---
book: 10
number: 81
id: "X.81"
kind: "theorem"
uses: ["[[book-10/proposition-75]]", "[[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-79]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_2.81"
license: "CC-BY-SA-4.0"
---

# X.81

*To a second apotome of a medial straight line only one medial straight line can be annexed which is commensurable with the whole in square only and which contains with the whole a medial rectangle*.

## Proof

Let *AB* be a second apotome of a medial straight line and *BC* an annex to *AB*; therefore *AC*, *CB* are medial straight lines commensurable in square only and such that the rectangle *AC*, *CB* which they contain is medial. [[book-10/proposition-75|X. 75]]

I say that no other medial straight line can be annexed to *AB* which is commensurable with the whole in square only and which contains with the whole a medial rectangle.

For, if possible, let *BD* also be so annexed; therefore *AD*, *DB* are also medial straight lines commensurable in square only and such that the rectangle *AD*, *DB* which they contain is medial. [[book-10/proposition-75|X. 75]]

Let a rational straight line *EF* be set out, let *EG* equal to the squares on *AC*, *CB* be applied to *EF*, producing *EM* as breadth, and let *HG* equal to twice the rectangle *AC*, *CB* be subtracted, producing *HM* as breadth; therefore the remainder *EL* is equal to the square on *AB*, [[book-2/proposition-7|II. 7]] so that *AB* is the side of *EL*.

Again, let *EI* equal to the squares on *AD*, *DB* be applied to *EF*, producing *EN* as breadth.

But *EL* is also equal to the square on *AB*; therefore the remainder *HI* is equal to twice the rectangle *AD*, *DB*. [[book-2/proposition-7|II. 7]]

Now, since *AC*, *CB* are medial straight lines, therefore the squares on *AC*, *CB* are also medial.

And they are equal to *EG*; therefore *EG* is also medial. [[book-10/proposition-15|X. 15 and 23, Por.]]

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

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

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

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

And, since *AC*, *CB* are commensurable in square only, therefore *AC* is incommensurable in length with *CB*.

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

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

And *EG* is equal to the squares on *AC*, *CB*, while *GH* is equal to twice the rectangle *AC*, *CB*; therefore *EG* is incommensurable with *HG*.

But, as *EG* is to *HG*, so is *EM* to *HM*; [[book-6/proposition-1|VI. 1]] therefore *EM* is incommensurable in length with *MH*. [[book-10/proposition-11|X. 11]]

And both are rational; therefore *EM*, *MH* are rational straight lines commensurable in square only; therefore *EH* is an apotome, and *HM* an annex to it. [[book-10/proposition-73|X. 73]]

Similarly we can prove that *HN* is also an annex to it; therefore to an apotome different straight lines are annexed which are commensurable with the wholes in square only: which is impossible. [[book-10/proposition-79|X. 79]]

Therefore etc. Q. E. D.