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ment of the orbit, that is, by what was lately seen, as the 

 force is to the velocity.* 



But also, the radius of curvature of the hodograph is to 

 the half chord of curvature of the same curve, as the radius 

 vector of the orbit is to the perpendicular let fall from the 

 fixed centre on the tangent to the same orbit; therefore, by 

 compounding equal ratios, we obtain this other proportion : 

 the radius of curvature of the hodograph is to the radius vec- 

 tor of the orbit, as the rectangle under the same radius vector 

 and the force is to the rectangle under the velocity and the 

 perpendicular, or to the constant parallelogram under the vec- 

 tors of position and velocity. If, therefore, the law of the in- 

 verse square hold good, so that the second and third terms of 

 this last proportion vary inversely as each other, while the 

 fourth term remains unchanged, the first term must be also 

 constant; that is, with Newton's law of force (supposed here 

 to act towards a fixed centre), the curvature of the hodograph 

 is constant: and, consequently, this curve, having been already 

 seen to be plane, is now perceived to be a circle, of which the 

 radius is equal to the attracting mass divided by the constant 

 double areal velocity in the orbit. Reciprocally, we see, that 

 no other law of force would conduct to the same result : so 

 that the Newtonian law may be characterized as being the 

 Laiv of the Circular Hodograph. 



Another mode of arriving at the same simple but impor- 

 tant result, is to observe, that because the radius of curvature of 

 the hodograph is equal to the element of that curve, divided 

 by the angle between the tangents at its extremities, or (in 

 the case of a central force) by the angle between the two cor- 

 responding vectors of the orbit, which angle is equal to the 



* By an exactly similar reasoning, the following known proportion may 

 be proved anew, namely, that the force is to the velocity as that velocity is 

 to the half chord of curvature of the orbit, whatever the law of central fores 

 may be. 



