Converging Series for the Circle . 47-g 
labour naturally attending it. His method is explained 
in Mr. maseres’s Appendix to his Diflertation on the 
the of the Negative Sign in Algebra; and I have given 
an anal-ylis oi it, or a conjecture concerning the manner 
in which it is probable Mr. machin dilcovered it, in my 
Treatife on Menfuration; which, I believe, are the only 
two books in which that method has been explained, as 
I never had feen it explained by any, till I met with 
Mr. maseres’s book abovementioned on the Ufe of the 
Negative Sign. For though the feries’ dilcovered by that 
method were publifhcd by Mr. jones, in his Synopjis 
Palmar iorum Mathefeos , which was printed in the year 
1706, he has given them merely by themfelves, with- 
out the leaii hint of the manner in which they were ob- 
tained. The refult fhews, that the proportion of the dia- 
meter to the circumference is equal to that of 1 to qua- 
druple the fum of the two feries’, 
1 
5-5 + 
5 2 39 + 
1 % 1 
7 -5 6 + ~9.5 S 
1 1 
7- 2 39° 9 ^ 39 " 
> See. 
’ See. 
The flower of which converges almoft thrice as fail as 
Dr. halley’s raifed from the tangent of 30°. The latter 
of thefetwo feries converges if ill a great deal quicker; but 
then the large incom polite number 2 3 9, by the reciprocals 
of the powers of which the feries converges, occalions 
fuch long, tedious divifions,as to counter-balance its quick- 
nefsof convergency; fo that the former feries is fummed, 
with rather more eafe than the latter, to the fame num- 
ber 
