loiowledge of planetary relations, our celestial observations have in turn taught us the law of the variation of weight, imperceptible in terrestrial phenomena. Men had always conceived weight to be an inalterable j^roperty of bodies, finding that no metamorphosis, — not even from life to death, — made any change in the weight of a body, while it remained entire. This was the one particular in Avhich men might suppose they had found the Absolute. In a moment, the Newtonian demonstration overthrew this fast-rooted notion, and showed that weight was a relative quality,— not under the circumstances in Avhich it had hitherto been observed, but under the new one, — the position of the observed body in the system, — its distance from the centre of the earth. The human mind could hardly have sought out this fact directly: but, once revealed in the course of astronomical study, the verification easily followed; and experiments on our globe, in the vertical direction, and yet more in the horizontal, have established the reality of the law, by experiments too delicate, from the necessity of the case, to be appreciable, if we had not known beforehand what differences must be found to exist. It is to express briefly the identity between S""i?nob-" weight and the accelerating force of the iectionable. planets that the happy term Gravitation has been devised. This tenn has every merit.
It expresses a simple fact, without any reference to the nature or cause of this universal action. It affords the only explanation which positive science admits; that is, the connection between certain less known facts and other better known facts. Since the creation of this term, there has been no excuse for the continued use of the word attraction. It is desirable to avoid pedantry in language; but it is of high importance to preserve pure the positive character of so fundamental a conception as this, by using a term which expresses exactly what we know, and dismissing one which assumes what is lourely fanciful, and wholly incorrect. Attraction is a drawing toivtirds. Now, when we draw anything towards us, the distance is of no importance: the same force draws the same body with equal ease three feet or thirty feet, which is directly contradictory to the facts of gravitation. Our business is with PRIMARY AND SECONDARY GRAVITATION. 197 the fact of tlie action, and not at all witli its nature. It was the use of this metaphysical term, it now appears, which occasioned the opposition that the Newtonian theory encountered so long, and especially in France. Descartes had, by laborious efforts, banished the notions of occult qualities, which he perceived to be so fatal to science; and in this theory of attraction, his followers saw a falling back into the old metaphysical delusions. We perceive this in the writings of John Bernouilli and Fontenelle: and it appears that the clear and positive scientific intellect of France did good service in stripping off from the sublime discovery of Newton the metaphysical aj^pearance which obscured its reality for a time.
One more consideration remains to be ad- Gravitation is verted to. "We have regarded the heavenly that of molebodies thus far as points, without reference ^ules.
to their forms and dimensions. But as it is proved that the intensity of the action of the sun on the planets, and of the planets on their satellites, is proportioned to the mass of the body acted upon, it is clear that the force operates directly only on molecules, which are all independently affected by it; and equally, their distance being the same. The gravitation of molecules is therefore the only real one; and that of masses is simply its mathematical result. In the mathesnatical study of motions however it is necessary to have a conception of a single force, instead of such an infinity of elementary actions: and hence arises that preliminary part of celestial mechanics which consists in compounding in one result all the mutual gravitation of the molecules of two stars. Newton founded this portion, with all the rest; and the two theorems which he established for the purpose still remain the commonest expression of this important theory. He proved that if the stars were truly spherical, and their strata were homogeneous, the gravitation of their particles would be so balanced that the bodies might be treated as points, in the study of their motions of translation. But the irregularity of their forms, however slight, must be considered in the theory of their rotations, to which these theorems cease to be applicable. For any other form than the sphere, the problem becomes very complicated; and the analytical difficulties can be surmounted only by approximation, notwithstanding all the perfections introduced into the theory in recent times. And unless we could also learn what is the law of density in the interior of the stars, — a kind of knowledge which seems to be for ever beyond our reach, — we cannot attain a perfect solution.
The fundamental law of Rational Me- "•ravitatioii chanics, which declares the necessary equality of action and reaction, shows that gravitation must be mutual, — that the sun must tend towards the planets, and the planets towards their satellites. The extreme inequality of the masses renders the ascertainment of the inverse gravitation extremely difficult; yet its reality is established by various secondary phenomena. The gravitation of the planets towards each other is a necesary part of the whole conception; but it was not mathematically demonstrated till Newton's successors deduced from it an exact explanation of the perturbations observed in the principal motions of the planets. Their labours have established secondary gravitation as positively as the primary.
Thus has every kind of proof concurred to establish that great fundamental law which is the noblest result of our aggregate studies of nature. All the molecules of our system gravitate towards each other, in proportion to their masses, and inversely to the squares of their distances.
