Download Algebraic Geometry Santa Cruz 1995: Summer Research by David R. Morrison, Janos Kolla Summer Research Institute on PDF

By David R. Morrison, Janos Kolla Summer Research Institute on Algebraic Geometry

ISBN-10: 0821808958

ISBN-13: 9780821808955

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Additional resources for Algebraic Geometry Santa Cruz 1995: Summer Research Institute on Algebraic Geometry, July 9-29, 1995, University of California, Santa Cruz (Proceedings of Symposia in Pure Mathematics) (Pt. 2)

Example text

Since, by hypothesis, AB:BD (= BD:BE) = BE:BG, the rectangles BZ, 8B, are similar; hence the diagonal 8Z passes through B. [iv] Draw diagonals BI, AG, intersecting at K, and join KD, KE (Fig. 5b). Since ABGI is a rectangle, the circle centered on K with radius KB will pass through all four of its vertices. Let EK meet this circle at M, and let its extension meet the circle at L. Since EBA, EML are secants to the circle drawn from E, BE'EA = ME'EL (Elem. III 36). Since, further, MK = KL, LE'EM + MK2 = EK2 (Elem.

10 Thus, neither the result nor, it appears, even the text of this appended lemma are original with Pappus. This passage thus provides remarkable insight into the level of result for which Pappus is at pains to take credit. Since the account by Eutocius is thus seen to be independent of Pappus, and since both accounts are in literal agreement throughout extended passages, each commentator must have made an exact transcription from the book of Nicomedes. We have already seen, in the comparisons ofHM and HP in the preceding chapter, an instance of Pappus' literal transcription from a source.

For another reason, even if we took AJ as a composite, Philoponus was hardly the one to produce such a synthesis of geometric sources. 30 His commentary on the passage of the Posterior Analytics that has provided our text of AJ actually turns on a misunderstanding of the mathematical issue, for Aristotle's passage is not about cube duplication at all. For when Aristotle asserts that it is not the task of geometry to prove that "two cubes are a cube;' he must refer to the separation of geometry from arithmetic, and the priority of the latter in the investigation of the named theorem-that the product of two cubic integers is itself a cubic integer.

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Algebraic Geometry Santa Cruz 1995: Summer Research Institute on Algebraic Geometry, July 9-29, 1995, University of California, Santa Cruz (Proceedings of Symposia in Pure Mathematics) (Pt. 2) by David R. Morrison, Janos Kolla Summer Research Institute on Algebraic Geometry


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