Mr. vince’s new Method of 
39 ° 
PART 
I. 
L E M. I. 
Let r be any whole number , and then the fluent of 
X 
I +X 
■can always be exhibited by circular arcs and logarithms ; but 
when x = i , the fluent of the fame fluxion will be exprejjed by the 
infinite feries i + — - — f &c. the Cum of this femes 
' r+ i 2 r + i y + i J J J 
therefore can always be found by circular arcs and logarithms. 
L E M. II. 
a -f b 
4* x r-p i . 4* i 
— &c. 
To find the fum of the infinite feries — L— _ — _J l 
J i . r+i r+i . 
a+ ib 
+ 
2r + i . 3r+ i 
Affume i — — — — l 
r+ I 2r+ -i 3> -{- i 
r-f I 2^+1 
4- &e. . — S ; therefore, 
4 - &e. • • . = S. 
(A). =U + = 
1*^ + 1 r -\- 1 . 2r + 1 2r -f I • 3/ 4* i 
In the firft feries, add together the 1 ft and 2d, the 2d and 3d, 
&c. &c. terms, and the refulting feries will evidently be equal 
to twice that feries minus the firft term ; therefore, 
(B), 
1 • 7 + 1 r + 1 . 2t -f 1 2' + 1 . 3 ' T 1 
Now A) ' = L, + .^- 4 + ~ 
V r J r 1 . r 4 -i / + I . 2/ 
— &C. ... = 2S - I . 
«+7 
or 
/ 4 - 1 • 2/ + 1 2r4- 1 . 3' 4- 1 
4-1 2r 4 - I . 3' 4 - 1 
4 -&C. . . . ^ 
4 ~ ... — - , 
+ - ~ * =L=z 4" s===5=r-^==s=== 
r 1 . r 4- 1 r 4- 1 .2/4-1 
- &C .... 
N 
ow 
