Though we can not prove numerical laws with perfect accuracy, it would be a great mistake to suppose that there is any inexactness in the laws of nature. We may even discover a law which we believe to represent the action of forces with perfect exactness. The mind may seem to pass in advance of its data, and choose out certain numerical results as absolutely true. We can never really pass beyond our data, and so far as assumption enters in, so far want of certainty will attach to our conclusions; nevertheless we may sometimes rightly prefer a probable assumption of a precise law to numerical results, which are at the best only approximate. We must accordingly draw a strong distinction between the laws of nature which we believe to be accurately stated in our formulas, and those to which our statements only make an approximation, so that at a future time the law will be differently stated.
The law of gravitation is expressed in the form F = Mm/D2, meaning that gravity is proportional directly to the product of the gravitating masses, and indirectly to the square of their distance. The latent heat of steam is expressed by the equation log F = a + bαt + cβt, in which are five quantities a, b, c, α, β, to be determined by experiment. There is every reason to believe that in the progress of science the law of gravity will remain entirely unaltered, and the only effect of further inquiry will be to render it a more and more probable expression of the absolute truth. The law of the latent heat of steam on the other hand, will be modified by every new series of experiments, and it may not improbably be shown that the assumed law can never be made to agree exactly with the results of experiment.
Philosophers have not always supposed that the law of gravity was exactly true. Newton, though he had the highest confidence in its truth, admitted that there were motions in the planetary system which he could not reconcile with the law. Euler and Clairaut who were, with D’Alembert, the first to apply the full powers of mathematical analysis to the theory of gravitation as explaining the perturbations of the planets, did not think the law sufficiently established to attribute all discrepancies to the errors of calculation and observation. They did not feel certain that the force of gravity exactly obeyed the well-known rule. The law might involve other powers of the distance. It might be expressed in the form
F = . . . + a/D + b/D2 + c/D3 + . . .
and the coefficients a and c might be so small that those terms would become apparent only in very accurate comparisons with fact. Attempts have been made to account for difficulties, by attributing value to such neglected terms. Gauss at one time thought the even more fundamental principle of gravity, that the force is dependent only on mass and distance, might not be exactly true, and he undertook accurate pendulum experiments to test this opinion. Only as repeated doubts have time after time been resolved in favour of the law of Newton, has it been assumed as precisely correct. But this belief does not rest on experiment or observation only. The calculations of physical astronomy, however accurate, could never show that the other terms of the above expression were absolutely devoid of value. It could only be shown that they had such slight value as never to become apparent.
There are, however, other reasons why the law is probably complete and true as commonly stated. Whatever influence spreads from a point, and expands uniformly through space, will doubtless vary inversely in intensity as the square of the distance, because the area over which it is spread increases as the square of the radius. This part of the law of gravity may be considered as due to the properties of space, and there is a perfect analogy in this respect between gravity and all other emanating forces, as was pointed out by Keill.[377] Thus the undulations of light, heat, and sound, and the attractions of electricity and magnetism obey the very same law so far as we can ascertain. If the molecules of a gas or the particles of matter constituting odour were to start from a point and spread uniformly, their distances would increase and their density decrease according to the same principle.
Other laws of nature stand in a similar position. Dalton’s laws of definite combining proportions never have been, and never can be, exactly proved; but chemists having shown, to a considerable degree of approximation, that the elements combine together as if each element had atoms of an invariable mass, assume that this is exactly true. They go even further. Prout pointed out in 1815 that the equivalent weights of the elements appeared to be simple numbers; and the researches of Dumas, Pelouze, Marignac, Erdmann, Stas, and others have gradually rendered it likely that the atomic weights of hydrogen, carbon, oxygen, nitrogen, chlorine, and silver, are in the ratios of the numbers 1, 12, 16, 14, 35·5, and 108. Chemists then step beyond their data; they throw aside their actual experimental numbers, and assume that the true ratios are not those exactly indicated by any weighings, but the simple ratios of these numbers. They boldly assume that the discrepancies are due to experimental errors, and they are justified by the fact that the more elaborate and skilful the researches on the subject, the more nearly their assumption is verified. Potassium is the only element whose atomic weight has been determined with great care, but which has not shown an approach to a simple ratio with the other elements. This exception may be due to some unsuspected cause of error.[378] A similar assumption is made in the law of definite combining volumes of gases, and Brodie has clearly pointed out the line of argument by which the chemist, observing that the discrepancies between the law and fact are within the limits of experimental error, assumes that they are due to error.[379]
Faraday, in one of his researches, expressly makes an assumption of the same kind. Having shown, with some degree of experimental precision, that there exists a simple proportion between quantities of electrical energy and the quantities of chemical substances which it can decompose, so that for every atom dissolved in the battery cell an atom ought theoretically, that is without regard to dissipation of some of the energy, to be decomposed in the electrolytic cell, he does not stop at his numerical results. “I have not hesitated,” he says,[380] “to apply the more strict results of chemical analysis to correct the numbers obtained as electrolytic results. This, it is evident, may be done in a great number of cases, without using too much liberty towards the due severity of scientific research.”
The law of the conservation of energy, one of the widest of all physical generalisations, rests upon the same footing. The most that we can do by experiment is to show that the energy entering into any experimental combination is almost equal to what comes out of it, and more nearly so the more accurately we perform the measurements. Absolute equality is always a matter of assumption. We cannot even prove the indestructibility of matter; for were an exceedingly minute fraction of existing matter to vanish in any experiment, say one part in ten millions, we could never detect the loss.