---
book: 8
number: 20
id: "VIII.20"
kind: "theorem"
uses: ["[[book-7/proposition-33]]", "[[book-7/proposition-20]]", "[[book-7/proposition-17]]", "[[book-7/proposition-13]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:8.prop.20"
license: "CC-BY-SA-4.0"
---

# VIII.20

*If one mean proportional number fall between two numbers, the numbers will be similar plane numbers.*

## Proof

For let one mean proportional number *C* fall between the two numbers *A*, *B*; I say that *A*, *B* are similar plane numbers.

Let *D*, *E*, the least numbers of those which have the same ratio with *A*, *C*, be taken; [[book-7/proposition-33|VII. 33]] therefore *D* measures *A* the same number of times that *E* measures *C*. [[book-7/proposition-20|VII. 20]]

Now, as many times as *D* measures *A*, so many units let there be in *F*; therefore *F* by multiplying *D* has made *A*, so that *A* is plane, and *D*, *F* are its sides.

Again, since *D*, *E* are the least of the numbers which have the same ratio with *C*, *B*, therefore *D* measures *C* the same number of times that *E* measures *B*. [[book-7/proposition-20|VII. 20]]

As many times, then, as *E* measures *B*, so many units let there be in *G*; therefore *E* measures *B* according to the units in *G*; therefore *G* by multiplying *E* has made *B*.

Therefore *B* is plane, and *E*, *G* are its sides.

Therefore *A*, *B* are plane numbers.

I say next that they are also similar.

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

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

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

And alternately, as *D* is to *F*, so is *E* to *G*. [[book-7/proposition-13|VII. 13]]

Therefore *A*, *B* are similar plane numbers; for their sides are proportional. Q. E. D.