Discovery of Neptune
The planet Neptune was mathematically predicted before it was directly observed. With a prediction by Urbain Le Verrier, telescopic observations confirming the existence of a major planet were made on the night of September 23–24, 1846, at the Berlin Observatory, by astronomer Johann Gottfried Galle (assisted by Heinrich Louis d'Arrest), working from Le Verrier's calculations. It was a sensational moment of 19th-century science, and dramatic confirmation of Newtonian gravitational theory. In François Arago's apt phrase, Le Verrier had discovered a planet "with the point of his pen".
In retrospect, after it was discovered, it turned out it had been observed many times before but not recognized, and there were others who made various calculations about its location which did not lead to its observation. By 1847, the planet Uranus had completed nearly one full orbit since its discovery by William Herschel in 1781, and astronomers had detected a series of irregularities in its path that could not be entirely explained by Newton's law of universal gravitation. These irregularities could, however, be resolved if the gravity of a farther, unknown planet were disturbing its path around the Sun. In 1845, astronomers Urbain Le Verrier in Paris and John Couch Adams in Cambridge separately began calculations to determine the nature and position of such a planet. Le Verrier's success also led to a tense international dispute over priority, because shortly after the discovery George Airy, at the time British Astronomer Royal, announced that Adams had also predicted the discovery of the planet. Nevertheless, the Royal Society awarded Le Verrier the Copley medal in 1846 for his achievement, without mention of Adams. The Royal Society, however, also awarded Adams the Copley medal in 1848.
Neptune is too dim to be visible to the naked eye: its apparent magnitude is never brighter than 7.7. 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. 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.
Galileo's drawings show that he observed Neptune on December 28, 1612, and again on January 27, 1613; on both occasions, Galileo mistook Neptune for a fixed star when it appeared very close (in conjunction) to Jupiter in the night sky. Historically it was thought that he believed it to be a fixed blue star, and so he is not credited with its discovery. At the time of his first observation in December 1612, it was stationary in the sky because it had just turned retrograde that very day; because it was only beginning its yearly retrograde cycle, Neptune's motion was thought to be too slight, and its apparent size too small, to clearly appear to be a planet in Galileo's small telescope. However, in July 2009 University of Melbourne physicist David Jamieson announced new evidence suggesting that Galileo was indeed aware that he had discovered something unusual about this "star". Galileo, in one of his notebooks, noted the movement of a background star (Neptune) on January 28 and a dot (in Neptune's position) drawn in a different ink suggests that he found it on an earlier sketch, drawn on the night of January 6, suggesting a systematic search among his earlier observations. However, so far there is neither clear evidence that he identified this moving object as a planet, nor that he published these observations of it. There is no evidence that he ever attempted to observe it again.
In 1847, Sears C. Walker of the U.S. Naval Observatory searched historical records and surveys for possible prediscovery sightings of the planet Neptune. He found that observations made by Lalande's staff at the Paris Observatory in 1795 were in the direction of Neptune's position in the sky. In the catalog observations for May 8 and again on May 10 of 1795 a star was observed in the approximate position expected for Neptune. The uncertainty of the position was noted with a colon. This notation was also used to indicate an observation error so it was not until the original records of the observatory were reviewed that it was established with certainty that the object was Neptune and the position error in the observations made two nights apart was due to the planet's motion across the sky. The discovery of these records of Neptune's position in 1795 led to a better calculation of the planet's orbit.
John Herschel almost discovered Neptune the same way his father, William Herschel, had discovered Uranus in 1781, by chance observation. In an 1846 letter to Wilhelm Struve, John Herschel states that he observed Neptune during a sweep of the sky on July 14, 1830. Although his telescope was powerful enough to resolve Neptune into a small blue disk and show it to be a planet, he did not recognize it at the time and mistook it for a star.
Irregularities in Uranus' orbit
In 1821, Alexis Bouvard had published astronomical tables of the orbit of Uranus, making predictions of future positions based on Newton's laws of motion and gravitation. Subsequent observations revealed substantial deviations from the tables, leading Bouvard to hypothesize some perturbing body. These irregularities or "residuals", both in the planet's ecliptic longitude and in its distance from the Sun, or radius vector, might be explained by a number of hypotheses: the effect of the Sun's gravity, at such a great distance might differ from Newton's description; or the discrepancies might simply be observational error; or perhaps Uranus was being pulled, or perturbed, by an as-yet undiscovered planet.
