Projective Geometry, Volume 1

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This seems to be Vol.1
Not Vol.2

Contents

The complete npoint 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 PAGK 24 General theory of correspondence Symbolic treatment
64
The notion of a group 66
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
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 conics
126
Desarguess theorem on conics
127
Pencils and ranges of conics Order of contact
128
Addition of points 49 Multiplication of points
144
The commutative law for multiplication 51 The inverse operations
148
The abstract concept of a number system Isomorphism
149
Nonhomogeneous coordinates
150
The analytic expression for a projectivity in a onedimensional priir l
152
Von Staudts algebra of throws
157
SECTION PAGE 56 The cross ratio
159
Coordinates in a net of rationality on a line
162
Homogeneous coordinates on a line
163
Projective correspondence between the points of two different lines
166
CHAPTER VII
169
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 K
254
Taylors theorem Polar forms
255
SECTION PAGE 90 Invariants and covariants of binary forms
257
Ternary and quaternary forms and their invariants
258
Proof of Proposition K
261
CHAPTER X
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 definitions of conies
285
Pairs of conics
289
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 Plucker line coordinates
327
Linear families of lines
329
Interpretation of line coordinates as point coordinates in S3
331
INDEX
335
Copyright

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Page 95 - If a projectivity leaves each of three distinct points of a line invariant, it leaves every point of the line invariant.
Page 228 - OlOjO ,8 is therefore self-polar. COROLLARY 1. Given any two involutions, there exists a third involution 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 involutions clearly satisfies the condition of the theorem for each of the latter. COROLLARY 2. Three involutions each of which is harmonic to the other two constitute, together with...
Page 4 - V there must be a third element in each of the m-classes AB, BC, CA, and by Assumption II these elements must be distinct from each other and from A, B, and C. Let the new elements be D, E, G, so that each of the triples ABD, BCE, CAG belongs to the same m-class. By Assumption III the m-classes AE and BG, which are distinct from 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...
Page 4 - VI contains at least seven elements. Now, making use of Assumption VII, we find that the m-classes thus far obtained contain only the elements mentioned. The m-classes CD and AEF have an element in common (by Assumption III) which cannot be A or E, and must therefore (by Assumption VII) be F. Similarly, ACG and the m-class DE have the element G in common. The seven elements A, B...
Page 3 - S and m-class, and one undefined relation, belonging to a class. The undefined terms, moreover, are entirely devoid of content except such as is implied in the assumptions. Now the first question to ask regarding a set of assumptions is: Are they logically consistent? In the example above, of a set of assumptions, the reader will find that the assumptions are all true statements, if the class S is interpreted to mean the digits 0, 1, 2, 3, 4, 5, 6 and the m-classes to mean the columns in the following...
Page 3 - S is interpreted to mean the digits 0, 1, 2, 3, 4, 5, 6 and the m-classes to mean the columns in the following table : 0123456 (1) 1234560 3456012 This interpretation is a concrete representation of our assumptions. Every proposition derived from the assumptions must be true of this system of triples. Hence none of the assumptions can be logically inconsistent with the rest ; otherwise contradictory statements would be true of this system of triples. Thus, in general, a set of assumptions is said...
Page 269 - The determinant of the product of two matrices (collineations) is equal to the product of the determinants of the two matrices (collineations).
Page 3 - 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 be consistent if a single concrete representation of the assumptions can be given.* Knowing our assumptions to be consistent, we may proceed to...
Page 27 - 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 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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