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A Mathematician and His Mathematical Work: Selected Papers by Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

By Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

Those chosen papers of S.S. Chern speak about themes similar to fundamental geometry in Klein areas, a theorem on orientable surfaces in 4-dimensional house, and transgression in linked bundles

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Example text

3, let y D jBDj. Triangle ABC is similar both to triangle ABD and to triangle ACD (because corresponding angles are equal), so their corresponding sides are proportional. a y/. a y/ D a2 ay D b 2 . So a2 c 2 D b 2 . The proof above is based on the same diagram used to prove Proposition VI,31. But neither the statement of the Pythagorean Theorem in 3. (the form in which it is usually stated), nor the proof just given, makes any reference to areas; only length relationships are involved. Thus both the meaning of the proposition stated in 3.

7 is used to illustrate proposition II,4 of Euclid’s Elements. So the question again arises, could not Euclid have proved proposition I,47 more simply by using those diagrams? Heath thought that the only objection to that idea was that such dissection proofs had “no specifically Greek” character (Heath 1956, vol. I, p. 355). Knorr, however, considered that objection “unjust” (Knorr 1975, p. 178 and fn 18 thereto, pp. 204–5). 7 requires spatial translation of the constituent triangles, an operation not justified by Euclid’s axioms, also does not hold up to scrutiny, for two reasons.

Yanney and James A. Calderhead (Yanney and Calderhead 1896–9) that appeared in vols. 3 through 6 of the American Mathematical Monthly, each entitled “New and old proofs of the Pythagorean theorem”; the book (Loomis 1940) by Elisha S. Loomis, first published in 1927 and reprinted in 1968 by the National Council of Teachers of Mathematics; and the geometry web pages maintained by Alexander Bogomolny (Bogomolny 2012). The first of those references nominally presents 100 different proofs; the second, 367; and the third, 96; but the qualifier ‘nominally’ is important for several reasons.

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