trical construction, the problem to which Kepler was thus led, namely, "to divide the area of a semicircle in a given ratio, by a line drawn from any point of the diameter." This is still termed "Kepler's Problem," and is, in fact, incapable of exact geometrical solution. As, however, the calculation can be performed, and, indeed, was performed by Kepler himself, with a sufficient degree of accuracy to show that the elliptical hypothesis is true, the insolubility of this problem is a mere mathematical difficulty in the deductive process, to which Kepler's inductions gave rise.

Of Kepler's physical reasonings we shall speak more at length on another occasion. His numerous and fanciful hypotheses had discharged their office, when they had suggested to him his many lines of laborious calculation, and encouraged him under the exertions and disappointments to which these led. The result of this work was the formal laws of the motion of Mars, established by a clear induction, since they represented, with sufficient accuracy, the best observations. And we may allow that Kepler was entitled to the praise which he claims in the motto on his first leaf. Ramus had said that if any one would construct an astronomy without hypothesis he would be ready to resign to him his professorship in the University of Paris. Kepler quotes this passage, and adds, "it is well, Ramus, that you have run from this pledge, by

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quitting life and your professorships; if you held it still, I should, with justice, claim it." This was not saying too much, since he had entirely overturned the hypothesis of eccentrics and epicycles, and had obtained a theory which was a mere representation of the motions and distances as they were observed.

13 Ramus perished in the Massacre of St. Bartholomew.

CHAPTER V.

SEQUEL TO THE EPOCH OF KEPLER. RECEPTION, VERIFICATION, AND EXTENSION OF THE ELLIPTICAL THEORY.

Sect. 1.-Application of the Elliptical Theory to the Planets.

THE

strong probability, This was made in

HE extension of Kepler's discoveries concerning the orbit of Mars to the other planets, obviously offered itself as a and was confirmed by trial. the first place upon the orbit of Mercury; which planet, in consequence of the largeness of its eccentricity, exhibits more clearly than the others the circumstances of the elliptical motion. These and various other supplementary portions of the views to which Kepler's discoveries had led, appeared in the latter part of his Epitome Astronomic Copernicana, published in 1622.

The real verification of the new doctrine concerning the orbits and motions of the heavenly bodies was, of course, to be found in the construction of tables of those motions, and in the continued comparison of such tables with observation. Kepler's discoveries had been founded, as we have

seen, principally on Tycho's observations. Longomontanus (so called as being a native of Langberg in Denmark,) published in 1621 in his Astronomia Danica, tables founded upon the theories as well as the observations of his countryman. Kepler1 in 1627 published his tables of the planets, which he called Rudolphine Tables, the result and application of his own theory. In 1633, Lansberg, a Belgian, published also Tabula Perpetuæ, a work which was ushered into the world with considerable pomp and pretension, and in which the author cavils very keenly at Kepler and Brahe. We may judge of the impression made upon the astronomical world in general by these rival works, from the account which our countryman Jeremy Horrox has given of their effect on him. He had been seduced by the magnificent promises of Lansberg, and the praises of his admirers, which are prefixed to the work, and was persuaded that the common opinion which preferred Tycho and Kepler to him was a prejudice. In 1636, however, he became acquainted with Crabtree, another young astronomer, who lived in the same part of Lancashire. By him Horrox was warned that Lansberg was not to be depended on; that his hypotheses were vicious, and his observations falsified or forced into agreement with his theories. He then read the works and adopted the opinions of Kepler; and after some hesitation which he felt at the thought of attacking the object of his Rheticus, Narratio, p. 98.

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former idolatry, he wrote a dissertation on the points of difference between them. It appears that, at one time, he intended to have offered himself as the umpire who was to adjudge the prize of excellence among the three rival theories of Longomontanus, Kepler and Lansberg; and, in allusion to the story of ancient mythology, his work was to have been called Paris Astronomicus; we easily see that he would have given the golden apple to the Keplerian goddess. Succeeding observations confirmed his judgment and the Rudolphine Tables, thus published seventy-six years after the Prutenic, which were founded on the doctrines of Copernicus, were for a long time those universally used.

Sect. 2.—Application of the Elliptical Theory to the Moon.

THE reduction of the moon's motions to rule was a harder task than the formation of planetary tables, if accuracy was required; for the moon's motion is affected by an incredible number of different and complex inequalities, which, till their law is detected, appear to defy all theory. Still, however, progress was made in this work. The most important advances were due to Tycho Brahe. In addition to the first and second inequalities of the moon (the Equation of the Center, known very early, and the Erection which Ptolemy had discovered), Tycho proved that there was another inequality, which he