---
book: 10
number: 85
id: "X.85"
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_3.85"
license: "CC-BY-SA-4.0"
---

# X.85

*To find the first apotome*.

## Proof

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

Let two square numbers *DE*, *EF* be set out, and let their difference *FD* not be square; therefore neither has *ED* to *DF* the ratio which a square number has to a square number.

Let it be contrived that, as *ED* is to *DF*, so is the square on *BG* to the square on *GC*; [[book-10/proposition-6|X. 6, Por.]] therefore the square on *BG* is commensurable with the square on *GC*. [[book-10/proposition-6|X. 6]]

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

And, since *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 *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 first apotome.

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

Now since. as *ED* is to *FD*, so is the square on *BG* to the square on *GC*, therefore also, convertendo, [[book-5/proposition-19|v. 19, Por.]] as *DE* is to *EF*, so is the square on *GB* to the square on *H*.

But *DE* has to *EF* the ratio which a square number has to a square number, for each is square; therefore the square on *GB* also 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 a straight line commensurable in length with *BG*.

And the whole *BG* is commensurable in length with the rational straight line *A* set out.

Therefore *BC* is a first apotome. [[book-10/definitions#Definition 1 (part 3)|X. Deff. III. 1]]

Therefore the first apotome *BC* has been found. (Being) that which it was required to find.