---
book: 10
number: 25
id: "X.25"
kind: "theorem"
uses: ["[[book-10/proposition-22]]", "[[book-6/proposition-1]]", "[[book-10/proposition-11]]", "[[book-10/proposition-19]]", "[[book-5/proposition-11]]", "[[book-6/proposition-17]]", "[[book-10/proposition-21]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.25"
license: "CC-BY-SA-4.0"
---

# X.25

*The rectangle contained by medial straight lines commensurable in square only is either rational or medial*.

## Proof

For let the rectangle *AC* be contained by the medial straight lines *AB*, *BC* which are commensurable in square only; I say that *AC* is either rational or medial.

For on *AB*, *BC* let the squares *AD*, *BE* be described; therefore each of the squares *AD*, *BE* is medial.

Let a rational straight line *FG* be set out, to *FG* let there be applied the rectangular parallelogram *GH* equal to *AD*, producing *FH* as breadth, to *HM* let there be applied the rectangular parallelogram *MK* equal to *AC*, producing *HK* as breadth, and further to *KN* let there be similarly applied *NL* equal to *BE*, producing *KL* as breadth; therefore *FH*, *HK*, *KL* are in a straight line.

Since then each of the squares *AD*, *BE* is medial, and *AD* is equal to *GH*, and *BE* to *NL*, therefore each of the rectangles *GH*, *NL* is also medial.

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

And, since *AD* is commensurable with *BE*, therefore *GH* is also commensurable with *NL*.

And, as *GH* is to *NL*, so is *FH* to *KL*; [[book-6/proposition-1|VI. 1]] therefore *FH* is commensurable in length with *KL*. [[book-10/proposition-11|X. 11]]

Therefore *FH*, *KL* are rational straight lines commensurable in length; therefore the rectangle *FH*, *KL* is rational. [[book-10/proposition-19|X. 19]]

And, since *DB* is equal to *BA*, and *OB* to *BC*, therefore, as *DB* is to *BC*, so is *AB* to *BO*.

But, as *DB* is to *BC*, so is *DA* to *AC*, [[book-6/proposition-1|VI. 1]] and, as *AB* is to *BO*, so is *AC* to *CO*; [*id*.] therefore, as *DA* is to *AC*, so is *AC* to *CO*.

But *AD* is equal to *GH*, *AC* to *MK* and *CO* to *NL*; therefore, as *GH* is to *MK*, so is *MK* to *NL*; therefore also, as *FH* is to *HK*, so is *HK* to *KL*; [[book-6/proposition-1|VI. 1]], [[book-5/proposition-11|V. 11]] therefore the rectangle *FH*, *KL* is equal to the square on *HK*. [[book-6/proposition-17|VI. 17]]

But the rectangle *FH*, *KL* is rational; therefore the square on *HK* is also rational.

Therefore *HK* is rational.

And, if it is commensurable in length with *FG*, *HN* is rational; [[book-10/proposition-19|X. 19]] but, if it is incommensurable in length with *FG*, *KH*, *HM* are rational straight lines commensurable in square only, and therefore *HN* is medial. [[book-10/proposition-21|X. 21]]

Therefore *HN* is either rational or medial.

But *HN* is equal to *AC*; therefore *AC* is either rational or medial.

Therefore etc.