1878.] Mr Glaisher, On factor tables. 127 



With regard to the calculation of this million Burckhardt states 

 that the library of the Institute is in possession of a manuscript 

 containing least factors of numbers from 1 to 1,008,000, and that he 

 has carefully compared this table with Chernac's, adding himself 

 the 12,000 which were wanting in the mauuscript. This work 

 was much more laborious than he expected, " soit que M. Schen- 

 mark n'elit pas du se servir de la methode d'Euler, soit que les 

 cinq Aleves et amis qui s'etaient joints au savant professeur de 

 I'universite de Lund pour cet ouvrage, n'y aient pas mis toute 

 I'attention qu'il exige." When the manuscript differed from 

 Chernac's table, and the number had a factor, it was easy to 

 decide which was right; but when the number was prime the 

 examination was much more troublesome, for it was not sufficient 

 to show that the factor indicated was not right: it was necessary 

 to prove that the number was really prime. In this way he was 

 obliged to examine 236 numbers, between 400,000 and the end of 

 the million. For similar disagreements met with between 1 and 

 400,000, he consulted Vega's tables, which always confirmed 

 Chernac. The errors found in Chernac have been referred to in 

 §14. 



The only other mention of Schenmark's manuscript that 

 I have seen is contained in a note in Lambert's Briefweclisel 

 (1785) by Bernoulli, who states that it was taken by Lexell from 

 Lund to St Petersburg and laid before the Academy there\ It 

 must therefore have been taken from St Petersburg to Paris, and 

 placed in the library of the Institute between 1785 and 1811. 



It is rather curious that Burckhardt, who devoted so much 

 labour to factor tables, and to whom we owe the best mode of 

 constructing and printing them, should have prefixed such brief 

 and meagre introductions to his volumes. 



I have not been able to find any further reference to Burck- 

 hardt's nearly complete manuscript of the fourth, fifth and sixth 

 millions, and do not know what became of it. 



Five errata in Burckhardt are given in Dase (seventh million), 

 but of these four occur in the introduction, and are of little conse- 

 quence. Only one relates to the tables, and that is due to a 19 that 

 has slipped back in the printing. In forming the factor table for 

 the fourth million (§ 20) my father found another error in Burck- 

 hardt's tables, viz. the number 3,026,279 is prime; but Burck- 

 hardt gives a least divisor 79. 



At the end of the first million Burckhardt has half a page to 

 spare, and this he devotes to a table of the number of figures in 



1 The sentence in full is "Dahin gehort auch Herm Prof. Schenmarks ra 

 Lund noch ungedruckte Tafel, die, wie mir aus schriftlichen Nachrichten bekannt 

 ist, bis auf eine Million sich erstreckt, und von H. rn Prof. Lexell bey seiner 

 letztern Eeise uber Lund nach Petersburg gebracbt und der Kais. Akademie der 

 Wissensch. Torgelegt wordeu ist." (Briefivechsel, v. p. 140.) 



