---
book: 9
number: 18
id: "IX.18"
kind: "construction"
uses: ["[[book-9/proposition-16]]", "[[book-7/proposition-19]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.18"
license: "CC-BY-SA-4.0"
---

# IX.18

*Given two numbers, to investigate whether it is possible to find a third proportional to them.*

## Proof

Let *A*, *B* be the given two numbers, and let it be required to investigate whether it is possible to find a third proportional to them.

Now *A*, *B* are either prime to one another or not.

And, if they are prime to one another, it has been proved that it is impossible to find a third proportional to them. [[book-9/proposition-16|IX. 16]]

Next, let *A*, *B* not be prime to one another, and let *B* by multiplying itself make *C*.

Then *A* either measures *C* or does not measure it.

First, let it measure it according to *D*; therefore *A* by multiplying *D* has made *C*.

But, further, *B* has also by multiplying itself made *C*; therefore the product of *A*, *D* is equal to the square on *B*.

Therefore, as *A* is to *B*, so is *B* to *D*; [[book-7/proposition-19|VII. 19]] therefore a third proportional number *D* has been found to *A*, *B*.

Next, let *A* not measure *C*; I say that it is impossible to find a third proportional number to *A*, *B*.

For, if possible, let *D*, such third proportional, have been found.

Therefore the product of *A*, *D* is equal to the square on *B*.

But the square on *B* is *C*; therefore the product of *A*, *D* is equal to *C*.

Hence *A* by multiplying *D* has made *C*; therefore *A* measures *C* according to *D*.

But, by hypothesis, it also does not measure it: which is absurd.

Therefore it is not possible to find a third proportional number to *A*, *B* when *A* does not measure *C*. Q. E. D.