192 Mr. J. W. L. Glaisher on Arithmetical Irrationality. 



are commensurable with a right angle, are irrational; but, as 

 far as I know, no one has attempted to prove this ; and the same 

 may be said of many similar properties. 



The most general theorems of this class with which I am 

 acquainted were proved by Lambert in the Berlin Memoirs for 

 1761 ; he has there shown that the tangent of every rational arc 

 (viz. arc commensurable with the unit arc, equal to radius) is 

 irrational, and that the tangents of all angles commensurable 

 with a right angle, except tan 45°, are irrational. From the first 



of these results it follows that ir is irrational (since tan — =1); 



and to establish this was the main object Lambert had in view 

 in undertaking his investigation. It also follows, though not 

 quite so simply, that ir 2, is irrational ; but I believe no one has 

 ever shown that it 3 or any higher power of ir is so too ; so that, 

 as far as proof is concerned, tt might be the nth root of a rational 

 quantity, though, as Legendre has remarked, there is very little 

 doubt that it is not the root of any algebraical equation with 

 rational coefficients. 



Lambert's principle consists in developing the quantity which 

 is to be proved irrational into a continued fraction ; and his 

 result, stated in what is apparently the most generalized form it 

 admits of, viz. that given to it by Legendre in the notes to his 

 ' Geometry/ is that if in the continued fraction 





/3 



2 + « 3 +&c. 

 (extended to infinity) — ■> — . . ., regarded as fractions (a v a 2 , . . . 



f3 u {3c>, . . . all integers), be all less than unity, then, whether 

 fin fi<2> • • • be all positive or all negative, or some positive and 

 some negative, the value of the continued fraction is irrational. 



It is clear that the expansion of the quantity into a continued 

 fraction is the most natural way of attacking the question, as the 

 process is identical in principle with that of finding the greatest 

 common measure in arithmetic. 



Besides the theorems cited above with regard to tangents, 

 Lambert showed that the hyperbolic logarithm (viz. logarithm 

 to base .2*7182818 . . .) of every rational number was irrational; 

 and the corresponding theorem when the base of the system is 

 a rational number is evident ; for, to take the common base, we 

 see at sight that 10*=N (an integer) can only be satisfied by a 

 rational value of x when N is a power of 10. 



There is another method by which the irrationality of a series 

 can be proved; but it is of exceedingly limited application. I 



