---
book: 8
number: 13
id: "VIII.13"
kind: "theorem"
uses: ["[[book-7/proposition-14]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:8.prop.13"
license: "CC-BY-SA-4.0"
---

# VIII.13

*If there be as many numbers as we please in continued proportion, and each by multiplying itself make some number, the products will be proportional; and, if the original numbers by multiplying the products make certain numbers, the latter will also be proportional.*

## Proof

Let there be as many numbers as we please, *A*, *B*, *C*, in continued proportion, so that, as *A* is to *B*, so is *B* to *C*; let *A*, *B*, *C* by multiplying themselves make *D*, *E*, *F*, and by multiplying *D*, *E*, *F* let them make *G*, *H*, *K*; I say that *D*, *E*, *F* and *G*, *H*, *K* are in continued proportion.

For let *A* by multiplying *B* make *L*, and let the numbers *A*, *B* by multiplying *L* make *M*. *N* respectively.

And again let *B* by multiplying *C* make *O*, and let the numbers *B*, *C* by multiplying *O* make *P*, *Q* respectively.

Then, in manner similar to the foregoing, we can prove that *D*, *L*, *E* and *G*, *M*, *N*, *H* are continuously proportional in the ratio of *A* to *B*, and further *E*, *O*, *F* and *H*, *P*, *Q*, *K* are continuously proportional in the ratio of *B* to *C*.

Now, as *A* is to *B*, so is *B* to *C*; therefore *D*, *L*, *E* are also in the same ratio with *E*, *O*, *F*, and further *G*, *M*, *N*, *H* in the same ratio with *H*, *P*, *Q*, *K*.

And the multitude of *D*, *L*, *E* is equal to the multitude of *E*, *O*, *F*, and that of *G*, *M*, *N*, *H* to that of *H*, *P*, *Q*, *K*; therefore, *ex acquali*, as *D* is to *E*, so is *E* to *F*, and, as *G* is to *H*, so is *H* to *K*. [[book-7/proposition-14|VII. 14]] Q. E. D.