In the Elements, Euclid states and proves twice that (in modern notation)
$$\boxed{\text{from } \frac{a}{b} = \frac{c}{d} \text{ it follows that } \frac{a}{b} = \frac{a-c}{b-d}}$$
– first for magnitudes (lengths), then for numbers.
Note that the proper propositions are almost identical in the Greek original:
So what does it mean that Euclid decidedly treats magnitudes (lengths) separately from numbers, even though he was obviously aware that they are (or behave) very much the same?
Furthermore, in the two proofs he makes use of different definitions, i.e. axioms, i.e. theories – but arrives at the very same result. What does this tell us about the two theories?
Best Answer
To Euclid, number meant whole number, or ratio of whole numbers (fraction). On the other hand, length was just that: the length of a line segment. It had been shown that some lengths were not "numbers" in Euclid's sense, in particular $\sqrt{2}$, the length of a diagonal of a unit square. So, not all lengths were numbers.
On the other hand, multiplication of lengths was very limited: it only made sense to multiply two or three lengths together, since objects were only $2$- or $3$-dimensional! On the other hand, one could multiply as many numbers as desired, since they were separate entities, not bound by geometry.
Therefore, to the ancient Greeks, numbers and lengths were completely separate things: they had different uses and behaved differently. So, Euclid had to treat them separately when proving properties about them.