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Topic: [HM] Does Euclid recognize a ratio between equal numbers or equal magnitudes?
Replies: 13   Last Post: Feb 8, 2006 5:53 PM

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 Robert (Bob) Eldon Taylor Posts: 101 Registered: 12/3/04
[HM] Does Euclid recognize a ratio between equal numbers or equal magnitudes?
Posted: Jan 7, 2006 10:55 AM

Dear Julio and All,

Euclid does not explicitly define ratios of numbers, but presumably the
definition would be similar to that for ratios of magnitudes, which are
essentially components of proportions. For numbers he defines
proportions in book VII Definition 20:

Numbers are proportional when the first is the same multiple, or the
same part, or the same parts, of the second that the third is of the fourth.

Now of any two _unequal_ numbers one can be said to be part, parts or a
multiple of the other, but there seems to be no room here for equal
numbers. Thus he has already said (in defs 3, 4 and 5) "A number is a
part of a number, _the less of the greater_, when it measures the
greater; but parts when it does not measure it."
And "The _greater number_ is a multiple of the less when it is measured
by the less."

Thus Euclid does not seem to recognize
3 is to 3 as 2 is to 2
as a valid proportion, and therefore 2 to 2 as a valid ratio.

A careful reading of the definitions in Book V leads me to the same
conclusion for magnitudes.

Is this an oversight on the part of Euclid? Euclid occasionally ignores
his own rules. Does he do so in this case? Khayyam, in his commentary
on Euclid's Elements, does not recognize this restriction. Do other
commentators?

Regards,
Bob
Robert Eldon Taylor
philologos at mindspring dot com