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Re: reason for "s = distance traveled"




On Mon, 25 Aug 1997 20:06:13 -0500 "JACK L. URETSKY (C) 1996; HEP DIV.,
ARGONNE NATIONAL LAB, ARGONNE, IL 60439" <JLU@hep.anl.gov> writes:
Hi all-
In response to:
**********************************************************************
At 19:42 8/24/97 EDT, Tom Wayburn wrote:
...
I would have wished to consult an English translation of
"Principia..." where I fancy Newton may possibly have used
the symbol; but I have no copy.

It would be particularly pleasing if he used s as a reasonable
initial letter for a term in his written language; something like
s - secta (path, method etc.) but this is wild speculation.

...
Leibnitz and Gauss are candidates for first use of
lower-case ess for arclength.
... The style of Newton is not entirely accessible, is it?

Tom

It is not, as Chandrasekhar found to his cost in the final
year of his life on the publication of his review of Newton's
major works.
*************************************************************
I think that Newton can safely be eliminated as the source.
The Principia was written for people who understood geometry, so
all curves are identified by their end points (as in, "the curve AB").
***I believe you (about Newton), even if I weren't convinced that ess
is the initial of a non-English word. It could be Latin, cf., Brian. -
TLW***
Sommerfeld (Lectures, Vol. I) does not seem to use it,
although
he introduces an "s" as a complex parameter in discussing Foucalt's
pendulum (Sec. 31). *****Mechanics is a subtopic of but not all of
differential geometry. No, I said that to get a rise out of the
professionals. They are simply different. Mechanics is likely to stick
with time since the curves are trajectories - often. - TLW****
I don't find Einstein using ds prior to his 1911 paper.
Minkowski,
in his 1908 lecture used c*dtau where tau stands for time interval.
It would have been natural to replace c*dtau by ds, s standing for
space. ***or 'space curve'?****
That's the best that I can do from home, I think. The
relativity
papers are in "The Principles of Relativity", by Einstein and Others.
Regards,
Jack
*************************************************************************
Dear Jack, I would imagine this is an old tradition rather than a modern
one. For example, J. J. Stoker used it to parametrize everything to do
with a space curve (and he took a delight in all things old - praising
Fritz John lavishly on one occasion for "using really nothing more than
calculus to work his wonders". [quoted approximately from a 28-year-old
memory - except that I remember almost every word he spoke and I hear it
in my mind's ear in his Ohio voice, " ex, wha, and zee", modified by a
European education]) Whereas Sternberg rarely uses it in *Differential
Geometry*; however, on p. 212 for Theorem 7.4, when he needs the
Euclidean metric on E^n, which is to be the argument of a function,
f*(ds), mapping it into the Riemannian metric, g = f*(ds), that exists on
every differentiable manifold (according to the theorem), he writes
(ds)^2 = (dx_1)^2 + (dx_2)^2 + ... + (dx_n)^2 in keeping with tradition.

I think that its suppression in the more advanced (abstract)
treatments speaks to its venerability rather than a recent birth in the
cradle of a Feynman or Einstein. I shall continue to look at Leibnitz
and Gauss.
I believe that in cases where lower-case tee is not explicitly displayed,
ess is there but simply not displayed. In the magnificent bouquet of
symbols needed to designate bundles, sprays, tubes, immersions,
coverings, isomorphisms, groups, partitions of unity, etc., the
parameter, s, can be taken for granted in many situations and needn't be
shown. (Now, I am indulging shamelessly in speculation concerning text
that I have read only cursorily and have not analyzed to check my wild
and dangerous conjecture. I'd still bet 25 cents that I'm right in many
cases.]

Who will rush to the library and look at the history of mathematics
section. I have not made it my business to collect history books - even
on the crucial and oft-neglected history of mathematics. (Contrariwise,
some didactic writers don't even present the material in the order of
discovery.)

I, speaking for myself, appreciate your interest and wisdom.

Regards / The Amateur