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 aud 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 convenient manner. It is one prime object of the experi menter, therefore, to obtain a regular and governable supply of the force which he is investigating. To determine correctly the efficiency of windmills, when the natural winds were constantly varying in force, would be exceedingly difficult. Smeaton, therefore, in his experiments on the subject, created a uniform wind of the required force by moving his models against the air on the extremity of a revolving arm. The velocity of the wind could thus be rendered greater or less, it could be maintained uniform for any length of time, and its amount could be exactly ascertained. In determining the laws of the chemical action of light it would be out of the question to employ the rays of the sun, which vary in intensity with the clearness of the atmosphere, and with every passing cloud. One great difficulty in photometry and the investigation of the chemical action of light consists in obtaining a uniform and governable source of light rays.2 Fizeau's method of measuring the velocity of light enabled him to appreciate the time occupied by light in travelling through a distance of eight or nine thousand metres. But the revolving mirror of Wheatstone subsequently enabled Foucault and Fizeau to measure the velocity in a space of four metres. In this latter method there was the advantage that various media could be substituted for air, and the temperature, density, and other conditions of the experiment could be accurately governed and measured. Measurement of the Variable. There is little use in obtaining exact measurements of an effect unless we can also exactly measure its conditions. It is absurd to measure the electrical resistance of a piece of metal, its elasticity, tenacity, density, or other physical qualities, if these vary, not only with the minute. impurities of the metal, but also with its physical condition. If the same bar changes its properties by being 1 Philosophical Transactions, vol. li. p. 138; abridgment, vol. xi. P. 355. 2 See Bunsen and Roscoe's researches, in Philosophical Transactions (1859), vol. cxlix. p. 880, &c., where they describe a constant flame of carbon monoxide gas. heated and cooled, and we cannot exactly define the state in which it is at any moment, our care in measuring will be wasted, because it can lead to no law. It is of little use to determine very exactly the electric conductibility of carbon, which as graphite or gas carbon conducts like a metal, as diamond is almost a non-conductor, and in several other forms possesses variable and intermediate powers of conduction. It will be of use only for immediate practical applications. Before measuring these we ought to have something to measure of which the conditions are capable of exact definition, and to which at a future time we can recur. Similarly the accuracy of our measurement need not much surpass the accuracy with which we can define the conditions of the object treated. The speed of electricity in passing through a conductor mainly depends upon the inductive capacity of the surrounding substances, and, except for technical or special purposes, there is little use in measuring velocities which in some cases are one hundred times as great as in other cases. But the maximum speed of electric conduction is probably a constant quantity of great scientific importance, and according to Prof. Clerk Maxwell's determination in 1868 is 174,800 miles per second, or little less than that of light. The true boiling point of water is a point on which practical thermometry depends, and it is highly important to determine that point in relation to the absolute thermometric scale. But when water free from air and impurity is heated there seems to be no definite limit to the temperature it may reach, a temperature of 180° Cent. having been actually observed. Such temperatures, therefore, do not require accurate measurement. All meteorological measurements depending on the accidental condition of the sky are of far less importance than physical measurements in which such accidental conditions do not intervene. Many profound investigations depend upon our knowledge of the radiant energy continually poured upon the earth by the sun; but this must be measured when the sky is perfectly clear, and the absorption of the atmosphere at its minimum. The slightest interference of cloud destroys the value of such a measurement, except for meteorological purposes, which are of vastly less generality and importance. It is seldom useful, again, to measure the height of a snow-covered mountain within a foot, when the thickness of the snow alone may cause it to vary 25 feet or more, when in short the height itself is indefinite to that extent.1 Maintenance of Similar Conditions. Our ultimate object in induction must be to obtain the complete relation between the conditions and the effect, but this relation will generally be so complex that we can only attack it in detail. We must, as far as possible, confine the variation to one condition at a time, and establish a separate relation between each condition and the effect. This is at any rate the first step in approximating to the complete law, and it will be a subsequent question how far the simultaneous variation of several conditions modifies their separate actions. In many experiments, indeed, it is only one condition which we wish to study, and the others are interfering forces which we would avoid if possible. One of the conditions of the motion of a pendulum is the resistance of the air, or other medium in which it swings; but when Newton was desirous of proving the equal gravitation of all substances, he had no interest in the air. His object was to observe a single force only, and so it is in a great many other experiments. Accordingly, one of the most important precautions in investigation consists in maintaining all conditions constant except that which is to be studied. As that admirable experimental philosopher, Gilbert, expressed it,2 "There is always need of similar preparation, of similar figure, and of equal magnitude, for in dissimilar and unequal circumstances the experiment is doubtful.” In Newton's decisive experiment similar conditions were provided for, with the simplicity which characterises the highest art. The pendulums of which the oscillations were compared consisted of equal boxes of wood, hanging by equal threads, and filled with different substances, so that the total weights should be equal and the centres of oscillation at the same distance from the points of suspension. 1 Humboldt's Cosmos (Bohn), vol. i. p. 7. Hence the resistance of the air became approximately a matter of indifference; for the outward size and shape of the pendulums being the same, the absolute force of resistance would be the same, so long as the pendulums vibrated with equal velocity; and the weights being equal the resistance would diminish the velocity equally. Hence if any inequality were observed in the vibrations of the two pendulums, it must arise from the only circumstance which was different, namely the chemical nature of the matter within the boxes. No inequality being observed, the chemical nature of substances can have no appreciable influence upon the force of gravitation.1 A beautiful experiment was devised by Dr. Joule for the purpose of showing that the gain or loss of heat by a gas is connected, not with the mere change of its volume and density, but with the energy received or given out by the gas. Two strong vessels, connected by a tube and stopcock, were placed in water after the air had been exhausted from one vessel and condensed in the other to the extent of twenty atmospheres. The whole apparatus having been brought to a uniform temperature by agitating the water, and the temperature having been exactly observed, the stopcock was opened, so that the air at once expanded and filled the two vessels uniformly. The temperature of the water being again noted was found to be almost unchanged. The experiment was then repeated in an exactly similar manner, except that the strong vessels were placed in separate portions of the water. Now cold was produced in the vessel from which the air rushed, and an almost exactly equal quantity of heat appeared in that to which it was conducted. Thus Dr. Joule clearly proved that rarefaction produces as much heat as cold, and that only when there is disappearance of mechanical energy will there be production of heat.2 What we have to notice, however, is not so much the result of the experiment, as the simple manner in which a single change in the apparatus, the separation of the portions of water surrounding the air vessels, is made to give indications of the utmost significance. 1 Principia, bk. iii. Prop. vi. 2 Philosophical Magazine, 3rd Series, vol. xxvi. p. 375. |