Fig. 70.
No. 1. Kepler's second law (equable description of areas) proves that each planet is acted on by a force directed towards the sun as a centre of force.
The equable description of areas about a centre of force has already been fully, though briefly, established. ([p. 175.]) It is undoubtedly of fundamental importance, and is the earliest instance of the serious discussion of central forces, i.e. of forces directed always to a fixed centre.
We may put it afresh thus:—OA has been the motion of a particle in a unit of time; at A it receives a knock towards C, whereby in the next unit it travels along AD instead of AB. Now the area of the triangle CAD, swept out by the radius vector in unit time, is ½bh; h being the perpendicular height of the triangle from the base AC. ([Fig. 70.]) Now the blow at A, being along the base, has no effect upon h; and consequently the area remains just what it would have been without the blow. A blow directed to any point other than C would at once alter the area of the triangle.
One interesting deduction may at once be drawn. If gravity were a radiant force emitted from the sun with a velocity like that of light, the moving planet would encounter it at a certain apparent angle (aberration), and the force experienced would come from a point a little in advance of the sun. The rate of description of areas would thus tend to increase; whereas in reality it is constant. Hence the force of gravity, if it travel at all, does so with a speed far greater than that of light. It appears to be practically instantaneous. (Cf. "Modern Views of Electricity," § 126, end of chap. xii.) Again, anything like a retarding effect of the medium through which the planets move would constitute a tangential force, entirely un-directed towards the sun. Hence no such frictional or retarding force can appreciably exist. It is, however, conceivable that both these effects might occur and just neutralize each other. The neutralization is unlikely to be exact for all the planets; and the fact is, that no trace of either effect has as yet been discovered. ([See also p. 176.])
The planets are, however, subject to forces not directed towards the sun, viz. their attractions for each other; and these perturbing forces do produce a slight discrepancy from Kepler's second law, but a discrepancy which is completely subject to calculation.
No. 2. Kepler's first law proves that this central force diminishes in the same proportion as the square of the distance increases.