---
book: 10
number: 67
id: "X.67"
kind: "theorem"
uses: ["[[book-10/proposition-37]]", "[[book-10/proposition-38]]", "[[book-5/proposition-19]]", "[[book-10/proposition-11]]", "[[book-10/proposition-23]]", "[[book-5/proposition-11]]", "[[book-5/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.67"
license: "CC-BY-SA-4.0"
---

# X.67

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

## Proof

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

For, since *AB* is bimedial, let it be divided into its medials at *E*; therefore *AE*, *EB* are medial straight lines commensurable in square only. [[book-10/proposition-37|X. 37, 38]]

And let it be contrived that, as *AB* is to *CD*, so is *AE* to *CF*; 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*, *EB* are also commensurable with *CF*, *FD* respectively. [[book-10/proposition-11|X. 11]]

But *AE*, *EB* are medial; therefore *CF*, *FD* are also medial. [[book-10/proposition-23|X. 23]]

And since, as *AE* is to *EB*, so is *CF* to *FD*, [[book-5/proposition-11|V. 11]] and *AE*, *EB* are commensurable in square only, *CF*, *FD* are also commensurable in square only. [[book-10/proposition-11|X. 11]]

But they were also proved medial; therefore *CD* is bimedial.

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

For since, as *AE* is to *EB*, so is *CF* to *FD*, therefore also, as the square on *AE* is to the rectangle *AE*, *EB*, so is the square on *CF* to the rectangle *CF*, *FD*; therefore, alternately, as the square on *AE* is to the square on *CF*, so is the rectangle *AE*, *EB* to the rectangle *CF*, *FD*. [[book-5/proposition-16|V. 16]]

But the square on *AE* is commensurable with the square on *CF*; therefore the rectangle *AE*, *EB* is also commensurable with the rectangle *CF*, *FD*.

If therefore the rectangle *AE*, *EB* is rational, the rectangle *CF*, *FD* is also rational, [and for this reason *CD* is a first bimedial]; [[book-10/proposition-37|X. 37]] but if medial, medial, [[book-10/proposition-23|X. 23, Por.]] and each of the straight lines *AB*, *CD* is a second bimedial. [[book-10/proposition-38|X. 38]]

And for this reason *CD* will be the same in order with *AB*. Q. E. D.