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 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.]

_______
/       \
/         \
|           |
|___________|
Q|  E  C  A  |P
|           |
\         /
\_______/

[Illustration:  Fig. 2.]

___M___
/___|\__\
//  N|\\ \\
|/    | \\ \|
|_____|__\\_|
Q| E   C   A |P
|\    |    /|
\\___|___//
\___|___/

[Transcriber’s Note:  Approximate renditions of these figures are provided.  Fig. 1 is a circle.  Fig. 2 is a circle which contains an ellipse, tangent to the circle at Q and P. Line segments from M (on the circle) and N (on the ellipse) meet at point A.]

Fig. 1.—­In Ptolemy’s excentric theory, A may be taken to represent the earth, C the centre of a planet’s orbit, and E the equant, P (perigee) and Q (apogee) being the apses of the orbit.  Ptolemy’s idea was that uniform motion in a circle must be provided, and since the motion was not uniform about the earth, A could not coincide with C; and since the motion still failed to be uniform about A or C, some point E must be found about which the motion should be uniform.