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 
muft be kept in mind, that one principal object of this paper is 
to employ only the firft 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 firft feries. 
from a known formula, the invention of which is attributed to 
Ever. It is this, ; 
a = fina fee ~ ee at, &c. * 
From this expreflion, by the theory of logarithms, we get 
log a= log fin a + log fec 5 a+ log fec= a + log fee 54-4, &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 
Am pe tan + a+ 7 tan pot geny a+, &e. 
a tana 
which is the feries in queftion. 
34. I now proceed to the inveftigation of formule for the 
quadrature of the hyperbola, and as the principles from which 
they. 
* Turs formula, although very elegant as an analytical transformation, does 
not feem to admit of being applied with advantage to the reétification. of an arch, 
on account of the great number of faétors of the produé which would be requi- 
red to give a refult tolerably correct. 
