---
book: 10
number: 42
id: "X.42"
kind: "theorem"
uses: ["[[book-10/proposition-36]]", "[[book-2/proposition-4]]", "[[book-10/proposition-21]]", "[[book-10/proposition-26]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.42"
license: "CC-BY-SA-4.0"
---

# X.42

Let *AB* be a binomial straight line divided into its terms at *C*; therefore *AC*, *CB* are rational straight lines commensurable in square only. [[book-10/proposition-36|X. 36]]

## Proof

I say that *AB* is not divided at another point into two rational straight lines commensurable in square only.

For, if possible, let it be divided at *D* also, so that *AD*, *DB* are also rational straight lines commensurable in square only.

It is then manifest that *AC* is not the same with *DB*.

For, if possible, let it be so.

Then *AD* will also be the same as *CB*, and, as *AC* is to *CB*, so will *BD* be to *DA*; thus *AB* will be divided at *D* also in the same way as by the division at *C*: which is contrary to the hypothesis.

Therefore *AC* is not the same with *DB*.

For this reason also the points *C*, *D* are not equidistant from the point of bisection.

Therefore that by which the squares on *AC*, *CB* differ from the squares on *AD*, *DB* is also that by which twice the rectangle *AD*, *DB* differs from twice the rectangle *AC*, *CB*, because both the squares on *AC*, *CB* together with twice the rectangle *AC*, *CB*, and the squares on *AD*, *DB* together with twice the rectangle *AD*, *DB*, are equal to the square on *AB*. [[book-2/proposition-4|II. 4]]

But the squares on *AC*, *CB* differ from the squares on *AD*, *DB* by a rational area, for both are rational; therefore twice the rectangle *AD*, *DB* also differs from twice the rectangle *AC*, *CB* by a rational area, though they are medial [[book-10/proposition-21|X. 21]] : which is absurd, for a medial area does not exceed a medial by a rational area. [[book-10/proposition-26|x. 26]]

Therefore a binomial straight line is not divided at different points; therefore it is divided at one point only. Q. E. D.