92 FROM NEBULA TO NEBULA 



of the earth, the surface will fall away from this line of sight at 

 the rate of about eight inches in the first mile, twenty-four inches 

 more in the second mile, and so on. In five miles the fall will 

 amount to sixteen feet. In ten miles, in addition to this sixteen 

 feet, three times that amount will be added, and so on, the law be- 

 ing the same with that of a falling body. Now, let A C be a high 

 steep mountain, from the summit of which a cannon-ball is fired 

 in the horizontal direction C E. The greater the velocity with 

 which the shot is fired, the farther it will go before it reaches the 

 ground. Suppose, at length, that we should fire it with a velocity 

 of five miles a second, and that it should meet with no resistance 

 from the air. Suppose e to be the point on the line five miles 

 from C. Since it would reach this point in one second, it fol- 

 lows, from the law of falling bodies just cited, that it will have 

 dropped sixteen feet below e. But we have just seen that the 

 earth itself curves away sixteen feet at this distance. Hence, 

 the shot is no nearer the earth than when it was fired. During 

 the next second, while the ball would go to E, it would fall forty- 

 eight feet more, or sixty-four feet in all. But here, again, the 

 earth has still been rounding off, so the distance D B is sixty- four 

 feet. Hence, the ball is still no nearer the earth than when it was 

 fired, although it has been dropping away from the line in which 

 it was fired exactly like a falling body. Moreover, meeting with 

 no resistance, it is still going on with undiminished velocity ; and, 

 just as it has been falling for two seconds without getting any 

 nearer the earth, so it can get no nearer in the third second, nor 

 in the fourth, nor in any subsequent second ; but the earth will 

 constantly curve away as fast as the ball can drop. Thus the 

 latter will pass clear round the earth, and come back to the first 

 point C, from which it started, in the direction of the arrow, with- 

 out any loss of velocity. The time of revolution will be about an 

 hour and twenty- four minutes, and the ball will thus keep on re- 

 volving round the earth in this space of time. In other words, 

 the ball will be a satellite of the earth, just like the moon, only 

 much nearer, and revolving much faster. 



Our next step is to extend gravitation to other bodies than 

 the earth. The planets move around the sun as the moon does 

 around the earth, and must, therefore, be acted on by a force di- 

 rected towards the sun. This force can be no other than the 

 gravitation of the sun itself. A very simple calculation from 

 Kepler's third law shows that the force with which each planet 

 thus gravitates towards the sun is inversely as the square of the 

 mean distance of the planet. 



Only one more step is necessary. What sort of an orbit will 

 a planet describe if acted on by a force directed towards the sun, 

 and inversely as the square of the distance? A very simple 

 demonstration will show that, no matter what the law of force, if 



