1888-89.] Dr E. Sang on Fundamental Tables. 
249 
Notice of Fundamental Tables in Trigonometry and 
Astronomy, arranged according to the Decimal 
Division of the Quadrant. By E. Sang, LL.D. 
(Read June 3, 1889.) 
Canon of Sines. 
In January of 1878, there was laid on the Society’s table the 
Canon of Sines to each fifth minute of the decimal division of the 
quadrant, computed to thirty-three for thirty places ; along with a 
detailed record of every step in the process. During the years 
1880-81, this work was continued for each single minute, but only 
to eighteen for fifteen places, and the record thereof to fifteen places 
is now submitted. When the rejected figures were from 497 to 503 
a mark of interrogation is recorded, and it is believed that not a 
single error exists in the work. The arrangement of the sines with 
their first and second differences in position enables us instantly to 
detect an error. 
That fifteen places suffice for all possible practical purposes, is 
made clear by this consideration, that the Earth’s distance from the 
Sun, measured in inches, is represented by the number 6, twelve 
removes from the unit’s place, that is 6 000 000 000 000 ; and that, 
if we take this as the radius of our circle, the figure 1 in the fifteenth 
decimal place will represent ’006 or the 170th part of an inch. 
Thus the present canon gives, on this circle, the co-ordinates of the 
ten thousand points in the quadrant, each true to within the three- 
hundredth part of an inch. 
The process followed in this work differs, in one very important 
respect, from that used by previous computers. The sine of the 
smallest tabular arc has hitherto been found indirectly by help of 
repeated bisections ; in the present work the quinquisection of the arcs 
has been accomplished directly by the solution of the appropriate 
equations of the fifth degree, according to the method described in 
my treatise “ On the Solution of Equations of all Orders.” The 
ease and rapidity of this method are well shown by the recorded 
details of the work for the various equations, to thirty decimal places. 
A table of one thousand multiples of 2 ver. V having been pre- 
