Mehran- Welcome back.
1. The idea that the laws are the same everywhere goes back to
before Newton, but I don't know who was the first to make it explicit.
Even the idea that the laws should look the same to observers in
constant motion with respect to each other is old. It was first clearly
stated by Galileo, and put in more precise form by Descartes.
2. It's a principle that fits with many cosmological observations.
Generally speaking, we don't do proofs of basic principles. We just see
if they work.
It's also a practical feature of what physicists do when seeking
to understand what the laws of physics are. If a "law" is found to
describe nature in some places but not others, it loses its status as a
"law of nature" and at best becomes a "local approximation". Instead,
we'd try to find out what it is that makes stuff behave differently in
one place than another, and formulate laws that describe the behavior
everywhere under as common and general a description as possible. One
very famous example of this is Newton's realization that the same law
of gravity applied to objects falling on the Earth as makes the planets
go around the Sun and the Moon around the Earth.
3. Really on any space whose curvature is finite, one gets the same
limiting circumference/diameter ratio, our old pi, for small circles.
For other circles, the ratio can be smaller (say on a sphere) or bigger (say on a saddle).
And beware of the 'proof'. Are you sure that your assumptions (trigonometry) don't already contain the thing you want to prove?
I'm not sure how to answer the question about "why 3.14159...etc?"
The reason that you get the same vale for all sufficiently small
circles is that if the curvature is finite then on a small enough scale
everything looks like flat space. Then similar figures exist on
different scales. The ratio has to then come out to be some particular
number, and any rigorous argument giving how to obtain that number
(going back to Archimedes at least) is an equally good way of
explaining it.
We're not too keen on redefining pi based on the curvature of space
and how big your circle is. It's better to leave pi as the number it
is, and just say that the relationship of the circumference of a circle
to its diameter involves other numbers than pi in curved space. (pi has
lots of other uses than just talking about circles and spheres,
although I suppose they are all related in some way).
Mike W. and Tom
(published on 10/22/2007)
