The argument on which we are to enter is, it must be confessed, somewhat subtle, but its cogency is irresistible. For this argument we are indebted to one of the great founders of the nebular theory. It was given by Kant himself in his famous essay.
We will commence with a preliminary point which relates to elementary mechanics. It may, however, help to clear up a difficult point in our argument if I now state some well-known principles in a manner specially adapted for our present purpose.
Let us think of two bodies, A and S, and, for the sake of clearness, we may suppose that each of these bodies is a perfect sphere. We might think of them as billiard balls, or balls of stone, or balls of iron. We shall, however, suppose them to be formed of material which is perfectly rigid. They may be of any size whatever, large or small, equal or unequal. One of them may be no greater than a grain of mustard-seed, and the other may be as large as the moon or the earth or the sun. Let us further suppose that there is no other body in the universe by which the mutual attraction of the two bodies we are considering can be interfered with. If these two bodies are abandoned to their mutual attraction, let us now see what the laws of mechanics assure us must necessarily happen.
Fig. 45.—A Spiral presented Edgewise (n.g.c. 4631; in Coma Berenices).
(Photographed by Dr. Isaac Roberts, F.R.S.)
Let A and S be simply released from initial positions of absolute rest. In these circumstances, the two points will start off towards each other. The time that must elapse before the two bodies collide will depend upon circumstances. The greater the initial distance between the two balls, their sizes being the same, the longer must be the interval before they come together. The relation between the distance separating the bodies and the time that must elapse before they meet may be illustrated in this way. Suppose that two balls, both starting from rest at a certain distance, should take a year to come together by their mutual attraction, then we know that if the distance of the two balls had been four times as great eight years would have to elapse before the two balls collided. If the distances were nine times as great then twenty-seven years would elapse before the balls collided, and generally the squares of the times would increase as the cubes of the distances. In such statements we are supposing that the radii of the balls are inconsiderable in comparison with the distances apart from which they are started. The time occupied in the journey must also generally depend on the masses of the two bodies, or, to speak more precisely, on the sum of the masses of the two bodies. If the two balls each weighed five hundred tons, then they would take precisely the same time to rush together as would two balls of one ton and nine hundred and ninety-nine tons respectively, provided the distances between the centres of the two balls had been the same in each case. If the united masses of the two bodies amounted to four thousand tons, then they would meet in half the time that would have been required if their united masses were one thousand tons, it being understood that in each case they started with the same initial distance between the centres.
Instead of simply releasing the two bodies A and S so that neither of them shall have any impulse tending to make it swerve from the line directly joining them, let us now suppose that we give one of the bodies. A, a slight push sideways. The question will be somewhat simpler if we think of S as very massive, while A is relatively small. If, for instance, S be as heavy as a cannon-ball, while A is no heavier than a grain of shot, then we may consider that S remains practically at rest during the movement. The small pull which A is able to give will produce no more than an inappreciable effect on S. If the two bodies come together, A will practically do all the moving.
Fig. 46.—The Plane of a Planet’s
Orbit.
We represent the movement in the adjoining figure. If A is started off with an initial velocity in the direction A T, the attraction of S will, however, make itself felt, even though A cannot move directly towards S. The body will not be allowed to travel along A T; it will be forced to swerve by the attraction of S; it will move from P to Q, gradually getting nearer to S. To enter into the details of the movement would require rather more calculation than it would be convenient to give here. Even though S is much more massive than A, we may suppose that the path which A follows is so great that the diameter of the globe S is quite insignificant in comparison with the diameter of the orbit which the smaller body describes. We shall thus regard both A and S as particles, and Kepler’s well-known law, to which we so often refer, tells us that A will revolve around S in that beautiful figure which the mathematician calls an ellipse. For our present purpose we are particularly to observe that the movement is restricted to a plane. The plane in which A moves depends entirely on the direction in which it was first started. The body will always continue to move in the same plane as that in which its motion originally commenced. This plane is determined by the point S and the straight line in which A was originally projected. It is essential for our argument to note that A will never swerve from its plane so long as there are not other forces in action beside those arising from the mutual attractions of A and S. The ordinary perturbations of one body by the action of others need not here concern us.