Re: reason for "s = distance traveled"

[twayburn@juno.com (Thomas L Wayburn)]

On Sun, 24 Aug 1997 11:56:11 -0400 Robert Kern Curtis
<bcurtis@email.njin.net> writes:
> It is used because "d" is usually reserved for derivatives. Some texts
> use x, many use s, and a few (unfortunately) use d--but not those
> which are calculus based.
>
> Bob Curtis
Hi Bob,

 This does not answer Inge's excellent question:  *WHY* do we use
*lower-case ess* for arclength and *ds* for an infinitesimal element of
length in any metric, I think.  Its use for length and distance, etc. in
other contexts undoubtedly springs from this (nearly) universal fact of
mathematical life.  First came its use for arclength in differential
geometry and then came other similar uses.   We must find (1) who was the
first to use *lower-case ess* for arclength and (2) why he chose
*lower-case ess* rather than, for example, *lower-case ell*, which
actually was a unit of length (equal to about 45 inches) in ancient
times.  (I think I might have expressed myself a little better in this
note, Inge.)
 
Lower-case ell is the twelfth letter of the Latin alphabet as used in the
United States.   It is written 'l' or in upper case form 'L', but let us
spell out the names of letters as words from now on, as in "lower-case
ess" or "upper-case tee", which is written T.  Normally, quotes are not
needed.  Let's avoid quotes.  Sometimes italics are useful to distinguish
the symbol under discussion - as in *T* or *s*.   But lower-case ess is
the letter we shall be referring to and it is clear what we mean by those
words in this context - even without italics.   

Bye for now / Tom

P.S.  Note that ess is spelled with the first three letters in the word
*essential*, which means fundamental, intrinsic, basic, inherent, vital. 
   I find that interesting, but what can we do?  It might turn out to be
relevant, but we have to solve the problem to find out!   Remember,
lower-case ess is used as the parameter for the equations of a curve in
space and everything else to do with a curve in space - the essential
triad of v^1 = unit tangent vector, v^2 = principal normal vector, and
v^3 = binormal vector, from which we can deduce the torsion and  the
curvature - are all parameterized by arclength.  The principal triad of
unit vectors is obtained from the celebrated Frenet Equations, which are
the centerpiece in the theory of curves in space.  Everything worth
knowing can be found from the Frenet Equations - except *why lower-case
ess*.