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.
We may also see from it that gravity does not travel like light, so as to take time on its journey from sun to planet; for, if it did, there would be a sort of aberration, and the force on its arrival could no longer be accurately directed to the centre of the sun. (See Nature, vol. xlvi., p. 497.) It is a matter for accuracy of observation, therefore, to decide whether the minutest trace of such deviation can be detected, i.e. within what limits of accuracy Kepler's second law is now known to be obeyed.
I will content myself by saying that the limits are extremely narrow. [Reference may be made also to [p. 208.]]
Thus then it became clear to Newton that the whole solar system depended on a central force emanating from the sun, and varying inversely with the square of the distance from him: for by that hypothesis all the laws of Kepler concerning these motions were completely accounted for; and, in fact, the laws necessitated the hypothesis and established it as a theory.
Similarly the satellites of Jupiter were controlled by a force emanating from Jupiter and varying according to the same law. And again our moon must be controlled by a force from the earth, decreasing with the distance according to the same law.
Grant this hypothetical attracting force pulling the planets towards the sun, pulling the moon towards the earth, and the whole mechanism of the solar system is beautifully explained.
If only one could be sure there was such a force! It was one thing to calculate out what the effects of such a force would be: it was another to be able to put one's finger upon it and say, this is the force that actually exists and is known to exist. We must picture him meditating in his garden on this want—an attractive force towards the earth.
If only such an attractive force pulling down bodies to the earth existed. An apple falls from a tree. Why, it does exist! There is gravitation, common gravity that makes bodies fall and gives them their weight.
Wanted, a force tending towards the centre of the earth. It is to hand!