---
book: 8
number: 15
id: "VIII.15"
kind: "theorem"
uses: ["[[book-8/proposition-11]]", "[[book-8/proposition-12]]", "[[book-8/proposition-7]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:8.prop.15"
license: "CC-BY-SA-4.0"
---

# VIII.15

*If a cube number measure a cube number, the side will also measure the side; and, if the side measure the side, the cube will also measure the cube.*

## Proof

For let the cube number *A* measure the cube *B*, and let *C* be the side of *A* and *D* of *B*; I say that *C* measures *D*.

For let *C* by multiplying itself make *E*, and let *D* by multiplying itself make *G*; further, let *C* by multiplying *D* make *F*, and let *C*, *D* by multiplying *F* make *H*, *K* respectively.

Now it is manifest that *E*, *F*, *G* and *A*, *H*, *K*, *B* are continuously proportional in the ratio of *C* to *D*. [[book-8/proposition-11|VIII. 11, 12]]

And, since *A*, *H*, *K*, *B* are continuously proportional, and *A* measures *B*, therefore it also measures *H*. [[book-8/proposition-7|VIII. 7]]

And, as *A* is to *H*, so is *C* to *D*; therefore *C* also measures *D*. [[book-7/definitions#Definition 20|VII. Def. 20]]

Next, let *C* measure *D*; I say that *A* will also measure *B*.

For, with the same construction, we can prove in a similar manner that *A*, *H*, *K*, *B* are continuously proportional in the ratio of *C* to *D*.

And, since *C* measures *D*, and, as *C* is to *D*, so is *A* to *H*, therefore *A* also measures *H*, [[book-7/definitions#Definition 20|VII. Def. 20]] so that *A* measures *B* also. Q. E. D.