And, as many times as C measures A, so many units let there be in D; and, as many times as C measures B, so many units let there be in E.

Since then C measures A according to the units in D, while the unit also measures D according to the units in it, therefore the unit measures the number D the same number of times as the magnitude C measures A; therefore, as C is to A, so is the unit to D; [[VII. Def. 20](elem.7.def.20)] therefore, inversely, as A is to C, so is D to the unit. [cf. [V. 7, Por.](elem.5.7.p.1)]

Again, since C measures B according to the units in E, while the unit also measures E according to the units in it, therefore the unit measures E the same number of times as C measures B; therefore, as C is to B, so is the unit to E.

But it was also proved that, as A is to C, so is D to the unit; therefore, ex aequali, as A is to B, so is the number D to E. [[V. 22](elem.5.22)]

Therefore the commensurable magnitudes A, B have to one another the ratio which the number D has to the number E. Q. E. D.