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.”[404] 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.
Having numerical results we are always at liberty to invent any kind of mathematical formula we like, and then try whether, by the suitable selection of values for the unknown constant quantities, we can make it give the required results. If ever we fall upon a formula which does so, to a fair degree of approximation, there is a presumption in favour of its being the true function, although there is no certainty whatever in the matter. In this way I discovered a simple mathematical law which closely agreed with the results of my experiments on muscular exertion. This law was afterwards shown by Professor Haughton to be the true rational law according to his theory of muscular action.[405]
But the chance of succeeding in this manner is small. The number of possible functions is infinite, and even the number of comparatively simple functions is so large that the probability of falling upon the correct one by mere chance is very slight. Even when we obtain the law it is by a deductive process, not by showing that the numbers give the law, but that the law gives the numbers.
In the second way, we may, by a survey of the numbers, gain a general notion of the kind of law they are likely to obey, and we may be much assisted in this process by drawing them out in the form of a curve. We can in this way ascertain with some probability whether the curve is likely to return into itself, or whether it has infinite branches; whether such branches are asymptotic, that is, approach infinitely towards straight lines; whether it is logarithmic in character, or trigonometric. This indeed we can only do if we remember the results of previous investigations. The process is still inversely deductive, and consists in noting what laws give particular curves, and then inferring inversely that such curves belong to such laws. If we can in this way discover the class of functions to which the required law belongs, our chances of success are much increased, because our haphazard trials are now reduced within a narrower sphere. But, unless we have almost the whole curve before us, the identification of its character must be a matter of great uncertainty; and if, as in most physical investigations, we have a mere fragment of the curve, the assistance given would be quite illusory. Curves of almost any character can be made to approximate to each other for a limited extent, so that it is only by a kind of divination that we fall upon the actual function, unless we have theoretical knowledge of the kind of function applicable to the case.