---
book: 10
number: 52
id: "X.52"
kind: "construction"
uses: ["[[book-10/proposition-6]]", "[[book-10/proposition-9]]", "[[book-10/proposition-36]]", "[[book-5/proposition-19]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.52"
license: "CC-BY-SA-4.0"
---

# X.52

*To find the fifth binomial straight line*.

## Proof

Let two numbers *AC*, *CB* be set out such that *AB* has not to either of them the ratio which a square number has to a square number; let any rational straight line *D* be set out, and let *EF* be commensurable with *D*; therefore *EF* is rational.

Let it be contrived that, as *CA* is to *AB*, so is the square on *EF* to the square on *FG*. [[book-10/proposition-6|X. 6, Por.]]

But *CA* has not to *AB* the ratio which a square number has to a square number; therefore neither has the square on *EF* to the square on *FG* the ratio which a square number has to a square number.

Therefore *EF*, *FG* are rational straight lines commensurable in square only; [[book-10/proposition-9|X. 9]] therefore *EG* is binomial. [[book-10/proposition-36|X. 36]]

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

For since, as *CA* is to *AB*, so is the square on *EF* to the square on *FG*, inversely, as *BA* is to *AC*, so is the square on *FG* to the square on *FE*; therefore the square on *GF* is greater than the square on *FE*.

Let then the squares on *EF*, *H* be equal to the square on *GF*; therefore, convertendo, as the number *AB* is to *BC*, so is the square on *GF* to the square on *H*. [[book-5/proposition-19|V. 19, Por.]]

But *AB* has not to *BC* the ratio which a square number has to a square number; therefore neither has the square on *FG* to the square on *H* the ratio which a square number has to a square number.

Therefore *FG* is incommensurable in length with *H*; [[book-10/proposition-9|X. 9]] so that the square on *FG* is greater than the square on *FE* by the square on a straight line incommensurable with *FG*.

And *GF*, *FE* are rational straight lines commensurable in square only, and the lesser term *EF* is commensurable in length with the rational straight line *D* set out.

Therefore *EG* is a fifth binomial straight line. Q. E. D.