PROJECTIVE GEOMETRY
BY
OSWALD VEBLEN
MIOFMSSOR OV MATHKMATIOS, PUINOBTON UNIVERSITY
AND
JOHN WESLEY YOUNG
I'UOl l.sMW HP MATUKMATICH, DAHTMOUTII COLLEGE
VOLUME I
GINN AND COMPANY
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ATLANTA DALLAS COLUMBUS * SAN FRANCISCO
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PREFACE
Geometry, which had boon 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 woj'k 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 m 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 elemeh-
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
dilnoulty 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 projectivo geometry." It will be found that the
theorems selected on this basis of logical simplicity are also elemen-
tary in the HGIUJO 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 projective geometry,
because it is through the latter that the other geometric disciplines
are most readily coordinated* Since it is more natural to derive
m
iv PREFACE
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 projoctive 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 pedagogically as
well as scientifically by maintaining the sharp distinction between
the projective and the metric.
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 mterchangeability of analytic and synthetic
methods, etc. Between two methods of treatment we have choser
the more conventional in all cases where a new method did nol
seem to have unquestionable advantages. We have tried also t<
PBEFACE 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 boon 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.
0. VEBLEIST
J. W. YOUNG
August, 1910
In the second impression we have corrected a number of typo-
graphical and other errors We have also added (p, 343) two
pages of "Notes and Corrections" dealing with inaccuracies or
obscurities which could not be readily dealt with in the text. We
wish to express our cordial thanks to those readers who have kindly
called our attention to errors and ambiguities.
o.v.
J.W.Y.
August, 1016
CONTENTS
INTRODUCTION
SffiOTION TAQB
I 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 fiist assumption of extension .... 18
{) The tlueo-space 20
10. The remaining assumptions of extension for a space of three dimensions . 24
11. The principle of duality 20
12. The theorems of alignment for a space of n dimensions 29
CHAPTER II
PROJECTION, SECTION, PERSPEQTIVITY, ELEMENTARY CONFIGURATIONS
IS, Projection, section, porspectivity 84
14. The complete n-pomt, etc. . . SO
15. OoniigmatioiiB , . . 88
10. Tho DoHttrguos configuration 89
17. Perspective tetrahedra 48
18. Tho quadrangle-quadrilateral configuration 44.
19. The fundamental theorem on quadrangular sets 47
20. Additional remarks concerning the Desargues configuration 61
CHAPTER III
FKOJEOTIVITIES OB 1 THE PRIMITIVE GEOMETRIC FORMS OF ONE, TWO,
AND THREE DIMENSIONS
21. The nine primitive geometric forms .55
22. Perspectovity and projectivity . 66
. The projectivity of one-dimensional primitive forms .,,.*.,... 59
vii
viii CONTENTS
SECTION VAGI6
24. General theory of coriespondence. Symbolic tieatment (54
26. The notion of a group 00
26. Groups of correspondences. Invariant elements and figures .... 07
27. Gioup properties of projectivities (58
28. Projective transformations of two-dimensional forms 71
20 Projective collineations of three-dimensional forms 76
CHAPTER IV
HARMONIC CONSTRUCTIONS AND THE FUNDAMENTAL THEOREM OF
PROJECTIVE GEOMETRY
80. The projoctivity of quadrangular sets 70
81. Harmonic sets , 80
- 32. Nets of rationality on a line 81
* 88. Nets of rationality in the plane 8(5
34 Nets of rationality in space 80
35, The fundamental theorem of projectivity 1)3
86. The configuration of Pappus Mutually inscribed and ciicumscribed tri-
angles 08
37. Construction of projectivitios on one-dimensional forms 100
38. Involutions 102
30. Axis and center of homology 103
40. Types of collineations in the plane 10(5
CHAPTER V
CONIC SECTIONS
41. Definitions, Pascal's and Brianchon'fl theorems 100
42. Tangents. Points of contact . . . 112
43. The tangents to a point conic form a line conic 110
44. The polar system oJt a conic 120
45. Degenerate conies 12(1
46. Desargues'a theorem on comes 127
47. Pencils and ranges of conies. Order of contact . . . , 128
CHAPTER VI
ALCiEBlU OF POINTS AND ONE-DIMENSIONAL COORDINATE SYSTEMS
48. Addition of points 141
40. Multiplication of points 144
60. The commutative law for mulLiplication 148
61. The inverse operations ,,..,,,, ,,,., 148
62. The abstract concept of a number system. Isomorphism ....... 149
68. 1 Nonhomogeneous coordinates 160
64. The analytic expression for a projectivity in a one-dimensional primitive
form ....... . * , . , , 162
66. Von Staudt'iS algebra of throws , 167
CONTENTS ix
SECTION PAOK
68 The cross ratio . 15$)
57 Coordinates m a not of rationality on a line . 102
68. Homogeneous; coordinates on a lino . . 103
60. Protective correspondence between the points of two diffcient lines . , 100
CHAPTER VII
COORDINATE SYSTEMS IN TWO- AND THREE-DIMENSIONAL FORMS
00. Nonhomogeneous coordinates in a plane . . . . ... .109
01 Simultaneous point and line coordinates . ... . 171
02. Condition that a point toe on a line . . . . . 172
68. Homogeneous coordinates in the plane . ... ,174
04. The lino on two points. The point on two lines 180
06. Pencils of points and lines. Projoctivity . . . ... 181
00. The equation of a conic ..... . . 185
07. Linear transfoimations, in a plane ... . . . 187
08. Collmeations between two different planes . . 100
60. Nonhomogeneous coordinates m space . . . .... 190
70 Homogeneous cooidmates in space ... . . ... 194
71. Linear transformations in space . 199
72. Finite spaces .... . . 201
CHAPTER VIII
PKOcIECTIVITlKS IN ONE- DIMENSIONAL FORMS
73. Characteristic throw and cross ratio . J . . ... . . 205
74. Protective piojeclwtiea .... ... 208
75. Groups of projectivities on a lino . . 200
70. Projectivo transformations between conies 212
77. Projectivities on a conic 217
78. Involutions . 221
70, Involutions associated with a given projectivity 225
80 Harmonic transformations 280
81. Scale on a conic 281
82. Parametric representation of a conic 284
CHAPTER IX
GEOMETRIC CONSTRUCTIONS. INVARIANTS
88, The degree of a geometric problem 236
84. The intersection of a given lino with a given conic 240
86. Improper elements. Proposition Ka 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
SECTION TAGK
' 00. Invariants and covanants of binaiy forms . 257
' 91. Ternary and quaternary forms and their invariants . . . 258
1 92. Pioof of Imposition K n 260
CHAPTER X
PEOJECTIVE TRANSFORMATIONS OP TWO-DIMENSIONAL FORMS
93. Correlations between two-dimensional forms 262
4 94. Analytic representation of a correlation between two planes , . . 260
95 General piojoctive gioup. Representation by matrices . . . 268
90. Double points and double lines of a collmeation in a plane . , 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 . . . .298
104. Tho polar system of a regains 300
105. Protective conies .... . . . 804
106. Linear dependence of lines 311
107. The linear congruence . 812
108. The linear complex . . , . * 819
109. The Plucker line coordinates . ... 827
110. Linear families of lines 829
111. Interpretation of line cobrdinates as point coordinates in SB 381
INDEX 885
PKOJEOTIVE GEOMETRY
INTRODUCTION
1. Undefined elements and unproved propositions. Geometry deals
with the pioperties 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 piopositions
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 unproved ; 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 "be 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 "be derived ly the methods of formal logic. Moreover, since we
assumed the point of view of formal (i.e 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
1
2 INTBODUCTIOtf
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 assumption as more expressive of their real logical character.
We understand the term a mathematical science to mean any net
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, whore 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,B,0, . . . 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 B are distinct elements of S, there is at least one
m-class containing loth A and B t
* Synonyms for doss are s&t, aggregate, assemblage, totality; in German, Menge;
, .
t <?f. B.Jftutoll, The [Principles of Mathematics, Cambridge, 1008 j and L. Cou-
turat, Lea prlncjpea des tnathe'inatiques, Paris, 1905,
t A class S' is said to be a subclass of another class S, if every element of ' is
an element of S.
2] A MATHEMATICAL SCIENCE 3
II. If A and 1> are distinct elements of S, there in not more, than
one in-class containing both A and B.
III. Any two m-dasses have at least one dement of S in, rownwn.
IV There exists at least one m-dass
V. Every m-dass contains at least three elements of S.
VI. All the elements of S do not belong to the same in-class.
VII. No m-dass 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-dass, 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 ahove, 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 he 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 be consistent if a
single concrete representation of the assumptions ean le given.*
Knowing our assumptions to be consistent, we may proceed to de-
rive some of tn*e theorems of the mathematical science of which they
are the basis :
Any two distinct elements of S determine one and only one in-class
containing loth 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 domain 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 Geometric, 2d ed., Leipzig (1903), p. 18, and
Verhandlungen d. III. intern, math. Kongresses zu Heidelberg, Leipzig (1004),
P. 174; Padoa, L'Enseignement mathe'matique, Vol. V (1903), p. 86,
4 INTRODUCTION
Tlie w-class containing the elements A and B may conveniently
be denoted by the symbol AB
Any two m-classcs have one and only one dement of S in common
(Assumptions II, III).
There exist three elements of S wliioJi are not all in the same
m-class (Assumptions IV, V, VI).
In accordance with, the last theorem, let A, B, G he three elements
of S not in the same m-class. By Assumption V there must he a
third element in each of the w-classes AB> BC, CA, and by Assump-
tion II these elements must be distinct from each other and from
A, B } and C Let the new elements be D, J&, G, so that each of
the triples ABD, BG1S, OAG belongs to the same m-class. By
Assumption III the m-classes AJS and BG, which are distinct from
all the m-classes thus far obtained, have an element of S in common,
which, by Assumption II, is distinct from those hitherto mentioned ;
let it be denoted by F, so that each of the triples ASF and BFG
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 I 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 #i-classes
GD and A1SF have an element in common (by Assumption III)
which cannot be A or J8, and must therefore (by Assumption VII)
be Jf. Similarly, AGO- and the m-class DE have the element G in
common. The seven elements A, B, C, D, JS, J?, G have now been
arranged into m-classes according to the table
A B D JS F a
(!') B D JS F G A
X> JE F G A B C
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 1 ) is obtained from (l)by
replacing by A, I by B t 2 by 0, etc. We can show, furthermore,
that S can. contain no other elements than J, J9, G, , JB> 2?, G. For
suppose there were another element, T. Then, by Assumption III,
2] CATEGORICALNESS 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
m-class ; it cannot be F, for then AFTJS would all belong to the same
m-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 m,-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 ex-
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 (!') 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 19 and that to every
m-class of S x there corresponds an m-class of S 2 , 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 INTRODUCTION
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 1 VII
This leads to the following fundamental notion :
A set of assumptions is said to ~b& categorical, if there is essentially
only one system for which the assumptions are vahd ; i e. if any two
such systems may be 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 i-classes, every column constituting one m-class :
1 2 3 4 5 6 -7 8 9 10 11 12
123466789 10 11 12
^3 4 5 6789 10 11 12 1 2
9 10 11 12 012345678
Hence 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, Q,
* The isomorphism of Systems (1) and (!') 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 6 4 different ways.
2,3] IDEAL ELEMENTS 7
and the m-olasses to consist of the pairs AJ3, BC, CA, then it is
cleflx 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. Of. exam-
ple of preceding section.
8 INTRODUCTION
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
line* point.
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 m 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 a point 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 Imes 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 mfinity, 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 pomts 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, he 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 tune
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, 2). 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 primarily to the real case
A plane is the set of all points (number triads) which satisfy a
single linear equation
ax + ly + cz + d = 0.
A line is the set of all points which satisfy two linear equations,
.3 + ^ = 0,
* Such knowledge is not presupposed elsewhere in this book, except m the case
of consistency proofs. The elements of analytic geometry are indeed developed
from the beginning (cf Chaps. VI, VII)
10 INTRODUCTION [INTBOD.
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
i *Y* OJ %
where d = v as 2 + y* + g 2 , and Z = -jm = ^>w = -- The origin itself
add
may be denoted by (0, 0, 0, Je), where 7c is arbitrary. Moreover, any
four numbers (x v x s , ss a , x 4 ) (x *f* 0), proportional respectively to
1\
I, m, n, ) i will serve equally well to represent the point (#, y, 3),
d/
provided we agree that (x v x z , x s , x^ and (cx v cx^ cx 8 , cx 4 ) represent
the same point for all values of c different from 0. For a point
(x, y, z) determines
ex , cy
= cl, x z f y ==cm,
CZ
where c is arbitrary (c= 0), and (sc v x z , x a , x^ determuies
/i \ "\ n "'
(1) % = -*> y*=-*> = >
w 4 a 4 . a 4
provided a? 4 ^ 0.
We have not assigned a meaning to (x v oc z , x a , x^ when x t = 0, but
it is evident that if the point ( cl, cm, en, -. } moves away from the
\ d l
origin an unb'imted distance on the line whose direction cosines are
I, 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, m this section, supposing
known regarding points, lines, etc., at infinity, but aye placing ott*selye|f on tftfc
basis of elementary geometry.
4] CONSISTENCY OF IDEAL ELEMENTS 11
cosines I, 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 1} x z ,
x s , 0) as a point at infinity or an ideal point We have thus associ-
ated with every set of four numbers (x v sc z , aj g , # 4 ) 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 ; their ordinary Cartesian coordinates are given by the
equations (1). The ideal points are those for which # 4 = 0. The num-
bers (x v x z , x s , ( 4 ) we call the homogeneous coordinates of the point.
We now define a plane to be the set of all points (x v x & , # 8> o; 4 )
which satisfy a linear homogeneous equation .
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 z , x a> # 4 ) which satisfy two distinct linear
homogeneous equations :
a i aj i+ &A+ C A+ ^A=
a z x v + Z> 2 # 2 H- c 2 x s + 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 a; 4 = constitute a line Such a line we call a hne 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 pomt with
coordinates (x^ OJ 2 , # 8 , 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 protective space of three
dimensions. It may be defined, in a way which, allows it to be either
real or complex, as consisting of ;
12 INTRODUCTION
Points : All sets of four numbers (x v x z , x s , x^) t except the set
(0, 0, 0, 0), where (cx v cx z , cx z , cx^ is regarded as identical with
(x v x z) x s , C 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 proactive space cannot involve contradictious 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 x^ 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 tile 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 in and m' 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 f as follows ; To
every point A on m let correspond that point A 1 on <m/ in which m 1
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] PBOJECTIVB AISTD METEIC GEOMETRY 13
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 in' be transformed into the
points of a third line m" by a perspective transformation with a new
center Q ; and if tins 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 w , 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 projwtwe, and the points of m are said
FIG 2
to have been transformed into the points of m 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 winch tins totality of lines meets
another plane TT' will form a new figure, such that to every point of
TT will correspond a unique point of IT', and to every line of ir will
correspond a unique line of IT' We say that the figure in TT has been
transformed into the figure in TT' by & perspective transformatwn 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 protective transformation. In projective geometry two figures
that may be made to correspond to each other by means of a projee-
tive transformation are not regarded as different. In other words,
14 INTRODUCTION
protective geometry is concerned twtli those properties of Jtryi^ r&s that
are left unchanged when the figures fire subjected to ft >voje^ve
transformation.
It is evident that no properties that involve essentially th.e notion
of measurement can have anyplace 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 Staxidb called
this science Q-eometrie der Zarje
In regard to the points and lines at infinity, we can now see why
they cannot be treated as in any way different from tlie 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 ra', and
conversely. And m the example given of a perspective tx*ansforma-
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 tlie ordinary
and the ideal elements of space.
* The theorems of metric geometry may however be regarded as special cases
of projectire theoiems.
CHAPTER I
THEOREMS OF ALIGNMENT AND THE PRINCIPLE OF DUALITY
6. The assumptions of alignment. In the following treatment of
projective geometry we have chosen the point and the line as unde-
fined elements We consider a class (of, 2, p. 2) the elements of
which we call points, and certain undefined classes of points which
we call hncs 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 be 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, I, c, I, etc , as names for lines. If A and B denote the same
point, this will be expressed by the relation A = JS, if they repre-
sent distinct points, by the relation A = B If A = J3, 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
15
16 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I
of exhibiting in its most elegant form one of tbo most far-reaching
theorems of protective geometry (Theorem 11). Two lines which have
a point in common are said to intersect in or to meet in that point, or
to be 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
loth 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 X> and
E (D = -2?) are points such that B, 0, D are on a hne and 0, A, JS
are on a line, there is a point F
such that A, B, F are on a line
and also Z>, JS t F are on a hne
(fig 3).f
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 tn 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.
t 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 anrl lines of ordinary Euclidean geometry (this requires the notion of ideal
points , cf, 8, p, 8). Their function is not merely to exhibit one of the many
possible concrete representations, but also to help keep an 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 OUT 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 hne.
(Al,A2)t
The line determined by the points A, B (A 3 s 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 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, B, C are points not all on
the same line, the line pining any point D on the line BO to any
point JS (D = E) on the line CA meets the hne 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, Q, JS are three points not on
the same line, and I is a line joining Q and R, the class S 2 of all
points on the lines joining P to the points of I is called the plane
determined by P and I,
We shall use the small letters of the Greek alphabet, a, /3, 7, 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 L
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 gam m generality as long as the fundamental assump-
tions are not categorical (of. p. 6). In the present treatment our assumptions are
not made categorical until very 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,
JPw. 4
18 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I
1 If both A and B are 011 I, or if the hue AJB contains P, the
theorem is immediate.
2. Suppose A is on I, B not on l t and AS does not contain _P (fig. 4).
Since ^ is a point of TT, there is a point B' on Z colliuear with 13 and P.
If C be any point on AB, the line
joining C on .^J? to P on BB'
will have a point T in common
with AB' = l (A 3). Hence is a
point of TT.
3. Suppose neither 4 nor B is
on 2 and that .4.2? does not con-
tain P (fig 5) Since A and .S are
points of TT, there exist two points
A and 5' on I coUinear with A, P and B, P respectively. The line join-
ing A on A'P to B on P' has a point Q in common with &'A f (A 3).
Hence every point of the line AB = A Q
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 "be in or to
he on the plane; the plane is said to
pass through, or to contain the line,
or wo may also say the plane is on the
line. Further, a point of a plane is said
to be 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, E, 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 v are on a common
point. (A, EO)
. 6
ASSUMPTION OF EXTENSION
19
EIG. C
Proof, Let the plane TT be determined by the point P and the line I,
and let a and & be two distinct lines of TT.
1. Suppose a coincides with I (fig 6). If I contains P, any point
B of & (E 0) is collinear with P and
some point of l=a, winch proves the
theorem when & contains P If 5 does
not contain P, there exist on 5 two
points A and B not on I (E 0), and
since they are points of TT, they are
collinear with P and two points A'
and B f of I respectively. The line
joining A on A'P to P on PP/ has a
point R in common with .4'P/ (A 3)
i.e. 1 = a and 5 have a point in common. Hence every line in the plane
TT has a point in common with I.
2 Let a and & both be distinct
fromJ. (i) Let a contain P (fig. 7)
The line joining P to any point
B of 5 (E 0) has a point B' in com-
mon with I (Case 1 of this proof)
Also the lines a and & have points
A' and ^ respectively in common
with I (Case 1). Now the line
A'P a contains the points A' of
RB' and P of .B'P, and hence has a point A in common with BR &.
Hence every line of IT has a, point
in common with any line of TT
through P. (u) Let neither a nor
& contain P (fig. 8). As before,
a and & meet I in two points (?
and JR respectively. Let JB' be a
point of Z distinct from Q and 12
(E 0). The line PP/ then meets
a and & in two points A and B
respectively (Case 2, (i)). If
A ~ B, the theorem is proved. If A ^ J5, the line & has the point
R in common with QB 1 and the point B in common with B'A, and
hence has a point in common with AQ *= a (A 3).
Pio 7
20 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I
THEOREM 6 The plane a determined "by a line I and a point P is
identical with the plane /3 determined ly a line m and a point Q,
provided in and Q are on a (A, E 0)
Proof. Any point JB of ft is coH.in.ear 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 <x (Theorem 4).
1 Hence every point of /S is a point
A Q of a. Conversely, let B be any point
of of. The line B Q meets m in a
point (Theorem 5). Hence every
point of a is also a point of /3.
COROLLARY. There is one and only
one plane determined "by three non-
collinear points, or by a line and a
point not on the line, or "by two inter-
Fm< 9 secting lines. (A, EO)
The data of the corollary are all equivalent by virtue of E 0. We
will denote by ABO the plane determined by the points A, J5, (7;
by aA the plane determined by the line a and the point A, etc.
THEOREM 7. Two distinct planes which are on two common points
A, B (A 3 s B) are on all the points of the line A3, and on no other com-
mon points. (A, E 0)
Proof. By Theorem 4 the line AS lies in each of the two planes,
which proves the first part of the proposition. Suppose 0, not on AB,
were a point common to the two planes. Then the plane determined by
At JB, 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. The three-space. DEFINITION. If P, Q, R, T are four points
not i the same plane, and if TT is a plane containing Q, JK, and 3T,
the class S 8 of all points on the lines joining P to the points of IT is
called the space of three dimensions, or the three-space deteraiiaed
by P and TT,
If a point belongs to a three-space or is a point of a three-space, it
is said to "be in or to lie in or to le on the three-space. If all the points
of a Hue or plane are points of a three-space S# the line or plane is said
]
THE THREE-SPACE
21
FIG. 10
to lie in or to l>e in or to le on the S 8 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 IT.
THEOREM 8 If A and B are didmct points on a three-space S 8 ,
every point on the line AB is on S a (A)
Proof. Let S 3 Le determined by
a plane TT and a point P.
1. If A and B are both in ir t 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
B' (= A] of TT collmear with B
and P (def ). The line joining any point M on AB to P on .SJS' has
a point M' in common with BfA (A 3). But .M 7 is a point of TT, since
it is a point of AB 1 . Hence If is a point of S 3 (def.).
4. Let neither A nor B he in TT, and let AB not contain P (fig. 11).
The lines PA and PB meet TT in
two points A' and J3' respectively.
But the line joining A on A'P to
J5 on PB' has a point (7 in common
with B'A 1 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, m connection
with E and Theorem 4, the fol-
lowing corollary; m '
COROLLARY 1 If S 8 is a three-space determined "by a point P and a
plane TT, then IT and any line on S a but not on IT are on one and only
one common point. (A, E 0)
COROLLARY 2 Every point on any plane determined ly three non-
collinear points on a three-space S 8 is on S 8 . (A)
22 THEOBEMS OP ALIGNMENT AND DUALITY LOHAP I
Proof As before, let the three-space be determined by -TT and JP,
and let the three noncollinear points be A, B, C. Every point of the
line JBC is a, point of S 3 (Theorem 8), and every point of tho piano
ABC* is collmear with A and some point of BC
COROLLAEY 3 If a three-space S 8 is determined fry a point P and
a, plane TT, then TT and any plane on S 3 distinct from ir arc, 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 iu two
distinct points, which are also
points of the plane of the lines
(Cor. 1). The result then follows
from Theorem 7.
THEOREM $ If a plane a and
a Ime a not on a are on the same
three-space S a , then a and a are
on one and only one common point.
(A,EO)
Proof. Let S a be determined by
die plane TT and the point P.
1. If a coincides with TT, the theo-
rem reduces to Cor. 1 of Theorem 8.
2. If a is distinct from TT, it has
a line I in common with IT (Theorem 8, Cor 3). Let A be any point
on a not on I (EO) (fig. 12) The planed, 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 hne. (A, E 0)
The proof is similar to that of Theorem 8, Cor. 3, and is left as an
exercise.
COEOLLARY 2 Conversely, if two planes are on a common line, there
exists a three-space on loth. (A, E 0)
, * Tfce proof can evidently fce so worded as not to imply Theorem 0,
Pio. 12
9] THE THREE-SPACE 23
Proof If the planes a and ]3 are distinct and have a line / in
common, any point P of fi 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 which arc not on a
common line are on one and only one common point (A, E 0)
Proof. This follows without difficulty 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 m common, they are said to intersect or meet in the point.
COROLLARY 4. If a, ft, <y are three distinct planes on the same S 8
but not on the same hne, and if a line I is on each of two planes p, v
which are on the lines fty and <ya respectively, then it ^s on a plane \
which is on the hne a/3 (A, E 0)
Proof. By Cor. 3 the planes a,
fi, 7 have a point P in common,
so that the lines /&y, rya, aj3 all
contain P. The line I, being com-
mon to planes through (3y and ya,
must pass through P, and the
lines I and aft therefore intersect
in P and hence determine a plane
X (Theorem 6, Cor,).
THEOREM 10. Tlie, three-space
c , , .77 7 7 FIG- 18
S s determined oy a plane TT and
a point P is identical with the three-space S determined "by a plane
TT' and a point P 1 , provided IT' and P' are on S s , (A, E 0)
Proof. Any point A of S^ (fig 13) is collinear with P' and some
point A 1 of TT'; but P' and A' are both points of S 8 and hence A is a
point of S 8 (Theorem 8). Hence every point of S' B is a point of S 8 .
Conversely, if A is any point of S 3 , the line AP' meets TT' in a point
(Theorem 9). Hence every point of S 8 is also a point of S^.
COEOLLAEY. There is one and only one three-space on four given
points not on the same plane, or a plane and a, point not on the plane t
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 the 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 1. There exists at least one line.
E2. 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 8 .
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 (of Ex. 4, p. 25). The treatment for the n-diinensional
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 hues 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 S s is a three-space, every plane is on jS z .
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 I, I'.
2 Prove that any two lines, each of which meets thiee 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 eveiy three-space is n 8 + n 2 + n + 1;
iv. the number of lines on a three-space is (n 2 + 1) (ra 2 + n + 1);
v. the number of lines meeting any two skew lines on a three-space is
(n + 1) 2 ,
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 8 is a three-space on Q, R, S t T, the class S 4 of all points on the
lines joining P to the points of S 8 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.
11. Every line on a four-space PQRST which is not on the three-space
QHST has one and only one point in common with the three-space
iii Every point on any plane determined by thiee noncollmear points on
a four-space is on the four-space.
iv. Eveiy 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 line in common with S 8 , provided the plane is not on S 8 .
vi. Every three-space on a four-space determined by a point P and a three-
space S 8 has one and only one plane in common with S 8 , provided it does
not coincide with S_
26 THEOREMS OF ALIGNMENT AND DUALITY [OKA..I
vii If a three-space S 3 and a plane a not on S 3 aie on the same four-space,
S 3 and a have one and only one line in common.
' vm If a three-space S 3 and a line I not on S,, ai e on the same four-space,
S 8 and I have one and only one point in common
ix Two planes on the same foui -space but not on the same tluee-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
xh The four-space S 4 deteimined by a three-space S_j and a point P is
identical with the four-space deteimined by a three-space 83 and a point P',
provided B' 3 and P' aie on S 4 .
5 On the assumption that a line contains n + 1 points, extend the icsults
of Ex 3 to a fom -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 sym-
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 following 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 let, 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 A and THEOREM 9, COB, 1. I aand/3
S are distinct points, there is one are distinct planes, there is one and
and only one line on A and JB, only one line on a and ;&*
* 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, howevei, that the assumption of closure is essential in the theorem to be proved,
11]
THE PRINCIPLE OF DUALITY
ASSUMPTION A 3. If A, B, are
points not all on the same line, and
I) and E (D & E} are points such
that B, C, D are on a line and C,
A, E are on a line, then there is a
point F such that A, B, F are on a
line and also D, J2, 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 s is a
three-space, every point is on S 3 .
THEOREM 9, COR. 4. If a, fi, y
are planes not all on the same line,
and /* and v(^^v) are planes such
that ft, <y, p are on a line and y, a, v
are on a line, then there is a plan e X
such that a, ft, X are on a line and
also ft, 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
COR. 4, p. 25. All planes are
not on the same point.
COR. 5, p. 25. If S 3 is a three-
space, every plane is on S 3
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, R are points not on
the same line, and Hs a line on
Q and JS, the class S 2 of all
points such that every point of
S a is on a line with P and some
point on I is called the plane
determined by P and I.
If X, /*, v are planes not on the
same line, and I is a line on p
and v, the class B 2 of all planes
such that every plane of B a is on
a line with X and some plane on
I is called the lunette determined
by X and I.
Now it is evident that, since X, /*, 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 THEOREMS OF ALIGNMENT AND DUALITY [CHAP.I
With the aid of these observations we are now ready to establish
the so-called principle of duality .
THEOREM 11 THE THEOREM 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 "on" terminology, when the words "point" and "plane"
are interchanged (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 I, says that a plane TT' is one of the class of planes on a line I',
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' contains a plane TT', i e that P' 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 IT 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 m 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 plane 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
and lines through a point remains valid, if stated in the " on " termi-
nology, when the words "plane" and " line" are interchanged (A, E)
The piinciple of duality was first stated explicitly by Gergonne (1826), but
was led up to by the writings of Poncelet and others duimg 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 tieatment of this principle in many standard
texts is far from convincing. The method of formal infeience from explicitly
stated assumptions makes the theoiems 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 foiming the duals
of given propositions. It is therefore suggested as an exercise that he state
the duals of each of the theorems and coiollanes in this chapter. Pie 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
that he dualize several of the proofs by writing out m order the duals of each
proposition used in the pi oofs 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 lemains valid when stated in the " on " terminology, if the words
point and three-space and the words hne 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
"* Ibis section may be omitted on a first reading.
522 (
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 EO, for a space of any number of dimensions.
In this section, then, we make use of Assumptions A and E only.
DEFINITION. If J, P i} P it - , P n are n + 1 points not on the same
(n l)-space, and S n _ 1 is an (n l)-space on J?, -2^, . , J, the class
S n of all points on the lines joining P to the points of S n _ x is called
the n-space determined by % and S M _ r
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 u-space, calling a plane a
2-space, a line a 1-space, and a point a 0-space, when tins is convenient.
S is then a symbol for a point.
DEFINITION. An S r is on an S, and an S 4 is on an S, (r < t), pro-
vided that every point of S 7 is a point of S ( .
DEFINITION. Is points arc said to be independent, if there is no S l _ 2
which contains them all
Corresponding to the theorems of 6-9 we shall now establish
the propositions contained in the* following Theorems S n l, S, t 2,
S B 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 ^ = 3, 4, ,/ 1 ; i e we assume that the
propositions contained in Theorem S n _ 1 l, a, b, c, d, c, f have been
proved, and derive Theorem S n l, a, , / 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 le defined "by the point R and the
(n l)-space R H-1 .
a There is an n-space on any n + 1 independent points.
J. Any line on two points of S n has one point in common with R n _]_,
and is on S n .
c. Any S r (r< n} on r + 1 independent points of S B is on S n .
d. Any S r (r < n} on r + 1 independent points of S n has an S r _ 1 in
common with R f ,_ 1 , provided the r + 1 points are not all on R H _ r
e. Any line I on two, points of S n has one point in common with
an V S n-i on s *-
/. If T a and T ll _ 1 (T not on T B _ 1 ) are any point and any
(n T)-$pace respectively of the n-space determined by R and R B _i,
the latter n'Space is the same as that determined by T and T n l r
12] SPACE OF N DIMENSIONS 31
Proof, a. Let the n + 1 independent points be P , P I} - , P n . Then
the points P v P 2 > -, P n 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,^ containing all the points J%, J%, - > JE? t , con-
trary to the hypothesis that they are independent. Hence, by Theorem
S^l a, there is an S B _ 1 on the points J I, , J^; and this S n _ 1
with J^ determines an n-space which is on the points JJ}, J%, P 2 , , Jf t .
1. If the line I is on R or R B _ 1} the proposition is evident from the
definition of S n . If I is not on R or R n _ v let A and jB be the given
points of I which are 011 S B The lines R A and R B then meet R n-1
in two points A' and ' respectively. The line I then meets the two
lines JB'R , R A'; 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 n _ l 1 5 To
show that every point of I is on S B , consider the points A, A 1 , P. Any
line joining an arbitrary point Q of I to R , meets the two lines PA
and AA', and hence, by Assumption A3, meets the third line A'P.
But every point of A'P is on R B-1 (Theorem S n _ 1 l 5), and hence Q
is, by definition, a point of S B
c. This may be proved by induction with respect to r. Tor r = 1 it
reduces to Theorem S n l 5 If the proposition is true for r Jc 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 i _ 1 determined by the remaining 7<3 points. But under the
hypothesis of the induction this S jl _ 1 is on S n , and hence, by Theorem
S n l 5, all points of S A are on S n .
d. Let r + 1 independent points of S n be -TjJ, J, , P r and let 1% be
not on R M _ X Each of the lines J.Z(7e = l,- -,r) has a point Q K in
common with R ;i-1 (by S H 1&). The points Q v Q^, '-, Q r are inde-
pendent; for if not, they would all be on the same S,_ 2 , which,
together with P^, would determine an S^_ t containing all the pc ; ^ta
P k (by S r ^ x l &). Hence, by S, , t l a, there is an S r _ 1 on Q v Q z , , Q r
which, by c, is on both S, and S B .
e. "We will suppose, first, that one of the given points is R Let
the other be A. By definition I then meets R n _ 1 m a point A', and, by
S W _ J 1 J, in only one such point If R is on S B _ P no proof is required
for this ease. Suppose, then, that R is not on S B _ lr and let be any
point of S a _ r The line R C7 meets R B _x in a point C' (by definition).
B^ d, S n _ 1 has in common with R^j an (n 2)-space, S w _ 2 , and, by
32 THEOREMS OF ALIGNMENT AND DUALITY [CHAP. I
Theorem S^le, this has in common with the line A'C" at least
one point D'. All points of the line D'O are then on $- by S^jl &.
Now the line I meets the two lines C'D' and C0 r ; hence it, meets the
line CD' (Assumption A3), and has at least one point on S n _ r
We will now suppose, secondly, that both of the given points are
distinct from R . Let them be denoted by A and B, and suppose that
R is not on S M _ 1 . By the case just considered, the lines R Q A and
R Q B meet S n _ i in two points A' and B' respectively. The line /, which
meets R -4' and R B' must then meet A'JB' in a point winch, by
Theorem S n _ 1 l'b, is on S n _^
Suppose,. finally, that R is on S n _ 1} still under the hypothesis that I
is not on R . By d, S n _ l meets R n _ x in an (n 2)-space Q /( _ 2 , and
the plane R Q l meets R n-1 in a line I'. By Theorem S fl _ 1 l e, I 1 and
Q n _ 2 have ui common at least one point P. Now the lines I and R P
are on the plane R Z, and hence have in common a point Q (by Theorem
S 2 1 e Theorem 5). By S^l Z the point Q is common to S n _ 1 and I.
f. Let the w-space determined by T and T n _ x be denoted by T n .
Any pouit of T B is on a hne joining T with some point of T B _ r
Hence, by &, every point of T M is on S n Let P be any point of S n
distinct from T . The luie T JP meets T,^ in a point, by e. Hence
every point of S n is a point of T n .
COROLLARY. Onn + 1 independent points there is one and but one S M .
This is a consequence of Theorem S B 1 a and S B 1/. The formal
proof is left as an exercise
THEOREM S n 2 An S r and an S^ having in common an S v , lut
not an S /J+1 , are on a common S r+L _ p and are not both on ilte same
Proof. Itk*=p, S k is on S r . If k >p, let J? be a point on S* not
on S p Then J* and S r determine an S rH lf and J$ and S p an S p hlj
such that S p+1 is contained in S,. +1 and S^ If & >p + 1, let J^ be a
point of S* not on S p+r Then P z and S r+1 determine an S r + s , while
P z and S p+1 determine an S J)+2 , which is on S rH . 2 and S A . This process
can be continued until there results an S p+i containing all the points
ofSj,. By Theorem S tt l, Cor, we have i= "k p. At this stage in the
process we obtain an S r4 . ft _ p which contains both S r and S k .
The argument just made shows that 2J, Jj|, . . . , P k _ p) together with
any set Q v Q a , . ., Q r+1 , of r + 1 independent points of S r , constitute
12] SPACE OF N DIMENSIONS 33
a set of r + Jc p + 1 independent points, each of which is either in
S, or S i If S r and S k were both on an S B , where n < r + 1$ p> these
could not be independent.
THEOREM S n 3. An S r and an S h contained in an S n arc ~botli on the
same S, +i _ n .
Proof. If there were less than r + 7c n -f- 1 independent points
common to S r and S^, say r+kn points, they would, by Theorem S B 2 ;
determine an S 5 , where gr = r + k (r + k n 1) = n + 1.
Theorems S B 2 and S n 3 can be remembered and applied very easily
by means of a diagram in winch S n is represented by n + 1 points.
Thus, if n 3, we have a set of four points That any two S 2 's have
an S 1 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 Jc + 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 :
Every line contains two and only two 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 E3 ; may
be replaced by the following :
En. Not all points are on the same S A , if Jc < n.
En' If S ts an S n , all points are on S.
For 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 Sj;^ and S n _ t _ ^
(counting a point as an S ), for all 7c's from to n 1 If n is odd,
there is a self-dual space in Sj if n is even, S n contains no self-dual
space.
EXERCISES
1. State and prove the theorems of duality in S 6 , in S n .
2. If m + 1 is the number of points on a line, how many S^'s aie there in
anS n ?
* 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
transnmte numbers.
* Exercises marked # are of a more advanced or difficult character.
CHAPTER II
PROJECTION, SECTION, PERSPECTIVES. 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 tlie 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 winch the relation
" on " has been defined. The theorems of the preceding chapter are to
be regarded as expressing the fundamental properties of this relation. f
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 IT determines with
P a line, and every line of F not IT a line, and every line of F not
on P determines with P a plane, on IT 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.
34
13] PROJECTION, SECTION, PEKSPECTIVITY 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 I 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 v 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 x corresponds (of. 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 a>, such
such that aH the projectors are that all the traces of either
36 PKOJECTION, SECTION, PERSPECTTVITY [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 a> is called the plane
twity. of perspectivity.
DEFINITION. If any two homologous hues 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 L 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', distinct from a, is to form the section by a' 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 I', distinct from I but in the same plane with I
and 0, is to form the section by I' of the projection of the set of points
from
Clearly in either case the two figures are perspective from 0, pro-
vided is not on either of the planes a, a' or the lines I, V.
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 -l)/2
lines joining every pair of the points and the n(n T)(n2)/& 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 w-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 Poncelet ; cf . his Traite" des proprie'te's projectives des
figures, Paris, 1822.
14] N-POINT, JV-PLASTE, JV-LIKE 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 fate*,
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 w-point. The plane dual of a complete
plane w-point is called a complete plane n-hne. It has n sides and
n(n T)/2 vertices. The simplest complete plane w-point consists of
three vertices and three sides and is called a triangle.
DEFINITION. A simple space n-point is a set of. n -paints ^P^P^ ,P n
taken in a certain order, in which no four consecutive points are
coplanar, together with the n lines %P Z> P z P a , , P n P^ joining suc-
cessive points and the n planes P^P Z P &) , P,A^ determined by
successive lines. The points, lines, and planes are called the lertices,
edges, and faces respectively of the figure The space dual of a simple
space w-point is a simple space n-plane
DEFINITION A simple plane n-point is a set of n points P V P Z) P & ,-- P n
of a plane taken in a certain order in which no three consecutive points
are collinear, together with the n lines P& P Z P 3 , , P n %. 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 -point is
called a simple plane n-line.
Evidently the simple space w-point and the simple space w-plane are
identical figures, as likewise the simple plane n-point and the simple
plane ra-line. Two sides of a simple rc-line which meet in one of its
vertices are adjacent. Two vertices are adjacent if in the dual relation.
Two vertices of a simple w-point J?J?> J? (ft even) are opposite if, ID
the order P& - 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 JJP f - - P n , as many vertices follow the
side and precede the vertex as follow the vertex and precede the side.
38 PROJECTION, SECTIOK, PEKSPECTIVITY
The space duals of the complete plane w-point and the complete plane
w-line are the complete n-plane on a point and the complete n-line on n
point respectively. They are the projections from a point, of the piano
w-line and the plane n-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 13 of lines and also on the same num-
ber a 13 of planes ; each line is on the same number a al of points and the
same number a aa of planes , and each plane is on the same number a al
of points and the same number S2 of lines.
A configuration may conveniently be described by a square matrix :
123
point line plane
1 point a n a la a 18
2 line a 21 a 22 a 2S
3 plane a 81 a S2 a 88
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 3= j) gives the number of elements of the yth kind
on every element of the itli kind. The numbers a u , a 22 , a 88 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
433
262
334
ic, io] CONFIGURATIONS 3 9
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
3 2
2 3
Since all the numbers referring to planes are of no importance m
case of a plane figure, they are omitted from the symbol for a plane
configuration.
In general, a complete plane w-point is of the symbol
n n 1
2 in(n-l)
and a complete space w-point of the symbol
n n-l (jj-l)(n-2)
2 n(n-l) n-2
3 3 $n(-l)(n-2)
Further examples of configurations are figs 14 and 15, regarded as
plane figures.
EXERCISE
Prove that the nunibeis in a configuration symbol must satisfy the condition
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
546
2 ifo"Y~
3 i 3 10
i
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
40 PROJECTION, SECTION, PEESPECTIVITY [CHAP, n
the plane, and every plane gives rise to a line. The configuration m
the plane lias then the symbol
10 3
3 10
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, C and of two points O lt O z
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 0^ on a.
A triangle 4 Z B Z C Z> the projection of the triangle ABC from 2 on a.
The trace of the line 6> X S .
The traces A S) B z , <7 8 of the lines BC, CA, AB respectively.
The trace of the plane ABC, which contains the points A 8 , JB a) C y
The traces of the three planes AO^, BO^, 00f) v which contain
respectively the triples of points OA^A^ OB^B^ OC^ V
The configuration may then be considered (in ten ways) as consist-
ing of two triangles A^B^C^ and A Z B Z C Z) perspective from a point and
I ifl] THEOREM OF DESAEGUES 41
having homologous sides meeting in three collinear points A 9> J3 S , C s .
These considerations lead to the following fundamental theorem :
THEOREM 1. THE TIIEOEEM OF DESAEGUES * If two triangles in the
same plane are perspective from a point, the three pairs of homologous
sides meet 'in colhnear points, ie the triangles are perspective from
a line (A, E)
Proof Let the two triangles he A^^C^ and A Z B Z C Z (fig. 14), the
lines A^A Z , B^B^, C^ meeting in the point Let B^A V B Z A Z inter-
sect in the point (7 a ; A^C V A Z C Z in J5 8 ; B^C V B Z C Z m A y It is required
to prove that A a , B s , C s are collinear Consider any hue through
which is not in the plane of the triangles, and denote l)y O v O a any
two distinct points on this line other than Since the lines A Z Z
and Afli he in the plane (A^A V 0^0^, they intersect in a point A.
Similarly, B^O^ and B. 2 Z intersect in a point B, and likewise G^O^ and
C Z Z in a point C. Thus ABCO^, 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 m a plane
with A 1 B V and also m a plane with A 2 B 2 , the lines A^ v A 2 B a , and
AB meet in C s So also A^C V A 2 Z , and AC meet in B s , and B^C V
B Z C Z , and BC meet m A & Since A s , B z , C s lie m the plane ABO and
also in the plane of the triangles A^B^C^ and A Z B Z C V they are collinear.
TIIEOEEM 1' If two triangles in the same plane are perspective
from a line > the lines joining pairs of homologous vertices are con-
current; ie 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 Z) J%, P, P 6
(fig. 15) The trace of the line IZ in the plane section is then
naturally denoted by P lz , in general, the trace of the line $% by Jg
fa j = 1, 2, 3, 4, 5, i =/). Likewise the trace of the plane J^J may
* Gorard Desargues, 1593-1662.
42 PROJECTION, SECTION, PEBSPECTIVITY [CHAP, n
be denoted by Z yA (i, j, k = I, 2, 3, 4, 5). This notation makes it pos-
sible to tell 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 J^ and J% k is the point Jf t ; two points are
on the same line if and only if they have a suffix in common, etc.
/ * \ x v
/ X N N
/ \
B/
3 / v V X t
?^V-\.\\B//
/
/
/
/
/
/
/
/
1 PU
&
ElO. 15
EXERCISES
1. Prove Theorem V without making use of the principle of duality.
2. If two complete n-points in different planes are perspective from a point,
the pairs of homologous sides intersect in colhnear 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 n-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 n-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^A
of Ex. 5 (cf . fig. 15, regarded as a plane figure).
IT]
PERSPECTIVE TETBAHEDKA
17. Perspective tetrahedra. As an application of the 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, the six
pairs of homologous edges intersect in coplanar points, and the four
pairs of homologous faces intersect in coplanar lines ; i.e, the tetra-
hedra are perspective from a plane. (A, E)
FIG 10
Proof. Let the two tetrahedra be J?-J?.ZJ and P^P^'P^P^, and let
the lines jgJJ', P Z PJ, P 6 P t r , PPJ meet in the center of perspectivity 0.
Two homologous edges P^ and P t 'Pj' then clearly intersect ; call the
point of intersection P v . The points P n) P 18 , P a!t lie on the same line,
since the triangles P^P^ and P^P^P^ are perspective from (The-
orem 1, Cor.). By similar reasonmg applied to the other pairs of
perspective triangles we find that the following triples of points are
collinear :
T> T> p . p p P-P PP'P P P
fta* *i8> %} -na* J u> *5A) **.*> f u> ^s*? -^asj J u> -*s4 p
The first two triples have the point P ia in common, and hence
determine a plane; each of the other two triples has a point in
44 PROJECTION, SECTION, PBSSPEOTIVITY [CHAP n
common with each of the first two Hence all the points P lt lie m
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.
THEOKEM 2' If two tetrahedra are perspective from a plane, the
hues joining pairs of homologous vertices are concurrent, as likewise
the planes determined ly pairs of homologous edges ; i.c. 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 o the last two Iheoiems.
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 collmear, the tri- onal lines are not concurrent, the
angle formed hy 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 throe 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 are
collmear and the diagonal lines of a complete quadrilateral concur-
rent, as may readily be verified Two perspective triangles cannot
exist in such a plane, and hence the Pesargties theorem becomes
* In general, the intersection of two sides of a complete plane ra-pomt which do
not have a vertex in common is called a diagonal point of the ?i-pomt, and the line
joining two vertices of a complete plane 71-line which do not lie on the same side
is called a diagonal line of the n-hue. A complete plane n-pomt (n-lmo) then has
n (n -!)(- 2) (n-3)/8 diagonal points (lines). Diagonal points and lines are
sometimes called false vertices and/a&e sides respectively.
w] ASSUMPTION JI 45
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 . The 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 v FlG
further assumptions than A and E,
understanding that in order to give our theorems meaning there must
l)e 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 confirmation.
A complete space five-point may evidently be regarded (in five ways)
as a tetrahedron and a complete four-line at a pomt 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 a> J% s , J? 4 , -Z? 6 and the corresponding
quadrilateral is Z 284 , / 286 , l ai& , l m
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 theie 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 n + 1
t To construct a figure IB to determine its elements in terms of certain given
elements.
46 PROJECTION, SECTION, PEKSPECTIVITY [CHAP, n
Let P t , P z , PS, P be the vertices of the given complete quadrangle,
and let Z> 12 , Z> 13 , D u be the vertices of the diagonal triangle, D 12 being
on the side P^, D 18 on the side JflJ, and Z> lt on the side Ifi* (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 perspectimty "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 D 12 , D 13 , D u .
It may readily be verified, by selecting two perspective triangles,
that the figure just formed is, indeed, a Desargues configuration. This
special case of the Desargues configuration is called the quadrangle-
quadrilateral 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 tho same triangle , that is, P is obtained from p by
a construction dual to that used in deriving p from P. Fiom this theorem it
follows that the relation between the quadrangle and quadiilateral 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 remaining 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] QUADRANGULAR SETS 47
configuration as mutual, thai is, if eithei is given, the other is determined.
For a leason which will be evident latei, eithei is called a covariant of the
other.
2 Show that the configuration consisting of two peispective tetrahedia,
their center and plane of perspectivity, and the piojectors and tiaces may be
regarded in six ways as consisting of a complete 5-point P 12 , P 1<p P u , P 16 , P 18
and a complete 5-plane 7r 8468 , v atM , v aasn , w sa4( ,, Tr^ M6 , the notation being
analogous to that u&ed on page 41 foi the Desargues configuration. Show
that the edges of the 5-plane are on the faces of the 5-point.
3. If Pj, P 2 , P 8 , P 4 , P 6 , are veitices of a complete space 5-point, thu ton
points Z> y , in which an edge p lf meets a face P L P t P m (/, j, !c, I, m all distinct),
are called diagonal point*. The tetrahedia P 2 P 8 P 4 P C and Z> 12 Z) 18 D U D 1S are per-
spective with PJ as center Their plane of perspectivity, ir^ is called the polar
of Pj with regard to the four veitices. In like nmnnei, the points P 2 , P 8 , P 4 , P B
deteimme their polar planes 7r 2 , TT S , ir v ir s Piove that the 5-poiut and the polai
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
confirmation ; and that the dual of the above construction applied to the 5-plane
detei mines the original 5-point.
4. If P is the pole of TT with regard to the tetrahedron A^l^A^A^, then is TT
the polai of P with regaid to the same tetrahedion?
19. The fundamental theorem on quadrangular sets.
THEOREM 3 If two complete quadrangles P t P z P 9 P and P^P^'P^Pl
correspond P^ to P^', P z to P^', etc. in sitch 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)
Tliis 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
P^, P^, P^, PtP a , P& be the five sides which, by hypothesis,
meet their homologous sides %'PJ, P^P^', P^P^ 1 , P a 'P & f f P^PJ in points
of I (fig. 19) We must show that P a P^ and P^'PJ meet in a point
of I, The triangles P^P^ and P^P z 'Pt are, by hypothesis, perspec-
tive from. l\ as also the triangles P^P^ and P^PJPJ. Each pair is
therefore (Theorem 1') perspective from a point, and this point is in
each case the intersection of the lines P^' and P S P Z '. 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 I But P 8 P and J'J' are
48 PKOJECTION, SECTION, PKRSPECTIVITY [CHAP. II
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 tbo five given points
j. 19
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 /. This
completes the proof of the theorem.
NOTE 1 It should be noted that the theorem is still valid if the line I con-
tains one or moie of the diagonal points of the quadrangles. The case in which
I contains two diagonal points is of particular importance arid will be discussed
in Chap. IV, 31.
NOTE 2. It is of importance to note in how far the quadrangle P{PPP^
is determined when the quadi angle P 1 / > a P 8 P 4 and the line I are given It may
be readily veiified that in such a case it is possible to choose any point P{ to
correspond to any one of the vertices P lt P 2 , P 8 , P 4 , say 7^; and that if m is
any line of the plane ZP X ' (not passing through P{) which meets one of the sides,
say a, of P^P^ (not passing through P x ) in a point o I, then m may be
chosen as the side homologous to o. But then the remainder of the figure is
uniquely determined.
* This evidently exists whenever the theorem is not trivially obvious.
19] QUADRANGULAR SETS 49
THEOREM 3' If two complete quadrilaterals a 1 a 2 a g a 4 and a(a' z a' a a[
correspond a 1 to a[, a s to a'%, etc in such a way that Jive 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 said to form a triangle triple,
in the latter a point triple of hues In a quadrangular 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 m 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
where ABC is a point triple and D2SF is a triangle triple, and
where A and D t B and JS, and G 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, E or C, 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 nse 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 m a apace in -which the diagonal points
of a complete quadrangle are collmear.
t It should "be 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. (of. 30, Chap IV),
50
PKOJECTI03ST, SECTION, PERSPECTIVE Y [CHAP. H
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.
21
With the notation for quadraagular sets defined above, the last
theorem leads to the following
COEOLLARY. If all lut one of the points of a quadrangular sot Q (AJBC,
DJSJ?) 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] DESARGUES 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 winch is the section by a 2-spaco of
a complete 5-point in a 3-space, the configuration of two perspective tetra-
hedra which may be considered as the section by a 3-space of a complete G-point
in a 4 -space aie all special cases of the section by an n-space of a complete
(n + 3)-pomt in an (n + l)-space The theorems which we have developed foi
the thiee cases here consideied are not wholly parallel. The leader will find
it an entertaining and far fioin trivial exeicise to develop the analogy in full
EXERCISES
1 A necessary and sufficient condition that three lines containing the ver-
tices of a tuangle shall be concuirent is that their intei sections P, Q, R with
a line Z form, with intersections E, F, G of corresponding sides of the tuangle
with Z, a quadrangular set Q(PQR, EFG)
2 If on a given transveisal line two quadrangles determine the same quad-
i angular set and are similarly placed, their diagonal triangles aie perspective
from the center of peispectivity of the two quadrangles
3 The polars of a point P on a line I with legard to all tuangles which
meet I in three fixed points pass through a common point P' on /
4. In a plane TT let there be given a quadrilateral a 1} a 2 , a s , 4 and a point
not on any of these lines Let A v A z , A s , A be any tetrahedron whose four
faces pass through the lines a lt a z , a s) a 4 respectively. The polar planes of
with respect to all such tetiahedra pass through the same line of it.
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, P^P^P^ 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 ^AP 8i P ts P &1 and P i8 P 85 P ss P^P 4r 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 l m 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 PROJECTION, SECTION, PEBSPECTIVITY [CHAP. 11
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 polo, and polar
are a vertex and its opposite side, eg. 7? 2 and l^I^
The Desargues configuration is one o a class of configurations
having similar properties. These configurations have been studied
by a numher of writers * Some of the theorems contained in these
memoirs appear in the exercises below
EXERCISES
In discussing these exercises the existence should l>e amnmcfl of a sufficient number
of point* on each line so that the fitjwes in question do not degenerate. In some ctisea
it may also ~be assumed that the diagonal points of a complete tjiinifranyle m e not
colhnear Without these assumptions ow theoiems are true, mth'crf, but Inoittl
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 coutaimng tins edge ?
also by a plane containing two or thiee such points?
