---
book: 13
number: 6
id: "XIII.6"
kind: "theorem"
uses: ["[[book-13/proposition-1]]", "[[book-10/proposition-6]]", "[[book-10/proposition-9]]", "[[book-10/proposition-73]]", "[[book-6/proposition-17]]", "[[book-10/proposition-97]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:13.prop.6"
license: "CC-BY-SA-4.0"
---

# XIII.6

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

## Proof

Let *AB* be a rational straight line, let it be cut in extreme and mean ratio at *C*, and let *AC* be the greater segment; I say that each of the straight lines *AC*, *CB* is the irrational straight line called apotome.

For let *BA* be produced, and let *AD* be made half of *BA*.

Since then the straight line *AB* has been 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*. [[book-13/proposition-1|XIII. 1]]

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*. [[book-10/proposition-6|X. 6]]

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; [[book-10/definitions#Definition 4|X. Def. 4]] therefore *CD* is also rational.

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*; [[book-10/proposition-9|X. 9]] therefore *CD*, *DA* are rational straight lines commensurable in square only; therefore *AC* is an apotome. [[book-10/proposition-73|X. 73]]

Again, since *AB* has been cut in extreme and mean ratio, and *AC* is the greater segment, therefore the rectangle *AB*, *BC* is equal to the square on *AC*. [[book-6/definitions#Definition 3|VI. Def. 3]], [[book-6/proposition-17|VI. 17]]

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; [[book-10/proposition-97|X. 97]] therefore *CB* is a first apotome.

And *CA* was also proved to be an apotome.

Therefore etc. Q. E. D.