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XVII. Of the Rectification of the Conic Sections. By the Rev. 
John Hellins, B. D. F. R. S. and Vicar of Potter* s-Pury, in 
Northamptonshire. 
Read July 8, 1802. 
PART I. 
Of the Rectification of the Hyperbola : containing several new 
Series for that Purpose ; together with the Methods of computmg 
the constant Quantities by zvhich the ascending Senes differ from 
the descending ones . 
INTRODUCTION. 
JT he conic sections are a part of geometry so requisite in men- 
suration, in optics, astronomy, and other branches of natural 
philosophy, that the properties of these curves have been much 
studied in the course of the last hundred and fifty years ; and 
, there is hardly a writer on fluxions, of any note, who has not 
treated of their rectification. It may therefore seem, that little 
is now left to the industry of the present and future generations, 
in this part of the mathematics, but the proper application of 
theorems already investigated. Yet, while we admire the skill, 
and praise the industry, of those who have discovered new 
truths, or thrown new light on old ones, within that period, we 
shall do well to recollect, that it is now no more tnan one hun- 
dred and thirty-seven years, since the two great discoveries of 
fluxions and infinite series were made by Sir Isaac Newton ; 
and that the observation of the late Mr. Emerson, respecting 
