---
book: 8
number: 5
id: "VIII.5"
kind: "theorem"
uses: ["[[book-8/proposition-4]]", "[[book-7/proposition-17]]", "[[book-7/proposition-14]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:8.prop.5"
license: "CC-BY-SA-4.0"
---

# VIII.5

*Plane numbers have to one another the ratio compounded of the ratios of their sides.*

## Proof

Let *A*, *B* be plane numbers, and let the numbers *C*, *D* be the sides of *A*, and *E*, *F* of *B*; I say that *A* has to *B* the ratio compounded of the ratios of the sides.

For, the ratios being given which *C* has to *E* and *D* to *F*, let the least numbers *G*, *H*, *K* that are continuously in the ratios *C* : *E*, *D* : *F* be taken, so that, as *C* is to *E*, so is *G* to *H*, and, as *D* is to *F*, so is *H* to *K*. [[book-8/proposition-4|VIII. 4]]

And let *D* by multiplying *E* make *L*.

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

But, as *C* is to *E*, so is *G* to *H*; therefore also, as *G* is to *H*, so is *A* to *L*.

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

But, as *D* is to *F*, so is *H* to *K*; therefore also, as *H* is to *K*, so is *L* to *B*.

But it was also proved that, as *G* is to *H*, so is *A* to *L*; therefore, ex aequali, as *G* is to *K*, so is *A* to *B*. [[book-7/proposition-14|VII. 14]]

But *G* has to *K* the ratio compounded of the ratios of the sides; therefore *A* also has to *B* the ratio compounded of the ratios of the sides. Q. E. D.