granted
that the centre was at some point on the line called the excentricity
(see Figs. 1, 2).

A marked improvement in residuals was the result of this step, proving,
so far, the correctness of Ptolemy's principle, but there still remained
discordances amounting to eight minutes of arc. Copernicus, who had no
idea of the accuracy obtainable in observations, would probably have
regarded such an agreement as remarkably good; but Kepler refused to
admit the possibility of an error of eight minutes in any of Tycho's
observations. He thereupon vowed to construct from these eight minutes a
new planetary theory that should account for them all. His repeated
failures had by this time convinced him that no uniformly described
circle could possibly represent the motion of Mars. Either the orbit
could not be circular, or else the angular velocity could not be
constant about any point whatever. He determined to attack the "second
inequality," i.e. the optical illusion caused by the earth's annual
motion, but first revived an old idea of his own that for the sake of
uniformity the sun, or as he preferred to regard it, the earth, should
have an equant as well as the planets. From the irregularities of the
solar motion he soon found that this was the case, and that the motion
was uniform about a point on the line from the sun to the centre of the
earth's orbit, such that the centre bisected the distance from the sun
to the "Equant"; this fully supported Ptolemy's principle. Clearly then
the earth's linear velocity could not be constant, and Kepler was
encouraged to revive another of his speculations as to a force which was
weaker at greater distances. He found the velocity greater at the nearer
apse, so that the time over an equal arc at either apse was proportional
to the distance. He conjectured that this might prove to be true for
arcs at all parts of the orbit, and to test this he divided the orbit
into 360 equal parts, and calculated the distances to the points of
division. Archimedes had obtained an approximation to the area of a
circle by dividing it radially into a very large number of triangles,
and Kepler had this device in mind. He found that the sums of successive
distances from his 360 points were approximately proportional to the
times from point to point, and was thus enabled to represent much more
accurately the annual motion of the earth which produced the second
inequality of Mars, to whose motion he now returned.