The first of Kepler's laws proves that the accelerating force of each planet is constantly directed towards the sun. The accelerating force, however great it may be supposed, does not at all affect the magnitude of the area which would })e described in a given time by the vector radius of the planet, in virtue of its velocity, if its direction passes exactly through the sun, while it would inevitably change it on any other supposition. Thus, the permanence of this area, — the first general datum of observation, — discloses the law of direction. The great difficulty of the problem, 192 POSITIVE PHILOSOPHY.
gloriously solved by Newton, lies iu the discovery, by means of Kepler's otlier two theorems, of the law of the intensity of this action, which we sj^eak of as exercised by tlie sun on the planets.
When Newton began to work on this conception, he took Kepler's third law as his basis, supposing the orbits, as he might do for such a purpose, to be circular and uniform. The solar action, equal, and opposed to the centrifugal force of the planet, thus became necessarily constant at the different points of the orbit, and could not vary but in passing from one planet to another. This variation between one planet and another was provided for by the theorems of Hiiyghens relating to the centrifugal force in the circle. This force being in proportion to the relation between the radius of the orbit and the square of the periodic time, must vary from one star to another in- versely to the square of its distance from the sun, in virtue of the permanence which Kepler showed to exist of the relation between the cube of this distance and this same sqiiare of the periodic time, for all the planets. It was this mathematical consideration which put Newton in the way of his great discovery, and not any metaphysical reasonings, such as prevailed before it, and which probably never entered his mind, one way or another.
There remained the difficulty of explaining how this law of the variation of the solar action agreed with tho geome- trical nature of the orbits, as exhibited by Kepler. The elliptical orbit 2:>resented tAvo remarkable points, ^ — the aphe- lion and the perihelion, in which the centrifugal force was directly opposed to the action of the sun, and consequently equal to it; and the change in this action there must be at the same time more marked. The curve of the orbit was evidently identical at these two j^oints; the action then had simply to be measured, according to Huyghens' theorems, by the square of the corresponding velocity. Thence, it was easily deduced, from Kepler's first kiw, that the de- crease of the solar action, from the perihelion to the aphelion, must be inversely to the square of the distance. Here was a full confirmation of the law which related to the different planets by an exact comparison between the two principal positions of each of them. Still, however.
tlie elliptical motion had not been considered. Any other curve would, thus far, have served as well as the ellipse, provided its two extremities had shown an equal curvature. The remaining portion of the demonstration, — the measure- ment of the solar action throughout the extent of the orbit, — is to be obtained only by transcendental analysis. The process is necessary for carrying on the comparison of the solar action and the centrifugal force; and the theory of the curvature of the ellipse is required. Huyghens made a. near approach to the principle of this great process; but it could not be completed without the aid of the differential analysis, of which Newton was the inventor, as well as Leibnitz. By the aid of this analysis, the force of the solar action in all parts of the orbit is easily ^^, estimated, m various ways; and it is found demonstration to vary inversely to the square of the distance, and that it is indej^endent of the direction. Furthermore, the same method shows, in accordance with Kepler's third law, that the action varies in proportion to distance alone; so that the sun acts upon all the planets alike, whatever may be their dimensions, their distance only being the cir- cumstance to be considered. Thus Newton completed his demonstration of the fundamental law that the solar action is, in every case, proportionate, at the same distance, to the mass of the planet; in the same way that, by the identity of the fall of all terrestrial bodies in a vacuum, or by the precise coincidence of their oscillations, proof had already been obtained of the proportion between their weight and their masses. We thus see how the three laws of Kepler have concurred in establishing, according to the rules of rational mechanics, this fundamental law of nature. The first shows the tendency of all the planets towards the sun; the second shows that this tendency, the same in every direction, changes with the distance from the sun, inversely to its square; and the third teaches that this action is always simply proportionate, the distance being- equal, to the mass of each planet. In accordance with the laws of Kepler, which relate to the whole interior of our system, the same theory applies to tte connection between the satellites and their planets.
Newton thought it necessary to complete his demonstra- I. o 194 rOSITIVE PHILOSOPHY.
"tion by presenting it in an inverse manner; that is, by determining a priori the planetary motions which must result from such a dynamic law. The process brought him back, as it must do, to Kepler's laws. Besides furnishing some means of simplifying the study of these motions, this labour proved that, whereas, by Kepler's laws, the orbit might have had more figures than one, the ellipse was the only one possible under the Newtonian law.
