in reality, are twenty-two years of labor to him who is about to become the legislator of worlds; who shall inscribe his name in ineffaceable characters upon the frontispiece of an immortal code; who shall be able to exclaim in dithyrambic language, and without incurring the reproach of any one, "The die is cast; I have written my book; it will be read either in the present age or by posterity, it matters not which; it may well await a reader, since God has waited six thousand years for an interpreter of his works!"* To investigate a physical cause capablə of making the planets revolve in closed curves; to place the principle of the stability of the universe in mechanical forces, and not in solid supports, such as the spheres of crystal which our ancestors had dreamed of; to extend to the revolutions of the heavenly bodies the general principles of the mechanics of terrestrial bodies, such were the questions which remained to be solved after Kepler had announced his discoveries to the world.

Very distinct traces of these great problems are perceived here and there among the ancients as well as the moderns, from Lucretius and Plutarch down to Kepler, Bouillaud, and Borelli. It is to Newton, however, that we must award the merit of their solution. This great man, like several of his predecessors, conceived the celestial bodies to have a tendency to approach toward each other in virtue of an attractive force, deduced the mathematical characteristics of this force from the laws of Kepler, extended it to all the material molecules of the solar system, and developed his brilliant discovery in a work which, even in the present day, is regarded as the most eminent production of the human intellect.

The heart aches when, studying the history of the sciences, we perceive so magnificent an intellectual movement effected without the co-operation of France. Practical astronomy increased our inferiority The means of investigation were at first inconsiderately intrusted to foreigners, to the prejudice of Frenchmen abounding in intelligence and zeal. Subsequently intellects of a superior order struggled with courage, but in vain, against the unskillfulness of our artists. During this period Bradley, more fortunate, on the other side of the Channel, immortalized himself by the discovery of aberration and nutation.

The contribution of France to these admirable revolutions in astronomical science consisted, in 1740, of the experimental determination of

* These celebrated laws, known in astronomy as the laws of Kepler, are three in number. The first law is, that the planets describe ellipses around the sun in their common focus; the second, that a line joining the planet and the sun sweeps over equal areas in equal times; the third, that the squares of the periodic times of the planets are proportional to the cubes of their mean distances from the sun. The first two laws were discovered by Kepler in the course of a laborious examination of the theory of the planet Mars; a full account of this inquiry is contained in his famous work De Stella Martis, published in 1609. The discovery of the third law was not effected until several years afterward. Kepler announced it to the world in his treatise on Harmonics, (1628.) The passage quoted below is extracted from that work.-TRANSLATOR.

the spheroidal figure of the earth, and of the discovery of the variation of gravity upon the surface of our planet. These were two great results; our country, however, had a right to demand more: when France is not in the first rank she has lost her place."

This rank, which was lost for a moment, was brilliantly regained, an achievement for which we are indebted to four geometers. When Newton, giving to his discoveries a generality which the laws of Kepler did not imply, imagined that the different planets were not only attracted by the sun, but that they also attract each other, he introduced into the heavens a cause of universal disturbance. Astronomers could

then see at the first glance that in no part of the universe, whether near or distant, would the Keplerian laws suffice for the exact representation of the phenomena; that the simple, regular movements with which the imaginations of the ancients were pleased to endue the heavenly bodies. would experience numerous, considerable, perpetually changing pertur bations.

To discover several of these perturbations, to assign their nature, and in a few rare cases their numerical values, such was the object which Newton proposed to himself in writing the Principia Mathematica Philosophiæ Naturalis.

Notwithstanding the incomparable sagacity of its author, the Prin cipia contained merely a rough outline of the planetary perturbations. If this sublime sketch did not become a complete portrait, we must not attribute the circumstance to any want of ardor or perseverance; the efforts of the great philosopher were always superhuman; the questions which he did not solve were incapable of solution in his time. When the mathematicians of the Continent entered upon the same career, when they wished to establish the Newtonian system upon an incontrovertible basis, and to improve the tables of astronomy, they actually found in their way difficulties which the genius of Newton had failed to surmount.

