Is equivalence sufficient for the equivalence relation proportion?

Question

In Euclid Elements Book 5 Definition 5 it defines an equivalence relation for proportions. $A:B = C:D$ if when $mA<=>nB \Rightarrow mC<=>nD$ where $m,n \in \mathbb{N}$ and $<=>$ are three statements corresponding to being less than, equal to, or greater than, e.g. $mA<nB \Rightarrow mC<nD$.

The question is: isn't $mA=nB \Rightarrow mC=nD$ sufficient in expressing the sameness of proportions? By increasing or decreasing the gains $m,n$ I can get the other two inequality conditions easily.

Example: to determine if $2:4=6:12$ I find an $m,n$ such that $2m=4n$ and $6m=12n$. $m=2, n=1$ works and I can easily prove the other two inequality conditions so the two proportions express the same thing.

To rephrase the question, why not define the equivalence relation with just the equality and save a lot of work when demonstrating sameness between two proportions?

Note: this is trivial with fractions but we're not allowed to use fractions. Is there something I am then missing when we are restricted to proportions?

Answer 1

See Commentary to Book V:

Given a proportion that says a ratio of lines equals a ratio of numbers, for instance, $A : B = 8 : 5$, we have two interpretations. One is that there is a shorter line $CA = 8C$ while $B = 5C$. This interpretation is the definition of proportion that appears in Book VII. A second interpretation is that $5 A = 8 B$. Either interpretation will do if one of the ratios is a ratio of numbers, and if $A : B$ equals a ratio of numbers that A and B are commensurable, that is, both are measured by a common measure.

Many straight lines, however, are not commensurable. If A is the side of a square and B its diagonal, then A and B are not commensurable; the ratio $A : B$ is not the ratio of numbers. This fact seems to have been discovered by the Pythagoreans, perhaps Hippasus of Metapontum (ca.530 – ca.450 BC), some time before 400 B.C.E., a hundred years before Euclid’s Elements.

The difficulty is one of foundations: what is an adequate definition of proportion that includes the incommensurable case? The solution is that in V.Def.5. That definition, and the whole theory of ratio and proportion in Book V, are attributed to Eudoxus of Cnidus (died ca.355 BC).