---
book: 8
number: 18
id: "VIII.18"
kind: "theorem"
uses: ["[[book-7/proposition-13]]", "[[book-7/proposition-17]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:8.prop.18"
license: "CC-BY-SA-4.0"
---

# VIII.18

*Between two similar plane numbers there is one mean proportional number; and the plane number has to the plane number the ratio duplicate of that which the corresponding side has to the corresponding side.*

## Proof

Let *A*, *B* be two similar plane numbers, and let the numbers *C*, *D* be the sides of *A*, and *E*, *F* of *B*.

Now, since similar plane numbers are those which have their sides proportional, [[book-7/definitions#Definition 21|VII. Def. 21]] therefore, as *C* is to *D*, so is *E* to *F*.

I say then that between *A*, *B* there is one mean proportional number, and *A* has to *B* the ratio duplicate of that which *C* has to *E*, or *D* to *F*, that is, of that which the corresponding side has to the corresponding side.

Now since, as *C* is to *D*, so is *E* to *F*, therefore, alternately, as *C* is to *E*, so is *D* to *F*. [[book-7/proposition-13|VII. 13]]

And, since *A* is plane, and *C*, *D* are its sides, therefore *D* by multiplying *C* has made *A*.

For the same reason also *E* by multiplying *F* has made *B*.

Now let *D* by multiplying *E* make *G*.

Then, since *D* by multiplying *C* has made *A*, and by multiplying *E* has made *G*, therefore, as *C* is to *E*, so is *A* to *G*. [[book-7/proposition-17|VII. 17]]

But, as *C* is to *E*, so is *D* to *F*; therefore also, as *D* is to *F*, so is *A* to *G*.

Again, since *E* by multiplying *D* has made *G*, and by multiplying *F* has made *B*, therefore, as *D* is to *F*, so is *G* to *B*. [[book-7/proposition-17|VII. 17]]

But it was also proved that, as *D* is to *F*, so is *A* to *G*; therefore also, as *A* is to *G*, so is *G* to *B*.

Therefore *A*, *G*, *B* are in continued proportion.

Therefore between *A*, *B* there is one mean proportional number.

I say next that *A* also has to *B* the ratio duplicate of that which the corresponding side has to the corresponding side, that is, of that which *C* has to *E* or *D* to *F*.

For, since *A*, *G*, *B* are in continued proportion, *A* has to *B* the ratio duplicate of that which it has to *G*. [[book-5/definitions#Definition 9|V. Def. 9]]

And, as *A* is to *G*, so is *C* to *E*, and so is *D* to *F*.

Therefore *A* also has to *B* the ratio duplicate of that which *C* has to *E* or *D* to *F*. Q. E. D.