---
book: 10
number: 87
id: "X.87"
kind: "construction"
uses: ["[[book-10/proposition-6]]", "[[book-10/proposition-9]]", "[[book-10/proposition-73]]", "[[book-5/proposition-22]]", "[[book-5/proposition-19]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_3.87"
license: "CC-BY-SA-4.0"
---

# X.87

*To find the third apotome*.

## Proof

Let a rational straight line *A* be set out, let three numbers *E*, *BC*, *CD* be set out which have not to one another the ratio which a square number has to a square number, but let *CB* have to *BD* the ratio which a square number has to a square number.

Let it be contrived that, as *E* is to *BC*, so is the square on *A* to the square on *FG*, and, as *BC* is to *CD*, so is the square on *FG* to the square on *GH*. [[book-10/proposition-6|X. 6, Por.]]

Since then, as *E* is to *BC*, so is the square on *A* to the square on *FG*, therefore the square on *A* is commensurable with the square on *FG*. [[book-10/proposition-6|X. 6]]

But the square on *A* is rational; therefore the square on *FG* is also rational; therefore *FG* is rational.

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

Again, since, as *BC* is to *CD*, so is the square on *FG* to the square on *GH*, therefore the square on *FG* is commensurable with the square on *GH*. [[book-10/proposition-6|X. 6]]

But the square on *FG* is rational; therefore the square on *GH* is also rational; therefore *GH* is rational.

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

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

I say next that it is also a third apotome.

For since, as *E* is to *BC*, so is the square on *A* to the square on *FG*, and, as *BC* is to *CD*, so is the square on *FG* to the square on *HG*, therefore, *ex aequali*, as *E* is to *CD*, so is the square on *A* to the square on *HG*. [[book-5/proposition-22|V. 22]]

But *E* has not to *CD* the ratio which a square number has to a square number; therefore neither has the square on *A* to the square on *GH* the ratio which a square number has to a square number; therefore *A* is incommensurable in length with *GH*. [[book-10/proposition-9|X. 9]]

Therefore neither of the straight lines *FG*, *GH* is commensurable in length with the rational straight line *A* set out.

Now let the square on *K* be that by which the square on *FG* is greater than the square on *GH*.

Since then, as *BC* is to *CD*, so is the square on *FG* to the square on *GH*, therefore, convertendo, as *BC* is to *BD*, so is the square on *FG* to the square on *K*. [[book-5/proposition-19|V. 19, Por.]]

But *BC* has to *BD* the ratio which a square number has to a square number; therefore the square on *FG* also has to the square on *K* the ratio which a square number has to a square number.

Therefore *FG* is commensurable in length with *K*, [[book-10/proposition-9|X. 9]] and the square on *FG* is greater than the square on *GH* by the square on a straight line commensurable with *FG*.

And neither of the straight lines *FG*, *GH* is commensurable in length with the rational straight line *A* set out; therefore *FH* is a third apotome. [[book-10/definitions#Definition 3 (part 3)|X. Deff. III. 3]]

Therefore the third apotome *FH* has been found. Q. E. D.