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analytic assumptions axis called Chap collinear complete quadrangle configuration congruence conies of Type conjugate points construction contains coplanar Corollary correlation correspondence 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 let the lines line at infinity line conic line joining linear linearly dependent lines meeting m-class n-point number system one-dimensional form pairs of homologous pairs of points Pascal's theorem pencil of lines pencil of points perspective planar point conic point of intersection points and lines points of contact polar projective collineation projective geometry projective transformation Proof propositions quadrangular set quadrilateral rationality regard regulus relation represented respectively satisfied self-polar sides skew lines symbol tetrahedron Theorem 13 three points three-space tion uv u2 vertex vertices
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 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,...