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Four Quadrant Representation of Electricity
by Eric P. Dollard
Published by A&P Electronic Media
Liberty Lake, Washington
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Cover Layout:

Aaron Murakami A&P Electronic Media Liberty Lake, Washington

Interior Images by: Eric Dollard & Other Sources Internet sources, used with attribution.

Transcribed from the author's notebooks by Jeffery Moe http://myplenum.com Final edit and layout by Peter Lindemann

Copyright 2013 A & P Electronic Media
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Book Version
Original Version 1.0 - Release Date November 7, 2013
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Table of Contents

Introduction . . . . . . . . page 10

(1.1) Objectives

page 10

(1.2) Subject Division

page 11

(1.3) Misnomers

page 12

(1.4) Origin of Subject

page 13

(1.5) Initiation

page 14

Section One . . . . . . .
Metaphysical Aspects of Quadrapolar Versor Algebra
Part I Pythagoras of Samos . . . .
[1] Pythagoras (1.1) Introduction (1.2) Pythagorean Theorem (1.3) Island of Samos (1.4) Origin of Mathematics (1.5) The Pythagoreans
[2] Ancient Concepts (2.1) Chaldean Number System (2.2) Geometric Manifestation

. page 15
. page 16
page 17 page 17 page 17 page 18 page 29 page 31 page 33 page 34 page 36

Part II Van Tassel of Giant Rock . . . . . page 42

[1] Van Tassel and His Integratron

page 43

(1.1) Four Quadrant Theory

page 43

(1.2) The Integratron

page 46

(1.3) Integratron Operation

page 47

(1.4) Epilog

page 61

Section Two . . . . . . .
Mathematical Aspects of Quadrapolar Versor Algebra
Part I The Knowledge of Electrical Phenomena .
[1] Alternating Current Machines (1.1) Rotating A. C. Apparatus (1.2) Arithmetic Expression

. page 63
. page 64
page 64 page 65 page 65

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(1.3) Algebraic Expression (1.4) Higher Order Expression [2] The Long Line Problem (2.1) Electric Transmission (2.2) Electric Wave Theory [3] Space Algebra and Electricity (3.1) Cartesian and Quaternion Systems (3.2) Michael Faraday (3.3) James Clerk Maxwell (3.4) Oliver Heaviside (3.5) Quaternions and Maxwell (3.6) Lost in Space (3.7) Engineering Reality

page 65 page 66 page 68 page 68 page 69 page 71 page 72 page 76 page 80 page 85 page 93 page 95 page 97

Part II Symbolic Representation of Cyclic Phenomena . page 105

[1] Primordial Symbolic Representation

page 106

(1.1) Nature

page 107

(1.2) Quadrapolar Geometry

page 108

(1.3) The Lunar Cycle

page 109

[2] Lunar Versors

page 110

(2.1) The Four Lunar Phases

page 110

(2.2) Exponents of the Phases

page 111

[3] Quadrapolar Relations

page 118

(3.1) Quadrapolar Arithmetic

page 118

(3.2) Real and Imaginary Lines

page 119

Part III The Versor Operator j . . . . . page 123

[1] Rotational Operation

page 124

(1.1) The Unit Rotation

page 125

(1.2) Bipolar Operations

page 128

[2] Real and Imaginary Operators

page 129

(2.1) Arithmetic Operations

page 129

(2.2) Alternating Current Analog

page 131

(2.3) Four Versor Operators

page 132

(2.4) The Rotational Operator

page 137

(2.5) Versor Operations

page 137

(2.6) The Bipolar Operator

page 142

(2.7) The Alternating Operator

page 143

(2.8) Real and Reactive Power

page 144

[3] The Quadrapolar Versor Operator

page 146

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(3.1) Bipolar Operation (3.2) Fourth Order Expression [4] Duo-Binary Representation (4.1) Bipolar Real and Bipolar Imaginary (4.2) The Exponential Series (4.3) Roots of a Unit (4.4) Lunar Representation

page 146 page 146 page 150 page 150 page 150 page 151 page 151

Part IV Versor Relations Between Sun, Moon, and Earth .
[1] Versor Attributes of Light and Dark (1.1) Quadrapolar Aspects (1.2) Positions of the Sun and Moon (1.3) Co-Axial Systems (1.4) Quadrature Relations (1.5) Ratios of Light and Dark
[2] Solar-Lunar Co-Axial Versors (2.1) The Pair of Cycles (2.2) Solar-Lunar Cycles (2.3) Counter Rotational Cycles (2.4) Alternating Current Analog

page 152
page 153 page 153 page 154 page 156 page 156 page 159 page 160 page 160 page 161 page 163 page 164

Part V

The Lunar Phases
[1] Light and Dark on the Lunar Face (1.1) Proportion of Light and Dark (1.2) Positions of the Sun, Moon, and Earth
[2] Electric Analogs (2.1) Lunar and Electric Phases (2.2) Proportion of Energy (2.3) Cycle of Light and Dark
[3] Co-Axial Versor Form (3.1) Energy Relations

page 165
page 166 page 166 page 167 page 170 page 173 page 173 page 173 page 175 page 175

