|Taking jointly the ratio u : v yields the ratio (u + v):v. Taking separately the ratio (u + v):v returns the ratio u : v. Taking the ratio (u + v):v in conversion yields the ratio (u + v):u.
These conversions are only important when the ratios are in proportions.
|The following three proportions are shown to be equivalent in propositions V.17 and V.18.
2. (u + v) : u = (x + y) : x.
3. u : v = x : y.
Proposition V.17 and V.18 show proportions 1 and 3 are equivalent. That means proportion 2 and the inverse of 3, v : u = y : x, are also equivalent. And of course, 3 and its inverse are equivalent, so all three proportions are equivalent.
Furthermore, when all the magnitudes are of the same kind, then the alternate proportions are also equivalent by V.16 making six equivalent statements.
5. (u + v) : (x + y) = u : x
6. u : x = v : y
Proposition V.19 goes on to say that 4 implies 5, and its corollary says 1 implies 2.
Heath translates “taken jointly,” “taken separately,” and “in conversion” by the Latin words componendo, separando, and convertendo, respectively.