but, as he himself says, on physical or metaphysical grounds.  In 1595, having more leisure from lectures, he turned his speculative mind to the number, size, and motion of the planetary orbits.  He first tried simple numerical relations, but none of them appeared to be twice, thrice, or four times as great as another, although he felt convinced that there was some relation between the motions and the distances, seeing that when a 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 pounced upon this idea and ultimately evolved the following scheme.  “The earth is the sphere, the measure of all; round it describe