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KEP making a diagram in his lecture-room, he observed the relation between the radius of a circle inscribed in a triangle and that of a circle inscribed round it (as one to two), which appeared to him identical with the orbits of Jupiter and Saturn. He used the triangle of Jupiter and Saturn because "they are the first planets." He next tried the second distance between Jupiter and Mars with a square, the third with a pentagon, and the fourth with a hexagon, but this hypothesis was equally unsuccessful. Taking up the idea that solid bodies ought to be used for solid orbits, he was led to the following conceit:—"The earth is the circle, the measurement of all. Round it describe a dodecahedron; the circle including this will be Mars. Round Mars describe a tetrahedron; the circle including this will be Jupiter. Describe a cube round Jupiter; the circle including this will be Saturn. Now inscribe in the Earth an icosahedron; the circle inscribed in it will be Venus. Inscribe an octahedron in Venus; the circle inscribed in it will be Mercury." Elated with this conceit, which rudely agreed with the numbers given by Copernicus, Kepler declared that he valued it more than the possession of the electorate of Saxony. These extraordinary calculations, with an unsuccessful attempt to discover the relation between the periodic times and the distance of the planets, were published in his "Prodromus Dissertationum Cosmographicorum," which appeared at Tübingen in 1596, and which was commended by Galileo and Tycho; but the latter judiciously advised him "to obtain a solid foundation for his views from actual observations, and then by ascending from these to strive to reach the causes of things."

In 1592 Kepler paid his addresses to Barbara Muller von Muleckh, a young lady of nineteen, but the marriage was not agreeable to her parents; and when it was again proposed in 1596, they refused their consent till Kepler produced a proof of his nobility. In 1597 the marriage took place, the lady having been a widow for the second time. Kepler's salary being very small, and his wife's fortune less than was expected, he was involved in pecuniary difficulties and disputes with her relations. The religious troubles in Styria drove him and his wife into Hungary, from which he was recalled in 1599 by the states of Styria, in order to resume his duties in the university. In 1600 he paid a visit to Tycho Brahe, then living at Benach, near Prague, an exile from his country; and during his residence there he had agreed to become assistant to Tycho for two years, with a salary of one hundred florins, provided he could retain the salary of his chair at Grätz. In terms of this arrangement Kepler and his wife set out for Prague; but having been attacked on the road by a quartan ague which lasted seven months, his funds were exhausted, and he was supported entirely by the bounty of Tycho. In 1601 Kepler was introduced by Tycho to the Emperor Rodolph, who conferred upon him the title of imperial mathematician on the condition of his assisting Tycho; and the two astronomers agreed to combine their talents in the computation of new astronomical tables, to be called the Rudolphine Tables, from the generosity of the emperor who had promised to defray the expense of them.

