---
book: 10
number: 26
id: "X.26"
kind: "theorem"
uses: ["[[book-10/proposition-22]]", "[[book-10/proposition-20]]", "[[book-10/proposition-13]]", "[[book-10/proposition-11]]", "[[book-10/proposition-6]]", "[[book-2/proposition-4]]", "[[book-10/proposition-16]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.26"
license: "CC-BY-SA-4.0"
---

# X.26

*4 medial area does not exceed a medial area by a rational area*.

## Proof

For, if possible, let the medial area *AB* exceed the medial area *AC* by the rational area *DB*, and let a rational straight line *EF* be set out; to *EF* let there be applied the rectangular parallelogram *FH* equal to *AB*, producing *EH* as breadth, and let the rectangle *FG* equal to *AC* be subtracted; therefore the remainder *BD* is equal to the remainder *KH*.

But *DB* is rational; therefore *KH* is also rational.

Since, then, each of the rectangles *AB*, *AC* is medial, and *AB* is equal to *FH*, and *AC* to *FG*, therefore each of the rectangles *FH*, *FG* is also medial.

And they are applied to the rational straight line *EF*; therefore each of the straight lines *HE*, *EG* is rational and incommensurable in length with *EF*. [[book-10/proposition-22|X. 22]]

And, since [*DB* is rational and is equal to *KH*, therefore] *KH* is [also] rational; and it is applied to the rational straight line *EF*; therefore *GH* is rational and commensurable in length with *EF*. [[book-10/proposition-20|X. 20]]

But *EG* is also rational, and is incommensurable in length with *EF*; therefore *EG* is incommensurable in length with *GH*. [[book-10/proposition-13|X. 13]]

And, as *EG* is to *GH*, so is the square on *EG* to the rectangle *EG*, *GH*; therefore the square on *EG* is incommensurable with the rectangle *EG*, *GH*. [[book-10/proposition-11|X. 11]]

But the squares on *EG*, *GH* are commensurable with the square on *EG*, for both are rational; and twice the rectangle *EG*, *GH* is commensurable with the rectangle *EG*, *GH*, for it is double of it; [[book-10/proposition-6|X. 6]] therefore the squares on *EG*, *GH* are incommensurable with twice the rectangle *EG*, *GH*; [[book-10/proposition-13|X. 13]] therefore also the sum of the squares on *EG*, *GH* and twice the rectangle *EG*, *GH*, that is, the square on *EH* [[book-2/proposition-4|II. 4]], is incommensurable with the squares on *EG*, *GH*. [[book-10/proposition-16|X. 16]]

But the squares on *EG*, *GH* are rational; therefore the square on *EH* is irrational. [[book-10/definitions#Definition 4|X. Def. 4]]

Therefore *EH* is irrational.

But it is also rational: which is impossible.

Therefore etc. Q. E. D.