actions and impediments from without being excluded. For,

If the bodies A and B (fig. 33) act upon each o other, the motion produced in each will be equal, (22, R) and the ratio of CA to C B will confequently remain the fame, whether they approach to, or recede from each other. The ftate of c will not therefore be changed by their mutual actions. If the third body ɛ be added to the system, the center D, for the fame reafon, will not be changed, as to its state of motion or reft whether E acts upon c or not: and the fame may be proved of any number of bodies.

Since then the state of the center of gravity of R any fyftem of bodies, as to rest, or uniform direct motion, is not affected either by the motions or mutual actions of the bodies of which it is composed, external actions or impediments being excluded, it is plain that the fame law holds good in the motion of a system of bodies as is obferved by a fingle body. For the progreffive motion of a single body, or of a fyftem of bodies, must be estimated by the motion of the center of gravity.

Hence it is that the center of gravity of the earth is not affected by the motions on its furface, or in its bowels. When a projectile, a cannon

ball for inftance, is thrown upwards, the projecting

force reacting on the earth, caufes it to move in the contrary direction; but as the motions are equal, the center of gravity remains the fame.

The

T

U

The motions and actions of bodies upon each other in a space that is carried uniformly forward, are the fame as if that space were at rest.

upon

each

For the motions and actions of bodies other depend on their relative motion, the velocity of which is the sum of their absolute velocities, when they are moved in oppofite directions, or their dif ference when they move in the fame direction. And this fum or difference is not altered by an equal velocity impreffed on all the bodies in the fame or a parallel direction, as in the prefent cafe: fince, when two bodies move in contrary directions, in a' space carried uniformly forward, the velocity added to that body, with whofe motion the motion of the space confpires, is exactly equal to the velocity deftroyed in the other body, whofe motion is oppofed by that of the fpace; and when the bodies move in the fame direction, an equal velocity being added to, or destroyed in both, the difference is v likewife unaltered. This is likewife confirmed by daily experience; motions performed on board a fhip under fail are the fame as if the ship were at anchor; except fo far as they may be disturbed by the irregular toffing of the waves, which affects them fucceffively, as much in one direction as another. A fleet of fhips carried by an uniform

.* Space being in its own nature immoveable, the expreffion is here improper; but it conveys a clear idea of the propofition in concise terms, though we can form no idea of bodies included in a fpace being acted upon by that space. The space here mentioned is merely ideal, may be called relative, and is defined to be a moveable dimenfion.

current,

current, either preferve the fame relative positions, or approach to, or recede from, each other in the fame manner as they would if no fuch current exifted. And the motions of bodies at the furface of the earth are no otherwife affected by its revolution on its axis than as the revclution is not rectilinear, the effects of which, though confiderable, are not enough fo to fall under common observation.

This propofition is likewife true, if the motion w of the space be uniformly accelerated, or, which is the fame thing, if all the bodies be constantly acted upon by parallel forces which act equally, according to their maffes, on each of them.

For fuch forces will caufe all the bodies to move x with the fame acceleration, and to describe equal fpaces in the fame direction with each other. They will not therefore change their relative motions or fituations.

CHAP.

CHA P. VII.

OF PENDULUMS.

THE bodies spoken of in the present chapter are fuppofed to move without rotation, friction, or refistance from the air, or any other medium; neither are the magnitudes of the bodies brought into confideration.

It has already been shewn (65, v), that the force of a body to defcend along an inclined plane is to the whole force of its gravity as the height of the z plane to its length. If the body be at liberty to descend on the plane, the first-mentioned force will, by its conftant and equal action (27, D) produce an uniform acceleration.

The spaces described from the beginning on a given inclined plane are (29, G) as the squares of A the times; that is to fay, the times of description

of inclined planes of the fame inclination are as the fquare roots of their lengths.

B The final velocities (36, H) of accelerated motions being equal when the forces are inversely as the spaces paffed through, and the length of an inclined plane being to its height in this fame ratio of the forces, by which a body would descend along the plane, or fall freely through its height, it folc lows that the final velocity acquired by a body that descends along such a plane is equal to the final velocity it would acquire by falling freely through Dits height. Hence alfo the final velocity is always

equal

equal when a body has fallen through an inclined plane of a given height, whatever may be its length.

The times of acquiring this given velocity, or of E paffing over the whole lengths, will be inversely as the forces (35, G); that is to fay, directly as the lengths when the heights are equal (65, v).

Bodies that defcend from a given height al- F ways acquire the fame final velocity, whether they defcend along a fingle plane or many. Let EG (fig. 34) represent an horizontal, and A G ́a perpendicular line; and fuppofe a body to defcend along the inclined planes AB, BC, CD, DE; continue A B to F in the line E G, and draw the lines, BK, CI, DH, parallel to the horizon. The body, after paffing through A B and B C, will have acquired a velocity equal to the velocity it would have acquired fimply by defcending along в c or (80, D) along B L, added to the velocity it had at B: therefore the velocity at c, after paffing through the planes A B, BC, is the fame as would have been acquired by defcending from the fame height AI through a fingle plane a L. The fame reasoning may be extended to prove, that when the body has arrived at D, it will have the fame velocity as it would have acquired by defcending in one plane A M of the fame height A H: and fo forth for any number of planes whatever.

Since the planes along which a body may pafs, in defcending from a given height, are not limited either in number or magnitude, we may affume them to be indefinitely fmall, and indefinitely numerous.

VOL. I.

G

They