348 Proceedings of the Royal Society 
fifth step with the previously computed test values. In this way 
immunity from error was obtained, excepting in the last places, 
where small errors are inevitable. 
In performing the multiplication sinwa.2 vera stopping at the 
thirty-third place, the last figure of each partial product may err in 
excess or in defect by J; now it is possible, though not likely, that 
all of these errors maybe on one side, and therefore there is a 'possible 
error in the last place of the total product, of half as many units 
as there are lines in the multiplication. By using the table of 
multiples up to one thousand, we reduce the number of lines to one 
third, and therefore the possible amount of residual error in the 
same ratio ; so that the auxiliary table both saves labour and 
augments the exactitude of the result. 
These final-place errors were corrected at each fifth step by 
altering the last figures of the second differences, and thus the 
accumulation of those errors was prevented. In the whole calcu- 
lation of the sines of the quarter degrees, it was not found necessary 
to alter any one second difference to so much as the limit of possible 
error, and therefore we may hold that the manuscript table of the 
sines of these arcs is absolutely correct to within the prescribed 
degree of precision, namely, unit in the thirtieth decimal place. 
The next quinquisection, conducted in the same way, gave the 
sine and cosine of 5'; the functions of whose multiples were ob- 
tained as before, and compared, at each fifth step, with the previous 
work. The table of sines to every fifth minute is already well 
advanced. 
The third quinquisection gave the sine and cosine of the single 
minute of the decimal division. A table of the multiples of 2 ver 1' 
has been constructed up to 1000, and has been used in forming a 
good beginning of the canon of sines to each single minute. 
For the purpose of preventing all error in the record of these 
from the 
duplicate scroll calculations, and the successive first differences 
and sines were thence recomputed on the record sheet. Since any 
error in copying, or even in the original computation, was neces- 
sarily continued and extended into the after part of the record, its 
detection was rendered certain, so that the recorded results may be 
implicitly relied on. To make the record more secure, each page 
calculations, the second differences only were copied 