Part VI Ratios, Projections, and the Sine and Cosine Functions
[1] Projections of the Cycle (1.1) Image of Light and Dark (1.2) Lunar shadow (1.3) Sine and Cosine (1.4) Shadows (1.5) Co-Operation of the Sine and Cosine Functions

page 177
page 178 page 178 page 179 page 181 page 184 page 184

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Section Three . . . . . . .
Practical Aspects of Quadrapolar Versor Algebra

Part I

The Method of Descartes . . . .
[1] The Cartesian Co-ordinate System (1.1) Sine and Cosine (1.2) Symmetrical Co-ordinate Systems (1.3) Latitude and Longitude (1.4) Co-ordinate Derived Positions (1.5) Pythagorean Ratios
[2] Ratios Derived from the Unit Radius (2.1) Positions in Terms of Co-ordinates (2.2) Radius of Unit Value (2.3) Electrical Analog (2.4) Angular and Rectangular Representation

Part II The Method of Steinmetz . . . .
[1] Application of Versor Algebra (1.1) Symbolic Representation (1.2) Steinmetz' Method (1.3) Promethean Myth
[2] Steinmetz the Wizard (2.1) Experiments (2.2) The Lizard (2.3) His Origin (2.4) The Educator

Part III The Rotary Field of Nikola Tesla . .
[1] The four Quadrant Archetype (1.1) Quadrantal Natural Laws (1.2) Quadrantal Attributes and Archforms
[2] The Synchronous Machine (2.1) The Invention (2.2) Real and Imaginary Currents (2.3) Rotating Transformer (2.4) Synchronous Motor (2.5) Space-Time Phase

. page 186
. page 187
page 188 page 188 page 192 page 196 page 199 page 200 page 203 page 203 page 204 page 233 page 234
. page 235
page 236 page 236 page 236 page 238 page 239 page 239 page 240 page 241 page 243
. page 245
page 246 page 246 page 247 page 253 page 253 page 256 page 257 page 258 page 259

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[3] Motor Operation (3.1) Basic Motor layout (3.2) The Four Motor Poles (3.3) Sine and Cosine Functions (3.4) Rotating Magnetic Pole
[4] The Vision of Nikola Tesla (4.1) Goethean Vision (4.2) Rotating Pole (4.3) Four Phase A. C. (4.4) Epilog

page 261 page 261 page 262 page 263 page 265 page 268 page 268 page 270 page 270 page 271

