Drexel dragonThe Math ForumDonate to the Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.

Math Forum » Discussions » Inactive » Historia-Matematica

Topic: [HM] Euclid on ratios between numbers versus magnitudes
Replies: 4   Last Post: Apr 10, 2006 10:57 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
David Pengelley

Posts: 61
Registered: 12/3/04
[HM] Euclid on ratios between numbers versus magnitudes
Posted: Jan 13, 2006 4:47 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

Was [HM] Does Euclid recognize a ratio between equal numbers or equal

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

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum 1994-2015. All Rights Reserved.