Book 10
96
But, as AG is to GD, so is AK to KD; [VI. 1] therefore AK is incommensurable with KD. [X. 11]
96
Now let the square LM be constructed equal to AI, and let NO equal to FK, and about the same angle, be subtracted; therefore the squares LM, NO are about the same diameter. [VI. 26]
96
Let PR be their diameter, and let the figure be drawn.
96
Then in manner similar to the above we can prove that LN is the side of the area AB.
96
I say that LN is a straight line which produces with a medial area a medial whole.
96
For, since AK was proved medial and is equal to the squares on LP, PN, therefore the sum of the squares on LP, PN is medial.
96
Again, since DK was proved medial and is equal to twice the rectangle LP, PN, twice the rectangle LP, PN is also medial.
96
And, since AK was proved incommensurable with DK, the squares on LP, PN are also incommensurable with twice the rectangle LP, PN.
96
And, since AI is incommensurable with FK, therefore the square on LP is also incommensurable with the square on PN; therefore LP, PN are straight lines incommensurable in square which make the sum of the squares on them medial, twice the rectangle contained by them medial, and further the squares on them incommensurable with twice the rectangle contained by them.
96
Therefore LN is the irrational straight line called that which produces with a medial area a medial whole; [X. 78] and it is the side of the area AB.
96
Therefore the side of the area is a straight line which produces with a medial area a medial whole. Q. E. D.
PROPOSITION 97.
97
The square on an apotome applied to a rational straight line produces as breadth a first apotome.
97
Let AB be an apotome, and CD rational, and to CD let there be applied CE equal to the square on AB and producing CF as breadth; I say that CF is a first apotome.
97
For let BG be the annex to AB; therefore AG, GB are rational straight lines commensurable in square only. [X. 73]
97
To CD let there be applied CH equal to the square on AG, and KL equal to the square on BG.
