who have any practical knowledge of astronomy. They are addressed especially to the general reader, who might be led by misrepresentation to form inaccurate views on the subject. Men of science have all been delighted and astonished that the planet was discovered so soon. Only those who make the cause of science subservient to the miserable gratification of personal feeling have sought to indulge in opprobrious language, because the discovery did not take place sooner. Such persons, true to their instincts, would have been the first to hold up to public ridicule the credulity of the planet hunters, and to raise the charge of a wasteful expenditure of the country's resources, if the speedy discovery of the disturbing body had not indicated a different, though equally unjustifiable, ground of obloquy.

It would be an invidious task to institute a comparison between the respective merits of Le Verrier and Adams in connexion with the immortal discovery, of which we have endeavoured to give some account in the preceding pages. We are of opinion that the labours of both these geometers are equally calculated to excite admiration. The annals of science do not contain a brighter page than that which records the progress of M. Le Verrier's labours as he advances from the ill-defined irregularities of Uranus to the precise position of the undiscovered planet. The care with which he scrutinizes every fact; the vigilance he exhibits in detecting every imaginable source of error, and the thorough manner in which he sifts all its parts; the ingenuity and conclusiveness of his methods, and his indomitable perseverance in calculation-indicate, in a high degree, the possession of those qualities which constitute the main elements of success in all researches relating to the physico-mathematical sciences. Nor does a review of Mr. Adams' labours offer a less pleasing picture. We see the obscure undergraduate, while his attention is yet distracted by the routine of academic discipline, seizing with the happy intuition of genius the true theory of Uranus, and forming the bold resolution of tracing it to its final results. The constancy with which he afterwards struggles to effect this object, notwithstanding the manifest disadvantages of his position, is equalled only by the masterly character of his analytical researches and the brilliant termination to which he conducts them. It is gratifying to reflect that the labours of both Le Verrier and Adams, in connexion with the perturbations of Uranus, are so completely dissociated, that no danger of a mis-statement of facts can exist in the discussion of their relative merits. Differences of opinion on this last point will, no doubt, always prevail; but we are confident that future ages will concur with the present in awarding to each geometer the tribute of unqualified admiration.

The discovery of the planet Neptune (for such is the name by which Astronomers have agreed to distinguish the Trans-Uranian member of the Solar System) marks an important epoch in the history of physical astronomy. Hitherto the object of the geometer had been to unfold by a deductive process the principles of perturbative influence, and to explain by them the various phenomena of the planetary motions. It was only in the determination of the masses of the planets, and in assigning the ratios of their polar and equatorial axes, that the order of inquiry was reversed, and an accurate knowledge of the disturbing influence was sought to be established by reasoning upwards from its observed effects. In each of these cases, however, the equations of condition are of extreme simplicity, and the value of the final results is dependent much less upon the skill of the geometer than upon the accuracy of the fundamental observations. The idea of submitting to a similar treatment the problem of three bodies does

not appear to have occurred to any of the great geometers whose names are associated with the development of the theory of gravitation. Nor is this circumstance calculated to excite surprise, for the actual state of Physical Astronomy had not yet demanded such an advanced step. It was only when all the consequences resulting from the mutual action of the planets already known had been fully deduced, and the outstanding irregularities had assumed the form of residual phenomena depending on some foreign influence, that further speculation suggested the expediency of inverting the usual order of investigation. It is manifest from this circumstance, that the complete establishment of the formulæ of planetary perturbation must have preceded any attempt to ascend from the effects produced by an unseen planet to the determination of its actual position. The accomplishment of this latter object is therefore an indication of the highlyadvanced state of physical astronomy, since it implies not only that the difficulties peculiar to the inverse problem of perturbation have been successfully overcome, but also that the irregularities occasioned by the mutual action of the planets have been deduced from the principles of the Newtonian theory, and have, in all instances, been found to accord with the results derived from observation. This remark does not, of course, apply to the planet which theory has recently revealed to us, since a considerable time must elapse before an accurate knowledge of the inequalities of its motion can be obtained. It will then be an interesting point to ascertain whether these inequalities do not in their turn afford indications of the existence of a still more remote member of the solar system. The astronomer is thus led to speculate on the theoretical discovery of planets, reflecting too feeble a light on account of their immense distance from the sun, to be ever visible, even by the aid of the most powerful telescopes. It is to be hoped, that notwithstanding the abundant harvest, which has been already reaped in Celestial Mechanics, that magnificent region is destined still to afford more profitable-fields for the application of the resources of analysis than that which the imagination here suggests.

