For the proper understanding of Kepler's Third Law we must note that the "mean distance" which appears in it is one half of the long diameter of the orbit and that the "periodic time" means the number of days or years required by the planet to make a complete circuit in its orbit. Representing the first of these by _a_ and the second by _T_, we have, as the mathematical equivalent of the law,
where the quotient, _C_, is a number which, as Kepler found, is the same for every planet of the solar system. If we take the mean distance of the earth from the sun as the unit of distance, and the year as the unit of time, we shall find by applying the equation to the earth's motion, _C_ = 1. Applying this value to any other planet we shall find in the same units, _a_ = _T_^{ 2/3}, by means of which we may determine the distance of any planet from the sun when its periodic time, _T_, has been learned from observation.
EXERCISE 18.--Uranus requires 84 years to make a revolution in its orbit. What is its mean distance from the sun? What are the mean distances of Mercury, Venus, and Mars? (See Chapter III for their periodic times.) Would it be possible for two planets at different distances from the sun to move around their orbits in the same time?
A circle is an ellipse in which the two foci have been brought together. Would Kepler's laws hold true for such an orbit?
34. NEWTON'S LAWS OF MOTION.--Kepler studied and described the motion of the planets. Newton, three generations later (1727 A. D.), studied and described the mechanism which controls that motion. To Kepler and his age the heavens were supernatural, while to Newton and his successors they are a part of Nature, governed by the same laws which obtain upon the earth, and we turn to the ordinary things of everyday life as the foundation of celestial mechanics.
Every one who has ridden a bicycle knows that he can coast farther upon a level road if it is smooth than if it is rough; but however smooth and hard the road may be and however fast the wheel may have been started, it is sooner or later stopped by the resistance which the road and the air offer to its motion, and when once stopped or checked it can be started again only by applying fresh power. We have here a familiar illustration of what is called