Newton did not stop with showing the manner of the earth's attraction upon bodies on or near its surface; he proved that the earth attracted the moon also, and thus retained it in its orbit. To understand this we must notice another fact concerning the manner in which gravitation acts. Its force varies with distance. Experiment followed by mathematical demonstration, has proved that the variation of the attraction is inversely proportional to the square of the distance. This simply means that if the distance between the two bodies concerned is doubled, the force of attraction will be diminished four times, 4 being the square of 2; and that if the distance is halved, the force will be increased fourfold. Increase the distance three times, and the force diminishes nine times; diminish the distance three times, and the force increases nine times, because 9 is the square of 3, and, as we have said, the force varies inversely, or contrarily, to the change of distance. Knowing this, Newton computed what the force of the earth's attraction must be on the moon, and he found that it was just sufficient to keep the latter moving round and round the earth. But why does not the moon fall directly to the earth? Because the moon had originally another motion across the direction of the earth's attraction. How it got this motion is a question into which we cannot here enter, but, if it were not attracted by the earth (or by the sun), the moon would travel in a straight line through space, like a stone escaping from a sling. The force of the attraction is just sufficient to make the moon move in an orbital path about the earth. [Illustration: _Fig. 5. How the Earth Controls the Moon._ Let C be the centre of the earth and M that of the moon. Suppose the moon to be moving in a straight line at such a velocity that it will, if not interfered with, go to A in one day. Now suppose the attraction of the earth to act upon it. That attraction will draw it to M′. Again suppose that at M′ the moon were suddenly released from the earth's attraction; it would then shoot straight ahead to B in the course of the next day. But, in fact, the earth's attraction acts continually, and in the second day the moon is drawn to M″. In other words the moon is all the time falling away from the straight line that it would pursue but for the earth's attraction, and yet it does not get nearer the earth but simply travels in an endless curve round it. ]