**Newton****’s
approach to Kepler’s area law**

First, consider an observer at point *S* and a body
moving under its own inertia along the line *ABC…*. For equal time
intervals, the segments *AB*, *BC*, etc are equal in length.
Furthermore, the areas of the successive triangles *SAB, SBC,* etc are
also equal, since the base of each triangle is the same and the heights are
also the same (the distance of *S* from the line *ABC*).

Figure 1

Now suppose that when the body reaches point *B* it
receives an (instantaneous) impulse directed along the line from *B* to *S*
(i.e. an impulse resulting from some sort of central force). The velocity of
the body, but not its position, will change, and the body will take a new path
to point *C*, as shown below:

Figure 2

In this figure, point *c* now marks the place where the
body *would have* gone in absence of the impulse (as in the first figure),
point *b* is where it would go due solely to the impulse, and point *C*
is where it actually does go under the combined influence of both effects.
However, since *BbCc* is a parallelogram (due to the vector-sum nature of
the velocities), triangles *SBC* and *SBc* have the same (common)
base and the same height (since points *C* and *c* are equidistant from
the line *SB*), and hence the same area, which is also the same area as
triangle *SAB, *as shown above. The procedure can be repeated
indefinitely, and all succeeding triangles will have the same area as the
initial triangle *SAB*. Furthermore, in the limit that the time interval
between impulses becomes infinitesimal, the path of the body will become a
continuous curve.

In this way Newton proves that in the presence of a central
force, or no force, a moving body sweeps out equal areas in equal times. And as
the story goes, Newton was so unimpressed with Kepler’s understanding of the
area law, and/or so impressed with his own, that he did not even mention Kepler
by name in Book 1 of the *Principia*.

The accompanying animation will show Figure 1 if you slide the impulse to zero, and Figure 2 for a finite impulse.