The comparison between mathematical theory and direct observation has never been carried out to any advantage, — • all the measurements having been taken in the ports, or near the shore. The tides in such places are very indirect; and they cannot properlv represent the regular tides from which they issue, their force being chiefly determined by the form of the soil, — at the bottom as well as on the surface, — and even perhaps affected by its structure. These are incidents which cannot enter into mathematical estimates; and to tliem we must doubtless refer the vast differences in the height of the tides at the same time, and in nearly the same place, — as, for instance, the tides of Bristol and Liverpool, of Granville and Dieppe. The only way of making an effectual direct observation would be to note the phenomena of the tides in a very small island, at the equator, and thirty degrees at least from any continent, for such a course of years as would allow of repeated record of variations as repeatedly foreseen. In this way, and in no other, might the mathematical theory of the tides be verified and perfected.
Whatever may be the uncertainty with regard to some of the data of this great theory, it has that conclusive sanction, — the fulfilment of its previsions; — a fulfilment so exact as to guide our conduct; and this, as we know, is the true end of all science. The principal local circumstances, except the winds, being calculable, it has been found practicable to assign for each port the mean height of the tides and their times; and thus have mathematical determinations been proved to be sufficiently conformable to reality, and a class of phenomena which, a century ago, were regarded as inexplicable, have been referred to invariable laws, and showTi to be as little arbitrary as anything else.
Such are the jihilosophical characteristics of the three great questions which compose the statical department of Celestial Mechanics. We must next look into the dynamical department, as rej^resented by the phenomena of our svstem.
CHAPTER Y.
CELESTIAL DYNAMICS.
we nave seen, deterniinea by the gravitation of each of them towards the focus of its orbit. The regularity of this movement must be impaired by the mutual gravitation of the bodies of the system. The most striking of these derangements were observed by the School of Alexandria, m the first days of Mathematical Astronomy; others have been observed, in proportion as our knowledge became more precise; and now, all are explained with such completeness by the theory of gravitation, that the smallest perturbations are known before they are observed. This is the last possible test and triumph of the Newtonian system.
There are, as Lagrange pointed out, two principal kinds of perturbations, which differ as much in their mathematical theory as in the circumstances which constitute them; instantaneous changes, from shocks or explosions, and gradual changes or perturbations, properly so called, caused by secondary gravitation, requiring time. The first kind may never have taken place in our t ^ j.
system; but it is necessary to consider it, not only because it is of possible occurrence, but because it is a necessary preliminary to the study of the other kind, — the gradual perturbations being treated theoretically as a series of little shocks.
The first case is easy of treatment. No collision or explosion would affect Kepler's laws: and, if the form of the orbit was altered, the accelerating forces would remain the same; and thus, the new variation once understood, our calculations might proceed as before. Supposing a collision between two planets, or the breakage of one planet into several fragments by an internal explosion; there might be any variations whatever in the astronomical elements of their elliptical movement; but there are two relations which are absolutely unalterable, and which misfht, in my opinion, generally enable us to establish the reality of such an event at any period whatever: these are the essential ]n'operties of the continuous motion of the centre of gravity, and the in variableness of the sum of the areas, — both resting on that gi'eat law of the equality of action and reaction to Avhich all changes must conform. From these must result two important equations between the masses, the velocities, and the positions of the two bodies, or the two fragments of the same body, considered before and after the event. No indication at present leads us to suppose that the case of collision has ever occurred in our system; and it is evident that such an encounter, though not mathematically impossible, woitM be very difficult. But it is far otherwise with regard to explosions.
The little planets discovered between Mars and Jupiter have mean distances and periodic times so nearly identical, that Mr. Olbers has conjectured that they once formed a single planet, which had exj^loded into fragments. Lagrange added a supposition, from the irregularity of their form, that the event must have happened after the consolidation of the primitive planet. Wlien their masses become known. I think this conjecture may be subjected to mathematical proof, — in this way. By calculating the positions and successive velocities of the centre of gravity of the system of these four planets, we might, if they had such an origin, retrace the principal motion of the primitive planet. If we should then find this centre of gravity describing an ellipse round the sun as a focus, and its vector radius tracing areas proportioned to the times, this event would Tie as completely established as any fact that we have not witnessed. We have not yet the materials for such a test; biit it is interesting to see how celestial mechanics mav establish, in a positive manner, events like these which nppear to have left no evidence behind them. It is obvious tliat the instantaneous character of such a change must [)reclude our fixing any date for it, since the phenomena would be precisely the same, whether the explosion were recent or long ago. It is otherwise with regard to perturbations, properly so called.
