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XVI. Dr. Halley's Quadrature of the Circle improved : being 
a Transformation of his Series for that Purpose to others 
which converge by the Powers of 80. By the Rev. John Hel- 
lins. Vicar of Potter s Pury, in Northamptonshire. Commu- 
nicated by Nevil Maskelyne, D.D. F.R.S. and Astrono- 
mer Royal. 
Read May 15, 1794,. 
2 * -^ R Halley’s method of computing the ratio of the dia- 
meter of the circle to its circumference was considered by him- 
self, and other learned mathematicians, as the easiest the pro- 
blem admits of. And although, in the course of a century, 
much easier methods have been discovered, still a celebrated 
mathematician of our own times has expressed an opinion, 
that no other aliquot part of the circumference of a circle can 
be so easily computed by means of its tangent as that which 
was chosen by Dr. Halley, viz. the arch of go degrees. This 
opinion, whether it be just or not, I shall not now inquire; 
my present design being to show, how the series by which 
Dr. Halley computed the ratio of the diameter to the cir- 
cumference of the circle, may be transformed into others of 
swifter convergency, and which, on account of the successive 
powers of ^ which occur in them, admit of an easy summa- 
tion. 
2. This transformation is obtained by means of different 
forms in which the fluents of some fluxions may be expressed. 
mdccxciv. F f 
