187 
1907-8.] Dr Edward Sang’s Tables. 
reaching so far as to unit in the fourteenth place — a degree of accuracy far, 
very far, beyond what can ever be required in any practical matter. 
“ In the compilation of the trigonometrical canon the same precautions 
were taken for securing the accuracy of the results. In the usual way, by 
means of the extraction of the square root, the quadrant was divided into 
ten equal parts, and the sines of these computed to thirty-three, for thirty 
places. These again were bisected thrice, thus giving the sine of each 
eightieth part of the quadrant ; all the steps of the process being recorded. 
“ The quinquesection of these parts was effected by help of the method 
of the solution of equations of all orders, published by me in 1829 ; and 
the computation of the multiples of those parts was effected by the use of 
the usual formula for second differences. A table of the multiples of 2 ver. 
00 c 25' was made to facilitate the work, and the sines, first differences, and 
second differences were recorded in such a way as to enable one instantly 
to examine the accuracy. The same method of quinquesection was again 
repeated, and the computation of the canon to each fifth minute was effected 
by help of a table of one thousand multiples of 2 ver. 00 c 05', the record 
being given to thirty-three places, the verification being examined at every 
fifth place. In this work there is no likelihood of a single error having 
escaped notice. 
“ For the third time this method of quinquesection was applied in order 
to obtain the sines of arcs to a single minute. A table of one thousand 
multiples of 2 ver. 00 c 01' was computed to thirty-three places, but in the 
actual canon it was judged proper to curtail these, and the calculations 
were restricted to eighteen decimals on the scroll paper. In the actual 
canon as transcribed, only fifteen places are given. In all cases the 
function, its first difference, and its second difference are given in position 
ready for instantaneous examination ; and the whole is expected to be free 
of error excepting in the rare cases where the rejected figures are 500 — 
these cases being duly noted. 
“For the computation of the canon of logarithmic sines the obvious 
process is to compute each one of its terms from the actual sine, by help of 
the table of logarithms ; but this process does not possess the great 
advantage of self-verification, and attempts have been made to obtain a 
better one. Formulse indeed have been given for the computation of the 
logarithmic sine without the intervention of the sine itself, but when we 
come to apply these formulse to actual business we find that they imply a 
much greater amount of labour than the natural process does ; and after 
all, they are only applicable to the separate individual cases. 
“ Nepair, as is well known, arranged his computations of the logarithms 