Appendices . . . . . . . . page 273

[1] In the Beginning, Versors

page 274

[2] Four Quadrant Energy Exchange

page 286

[3] Quadrapolar Resonant Circuit

page 320

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Introduction
1.1 Objectives The objective of the following writing is to provide an introduction of Versor Algebra and its application to the Four Phase Cycle. This will be directed to the common understanding rather than that of the engineer or mathematician.
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Versor algebra is well suited for the inner workings of the Alternating Current Cycle. The cycle is the fundamental dimension of alternating current, given as Per Second. This is a dimension of change. The cycle and its phases are best represented by versor symbolism. This symbolic representation in quadrapolar form is the fundamental geometric algebra in natural and electrical cycles of revolution.
What will in essence be represented here is a doorway into “Another Dimension” to use the words of physics. The aim is to cross the border from the “Real” into the “Imaginary”, to cross over into the “Other Side”. The “Red Pill” to take us down the “Rabbit Hole” will be the Square Root of Negative One.
1.2 Subject Division This subject will be presented in three sections:
I) Metaphysical II) Mathematical II) Practical
The first section will relate this subject to the purely metaphysical. Such concepts have become foreign to the modern mind, to be regarded as delusions or works of fantasy. Moreover, the metaphysical is rejected with a singular fervor by modernists. However, such a form of understanding serves as the foundation of music, mathematics, and the creative process itself.
The second section is strictly mathematical, but no attempt is made to develop practical formulae. The aim of the second section is to develop an archetypal view of the cyclic process, this given in terms of the Lunar Cycle. The concept of a Quadrantal Versor is developed in this section.
The third section is directed towards a more practical understanding of the Quadrantal Versor form of the representation and its application to Four Quadrant Geometries. The specific application in the third section is the
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symbolic algebra of Carl Proteus Steinmetz, and the synchronous machine of Nikola Tesla. The work of these two pioneers is the progenesis of today’s alternating current technology.
1.3 Misnomers Three misnomers exist in any discussion or presentation of electric phenomena. First is the notion of Current Flow, second is that of an Alternating Current, and the third is the use of the term Imaginary. In order not to confuse matters, these terms will continue to be used in the following writings. This will however be done with qualification, as follows:
I) Current Flow: The pre-historic concept of Electric Current Flow is a persistent holdover from the ideas prevalent before the discoveries of Faraday and Maxwell. This current in reality is no more than a mathematical fiction, derived from the notions of physics. This misconception works great harm into the functional understanding of electricity.
II) Alternating Current: The machines of Nicola Tesla are not based upon Alternating Current, but upon a Rotating Continuous Current. This is the novelty of Tesla’s discovery. Alternating Current is but a shadow, or projection of the Rotating Electric Wave. A.C. is a Direct Current undergoing cyclic reversals. With an Alternating Current rotation is not possible, a situation facing motor builders prior to the discoveries of Nikola Tesla.
III) Imaginary: The term Imaginary is a holdover from the view of mathematics that the Square Root of Negative One is an algebraic impossibility. However, Imaginary Currents can act with as much force as Real Currents, but the laws of behavior can be very much different.
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1.4 Origin of Subject The origin of the Versor System of representation as presented in the following writings begins with Dr. Alexander MacFarlane, University of Texas in Austin. His two groundbreaking papers on the subject of Versor Algebra are:
I) The Imaginary of Algebra II) The Principles of the Algebra of Physics
Three subsequent papers were written by me, as an extension of MacFarlane’s writings:
I) Symbolic Representation Of The Alternating Electric Wave II) Symbolic Representation Of The Generalized Electric Wave In The
Time Domain III) Symbolic Representation Of Induction In Space
These three papers are an adaption of MacFarlane’s view through the methods of Steinmetz and Kennelly. Carl Steinmetz developed his A.C. symbolic geometry for General Electric, and Arthur Kennelly developed his D.C. symbolic geometry for Edison Electric. Space algebra found no further development after the initial work of Dr. MacFarlane. Induction in Space is a preliminary attempt to overcome the crippling limitation of Maxwell’s Quaternions and Heaviside’s vector calculus, both useless in transformer analysis.
1.5 Initiation The writings that now follow are an attempt towards a reduction of the subject to a more common level of understanding. As a mathematical subject, it is by its very nature an abstract one, it lives in unseen dimensions of the unknowable, as does electricity. Music, mathematics and electricity exist in this type of co-referencing triad.
This abstract character will be difficult for the layman to grasp, it may be impossible for some. However, the subject is here presented in a
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methodical and repetitive form, this given as new names for known elementary ideas. For those who can hold on, the image in the mind of Nikola Tesla, which revealed a rotating magnetic field, will give rise to itself again in the mind of the reader. The reader must continue on in the “Spirit of a bold sense of curiosity for the adventure ahead”
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Section One The Metaphysical Aspects of Quadrapolar Versor Algebra
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(I) Pythagoras of Samos
Figure 2
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[1] Pythagoras 1.1 Introduction Any endeavor of this nature must begin with Pythagoras of Samos. Pythagoras is the grandfather of mathematics and music. His discovery of most significance to the engineer is the Pythagorean Theorem of Squares. This theorem plays an important role in Alternating Current Theory. 1.2 Theorem
PYTHAGOREAN THEOREM OF SQUARES Figure 3
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The theorem states a relation which allows for the addition of Real & Imaginary quantities1, such as current or E.M.F. Shown is a Real Length, b, it is added to Imaginary Length, a. The sum of the Real Length and the Imaginary Length, is given by the resultant, or Complex Length, c. By stacking unit squares of equivalent total side length, it is found that the complex sum square is the sum of the square of a and the square of b. The resulting complex length, c, is then the square root of the square, that is the square root of c squared. This square root, a second order relation, is the source of a particular complication as will be seen later on. 1.3 Island of Samos
ISLAND OF SAMOS Figure 4
1 Real here is a new word for horizontal, Imaginary a new word for vertical.
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The island of Samos was, and still is, a small fishing village. Its location is obscure, today a location never heard about. In the era of Pythagoras its location was central to the great cultures of antiquity. Samos was the crossroads of the ideas of these cultures.
LOCATION OF SAMOS Figure 5
Pythagoras lived in a world very much different than the modernistic world of today. To comprehend the world of Pythagoras is at best unlikely. His world was a musical world, an infinite universe manifesting in geometric ratios.
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Figure 6
20

Figure 7
21

Figure 8
22

Figure 9
23

Figure 10 Shown is the Geometric Pattern of the Infinite.
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Shown is the geometric pattern of music, and its analog to Electric Form. Figure 11
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Pythagorean Analog of Heaviside Telegraph Expression Figure 12
26

