Comte - The positive philosophy. Vol. 1

Positive Philosophy/201

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.

In the study of motions of translation, the planets must be treated as if they were condensed in their centres of gravity.

The direct method, the only rational one, of calculating the differential equations of the motion of any one planet, under the influences 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 what constitutes in general the celebrated 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 very 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 tarn 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 planets, 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 supposed 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 few, very far from each other, and very unequal is 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

202/Auguste Comte

extremely varied, since a great number of bodies must have powerfully acted in each disturbance. 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 parts, answering, as in a former case, to the planets, satellites and comets. Rigorously speaking, we ought to male it 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 the system itself, which is the true focus of planetary motion, and round which the sun itself must oscillate, in directions which vary according to the positions of the planets. This point 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 purposes, 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, under another point of view, by celestial geometry.

The simplest problem is here, as before that of the planets, and for the same reasons,—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 than a ten thousandth part of his, while the mass of our moon is a sixtyeighth part of that of the earth. Jupiter’s circulation is sensibly affected

Positive Philosophy/203

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 more complicated than that of the planets, 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 then all those to which their planet is liable. The founders of Celestial Mechanics were long perplexed, for instance, by 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 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 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 planet; 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. For 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 approximate 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 difficulties 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 even the satellites, have always the same relations, the variation being only in the intensity. The perturbation which,

204/Auguste Comte

in every 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 are 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 perplex the study of their perturbations. These are the incidents which malice 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 be Lexell. This comet had then a revolution of less than six years: but it has never appeared since, having been entirely deranged 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 proof 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 no 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 the perturbations proper to 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

Positive Philosophy/205

seen, it ii 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 on the surface of a planet, that we might a priori assert this stability wherever life is possible, front 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 body, they could in no way affect us.

If the planets were perfect spheres, the total gravitation of their 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 befog caused 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 our own planet the precession of the equinoxes, modified lay the notation, results from the action of the other bodies of our system,— especially of the sun and moon,—upon our equatorial protuberance. 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 ant moon are the only bodies whose influence meal be considered. Further, the extent of the deviation depends on 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 perceptible to direct observation—These are the general causes which determine 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 planets 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 degree 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 consideration of the highest interest,—that remarkable equality between the duration of this ro-

206/Auguste Comte

tation and that of their circuit round their planet, by which they present always the same hemisphere, except from those very small oscillations called librations, whose law is well understood. The fact is absolutely certain only with regard to the moon; but our mechanical principles justify 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 natural tendency in the satellite to return perpetually to the same position. If it is thus with the moon, there is every reason to suppose 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 system, 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 necessarily be independent of all their variations. invariable —Among several planes which have been proposed, differing 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 may 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 o¢ cur 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 were within our reach.

We end this study of perturbations with a 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 periodical and their period is usually very long, while their extent is very small; so that the whole of our planetary system can

Positive Philosophy/207

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 the 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 perceive a special and most marked stability with regard to the elements which are concerned in the continued existence of living beings.—Such are the sublime theorems of natural philosophy for which humanity is 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 very narrow limits.

The Mathematical theory of celestial mechanics has taken no notice, thus far, of the 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 should 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 he periodical, and must always he 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 increasing 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 they proceeded: and thus

208/Auguste Comte

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 recognize 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, 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. Some 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 insignificant, 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. [M. Comte estimates too lightly the indications of a medium given by Encke’s comet.—J. P. N.]

In our geometrical review we saw, by the agreement of astronomical tables with direct observation, that our system is independent of all that lies outside. This incontestable truth is confirmed by the mechanical view. If 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 planets 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 encyclopaedical law which is the basis of this work,—that the most general phenomena rule the most particular, without being in any degree reciprocally influenced. Thus our astronomical phenomena regulate those of our own globe,—whether physical, chemical, physiological, or social. Yet here we find that the phenomena of the universe have no influence over those of the solar system. There is no difficulty about this to persons who, like myself, admit that

Positive Philosophy/209

our researches 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 apparent, 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.

We see however that besides the sublime direct knowledge afforded to us, celestial of dynamics has powerfully contributed to perfect 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, past 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 period, 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 in regard to cosmogony.

Chapter VI

Sidereal Astronomy And Cosmogony

The only branch of Sidereal Astronomy which appears to admit of exact study 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 distance 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

210/Auguste Comte

separate them. The relative movements of these stars tend to deceive us as to their precise multiple character, 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 be sure whether they form a system any more than any other two stars combined by chance in the heavens. Because a few incontestable examples are before us of a binary system, in which the smaller circulates round the larger, it is anything but philosophical to conclude 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 analogs is merely the imperfection of our investigations. No astronomer 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 ground for presuming their mutual revolutions, 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. [M. Comte quite underrates the importance of the phenomena of the multiple stars. The orbits of a very considerable number are now distinctly ascertained, and the laws of motion in their orbit. The