---
book: 10
number: 103
id: "X.103"
kind: "theorem"
uses: ["[[book-10/proposition-73]]", "[[book-6/proposition-12]]", "[[book-5/proposition-12]]", "[[book-10/proposition-11]]", "[[book-10/proposition-13]]", "[[book-5/proposition-16]]", "[[book-10/proposition-14]]", "[[book-10/proposition-12]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_3.103"
license: "CC-BY-SA-4.0"
---

# X.103

*A straight line commensurable in length with an apotome is an apotome and the same in order*.

## Proof

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

For, since *AB* is an apotome, let *BE* be the annex to it; therefore *AE*, *EB* are rational straight lines commensurable in square only. [[book-10/proposition-73|X. 73]]

Let it be contrived that the ratio of *BE* to *DF* is the same as the ratio of *AB* to *CD*; [[book-6/proposition-12|VI. 12]] therefore also, as one is to one, so are all to all; [[book-5/proposition-12|V. 12]] therefore also, as the whole *AE* is to the whole *CF*, so is *AB* to *CD*.

But *AB* is commensurable in length with *CD*.

Therefore *AE* is also commensurable with *CF*, and *BE* with *DF*. [[book-10/proposition-11|X. 11]]

And *AE*, *EB* are rational straight lines commensurable in square only; therefore *CF*, *FD* are also rational straight lines commensurable in square only. [[book-10/proposition-13|X. 13]]

Now since, as *AE* is to *CF*, so is *BE* to *DF*, alternately therefore, as *AE* is to *EB*, so is *CF* to *FD*. [[book-5/proposition-16|V. 16]]

And the square on *AE* is greater than the square on *EB* either by the square on a straight line commensurable with *AE* or by the square on a straight line incommensurable with it.

If then the square on *AE* is greater than the square on *EB* by the square on a straight line commensurable with *AE*, the square on *CF* will also be greater than the square on *FD* by the square on a straight line commensurable with *CF*. [[book-10/proposition-14|X. 14]]

And, if *AE* is commensurable in length with the rational straight line set out, *CF* is so also, [[book-10/proposition-12|X. 12]] if *BE*, then *DF* also, [id.] and, if neither of the straight lines *AE*, *EB*, then neither of the straight lines *CF*, *FD*. [[book-10/proposition-13|X. 13]]

But, if the square on *AE* is greater than the square on *EB* by the square on a straight line incommensurable with *AE*, the square on *CF* will also be greater than the square on *FD* by the square on a straight line incommensurable with *CF*. [[book-10/proposition-14|X. 14]]

And, if *AE* is commensurable in length with the rational straight line set out, *CF* is so also, if *BE*, then *DF* also, [[book-10/proposition-12|X. 12]] and, if neither of the straight lines *AE*, *EB*, then neither of the straight lines *CF*, *FD*. [[book-10/proposition-13|X. 13]]

Therefore *CD* is an apotome and the same in order with *AB*. Q. E. D.