XB RG Figure 13
27

XG
RB
Figure 14
(XB + RG) + j (RB – XG)
(Heaviside Telegraph Expression)
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In the Pythagorean mind, everything is linked canonically in a cosmic series of octaves, these spanning from the infinitesimal to the infinite, no end in either direction. Each entity in the universe has its niche in a particular span of octaves; everything is a part of everything else. Nothing was isolated or separate in the reasoning of the Pythagorean. 1.4 Origins of Mathematics
Figure 15
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Pythagoras and his followers were hermetic occultists, a secret brotherhood. Theirs alone were the sacred secrets of ancient Aegypt. The Pythagoreans are considered the founding fathers of mathematics, but may actually have delivered to the world the mathematics as a Promethean endeavor. For the most part, it can be said that the knowledge of mathematics begins with Pythagoras of Samos, 582 B.C. As one would expect, Pythagoras was killed by political operatives and his followers were scattered. Hereby the Tale of Prometheus holds true.
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1.5 The Pythagoreans
Aristotle
Figure 16
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Aristotle, 384 B.C., can be considered the grandfather of science. In his book titled Metaphysics, three hundred years after Pythagoras, he wrote:
I) The Pythagoreans, as they are called, devoted themselves to mathematics.
II) They were the first to advance this study, and having been brought up upon it they thought its principles were the principles of all things.
III) Since the principles of numbers are by nature first, and in numbers they seemed to see many resemblances to things that exist or come into being.
IV) Since again, they say the attributes and ratios of the musical scales were expressible in numbers.
V) Since, then, all other things seemed in their whole nature to be modeled after numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things and the whole heaven to be a musical scale and a Number.
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[2] Ancient Concepts
Figure 17
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2.1 Chaldean Number System In the era of Pythagoras the concept of numbers was much deeper than that of mere numeration or counting. Numbers were considered symbolic arch forms. This was the philosophy of the ancient Caldeans.
Figure 18
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612-539 BC Figure 19
To the Chaldeans, each number contained an intrinsic arch form. This arch form directs the formative forces, which in turn act upon the Primordial Aether. The Aether is hereby molded into concrete reality. This is in line with the reasoning of the Pythagoreans and to them numbers were considered Divine Ratios.
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2.2 Geometric Manifestation
Figure 20
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Figure 21
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Figure 22 The numerical arch form for electricity is the number Four. Electricity establishes itself in quadrapolar relationships, it manifests in Four Phases and other quadrature configurations. This can be seen in its concrete arch forms as motors, transformers, transmission lines and etc.
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Figure 23 Any variety of quadrapolar arch forms are possible, some more “Electrical” than others. Here we are concerned specifically with alternating current, or properly with the alternating electric wave, of which so called A.C. is a projection. These waves exist in cycles of revolution, this progressing through the Four Phases of the cycle. It is a rotational phenomenon in cycles per unit time, such as the common 60 cycles per second.
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Figure 24
The primitive arch forms shown in the figure are various cultural expressions of the rotational Four Phase Cycle. The offsets at the ends of the Four poles give the sense, or direction, of the cyclic rotation; that is, if it is clockwise, or counter-clockwise.
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Figure 25
The primitive arch form most suited for A.C. theory is shown in this figure. This consists of a pair of Four pole forms, each rotating in an opposite direction. The light is rotating in one direction; the dark is rotating in the other direction. This light-dark relationship is in direct analogy with the magnetic-dielectric relationship in the Alternating Electric Wave. Each time the poles of the counter-rotating forms coincide or overlap, a specific aspect of the cycle manifests, this occurring four times per cycle, giving the four cyclic aspects:
I) Energy Production II) Energy Return and Storage III) Energy Consumption IV) Energy Storage and Return
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(II) Van Tassel of Giant Rock
GEORGE VAN TASSEL 1910-1978 Figure 26
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[1] Van Tassel and his Integratron 1.1 Four Quadrant Theory The ancient symbolic arch forms were adapted into the Four Quadrant Theory of George Van Tassel, the designer of the Integratron. Van Tassel saw the Earth as an enormous electric Generator and sought to duplicate its actions with his Integratron.
Figure 27
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This diagram explains his Four Quadrant Theory as it is manifested in the actions of the Earth’s rotation.
Figure 28 Here shown is the Earth-Sun relation, this in a Four Quadrant expression.
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Figure 29 Shown is the lunar cycle. The Lunar cycle in its Four Quadrant arch type. This will be extensively treated in later chapters.
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The Four Quadrant Theory of Van Tassel portrays an Aether-Dynamical System, this involving the Earth, Sun, and Moon. The quadrapolar dynamic brings into being the seasons, tides, and life itself on Earth. This conception of George Van Tassel bears a remarkable resemblance to that Four Quadrant understanding of the Aboriginal American. Van Tassel’s theory and the figures shown are given in his book When Stars Look Down. 1.2 The Integratron
Figure 30
