A similar question actually occurs in the case of the moon's motion. We have no record that any other portion of the moon was ever visible to men than such as we now see. This fact sufficiently proves that within the historical period the rotation of the moon on its own axis has coincided with its revolutions round the earth. Does this coincidence prove a relation of cause and effect to exist? The answer must be in the negative, because there might have been so slight a discrepancy between the motions that there has not yet been time to produce any appreciable effect. There may nevertheless be a high probability of connection.

The whole question of the relation of quantities thus resolves itself into one of probability. When we can only rudely measure a quantitative result, we can assign but slight importance to any correspondence. Because the brightness of two stars seems to vary in the same manner, there is no considerable probability that they have any relation with each other. Could it be shown that their periods of variation were the same to infinitely small quantities it would be certain, that is infinitely probable, that they were connected, however unlikely this might be on other grounds. The general mode of estimating such probabilities is identical with that applied to other inductive problems. That any two periods of variation should by chance become absolutely equal is infinitely improbable; hence if, in the case of the moon or other moving bodies, we could prove absolute coincidence we should have certainty of connection. With approximate measurements, which alone are within our power, we must hope for approximate certainty at the most.

The principles of inference and probability, according to which we treat causes and effects varying in amount, are exactly the same as those by which we treated simple experiments. Continuous quantity, however, affords us an infinitely more extensive sphere of observation, because every different amount of cause, however little different, ought to be followed by a different amount of effect. If we can measure temperature to the one-hundredth part of a degree centigrade, then between o° and 100° we have

1 Laplace, System of the World, translated by Harte, vol. ii. p. 366.

10,000 possible trials. If the precision of our measurements is increased, so that the one-thousandth part of a degree can be appreciated, our trials may be increased tenfold. The probability of connection will be proportional to the accuracy of our measurements.

When we can vary the quantity of a cause at will it is easy to discover whether a certain effect is due to that cause or not. We can then make as many irregular changes as we like, and it is quite incredible that the supposed effect should by chance go through exactly the corresponding series of changes except by dependence. If we have a bell ringing in vacuo, the sound increases as we let in the air, and it decreases again as we exhaust the air. Tyndall's singing flames evidently obeyed the directions of his own voice; and Faraday when he discovered. the relation of magnetism and light found that, by making or breaking or reversing the current of the electro-magnet, he had complete command over a ray of light, proving beyond all reasonable doubt the dependence of cause and effect. In such cases it is the perfect coincidence in time between the change in the effect and that in the cause which raises a high improbability of casual coincidence.

It is by a simple case of variation that we infer the existence of a material connection between two bodies moving with exactly equal velocity, such as the locomotive engine and the train which follows it. Elaborate observations were requisite before astronomers could all be convinced that the red hydrogen flames seen during solar eclipses belonged to the sun, and not to the moon's atmosphere as Flamsteed assumed. As early as 1706, Stannyan noticed a blood-red streak in an eclipse which he witnessed at Berne, and he asserted that it belonged to the sun; but his opinion was not finally established until photographs of the eclipse in 1860, taken by Mr. De la Rue, showed that the moon's dark body gradually covered the red prominences on one side, and uncovered those on the other; in short, that these prominences moved precisely as the sun moved, and not as the moon moved.

Even when we have no means of accurately measuring the variable quantities we may yet be convinced of their connection, if one always varies perceptibly at the same time as the other. Fatigue increases with exertion;

hunger with abstinence from food; desire and degree of utility decrease with the quantity of commodity consumed. We know that the sun's heating power depends upon his height of the sky; that the temperature of the air falls in ascending a mountain; that the earth's crust is found to be perceptibly warmer as we sink mines into it; we infer the direction in which a sound comes from the change of loudness as we approach or recede. The facility with which we can time after time observe the increase or decrease of one quantity with another sufficiently shows the connection, although we may be unable to assign any precise law of relation. The probability in such cases depends upon frequent coincidence in time.

Empirical Mathematical Laws.

It is important to acquire a clear comprehension of the part which is played in scientific investigation by empirical formulæ and laws. If we have a table containing certain values of a variable and the corresponding values of the variant, there are mathematical processes by which we can infallibly discover a mathematical formula yielding numbers in more or less exact agreement with the table. We may generally assume that the quantities will approximately conform to a law of the form

y = A + B x + C x2,

We can

in which is the variable and y the variant. then select from the table three values of y, and the corresponding values of x; inserting them in the equation, we obtain three equations by the solution of which we gain the values of A, B, and C. It will be found as a general rule that the formula thus obtained yields the other numbers of the table to a considerable degree of approximation.

In many cases even the second power of the variable will be unnecessary; Regnault found that the results of his elaborate inquiry into the latent heat of steam at different pressures were represented with sufficient accuracy by the empirical formula

λ = 6065 +0.305 t,

in which is the total heat of the steam, and t the tem

perature. In other cases it may be requisite to include. the third power of the variable. Thus physicists assume the law of the dilatation of liquids to be of the form &1 = a t + b t2 + c t3,

and they calculate from results of observation the values of the three constants a, b, c, which are usually small quantities not exceeding one-hundredth part of a unit, but requiring to be determined with great accuracy. Theoretically speaking, this process of empirical representation might be applied with any degree of accuracy; we might include still higher powers in the formula, and with sufficient labour obtain the values of the constants, by using an equal number of experimental results. The method of least squares may also be employed to obtain the most probable values of the constants.

In a similar manner all periodic variations may be represented with any required degree of accuracy by formule involving the sines and cosines of angles and their multiples. The form of any tidal or other wave may thus be expressed, as Sir G. B. Airy has explained. Almost all the phenomena registered by meteorologists are periodic in character, and when freed from disturbing causes may be embodied in empirical formula. 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, I need not further enter into them.

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

1 Chemical Reports and Memoirs, Cavendish Society, p. 294. 2 Jamin, Cours de Physique, vol. ii. p. 38.

3 On Tides and Waves, Encyclopædia Metropolitana, p. 366*. • Encyclopædia Britannica, art. Meteorology. Reprint, §§ 152-156.

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 not 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 οι 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." 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.

"1

Given the quantities, we want the function of which

1 Lagrange, Leçons sur le Calcul des Fonctions, 1806, p. 4.