motion of the orbits of the planets. He first tried if
the size of the planets' orbits, or the difference of their sizes, had
any regular proportion to each other. Finding no proof of this, he
inserted a new planet between Mars and Jupiter, and another between
Venus and Mercury, which he supposed might be invisible from their
smallness; but even with these assumptions the distances of the planets
exhibited no regular progression. Kepler next tried if these distances
varied as the cosines of the quadrant, and if their motion varied as
the sun's, the sine of 90 representing the motion at the sun, and the
sine of 0 deg. that at the fixed stars; but in this trial he was also
disappointed.

Having spent the whole summer in these fruitless speculations, and
praying constantly to his Maker for success, he was accidentally drawing
a diagram in his lecture-room, in July 1595, when he observed the
relation between the circle inscribed in a triangle, and that described
round it; and the ratio of these circles, which was that of 1 to 2,
appeared to his eye to be identical with that of Jupiter's and Saturn's
orbits. Hence he was led to compare the orbits of the other planets'
circles described in pentagons and hexagons. As this hypothesis was as
inapplicable to the heavens as its predecessors, Kepler asked himself in
despair, "What have _plane_ figures to do with _solid_ orbits? Solid
bodies ought to be used for solid orbits." On the strength of this
conceit, he supposed that the distances of the planets were regulated by
the sizes of the five regular solids described within one another. "The
Earth is the circle, the measurer of all. Round it describe a
dodecahedron; the circle including this will be Mars. Round Mars
describe a tetrahedron; the circle including this will be Jupiter.
Describe a cube round Jupiter; the circle including this will be Saturn.
Then inscribe in the Earth an icosahedron; the circle described in it
will be Venus. Inscribe an octohedron in Venus; the circle inscribed in
it will be Mercury."

This discovery, as he considered it, harmonized in a very rude way with
the measures of the planetary orbits given by Copernicus; but Kepler was
so enamoured with it, that he ascribed the differences to errors of
observation, and declared that he would not renounce the glory of having
made it for the whole Electorate of Saxony.

In his attempt to discover the relation between the periodic times of
the planets and thei