as of another: Secondly; that out of an infinite number of possible laws, those which were admissible for the purpose of supporting the heavenly motions, lay within certain narrow limits: Thirdly; that of the admissible laws, or those which come within the limits prescribed, the law that actu ally prevails is the most beneficial. So far as these propositions can be made out, we may be said, I think, to prove choice and regulation; choice, out of boundless variety; and regulation, of that which, by its own nature, was, in respect of the property regulated, indifferent and indefinite.

First then, attraction, for anything we know about it, was originally indifferent to all laws of variation depending upon change of distance, i. e. just as susceptible of one law as of another. It might have been the same at all distances; it might have increased as the distance increased: or it might have diminished with the increase of the distance, yet in ten thousand different proportions from the present; it might have followed no stated law at all. If attraction be what Cotes, with many other Newtonians, have thought it to be, a primordial property of matter, not dependent upon, or traceable to, any other material cause; then, by the very nature and definition of a primordial property, it stood indifferent to all laws. If it be the agency of something immaterial, then also, for anything we know of it, it was indifferent to all laws. If the revolution of bodies round a centre depend upon vortices, neither are these limited to one law more than another.

There is, I know, an account given of attraction, which should seem, in its very cause, to assign to it the law which we find it to observe; and which, therefore, makes that law, a law, not of choice, but of necessity: and it is the account, which ascribes attraction to an emanation from the attracting body. It is probable, that the influence of such an emanation will be proportioned to the spissitude of the rays of which it is composed; which spissitude, supposing the rays to issue in right lines on all sides from a point, will be reciprocally as the square of the distance.*

*Let the light of a candle fall upon a square object like A B C D, Fig. 4, Plate XXXIX, and if a screen be placed parallel to the object and at double the distance, the shadow E F G H, received upon it, will be four times the size of the object itself. For the rays passing in straight lines by the angles A, B, C, D, the sides E F, F G, GH, HE, must be each double of A B, B C, C D, DA: therefore the shadow may be divided into four squares each equal in size to the object. At three times the distance from the candle, the sides of the shadow would each be three times as large as the sides of the object, and its area would, therefore, contain

The mathematics of this solution we do not call in question the question with us is, whether there be any sufficient reason to believe, that attraction is produced by an emanation. For my part, I am totally at a loss to comprehend low particles streaming from a centre should draw a body towards it. The impulse, if impulse it be, is all the other way. Nor shall we find less difficulty in conceiving a conflux of particles, incessantly flowing to a centre, and carrying down all bodies along with it, that centre also itself being in a state of rapid motion through absolute space: for by what source is the stream fed, or what becomes of the accumulation? Add to which, that it seems to imply a contrariety of properties, to suppose an ethereal fluid to act, but not to resist; powerful enough to carry down bodies with great force towards a centre, yet, inconsistently with the nature of inert matter, powerless and perfectly yielding with respect to the motions which result from the projectile impulse. By calculations drawn from ancient notices of eclipses of the moon, we can prove that, if such a fluid exist at all, its resistance has had no sensible effect upon the moon's motion for two thousand five hundred years. The truth is, that, except this one circumstance of the variation of the attracting force at different distances agreeing with the variation of the spissitude, there is no reason whatever to support the hypothesis of an emanation; and, as it seems to me, almost insuperable reasons against it.

(*) II. Our second proposition is, that whilst the possible laws of variation were infinite, the admissible laws, or the laws compatible with the preservation of the system, lie within narrow limits. If the attracting force had varied according to any direct law of the distance, let it have been what it would, great destruction and confusion would have taken place. The direct simple proportion of the distance would, it is true, have produced an ellipse; but the perturbing forces would have acted with so much advantage, as to be continually changing the dimensions of the ellipse, in a manner inconsistent with our terrestrial

nine times the space. For the same reason if the distance be increased four, five, or six times, the area of the shadow will contain sixteen, twenty

five, or thirty-six squares, each equal to the object. Now the quantity of light which falls upon the object would, if it had not been intercepted. have spread over that part of the screen, which is occupied by the shadow; and as the surface is increased, over which a certain quantity of rays is spread, in the same ratio their spissitude or density will be diminished; consequently this spissitude will be reciprocally as the squares of the distances.-Paxton. T*

creation For instance; if the planet Saturn, so large and so remote, had attracted the earth, both in proportion to the quantity of matter contained in it, which it does; and also in any proportion to its distance; i. e. if it had pulled the harder for being the farther off, (instead of the reverse of it,) it would have dragged out of its course the globe which we inhabit, and have perplexed its motions, to a degree incompatible with our security, our enjoyments, and probably our existence. Of the inverse laws, if the centripetal force had changed as the cube of the distance, or in any higher proportion, that is (for I speak to the unlearned,) if, at double the distance, the attractive force had been diminished to an eighth part, or to less than that, the consequence would have been, that the planets, if they once began to approach the sun, would have fallen into his body; if they once, though by ever so little, increased their distance from the centre, would forever have receded from it. The laws, therefore, of attraction, by which a system of revolving bodies could be upholden in their motions, lie within narrow limits, compared with the possible laws. I much underrate the restriction, when I say that, in a scale of a mile, they are confined to an inch. All direct ratios of the distance are excluded, on account of danger from perturbing forces; all reciprocal ratios, except what lie beneath the cube of the distance, by the demonstrable consequence, that every the least change of distance would, under the operation of such laws, have been fatal to the repose and order of the system. We do not know, that is, we seldom reflect, how interested we are in this matter. Small irregularities may be endured; but changes within these limits being allowed for, the permanency of our ellipse is a question of life and death to our whole sensitive world.

