14 REPORT — 1871. 



Legendre proves a theorem which is easily seen to follow directly from Lambert's 

 mode of investigation, viz. that if in the continued fraction 



/3, /3, H, 



(extended to infinity), 



^',^ . . . , regarded as fractions (a i . , ./3i . . ., all integers), be all less than unity. 



a, a., 



then, whether /3j, /3^ • . . be all positive or all negative, or some positive and some 

 negative, the value of the continued fraction is in-ational. He also remarks that 



w^ must be irrational ; for if it =— , we should have, from (i), since tan 7r=0, 



„_ m m m 



6m— 7— »»-&c. ' 



and as after some value of /• the fractions — , , &c. must be less than 



(2/-+l)w' 2/-+.3 



imity, 3 must be irrational if m and n are integers, whence ir'^ is irrational. 



The expression of tan v in the form of a continued fraction Lambert obtained by 



treating sin v or v — - — ^-—^ -{-. . . and cos v or 1 — -J!_^-|- . . . in a manner analogous 



to that in which the greatest common measure of two numbers is found in arith- 

 metic ; and Legendre deduced it from a more general theorem he had proved with 

 regard to the conversion of the ratio of two series into a continued fraction. 

 11 may be obtained very simply by forming the differential equation corresponding 

 to y = Acos(V2a;+B), viz. 



whence y(>) + (2t+l)y(«+i)-l-2a;y(«+2)=0 by application of Leibnitz's theorem. 

 From this we have 



y i+2xy- 



y' 



y" _ -1 

 ^ 3+2^?^ 

 y" 



&c.; 

 therefore 



whence, after determining B by putting ir=0 and writing >^(2x) = v, 



V v^ r 



tan ?)= , „ — . ^ — 

 1_3- 5-&C. 



That Lambert's proof is perfectly rigorous and places the ftict of the irra- 

 tionality of TT beyond all doubt, is evident to every one who examines it carefully ; 

 and considering the small attention that had been paid to continued fractions pre- 

 viously to the time at which it was written, it cannot but be regarded as a very 

 admirable work. 



From the continued fraction 



g"+l_ 1 1 1 1 



2 1- 2rt-i+ &^-l+ 10«-i-|-&c,' 



Lambert showed, in the same memoii', that e" is irrational, so that the Napierian 

 logarithm of every rational number is in-ational. 



1 



We can obtain a little more information about the iiTationality of e', for we have 



^—\ _ 1 1 1 



2 2x— 1+ 6x+ 10jr-|-&c. ■ 



