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[HM] Euclid on ratios between numbers versus magnitudes
Posted:
Jan 13, 2006 4:47 PM
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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 (davidp@nmsu.edu)
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
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