Book 10
14
Let A, B, C, D be four straight lines in proportion, so that, as A is to B, so is C to D; and let the square on A be greater than the square on B by the square on E, and let the square on C be greater than the square on D by the square on F; I say that, if A is commensurable with E, C is also commensurable with F, and, if A is incommensurable with E, C is also incommensurable with F.
14
For since, as A is to B, so is C to D, therefore also, as the square on A is to the square on B, so is the square on C to the square on D. [VI. 22]
14
But the squares on E, B are equal to the square on A, and the squares on D, F are equal to the square on C.
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Therefore, as the squares on E, B are to the square on B, so are the squares on D, F to the square on D; therefore, separando, as the square on E is to the square on B, so is the square on F to the square on D; [V. 17] therefore also, as E is to B, so is F to D; [VI. 22] therefore, inversely, as B is to E, so is D to F.
14
But, as A is to B, so also is C to D; therefore, ex aequali, as A is to E, so is C to F. [V. 22]
14
Therefore, if A is commensurable with E, C is also commensurable with F, and, if A is incommensurable with E, C is also incommensurable with F. [X. 11]
14
Therefore etc.3, 5, 8, 10. Euclid speaks of the square on the first (third) being greater than the square on the second (fourth) by the square on a straight line commensurable (incommensurable) with itself (ἑαυτῇ), and similarly in all like phrases throughout the Book. For clearness' sake I substitute the first, the third, or whatever it may be, for itself in these cases.
PROPOSITION 15.
15
If two commensurable magnitudes be added together, the whole will also be commensurable with each of them; and, if the whole be commensurable with one of them, the original magnitudes will also be commensurable.
15
For let the two commensurable magnitudes AB, BC be added together; I say that the whole AC is also commensurable with each of the magnitudes AB, BC.
