the earth.  From its extreme distance from the earth, at A, Fig. 2, it rushes with increasing speed nearly a quarter of a 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 [Page 11] of our night-wandering orb has just been explained by Professor Hansen, of Gotha, as a curious result of the attraction of Venus.

[Illustration:  Fig. 3.—­Changes of orbit by mutual attraction.]

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 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 [Page 12] 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.