---
book: 8
number: 26
id: "VIII.26"
kind: "theorem"
uses: ["[[book-8/proposition-18]]", "[[book-7/proposition-33]]", "[[book-8/proposition-2]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:8.prop.26"
license: "CC-BY-SA-4.0"
---

# VIII.26

*Similar plane numbers have to one another the ratio which a square number has to a square number.*

## Proof

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

For, since *A*, *B* are similar plane numbers, therefore one mean proportional number falls between *A*, *B*. [[book-8/proposition-18|VIII. 18]]

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

And since, as *D* is to *F*, so is *A* to *B*, and *D*, *F* are square, therefore *A* has to *B* the ratio which a square number has to a square number. Q. E. D.