36 



EEPOKT — 1880. 



The following table contains the two longest successions of composite 

 numbers met with in each of the five millions : 



In the ' Philosophical Magazine ' for August, 1854, the late Mr. 

 C. J. Hargreave determined the number of primes inferior to 5,000,000 

 at 348,527. His method, which is there described, consisted in calculating 

 the number of numbers which are the products of two prime factors, of 

 three prime factors, &c., and thus determining the total number of com- 

 posite numbers between the limits in question. The number of primes 

 in the five millions obtained by enumeration from the tables is 348,515. 

 This includes unity as a prime, and it appears that Hargreave excluded 

 unity, so that if it be included, his number would become 348,628, which 

 differs by 13 from the number obtained from the tables. 



IV. Comparison of the immhers of Primes counted with the Values given by 

 Legendre's and Gauss's Formula;. 



Legendre's formula for the number of primes inferior to a given 

 number x is 



! log aj- 1-08366 



This expression Legendre published in the second edition of his ' Theorie 

 des Nombres ' (Part iv. 1808), and he there gave a table containing com- 

 parisons between the numbers obtained from it and the numbers obtained 

 by counting up to 400,000. This table Legendre subsequently extended 

 in 1816, after the publication of Chernac's ' Cribrum Arithmeticum,' to 

 1,000,000. It does not appear why Legendre assigned the value 1-08366 

 to the constant which occurs in his formula, but it is probable that this 

 value was originally determined so as to render very close the agreement 

 with the numbers counted in the earlier enumerations, and as the formula 

 still continued to yield good results as far as the later enumerations ex- 

 tended, no attempt was made to improve the value at first assigned to it. 

 The logarithm-integral li x, -where li x denotes the integral, 



/'*x dx 



^ d log* 



