Natural Constants.
Having acquired accurate measuring instruments, and decided upon the units in which the results shall be expressed, there remains the question, What use shall be made of our powers of measurement? Our principal object must be to discover general quantitative laws of nature; but a very large amount of preliminary labour is employed in the accurate determination of the dimensions of existing objects, and the numerical relations between diverse forces and phenomena. Step by step every part of the material universe is surveyed and brought into known relations with other parts. Each manifestation of energy is correlated with each other kind of manifestation. Professor Tyndall has described the care with which such operations are conducted.[229]
“Those who are unacquainted with the details of scientific investigation, have no idea of the amount of labour expended on the determination of those numbers on which important calculations or inferences depend. They have no idea of the patience shown by a Berzelius in determining atomic weights; by a Regnault in determining coefficients of expansion; or by a Joule in determining the mechanical equivalent of heat. There is a morality brought to bear upon such matters which, in point of severity, is probably without a parallel in any other domain of intellectual action.”
Every new natural constant which is recorded brings many fresh inferences within our power. For if n be the number of such constants known, then 1/2 (n2—n) is the number of ratios which are within our powers of calculation, and this increases with the square of n. We thus gradually piece together a map of nature, in which the lines of inference from one phenomenon to another rapidly grow in complexity, and the powers of scientific prediction are correspondingly augmented.
Babbage[230] proposed the formation of a collection of the constant numbers of nature, a work which has at last been taken in hand by the Smithsonian Institution.[231] It is true that a complete collection of such numbers would be almost co-extensive with scientific literature, since almost all the numbers occurring in works on chemistry, mineralogy, physics, astronomy, &c., would have to be included. Still a handy volume giving all the more important numbers and their logarithms, referred when requisite to the different units in common use, would be very useful. A small collection of constant numbers will be found at the end of Babbage’s, Hutton’s, and many other tables of logarithms, and a somewhat larger collection is given in Templeton’s Millwright and Engineer’s Pocket Companion.
Our present object will be to classify these constant numbers roughly, according to their comparative generality and importance, under the following heads:—
(1) Mathematical constants.
(2) Physical constants.
(3) Astronomical constants.
(4) Terrestrial numbers.
(5) Organic numbers.
(6) Social numbers.
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]).