---
book: 8
number: 25
id: "VIII.25"
kind: "theorem"
uses: ["[[book-8/proposition-19]]", "[[book-8/proposition-8]]", "[[book-8/proposition-23]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:8.prop.25"
license: "CC-BY-SA-4.0"
---

# VIII.25

*If two numbers have to one another the ratio which a cube number has to a cube number, and the first be cube, the second will also be cube.*

## Proof

For let the two numbers *A*, *B* have to one another the ratio which the cube number *C* has to the cube number *D*, and let *A* be cube; I say that *B* is also cube.

For, since *C*, *D* are cube, *C*, *D* are similar solid numbers.

Therefore two mean proportional numbers fall between *C*, *D*. [[book-8/proposition-19|VIII. 19]]

And, as many numbers as fall between *C*, *D* in continued proportion, so many will also fall between those which have the same ratio with them; [[book-8/proposition-8|VIII. 8]] so that two mean proportional numbers fall between *A*, *B* also.

Let *E*, *F* so fall.

Since, then, the four numbers *A*, *E*, *F*, *B* are in continued proportion, and *A* is cube, therefore *B* is also cube. [[book-8/proposition-23|VIII. 23]] Q. E. D.