MR SANG'S NEW TABLE OF LOGARITHMS TO 200 000. 525 



return, is only equalled by the scrupulous carefulness of the execution. For 

 many years, and in various branches of research, I have habitually used the 

 Table des Diviseurs, and only in one instauce have found a fault, — that fault 

 having been caused by a displacement of the types in the process of printing. 



So long as the primes were under 1000, their logarithms were compared 

 with those given by Callet in his Tables Portatives to 60 places ; and the 

 coincidence was held as a sufficient check. In no one instance was an error 

 found in Callet. Afterwards, however, each logarithm was computed twice, 

 generally once from a multiple ending in 01, and once from another ending in 

 99. At times the divisors were enormously large ; thus, for the prime 653 the 

 divisor 249 000 000 was used. In such cases the advantage of the second result 

 was lost, since it would have been a matter of great labour to have found the 

 divisors of 248 999 999. It would be still more laborious in the case of 

 7 580 000 001 attending the computation of the logarithm of 1277. 



The computation of the logarithms of all the primes in succession to above 

 2000 was thus carried on, and those of many other primes incidentally found ; 

 these and the logarithms of their multiples up to 10 000 to twenty-eight places, 

 having been written in their places, a sufficient groundwork was obtained for a 

 table to fifteen places, beginning at 100 000, since the differences of the third 

 order there count only in the sixteenth place. 



Paper having been ruled, and the lines numbered from 100 000 to 150 000, 

 the logarithms, but only to fifteen places, of the products of the numbers already 

 found, were inscribed in their proper places ; the first and second differences of 

 these were taken wherever they happened to be sufficiently grouped together, 

 and the gaps were then filled up by means of second differences. 



These interpolations were easily accomplished, because, since the third 

 differences are less than units in the fifteenth place, the progress of the second 

 differences could be estimated. It was enough, then, to make trials with the 

 last three figures of each order of differences ; and after a little practice these 

 trials were quickly made. The final figures of the second differences having 

 thus been found, the others were obtained, and the first differences with the 

 logarithms themselves were found by subtraction and addition. 



In this way a table of fifteen-place logarithms for all numbers from 100 000 

 to 150 000 was formed. 



To all who are familiar with extensive tabular work, it must be apparent 

 that these results, though trustworthy on the whole, could not be depended 

 upon as accurate in each item. A wrong figure may have been written, and 

 yet, on account of the consecutiveness of the differences, may not have been 

 detected by the subtraction. It became necessary, therefore, to revise the 

 whole in such a way as to preclude this source of error. 



For this purpose a new set of ruled pages were consecutively numbered, and 



VOL. XXVI. PART. III. 6 X 



