Kitap 10
94
But, as AF is to EG, so is AI to EK, and, as EG is to FG, so is EK to FK; [VI. 1] therefore EK is a mean proportional between AI, FK. [V. 11]
94
But MN is also a mean proportional between the squares LM, NO, and AI is equal to LM, and FK to NO; therefore EK is also equal to MN.
94
But DH is equal to EK, and LO is equal to MN; therefore the whole DK is equal to the gnomon UVW and NO.
94
Since, then, the whole AK is equal to the squares LM, NO, and, in these, DK is equal to the gnomon UVW and the square NO, therefore the remainder AB is equal to ST, that is, to the square on LN; therefore LN is the side of the area AB.
94
I say that LN is the irrational straight line called minor.
94
For, since AK is rational and is equal to the squares on LP, PN, therefore the sum of the squares on LP, PN is rational.
94
Again, since DK is medial, and DK is equal to twice the rectangle LP, PN, therefore twice the rectangle LP, PN is medial.
94
And, since AI was proved incommensurable with FK, therefore the square on LP is also incommensurable with the square on PN.
94
Therefore LP, PN are straight lines incommensurable in square which make the sum of the squares on them rational, but twice the rectangle contained by them medial.
94
Therefore LN is the irrational straight line called minor; [X. 76] and it is the side of the area AB.
94
Therefore the side of the area AB is minor. Q. E. D.
PROPOSITION 95.
95
If an area be contained by a rational straight line and a fifth apotome, the side of the area is a straight line which produces with a rational area a medial whole.
95
For let the area AB be contained by the rational straight line AC and the fifth apotome AD; I say that the side of the area AB is a straight line which produces with a rational area a medial whole.
