---
book: 10
number: 74
id: "X.74"
kind: "theorem"
uses: ["[[book-2/proposition-7]]", "[[book-10/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.74"
license: "CC-BY-SA-4.0"
---

# X.74

*If from a medial straight line there be subtracted a medial straight line which is commensurable with the whole in square only, and which contains with the whole a rational rectangle, the remainder is irrational. And let it be called a* *first apotome of a medial* *straight line*.

## Proof

For from the medial straight line *AB* let there be subtracted the medial straight line *BC* which is commensurable with *AB* in square only and with *AB* makes the rectangle *AB*, *BC* rational; I say that the remainder *AC* is irrational; and let it be called a *first apotome of a medial* straight line.

For, since *AB*, *BC* are medial, the squares on *AB*, *BC* are also medial.

But twice the rectangle *AB*, *BC* is rational; therefore the squares on *AB*, *BC* are incommensurable with twice the rectangle *AB*, *BC*; therefore twice the rectangle *AB*, *BC* is also incommensurable with the remainder, the square on *AC*, Cf. [[book-2/proposition-7|II. 7]] since, if the whole is incommensurable with one of the magnitudes, the original magnitudes will also be incommensurable. [[book-10/proposition-16|X. 16]]

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

And let it be called a *first apotome of a medial* straight line.