But, secondly, as a matter of fact no planet does move in a perfect ellipse, or manifest the truth of Kepler’s laws exactly. The law of gravity prevents its own results from being clearly exhibited, because the mutual perturbations of the planets distort the elliptical paths. Those laws, again, hold exactly true only of infinitely small bodies, and when two great globes, like the sun and Jupiter, attract each other, the law must be modified. The periodic time is then shortened in the ratio of the square root of the number expressing the sun’s mass, to that of the sum of the numbers expressing the masses of the sun and planet, as was shown by Newton.[374] Even at the present day discrepancies exist between the observed dimensions of the planetary orbits and their theoretical magnitudes, after making allowance for all disturbing causes.[375] Nothing is more certain in scientific method than that approximate coincidence alone can be expected. In the measurement of continuous quantity perfect correspondence must be accidental, and should give rise to suspicion rather than to satisfaction.
One remarkable result of the approximate character of our observations is that we could never prove the existence of perfectly circular or parabolic movement, even if it existed. The circle is a singular case of the ellipse, for which the eccentricity is zero; it is infinitely improbable that any planet, even if undisturbed by other bodies, would have a circle for its orbit; but if the orbit were a circle we could never prove the entire absence of eccentricity. All that we could do would be to declare the divergence from the circular form to be inappreciable. Delambre was unable to detect the slightest ellipticity in the orbit of Jupiter’s first satellite, but he could only infer that the orbit was nearly circular. The parabola is the singular limit between the ellipse and the hyperbola. As there are elliptic and hyperbolic comets, so we might conceive the existence of a parabolic comet. Indeed if an undisturbed comet fell towards the sun from an infinite distance it would move in a parabola; but we could never prove that it so moved.
Substitution of Simple Hypotheses.
In truth men never can solve problems fulfilling the complex circumstances of nature. All laws and explanations are in a certain sense hypothetical, and apply exactly to nothing which we can know to exist. In place of the actual objects which we see and feel, the mathematician substitutes imaginary objects, only partially resembling those represented, but so devised that the discrepancies are not of an amount to alter seriously the character of the solution. When we probe the matter to the bottom physical astronomy is as hypothetical as Euclid’s elements. There may exist in nature perfect straight lines, triangles, circles, and other regular geometrical figures; to our science it is a matter of indifference whether they do or do not exist, because in any case they must be beyond our powers of perception. If we submitted a perfect circle to the most rigorous scrutiny, it is impossible that we should discover whether it were perfect or not. Nevertheless in geometry we argue concerning perfect curves, and rectilinear figures, and the conclusions apply to existing objects so far as we can assure ourselves that they agree with the hypothetical conditions of our reasoning. This is in reality all that we can do in the most perfect of the sciences.
Doubtless in astronomy we meet with the nearest approximation to actual conditions. The law of gravity is not a complex one in itself, and we believe it with much probability to be exactly true; but we cannot calculate out in any real case its accurate results. The law asserts that every particle of matter in the universe attracts every other particle, with a force depending on the masses of the particles and their distances. We cannot know the force acting on any particle unless we know the masses and distances and positions of all other particles in the universe. The physical astronomer has made a sweeping assumption, namely, that all the millions of existing systems exert no perturbing effects on our planetary system, that is to say, no effects in the least appreciable. The problem at once becomes hypothetical, because there is little doubt that gravitation between our sun and planets and other systems does exist. Even when they consider the relations of our planetary bodies inter se, all their processes are only approximate. In the first place they assume that each of the planets is a perfect ellipsoid, with a smooth surface and a homogeneous interior. That this assumption is untrue every mountain and valley, every sea, every mine affords conclusive evidence. If astronomers are to make their calculations perfect, they must not only take account of the Himalayas and the Andes, but must calculate separately the attraction of every hill, nay, of every ant-hill. So far are they from having considered any local inequality of the surface, that they have not yet decided upon the general form of the earth; it is still a matter of speculation whether or not the earth is an ellipsoid with three unequal axes. If, as is probable, the globe is irregularly compressed in some directions, the calculations of astronomers will have to be repeated and refined, in order that they may approximate to the attractive power of such a body. If we cannot accurately learn the form of our own earth, how can we expect to ascertain that of the moon, the sun, and other planets, in some of which probably are irregularities of greater proportional amount?
