---
book: 10
number: 10
id: "X.10"
kind: "construction"
uses: ["[[book-10/proposition-6]]", "[[book-10/proposition-9]]", "[[book-10/proposition-11]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.10"
license: "CC-BY-SA-4.0"
---

# X.10

*To find two straight lines incommensurable, the one in length only, and the other in square also, with an assigned straight line*.

## Proof

Let *A* be the assigned straight line; thus it is required to find two straight lines incommensurable, the one in length only, and the other in square also, with *A*.

Let two numbers *B*, *C* be set out which have not to one another the ratio which a square number has to a square number, that is, which are not similar plane numbers; and let it be contrived that, as *B* is to *C*, so is the square on *A* to the square on *D* —for we have learnt how to do this— [[book-10/proposition-6|X. 6, Por.]] therefore the square on *A* is commensurable with the square on *D*. [[book-10/proposition-6|X. 6]]

And, since *B* has not to *C* the ratio which a square number has to a square number, therefore neither has the square on *A* to the square on *D* the ratio which a square number has to a square number; therefore *A* is incommensurable in length with *D*. [[book-10/proposition-9|X. 9]]

Let *E* be taken a mean proportional between *A*, *D*; therefore, as *A* is to *D*, so is the square on *A* to the square on *E*. [[book-5/definitions#Definition 9|V. Def. 9]]

But *A* is incommensurable in length with *D*; therefore the square on *A* is also incommensurable with the square on *E*; [[book-10/proposition-11|X. 11]] therefore *A* is incommensurable in square with *E*.

Therefore two straight lines *D*, *E* have been found incommensurable, *D* in length only, and *E* in square and of course in length also, with the assigned straight line *A*.]