PLANETARY PERTURBATIONS. 213 Lagrange believed tliat these exj^losions had been frequent in our system, and that this was the true explanation of comets, judging from the greatness of their eccentricity and inclmatiou, and the sniallness of their masses. We have only to conceive that a planet may have burst into two very unequal fragments, the larger of which would proceed pretty nearly as before, while the smaller must describe a very long ellipse, much inclined to the eclij^tic. Lagrange showed that the amount of impulsion necessary for this change is not great; and that it is less in proportion as the primitive planet is remote from the sun. This opinion is far from having been demonstrated; but it appears to me more satisfactory than any other that has been proposed on the subject of comets.
The important and difficult subject of per- p. i i turbations is the principal object of celes- turbatious tial mechanics, for the perfecting of astronomical tables. They are of two classes; the one relating to motions of translation, the other of rotation. The latter are, as before, the most difficult: but the motions of rotation are less altered than the other class, within our own system; and they are less important to be known.
planets must be treated as it tliey were con- of translation densed in their centres of gravity.
The direct method, the only rational one, of calculating the diiferential equations of the motion of any one planet, under the iutlaences of all the rest, is impracticable, from the unmanageable complication of the problem. It would make an inextricable analytical enigma. Geometers have therefore been obliged to analyze directly the motion of each planet round that which is its focus, taking for modification only one at a time. This is p,,, r what constitutes in general the celebrated three bodies problem of three bodies, though this denomination was at first employed only for the theory of the moon. It is easy to see what circumvolutions are involved in this method, since the modifying body, being in its turn modified by others, compels a return to the study of the primitive body, to understand its perturbations. The determination of the motions of the whole of our system must, by its vei'y nature, be a single problem. It is the imperfection of our analysis which obliges us to divide it into detached problems, and to overload our formulas with multiplied modifications. The elementary problem of two bodies, — one of these even being regarded as fixed, — is the only one that we are capable of bringing to a solution; the problem of the elliptical motion, represented by Kepler's laws; and here the calculations are extremely laborious. It is to this type that geometers have to refer the motions of the 2>l'mets, by extremely complicated approximations, accumulating the perturbations separately produced by every body that can be supposed to exert any influence; and these perturbations prescribe the series required for the integration of the equations belonging to the case of the three bodies.
Then follows the task of choosing the perturbations which have to enter into the estimate. The law of gravitation enables us to compare the secondary influences involved in each case, — the masses of all within our own system being supjiosed to be known. It is a favourable circumstance to mathematical research that our system is constituted of bodies of very small mass in comparison with the sun (making the perturbations extremely small); moreover, very feAV, very far from each other, and very unequal in mass; the result of all which is that, in almost every case, the principal motion is modified by only one body. If the contrary had been the case, the perturbations must have been very great, and exti'emely varied, since a great number of bodies must have powerfully acted in each distiirbance. Celestial Mechanics must then, we should think, have presented an inextricable complication, being incapable of reduction to the problem of three bodies.
The study of modified motions divides itself into three pai'ts, answering, as in a former case, to the planets, satellites and comets. Rigorously speaking, we ought to make a fourth case of the sun, which cannot here be regarded as motionless, because the planets react upon it. In fact, we cannot allow ourselves to consider any point within the system ^,. as motionless, except the centre of gravity of Solar Svsteiii ^^^^ system itself, whicli is the true focus of planetary motion, and round which the sun THE THREE PROBLEMS. 215 itself must oscillate, in directions which vary according to the positions of the planets. This jjoint is always between the centre and the surface of the sun. But we cannot approach nearer to the fact than this: we shall probably never be able to indicate this centre precisely; and it is enough for practical pi;rposes, and necessary to them, to consider the sun as fixed, except as to its rotary motion. The same conclusion must be come to with regard to the planets and their satellites, — even in the case of the earth and moon, where the variations of the primary body are greatest. The centre of gravity falling within the mass of the primary body, its variations from that centre may be neglected as having no appreciable influence on the motion of translation; and thus, celestial mechanics presents, in this branch, no other problems than those treated, vinder another point of view, by celestial geometry.
