And, since AE is incommensurable in length with ED, while AE is commensurable with AG,
and ED is commensurable with EF, therefore AG is incommensurable with EF; [X. 13] so that AH is also incommensurable with EL, that is, SN is incommensurable with MR, that is, PN with NR, [VI. 1, X. 11]
that is, MN is incommensurable in length with NO. But MN, NO were proved to be both medial and commensurable in square; therefore MN, NO are medial straight lines commensurable in square only.
I say next that they also contain a rational rectangle. For, since DE is, by hypothesis, commensurable with each of the straight lines AB, EF, therefore EF is also commensurable with EK. [X. 12] And each of them is rational;
therefore EL, that is, MR is rational, [X. 19] and MR is the rectangle MN, NO. But, if two medial straight lines commensurable in square only and containing a rational rectangle be added together, the whole is irrational and is called a first bimedial straight line. [X. 37]
Therefore MO is a first bimedial straight line. Q. E. D. 1
1 39. Therefore BA, AG and BA, GE are pairs of rational straight lines commensurable in square only. The text has “Therefore BA, AG, GE are rational straight lines commensurable in square only,” which I have altered because it would naturally convey the impression that any two of the three straight lines are commensurable in square only, whereas AG, GE are commensurable in length (I. 18), and it is only the other two pairs which are commensurable in square only.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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