Book 10
6
Further it measures A also; therefore C measures A, B.
6
Therefore A is commensurable with B.
6
Therefore etc.
6
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.
6
And, if a mean proportional be also taken between A, F, as B,
6
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.]
6
But, as A is to F, so is the number D to the number E; therefore it has been contrived that, as the number D is to the number E, so also is the figure on the straight line A to the figure on the straight line B. Q. E. D.
PROPOSITION 7.
7
Incommensurable magnitudes have not to one another the ratio which a number has to a number.
7
Let A, B be incommensurable magnitudes; I say that A has not to B the ratio which a number has to a number.
7
For, if A has to B the ratio which a number has to a number, A will be commensurable with B. [X. 6]
7
But it is not; therefore A has not to B the ratio which a number has to a number.
7
Therefore etc.
PROPOSITION 8.
8
If two magnitudes have not to one another the ratio which a number has to a number, the magnitudes will be incommensurable.
