To find the second apotome.

Set out a rational straight line *A*, and let *GC* be commensurable in length with *A*. Then *GC* is rational. Set out two square numbers *DE* and *EF*, and let their difference *DF* not be square.

Now let it be contrived that *FD* is to *DE* as the square on *CG* is to the square on *GB*.

Then the square on *CG* is commensurable with the square on *GB*.

But the square on *CG* is rational, therefore the square on *GB* is also rational. Therefore *BG* is rational. And, since the square on *GC* does not have to the square on *GB* the ratio which a square number has to a square number, therefore *CG* is incommensurable in length with *GB*.

And both are rational, therefore *CG* and *GB* are rational straight lines commensurable in square only. Therefore *BC* is an apotome.

I say next that it is also a second apotome.

Let the square on *H* be that by which the square on *BG* is greater than the square on *GC*.

Since the square on *BG* is to the square on *GC* as the number *ED* is to the number *DF*, therefore, in conversion, the square on *BG* is to the square on *H* as *DE* is to *EF*.

And each of the numbers *DE* and *EF* is square, therefore the square on *BG* has to the square on *H* the ratio which a square number has to a square number. Therefore *BG* is commensurable in length with *H*.

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 commensurable in length with *BG*.

And *CG*, the annex, is commensurable with the rational straight line *A* set out, therefore *BC* is a second apotome.

Therefore the second apotome *BC* has been found.

Q.E.F.