termed the mural circle.  These two inequalities of very different magnitudes connected with the cause which produces them by analytical combinations of totally different kinds have, however, both conducted to the same value of the ellipticity.  It must be borne in mind, however, that the ellipticity thus deduced from the movements of the moon, is not the ellipticity corresponding to such or such a country, the ellipticity observed in France, in England, in Italy, in Lapland, in North America, in India, or in the region of the Cape of Good Hope, for the earth’s materials having undergone considerable upheavings at different times and in different places, the primitive regularity of its curvature has been sensibly disturbed by this cause.  The moon, and it is this circumstance which renders the result of such inestimable value, ought to assign, and has in reality assigned the general ellipticity of the earth; in other words, it has indicated a sort of mean value of the various determinations obtained at enormous expense, and with infinite labour, as the result of long voyages undertaken by astronomers of all the countries of Europe.

I shall add a few brief remarks, for which I am mainly indebted to the author of the Mecanique Celeste.  They seem to be eminently adapted for illustrating the profound, the unexpected, and almost paradoxical character of the methods which I have just attempted to sketch.

What are the elements which it has been found necessary to confront with each other in order to arrive at results expressed even to the precision of the smallest decimals?

On the one hand, mathematical formulae, deduced from the principle of universal attraction; on the other hand, certain irregularities observed in the returns of the moon to the meridian.

An observing geometer who, from his infancy, had never quitted his chamber of study, and who had never viewed the heavens except through a narrow aperture directed north and south, in the vertical plane in which the principal astronomical instruments are made to move,—­to whom nothing had ever been revealed respecting the bodies revolving above his head, except that they attract each other according to the Newtonian law of gravitation,—­would, however, be enabled to ascertain that his narrow abode was situated upon the surface of a spheroidal body, the equatorial axis of which surpassed the polar axis by a three hundred and sixth part; he would have also found, in his isolated immovable position, his true distance from the sun.

I have stated at the commencement of this Notice, that it is to D’Alembert we owe the first satisfactory mathematical explanation of the phenomenon of the precession of the equinoxes.  But our illustrious countryman, as well as Euler, whose solution appeared subsequently to that of D’Alembert, omitted all consideration of certain physical circumstances, which, however, did not seem to be of a nature to be neglected without examination.  Laplace has supplied this deficiency.  He has shown that the sea, notwithstanding its fluidity, and that the atmosphere, notwithstanding its currents, exercise the same influence on the movements of the terrestrial axis as if they formed solid masses adhering to the terrestrial spheroid.