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 circula- tion 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.
216 POSITIVE PHILOSOPHY.
on account of the instability of the focus of the princi- pal 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 per- turbations 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 disturbing- body 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 satel- lite, 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 ex- treme 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 circum- stances, 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 im- possible; 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 218 POSITIVE PHILOSOPHY.
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 modifica- tions, 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 ecjua- torial 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 con- sidered. 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 under- goes, 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 ten- dency 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 move- ments 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 deter- 220 POSITIVE PHILOSOPHY.
mine, 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 accom- plished 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, pro- duced 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 corre- sponding 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 come- tary 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 mathe- matical tables with observed facts shows that the resistance is imperceptible in degree; yet, as it is manifestly impos- sible 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 dimi- nishes, 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 per- turbations properly so called. These are among the incon- testable 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 222 POSITIVE PHILOSOPHY.
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 con- firmed 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 occa- sions our tides. We may therefore pronounce the indepen- dence 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 imperfec- tion which results from the insufficiency of our mathe- matical 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 observa- tion; and this in regard not only to bodies whose mechani- cal theory is but just initiated, but with regard to some with which we are best acquainted.
We see however that besides the sublime Achievements direct knowledge afforded to us, celestial of Celestial dynamics has powerfully contributed to per- Dynamics. feet the whole body of astronomical theories in regard to their definitive aim, — the exact prevision of the state of the heavens at any period whatever, jmst or future. Kepler's laws might suffice to determine the state of our system for a short time, proper data being chosen; but if we wish to extend the inquiry, back or forwards, to any considerable I^eriod, we find the most perfect theory of perturbations absolutely necessary. It is to celestial dynamics that we owe our power of ranging up and down the centuries, to fix the precise moments of various celestial phenomena, such as eclipses, with certainty, and with a minuteness only inferior to that which is possible in the case of present events.
Though we have, according to my view, completed our consideration of astronomical science, it would be felt to be a great omission if we passed over altogether what is now called Sidereal Astronomy. We will therefore see how much there is that we can conceive to be positive iu regard to cosmogony.
224 CHAPTER VI.
SIDERIAL ASTRONOMY AND COSMOGONY.
^ J- which appears to admit ot exact stuclv is that of the relative motions of the Multiple Stars, first discovered by Herschell. By multiple stars astronomers understand stars very near each other, whose angular dis- tance never exceeds a half minute, and which, for this reason, appear to be one, not only to the naked eye, but to ordinary telescopes, only the most powerful lenses being able to separate them. The relative movements of these stars tend to deceive us as to their precise multiple cha- racter, as, for instance, by mutual occultations, which do not permit us to separate them. Among some thousands of multiple stars registered in the catalogues, before the southern heavens had been really explored, almost all were only double, and we have found none which are more than triple, — a circumstance which may be owing solely to the imperfection of our telescopes, as we knew of none but single stars before Herschell's time. However interesting the study of them is, they constitute only a particular case in the universe, as the intervals of the stars which compose them are probably much smaller than those which divide the suns of the universe, so that the study of their relative motions does not lead us up to any of the great general phenomena of the heavens, and the speciality would be more conspicuous if astronomers did as I think they ought. — form their catalogues of those double stars only whose motions they have fully established. With regard to others, we cannot be sure whether their duality is a real relation or an accident. Knowing nothing whatever of their interval, or of the distance of either of them from us, we cannot Ite sure whether they form a system any more than any other two stars combined by chance in the MULTIPLE STARS. 225 heavens. Because a few incontestable examples are before us of a binary system, in wliicli the smaller circulates round the larger, it is anything- but philosophical to con- clude the same to be the case with the whole multitude of double stars, some of which may appear so merely through an accident of position, apparent only to our own system. Analogy is not applicable here; as what looks like analogy is merely the imperfection of our investigations. No astro- nomer would venture to assert that if our telescopes were what they may one day become, we might not find between stars now apparently independent a multitude of clustered intermediate stars which should render the case of duality almost general. The apparent nearness would not then be a sufficient gi'ound for presuming their mutual revolu- tions, because it is in virtue of their very small number that analogy now suggests that presumption. The only positive study in sidereal astronomy is that of the known relative motions of certain double stars, at present not more than seven or eight in number. We could never hope to assign with accuracy their orbits, or their periodic times, or any solid basis for dynamical conclusions.^ The importance of such inquiries is much diminished by the consideration that our system, which, in such a case, means our sun, belongs to no groups of the kind, — either investi- gated or merely pointed out. This circumstance seems to me not at all accidental; for, if our system made a part of a double star, which it is not difficult to imagine, it would pi'obably be impossible for us ever to be aware of the other part of such a duality, because in the direction of the sun it would be so near that its light would be lost to us in that of our sun. Such a case might, however, have a scientific interest for us, not only as elucidating the dis- placements of our system, but as allowing such great pre- cision, as might arise from the position of the inquirer on one of the stars of the couple.