34 NEWTON S PKINCIPIA. 



of the evanescent arcs ; and the same holds true if instead 

 of two arcs in the same curve, we take two arcs in dif 

 ferent but similar curves.* 



From these propositions ajiother follows plainly, and its 

 consequences are most extensive and important. If two 

 or more bodies move in circular orbits (or trajectories) 

 with an equable motion, they are retained in those paths 

 by forces tending towards the centres of the circles ; and 

 those forces are in the direct proportion of the squares of 

 the arcs described in a given time, and in the inverse pro 

 portion of the radii of the circles. 



First of all it is plain, by the fundamental proposition, 

 that the forces tend to the centres S, s, because the sectors 

 A S B and PBS being as the arcs A B, B P, and the 

 sectors a s b, p b s, as the arcs a b, b p, which arcs being 

 all as the times, the areas are proportional to those times 

 of describing them, and therefore S and s are the centres 

 of the deflecting forces. Then, drawing the tangents A C, 

 a c, and completing the parallelograms D C, d c, the diago 

 nals of which coincide with the evanescent arcs A B, a b 9 

 we have the centripetal forces in A and a, as the versed 

 sines A D, a d. But because A B P and a b p are right 

 angles (by the property of the circle), the triangles A D B, 

 A P B, and a d b, a p b, are respectively similar to one 

 another. Wherefore A D : A B :: A B : AP and AD 



AB 2 i n j ab 2 



= --; and in like manner a d = , or. as the evan- 

 AP ap 



A B 2 



escent arcs coincide with the chords, A D = arc . ^ and 



XX JL 



a IP 



6/=arc . Now these are the properties of any arcs de 

 scribed in equal times ; and the diameters are in the pro- 



* If B C, D E, are bisected, the proportion is found with the halves 

 of V B, D d- t and that is the same proportion with the whole versed sines. 



