X.5

Commensurable magnitudes have to one another the ratio which a number has to a number.

Let A, B be commensurable magnitudes; I say that A has to B the ratio which a number has to a number.

For, since A, B are commensurable, some magnitude will measure them.

Let it measure them, and let it be C.

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] therefore, inversely, as A is to C, so is D to the unit. [cf. V.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]

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