Proposition 2

Proposition 2 Theorem 2

Every body that moves in any curve line described in a plane, and by a radius, drawn to a point either immovable, or moving forward with an uniform rectilinear motion, describes about that point areas proportional to the times, is urged by a centripetal force directed

CASE 1

For every body that moves in a curve line, is (by Law 1) turned aside from its rectilinear course by the action of some force that impels it.

That force by which the body is turned off from its rectilinear course, and is made describe, in to equal times, the equal least triangles SAB, &c., about the point S 1, SBC, SCD, immovable

XL. Book Law II), acts (by Prop. Elem. and in the place B, according to the direction of a line par- to that point K allel cC. that accordii g is, in the direction of the line to the direction of a line parallel to [BOOK BS. and dD, that is, f. in the place C, in the direction of the line CS, (fee.; and therefore acts always in the direction of lines

tending to the immovable point S.

(by Cor. 5 of the Laws) it is indifferent whether the su- which a body describes a curvilinear figure be quiescent, or moves together with the body, the figure described, and its point S, uniformly

Case 2.

perfices in forward in right lines.

COR. 1. In non-resisting spaces or mediums, if the areas are not propor tional to the times, the forces are not directed to the point in which the but deviate therefrom in. consequently or towards the parts to radii meet ; which the motion is directed, if the description of the areas is accelerated ; but in antecedentia, if retarded. COR. 2. And even in resisting mediums, if the description of the areas is accelerated, the directions of the forces deviate from the point in which the radii meet, towards the parts to which the motion tends.

SCHOLIUM

A body may be urged by a centripetal force compounded of several forces. The resulting force tends to the Point S.

But if any force acts perpetually in the direction of lines perpendicular to the surface, this force will make the body to deviate from the plane of its motion. But it will neither augment nor diminish the quantity of the described surface and is therefore to be neglected in the composition of forces.

Proposition 3 Theorem 3

Every body, that by a radius drawn to the centre of another body, howsoever moved, describes areas about that centre proportional to iJie times, is urged by a force compounded out of the centripetal force Bending fo that other body, and of all the accelerative force by which that other is impelled. body Let L represent the one, and T the other body ; and (by Cor. both bodies are urged in the direction of parallel equal and contrary to that by which the second body if L of the Laws) lines, by a ne T force is tinned, the first the same areas as will go on to describe about the other body body was urged will be now before but the force by which that other body destroyed by an equal and contrary force; and therefore (by Law I.) that other body T, now left to itself, will either rest, or move uniformly forward L and the first body in a right line impelled by the difference of the will force the that is, forces, remaining, go on to describe about the other by to the And times. therefore (by Theor. II.) the areas proportional body : T difference ;f the forces is directed to the other body T as its centre.

Hence if drawn to the other body T, and from the whole force, by which the one body L, by a radius describes areas proportional to the times the 107 ; urged (whether that force is simple, or, according to several forces), we subduct (by the same Cor.) that whole accelerative force by which the other body is urged the who_e remaining force by which the first body is urged will tend to the firr.t Cor. body is 2 of the Laws, compounded out of ; ther body T, as its centre.

Corollary 2

If these areas are proportional to the times nearly, the re nearly. maining force will tend to the other body

Corollary

Vvice versa, if the remaining force tends nearly to the other will be those areas nearly proportional to the times. body T,

Corollary 4

If the body L, by a radius drawn to the other body T, describes areas, which, body

compared with the times, are very unequal and that other the rest, or moves uniformly forward in a right line be either at is either none action of the centripetal force tending to that other body at all, or it is mixed and compounded with very powerful actions of other forces : and the whole force compounded of them all, if they are many, is The same thing ob directed to another (immovable or moveaJble) centre. tains, when the other body is moved by any motion whatsoever provided ; that centripetal force is taken, w hich remains after subducting that whole force acting upon that other body T. r

SCHOLIUM

Because the equable description of areas indicates that a centre is re spected by that force with which the body is most affected, and by which it its rectilinear motion, and retained in its orbit why not be allowed, in the following discourse, to use the equable de scription of areas as an indication of a centre, about which all circular is drawn back from may we motion is performed in free spaces ?