---
book: 10
number: 64
id: "X.64"
kind: "theorem"
uses: ["[[book-10/proposition-40]]", "[[book-10/proposition-22]]", "[[book-10/proposition-20]]", "[[book-10/proposition-13]]", "[[book-10/proposition-36]]", "[[book-10/proposition-18]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.64"
license: "CC-BY-SA-4.0"
---

# X.64

*The square on the side of a rational plus a medial area applied to a rational straight line produces as breadth the fifth binomial*.

## Proof

Let *AB* be the side of a rational plus a medial area, divided into its straight lines at *C*, so that *AC* is the greater; let a rational straight line *DE* be set out, and let there be applied to *DE* the parallelogram *DF* equal to the square on *AB*, producing *DG* as its breadth; I say that *DG* is a fifth binomial straight line.

Let the same construction as before be made.

Since then *AB* is the side of a rational plus a medial area, divided at *C*, therefore *AC*, *CB* are straight lines incommensurable in square which make the sum of the squares on them medial, but the rectangle contained by them rational. [[book-10/proposition-40|X. 40]]

Since then the sum of the squares on *AC*, *CB* is medial, therefore *DL* is medial, so that *DM* is rational and incommensurable in length with *DE*. [[book-10/proposition-22|X. 22]]

Again, since twice the rectangle *AC*, *CB*, that is *MF*, is rational, therefore *MG* is rational and commensurable with *DE*. [[book-10/proposition-20|X. 20]]

Therefore *DM* is incommensurable 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]]

I say next that it is also a fifth binomial straight line.

For it can be proved similarly that the rectangle *DK*, *KM* is equal to the square on *MN*, and that *DK* is incommensurable in length with *KM*; therefore the square on *DM* is greater than the square on *MG* by the square on a straight line incommensurable with *DM*. [[book-10/proposition-18|X. 18]]

And *DM*, *MG* are commensurable in square only, and the less, *MG*, is commensurable in length with *DE*.

Therefore *DG* is a fifth binomial. Q. E. D.