---
book: 10
number: 63
id: "X.63"
kind: "theorem"
uses: ["[[book-10/proposition-39]]", "[[book-10/proposition-20]]", "[[book-10/proposition-22]]", "[[book-10/proposition-13]]", "[[book-10/proposition-36]]", "[[book-6/proposition-1]]", "[[book-10/proposition-11]]", "[[book-10/proposition-18]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.63"
license: "CC-BY-SA-4.0"
---

# X.63

*The square on the major straight line applied to a rational straight line produces as breadth the fourth binomial*.

## Proof

Let *AB* be a major straight line divided at *C*, so that *AC* is greater than *CB*; let *DE* be a rational straight line, and to *DE* let there be applied the parallelogram *DF* equal to the square on *AB* and producing *DG* as its breadth; I say that *DG* is a fourth binomial straight line.

Let the same construction be made as before shown.

Then, since *AB* is a major straight line divided at *C*, *AC*, *CB* are straight lines incommensurable in square which make the sum of the squares on them rational, but the rectangle contained by them medial. [[book-10/proposition-39|X. 39]]

Since then the sum of the squares on *AC*, *CB* is rational, therefore *DL* is rational; therefore *DM* is also rational and commensurable in length with *DE*. [[book-10/proposition-20|X. 20]]

Again, since twice the rectangle *AC*, *CB*, that is, *MF*, is medial, and it is applied to the rational straight line *ML*, therefore *MG* is also rational and incommensurable in length with *DE*; [[book-10/proposition-22|X. 22]] therefore *DM* is also incommensurable in length with *MG*. [[book-10/proposition-13|X. 13]]

Therefore *DM*, *MG* are rational straight lines commensurable in square only; therefore *DG* is binomial. [[book-10/proposition-36|X. 36]]

It is to be proved that it is also a fourth binomial straight line.

In manner similar to the foregoing we can prove that *DM* is greater than *MG*, and that the rectangle *DK*, *KM* is equal to the square on *MN*.

Since then the square on *AC* is incommensurable with the square on *CB*, therefore *DH* is also incommensurable with *KL*, so that *DK* is also incommensurable with *KM*. [[book-6/proposition-1|VI. 1]], [[book-10/proposition-11|X. 11]]

But, if there be two unequal straight lines, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into incommensurable parts, then the square on the greater will be greater than the square on the less by the square on a straight line incommensurable in length with the greater; [[book-10/proposition-18|X. 18]] therefore the square on *DM* is greater than the square on *MG* by the square on a straight line incommensurable with *DM*.

And *DM*, *MG* are rational straight lines commensurable in square only, and *DM* is commensurable with the rational straight line *DE* set out.

Therefore *DG* is a fourth binomial straight line. [[book-10/definitions#Definition 4 (part 2)|X. Deff. II. 4]] Q. E. D.