Mr. hutton m 
480 
ber of places of figures. Mr. jones, in his Synopjis , 
mentions other femes’ befides this, which he had received 
from Mr. machin for the fame purpofe, and drawn from 
the fame principle. But we may conclude this to be the 
beft of them all, as he did not publifh any other befules 
it. 
M. euler too, in his Introdudiio in Analyfm Infini - 
torum , by a contrivance fomething like Mr. machin’s, 
difcovers, that j and j are the tangents of two arcs, the 
fum of which is juft 45 0 ; and that, therefore, the dia- 
meter is to the circumference as 1 to quadruple the fum 
of the two feries’, 
I 
+ 
1 
I 
+ 4 ’ See. 
3-4 
5 - 4 " 
7 4 ' 
9*4 
1 
1 
1 
1 
— + 
— 
+ — V See. 
3-9 
5 - 9 ' 
7 - 9 ' 
9-9 
Both which feries’ converge much fafter than Dr. Hal- 
ley’s, and are yet at the fame time made to converge by 
the powers of numbers producing only fhort divifions; 
that is, divifions performed in one line, or without writing- 
down any thing belides the quotients. 
I come now to explain my own method, which, in- 
deed, bears fome little refemblance to the methods of 
machin and euler; but then it is more general, and 
difcovers, as particular cafes of it, both the feries’ of thofe 
gentlemen and many others, fome.of which are fitter for 
this purpofe than theirs are. 
This method then confifts in finding out fuch fmall 
arcs, as have for tangents fome fmall and fimple vulgar 
fractions (the radius being denoted by 1 ),and fuch allb that 
fome 
