Kitap 10
92
Since then AI, FK are medial and are equal to the squares on LP, PN, the squares on LP, PN are also medial; therefore LP, PN are also medial straight lines commensurable in square only.
92
And, since the rectangle AF, FG is equal to the square on EG, therefore, as AF is to EG, so is EG to FG, [VI. 17] while, 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]
92
But MN is also a mean proportional between the squares LM, NO, and AI is equal to LM, and FK to NO; therefore MN is also equal to EK.
92
But DH is equal to EK, and LO equal to MN; therefore the whole DK is equal to the gnomon UVW and NO.
92
Since then the whole AK is equal to LM, NO, and, in these, DK is equal to the gnomon UVW and NO, therefore the remainder AB is equal to TS.
92
But TS is the square on LN; therefore the square on LN is equal to the area AB; therefore LN is the side of the area AB.
92
I say that LN is a first apotome of a medial straight line.
92
For, since EK is rational and is equal to LO, therefore LO, that is, the rectangle LP, PN, is rational.
92
But NO was proved medial; therefore LO is incommensurable with NO.
92
But, as LO is to NO, so is LP to PN; [VI. 1] therefore LP, PN are incommensurable in length. [X. 11]
92
Therefore LP, PN are medial straight lines commensurable in square only which contain a rational rectangle; therefore LN is a first apotome of a medial straight line. [X. 74]
92
And it is the side of the area AB.
92
Therefore the side of the area AB is a first apotome of a medial straight line. Q. E. D.
PROPOSITION 93.
93
If an area be contained by a rational straight line and a third apotome, the side of the area is a second apotome of a medial straight line.
93
For let the area AB be contained by the rational straight line AC and the third apotome AD; I say that the side of the area AB is a second apotome of a medial straight line.