The death of Tycho in October, 1601, put a stop to this important arrangement; but Kepler was appointed principal mathematician to the emperor, with an ample salary, to be paid in March, 1601. This promise, however, was not fulfilled; and from the non-payment of his salary he was obliged to postpone the Rudolphine tables, to devote his time to the completion of other works, and even to cast nativities; though it appears from other parts of his writings that he held astrology in contempt. In 1602 he published at Prague his work, "De Fundamentis Astrologiæ;" and in 1604, at Frankfort, his "Paralipomena ad Vitellionem"—a work containing much new and interesting information on dioptrics and vision. In 1605 he published his "Epistola de Solis Deliquio," and in 1606 his treatise "De Stella nova," the new star which appeared in 1604 in Serpentarius, and which, like that of 1572, rivalled even Venus in lustre. In 1609 Kepler published at Prague his greatest work, entitled "Astronomia Nova, sen physica cælestis tradita commentariis de motibus stellæ Martis." In 1601, when residing with Tycho, he had begun the researches contained in this volume. He had failed in every attempt to represent the observations on Mars by a uniform motion in a circular orbit, and by the aid of the cycles and epicycles, as employed by Copernicus in explaining the planetary inequalities; and having, after many abortive speculations, conjectured that the orbit of the planet might be of an oval form, he was led to the conclusion that Mars moved in an elliptical orbit, with the sun placed in one of its foci. His knowledge of the conic sections enabled him to determine the major and minor axis of the ellipse; and by comparing the times in which the planet describes any portion of its orbit with the time of a whole revolution, or the time of describing any other portion of it, he found that these numbers were always to one another as the areas contained by lines drawn from the sun's centre, or focus of the ellipse, to the extremities of the respective portions or arches of the orbit; or, more precisely, that the radius vector or line joining the sun's centre and Mars describes equal areas in equal times. Kepler's third law, that the squares of the periodic times are proportional to the cubes of the mean distances of the planets from the sun, was not discovered till nine years after he had discovered the other two laws. It was published in 1619 at Linz, in his "Harmonia Mundi," which was dedicated to James I. of England. The law, he tells us, first entered his mind on the 8th March, 1618; but having committed a mistake in his calculations, he resumed the subject on the 19th May, and placed beyond a doubt the absolute conformity with observation of a law which for seventeen years he had laboured to discover. Having in his "Supplement to Vitellio" thrown much light both upon geometrical and physiological optics, he resumed the study of it; and in 1611 he published at Frankfort his "Dioptrica," with an appendix on the use of optics in philosophy. In this excellent work, which was reprinted in London in 1653, he explained the principle of the telescope, and described the astronomical telescope with two convex lenses, which was his own invention, and which was greatly superior to that of Galileo, from its admitting in front of the eye-glass micrometer wires for measuring distances in the heavens. He proved that spherical surfaces cannot converge rays to a single focus; and he conjectured what Descartes afterwards proved, that this property might be possessed by surfaces having the figures of some of the conic sections. When Kepler presented to the Emperor Rodolph his "Astronomia Nova," he jocularly stated to him that a similar attack on the other planets required the sinews of war; but the emperor had other enemies than planets to overcome, and could not afford to science what he gave to war. Kepler's wife was seized in 1610 with fever, epilepsy, and phrenitis, and three of his children were attacked with the small-pox. His favourite son died of the complaint; and while Kepler was in the midst of poverty and domestic affliction, Prague was oppressed with Austrian troops, who had introduced the plague into the city. Disappointed in obtaining the arrears of his salary, Kepler went to Linz in the hope of being appointed to the mathematical chair, which was then vacant in that university; but the emperor encouraged him to remain at Prague, with the promise that the arrears due to him would be paid from Saxony. This promise, however, was not fulfilled, and it was not till the death of Rodolph in 1612 that the arrears were paid.

The Emperor Mathias permitted Kepler to accept the professorship at Linz, and continued him in the office of imperial mathematician; but though his pecuniary difficulties were thus to a great extent removed, his domestic calamities were increased by the death of his wife. He had now a son and a daughter, both of tender age, who required a mother's care; and being wholly engrossed with his private studies and the duties of his new charge, he found it necessary to have another parent to his children. The narrative of his search for a wife is one of the most curious chapters in his history. No fewer than eleven ladies were presented to his choice, and in a jocular letter to Baron Strahlendorf, he has described their different characters, and the various negotiations which preceded his marriage. He had commissioned his friends to find for him a suitable companion; but he took the advice of none of them, and married a girl of humble station, whose person, manners, and excellent education he considered better than a good dowry. The marriage seems to have taken place in 1614, as he mentions it in his book entitled "Nova Stereometria," which appeared at Linz in 1615. On this occasion he stocked his cellar with a few casks of wine; but finding that the merchant in measuring their contents had made no allowance for the bulging part of the casks, he was induced to study the subject and publish his work on guaging, which contains the earliest specimens of the modern analysis. In the year 1617 Kepler published at Linz his "Ephemerides" for 1617-20, the Ephemeris for 1620 having been