292 NEW SERIES for the 



and here the number of terms compofing the feries in the pa- 

 renthefis is n. 



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 fee 2 — will become equal to rad 2 ; now 2« tan — will be- 



\ 



come a, (Art. 6. and 7.) therefore 2 3 «tan 3 — will become a* : 



f 2 CI 



hence, fubftituting -j for ■ §1_ in our equation, and tran- 



a 2 3 *tan>* 



2 n 



fpofing, we get at laft 



fee' 1 a 



f- * r tan I afec* - a+* tan ? a fee 2 - a 





tan 3 a 8 2 2 . 8 2 4 



-f- ±- tan \ a fee 2 \ a -f, &c. 



o 3 o o 



and this is the third feries which I propofed to inveftigate for 

 the rectification of an arch of a circle. 



25. The feries we have jufl 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- 

 trad 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, deferring of notice than either of them. 



26.. The 



