I 210 ] 
^For if, by the preceding Corollaries , a regular 
^re&ilinear Polygon be infcribed within, and another 
without, the Arch A y each having half fo many Sides 
as is expreffed by the Number n ; then will the former 
of thefe Quantities be the Length of the Bow of the 
circumfcribed Polygon, (or the Sum of ail its Sides) 
which is always bigger, and the latter will be the 
Length of the Bow of the infcribed Polygon, which 
is always lefs, than the Arch of the Circle; how^grcat 
foever the Number n be taken. 
Corollary i. Hence the Series’s for the Re&ification 
of the Arch of a Circle may be derived. 
For by converting the Binomials into the Form of 
a Series, that the fi&itious Quantities, £, t, v may be 
deftroyed ; it will appear, that no Number n can be 
taken fo large as to make the infcribed Polygon fo 
big, or the circumfcribed fo little as the Series. 
r r 
3 *3 + 
— LZl-f. &c. in one Cafe, or its Equal 
jzSjz 1 
t ^ Z? 
. . — t 5+ &c. in the other Cafe. 
3 r 1 5 r* yt 
Wherefore fmee the Quantity denoted by the Sum 
of the Terms in either of thefe Series's is always 
bigger than any infcribed Polygon, and always lefs 
than any circumfcribed, it muft therefore be equal to 
the Arch of the Circle. 
Corollar y 2 . If, in the firft of the above Series’s, 
the Hoot Vrr—yy, be extra&ed and fubftituted for z, 
there will arife the other Series of Sir Ifaac New- 
ton, for giving the Arch from the Sine; namely. 
, 7 s . 
y + — ~r 
17 
6r* 
I 
40 r 
7 3 , 
1L 
112 7 
3-3 
)r 
777^+ or otherwife, 
[ ■ VMA 
1.2.3 r 2 1.2.34.5. r^‘1.2.^.^.6.7. r(> 
x 76+i^‘ : - 
SCHO- 
