TRANSACTIONS OF THE SECTIONS. 15 



Now any continued fraction in which all the numerators are unity and all the 

 denominators are positive integers must circulate if it be the derelopment of an ex- 

 pression of the form A+BVC ; so that we see that e*, when x is integral, cannot 

 be of this fonn. 



Taking the expression for the tangent in the form 



cotl= _1_ J_ _1_ 1. _i * 



X X—1+ 1+ 3«— 2+ 1+ 5a— 2+&C. 



we see that when x is an integer, cot — is irrational, but cannot be of the form 

 A+BVC 



Al • 1 ,/, 1\ 1 V(l + COt^-) n 



Also, since cosec - = V ( l+cot=i ), and 8ec-= _J^ i/, sini and 



X \ xf X xl« 



cot — 



1 * 



COS - cannot either of them be rational unless coti is of the form B\/0, which 

 not being the case, sin- and cos - are irrational, and cannot be of the form B y'C. 



Since cos- =2cos-_-l, cos- cannot be rational unless cos i =6^0, which 



X X X X 



would require that cot 1 = /( , ^pan )' * "^^^^ '^^^^^ ^^ ^^"^'^ &\^ovra. it cannot 



2 

 have J so that cos - is irrational. Similar results hold good for the hyperbolic 



* i _' i _1 ? 2 



sine and cosine ; thatis tosay, i(ei+e '), K^r'-e '),and i(e^+e~^) arein-ational. 



It may be remarked that it is easy to show that sin - is incommensurable from 

 the series 5 for if sin- = ?, then {q even, as of course we may take it) 

 jo_l_ 1 ?_j 1 ? 1 



9 'x 1^^73^^+-(-> l.2..(5-l>.-> + ^"^\.2..(y4l>'-^^'^"' 

 whence, multiplying both sides by (1 . 2. .. 5),r?-i, 



p(l . 2. ..(,-1)),.-= i..e,e, +(-)l(gJi^ - ^-p^^^^^^j^ . ..)_ 



and the series on the right-hand side must be intermediate in value to ^ 



1 (9 + 1)** 



and 7 — nrv? — ToTT — I o, 1 ) ^°d is therefore fractional ; thus we have 



integer = integer + fraction 



if sin- is commensurable. An exactly similar method proves the in-ationality of 



1 - -- 2 ? -? 



cos-, \{f — e '), &c., but gives no result when applied to cos- or \{e^-\.e *). It is 



probably true that^ both the sine and cosine of every rational arc are irrational, 

 though no proof of this has, I believe, been given ; and there is, as Legendre has 

 remarked, very little doubt that tt is not only not the square root of a rational quan- 

 tity, but also not even the root of any algebraical equation with rational coeffi- 

 cients, although the demonstration of this seems difficult. Similai- remarks may be 

 made with respect to e. 



* This expression can be deduced from (i) by transfoi-ming the terms of the latter thus: 



A-i=A-l+ 1 



B 1 + -L' 



^B-1 



