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 sign, so he determined to try whether an oval orbit would fit better, following a suggestion made by Purbach in the case of Mercury, whose orbit is even more eccentric than that of Mars, though observations were too scanty to form the foundation of any theory. Kepler gave his fancy play in the choice of an oval, greater at one end than the other, endeavouring to satisfy some ideas about epicyclic motion, but could not find a satisfactory curve. He then had the fortunate idea of trying an ellipse with the same axis as his tentative oval. Mars now appeared too slow at the apses instead of too quick, so obviously some intermediate ellipse must be sought between the trial ellipse and the circle on the same axis. At this point the "long arm of coincidence" came into play. Half-way between the apses lay the mean distance, and at this position the error was half the distance between the ellipse and the circle, amounting to .00429 of a radius. With these figures in his mind, Kepler looked up the greatest optical inequality of Mars, the angle between the straight lines from Mars to the Sun and to the centre of the circle.[3] The secant of this angle was 1.00429, so that he noted that an ellipse reduced from the circle in the ratio of 1.00429 to 1 would fit the motion of Mars at the mean distance as well as the apses. [Footnote 3: This is clearly a maximum at AMC in Fig. 2, when its tangent AC/CM = the eccentricity.]