and 1769, on which occasions France-not to speak of stations in Europe-was represented at the isle of Rodrigo by Pingré; at the isle of San Domingo by Fleurin; at California by the Abbé Chappe; at Pondicherry by Legentil. At the same epochs England sent Maskelyne to St. Helena; Wales to Hudson's Bay; Mason to the Cape of Good Hope; Captain Cooke to Otaheite, &c. The observations of the southern hemisphere, compared with those of Europe, and especially with the observations made by an Austrian astronomer, Father Hell, at Wardhus, in Lapland, gave, for the distance of the sun, the result which has since figured in all treatises on astronomy and navigation.

No government hesitated in furnishing academies with the means, however expensive they might be, of conveniently establishing their observers in the most distant regions. We have already remarked that the determination of the contemplated distance appeared to demand imperiously an extensive base; for small bases would have been totally inadequate to the purpose. Well, Laplace has solved the problem numerically, without a base of any kind whatever. He has deduced the distance of the sun from observations of the moon made in one and the same place!

The sun is, with respect to our satellite, the cause of perturbations which evidently depend on the distance of the immense luminous globe from the earth. Who does not see that these perturbations would diminish if the distance increased; that they would increase, on the contrary, if the dis tance diminished; that the distance finally determines the magnitude of the perturbations?

Observation assigns the numerical value of these perturbations; theory, on the other hand, unfolds the general mathematical relation, which connects them with the solar parallax, and with other known elements. The determination of the mean radius of the terrestrial orbit then becomes one of the most simple operations of algebra. Such is the happy combination by the aid of which Laplace has solved the great, the celebrated problem of parallax. It is thus that the illustrious geometer found for the mean distance of the sun from the earth, expressed in radii of the terrestrial orbit, a value differing only in a slight degree from that which was the fruit of so many troublesome and expensive voyages. According to the opinion of very competent judges, the result of the indirect method might not impossibly merit the preference.*

The movements of the moon proved a fertile mine of research to our great geometer. His penetrating intellect discovered in them unknown treasures. He disentangled them from everything which concealed them from vulgar eyes with an ability and a perseverance equally worthy of "Mayer, from the principles of gravitation, (Theoria Lunæ, 1767,) computed the value of the solar parallax to be 7".8. He remarked that the error of this determination did not amount to one-twentieth of the whole, whence it followed that the true value of the parallax could not exceed 8".2. Laplace, by an analogous process, determined the parallax to be 8.45. Encke, by a profound discussion of the observations of the transits of Venus in 1761 and 1769, found the value of the same element to be 8.5776.-TRANSLATOR.

admiration. The reader will excuse me for citing another of such examples.

The earth governs the movements of the moon. The earth is flattened; in other words, its figure is spheroidal. A spheroidal body does not attract like a sphere. There ought, then, to exist in the movement, I had almost said in the countenance, of the moon a sort of impression of the spheroidal figure of the earth. Such was the idea as it originally oc curred to Laplace.

It still remained to ascertain (and here consisted the chief difficulty) whether the effects attributable to the spheroidal figure of the earth were sufficiently sensible not to be confounded with the errors of observation. It was accordingly necessary to find the general formula of perturbations of this nature, in order to be able, as in the case of the solar parallax, to eliminate the unknown quantity.

The ardor of Laplace, combined with his power of analytical research, surmounted all obstacles. By means of an investigation which demanded the most minute attention, the great geometer discovered in the theory of the moon's movements two well-defined perturbations depending on the spheroidal figure of the earth. The first affected the resolved element of the motion of our satellite, which is chiefly measured with the instrument known in observatories by the name of the transit instrument; the second, which operated in the direction north and south, could only be effected by observations with a second instrument, 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 labor, 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 Mécanique Céleste. They seem to be eminently adapted for illustrating the profound, the unexpected, and almost par adoxical 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 formulæ 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 of which the principal astronomical instruments are made to moveto 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, im movable 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 neg lected 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.

Do the extremities of the axis around which the earth performs an entire revolution once in every twenty-four hours correspond always to the same material points of the terrestrial spheroid? In other words, do the poles of rotation, which from year to year correspond to different stars, undergo also a displacement at the surface of the earth?

In the case of the affirmative, the equator is movable as well as the poles; the terrestrial latitudes are variable; no country during the lapse of ages will enjoy, even on an average, a constant climate; regions the most different will, in their turn, become circumpolar. Adopt the contrary supposition, and everything assumes the character of an admirable permanence.

The question which I have just suggested, one of the most important in astronomy, cannot be solved by the aid of mere observation, on account of the uncertainty of the early determinations of terrestrial latitude. Laplace has supplied this defect by analysis. The great geometer has demonstrated that no circumstance depending on universal gravitation can sensibly displace the poles of the earth's axis relatively to the surface of the terrestrial spheroid. The sea, far from being an ob stacle to the invariable rotation of the earth upon its axis, would, on the

contrary, reduce the axis to a permanent condition in consequence of the mobility of the waters and the resistance which their oscillations experience.

