o an incidence of 90.

Two years before the publication of his Dioptrics, viz. in 1609, Kepler
had given to the world his great work, entitled "The New Astronomy, or
Commentaries on the Motions of Mars." The discoveries which this volume
records form the basis of physical astronomy. The inquiries by which he
was led to them began in that memorable year 1601, when he became the
colleague or assistant of Tycho. The powers of original genius were then
for the first time associated with inventive skill and patient
observation; and though the astronomical data provided by Tycho were
sure of finding their application in some future age, yet without them
Kepler's speculations would have been vain, and the laws which they
enabled him to determine would have adorned the history of another
century. Having tried in vain to represent the motion of Mars by an
uniform motion in a circular orbit, and by the cycles and epicycles with
which Copernicus had endeavoured to explain the planetary inequalities,
Kepler was led, after many fruitless speculations,[47] to suppose the
orbit of the planet to be oval; and, from his knowledge of the conic
sections, he afterwards determined it to be an ellipse, with the sun
placed in one of its foci. He then ascertained the dimensions of the
orbit; and, by a comparison of the times employed by the planet to
complete a whole revolution or any part of one, he discovered that the
time in which Mars describes any arches of his elliptic orbit, were
always to one another as the areas contained by lines drawn from the
focus or the centre of the sun to the extremities of the respective
arches; or, in other words, that the radius vector, or the line joining
the Sun and Mars described equal areas in equal times. By examining the
inequalities of the other planets he found that they all moved in
elliptic orbits, and that the radius vector of each described areas
proportional to the times. These two great results are known by the name
of the first and second laws of Kepler. The third law, or that which
relates to the connexion between the periodic times and the distances of
the planets, was not discovered till a later period of his life.

[47] An interesting account of the steps by which Kepler proceeded
will be found in Mr Drinkwater Bethune's admirable Life of Kepler,
in the Library of Useful Knowledge.

When Kepler presented to Rudolph the volume which contained these fine
discoveries, he remin