Our present object will be to classify these constant numbers roughly, according to their comparative generality and importance, under the following heads:—
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 significance of which has already been pointed out (p. 182). 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 ε, being equal to the infinite series 1 + 1/1 + 1/1.2 + 1/1.2.3 + 1/1.2.3.4 +...., and thus consisting of the sum of the ratios between the numbers of permutations and combinations of 0, 1, 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 measurement 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 π, usually employed as expressing the ratio of the circumference and the diameter of a circle; from π 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 Bernoulli’s numbers, tables of the error function,25 which latter are indispensable not only