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.3 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.