t 3* ] 
IV. ^ Supplement to the Third Part of the Paper on the Sum** 
ftiation of infinite Series , in the Philolophical Tran factions 
for the Tear 1782. By the Rev. S. Vince, M. A.; 
■.communicated by Nevil Maikelyne, D. D . F* R. &, and 
Afironomer Roy ah 
[Read November 25, 1784. 
T HE reafonlng in the third part of my paper on the Sum- 
mation of infinite Series having been mifunderftood, I 
have thought it proper to offer to the Royal Society the follow- 
ing explanation. When I propofed, for example, to fum the 
feries \ - % + f ~ See. fine fine> I wanted to find fome quantity 
which, by its expanfion, would produce that feries, and that 
quantity I called its fum ; not {as I conceived muff have been 
evident to every one) in the common acceptation of that word, 
that the more terms we take, the more nearly we fhould 
approach to that quantity, and at laft arrive nearer to it than 
by any affign able difference, for there manifeffly can be no fuch 
quantity ; but as being a quantity from which the feries muff 
have been deduced by expanfion, which quantity I found to be 
~-§+H. L. 2. If therefore in the folution of any problem, 
the conclufion, whofe value I want, is exprefled by the above 
-feries, and which arofe from the neceffity of expanding fome 
quantity in the preceding part of the operation, furely no one 
nan deny but that I may fubftitute for it - \ +H. L. 2. For 
whatever quantity it was, which by its expanfion produced at 
firft 
