =
represents a point conic (proper or improper) which may, liowever,
degenerate ; and, conversely, every point conic may be represented "by
an equation of this form. (A, E, P, H , K 2 )

THEOEEM 4'. Every equation of the form

AX + A ^l + A **< + 2 -W 2 + 2 4As + 2 As Vs =
represents a line conic (proper or improper) which, may, however, de-
generate; and, conversely, every line conic may be represented "by an
equation of this form. (A, E, P, H , K 2 )

86. Problems of the second degree. "We have seen in 83 that
any problem of the first degree can be solved completely by means
of linear constructions ; but that a problem of degree higher than the
first cannot be solved by linear constructions alone. In regard to
problems of the second degree in a plane, however, it was seen in
84 that any such problem may be reduced to the problem of find-
ing the points of intersection of an arbitrary line in the plane with
a particular conic in the plane. This result we may state in the
following form:

THEOEEM 5. Any problem of the second degree in a plane may be
solved by linear constructions if the intersections of every line in the
plane with a single conic in this plane are assumed "known. (A, E,
P, H , K 2 )

In the usual representation of the protective geometry of a real plane by
means of points, lines, etc., drawn with a pencil, say, on a sheet of paper, the
linear constructions, as has already been noted, are those that can be per-
formed with the use of a straightedge alone. It will be shown later that any

246 GEOMETRIC CONSTEUCTIONS [CHAP.IX

conic in the real geometry is equivalent protectively to a circle. The instru-
ment usually employed to draw circles is the compass. It is then clear that
in this representation any problem of the second degree can be solved ly means of
a straightedge and compass alone. The theorem just stated, however, shows that
if a single circle is drawn once for all in the plane, the straightedge alone
suffices for the solution of any problem of the second degree in this plane.
The discussion immediately following serves to indicate briefly how this may
be accomplished.

We proceed to show how this theorem may be used in the solution
of problems of the second degree. Any such problem may be reduced
more or less readily to the first of the following :

PROBLEM 1. To find the double points of a projectivity on a line of
which three pairs of homologous points are given. We may assume

PA' B' C Q B A

FIG. 98

that the given pairs of homologous points all consist of distinct points
(otherwise the problem is linear). In accordance with Theorem 5,
we suppose given a conic (in a plane with the line) and assume
known the intersections of any line of the plane with this conic. Let
be any point of the given conic, and with as center project the
given pairs of homologous points on the conic (fig. 98). These define
a projectivity on the conic. Construct the axis of this projectivity
and let it meet the conic in the points P, Q. The lines OP, OQ then
meet the given line in the required double points.

PROBLEM 2. To find the points of intersection of a given line with
a conic of which five points are given. Let A, B, C 9 D 9 H be the given
points of the conic. The conic is then defined by the projectivity
D(A 9 B, C)j^E(A 9 B, 0) between the pencils of lines at D and E.

SEXTUPLY PEESPECTIVE TBIANGLES

247

This projectivity gives rise to a projectivity on the given line of
which three pairs of homologous points are known. The double
points of the latter projectivity are the points of intersection of the
line with the conic. The problem is thus reduced to Problem 1.

PROBLEM 3. We have seen that it is possible for two triangles in
a plane to be perspective from four different centers (ef. Ex. 8, p. 105).
The maximum number of ways in which it is conceivable that two
triangles may be perspective is clearly equal to the number of per-
mutations of three things three at a time, ie. six. The question then
arises, Is it possible to construct two triangles that are perspective from
six different centers? Let the two triangles be ABC and A!ffG^ and let

x l = 0, x z = 0, x s =

be the sides of the first opposite to A, B, C respectively. Let the
sides of the second opposite to -4', B f , C' respectively be

, = 0,

Fas, = 0,

The condition for ABC = A f B r C f is that the points of intersection of

corresponding sides be collinear, ie.

(i)

01-1
-F 1
-J' 1

In like manner, the condition for BCA~ A r B r C f is

(2)

- 1" I'
-101
-V 1

From these two conditions follows

-k" U
-I" 1
1-10

which is the condition for CAB = A f B f C f . Hence, if two triangles are
in the relations ABC = A r r C f and BCA = A'B'C J , they are also in

A A

the relation CAB = A r B f C f . Two triangles in this relation are said to
be triply perspective (cf. Ex. 2, p. 100). The 'domain of rationality
defined by the data of our problem is clearly

248 GEOMETRIC CONSTRUCTIONS [CHAP.IX

Since numbers in this domain may be found which satisfy equations
(1) and (2), the problem of constructing two triply perspective tri-
angles is linear.

The condition for ACB = A'B'C' is

(3) V-V'=Q.

If relations (1), (2), and (3) are satisfied, the triangles will be per-
spective from four centers. Let Jc be the common value of W and l n
(3), and let I be the common value of V and W (1). Relation (2) then
gives the condition k* I = 0. The relations

then define two quadruply perspective triangles. The problem of
constructing two such triangles is therefore still linear.

If now we add the condition for CBA = A'B ! C' 9 the two triangles
will, by what precedes, be perspective from six different centers. The
latter condition is

(4) A"J'-Z"=0.

With the preceding conditions (1), (2), (3) and the notation adopted
above, this leads to the condition

L3 _ 78 _ I

A; (/ = JL

The equation IP 1 = is, however, reducible in K ; indeed, it is

equivalent to

-1=0, # + * + != 0.

The first of these equations leads to the condition that A ( , J3 f , C r are
collinear, and does not therefore give a solution of the problem. The
problem of constructing two triangles that are sextuply perspective
is therefore of the second degree. The equation

has two roots w, w* (proper or improper and, in general,* distinct).
Hence our problem has two solutions. One of these consists of the

triangles

8 == 0, X L + w*% 2 + wx^= 0.

* They can coincide only if the number system is such that 1 + 1 + 1 = 0; e.g. m
a finite space involving the modulus 3.

86]

SEXTUPLY PERSPECTIVE TRIANGLES

249

Two of the sides of the second triangle may be improper.* The
points of intersection of the sides of one of these triangles with the
sides of the other are the following nine points :

(0, -1, 1) ( 0, w*, -w) ( 0, w, -w 2 )

(5) (-1, 0, 1) (~w\ 0, 1 ) (-to, 0, 1 )

(-1, 1, 0) ( w, -1, ) ( < -1, )

They form a configuration

which contains four configurations

of the kind studied in 36, Chap. IV. All triples of points in the
same row or column or term of the determinant expansion of their
matrix are collinear.f If one line is omitted from a finite plane (in
the sense of 72, Chap. VII) having four points on each line, the
remaining nine points and twelve lines are isomorphic with this
configuration.

EXERCISES

The problems in a plane given below that are of the second degree are to be solved
by linear constructions ^ with the assumption that the points of intersection of any line
in the plane with a given fixed conic in the plane are known : i.e. "with a straight-
edge and a given circle in tJie plane"

1. Construct the points of intersection of a given line with a conic deter-
mined by (i) four points and a tangent through one of them ; (ii) three points
and the tangents through two of them ; (iii) five tangents.

2. Construct the conjugate pair common to two involutions on a line.

3. Given a conic determined by five points, construct a triangle inscribed
in this conic whose sides pass through three given points of the plane.

* It may be noted that in the ordinary real geometry two sides of the second
triangle are necessarily improper, so that in this geometry our problem has no
real solution.

t They all lie on any cubic curve of the form x* + xj + xj -j- 3 \xix^x 9 = for
any value of X, and are, in fact, the points of inflexion of the cubic. This configura-
tion forms the point of departure for a variety of investigations leading into many
different branches of mathematics.

250 GEOMETRIC CONSTRUCTIONS

4. Given a triangle A^B Z C 2 inscribed in a triangle A l B 1 C l . In how
many ways can a triangle A 3 B S C 3 be inscribed in A 2 B Z C 2 and circumscribed
to A^C^t Show that in one case, in which one vertex of A S B^C S may be
chosen arbitrarily, the problem is linear (cf. 36, Chap. IV) ; and that in
another case the problem is quadratic. Show that this problem gives all COn-

rt g

figurations of the symbol . Give the constructions for all cases (cf.

S. Ivantor, Sitzungsberichte dermathematisch-naturwissenschaftlichen Classe
der Kaiserlichen Akademie der Wissenschaften zu Wien, Vol. LXXXIV
(1881), p. 915).

5. If opposite vertices of a simple plane hexagon PiP 2 P s P 4 P 5 P G are on
three concurrent lines, and the lines P-JP^ PJP^ Pf^ are concurrent, then the
lines P 2 P 3 , PPV P Q Pi are also concurrent, and the figure thus formed is a
configuration of Pappus.

6. Show how to construct a simple ?i-point inscribed in a given simple
n-point and circumscribed to another given simple n-point.

7. Show how to inscribe in a given conic a simple n-point whose sides
pass respectively through n given points.

8. Construct a conic through four points and tangent to a line not meeting
any of the four points.

9. Construct a conic through three points and tangent to two lines not
meeting any of the points.

10. Construct a conic through four given points and meeting a given line
in two points harmonic with two given points on the line.

11. If A is a given point of a conic and A r , Y are two variable points of the
conic such that AX, A Y always pass through a conjugate pair of a given
involution on a line I, the line XY will always pass through a fixed point B.
The line AB and the tangent to the conic at A pass through a conjugate pair
of the given involution.

12. Given a collineation in a plane and a line which does not contain a
fixed point of the collineation ; show that there is one and only one point on
the line which is transformed by the collineation into another point on the line.

13. Given four skew lines, show that there are in general two lines which
meet each of the given four lines ; and that if there are three such lines, there
is one through every point on one of the lines.

14. Given in a plane two systems of five points A^A^A^A^A^ and
BiB^B^B^ ; given also a point X in the plane, determine a point Y such
that we have X (A^^A^A^A^j- F(B 1 J5 3 B 8 J5 4J B 6 ). In general, there is one
and only one such point F. Under what condition is there more than one?
(R. Sturm, Mathematische Annalen, Vol. I (1869), p. 533.*)

* This is a special case of the so-called problem of projectivity. For references
and a systematic treatment see Sturm, Die Lehre von den geometrischen Ver-
wandtschaften, Vol. I, p. 348.

37]

USTVABIANTS

251

87. Invariants of linear and quadratic binary forms. An expres-
sion of the form a^ -h a z x 2 is called a linear "binary form in the
two variables x v x 2 . The word linear refers to the degree in the
variables, the word Unary to the number (two) of the variables. A
convenient notation for such a form is a^ The equation

defines a unique element A of a one-dimensional form in which a
scale has been established, viz. the element whose homogeneous co-
ordinates are (x v x 2 ] = (a v a x ). If l x = \x^ + 6 2 # 2 is another linear
binary form determining the element B, say, the question arises
as to the condition under which the two elements A and B coincide.
This condition is at once obtained as the vanishing of the determinant
A formed by the coefficients of the two forms ; i.e. the elements A
and B will coincide if and only if we have

A =

a, a,

= 0.

Now suppose the two elements A and B are subjected to any pro-
jective transformation II :

=0.

The forms a x and l x will be transformed into two forms a x , and & a ',
respectively, which, when equated to 0, define the points A r 3 B r into
which the points A, B are transformed by II. The coefficients of

are readily calculated as

the forms a x , 9 l x , in terms of those of a x) l
follows:

which gives
Similarly, we find

=a (aa l

i -h

Now it is clear that if the elements -4, B coincide, so also will the
new elements A 1 , B 1 coincide. If we have A = 0, therefore we should

also have A' =
We have

= 0. That this is the case is readily verified.

A' =

252 GEOMETEIC CONSTRUCTIONS [CHAP, ix

by a well-known theorem in determinants. Tins relation may also

be written

a ft

A' =

7 8

The determinant A is then a function of the coefficients of the forms
a *> b*> wikh tt 16 property that, if the two forms are subjected to a lin-
ear homogeneous transformation of the variables (with nonvamshing
determinant), the same function of the coefficients of the new forms
is equal to the function of the coefficients of the old forms multiplied
by an expression which is a function of the coefficients of the trans-
formation only. Such a function of the coefficients of two forms is
called a (simultaneous) invariant of the forms.

Suppose, now, we form the product a x - l x of the two forms a x , l a ,.
If multiplied out, this product is of the form

a = # 2 + 2 a

Any such form is called a quadratic Unary form. Under Proposi-
tion K 2 every such form may be factored into two linear factors
(proper or improper), and hence any such form represents two ele-
ments (proper or improper) of a one-dimensional form. These two
elements will coincide, if and only if the discriminant D a = a^
a n ' a w ^ ^ e quadratic form vanishes. The condition D a =s there-
fore expresses a property which is invariant under any projectivity.
If, then, the form a% be subjected to a projective transformation, the
discriminant D a , of the new form a r x must vanish whenever D n van-
ishes. There must accordingly be a relation of the form D a , = k - D a .
If &l be subjected to the transformation II given above, the coefficients
new form aj are readily found to be

2
(1) < - a^aft + a u (a$ + /3y) + a

By actual computation the reader may then verify the relation

a /3 3
7 8

The discriminant D a of a quadratic form a 2 is therefore called an
invariant of the form.

87] INVARIANTS 253

Suppose, now, we consider two binary quadratic forms

Each of these (under JK 2 ) represents a pair of points (proper or im-
proper). Let us seek the condition that these two pairs be harmonic.
This property is invariant under protective transformations ; we may
therefore expect the condition sought to be an invariant of the two
forms. We know that if a v a 2 are the nonliomogeneous coordinates
of the two points represented by a* = 0, we have relations

= ^ss, a +a = 2 *V

a n " a n

with similar relations for the nonhomogeneous coordinates & 1? 6 2 of
the points represented by &| = 0. The two pairs of points a v a z ; l v 6 2
will be harmonic if we have (Theorem 13, Cor. 2, Chap. VI)

* = -!.

This relation may readily be changed into the following :

which, on substituting from the relations just given, becomes

1 -2o 1 A.= 0.

This is the condition sought. If we form the same function of the
coefficients of the two forms a^ 9 V x * obtained from a*, I* by subjecting
them to the transformation II, and substitute from equations (1), we
obtain the relation

In the three examples of invariants of binary forms thus far
obtained, the function of the new coefficients was always equal to
the function of the old coefficients multiplied by a power of the
determinant of the transformation. This is a general theorem regard-
ing invariants to which we shall refer again in 90, when a formal
definition of an invariant will be given. Before closing this section,
however, let us consider briefly the cross ratio B (a-fl 29 &A) f ^ e
two pairs of points represented by a% = 0, J = 0. This cross ratio

254 GEOMETRIC CONSTRUCTIONS [CHAP, ix

is entirely unchanged when the two forms are subjected to a pro-
jective transformation. If, therefore, this cross ratio be calculated in
terms of the coefficients of the two forms, the resulting function of
the coefficients must be exactly equal to the same function of the
coefficients of the forms a^ b f x ; the power of the determinant referred
to above is in this case zero. Such an invariant is called an absolute
invariant; for purposes of distinction the invariants which when
transformed are multiplied by a power = of the determinant of
the transformation are then called relative invariants.

EXERCISES

1. Show that the cross ratio Bs (a 1 <7 2 , b^ referred to at the end of the
last section is

and hence show, by reference to preceding results, that it is indeed an absolute
invariant.

2. Given three pairs of points defined by the three binary quadratic forms
#2 Q } jj = o, c = ; show that the three will be in involution if we have

H 12 22 _

Hence show that the above determinant is a simultaneous invariant of the
three forms (cf. Ex. 13, p. 230).

88. Proposition K^ If we form the product of n linear binary

forms a x a' x a'J a~ l \ we obtain an expression of the form

An expression of this form is called a "binary homogeneous form or
quant of the nth degree. If it is obtained as the product of n linear
forms, it will represent a set of n points on a line (or a set of n ele-
ments of some one-dimensional form).

If it is of the second degree, we have, by Proposition K 2 , that there
exists an extended space in which it represents a pair of points. At
the end of this chapter there will be proved the following generali-
zation of K 2 :

88,89] PBOPOSITION K n 255

PROPOSITION K n . If a, a l x) - are a finite number of Unary "homo-
geneous forms whose coefficients are proper in a space S which satisfies
Assumptions A, E, P, there exists a space S', of which S is a sub-
space, in the number system of which each of these forms is a product
of linear factors.

As in 85, S' is called an extended space, and elements in S' but
not in S are called improper elements. Proposition K n thus implies
that an equation of the form a * = can always be thought of as
representing n (distinct or partly coinciding) improper points in an
extended space in case it does not represent any proper points.

Proposition K n could be introduced as an (not independent) assump-
tion in addition to A, E, P, and H . Its consistency with the other
assumptions would be shown by the example of the ordinary com-
plex space in which it is equivalent to the fundamental theorem
of algebra,

89. Taylor's theorem. Polar forms. It is desirable at this point
to borrow an important theorem from elementary algebra.

DEFINITION. Given a term Ax" of any polynomial, the expression
nAx?~ l is called the derivative of Ax? with respect to x t in symbols

Ax* = nAx?-\

The derivative of a polynomial with respect to x i is, by definition, the

sum of the derivatives of its respective terms.

o
This definition gives at once A == 0, if A is independent of # .

u%i

Applied to a term of a binary form it gives

With this definition it is possible to derive Taylor's theorem for the
expansion of a polynomial. *We state it for a binary form as follows :
Given the binary form

* For the proof of this theorem on the basis of the definition just given, cf . Fine,
College Algebra, pp. 460-462.

256 GEOMETEIC CONSTRUCTIONS [CHAP.IX

If herein we substitute for x v x 2 respectively the expressions x l + \y v
t + *# 2 > we obtain,

/ n o

+ *

Here the parentheses are differential operators. Thus

where -^ means means rH etc. It is readily

daf ^iL^iJ ^2^1 && 2 |_teJ

proved for any term of a polynomial (and hence for the polynomial
itself) that the value of such a higher derivative as d z f/dx z 8x l is
independent of the order of differentiation ; i.e. that we have

a 2 / = a 2 /

DEFINITION. The coefficient of X in the above expansion, viz.
y l df/8x 1 + 2/2#//d# 2 is called the first polar form of (y l} y z ) with
respect to f (x v x z ) ; the coefficient of X 2 is called the second ; the
coefficient of \ n is called the nth polar form of (y v # 2 ) ivitli respect
to the form f. If any polar form be equated to 0, it represents a set
of points which is called the first) second, , nth polar of the point
(y 1? y 3 ) with respect to the set of points represented by f (x v # 2 ) = 0.

Consider now a binary ioim.f(x v cc 2 ) = and the effect upon it of
a projective transformation

If we substitute these values in / (x v o? 3 ), we obtain a new form
&(8l> %) A point (x v x z ) represented by/(^, o? 2 )= will be trans-
formed into a point (a?/, x$ represented by the form F(x' v x)= 0.
Moreover, if the point (y l9 y 3 ) be subjected to the same projectivity,
it is evident from the nature of the expansion given above that the
polars of (y l9 y z ) with respect to f(x v # 2 ) = are transformed into
the polars of (y[, y a ') with respect to F(x[, x) = 0.

89,90] INVAKLAJSTTS 257

We may summarize the results thus obtained as follows :

THEOREM 6. If a Unary form f is transformed ly a protective
transformation into the form F, the set of points represented byf=Q
is transformed into the set represented ly F=Q. Any polar of a
point (y v y^ with respect to f = is transformed into the correspond-
ing polar of the point (y[, y) with respect to J?= 0.

The following is a simple illustration of a polar of a point with
respect to a set of points on a line.

The form x^ = represents the two points whose nonhomo-
geneous coordinates are and oo respectively. The first polar of any
point (y v y^) with respect to this form is clearly y^ & + y^ = 0, and
represents the point ( y v y z ) ; in other words, the first polar of a
point P with respect to the pair of points represented by the given
form is the harmonic conjugate of this point with respect to the pair.

EXERCISE

Determine the geometrical construction, of the (n l)th polar of a point
with respect to a set of n distinct points on a line (cf. Ex. 3, p. 51).

90. Invariants and covariants of binary forms. DEFINITION. If a
binary form a = a Q x" + na l x"~~ l x z + + a> n %S be changed by the

transformation , _

_ ,
*

into a new form A = A^x[ n + A$*~' L xl H ----- h A n x n , any rational
function I(a Q) a v - , a n ) of the coefficients such that we have

I(A 4 V ..-, A n ) = <j>(a, A 7, 8) I(a Q> a v , a n )
is called an invariant of the form a. A function

tfK^, -.,.; x lt x z )

of the coefficients and the variables such that we have
C(A A v .>.,A n \ x[, xf) ^ (a, /3, 7, 8) . C(a , a v -.- 9 a n ; x v x^)

is called a covariant of the form a. The same terms apply to func-
tions I and C of the coefficients and variables of any finite number
of binary forms with the property that the same function of the
coefficients and variables of the new forms is equal to the original
function multiplied by a function of a, & 7, S only ; they are then
called simultaneous invariants or covariants.

258 GEOMETRIC CONSTRUCTIONS [CHAP.IX

In 87 we gave several examples of invariants of binary forms,
linear and quadratic. It is evident from the definition that the con-
dition obtained ly equating to any invariant of a form (or of a
system of forms] must determine a property of tlie set of points
represented by the form (or forms) which is invariant under a pro-
jective transformation. Hence the complete study of the projective
geometry of a single line would involve the complete theory of invari-
ants and covariants of binary forms. It is not our purpose in this
book to give an account of this theory. But we will mention one
theorem which we have already seen verified in special cases.

Tlie functions <(#, /3, y, 8) and -^(a, /3, % 8) occurring in the
definition above are always powers of tlie determinant aS /3<y of
the projective transformation in question.*

Before closing this section we will give a simple example of a cova-
riant. Consider two binary quadratic forms a*, b* and form the new
quantic

c a * = K & i - <* A) x i + (<*>& - & A) ^ A + ( a A - a A) x l-

By means of equations (1), 87, the reader may then verify without
difficulty that the relation

holds, which proves C ab to be a covariant. The two points represented
by C& = are the double points (proper or improper) of the involu-
tion of which the pairs determined by a* = 0, b* = are conjugate
pairs. This shows why the form should be a covariant.

EXERCISE

Prove the statement contained in the next to the iast sentence.

91. Ternary and quaternary forms and their invariants. The remarks
which have been made above regarding binary forms can evidently be
generalized. A p-aryform of the nth degree is a polynomial of the nth
degree homogeneous in p variables. When the number of variables is
three or four, the form is called ternary or quaternary respectively.
The general ternary form of the second degree when equated to zero
has been shown to be the equation of a conic. In general, the set of
points (proper and improper) in a plane which satisfy an equation

- =

* For proof, cf M for example, Grace and Young, Algebra of Invariants, pp. 21, 22.

01] INVARIANTS 259

obtained by equating to zero a ternary form of the nth degree is
called an algebraic curve of the nth degree (order). Similarly, the set
of points determined in space by a quaternary form of the nth degree
equated to zero is called an algebraic surface of the nth degree.

The definitions of invariants and covariants of jp-ary forms is pre-
cisely the same as that given above for binary forms, allowance being
made for the change in the number of variables. Just as in the
binary case, if an invariant of a ternary or quaternary form vanishes,
the corresponding function of the coefficients of any protectively
equivalent form also vanishes, and consequently it represents a prop-
erty of the corresponding algebraic curve or surface which is not
changed when the curve or surface undergoes a projective transforma-
tion. Similar remarks apply to covariants of systems of ternary and
quaternary forms.

Invariants and covariants as defined above are with respect to the
group of all projective collineations. The geometric properties which
they represent are properties unaltered by any projective collineatiou.
Like definitions can of course be made of invariants with respect to
any subgroup of the total group. Evidently any function of the
coefficients of a form which is invariant under the group of all col-
lineations will also be an invariant under any subgroup. But there
will in general be functions which remain invariant under a subgroup
but which are not invariant under the total group. These correspond
to properties of figures which are invariant under the subgroup with-
out being invariant under the total group. We thus arrive at the
fundamental notion of a geometry as associated with a given group,
a subject to which we shall return in detail in a later chapter.

