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, " 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. Successive Approximations to Natural Conditions. When we examine the history of scientific problems, we find that one man or one generation is usually able to make but a single step at a time. A problem is solved for the first time by making some bold hypothetical simplification, upon which the next investigator makes hypothetical modifications approaching more nearly to the truth. Errors are successively pointed out in previous solutions, until at last there might seem little more to be desired. Careful examination, however, will show that a series of minor inaccuracies remain to be corrected and explained, were our powers of reasoning sufficiently great, and the purpose adequate in importance. 1 Experimental Researches in Electricity, vol. i. p. 246. H H Newton's successful solution of the problem of the planetary movements entirely depended at first upon a great simplification. The law of gravity only applies directly to two infinitely small particles, so that when we deal with vast globes like the earth, Jupiter, and the sun, we have an immense aggregate of separate attractions to deal with, and the law of the aggregate need not coincide with the law of the elementary particles. But Newton, by a great effort of mathematical reasoning, was able to show that two homogeneous spheres of matter act as if the whole of their masses were concentrated at the centres; in short, that such spheres are centrobaric bodies (p. 364). He was then able with comparative ease to calculate the motions of the planets on the hypothesis of their being spheres, and to show that the results roughly agreed with observation. Newton, indeed, was one of the few men who could make two great steps at once. He did not rest contented with the spherical hypothesis; having reason to believe that the earth was really a spheroid with a protuberance around the equator, he proceeded to a second approximation, and proved that the attraction of the protuberant matter upon the moon accounted for the precession of the equinoxes, and led to various complicated effects. But, (p. 459), even the spheroidal hypothesis is far from the truth. It takes no account of the irregularities of surface, the great protuberance of land in Central Asia and South America, and the deficiency in the bed of the Atlantic. To determine the law according to which a projectile, such as a cannon ball, moves through the atmosphere is a problem very imperfectly solved at the present day, but in which many successive advances have been made. So little was known concerning the subject three or four centuries ago that a cannon ball was supposed to move at first in a straight line, and after a time to be deflected into a curve. Tartaglia ventured to maintain that the path was curved throughout, as by the principle of continuity it should be; but the ingenuity of Galileo was required to prove this opinion, and to show that the curve was approximately a parabola. It is only, however, under forced hypotheses that we can assert the path of a projectile to be truly a parabola: the path must be through a perfect vacuum, where there is no resisting medium of any kind; the force of gravity must be uniform and act in parallel lines; or else the moving body must be either a mere point, or a perfect centrobaric body, that is a body possessing a definite centre of gravity. These conditions. cannot be really fulfilled in practice. The next great step in the problem was made by Newton and Huyghens, the latter of whom asserted that the atmosphere would offer a resistance proportional to the velocity of the moving body, and concluded that the path would have in consequence a logarithmic character. Newton investigated in a general manner the subject of resisting media, and came to the conclusion that the resistance is more nearly proportional to the square of the velocity. The subject then fell into the hands of Daniel Bernoulli, who pointed out the enormous resistance of the air in cases of rapid movement, and calculated that a cannon ball, if fired vertically in a vacuum, would rise eight times as high as in the atmosphere. In recent times an immense amount both of theoretical and experimental investigation has been spent. upon the subject, since it is one of importance in the art of war. Successive approximations to the true law have been made, but nothing like a complete and final solution has been achieved or even hoped for.1 It is quite to be expected that the earliest experimenters in any branch of science will overlook errors which afterwards become most apparent. The Arabian astronomers determined the meridian by taking the middle point between the places of the sun when at equal altitudes on the same day. They overlooked the fact that the sun has its own motion in the time between the observations. Newton thought that the mutual disturbances of the planets might be disregarded, excepting perhaps the effect of the mutual attraction of the greater planets, Jupiter and Saturn, near their conjunction.2 The expansion of quicksilver was long used as the measure of temperature, no clear idea being possessed of temperature apart from some of its more obvious effects. Rumford, in the first experiment leading to a determination of the mechanical 1 Hutton's Mathematical Dictionary, vol. ii. pp. 287–292. equivalent of heat, disregarded the heat absorbed by the apparatus, otherwise he would, in Dr. Joule's opinion, have come nearly to the correct result. It is surprising to learn the number of causes of error which enter into the simplest experiment, when we strive to attain rigid accuracy. We cannot accurately perform the simple experiment of compressing gas in a bent tube by a column of mercury, in order to test the truth of Boyle's Law, without paying regard to-(1) the variations. of atmospheric pressure, which are communicated to the gas through the mercury; (2) the compressibility of mercury, which causes the column of mercury to vary in density; (3) the temperature of the mercury throughout the column; (4) the temperature of the gas, which is with difficulty maintained invariable; (5) the expansion of the glass tube containing the gas. Although Regnault took all these circumstances into account in his examination of the law, there is no reason to suppose that he exhausted the sources of inaccuracy. The early investigations concerning the nature of waves in elastic media proceeded upon the assumption that waves of different lengths would travel with equal speed. Newton's theory of sound led him to this conclusion, and observation (p. 295) had verified the inference. When the undulatory theory came to be applied at the commencement of this century to explain the phenomena of light, a great difficulty was encountered. The angle at which a ray of light is refracted in entering a denser medium depends, according to that theory, on the velocity with which the wave travels, so that if all waves of light were to travel with equal velocity in the same medium, the dispersion of mixed light by the prism and the production of the spectrum could not take place. Some most striking phenomena were thus in direct conflict with the theory. Cauchy first pointed out the explanation, namely, that all previous investigators had made an arbitrary assumption for the sake of simplifying the calculations. They had assumed that the particles of the vibrating medium are so close together that the intervals. are inconsiderable compared with the length of the wave. 1 Jamin, Cours de Physique, vol. i. pp. 282, 283. This hypothesis happened to be approximately true in the case of air, so that no error was discovered in experiments on sound. Had it not been so, the earlier analysts would probably have failed to give any solution, and the progress of the subject might have been retarded. Cauchy was able to make a new approximation under the more difficult supposition, that the particles of the vibrating medium are situated at considerable distances, and act and react upon the neighbouring particles by attractive and repulsive forces. To calculate the rate of propagation of disturbance in such a medium is a work of excessive difficulty. The complete solution of the problem appears indeed to be beyond human power, so that we must be content, as in the case of the planetary motions, to look forward to successive approximations. All that Cauchy could do was to show that certain quantities, neglected in previous theories, became of considerable amount under the new conditions of the problem, so that there will exist a relation between the length of the wave, and the velocity at which it travels. To remove, then, the difficulties in the way of the undulatory theory of light, a new approach to probable conditions was needed.1 In a similar manner Fourier's theory of the conduction and radiation of heat was based upon the hypothesis that the quantity of heat passing along any line is simply proportional to the rate of change of temperature. But it has since been shown by Forbes that the conductivity of a body diminishes as its temperature increases. All the details of Fourier's solution therefore require modification, and the results are in the meantime to be regarded as only approximately true.2 We ought to distinguish between those problems which are physically and those which are merely mathematically incomplete. In the latter case the physical law is correctly seized, but the mathematician neglects, or is more often unable to follow out the law in all its results. The law of gravitation and the principles of harmonic or undulatory movement, even supposing the data to be correct, 1 Lloyd's Lectures on the Wave Theory, pp. 22, 23. 2 Tait's Thermodynamics, p. 10. |