, 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).

He announced that (1) The planet describes an ellipse, the sun being in
one focus; and (2) The straight line joining the planet to the sun
sweeps out equal areas in any two equal intervals of time. These are
Kepler's first and second Laws though not discovered in that order, and
it was at once clear that Ptolemy's "bisection of the excentricity"
simply amounted to the fact that the centre of an ellipse bisects the
distance between the foci, the sun being in one focus and the angular
velocity being uniform about the empty focus. For so many centuries had
the fetish of circular motion postponed discovery. It was natural that
Kepler should assume that his laws would apply eq