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 areas is uniform, that point and no other must be the centre of all the force there is. If there be no force, as in Fig. 59, well and good, but if there be any force however small not directed towards _S_, then the rate of description of areas about _S_ cannot be uniform. Kepler, however, says that the rate of description of areas of each planet about the sun is, by Tycho's observations, uniform; hence the sun is the centre of all the force that acts on them, and there is no other force, not even friction. That is the moral of Kepler's second law.