Discovery of Neptune

The planet Neptune is a planet which has been claimed to have been discovered through mathematics alone. Discussions on this topic revolve around a claim that the position of Neptune was predicted and discovered through Newton's Laws of Gravitation, and that it is therefore a confirmation of Newton's theory.

Writer and astronomer Jeffrey Bennett describes this claim (Archive):

Perturbation Theory
According to some sources we find that Neptune was actually discovered, at least in part, on the basis of Perturbation Theory. Perturbation Theory predictions do not use the full laws of Newton, but are based on patterns.

https://www.encyclopediaofmath.org/index.php/Perturbation_theory (Archive)

From Perturbation Theory in Celestial Mechanics (Archive) by Dr. Alessandra Celletti we read:

See Astronomical Prediction Based on Patterns for information on Perturbation Theory.

Pre-Discovery Observations
Curiously, according to a depiction of its discovery it is described that Neptune was "pre-discovered" and that these observations were "important in accurately determining the orbit of Neptune."

https://en.wikipedia.org/wiki/Discovery_of_Neptune (Archive)

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]

Discovered By Luck
https://earthsky.org/human-world/today-in-science-discovery-of-neptune (Archive)

http://www.helas.gr/conf/2011/posters/S_5/dallas.pdf (Archive)

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:

Further Reference

 * The Planet Neptune in Earth Not a Globe