Re: Does the Calculus rest on Euclid?

"Nathan" <ntspam2@xxxxxxxxxxxx> writes:

Lee Rudolph wrote:

Salas, Hille, and Etgen is what we use lately, but I'm sure their
treatment is the most widespread. It starts like this:
"An angle in the plane is generated by rotating a ray (half-line) about
its endpoint."
So that's "the *definition* of an angle." Now, Nathan may have been
meaning to say that nowadays "this" (HvA's assertion, or something
derived from it) "is the *definition* of" NOT an angle, but of the
measure of an angle. That is, indeed, with r = 1, the definition in
SHE. It's deplorable, but there we are: stuck.

Ah, I see. So you were just being pedantic.

Certainly I was being pedantic. (I prefer to think of it as
"being accurate", and under either name I think it's the right
way to talk about mathematics when one can't trust everyone in
one's intended audience--in this case, HvA--to be either willing,
able, or knowledgeable enough to correctly understand all the
meaningful but inaccurate shorthand, and so on, that don't mislead
more adept readers-of-mathematics.) But I wasn't *just* being
pedantic (nor was I being *just* pedantic).

Yes, the equation s = r * theta is nowadays used to define the
measure of an angle. As I said, it's not derived from anything.

And as I said, that's deplorable (because it rests on--even if it
is not "derived from"--the notion of arc length, which is far subtler
than that of area, from which it could equally well be derived; note
that HvA himself seems to find arc length trickier than area, and
who can blame him?).

....Although, to be fair, the notion of length of a *circular* arc
is much less subtle than the general notion.

Lee Rudolph
.

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