194 JVfr. maseres’s Method of 
and r from s, and s from t, and fo on; and he feems alfo 
to fuppofe the differences q-p , r-q, s—r , t-s, See. to 
form an increafing progreffion, and every fubfequent 
order of differences to form likewife an increafing pro- 
greffion, and accordingly fubtraefs q-p from r-q, and 
r—q from s-r, and s—r from t-s, and fo on; whereas in 
the foregoing feries a— b x+cx x-dx^’+ex*- fx'+gx**— bx 1 
+ See. the numbers a, b, c, d, e, f g, h, See. are fuppofed 
to form a decreafing progreffion of terms, as they are 
moft commonly found to do in the feriefes that occur in 
the folution of mathematical or philofophical problems* 
Examples of the. ufefulnefs of the foregoing differential fe- 
ries in finding the values of infinite feriefes whofe terms 
decreafe very ffozvly. 
Computations of the lengths of circular arcs by means of 
infinite feriefes derived from their tangents-. 
Art. 7. It is well known, that if r be put for the 
radius of a circle, and t for the tangent of any arch in it 
that is not greater than 45 °, the magnitude of the arch 
whole tangent is t will be expreffed by the infinite feries 
r r 
’ — “ 4 : 
7 r 9 r 
t 1 ' 
13 - 4 ; 
Hr 1 3 r 
15 r 
— + Sec. This fe- 
ries converges with, great fwiftnefs when the tangent is 
1 much 
