C J 47 3 
VII. The Application of a Method of Differences to the Species 
of Series whose Sums are obtained by Mr. Landen, by the Help 
of impossible Quantities. By Mr. Benjamin Gompertz. 
Communicated by the Rev. Nevil Maskelyne, D. D. Astro- 
nomer Royal , F. R. S. 
Read February 13, 1806?. 
Having some years back, when reading the learned Mr. 
Landen 's fifth Memoir, discovered the manner of applying 
a method of differences* to the species of series, whose sums 
are there obtained by the help of impossible quantities, and 
having since extended that application, I now venture to offer 
it to the consideration of others. 
The practice of this method, in most cases, appears to me 
extremely simple ; and on that account, I am almost induced 
to imagine, that they have already been considered by mathe- 
maticians ; indeed since the greatest part of this Paper was 
written, I met with Euler's mstitutiones Calculi integralis ; 
two simple series are in that work summed by multiplications 
similar to those employed in the investigation of the principal 
theorems contained in this Paper ; but whether that learned 
mathematician has farther pursued the method, in that or in 
any other work, I have not as yet been able to ascertain. 
I have purposely considered some of the series summed by 
Mr. Landen, to afford an opportunity of comparing both the 
results and methods ; and because the series may have parti- 
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