In a tragedy for science (if not for society in general)
whose outlines we are only now beginning to appreciate, after Maxwell's
death, two other 19th Century "mathematical physicists" -- Oliver Heaviside
and William Gibbs -- "streamlined" Maxwell's original equations down
to *four* simple (if woefully incomplete!) expressions. Because
Heaviside openly felt the quaternions were "an abomination" -- never
fully understanding the linkage between the critical scalar and vector
components in Maxwell's use of them to describe the potentials of empty
space ("apples and oranges," he termed them) -- he eliminated over *200
quaternions* from Maxwell's original theory in his attempted "simplification."

[Oliver Heaviside,
described by **Scientific American** (Sept. 1950) as "self-taught
and ... never connected with any university ... had [however] a remarkable
and inexplicable ability (which was possessed also by Newton and Laplace
...) to arrive at mathematical results of considerable complexity without
going through any conscious process of proof ..." According to other
observers, Heaviside actually felt that Maxwell's use of quaternions
and their description of the "potentials" of space was "... *mystical,
and should be murdered from the theory* ..." which -- by drastically
editing Maxwell's original work after the latter's untimely death (from
cancer), excising the scalar component of the quaternions and eliminating
the hyperspatial characteristics of the directional (vector) components
-- Oliver Heaviside effectively accomplished singlehanded.]

This means, of course, that the four surviving "classic"
Maxwell's Equations -- which appear in every electrical and physics
text the world over, as *the* underpinnings of *all* 20th
Century electrical and electromagnetic engineering, from radio to radar,
from television to computer science, if not inclusive of every "hard"
science from physics to chemistry to astrophysics that deals with electromagnetic
radiative processes -- never appeared in *any* original Maxwell'
paper or treatise! They are, in fact--

"Heaviside's equations!"

Lest anyone doubt this is the case, they merely have
to read a highly revealing paper on the subject by another renowned
British mathematical physicist of *this* century, Sir Edmund Whittaker,
titled simply "Oliver Heaviside" (*Bulletin of the Calcutta Mathematical
Society*, Vol. 20, 1928-29, p.202); or, another overview of Heaviside
by Paul J. Nahin, "Oliver Heaviside: Sage in Solitude" (IEEE Press,
New York, 1988, p.9, note 3.).

The end result was that physics lost its promising theoretical
beginnings to becoming truly "hyperdimensional" physics ... *over
a century ago* ... and all that that implies.

Georg Bernard Riemann mathematically initiated the 19th
Century scientific community (if not the rest of Victorian society)
into the "unsettling" idea of "hyperspace," on June 10, 1854. In a seminal
presentation made at the University of Gottinggen in Germany, Riemann
put forth the first mathematical description of the possibility of "higher,
unseen dimensions ..." under the deceptively simple title: "On the Hypotheses
Which Lie at the Foundation of Geometry."

Riemann's paper was a fundamental assault on the 2000-year
old assumptions of "Euclidian Geometry" -- the ordered, rectilinear
laws of "ordinary" three dimensional reality. In its place, Riemann
proposed a *four-dimensional* reality (of which our 3-D reality
was merely a "subset"), in which the geometric rules were radically
different, but also internally self-consistent. Even more radical: Riemann
proposed that the basic laws of nature in 3-space, the three mysterious
forces then known to physics -- electrostatics, magnetism and gravity
-- were all fundamentally *united* in 4-space, and merely "looked
different" because of the resulting "crumpled geometry" of our three-dimensional
reality ...

In terms of actual physics, Riemann was suggesting something
clearly revolutionary: a major break with Newton's "force creates action-at-a-distance"
theories of the time, which had been proposed to explain the "magical"
properties of magnetic and electrical attraction and repulsion, gravitationally-curved
motions of planets ... and falling apples, for over 200 years; in place
of Newton, Riemann was proposing that such "apparent forces'" are a
direct result of objects moving through 3-space "geometry" ...* distorted*
by the intruding geometry of "4-space!"

