1878.] Mr Glaisher, On factor tables. 12 o 



2, 3 or 5, from 1,020,000 to 2,028,000, on 112 quarto pages\ there 

 being thus least factors for 9,000 numbers on each page. Omitting 

 headings, each page contains 80 lines and, omitting the left- 

 hand argument column, 30 columns. The first three figures of 

 the argument are taken from the top of the page, the next two 

 from the heading of the column, and the last two from the argu- 

 ment column. Each page is divided into three portions by hori- 

 zontal lines, the end-figures in the argument column for the first 

 being 01, 07,... 97, for the second 01, 03,... 99, and for the third 



03, 09,... 99. The reason for this is clear, for if we throw out the 

 multiples of 2, 3, and 5 from 1, 2, 3,... 99 we have left 1, 7,... 97, if 

 from 101, 102,.. .199 we have left 101, 103,. ..199, and if from 201, 

 202,... 299 we have left 203, 209,... 299; after which the cycle 

 repeats. 



Thus, leaving out column headings, the page contains 80 lines 

 corresponding to the 80 values of r for which 300^' + r repre- 

 sents numbers not divisible by 2, 3, or 5, viz. on the first line are 

 numbers of the form 300^ +1, on the second numbers of the 

 form 3002- -1-7,... on the last numbers of the form SOO^' 4- 299. 

 It is interesting to notice the connexion between this arrange- 

 ment and that adopted by Felkel and Euler. Felkel divides each 

 page into two half-pages, each half-page containing 8 columns 

 corresponding to the values of r when multiples of 2, 3, or 5 

 are thrown out from 30^' + r. If then we imagine Felkel's page 

 widened so as to contain ten groups of eight columns instead 

 of only two, we should have 80 columns on a page, and if the 

 arguments be supposed to run along each line across the whole 

 page, we obtain Burckhardt's arrangement except that lines and 

 columns are interchanged. The improvement effected by the 

 change is considerable, as the arguments in Burckhardt are 

 found at once with great ease, and the troublesome double 

 entry process in. Felkel is avoided. Burckhardt's arrangement is 

 the same as that adopted by Lambert for his table to 10,200 

 in the Beytrdge, except that the table is there printed on a fold- 

 ing sheet, while here a quarto page contains it. The tripartite 

 system of Lambert was followed by Vega, who however could not 

 place the three groups upon his octavo page, but had only room 

 for one and a half^. This produces an awkward dislocation, not 

 very easy to explain briefly, but the effect of which is that, as the 

 headings of the columns in each group are not consecutive numbers, 



1 The size is that of an ordinary quarto, though the vohxme is technically 

 a folio. 



2 In Hiilsse's edition of Vega (1840) two of the three groups are contained 

 on the same page. I have referred to the irregularity of progression in this 

 table in a paper "Eemarks on logarithmic and factor tables, with special reference 

 to Mr Drach's suggestions." Messenger of Mathematics, vol, iii. pp. 7 — 12, (May, 

 1873.) 



