If A, B are any magnitudes (length, area, volume) we would like to be able to discuss the ratio between A and B. Now this ratio is not a number (in the classical sense), and saying that it's a real number is not taking into account the massive amount of theory needed to truly establish the real numbers (Dedekind cuts, completeness, etc). So what we can do is define when two ratios are equal. So, we don't "really" define what A:B is, but we should be able to tell when A:B = C:D
So how do we do that? Paul suggested defining it by A:B = C:D iff AD = BC. Now, what is AD? If A and D are lengths, then AD is an area. If A is a length and D is an area, then AD is a volume, but if A and D are areas, then AD is meaningless.
So this isn't a good definition.
Eudoxus' definition, given in Euclid (http://aleph0.clarku.edu/~djoyce/java/elements/bookV/defV5.html), defines it the following way:
A:B = C:D iff for any (natural) numbers n, m is is true that:
nA < mB iff nC < mD,
nA = mB iff nC = mD, and
nA > mB iff nC > mD.
Interestingly enough, this definition is equivalent to Dedekind cuts in the way we define real numbers (http://en.wikipedia.org/wiki/Dedekind_cuts).
This is all very nice, and Euclid uses this definition to prove many things, for example that the area of a circle is proportional to the area of the square on its radius (http://aleph0.clarku.edu/~djoyce/java/elements/bookXII/propXII2.html).
But Paul's definition seemed to be good, at least in the cases that all the magnitudes are lengths. So let's try to prove that the two definitions are equivalent in this case:
Question
Assume that A, B, C, D are lengths. Prove that A:B = C:D (in Eudoxus's definition) iff AD = BC.