conjunctions, he drew quasi-triangles in a circular zodiac showing the
slow progression of these points of conjunction at intervals of just
over 240° or eight signs. The successive chords marked out a smaller
circle to which they were tangents, about half the diameter of the
zodiacal circle as drawn, and Kepler at once saw a similarity to the
orbits of Saturn and Jupiter, the radius of the inscribed circle of an
equilateral triangle being half that of the circumscribed circle. His
natural sequence of ideas impelled him to try a square, in the hope that
the circumscribed and inscribed circles might give him a similar
"analogy" for the orbits of Jupiter and Mars. He next tried a pentagon
and so on, but he soon noted that he would never reach the sun that way,
nor would he find any such limitation as six, the number of "possibles"
being obviously infinite. The actual planets moreover were not even six
but only five, so far as he knew, so he next pondered the question of
what sort of things these could be of which only five different figures
were possible and suddenly thought of the five regular solids.[2] He
immediately pounced upon this idea and ultimately evolved the following
scheme. "The earth is the sphere, the measure of all; round it describe
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