It is clear that Maxwell and other "giants" of 19th Century
physics (Kelvin, for one), as well as an entire contemporary generation
of 19th Century mathematicians (like Cayle, Tait, etc.) , took Riemann's
ideas very much to heart; Maxwell's original selection of *4-space
quaternions* as the mathematical operators for his force equations
and descriptions of electrical and magnetic interaction, clearly demonstrate
his belief in Riemann's approach; and, his surprising literary excursions
into poetry -- vividly extolling the implications of "higher-dimensional
realities" ... including musings on their relationship to the ultimate
origin of the* human soul* (above) -- emphatically confirm this
outlook.

So, how can modern "hyperdimensional physicists" -- like
Michio Kaku, at City College of the
City University of New York -- representative of an entirely new
generation of physical scientists now reexamining these century-old
implications of "hyperspatial geometries" for generating the basic laws
of Reality itself, almost casually claim:

"... In retrospect, Riemann's famous lecture was popularized to a
wide audience via* mystics, philosophers and artists, but did little
to further our understanding of nature* ... First, there was no
attempt to use hyperspace to simplify the laws of nature. Without
Riemann's original guiding principle -- that the laws of nature become
simple in higher dimensions -- scientists during this period were
groping in the dark. Riemann's seminal idea of using geometry -- that
is, crumpled hyperspace -- to explain the essence of a a force' was
forgotten during those years ... The mathematical apparatus developed
by Riemann became a province of pure mathematics, contrary to Riemann's
original intentions. *Without field theory*, you cannot make
any predictions with hyperspace [emphasis added]..."

-- M. Kaku, "Hyperspace"

[ Doubleday (Anchor Books): New York, 1994]

Kaku's statement belies the entire "modern" outlook on
19th Century physics, and leaves the distinct impression of an apparently
unconscious "bias" similar to Heaviside's, regarding Maxwell's actual
treatment of such matters; certainly, in completely ignoring Maxwell's
true discussion of the importance of the underlying *four-dimensional*
"scalar potentials" for creating such "fields." And remember: Heaviside
also thought of such "potentials" as ... "mystical ..."

The use of little-known Hamiltonian 4-space quaternions,
to represent the effect of "scalar potentials" on electric charges (as
opposed to Heaviside's *vectorial* descriptions of direct "electric
force fields") obviously have led to great confusion; because ... Maxwell's
"scalar potentials" are, of course, nothing short of exactly what Riemann
initially proposed--

*Quantifiable* "geometric spatial distortions" ...
the exact marriage of hyperspatial geometry and field theory that Kaku
and others mistakenly believe (because they're basing their analysis
on Heaviside's surviving vectorial version of Maxwell's original "Equations")
is totally *missing* from this greatest achievement of 19th Century
physics!

The major source of confusion surrounding Maxwell's actual
Theory, versus what Heaviside reduced it to, is its math -- a notation
system perhaps best described by H.J. Josephs ("The Heaviside Papers
found at Paignton in 1957," *Electromagnetic Theory by Oliver Heaviside*,
Including an account of Heaviside's unpublished notes for a fourth volume,
and with a forward by Sir Edmund Whittaker, Vol. III, Third Edition,
Chelsea Publishing Co., New York, 1971).

According to Josephs:

"Hamilton's algebra of quaternions, unlike Heaviside's
algebra of vectors, is not a mere abbreviated mode of expressing Cartesian
analysis, *but is an independent branch of mathematics with its
own rules of operation and its own special theorems. A quaternion
is, in fact, a generalized or hypercomplex number* ... [emphasis
added]"

And, you will remember, in 1897 Hathaway published a
paper specifically identifying these hypercomplex numbers as "... numbers
in *four-dimensional space*" (above). Thus, modern physics' apparent
ignorance of Maxwell's 19th Century success -- a mathematically-based,
*four-dimensional* "field-theory" -- would seem to originate from
a basic lack of knowledge of the true nature of Hamilton's quaternion
algebra itself!

