Book 10
93
I say that LN is a second apotome of a medial straight line.
93
For, since AI, FK were proved medial, and are equal to the squares on LP, PN, therefore each of the squares on LP, PN is also medial; therefore each of the straight lines LP, PN is medial.
93
And, since AI is commensurable with FK, [VI. 1, X. 11] therefore the square on LP is also commensurable with the square on PN.
93
Again, since AI was proved incommensurable with EK, therefore LM is also incommensurable with MN, that is, the square on LP with the rectangle LP, PN; so that LP is also incommensurable in length with PN; [VI. 1, X. 11] therefore LP, PN are medial straight lines commensurable in square only.
93
I say next that they also contain a medial rectangle.
93
For, since EK was proved medial, and is equal to the rectangle LP, PN, therefore the rectangle LP, PN is also medial, so that LP, PN are medial straight lines commensurable in square only which contain a medial rectangle.
93
Therefore LN is a second apotome of a medial straight line; [X. 75] and it is the side of the area AB.
93
Therefore the side of the area AB is a second apotome of a medial straight line. Q. E. D.
PROPOSITION 94.
94
If an area be contained by a rational straight line and a fourth apotome, the side of the area is minor.
94
For let the area AB be contained by the rational straight line AC and the fourth apotome AD; I say that the side of the area AB is minor.
