---
book: 8
number: 27
id: "VIII.27"
kind: "theorem"
uses: ["[[book-8/proposition-19]]", "[[book-7/proposition-33]]", "[[book-8/proposition-2]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:8.prop.27"
license: "CC-BY-SA-4.0"
---

# VIII.27

*Similar solid numbers have to one another the ratio which a cube number has to a cube number.*

## Proof

Let *A*, *B* be similar solid numbers; I say that *A* has to *B* the ratio which a cube number has to a cube number.

For, since *A*, *B* are similar solid numbers, therefore two mean proportional numbers fall between *A*, *B*. [[book-8/proposition-19|VIII. 19]]

Let *C*, *D* so fall, and let *E*, *F*, *G*, *H*, the least numbers of those which have the same ratio with *A*, *C*, *D*, *B*, and equal with them in multitude, be taken; [[book-7/proposition-33|VII. 33]] or [[book-8/proposition-2|VIII. 2]] therefore the extremes of them *E*, *H* are cube. [[book-8/proposition-2|VIII. 2, Por.]]

And, as *E* is to *H*, so is *A* to *B*; therefore *A* also has to *B* the ratio which a cube number has to a cube number. Q. E. D.