We have not yet formally considered any processes of reasoning which have for their object to disclose laws of nature expressed in quantitative equations. We have been inquiring into the modes by which a phenomenon may be measured, and, if it be a composite phenomenon, may be resolved, by the aid of several measurements, into its component parts. We have also considered the precautions to be taken in the performance of observations and experiments in order that we may know what phenomena we really do measure, but we must remember that, no number of facts and observations can by themselves constitute science. Numerical facts, like other facts, are but the raw materials of knowledge, upon which our reasoning faculties must be exerted in order to draw forth the principles of nature. It is by an inverse process of reasoning that we can alone discover the mathematical laws to which varying quantities conform. By well-conducted experiments we gain a series of values of a variable, and a corresponding series of values of a variant, and we now want to know what mathematical function the variant is as regards the variable. In the usual progress of a science three questions will have to be answered as regards every important quantitative phenomenon:—
(1) Is there any constant relation between a variable and a variant?
(2) What is the empirical formula expressing this relation?
(3) What is the rational formula expressing the law of nature involved?
Probable Connection of Varying Quantities.
We find it stated by Mill,1 that “Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation.” This assertion may be considered true when it is interpreted with sufficient caution; but it might otherwise lead us into error. There is nothing whatever in the nature of things to prevent the existence of two variations which should apparently follow the same law, and yet have no connection with each other. One binary star might be going through a revolution which, so far as we could tell, was of equal period with that of another binary star, and according to the above rule the motion of one would be the cause of the motion of the other, which would not be really the case. Two astronomical clocks might conceivably be made so nearly perfect that, for several years, no difference could be detected, and we might then infer that the motion of one clock was the cause or effect of the motion of the other. This matter requires careful discrimination. We must bear in mind that the continuous quantities of space, time, force, &c., which we measure, are made up of an infinite number of infinitely small units. We may then meet with two variable phenomena which follow laws so nearly the same, that in no part of the variations open to our observation can any discrepancy be discovered. I grant that if two clocks could be shown to have kept exactly the same time during any finite interval, the probability would become infinitely high that there was a connection between their motions. But we can never absolutely prove such coincidences to exist. Allow that we may observe a difference of one-tenth of a second in their time, yet it is possible that they were