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The Four Quadrant Theory of George Van Tassel describes his Integratron. It was a large magneto-dielectric apparatus, which claimed to have the largest rotary electro-static generator in existence. This machine created a potential of hundreds of thousands of volts. The Integratron was a domed structure 55 feet in diameter and 30 feet in height. It had the appearance of an alien space craft. In fact, Van Tassel claimed the instructions for the Integratron were provided by the extraterrestrials. Accordingly the ideas of George Van Tassel are not taken seriously. Here the Integratron serves as an introductory example of a Four Quadrant Electro-Dynamic apparatus, which of whatever origin or purpose, must never the less possess a frequency and an impedance as do all forms of electric systems. 1.3 Integratron Operation
INTEGRATRON – LOWER LEVEL Figure 31
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INTEGRATRON – UPPER LEVEL Figure 32
The Integratron was composed of two levels, a lower level and an upper level. The lower level was a cylindrical form, this with a central hollow core, or column. The upper level was a void space enclosed by a hemispherical wooden dome. The inner surface of the dome was covered in thin aluminum sheeting. The outside of the dome was painted with a special dielectric paint. The wooden dome enclosed the metallic foil as a dielectric coating. This hemispherical structure served as a terminal condenser, similar to that found in designs of Nikola Tesla. The admittance of this terminal coupled the Integratron into the Earth-Ionosphere condenser. Because of the dielectric stresses involved, the Integratron building was of non-metallic construction, no nails or screws.
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INTEGRATRON MODEL – BOTH LEVELS Figure 33
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INTEGRATRON – CEILING TRANSFORMER WINDINGS & CENTRAL COLUMN
Figure 34
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INTEGRATRON – TRANSFORMER WINDING DIAGRAM Figure 35
On the ceiling of the lower level was wound a large transformer. This unique transformer consisted of a pair of flat spiral coils, in the manner of Nikola Tesla. Each coil was wound with 5000 feet of wire, divided into 90 turns. The two coils were wound in opposite directions, one clock-wise, the other counter-clock-wise. One coil was wound with insulated 14 gauge iron wire, the other wound with insulated10 gauge copper wire. Each coil had an inside terminal at the central column, and an outer terminal that ran to the outer periphery of the hemispherical dome. The terminals of the iron coil were situated opposite to the terminals of the copper coil. The axis formed by the iron vs. copper terminal arrangement was in a particular alignment with the poles of the Earth, a 22.5 degree space phase displacement. This Pi over Eight angle was important in the operation of the Integratron, this as a certain “Angle of Hysteresis”.
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On the same plane as the transformer, and around the outside of the dome, was the electro-static rotor.
INTEGRATRON – ELECTROSTATIC GENERATOR Figure 36
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INTEGRATRON – ELECTROSTATIC TERMINAL PLACEMENT Figure 37
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INTEGRATRON – ELECTROSTATIC TERMINAL Figure 38
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INTEGRATRON – ELECTROSTATIC TERMINAL PLACEMENTS Figure 39
INTEGRATRON – QUADRAPOLAR LAYOUT Figure 40
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INTEGRATRON BASED ON BASIC DIROD GENERATOR Figure 41
This was an electro-static generator commonly known as a Di-Rod machine. It is in use in many designs, serving as a motor or generator and has the advantage of not needing two counter-rotating rotors; this greatly simplifying large scale units such as the Integratron.
The rotor was to be driven by compressed air supplied through dielectric piping from a 100 horsepower air compressor. The compressor was in a shack distant from the electric field of the Integratron.
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This rotating generator had Four poles, two coincided with the transformer terminals; the other two coincided with the terminals on the hemispherical condenser. The transformer pair of terminals exists in space quadrature with the condenser pair of terminals, all in a Pi over Eight angular relation to the poles of the Earth. Pi over Eight, or 22.5 degrees is the 16th order division of the circle. A four pole system of triggered spark gaps commutated the rotor, the hemispherical condenser, and the transformer. The spark gaps were triggered by air through dielectric pipes from a remote time function generator.
INTEGRATRON – TRANSFORMER DETAIL CENTRAL COLUMN CONDENSER (RED) Figure 42
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INTEGRATRON – TRANSFORMER & CENTRAL COLUMN CONDENSER (YELLOW)
Figure 43
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INTEGRATRON – CONDENSER NETWORK ON COLUMN (RED) Figure 44
The iron and copper inner terminals of the transformer each terminated into a zigzag condenser structure which was configured on the outside of the central column. This column is hollow, enclosing a void space. This void is accessible from a hatch on the upper floor. The iron-copper zigzag condenser enclosed this void space, one on each side as pair of condenser plates. The void space between the plates hereby became the seat of the dielectric activity.
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INTEGRATRON – UPPER LEVEL DECK HATCH DOWN INTO CENTRAL COLUMN VOID Figure 45
In the upper level a special network was lowered through the opening in the floor, down into the void. The network then operated within the ironcopper condenser dielectric flux. This network is the mystery never revealed by the extra-terrestrials. It was said that work on the Integratron stopped at this point, however the immense difficulties in driving the rotor may be the cause; it is an unknown.
The Integratron was to be brought into synchronism with the electrodynamic Earth. Its space phase position, and its time phase function generator aligned the Integratron with the time position of the Earth. The Integratron can be seen as a form of synchronous condenser.
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The general layout of the Integratron would suggest the transfer of electric energy between its two basic ports, one is the energy taken from or delivered to the Earth-Ionosphere condenser, the other is the energy into or out of the field of induction enclosed in the void of the central column. 1.4 Epilog
INTEGRATRON – VAN TASSEL & ASSOCIATES Figure 46
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The objective of the Integratron, by Van Tassel’s account, was to establish an intense Four Quadrant Field of electric induction. This is shown in his book. He claimed that out of phase operation of this field would result in a violent discharge of similar magnitude to a large H-Bomb.
