Such is the purely approximate character of all our operations that it is not uncommon to find the theoretically worse method giving truer results than the theoretically perfect method. The common pendulum which is not isochronous is better for practical purposes than the cycloidal pendulum, which is isochronous in theory but subject to mechanical difficulties. The spherical form is not the correct form for a speculum or lense, but it differs so slightly from the true form, and is so much more easily produced mechanically, that it is generally best to rest content with the spherical surface. Even in a six-feet mirror the difference between the parabola and the sphere is only about one ten-thousandth part of an inch, a thickness which would be taken off in a few rubs of the polisher. Watts’ ingenious parallel motion was intended to produce rectilinear movement of the piston-rod. In reality the motion was always curvilinear, but for his purposes a certain part of the curve approximated sufficiently to a straight line.
Approximation to Exact Laws.
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
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