Peter Wolff's Breakthroughs in Mathematics PDF
By Peter Wolff
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There can be no quarreling with the result. We may wonder, however, about the legitimacy of Euclid’s method of proof. HOW valid is the method of superimposition as geometrical proof? The reader may recall that in connection with Proposition 2 we pointed out that geometrical entities like points and Iines are not physical things and that they cannot simply be picked up and moved about in space. Here, however, Euclid does this very thing. If it is legitimate here, why wasn’t it legitimate in Proposition 21 If Euclid had allowed himself that method earlier, the whole cumbersome method of construction in Proposition 2 could have been eliminated.
Postulate 2 might also be abandoned and the following substituted: No line can be indefinitely extended; ail lines are tite. Strange as these postulates may seem, they would serve (with certain exceptions) for the geometry that can be studied on the surface of a sphere. ” Any two points on a sphere (except end points of a diameter) can THE BEGINNINGS OF GEOMETRY 49 be joined by a unique great circle (a great circle is one whose center is at the center of the sphere); but all great circles are finite in length and ah are equally long.
This, together with Common Notion 1, is sullicient to show that all three sides of the triangle are of the same length. A purist could raise some objections to Euclid’s procedure. For instance, how do we know that the two circles, one with the center at A, and the other with the center at B, meet at all? And if they do meet, is Euclid correct in assuming, as he obviously does, that they meet in a point? This latter fact could probably be proved from the definition of “line” as “breadthless length”; but Euclid certainly does not do it.