PROPOSITION 17.
If there be two unequal straight lines, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are commensurable in length, thenthe square on the greater will be greater than the square on the less by the square on a straight line commensurable with the greater. And, if the square on the greater be greater than the square on the less by the square on a straight line commensurable with
the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it will divide it into parts which are commensurable in length. Let A, BC be two unequal straight lines, of which BC is
the greater, and let there be applied to BC a parallelogram equal to the fourth part of the square on the less, A, that is, equal to the square on the half of A, and deficient
by a square figure. Let this be the rectangle BD, DC, [cf. Lemma] and let BD be commensurable in length with DC; I say that the square on BC is greater than the square on A by the square on a straight line commensurable with BC.
For let BC be bisected at the point E, and let EF be made equal to DE. Therefore the remainder DC is equal to BF. And, since the straight line BC has been cut into equal parts at E, and into unequal parts at D,
therefore the rectangle contained by BD, DC, together with the square on ED, is equal to the square on EC; [II. 5] And the same is true of their quadruples; therefore four times the rectangle BD, DC, together with four times the square on DE, is equal to four times the square
on EC. But the square on A is equal to four times the rectangle BD, DC; and the square on DF is equal to four times the square on DE, for DF is double of DE.
And the square on BC is equal to four times the square on EC, for again BC is double of CE. Therefore the squares on A, DF are equal to the square on BC, so that the square on BC is greater than the square on A by
the square on DF. It is to be proved that BC is also commensurable with DF. Since BD is commensurable in length with DC, therefore BC is also commensurable in length with CD. [X. 15] But CD is commensurable in length with CD, BF, for
CD is equal to BF. [X. 6] Therefore BC is also commensurable in length with BF, CD, [X. 12] so that BC is also commensurable in length with the remainder FD; [X. 15]
therefore the square on BC is greater than the square on A by the square on a straight line commensurable with BC. Next, let the square on BC be greater than the square on A by the square on a straight line commensurable with BC, let a parallelogram be applied to BC equal to the fourth part
of the square on A and deficient by a square figure, and let it be the rectangle BD, DC. It is to be proved that BD is commensurable in length with DC. With the same construction, we can prove similarly that
the square on BC is greater than the square on A by the square on FD. But the square on BC is greater than the square on A by the square on a straight line commensurable with BC. Therefore BC is commensurable in length with FD,
so that BC is also commensurable in length with the remainder, the sum of BF, DC. [X. 15] But the sum of BF, DC is commensurable with DC, [X. 6] so that BC is also commensurable in length with CD; [X. 12] and therefore, separando, BD is commensurable in length
with DC. [X. 15] Therefore etc. 1
