number of points which can be placed on one spherical surface, at a distance from each other equal to the radius of the sphere, is thirteen; it follows, therefore, that if the stars of the first magnitude are so placed, our sun being at the centre of the spherical surface, the equilibrium will subsist and, in this case, there should appear only thirteen such stars. Then supposing the stars of the second magnitude to be on the surface of a sphere having twice the radius of the first sphere, such surface would contain fifty-two points at distances from each other equal to the distance of the surface of this sphere from that of the first; and hence it is concluded that there should appear but 52 stars of the second magnitude; for a like reason there should be but 117, of the third, and so on, to those of the least visible magnitude: and though it is not probable that the stars are disposed exactly on the surfaces of such spheres, yet there is nothing to disprove the opinion that the stars which appear to us of different magnitudes are situated nearly at the distances from us which the hypothesis assigns. The stars of the first two magnitudes are, very nearly, equal in number to those supposed by Dr. Halley, but we are scarcely able to form a correct opinion of the number which should be classed under the respective magnitudes beyond the first and second.

The proper motions first suspected by Dr. Halley from a comparison of the latitudes of certain stars, observed by himself, with the latitudes determined by Ptolemy and Tycho Brahe, have been already verified in many of the principal stars; and we have mentioned the assertion of Sir William Herschell that some of these motions seemed to indicate a movement of the whole solar system in space: but, whether this movement is progressive or performed in a circular manner about some unknown centre of attraction, the observations are not yet sufficiently sure to allow an opinion to be formed; and it must also be remarked that many of the observed changes of place in the stars are quite inconsistent with either hypothesis: the complete determination of this question must, therefore, be left to the astronomers of a future age.

The ancient catalogues of stars being extremely deficient in accuracy, Flamstead, between 1668 and 1718, by means of the

transit instrument and mural circle with which the Greenwich observatory was then furnished, fixed the position of three thousand stars with respect to their right ascensions and declinations: within that interval of time Dr. Halley made a voyage to St. Helena, partly for the purpose of obtaining the places of the stars in the southern region of the heavens; and, though the atmosphere about the island was unfavourable for astronomical observations, he succeeded in determining the positions of more than 350 stars, but the honour of making a complete catalogue of the southern stars was reserved for La Caille, who accomplished this great task from observations made in the years 1751 and 1752, during his stay at the Cape of Good Hope. Subsequently, an extensive catalogue, containing the places of fifty thousand stars, in both hemispheres, which it cost the labour of ten years to complete, was made by M.M. Le Français and Jerome Lalande; and we now possess in the Berlin catalogue of M. Bode, which contains 17,240 stars with their right ascensions and declinations, computed for the first day of January, 1801, and the annual variations of those elements.

CHAPTER XXIV.

THE TRANSCENDENTAL ANALYSIS EMPLOYED IN PHYSICAL

ASTRONOMY.

At what time the planetary theory was first investigated analytically.— Cause of the revolution of a smaller body about a greater.-The problem of Three Bodies applied by Clairaut and others to the investigation of the lunar inequalities.-The cause of the moon's acceleration explained by La Place. The inequalities of the mean motions of Jupiter and Saturn were investigated by Euler and others.-The figure of the earth and the constancy of its time of rotation determined by La Place.-The variations of the precession, the obliquity of the ecliptic, and the length of the year, shewn. The figure of the moon determined by La Grange.-The permanency of the planetary system.

