The following is as given in Sir Thomas L. Heath's translation, which can be found in the book The Thirteen Books of The Elements, Vol. 2.
|Book I||Book II||Book IX|
|Definitions, Postulates, and Common Notions||Definitions||Proposition 20|
|Proposition 1, Proposition 3,||Proposition 14||Proposition 36|
|Proposition 5, Proposition 6,|
|Proposition 29, Proposition 47|
If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect.
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,
For, however many A, B, C, D are int multitude, let so many E, HK, L, M be taken in double proportion beginning from E;
Therefore the product of E, D is FG;
Therefore A by multiplying M has made 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, E are continuously double of each other; therefore E, HK, L, M, FG are continuously proportional in double proportion.
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]
Therefore, as NK is to E, so is OG to M, L, KH, E.
And NK is equal to E;
But FO is also equal to E,
Therefore the whole FG is equal to E, HK, L, M and A, B, C, D and the unit.
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.
For, if possible, let some number P measure FG,
And, as many times as P measures FG, so many units let there be in Q;
But, further, E has also by multiplying D made FG; therefore, as E is to Q, so is P to D.
And, since A, B, C, D are continuously proportional beginning from an unit,
And, by hypothesis, P is not the same with any of the numbers A, B, C;
But, as P is to D, so is E to Q;
And E is prime;
Therefore E, Q are prime to one another.
But primes are also least, [VII. 21]
But D is not measured by any number except A, B, C;
Let it be the same with B.
And, however many B, C, D are in multitude, let so many E, HK, L be taken beginning from E.
Now E, HK, L are in the same ratio with B, C, D;
Therefore the product of B, L is equal to the product of D, E. [VII. 19]
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.
Therefore, as Q is to B, so is L to P. [VII. 19]
And Q is the same with B;
Therefore no number will measure FG except A, B, C, D, E, HK, L, M and the unit.
And FG was proved equal to A, B, C, D, E, HK, L, M and the unit;