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demonfirations here given are general, and deduced from the 

 circle by help of th_ do rine of combinations. 



As the hype bola has been fo frequently ufed to demonftrate 

 properties of the circle, I have fubjoined a theorem by which 

 the conneciion of multiple circular areas, and multiple hy- 

 perbolic areas is more fully apparent than by any other that 

 I have met with, and from whence by the dodlrine of combina- 

 tions, theorems may be deduced for hyperbolic areas fimilar to 

 thofe of the circle. 



I. Theorem. Let s and c be the fine and cofine of any arc a, 

 then, radius being unity, and « any whole number. 



I . The line ot u a = n c t c s + &c. 



I. 2. 3 



" n. n — I n_2 2 

 ^ 2. The cofine ot na = c — . s + &c. 



In each the powers of s increafe by 2, and thofe of c diminifli 

 by 2, till the laft becomes i or o. In the fine the coefficient of 



n — V V • 



C J- = + «. ;? — I » 2 



(to V terms 

 + when ""—I 



V ' ""'•" IS even 



2 



« — v <w 



and — when odd. And in the cofine the coefficient of <: •<■ — + 



— — + when ■ — is even and — when odd. 



I. 2. ^. V 2 



Demonftration 



