To find the sixth apotome.

Set out a rational straight line *A*, and set out three numbers *E*, *BC*, and *CD* not having to one another the ratio which a square number has to a square number, and further let *CB* also not have to *BD* the ratio which a square number has to a square number.

Let it be contrived that *E* is to *BC* as the square on *A* is to the square on *FG*, and *BC* is to *CD* as the square on *FG* is to the square on *GH*.

Now since *E* is to *BC* as the square on *A* is to the square on *FG*, therefore the square on *A* is commensurable with the square on *FG*.

But the square on *A* is rational, therefore the square on *FG* is also rational. Therefore *FG* is also rational.

Since *E* does not have to *BC* the ratio which a square number has to a square number, therefore neither does the square on *A* have to the square on *FG* the ratio which a square number has to a square number, therefore *A* is incommensurable in length with *FG*.

Again, since *BC* is to *CD* as the square on *FG* is to the square on *GH*, therefore the square on *FG* is commensurable with the square on *GH*.

But the square on *FG* is rational, therefore the square on *GH* is also rational. Therefore *GH* is also rational.

Since *BC* does not have to *CD* the ratio which a square number has to a square number, therefore neither does the square on *FG* have to the square on *GH* the ratio which a square number has to a square number. Therefore *FG* is incommensurable in length with *GH*

And both are rational, therefore *FG* and *GH* are rational straight lines commensurable in square only. Therefore *FH* is an apotome.

I say next that it is also a sixth apotome.

Since *E* is to *BC* as the square on *A* is to the square on *FG*, and *BC* is to *CD* as the square on *FG* is to the square on *GH*, therefore, *ex aequali*, *E* is to *CD* as the square on *A* is to the square on *GH*.

But *E* does not have to *CD* the ratio which a square number has to a square number, therefore neither does the square on *A* have to the square on *GH* the ratio which a square number has to a square number. Therefore *A* is incommensurable in length with *GH*. Therefore neither of the straight lines *FG* nor *GH* is commensurable in length with the rational straight line *A*.

Now let the square on *K* be that by which the square on *FG* is greater than the square on *GH*.

Since *BC* is to *CD* as the square on *FG* is to the square on *GH*, therefore, in conversion, *CB* is to *BD* as the square on *FG* is to the square on *K*.

But *CB* does not have to *BD* the ratio which a square number has to a square number, therefore neither does the square on *FG* have to the square on *K* the ratio which a square number has to a square number, therefore *FG* is incommensurable in length with *K*.

And the square on *FG* is greater than the square on *GH* by the square on *K*, therefore the square on *FG* is greater than the square on *GH* by the square on a straight line incommensurable in length with *FG*.

And neither of the straight lines *FG* nor *GH* is commensurable with the rational straight line *A* set out. Therefore *FH* is a sixth apotome.

Therefore the sixth apotome *FH* has been found.

Q.E.F.