He specially attacked the motion of the planet Mars, because that was sufficiently rapid in its changes for a considerable collection of data to have accumulated with respect to it. He tried all manner of circular orbits for the earth and for Mars, placing them in all sorts of aspects with respect to the sun. The problem to be solved was to choose such an orbit and such a law of speed, for both the earth and Mars, that a line joining them, produced out to the stars, should always mark correctly the apparent position of Mars as seen from the earth. He had to arrange the size of the orbits that suited best, then the positions of their centres, both being supposed excentric with respect to the sun; but he could not get any such arrangement to work with uniform motion about the sun. So he reintroduced the equant, and thus had another variable at his disposal--in fact, two, for he had an equant for the earth and another for Mars, getting a pattern of the kind suggested in Fig. 29. The equants might divide the line in any arbitrary ratio. All sorts of combinations had to be tried, the relative positions of the earth and Mars to be worked out for each, and compared with Tycho's recorded observations. It was easy to get them to agree for a short time, but sooner or later a discrepancy showed itself. [Illustration: FIG. 29.--_S_ represents the sun; _EC_, the centre of the earth's orbit, to be placed as best suited; _MC_, the same for Mars; _EE_, the earth's equant, or point about which the earth uniformly revolved (_i.e._ the point determining the law of speed about the sun), likewise to be placed anywhere, but supposed to be in the line joining _S_ to _EC_; _ME_, the same thing for Mars; with _?ME_ for an alternative hypothesis that perhaps Mars' equant was on line joining _EC_ with _MC_.] I need not say that all these attempts and gropings, thus briefly summarized, entailed enormous labour, and required not only great pertinacity, but a most singularly constituted mind, that could thus continue groping in the dark without a possible ray of theory to illuminate its search. Grope he did, however, with unexampled diligence. At length he hit upon a point that seemed nearly right. He thought he had found the truth; but no, before long the position of the planet, as calculated, and as recorded by Tycho, differed by eight minutes of arc, or about one-eighth of a degree. Could the observation be wrong by this small amount? No, he had known Tycho, and knew that he was never wrong eight minutes in an observation.