by meteorologists are periodic in character, and when freed from disturbing causes may be embodied in empirical formulæ. Bessel has given a rule by which from any regular series of observations we may, on the principle of the method of least squares, calculate out with a moderate amount of labour a formula expressing the variation of the quantity observed, in the most probable manner. In meteorology three or four terms are usually sufficient for representing any periodic phenomenon, but the calculation might be carried to any higher degree of accuracy. As the details of the process have been described by Herschel in his treatise on Meteorology,6 I need not further enter into
The reader might be tempted to think that in these processes of calculation we have an infallible method of discovering inductive laws, and that my previous statements (Chap. VII.) as to the purely tentative and inverse character of the inductive process are negatived. Were there indeed any general method of inferring laws from facts it would overturn my statement, but it must be carefully observed that these empirical formulæ do not coincide with natural laws. They are only approximations to the results of natural laws founded upon the general principles of approximation. It has already been pointed out that however complicated be the nature of a curve, we may examine so small a portion of it, or we may examine it with such rude means of measurement, that its divergence from an elliptic curve will not be apparent. As a still ruder approximation a portion of a straight line will always serve our purpose; but if we need higher precision a curve of the third or fourth degree will almost certainly be sufficient. Now empirical formulæ really represent these approximate curves, but they give us no information as to the precise nature of the curve itself to which we are approximating. We do not learn what function the variant is of the variable, but we obtain another function which, within the bounds of observation, gives nearly the same values.
Discovery of Rational Formulæ.
Let us now proceed to consider the modes in which from numerical results we can establish the actual relation between the quantity of the cause and that of the effect. What we want is a rational formula or function, which will exhibit the reason or exact nature and origin of the law in question. There is no word more frequently used by mathematicians than the word function, and yet it is difficult to define its meaning with perfect accuracy. Originally it meant performance or execution, being equivalent to the Greek λειτουργία or τέλεσμα. Mathematicians at first used it to mean any power of a quantity, but afterwards generalised it so as to include “any quantity formed in any manner whatsoever from another quantity.”7 Any quantity, then, which depends upon and varies with another quantity may be called a function of it, and either may be considered a function of the other.
Given the quantities, we want the function of which they are the values. Simple inspection of the numbers cannot as a general rule disclose the function. In an earlier chapter (p. 124) I put before the reader certain numbers, and requested him to point out the law which they obey, and the same question will have to be asked in every case of quantitative induction. There are perhaps three methods, more or less distinct, by which we may hope to obtain an answer:
(1) By purely haphazard trial.
(2) By noting the general character of the variation of the quantities, and trying by preference functions which give a similar form of variation.
(3) By deducing from previous knowledge the form of the function which is most likely to suit.