Book 9
36
But primes are also least, [VII. 21] and the least numbers measure those which have the same ratio the same number of times, the antecedent the antecedent and the consequent the consequent; [VII. 20] and, as E is to Q, so is P to D; therefore E measures P the same number of times that Q measures D.
36
But D is not measured by any other number except A, B, C; therefore Q is the same with one of the numbers A, B, C.
36
Let it be the same with B.
36
And, however many B, C, D are in multitude, let so many E, HK, L be taken beginning from E.
36
Now E, HK, L are in the same ratio with B, C, D; therefore, ex aequali, as B is to D, so is E to L. [VII. 14]
36
Therefore the product of B, L is equal to the product of D, E. [VII. 19]
36
But the product of D, E is equal to the product of Q, P; therefore the product of Q, P is also equal to the product of B, L.
36
Therefore, as Q is to B, so is L to P. [VII. 19]
36
And Q is the same with B; therefore L is also the same with P; which is impossible, for by hypothesis P is not the same with any of the numbers set out.
36
Therefore no number will measure FG except A, B, C, D, E, HK, L, M and the unit.
36
And FG was proved equal to A, B, C, D, E, HK, L, M and the unit; and a perfect number is that which is equal to its own parts; [VII. Def. 22] therefore FG is perfect. Q. E. D.
