Mayer s tables, thus corrected, were published in 1784, and for a long time continued to be the most accurate that
had appeared.The solution of the problem of three bodies by Clairaut, D Alembert, and Euler, gave rise to many other import ant works relative to the theory of the moon, into the merits of which, however, our limits will not permit us to enter. Thomas Simpson, "Walmesley, Frisi, Lambert, Schulze, and Matthew Stewart treated the subject with more or less success ; but the complete explication of the theory of the lunar and planetary perturbations was re served for two mathematicians, whose discoveries perfected the theory of gravitation, and explained the last inequalities which remained to be accounted for in the celestial motions, Lagrange and Laplace.
In the year 1764, the Academy of Sciences of Paris, which had so successfully promoted the great efforts that had already been made to perfect the theory of attraction, proposed fur the subject of a prize the theory of the libration agraage. of the moon. Lagrange had the honour of carrying off the prize ; but although he treated the subject in a manner altogether new, and with extraordinary analytical skill, he did not on this occasion arrive at a complete solution of the problem. In 17GG he obtained another prize for a theory of Jupiter s satellites. In the admirable memoir which Lagrange presented to the academy on this subject, he included in the differential equations of the disturbed motion of a satellite the attracting force of the sun, as well as of all the other satellites, and thus, in fact, had to consider a problem of six bodies. His analysis of this problem is remarkable, inasmuch as it contained the first general method Avhich was given for determining the variations which the mutual attractions of the satellites produce in the forms and positions of their orbits, and pointed out the plan which has since been so successfully followed in the treatment of similar questions.
Of all the grand discoveries by which the name of Lagrange has been immortalised, the most remarkable is that of the invariability of the mean distances of the planets from the sun. We have already mentioned that Euler had perceived that the inequalities of Jupiter and Saturn, in consequence of their mutual actions, are ulti mately compensated, though after a very long period. In prosecuting this subject, which Euler had left imperfect, Laplace had discovered that, on neglecting the fourth powers in the expressions of the eccentricities and inclina tions of the orbits and the squares of the disturbing masses, the mean motions of the planets and their mean distances from the sun are invariable. In a short memoir of 14 pages, which appeared among those of the Berlin Academy for 177G, Lagrange demonstrated generally, and by a very simple and luminous analysis, that whatever powers of the eccentricities and inclinations are included in the calculation of the perturbations, no secular inequa lity, or term proportional to the time, can possibly enter into the expression of the greater axis of the orbit, or, consequently, into the mean motion connected with it by the third law of Kepler. From this conclusion, which is a necessary consequence of the peculiar conditions of the planetary system, it results that all the changes to which the orbits of the planets are subject iu consequence of their reciprocal gravitation, are periodic, and that the system contains within itself no principle of destruction, but is calculated to endure for ever.
In 1780 Lagrange undertook a second time the subject of the moon s libration ; and it is to the memoir which he now presented to the Berlin Academy that we must look for the complete and rigorous solution of this difficult problem, which had not been resolved before in a satis factory manner, cither on the footing of analysis or observation. In the same year he obtained the prize of the Academy of Sciences on the subject of the perturba tions of comets. In 1781 he published, in the Berlin Memoirs, the first of a series of five papers on the seculai and periodic inequalities of the planets, which together formed by far the most important work that had yet appeared on physical astronomy since the publication of the Principia. This series did not, properly speaking, contain any new discovery, but it embodied and brought into one view all the results and peculiar analytical methods which had appeared in his former memoirs, and contained the germs of all the happy ideas which he afterwards developed in the M&canique Analytique.
On account of the brilliant discoveries and important labours which we have thus briefly noticed, Lagrange must be considered as one of the most successful of those illustrious men who have undertaken to perfect the theory of Newton, and pursue the principle of gravitation to its remotest consequences. But the value of his services to science is not limited to his discoveries iu physical astronomy, great and numerous as these were. After Euler, he has contributed more than any other to increase the power and extend the applications of tho calculus, and thereby to arm future inquirers with an instrument of greater efficiency, by means of which they may push their conquests into new and unexplored fields of discovery.
With the name of Lagrange is associated that of Laplace, their rival labours dividing the admiration of the scientific world during half a century. Like Xewton and Lagrange, Laplace raised himself at an early age to the very highest rank in science. Before completing his 24th year, he had signalised himself by the important discovery of the invaria bility of the mean distances of the planets from the sun, on an hypothesis restricted, indeed, but which, as we have already mentioned, was afterwards generalised by Lagrange. About the same time he was admitted into the Academy of Sciences, and thenceforward devoted himself to tho development of the laws which regulate the system of tho world, and to the composition of a series of memoirs on the most important subjects connected with astronomy and analysis. His researches embraced the whole theory of gravitation ; and he had the high honour of perfecting what had been left incomplete by his predecessors.
Among the numerous inequalities which affect the motion of the moon, one still remained which no philosopher as yet had been able to explain. This was the acceleration of the mean lunar motion, which had been first suspected by Halley, from a comparison of the ancient Babylonian observations, recorded by Hipparchus, with those of Alba- tegni and the moderns. The existence of the acceleration had been confirmed by Dunthorne and Mayer, and its quantity assigned at 10" in a century, but the cause of it remained doubtful. Lagrange demonstrated that it could not be occasioned by any peculiarity in the form of tho earth ; Bossut ascribed it to the resistance of the medium in which he supposed the moon to move ; and Laplaco himself at first explained it on the supposition that gravity is not transmitted from one body to another instantaneously, but successively, in the manner of sound or light. Having afterwards remarked, however, in the course of his re searches on Jupiter s satellites, that the secular variation of the eccentricity of the orbit of Jupiter occasions a secular variation of the mean motii ns ( f the satellites, he hastened to transfer this result to the moon, and had the satisfaction to find that the acceleration observed by astronomers is occasioned by the secular variation of the eccentricity of the terrestrial orbit. This conclusion has, however, been partly invalidated by the recent researches of Adams of Cambridge.
