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out regard to the resistance of the medium that fills the 

 space in which they move ; and it is principally devoted 

 to the consideration of motions in orbits determined by 

 centripetal forces, and to examining the attraction of bodies. 

 The Second book treats of the resistance of fluids chiefly 

 as affecting the motions of bodies that move in them. 

 The Third book contains the application of the principles 

 thus established to the motions, attractions, and figures 

 of the heavenly bodies. 



The fundamental proposition, as it may justly be termed, 

 of the whole system, is one which Newton s predecessors 

 may be said to have nearly reached; which Kepler, had 

 he been more inclined to trust demonstration than em 

 pirical observation, probably would have attained ; and 

 which Galileo would certainly have discovered had he con 

 templated the facts discovered by Kepler, particularly his 

 second law * : The proposition 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 in a straight line, the areas or triangles 



* See the historical notice above respecting this second law, viz., that the 

 planets describe areas proportional to the times by their radii vectores. 



