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Meanwhile his official duties evidently left him a good deal of leisure, which he spent with characteristic energy in acquiring as thorough a knowledge as possible of astronomy, and in speculating on the subject.
According to his own statement, “there were three things in particular, viz. the number, the size, and the motion of the heavenly bodies, as to which he searched zealously for reasons why they were as they were and not otherwise”; and the results of a long course of wild speculation on the subject led him at last to a result with which he was immensely pleased—a numerical relation connecting the distances of the several planets from the sun with certain geometrical bodies known as the regular solids (of which the cube is the best known), a relation which is not very accurate numerically, and is of absolutely no significance or importance.84 This discovery, together with a detailed account of the steps which led to it, as well as of a number of other steps which led nowhere, was published in 1596 in a book a portion of the title of which may be translated as The Forerunner of Dissertations on the Universe, containing the Mystery of the Universe, commonly referred to as the Mysterium Cosmographicum. The contents were probably much more attractive and seemed more valuable to Kepler’s contemporaries than to us, but even to those who were least inclined to attach weight to its conclusions, the book shewed evidence of considerable astronomical knowledge and very great ingenuity; and both Tycho Brahe and Galilei, to whom copies were sent, recognised in the author a rising astronomer likely to do good work.
137. In 1597 Kepler married. In the following year the religious troubles, which had for some years been steadily growing, were increased by the action of the Archduke Ferdinand of Austria (afterwards the Emperor Ferdinand II.), who on his return from a pilgrimage to Loretto started a vigorous persecution of Protestants in his dominions, one step in which was an order that all Protestant ministers and teachers in Styria should quit the country at once (1598). Kepler accordingly fled to Hungary, but returned after a few weeks by special permission of the Archduke, given apparently on the advice of the Jesuit party, who had hopes of converting the astronomer. Kepler’s hearers had, however, mostly been scattered by the persecution, it became difficult to ensure regular payment of his stipend, and the rising tide of Catholicism made his position increasingly insecure. Tycho’s overtures were accordingly welcome, and in 1600 he paid a visit to him, as already described (chapter V., § 108), at Benatek and Prague. He returned to Gratz in the autumn, still uncertain whether to accept Tycho’s offer or not, but being then definitely dismissed from his position at Gratz on account of his Protestant opinions, he returned finally to Prague at the end of the year.
138. Soon after Tycho’s death Kepler was appointed his successor as mathematician to the Emperor Rudolph (1602), but at only half his predecessor’s salary, and even this was paid with great irregularity, so that complaints as to arrears and constant pecuniary difficulties played an important part in his future life, as they had done during the later years at Gratz. Tycho’s instruments never passed into his possession, but as he had little taste or skill for observing, the loss was probably not great; fortunately, after some difficulties with the heirs, he secured control of the greater part of Tycho’s incomparable series of observations, the working up of which into an improved theory of the solar system was the main occupation of the next 25 years of his life. Before, however, he had achieved any substantial result in this direction, he published several minor works—for example, two pamphlets on a new star which appeared in 1604, and a treatise on the applications of optics to astronomy (published in 1604 with a title beginning Ad Vitellionem Paralipomena quibus Astronomiae Pars Optica Traditur ...), the most interesting and important part of which was a considerable improvement in the theory of astronomical refraction (chapter II., § 46, and chapter V., § 110). A later optical treatise (the Dioptrice of 1611) contained a suggestion for the construction of a telescope by the use of two convex lenses, which is the form now most commonly adopted, and is a notable improvement on Galilei’s instrument (chapter VI., § 118), one of the lenses of which is concave; but Kepler does not seem himself to have had enough mechanical skill to actually construct a telescope on this plan, or to have had access to workmen capable of doing so for him; and it is probable that Galilei’s enemy Scheiner (chapter VI., §§ 124, 125) was the first person to use (about 1613) an instrument of this kind.