[Apparently, unless a "hyperdimensional theory" is
narrowly expressed in terms of a separate technique Riemann himself
invented for his own N-dimensional mapping -- the so-called "metric
tensor" -- modern physicists don't seem to be able to recognize it
as a valid higher-dimensional model ... not even when it was written
in its own, specifically-designed, *four-dimensional* mathematical
notation! (Riemann's "metric tensor," BTW, is essentially a graphical
checkerboard composed, for a 4-space description, of 16 numbers defining,
for instance, field strength at each point in that four-dimensional
space. It is NOT written in quaternions.)

And, unless you track down an original 1873 copy of
Maxwell's "*Treatise*," there is no easy way to verify the existence
of Maxwell's "hyperdimensional" quaternion notation; for, by 1892,
the Third Edition incorporated a "correction" to Maxwell's original
use of "scalar potentials" (contributed by George Francis Fitzgerald
-- whom Heaviside heavily admired) -- thus removing a crucial distinction
between 4-space "geometric potential," and a 3-space "vector field,"
from all subsequent "Maxwellian theory." Which is why Kaku apparently
doesn't realize that Maxwell's original equations *were*, in
fact, the first *geometric 4-space field theory* ... expressed
in specific 4-space terms ... the language of quaternions!

Just another measure of Heaviside's effectiveness ...]

One of the difficulties of proposing a "higher dimension"
is that, inevitably, people (and scientists are people!), will ask:
"Ok, where is it? Where is the fourth dimension' ..?"

One of the most persistent objections to the 4-space
geometries of Riemann, Cayley, Tait ... and Maxwell, was that no experimental
proof of a "fourth dimension" was readily apparent; one of the more
easily understandable aspects of "higher dimensionality" was that, a
being from a "lower dimension" (a two-dimensional "Flatlander," for
instance) entering our "higher" three-dimensional reality, would appear
to vanish instantly from the lower-dimensional world (and, consequently,
appear just as suddenly in the higher dimension -- but geometrically
distorted.) When she returned to her own dimension, she would just as
"magically" reappear ...

Unfortunately (or fortunately, depending on your perspective
...) to the scientific mind, people in our dimension don't just "turn
a corner one day ... and promptly vanish into Riemann's fourth dimension.'"
While mathematically derivable and beautifully consistent, to "experimentalists"
(and all real science ultimately has to be based on verifiable, independently
repeatable experiments) there seemed no testable, physical *proof*
of "hyperdimensional physics."

Thus "hyperspace"-- as a potential solution to unifying
the major laws of physics -- after Maxwell's death, and the major rewriting
of his Theory, quietly disappeared ... not to resurface for almost *half
a century* ...

Until April of 1919.

At that time, a remarkable letter was delivered to one "Albert
Einstein." Written by an obscure mathematician at the University
of Konigsberg in Germany, Theodr Kaluza, the letter's first few lines
offered a startling solution (at least, to Einstein -- unknowing of
Maxwell's original quaternion equations) to one of physics' still most
intractable problems: the mathematical unification of his own theory
of gravity with Maxwell's theory of electromagnetic radiation ... via
introduction of a *fifth* dimension. (Because Einstein, in formulating
the General and Special Theory of Relativity in the intervening years
since Riemann, had already appropriated *time* as the "fourth dimension,"
Kaluza was forced to specify his additional spatial dimension as "the
fifth." In fact, this was the *same spatial dimension* as the 4-space
designations used by Maxwell and his colleagues in their models ...
over 50 years before.)

Despite its stunning (Einstein mulled over the paper's
implications for more than two years, before finally supporting its
scientific publication) mathematical success, in apparently -- finally
-- uniting "gravity" and "light," the same question, "OK, where is it?"
was asked of Kaluza as had been asked of Riemann, over 60 years before;
because, there was no overt experimental proof (for instance, people
and things up and "disappearing") of the physical existence of another
spatial dimension. To which Kaluza this time had a very clever answer:
he proposed that this "fourth dimension" -- unlike the other three we
are familiar with -- somehow had collapsed down to a tiny circle ...
"smaller than the smallest atom ..."