The operating flux of the Integratron, as an induction generator, would exist in the lower level. This operating flux was established by the transformer structure. This flux would have a profound effect on objects put within it.
Two doors enter the Integratron lower level, these in quadrature with respect to each other. These were to facilitate human passage through one specific quadrant of the Four Quadrant Electric Field. Someone walking through this quadrant was expected to be biologically rejuvenated. The opposite quadrant would have the opposite effect; biological destruction.
It is possible that the Integratron would somehow cancel out the dimensions of time or of space. The cancellation of space, or the space scalar condition, would allow for moving objects from one point to another point without traversing the distance in between; a new type of radio. In common language, walking into one door of the Earth Integratron leads to walking out of the other door of the Integratron of a distant planet.
At its least, the Integratron is an interesting extra-terrestrial fairy tale, a myth intended here to provoke thought into a mode of fantastic possibilities, the prelude to invention The Integratron provides an example of a Four Quadrant Theory, and an electro-dynamic apparatus constructed upon that theory.
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Section Two Mathematical Aspects of Quadrapolar
Versor Algebra
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(I) The Knowledge of Electric Phenomena
[I] Alternating Current Machines
AC/DC MOTOR GENERATOR SET Figure 47
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1.1 Rotating A.C. Apparatus
Shown in the figure is an alternating current synchronous motor driving a direct current generator. The A.C. machine is in the foreground. This motor-generator set utilizes the rotary force developments of Nikola Tesla. The A.C. machine is an invention of Tesla, the D.C. machine is his improvement upon the designs of Thomas Edison.
1.2 Arithmetic Expression
To develop an engineering understanding of the inventions of Nikola Tesla, certain mathematical relations are required. These relations are of a higher order than is commonly understood. Conventional mathematics is rooted in a dualistic, or bipolar, form of expression. Plus and minus serve as the poles in common electrical configurations, but this is not so with the configurations of Nikola Tesla in his synchronous machine.
Two arithmetic operations are derived from plus and minus; these are addition and subtraction, respectively. In bipolar symbolic representation, two operators are defined, positive one and negative one. In symbolic terminology, positive one is a real number and negative one is an imaginary number. Subtraction, or negative one, is considered here and imaginary operation by reason of subtraction existing beyond the bounds of common additive numeration, such as counting with the fingers (digits). This line of reasoning can be carried further into multi-polar symbolic representation, such as is required for poly-phase A.C systems. This is the principle objective of these writings.
1.3 Algebraic Expression
Second order algebraic expressions are also bipolar relations in that they always resolve into two solutions, or a pair of roots. Typically one solution is positive and the other is negative; the two poles of the second order expression. For example, the square root of positive one is a second order expression since it is the “Second Root” of one. This expression then must
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have two roots, positive one, the real root, and negative one, the imaginary root. It is common place to neglect the imaginary root, leaving only the real, that is, just one. Everyone has been taught this way, the square root of one is just one. This so-called “Fact” creates a source of misunderstanding in how to properly represent the roots of the unit, the basis of Versor Algebra. In its proper representation, the square root of positive one defines the bipolar arithmetic operator, that is one to the one half power, its roots being positive one, addition, and negative one, subtraction. The exponent one half is the exponential representation of the bipolar arithmetic operator.
1.4 Higher Order Expressions
4TH ORDER ALGEBRAIC EXPRESSION Figure 48
The algebraic expressions for alternating current are fourth order. This creates a complication in expression. The rotating machines of Nikola Tesla have four phases or sets of poles. Direct current machines only have two poles, plus and minus; for A.C. machines this is only half the equation. When A.C. is represented by a second order expressions its solutions become prefixed by the square root of negative one. This is considered an impossible solution by conventional mathematical reasoning. The common concept of the imaginary number has its origin in the square root of negative one. This is the Imaginary of Algebra as discussed by Dr. MacFarlane.
To properly understand alternating current requires solutions to fourth order algebraic expressions. This gives four roots, or solutions, one for each of the Four Phases.
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The second order expression yields its pair of roots via what is known as the quadratic equation; this being a sort of calculating tool. No such equation exists for the fourth order expression. In fact no definite solutions exist for any algebraic expression higher than the second order. Solutions must be found via trickery and guesswork. Hereby, electrical theory is mired in bipolar mathematics, as are the mathematicians themselves. Engineers must find their way out of this dilemma, in the manner of Steinmetz. Since mathematicians cannot assist, nature will provide the guidance. To understand the electrical developments of Nikola Tesla and other situations of higher order, a “New Math” is required. This will find its beginning in the originator of mathematics, Pythagoras, and his ancient arch forms will serve as a guide. The objective here is the development of quadrapolar mathematics for the representation of the electric phenomenon.
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[2] The Long Line Problem
TWO DOUBLE CIRCUIT 115KV, 3-PHASE AC TRANSMISSION LINES Figure 49
2.1 Electric Transmission Shown in the figure is a pair of double circuit, 115 kilovolt, 3 phase transmission lines. This is the power of one hundred locomotives; this power moving near the speed of light flows in each circuit. For all the activity, no visible presence exists of this tremendous flow of electrical energy. The polyphase A.C. System of Nikola Tesla allows for the transmission of great quantities of energy over very long distances. The polyphase system is used world-wide, but for all its grandeur, for all the self edification if its educators and administrators, very little is actually known about electrical transmission. A rigorous solution for the transmission lines shown in the figure is an insurmountable mathematical task, the work of a lifetime. In
