---
book: 10
number: 78
id: "X.78"
kind: "theorem"
uses: ["[[book-10/proposition-35]]", "[[book-2/proposition-7]]", "[[book-10/proposition-22]]", "[[book-6/proposition-1]]", "[[book-10/proposition-11]]", "[[book-10/proposition-73]]", "[[book-10/proposition-20]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.78"
license: "CC-BY-SA-4.0"
---

# X.78

*If from a straight line there be subtracted a straight line which is incommensurable in square with the whole and which with the whole makes the sum of the squares on them medial, twice the rectangle contained by them medial, and further the squares on them incommensurable with twice the rectangle contained by them, the remainder is irrational; and let it be called* *that which produces with a medial area a medial whole.*

## Proof

For from the straight line *AB* let there be subtracted the straight line *BC* incommensurable in square with *AB* and fulfilling the given conditions; [[book-10/proposition-35|X. 35]] I say that the remainder *AC* is the irrational straight line called *that which produces with a medial area a medial whole*.

For let a rational straight line *DI* be set out, to *DI* let there be applied *DE* equal to the squares on *AB*, *BC*, producing *DG* as breadth, and let *DH* equal to twice the rectangle *AB*, *BC* be subtracted.

Therefore the remainder *FE* is equal to the square on *AC*, [[book-2/proposition-7|II. 7]] so that *AC* is the side of *FE*.

Now, since the sum of the squares on *AB*, *BC* is medial and is equal to *DE*, therefore *DE* is medial.

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 twice the rectangle *AB*, *BC* is medial and is equal to *DH*, therefore *DH* is medial.

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

And, since the squares on *AB*, *BC* are incommensurable with twice the rectangle *AB*, *BC*, therefore *DE* is also incommensurable with *DH*.

But, as *DE* is to *DH*, so also is *DG* to *DF*; [[book-6/proposition-1|VI. 1]] therefore *DG* 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]]

And *FH* is rational; but the rectangle contained by a rational straight line and an apotome 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 *that which produces with a medial area a medial whole*. Q. E. D.