Well, it seems that all the time that you have spent
running around the world giving lectures, signing books
and chasing taxi cabs has gotten you in a lot of trouble,
you've lost your faculty position at the Grand Academy of
Lagado. Never mind, you always seem to land on your feet.
This time you net a nice job as a life guard at Malibu
beach. Of course, being a life guard has its disadvantages.
Occasionally you have to actually save a drowning swimmer.
The essential element here is time. You want to get to the
swimmer as fast as possible. However, recognizing that
your running speed v_r is considerably faster than your
swimming speed v_s, the best path to take to the swimmer
is not particularly obvious. The figure below gives a birds
eye view of the beach illustrating two possible strategies:
L <--- Life Guard
@
@#
@ #
@ # BEACH
@ #
@ #
@ #
@ # B
@ # |
@ # |
@ #|
@ |#
@ | #
@ | #
__________________@______A______#______________
@ | #
@ | #
@ | #
@ C #
@ #
@ #
@ #
WATER @ #
@ #
@ #
@ #
@ #
@#
Swimmer ----> S
The path @ @ @ shows the path of least distance while
the path # # # is path that minimizes the time spent
at the smaller speed v_s. Unfortunately, this is the
path of maximum total length (for all sensible paths
that is). Which to choose? Bear in mind that one must
decide instantly since time is the most important
factor.
It turns out that the optimal path is somewhere in
between the two shown in the figure. One goes in a
straight line from L to A at v_r, then in a straight
line from A to S at v_s. This is an easy optimal
problem to solve and is best described in terms of
the angles that the two straight lines LA and AS
make with the vertical, i.e. angle LAB = alpha_r
and angle CAS = alpha_s. The optimal path is
described by the relation:
sin(alpha_s) v_s
------------ = --- (1)
sin(alpha_r) v_r
To further appreciate the problem for which this is
an analogy, assume that you have just arrived at the
beach when the emergency arose and that you have no
idea what the water conditions are, thus you do not
know in advance what your swimming speed v_s is. But
the appropriate point A depends on v_s. How can you
possibly decide on an optimal path?
OK, I think a lot of people have recognized equation (1)
as Snell's law giving the ratio of the angle of refraction
to the angle of incidence in terms of the ratio of the
velocity of light in two dissimilar media. Perhaps not
everyone realized that light travels along a path that
minimizes the travel time or appreciated the implications
of this observation.
It was Fermat who first arrived at Snell's law by way of
the principle of least time:
<"The reward of my work has been most extraordinary, most
<unexpected, and the most fortunate that I have ever
<obtained. ... I was so surprised by a happening that was
<so little expected that I only recovered from my astonishment
<with difficulty." -- Fermat
But Fermat was vehemently opposed on account of the obvious
teleological implications of his principle. A prominent
Cartesian, Clerselier, wrote to Fermat that his principle
was:
<"...a principle which is moral and in no way physical;
<which is not, and which cannot be, the cause of any
<effect in Nature. ... That path, which you reckon the
<shortest because it is the quickest, is only a path of
<error and bewilderment, which Nature in no way follows
<and cannot intend to follow ..." Clerselier
I have a number of things in mind with the examples
I've given (I hope I haven't been too boring :).
One is to counter what seems to be a common view
among many that teleological explanations were
common and acceptable before Darwin. The Fermat
controversy occurred in the early 1610's, more
than 250 years before the Origin was published.
Brian Harper
Associate Professor
Applied Mechanics
The Ohio State University
"It is not certain that all is uncertain,
to the glory of skepticism." -- Pascal