EXERCISES

1. Define by analogy with the developments of 80, the n - 1 polars of a
ternary or quaternary form of the nth degree.

2. Regarding a triangle as a curve of the third degree, show that the second
polar of a point with regard to a triangle is the polar line defined on page 46.

3. Generalize Ex. 2 in the plane and in space, and dualize.

^11 ^12 ^13

4. Prove that the discriminant a la a 22 a 23 of the ternary quadratic form
is an invariant. What is its geometrical interpretation? Cf. Ex., p. 187.

260 GEOMETEIC CONSTRUCTIONS [CHAP. IX

92. Proof of Proposition K n . Given a rational integral function

<f> (x) = a tf + a^"- 1 -{ ----- h a n , a^ 0,

whose coefficients belong to a given field F, and which is irreducible in
F, there exists a field F', containing F, in which the equation <f> (x) =
has a root.

Let f(x) be any rational integral function of x with coefficients in
F, and let / be an arbitrary symbol not an element of F. Consider
the class F^ = [/(/)] of all symbols f(j)> where [/(&)] is the class of
all rational integral functions with coefficients in F. We proceed to
define laws of combination for the elements of Fj which render the
latter a field. The process depends on the theorem * that any poly-
nomial f(x) can be represented uniquely in the form

where q(x) and r(x) are polynomials belonging to F, i.e. with
coefficients in F, and where r(x) is of degree lower than the degree
n of <j> (x). If two polynomials f l9 / 2 belonging to F are such that
their difference is exactly divisible by <f>(x), then they are said to be
congruent modulo <f>(x), in symbols / t = / 2 , mod. <p(x).

1. Two elements /^j), / 2 (/) of F, are said to be equal, if and only
if/!^) andj^as) are congruent mod. <p(x). By virtue of the theorem
referred to above, every element /(/) of F y is equal to one and only
one element f f (j) of degree less than n. We need hence consider only
those elements f(j) of degree less than n. Further, it follows from
this definition that <f> (/) = 0.

2. If /!(*)+/, (a) s/ t (a), mod * (as), then/ 1 (y)+/ 2 (y)-/ 8 (/).

3. If /, (x) /, (x) s/ 8 (*), mod. * (x), then / t (/) ./, (/) =/ 3 (/).
Addition and multiplication of the elements of F y having tlius

been defined, the associative and distributive laws follow as immedi-
ate consequences of the corresponding laws for the polynomials /(#).
It remains merely to show that the inverse operations exist and are
unique. That addition has a unique inverse is obvious. To prove
that the same holds for multiplication (with the exception of 0) we
need only recall f that, since <(#) and any polynomial f(x) have no
common factors, there exist two polynomials h(x) and Jc(x) with
coefficients in F such that

A (a?) -/() + ft (a?). ^ () = !.
* Fine, College Algebra, p. 156. f Fine, loc. cit., p. 208.

92] PEOOF OF K n 261

This gives at once h (/) /(/) = 1,

so that every element /(/) distinct from has a reciprocal. The class
F y is therefore a field with respect to the operations of addition and
mutiplication defined above (cf. 52), such that <(/)= 0. It follows
at once* that xj is a factor of $(x) in the field F, which is there-
fore the required field F'. The quotient <f>(x)/(xj) is either irre-
ducible in Fj, or, if reducible, has certain irreducible factors. If the
degree of one of the latter is greater than unity, the above process may
be repeated leading to a field F^,,/' being a zero of the factor in
question. Continuing in this way, it is possible to construct a field
^7,y,...,/ m) > w ^ ere fl&^w 1, in which <f>(x) is completely reducible,
i.e. in which <(#) may be decomposed into n linear factors. This
gives the following corollary :

Given a polynomial <f> (x) "belonging to a given field F, there exists a
field F 7 containing F in which <j> (x) is completely reducible.

Finally, an obvious extension of this argument gives the corollary :

Given a finite number of polynomials each of which belongs to a
given field F, there exists a field F', containing F, in which each of the
given polynomials is completely reducible.

This corollary is equivalent to Proposition K n . For if S be any
space, let F be the number system on one of its lines. Then, as in
the Introduction (p. 11), F' determines an analytic space which is
the required space S' of Proposition K n .

The more general question at once presents itself: Given a field
F, does there exist a field F, containing F, in which every polynomial
belonging to F is completely reducible ? The argument used above
does not appear to offer a direct answer to this question. The ques-
tion has, however, recently been answered in the affirmative by an
extension of the above argument which assumes the possibility of
" well ordering " any class, f

EXERCISE

Many theorems of this and other chapters are given as dependent on
A, E, P, H , whereas they are provable without the use of H . Determine
which theorems are true in those spaces for which H is false.

* Fine, College Algebra, p. 1G9.

t Cf. E. Steinitz, Algebraische Theorie der Kbrper, Journal fur reine u. ange-
wandte Mathematik, Yol. CXXXVII (1909), p. 167 ; especially pp. 271-286.

CHAPTER

PROJECTIVE TRANSFORMATIONS OF TWO-DIMENSIONAL FORMS

93. Correlations between two-dimensional forms. DEFINITION. A
projective correspondence between the elements of a plane of points
and the elements of a plane of lines (whether they be on the same
or on different bases) is called a correlation. Likewise, a projective
correspondence between the elements of a bundle of planes and the
elements of a bundle of lines is called a correlation.^

Under the principle of duality we may confine ourselves to a con-
sideration of correlations between planes. In such a correlation, then,
to every point of the plane of points corresponds a unique line of the
plane of lines ; and to every pencil of points in the plane of points
corresponds a unique projective pencil of lines in the plane of lines.
In particular, if the plane of points and the plane of lines are on the
same base, we have a correlation in a planar field, whereby to every
point P of the plane corresponds a unique line p of the same plane,
and in which, if Jf, P& P& P are collinear points, the corresponding
lines p v p 2 , p z) p^ are concurrent and such that

That a correlation T transforms the points [P] of a plane into the
lines [p] of the plane, we indicate as usual by the functional notation

The points on a line I are transformed by F into the lines on a
point L. This determines a transformation of the lines [I] into the
points [], which we may denote by T', thus:

T'(l) = L.

That F ; is also a correlation is evident (the formal proof may be
supplied by the reader). The transformation T r is called the correla-
tion induced by T. If a correlation T transforms the lines [I] of a

* All developments of this chapter are on the basis of Assumptions A, E, P, and
H . Cf . the exercise at the end of the last chapter.

t The terms reciprocity and dudlity are sometimes used in place of correlation.

262

93] CORRELATIONS 263

plane into the points [L] of the plane, the correlation which trans-
forms the points [ll r ] into the lines [LL f ] is the correlation induced
by T. If F is induced by F, it is clear that T is induced by P.
For if we have

we have also

and hence the induced correlation of F transforms P 2 into p^ etc.

That correlations in a plane exist follows from the existence of the
polar system of a conic. The latter is in fact a projective transforma-
tion in which to every point in the plane of the conic corresponds a
unique line of the plane, to every line corresponds a unique point,
and to every pencil of points (lines) corresponds a projective pencil
of lines (points) (Theorem 18, Cor., Chap. V). This example is, how-
ever, of a special type having the peculiarity that, if a point P corre-
sponds to a line p, then in the induced correlation the line p will
correspond to the point P ; i.e. in a polar system the points and lines
correspond doubly. This is by no means the case in every correlation.

DEFINITION. A correlation in a plane in which the points and
lines correspond doubly is called a polarity.

It has been found convenient in the case of a polarity defined by
a conic to study a transformation of points into lines and the induced
transformation of lines into points simultaneously. Analogously, in
studying collineations we have regarded a transformation T of points
P v P^ P B , P into points _Zf, P^ PJ 9 P^ and the transformation T' of
the lines P& P 2 P B , P B P^ 1J1> into the lines IJ'JJ', PjPj, #2J', #?' as
the same collineation. In like manner, when considering a trans-
formation of the points and lines of a plane into its lines and points
respectively, a correlation F operating on the points and its induced
correlation T 1 operating on the lines constitute one transformation of
the points and lines of the plane. For this sort of transformation we
shall also use the term correlation. In the first instance a correlation
in a plane is a correspondence between a plane of points (lines) and
a plane of lines (points). In the extended sense it is a transformation
of a planar field either into itself or into another planar field, in
which an element of one kind (point or line) corresponds to an ele-
ment of the other kind.

264 TWO-DIMENSIONAL PBOJECTIVITIES [CHAP.X

The following theorem is an immediate consequence of the defini-
tion and the fact that the resultant of any two protective correspond-
ences is a projective correspondence.

THEOREM 1. The resultant of two correlations is a projective col-
lineation, and the resultant of a correlation and a projective collinea-
tion is a correlation.

We now proceed to derive tins fundamental theorem for correlations
between two-dimensional forms.

THEOREM 2. A correlation letween two two-dimensional primitive
forms is uniquely defined when four pairs of homologous elements are
given, provided that no three elements of either form are on the same
one-dimensional primitive form.

Proof. Let the two forms be a plane of points a and a plane of
lines #'. Let C* be any conic in a r , and let the four pairs of homol-
ogous elements be A 9 3, C, D in a and a', V, c', dl in a!. Let A!, B f ,
C\ D f be the poles of a f , V, c f , d r respectively with respect to C 2 . If
the four points A, B, C, D are the vertices of a quadrangle and the
four points A', B r , C r , D* are likewise the vertices of a quadrangle
(and this implies that no three of the lines a r , V, c r , d f are concurrent),
there exists one and only one collineation transforming A into A r , B
into B f , C into C 1 , and D into D f (Theorem 18, Chap. IV). Let tins
collineation be denoted by T, and let the polarity defined by the conic
C 2 be denoted by P. Then the projective transformation F which is
the resultant of these two transforms A into a r , B into V, etc. More-
over, there cannot be more than one correspondence effecting this
transformation. For, suppose there were two, F and F r Then the
projective correspondence I^" 1 F would leave each of the four points
A, B, C, D fixed; i.e. would be the identity (Theorem 18, Chap. IY).
But this would imply F x = F.

THEOREM 3. A correlation which interchanges the vertices of a
triangle with the opposite sides is a polarity.

Proof. Let the vertices of the given triangle be A, B, C, and let
the opposite sides be respectively a, I, c. Let P be any point of the
plane ABC which is not on a side of the triangle. The line p into
which P is transformed by the given correlation F does not, then, pass
through a vertex of the triangle ABC. The correlation F is deter-
mined by the equation F (ABCP) = alcp, and, by hypothesis, is such

93]

COEKELATIONS

265

that F (ale) ABC. The points [Q] of c are transformed into the
lines [g] on C, and these meet c in a pencil [Q r ] projective with [Q]
(fig. 99). Since A corresponds to B and to A in the projectivity
[Q] ~^[Q f ], this projectivity is an involution L The point Q in which

CP meets c is transformed by F into a line on the point cp\ and
since Q and cp are paired in I, it follows that cp is transformed
into the line CQ Q CP. In like manner, Ip is transformed into BP.
Hence p = (cp, lp] is transformed into P = (CP, BP).

THEOBEM 4. Atiy projective collineatwn, II, m a plane, <x, is the
product of two polarities.

Proof, Let Aa be a lineal element o! a, and let
H (4a) =* jl V, n (4W) A" a".

Unless II is perspective, Aa may be so chosen that A 9 A f , A n are not
collinear, aa r a fr are not concurrent, and no line of one of the three
lineal elements passes through the point of another. In this case there
exists a polarity P such that J?(AA'A rr ) = a"a'a, namely the polarity-
defined by the conic with regard to which AA"(aa ff ) is a self-polar tri-
angle and to which o! is tangent at A f . If II is perspective,,the existence
of P follows directly on choosing Aa, so that neither A nor a is fixed.
We then have

and hence the triangle AA[(aaf) is self-reciprocal. Hence (Theorem 3)
PIT ss Pj is a polarity, and therefore II = PP r

266 TWO-DIMENSIONAL PBOJECTIVITIES [CHAI>. x

94. Analytic representation of a correlation between two planes.
Bilinear forms. Let a system of simultaneous point-and-line coordi-
nates be established in a planar field. We then have

THEOREM 5. Any correlation in a plane is given as a transforma-
tion of points into lines by equations of the form

(1) pul = a^ +

where the determinant A of the coefficients a l} is different from zero.
Conversely, every transformation of this form in which the determinant
A is different from zero represents a correlation.

The proof of this theorem is completely analogous to the proof of
Theorem 8, Chapter VII, and need not be repeated here.
As a corollary we have

COROLLARY 1. The transformation pu[~x v pu^x^ pu! } =x s in
a plane represents a polarity in which to every side of the triangle of
reference corresponds the opposite vertex.

Also, if (u[, ii' 2) u^) be interpreted as line coordinates in a plane
different from that containing the points (x lf # 2 , x s ) (and if the num-
ber systems are so related that the correspondence X 1 = X between
the two planes is projective), we have at once

COROLLARY 2. The equations of Theorem 5 also represent a correla-
tion between the plane of (x v w z > #3) and the plane of (%', ul, v. 8 ').

Returning now to the consideration of a correlation in a plane
(planar field), we have seen that the equations (1) give the coordi-
nates (u[, tf a ', ui) of the line u'= T (X), which corresponds to the
point -3T= (x v % z , a? 8 ). By solving these equations for x i9

we obtain the coordinates of Z= T" 1 (vf) in terms of the coordinates
u[ of the line to which Xis homologous in the inverse correlation T"\
If, however, we seek the coordinates of the point -5T' = T (u] which
corresponds to any line u in the correlation T, we may proceed as
follows :

COKBELATI02TS 267

Let the equation of the point JT' = (&[, x! 2 , x' A ) in line coordinates be

= 0.

Substituting in this equation from (1) and arranging the terms as a
linear expression in x v x 2) # 3 ,

= 0,
we readily find

(3)

The coordinates of X ! in terms of the coordinates of u are then
given by

(4)

This is the analytic expression of the correlation as a transformation
of lines into points ; i.e. of the induced correlation of F. These equa-
tions clearly apply also in the case of a correlation between two
different planes.

It is perhaps well to emphasize the fact that Equations (1) express T as a
transformation of points into lines, while Equations (4) represent the induced
correlation of lines into points. Since we consider a correlation as a trans-
formation of points into lines and lines into points, T is completely represented
by (1) and (4) taken together. Equations (2) and (3) taken together repre-
sent the inverse of T.

Another way of representing F analytically is obtained by observ-
ing that the point (x v x 2 , # 8 ) is transformed by F into the line whose
equation in current coordinates (x[, x, x) is

or,

(5) (a^ + a 12 2

= 0.

The left-hand member of (5) is a general ternary bilinear form. We
have then

COROLLARY 3. Any ternary bilinear form in which the determinant
A is different from zero represents a correlation in a plane.

268 TWO-DIMENSIONAL PEOJECTIVITIES [CHAP.X

95. General projective group. Representation by matrices. The

general protective group of transformations in a plane (which., under
duality, we take as representative of the two-dimensional primitive
forms) consists of all projective collineations (including the identity)
and all correlations in the plane. Since the product of two collinea-
tions is a collineation, the set of all projective collineations .forms a
subgroup of the general group. Since, however, the product of two
correlations is a collineation, there exists no subgroup consisting
entirely of correlations.*

According to the point of view developed in the last chapter, the
projective geometry of a plane is concerned with theorems which
state properties invariant under the general projective group in the
plane. In particular, the principle of duality may be regarded as a
consequence of the presence of correlations in this group.

Analytically, collineations and correlations may be regarded as
aspects of the theory of matrices. The collineation

may be conveniently represented by the matrix A of the coefficients a t/ :

The product of two collineations A = (a v ) and B = (&J is then given
by the product of their matrices :

the element of the ith row and the yth column of the matrix BA
being obtained by multiplying each element of the ith row of B by the
corresponding element of the^th column of A and adding the products
thus obtained. It is clear that two collineations are not in general
commutative.

* A polarity and the identity fcym a group ; but this forms no exception to the
statement just made, since the identity must be regarded as a collineation.

MATEICES 269

Of the two matrices

/ a il <*12 13\ /*n u ai\

^31 ^32 <W \^13 ^23 ^33/

either of which is obtained from the other by interchanging rows and
columns, one is called the conjugate or transposed matrix of the
other. The matrix

/ A A 4 \

I -fX-. i L n i uTjL - I

is called the adjoint matrix of the matrix A. The adjoint matrix is
clearly obtained by replacing each element of the transposed matrix
by its cof actor. Equations (2) of 67 show that the adjoint of a
given matrix represents the inverse of the collineation represented "by
the given matrix. Indeed, by direct multiplication,

Kl ^22 ^23 MlS 4*2 As H

and the matrix just obtained clearly represents the identical col-
lineation. Since, when a matrix is thought of as representing a
collineation, we may evidently remove any common factor from all
the elements of the matrix, the latter matrix is equivalent to the
so-called identical matrix,*

II 0\
1 .

\0 I/

Furthermore, Equations (3), 67, show that if a given matrix
represents a collineation in point coordinates, the conjugate of the
adjoint matrix represents the same collineation in line coordinates.
Also from the representation of the product of two matrices just
derived, ' follows the important result:

The determinant of the product of two matrices (collineations) is
equal to the product of the determinants of the two matrices (col-
lineations).

* In the general theory of matrices these two matrices are not, however, re-
garded as the same. It is only the interpretation of them as collineations which
renders them equivalent.

270 TWO-DIMENSIONAL PEOJECTIVITIES [CHAP.X

From what has just been said it is clear that a matrix does not
completely define a collineatiou, unless the nature of the coordinates
is specified. If it is desired to exhibit the coordinates in the nota-
tion, we may write the collineation ?/ = 20 v a ; in the symbolic form

The matrix (a v ) may then be regarded as an operator transforming
the coordinates x=s(x l9 # a , # 8 ) into the coordinates #' = (#/, h # 8 ). If
we place a v = a jl9 the matrix conjugate to (aj is (a,,). Also by plac-
ing JT = -4,,, the adjoint matrix of (c^) is ( Jj. The inverse of the
above collineation is then written

Furthermore, the collineation #'= (a v )# is represented in line coordi-
nates by the equation

This more complete notation will not be found necessary in gen-
eral in the analytic treatment of collineations, when no correlations
are present, but it is essential in the representation of correlations
by means of matrices.

The correlation (1) of 94 may clearly be represented symbolically

by the equation

'(0 t ,)0,

where the matrix (a^) is to be regarded as an operator transforming
the point & into the line vf. This correlation is then expressed as a
transformation of lines into points by

The product of two correlations u f ~ (a iy ) x and %'= (B v ) x is there-
fore represented by

d**(Bv)(av)x

(d. Equations (4), 94), or by

Also, the inverse of the correlation u 1 = (a ) x is given by

^ = (l v )<
or by

TYPES OF COLLISrEATIOlSrS

271

EXERCISE

Show that if [II] is the set of all collineations in a plane and I\ is any
correlation, the set of all correlations in the plane is [Orj, so that the two
sets of transformations [II] and [Iirj comprise the general projective group
in the plane. By virtue of this fact the subgroup of all projective collineations
is said to be of index in the general projective group.*

96. Double points and double lines of a collineation in a plane.

Referring to Equations (1) of 67 we see that a point (x v x z , x s )
which is transformed into'itself by the collineation. (1) must satisfy
the equations

which, by a simple rearrangement, may be written

(1)

If a point (x v C 2 , # 3 ) is to satisfy these three equations, the deter-
minant of this system of equations must vanish ; i.e. p must satisfy
the equation

(2)

= 0.

Tliis is an equation of the third degree in p, which cannot have more
than three roots in the number system of our geometry.

Suppose that /> t is a root of this equation. The system of equa-
tions (1) is then consistent (which means geometrically that the
three lines represented by them pass through the same point), and
the point determined by any two of them (if they are independent,
i.e. if they do not represent the same line) is a double point Solving
the first two of these equations, for example, we find as the coordi-
nates (% v x z> o? 3 ) of a double point

(3) iv-v^s 8 *

2 Pi

*u-/>i

& 21

*13

* A subgroup [II] of a group is said to he of index n, if there exist n 1 trans-
formations T l (i ss 1, 2, . . . n - l),-such that the n - 1 sets [nTf] of transformations
together with the set [II] contain all the transformations of the group, while no two
transformations within the same set or from any two sets are identical.

272 TWO-DIMENSIONAL PEOJECTIVITIES [CHAP, x

which represent a unique point, unless it should happen that all the
determinants on the right of this equation vanish. Leaving aside
this possibility for the moment, we see that every root of Equation
(2), which is called the characteristic equation of the collineatioii (or
of the representative matrix), gives rise to a unique double point.
Moreover, every double point is obtainable in this way. This is the
analytic form of the fact already noted, that a collineation which is
not a Jiomology or an elation cannot have more than three do Me
points, unless it is the identical collineation.

If, however, all the determinants on the right in Equations (3)
vanish, it follows readily that the first two of Equations (1) represent
the same line. If the determinants formed analogously from the last
two equations do not all vanish, we again get a unique double point ;
but if the latter also vanish, then all three of the equations above
represent the same line. Every point of this line is then a double point,
and the collineation must be a homology or an elation. Clearly this
can happen only if p l is at least a double root of Equation (2) ; for
we know that a perspective collineation cannot have more than one
double point which is not on the axis of the collineation.

A complete enumeration of the possible configurations of double
points and lines of a collineation can be made by means of a study
of the characteristic equation, making use of the theory of elementary
divisors.* It seems more natural in the present connection to start
with the existence of one fixed point (Proposition K 8 ) and discuss
geometrically the cases that can arise.

By Theorem 4 a collineation is the product of two polarities. Hence,
any double point has the same polar line in both polarities, and that
polar line is a double line. Hence the invariant figitre of douUc points
and lines is self-dual.

Four points of the plane, no three of which are collineor, cannot
be invariant unless the collineation reduces to the identity. If three
noncollinear points are invariant, two cases present themselves. If
the collineation reduces to the identity on no side of the invariant
triangle, the collineation is of Type I (cf. 40, Chap. IV). If the
collineatioia is the identity on one and only one side of the invariant
triangle, the collineation is of Type JF.f If two distinct points are

* Cf. Baeher, Introduction to Higher Algebra, Chaps. XX and XXI.

t If it is the identity on more than one side, it is the identical collineation.

no]

TYPES OF COLLDTEATIONS

273

invariant, but no point not on the line I joining these two is invariant,
two possibilities again arise. If the collineation does not leave every
point of this line invariant, there is a unique other line through one
of these points that is invariant, since the invariant figure is self-dual.
The collineation is then of Type II. If every point of the line is
invariant, on the other hand, all the lines through a point of the
line I must be invariant, since the figure of invariant elements is
self-dual. The collineation is then of Type V.

If only one point is fixed, only one line can be fixed. The collinea-
tion is then parabolic both on the line and on the point, and the
collineation is of Type III.

We have thus proved that every collineation different from the
identity is of one of the five types previously enumerated. Type I
may be represented by the symbol [1, 1, 1], the three 1's denoting
three distinct double points. In Type IV there are also three distinct
double points, but all points on the line joining two of them are fixed
and Equation (1) has one double root. Type 7Fis denoted by [(1, 1), 1].
In Type //, as there are only two distinct double points, Equation
(1) must have a double root and one simple root. This type is ac-
cordingly denoted by the symbol [2, 1], the 2 indicating the double
point corresponding to the double root. Type Fis then naturally repre-
sented by [(2, 1)], the parentheses again indicating that every point
of the line joining the two points is fixed. Type III corresponds to a
triple root of (1), and may therefore be denoted by [3]. We have
then the following :

THEOREM 6. Every protective collineation in a plane is of one of
the following five types :

[1,1,1]

[(1.1). 1]

[2,1]

[(2, 1)]

[3]

In this table the first column corresponds to three distinct roots
of the characteristic equation, the second column to a double root,
the third column to a triple root. The first row corresponds to the
cases in which there exist at least three double points which are

274 TWO-DIMENSIONAL PBOJEOTIVITIES [CHAP, x

not collinear ; tlie second row to the case where there exist at least
two distinct double points and all such points are on the same line ;
the third row to the case in which there exists only a single -double
point.