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fact, no rigorous solution exists even for the simple open wire telephone pair on glass insulators. Only a mind like that of Oliver Heaviside could grasp the complexity of such a seemingly simple problem. Any transmission theory that exists can be traced back to the writings of Heaviside, but unfortunately he was not a good teacher of his ideas. Most of the complexity in the long line problem is introduced through the distortion caused by electronic actions within the metallic “conductors”. This creates a serious imbalance between magnetic and dielectric energies.
2.2 Electric Wave Theory
What do we really know? The deficiency of knowledge is aptly stated by Ernst Guillimen in his book Communications Networks, Vol. 2, the first chapter, Engineering Solutions to the Long Line Problem. He describes the line calculations for the telephone pair as “A huge mathematical task”. This is the task he undertook in the writing of his book.
Guillimen explains how the equations that have become so commonplace are in actuality only certain terms of more complex, and less understood, equations. These are never mentioned, swept under the carpet so to speak. He goes on to explain that out of all the possible orders of electric waves, only three are acknowledged to exist. One is the guided, or transmitted wave, bounded in the space enclosed by the so-called conductors; second is the radiated wave, escaping the conductor boundaries; third is the dissipated wave, which sinks into the conductor, expending its energy as heat. No other electric waves are reasoned to exist.
For the electric waves that are recognized, their understanding is derived from the Maxwell-Heaviside Electro-magnetic Theory. This theory finds its origin in the writings of Michael Faraday. Faraday stated:
“The seat of electricity is not within the conductor, but is in the space surrounding it”.
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In other words, electricity is not a property of matter, it is a property of the aether. This view is not popular in the common way of thinking today, particularly in the minds of physicists, who are endeared to matter in an Einsteinian space.
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[3] Space Algebra and Electricity
ALGEBRA OF SPACE Figure 50
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3.1 Cartesian and Quaternion Systems The symbolic representation of dimensions in space, or space algebra, is a relatively recent development. It followed closely the development of the understanding of electricity, in fact, the two grew together, one aiding the other at times. But not at all times, and the stunted growth of space algebra retarded progress in the development of the understanding of electricity.
CARTESIAN COORDINATE SYSTEM Figure 51
Space algebra, as it is understood, was not easy to arrive at. It required evolutionary contributions from the greatest minds in mathematics. The process finds its origins in the work of Rene Descartes, 1596 - 1650. Developments continue with Leonhard Euler, 1707 to 1783, and then JeanRobert Argand, 1768 to 1822. Many others contributed and the effort came to rest with Alexander MacFarlane in 1892, and Oliver Heaviside in 1891.
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From that point onward space algebra became retrograde, dissolving into the Theory of Relativity, where space became confounded in the dimensional ratio of velocity, and sank beneath the waves.
Figure 52
Principle in the development of space algebra is the concept of the plane vector, originating in the Argand diagram, 1806. A working space algebra was defined within the bounds of two independent coordinate systems, these confined to a plane. This space algebra permits the representation of a physical quantity existing in terms of a plane vector, an entity divisible into two quadrature components. One component is parallel to what is called the Real Line, the other component parallel to what is called the Imaginary Line. These terms will be defined later on. In this manner the physical laws of plane vectors can be written in an algebraic expression.
A logical next step was to extend this representation of physical quantities into what is called "Three Dimensional Space". Here is where the seemingly insurmountable complications begin. Foremost is the notion of Descartes that there is but one dimension of space, and that is space itself. Coordinates are not actual dimensions but represent "Mathematical Fictions" to use the words of Carl Steinmetz.
Principle in the effort of extending the algebra of space into a tridimensional relationship was Sir William Rowan Hamilton, 1805 - 1865.
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SIR WILLIAM ROWAN HAMILTON
Figure 53 Hamilton took on the task of trying to retain the laws of the Cartesian plane and then extend them into the realm of three mutually orthogonal axes. The laws of the Hamiltonian system of representation became known as quaternions. It serves as a first attempt towards a more general space algebra.
QUATERNION RULES OF MULTIPLICATION Figure 54
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VERSOR SYSTEM VS. QUATERNION SYSTEM Figure 55
The Cartesian plane representation conforms to the laws of algebra, the Hamiltonian quaternion does not. Two breakdowns occur, first, the square of a quantity is negative, but in physical form it is positive, second is that the terms of a quaternion expression cannot be interchanged. This is to say that A times B is not equal to B times A. This violates a most fundamental law of algebra. No space algebra has as of yet overcome this problem, all systems violate what is known as the commutative principle. The overall complication with space algebra suggests the view of Kant; "Algebra is the science of time, geometry is the science of space". Is a space algebra possible?
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3.2 Michael Faraday
A workable knowledge of electricity begins with Benjamin Franklin, 1706 to 1790. His important contribution was to find that electricity manifests in a bipolar form, it is a polar phenomena. He perceived this polarity to exist in a single "Electric Fluid", a unified electric medium. Through his understanding the "Leyden Jar" was transformed by him into what is known as an electro-static condenser.

