  Ironically, extant Greek mathematics shows no traces of an Aristotelian universal mathematics. The theory of ratio for magnitudes in Euclid, Elements v is completely separate from the treatment of ratio for number in Elements vii and parts of viii, none of which appeals to v, even though almost all of the proofs of v could apply straightforwardly to numbers. For example, Euclid provides separate definitions of proportion (v def. 5, and vii def. 20). Compare the rule above (alternando),which is proved at v.16, while the rule follows trivially for numbers from the commutivity of multiplication and vii.19: ad = bc ⇔ a : b = c : d.