62 NEWTON S PRINCIPIA. 



the greater axis does not depend at all upon the direction 

 of the tangential or projectile force, but only upon its 

 quantity, the direction influencing the length of the lesser 

 axis alone. 



Lastly, it may be observed, that as these latter pro 

 positions give a measure of the velocity in terms of the 

 radius vector and perpendicular to the tangent for each of 

 the conic sections, we are enabled by knowing that ve 

 locity in any given case where the centripetal force is 

 inversely as the square of the distance, and the absolute 

 amount of that force is given, as well as the direction of 

 the projectile force and the point of the projection, to 

 determine the parameters and foci of the curve, and also 

 which of the conic sections is the one described with that 

 force. For it will be a parabola, an hyperbola, or an ellipse, 

 according as the expression obtained for /&amp;gt; 2 (the square of the 

 perpendicular to the tangent) is as the radius vector, or in 

 a greater proportion, or in a less proportion. This is the 

 problem above referred to, which John Bernouilli had en 

 tirely overlooked, when he charged Sir Isaac Newton with 

 having left unproved the important theorem respecting 

 motion in a conic section, which is clearly involved in its 

 solution. 



Before leaving this proposition, it is right to observe 

 that the two last of its corollaries give one of those sa 

 gacious anticipations of future discovery which it is in 

 vain to look for anywhere but in the writings of this great 

 man.* He says, that by pursuing the methods indicated 



in the investigation, we may determine the variations im 

 pressed upon curvilinear motion by the action of disturbing, 

 or, what he terms, foreign forces; for the changes intro- 



* See a singular anticipation respecting dynamics, by Lord Bacon, in De 

 Ang. Lib. III., under the head Translation of Experiments. It was pointed 

 out to me by my learned friend B. Montague. 



