It was for a long time supposed, that a body would move more rapidly in a direct line, not being vertical, than by any other course ; but it has been ascertained, that any curve, not exceeding 60 degrees, is a quicker descent than its chord ; and that a curve of 90 degrees is a quicker de scent than any tangent laying between the same parallels. Thus,. in fig. 9, the curve A 11 C is of quicker descent than the chord A C, and the curve M A BC gives a quicker descent than the tangent F G, or H I, or E L, laying between the same parallels, M K and C 1).
With regard to the motions of projec tiles, we refer the reader to that article.
Of central forces we have already given an article, but shall observe here, in ad dition, that all bodies, when put in mo tion, would preserve their respective ve locities, and their original directions, were they not acted upon by other forces. When a force acts equally for a limited distance, and then is superseded by the actions of another force, the body will de scribe a polygon in its track : but if the original force be gradually weakened, the body will then describe a curve, bending towards the centre of attraction, resulting from the operation of a deflect ing force, which, by its pressure, causes the body to bend from its original direc tion. When a body is constantly attract ed towards a centre, it is under the influ ence of a centripetal force; and when it is disposed to fly from that centre, it is under the bias of a centrifugal force. These two latter constitute what are termed central forces. The projectile force is the original direction of an im pelled body, forming a tangent with the curve occasioned by the deflecting force. The track of a body under the influence of a centripetal force, is called its trajec tory, or orbit. The radius vector is a line drawn from the centre to which the force is referred, or wherein it is supposed to act, to any point in the trajectory where the body is found. A body moving regn. larly on a trajectory, or orbit, which re turns into itself; is,. on its return to the incipient point whence the motion began, said to have made a period ; and the time occupied is called its periodical time. It must be understood, that a body can nei ther set itself in motion, nor avert its own course ; such effects must be the result of forces exteriorly applied ; also we must state, that the motion of each body is na turally in a right line, but by the impulse of some one or more powers its course will deviate into a curve. Thus, a peb
ble in a sling, or a glass full of water placed within a hoop that is turned swift ly round, will follow, the course of the sling or hoop, respectively ; but when liberated, or improperly checked, they will fly off in a right line, which they must preserve if not opposed by the air, &c.
We invariably find, that, when a boat is pushed off from the shore, a certain bias towards the place quitted is felt by every person on board. If any thing should be overset at that instant, unless pressed towards any other point, it will fall towards that shore. On arriving at the opposite bank, if the boat is allowed to run against it, a disposition to fall to wards that bank will be manifested by every person, and by every matter at li berty, within the boat. Hence we find. that all bodies at rest are disposed to re main so ; and that, when bodies are set in motion, they would continue to move, were they not obstructed by either a me chanical, or an invisible, agent. All bo dies moving in orbits have a disposition to fly out of them ; and those which de scribe orbits of the smallest diameter have rotatory motions quicker than those which take a greater range. If one body moves round another, both will describe curves round their common centre of gravity. The centrifugal force of a re volving body is in direct proportion to the quantity of matter multiplied into the velocity. 'I'he centripetal forces in cir cles are as the squares of the velocities directly, and of the radii inversely : therefore, when the centripetal force, and the distance from the centre, are gi ven, the velocity is given. The mutual attraction of bodies does not affect their centre of gravity ; and if, while two bo dies act on each other, they be projected in opposite and parallel directions, with velocities in proportion to their respec tive distances from the centre of gravity, they will describe similar figures around that centre. It is on these principles (which involve an immense collection of cases and circumstances), that the science of astronomy, and whatever relates to the wonderful correspondence we observe in all the operations of the grand universal system, is founded.