Fig. 60.
The fact that a body might be projected forward fast enough to keep it from falling into the earth is evident from Fig. 61. AB represents the level surface of the ocean, C a mountain from the summit of which a cannon-ball is supposed to be fired in the direction CE. AD is a line parallel with CE; DB is a line equal to the distance between the two parallel lines AD and CE. This distance is equal to that over which gravity would pull a ball towards the centre of the earth in a minute. No matter, then, with what velocity the ball C is fired, at the end of a minute it will be somewhere on the line AD. Suppose it were fired fast enough to reach the point D in a minute: it would be on the line AD at the end of the minute, but still just as far from the surface of the water as when it started. It will be seen, that, although it has all the while been falling towards the earth, it has all the while kept at exactly the same distance from the surface. The same thing would of course be true during each succeeding minute, till the ball came round to C again, and the ball would continue to revolve in a circle around the earth.
Fig. 61.
50. The Form of a Body's Orbit depends upon the Rate of its Forward Motion.—If the ball C were fired fast enough to reach the line AD beyond the point D, it would be farther from the surface at the end of the second than when it started. Its orbit would no longer be circular, but elliptical. If the speed of projection were gradually augmented, the orbit would become a more and more elongated ellipse. At a certain rate of projection, the orbit would become a parabola; at a still greater rate, a hyperbola.
51. The Moon held in her Orbit by Gravity.—Newton compared the distance mN that the moon is drawn to the earth in a given time, with the distance a body near the surface of the earth would be pulled toward the earth in the same time; and he found that these distances are to each other inversely as the squares of the distances of the two bodies from the centre of the earth. He therefore concluded that the moon is drawn to the earth by gravity, and that the intensity of gravity decreases as the square of the distance increases.
Fig. 62.
52. Any Body whose Orbit is a Conic Section, and which moves according to Kepler's Second Law, is acted upon by a Force varying inversely as the Square of the Distance.—Newton compared the distance which any body, moving in an ellipse, according to Kepler's Second Law, is drawn towards the sun in the same time in different parts of its orbit. He found these distances in all cases to vary inversely as the square of the distance of the planet from the sun. Thus, in Fig. 62, suppose a planet would move from K to B in the same time that it would move from k to b in another part of its orbit. In the first instance the planet would be drawn towards the sun the distance AB, and in the second instance the distance ab. Newton found that AB : ab = (SK)2 : (Sk)2. He also found that the same would be true when the body moved in a parabola or a hyperbola: hence he concluded that every body that moves around the sun in an ellipse, a parabola, or a hyperbola, is moving under the influence of gravity.
[Transcriber's Note: In Newton's equation above, (SK)2 means to group S and K together and square their product. In the original book, instead of using parentheses, there was a vinculum, a horizontal bar, drawn over the S and the K to express the same grouping.]