The simplest problem is here, as before, p 1 1 f that of the planets, and for the same reasons, ^]jg pianets — the smallness of their eccentricities, and of the inclinations of their orbits. There is also a considerable uniformity of perturbations, since each planet remaining in the same regions of the sky, continues in the same mechanical relations, though their intensity varies within certain limits. The least privileged of these bodies in these matters is unhappily our own planet, on account of the heavy satellite which escorts it so closely, and to which its chief perturbations are due; though this does not save it from being sensibly troubled by others, at the period of opposition, and especially by such a mass as that of Jupiter. No other planet with satellites, not even Jupiter, is in so unfavourable a case; for Jupiter's motion could not be very much deranged by the action of his satellites, however near in position, since the mass of the largest is less tlian a ten-thousandth part of his, while the mass of our moon is a sixty-eighth part of that of the earth. Jupiter's circulation is sensibly affected by Saturn alone. The simplest case of all seems to be that of Uranus, from its being the last planet, and very remote from the next; and its six satellites do not appear to trouble its motion.
The problem of the satellites is necessarily Piolilem of more complicated than that of the planets, the Satellites.
on account of the instability of the focus of the principal motion, as in celestial geometry. Besides their own perturbations, the satellites have reflected, upon them all those to which their planet is liable. The founders of Celestial Mechanics were long perplexed, for instance, bj the perpetual acceleration of the mean motion of the moon; it was considered inexplicable, till Laplace discovered its cause in the slight variation to which the eccentricity of the earth's orbit is subject. In regard to the direct perturbations of the satellites, there is an essential distinction between the case of one, and that of several satellites. In the first, — the single case of our moon, — the disturbingbody is the sun, on account of its unequal action on the planet and the satellite. If the difficulties arising out of this position are greater than in the case of any other satellite, it is partly because the case more immediately concerns us, and because our opportunities of observation disclose more fully the imperfection of our means. For, in the mathematical point of view, there must be more complexity in the case of several satellites; all that is true in regard to one being true in regard to each one, with the addition of the mutual action of the members of the group. Their perturbations are reduced by the preponderating size of their j^lanet; but from there being so many of them, of such nearly equal sizes and direction, and all so close together, the difficulty of calculating their motions is so great that the only theory as yet established is that of the satellites of Jupiter. I'or the motions of three of them, Laplace found means completely to account. Those of Saturn and Uranus are known only geometrically, we having not even an ai:>proximate estimate of their masses. It is to be remembered, however, that we do not need so perfect a knowledge of them as of the moon; and that a much less exact theory will suffice for them than for the moon, whose slightest irregularity is very evident to us.
.. The comets intervene to increase our diflBthe Comets culties about the satellites. From the extreme prolongation of their orbits, and their inclination in all directions, comets are in a state of ever variable mechanical relations, from the number of bodies that they approach in their course; whilst the planets, and PROBLEM OF THE COMETS. 217 eveQ the satellites, have always the same relatious, the variation being only in the intensity. The perturbation which, in evei'y other case, bears a very small proportion to the gravitation, may, in the case of comets, exceed it: so that it is conceivable that a comet might be diverted from its orbit, and become a satellite, when it passes near so considerable a body as Jupiter, Saturn, or even Uranus. Besides the eccentricities of comets, there ai'e other circumstances, such as their small weight, and their possible loss of weight by parting with some of their atmosphere to the bodies they approach, which tend to 2)erj)lex the study of their perturbations. These are the incidents which make it so difficult to foresee exactly the return of these little bodies. When we have studied them so long and so laboriously as to have, to the best of our belief, mastered their case, we find that their periods are entirely changed through one omitted circumstance. A memorable example of this was the comet of 1770, calculated by Lexell. This comet had then a revolution of less than six years: but it has never apj^eared since, having been entirely dei'anged by passing too near Jupiter. The imperfection of our knowledge about these small bodies is from the same cause that renders them of very little consequence to us.
From their vast distances, their action upon any one body of the system is little more than momentary; and their lightness prevents even the satellites from being affected by their passage. The passage of the comet of 1770 among the satellites of Jupiter proved this, in a striking manner. Their tables, constructed beforehand, without any idea of such an incident, perfectly agreed with direct observations; a j^roof that the intrusion of the comet did not sensibly affect their motions. There is, therefore, no more occasion for the puerile fears of our day than for the religious terrors of former times, in regard to the passage of comets. Their collision with the earth is all but impossible; and they could not otherwise be felt at all. Their mere approach, however near, could have no other effect than to raise somewhat the corresponding tide. If a comet could pass two or three times nearer to us than the moon (which no known comet could do) its very small mass could produce ncj other effect than an imperceptible rise of the tides. We have therefore no immediate and practical reason to regret the imperfection of our cometary theories.
Passing from the pertiirbations proper to o/rotatk)n^'^^ motions of translation, we must notice those belonging to rotation.
