In the system of Copernicus, which Kepler ardently adopted, the earth revolves around the sun. Now, observation having shown that the sun remains seven or eight days longer in the northern than in the southern signs of the Zodiac, we must of necessity admit that the sun, instead of being situated in the centre of the terrestrial orbit, occupies a point _outside_ that centre, in such a manner that the earth must sometimes be nearer to, and sometimes farther from, the sun. The distance by which it departs from the centre of its orbit, which Copernicus, like the ancients, supposed to be circular, is called its _eccentricity_. Astronomers were long preoccupied with the idea of seeking in this _eccentricity_ a point where the movements should appear equal. This point was the centre of the _equant_,--a name given to the eccentric circle described from the point of equality or from the centre of the mean movements. Now, let us recall the principal condition of the problem which Kepler had undertaken to solve. This condition required that the straight line drawn from the centre of our globe to the centre of the sun,--in a word, that the _vector radius_, as it is called, should describe around the sun certain angles, whose variability should agree with the results of observation. Starting from this point, Kepler found that, for certain positions of Mars (in the _aphelion_ and _perihelion_, corresponding to the _minimum_ and _maximum_ of velocity), the centre of the orbit, always supposing it to be circular, divided into two equal parts (or _bisected_) the total eccentricity: in other words, that it exactly occupied the middle between the centre of the eccentric and the _equant_ of Ptolemæus; but it did not appear to him necessary to bisect it in other positions, intermediate between those of the aphelion and the perihelion. He established that the differences in longitude amounted to eight or nine minutes. Now, observations so exact as those of Tycho were altogether incompatible with such great error.[76] Therefore, the geometrical hypothesis which gave these errors was false; the orbit of Mars could not be a circle, and to save these eight or nine minutes, furnished by observation but in disaccord with theory, it would be needful to recommence all the calculations of astronomy. This conclusion, not less legitimate than daring, supplied Kepler with the first decisive step in the task he had undertaken. This is not the place to relate all the essays and miscarriages through which this man of genius passed before finally completing his discovery of the rules that bear his name. But we may put before the reader the construction which led to them.