Book 8
3
For let two numbers E, F, the least that are in the ratio of A, B, C, D, be taken, [VII. 33] then three others G, H, K with the same property; and others, more by one continually, [VIII. 2] until the multitude taken becomes equal to the multitude of the numbers A, B, C, D.
3
Let them be taken, and let them be L, M, N, O.
3
Now, since E, F are the least of those which have the same ratio with them, they are prime to one another. [VII. 22]
3
And, since the numbers E, F by multiplying themselves respectively have made the numbers G, K, and by multiplying the numbers G, K respectively have made the numbers L, O, [VIII. 2, Por.] therefore both G, K and L, O are prime to one another. [VII. 27]
3
And, since A, B, C, D are the least of those which have the same ratio with them, while L, M, N, O are the least that are in the same ratio with A, B, C, D, and the multitude of the numbers A, B, C, D is equal to the multitude of the numbers L, M, N, O, therefore the numbers A, B, C, D are equal to the numbers L, M, N, O respectively; therefore A is equal to L, and D to O.
3
And L, O are prime to one another.
3
Therefore A, D are also prime to one another. Q. E. D.
PROPOSITION 4.
4
Given as many ratios as we please in least numbers, to find numbers in continued proportion which are the least in the given ratios.
