If two magnitudes have to one another the ratio which a number has to a number, the magnitudes will be commensurable.

If two magnitudes have to one another the ratio which a number has to a number, the magnitudes will be commensurable. For let the two magnitudes A, B have to one another the ratio which the number D has to the number E; I say that the magnitudes A, B are commensurable. For let A be divided into as many equal parts as there are units in D, and let C be equal to 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, 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. Therefore etc. PORISM. From this it is manifest that, if there be two numbers, as D, E, and a straight line, as A, 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 be also taken between A, F, as B, as A is to F, so will the square on A be to the square on B, that is, as the first is to the third, so is the figure on the first to that which is similar and similarly described on the second. [VI. 19, Por.] |