“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. 60[88] 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. [Illustration: FIG. 60.—Kepler’s second law.] 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 importance, together with innumerable digressions and quaint comments on the progress of the inquiry, was published in 1609 in a book of considerable length, the _Commentaries on the Motions of Mars_.[89] [Illustration: FIG. 61.—Diagram used by Kepler to establish his laws of planetary motion. From the _Commentaries on Mars_.]