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. 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. VII. 14

But A, C are prime, primes are also least, VII. 21 and the least numbers measure those which have the same ratio, the antecedent the antecedent and the consequent the consequent. 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; 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. 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.