SCIENCE 



Friday, January 23, 1914 



CONTENTS 



The American Association for the Advance- 

 ment of Science: — 



The Influence of Fourier's Series upon the 

 Development of Mathematics: Edwakd B. 

 Van Vleck 113 



University Registration Statistics: Professoe 

 Eddolf Tombo, Jr 124 



The Massachusetts Institute of Technology 

 and Harvard University 132 



Scientific Notes and Neivs 135 



Uni/versity and Educational News 138 



Discussion and Correspondence : — 



Columbium versus Niobium: Dr. F. W. 

 Clarke. The Cytological Time of Muta- 

 tion in Tobacco : Professor W. E. Castle . . 139 



Scientific Books: — 



Dadourian's Analytical Mechanics: Pro- 

 fessor E. W. Eettger. McCulloh on the 

 Conservation of Water: Professor E. L. 

 Daughertt. Scripture on Stuttering and 

 Lisping : Professor Stevenson Smith .... 140 



Special Articles: — 



Some Physiological Observations regarding 

 Plumage Patterns: Professor Eaymond 

 Pearl and Alice M. Boring 143 



The American Society for Pharmacology and 

 Experimental Therapeutics: Dr. John Auer 144 



Societies and Academies: — 



The Anthropological Society of Washing- 



ton: Dr. Daniel Folkmar 



Mas. intended for publication and books, etc.. intended for 

 roTiew should be sent to Professor J. McKeen Cattell, Garrison- 

 On-Hudson. N. Y. 



THE INFLUENCE OF FOUBIES'S SEBIES 



UPON THE DEVELOPMENT OF 



MATHEMATICS-^ 



In selecting a subject for to-day's address 

 I have had the difficult task of interesting 

 two distinct classes of men, the astronomer 

 and the mathematician. I have therefore 

 chosen a topic which, I trust, will appeal to 

 both — trigonometric series. Though I pro- 

 pose to treat it only in its mathematical 

 aspects, I shall try to do so in a broad way, 

 tracing its general influence upon the trend 

 of mathematical thought. 



As you know, the theory of the infinite 

 trigonometric series, 

 (I.) f{x) = i a„ -f (Oi cos a; -f- &i sin x) 



4- (.<h cos 2x + 63 sin 2a;) + ••• 



is different ah initio from that of the power 

 series, 



P{x) =:c„ -f Ci{x — a) -|- (Xiix — ay -|- ... . 



For the latter the fundamental element 

 is x", of which the graph is, for positive x, 

 a monotone increasing function, wholly reg- 

 ular, without peculiarities of any sort. It 

 is therefore in no way surprising that the 

 power series obtained by combining terms 

 of form CnX" define the most civilized mem- 

 bers of mathematical society — the so-called 

 analytic functions — which are most orderly 

 in their behavior, being continuous through- 

 out their "domains," possessing deriva- 

 tives of all orders and a Taylor's series at 

 every point; and so forth. On the other 

 hand, the graph of sin ma; or eosnx is a 

 wave curve with crests and troughs, whose 

 number in any x interval increases indefi- 



1 Address of the vice-president of Section A — 

 Mathematics and Astronomy, American Associa- 

 tion for the Advancement of Science, Atlanta, 

 1913. 