THE geometrical analysis which Newton, following the example of the ancient mathematicians, had adopted in his investigations being almost immediately after his death abandoned, the fluxionary, or differential calculus, which had been discovered by that great philosopher and, perhaps independently of him, by the celebrated Leibnitz, was zealously cultivated both in this country and on the continent; and, having been brought to a highly-improved state, it was applied to the solution of problems relating to the phenomena of the heavens. More than half a century, however, had elapsed since the publication of the Principia without any attempt being made to repeat or extend the researches commenced by its illustrious author; but, about the middle of the eighteenth century, a number of learned men, as if by a common impulse, embracing the law of gravitation proposed by Newton, applied themselves to the task of forming on it, as a basis, by the new analysis alone, a complete theory of the planetary movements: among these were Clairaut, Euler, and D'Alembert, who, in 1747, apparently without any knowledge of each other's intentions, investigated the curve which would be described by a body when urged by an impulsive force in a given direction, attracted by a second

body with a force varying, inversely, as the square of the distance, and having its path disturbed by the attraction of a third body acting on both at a finite distance from either.

In the Newtonian hypothesis, all the bodies of the solar system are supposed to attract each other, mutually, according to the same law, and the first step in the enquiry concerning the consequences of the principle, appears to have been that of determining for what reason any one or more should revolve about another, and why this last should not revolve about either of the others; why, for example, the moon should revolve about the earth rather than the earth about the moon, or why both of these should revolve about the sun rather than the sun about them; and in answer to this question, it was rightly alleged that, since the attractive principle resides in every particle of matter, the greater bodies must necessarily exert the greater influence, and cause a greater movement than can be communicated by those which are smaller. When, however, two bodies of unequal magnitude, as the earth and moon, attract each other, the common centre of gravity of both is at a distance from either, inversely proportional to the magnitudes, and the attraction of each causes the other to revolve about this centre of gravity; so that the moon, which is the smaller body, being at the greatest distance from the centre of motion, describes an orbit which necessarily includes that described by the earth about the same point. The sun, also, being vastly greater than all the circumvolving planets is, for a like reason, able to make but a very small movement about the common centre of the system, when compared with that motion which is performed by any one of the planets.

The investigations of Newton had clearly shewn it to be a necessary result of his law of gravitation, that the orbit of any body revolving about a centre of attraction should be a mathematical ellipse, the attraction of the other bodies of the system being excluded: now, delicate observations had shewn that this is neither the figure of the lunar orbit nor of the orbit described by any planet, or by the common centre of gravity of a planet and his satellites; but it was reasonable to suppose that, if the planetary bodies reciprocally gravitate towards each other by

the same law of attraction, such law would suffice to explain the deviations of the orbits from the elliptical figure; consequently, it would be possible, in given circumstances, to compute, from it, the amount of the deviation and of the inequalities produced in the motion of a planet; and this is what has been, at length, satisfactorily proved by the mathematicians of the last and present ages. If, however, the investigation of the figure of an undisturbed orbit; or, in other words, if a theory of two bodies, was thought to be difficult, much more so must have appeared that which relates to the attractions of three, or a greater number of bodies. In fact, the problem of Three Bodies, as it was called, presents, if considered in all its generality, difficulties which all the powers of the modern analysis are not able to overcome; but these have been considerably diminished by supposing one of the attracting bodies to be very superior in mass to either of the others, as is the case with the sun and planets; or very remote, compared with the distance between the others, as is the case with the sun when disturbing the

moon.

The variations, caused by the attraction of the sun, in the movements of the moon and in the figure of her orbit, are far greater than those in the movement and orbit of the earth or of any one planet by the attractions of the others, and, on that account, the correct determination of them from the assumed law of gravitation is, evidently, the best test of the truth of the law; accordingly, the theory of the moon was almost the first object to which the continental mathematicians applied the powers of the differential analysis. Agreeably to the method of Newton, which has been already explained, and indeed it was not possible that any other should be pursued, the disturbing force, both in the lunar and planetary theories, was conceived to be decomposed into two forces, one acting in the direction of a tangent to the path or orbit of a disturbed body, and the other in the direction of its radius vector: but, from this point, the courses of the investigations conducted by Newton, and by the foreign disciples of his school, diverged from each other, to meet only at the conclusion of the enquiry.

In 1747, both Clairaut and D'Alembert, when treating in a