His most famous achievement was predicting the existence and position of Neptune, using only mathematics. The calculations were made to explain discrepancies with Uranus's orbit and the laws of Kepler and Newton. At the same time, but unknown to each other, the same calculations were made by Urbain Le Verrier. Le Verrier would assist Galle in locating the planet (September 1846); which was found within 1° of its predicted location, a point in Aquarius. (There was, and to some extent still is, some controversy over the apportionment of credit for the discovery; see Discovery of Neptune.)
He was Lowndean Professor at the University of Cambridge for thirty-three years from 1859 to his death. He won the Gold Medal of the Royal Astronomical Society in 1866. In 1884, he attended the International Meridian Conference as a delegate for Britain.
A crater on the Moon is jointly named after him, Walter Sydney Adams and Charles Hitchcock Adams. Neptune's outermost known ring and the asteroid 1996 Adams are also named after him. The Adams Prize, presented by the University of Cambridge, commemorates his prediction of the position of Neptune. His personal library is now in the care of Cambridge University Library.
John attended the Laneast village school where he acquired some Greek and algebra. From there, he went, at the age of twelve, to Devonport, where his mother's cousin, the Rev. John Couch Grylls, kept a private school. There he learned classics but was largely self-taught in mathematics, studying in the Library of Devonport Mechanics' Institute and reading Rees's Cyclopaedia and Samuel Vince's Fluxions. He observed Halley's comet in 1835 from Landulph and the following year started to make his own astronomical calculations, predictions and observations, engaging in private tutoring to finance his activities.
In 1836, his mother inherited a small estate at Badharlick and his promise as a mathematician induced his parents to send him to the University of Cambridge, and in October 1839 he entered as a sizar at St John's College. He graduated with a B.A. in 1843 as the senior wrangler and first Smith's prizeman of his year.
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. Adams learnt of the irregularities while still an undergraduate and became convinced of the "perturbation" theory. 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. On the 3rd of July 1841, he noted his intention to work on the problem.
After his final examinations in 1843, Adams was elected fellow of his college and spent the summer vacation in Cornwall calculating the first of six iterations. While he worked on the problem back in Cambridge, he tutored undergraduates, sending money home to educate his brothers, and even taught Mrs Ireland, his bedmaker, to read.
Supposedly, Adams communicated his work to James Challis, director of the Cambridge Observatory, in mid-September 1845 but there is some controversy as to how. On 21 October 1845, Adams, returning from a Cornwall vacation and without appointment, twice called on Astronomer Royal George Biddell Airy in Greenwich. Failing to find him at home, Adams reputedly left a manuscript of his solution, again without the detailed calculations. Airy responded with a letter to Adams asking for some clarification. It appears that Adams did not regard the question as "trivial", as is often alleged, but he failed to complete a response. Various theories have been discussed as to Adams's failure to reply, such as his general nervousness, procrastination and disorganisation.
Meanwhile, Urbain Le Verrier, on 10 November 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. On reading Le Verrier's memoir, Airy was struck by the coincidence and initiated a desperate race for English priority in discovery of the planet. The search was begun by a laborious method on 29 July. Only after the discovery of Neptune on 23 September 1846 had been announced in Paris did it become apparent that Neptune had been observed on 8 and 12 August but because Challis lacked an up-to-date star-map it was not recognized as a planet.
A keen controversy arose in France and England as to the merits of the two astronomers. As the facts became known, there was wide recognition that the two astronomers had independently solved the problem of Uranus, and each was ascribed equal importance. However, there have been subsequent assertions that "The Brits Stole Neptune" and that Adams's British contemporaries retrospectively ascribed him more credit than he was due. But it is also notable (and not included in some of the foregoing discussion references) that Adams himself publicly acknowledged Le Verrier's priority and credit (not forgetting to mention the role of Galle) in the paper that he gave 'On the Perturbations of Uranus' to the Royal Astronomical Society in November 1846:-
Adams held no bitterness towards Challis or Airy and acknowledged his own failure to convince the astronomical world:
In 1852, he published new and accurate tables of the moon's parallax, which superseded Johann Karl Burckhardt's, and supplied corrections to the theories of Marie-Charles Damoiseau, Giovanni Antonio Amedeo Plana, and Philippe Gustave Doulcet.
