---
book: 9
number: 5
id: "IX.5"
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.5"
license: "CC-BY-SA-4.0"
---

# IX.5

*If a cube number by multiplying any number make a cube number, the multiplied number will also be cube.*

## Proof

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

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

Now, 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 *D*, *C* are cube, they are similar solid numbers.

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

And, as *D* is to *C*, so is *A* to *B*; therefore two mean proportional numbers fall between *A*, *B* also. [[book-8/proposition-8|VIII. 8]]

And *A* is cube; therefore *B* is also cube. [[book-8/proposition-23|VIII. 23]]