UNIVERSITY OF VIRGINIA PUBLICATIONS 



BULLETIN OF THE PHILOSOPHICAL SOCIETY 



SCIENTIFIC SECTION 



Vol. I, No. 13, pp. 319-330 January, 1913 



THE EVOLUTIONARY CONSTRUCTION OF THE IMAGINARY 

 POWER OF A NUMBER AND ITS EXPRESSION AS THE 

 EXPONENTIAL FUNCTION.* 



W. H. ECHOLS 



1. In the preface to a most inspiring book, the first edition of his 

 Introduction a la Theorie des Fonctions d'un Variable, M. Jules Tannery 

 remarks : 



On pent constituer entierement I'Analyse avec la notion de nombre entier et 

 les notions relatives a 1' addition des nombres entiers; il est inutile de faire appel S, 

 aucun autre postulat, k auoune autre donn4e de 1' experience; .... II va 

 sans dire que j'ai degager la definition des fonctions circulaires de toute consideration 

 geometrique. 



When he attempts however to evolve the circular functions from the 

 functional law of the addition formulae he finds it necessary to make the 

 additional assumption that the limit of the quotient of sin a; to a; is unity 

 when X converges to zero, and apparently on account of the necessity for 

 this assumption he abandons the analytical design of these functions in the 

 second edition of his book and resorts as do others to their geometrical 

 definitions, or boldly and artificially assumes them in analytic form. 



The circular functions appear as necessarily and naturally born into 

 existence in the construction of the imaginary power in exactly the same 

 way that the hyperbolic functions are created in the construction of the 



* This note is extracted from a lithographed edition (1903) of the writer's lectures 

 on the Introduction to Theorj' of Functions. Much of it is of course not new but it 

 is believed the novelty of treatment is. interesting. Presented to the Scientific Sec. 

 Philos. Socy., January, 1913, in its present form. 



319 