CHAPTER XIII.

The Elements of the Planet Neptune deduced from Observation.-They are found to be Discordant with the Results of Theory. The cause of Discordance assigned.—The Planet observed by Lalande.-Theory of its Perturbations.-Researches on the Value of its Mass.-Uncertainty respecting this Element.-Researches of M. Hansen on the Lunar Theory. Conclusion of the History of Physical Astronomy.

As soon as astronomers received intelligence of the discovery of the Planet Neptune, the new member of the solar system was regarded with intense interest, and accurate observations of it were made both in Europe and America. When the elements of the orbit were calculated from these observations, a comparison of the results with those assigned by the theories of Le Verrier and Adams led to rather unexpected conclusions. Although the orbits assigned by these geometers to the disturbing planet did not differ materially from each other, they both, on the other hand, exhibited

a very marked discordance with the real orbit, as indicated by observation. It was found that the orbit in which the planet actually revolved was much smaller than either of those deduced from theory, and that, instead of being very eccentric, it approached very nearly to a circular form. The following are the elements calculated from observation by Mr. Walker of Washington, U.S.:

Mean distance

30.0363

M. Long. January 1, 1847; M. T. Greenwich 328° 32′ 44′′.20
Eccentricity

Long. of Perihelion

Long. of Ascending Node.

Inclination

Mean Daily Motion
Periodic Time

.00871946 47° 12′ 6′′.50 130 4 20 .81

1 46 58 .97

21′′.55448

161.6181 trop. years.

Elements of the planet's orbit were also calculated by other astronomers, and the results were found to agree very nearly with those above given. It appears that the mean distance, instead of being nearly double the mean distance of Uranus, amounts only to about two-thirds of it. The law of Bode, therefore, which is so remarkably applicable to the other members of the planetary system, totally fails in this case. The general discordance of the elements with those severally assigned by the two geometers who were led to the theoretical discovery of the planet, at first occasioned considerable surprise, and it was suspected that some difficulty would be experienced in rendering a satisfactory account of its origin. A little reflection, however, served to arrive at clearer views on the subject. In order that the reader may understand how elements so remote from the truth as those of Le Verrier and Adams, could have sufficed to effect the theoretical discovery of the planet, it is necessary to form a distinct conception of the nature of the problem, by the solution of which these geometers were conducted to their respective results. The data of this problem were the observed perturbations of Uranus, and the main object to be accomplished was to determine the position in the zodiac occupied at any assigned instant by the disturbing body so as to arrive at its actual discovery. Now, the derangement occasioned in the motion of any planet by the action of another planet upon it depends on the intensity and direction of the disturbing force at each instant; and these again depend on the mass of the disturbing body, and on its distance and longitude with respect to the sun. It is manifest, therefore, that only those elements which will accurately assign the distance and longitude of the disturbing body will render a complete account of the perturbations of Uranus. But if the values of the heliocentric co-ordinates should not be absolutely correct, still, if they approach with tolerable approximation to the true values, it is not difficult to perceive that by a due adjustment of the mass, the intensity and direction of the disturbing force will be represented with a corresponding degree of precision. Under such circumstances the anomalies of the disturbed planet will be accounted for with nearly as great fidelity as if the disturbing planet were in its true place, for the error of perturbation is obviously of an order inferior to the error in the place of the disturbing body. Now, although it is impossible permanently to represent, even with tolerable accuracy, the heliocentric co-ordinates of a body revolving in an elliptic orbit by means of any elements which differ from

the true elements, still, when the question refers only to a section of the orbit, this object may be accomplished by employing indefinite combinations of elements, differing very considerably from each other. This will be readily understood when it is borne in mind that the theory of elliptic motion assigns four arbitrary constants, which may be modified in a variety of ways, so as to answer the purpose of mutual correction; and that on account of the smallness of the arc described by the body, the outstanding errors, which inevitably exist in all such cases, are not allowed time to develope themselves to any serious extent. In the case of Neptune disturbing Uranus, the perturbations are sensible only a little before and after conjunction. Throughout the whole period embraced between 1690, the year of the earliest observation of Uranus, and the commencement of the present century, the action of the disturbing planet has been quite inappreciable; and consequently the tabular errors of Uranus for that period may be considered as wholly explicable by the errors of the elliptic elements. The last conjunction of the two planets took place in the year 1822, and the action of the disturbing planet was sensible only during about twenty years anterior to that event, and the same number of years subsequent to it. It is manifest, therefore, that any elements which will afford a pretty accurate representation of the heliocentric co-ordinates of Neptune during the present century, will account with sufficient fidelity for the perturbations of Uranus during the same period. Conversely, if the perturbations are faithfully accounted for, we may conclude that the theory is capable of representing the co-ordinates of the disturbing planet with considerable precision, and that they may be employed with confidence for the purpose of its actual discovery. That the theories of Le Verrier and Adams were capable of so representing the co-ordinates of Neptune during the whole period when its action was sensible, may be seen from the following table, which exhibits the actual and theoretical values of the longitude and radius vector for the beginning of each of the years specified between the years 1800 and 1860 :—