He had hoped that this work would leverage him into the vacant post as superintendent of HM Nautical Almanac Office but John Russell Hind was preferred, Adams lacking the necessary ability as an organiser and administrator.
In 1820, at the insistence of the Académie des sciences, Damoiseau, Plana and Francesco Carlini revisited Laplace's work, investigating quadratic and higher-order perturbing terms, and obtained similar results, again addressing only a radial, and neglecting tangential, gravitational force on the Moon. Hansen obtained similar results in 1842 and 1847.
In 1853, Adams published a paper showing that, while tangential terms vanish in the first-order theory of Laplace, they become substantial when quadratic terms are admitted. Small terms integrated in time come to have large effects and Adams concluded that Plana had overestimated the secular acceleration by approximately 1.66" per century.
At first, Le Verrier rejected Adams's results. In 1856, Plana admitted Adams's conclusions, claiming to have revised his own analysis and arrived at the same results. However, he soon recanted, publishing a third result different both from Adams's and Plana's own earlier work. Delaunay in 1859 calculated the fourth-order term and duplicated Adams's result leading Adams to publish his own calculations for the fifth, sixth and seventh-order terms. Adams now calculated that only 5.7" of the observed 11" was accounted for by gravitational effects.
Later that year, Philippe Gustave Doulcet, Comte de Pontécoulant published a claim that the tangential force could have no effect though Peter Andreas Hansen, who seems to have cast himself in the role of arbitrator, declared that the burden of proof rested on Pontécoulant, while lamenting the need to discover a further effect to account for the balance. Much of the controversy centred around the convergence of the power series expansion used and, in 1860, Adams duplicated his results without using a power series. Sir John Lubbock also duplicated Adams's results and Plana finally concurred. Adams's view was ultimately accepted and further developed, winning him the Gold Medal of the Royal Astronomical Society in 1866. The unexplained drift is now known to be due to tidal acceleration.
In 1858 Adams became professor of mathematics at the University of St Andrews, but lectured only for a session, before returning to Cambridge for the Lowndean professorship of astronomy and geometry. Two years later he succeeded Challis as director of the Observatory, a post Adams held until his death.
The great meteor shower of November 1866 turned his attention to the Leonids, whose probable path and period had already been discussed and predicted by Hubert Anson Newton in 1864. Newton had asserted that the longitude of the ascending node, that marked where the shower would occur, was increasing and the problem of explaining this variation attracted some of Europe's leading astronomers.
Using a powerful and elaborate analysis, Adams ascertained that this cluster of meteors, which belongs to the solar system, traverses an elongated ellipse in 33.25 years, and is subject to definite perturbations from the larger planets, Jupiter, Saturn and Uranus. These results were published in 1867.
Some experts consider this Adams's most substantial achievement. His "definitive orbit" for the Leonids coincided with that of the comet 55P/Tempel-Tuttle and therefore suggested the, later widely accepted, close relationship between comets and meteors.
Over a period of forty years, he periodically addressed the determination of the constants in Carl Friedrich Gauss's theory of terrestrial magnetism. Again, the calculations involved great labour, and were not published during his lifetime. They were edited by his brother, William Grylls Adams, and appear in the second volume of the collected Scientific Papers. Numerical computation of this kind might almost be described as his pastime. He calculated the Euler–Mascheroni constant, perhaps somewhat eccentrically, to 236 decimal places and evaluated the Bernoulli numbers up to the 62nd.
Adams had boundless admiration for Newton and his writings and many of his papers bear the cast of Newton's thought. In 1872, Isaac Newton Wallop, 5th Earl of Portsmouth donated his private collection of Newton's papers to Cambridge University. Adams and G. G. Stokes took on the task or arranging the material, publishing a catalogue in 1888.
The post of Astronomer Royal was offered him in 1881, but he preferred to pursue his peaceful course of teaching and research in Cambridge. He was British delegate to the International Meridian Conference at Washington in 1884, when he also attended the meetings of the British Association at Montreal and of the American Association at Philadelphia.