2 Given a simple pentagon in a plane, construct another pentagon in the
same plane, whose veitices lie on the sides of the first anil whoso aides con-
tain the veitices of the fiist (cf p 51) Is the second uniquely determined
when the fiist and one side of the second are given?
3 If two sets of thiee points A, B, C and A', D', C" on two coplanar lines
I and I" lespectively are so related that the hues A A', /?', CC" are concurrent,
then the points of intersection of the paiis of lines AB' and DA', BC f and CB',
CA' and A C' are collmeai with the point IV. The line thus dotarmmod is called
the polar of the point (A A', BB") with respect to I and I'. [Duo-hue
4 Using the theorem of Ex. 3, give a construction for a hue joining any
given point in the plane of two lines I, I' to the point of iuterwaotion of I, I'
without making use of the latter point
5. Using the definition in Ex 8, show that if the point P f is on the polar p
of a point P with lespect to two lines /, /', then the point P ia on the polar/
of P f with, respect to /, V
6. If the vertices A v A 2 , A a , A 4 of & simple plane quadrangle are respec-
tively on the sides a v a a ,a s , a 4 of a simple plane quadrilateral, and if the inter-
section of the pair of opposite sides A l A a , A S A 4 is on the line joining the pair
of opposite points a^, a 2 n a , the remaining pair of opposite sides of the quad-
rangle will meet on the line pining the remaining pair of opposite vertices of
the quadrilateral. Dualize.
* A Cayley, Collected Works, Vol. I (1846), p. 817. 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 U005}
p. 684 ft\ n
20] EXEECISES 53
7. If two complete piano w-poiuts A v A v , A n and A^, A^, , A,' t arn
so related that the side A^i^ and tho u>uwimng 2 ( 2) sides passing through
A : and A z meet the coi responding sides of the other ?i-point in points of a lino /,
the remaining pairs of homologous sides of the two n-pomls meet on I and the
two n-pomts aie peispective fioiu a point. Dualize.
8. If five sides of a complete quadrangle A^^A^A^ pass thiough five
vertices of a complete quadnlatei al a^a^ m such a way that A^ s is on
a a a 4 , A Z A S on a 4 a lt etc., then, the sixth side of the quadrangle passes through
the sixth vertex of the quadnlateial. Dualize.
9. If on each of three ooncuri ent lines a, b, c two points are given, A j , A 3
on a; B lt B z on &; C v C z on c, there can be formed four pairs of triangles
A i B j C k (i,j, k = l, 2) and the pairs of corresponding sides meet in six points
which are the vertices of a complete quadnlateral (Veronese, Atti del Lincei,
1876-1877, p 649).
10 With nine points situated in sets of thiee on three concurrent lines
aie foimed 36 sets of thiee peispective 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,
giving a confirmation
36 3
(Veronese, loc. cit ).
4 27
11. A plane section of a 6-pomt in space can be considered as 3 triangles
perspective in pairs from 3 collmear points with coriesponding sides meeting
in 3 collmear points
12. A plane section of a 6-pomt in space can be considered as 2 perspective
complete quadi angles with corresponding sides meeting in the vertices of a
complete quadrilateral.
13 A plane section of an w-point m space gives the configuration *
n 2
ti^fi
which maybe considered (in n C n _ L ways) as a set of (n k) ^-points pei&pective
C M JL 2
in pairs from n -iPz points, which form a configuration "-- a ~ " and
the points of intersection of corresponding sides form a configuration
14. A plane section of a 7-point in space can be considered (m 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 [0 ways)
as composed of five 11-pomts cyclically circumscribing each other.
16 A plane section of an n-point in space for n prime can be considered
(in [ 2 ways) as ~ simple n-points cyclically circumscribing each other.
A
* The symbol n C r is used to denote the number of combinations of n things
taken r at a time.
54 PROJECTION, SECTION, PERSPECTIVITY [CHAP II
17 A plane section of a 6-point in space gives (in six ways) a 5-point "whose
10 3
sides pass tluough the points of. a configuiation
3 10
18, A plane section of an n-pomt in space gives a complete (?? l)-point
/-f ., _ O
whose sides pass thiough the points of a configuiation "~ 1 a
3 n-1^3
* 19 The n-space section of an ?-point (m n + 2) 111 an (n + l)-space can be
considered in th e n-space as (m A*) ^-points (in m C m _ A ways) perspective in pan s
from the vertices of the n-space section of one (ni A)-pomt, the r-spaces of
the fc-point figures meet in (r l)-spaces (r = 1, 2, , n 1) which foim the
n-space section of a fc-point.
*20. The figure of two peispective (n + 1) -points in an n-space separates
(in n + 8 ways) into two dual figures, respectively an (n + 2)-point cucum-
scribing the figure of (n + 2) (n l)-spaces.
*21. The section by a 3-space of an n-pomt in 4-space is a configuration
n C 2 n-2 tt _5sC 2
3 ,A n-8 .
6 4 B C 4
The plane section of this configuration is
C n-8
22. Let there be three points on each of two concurrent lines Z lf l z . The
nine lines joining points of one set of three to points of the other determine
six triangles whose vertices are not on / a or J a , The point of intersection of l
and l z 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 intei section of each side of the given
triangles with a line joining a fixed point of the axis of peispectivity 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 coiresponding plane faces meet in the
lines, and the corresponding 8-space faces in the planes of a complete 5-plane
in a 3-space,
* #6. If two (n + l)-points in an n-space are perspective from a point,
their corresponding r-spaces meet in (r l)-spaces which lie in the same
(n l)-space (r = l, 2 , 1) and form a complete configuration of
(n + 1) (n 2)-spaces in (n l)-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 m 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 acme 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, wo 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 cases 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 m space each as a single form. We have fol-
lowed the usage of Enriques, Vorlesungen iiber Projektive Geometrle.
22] PEESPECTIVITY 57
central with center P; if the third form is a pencil of points or ft pencil
of planes with axis I, the perspectivity is said to bo a^nal with axis /,
As examples of tins definition we mention the following: Two
pencils of points on skew lilies 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 arc peispective, if every two homologous pianos
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; i.e. 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 [<?].
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 [$], that each Q is on a line r of the flat pencil j>], and
that the pencil |>] is perspective by the axis a with the flat pencil [s].
58 PRIMITIVE GEOMETRIC FORMS [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: = ABC D
A
implies that 1 and A, 2 and 3, 3 and C, 4 and Z> are homologous
DEFINITION.* Two one-dimensional primitive forms [or] and [a- 1 ] (of
the same or different kinds) are said to be protective, provided there
exists a sequence of forms [r], |V], >, [T (B) ] such that
Co-1 __ r T -i _ r r n _ . _ rynri _ r n
1 J A L J A L J A A L J A L J *
The correspondence thus established between [a-] and |V] is called
a protective correspondence or projectivity, or also a protective trans-
formation. Any element a- is said to be projected into its homologous
element or 1 by the sequence of perspectivities.
Thus a projectivity is the resultant of a sequence of perspectivities
It is evident that [cr] and |V ] 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 JB, A, D, 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 cr of a
form homologous with an element a- 1 of another or the same form,
we will sometimes denote this by the relation 7r(o-) = <r'. In this
case we may say the projectivity transforms a- into cr'. Here the
symbol I JT( ) is used as a functional symbol f acting on the variable^
tr, which represents any one of the elements of a given form.
* This is Poncelet'0 definition of a projectivity.
t Just like F(x), sin (a), log(), etc.
t The definition of variable is " a symbol x which represents any one of a class
of elements [*] " It is m this sense that we speak of " a variable point."
23]
PEOJECTIVITY
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', J5', C'
three points of another line I', then A can le projected into A f , B into
B', and C into C' ly means of two centers of perspectimty, (The lines
may be in the same or in different planes.) (A, E)
Proof. If the points in any one of the pairs AA', BB', or CO' 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 S be any point
of the line AA', distinct
from A and A r (fig 22).
Prom S project A, B, C
on any line I" distinct
from I and I', but con-
taining A' and a point
of I If B", C" are the
points of I" correspond-
ing to B, C respectively,
the point of intersection S' of the lines B'B" and C'C" is the second
center of perspectivity. This argument holds without modification,
if one of the points A, B, C comcides with one of the points A 1 , B' t C'
other than its corresponding point.
COROLLARY 1. IfAfB^ and A', B', G r are on the same line, three
centers of perspectivity are sufficient to project A, B, into A', B 1 , C f
respectively, (A, E)
COROLLARY 2. Any three distinct elements of a one-dimensional
primitive form are projectile 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 GEOMETRIC FORMS [CHA*. m
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 projectimty ABCD-^BADC holds for any four
distinct points A, B, C, D of a line. (A, E)
Proof. From a point S, not on the line I = AB, project ABCD into
AB'C'D 1 on a line V through A and distinct from I (fig 23). From D
project AB'C'D' on the line SB. The last four points will then project
into BADO by means of the center C'. In fig. 23 we have
S D C'
AB CD = AB' C'D' = BB' C"S = BAD C.
A A A
It is to be noted that a geometrical ordei 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 o the
present assumptions alone The theo-
rem meiely states that the corievpond-
ence obtained by interchanging any two
of four colhnear 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; if the pairs are AB, CD) this is denoted by
T(AB, CD) Two throws are said to be equal, provided they are
protective ; in symbols, T (AB, CD] T (A'B', C'D'), provided we have
ABCDj;A'B'C'D'.
The last theorem then states the equality of throws ;
T(AB, CD) = T(BA, JDC)T(CZ>, AB) = J(DC, BA).
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 protective transformations which will effect any
-one of the six permutations of these three elements.
23] PROJECTIVITY 61
THEOREM 2'. If 1, @, 8, 4- an any four distinct elements of a one-
dimensional primitive form, there exist protective transformations
which will transform 1234 into any one of the following permuta-
tions of itself: 1234, 2143, 3412, 4321.
A protective 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 restilt that any project/we
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 1 ], [P"] are pencils of points on three distinct'
S , 8 r
concurrent lines I, I', I" respectively, such that [P] == [P'} and [JP '] ==
S"
[P"], then likewise [P] = [P"], and the three centers of perspectivity
S, S' S" are collinear. (A, E)
S"
Proof. Let be the common point of the lines l } V, I". If P i} P z , P s
are three points of [P], and P^'Ps' and P^'P^P^" the corresponding
points of [JP'], [P' r ] (fig. 24), it is clear that the triangles -Z^'-ZJ",
P^P^PX", P S P 8 'P 8 " are perspective from 0* By Desargues's theorem
(Theorem 1, Chap II) homologous sides of any pair of these three
triangles meet in collinear points. The conclusion 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.
62 PRIMITIVE GEOMETRIC FORMS [CHAP.III
COROLLARY. If n concurrent 'lines l lt l z) l s , , l n are connected ly
perspectivities [JJ] =i? [J>] = [J] =t ^-' ft], and */ ^ and l n
are distinct lines, then we have [J%] = [j] (A, E)
JProo/. This follows almost immediately from tlie theorem, except
when it happens that a set of four successive lines of the set ljl a l t > . 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 l v l s , l on the hypothesis that ^= l v l 2 l t
(fig. 25.) Let l v l z meet in 0, and let the line S 12 S 2a meet l t in A
A l = A, B, B 3 C, a
and l z in A z \ let A^A^ and A be the corresponding points of l s and
l t respectively, Further, let S v B z , 3^ B and C v C v C v C be any
other two sequences of corresponding points in the perspectivities.
Let SH be determined as the intersection of the lines A^A^ and B^B V
The two quadrangles S 1Z S^B Z C Z and S^S^S^ have five pairs of
homologous sides meeting l^l t in the points OA^B z Q y Hence
the' side S u C t meets ^ in O l (Theorem 3, Chap. II).
THEOREM 4. If [JJ, [P z ], [P] are pencils of points on distinct
a
Imes l v l v I respectively, such that [J] = [P] == [J], and if [P 1 ] is
the pencil of points on any line V containing the intersection of l v I
and also a point of l s) "but not containing $,, then there esoists a point
81 on S&, such that [P,] [P'] [jg. (A, E)
23] PEOJECTIVITY 63
Proof. Clearly we have
K] = F] = [^fK].
But by the preceding theorem and the conditions on V we have
SI
] [ p/ l> where S[ is a point of 8^ Hence we have
81 8.
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 skew
lines is equivalent to (and may be replaced by] two central perspectivities
(A,E)
Tor let [P], [P 1 ] 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 S[P], S'[P'"\ * are so related that pairs of homol-
ogous lines intersect in points of the line common to the planes of the
two pencils S[P] and S'[P']> since each pair of homologous lines lie,
by hypothesis, in a plane of the axial pencil s[P]=s[P 7 ].
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 pro jectivity 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 protective, there esdsts a pencil of points [Q] and two points S, S r
S S 1
such that we have [P] = [Q] = [P']. (A, E)
Proof, By hypothesis and the two preceding lemmas we have a
sequence of perspectivities
# #, 8. &
* Given a class of elements [P]; the symbol S[P] is used to denote the class
of elements SP determined by a given element 8 and any element of [P]. Hence,
if [P] is a pencil of points and S a point not in [P], 8 [P] is a pencil of lines -with
center S ; if s is a line not on any P, 8 [P] is a pencil of planes 'with aaas 8.
64 PEIMITIVE GEOMETEIC FOEMS [CHAP, m
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 l t l v l v l v , V of the pencils
[PI [PI, [Pa] , K] , [-?'] are coincident ; for Theorem 4 may evidently
be used to replace any l k (= l t ) by a line l' k (& ty. Now let l[ be the
line joining the points # t and lj a , and let us suppose that it does not
contain the center 8 t (fig 26). If then [P^] is the pencil of points
on l[, we may (by Theorem 4) replace the given sequence of per-
% , % Sir 8 *
spectivities by [P] = [P/] = [P^\ == [J?] ^ and this sequence
may in turn be replaced by
I, tfj $ 8 t
(Theorem 3). If S s is on the line
joining Hi an d lJa> we mav replace
^ by any line % through the inter-
section of ljl a which meets I and
p JG 2 Q does not contain the point # x (The-
orem 4). The line joining Z 2 8 to
l$l does not contain the pomt S" which replaces 8 y For, since /S 2 is
on the line joining 8 Z 2 to ll v the points l a l z and 11^ are homologous
points of the pencils [P 8 ] and [P] ; and if S'J were on the line join-
ing l a l z to UC, the point l a l z would also be homologous to tt('. 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
8 S'
As a consequence we have the important theorem :
THEOREM 6. Any two protective pencils of points on skew lines are
axially perspective (A, E)
Proof, The axis of the perspectivity is the line 88' of the last
theorem.
24. General theory of correspondence. Symbolic treatment. In
preparation for a more detailed study of protective (and other) corre-
spondences, we will now develop certain general ideas applicable to
24] 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
Sj by the element homologous to it in a correspondence A, the sys-
tem S x is transformed into a system S 2 , we express this by the relation
A(S 1 ) = S 2 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(Sj)= S 2 and B(S 2 ) = S 3 , the single corre-
spondence r which transforms S t into S s is called the resultant or
product of A by B; in symbols S 8 = B (S 2 ) = B (A(S 1 )) ** BA (S^, or,
more briefly, BA = T Similarly, for a succession of more than two
correspondences.
II Two successions of correspondences A m A m-1 Aj and
"BqBq-i ' Bj_ have the same resultant, or their products are equal,
provided they transform S into the same S'; in symbols, from the
relation
foUows A.A..! A 1 -B.B a _ 1 ...B
1 .
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 t into S 2 , the corre-
spondence which transforms S 3 into Sj is called the inverse of A and is
represented by A" 1 , i.e. if we have A (SJ = 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
* In this section, we have followed to a considerable extent the treatment given
by H. Wiener, ^enohte der K sachsischen Gesellschaft der Wisspnschaften, Leipzig,
Vol. XLII (1890), pp. 249-252. ........
66 PRIMITIVE GEOMETEIG FORMS [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 X )=S 2 , B(S a )=S 8 , F(S 8 )=S 4 follows at once BA(S 1 )=S S ,
whence F(BA) (SJ =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" 1 **! (of. 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^A = 1, BA- 1 = 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 AAA = A". 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, I,
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 field by Independent Postulates, Transactions of the
American Mathematical Society, Vol. VI (1005), p. 109.
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 t of acting with the operation o on the pair in the order
given * is a uniquely determined element of G.
G 2. The relation (aob)oc = ao(boc) holds for any three (equal or
distinct} elements a, o, c of G.
G 3 There occurs in G an element i, such that the relation aoi = a
holds for every element a of G
G4. For every element a in G there exists an element a' satisfying
the relation aoa'= i.
From the above set of postulates follow, as theorems, the following
The relations aoa'=i and a o i = a imply respectively the relations
a'oa = 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' = i is called an inverse element of a.
There is only one identity element in G.
For every element a of G 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 alelian} :
G 5. The relation aoo = 'boa is satisfied for every pair of ele-
ments a, & 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 a o & and 6 o a are not necessarily identical. The operation o simply defines
a correspondence, whereby to every pair of elements o, b in G m a given order corre-
sponds a unique element; this element is denoted by ao&.
68 PRIMITIVE GEOMETRIC FORMS [CHAP. Ill
an element which is transformed into itself by A is said to be an
invanant 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 :
TJie set of all correspondences in a group G which leave 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
denned 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
THEOEEM 7 The inverse of any projectivity and the resultant of
any two projectimhes 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 m 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
protective 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 1') 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] GEOTJP OF PEOJECTIVITIES C9
EXAMPLE 1. A group of projectivities leaving each of two given
points invariant Let M, 2f be two distinct points on a line I, and
let m, n be any two lines through M, N respectively and coplanar
with I (fig. 27). On m let there be an arbitrary given point S. If S t
is any other point on m and not on I or n, the points S, ^ together
with, the line n define a projectivity TT I on I as follows : The point
7r l (A) = A' homologous to any point A of Hs obtained by the two
o a
perspectivities [A] == [A^\ ~= [A 1 ], where [Aj] is the pencil of points
on n Every point $, then, if not on I or n, defines a unique pro-
jectivity TTJ we are to show that the set of all these projectivities ?r t
forms a group. We show first that the product
of any two TT I} ir z is a uniquely determined pro-
jectivity 7r s of the set (fig. 27).
In the figure, A 1 = ir^ (A)
and A"= ir z (A'} have been
s ^
n '"""
constiucted. The point S 8 giving A" directly from A by a similar con-
struction is then uniquely determined as the intersection of the lines
A" Ay m. Let B be any other point of I distinct from M, N, and let
B'= ir^JB) and B"~ ir t (jB') be constructed ; we must show that we have
J?"=7r 8 (JJ). We recognize the quadrangular set Q(MS'A' t NA"") as
defined by the quadrangle SS^A^ But of this quadrangular set all
points except B" are also obtained from the quadrangle S^B^A^
whence the line 8^ determines the point B" (Theorem 3, Chap II)
It is necessary further to show that the inverse of any projectivity in
the set is ia the set Tor this purpose we need simply determine 2
as the intersection of the line AA S 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. Ill
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'
and Q, Q' 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 pro]ectivity which
will transform any point A of I into any
other point B of I, provided only that
A and B are distinct from
Jf and N By virtue of
the latter property the &
group is
to be transitive.
EXAMPLE 2. Commutative projectivities. Let M be a point of a
line I, and let m, m' be any two lines through M distinct from I, but
in the same plane with I (fig. 28.) Let S be a given point of m, and
let a projectivity ^ be defined by another pomt S l of m which deter-
o a
mines the perspectivities [A] = [A^] == [A'], where [Aj] is the pencil
of points on m'. Any two projectimties defined m, this way by points $ {
are commutative Let 7r 2 be another such projectivity, and construct
the points A'ir^A), A 1 ' ir z (A'}, and A[**7r z (A). The quadrangle
SS^A^ gives Q(MAA' t MA"A[}\ and the quadrangular set determined
on I by the quadrangle SS^A'^ has the first five points of the former
m the same positions in the symbols. Hence we have ir^Af) = A", and
therefore TT^ = Tr,pr v
EXERCISES
1. Show that the set of all projectivities v t of Example 2 above forms a
group, which is then a commutative group.
2. Show that the projeotivity ^ 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-DIMENSIONAL PKOJECTIVITIES 71
pair (distinct fiom M 01 JV") in common Investigate the general question as
to how far the consti notion may be modified so as still to preserve the propo-
sition that the projectivitaes aie determined by the double points M, N 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 piojectivities of Example. 1.
Why does it fail to show that any two of the lattei aie commutative? State
the space and plane duals of the two examples.
5 ABCD is a tetrahedron and a, /3, y, 8 the faces not containing A,B,C,D
respectively, and / is any line not meeting an edge. The planes (I A, IB, 1C, ID)
are protective with the points (la, I ft, ly, IS).
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 piojectivities among these
ten sets of five points each
28. Projective transformations of two-dimensional forms.
DEFINITION. A protective 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 protective one-dimensional form of the other.
DEFINITION. A collineation is any (1, 1) correspondence between
two two-dimensional or 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. protective colhne-
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
planes TT, TT' 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 & collineation must be projective will appear later.
72 PRIMITIVE GEOMETEIC FOBMS [CHAP m
i.e. which transforms every element of a plane into an element of the
same plane, is the following :
DEFINITION. A perspective collineation, in a plane is a projective
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 TT X be any plane through o distinct from -TV.
Let O v O z be any two points collinear with and in neither of the
planes TT, 7r r The perspective collineation is then obtained by the
two perspectivities [P] == [PJ == [P r ], where P is any point of TT and
P v P' are points of ir^ 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 projective. 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 Q,
THEOREM 9. A perspective col~
lineation in a plane is uniquely
defined if the center, axis, and any
two homologous points (not on the
anois or center) are given, with the
single restriction that the homol-
ogous points must be 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' be the given pair of homologous points collinear with 0. The
28] TWO-DIMENSIONAL PKOJECTIVITIES 73
point }S r homologous to any point B of the plane is then determined.
We may assume B to be distinct from 0, A and not to be on o,
B 1 is on the line OB, and if the line AB meets o in C, then, since G
is invariant by definition, the line AB = A C is transformed into A' C.
B' is then determined as the intersection of the lines OB and A'C.
This applies unless B is on the line AA'; in this case we determine
as above a pair of homologous points not on AA', and then use the
two points thus determined to construct B'. 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 6> x be any point on neither of the planes TT, TT X If the line AO^
meets TT X in A v the line A'A\ meets 00 t in a point O z . The perspec-
tive colbneation determined by the two centers of perspectivity O v O a
and the plane TT X then has 0, o as center and axis respectively and A, A'
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 aoois 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
leing the center of the collineation and the axis 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 projective.
COROLLARY 4 If H is a perspective collineation such that H(0) = C>,
H(0) = o } H(.4) =tA r , H(5) = B' where A, A', B, B 1 are collinear with
a point JT of o, then we have Q(OAB, KB' A'}. (A, E)
Proof. If is any point not on AA' and H() = C", the lines AC
and A'C' meet in a point L of o, and BC and B'C 1 meet in a point M
of o ; and the required quadrangle is CC'LM (of fig. 32, p. 77),
74
PRIMITIVE GEOMETRIC FOEMS
[CHAP. Ill
THEOREM 10. Any complete quadrangle of a plane can be trans-
formed into any complete quadrangle of the same or a different plane
by a projective colhneation which, if the quadrangles are in the same
plane, is the resultant of a finite number of perspective collineatwns.
(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' respectively. We show first that
there exists a collineation leaving any three vertices, say A' } B', C', of
Fro. 30
the quadrangle A'B'C'D 1 invariant and transforming into the fourth,
D', any other point Z> 8 not on a side of the triangle A'B'C'fig. 30). Let
jD be the intersection of A'D &) B'D' and consider the homology with
center A 1 and axis B'C' transforming 2) a into 35. Next consider the
homology with center B' and axis C'A' transforming U into D', Both
these homologies exist by Theorem 9. The resultant of these two
homologies is a oollineation leaving fixed A', B', C' and transforming
Z> 8 into D 1 . (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 f and let <? t be
any line not passing through A or A'. By Theorem 9 there exists a
28,29] THBBE-DIMENSIONAL PKOJECT1VITIES 75
perspective collmeation ir l transforming A to A' and having and o l
as center and axis. Let B v C v J) : be points such that
In like manner, let o z be any line through A' not containing B or
B' and let 2 be any point on the line B^B*. Let 7r a be the perspec-
tive collmeation with axis o 3 , center a , and transforming J^ to B'.
Let ,= TT^) and a = TT,^) Here
Now let O a be any point on the line C Z C' and let ir a be the per-
spective collmeation which has A'B'o a for axis, O a for center, and
transforms C t to C". The existence of 7r a follows from Theorem. 9 as
soon as we observe that C' is not on the line A'B', by hypothesis,
and C a is not on A'JB'; because if so, C l would be on A'B^ and there-
fore C would be on AB. Let 7r 3 (Z> a ) = Z> 8 . It follows that
The point D a camiot be on a side of the triangle A'B'C' because
then JD Z would be on a side of A'B'C Z> and hence JD^ on a side of
A'B^CV 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
= A'JB'C'JD'.
The resultant Tr^TTg-Tr.^ of these four collmeations clearly transforms
A, B, C, D into A 1 , B', C", D' respectively If the quadrangles are in
different planes, we need only add a perspective transformation between
the two planes.
COROLLAEY There exist protective collinoations in a, plane which
will effect any one of the possible @4> permutations of the vertices of
a complete quadrangle in the plane. (A, E)
29. Protective collmeations of three-dimensional forms. Protective
collmeations in a three-dimensional form have been defined at the
beginning of 28.
DEFINITION. A protective collineation in space which leaves inva-
riant every point of a plane CD and every plane on a point is called a
perspective collineation. The plane e is called the plane ofperspectivity;
the point is called the center. If is on w, the collineation is said
to be an elation in space ; otherwise, a homology in space.
Y6 PEIMITIVE G-EOMETEIO FOBMS [CHAP, in
THEOREM 11. IfOis any point and o> any plane, there, exists one
and only one perspective collineation in space having O, co jfor center
and plane of perspe^mty respectively, which transforms cct^y point A
(distinct from and not on to) into any other point A' (distinct from
and not on o>) cottinear with AO. (A, E)
Proof. "We show first that there cannot be more than, one per-
spective coUineation satisfying the conditions of tlie theorem, by
showing that the point B 1 homologous to any point JS is iiniquely
Pro 31
determined by the given conditions. We may assume IS not on a>
and distinct from and A. Suppose first that B is not cm the lino
AO (fig. 31). Since BO is an invariant line, J?' is on J3O; and if
the line AB meets o> in L, the line A B AL is transfer JUG J into
the line A f L. Hence B' is determined as the intersection of JiO
and A'L. There remains the case where B is on A O aixd distinct
from A and (fig 32). Let C, C' be any pair of homologous points
not on AO, and let AC and BC meet to in L and M respectively.
The line MB - MO is transformed into M C 1 , and the point l? r is then
determined as the intersection of the lines BO and MC f . That this
point is independent of the choice of the pair O t C 1 now follows
from the fact that the quadrangle MLCC' gives the quadrangular
set Q(KAA ( , OB'B), where JT is the point in which AO meets a>
(Xmay coincide with without affecting the argument). The point
B' is then uniquely determined by the five points O, JC, A, A' t B.
The correspondence defined by the construction in the paragraph
above has been proved to be one-to-one throughout. On tb.e line AO
it is projective because of the perspectivities (fig. 32)
29] THEEE-DIMENSIONAL PKOJECTIVITIES 77
On OJB, any other line through 0, it is proactive because of the per-
spectivities (fig. 31) . ,,
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 Erom this it follows
K
PIG 32
that any one-dimensional form is transformed into a protective form,
so that the correspondence which has been constructed satisfies the
definition of a projective collineation.
TIIEOKEM 12. Any complete five-point in space can be transformed
into any other complete five-point in space by a projective collineation
which is the resultant of a finite number of perspective collineations. (A,E)
Proof. Let the five-points be ABODE and A'B'C'D'E' respectively.
We will show first that there exists a collineation leaving A'JB'C'D'
invariant and transforming into E 1 any point J$ not coplanar with
three of the points A'B'C'D'. Consider a homology having A'B'C' as
plane of perspectivity and D f as center. Any such homology trans-
forms E into a point on the line JE Q D' Similarly, a homology with
plane A'B'D' and center C' transforms IS' into a point on the line E'C'.
If H D' and E'0 ! intersect in a point JS V the resultant of two honiol-
ogies of the kind described, of which the first transforms E Q into JS? t
and the second transforms E^ into M' } leaves A'B'C'D 1 invariant and
transforms J into E'. If the lines JS D' and E'C' are skew, there
is a line through ' meeting the lines J^-D' and E'C' respectively
78 PEIMITIVE GEOMETRIC FOEMS [CHAP, in
in two points JS t and IS 2 . The resultant of tlie three homologies, of
which the first has the plane A'B'C' and center D' and transforms
JS to JB V of which the second has the plane A'C'D' and center B'
and transforms J& t to E v and of which the tliird has the plane A'B'D 1
and center C' and transforms E z to JS', is a collineation leaving A'B'C'D'
invariant and transforming JS to E'. 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 protective collineations which will effect
any one of the possible 120 permutations of the vertices of a complete
Jive-point in space. (A, E)
EXERCISES
1 Prove the existence of perspective colhneations in a plane without
making use of any points outside the plane
2 Discuss the figure formed by two triangles which aio homologous
undei an elation. How is this special form of the Desarguos 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 O, show that wo 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. Wnte 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 comrmitalave ?
8. Prove that two elationa in a piano 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 j (g) any plane; (8) any bundle of lines; (4) * tetrahedron} (5) a
complete i-ve^oint in space.
11. The set of all collineations in space (in a plane) form a group.
12. The set of all protective collineations in space (in a plane) form a group,
, 13 Show that under certain conditions the configuration of two p wp&otive
tetrahedra is left invariant by 120 oollmeations (of, Ex. 8, p. 4(7), >
CHAPTER IV
HARMONIC CONSTRUCTIONS AND THE FUNDAMENTAL THEOREM
OF PROJECTIVE 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 011
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.
The 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 hues in question (fig. 33). Let the remaining three
sides of such a quadrilateral be a, 6, c. Let the points 5c, cat, and ab
* 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.
79
80 THE FUNDAMENTAL THEOREM [CHAP. IV
be denoted by A, J?, and C respectively. The sides of the quadrangle
PABCmGQt I in the same points as the lines of the quadrangular set
of lines.
COROLLARY. A set of collinear points which is protective with a
quadrangular sot is a quadrangular set (A, E)
THEOREM 1'. 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 1". The section "by 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 projective with a quadrangular set, it is itself a quadrangular set.
(A,E)
31. Harmonic sets. DEFINITION. A quadrangular set Q (12 3, 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 harmomc conjugate of 4 (or 3) with respect
to 1 and 2.
From this definition we see that in a harmonic set of points
\\(AC, JBJD), the points A and are diagonal points of a complete
FIG. 84 FIG. 85
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] HARMONIC 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 G (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 collmear, any
three points of a line foim a harmonic set and any point is its own harmonic
conjugate with regaid to any two points collmear with it Theorems on hai-
monic sets aie therefoie tuvial in those spaces for which Assumption Jf is
not tine. We shall therefoie base oiu leasonmg, in this and the following
two sections, on Assumption If 0) though most of the theorems are obviously
true also in case H is false. This is why some of the theoiems aie labeled as
dependent on Assumptions A and E, whereas the proofs given involve H also.
The corollary of Theorem 3, Chap. II, when applied to harmonic
sets yields the following:
THEOREM 2 The harmonic conjugate of an element with respect to
two other 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 projective 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 III, 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 role and may be interchanged,*
* The corresponding theorem for the more general expression Q (128, 456)
cannot be derived without the use of an additional assumption (cf. Theorem 24,
Chap. IV).
82 THE FUNDAMENTAL THEOREM [CHAF.IV
THEOREM 5. Given two harmonic sets H (12, 34) and H (1'2', 3'4'),
there exists a projectimty such that 1234 -^ 1'2'3'4'. (A, K)
Bw/. 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). Tliis is the converse of Theorem 3, Cor.
COROLLARY 1. If H (12, 34) and H (12', 3'4') arc two harmonic sets
of different one-dimensional forms having the element 1 <iu common,
we toe 1234 =12'3'4'. (A, E)
For under the hypotheses of the corollary the pro] ectivity 1 2 3 -^ 1'2 '3'
of the preceding proof may be replaced by the perspectivity 123 12'3'.
COROLLARY 2 If H (12, 34) 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 (cf. Theorem 27, Cor. 5)
We see as a result of the last corollary and Theorem 2, Chap. ITI,
that if we have H (12, 34), there exiRt 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), wo have likewise H (12, 43),
H(21, 34), H(21, 43), H(34, 12), H(34, 2]),H(43, 12), H (43, 21).
THEOREM 6. The two sides of a complete quadrangrle wfiieh meet in
a diagonal point are harmonic conjugates witJb 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 tho
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 jEorm 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 eight-group*
31]
HARMONIC SETS
83
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 hues is
harmonic with a pair of planes, if p IG 30
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 Piove Theorem 4 dhectly fiom a figure without using Theorem 2,
Chap. III.
2. Prove Theorem 5, Coi. 2, directly fiom a nguie.
3 Through a given point in a plane constiuct a line which passes through
the point of intersection of two given lines in the plane, without making use
of the lattei point.
4. A line meets the sides of a triangle ABC in the points A v B I} C l} and
the harmonic conjugates A z , B z , C z of these points with respect to the two
vertices on the same side are deteimmed, so that we have W(AB, C^C^),
H(pC,A l A t '),ainiaLH(CA,B 1 B t ). Show that A v B z , C a , B v C v A 9 \C lt A st B 9
are collinear; that AA Z , BB Z , CC % are concurrent; and that AA S , BB V CC V
AA l} BB Z , CC^ AA V BB V CC Z aie also concurrent.
5. If each of two sides AB, BC of a triangle ABC meets a pair of opposite
edges of a tetrahedron in two points which are harmonic conjugates with
respect to A, 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 v P v P 8 , 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 FUNDAMENTAL THEOREM [CHAP IV
quadrangle is constructed and these six points are denoted by P{, Pg, Pg, P^,
Pj, Pg respectively The thiee lines joining the pairs of the latter points
which lie on opposite sides of the quadi angle meet m a point, P, which in the
hanuomc conjugate of each of the points in which these thioe lines meet /
with icspect to the pans of points P' defining the lines.
7 Defining the polai line of a point with itjspoet to a pan of hues as the
haimomc conjugate line of the point with i eg aid to the pan of lines, piove
that the three polar lines of a point as to the pairs of lines of a tuangle foim
a triangle (called the cogredient tiiangle) peispective to the given tuanglo
8. Show that the polar line defined in Ex 7 is the same as the polai line
denned m Ex. 3, p. 02.
9. Show that any line through a point and meeting two intersecting
lines I, I' meets the polai of with respect to I, I' in a point which is the
liairnonic conjugate of with lespect to the points in which the line through
meets Z, I'
10 The axis of perspectivity of a triangle and its cogredient triangle is the
polar line (cf . p 4G) of the triangle as to the given point.
11 If two triangles arc peispective, the two polai lines of a point on then
axis of perspectivity meet on the axis of perspectivity
112 If the lines joining corresponding vertices of two n-hnes meet in a point,
the points of intersection of coiiespondmg 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)-lmes of an n-hne in a plane form an n-hne (the cogredient n-line)
whose sides meet the coriesponding sides of the given n-lme in the points of
a line p. The line p is called the polar of P as to the n-line *
14. (Geneialization of Ex. 11.) If two n-lines are perspective, the two
polar linos 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 n-dimensional space. These theorems are fundamental for the con-
struction of polars of algebuue curves and suifaces of the n-tlx degree.
32. Nets of rationality on a line. DEFINITION A point P of a line
is said to be harmonically related to three given distinct points A, ft, C
of the line, provided P is one of a sequence of points A, S, 0, B v // 2 , JG",,
of the line, finite in number, such that H^ is the harmonic conju-
gate of one of the points A, J3, C with respect to the other two, and
such that every other point H t is harmonic with three of the set A, B, 0,
JF V JST 2 , . -, J^_ v The class of all points harmonically related to three
distinct points A, J3, C on a line is called the one-dimensional net oj
rationality defined by A> #, G\ it is denoted by R(JLBC) t A net oj
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 ax
n-line.
32] 3STETS OF RATIONALITY 85
THEOREM 7. If A, B, 0, D and A', B', C', >' are respectively points
of two lines such that ABCD-rA'B'C'D', and ifD is harmonically
related to A, B, C, then D' is harmonically related to A' } B' } C'. (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, projective with a set of points J/J such that every harmonic set
of points of the sequence A, B, C, H v H 2 , , D is homologous with a
harmonic set of the sequence .4', B', C', H[, H' Z) -,!>' (Theorem 3, Cor ).
COROLLARY. If a class of points on a line is protective with a net
of rationality on a line, it is itself a net of rationality.
THEOREM 8. If K, L, M are three distinct points of R (AB C), A, B, C
are points of R (KLM ) (A, E)
Proof. From the projectivity ABGK-^ BAKC follows, by Theorem?,
that C is a point of R (ABK). 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)
= R (AMK) = R (KLM).
COROLLARY. A net of rationality on a line is determined by any
distinct three of its points.
THEOREM 9. If all lut 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^\ J5,jB 1? ' C, C v so that B 3= B^\
and suppose that the first five of these are points of a net of rationality
We must prove that C^ is a point of R. Let the pair of lines US and
PQ meet in B'. We then have
8
Since A is in R(BGB l ) > it follows from this projectivity, in view of
Theorem 7, that C^ is in R (BA&) *= R.
DEFINITION. A point P of a line is said to be guadrangularly
related to three given distinct points A } B, C of the line, provided
86 THE FUNDAMENTAL THEOREM [CHAP. IV
P is one of a sequence of points A, , C, ff v /Z" a , J/ 8 , . . of the line,
finite in number, such that 11^ is the harmonic conjugate of one o
the points A, B, C with respect to the other two, and such that overy
other point S t is one of a quadrangular set of which the other tivo
belong to the set A, JB, C, H v // 2 , , #",_!
PIG- 37
COROLLARY The class of all points quadrangularly related to three
distinct collinear points A, B, C is R (AJiC). (A, E)
From the last corollaiy it is plain that R (ABC) consists of all points that
can "be constructed from A , J3, C by means of points and linos alone ; that
to say, all points whose existence can be infeired from Assumptions A, E, II,,,
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 nil
points constructible, by means o points and lines, out of four points A , 72, ( ', /),
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 u R B
having a point in common, provided it is the intersection of two lines
each of which joins a point of R t to a distinct point of R a . A line is
said to be rationally related to R! and R 8 , provided ifc joins two points
that are rationally related to them. The set of ajl points and lines
rationally related to R v R fl is called the net &f rUonM$ fy a $law
(or of two dimensions) determined by R v f^j, it is alii oSbUM the
planar net denned by R w R a .
Prom this definition it follows directly that all the points of R x
and R a are points of the pknax net denned by R R, ,
33] NETS OF RATIONALITY 87
THEOREM 10. Any line of the planar net R 2 defined ly R v R 2 meets
R,andR z . (A, E)
Proof We prove first thai il a line of the planar net R a meets R v
it meets R 2 . Suppose a line I meets R x in Aj it then contains a second
point P of Rl By definition, through P pass two lines, each of which
joins a point of R 1 to a distinct point of R 2 . If li& one of these lines,
the proposition is proved ; if these lines are distinct from /, let them
meet Rj and R 2 respectively in the points B v B z and %, P z (fig 38).
If is the common point of R 19 R 2 , we then have
OA^P^OA^P,,
where A 2 is the point in which I meets the line of R 2 Hence A 3 is a
point of R 2 (Theorem 7).
Now let I be any line of the net R 2 , and let P, Q be two points
of the net and on I (del). 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 1 to a distinct point of R 2 , pass through
P; let them meet R x in A v B v and R 2 in A v B z respectively (fig 38).
Let the lines QA : and QB l meet R 2 in A[ and J? a ' respectively (first case).
O At A, B' t B t P t
Then if I meets the lines of R l and R 2 in J% and P% respectively, the
quadrangle PQA^B^ gives rise to the quadrangular set Q(^ 2J B a ,
QB'zA'J) 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)
gg
THE FUNDAMENTAL THEOREM
[CHAP. IV
B,
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 x , R 3 defining the planar net. In the latter
case let the two lines of the planar net be l v l a and suppose l a passes
through 0, while l^ meets R 1? R 2 in A v A z respectively (fig. 39). If the
point of intersection P of y, were not a point of the planar net, l a
would, by definition,
contain a point Q of
the planar net, dis-
tinct from O and P.
The lines QA l and
QA Z would meet R 3
and Rj in two points
J5 2 and 7> x respec-
tively. The point C s
m which the line
PB^ met the line of
89 R a would then be the
harmonic conjugate
of JB 2 with respect to and A z (through the quadrangle JPQA^);
<7 2 would therefore be a point of R a , and hence P would be a
point of the planar net, being the intersection of the lines A^A Z
and B^Cy
THEOREM 12. The points of a planar net R 3 on a line of t7ie planar
net form a linear net. (A, E)
Proof. Let the planar net be defined by the linear nets R y R a and
let I be any line of the planar net. Let P be any point of the planar
net not on I or Rj or R 2 . The lines joining P to the points of R 2 on I
meet R^ and R 2 by Theorems 10 and 11. Hence P is the center of
a perspectivity which makes the points of R 2 ou I perspective with
points of R x or R 2 . Hence the points of I belonging to the planar not
form a linear net. (Theorem 7, Cor.)
COBOLLAKY. The planar net R* defined "by two linear nets R p R 9 is
identical with the planar net R 2 a defined "by two linear nets R 8 , R 4 , pro-
vided R 8 , R 4 are linear nets in R x s . (A, E)
For every point of R t a is a point of R a 8 by the above theorem, and
every point of R a * is a point of R t a by Theorem 10.
33,34] NETS OF BATIONALITY 89
EXERCISE
If A, B, C, D are the veitices of a complete quadrangle, there is one and
only one planar net of rationality containing them ; and a point P boloiigw to
this net if and only if P is one of a sequence of points ABCD1)^D Z , iinile
in number, such that D i is the intei section of two sides of the original quad-
rangle and such that eveiy othei point D f is the intersection of two lines join-
ing pans of points of the set ABCDD^ 2? t -i-
34. Nets of rationality in space. DEFINITION. A point is said to
be rationally related to two planar nets R*, R 2 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|
A line is said to be rationally related to R 2 , R 2 2 , 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 2 2 is called the
net of rationality in space (or of three dimensions] determined by
R 2 , R 2 , it is also called the spatial net defined Tby 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 a 2 is a point of the spatial net R 3 defined by R/, R 2 2 (the
definition applies equally well to the points of the linear net common
to R 2 , R, 2 ) ; 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 2 to a distinct point P of
W meets R*. (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 . We may assume these
lines distinct from the line PA V since otherwise the lemma is proved.
Let the two lines through P meet Rf, R 2 3 in JB V B z and C v C z respec-
tively (fig. 40). If A v v C l are not collinear, the planes PA^B^ and
PA^CI meet R. 2 in the lines A l S l and A^C y respectively, which meet
the linear net common to R 2 , R a 2 in two points S, T respectively
(Theorems 11, 12). The same planes meet the plane of R 2 in the lines
ASB 2 and TO Z respectively, which are lines of R 2 2 , since $ T are points
of R|. These lines meet in a point A z of R a 2 (Theorem 11), which
is evidently the point in which the line PA^ meets the plane of R 2 a .
If A v B v C l are collinear, let A e be the intersection of PA V with the
90 THE FUNDAMENTAL THEOREM
plane of R 2 2 and 8 the intersection of A,B^ with the linear not
common to R and R| Since A, is in R (7^), the
implies thai, J a is in
in
THEOREM 13. Any tit.
RfandR*. (A, E)
o/ the spatial net R"
A
Rf, R.f
FIG. 41
Proof. By definition the given lipe I contains two points A and B
of the net R* (fig, 41)', It A or JS? is on R.? or R a 8 , the theorem raducea
to the lemma. If not, let J% he a point of R^, and A and J5 a the points
in whioh, by the lemma, %A und $B meet R a a j
NETS OF RATIONALITY
91
point of R 2 not in the plane P^AB, and let P^A and P^'B meet R 2 2 in A[
and J3' 2 . The lines A a B z and A'^Bl meet in a point of R 2 (Theorem 11),
and this point is the point of intersection of / with the plane of R 2 a .
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* and R 2 2 the planar nets defining
the spatial net R', and L^ and L 2 the points in which (Theorem 13)
I meets R x 2 and R| (L^ and L z may coincide). Let A i be any point of
Rf not on I or on R 2 2 , and S the point in which A^ meets the linear
net common to R 2 and R a 2 (fig. 42). If L : and L 2 are distinct, the lines
FIG. 42
FIG. 43
SLj_ and SL Z meet R 2 and R 2 2 in linear nets (Theorem 12); and, by
Theorem 13, a hue joining any point P of R 8 on I to A^ meets each
of these linear nets. Hence all points of R 8 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 i of the linear net on SL a
upon I are points of R 8 . Hence the points of R 8 on I are a linear net
If Li L^ S, then, by definition, there is on I a point A of R 8 , and
the line AA 1 meets R| in a point A z (fig. 43). The lines 8A^ and SA^
meet R and R a 2 in linear nets R t and R 2 by Theorem 12. If 3^ is
any point of R t other than A v the line AB meets R 2 2 in a point J3 8 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 8 .
Moreover, any point of R 8 on I, when joined to A v meets R 2 2 by Theo-
rem 13, and hence belongs to the planar net determined by Rj and R a .
Hence, in this case also, the points of R 8 on I constitute a linear net.
92 THE FUNDAMENTAL THEOREM [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 noncollmear points .4,1?, C of
R* and the three lines AB, BC, CA meet the planar nets R and R a 2 ,
which determine R 8 , in points of two linear nets R, and R 2 , consisting
entirely of points of R 8 . 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 8 . Moreover, any hue joining a point of R" in a to A
or B or C must, by Theorem 13, meet R x and R a and hence be in R fl .
Hence all points and lines of R 3 on a are points and lines of R a . Thw
completes the proof except in case R x = R,, which case is left as an
exercise.
COROLLARY 1. A net of rationality in space is a space satisfying
Assumptions A and E, */ "hne" le interpreted an "linear nd" and
"plane" as "planar net" (A, E)
Tor all assumptions A and E, except A3, 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 lino),
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,S, C,D,E are the vertices of a complete Bpace fivci-point, thoro 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 ARCDKJJi , finite in
number, such that J x is the point of intersection of three faoun of the original
five-point and every other pomt I t is the mlcmujction of three distinct pianos
through triples of points of the set ABCDEI^ / t _j..
2 Show that a planar net is determined if three noncollinear points anl a
line not passing through any of these points are 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 pomt in common.
34,36] THE FUNDAMENTAL THEOREM 93
4. Show that a set of points and lines which is piojective with a planar
net is a planar net
5 A line joining a point P of a planar net to any point not in the nut, 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 111 the same plane do not in geneial belong to
the same planar net
7 Discuss the determination of spatial nets by points and planes, similarly
to Exs. 2, 3, and G.
8 Any class of points piojective with a spatial net is itself a spatial net.
9 If a perspective colhneation (homology 01 elation) in a plane with
centei A and axis I leaves a net of rationality in the plane invariant, the
net contains A and I
10 Prove the conespondmg proposition for a net of rationality in space
invariant under a perspective tiansfonnation.
11 Show that two linear nets on skew lines always belong to some spatial
net, in fact, that the uunibei of spatial nets containing two given hnea"
nets on skew lines is the same as the numbei 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 111 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 III) 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 projective
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 THEOREM [CHAP. IV
then 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 ly these points,
This follows at once from the fact that if three points are left
invariant hy 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 FROJISCTIVITY FOR A
NET OF RATIONALITY ON A LINE. If A, B, C, D are distinct points of
a linear net of rationality, and A', B', 0' are any three distinct points
of another or the same linear net, then for any projcstimtias giving
AB CD - A'B'C'D' and AB CD -^ A'B'C'D[, we have D' = D[ (A, E)
Proof. If IT, TJ-! are respectively the two pro jectivities of the theorem,
the projectivity TTyrr" 1 leaves A'B'C' fixed and transforms D' into D[.
Since D' is harmonically related to A', B', C' (Theorem 7), the theorem
follows from the lemma.
This theorem gives the answer to the question proposed m its
relation to the transformation of the points of a linear net. The
corresponding proposition for all the points of a line, i.e. 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 act txtefe It has
* Different, of course, frpro ordinary space; "rational sipaoss" (ff/.p. 08 and
the next footnote) are examples in which continuity does not exist; " finite spaces, 1 *
of which examples are given in the introduction ( 2), are spaces in which neither
order nor continuity xist,
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
TIIEOEEM 17. THE FUNDAMENTAL THEOREM OF PROJECTIVE GEOM-
ETRY, f Ifl, 2, 3,4 are any four elements of a one-dimensional primitive
form, and 1', 2', 3' are any three elements of another or the same one-
dimensional primitive form, then for any projectivities giving 1234~j^
1>>3'4' and 1234 ^ l'%'3'4[, we h a 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 two pencils of points on different lines are projective
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 projective 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 coi re-
spond to the points of the line whose coSrdmates, when represented analytically, are
rational numbers. This consideration should make the last assumption almost, if
not quite, as intmtionally 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 EncyklopaMie articles on Projective Geometry and
Foundations of Geometry. It is associated with the names of von Staudt, Klein,
Zeuthen, Luroth, Darbous, P. Schur, Pieri, Wiener, Hilbert. Of. also 50, Chap. VI.
96 THE FUNDAMENTAL THEOREM [CHAP iv
Proof. For if is the self-corresponding point, and AA' and BB'
are any two pairs of homologous points distinct from 0, the perspec-
tivity whose center is the intersection of the lines AA', BB' is a
projectivity between the two lines which has the three pairs of
homologous points 00, A A', BB', 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 protective transformation which leaves invariant each
j; * j- f our j. j> a plane three . 7 . 7 , , 7
of a set of points of no , of which belong to the same
J J jive * J space four J J
line 7 . . , . , _, the plane . , ,-, T ,.
, leaves invariant every point of * (A, E. P)
plane a * J space. \ > > /
Proof. If A, B, C, D are four points of a plane no three of which
are collinear, a proactive transformation leaving each of them inva-
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 AH,
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, 7r t having the given
pairs of homologous points. The collineation ir^" 1 is then, by the
lemma, the identical collineation in one of the planes. This gives at
once TTjSs TT, contrary to the hypothesis.
* We confine the statement to the case of the oollineation for the s&ke of sim-
plicity of enunciation. Projeotive transformations which are not ColUmeations will
he discussed in detail later, at which time attention will he called explicitly to the
fundamental theorem,
35] THE FUNDAMENTAL THEOREM 97
By precisely similar reasoning we liave :
THEOEEM 19 A projectile colhneation in space is uniquely deter-
mined when five pairs of homologous points are given, provided no
four of either set of five points are in the same plane. (A, E, P)
The fundamental theorem deserves its name not only because so
large a part of protective geometry is logically connected with it, but
also because it is used explicitly in so many arguments. It is indeed
possible to announce a general 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 , l a pass through a point, find two
other lines m v m z such that the four points m^, m t l 2 , m 1 Z 8 , m 1 m z may
be shown to be protective with the four points m a l v m z l z) m z l s , m z m v
respectively. Then, since m this pro^ectivity the point m 1 m 2 is self-
corresponding, the three lines l v l z) 1 8 joining corresponding points
are concurrent (Theorem 17, Cor.) The dual of this method appears
when three points are to be shown colhnear. 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, i.e. 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 protective space. It will appear in Chap. YI that there
exist spaces in which this assumption is not valid. Such a space,
i.e. a space satisfying Assumptions A and E but not P, we will call
an improperly protective space
From Theorem 15, Cor. 1 and Lemma 1, we then have
THEOREM 20. A net of rationality in space is a properly protective
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).
98 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 projective
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, are any three distinct points of a line I,
and A', B', C' any three distinct points of another line I 1 maetinff I,
the three points of intersection of the pairs of hnes AB' and A'li, ISC'
and B'C, CA' and C'A
are collinear (A, E, P)
Proof. Let the three points of intersection referred to in the theorem
be denoted by C", A", B" respectively (fig. 44). Let the line Ji"C"
meet the line B' C in a point Z> (to be proved identical with A");
also let B"C" meet I 1 in A v the line A'B meet A C 1 in JB V the line AB'
meet A'C in B[. We then have the following perspectivities :
A'CP'^JB = A'B[B"C = A"B"D.
By the principle of projectivity then, since in the projectivity thus
established C" is self-corresponding, we conclude that the three lines
A^A 1 , "B V JOB meet in the point C', Hence D is identical with A",
and A", B", C" are collinear
It should be noted that the figure of the last theorem 'is a con-
figuration of the symbol
9 8
3 9
36] COJSTFIGUKATION OF PAPPUS , * 99
It is known as the configuration of Pappus.* It should also be notetlv I Q;
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 (of.
Chap. II, 14), the last theorem may be stated in the following form .