It was once a great jjerplexity to some exnlaiiied peo])le, which others could not satisfactorily explain, that when the planet is travelling towards its aphelion we cannot say that it tends towards the sun. But the difficulty arose out of the use of inap- propriate language. The question is, not whether the planet is nearer to the sun than it lately was, but whether it is nearer than it would have been without the force that sends it forward. It is always tending towards the sun to the utmost that is allowed by the other force to which it is subjected. The orbit is always concave towards the sun; and it would evidently have been insurmountable if the trajectory could have been convex. In the same way, when a bomb ascends, its weight is not suspended or reversed: it always tends towards the earth, and is, in fact, falling towards it more rapidly every moment, even if ascending, because it is eveiy moment further below the point at which it would have been but for the action of the earth upon it; and its trajectory is always concave to the ground.
Hqxiw Attrac- I have thus far carefully avoided giving tion inadmis- any name to the tendency of the planets ^^^^®- towards the sun, and of the satellites towards the planets. To call it attraction would be misleading; and we, in triith, can know nothing of its nature. All that we know is that these bodies are conne(^ted, and that their •effect upon each other is mathematically calculable. It is by quite another property of Newton's great discovery that this effect is explained, in the true sense of the word, — that is, comprehended from its conformity with the ordinary phenomena which gravity continually produces on the sur- face of our globe. Let us now see what this property of the discovery is.
EXTENT OF THE DEMONSTRATION. 195 We owe a great deal to the moou. If the earth had uo satellite, we might calculate the celestial motions by the rules of dynamics, but we could not connect them with those which are under our immediate observation. It is the moon which affords this connection by enabling us to establish the identity of its tendency towards the earth with weight, properly so called; and from this knowledge, we have risen to the view that the mutual action of the heavenly bodies is nothing else than weight properly generalized; or, putting it the other way, that weight is only a particular case of the general action. The case of the moon is susceptible of the most precise testing. The data are known; and by dynamical analysis, the intensity of the action of the earth upon the moon is exactly ascer- tainable. We have only to suppose the moon close to the earth, with the due increase of this intensity, inversely to the square of the distance, and compare it with the inten- sity of weight on the earth, as manifest to us by the fall of Ijodies, or by the pendulum. A coincidence between the two amounts to proof; and we have, in fact, mathematical demonstration of it. It was in pursuing this method of proof that Newton evinced that philosophical severity which we find so interesting in the anecdote of his long delay, because he could not establish the coincidence, while con- fident that he had discovered the fact. He failed for want of an accurate measurement of a degree on the earth's sur- face; and he put aside this important part of his great conception till Picard's measurement of the earth enabled him to establish his demonstration.
The identity of weight and the moon's ten- + f +i dency towards the earth places the whole of demonstration celestial mechanics in a new light. It shows us the motions of the stars as exactly like that of projectiles which we have under our immediate observation. If we could start our projectiles with a sufficient and continuous force, we should, except for the resistance of the air, find them the models of the planetary system: or, in other words, astronomy has become to us an artillery problem, simplified by the absence of a resisting medium, but com- plicated by the variety and plurality of weights. — If our observation of weight on our globe has helped us to a 196 POSITIVE PHILOSOPHY.
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 fol- lowed; and experiments on our globe, in the vertical direc- tion, 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 dis- missing 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 mole- bodies 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 indepen- dently 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 esta- blished for the purpose still remain the commonest expres- sion 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 198 POSITIVE PHILOSOPHY.
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 mathe- matically 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 enter- taining 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 con- ceive 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 inevit- able variations, stable enough to give steadiness and conli- dence 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 philoso- phical progress, and of the general education of human reason. Descartes could not rise to a mechanical concep- tion 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 assimi- late 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.
200 CHAPTER IV.
CELESTIAL STATICS.
Consumma- "I/KEPLER'S laws connected celestial phe- tion 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 mathe- matical explanation of celestial phenomena, and the per- . fecting of their study. For this purpose, siderations ^^ "^'^^^ recur to our former division or sub- jects, and contemplate the phenomena of planets as immovable first, and of planets in motion afterwards; the statical j^heuomena first, and the dyna- mical afterwards.
To know the mutual gravitation of the heavenly bodies, we must know their masses. Such knowledge once ap- peared inaccessible from its very nature; but the New- tonian 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 state- ment 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 dis- tance; 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 con- 202 POSITIVE PHILOSOPHY.
sidered 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 any- thing, 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 con- tents, 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 them- selves 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.
SECTION I.
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 plumb- line, however slightly, if it be brought close enough, sub- jecting 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 occa- sioned 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 uncer-