What's wrong with Euclid Book V
Bulletin of the London Mathematical Society
, Volume 40 (1) – Feb 1, 2008
Abstract
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.
