oe 
306 NEW SERIES for the 
Now, the rectilineal fpace Q_is evidently lefs than the hy- 
perbolic fector s ; but x may be conceived fo great that the dif- 
ference between Q and s fhall be lefs than any aflignable {pace, 
as it is eafy to demonftrate upon principles ftritly geometrical ; 
therefore, if we fuppofe » indefinitely great, then Q becomes 
s; and as, upon this hypothefis, the feries goes on ad infinitum, 
we have 
alae I I I I I I a 
—(tan 55+ rhage wile a tan 5+, &e.) | 
~ tans 
which is our firft feries for the quadrature of an hyperbolic 
pee I z I : > 
feGor. And as Gas =20e=s + Zi tan = S, by refolving this 
equation in refpect of tan ; S, we get the formula 
by which the feries of quantities tan 7 5, tan ; s, &c. may be 
od . 
ord s 
deduced from tans = SEs? and from one another. 
37. Tuis expreflion for an hyperbolic fector is perfeétly 
fimilar in its form to that given in Art. 7. for an arch of a 
circle. It may, however, be transformed into another better 
adapted to calculation, by means of a property of the hyperbo- 
la to which there is no corref{ponding property of the circle, or 
at leaft none that can be expreffed without employing the fign 
Y“—t. The property alluded to may be deduced from the 
known 
Pir acd 