COROLLARY. If a simple hexagon be 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. //
A, t B z C z "be a triangle
inscribed in a triangle
A^BjCv 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^ [5] the pencil
with center 2^; and [c] the pencil with center C f 1 (fig. 45) Consider the
3.A Z J3 Z C,
perspectivities [a] === [&] === [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 Z A S ' for the lines
AjC a and CjO z are clearly homologous, as also the lines A^A Z and C^A Z .
Any three homologous lines of the perspective pencils [a], [b], [c] then
form a triangle which is circumscribed about -4 1 J? 1 C' 1 and inscribed
in A S S Z C Z .
* Pappus, of Alexandria, hyed about 840 A.I>. A special case of this theorem may
he proved without the use of the fundamental theorem (cf. Ex 3, p. 62).
t In this form it is a special case of Pascal' s theorem on conic sections
(cf. Theorem 8, Chftp, V).
100 THE FUNDAMENTAL THEOREM [CHAP IV
EXERCISES
1. Given a triangle ABC and two distinct points A', B' t determine a point C"
such that iheline$AA',BB', C6 v aie concimeiit, and also the lines AB',BC f , CA'
aie concurrent, i e. such that the two tuanglcs aie perspective from two dif-
feient points The two triangles aie then said to be doubly perspective
2 If two triangles ABC and A'&C' aie doubly perspective in such a way
that the vertices A, B, C are homologous with A', R', C' respectively in one
peispectivity and with B', C', A' respectively in the other, they will also be per-
spective fiom a thiid point m such away that./!, B, Care homologous respec-
tively with C", A', J3'; i e. they will be triply perspective.
3. Show that if A", B", C" aie the centers of perspectivity for the triangles
in Ex. 2, the three triangles ABC, A'B'C', A"B"C" aie so related that any two
are triply perspective, the centers of peispectivity being in each case the veitices
of the remaining tiiangle. The nine veitices of the three triangles form the
points of a configuration of Pappus.
4. Dualize Ex. 3.
37. Construction of projectivities on one-dimensional forms.
THEOREM 23. A necessary and sufficient condition for tlie projectimty
on a line MNAB -^ MNA< B' (M 1= N) is Q (JfAJB, NB'A'). (A, E, P)
Proof. Let n be any line on N not passing through A (fig. 46). Let 1
be any point not on n or on MA, and let A^ and JB l be the intersections
respectively of O^A and O^B with n. Let O z be the intersection of A'A l
andjB'5 r Then Q Q
NAB =1 NA.B. = NA'J* 1 .
A l * A
By Theorem 17 the pro j activity so determined on the line AM is the
same as MNAB
The only possible double points of the projectivity are N" and the
intersection of AN with O t 2 . Hence 0^ passes through M, and
Q (MAB, NB'A') is determined by the quadrangle OjO^AjBy
37] ONE-DIMENSIONAL PKOJECTIVITIES 101
Conversely, if Q(MAB, NB'A') we have a quadrangle O^A^B V
and lienoe ~
and by this construction Mia self-corresponding, so that
MNAB-j^MNA'B'.
If m the above construction we have M=N, we obtain a projeo-
tivity with the single double point M N
DEFINITION A projectivity on a one-dimensional primitive form
with a single double element is called parabolic. If the double ele-
ment is M, and AA', 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 MMAB-^MMA'B' is Q(MAB, MB' A'). (A, E, P)
THEOREM 24 If we have
Q(ABC, A'B'C'},
we have also Q (A'B'C', ABC]
Proof By the theorem above,
Q(ABC, A'B'C')
implies AA'BC^AA'C'B',
which is the inverse of A'AB'G' -^ A' A OB,
which, by the theorem above, implies
Q (A'B'C 1 , ABC),
The notation Q (ABO, A'B'C 1 ) implies that A, B, 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', C" 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 P^P^ and Q^Q Z Q Z Q^ are so related
J^ to Q v PZ to Q z , etc that five of the sides P i P J (i,j 1^,3,4',
i =/) meet the five sides of the second which are opposite to Q i Q J in points
of a hne I, the remaining sides of the two quadrangles meet on I, (A,E,P)
102 THE FUNDAMENTAL THEOREM [CHAP rv
Proof. The sides of the first quadrangle meet /ma quadrangular
set Q (P u P ia P w P 8i P 2l P 2 ,) , hence Q (P^PA P 1S PM. But, by hypoth-
esis, five of the sides of the second quadrangle pass thiough these
points as follows : Q$ z through P w $ x $ 3 through P M , Q t Q t through P 23 ,
Q s Qi through P^, $ 4 $ 2 through JJ 8 , Q a Q z through P^ 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, constiuct the othei double point.
2. If a, b, c are thiee nonconcuirent lines and A', B', C f are three collmear
points, give a construction for a triangle "whose vertices A , B, C are lespectively
on the given lines and whose sides SC } CA, AB pass respectively through the
given points. What happens when the three lines a, 6, 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',
then it also transforms A' into A; this is expressed by the phrase that
the points A, A 1 correspond to each other doubly. 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 tr on the line we have the relations ir (A) A 1
and i rr(A')=iA > the projectivity is an involution. (A, E, P)
Proof. For suppose P is any other point on the line (not a double
point of TT), and suppose 7r(P) = P'. There then exists a projectivity
g ivin S AA'PP'-^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', A, P, P'* But
in this projectivity P' is transformed into P. Thus every pair of
homologous points corresponds doubly.
CORQLLAJRY. 'An involution is completely determined when two pairs
of conjugate point* are gwen. (A, E, P)
38,39] INVOLUTIONS 103
THEOREM 27. A necessary and sufficient condition that three pair?
of points A, A' ; B, B' ; C, C' "be conjugate pairs of an involution is
Q(ABQ,A'B'C'). (A, E, P)
Proof. By hypothesis we have
AA'BC-^A'AB'O 1 .
By Theorem 2, Chap. Ill, we also have
which, with the first projectivity, gives
AA'BC^ AA'C'B'.
A necessary and sufficient condition that the latter projectivity hold
is Q(ABC, A'B'C'} (Theorem 23).
COROLLARY 1 If an involution lias double points, they are harmonic
conjugates with respect to every pair of the involution. (A, E, P)
For the hypothesis A^A',B = B' gives at once H (AB, CO') as the
condition of the theorem.
COROLLARY 2 An involution is completely determined when two
doulle points are given, or when one doulle point and one pair of
conjugates are given (A, E, P)
COROLLARY 3. If M,N are distinct double points of a projectivity
on a line, and A, A'; B, B' are any two pairs of homologous elements,
the pairs M, N; A, B 1 ; A', 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, H fl , P)
COROLLARY 5, The projectivity ABCD-^ABDC letween four dis-
tinct points of a line implies the relation H (AB, CD}. (A, E, P)
For 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 homology.
THEOREM 28. If [A] and [B] THEOREM 28'. If [1] and [m]
are any two protective pencils are any two protective pencils of
of points in the same plane on lines in the same plane on distinct
* This relation is sometimes expressed by saying, "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 m involution, but in general three pairs are not.
104
THE FUNDAMENTAL THEOREM
[CHAP IV
distinct hnes l v Z 2 , there exists a
line I such that if A v B^ and A z , J? 2
are any two pairs of homologous
points of the two pencils, the lines
A 1 B n and A Z B^ intersect on L
(A,E,P)
DEFINITION The line I is called
the axis of homology of the two
pencils of points.
points S v S z , there exists a point S
such that if a v l^ and 2 , \ are
any two pairs of homologous linos
of the two pencils, the points aji^
and aj) 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 [B] -^ [A] follows A^B^B^A] (fig 47). But in this pro-
jjectivity the line A^ is self-corresponding, so that (Theorem 17, Cor )
A t
the two pencils are perspective. Hence pairs of corresponding lines
meet on a line I ; e.g. the lines A^B^ and J^A n meet on I as well as
A^ and B^. To prove our theorem it remains only to show that
B Z A^ and A Z B S also meet on I. 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 [1], [m] are not
perspective, the axis of homology is perspective, the center of homology
the line joining the points homol-
ogous with the point IJ, Z regarded
first as a point of l : and then as
a point of l a .
I
is the point of intersection of the
lines homologous with the line 8^
regarded first as a line of [I] and
then as a line of [m].
For in the perspectivity AB]~B l [A] the line ^ corresponds to
Bi(ll^)t and hence the point l^l z corresponds to M t in the projectivity
[B] ~ [A] Similarly, ll z corresponds to l^ y
39] CENTEK AND AXIS OF HOMOLOGY 105
EXERCISES
1 Tlieie is one and only one piojectivity of a one-dimensional form leaving
invariant one and only one element 0, and tiansfoinung a given other element
A to an clement B
2. Two piojective langes on skew lines are always perspective
3 Prove Cor 5, Theoiem 27, without using the notion of involution
4. If MNAB-^MNA'B', then MNAA'-^MNBB'
5 If P is any point of the axis of homology of two piojective ranges
[A"]-r [.5], then the piojectivity P[A~\-r- P[B~\ is an involution. Dualize.
6. Call the faces of one tetrahedron %, a 2 , a 8 , a 4 and the opposite veitices
A^A^A^Ai respectively, and similarly the faces and vertices of another tetra-
hedron fa, fa, fa, fa and B v B z , B a , B+ If A lt A z , A a> A 4 lie on fa, fa, fa, fa
respectively, arid B\ lies on oj, JB 2 on 03, B 3 on a 3 , then B lies on o 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 pimciple of pro-
jectivity.
8 Given a tiiangle ABC and a point A', show how to construct two points
B', C f such that the tuangles ABC and A'B'C' are peispective from four
different centers.
9. If two triangles A^B^C^ and A Z B Z C Z aie peispective, the three points
(AiB v A Z BJ = C 3 , (A&, A 2 C,) = B 8 , (^C 2 , B Z CJ = A v
if not collineai, form a triangle perspective with the first two, and the three
centers of perspectivity aie collinear.
* 10. (a) If TT is a piojectivity m a pencil of points [A] on a line with inva-
riant points A lt AV and if \L"\, [M] are the pencils of points on two lines Z, m
through A lt A s respectively, show by the methods of Chap. Ill that there exist
three points S v S z , S a such that we have
S-,
where ir(A)-A'; that S v S^A a are collinear; and that S^S^Aj^ are collinear.
(&) Using the fundamental theorem, show that there exists on the line S 1 A 3
a point S such that we have
(c) Show that (7;) 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 A t > A z , giving TT (A) - A' in a pencil of points [A], and [L] is a pencil
of points on a line I through A v there exist two points S v S z such that
10 8 THE FUNDAMENTAL THEOREM [CHAP.IV
.11. Show that Assumption P could be replaced by the corollary of
Th nrShow that Assumption P could be replaced by the following. If we
have a projectivity in a pencil of points defined by the perspective.
and [M] is the pencil of points on the line S&, there exist on the base of []
two points S, S such that we have also
40 Types of collineations in the plane. We have seen in the
proof of Theorem 10, Chap III, that if 0,0,0, is any triangle, there
exists a coUineation II leaving O v O z> and 0. invariant, and trans-
forming any point not on a side of the triangle into any other such
o
]f 10. 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
S B and 0^. Hence, if Hj is a homology with center O l and axis
S 8 , which determines the same projectivity as II on the line O t O g ,
and if H a is a homology with center O a and axis 0^ which deter-
mines the same projectivity as II on. the line 2 8 , then it is evident
* U =? HH
40] TYPES OF COLLINEATIONS 107
It is also evident 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 homology is said to be of Type I,
and is denoted by Diagram I (fig. 48)
EXERCISE
Piove that two homologies with the same center and axis aie commutative,
and hence that two projectivities of Type I with the same invariant figuie aie
commutative.
Consider the figure of two points O v O a and two lines o v o s , such
that O t and O z are on o v and o x and o 2 are on O r A collineation II
which is the product of a homology H, leaving a and 2 invariant,
and an elation E, leaving O l and o i 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(A)=B t
this fixes the transformation (with two distinct double lines) among
the lines at O v and the parabolic transformation among the lines at O s>
and thus determines II completely. Cleaily if II is not to reduce to a
homology or an elation, the line AB must not pass through t or O g .
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 //(fig 48).
EXERCISE
Two protective collineations of Type 71, 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 Eb 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 a. Let E t be an elation with center
and axis OA, which transforms the point (oa) to the point (06). Let E 2
be an elation of center (AS, o) and axis o, which transforms A to B.
Then II = E s E t has evidently no other invariant elements than and o
and transforms Aa to J55.
108 THE FUNDAMENTAL THEOREM [CHAP IV
Suppose that another projectivity II' would transfer A a to />'& with
Oo as only invariant elements The transformation II' would evidently
have the same effect on the lines of and points of o as II. Hence
n'H" 1 would be the identity or an elation. But as n'TI" 1 ^) = B it
would be the identity. Hence II is the only pro]ectivity which trans-
forms Aa to Bb 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 V.
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 fly nro F,
if of Type I or II will transform any point P not on a, line of F into
any other suoh 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 in not on
a point, or P on a line, of F into any other such dement Qg ; 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 i61e of Assumption P is well illustiated by this theorem. In case of
each of the fust three types the existence of the required collineation wan proved
by means of Assumptions A and E, togethei with the existence of a sufficient
number of points to effect the construction. But its ttwV/www* was eHtabliwluul
only by means of Assumption. P In case of Types IV and V, both existence
and uniqueness follow from Assumptions A and E.
EXERCISES
1. State the dual of Theorem 20.
2. If the number of points on a lino is p + 1, the number of collineations
with a given invariant figure is as follows :
Type/, 0>-2)Cp-8).
Type/7, (p- 8) (/-!).
Type J/J, j,(j-l).
Type IV, p - 2.
TypeF, ;>-!.
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 8 of Type ///; and oo 1 of Types /Fand V.
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
FIG. 40 FIG, 60
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.
TIIEOEEM 1. The section of a cone of lines ~by a plane not on the
vertex of the cone is a point conic The section of a cone of planes ly
a plane not on the vertex is a line conic.
Now let A^ and B^ be the centers of two flat pencils defining a
point conic. They are themselves, evidently, points of the conic, for the
line A^ regarded as a line of the pencil on A corresponds to some
other line through B l (since the pencils are, by hypothesis, projective
* All the developments of this chapter are on the basis of Assumptions A, E, P,
and HQ.
t A fifth one-dimensional form a self-dual form of lines in space called the
regulus will be defined in Chap. XI This definition of the first four one-dimen-
sional forms of the second degree is due to Jacob Sterner (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 v The conic is clearly determined by any other three of its
pomts, say A v JB Z , C z , because the projectiviLy of the pencils is then
determined by
^W*M-*BJ4**M
(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 lino I through 7? . Jf the
Ime I is met by the lines A^ A^C Z> ^A y J3 l C s in the points JS, T, U> A
c,
"*..."
FIG. 51
respectively (fig. 51), we have, by hypothesis, SB^U^A. The other
double point of this projectivity, which we will call C v ia given by tho
quadrangular set Q(B Z ST, C,AU} (Theorem 23, Chap. IV), A quad-
rangle which determmes it may be obtained as follows : Let tho lines
A Z B V and A,B Z meet in a point 0, and the lines AC and A C in a
point*; then therequired quadrangle is^C^.and C, is determined
as the intersection of A Z B with I w-wwmrnea
Ci
" U A **
?.
This means
* * be the
z , B 9 , <7 ? , C, are points of a point
41] PASCAL'S THEOREM 111
determined "by two protective pencils on A and B v if and only if the
threepoints C=(A i B & )(A s B 1 ),B = (A 1 C z )(A,C 1 ),A = (B 1 C s )(B 2 C 1 ) are
collinear. The three points in question are clearly the intersections
of pairs of opposite sides of the simple hexagon A^B^G^B^Cy
Since A v B v C l may be interchanged with A z , B z , C z respectively
in the above statement, it follows that A v B v C v C a are points of a
conic determined by projective pencils on A z and J3 2 . Thus, if C 1 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. STEINER'S THEOREM. If A and B are any two given
points of a conic, and P is a variable point of this come, we 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 six points, no three of which are collinear, be points of
the same conic is that the three pairs of opposite sides of a simple
hexagon of which they are vertices shall meet in collinear points f
The plane dual of this theorem is
THEOREM 3'. BRIANCHON'S THEOREM. FJie necessary and sufficient
condition that six lines, no three of which are concurrent, be lines of
a line conic is that the lines joining the three pairs of opposite vertices
of any simple hexagon of which 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 plane of a line conic cannot be on
more than two lines of the conic
* Theorem 8 was proved by B Pascal m 1640 when only sixteen years of age
He proved it first for the circle and then obtained it for any conic by projection
and section This is one of the earliest applications of this method Theorem 3'
was first given by C J. Bnanchon m 1806 (Journal de 1'Ecole Polytechmque,
Vol VI, p 801),
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 point of a
hexagon whose sides belong to a line come.
112 CONIC SECTIONS [UIUP V
Also as immediate corollaries of these theorems we have
THEOREM 4 There is one and only one point conic, eantitimity five.
given points of a plane no three of which are collinmr.
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 Buanchon's theorem without making uao of tho principle of
duality. . , .. . , . ,
3 A necessaiy and sufficient condition that six pomls, no lhnu of which
are collmeai, be points of a point conic, is that thoy lo tho points of mk-i-
section (a&')> (6c / ), (O ( fifl ')> W> (0 of tho &1(lcs rt > 6 ' c aml "' '''' '"' of iwo
perspective tnangles, in which a and a', I and //, c ami tf aio homologous.
42. Tangents. Points of contact. DEFINITION. A lino p in the
plane of a point conic which meets the point conic iu ono and only
one point P is called a tangent to the point conic! at P. A point 1> in
the plane of a line conic through which pasnofl ono and only ono lino
p of the line conic is called a point of contact of the lino conic on p.
THEOREM 5 Through any point of a point conic there is one and
only one tangent to the point conic
Proof. If P Q is the given point of the point conic, and I{ is any
other point of the point conic, while P is a variable point ot this
conic, we have, by Theorem 2,
Any line through P meets its homologous line of tho poiujil on I{ in
a point distinct from P , except when its homologous lino in I{1^.
Since a projectivity is a one-to-one correspondence, thero i,s only ono
line on P which has P^ as its homologous lino,
THEOREM 5'. On any line of a line conic there fa one and only OHG
point of contact of the line conic.
This is the plane dual of the preceding theorem.
EXERCISE
Give the space duals of the preceding doBniticraa anil theorems,
Eeturning now to the construction in the preceding section for the
points of a point conic containing five given points, we recall that
42] TANGENTS 113
the point of intersection (7 X of a line I through J? 2 was determined by
the quadrangular set Q^ST, Q^AU}. The points J5 3 and C t can,
by the preceding theorem, coincide on one and only one of the lines
through B z * For this particular line I, A becomes the intersection
o
U
FIG 52
of the tangent at 7? 2 with BQ, and the collmearity of the points A, B, G
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 undei stood that the theoiem 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 JfJ, P^, P^ P^ Jj? of the
point conic (fig. 53). By this theorem the tangent j a at J% must be
* As explained in the fine print on page 110, this occurs when I passes through
the point of intersection of SiCi with the line joining C = (AiB z ) (A z Bi) and
114 CONIC SECTIONS [CHAP.V
such that the points Pl (P^ = 4 (-M -5) = *> *nd ^) (W) =
are collmear. But ^ and C are determined by J?, P z> P 8) P it P 6 , and
hence p 1 is the line joining P to the intersection of the lines *>"
and
FIG. 63
In like manner, if P v P z , P 3 , P i3 and p l are given, to construct the
point J on any line I through P of a point conic containing P v P Z) P a , P
and of which p^ is the tangent at P v we need only determine the points
A =j 1 (^), B = l(P l P a ), and C = (AB) (P a P B )-, then Jja meets I in J
(fig 53).
. 64
In case I is the tangent y t at ^ } ^ coincides with P< and the fol-
lowing points are collmear (fig. 54) :
42] TANGENTS 115
Hence we have the following theorem :
THEOREM 7 If the vertices P v J\, P s , P 4 of a simple quadrangle are
points of a point conic, the tangent at I( and the ftido /iJ/J, the, tanynit
at PI and the side P : P Z , and the pair of sides 7J./J and 7j/j meet in three
collinear points.
If P i} P z , P & , P & and the tangent p l at P t are given, the construction
determined by Theorem 6 for a point P of the point conic on a line I
through P 8 is as follows (fig. 53): Determine = (P t P 6 ) (PP,), A **$
and B = (A C)(P^ ; then P & B meets I in P v
In case I is the tangent at P s , P^ coincides with P & and we have the
result that C = (P l P,)(P & P a ), A=p 1 p s> B = (P^(P,P^ are collinear
points, which gives
FIG. 65
THEOREM 8. If the vertices of a complete quadrangle are points of
a point conic, the tangents at a pair of vertices meet in a point 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 JF, JJ, P 8 and the given tangents be p# p r Let I be any line
through P v If PH is the other point in which I meets the point conic,
the points A = p l (P a P t ), =p^ (P l P i ), and C = (P Z P 8 ) (P^) are collinear.
Hence, if C=l(P l P i ) and S=p i (AC) ) then P z is the intersection of I
with Pi*
In case I is the tangent p s at P z , the points P% and P B coincide, and
the points
CONIC SECTIONS [CHAP.V
are collinear Hence the two triangles P.P.P, and p,p,p, are per-
spective, and we obtain as a last specialization of Pascal's theorem
( fi g- 56 )' . , f . , .
THEOREM 9. A triangle whose vertices are points of a, point conic
is perspective with the triangle formed ly the tangents at these points,
the tangent at any vertex levng homologous with the side of the first
triangle which does not contain this vertex.
COROLLARY. If P v P s , P are three points of a point conic, the lines
P&PV P a % are harmonic with the tangent at P s and the line Joining P 3
to the intersection of the tangents at JJ and P
Proof. This follows from the definition of a harmonic set of lines,
on considering the Quadrilateral P^A, AB, BP V PJl (fig. 56).
FIG. 56
43. The tangents to a point conic form a line conic. If P v 7>, P 6 , I*
are points of a point conic and p v p z , p s , p t are the tangenta to the
conic at these points respectively, then (by Theorem 8) the lino join-
ing the diagonal points (P : P Z ) (P S P^ and (P^ (/>/*) contains the inter-
section of the tangents Pv p 9 and also the intersection of p a , p^ This
line is a diagonal line not only of the quadrangle P^I^J^ but also of
the quadrilateral PlPsPtPt . Theorem 8 may "therefore be stated in
the form:
THEOREM 10. The 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] TANGENTS 117
Proof. Let P if P 2) P 8 be any three fixed points on a conic, and let P
be a variable point of this conic. Let p v p.,, p a , p bo respectively the
tangents at these points (fig 57) By the corollary of Theorem 28,
Chap. IV, P^ is the axis of homology of the projcctivity between the
pencils of points on p i and p a denned by
P l(PlP) (PiPt) 7- (PiPjWPiP*)'
But by Theorem 10, if Q=(P i P 2 ) (P S P), the points pp z> p L p a , and Q are
collmear. For the same reason the points p z p a , pp v Q are collmear.
It follows, by Theorem 28, Chap. IV, that the homolog of the variable
FIG. 67
point p t p is p a p; ie. p is the line joining pairs of homologous points
on the two lines p v p v so that the totality of the lines p satisfies the
definition of a line conic.
COROLLARY. The center of homology of the projectivity P^ [P] -^ P z [P]
determined ly the points P of a point conic containing P v P z is the
intersection of the tangents at P v P a . The axis of homology of the
projectivity p l [p] -^p z [p] determined "by the lines p of a line conic
containing the lines p v p z is the line joining the points of contact
f Pv Pi-
THEOREM 12. If P : is a fixed and P a variable point of a point
conic, and p v p are the tangents at these two points respectively f then
CONIC SECTIONS [CHAP. V
Proof. Using the notation of the proof of Theorem 11 (fig D7),
weliave
where Q is always on P& But we also have
and, by Theorem 11, 1?M T^
Combining these projectivities, we have
The plane dual of Theorem 11 states that the points of contact, of
a hne come form a, point come. In view of these two theorems and
their space duals we now make the following
DEFINITION. A eonic section or a conic is the figure formed by a
point conic and its tangents. A cone is the figure formed by a cono
of lines and its tangent planes.
The figure formed by a line conic and its points of contact is then
hkewise 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 \w a
set of theorems of the same consequence for point conies as for line
conies. We content ourselves with restating Brianchon's theyrem
(Theorem 3') from this point of view.
BRIANCHON'S THEOREM. If tlie 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 conio is left unchanged,
while the words point (of a conic) and tangent (of a conic) are inter-
changed. We shall also, in the future, make use of the phrase a 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.
43] TANGENTS 119
EXERCISES
1 State the plane and space duals of the special caaoH of Pascal's theorem.
2 Constiuct a conic, given (1) five tangents, (2) foni Undents ami tho
point of contact of one of them, (3) three tangents and the points of contuoti
of two of them.
3 ABX is a triangle whose veitices ate on a conic, and , li t j; are the tnn-
gents at A, B, X lespectively If A, B are given points and X is vanablo,
deteimine the locus of (1) the centei of perspectivifcy of tho triangles ABX
and d)i , (2) the axis of peispectivity.
4 A", Y, Z are the vertices of a variable triangle, such that X, Y ai e always
on two given lines a, I lespectively, -while the sides XY, ZX, ZY always pass
through thiee given points P, A s B respectively Show that the loons of tho
poi nt Z is a point conic containing A,B,D = (aJi) , M (A P) 1), and N = (BP) a
(Maclaurin's theoiem). Dualize. (The plane dual of this theorem is known
as the theorem of Braikenridge.)
5. If a simple plane n-pomt varies in such a way that its sides always pass
thiough n given points, while n 1 of its vertices aie always on n 1 given
lines, the ?ith veitex describes a conic (Poncelet).
6 If the vertices of two triangles are on a conic, the six sides of these two
tiiangles are tangents of a second conic, and conveisely. Corresponding to
every point of the fiist conic there exists a triangle having this point as a
vertex, whose other two veitices aie also on the fiist conic and whose sides
are tangents to the second conic Dualize.
7. If two tiiangles in the same plane aie perspective, the points m which
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 nou-
homologous vertices of the other are tangents to another conic.
8. If A, B, C, D be the vertices of a complete quadiangle, whose sides
AB, AC, AD, BC, BD, CD are cut by a line in the points P, Q, R, S t T, V
respectively, and if E t F, G, K, Z, M 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, <7, K,L,M
are on a come which also passes thiough the diagonal points of the quadrangle
(Holgate, Annals of Mathematics, Ser. 1, Vol VII (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, Vol. VII (1808), 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 the quadrangle which meet on A A', M is a variable point
120
CONIC SECTIONS
[CHAP
on m, the lines BM, B'M meet n in the points JV, 2V' respectively; th e lines
AN, A'N' meet in a point P. Show that the locus of the lines PM is a line
come, which contains the lines m, p = P (n, BB"), and also the hues /I , 1 ' /;/,'
.4'.B', -4.C (Amodeo, Lezioni di Geomelna Projettiva, Naples (1903), p. ;J3l),
12. Use the lesult of Ex 11 to give a contraction of a line conic dek'i-
nimed 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, b, c are the sides of a triangle whose vertices aie on a came and
m, m' are two lines meeting on the conic which meet a, ft, c in the points A /> C
and A' t J5', C' respectively, and which meet the conic again in JV, 2V' respec-
tively, we have ABCN-^A'B'C'N' (cf Ex 6).
14. If A, J5, C, D are points on a conic and a, ?>, c, d are the tail gents to
the conic at these points, the four diagonals of the simple quadrangle A BCD
and the simple quadrilateral abed aie concurrent
44. The polar system of a conic.
THEOREM 13. I/P is apoint in
the plane of a come, Intt not on the
conic, the points of intersection of
the tangents to the conic at all the
THEOREM 13'. Ifp is a line in the
plane of a conic, Imt not tangent to
the conic, the lines joining the points
of contact of pairs of tangents to the
pairs of points which are collmear conic which meet on p gag* tit. rough
with Pare on a line, winch also con- a point P, through which pans (duo
tains the liarmonic conjugates ofP the harmonic conjugates of p with
with respect to these pairs of points, respect to these pairs of tangents.
. 58
7(ng.*>).
44] POLAR SYSTEM 121
respectively, the line D^ passes through the intersection Q of p v p z
(Theorem 8). Moreover, the point P' in which DJ>^ meets 7J/JJ is the,
harmonic conjugate of P with respect to 1{, K (Theorem 6, Chap. IV).
This shows that the line 1)^= QP 1 is completely determined by the
pair of points P v P z Hence the same line QP 1 is obtained liy replacing
P 8 , PI by any other pair of points on the conic collinear with P, and
distinct from P v P v 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, hut not on the plane of a conic, but not tangent
conic, is called the polar of P to the conic, is called the pole of 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 JP.
THEOREM 14. The line joining THEOEEM 14'. The point of
two diagonal points of any com- intersection of two diagonal hnes
plete quadrangle whose vertices of any complete quadrilateral
are points of a conic is the polar whose 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'. The 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 points in which p meets the conic,
through P, if such tangents exist if such points exist.
Proof. Let P^ be the point of contact of a tangent through P, and
let P z , Pj be any pair of distinct points of the conic collinear with P.
The line through P l and the intersection of the tangents at P >2 , Pj
meets the line P 9 P S in the harmonic conjugate of P with respect to
P 8 , % (Theorem 9, Cor.). But the line thus determined is the polar of P
(Theorem 1 3) This proves Theorem 1 5 Theorem 1 5 ' is its plane dual
THEOREM 16 If p is the polar of a point P with respect to a conic,
P is the pole ofp with respect to the same conic.
122 CONIC 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 Hi* polar of a point P passes iliromjli, a point Q 9
the polar of Q passes through P.
Proof If P or Q is on the conic, the theorem is equivalent to
Theorem 15. If neither P nor Q is on the conic, let PP l be a line
~
Fie. 59
meeting the conic in two points, P^, P v If one of the lines J^Q, J\Q
is a tangent to the come, the other is also a tangent (Theorem 13);
the line J*j = P^P is then the polar of Q, which proves the theorem
under this hypothesis. If, on the other hand, the lines P^Q, QQ meet
the conic again in the points P s , P respectively (fig. 59), the point
(P^) (P a PJ is on the polar of Q (Theorem 14). By Theorems 13 and 1 4
the polar of (P^) (P S P) contains the intersection of the tangents at
P v P} and the point Q. By hypothesis, however, and Theorem 13, the
polar of P contains these points also. Hence we have (1J J3) (^J<) = P>
which proves the theorem.
COROLLARY 1. If two vertices of a triangle are the poles of their
opposite sides vrith 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 15ne pn the pole
44] POLAR 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 come
COROLLARY 2 The diagonal triangle of a complete quadrangle whose
vertices are on a conic, or of a complete quadrilateral whose sides are
tangent to a conic, is self-polar with regard to the conic; 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 g. In the
special case where q is a tangent to the conic, we have already seen
(Theorem 12) that we have
m-zW-
If Q is not on q, let A (fig. 60) be a fixed point on the come, a the
tangent at A, X 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 (ass). Hence we have
[ZlAW
If P' is the point pq, this gives
But since the polar of JP' 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 ol the last century in connection with the principle
of duality.
124 CONIC SECTIONS t0.ir V
18 On any I * tan^nt to a fl*x < 0" P*ir,
a "
, f - ,
It points, these points are tU ao^U point* of ***.
COBOLIARY As a yot f <"*" a V"^ f ?" ' ' 2 ar
to My tonie varies over a prj* P** / *"
FIG. 60
The pairing of the points and lines of a plane brought
about by associating with every point its polar and with ovory 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 <z and 6 are two nonconjugate lines in a polar system, every point A
of a has a conjugate point B on Z>. The pencils of points [/I] and [#] are
protective; they are perspective if and only if a and & intorseob on the conies
of the polar system
4. Let A he a point and 6 a line not the polar ef 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 Ib, the locus of P is a conic
passing through A and the pole J3 of &, unless the line AB is tangent to the
POLAE SYSTEM
conic, in which case the locus of P is a line. If A B is not tangent, to the conic,
the locus of P also passes through the points in which ft meets the given conic
(if such points exist), and also through the points of contact of the tangents to
the given conic through A (if such tangents exist). Dualize (Keye-IIolgate,
Geometiy of Position, p 106)
5. If the vertices of a tuangle 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 thiough
the pole of the third side (von Staudt).
6 If two lines conjugate with respect to a conic meet the conic in two
pans of points, these pairs are projected from any point on the conic by a
haimomc set of lines, and the tangents at these pairs of points meet any
tangent in a haimomc set of points
7 With a given point not on a given conic as center and the polar of this
point as axis, the conic is tiansfoimed into itself by a homology of period two.
8. The Pascal line of any simple hexagon whose veitices are on a conic is
the polar with respect to the conic of the Bnanchon point of the simple hexagon
whose sides aie 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 ai e harmonic with A , J5.
10 If in a plane theie are given two conies C* and C 2 2 , and the polars of
all the points of C 3 with respect to C 2 2 are detei mined, these polars aie 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 thud conic
12 Given five points of a conic (or four points and the tangent thi ough
one of them, or any one of the other conditions determining a conic), show
how to construct the polar of a given point with respect to the conic.
13. If two pans of opposite sides of a complete quadrangle are pairs of
conjugate lines with respect to a conic, the third pair of opposite sides are
conjugate with lespect to the conic (von Staudt)
14 If each of two triangles in a plane is the polar of the other with i espect
to a conic, they aie peispective, 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 thiid
conic (Steinei).
16. Kegarding the Desargues configuration as composed of a quadrangle
and a quadnlateral 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 lines AM, BM joining them to an arbitrary poinb of the conic meet the
latter again in the points C, D respectively. The lines AD, BC will then meet
on the come, and the lines CD and AB aie conjugate. Dualise.
126 CONIC SECTIONS [OBAV. V
45. Degenerate conies. For a variety of reasons it is dcsiniblo to
* 1 I
regard two coplanar lines or one line (thought of as two eomcuK'nt
lines) as degenerate cases of a point conic; and dually to re#aw
two points or one point (thought of as two coincident points) i
degenerate cases of a line conic This conception makes it poHwilw*
to leave out the restriction as to the plane of section in Thooivm 1.
For the section of a cone of lines by a plane through the vurtcx if
the cone consists evidently of two (distinct or coincident) liiun, i.e.
of a degenerate point conic; and the section of a cone of plaiiow by
a plane through the vertex of the cone is the figure formed by floiw
or all the lines of a flat pencil, i.e a degenerate line conic.
EXERCISE
Dualize in all possible ways the degenerate and nondegonorate cani'B of
Theorem 1.
Historically, the first definition of a conic section was given by tho andont
Greek geometeis (e.g. Mensechmus, about 350 B.C ), who defined them UN tin*
plane sections of a "light circular cone." Jn a later chapter \vo mil show
that in the " geometry of reals " any nondegeneiate point conic is pi'ojcetivuly
equivalent to a circle, and thus that for the ordinary geometry tha modern
protective definition given in 41 is equivalent to the old definition, We luv
here using one of the modern definitions because it can be applied before devel-
oping the Euclidean metric geometiy.
Degenerate comes would be included in our definition (p. 100), if
we had not imposed the restriction on the generating protective
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 perspeotivity 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 (of. in particular the
corollary). The polar of a point with regard to a degenerate conic
consisting of two lines is the harmonic conjugate^ tjie point with
respect to the two lines (cf. the deftniM0;a, ,p. 84, $x. 7).^ Htoo* tha
polar system of a degenerate co&ic
points) determines an involution,
-45,46] THEOREM OF DESARGUES
EXERCISES
1 State Brianchon's theorem (Theorem 3') for the caso of a degenerate
line conic consisting of two points
2. Examine all the theoiems of the preceding sections with refoienoo to
their behavior when the conic in question becomes degenerate.
46. Desargues's theorem on conies.
TIIEOEEM 19. If the vertices of a complete quadrangle are on a conic
which meets a line in two points, the latter are a pair in the invo-
lution determined on the line by the pairs of opposite sides of the
quadrangle.*
Proof Reverting to the proof of Theorem 2 (fig. 51), let tho line
meet the conic in the points I? 2 , <7 X and let the vortices of the quad-
rangle be A v A a> 23 V O a . This quadrangle determines on the line an
involution in which 8, A and T, U are conjugate pairs. But in the
proof of Theorem 2 we saw that the quadrangle A^A^BG determines
Q(B S ST, C^AU}. Hence the two quadrangles determine the same
involution on the line, and therefore # 2 , C l are a pair of the involution
determined "by the quadrangle A^AyB^Cy
Since the quadrangles A^A^B^C^ and A^A Z BO 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 :
COKOLLARY. 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 tangent is a double
point.
The Desargues theorem leads to a slightly different form of statement foi
the construction of a conic through five given points Oil 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 JBCP^ 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 J^P & and those
* First given by Desargues in 1639; of. CBuvres, Paris, Vol. I (1864), p. 188',
12 g CONIC SECTIONS [CHAP, v
determined by 2>J?, %P 6 are two other pairs. This gives the following
special case of the theorem of Desargues :
THEOREM 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 ly the pair of points in which two sides of the triangle
meet I 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 teims of this theoiem we may state the construction of a conic thiough
four points and tangent to a line thiough one of them as follows . On any lino
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 pan. 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 hue 2j. The quad-
rangle ABP^ determines on I an involution in. which 1%, P a are one
pair, in which the tangents at P v P determine another pair, and in
which the line P i P i determines a double point. Hence we have
THEOREM 21. If a line I meets a conic in two points and P v 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^ and through a pair of conjugate points of which pass the
tangents at P I} 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 I is 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 AND 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
through a vertex of the deter-
mining quadrangle] is met "by the
conies of a pencil of Type I in the
pairs of an involution.*
DEFINITION. The set of all conies
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
through any point (not on a side
of the determining quadrilateral)
to the conies of a range of Type /
are the pairs of an involution.
FIG 61
FIG. 62
Fia 68
COROLLABY. 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 gwen pencil of Type L
FIG. 64k
COROLLARY. 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 given range of Type L
* This form of Desargues's theorem is due to Ch. Sturm, Atmales de Math<5ma-
t The vertices of the quadrangle are regarded as exceptional points.
130
CONIC SECTIONS
[CHAP 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 comes
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 liy 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; a,nd
tangent to a general line in the
plane there are two or no conies
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.
of the pencil.
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 "by 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. The
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 {of, Bare. 2, 4, 7, below).
FIG. 65
47] PENCILS AND RANGES 131
any line in a double point of the involution determined on that hue.
And the point of intersection of the common tangents is joined to any
point by a double line of the involution determined at that point
GOKOLLAHY. Through any general point or tangent 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 J? The
diagonal triangle of the fundamental quadrangle (quadnlateral) of the pencil
(range) is the only triangle which is self-polar with respect to two conies of
the pencil (range).
2. Let A z and J3 2 be any two comes of a pencil of Type I, and let P be any
point in the plane of the pencil. If p is the polar of P with respect to A s , and
P' is the pole of p with respect to U 2 , the coirespondence thus established
between [P] and [P'] is a piojective colhneation of Type 7, whose invariant
triangle is the diagonal triangle of the fundamental quadrangle. Do all pio-
jective collineations thus determined by a pencil of comes of Type I form a
gioup? Dualize.
3. -What are the degenerate conies of a pencil or range of Type III
4 Let a pencil of comes of Type 7/be determined by a triangle ABC and
a tangent a through A. Further, let o" be the harmonic conjugate of a with
respect to AB and A C, 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 JL
5 What are the degenerate comes of a pencil or range of Type IV 1
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
comes 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/ V 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 denned
as soon as two conies of the pencil (range) are given ; (2) the comes
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 denned which likewise have these properties. These new systems
* The remainder of this section may be omitted on a first reading.
CONIC SECTIONS I
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 ly a projective collineation in
the plane of the conic into a conic such, 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 coUmeation
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 m 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* at a point P Q ,
and Q is any point of A z , then through any point on the plane of A z
out not on A z or p ,
there is one and only
one conic B* tJirough
P Q and Q, tangent to
p , and such that there
is no point of p Qf ex-
cept J, having the same
polar with regard to
loth A* and B\
Proof. If P' is any point of the plane not on p or A*, let JP be
the second point in which P P' meets A 2 (fig. 66) There is one and
only one elation with center P and axis P Q changing P into P'
(Theorem 9, Chap. III). This elation (by the lemma above) changes
A* into another conic B z through the points P Q and Q and tangent
to #,. The lines through P are unchanged by the elation, whereas
their poles (on_p ) are subjected to a parabolic projectivity. Hence
no point cm A (distinct from J>) has the same polar with regard to A*
as with regard to B\ Since A* is transformed into & by an elation,
the two conies can have no other points in common than JR and Q.
47]
PENCILS AND RANGES
133
That there is only one conic B* through P' satisfying the con-
ditions of the theorem is to be seen as follows Let QP meet p Q in
S, and QP' meet p in S' (fig. 66). The point S has the same polar
with regard to A z as S' with regard to any conic J5 3 , since this polar
must be the harmonic conjugate of p Q with regard to P Q and P Q P.
Let p be the tangent to A 2 at P and p' be the tangent to J? 2 at P',
and let p and p' meet p in T and 2 1 ' respectively. The points
Em 07
T and T' have the same polar, namely P P, with regard to A* and
any conic B z . By the conditions of the theorem the projectivity
must be parabolic Hence, by Theorem 23, Cor , Chap. IV,
Q(P ST, P T'S')
Hence p and p' must meet on P^Q in a point R so as to form the quad-
rangle RQPP'. This determines the elements P , Q, P',p Q , p 1 ot B*,
and hence there is only one possible conic B*.
COROLLARY 1. Tlie conies A s and B z 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 JB a
meets them in pairs of an involution in which the points of intersection
of I with P^Q and p are conjugate
Proof, Let I meet A* in N and 2T V B*ia.L and L v P QinM, and
p in Jfj (fig. 67). Let 7f and JT X be the points of A* which are trans-
formed by the elation into L and L respectively. By the definition of
an elation JT and Jf x are collinear with M, while JTis on the hne ZP
and JTj on L& Let KN^ meet p^ in J8, and ^TJ? meet JT^ in &
134 CONIC SECTIONS [CHAP V
Then since N, K, N v K, are on the conic to which PQ is tangent at J, we
have'by Theorem 6, applied to the degenerate hexagon ^KXN^
that S L and S are collinear. Hence the complete quadrilateral
SB KN V KK I has pairs of opposite vertices on P Q M and P,M V J' N
and P,N V 3/and % He ce ( MNL > ^Wi)-*
DEFINITION The set of all conies DEFINITION The set of all conies
through a point Q and tangent to tangent to a line q and tangent to
a line p at a point P Q> and such
that no point of p except P Q has
the same polar with, regard to two
conies of the set, is called a pencil
of comes of Type III (fig- 68)
a line p at a point P , 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 09).
Fiu 09
Two conies of such a pencil (range) are said to have contact of the
second order, or to osculate, at P Q .
Corollary 2 of Theorem 25 now gives at once:
THEOREM 26. Any line in the
plane of a pencil of conies of
Type III, which is not on either of
the common points of the pencil, is
met T)y the conies of the pencil in the
pairs of an involution. Through
any point in the plane except the
common points there is one and
only one conic of the pencil; and
tangent to any line not through
either of the common points there
are two or no conies of the pencil.
20'. Through any point
in the plane of a range of conies of
Type III, which is not on sitlwr of
the common tangents of the range,
the tangents to the cornea of the pen-
cil are the pairs of an involution.
Tangent to any line in th c plane, e,<3-
cept the common tungcuts there is
one andonli/ona 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
JOTi, NNi, P Q, such, that the lines joining them meet in a point JMT, are projected
from any point of the conic l>y a quadrangular set of lines (Theorem 10, Chap, VIII),
47]
PENCILS AND RANGES
135
The pencil Is determined by
the two common points, the com-
mon tangent, and one conic of the
pencil.
The range is determined by
the two common tangents, the
common point, and one conic of
the range.
EXERCISES
1 What are the degenerate comes of this pencil and range ?
2 Show that the colhneation obtained by making coi respond to any point P
the point P' which has the same polar p with regard to one given conic of the
pencil (lange) that P has with regard to anothei given conic of the pencil (range)
is of Type JIJ.
THEOREM 27. If a hue p Q is tangent to a conic A* at a point P ,
there is one and only one conic tangent to p 6 at P Q and passing
through any other point P' of the plane of A z not on p Q or A*
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 P is center and p Q axis,
changing P to P'. This elation changes A* into a conic J5 2 through
FIG. 70
P', and is such that if q is any tangent to A* at a point Q, then c[ is
transformed to a tangent q f of JB* passing through qp Q , and Q is trans-
formed into the point of contact Q' of %', collinear with Q and J
Hence there is one conic of the required type through P'.
That there is only one is evident, because if I is any line through P',
any conic j# 8 must pass through the fourth harmonic of P' with regard
1o Ipt and the polar of Ip as to A z (Theorem 13), By considering two
lines I we thus determine enough points to fix JB*.
COROLLARY 1. By duality there is one and only one conic JS a tangent
to any line not passing through P^
136 CONIC SECTIONS [CHAP, v
COROLLARY 2. Any line I not on P Q which meets A z and B z meets
them in pairs of an involution one double point of which ^s Ip 0) and
the other the point of I conjugate to Ip with respect to A 2 . A dual
statement holds for any point L not on p .
COROLLARY 3. The conies A* and B z can have no other point in
common than P Q and no other tangent in common than p .
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 comes tangent to a given line p Q at a
given point P Q , 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 P ,
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 by 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 whicJi 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 come 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 comes of a pencil of Type VI
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 tho 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 come C a being
given four points of C z and a conjugate of A with regard to <7 a .
47] PENCILS AND RANGES 137
5. Given an involution I on a line I, a pair of points A and A' on I not
conjugate in I, and any othei point JB on /, construct a point B' such that A
and A' and J3 and jB' are pans of an involution I' whose double points are a
pair in I The involution I' may also be desciibed as one which is commu-
tative -with I, or such that the product of I and I" is an involution.
6 There is one and only one conic through thiee points and having a
given point P and line p as pole and polar.
7. The comes through three points and having a given pair of points as
conjugate points form a pencil of comes.
MISCELLANEOUS EXERCISES
I. If and o are pole and polar with regard to a conic, and A and B are
two points of the come collinear with 0, 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 plane five-point ABODE. The locus of all points X
such that
X(BCDE) A(BCDE)
is a conic. A
3 Given two projective nonperspective pencils, [ja] and [<?]. Every line I
upon which the projectivity Z[j>]-T-Z[0] is mvolutoric passes thiough 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 he 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 lespect to which a given triangle is self -con jugate, 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', Qf
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, I will meet the conic and
the pairs of opposite sides in conjugate pairs of an involution. Dualize.
II. 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. XVH).
138 CONIC SECTIONS [CHAP V
12 If a vauable simple 7i-lme (n even) is inscubed in a conic in suclx a way
that n 1 of its sides pass thiough n 1 fixed collinear points, then the otliei
side passes thiough anothei fixed point of the same line Puah/e this theorem
13. If two comes inteisect in two points A, B (01 aie tangent at a point A)
and two lines through A and B lespectivoly (01 thiough the point oi contact
A) meet the conies again in 0, 0' and L, L', then the lines OL and O'L' meet
on the line joining the lemaming points of intellection (i existent) of the
two conies.
14. If a conic C 2 passes through the vertices of a triangle which is self-
polar with lespect to another conic /C 2 , theie is a tuangle msciibed in C 2 and
self -polar with regard to K 2 , and having one vertex at any point of C 2 The
lines -which cut C 2 and K z in two pairs of points which are haimonically con-
jugate to one another constitute a line conic Cg, which is the polai leciprocal
of C 2 with regard to 1C 2 (Ciemona, Chap. XXII).
15. If a vanahle tuangle is such that two of its sides pass lespectively
thiough 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 f , 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 vauable point on a conic containing A, B, C, and I is a vari-
able line thiough P such that all throws T (PA, PB, PC, I) are projective,
then all lines I meet in a point of the conic (Schroter, Jouinal fiir die reine uud
angewandte Mathematik, Vol. LXII, p. 222).
17. Given a fixed come and a fixed line, and three fixed points A , B, C 011
the conic, let P be a variable point on the conic and let PA, PB, PC meet
the fixed line m A', B', C' If is a fixed point of the plane and (0.1", PB") = K
and (JTC") = I, then K descubes a conic and I a pencil of lines whose center is
on the conic described by K (Schiotei, loc. cit.).
18. Two tiiangles 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 Schioter, 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 figmo thus associated
with any six points of a conic is called the hexafframmum myttticum.* Piove the
following properties of the hexagranimum. mystictun :
i. The Pascal lines of the thi ee hexagons PiPgPBP^PgP,,, PiP 4 P^P 6 P 6 P z ,
and P 1 P P 8 P 8 P 6 P 4 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 Stemer-Rchrbter, Vorlesungen liber Synthetische
Geometne, Vol II, 28 , Salmon, Conic Sections m the Notes ; Christine Ladd,
American Journal of Mathematics, Vol, II (1879), p 1.
47] ' EXERCISES 13 Q
iii. Fiom a given simple hexagon five others are obtained by permuting
in all possible ways a set of thiee vertices no two of which are adjacent. The
Pascal lines of these six hexagons pass through two Steinei points, which ai*e
called conjugate Sterner points. The 20 Steiner points fall into ten pahs of
conjugates.
iv The 20 Steiner points lie by fours on 15 lines called Steiner lines.
v What is the symbol of the configui ation 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 polai with
respect to each of them If these polars intersect iu A', the points A, A' aie
conjugate with respect to both comes. The polars of A' likewise meet in A.
In this way every point in the plane is paired with a unique other point. By
the dual piocess every line in the plane is paned with a unique line to which
it is conjugate with respect to both comes Show that in this correspondence
the points of a line correspond in geneial to the points of a conic. All such
comes which correspond to lines of the plane have in common a set of at most
three points. The polais of eveiy such common point coincide, so that to each
of them is made to correspond all the points of a line. They foim the excep-
tional elements of the coirespondence. Dualize (Reye-PIolgate, p. 110).*
23 If in the last exercise the two given comes 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 thiough 0. Then a
one-to-one correspondence may be established among the lines thiough O by
associating with every such line o its conjugate a' with respect to the cone
lying in the plane au If, then, a descubes a plane tr, of will desciibe a cone of
lines passing thiough u and through the polar line of IT, 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. Ill) *
25. Two conies are determined by the two sets of five points A^B,C,D,E
and A , B, C, H, K. Construct the fourth point of intersection of the two conies
(Castelnuovo, Leziom di Geometna, p 391).
26. Apply the result of the preceding Exercise to construct the point P such
that the set of lines P(A, J3, 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
quadratic correspondences.
140 CONIC SECTIONS [CHAP.V
28. Two sets of five points A, B, C, D, E and A, B, If, K, L determine
two comes which intersect again in two points X, Y. Construct the line XY
and show that the points X, Y are the double points of a certain involution
(Castelnaovo, loc. cit.).
29. If three conies pass through two given points A, B and the three pairs
of comes 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 theoiem of Desargnes * The conies
passing through thiee 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 pan 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 / 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 A, JB, C, then 7? passes through a fixed
point Q, (Cayley, Collected Works, Vol I, p. 361).
33. If two comes are inscribed in a triangle, the six points of contact aie
on a third conic.
34. Any two vertices of a triangle circumscribed to a conic are separated
harmonically by the point of contact of the side containing them and the point
where this side meets the line joining the points of contact of the other sides.
CHAPTER YI
ALGEBRA OF POINTS AND ONE-DIMENSIONAL COORDINATE
SYSTEMS
48. Addition of points. That analytic methods may be introduce!
into geometry on a strictly projective basis was first shown by voi
Staudt.* The point algebra on a line which is defined in this chapte
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 accoum
of which will be found in 55. The original method of von Staudi
has, however, been considerably clarified and simplified by moderr
researches on the foundations of geometry f All the definitions ano
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 l t and on I three distinct (arbitrary) fixed points which
for convenience and suggestiveness we denote by P Q , P lt P a) we define
two one-valued operations $ on pairs of points of I with reference to
the fundamental points P , P^ P n The fundamental points are said
to determine a scale on I.
DEFINITION. In any plane through I let l n and IL be any two lines
through P n) and let Z be any line through JfJ meeting Z and IL in
points A and A' respectively (fig 71). Let P x and P y be any two points
of I, and let the lines P X A and P V A' meet Z and Z* m the points X
and Y respectively. The point JJ +f , in which the line JET meets I, is
called the sum of the points P x and P y (in symbols P^+P^P^^ in
* K G. 0. von Staudt (1798-1867), Beitiage zur Geometne derLage, 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, TJeber emen Geometrischen Calcul, Acta
Mathematica, Vol. XXIX, p 1.
| By a one-valued operation o on a pair of points A, B is meant any process
whereby with every pair 4, B is associated a point (7, which is nmque provided
the order of A, B is given j in symbols AoB = C Here "order" has no geo-
metrical significance, but implies merely the formal difference of AoB and Bo A
TiAQBs=BoA, the operation is commutatwe; if (J.oJB)oC= Ao(BoG), the opera-
tion is associative,
HI
142
ALGEBRA OF POINTS
[CllAl?. VI
the scale P , P v P a 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 }>, Q(/i/J7>,
) is a necessary and sufficient condition for the (^nudity
Z + r ( A > E )
This follows immediately from the definition, AA'Y being a
quadrangle which determines the given quadrangular set.
