Kitap 13
6
And, since the square on CD has not to the square on DA the ratio which a square number has to a square number, therefore CD is incommensurable in length with DA; [X. 9] therefore CD, DA are rational straight lines commensurable in square only; therefore AC is an apotome. [X. 73]
6
Again, since AB has been cut in extreme and mean ratio, and AC is the greater segment, therefore the rectangle AB, BC is equal to the square on AC. [VI. Def. 3, VI. 17]
6
Therefore the square on the apotome AC, if applied to the rational straight line AB, produces BC as breadth.
6
But the square on an apotome, if applied to a rational straight line, produces as breadth a first apotome; [X. 97] therefore CB is a first apotome.
6
And CA was also proved to be an apotome.
6
Therefore etc. Q. E. D.
PROPOSITION 7.
7
If three angles of an equilateral pentagon, taken either in order or not in order, be equal, the pentagon will be equiangular.
7
For in the equilateral pentagon ABCDE let, first, three angles taken in order, those at A, B, C, be equal to one another; I say that the pentagon ABCDE is equiangular.
7
For let AC, BE, FD be joined.
