i<)$ Mr, Hellins’s new Method of computing 
13. The values of the several parts, into which the proposed 
series has been resolved, being now so far obtained that we 
have only to multiply each by its proper factor, viz. the nu- 
merical value of a: 8 , a; 16 , x 3Z , &c. and add the products together, 
to get the sum of it ; this, therefore, is now to be performed. 
And, in this part of the calculation, several multiplications may 
be saved, and no larger factor than x 8 be used, by attending to 
the method described by Sir Isaac Newton, in his Tract De 
Analysi per JEquationes infinitas ; p. 10. of Mr. Jones’s edition 
of Sir Isaac s Tracts ; or p. 270. Vol. I. of Bishop Horsley’s 
edition of all his works. The manner in which this is to be 
done will appear, by collecting the several parts from the pre- 
ceding Articles, and exhibiting them in one view, thus : 
A + Bx 8 -f- Cx 16 + « + r+TTxx 3Z + J+Fx 
x 43 = the sum of the proposed series. Now, 
1st. Calling ^ + F, and multiplying by x s , we have %' x* 
= 0-26584,70242 x 0-43046,721 = 0-11443,84267,9. 
2dly. Putting y -|- z' x 8 = z", and multiplying by x\ we get 
z"x S =z 0-15172,24360,1 X 0-43046,721 = 0-06531,15337,2. 
3dly. Putting £ -j- E -j- z"x 8 ~ z"\ and multiplying by x s , we 
get z'"x 8 = 0-20827,0 1655,4 x °'43 046,721 = 0-08965,34770,9. 
4thly. Putting a -j- D -{- z"'x 8 =z iv , and multiplying by a: 8 , we 
get z iy x 8 = 0-28046, 43895, 9x0.43046, 72 i = o- 12073, 07232, 90. 
5thly. Putting C -f z iv x 8 = z v , and multiplying by x 8 , we get 
z y x 8 = 0-38085,19524,75 x 0-43046,721 == 0-16394,42774,05. 
6thly. Putting B -f- £ v x 8 = z vi , and multiplying by x 8 , we get 
s Ti x 8 = 0-60806,50444,24 x 0-43046,721 = 0-26175,20631,73. 
Lastly, to this product add A - = 2-04083,30298,21, 
and we have the value of the series proposed = 2*30258,50929,94, 
which is true to twelve places of figures. 
