the rate of description of areas is
equable. It proves, in fact, that the sun is the attracting body, and
that no other force acts.

For first of all if the first law of motion is obeyed, _i.e._ if no
force acts, and if the path be equally subdivided to represent
equal times, and straight lines be drawn from the divisions to any
point whatever, all these areas thus enclosed will be equal,
because they are triangles on equal base and of the same height
(Euclid, I). See Fig. 59; _S_ being any point whatever, and _A_,
_B_, _C_, successive positions of a body.

Now at each of the successive instants let the body receive a
sudden blow in the direction of that same point _S_, sufficient to
carry it from _A_ to _D_ in the same time as it would have got to
_B_ if left alone. The result will be that there will be a
compromise, and it will really arrive at _P_, travelling along the
diagonal of the parallelogram _AP_. The area its radius vector
sweeps out is therefore _SAP_, instead of what it would have been,
_SAB_. But then these two areas are equal, because they are
triangles on the same base _AS_, and between the same parallels
_BP_, _AS_; for by the parallelogram law _BP_ is parallel to _AD_.
Hence the area that would have been described is described, and as
all the areas were equal in the case of no force, they remain equal
when the body receives a blow at the end of every equal interval of
time, _provided_ that every blow is actually directed to _S_, the
point to which radii vectores are drawn.

[Illustration: FIG. 60.]

[Illustration: FIG. 61.]

It is instructive to see that it does not hold if the blow is any
otherwise directed; for instance, as in Fig. 61, when the blow is
along _AE_, the body finds itself at _P_ at the end of the second
interval, but the area _SAP_ is by no means equal to _SAB_, and
therefore not equal to _SOA_, the area swept out in the first
interval.

In order to modify Fig. 60 so as to represent continuous motion and
steady forces, we have to take the sides of the polygon _OAPQ_,
&c., very numerous and very small; in the limit, infinitely
numerous and infinitely small. The path then becomes a curve, and
the series of blows becomes a steady force directed towards _S_.
About whatever point therefore the rate of description of a