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.8
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.
When we have once obtained what we believe to be the correct form of function, the remainder of the work is mere mathematical computation to be performed infallibly according to fixed rules,9 which include those employed in the determination of empirical formulæ (p. 487). The function will involve two or three or more unknown constants, the values of which we need to determine by our experimental results. Selecting some of our results widely apart and nearly equidistant, we form by means of them as many equations as there are constant quantities to be determined. The solution of these equations will then give us the constants required, and having now the actual function we can try whether it gives with sufficient accuracy the remainder of our experimental results. If not, we must either make a new selection of results to give a new set of equations, and thus obtain a new set of values for the constants, or we must acknowledge that our form of function has been wrongly chosen. If it appears that the form of function has been correctly ascertained, we may regard the constants as only approximately accurate and may proceed by the Method of Least Squares (p. 393) to determine the most probable values as given by the whole of the experimental results.
In most cases we shall find ourselves obliged to fall back upon the third mode, that is, anticipation of the form of the law to be expected on the ground of previous knowledge. Theory and analogical reasoning must be our guides. The general nature of the phenomenon will often indicate the kind of law to be looked for. If one form of energy or one kind of substance is being converted into another, we may expect the law of direct