bodies, is the point about which they would equiponderate, or rest in any position.

Illus. If the centres of gravity of two A or more bodies, A and B, be connected by the right line A в, the distances ac and

B

B C, from the common centre of gravity c, are reciprocally as the weights of the bodies A and B; that is, A C: B C::

B: A.

PROP. I. If a body by an uniform_motion, describe one side of a parallelogram, in the same time that it would describe the adjacent side by an accelerative force; this body, by the joint action of these forces, would describe a curve, terminating in the opposite angle of the parallelogram.

A EK B

F

G

I

297. Let ABDC be a parallelogram, and suppose the body a to be carried through AB by an uniform force in the same time that it would be carried through Ac by an accelerative force, then by the joint action of these forces, the body would describe a curve AGID. For, by the preceding illustration, (Art. 293.) if the spaces AE, EK, and KB, be proportional to each other,

H

the spaces AF, FH, and HC, will be in the same proportion, and the line AGID will be a straight line when the body is acted upon by uniform forces; but in this example, the force in the direction AB being uniform, would cause the body to move over equal spaces AE, EK, and KB, in equal portions of time: while the accelerative force in the direction AC, would cause the body to describe spaces AF, FH, and HC, increasing in magnitude in equal successive portions of time, hence the parallelograms AEGF, AKIH, &c. are not about the same diagonal, therefore AGID is not a straight line, but a curve.

PROP. II. The curvilinear motions of the planets arise from the uniform projectile_motion of bodies in straight lines, and the universal power of attraction which draws them off from these lines.

given time, will carry it from A to F in a successive and equal portion of time, and so on; there being nothing either to obstruct or alter its motion. But, if, when the projectile force has carried the body to A, another body as s, begins to attract it, with a power duly adjusted and perpendicular to its motion at A, it will be drawn from the straight line EAF, and revolve about s in the circle* AGOOA. When the body E arrives at o, or any other part of its orbit, if the small body M, within the sphere of 's attraction, be projected, as in the straight line мn, with a force perpendicular to the attraction of E, it will go round the body E, in the orbit m, and accompany E in its whole course round the body s.-Here s may represent the sun, E the earth, and м the moon.

If the earth at A be attracted towards the sun at s, so as to fall from A to H by the force of gravity alone in the same time which the projectile force singly would have carried it from A to F; by the combined action of those forces it will describe the curve AG; and if the velocity with which E is - projected from A, be such as it would have acquired by falling from A to v (the half of As,) by the force of gravity alone †, it will revolve round s in a circle.-Keith.

* If any body revolve round another in a circle, the revolv-. ing body must be projected with a velocity equal to that. which it would have acquired by falling through half the radius of the circle, towards the attracting body. Emerson's Cent. Forces, Prop. ii.

A body, by the force of gravity alone, falls 16 feet.in

PROP. III. If one body revolve round another (as the earth round the sun), so as to vary its distance from the centre of motion, the projectile and centripetal forces must each be variable, and the path of the revolving body will differ from a circle.

299. Thus, if while a projectile force would carry a planet from A to F, the sun's attraction at s would bring it from C A to H, the gravitating power would be too great for the projectile force; the planet, therefore, instead of proceeding in the circle ABC (as in the preced- n R ing article) would describe the curve AO,

m

G F

B

/P

E

H

n

M

2

2

and approach nearer to

D

the sun; so being less

than SA. Now, as the eentripetal force, or gravitating power, al

ways increases as the square of the planet's distance from the sun diminishes *, when the planet arrives at o the centripetal force will be increased, which will likewise increase the velocity of the planet, and accelerate its motion from o to v; so as to cause it to describe the arcs OP, PQ, QR, RD, DT, TV, successively increasing in magnitude, in equal portions of time. The motion of the planet being thus accelerated, it gains such a centrifugal force, or tendency to fly off at v, in the line vw, as overcomes the sun's attraction; this centrifugal or projectile force being too great to allow the pla net to approach nearer the sun than it is at v, or even to move round the sun in circle t a b c d, &c. it flies off in the curve XZMA, with a velocity decreasing as gradually from v to A, as if it had returned through the arcs VT, TD, DR, &c. to A, with

the first second of time, and acquires a velocity which will carry it uniformly through 32 feet in each succeeding second. This is proved experimentally, by writers on mecha

nics.

* Newton's Princip. Book III. Prop. ii.

the same velocity which it passed through these arcs im its motion from A, towards v. At A the planet will have acquired the same velocity it had at first, and thus by the centrifugal and centripetal forces it will continue to move round s.

Two very natural questions may here be asked; viz. why the action of gravity, if it be too great for the projectile force at o, does not draw the planet to the sun at s? and why the projectile force at v, if it be too great for the centripetal force, or gravity, at the same point, does not carry the planet farther and farther from the sun, till it is beyond the power of his attraction?

First. If the projectile force at A were such as to carry the planet from A to G, double the distance, in the same time that it was carried from a to F, it would require four times as much gravity to retain it in its orbit, viz. it must fall through AI in the time that the projectile force would carry it from A to G, otherwise it would not describe the curve AOP. But an increase of gravity gives the planet an increase of velocity, and an increase of velocity, increases the projectile force; therefore, the tendency of the planet to fly off from the curve in a tangent P m, is greater at P than at o, and greater at Q than at P, and so on; hence, while the gravitating power increases, the projectile power increases, so that the planet cannot be drawn to the sun.

Secondly. The projectile force is the greatest at, or near, the point v, and the gravitating power is likewise the greatest at that point. For if as be double of vs, the centripetal force at v will be four times as great as at A, being as the square of the distance from the sun. If the projectile force at v be double of what it was at A, the space vw, which is the double of AF, will be described in the same time that AF was described, and the planet will be at x in that time. Now, if the action of gravity had been an exact counterbalance for the projectile force during the time mentioned, the planet would have been at instead of x, and it would describe the circle, t, a, b, c, &c.; but the projectile force being too powerful for the centripetal force, the planet recedes from the sun at s, and ascends in the curve xzм, &c. Yet, it cannot fly off in a tangent in its ascent, because its velocity is retarded, and consequently its projectile force is diminished, by the action of gravity. Thus, when the planet arrives at z, its tendency to fly off in a tangent zn, is just as much retarded, by the action of gravity as its motion was accelerated thereby at Q, therefore it must be retained in its orbit. Keith on the Globes.

* Ferguson's Astronomy, Art. 153.

CHAPTER XII.

CHRONOLOGICAL SCIENCE.

SECTION I.

OF TIME AND ITS PARTS.

300. CHRONOLOGY is that science which treats of time, and shews its different measures or computations, as they have been observed by different nations.

By chronology we are enabled truly to date the beginning and end of the reigns of princes, the births and deaths of eminent persons, the revolutions of empires, and kingdoms, battles, sieges, or any other remarkable events. Without this useful science, that is to say, without distinguishing the times of events as clearly as the nature of the case will admit, history would be little better than a heap of confusion, destitute of light, order, or beauty.

201. Time. Its usual divisions are years, months, weeks, days, hours, minutes, and seconds; besides periods, centuries, and cycles.

A year is the most complete period of time, being that in which all the seasons return in succession, and begin anew.

It is that space of time wherein the earth finishes its course round the sun, returning to the same point from which it departed.

This consists of three hundred and sixty-five days, five hours, and forty-nine minutes; and is called the tropical, natural, or solar.

But that period of time in which the sun having departed from any fixed star, returns to the same again, is called the sidereal year, and contains three hundred and sixty-five days, six hours, ten minutes.