The Bode-Titius relationship may give us a key to the answer. Titius and Bode, two German astronomers, individually, concurrently, discovered this relationship in the eighteenth century. If we take orbit numbers, or ring numbers, of the planets through Saturn the ring numbers being 0, 1, 2, 4, 8, 16, 32, multiply each number by 3, add 4 to each result, divide each by 10, the series becomes 0.4, 0.7, 1.0, 1.6, 2.8, 5.2, and 10.0. These numbers, excepting and skipping 2.8, represent the relative distances of the then known planets from the Sun – Mercury, Venus, Earth, Mars, 2.8, Jupiter, and Saturn – with 2.8 representing no known planet at that time for that distance.

When the planet Uranus was discovered in 1781, it fit right into the series at 19.6; the “law” seemed strengthened, and an intense search was initiated for anything that might be at the 2.8 distance. In 1801 the little planetoid Ceres was discovered at 2.8; by 1945 more than 1,500 more were found on the same orbit. It has been well established as the ring of minor planets, or planetoids, or asteroids.

In 1846 the planet Neptune was discovered – and it seemed to disobey the rules set down by the Bode-Titius relationship. It should have been at 38.8 on the relative distance scale – but it was closer to 29.2.

In 1930 the planet Pluto was discovered, and the Bode-Titius “law” seemed to fall apart completely. Pluto was found close to 38.8, where Neptune was supposed to be, whereas the “law” seemed to indicate that Pluto should be at 77.2.

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Since that time the relationship, commonly known as “Bode’s Law”, has been regarded in astronomy as nothing more than an insignificant curiosity.

Perhaps a new look at Bode’s Law is in order. If so much of it is correct, then the part of it which appears to be erroneous seems to be so only because of our lack of understanding of the basics involved.

First, instead of using relative numbers, we shall work with ring numbers, or orbit numbers. The first progression (0, 1, 2, 4, 8, 16, 32, etc.) represents these numbers. Also, instead of this progression – which is geometric except for the zero – let’s fill in all of the numbers, making a true arithmetic progression. The numbers will be 0, 1, 2, 3, 4, 5, 6, 7, 8, – so on to 256.

Now in this progression the ring numbers 0, 1, 2, 4, 8, 16, 32, 64, 128, and 256 can be regarded as fundamental rings. All other rings can be regarded as harmonic rings. Between any two fundamentals, the ring which lies halfway between is the first harmonic; any ring which lies halfway between a fundamental and a first harmonic is a second harmonic; any ring halfway between a second harmonic and a first harmonic, or halfway between a second harmonic and a fundamental, is a third harmonic, and so on.

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