| Given two numbers, to investigate whether it is possible to find a third proportional to them. | ||
| Let A and B be the given two numbers. It is required to investigate whether it is possible to find a third proportional to them. | ||
| Now A and B are either relatively prime or not. And, if they are relatively prime, it was proved that it is impossible to find a third proportional to them. | IX.16 | |
| Next, let A and B not be relatively prime, and let B multiplied by itself make C. Then A either measures C or does not measure it.
First, let it measure it according to D, therefore A multiplied by D makes C. But, further, B multiplied by itself makes C, therefore the product of A and D equals the square on B. |
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| Therefore A is to B as B is to D, therefore a third proportional number D has been found to A and B. | VII.19 | |
| Next, let A not measure C.
I say that it is impossible to find a third proportional number to A and B. If possible, let D be such third proportional. Then the product of A and D equals the square on B. But the square on B is C, therefore the product of A and D equals C. Hence A multiplied by D makes C, therefore A measures C according to D. But, by hypothesis, it also does not measure it, which is absurd. Therefore it is not possible to find a third proportional number to A and B when A does not measure C. | ||
| Q.E.D. | ||
Book IX Introduction - Proposition IX.17 - Proposition IX.19.