The Titius-Bode Law is an apparent relationship between the the orbits of the planets about the Sun. It was published in 1766 by Johann Titius
but it did not gain acceptance until it was popularised by Johann Bode
in the 1770s. The name is a misnomer as it is not a "law" as such; it does not give accurate results and has no basis in theory.
The "law" goes as follows:
Take the series of numbers 0, 3, 6, 12, 24, 48, 96, ...
Add 4 to each number.
Divide each number by 10. The resulting numbers are (approximately) the orbital distances of the planets in AU.
The relationship predicts a "missing planet" between Mars and Jupiter. It was the search for this that resulted in the discovery of the asteroid belt. In fact, as the asteroid belt fits the law so well, some suggest that perhaps it is the remains of a planet which was destroyed in the early days of the solar system.
The law also fails when applied to the outer planets. After Uranus, it predicts that the next two planets should have orbital distances of 38.8 and 77.2 AU. In fact Neptune and Pluto are much closer, at 30.06 and 39.44 AU respectively.
Most modern scientists believe that the Titius-Bode law is no more than a statistical fluke and that it has no basis in any physical law. However, the recent discovery of extrasolar planets has lent weight to the idea, as some of these systems seem to obey similar relationships.