And, since C measures A according to the units in E, therefore E also measures A according to the units in C. [VII. 16] For the same reason E also measures B according to the units in D. [VII. 16] Therefore E measures A, B which are prime to one another: which is impossible. [VII. Def. 12] Therefore there will be no numbers less than A, B which are in the same ratio with A, B. Therefore A, B are the least of those which have the same ratio with them. Q. E. D.
And, since C measures A according to the units in E, therefore E also measures A according to the units in C. [VII. 16] For the same reason E also measures B according to the units in D. [VII. 16] Therefore E measures A, B which are prime to one another: which is impossible. [VII. Def. 12] Therefore there will be no numbers less than A, B which are in the same ratio with A, B. Therefore A, B are the least of those which have the same ratio with them. Q. E. D.
Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.
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