^. r ii I dare not, as many do, confidently extend Ig^^^, the application or this law to the entire universe. There can be no objection to entertaining it analogically till we obtain some knowledge of the mechanism of the sidereal heavens; but we must remember that we have not yet that knowledge, and that we cannot promise ourselves that we ever shall. Without the phenomena of our own system, the theory of its motions would be only an intellectual exercise and sport: there can be no positive science apart from phenomena, and of the phenomena of the universe beyond our own system we are not in scientific possession.^ It must be understood that ^ M. Comte omits here all notice of sncli positive api)lications as Ave are al)le to make in Sidereal astronomy. He takes no notice of the fact tliat the motion of the multii)Ie stars in elliptical orhits, OPERATION OF NEWTON's DISCOVERY. 199 I advocate simply a suspensiou of judgmeut where there is no groiind for either affirmation or denial. I merely desire to keep in view that all our positive knowledge is relative; and, in my dread of our resting in notions of anything absolute, I would venture to say that I can conceive of such a thing as even our theory of gravitation being hereafter superseded. I do not think it probable; aud the fact will ever remain that it answers completely to our present needs. It sustains us, up to the last point of precision that we can attain. If a future generation should reach a greater, and feel, in consequence, a need to construct a new law of gravitation, it will be as true as it now is that the Newtonian theory is, in the midst of inevitable variations, stable enough to give steadiness and conlidence to our understandings. It will appear hereafter how inestimable this theory is in the interpretation of the 2:)henomena of the interior of our system. We already see how much we owe to it, apart from all specific knowledge which it has given us, in the advancement of our philosophical progress, and of the general education of human reason. Descartes could not rise to a mechanical conception of general phenomena without occupying himself with a baseless hypothesis about their mode of production.
This was, doubtless, a necessary process of transition from the old notions of the absolute to the positive view; but too long a continuance in this stage would have seriously impeded human progress. The Newtonian discovery set us forward in tlae true positive direction. It retains Descartes' fundamental idea of a Mechanism, but casts aside all inquiry into its origin and mode of production. It shows practically how, without attempting to penetrate into the essence of phenomena, we may connect and assimilate them, so as to attain, with precision and certainty, the true end of our studies, — that exact prevision of events which a priori concej)tions are necessarily unable to supply.
and in accordance with Kepler's law of the velocities, demonstrates the exij^tence of a law of force, according to the inverse square of the distance. — J. P. N.
CELESTIAL STATICS.
Consumma- "I/KEPLER'S laws connected celestial phetion by i-*- nomena to a certain degree, before Newton. Newton's theory was jjropounded: but they left this imperfection, — that phenomena which ranked tinder two of these laws had no necessary connection with each other. Newton brought under one head all the three classes of general facts, uniting them in one more general still; and since that time we have been able to perceive exactly the relation between any two of the j^henomena which are all connected with the common theory. As far as we can see, there is nothing more to gain in this direction.
We have seen what this great conception is in itself. We have now to observe its a])plication to the mathematical explanation of celestial phenomena, and the per-. fecting of their study. For this purpose, siderations ^^ "^'^^^ recur to our former division or subjects, and contemplate the phenomena of planets as immovable first, and of planets in motion afterwards; the statical j^heuomena first, and the dynamical afterwards.
To know the mutual gravitation of the heavenly bodies, we must know their masses. Such knowledge once appeared inaccessible from its very nature; but the Newtonian theory has put it within our power, and furnished us with a wholly new set of ideas about these bodies. There are tliree ways in whicli the inquiry has been pro-First method secuted, all differing from each other, both of inquiry in generality and in simplicity. The first into masses. method, the most general, the only one in fact which is applicable to all cases, is the most difficult. It consists in analyzing the special share of each body in STATICAL INQUIRIES. 201 the pertui'bations observed in the principal motions of another, — both of translation and rotation. Here two elements are concerned, — the distance, and the mass of the star in question. The first is well known, the other is not; and only an approximate determination is possible.
It is difficult to apportion the shares in the action; and geometers place little dependence on the computation of masses obtained by this method, in comparison with that obtained by either of the others.
is that which JNewton employed witli regard to planets that had a satellite; that of comparing the motion of the satellite round the planet with that of the planet round the sun. The law which determines the action by the distance being compared, in its results, in the two cases, gives the relation of the masses of the suii and the planet. The mass of Jupiter, determined by Newton in this way, has undergone little change of statement by methods since employed; and what difference there is is almost wholly owing to the data of the process being now better known.
