ON MATHEMATICAL TABLES, 37 



says : " Non solum iiiveni formam omnes divisores nuraerorum excepto maxi- 

 mo, ab 1 usque 1,008,000 in spatio 42 plagularum rcprffiscntandi, vcruni etiam 

 rcipsa opus spatio 16 meusium usque ad 2,016,000 coiifeci, annoque 1785 

 .... ad 5,000,000 usque continuavi." (See also p. yII of the ' Introductio lu- 

 tcrpretis '). 



Since writing the above description of Felkel, I have examined (in the 

 Graves Library) a far more complete copy, which contains probably all that 

 Tclkel ever printed. There are three parts (bound together). The first is the 

 same as that described above, and extends to 144,000 ; the second part 

 (with fresh pagination) extends from 144,001 to 336,000 (pp. 2-03) ; we 

 then have 'Tabula Factorum pars III exhibens factores numerorum ab 

 336,001 usque 408,000,' occupying pp. 65-87. The table thus gives factors 

 as far as 408,000. The words " 336,001 usque 408,000 " have clearly ori- 

 ginally stood " 144,001 usque 366,000 ;" but the latter numbers have been 

 stamped out and the former printed over them. There is a note in the work 

 in the handwriting of Mr. Graves's librarian, which, referring to Gauss's 

 remark quoted above, proceeds : — " This copy contains 3 parts and gives the 

 factors of all numbers up to 408,000 ; such a copy is perhaps unique." 

 Gauss stated that all the copies were destroyed except a few, which extended 

 to 330,000 ; so that there can be no doubt that tlie Graves copy, extending 

 to 408,000, must be, to say the least, excessively rare. 



It should be added that the title and preface to the Graves copy are in 

 Latin, while the lloyal Society's copy has them in German (Poggendorff 

 also quotes the title in German with date 1777) ; the preface is dated April 1, 

 1777, although the titlepage bears date 1776. In the Graves copy some 

 errata in Part I. are given. 



For several reasons Felkel's connexion with numerical tables is a curious 

 one, and the record of his life would be interesting. "VVe have seen (in some 

 work of reference) a number of mechanical contrivances assigned to him as 

 their inventor. 



Chernac, 1811. In a thick quarto are given all the simple divisors of 

 numbers from 1 to 1,020,000 (multiples of 2, 3, and 5 being excluded). 

 This book was found by Burckhardt (who subsequently published the same 

 table, the least divisor only being given) to be very acciu-ate ; he detected only 

 38 errors (he has given them in the preface to his first million), of which only 

 9 are due to the author, the remaining 29 having been caused by the slipping 

 &c. of type in the printing. 



Hutton's Phil, and Math. Diet. 1815. In vol. ii. pp. 236-238 (Art. 

 ' Prime Numbers ') is a table giving the least divisor of all numbers from 1 to 

 10,000, multiples of 2 and 5 being omitted. 



Burckhardt (First Million), 1817. Least divisors of every number to 

 1,020,000. The library of the Institute contained a manuscript (calculated 

 by Schenmarck ?) giving the least divisor of numbers to 1,008,000 ; Burck- 

 hardt therefore computed the next 12,000 himself, and compared the manu- 

 script with Cheenac — a laborious work, as when a wrong divisor was given, 

 Burckhardt had to satisfy himself if the number was really prime, as was 

 the case in 236 instances. For primes less than 400,000 he referred to Ycga 

 (see Vega's ' Tabula,' 1797, Vol. II. T, I., and Hulsse's Jega, 1840, T. V.). 

 Ojily 38 errors were found in Chebnac. On the last page is a small table con- 

 taining the number of figures in the periods of the reciprocals of 794 primes 

 below 9901 (779 of which are below 3000). Burckhardt mentions in the preface 

 that he has nearly completed the manuscript of the fourth, fifth, and sixth 

 millions, which will be published, if the sale of the first three millions is 



