Experiments may be of two kinds, experiments of simple fact, and experiments of quantity. In the first class of experiments we combine certain conditions, and wish to ascertain whether or not a certain effect of any quantity exists. Hooke wished to ascertain whether or not there was any difference in the force of gravity at the top and bottom of St. Paul’s Cathedral. The chemist continually performs analyses for the purpose of ascertaining whether or not a given element exists in a particular mineral or mixture; all such experiments and analyses are qualitative rather than quantitative, because though the result may be more or less, the particular amount of the result is not the object of the inquiry.
So soon, however, as a result is known to be discoverable, the scientific man ought to proceed to the quantitative inquiry, how great a result follows from a certain amount of the conditions which are supposed to constitute the cause? The possible numbers of experiments are now infinitely great, for every variation in a quantitative condition will usually produce a variation in the amount of the effect. The method of variation which thus arises is no narrow or special method, but it is the general application of experiment to phenomena capable of continuous variation. As Mr. Fowler has well remarked,1 the observation of variations is really an integration of a supposed infinite number of applications of the so-called method of difference, that is of experiment in its perfect form.
In induction we aim at establishing a general law, and if we deal with quantities that law must really be expressed more or less obviously in the form of an equation, or equations. We treat as before of conditions, and of what happens under those conditions. But the conditions will now vary, not in quality, but quantity, and the effect will also vary in quantity, so that the result of quantitative induction is always to arrive at some mathematical expression involving the quantity of each condition, and expressing the quantity of the result. In other words, we wish to know what function the effect is of its conditions. We shall find that it is one thing to obtain the numerical results, and quite another thing to detect the law obeyed by those results, the latter being an operation of an inverse and tentative character.
The Variable and the Variant.
Almost every series of quantitative experiments is directed to obtain the relation between the different values of one quantity which is varied at will, and another quantity which is caused thereby to vary. We may conveniently distinguish these as respectively the variable and the variant. When we are examining the effect of heat in expanding bodies, heat, or one of its dimensions, temperature, is the variable, length the variant. If we compress a body to observe how much it is thereby heated, pressure, or it may be the dimensions of the body, forms the variable, heat the variant. In the thermo-electric pile we make heat the variable and measure electricity as the variant. That one of the two measured quantities which is an antecedent condition of the other will be the variable.
It is always convenient to have the variable entirely under our command. Experiments may indeed be made with accuracy, provided we can exactly measure the variable at the moment when the quantity of the effect is determined. But if we have to trust to the action of some capricious force, there may be great difficulty in making exact measurements, and those results may not be disposed over the whole range of quantity in a