222 Mr. R. Templeton on a Method of 



M. Lefort continues, " Under this double relation it would be 

 undoubtedly of much interest to have these Tables published; 

 but it would appear still more desirable to make them first serve 

 for an edition to eight places of decimals, which would be of 

 commodious form and not necessitate the employment of second 

 differences. Such Tables would be of extreme value in certain 

 calculations in geodesy, astronomy, and compound interest." 

 M. Lefort would like to see more extensive tables of logarithms 

 published, but is reasonably enough frightened at columns of 

 differences. There are, no doubt, many persons who will sym- 

 pathize in M. Lefort' s desire to have the facilities which loga- 

 rithms afford for verifying complex formula? with a greater 

 amount of accuracy than the common tables permit, or who 

 may require logarithms to ten or twelve places for application 

 to geodetic or other computations, a preference being given 

 at the present day to exact operations rather than to the ap- 

 proximative which were so much in vogue at the early part of 

 the century. It will be presently shown that these facilities are 

 accessible without adopting the suggestion of M. Lefort. A 

 very compact table of not greater bulk than would add three- 

 tenths of an inch to the thickness of Hutton's volume would 

 put it in the power of anyone accustomed to logarithmic cal- 

 culations to find these numbers very readily, if only they would 

 consent to use logarithms also for finding the corrections to be 

 applied to the tabular numbers. 



Let the common logarithmic formula be thus transformed : 



^rf^-^ + fel 



° x lx 2 x J 



x L 2 x J 



, , f. Mn IMn 5 Mrc 2 lMn 3 1 

 = log-i<{tog-- g — + ___--^ + &c^ 



. I\ Mifc . i Mn\ 



= log- 1 j_iog— - 2 — / ver y nearl y- 



So, to find from log# the value of log (x-\-n), we have but to 



M?z 



find the log — , subtract from it one-half of the first few figures 



x 



of the number answering to it, and at the same opening of the 

 Tables find the number of which the remainder is the logarithm. 

 This number will be the quantity to be added to log x to obtain 

 the logarithm of (x + n). Suppose the logarithm of 3141 5926535 

 to be required; here #=3141, and w = 592 &c, which, until 

 the required logarithm be found, is to be considered a decimal 



