47 8 Mr. Hutton on 
circumference is = 6 Vy x : i- — + ^~— 3 +— ’ See. which 
feries is, to be fure, very fimple ; but its rate of converging 
is not very great, on which account a great many terms 
muft he ufed to compute the circumference to many 
places of figures. By this very feries, however, the in- 
duftrious Mr. sharp computed the circumference to 72 
places of figures; Mr. maciiin extended it to 100 ; and 
M. de lagney, Hill by the fame feries, continued it to 
128 places of figures. But although this feries, from 
the 1 2 th part of the circumference, does not converge 
very quickly, it is, perhaps, the befi: aliquot part of the 
circumference which can be ufed for this purpofe ; for 
when fmaller arcs, which are exadf aliquot parts, are 
ufed, their tangents, although fmaller, are fo much more 
complex, as to render them, on the w r hole, more operofe 
in the application;, this will eafily appear, by infpedting 
lome inftances, that have been given by Mr. gardiner, 
in his editions of sherwin’s Tables. One of thefe me- 
thods is from the arc of 18 0 , the tangent of which is 
>/ 1 - 2^ j another is from the arc of 2 2°{, the tangent 
of which is V2— 1 ; and a third is from the arc of 15 0 , 
the tangent of which is 2-V3. All of which are evi- 
dently too complex to afford an eafy application to the 
general feries. 
In order to a frill farther improvement of the method 
by the above general feries, Mr. machin, by a very lin- 
gular and excellent contrivance, has greatly reduced the 
labour 
