Applicable geometry: global and local convexity by Heinrich W Guggenheimer

By Heinrich W Guggenheimer

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Using in essence the to prove the Fifth Postulate directly, following argument: Fisure Given two such a in two lines way 19 AP\ and A\B\ (Fig. 19) cut by the transversal AA\ sum of angles P\AA\ and AA\E\ is less than it is to be proved that AP\ and A\Bi meet if suffi- that the right angles, ciently produced. AB so Construct A% is that angle BAA\ is equal to angle B\A\A^ where AA\ produced through A\. Then AP\ will lie within a point on angle BAAi, since angle P\AA\ is less than angle BiAiA*. AP 29 AP3y , AP n so that angles P\AP^ P*AP^ .

9 A n ~\A n that line. Then the infinite strips BAA\B\ 9 . . , , B\AiA 2 Bz 9 , Bn -iA n -\A n Bn can be superposed and thus have equal areas, each to the area of the infinite strip BAA n B n divided by n. Since equal the infinite sector BAPn includes the infinite strip BAA n B n , it follows that the area of the sector BAP\ is greater than that of the strip THE FIRM POSTULATE i, 43 and therefore AP\ must intersect A\R\ if produced suffi- ciently far. The fallacy lies in treating infinite magnitudes as though they In the first place, the idea of congruence as used above for infinite areas has been slurred over and not even defined.

Lambert. 23. Germany, a little later, Johann Hemrich Lambert (1718-1777) came close to the discovery of Non-Euclidean Geometry. His investigations on the theory of parallels were stimulated by a disIt sertation by Georgius Simon Kliigel which appeared in 1763. first to express some doubt about the was the that Kliigel appears In also Fifth Postulate. possibility of proving the There is a striking resemblance between Saccheri's Eucltdes Vtndtcatus and Lambert's Theone der Parallelhmen^ which was written m Lambert chose for his funda- 1766, but appeared posthumously.

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