414 PHILOSOPHICAL TRANSACTIONS. [aNNO 1794, 



mena dependent on this more important organ of our body, if any part possesses 

 a pre-eminence in nature, I also hope it may, in abler hands, admit of some 

 practical application, in alleviating the diseases to which its delicate organization 

 so particularly exposes it.* 



XJ^I. Dr. Halley's Quadrature of the Circle Improved: being a Transforviation 

 of his Series for that Purpose to others ivhich Converge by the Poivers of 80. 

 By the Rev. John Hellins, Vicar of Potter s Pury, Northamptonshire. 

 p. 217. 



Dr. Halley's method of computing the ratio of the diameter of the circle to its 

 circumference was considered by himself, and other learned mathematicians, as 

 the easiest the problem admits of. And though, 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 circum- 

 ference 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 30 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 circumference of the circle, may be transformed into others of swifter con- 

 vergency, and which, on account of the successive powers of -^ which occur in 

 them, admit of an easy summation. 



This transformation is obtained by means of different forms in which the fluents 

 of some fluxions may be expressed. 



Thus, the fluent of .s = — - -\ _ , &c. which 



series, being of the simplest form which the fluent seems to admit, was first dis- 

 covered, and probably is the most generally useful. But it has also been found, 

 that the fluent of the same fluxion may be expressed in series of other forms, 

 which, though less simple than that above written, yet have their particular ad- 

 vantages. Among those other forms of series which the fluent admits of, that 

 which suits the present purpose is 



TO (1 - X") m(m + «)•(!- ■!")* m(m + ;i)7(w + 2/0.(1 - x»)' ~" whlch, tO 



* Since the above pages have been written, I have found, on consulting some of the earliest 

 writers, that the eftects of tlie external muscles did not altogether escape tlieij' attention ; at the same 

 time they had no distinct idea of their action : 1 must therefore disclaim the originality of the tliought, 

 though I had never met with it before the circumstances already noticed, of the insufficiency of tlie 

 iris, had suggested it. If however, I have succeeded in pointing out the precise action of those 

 muscles, and its application to the general principles of vision, in which 1 believe 1 have never been 

 anticipated, it will be the height of my wishes. — Orig. 



