## Projective Geometry, Volume 1Ginn, 1916 - Geometry, Projective |

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### Common terms and phrases

A₁ assumptions axis b₁ b₂ Brianchon's theorem called Chap collinear complete quadrangle configuration congruence conics of Type conjugate points construction contains coördinates COROLLARY correspondence cross ratio defined definition denoted Desargues configuration diagonal points directrices distinct points double point Dualize elements equation figure fixed point flat pencil follows given points harmonic conjugate harmonic set Hence homologous lines homologous points invariant involution let the lines line at infinity line conic line joining linear lines meeting n-point number system one-dimensional form pairs of homologous pairs of points parabolic Pascal's theorem pencil of lines pencil of points planar plane dual point conic point of contact point of intersection points and lines projective collineation projective geometry projective transformation Proof quadrangular set quadrilateral rationality regard regulus relation respectively satisfied self-polar sides skew lines symbol tangent tetrahedron Theorem 13 three points three-space triangle vertex vertices x₁

### 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 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, C, D...

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...

Page 3 - S is interpreted to mean the digits 0, 1, 2, 3, 4, 5, 6 and the ra-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 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 - 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 7 - C, and the m-classes to consist of the pairs AB, BC, CA, then it is clear that Assumptions I, II, III, IV, VI, VII are true of this class S, and therefore that any logical consequence of them is true with this interpretation. Assumption V, however, is false for this class, and cannot, therefore, be a logical consequence of the other assumptions. In like manner, other examples can be constructed to show that each of the Assumptions I-VII is independent of the remaining ones.

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 5 - 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 46 - Prf (fig. 18). We observe first that the diagonal triangle is perspective with each of the four triangles formed by a set of three of the vertices of the quadrangle, the center of perspectivity being in each case the fourth vertex. This gives rise to four axes of perspectivity (Theorem 1), one corresponding to each vertex of the quadrangle.* These four lines clearly form the sides of a complete quadrilateral whose diagonal triangle is Du, Dw Dl,.