While thinking and pondering for weeks and months over this new dilemma and complication of difficulties, till his brain reeled, an accidental ray of light broke upon him in a way not now intelligible, or barely intelligible. Half the extreme breadth intercepted between the circle and oval was 429/100,000 of the radius, and he remembered that the "optical inequality" of Mars was also about 429/100,000. This coincidence, in his own words, woke him out of sleep; and for some reason or other impelled him instantly to try making the planet oscillate in the diameter of its epicycle instead of revolve round it--a singular idea, but Copernicus had had a similar one to explain the motions of Mercury. [Illustration: FIG. 31.--Mode of drawing an ellipse. The two pins _F_ are the foci.] Away he started through his calculations again. A long course of work night and day was rewarded by finding that he was now able to hit off the motions better than before; but what a singularly complicated motion it was. Could it be expressed no more simply? Yes, the curve so described by the planet is a comparatively simple one: it is a special kind of oval--the ellipse. Strange that he had not thought of it before. It was a famous curve, for the Greek geometers had studied it as one of the sections of a cone, but it was not so well known in Kepler's time. The fact that the planets move in it has raised it to the first importance, and it is familiar enough to us now. But did it satisfy the law of speed? Could the rate of description of areas be uniform with it? Well, he tried the ellipse, and to his inexpressible delight he found that it did satisfy the condition of equable description of areas, if the sun was in one focus. So, moving the planet in a selected ellipse, with the sun in one focus, at a speed given by the equable area description, its position agreed with Tycho's observations within the limits of the error of experiment. Mars was finally conquered, and remains in his prison-house to this day. The orbit was found. [Illustration: FIG. 32.] In a paroxysm of delight Kepler celebrates his victory by a triumphant figure, sketched actually on his geometrical diagram--the diagram which proves that the law of equable description of areas can hold good with an ellipse. The above is a tracing of it. Such is a crude and bald sketch of the steps by which Kepler rose to his great generalizations--the two laws which have immortalized his name. All the complications of epicycle, equant, deferent, excentric, and the like, were swept at once away, and an orbit of striking and beautiful properties substituted. Well might he be called, as he was, "the legislator," or law interpreter, "of the heavens." [Illustration: FIG. 33.--If _S_ is the sun, a planet or comet moves from _P_ to _P_1_, from _P_2_ to _P_3_, and from _P_4_ to _P_5_ in the same time; if the shaded areas are equal.]