---
book: 10
number: 66
id: "X.66"
kind: "theorem"
uses: ["[[book-10/proposition-36]]", "[[book-6/proposition-12]]", "[[book-5/proposition-19]]", "[[book-10/proposition-11]]", "[[book-5/proposition-11]]", "[[book-5/proposition-16]]", "[[book-10/proposition-14]]", "[[book-10/proposition-12]]", "[[book-10/proposition-13]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.66"
license: "CC-BY-SA-4.0"
---

# X.66

*A straight line commensurable in length with a binomial straight line is itself also binomial and the same in order*.

## Proof

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

For, since *AB* is binomial, let it be divided into its terms at *E*, and let *AE* be the greater term; therefore *AE*, *EB* are rational straight lines commensurable in square only. [[book-10/proposition-36|X. 36]]

Let it be contrived that, as *AB* is to *CD*, so is *AE* to *CF*; [[book-6/proposition-12|VI. 12]] therefore also the remainder *EB* is to the remainder *FD* as *AB* is to *CD*. [[book-5/proposition-19|V. 19]]

But *AB* is commensurable in length with *CD*; therefore *AE* is also commensurable with *CF*, and *EB* with *FD*. [[book-10/proposition-11|X. 11]]

And *AE*, *EB* are rational; therefore *CF*, *FD* are also rational.

And, as *AE* is to *CF*, so is *EB* to *FD*. [[book-5/proposition-11|V. 11]]

Therefore, alternately, as *AE* is to *EB*, so is *CF* to *FD*. [[book-5/proposition-16|V. 16]]

But *AE*, *EB* are commensurable in square only; therefore *CF*, *FD* are also commensurable in square only. [[book-10/proposition-11|X. 11]]

And they are rational; therefore *CD* is binomial. [[book-10/proposition-36|X. 36]]

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

For 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 with the rational straight line set out, *CF* will also be commensurable with it, [[book-10/proposition-12|X. 12]] and for this reason each of the straight lines *AB*, *CD* is a first binomial, that is, the same in order. [[book-10/definitions#Definition 1 (part 2)|X. Deff. II. 1]]

But, if *EB* is commensurable with the rational straight line set out, *FD* is also commensurable with it, [[book-10/proposition-12|X. 12]] and for this reason again *CD* will be the same in order with *AB*, for each of them will be a second binomial. [[book-10/definitions#Definition 2 (part 2)|X. Deff. II. 2]]

But, if neither of the straight lines *AE*, *EB* is commensurable with the rational straight line set out, neither of the straight lines *CF*, *FD* will be commensurable with it, [[book-10/proposition-13|X. 13]] and each of the straight lines *AB*, *CD* is a third binomial. [[book-10/definitions#Definition 3 (part 2)|X. Deff. II. 3]]

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* is also 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 with the rational straight line set out, *CF* is also commensurable with it, and each of the straight lines *AB*, *CD* is a fourth binomial. [[book-10/definitions#Definition 4 (part 2)|X. Deff. II. 4]]

But, if *EB* is so commensurable, so is *FD* also, and each of the straight lines *AB*, *CD* will be a fifth binomial. [[book-10/definitions#Definition 5 (part 2)|X. Deff. II. 5]]

But, if neither of the straight lines *AE*, *EB* is so commensurable, neither of the straight lines *CF*, *FD* is commensurable with the rational straight line set out, and each of the straight lines *AB*, *CD* will be a sixth binomial. [[book-10/definitions#Definition 6 (part 2)|X. Deff. II. 6]]

Hence a straight line commensurable in length with a binomial straight line is binomial and the same in order. Q. E. D.