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