With every collineation in a plane are associated certain projec-
tivities on the invariant lines and in the pencils on the invariant
points. In case the collineation is of Type /, it is completely deter-
mined if the projectivities on two sides of the invariant triangle are
given. There must therefore be a relation between the projectivities
on the three sides of the invariant triangle (cf. Ex. 5, p. 276). In a
collineation of Type II the projectivity is parabolic on one of the
invariant lines but not on the other. The point in which the two
invariant lines meet may therefore be called singly parabolic. The
collineation is completely determined if the projectivities on the
two invariant lines are given. In a collineation of Type /// the pro-
jectivity on the invariant line is parabolic, as likewise the projectivity
on the invariant point. The fixed point may then be called doubly
parabolic. The projectivities on the invariant lines of a collineation
of Type V are parabolic except the one on the axis which is the
identity. The center is thus a singly parabolic point. In the table
of Theorem 6 the symbols 3, 2, and 1 may be taken to indicate
doubly and singly and nonparabolic points respectively.*

We give below certain simple, so-called canonical forms of the
equations defining collineations of these five types.

Type L Let the invariant triangle be the triangle of reference.
The collineation is then given by equations of the form

in which a lv # 22 , & 33 are the roots of the characteristic equation and
must therefore be all distinct.

Type IV, Homology. If the vertices of the triangle of reference
are taken as invariant points, the equations reduce to the form written
above; but since one of the lines 3^= 0, # 3 = 0, # 3 = is pointwise

* For a more detailed discussion of collineations, reference may "be made to
Newson, A New Theory of Collineations, etc., American Journal of Mathematics,
Vol. XXIV, p. 109.

J] TYPES OF COLLIKEATIONS 275

invariant, we must have either a 22 = a ss or a ss = a n or & n = a Z2 . Thus
the homology, may be written

A harmonic homology or reflection is obtained by setting & 83 = 1.

Type IL The characteristic equation has one double root, p^ = p ,
say, and a simple root /3 3 . Let the double point corresponding to
Pi Pz ^ e ^i (0> 0, 1), let the double point corresponding to p z be
?7g = (1, 0, 0), and let the third vertex of the triangle of reference
be any point on the double line u z corresponding to /> 3 , which line
will pass through the point U r The collineation is then of the form

since the lines ^=0 and a? 2 = are double lines and (1, 0, 0) is a
double point. The characteristic equation of the collineation is clearly

and since this must have a double root, it follows that two of the
numbers a n> 22 , a 33 must be equal. To determine which, place
p = a 00 ; using the minors of the second row, we find, as coordinates
of the corresponding double point,

(0, (a n - a 23 ) (a 22 - a S3 ), a 32 (a n - a M ) ),

which is U v and hence we have #. 22 =ft 83 . The collineation then is
of Type II, if a xl ^ a 2Z . Its equations are therefore

where a 32 3=- and a n = a 22 .

jf^e /JJ. The characteristic equation has a triple root, p 1 = p 2 = P z
say. Let U : = (0, 0, 1) be the single double point, and the line x l = be
the single double line. With this choice of coordinates the collineation

has the form ,

P x i a 'n^i>

276 TWO-DIMENSIONAL PROJECTIVITIES [OHAP.X

By writing the characteristic equation we find, in view of the fact
that the equation has a triple root, that a n =^ 25J =^ 33 . The form of
the collineation is therefore

where the numbers a 21 , a sz must be different from 0. *

Type F, Elation. Choosing (0, 0, 1) as center and ^=0 as axis,
the equations of the collineation reduce to the form given for Type ///,
where, however, a 33 must be zero in order that the line 0^= be
pointwise invariant. The equations for Type II also yield an elation
in case & n = a 22 . Thus an elation may be written

EXERCISES

1. Determine the collineation which transforms the points A = (0, 0, 1),
B = (0, 1, 0), C= (1, 0, 0), D = (1, 1, 1) into the points E, C, D, A respec-
tively. Show that the characteristic equation of this collineation is (p I)
(p 2 + 1) = 0, which in any field has one root. Determine the double point
and double line corresponding to this root. Assuming the field of numbers to
be the ordinary complex field, determine the coordinates of the remaining two
double points and double lines. Verify, by actually multiplying the matrices,
that this collineation is of period 4 (a fact which is evident from the defini-
tion of the collineation).

2. With the same coordinates for A, JS, C, D determine the collincation
which transforms these points respectively into the points Z>, A , 7>, C. The
resulting collineation must, from this definition, be a homology. Why? De-
termine its center and its axis. By actual multiplication of the matrices
verify that its square is the identical collineation.

3. Express each of the collineations in Exs. 1 and 2 in terms of line
coordinates.

4. Show that the characteristic cross ratios of the one-dimensional projec-
tivities on the sides of the invariant triangle of the collineation x'z=ax v
x^ = bx z , z = c# 3 are the ratios of the numbers , /;, c. Hence show that tho
product of these cross ratios is equal to unity, the double points being taken
around the triangle in a given order.

5. Prove the latter part of Ex. 4 for the cross ratios of tho projectivities
on the sides of the invariant triangle of any collineation of Type /.

TYPES OF COLLINEATIOjtfS 277

6. Write the equations of a collineation of period 3;4;5;-..;n;-...

7. By properly choosing the system of nonhomogeneous coordinates any
collineation of Type / may be represented by equations x' = ax, y' = ly. The
set of all collineations obtained by giving the parameters a, b all possible
values forms a group. Show that the collineations #' = axj ?/' = a r y, where r
is constant for all collineations of the set, form a subgroup. Show that every
collineation of this subgroup leaves invariant every curve whose equation is
y = cx r 9 where c is any constant. Such curves are called path curves of the
collineations.

8. If P is any point of a given path curve, p the tangent at P, and
A y B, C the vertices of the invariant triangle, then & (p, PA, PJ3, PC) is a
constant.

9. For the values r = 1, 2, -| the path curves of the collineations of the
subgroup described in Ex. 7 are conies tangent to two sides of the invariant
triangle at two vertices.

10. If r = 0, the subgroup of Ex. 7 consists entirely of homologies.

11. Prove that any collineation of Type / may be expressed in the form

x' = k(ax + by),
/ = *(& -ay),
with the restriction a 2 + b 2 = 1.

12. Prove that any collineation can be expressed as a product of collinea-
tions of Type /.

13. Let the invariant figure of a collineation of Type // be A, JB, I, m,
where I = AB, B = Im. The product of such a collineation by another of
Type // with invariant figure A', B, Z, m' is in general of Type //, but may
be of Types /// or 1 V. Under what conditions do the latter cases arise ?

14. Using the notation of Ex. 13, the product of a collineation of Type II
with invariant figure A, J5, Z, m by one with invariant figure -4,,J5', Z, m' is
in general of Type II, but may be of Types /// or IV. Under what conditions
do the latter cases arise ?

15. Prove that any collineation can be expressed as a product of collinea-
tions of Type II.

16. Two collineations of Type III with the same invariant figure are not
in general commutative.

17. Any protective collineation can be expressed as a product of collinea-
tions of Type ///.

18. If II is an elation whose center is C, and P any point not on the
axis, then P and C are harmonically conjugate with respect to n~ 1 ( J P)
andH(P).

19. If two coplanar conies are projective, the correspondence between the
points of one and the tangents at homologous points of the other determines
a correlation*

20. If in a collineation between two distinct planes every point of the
line of intersection of the planes is self-corresponding, the planes are per-
spective.

278 TWO-DIMENSIONAL PROJEOTIV1TIES [CHAP, x

21. In nonhomogeneous coordinates a collineation of Type / with fixed
points (o lf a a ), (b l9 6 a ) (c v c 2 ) may be written

a; y 1 x y 1
! #2 1 t ! a 2 1 cr 2

(? 1 c . 2 ^ fc'c l c ~ I /j'^

Type II may be w

x y 1 ' ^ a? y 1
a x ij 1 1 ! n a 1 1

b 1% 1 /J />i & 2 1 ^*

Cj c 2 1 k' ci c z 1 /t'
ritten

ar y 1 2: y 1
i /i ft ~\ ft

/^ /; 3 1 ^j />! ^ 2 1 AV> a

and Type III may

f __

a? y 1 ?/ a,- y 1
i i 11
#! 2 1 1 rtj_ a 2 J. J.

Z^ J 2 1 A' bi h n 1 X:

! 5 2 f *! 4 t

be written
2- y 1

w i 20 s (a* 2 + 2j8J)a 1 + 2as 1 ? + W' 1

a; y 1

^ x w a 0^ + 2^
a; y 1

?^ w a (a 2 + 2^) 3 + 2a.V + ?a>

y x y 1

&* 1 3 o

u\ w 9 ai 2 +2/fr

97. Double pairs of a correlation. We inquire now regarding the
existence of double pairs of a correlation in a plane. By a double pair
is meant a point X and a line u such that the correlation transforms
X into u and also transforms u into X; in symbols, if F is the cor-
relation, such that r(Z)=w and F(w)=:X. We have already seen
(Theorem 3) that if the vertices and opposite sides of a triangle are
double pairs of a correlation, the correlation is a polarity.

We may note first that the problem of finding the double pairs of
a correlation is in one form equivalent to finding the double elements

07] DOUBLE PAIRS OF A CORRELATION 279

of a certain eollineation. In fact, a double pair X, u is such that
F(JL")=^ and r 2 (J5T) = r(w)=JT, so that the point of a double pair
of a correlation F is a double point of the coUineation F 2 Similarly,
it may be seen that the lines of the double pairs are the double lines
of the eollineation F 2 . It follows also from these considerations that
F is a polarity, if F 2 is the identical eollineation.

Analytically, the problem of determining the double pairs of a
correlation leads to the question : For what values of (x v # 2 , # 8 ) are
the coordinates

of the line to which it corresponds proportional to the coordinates

a x +

of the line which corresponds to it in the given correlation ? If p is
the unknown factor of proportionality, this condition is expressed by
the equations

K - P a u) x i + (* - P a n) + ( a is - P a ^ x* = 3

pa S2 ) ^ 8 = 0,

which must be satisfied by the coordinates (x lf # 2 , a? 8 ) of any point
of a double pair. The remainder of the treatment of this problem is*
similar to the corresponding part of the problem of determining the
double elements of a eollineation ( 96). The factor of proportionality
p is determined by the equation

*p a *l

(2)

which is of the third degree and has (under Proposition K 2 ) three
roots, of which one is 1, and of which the other two may be proper
or improper. Every root of this equation when substituted for p in
(1) renders these equations consistent. The coordinates (x v x v # 8 )
are then determined by solving two of these.

If the reciprocity in question is a polarity, Equations (1) must be
satisfied identically, i.e. for every set of values (x v # 2 , x 3 ). This would
imply that all the relations

S-/>^-0 (*,/=!, 2, 3)

are satisfied.

280 TWO-DIMENSIONAL PEOJECTIVITIES [CHAP. X

Let us suppose first that at least one of the diagonal elements of the
matrix of the coefficients (a i} ) be different from 0. If this be a lv the
relation & u -/^n O gives at once /> = !; and this value leads at
once to the further relations

% = V ft/ = 1,2, 3).

The matrix in question must then be symmetrical. If, on the other
hand, we have a u = a^ = a^ = 9 there must be some coefficient a^
different from 0. Suppose, for example, # 12 * 0. Then the Delation
a 12 Aa 21 = shows that neither k nor # 21 can be 0. The substitution
of one in. the other of the relations a 12 =&& 21 and a^ Jea^ then gives
Jc* = 1, or Jc = 1. The value 7c = 1 again leads to the condition that
the matrix of the coefficients be symmetrical The value & = 1
gives a K = 0, and # y = <^, which would render the matrix skew
symmetrical. The determinant of the transformation would on this
supposition vanish (since every skew-symmetrical determinant of odd
order vanishes), which is contrary to the hypothesis. The value
Jc BBS 1 is therefore impossible. We have thus been led to the fol-
lowing theorem:

THEOREM 7. TJie necessary and sufficient condition that a reci-
procity in a plane le a polarity is that the matrix of its coefficients
le symmetrical.

If the coordinate system is chosen so that the point which corre-
sponds to p =s 1 in Equation (2) is (1, 0, 0), it is clear that we must
have a zl a 12 and # 81 = fl ia . If the line corresponding doubly to
(1, 0, 0) does not pass through it, the coordinates [1, 0, 0] may be
assigned to this line. The equations of the correlation thus assume
the form

(3)

and Equation (2) reduces to

(4)

= 0.

The roots, other than 1, of this equation clearly correspond to points
on [1, 0, 0], Choosing one of these points (Proposition K 2 ) as (0, 0, 1),
we have either # 28 = a S2 , which would lead to a polarity, or # 88 = 0.

97] DOUBLE PAIES OF A COEBELATION 281

In the latter case it is evident that (4) has a double root if # 32 = a 23 ,
but that otherwise it has two distinct roots. Therefore a correlation
in which (1, 0, 0) and [1, 0, 0] correspond doubly, and which is not
a polarity, may be reduced to one of the three forms :

(a = 0, 1 = 0)

K= -

The squares of these correlations are collineations of Types /, //, IV
respectively.

If the line doubly corresponding to (1, 0, 0) does pass through it,
the coordinates [0, 1, 0] may be assigned to this line, and the equa-
tions of the correlation become

Equation (2) at the same time reduces to

(i-p)*-o,

and the square of the correlation is always of Type III. There are
thus five types of correlations, the polarity and those whose squares
are collineations of Types /, II, III, IV.

EXERCISES *

1. The points which lie upon the lines to which they correspond in a cor-
relation form a conic section C 2 , and the lines which lie upon the points to
which they correspond are the tangents to a conic K 2 . How are C 2 and E?
related, in each of the five types of correlations, to one another and to the
doubly corresponding elements ?

* On the theory of correlations see Seydewitz, Archiv der Mathematik, 1st series,
Vol. VIII (1846), p. 32 ; and Schr5ter, Journal fur die reine und angewandte Mathe-
matik, Vol. LXXVII (1874), p. 105.

282 TWO-DIMENSIONAL PROJECTIVITIES [CHAP. X

2. If a line a does not lie upon the point A' to which it corresponds in a
' correlation, there is a projectivity between the points of a and tho points in

which their corresponding lines meet a. In the case of a polarity this pro-
jectivity is always an involution. In any other correlation the linos upon
which this projectivity is involutoric all pass through a unique fixed point 0.
The line o having the dual property corresponds doubly to 0. The doublo
points of the involutions on the lines through are on the conic ( '-, and th< k
double lines of the involutions on the points of /C 2 are tangent to A" 2 . and o
are polar with respect to C 2 and K z . If a correlation determines involutions
on three nonconcurrent lines, it is a polarity.

3. The lines of K 2 through a point P of C 2 are the line which is transformed
into P and the line into which P is transformed by the given correlation.

4. In a polarity C 2 and K* are the same conic.

5. A necessary and sufficient condition that a collineation be the product of
two reflections is the existence of a correlation which is left invariant by the
collineation.*

98. Fundamental conic of a polarity in a plane. We have just
seen that a polarity in a plane is given by the equations

(1)

DEFINITION. Two homologous elements of a polarity in a plane are
called pole and polar, the point being the pole of the line and the
line being the polar of the point. If two points are so situated that
one is on the polar of the other, they are said to be conjugate.

The condition that two points in a plane of a polarity he conju-
gate is readily derived. In fact, if two points P = (M V x z , $ a ) and
p* '=(#/, # a ', x) are conjugate, the condition sought is simply that
the point P ! shall be on the line p f = \u[, u[, M 8 '], the polar of J\\ i.e.
u[x{ + u&l + u&l = 0. Substituting for u[, ut, %[ their values in
terms of x v x^ x s from (1), we obtain the desired condition, via. :

(2) a^xl +

As was to be expected, this condition is symmetrical in the coordi-
nates of the two points P and P f . By placing x{ = as, we obtain the

* This is a special case of a theorem of Dunham Jackson, Transactions of the
American Mathematical Society, Vol. X (1909), p. 479,

POLAR SYSTEM 283

condition that the point /' be self -conjugate, ie. that it be on its polar.
We thus uUuiu the result;

THKOUBM 8. 77/0 sdfaonjugttte print* of the polarity (1) are on
the- MUM whoso equation is

ft, r<f + <typ* + a^l + 2 a^v,, + 2 fl w avcfr + 2 a 2g aj a a? 8 = ;
( 9 conversely, every point of this conic is sdf -conjugate.

This conic is called the fundamental conic of the polarity. All of
it points may be improper, but it can never degenerate, for, if so,
the determinant v | would have to vanish (cf. Ex., p. 187). By
duality we obtain

THEOREM 8 r . The self-c,o>iju<jate lines of tlie polarity (1) are lines
of the, conic

(4) A^ 4- A^l 4- J S X + 2' A^u^ 4- 2 A^n^ 4- 2 ^ 23 w 3 ^ 3 = ;
<mrf, emwrMly, every line of this conic is self-conjugate.

Every point A' of the conic (3) corresponds in the polarity (1) to
the tangent to (3) at Jif. For if not, a point A of (3) would be polar
to a line a through A and meeting (3) also in a point B> B would
then be polar to a line I through B, and hence the line a^AB
would, by the definition of a polarity, be polar to ab = B. This would
require that a correspond both to A and to B.

If now we recall that the polar system of a conic constitutes a
polarity (Theorem 18, Cor., Chap. V) in which all the points and
lines of the conic, and only these, are self-conjugate, it follows from
the above that every polarity is given by the polar system of its
fundamental conic. This and other results following immediately
from it are contained in the following theorem :

THEOREM 9. Mrery polarity w the polar &y$t&ni of a conic, the
fwudaviental conic of the polarity. The self-conjugate points are
the points and the self -conjugate lines are the tangents of this conic.
Jtfvcry yoU and polar pair are pole and polar with respect to the
fundamental conin.

This establishes that Equation (4) represents the same conic as
Equation (3). The last theorem may be utilized to develop the ana-
lytic expressions for poles and polars, and tangents to a conic. This
we take up in the next section.

284 TWO-DIMENSIONAL PEOJECTIVITIES |CHAI>. x

99. Poles and polars with respect to a conic. Tangents. We

have seen that the most general equation of a conic in point coor-
dinates may be written

(1) a n x%+ & 22 # 2 2 + ^as&'s + 2 a u x l x 2 + 2 a 1 ^x 1 x s + 2 ^ 23 ^ 2 ^ 3 = 0.

The result of the preceding section shows that the equation of the
same conic in line coordinates is

(2) 4X + A^ul + A^ul + 2 A^u, + 2 A^u, + 2 A n u t u s = 0,
where A, is the cofactor of a,, in the determinant

This result may also be stated as follows :

THEOREM 10. The necessary and sufficient condition that the line
w 1 a? 1 +^ a a; 3 + W 8 a? a =s le tangent to the conic (1) is that JSyitation (2)
"be satisfied.

COKOLLAKY. This condition may also le written in the form

u, u n u 9

= 0.

Equation (2) of the preceding section expresses the condition that
the points (x v # a , # 3 ) and (#/, x 9 ojj) be conjugate with respect to the
conic (1). If in this equation (a?/, x, &) be supposed given, while
(x v # 2 , # 3 ) is regarded as variable, this condition is satisfied by all the
points of the polar of (x[, x^, x^) with respect to the conic and by no
others. It is therefore the equation of this polar. When arranged
according to the variable coordinates x it it becomes

(3) (*!!/+ ajX+ *vf&*i + (*vPi + * M atf + ^<)^2

, ;

while if we arrange it according to the coordinates #?/, it becomes
(4) (a lA + a lz x, + a lA ) x[ + (a, A + ^ 2 A + a^ <

== 0.

Now it is readily verified that the latter of these equations may
be derived from the equation (1) of the conic by applying to the
left-hand member of this equation the polar operator

oi),ioo] VARIOUS DEFINITIONS OF CONICS 285

( 89) and dividing the resulting equation by 2. Furthermore, if

we define the symbols ~~ > -^ > -^ to be the result of substituting
dx l dx^ 0a? 8

(^/, 2 ', # 3 ') for (ajj, a? a , a? 8 ) in the expressions -^-> ~^> -^- (/ being any

</i#.| ve/t'g </cH2g

polynomial in x v % 2 , X Q ), it is readily seen that Equation (3) is
equivalent to

where now/ is the left-hand member of (1).
This leads to the following theorem :

THEOREM 11. Iff^Qis the equation of a conic in homogeneous
point coordinates, the equation of the polar of any point (%{, x 9 x$ is
given ly either of the equations

= ( ) or Xif + x

8x 8 *dx[ z d% o%

If the point (%[, x^ x) is a point on the conic, either of these equa-
tions represents the tangent to the conic f = at this point.

100. Various definitions of conies. The definition of a (point)
conic as the locus of the intersections of homologous lines of two
protective flat pencils in the same plane was first given by Steiner in
1832 and used about the same time by Ghasles. The considerations
of the preceding sections at once suggest two other methods of defi-
nition, one synthetic, the other analytic. The former begins by the
synthetic definition of a polarity (cf. p. 263), and then defines a point
conic as the set of all self-conjugate points of a polarity, and a line
conic as the set of all self -con jugate lines of a polarity. This defini-
tion was first given by wn Staudt in 1847. From it he derived the
fundamental properties of conies and showed easily that his definition
is equivalent to Steiner's. The analytic method is to define a (point)
conic as the set of all points satisfying any equation of the second
degree, homogeneous in three variables % v x z , x 3 . This definition (at
least in its nonhomogeneous form) dates back to Descartes and Fermat
(1637) and the introduction of the notions of analytic geometry.

286 TWO-DIMENSIONAL PBOJECTIVITIBB [CHAP. X

The oldest definition of conies is due to the ancient Greek geometers, who
defined a conic as the plane section of a circular cone. This definition involves
metric ideas and hence does not concern us at this point. We will return to it
later. It is of interest to note in passing, however, that from this definition
Apollonius (about 200 A.D.) derived a theorem equivalent to the one that the
equation of a conic in point coordinates is of the second degree.

The reader will find it a valuable exercise to derive for himself
the fundamental properties of polarities synthetically, and thence to
develop the theory of conies from von Staudt's definition, at least so
far as to show that his definition is equivalent to Steiner's. It may
be noted that von Staudt's definition has the advantage over Steiner's
of including, without reference to Proposition K 2 , conies consisting
entirely of improper points (since there exist polarities which have
no proper self-conjugate points). The reader may in this connection
refer to the original work of von Staudt, Die Geometric der Lage,
Niirnberg (1847)'; or to the textbook of Enriques, Vorlesungen liber
projective G-eometrie, Leipzig (1903).

EXERCISES

1. Derive the condition of Theorem 10 directly by imposing tho condition
that the quadratic which determines the intersections of the given line with
the conic shall have equal roots. What is the dual of this theorem ?

2. Verify analytically the fundamental properties of poles and polars with
respect to a conic (Theorems 13-18, Chap. V).

3. State the dual of Theorem 11.

4. Show how to construct the correlation between a plane of points and a
plane 'of lines, having given the homologous pairs A, of\ B, &'; C, cf\ 7>, d'.

5. Show that a correlation between two planes is uniquely determined if
two pencils of points in one plane are made projective respectively with two
pencils of lines in the other, provided that in this projectivity the point of
intersection of the axes of the two pencils of points corresponds to the lino
joining the two centers of the pencils of lines.

6. Show that in our system of homogeneous point and line coSrdinaten the
pairs of points and lines with the same coordinates are poles and polars with
respect to the conic x * + #| + xj = 0.

7. On a general line of a plane in which a polarity has boon defined the
pairs of conjugate points form an involution the double points of which are
the (proper or improper) points of intersection of the line -with the funda-
mental conic of the polarity,

8. A polarity in a plane is completely defined if a self-polar triangle is
given together with one pole and polar pair of which the point is not on a
side nor the line on a vertex of the triangle.

f 100, 101] PAIRS OF CONICS 287

9. Prove Theorem tt analytically.

10. G iven a simple plane pentagon, there exists a polarity in which to each
vertex corresponds the opposite side.

11. The three points A', />", 6" on the sides BC, CA 9 AI)otQ, triangle that
aro conjugate in a polarity to the vertices A> />, C respectively are collinear
(cf. Ex. liJ, p. 125).

