Y OF 
ee ae 8 how Laplace demonstrated 
that the solar system can experience only small periodic 
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 com- 
A. 
pleted 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 Ps 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°. 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°, and the angle de- 
scribed by Saturn is represented by P/ s/ Q/; consequently 360° 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°. In the same way it 
may be shown that the two planets will come into conjunction again 
at R, when Saturn has described another are of 240°. Finally, when 
Saturn has advanced through a third are of 240°, the two planets will 
