A great advance in lunar theory was made by Peter Andreas Hansen (1795-1874) of Gotha, who published in 1838 and 1862-64 the treatises commonly known respectively as the Fundamenta[164] and the Darlegung,[165] and produced in 1857 tables of the moon’s motion of such accuracy that the discrepancies between the tables and observations in the century 1750-1850 were never greater than 1″ or 2″. These tables were at once used for the calculation of the Nautical Almanac and other periodicals of the same kind, and with some modifications have remained in use up to the present day.
A completely new lunar theory—of great mathematical interest and of equal complexity—was published by Charles Delaunay (1816-1872) in 1860 and 1867. Unfortunately the author died before he was able to work out the corresponding tables.
Professor Newcomb of Washington ([§ 283]) has rendered valuable services to lunar theory—as to other branches of astronomy—by a number of delicate and intricate calculations, the best known being his comparison of Hansen’s tables with observation and consequent corrections of the tables.
New methods of dealing with lunar theory were devised by the late Professor John Couch Adams of Cambridge (1819-1892), and similar methods have been developed by Dr. G. W. Hill of Washington; so far they have not been worked out in detail in such a way as to be available for the calculation of tables, and their interest seems to be at present mathematical rather than practical; but the necessary detailed work is now in progress, and these and allied methods may be expected to lead to a considerable diminution of the present excessive intricacy of lunar theory.
287. One special point in lunar theory may be worth mentioning. The secular acceleration of the moon’s mean motion which had perplexed astronomers since its first discovery by Halley (chapter X., [§ 201]) had, as we have seen (chapter XI., [§ 240]), received an explanation in 1787 at the hands of Laplace. Adams, on going through the calculation, found that some quantities omitted by Laplace as unimportant had in reality a very sensible effect on the result, so that a certain quantity expressing the rate of increase of the moon’s motion came out to be between 5″ and 6″, instead of being about 10″, as Laplace had found and as observation required. The correction was disputed at first by several of the leading experts, but was confirmed independently by Delaunay and is now accepted. The moon appears in consequence to have a certain very minute increase in speed for which the theory of gravitation affords no explanation. An ingenious though by no means certain explanation was suggested by Delaunay in 1865. It had been noticed by Kant that tidal friction—that is, the friction set up between the solid earth and the ocean as the result of the tidal motion of the latter—would have the effect of checking to some extent the rotation of the earth; but as the effect seemed to be excessively minute and incapable of precise calculation it was generally ignored. An attempt to calculate its amount was, however, made in 1853 by William Ferrel, who also pointed out that, as the period of the earth’s rotation—the day—is our fundamental unit of time, a reduction of the earth’s rate of rotation involves the lengthening of our unit of time, and consequently produces an apparent increase of speed in all other motions measured in terms of this unit. Delaunay, working independently, arrived at like conclusions, and shewed that tidal friction might thus be capable of producing just such an alteration in the moon’s motion as had to be explained; if this explanation were accepted the observed motion of the moon would give a measure of the effect of tidal friction. The minuteness of the quantities involved is shewn by the fact that an alteration in the earth’s rotation equivalent to the lengthening of the day by 1∕10 second in 10,000 years is sufficient to explain the acceleration in question. Moreover it is by no means certain that the usual estimate of the amount of this acceleration—based as it is in part on ancient eclipse observations—is correct, and even then a part of it may conceivably be due to some indirect effect of gravitation even more obscure than that detected by Laplace, or to some other cause hitherto unsuspected.
288. Most of the writers on lunar theory already mentioned have also made contributions to various parts of planetary theory, but some of the most important advances in planetary theory made since the death of Laplace have been due to the French mathematician Urbain Jean Joseph Leverrier (1811-1877), whose methods of determining the distance of the sun have been already referred to ([§ 282]). His first important astronomical paper (1839) was a discussion of the stability (chapter XI., [§ 245]) of the system formed by the sun and the three largest and most distant planets then known, Jupiter, Saturn, and Uranus. Subsequently he worked out afresh the theory of the motion of the sun and of each of the principal planets, and constructed tables of them, which at once superseded earlier ones, and are now used as the basis of the chief planetary calculations in the Nautical Almanac and most other astronomical almanacs. Leverrier failed to obtain a satisfactory agreement between observation and theory in the case of Mercury, a planet which has always given great trouble to astronomers, and was inclined to explain the discrepancies as due to the influence either of a planet revolving between Mercury and the sun or of a number of smaller bodies analogous to the minor planets ([§ 294]).
Researches of a more abstract character, connecting planetary theory with some of the most recent advances in pure mathematics, have been carried out by Hugo Gyldén (1841-1896), while one of the most eminent pure mathematicians of the day, M. Henri Poincaré of Paris, has recently turned his attention to astronomy, and is engaged in investigations which, though they have at present but little bearing on practical astronomy, seem likely to throw important light on some of the general problems of celestial mechanics.
289. One memorable triumph of gravitational astronomy, the discovery of Neptune, has been described so often and so fully elsewhere[166] that a very brief account will suffice here. Soon after the discovery of Uranus (chapter XII., [§ 253]) it was found that the planet had evidently been observed, though not recognised as a planet, as early as 1690, and on several occasions afterwards.
When the first attempts were made to compute its orbit carefully, it was found impossible satisfactorily to reconcile the earlier with the later observations, and in Bouvard’s tables (chapter XI., [§ 247], note) published in 1821 the earlier observations were rejected. But even this drastic measure did not cure the evil; discrepancies between the observed and calculated places soon appeared and increased year by year. Several explanations were proposed, and more than one astronomer threw out the suggestion that the irregularities might be due to the attraction of a hitherto unknown planet. The first serious attempt to deduce from the irregularities in the motion of Uranus the position of this hypothetical body was made by Adams immediately after taking his degree (1843). By October 1845 he had succeeded in constructing an orbit for the new planet, and in assigning for it a position differing (as we now know) by less than 2° (four times the diameter of the full moon) from its actual position. No telescopic search for it was, however, undertaken. Meanwhile, Leverrier had independently taken up the inquiry, and by August 31st, 1846, he, like Adams, had succeeded in determining the orbit and the position of the disturbing body. On the 23rd of the following month Dr. Galle of the Berlin Observatory received from Leverrier a request to search for it, and on the same evening found close to the position given by Leverrier a strange body shewing a small planetary disc, which was soon recognised as a new planet, known now as Neptune.
It may be worth while noticing that the error in the motion of Uranus which led to this remarkable discovery never exceeded 2′, a quantity imperceptible to the ordinary eye; so that if two stars were side by side in the sky, one in the true position of Uranus and one in the calculated position as given by Bouvard’s tables, an observer of ordinary eyesight would see one star only.