>>12400561I did not say that E must be an integer. It is not a real number, either. It is a magnitude, which might be a length, volume, time, or weight.
The proposition says "If there are any number of magnitudes whatsoever (which are) equal multiples, respectively, of some (other) magnitudes ..."
Multiples was defined as follows: "The greater is a multiple of the less when it is measured by the less." "Measured" in this context means that I can combine the lesser magnitude some natural number of times to equal the greater multitude, the way one might measure a distance with a rod or a volume with a measuring cup. So a "multiple" as it is being used means a magnitude multiplied by a natural number.
If a "multiple" meant multiplying the magnitude by a real number, then none of the proofs in Book V would make sense.
In fact, the whole point of Book V is to report on the work of Eudoxus, who had come up with the first theory of proportions that worked for incommensurable ratios, so it would make no sense to assume at the outset that the reader understood what multiplying a magnitude by a real number meant.
