| If as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect. | |
| Let as many numbers as we please, A, B, C, and D, beginning from a unit be set out in double proportion, until the sum of all becomes prime, let E equal the sum, and let E multiplied by D make FG.
I say that FG is perfect. For, however many A, B, C, and D are in multitude, take so many E, HK, L, and M in double proportion beginning from E. | |
| Therefore, ex aequali A is to D as E is to M. Therefore the product of E and D equals the product of A and M. And the product of E and D is FG, therefore the product of A and M is also FG. | VII.14
VII.19 |
| Therefore A multiplied by M makes FG. Therefore M measures FG according to the units in A. And A is a dyad, therefore FG is double of M.
But M, L, HK, and E are continuously double of each other, therefore E, HK, L, M, and FG are continuously proportional in double proportion. | |
| Subtract from the second HK and the last FG the numbers HN and FO, each equal to the first E. Therefore the excess of the second is to the first as the excess of the last is to the sum of those before it. Therefore NK is to E as OG is to the sum of M, L, KH, and E. | IX.35 |
| And NK equals E, therefore OG also equals M, L, HK, E. But FO also equals E, and E equals the sum of A, B, C, D and the unit. Therefore the whole FG equals the sum of E, HK, L, M, A, B, C, D, and the unit, and it is measured by them.
I say also that FG is not measured by any other number except A, B, C, D, E, HK, L, M, and the unit. If possible, let some number P measure FG, and let P not be the same with any of the numbers A, B, C, D, E, HK, L, or M. And, as many times as P measures FG, so many units let there be in Q, therefore Q multiplied by P makes FG. | |
| But, further, E multiplied by D makes FG, therefore E is to Q as P is to D. | VII.19 |
| And, since A, B, C, and D are continuously proportional beginning from a unit, therefore D is not measured by any other number except A, B, or C. | IX.13 |
| And, by hypothesis, P is not the same with any of the numbers A, B, or C, therefore P does not measure D. But P is to D as E is to Q, therefore neither does E measure Q. | VII.Def.20 |
| And E is prime, and any prime number is prime to any number which it does not measure. Therefore E and Q are relatively prime. | VII.29 |
| But primes are also least, 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, and E is to Q as P is to D, therefore E measures P the same number of times that Q measures D. | VII.21
VII.20 |
| But D is not measured by any other number except A, B, or C, therefore Q is the same with one of the numbers A, B, or C. Let it be the same with B.
And, however many B, C, and D are in multitude, take so many E, HK, and L beginning from E. | |
| Now E, HK, and L are in the same ratio with B, C, and D, therefore, ex aequali B is to D as E is to L. | VII.14 |
| Therefore the product of B and L equals the product of D and E. But the product of D and E equals the product of Q and P, therefore the product of Q and P also equals the product of B and L. | VII.19 |
| Therefore Q is to B as L is to P. 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. | VII.19 |
| Therefore no number measures FG except A, B, C, D, E, HK, L, M, and the unit. | |
| And FG was proved equal to the sum of A, B, C, D, E, HK, L, M, and the unit, and a perfect number is that which equals its own parts, therefore FG is perfect. | VII.Def.22 |
| Therefore, if as many numbers as we please beginning from a unit are set out continuously in double proportion until the sum of all becomes prime, and if the sum multiplied into the last makes some number, then the product is perfect. | |
| Q.E.D. | |
Book IX Introduction - Proposition IX.35 - Book X Introduction.