ON MATHEMATICAL TABLES. 173 



a gap of three millions between the third and seventh millions, which 

 it was very desirable to fill up not only for the sake of completing the 

 table up to nine millions, but also in order to render more useful the 

 millions already published. Accordingly, Professor Cayley, the chair- 

 man of the Committee, wrote to Professor Kummer, the secretary of the 

 Mathematical Section of the Berlin Academy, asking if there were any 

 chance of the publication of the manuscript ; and Professor Kummer, in a 

 letter dated April 29th, 1877, replied that it had been examined on a 

 former occasion, and found to be so inaccurate that " the Academy was 

 convinced that the publication would never be advisable." The calcula- 

 tion was then at once commenced by Mr. James Glaisher, with the assis- 

 tance of two computers, and has been continued without interruption since. 

 The fourth million is completed and ready for press, and some progress 

 has been made with the fifth and sixth millions, which are being calcu- 

 lated together, and which will be completed, it is believed, by the meeting 

 of the Association at Sheffield. The tenth million has not been published. 

 It remained in the possession of the widow of Dr. Rosenberg till the early 

 part of the present year, when it was presented by her to the Berlin 

 Academy. 



The method employed in the calculation of the fourth million, and by 

 which the fifth and sixth millions are being calculated, is practically 

 the same as that which was invented by Burckhardt, and was adopted by 

 Dase. As the method is a very remarkable one, and as no description 

 of it (with the exception of a brief notice by Burckhardt himself) has 

 been published, the following account of it is given here : — 



A form was lithographed (Plate rV\), having 78 vertical lines and 81 

 horizontal lines (besides several other lines used for headings, <fec.) ; it is 

 thus divided into 77 x 80 oblong spaces which may for convenience be 

 called squares. The eighty rows are numbered, at the extreme left of the 

 sheet, 01,07... 97; 01,03. ..99; 03,09,. ..99; there being two white spaces 

 separating the hundreds. This is the same as in Burckhardt's or Dase's 

 tables, each column representing 300 numbers. The advantage of having 77 

 columns is that the 7's and ll's are lithographed on the form and have not 

 to be determined and inserted by hand. Thus if 77 consecutive columns 

 of Burckhardt's tables be taken, and all the headings and tabular results 

 except 7's and ll's be supposed to be removed, we have a representation of 

 the form. The form actually used was constructed to begin from 3,000,000, 

 so that for the exact representation of it we are to commence with the 

 column headed 201 on p. 3 of Burckhardt's table (i.e., the 68th column). 



Since each sheet corresponds to 77 X 300 numbers, a million -occupies 

 about 43^ sheets, and as on each sheet the number of 7's lithographed is 

 880, and the number of ll's is 480, it follows that, by adopting a form 

 which permits the 7's and ll's to be lithographed, about 59,000 entries 

 are saved in each million ; and, what is even more important, the accuracy 

 of these 59,000 tabular results is assured. 



The squares to which the least factor 13 belongs were obtained as fol- 

 lows : Find the numbers between 3,000,000 and 3,000,000 + 13 x 300, 

 which are divisible by 13, but not by 2, 3, or 5. Take 13 consecutive 

 columns of any blank form and cut them off from the rest of the form ; 

 then, supposing the first column to correspond to the column headed 

 3,000,000, make a mark in the squares that correspond to the multiples of 

 13, previously found, and cutout the squares so marked. We thus have 

 a group of 13 columns, from which a number of squares (80) have been 



