202 SCIENCE PROGRESS 



accurate to 10, 12, or 14 figures. This has been supplied by 

 Edward Sang within the limits just named. 



It would not however be necessary, as pointed out by 

 Prof. Andoyer, to publish more than a certain part of these 

 tables in order to place in the hands of the mathematical world 

 a fundamental table of logarithms to 12 or 14 figures. It would 

 be sufficient to tabulate the logarithms of numbers from 100,000 

 to 200,000, for by simple division any number can be reduced to 

 a number beginning with the figure " one." 



The simplest and certainly the most accurate way to publish 

 Sang's tables would be to reproduce the original manuscript 

 pages as line engravings by photography. This was indeed 

 the method which first suggested itself to Dr. Burgess and 

 myself when Dr. Sang's manuscript volumes were consigned 

 to the care of the Royal Society of Edinburgh. And various 

 considerations have recently made some of us regard with 

 increasing favour this method of utilising to the full the 

 fundamentally important parts of Sang's manuscript volumes. 



It should be noted in the first place that the manuscripts have 

 been prepared with extreme care, the figures being beautifully 

 written and entered with great neatness in appropriate spaces in 

 specially ruled paper. The rulings are differently spaced ac- 

 cording to the nature of the table. For example, in the table of 

 the logarithms of integer numbers from 1 to 10,000, there are 

 twenty-five numbers on each page with their logarithms to 28 

 figures, covering, therefore, 400 manuscript pages in all. This, 

 which forms the basis of the whole, could be reproduced by 

 a slight reduction in 100 pages of 100 numbers to the page. 



The logarithms to 15 figures of numbers from 100,000 to 

 200,000 are also arranged in the manuscript volumes twenty-five 

 to the page, and contain the first and second differences entered 

 in specially prepared ruled paper. Here, also, by a similar 

 reduction four of the manuscript pages could be reproduced 

 as one page containing 100 numbers with their logarithms and 

 first and second differences. The whole could be reproduced in 

 a large quarto volume of 1,000 pages. 



It should be noted that the final tabulations were made by 

 Dr. Sang after three verifications, so that it is almost impossible 

 for any error of tabulation to exist, the only possible errors being 

 those which result from the method of interpolation, and which 

 rarely reach three units in the fifteenth figure. 



