1879.] 



the number of primes between limits. 



801 



The results of the numeration for the first three millions are 

 taken from p. 48 of the present volume, and those for the fourth 

 million from p. 44 of the Introduction to my father's "Factor 

 table for the fourth million" (1879), 



The last column, headed A^, contains the difference between 

 each value of A and the next. 



The factor tables for the fifth and sixth millions are not yet 

 pubHshed, so that the foregoing table cannot at present be extended 

 to 9,000,000. 



§ 6. It is not difficult to see that it is possible to determine 

 the number of primes inferior to any given number without 

 actually forming a factor table to this extent, and counting the 

 primes in it. This can be effected by means of the known formula 

 for the number of numbers inferior to x and not divisible by any 

 given primes _Pj, />, . . .jj„; there are also other methods which are more 

 suitable in the case of the multiples of large primes*. These pro- 

 cesses are very laborious, but they have been applied by Hargreave 

 and Meissel independently to find the value of (/> (x) for 



X = 10,000,000, 

 and by the latter alone in the case of 



^=100,000,000. 



Hargreave's investigations are contained in the Philosophical 

 Magazine for 18.54-f-; he there gives 664,632 as the number of 

 primes (excluding unity) up to 10,000,000. This does not agree 

 with the result given by Meissel in the second volume (1870) of 

 the Mathematische Annalen% which is 664,579. In a paper j in 

 the third volume (1871) of the same journal Meissel calculates 

 the number of primes inferior to 100,000,000 at 5,761,460. 



The values of A determined from these results are shown below : 

 TABLE IV. 



* This subject is referred to in more detail on p. 34 of- the Introduction to the 

 Fourth Million ; where also references to the papers in which these methods have 

 been used are given. 



t On the law of prime numbers, Ser. 4, t. viii. pp. 114 — 122. 



§ Veber die Bestimmung der Primzahlenmenge innerhalb gegebener Grenzen, 

 pp. 636—642. 



J Berechnung der 3Ienge von PrimzaMen, welche innerhalb der ersten hundert 

 MilUonen natiirlichen Zahlen vorkommen, pp. 528 — 525. 



