gap appeared in one series, there was a corresponding gap in the
other. These gaps he attempted to fill by hypothetical planets between
Mars and Jupiter, and between Mercury and Venus, but this method also
failed to provide the regular proportion which he sought, besides being
open to the objection that on the same principle there might be many
more equally invisible planets at either end of the series. He was
nevertheless unwilling to adopt the opinion of Rheticus that the number
six was sacred, maintaining that the "sacredness" of the number was of
much more recent date than the creation of the worlds, and could not
therefore account for it. He next tried an ingenious idea, comparing the
perpendiculars from different points of a quadrant of a circle on a
tangent at its extremity. The greatest of these, the tangent, not being
cut by the quadrant, he called the line of the sun, and associated with
infinite force. The shortest, being the point at the other end of the
quadrant, thus corresponded to the fixed stars or zero force;
intermediate ones were to be found proportional to the "forces" of the
six planets. After a great amount of unfinished trial calculations,
which took nearly a whole summer, he convinced himself that success did
not lie that way. In July, 1595, while lecturing on the great planetary
conjunctions, he drew quasi-triangles in a circular zodiac showing the
slow progression of these points of conjunction at intervals of just
over 240 deg. 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
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