---
book: 10
number: 111
id: "X.111"
kind: "theorem"
uses: ["[[book-10/proposition-97]]", "[[book-10/proposition-60]]", "[[book-10/proposition-12]]", "[[book-10/proposition-15]]", "[[book-10/proposition-13]]", "[[book-10/proposition-73]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_3.111"
license: "CC-BY-SA-4.0"
---

# X.111

*The apotome is not the same with the binomial straight line*.

## Proof

Let *AB* be an apotome; I say that *AB* is not the same with the binomial straight line.

For, if possible, let it be so; let a rational straight line *DC* be set out, and to *CD* let there be applied the rectangle *CE* equal to the square on *AB* and producing *DE* as breadth.

Then, since *AB* is an apotome, *DE* is a first apotome. [[book-10/proposition-97|X. 97]]

Let *EF* be the annex to it; therefore *DF*, *FE* are rational straight lines commensurable in square only, the square on *DF* is greater than the square on *FE* by the square on a straight line commensurable with *DF*, and *DF* is commensurable in length with the rational straight line *DC* set out. [[book-10/definitions#Definition 1 (part 3)|X. Deff. III. 1]]

Again, since *AB* is binomial, therefore *DE* is a first binomial straight line. [[book-10/proposition-60|X. 60]]

Let it be divided into its terms at *G*, and let *DG* be the greater term; therefore *DG*, *GE* are rational straight lines commensurable in square only, the square on *DG* is greater than the square on *GE* by the square on a straight line commensurable with *DG*, and the greater term *DG* is commensurable in length with the rational straight line *DC* set out. [[book-10/definitions#Definition 1 (part 2)|X. Deff. II. 1]]

Therefore *DF* is also commensurable in length with *DG*; [[book-10/proposition-12|X. 12]] therefore the remainder *GF* is also commensurable in length with *DF*. [[book-10/proposition-15|X. 15]]

But *DF* is incommensurable in length with *EF*; therefore *FG* is also incommensurable in length with *EF*. [[book-10/proposition-13|X. 13]]

Therefore *GF*, *FE* are rational straight lines commensurable in square only; therefore *EG* is an apotome. [[book-10/proposition-73|X. 73]]

But it is also rational: which is impossible.

Therefore the apotome is not the same with the binomial straight line. Q. E. D.