292: NEW SERIES for the 
and here the number of terms Campane the feries in; the pa- 
renthefis is 7. ; 
LET us now conceive the feries to go on ad infinitum, fo that 
n may be confidered as indefinitely great, then it is manifeft, 
that fec* < will become equal to, rad’ 5, now 2 tan = will be- 
Tom aes . 
come 4, (Art. 6. and 7.) therefore 23"tan3 will become a? 3. 
t fec’ & 
hence, fubftituting at for 2". in our equation, and tran- 
2 3™han? — a . 
Qn 
{pofing, we get at laft 
fec”a L 
\~ - \ ih = Ly oF = 
[eet g BS fec’ = sats z tans a fc" am - 
I 
as 
bak a tang ta fect X aby 8 ails 
and this is the third feries which I propofed to agi for 
the rectification ofan arch of a circle. 
25. THE feries we have juft now found, is evidently of a 
very fimple form; it alfo converges pretty faft, each term be- 
ing lefs. than the 16th of that which precedes it. As, however, 
to apply it to actual calculation, it will be neceflary to ex- 
tra@t the cube-root of a number, which is an operation of con- 
fiderable labour when the root is to be found to feveral figures, . 
perhaps, confidered as a practical rule, this third formula is in- 
ferior to the two former. But if, on’ the’ other hand, we re- 
gard it merely as an elegant analytical theorem, it does not 
feem lefs. deferying of notice tham either ef them., 
26. THE 
