method in their calculations from
Tycho's twenty years' observations. Their aim was to find a position of
the "equant," such that these observations would show a constant angular
motion about it; and that the computed positions would agree in latitude
and longitude with the actual observed positions. When Kepler arrived he
was told that their longitudes agreed within a couple of minutes of arc,
but that something was wrong with the latitudes. He found, however, that
even in longitude their positions showed discordances ten times as great
as they admitted, and so, to clear the ground of assumptions as far as
possible, he determined to use true oppositions. To this Tycho objected,
and Kepler had great difficulty in convincing him that the new move
would be any improvement, but undertook to prove to him by actual
examples that a false position of the orbit could by adjusting the
equant be made to fit the longitudes within five minutes of arc, while
giving quite erroneous values of the latitudes and second inequalities.
To avoid the possibility of further objection he carried out this
demonstration separately for each of the systems of Ptolemy, Copernicus,
and Tycho. For the new method he noticed that great accuracy was
required in the reduction of the observed places of Mars to the
ecliptic, and for this purpose the value obtained for the parallax by
Tycho's assistants fell far short of the requisite accuracy. Kepler
therefore was obliged to recompute the parallax from the original
observations, as also the position of the line of nodes and the
inclination of the orbit. The last he found to be constant, thus
corroborating his theory that the plane of the orbit passed through the
sun. He repeated his calculations no fewer than seventy times (and that
before the invention of logarithms), and at length adopted values for
the mean longitude and longitude of aphelion. He found no discordance
greater than two minutes of arc in Tycho's observed longitudes in
opposition, but the latitudes, and also longitudes in other parts of the
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