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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.

Nor should such refusal be considered perverse; there are many known instances where the “obvious truth” turned out to be false. ) Instead to of intuition, the geometer relies on proof or demonstration convince himself of the truth of a geometrical proposition. This is a second, and a much more important, task of geometry. In his Elements, Euclid brings definition and proof, order and precision, to the entire geometrical area. Euclid is neither the first nor the greatest geometer who ever lived.

IO], and let BE be joined and produced in a straight line to F; let EF be made equal to BE II. II, and let AC be drawn through to G. [Post. 21. Then, since AE is equal to EC, and BE to EF, the two sides 32 BREAKTHROUGHS IN MATHEMATICS AE, EB are equal to the two sides CE, EF respectively; and the angle AEB is equal to the angle FEC, for they are vertical angles. [I. 151 Therefore the base AB is equal to the base FC, and the triangle ABE is equal to the triangle CFE, and the remaining angles are equal to the remaining angles respectively, namely those which the equal sides subtend; II.