Page:Popular Astronomy - Airy - 1881.djvu/240

226 round it is inversely proportional to the squares of their distances.

But, though it is thus established that the attraction of the central body on the different bodies follows that law, is it true that its attraction on the same body alters in the inverse proportion of the square of the distances, when the distance of that body is altered? It is quite certain. But so difficult are the mathematical operations by which this is proved, that I can do little more than refer you (as I have done once before) to the results. The following, however, are the principal steps.

The planets do not revolve in circles but in ellipses, (see page 126,) and therefore, the distance of each planet from the sun undergoes considerable alteration. Kepler's second law of planetary motion was this: that if we draw a line from the sun to a planet, that line passes over equal areas in every successive hour; that area not being the same for different planets, but being constantly the same for the same planet; or, which is the same thing, it describes areas proportional to the times. Now, if we assume the first and second laws of motion to be true, we find that this equal description of areas compels us to admit that the planet is attracted towards the sun; but it does not give us any information as to the law of the attractive force. But the circumstance that the planets move in ellipses, with the sun in one focus of each ellipse, settles this question. It has already been proved that the attractive force must be directed to the sun, that is, to the focus of each ellipse; and then it is proved by a mathematical investigation that if a planet moves in an ellipse, and if the force is directed to the focus of the ellipse, that force in different parts of the orbit must be inversely as the square of the