45° Mr. Hellins on the Rectification 
consideration in the year 17 70, when a problem requiring the 
rectification of a large portion of an equilateral hyperbola was 
proposed in a periodical work; which problem I then solved by 
means of descending series; but, for want of an easy method 
of correcting the fluent so found, I laid it aside for the exercise 
of maturer judgment. Afterwards, the subject was resumed at 
different times, as leisure permitted, and put into nearly the 
same form in which it now appears, in the year 1795; since 
which time, till now, the duties of my station, and unexpected 
occurrences, have left me no opportunity to revise my papers. 
As the investigation of the following theorems is very obvious 
and easy, I thought it probable that they might have been dis- 
covered by some other person before me ; yet, upon perusal of 
Dr. Hutton's mathematical and philosophical Dictionary , lately 
published, I find but one of them. And since, in the compilation 
of that work, as the learned and industrious author professes, 
“ Not only most of the encyclopedias already extant , and the 
“ various transactions of the learned societies throughout Europe , 
“ have been carefully consulted , but also all the original works , of 
“ any reputation , which have hitherto appeared upon these sub - 
“ jectsf * I therefore conclude that all the rest are new. 
The subject of this Paper is naturally divided into three sec- 
tions : the first containing the investigations of the several 
series; the second, the methods of computing the constant 
quantities by which the ascending series differ from the de- 
scending ones ; and the third, examples of their use, by way of 
illustration. But, for more convenient reference, I have further 
divided it into articles, or minor sections. 
* See Dr. Hutton’s Address to the public, on the publication of the first part of 
the Dictionary above mentioned, in 1795 ; and preface to the Dictionary. 