Adams learned of the irregularities while still an undergraduate and became convinced of the "perturbation" hypothesis. Adams believed, in the face of anything that had been attempted before, that he could use the observed data on Uranus, and utilising nothing more than Newton's law of gravitation, deduce the mass, position and orbit of the perturbing body.
In modern terms, the problem is an inverse problem, an attempt to deduce the parameters of a mathematical model from observed data. Though the problem is a simple one for modern mathematics after the advent of electronic computers, at the time it involved much laborious hand calculation. Adams began by assuming a nominal position for the hypothesised body, using the empirical Bode's law. He then calculated the path of Uranus using the assumed position of the perturbing body and calculated the difference between his calculated path and the observations, in modern terms the residuals. He then adjusted the characteristics of the perturbing body in a way suggested by the residuals and repeated the process, a process similar to regression analysis.
On 13 February 1844, James Challis, director of the Cambridge Observatory, requested data on the position of Uranus, for Adams, from Astronomer Royal George Biddell Airy at the Royal Observatory, Greenwich. Adams certainly completed some calculations on 18 September 1845.
Supposedly, Adams communicated his work to Challis in mid-September 1845 but there is some controversy as to how. The story and date of this communication only seem to have come to light in a letter from Challis to the Athenaeum dated 17 October 1846. However, no document was identified until 1904 when Sampson suggested a note in Adams's papers that describes "the New Planet" and is endorsed, in handwriting not Adams's, with the note "Received in September 1845". Though this has often been taken to establish Adams's priority, some historians have disputed its authenticity, on the basis that "the New Planet" was not a term current in 1845, and on the basis that the note is dated only after the fact by someone other than Adams. Further, the results of the calculations are different from those communicated to Airy a few weeks later. Adams certainly gave Challis no detailed calculations and Challis was unimpressed by the description of his method of successively approximating the position of the body, being disinclined to start a laborious observational programme at the observatory, remarking "while the labour was certain, success appeared to be so uncertain."
Meanwhile, Urbain Le Verrier, on November 10, 1845, presented to the Académie des sciences in Paris a memoir on Uranus, showing that the pre-existing theory failed to account for its motion. Unaware of Adams's work, he attempted a similar investigation, and on June 1, 1846, in a second memoir presented to a public meeting of the Académie, gave the position, but not the mass or orbit, of the proposed perturbing body. Le Verrier located Neptune within one degree of its discovery position.
Upon receiving in England the news of Le Verrier's June prediction, George Airy immediately recognized the similarity of Le Verrier's and Adams' solutions. Up until that moment, Adams' work had been little more than a curiosity, but independent confirmation from Le Verrier spurred Airy to organize a secret attempt to find the planet. At a July 1846 meeting of the Board of Visitors of the Greenwich Observatory, with Challis and Sir John Herschel present, Airy suggested that Challis urgently look for the planet with the Cambridge 11.25 inch equatorial telescope, "in the hope of rescuing the matter from a state which is ... almost desperate". The search was begun by a laborious method on 29 July. Adams continued to work on the problem, providing the British team with six solutions in 1845 and 1846  which sent Challis searching the wrong part of the sky. Only after the discovery of Neptune had been announced in Paris and Berlin did it become apparent that Neptune had been observed on August 8 and August 12 but because Challis lacked an up-to-date star-map, it was not recognized as a planet.
Discovery observation: September 24, 1846
Le Verrier was unaware that his public confirmation of Adams' private computations had set in motion a British search for the purported planet. On 31 August, Le Verrier presented a third memoir, now giving the mass and orbit of the new body. Having been unsuccessful in his efforts to interest any French astronomer in the problem, Le Verrier finally sent his results by post to Johann Gottfried Galle at the Berlin Observatory. Galle received Le Verrier's letter on 23 September and immediately set to work observing in the region suggested by Le Verrier. Galle's student, Heinrich Louis d'Arrest, suggested that a recently drawn chart of the sky, in the region of 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.
Neptune was discovered just after midnight, after less than an hour of searching and less than 1 degree from the position Le Verrier had predicted, a remarkable match. After two further nights of observations in which its position and movement were verified, Galle replied to Le Verrier with astonishment: "the planet whose place you have [computed] really exists" (emphasis in original). The discovery telescope was an equatorial mounted achromatic refractor by Joseph Fraunhofer's firm Merz und Mahler.
On the announcement of the discovery, Herschel, Challis and Richard Sheepshanks, foreign secretary of the Royal Astronomical Society, announced that Adams had already calculated the planet's characteristics and position. Airy, at length, published an account of the circumstances, and Adams's memoir was printed as an appendix to the Nautical Almanac. However, it appears that the version published by Airy had been edited by the omission of a "crucial phrase" to disguise the fact that Adams had quoted only mean longitude and not the orbital elements.
