Book 9
17
But it also measures itself; therefore A measures A, D which are prime to one another : which is impossible.
17
Therefore D will not be to any other number as A is to B. Q. E. D.
PROPOSITION 18.
18
Given two numbers, to investigate whether it is possible to find a third proportional to them.
18
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.
18
Now A, B are either prime to one another or not.
18
And, if they are prime to one another, it has been proved that it is impossible to find a third proportional to them. [IX. 16]
18
Next, let A, B not be prime to one another, and let B by multiplying itself make C.
18
Then A either measures C or does not measure it.
18
First, let it measure it according to D; therefore A by multiplying D has made C.
18
But, further, B has also by multiplying itself made C; therefore the product of A, D is equal to the square on B.
18
Therefore, as A is to B, so is B to D; [VII. 19] therefore a third proportional number D has been found to A, B.
18
Next, let A not measure C; I say that it is impossible to find a third proportional number to A, B.
18
For, if possible, let D, such third proportional, have been found.
18
Therefore the product of A, D is equal to the square on B.
18
But the square on B is C; therefore the product of A, D is equal to C.
18
Hence A by multiplying D has made C; therefore A measures C according to D.
18
But, by hypothesis, it also does not measure it: which is absurd.
18
Therefore it is not possible to find a third proportional number to A, B when A does not measure C. Q. E. D.
PROPOSITION 19.
19
Given three numbers, to investigate when it is possible to find a fourth proportional to them.
19
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.
