lly I may remark that it is not known even now: a crude
empirical statement known as Bode's law--see page 294--is all that has
been discovered.)

Once more, the further the planet the slower it moved; there seemed to
be some law connecting speed and distance. This also Kepler made
continual attempts to discover.

[Illustration: FIG. 26.--Orbits of some of the planets drawn to scale:
showing the gap between Mars and Jupiter.]

One of his ideas concerning the law of the successive distances was
based on the inscription of a triangle in a circle. If you inscribe in a
circle a large number of equilateral triangles, they envelop another
circle bearing a definite ratio to the first: these might do for the
orbits of two planets (see Fig. 27). Then try inscribing and
circumscribing squares, hexagons, and other figures, and see if the
circles thus defined would correspond to the several planetary orbits.
But they would not give any satisfactory result. Brooding over this
disappointment, the idea of trying solid figures suddenly strikes him.
"What have plane figures to do with the celestial orbits?" he cries out;
"inscribe the regular solids." And then--brilliant idea--he remembers
that there are but five. Euclid had shown that there could be only five
regular solids.[4] The number evidently corresponds to the gaps between
the six planets. The reason of there being only six seems to be
attained. This coincidence assures him he is on the right track, and
with great enthusiasm and hope he "represents the earth's orbit by a
sphere as the norm and measure of all"; round it he circumscribes a
dodecahedron, and puts another sphere round that, which is approximately
the orbit of Mars; round that, again, a tetrahedron, the corners of
which mark the sphere of the orbit of Jupiter; round that sphere, again,
he places a cube, which roughly gives the orbit of Saturn.

[Illustration: FIG. 27.--Many-sided polygon or approximate circle
enveloped by straight lines, as for instance by a number of equilateral
triangles.]

On the other hand, he inscribes in the sphere of the earth's orbit an
icosahedron; and inside the sphere determined by that, an octahedron;
which figures he takes to inclose the spheres of Venus and of Mercury
respectively.

The imagined discovery is purely fictitious and accidental. First of
all, eight planets are now known; and secondly, their real distances
agree only very approximately with Kepler's hypothesis.

[Ill