I. THEORY OF THE SOLAR SYSTEM.
35. Members of the Solar System.—The solar system is composed of the sun, planets, moons, comets, and meteors. Five planets, besides the earth, are readily distinguished by the naked eye, and were known to the ancients: these are Mercury, Venus, Mars, Jupiter, and Saturn. These, with the sun and moon, made up the seven planets of the ancients, from which the seven days of the week were named.
The Ptolemaic System.
36. The Crystalline Spheres.—We have seen that all the heavenly bodies appear to be situated on the surface of the celestial sphere. The ancients assumed that the stars were really fixed on the surface of a crystalline sphere, and that they were carried around the earth daily by the rotation of this sphere. They had, however, learned to distinguish the planets from the stars, and they had come to the conclusion that some of the planets were nearer the earth than others, and that all of them were nearer the earth than the stars are. This led them to imagine that the heavens were composed of a number of crystalline spheres, one above another, each carrying one of the planets, and all revolving around the earth from east to west, but at different rates. This structure of the heavens is shown in section in Fig. 49.
Fig. 49.
37. Cycles and Epicycles.—The ancients had also noticed that, while all the planets move around the heavens from west to east, their motion is not one of uniform advancement. Mercury and Venus appear to oscillate to and fro across the sun, while Jupiter and Saturn appear to oscillate to and fro across a centre which is moving around the earth, so as to describe a series of loops, as shown in Fig. 50.
Fig. 50.
The ancients assumed that the planets moved in exact circles, and, in fact, that all motion in the heavens was circular, the circle being the simplest and most perfect curve. To account for the loops described by the planets, they imagined that each planet revolved in a circle around a centre, which, in turn, revolved in a circle around the earth. The circle described by this centre around the earth they called the cycle, and the circle described by the planet around this centre they called the epicycle.
38. The Eccentric.—The ancients assumed that the planets moved at a uniform rate in describing the epicycle, and also the centre in describing the cycle. They had, however, discovered that the planets advance eastward more rapidly in some parts of their orbits than in others. To account for this they assumed that the cycles described by the centre, around which the planets revolved, were eccentric; that is to say, that the earth was not at the centre of the cycle, but some distance away from it, as shown in Fig. 51. E is the position of the earth, and C is the centre of the cycle. The lines from E are drawn so as to intercept equal arcs of the cycle. It will be seen at once that the angle between any pair of lines is greatest at P, and least at A; so that, were a planet moving at the same rate at P and A, it would seem to be moving much faster at P. The point P of the planet's cycle was called its perigee, and the point A its apogee.
Fig. 51.
As the apparent motion of the planets became more accurately known, it was found necessary to make the system of cycles, epicycles, and eccentrics exceedingly complicated to represent that motion.
The Copernican System.
39. Copernicus.—Copernicus simplified the Ptolemaic system greatly by assuming that the earth and all the planets revolved about the sun as a centre. He, however, still maintained that all motion in the heavens was circular, and hence he could not rid his system entirely of cycles and epicycles.
Tycho Brahe's System.
40. Tycho Brahe.—Tycho Brahe was the greatest of the early astronomical observers. He, however, rejected the system of Copernicus, and adopted one of his own, which was much more complicated. He held that all the planets but the earth revolved around the sun, while the sun and moon revolved around the earth. This system is shown in Fig. 52.
Fig. 52.
Kepler's System.
41. Kepler.—While Tycho Brahe devoted his life to the observation of the planets. Kepler gave his to the study of Tycho's observations, for the purpose of discovering the true laws of planetary motion. He banished the complicated system of cycles, epicycles, and eccentrics forever from the heavens, and discovered the three laws of planetary motion which have rendered his name immortal.
42. The Ellipse.—An ellipse is a closed curve which has two points within it, the sum of whose distances from every point on the curve is the same. These two points are called the foci of the ellipse.
Fig. 53.
One method of describing an ellipse is shown in Fig. 53. Two tacks, F and F', are stuck into a piece of paper, and to these are fastened the two ends of a string which is longer than the distance between the tacks. A pencil is then placed against the string, and carried around, as shown in the figure. The curve described by the pencil is an ellipse. The two points F and F' are the foci of the ellipse: the sum of the distances of these two points from every point on the curve is equal to the length of the string. When half of the ellipse has been described, the pencil must be held against the other side of the string in the same way, and carried around as before.
The point O, half way between F and F', is called the centre of the ellipse; AA' is the major axis of the ellipse, and CD is the minor axis.
43. The Eccentricity of the Ellipse.—The ratio of the distance between the two foci to the major axis of the ellipse is called the eccentricity of the ellipse. The greater the distance between the two foci, compared with the major axis of the ellipse, the greater is the eccentricity of the ellipse; and the less the distance between the foci, compared with the length of the major axis, the less the eccentricity of the ellipse. The ellipse of Fig. 54 has an eccentricity of 1/8. This ellipse scarcely differs in appearance from a circle. The ellipse of Fig. 55 has an eccentricity of 1/2, and that of Fig. 56 an eccentricity of 7/8.
Fig. 54.
Fig. 55.
Fig. 56.
44. Kepler's First Law.—Kepler first discovered that all the planets move from west to east in ellipses which have the sun as a common focus. This law of planetary motion is known as Kepler's First Law. The planets appear to describe loops, because we view them from a moving point.
The ellipses described by the planets differ in eccentricity; and, though they all have one focus at the sun, their major axes have different directions. The eccentricity of the planetary orbits is comparatively small. The ellipse of Fig. 54 has seven times the eccentricity of the earth's orbit, and twice that of the orbit of any of the larger planets except Mercury; and its eccentricity is more than half of that of the orbit of Mercury. Owing to their small eccentricity, the orbits of the planets are usually represented by circles in astronomical diagrams.
Fig. 57.
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 P1 in the same time that it would move from P2 to P3, or from P4 to P5; 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.
Fig. 58.
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.
Fig. 59.
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.
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.]
Fig. 63.
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.
Fig. 64.