330 THE PRINCIPLES OF SCIENCE. [CHAP. 



Mathematical Constants. 



At the head of the list of natural constants must come 

 those which express the necessary relations of numbers to 

 each other. The ordinary Multiplication Table is the 

 most familiar and the most important of such series of 

 constants, and is, theoretically speaking, infinite in extent. 

 Next we must place the Arithmetical Triangle, the sig- 

 nificance of which has already been pointed out (p. 1 82). 

 Tables of logarithms also contain vast series of natural 

 constants, arising out of the relations of pure numbers. 

 At the base of all logarithmic theory is the mysterious 

 natural constant commonly denoted by e, or e, being 



equal to the infinite series i + [-+ ^ 2+ _1_ +_L- +...., 



and thus consisting of the sum of the ratios between the 

 numbers of permutations and combinations of o, i, 2, 3, 

 4, &c. things. Tables of prime numbers and of the factors 

 of composite numbers must not be forgotten. 



Another vast and in fact infinite series of numerical 

 constants contains those connected with the measure- 

 ment of angles, and embodied in trigonometrical tables, 

 whether as natural or logarithmic sines, cosines, and 

 tangents. It should never be forgotten that though 

 these numbers find their chief employment in connection 

 with trigonometry, or the measurement of the sides of a 

 right-angled triangle, yet the numbers themselves arise 

 out of numerical relations bearing no special relation to 

 space. Foremost among trigonometrical constants is the 

 well known number TT, usually employed as expressing 

 the ratio of the circumference and the diameter of a 

 circle ; from TT follows the value of the arcual or natural 

 unit of angular value as expressed in ordinary degrees 

 (p. 306). 



Among other mathematical constants not uncommonly 

 used may be mentioned tables of factorials (p. 179), tables 

 of Bernouilli's numbers, tables of the error function, 1 

 which latter are indispensable not only in the theory of 

 probability but also in several other branches of science. 



1 J. W. L. Glaisher, Philosophical Magazine, 4th. Series, vol. xlii. 

 p. 421. 



