tainty about the weight of the mountain, which can be calculated only from its volume.
When Coulomb had invented his Torsion Balance, in- tended 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 com- puted. 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 occu- pying 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 204 POSITIVE PHILOSOPHY.
know not, further than that it must be something con- sistent with the condition of superior density.
SECTION II.
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 pheno- mena 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 sur- mounted. 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 206 POSITIVE PHILOSOPHY.
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.
SECTION III.
THE TIDES.
There remains the question of the tides, — the last im- portant 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. More- over, 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 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 after- wards 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 know- ledge 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 gravi- tate 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 dis- tance 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 calcu- lated 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 ques- tion; but the most striking part has yet to be cousidei'ed, — the periodical rise and fall. It is the diurnal motion of 208 POSITIVE PHILOSOPHY.
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, conse- quently, 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 great- ness 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 inac- cessible 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 con- siderations 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."
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 esti- 210 POSITIVE PHILOSOPHY.
mates; 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 conti- nent, 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 circum- stances, 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 invari- able 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 dyna- mical department, as rej^resented by the phenomena of our svstem.
211 CHAPTER Y.
CELESTIAL DYNAMICS.
we nave seen, deterniinea by the gravita- tion 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 ex- plained 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 mathe- matical 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 explo- sion 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 212 POSITIVE PHILOSOPHY.
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 re- action 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 sup- pose 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 suc- cessive 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 astrono- mical 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 rota- tion 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 denomi- nation 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 214 POSITIVE PHILOSOPHY.
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 gravita- tion 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 circum- stance to mathematical research that our system is consti- tuted 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 pre- sented 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, satel- lites 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.