propositions, contrast strongly with the loose and imperfectly
supported explanations of all his predecessors; and the indulgent
reader will excuse the devotion of a few lines to an example of the
ingenuity and beauty of his methods.

It may seem a hopeless task to find out the true paths of Mars and the
earth (at that time when their shape even was not known) from the
observations giving only the relative direction from night to
night. Now, Kepler had twenty years of observations of Mars to deal
with. This enabled him to use a new method, to find the earth's
orbit. Observe the date at any time when Mars is in opposition. The
earth's position E at that date gives the longitude of Mars M. His
period is 687 days. Now choose dates before and after the principal
date at intervals of 687 days and its multiples. Mars is in each case
in the same position. Now for any date when Mars is at M and the earth
at E3 the date of the year gives the angle E3SM. And the
observation of Tycho gives the direction of Mars compared with the
sun, SE3M. So all the angles of the triangle SEM in any of these
positions of E are known, and also the ratios of SE1, SE2, SE3,
SE4 to SM and to each other.

For the orbit of Mars observations were chosen at intervals of a year,
when the earth was always in the same place.

[Illustration]

But Kepler saw much farther than the geometrical facts. He realised
that the orbits are followed owing to a force directed to the sun; and
he guessed that this is the same force as the gravity that makes a
stone fall. He saw the difficulty of gravitation acting through the