|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.|
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.
Book X Introduction - Proposition X.5 - Proposition X.7.