| To find as many numbers as are prescribed in continued proportion, and the least that are in a given ratio. | ||
| Let the ratio of A to B be the given ratio in least numbers.
It is required to find numbers in continued proportion, as many as may be prescribed, the least that are in the ratio of A to B. Let four be prescribed. Multiply A by itself to make C, and by B to make D. Multiply B by itself to make E. Also multiply A by C, D, and E to make F, G, and H, and multiply B by E to make K. | ||
| Now, since A multiplied by itself makes C, and multiplied by B makes D, therefore A is to B as C is to D. | VII.17 | |
| Again, since A multiplied by B makes D, and B multiplied by itself makes E, therefore the numbers A and B multiplied by B make the numbers D and E respectively. | ||
| Therefore A is to B as D is to E. But A is to B as C is to D, therefore C is to D as D is to E. | VII.18 | |
| And, since A multiplied by C and D makes F and G, therefore C is to D as F is to G. | VII.17 | |
| But C is to D as A is to B, therefore A is to B as F is to G. | ||
| Again, since A multiplied by D and E makes G and H, therefore D is to E as G is to H. But D is to E as A is to B. Therefore A is to B as G is to H. | VII.17 | |
| And, since A and B multiplied by E make H and K, therefore A is to B as H is to K. But A is to B as F is to G, and as G is to H. Therefore F is to G as G is to H, and as H is to K. | VII.18 | |
| Therefore C, D, and E, and F, G, H, and K are proportional in the ratio of A to B.
I say next that they are the least numbers that are so. | ||
| Since A and B are the least of those which have the same ratio with them, and the least of those which have the same ratio are relatively prime, therefore A and B are relatively prime. | VII.22 | |
| And the numbers A and B multiplied by themselves respectively make the numbers C and E, and multiplied by the numbers C and E respectively make the numbers F and K, therefore C and E and F and K are relatively prime respectively. | VII.27 | |
| But, if there are as many numbers as we please in continued proportion, and the extremes of them are relatively prime, then they are the least of those which have the same ratio with them. Therefore C, D, and E and F, G, H and K are the least of those which have the same ratio with A and B. | VIII.1 | |
| Q.E.D. | ||
Book VIII Introduction - Proposition VIII.1 - Proposition VIII.3.