---
book: 10
number: 73
id: "X.73"
kind: "theorem"
uses: ["[[book-10/proposition-11]]", "[[book-10/proposition-15]]", "[[book-10/proposition-6]]", "[[book-2/proposition-7]]", "[[book-10/proposition-13]]", "[[book-10/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.73"
license: "CC-BY-SA-4.0"
---

# X.73

*If from a rational straight line there be subtracted a rational straight line commensurable with the whole in square only, the remainder is irrational; and let it be called* *an apotome*.

## Proof

For from the rational straight line *AB* let the rational straight line *BC*, commensurable with the whole in square only, be subtracted; I say that the remainder *AC* is the irrational straight line called *apotome*.

For, since *AB* is incommensurable in length with *BC*, and, as *AB* is to *BC*, so is the square on *AB* to the rectangle *AB*, *BC*, therefore the square on *AB* is incommensurable with the rectangle *AB*, *BC*. [[book-10/proposition-11|X. 11]]

But the squares on *AB*, *BC* are commensurable with the square on *AB*, [[book-10/proposition-15|X. 15]] and twice the rectangle *AB*, *BC* is commensurable with the rectangle *AB*, *BC*. [[book-10/proposition-6|X. 6]]

And, inasmuch as the squares on *AB*, *BC* are equal to twice the rectangle *AB*, *BC* together with the square on *CA*, [[book-2/proposition-7|II. 7]] therefore the squares on *AB*, *BC* are also incommensurable with the remainder, the square on *AC*. [[book-10/proposition-13|X. 13, 16]]

But the squares on *AB*, *BC* are rational; therefore *AC* is irrational. [[book-10/definitions#Definition 4|X. Def. 4]]

And let it be called an *apotome*. Q. E. D.