$ 140] The Discovery of the Elliptic Mot ion of Mars 185 



he tried the simplest known oval curve, the ellipse,* 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 plnnet 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. 6ot 

 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. 



Thus if, in the figure, s and H are the foci, and P, Q are any two 



FIG. 59. An ellipse. 



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. 



f 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 \ in the figure, that of Mars being fa. 



