In spite of these embarrassments, and the occupation of his time in the practice of astrology, Kepler found leisure for his favourite pursuits. No adverse circumstances were capable of extinguishing his scientific ardour, and whenever he directed his vigorous mind to the investigation of phenomena, he never failed to obtain interesting and original results. Since the death of Tycho, his attention had been much occupied with the subject of refraction and vision; and, in 1606, he published the result of his researches in a work, entitled "A Supplement to Vitellio, in which the optical part of astronomy is treated, but chiefly on the artificial observation and estimation of diameters, and of the eclipses of the Sun and Moon." Astronomers had long been perplexed with the refraction of the atmosphere, and so little was known of the general subject, as well as of this branch of it, that Tycho believed the refraction of the atmosphere to cease at 45 deg. of altitude. Even at the beginning of the second century, Claudius Ptolemy of Alexandria had unravelled its principal mysteries, and had given in his Optics a theory of astronomical refraction more complete than that of any astronomer before the time of Cassini;[46] but the MSS. had unfortunately been mislaid, and Alhazen and Vitellio and Kepler were obliged to take up the subject from its commencement. Ptolemy had not only determined that the refraction of the atmosphere had gradually increased from the zenith to the horizon, but he had measured with singular accuracy the angles of refraction for water and glass, from a perpendicular incidence to a horizontal one. [46] Cassini was born in 1625, and died in 1712. Kepler treated this branch of science in his own peculiar way, "hunting down," as he expressed it, every hypothesis which his fertile imagination had successively presented to him. In his various attempts to discover the law of refraction, or a measure of it, as varying with the density of the body and the angle of incidence of the light, he was nearer the goal, in his first speculation, than in any of the rest; and he seems to have failed in consequence of his not separating the question as it related to density from the question as it related to incidence. "I did not leave untried," says he, "whether, by assuming a horizontal refraction according to the density of the medium, the rest would correspond to the sines of the distances from a vertical direction, but calculation proved that it was not so: and, indeed, there was no occasion to have tried it, for thus the _refraction would increase according to the same law in all mediums, which is contradicted by experiment_."