Davis carried out a reworking of chapter 32 of Kepler's Astronomia Nova in order to explore the
so called "distance law." Davis finds there is no foundation for such
a law there or subsequently. Instead, Davis shows that Kepler's innovative
approached allowed for a consideration of the component of the velocity which
is precisely perpendicular to the Sun-planet line which is the exact equivalent
of the area law according to modern standards. Davis holds that no other
interpretation would have been compatible with Kepler's Aristotelian
principles.
What I find interesting in the work for this chapter of
Kepler is the treatment of time and how it differs from that of Newton. For
Newtonian time, it is easy to arrange it to vary uniformly, but Keplerian form
has it straightforward to be represented geometrically. I see this as being a
very key feature for making Kepler's laws appear. What is really nice is that
it is possible to transition between both styles.
It is from this treatment of the times and ellipses that it
is possible to derive the area law, and it is what Davis does. It flows cleanly
in a way that any student that is versed in calculus II would be able to
follow.
For students and to help them with this, I think it is
important to focus on where Davis talks about the exactitude and usage of
approximations. Choosing when to be exact and when to be approximate is very
important for a physicist trying to model the world. To me, this screams of a
time when intuition needs to be properly trained so that it is properly
employed; an understanding of something along the lines of when it is viable to
use the small angle approximation or to keep it exact. Based on the problem at
hand, choosing one will simplify it while the other will bog down any
calculations and deny an analytical solution.
Further touching on Davis's astonishment that the existence
of a so called "distance law," it is interesting that such persisted.
This reinforces my feeling that understanding exactly what an article or
textbook says is very important for the budding physicist because gaining a
false notion can be detrimental to the development of their tailored intuition.
Davis*, A. E. L. (1992), Kepler's ‘Distance Law’ - Myth not Reality. Centaurus, 35: 103–120. doi:10.1111/j.1600-0498.1992.tb00872.x
