PROPOSITION 19.
Given three numbers, to investigate when it is possible to find a fourth proportional to them.
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.