What's wrong with Euclid Book V

Not having the real numbers, Euclid defined ratios abstractly in Book V for use in geometric theorems, but failed to define ratios of ratios, in effect blocking certain further developments of Greek mathematics. We introduce a new axiom that can be used instead of subtraction, not only to prove the...

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Bibliographic Details
Published in:The Bulletin of the London Mathematical Society 2008-02, Vol.40 (1), p.1-17
Main Author: Zeeman, E. C.
Format: Article
Language:English
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Summary:Not having the real numbers, Euclid defined ratios abstractly in Book V for use in geometric theorems, but failed to define ratios of ratios, in effect blocking certain further developments of Greek mathematics. We introduce a new axiom that can be used instead of subtraction, not only to prove the propositions of Book V but also to define ratios of ratios.
ISSN:0024-6093
1469-2120
DOI:10.1112/blms/bdm104