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 distance diminished; that the distance finally determines the magnitude of the perturbations?

Observation a.s.signs 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 ill.u.s.trious 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.[36]

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 every thing which concealed them from vulgar eyes with an ability and a perseverance equally worthy of 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 occurred 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 quant.i.ty.

The ardour of Laplace, combined with his power of a.n.a.lytical 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 a.n.a.lytical 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 circ.u.mstance which renders the result of such inestimable value, ought to a.s.sign, and has in reality a.s.signed 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 ill.u.s.trating 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 princ.i.p.al 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 surpa.s.sed 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 ill.u.s.trious countryman, as well as Euler, whose solution appeared subsequently to that of D'Alembert, omitted all consideration of certain physical circ.u.mstances, 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 ma.s.ses 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 lat.i.tudes 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 circ.u.mpolar.

Adopt the contrary supposition, and every thing a.s.sumes 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 lat.i.tude.

Laplace has supplied this defect by a.n.a.lysis. The great geometer has demonstrated that no circ.u.mstance 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 obstacle 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 circ.u.mferences, at angular distances of a semi-circle, a circle and a half, &c., from the position which it would otherwise occupy. If the movement of rotation did not partic.i.p.ate 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 for ever. 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 a.n.a.lysis. It would have been very difficult to have discovered by synthesis truths so profoundly enveloped in the complex action of a mult.i.tude of forces.

We should be inexcusable if we omitted to notice the high importance of the labours of Laplace on the improvement of the lunar tables. The immediate object of this improvement was, in effect, the promotion 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 ardour always youthful and which communicated itself to able coadjutors, 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 existing tables of the moon ent.i.tles Laplace to be ranked among the benefactors of humanity.[37]

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 labour 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 a.n.a.lysis.

In the present day, the nautical ephemerides contain, several years in advance, the indication of the times of the eclipses and reappearances 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 a.n.a.logous to those which the planets experience. The rapidity of the revolutions unfolds, in a sufficiently short s.p.a.ce 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 ill.u.s.trious countryman was enabled to determine their ma.s.ses. 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.

Let us cite two or three of the laws of Laplace:--

If we add to the mean longitude of the first satellite twice that of the third, and subtract from the sum three times the mean longitude of the second, the result will be exactly equal to 180.

Would it not be very extraordinary if the three satellites had been placed originally at the distances from Jupiter, and in the positions, with respect to each other, adapted for constantly and rigorously maintaining the foregoing relation? Laplace has replied to this question by showing that it is not necessary that this relation should have been rigorously true at the origin. The mutual action of the satellites would necessarily have reduced it to its present mathematical condition, if once the distances and the positions satisfied the law approximately.

This first law is equally true when we employ the synodical elements. It hence plainly results, that the first three satellites of Jupiter can never be all eclipsed at the same time. Bearing this in mind, we shall have no difficulty in apprehending the import of a celebrated observation of recent times, during which certain astronomers perceived the planet for a short time without any of his four satellites. This would not by any means authorize us in supposing the satellites to be eclipsed. A satellite disappears when it is projected upon the central part of the luminous disk of Jupiter, and also when it pa.s.ses behind the opaque body of the planet.

The following is another very simple law to which the mean motions of the same satellites of Jupiter are subject:

If we add to the mean motion of the first satellite twice the mean motion of the third, the sum is exactly equal to three times the mean motion of the second.[38]

This numerical coincidence, which is perfectly accurate, would be one of the most mysterious phenomena in the system of the universe if Laplace had not proved that the law need only have been approximate at the origin, and that the mutual action of the satellites has sufficed to render it rigorous.

The ill.u.s.trious geometer, who always pursued his researches to their most remote ramifications, arrived at the following result: The action of Jupiter regulates the movements of rotation of the satellites so that, without taking into account the secular perturbations, the time of rotation of the first satellite plus twice the time of rotation of the third, forms a sum which is constantly equal to three times the time of rotation of the second.

Influenced by a deference, a modesty, a timidity, without any plausible motive, our artists in the last century surrendered to the English the exclusive privilege of constructing instruments of astronomy. Thus, let us frankly acknowledge the fact, at the time when Herschel was prosecuting his beautiful observations on the other side of the Channel, there existed in France no instruments adapted for developing them; we had not even the means of verifying them. Fortunately for the scientific honour of our country, mathematical a.n.a.lysis is also a powerful instrument. Laplace gave ample proof of this on a memorable occasion when from the retirement of his chamber he predicted, he minutely announced, what the excellent astronomer of Windsor would see with the largest telescopes which were ever constructed by the hand of man.

When Galileo, in the beginning of the year 1610, directed towards Saturn a telescope of very low power which he had just executed with his own hands, he perceived that the planet was not an ordinary globe, without however being able to ascertain its real form. The expression _tri-corporate_, by which the ill.u.s.trious Florentine designated the appearance of the planet, implied even a totally erroneous idea of its structure. Our countryman Roberval entertained much sounder views on the subject, but from not having inst.i.tuted a detailed comparison between his hypothesis and the results of observation, he abandoned to Huyghens the honour of being regarded as the author of the true theory of the phenomena presented by the wonderful planet.

