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). 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