ig6 Dr, Robertson’s expeditious methods of calculating 
expressed, I adopted in this example the same data with Mr. 
Ivory : see Transactions of the Royal Society of Edinburgh, 
Vol. V. page 23 6. 
The preceding method bears a nearer resemblance to that 
given by Keill, in his Astronomy, than to any other of which 
I know. Adapting his manner of proceeding to the figure here 
used, he puts y = EM, e == sin. AM,/ cos. AM, and g = CS. 
The series expressing the sine of AE, is therefore equal to 
e - — fy — — — J- — + See. But the radius, which is 1, is 
to the sine of AE as CS or g is to ST or EM, that is to y. 
Consequently y —ge «— gfy — -j- s -^- -j- See. and there- 
fore ge — y -j - gfy 4 - — — Sec. 
o J 1 a// 8 2 2.3 2.3.4 
By reversing this he obtains a series, which, omitting the 
numbers in the coefficients, converges as the powers of t + —j 
°r 1 "+ S cs This degree of convergency to the value 
ofjy in the foregoing examples is as follows. 
In the first example as the powers of .2147372, 
In the second example as the powers of .2553020, 
In the third example as the powers of .1086700. 
In the third method which has been here investigated, the 
series converges to the value of SMH as the powers of 
or CM S-C S x s.n CSM This degree of con vergency in 
the foregoing examples is as follows. 
In the first example as the powers of .0018209, 
In the second example as the powers of .0250438, 
In the third example as the powers of .0055358. 
The third method, by which the three foregoing examples 
