#### PROPOSITION 88.

To find the fourth apotome.

Let a rational straight line A be set out, and BG commensurable in length with it; therefore BG is also rational.

Let two numbers DF, FE be set out such that the whole DE has not to either of the numbers DF, EF the ratio which a square number has to a square number.

Let it be contrived that, as DE is to EF, so is the square on BG to the square on GC; [X. 6, Por.] therefore the square on BG is commensurable with the square on GC. [X. 6]

But the square on BG is rational; therefore the square on GC is also rational; therefore GC is rational.

Now, since DE has not to EF the ratio which a square number has to a square number, therefore neither has the square on BG to the square on GC the ratio which a square number has to a square number; therefore BG is incommensurable in length with GC. [X. 9]

And both are rational; therefore BG, GC are rational straight lines commensurable in square only; therefore BC is an apotome. [X. 73]

Now let the square on H be that by which the square on BG is greater than the square on GC.

Since then, as DE is to EF, so is the square on BG to the square on GC, therefore also, convertendo, as ED is to DF, so is the square on GB to the square on H. [v. 19, Por.]

But ED has not to DF the ratio which a square number has to a square number; therefore neither has the square on GB to the square on H the ratio which a square number has to a square number; therefore BG is incommensurable in length with H. [X. 9]

And the square on BG is greater than the square on GC by the square on H; therefore the square on BG is greater than the square on GC by the square on a straight line incommensurable with BG.

And the whole BG is commensurable in length with the rational straight line A set out.

Therefore BC is a fourth apotome. [X. Deff. III. 4]

Therefore the fourth apotome has been found. Q. E. D.