540 
Proceedings of the Royal Society 
of tlie previous comj3utations (themselves similarly checked) for the 
above eight primes 11, 599, 742 ; 7, 19, 719; 7, 331, and also for 
the prime divisors of 3213, 6276, 3693, namely, 17, 523, 1231. In 
this way the whole work is bound together by an intimate inter- 
lacing of tests. The search for the appropriate formulae was greatly 
facilitated by Burckhardt’s admirable “ Table des Diviseurs,” but 
the recent extensions of that table by Dase and by Glaisher would 
have been most welcome. 
By the combination of these primes and by interpolation to 
second differences, the logarithms, to 15 places, of all numbers from 
100 000 to 370 000 have been computed. The actual calculations 
are contained in the twenty-seven volumes herewith presented, and 
the transfers in nine. 
These logarithms are necessarily liable to residual errors, whose 
amount, however, cannot exceed three units in the fifteenth place. 
Among a large number of verifications, made for other purposes, no 
error exceeding two units has been found. 
They are accompanied by the first and second differences — 
differences of the third order would only appear in the sixteenth 
place even at the beginning of the canon. 
By help of these differences we can interpolate the logarithm of a 
number of more than six effective figures ; the w'ork consisting of 
three multiplications. For the converse operation, that of computing 
the number corresponding to a logarithm not found in the table, we 
need to resolve an equation of the second degree. jSTow the first 
differences have teyi, the second difference have five effective figures, 
and therefore, when the utmost precision is required, either of these 
interpolations is necessarily laborious. 
For the purposes of shortening the work, and of avoiding the 
solution of the quadratic equation, the expedient used by ilepair in 
the computation of his original canon miri ficus ^ is had recourse 
in a form modified to suit the present circumstances. 
To get the logarithm of a number not in the table, it is enough to 
discover that of the ratio which it bears to the tabulated number 
immediately below it. ISiow this ratio itself is easily found by 
division, and, in our present case, is expressed by unit followed at an 
interval of at least five blanks, by other figures ; its greatest possible 
value is 1,00001. In the volume marked Auxiliary taible the 
