---
book: 10
number: 80
id: "X.80"
kind: "theorem"
uses: ["[[book-10/proposition-74]]", "[[book-2/proposition-7]]", "[[book-10/proposition-15]]", "[[book-10/proposition-23]]", "[[book-10/proposition-26]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.80"
license: "CC-BY-SA-4.0"
---

# X.80

*To a first apotome of a medial straight line only one medial straight line can be annexed which is commensurable with the whole in square only and which contains with the whole a rational rectangle*.

## Proof

For let *AB* be a first apotome of a medial straight line, and let *BC* be an annex to *AB*; therefore *AC*, *CB* are medial straight lines commensurable in square only and such that the rectangle *AC*, *CB* which they contain is rational; [[book-10/proposition-74|X. 74]] I say that no other medial straight line can be annexed to *AB* which is commensurable with the whole in square only and which contains with the whole a rational area.

For, if possible, let *DB* also be so annexed; therefore *AD*, *DB* are medial straight lines commensurable in square only and such that the rectangle *AD*, *DB* which they contain is rational. [[book-10/proposition-74|X. 74]]

Now, since the excess of the squares on *AD*, *DB* over twice the rectangle *AD*, *DB* is also the excess of the squares on *AC*, *CB* over twice the rectangle *AC*, *CB*, for they exceed by the same, the square on *AB*, [[book-2/proposition-7|II. 7]] therefore, alternately, the excess of the squares on *AD*, *DB* over the squares on *AC*, *CB* is also the excess of twice the rectangle *AD*, *DB* over twice the rectangle *AC*, *CB*.

But twice the rectangle *AD*, *DB* exceeds twice the rectangle *AC*, *CB* by a rational area, for both are rational.

Therefore the squares on *AD*, *DB* also exceed the squares on *AC*, *CB* by a rational area. which is impossible, for both are medial [[book-10/proposition-15|X. 15 and 23, Por.]], and a medial area does not exceed a medial by a rational area. [[book-10/proposition-26|X. 26]]

Therefore etc. Q. E. D.