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139. It has already been mentioned (chapter V., § 108) that when Tycho was dividing the work of his observatory among his assistants he assigned to Kepler the study of the planet Mars, probably as presenting more difficulties than the subjects assigned to the others. It had been known since the time of Coppernicus that the planets, including the earth, revolved round the sun in paths that were at any rate not very different from circles, and that the deviations from uniform circular motion could be represented roughly by systems of eccentrics and epicycles. The deviations from uniform circular motion were, however, notably different in amount in different planets, being, for example, very small in the case of Venus, relatively large in the case of Mars, and larger still in that of Mercury. The Prussian Tables calculated by Reinhold on a Coppernican basis (chapter V., § 94) were soon found to represent the actual motions very imperfectly, errors of 4° and 5° having been noted by Tycho and Kepler, so that the principles on which the tables were calculated were evidently at fault.
The solution of the problem was clearly more likely to be found by the study of a planet in which the deviations from circular motion were as great as possible. In the case of Mercury satisfactory observations were scarce, whereas in the case of Mars there was an abundant series recorded by Tycho, and hence it was true insight on Tycho’s part to assign to his ablest assistant this particular planet, and on Kepler’s to continue the research with unwearied patience. The particular system of epicycles used by Coppernicus (chapter IV., § 87) having proved defective, Kepler set to work to devise other geometrical schemes, the results of which could be compared with observation. The places of Mars as seen on the sky being a combined result of the motions of Mars and of the earth in their respective orbits round the sun, the irregularities of the two orbits were apparently inextricably mixed up, and a great simplification was accordingly effected when Kepler succeeded, by an ingenious combination of observations taken at suitable times, in disentangling the irregularities due to the earth from those due to the motion of Mars itself, and thus rendering it possible to concentrate his attention on the latter. His fertile imagination suggested hypothesis after hypothesis, combination after combination of eccentric, epicycle, and equant; he calculated the results of each and compared them rigorously with observation; and at one stage he arrived at a geometrical scheme which was capable of representing the observations with errors not exceeding 8′.85 A man of less intellectual honesty, or less convinced of the necessity of subordinating theory to fact when the two conflict, might have rested content with this degree of accuracy, or might have supposed Tycho’s refractory observations to be in error. Kepler, however, thought otherwise:—
“Since the divine goodness has given to us in Tycho Brahe a most careful observer, from whose observations the error of 8′ is shewn in this calculation, ... it is right that we should with gratitude recognise and make use of this gift of God.... For if I could have treated 8′ of longitude as negligible I should have already corrected sufficiently the hypothesis ... discovered in chapter XVI. But as they could not be neglected, these 8′ alone have led the way towards the complete reformation of astronomy, and have been made the subject-matter of a great part of this work.”86
140. He accordingly started afresh, and after trying a variety of other combinations of circles decided that the path of Mars must be an oval of some kind. At first he was inclined to believe in an egg-shaped oval, larger at one end than at the other, but soon had to abandon this idea. Finally he tried the simplest known oval curve, the ellipse,87 and found to his delight that it satisfied the conditions of the problem, if the sun were taken to be at a focus of the ellipse described by Mars.
It was further necessary to formulate the law of variation of the rate of motion of the planet in different parts of its orbit. Here again Kepler tried a number of hypotheses, in the course of which he fairly lost his way in the intricacies of the mathematical questions involved, but fortunately arrived, after a dubious process of compensation of errors, at a simple law which agreed with observation. He found that the planet moved fast when near the sun and slowly when distant from it, in such a way that the area described or swept out in any time by the line joining the sun to Mars was always proportional to the time. Thus in fig. 6088 the motion of Mars is most rapid at the point A nearest to the focus S where the sun is, least rapid at A′, and the shaded and unshaded portions of the figure represent equal areas each corresponding to the motion of the planet during a month. Kepler’s triumph at arriving at this result is expressed by the figure of victory in the corner of the diagram (fig. 61) which was used in establishing the last stage of his proof.
141. Thus were established for the case of Mars the two important results generally known as Kepler’s first two laws:—
1. The planet describes an ellipse, the sun being in one focus.
2. The straight line joining the planet to the sun sweeps out equal areas in any two equal intervals of time.
The full history of this investigation, with the results already stated and a number of developments and results of minor
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