0.4 + 0.3 · 2n
| Planet | Calculation | Prediction | Actual distance |
| Mercury | 0.4 + 0 · 0.3 | 0.4 | 0.39 |
| Venus | 0.4 + 1 · 0.3 | 0.7 | 0.72 |
| Earth | 0.4 + 2 · 0.3 | 1.0 | 1.00 |
| Mars | 0.4 + 4 · 0.3 | 1.6 | 1.52 |
| Ceres | 0.4 + 8 · 0.3 | 2.8 | 2.77 |
| Jupiter | 0.4 + 16 · 0.3 | 5.2 | 5.20 |
| Saturn | 0.4 + 32 · 0.3 | 10.0 | 9.54 |
| Uranus | 0.4 + 64 · 0.3 | 19.6 | 19.19 |
| Neptune | 0.4 + 128 · 0.3 | 38.8 | 30.07 |
Uranus[2] was discovered after Bode's publication, but the rule likely played no role in that discovery. There was no reason then to suppose that Saturn wasn't both the outermost planet and the last entry in the table. Ceres[3] is considered to be a major triumph of the Bode-Titius rule, Neptune[4] its major defeat.
Though the Bode-Titius rule played no part in the discovery of Uranus, that discovery was considered to be strong support for the validity of the rule. Science rightly favors theories which are a good fit for new data discovered after a theory was developed.
Ceres was discovered by chance, not by application of the Bode-Titius rule. Nevertheless, its orbit fit the rule so perfectly that there had been active search for a planet at that distance and the discovery was considered to be another vindication.
The Bode-Titius rule was used in the calculations that led to the discovery of Neptune. Only later was its orbit calculated precisely and found not to fit.
Friedrich Hegel published a philosophical objection to the Bode-Titius rule in 1801. His objection, which sounds strange to modern ears, was that no philosophy could agree with an attempt to express the distances of the planets numerically. Hegel's apparent distaste for such a mathematical expression may have prevented him from seeing what I think is the most fundamental flaw in the Bode-Titius rule.
Science favors theories that predict future discoveries, but it most emphatically does not favor theories that find a correlation without proposing a cause. The best objection to "Bode's Law", other than mine, has always been that no reason for the relation has ever been found.
Willy Ley, one of the greatest popular science writers of all time, wrote (see Ley 1966)
Professor Bode could not explain why the rule worked ... but anybody who could add and multiply had no doubt that it did work.But anybody who could raise two to the power of a negative integer should have had no doubt that the rule fails miserably for the case of Mercury. Working our way inward from Mars, the values for the number n in the formula should be 2 (Mars), 1 (Earth), 0 (Venus), -1 (Mercury). Since 2 raised to the power of -1 equals 0.5, the table entry for Mercury should be 0.4 + 0.5 · 0.3, giving a predicted distance of .55 au, not even close to the actual value of .39 au.
2. William Herschel discovered Uranus in 1781, nine years after Bode's publication. He wasn't looking for a planet, he was attempting to measure the distances of stars. While doing this, he noticed a visible disc too large to be a star, which turned out to be Uranus.
3. Ceres was the first asteroid to be discovered, by Giuseppe Piazzi in 1801. He actually first saw it on New Year's Eve in 1800, but didn't know it wasn't a star until the following night when he saw that it had moved. Thousands more asteroids have been discovered since then, the majority of them in similar orbits at similar distances from the Sun.
4. Neptune was first seen by Johann Galle in 1846, but credit for its discovery is given to Urbain Leverrier who predicted its position and told Galle where to look. The position was predicted earlier by John Couch Adams, but for various, mostly valid reasons (see Ley 1966, chapter 17) nobody looked for it based on his prediction.
Watchers of the Skies, Willy Ley, The Viking Press, 1966
A History of Astronomy, A. Pannekoek, Dover Publications, 1989