By A. Heyting, N. G. De Bruijn, J. De Groot, A. C. Zaanen

Bibliotheca Mathematica: a chain of Monographs on natural and utilized arithmetic, quantity V: Axiomatic Projective Geometry, moment variation specializes in the foundations, operations, and theorems in axiomatic projective geometry, together with set concept, prevalence propositions, collineations, axioms, and coordinates. The book first elaborates at the axiomatic technique, notions from set concept and algebra, analytic projective geometry, and prevalence propositions and coordinates within the airplane. Discussions concentrate on ternary fields connected to a given projective aircraft, homogeneous coordinates, ternary box and axiom procedure, projectivities among traces, Desargues' proposition, and collineations. The publication takes a glance at occurrence propositions and coordinates in area. themes contain coordinates of some extent, equation of a aircraft, geometry over a given department ring, trivial axioms and propositions, 16 issues proposition, and homogeneous coordinates. The textual content examines the basic proposition of projective geometry and order, together with cyclic order of the projective line, order and coordinates, geometry over an ordered ternary box, cyclically ordered units, and basic proposition. The manuscript is a important resource of knowledge for mathematicians and researchers drawn to axiomatic projective geometry.

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In other wordsili A1A2A3A^ÌSSL quadrangle as in the definition above, and if B1B2 B3 B4 is a second quadrangle suchthat B1B2nB3B4=P9 B1BAn B2B3 = P'9 In B1BZ= Q9 then In B2Bé = Q'. PROOF. First of all, we prove Th. 2 in two special cases, in which a vertex of the first quadrangle coincides with the cor responding vertex of the second. Case a). The coinciding vertices are joined to Ç, dD\ can be applied to the triangles P'A2B2 and P' € A 4 B 4 , P € A 2 B2, and the points of intersection Al9 A2P' n AAP = A39 B2P' n B^P = B3 are follows that PP\ A2A^ and B2BA are concurrent; passes through Q'.

Obviously S3' is a proof of Θ' from V2, Th. 1, V i a and Th. 2. B u t Th. 1 and Th. 2 can in their turn be derived from V I , V2, V3, so t h a t Θ' can be derived from VI, V2, V3. 3. If in a theorem of $ we interchange the words "point" and "line", we obtain again a theorem of 5β. Two theorems which change into each other if we interchange "point" and "line", are called dual. 3 expresses t h a t the duality principle is valid for iß. *) If we speak of two points, this does not mean that the points must be dif ferent.

2 E x e r c i s e . Show by a counterexample t h a t the assertion in D n need not be true if Ax = A2 = Az. Dual of D e s a r g u e s ' Proposition {dDn). Let two t r i l a t e r a l αχα2α3 and b1b2b3 be given, such t h a t corresponding sides as well as corresponding vertices are different. If corresponding sides intersect in points which are incident with a line I, then the lines connecting corresponding vertices are incident with a point 0 . I t is clear t h a t dDu is also a converse of Du.