20 Quaternion Equations
James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field"
Royal Society Transactions, Vol. CLV, 1865, p 459 ); Orally read
Dec. 8, 1864. [ MS-Word.doc ]
Andre Waser : On the Notation of Maxwell's Field Equations [ PDF
The 1873 edition of A Treatise on Electricity
& Magnetism contains the 20 Quaternion Equations that later were
rewritten --- censored --- by Oliver Heaviside, et al.. These equations
reconcile relativity with modern quantum physics and help to explain "free
energy" and anti-gravity.
Table of Contents:
Volume 1: 1
~ 2 ~ 3
~ 4 ~ 5
~ 6 ~ 7
~ 8 ~9
~ 10 ~ 11
~ 12 ~ 13
Volume 2: 1
~ 2 ~ 3
~ 4 ~ 5
~ 6 ~ 7
~ 8 ~ 9
~ 10 ~ 11
~ 12 ~ 13
~ 14 ~ 15
~ 16 ~ 17
~ 18 ~ 19
Links to the complete copies in the Posner
Collection at Carnegie Mellon University:
The complete copies also are included
Rex Research website CD.
Maxwell's Quaternion Equations
Col. Tom Bearden
Maxwell's original theory was published as:
James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field",
Society Transactions, Vol. CLV, 1865, p 459. The paper was orally read
Dec. 8, 1864.
[ MS-Word.doc ]
It is also published in The Scientific Papers of James Clerk Maxwell,
2 vols. bound as one, edited by W. D. Niven, Dover, New York, 1952, Vol.
1, p. 526-597. Two errata are given on the unnumbered page prior to page
1 of Vol. 1.
In this paper Maxwell presented his seminal theory of electromagnetism,
containing 20 equations in 20 unknowns. His equations of the electromagnetic
field are given in Part III, General Equations of the Electromagnetic
Field, p. 554-564. On p. 561, he lists his 20 variables. On p. 562,
he summarizes the different subjects of the 20 equations, being three equations
each for magnetic force, electric currents, electromotive force, electric
elasticity, electric resistance, total currents; and one equation each
for free electricity and continuity. In the paper, Maxwell adopts the approach
of first arriving at the laws of induction and then deducing the mechanical
attractions and repulsions.
A copy of the original Maxwell paper can easily be obtained for about
$15 from Amazon etc. It is:
James Clerk Maxwell, The Dynamical Theory of the Electromagnetic
Field, edited by Thomas F. Torrance, Wipf and Stock Publishers, Eugene,
Oregon, 1996. This booklet, which sells for about $15, contains Maxwell's
original 1865 dynamical theory paper and some additional commentaries.
Here's what Barrett --- a nationally known electrodynamicist and
one of the co-founders of ultrawideband radar --- has to say about Maxwell's
"In the case of electromagnetism, the theory was first simplified
before being frozen. Maxwell expressed electromagnetism in the algebra
of quaternions and made the electromagnetic potential the centerpiece of
his theory. In 1881 Heaviside replaced the electromagnetic potential field
by force fields as the centerpiece of electromagnetic theory. According
to him, the electromagnetic potential field was arbitrary and needed to
be "assassinated" (sic). A few years later there was a great debate between
Heaviside and Tate about the relative merits of vector analysis and quaternions.
The result was the realization that there was no need for the greater physical
insights provided by quaternions if the theory was purely local, and vector
analysis became commonplace.
The vast applications of electromagnetic theory since then were made
using vector analysis. Although generations of very effective students
were trained using vector analysis, more might be learned physically by
returning, if not to quaternions, to other mathematical formulations in
certain well-defined circumstances. As examples, since the time when the
theoretical design of electromagnetism was frozen, gauge theory has been
invented and brought to maturity and topology and geometry have been introduced
to field theory. Although most persons view their subject matter through
the filter of the mathematical tools in which they are trained, the best
mathematical techniques for a specific analysis depend upon the best match
between the algebraic logic and the underpinning physical dynamics of a
theoretical system." [Terence W. Barrett and Dale M. Grimes, Preface, p.
vii-viii, in Advanced Electromagnetism: Foundations, Theory and
Applications, Terence W. Barrett and Dale M. Grimes (eds.), World Scientific,
Maxwell died in 1879 of stomach cancer.