The remarks which I have just made with respect to the position of the terrestrial axis are equally applicable to the time of the earth's rotation, which is the unit, the true standard of time. The importance of this element induced Laplace to examine whether its numerical value might not be liable to vary from internal causes, such as earthquakes and volcanoes. It is hardly necessary for me to state that the result obtained was negative.

The admirable memoir of Lagrange upon the libration of the moon seemed to have exhausted the subject. This, however, was not the

case.

The motion of revolution of our satellite around the earth is subject to perturbations, technically termed secular, which were either unknown to Lagrange or which he neglected. These inequalities eventually place the body, not to speak of entire circumferences, at angular distances of a semicircle, a circle and a half, &c., from the position which it would otherwise occupy. If the movement of rotation did not participate in such perturbations, the moon in the lapse of ages would present in succession all the parts of its surface to the earth.

This event will not occur. The hemisphere of the moon which is actually invisible will remain invisible forever. Laplace, in fact, has shown that the attraction of the earth introduces into the rotatory motion of the lunar spheroid the secular inequalities which exist in the movement of revolution.

Researches of this nature exhibit in full relief the power of mathematical analysis. It would have been very difficult to have discovered by synthesis truths so profoundly enveloped in the complex action of a multitude of forces.

We should be inexcusable if we omitted to notice the high importance of the labors of Laplace on the improvement of the lunar tables. The immediate object of this improvement was, in effect, the promo tion of maritime intercourse between distant countries; and, what was indeed far superior to all considerations of mercantile interest, the preservation of the lives of mariners.

Thanks to a sagacity without parallel, to a perseverance which knew no limits, to an ardor always youthful, and which communicated itself to able co-adjutors, Laplace solved the celebrated problem of the longitude more completely than could have been hoped for in a scientific point of view, with greater precision than the art of navigation, in its utmost refinement, demanded. The ship, the sport of the winds and tempests, has no occasion, in the present day, to be afraid of losing itself in the immensity of the ocean. An intelligent glance at the starry vault indicates to the pilot, in every place and at every time, his distance from the meridian of Paris. The extreme perfection of the ex

isting tables of the moon entitles Laplace to be ranked among the benefactors of humanity.*

In the beginning of the year 1611, Galileo supposed that he found in the eclipses of Jupiter's satellites a simple and rigorous solution of the famous problem of the longitude, and active negotiations were immediately commenced with the view of introducing the new method on board the numerous vessels of Spain and Holland. These negotiations failed. From the discussion, it plainly appeared that the accurate observation of the eclipses of the satellites would require powerful telescopes; but such telescopes could not be employed on board a ship tossed about by the waves.

The method of Galileo seemed, at any rate, to retain all its advantages when applied on land, and to promise immense improvements to geography. These expectations were found to be premature. The movements of the satellites of Jupiter are not by any means so simple as the immortal inventor of the method of longitudes supposed them to be. It was necessary that three generations of astronomers and mathematicians should labor with perseverance in unfolding their most considerable perturbations. It was necessary, in fine, that the tables of those bodies should acquire all desirable and necessary precision, that Laplace should introduce into the midst of them the torch of mathematical analysis.

In the present day, the nautical ephemerides contain, several years in advance, the indication of the times of the eclipses and re-appearances of Jupiter's satellites. Calculation does not yield in precision to direct observation. In this group of satellites, considered as an independent system of bodies, Laplace found a series of perturbations analogous to those which the planets experience. The rapidity of the revolutions unfolds, in a sufficiently short space of time, changes in this system which require centuries for their complete development in the solar system. Although the satellites exhibit hardly an appreciable diameter even when viewed in the best telescopes, our illustrious countryman was enabled to determine their masses. Finally, he discovered certain simple relations of an extremely remarkable character between the movements of those bodies, which have been called the laws of Laplace. Posterity will not obliterate this designation; it will acknowledge the propriety of inscribing in the heavens the name of so great an astronomer beside that of Kepler.

*The theoretical researches of Laplace formed the basis of Burckhardt's Lunar Tables,. which are chiefly employed in computing the places of the moon for the Nautical Almanac and other ephemerides. These tables were defaced by an empiric equation, suggested for the purpose of representing an inequality of long period which seemed to affect the mean longitude of the moon. No satisfactory explanation of the origin of this inequality could be discovered by any geometer, although it formed the subject of much toilsome investigation throughout the present century, until at length M. Hanson found it to arise from a combination of two inequalities due to the disturbing action of Venus. The period of one of these inequalities is 273 years, and that of the other is 239 years. The maximum value of the former is 27".4, and that of the latter is 23.2.--TRANSLATOR.