the way of reducing and palliating these differences, and that, with such success, that up to the year 1804 it might have been safely asserted that positively no ground whatever existed for suspecting any disturbing influence. But with this epoch an action appears to have commenced, and gone on increasing, producing an acceleration of the motion in longitude, in consequence of which, Uranus continually gains on its elliptic place, and continued to do so till 1822, when it ceased to gain, and the excess of longitude was at its maximum, after which it began rapidly to lose ground, and has continued to do so up to the present time. It is perfectly clear, then, that in this interval some extraneous cause must have come into action which was not so before, or not in sufficient power to manifest itself by any marked effect, and that that cause must have ceased to act, or rather begun to reverse its action, in or about the year 1822, the reverse action being even more energetic than the direct.

(767.) Such is the phænomenon in the simplest form we are now able to present it. Of the various hypotheses formed to account for it, during the progress of its developement, none seemed to have any degree of rational probability but that of the existence of an exterior, and hitherto undiscovered, planet, disturbing, according to the received laws of planetary disturbance, the motion of Uranus by its attraction, or rather superposing its disturbance on those produced by Jupiter and Saturn, the only two of the old planets which exercise any sensible disturbing action on that planet. Accordingly, this was the explanation which naturally, and almost of necessity, suggested itself to those conversant with the planetary perturbations who considered the subject with any degree of attention. The idea, however, of setting out from the observed anomalous deviations, and employing them as data to ascertain the distance and situation of the unknown body, or, in other words, to resolve the inverse problem of perturbations, "given the disturbances to find the orbit, and place in that orbit of the disturbing planet," appears to have occurred only to two mathematicians, Mr. Adams in England and M. Leverrier in France, with

sufficient distinctness and hopefulness of success to induce them to attempt its solution. Both succeeded, and their solutions, arrived at with perfect independence, and by each in entire ignorance of the other's attempt, were found to agree in a surprising manner when the nature and difficulty of the problem is considered; the calculations of M. Leverrier assigning for the heliocentric longitude of the disturbing planet for the 23d Sept. 1846, 326° 0′, and those of Mr. Adams (brought to the same date) 329° 19', differing only 3o 19'; the plane of its orbit deviating very slightly, if at all, from that of the ecliptic.

(768.) On the day above mentioned a day for ever memorable in the annals of astronomy-Dr. Galle, one of the astronomers of the Royal Observatory at Berlin, received a letter from M. Leverrier, announcing to him the result he had arrived at, and requesting him to look for the disturbing planet in or near the place assigned by his calculation. He did so, and on that very night actually found it. A star of the eighth magnitude was seen by him and by M. Encke in a situation where no star was marked as existing in Dr. Bremiker's chart, then recently published by the Berlin Academy. The next night it was found to have moved from its place, and was therefore assuredly a planet. Subsequent observations and calculations have fully demonstrated this planet, to which the name of Neptune has been assigned, to be really that body to whose disturbing attraction, according to the Newtonian law of gravity, the observed anomalies in the motion of Uranus were owing. The geocentric longitude determined by Dr. Galle from this observation was 325° 53', which, converted into heliocentric, gives 326° 52', differing 0° 52′ from M. Leverrier's place, 2° 27′ from that of Mr. Adams, and only 47' from a mean of the two calculations.

(769.) It would be quite beyond the scope of this work, and far in advance of the amount of mathematical knowledge we have assumed our readers to possess, to attempt giving more than a superficial idea of the course followed by these geometers in their arduous investigations. Suffice it to say, that it consisted in regarding, as unknown quantities, to be determined, the

