PROPOSITION 29.
To find two rational straight lines commensurable in square only and such that the square on the greater is greater than the square on the less by the square on a straight line commensurable in length with the greater.
For let there be set out any rational straight line
AB, and two square numbers
CD,
DE such that their difference
CE is not square; [
Lemma 1] let there be described on
AB the semicircle
AFB, and let it be contrived that,
as DC is to CE, so is the square on BA to the square on AF. [X. 6, Por.]
Let
FB be joined.
Since, as the square on
BA is to the square on
AF, so is
DC to
CE, therefore the square on
BA has to the square on
AF the ratio which the number
DC has to the number
CE; therefore the square on
BA is commensurable with the square on
AF. [
X. 6]
But the square on
AB is rational; [
X. Def. 4] therefore the square on
AF is also rational; [
id.] therefore
AF is also rational.
And, since
DC has not to
CE the ratio which a square number has to a square number, neither has the square on
BA to the square on
AF the ratio which a square number has to a square number; therefore
AB is incommensurable in length with
AF. [
X. 9]
Therefore
BA,
AF are rational straight lines commensurable in square only.
And since, as
DC is to
CE, so is the square on
BA to the square on
AF, therefore,
convertendo, as
CD is to
DE, so is the square on
AB to the square on
BF. [
V. 19, Por.,
III. 31,
I. 47]
But
CD has to
DE the ratio which a square number has to a square number: therefore also the square on
AB has to the square on
BF the ratio which a square number has to a square number; therefore
AB is commensurable in length with
BF. [
X. 9]
And the square on
AB is equal to the squares on
AF,
FB; therefore the square on
AB is greater than the square on
AF by the square on
BF commensurable with
AB.
Therefore there have been found two rational straight lines
BA,
AF commensurable in square only and such that the square on the greater
AB is greater than the square on the less
AF by the square on
BF commensurable in length with
AB.