pounding of the motion of the observer with the motion of the planet observed rendered the problem very difficult.
Copernicus furnished the key, by showing that the sun and not the earth is the center of the solar system. Tycho Brahe soon followed, and furnished an extensive series of accurate observations that afforded Kepler the material upon which he based his studies that developed those immortal laws defining the forms of the orbits of the planets, the character of their motions, and the relation between the dimensions of their orbits and their periods of revolution.
It remained for Newton to discover the existence of the law of universal gravitation, of which Kepler's laws are an immediate sequence.
Thus the secrets of the motions of the planets were explained. But comets, those erratic visitants of our system, whose advent in olden time filled the mind with universal awe, were still an unfathomed mystery. Suddenly they would blaze out in the sky, and as suddenly pass out of sight, and no astronomer could tell whence they came or whither they went, or the laws which governed their motions.
Newton first showed that comets also were obedient to the attraction of gravitation. He demonstrated this fact by means of the comet of 1680. The orbit of this comet he found not to differ perceptibly from a parabola.
After Newton, Edmund Halley, from a careful study of the comets of 1531, 1607, and 1682, ventured the assertion that these were only different appearances of one and the same body, whose period of revolution was about seventy-five years. Halley, consequently, predicted a reappearance of this comet in 1759. This comet was shown to move in a very elongated ellipse. In accordance with prediction, reappearances of this comet occurred in 1759 and 1835.
Since the time of Newton all the comets which have come to view have been submitted to a careful study. To determine the orbit of any newly discovered member of our system, it is necessary that its direction in space from the earth at three dates, as nearly equidistant as may be, should be determined by observation.
The data for the problem are, then, as follows: the positions of the earth with reference to the sun at three different dates, and the positions of the heavenly body with reference to the earth at the same dates. The unknown elements which describe the character of the orbit and its position in space are as follows:
I. The mean longitude of the body at any convenient epoch.
II. The semi-major axis of the orbit.
III. The eccentricity of the orbit.
IV. The longitude of the perihelion.
V. The longitude of the ascending node.
VI. The inclination between the orbit-plane and the plane of the earth's orbit.
