little doubt that had it not been for his self-imposed exile, his observations would not have come into Kepler's hands, and their great value might have been lost. In any case it was at Uraniborg that the mass of observations was produced upon which the fame of Tycho Brahe rests. His own discoveries, though in themselves the most important made in astronomy for many centuries, are far less valuable than those for which his observations furnished the material. He discovered the third and fourth inequalities of the moon in longitude, called respectively the variation and the annual equation, also the variability of the motion of the moon's nodes and the inclination of its orbit to the ecliptic. He obtained an improved value of the constant of precession, and did good service by rejecting the idea that it was variable, an idea which, under the name of trepidation, had for many centuries been accepted. He discovered the effect of refraction, though only approximately its amount, and determined improved values of many other astronomical constants, but singularly enough made no determination of the distance of the sun, adopting instead the ancient and erroneous value given by Hipparchus. His magnificent Observatory of Uraniborg, the finest building for astronomical purposes that the world had hitherto seen, was allowed to fall into decay, and scarcely more than mere indications of the site may now be seen. CHAPTER IV. KEPLER JOINS TYCHO. The association of Kepler with Tycho was one of the most important landmarks in the history of astronomy. The younger man hoped, by the aid of Tycho's planetary observations, to obtain better support for some of his fanciful speculative theories, while the latter, who had certainly not gained in prestige by leaving Denmark, was in great need of a competent staff of assistants. Of the two it would almost seem that Tycho thought himself the greater gainer, for in spite of his reputation for brusqueness and want of consideration, he not only made light of Kepler's apology in the matter of Reymers, but treated him with uniform kindness in the face of great rudeness and ingratitude. He begged him to come "as a welcome friend," though Kepler, very touchy on the subject of his own astronomical powers, was afraid he might be regarded as simply a subordinate assistant. An arrangement had been suggested by which Kepler should obtain two years' leave of absence from Gratz on full pay, which, because of the higher cost of living in Prague, should be supplemented by the Emperor; but before this could be concluded, Kepler threw up his professorship, and thinking he had thereby also lost the chance of going to Prague, applied to Maestlin and others of his Tübingen friends to make interest for him with the Duke of Wurtemberg and secure the professorship of medicine. Tycho, however, still urged him to come to Prague, promising to do his utmost to secure for him a permanent