Every person knows, in the present day, that Saturn consists of a globe about 900 times greater than the earth, and a ring. This ring does not touch the ball of the planet, being everywhere removed from it at a distance of 20,000 (English) miles. Observation indicates the breadth of the ring to be 54,000 miles. The thickness certainly does not exceed 250 miles. With the exception of a black streak which divides the ring throughout its whole contour into two parts of unequal breadth and of different brightness, this strange colossal bridge without piles had never offered to the most experienced or skilful observers either spot or protuberance adapted for deciding whether it was immovable or endued with a movement of rotation.

Laplace considered it to be very improbable, if the ring was immovable, that its const.i.tuent parts should be capable of resisting by their mere cohesion the continual attraction of the planet. A movement of rotation occurred to his mind as const.i.tuting the principle of stability, and he hence deduced the necessary velocity. The velocity thus found was exactly equal to that which Herschel subsequently deduced from a course of extremely delicate observations.

The two parts of the ring being placed at different distances from the planet, could not fail to experience from the action of the sun, different movements of rotation. It would hence seem that the planes of both rings ought to be generally inclined towards each other, whereas they appear from observation always to coincide. It was necessary then that some physical cause should exist which would be capable of neutralizing the action of the sun. In a memoir published in February, 1789, Laplace found that this cause must reside in the ellipticity of Saturn produced by a rapid movement of rotation of the planet, a movement the existence of which Herschel announced in November, 1789.

The reader cannot fail to remark how, on certain occasions, the eyes of the mind can supply the want of the most powerful telescopes, and lead to astronomical discoveries of the highest importance.

Let us descend from the heavens upon the earth. The discoveries of Laplace will appear not less important, not less worthy of his genius.

The phenomena of the tides, which an ancient philosopher designated in despair as _the tomb of human curiosity_, were connected by Laplace with an a.n.a.lytical theory in which the physical conditions of the question figure for the first time. Accordingly calculators, to the immense advantage of the navigation of our maritime coasts, venture in the present day to predict several years in advance the details of the time and height of the full tides without more anxiety respecting the result than if the question related to the phases of an eclipse.

There exists between the different phenomena of the ebb and flow of the tides and the attractive forces which the sun and moon exercise upon the fluid sheet which covers three fourths of the globe, an intimate and necessary connection from which Laplace, by the aid of a series of twenty years of observations executed at Brest, deduced the value of the ma.s.s of our satellite. Science knows in the present day that seventy-five moons would be necessary to form a weight equivalent to that of the terrestrial globe, and it is indebted for this result to an attentive and minute study of the oscillations of the ocean. We know only one means of enhancing the admiration which every thoughtful mind will entertain for theories capable of leading to such conclusions. An historical statement will supply it. In the year 1631, the ill.u.s.trious Galileo, as appears from his _Dialogues_, was so far from perceiving the mathematical relations from which Laplace deduced results so beautiful, so unequivocal, and so useful, that he taxed with frivolousness the vague idea which Kepler entertained of attributing to the moon's attraction a certain share in the production of the diurnal and periodical movements of the waters of the ocean.

Laplace did not confine himself to extending so considerably, and improving so essentially, the mathematical theory of the tides; he considered the phenomenon from an entirely new point of view; it was he who first treated of the stability of the ocean. Systems of bodies, whether solid or fluid, are subject to two kinds of equilibrium, which we must carefully distinguish from each other. In the case of stable equilibrium the system, when slightly disturbed, tends always to return to its original condition. On the other hand, when the system is in unstable equilibrium, a very insignificant derangement might occasion an enormous dislocation in the relative positions of its const.i.tuent parts.

If the equilibrium of waves is of the latter kind, the waves engendered by the action of winds, by earthquakes, and by sudden movements from the bottom of the ocean, have perhaps risen in past times and may rise in the future to the height of the highest mountains. The geologist will have the satisfaction of deducing from these prodigious oscillations a rational explanation of a great mult.i.tude of phenomena, but the public will thereby be exposed to new and terrible catastrophes.

Mankind may rest a.s.sured: Laplace has proved that the equilibrium of the ocean is stable, but upon the express condition (which, however, has been amply verified by established facts), that the mean density of the fluid ma.s.s is less than the mean density of the earth. Every thing else remaining the same, let us subst.i.tute an ocean of mercury for the actual ocean, and the stability will disappear, and the fluid will frequently surpa.s.s its boundaries, to ravage continents even to the height of the snowy regions which lose themselves in the clouds.

Does not the reader remark how each of the a.n.a.lytical investigations of Laplace serves to disclose the harmony and duration of the universe and of our globe!

It was impossible that the great geometer, who had succeeded so well in the study of the tides of the ocean, should not have occupied his attention with the tides of the atmosphere; that he should not have submitted to the delicate and definitive tests of a rigorous calculus, the generally diffused opinions respecting the influence of the moon upon the height of the barometer and other meteorological phenomena.