through a quadruple space in a double time; fo that the moon defcending freely would neceffarily fall four times as far in two minutes as in one minute; that is, through four times as much space in one minute at her leaft diftance, as at her greateft dif tance in the fame time.

But the forces with which heavy bodies defcend are in the fame proportion as the spaces defcribed, in confequence of thofe forces, in equal fmall parts of time; confequently the force which acts at the leaft diftance is quadruple that which acts at a greater diftance, when the latter is fuppofed to be double the former; or the forces are as 4 to 1, when the diftances are as 1 to 2. force therefore which acts upon the moon, and bends her into a curvilinear orbit, increases as the distance from the center of the earth decreases, fo as to be quadruple at half that distance.

The

In the fame manner it is fhewn, that if her leaft diftance was the third part only of her greatest distance, her velocity would be triple at the least diftance, to preferve the equability of the areas defcribed by a ray drawn from her to the center of the earth; and that the would be acted upon there by a power, which would have the fame effect in one minute, as in three minutes at her greateft diftance; fo that if fhe was allowed to defcend freely from each diftance, fhe would fall 9 times as far from the leaft diftance as from the greatest in the fame time; confequently the power itself which caufes her to defcend would be nine times greater at the third part of the distance, or the distances being as 1 to 3, the force would be as 9 to 1, or inverfely as the fquares of the diftances.

In the fame manner it appears, that when the greatest and leaft diftances are fuppofed to be in any proportion of a greater to a lefs number, the velocities of the revolving planet are in the inverse

ratio of the fame numbers; and that the powers which deflect or bend it's motion into a curve, are in the inverfe ratio of thofe numbers.

To confider this in general, let T, fig. 5, pl. 15, reprefent the center of the earth, A LP the moon's elliptical orbit, A the apogeum, P the perigæum, AH and P K the tangents at thofe points, A M and PN any small arcs defcribed by the moon in equal times at those distances, M H, NK, the fubtenfes of the angles of contact, terminated by the tangents in H and K; then MH and NK will be equal to the spaces that would be defcribed by the moon, if allowed to fall freely from the refpective places A and P in equal times; and will be in the fame proportion to each other, as the powers which act upon the moon, and inflect her course at thofe places.

Let A m be taken equal to P N, and m h parallel to AP meet the tangent at A in h; now as the curvature of the ellipfe is the fame at A as at P, m h is equal to K N; and if the moon was to fall freely from the places P and A towards the earth, her gravity would have a greater effect at P than at A, in equal times, in proportion as mh is greater than MH. But mh is the space which the moon would defcribe freely by her gravity at A, in the time which mh would be defcribed by her projectile motion at A, and M H is the space through which he would defcend freely by her gravity at A in the time in which A H would be defcribed by her projectile motion; and these spaces being as the fquares of the times, it follows, that mh is to MH, as the fquare of A h to the fquare of A H, or (because of the equality of the areas TAH, TPK) as the fquare of T P to the fquare of T A.

Therefore the gravity at P. is to the gravity at A, as the fquare of T A to the fquare of TP; that

s, the gravity of the moon towards the earth increases in the fame proportion, as the fquare of the distance from the center of the earth decreases.

Sir Ifaac Newton fhews the univerfality of this law, in all her diftances, from the direction of the power that acts upon her, and from the nature of the ellipfis, the line which the defcribes in her revolution; and it follows from the properties of this curve, that if you take fmall arcs defcribed by the moon in equal times, the space by which the extremity of any arc defcends towards the earth below it's tangent at the other extremity, is always greater in proportion as the fquare of the distance from the focus is lefs; from which it follows, that the power which is proportional to this space obferves the fame proportion.

The moon's orbit, according to aftronomers, differs not much from a circle of a radius equal to 60 times the femi-diameter of the earth; and the circumference of her orbit is therefore about 60 times the circumference of a great circle of the earth.

From this the circumference of the moon's orbit is easily computed, and as the finishes her revolution in 27 days, 7 hours, and 43 minutes, it is also easy to calculate what arc fhe describes in one minute.

The next thing is to compute how much this arc of one minute is deflected below a tangent drawn at the other end: now geometricians prove, that this space is nearly a third proportional to the diameter of her orbit, and the arc the defcribes in a minute; whence by an eafy calculation this space is found to be about 16 feet 1 inch.

But you have feen that this space was described in, confequence of her gravity, or tendency towards the earth, which is therefore a power, that at the distance of 60 femi-diameters of the earth,

is

is able to make her defcend in one minute throug 16 feet 1 inch.

Now, as this power increafes as fhe approaches the earth, let us fee what it's force would be at the furface thereof; and for this purpose, let us fuppofe her to defcend fo low in her orbit as at her least distance to pafs by the furface of the earth; fhe would then be 60 times nearer to the center of the earth, and move with a velocity 60 times greater, that the areas defcribed by a line drawn from her to that center in equal times, might ftill continue equal.

The moon therefore, paffing by the earth at her loweft ebb, would defcribe an arc in one fecond of time, (the 60th part of a minute) equal to that the defcribes in one minute at her prefent mean distance, and would fall as much below the tangent at the beginning of the arc in a fecond, as fhe falls from the tangent at her mean diftance in a minute; that is, fhe would fall near the furface of the earth 16 feet 1 inch in one fecond of time.

Now this is exactly the fame fpace, through which all heavy bodies are found by experience to defcend by their gravity near the furface of the earth. The moon, therefore, would defcend at the furface of the earth with the fame velocity, and every way in the fame manner, as heavy bodies fall towards the earth; and the power which acts upon the moon, agreeing in direction and force with the gravity of heavy bodies, and acting inceffantly every moment, as their gravity does, they muit be of the fame kind, and proceed from the fame canse.

Thus Sir Ifaac Newton fhewed, that the power of gravity is extended to the moon; that the is heavy, as all bodies belonging to the earth are found to be; and that he is retained in her orbit

by

by the fame caufe which occafions a stone, a bullet, or any other projectile, to describe a curve in the air. If the moon or any part of her were brought down to the earth, and projected in the fame line, and with the fame velocity as a terreftrial body, it would move in the fame curve. On the other hand, if any body was carried from our earth to the diftance of the moon, and was projected in the fame direction and with the fame velocity with which the moon is moved, it would proceed in the fame orbit which the moon defcribes, and with the fame velocity. Thus the moon is a projectile, and the motion of every projectile gives an image of the motion of a fatellite

or moon.

That the primary planets are heavy bodies, and gravitate towards the fun; and that the fecondary planets gravitate towards their refpettive primaries:

Obfervation proves, that each of the primary planets bend their path about the center of the fun, are accelerated as they approach to him, and are retarded as they recede from him, always defcribing equal areas in equal times; from whence it follows, that the power by which they are deflected must be directed to the fun. This power alfo varies always in the fame manner as the gravity of the moon towards the earth.

The fame reasoning, by which the gravity of the moon towards the earth at her greatest and leaft diftances were compared together, may be applied in comparing the powers which act on any primary planet at it's greatest and leaft diftances from the fun; and it will appear, that thefe powers increafe as the fquare of the distances from the fun decrease.

VOL. IV.

T

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