A keen controversy arose in France and England as to the merits of the two astronomers. There was much criticism of Airy in England. Adams was a diffident young man who was naturally reluctant to publish a result that would establish or ruin his career. Airy and Challis were criticised, particularly by James Glaisher, as failing to exercise their proper role as mentors of a young talent. Challis was contrite but Airy defended his own behaviour, claiming that the search for a planet was not the role of the Greenwich Observatory. On the whole, Airy has been defended by his biographers. In France the claims made for an unknown Englishman were resented as detracting from the credit due to Le Verrier's achievement.
The Royal Society awarded Le Verrier the Copley medal in 1846 for his achievement, without mention of Adams, but Adams's academic reputation at Cambridge, and in society, was assured. As the facts became known, some British astronomers pushed the view that the two astronomers had independently solved the problem of Neptune, and ascribed equal importance to each. But Adams himself publicly acknowledged Le Verrier's priority and credit (not forgetting to mention the role of Galle) in the paper that he gave to the Royal Astronomical Society in November 1846:
I mention these dates merely to show that my results were arrived at independently, and previously to the publication of those of M. Le Verrier, and not with the intention of interfering with his just claims to the honours of the discovery ; for there is no doubt that his researches were first published to the world, and led to the actual discovery of the planet by Dr. Galle, so that the facts stated above cannot detract, in the slightest degree, from the credit due to M. Le Verrier.— Adams (1846) 
The criticism was soon afterwards made, that both Adams and Le Verrier had been over-optimistic in the precision they claimed for their calculations, and both had, by using Bode's law, greatly overestimated the planet's distance from the sun. Further, it was suggested that they both succeeded in getting the longitude almost right only because of a "fluke of orbital timing". This criticism was discussed in detail by Danjon (1946)  who illustrated with a diagram and discussion that while hypothetical orbits calculated by both LeVerrier and Adams for the new planet were indeed of very different size on the whole from that of the real Neptune (and actually similar to each other), they were both much closer to the real Neptune over that crucial segment of orbit covering the interval of years for which the observations and calculations were made, than they were for the rest of the calculated orbits. So the fact that both the calculators used a much larger orbital major axis than the reality was shown to be not so important, and not the most relevant parameter.
The new planet, at first called "Le Verrier" by François Arago, received by consensus the neutral name of Neptune. Its mathematical prediction was a great intellectual feat, but it showed also that Newton's law of gravitation, which Airy had almost called in question, prevailed even at the limits of the Solar System.
I could not expect however that practical astronomers, who were already fully occupied with important labours, would feel as much confidence in the results of my investigations, as I myself did.
By contrast, Le Verrier was arrogant and assertive, enabling the British scientific establishment to close ranks behind Adams while the French, in general, found little sympathy with Le Verrier. In 1874–1876, Adams was President of the Royal Astronomical Society when it fell to him to present the RAS Gold Medal of the year to Le Verrier.
The conventional wisdom that Neptune's discovery should be "credited to both Adams and Le Verrier" has recently been challenged putting in doubt the accounts of Airy, Challis and Adams in 1846.
In 1999, Adams's correspondence with Airy, which had been lost by the Royal Greenwich Observatory, was rediscovered in Chile among the possessions of astronomer Olin J. Eggen after his death. In an interview in 2003, historian Nicholas Kollerstrom concluded that Adams's claim to Neptune was far weaker than had been suggested, as he had vacillated repeatedly over the planet's exact location, with estimates ranging across 20 degrees of arc. Airy's role as the hidebound superior willfully ignoring the upstart young intellect was, according to Kollerstrom, largely constructed after the planet was found, in order to boost Adams's, and therefore Britain's, credit for the discovery. A later Scientific American article by Sheehan, Kollerstrom and Waff claimed more boldly "The Brits Stole Neptune" and concluded "The achievement was Le Verrier's alone."
Neptune discovery telescope
The telescope, at New Berlin Observatory (1835–1913), that discovered Neptune was an achromatic refractor with an aperture of 9 Paris inches (9.6 English inches, or 24.4 cm). Made by the late Joseph Fraunhofer's firm, Merz und Mahler, it was a high-performance telescope of its era, with one of the largest achromatic doublets available and a finely made equatorial mount, with a clock drive to move the 4 m (13.4′) main tube at the same rate as Earth's rotation. Eventually the telescope was moved to the Deutsches Museum in Munich, Germany, where it can still be seen as an exhibit.
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