---
book: 10
number: 86
id: "X.86"
kind: "construction"
uses: ["[[book-10/proposition-6]]", "[[book-10/proposition-9]]", "[[book-10/proposition-73]]", "[[book-5/proposition-19]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_3.86"
license: "CC-BY-SA-4.0"
---

# X.86

*To find the second apotome*.

## Proof

Let a rational straight line *A* be set out, and *GC* commensurable in length with *A*; therefore *GC* is rational.

Let two square numbers *DE*, *EF* be set out, and let their difference *DF* not be square.

Now let it be contrived that, as *FD* is to *DE*, so is the square on *CG* to the square on *GB*. [[book-10/proposition-6|X. 6, Por.]]

Therefore the square on *CG* is commensurable with the square on *GB*. [[book-10/proposition-6|X. 6]]

But the square on *CG* is rational; therefore the square on *GB* is also rational; therefore *BG* is rational.

And, since the square on *GC* has not to the square on *GB* the ratio which a square number has to a square number, *CG* is incommensurable in length with *GB*. [[book-10/proposition-9|X. 9]]

And both are rational; therefore *CG*, *GB* are rational straight lines commensurable in square only; therefore *BC* is an apotome. [[book-10/proposition-73|X. 73]]

I say next that it is also a second apotome.

For let the square on *H* be that by which the square on *BG* is greater than the square on *GC*.

Since then, as the square on *BG* is to the square on *GC*, so is the number *ED* to the number *DF*, therefore, convertendo, as the square on *BG* is to the square on *H*, so is *DE* to *EF*. [[book-5/proposition-19|V. 19, Por.]]

And each of the numbers *DE*, *EF* is square; therefore the square on *BG* has to the square on *H* the ratio which a square number has to a square number; therefore *BG* is commensurable in length with *H*. [[book-10/proposition-9|X. 9]]

And the square on *BG* is greater than the square on *GC* by the square on *H*; therefore the square on *BG* is greater than the square on *GC* by the square on a straight line commensurable in length with *BG*.

And *CG*, the annex, is commensurable with the rational straight line *A* set out.

Therefore *BC* is a second apotome. [[book-10/definitions#Definition 2 (part 3)|X. Deff. III. 2]]

Therefore the second apotome *BC* has been found. Q. E. D.