1879 ] the number of primes between limits. 297 



Legendre then gives a table in which the values obtained from 

 the formula are compared with the numbers actually counted up 

 to 400,000. The interval in this table is 10,000 from to 100,000 

 and 50,000 from 100,000 to 400,000 ; so that the table shows the 

 number of primes obtained by the formula and by counting from 

 to 10,000, from to 20,000, &c.; the enumerations were made 

 from the factor table and list of primes in Vega's Tabulce, which 

 extend from to 400,031. After the table Legendre remarks 

 " II est impossible qu'une formule represente plus fidelement une 

 sdrie de nombres d'une aussi grande etendue et sujette ndcessaire- 

 ment a de frdquentes anomalies"; but he does not give any 

 definite information with regard to the manner in which he was 

 led to the formula, or to assign the -value 1"08866 to the constant. 

 Legendre does indeed, at the end of the Chapter, indicate certain con- 

 siderations from which he deduces that the average distance between 

 two primes at the point x in the series of numbers is of the form 

 A log sc + B, and thence that the number of primes inferior to x is 

 approximately equal to 



Alogw + B — A' 



"ce qui s'accorde avec la formule g^nerale donn^e ci-dessus, en 

 prenant A = l, i? = — O'OS.366"; the reasoning however by which 

 the average interval A log x + B is obtained is vague and unsatis- 

 factory. 



It would thus appear that Legendre, having been led by the 

 analytical considerations just mentioned or otherwise, to a formula 

 of the form 



Alogx — B' 



determined the values of A and B empirically by means of the 

 enumerations ; the value of A would be at once found to be unity, 

 and since in the table the number of primes up to 10,000 is 

 given as 1230 both by the formula and the enumeration, and the two 

 numbers are identical for no other value of x, it would seem that 

 probably the constant was mainly determined from the value 

 ^ = 10,000; see § 13. 



In 1816 Legendre published a supplement to the Theorie des 

 Nomhres, which contains (pp. 61, 62) an addition to the Chapter on 

 the distribution of primes. This principally consists of a con- 

 tinuation of the table from 400,000 to 1,000,000 at intervals of 

 50,000. The enumerations were made from Chernac's Crihrum 

 Arithmeticum, pubKshed in 1811, which gives all prime factors of 

 numbers up to 1,020,000. 



In the third edition of the Theorie des Nomhres (2 vols., 1830) 

 the Chapter on the distribution of primes remains substantially 



