ON MATHEMATICAL TABLES. 177 



of the sieves (the 307-sieve is 10 feet 6 inches in length, and the 

 499-sieve 17 feet 1 inch) is productive of great inconvenience, and would 

 also necessitate very great accuracy and care in the lithographing and 

 printing of the sheets, so that the squares should correspond exactly, over 

 so great a distance ; and it seemed surprising that Burckhardt should 

 have continued the sieve method so far. But this was on the supposition 

 that the portions of the sieve would he all fixed together, so that it would 

 consist of one long sheet. Experience, however, soon showed that nothing 

 was gained by fixing the sheets together, and in fact that it was a positive 

 inconvenience to do so. The sheets forming the sieve were numbered 1, 

 2, 3, &c, and all that was requisite was to use sheet 1 first, then sheet 2, 

 then sheet 3, then sheet 1 again (if the sieve consisted of only 3 sheets), 

 and so on ; in fact, the long sieves were found to be quite as easy to use 

 as the smaller ones. Above 307, however, it seemed to be scarcely worth 

 while to construct the sieves, as so little use was made of them, and as the 

 multiple method was preferable in consequence of the verification afforded 

 by it. 



The mode of work was as follows : The entries were made by the 

 sieves, and one multiple of $ obtained from each position of the £>-sieve 

 was divided out by p, in order to verify that the sieve was always rightly 

 placed ; this verification was employed for each position of every sieve. The 

 numbers were then examined by Mr. Glaisher himself by the sieves. They 

 were then examined a third time by the sieves, and every number ticked. 

 The least factors obtained by the multiple method were read out and 

 entered on the sheets ; and they were subsequently read out again in a 

 different manner and ticked. Any numbers found unticked were after- 

 wards specially examined. The proofs of the table when printed will be 

 read with the original calculations of numbers by the multiple method. 



On the whole the method of construction is a very perfect one. It has 

 been explained in some detail, because Burckhardt contents himself with 

 a very brief sketch occupying only two paragraphs ; and the process is 

 sufficiently interesting to deserve a more complete account. Bach sieve, 

 as stated, has 80 squares cut out, one in each line ; though of course, as 

 there are only 80 squares cut out, whatever be the length of the sieve, 

 many of the columns on the longer sieves are left intact. The patterns 

 formed by the holes in the sieves were very curious, some being very 

 regular, while in others the holes were very scattered, and no two were 

 much alike. The sieves for 149 and 151 were remarkable, the holes 

 running steadily up in the one case and steadily down in the other.* The 

 reason for this is that these numbers are nearly equal to the half of 300, 

 the difference between two adjacent squares in the same line, so that 

 numbers distant from one another by even multiples of 150 are in the 

 same line. For a similar reason the holes in the sieves for 59 and 61, and 

 29 and 31, show a steady ascent and descent. When the sieve for 23 is 

 laid in its proper position on any one of the sheets a slightly ascending 

 row of 13's (including some 7's and ll's) is seen through the holes ; this 

 is connected with the fact that 13 x 23 = 299, and differs from 300 by 1 

 only. Similarly, when the 43-sieve is laid on the sheets a slightly de- 

 scending row of 7's is seen, as 7 X 43 = 301, and other instances of the 

 same kind were remarked. It may be observed that when the pattern 



* Several of the sieves, including those for 149 and 151, were exhibited to the 

 Section at the Meeting at Dublin. 



1878. N 



