1877.] 



of the Primes in BurcJchardt' s Tables. 



51 



400''' chiliad, which agrees with the number in Table A, it is clear 

 that he must have used the correct value in his calculation. Meissel 

 gives the number of primes in each group of 100,000 in the first 

 million, and his values agree with those in Table A. In the first 

 100,000 however, he gives the number of primes as 9,592, while the 

 number in Table A is 9,593; but this discrepancy is no doubt 

 due to the fact that he has not counted 1 and 2 as primes, for 

 in Gauss's Table the number of primes in the first chiliad is given 

 as 168, and Meissel accepts this value; but if 1 and 2 be both 

 counted as primes the number is 169. Meissel obtained the 

 number of primes in the first million both by counting from 

 Burckhardt's Tables, and also by means of an analytical process 

 of his own, and as the results obtained by him agree with those 

 given in this paper, there can be no doubt of the accuracy of the 

 enumeration as far as the first million is concerned. It does not 

 seem that Meissel has given any results relating to the counting of 

 primes from Burckhardt's or Dase's Tables in the second or 

 higher millions; but by his analytical process he has calculated 

 the number of primes in the first ten millions, and also in the 

 first hundred millions^. 



It would occupy too much space to give in detail the errata 

 in the Tables for the second and third millions that appear in 

 Gauss's Werke, but the following list contains the values given in 

 Gauss for the number of primes in each group of 100,000 and the 

 values found by the present enumeration. 



Second Million. 



1 " Berecliuimg cler Menge von Primzalilen, welclie innerhalb der ersten Huiidert 

 Mllionen natiirlicher Zalilen vorkommen." Matliematische Annakn, t. iii. (1871), 

 pp. 523-525. 



