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Bulletin of the London Mathematical Society Advance Access originally published online on February 22, 2008
Bulletin of the London Mathematical Society 2008 40(1):1-17; doi:10.1112/blms/bdm104
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© 2008 London Mathematical Society

What's wrong with Euclid Book V

E. C. Zeeman

Hertford College
Oxford, OX1 3BW
United Kingdom
23 High Street
Woodstock, OX20 1TE
United Kingdom

Received 3 March 2007.

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.

2000 Mathematics Subject Classification 01A20.


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