---
book: 10
number: 18
id: "X.18"
kind: "theorem"
uses: ["[[book-10/proposition-16]]", "[[book-10/proposition-6]]", "[[book-10/proposition-13]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.18"
license: "CC-BY-SA-4.0"
---

# X.18

*If there be two unequal straight lines, and to the greater there be applied a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, and if it divide it into parts which are incommensurable, the square on the greater will be greater than the square on the less by the square on a straight line incommensurable with the greater.*

## Proof

*And, if the square on the greater be greater than the square on the less by the square on a straight line incommensurable with the greater, and if there be applied to the greater a parallelogram equal to the fourth part of the square on the less and deficient by a square figure, it divides it into parts which are incommensurable*.

Let *A*, *BC* be two unequal straight lines, of which *BC* is the greater, and to *BC* let there be applied a parallelogram equal to the fourth part of the square on the less, *A*, and deficient by a square figure. Let this be the rectangle *BD*, *DC*, cf. [[book-10/proposition-16|Lemma before X. 17]] and let *BD* be incommensurable in length with *DC*; I say that the square on *BC* is greater than the square on *A* by the square on a straight line incommensurable with *BC*.

For, with the same construction as before, we can prove similarly that the square on *BC* is greater than the square on *A* by the square on *FD*.

It is to be proved that *BC* is incommensurable in length with *DF*.

Since *BD* is incommensurable in length with *DC*, therefore *BC* is also incommensurable in length with *CD*. [[book-10/proposition-16|X. 16]]

But *DC* is commensurable with the sum of *BF*, *DC*; [[book-10/proposition-6|X. 6]] therefore *BC* is also incommensurable with the sum of *BF*, *DC*; [[book-10/proposition-13|X. 13]] so that *BC* is also incommensurable in length with the remainder *FD*. [[book-10/proposition-16|X. 16]]

And the square on *BC* is greater than the square on *A* by the square on *FD*; therefore the square on *BC* is greater than the square on *A* by the square on a straight line incommensurable with *BC*.

Again, let the square on *BC* be greater than the square on *A* by the square on a straight line incommensurable with *BC*, and let there be applied to *BC* a parallelogram equal to the fourth part of the square on *A* and deficient by a square figure. Let this be the rectangle *BD*, *DC*.

It is to be proved that *BD* is incommensurable in length with *DC*.

For, with the same construction, we can prove similarly that the square on *BC* is greater than the square on *A* by the square on *FD*.

But the square on *BC* is greater than the square on *A* by the square on a straight line incommensurable with *BC*; therefore *BC* is incommensurable in length with *FD*. so that *BC* is also incommensurable with the remainder, the sum of *BF*, *DC*. [[book-10/proposition-16|X. 16]]

But the sum of *BF*, *DC* is commensurable in length with *DC*; [[book-10/proposition-6|X. 6]] therefore *BC* is also incommensurable in length with *DC*, [[book-10/proposition-13|X. 13]] so that, separando, *BD* is also incommensurable in length with *DC*. [[book-10/proposition-16|X. 16]]

Therefore etc.

[Lemma. Since it has been proved that straight lines commensurable in length are always commensurable in square also, while those commensurable in square are not always commensurable in length also, but can of course be either commensurable or incommensurable in length, it is manifest that, if any straight line be commensurable in length with a given rational straight line, it is called rational and commensurable with the other not only in length but in square also, since straight lines commensurable in length are always commensurable in square also.

But, if any straight line be commensurable in square with a given rational straight line, then, if it is also commensurable in length with it, it is called in this case also rational and commensurable with it both in length and in square; but, if again any straight line, being commensurable in square with a given rational straight line, be incommensurable in length with it, it is called in this case also rational but commensurable in square only.]