430
CONSTRUCTION OF
TRACT 21 ,
this purpose, he directs to make use of some of his own infi-nite series: thus, by them will be found 1 \57079&c for thequadranta! arc, the square of which is 2*4694&c ; divide thissquare by the square of the number expressing the ratio of90 degrees to the angle a, calling the quotient then 3 or
&c, will give
4 terms of this series 1 --- + ~ -~ + 1(m0the cosine of that angle a. Thus we may first find an angleof 5 degrees, and thence the table be computed to the seriesof every 5 degrees ; then these interpolated to degrees orhalf degrees by the same method, and these interpolatedagain ; and so on as far as necessary. But two-thirds of thetable being computed in this manner, the remaining third willbe found by addition or subtraction only, as is well known.
Various other improvements in logarithms and trigonome-try are owing to the same excellent personage ; such as, theseries for expressing the relation between circular arcs andtheir sines, cosines, versed-sines, tangents, &c ; namely, thearc being a, the sine s, the versed-sine v, cosine c, tangent t,radius 1, then is
a
~
S +
1
+
3 c5?o S
+
5 ? 7
TT 2V
+
rf ra'S 9 "f"
I
3
5
9
a
V* +
i* s
+
+
5 h) iTTS V
+
+ &c.
a
=
t -
+
¥ s
—
l fl
y L
+
V 9
— &c.
s
=
a —
i a *
+
rib« s
—
T al
+
T'5bVro a ' )
— &c.
c
=
1 -
+
—
i rfi
'rrt> a
+
ToTIb’^ 8
— &c.
V
=
1 /7 1 - I
% u Y 4 (t
+
7a#
—
I ^.8
+
r
- &c.
t
=
a -(- fa 3
+
T7 u
+
Tr'r
+
® 2.
+ &c.
Of Dr. IIalley's Method.
Many other improvements in the construction of loga-rithms are also derived Irom the same doctrine of fluxions, aswe shall show hereafter. In the mean time proceed we tothe ingenious method of the learned Dr. Edmund Halley ,secretary to the Royal Society , and the second astronomerroyal, having succeeded Mr. Ilamsteed in that honourableoffipe in the year 1719, at the Royal Observatory at Green wich , where he died the 14th January 1742, in the 86th year