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a dodecahedron; the sphere including this will be Mars.
Round Mars describe a tetrahedron; the sphere including
this will be Jupiter. Describe a cube round Jupiter;
the sphere including this will be Saturn. Now,
inscribe in the earth an icosahedron, the sphere inscribed
in it will be Venus: inscribe an octahedron in
Venus: the circle inscribed in it will be Mercury.”
With this result Kepler was inordinately pleased,
and regretted not a moment of the time spent in obtaining
it, though to us this “Mysterium Cosmographicum”
can only appear useless, even without the more recent
additions to the known planets. He admitted that
a certain thickness must be assigned to the intervening
spheres to cover the greatest and least distances
of the several planets from the sun, but even then
some of the numbers obtained are not a very close fit
for the corresponding planetary orbits. Kepler’s
own suggested explanation of the discordances was
that they must be due to erroneous measures of the
planetary distances, and this, in those days of crude
and infrequent observations, could not easily be disproved.
He next thought of a variety of reasons why the five
regular solids should occur in precisely the order
given and in no other, diverging from this into a subtle
and not very intelligible process of reasoning to
account for the division of the zodiac into 360 deg..
The next subject was more important, and dealt with
the relation between the distances of the planets and
their times of revolution round the sun. It was
obvious that the period was not simply proportional
to the distance, as the outer planets were all too
slow for this, and he concluded “either that
the moving intelligences of the planets are weakest
in those that are farthest from the sun, or that there
is one moving intelligence in the sun, the common centre,
forcing them all round, but those most violently which
are nearest, and that it languishes in some sort and
grows weaker at the most distant, because of the remoteness
and the attenuation of the virtue”. This
is not so near a guess at the theory of gravitation
as might be supposed, for Kepler imagined that a repulsive
force was necessary to account for the planets being
sometimes further from the sun, and so laid aside the
idea of a constant attractive force. He made
several other attempts to find a law connecting the
distances and periods of the planets, but without success
at that time, and only desisted when by unconsciously
arguing in a circle he appeared to get the same result
from two totally different hypotheses. He sent
copies of his book to several leading astronomers,
of whom Galileo praised his ingenuity and good faith,
while Tycho Brahe was evidently much struck with the
work and advised him to adapt something similar to
the Tychonic system instead of the Copernican.
He also intimated that his Uraniborg observations
would provide more accurate determinations of the
planetary orbits, and thus made Kepler eager to visit
him, a project which as we shall see was more than