Book 9
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.
19
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.
19
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]
