---
book: 9
number: 19
id: "IX.19"
kind: "construction"
uses: ["[[book-9/proposition-17]]", "[[book-7/proposition-14]]", "[[book-7/proposition-21]]", "[[book-7/proposition-20]]", "[[book-7/proposition-19]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.19"
license: "CC-BY-SA-4.0"
---

# IX.19

*Given three numbers, to investigate when it is possible to find a fourth proportional to them.*

## Proof

Let *A*, *B*, *C* be the given three numbers, and let it be required to investigate when it is possible to find a fourth proportional to them.

Now either they are not in continued proportion, and the extremes of them are prime to one another; or they are in continued proportion, and the extremes of them are not prime to one another; or they are not in continued proportion, nor are the extremes of them prime to one another; or they are in continued proportion, and the extremes of them are prime to one another.

If then *A*, *B*, *C* are in continued proportion, and the extremes of them *A*, *C* are prime to one another, it has been proved that it is impossible to find a fourth proportional number to them. [[book-9/proposition-17|IX. 17]]

†Next, let *A*, *B*, *C* not be in continued proportion, the extremes being again prime to one another; I say that in this case also it is impossible to find a fourth proportional to them.

For, if possible, let *D* have been found, so that, as *A* is to *B*, so is *C* to *D*, and let it be contrived that, as *B* is to *C*, so is *D* to *E*.

Now, since, as *A* is to *B*, so is *C* to *D*, and, as *B* is to *C*, so is *D* to *E*, therefore, ex aequali, as *A* is to *C*, so is *C* to *E*. [[book-7/proposition-14|VII. 14]]

But *A*, *C* are prime, primes are also least, [[book-7/proposition-21|VII. 21]] and the least numbers measure those which have the same ratio, the antecedent the antecedent and the consequent the consequent. [[book-7/proposition-20|VII. 20]]

Therefore *A* measures *C* as antecedent antecedent.

But it also measures itself; therefore *A* measures *A*, *C* which are prime to one another: which is impossible.

Therefore it is not possible to find a fourth proportional to *A*, *B*, *C*.†

Next, let *A*, *B*, *C* be again in continued proportion, but let *A*, *C* not be prime to one another.

I say that it is possible to find a fourth proportional to them.

For let *B* by multiplying *C* make *D*; therefore *A* either measures *D* or does not measure it.

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

But, further, *B* has also by multiplying *C* made *D*; therefore the product of *A*, *E* is equal to the product of *B*, *C*; therefore, proportionally, as *A* is to *B*, so is *C* to *E*; [[book-7/proposition-19|VII. 19]] therefore *E* has been found a fourth proportional to *A*, *B*, *C*.

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

For, if possible, let *E* have been found; therefore the product of *A*, *E* is equal to the product of *B*, *C*. [[book-7/proposition-19|VII. 19]]

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

Therefore *A* by multiplying *E* has made *D*; therefore *A* measures *D* according to *E*, so that *A* measures *D*.

But it also does not measure it: which is absurd.

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

Next, let *A*, *B*, *C* not be in continued proportion, nor the extremes prime to one another.

And let *B* by multiplying *C* make *D*.

Similarly then it can be proved that, if *A* measures *D*, it is possible to find a fourth proportional to them, but, if it does not measure it, impossible. Q. E. D.