Fig. 6.5. Successive conjunctions of Jupiter and Saturn. Given a conjunction at 1, the next will be at 2, and so on.

on a circle and joined them up (see Figure 6.5) and noticed that the ratio of the radii of the outer circle to the inner circle that is formed by this procedure looked remarkably like the ratio of the sizes of the orbits of Saturn and Jupiter. The inner circle is very close to being inscribed inside an equilateral triangle and, if this were the case, the ratio of the radii would be 2, which was fairly close to the ratio of Saturn's distance to that of Jupiter (about 1.8 in the Copernican theory) and he tried to find similar relationships involving different polygons (squares, pentagons and so on) and match them to other planetary ratios. This did not work either, but 'the end of this useless attempt was also the beginning of the last, and successful one'.

In all Kepler's previous theorizing there was nothing to explain why there

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