12. Show that a polarity is completely determined when the two involutions
o conjugate points on two conjugate lines are given.

13. Construct the polarity determined by a self-polar triangle ABC and an
involution of conjugate points on a line.

14. Construct the polarity determined by two pole and polar pairs A, a and
/>, // and one pair of conjugate points C, C'.

15. If a triangle RTU is self-polar with regard to a conic C 2 , and A is any
point of G' 2 , there are three triangles having A as a vertex which are inscribed
to C Y2 and circumscribed to STU (ISturm, Die Lehre von den geometrischen
Verwandtschaften, Vol. I, p. 147).

101, Pairs of conies. If two polarities, i.e. two conies (proper or
improper), are given, their product is a collineatiorx which leaves
invariant any point or line which has the same polar or pole with
regard to both conies. Moreover, any point or line which is not left
invariant by this collineation must have different polars or poles
with regard to the two conies. Hence the points and lines which
have the same polars and poles with regard to two conies in the
same plane form one of the five invariant figures of a nonidentical
collineation.

Type I. If the common self-polar figure of the two conies is of
Type /, it is a self-polar triangle for both conies. Since any two conies
are protectively equivalent (Theorem 9, Chap. VIII), the coordinate
system may be so chosen that the equation of one of the conies, Jt a , is

(1) itf-asJ + atf^O.

With regard to this conic the triangle (0, 0, 1), (0, 1, 0), (1, 0, 0) is
self-polar. The general equation of a conic with respect to which this
triangle is self-polar is clearly

(2) a ^ a^l + # A 2 = 0.

An equation of the form (2) may therefore be taken as the equation
of the other conic, # 2 , provided (1) and (2) have no other self-polar
elements than the fxindamental triangle. Consider the set of conies

(8) a^?- &A 2 -h o^ + XOaf- *?+) = 0.

288

TWO-DIMENSIONAL PRO JBCTIVITIBS [CHAP. X

The coordinates of any point which satisfy (1) and (2) also satisfy (3),
Hence all conies (3) pass through the points common to * and !> >a .
For the value X = a s> (3) gives the pair of lines
(4) (a, - a,) x* - fa - a,) v* = 0,

which intersect in (0, 0, 1). The points of intersection of these lines
with (1) are common to all the conies (3).

The lines (4) are distinct, unless a^ a^ or <x 2 = <V But if ] = ,,,
any point (xf, 0, x) on the line # 2 =0 has the polar x^ + x^ =
both with regard to (1) and with regard to (2). The self-polar figure
is therefore of Type IV. In order that this figure be of Type 7, the
three numbers a v a z> a z must all be distinct. If this condition is
satisfied, the lines (4) meet the conies (3) in four distinct points.

FIG. 100

The actual construction of the points is now a problem of the second
degree. We have thus established (fig. 100)

THEOREM 12. If two conies have a common self -polar triangle (and
no oilier common self -polar pair of point and line), they intersect in
four distinct points (proper or improper). Any two conies of the
pencil determined "by these points ham the same self -polar triangle.
Dually, two such conies have four common tangents, and any two

wi]

PAIRS OF CONICS

289

B=(010)

eowks of tlie rttnye daterwiucd "by those common taiigents have the same
self-j>ol(tr tri<mgk.

CoKoiJAuy. Any pencil of comas of Type I can be represented by *

(5) (^- afl + X (-) =0,

the four common points being in this case (1,1, 1), (1, 1, 1), (1, 1, 1),
1,1,1).

//. When the
common self-polar figure
is of Type //, one of the
points lies on its polar,
and therefore, this polar is
a tangent to eacli of the
conies A*, jB fl . Since two
tangents cannot intersect
in a point of contact, the
two lines of the self-polar
figure are not both tan-

gents. Hence the point JB p IGf 101

of the self-polar figure

which is on only one of the lines is the pole of the line I of the figure
which is on only one of the points (fig. 101), and the line a on the two
points is tangent to both conies at the point A which is on the two lines,
Choose a system of coordinates with ^ = (1,0, 0), a=[0, 0, 1],
#= (0, 1, 0), and i= [0, 1, 0]. The equation of any conic being

2 *=* >

the condition that A be on the conic is ^=0 ; that & then be tan-
gent is Z> 8 = 0; that I then be the polar of J3 is ^=0. Hence the
general equation of a conic with the given self-polar figure is

(6) <ytf + <V* + 2

* Equation (6) is typical for a pencil of conies of Type I, and Theorem 12 is a
sort of converse to the developments of 47, Chap, V. The reader will note that
if the problem of finding the points of intersection of two conies is set up directly,
it is of the fourth degree, but that it is here reduced to a problem of the third
degree (the determination of a common self-polar triangle) followed by two quad-
ratic constructions. This corresponds to the well-known solution of the general
biquadratic equation (cf. Fine, College Algebra, p. 486). For a further discussion
of the analytic geometry of pencils of conies, cf . Clebsoh-Lindemann, Vorlesungen
iiber Geometrie, 2d ed., Vol. I, Part I (1906), pp. 212 ff.

290 TWO-DIMENSIONAL PROJBOTIVITIES LCHAP. x

Since any two conies are projectively equivalent, A 1 may be chosen
to be

(7) +;+ 2 3^=0.

The equation of B* then has the form (6), with the condition that
the two conies have no other common self-polar elements. Since the
figure in which a is polar to A and b to B can only reduce to Types
IV or F, we must determine under what conditions each point on a
or each point on I has the same polar with regard to (6) and (7).
The polar of (x[, x, A- 8 ') with regard to (6) is given by

1^3= 0-

Hence the first case can arise only if & 2 = & 2 ; and the second only
if a = 6 .

> s

Introducing the condition that a z , 8 , 5 2 are all distinct, it is then
clear that the set of conies

af* + a,x* + 2 l&Xi + X (xj + ^ + 2 x A ) =
contains a line pair for X = # 2 , viz. the lines

Hence the conies have in common the points of intersection with (7)

of the line

(o,-a a ) 0,+ 2(5,- ^=0.
This gives

THEOREM 13. If two conies have a common self -polar figure of
Type II, tliey have three points in common and a common tangent at
one of them. Diially, tliey have three common tangents and a common
point of contact on one of the tangents. The two conies determine a,
pencil and also a range of conies of Type II.

COROLLARY. Any pencil of conies of Type II may le represented
ly the equation x% # 8 2 + \x^ = 0. The conies of this pencil all pass
through the points (0, 1, 1), (0, 1, 1), (1, 0, 0) and are tangent to
# 3 = 0.

Type III. When the common self-polar figure is of Type ///, the
two conies evidently have a common tangent and a common point
of contact, and only one of each. Let the common tangent be # 8 =s 0,
its point of contact be (1, 0, 0), and let A 2 be given by

(8) ^+

PAIRS OF CWICS 291

The general equation of a conic tangent to # 3 = at (1, 0, 0) is
( 8) <Vtt? + <v + 2 MA + 2 MA = 0,

with regard to which the polar of any point (#/, &/, 0) on # 3 = is
given l)y
( 1 0)

This will be identical with the polar of (a x^ 0) with regard to A*
for all values of as/, ai if &,,= 2 and \*= 0. Since (1, 0, 0) only is to
have the same polar with regard to both conies, we impose at least
one of the conditions & 2 =tf 2 , ^=0. The line (10) will now be
identical with the polar of (8) for any point (x[, #/, 0) satisfying the
condition

This quadratic equation must have only one root if the self-polar figure
is to be of Type ///. This requires & 2 = a a , and as J 2 , a 2 cannot both
l)e unless (9) degenerates, the equation of B* can be taken as

(11) xl + 2 x^ + off + 2 MA = 0, (B 1 *= 0).

A~(lQO) a=[ooi]

The conies (8) and (11) now evidently have in common the points of
intci'HGction of (8) with the
line pair

8 a; 8 fl + 2^8^8=0,

and no other points. Since

$*% = is a tangent, this gives

two common points. If the

second common point is taken

to be (0, 0, 1), the set of

conies which have in com-

mon the points (0, 0, 1) and (1, 0, 0) = A and the tangent a at A 9

and no other points, may be written (fig. 102)

(1 2) <* + 2 fa + \% 2 x s = 0.

THEOREM 14. If two conies have a common self-polar figure of
Type III t thfiy have two points in common and a common tangent
at one of them, and one other common tangent. They determine a
pencil and a range of conies of Type III.

292 TWO-DIMENSIONAL PROJECTIVITIES [OHAP.X

COROLLARY. A pencil of conies of Type III can "be represented ly
the equation x%+ 2 #3^ + \a? 2 # 3 = 0.

Type IV. When the common self-polar figure is of Type 7F, let the
line of fixed points be X B = and its pole be (0, 0, 1). The coordinates
being chosen as they were for Type /, the conic A* has the equation

X^ - XQ + #?g = U J

and any other conic having in common with A* the self-polar tri-
angle (1, 0, 0), (0, 1, 0), (0, 0, 1) has an equation of the form

2== 0.

The condition that every point on # 8 = shall have the same polar
with regard to this conic as with regard to A* is a l = a y Hence B
may be written

Any conic of this form has the same tangents as A 2 at the points
(1, 1, 0) and (1, 1, 0) (fig. 103). Hence, if X is a variable parameter,

the last equation represents
a pencil of conies of Type IV
according to the classification
previously made.

THEOREM 1 5. If two conies
l la w a common self-polar
figure of Type IV, they have

Fio 103 two points in common and

the tangents at these points,

They determine a pencil (which is also a range) of conies of Type IV.
COKOLLARY. A pencil of conies of Type IV may le represented ly

the equation

and also by the equation

Type V. When the common self-polar figure is of Type F, let the
point of fixed lines be (1, 0, 0) and the line of fixed points be # 8 = 0.
As in Type ///, let J? be given by

(8) ^+2^3=0.

We have seen, in the discussion of that type, that all points of x^
have the same polars with respect to (8) and (9), if in (9) we have

101] PAIES OF CONICS 293

Z> a ss a% and l^ 0. Hence, if A* and J? 2 are to have a common self-

polar figure of Type V, the equation of & must have the form

(13) a.(x*+2x A )+a^*~Q.

Prom the form of equations (8) and

(13) it is evident that the conies have

in common only the point (1, 0, 0) and

the tangent X B = 0, and that every point

on # 8 = has the same polar with ro- o,= [o0lj A = (\wo)

spect to both conies (fig. 104). Hence

they determine a pencil of Type V.

THEOREM 16. If two conies have a common self -polar figure of
Type V, they have a lineal element (and no other elements) in com-
mon and determine a pencil (which is also a range) of conies of
Type V according to the classification already given.

COROLLAHY. A pencil of conies of Type V can le represented ly the

equation

1 *

As an immediate consequence of the corollaries of Theorems 12-16
we have

THEOREM 17. Any pencil of conies may le written in the form

where f~ and #= are the equations of two conies (degenerate or
not) of the pencil.

EXERCISES

1. Prove analytically that the polars of a point P with, respect to the
conies of a pencil all pass through a point 'Q. The points P and Q are double
points of the involution determined by the conies of the pencil on the line PQ.
Give a linear construction for Q (cf. Ex. 3, p. 13 C). The correspondence
obtained by letting every point P correspond to the associated point Q is a
< < quadratic birational transformation. ' ' Determine the equations representing
this transformation. The point Q, which is conjugate to P with regard to all
the conies of the pencil, is called the conjugate of P with respect to the pencil.
The locus of the conjugates of the points of a line with regard to a pencil of
conies is a conic (cf. Ex. 81, p. 140).

2. One an<J only one conic passes through four given points and has two
given points as conjugate points, provided the two given points are not con-
jugate with respect to all the conies of the pencil determined by the given
set of four. Show how to construct this conic.

294 TWO-DIMENSIONAL PEOJECTIVITIES [CHAP.X

3. One conic in general, or a pencil of conies in a special case, passes
through three given points and has two given pairs of points as conjugate
points. Give the construction.

4. One conic in general, or a pencil of conies in a special case, passes
through two given points and has three pairs of given points as conjugate
points ; or passes through a given point and has four pairs of given points as
conjugate points ; or has five given pairs of conjugate points. Give the cor-
responding constructions for each case.

102. Problems of the third and fourth degrees.* The problem of
constructing the points of intersection of two cdhics in the same
plane is, in general, of the fourth degree according to the classifi-
cation of geometric problems described in 83. Indeed, if one of
the coordinates be eliminated between the equations of two conies,
the resulting equation is, in general, an irreducible equation of the
fourth degree. Moreover, a little consideration will show that any
equation of the fourth degree may be obtained in this way. It
results that every problem of the fourth degree in a plane may
be reduced to the problem of constructing the common points (or
by duality the common tangents) of two conies. Further, the prob-
lem of finding the remaining intersections of two conies in a plane
of which one point of intersection is given 1 , is readily seen to be of
the third degree, in general; and any problem of this degree can be
reduced to that of finding the remaining intersections of two conies
of which one point of intersection is known. It follows that any
problem of the third or fourth degree in a plane may be reduced
to that of finding the common elements of two conies in the
plane, f

A problem of the fourth (or third) degree cannot therefore be
solved by the methods sufficient for the solution of problems of the
first and second degrees (straight edge and compass). | In the case
of problems of the second degree we have seen that any such prob-
lem could be solved by linear constructions if the intersections of

* In this section wo have made use of Amodeo, Lezioni di Ooomctria Projettiva,
pp. 40(5, 437. Some of the exercises are taken from the same book, pp. 448-451.

1 Moreover, we have soon (p, 28!), footnote) that any problem of the fourth
degree may be reduced to one of the third degree, followed by two of the floc.ond
degree.

| "With the usual representation of the ordinary real geometry \vo should require
an instrument to draw conies.

H)2] TI1IJID AND FOURTH DEGKEE PROBLEMS 295

every line in the plane with a fixed conic in that plane were assumed
known. Similarly, any problem of the fourth (or third) degree can
he solved by linear and quadratic constructions if the intersections
of every conic in the plane with a fixed conic in this plane are
assumed known. This follows readily from the fact that any conic
in the plane can "be transformed by linear constructions into the
fixed conic. A problem of the third or fourth degree in a plane
will then, in the future, be considered solved if it has been reduced
to the finding of the intersections of two conies, combined with
any linear or quadratic constructions. As a typical problem of the
third degree, for example, we give the following:

To find the double points of a nonpcrspective collineation in a plane
'which is* determined ly four pairs of homologous points.

Solution. When four pairs of homologous elements are given, we
can construct linearly the point or line homologous with any given
point or line in the plane. Let the collineation be represented by II,
and lot A be any point of the plane which is not on an invariant
line. Let U(A)=A r and II (A 1 )** A". The points A, A 1 , A" are then
not collinear. The pencil of lines at
A is protective with the pencil at
A 1 , and these two projective pencils
generate a conic C 2 which passes
through all the double points of II,
and which is tangent at A' to the
line A r A" (fig. 105). The conic <7 2 is
transformed by the collineation II
into a conic C? generated by the pro-
jective pencils of lines at A f and A". FlQ " 10
6\ a also passes through A! and is tangent at this point- to the line
A A 1 . The double points of II are also points of C*. The point A f
is not a double point of II by hypothesis. It is evident, however,
that every other point common to the two conies C 2 and C* is a
double point.

If (7 s and C? intersect again in three distinct points Z, H> N 9 the
latter form a triangle and the collineation is of Type /. If <7 2 and C
intersect in a point N, distinct from A f , and are tangent to each other
at a third point L=zM, the collineation has M, N for double points

296 TWO-DIMENSIOKAL PEOJECTIVITIES [CHAP.X

and the line MN and the common tangent at M for double lines
(fig, 106); it is then of Type //. If, finally, the two couics have
contact of the second order at a point L = M=* N, distinct from A r ,
the collineation has the single double line which is tangent to the
conies at this point, and is of Type III (fig. 107).

, 106 tfw. 107

EXERCISES

1. Give a discussion of the problem above without making at the outset
the hypothesis that the collineation is nonperspective.

2. Construct the double pairs of a correlation in the piano, which is not
a polarity.

3. Given two polarities in a plane, construct their common polo and
polar pairs.

4. On a line tangent to a conic at a point A is given an involution I, and
from any pair of conjugates P, P f of I are drawn the second tangents p, p' to
the conic, their points of contact being Q, Qf respectively. Show that the IOCUH
of the point pp' is a line, /, passing through the conjugate, A', of A in the invo-
lution I ; and that the line QQ' passes through the pole of I with respect to
the conic.

5. Construct the conic which is tangent at two points to a given conic and
which passes through three given points. Dualize.

6. The lines joining pairs of homologous points of a noninvolutoric pro-
jectivity on a conic A* are tangent to a second conic B* which is tangent to
A 2 at two points, or which hyperosculates A*.

7. A pencil of conies of Type // is determined by three points A, B, C
and a line c through C. What is the locus of the points of contact of tho
conies of the pencil with the tangents drawn from a givn point P of <??

8. Construct the conies which pass through a given point P and which are
tangent at two points to each of two given conies.

9. If /= 0, g s= 0, h = are the equations of thrcd couicH in a piano not
belonging to the same pencil, the system of conies given by the equation

JOiiJ THIRD AND FOURTH DEGREE PROBLEMS 297

X, ju,, v being variable parameters, is called a bundle of conies. Through every
point of the plane passes a pencil of conies belonging to this bundle ; through
any two distinct points passes in general one and only one conic of the bundle.
If the conies /, r/, h have a point in common, this point is common to all the
conies of the bundle. Give a nonalgebraic definition o a bundle of conies.

10. Tho set of all conies in u plane passing through the vertices of a triangle
form a bundle. Tf the equations of the sides of this triangle are I = 0, m = 0,
n = 0, show that the bundle may be represented by the equation

Xwn + jjinl + vim = 0.

What aro the degenerate conies oi this bundle ? *

11. The not of all conies in a plane which have a given triangle as a self-
polar triangle forms a bundle. If the equations of the sides of this triangle are
/ = 0, m 0, n = 0, show that the bundle may be represented by the equation

A/ 2 4- /im a + w* = 0.

What aro the degenerate conies of this bundle?

12. The conies of the bundle described in Ex. 11 which pass through a
general point P of the plane pass through the other three vertices of the
quadrangle, of which one vertex is P and of which the given triangle is the
diagonal triangle. What happens when P is on a side of the given triangle ?
Dualize.

13. The reflections whose centers and axes are the vertices and opposite
aides of a triangle form a commutative group. Any point of the plane not
on a side of the triangle is transformed by the operations of this group into
the other three vertices of a complete quadrangle of which the given triangle
is the diagonal triangle. If this triangle is taken as the reference triangle,
what are the equations of transformation ? What conies are transformed into
themselves by the group, and how is it associated with the quadrangle-
quadrilateral configuration ?

14. The necessary and sufficient condition that two reflections be com-
mutative Is that the center of each shall be on the axis of the other.

15. The invariant figure of a collineation may be regarded as composed of
two lineal elements, the five types corresponding to various special relations
between the two lineal elements.

16. A correlation which transforms a lineal element Aa into a lineal
clement 7#; and also transforms Bl into -la is a polarity.

17. How many collineations and correlations are in the group generated
by the. reflections whoso centers and axes are the vertices and opposite sides
of a triangle and a polarity with regard to which the triangle is self -polar ?

* In connection with this and the two following exercises, cf . Castelnuovo,
Lessioni di Geornetria Analitica e Projettiva, Vol. I, p. 395.

CHAPTER XI*

FAMILIES OF LINES

103- The regulus. The following theorem, on which depends the
existence of the figures to be studied in this chapter, IKS logically

equivalent (in the presence of Assump-
tions A and E) to Assumption P. 1 1
might have been used to replace that
assumption.

THEOREM 1. If l v 1 S9 1 9 are three,
mutally skew lines, and if m v w 2 , w 8 ,
w 4 are four lines each of which meets
each of the lines l v /, 7 3 , then any line l^
which meets three of the lines m v m^
m s , m^ also meets the fourth.

Proof. The four planes l$n v Ijn^
Z x m 3 , lp\ of the pencil with axis l l are
perspective through the pencil of points
on 1 3 with the four planes l z m v / a w fi ,
fy j h> ^2 m 4 ^ ^ P enc il with axis / a
(fig. 108). For, by hypothesis, the lines
of intersection m v w 2 , m^ m^ of the
pairs of homologous planes all meet / 8 .
The set of four points in which the four planes of the pencil on ^
meet Z 4 is therefore projective with the set of four points in which
the four planes of the pencil on 2 meet Z 4 . But l meets three of the
pairs of homologous planes in points of their lines of intersection,
since, by hypothesis, it meets three of the lines m v m 2 , w 3 , w 4 . Hence
in the projectivity on l there are three invariant points, and hence
(Assumption P) every point is invariant. Hence l meets the remain-
ing line of the set m v m^ m v m.

* AH the developments of this chapter are on the basis of Assumptions A, E, P, HO.
But see the exercise oa page 201.

208

108

SW] THE EEG-ULUS 299

DEFINITION. If l v Z 2 , l s are three lines no two of which are in the
Banie plane, the set of all lines which meet each of the three given lines
is called a regulus. The lines l v ,, 1% are called directrices of this regulus.

It is clear that no two lines of a regains can intersect, for other-
wise two of the directrices would lie in a plane. The next theorem
follows at once from the last.

THEOREM 2. If l v / 2 , Z 8 are three lines of a regulus of which
m v v/6 a , w 8 are directrices, m v m 2 , m 3 are lines of the reyidus of which
l v l, / 8 arc directrices.

It follows that any three lines no two of which lie in a plane are
directrices of one and only one regulus and are lines of one and only
one regulus.

DEFINITION. Two reguli which are such that every line of one
meets all the lines of the other are said to be conjugate. The lines of
a regulus are called its generators or rulers ; the lines of a conjugate
regulus are called the directrices of the given regulus.

THEOREM 3, Every regulus has one and only one conjugate regulus.

This follows immediately from the preceding. Also from the proof
of Theorem 1 we have

THEOREM 4. Tlie correspondence THEOREM 4'. The correspond-
established ly the lines of a regu- ence established "by the lines of a
lux between the points of two lines regulus between the planes on any
of its conjugate regulus is projec- two lines of its conjugate regulus
tivc. is projectile.

THEOR*^ 5. The set of all lines THEOREM 5'. The set of all lines of
joining pairs of homologous points intersection of pairs of homologous
of two projectioe pencils of points planes of two projectile pencils of
on #7iYW lines is a regulus. planes on skew lines is a regulus.

Proof. We may confine ourselves to the proof of the theorem on
the left. By Theorem 6, Chap. Ill, the two pencils of points are
perspective through a pencil of planes. Every line joining a pair of
homologous points of these two pencils, therefore, meets the axis of
the pencil of planes. Hence all these lines meet three (necessarily
skew) lines, namely the axes of the two pencils of points and of the
pencil of planes, and therefore satisfy the definition of a regulus.
Moreover, every line which meets these three lines joins a pair of
homologous points of the two pencils of points.

300 FAMILIES OF LINES LOUAP. XI

THEOREM 6. If [p] are tlie lines of a rcyidus tt,nd q in u* dirwkrir,
of tlie regulus, the pencil of points gfj?] 'is projcvtire with, tlie, pMwil
of planes q [p].

Proof. Let g[ be any other directrix. By Theorem 4 the pencil of
points q[p] is perspective with the pencil of points j'jjp]. But each
of the points of this pencil lies on the corresponding piano <jj>.
Hence the pencil of points < f [p] is also perspective with the pencil
of planes q [p].

EXERCISES

1. Every point which is on a line of a rogulus is also on a line of its
conjugate regains.

2. A plane which contains one line of a regnlus contains also a line of its
conjugate regains.

3. Show that a regains is uniquely denned by two of its lines and throe
of its points,* provided no two of the latter are coplanar with either of the
given lines.

4. If four lines of a regains cut any line of the conjugate regains in points
of a harmonic set, they are cut by every such line in points of a harmonic
set. Hence give a construction for the harmonic conjugate of a line oi! a
regains with respect to two other lines of the regains.

