If a rational straight line is cut in extreme and mean ratio, then each of the segments is the irrational straight line called apotome.

Let *AB* be a rational straight line cut in extreme and mean ratio at *C,* and let *AC* be the greater segment.

I say that each of the straight lines *AC* and *CB* is the irrational straight line called apotome.

Produce *BA,* and make *AD* half of *BA.*

Since, then, the straight line *AB* is cut in extreme and mean ratio, and to the greater segment *AC* is added *AD* which is half of *AB,* therefore the square on *CD* is five times the square on *DA.*

Therefore the square on *CD* has to the square on *DA* the ratio which a number has to a number, therefore the square on *CD* is commensurable with the square on *DA*

But the square on *DA* is rational, for *DA* is rational being half of *AB* which is rational, therefore the square on *CD* is also rational. Therefore *CD* is also rational.
X.Def.4

And, since the square on *CD* has not to the square on *DA* the ratio which a square number has to a square number, therefore *CD* is incommensurable in length with *DA.* Therefore *CD* and *DA* are rational straight lines commensurable in square only. Therefore *AC* is an apotome.

Again, since *AB* is cut in extreme and mean ratio, and *AC* is the greater segment, therefore the rectangle *AB* by *BC* equals the square on *AC.*

Therefore the square on the apotome *AC,* if applied to the rational straight line *AB,* produces *BC* as breadth. But the square on an apotome, if applied to a rational straight line, produces as breadth a first apotome, therefore *CB* is a first apotome. And *CA* was also proved to be an apotome.

Therefore, *if a rational straight line is cut in extreme and mean ratio, then each of the segments is the irrational straight line called apotome.*

Q.E.D.