COROLLAEY 1. If P x is any point of I, we have P x +P<> Ps + P K ~P x ,
andP^P m ^P m +P m ^^(P m ^P m ), (A,E)
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 w) P a . (A, K)
This follows from the theo-
rem above and the corollary of
* The historical origin of this con-
struction will be evident on inspection
of the attached figure This ia 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 P y along the line J,
which brings its initial point into coincidence with the terminal point of the vector
PP*, which is the ordinary construction for the sum of two vectors oa a Itoe,
ADDITION 143
Theorem 3, Chap II, in case P x and P y are distinct from 7 and /;.
If one of the points P x , P y coincides with P (} or P w) it follows from
Corollary 1
COEOLLARY 3. The operation of addition is associative; i.e
for any three points P x> P u , P g for which the above expressions are
defined. (A, E)
Proof (fig. 73). Let P^ + Py he determined as in the definition by
means of three lines L, V n , 1 Q and the line XY. Let the line P Q Y be
denoted by ', and by means of l a , IL, IQ construct the point (P x +P y ) -f P z)
which is determined by the line ZZ t say. If now the point P v -h 7*
be constructed by means of the lines l a> IJ,, IQ, and then, the point
P J , + (P y + P z ) be constructed by means of the lines , /, 1 0> it will be
seen that the latter point is determined by the same line XZ,
COROLLARY 4 The operation of addition is commutative; i.&.
<K "T* ^y "~ -*y "T~ J~x
for every pair of points P y> P v 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 a!> P v P 0) P n P^+y), which by the
theorem implies that P v + P x = P a + ^ But, by definition,
Hence P y + P x
ALGEBRA OF POINTS [CHAP VI
THEOREM 2. Any three points P x> P y , P a (P a *&) satisfy the relation
ie. the correspondence established ly malting each point P r of I corre-
spond to P x ' = P x + P a > where P a (* P.) is any fixed point of I, is projectile.
(A,E) * .
Proof The definition of addition (fig 71) gives tins projectivity as
the result of two perspectivities : *
The set of all pi oj activities determined by all possible choices of P a in the
formula P' x = P x +P<t is the group described in Example 2, p. 70. Thu stun of
two points P a and P 6 might indeed have been defined as the point into which
P b is transformed when P is tiansformed into P by a projectivil-y of this
group. The associative law for addition would thus appear as a special oaao
of the associative law which holds for the composition of correspondence's in
general ; and the commutative law for addition would be a consequence of tho
commutativity of this particular gioup,
Po
49. Multiplication of points. DEFINITION In any plane through /
let J , l v l v be any three lines through P , P v P m respectively, and let J t
meet Z and 4, in points A and B respectively (fig. 74). Let P x) P u be any
two points of I, and let the lines P X A and P V B meet C and / in the points
X and T respectively. The point P XJI in which the line JET meets I is
-^ *J? 1**^. ** 71 corres P d to the notation of this theorem, P v must T>e
identified with P a , ' *
49] MULTIPLICATION 145
called the product ofP x ly P y (m symbols JP P y = PJ m the scale P Q , P lt P a
on I. The operation of obtaining the product of two points is called
multiplication * Each of the points P x , P v is called a factor of the
product .% P y .
THEOEEM 3 If P x and P y are any two points of I distinct from
PO> ^v P> Q(^-^J P<P v P X u) w necessary and sufficient for the equality
P x P V = P XV . (A,E)
This follows at once from the definition, AXJBY being the defining
quadrangle.
COROLLARY 1 For any point P x (^=P a> ) on I we have the relations
P ' P = P . P = P ' P P = P P P p p p T> r> f-p, r>\
*! ** '-at -*! - l x> J t x - L x J -Q J: o> - c *> J: x J-y *> K> (J^ = 1 Q ).
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 9 P a and P m P . (A, E)
This follows from Corollary 1, if one of the points P x , P u coincides
with P , P v or j. Otherwise, it follows from the corollary, p. 50, in
connection with the above theorem.
B
* The origin of this construction may also be seen m a simple construction of
metric Euclidean geometry, which results from the construction of the definition
by letting the line l u be the "line at infinity" (of. p 8) In the attached figure
which gives this metric construction we have readily, from similar triangles, the
proportions:
which, on taking the segment P Pi=l, gives the desired result PoP^^
146 ALGEBRA OF POINTS [CHAP VI
COROLLAKY 3 The operation of multiplication is associative, i.e we
have (P v 'P y )'P z = P 3 (P v P s ) for every three points P x , P v , P s 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 v be constructed
J?IG 70
as in the definition by means of three fundamental lines 1 0> l
the point P xy being determined by the line XY. Denote the line
by l[, and construct the point P^ P e = (P f J;J) - P g) using the linos Z , J|, ?,
as fundamental Further, let the point P a J = / be constructed by
means of the lines 1 0) l(, I*, and then let P x - P^ P X > (P v - 2*) be con-
structed by means of 1 Q) l v /. It is then seen that the points I* J^
and P xa P g 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 J and J 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 coin
imitative law for multiplication as in the case of addition.
An important theorem analogous to Theorem 2 is, however, inde-
pendent of Assumption P Et is as follows
THEOREM 4 If the relation P^-P^P^ holds letween any three
points PI, P v , P IU on I distinct from />, we have P M P^P X -^ P^P^
and also P^P^P^Py-j^P^P^P^; ie. the correspondence established by
making each point P x of I correspond to J^'=^ -P a (or to Pj = P a 'P,),
where P a is any fixed point of I distinct from P Q) is protective. (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 deteimined by all choices of P a in the for-
mula P f x T> x P a 1S t ne group described in Example 1, p 69 The proper-
ties of multiplication may be legarded as properties of that group in the same
way that the piopeities of addition aiise fiom the gioup descubed 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 v , P s are any three points on I (for which the operations
"below are defined), we have
Proof Place
+?=+* -S'-S-J
By Theorem 4 we then have
But by Theorem 1 we also have Q(P a ,P x P 0> %,%%+) Hence, by
Theorem 1, Cor., Chap IV, we have Q(^ ^A+a,)) which,
by Theorem 1, implies % + %=%&+ The relation
is proved similarly.
14g ALGEBBA OF POINTS [CHAP VI
50 The commutative law for multiplication. With the aid of
Assumption P we will now derive finally the commutative law for
multiplication :
THEOREM 6. The operation of multiplication in commutative, ; i.e.
we have P x P = P y % f or ever y P air f ^ oin ^ J * ^ l ^ T wllich
these two products are defined. (A, E, P)
Proof. Let us place as before J - P = P xy , and P v - P, = P vr . Then, by
the first relation of Theorem 4, and interchanging the points P A , P yf
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. IV) is equivalent to
Assumption P, it follows (cf. 3, Vol. II) that :
THEOREM 7. Assumption Pis necessary and sufficient for the, com-
mutative law for multiplication.* (A, E)
51. The inverse operations. DEFINITION. Given two points J' tt Jj
on /, the operation determining a point P x satisfying the rotation
j4-JP=Jjis called subtraction; in symbols /jJ--J* = JP. The point
P x is called the difference of .ZjJ from P a . Subtraction is the iiwne of
addition.
The construction for addition may readily be reversed to give a con-
structionf or subtraction. The preceding theorems on addition then give:
THEOREM 8 Subtraction is a one-valued operation for every pair
of points P , P b on I, except the pair P M> P n , (A, K)
COROLLARY. We have in particular J-~JP==:.7jJ for every point
nl (A, E)
* The existence of algebras in which multiplication is not commutative ifl then
sufficient to establish the fact that Assumption P is independent of the previous
Assumptions A and E Por in order to construct a system (cf . p. (5) 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 uae as a
baswa nbncomnratative number system, e g. the system of quaternions. That the
fundamental theorem of protective geometry is equivalent to the commutative
fcw for multiplication was first established by Hilbert, who, in MB Foundations of
Crfometry, showed that the commutative law is equivalent to the theorem of Pappus
mStheo The ktter M eaSily Seen t0 b6 ec * ulvalent to to 6 funda "
so, si, 52] ABSTRACT NUMBER SYSTEM 149
DEFINITION. Given two points P at P b on l\ the point P x determined by
the relation P a P x = P b is called the quotient of P h by P u (also the ratio
of P b to P a ) ; in symbols P b /P a = P^ or P b : P a - P, The operation deter-
mining P b /P a is called division; 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 onl except the pairs P^ P Q and J, P n (A, E)
COROLLARY. We have in particular P a /P a = P i} PjP a =^P^, %/% = %,
etc., for every point P a onl distinct from P and P n . (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 (cl Chap. Ill, p. 66), the general con-
cept of a number system is descnbed 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 o
respectively, exist and operate on pairs of elements of N under the
following conditions:
1. The set N forms a group with respect to .
2 The set N forms a group with respect to o, except that if i + is
the identity element of N with respect to , no inverse with respect
to o exists for *' + .f If a is any element of N, a o i + = i + O a = i + .
3. Any three elements a, &, c of N satisfy the relations a (b e)
= (aofy(aoc) and (& c) o a, = (& o a) (c o a)
The elements of a number system are called numbers, the two oper-
ations and o are called addition and multiplication respectively.
If a number system, forms commutative groups with respect to both
addition and multiplication, the numbers are said to form b
* What we have defined is more precisely right-handed division The left-handed
quotient is defined similarly as the point P* determined by the relation Pa, P = 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 (jp a prime), etc.
150 ALGEBRA OF POINTS [CHAP.VI
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.
The whole terminology of this 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.
DEFINITION. 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)
isomorphic.*
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. Nonhomogeneous 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 &
be excluded from the set. In this group 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.
03] COORDINATES 151
point; !$> has been excluded, forms a group with respect to multipli-
cation, except that no inverse with respect to multiplication exists
for J^; J^ 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
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 6. 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 The set of all points on a line on which a scale lias
"been established, and from which the point J is excluded, forms a field
with respect to the operations of addition and multiplication previously
defined. (A, E, P)
This provides a new way of regarding a point, viz., that of regarding
a point as a number of a number system This conception of a point
will apply to any point of a line except the one chosen as &. 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, ft, c, > , 7, &, , as, y, z t - which
is isomorphie with 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 m 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 J must
correspond to 0, the point J 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 ALGHEBBA OF POINTS [CHAP VI
point will correspond a number (except to J), and to every number
of the field will correspond a point In tins way every point of tlie
line (except j) is labeled by a number. This number is called the
(nonliornogeneous) 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 x"
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 , l v l n of a flat
pencil determine a scale in the pencil of lines; and three planes
a , a v a 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 Q ,P^P a } and let the resulting field of points
on a line be made isomorphic with a field of numbers 0, 1, a, , so
that J corresponds to 0, P^ to 1, and, in general, P a to a. For the
exceptional point P*, let us introduce a special symbol oo 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 protective correspondence between
these points. Let x be the coordinate of any point of L We have seen
that if the point whose coordinate is as is made to correspond to either
of the points
(I) a/ # + , (a^oo)
or (II) x'-ax,
54] LINEAR FRACTIONAL TRANSFORMATION 153
where a is the coordinate of any given point on I, 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
(III) a/-i,
the resulting correspondence is likewise projective. For we clearly
have the following construction for the point l/.u (fig. 77} With the
same notation as before for the construction of the product of two
numbers, let the line scA meet /, in JT. If Y is determined as the
intersection of IX with 1 0) the line BY determines on I a point #',
such that scx'=l, by definition. We now have
M. JL
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 x'~ go + a, from its definition, leaves the point J&,
which we associated with oo, invariant. We therefore place oo -f 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 so + a gives at once a = 0, if $ = oo. Further, by prop-
erly choosing a, any point on can be made to correspond to any point x f ;
154 ALGEBRA OF POINTS [CHAP. VI
but when one such pair of homologous points is assigned in addition
to the double point oo, the projectivity is completely determined.
The resultant or product of any two projectivities '=,'+ ami
x l =x + b is clearly x' = x + (a-\- 6). Two such projectivities are
therefore commutative.
The projectrvity os 1 = ax, from its definition, leaves the points and co
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 co = a co (a = 0) Here, also, by properly choosing a, any
point a? can be made to correspond to any point ai', but 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 1 = ax and x' = lx is clearly x' = (ab) ss, so that any two projectivities of
this type are also commutative (Theorem 6).
Finally, the projectivity x' l/x, by its definition, makes the
point co correspond to and the point to oo. We are therefore led
to assign to the symbol co the following further properties : ] /oo = 0,
and 1/0 = co. This projectivity leaves 1 and 1 (denned as 1)
invariant Moreover, it is an involution because the resultant of two
applications of this projectivity is clearly the identity; i.e. if the
projectivity is denoted by TT, it satisfies the relation, TT* =s 1.
THEOREM 11. Any projectivity on a line is the product of projec-
tivities of the three types (/), (//), and (III), and may "be expressed
vn, the form
(1) *-" + *
v ' ax + d
Conversely, every equation of this form represents a projectimty, if
ad-bo^Q (A, E, P)
Proof. We will prove the latter part of the theorem first. 1C wo
suppose first that c ^ 0, we may write the equation of the given
transformation in the form
&-*?
(2) a/ = ^+ V
v c w + d
This shows first that the determinant ad lc must be different from
, otherwise the second term on the right of (3) would vanish, which
54] LINEAR FRACTIONAL TRANSFORMATION 155
would make every x correspond to the same point a/c, while a pro-
jectivity is a one-to-one correspondence. Equation (3), moreover,
shows at once that the correspondence established by it is the result-
ant of the five :
/7 ft< A /
[O -- ) # 8 , of = x , -f
\ c /
Uc Q, and ad = 0, this argument is readily modified to show that
the transformation of the theorem is the resultant of projectivities of
the types (/) 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 by an equation s/ = To do this simply, it is desirable
to determine first what point is made to correspond to the point oo by
this projectivity. If we follow the course of this point through the
five projectivities mto which we have j'ust 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 GO,
viz, , ,
ax + o a
x = GO.
cx + d o
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, co are transformed into the points p, q, r respectively.
Then the transformation
clearly transforms into p t 1 into #, and, by virtue of the relation
just developed for oo, it also transforms co into r. It is, moreover, of the
form of (1) The determinant ad "be is in this case (# p) (r(i)(r p},
which is clearly different from zero, if p, q, r are all distinct. This
transformation is therefore the given projectivity.
COROLLARY 1, The projectivity so r *= a/x(a = 0, or oo) transforms
oo and co into 0. (A, E, P)
156 ALGEBBA OF POINTS [CHAP. VI
For it is the resultant of the two projectivities, a^-l/a and
a'= ax of which the first interchanges and oo, while the second
leaves them both invariant. We are therefore led to define the tymbok
a/Q and a/cc as equal to oo and respectively, when a is neither
nor oo
COROLLARY 2. Any projectivity leaving the point oo invariant may
le expressed in the form x' = ax + 1. (A, E, P)
COROLLARY 3 Any projectivity may le expressed analyticalli/ tyj
the -bilinear equation cxx' + dx'-ax-l - 0; and conversely, any
Ulmear equation defines a projective correspondence letwwn its two
variables, provided adl>c=f= 0. (A, E, P)
COROLLARY 4 If a projectivity leaves any points invariant, the
coordinates of these double points satisfy the quadratic equation
cy ?j r (d-a)x-'b=Q. (A, E, P)
DEFINITION. A system of mn numbers arranged in a rectangular
array of m rows and n columns is called a matrix, if m s= n, it is
called a square matrix of order n *
The coefficients f a ^ j of the projective transformation (1) form a
V /
square matrix of the second order, which may bo conveniently used to
denote the transformation Two matrices \ } , } and ( a , L } repre-
yc (if \c a/
sent the same transformation, if and only if a : a' I ; I' s c : c' ~ d : d 1 .
The product of two projectivities
ax + 5 ,
aud so" ~ TT.
cas + d 1
is given by the equation
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
fa' V\/a l\ faa' + cb 1
\c' d')\c d) \ac r +cd'
. the P 11110 ^ 1 Properties of
r Algebra, pp. 20 ff.
54,55] THBOWS 157
Tins gives, in connection with the result just derived,
THEOREM 12. The product of two projcctivities
7T , ,
.c a
is represented ly the product of their matrices , in symbols,
COROLLARY 1. The determinant of the product of two projectivities
is equal to the product of their determinants (A, E, P)
COROLLARY 2. The inverse of the projectivity TT = ( , ) is given
\ G w
"by vr" 1 = ( ~" ) = ( r> T) ) where A, B> C, D are the cof actors
\ 6 a I \J5 JLs I
\ / \ / -T
of a, &, c, d respectively in the determinant
c d
(A,E,P)
This follows at once from Corollary 3 of the last theorem by inter-
changing a?, x' We may also verify the relation by forming the
product 7r~V = ( T~ , , ), which transformation is equiva-
/i \ \ ad ~ lc '
lent to ( n .j I The latter is called the identical matrix
CPROLLARY 3. Any involution is represented ly ( a ), that is
\6 ~ a I
ax 4- 1)
ly a/ = ~ , with the condition that a* + Ic =f= (A, E, P)
Cw * ' Cfi
55. Von Statidt's algebra of throws. We will now consider the
number system of points on a line from a slightly different point of
view On p. 60 we denned a throw as consisting of two ordered
pairs of points on a line ; and denned two throws as equal when they
are projective. The class of all throws which are projective (i.e; 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 f J?, C? of a throw and their places in the symbol
T(AB, 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 one of the (projective) throws which define it. We
shall also denote marks by the small letters of the alphabet. And so,
since the equality sign ( ) indicates that the two symbols between
ALGEBKA OF POINTS [CHAr. VI
which it stands denote the same thing, we may write T(AB t
a = l if a, b, T(AB, CD) are notations for the same mark. Thus
T(AB, OZ>)'=T(2L4, VC} = 7(CD, AB)=>T(I>C, J3A) 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 foUowing sets of two ordered pairs, where A, Jt, are
distinct. The set of all throws of the type T (AB, CA) is called a
mark and denoted by co; the set of all throws of the type T(AJ!, OB)
is called a mark and is denoted by ; the set of all throws of the type
T(AB, CO) is a mark and is denoted by 1. It is readily seen that
if J, J?, J are any three points of a line, there exists for every point
P of the line a unique throw T (Jg J, J? 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 J; the mark to /% and
the mark 1 to P.
DEFINITION. Let T(AJ3, CXJ be a throw of the mark , and let
T(AB, <7D 2 ) be a throw of the mark &; then, if Z> 8 is determined by
Q(AD 1 B, AD Z D S ), the mark c of the throw T(A, CD 9 ) is called the
sum of the marks a and 5, and is denoted by a + & ; in symbols,
a + b = c. Also, the point D^ determined by Q(AI> i C } JSD^) deter-
mines a mark with the symbol T(AjB, CD' 6 ) ** o f (say), which is called
the product of the marks a and & ; in symbols, ab = c'. As to the
marks and 1, to which these two definitions do not apply, we define
further: a+0 = + a = a, a-0 = 0-a = 0, and a 1 a 1 . a a.
Since any three distinct points A, J3, C may be projected into a fixed
triple R, %, J, it follows that the operation of adding or multiplying
marks may be performed on their representative throws of the form
T(Z,P IIP). By reference to Theorems 1 and 3 it is then clear that
the class of all marks on a line (except co) 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
5M,Bfl] CROSS TUT 10 159
given it here partly on account of its historical importance; partly
because it gives a concrete example of a number system isomorphio
with the points of a lino*; and partly because ifc gives a natural
introduction to the fundamental concept of the cross ratio of four
points. This we proceed to derive in the noxt section.
56. The 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 tins section apply also
to the other one-dimensional primitive forms, i.e. the pencil of lines
and the pencil of planes. With 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 projective with the
first, and is therefore an invariant under any protective transforma-
tion, i.e. a property oC the throw that is not changed when the throw
is replaced by any protective throw This number is called the cross
rutin of the throw. Jt is also called the double rtttw or the anhar-
mtfiu'ft ratio. The reason for these names will appear presently.
hi general, four given points yivG rise to si,u rffj/tirunt cross ratios.
For the 24 possible permutations of the letters in the symbol
T(AJi, CD} fall into sets of four which, by virtue of Theorem 2,
Chap. JIT, have the same cross ratios. In the array below, the per-
mutations in any line are projective with each other, two permuta-
tions of different lines being in general not projective :
AH, CD
HA, 2)0
DC, JL-L
CD, AK
AB, DG
MA, CD
CD, HA
DC, AB
AC, HD
CA, DB
DK, CA
rw, AC
AC, DJ*
CA, IW
IW, CA
DB, AC
AT), HC
DA, CJi
cj?, DA
JiC, AD
AD, Cli
DA, BC
BC, DA
CH, AD
If, however, the four points form a harmonic set H (AB, CD), the
throws T(AB, CD) and T(AH, 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, gim rise to only
three cross ratios. The values of these cross ratios are readily seen
* Ct 53. Here, with every point of a line on which a scale has been estab-
lished, is associated a mark which, is the coordinate of the point.
ALGEBRA OF POINTS [CHAP. VI
to b e __ i l 2 respectively, for the constructions of our number
system give at once H (Jg, J^PJ, H (J, 7>7?), and H (/;, />/.{).
We now proceed to develop an analytic expression for the cross
ratio B (xfa t x s x t ) of any four points on a line (or, in general, of any
four elements of any one-dimensional primitive form) whoso coordi-
nates in a given scale are given. It seems desirable to proewlo 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 R (x^, -' a .r 4 ) of elements ;>\, ;>' s , >!.,, .'^
of any one-dimensional form is, if x i} sc v u\ arc dLstiiu-t, Llw coordi-
nate X, of the element of the form into which a; 4 i,s tran Rftiriuod by
the projectivity which transforms ss v x r a into co, 0, I re,spt3(5tivly ;
le the number, X, defined by the projectivity ^j.,.'/',,,^ A ooOlX. If
two of the elements x v x z , x 3 coincide and 4 is distinct from all of
them, we define R (xjs z , xjsj as that one of $ (^,r 1( ,*y g ), Ijb (,/
sc,% ), $ (x.x si x n x,), for whicli the first three elements tiro
""I 2/' ^^ \ 4 8' S 1' *
THEOREM 13 IVie cross ratio "& (x^, x^ of the, four
whose coordinates are respectively x v x z) x z> x is yircn 1>y the, mltttitm
(A,E,P) ^"^ ^""^
Proof. The transformation
jKj a? a3 g a?
is evidently a projectivity, since it is reducible to the form of a
linear fractional transformation, viz..
in which the determinant (x^ x^ (x z as,) (a? 8 ^) is not zoro, pro-
vided the points % v x z) x s are distinct. This projectivity transforms
ay ss 2 , x z into co, 0, 1 respectively. By definition, therefore, this pro-
jectivity transforms 4 into the point whose coordinate is the cross
ratio in question, i.e. into the expression given in the theorem. If
a^ 2 , x s are not all distinct, replace the symbol B (0^, 8 a? 4 ) by one
of its equal cross ratios B (x z x v a; A ), etc. ; on of these must have
the first three elements of the symbol distinct, elnoe in a cross ratio
of four points at least three must be distinct (d$f,).
56] CROSS RATIO 161
COROLLARY 1. We have in particular
Tfr (x,x v x s x^ = oo, B (x^ z , x s x s ) = 0, and B (x^ t x a x s ) = 1,
if x v x z , x 9 are any three distinct elements of the form. (A, E)
COROLLARY 2. The cross ratio of a harmonic set H (x^, x a x t ) is
B (a:^, x a x t ) = 1, for we have H (GO 0, 1 1) (A, E, P)
COROLLARY 3 If B (x^, a? 8 oj 4 ) = X, tf/ie other Jive cross ratios of the
throws composed of the four elements w v x a , x s , x are
B (x^, a A) = ^ > ^ (0i4 ^a^a) = ~Y-
$ (aj.-a:.. a;,,a3,) = 1 X.
V * " 2 *'
1a , ^
(A, E, P)
The proof is left as an exercise.
COROLLARY 4 If x v x^, x a , x^ form a harmonic set H (x^ x^x^,
we have
2 = ! + ! .
i 1 i *v* ^y* .--i 'y 'v* . ___ o"
Wo ^^ W- l4>a ^^ <**.* W/j ^^ bt*.|
(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', &', c' are any three other distinct
elements of the same form, tJien the correspondence established "by the
relation Tfr (ab } ex) = B (a'&', c'x 1 ) is projective. (A, E, P)
Proof. Analytically this relation gives
a c "b x
a x & o a'' b' - G'
which, when expanded, evidently leads to a bilinear equation in
the variables x, x', which defines a projective correspondence by
Theorem 11, Cor. 3.
That the cross ratio
ffi i ~" ^a . x a_~~_ x $
1 "**** A S *""" *4
is invariant under any projective transfoimation may also be verified directly
by observing that each of the three types (I), (II), (III) of projectivities on
pp. 152, 158 leaves it invariant That every projectivity leaves it invariant
then follows from Theorem 11.
162 ALGEBRA OF POINTS [CHAP. VI
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 tho
following theorem :
THEOREM 14 The coordinates of the points of the net of rationality
R(j=>jj)/om a number system, or field, which consists of all numbers
each of which can be obtained by a finite number of rational algebraic
operations on and 1, and only these (A, E)
Proof. By Theorem 14, Chap. IV, the linear net is a lino of the
rational space constituted by the points of a three-dimensional not of
rationality. By Theorem 20, Chap. IV, this three-diimmsional not in
a properly projective space. Hence, by Theorem 10 of the present
chapter, the numbers associated with R(Olco) form a field.
All numbers obtainable from and 1 by the operations of addi-
tion, subtraction, multiplication, and division are in R(Olco), bueanso
(Theorem 9, Chap. IV) whenever x and y are in R (Olco) the, quadran-
gular sets determining x + y, xy, % y, ly have live out of six
elements in R(Oloo). 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(Olco)
with respect to two others, b, c, can be obtained by a finite number
of rational operations on a, b, c. This fact is a consequence of Theo-
rem 13, Cor 2, which shows that sc is connected with a, b, c by the
relation
(x b)(a c) + (x c) (&) 0.
Solving this equation for x, we have
2 be - ab ao
x ~- , >
2a 5 o
a number * which is clearly the result of a finite number of rational
operations on a, 5, 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 st cannot be indeterminate unless 6 s= c,
57,58] HOMOGENEOUS CQOKDINATES 163
three points on a line. If in that triple-system geometry wo perform the con-
struction foi the number 1 + 1 on any line in which we have assigned the
numbeis 0, 1, oo to the thiee points of the hue in any way, it will be found
that this consti uction yields the point Thit> is due to the fact previously
noted that in that geometiy the diagonal points of a complete quadi angle
are collmeai In every geometry to which Assumptions A, E, P apply we
may constiuct the points 1 + 1,1 + 1 + 1,, , thus foiimng a sequence of
points which, with the usual notation foi these sums, we may denote by 0, 1,
2, 3, 4, . Two possibilities then present themselves, either the points
thus obtained aie all distinct, in which case the net R (Olw) contains all the
ordinary lational numbers ; or some point of this sequence coincides with one
of the pieceding points of the sequence, in which case the numbei of points
in a net of lationality is finite. We shall consider this situation in detail in
a later chaptei, and will then add furthei assumptions Here it should be
emphasized that our lesults hitheito, and all subsequent results depending only
on Assumptions A, E, P, are valid not only in the ordmaiy real or complex
geometries, but in a much more general class of spaces, which are chaiacter-
ized meiely 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 ., as the coordinate of which we introduced a symbol
ao with exceptional properties, often proves troublesome, and is, more-
over, contrary to the spirit of projective geometry in which the points
of a line are all equivalent ; indeed, the choice of the point R was
entirely arbitrary. It is exceptional only in its relation to the opera-
tions of addition, multiphcation, etc., which we have denned 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 thret
points to be the points J, J, J ; 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 # a ) of this system in a given order,
such that if x is the (nonhomogeneous) coordinate of any point dis-
tinct from J&, the pair (x v C 2 ) associated with the point on satisfies the
relation x Jx y With the point J we associate any pair of the
form (k, 0), where Tc is any number (Jc 3= 0) of the number system
isomorphic with the line To every point of the line corresponds a pair
of numbers, and to every pair of numbers in the field, except the pair
ALGEBRA OF POINTS [OHAP.VI
(0, 0), corresponds a unique point 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 v x z ) and (mx v mx z ) represent the same point for all
values of m different from The point J is characterized by the
fact that X T ; the point J& by the fact that # 2 = ; and the point
J;* by the fact that a^ = x z .
THEOREM 15. In homogeneous coordinates a projectimty on a line is
represented ly a linear homogeneous transformation in two variables,
m P x>=ax,+ lx z , (ad-lo+Q)
v ' pa% == cx^ + dx 2 ,
where p is an arbitrary factor of proportionality. (A, E, P)
Proof. By division, this clearly leads to the transformation
(2) r - fla + ^'
w ex + d
provided a? 2 ' and x z are both different from 0. If a3 a = 0, the trans-
formation (1) gives the point (#/, x) = (a, c) ; i e. the point ^ =
(1, 0) is transformed by (1) into the point whose nonhomogeneous
coordinate is a/c And if C 2 '=0, we have in (1) ( v % a )~(d, c);
ie (1) transforms the point whose nonhomogeneous coordinate is
d/c into the point JS. By reference to Theorem 11 the validity
of the theorem is therefore established.
( 7i\
, J 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 (x v se s ) must satisfy the
equations x
These equations are compatible only if the determinant of the system
vanishes. This leads to the equation
a p
G
HOMOGENEOUS COOEDI1TATES
165
for the determination of the factor of proportionality p. This equa-
tion is called the characteristic equation of the matrix representing
the projectivity. Every value of p satisfying this equation then leads
to a double point when substituted in one of the equations (3) ; viz ,
if p v be a solution of the characteristic equation, the point
(x v x z ) = (-&,- Pl ) = (d- Pi> - c)
is a double point.*
In homogeneous coordinates the cross ratio 1^(AB, CD) of four
points A = (a v a a ), B = (& & 3 ), C = (c v c 2 ), D = (d v dj is given by
(ae) m (be)
where the expressions (ac), etc, are used as abbreviations for a^a^,
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 x s ) of a point, in addition to the defining relation x l /x 2 =x,
where x is the nonhomogeneous coordinate of the pomt in question.
We choose the relation a, + = 1 If this relation is satisfied,
1 -1
1
1 -1
1
1
1
1 -1
1
1 -1
-
1
1
1
loo,
Thus homogeneous coordinates subject to the condition x^ + x^ I
can be defined by choosing three points A, B, arbitrarily, and letting
jCj, = B (AB, CZ) and 03 2 = B (AC, J?-3T) The ordinary homogeneous
coordinates would then be defined as any two numbers proportional
to these two cross ratios.
* This point is indeterminate only if 6 = c = and. a = d The projeotwty as
then the identity.
ALGEBRA OF 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-dimeusioual 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
os, and that of any point of m by y The question then arises as to
how a projective correspondence between the point 02 and the point y
may be expressed analytically. It is necessary, first of all, to givo a
meaning to the equation y x. In other words : What is meant by say-
ing that two points # on I, 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 I. "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%, J%, JS, 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 pomts J, J, J just mentioned, then we should Icnow
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 uses of
the projectivity which carries the fundamental points 0, 1, co of I
into the fundamental points 0, 1, oo of m, and to assign tlio coordinate
a; 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 isomorphw by this process follows
directly from Theorems 1 and 3 in connection with Theorem 1, Cor.,
Chap. IY. We may now readily prove the following theorem :
* This is shown by the fact that the field of all ordinary complex numbers can
DO isomorphic with itself not only by mafcmg each number correspond to itself, but
also by making each number a + ib correspond to its conjugate a - #>.
69] EXEECISES 167
THEOEEM 16. Any protective correspondence /between the points [x~\
and [y] of two distinct hnes may le represented analytically l>y the
relation y = x ly properly choosing the coordinates on the two lines
Jf the coordinates on the two hnes are so related that the relation
y = x represents a projective correspondence, then any projective cor-
respondence "between the points of the two lines is given ly a relation
ax+l , ., , m
y = ' > (ad oc3= 0).
y cx + d v '
(A, E, P)
Proof. The first part of the theorem follows at once from the pre-
ceding discussion, since any projectivity is determined Tby 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 IT be any given projective transformation of the
points of the line I into those of m, and let TT O be the projectivity
y = x, regarded as a transformation from m to I. The resultant
7r 7r = 7r x is a proj'ectivity on /, and may therefore be represented by
x' = (ax + 1) /(ex + d] Since TT = w ~ 1 '7r 1 , 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 consti uctions for the sum and the pioduct of two lines of a pencil
of lines in which a scale has been established.
3 Develop the point algelna 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 III, show that addition and multi-
plication may be defined as follows . As before, choose three points P , P x ,
PW on a line Z as fundamental points, and let any line through P be labeled
Z,. Then the sum of two numbers P a and P y is the point P x + y into which P v
is transformed by the elation with axis Z, and center P w which transforms
P into P,,.; and the product P x P v is the point P w into which P y is trans-
formed by the homology with axis Z w and center P which transforms P x into
PX. Develop the point algebra on this basis without using Assumption P,
except in the proof of the commutativity of multiplication.
168 ALGEBRA. OF POINTS [CHAP. VI
5 If the relation ax = ly holds between four points a, &, x, y of a lino,
show that we have Q(Qba, coyr). Is Assumption P necessary for this losult 1 ?
6 Piove by duect computation that the expiession l~ x & ?.'g "", >' liS
*i-J4 x s -J>i
unchanged in value when the foui points x lt x z , x s , x are subjected to any
linear fractional transformation x r ? x - -
ex + a
7. Prove that the transformations
\t \ V w -I i \t 1 \/ A, .., X 1
A A, A i A = 1 A, A = > A = r > A =
A 1 A A -~ Z X
form a group. What aie the periods of the vanous transformations of tins
group? (Cf Theorem 13, Cor, 3.)
8. If A, B, C, P v P z , , P n are any n + B points of a line, show that
every cross latio of any four of these points can be expressed rationally in
terms of the n cross ratios \= ft (AB, CP { ), = 1,2, , n. When n = 1
this reduces to Theoiem 13, Cor. 3. Discuss m detail the case n = 2,
9. If T% (XjX z , x s x) = X, show that
1-X _ 1 X__
a; 8 a, 4 a; 3 x z x s x^
The relation of Coi. 3 of Theoiem 13 is a special case of this relation.
10. Show that if B (AB, CD) = ft (AS, DC), the points fonn a harmonic
set H (AB, CD)
11. If the cioss ratio R (A B, CD) - X satisfies the equation X 2 - X + 1 = 0,
then $ (AB, CD) = T&(AC, DD) = TJ (AD, J3C) = X,
and $ (AB, DC) = $ (A C, BD) = # (.4 D, CU) = - X a .
12. If A, B, X, Y, Z are any five distinct points oil a line, show that
B (AB, XY) B (AB, TZ) U (AB, ZX) = 1.
13. State the eorollaues of Theorem 11 lu homogeneous coordinates.
14. By direct computation show that the two methods of determining the
double points of a projectivity described in 54 and 58 arc equivalent
15 If Q(ABC,XYZ),ibsa.
B (AX, YC) + T* (BY, ZA) + # (CZ, X7i) , 1.
16. If M v J/ 2J M t are any thiee points in the plane of a line / but, not on
/, thecross ratios of the lines I, PM PM Z , PM a aro dlilmMil for ay two
points P on / ^
17. If A, B are any two fixed points on a line Z, and A", F are two variable
that B (AB, XY) is constant, the set m to prc,jcti ve with tha
CHAPTER VII
COORDINATE SYSTEMS IN TWO- AND THREE-DIMENSIONAL*
FORMS
60. Nonhomogeneous 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 he the
0-pomt 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,
kyO,, 1, GO,; and
on the other line,
which we will call
the y-axis t by O y ,
l v , co y . Let the
line cOjjOOj, be de-
noted by /.
Now let P be any point in the plane not on Z. Let the lines Poo 9
and Poo.,, meet the re-axis and the ^-axis m points whose nonhomoge-
neous coordinates are a and 5 respectively, in the scales just estab-
lished The two numbers a, 1 uniquely determine and are uniquely
determined by the point P. Thus every point in the plane not on l n
is represented by a pair of numbers ; and, conversely, every pair of
numbers of which one belongs to the scale on the oj-axis and the
other to the scale on the y-axis determines a point in the plane (the
pair of symbols GO,., oo y being excluded). The exceptional character
of the points on I* will 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
78
170 COORDINATE SYSTEMS [CHAP vn
the point co 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 nonhomogeneoiis coordinates of P with reference to
FIG 79
the two scales on the so- and the 2/-axes. The point P is then repre-
sented analytically by the symbol (a, 6). The number a is called the
x-Goordinate or the abscissa of the point, and is always written first
in the symbol representing the point; the number Z> is called the
y-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 , !, oo tt ; and on the other point, which
we will call the it-center, by 0,,, !, QO B . Let the point of intersection
of the lines oo tt , oo be denoted by R> (fig. 79).
Now let I be any lin in the plane not on J. Let the points Zoo,,
and l<x> v be on the lines of the ^-center and the 0-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 I. 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 v-center determines a line in the plane (the
parr of symbols co w , co tf being excluded). The exceptional character
CO, 61]
COUEDINATES IN A PLANE
171
of the lines on & will also be removed presently. The two numbers
just described, determining the line /, are called the norihomogeneous
coordinates of I with reference to the two scales on the u- and
^-centers. The line I is then represented analytically by the symbol
[m, ri]. The number m is called the u-coordinate of the line, and is
always written first in the symbol just given ; the number n is called
the v-coordviiate 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, y) ; a variable line by the symbol [u, v~\. 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 l n and the point J 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
JIG. 80
0, U", V. Let the lines OU and OF be the y- and #-axes respectively,
and in establishing the scales on these axes let the points U, V be
the points <, oo,,, respectively (fig. 80). Further, let the points V, V
be the 'it-center and the ^-center respectively, and in establishing the
COORDINATE SYSTEMS [CHAP VH
scales on these centers let the lines UO, VO be the lines co tt , co u
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
1. on the 0-axis and a point 1, on the y-axis (distinct, of course, from
the points 0, V, F). The scales on the axes now being determined,
we determine the scales on the centers as follows : Let the line on
U and the point - 1, on the a;-axis be the line l w ; and let the lino
on V and the point l g on the y-axis be the line !. All the scales
are now fixed. Let nr be the projectivity ( 59, Chap. VI) between
the points of the cc-axis and the lines of the it-center in "which points
and lines correspond when their so- and w-codrdinates respectively
are the same. If IT' is the perspectivity in which every line on the
w-center corresponds to the point in which it meets the a-axis, the
product TT'TT transforms the #-axis into itself and interchanges and
co.,., and 1. and 1^ Hence TT'TT is the involution x' 1/ai Hence
it follows that the line on U whose coordinate is u is on the point of
the 3B-aocis whose coordinate is 1/u; and the point on the x-axfa
whose coordinate is x is on the line of the it-center whose coordinate
is 1/sB. This is the relation between the scales on the awixis and
the w-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-as&is whose coordinate is -~l/v;
and the point of the y-axis whose coordinate is y is on that line of the
^-center whose coordinate is l/i/.
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 I has coordinates [m, ri\. The condition
that P be on I is now readily obtainable. Let \is suppose, first, that
none of the coordinates a, 6, 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/wi, and meets the y-axia in a
point whose y-coordinate is 1/n (fig 81). Also, by definition, the
line joining P = (a, 5) to U meets the aj-axis in a point whose #-eob*r~
dinate is a ; and the line joining P to V meets the y-axis in a point
whose y-coordinate is & If P is on l t we clearly have the following
perspectivity :
62]
(1)
Hence we have
(2)
COORDINATES IN A PLANE
173
- l -o,
m
i 0,oo A
n y /
which, when expanded (Theorem 13, Chap. VI), gives for the desired
condition
(3) ma, + rib + 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. VI)
Oaca x -r- Ooo 6.
m A n
But since this projectivity has the self-c'orrespondmg element O, it
is a perspectivity which leads to relation (1). But this implies that
P is on L
81
If now a = (b 3* 0), we have at once 5 = 1/n ; and if 6 = (a =5*= 0),
we have likewise a = 1/m for the condition that P be on I But
each of 'these relations is equivalent to (3) when = and 6=0
respectively. The combination a, = 0, 6 = gives the origin wliich.
is never on a line [m, n] where m 3= GO ^ 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
COORDINATE SYSTEMS [CHAP.VII
THEOREM 1 The necessary and sufficient condition that a point
P = (a, 1) le on a line I = [m, n] is that the relation ma + nb + 1 =
be satisfied.
DEFINITION. The equation DEFINITION. The equation
which is satisfied by the coordi- which is satisfied by the coordi-
nates of aU 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.
COROLLARY 1. The point equa- OOHOLLAIIY 1'. The line equa-
tion of the line [m, n] is tion of the point (a, 5) is
EXERCISE
Derive the condition of Theoiem 1 by dualizing the proof given.
63. Homogeneous coordinates in the plane. In the analytic repre-
sentation of points and lines developed m the preceding sections the
points on the line TJV o 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 iionhomogeneous coordinate x of
a point on a line by a pair of numbers x v # 2 whose ratio $j/K 2 was
equal to x(x = eo), and such that a? a = when x = co.
A similar system of homogeneous coordinates can be established for
the plane. Denote the vertices 0, U, V of any triangle, which we will
call the triangle of reference, by the " coordinates " (0, 0, 1), (0, ] , 0),
(1, 0, 0) respectively, and an arbitrary point T t not on a side of tho
triangle of reference, by (1, 1, 1). The complete quadrangle QUVT
is called the/rame of reference* of the system of coordinates to bo
established The three lines UT, VT, OT meet the other sides of the
triangle of reference in points which we denote by 3 e (I, 0, 1),
l tf = (0, 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 a? a , :). Observe first that wo 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 07; and (0, 1, 0), (1, 1, 0), (1, 0, 0) on UV. The coordi-
nates which we have assigned to these points are all of the form
(x v # 2 , 8 ). The three points on OU are characterized by the fact that
#! = 0. Fixing attention on the remaining coordinates, we choose the
points (0, 0, 1), (0, 1, 1), (0, 1, 0) as the fundamental points (0, 1),
(1, 1), (1, 0) of a system of homogeneous coordinates on the line OU.
If in this system a point has coordinates (I, m), 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 (7c, 0, m) and (Is, I, 0) respectively. We have thus assigned
coordinates of the form (x lf x 2> ss 3 ) to all 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 x : = 0, x s = 0, x 8 = 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 V and U be (k, 0, n) and (0, I', n') respectively. Since
under the hypothesis none of the numbers k, n, I', n' is zero, it is
clearly possible to choose three numbers (ss v x z , x s ) such that aj x : x a
= k:n, and x z : x z = I' . n' We may then denote P by the coordinates
(ac v J 2 , x s ). To make this system of coordinates effective, however,
we must show that the same set of three numbers (x,. x., x.) can be
J,* & o/
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 v a? 2 , x a ) from on the line UV is the point (% v x z , 0). Since
this is clearly true of the point ^ = (1, 1, 1), we assume P distinct
from T. Since the numbers x v x a , x 9 are all different from 0, let us
place xs 1 :x^ = x, and x^:x & = y, so that x and y are the nonhomoge-
neous coordinates of (x v 0, x s ) and (0, (c a , B 8 ) respectively in the scales
on 0V and OU defined by = 0, !, F co,, and = O y , l y , U= oo t .
Finally, let OP meet UV in the point whose nonhomogeneous coor-
dinate in the scale defined by Z7= O e , 1 B , V=s oq, is g; and let OP
meet the line 1J7 in A. We now have
V
co 1 g = 1 TA = en 1 H.
* * A * A
COORDINATE SYSTEMS [CHAI>.VII
where C is the point in which VA meets OU. This projectivity
between the hnes UV and OU transforms 3 into , oo 8 into 0,,, and
l z into 1 . It follows that C has the coordinate 1/s in the scale on
U. We have also ^
,, f
which gives .
, = B KO,, Itf - ( ,0,, - y\ - *?/-
\ U/
Substituting x = OL : a? 8 , and y = a a : OJ B , this tfivoa tlio dcvsirod rolation
2 = c a. The results of this discussion may l>e summarised as
1 *
follows :
FIG. 82
THEOREM 2. DEFHSTITION. If P is any point not on a side of the
triangle of reference OUT, there exist three numbers x v # a , cc 8 (all dif-
ferent from 0) such that the projections of P from the wrtiees 0, V,
V on the opposite sides have coordinates (x^ z) 0), (as v 0, ,'W B ), (0, J a , <r 8 )
respectively. These three numbers we catted the homogeneous coordi-
nates of P, and P is denoted ly (x v 03 2 , B ). Any set of three numbers
(not all equal to 0) determine uniguely 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 (to v 0, a s ) and V to (0, ai a> a$
meet in a point whose coordinates by the reasoning above are (^aj^a?,),
63] HOMOGENEOUS COORDINATES 177
COROLLARY. The coordinates (x lt x z , # 8 ) and (Jcx i} kx z , k% 8 ) determine
the same point, if k is not
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 u z ], it is
denoted in the planar system by \u v u z , 0] In like manner, to the
lines on the other vertices are assigned coordinates of the forms
[0, u & , u s ] and [u v 0, u s ~\ respectively As the plane dual of the
theorem and definition above we then have at once
THEOREM 2'. DEFINITION. If I is any line not on a vertex of tJie
triangle of reference, there exist three numbers u v u z , u & 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 ly the lines
[u v u z) 0], [u v 0, u a ], [0, u s , u a ]. These three numbers are called the
homogeneous coordinates of I, and I is denoted by \u v u a , u a ]. Any
set of three numbers (not all zero} determine uniquely a line whose
coordinates they are.
Homogeneous point and hue coordinates may be put into such
a relation that the condition that a point (x v o? 2 , % s ) be on a line
[u v u z , u s ] is that the relation u^ + w 2 03 3 + u s x s =* be satisfied. We
have seen that if (x v x z , x s ) is a point not on a side of the triangle of
reference, and we place x = xjx z , and y = x z /ac z , the numbers (x, y)
are the nonhomogeneous coordinates of the point (x v x z , x s ) referred
to OF as the a-axis and to OU as the y-axis of a system of nonho-
mogeneous coordinates in which the point T=(l, 1, 1) is the point
(1, 1) (0, V", V being used in the same significance as in the proof of
Theorem 2). By duality, if [u v u a> u a ] is any line not on any vertex
of the triangle of reference, and we place u = uju^ and v = u z /u 9 ,
the numbers [u, v} are the nonhomogeneous coordinates of the line
u u a , % referred to two of the vertices of the triangle of reference
178
COORDINATE SYSTEMS
[('HAP. VII
as Z7-center and F-center respectively, and in which the line [1, 1, 1]
is the line [1, 1] It now, we superpose these two systems of nonhom-
ogeneous coordinates m the way described in the preceding section,
the condition 'that the point (x, y) be on the line [u, v] is that the
relation use + vy + 1 == 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 w
the Z7-center, (1, 0, 0) = V is the F-center, and (0, 0, 1) => is the third
. 88
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 x a = is the line [0, 1, 0], and
the line for which x z = 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 o^ = 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 tlie preceding section
that the condition that (x v os s) aj s ) be, j>n ,the,Kne [it ^, J t is
+ 8 a? 8 0, if none pf
1 7 $.
cs] HOMOGENEOUS COORDINATES 179
is zero. To see that the same condition holds also when one (or more)
of the coordinates is zero, we note lirst that the points (0, 1^ ij
( 1, 0, 1), and ( 1, 1, 0) are collmear. They are, in fact (fig 83), on
the axis of perspectivity of the two perspective triangles OUV and
la-lj,!^ the center of perspectivity being T. 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 x z , 0) of the line
x a = and the traces (a/, a?/, 0) of the lines with the same coordinates,
and this involution is given by the equations
In other words, the line [u v 2 , 0] passes through the point (~u z , u lf 0).
Any other point of this line (except (0, 0, 1)) has, by definition, the
coordinates ( w 2 , u v x & ) Hence all points (x v sc z , # 8 ) of the line
[u v 2 , 0] satisfy the relation u^ + u z ss 2 + u a x s = 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 (x v x z> OJ 3 )
le on a line [u v w a , u a ] is that the relation u^ + u^x a + u s x a = O fe
satisfied
COROLLARY. The equation of a line through the origin of a system,
of nonliomogeneous coordinates is of the form mx + 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 repiesented by (a x , a 2 , a 8 ) and (l v & 2 , 6 g ) if and. only
if the two-rowed determinants of the matrix /?* 2 J 8 \ are all zero
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 ?
lgo
COOBDINATE SYSTEMS
64. The line on two points. The point on two lines. Given two
points, A = (a v a,, 8 ) and 3 = (6 V \, \), 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 v m s ] and n -
\n,n , J to fc"* tne coordinates * the P 0111 *' ^ in tersection of the
two lines.
THEOEEM 4. The equation of THEOREM 4'. The equation of
the line joining the points (a^a^) the point of intersection of the
and (b v l z , & a ) is IMM& [m v m v m a ] and [n v n v n s ] is
= 0.
= 0.
Proof. When these determinants are expanded, we get
n a
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 :
COROLLARY 1'. The coordinates
of the point of intersection of the
lines [m 13 m z , m s ], [n v n^, n^] are
COROLLARY 1. The coordinates
of the line joining the points
(a v a s) a s ), (& & 2 , & 3 ) are
Ms
yt * M ftn
W^B
n
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.
Pi
EXAMPLE. Let us verify the theorem of Desargues (Theorem 1, Chap. II)
analytically. Choose one of the two perspective triangles aa triangle of refer-
ence, say A' = (0, 0, 1), B f = (0, 1, 0), C" = (1, 0, 0), and let the center of per-
spectivity he P - (1, 1, 1). If the other triangle is ABC, we may place
64,63] PROJECTIYE PENCILS
A = (1, 1, ), I? = (1, ft, 1), C = (c, 1, 1) , foi the equation of the lm e p^ f
is x 1 Zy , and since -4 is, by hypothesis, on this line, its fust two coordi-
nates must be equal, and may thereinto bo assumed equal to 1, the thud
cooidinate is aibitiary Similaily foi the other points. Now, from the above
theoiems and then coiollanes we leadily obtain in succession the following :
The coordinates of the line A'B' are [1, 0, 0].
The coordinates of the line AB are [1 aft, a 1, b 1].
Hence the coordinates of their intersection C" are
C"' = (0, 1-6, a- 1).
Similaily, we find the coordinates of the intersection A" of the lines B'C', EC
tobe 4" = (l-c, 6-1, 0);
and, finally, the coordinates of the intersection B" of the lines C'A', CA to be
J3" = (c-l,0, 1-a)
The points A", B", C" are leadily seen to satisfy the condition foi 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 cooidinates Show that the work may be made identical, step
for step, with that above, except foi the interpretation of the symbols.
2 Show that the system of coordinates may be so chosen that a quadi angle-
quadiilateral configuration as represented by all the sets of coordinates that
can be foimed 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 "be, repre- pencil of lines may "be represented
sented "by "by
P = (\ 1 + \\, \a z + Xj&j, p = [/y^
where A = (a^ a z , a a ) and JB = where m = \m v m 2 , m a ] and n =
(b v & 2 , & 8 ) are any two distinct [n v n z> n 8 ] 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 (a^, x z) a; 8 ) of the pencil of
points on the line AB satisfies the relation
182 COORDINATE SYSTEMS
(1)
0.
We may then determine three numbers p, X 2 , X/, sucn that we have
(2) p^XX+V^- (*1, 2, 3)
The number p cannot be under the hypothesis, for then we should
have from (2) the proportion a^ : a s : a s = 5j : 5 2 : & 8 , which would imply
that the points A and B coincide. We may therefore divide by p.
Denoting the ratios X 2 //o and X///D by X 2 and X x , 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 lasc
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 \/X a is called the
parameter of the point it determines. It is here written in liomoga-
neous form, which gives the point A for the value \ = 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 1 +X5 1J a 2 +X& 3 , 3 +X5 3 ),
which is obtained from the preceding by dividing by \ and replacing
\/\ by X. In this representation the point B corresponds to the
value X = oo. We may also speak of any point of tlie pencil under
this representation as the point X X :X 2 or the point X when it corre-
sponds to the value X a /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 + \J3 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 + ^n 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 + \3 be
on the line /* of a pencil of lines m 4- /w. By Theorem 3 tlxe condition
that the point X be on the line /i is the relation
PEOJECTIVE PENCILS 183
a.4-X& 1 )=0.
When expanded this relation gives
1=8 t=8
= 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 Cp\ + Dp A\ JB =
which may also be written *
(1) ,
v ' p
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 protective correspondence between two one-dimen-
sional primitive forms in the plane is obtained by establishing a
protective relation x + /3
/*= >"[x (aS
f/A, -J- O
between the parameters p, \ of the two forms.
In particular we have
COROLLARY 1. Any projectivity in a one-dimensional primitive
form in the plane is given by a relation of the form
where \ is the parameter of the form.
* The determinant
X and /j. is (1, 1).
AS
CD
does not vanish because the correspondence between
184 COORDINATE SYSTEMS [CHAP.VH
COROLLARY 2. If \, X 2 , X 3 , X 4 are the parameters of four claiwnttt
AV A z , A z , A of a one-dimensional primitive form., the cross ratio
B (AiA z , AyA]) is given T>y
A projectivity between two different one-diniensional forms may
be represented in a particularly simple form by a judicious choice oC
the base elements of the parametric representation. To fix ideas, k-t
us take the case of two protective pencils of points. Choose any two
distinct points A, B of the first pencil to be the base points, and lot
the homologous points of the second pencil be base pomln of the
latter. Then to the values X = and X = GO of the first pencil must
correspond the values p = and p = oo respectively of the, second.
