Book 9
36
For let as many numbers as we please, A, B, C, D, beginning from an unit be set out in double proportion, until the sum of all becomes prime, let E be equal to the sum, and let E by multiplying D make FG; I say that FG is perfect.
36
For, however many A, B, C, D are in multitude, let so many E, HK, L, M be taken in double proportion beginning from E; therefore, ex aequali, as A is to D, so is E to M. [VII. 14]
36
Therefore the product of E, D is equal to the product of A, M. [VII. 19]
36
And the product of E, D is FG; therefore the product of A, M is also FG.
36
Therefore A by multiplying M has made FG; therefore M measures FG according to the units in A.
36
And A is a dyad; therefore FG is double of M.
36
But M, L, HK, E are continuously double of each other; therefore E, HK, L, M, FG are continuously proportional in double proportion.
36
Now let there be subtracted from the second HK and the last FG the numbers HN, FO, each equal to the first E; therefore, as the excess of the second is to the first, so is the excess of the last to all those before it. [IX. 35]
36
Therefore, as NK is to E, so is OG to M, L, KH, E.
36
And NK is equal to E; therefore OG is also equal to M, L, HK, E.
36
But FO is also equal to E, and E is equal to A, B, C, D and the unit.
36
Therefore the whole FG is equal to E, HK, L, M and A, B, C, D and the unit; and it is measured by them.
36
I say also that FG will not be measured by any other number except A, B, C, D, E, HK, L, M and the unit.
