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.1

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

1 Haughton, Principles of Animal Mechanics, 1873, pp. 444-450. Jevons, Nature, 30th of June, 1870, vol. ii. p. 158. See also the experiments of Professor Nipher, of Washington University, St. Louis, in American Journal of Science, vol. ix. p. 130, vol. x. p. 1; Nature, vol. xi. pp. 256,276.

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, 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. 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 1 Jamin, Cours de Physique, vol. ii. p. 50.

If it appears

form of function has been wrongly chosen. 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 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 of a force influences its action, trigonometrical functions enter. Any 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 desirable to call in the aid of space-representations. Every equation involving two variable quantities corresponds to some kind of plane curve, and every plane curve may be represented symbolically in an equation 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 prepare a paper divided into equal rectangular spaces, a convenient size for the spaces being one-tenth of an inch square. The values of the variable being marked off on 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 value 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 a curved line be drawn through all the points or ends of the ordinates, it will probably exhibit irregular inflections, owing to the errors which affect the numbers. But, when the results are numerous, it becomes apparent which results are more divergent than others, and guided by a so-called sense of continuity, it is possible to trace a line among the points which will approximate to the true law more nearly than the points themselves. The accompanying figure sufficiently explains itself.

Perkins employed this graphical method with much care in exhibiting the results of his experiments on the compression of water. The numerical results were marked

1

upon a sheet of paper very exactly ruled at intervals of one-tenth of an inch, and the original marks were left in order that the reader might judge of the correctness of the curve drawn, or choose another for himself. Regnault carried the method to perfection by laying off the points with a screw dividing engine; and he then formed a table of results by drawing a continuous curve, and measuring its height for equidistant values of the variable. Not only does a curve drawn in this manner enable us to infer numerical results more free from accidental errors than any of the numbers obtained directly from experiment, but the form of the curve sometimes indicates the class of functions to which our results belong.

Engraved sheets of paper prepared for the drawing of curves may be obtained from Mr. Stanford at Charing Cross, Messrs. W. and A. K. Johnston, of London and Edinburgh, Waterlow and Sons, Letts and Co., and probably other publishers. When we do not require great accuracy, paper ruled by the common machine-ruler into equal squares of about one-fifth or one-sixth of an inch square will serve well enough. I have met with engineers' and surveyors' memorandum books ruled with one-twelfth inch squares. When a number of curves have to be drawn, I have found it best to rule a good sheet of drawing paper with lines carefully adjusted at the most convenient distances, and then to prick the points of the curve through it upon another sheet fixed underneath. In this way we obtain an accurate curve upon a blank sheet, and need only introduce such division lines as are requisite to the understanding of the curve.

In some cases our numerical results will correspond, not to the height of single ordinates, but to the area of the curve between two ordinates, or the average height of ordinates between certain limits. If we measure, for instance, the quantities of heat absorbed by water when raised in temperature from o° to 5°, from 5° to 10°, and so on, these quantities will really be represented by areas of the curve denoting the specific heat of water; and since the specific heat varies continuously between every two points of temperature, we shall not get the correct curve

1 Jamin, Cours de Physique, vol. ii. p. 24, &c.