In this case the relation of Theorem 6, however, assumes the form
p = 1c\. Hence, since the same argument applies to any diwtmot
forms, we have
COROLLARY 3. If two distinct protective one-dimensional primitive
forms in the plane are represented parametricalli/ so that thfi Itrw
elements form two homologous pairs, the projectivity is represented li/
a relation of the form p = 7cX oetween the parameters p, X of the two
forms.
This relation may be still further simplified Taking again the pruso
discussed above of two projective pencils of points, wo have seen that,
in general, to the point (^+ o v a a + & 2 , a, s + & 8 ), io. to X *= 1, rorro-
sponds the point + &/, a a '+fc& a ', aj+Mh io. the point p**k.
Suice the point ' = (&/, &,/, & a ') is also represented by the sot of wil>wU-
nates (&/, W Zi Jebj), it follows that if we choose the late values f r tho
coordmates of the base point ', to the value X 1 will <;oimsp<md
the value /* = 1, and hence we have always /* X. In other wordn,
we have
COROLLARY 4, If two distinct one-dimensional forms im project far,
the Use elements may le so chosen that the parwnefrra of any two
homologous elements are equal
Before closing this section it seems desirable to call atUmlion
explicitly to the forms of the equation of any line of a ponc.il 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^ + m,x, + m& and
65,66] EQUATION OF A CONIC 2.85
n = njS&i + n z x z + n s aj s , it follows from the first theorem mentioned
that the equation of any hue of the pencil whose center is the inter-
section of the lines m 0, n = is given by an equation of the form
m + fm = 0. Similarly, the equation of any point of the line joining
A a^ + a 2 w a + a & u s = and B = \u^ + \u z + \u 9 = is of the
form A +\B = 0.
66. The equation of a come. The results of 65 lead readily to
the equation of a conic. By this is meant an equation m point (line)
coordinates which is satisfied by all the points (lines) of a conic, and
by no others. To derive this equation, let A, B be two distinct points
on a conic, and let
m m^ + m 2 aJ 2 -f- m, 3 x 3 = 0,
(1) n = n^ + ?i 2 # 2 + 7t 3 # s = 0,
be the equations of the tangent at A, the tangent at B, and the line
AB 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 p, n at B respectively This projectivity between
the pencils
m + \p**Q,
^ ' p + pn =
is given (Theorem 6, Cor. 3) by a relation
(3) p = &X
between the parameters p, \ 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 /*, \ between the last three relations. The result of this
elimination is
(4) p*-7cmn=Q,
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 /* = 7e\ If m, n, p are fixed, the condition
that the conic (4) shall pass through a point (a v a 2 , a s ) is a linear
equation in L Hence we have
186
COORDINATE SYSTEMS
[CHAP VII
THEOREM 7. If m 0, n 0,
jp = are the equations of two
distinct tangents of a conic and
the line joining their points of con-
tact respectively, the point equa-
tion of the conic is of the form
p z lcm,n 0.
The coefficient k is determined "by
any third point on the conic. Con-
versely, the points whicJi satisfy
an equation of the above form
constitute a conic of which m =
and n = are tangents at points
on p = 0.
COROLLARY. By properly cJwos-
ing the triangle of reference, the
point equation of any conic may
be put in the form
where x l = 0, # 8 = are two tan-
gents, and # a = is the line join-
ing their points of contact.
THEOREM 7 ' If A = 0, B = 0,
C=Q-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
The coefficient k is determined by
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
JB=Q are points of contact of the
tangents through = 0.
COROLLARY. By properly choos-
ing the triangle of reference, the
line equation of any conic may
be put in the form
where u^ 0, u a = are two points,
and u z = 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 v u e ] be tangent to the conic
This condition is equivalent to the condition that the line whose
equation is
shall have one and only one point in common with the conic. Elimi-
nating a; 8 between this equation and that of the conic, the points
common to the line and the conic are determined by the equation
U^X* + Ufttti + W 8 OJ 2 a 0.
The roots of this equation are equal, if and only if we have
ul 4 UU = 0.
66,67] LINEAR TKANSFOEMATIOlSrS 187
Since this is the line equation of all tangents to the conic, and since
it is of the form given m Theorem 1', Cor , above, we have here a new
proof of the fact that the tangents to a point conic form a line conic
(cf Theorem 11, Chap V).
When the linear expressions for m, n, p are substituted in the equa-
tion p z kmn of any conic, there results, when multiplied out, a
homogeneous equation of the second degree in x v x z> x s> which may
be written in the form
(1) a u x? + a^xl + a 8 X + 2 a^x^ + 2 a ia x t x s + 2 a sa x z x a = 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
Show that the conic
a n x? + a 22 r| + a 33 xj + 2 a^x^ + 2 a^or, + 2 a zs x z x & =
degenerates into (distinct or coincident) stiaight 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
Such a transformation transforms any point (x v x z , as s ) of the plane
into a unique point (as/, a? a ', iC 8 ') of the plane. Reciprocally, to every
pomt x' will correspond a unique point x, provided the determinant
of the transformation
a n a i2 a is
A = * a 22 a zs
a 8l a sz a S8
is not 0. For we may then solve equations (1) for the ratios O3 t : oj a : x t
in terms of a?/: 2 ': # 8 ' as follows :
(2)
188 COORDINATE SYSTEMS [C.UAP.VU
here the coefficients A rj are the cofactors of the elements a ti mspoc-
tively m the determinant A.
Further, equations (1) transform every line in the plane into a
unique line In fact, the points x satisfying the equation
W 1 1 +tt 2 # 2 +w 3 a j 8= !0
are, by reference to equations (2), transformed into points a/ satisfy-
ing the equation
3 ) x[ + (A zl u { + A s ^ + A w u,} ^
x =* 0,
which is the equation of a line. If the coordinates of this new lino be
denoted by [< < fl, we clearly have the following relations between
the coordinates [u v u v u 3 ] of any line and the coordinates [< < <|
of the line into which it is transformed by (1) :
(3) a-u^A^
fful = A^
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 protective pencil of lines But it is clear that if
m = and n are the equations of any two lines, and if (1) trans-
torms them respectively into the lines whose equations aro m' =
and H/=0, any line m + \nQ is transformed into m'4-Xw' 0,
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
collineation, we wiH now show conversely that every projective
collmeation in a plane may be represented by equations of the form
(1) To this end we recall that every such collineation is completely
rlef.A-nmnp.fi 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 TRANSFORMATIONS
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 a,), B = (b v \, &,), = (o v c 2 , c 8 ), and D = (d v d v dj, 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 s> a^) 3 W e
must have ,. _ -
a w = \a v 28 =Xa 2 , a 83 =Xa s ,
\ 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 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 z , d 3 ), the following
relations must hold :
Placing p = 1 and solving this system of equations for v, p, X, we
obtain the coefficients a y of the transformation. This solution is
unique, since the determinant of the system is not zero. Moreover,
none of the values X, /A, v will be ; for the supposition that v = 0,
for example, would imply the vanishing of the determinant
which in turn would imply that the three points D, B, A are collinear,
contrary to the hypothesis that the four points A, B, (7, D form a
complete quadrangle.
Collecting the results of this section, we have
THEOREM 8. Any projective calUneation in the plane may "be repre-
sented in point coordinates "by equations of form, (1) or in line coordi-
nates "by equations of form (3), and ^n each case the determinant of
IQQ COORDINATE 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.
OOKOLLAEY 1. In nonhomogeneous point coordinates the equations of
a protective collineation are
. a..x
X
-
a a ^ .
GOROLLAEY 2. If the singular line of tJie system of nonhomogeneous
point coordinates is transformed into itself, these equations can be
written saf=ax
y' =
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 j3, and let a system of coordinates be
established in each, the point coordinates in a being (x v x z , # 8 ) and
the point coordinates in (3 being (y v y z , y 8 ). 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
2/1=^ y>-. 2/8 = ^
is projective. 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 t say, into a point X', and the pro-
jectivity Y X 1 between the two planes. The analytic form of any
projective collineation "between 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 collineatwn between 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,, #" oo a>
COORDINATES IN SPACE
191
y=<x> y , W=co g (fig. 84) ; and on the lines 0.^, 0^, O z oo g> called
respectively the oc-axis, the y-axis, the z-axis, establish three scales by
choosing the points l a , l y , l a . The planes Ooo^cx> v , Ooo^oOg, Ooo co 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^co^, which
is called the singular plane of the coordinate system, the plane
P co^coj. meets the a>axis in a point whose nonhomogeneous coordinate
in the scale (0.,., l x , GO,,) we call a. Similarly, let the plane
meet the y-axis in a point
whose nonhomogeneous
coordinate in the scale
(O y , l y , co y ) is I ; and let
the plane Pco^ meet the
#-axis in a point whose
nonhomogeneous coordi-
nate in the scale (0,., l e , co a )
is c. The numbers a, 5, c
are then the nonhomo-
geneous x-, y-, and z-coor- 1*
dinates of the point P
Conversely, any three
numbers a, 5, c determine
three points A, B, on
the so-, y-, and #-axes respectively, and the three planes ^too,
Coo a co y meet in a point P whose coordinates are a, &, 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, &, 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, UW, UV a scale by choosing the plane UVW as
the zero plane M = () = O w in each of the scales, and letting the planes
VW, OUW,OU7\)ethe planes oo tt , oo v , oo w respectively. In the ^scale
. 84
192 COORDINATE SYSTEMS [CHAP. VIS
let that plane through VW be the plane 1 M , which meets the ,^-axis
in the point I,. Similarly, let the plane ! meet the y-axis in the
point \ y \ and let the plane l w meet the z-axis in the point 1 g
The it-scale, v-scale, 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 a:-, y-, and #-axes in
three points L, M, N which determine in the w-, v-, and w-scalea planes
whose coordinates, let us say, are I, 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, b, c) be on the plane [I, m, n] is that the relation
la + mb + ne + l=Q be satisfied. It follows readily, as in the planar
case, that the plane [I, m, n~\ meets the %-, y-, and #-axes in points
whose coordinates on these axes are l/l, l/m, and 1/n respec-
tively.* In deriving the above condition we will suppose that the
plane IT [I, m, n] does not contain two of the points Z7, V t W> leav-
ing the other case as an exercise for the reader. Suppose, then, that
U=*GQ X and F= oo ff are not on TT. By projecting the y-plane with
U as center upon the plane -rr, and then projecting TT with V as center
on the a-plane, we obtain the following perspectivities :
[(.y.)]|[toy,^][too,i)],
where (x, y, s) represents any point on TT, The product of those two
perspectivities is a projectivity between the ys-plane and the ire-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
ys-plane by &', this projectivity is represented (according to Theorem
8, Gor. 2, and 68) by relations of the form
(1) l^ X+ lZ + Cv
We proceed to determine the coefficients n v l v Gy The point of
intersection of * with the y-axis is (0, -1/ TO , 0), and to clearly
69] COORDINATES IN SPACE
transformed by the projectivity in question into the point (0, 0, 0)
Hence (1) gives -
c =
m
The point of intersection of TT with the a-axis is, if %=(), (0, 0, 1/ri)
and is transformed into itself. Hence (1) gives
n m
1 m
If n = 0, we have at once \ = 0.
Finally, the point of intersection of TT with the a-axis is ( 1/Z, 0, 0),
and the transform of the point (0, 0, 0) Hence we have
__. _i _ n
7 >
It ffli
1 m
Hence (1) becomes y = x z
m m m
a relation which must be satisfied by the coordinates (x, y, 2) of any
point on TT. This relation is equivalent to
Ix + my + nz + 1 = 0.
Hence (a, 5, c) is on [I, m, n], if
(2) la + mb + nc + 1 = 0.
Conversely, if (2) is satisfied by a point (a, &, c), the point (0, 6, c)= _P
is transformed by the projectivity above into (a, 0, c) = $, and hence
the lines P U and Q V which meet in (a, b, c) meet on TT.
DEFINITION. An equation which DEFINITION. An equation which
is satisfied by all the points (x, y, 3) 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, ri\ is of the point (a, b, c) is
Ix + my H- nz + 1 = 0, au + bv + cw + 1 = Q.
19 4 COORDINATE SYSTEMS [CHAP, vn
70. Homogeneous coordinates in space. Assign to the vertices 0, Z7,
F, 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, V, F, 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, F, 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, 0),
(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, x v x a , o? 4 ), if
its coordinates in the planar system ]ust indicated are (aj a , J 8 , o& 4 ). In
like manner, to the points of the other three faces of the tetrahedron of
reference we assign coordinates of the forms (a^, 0, x s , x t ), (x v x z , 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 l = 0, x z = 0, # 8 = 0, 03 4 =
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 0^=0 and x z = 0. Regarding this edge as a line of M^ 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 a! 3 =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 != and
a? a = 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 x at 8 , 4 , oU
di/erentfrom zero, such that the projections of P from, the four vertices
(1, 0, 0, 0), (0, 1, 0, 0) t , (0, 0, 1, 0), (0, 0, 0, 1) respectively upon
70] COORDINATES IN" SPACE 195
opposite faces are (0, 2 , a> 8 , a? 4 ), fo, 0, a? 8 , osj, (a^, a 2 , 0, a? 4 ), (a^, as,, a;,, 0)
57wse four numbers are called the Jwmof/eneous coordinates of P and
P is denoted ly (x v x a , o& a , a; 4 ) Any ordered set of four numbers, 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 s , B 4 ), which is not an edge of the tetrahedron of refer-
ence, and such therefore that none of the numhers a? 2 , as 8 , zc 4 is zero.
Likewise the line joining P to (0, 1, 0, 0) meets the opposite face in
a point (as/, 0, #', aJ 4 ), such that none of the numhers x[, x^ %[ is zero.
But the plane P(l, 0, 0, 0) (0, 1, 0, 0) meets a; 1 = in the line joining
(0, 1, 0, 0) to (0, # 2 , SG S , a: 4 ), and meets x z in the line joining
(1, 0, 0, 0) to (a?./, 0, xl, a?/). By the analytic methods already devel-
oped for the plane, the first of these lines meets the edge common
to x : = and x a = in the point (0, 0, x a , # 4 ), and the second meets
it in the point (0, 0, a?/, 03 4 ). But the points (0, 0, D S , x 4 ) and
(0, 0, xl , xl) are identical, and hence, by the preceding paragraph, we
have a? 8 /a2 4 = #//#/ Hence, if we place x l = x[xjx[ y the point
(x^ 0, a, xl} is identical with (x v 0, x a , x t ). The line joining P to
(0, 0, 1, 0) meets the face x a = in a point (a?/', # 2 ", 0, #/'). By the
game reasoning as that above it follows that we have x / x^ *= xj x^
and x" /xl' = xjx^ so that the point (x", x r , 0, a? 4 ") is identical with
(x v 03 a , 0, a? 4 ). Einally, the line joining P to (0, 0, 0, 1) meets the face
a; 4 =s in a point which a like argument shows to be (x v a; 2 , x 3 , 0).
Conversely, if the coordinates (x v x v x a , # 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 s) x s , 03 4 ) and (0, 1, 0, 0) to (x lt 0, o) 8 , * 4 ) are in the plane
(1, 0, 0, 0) (0, 1, 0, 0) (0, 0, x 9 , aJ 4 ), and hence meet in a point which,
by the reasoning above, has the coordinates (x v x z , x a , x^
COEOLLABY. Tlie notations (x v x z ,x 3 ,xj and (kx v 7ex a , Jex s , Tcx^
denote the same point for any value of TV 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], [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
i * > I
}
I
196 COOKDEKTATE 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 8 , wj, if
its coordinates in the bundle are \u v u s , wj. In like manner, to the
planes on the other vertices are assigned coordinates of the forms
[u v 0, w 8 , wj, [u v u a , 0, wj, [u v u a , u 3> 0], The space dual of the last
theorem then gives :
THEOEEM 10'. DEFINITION. Ifwis any plane not on a vertex of the
tetrahedron of reference, there exist four numbers u v u z , u z> w 4 , all 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 verities ly the planes [0, u z , u 3 , wj, \u v 0, w 3 , 4 ], [u v u z , 0, wj,
\u v u z ,u s , 0] These four numbers are catted the homogeneous coordinates
of IT, and IT is denoted ly [u v u z , u s , wj. Any ordered set of four num-
bers, not all zero, determine uniquely a plane whose coordinates tJiey 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 os z , x s , x 4 ) be on a plane [u v u^, u s , i*J. This
condition -will turn out to be
!! + U 2 3S a + U 3 3S a + U& = 0.
We note first that in a system of point coordinates as described above
the six 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, l) 1S projected by the line joming 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 (ef Ex 3
p. 47). Now choose " '
as the plane [1, 0, 0, 0] the plane ^ = 0,
as the plane [0, 1, 0, 0] the plane x z ~ 0,
as the plane [0, 0, 1, 0] the plane a 8=s = 0,
as the plane [0, 0, 0, 1] the plane x^ 0,
TO] COORDINATES IN SPACE 197
3 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],
3, 0, 1, 0], and [1, 1, 1, 0] through the vertex F" 4 , say, whose point
3ordmates are (0, 0, 0, 1), meet the opposite face # 4 = in lines
hose equations in that plane are
x^ =0, x z 0, tf s =0, x l + a3 a 4- % s = 0.
ence the first three coordinates of any plane \u v u z> u s> 0] on F" 4
'e the line coordinates of its trace on x = 0, in a system so chosen
tat the point (x v x z) os s ) is on the line [u v u 2 , u s ] if and only if the
'lation ufii + u z x z + u s x s = is satisfied Hence a point (x v x z , x s , 0)
3s on a plane [u v u z , u a> 0] if and only if we have u 1 x i +u z x Si +
t x s = 0. But any point (x v x z , % 8 , 4 ) on the plane [u v u z , u a , 0] has,
r definition, its first three coordinates identical with the first three
ordinates of some point on the trace of this plane with the plane
= 0. Hence any point (x v x z , #, # 4 ) on [u v u z , u s , 0] satisfies the
ndition u^ + w 2 # a + u s x s + 4 a? 4 = Applying this reasoning to
ch of the four vertices of the tetrahedron of reference and dualizing,
3 find that ^f one coordinate of [it, v u z , u z , wj is zero, the necessary
id sufficient condition that this plane contain a point (x v x z , a? a , 3 4 )
that the relation
U^ + Ufa + U a X 3 + W 4 # 4 =
satisfied; and if one coordinate of (x v x z , % a , 4 ) is zero, the neces-
ry and sufficient condition that this point "be on the plane [?^,w 2 ,?t a ,wj
likewise that the relation just given "be satisfied
Confining our attention now to points and planes no coordinate of
lich. is zero, let xjx^x, xjx^y, xjx^z, and let uju^u,
/Ut=*v, uju^w. Since x, y, * are the ratios of homogeneous
ordinates on the lines x z = x & = 0, x^ = x s = 0, and ^ = a; 2 = respec-
r ely, they satisfy the definition of nonhomogeneous coordinates
,-en in 69. And since the homogeneous coordinates have teen
chosen that the plane (u v u z , u a , w 4 ) meets the line o2 2 = x s = in
3 point ( u^ 0, 0, w x ) = ( l/u, 0, 0, 1), it follows that w, v, w are
nhomogeneous plane coordinates so chosen that a point (x, y, z),
ne of whose coordinates is zero, is on a plane [u, v, w\ none of
lose co8rdmates is zero, if and only if we have (Theorem 9)
was + vy + we + 1 = ;
198 COOEDINA.TE SYSTEMS [CHAP VII
that is, if and only if we have
U& + u z x s + u s x s + u 4 is t =
This completes for all cases the proof of
THEOREM 11. The necessary and sufficient condition that a point
(x v x v # 8 , 4 ) le on a plane [u v u z , u s , uj is that the relation
upi + tt a afe + u s x g + u^
le 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 = (ct 15 a z , a 3 , a 4 ),
Dualize
2 The necessary and sufficient condition that four points A, B, C, D be
coplanar is the vanishing of the determinant
3. The necessary and sufficient condition that three points A) B, C be
collinear is the vanishing of the thiee-rowed determinants of the matrix
4. Any point of a pencil of points containing A and B may be represented by
P X + X}
5 Any plane of a pencil of planes containing m = [m x , m a , w 8 , w 4 ] and
" = Oti n s> n 8> "4] ma y be represented by
+ Xjj^, Xgma + A^, X^wij, + X^g, X 2 m 4 + X^J,
6 Any projectivity between two one-dimensional primitive forma (of points
or planes) in space Is expressed by a relation between their parameters X, /*
of the form
oX + ft
^"^xTs"
If the base elements of the pencil are homologous, this relation reduces to
.
' V v *-'/*'
70,71] LINEAR TRANSFORMATION " ^199 1
<**',- -^
7 If \ v A 2 , A s , A 4 aie the parameters of four points or planes of a pencil K " -~.^
then ciobs ratio is *"
l ~ <t 2 "~ 4
8 Any point (plane) of a plane of points (bundle of planes) containing
the noncollineai points A, JB, C (planes a, /8, y) may be repiesented by
P - (A^ + A^ + A 8 c T , kfy + Xgb z + A 8 c a , A^a + A 2 Z> 8 + A 8 c s , A^ + A,i 4 + A 8 c 4 ).
9. Derive the equation of the polar plane of any point with regard t<? the
tetrahedron of reference
10. Derive the equation of a cone.
*11. Derive nonhomogeneous and homogeneous systems of codrdinates in
a space of four dimensions.
71. Linear transformations in space. The properties of a linear
transformation in space
+ ax
"IS^S
(i) p a*z. f .~ J "~ J "
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. :
p'x s
^ ' ' B + AX,
X + AX + AX>
where the coefficients ^ are the cofactors of the elements a y respec-
tively in the determinant A.
The transformation is evidently a collineation, as it transforms the
P
into the plane
*
+ A *
200 COORDINATE SYSTEMS |C"Ai-. VH
Hence the collineation (1) produces on the planes of space the trans-
formation
' '
To show that the transformation (1) is productive consider any
pencil of planes
+ & a 2 + V 8 + ^ 0,
In accordance with (2) this pencil is transformed into a pencil of the
form
(o/jBi + a/a^ + a 3 'a? 8 + a 4 'as 4 ) + X (&/! + l& 9 + &'!, + &/,*; 4 ) = 0,
and these two pencils of planes are protective (Ex. 6, p. 108).
Finally, as in 67, we see that there is one and only 0110 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 eomplutu
five-point in space. Since this transformation is a protective collinwi-
tion, and since there is only one projeotivo collinoatioii transforming
one five-point into another (Theorem 19, Chap. IV), it fullowH that
every projective collmeation in space may he represented by a liiuwr
transformation of the form (1). This gives
THEOREM 12. Any projective collinmtion of ^mea mmj 1w repre-
sented in point coordinates by equations of the, /upin, (1), or in yfana
coordinates by equations of the form (3). lib cttftji, anac the drtaruMMitt
of the transformation is different from sort). $ww/wj/y, // trans-
formation of this form in which, the det&nwtoumt is- differ ant front, aero
represents a projective collineation of space.
COROLLARY 1. In nonhomogeneons point covrdinfttrtt a prtyeetiw
collineation is represented ly the linear fraot/imml cij>iuttwn8
y i = V + a^y + orft + a^
''
vn which the determinant A is di/erentfrom zero.
71,72] FINITE SPACES 201
OOEOLLAEY 2. If the singular plane of the nonhomoffeneous system
is transformed into itself, these equations reduce to
sd = a^ + a z y + a# + a v ^ a z 8
y / & 1 aj + & a y+6 4l a + J 4 , \ \ \ ^ 0.
' = c^x + c z y + c<p -f- <? 4 , e i G z c s
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-dimension al 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 (as v C 2 , 03 8 ), it being understood that the sets
(x v # 2 , x 8 ) and (ksc v Tcx z> kx a ) are equivalent for all values of k 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
16^+ u s x z + u a x a = 0, the set of all lines being obtained by giving
the coefficients (coordinates) \u v u z , u s ] all possible values in the
number system (except [0, 0, 0]), with the obvious agreement that
\u v u z , w 8 ] and \ku v ku z) ku & ] represent the same line (/=()). 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 w, n'
(positive, negative, or zero) are said to be congruent modulo p, written n=n', mod. p,
if the difference n n" is divisible by jp. 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 by addition^ subtraction,
202 COORDINATE SYSTEMS [CHAP.VII
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, JF, G, as f ollows : A = (0, 0, 1), B = (0, 1, 0),
C = (l, 0, 0), D = (0, 1, 1), JP-(1, 1, 0), ^ = (1, 1, 1), = (1, 0, 1).
The reader will readily verify that these seven points are arranged
in lines according to the table
A B D IS F &
C D E F G A
D E F G A B C,
each column constituting a line For example, the line o^ = 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 "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 piojective pencils of lines in a plane have
a self-coriesponding line, they are perspective. (This is equivalent to Assump-
tion P )
3 Show that the lines whose equations aie a^ + A..e 2 = 0, x z + it% z = 0, and
x a + vxj = are concui rent if X/xv = 1 j and that they meet the opposite
sides of the triangle of reference respectively in collinear points, if Xp> = 1
4 Find the equations of the lines joining (c v c a , c 8 ) to the four points
(1, 1, 1)> and determine the cross ratios of the pencil.
and muttvphcation, 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 1 + 8 = 4, 2 + 8 = (since 5 = 0, mod. 6), 1 4 = 2, 2 8 = 1, etc.
Furthermore, if a, 6 are any two elements of this field (a 5* 0), there is a unique
element determined by the congruence o=6, mod, jp; 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 = 8.
* For references and a further discussion of finite projective geometries see a
paper by Veblen and W. H, Bussey, Finite Projective Geometries, Transactions
of the American Mathematical Society, Vol. VII (1906), pp, 241-260 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] EXEECISES 203
5. Show that the throw of lines determined on (c lf c a , c a ) by the four
points (1, 1, 1) is projective with (equal to) the throw of lines determined
on (&j, Z> a , & 3 ) by the points (a 19 a 2 , a 8 ), if the following relations hold:
and that the six cross ratios are a 2 /a 3 , a^/a^, a^/a z , a s /a 2 , (t^/a^
2 /a 1 (C. A. Scott, Mod. Anal. Geom , p 50).
6 Wiite the equations of transformation for the five types of planai col-
lineations desciibed in 40, Chap. IV, choosing points of the tuangle of
refeience as fixed points
7. Generalize Ex. 6 to space
8 Show that the set of values of the paiameter A. of the pencil of lines
wi + An = is isomorphic with the scale determined in this pencil by the lines
for which the fundamental lines are respectively the lines A. = 0, 1, oo
9 Show directly fiom the discussion of 61 that the points whose non-
homogeneous cooidinates x, y satisfy the equation y = x aie on the line joining
the ongin to the point (1, 1).
10 Theic is then established on this line a scale whose fundamental points
are respectively the origin, the point (1,1), and the point in which the line m eefcs
the line .. The lines joining any point P in the plane to the points oo y , oo x
meet the line y x in two points whose coordinates in the scale just determined
aie the nonhomogeneous coordinates of P, so that any point in the plane
(not on L,) is lepresented by a pair of points on the line y = x. Hence, show
that in general the points (x, y) of any line in the plane deteimine on the
line y x& projectivity with a double point on Z ; and hence that the equa-
tion of any such line is of the foini y = ax + b. What lines are exceptions to
this proposition?
11. Discuss the modular plane geometiy in which the modulus is p = 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 geneial that the modular projective plane with modulus p
contains p* -f p + 1 points and the same number of lines ; and that there are
p + 1 points (lines) on every line (point).
13. The diagonal points of a complete quadrangle in. a modular plane pio-
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 1 4, 2 8, 5 10 by cyclic permutations. (For a
204 COOEDINATE 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) aie 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 foui 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 lesults of Ex. 17 hold without modification.
* 19. Derive homogeneous and nonhoniogeneous cooidinate systems for
a space of n dimensions, and establish the formulas for an n -dimensional
projective collmeation.
CHAPTER VIII
PROJECTIVITIES IN ONE-DIMENSIONAL FORMS*
73. Characteristic throw and cross ratio.
THEOREM 1. If M, N are double points of a projectvoity on a, line,
and AA', JBB' are any two pairs of homologous points (ie. if
MNAB % MNA'B'}, then MNAA' -% MNBB 1 .
Proof. Let S, S' be any two distinct points on a line through
M (fig. 85), and let the lines SA and S'A' meet in A", and SB and
S
S'B' meet in B". The points A", B", JVare then collinear (Theorem 23,
Ohap. IV). If the line A"B" meets SS' in a point Q, we have
A" B"
MNAA' == MQ88' === MNBB 1 .
This proves the theorem, which may also he stated as follows :
The throws consisting of the pair of double points in a given order
and any pair of homologous points are all egual.
DEFINITION. The throw T(MN, A A'), consisting of the double points
and a pair of homologous points of a projectivity, is called the charac-
teristic throw of the projeotivity ; and the cross ratio of this throw
is called the characteristic cross ratio of the projectivity.f
* All the developments of this chapter are on the basis of Assumptions .A T? P TT
t Since the double points enter symmetrically, the throws T / '"" r
T (JVlf, AA') may be used equally well for the characteristic th
spending cross ratios Jfr (Jf JT, A A') and Bs (N3f t A A') are reoip""
(of. Theorem 13, Cor. 3, Chap. VI).
20F
206 ONE-DIMENSIONAL PROJECTIVITIES [CHAP vin
COEOLLAEY 1. A projectivity on a line with two given distinct
double points is uniquely determined by its characteristic throw or
cross ratio.
COBOLLAEY 2. The characteristic cross ratio of any involution with
double points is I.
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
k is the characteristic cross ratio of a projectivity on a line, we have
x'm f x in _ ,
x'n x n
for every pair of homologous points x, x' 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:
COKOLLAEY 3 Any projectivity on a line with two distinct doulle
points m, n may be represented by the equation
x r m __ , x m
x'n x n'
*', x being any pair of homologous points.
For when cleared of fractions this is a bilinear equation in x' } 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 h These considerations offer an analytic proof of Theo-
rem 1, for the case when the double points M, JVare 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 :
THEOEEM 2. If in a parabolic projectivity with double point M the
points AA! and SB' are two pavrs of homologous points, the parabolic
projectivity with double point M which puts A into 3 also puts A'
into B'. *
COBOLLAEY. The characteristic cross ratio of any parabolic
timty is unity.
73] CHARACTERISTIC! THROW 207
The characteristic cross ratio together with the double point is
therefore not sufficient to characterize a paiabolic projectivity com-
pletely Also, the analytic form for a projectivity with double points
TO, 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 ivith double, point M trans-
forms a point A into A' and A' into A 1 ', the pair of points A, A" is
harmonic with tlie pair A'M; i.e we have H (MA 1 , AA").
Proof By Theorem 23, Chap IV, Cor., we have Q(MAA', MA" A').
Analytically, if the coordinates of M, A, A', A" are m, x, x 1 , ss"
respectively, we have, by Theorem 13, Cor. 4, Chap. VI,
m js iii sd' m
This gives
1 __ J^^ 1 __ 1
a/ 111 x m ,il' m x' m
which shows that if each member of this equation be placed equal to
t, the relation
(1)
~
x'm x m
is satisfied by every pair of homologous points of the sequence obtained
by applying the projectivity successively to the points A, A', A", .
It is, however, readily seen that this relation is satisfied by every pair
of homologous points on the line. For relation (1), when cleared of
fractions, clearly gives a bilinear form in t 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, 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 :
COROLLARY 1. Any parabolic projectivity with a double point, M,
may "be represented ly the relation (1).
DEFINITION. The number t is called the characteristic constant of
the projectivity (1).
208 ONE-DIMENSIONAL PBOJEOTIVITIES [CHAP VIII
COROLLARY 2. Conversely, if a projectivity with a double point
M transforms a point A into A', and A' into A", such that we have
H (MA', AA"), the projectivity is parabolic.
Proof The double point M and the two pairs of homologous
points A A', A' A" 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 lino I,
and let ir^ be a projectivity transforming the points of I into the
points of another or the same line I'. The projectivity -n-^TTf 1 is then
a projectivity on I'. Tor wf 1 transforms any point of I' into a point
of I, IT transforms this point into another point of I, which m turn is
transformed into a point of V by w r Thus, to every point of V is made
to correspond a unique point of I', and this correspondence is projec-
tive, since it is the product of protective correspondences. Clearly,
also, the projectivity ^ transforms any pair of homologous points of
TT into a pair of homologous points of w^TTf 1 .
DEFINITION. The projectivity WjWTrf 1 is called the transform of IT
ly 7r x ; two projectivities are said to be protective 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 ir^ transforming TT into TT' transforms
the double points M, N of TT into the double points M 1 , N' of ir 1 , and
also transforms any pair of homologous points A, A v of TT into a pair
of homologous points A 1 , A^ of TT' ; i e.
But this states that their characteristic throws are ecjual
74,75] GROUPS ON A LINE 209
The condition is also sufficient; for if it is satisfied, tho projec-
tivity TT, defined by
clearly transforms TT into TT'.
COROLLAEY. Any two involutions with double points are conjugate.
THEOREM 5. Any two parabolic projectivities are conjugate.
Proof. Let the two parabolic projectivities be defined by
ir(MMA}=*MMAv and ir'(M'M' A 1 ) - M'M'Af.
Then the projectivity 71^ defined by
clearly transforms TT into IT'.
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 ir^-ir- 1 = G / ;
and G' is said to be the transform of G "by ir f
We have already seen (Theorem 8, Chap III) 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. The set of all projectivities leaving a given point of the line
invariant.
Any two groups of this type are conjugate Tor 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
#'= asa + b.
2. The set of all projectivities leaving two given distinct points
invariant.
Any two groups of this type are conjugate. For 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 PROJECTIVITIES [CHAP vm
into a projectivity of the other Analytically, if x lt x a are the two
invariant points, the group consists of all projectivities of the farm
The product of two such projectivities with multipliers & and lc' is
clearly given by
,
tAs ^ v(J*
30
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 gioup consists of all piojccs-
tivities of the form yf 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
1 - l 4-f
- - -f- (,
/yi ^ M / M _ M
tfj |/- |A/ "- tAj
where x^ is the common double point (Theorem 3, Cor. 1). If
~~j ^ | and . ~ ___ _j_ j
x' x l jj x ar ajj ta ^ a
are two projectivities of this set, the product of the first by the second
is given by 11
r'r ~ v v +t i +i *>
tb *""* iA/- tfa " tXj't
which is clearly a projectivity of the set. It showa, 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 pi oof we have given. The theorem has already been proved
without thia assumption in Example 2, p. 70,
75] GEOUPS ON A LINE 211
Any two groups of this type are conjugate. For every projectmty
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 wider G if it is transformed into itself by
every transformation of G; ie 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 protective group on the line.
We may now give an example of a self-conjugate subgroup.
The set of all parabolic projectivities in a group of Type 1 above 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. Wiite the equations of all the protective transformations which permute
among themselves (a) the points (0,1), (1,0), (1,1), (&) the points (0, 1),
(1,0), (1,1), (a, ft), (<0 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 of = (ac + Z)/(ca; + d) having two distinct double ele-
ments be written in the form of Cor. 3, Theorem 1, show that
^t^ (1 + fr) 2 (a + <*) 2
that v . = , /
k ad be
3. If a parabolic projectivity xf = (ax + V)/(cx + d) be written in the form
of Theorem 8, Cor. 1, show that m = (a d)/2 c, and t 2 c/(a + 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
a; 1
* ^ i
ar a fcc 9 1
ar= x 1
a? 1 1
212 ONE-DIMENSIONAL PEOJECTTVITIES [CKAP. VIII
5. Show tli.it the parabolic project! vity oE Theorem 8, Cor. 1, may be
written in the form
a; 1
1 1
I te. + l
1M* *
* ~ x 1
,'! 1 1
1 t
6. If by means of a suitably chosen transfoimation of a group any of the
elements transformed may be tiansformed into any othei element, the group
is said to be tramitwe. If by a suitably chosen traiibfoimation of a group any
set of n distinct elements may be tiansformed into any othei set of n distinct
elements, and if this is not true for all sets of n + 1 distinct elements, the
gioup is said to be n-ply transitive. Show that the general projective group on
a line is tuply transitive, and that of the subgroups listed in 75 the first
is doubly tiansitive and the other two are simply tiansitive.
7. If two projectivities on a line, each having two distinct double points,
have one double point in common, the characteristic cross latio of then prod-
uct is equal to the product of theii 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 now light.
As typical for the one-dimensional forms of the second degree wo
choose the conic. The corresponding theorems for the cone then
follow by the principle of duality.
Let TT X be a projective collmeation between two planes a, <x v and
let <7 2 be any conic in cc. Any two projective pencils of lines in a
are then transformed by ^ into two projective pencils of lines in <x lt
such that any two homologous lines of the pencils in <x are trans-
formed into a pair of homologous lines in t\ ; for if IT be the projoc-
tivily between the pencils in a, ^Tm-f 1 will be a projectivily between
the pencils in ^ (cf. 74). Two projective pencils of lines generating
the conic C? 2 thus correspond to two pencils of linos in ^ generating
a conic Cf. The transformation ^ then transforms every point of (7 a
into a unique point of 0%, Similarly, it is seen that ^ transforms
every tangent of <7 a into a unique tangent of C*,
DEFINITION. Two conies are said to be protective if to every point of
one corresponds a point of the other, and to every tangent of one
70]
TRANSFORMATION OF CONICS
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 comes.
Two conies in dii'feient pianos are piojective, for example, if one is the pro-
jection of the other from a point on neithei 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 protective with a third are
projeetive,
Proof. This is an immediate consequence of the definition and the
fact that the resultant of two collmeations is a collineation.
We proceed now to prove the fundamental theorem for projec-
tivities between two comes.
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
C'
PIG 80
Proof. Let C'\ C f 1 a be the two conies (fig. 86), and let A, 3, be
three points of C z , and A', B', C' the corresponding points of Cf. Let
P and P' be the poles of AB and A'B' with respect to C* and C\
respectively. If now the collineation IT is defined by the relation
7r(ABCP)=A r J3'C'P' (Theorem 18, Chap. IV), it is clear that the
conic <7 a is transformed by ir into a conic through the points A', B 1 , C',
with tangents A'P' and B'P r > This conic is uniquely determined by
these specifications, however, and is therefore identical with C7 x a . The
collineation tr then transforms (7 a into C* in such a way that the
points A, B, are transformed into A', B', C' respectively. Moreover,
214 ONE-DIMENSIONAL PKOJECTIVITIES [CHAP VIII
suppose TT' were a second collmeation transforming C z into C? in the
way specified Then w'-V would be a collineation leaving A, B, C, P
invariant , i.e TT = TT'.
The argument applies equally well if A'J3'C' are on the conic C*,
i e. when the two conies C' 2 , C* coincide. In this case the projectivity
is on the conic 0. 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 come 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' are THEOREM 10'. If a and I' are
any two points of two protective any two tangents of two projectile
conies C z and Cf respectively, the conies C* and Q% respectively, the
pencils of lines with centers at A pencils of points on a and &' are
and B'are protective if every pair projective 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 1 be
the point of (7* homologous with A. The collmeation which generates
the projectivity between the comes then makes the pencils of lines at
A and A' projective, 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' is projective with that at A f if they correspond in
such a way that pairs of homologous lines intersect on Of (Theorem
2, Chap. V). This establishes a projective correspondence between
the pencils at A and B' 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 a at A
corresponds to the line of the pencil at B 1 passing through A 1 .
76]
TRANSFORMATION OE CONICS
215
COROLLARY. Conversely, if two
conies correspond in such a way
that every pair of homologous
points is on a pair of homologous
lines of two projectwe pencils of
lines whose centers are on the
conies, they are projectile.
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 protective petals of
points whose axes are tangents of
the conies, they are protective.
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 lines of the pencils meet the comes. 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 comes 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 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 ol 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.
DEFINITION. A line 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
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-DIMENSIOSTAX PROJEGTIVITIES [CHA* vm
perspective with the same pencil of lines or with the same pencil of
points are proiective by Theorem 10, Cor., etc. ^
Another illustration of this extension of the notion of perspectivity
leads readily to the f oUowing important theorem :
THEOREM 11 Two conies whicJi are not in tU same plane and have
a common tangent at a point A are sections of one and the same cone.
Proof. If the two conies C\ Of (fig 87) are made to correspond
in such'a way that every tangent * of one is associated with that
tangent x 1 of the other
which meets re in a
point of the common
tangent a of the conies,
they are projective,
Tor the tangents of
the conies are then,
perspective with the
FlG> 87 same pencil of points
(cf Theorem 10', Cor.). Every pair of homologous tangents of the two
comes determines a plane. If we consider the point of intersection
of three of these planes, say, those determined by the pairs of tangents
W cc' dd', and project the conic Cf on the plane of <7 2 from 0, there
results a conic in the plane of 0\ This conic has the lines &, c, d for
tangents and is tangent to a at .4; it therefore coincides with C
(Theorem 6', Chap. V). The two conies C\ 0} 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. , . j. u
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 comes have three self-corresponding points, they are
perspective with a common pencil of lines.
4. If two projective comes have four self-corresponding elements, tney
coincide.
5. State the space duals of the last two propositions.
It will he seen later that this tbaopfe leads to the proposition, that any conic
may he ohtained as the projection of a oWftj tangent t<|> It In, WteW* P^e,
' '
76, 77]
PEOJECTIVITIES ON 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 thiee lines of the pencil pass through then
homologous points on the conic (Hint. Con&idei the points of intersection of
the given conic with the conic generated by the given pencil and a pencil of
lines pei&pective with the given conic ) Dualize.
7. The homologous lines of a hue conic and a projective pencil of lines in
the same plane intersect in points of a " cuive 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 oider" such that any plane has
at most thiee points in common with it.
9. Dualize the last two propositions
77. Projectivities on a conic. We have seen that two projective
conies may coincide (Theorems 8-10), in which case we obtain a
projective correspondence among the points or the tangents of the
TIG. 88
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' are any THEOREM 12' If a, a' are any
two distinct "homologous points of two distinct homologous tangents
a projectivity on a conic, and]) } B'; of a projectivity on a conic, and
C, C f ; etc , are any other pairs of I, V ; c, c'; etc., are any other pairs
218 ONE-DIMENSIONAL PEOJEGTIVITIES [OKAP.VIII
homologous points, the lines A'B of honioloyouit twiffentH, the
and AJB', A'G and AC', etc, meet a'b and ah', a'n <md tin', de., nre
in points of the same hue ; and eolFuiear with the, mine point,;
this line is independent of the pair and tins point is independent, of
AA' chosen. Mw pw ' /.
Proof The pencils of lines A' (ABC- ) and A(A'Ji'C'' - ) are pro-
jeotive (Theorem 10), and since they have a solf-comjspondinju' line
AA', they are perspective, and the pairs of homologous lines of these
two pencils therefore meet m the points of a line (iig. 88). This
proves the first part of the theorem on the left. That the line thus
determined is independent of the homologous pair AA 1 chosen then
follows at once from the fact this line is the Pascal lino of the simple
hexagon AB'QA'BC', so that the lines JB'C and JiQ' 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 3 ; . A (nonidenti-
projectiwty on a conic is uniquely cal) projectivity on a conic is
determined when the axis of pro- uniquely determined when the
jectimty 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' } to get the point P' homologous with any point P on the conic j
join PtoA'; the point P' 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' t to construct the tangent / homolo-
gous with any tangent p ; the line joining the point a'p to the center
meets a in a point of jt/.
COROLLAKY 2 Every dvulle CoBOLLABT 2'. flv&ry double
point of a projectimty on a conic line of a pro/eetiwty on a, conic
is on the axis of the projectfoity ; contains the center of the projec-
and, conversely, every point com^ tivity; anfc,
mon to the axis and the conic is ff ent
a doulle point.
77]
PEOJECTIVITIES ON A CONIC
219
COROLLARY 3. A projectivity COROLLAEY 3'. A projectivity
among the points on a come 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 projectimty of tangents on a conic
and the axis of the corresponding projectimty of points are pole and
polar with respect to the conic.
Proof Let AA' } BB', CO' (fig. 89) be three pairs of homologous
points (AA 1 being distinct), and let A'B and AB' } A'C and AG', meet
in points R and 8 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', B and
A, B' respectively ; and the polar of 8 is determined by the pairs of
tangents at A', C and A, C' respectively (Theorem 13, Chap. V). The
pole of the axis RS is then determined as the intersection of these
220 ONE-DIMENSIONAL PEOJECTIVITIES [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 collmeation 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 (of. 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 collmeation ; and any two
points or lines of the conic may be chosen as a homologous pair of
the collineation. The collmeation 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 come in its plane invariant is of Type I if it has two
double points on the conic, unless it is of period two, 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 protective 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 projective 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 isomorphic it 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 flat pencil, since it is clear that in the corre-
spondence described between the projectrvities 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 protective
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, i.e. by the
relation I 2 = 1 (I = 1), where I represents an involution. This relation
is clearly equivalent to the other, I = I~ 1 (I=f= 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 1 and A' 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 TT* (A) = A "holds for a, single element A (not a
double element of IT) of a one-dimensional form, the projectivity IT 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 conju-
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' and B, B 1 are any two pairs of
"homologous elements of the projectivity, the pairs of elements MN, AJB'
A'B are three pairs of an involution (Theorem 27, Cor. 3, Chap. IV),
5. If M, J?V 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.VIII
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 by a "bilinear form
cxx'a(x + #') 6 = 0, or "by the transformation
ex a
(Theorem 12, Cor. 3, Chap. VI).
8. An involution with double elements m, n may "be represented "by
the transformation
(Theorem 1, Cors. 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 be transformed into a
projeotivity 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
*V to the case of an involu-
tion of points on a conic.
THEOREM 16. The lines
joining the conjugate points
of an involution on a conic
all pass through the center
of the involution.
Proof. Let ^U' (fig. 90)
be any conjugate pair (A
not a double point) of an
involution of points on a
conic C*. The line A A' is then an invariant line of the coUioeation. gener-
ating the involution. Every line joining a pair of <iisti:ao!t conjugate
points of the involution is therefore invariant, and the generating
collineation must be a perspective collineatiqni, sinjce a&^joajjlfiaeatian
leaving four lines invariant is either $ertp<|trvje bl rib* I Id^iiithr
1 ' F ! * ' * j] i> j j i i i i
. 90
78] INVOLUTIONS 223
(Theorem 9, Cor. 3, Chap III) It remains only to show that the
center of this perspective colhueation is the center of the involution.
Let B, B' (B not a double point) be any other conjugate pair of the
involution, distinct from A, A'. Then the lines AB' and A'B inter-
sect on the axis of the involution. But since B, B' correspond to each
other doubly, it follows that the lines AB and A'B' also intersect
on the axis. This axis then joins two of the diagonal points of the
quadrangle AA'BB' The center of the perspective collineation is
determined as the intersection of the lines A A' and BB', le 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 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 projective 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, A' and B, B' are given, the lines
AA' and BB' determine the center of the involution The conjugate
of any other point G 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 projeotivity in a one-dimensional form may be
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 PROJECTIVITIEtf [CUAP vm
Further, let 11(4) = A' and ft (A 1 ) = A" Then, if I, is the involution
of which A' is a double point and of which A A" is a conjugate pair
(Prop. 6, p. 222), we have
so that in the projectivity I^II the pair AA 1 corresponds to itself
doubly. Ii II is therefore an involution (Prop. 1, p. 221). If it he
denoted by I,, we have I 1 -n = I 2 , or II ^ I 2 , which was to be
proved
This proof gives at once :
COROLLARY 1 Any projectivity IT may be represented ff Me prod-
uct of two involutions, 11 1^-1^ either of which (but not both) has
an arbitrary po^nt (not a double point of II) for a double point.
Proof We have seen above that the involution I x may have an
arbitrary point (A 1 ) for a double point If 111 the above argument we
let I 2 be the involution of which A' is a double point and AA" is a
conjugate- pair, we have II l a (A'A") = A" A!] whence II I 2 is an invo-
lution, say I r We then have II = I x 1 2 , in which I 8 lias the arbitrary
point A' for a double point.
The argument given above for the proof of the theorem applies
without change when A =A", 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^Ij-Ig, where Ij
is determined as the involution of which A, A 1 are double points. We
have then I t I = I 2 , from which follows, by taking the inverse of both
sides of the equality, M^I-i^I,, or I^I-My or I^M^-I,
As an immediate corollary of the preceding we have
COROLLARY 3. The product of two involutions with double points
A, A' and B t B' respectively transforms into itself the involution in
which A A' 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 harmonio it their
product is an involution.
s 78, 79] INVOLUTIONS 225
THEOREM 18. Two harmonic involutions are commutative.
Proof. If Ij, I 2 are harmonic, we have, by definition, I t I 2 = T 8 , where
I s is an involution. This gives at once the relations ^ I a I 3 = 1 and
^i 'a = *%' *r
COEOLLAEY Conversely, if two distinct involutions arc commutative)
ttiey are Jiarmonic.
For from the relation I x - I a = I a - I t follows (I x . I 2 ) 2 = 1 ; i.e. I x . I 2
is an involution, since I x Ij^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
0,
Proof. Let the given projec- ^
tivity be II = I 2 \ ; I v I 2 being
two involutions. Let O v O z be
the centers of I t , 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 I x or I a
and which are not a conjugate
pair of I x or I 2 . If, then, we
have II (AB) = A'B', we have, by ^ 91
hypothesis, ^(AJS) A^ and
I a ( J 4 1 J5 1 ) = A'B'-, A v 2?j being uniquely determined points of the conic,
such that the lines AA V BB^ intersect m O l and the lines A^A', B^B'
intersect in 2 . The Pascal line of the hexagon AA^A'BB^B' then
passes through O v O s and the intersection of the lines AB' and A'B.
But the latter point is a point on the axis of U. This proves the theorem.
226 ONE-DIMENSIONAL PROJECT IV IT IKS [CirAi-. vm
COEOLLAEY. A projectivity on a conic, is the, product of two involu-
tions, the center of one of which may be any arbitrary point (not, a
double point] on the axis of tlie projectivity ; the cantor 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. 93). Let O l bo any point on tho
axis not a double point of II, and let I t be the involution of which
O t is the center. If, then, I l (A)^A v the center () a of tho involution
I 2 such that n = I 2 I x is clearly determined as the intersection of the
line A^A' with the axis. For by the theorem tho product 7 U / x is a
projectivity having I for an axis, and it has the points A, A 1 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 tho first) is
obvious.
THEOEEM 20. There is one and only one involution commutative
with a given nonparabolic noninvolutoric pro/ectimti/. If the projec,-
timty 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 IT ; ie, such that we
have H . I = I. H. This is equivalent to EM IT" 1 I. That is to say,
I is transformed into itself by IL 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 H, I is trans-
formed into itself by the collineation generating II, II. I. II"" 1 " I.
Hence 11.1 = 1-11. Hence I is the one and only involution commu-
tative with H.
COROLLARY 1. There is no involution commutative with a parabolic
projectivity.
DEFINITION. The involution commutativf ;with a given nonpars
bolic noninvolutoric projectivity is called ^^VQlution Mon to
the given prajectivity. An involution fe&yuft to
.,11!
79] INVOLUTIONS 227
COHOLLAKY 2 If a nonparabolic projectivity has double points, the
involution belonging to the projectivity has the same do-Me 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 1 = 11 and n-I-n- 1 ^!. Conversely, from the equation
n.i-n- 1 follows 11.1=1-11.
THEOREM 21. The necessary and sufficient condition that two invo-
lutions on a conie be harmonic is that their centers be conjugate with
respect to the conic.
Proof. The condi- 3 A
tion is sufficient For
let I x , I a be two invo-
lutions on the conic
whose centers O v O z
respectively are con-
jugate with respect
to the conic (fig. 92).
Let A be any point Fio> 92
of the conic not a
double point of either involution, and let l l (A)=A 1 and
If, then, Ij_ (A') A[, the center 0^ is a diagonal point of the quadrangle
AA^A'A^, and the center O z is on the side A^A!* Since, by hypothesis,
O s is conjugate to O x with respect to the conic, it must be the diago-
nal pomt on A^', i.e. it must be collinear with AA[. We have then
I z <\i(AA') = A' A, i.e. the projectivity I^L^ is an involution I 8 , The
center 8 of the involution I 8 is then the pole of the line 0^ with
respect to the conic (Theorem 19). The triangle O a O a 8 is therefore
self-polar with respect to the conic. It follows readily also that the
condition is necessary. For the relation L^ -12 = 13 leads at once to
the relation I^ I^l,,. If O v 2 , O a are the centers respectively of
the involutions l v I 2 , I 8 , the former of these two relations shows
(Theorem 19) that 8 is the pole of the line C^; while the latter
shows that 2 is the pole of the line X 8 . The triangle O^0 z t is
therefore self-polar.
228 ONE-DIMENSIONAL PROJECTIVITIES [CHAP VIII
OOEOLLAEY 1. Given any two involutions, there exists a third invo-
lution which is harmonic with each of the given involutions
For if we take the 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 mvolutions clearly satisfies the condition of the
theorem for each of the latter
GOEOLLAEY 2 Three involutions each of which is harmonic to the
other two constitute, together with the identity, a group.
COEOLLAKY 3. The centers of all involutions in a pencil of involu-
tions are collinear.
THEOEEM 22. The set of all projectivities to which belongs the same
involution I forms a commutative group.
Proof If II, H! are two projectivities to each of which belongs the
involution I, we have the relations I-HM = II and I E^ I = n r
from which follows I -IE" 1 I^II" 1 and, by multiplication, the rela-
tion I-n M-H! l^l-IL'U^l^U'Uy 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 projeotivities
given on a conic. Let A
be any point on this
conic, and let II (A) = A 1
and II 1 (.4 / ) Al t so that
TL.>Tl(A) = A' Since the
PIG. 93 x . , .. 1 T . ,
same involution I belongs,
by hypothesis, both to II and H v these two projectivities have the
same axis ; let it be the line I (fig. 93). The point TL^A) A t is now
readily determined (Theorem 12) as the intersection with the conic of
the line joining A' to the intersection of the line AA[ with the axis I.
