41 i Mr. vince’s new Method of 
formed by reducing two terms into one, has its numerator inde- 
pendent ot the value of, n , it is manifeft, that the numerators of 
that feries will be all equal. Now, if a feries be afTumed, the 
numerators of whofe terms are unity, and in every other 
reflect the fame as the feries in this proportion, that is, if 
— ...1 = — be two fucceffive terms of a feries, it is ma- 
zv -j- nv iu + n + a . v 
nifeft, that if every two terms of this feries be reduced 
into one, the general term of the refulting feries will be 
— = . . where the numerator is a confhunt quantity 
w -f- nv X w + n -{- a . v 
— av ; conlequently the fum of the feries whofe general term is 
a :x — i s to the fum of the feries whofe general term 
tv -f- nv xw + n + a . v 
is ~ aa ns vx — wz to - v, or in a given ratio ; 
vj f nv x iv 4- n + a . v 
whenever, therefore, the fum of the latter feries can be found, 
the fum of the former can be found, and conlequently, after 
proper correction, the fum of the feries in this propofition can 
be found. 
Hence, therefore, in the two cafes given above in whatever 
arithmetic progreffion the numerators may proceed, the fum of 
the former can always be exprefTed by circular arcs, and the 
latter by the hyp. log. of 2 . 
Hence alio, as it appears from lem. i. part the firft, that 
the fum of the feries * — -y — L_ ~ — — + &c. can always 
i r-f l n r + i y + i 
l)e exp relied by circular arcs, and logarithms, it is manifeft, 
that if the numerators form any arithmetic progreffion, the 
fum of Inch ieries may be found by this proportion, and will 
always be exhibited by circular arcs and logarithms. 
Befides 
