The only difference in the apparent motions of the inferior planet is that resulting from the position of this planet with respect to the sun. When P, is viewed from E 4 (fig. JO), it is seen directly opposite the sun ; but E, viewed from Pj lies in the same direction as the sun, and is there fore invisible. This corresponds with the observed fact that Venus and Mercury are in conjunction with the sun, not only in the middle of their advaacing arcs, but also in
the middle of their arc of retrogradation.But although the Copernican theory explains the general features of planetary motion, it could not, as originally advanced, explain those features which had rendered neces sary the eccentrics and the subordinate epicycles of the Ptolemaic system. It was known to Copernicus that the earth does not move uniformly in a circle around the sun as centre, but on an eccentric path with varying velocity. He might, therefore, reasonably assume that the other plant ts have paths similarly eccentric, and move with vary ing velocities. But he thought it necessary to explain the planetary motions by uniform motion in circles, using such contrivances to save appearances as the Ptolemaic system had rendered familiar to astronomers. Suppose, for example, that S (fig. 21) is the sun, A the place of a planet when at its greatest distance from the sun, and A its place when nearest to the sun, C being the bisection of AA ; draw a circle aba b with centre C, and any radius less than CA, and with a as centre draw circle AKK ; then if a point revolve round the circle AKK in that direction in the same time that the centre of this circle travels round the circle aba b in that direction, the point will trace out an ellipse, having C as centre. This is easily proved. For let Ca=11, aA=r, and put CA=R + r=a and CK=E, - r=b=CB. When the moving point is at P, let the centre of the small circle be at p. The angular velocities being equal, pC and p~P are inclined to aC at the same angle ; let this be a. Then the co-ordinates of P parallel to CA and C6 are—
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x=(R + r) cos. a=a COS. a y=(II - r) sin. a=I sin. a Whence Wx~ + a-y-=o 2 6 2 , the equation to an ellipse having C as centre, CA and CB as semi-axes. S will not be the focus of this ellipse, unless SB=CA ; and even then the velocities will not be those observed of a planet re volving around S in the elliptic orbit ABA B for the time from A to B will be one-fourth of the period, whereas in the case of a planet the time from A to B bears to the period the ratio of the area ASB to the area of the ellipse, a ratio exceeding one-fourth. Nevertheless, observed appearances were to some degree explained by the motion illustrated in fig. 21, seeing that at A, or when farthest from the centre S, the tracing point moves most slowly, having there the difference of the velocities due to the two circular motions ; while at A the point moves most quickly, having there the sum of these velocities. And when Copernicus advanced his theory, observations had not been made with sufficient exactness to prove the insufficiency of such an explanation in any case save that of the moon s motion round the earth.|1}}
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Fig. 21.
Tycho Brahe, however, having completed a series of observations of Mars, the nearest planet moving on a manifestly eccentric orbit, Kepler tested the theory of Laws of Copernicus in order to ascertain whether any ellipse, described as ABA B in lig. 21, could account for the observed positions of the planet. It will be seen that, while the points S, A, and A in his inquiry were fixed, and therefore C also fixed, the point a might be taken nearer or further from A within certain limits, these limits being determined by the observed fact, that when near B and B the planet was not nearer to C by a distance B6 exceeding a certain moderate amount, such as the probable error of Tycho Brahe s observations permitted Kepler to assume.
It was after trying nineteen such arrangements, and rejecting them one after the other as he found them dis proved by Tycho Brahe s observations, that Kepler was led at last to abandon the attempt to explain the motions of Mars by combining circular uniform motions. Passing to the ellipse, as the curve which Mars appeared to follow, and testing various empiric laws of motion in an elliptic orbit, he at length lighted upon the actual relation, pre sented in his first two laws as true for all the planets, though actually proved only in the case of Mars. The laws are these—
1. Every planet moves in an elliptical orbit, in one focus of which the sun is situated.
2. The line drawn from the sun to a planet, or the radius-vector of the planet, sweeps over equal areas in equal times.
The second law may be thus illustrated. Let ABA B (fig. 22) be the elliptic path of a planet about the sun S, in. the focus of the ellipse whose axes are ACSA and BCB . Let P be the period in which the planet performs the com plete circuit of its orbit, and let T be the time occupied by the planet in traversing any arc pp of its orbit. Then joining Sp, Sp,
T : P : : sectorial area pSp : area of the ellipse ABA B.
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Fig. 22.
