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 makes a speciality of the rules, operations, and theorems in axiomatic projective geometry, together with set conception, occurrence propositions, collineations, axioms, and coordinates. The ebook first elaborates at the axiomatic technique, notions from set thought and algebra, analytic projective geometry, and prevalence propositions and coordinates within the aircraft. Discussions specialize in ternary fields connected to a given projective aircraft, homogeneous coordinates, ternary box and axiom procedure, projectivities among strains, Desargues' proposition, and collineations. The e-book takes a glance at prevalence propositions and coordinates in area. subject matters comprise coordinates of some degree, equation of a airplane, 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 useful resource of information for mathematicians and researchers drawn to axiomatic projective geometry.

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Similarly, B3G3 contains P . I t remains to be shown t h a t B2G2 = B3G3. To prove this, we remark that, by the quadrangle B2G2OC39 H29 Q is harmonic with D29 F2; similarly H39 R is harmonic with D39 F3. Projecting the points D2F2QH2 from P on AXR9 we see by Th. 5 t h a t H2H3 contains P . B u t H2 is on B2G2 and H3 is on B3G3, which both contain P . I t follows t h a t B2G2 = B3G39 so t h a t S 2 ^ 3 contains P . Note the snake in the grass where we project the points of AXQ from P on ^ P . This is only possible if P is outside A1Q and outside A±R9 which we easily prove if P Φ Q and P Φ R.

The proof is left to the reader. Note that only axiom Vlb is needed ! 2. In *β: Λ contains at least four lines such that no three of them are incident with one and the same point. 26 INCIDENCE PROPOSITIONS IN THE PLANE Chap. 2 By V3 we can find four points A19 A2, A3, AA, such three of them are incident with one and the same line. ) 1 ). 2. PROOF. t h a t no These t h e four theorem A. Fig. 1. Suppose t h a t ΑλΑ2, ASA^9 A1AS were incident with a point P. ). I t follows from Th. 1. t h a t P = Ax.

Let P , X', F be the projections of P, P ' , Q from 5 on P X \ Let Z = S P n P ' F . The quadrangle SFX'Z shows that ZX' passes through Q'{=Q). Then the quadrangle SP'QZ shows that V is its own harmonic conjugate with respect to P , X'. The second part of the proof is similar. 5 show that the theory of harmonic pairs can be based entirely upon D9; Dyx, or even Z>10, is not needed. 54 INCIDENCE PROPOSITIONS IN THE PLANE «v &r Chap. 4. FIRST QUADRANGLE PROPOSITION, HARMONIC PAIRS 55 However, the next theorem shows t h a t D9 is not much weaker than D 1 0 .