306 NEW SERIES for the 



Now, the rectilineal fpace Q^ is evidently lefs than the hy- 

 perbolic feclor s ; but n may be conceived fo great that the dif- 

 ference between Q^and s (hall be lefs than any affignable fpace, 

 as it is eafy to demonftrate upon principles ftri&ly geometrical ; 

 therefore, if we fuppofe n indefinitely great, then (^becomes 

 s j and as, upon this hypothesis, the feries goes on ad infinitum, 

 we have 



- = (tan - s -f- - tan - s -f - tan - s -f - tan —. s -f, &c.) 



s tan j ..:>_, 2 24 4 8 8 10 



which is our firffc feries for the quadrature of an hyperbolic 



fector. And as =? = ns + ~ tan - S, by refolving this 



tan S 2 tan ~ S 1 % 7 b 



equation in refpect of tan S, we get the formula 



tan- S r: — ^-— — J — * 1, 



2 tan S v tan 2 S ' 



by which the feries of quantities tan _ j, tan - s, &c. may be 



2 < 4 



deduced from tan j- =: — - — , and from one another. 



ab 



37. This expreflion for an hyperbolic feclor is perfectly 

 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 correfponding property of the circle, or 

 at leaft none that can be exprelfed without employing the fign 



s/ — 1. The property alluded to may be deduced from the 



known 



