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 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