Kitap 10
26
And, since [DB is rational and is equal to KH, therefore] KH is [also] rational; and it is applied to the rational straight line EF; therefore GH is rational and commensurable in length with EF. [X. 20]
26
But EG is also rational, and is incommensurable in length with EF; therefore EG is incommensurable in length with GH. [X. 13]
26
And, as EG is to GH, so is the square on EG to the rectangle EG, GH; therefore the square on EG is incommensurable with the rectangle EG, GH. [X. 11]
26
But the squares on EG, GH are commensurable with the square on EG, for both are rational; and twice the rectangle EG, GH is commensurable with the rectangle EG, GH, for it is double of it; [X. 6] therefore the squares on EG, GH are incommensurable with twice the rectangle EG, GH; [X. 13] therefore also the sum of the squares on EG, GH and twice the rectangle EG, GH, that is, the square on EH [II. 4], is incommensurable with the squares on EG, GH. [X. 16]
26
But the squares on EG, GH are rational; therefore the square on EH is irrational. [X. Def. 4]
26
Therefore EH is irrational.
26
But it is also rational: which is impossible.
26
Therefore etc. Q. E. D.
PROPOSITION 27.
27
To find medial straight lines commensurable in square only which contain a rational rectangle.
27
Let two rational straight lines A, B commensurable in square only be set out; let C be taken a mean proportional between A, B, [VI. 13] and let it be contrived that, as A is to B, so is C to D. [VI. 12]