The ellipsoid bodies of our system must, whether they began or not, have ended, sooner or later, with turning round one of their axes, — and that one the most stable, — that of their smallest diameter: for, as we have seen, it is their rotation that has produced their deviation from a perfectly spherical form, and determined the direction favourable to stability. The regularity of this rotation is evidently so indispensable to the existence of living bodies rp,,,. on the surface of a planet, that we might a priori assert this stability wherever life is possible, from the time when it became possible. But, stable as each planet is in itself, its mutual gravitation with others must introduce certain secondary modifications, the bearing of which must be upon the direction of its axis in space. It is only with regard to the earth that these modifications concern us; for however great they might be in any other l)ody, they could in no way affect us.
If the planets were perfect s])heres, the total gravitation of tlieir particles must pass through their centres of gravity; and thus, it is only through their slight failure in sphericity that they can act at all upon one another's rotation; that failure being caiised by the rotation itself. "We see here how the same necessity which secures the stability of the rotations, with regard to their duration and their poles, determines, from another point of view, the inevitable alteration of the parallelism of their axes. — In oiar own jdanet the precession of the equinoxes, modified by the nutation, results from the action of the other bodies of our system, — especially of the sun and moon, — upon our ecjuatorial protulierance. The power of each body is, as in the case of the tides, in the direct ratio of its mass, and inversely to the cube of its distance; so that the sun and moon are the only bodies whose influence need be considered. Further, the extent of the deviation depends on PERTURBATIONS OF ROTATION. 219 the mass and magnitude of the earth, on the time of its rotation, on its degree of flattening, and on the obliquity of the ecliptic. The intensity of the influence must vary, as in the case of the tides, with the variable distance of the sun from the earth, and yet more of the moon; but the want of uniformity is too slight to be j^ereeptible to direct observation. — These ai'e the general causes which dptermine the small changes which the rotation of our globe undergoes, in regard to the direction of its axis in space. — The case of the other ]ilanets bears a general likeness to that of the earth, varied according to the different inclinations of their axes to their orbits, their position, their mass. their size, the duration of their rotation, and the decree of their flattening at the poles. On all these grounds, the perturbations of Mars are the most remarkable.
The rotation of the satellites presents one m, ^, ^y. ^ consideration of the highest interest, — that remarkable equality between the duration of this rotation and that of their circuit round their planet, by which they present alwavs the same hemisphere, except from those verv small oscillations called librations, whose law is well understood. The fact is absoliTtelv cei'tain only with regard to the moon; but our mechanical principles pistifv our erecting it into a general law of all the satellites. Lagrange has shown that it results from the preponderance that, by the action of the planet, the nearer hemisphere must acquire at the outset, whence arises a natiiral tendency in the satellite to return perpetually to the same position. If it is thus with the moon, there is every reason to sup]>ose the same fact with regard to satellites belonging to heavier planets, to which they are proportionally nearer.
Such are the various kinds of perturbations produced in the movements of the bodies of our svstem, by their mutual action. This study may be simplified and rendered much more exact, by the device of referring all these movements to a plane whose position must neces- Pevire of an sarily be independent of all their variations. invariable — Among several planes which have been i>laTie.
proposed, difl^ei'ing in their degrees of variableness, M. Poinsot has discovered one which is the only truly invariable one, but which is extremely difficult to determine, since it requires not only an estimate of the planetary masses, but data dependent on the mathematical law of the interior density of the heavenly bodies, — a law which is still very hypothetical. The theory is complete; but its precise application is at present impossible. Whatever m.ay be the practical difficulties, we cannot but feel a deep interest in seeing how Celestial Mechanics has accomplished the fixing of an invariable plane in the midst of all the interior perturbations of our system, as Newton had first recognized an inalterable velocity, — that of the centre of general gravity. These are the only two elements in our system which are rigorously independent of all the events that can occur in its interior; — of even the vastest commotions that our imagination can suggest. Such variations as they can be conceived to have could relate only to the most general phenomena of the universe, produced by the mutual action of different suns, of which they would afford us the clearest manifestation, if such knowledge wei*e within our reach.
We end this study of perturbations with our system ^ recognition of the stability of our own system, in regard to all its most important constituent bodies. Setting aside the comets, all the variations whatever of any perceptible value are pei'iodical; and their period is usually very long, while their extent is very small; so that the whole of our planetary systeni can only oscillate with extreme slowness round a mean state, from which it deviates very little. Through all starry changes the translations of our planets present the almost rigorous invariableness of the great axes of their elliptical orbits, and of the duration of their sidereal revolutions: and their rotation shows a regularity even more perfect, in its duration, in its poles, and even, though in a somewhat smaller degree, in tlie inclination of its axis to the corresponding orbit. We know, for instance, that from the time of Hipparchus, the length of the day has not varied the hundredth part of a second. Amidst all this general regularity, we jaerceive a special and most marked stability with regard to the elements Avhich are concerned in the continued existence of living beings. — Such are the sublime theorems of natural philosophy for which humanity is STABILITY OF THE SOLAR SYSTEM. 221 indebted to the sum of the great works executed in the last century by the successors of Newton.
