Adams started with the assumptions, not improbable, that the orbit of the unknown planet was a circle, and that its distance from the sun was twice that of Uranus. This latter assumption was in accord with the so-called "Bode's Law," which taught that a simple numerical relationship exists between the planetary distances (4, 7, 10, 16, 28, 52, 100, 196), and that the planets as they lie more remote from the sun tend to be more nearly double the distance of the next preceding. Adams was encouraged, by his first attempt, to undertake a more precise determination.
On his behalf Professor Challis of Cambridge applied to Astronomer Royal Airy, who furnished the Reductions of the Planetary Observations made at Greenwich from 1750 till 1830. In his second endeavor Adams assumed that the unknown planet had an elliptical orbit. He approached the solution gradually, ever taking into account more terms of the perturbations. In September, 1845, he gave the results to Challis, who wrote to Airy on the 22d of that month that Adams sought an opportunity to submit the solution personally to the Astronomer Royal. On the 21st of October, 1845, the young mathematician, twice disappointed in his attempt to meet Airy, left at the Royal Observatory a paper containing the elements of the new planet. The position assigned to it was within about one degree of its actual place.
On November 5 Airy wrote to Adams and, among other things, inquired whether the solution obtained would account for the errors of the radius vector as well as for those of heliocentric longitude. For Airy this was a crucial question; but to Adams it seemed unessential, and he failed to reply.
By this time a formidable rival had entered the field. Leverrier at the request of Arago had undertaken to investigate the irregularities in the tables of Uranus. In September of the same year Eugène Bouvard had presented new tables of that planet. Leverrier acted very promptly and systematically. His first paper on the problem undertaken appeared in the Comptes Rendus of the Académie des Sciences November 10, 1845. He had submitted to rigorous examination the data in reference to the disturbing influence of Jupiter and of Saturn on the orbit of Uranus. In his second paper, June 1, 1846, Leverrier reviewed the records of the ancient and modern observations of Uranus (279 in all), subjected Bouvard's tables to severe criticism, and decided that there existed in the orbit of Uranus anomalies that could not be accounted due to errors of observation. There must exist some extraneous influence, hitherto unknown to astronomers. Some scientists had thought that the law of gravitation did not hold at the confines of the solar system (others that the attractive force of other systems might prove a factor), but Leverrier rejected this conception. Other theories being likewise discarded he asked: "Is it possible that the irregularities of Uranus are due to the action of a disturbing planet, situated in the ecliptic at a mean distance double that of Uranus? And if so, at what point is this planet situated? What is its mass? What are the elements of the orbit which it describes?" The conclusion reached by the calculations recorded in this second paper was that all the so-called anomalies in the observations of Uranus could be explained as the perturbation caused by a planet with a heliocentric longitude of 252° on January 1, 1800. This would correspond to 325° on January 1, 1847.
Airy received Leverrier's second paper on June 23, and was struck by the fact that the French mathematician assigned the same place to the new planet as had Adams in the preceding October. He wrote to Leverrier in reference to the errors of the radius vector and received a satisfactory and sufficiently compliant reply. At one time the Astronomer Royal had felt very skeptical about the possibility of the discovery which his own labors had contributed to advance. He had always, to quote his own rather nebulous statement, considered the correctness of a distant mathematical result to be the subject of moral rather than of mathematical evidence. Now that corroboration of Adams's results had arrived, he felt it urgent to make a telescopic examination of that part of the heavens indicated by the theoretical findings of Adams and Leverrier. He accordingly wrote to Professor Challis, July 9, requesting him to employ for the purpose the great Northumberland equatorial of the Cambridge Observatory.
Professor Challis had felt, to use his own language, that it was so novel a thing to undertake observations in reliance upon merely theoretical deductions, that, while much labor was certain, success appeared very doubtful. Nevertheless, having received fresh instructions from Adams relative to the theoretical place of the new planet, he began observations July 29. On August 4 in fixing certain reference points he noted, but mistook for a star, the new planet. On August 12, having directed the telescope in accordance with Adams's instructions he again noted the same heavenly body, as a star. Before Challis had compared the results of the observation of August 12 with the results of an observation of the same region made on July 30, and arrived at the inference that the body in question, being absent in the latter observation, was not a star but a planet, the prize of discovery had fallen into the hands of another observer.
On August 31 had appeared Leverrier's third paper, in which were stated the new planet's orbit, mass, distance from the sun, eccentricity, and longitude. The true heliocentric longitude was given as 326° 32' for January 1, 1847. This determination placed the planet about 5° to the east of star δ of Capricorn. Leverrier said it might be recognized by its disk, which, moreover, would subtend a certain angle.
The systematic and conclusive character of Leverrier's research, submitted to one of the greatest academies of science, carried conviction to the minds of astronomers. The learned world felt itself on the eve of a great discovery. Sir John Herschel, in an address before the British Association on September 10, said that the year past had given prospect of a new planet. "We see it as Columbus saw America from the shores of Spain. Its movements have been felt trembling along the far-reaching line of our analysis with a certainty hardly inferior to ocular demonstration."
On September 18 Leverrier sent a letter to Dr. Galle, of the Berlin Observatory, which was provided with a set of star maps, prepared at the instance of Bessel. Galle replied one week later. "The planet, of the position of which you gave the indication, really exists. The same day that I received your letter [September 23] I found a star of the eighth magnitude, which was not inscribed in the excellent map (prepared by Dr. Bremiker) belonging to the collection of star maps of the Royal Academy of Berlin. The observation of the following day showed decisively that it was the planet sought." It was only 57' from the point predicted.