The third method is the most direct and rp, •.,.,, simple of all; but it is the most restricted, as it is necessarily confined to the planet inhabited by the observer. It consists in estimating the relative masses by the comparison of the weights which they produce. If we knew the mass of any j^lanet, we should know what would be the weight of things on its surface, or at a given distance; and reciprocally, the weight being known, we are able to estimate the mass. With the pendulum, we have measured terrestrial weight with absolute precision; and, diminishing it, inversely to the square of the distance, we shall know its value at the distance of the sun. We have then only to compare it with the amount, before well known, which expresses the sun's action upon the earth, to find immediately the relation of the mass of the earth to that of the sun. With regard to every other planet, on the contrary, it must be the estimate of its mass which would yield that of its corresponding gravity. All these methods being practicable in the case of the earth, its mass, in comparison with that of the sun, must be considered the best known of all within our system. The mass of the moon, and that of Jujiiter, are now estimated almost as perfectly; and those of Saturn and Uranus come next. We are less sure about the other three which have been calculated, — Mercviry, Venus, and Mars; though the uncertainty about them cannot be very!:^reat. Of the telescopic planets and the comets we know scarcely anything, owing to their extreme smailness, which precludes their exerting any sensible influence on perturbations.
Comets pass, during their prodigious course, near very small stars, such as the satellites of Jupiter and Saturn, without producing any perceptil)le derangement. As for the satellites, we have no knowledge except of the moon, and approximately, of those of Jupiter. No comparison of results has as yet exhibited any harmony whatever between them. The only essential circumstance which they present is the vast superiority of the size of the sun to the whole contents of the system. Those entire contents, if thrown together, would scai'cely amount to a thousandth part of the mass of the sun. Looking abroad from the sun, we see alternating, without any visible order, here decreasing, there increasing masses. We might have supposed, a priori, as Kepler did, that the masses were regularly connected with the volume < (which are themselves ii-regular however), so that the mean densities should be continually less in mathematical j^roportion to their distances fi om the sun. But, independently of this numerical law, which is never exactly ol>served, the simple fact of the decrease of density ] resents some exceptions, in regard to Uranus, among others. No rational ground can be assigned for this.
WEIGHT OF THE EARTH.
These are the means by which the masses of the bodies of our system are ascertained. The remaining process is to bring them into relation with our estimates of weight, by ascertaining the Weight of the eartii.
WEIGHT OF THE EARTH. 203 total weight of the earth. Bouguer was the first wlio distinctly perceived the possibility of such an estimate, during his scientific expedition to Peru, when he found that the neighbourhood of vast mountains slightly affected the direction of weight. We see how, in accordance with the law of gravitation, a considerable mass, regarded as condensed in its centre of gravity, may affect the plumbline, however slightly, if it be brought close enough, subjecting it to a secondary gravitation, which affords data for a comparison between the action of the earth and that of the mountain. By this, some estimate may be formed of the proportion of the mountain to the globe. In the time of Bouguer science was not advanced enough to admit of more than the conception of how the thing could be done. Half a century later, Maskelyne observed the mountain Schehallion in Scotland, and found that it occasioned an alteration in the natural direction of weight of from five to six seconds; and Hutton deduced from this that the weight of the earth is equal to four and a half times that of a similar volume of distilled water at its maximum of density. Anything like exactness, however, is out of the question while there must be so much uncertainty about the weight of the mountain, which can be calculated only from its volume.
When Coulomb had invented his Torsion Balance, intended to measure the smallest forces, Cavendish saw how the earth might be weighed by comparing it, by means of this balance, with artificial masses which might be computed. By his immortal experiments, he discovered the miean density of our globe to be five and a half times equal to that of water; whence we can, if we think proper, deduce the weight of the earth in cwts. and tons. — We thus obtain, among other advantages, some insight into the constitution of our globe, which by its positivity, puts to flight many fanciful notions. The density of the parts near the surface is so far below the average, — water occupying much space, for instance, — that the density nearer the centre must be much above the average. This is in accordance with the indications of Celestial Mechanics; and it furnishes us with one condition of the interior of the globe. There can be no void there. What there is we Form of the know not, further than that it must be something consistent with the condition of superior density.
FORM OF THE PLANETS.
The next great statical inquiry relates to iDlanets ^^ ^^® form of the heavenly bodies, as deduced from the theory of their equilibrium.
Geometers suppose the planetary bodies to have been originally fluid, because their equilibrium can thus consist with only one form; whereas, if they had been always solid, as our earth is now, their equilibrium might have been compatible with any form whatever. Several phenomena indicate this supposition, and it agrees remarkably with the whole of our direct observations.