In a further way the science of physical astronomy is merely approximate and hypothetical. Given homogeneous ellipsoids acting upon each other according to the law of gravity, the best mathematicians have never and perhaps never will determine exactly the resulting movements. Even when three bodies simultaneously attract each other the complication of effects is so great that only approximate calculations can be made. Astronomers have not even attempted the general problem of the simultaneous attractions of four, five, six, or more bodies; they resolve the general problem into so many different problems of three bodies. The principle upon which the calculations of physical astronomy proceed, is to neglect every quantity which does not seem likely to lead to an effect appreciable in observation, and the quantities rejected are far more numerous and complex than the few larger terms which are retained. All then is merely approximate.
Concerning other branches of physical science the same statements are even more evidently true. We speak and calculate about inflexible bars, inextensible lines, heavy points, homogeneous substances, uniform spheres, perfect fluids and gases, and we deduce a great number of beautiful theorems; but all is hypothetical. There is no such thing as an inflexible bar, an inextensible line, nor any one of the other perfect objects of mechanical science; they are to be classed with those mythical existences, the straight line, triangle, circle, &c., about which Euclid so freely reasoned. Take the simplest operation considered in statics—the use of a crowbar in raising a heavy stone, and we shall find, as Thomson and Tait have pointed out, that we neglect far more than we observe.[376] If we suppose the bar to be quite rigid, the fulcrum and stone perfectly hard, and the points of contact real points, we may give the true relation of the forces. But in reality the bar must bend, and the extension and compression of different parts involve us in difficulties. Even if the bar be homogeneous in all its parts, there is no mathematical theory capable of determining with accuracy all that goes on; if, as is infinitely more probable, the bar is not homogeneous, the complete solution will be immensely more complicated, but hardly more hopeless. No sooner had we determined the change of form according to simple mechanical principles, than we should discover the interference of thermodynamic principles. Compression produces heat and extension cold, and thus the conditions of the problem are modified throughout. In attempting a fourth approximation we should have to allow for the conduction of heat from one part of the bar to another. All these effects are utterly inappreciable in a practical point of view, if the bar be a good stout one; but in a theoretical point of view they entirely prevent our saying that we have solved a natural problem. The faculties of the human mind, even when aided by the wonderful powers of abbreviation conferred by analytical methods, are utterly unable to cope with the complications of any real problem. And had we exhausted all the known phenomena of a mechanical problem, how can we tell that hidden phenomena, as yet undetected, do not intervene in the commonest actions? It is plain that no phenomenon comes within the sphere of our senses unless it possesses a momentum capable of irritating the appropriate nerves. There may then be worlds of phenomena too slight to rise within the scope of our consciousness.
All the instruments with which we perform our measurements are faulty. We assume that a plumb-line gives a vertical line; but this is never true in an absolute sense, owing to the attraction of mountains and other inequalities in the surface of the earth. In an accurate trigonometrical survey, the divergencies of the plumb-line must be approximately determined and allowed for. We assume a surface of mercury to be a perfect plane, but even in the breadth of 5 inches there is a calculable divergence from a true plane of about one ten-millionth part of an inch; and this surface further diverges from true horizontality as the plumb-line does from true verticality. That most perfect instrument, the pendulum, is not theoretically perfect, except for infinitely small arcs of vibration, and the delicate experiments performed with the torsion balance proceed on the assumption that the force of torsion of a wire is proportional to the angle of torsion, which again is only true for infinitely small angles.
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.