This might come down to details of how you explain it: iirc Apostol took those basic moves (axioms) and calculated what the formula would have to be, rather tha

This might come down to details of how you explain it: iirc Apostol took those basic moves (axioms) and calculated what the formula would have to be, rather than starting with a formula and checking that it has the properties of a signed volume. But I don't know, it's been a very long time for me. It's good to have a variety of approaches to the subject.