192 Proceedings of the Royal Society of Edinburgh. [Sess. 
By the extraction of the square root and repeated bisections, the 
quadrant was divided into 80 parts, and the sines of the multiples of 
•01 c 25' were computed. 
Thereafter the sines and cosines of 00 c 25' and of 01 c 25 ' were got by the 
direct resolution of the appropriate equations of the fifth degree, and were 
compared with those which had been got in computing the recurring 
functions of submultiples of n r, the steps of which are copied into this 
record. 
By help of 100 multiples of 2 ver. 25', and of 1000 multiples of 2 ver. 5', 
a table of sines of arcs differing by 25', and thereafter one of arcs differing 
by 5', were computed on the actual pages. 
Although these have the appearance of being interpolations, they are 
truly independent computations, the use of the preceding work preventing 
mistakes, as well as the accumulation of the minute errors due to the 
rejection of figures beyond the 33rd place. 
Nos. 41, 42. Canon of Sines, Parts II. 
These volumes contain the sines to 15 places of arcs differing by 1' 
(centesimal division) with their first and second differences, the computation 
having been facilitated by a table of 1000 multiples of 2 ver. T. 
The table has been bound in two parts, for the convenience of referring 
to the sine and to the cosine of an arc. (Also in transfer duplicate.) 
No. 43. Log Sines and Tangents. 
The log sines from 100° 00" down to 50° 00' are here given to 15 places, 
with their first, second, and third differences. They were computed 
directly from the Canon of Sines by the 15-place table of logarithms 
from 100 000 to 370 000, and by use of the auxiliary table. 
The log sines from 50 c 00' to 0 C 00' were derived from the preceding, 
according to the formula — 
sin a = \ sin 2 a. sec a , 
using the first differences only. 
The log tangents from 50 c 00' to 0 C 00' were obtained from the preceding 
log sines, using only the first differences. 
Upwards of two million eight hundred thousand figures were written 
for the completion of this volume. (Also in transfer duplicate.) 
No. 44. Sines in Degrees. 
This volume contains the values of the sines measured, not in parts of 
the radius, but in parts of the quadrant, and given to the ten-thousandth 
