This then is the meaning of the first and third laws of Kepler. What about the second? What is the meaning of the equable description of areas? Well, that rigorously proves that a planet is acted on by a force directed to the centre about which 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.
Fig. 60.
Fig. 61.