The follow up (correct me on the details here) is that they later measured Mercury’s orbit to a high precision, estimated that some body was disturbing Mercury’s orbit, and went looking again to repeat Sir Isaac’s miraculous math. But… nothing was there. It turned out Mercury orbits at relativistic speeds and Newton’s model wasn’t taking that into account.
Vaguely remember this from Kuhn’s Structure of Scientific Revolutions
In the case of Neptune, Le Verrier hit the nail right on its head (where others suggested a limit to Newton's theory of gravity). This tremendous feat gave him all the reason to search for other perturbations around the sky to find unknown objects.
So, by 1859 he was the first to find Mercury not completely adhering to Newtonian mechanics (anomalous rate of precession of the perihelion[0]), therefore he postulated smaller unseen objects in its vicinity, even naming a possible planet Vulcan.
But there, he was wrong and Newton's law really hit a limit here at such proximity to the sun.
Ironically not at "long distances" as suggested in the anomaly of Uranus' orbit. Also, Einstein within his GR framework correctly predicted the value of deflection of "starlight passing very close to the sun" in 1915 which was validated by the measurements of Eddington taking advantage of a total eclipse in 1919 --- a natural occurring coronagraph (a big moon) of the Sun from Earth's view ;)
P.S.: I forgot to mention D'Arrest[1] role in finding Neptune in the sky:
> While still a student at the University of Berlin, d'Arrest was party to Johann Gottfried Galle's search for Neptune. On 23 September 1846, he suggested that a recently drawn chart of the sky, in the region of Urbain Le Verrier's predicted location, could be compared with the current sky to seek the displacement characteristic of a planet, as opposed to a stationary star.
[0]https://en.m.wikipedia.org/wiki/Tests_of_general_relativity#...
[1]https://en.m.wikipedia.org/wiki/Heinrich_Louis_d%27Arrest#Bi...
Meanwhile, my same country mail takes up to 2 weeks to arrive. No wonder I'm not making any ground breaking advances.
But, yes - if you will - the area of the solar system consequently has more than doubled; which has nicer ring to it :)
To be exact if my numbers are correct:
πab = πa² √(1-e²)
Uranus (a = 19.19126 AU; e = 0.04717)
⇒ ~ 1155.755 AU²
Neptune (a = 30.07 AU; e = 0.008678 (what a neat circle!))
⇒ ~ 2840.54 AU²
i.e. approx. 2.46x the former area
Tomorrow (23rd) marks the 176th anniversary of the discovery of Neptune (~ 1.07 years on Neptune; 164.8 Earth years).
Johann Gottfried Galle an astronomer at the Berlin observatory found Neptune at the behest of Urbain Le Verrier who calculated its position by hypothesizing that some body must perturb Uranus orbit to explain the distinct deviation from the predicted motion by Newton's law of universal gravitation.
Le Verrier couldn't find a single French astronomer willing to take a look out at the sky after presenting his extensive calculations to the French Academy on 31 August. On 18 September he then wrote a letter to the German astronomer Galle at the Berlin observatory.
After receiving the letter 5 days later Galle immediately set up his telescope the same night and found a +8 mag star (within 1 degree of the Le Verrier's calculation) not listed on the official Prussian star chart. The next night he managed to measure the star itself moving by 4 arcseconds, finally confirming Le Verrier prediction of a new planet (and to this day last).
At ~30 AU (30x the distance sun-earth; Uranus being at ~20 AU) the discovery instantaneously made the solar system 1.5x bigger ;)