There being several magnitudes, and other (magnitudes) of equal number to them, (which are) also in the same ratio taken two by two, a ratio via equality (or ex aequali) occurs when as the first is to the last in the first (set of) magnitudes, so the first (is) to the last in the second (set of) magnitudes. Or alternately, (it is) a taking of the (ratio of the) outer (magnitudes) by the removal of the inner (magnitudes).
In other words, if \(\alpha_1,\ldots,\alpha_n\) and \(\beta_1,\ldots,\beta_n\) are positive real numbers such that the following corresponding pairs of numbers have the same ratio: \[\frac{\alpha_1}{\alpha_{2}}=\frac{\beta_1}{\beta_{2}},\quad\frac{\alpha_2}{\alpha_{3}}=\frac{\beta_2}{\beta_{3}},\quad\ldots,\quad\frac{\alpha_{n-1}}{\alpha_{n}}=\frac{\beta_{n-1}}{\beta_{n}}\] then the ratio ex aequali or (the ratio via equality) is the equation \[\frac{\alpha_1}{\alpha_{n}}=\frac{\beta_1}{\beta_{n}}.\]
Proofs: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Propositions: 32 33 34 35 36 37