By Yum-Tong Siu

ISBN-10: 3540052941

ISBN-13: 9783540052944

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**Extra resources for Gap-Sheaves and Extension of Coherent Analytic Subsheaves **

**Sample text**

As to proofs of the first part, the second proof is direct while the first proof is a proof by reductio. As to the second part, Gauss’s proof extended that of Euclid; the second, sixth and seventh proofs, and especially the ninth, exhibit various forms of simplification; and the third and fourth proofs introduce a concept (that of representing the greatest common divisor of two integers as a linear combination of them) that is foreign to all the others, while the fifth proof deliberately avoids using that concept.

2 Further consequences and extensions 35 Fig. 7 combined Fig. 11 Three scaled copies fitted together 7. 11. Readers may judge for themselves which, if any, of the seven proofs above is simplest or most perspicuous. 2 Further consequences and extensions Euclid’s Proposition VI,31 extended Proposition I,47 to ‘arbitrary’ similar figures described on the sides of a right triangle. In another direction, the Law of Cosines provides an extension of I,47 to arbitrary triangles. It includes the Pythagorean Theorem as a special case and also implies its converse.

The most egregious example is Loomis’s ‘algebraic’ proof 16, apparently taken over uncritically from Yanney’s and Calderhead’s proof X. 1 Several other fallacious proofs from those two earlier sources are also cited on Bogomolny’s web site. Loomis’s book is problematical on other grounds as well. The very idea, for example, of distinguishing proofs according to the diagrams used to represent them seems fatally flawed, both because the same diagram, interpreted differently, may be used to represent conceptually distinct arguments, and because, conversely, some arguments can be represented by more than one distinct diagram.

### Gap-Sheaves and Extension of Coherent Analytic Subsheaves by Yum-Tong Siu

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