50 Mr Glaisher, Account of an Enumeration [Mar. 12, 



sufficient patience for a continuous enumeration of the whole 

 million, he often employed unoccupied quarters of an hour in 

 counting here and there a chiliad. Finally the work was laid 

 aside without the million being made complete, but later on. 

 Gauss "made use of Goldschmidt's industry, to fill in the gaps that 

 were left in the first million, and to extend the enumeration 

 further by means of Burckhardt's Tables." Dr Sobering, in his 

 notes at the end of the volume (p. 520) states that the Table for 

 the first million was in Gauss's own handwriting, and that the 

 Tables for the second and third millions were in Goldschmidt's 

 handwriting. It is thus clear that the enumeration could not be 

 a very accurate one: in fact, the books used were themselves by 

 no means free from error. Gauss states that the chiliad in 

 Lambert's Supplementa between 101,000 and 102,000 "swarmed 

 with errors;" seven composite numbers in this chiliad were given 

 as primes, and two primes were omitted. Also Chernac at the end 

 of his Cribrum gives a list of 20 composite numbers that are con- 

 tained in Vega's list of primes, and of 22 primes that are omitted. 

 Gauss did not himself publish his results, and was, no doubt, 

 perfectly aware that they were not to be depended upon as 

 accurate. 



In the first million (in which the enumeration was made chiefly 

 by Gauss himself), the number of primes in each chiliad is given ; 

 but the results for the other two millions are arranged in a similar 

 manner to Tables A, JB, G at the end of the present paper. 

 There are, in all, 22 such Tables, one for each group of 100,000 

 in the second and third millions, and one for the whole of the 

 second million similar to B, and one for the third million 

 similar to C: this part of the enumeration was made, as mentioned 

 above, by Goldschmidt. 



When, in 1873, I compared my results with the Tables in 

 Gauss's Werke, I found many discrepancies : but in the notes to 

 the new edition of the Werke (1876) nineteen errata, found by 

 Dr Meissel, of Iserlohn, are pointed out in Gauss's first million, 

 and when these are corrected the values agree entirely with my 

 OAvn, except in one instance, viz. the number of primes in the 

 354*^ chiliad should be 76 instead of 79. 



Recently, since the completion of my enumeration, Professor 

 H. J. S. Smith called my attention to Meissel's paper, "Ueber die 

 Bestimmung der Primzahlenmenge innerhalb gegebener Grenzen" 

 (Mathematische Annalen, t. ii. (1870), pp. 636 — 642), in which are 

 given the errata in the first million that are reproduced in the 

 second edition of Gauss's Werke. I find that the error in the 

 354* chiliad is not noticed by Meissel in his paper, but as he 

 assigns 7,863 as the number of primes between the 300*^ and the 