In the 1880s, several scientists --- Heaviside, Gibbs, Hertz etc.
--- strongly assaulted the Maxwellian theory and dramatically reduced it,
creating vector algebra in the process. Then circa 1892 Lorentz arbitrarily
symmetrized the already seriously constrained Heaviside-Maxwell equations,
just to get simpler equations easier to solve algebraically, and thus to
dramatically reduce the need for numerical methods (which were a "real
bear" before the computer). But that symmetrization also arbitrarily discarded
all asymmetrical Maxwellian systems - the very ones of interest to us today
if we are seriously interested in usable EM energy from the vacuum.
So anyone seriously interested in potential systems that accept and
use additional EM energy from the vacuum, must first violate the Lorentz
symmetry condition, else all his efforts are doomed to failure a priori.
We point out that quaternion algebra has a higher group symmetry
than either vector algebra or tensor algebra, and hence it reveals much
more EM phenomenology and dynamics than does EM in vector or tensor form.
Today, the tremendously crippled Maxwell-Heaviside equations ---
symmetrized by Lorentz --- are taught in all our universities in the electrical
engineering (EE) department. Note that the EE professors still dutifully
symmetrize the equations, following Lorentz, and thus they continue to
arbitrarily discard all asymmetrical Maxwellian systems. Hence none of
them has the foggiest notion of how to go about developing an "energy from
the vacuum" system, which is asymmetrical a priori.
The resulting classical electromagnetics and electrical engineering
(CEM/EE) model taught in all our university EE departments also contains
very serious falsities. Most of modern physics, such as special and general
relativity, quantum field theory, etc., has been developed since the 1880s
and 1890s fixating of the symmetrized Maxwell-Heaviside equations. A paper
gathering together a listing these serious flaws and giving proper citations,
is T. E. Bearden, "Errors and Omissions in the CEM/EE Model," available
for free downloading at:
This paper also shows a magnetic Wankel engine (suppressed from the
world market) that can be built by any electrical engineering department
or physics department, and then tested at COP>1.0 to one's heart's content.
The magnetic Wankel system is also easily close-looped for self-powering
(where all its input energy is freely furnished by the vacuum, and the
operator need furnish none of the input energy at all --- thus providing
fuel free, continuous use of the energy from the vacuum, at will.
In the hard physics literature, rigorous proof that eliminating the
arbitrary Lorentz condition provides systems having free additional energy
currents from the vacuum is given by M. W. Evans et al., "Classical Electrodynamics
without the Lorentz Condition: Extracting Energy from the Vacuum," Physica
Scripta, Vol. 61, 2000, p. 513-517. Evans' own O(3) model is very advanced,
and it also directly specifies mechanisms for an EM system receiving and
using excess energy freely from the vacuum.
Fortunately, today some scientists have turned again to higher group
symmetry algebras in which EM is expressed. These higher group symmetry
electrodynamics theories then show far more EM phenomenology than the standard
CEM/EE model used in electrical power engineering.
Anyway, that gives you a brief overview of the Maxwell theory, and
the rather sharp curtailment of it that has become the accepted but very
crippled model for electrical engineering. Specifically, it is that crippled
model and its continued propagation and use that is directly responsible
for the increasing energy crisis worldwide, and our dependence on conventional
We do point out that the original Maxwell quaternion and quaternion-like
theory of 1865 also contained errors, by the physics that has been learned
since then. One of those errors was Maxwell's assumption of the material
ether, an ether which was falsified experimentally in 1887 after Maxwell
was already dead. But the present CEM/EE model still assumes that same
old material ether, more than a century later.
