Part 9

45. Kepler's Second Law.--Kepler next discovered that a planet's rate of motion in the various parts of its orbit is such that a line drawn from the planet to the sun would always sweep over equal areas in equal times. Thus, in Fig. 57, suppose the planet would move from P to P^1 in the same time that it would move from P^2 to P^3, or from P^4 to P^5; then the dark spaces, which would be swept over by a line joining the sun and the planet, in these equal times, would all be equal.

A line drawn from the sun to a planet is called the radius vector of the planet. The radius vector of a planet is shortest when the planet is nearest the sun, or at perihelion, and longest when the planet is farthest from the sun, or at aphelion: hence, in order to have the areas equal, it follows that a planet must move fastest when at perihelion, and slowest at aphelion.

Kepler's Second Law of planetary motion is usually stated as follows: The radius vector of a planet describes equal areas in equal times in every part of the planet's orbit.

46. Kepler's Third Law.--Kepler finally discovered that the periodic times of the planets bear the following relation to the distances of the planets from the sun: The squares of the periodic times of the planets are to each other as the cubes of their mean distances from the sun. This is known as Kepler's Third Law of planetary motion. By periodic time is meant the time it takes a planet to revolve around the sun.

These three laws of Kepler's are the foundation of modern physical astronomy.

The Newtonian System.

47. Newton's Discovery.--Newton followed Kepler, and by means of his three laws of planetary motion made his own immortal discovery of the law of gravitation. This law is as follows: Every portion of matter in the universe attracts every other portion with a force varying directly as the product of the masses acted upon, and inversely as the square of the distances between them.

48. The Conic Sections.--The conic sections are the figures formed by the various plane sections of a right cone. There are four classes of figures formed by these sections, according to the angle which the plane of the section makes with the axis of the cone.

OPQ, Fig. 58, is a right cone, and ON is its axis. Any section, AB, of this cone, whose plane is perpendicular to the axis of the cone, is a circle.

Any section, CD, of this cone, whose plane is oblique to the axis, but forms with it an angle greater than NOP, is an ellipse. The less the angle which the plane of the section makes with the axis, the more elongated is the ellipse.

Any section, EF, of this cone, whose plane makes with the axis an angle equal to NOP, is a parabola. It will be seen, that, by changing the obliquity of the plane CD to the axis NO, we may pass uninterruptedly from the circle through ellipses of greater and greater elongation to the parabola.

Any section, GH, of this cone, whose plane makes with the axis ON an angle less than NOP, is a hyperbola.

It will be seen from Fig. 59, in which comparative views of the four conic sections are given, that the circle and the ellipse are closed curves, or curves which return into themselves. The parabola and the hyperbola are, on the contrary, open curves, or curves which do not return into themselves.

49. A Revolving Body is continually Falling towards its Centre of Revolution.--In Fig. 60 let M represent the moon, and E the earth around which the moon is revolving in the direction MN. It will be seen that the moon, in moving from M to N, falls towards the earth a distance equal to mN. It is kept from falling into the earth by its orbital motion.

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.

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.

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.

53. The Force that draws the Different Planets to the Sun Varies inversely as the Squares of the Distances of the Planets from the Sun.--Newton compared the distances jK and eF, over which two planets are drawn towards the sun in the same time, and found these distances to vary inversely as the squares of the distances of the planets from the sun: hence he concluded that all the planets are held in their orbits by gravity. He also showed that this would be true of any two bodies that were revolving around the sun's centre, according to Kepler's Third Law.

54. The Copernican System.--The theory of the solar system which originated with Copernicus, and which was developed and completed by Kepler and Newton, is commonly known as the Copernican System. This system is shown in Fig. 64.

II. THE SUN AND PLANETS.

I. THE EARTH.

Form and Size.

55. Form of the Earth.--In ordinary language the term horizon denotes the line that bounds the portion of the earth's surface that is visible at any point.

(1) It is well known that the horizon of a plain presents the form of a circle surrounding the observer. If the latter moves, the circle moves also; but its form remains the same, and is modified only when mountains or other obstacles limit the view. Out at sea, the circular form of the horizon is still more decided, and changes only near the coasts, the outline of which breaks the regularity.

Here, then, we obtain a first notion of the rotundity of the earth, since a sphere is the only body which is presented always to us under the form of a circle, from whatever point on its surface it is viewed.

(2) Moreover, it cannot be maintained that the horizon is the vanishing point of distinct vision, and that it is this which causes the appearance of a circular boundary, because the horizon is enlarged when we mount above the surface of the plain. This will be evident from Fig. 65, in which a mountain is depicted in the middle of a plain, whose uniform curvature is that of a sphere. From the foot of the mountain the spectator will have but a very limited horizon. Let him ascend half way, his visual radius extends, is inclined below the first horizon, and reveals a more extended circular area. At the summit of the mountain the horizon still increases; and, if the atmosphere is pure, the spectator will see numerous objects where from the lower stations the sky alone was visible.

This extension of the horizon would be inexplicable if the earth had the form of an extended plane.