5. Two distinct reguli can have in common at most two distinct linen.

6. Show how to construct a regnlus having in common with a given
regains one and but one ruler.

104. The polar system of a regulus. A plane meets every line of
a regulus in a point, unless it contains a line of the regulus, in which
case it meets all the other lines in points that are collinear. Since
the regulus may be thought of as the lines of intersection of pairs of
homologous planes of two projective axial pencils (Theorem fi'), the
section by a plane consists of the points of intersection of pairs of
homologous lines of two projective flat pencils. Hence the section
of a regulus by a plane is a point conic, and the conjugate regulun
has the same section. By duality the projection of a regulus and its
conjugate from any point is a cone of planes.

The last remark implies that a line conic is the "picture " in a piano of
a regulus and its conjugate. For such a picture in clearly a plant) Hoctiou of
the projection of the object depicted from the eye of an observer. Fig. 108
illustrates this fact.

* By a point of a regains is meant any point on a line of the regains.

iw] THE REGULUS 301

The section of a regiilus by a plane containing a line of the regu-
lus is a degenerate conic of two lines. The plane section can never
degenerate into two coincident lines because the lines of a regulus
and its conjugate are distinct from each other. In like manner, the
projection from a point on a line of the regulus is a degenerate cone
of planes consisting of two pencils of planes whose axes are a ruler
and a directrix of the regulus.

DEFINITION. The class of all points on the lines of a regulus is
called a surface of the second order or a guadric surface. The planes
on the lines of the regulus are called the tangent planes of the sur-
face or of the regulus. The point of intersection of the two lines of
the regulus and its conjugate in a tangent plane is called the point
of contact of the plane. The lines through the point of contact in a
tangent plane are called tangent lines, and the point of contact of the
plane is also the point of contact of any tangent line.

The tangent lines at a point of a quadric surface include the lines
of the two conjugate reguli through this point and all other lines
through this point which meet the surface in no other point Any
other line, of course, meets the surface in two or no points, since a
plane through the line meets the surface in a conic. The tangent
lines are, by duality, also the lines through each of which passes only
cue tangent plane to the surface.

THEOREM 7. TJw tangent planes at the points of a plane section of
a quadric surface pass through a point and constitute a cone of planes.
Dually, the points of contact of the cone of tangent planes through a
point are coplanar and form a point conic.

Proof. It will suffice to prove the latter of these two dual theorems.
Let the vertex P of the cone of tangent planes be not a point of the
surface. Consider three tangent planes through P, and their points of
contact. The three lines from these points of contact to P are tan-
gent lines o the surface and hence there is only one tangent plane
through each of them. Hence they are lines of the cone of lines asso-
ciated with the coue of tangent planes. Let TT be the plane through
their points of contact. The section by TT of the cone of planes through
P is therefore the conic determined by the three points of contact
and the two tangent lines in which two of the tangent planes meet
TT. The plane TT, however, meets the regulus in a conic of which the
three points of contact are points. The two lines of intersection with

302 FAMILIES OF LINES LULU-. XI

TT of two of the tangent planes througli P are tangents to this conic,
because they cannot meet it more than one point each. The section
of the surface and the section of the cone ot planes then have three
points and the tangents through two of them in common. Hence these
sections are identical, which proves the theorem when P IN not on
the surface.

If P is on the surface, the cone of planes degenerates into two lines
of the regulus (or the pencils of planes on these lines), and the points
of contact of these planes are all on the same two lines. Hence the
theorem is true also in this case.

DEFINITION. If a point P and a plane TT are so related to a regulus
that all the tangent planes to the regulus at points of its section
by TT pass through P (and hence all the points of contact of tangent
planes through P are on TT), then P is called the pole of vr and TT the
polar of P with respect to the regulus.

COROLLARY. A tangent plane to a regulus is the polar of its point
of contact.

THEOREM 8. The polar of a point P not on a regidus contains all
points P 1 such that the line PP 1 meets the siwface in two points which
are harmonic conjugates with respect to P, PJ

Proof. Consider a plane, a, through PP' and containing two lines
a, Z> of the cone of tangent lines through P. This plane meets the
surface in a conic <7 2 , to which the lines a f I are tangent. As the polar
plane of P contains the points of contact of a and ft, its section by a
is the polar of P with respect to <7 2 . Hence the theorem follows
as a consequence of Theorem 13, Chap. V,

THEOREM 9. The polar of a point of a plane TT with respect to a
regulus meets TT in the 'polcvr line of this point with regard to the conic
which is the section of the regulus ly TT.

Proof. By Theorem 8 the line in which the polar plane meets TT
has the characteristic property of the polar line with respect to a conic
(Theorem 13, Chap. V). This argument applies equally well if the
conic is degenerate. In this case the theorem reduces to the following

COROLLARY. The tangent lines of a regulus at a point on it, are
paired in an involution the double lines of which are the ruhr (tn,d
directrix through that point. Each line of a pair contains the polar
points of all the planes on the other line.

w*J THE EEGULUS 303

THKOKBM 10. The polar s with regard to a regulus of the points of
a line I are an axial pencil of planes projeetive with the pencil of
points on L

Proof. In case the given line is a line of the regulus this reduces
to Theorem 6. In any other case consider two planes through I. In
each plane the polars of the points of I determine a pencil of lines
projeetive with the range on L Hence the polars must all meet the
line joining the centers of these two pencils of lines, and, being per-
spective with either of these pencils of lines, are projeetive with the
range on /.

DMFINITTON. A line V is polar to a line I if the polar planes of the
points of I meet on V. A line is conjugate to I if it meets V. A point
JP r is conjugate to a point P if it is on the polar of P. A line p is
conjugate to P if it is on the polar of P. A plane TT' is conjugate to
a plane TT if TT' is on the pole of ir. A line p is conjugate to TT if it
is on the pole of TT.

EXERCISES

Polar points and planes with respect to a regulm are denoted "by corresponding
capital Roman and small Greek letters. Conjugate elements of the same kind are
denoted by the same letters wilh primes.

1. If TT is on It, then 7* is on g.

2. If Ms polar to /, then /is polar to /.

3. If one clement (point, line, or plane) is conjugate to a second element,
then the second element is conjugate to the first.

4. If two lines intersect, their two polar lines intersect.

5. A ruler or a directrix of a regulus is polar to itself. A tangent line is
polar to its harmonic conjugate with regard to the ruler and directrix through
its point of contact. Any other line is skew to its polar.

6. The points of two polar lines are conjugate.

7. The pairs of conjugate points (or planes) on any line form an involu-
tion the double points (planes) of which (if existent) are on the regulus.

8. The conjugate lines in a flat pencil of which neither the center nor the
plane is on the regulus form an involution.

9. The line of intersection of two tangent planes is polar to the line
joining the two points of contact.

10. A line of! the regulus which meets one of two polar lines meets the other.

11. Two one- or two-dimensional forms whose bases are not conjugate or
polar are projeetive if conjugate elements correspond.

12. A line / is conjugate to I' if and only if some plane on / is polar to
some point on /'.

304 FAMILIES OF LINES LO.IAP. xi

13. Show that there are two (proper or improper) lines ?, ,s nutting two
given linos and conjugate to them "both. Show also that r is the polar of .

14. Jf , b, c are three generators of a regains and ', //, <' three of the con-
jugate rcgulus, then the three diagonal lines joining the points

(?;<?') and (Zi'e),
(c'a) and (c'),
(&') and (a'fi)

meet in a point tf which is the pole of a plane containing the lines of intersec-
tion of the pairs of tangent planes at the same vertices.

15. The six lines a, &, c, a', ft', c' of Ex. 14 determine the following trios
of simple hexagons

(bc'aVca') , (Aa'aeW) , (W/aa'cc") ,
(&c'aa'cft') , (bb'ac'ca') , (6a W/cc') .

The points S determined by each trio of hexagons are collinear, and the two
lines on which they lie are polar with regard to the quadric surface.*

16. The section of the figure of Ex. 14 by a plane leads to the Pascal
and Brianchon theorems ; and, in like manner, Ex. 15 leads to the theorem
that the 60 Pascal lines corresponding to the GO simple hexagons formed
from 6 points of a conic meet by threes in 20 points which constitute 10
pairs of points conjugate with regard to the conic (cf. Ex. 10, p.

105. Protective conies. Consider two sections of a regains by
planes which are not tangent to it. These two conies are both per-
spective with any axial pencil of a pair of axial pencils which generate
the regulus (cf. 76, Chap. VIII). The correspondence established
between the conies by letting correspond pairs of points which lie on
the same ruler is therefore projective. On the line of intersection, I,
of the two planes, if it is not a tangent line, the two conies determine
the same involution I of conjugate points. Hence, if one of them inter-
sects this line in two points, they have these two points in common.
If one is tangent, they have one common point and one common
tangent. The projectivity between the two conies fully determines a
projectivity between their planes in which the HBO I is transformed
into itself. The involution I belongs to the projectivity thus deter-
mined on I The converse of these statements leads to a theorem
which -is exemplified in the familiar string models:

THEOREM 11. The lines joining corresponding points of 9 two pro-
jective conies in different planes form a reyul'U$ t provided GM two
conies determine the same involution, T, of conjugate points on th.Q

* Cf. Saimia, Lezioni di Geometria Projettiva (Naples, 1805), pp.

10SJ

PKOJECTIVE CONICS

305

lin,e of intersection, I, of the two planes; and provided the collineation
between the two pkmes determined ly the correspondence of the conies
transforms I into itself ly a projectiviky to which I belongs (in par-
ticular, 'if the conies meet in two points which are self -corresponding
in> the.

Proof. Let L be the pole with regard to one conic of the line of
intersection, /, of the two planes (fig. 109). Let A and B be two

points of this conic collinear with L and not on L The conic is gen-
erated by the two pencils A [P] and B [P r ] where P and P f are con-
jugates in the involution I on I (cf. Ex. 1, p. 137). Let A and
B be the points homologous to A and B on the second conic, and let
2 be the point in which the second conic is_ met by the plane con-
taining A, A> and the tangent at A\ and let B be the point in which
the second conic is met by the plane of B, B, and the tangent at B.

The line AB contains the pole of I with regard to the second conic
because this line is projective with --4-??. Since the tangents to the
first conic at A and B meet on Z, the complete quadrangle AABB has
one diagonal point, the intersection of A A and BB, on I ; hence the

306 FAMILIES OF LINES l<!iui.xi

opposite side of the diagonal triangle pauses through the polo of /,
Hence it intersects AB in the pole of /. But the intersection of All
with An is on this diagonal line. Hence All uieetH AB in the pole
of /. Hence the pencils A[P] and B[P f "\ generate the second conic.
Hence, denoting by a and 6 the lines A A and BB, the pencils of planes
a [P] and I [P f ] are protective and generate a regulus of which the
two conies are sections.

The projectivity between the planes of the two conies established
by this regulus transforms the line I into itself by a projectivity to
which the involution I belongs and makes the point A correspond
to A. The projectivity between two conies is fully determined by
these conditions (cf. Theorem 12, Cor. 1, Chap. VIII). Hence the
lines of the regulus constructed above join homologous points in the
given projectivity. Q.E.D.

It should be observed that if the two conies are tangent to I, the
projectivity on I fully determines the projectivity between the two
conies. For if a point P of I corresponds to a point Q of /, the unique
tangent other than I through P to the first conic must correspond to
the tangent to the second conic from Q. If the projectivity between
the two conies is to generate a regulus, the projectivity on I must be
parabolic with the double point at the point of contact of the conies
with L For if another point D is a double point of the projectivity
on I, the plane of the tangents other than I, through D to the two
conies meets each conic in one and only one point, and, as these
points are homologous, contains a straight line of the locus generated.
As this plane contains only one point on either conic, it meets the
locus in only one line, whereas a plane meeting a regulus in one
line meets it also in another distinct line.

Since the parabolic projectivity on I is fully determined by the
double point and one pair of homologous points, the projectivity be-
tween the two conies is fully determined by the correspondent of one
point, not on Z, of the first conic.

To show that a projectivity between the two conies which is para-
bolic on I does generate a regulus, let A be any point of the first
conic and A J its correspondent on the second (fig. 110). Let the
plane of A 1 and the tangent at A meet the second conic in A n .
Denote the common point of the two conies by , and consider the

> 105 J

PEOJECTIVE CONICS

307

FIG. 110

two conies as generated by the flat pencils at A and B and at A"
and 11. The correspondence established between the two flat pencils
at 7> by letting correspond lines joining B to homologous points of
the two conies is perspective because the line I corresponds to itself.
Hence there is a pencil of
planes whose axis, Z>, passes
through 7* and whose planes
contain homologous pairs
of lines of the flat pencils
at 7A The correspondence
established in like manner
between the flat pencil at A
and the flat pencil at ./I" may
bo regarded as the product
of the projectivity between
the two planes, which car-
ries the pencil at A to the
pencil at A', followed by
the projectivity between the
pencils at A' and A n generated by the second conic. Both of these
projectivities determine parabolic projectivities on I with B as inva-
riant point. Hence their product determines on I either a parabolic
projectivity with B as invariant point or the identity. This product
transforms the tangent at A into the line A n A*. As these lines meet
/ in the same point, the projectivity determined on I is the identity.
Hence corresponding lines of the projective pencils at A and A n meet
on Z, and hence they determine a pencil of planes whose axis is a=AA lf .
The axial pencils on a and & are projective and hence generate a
regulus the lines of which, by construction, pass through homologous
points of the two conies. We are therefore able to supplement
Theorem 11 by the following

COROLLARY 1. The lines joining corresponding points of two pro-
jectwe conies in different planes form a regulus, if the two conies
have a common tangent and point of contact and the projectivity
determined between the two planes ly the projectivity of the conies
transforms their common tangent into itself and has the common
point of the two conies as its only fixed point.

808 FAMILIES OB 1 LINES [CHAP. XI

Tlu. generation of a ivguiuH by projective ranges of points on skew
linen may bo regarded UH a degenerate case of this theorem and eor-
olluty, A further degenerate cane is stated in the first exercise,

The proof of Theorem 11 given above is more complicated than it would
lutve been if, under Proposition K 2 , we had made use of the points of intor-
Hwtioii of the lino I with the two conies. But since the discussion of linear
families of linen in the following section employs only proper elements and
depends in part on this theorem, it seems more satisfactory to prove this
theorem as we have done. It is of course evident that any theorem relating
entirely to proper elements of space which is proved with the aid of Proposi-
tion, K n can also be proved Ly an argument employing only proper elements.
The latter form of proof is often much more difficult than the former, but it
often yields more information as to the constructions related to the theorem.

These results may be applied to the problem of passing a quadric
surface through a given set of points in space. Proposition K 2 will be
used in this discussion so as to allow the possibility that the two con-
jugate reguli may be improper though intersecting in proper points.

COROLLARY 2. If three planes a, fi, j meet in three lines a = /3%
6 = yor, e = a/3 and contain three conies A 2 , *, C 2 , of which Jf and C' 2
meet in two points P, P r of a, (? and A 2 meet in two points Q, Q r of I,
and A 2 and B* meet in two points E, It' of e, then there is one and lut
one qiiadric surface * containing the points of the three conic,*.

Proof. let M be any point of <7 2 . The conic # 2 is projected from
M by a cone which meets the plane a in a conic which intersects A*
in two points, proper or improper or coincident, other than H and Ji 1 .
Hence there are two lines m, m f , proper or improper or coincident,
through M which meet both A* and B*. The projectivity determined
between A* and JB* by either of these lines generates a regulus, or,
in a special case, a cone of lines, the lines of which must pass through
all points of <7 2 because they pass through P, P', Q r , Q, and M t all of
which are points of <7 2 .

The conjugate of such a regulus also contains a line through M
which meets both A* and B*. Hence the lines m and m' determine
conjugate reguli if they are distinct. If coincident they evidently de-
termine a cone. The three conies being proper, the quadrio miittt con-
tain proper points even though the lines rn, m f are improper.

* In this corollary and in Theorem 12 the term quadric surface must be taken
to include the points on a cone of lines as a special case.

1031

QUADEJU THROUGH NINE POINTS

309

If six points 1, 2, , 4, 5, 6 are given, no four of which are col-
linear,* there evidently exist two planes, cc and /?, each containing
three of the points and having none on their line of intersection.

FIG. ill

Assign the notation so that 1, 2, 3 are in #. A quadric surface which
contains the six points must meet the two planes in two conies A 2 ,
If which meet the line #/3=c in a common point-pair or point of
contact ; and every point-pair, proper or improper or coincident, of c
determines such a pair of conies.

Let us consider the problem .of determining the polar plane & of
an arbitrary point on the line c. The polar lines of with regard
to a pair of conies JL and B* meet c in the same point and hence
determine o>. If no two of the points 1, 2, 3, 4, 5, 6 are collinear
with 0, any line I in the plane a determines a unique conic A 2 with
regard to which it is polar to 0, and which passes through 1, 2, 3.
A* determines a unique conic 2? a which passes through 4, 5, 6 and
meets c in the same points as -4 2 ; and with regard to this conic

* The construction of a quadric surface through nine points by the method used
in the text is given in Rohn'and Papperitz, Darstellende Geometric, Vol. II
(Leipzig, 1806), 676, 677.

310 FAMILIES OF LINES [CHAP. XI

has a polar line m. Thus there is established a one-to-one corre-
spondence II between the lines of a and the lines of fi. This corre-
spondence is a collineation. ' For consider a pencil of linen [I] in a.
The conies J 2 determined by it form a pencil. Hence the point-pairs
in which they meet c are an involution. Hence the conies 7> a deter-
mined by the point-pairs form a pencil, and hence the lines [w] form
a pencil. Since every line I meets its corresponding line m on c, the
correspondence II is not only a collineation but is a pernpectivity,
of which let the center be (7. Any two corresponding lines I and m
are coplanar with C. Hence the polar planes of wiih regard to
qiiadrics through 1, 2, 3, 4, 5, 6 are the planes on G.

This was on the assumption that no two of the points 1, 2, 3, 4, 5, 6
are collinear with 0. If two are collinear with 0, every polar plane
of must pass through the harmonic conjugate of with regard to
them. This harmonic conjugate may be taken as the point C.

Now if nine points are given, no four being in the same plane, the
notation may be assigned so that the planes a 123, /? = 456, 7 = 780
are such that none of their lines of intersection a = 0% "b ~ yet, c = a/3
contains one of the nine points. Let be the point <xj3y (or a point
on the line aft if a, ft, and 7 are in the same pencil). By the argu-
ment above the polars of with regard to all quadrics through the
six points in a and yS must meet in a point C. The polars of with
regard to all quadrics through the six points in ft and 7 must simi-
larly pass through a point A, and the polars with regard to all quad-
rics through the six points in 7 and a must pass through a point /?.

If A, B, and G are not collinear, the plane <o=xAnC must be the
polar of with regard to any quadric through the nine points. The
plane <a> meets a, ft, and 7 each in a line which must be polar to
with regard to the section of any sucli quadric. But this determines
three conies A* in a, B* in y8, and (7 2 in 7, which meet by pairs in
three point-pairs on the lines a, 5, c. Hence if a, /3, 7 are not in the
same pencil, it follows, by Corollary 2, that there is a unique quadric
through the nine points. If a, /9, 7 have a line in common, the three
conies A 2 , z , C* meet this line in the same point-pair. Consider a
plane <r through which meets the conies A*, I? 2 , C 2 in three point-
pairs. These point-pairs are separated harmonically by and the
trace, $, oh <r of the plane G>. Hence they lie on a conic 7> a , which,
with A 2 and P 2 , determines a unique quadric. The section of this

ion, iixij LINEAH DEPENDENCE 0V LINES 311

quadrie by the plane 7 lias in common with # 2 two point-pairs and
the polar pair 0, a. Hence the quadric has G 1 as its section by 7.

Tu case A, 7?, and G are collinear, there is a pencil of planes CD which
meet them. There is thus determined a family of quadrics which is
called a pwml and is analogous to a pencil of conies. In case A, B,
and G coincide, there is a bundle of possible planes a> and a quadric is
determined for each one. This family of quadrics is called a bundle.
Without inquiring at present under what conditions on the points
1, 2, , 9 these cases can arise, we may state the following theorem :

THEOREM 1 2. Through nine points no four of which are collinear
there JMMMS one quadric surface or a pencil of yuadrics or a bundle
of qit,adric8.

EXERCISES

1. The lines joining homologous points of a projective conic and straight
lino form a rcguliiH, provided the line meets the conic and is not coplanar
with it, and their point of intersection is self -corresponding.

2. State the duals of Theorems 11 and 12.

3. Show that two (proper or improper) 'conjugate reguli pass through two
(ionics in different planes having two points (proper or improper or coincident)
in common and through a point not in the plane of either conic. Two such
conies and a point not in either plane thus detenninc one quadric surface.

4. Show how to construct a regulus passing through six given points
and a given line.

106. Linear dependence of liaes. DEFINITION. If two lines are co-
plauar, the lines of the flat pencil containing them both are said to
be linearly dependent on them. If two lines are skew, the only lines
linearly dependent on them are the two lines themselves. On three
skew lines are linearly dependent the lines of the regulus, of which
they are rulers. If l v l^ - , l n are any number of lines and m v m z ,--, m k
are lines such that ^\ is linearly dependent on two or three of l v Z 2 , - - - , Z n ,
and w 2 is linearly dependent on two or three of l v 1 2 , , l n , m v and
so on, m k being linearly dependent on two or three of l v l v , l n , m lt
m< 2 , - , tn, kmml , then m k is said to be linearly dependent on l v l%> , l^
A set of n lines no one of which is linearly dependent on the n~l
others is said to be linearly independent.

As examples of these definitions there arise the following cases of
linear dependence of lines on three linearly independent lines which
may be regarded as degenerate cases of the regulus. (1) If lines a

312 FAMILIES OF LINES . [OIIAP.XI

and I intersect in a point P, and a line c skew to both of them meets
their plane in a point Q, then in the first place all lines of the pencil
ctb are linearly dependent on a, &, and e ; since the line QP is in this
pencil, all lines of the pencil determined by QP and c are in the sot.
As these pencils have in common only the line QP and do not con-
tain three mutually skew lines, the set contains no other linen.
Hence in this case the lines linearly dependent on a, "b, c are the flat
pencil ab and the flat pencil (c, QP). (2) If one of the lines, as a, meets
both of the others, which, however, are skew to each other, the set of
linearly dependent lines consists of the flat pencils ab and UG. This
is the same as case (1). (3) If every two intersect but not all in the
same point, the three lines are coplanar and all lines of their plane
are linearly dependent on them. (4) If all three intersect in the same
point and are not coplanar, th bundle of lines through their common
point is linearly dependent on them. The case where all three are
concurrent and coplanar does not arise because three such lines are
not independent.

This enumeration of cases may be summarized as follows :

THEOREM 13. DEFINITION. The set of all lines linearly dependent
on three linearly independent lines is either a regidus, or a "bundle, of
lines, or a plane of lines, or two flat pencils having diffc-rcnt centers
and planes hot a common line. The last three sets of lines are called
degenerate reguli.

DEFINITION. The set of all lines linearly dependent on four linearly
independent lines is called a linear congruence. The set of all linen
linearly dependent on five linearly independent lines is called a linear
complex*

107. The linear congruence. Of the four lines a, ?;, 0, d upon
which the lines of the congruence are linearly dependent, ft, v, d
determine, as we have just seen, either a regulus, or two Hat pencils
with different centers and planes but with one common lino, or a
bundle of lines, or a plane of lines. The lines 7>, 0, d can of courso be
replaced by any three which determine the same rcgulus or degen-
erate regulus as &, c, d.

* The terms congruence and complex are general terms to denote two- and three-
parameter families of lines respectively, For example, all lines mooting a curve or
all tangents to a surface form a complex, while all lines meeting two curves or all
common tangents of two surfaces are a congruence.

i<>7] THE LINEAll CONGRUENCE 313

So in case 5, c, d determine a nondegenerate regulus of which a is
not a directrix, the congruence can be regarded as determined by four
mutually skew lines. In case a is a directrix, the lines linearly de-
pendent on a, ?>, c, d clearly include all tangent lines to the regulus
bud, whose points of contact are on a. But as a is in a flat pencil
with any tangent whose point of contact is on a and one of the
rulers, the family of lines dependent on a, 5, c, d is the family de-
pendent on ft, c, d and a tangent line which does not meet &, c, d. Hence
in either case the congruence is determined by four skew lines.

