---
book: 5
number: 11
id: "V.11"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:5.prop.11"
license: "CC-BY-SA-4.0"
---

# V.11

*Ratios which are the same with the same ratio are also the same with one another*.

## Proof

For, as *A* is to *B*, so let *C* be to *D*, and, as *C* is to *D*, so let *E* be to *F*; I say that, as *A* is to *B*, so is *E* to *F*.

For of *A*, *C*, *E* let equimultiples *G*, *H*, *K* be taken, and of *B*, *D*, *F* other, chance, equimultiples *L*, *M*, *N*.

Then since, as *A* is to *B*, so is *C* to *D*, and of *A*, *C* equimultiples *G*, *H* have been taken, and of *B*, *D* other, chance, equimultiples *L*, *M*, therefore, if *G* is in excess of *L*, *H* is also in excess of *M*, if equal, equal, and if less, less.

Again, since, as *C* is to *D*, so is *E* to *F*, and of *C*, *E* equimultiples *H*, *K* have been taken, and of *D*, *F* other, chance, equimultiples *M*, *N*, therefore, if *H* is in excess of *M*, *K* is also in excess of *N*, if equal, equal, and if less, less.

But we saw that, if *H* was in excess of *M*, *G* was also in excess of *L*; if equal, equal; and if less, less; so that, in addition, if *G* is in excess of *L*, *K* is also in excess of *N*, if equal, equal, and if less, less.

And *G*, *K* are equimultiples of *A*, *E*, while *L*, *N* are other, chance, equimultiples of *B*, *F*; therefore, as *A* is to *B*, so is *E* to *F*.

Therefore etc. Q. E. D.