he tried the simplest known oval curve, the ellipse,[1] 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[2] 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
- ↑ An ellipse is one of several curves, known as conic sections, which can be formed by taking a section of a cone, and may also be defined as a curve the sum of the distances of any point on which from two fixed points inside it, known as the foci, is always the same.
Fig. 59.—An ellipse.
Thus if, in the figure, s and h are the foci, and p, q are any two points on the curve, then the distances s p, h p added together are equal to the distances s q, q h added together, and each sum is equal to the length a a' of the ellipse. The ratio of the distance s h to the length a a' is known as the eccentricity, and is a convenient measure of the extent to which the ellipse differs from a circle.
- ↑ The ellipse is more elongated than the actual path of Mars, an accurate drawing of which would be undistinguishable to the eye from a circle. The eccentricity is 13 in the figure, that of Mars being 110.
