---
book: 10
number: 68
id: "X.68"
kind: "theorem"
uses: ["[[book-10/proposition-39]]", "[[book-5/proposition-11]]", "[[book-10/proposition-11]]", "[[book-5/proposition-16]]", "[[book-5/proposition-18]]", "[[book-6/proposition-20]]", "[[book-10/proposition-23]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.68"
license: "CC-BY-SA-4.0"
---

# X.68

*A straight line commensurable with a major straight line is itself also major*.

## Proof

Let *AB* be major, and let *CD* be commensurable with *AB*; I say that *CD* is major.

Let *AB* be divided at *E*; therefore *AE*, *EB* 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]]

Let the same construction be made as before.

Then since, as *AB* is to *CD*, so is *AE* to *CF*, and *EB* to *FD*, therefore also, as *AE* is to *CF*, so is *EB* to *FD*. [[book-5/proposition-11|V. 11]]

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

And since, as *AE* is to *CF*, so is *EB* to *FD*, alternately also, as *AE* is to *EB*, so is *CF* to *FD*; [[book-5/proposition-16|V. 16]] therefore also, componendo, as *AB* is to *BE*, so is *CD* to *DF*; [[book-5/proposition-18|V. 18]] therefore also, as the square on *AB* is to the square on *BE*, so is the square on *CD* to the square on *DF*. [[book-6/proposition-20|VI. 20]]

Similarly we can prove that, as the square on *AB* is to the square on *AE*, so also is the square on *CD* to the square on *CF*.

Therefore also, as the square on *AB* is to the squares on *AE*, *EB*, so is the square on *CD* to the squares on *CF*, *FD*; therefore also, alternately, as the square on *AB* is to the square on *CD*, so are the squares on *AE*, *EB* to the squares on *CF*, *FD*. [[book-5/proposition-16|V. 16]]

But the square on *AB* is commensurable with the square on *CD*; therefore the squares on *AE*, *EB* are also commensurable with the squares on *CF*, *FD*.

And the squares on *AE*, *EB* together are rational; therefore the squares on *CF*, *FD* together are rational.

Similarly also twice the rectangle *AE*, *EB* is commensurable with twice the rectangle *CF*, *FD*.

And twice the rectangle *AE*, *EB* is medial; therefore twice the rectangle *CF*, *FD* is also medial. [[book-10/proposition-23|X. 23, Por.]]

Therefore *CF*, *FD* are straight lines incommensurable in square which make, at the same time, the sum of the squares on them rational, but the rectangle contained by them medial; therefore the whole *CD* is the irrational straight line called major. [[book-10/proposition-39|X. 39]]

Therefore a straight line commensurable with the major straight line is major. Q. E. D.