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CENTRAL FORCES AND LAW OF THE UNIVERSE 

 ANALYTICALLY INVESTIGATED. 



THE fundamental proposition of the whole Newtonian system 

 is this. If a body is driven by any single impulse or force of 

 projection, and is also drawn continually by another force so 

 as to revolve round a fixed centre, the radius vector, or line 

 drawn from the body to that centre, describes areas which 

 are in the same fixed plane, and are always proportional to 

 the times of the body's motion ; and conversely, if any body 

 which moves in any curve described in a plane so that the 

 radius vector to a point either fixed or moving uniformly in 

 a straight line, describes areas proportional to the times of 

 the body's motion, that body is acted on by a centripetal force 

 tending towards and drawing it to the point. 



To prove this, we have to consider that if a body moves 

 equably on a straight line, the areas or triangles which are 

 described by a line drawn from it to any point are propor- 

 tional to the portions of the straight line through which the 

 body moves (that is to the time, since, moving equably, it 

 moves through equal spaces in equal times), because those 

 triangles, having the same altitude, are to one another in the 

 proportions of their bases. S being the point and AO the 

 line of motion, SAB is to SB c as A B to B c. If then at 

 B a force acts in the line S B, drawing the body towards S, 

 it will move in the diagonal B C of a parallelogram of which 

 the sides are B c and B V, the line through which the deflecting 

 force would make it move if the motion caused by the other 

 force ceased. C c therefore is parallel to V B, and the triangle 



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