302 NEW SERIES for the 



From this feries, by a like mode of proceeding, we may de- 

 duce our third feries, and thence, again, our fourth, and fo on : 

 but this mode of inveftigation, although very fimple, is certain- 

 ly lefs elementary than that which we have followed. And it 

 mud be kept in mind, that one principal objed of this paper is 

 to employ only the firfl principles of geometry and analyfis in 

 treating of the fubjects announced in its title. 



33. By a mode of deduction differing but little from that 

 employed in the laft article, we may even derive our firfl feries 

 from a known formula, the invention of which is attributed to- 

 Euler. It is this, 



a zz fin a fee - a fee - a fee - a 4-, &c. * 

 2 4 8 



From this expreflion, by the theory of logarithms, we get 



log a = log fin a + log fee - a -f- log fee - a -f log fee \ a -f-, & c . 



we have now only to take the fluxions of all the terms, and re- 

 ject da, which is found in each, and the refult is 



- = + - tan - a-\- - tan - a -{- - tan - a -f , &c 



a tan# 2 2 4 4 8 8 



which is the feries in queflion. 



34. I now proceed to the inveftigation of formulae for the 

 quadrature of the hyperbola, and as the principles from which 



they 



* This formula, although very elegant as an analytical transformation, does 

 not feem to admit of being applied with advantage to the rectification of an arch, 

 on account of the great number of factors of the product which would be requi- 

 red to give a refult tolerably correft. 



