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. FIG. 2.--This is not drawn to scale, but is intended to illustrate Kepler's modification of Ptolemy's excentric. Kepler found velocities at P and Q proportional not to AP and AQ but to AQ and AP, or to EP and EQ if EC = CA (bisection of the excentricity). The velocity at M was wrong, and AM appeared too great. Kepler's first ellipse had M moved too near C. The distance AC is much exaggerated in the figure, as also is MN. AN = CP, the radius of the circle. MN should be .00429 of the radius, and MC/NC should be 1.00429. The velocity at N appeared to be proportional to EN ( = AN). Kepler concluded that Mars moved round PNQ, so that the area described about A (the sun) was equal in equal times, A being the focus of the ellipse PNQ. The angular velocity is not quite constant about E, the equant or empty focus, but the difference could hardly have been detected in Kepler's time.