Modern vs Euclid's definition of equivalent ratios

[logicgate]

TL;DR Summary

I'm trying to understand the relationship between Euclid's definition of equivalent ratios and our modern understanding of it.

Euclid defines equivalent ratios as the following : "Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order."
I can understand Euclid's definition of equivalent ratios but what I don't understand is how it relates to our current definition of equivalent ratios which states that two ratios a/b and c/d are equivalent if and only if their cross products are equal.

 

[FactChecker]

Two consecutive "equimultiplies", first by $b\ne 0$, then by $d\ne 0$.
$a/b=c/d \iff a=bc/d \iff da=bc$