Book 10
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.
8
For let the two magnitudes A, B not have to one another the ratio which a number has to a number; I say that the magnitudes A, B are incommensurable.
8
For, if they are commensurable, A will have to B the ratio which a number has to a number. [X. 5]
8
But it has not; therefore the magnitudes A, B are incommensurable.
8
Therefore etc.
PROPOSITION 9.
9
The squares on straight lines commensurable in length have to one another the ratio which a square number has to a square number; and squares which have to one another the ratio which a square number has to a square number will also have their sides commensurable in length. But the squares on straight lines incommensurable in length have not to one another the ratio which a square number has to a square number; and squares which have not to one another the ratio which a square number has to a square number will not have their sides commensurable in length either.
