Book 10
40
For let two straight lines AB, BC incommensurable in square, and fulfilling the given conditions [X. 34 ], be added together; I say that AC is irrational.
40
For, since the sum of the squares on AB, BC is medial, while twice the rectangle AB, BC is rational, therefore the sum of the squares on AB, BC is incommensurable with twice the rectangle AB, BC; so that the square on AC is also incommensurable with twice the rectangle AB, BC. [X. 16 ]
40
But twice the rectangle AB, BC is rational; therefore the square on AC is irrational.
40
Therefore AC is irrational. [X. Def. 4 ]
40
And let it be called the side of a rational plus a medial area. Q. E. D.
PROPOSITION 41.
41
If two straight lines incommensurable in square which make the sum of the squares on them medial, and the rectangle contained by them medial and also incommensurable with the sum of the squares on them, be added together, the whole straight line is irrational; and let it be called the side of the sum of two medial areas.
41
For let two straight lines AB, BC incommensurable in square and satisfying the given conditions [X. 35 ] be added together; I say that AC is irrational.
41
Let a rational straight line DE be set out, and let there be applied to DE the rectangle DF equal to the squares on AB, BC, and the rectangle GH equal to twice the rectangle AB, BC; therefore the whole DH is equal to the square on AC. [II. 4 ]
41
Now, since the sum of the squares on AB, BC is medial, and is equal to DF, therefore DF is also medial.
41
And it is applied to the rational straight line DE; therefore DG is rational and incommensurable in length with DE. [X. 22 ]
41
For the same reason GK is also rational and incommensurable in length with GF, that is, DE.
41
And, since the squares on AB, BC are incommensurable with twice the rectangle AB, BC, DF is incommensurable with GH; so that DG is also incommensurable with GK. [VI. 1 , X. 11 ]