If one of the four skew lines meets the regulus determined by the
other three in two distinct points, JP, Q, the two directrices p, q
through these points meet all four lines. The line not in the regulus
determines with the rulers through P and Q, two flat pencils of lines
which join P to all the points of q, and Q to all the points of p.
From this it is evident that all lines meeting loth p and q are linearly
dtyenflvit on Ike given four. For if 1\ is any point on p, the line
I\Q and the ruler through P l determine a flat pencil joining P to
all the points of q; similarly, for any point of q. No other lines
can be dependent on them, because if three lines of any regulus
meet p and q, so do all the lines.

If one of the four skew lines is tangent to the regulus determined
by the other three in a point P, the family of dependent lines in-
cludes the regulus and all lines of the flat pencil of tangents at P.
Hence it includes the directrix p through P and hence all the tangent
lines whose points of contact are on p. By Theorem 6 this family
of lines can be described as the set of all lines on homologous pairs
in a certain projectivity II between the points and planes of p. Any
two lines in this set, if they intersect, determine a flat pencil of lines
in the set. Any regulus determined by three skew lines I, m, n of
the set determines a projectivity between the points and planes on p,
but this projectivity sets up the same correspondence as II for the
three points and planes determined by I, m, and n. Hence by the
fundamental theorem (Theorem 17, Chap. IV) the projectivity deter-
mined by the regulus Imn is the same as H, and all lines of the
regulus are in the set. Hence, when one of four skew lines is tangent
to the rqjulus of the other three, the family of depende?it lines consists
of a regulus and all lines tangent to it at points of a directrix. The
directrix is itself in the family.

314

FAMILIES OF LINES

F. XI

If no one of the four skew lines meets the regulas of the other
three in a proper point, we have a case studied more fully below.

In case 5, c } d determine two flat pencils with a common line, u.
may meet the center A of one of the pencils. The linearly dependent
lines, therefore, include the bundle whose center is A. The plane of
the other flat pencil passes through A and contains throe noucon-
current lines dependent on a, &, e, d. Hence the family of lines also
includes all lines of this plane. The family of all lines through a
point and all lines in a plane containing this point has evidently
no further lines dependent on it. This is a degenerate case of a con-
gruence. If a is in the plane of one of the flat pencils, we have, by
duality, the case just considered. If a meets the common line of the
two flat pencils in a point distinct from the centers, the two flat
pencils may be regarded as determined by their common line d r and
by lines V and c f , one from each pencil, not meeting a. Hence the
family of lines includes those dependent on the regulus aW and its
directrix d f . This case has already been seen to yield the family of all
lines of the regulus aW and all lines tangent to it at points of d(

Fia. 112

If a does not meet the common line, it meets the plants of tho
two pencils in points and D. Call the centers of the pencils A and
B (fig. 112). The first pencil consists of the lines dependent on AD
and AB, the second of those dependent on An and BC. AH (ID is
the line a, the family of lines is seen to consist of the lines which
are linearly dependent on AB, BC, CD, DA. Since any point of Bl)
is joined by lines of the family to A and C, it is joined by lines of

107 J THE LINEAll CONGRUENCE 315

tho family to every point of AC. Hence this case gives the family
of all lines meeting both AC and BD.

li\ caiso ft, 0, d determine a bundle of lines, a, being independent of
them, does not pass through the center of the bundle. Hence the
family of dependent lines includes all lines of the plane of a and the
center of the bundle as well as the bundle itself.

Lastly, if &, c, d are coplanar, we have, by duality, the same case as
if I, e, d were concurrent. We have thus proved

THEOREM 14. A linear congruence is cither (1) a set of lines
linearly dependent on four linearly independent skew lines, such that
no otic of them meets tlic rcgulus containing the other three in a proper
point; or (2) it is the set of all lines meeting two skew lines; or (3)
it is the set of all rulers and tangent lines of a given regulus which
meet a Ji<ml directrix of the regulus ; or (4) it consists of a bundle
of lines and a plane of lines, the center of the bundle being on the
plane.

DEFINITION. A congruence of the first kind is called elliptic; of the
second kind, hyperbolic ; of the third kind, parabolic ; of the fourth
kind, degenerate. A line which has points in common with all lines
of a congruence is called a directrix of the congruence.

COHOLLAKY. A parabolic congruence consists of all lines on corre-
sponding points and planes in a projectivity between the points and
planes on a line. The directrix is a line of the congruence.

To study the general nondegenerate case, let us denote four linearly
independent and mutually skew lines on which the other lines of the
congruence depend by a, b, c, d, and let TT I and 7r 2 be two planes in-
tersecting in a. Let the points of intersection with ^ and ?r 2 of b, c,
and d be 7^, C v and D l and J? 2 , O v and > z respectively. By letting
the complete quadrilateral a, B&, G^ D^ correspond to the
complete quadrilateral a, J? 2 (7 2 , 2 Z> 2 , Z> 2 # 2 , there is established a
protective collineation II between the planes ir^ and TT Z in which
the lines &, a, d join homologous points (fig. 113).

Among the lines dependent on a, b, c, d are the lines of the reguli
ale, acd, adb, and all reguli containing a and two Knes from any
of these three reguli. But all such reguli meet ^ and TT Z in lines
(e.g. B^D V ^ 2 # 2 ) because they have a in common with ^ and
7T 2 , Furthermore, the lines of the fundamental reguli join points

316

FAMILIES OF LIKES

which correspond in II (Theorem 5 of this chapter and Theorem 18,
Chap. IV), Hence the reguli which contain a and linos nhown by
means of such reguli to be dependent on a, i, c, d arc those gen-
erated by the projectivities determined by II between lines of TT JL
and 7r 2 .

d fi*

rio. 113

Now consider reguli containing triples of the lines already shown
to be in the congruence, but not containing a. Three such linos, /,
m, n, join three noncollinear points L v M lt ^ of ir l to the points
L 2 , M ZJ jV 2 of 7r 3 which correspond to them in the collineation II. The
regulus containing Z, m> and n meets TT^ and TT S in two conies which
are protective in such a way that L v M v N^ correspond to &, Jf 2 , JV 9 .
The projectivity between the conies determines a projectivity between
the planes, and as this projectivity has the same effect an II on the
quadrilateral composed of the sides of the triangle L^M^ and the
line a, it is identical with II. Hence the lines of the regulus I, m, n
join points of 7r t and 7r 2 which are homologous under II and are
therefore among the lines already constructed.

Among the lines linearly dependent on the family thus far con-
structed are also such as appear in flat pencils containing two inter-
secting lines of the family. If one of the two lines is a, the other
must meet a in a double point of the projectivity determined on a by
II. If neither of the two lines is a, they must meet 7r a and 7T 3 , the
first in points JFJ, JEJ and the second in points Q v <? 2 , and these four

107J THE LINEAK CONGRUENCE 317

points are clearly distinct from one another. But as the given lines of
the congruence, JJJJ and Q^, intersect, so must also the lines P^ and
Wa ' rrr \ an( l ^a intersect, and the projectivity determined between
l[Qi and Jj() a by II is a perspectivity. Hence the common point of
./^ and JJ(J a IH a point of # and is transformed into itself by II.
Hence, if lines of the family intersect, II has at least one double point
on a, which means, by 105,* that the line a meets the regulus led
and the congruence has one or two directrices. Thus two lines of a
nondcgenerate congruence intersect only in the parabolic and hyper-
bolic cases ; and from our previous study of these cases we know that
the lines of a congruence through a point of intersection of two lines
form a flat pencil.

We have thus shown that all the lines linearly dependent on
a, I, c, d, with the exception of a flat pencil at each double point of
the projectivity on a, are obtained by joining the points of TT X and 7r 2
which are homologous under II. From this it is evident that any four
linearly independent lines of the congruence could have been taken
as the fundamental lines instead of a, 6, c, d. These two results are
summarized as follows :

THKOKKM 15. All the lines of a linear congruence are linearly
dependent on any linearly independent four of its lines. No lines not
in the congruence are linearly dependent on four such lines.

THEOREM 16. If two planes meet in a line of a linear congruence
and neither contains a directrix, the other lines of the congruence meet
the planes in homologous points of a projectivity. Conversely, if two
planes are protective in such a way that their line of intersection cor-
responds to itself, the lines joining homologous points are in the same
linear congruence.

* If there are two double points, J0, JP, on a, the conic BiCiDiEF must be trans-
formed by H into -the conic BaOtDtEFi and the lines joining corresponding points of
these conies must form a regulus contained in the congruence. As JS? and 3? are
on lines of the regulus 6c<Z, there are two directrices jp, q of this regulus which
meet *$ and JP respectively. The lines p and g meet all four of the lines a, &, c, d.
Hence they meet all lines linearly dependent on a, 6, c, d.

In the parabolic case the regulus bed must be met by a in the single invariant
point IT of the parabolic projectivity on a, because the conic tangent to a at IT and
passing through BtCiDi must be transformed by II into the conic tangent to a at JET
and passing through JfeCiD* ; and the'lines joining homologous points of these conies
must form a regulus contained in the congruence. As JET, a point of a, is on a line
of the regulus &ctf, there is one and only one directrix p of this regulus which meets
all four of a, 6, c, d and hence meets all lines of the congruence.

318 FAMILIES OF LINES [CHAIN XI

The dual of Theorem 16 may be stated in the following form:
THEOREM 17. From two points* on the stmw line, of a, linear congru-
ence the latter is projected Ity two projecMoe "biuutlM of pitmen, (^in-
versely, two bundles of planes projectile in such a wj/ Unit the. line,
joining their centers is self -corresponding, generate a Iw.ar MnyruMiec.

DEFINITION. A regulus all of whose rulers are in, a congruence is
called a regulus of-the congruence and is said to le in or to le con-
tained in the congruence.

COROLLARY. If three lines of a regulus are in a congruence, the
regulus is in the congruence.

In the hyperbolic (or parabolic) case the regulus led (in the notation
already used) is met by a in two points (or one point), its points of
intersection with the directrices (or directrix). In the elliptic case the
regulus bed cannot be met by a in proper points, because if it were,
the projectivity II, between TT X and 7r 2 , would have these points as
double points. Hence no line of the congruence meets a regulus of
the congruence without being itself a generator. Hence through each
point of space, without exception, there is one and only one line of
the congruence. The involution of conjugate, points of the regulus
led on the line a is transformed into itself by II, and 'the same must
be true of any other regulus of the congruence, if it does not con-
tain a. Since there is but one involution transformed into itself by a
noninvolutoric projectivity on a line (Theorem 20, Chap. VIII), we
have that the same involution of conjugate points is determined on
any line of the congruence by all reguli of the congruence whicli do
not contain the given line. This is entirely analogous to the hyper-
bolic case, and can be used to gain a representation in terms of proper
elements of the improper directrices of an elliptic congruence*

The three kinds of congruences may be characterized as follows :

THEOREM 18. In a parabolic linear congruence each line is tangent
at a fixed one of its points to all reguli of the congruence of which it u
not a ruler. On each line of a hyperbolic or elliptic congruence all reguli
of the congruence not containmg the given line determine the same
involution of conjugate points. Through each point of space there is
one and only one line of an elliptic congruence, for hyperbolic and
parabolic congruences this statement i$ true except for points on a
directrix.

i<>7, iosj TJIE LINEAR COMPLEX 319

EXERCISES

1. All lines of a congruence can bo constructed from four lines by means
of reguli all of which have two given lines in counnon.

2. (liven two involutions (both having or both not having double points)
ou two skew line,H. Through each point of apace there are two and only two
lines which are axes of pern] activity projecting one involution into the other,
i.e., Much that two planes through conjugate pairs of the first involution pass
through a conjugate pair of the second involution. These lines constitute

tWO COngrUetKHiH.

3. All lines of a congruence meeting a line not in the congruence form
a regular,

4. A linear congruence i self-polar with regard to any regulus of the
congruence.

5. A degenerate linear congruence consists of all linos meeting two inter-
secting linos.

108* The linear complex. THTCOKEM 19. A linear complex con-
mix of ail linw linearly dependent on the edges of a simple skew

Proof, By definition ( 106) the complex consists of all lines
linearly dependent on five independent lines. Let a be one of these
which does not meet the other four, V, c f , $, e f . The complex consists of
all linos dependent on # and the congruence Vctdld. If this con-
gruence is degenerate, it consists of all lines dependent on three sides
o a triangle ode and a line I not in the plane of the triangle
(Theorems 14, 15). As 6 may be any line of a bundle, it may be
chosen so as to meet a c may be chosen so as to meet I, and e may
be so chosen as to meet a. Thus in this case the complex depends
on five lines a, J, c, d, e not all coplanar, forming the edges of a simple
pentagon.

If the congruence is not degenerate, the four lines 6", c rr , d", e" upon
which it depends may (Theorem 15) be chosen so that no two of
them intersect, but so that two and only two of them, W and e n ,
meet a. Thus the complex consists of all lines linearly dependent
on the two flat pencils aW and ae" and the two lines c" and d". Let
& and e be the lines of these pencils (necessarily distinct from each
other and from a) which meet c n and d" respectively. The complex
then consists of all lines dependent on the flat pencils e&&, "be", ae, ed".

* The edges of a simple skew pentagon are five lines in a given order, not all
coplanar, each line intersecting its predecessor and the last meeting the first.

20 FAMILIES OF LINKS [(iiur. xi

.Finally, let c and d be two intersecting linos distinct from // ami f,
which are in the pencils bo n and #F'. The complex consists of all lines
linearly dependent on the flat pencils (tf> 9 be-, cd, <l< l t w. Not all tlio
vertices of the pentagon dbcdc can be ooplanar, because then all the
lines would be in the same degenerate congruence.

THEOREM 20. DEFINITION. There are two dawn of wm>j*lc t m #m'h
that all contylettes of cither clans are projecti'veli/ MjKiwtleuf. A c<nn,
plc& of one class consists of a line and all linw of xptw, which, meet
it. These are called special complexes. A complex of the, ofhe.r cJtttttt
is catted general. No four vertices of a pentagon which determines it
are coplanar.

Proof. Given any complex, by the last theorem there is at least
one skew pentagon abode which determines it. If there is a line /
meeting the five edges of this pentagon, this line must meet all lines
of the complex, because any line meeting three linearly independent
lines of a regains (degenerate or not) meets all lines of it. Moreover,
if the line I meets a and I as well as c and rf, it must either join
their two points of intersection or be the line of intersection of their
common planes. If I meets e also, it follows in either case thai four
of the vertices of the pentagon are coplanar, two of them being on c.
(That all five cannot be coplanar was explained at the end of the
last proof.) Conversely, if four of the five vertices of the skew
pentagon are coplanar, two and only two of its edges are not in this
plane, and the line of intersection of the plane of the two edges with
the plane of the other three meets all five edges.

Hence, if and only if four of the five vertices are wplwmr, there. r,r-
ists a line meeting the Jive linen. Since any two skew pentagons are pro-
jectively equivalent, if no four vertices are coplanar (Theorem 1U,
Chap. Ill), any two complexes determined by such pentagons are
protectively equivalent. Two simple pentagons are also equivalent
if four vertices, but not five, of each are coplanar, because any simple
planar four-point can be transformed by a collineation of space into
any other, and then there exists a collineation holding the plane
of the second four-point pointwise invariant and transforming any
point not on the plane into any other point not on the plane. There-
fore all complexes determined by pentagons of this kind are pro jec-
tively equivalent. But these are the only two kinds of skew pentagons.
Hence there are two and only two kinds of complexes.

THE LINE All COMPLEX

321

hi ease four vertices of the pentagon are coplanar, we have seen
that there is a line / meeting all its edges. Since this line was
detiH'mmed as the intersection of the plane of two adjacent edges
with the plane of the other three, it contains at least two vertices.
It ('.winot contain three vertices because then all five would be
<*.o]>luiwr. AH one of the two planes meeting on I contains three
independent lines, all lines of that plane are lines of the complex.
Tim line / ilnel is therefore in the complex as well as the two lines
of the other plane. Hence all lines of both planes are in the complex.
II once all lines meeting / are in the complex. But as any regulus
three of whose lines meet / has all its lines meeting Z, the complex
IioM the requirements stated in the theorem for a special complex.

Fi<*. 114

A more definite idea of the general complex may be formed as
follows. Let piPiPdW* (% 11 ' i ) ke a simple pentagon upon whose
edges all lines of the complex are linearly dependent. Let q be the
line of the flat pencil jp 8 jp 4 which meets p v and let ft be the point of
intersection of q and p r Denote the vertices of the pentagon by P 12 ,
& ^H> / ?&i> ^ ie subscripts indicating the edges which meet in a
given vertex.

The four independent lines p^^y determine a congruence of lines
all of which are in the complex and whose directrices are # = jftj^ 8
and a f a* 7f 2 / 8 V Iu like manner, gp^p 6 p^ determine a congruence whose
directrices are l RP^ and V^=P M P 6l . The complex consists of all
lines linearly dependent on the lines of these two congruences. The

322

FAMILIES OF LINES

[('HAP. XI

directrices of the two congruences intersect at R and /,5 t respectively
and determine two planes, al = p and aW^vr, which meet OH <y.

Through any point P of space not on /> or TT then 1 , are two linos
1 9 m, the first meeting a and a r , and the second meeting h and //
(fig. 115). All lines in the ilat pencil Ini are in the complex by deii-
nition. This fiat pencil meets p and TT in two perspective ranges of

FIG. 115

points and thus determines a projectivity between the flat pencil a<b
and the flat pencil a'V 9 in which a and a f , & and V correspond and q
corresponds to itself. The projectivity thus determined between the
pencils ab and aV is the same for all points P> because a< 9 ft, q always
correspond to a r , V, q f . Hence the complex contain,* all lines in the
flat pencils of lines which meet homologous lines in the proJMtiwity

determined ly

alq -ft a'Vq.

Denote this set of lines ly S. We have seen that it has the properly
that all its lines through a point not on p or TT are coplanar. If a
point P is on p but not on q, the line Pit has a corresponding linojp/
in the pencil a'V and hence S contains all lines joining /* to points
of p f . Similarly, for points on TT bxit not on q. By duality every plane
not on q contains a flat pencil of lines of S.

Each of the flat pencils not on q has one line meeting q. Hence
each plane of space not on q contains one and only one line of S
meeting j. Applying this to the planes through P H not contain-
ing 2, we have that any line through P^ and not on p is not in the

i 108 J THE LINEAR COMPLEX 323

sot S. Let I be any such line. All lines of S in each plane through
/ form a flat pencil P, and the centers of all these pencils lie on a line
/', because all lines through two points of I form two flat pencils each
of which contains a line from each pencil P. Hence the lines of S
meeting I form a congruence whose other directrix V evidently lies on
p. The point of intersection of V with q is the center of a flat pencil
of lines of S all meeting L Hence all lines of the plane lq form a flat
pencil. Since I is any line on P 34 and not on TT, this establishes that
uaeh plane and, by duality, each point on q, as well as not on q, con-
tains a Hat pencil of lines of S.

We can now prove that S contains no lines not in the complex.
To do 80 we have to show that all lines linearly dependent on lines
of S are in S. If two lines of S intersect, the flat pencil they deter-
mine is by definition in S. If three lines m v m 29 m B of S are skew to
one another, not more than two of the directrices of the regulus con-
taining them are in S. For if three directrices were in S, all the tan-
gent line** at points of these three lines would be in S, and hence any
piano would contain three nonconcurrent lines of S. Let I be a
directrix of the regulus m^m^m^ which is not in S. By the argu-
ment made in the last paragraph all lines of S meeting I form a con-
gruence. But this congruence contains all lines of the regulus mjn^n^
and hence all lines of this regulus are in S. Hence the set of lines ,S
is identical with the complex.

TIIKOREM 21 (SYLVESTER'S THEOREM*). If two protective flat pencils
with different centers and planes have a line q in common which is
Mlfaorretipontliny, all lines meeting homologous pairs of lines in these
two pencils are in the same linear complex. This complex consists of
these lines together with a parabolic congruence whose directrix is q.

Proof. This has all been proved in the paragraphs above, with the
exception of the statement that q and the lines meeting q form a
linear congruence. Take three skew lines of the complex meeting q ;
they determine with q a congruence C all of whose lines are in the
complex. There cannot be any other lines of the complex meeting q,
because there would be dependent on such lines and on the congru-
ence C all lines meeting q, and hence all lines meeting q would be in
the given complex, contrary to what has been proved above.

* Cf. Coinptes Kendus, Vol. LH (1861), p. 741.

324 FAMILIES OF LINKS [CUAI>.XI

Another theorem proved in the discussion above is:

THEOREM 22. DEFINITION OF NULL MYWTKM. All the lhw& of a
linear complex which pans through a point, /* lie, in <i plans. TT, and all
the lines which lie in a plane TT jprwwf tlirowgh a 'point /*. In MM, of
a special complex, exception must be made of the points and pht.nw on
the directrix. The point P is called the null point of the plane TT and
TT is called the null plane of P with regard to the wm-pliw. The cor-
res^^ondence between the points and planes of space thus established is
called a null system or null polarity.

Another direct consequence, remembering that there are only two
kinds of complexes, is the following :

THEOREM 23. Any five linearly independent lines are in one and
only one complex. If the edges of a simple pentagon are in a given
complex, the pentagon is skew and its edges linearly independent. If
the complex is general, no four vertices of a simple pentagon of its
lines are eoplanar.

THEOREM 24. Any set of lines, K, in space such that the lines of the
set on each point of space constitute a flat pencil is a linear complex.

Proof, (a] If two lines of the set K intersect, the set contains (ill
lines linearly dependent on them, by definition.

(6) Consider any line a not in the given set K. Two points A, B on
a have flat pencils of lines of K on different planes ; for if the planes
coincided, every line of the plane would, by (a), be a line of K. Hence
the lines of K through A and B all meet a line a 1 skew to a. Prom
this it follows that all the lines of the congruence whoso directrices
are a, a' are ii^ K. Similarly, if & is any other line not in K but meet-
ing a, all lines of K which meet & also meet another line ?/. More-
over, since any line meeting a, &,and V is in K and hence also meets
a 1 , the four lines a, a 1 , &, V lie on a degenerate regains consisting of the
flat pencils aa r and W (Theorem 13). Let q (fig. 115) bo the common
line of the pencils al and a!V. Through any point o sjwce not on one
of the planes ab and a'U there are three eoplanar lines of K which
meet q and the pairs aa r and W. Hence K consists of lines mooting
homologous lines in the projectivity

and therefore is a complex by Theorem 21.

THE LINEAR COMPLEX 325

OOROLLAKY. Any (1,1) correspondence between the points and the
planes of space such that each point lies on its corresponding plane
in a null system.

THEOREM 25. Two linear complexes have in common a linear
congruence.

Proof, At any point of space the two flat pencils belonging to the
two complexes have a line in common. Obviously, then, there are
three linearly independent lines l v L 2 , Z 3 common to the complexes.
All lines in the regulus / i y a are, by definition, in each complex. But
as there are points or planes of space not on the regulus, there is a
line 1 4 common to the two complexes and not belonging to this regulus.
All lines linearly dependent on l v Z 2 , Z 3 , 1 4 are, by definition, common
to the complexes and form a congruence. No further line could be
common or, by Theorem 23, the two complexes would be identical

COROLLARY 1. The lines of a complex meeting a line I not in the
complex form a hyperbolic congruence.

Proof. The line is the directrix of a special complex which, by the
theorem, has a congruence in common with the given complex. The
common congruence cannot be parabolic because the lines of the first
complex in a plane on I form a flat pencil whose center is not on I,
since I is not in the complex.

COROLLARY 2. The lines of a complex meeting a line I of the com-
plex form a parabolic congruence.

Proof. The centers of all pencils of lines in this congruence must
be on I because I is itself a line of each pencil.

