---
book: 10
number: 89
id: "X.89"
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_3.89"
license: "CC-BY-SA-4.0"
---

# X.89

*To find the fifth apotome*.

## Proof

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

Let two numbers *DF*, *FE* be set out such that *DE* again has not to either of the numbers *DF*, *FE* the ratio which a square number has to a square number; and let it be contrived that, as *FE* is to *ED*, so is the square on *CG* to the square on *GB*.

Therefore the square on *GB* is also rational; [[book-10/proposition-6|X. 6]] therefore *BG* is also rational.

Now since, as *DE* is to *EF*, so is the square on *BG* to the square on *GC*, while *DE* has not to *EF* the ratio which a square number has to a square number, therefore neither has the square on *BG* to the square on *GC* the ratio which a square number has to a square number; therefore *BG* is incommensurable in length with *GC*. [[book-10/proposition-9|X. 9]]

And both are rational; therefore *BG*, *GC* 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 fifth 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 *DE* to *EF*, therefore, convertendo, as *ED* is to *DF*, so is the square on *BG* to the square on *H*. [[book-5/proposition-19|V. 19, Por.]]

But *ED* has not to *DF* the ratio which a square number has to a square number; therefore neither has the square on *BG* to the square on *H* the ratio which a square number has to a square number; therefore *BG* is incommensurable 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 *GB* is greater than the square on *GC* by the square on a straight line incommensurable in length with *GB*.

And the annex *CG* is commensurable in length with the rational straight line *A* set out; therefore *BC* is a fifth apotome. [[book-10/definitions#Definition 5 (part 3)|X. Deff. III. 5]]

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