In 1926, another essentially unknown mathematician, Oskar Klein, was
investigating the peculiar implications of Kaluza's ideas in the context
of the newly-invented atomic theory of "quantum mechanics." [Klein was
a specialist in the truly arcane field of mathematical topology -- the
*higher dimensional surfaces* of objects; the twisted 3-D topology
of the 2-D surface of a "Klein
Bottle" is named specifically in his honor]. Quantum mechanics had
just been proposed a year or so before Klein's further topological investigation
of Kaluza's ideas, by Max Planck and many others rebelling against perceived
limitations of Maxwell's (remember, heavily sanitized by Gibbs and Heaviside)
classical Electromagnetic Theory. The "quantum mechanics " theory would
eventually become a highly successful (if bizarre, by common-sense standards)
non-geometric effort to describe interactions between "fundamental particles,"
exchanging "forces" through discrete "quantitized" particles and energy
in the sub-atomic world. Eventually, combining the two inquiries, Klein
theorized that, if it truly existed, Kaluza's new dimension likely had
somehow collapsed down to the "Planck length" itself -- supposedly the
smallest possible size allowed by these fundamental interactions. However,
that size was only about ... 10-33 cm long!

Thus, the main obstacle to experimental verification of the Kaluza-Klein
Theory (and the reason why people simply didn't "walk into the fourth
dimension"), was that quantum mechanics calculations affirmed that the
*only way* to physically probe such an infinitesimally tiny dimension
was with a new machine ... an "atom
smasher." There was only one small "technical" problem ...

The energy required would *exceed the output of all the power plants
on Earth* ... and then some!

Thus, the brief "blip" of new interest in "hyperdimensional physics"
-- the discussions of Kaluza-Klein among physicists and topologists
-- "dropped through the floor" by the 1930's. This occurred both because
of Klein's "proof" of the apparent *impossibility* of any direct
experimental verification of additional dimensions ... and because of
the dramatic revolution then sweeping the increasingly technological
world of Big Science--

The flood of "verifications"gushing forth from atom smashers
all around the world, feverishly engaged in probing the new area the
experimentalists apparently *could* verify: the multiplying populations
of "fundamental particles"spawned by the bizarre mathematical world
(even more bizarre than "N-dimensions") of Quantum Mechanics.

30 more years would pass ... before (almost by mathematical
"accident") in 1968, the current mainstream "flap" of renewed scientific
interest in "hyperspace" would be, like the legendary Phoenix, "magically"
reborn -- a theory now known as "Superstrings" ... in which fundamental
particles, and "fields," are viewed as hyperspace vibrations of infinitesimal,
*multi-dimensional strings* ... From those relatively inauspicious
beginnings, stretching across more than 60 years, the current focus
of scientific research papers on "hyperspace" -- from continued research
into updated versions of the old "Kaluza-Klein Theory"; to discussions
of the much newer "Supergravity" hyperspace unification model; to the
exotic "String Theory" itself -- has grown geometrically (over 5000
papers by 1994 alone, according to Michio Kaku -- see above). This much
attention to a subject involving realities you can't even *see*,
represents nothing short of a fundamental psychological revolution sweeping
across a major segment of the worldwide scientific community.

For most physicists currently interested in the problem,
the "Superstring" hyper-dimensional model has overwhelming advantages
over all its predecessors. Besides effectively unifying all the known
forces of the Universe ... from electromagnetism to the nuclear force
... in a literally beautiful "ultimate"picture of Reality, it also makes
a *specific prediction* about the total number of N-dimensions
that can form:

"Ten" (or "26," depending on the rotation of the "strings").

The bad news is: *they can't be tested either* ...

As all *ten dimensions* are curled up (in the model)
inside the same experimentally unreachable "Planck length" which spelled
the scientific demise of the original Kaluza-Klein ...

Impasse.

This, then is the current situation.

The "hottest" mainstream scientific theory to come along
in more than half a century, the next best thing to a "Theory of Everything"
(and seriously attempting to become precisely *that* ...), is not
only a Hyperdimensional Model of Reality ... it is *another* one
which, by its fundamental nature--

*Can't* scientifically be tested!

While a "hyperdimensional model" which can be tested
easily -- as this paper will unequivocally show -- for over a 100 years
has been systematically ignored.

Is it just us ... or is there something truly *wrong* with this
picture?