Physical sciences: Bode's Law and the Missing Planet

EXACTLY two hundred years ago, Titius¹ published a mnemonic for the mean distances of the planets from the Sun. His rule was where α _i is the major semi-axis of the orbit of the ith planet from the Sun. Titius's law represents the distances of the then known planets with an accuracy of a few per cent, provided that (i) for Mercury, we take i= -∞ instead of i=1 and (ii) the orbital i=5 is left vacant. The law made three valid predictions. Uranus (discovered by William Herschel in 1781) fits the orbital i=8. After the discovery of Uranus, Bode publicized the law, which became known as Bode's Law. The search for a planet for i=5 culminated in the discovery of the first asteroid, Ceres, in 1801. Recognition of a similar law for the satellite system of Saturn led to the discovery of Hyperion in 1848.