Though Kepler does not derive this curious quantity, a simple analysis shows that he was calculating the angular speed rn that satisfies the relations rn/rnp = rp/r2 and = r2/r2, where r = (ra + rp)/2 is the mean distance. If we add and multiply these relations together we can derive

and if we then combine these relations, we obtain rn = 2G2 /(A + G). On the assumption that A and G are close together, we have

« = 2G2(2G + A - G)-1 = G ^ 1 + A2gG) ^ G - ±(A - G).

See Brackenridge (1982) for an analysis.

Table 6.3. The accuracy of Kepler's third law based on Kepler's data.

Period T (years) Relative distance a T2 a3

Mercury 0.242 0.388 0.0584 0.0580

Venus 0.616 0.724 0.3795 0.3795

Earth 1.000 1.000 1.000 1.000

Mars 1.881 1.524 3.540 3.538

Jupiter 11.86 5.200 140.61 140.73

Saturn 29.33 9.510 860.08 867.69

The third law brought the whole of Copernican planetary theory together. While Copernicus had improved on Ptolemy in that the relative planetary distances could be determined from his theory by observation, there was no reason in Copernican theory for these distances being what they are. As far as Kepler was concerned, his third law provided that final link, and at the same time further justified his belief that the nature of the Universe could be described by simple mathematical relationships. The third, or harmonic, law was not given much emphasis in the Harmony of the World; in particular, Kepler did not publish a table of values showing the accuracy with which the third law relates the periods and distances of the planets. However, he must have performed these

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