Fig. 2. million of miles toward the sun, feeling its attraction increase with every mile until it reaches B; then it is retarded in its speed, by the same attraction, as it climbs back its quarter of a million of miles away from the sun, in defiance of its power, to C. All the while the invisible elastic force of the earth is unweariedly maintained; and though the moon's distances vary over a range of 31,355 miles, the moon is always in a determinable place. A simple revolution of one world about another in a circular orbit would be a problem of easy solution. It would always be at the same distance from its centre, and going with the same velocity. But there are over sixty causes that interfere with such a simple orbit in the case of the moon, all of which causes and their disturbances must be considered in calculating such a simple matter as an eclipse, or predicting the moon's place as the sailors guide. One of the most puzzling of the irregularities of our night-wandering orb has just been explained by Professor Hansen, of Gotha, as a curious result of the attraction of Venus.
Take a single instance of the perturbations of Jupiter and Saturn which can be rendered evident. The times of orbital revolution of Saturn and Jupiter are nearly as five to two. Suppose the orbits of
Fig. 3.—Changes of orbit by mutual attraction. the planets to be, as in Fig. 3, both ellipses, but not necessarily equally distant in all parts. The planets are as near as possible at 1, 1. Drawn toward each other by mutual attraction, Jupiter's orbit bends outward, and Saturn's becomes more nearly straight, as shown by the dotted lines. A partial correction of this difficulty immediately follows. As Jupiter moves on ahead of Saturn it is held back—retarded in its orbit by that body; and Saturn is hastened in its orbit by the attraction of Jupiter. Now greater speed means a straighter orbit. A rifle-ball flies nearer in a straight line than a thrown stone. A greater velocity given to a whirled ball pulls the elastic cord far enough to give the ball a larger orbit. Hence, being hastened, Saturn stretches out nearer its proper orbit, and, retarded, Jupiter approaches the smaller curve that is its true orbit.
But if they were always to meet at this point, as they would if Jupiter made two revolutions to Saturn's one, it would be disastrous. In reality, when Saturn has gone around two-thirds of its orbit to 2, Jupiter will have gone once and two-thirds around and overtaken Saturn; and they will be near again, be drawn together, hastened, and retarded, as before; their next conjunction would be at 3, 3, etc.
Now, if they always made their conjunction at points equally distant, or at thirds of their orbits, it would cause a series of increasing deviations; for Jupiter would be constantly swelling his orbit at three points, and Saturn increasingly contracting his orbit at the same points. Disaster would be easily foretold. But as their times of orbital revolutions are not exactly in the ratio of five and two, their points of conjunction slowly travel around the orbit, till, in a period of nine hundred years, the starting-point is again reached, and the perturbations have mutually corrected one another.
For example, the total attractive effect of one planet on the other for 450 years is to quicken its speed. The effect for the next 450 years is to retard. The place of Saturn, when all the retardations have accumulated for 450 years, is one degree behind what it is computed if they are not considered; and 450 years later it will be one degree before its computed place—a perturbation of two degrees. When a bullet is a little heavier or ragged on one side, it will constantly swerve in that direction. The spiral groove in the rifle, of one turn in forty-five feet, turns the disturbing weight or raggedness from side to side—makes one error correct another, and so the ball flies straight to the bull's-eye. So the place of Jupiter and Saturn, though further complicated by four moons in the case of Jupiter, and eight in the case of Saturn, and also by perturbations caused by other planets, can be calculated with exceeding nicety.
The difficulties would be greatly increased if the orbits of Saturn and Jupiter, instead of being 400,000,000 miles apart, were interlaced. Yet there are the orbits of one hundred and ninety-two asteroids so interlaced that, if they were made of wire, no one could be lifted without raising the whole net-work of them. Nevertheless, all these swift chariots of the sky race along the course of their intermingling tracks as securely as if they were each guided by an intelligent mind. They are guided by an intelligent mind and an almighty arm.
Still more complicated is the question of the mutual attractions of all the planets. Lagrange has been able to show, by a mathematical genius that seems little short of omniscience in his single department of knowledge, that there is a discovered system of oscillations, affecting the entire planetary system, the periods of which are immensely long. The number of these oscillations is equal to that of all the planets, and their periods range from 50,000 to 2,000,000 years,