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 unduly favoured. His genius, in
his case certainly "an infinite capacity for taking pains," enabled him
out of his medley of hypotheses, mainly unsound, by dint of enormous
labour and patience, to arrive thus at the first two of the laws which
established his title of "Legislator of the Heavens".

FIGURES EXPLANATORY OF KEPLER'S THEORY OF THE MOTION OF MARS.

[Illustration: FIG. 1.]