Elements of the History of Mathematics by Nicolas Bourbaki has short historical essays. These are relevant to some of the topics discussed here. They take the position that Euclid and the Greeks treated rational numbers as operators on the natural numbers.
So p/q operating on n is defined as p*(n/q) and is only defined on the set of multiples of q.
Whether this is what Euclid did or not, this approach has the disadvantage that the domain of 1/2 and 2/4 are different. If one multiples the numerator times the test number n, then the two domains are the same.