the variant is of the variable, but we obtain another function which, within the bounds of our observation, gives nearly the same series of 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 may exhibit the reason or exact character 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 generalized it so as to include 'any quantity formed in any manner whatsoever from another quantitys.' 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. It may first of all be pointed out that simple inspection of the numbers cannot as a general rule disclose the function. In an earlier part of this work (vol. i. p. 142) 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

g Lagrange, Leçons sur le Calcul des Fonctions,' 1806, p. 4. VOL. II.

I

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 certain numerical results we are always at perfect 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 happened to discover a simple mathematical law which closely agreed with the results of certain 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 h

But the chance of succeeding in this manner is usually very small. The number of possible functions is certainly infinite, and even the number of comparatively simple functions is so very large that the probability of falling upon the correct one by mere chance is very slight. Let the reader observe that even when we can thus 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 place, 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, as will be presently considered. We can in this way ascertain with some probability whether the curve is likely to

h Haughton, 'Principles of Animal Mechanics,' 1873, pp. 444-450. Nature, 30th of June, 1870, vol. ii. p. 158.

be a closed one, or whether it has infinite branches; whether such branches are asymptotic, that is, approach indefinitely 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 gave 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 complete 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 can 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, which include those employed in the determination of empirical formulæ (vol. ii. p. 110). 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 must form by means of them as many equations as there are constant quantities to be determined. The solution of these equa tions will then give us the constants required, and having i See Jamin, Cours de Physique,' vol. ii. p. 50.

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 (vol. i. p. 458) 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 simple proportion. In one distinct class of cases the effect already produced influences the amount of the ensuing effect, as for instance in the cooling of a heated body, when the law will be of an exponential form. When the direction in which a force acts influences its action, trigonometrical functions must of course enter. Any force or influence which spreads freely through tridimensional space will be subject to the law of the inverse square of the distance. From such considerations we may sometimes arrive deductively and analogically at the general nature of the mathematical law required.

The Graphical Method.

In endeavouring to discover the mathematical law obeyed by experimental results it is often necessary,

and almost always desirable, to call in the aid of spacerepresentations. Every equation involving two variable quantities corresponds to some kind of plane curve, and every plane curve may be represented symbolically in an equation of a more or less complex character, containing two unknown quantities. Now in an experimental research we obtain a number of values of the variant corresponding to an equal number of values of the variable; but all the numbers are affected by more or less error, and the values of the variable will often be irregularly disposed. Even if the numbers were absolutely correct and disposed at regular intervals, there is, as we have seen, no direct mode of discovering the law, but the difficulty of discovery is much increased by the uncertainty and irregularity of the results.

Under such circumstances, the best mode of proceeding is to procure or prepare a paper divided into small equal rectangular spaces, a convenient size for the spaces being one-tenth of an inch square. The values of the variables being marked off along the scale formed by the lowest horizontal line, a point is marked for each corresponding value of the variant perpendicularly above that of the variable, and at such a height as corresponds to the amount of the variant.

The exact scale of the drawing is not of much importance, but it may require to be adjusted according to circumstances, and different values must often be attributed to the upright and horizontal divisions, so as to make the variations conspicuous, but not excessive. If now a curved line be drawn through all the extremities of the ordinates, it will probably exhibit many irregular inflections, owing to the errors which affect all the numbers. But, when the results are numerous, it soon becomes apparent which results are more divergent than others, and guided by a so-called sense of continuity, it becomes pos