mass, and all the elements of the unknown planet (supposed to revolve in the same plane and the same direction with Uranus), except its major semiaxis. This was assumed in the first instance (in conformity with "Bode's law," art. (505.), and certainly at the time with a high primâ facie probability,) to be double that of Uranus, or 38-364 radii of the Earth's orbit. Without some assumption as to the value of this element, owing to the peculiar form of the analytical expression of the perturbations, the analytical investigation would have presented difficulties apparently insuperable. But besides these, it was also necessary to regard as unknown, or at least as liable to corrections of unknown magnitude of the same order as the perturbations, all the elements of Uranus itself, a circumstance whose necessity will easily be understood, when we consider that the received elements could only be regarded as provisional, and must certainly be erroneous, the places from which they were obtained being affected by at least some portions of the very perturbations in question. This consideration, though indispensable, added vastly both to the complication and the labour of the inquiry. The axis (and therefore the mean motion) of the one orbit, then, being known very nearly, and that of the other thus hypothetically assumed, it became practicable to express in terms, partly algebraic, partly numerical, the amount of perturbation at any instant, by the aid of general expressions delivered by Laplace in his "Mécanique Céleste" and elsewhere. These, then, together with the corrections due to the altered elements of Uranus itself, being applied to the tabular longitudes, furnished, when compared with those observed, a series of equations, in which the elements and mass of Neptune, and the corrections of those of Uranus entered as the unknown quantities, and by whose resolution (no slight effort of analytical skill) all their values were at length obtained. The calculations were then repeated, reducing at the same time the value of the assumed distance of the new planet, the discordances between the given and calculated results indicating it to have been assumed too large when the results were found to agree better, and the solutions to be, in fact, more satisfactory.

Thus, at length, elements were arrived at for the orbit of the unknown planet, as below.

The elements of M. Leverrier were obtained from a consideration of the observations up to the year 1845, those of Mr. Adams, only as far as 1840. On subsequently taking into account, however, those of the five years up to 1845, the latter was led to conclude that the semiaxis ought to be still much further diminished, and that a mean distance of 33.33 (being to that of Uranus as 1:0.574) would probably satisfy all the observations very nearly.*

(770.) On the actual discovery of the planet, it was, of course, assiduously observed, and it was soon ascertained that a mean distance, even less than Mr. Adams's last presumed value, agreed better with its motion; and no sooner were elements obtained from direct observation, sufficiently approximate to trace back its path in the heavens for a considerable interval of time, than it was ascertained to have been observed as a star by Lalande on the 8th and 10th of May, 1795, the latter of the two observations, however, having been rejected by him as faulty, by reason of its non-agreement with the former (a consequence of the motion of the planet in the interval). From these observations, combined with those since accumulated, the elements calculated by Prof. Walker, U.S., result as follows:Epoch of Elements

Mean longitude at Epoch
Semiaxis major

Jan. 1. 1847, M. noon, Greenwich. 328° 32' 44" 2

In a letter to the Astronomer Royal, dated Sept. 2. 1846,-i. e. three

weeks previous to the optical discovery of the planet.

(771.) The great disagreement between these elements and those assigned either by M. Leverrier or Mr. Adams wil not fail to be remarked; and it will naturally be asked how it has come to pass, that elements so widely different from the truth should afford anything like a satisfactory representation of the perturbation in question, and that the true situation of the planet in the heavens should have been so well, and indeed accurately, pointed out by them. As to the latter point, any one may satisfy himself by half an hour's calculation that both sets of elements do really place the planet, on the day of its discovery, not only in the longitudes assigned in art. 763., i. e. extremely near its apparent place, but also at a distance from the Sun very much more approximately correct than the mean distances or semiaxes of the respective orbits. Thus the radius vector of Neptune, calculated from M. Leverrier's elements for the day in question, instead of 36.1539 (the mean distance) comes out almost exactly 33; and indeed, if we consider that the excentricity assigned by those elements gives for the perihelion distance 32·2634, the longitude assigned to the perihelion brings the whole arc of the orbit (more than 83°), described in the interval from 1806 to 1847 to lie within 42° one way or the other of the perihelion, and therefore, during the whole of that interval, the hypothetical planet would be moving within limits of distance from the sun, 32.6 and 33.0. The following comparative tables of the relative situations of Uranus, the real and hypothetical planet, will exhibit more clearly than any lengthened statement, the near imitation of the motion of the former by the latter within that interval. The longitudes are heliocentric.*

The calculations are carried only to tenths of degrees, as quite sufficient for the object in view.