Book 10
54
And, since AE is incommensurable in length with ED, while AE is commensurable with AG, and DE is commensurable with EF, therefore AG is also incommensurable with EF, [X. 13] so that AH is also incommensurable with EL. [VI. 1, X. 11]
54
But AH is equal to SN, and EL to MR; therefore SN is also incommensurable with MR.
54
But, as SN is to MR, so is PN to NR; [VI. 1] therefore PN is incommensurable with NR. [X. 11]
54
But PN is equal to MN, and NR to NO; therefore MN is incommensurable with NO.
54
And the square on MN is commensurable with the square on NO, and each is rational; therefore MN, NO are rational straight lines commensurable in square only.
54
Therefore MO is binomial [X. 36] and the side of AC. Q. E. D. 2. side. I use the word side in the sense explained in the note on X. Def. 4 (P. 13 above), i.e. as short for side of a square equal to. The Greek is ἡ τὸ χωίον δυναμένη.
PROPOSITION 55.
55
If an area be contained by a rational straight line and the second binomial, the side of the area is the irrational straight line which is called a first bimedial.
55
For let the area ABCD be contained by the rational straight line AB and the second binomial AD; I say that the side of the area AC is a first bimedial straight line.
55
For, since AD is a second binomial straight line, let it be divided into its terms at E, so that AE is the greater term; therefore AE, ED are rational straight lines commensurable in square only, the square on AE is greater than the square on ED by the square on a straight line commensurable with AE, and the lesser term ED is commensurable in length with AB. [X. Deff. II. 2]
55
Let ED be bisected at F, and let there be applied to AE the rectangle AG, GE equal to the square on EF and deficient by a square figure; therefore AG is commensurable in length with GE. [X. 17]
