Since then there are in A as many magnitudes equal to C as there are units in D, whatever part the unit is of D, the same part is C of A also;
therefore, as C is to A, so is the unit to D. [VII. Def. 20]
But the unit measures the number D;
therefore C also measures A.
And since, as C is to A, so is the unit to D, therefore,
inversely, as A is to C, so is the number D to the unit. [cf. V. 7, Por.]
Again, since there are in F as many magnitudes equal to C as there are units in E,
therefore, as C is to F, so is the unit to E. [VII. Def. 20]
But it was also proved that, as A is to C, so is D to the unit;
therefore, ex aequali, as A is to F, so is D to E. [v. 22]
But, as D is to E, so is A to B;
therefore also, as A is to B, so is it to F also. [V. 11]
Therefore A has the same ratio to each of the magnitudes B, F;
therefore B is equal to F. [V. 9]
But C measures F;
therefore it measures B also.
Further it measures A also;
therefore C measures A, B.
Therefore A is commensurable with B.
Complete text
Book I