During the time that Euler was employed on the problem of the Moon, he composed his excellent paper on the theory of the motions of Jupiter and Sa turn, which obtained the prize of the Academy of Sciences at Paris for 1748. This problem is of the same nature as that of the motions of the Moon. Saturn and Jupiter reciprocally disturb the elliptical motion, which each ought to have separately round the Sun. The researches of Euler on this subject are remarkable for a profound analysis, and for several series of a kind absolutely new. Nevertheless, as the difficulty and immense extent of the calculations, which such a question demanded, did not allow him to carry his theory to perfection at once, the academy of sciences proposed the same subject anew for the prize of 1750, and again postponed it to 1752 with a double prize.

Euler sent a second paper, to which this prize was awarded. It is founded on a method in many respects new. In the former the author had been led to approximations, of the sufficiency of which some doubts might be entertained: for the number of inequalities being as it were infinite, those, which he had determined, depended according to his method on other inequalities, which he had neglected; and this rendered their values incomplete, and even a little uncertain. The paper of 1752 is more perfect in this respect. It It separates and more fully unfolds the inequalities, which are to be discovered in succession; and thus the analytical formule, to which it leads, are more simple, and more easily applicable to observations. The author has not treated anew

the inequalities, which effect the line of the nodes, and the mutual inclination of the orbits of the two planets, this part of the subject having been completely developed in the paper of 1748.

The academy of sciences at Paris having proposed for the prize of 1754, and afterward for the double prize of 1756, the theory of the inequalities, which the planets may occasion in the motions of the Earth, Euler was, likewise, equally successful in this. He began by giving general formulæ, for determining the alterations, which the primary planets mutually occasion in the motions of each other round the Sun. Not to render the question unnecessarily complicated, by introducing into it terms that might be neglected, he considers only two planets at a time; and he determines the alterations, that the elliptical motion of the one round the Sun must undergo from the attraction of the other: alterations, which, being very small, would produce only infinitely small quantities of the second order, if they were combined with those, which might arise from other planets. He then applies this general theory to the subject proposed: he analyzes successively, and in order, the alterations, which Saturn, Jupiter, Mars, and Venus produce in the motion of the Earth: and he finds, that their general effect is, to occasion the aphelion of the Earth to advance in the order of the signs, to vary the obliquity of the ecliptic, and alter the latitude and longitude of the Sun, &c. The action of the Moon on the Earth's, orbit Euler did not take into

consideration, either because he deemed it to make no part of the problem proposed by the academy, or because

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because d'Alembert had already treated this question in the second volume of his Inquiries into the Mundane System, published in 1754.

Clairaut, in a memoir read before the Academy of Sciences at Paris in 1757, and printed by anticipation in the volume of 1754, applied his method for the problem of three bodies to the motions of the Earth. And to the perturbations considered by Euler he added the action of the Moon, which tended to make the theory complete.

Mayer the celebrated astronomer, who was likewise a skilful geometrician, constructed new lunar tables, partly from Euler's theory, and partly from observations, which were more accurate than any that had before appeared. A. D. 1754, 1759. Clairaut also constructed very good ones from his own theory, in 1764. Tables of this kind, requiring a number of scrupulous attentions, and a choice of the most excellent observations, on which the data of the problem depend, cannot be too frequently renewed or corrected.

Notwithstanding the efforts of geometricians, the theory of the Moon still remained imperfect in certain respects. By observations alone, dextrously combined, Clairaut and Mayer had obtained several results, which would have gone near to destroy the system of gravitation. The chief cause of these difficulties arose from attributing to the Moon an orbit movable on the plane of the ecliptic; and making an angle with this plane, which varied from one instant to another; so that, to know the true

place of the Moon, or it's longitude and latitude, it was necessary first to determine the intersection of the lunar orbit with the ecliptic, or the line of the nodes, and afterward the inclination of the two orbits; which led to a great number of equations, some of which were uncertain, or precarious.

In 1769, Euler, contemplated the questions in a new point of view, and arrived at a more simple, clear, and accurate solution, than any before known. He determined the true place of the Moon, by referring it to three normal coordinates, two of which are in the plane of the ecliptic, and the third perpendicular to it. The values of these coordinates are determined at every instant by equations founded on eight sorts of quantities, four of them constant, and four variable. The constant quantities are the mean eccentricity of the lunar orbit, the mean inclination of this orbit to the plane of the ecliptic, the mean eccentricity of the Earth's orbit, and the ratio between the Earth's mean distance from the Sun and the Moon's mean distance from the Earth. The variable quantities are the four angles proportional to the time; namely, the mean elongation of the Moon, the mean anomaly of the Moon, the mean argument of the latitude of the Moon, and the mean anomaly of the Sun. These are the bases, on which all the equations of the inequalities of the Moon are established. By these also they are distributed into different classes, and the calculations are performed separately, so that we have no reason to fear that an errour committed in one part will affect the rest.

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On this theory Euler constructed new lunar tables, in which the number of equations is less, and which are more convenient for use, than those formed according to the ancient methods. This vast labour is the object of a particular work, printed at Petersburg in 1772, under the following title: Theoria Motuum Lunæ, nova Methodo pertractata. As the author was at this time nearly blind, three of his most illustrious scholars, John Albert Euler, his son, Lewis Krafft, and John Lexel, executed or verified the calculations. The Academy of Sciences at Paris, having proposed for the prize subjects of 1770 and 1772 the improvement of the theory of the Moon, awarded the prizes in one instance, and part of it in the other, to the two papers sent by Euler, in which his new theory was still farther simplified.

The Moon is not the only satellite, of which the motion has been considered. The principle of gravitation has been successfully applied to the inequalities of the satellites of Jupiter likewise.

After it was known, that comets are bodies perfectly similar to the planets, and subjected to the same laws of motion round the Sun, the researches into the inequalities of the planets could not fail to be extended to the comets; particularly as the comet of Halley offered a direct application of these new calculations. This astronomer had found, that, in consequence of the attraction of Jupiter, the period of the comet in question beginning in 1682 would be somewhat more than a year longer than it's preceding period; but the state of geometry in his time did not allow him, to make the computation with all