In
Proposition I of the Principia Newton derives Kepler's Law of Areas
on the assumption that a revolving body is subject to a centripetal force
directed to a fixed point. Newton's demonstration employs equal finite
time intervals, after each of whlch the force gives a finite impulse to
the body towards center S. During one interval, from A to B, the motion
remains the same; then at B it is suddenly changed by the additional motion
BV. In the second interval, instead of continuing the motion along Bc=AB
the body follows the resulting path BC. Since the areas SAB and SBc are
equal, and because impulse BV is directed toward S, areas SBc and SBC are
equal, and area SBC is equal to SAB, which holds for each succeeding interval:
(Every next triangular area is equal to the preceding one). Hence all triangular
areas described in equal intervals are equal, and will lie all in the same
place. This holds also when the time intervals are taken ever smaller and
their number ever greater in the same rate until, finally, we have a continuously
acting force and a curved orbit, for which the areas described are proportional
to the time used.
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