---
book: 10
number: 112
id: "X.112"
kind: "theorem"
uses: ["[[book-6/proposition-16]]", "[[book-5/proposition-16]]", "[[book-5/proposition-14]]", "[[book-5/proposition-17]]", "[[book-5/proposition-12]]", "[[book-5/proposition-11]]", "[[book-10/proposition-36]]", "[[book-6/proposition-22]]", "[[book-10/proposition-11]]", "[[book-10/proposition-15]]", "[[book-10/proposition-20]]", "[[book-10/proposition-73]]", "[[book-10/proposition-14]]", "[[book-10/proposition-12]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_3.112"
license: "CC-BY-SA-4.0"
---

# X.112

*The square on a rational straight line applied to the binomial straight line produces as breadth an apotome the terms of which are commensurable with the terms of the binomial and moreover in the same ratio; and further the apotome so arising will have the same order as the binomial straight line*.

## Proof

Let *A* be a rational straight line, let *BC* be a binomial, and let *DC* be its greater term; let the rectangle *BC*, *EF* be equal to the square on *A*; I say that *EF* is an apotome the terms of which are commensurable with *CD*, *DB*, and in the same ratio, and further *EF* will have the same order as *BC*.

For again let the rectangle *BD*, *G* be equal to the square on *A*.

Since then the rectangle *BC*, *EF* is equal to the rectangle *BD*, *G*, therefore, as *CB* is to *BD*, so is *G* to *EF*. [[book-6/proposition-16|VI. 16]]

But *CB* is greater than *BD*; therefore *G* is also greater than *EF*. [[book-5/proposition-16|V. 16]], [[book-5/proposition-14|V. 14]]

Let *EH* be equal to *G*; therefore, as *CB* is to *BD*, so is *HE* to *EF*; therefore, separando, as *CD* is to *BD*, so is *HF* to *FE*. [[book-5/proposition-17|V. 17]]

Let it be contrived that, as *HF* is to *FE*, so is *FK* to *KE*; therefore also the whole *HK* is to the whole *KF* as *FK* is to *KE*; for, as one of the antecedents is to one of the consequents, so are all the antecedents to all the consequents. [[book-5/proposition-12|V. 12]]

But, as *FK* is to *KE*, so is *CD* to *DB*; [[book-5/proposition-11|V. 11]] therefore also, as *HK* is to *KF*, so is *CD* to *DB*. [id.]

But the square on *CD* is commensurable with the square on *DB*; [[book-10/proposition-36|X. 36]] therefore the square on *HK* is also commensurable with the square on *KF*. [[book-6/proposition-22|VI. 22]], [[book-10/proposition-11|X. 11]]

And, as the square on *HK* is to the square on *KF*, so is *HK* to *KE*, since the three straight lines *HK*, *KF*, *KE* are proportional. [[book-5/definitions#Definition 9|V. Def. 9]]

Therefore *HK* is commensurable in length with *KE*, so that *HE* is also commensurable in length with *EK*. [[book-10/proposition-15|X. 15]]

Now, since the square on *A* is equal to the rectangle *EH*, *BD*, while the square on *A* is rational, therefore the rectangle *EH*, *BD* is also rational.

And it is applied to the rational straight line *BD*; therefore *EH* is rational and commensurable in length with *BD*; [[book-10/proposition-20|X. 20]] so that *EK*, being commensurable with it, is also rational and commensurable in length with *BD*.

Since, then, as *CD* is to *DB*, so is *FK* to *KE*, while *CD*, *DB* are straight lines commensurable in square only, therefore *FK*, *KE* are also commensurable in square only. [[book-10/proposition-11|X. 11]]

But *KE* is rational; therefore *FK* is also rational.

Therefore *FK*, *KE* are rational straight lines commensurable in square only; therefore *EF* is an apotome. [[book-10/proposition-73|X. 73]]

Now the square on *CD* is greater than the square on *DB* either by the square on a straight line commensurable with *CD* or by the square on a straight line incommensurable with it.

If then the square on *CD* is greater than the square on *DB* by the square on a straight line commensurable with *CD*, the square on *FK* is also greater than the square on *KE* by the square on a straight line commensurable with *FK*. [[book-10/proposition-14|X. 14]]

And, if *CD* is commensurable in length with the rational straight line set out, so also is *FK*; [[book-10/proposition-11|X. 11, 12]] if *BD* is so commensurable, so also is *KE*; [[book-10/proposition-12|X. 12]] but, if neither of the straight lines *CD*, *DB* is so commensurable, neither of the straight lines *FK*, *KE* is so.

But, if the square on *CD* is greater than the square on *DB* by the square on a straight line incommensurable with *CD*, the square on *FK* is also greater than the square on *KE* by the square on a straight line incommensurable with *FK*. [[book-10/proposition-14|X. 14]]

And, if *CD* is commensurable with the rational straight line set out, so also is *FK*; if *BD* is so commensurable, so also is *KE*; but, if neither of the straight lines *CD*, *DB* is so commensurable, neither of the straight lines *FK*, *KE* is so; so that *FE* is an apotome, the terms of which *FK*, *KE* are commensurable with the terms *CD*, *DB* of the binomial straight line and in the same ratio, and it has the same order as *BC*. Q. E. D.