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I 
XVIII. On some new Methods of investigating the Sums of 
several Classes of infinite Series. By Charles Babbage, Esq. 
A. M. F. R. S. 
Read April 1, 1819. 
The processes which it is the object of this paper to explain, 
were discovered several years since ; but certain difficulties 
connected with the subject, which I was at that time unable 
to explain, and which were equally inexplicable to several of 
my friends, to whom I had communicated these methods, 
induced me to defer publishing them, until I could offer some 
satisfactory solution. 
These observations refer more particularly to the second 
method which I have detailed in this paper, and which may 
not inappropriately be called the method of expanding hori- 
zontally and summing vertically. Some traces of this method 
may, perhaps, be found in former writers, and particularly 
in a paper by Professor Vince, “ On the Summation of Se- 
ries," printed in the Philosophical Transactions for 1791 ; but 
there exists this peculiarity in that which I have employed, that 
after a certain number of the vertical columns are summed, 
all the remainder either vanish, or else have some common 
factor. This method, which I employed about the year 181s, 
gave the values of a variety of series whose sums had not 
hitherto been known, most of which were apparently correct, 
but some of the consequences which followed were evidently 
erroneous. About this time, Mr. Herschel, to whom I had 
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