---
book: 10
number: 105
id: "X.105"
kind: "theorem"
uses: ["[[book-10/proposition-76]]", "[[book-10/proposition-13]]", "[[book-5/proposition-12]]", "[[book-5/proposition-16]]", "[[book-6/proposition-22]]", "[[book-5/proposition-18]]", "[[book-10/proposition-11]]", "[[book-10/proposition-23]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_3.105"
license: "CC-BY-SA-4.0"
---

# X.105

*A straight line commensurable with a minor straight line is minor*.

## Proof

Let *AB* be a minor straight line, and *CD* commensurable with *AB*; I say that *CD* is also minor.

Let the same construction be made as before; then, since *AE*, *EB* are incommensurable in square, [[book-10/proposition-76|X. 76]] therefore *CF*, *FD* are also incommensurable in square. [[book-10/proposition-13|X. 13]]

Now since, as *AE* is to *EB*, so is *CF* to *FD*, [[book-5/proposition-12|V. 12]], [[book-5/proposition-16|V. 16]] therefore also, as the square on *AE* is to the square on *EB*, so is the square on *CF* to the square on *FD*. [[book-6/proposition-22|VI. 22]]

Therefore, componendo, as the squares on *AE*, *EB* are to the square on *EB*, so are the squares on *CF*, *FD* to the square on *FD*. [[book-5/proposition-18|V. 18]]

But the square on *BE* is commensurable with the square on *DF*; therefore the sum of the squares on *AE*, *EB* is also commensurable with the sum of the squares on *CF*, *FD*. [[book-5/proposition-16|V. 16]], [[book-10/proposition-11|X. 11]]

But the sum of the squares on *AE*, *EB* is rational; [[book-10/proposition-76|X. 76]] therefore the sum of the squares on *CF*, *FD* is also rational. [[book-10/definitions#Definition 4|X. Def. 4]]

Again, since, as the square on *AE* is to the rectangle *AE*, *EB*, so is the square on *CF* to the rectangle *CF*, *FD*, while the square on *AE* is commensurable with the square on *CF*, therefore the rectangle *AE*, *EB* is also commensurable with the rectangle *CF*, *FD*.

But the rectangle *AE*, *EB* is medial; [[book-10/proposition-76|X. 76]] therefore 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 the sum of the squares on them rational, but the rectangle contained by them medial.

Therefore *CD* is minor. [[book-10/proposition-76|X. 76]] Q. E. D.