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
in which x is the variable and y the variant. We can 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
in which λ is the total heat of the steam, and t the temperature.3 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
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.4 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 formulæ 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.5 Almost all the phenomena registered