---
book: 10
number: 110
id: "X.110"
kind: "theorem"
uses: ["[[book-10/proposition-22]]", "[[book-6/proposition-1]]", "[[book-10/proposition-11]]", "[[book-10/proposition-73]]", "[[book-10/proposition-93]]", "[[book-10/proposition-96]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_3.110"
license: "CC-BY-SA-4.0"
---

# X.110

*If from a medial area there be subtracted a medial area incommensurable with the whole, the two remaining irrational straight lines arise, either a second apotome of a medial straight line or a straight line which produces with a medial area a medial whole*.

## Proof

For, as in the foregoing figures, let there be subtracted from the medial area *BC* the medial area *BD* incommensurable with the whole; I say that the side of *EC* is one of two irrational straight lines, either a second apotome of a medial straight line or a straight line which produces with a medial area a medial whole.

For, since each of the rectangles *BC*, *BD* is medial, and *BC* is incommensurable with *BD*, it follows that each of the straight lines *FH*, *FK* will be rational and incommensurable in length with *FG*. [[book-10/proposition-22|X. 22]]

And, since *BC* is incommensurable with *BD*, that is, *GH* with *GK*, *HF* is also incommensurable with *FK*; [[book-6/proposition-1|VI. 1]], [[book-10/proposition-11|X. 11]] therefore *FH*, *FK* are rational straight lines commensurable in square only; therefore *KH* is an apotome. [[book-10/proposition-73|X. 73]]

If then the square on *FH* is greater than the square on *FK* by the square on a straight line commensurable with *FH*, while neither of the straight lines *FH*, *FK* is commensurable in length with the rational straight line *FG* set out, *KH* is a third apotome. [[book-10/definitions#Definition 3 (part 3)|X. Deff. III. 3]]

But *KL* is rational, and the rectangle contained by a rational straight line and a third apotome is irrational, and the side of it is irrational, and is called a second apotome of a medial straight line; [[book-10/proposition-93|X. 93]] so that the side of *LH*, that is, of *EC*, is a second apotome of a medial straight line.

But, if the square on *FH* is greater than the square on *FK* by the square on a straight line incommensurable with *FH*, while neither of the straight lines *HF*, *FK* is commensurable in length with *FG*, *KH* is a sixth apotome. [[book-10/definitions#Definition 6 (part 3)|X. Deff. III. 6]]

But the side of the rectangle contained by a rational straight line and a sixth apotome is a straight line which produces with a medial area a medial whole. [[book-10/proposition-96|X. 96]]

Therefore the side of *LH*, that is, of *EC*, is a straight line which produces with a medial area a medial whole. Q. E. D.