## Projective Geometry, Volume 1 |

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### Contents

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### Common terms and phrases

analytic assumptions axis called Chap collinear complete quadrangle configuration congruence conics of Type conies conjugate points construction contains coplanar Corollary correlation corresponding cross ratio defined definition denoted Desargues configuration diagonal points directrices distinct points double point Dualize elements equation figure fixed point flat pencil follows four points given points harmonic conjugate Hence homogeneous coordinates homologous lines homologous points invariant involution line at infinity line conic line joining linear linearly dependent lines meeting m-class meets the line n-point number system one-dimensional form pairs of homologous pairs of points Pascal's theorem pass pencil of lines pencil of points planar point conic point of intersection points and lines points of contact projective collineation projective geometry projective transformation Proof propositions quadrangular set quadrilateral rationality regard regulus relation represented respectively satisfied self-polar Show sides skew lines symbol tetrahedron Theorem 13 three points three-space tion vertex vertices xv x2

### Popular passages

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.