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ment of the orbit, that is, by what was lately seen, as the 
Sorce 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 
Law 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 force 
may be. 
