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.  Three points are sufficient to define a circle, so he took three observed positions of Mars and found a circle; he then took three other positions, but obtained a different circle, and a third set gave yet another.  It thus began to appear that the orbit could not be a circle.  He next tried to divide into 360 equal parts, as he had in the case of the earth, but the sums of distances failed to fit the times, and he realised that the sums of distances were not a good measure of the area of successive triangles.  He noted, however, that the errors at the apses were now smaller than with a central circular orbit, and of the opposite