In like manner, II (A^ is determined as the intersection with the
conic of the line joining A to the intersection of the line A^A' with
the axis I. Hence II (A = A{, and hence II - TL^A) *= A[.
It is noteworthy that when the common axis of the projectivities
of this group meets the conic in two points, which ,ar6 tliein* common
double points of all the projectivities of the group, tq (rcojip 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 come 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 doulle
point} in common if and o\dy if the product of the two involutions
lias two double points (or is paritbolw]
Proof Tins follows at once if the involutions are taken on a conic
For a common conjugate pair (or double point) must be on the line
jonnug 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 coiollanes of the last two sections.
2. The product of two involutions on a conic is paiabohc if and only if the
line joining the centeis of the involutions is tangent to the conic. Dualize.
3. Any involution of a pencil is uniquely determined when one of its con-
jugate pahs is given
4. Let II be a noninvolutoiic piojectivity, and let I be the involution be-
longing to II; further, let II (A A') = A' A", A being any point on which the
piojectivity operates which is not a double point, and let I(A") = A Show,
by taking the piojectivity on a come, that the points A'A[ are harmonic
with the points A A".
5. Derive the theoiem of Ex. 4 directly as a corollaiy of Prop. 4, p. 221,
assuming that the projectivity II has two distinct double points.
6 From the theorem of Ex. 4 show how to construct the involution be-
longing to a projectivity II 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 a 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 haimonic with I.
10. Show that any two piojectivities TI lt II a may be obtained as the
product of involutions in the form IL^ =1 I t , II 2 = I 3 I; and hence that the
product of the two projectivities is given by n^f!]. = I 3 > I 1
11. Show that a projectivity II I Ij may also be written II = I a -I, I 2
being a uniquely determined involution , and that in this case the two invo-
lutions I I are distinct xmless IT is involutoric.
230 ONE-DIMENSIONAL PEOJEGTIVITIES [OHAP.VIII
12. Show that if I x , I 2 , I 3 are three involutions of the same pencil, the
relation (I t I 2 I 8 ) a = 1 must hold
13 If act?, W, cc f aie the cooidmates of three paiis of points m involution,
_,, , a' 1) V c c' a -
show that r = !
a' c b a c'b
80. Harmonic transformations. The definition of harmonic involu-
tions m the section above is a special case of a more general notion
which can be 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 ) 2 = 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 mid 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 Wissensoliaften,
Leipzig, 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
(ABr) 2 = 1, ABIV 1, they are all tlneo harmonic to the transformation AB.
3. If a transformation S is the product of two involutono transformations
A, B (i.e 5 = AB) and T is an involutoric transfoimation harmonic to S, then
we have (ABr) 2 = l
4. If .4, B, C, A", B', <7 are six points of a line, the involutions A, B, T,
such that T(AA*) = B'B, A(5B') = C"C, B(CC") = A'A t 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 8, Chap. V).
5. The set of involutions of a one-dimensional form which are harmonic
to a given nonparabolac 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 o be a fixed point of a line I, and let Q be called the fnUpofa* of a
S^art to^ and's ' pr vided that C is ^ e harmonic oopfnptt of with
from O, and ASf have the same mid-point as^^A 'JG" ^ib^Hmi
mjd-pomt as src t them CA' irSl have the sa^e 4^p]Lj ^<f|: \
80,81] SCALE ON A CONIC 231
7. Any two involutions of tho same one-diiuonsional form determine a
pencil of involutions. Given two involutions A, B and a point J/, show how
to construct the olhtu 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 -\\ith regard to the conies
of any pencil o conies in a plane with / ioini a pencil of involutions.
9. If two nonpaiabohc piojcctivities aie commutative, the involutions
belonging to them coincide, unless both proj activities aie 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(^L)J -T- [II (JB)].
11. A conic through two of the foui common points of a pencil of conies
of Type / meets the conies of the pencil in pans of an involution. Extend
this theorem to the other types of pencils of comes. 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 (Stuiin, Die Lehie von den Geometiischen Verwandtschaften,
Vol I, p. 149).
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 are remarkably simple. As in the case on the
line, let 0, 1, co be any three arbitrary distinct points on a conic C z .
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 <x>) are defined 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, 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 t
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 so + y is then determined as the intersection
with the conic of the line joining the center to the point 0. Similarly,
232 ONE-DIMENSIONAL PEOJECTIYITIES [Ciur. Vill
to obtain the product of the points x, y we determine the center of the
involution as the intersection of the lines Ooo and asy. The point s, y
is then the intersection with the conic of the line joining this center to
FIG 04
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 propeities of addition and subtraction, multiplication
and division It is clear from this consideration that tJie points of a
come form a field with reference to the operations junt defined. This
fact will be found of use in tlie 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. 96
preceding discussion that if a numl^er to satisfies the equation 05 8 a>
the tangent to the conic at the point a must pass through the inter-
section of the lines Oco and 1 a (fig, 00). 4. number a* wifl therefore
Twee a square root in the field if'anjL only if a, tangent w% |e foatofi'to
* ? ' '
81]
SCALE ON A CONIC
233
the conic from the intersection of the lines Ooo and la; and, conversely,
if the number a has a square root in the field, a tangent can le 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 Ooo and la. Since this tangent meets the conic
in a point which is the harmonic conjugate of x with respect to Ooo,
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, ie whenever ZT is satisfied.
We may use these considerations to derive the following theorem,
which will be used later
THEOKEM 24. If A A', BE' are any two distinct pairs of an involu-
tion, there exists one and only one pair CC' distinct from BB' such
that the cross ratios
&(AA', BB') and
B (AA' } CC') are
equal.
Proof. Let the ^^ L n _<?
involution be taken
on a conic, and let
the pairs AA f and
BB' be represented
by the points Ooo
and la respectively (fig. 96). Let xx' be any other pair of the invo-
lution. We then have, clearly from the above, xx l = a. Further, the
cross ratios in question give
]J(Oco, la) = i, B(0oo, xx')*=-,.
a x ' x'
These are equal, if and only if x' = ax, or if xx 1 = ax*. But this implies
the relation a ax*, and since we have a =j= 0, this gives a? 2 = 1. The
only pair of the involution satisfying the conditions of the theorem
is therefore the pair CC f = 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 Tbe repiesented by the equation afx = a.
96
234
ONE-DIMENSIONAL PROJECTIVITIES [CUAP.VIII
S The eauation of Ex. 13, p. 230, is the condition that the linen joining
the three pairs of points aa>, W, f on a conic aie oououironl.
V Show that if the involution ** = a has a conjugate pa.r W such that
the cross ratio B(0oo, W) has the value X, the nnmlHi a\ has a Hquaro root
in the field
82 Parametric representation of a conic. Let a scale be established
on a 'come C* by choosing three distinct points of the conic as the
fundamental points, say, - 0, Jf - oo, A - 1 Then let us establish
a system of nonhomogeneous point coordinates in the plane of the
3 conic as follows: Let
N the line OM be the x-
axis, with as origin
and M as oo^. (fig, 07).
Let the tangents at
and M to the conic
meet in a point N, and
let the tangent ON be
the y-axis, with N as
coj,. Finally, let the
point A be the point
(1, 1), so that the line
AN meets the a?-axis
in the point for which
te 1, and AM meets
the y-axis in the point for which y =* 1. Now let P X be any point
on the conic. The coordinates (x t y} of P are determined by the
intersections of the lines JPJVand PIT with the o:-axis and the #-axis
respectively. We have at once the relation
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 colHniSar with N and the
point 1 on the oj-axis, and the points X, X on the conic are eolliaear
with JVand the point so on the a?4m S}nc$ the latter point is al^o
on the line joining and oo on ? t|ie oon,ic, the OQ$str;Ucii<Jtt for multi-
plication on the conic show^tliati anv, line tntouizh !thi Itiiblat &
. 97
82] PARAMETRIC REPRESENTATION 235
the x-axis meets the conic (if at all) m two points whose product is
constant, and hence equal to A 2 . The line joining the point x on the
aj-axis to the point 1 on the conic therefore meets the conic again in
the point A 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, <x>, 1, x on the
tf-axis. This gives the relation
a; = A 2 .
We may now readily express these relations in. homogeneous form
If the triangle OHN is taken as triangle of reference, ON heing
x l = 0, OM being # a 0, and the point A being the point (1,1, 1),
we pass from the nouhomogeneous to the homogeneous by simply
placing x = %i/x z , y = M a /% s - The points of the come <7 a may then
le represented by the relations
(1) ^rtfgitfgSsX^X: 1
This agrees with our preceding results, since the elimination of X
between these equations gives at once
a; 2 -aV 3 =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 = co, is exceptional in this equation. This exceptional
character is readily removed by writing the parameter X homogene-
ously X = X 1 -X 2 . Equations (1) then readily give
THEOREM 25. A conic may le represented analytically "by the equa-
tions cc l ;x, i :x s = X 2 . \\ z : X, 2 .
This is called a parametric representation of a conic.
EXERCISES
1. Show that the equation of the line joimng two points A t , A 2 on the conic
(1) above is a: t (A x + A 2 ) jr a + AiA.^ = , and that the equation of the tan-
gent to the conic at a point A x is x l 2 Ajo: 2 + \iZ s 0. Dualize.
5J. Show that any collineation leaving the conic (1) invariant is of the form
atf J ar 2 ' : z,' - a 2 *, + 2 a#r g + /8%? B : ay^ + (08 + y) x z + @8x 3 y*^ + 2 ySo; 2 + S V
(Hint Use the parametric representation of the conic and let the projectivity
generated on the conic by the collineation be Af = oA 4 + /3A a , A.J = y\i + SA,.)
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* 1 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 constructions, or by constructions 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
pan- of linear equations; and the linear operations m 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 problem or a proUem
of the first degree. Any such problem has, if determinate, one and
only one solution.
In the usual representation of the ordinary leal project geometry in a
plane by means of points and hues dia*n, let us say, with a pencil on a sheet
* u P6r A Constructions are evidently those that can be carried
out by the use of a straightedge alone There is no familmr mechanical
to tse o dvenel e ? ^ T* 3 M A& *** as a Despondence whereby
83] DEGREE OF A GEOMETRIC PROBLEM 237
device for di awing lines and planes in space. But a pictuie (which is the
section by a plane of a projection fioin a point) of the lines and points of
intei section of linearly constructed planes may bo constructed with a stiaight-
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
hue of which three pairs of homologous points are given; (I) the
determination of the sixth point of a quadrangular set of which five
points are given ; () 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. 407 ff.
238 GEOMETRIC CONSTRUCTIONS [OHAP.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, 6, 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,
K = [1, a, 6, c, ],
which we will call the domain of rationality defined ly the data f 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 y, we obtain two equations,
/i(*)-0, /,(a,y) = 0,
the first containing ss 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 tins 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) x n + ttjaf-H a a a3 B - 2 + . . . + a n - 0,
in which the coefficients a v # 2 , . . ., 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 coordinates are used, a, &, c, denote the, mutual ratios
of the codrdmates of the given elements.
t A moment's consideration will show that the points whose coBrdinates 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 co$strjn&t!ois ; j^e, such
that *A,BiO are any three points of the class, the harmonid tofriotoa ft A With
respect to J5 and C is a pomt of the class. , , * \ !
83] DEGREE OF A GEOMETRIC PROBLEM 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 fa, fa are two polynomials of the kind indicated, and of degrees
Tij and n z respectively (n t + n s = n). Equation (1) is then equivalent
to the two equations
0i(HO, fc(3)-0.
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
Wj < 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 n l and one of degree n v In speaking of
a problem of the %th 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 m 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 rx
84. The intersection of a given line with a given conic. Given a
conic denned, let us say, by three points A, B, C 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 C-axis one
of the given tangents to the conic, as ^-axis the line joining the points
A and B, 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) s? px # = 0.
This equation is not m 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 x 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 + rp) (x r)=Q.
The problem of finding the remaining point of intersection then
Depends merely on the solution of the linear equation
* 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 aj *= c, we should' merely
have to choose the other tangent as x-axis to bring the problem into the form here
assumed.
84, 85]
PROPOSITION
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 tins later ( 86).
85. Improper elements. Proposition K a . 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.
"Pat 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, suoh that each of the elements a, v a v , is a
square in F'. This is, of course, less general than the theorem that
a field F' exists in which 4very, element frf F is a square, but it is
sufficiently general fr^marm gattf " - ; ' - J - -'- " - - - <
242 GEOMETRIC CONSTRUCTIONS [CHAP ix
PROPOSITION K 2 If any finite number of involutions are given in
a space S satisfying Assumptions A, E, P, there exists a space S' of
which, S is a subspace,* suck 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 a )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' 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.
COROLLAHY 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 s ) -
COROLLARY 2. JUiery quadratic equation with proper coefficients has
two roots which, if distinct, are both proper or both improper. (A, E,
P, H , K a )
* We use the word sulspace to mean any space, every point of "which is a point
of the space of which it is a suhspace 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 . -f Cf . Ex , p. 261.
80] PROPOSITION K 2 243
For the double points of any projectivity satisfy an equation of
the form cj? + (if <i) ,r b = (Theorem 1 J , Our 4, Chap VI), and
any quadratic equation may bo put into this form.
TIIKOIIKM 2 Any two involution* in, tlie mint one-dimensional form
have a conjugate pair in common, which may le proper or improper.
(A, E, P, 1I , 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 improper, which is harmonic with any given conjugate pair. (A,
E, P, H 0) K a )
For the involution which lias 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 v 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 perspeetivity being 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 g )
Proof. If the point of intersection of the two lines be a double
point of each of the involutions, let Q and M be an arbitrary pair
of one involution and Q 1 and Jtf an arbitrary pair of the other involu-
tion The point of intersection of the lines 'QQ' and MR' 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 ' of the other (these double points
are proper or else exist in an extended space S f which exists by
Proposition K 2 ). Also let .Wand JZV"' be the conjugates of in the two
involutions. Then by the same argument as before, the point of
intersection of the lines MM 1 , $W 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
_ 2 1 2 J_ O /V rv m 1 O /v rv /yi -I O /v /y /v U1TTT . A
/"I \ /V /* _L_ * I C(/nnfX' "*T -" w-tnwJiXsn "7~ ^J "^IR^I^ffl "T" " ^Qfl j! 8 """"" J
but it was not shown that every equation of this form represents a
conic The line ^ = contains the point (0, # 2 , o; 8 ) satisfying (1),
provided the ratio x z : x t satisfies the quadratic equation
Similarly, the lines x z = and a; 8 = contain points of the locus
defined by (1) 3 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 winch is therefore of the form
(3) ax* + c^ajjj H- c z x^ + c & x^x z = 0.
If c x = 0, the points satisfying (3) lie on the two lines
! = 0, ax i + c a x a + c s x 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 = 0, the trans-
formation
* Proposition Ka 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 K 2
85, so] PBOBLEMS OF THE SECOND DEGREE 245
transforms the points (x v x y x s ) satisfying (3) into points (.-/, xj, ,r a ')
satisfying
(5 ) L - A\ ,,[ + (,X + v /) 4 = 0.
\ by /
But (5) is in the form which was proved in Theorem 7, 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 eguatwn of the form
a u aif + a,yp* 4- a w x% + 2 a^x^ + 2 a u x^ + 2 a z& x z x s =
represents a point come (proper or improper) which may, however,
degenerate ; and, conversely, every point conie may le represented T)y
an equation of this form. (A, E, P, H , K 2 )
THEOREM 4' Every equation of the form
A u uf + A^ 4- 4 M * + 2 -/!!, + 2 A u UjU t + 2 A^uja^ =
represents a lim conic (proper or improper} which may, however, de-
(jenerate ; and, conversely, every line conie may "be represented "by an
equation of this form, (A, E, P, II , K a )
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 bo reduced to the problem of find-
ing the points of intersection of an arbitrary lino in the plane with
a particular conic in the plane. This result we may state in the
following form:
THEOREM 5. Any problem of the second degree in a plane may le
solved "by linear constructions if the intersections of every line in the
plane with a single conic in this plane are assumed Jcnown. (A, E,
P, H OI K 2 )
In the xisual 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
GKEOMETEIC CONSTRUCTIONS
[CHAP. IX
conic in. the leal 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 lepresentation any pioUem of the second degree can be solved 1>y ineani> of
a stiaightedye and compass alone The theoiem just stated, however, shows that
if a single circle is drawn once foi 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 tliis 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 f ollowing :
PROBLEM 1. To find the double points of a projectimty on a line of
which three pairs of homologous points are given. We may assume
A
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 assunu
known the intersections of any line of the plane with this conic. Le
be any point of the given conic, and with as center project lh<
given pairs of homologous points on the conic (fig. 98). These defim
a projectivity on the conic. Construct the axis of this projectivit;
and let it meet the conic in the points P, Q The lines OP, OQ thei
meet the given line in the required double points.
PROBLEM 2, To find the points of intersection of a given line wih
a conic of which five points are given. Let A, B, C, JO, E be the give
points of the conic The conic is then defined by the projectivit
D(A, 3, G)-^E(A, B, C) between the pencils of lines at D and J5
86] SEXTUPLY PERSPECTIVE TEIANGLES 247
This projeclivily 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.
PKOJILTCM 3. We have seen that it is possible for two triangles in
a plane to be perspective from four different centers (cf. 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, 7s it poimille to construct two triangles that are perspective from
six different centers? Let the two triangles be ABC and A'JJ'C', and let
B! =0, $ a = 0, <B, =
be the sides of the first opposite to A, 7?, respectively. Let the
sides of the second opposite to A', J3', C' respectively be
s + # 3 =0, x 1 + //ajg + F.e B = 0, x l ,+ l'x z + 1\ = 0.
The condition for ABC~A'B'C' is that the points of intersection of
corresponding sides be collmear, i e.
1 -1
(1) -F 1 = F-Z'0.
-P 1
In like manner, the condition for JBCA ^- A'B'C' is
-l V
(2) -101 =H"-^=0.
~/y i o
From these two conditions follows
-F ti
~l" 1 -H^-F-O,
1 -1
which is the condition for CAB ~ A'B'C'. Hence, if two triangles are
in the relatwna ABC ~~ A'B'C' and BCA~~A'JB'G', they are also in
the relation CAB A'B'C 1 * Two triangles in this relation are said to
A
be triply perspective (cf. Ex. 2, p, 100)*, The domain of rationality
defined by the data of our problem; is clearly
248 GEOMETEIC CON STKUC TICKS [CHAP. IX
Since numbers in this domain may be found winch satisfy equutionH
(1) and (2), the problem of constructing two triply perspective tri-
angles is linear
The condition for ACB == A'H'C' is
A
(3) L J -l=
If relations (1), (2), and (3) are satisfied, the triangles will be per-
spective from four centers Let k be the common value of // and I"
(3), and let I be the common value of I' and V (1) Relation (2) then
gives the condition Jc 2 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', the two triangles
will, by what precedes, be perspective from six different centers The
latter condition is
(4) F2'-Z"=0.
With the preceding conditions (1), (2), (3) and the notation adopted
above, this leads to the condition
The equation k s 1 = is, however, reducible in K; indeed, it w
equivalent to
The first of these equations leads to the condition that A', 7?', C' are
eollinear, 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
^+^=0, a^ + ias. + wfyasO, i + ^a 4- ww =
* They can coincide only if the number system IB such that 1 + 1 + 1 = 0: e.g. in
a finite space involving the modulus 3.
80] 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 1 nine points :
(0, -1,1) (0, w*, -10) { 0, w t -w*)
(5) (-1, 0, 1) (-10*, 0, 1 ) (-w, 0, 1 )
(-1, 1, 0) ( 10, -1, ) (w\ -1, )
They form a configuration
4
3 12
which contains four configurations
3
3
of the land 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 problem in a plane ffiven Mow that are of the second degree are to be solved
1>H linear construct ions, with the amtmplum that the points of intersection of any line
in the plane with a given faed conic in the plane are known; i.e. "with a straiglit-
cilye and a y wn circle in the plane"
1. Construct the points of intersection of a given lino with a conic deter-
mined by (i) four points and a tangent through one of them ; (a) three points
unrl the tangents through two of thorn ; (in) 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 * + | + aJ 8 s + fiXa^o* = for
any value of X, and are, in fact, the points of inflexion of the cubic. This configura-
tion forms the point of departure tor a variety of investigations leading into many
different branches of mathematicSi v
250 GEOMETKIC CONSTRUCTIONS [CHAP. IX
4. Given a triangle A Z B Z C 2 inscnbed in a tnangle A v K l C r In how
many ways can a tnangle A 9 B s C a be inscubed in A z B^C a and cncumsciibod
to A^C^i Show that in one case, in which one vertex of A a B a C. t may be
chosen arbitiarily, the problem is hneai (cf. 3(3, Chap IV) , ami that, in
another case the problem is quadratic. Show that this pioblem gives all con-
figurations of the symbol
9 3
3 9
Give the consti actions 01 all cases (c.
S. Kantor, Sitzungsbenchte der inathematisch-naturwissenschafthcheu Clnaae
der Kaiserhchen Akademie der Wissenschaften zu Wien, Vol. LXXXIV
(1881), p 915)
5. If opposite vertices of a simple plane hexagon P]P s ,P a P lt P B P a arc on
three concurrent lines, and the lines P^P^ P S P^ P 5 P 6 ai e concurrent, then the
lines P Z P S , PiP 6 > P 6 Pi are also concurrent, and the figure thus formed is a
configuration of Pappus
6. Show how to constiuct a simple n-point iusciibed in a given simple
n-point and circumscribed to another given simple n-point.
7 Show how to inscnbe 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 thiough thiee points and tangent to two lines not
meeting any of the points.
10 Constiuct a conic thiough 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 X, Y are two variable points of the
conic such that AX, AY always pass through a conjugate pan of a given
involution on a line I, the line JTFwill always pass thiougli a fixed point J3.
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 thei e 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 hues which
meet each of the given four lines ; and that if there are three such lines, theie
is one through every point on one of the lines,
14. Given in a plane two systems of five points A^^A^A^A^ and
B 1 B Z B 9 B^B 6 , given also a point X in the plane, determine a point Y such
that we have XtA-LAiAiAiA^-j-Y^BiByBzBiBt). In general, there is one
and only one such point Y. Under what condition is there more than one ?
(R. Sturm, Mathematische Annalen, Vol. I (1869), p, 583,*)
* This is a special case of the so-called problem of projectwity, 3Tor references
and a systematic treatment see Sturm, Die Lehre von den geometrischen Ver-
wandtschaften, Vol. I, p 848.
87]
INVARIANTS
251
87. Invariants of linear and quadratic binary forms. An expres-
sion of the form a^ + <V a is called a linear binary form, in the
two variables x v x z 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 <\. 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 as a ) = (a z> a x ). If & = \x^ + & a # 2 is another linear
buiary 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
0.
Now suppose the two elements A and B are subjected to any pro-
jective transformation II :
a @
ry 8
The forms a x and 1 9 will be transformed into two forms and &
respectively, which, when equated to 0, define the points A 1 , B' into
which the points A, B are transformed by II. The coefficients of
the forms <
follows ;
^, in terms of those of a s) l x arj readily calculated as
which gives
Similarly, we find
a 1 -f- 5a 2 .
Now it is clear that if the elements J, J3 coincide, so also will the
new elements A', B' coincide. If we have A 0, therefore we should
also have A'
We have
a/
0. That this is the case is readily verified.
252 GEOMETRIC CONSTRUCTIONS [CHAP.IX
by a well-known theorem in determinants. This relation may also
be written
a ft
A' =
8 ' A -
The determinant A is then a function of the coefficients of the forms
a x) l x) with the property that, if the two forms are subjected to a lin-
ear homogeneous transformation of the variables (with nonvaiiishmg
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 f - &, of the two forms a^ & a .
If multiplied out, this product is of the form
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 zz of the quadratic form vanishes. The condition D a = 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 must vanish whenever J) n van-
ishes. There must accordingly be a relation of the form Z> n , = Jc D n .
If a% be subjected to the transformation II given above, the coefficients
a n> ia> a aa of fcne new form ^ are readily found to be
By actual computation the reader may then verify the relation
a $
7 8
The discriminant D a of a quadratic form 0% is therefore called an
invariant of the form.
87] INVARIANTS 253
Suppose, now, we consider two binary quadratic forms
Each of these (under jfiQ 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 projective transformations ; we may
therefore expect the condition sought to be an invariant of the two
forms. We know that if a v & a are the nonliomogeneous coordinates
of the two points represented by a = 0, we have relations
2B 2 & 13
a^ ct, z > ttj + a a - >
with similar relations for the nonhomogeneous coordinates & p & 2 of
the points represented by &J = 0. The two pairs of points a v a a ; \, 5 2
will be harmonic if we have (Theorem 13, Cor. 2, Chap. VI)
i
This relation may readily be changed into the following :
which, on substituting from the relations just given, becomes
This is the condition sought. If we form the same function of the
coefficients of the two forms a^ 8 , V* obtained from a t 1% 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^a,, Zj.6 a ) of the
two pairs of points represented by a* 0, &* *= 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 J!imctioii of the
coefficients of the forms a[, 5j , 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 ^nvariants.
EXERCISES
1 Show that the cross latio Ijfc (cjOj, Va) 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 pans of points denned by the three binary quadratic forms
as = 0, 1% = 0, cj = ; show that the three will be in involution if we have
= 0.
Hence show that the above determinant is a simultaneous invariant of the
three forms (cf. Ex 13, p. 230).
88. Proposition E^. If we form the product of n linear binary
forms a x 'a' x -a'J' - > a*" 1 *, we obtain an expression of the form
An expression of this form is called a Unary homogeneous form or
guantic 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 BJ 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-
sation of K a :
88,89] PKOPOSITION K a 255
PROPOSITION K n . If a^ a^ -are a finite number of binary 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 w " = 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
o?~ l is called the derivative of Ax? with respect to x k in symbols
The derivative of a polynomial with respect to a?, is, by definition, the
sum of the derivatives of its respective terms.
Q
This definition gives at once f-A = 0, if A is independent of av
o^i
Applied to a term of a binary form it gives
nk3%' l se? t -j- ^xf
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 f ollows :
Given the binary form
~\ -f- * ^~~ 1 } A W ~ X
* For the proof of this theorem on the basis of the definition just given, of. Fine,
College Algebra, pp.; <I|Q~462.
256 GEOMETRIC CONSTRUCTIONS [OUAP.IX
If herein we substitute for x v x a respectively the expressions ,)\ + X?/ ; ,
, + X?/ 2 > we obtain,
3
Here the parentheses are differential operators. Thus
ay a ran ay a ra/i . T . ,,
where ~ means - M- r-~- means - - - U etc It is readily
a#i ^iL^J ^a^i ^ /J L^iJ
proved for any term of a polynomial (and hence for the polynomial
itself) that the value of such a higher derivative as ^ 2 //a, a a. 1 is
independent of the order of differentiation; i.e. that we have
a 2 / = a 3 /
aCj&Bj Sas^jj,
DEFINITION The coefficient of X in the above expansion, viz,
yfif/dx^ + y^f/dx z is called the first polar form of (y v y a ) with
respect to f (x v x a ) ; the coefficient of X 2 is called the wfitml ; the
coefficient of X" is called the nth polar form of (y v y^ with rasped
to the form f If any polar form be equated to 0, it represents a set
of points which is called il\Q first, second, , nth polar of the point
(y v y s ) with respect to the set of points represented %/ (^, jr a ) ~ 0.
Consider now a binary form/ (x v se z ) = and the effect upon it of
a projective transformation
If we substitute these values in / (oz x , x s ), we obtain a new form
F(x[, 0$. A point (x v ss z ) represented by/( 1 , aJ 8 )= will bo trans-
formed into a point (as/, aj a ') represented by the form y(sp[ t ii)0.
Moreover, if the point (y v y,) be subjected to the same projectivity,
it is evident from the nature of the expansion given above that the
polars of (y lf g/ a ) with respect to f(x v o} 2 ) = are transformed into
the polars of (y/, yj) with respect to ff(es[ t %) = 0.
& so, oo] INVARIANTS 257
"We may summarize the results thus obtained as follows :
THEOREM 6 If a Unary form f is transformed ly a projective
transformation into the form F, the set of points represented by f= Q
is transformed into the set represented ly ^=0. Any polar of a
point (y v 7/ a ) with respect tof=Q is transformed into the correspond-
ing polar of the point (y[, v/ a ') with respect to F=
The following iw a simple illustration of a polar of a point with
respect to a set of points on a line
The form ayj a = represents the two points whose nonhomo-
geneous coordinates are and co respectively. The first polar of any
point (y v y a ) with respect to this form is clearly y^ + y z x : = Q, and
represents the point (y v y^)] in other words, the first polar of a
point P with respect to the pair of points represented by the given
form is the harmonic conjugate of this point with respect to the pair.
EXERCISE
Determine the geomeh ical consti notion of the ( l)tih polar of a point
with lespoct to a sot of n distinct points on a line (of. Ex 3, p 51).
90. Invariants and covariants of binary forms. DEFINITION. If a
binary form a = a v a% + na^~ l x z -\ ----- h aa be changed by the
transformation / _ ,
l
into a new form A$ ~ A ^ n + ^X"~X + ---- 1" 4X B an y rational
function f(n Q) a v , a n ) of the coefficients such that we have
I(A Q , A v , 4.) = ^(, A % 8) /(a,, a v . - , oj
is called an invariant of the form ". A function
C(a 9t a v -.,a n ; as lf x s )
of the coefficients and the variables such that we have
C(A 0) -4,-. -,4,; /, <) = f (, &7, SJ-C'K, ^,...,^5 fl^, aO
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 , & 7, $ only; they are" then
called siwrnUaneous invariants or eovariants,
258 GEOMETRIC CONSTRUCTIONS [OIIAP. 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 "by equating to any invariant of a form (or of a
system of forms) must determine a property of the st't of points
represented by the forin (or forms) which in invariant 'tinder a pro-
jective transformation. Hence the complete study of the projeetive
geometry of a single line would involve tl*e 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.
The functions <f>(a, /3, 7, 8) and ty(a, ft, 7, 8) occnrrvny in (he,
definition above are always powers of the determinant S--/:ty of
the projectile transformation in question.*
Before closing this section we will give a simple example of a oovu-
riant. Consider two binary quadratic forms a?, b% and form the new
quantic
c * = Wi ~ A) x i + K & 3 - 'A) A + (a A ~ <* A)
By means of equations (1), 87, the reader may then verify without
difficulty that the relation
holds, which proves <7 n6 to be a covariant. The two points represented
by <?6 = are the double points (proper or improper) of the involu-
tion of which the pairs determined by a* = 0, V* = are conjugate
pairs. This shows why the form should be a covariaut.
EXERCISE
Piove the statement contained in the next to the last sentence.
91 . Ternary and quaternary foi ms 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 wth
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 xero
lias 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, of., for example, Grace and Young, Algebra of Invariants, pp, 21,
9 i] INVARIANTS 259
obtained by equating to zero a ternary form of the wth 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 wth degree
equated to zero is called an algebraic surface of the nth degree
The definitions of invariants and covanants of p-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 projectively
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 protective transforma-
tion. Similar remarks apply to covanants of systems of ternary and
quaternary forms.
Invariants and covariants as defined above are with respect to the
group of all protective collineations. The geometric properties which
they represent are properties unaltered by any projective collmeation.
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 89, 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.
4. Prove that the discriminant a\l aJJ o| of the ternary quadratic form
is an ^variant. What is its geometrical interpretation? Cf. Ex,, p J87-
260 GEOMETRIC CONSTRUCTIONS [CHAP IX
92. Proof of Proposition K^. Given a rational integral function
<jE> (x) = a ,e n + ap*- 1 + + n ni 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 $ (x) =
has a root
Let/(#) 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 /(/), where [/(as)] is the class of
all rational integral functions with coefficients in F. We proceed to
define laws of combination for the elements of F^ 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, ie. with
coefficients in F, and where r(x) is of degree lower than the degree
n of < (x) If two polynomials f lt / a belonging to F are such that
their difference is exactly divisible by < (x), then they are said to be
congruent modulo <f)(x), in symbols / t = / 2 , mod. <>(#).
1 Two elements f^j), / 2 (/) of F, are said to be equal, if and only
\lfi(x} and/ 2 (#) are congruent mod <(e). By virtue of the theorem
referred to above, every element f(j) of F f is equal to one and only
one element /' (/) of degree less than n. We need hence consider only
those elements /(/) of degree less than n. Further, it follows from
this definition that $ (/) = 0.
2. If / t (x) +/, (x) =/ 3 (as), mod. (*), then /, (/) + /, (/) -/, (/)-
3- VfM /, (^^/.(a!), mod. 0(0?), then /,(;)/.(/)=/ 0>
Addition and multiphcation of the elements of F^ having thus
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 $(x) and any polynomial /(x) have no
common factors, there exist two polynomials h(x) and k(x) with
coefficients in F such that
* Fine, College Algebra, p. 156. f Kne, loc, cit., p. 208,
92] PROOF OF K n 261
This gives al once Ji (j) f(j) 1,
so that every element f(J) distinct from has a reciprocal. The class
F, is therefore a field with respect to the operations of addition and
nmtiplieation defined above (of. 52), such that <j>(j)= 0. It follows
at once* that xj is a factor of <(#) in the field F,, which is there-
fore the required field F'. The quotient <j>(x)/(x j) is either irre-
ducible in F J} 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 zeio of the factor m
question Continuing in this way, it is possible to construct a field
F ., j w , where m ^jin 1, in which <f> () is completely reducible,
i.e. in which <j>(%) may be decomposed into n linear factors This
gives the following corollary :
Given a polynomial < (,#) belonging to a given* field F, there exists a
field F' 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 theoiems of this and other chapters are given as dependent on
A, E, P, H 0> whereas they are provable without the use of H . Determine
which theorems are true in those spaces for which H is false.
* Erne, College Algebra, p. 169,
t Cf. B. Steinitz, Algebraisclie Theorie der Kbrper, Journal fur reme u. ange-
wandte Matheraatik, Vol. CXXXVII (1909), p. 107 ; especially pp. 271-286.
CHAPTER X*
PROJECTIVE TRANSFORMATIONS OF TWO-DIMENSIONAL FORMS
93. Correlations between two-dimensional forms. DEFINITION. A
protective 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 protective
correspondence between the elements of a bundle of planes and the
elements of a bundle of lines is called a correlation f
"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 Jjj*, P z , P 8 , P are collinear points, the corresponding
lines p v p s> p a , PI 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 T into the lines on a
point L This determines a transformation of the lines [Z] into the
points [], which we may denote by F', thus:
F(i)-i
That F' is also a correlation is evident (the formal proof may be
supplied by the reader). The transformation F is called the correla-
tion induced by I\ If a correlation F 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 duality are sometimes used in place of correlation.
262
93] COREELATIONS 263
plane into the points [L] of the plane, the correlation which trans-
forms the points [IV] into the lines [LL 1 ] is the correlation induced
by T. If T' is induced by F, it is clear that T is induced by I".
For if we have
we have also
and hence the induced correlation of I" transforms P z into jp a , 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 m 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 P v P,, J into points JJ' f #, If, J?, and the transformation T' of
the lines jJJ, 2J2J, PA P& into the lines ##, P 2 'P 8 ', tftf, -W BB
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 T 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
;ii H
204 TWO-DIMENSIONAL PEOJECTIVITIES [CHAP.X
The following theorem is an immediate consequence of the defini-
tion and the fact that the resultant of any two projective correspond-
ences is a protective correspondence.
THEOREM 1 Tlie resultant of two correlations is a projective col-
Iwieatwn, and the resultant of a correlation and a prujeetvve eolhnea-
tion is a correlation.
We now proceed to derive Hie fundamental theorem for correlations
between two-dimensional forms
THEOREM 2. A correlation between two two-dimensional primitive
forms is uniquely defined when four pairs of homologous elements are
(/iven, 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 a 1 . Let C' a be any conic in a', and let the four pairs of homol-
ogous elements be A, B, G, D in a and a', I', c', d' in a'. Let A', B',
C', D' be the poles of a', &', e', d' respectively with respect to C z . If
the four points A, 73, 0, D are the vertices of a quadrangle and the
four points A', B', C', D' are likewise the vertices of a quadrangle
(and this implies that no three of the lines a', V, c', d' are concurrent),
there exists one and only one collmeation transforming A into A', B
into B', G into C', and D into D' (Theorem 18, Chap. IV). Let this
collmeation be denoted by T, and let the polarity denned by the conic
C 2 be denoted by P. Then the projective transformation T which is
the resultant of these two transforms A into a', B into &', etc. More-
over, there cannot be more than one correspondence effecting this
transformation. For, suppose there were two, T and T r Then the
projective correspondence I\- 1 T would leavo each of the four points
A, B, C, D fixed ; i.e. would be the identity (Theorem ] 8, Chap. IV).
But this would imply I\ = 37.
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, &, c. Let P be any point of the
plane ABO which is not on a side of the triangle. The line p ink
which P is transformed by the given correlation T does not, then, pas$
through a vertex of the triangle ABO. The correlation T is deter
mined by the equation F (ABCP) = abcp, and, by hypothesis, is suol
93] COREELATIONS 265
that T (ale) = ABC The points [Q] of c are transformed into the
lines [17] on C, and these meet in a pencil [Q 1 ] projective with [Q]
(fig 99) Since A corresponds to B and B to A in the projectivity
[Q] -fr[Q'], this projectivity is an involution T. The point Q in which
FIG. 90
CP meets c is transformed by T into a line on the point cp; and
since Q Q and cp are paired in I, it follows that cp is transformed
into the line CQ CP. In like manner, lp is transformed into J5P.
Hence p = (<$>, &2?) is transformed into P = (CP, BP).
THEOREM 4. Any projective collineation, IT, in a plane, a, is the
product of two polarities.
Proof. Let Aa be a lineal element of a, and let
H (Aa) = A'a', H (A 1 a') = A" a".
Unless II is perspective, Aa may be so chosen that A, A', A" are not
collinear, aa'a" 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 P(AA'A") a"a f a, namely the polarity
denned by the conic with regard to which AA n (aa") is a self-polar tri-
angle and to which a f is tangent at A'. 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'(aa') is self-reciprocal Hence (Theorem 3)
PII P x is a polarity, and therefore II P^.
266 TWO-DIMENSIONAL PKOJECTIVITIES [CHAP. X
94. Analytic representation of a correlation between two planes.
Bilinear forms. Let a system of simultaneous pomt-and-hhe 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 ly equations of the form
pu^= a n x t -\- a lz x a + a ls x 3 ,
(1) pul = a^ + a 22 zc a + a^x 3 ,
where the determinant A of the coefficients a lt is different froin 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 l} pu^ = x s> pu(=x 3 in
a plane represents a polarity in which to every side of the triani/le of
reference corresponds the opposite vertex.
Also, if (u[, u' 2> u a ) be interpreted as line coordinates in a plane
different from that containing the points (x lt x z , ; 8 ) (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 x z , J 8 ) and the plane of (u[, u^, u)
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[, u.!,, u s ) of the line u'= F (X], which corresponds to the
point X (x v # 2 , a? 3 ) By solving these equations for x t ,
(2)
we obtain the coordinates of JT= P" 1 (u'} in terms of the coordinates
u' of the line to which JTis homologous in the inverse correlation T~\
If, however, we seek the coordinates of the point X' = F (u) which
corresponds to any line u in the correlation. F, we may proceed as
follows :
94] CORRELATIONS 267
Let the equation of the point Z' = (A/, ic a ', x) in line coordinates be
It/it!/ + 16} X! 2 + WgiCg = 0.
Substituting in this equation from (1) and arranging the terms as a
linear expression in x v x a , x a ,
at f\ -I- /v L ji. fi
we readily find
fV\ M/ /y ft JL, ft f>t JL ft <y '
The coordinates of Jl r ' in terms of the coordinates of u are then
given by
1 " " 11 1 *^ 12 2 ' 1JJ S*
/ j \ t A
i n \ lit* yi /
V*/ Vl "a ""-21 '
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 con elation 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 (8) taken together repre-
sent the inverse of T.
Another way of representing T analytically is obtained by observ-
ing that the point (x v x v # 8 ) is transformed by F into the line whose
equation in current coordinates (x[, x^ #/) is
or,
(5) (a u o!i + a la flj a + o^) / + (a.^4- a 23 3 + <V 8 ) ^
+ ax + ** - 0-
The left-hand member of (5) is a general ternary "bilinear form. We
have then
COROLLARY 3. Any ternary "bilinear form in wbicJi the determinant
"" ^^"entlrgr^ &ro, represents a, correlation m a plane.
1 * > ii * * , , - , , t
. i ' * * ' x 5 * i , ' ? ' , 5 ' 5 , ! * l< ;
1 * I I ' ' r , ! ' ' , ' - \ ? ,1 i t 1 S ? I i ,* I t t '
268 TWO-DIMENSIONAL PKOJECTIVITIES [CHAP x
95. General projective group. Representation by matrices. The
general projective group of transformations in a plane (which, under
duality, we take as representative of the two-dimensional primitive
forms) consists of all projective collmeations (including the identity)
and all correlations in the plane. Since the product of two collmea-
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 le 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
x' t =
.7=1
may be conveniently represented by the matrix A of the coefficients a tj :
The product of two collineations A = (a y ) and B = (&^) is then given
by the product of their matrices :
Z>
At
the element of the *th row and the t /th column of the matrix BA
being obtained by multiplying each element of the tth 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 form a group ; but this forma no exception to the
statement just made, since the identity must be regarded as a collineation.
MATRICES 269
Of the two matrices
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 A
r-*i3 -"as M wi
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 eollineation represented "by
the given matrix. Indeed, by direct multiplication,
' a u a is a u '4 U Ai -V '^00'
a ai a aa a as - 4 ia ^22 As = A
,8i tf 3a "MI A 4s 4*1 ,00^,
and the matrix just obtained clearly represents the identical col-
Imeation. Since, when a matrix is thought of as representing a
eollineation, 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,*
1 0'
010.
, !;
Furthermore, Equations (3), 67, show that if a given matrix
represents a eollineation in point coordinates, the conjugate of the
adjoint matrix represents the same eollineation 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 PEOJECTIYITIES [CHAP.X
From what has ]ust been said it is clear that a matrix does not
completely define a collineation, 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 x[ = 2a^ in the symbolic form
The matrix (a y ) may then be regarded as an operator transforming
the coordinates x = (x v x s , x s ) into the coordinates xf = (x[, x[ t x^) If
we place d tj = a jl} the matrix conjugate to (a if ) is (a,,) Also by plac-
ing I lj = A Jl > the adjoint matrix of (a tj ) is (A t] ). The inverse of the
above collineation is then written
Furthermore, the collineation x 1 (a tj )x is represented in line coordi-
nates by the equation
% ' = (A*) W -
This more complete notation will not be found necessary in gen-
eral in the analytic treatment of collmeations, 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
'=K)0,
where the matrix (a v ) is to be regarded as an operator transforming
the point x into the line u r . This correlation is then expressed as a
transformation of lines into points by
The product of two correlations u' (a^ x and '= (Z^) x is there-
fore represented by
' = WK)
(cf. Equations (4), 94), or by
' = &,) (4,)*
Also, the inverse of the correlation u' = (a^os is given by
orby "&>*
95,96]
TYPES OF COLLINEATIONS
271
EXERCISE
Show that if [II] is the set of all colhneations in a plane and I\ is any
con elation, the set of all correlations in the plane is [Iirj, so that the two
sets of transformations [II] and [IH\] comprise the general protective gioup
in the plane By vntue of this fact the subgroup of all protective colhneations
is said to be of index 2 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^ X 9 )
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 (io lt x z , x s ) 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.
This 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 /? x 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 (x v x z) # 8 ) of a double point
(3)
"a&
$i <&ii~""i
*21
* A subgroup [II] of a group is said to be of index n, if there exist n 1 trans-
formations Ti (i = 1, 2 r . . . n - 1), such that the n - 1 sets [nr f ] of transformations
together with the set [II] contain, all the transf prmaMons of the group, while no two
transformations within the same set or from a$y two sets aie identical.
272 TWO-DIMENSIONAL PEOJECTIVITIES [CHAP X
winch 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 collmeation (or
of the representative matrix), gives rise to a unique double point.
Moreover, every double point is obtainable in tins way. This is the
analytic form of the fact already noted, that a collineation which is
not a homology or an elation cannot have more than three doulle
points, unless it is the identical collineation
If, however, all the determinants on the right m 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 collmeation 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 collmeation 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 3 ) 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 figure of double points
and lines is self-dual.
Eour points of the plane, no three of which are collinear, cannot
be invariant unless the collineation reduces to the identity. If three
noncollmear points are invariant, two cases present themselves. If
the collineation reduces to the identity on no side of the invariant
triangle, the collmeation is of Type I (cf. 40, Chap. IV). If the
collmeation is the identity on one and only one side of the invariant
triangle, the colhneation is of Type JFif If two distinct points are
* Cf BOcher, Introduction to Higher Algebra, Chaps XX and XXI.
t If it is the identity on more than one side, it is the identical collineation.
00] TYPES OF COLLnSTEATIONS 273
invariant, but no point not on the line I joining these two is invariant,
two possibilities again arise If the collmeation 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 //. 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], 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 (3) has one double root. Type IV i& denoted by [(1, 1), 1].
In Type II, 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 pro/active collineatwn in a, plane is of one of
the following Jim types :
[1,1,1] [(1,1),!]
[2. 1] [& 1)]
In this table the first column corresponds to three distinct roots
of the characteristic equation, the second colximn. 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 PEOJECTIVITIES [CHAP.X
not collmear ; the 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 (of. Ex. 5, p 276). In a
collineation of Type // 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 III 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 collmeations of these five types.
Type I. Let the invariant triangle be the triangle of reference.
The collineation is then given by equations of the form.
/>.<= a ss x
in which a n , a zz , a ss 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 x 1 = 0, # 2 = 0, x a = is pointwise
* For a more detailed discussion of collineataons, reference may be made to
Newson, A New Theory of Collmeations, etc., American Journal of Mathematics,
Vol. XXIV, p 109.
90] TYPES OF GOLLINEATIONS 275
invariant, we must have either a aa - a ss or ff !13 = a n or a n = a Z2 Thus
the homology may he written
A harmonic homology or reflection is obtained by setting a aa = ~l.
jfyjpe JJ. The characteristic equation has one double root, p l = p
say, and a simple root p 3 Let the double point corresponding to
Pi~P* be ^=(0, 0, 3), let the double point corresponding to p s be
Z7 8 = (l, 0, 0), and let the third vertex of the triangle of reference
be any point on the double line w 8 corresponding to p a , which line
will pass through the point Z7 r The collineatiou is then of the form
z)
since the lines 0^= and x%= are double lines and (1, 0, 0) is a
double point. The characteristic equation of the collineation is clearly
K-P)K~/ ) )K3~^=
and since this must have a double root, it follows that two of the
numbers u , a2 , a m must be equal. To determine which, place
P = a M ) using the minors of the second row, we find, as coordinates
of the corresponding double point,
(0, (a u - a 2a ) (a ea - ), a ss (a u - a 22 ) ),
which is U v and heuce we have a 2a == a as . The collineation then is
of Type II, if u = a 22 . Its equations are therefore
where a 3a * and a u ^ a a2 .
Type III. The characteristic equation has a triple root, p^p^ p & ,
say. Let U^ (0, 0, 1) be the single double point, and the line x^ be
the single double line. With this choice of cobrdinates the collineation
has the form ,
276 TWO-DIMENSIONAL PROJECTIVITIES [CHAP X
By writing the characteristic equation we find, in view of the fact
that the equation has a triple root, that a n = 22 = a ay The form of
the collineation is therefore
where the numbers a 21 , a S2 must be different from
Type F, Elatwn 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 82 must be zero in order that the line x l be
pointwise invariant. The equations for Type II also yield an elation
in case a u = 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 B, C, D, A respec-
tively Show that the chaiacteiistic equation of this collineation is (p 1)
(p 2 + 1) = 0, which in any field has one loot. Deteimine the double point
and double line corresponding to this root Assuming the field of numbers to
be the ordiuaiy complex field, determine the cooidmates of the remaining two
double points and double lines. Veiify, by actually multiplying the matiices,
that this collineation is of period 4 (a fact which is evident from the defini-
tion of the collineation).
2 With the same cooidmates for A, B, C, D determine the collineation
which transforms these points respectively into the points ft, A , D, C. The
le&ultmg collineation must, from, this definition, be a homology. Why? De-
termine its center and its a,xis By actual multiplication of the matrices
veufy that its squaie is the identical collineation.
3. Express each of the collineations in Exs. i and 2 in terms of line
coordinates
4. Show that the characteristic cross ratios of the one-dimensional projcc-
tivities on the sides of the invariant triangle of the collineation a % ' = ar 1 ,
x l ~ ^a> x l = CX 3 are *^ e ia faos of the numbers a, b, c. Hence show that the
product of these cross ratios is equal to unity, the double points being taken
around the tuangle in a given oider.
5 Prove the latter part of Ex. 4 for the cross ratios of the projectivities
on the sides of the invanant triangle of any collineation of Type /.
an] TYPES OF COLLINEATIOKS 277
6 Write the equations of a collineation of period 3 ; 4 , 5 , .; n ;
7 By properly choosing the system of nonhomogenoous cooidmatcs any
collmcatHm of Type / iiuy be iopi osented by equations ?' = rw , / = by The
.sot ol all colhneatums obtained by giving the parameter* n, b all possible
values J'oiuiH <i gioup. Show that the eollmeatiom *' = <u, y' - a'i/, -where r
is constant foi all eollinejitions of the set, form a subgroup Show that eveiy
collineation of this subgroup leaves invanant every ciuve whose equation is
?/ = cjf, where c is any constant. Such curves aie oalled path cwves of the
colhneations.
8. If P is any point of a given path curve, p the tangent at P, and
A,B, C the vortices of the invariant tiiaugle, then Ijfc (;;, 7M, PB, PC) is a
constant.
9. For the values ? =- 1, 2, % the path ciuves 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 expiessed in the form
with the restriction a -M 2 = 1
12. Prove that any collineation can be expressed as a product of colhnea-
( tions of Type 7.
13. Let the invariant figure of a collineation of Type 77 be A, B, 1, m,
where l AB, B lm. Tho product of such a collmeation by another of
j Type 77 with invariant figure A', B, I, m' is in general of Type 77, but may
1 be of Types 777, TV, 01 V. Under what conditions do tho latter cases anse ?
14. Using the notation of Ex. 13, the product of a collineation of Type II
with invariant figure A , B, I, m by one with invariant figure A , B' t I, m' is
in general of Type 77, but may be of Types 777 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 77.
16. Two collineations of Type 777 with the same invariant figure are not
in general commutative.
17. Any protective collineation can be expressed as a pioduct of collinea-
tions of Type 777.
18. If II is an elation whose center is C, and P any point not on tho
axis, then P and C are harmonically conjugate with respect to n- 1 (P)
and II (7>),
19. If two coplanar conies are projoetive, tho 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.
t
3
278
TWO-DIMENSIONAL PROJEOTIVITIES [CHAP X
21 In nonhomogeneous coordinates a collineation of Type I with fixed
points (a v a a ), (b v 6 2 ) (c v c 2 ) may be wntten
x y 1
a; ?/ 1
,_
X ~
x y I '
! a 1 1
a: # 1
Oj a 2 1 1
l> \ k
Type II may be -written
X'=
s
s x s 2
and Type III may be written
or y 1
t
x y
a a
1
!
z
1
t
1
1 1
i
x y
a t a 2
S s
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 T is the cor-
relation, such that T(X) = u and r(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
97] DOUBLE PAIRS OF A CORRELATION 279
of a certain collineation. In fact, a double pair X, u is such that
T(X)=u and F a (JtT) = F (w) = X, so that the pomt of a double pair
of a correlation F is a double point of the colhneation F a . Similarly,
it umy be seen that the lines of the double pairs are the double lines
of the collmeatiou F 3 It follows also from these considerations that
F is a polarity, if F a is the identical collineation.
Analytically, the problem of determining the double pairs of a
correlation leads to the question For what values of (x v % s , x 3 ) are
the coordinates
of the line to which it corresponds proportional to the coordinates
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
(1) K~^^^+K~p 2a )a3 2 +(ff 23 --/)a 82 )a; 3 == 0,
which must be satisfied by the coordinates (x lt x v Kg) 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 collineation ( 96). The factor of proportionality
p is determined by the equation
a u
(2) *ai
which is of the third degree and has (under Proposition K a ) three
roots, of which one is 3 , 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 % a; 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 lt ss z) a? 8 ), This would
imply that all the relations
S-/ K ** aa <&/=!, 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 ) be different from 0. If this be a n , the
relation a u pa u =Q gives at once p = l; and this value leads at
once to the further relations
% = S> (v = l, 2, 3).
The matrix in question must then be symmetrical If, on the other
hand, we have a u a zz = a 6a 0, there must be some coefficient v
different from 0. Suppose, for example, a ia =f= Then the relation
a 12 &a 21 = shows that neither Ic nor a sl can be The substitution
of one in the other of the relations a lz = ka zl and 21 = ka lz then gives
& 2 = 1, or Jc = 1. The value Jo = 1 again leads to the condition that
the matrix of the coefficients be symmetrical The value Jc = -1
gives a h =0, and #= a j{ , 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
& = -1 is therefore impossible. We have thus been led to the fol-
lowing theorem:
THEOEEM 7. The necessary and sufficient condition that a reci-
procity in a plane be a polarity is that the matrix of its coefficients
"be symmetrical.
