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. 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] 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] Therefore etc.1
1 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.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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