The general cause of these important results lies in the small eccentricity of all the principal orbits, and the small divergence of their planes. If the planets had had cometary orbits and planes, there would have been no regularity — no periodicity, — and, we may add. no life upon their surface, No planets can be habitable but such as have their oscillations restricted within verv narrow limits.
The Mathematical theory of celestial me-.
chanics has taken no notice, thus far, of the a' Medium resistance of any general medium, in which these motions are proceeding^. The conformity of our mathematical tables with observed facts shows that the resistance is imperceptible in degree; yet, as it is manifestly impossible that it shovild be null, the geometers have endeavoured to prepare beforehand a general analysis of it. Considered apart from its intensity, this action is of a totally different nature from that of perturbations, though gradual like them: for it cannot be periodical, and must always be exercised in the same direction, so as continually to diminish all velocities, and the more the greater they are. It cannot alter the positions of the orbits, but can by possibility affect only their dimensions, and periodic times, and the duration of rotations: that is, it affects the elements which are spared by the perturbations. Thus, the rotations must become slower, the orbits must grow smaller and rounder, and their periodic times shorter; because, as velocity diminishes, the solar action must become more powerful, and these effects are not only continuous, but always increasino' in rapidity. So, in a future too remote to be assigned, all the bodies of our system must be united to the solar mass, from which it is probable that thev proceeded: and thus the stability of the system is simply in relation to the perturbations properly so called. These are among the incontestable indications of Celestial Mechanics.
As yet. we practically fail to i*ecognize the effect of a resisting medium. We neither trace its operations, nor should know how to calculate it if we could trace it. Whenever we do, it will be by the study of comets; for their small mass, and the great surface which they present to the action of the medium when their atmospheres are widely diffused, must render its resistance much more appreciable than in the case of planets, — their velocity being besides naturally at its maximum at the moment of this expansion. iSome contemporary astronomers believe that they have established the effect of this resistance in regard to one or two comets. Hitherto the study of these bodies seems to be only negatively useful, to prevent the return of the absurd terrors which they formerly occasioned. We now see that there is no body in our system, however insigniticant, whose theory may not offer to us a direct and positive interest, since we may owe to comets the knowledge of one of the most important general laws of the system to which we belong, and that which, in a remote future, must chiefly rule its destinies.^ Independence In our geometrical review we saw, by the of tlie solar agreement of astronomical tables with direct system. observation, that our system is independent of all that lies outside. This incontestable truth is confirmed by the mechanical view. K our system gravitated towards any of the suns outside, the action of other suns would nearly neutralize the tendency. Again, it would be only by an unequal action of those suns upon our plantets that any change could be occasioned. Again, the vast distances would, according to our law of gravitation, make the action of remote suns imperceptible. The nearest body, if a million times heavier than our system, would produce an effect incalculably smaller than the action which occasions our tides. We may therefore pronounce the independence of our system to be perfectly certain. I notice this because we seem to find here the only exception to the great encyclopedical law which is the basis of this work, — that the most general phenomena rule tiie most particular, without being m any degree reciprocally influenced. Thus our astronomical phenomena regulate those of our own globe, — whether physical, chemical, physiological, or social. )[et here we find that the phenomena of the universe have no influence over those of the solar system. There is no difliculty about this to persons who, like myself, admit ^ M. Comte estimates too lightly the indications of a medium given by Enckes comet. — J. P. N.
ACHIEVEMENTS OF CELESTIAL DYNAMICS. 223 that our researclies are limited by the boundaries of our own system, and that positive knowledge cannot go beyond it. The study of the universe forms no part of natural philosophy: a truth which will become more apj^arent, and be seen to be more important the further our studies extend.
At the close of this brief review of celestial dynamics, we see that, great as are the achievements since Newton's time, we are reminded in many directions of the imperfection which results from the insufficiency of our mathematical analysis. In the execution of astronomical tables it has to borrow from celestial geometry other aid than the estimate of indispensable data, derived from direct observation; and this in regard not only to bodies whose mechanical theory is but just initiated, but with regard to some with which we are best acquainted.