(snip)
(snip, I wrote)
There is an interesting book called "Feynman's lost lecture."
Feynman once gave a special lectures (not part of the regular
class series) on Newton's derivation of elliptical orbits.
While Newton may have done it himself using Calculus, his
explanation at the time used conic sections (a popular subject
at the time). Feynman rederived Newton's explanation in his
lecture. Unlike all the other lectures, there was no transcript,
only his (very abbreviated) notes. Goodstein, then, rederived
Feynman's rederivation of Newton and wrote a book about it.
Long before Copernicus' time, the orbits of the planets were known with
sufficient precision to rule out circles. You can approximate an ellipse
with arbitrarily high precision with a sufficiently large number of
epicycles, which is what they did. They had sufficiently precise
measurements to require several levels of epicycles.
I don't remember by now how well it was known, and when. My first
thought learning about epicycles was that the first one should
obviously be from the planets (approximately) orbiting the sun
instead of the earth. If the orbits were otherwise circular,
that it should have been obvious.
Except for (the non-planet now) pluto, though, most are darn close
to circles.
Also, to get even farther off subject, note that with all the
measurements that Kepler and such made, they had no idea of the
actual distance involved. All the measurements were relative.
The recent transit of Venus revived discussion on the way the
first measurement of the astronomical unit was done.
It's an important insight that the shape of the orbit is an
ellipse, and even more, to derive how the speed of the planet
varies as it moves around it's orbit. However, no real orbit
is exactly a Keplerian ellipse, any more than any real orbit
is an exact fit to a finite number of epicycles.
And if the study of conic sections wasn't popular at the time,
he might not have figured that one. If the relation between
radius and orbit was a power like 17/9, he might not have
figured that one, either.
For real orbits, you have to consider the fact that there's
more than two massive bodies involved, and that the bodies
are, in general, not exactly spherically symmetrical.
There are no exact analytic solutions for the more general
cases, though perturbation analysis can be used to analytically
derive low-order corrections.
Fortunately, it is close enough most of the time.
The high-accuracy orbital predictions are done numerically,
rather than analytically. Of course, the data available at
that time was insufficiently accurate to measure any of
those corrections, but the need for them was implicit in
the theory of Gravity.
Yes, and the important part of Newton's theory was the universal
part. That the same law worked not just for falling apples, but
moons and planets.
But I always liked the Galileo orgument on why acceleration should
be independent of mass, unlike was commonly believed at the time.
If you take two large masses and tie them together with a thin
thread (so it is now one mass twice as big) should they then fall
twice as fast? If you still aren't convinced, use a thinner thread.
That's the key difference between the epicycle approach and the
gravitational one. The theory of gravity can be used to determine
what the corrections to the simple approximations must be.
The epicycle approach is simply a curve fitting exercise - the
more epicycles you use, the better the fit to the curve, but
there's no underlying theory to predict the size, location,
or period of the next epicycle.
Well, there is also that the circle was the perfect shape, and
planets should have perfect orbits. Why would god give them any
less than perfect shape? In addition, note that Kepler thought
that the orbital radius should be determined by the radii of
inscribed and circumscribed spheres on the five regular polyhedra.
-- glen
