---
book: 10
number: 75
id: "X.75"
kind: "theorem"
uses: ["[[book-10/proposition-28]]", "[[book-2/proposition-7]]", "[[book-10/proposition-15]]", "[[book-10/proposition-23]]", "[[book-10/proposition-22]]", "[[book-10/proposition-11]]", "[[book-10/proposition-6]]", "[[book-10/proposition-13]]", "[[book-6/proposition-1]]", "[[book-10/proposition-73]]", "[[book-10/proposition-20]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.75"
license: "CC-BY-SA-4.0"
---

# X.75

*If from a medial straight line there be subtracted a medial straight line which is commensurable with the whole in square only, and which contains with the whole a medial rectangle, the remainder is irrational; and let it be called a* *second apotome of a medial* *straight line*.

## Proof

For from the medial straight line *AB* let there be subtracted the medial straight line *CB* which is commensurable with the whole *AB* in square only and such that the rectangle *AB*, *BC*, which it contains with the whole *AB*, is medial; [[book-10/proposition-28|X. 28]] I say that the remainder *AC* is irrational; and let it *be called a second apotome of a medial* straight line.

For let a rational straight line *DI* be set out, let *DE* equal to the squares on *AB*, *BC* be applied to *DI*, producing *DG* as breadth, and let *DH* equal to twice the rectangle *AB*, *BC* be applied to *DI*, producing *DF* as breadth; therefore the remainder *FE* is equal to the square on *AC*. [[book-2/proposition-7|II. 7]]

Now, since the squares on *AB*, *BC* are medial and commensurable, therefore *DE* is also medial. [[book-10/proposition-15|X. 15 and 23, Por.]]

And it is applied to the rational straight line *DI*, producing *DG* as breadth; therefore *DG* is rational and incommensurable in length with *DI*. [[book-10/proposition-22|X. 22]]

Again, since the rectangle *AB*, *BC* is medial, therefore twice the rectangle *AB*, *BC* is also medial. [[book-10/proposition-23|X. 23, Por.]]

And it is equal to *DH*; therefore *DH* is also medial.

And it has been applied to the rational straight line *DI*, producing *DF* as breadth; therefore *DF* is rational and incommensurable in length with *DI*. [[book-10/proposition-22|X. 22]]

And, since *AB*, *BC* are commensurable in square only, therefore *AB* is incommensurable in length with *BC*; therefore the square on *AB* is also incommensurable with the rectangle *AB*, *BC*. [[book-10/proposition-11|X. 11]]

But the squares on *AB*, *BC* are commensurable with the square on *AB*, [[book-10/proposition-15|X. 15]] and twice the rectangle *AB*, *BC* is commensurable with the rectangle *AB*, *BC*; [[book-10/proposition-6|X. 6]] therefore twice the rectangle *AB*, *BC* is incommensurable with the squares on *AB*, *BC*. [[book-10/proposition-13|X. 13]]

But *DE* is equal to the squares on *AB*, *BC*, and *DH* to twice the rectangle *AB*, *BC*; therefore *DE* is incommensurable with *DH*.

But, as *DE* is to *DH*, so is *GD* to *DF*; [[book-6/proposition-1|VI. 1]] therefore *GD* is incommensurable with *DF*. [[book-10/proposition-11|X. 11]]

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

But *DI* is rational, and the rectangle contained by a rational and an irrational straight line is irrational, deduction from [[book-10/proposition-20|X. 20]] and its ’side’ is irrational.

And *AC* is the ’side’ of *FE*; therefore *AC* is irrational.

And let it be called *a second apotome of a medial* straight line. Q. E. D.