uniform motion in a circle must be provided, and since the motion was
not uniform about the earth, A could not coincide with C; and since the
motion still failed to be uniform about A or C, some point E must be
found about which the motion should be uniform.

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