Five geometers, Clairaut, Euler, D'Alembert, Lagrange, and Laplace, shared between them the world of which Newton had disclosed the existence. They explored it in all directions, penetrated into regions which had been supposed inaccessible, pointed out there a multitude of

* The spheroidal figure of the earth was established by the comparison of an arc of the meridian that had been measured, in France, with a similar arc measured in Lapland, from which it appeared that the length of a degree of the meridian increases from the equator toward the poles, conformably to what ought to result upon the supposition of the earth having the figure of an oblate spheroid. The length of the Lapland arc was determined by means of an expedition which the French government had dispatched to the north of Europe for that purpose. A similar expedition had been dispatched from France about the same time to Peru, in South America, for the purpose of measuring an arc of the meridian under the equator, but the results had not been ascertained at the time to which the author alludes in the text. The variation of gravity at the surface of the earth was established by Richer's experiments with the pendulum at Cayenne, in South America, (1673–’74,) from which it appeared that the pendulum oscillates more slowly, and consequently the force of gravity is less intense, under the equator than in the latitude of Paris.-TRANSLATOR.

phenomena which observation had not yet detected-finally-and it is this which constitutes their imperishable glory-they reduced under the domain of a single principle, a single law, everything that was most refined and mysterious in the celestial movements. Geometry had thus the boldness to dispose of the future. The evolutions of ages are scrupulously ratifying the decisions of science.

We shall not occupy our attention with the magnificent labors of Euler. We shall, on the contrary, present the reader with a rapid analysis of the discoveries of his four rivals, our countrymen.* If a celestial body-the moon, for example-gravitated solely toward the center of the earth, it would describe a mathematical ellipse. It would strictly obey the laws of Kepler, or, which is the same thing, the principles of mechanics expounded by Newton in the first sections of his immortal work.

Let us now consider the action of a second force. Let us take into account the attraction which the sun exercises upon the moon. In other words, instead of two bodies, let us suppose three to operate on each other. The Keplerian ellipse will now furnish merely a rough indication of the motion of our satellite. In some parts the attraction of the sun will tend to enlarge the orbit, and will in reality do so. In other parts, the effect will be the reverse of this. In a word, by the introduction of a third attractive body, the greatest complication will succeed to a simple, regular movement upon which the mind reposed with complacency.

If Newton gave a complete solution of the question of the celestial movements in the case wherein two bodies attract each other, he did not even attempt an analytical investigation of the infinitely more difficult problem of three bodies. The problem of three bodies, (this is the name by which it has become celebrated,) the problem for determining the movement of a body subjected to the attractive influence of two other bodies, was solved for the first time by our countryman Clairaut. From this solution we may date the important improvements of the lunar tables effected in the last century.

The most beautiful astronomical discovery of antiquity is that of the precession of the equinoxes. Hipparchus, to whom the honor of it is due, gave a complete and precise statement of all the consequences

*It may, perhaps, be asked why we place Lagrange among the French geometers. This is our reply: It appears to us that the individual who was named Lagrange Tournier-two of the most characteristic French names which it is possible to imagine-whose maternal grandfather was M. Gros, whose paternal great grandfather was a French officer, a native of Paris, who never wrote except in French, and who was invested in our country with high honors during a period of nearly thirty years, ought to be regarded as a Frenchman, although born at Turin.-AUTHOR.

+ The problem of three bodies was solved independently about the same time by Euler, D'Alembert, and Clairaut. The two last-mentioned geometers communicated their solutions to the Academy of Sciences on the same day-November 15, 1747. Euler had already, in 1746, published tables of the moon, founded on his solution of the same problem, the details of which h subsequently published, in 1753.-TRANslator,

which flow from this movement. Two of these have more especially attracted attention.

By reason of the precession of the equinoxes, it is not always the same groups of stars, the same constellations, which are perceived in the heavens at the same season of the year. In the lapse of ages the constellations of winter will become those of summer, and reciprocally.

By reason of the precession of the equinoxes, the pole does not always occupy the same place in the starry vault. The moderately bright star which is very justly named in the present day the pole star, was far removed from the pole in the time of Hipparchus; in the course of a few centuries it will again appear removed from it. The designation of pole-star has been and will be applied to stars very distant from each other.

When the inquirer, in attempting to explain natural phenomena, has the misfortune to enter upon a wrong path, each precise observation throws him into new complications. Seven spheres of crystal did not suffice for representing the phenomena as soon as the illustrious astronomer of Rhodes discovered precession. An eighth sphere was then wanted to account for a movement in which all the stars participated at the same time.

