346 
Proceedings of the Royal Society 
effective figures is less than the total number of figures in the cal- 
culations; we have to extract the square root of its half, which 
square root, can only be true to as many effective figures, and thus 
cannot be extended to the full number. If, then, we desire to 
obtain accuracy to a specified number of decimal places, we must 
extend our first calculations far beyond. Whereas the sine of an 
odd sub-multiple is deduced directly from the sine of the whole 
arc, and the computation is such as to retain the full number of 
effective figures. Hence we must proceed by first making all the 
requisite bisections, and thereafter the quinquisections. 
There is, however, an obvious exception. The sines of 20 c , 40 c , 
60 c and 80 c are obtained by the solution of quadratic equations, so 
that this quinquisection naturally comes first; the bisections after- 
wards, and then the remaining quinquisections. 
Having prescribed unit in the thirtieth decimal place as the 
limit of inexactitude, and taken three figures more to obviate the 
accumulation of the minute errors inevitable in all approximate 
work, I computed the sines of these four arcs to thirty-three 
places. 
The bisection of the quadrant gave the sine of 50 c ; that of 
60 c gave for 30 c and for its complement 70 c ; that of 20 c gave for 
10 c and 90 c , thus completing the table for each tenth part of the 
quadrant. 
In the same way the bisections of 10 c , 30°, 50 c , 70°, and 90 c gave 
the sines of the intermediate fifth degrees. 
The bisections were continued in this way until the quadrant 
was divided into 80 equal parts. 
In performing this work, each of the roots 
sina= cos 2a} , cosa= \Z{1 + J cos 2a} 
was extracted twice, and sin a was also found from the division 
sina = 
sin 2a 
2 cos a y 
by which means the values of sin a and cos a were securely got, 
and the previous computations of sin 2a, cos 2 a again verified. 
There were, as a matter of course, small errors in the last places, 
but the results, to the thirtieth place, were made sure. 
