Fig. 18.—Epicycle and deferent.
33. Kepler's laws.—Two generations later came Kepler with his three famous laws of planetary motion:
I. Every planet moves in an ellipse which has the sun at one of its foci.
II. The radius vector of each planet moves over equal areas in equal times.
III. The squares of the periodic times of the planets are proportional to the cubes of their mean distances from the sun.
These laws are the crowning glory, not only of Kepler's career, but of all astronomical discovery from the beginning up to his time, and they well deserve careful study and explanation, although more modern progress has shown that they are only approximately true.
Exercise 17.—Drive two pins into a smooth board an inch apart and fasten to them the ends of a string a foot long. Take up the slack of the string with the point of a lead pencil and, keeping the string drawn taut, move the pencil point over the board into every possible position. The curve thus traced will be an ellipse having the pins at the two points which are called its foci.
In the case of the planetary orbits one focus of the ellipse is vacant, and, in accordance with the first law, the center of the sun is at the other focus. In [Fig. 17] the dot, inside the orbit of Mercury, which is marked a, shows the position of the vacant focus of the orbit of Mars, and the dot b is the vacant focus of Mercury's orbit. The orbits of Venus and the earth are so nearly circular that their vacant foci lie very close to the sun and are not marked in the figure. The line drawn from the sun to any point of the orbit (the string from pin to pencil point) is a radius vector. The point midway between the pins is the center of the ellipse, and the distance of either pin from the center measures the eccentricity of the ellipse.
Draw several ellipses with the same length of string, but with the pins at different distances apart, and note that the greater the eccentricity the flatter is the ellipse, but that all of them have the same length.