1842.] Compendious Logarithmic Tables. 43 



Each table may be made to do all that can be done by the other, 

 but not with equal convenience. The first table giving at sight the 

 logarithm to a number of 3 figures, and the second giving at sight the 

 number to a logarithm of 3 figures, the proportional parts for the 4th, 

 and subsequent figures are additive : but if a number to a given logari- 

 thm be sought by means of the first table, or if the logarithm to a given 

 number be sought by means of the second, one subtraction is required 

 for each figure in the proportional part : and subtraction, though a 

 simple operation, is by no means so short or so easy as addition, and 

 hence the advantage of using both tables instead of either exclusively. 



Great care has been taken to make these tables correct in every case 

 to the nearest unit in the last figure. The first table was taken from 

 Lalande, whose tables are known to be correct, and has been rigidly 

 compared in every figure. The second table was made by means of the 

 common table in Callet, and after having been written out, it was ex- 

 amined by reading out the number to every logarithm, using Babbage's 

 tables. When it was uncertain in this way whether the number was 

 more or less than 5 in the 6th place, it has been determined by a 

 calculation carried to ten figures. 



For finding the common differences and proportional parts the follow- 

 ing method was used. Having determined to give these for the middle 

 between columns 4 and 5 of each table, those for the first table were 

 thus found. M denoting as usual the common modulus, and N the 

 number in the first column of the Table, the common difference has 

 been taken by the formula _^ ; and the proportional parts by its 

 decimal products, taking care to make each true to the nearest unit in 

 the 5th place of decimals. 



For the second table the principal was the same, but the process vast- 

 ly simpler. By the common differential formula d logN=^^ ; (the 

 same as that used above) ; from this we have d N=N. d . logN : in which M 



M 



and d. log N. being constant, the only variable is N, and by the nature 

 of the Table log N. in each successive line increases by unit in its 2nd 

 figure. Hence the logarithm of d N being calculated for any one line, 

 it is found for each succeeding line by adding one to its 2nd digit, 

 and the common difference, or the number corresponding to this cal- 

 culated logarithm being found more or less nearly in some column in 



