---
book: 9
number: 4
id: "IX.4"
kind: "theorem"
uses: ["[[book-9/proposition-3]]", "[[book-7/proposition-17]]", "[[book-8/proposition-19]]", "[[book-8/proposition-8]]", "[[book-8/proposition-23]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.4"
license: "CC-BY-SA-4.0"
---

# IX.4

*If a cube number by multiplying a cube number make some number, the product will be cube.*

## Proof

For let the cube number *A* by multiplying the cube number *B* make *C*; I say that *C* is cube.

For let *A* by multiplying itself make *D*; therefore *D* is cube. [[book-9/proposition-3|IX. 3]]

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

And, since *A*, *B* are cube numbers, *A*, *B* are similar solid numbers.

Therefore two mean proportional numbers fall between *A*, *B*; [[book-8/proposition-19|VIII. 19]] so that two mean proportional numbers will fall between *D*, *C* also. [[book-8/proposition-8|VIII. 8]]

And *D* is cube; therefore *C* is also cube [[book-8/proposition-23|VIII. 23]] Q. E. D.