them the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures A the same number of times that G measures D. Now, as many times as E measures A, so many units let there be in N. Therefore N by multiplying E has made A. But E is the product of H, K; therefore N by multiplying the product of H, K has made A. Therefore A is solid, and H, K, N are its sides. Again, since E, F, G are the least of the numbers which have the same ratio as C, D, B, therefore E measures C the same number of times that G measures B. Now, as many times as E measures C, so many units let there be in O. Therefore G measures B according to the units in O; therefore O by multiplying G has made B. But G is the product of L, M; therefore O by multiplying the product of L, M has made B. Therefore B is solid, and L, M, O are its sides; therefore A, B are solid. I say that they are also similar. For since N, O by multiplying E have made A, C, therefore, as N is to O, so is A to C, that is, E to F. [VII. 18] But, as E is to F, so is H to L and K to M; therefore also, as H is to L, so is K to M and N to O. And H, K, N are the sides of A, and O, L, M the sides of B. Therefore A, B are similar solid numbers. Q. E. D.
them the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore E measures A the same number of times that G measures D. Now, as many times as E measures A, so many units let there be in N. Therefore N by multiplying E has made A. But E is the product of H, K; therefore N by multiplying the product of H, K has made A. Therefore A is solid, and H, K, N are its sides. Again, since E, F, G are the least of the numbers which have the same ratio as C, D, B, therefore E measures C the same number of times that G measures B. Now, as many times as E measures C, so many units let there be in O. Therefore G measures B according to the units in O; therefore O by multiplying G has made B. But G is the product of L, M; therefore O by multiplying the product of L, M has made B. Therefore B is solid, and L, M, O are its sides; therefore A, B are solid. I say that they are also similar. For since N, O by multiplying E have made A, C, therefore, as N is to O, so is A to C, that is, E to F. [VII. 18] But, as E is to F, so is H to L and K to M; therefore also, as H is to L, so is K to M and N to O. And H, K, N are the sides of A, and O, L, M the sides of B. Therefore A, B are similar solid numbers. Q. E. D.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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