Clairaut history of the parallel postulate.pptx

A Clairaut's history of the parallel postulate

1. Clairaut History of the parallel postulate

2. He was a leading French geometer. Like Wallis, he did not try to prove the parallel postulate in neutral geometry but replaced it in his 1741 text Eléments de géometrie with another axiom. Alexis Claude Clairaut (1713-1765)

3. CLAIRAUT’S AXIOM Rectangles exist.

4. Proof : If we assume the latter, then the existence of rectangles follows easily from Proposition 4.11 and Theorem 4.7. Conversely, assume Clairaut’s axiom. Then by Theorem 4.7, all triangles have angle sum 180°. And by introducing a diagonal, all convex quadrilaterals have angle sum 360°. Return to Proclus’ argument as illustrated in Figure 5.2. Let S be the foot of the perpendicular from Y to segment PQ. S is on the same side of m as Y and Q because segment SY is parallel to m (Corollary 1 to Theorem 4.1). Moreover, rectangle PXYS, which has three right angles, is now known to be a rectangle. You can easily prove (Exercise 4) that opposite sides of a rectangle are congruent, so PS = XY. By Aristotle’s axiom (Chapter 3), Y can be chosen on the given ray of n so that XY>PO. Then PS>PQ and P*Q*S. As above, Y is on the same side of/as S, hence on the opposite side of/ from P. Therefore/meets at some point between P and Y.

5. Thank You!