---
book: 10
number: 6
id: "X.6"
kind: "theorem"
uses: ["[[book-5/proposition-7]]", "[[book-5/proposition-22]]", "[[book-5/proposition-11]]", "[[book-5/proposition-9]]", "[[book-6/proposition-19]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:10.prop_1.6"
license: "CC-BY-SA-4.0"
---

# X.6

*If two magnitudes have to one another the ratio which a number has to a number*, *the magnitudes will be commensurable*.

## Proof

For let the two magnitudes *A*, *B* have to one another the ratio which the number *D* has to the number *E*; I say that the magnitudes *A*, *B* are commensurable.

For let *A* be divided into as many equal parts as there are units in *D*, and let *C* be equal to one of them; and let *F* be made up of as many magnitudes equal to *C* as there are units in *E*.

Since then there are in *A* as many magnitudes equal to *C* as there are units in *D*, whatever part the unit is of *D*, the same part is *C* of *A* also; therefore, as *C* is to *A*, so is the unit to *D*. [[book-7/definitions#Definition 20|VII. Def. 20]]

But the unit measures the number *D*; therefore *C* also measures *A*.

And since, as *C* is to *A*, so is the unit to *D*, therefore, inversely, as *A* is to *C*, so is the number *D* to the unit. cf. [[book-5/proposition-7|V. 7, Por.]]

Again, since there are in *F* as many magnitudes equal to *C* as there are units in *E*, therefore, as *C* is to *F*, so is the unit to *E*. [[book-7/definitions#Definition 20|VII. Def. 20]]

But it was also proved that, as *A* is to *C*, so is *D* to the unit; therefore, ex aequali, as *A* is to *F*, so is *D* to *E*. [[book-5/proposition-22|v. 22]]

But, as *D* is to *E*, so is *A* to *B*; therefore also, as *A* is to *B*, so is it to *F* also. [[book-5/proposition-11|V. 11]]

Therefore *A* has the same ratio to each of the magnitudes *B*, *F*; therefore *B* is equal to *F*. [[book-5/proposition-9|V. 9]]

But *C* measures *F*; therefore it measures *B* also.

Further it measures *A* also; therefore *C* measures *A*, *B*.

Therefore *A* is commensurable with *B*.

Therefore etc.

Porism. From this it is manifest that, if there be two numbers, as *D*, *E*, and a straight line, as *A*, it is possible to make a straight line [*F*] such that the given straight line is to it as the number *D* is to the number *E*.

And, if a mean proportional be also taken between *A*, *F*, as *B*,

as *A* is to *F*, so will the square on *A* be to the square on *B*, that is, as the first is to the third, so is the figure on the first to that which is similar and similarly described on the second. [[book-6/proposition-19|VI. 19, Por.]]

But, as *A* is to *F*, so is the number *D* to the number *E*; therefore it has been contrived that, as the number *D* is to the number *E*, so also is the figure on the straight line *A* to the figure on the straight line *B*. Q. E. D.