orbit were much more discordant, and he found to his chagrin that four
years' work was practically wasted. Before making a fresh start he
looked for some simplification of the labour; and determined to adopt
Ptolemy's assumption known as the principle of the bisection of the
excentricity. Hitherto, since Ptolemy had given no reason for this
assumption, Kepler had preferred not to make it, only taking for 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