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.]

It is often said that a coincidence like this only happens to somebody
who "deserves his luck," but this simply means that recognition is
essential to the coincidence. In the same way the appearance of one of a
large number of people mentioned is hailed as a case of the old adage
"Talk of the devil, etc.," ignoring all the people who failed to appear.
No one, however, will consider Kepler un