duly 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.

Kepler's improved determination of the earth's orbit was obtained by
plotting the different positions of the earth corresponding to
successive rotations of Mars