The Discovery of  Neptune

Soon after discovery of Uranus, astronomers realized that this planet had been observed numerous times, as far back as 1690, but was never recognized as a planet.  By early in the 19th century, it was clear that the historical and new observations of Uranus could not be reconciled with a single elliptical orbit, as demanded by Kepler's and Newton's laws.  The errors in the observed position of Uranus were never more than 2 minutes of arc off from those predicted, but even this small error is much larger than observational error. Clearly the observations were not the source of the error.  So either the law of gravity is wrong (or incomplete) or there is some other physical law waiting to be discovered or ...?

In the period from 1843 to 1845, the English astronomer John Couch Adams analyzed all  the existing Uranus data.  In October, 1845, he proposed an answer: there is another massive planet out there.  The gravitational tug of that planet influences Uranus' orbit.  He predicted where and how big that planet must be.  British Astronomer Royal, Sir George Airy, didn't take the prediction very seriously and didn't push his staff to make the necessary observations.

In 1846, Frenchman Urbain Le Verrier independently made the same mathematical prediction as Adams.  He sent his calculations to Johann Galle at the Berlin Obsevatory.  Galle received the letter from Leverrier on the afternoon of Sept 23, 1846.  Within 30 minutes of looking, he found Neptune that very night, exactly (within two minutes of arc) where he'd been advised to look.

(As it turns out, Galileo himself observed Neptune twice  (12/24/1612, 1/28/1613) but didn't recognize that it wasn't a star when it was in the same field of view for him as Jupiter.)

Again, we have an incredible triumph of the law of Gravity.  Newton is king.  Long live Newton!  And what of the Titius-Bode "law"? Neptune has a semi-major axis of 30.11 AU.

 Planet number Add 4 Divide by 10 Actual a (AU) Planet? 0 4 0.4 0.39 Mercury 3 7 0.7 0.72 Venus 6 10 1.0 1.00 Earth 12 16 1.6 1.52 Mars 24 28 2.8 2.6-2.8 Ceres,Pallas, Vesta, Juno, etc. 48 52 5.2 5.2 Jupiter 96 100 10.0 9.56 Saturn 192 196 19.6 19.22 Uranus 384 388 38.8 30.11 Neptune 768 772 77.2 ... ...

This result is inauspicious for the Titius-Bode "law."  Neptune is in the wrong place by quite a large margin.  At this point, a sensible approach would be to toss out the Titius-Bode "law" as a curious piece of numerology.  It clearly is not a fundamental law of physics. It is not a good theory with predictive power.  However, it does lead us to ask interesting questions, such as "why are the planets spaced the way that they are?"

...  the discovery of Pluto ...