Fig. 2.—­This is not drawn to scale, but is intended to illustrate Kepler’s modification of Ptolemy’s excentric.  Kepler found velocities at P and Q proportional not to AP and AQ but to AQ and AP, or to EP and EQ if EC = CA (bisection of the excentricity).  The velocity at M was wrong, and am appeared too great.  Kepler’s first ellipse had M moved too near C. The distance AC is much exaggerated in the figure, as also is MN.  An = CP, the radius of the circle.  MN should be .00429 of the radius, and Mc/NC should be 1.00429.  The velocity at N appeared to be proportional to en ( = an).  Kepler concluded that Mars moved round PNQ, so that the area described about A (the sun) was equal in equal times, A being the focus of the ellipse PNQ.  The angular velocity is not quite constant about E, the equant or empty focus, but the difference could hardly have been detected in Kepler’s time.

Kepler’s improved determination of the earth’s orbit was obtained by plotting the different positions of the earth corresponding to successive rotations of Mars, i.e. intervals of 687 days.  At each of these the date of the year would give the angle MSE (Mars-Sun-Earth), and Tycho’s observation the angle MES.  So the triangle could be solved except for scale, and the ratio of Se to SM would give the distance of Mars from the sun in terms of that of the earth.  Measuring from a fixed position of Mars (e.g. perihelion), this gave the variation of Se, showing the earth’s inequality.  Measuring from a fixed position of the earth, it would give similarly a series of positions of Mars, which, though lying not far from the circle whose diameter was the axis of Mars’ orbit, joining perihelion and aphelion, always fell inside the circle except at those two points.  It was a long time before it dawned upon Kepler that the simplest figure falling within the circle except at the two extremities of the diameter, was an ellipse, and it is not clear why his first attempt with an ellipse should have been just as much too narrow as the circle was too wide.  The fact remains that he recognised suddenly that halving this error was tantamount to reducing the circle to the ellipse whose eccentricity was that of the old theory, i.e. that in which the sun would be in one focus and the equant in the other.

Having now fitted the ends of both major and minor axes of the ellipse, he leaped to the conclusion that the orbit would fit everywhere.

The practical effect of his clearing of the “second inequality” was to refer the orbit of Mars directly to the sun, and he found that the area between successive distances of Mars from the sun (instead of the sum of the distances) was strictly proportional to the time taken, in short, equal areas were described in equal times (2nd Law) when referred to the sun in the focus of the ellipse (1st Law).