Theory of Adams*.

Planet Neptune.

Theory of
Le Verrier.

1st Approximation.

2nd Approximation.

Year. Longitude. Rad. Vec. Longitude. Rad. Vec. Longitude. Rad. Vec. Longitude. Rad, Vec.

Mr. Adams' theory is founded upon a method of successive approximation. A certain value of the mean distance is first assumed, and by a comparison of the results calculated by its aid with those derived from observation, an indication is obtained of the direction in which the error in the mean distance lies, and also of its probable magnitude. The solution is then repeated with a new value of the mean distance, suggested by the original solution, and a further approximation is obtained. This is the method of

It is manifest that any two corresponding co-ordinates of either of the theories contained in the above table might have been employed with success in searching for the planet. Even twenty years before conjunction the error of its position as assigned by Adams' first approximation amounts to little more than 10°, a quantity which falls considerably within the range of search proposed by the Astronomer Royal to Professor Challis, in the month of July, 1846. The reader will not fail to remark, that the longitudes are generally represented with greater accuracy than the distances. This circumstance admits of easy explanation. The intensity of the disturbing force in any given configuration of the two planets depends on the mass and distance of the disturbing planet. Now, if the distance be made too great, the disturbing force will be enfeebled in a corresponding degree; but this effect may be obviated by a suitable increase of the mass. This is precisely what happens in the theories of Le Verrier and Adams. The distances are all too great in both theories; but, on the other hand, the mass in each case is considerably enlarged beyond its true value. The perturbations of Uranus may therefore be accounted for, in so far as the intensity of the disturbing force is concerned, even although the radius vector of the theoretical planet should be considerably erroneous. The mass, however, exercises no influence in determining the direction of the disturbing force; and if the latter element be erroneous to any great extent, the perturbation will be necessarily erroneous also, since it cannot derive compensation from any other source. Hence arises the necessity of a comparatively higher degree of accuracy in the representation of the longitudes of the theoretical planet.

It

When the elements of Neptune were determined with a degree of precision sufficient to enable astronomers to trace its motion through the anterior part of its orbit, attempts were made to ascertain whether it had been observed on any occasion previous to its discovery as a planet by Dr. Galle. It was soon found by Dr. Petersen, of Altona, and Mr. S. Walker, of Washington, that a star in the Histoire Celeste of Lalande, observed May 10, 1795, and since missing, could be no other than the planet Neptune. The place of the star being marked doubtful, the French astronomers were induced to examine the original manuscripts of the Histoire Celeste, which are deposited in the Royal Observatory of Paris. An inspection of the observation established the identity of the planet with the recorded star, and disclosed an additional fact of extreme interest. appeared that the planet was also observed on the 8th May, 1795, and that its right ascension and declination were regularly recorded, although they were not subsequently inserted in the printed catalogue. The discordance of the two positions May 8-10, in a case where identity was looked for, beyond doubt suggested to Le François Lalande a suspicion of the accuracy of his observations, and induced him to suppress altogether the observation of May 8. A comparison of the two observations clearly investigation generally employed in physical astronomy. In his first solution, Mr. Adams assumed that the mean distance of the hypothetic planet was equal to 38.4; in his second solution he made it 37.5; and in his communication to the Astronomer Royal, dated September 2, 1846, he stated, as the result of further discussion, that 33.6 would probably be a very near approximation. The actual mean distance in fact is 30.04.

Thus we see that Mr. Adams was fairly on the track of the true orbit. The method employed by M. Le Verrier is of a more ambitious character, but, unfortunately, it is not adequate to meet all the difficulties of such abstruse enquiries, and in the present instance it had the effect of betraying M. Le Verrier into error with respect to the limits of the mean distance and the other elements of the orbit.