Copernicus having deprived the earth of its alleged immobility, gave a very simple explanation of the most minute circumstances of precession. He supposed that the axis of rotation does not remain exactly parallel to itself; that in the course of each complete revolution of the earth around the sun the axis deviates from its position by a small quantity; in a word, instead of supposing the circumpolar stars to advance in a certain way toward the pole, he makes the pole advance toward the stars. This hypothesis divested the mechanism of the universe of the greatest complication which the love of theorizing had introduced into it. A new Alphonse would have then wanted a pretext to address to his astronomical synod the profound remark, so erroneously interpreted, which history ascribes to the King of Castile. If the conception of Copernicus improved by Kepler had, as we have just seen, introduced a striking improvement into the mechanism of the heavens, it still remained to discover the motive force which, by altering the position of the terrestrial axis during each successive year, would cause it to describe an entire circle, of nearly 500 in diameter, in a period of about 26,000 years.

Newton conjectured that this force arose from the action of the sun and moon upon the redundant matter accumulated in the equatorial regions of the earth; thus he made the precession of the equinoxes depend upon the spheroidal figure of the earth; he declared that upon a round planet no precession would exist.

All this was quite true, but Newton did not succeed in establishing it by a mathematical process. Now this great man had introduced into philosophy the severe and just rule: Consider as certain only what has

been demonstrated. The demonstration of the Newtonian conception of the precession of the equinoxes was, then, a great discovery, and it is to D'Alembert that the glory of it is due.* The illustrious geometer gave a complete explanation of the general movement in virtue of which the terrestrial axis returns to the same stars in a period of about 26,000 years. He also connected with the theory of gravitation the perturbation of precession discovered by Bradley, that remarkable oscillation which the earth's axis experiences continually during its movement of progression, and the period of which, amounting to about eighteen years, is exactly equal to the time which the intersection of the moon's orbit with the ecliptic employs in describing the 360° of the entire circumference.

Geometers and astronomers are justly occupied as much with the figure and physical constitution which the earth might have had in remote ages as with its present figure and constitution.

As soon as our countryman Richer discovered that a body, whatever be its nature, weighs less when it is transported nearer the equatorial regions, everybody perceived that the earth, if it was originally fluid, ought to bulge out at the equator. Huyghens and Newton did more; they calculated the difference between the greatest and least axes, the excess of the equatorial diameter over the line of the poles.t

The calculation of Huyghens was founded upon hypothetic properties of the attractive force which were wholly inadmissible; that of Newton upon a theorem which he ought to have demonstrated. The theory of

It must be admitted that M. Arago has here imperfectly represented Newton's labors on the great problem of the precession of the equinoxes. The immortal author of the Principia did not merely conjecture that the conical motion of the earth's axis is due to the disturbing action of the sun and moon upon the matter accumulated around the earth's equator; he demonstrated by a very beautiful and satisfactory process that the movement must necessarily arise from that cause; and although the means of investigation, in his time, were inadequate to a rigorous computation of the quantitative effect, still his researches on the subject have been always regarded as affording one of the most striking proofs of sagacity which is to be found in all his works.-TRANSLATOR.

+ It would appear that Hooke had conjectured that the figure of the earth might be spheroidal before Newton or Huyghens turned their attention to the subject. At a meeting of the Royal Society on the 28th of February, 1678, a discussion arose respecting the figure of Mercury, which M. Gallet, of Avignon, had remarked to be oval on the occasion of the planet's transit across the sun's disk on the 7th of November, 1677. Hooke was inclined to suppose that the phenomenon was real, and that it was due to the whirling of the planet on an axis "which made it somewhat of the shape of a turnip, or of a solid made by an ellipsis turned round upon its shorter diameter." At the meeting of the society on the 7th of March, the subject was again discussed. In reply to the objection offered to his hypothesis on the ground of the planet being a solid body, Hooke remarked that "although it might now be solid, yet that at the beginning it might have been fluid enough to receive that shape; and that although this supposition should not be granted, it would be probable enough that it would really run into that shape and make the same appearance; and that it is not improbable but that the water here upon the earth might do it in some measure by the influence of the diurnal motion, which, compounded with that of the moon, he conceived to be the cause of the tides." (Journal Book of the Royal Society, vol. vi, p. 60.) Richer returned from Cayenne