However, the facts of the cosmos were on his side, but the calculations essential in testing his various hypotheses were of the most tedious nature, because logarithms were not yet known in his day. His first discovery was that the orbit of Mars is certainly not a circle, but oval or elliptic in figure. And the sun, he soon found, could not be in the center of the ellipse, so he made a series of trial calculations with the sun located in one of the foci of the ellipse instead. Then he found he could make his calculated places of Mars agree quite perfectly with Tycho Brahe's observed positions, if only he gave up the other ancient requisite of perfectly uniform motion. On doing this, it soon appeared that Mars, when in perihelion, or nearest the sun, always moved swiftest, while at its greatest distance from the sun, or aphelion, its orbital velocity was slowest. Kepler did not busy himself to inquire why these revolutionary discoveries of his were as they were; he simply went on making enough trials on Mars, and then on the other planets in turn, to satisfy himself that all the planetary orbits are elliptical, not circular in form, and are so located in space that the center of the sun is at one of the two foci of each orbit. This is known as Kepler's first law of planetary motion. The second one did not come quite so easy; it concerned the variable speed with which the planet moves at every point of the orbit. We must remember how handicapped he was in solving this problem: only the geometry of Euclid to work with, and none of the refinements of the higher mathematics of a later day. But he finally found a very simple relation which represented the velocity of the planet everywhere in its orbit. It was this: if we calculate the area swept, or passed over, by the planet's radius vector (that is, the line joining its center to the sun's center) during a week's time near perihelion, and then calculate the similar area for a week near aphelion, or indeed for a week when Mars is in any intermediate part of its orbit, we shall find that these areas are all equal to each other. So Kepler formulated his second great law of planetary motion very simply: the radius vector of any planet describes, or sweeps over, equal areas in equal times. And he found this was true for all the planets. But the real genius of the great mathematician was shown in the discovery of his third law, which is more complex and even more significant than the other two--a law connecting the distances of the planets from the sun with their periods of revolution about the sun. This cost Kepler many additional years of close calculation, and the resulting law, his third law of planetary motion is this: The cubes of the mean or average distances of the planets from the sun are proportional to the squares of their times of revolution around him.