Jupiter and Saturn must have presented to the earliest observers of the celestial motions less difficulties than the interior planets. The first things noted, undoubtedly, were, that the first made a circuit of the heavens in about twelve and the second in about thirty years. Then the retrograde motion, at the time of opposition, and its extent would be perceived. The slowness and steadiness of the motion would naturally suggest the hypothesis of circular motion, but it was certainly reserved for a later and more philosophic age to explain the later-observed phenomenon by means of an epicycle.
The earliest tables of the motions of Jupiter and Saturn, as well as those of the other large planets which have come down to us, are those contained in the Syntaxis of Claudius Ptolemy. The annual parallax is there taken into account by one epicycle and the proper eccentricity of the orbit by a second. This, in the main, is the character of all the tables of the planets until the publication of Kepler's Tabulae Rudolphinae in 1627, where, for the first time, the equation of the centre is derived from an elliptic formula, and we pass from heliocentric to geocentric positions in the modern way. From Kepler onwards the fact of the deviation of Jupiter and Saturn from a purely elliptic theory was recognized. Many attempts were made to better the theory; but it was found that no observations, embracing a long period of time, could be satisfied by elliptic elements varying proportionally to the time. Halley seems to have been the most successful in his tables; he adopted terms in the mean longitudes varying as the square of the time.
It was not until 1748 that any computation of the perturbations of Jupiter and Saturn, in accordance with the theory of gravitation, was undertaken. This was by Euler. He appears to have limited himself to the terms which have the mean elongation of the planets from each other as their argument. Later the terms factored by the simple power of the eccentricities were added by himself, Lalande, Lagrange, Bailly and Lambert. But these terms not bringing about a reconciliation between observation and theory, Lagrange and Laplace were led to make their notable researches on the possibility of secular equations in the mean motions of the planets. At length the whole difficulty with Jupiter and Saturn was removed by Laplace's discovery of the great inequalities in 1786.
Delambre almost immediately constructed tables for these planets which far exceeded in accuracy any previously possessed. They are those which appear in the third edition of Lalande's Astronomie. This great success seems to have stirred up Laplace and his collaborators to pushing the approximations still further. On the publication of the third volume of the Mécanique Celeste, terms of the fifth order with respect to the eccentricities and mutual inclination, as well as some of two dimensions with respect to disturbing forces, had been added to the coefficients of the great inequalities. That these advances might be utilized Bouvard constructed tables of the planets founded on observed oppositions from 1747 to 1803. The formulae used are very nearly those given in the Mécanique Celeste," Tom. III. These tables were published by the Bureau des Longitudes in 1808. It was discovered, however, that the terms of the fifth order, mentioned above, had been taken with the wrong sign. This led Bouvard to prepare a new edition of his tables, which appeared in 1821, and in which this error was rectified, and the observations employed in the discussion extended to 1814. Although Bouvard himself speaks in admiration of the small residuals shown by the comparison of his theory with the observations, yet a glance shows their tendency to a systematic character, and this, too, with observations rather rudely reduced.
Plana undertook, shortly after, to compute the portions of the great inequalities which arise from considering the square of the disturbing force. The results he obtained failed to satisfy an equation of condition which Laplace had employed in his investigation. After some discussion Laplace abandoned his equation and substituted for it another, which Plana's results were as far from satisfying as before. Pontécoulant then, taking up the subject, discovered that Laplace's results had been taken with the wrong sign, and that Plana had made errors of some importance in his investigation. When these oversights had been corrected the different results were brought into tolerable agreement.
However, the failure of Bouvard's tables to better represent observations, and his getting for the mass of Jupiter a value so much smaller than was shortly after obtained from the action of this planet on the asteroids and on its own satellites, can not be explained by this error of sign. It is somewhat singular that no one has yet pointed out the real cause, which, it seems, must be either some error in the coefficients of his formulae or some error in putting his equations in tables.
Neither Laplace's, Plana's, nor Pontécoulant's determination of these second-order terms can be regarded as anything else than a very rude and inadequate approximation.
Hansen had, a short time previous, imagined a new method of treating perturbations. In the Mécanique Céleste, Laplace had determined all long-period inequalities as if they were to be applied to the mean longitude, and had so directed they should, while the short-period ones were derived as if they were to be added to the true longitude. There is, therefore, a want of congruity, and even of rigour, in this way of proceeding. For Laplace has nowhere shown how these two modes of application can be employed in unison. It is plain there would be as many methods of perturbations as there were opinions as to the dividing line separating long from short-period inequalities. These imperfections no doubt attracted the attention of Hansen, whose thought must have been: Since it is advantageous to apply the long-period terms to the mean longitude, and indifferent whether the short-period ones are applied to the mean or true, why not apply all to the mean, and, moreover, compute the radius-vector and latitude with this equated quantity? Then the additional quantities necessary to complete the values of the latter co-ordinates would be, for the first, a function of three variations of the elements, and for the second, a function of two only. This, undoubtedly, was the origin of Hansen's new method.
