The time taken from one opposition of Mars to the next is decidedly unequal at different parts of his orbit, so that many oppositions must be used to determine the mean motion.  The ancients had noticed that what was called the “second inequality,” due as we now know to the orbital motion of the earth, only vanished when earth, sun, and planet were in line, i.e. at the planet’s opposition; therefore they used oppositions to determine the mean motion, but deemed it necessary to apply a correction to the true opposition to reduce to mean opposition, thus sacrificing part of the advantage of using oppositions.  Tycho and Longomontanus had followed this 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 granted that the centre was at some point on the line called the excentricity (see Figs. 1, 2).