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 ASTRONOMY 45 from the sun. He therefore adopted a modifi- cation of a system once held by the Egyptians, regarding the earth as the centre around which the sun revolves, while the j>laiiets revolve around the sun as a subordinate centre. Al- though this was a retrogression, astronomy owes a debt of gratitude to Tycho Brahe for the observations by which lie endeavored to put the Copernican theory to the test. His observations of Mars, in particular, enabled Kepler to remove for ever from astronomy the cycles and epicycles, centrics and eccentrics of the old systems. Endeavoring to explain the motions of Mars on the Copernican theory, Kepler found himself baffled so long as he ad- hered to circular and uniform motions so com- bined as to produce epicyclic paths. He was thus led to try whether the ellipse would bet- ter explain the movements of Mars. After long and patient study ho was able in 1609 to establish his first two laws, and nine years later his third law. The three laws are as follows: 1. Every planet describes an ellipse about the sun, this orb occupying one focus of each such ellipse. 2. If a line be supposed continually drawn from the sun to any given planet, this line will sweep over equal areas in equal times. 3. The squares of the periodic times of the planets are proportional to the cubes of their mean distances. In the mean time the telescope had been invented, and when less than one year had passed after the publication of the first two laws of Kepler, Galileo had made a series of observations tending to illustrate if not even to demonstrate the truth of the Co- pernican system. In particular his discovery of the satellites of Jupiter, and the recognition of the motions of these orbs around their pri- mary, was felt even by the enemies of the new theory to be strikingly in its favor. Here was a system in which the motions of the earth and planets around the sun seemed pictured in miniature. The discovery of the phases of Ve- nus was also regarded as a serious blow to the Ptolemaic system. The invention of the tele- scope supplied also the means of determining the places and therefore the motions of the celestial bodies with a degree of accuracy which had hitherto been unattainable. He- velius indeed endeavored to make a stand against the innovation, adhering until the end of his career to the methods used by the an- cients. But gradually the telescope prevailed, and the way was thus prepared for the re- searches of Newton, whose discovery of the law of gravitation would never have been ad- mitted but for the evidence in its favor attained by means of telescopic observations. In par- ticular, the measurement of the earth's dimen- sions with the requisite accuracy could not have been accomplished without telescopic ob- servations of star places ; and Newton would have been unable to show that the moon is re- tained in her orbit by the same force which draws objects to the earth's surface, had not accurate measurements of the earth been ob- tained by Picard. We know in fact that New- ton was led by erroneous ideas of the earth's dimensions to abandon the theory of gravita- tion for nearly 20 years. Returning to his re- searches in 1680, when news of Picard's results had reached him, Newton was able to establish the theory of gravitation on a firm and stable basis. He showed that the moon is drawn to the earth by terrestrial gravity, diminished at the moon's distance in the same degree that the square of that distance exceeds the dis- tance of points on the earth's surface from the earth's centre. He proved that when the force of attraction diminishes according to the law of the inverse square, the attracted body will obey all the laws of Kepler in its motions around the attracting orb. Then he extended his inquiries to the mutual perturbations of bodies so moving. Taking the moon as an in- stance of the effects of perturbation, he showed how several peculiarities in her motions which had hitherto seemed inexplicable are caused by the sun's perturbing action on the moon, that is, by the excess or defect of his action on the moon in different parts of her orbit, as com- pared with his action on the earth. Pursuing his researches, he showed how the precession of the equinoxes can be accounted for by the law of gravitation ; he formed and discussed two theories of the tides ; he solved the prob- lem presented by the oblateness of the earth's figure. Half a century passed before any at- tempts were made to extend the reasoning of the Principia, or to develop the views of its author. During this half century British mathematicians were chiefly engaged in de- fending, continental mathematicians in attack- ing, the principle of universal gravitation. But in 1745 Euler and Clairaut began to ap- ply the new methods of mathematical anal- ysis to the problems discussed by Newton. Clairaut succeeded in explaining the lunar evection, which had foiled Newton ; and this success encouraged continental astronomers to devote their powers to the investigation of the problems presented by the celestial motions. They mastered one after another the difficulties of the lunar and planetary perturbations. The analytical researches of Lagrange and Laplace, and in particular the discovery (independently made by both) of the great laws on which the stability of the planetary system depends, are only inferior to the discovery of the law of gravitation itself in interest and importance. It would be difficult to say which of these two geometers displayed the greater powers of analytical research. If the genius of Lagrange was the more profound, yet Laplace's labors led to more important practical results, and in discovering the real interpretation of the " long inequality " of Jupiter and Saturn he mastered a problem which had foiled his great rival. Yet another noble achievement of Laplace's must be mentioned hJs interpretation of the secu- lar acceleration of the moon's mean motion. In recent times it has been shown indeed by