He determined to apply it to Jupiter and Saturn, and his memoir, crowned by the Berlin Academy, must be regarded as the earliest example of an adequate treatment of perturbations of the second order with respect to disturbing forces. In all previous investigations it is impossible to form a conception of the probable magnitude of the terms passed over on account of the habit of the investigator of selecting here and there a term to be computed. But in Hansen the continuity in the computed terms enables one to form a fair judgment as to the importance of those neglected. However, Saturn alone is treated with a fair degree of completeness. The expressions for Jupiter are limited to the terms arising from the first power of the disturbing force. Had this theory of Saturn been completed by the addition of the terms due to the action of Uranus and the whole compared with the observations more carefully reduced, as they then could have been by the aid of Bessel's Tabulae Regiomontanae, very excellent tables would have been obtained. But Hansen seems to have been carried away with the ambition of applying his peculiar method of treatment to the lunar theory.
A long period of over forty years now elapsed without anything being contributed to the theories of Jupiter and Saturn, for the expressions of the perturbations given in Pontécoulant's Théorie Analytique du Systeme du Monde, beyond the correction of the error of sign in the second-order terms of the great inequalities, do not seem to be in anything more perfect than those found in the Mécanique Céleste ...
Hansen, in 1875, published a memoir on Jupiter. But here, deserting his earlier notions on the lack of convergence in algebraical developments, he confines himself to calculating the easier terms of the co-ordinates. Hence this memoir can not be regarded as advancing much our knowledge of the subject.
In the years 1874 to 1876 appeared Le Verrier's investigation, concluding with the tables which are at present employed for all the European ephemerides. The method followed is that of attributing the perturbations to the six elements of the Keplerian ellipse; and, contrary to the mode followed in his earlier planetary theories, these are also the quantities tabulated ...
The desirableness of a new investigation of the subject has been generally admitted, and fault has been found with the amount of labour required to deduce positions of the planets from Le Verrier's tables. But I had not these inducements to take up the subject when I began work, for these tables were then unpublished. The long interval which occurred between the publication of Le Verrier's theory of Mars and the appearance of anything from him on Jupiter and Saturn was the occasion of leading me to consider the undertaking. On making known to the Superintendent of the American Ephemeris my desire to take up the problem I was relieved from all other routine work, and supplied with the assistance necessary to duplicate all my computations which required this safeguard against error. It was desired to abandon the use of the antiquated tables of Bouvard, and it appeared uncertain when Le Verrier would publish his.
The plan, therefore, was to form theories of Jupiter and Saturn which would be practically serviceable for a space of three hundred years on each side of a central epoch taken near the centre of gravity of all the times of observation; theories whose errors in this interval would simply result not from neglected terms in the developments, but from the unavoidable imperfections in the values of the arbitrary constants and masses adopted from the indications of observation.
Such were the considerations which influenced the adoption of the course to be followed. As there was no desire to lose time by forming a special method of treatment for the problem in hand it was decided to employ the method of Hansen, with such slight modifications as the exigencies of the case might suggest. On account of the presence of the great inequalities this method seemed to me to give expressions best suited to tabulation. The latest form of this method appears in Hansen's memoir entitled Auseinandersetzung, etc. The employment of the eccentric anomaly of the planet whose co-ordinates are sought as the independent variable undoubtedly augments the convergence of the series; but the adoption of this mode of proceeding would bring about the use of two independent variables, one of the co-ordinates of Jupiter, another for those of Saturn. As the developments have to be pushed to terms of three dimensions with respect to disturbing forces the heaviest part of the labour consists in forming products of periodic series, one of which belongs to Jupiter, the other to Saturn; and as integration can not be performed unless these products are transformed so as to involve but one variable we should have an endless series of transformations to make. It, therefore, seems a necessity to have a single independent variable for the whole work. In consequence, the final form adopted for all the periodic series is in terms of the mean anomalies, so that the time is always the independent variable. Fortunately, very slight and readily perceived changes only are necessary in the formulae of the Auseinandersetzung to render them applicable to the modified mode of proceeding.