---
book: 7
number: 19
id: "VII.19"
kind: "theorem"
uses: ["[[book-7/proposition-17]]", "[[book-7/proposition-18]]", "[[book-5/proposition-9]]", "[[book-5/proposition-7]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.19"
license: "CC-BY-SA-4.0"
---

# VII.19

*If four numbers be proportional, the number produced from the first and fourth will be equal to the number produced from the second and third; and, if the number produced from the first and fourth be equal to that produced from the second and third, the four numbers will be proportional.*

## Proof

Let *A*, *B*, *C*, *D* be four numbers in proportion, so that, as *A* is to *B*, so is *C* to *D*; and let *A* by multiplying *D* make *E*, and let *B* by multiplying *C* make *F*; I say that *E* is equal to *F*.

For let *A* by multiplying *C* make *G*.

Since, then, *A* by multiplying *C* has made *G*, and by multiplying *D* has made *E*, the number *A* by multiplying the two numbers *C*, *D* has made *G*, *E*.

Therefore, as *C* is to *D*, so is *G* to *E*. [[book-7/proposition-17|VII. 17]]

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

Again, since *A* by multiplying *C* has made *G*, but, further, *B* has also by multiplying *C* made *F*, the two numbers *A*, *B* by multiplying a certain number *C* have made *G*, *F*.

Therefore, as *A* is to *B*, so is *G* to *F*. [[book-7/proposition-18|VII. 18]]

But further, as *A* is to *B*, so is *G* to *E* also; therefore also, as *G* is to *E*, so is *G* to *F*.

Therefore *G* has to each of the numbers *E*, *F* the same ratio; therefore *E* is equal to *F*. cf. [[book-5/proposition-9|V. 9]]

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

For, with the same construction, since *E* is equal to *F*, therefore, as *G* is to *E*, so is *G* to *F*. cf. [[book-5/proposition-7|V. 7]]

But, as *G* is to *E*, so is *C* to *D*, [[book-7/proposition-17|VII. 17]] and, as *G* is to *F*, so is *A* to *B*. [[book-7/proposition-18|VII. 18]]

Therefore also, as *A* is to *B*, so is *C* to *D*. Q. E. D.