A little history (I)
After thousands of years of observation, the list of known planets seemed to end with Saturn. How did this list come to be expanded?
Although dim, Uranus is visible to the naked eye and, in fact, it was observed before being identified as a planet. In 1690 John Flamsteed observed it several times. Between 1750 and 1769 Pierre Lemonnier observed it twelve times.
These earlier observers may have misidentified it as a star.
Initially William Herschel observed it on 13/03/1781 and reported it as a comet. The Astronomer Royal, whom Herschel had notified, identified it as a planet.
By 1783 Herschel himself acknowledged that Uranus was a planet.
One could say that Herschel's achievement was more related to the correct identification of Uranus' true nature and that it largely was a serendipitous event.
A little history (II)
Unlike Uranus, Neptune is essentially invisible to the naked eye.
It may have been observed through telescope by Galileo (1613), Jerome Lalande (1795) and John Herschel (1830). However, like Uranus before, it was probably mistaken for a star, and, in any case, its true nature was never communicated by its would-be discoverers.
But the histories of Neptune and Uranus are related in a more fundamental way.
After Uranus' discovery, studies were conducted to predict its orbit. But these studies suffered from a curious problem: after a while, they would become increasingly imprecise. The observations, in other words, falsified the predictions.
Discrepancies between Uranus' actual position and Alexis Bouvard's 1821 predictions could be explained in at least three ways:
(1) Perhaps gravity, at such great distances, operated in a manner slightly different to what Newton described;
(2) Maybe Bouvard's predictions were based on observations containing systematic measurement errors;
(3) A yet undiscovered planet could be perturbing Uranus' orbit.
Two researchers, John Couch Adams and Urbain Le Verrier, independently decided to pursue the third explanation.
By early 1846 both researchers had produced at least some calculations, based on Newton's gravitation; and, being finally aware of each other's work, commissioned the Greenwich and Berlin observatories, respectively, to conduct the astronomical search.
The exact content of the calculations made by Adams and Le Verrier, as well as the details of the search conducted at Greenwich and Berlin, are subject to controversy.
The fact is that on 24/09/1846, after less than an hour search, the Berlin Observatory reported the planet had been found, very near where Le Verrier had predicted.
Adams recognized Le Verrier's priority. Apparently, the most recent opinion prevailing among historians of astronomy favors Le Verrier.
Discussion
Unlike the discovery of Uranus, where serendipity played such a clear role, the discovery of Neptune was hailed as a major triumph for Newtonian physics.
By correct deduction on the basis of current knowledge, Le Verrier and Adams had produced new knowledge. The apparent inexactitude in the orbit of Uranus had been explained: it had very little to do with systematic measurement errors and nothing to do with an erroneous appreciation of gravity by Newton.
However, I would like to point to an asymmetry between knowledge acquired through observation and knowledge acquired through deduction, at one hand; at the other hand, I would like to call the attention to a failure of some views on the philosophy of science.
The existence and true nature of Uranus was established by observation. That of Neptune was also established by observation.
Whoever rightfully deserves be credited with the correct prediction of the orbit, mass and angular speed of Neptune, was only formulating a hypothesis. That's why they required the collaboration of the observatories of Greenwich and Berlin.
In this sense, there is an asymmetry between knowledge acquired through observation and that acquired through deduction: deduction is neither sufficient nor necessary to determine the reality of a phenomenon.
It's not necessary, because discovery of a phenomenon can happen at least through serendipitous events, as in Uranus' case.
It's not sufficient, because predictions and hypothesis could be wrong, even if they are validly derived from acknowledged principles.
This last point was illustrated by the searches by Greenwich and Berlin. An even better illustration, however, comes also from the history of astronomy: the Ptolemaic system. A treatment of this episode falls outside the scope of this piece.
The second point I would like to make is that an extreme understanding of Popper's falsificationism would probably have discarded Newton's gravity, on the grounds of the discrepancy between the predicted orbit of Uranus and its actual orbit.
Contrary to Popper's view, Newtonian gravity gained credibility, and rightly so, when its apparent failure was explained through a discovery of a new planetary body.
NOTE: Images and information from Wikipedia.
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