DEFINITION. A line I is a polar to a line V with regard to a
complex or null system, if all lines of the complex meeting I also
meet V.

COROLLARY 3. If I is polar to I', V is polar to I A line is polar
to itself, if and only if it is a line of the complex.

THEOREM 26. A null system is a protective correspondence "between
the points and planes of space.

Proof. The points on a line I correspond to the planes on a line V
by Corollaries 1 and 2 of the last theorem. If I and V are distinct,
the correspondence between the points of I and planes of V is a per-
spectivity. If l*=V, the correspondence is projective by the corollary
of Theorem 14.

320 FAMILIES OF LINES l<'n.u>. XI

EXERCISES

1. If a point P is on a plane p, the null piano TT of /'is on the null point It of p.

2. Two pairs of lines polar with regard to the same, null system are uhvuys in
the same regulus (degenerate, if a line of one, pair meets a lino of the other pair).

3. If a lino / meets a line m, the polar of I moots the polar of nt.

4. Pairs of lines of the, regains in Ex. "2 which are, polar with regard to
the complex are met by any directrix of the regains in pairs of points of an
involution. Tims the complex determines an involution among the linos of
the regnlus.

5. Conversely (Theorem of Chasie,s), the lines meeting conjugate pairs of
lines in an involution on a regulus are in the same complex. Show that
Theorem 21 is a special case of this.

6. Find the lines common to a linear complex and a regulus not in the
complex.

7. Three skew lines , I, m determine one and only one complex contain-
ing k and having I and m as polars of each other.

8. If the number of points on a line is + 1, how many reguli, how many
congraences, how many complexes are there in space? How many lines are
there in each kind of regulus, congruence, complex ?

9. Given any general complex and any tetrahedron whose faces are not
null planes to its vertices. The null planes of the vertices constitute a second
tetrahedron whose vertices lie on the planes of the first tetrahedron. The
two tetrahedra are mutually inscribed and circumscribed each to the, other*
(cf. Ex. G, p. 105).

10. A null system is fully determined by associating with the tlirco vertices
of a triangle three planes through these vertices and having their ono common
point in the plane of the triangle but not on one of its sides.

11. A tetrahedron is self-polar with regard to a null system if two opposite
edges are polar.

12. Every line of the complex determined by a pair of Mobius tolrahodra
meets their faces and projects their vertices in projectivo throws of points and
planes.

13. If a tetrahedron T is inscribed and circumscribed to 7\ and also to 7^,
the lines joining corresponding vertices of !Z\ and T, 2 and the linos of intersec-
tion of their corresponding planes are all in the same complex.

14. A null system is determined by the condition that two pairs of lines
of a regulus shall be polar.

15. A linear complex is self-polar with regard to a regnlus all of whoso
lines are in the complex.

16. The lines from which two projective pencils of points on skew linos
are projected by involutions of planes are all in the same complex* Dunllxo.

* This configuration was discovered by Mobius, Journal fur Mathomatik, Vol. HI
(1828), p. 273. Two tetrahedra in this relation are known as Mobius tctrahwlra,

iw] LINE COORDINATES 327

109. The Pliicker line coordinates. Two points whose coordinates are

determine a line L The coordinates of the two points determine six
numbers

which are known as the Pluckw coordinates of the line. Since the
coordinates of the two points are homogeneous, the ratios only of the
numbers p lf are determined. Any other two points of the line deter-
mine the same set of line coordinates, since the ratios of the jp y 's are
evidently unchanged if (& v x, # 3 , # 4 ) is replaced by (x l + \y v o? 2 -
The six numbers satisfy the equation*

Thin is evident on expanding in terms of two-rowed minors the
identity

x l

2/4

= o.

(Jbnversely, if any six numbers, jp , are given which satisfy Equa-
tion (1), then two points P^fa, %%> 8 , 0), Q~(y v 0, y 8 , y 4 ) can be
determined such that the numbers p tf are the coordinates of the line
PQ. To do this it is simply necessary to solve the equations

which are easily seen to be consistent if and only if

Hence we have

THEOREM 27. Every line of space determines and is determined
ly the ratios of six niwibers jp u , jp 18 , jp 14> p$v p&, p 2S subject to the

* Notice that in Equation (1) the number of inversions in the four subscripts of
any term is always even.

328

condition

FAMILIES OF LINES [<',HAI>, XI

! 0, Mtch thttt. if (,','p ,-r5 a , .", , 4 ) <r/f

it; 8 ^'4

^ y/4

i'w

^=~"

J'u-

'*'4

?/,!

OOROLLAEY. Four independent coordinates determine, a line.
In precisely similar manner two planes (u, v n^ w s , v^) and (v\, 7? a , v rt , v 4
determine six numbers such that

2 12 ==

The quantities 2 V . satisfy a theorem dual to the one just proved for
the jp v 's.

THEOREM 28. I7^e jp and % coordinates of a line are connected 1>y
the equations jp 12 : _p ls : jp u : p 34 : ^ 42 : ^ 23 = J M : <7, 2 : JJT W : y la : </ u : y w .

Proof. Let the j? coordinates he determined l)y the two points
(x v XM x 3 , x^j (y v y z , y^ y 4 ), and the q coordinates by the two planes
(v, v u# u a , u 4 ), (v v v> 2> -z; 8 , i? 4 ). These coordinates satisfy the four equations

u^ + Ufa + Ufa + Ufa = 0,
V& + vfa + vfa + v& = 0,

^2/1 + ^2/2 + ^ 8 + ^4 = 0,
^1^1 + ^8 + ^#8 + '^4^4 = '

Multiplying the first equation by # x and the second by W L and adding,
we obtain

In like manner, from the third and fourth equations wo obtain

Combining the last two equations similarly, we obtain

or, * = *.

<?14 Pi*

By similar combinations of the first four equations we find

i* n u

: Pu :

: Si* :

io, noj

LINE COORDINATES

329

EXERCISE

Given tlie tetrahedron of reference, the point (1, 1, 1, 1), and a line I,
determine six sets of four points each, whose cross ratios are the coordinates
of/.

110. Linear families of lines. THEOREM 29. The necessary and
tni,f/i('.ieti,t condition that two lines p and p f intersect, and hence are
coplanar, is

where p (j are the coordinates of p and p!j of p 1 .

Proof. If the first line contains two points x and y, and the second
two points x 1 and y r , the lines will intersect if and only if these four
points are coplanar ; that is to say, if and only if

ffi ft ?/

./ ,/ ,/

'''i '^a '*'8

';/' <?/' <?/'

yi 2^2 ya

uP* + J'sJ'u

TIIEOHKM 30. Aflat pencil of lines consists of the lines whose coordi-
nates are Xp,. ; +/*p^> if p und p f are two lines of the pencil.

Proof, The lines p and p f intersect in a point A and are perspec-
tive with a range of points <7+XZ>. Hence their coordinates may be
written

which may be expanded in the form

THEOREM 31. The lines whose coordinates satisfy one linear
eqitation

form a linear complex. Those whose coordinates satisfy two independ-
ent linear equations form a linear congruence, and those satisfying
three independent linear equations form a regulus. Four independent
linear equations are satisfied ly two (distinct or coincident] Iines 9
which may 'be improper.

330

FAMILIES OF LINES

[CHAP. XI

Proof. If (\, Z^, J 8 , 6 4 ) is any point of space, the points (M V w a , ,f,,, ,f 4 )
which lie on lines through 1 19 J a , 6 8 , 6 4 satisfying (1) must satisfy

KX.X^^X,^

( '4 '"'2

= 0,

(2)

^ A) t/; a

which is the equation of a plane. Hence the family of lines repre-
sented by (1) has a flat pencil of lines at every point of space, and so,
by Theorem 24, is a linear complex.

Since two complexes have a congruence of common lines, two linear
equations determine a congruence. Since a congruence and a complex
have a regulus in common, three linear equations determine a regulus.

If the four equations

are independent, one of the four-rowed determinants of their cooili-
cients is different from zero, and the equations have solutions of the

If one of these solutions is to represent the coordinates of' a line, it
must satisfy the condition

which gives a quadratic equation to determine X//A. Hence, by Propo-
sition K 2 , there are two (proper, improper, or coincident) lines whose
coordinates satisfy four linear equations.

COKOLLARY 1. TJie lines of a regulus are of the form

where p r , p fr , p nr are lines of the reyulus. In like manner, (lie Un&x of
a congruence are of the form

* Cf . Bocher, Introduction to Higher Algebra, Chap. IV.

>, m] LINE COORDINATES 331

of tt complex of the form

All of these formulas must le taken in connection with

CouoLLAiiY 2. As a transformation from points to planes the null
determined "by the complex whose equation is

/ -y xv m I C\

The first of these corollaries simply states the form of the solu-
tions of systems of homogeneous linear equations in six variables.
The second corollary is obtained by inspection of Equation (2) the
coefficients of which are the coordinates of the null plane of the
point (l v & 2 , 5 8 , & 4 ).

Corollary 1 shows that the geometric definition of linear dependence of
lim*H givon in thin chapter corresponds to the conventional analytic concep-
tion of linear dependence.

111. Interpretation of line coordinates as point coordinates in S 5 .

It may be shown without difficulty that the method of introducing
homogeneous coordinates in Chap. VII is extensible to space of any
number of dimensions (cf. Chap. I, 12). 'Therefore the set of all sets
of six numbers

can be regarded as homogeneous point coordinates in a space of five
dimensions, S 6 . Since the coordinates of a line in S s satisfy the
quadratic condition

they may be regarded as forming the points of a quadratic locus or
spread,* L 4 a , in S 6 . The lines of a linear complex correspond to the
points of intersection with this spread of an S 4 that is determined by
one linear equation. The lines of a congruence correspond, therefore,
to the intersection with L 4 2 of an S 3 , the lines of a regulus to the

* This is a generalization of a conic section.

332 FAMILIES OF LINES [<hui>. xi

intersection with Lj 2 of au S 2 , and any pair of linos to the intersee-
tion with L* of an S r

Any point (^ p(. l9 p{, |4, }>&,}>&) of S c has as its polar*' S 4 , with
regard to L 4 2 ,

(2) l*Ll\i + pJaPu + 2>*2>u + P/a^M + 1>l* Vi* + u ?' = (] >

.which is the equation of a linear complex in the original S. r
any point in S 5 cttn be thought of tin re.preMnting the, winylcs, of I
represented Inj the points of S 5 in whitfi its poltir S 4 meets L.j*.

Since a line is represented by a point on L 4 a , a special complex is
represented by a point on L^ 2 , and all the lines of the special complex
by the points in which a tangent S 4 meets L*.

The points of a line, # + X&, in S 5 represent a set of complexes
whose equations are

(3) (

and all these complexes have in common the congruence common to
the complexes a and &. Their congruence, of course, consists of the
lines of the original S 3 represented by the points in which L* is met
by the polar S 3 of the line a + X&.

A system of complexes, a + \l, is called a pencil of ctMi'pliWx, and
their common congruence is called its Iww or laml MMfrwnw. It
evidently has the property that the null planes of any point with
regard to the complexes of the pencil form an axial pencil whose
axis is a line of the basal congruence. Dually, the null points of
any plane with regard to the complexes of the pencil form a rnngo
of points on a line of the basal congruence.

The cross ratio of four complexes of a pencil may be defined as
the cross ratio of their representative points in S 6 . From the. form of
Equation (3) this is evidently the cross ratio of the four null planes
of any point with regard to the four complexes.

A pencil of complexes evidently contains the special complexes
whose directrices are the directrices of the basal congruence. Ilenco

* Equation (2) may be taken as the definition of a polar S 4 of a point with
regard to L|. Two points are conjugate with regard to L* if the polar S 4 of one
contains the other. The polar S 4 's of the points of an S t - (i =s 1, 2, &, 4) all have an
S 4 _, in common which is called the polar S 4 ^ t - of the S;. These and other obvioiw
generalizations of the polar theory of a conic or a regains we take for granted
without further proof.

1J J LIKE COOEDINATES 333

there are two improper, two proper, one, or a flat pencil of lines which
are the directrices of special complexes of the pencil. These cases
arise as the representative line a + \l meets L 4 2 in two improper
points, two proper points, or one point, or lies wholly on L 4 2 . Two
points in which a representative line meets L 2 are the double points
of an involution the pairs of which are conjugate with regard to L 2 .

Two complexes p, p r whose representative points are conjugate
with regard to L.* are said to be conjugate or in involution. They
evidently satisfy Equation (2) and have the property that the null
points of any plane with regard to them are harmonically conjugate
with regard to the directrices of their common congruence. Any
complex a is in involution with all the special complexes whose
directrices are lines of a.

Let ff t be an arbitrary complex and & 2 any complex conjugate to
(in involution with) it. Then any representative point in the polar S 8
with regard to L* of the representative line a^a 2 represents a complex
conjugate to a l and a y Let cr, B be any such complex. The represent-
ative points of a v a a , a s form a self-conjugate triangle of L 4 2 . Any
point of the representative plane polar to the plane a^a^ with
regard to L 4 2 is conjugate to a^a^ Let such a point be 4 . In like
manner, a. $ and & can be determined, forming a self-polar 6-point of
LJ*, the generalization of a self-polar triangle of a conic section. The
Hix points are the representatives of six complexes, each pair of which
in in involution.

It can be proved that by a proper choice of the six points of refer-
ence in the representative S e , the equation of L 4 2 may be taken as any
quadratic relation among six variables. Hence the lines of a three-
space may be represented analytically by six homogeneous coordinates
subject to any quadratic relation. In particular they may be repre-
sented by (#i, a? 9 ,- , # 6 ), where

= 0.*

In this case, the six-point of reference being self-polar with regard
to L*, its vertices represent complexes which are two by two in
involution.

* Those are known as Klein's coordinates. Most of the ideas in the present sec-
tion are to be found in F. Klein, Zur Theorie der Liniencomplexe des ersten
zwoiten Grades, Mathematische Annalen, Vol. II (1870), p. 198.

334 FAMILIES OF LINKS Loiur. xi

EXERCISES

1. If a pencil of complcxe-H contains two special complexes, the. basal con-
gruence of the pencil is hyperbolic or elliptic, according as the special com-
plexes are proper or improper.

2. If a pencil of linear complexes contains only a single special complex,
the basal congruence is parabolic.

3. If all the complexes of a pencil of linear complexes are special, tlw
basal congruence is degenerate,

4j. Define a pencil of complexes as the system of all complexes having a
common congruence of lines and derive its properties synthetically.

5. The polars of a line with regard to the complexes of a pencil form
a regulus.

6. The null points of two planes with regard to the complexes of a pencil
generate two projective pencils of points.

7. If C = 0, C'= 0, C"' = are the equations of three linear complexes
which do not have a congruence in common, the equation C + AC" + i**C" =
is said to represent a "bundle of complexes. The lines common to the three
fundamental complexes C, C", C" of the bundle form a regulus, the con-
jugate regulus of which consists of all the directrices of the special com-
plexes of the bundle.

8. Two linear complexes Sa, 7j fr/ = and Sfyjft/ = are in invohition if and
only if we have

+ G * As

9. Using Klein's coordinates, any two complexes are given by
and SMi 0. These two are in involution if 2a^ = 0.

10. The six fundamental complexes of a system of Klein's coordinates
intersect in pairs in fifteen linear congruences all of whose directrices are dis-
tinct. The directrices of one of these congruences are lines of tho remaining
four fundamental complexes, and meet, therefore, the twelve directrices of
the six congruences determined by these four complexes.

INDEX

The numbers refer to pages

Abolian group, 67

Absoksa, 170

Abstract science-, 2

Addition, of points, 142, 231 ; theorems
on, 142-144 ; other definitions of, 1C7,
Em 3, 4

Adjacent sides or vertices of simple
ft-lino, 87

Algebraic curve, 259

Algebraic problem, 288

Algebraic surface, 259

Alignment, assumptions of, 10 ; consist-
ency of assumptions of, 17; theorems
of, for the, plane, 17-20 ; theorems of,
for #-space, 20-24 ; theorems of, for
4-space, 25, Ex. 4; theorems of, for
n-spaco, 29-33

Amodeo, E,, 120, 294

Anharmonic ratio, 159

Apollonius, 286

Associative law, for correspondences,
06 ; for addition of points, 143 ; for
null tit >li cation of points, 146

Assumption, H , 45; H , rOlo of, 81,
261; of projectivity, 95; of projec-
tivity, alternative forms of, 105, 106,
Exs. 10-12 ; 298

Assumptions, are necessary, 2; exam-
ples of, for a mathematical science,
2; consistency of, 3; independence
of, 6 ; categoricalnoss of, 6 j of align-
ment, 16; of alignment, consistency
of, 17; of extension, 18, 24; of clo-
sure, 24 ; for an n-space, 33

Axial pencil, 55

Axial perspectivity, 57

Axis, of perspectivity, 36; of pencil,
55 ; of perspective collineation, 72 ; of
homology, 104; of coSrdinates, 169,
191; of projectivity on conic, 218

Bawo, of piano of points or lines, 55 ; of
pencil of complexes, 332

Bilinear equation, "binary, represents
projectivity on a line, 156 ; ternary,
represents correlation in a plane, 267

Binary form, 251, 252, 254

B&cuor, M., 156, 272, 289, 330

Braikenriclge, 119

Brian chon point, 111

Brianchon's theorem, 111

Bundle, of planes or lines, 27, 55; of
conies, 297, Exs. 9-12; of quadrics,
311 ; of complexes, 334, Ex. 7

Burnside, W., 150

Bussey, W. H,, 202

Canonical forms, of collineations in
plane, 274-276; of correlations in a
plane, 281 ; of pencils of conies, 287-
293

Castelnuovo, G., 139, 140, 237, 297

Categorical set of assumptions, 6

Cayley, A., 52, 140

Center, of perspectivity, 36 ; of flat pen-
cil, 55 ; of bundle, 55 ; of perspective
collineation in plane, 72 ; of perspec-
tive collineation in space, 75; of
homolpgy, 104; of coordinates, 170;
of projectivity on conic, 218

Central perspectivity, 57

Characteristic constant of parabolic
projectivity, 207

Characteristic equation of matrix, 165

Characteristic throw and cross ratio, of
one-dimensional projectivity, 205, 211,
Exs. 2, 3, 4 ; 212, Exs. 5, 7 ; of involu-
tion, 206; of parabolic projectivity,,
206

Chasles, 125

Class, notion of, 2 ; elements of, 2 ; re-
lation of belonging to a, 2 ; subclass of
a, 2 ; undefined, 15 ; notation for, 57

Clebsch, A., 289

Cogredient n-line, 84, Ex. 13

Cogredient triangle, 84, Exs. 7, 10

Collineation, defined, 71 ; perspective, in
plane, 72 ; perspective, in space, 75 ;
transforming a quadrangle into a
quadrangle, 74 ; transforming a five-
point into a five-point, 77 ; transform-
ing a conic into a conic, 182 ; in plane,
analytic form of, 189, 190, 268 ; be-
tween two planes, analytic form of,
190 ; in space, analytic form of, 200 ;
leaving conic invariant, 214, 220, 235,
Ex. 2; is the product of two polar-
ities, 265; which is the product of
two reflections, 282, Ex. 5 ; double ele-
ments of, in plane, 271 ; character-
istic equation of, 272 ; invariant figure
of, is self-dual, 272

335

336

INDEX

Collineationw, types of, in plants 100,
278; associated with two conies oC
a pencil, i;U, EXH. 2, 4, 0; 135,
Kx. 2 ; 130, Ex. 2 ; group of, in piano,
208; represented by matriees, 208-
270; two, not in general commuta-
tive, 208 ; canonical forms of, 274-
276

Commutative C'orrospondoncc, 00

Commutative group, 07, 70, Kx. 1; 228

Commutative law of multiplication,
148

Commutative pro jectivi ties, 70, 210, 228

Compass, constructions with, 240

Complete Wrline, in plane, 37; on point, 38

Complete w-plane, in space, 37 ; on point,
38

Complete n-point, in space, 30 ; in piano,
37

Complete quadrangle and quadrilat-
eral, 44

Complex, linear, 312; determined by
skew pentagon, 319 ; general and spe-
cial, 320 ; determined by two projec-
tive flat pencils, 323 ; determined by
live independent lines, 324; deter-
mined by correspondence between
points and planes of space, 324 ; null
system of, 324 ; generated by involu-
tion on regains, 320, Ex. 6 ; equation
of, 329, 331

Complexes, pencil of, 332; in involu-
tion, 333 ; bundle of, 334, Ex. 7

Concrete representation or application
of an abstract science, 2

Concurrent, 10

Cone, 118 ; of lines, 109 ; of planes, 109 ;
section of, by plane, is conic, 109;
as degenerate case of quadric, 308

Configuration, 38; symbol of, 38; of
Desargues, 40, 51 ; quadrangle-quad-
rilateral, 44; of Pappus, 98, 249; of
Mobius, 320, Ex. 9

Congruence, linear, 312 ; elliptic, hyper-
bolic, parabolic, degenerate, 316; de-
termined by four independent lines,
317 ; determined by projective planes,
317; determined by two ^complexes,
325 ; equation of, 329, 330

Conic, 109, 118 , theorems on, 109-140 ;
polar system of, 120-124; equation
of, 185, 245; projectiyity on, 217;
intersection of lino with, 240, 242,
240 ; through four points and tangent
to line, 250, Ex. 8; through three
points and tangent to two linos, 250,
Ex. 9 ; through four points and moot-
ing given line in two points harmonic
with two given points, 250, Ex. 10;
determined by conjugate points, 293,
Ex. 2 ; 294, Exs. 3, 4

Conic section, 118

(Ionics, pencils and ranges of, 128-130,
287-293; projeetive, 212, ,*K)4

Conjugate groups, 209

Conjugate pair of involution, 102

Conjugate points (lines), with regard to
conic, 122; on line (point), form invo-
lution, 124; wifch regard to a pencil of
monies, U)0, Bx. ;J ; 140, Kx, JU ; 293,
Kx. 1

Conjugate projeotivitios, 208; condi-
tions for, 208, 209

Conjugate subgroups, 211

Consistency, of a sot of assumptions, # ;
of notion of elements at infinity, 9;
of assumptions of alignment, 17

Construct, 45

Constructions, linear (first degree), 230 ;
of second degree, 245, 249-250,
Exs.; of third and fourth degrees,
294-290

Contact, point of, of lino of lino conic,
112; of second order between two
conies, 134; of third order between
two conies, 130

Conwell, G. M., 204

Coordinators, non homogeneous, of points
online, 152; homogeneous, of points
on lino, 103; nonhomogeneous, of
points in plane, 109; iionhomogeno-.
OIIK, of linos in plane, 170; homogene-
ous, of point;* and linos in piano, 174;
in a bundle, 179, Ex. 3; of quadran-
gle-quadrilateral configuration, 181,
Ex. 2; nonhomogoneous, in space,
190; homogeneous, in space, 394;
Plttokor'H lino, 327 ; Klein's lino, MM

Coplanar, 24

Copunctal, 10

Correlation, between two-dimensional
forms, 202, 203; induced, 202; be-
tween two-dimensional forum deter-
mined by four pairs o homologous
elements', 204; which interchanges
vortices and sides of triangle is polar-
ity, 204 ; between two pianos, analytic
representation of, 200, 207; repre-
sented by ternary bilinear form, 207 ;
represented by matrices, 270; double
pairs of a, 278-281

Correlations and duality, 20R

Correspondence, as a logical term, 5j
perspective, 12; (1, 1) of two liguros,
35; general theory of, 04-0(1; idim-
tioal, 05; inverse of, 05; period of,
(JO; periodic or cyclic, 00; involutorio
or reflexive, 00 ; perspective between
two pianos, 71 ; quadratic, 139, KXH.
22, 24; 293, Ex. 1

Correspondences, resultant or product
of two, 05; associative law for, 00;
commutative, 60 ; groups of, 07; leav-
ing a figure invariant form a group, 08

