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Breakthroughs in Mathematics by Peter Wolff

By Peter Wolff

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But geometry, when conducted as a science, does not rely on intuition. A geometer would refuse to believe a statement of the THE BEGINNINGS OF GEOMETRY 43 kind made above until it had been proved. 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.

The answer is quite simple: he did not have to. A postulate, after all, is a sign of a sort of weakness. It constitutes a demand on the part of the geometer that something be granted him-either that something is true (as in Postulate 4) or that something can be done (as in Postulates l-3). If we do not grant the geometer’s postulates, he cannot force us to do so; on the other hand, we cannot then expect him to prove his geometrical propositions, either. The more postulates a geometer makes, the less surprising it becomes that he can prove many and complicated propositions.

The culmination of this group is Proposition 26, another congruency proposition which shows that two triangles are congruent if one side and two angles of one triangle are equal to the corresponding side and angles of the other triangle. Now it is time to look at Euclid’s work in some detail. We begin with the Definitions. It is quite easy to understand what definitions are and why they must precede the remainder of Euclid’s work. , he must tell us what these things are; otherwise we should know neither what he is talking about nor whether he is correct in his assertions.

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