NEWTON S PRINCIPIA. 



33 



S B c and SBC are equal, they are between the same 

 parallels, and c C is parallel to S B, and D d to S C ; 

 consequently the force which deflects acts in the lines SB 

 and S C, or towards the point, S. It is equally manifest 

 that the direction of the lines Be, C d, from which the 

 centripetal force deflects the body, is that of tangents to 

 the curve which the body describes, and that consequently 

 the velocity of the body is in any given point inversely 

 proportional to the perpendicular drawn from the centre to 

 the tangent; the areas of the triangles whose bases are 

 equal, being in the proportion of their altitude, that is, 

 of those perpendiculars, and those areas being by the pro 

 position, proportional to the times. 



There are several other corollaries to this important 

 proposition which deserve particular attention. B c and 

 D e are tangents to the curve at B and D respectively; 



v B 



B C and D E the arcs described in a given time; C c 

 and E e lines parallel to the radii vectores S B and S D 

 respectively ; and C V, E d parallel to the tangents. The 

 centripetal forces at B and D must be in the proportion of 

 V B and d D (being the other sides of the parallelograms 

 of forces) if the arcs are evanescent, so as to coincide with 

 the diagonals of the parallelograms V c and d e. Hence 

 the centripetal forces in B and D are as the versed sines 



D 