INDEX

337

Corresponding elements, 35; doubly,
102

Co variant, 257; example of, 258

Oemona, L., 187, 138

OroKH ratio, 151); of harmonic set, 150,
Hit ; definition of, 100 ; expression for,
100; iu UoiiH Aeneous coordinates,
105; theorems on, 107, 108, EXH. ;
characteristic, of projectivity, 205;
characteristic, of involution, 200 ; as
an invariant of two quadratic binary
forms, 251, Ex. 1 ; of four complexes,
o>2

Crows ratios, the six, defined by four ele-
ments, 101

Curve, of third order, 217, Exs. 7, 8, 0;
algebraic, 250

Cyclic e,orrespondence, 00

Darboux, <}., 05

Degenerate conies, 120

Degenerate regulus, 311

"Decree of geometric problem, 230

Derivative, 255

Desargucs, configuration of, 40, 51 ; the-
orem on perspective triangles, 41,
180; theorem on conies, 127, 128

Descartes, R., 285

Diagonal point (line), of complete quad-
rangle (quadrilateral), 44; of com-
plete u-point (n-line) in plane, 44

Diagonal triangle of quadrangle (quad-
rilateral), 41

Dirkson, L. K., 00

Difference of two points, 148

Differential operators, 250

Dimensions, space of three, 20 ; space of
n, 30 ; assumptions for space of ?i, 33 ;
space of live, 331

"Directrices, of a rogulus, 200 ; of a con-
gruence, 315; of a special complex,
IJ24

Distributive law for multiplication with
respect to addition, 147

Division of points, 140

Domain of rationality, 238

Double element (point, line, plane) of
correspondence, 08

Double pairs of a correlation, 97

Double* points, of a projeetivity on a
line satisfy a quadratic equation, 150 ;
of projectivity on a lino, homogeneous
coordinates of, 104; of projectivity
always exist in extended space, 242 ;
of projectivity on a line, construction
of, 240 ; of involution determined by
covariant, 258 ; and lines of collinea-
tion in plane, 271, 205

Doublo ratio, 150

Doubly parabolic point, 274

Duality, in three-space, 28; in plane,
29 j at a point, 259 j in four-space, 29,

Ex. ; a consequence of existence of

correlations, 208

Edge of %-point or ?i-plane, 36, 37
Elation, in plane, 72 ; iu space, 75
Element, undefined, 1; of a figure, 1;

fundamental, 1; ideal, 7; simple, of

space, 34; invariant, or double, or

lixed, 08 ; lineal, 107
Eleven-point, plane section of, 53, Ex. 15
Enriques, F., 50, 286
Equation, of line (point), 174; of conic,

185, 245; of plane (point), 193, 198;

reducible, irreducible, 239 , quadratic,

has roots in extended space, 242
Equivalent number systems, 150
Extended space, 242, 255
Extension, assumptions of, 18, 24

Face of ?i-point or n-plaue, 36, 37

format, P., 285

Field., 140 ; points on a line form a, 151 ;
finite, modular, 201 ; extended, in
which any polynomial is reducible, 200

Figure, 34

Fine, H. B., 255, 260, 261, 289

Finite spaces, 201

Five-point, plane section of, in space,
39 ; in space may be transformed into
any other by projective collineation,
77 ; diagonal points, lines, and planes
of, in space, 204, Exs. 16, 17, 18;
simple, in* space determines linear
congruence, 319

Five-points, perspective, in four-space,
54, Ex. 25

Fixed element of correspondence, 68

Flat pencil, 55

Forms, primitive geometric, of one, two,
and three dimensions, 55 ; one-dimen-
sional, of second degree, 109; linear
binary, 251; quadratic binary, 252;
of nth degree, 254 ; polar forms, 256 ;
ternary bilinear, represents correla-
tion in plane, 207

Four-space, 25, Ex. 4

Frame of reference, 174

Fundamental elements, 1

Fundamental points of a scale, 141, 231

Fundamental propositions, 1

Fundamental theorem of projectivity,
94-97, 213, 264

General point, 129

Geometry, object of, 1; starting point
of, 1 ; distinction between projective
and metric, 12 ; finite, 201 ; associated
with a group, 259

Gergonne, J. D., 29, 123

Grade, geometric forms of first, second,
third, 55

Group, 66 ; of correspondences, 67 ; gen-
eral projective, on line, 68, 209;

338

INDEX

examples of, 09, 70 ; commutative, 70 ;
general project! ve, in plane, 268

HO, assumption, 45 ; r61e of, 81, 261

Harmonic conjugate, 80

Harmonic honiology, 223

Harmonic involutions, 224

Harmonic set, 80-82 ; exercises on, 83,
84 ; cross ratio of, 159

Harmonic transformations, 230

Harmonically related, 84

Hesse, 125

Hessenberg, G., 141

Hexagon, simple, inscribed in two inter-
secting lines, 99 ; simple, inscribed in
three concurrent lines, 250, Ex. 5;
simple, inscribed in conic, 110, 111

Hexagram, of Pascal (hexagramma mys~
ticum), 138, Exs. 19-21; 304, Ex. 16

Hilbert, D., 3, 95, 148

Holgate, T. P., 119, 125, 139

Homogeneous coordinates in plane,
174

Homogeneous coordinates, in space, 11,
194 ; on line, 163 ; geometrical signifi-
cance of, 165

Homogeneous forms, 254

Homologous elements, 35

Homplogy, in plane, 72; in space, 75;
axis and center of, 104; harmonic,
223, 275 ; canonical lorm of, in plane,
274, 275

Hyperosculate, applied to two conies, 136

Ideal elements, 7

Ideal points, 8

Identical correspondence, 65

Identical matrix, 157, 269

Identity (correspondence), 65; element
of group, 67

Improper elements, 239, 241, 242, 255

Improper transformation, 242

Improperly protective, 97

Independence, of assumptions, 6 ; neces-
sary for distinction between assump-
tion and theorem, 7

Index, of subgroup, 271 ; of group of col-
lineations in general protective group
in plane, 271

Induced correlation in planar field, 262

Infinity, points, lines, and planes at, 8

Inscribed and circumscribed triangles,
98, 250, Ex. 4

Inscribed figure, in a conic, 118

Invariant, of two linear binary forms,
252 ; of quadratic binary forms, 252-
254, Ex. 1; of binary form of nth
degree, 257

Invariant element, 68

Invariant figure, under a correspond-
ence, 67 ; of collineation is self -dual,
272

Invariant subgroup, 211

Invariant triangle of collineation, rela-
tion between project! vities on, 274,
276, Ex. 5

Inverse, of a correspondence, 65; of
element in group, 67; of projectiyity
is a projectivity, 68; of projectivity,
analytic expression for, 157

Inverse operations (subtraction, divi-
sion), 148, 149

Involution, 102 ; theorems on, 102, 103,
124, 127-131, 133, 134, 136, 206, 209,
221-229, 242-243 ; analytic expression
for, 157, 222, 254, Ex. 2 ; character-
istic cross ratio of, 206 ; on conic, 222-
230 ; belonging to a projectivity, 220 ;
double points of, in extended space,
242 ; condition for, 254, Ex. 2 ; dou-
ble points of, determined by covari-
ant, 258 ; complexes in, 333

Involutions, any projectivity is product
of two, 223; harmonic, 224; pencil
of, 225; two, have pair in common,
243; two, on distinct lines are per-
spective, 243

Involutoric correspondence, 66

Irreducible equation, 239

Isomorphism, 6; between number sys-
tems, 150 ; simple, 220

Jackson, D., 282
Join, 16

Kantor, S., 250
Klein, P., 95, 333, 334

Ladd, C., 138

Lage, Geometrie der, 14

Lennes, N. J., 24

Lindemann, F., 289

Line, at infinity, 8 ; as undefined class

of points, 15 ; and plane on the same

three-space intersect, 22 ; equation of,

174; and conic, intersection of, 240,

246

Line conic, 109
Line coordinates, in plane, 171 ; in space,

327, 333

Lineal element, 107
Linear binary forms, 251 ; invariant of,

251
Linear dependence, of points, 30; of

lines, 311

Linear fractional transformation, 152
Linear net, 84
Linear operations, 236
Linear transformations, in plane, 187;

in space, 199
Lines, two, in same plane intersect,

18
Liiroth, J., 95

INDEX

339

Maclaurin, C., 119

MacNeish, H. P., 46

Mathematical science, 2

Matrices, product of, 156, 268 ; determi-
nant of product of two, 269

Matrix, as symbol for configuration, 38 ;
definition, 150; used to denote pro-

. jectivity, 15C; identical, 157, 269;
characteristic equation of, 165, 272;
conjugate, transposed, adjoint, 269;
as operator, 270

MensBchnuw, 126

Metric geometry, 12

Midpoint of pair of points, 230, Ex. 6

Mbbius tetrahedra, 105, Ex. 6; 326,
Ex. 9

Multiplication of points, 145, 231 ; the-
orems on, 145-148 ; commutative law
of, is equivalent to Assumption P,
148; other definitions of, 107, Exs.
3,4

n-lino, complete or simple, 37, 38; in-
scribed in conic, 138, Ex. 12

nrplane, complete in space, 37 ; on point,
38 ; simple in space, 37

n-poinl, complete, in space, 36 ; complete,
in a plane, 37; simple, in space, 37;
simple, in a plane, 37; plane section of,
in space, 53, Exs. 13, 16 ; 54, Ex. 18 ;
m-space section of, in (n + ] )-space,
54, Ex. 19 , section by three-space of,
in four-space, 54, Ex. 21; inscribed
in conic, 119, Ex. 5; 250, Ex. 7

n-points, in different planes and per-
spective from a point, 42, Ex. 2 ; in
same plane and perspective from a
line, 42, Ex. 4; two complete, in a
plane, 53, Ex. 7 ; two perspective, in
(w l)-spaco, theorem on, 54, Ex.
26 ; mutually inscribed and circum-
scribed, 250, Ex. 6

Net of rationality, on line (linear net),
84; theorems on, 85; in plane, 86;
theorems on, 87, 88, Exs. 92, 93; in
space, 89 ; theorems on, 89-92, Exs. 92,
93 ; in plane (space) left invariant by
perspective collineation, 93, Exs. 9,
10; in space is properly projective,
97; coordinates in, 162

Newson, II. B., 274

Nonhomogeneous coordinates, on a line,
162 ; in plane, 169 ; in space, 190

Null system, 324

Number system, 149

On, 7, 8, 15

Operation, one-valued, commutative, as-
sociative, 141 ; geometric, 236 ; linear,
236

Operator, differential, 256 ; represented
by matrix, 270 ; polar, 284

Opposite sides of complete quadrangle,
44

Opposite vertex and side of simple
7i-point, 37

Opposite vertices, of complete quadrilat-
eral, 44 ; of simple ?i-point, 37

Oppositely placed quadrangles, 50

Order, 60

Ordinate, 170

Origin of coordinates, 169

Osculate, applied to two conies, 134

Padoa, A., 3

Papperitz, E., 309

Pappus, configuration of, 98, 99, 100,

126, 148

Parabolic congruence, 315

Parabolic point of collineation in plane,
274

Parabolic projectivities, any two, are
conjugate, 209

Parabolic projectivity, 101; charac-
teristic cross ratio of, 206; analytic
expression for, 207 ; characteristic con-
stants, 207 ; gives H(MA', AA"), 207

Parametric representation, of points
(lines) of pencil, 182; of conic, 234; of
regulus, congruence, complex, 330, 331

Pascal, B., 36, 99, 111-116, 123, 120,

127, 138, 139

Pencil, of points, planes, lines, 55; of
conies, 129-136, 287-293; of points
(lines), coordinates of, 181 ; paramet-
ric representation of, 182 ; base points
of, 182 ; of involutions, 225 ; of com-
plexes, 332

Period of correspondence, 66

Perspective collineation, in plane, 71 ;
in space, 75 ; in plane defined when
center, axis, and one pair of homol-
ogous points are given, 72 ; leaving R z
(R*) invariant, 93, Exs. 9, 10

Perspective conic and pencil of lines
(points), 215

Perspective correspondence, 12, 13 ; be-
tween two planes, 71, 277, Ex. 20

Perspective figures, from a point or
from a plane, 35 ; from a line, 36 ; if
A, JB, C and A', B', C' on two coplanar
lines are perspective, the points (AB',
JR4/), (AC', CA^, and (BC f , CB') are
collinear, 62, Ex. 3

Perspective geometric forms, 56

Perspective n-lines, theorems on, 84,
Exs. 13, 14 ; five-points in four-space,
54, Ex, 25

Perspective (n + l)-points in w-space,
54, Exs. 20, 26

Perspective tctrahcdra, 43

Perspective triangles, theorems on, 41,
53, Exs. 9, 10, 11; 54, Ex. 23; 84,
Exs. 7, 10, 11; 246; sextuply, 246

340

INDEX

Perspectivity, center of, plane of, axis
ot, 36 ; notation for, 57 ; central and
axial, 57, between conic and pencil
of lines (points), 215
Fieri, M., 95

Hanar field, 55

Planar net, 86

Plane, at infinity, 8, defined, 17; deter-
mined uniquely by three noncollinear
points, or a point and line, or two in-
tersecting lines, 20 ; and line on same
three-space are on common point,
22 ; of perspectivity, 36, 75; of points,
55; of lines, 55; equation of, 193,
198

Plane figure, 34

Plane section, 34

Planes, two, on two points -4, B are on
all points of line AB, 20, two, on
same three-space are on a common
line, and conversely, 22 ; three, on a
three-space and not on a common
line are on a common point, 23

Plucker's line coordinates, 327

Point, at infinity, 8 ; as undefined ele-
ment, 15 ; and line determine plane,
17, 20; equation of, 174, 193, 198; of
contact of a line with a conic, 112

Point conic, 109

Point figure, 34

Points, three, determine plane, 17, 20

Polar, with respect to triangle, 46;
equation of, 181, Ex. 3 ; with respect
to two lines, 52, Exs. 3, 5 ; 84, Exs. 7,
9 ; with respect to triangle, theorems
on, 54, Ex. 22 ; 84, Exs. 10, 11 ; with
respect to n-line, 84, Exs. 13, 14 ; with
respect to conic, 120-125, 284, 285

Polar forms, 256 ; with respect to set of
tt-points, 256; with respect to regu-
lus, 302 ; with respect to linear com-
plex, 324

Polar reciprocal figures, 123

Polarity, in planar field, 263, 279, 282,
283 ; in space, 302 j null, 324

Pole, with respect to triangle, 46 ; with
respect to two lines, 52, Ex. 3 ; with
respect to conic, 120 ; with respect to
regulus, 302; with respect to null
system, 324

Poncelet, J. V., 29, 36, 58, 119, 123

Problem, degree of, 236, 238 ; algebraic,
transcendental, 288; of second de-
gree, 245 ; of projectivity, 250, Ex, 14

Product, of two correspondences, 65;
of points, 145, 231

Project, a figure from a point, 36 ; an
element into, 58, ABC can be pro-
jected into A'B'C', 59

Projection, of a figure from a point, 34

Protective collineation, 71

Projective comes, 212, 304

Projective correspondence or transfor-
mation, 13, 58 ; general group on line,
08 ; in plane, 268 ; of two- or three-
dimensional forms, 71, 152

Projective geometry distinguished from
metric, 12

Projective pencils of points on skew
lines are axially perspective, 64

Projective pro jectivi ties, 208

Projective space, 97

Projectivity, definition and notation for,
58; ABC-xA'B'C', 59; ABGD^
JSAJDO, 60 ; in one-dimensional forms
is the result of two perspectivities, 63 ;
if JET (12, 34), then 1234^1243, 82;
fundamental theorem of, for linear
net, 94 ; fundamental theorem of, for
line, 95; assumption of, 95; funda-
mental theorem of, for plane, 96 ; for
space, 97 ; principle of, 97 ; necessary
and sufficient condition for MNAB -r-
MNA'B' is Q(MAB, NB'A^, 100;
necessary and sufficient condition
for MMAB ^ MMA'B' is Q (MAB,
MB' A'), 101 ; parabolic, 101 ; ABCD
-K-ABDC implies H(AB, CD), 103;
nonhomogeneous analytic expression
for, 154-157, 206 ; homogeneous ana-
lytic expression for, 164; analytic
expression for, between points of dif-
ferent lines, 167 ; analytic expression
for, between pencils in plane, 183;
between two conies, 212-216; on
conic, 217-221; axis (center) of, on
conic, 218; involution belonging to,
226 ; problem of, 250, Ex. 14.

Projectivities, commutative, example of,
70 ; on sides of invariant triangle of
collineation, 274, 276, Ex. 5

Projector, 35

Properly projective, 97; spatial net is, 97

Quadrangle, complete, 44; quadrangle-
quadrilateral configuration, 46; sim-
ple, theorem on, 52, Ex. 6 ; complete,
and quadrilateral, theorem on, 53,
Ex. 8; any complete, may be trans-
formed into any other by projective
collineation, 74; opposite sides of,
meet line in pairs of an involution,
103 ; conies through vertices of, meet
line 111 pairs of an involution, 127 ;
inscribed in conic, 137, Ex. 11

Quadrangles, if two, correspond so that
five pairs of homologous sides meet
on a line Z, ,tho sixth pair meets on
, 47, perspective, theorem on, 68,
Ex. 12 ; if two, have same diagonal
triangle, their eight vertices are on
conic, 137, Ex. 4

Quadrangular set, 49, 79 ; of lines, 79 ; of
planes, 79

INDEX

341

Quadrangular section by transversal of
quadrangular set of lines is a quad-
rangular set of points, 79 ; of elements
protective with quadrangular set is
a quadrangular set, 80; Q(MAB,
NB'A')is the condition for MNAB-^
MNATK, 100 ; Q(MAB, MR' A') is the
condition for MM A B-j- M, M A'B', 101 ;
Q(ABC, A'B'V) implies Q(A'B'C',
ABC), 101 ; Q(ABC, A'B'C') is the
condition that A A', BB', Ware in in-
volution, 103 ; Q(P Pa-Po, P, PyPx+v)
is necessary and sufficient for P^ + P
= P a4 . y , 142; QCP.PoPi, P P,rP,v)
is necessary and sufficient for P x P y
= P^145

Quadrangularly related, 80

Quadratic binary form, 252; invariant
of, 252

Quadratic correspondence, 139, Exs.
22, 24

Quadric spread in S a , 331

Quadric surface, 301 ; degenerate, 308 ;
determined by nine points, 311

Quadrilateral, complete, 44 ; if two quad-
rilaterals correspond so that five of the
lines joining pairs of homologous ver-
tices pass through a point P, the line
joining the sixth pair of vertices will
also pass through P, 40

Quantic, 254

Quaternary forms, 258

Quotient of points, 149

Range, of points, 55 ; of conies, 128-136

Ratio, of points, 149

Rational operations, 149

Rational space, 98

Rationality, net of, on line, 84, 85 ; planar
net of, 80-88 ; spatial net of, 89-93 ;
domain of, 238

Rationally related, 80, 89

Reducible equation, 239

Reflection, point-line, projective, 223

Reflexive correspondence, 60

Regulus, determined by three lines, 298 ;
directrices of, 299 ; generators or
rulers of, 299; conjugate, 299; gen-
erated by projective ranges or axial
pencils, 299 ; generated by projective
conies, 304, 307 ; polar system of, 300 ;
picture of, 300 ; degenerate cases, 311 ;
of a congruence, 318

Related figures, 35

Resultant, of two correspondences, 65 ;
equal, 65; of two projectivities is a
projectivity, 68

Reye, T., 125, 139

Rohn, K, 309

Salmon, G., 138
Sannia, A., 304

Scale, defined by three points, 141, 231 ;
on a conic, 231

Schroter, H., 138, 281

Schur, F., 95

Science, abstract mathematical, 2 ; con-
crete application or representation of, 2

Scott, C. A., 203

Section, of figure by plane, 34 ; of plane
figure by line, 35 ; conic section, 109

Segre, C., 230

Self -conjugate subgroup, 211

Self -conjugate triangle with respect to
conic, 123

Self-polar triangle with respect to conic,
123

Set, synonymous with class, 2 ; quadran-
gular, 49, 79; of elements projective
with quadrangular set is quadrangu-
lar, 80*; harmonic, 80; theorems on
harmonic sets, 81

Seven-point, plane section of, 53, Ex. 14

Seydewitz, F., 281

Sheaf of planes, 55

Side, of ft-point, 37 ; false, of complete
quadrangle, 44

Similarly placed quadrangles, 60

Simple element of space, 39

Simple n-point, n-line, n-plane, 37

Singly parabolic point, 274

Singular point and line in nonhomoge-
neous coordinates, 171

Six-point, plane section of, 54, Ex, 17;
in four-space section by three-space,
54, Ex. 24

Skew lines, 24; projective pencils on,
are perspective, 105, Ex. 2 ; four, are
met by two lines, 250, Ex. 13

Space, analytic projective, 11 ; of three
dimensions, 20; theorem of duality
for, of three dimensions, 28 ; n-, 30 ;
assumption for, of n dimensions, 33 ;
as equivalent of three-space, 34;
properly or improperly projective,
97; rational, 98; finite, 201, 202;
extended, 242

Spatial net, 89, theorems on, 89-92;
is properly projective, 97

von Staudt, K. G. C., 14, 95, 125, 141,
151, 158, 100, 286

Steiner, J., 109, 111, 125, 138, 139, 285,
280

Steiner point and line, 138, Ex. 19

Steinitz, E., 261

Sturm, Ch., 129

Sturm, R., 231, 250, 287

Subclass, 2

Subgroup, 68

Subtraction of points, 148

Sum of two points, 141, 231

Surface, algebraic, 259; quadric, 301

Sylvester, J. J., 323

System affected by a correspondence, 65

342

INDEX

Tangent, to conic, 112

Tangents to a point conic form a line
conic, 116 ; analytic proof, 187

Taylor's theorem, 255

Ternary forms, 258; bilinear, repre-
sent correlation in a plane, 267

Tetrahedra, perspective, 43, 44 ; config-
uration of perspective, as section of
six-point in four-space, 54, Ex. 24:
Mbbius, 105, Ex. ; 326, Ex. 9

Tetrahedron, 37; four planes joining
line to vertices of, projective with
four points of intersection of line
with faces, 71, Ex, 5

Three-space, 20; determined uniquely
by four points, by a plane and a point,
by two nonintersecting lines, 23 ; the-
orem of duality for, 28

Throw, definition of, 60 ; algebra of, 141,
157; characteristic, of project! vity,
205

Throws, two, sum and product of, 158

Trace, 35

Transform, of one projectivity by an-
other, 208 ; of a group, 209

Transform, to, 58

Transformation, perspective, 18; pro-
jective, 13 ; of one-dimensional forms,
58; of two- and three-dimensional
forms, 71

Transitive group, 70, 212, Ex. 6

Triangle, 37; diagonal, of quadrangle
(quadrilateral), 44; whose sides pass-
through three given collinear points
and whose vertices are on three given
lines, 102, Ex. 2; of reference of
system of homogeneous coordinates

in plane, 174; invariant, of collinea-
tion, relation between projectivities
on sides of, 274, 270, Ex. 5

Triangles, perspective, from point arc
perspective from line, 41 ; axes of
perspectivity of three, in plane per-
spective from same point, are con-
current, 42, Ex, G; perspective, theo-
rems on, 53, Exs. 9, 10, 11 ; 105, Ex.
9 ; 116, 247 ; mutually inscribed and
circumscribed, 99; perspective, from
two centers, 100, Exs. 1, 2, 3; from
four centers, 105, Ex. 8 ; 138, Ex. 18 ;
from six centers, 246-248 ; inscribed
and circumscribed, 250, Ex. 4

Triple, point, of lines of a quadrangle,
49 ; of points of a quadrangular set, 49

Triple, triangle, of lines of a quadran-
gle, 49; of points of a quadrangular
set, 49

Triple system, 3

Undefined elements in geometry, 1

United position, 15

Unproved propositions in geometry, 1

Variable, 58, 150

Veblen, 0., 202

Veronese, G., 52, 53

Vertex, of w-points, 36, 37; of n-planes,

37; of fiat pencil, 55; of cone, 109;

false, of complete quadrangle, 44

Wiener, H., 65, 95, 230
Zeuthen, H. G., 95

Date Due

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