Also, after Maxwell published the first edition of his famous "Treatise.",
not much happened. He was soundly criticized for using the quaternion approach,
and even his own editor chastised him rather unmercifully for it. His attachment
to the potentials as primary was also roundly criticized, since almost
all theorists of the day believed that the potentials were simply mathematical
conveniences having no physical reality whatsoever. To them, the force
fields were the only physical reality in Maxwell's theory. Today, of course,
we know in the quantum theory that it is the potentials that are primary,
and the fields are derived from changes in the potentials.
The history of Maxwell's famous treatise is as follows: The publications
are James Clerk Maxwell, A Treatise on Electricity and Magnetism,
Oxford University Press, Oxford, 1873, Second Edition 1881 (Maxwell was
already dead), Third Edition, Volumes 1 and 2, 1891. Foreword to the second
edition was by Niven, who finished the work as Maxwell had dramatically
rewritten the first nine chapters, much new matter added and the former
contents rearranged and simplified. Maxwell died before finishing the rest
of the second edition. The rest of the second edition is therefore largely
a reprint from the first edition. The third edition edited by J. J. Thomson
was published in 1892, by Oxford University Press, and later was published
unabridged, Dover Publications, New York, 1954. J. J. Thomson finished
the publication of the third edition, and wrote a "Supplementary Volume"
with his notes. A summary of Maxwell's equations is given in Vol. II, Chapter
IX of the third edition. However, Maxwell had gone (in his second edition)
to some pains to reduce the quaternion expressions himself, and not require
the students to know the calculus of quaternions (so stated on p. 257).
We note that Maxwell did not finish the second edition, but died before
that. He actually had no hand at all in the third edition as to any further
changes. The Second edition (unfinished by Maxwell) was later finished
by Niven by simply adding the remaining material from the previous first
edition approved by Maxwell to that part that Maxwell had revised. The
printing of the first nine chapters of the third edition was already underway
when J. J. Thomson was assigned to finish the editing of the manuscript.
Indeed, as an example of a major error in the present CEM/EE model,
we know today that matter is a component of force, and therefore the EM
force fields prescribed in matter-free space by Maxwell and his followers
(and by all our electrical engineering departments today), do not exist.
The EM field in massless space is force-free, and is a "condition of space"
itself, as pointed out by Feynman in his three volumes of sophomore physics.
Specifically, speaking of the electric field Feynman states:
"...the existence of the positive charge, in some sense, distorts,
or creates a "condition" in space, so that when we put the negative charge
in, it feels a force. This potentiality for producing a force is called
an electric field." [Richard P. Feynman, Robert B. Leighton, and Matthew
Sands, The Feynman Lectures on Physics, Addison-Wesley, Reading,
MA, Vol. 1, 1964, p. 2-4].
He further states:
"We may think of E(x, y, z, t) and B(x, y, z, t) as giving the forces
that would be experienced at the time t by a charge located at (x, y, z),
with the condition that placing the charge there did not disturb the positions
or motion of all the other charges responsible for the fields." [ibid,
vol. II, p. 1-3.]
But the CEM/EE texts still teach that old force field in empty space.
However, Jackson --- a superb classical electrodynamicist of international
reknown --- at least points out that this dramatic error in the model is
just ignored. Jackson states:
"Most classical electrodynamicists continue to adhere to the notion
that the EM force field exists as such in the vacuum, but do admit that
physically measurable quantities such as force somehow involve the product
of charge and field." [J. D. Jackson, Classical Electrodynamics,
Second Edition, Wiley, 1975, p. 249].
Jackson does admit it and point out that this logical problem is
just ignored, for which he is to be highly commended. Most textbooks simply
do not even discuss it.
So at his death in 1879, Maxwell had already laboriously simplified
some 80% of his "Treatise" himself, to comply with the severe demands of
the publisher. The second edition of his book thus has the first 80% considerably
changed by Maxwell himself. The third edition contained the same theory
as the second edition essentially, but just with additional commentary.
It is this third edition that is widely available and usually referred
to as "Maxwell's theory".
Today, there is still a widespread belief that the third edition
represents Maxwell's original EM work and theory, in pristine form just
as created originally by Maxwell.