Was [HM] Does Euclid recognize a ratio between equal numbers or equal magnitudes?
Euclid's notion of a proportionality between four numbers, and thus his notion of ratio, is fundamentally different for magnitudes in Book V than it is for numbers in Book VII. The former is based on comparing "multiplying up", i.e., products agreeing, whereas the latter is based on "cancelling down", which is what is needed for studying prime factorization. As part of his development in Book VII, he essentially "proves" in VII.19 that these two notions agree for numbers; this is the key to uniqueness of prime factorization. But there is a very subtle and fundamental flaw in his development in Book VII, which has gone largely unnoticed, due to the intricacy of the development; and this flaw essentially assumes what he is trying to prove about uniqueness of prime factorization. I have an article about this appearing in the March 2006 American Mathematical Monthly, intended to flummox the reader a couple of times; it is titled "Did Euclid need the Euclidean algorithm to prove unique factorization", and is available on my web site, below. It is extraordinary that such a serious glitch exists in Euclid's development, and understanding it sheds some interesting light on several basic properties of the natural numbers, and their occurrence in more general monoids, especially in relation to the role of the Euclidean algorithm with which Euclid begins Book VII.
Best wishes, David Pengelley (email@example.com) Mathematics, New Mexico State University, Las Cruces, NM 88003 USA Tel: 505-646-3901=dept., 505-646-2723=my office; Fax: 505-646-1064 http://math.nmsu.edu/~davidp