NEWTON S PRINCIPIA. 47 



But the proposition is so important, that it may be 

 well to prove it, and to show that it is almost in terms 

 involved in the third corollary to Prop. VI. Book I. of 



the Principia. By that corollary F = -^-^ (C being 



the osculating circle s chord which passes through the 

 centre of forces). But drawing S Y, the perpendicular 

 to the tangent, and P C F through the centre of the 



circle, and joining V F, which is, therefore, parallel to 

 Y P, we have V P : P F :: S Y : S P or C : 2 R ::p : r 



and C = -, which substituted for C in the above 



equation, gives F = ^L^. 



It is remarkable that the circumstance of this formula 

 being thus involved in that of Sir Isaac Newton seems never 

 to have been observed by Keill, who, in the Philosophical 

 Transactions, xxvi. 74, gives a demonstration of it much 

 more roundabout, and as of a theorem which Demoivre had 

 communicated to him, adding, that Demoivre also informed 

 him of Sir Isaac Newton having invented a similar 

 method before. In fact, he had, above 20 years before, 

 given it in substance, though not in express terms, in the 

 Sixth Proposition, the addition of two lines to which would 

 at once have led to this formula. But, again, when John 



