| If two magnitudes have to one another the ratio which a number has to a number, then the magnitudes are commensurable. | ||
| Let the two magnitudes A and B have to one another the ratio which the number D has to the number E.
I say that the magnitudes A and B are commensurable. Divide A into as many equal parts as there are units in D, and let C equal one of them, and let F be made up of as many magnitudes equal to C as there are units in E. | ||
| 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 C is to A as the unit is to D. | VII.Def.20 | |
| But the unit measures the number D, therefore C also measures A. And since C is to A as the unit is to D, therefore, inversely, A is to C as the number D is to the unit. | V.7,Cor. | |
| Again, since there are in F as many magnitudes equal to C as there are units in E, therefore C is to F as the unit is to E. | VII.Def.20 | |
| But it was also proved that A is to C as D is to the unit, therefore, ex aequali, A is to F as D is to E. | V.22 | |
| But D is to E as A is to B, therefore A is to B as it is to F also. | V.11 | |
| Therefore A has the same ratio to each of the magnitudes B and F. Therefore B equals F. | V.9 | |
| But C measures F, therefore it measures B also. Further it measures A also, therefore C measures A and B.
Therefore A is commensurable with B. | ||
| Therefore, if two magnitudes have to one another the ratio which a number has to a number, then the magnitudes are commensurable. | ||
| Q.E.D. | ||
Corollary.From this it is manifest that, if there are two numbers as D and E, and a straight line as A, then it is possible to make a straight line F such that the given straight line is to it as the number D is to the number E. | ||
| And if a mean proportional is also taken between A and F, as B, then A is to F as the square on A is to the square on B, that is, the first is to the third as the figure on the first is to that which is similar and similarly described on the second. | V.19,Cor. | |
| But A is to F as the number D is to the number E, therefore the number D is to the number E as the figure on the straight line A is to the figure on the straight line B. | ||
The proof assumes that magnitudes are divisible. Not all magnitudes, however, are constructively divisible. For instance, a 60° angle cannot be trisected by a Euclidean construction. An alternate proof which does not depend on divisibility of magnitudes can be based on antenaresis.