Neptune

Perturbation Theory

https://en.wikipedia.org/wiki/Perturbation_theory#History

  “ Perturbation theory was investigated by the classical scholars—Laplace, Poisson, Gauss—as a result of which the computations could be performed with a very high accuracy. The discovery of the planet Neptune in 1848 by Urbain Le Verrier, based on the deviations in motion of the planet Uranus (he sent the coordinates to Johann Gottfried Galle who successfully observed Neptune through his telescope), represented a triumph of perturbation theory.^[9] ”

https://www.encyclopediaofmath.org/index.php/Perturbation_theory

  “ The discovery of the planet Neptune in 1848 by J. Adams and U. le Verrier, based on the deviations in motion of the planet Uranus, represented a triumph of perturbation theory. ”

Perturbation Theory does not use the full laws of Newton. See: Astronomical Prediction Based on Patterns.

Pre-Discovery Observations

https://en.wikipedia.org/wiki/Discovery_of_Neptune

Neptune is too dim to be visible to the naked eye: its apparent magnitude is never brighter than 7.7.[5] Therefore, the first observations of Neptune were only possible after the invention of the telescope. There is evidence that Neptune was seen and recorded by Galileo Galilei in 1613, Jérôme Lalande in 1795 and John Herschel in 1830, but none is known to have recognized it as a planet at the time.[6] These pre-discovery observations were important in accurately determining the orbit of Neptune. Neptune would appear prominently even in early telescopes so other pre-discovery observation records are likely.[7]

American Journal of Science and Arts

From Vol IV of the American Journal of Science and Arts, November, 1847, we read the following from astronomer Sears C. Walker:

  “ If we admit for the moment that my views are correct, then LeVerrier's announcement of March 29th is in perfect accordance with that of Professor Peirce of the 16th of the same month, viz. that the present visible planet Neptune is not the mathematical planet to which theory had directed the telescope. None of its elements conform to the theoretical limits. Nor does it perform the functions on which alone its existence was predicted, viz. those of removing that opprobrium of astronomers, the unexplained perturbations of Uranus.

We have it on the authority of Professor Peirce that if we ascribe to Neptune a mass of three-fourths of the amount predicted by LeVerrier, it will have the best possible effect in reducing the residual perturbations of Uranus below their former value; but will nevertheless leave them on the average two-thirds as great as before.

It is indeed remarkable that the two distinguished European astronomers, LeVerrier and Adams, should, by a wrong hypothesis, have been led to a right conclusion respecting the actual position of a planet in the heavens. It required for their success a compensation of errors. The unforeseen error of sixty years in their assumed period was compensated by the other unforeseen error of their assumed office of the planet. If both of them had committed only one theoretical error, (not then, but now believed to be such,) they would, according to Prof. Peirce's computations, have agreed in pointing the telescope in the wrong direction, and Neptune might have been unknown for years to come. ”

See Also

The Planet Neptune in Earth Not a Globe