In a curvilineal motion the change of direction is measured by the angle contained between the tangents to the curve.

A curvilineal motion is therefore always a compound motion; but the great bodies of this fyftem, as the planets, move round the fun in curve lines; on thefe principles there muft therefore be neceffarily two powers acting on them, one impelling them to move in a ftrait Fine, the other deflecting or bending them continually towards a

center.

You may therefore confider deflecting forces as always directed to or from a point; in the first case they are called centripetal forces, in the fecond cafe they are called centrifugal forces. In general, they are termed CENTRAL forces; and the point, through which their direction always paffes, is called the center of the forces.

Among the various curvilinear motions which may arife from the action of central forces, there is a circumftance in which they all agree, and which enables the mathematician to investigate the forces by which they are produced.

If a body moves in a curve line, A B C D E F, fig. 3, pl. 15, by means of a force always directed to a fixed point S, the curve is all in one plane, and the areas (AS B, ASC, AS D) defcribed by the ftrait line joining the body with the point S, are proportional to the times of defcription; i. e. equal areas are defcribed in equal times, unequal areas in unequal times. Thus the triangular areas ASB, BS C, CSD, &c. defcribed by the ftrait line joining the body with the point S, are proportional to the times of defcription.

Let the time be divided into equal parts, let the body beted on by an impulfe that will carry it from A he first given particle of time; then in th article it would go an equal fpace

S2

Space, and defcribe the line B c equal to the line

A B.

But when the body is arrived at B, let a deflecting (centripetal) force fo act upon it, that while it's firft impulfe would carry it to c, the deflecting force would carry it to V; complete the parallelogram, BV c c, and it is evident, from the doctrine of compound forces, that the body would in the fecond particle of time defcribe the diagonal BC.

Now as Cc is parallel to SV, the triangles SBC, SB, are between the fame parallel lines, and as fuch, are (by geometry proved to be) equal; for the fame reafon the triangles SCD, SEF, are proved to be equal to S B A.

If any number of these triangles be added together, the total fums, as ADS, FCS, will be proportional to the times wherein they are defcribed.

If the lines AB, BC, be continued round a center, they will form a polygon, and if the fides of the polygon be indefinitely increafed in number, and indefinitely decreased in length, they will form a curve, a circle, or an ellipfis: and the propofition will be true of thefe curves, that a line drawn from the center to a body in the circumference of the circle, or from the focus to a body in the circumference of the ellipfis, will fweep equal areas in equal times.

The power, therefore, directed towards the given point S has no effect on the magnitude of the area defcribed by the line fuppofed to be drawn from the body to that point. It may accelerate or retard the motion of the body, but affects not the area or space defcribed by the line. The line will fill continue to defcribe the fame fpaces in equal times, about the given point, as it would have done, if no new force had acted on the body, but it had been permitted to proceed uniformly in the line of projection,

As

As one impulfe towards the given point has no effect on the area or space defcribed by the ray or line from the body to that point, fo any number of fucceffive impulfes directed to the fame point can have no effect on the area; and if you suppose the power directed to that point, to act continually, it will bend the way of the body in motion into a curve, and may accelerate or retard it's velocity, but can never affect the area defcribed in a given time by a line fuppofed to be drawn from the body to the given point, which will always be of an invariable quantity, equal to that which would have been defcribed in the fame time, if the body had proceeded uniformly in a right line from the beginning of the motion.

The converfe of the foregoing propofition fhews, that if a body A defcribes a curve all in one plane, and if there be a point S fo fituated in this plane, that a line drawn therefrom to the circumference defcribes proportional areas in proportional times, then is the body urged round by a force tending towards that center.

In other words, the equable increase of the areas defcribed by a line drawn from a body to a given point, is an indication that the direction of the power that acts upon the body, and that deflects it into a curve, is directed to that point.

By the fame propofition we may illuftrate and explain the revolutions of the primary planets in elliptical orbits (not much differing from circles) round the fun, who is in one of the foci of each ellipfis.

Let the ellipfis ABCDEFGHIKLM, fig. 4, pl. 15, reprefent the orbit of a planet moving therein round the fun S, according to the order of the letters, the fun S being in one of the foci of the ellipfis; let the time of it's revolution be divided into any number of equal parts,

S 3

fuppofe

fuppofe 12, in moving from A through BCD, &c. the planet' approaches nearer the fun, and the central tendency continually increafing it's velocity, it goes through greater arcs in equal times, till it comes to G; from thence it's motion cortinually carries it to a greater diftance from the fun, and it defcribes in equal times fmaller and fmaller arcs, till it returns to A, from whence it proceeds as before.

Now the triangular fpaces paffed over by a line drawn from the planet to the center of the fun will be equal, because in the planet's going the firft half of the ellipfis from A to C, the arcs which may be confidered as the bafe of the mixed triangles defcribed in equal times, grow longer and longer, as the legs grow fhorter, fo as to preferve the equability of the triangular space: in the other half of the ellipfis in the planet's going from C to A, the arcs grow fhorter; but this is compensated by the greater length of the legs.

The fum of what has been proved is, 1. that the areas or fpaces revolving round an immoveable center are proportional to the times; and, 2d, that if a body revolving round a center describes about it areas proportional to the times, the body is actuated by a force directed to that

center.

But by Kepler's first law, we know "that the primary planets defcribe round the fun, and the fecondary planets defcribe round their refpective primary planets, arcas proportional to the times." From hence it is inferred, that the primary planets are retained in their orbits by forces which are always directed to the fun; and that the fecondary planets are retained in their oroits round their primary planets by forces which are always directed to thole primary planets.

Kepler's fecond law is, "That the orbits defcrib

ed

ed round the fun, and round the primary planets, are ellipfes, having the fun, or the primary planet, in the focus." From hence it is inferred, that the accelerating force, by which a planet is retained in the different parts of it's elliptical orbit, is inverfely proportional to the fquare of it's diftance from the fun, or from it's primary.

Kepler's third law is, "That the fquare of the periodic times of planets revolving round common centers, are proportional to the cubes of their mean diftances." From this it is inferred, that the forces, by which the planets are retained in their different orbits, are inversely proportional to the fquares of their diftances from the fun. The fame reasoning applies to the fatellites.

Hence it is also inferred, that the forces, by which different planets are retained in their different orbits, are not forces of different kinds, but the fame force operating at different distances.

The fecondary planets accompany the primary planets by the action of a force always directed to the fun, and inverfely proportional to the fquare of the distance from the fun.

That the moon is a heavy body, and gravitates towards the earth in the fame manner as terrestrial bodies.*

Sir Ifaac Newton, confidering that the power of gravity acts equally on all matter that is on or near the furface of the earth, that it is not fenfibly lefs on the tops of the highest mountains, that it affects the air and reaches upward to the utmost limits of the atmosphere, was induced to think it might be a more general principle, and extend to the heavens, fo as to affect the moon at least, which is the nearest to us of all the bodies in the fyftem. He afterwards extended this principle ftill S4 further Maclaurin's Sir Ifaac Newton's Difcoverics, p. 214 to 265.