Book 10
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]
