The origin of this famous inequality may be best understood by reference to the mode in which the disturbing forces operate.  Let P Q R, P’ Q’ R’ represent the orbits of Jupiter and Saturn, and let us suppose, for the sake of illustration, that they are both situate in the same plane.  Let the planets be in conjunction at P, P’, and let them both be revolving around the sun S, in the direction represented by the arrows.  Assuming that the mean motion of Jupiter is to that of Saturn exactly in the proportion of five to two, it follows that when Jupiter has completed one revolution, Saturn will have advanced through two fifths of a revolution.  Similarly, when Jupiter has completed a revolution and a half, Saturn will have effected three fifths of a revolution.  Hence when Jupiter arrives at T, Saturn will be a little in advance of T’.  Let us suppose that the two planets come again into conjunction at Q, Q’.  It is plain that while Jupiter has completed one revolution, and, advanced through the angle P S Q (measured in the direction of the arrow), Saturn has simply described around S the angle P’ S’ Q’.  Hence the excess of the angle described around S, by Jupiter, over the angle similarly described by Saturn, will amount to one complete revolution, or, 360 deg..  But since the mean motions of the two planets are in the proportion of five to two, the angles described by them around S in any given time will be in the same proportion, and therefore the excess of the angle described by Jupiter over that described by Saturn will be to the angle described by Saturn in the proportion of three to two.  But we have just found that the excess of these two angles in the present case amounts to 360 deg., and the angle described by Saturn is represented by P’ S’ Q’; consequently 360 deg. is to the angle P’ S’ Q’ in the proportion of three to two, in other words P’ S’ Q’ is equal to two thirds of the circumference or 240 deg..  In the same way it may be shown that the two planets will come into conjunction again at R, when Saturn has described another arc of 240 deg..  Finally, when Saturn has advanced through a third arc of 240 deg., the two planets will come into conjunction at P, P’, the points whence they originally set out; and the two succeeding conjunctions will also manifestly occur at Q, Q’ and R, R’.  Thus we see, that the conjunctions will always occur in three given points of the orbit of each planet situate at angular distances of 120 deg. from each other.  It is also obvious, that during the interval which elapses between the occurrence of two conjunctions in the same points of the orbits, and which includes three synodic revolutions of the planets, Jupiter will have accomplished five revolutions around the sun, and Saturn will have accomplished two revolutions.  Now if the orbits of both planets were perfectly circular, the retarding and accelerating effects of the disturbing force of either planet would neutralize each other in the course of a synodic revolution, and therefore