---
book: 10
number: 104
id: "X.104"
kind: "theorem"
uses: ["[[book-10/proposition-74]]", "[[book-10/proposition-75]]", "[[book-6/proposition-12]]", "[[book-5/proposition-12]]", "[[book-10/proposition-11]]", "[[book-10/proposition-23]]", "[[book-10/proposition-13]]", "[[book-5/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_3.104"
license: "CC-BY-SA-4.0"
---

# X.104

*A straight line commensurable with an apotome of a medial straight line is an apotome of a medial straight line and the same in order*.

## Proof

Let *AB* be an apotome of a medial straight line, and let *CD* be commensurable in length with *AB*; I say that *CD* is also an apotome of a medial straight line and the same in order with *AB*.

For, since *AB* is an apotome of a medial straight line, let *EB* be the annex to it.

Therefore *AE*, *EB* are medial straight lines commensurable in square only. [[book-10/proposition-74|X. 74, 75]]

Let it be contrived that, as *AB* is to *CD*, so is *BE* to *DF*; [[book-6/proposition-12|VI. 12]] therefore *AE* is also commensurable with *CF*, and *BE* with *DF*. [[book-5/proposition-12|V. 12]], [[book-10/proposition-11|X. 11]]

But *AE*, *EB* are medial straight lines commensurable in square only; therefore *CF*, *FD* are also medial straight lines [[book-10/proposition-23|X. 23]] commensurable in square only; [[book-10/proposition-13|X. 13]] therefore *CD* is an apotome of a medial straight line. [[book-10/proposition-74|X. 74, 75]]

I say next that it is also the same in order with *AB*.

Since, as *AE* is to *EB*, so is *CF* to *FD*, therefore also, as the square on *AE* is to the rectangle *AE*, *EB*, so is the square on *CF* to the rectangle *CF*, *FD*.

But the square on *AE* is commensurable with the square on *CF*; therefore the rectangle *AE*, *EB* is also commensurable with the rectangle *CF*, *FD*. [[book-5/proposition-16|V. 16]], [[book-10/proposition-11|X. 11]]

Therefore, if the rectangle *AE*, *EB* is rational, the rectangle *CF*, *FD* will also be rational, [[book-10/definitions#Definition 4|X. Def. 4]] and if the rectangle *AE*, *EB* is medial, the rectangle *CF*, *FD* is also medial. [[book-10/proposition-23|X. 23, Por.]]

Therefore *CD* is an apotome of a medial straight line and the same in order with *AB*. [[book-10/proposition-74|X. 74, 75]] Q. E. D.