To find the fifth apotome.

Set out a rational straight line *A*, and let *CG* be commensurable in length with *A*. Then *CG* is rational.

Set out two numbers *DF* and *FE* such that *DE* again has to neither of the numbers *DF* nor *FE* the ratio which a square number has to a square number, and let it be contrived that *FE* is to *ED* as the square on *CG* is to the square on *GB*.

Then the square on *GB* is also rational. Therefore *BG* is also rational.

Now since *DE* is to *EF* as the square on *BG* is to the square on *GC*, while *DE* does not have to *EF* the ratio which a square number has to a square number, therefore neither does the square on *BG* have to the square on *GC* the ratio which a square number has to a square number. Therefore *BG* is incommensurable in length with *GC*.

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

I say next that it is also a fifth 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 *DE* is to *EF*, therefore, in conversion, *ED* is to *DF* as the square on *BG* is to the square on *H*.

But *ED* does not have to *DF* the ratio which a square number has to a square number, therefore neither has the square on *BG* to the square on *H* the ratio which a square number has to a square number. Therefore *BG* is incommensurable 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 *GB* is greater than the square on *GC* by the square on a straight line incommensurable in length with *GB*.

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

Therefore the fifth apotome *BC* has been found.

Q.E.F.