PROJECTIVE GEOMETRY

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Contents

The threespace
20
The remaining assumptions of extension for a space of three dimensions
24
The principle of duality
26
The theorems of alignment for a space of n dimensions
29
CHAPTER II
34
The complete wpoint etc
36
Configurations
38
The Desargues configuration
39
Perspective tetrahedra
43
The quadranglequadrilateral configuration
44
The fundamental theorem on quadrangular sets
47
Additional remarks concerning the Desargues configuration
51
CHAPTER III
55
Perspectivity and projectivity
56
The projectivity of onedimensional primitive forms
59
SECTION pAGE 24 General theory of correspondence Symbolic treatment
64
vii
66
Groups of correspondences Invariant elements and figures
67
Group properties of projectivities
68
Projective transformations of twodimensional forms
71
Projective collineations of threedimensional forms
75
CHAPTER IV
79
Harmonic sets
80
Nets of rationality on a line
84
Nets of rationality in the plane
86
Nets of rationality in space
89
The fundamental theorem of projectivity
93
The configuration of Pappus Mutually inscribed and circumscribed tri angles
98
Construction of projectivities on onedimensional forms
100
Involutions
102
Axis and center of homology
103
Types of collineations in the plane
106
CHAPTER V
109
Tangents Points of contact
112
The tangents to a point conic form a line conic
116
The polar system of a conic
120
Degenerate conies
126
Desarguess theorem on conies
127
Pencils and ranges of conies Order of contact
128
CHAPTER VI
141
Multiplication of points
144
The commutative law for multiplication
148
The abstract concept of a number system Isomorphism
149
Nonhomogeneous coordinates
150
The analytic expression for a projectivity in a onedimensional primitive form
152
Von Staudts algebra of throws
157
SECTION PAGE 56 The cross ratio
159
Pencils of points and lines Projectivity
181
G6 The equation of a conic
185
Linear transformations in a plane
187
Collineations between two different planes
190
Homogeneous coordinates in space
194
Linear transformations in space
199
Finite spaces
201
CHAPTER VIII
205
Projective projectivities
208
Groups of projectivities on a line
209
Projective transformations between conies
212
Projectivities on a conic
217
Involutions
221
Involutions associated with a given projectivity
225
Harmonic transformations
230
Scale on a conic
231
Parametric representation of a conic
234
CHAPTER IX
236
The intersection of a given line with a given conic
240
Improper elements Proposition K2
241
Problems of the second degree
245
Invariants of linear and quadratic binary forms
251
Proposition Kn
255
SECTION PAGE 90 Invariants and covariants of binary forms
257
Ternary and quaternary forms and their invariants
258
Proof of Proposition Kn
260
PROJECTIVE TRANSFORMATIONS OF TWODIMENSIONAL FORMS 93 Correlations between twodimensional forms
262
Analytic representation of a correlation between two planes
266
General projective group Representation by matrices
268
Double points and double lines of a collineation in a plane
271
Double pairs of a correlation
278
Fundamental conic of a polarity in a plane
282
Poles and polars with respect to a conic Tangents
284
Various definitions6 of conies
285
Pairs of conies
287
Problems of the third and fourth degrees
294
CHAPTER XI
298
The polar system of a regulus
300
Projective conies
304
Linear dependence of lines
311
The linear congruence
312
The linear complex
319
The Pliicker line coordinates
327
Linear families of lines
329
Interpretation of line coordinates as point coordinates in S5
331
INDEX
335

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Page 16 - If A, B, C are points not all on the same line, and D and E (D ^ E) are points such that B, C, D are on a line and C, A, E are on a line, then there is a point F such that A, B, F are on a line and also D, E, F are on a line.
Page 95 - 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 assumption are immediate consequences of the assumption.
Page 5 - 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.
Page 86 - A. B, C, D, no three of which are collinear. This class of points is studied by an indirect method in the next section. 33. Nets of rationality in the plane. DEFINITION. A point is said to be rationally related to two noncollinear nets of rationality...
Page 4 - 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 AEF and BFG belong to the same m-class. No use has as yet been made of Assumption 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...
Page 4 - BCDEFGA DEFGABC 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 (!') is obtained from (1) by replacing 0 by A, 1 by B, 2 by C, etc. We can show, furthermore, that S can contain no other elements than A, B, C, D, E, F, G. For suppose there were another element, T. Then, by Assumption HI, 1 2] CATEGOBICALNESS 6 the m-classes TA and BFG would have an element in common.
Page 3 - S is interpreted to mean the digits 0,1, 2, 3, 4, 5, 6 and the w-classes to mean the columns in the following table: 0123456 (1) 1234560 3456012 This interpretation is a concrete representation of our assumptions. Every proposition derived from the assumptions must be true of this system of triples. Hence none of the assumptions can be logically inconsistent with the rest; otherwise contradictory statements would be true of this system of triples. Thus, in general, a set of assumptions is said to...
Page 2 - ... this latter procedure makes it possible to give a variety of interpretations to the undefined elements, and so to exhibit an identity of structure in different concrete settings. We shall now demonstrate six theorems, some of which may be regarded as trite consequences of our assumptions.
Page 3 - II. If A and B are distinct elements of S, there is not more than one m-class containing both A and B. III. Any two m-classes have at least one element of S in common. IV. There exists at least one m-class. V. Every m-class contains at least three elements of S. VI. All the elements of S do not "belong to the same m-class. VII. No m-class contains more than three elements of S. The reader will observe that in this set of assumptions we have just two undefined terms, viz., element of S and m-class,...
Page 266 - The proof of this theorem is completely analogous to the proof of Theorem 8, Chapter VII, and need not be repeated here.

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