## Jj

Fig. 8.5. The consistency of the inverse square law and elliptical orbits.

DC is parallel to BS. Now, since the area of SAD is twice the area of SAB, it follows that the triangles SAB and SBD have the same area. Also, since DC is parallel to BS, the areas of SBC and SBD are the same. Hence, area SAB = area SBC. Finally, the intervals of time are made shorter and shorter, a process Newton described as follows.

Now let the number of triangles be increased, and their width decreased indefinitely, and their ultimate perimeter [ABC] will... be a curved line; and thus the centripetal force by which the body is continually drawn back from the tangent of this curve will act uninterruptedly, while any areas described, [SABS and SACS], which are always proportional to the times of descriptions, will be proportional to those in this case.

Newton also proved the converse result: that any body moving according to the area law was under the influence of a net force that remained directed toward a fixed point. This general relation between motion subject to Kepler's second law and central forces was the basis of all Newton's subsequent work on orbital motion.

The logical next step was to investigate the motion of a body in an elliptical path under the action of a force directed toward one of the focuses of its orbit, and this is precisely what Newton did. His method37 is illustrated in Figure 8.5, in which we can think of S as representing the Sun, at one focus of the elliptical orbit, and P a planet. If the planet were moving inertially, it would do so along the tangent of the ellipse to R in a short time At, but the force acting along PS results in the planet actually moving to Q, where RQ is parallel to PS. Newton's argument then proceeds as follows. The magnitude of the force F is proportional to | RQ| and for a given force (from Galileo's results on uniformly accelerated motion) |RQ| is proportional to (At)2. Thus,

36 Newton, Principia, Book I, Proposition 1.

37 Described in Herivel (1965b).

Next, we can invoke the area law which says that At is proportional to the sector SP Q or, since At is small, to the area of the triangle SPQ. Combining this with the above proportion gives 38

where T is the foot of the perpendicular from Q to SP. So far, the argument is independent of the actual shape of the orbit. Newton then showed that for an ellipse, in the limit as Q approaches P, the ratio |RQ|/| QT |2 tends to 1/|LL'| which is constant and, hence, that F a 1 /1 SP |2. In this way, he managed to demonstrate that elliptical orbits with focal attraction were consistent with an inverse square law.

Another hint at the universality of gravitation came from analysis of cometary orbits. The appearance of some particularly bright comets during the period 1680-82 excited a great deal of interest in England, as had the comets of 1664 before them. The generally accepted view was that cometary motion was distinct from planetary motion. Wallis, for example, expounded Horrocks' cometary theory (straight lines modified by the magnetic action of the Sun), whereas Wren followed Kepler in believing that comets travelled in straight

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