If the coordinate system is chosen so that the point which corre-
sponds to p = 1 in Equation (2) is (1, 0, 0), it is clear that we must
have a zl = a lz and a 31 = a 18 . 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
pu[= a n x^
(3) K=
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 a zs = 8a , which would lead to a polarity, or a 88 = 0.
07] DOUBLE PAIRS OF A CORRELATION 281
In the latter case it is evident that (4) has a double root if a w = a t)
Imt that otherwise it has two distinct roots. Therefore a con elation
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 :
(Q = C = 1, rt = 0)
IV pul~ -x a , (a =0)
K=
The squares of these correlations are collmeations 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
K= n*
puf - %i + Ogfa + a 38 a; 8 , (a M ^ 0, 28 & a 82 )
pMg'sa 8a a +a 88 a! 8 ,
Equation (2) at the same time reduces to
Ml-?) 8 =0,
and the square of the correlation is always of Type ///. There are
thus five types of correlations, the polarity and those whose squares
are collmeations of Types J, //, J/J, IV.
EXERCISES *
1 The points which lie upon the lines to which they correspond in a cor-
relation form a conic section C 8 , ami the hues which lie upon the points to
which they correspond arc, the tangents to a conic A' 2 How are C 9 and K z
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, Arclnv der Mathematik, 1st series,
Vol VIII (1846), p. 82 ; and SchrSter, Journal ftir die reme und angewandte Mathe-
matik, Vol. ZJOCVII (1874), p. 105.
282 TWO-DIMENSIONAL PROJEOTIVITIES
2. If a line a does not lie upon the point A' to which it conesponds in a
coir elation, there is a projectivity between the points of a and the points in
which their corresponding lines meet a. In the case of a polarity this pio-
jectivity is always an involution. In any other correlation the lines upon
which this projectivity is involutoric all pass through a unique fixed point
The line o having the dual property coi responds doubly to The double
points of the involutions on the lines through aie on the conic C 2 , and the
double lines of the involutions on the points of K* are tangent to K* and o
are polar with respect to C 2 and /C a . If a correlation determines involutions
on three nonconcurrent lines, it is a polarity.
3. The lines of K z through a point P of C 2 are the line which is ti ansf ormcd
into P and the line into which P is transformed by the given coi relation.
4. In a polarity C 2 and K z aie the same conic
5. A necessary and sufficient condition that a colhneation be the pioduct of
two reflections is the existence of a correlation which is left invariant by the
colhneation *
98. Fundamental conic of a polarity in a plane. We have jusl
seen that a polarity m a plane is given by the equations
(1)
DEFINITION. Two homologous elements of a polarity in a plane ar
called pole and polar, the point being the pole of the line and th
line being the polar of the point. If two points are so situated the
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 be conji
gate is readily derived. In fact, if two points P = (x v x s , x s ) an
P f =z(xl, x'v C 8 ') are conjugate, the condition sought is simply tin
the point P' shall be on the line p'= [/, w 2 ', M B '], the polar of JP; i
u[x[ -f- ufaz + w 8 X = 0. Substituting for u[, u, u their values
terms of x v x z , 8 from (1), we obtain the desired condition, viz. :
(2)
4-
As was to be expected, this condition is symmetrical in the cobr
nates of the two points P and P'. By placing x[ = ic t we obtain 1
* This is a special case of a theorem of Dunham Jackson, Transactions of
American Mathematical Society, Vol. X (1909), p 479.
POLAB, SYSTEM. 283
condition that the point P be self-conjugate, i.e that it be on its polar.
We thus obtain the result .
THEOREM 8 The sdf -conjugate points of the polarity (1) are on
the conic whose equation is
(3) a n x? + a n ^x* + a ss x$ + 2 a^x z + 2 a^te, + 2 28 x^ = ;
and, conversely, every point of this come is self-conjugate.
This conic is called the fundamental conic of the polarity. All of
its points may be improper, but it can never degenerate, for, if so,
the determinant \a^\ would have to vanish (cf. Ex., p. 187). By
duality we obtain
THEOREM 8'. The self-conjugate lines of the polarity (1) are lines
of the conic
(4) A u u?+ A n uf+A n u*+ 2 ^ M 1 w a + 2 A n u^ + 2 A n u t u t - ,
and, conversely, every line of this conic is self-conjugate.
Every point JT of the conic (3) corresponds in the polarity (1) to
the tangent to (3) at X. For if not, a point A of (3) would be polar
to a line a through A and meeting (3) also in a point J5. B would
then be polar to a line 5 through B, and hence the line a = AB
would, by the definition of a polarity, be polar to ab = B. This woxild
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. Every polarity is the polar system of a conic, the
fundamental conic of the polarity. The self-conjugate points are
the points and the self-conjugate lines are the tangents of this conic.
Every pole and polar pair are pole and polar with respect to the
fundamental conic.
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 PROJECTIVITIES [CHAP 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 aj 1 2 + ^22^+ Vs 2 + 2 <WB + 2 VA+ 2 Va^ 0.
The result of the preceding section shows that the equation of the
same conic in line coordinates is
(2) 4X+ -4.X + 4X + 2 A u i tt + 2 4Xtt,+ 2 ^ 28 w 2 w s = 0,
where -4 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
u^+ u z oe 2 -\- u 3 x a = be tangent to the conic (1) is that Equatwn (2)
be satisfied.
COROLLARY. This condition may also be written in the form
a n a i2 a is u i
u : u a u s
Equation (2) of the preceding section expresses the condition that
the points (x v x z , # 8 ) and (x[, x}, x) be conjugate with respect to the
conic (1). If in this equation (#/, aj 2 , #') be supposed given, while
(x v a; 2 , C 3 ) is regarded as variable, this condition is satisfied by all the
points of the polar of (x[, x, a; 3 ') 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 l} it becomes
while if we arrange it according to the coordinates #/, it becomes
(4)
+ (! + a 28 a; 2 + 83 a5 8 ) = 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
i uo,ioo] VAEIOUS 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{ tog dx^
(ic/, a3 2 , ,Tg) for (IL\, ,r a , .c 3 ) m the expressions ~- >>- (f being any
9.^ t 3it' 2 oJ3 8
polynomial in a^, je 2 , .'e a ), it is readily seen that Equation (3) is
equivalent to
x <L + X $L + X K^Q
where now/ is the left-hand member of (1).
This loads to the following theorem :
TIIEOHKM ]]. IffQis the equation of a conic in homogeneous
point coordinates, the equation of the polar of awy point (./, x^ ac) is
given "by either of the equations
If the point (a;/, .r a ', a; a ') is a point on the come, 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)
conio as the locus of the intersections of homologous lines of two
projective flat pencils in the same plane was first given by Steiner in
1832 and used about the same time by Chasles. 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 (of. 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-conjugate lines of a polarity. This defini-
tion was first given by von 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 go v # 8 , a? 8 - This definition (at
least in its nonhomogeneous form) dates back to Descartes and format
(1637) and the introduction of the notions of analytic geometry.
286 TWO-DIMENSIONAL PEOJECTIVITIES [CHAP x
The oldest definition of conies is due to the ancient Greek geometeis, who
denned a conic as the plane section of a circular cone. This definition involves
metnc ideas and hence does not concern us at this point. We will leturn to it
later It is of interest to note in passing, howevei, that fiom this definition
Apollomus (about 200 B c ) derived a theorem equivalent to the one that the
equation of a conic in point cooidinates is of the second degiee.
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 Stemer's. It may
be noted that von Staudt's definition has the advantage over Stemer'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 Geometrie der Lage,
Nurnberg (1847) , or to the textbook of Ennques, Vorlesungen iiber
projective Geometrie, Leipzig (1903),
EXERCISES
1 Derive the condition of Theorem 10 dn ectly by imposing the condition
that the quadratic which determines the intei sections of the given line with
the conic shall have equal loots What is the dual of this theorem ?
2. Verify analytically the fundamental pioperties of poles and polais with
respect to a conic (Theorems 13-18, Chap. V)
3 State the dual of Theorem 11.
4. Show how to constiuct the correlation between a plane of points and a
plane of lines, having given the homologous pairs A , a'; B, I' ; C, c' ; D, d'
5 Show that a correlation between two pianos 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 coriesponds to the line
joining the two centers of the pencils of lines.
6. Show that in our system of homogeneous point and line cobrdmates the
pairs of points and lines with the same cobrdmates are poles and polars with
respect to the conic a:, 2 + a;., 2 + a; 2 = 0.
* a o
7 On a general line of a plane in which a polanty has been defined the
pairs of conjugate points form an involution the doxible 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 hue on a vertex of the triangle,
loo, 101] PAIRS OF CONICS 287
9. Prove Thcoicm 3 analytically
10. Given a simples plane pentagon, theie exists a polaiity in -which to each
vertex conesponds the opposite side
11 The three points A', B', C' on the Bides BC, CA, AB of a triangle that
are conjugate in a polanty to the veitices A, B, C respectively are collineai
(cf. Ex 13, p. 125).
12. Show that a polaiity is completely determined when the two involutions
of conjugate points on two conjugate lines are given.
13 Constiuct the polarity determined Ity a self-polar tuangle ABC and an
involution of conjugate points on a line.
14. Construct the polarity determined by two pole and polai pans A, a and
B, b and one pair of conjugate points C, C".
15 If a triangle STU is self-polai with regard to a conic C 2 , and A is any
point of C 2 , there are three triangles having A as a vertex which ate inscribed
to C 3 and circumscribed to STU (Sturm, 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 collineation 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 colhnealion 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 L If the common self-polar figure of the two conies is of
Type I, it is a self-polar triangle for both conies. Since any two conies
are protectively equivalent (Theorem 9, Chap. YIII), the coordinate
system may be so chosen that the equation of one of the conies, A*, is
(1) s-*+a, 9s!S 0.
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) &&1 ckfll + a 8 aj a 3 ss 0.
An equation of the form (2) may therefore be taken as the equation
of the other conic, J5 a , if (1) and (2) have no other common self-polar
elements than the fundamental triangle, Consider the set of conies
(3)
288
TWO-DIMENSIONAL PROJECTIVITIES [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 A z and B*.
For the value X = a s , (3) gives the pair of lines
(4) (a 1 - a s ) xl - (a 2 - a s ) x* = 0,
which intersect in (0, 0, 1). The points of intersection of these lines
with (1) are common to all the comes (3)
The lines (4) are distinct, unless ft 1 = a 3 or a z =a s But if a 1 = a 3 ,
any point (a;/, 0, a; 3 ') on the line x z has the polar x[x v + # 8 'a? 3 =
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 I, the
three numbers a v a 2> a s must all be distinct If this condition is
satisfied, the lines (4) meet the conies (3) m four distinct points.
#010)
(100)\
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 other common self-polar pair of point and line], tli&y intersect in
four distinct points (proper or improper] Any two conies of the
pencil determined ly these points have the same self-polar triangle.
Dually, two such conies have four common tangents, and any two
101]
PAIRS OF CONICS
289
conies of the range determined by these common tangents have the same
self -polar triangle
COKOLLAKY Any pencil of comes of Type I can be represented by *
the four common points being in this ease (1, 1, 1), (1, 1, 1), (1, 1, 1),
and ( 1, 1, 1).
Type II. When the
common self-polar figure
is of Type //, one of the ^t^/^^ _ (oi-
points lies on its polar, \ /'"^^ ~^~>^ *s:- a=: [ooi]
and therefore this polar is
a tangent to each of the
comes A 3 , 2> a . 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 B
of the self-polar figure
which IB on only one of the lines is the pole of the line 5 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 [Q, 0, 1],
B (0, 1, 0), and & = [0, 1, 0]. The equation of any conic being
A=(100)
101
the condition that A be on the conic is a i = ; that a then be tan-
gent is & 8 ; that I then be the polar of B is \ 0. Hence the
general equation of a conic with the given self-polar figure is
2 = 0.
(6)
+
* 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 thud
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. Clebsch-Lindemann, Vorlesungen
uber Geometne, 2d ed., Vol. I, Part I (1906), pp, 212 ft
290 TWO-DIMENSIONAL PROJECTIVITIES [CHAP x
Since any two conies are projectively equivalent, A 2 may be chosen
to be
(7) af+xi+Zx&^O.
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 V, we must determine under what conditions each point on a
or each point on & has the same polar with regard to (6) and (7).
The polar of (#/, # a ', a? 8 ') with regard to (6) is given by
= 0-
Hence the first case can arise only if a a = & 2 ; and the second only
if a a = & 2
Introducing the condition that a a , a a , 5 2 are all distinct, it is then
clear that the set of comes
OJB* + a a x* + 2 \x : x s + X (a + xf + 2 o^) =
contains a line pair for X = a z> viz the lines
(a, - a,) xl + 2 (&, - a 2 ) a;^ =
Hence the comes have in common the points of intersection with (7)
of the line
(a, o a ) 8 + 2 (& 2 a a ) a?! =
This gives
THEOREM 13 If two comes have a common self -polar figure of
Type II, they have three points in common and a common tangent at
one of them. Dually, they 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* x\ + Xo^ = 0. The conies of this pencil all pass
through the points (0, 1, 1), (0, 1, 1), (1, 0, 0) and are tangent to
a- 8 =0.
Type III. When the common self-polar figure is of Type III, 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 *** 0,
its point of contact be (1, 0, 0), and let A 2 be given by
( 8 ) as? +20^=0.
ioi] PAIRS OF CONIGS 291
The general equation of a conic tangent to # 3 = at (1, 0, 0) is
(9) (ijrl + #r% + 2 j^a^R, + 2 Igfcft = 0,
with regard to which the polar of any point (./, a 1 ,, 0) on x s = is
given by
( 1 0) X a; 2 + I 1 a^x 9 + fc^X = 0.
This will "be identical with the polar of (x[, aJ 2 ', 0) with regard to A z
for all values of x[, a?j, if & 2 = 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 Z> 3 = tt a , & a = 0. The line (10) will now be
identical with the polar of (8) for any point (x[ t #/, 0) satisfying the
condition
This quadratic equation must have only one root if the self-polar figure
is to be of Type IIL This requires & 2 = 2 , and as 5 a , a a cannot both
be unless (9) degenerates, the equation of J:? 8 can be taken as
(11) x* + 2 x& + a,r + 2 \x^ = 0, (& t * 0).
The conies (8) and (11) now evidently have in common the points of
intersection of (8) with the
line pair
and no other points. Since
# 8 = is a tangent, this gives
two common points. If the
second common point is taken A~(l 00) a=[o ll
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 t
and no other points, may be written (fig. 102)
THEOBEM 14. If two conies have a common self-polar figure of
Type III, they have two points in common and a common tangent
at one of them, and one other common tangent They determine a
pencil and a f range of conies of type IIL
292 TWO-DIMENSIONAL PROJECTIVITIES [CHAP. X
COROLLARY A pencil of conies of Type III can be represented "by
the equation x\ + 2 x^ + \x 2 x s 0.
Type IV. When the common self-polar figure is of Type IV, let the
line of fixed points be x a = and its pole be (0, 0, 1) The coordinates
being chosen as they were for Type /, the conic A* has the equation
a - n
and any other conic having in common with A 2 the self-polar tri
angle (I, 0, 0), (0, 1, 0), (0, 0, 1) has an equation of the form
The condition that every point on x s shall have the same polar
with regard to this conic as with regard to A* is a t = a z . Hence B
may be written -
Any conic of this form has the same tangents as A* 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.
/OfljT""--^ V / / J THEOREM 15. If two conies
^""^^ \ i / / j iave a common self-polar
figure of Type IV, they have
two 'points in common and
Fl - 103 n f * ^ *
the tangents at these points.
They determine a pencil (which is also a range) of conies of Type IV,
COROLLARY. A pencil of conies of Type IV may be represented ~by
the equation
, ,
xl 0.7 -f Xa3 8 2 = ;
and also by the equation
x? + \x z x s = 0.
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 a? 8 == 0.
As in Type ///, let -4 2 be given by
(8) ^+2^=0.
We have seen, in the discussion of that type, that all points of # 8 =
have the same polars with respect to (8) and (9), if in (9) we have
101] PAIRS OF CONICS 293
\ = a z and l) t = 0. Hence, IE A 3 and li 2 are to have a common self-
polar figure of Type V, tlio equation of If must have the form
(1 il) a a (atf + 2 r r a ) + a^* =
From the form of equations (8) and
(13) it is evident that the comes have
m common only the point (1, 0, 0) and
the tangent a, = 0, and that every point
on a? a = has the same polar with re- a=[00l] A = (<ioo)
spect to both conies (fig. 104). Hence,
they determine a pencil of Typo V.
THEOREM 16. If two conies have a common self-polar figure of
Type V, they have a lineal element (and no oilier elements) in com-
mon and determine a pencil (which is also a range) of conies of
Type V according to the classification already given.
COROLLARY. A pencil of conies of Type V can be represented, "by the
equation
As an immediate consequence of the corollaries of Theorems 12-16
we have
THEOREM 17. Any pencil of conies may be written in the form
where f= and ff=*Q are the equations of two conies (degenerate or
not) of the pencil.
EXERCISES
1. Prove analytically that tho polars of a point P with respect to the
conies of a pencil all pass through a point Q. Tho points P and Q a-ie double
points of the involution determined by the conies of tho pencil on the line PQ
Give a linear construction for Q (of. Ex. 3, p. 130). The coirespondence
obtained by letting every point P correspond to the associated point Q is a
"quadiatic birational transformation." Determine the equations representing
this transformation. The point <2, which is conjugate to P with regard to all
the conies of the pencil, is called the conjut/ale of P with respect to the pencil.
The locus of the conjugates of the points of a line with legard to a pencil of
conies is a conic (cf. Ex, 81, p. 140).
2. One and only one conic passes through four given points and has two
given pomes 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 geneial, 01 a pencil of conies in a special case, passes
thiough two given points and has thiee pans of given points as conjugate
points ; or passes thiough a given point and has four pairs of given points as
conjugate points ; or has five given pans of conjugate points. Give the cor-
responding constructions for each case
102. Problems of the third and fourth degrees.* The problem of
constructing tlie points of intersection of two conies 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 comes. Further, the prob-
lem of finding the remaining intersections of two conies in a plane
of which one point of intersection is given, 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 we have made use of Amodeo, Lezioni di Geometria Projettiva,
pp, 436, 437. Some of the exercises are taken from the same book, pp. 448-461.
t Moreover, we have seen (p. 289, footnote) that any problem of the fourth
degree may be reduced to one of the third degree, followed by two of the second
degree.
t With the usual representation of the ordinary real geometry we should require
an instrument to draw conies
102] THIRD AND FOURTH DEGREE 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
be solved by linear and quadratic constructions if the intersections
of every conic in the- plane with a iixod 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, noniperspcctive collineation in a, plane
which is determined "by 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 let A be any point of the plane which is not on an invariant
line. Let U(A)**A' and tt(A')**A". The points A t A', A" are then
not collinear. The pencil of lines at
A is protective with the pencil at
A', and these two protective pencils
generate a conic C a which passes
through all the double points of II,
and which is tangent at A' to the
line A' A" (fig. 105). The conic C* is
transformed by the collineation II
into a conic Cf generated by the pro-
jective pencils of lines at A' and A". Hia '
Cl also passes through A' and is tangent at this point to the line
AA', The double points of n are also points of <7 X 2 . The point A 1
is not a double point of II by hypothesis. It is evident, however,
that every other point common to the two conies 2 and 0* is a
double point.
If <7 2 and Cl intersect again in three distinct points L, M, N, the
latter form a triangle and the collineation is of Type /. If a and C?
intersect in a point N, distinct from A', and are tangent to each other
at a thfrd point Z^M, the oollineation has M, N for double points
296 TWO-DIMENSIONAL PKOJECTIVITIES [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 conies have
contact of the second order at a point L = M=N, distinct from A f ,
the collineation has the single double line which is tangent to the
comes at this point, and is of Type /// (fig. 107).
FIG 107
EXERCISES
1. Give a disciission of the problem above -without making at the outset
the hypothesis that the collineation is nonpeispective.
2 Construct the double pans of a correlation in the plane, which is not
a polanty.
3. Given two polarities in a plane, construct their common pole and
polar pahs
4. On a line tangent to a conic at a point A is given an involution I, and
fiom any pair of conjugates P, P' of I aie drawn the second tangents p, p' to
the conic, their points of contact being Q, Q' 1 espectively. Show that the locus
of the point pp' is a line, Z, passing through the conjugate, A', of A in the invo-
lution I, and that the line QQ' passes through the pole of I with lespect to
the conic.
5. Construct the conic which is tangent at two points to a given conic and
which passes through three given points Duali/e
6. The lines joining pairs of homologous points of a nomnvolutoric pro-
jectivity on a come A a aie tangent to a second conic # 3 which is tangent to
A 3 at two points, or which hyperosculates A*.
7. A pencil of conies of Type II is determined by three points A, JFS, C
and a line c through C. What is the locus of the points of contact of the
conies of the pencil with the tangents drawn from a given point P of c?
8. Construct the conies which pass through a given point P and which, are
tangent at two points to each of two given conic?.
9. If /= 0, r=Q, 7i = are the equations of thice conies in a plane not
belonging to the same pencil, the system of conies given by the equation
X/+ w + vli = 0,
102] THIRD AND FOURTH DEGEEE PROBLEMS 297
X, fj-, v being vaiiable parameters, is culled a luiitlle of cornea Thiough every
point of the plane passes a pencil of conies belonging i,o this bundle ; tlnough
any two disl.mei. points passes in geneml one and only one conic of the bundle
If the comes/, y, k have a point in common, this point is common to all the
conies of the bundle. Give a noualgebiaic definition of a bundle of comes.
10 The set of all comes in a plane passing through the vertices of a triangle
form a bundle. If the equations of the sides of tlas tiiaugle are I 0, m = 0,
n = 0, show that the bundle may be represented by the equation
A.HM + pid + vim = 0.
What are the degenerate comes o this bundle?*
11. The set of all conies in a plane which have a given tuangle as a self-
polai triangle forms a bundle. If the equations of the sides of this tuangle aie
I 0, m = 0, n = 0, show that the bundle may be represented by the equation
What are the degenerate conies of this bundle ?
12. The comes of the bundle described in Ex. 11 which pass through a
general point P of the plane pass tlnough the other three veitices 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 centeis and axes aie the vertices and opposite
sides of a triangle foim a commutative gioup. 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 tiiangle 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-
quadnlatrral configuration ?
14. The necessary and sufficient condition that two reflections be com-
mutative is that the centei of each shall be on the axis of the other.
15. The invariant figure of a colliueatioii 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
element Bb and also transforms Bb into Aa is a polarity
17 How many collmeatious and correlations are in the group geneiated
by the reflections whose centers and axes are the vertices and opposite sides
of a triangle and a polarity with regard to which the triangle is self-polai ?
* In connection with this and the two following exercises, of. Castelmiovo,
Lezioni di Geometria 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, is logically
equivalent (in the presence of Assump-
tions A and E) to Assumption P. It
might have been used to replace that
assumption
THEOREM 1. If l v l z) l & are three
mutually skew lines, and if m^ m a , m a ,
9?i 4 are four lines each of which meets
each of the lines l v l v l at then any line l
which meets three of the lines in v m z ,
w 8 , wi 4 also meets the fourth.
Proof The four planes ljn v l^n^
ljn z , Z 1 ??i 4 of the pencil with axis l t are
perspective through the pencil of points
on l s with the four planes l a m v Z 2 m 2 ,
l z m s , Z 2 m 4 of the pencil with axis l z
(fig. 108). For, by hypothesis, the lines
FIG 108
of intersection m v m a , m a , m^ of the
pairs of homologous planes all meet l a .
The set of four points in which the four planes of the pencil on l v
meet Z 4 is therefore protective with the set of four points in which
the four planes of the pencil on l z meet l v But 4 meets three of the
pairs of homologous planes in points of their lines of intersection,
since, by hypothesis, it meets three of the luxes m v m z , m^, w 4 . Hence
in the projectivity on Z 4 there are three invariant points, and hence
(Assumption P) every point is invariant Hence Z 4 meets the remain-
ing line of the set m v m z , m s , m 4
* All the developments of this chapter are on the basis of Assumptions A, B, P, HO.
But see the exercise on page 261
208
io.Jj THE KEGULUS \ 29Sb
DEFINITION. If l v l z> 1 A are throe lines no two of which are- in the
same plane, the set of all lines which meet each of the three given lines"
is called a regidus The line-s l v l z , l a are called directrices of this regulus.
Ic is clear that no two lines of a regulus can intersect, for other-
wise two of the directrices would lie in a plane. The next theorem
follows at once from the definition.
THEOREM 2. If l v a , l a arc three lines of a regulus of which
m v m a , m a are directrices, m v m 2 , m s are lines of the regulus of which
l v 1.,, l a are 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 ana 1 only one conjugate regulus.
This follows immediately from the preceding. Also from the proof
of Theorem 1 we have
THEOREM 4 TJie correspondence THEOREM 4'. The correspond^
established ly the lines of & regu- ence established "by the lines of a
lus "between the points of two lines regulus "between the planes on any
of its conjugate regulus is projec- two lines of its conjugate regulus
tive. is projective.
THEOREM 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 protective pencils of points planes of two protective pencils of
on skew lines is a regulus, planes on sJcew 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 hue which meets these three lines joins a pair of
homologous points of the two pencils of points.
300 FAMILIES OF LINES [CHAP XI
THEOREM 6. If [p] are the lines of a regulus and q is a directrix
of the regulus, the pencil of points q [p] is protective with the pencil
of planes q[p~\-
Proof Let q' be any other directrix By Theorem 4 the pencil of
points q[p] is perspective with the pencil of points q'[p]. But each
of the points of this pencil lies on the corresponding plane gp.
Hence the pencil of points q'[p] is also perspective with the pencil
of planes q[p~\.
EXERCISES
1. Every point -which is on a line of a regulus is also on a line of its
conjugate regulus
2. A plane which, contains one line of a legulus contains also a lino of its
conjugate regulus.
3 Show that a regulus is uniquely denned by two of its lines and thiee
of its points,* piovided no two of the latter aie coplanar with either of the
given lines
4 If four lines of a legulus cut any line of the conjugate regulus in points
of a harmonic set, they aie cut by every such line in points of a harmonic
set. Hence give a construction for the harmonic conjugate of a line of a
regulus with respect to two other lines of the legulus
5. Two distinct reguli can have in common at most two distinct lines.
6 Show how to construct a regulus having in common with a given
regulus one and but one luler
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 hues in points that are collmear. Since
the regulus may be thought of as the lines of intersection of pairs of
homologous planes of two protective axial pencils (Theorem 5'), 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 regulus
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 plane oi
a regulus and its conjugate For such a picture is clearly a plane section of
the pioj'ection of the object depicted from the eye of an observer. Fig. 108
illustrates this fact. ,
* By a point of a regulus is meant any point on a line of the regulus.
104] THE EEGULUS 30
The section of a regulus by a plane containing a line of the regi
lus is a degenerate conic of two lines. The plane section can neve
degenerate into two coincident lines because the hues of a reguli
and its conjugate are distinct from each other. In like manner, tt
projection from a point on a line of the regulus is a degenerate cor
of planes consisting of two pencils of planes whose axes are a rult
and a directrix of the regulus.
DEFINITION. The class of all points on the lines of a regulus
called a surface of the second order or a quadric surface. The plane
on the lines of the regulus are called the tangent planes of the su
face or of the regulus. The point of intersection of the two lines (
the regulus and its conjugate in a tangent plane is called the poit
of contact of the plane. The lines through the point of contact in
tangent plane are called tangent lines, and the point of contact of tt
plane is also the point of contact of any tangent line.
The tangent lines at a point of a quadric surface include the line
of the two conjugate roguli through this point and all other line
through this point which meet the surface in no other point. An
other line, of course, meets the surface in two or no points, since
plane through the line meets the surface in a conic. The tanger
lines are, by duality, also the lines through each of which passes onl
one tangent plane to the surface.
THEOREM 7 The tangent planes at the points of a, plane section o t
a quadric surface pass through a point and constitute a cone of plane
Dually, the points of contact of the cone of tangent planes through
point are coplanar and form a point conic.
Proof It will suffice to prove the latter of these two dual theorem
Let the vertex P of the cone of tangent planes be not a point of to
surface. Consider three tangent planes through P } and their points (
contact. The three lines from these points of contact to P are tai
gent lines of the surface and hence there is only one tangent plar
through each of them. Hence they are lines of the cone of lines ass<
ciated with the cone of tangent planes. . Let TT be the plane throug
their points of contact. The section by IT of the cone of planes throug
P is therefore the conic determined by the three points of conta<
and the two tangent lines in wljich two of the tangent planes me<
IT. The plane TT, however, meets the regulus in a conic of which, tl
three points of contact are points. The two lines of intersection wit
302 FAMILIES OF LIKES [CHAP. XI
TT of two of the tangent planes through P are tangents to this conic,
because they cannot meet it in more than one point each The section
of the surface and the section of the cone of 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 is not on
the surface.
If P is on the surface, the cone of planes degenerates into two lines
of the surface (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 TT 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 regulus contains all
points P' such that the hne PP' meets the surface in two points which
are harmonic conjugates with respect to P, P.'
Proof Consider a plane, a, through PP' and containing two lines
a, 1} of the cone of tangent lines through P. This plane meets the
surface in a conic <7 2 , to which the lines a, & are tangent. As the polar
plane of P contains the points of contact of a and &, 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 polar line of this point with regard to the conic
which is the section of the regulus ty ir.
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 ruler and
directrix through that point. Each line of a pair contains the polar
joints of all the planes on the other tine.
104] .THE EEGULUS 303
THEOREM 10 The polars with regard to a regulus of the points of
a line I are an axial pencil of planes protective 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
projective with the range on I. 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 protective with the
range on L
DEFINITION 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
P' 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 TT. A line p is conjugate to TT if it
is on the pole of TT.
EXERCISES
Polar points and planes witJi respect to a regulus are denoted by corresponding
capital Roman and small Greek letters. Conjugate elements of the same kind are
denoted by the same letters until primes
1 If IT is on JR, then P is on p.
2. If Hs polar to I, then lia polar to L
3. If one element (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 01 a duectrix 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 forma whose bases are not conjugate or
polar are projective if conjugate elements correspond.
12. A line I is conjugate to V if and only if some plane on I is polar to
some point on V. <
304 FAMILIES OF LINES [CHAP.XI
13 Show that theie are two (proper or improper) lines r, s meeting two,
given lines and conjugate to them both Show also that r is the polar of s.
14 If a, b, c aie thiee generators of a legulus and a', &', c' three of the con-
3Ugate regulus, then the three diagonal lines joining the points
(be') and (6'c),
(c'a) and (ca') 5
(a&') and (a'&)
meet in a point S 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, 6, c, a', &', c' of Ex. 14 determine the following tuos
of simple hexagons
(bc'db'ca"), (ba'ac'cb''), (bb'aa'cc'),
(bc'aa'cb"), (bb'ac'ca"), (ba'ab'cc').
The points S determined by each trio of hexagons aie colhnear, and the two
lines on which they lie are polar with legard to the quadnc suiface *
16 The section of the figuie of Ex 14 by a plane leads to the Pascal
and Biianchon theorems , and, in like manner, Ex 15 leads to the theoiem
that the 60 Pascal lines corresponding to the 60 simple hexagons foimed
fioni 6 points of a conic meet by threes in 20 points which constitute 10
pairs of points conjugate with regaid to the conic (cf. Ex. 19, p 138).
105. Protective conies. Consider two sections of a regulus by
planes which are not tangent to it. These two comes 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 comes 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 comes 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 line 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 two pro-
jettire comes in different planes form a regulus, provided the two
comes determine tJie same involution, I, of conjugate points on the
* Cf Sanaa, Lezioni di Geometria Projettiva (Naples, 1895), pp. 262-263.
105]
PROJECTIVE CONICS
305
line of intersection, I, of the two planes ; and provided the eollineation
between the two planes determined by the correspondence of the conies
transforms I into itself by a projectivtty to which I belongs (in par-
ticular, if the conies meet in two points which are self-corresponding
in the projcetimty).
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 I. The conic is gen-
erated by the two pencils A[P] and B[P'] where P and P' are con-
jugates m the involution I on I (of. Ex. 1, p. 137). Let A and
B be the points homologous to A and B on the second conic, and let
A be the point in which the second conic is met by the plane con-
taming A, A, and the tangent at A ; and let B be the point in which
the second conic is met by the plane of J?, B, and the tangent at B.
The line AB contains the pole of I with regard to the second conic
because this line is protective with AB. Since the tangents to_the
first conic at A and B meet on I, the complete quadrangle A ABB has
we diagonal point, the intersection, of A A and JBJB, on I ; hence the
306 FAMILIES OF LINES [CHAP. XI
opposite side of the diagonal triangle passes through, the pole of L
Hence it intersects AB in the pole of I But the intersection of AB
with AB is on this diagonal line. Hence AB meets AB in the pole
of L Hence the pencils A [P] and B [P 1 ] generate the second conic.
Hence, denoting by a and & the lines AA and BB, the pencils of planes
a[P] and &[P'] are protective and generate a regulus of which the
two conies are sections
The projectrvity between the planes of the two comes 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 2. The projectivity between two conies is fully determined by
these conditions (cf. Theorem 12, Cor. 1, Chap. VIII). Hence the
Imes 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
comes. For if a point P of I corresponds to a point Q of I, the unique
tangent other than I through P to the first conic must correspond to
the tangent to t>he second conic from Q. If the projectivity between
the two comes is to generate a regulus, the pfojectivity on I must be
parabolic with the double point at the point of contact of the comes
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
comes 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 comes is fully determined by the correspondent of one
point, not on Z, of the first conic
To show that a projeetivity between the two comes which is para-
bolic on I does generate a regulus, let A be any point of the first
conic and A' its correspondent on the second (fig. 110). Let the
plane of A' and the tangent at A meet the second conic in A".
Denote the common point of the two conies by JB, and consider the
105]
PROJECTIVE CONICS
307
two comes as generated by the flat pencils at A and B and at A"
and />'. The correspondence established between the two flat pencilf
at B by letting correspond lines joining B to homologous points ot
the two cornea is perspective because the line I corresponds to itself.
Hence there is a pencil of
planes whose axis, I, passes
through B and whose planes
contain homologous pairs
of lines of the flat pencils
at II The correspondence
established in like manner
between the flat pencil at A
and the flat pencil atyl"may
be 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" 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 11 A'. As these lines meet
I in the same point, the projectivity determined on I is the identity.
Hence corresponding lines of the projective pencils at A and A" meet
on I, and hence they determine a pencil of planes whose axis is a, = AA n
The axial pencils on a and 1) 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
110
OOROLLAEY 1. The lines joining corresponding points of two
jective 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 "by 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.
308 FAMILIES OF LINES [CHAP XI
The generation of a regulus by protective ranges of points on skew
lines may be regarded as a degenerate case of this theorem and cor-
ollary. A further degenerate case is stated in the first exercise
The proof of Theoiern 11 given above is moie complicated than it would
have been if, under Pioposition K 2 , we had made use of the points of mlei-
section of the line Z with the two conies But since the discussion of linear
families of lines in the following section employs only pioper elements and
depends in part on this theoiem, it seems more satisfactory to piove this
theorem as we have done It is of comse evident that any theoiem i elating
entirely to proper elements of space which is proved with the aid of Pi oposi-
tion K n can also be proved by an argument employing only pioper elements.
The latter form of pi oof is often much moie difficult than the foimer, but it
often yields more information as to the constructions related to the theoiem.
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.
COEOLLAEY 2. If three planes a, ft, 7 meet in three lines a y&y,
6 = ya, c = aft and contain three comes A 2 , B z , G*, of which B* and s
meet in two points P, P' of a, C 2 and A* i/ieet in two points Q, Q' of 1),
and A z and B 2 meet in two points R, R' of c, then there is one and but
one quadric surface * containing the points of the three conies.
Proof. Let M be any point of s . The conic B* is projected from
M by a cone which meets the plane a in a conic which intersects A 2
in two points, proper or improper or coincident, other than R and R'.
Hence there are two lines m, m 1 , proper or improper or coincident,
through M which meet both A z and B z . The projectivity determined
between A* and B z 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 C* because they pass through P, P', Q', Q, and M, all of
which are points of C*.
The conjugate of such a regulus also contains a line through M
which meets both A* and B*, Hence the lines m and m 1 determine
conjugate reguli if they are distinct. If coincident they evidently de-
termine a cone. The three conies being proper, the quadric must con-
tarn proper points even though the lines m, m' are improper.
* In this corollary and in Theorem 12 the term guadnc surface must be taken,
to include the points on a cone of lines as a special case.
100]
QUADEIC THROUGH NINE POINTS
309
If six points 1, 2, 3, 4, 5, 6 are given, no four of which are co-
plttnar,* there evidently exist two planes, <x and & each containing
three of the points and having none on their line of intersection.
PIG. ill
Assign the notation so that 1, 2, 3 are in a. A quadric surface which
contains the six points must meet the two planes m two conies A*,
& which meet the line afi^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 co of
an arbitrary point on the line c. The polar lines of with regard
to a pair of conies A* and B* meet c in the same point and hence
determine to. If no two of the points 1, 2, 3, 4, 5, 6 are collinaar
with 0, any line I in the plane a determines a unique conic A z with
regard to which it is polar to 0, and which passes through 3, 2, 3.
A* determines a unique conic J2 2 which passes through 4, 5, G and
meets c in the same points as A*-, and with regard to this conic O
* The construction oi a quadric surface through mne points by the method used
in the text is given in Eohn and Papperitz, Darstellende Geometrie, Vol II
(Leipzig, 1896), 676, 677 '
310 FAMILIES OF LINES [CHAP. XI
has a polar lino in. Thus there is established a one-to-one corre-
spondence II between the lines of a and the lines of ft. This corre-
spondence is a collmeation For consider a pencil of lines [I] in a.
The conies A* determined by it form a pencil. Hence the point-pairs
in which they meet c are an involution Hence the conies J5 2 deter-
mined by the point-pairs form a pencil, and hence the lines [m] form
a pencil. Since every line I meets its corresponding line m on c, the
correspondence II is not only a collmeation but is a perspectivity,
of which let the center be C. Any two corresponding lines I and m
are coplanar with C. Hence the polar planes of with regard to
yuadrics through 1, 2, 3, 4, 5, 6 are the planes on C.
This was on the assumption that no two of the points 1, 2, 3, 4, 5, 6
are colhnear with If two are colknear 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, /3 = 456, 7 = 789
are such that none of their lines of intersection a = /3<y, & = 70;, c = aft
contains one of the nine points Let be the point afty (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 13 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 B.
'If A, B, and C are not collinear, the plane (o ABO must be the
polar of with regard to any quadric through the nine points. The
plane to meets a, ft, and 7 each in a line which must be polar to
with regard to the section of any such quadric But this determines
three conies A* in a, B* in ft, and C* in 7, which meet by pairs in
three point-pairs on the lines a, &, c. Hence if a, ft, 7 are not in the
same pencil, it follows, by Corollary 2, that there is a unique quadnc
through the nine points. If a, ft, 7 have a line in common, the three
conies A z , B z , C* meet this line in the same point-pair. Consider a
plane a- through which meets the conies A*, B*, C* in three point-
pairs. These point-pairs are harmonically conjugate to and the
trace, s, on cr of the plane to. Hence they He on a conic Z> 2 , which,
with A* and B 2 , determines a unique quadric. The section of this
103,106] LINEAR DEPENDENCE 01? LINES
quadrio by the plane 7 has in common with C a two point-pairs and
the polar pair 0, s. Hence the quadric has 6' a as its section by y.
.. In case A, /?, and C are oollinear, there is a pencil of planes o> which
meet them. There is thus determined a family of quadrics which is
called a pencil and is analogous to a pencil of comes. In case A, ,
and C 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 coplanar
there passes one quadric surface or a pencil of quadrics or a faindle
of quadriGS.
EXERCISES
1. The lines joining homologous points of a projoctive conic and stiaight
line foim a regulus, provided the line meets the conic and is not coplanar
with it, and their point of intersection is self-uorresponding.
2. State the duals of Theorems 11 and 12.
3. Show that two (proper or impioper) conjugate leguli pass through two
conies in different planes having two points (proper or improper or coincident)
111 common and through a point not in the plane of either conic. Two such
comes and a point not in either plane thus determine one quadric surface.
4. Show how to consti uct a regulus passing through six given points
and a given line
106. Linear dependence of lines. DEFINITION. If two lines are co-
planar, 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 hnearly dependent the lines of the regulus, of which
they are rulers. Jll v l v -.*, l n are any number of lines and m v m a , , m K
are lines such that m^s linearly dependent on two or three of l v Z a , , ,
and m & is linearly dependent on two or three of l v l s , -, l n , m v and
so on, m k being linearly dependent on two or three of l v l^> , l n , m v
m z> - - ,'m k _ 1 , then m k is said to be linearly dependent on l v l z> * , l n .
A set of n lines no one of which is linearly dependent on the n 1
others is said to be Imearly 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 LUSE8 [CHAP XI
and b intersect in a point P, and a line c skew to both of them meets
their plane m a point Q, then in the first place all lines of the pencil
ab are linearly dependent on a, 6, and c ; since the line QP is in this
pencil, all lines of the pencil determined by QP and c are m the set
As these pencils have in common only the line QP and do not con-
tain three mutually skew lines, the set contains no other lines.
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 hues, as a, meets
both of the others, which, however, are skew to each other, the set df
linearly dependent lines consists of the flat pencils ab and ac 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 m the same
point and are not coplanar, the 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 regulus, or a bundle of
lines, or a plane of lines, or two Jlat pencils having different centers
and planes but a common tine. 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 congruenee The set of all lines
linearly dependent on five linearly independent lines is called a linear
complex *
107. The linear congruence. Of the four lines a, b, c, d upor
which the lines of the congruence are linearly dependent, b, c, d
determine, as we have just seen, either a regulus, or two flat pencils
with different centers and planes but with one common line, or a
bundle of lines, or a plane of lines. The lines &, e, d can of course be
replaced by any three which determine the same regulus or degen-
erate regulus as 5, c, d
* The terms congruence and complex are general terms to denote two- and three-
parameter families of lines respectively For example, all lines meeting a carve or
all tangents to a surface form a complex, wlnle all lines meeting two curves or all
common tangents of two surfaces are a corgruence.
107] THE LINEAK CONGRUENCE 313
So in case b, c, d determine a noudegenerate regulua of which a is
not a directrix, the congruence can bo regarded as determined by four
mutually skew lines. In case a is a directrix, the lines linearly de-
pendent on a, b, c, d clearly include all tangent lines to the regains
led, whose points of contact are on a. But as a is m a flat pencil
with any tangent whose point of contact is 011 a, and one of the
rulers, the family of lines dependent on a, b, c, d is the family de-
pendent on b, 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, P > 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 y
From this it is evident that all lines meeting both p and q are linearly
dependent on the given four. For if P v is any point on p, the line
P^Q and the ruler through J^ determine a flat pencil joining P x 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 H 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 II, and all lines of the
regulus are in the set. Hence* when one of four &kew lines is tangent
to the regulus of the other three, the family of dependent 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
[CHAP. XI
If no one of the four skew lines meets the regulus of the other
three m a proper point, we have a case studied more fully below.
1 In case I, c, d determine two fiat pencils with a common line, a
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 three noucon-
current lines dependent on a, I, c, 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' and
by lines V and c 1 , one from each pencil, not meeting a. Hence the
family of lines includes those dependent on the regulus ab'cf and its
directrix d'. 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'.
TIG 112
If a does not meet the common line, it meets the planes of the
two pencils in points C and D. Call the centers of the pencils A and
S (fig. 112). The first pencil consists of the lines dependent on AD
and AS, the second of those dependent on AS and SO. As CD is
the line a, the family of lines is seen to consist of the lines which
are linearly dependent on AS, SO, CD, DA Since any point of JBD
is joined by lines of the family to A and (7, it is joined by lines of
107] THE LINEAR CONGRUENCE
the family to every point of A C Hence this case gives the family
of all lines meeting both AC and HD.
In case 1), c, d determine a bundle of lines, ct, being independent of
them, does not pass through tho center of the bundle. Hence the
family of dependent hues includes all hues of the plane of a and the
center of the bundle as well as the bundle itself.
Lastly, if o, c, d are coplauar, we have, by duality, the same case as
if &, c, d were concurrent. We have thus proved
THEOREM 14. A linear congruence is either (1) a set of hues
linearly dependent on four linearly independent S?MW lines, such that
no one of them meets the regidus 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 regidus which
meet a fixed directrix of the reyulus , or (4) it consists of a bundle
of lines and a plane of lines, the center of the bundle leing 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.
COEOLLAEY. A parabolic congruence consists of all lines on corre-
sponding points and planes in a project ivity between the points and
planes on a, line. The directrix is a hne 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, I, c, d, and let 7r t and 7r a be two planes in-
tersecting in a. Let the points of intersection with TT I and ir z of &, c,
and d be JB V C v and D l and !?, <7 2 , and D B respectively By letting
the complete quadrilateral a, J)^0 V 0^, I> 1 7? 1 correspond to the
complete quadrilateral a, J3 S C Z , C Z 2) Z , J5 2 7? a , there is established a
projective colhneatioii II between the planea 77^ and 7r a in which
the lines 5, c, d join homologous points (fig. 113).
Among the lines dependent on a, 5, c, d are the lines of the reguli
ale, acd, ado, and all reguli containing a and two lines from any
of these three reguli. But all such reguli meet t rr l and ir z in lines
(e.g, -BjZ^, JB Z D Z ) because they have a in common with TJ^ and
7r z . Furthermore, the lines of the fundamental reguli join points
316
FAMILIES OF LINES
[CHAP XI
which correspond in II (Theorem 5 of this chapter and Theorem 18,
Chap. IV) Hence the reguh which contain a and lines shown by
means of such reguh to be dependent on a, I, c, d are those gen-
erated by the projectivities determined by II between lines of ^
and 7r 2
d tit,
\
FIG 113
Now consider reguli containing triples of the lines already shown
to be in the congruence, but not containing a Three such lines, I,
m, n, ]om three noncolhnear points L v M v N^ of ir^ to the points
L v M v N z of 7r 2 which correspond to them in the colhneation II The
regulus containing Z, m, and n meets ir^ and tr z in two conies which
are projective in such a way that L v M v N^ correspond to L z , Jf 2 , N v
The pro]ectivity between the conies determines a projectivity between
the planes, and as this projectivity has the same effect as II on the
quadrilateral composed of the sides of the triangle L^H^ and the
hne a, it is identical with U. Hence the lines of the regulus Imn
join points of w t and Tr 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 pro]ectivity determined on a by
H. If neither of the two lines is a, they must meet w-j and 7r 2 , the
first in points P v P a and the second in points Q v Q z> and these four
io?J THE LINEAR CONGRUENCE 317
points are clearly distinct from one another. But as the given lines of
the congruence, PJ^ and Q^Q^ intersect, so must also the lines P t Q 1 and
of 77*! and 7r a intersect, and the pro j activity determined between
and P$ z by II is a perspectivity. Hence the common point of
and P^Q Z is a point of a and is transformed into itself by H.
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, &, 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, b, c, d These two results are
summarized as follows :
THEOREM 15. All the lines of a linear congricence 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, JE7, F, on a, the conic BiCiDiJSF roust be trans-
formed by n into the conic BaOaD*EF, and the lines joining corresponding points of
these conies must form a regulus contained m the congruence. As JS and F are
on lines of the regulus bed, there are two directrices p, q of this regulus which
meet JS7 and F respectively, The lines p and q meet all four of the lines a, 6, 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 JET of the parabolic projectivity on a, because the conic tangent to a at U and
passing through BiCiDi must be transformed by II into the conic tangent to a at H
and passing through JBgC7 2 Z> 2 ; and the lines joining homologous points of these conies
must form a regulus contained in the congruence As IT, a point of a, is on a line
of the regulus bed, there is one and only one directrix jp of this regulus which meets
all four of a, 6, c, d and hence meets all lines of the congruence.
318 FAMILIES OF LINES [CHAP xi
The dual of Theorem 16 may be stated in the following form .
THEOREM 17. From two points on the same line of a linear congru-
ence the latter is projected by two projective bundles of planes. Con-
versely, two bundles of planes projective in such a way that the hne
joining their centers is self -corresponding, generate a linear congruence
DEFINITION A regulus all of whose rulers are in a congruence is
called a regulus of the congruence and is said to be in or to be 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 bed (in the notation
already used) is met by a in two pomts (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 pomts, because if it were,
the pro]ectivity II, between ir l 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
bed 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-
tarn a. Since there is but one involution transformed into itself by a
nomnvolutoric projectivity on a line (Theorem 20, Chap VIII), we
have that the same involution of conjugate pomts is determined on
any line of the congruence by all reguli of the congruence which do
not contain the given line This is entirely analogous to the hyper-
bolic case, and can be used to gam 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 hnear congruence each line is tangent
at a fixed one of its points to all reguli of the congruence of which it is
not a ruler On each hne of a hyperbolic or elliptic congruence all reguli
of the congruence not containing the given line determine the same
involution of conjugate points Tlirough each point of space there is
one and only one line of an elliptic congruence For hyperbolic and
parabolic congruences this statement is true except for points on a
directrix.
107, 108] THE LINKAK COMPLEX
EXERCISES
"r"
1. All lines of a congruence can be constuicted from four lines by means
of reguli all of which have two given lines in common.
2. Given two involutions (both having or both not having double points)
on two skew lines Thiough each point of space there are two and only two
lines which are axes of peispectivity projecting one involution into the othei,
i e. such that t-vo planes through conjugate pans of the fiist involution pass
through a conjugate pair of the second involution. These lines constitute
two congruences.
3. All lines of a congiucnce meeting a line not in the congiuence form
a regulus.
4. A linear congruence is self-polar with regaid to any regulus of the
congruence.
5. A degenerate linear congruence consists of all lines meeting two inter-
secting lines.
108. The linear complex. THEOREM 19. A linear complex con-
sists of all lines linearly dependent on the edges of a simple skew
pentagon.*
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, &', c', d', e'. The complex consists of
all lines dependent on a and the congruence I'c'd'e'. If this con-
gruence is degenerate, it consists of all lines dependent on three sides
of a triangle cde and a line & not in the plane of the triangle
(Theorems 14, 15). As & may be any line of a bundle, it may be
chosen so as to meet a ; e may be chosen so as to meet &, and e may
be so chosen as to meet a. Thus in this case the complex depends
on five lines a, I, c, d, e not all coplanar, forming the edges of a simple
pentagon.
If tlie congruence is not degenerate, the four lines &", c", 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, 6" and e",
meet a. Thus the complex consists of all lines linearly dependent
on the two flat pencils ab" and ae n and the two lines G" and d". Let
5 and e be the lines of these pencils (necessarily distinct from each
other and from a) which meet e" and d" respectively. The complex
then consists of all lines dependent on the flat pencils a&, le 1 ', ae, ed".
* The edges of a simple skew pentagon are five lines m a given order, not all
coplanar, each line intersecting its predecessor and the last meeting the first.
320 FAMILIES OF LINES [OHAP.XI
Finally, let c and d be two intersecting lines distinct from 6 and e,
which are in the pencils be" and ed". The complex consists of all lines
linearly dependent on the flat pencils aZ>, 6c, cd, de, ea. Not all the
vertices of the pentagon abcde can be coplanar, because then all the
lines would be in the same degenerate congruence.
THEOREM 20. DEFINITION There are two classes of complexes such
that all complexes of either class are protectively equivalent. A com-
plex of one class consists of a line and all lines of space which meet
it. These are called special complexes. A complex of the other class
is called 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 abcde which determines it If there is a ImeZ
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 regulus (degenerate or not) meets all lines of it. Moreover,
if the line I meets a and & as well as c and d, 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 that four
of the vertices of the pentagon are coplanar, two of them being on e.
(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 coplanar, there ex-
ists a line meeting the Jive lines Since any two skew pentagons are pro-
jectively equivalent, if no four vertices aie coplanar (Theorem 12,
Chap. Ill), any two complexes determined by such pentagons are
projectively 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 colhneation of space into
any other, and then there exists a collmeation holding the plane
of the second four-point pomtwise 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 projec-
tively equivalent But these are the only two kinds of skew pentagons
Hence there are two and only two kinds of complexes
108]
THE LINEAR COMPLEX
321
In case four vertices of the pentagon are ooplauar, we have seen
that there is a lino I meeting all its edges Since this line was
determined as the intersection of the plane of two adjacent edges
with the plane of the other three, it contains at least two vertices.
It cannot contain three vertices because then all five would be
coplanar. As one of the two planes meeting on I contains three
independent lines, all lines of that plane are lines of the complex.
The lino I itself is therefore in the complex as well as the two lines
of the other plane. Hence all lines o both planes are in the complex.
Hence all lines meeting I are in the complex But as any regulus
three of whose lines meet I has all its lines meeting I, the complex
satisfies the requirements stated in the theorem for a special complex.
FIG. 114
A more definite idea of the general complex may be formed as
follows Lot piPiPnPiPs (fig- 114) be a simple pentagon upon whose
edges all lines of the complex are linearly dependent. Let q be the
line of the flat pencil p s p^ which meets p v and let Jffi be the point of
intersection of q and p r Denote the vertices of the pentagon by 7 J ia ,
P 3 , P H , PW, P 6li the subscripts indicating the edges which meet in a
given vertex.
