t >4« ] 
IX, On Infinite Series . By Edward Waring, M. D. F. R. S. Luca- 
fian Profefior of Mathematics in the Univerfity of Cambridge. 
Read March 24, 1 79 x . 
1 , 1% /fERCATOR firffc publifhed the contiuation of the 
XV JL common method of divifion to an infinite feries of 
terms proceding according to the dimenfions of a variable quan- 
tity; Newton did the fame for the common method of ex- 
traction of roots. Dr. Barrow before applied the fame princi- 
ples in fome eafy examples to find the afymptotes of curves. 
2. The methods of divifion and extraction of roots were 
long before taught; but the continuation of them in in- 
finitum would have been ufelefs, as the areae of curves, whofe 
n 
ordinates are ax’" (where * denotes the abfcifs, and a, n, and m 
invariable quantities) had not been difcovered many years before 
the time of Mercator’s Publication, and conlequently it 
would have been of little ufe to transform an ordinate or fluxion,, 
whofe area or fluent is unknown, into another form, of which 
the area, &c. is equally unknown. 
3. Sir Isaac Newton extended the rule for railing a bino- 
rnial (to any affirmative power) to negative powers, the ex- 
traction of roots and fra&ional indexes, by applying the law 
of the feries for affirmative powers to them, and continuing it 
in infinitum. M. de Moxvre extracted the root, &c. of a 
multinomial by a feries of a fimilar nature ; but thefe methods 
