concerning Numbers, 4-1 



nature are based. Now in effecting such an extension, we 

 attach a notional continuity to expressions which in their 

 natural sense are not properly continuous, but proceed per 

 saltus ; and our power to do this arises from the circumstance, 

 that whatever discontinuity in the algebraical sense exists in 



expressionso^theforml.2.3..a;, 1+ — + — + .. +— , logl + 



log 2 + ..+ log .r, &c., such discontinuity is itself the sub- 

 ject of a regular and continuous law. The process is an in- 

 terpolation of all those algebraical forms which are necessary 

 to be interpolated, in order that expressions of this kind may 

 have continuous values for every value of x proceeding by 

 infinitely small increments, instead of proceeding by definite 

 equal increments. 



Now prime numbers are not naturally anomalous any more 

 than ordinals are ; they are the elements of the ordinals ; and 

 the infinite series of natural primes is the framework or ske- 

 leton of the series of natural ordinals, the texture superposed 

 upon such framework being manufactured by a law which 

 we can express. The whole series of ordinals 



1+2 + 3 + 4. + 5 + 6 + 7 + 8 + 9+10+114-12 + 



is formed by the continued multiplication of the several series 



2" + 2^ + 2^+23 + 2*+ 25 + 2^" + .. 

 3<^ + 3i + 3H3^ + 3'* + 3^ + 3« + .. 

 5f^ + 5^-\-5'^ + 5^ + 5'^+5^ + 5^ + ., 



^0 +pl + p2 ^^3 ^^4 ^pb ^pG^„ 



where the sign + is used in the ordinary sense for the pur- 

 pose of multiplication, but in the result merely means juxta- 

 position. This is sufficiently apparent from the fact that the 

 above product gives every possible combination of primes and 

 powers and multiples of primes, and cannot give the same 

 number twice, since no one prime or any power of it can 

 be equivalent to any other prime or any power of it. In 

 effect, in the first theorem we resolved the infinite series of 

 ordinals into 



(l-2)-Hl-3)->(l-5)-^..(l -;?)-'..., 



which is the algebraical form of the expression last above 

 w^ritten. 



Let us now denote the natural series of primes by 



