351 
of Edinburgh, Session 1876 - 77 . 
Id all such calculations we seek to arrive at the result by two 
independent processes. All the use, therefore, that can be made 
of Sharp’s tables, is to hold his work as constituting one of these 
processes ; a great use certainly, yet, at best, only half of what it 
might have been. 
Now, in the computation, to twenty-eight places, of the logar- 
ithms of the prime numbers, no error whatever was discovered 
among those given by Callet ; so here we have an instance of 
records, in themselves quite exact, and yet insufficient to obviate 
subsequent recomputation. 
The means of readily verifying the record are awanting ; these 
means must necessarily vary with the nature of the tabulated func- 
tions. 
In the volumes containing the computation of the logarithms to 
twenty-eight places of all primes below ten thousand, which was 
laid before the Society, the articulate steps of every calculation are 
recorded and indexed, so that if an error be suspected in any one 
logarithm, we have the means of instantly verifying the table, or 
of detecting the source of the error. Had such a record accom- 
panied Sharp’s admirable table, the need for subsequent re-calcula- 
tion would have been entirely obviated. 
The vast majority of tables have their arguments arranged with 
equal differences, consequently the functions progress gradually ; 
and, for the most part, these tables have been constructed by help 
of differences. It is then sufficient to record the differences along 
with the values of the functions. For the canon of sines I have 
found it convenient to place the first difference, with the sign + , 
and then the second difference, with the sign - , below the pre- 
ceding sine, as shown below : — 
Arc. 
Sine. 
l c 12' 
.01759 20113 41099 72108 93607 04490 
4 - 15 70551 06706 95090 77046 38118 
4 37940 65992 07711 07444 
1 .. .13 .01774 90664 47806 67199 70653 42608 
+ 15 70546 68766 29098 69335 30674 
4 41815 82853 79716 53726 
1 .. .14 .01790 61211 16572 96298 39988 73281 
VOL. IX. 