If the planets had no motion of rotation, their being perfectly spherical would accord with the equilibrium of their molecules: but the centrifugal force engendered by the rotation must necessarily modify the primitive form,, by altering, more or less, the direction, or +iw/^*i!,".,"]L? the intensity of weight, properly so called. Huyghens established this with regard to the direction, and Newton with regard to the intensity. We thus become easily assured of the general fact of the nearly spherical form of all the planets, and of their being slightly flattened at the jjoles: but, when we go further, and attempt to estimate their forms mathematically, and learu the precise degree of the flattening at the jioles, the question becomes one of transcendental analysis, and is involved in difficulty which can never be entirely surmounted. The inquiry involves a sort of vicious circle, which does not admit of a logical issue. In order to form an equation of the surface, Ave ought, by the law of equili- ,., brium of fluids, to know the weight of the Geometrical,1 j ^ ij.ii estimate molecules concei'ned; whereas, by the law of gravitation, this can be ascertained only through the knowledge of the form of the planet, and even of the mode of variation of its interior density. All that PLANETARY FORMS. 205 can be done is to discover whether the proposed form fulfils such and such conditions. Maclaurin discovered a theorem, highly valued bv geometers, which has become the basis of all our inquiries on this subject, and which shows that the ellipsoid of revolution precisely fulfils the conditions of equilibrium. But this supposes the structure of the body to be homogeneous; which it is not, in any case. The labours of geometers have however brought within very narrow limits the possible variations of the polar flattening. The result with regard to the earth is that the mathematical rule perfectly agrees with direct observation.
In the case of the planets we have another ^ ^.
resource. Their flattening attects certain perturbations.
phenomena of perturbation, by the study of which we obtain materials for an estimate. Altogether, the calculations and measurements agree more closely than we could have ventured to hope. The only case which seems to present a real exception is that of Mars, which, by its magnitude, its mass, and the time of its rotation, should be little more flattened than the earth; whereas, if the observations of Herschell are exact, it is almost as much so as Jupiter. — We must observe, moreover, that though, as Maclaurin has shown, equilibrium is compatible with the ellipsoid form, this form is not to be supposed the only one: — witness, in our own system, the rings of Saturn, which are a remarkable example to the contrary: and Laplace has demonstrated how these rings could, even in a fluid state, be in equilibrium.
The most useful consequence of the mathe- Indirect estimatical theory of the planetary forms is that mate of the it has established an important relation be- earth's form.
tween the value of the different degrees on the earth's surface and the intensity of the corresponding gravity, measured by the length of the seconds pendulum in different latitudes. We can thus, with great ease, multiply our indirect observations about the form of our globe; whereas the geometrical estimate of degrees is a long and laborious operation, which cannot be often repeated with due care.
But, generally speaking, the more indirect a measurement is, cceteris paribus, the more uncertain it is: and there remains the uncertainty arising from our ignorance of the law of interior density in our earth; so that our chief reliance should still be on mathematical measurement, ■conducted with due care.
Hydrostatic -A-n interesting question belonging to the tlieory of pla- hydrostatic theory of the planetary forms is netary forms, of the conditions of stability of equilibrium of the fluids which are collected on a part or the whole of the surface of the planets. Laplace shows this stability to depend, under all circumstances, on the density of the iluid being less than the mean density of the planet; a view established with regard to the earth by Cavendish's fine experiment.
THE TIDES.
There remains the question of the tides, — the last important inquiry under the head of celestial statics. Under the astronomical point of view, this is evidently a statical question, — the earth being, in that view, regarded as motionless: and it is not less a statical question in a mathematical view, because what we are looking at is the figure of the ocean during periods of equili- 8iTtkles° brium, without thinking of the motions which j^i'ocluced that equilibrium. Moreover, this inquiry naturally belongs to the study of the planetary forms.
A particular interest attaches to this qviestion, from its being the link l)etween celestial and terrestrial physics, — the celestial explanation of a great terrestrial phenomenon. — Descartes did much for us in establishing this. He failed to explain the phenomenon; but he cast aside the metaphysical conceptions which had i">revailed before, and •showed that there was a connection between the change of the tides and the motions of the moon; and this certainly Tielped to put Newton in the way of the true theory. As soon as it was known that the cause of the tides was to be looked for in the sky, the theory of gravitation was certain to afford its true explanation. Newton thex-efore gave out THE TIDES, 207 the simple principle that the unequal gravitation of the different parts of the ocean towards any one of the bodies of our system, and particularly towards the sun and moon, was the cause of the tides: and Daniel Bernouilli afterwards perfected the theory. The same theory answers for the atmosphere: but we had better study it in the case of the seas alone; on account of the uncertainty of our knowledge of the vast gaseous covering of our globe, whose diffused mass alm.ost defies precise observation.
