---
book: 10
number: 108
id: "X.108"
kind: "theorem"
uses: ["[[book-10/proposition-20]]", "[[book-10/proposition-22]]", "[[book-10/proposition-13]]", "[[book-10/proposition-73]]", "[[book-10/proposition-91]]", "[[book-10/proposition-94]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_3.108"
license: "CC-BY-SA-4.0"
---

# X.108

If from a rational area a medial area be subtracted, the side of the remaining area becomes one of two irrational straight lines, either an apotome or a minor straight line.

## Proof

For from the rational area *BC* let the medial area *BD* be subtracted; I say that the side of the remainder *EC* becomes one of two irrational straight lines, either an apotome or a minor straight line.

For let a rational straight line *FG* be set out, to *FG* let there be applied the rectangular parallelogram *GH* equal to *BC*, and let *GK* equal to *DB* be subtracted; therefore the remainder *EC* is equal to *LH*.

Since then *BC* is rational, and *BD* medial, while *BC* is equal to *GH*, and *BD* to *GK*, therefore *GH* is rational, and *GK* medial.

And they are applied to the rational straight line *FG*; therefore *FH* is rational and commensurable in length with *FG*, [[book-10/proposition-20|X. 20]] while *FK* is rational and incommensurable in length with *FG*; [[book-10/proposition-22|X. 22]] therefore *FH* is incommensurable in length with *FK*. [[book-10/proposition-13|X. 13]]

Therefore *FH*, *FK* are rational straight lines commensurable in square only; therefore *KH* is an apotome [[book-10/proposition-73|X. 73]], and *KF* the annex to it.

Now the square on *HF* is greater than the square on *FK* by the square on a straight line either commensurable with *HF* or not commensurable.

First, let the square on it be greater by the square on a straight line commensurable with it.

Now the whole *HF* is commensurable in length with the rational straight line *FG* set out; therefore *KH* is a first apotome. [[book-10/definitions#Definition 1 (part 3)|X. Deff. III. 1]]

But the side of the rectangle contained by a rational straight line and a first apotome is an apotome. [[book-10/proposition-91|X. 91]]

Therefore the side of *LH*, that is, of *EC*, is an apotome.

But, if the square on *HF* is greater than the square on *FK* by the square on a straight line incommensurable with *HF*, while the whole *FH* is commensurable in length with the rational straight line *FG* set out, *KH* is a fourth apotome. [[book-10/definitions#Definition 4 (part 3)|X. Deff. III. 4]]

But the side of the rectangle contained by a rational straight line and a fourth apotome is minor. [[book-10/proposition-94|X. 94]] Q. E. D.