If both pins were driven into the same hole, what kind of an ellipse would you get?
The Second Law was worked out by Kepler as his answer to a problem suggested by the first law. In [Fig. 17] it is apparent from a mere inspection of the orbit of Mercury that this planet travels much faster on one side of its orbit than on the other, the distance covered in ten days between the numbers 10 and 20 being more than fifty per cent greater than that between 50 and 60. The same difference is found, though usually in less degree, for every other planet, and Kepler's problem was to discover a means by which to mark upon the orbit the figures showing the positions of the planet at the end of equal intervals of time. His solution of this problem, contained in the second law, asserts that if we draw radii vectors from the sun to each of the marked points taken at equal time intervals around the orbit, then the area of the sector formed by two adjacent radii vectores and the arc included between them is equal to the area of each and every other such sector, the short radii vectores being spread apart so as to include a long arc between them while the long radii vectores have a short arc. In Kepler's form of stating the law the radius vector is supposed to travel with the planet and in each day to sweep over the same fractional part of the total area of the orbit. The spacing of the numbers in [Fig. 17] was done by means of this law.
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,
a3 ÷ T2 = C
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 = T2/3, 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
The first law of motion.—"Every body continues in its state of rest or of uniform motion in a straight line except in so far as it may be compelled by force to change that state." A gust of wind, a stone, a careless movement of the rider may turn the bicycle to the right or the left, but unless some disturbing force is applied it will go straight ahead, and if all resistance to its motion could be removed it would go always at the speed given it by the last power applied, swerving neither to the one hand nor the other.