Suppose the earth ioined to anv heavenlv „,, 1 J Y T • j-i 1 /i Ai )' Iheorvof the body by a line passing through the earth s ^jj^^, * centre. It is clear that the point of the earth's surface which is nearest the other body will gravitate towards it more, and the remoter point less, than the centre, inversely to the squares of their respective distances. The first point tends away from the centre: and the centre tends away from the second point; and in each case the fluid surface must rise; and in nearly the same degree in both cases. The effect must diminish in proportion to the distance from these points in any direction: and at a distance of ninety degrees it ceases. But there the level of the waters must be lowered because of the exhaustion in that place caused by the overHow elsewhere. And here enters a new consideration, difiicult to manage: — the changes in the terrestrial gravity of the waters, occasioned by their changes of level. — Thus the action of any heavenly body causes the ocean to assume the form of a spheroid, elongated in the direction of that body. Newton calculated the chief part of the phenomenon of the tide.'^ on the sup23osition of an ellipsoid of homogeneous structure, as he had done in estimating the effect of the centrifugal force on the earth's figure, substituting for the centrifugal force the difference between the gravitation of the centre of the globe and that of its surface next the proposed body. After that, Maclauriu's theorem served Daniel Bernouilli for a basis of an exact theory of the tides.
Thus far, we have regarded the tides only as if they were a fixed accumulation of waters under the proposed star. This is the mathematical basis of the whole question; but the most striking part has yet to be cousidei'ed, — the periodical rise and fall. It is the diurnal motion of our globe which causes this rise and fall, by carryiug the waters successively into all the positions in which the other body can raise or depress them. Hence arise the four nearly equal periodical alternations, when the two greatest elevations take place during the two passages of the heavenly body over the meridian of the place, and the lower levels at its rising and setting; the total period being precisely fixed by combining the terrestrial rotation with the proper daily movement of the heavenly body. The last indispensable element of the question is the valuation of the powers of the different heavenly bodies. This calculation is easily made from the difference between the gravitation of the centre of our globe and that of the extreme points of its surface next the observed body. Guided by the law of gravitation, we can determine which, among all the bodies of our system, are those which can participate in the phenomenon, and what is the sun by its immense mass, and the moon by its proximity, are the only ones which produce any ^.,, appreciable tides: that the action of the moon is trom two and a halt to three times more powerful than that of the sun; and that, consequently, when they act in opposite directions, that of the moon prevails; which explains the primary observation of Descartes about the coincidence of the tidal period with the lunar day.
. Thus far, we have considered only the influence*^ effect of a single heavenly body upon the tides; that is, the case of a simple and abstract tide. The complication is very great, when the action of two such bodies has to be considered. But the resources of science are sufficient to meet this case, — even deriving from it new means of estimating the mass of the sun and moon; — and also of calculating the modifications arising out of the various distances of the earth from either body; and again, of tracing the changes of direction caused by the diurnal movement of the proposed body, — whether in accordance with the earth's axis of rotation, or parallel with the equator, which makes the difference between the tides of our equinoctial and solstitial lunar PROOF OF THE THEORY. 209 months. As for the difference of the phenomenon in various climates, the consideration of latitude is the only one which affords much result. At the jioles, thei'e can of course he no other tides than such as are caused by the flux and reflux of waters elsewhere, as the earth has no rotation there. The eqviator must exhibit the tides at their extremes, not only on account of the diminished gravitation there, but yet more on account of the more complete diversity of the successive positions occupied by the waters durinsf the daily rotation. Elsewhere the greatness of the tide must vary in proportion to the force of the rotation.
The mathematical theory of the tides accords with direct observation to a degree exactitude which is really wonderful, considering how many hypotheses geometers must have recourse to, to make the questions calculable at all, and how many inaccessible data would be required to make an estimate thoroughly logical. It would not even be enough to know the extent and form of the bed of the ocean. Something beyond that in difficulty is required, — the true law of density, in the interior of the earth, as with regard to the figure of the planets. We ought to know too whether the interior strata are solid or fluid, in order to know whether they participate in tidal ]>henomena, aiid whether they therefore modify those at the surface or not. These considerations show the soundness of the advice given by one who was full of the true mathematical spirit, consisting above all in the relation of the concrete to the abstract, Daniel Bernouilli, who recommended geometers "not to urge too far the results of formulas, for fear of di'awing conclusions contrary to truth."