But, further, E has also by multiplying D made FG; therefore, as E is to Q, so is P to D.

[VII. 19]

And, since A, B, C, D are continuously proportional beginning from an unit,

therefore D will not be measured by any other number except A, B, C.

[IX. 13]

And, by hypothesis, P is not the same with any of the numbers A, B, C ;

therefore P will not measure D.

But, as P is to D, so is E to Q;

therefore neither does E measure Q.

And E is prime;

[VII. Def. 20]

and any prime number is prime to any number which it does

not measure.

Therefore E, Q are prime to one another.

But primes are also least,

[VII. 29]

[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 ;

and, as E is to Q, so is P to D;

[VII. 20]

therefore E measures P the same number of times that Q measures D.

But D is not measured by any other number except A, B, C;

therefore is the same with one of the numbers 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, ex aequali, as B is to D, so is E to L. [VII. 14] Therefore the product of B, L is equal to the product of D, E. 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.

And is the same with B;

therefore L is also the same with P:

[VII. 19]

[VII. 19]

which is impossible, for by hypothesis P is not the same with any of the numbers set out.

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;

and a perfect number is that which is equal to its own parts;

therefore FG is perfect.

If the sum of any number of terms of the series

I, 2, 22, 22-1

[VII. Def. 22]

Q. E. D.

be prime, and the said sum be multiplied by the last term, the product will be a "perfect" number, i.e. equal to the sum of all its factors.

Let 1 + 2 + 22 + + 2"-1 (= Sn) be prime;

(This is of course obvious algebraically, but Euclid's notation requires him to prove it.)

and

or

...

+ 2n−2Sn

Now, by 1x. 35, we can sum the series S + 2 Sn +
(2S-Sn): Sn = (2"1 Sn - Sn): (Sn + 2Sn + ... + 2"-2S).
Sn + 2Sn +2°S2+ ... + 2"-2S1 = 2"-1Sn - Sn
2"-1S = Sn+2Sn + 2aSn + ... + 2"-2 Sn + Sn

Therefore

and 2"-1 S, is measured by every term of the right hand expression.

It is now necessary to prove that 2-1S, cannot have any factor except those terms.

Suppose, if possible, that it has a factor x different from all of them,

Therefore

2n-1 Sn
Sn: m = x: 2n−1.

= x m.

[VII. 19]

Now 2"-1 can only be measured by the preceding terms of the series I, 2, 22,... 2′′-1,

[IX. 13]

And m = 2" ;

therefore x = = 2"-r-1, one of the terms of the series 1, 2, 22,... 2"-2: which

contradicts the hypothesis.

Therefore 2"-1S has no factors except

Sn, 2Sμ, 22Sn, ... 2"-2S, 1, 2, 22, ... 2"-1.

22 ...

INDEX OF GREEK WORDS AND FORMS.