(*) III. That the subsisting law of attraction falls within the limits which utility requires, when these limits bear so small a proportion to the range of possibilities upon which chance might equally have cast it, is not, with any appearance of reason, to be accounted for by any other cause than a regulation proceeding from a designing mind. But our next proposition carries the matter somewhat farther. We say, in the third place, that, out of the different laws which lie within the limits of admissible laws, the best is made choice of; that there are advantages in this particular law which cannot be demonstrated to belong to any other law; and, concerning some of which, it can be demonstrated that they do not belong to any other.

(*) 1. Whilst this law prevails between each particle of matter, the united attraction of a sphere, composed of that matter, observes the same law. This property of the law is necessary, to render it applicable to a system composed of spheres, but it is a property which belongs to no other law of attraction that is admissible. The law of variation of the united attraction is in no other case the same as the law of attraction of each particle, one case excepted, and that is of the attraction varying directly as the distance; the inconveniency of which law, in other respects, we have already noticed.

We may follow this regulation somewhat farther, and still more strikingly perceive that it proceeded from a designing mind. A law both admissible and convenient was requisite. In what way is the law of the attracting globes obtained?. Astronomical observations and terrestrial experiments show, that the attraction of the globes of the system is made up of the attraction of their parts; the attraction of each globe being compounded of the attractions of its parts. Now, the admissible and convenient law which exists, could not be obtained in a system of bodies gravitating by the united gravitation of their parts, unless each particle of matter were attracted by a force varying by one particular law, viz. varying inversely as the square of the distance; for, if the action of the particles be according to any other law whatever, the admissible and convenient law which is adopted could not be obtained. Here then are clearly shown regulation and design. A law both admissible and convenient was to be obtained: the mode chosen for obtaining that law was by making each particle of matter act. After this choice was made, then farther attention was to be given to each particle of matter, and one, and one only particular law of action to be assigned to it. No other law would have answered the purpose intended.

(*) 2. All systems must be liable to perturbations. And therefore, to guard against these perturbations, or rather to guard against their running to destructive lengths, is perhaps the strongest evidence of care and foresight that can be given. Now we are able to demonstrate of our law of

Let A, Fig. 5, Plate XXXIX, represent a sphere composed of particles, which mutually attract each other with a force, which varies reciprocally as the squares of the distances; their united attraction, on a similar particle P without the sphere, will be according to the same law. that is, the particle will be attracted towards the sphere with a force, which will also vary reciprocally as the square of C P, its distance from the centre of the sphere.-Paxton.

attraction, what can be demonstrated of no other, and what qualifies the dangers which arise from cross but unavoidable influences, that the action of the parts of our system upon one another will not cause permanently increasing irregularities, but merely periodical or vibratory ones; that is, they will come to a limit, and then go back again. This we can demonstrate only of a system, in which the following properties concur, viz. that the force shall be inversely as the square of the distance; the masses of the revolving bodies small, compared with that of the body at the centre; the orbits not much inclined to one another; and their eccentricity little. In such a system the grand points are secure. The mean distances and periodic times, upon which depend our temperature and the regularity of our year, are constant. The eccentricities, it is true, will still vary, but so slowly, and to so small an extent, as to produce no inconveniency from fluctuation of temperature and season. The same as to the obliquity of the planes of the orbits. For instance, the inclination of the ecliptic to the equator will never change above two degrees, (out of ninety,) and that will require many thousand years in performing.

It has been rightly also remarked, that if the great planets, Jupiter and Saturn, had moved in lower spheres, their influences would have had much more effect, as to disturbing the planetary motions, than they now have. While they revolve at so great distances from the rest, they act almost equally on the sun and on the inferior planets; which has nearly the same consequence as not acting at all upon either.

If it be said that the planets might have been sent round the sun in exact circles, in which case, no change of distance from the centre taking place, the law of variation of the attracting power would have never come in question, one law would have served as well as another; an answer to the scheme may be drawn from the consideration of these same perturbing forces. The system retaining in other respects its present constitution, though the planets had been at first sent round in exact circular orbits, they could not have kept them: and if the law of attraction had not been what it is, or, at least, if the prevailing law had transgressed the limits above assigned, every evagation would have been fatal: the planet once drawn, as drawn it necessarily must have been, out of its course, would have wandered in endless error.

V. What we have seen in the law of the centripetal force, viz. a choice guided by views of utility, and a choice