The four independent lines p^p^q determine a congruence of lines
all of which are in the complex and whose directrices are a^RP^
and a 1 = P^P^- In like manner, qp^p^p^ determine a congruence whose
directrices are l = RP^ and 6' = ^ 4 ^i The complex consists of all
lines linearly dependent on the lines of these two congruences. The
322
FAMILIES OF LINES
[CHAP. XI
directrices of the two congruences intersect at ft and P^ respectively
and determine two planes, ab = p and a'b' = ir, which meet on q.
Through any point P of space not on p or TT there are two lines
I, tn, the first meeting a and a', and the second meeting b and b'
(fig. 115) All lines in the flat pencil Im are in the complex by defi-
nition This flat pencil meets p and TT in two perspective ranges of
PIG 115
points and thus determines a projectivity between the flat pencil ab
and the flat pencil a'b', in which a and a', b and b' correspond and q
corresponds to itself. The projectivity thus determined between the
pencils ab and a'b' is the same for all points P, because a, b, q always
correspond to a', b', q 1 . Hence the complex contains all lines in the
flat pencils of lines which meet homologous lines in the projectivity
determined by
Denote this set of lines ly S. We have seen that it has the property
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 PR has a corresponding line p'
in the pencil a'b' and hence S contains all lines joining P to points
of p'. Similarly, for points on TT but 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 q. Applying this to the planes through P s not contain-
ing q } we have that any line through P sl and not on p is not in the
108] THE LINEAK COMPLEX 333
set S. Lei I be any such line. All lines of S in each plane through
I form a fiat pencil P, and the centers of all these pencils lie on a line
I', because all lines through two points of I form two flat pencils each
of which contains a line fiom each pencil P Hence the lines of S
meeting I form a congruence whose other directrix I 1 evidently lies on
p. The point of intersection of I 1 with q is the center of a flat pencil
of lines of S all meeting I Hence all lines of the plane Iq form, a flat
pencil Since I is any line on P 34 and nob on TT, this establishes that
each plane and, by duality, each point on q, as well as not on q, con-
tains a flat pencil of hues of S.
We can now prove that the complex contains no lines not in S
To do so we have to show that all lines linearly dependent on lines
of S are in S If two lines of S intersect, the Hat pencil they deter-
mine is by definition in S. If three lines m v m zi m^ 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 lines at points of these three lines would be in S, and hence any
plane would contain three nonconcurreut lines of S. Let I be a
directrix of the regulus >n\m^n v 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 tins congruence contains all linos of the regulus mjn$n, v
and hence all lines of this regulus are in S. Hence the set of lines S
is identical with the complex.
THEOREM 21 (SYLVESTEK'S THEOREM *). If two pro/active flat pencils
with different centers and planes have a line q in common wMeJi is
self -corresponding, 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 #.
Proof. This has all been proved in the paragraphs above, with the
exception of the statement that q and the lines meeting g form a
linear congruence Take three skew lines of the complex meeting q ,
they determine with j a congruence C all of whose Hues are in the
complex. There cannot be any other lines oC 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.
# Of. Comptes Bendus, Vol. UI (1861), p, 741.
324 FAMILIES OF LINES [CHAP. XI
Another theorem proved in the discussion above is :
THEOREM 22. DEFINITION OF NULL SYSTEM. All the lines of a
linear complex which pass through a point P lie in a plane TT, and all
the lines which lie in a plane IT pass through a point P. In case of
a special complex, exception must be made of the points and planes 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 complex. The cor-
respondence 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 coplanar.
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 contams all
lines linearly dependent on them, by definition
(&) Consider any hue 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 JB all meet a line a' skew to a. From
this it follows that all the lines of the congruence whose directrices
are a, a' are in K. Similarly, if b is any other line not in K but meet-
ing a, all lines of K which meet b also meet another line b'. More-
over, since any line meeting a, b, and b' is in K and hence also meets
a', the four lines a, a', b, b' he on a degenerate regulus consisting of the
flat pencils ab and a'b' (Theorem 13). Let q (fig. 115) be the common
line of the pencils ab and a'b'. Through any point of space not on one
of the planes ab and a'b' there are three coplanar lines of K which
meet q and the pairs aa' and bb'. Hence K consists of lines meeting
homologous lines in the pro;jectivity
gab - qa'b',
and therefore is a complex by Theorem 21.
108] THE LINEAR COMPLEX 325
COROLLARY. Any (1, 1) correspondence between the points and the
planes of space such that each point lies on its corresponding plane
is a null system.
THEOREM 25 Two hnear 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 111 common. Obviously, then, there are
three linearly independent lines l v l J, common to the complexes.
All lines in the regulus 1,1,1, are, by definition, in each complex. Bufc
as there are points or planes of space not 011 the regulus, there is a
line Z 4 common to the two complexes and not belonging to this regulus.
All lines linearly dependent on l lt Z 2 , Z 8 , J, 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 l f
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 and only if I and V are directrices of a
congruence of lines of the complex.
COROLLARY 3. If I is polar to V, 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 proactive 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,
32G FAMILIES OF LINES [CHAP. XI
EXERCISES
1 . If a point P is on a plane p , the null plane TT of P is on the null point Roip
2 Two pan s of lines polai with i egard to the same null system ai e always in
the same legulus (degenerate, if a line of one pan meets a line of the other pan)
3 If a line I meets a line m, the polai of Z meets the ^polar of in
4. Pairs of lines of the regulus in Ex. 2 which aie polar with i egard to
the complex aie met hy any dnectrix of the legulus in pans of points of an
involution. Thus the complex determines an involution among the lines of
the regulus.
5. Conversely (Theorem of Chasles), the lines meeting conjugate pans of
lines in an involution on a legulus aie in the same complex. Show that
Theoiem 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, I, m determine one and only one complex contain-
ing k and having I and m as polai s of each othei
8. If the numbei of points on a line is n + 1, how many reguli, how many
congruences, how many complexes are there in space? How many lines aie
there in each kind of regulus, congiuence, complex ?
9 Given any general complex and any tetiahedion whose faces ai'e not
null planes to its vertices The null planes of the veitices constitute a second
tetrahedron whose veitices be on the planes of the fiist tetiahedron The
two tetrahedia are mutually inscribed and ciicumscribed each to the other*
(cf . Ex. 6, p. 103).
10. A null system is fully determined by associating with the three vertices
of a tnangle three planes through these veitices and having their one common
point in the plane of the tiiangle but not on one of its sides
11 A tetrahedron is self -polar with regaid to a null system if two opposite
edges are polar.
12 Every line of the complex determined by a pair of Mobius tetrahedra
meets their faces and projects their vertices in protective throws of points and
planes
13 If a tetrahedron T is insciibed and ciicumscribed to 7\ and also to !T 2 ,
the lines joining corresponding vertices of T^ and T s and the lines 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 regulus all of whose
lines are in the complex.
16 The lines from which two projective pencils of points on skew lines
are projected by involutions of planes are all in the same complex Dualize
* This configuration was discovered by Mobius, Journal fur Mathematik, Vol. Ill
(1828), p 273 Two tetrahedra in this relation are known as Mobius tetrahedra.
109] LINE COOEDINATES 327
109. The Pliicker line coordinates. Two points whose coordinates are
(*!> Ej - 7 ' 3 > ^''t)
(2/1, y*> 2/ 3 > &)
determine a line L The coordinates of the two points determine six
numbers
#34 =
2/ a
2/ 3
2/8
2/4 2/2
2/
2/a 2/'
which are known as the Pluclwr coordinates of the line. Since the
coordinates of the two points are homogeneous, the ratios only of the
numbers p l} are determined Any other two points of the line deter-
mine the same set of line coordinates, since the ratios of the ^> f/ 's are
evidently unchanged if (x v j 2 , x &t aij is replaced by (jCj+X^,
The six numbers satisfy the equation*
== 0.
This is evident on expanding in terms of two-rowed minors the
identity
2/ 3
2/4 =
2/4
Conversely, if any six numbers, p ij} are given, which satisfy Equa-
tion (1), then two points P = (os v x z , x a> 0), Q~(y v 0, y n , y 4 ) can be
determined such that the numbers p (f 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. Jl^ery line of space determines and is determined
"by the ratios of six numbers p w p 18 , j9 :4 , p Mi jp 48 , p ag sitbjeot to the
* Notice that in Equation (1) the number of inversions in the four subscripts of
any term is always even
328
cond^t^on
\2/i> 2/2' y B>
FAMILIES OF LINES
[CHAP XI
are
= 0, such that if (x lt x z , x a , # 4 ) and
on ^e hne,
2/ 2
2/
as. %
-! ^4
O* 'T*
2 S
2/2 2/3
COEOLLAEY ^bwr independent coordinates determine a line.
In precisely similar manner two planes (w ls u z> u s> ^6 4 ) and (Vj^, v z , v s , -yj
determine six numbers such that
234 =
242 =
u,
The quantities q v satisfy a theorem dual to the one just proved for
the p v 's
THEOREM 28. TJie p and q coordinates of a line are connected 1y
the equations p 12 : p l3 : p u p a4 : p, z ' p 2a = ^ . q iz ' q za ' q lz : q ls : q^.
Proof Let the p coordinates be determined by the two points
(x v x zt x a , x^, (y v y z , y s> y 4 ), and the q coordinates by the two planes
(u v u 2 , u a> w 4 ), (v v v z> v 3 , v t ). These coordinates satisfy the four equations
U S X S + U^ = 0,
V 3 X S + Vft = 0,
4 = 0.
Multiplying the first equation by v 1 and the second by u^ and adding
we obtain .
In like manner, from the third and fourth equations we obtain.
Ziaft + 2is/ 8 + Su^= 0.
Combining the last two equations similarly, we obtain
or, 13 42.
ft* ^28
By similar combinations of the first four equations we find
JPl ' JPlS : Pl4 : ^84 : P42 : ^28= ^34 = ^42 = ^23 ! 2l2 : ?! : 14-
i(M) f iio]
LLIS'E COORDINATES
329
EXERCISE
Given the tetiahedron of reference, the point (1, 1, 1, 1), and a line ',
determine six sots of oui points each, whose cioss latioh aie the cooidmates
of/
110. Linear families of lines. THEOREM 29. The necessary and
sufficient condition that two lines p and p' intersect, and hence are
coplanar, is
PI K +p u p +PUPL +p*ipL+p a pU +v<>pL = o,
where p n are the coordinates of p ant ^ P'J f P'-
Proof. If the first line contains two points x and y, and the second
two points %' and y', the lines will intersect if and only if these foui
points are coplanar ; that is to say, if and only if
0==
1 l ^
y[
THEOREM 30. 'Aflat pencil of linen consists of the lines whose coordi-
nates are \p ti + pplj, if P and 2 J ' arft t' wo ^ nes f ^ >(i pnwil
Proof. The lines p and p' intersect in a point A and are perspec-
tive with a range of points \( 1 + pD. Hence their coordinates may be
written
ofp
' '
i, etc.
which may be expanded in the form
d, d.
THEOREM 31. The lines whose coordinates satisfy one linear
equation
(1) a w jp-h !#+ 14 2 7 n+ ^P 4 + JP4a+ ' 28 ^ 2S =
form a> linear complex. F/iose 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 T>y two (distinct or coincident] lines,
which may %e improper,
330
FAMILIES OF LIKES
[CHAP. XI
Proof If (& v & 2 , & 3 , & 4 ) is any point of space, the points fo, as v # 3 , # 4 )
which lie on lines through \, & 2 , 6,, 6 4 satisfying (1) must satisfy
rt !2
M
3
i ""'a
-M,
'\ "4
Cj # 4
= 0,
or
(2)
! A) l
! A
ft s A) s, + (- a i 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
+ <^ 2S = 0,
42 + <jPss= 0,
4 2 + <K= 0,
are independent, one of the four-rowed determinants of their coeffi-
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
ia**
which gives a quadratic equation to determine X//*. Hence, by Propo-
sition K 2 , there are two (proper, improper, or coincident) lines whose
coordinates satisfy four linear equations.
COROLLARY 1. The lines of a regulus are of the form
u-Jtere p f , p", p'" are lines of the regulus. In like manner, the lines of
a congruence are of the form,
* Of Bocher, Introduction to Higher Algebra, Chap IV.
i]0,m] LINE COORDINATES 331
and of a complex of the form
All of these formulas must be, taken in connection with
COROLLARY 2. As a transformation from points to planes the null
system determined Inj the complex whose equation is
is
w x =
I (\
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 (& 11( 5 2 , 6 3 , & 4 )
Coiollary 1 shows that the geometric definition of linear dependence of
lines given in this chapter corresponds to the conventional analytic concep-
tion of linear dependence.
111. Interpretation of line coordinates as point coordinates in S 6 .
It may be shown without difficulty that the method of introducing
homogeneous coordinates iii 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 8 satisfy the
quadratic condition
they may be ^garded as forming the points of a quadratic locus or
spread,* L 4 a , in S 5 . 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 \- of an S a , the lines of a regulus to the
* This is a generalization of a conic section.
$2 FAMILIES OF LINES [CHAP XI
intersection with L 4 2 of an S 2 , and any pair of lines to the mtersec-
Aon with L 2 of an S r
Any point (p' w p' ls , p' u , p' M , p' 4Z , p' 2S ) of S 5 has as its polar* S 4 , with
regard to L 2 ,
( 2 ) PLPu + PLPu + PaPu + PuPK + PuPv + PuP*** >
which is the equation of a linear complex in the original S 3 Hence
any point in S 5 can l>e thought of as representing the complex of hncs
represented ty the points of S 5 m which its polar S 4 meets L 4 2
Since a Ime is represented by a point on L 4 2 , 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 4 a .
The points of a line, a + X5, in S 5 represent a set of complexes
whose equations are
(3) (**+** ^P*+(**+*)p + = 0,
and all these complexes have ni common the congruence common to
the complexes a and &. Their congruence, of course, consists of the
lines of the onginal S 3 represented by the points in which L 4 2 is met
by the polar S 3 of the line a + X&
A system of complexes, a + X6, is called a pencil of complexes, and
then 1 common congruence is called its "base or lasal congruence. 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 range
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 J
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 Hence f M
* Equation (2) may be taken as the definition of a polar 84 of a point with
regard to l_ 4 a Two points are conjugate with regard to L 4 2 if the polar S 4 of one I
contains the other The polar S 4 's of the points of an S, (i = 1, 2, 8, 4) all have an J.
S 4 _, m common which is called the polar S 4 _t of the S, These and other obvious *
generalizations of the polar theory of a conic or a regulus we take for granted
without further proof. '
111] LINE COORDINATES 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 hue a + \l meets L 4 a m two improper
points, two proper points, or one point, or lies wholly on Lf Two
points in which a representative line meets L 4 2 are the double points
of an involution the pairs of which are conjugate with regard to L 2 .
Two complexes y, p' whose representative points are conjugate
with regard to Lf 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 a x be an arbitrary complex and 3 any complex conjugate to
(in involution with) it. Then any representative point in the polar S 8
with regard to L 4 a of the representative line a^ represents a complex
conjugate to a : and a . Let ct^ be any such complex. The represent-
ative points of a v 2 , form a self-conjugate triangle of L 4 3 Any
point of the representative plane polar to the plane a^a^ with
regard to L 4 a is conjugate to a^a^. Let such a point be a 4 . In like
manner, a 6 and a can be determined, forming a self-polar 6-point of
L 4 2 , the generalization of a self-polar triangle of a conic section The
six points are the representatives of six complexes, each pair of which
is in involution.
It can be proved that by a proper choice of the six points of refer-
ence in the representative S 6 , the equation of L* may be taken as any
quadratic relation among six variables, llenoe 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 (us v a? a) - , sn & ), where
0.*
In this case, the six-point of reference being self-polar with regard
to L 4 2 , its vertices represent complexes which are two by fcvo in
involution.
* These are known as Klein's coordinates. Most of the ideas in the present sec-
tion are to he found in F. filein, Zur Theorie der Linienooraplexe des ersteu und
sweiten Grades, Mathematisohe Annalen, Vol. H (1870), p. 198.
-334 FAMILIES OF LINES [CHAP. XI
EXERCISES
1 If a pencil of complexes contains two special complexes, the basal con-
giuence of the pencil is hypcibolic 01 elliptic, according as the special com-
plexes aie piopei or impropei
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 aie special, the
basal congruence is degenerate.
4 Define a pencil of complexes as the system of all complexes having a
common congiuence of lines and derive its piopeities synthetically
5. The polais of a line with legard to the complexes of a pencil form
a legulus.
6. The null points of two planes with regaid to the complexes of a pencil
generate two piojective pencils of points
7 If C= 0, C' 0, C"= aie the equations of three linear complexes
which do not have a congruence in common, the equation C + \C' + p.C" =
is said to lepresent a bundle of complexes The lines common to the thiee
fundamental complexes C, C", C" of the bundle form a legulus, the con-
jugate legulus of which consists of all the dnectuces of the special com-
plexes of the bundle.
8 Two lineal complexes ^a t) p t j = and "S,b tj p t j = aie m involution if and
only if we ha\e
9 Using Klein's cooidmates, any two complexes are given by 2a,a: t =
and 25&,z t = These two aie in involution if 2a,6 t =
10 The six fundamental complexes of a system of Klein's coordinates
inteisect in pans in fifteen linear congiuences all of whose directrices are dis;
tmct. The directrices of one of these congruences are lines- of the remaining
four fundamental complexes, and meet, theiefore, the twelve directrices of
the six congruences determined by these four complexes.
52/1
INDEX
Tllo nuniboia rofor U> pages
Abolian group, 07
Abscissa, 170
Abstract science, 2
Addit . J * .' J - H2 291 * 1
on, U I I! . I l i - ' o: 107,
EXH 8, 4
Adjacent sides or vortices of simple
n-hno, 87 ,
Algebraic curve, 250
Algebraic problem, 288
Algebraic surface, 25',)
Alignment, assumptions of, 1(5 ; consist-
' ency oi assumptions of, 17 j theorems
of, for the piano, 17-20 , theorems of,
for 3-space, 20-24 ; theorems of, for
4-space, 26, Ex. 4, theorems of, for
n-space, 20-33
Ainodoo, P., 120, 294
Anharmomc ratio, 160
Apollomus, 280
Associative law, for correspondences,
66, f oi'. addition of points, 143; for
multiplication of points, 140
Assumption, II , 45; IT , role of, 81,
261, of 4 n( ,*i.,,*v (>5; of projec-
tivity, i- ' . i \ ' - of, 105, 1Q6,
Exs. 10-12 , 208
Assumptions, are necessary, 2; exam-
ples of, for a mathematical science,
2; consistency of, 8; Independence
of f 6,; categdriealness of, ; of align-
ment, 16; of alignment, conHistenoy
1 of, 17; of extension, 18, 24; of clo-
sure, 24 , for an n-spaeo, 88
Axial pencil, 55
Axial pcrspectivity, 67
Axis, of porspeotivity, 86; of pencil,
J6 ; of perspective collinoation, 72 ; of
homology, 104; of coordinated, 160,
191'; of projectivity on conic, 218
Base, of plane of points or lines, 55 ; of
pencil * complexes, 882 ,
Bilinear equation, binary, represents
project! vity on a line, 150; ternary,
represents correlation in a plane, 267
Binary form, 261, 252, 254
BOcher, M., 166, 272, 289, 880
Braikenridge, 119
Brianchon point, 111
BriancJaon's theorem, 111
Bundle, of planes or linos, 27, 66; of
COUJCH, 207, BXH. 0-12, of quadrics,
811 ; oi complexes, 884, Ex 7
Bnniwdo, W., 150
, W. II., 202
Canonical forms, of collineations in
plane, 274-27U, of correlations in a
plane, 281 ; of pencils of comes, 287-
208
Castelnuovo, G., 130, 140, 287, 297
Categorical set of asHumptions, 6
Coyloy, A., 52, HO
Center, of pcrspeciivity, 36, of flat pen-
cil, 65, of bundle 1 , 55, of perspective
colliiioation in plant', 72 , oi perspec-
tive collinoation m space, 76; of
1i - 1 1 H.-. 1 n l rC f-drrtinates, 170;
(, i , . i . i . ,-c, 218
( i r -si -i- r -7
( . , i i , .- i i- -of parabolic
projeotivity, 207
Characteristic equation of matrix, 165
Characteristic throw and crow* ratio, of
one-dimensional projoctivity, 205, 211,
Exs, 2, 8, 4; 212, Exs. 6, 7 ; of involu-
tion, 20(1; of parabolic projectivity,
206
Chaslos, 125
Class, notion of, 2; elements of, 2; re-
lation of belonging to a, 2 ; subclass of
a, 2; undefined, 16; notation for, 57
Clebsoh, A., 280
Cogretliont n-lino, 84, Ex. 18
Cotfredieat triangle, 84, Exs. 7, 10
Gollineation, denned, 71 ; pewsppctivo, in
plane, 72 ; perspective, in space, 75 ;
transfonfting a quadrangle into a
quadrangle, 74 ; transforming a live-
point into a live-point, 77 ; Irannf orm-
imr a conic, into a conic, 182 ; In piano,
analytic form of, 381), 100, 2118; be-
tween two planes, analytic, form of,
190; in space, analytic- form of, 200;
-leaving oonio invariant, 214, 220, 285,
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 j invariant figure
of, is self-dual, 272
336
INDEX
Collmeations, types of, in piano, 106,
273 , associated with two comes of
a pencil, 131, Exs 2, 4, 6, 135,
Ex. 2 , 136, Ex 2 , group of, 111 plane,
268, represented by matnces, 268-
270 , two, not in general commuta-
tive, 268 , canonical forms of, 274-
276
Commutative coirespondence, 66
Commutative group, 67, 70, Ex 1, 228
Commutative law of multiplication,
148
Commutative projectivities, 70, 210, 228
Compass, constructions with, 246
Complete it-line, in plane, 37, on point, 38
Complete n-plane, in space, 37 , on point,
38
Complete n-pomt, in space, 36 , in plane,
37
Complete quadrangle and quadrilat-
eral, 44
Complex, linear, 312, determined by
skew pentagon, 319 , general and spe-
cial, 320 , determined by two piojec-
tive flat pencils, 323 , determined by
five independent lines, 324, deter-
mined by coirespondence between
points and planes of space, 324 , null
system of, 324 , generated by involu-
tion on regulus, 826, Ex. 5 , equation
of, 329, 331
Complexes, pencil of, 332 , in involu-
tion, 333 , bundle of, 834, Ex. 7
Concrete representation or application
of an abstract science, 2
Concurrent, 16
Cone, 118, of lines, 109, of planes, 109,
section of, by plane, is conic, 109,
as degenerate case of quadnc, 308
Configuration, 38, symbol of, 38, of
Desargues, 40, 51, quadrangle-quad-
rilateral, 44 , of Pappus, 98, 249 , of
Mobius, 326, Ex. 9
Congruence, linear, 312 ; elliptic, hyper-
bolic, paiabolic, degenerate, 315 , de-
termined by four independent lines,
317 , deteinuned by protective 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, projectivity on, 217,
intersection of line with, 240, 242,
246 , through four points and tangent
to line, 250, Ex 8 , through three
points and tangent to two lines, 250,
Ex. 9 , through four points and meet-
ing given line m two points harmonic
with two given points, 250, Ex 10,
determined by conjugate points, 293,
Ex. 2 , 294, Exs 3, 4
Conic section, 118
Comes, pencils and ranges of, 128-136,
287-293 , projective, 212, 304
Con]ugate groups, 209
Conjugate pair of involution, 102
Conjugate points (lines), with legard to
conic, 122 , on line (point), form invo-
lution, 124 , with regaid to a pencil of
comes, 136, Ex. 3 , 140, Ex. 31 , 293,
Ex 1
Conjugate projectivities, 208; condi-
tions for, 208, 209
Conjugate subgroups, 211
Consistency, of a set of assumptions, 8 ,
of notion of elements at infinity, 9 ,
of assumptions of alignment, 17
Construct, 45
Constructions, linear (first degree), 236 ,
of second degree, 245, 249-250,
Exs , of third and fourth degiees,
294-296
Contact, point of, of line of line conic,
112 , of second order between two
comes, 134, of third older between
two comes, 136
Conwell, G M , 204
Coordinates, nonhomogeneous, of points
on line, 152 , homogeneous, of points
on line, 163, nonhomogeneous, of
points in plane, 169, nonhomogene-
ous, of lines in plane, 170 , homogene-
ous, of points and lines in plane, 174,
in a bundle, 179, Ex 3 , of quadran-
gle-quadrilateral configuration, 181,
Ex. 2, nonhomogeneous, in space,
190 , homogeneous, m space, 194 ,
Pluckei's line, 327 , Klein's line, 333
Coplanar, 24
Copunctal, 16
Correlation, between two-dimensional
forms, 262, 263, induced, 262, be-
tween two-dimensional foims deter-
mined by four pairs of homologous
elements, 264, which interchanges
vertices and sides of tuangle is polar-
ity, 264 , between two planes, analytic
representation of, 266, 267, repie-
sented by ternaiy bilinear form, 267
represented by matuces, 270 , doubk
paiis of a, 278-281
Conelations and duality, 268
Correspondence, as a logical term, 5
perspective, 12 , (1, 1) of two figures
85, geneial theoiy of, 64-66; iden
tical, 65, inverse of, 65, period of
66, periodic or cyclic, 66, involutoric
or reflexive, 66 , peispective betweei
two planes, 71 , quadratic, 139, Exs
22, 24, 293, Ex. 1
Correspondences, resultant or produc
of two, 65, associative law for, 66
commutative, 66 , groups of, 67, leav
ing a figure invariant form a group, 6i
INDEX
337
Corresponding oloinouta, 85, doubly,
102
Covariant, 257 , example of, 258
Cremona, L., 187, 188
Gross latio, 15!), ot haimomc set, 150,
101 , clutuuticm of, 100 , expression foi,
100 , 111 homogeneous coimhnates,
105, theorems on, 107, 108, Exs ,
characteristic, ot piojoetivily, 205,
characteristic, of involution, 200, <is
an invariant, of two quadratic binary
forms, 254, Ex, 1 , of four complexes,
882
Crohs ratios, tlio BIX, defined by four ole-
monts, 101
Curvn, ot third order, 217, Exs. 7, 8, ;
algebraic, 250
Cyclic correspondence, 0(5
Darboux, U., 05
Degenerate conies, 120
Degenerate rogulus, 811
Degree of geometric problem, 280
Derivative, 255
Dusargues, configuration of, 40, 61 ; the-
orem on perspective triangles, 41,
180 ; theorem on conies, 127, 128
Dascartos, tt., 285
Diagonal point, (line), of complete quad-
rangh ("'.I' 1 lit *,.^ 44; of eom-
plote i -*! if--. ; piano, 44
Diagonal triangle ot quadrangle (quad-
rilateral), 44
Dickson, L. E., 00
Difference of two points, 148
Differential operators, 250
Dimensions, spaou of threw, 20 ; Bpaco of
n, 80 ; assumption!) for space of n, 88 ;
space of live, 881
Directrices, of a regulus, 200 j of a con-
gruence, 816 ; of a special complex,
324
Distributive law for multiplication with
respect to addition, 147
Division of points, 140
Domain of rationality, 288
Double element (point, line, plane) of
correspondence, 08
Double pairs of a correlation, 07
Double points, of a productivity on a
line satisfy a quadratic equation, 150 ;
of pro j activity on a line, homogeneous
coordinates of, 104; of projectivity
always exist in extended space, 242 j
of projectivity on a line, construction
of, 246 ; of involution determined by
covariant, 258 ; and lines of collinea-
tion in plane, 271, 295
Double ratio, 159
Doubly parabolic point, 274
Duality, in three-space, 28, in plane,
20; at a point, 20, in four-space, 29,
Ex. ; a consequence of existence of
correlations, 208
Edge of it-point or ri-plano, 30, 37
Elation, in plane, 72 , in space, 75
Element, undefined, 1 , of a figure, 1 ,
fundamental, 1, ideal, 7, bimplo, of
space, 84; Invariant, or double, or
fixed, 08 , lineal, 107
Eleven-point, plane section of, 58, Ex. 15
Enriquos, F., 50, 280
Equation, of hnu (point), 174, of come,
185,245; oi plane (point), 103, 198,
redui'ible, irreducible, 280; quadiatic,
has roots in extended space, 242
"Equivalent number systems, 150
Extonded space, 2 12, 255
Extension, assumptions of, 18, 24
Fact! of n-point or it-piano, 80, 37
Format, P., 285
Field, 140 , points on a line form a, 151 ,
finite, modular, 201, extended, m
which any polynomial is reducible, 200
Figure, 34
Fine, II. B., 255, 200, 201, 289
Finite spaces, 201
Five-point, piano section of, in space,
3D ; in space may be translormod into
any other by projootivo collineatiou,
77 , diagonal points, linos, and planes
of, in spaee, 204, Exs 16, 17, 18;
simple, in space determines linear
congruence, 810
Five-points, perspective, in four-space,
54, Ex. 25
Fixed element, of correspondence, 08
Flat pencil, 55
Forms. *"n>' j '"p .;/.<" e j ' IP of one, two,
and i.i- i i -i- ( - ">. one-dimen-
sional, of second degree, 109; linear
binary, 251 ; quadratic binary, 252 ,
of nth degree, 254; polar forms, 250 ;
ternary lulinoar, loproaents correla-
tion in piano, 207
Four-space, 25, Ex. 4
Frame of roi'cmmce, 174
Fundamental elements, 1
Fundamental points of a scale, 141, 281
Fundamental propositions, 1
Fundamental theorem of projectivity,
94-97, 218, 204
General point, 120
Geometry, object of, Ij starting point
of, 1 ; distinction between projjective
and metric, 12 ; Unite, 201 ; associated
with a group, 259
Gergonne, J. D., 29, 128
Grade, geometric forms of first second.
third, 55
Group, 66 ; of correspondences, 67 5 gen*
eral projective, on line, 68, 209;
338
INDEX
examples of, 69, 70 , commutative, 70 ,
general projective, in plane, 268
HO, assumption, 45 , r61e of, 81, 261
Harmonic conjugate, 80
Harmonic homology, 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, msciibed in
three concurient lines, 250, Ex 5,
simple, inscribed m conic, 110, 111
Hexagram, of Pascal (hexagramma mys-
ticum), 138, Exs 19-21, 304, Ex 16
Hilbeit, D , 3, 95, 148
Holgate, T. F , 119, 125, 139
Homogeneous coordinates in plane,
174
Homogeneous coordinates, m space, 11,
194 , on line, 163 , geometrical signifi-
cance of, 165
Homogeneous forms, 254
Homologous elements, 35
Homology, in plane, 72 , in space, 75 ,
axis and center of, 104, harmonic,
223, 275 , canonical f oim of, in plane,
274, 275
Hyperosculate, applied to two comes, 136
Ideal elements, 7
Ideal points, 8
Identical coirespondence, 65
Identical matrix, 157, 269
Identity (correspondence), 65, element
of group, 67
Improper elements, 239, 241, 242, 255
Improper transformation, 242
Impropeily proactive, 97
Independence, of assumptions, 6 , neces-
sary for distinction between assump-
tion and theorem, 7
Index, of subgroup, 271 ; of group of col-
hneations in general projective group
in plane, 271
Induced correlation in planar field, 262
Infinity, points, lines, and planes at, 8
Inscribed and circumscubed 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 bmaiy form of nth
degree, 257
Invariant element, 68
Invariant figure, under a correspond-
ence, 67 , of collineation is self -dual,
272
Invanant subgroup, 211
Invariant triangle of collineation, rela-
tion between projectivities on, 274,
276, Ex 5
Inverse, of a coriespondence, 65, of
element in gioup, 67 , of piojectivity
is a piojectivity, 68, of piojectivity,
analytic expiession for, 157
Inverse opeiations (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 latio of, 206 , on conic, 222-
230 , belonging to a proj activity, 226 ,
double points of, in extended space,
242 , condition foi, 254, Ex 2 , dou-
ble points of, 1 * determined by covan-
ant, 258 , complexes in, 333
Involutions, any projectivity is product
of two, 223 , haimonic, 224 , pencil
of, 225, two, have pair in common,
243, two, on distinct lines are per-
spective, 243
Involutoric coriespondence, 66
Irieducible equation, 239
Isomorphism, 6, between number sys-
tems, 150 , simple, 220
Jackson, D . 282
Join, 16
Kantoi, S , 250
Klein, F , 95, 333, 334
Ladd, C , 138
Lage, Geometne dei, 14
Lennes, N J , 24
Lmdemann, F , 289
Line, at infinity, 8 , as undefined class
of points, 15 , and plane on the same
three-space inteisect, 22 , equation of,
174, and conic, inteisection of, 240,
246
Line conic, 109
Line coordinates, in plane, 171 ; in space,
327, 833
Lineal element, 107
Linear binary forms, 251 , invariant of,
251
Linear dependence, of points, 30, of
lines, 311
Linear fractional tiansformation, 152
Linear net, 84
Linear operations, 236
Linear transformations, m plane, 187 ,
in space, 199
Lines, two, in same plane intersect,
18
Luroth, J , 95
INDEX
339
Maclaunn, C., 110
MacNoish, II E., 46
Mathematical science, 2
Matrices, product of, 156, 268 , determi-
nant ot pioduct of two, 269
Matrix, as symbol for configuration, 88 ,
definition, 15(5, used to denote pro-
jectivity, 156, identical, 157, 269,
oharaotoiiBtic equation of, 105, 272;
conjugate, transposed, adjoint, 209,
as operator, 270
Memeolmms, 120
Metric geometry, 12
Midpoint oi pair of points, 280, Ex. 6
Mobius tetrahodra, 105, Ex. 6; 826,
Ex.
Multiplication of points, 145, 281 , the-
oic'ins on, 145-J48; commutative law
of, is equivalent to* Aasmuption P,
148, other definitions of, 107, Exs.
8,4
n-line, complete or simple, 87, 88, in-
scribed in conic, 188, Ex. 12
n-plane, complete m space, 87 ; on point,
88 , simple in space, 87
n-point, complete, in space, 86 ; complete,
m a piano, 87, simple, iu space, 37,
simple, in a piano, 37; plane section of,
in space, 58, Exs. 18j 16 , 54, Ex. 18;
m-ftpace section of, in (n + 1) -apace,
54, Ex. Ifl , section by three-space of,
in four-space, 54, Ex. 21 ; inscribed
m came, 119, Ex 5; 250, Ex. 7
71-pomts, 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, 68, Ex. 7 ; two perspective, in
(n 1) -space, theorem on, 64, Ex.
26; mutually inscribed and circum-
scribed, 250, Ex.
Net of rationality, on line (linear net),
84; theorems on, 85; in plane, 86;
theorems on, 87, 88, Exs. 92, 98; in
space, 89 ; theorems on, 89-92, Exs. 02,
98 , in plane (space) left invariant by
perspective collineation, 08, Exs. 0,
10; in space is properly projeotive,
97; coordinates in, 162
Newson, H. B., 274
Nonhomogeneous coordinates, on a line,
152 ; in plane, 160; in space, 100
Null system, 824
Number system, 140
On, 7, 8, IS
Operation, one-valued, commutative, as-
sociative, 141; geometric, 2$0; linear,
280
Operator, differential, 2$0; represented
by matrix, 270 > pmar ? 284
Opposite sides of complete quadrangle,
44
Opposite veitex and side of simple
n-pomt, 37
Opposite veitices, of complete quadnlat-
eralj 44 , of simple n-point, 87
Oppositely placed quadi angles, 50
Order, 60
Ordinal e, 170
Origin of coordinates, 169
Osculate, applied to two conies, 184
Padoa, A., 3
Papperite, E , 800
Pappus, configuration of, 98, 99, 100,
126, 148
Parabolic congruence, 815
Parabolic point of collineation in plane,
274
Parabolic projoctivities, any two, are
conjugate, 209
Parabolic projoetlvity, 101 , charac-
teristic cross ratio of, 206; analytic
expression for, 207 ; chaiactenstic con-
stants, 207 ; gives II( MA', AA"), 207
Parametric representation, of points
(linos) of pencil, 182 ; of conic, 284; of
roguluH, congruence, complex, 880, 881
Pascal, B., 36, 1)0, 111-110, 128, 126,
127, 188, 189
Pencil, of points, planes, lines, 55; of
conies, 120-186, 287-208; of points
(lines), coordinates of, 181 , paiamet-
ric representation of, 182 ; base points
of, 182 ; of involutions, 225 ; of com-
plexes, 832
Period of correspondence, 06
Perspective collinoation, in plane, 71 ;
in space, 75 ; in plane donned when
center, axis, and one pair of homol-
ogous points are given, 72 , leaving JB a
( 8 ) invariant, 98, Exs. 9, 10
Perspective conic and pencil of lines
(points), 215
Perspective correspondence, 12, 18 ; be-
tween two planes, 71, 277, Ex. 20
Perspective iigures, from a point or
from a plane. 35 ; from a line, 80 ; if
A,B,C and A', B', C' on two coplanar
lines are perspective, the points (AB',
&A')> (AC', OA% and (B<7, CS') are
collinear, 52, Ex. 8
Perspective geometric forms, 56
Perspective n-lines, theorems on, 84,
Exs, 13, 14 ; five-points in four-space,
64, Ex. 25
Perspective (n + l)-points in n-spaoe,
64, Exs. SO, 26
3|erspectlve tetrahedra, 48
perspective triangles, theorems on, 41.
ft Exs. 0, 10, 11 5 64, Ex. 28; 84,
, EXS, 7, 10, 11 ; 246 ; sextuply, 246
340
IHDEX
Perspectivity, center of, plane of, axis
of , 36 , notation for, 57 , central and
axial, 57 , between conic and pencil
of lines (points), 215
Fieri, M , 95
Planar field, 55
Planar net, 86
Plane, at infinity, 8 , defined, 17 , deter-
mined uniquely by three noncollmear
points, or a point and line, or two in-
tersecting lines, 20 , and line on same
tluee-space are on common point,
22 , of perspectivity, 36, 75, of points,
55, of lines, 55, equation o, 193,
198
Plane figure, 34
Plane section, 34
Planes, two, on two points A, 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
thiee-space and not on a common
line aie 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 come, 109
Point figure, 34
Points, thiee, determine plane, 17, 20
Polar, with respect to triangle, 46,
equation of, 181, Ex 3 , with respect
to two lines. 52, Ess 3, 5 , 84, Exs. 7,
9 , with respect to tuangle, theorems
on, 54, Ex. 22 , 84, Exs 10, 11 , with
respect to n-lme, 84, Exs 13, 14 , with
respect to conic, 120-125, 284, 285
Polai forms, 256 , with lespect to set of
ji-pomts, 256, with lespect to regu-
lus, 302 , with respect to linear com-
plex, 324
Polar reciprocal figmes, 123
Polaiity, in planai field, 263, 279, 282,
283 , in space, 302 , 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, 824
Poncelet, J V , 29, 36, 58, 119, 123
Problem, degree of, 236, 238 , algebraic,
transcendental, 238, of second de-
giee, 243 , of projectivity, 250, Ex 14
Product, of two correspondences, 65,
of points, 145, 231
Project, a figure from a point, 86 , 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
Piojective conies, 212, 304
Project! ve correspondence or transfor-
mation, 13, 58 , general group on line,
68 , m 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 projectivities, 208
Projective space, 97
Projectivity, definition and notation for,
58, ABC-xA'B'Q', 59; ABCD-^-
BADG, 60 , in one-dimensional forms
is the lesultof twopeispectivities, 63 ,
if H (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 ot, 97 , necessaiy
and sufficient condition for MNAB -^-
MNA'E' is Q(MAB, NB'A*), 100,
necessary and sufficient condition
f 01 MMAB x MMA'B' is Q (MAB,
MB' A'), 101 , parabolic, 101 , ABCD
-rABDC implies H(AB, CD), 103,
nonhomogeneous analytic expiession
for, 154-157, 206 , homogeneous ana-
lytic expression for, 164, analytic
expiession for, between points of dif-
ferent lines, 167 , analytic expression
for, between pencils in plane, 183 ,
between two comes, 212-216, on
conic, 217-221, axis (centei) 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
Pioperlyprojective, 97, spatial net is, 97
Quadrangle, complete, 44, quadrangle-
quadrilateral configuiation, 46, sim-
ple, theorem on, 52, Ex. 6 , complete,
and quadnlateial, theoiem on, 53,
Ex 8, any complete, may be trans-
formed into any other by piojective
collineation, 74, opposite sides of,
meet line in pans of an involution,
103 , comes thiough vertices of, meet
line in 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 2, the sixth pair meets on
I, 47, perspective, theorem on, 53,
Ex. 12 , if two, have same diagonal
tuangle, their eight vertices are on
come, 137, Ex. 4
Quadrangular set, 49, 79 , of lines, 79 ; of
planes, 79
INDEX
341
Quadrangular section by transveihal of
quadrangular set, of linos is a quad-
rangulai set of points, 70 ; of elements
projoctive with quadrangular wL is
a quadrangular sot, 80; Q(MAH,
NB'A') is tho condition for MNA B-r
MNA'B' 100 , Q(MA B, Mli'A') is tho
condition for MMAB-^WMA'B', 101 ,
Q(ABC, A'Jl'C') iniplujK Q(A'irW,
ABC), 101, Q(ABV, A'W(J') is tlio
condition that ./I/I', />'/?', CO' are in in-
volution, 103 , Q(P* P.PQ, 7'/',Ai-)
is necessary and suthciont fin /', 4- /'
= /',+ 112; Q(/Vy>i, 2VV'.#)
is necessary and sufheiont for P x P v
= /*, M5
Quadrangularly related, 8(5
Quadratic binary ionn, 252; invariant
of, 2, r )2
Quadratic correspondence, 18!), 1C vs.
22, 24
Quartno spread in SB, 331
Quadno surface, 801 ; degenerate, .308 ;
determined by ninopomts, 311
Quadrilateral, complete, 44 , if two quad-
rilaterals correspond ho that live of tho
lines joining pairs of homologous vor-
tices pass through a point, P, tho lino
joining the .sixth pair of vertices will
also pass through P, 49
Quantic, 254
Quaternary forms, 258
Quotient of points, 140
Range, of points, 55; of conies, 128-180
Katio, of points, 140
Kational operations, 140
Rational space, 08
Rationality, not of, on lino, 84, 85 ; planar
net, of, 80-88 ; spatial not of, 80-05} 3
domain of, 288
Rationally related, 80, 80
Reducible equation, 280
Reflection, point-line, projoctivo, 228
Reflexive c-nvrfUTion-lc""" <KI
RcguluK, 1. 1 ,i" . i 3 I'-n ' i. - 208;
directrices of, 200 ; generators or
rulers of, 200; conjugate, 200; gen-
erated by projeoUve ranges or axial
pencils, 200 ; generated by projeetivo
comes, 804, 807 ; polar system of, 800 ;
picture of, 800 ; degenerate COHOS, 81 1 ,
of a congruence, 818
Related ilgnres, 85
Resultant, of two correspondences, 05;
equal, 65; of two projectivities is a
project! vity, 08
Reye, T,, 125, 180
Holm, K., 300
Salmon, G., 188
Sannia, A., 304
Scale, defined by Lliroo points, 141, 231 ,
on a conic, 28 % 1
Schrotoi, IT., 188, 281
Helmr, E , 05
ijuionuu, abstiact mathematical, 2, con-
crete application or representation of, 2
Scott, C A , 203
Suction, of iiguro by pUuiu, 34 , of piano
llgiue by line, 35, conic section, 100
Sogru, C., 280
Hiill'-conjugato Hiibgrou]i, 211
Sulf-eonjugato triauglo vvitli respect to
conn;, 128
Self-polar triangle with respect to come,
128
Set, synonymous with class, 2 , quadran-
gular, 40, 70 , ot elemental projoctivo
with quadrangular sot is quadrangu-
Icir, 80, harmonic, 80; theorems on
luuimmiu .sets, 81
Seven-point, piano section of, 53, Ex. 14
SuydowitK, If , 281
Sheaf of planes, 55
Side, of n-pomt, 87 ; false, of complete
s . quadrangles, 50
huupio cieuumr, of spaco, 30
Simple ?i-pomt, n-lino, n-plano, 37
Singly parabolic point, 274
Singular point and lino in nonhomoge-
neous eooulmates, 171
Six-point, plane section of, 54, Ex. ] 7 ;
in four-space suction by three-space,
54, Ex. 24
Skew lines, 24, projectivo pencils on,
are perspective, 105, Kx. Ji ; fom, are
met by two lines, 250, Ex. 18
Spat',0, nualytUi projectivo, 11 ; of three
dimensions, 20; theorem of duality
for, of three dimensions, 28, n-, 80;
assumption for, of n dimensions, 83 ;
as equivalent of three-space, 84;
properly or improperly projoctivo,
07; rational, 08; Unite, 201, 202;
extended, 242
Spatial not, 80; theorems on, 80-02;
IH properly protective, 07
von Htauilt, K. (3. C., 14, 05, 125, 141,
151, 158, 10, 28(J
Hteinor, J,, 100, 111, L25, 188, 180, 286,
280
SLoinor point and lino, 188, Ex. 10
SteiniU, K., 201
Sluriu, Oh., 120
Sturm, 11., 281, 250, 287
Subclass, 2
Subgroup, 08
Subtraction of points, 148
Sum of two points, 141, 281
Surface, algebraic, 260; quadrlo, 801
Sylvester, J. J., 828
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 foims, 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,
Mobius, 105, Ex. 6 , 326, Ex. 9
Tetiahedron, 37, four planes joining
line to vertices of, projective with
four points of inteiaection of line
with faces, 71, Ex 5
Three-space, 20, determined uniquely
by four points, by a plane and a point,
by two nomnteisectmg lines, 23 , the-
orem of duality foi, 28
Throw, definition of, 60 , algebra of, 141,
157, characteristic, of projectivity,
205
Throws, two, sum and product of, 158
Trace, 35
Transform, of one piojectivity by an-
other, 208 , of a group, 209
Transform, to, 58
Tiansfonnation, perspective, 13, pio-
jective, 13 , of one-dimensional foims,
58, of two- and thiee-dimensional
forms, 71
Tiansitive group, 70, 212, Ex 6
Triangle, 37; diagonal, of quadiangle
(quadrilateral), 44 , whose sides pass
through three given collmear points
and whose vertices are on three given
lines, 102, Ex 2, of leference of
system of homogeneous coordinates
in plane, 174, invariant, of collinea-
tion, i Ration between projectivities
on sides of, 274, 276, Ex. 5
Triangles, perspective, from point aie
perspective from line, 41 , axes of
perspectivity of three, in plane per-
spective from same point, are con-
current, 42, Ex. 6 , peispective, theo-
rems on, 53, Exs. 9, 10, 11 , 105, Ex.
9 ; 116, 247 ; mutually inscribed and
circumscribed, 99, perspective, fiom
two centers, 100, Exs. 1, 2, 8, from
four centers, 105, Ex 8 , 138, Ex 18 ,
fiom six centers, 246-248 , inscribed
and cucumscribed, 250, Ex. 4
Triple, point, of lines of a quadrangle,
49 , of points of a quadrangulai set, 49
Triple, triangle, of lines of a quadran-
gle, 49; of points of a quadrangulai
set, 49
Triple system, 3
Undefined elements in geometry, 1
United position, 15
Unproved propositions in geometry, 1
Variable, 58, 150
Veblen, O , 202
Veronese, G., 52, 53
Vertex, of ?i-pomts, 36, 37 , of n-planes,
37, of flat pencil, 55, of cone, 109,
false, of complete quadrangle, 44
Wiener, H , 65, 95, 230
Zeuthen, H. G,, 95
NOTES AND CORRECTIONS
Page 22. In the proof of Theorem 0, under the Loading 2, it is assumed that A
is not on a. But if -A. were on a, the theorem would, bo verified.
Page 34. In the definition of projection, after "P," in the last line on the page,
insert " , together with the lines and planes of F through P,".
Page 34. In the definition of section, after " TT," in the last line on the page, insert
" together with the lines and points of F on TT,".
Page 35. In the definition of section of a plane figure F by a line Z, the section
should include also all the points of F that are on Z
Page 44, line 6 from "bottom of page. The triple system referred to does not
of course, satisfy J 8 . It is not difficult, however, to build up a system of triples
which does satisfy all the assumptions A and JS. Such a finite S a would contain
15 "points" and 15 "planes" (of which the given triple system is one) and 35
"lines" (triples). See Ex. 8, p. 25, and Ex. 15, p. 203.
Page 47, Theorem 3. Add the restriction that the lino I must not contain a
vertex of either quadrangle.
Page 49 In the definition, of quadrangular set, after "a lino Z" insert " not
containing a vertex of the quadrangle,".
Page 62, Ex. 1. The latter part should road- "... of an edge [joining two
vertices of the five-point with the face containing the other three vertices ? "
Page 53, Exs. 14, 15, 10. The term circumscribed may be explicitly defined as
follows : A simple n-point is said to bo circumscribed to another simple n-point
if there is a one-to-one reciprocal correspondence between the lines of the first
n-pomt and the points of the second, such that each lino passes through its corre-
sponding point. The second n-point is then said to be inscribed m the first.
Page 53, Ex. 16. The theorem as stated is inaccurate. If w is the smallest
exponent for -which 2"*sl, mod. w, the vertices of the plane section may be
VL i T MM 1
divided into - simple n-points, which fall into cycles of m n-points
2 2 771
each, such that the w^points of each cycle circumscribe each other cyclically.
Thus, when n = 17, there are two cycles of 4 n-points, the n-points of each cycle
circumscribing each other cyclically.
Page 85, Theorem 9. If the quadrangular set contains one or two diagonal
points of the determining quadrangle, these diagonal points must be among the
five or four given points.
Page 88, Theorem 12, To complete the proof of this theorem the perspectivity
mentioned must be used in both directions i.e. it also makes the points of Bj or
E a perspective with the points of JB a on I.
Page 99, Theorem 22. See note to p. 58, Exs. 14, 15, 16.
Page 108, Theorem 29. tinder Type III, the proviso should be added that the
line PQ is not on the center of F and the point pg is not on the axis of F.
Page 119, Ex. 6. The latter part of this exercise requires a quadratic construc-
tion. Bee Cha$>. IX.
Page 137, Ex. 7 (Miscellaneous Exercises). The two points must not be collinear
With- a vetted ; oy, if eolUnear with a vertex, they must be harmonic with respect
to the vertex and the opposite side.
848
* ;
T *
> f r
If
344 NOTES AND CORRECTIONS
Page 165, last paragraph. The point ( 1, 1) forms an exception in the definition
of homogeneous coordinates subject to the condition x t + x a = 1. An exceptional
point (or points) will always exist if homogeneous coordinates aie subjected to a
nonhomogeneous condition
Page 168, Ex 10. The points A, B, G t D must be distinct
Page 182, bottom of page We assume that the center of the pencil of lines is
not on the axis of the pencil of points (cf . the footnote on p. 183)
Page 186 While the second sentence of Theorem 7 is literally correct, it may
easily be misunderstood If the left-hand member of the equation of one of the
lines m = 0, n = 0, or p = be multiplied by a constant p, the value of k may
be changed without changing the conic In fact, by choosing p properly, k may be
given an arbitrary value (5* 0) for any conic.
As pointed out in the review of this book by H. Beck, Archiv der Mathematik,
Vol. XVIII (1911), p. 85, the equation of the conic may be written as follows
Let (c^, a 2 , a s ) be an arbitrary point in the plane of the conic, and let
m x = mfa + m 2 a> 2 + m a x 3 ,
n x =
Px J
then the equation of the conic may be written
Jc 2 m a n a p% k^mxUx =
When the equation is written m this foim, there is one and only one conic for
Tf
every value of the ratio -i
k z
Page 301. The fiist sentence is not correct under om oiiginal definition of section
by a plane. We have accordingly changed this definition (cf note to p. 34)
Page 301. In the sentence before Theorem 7 the tangent hues referred to are
not lines of the quadnc surface
Page 303, Ex 5 The tangent line must not be a line of the surface.
Page 303, Ex. 7. The line must not be a tangent line.
Page 304. Theorem 11 should read " . . form a legulus or a cone of lines, pro-
vided . . ". In case the collineation between the planes of the comes leaves every
point of I invariant, the lines joining corresponding points of the two comes form
a cone of lines. In this case A = A and B = 2?, and the lines a and b intersect
Page 306, line 7 After "sections," insert ", unless a and 6 intersect, in which
case they generate a cone of lines " (cf . note to p 304).
Page 308, proof of Corollary 2. Let A\ be the projection on a of 5 2 from the
point Jf A\ might ha\e double contact with A z at E and JR', or might have con-
tact of the second order at R or H' However, if C a is not degeneiate, it is possible
to choose M for which neither of these happens For if all conies obtained from
[M ] had either of the above properties, they would form a pencil of comes of
which A 2 is one There would then exist a point M for which A\ and A* would
coincide C 2 would m this case have to contain three collmear points and would
then be degenerate.
Page 310, paragraph beginning "Now if nine points . . ". It is obvious that
no line of intersection of two of the planes a, )3, 7 will contain one of the nine
points, no matter how the notation is assigned.
Page 315, line 12 from bottom of page. Neither^ nor ir z must contain a directrix.
Page 319, Ex 2. If the two involutions have double points, the points on the
lines joining the double points are to be excepted in the second sentence.
NOTES AND CORKECTIONS 345
Pages 320, 321. In the proof of Theorem 20 the possibility that tliioe of the
vertices of the simple pentagon may be collmear is overlooked. Theiefoic the
third sentence of the last paragraph of page 320 and the third bontence of
page 321 are incorrect It is not haul to restate the proof correctly, as all the
facts needed are grven in the text, but thus restatement requires, several verbal
changes and rb therefore left as an excrci.se to the reader
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