Benjamin Franklin Figure 56

Hans Christian Ørsted

The next important step in electrical knowledge comes with Hans Oersted, 1777 to 1851. Oersted's ground breaking discovery was reached quite by accident. In the course of a laboratory demonstration he had aligned a compass with a current carrying wire, this in such a way as to have the current impel the needle of the compass to fall in line with the direction of current flow. No such alignment could be detected. In closing the demonstration Oersted moved the compass and its needle promptly

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aligned itself perpendicular to the flow of current in the wire. Here the direction of the needle is in space quadrature with the current impelling it. The first known quadrature aspect of electricity had presented itself to Oersted.
Michael Faraday 1791-1867 Figure 57
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In this phase of electrical history only the electro-static and the magnetostatic aspects of electricity were being considered. Dynamic, or Time Variant, aspects were still to be discovered. Numerous theories with incongruous mathematics abounded and began to solidly establish themselves. Michael Faraday 1791 to 1867, would significantly alter this scenario, with his discovery of the Law of Electro-Magnetic Induction, and his establishment of the concept of what he named the "Dielectric".
Oliver Heaviside wrote of Faraday as "The Prince of Experimentalists", Nikola Tesla once called Faraday "The Columbus of Electricity". Michael Faraday provided the primordial foundation for all existing future understanding of electricity and its practical application. He would be the inspiration for future renowned British theoreticians who would establish the greater part of electrical knowledge that exists today. With Faraday began The Golden Age of Electrical Science.
Two fundamental understandings were arrived at with Faraday's work. One was his discovery in 1837 that the charge capacity of an electro-static condenser increased when a dielectric material, such as glass, was placed within the space between the condenser metallic surfaces. The findings of Faraday were not received well by his contemporaries. He expanded these studies until it became obvious that electricity is not a property of matter, but rather a property of Space. This has been, and still is, and unpopular idea.
Within space, electricity was found to exhibit complex and revealing geometries. This gave the basis for the Faradic Lines of Force, these the polarization of his "Contiguous Particles of the Aether". In common expression this is the iron filings around the bar magnet.
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Figure 58
The particles of iron polarize into the shape and form that the Aether corpuscles take in Faraday's theory.
The need for a mathematical interpretation was now required. However, Faraday was not a mathematician, he was an experimenter, as was Ben Franklin.
Faraday's second important discovery was a step out of the conventional static, time invariant, condition of known electrical quantities. He observed that when the conditions for magnetism are caused to vary with respect to time, actions manifest in the current carrying conductor. He called this the "Electro-Tonic State", and this gave rise to an "Extra Current". This electrotonic condition is now known as an Electro-Motive Force, or E.M.F. for short. The variation of magnetism and the intensity of the E.M.F. are given in his Law of Electro-Magnetic Induction, later formulated by Clerk Maxwell. Through this process electricity and magnetism became interchangeable through the dimension of time, this leading to the electric generator, and the basis of all alternating current apparatus.
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3.3 James Clerk Maxwell
JAMES CLERK MAXWELL 1831-1879 Figure 59
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James Clerk Maxwell began the study of Faraday's work in 1850 at age 19. Maxwell was quite impressed with the concept of spatial fields of induction connecting physical bodies. Accordingly he sought the mathematical description that Faraday could not provide. It has been said that "Maxwell put the math into Faraday".
The conception of electricity in the time before Faraday was based upon its static condition, no motion in time or space, it was stationary. This condition was reasoned to exist within matter as electric fluids which soak into it. Faraday's discovery of a time variant resultant, the E.M.F., introduced the dynamical aspect of electricity. The E.M.F. was only actually half of the equation and Maxwell would find the other half.
Maxwell carried out research into the Faraday concept of the dielectric seeking out its various aspects. He then founded the discovery that would eventually jolt the scientific community; the ability of the dielectric material to concentrate an electro-static field was a function of the velocity of light in that dielectric. This discovery unified electricity and optics, but in a way not yet quite understood. This connection would occupy the minds of scientists for a century afterwards, ending with Albert Einstein.
Maxwell's continued research into Faraday's dielectric field resulted in what might not be his most profound discovery, but possibly his most important. His discovery was the companion to the Faraday Law of Electro-Magnetic Induction.
Maxwell discovered that when the condition for a dielectric field are altered, or caused to change with respect to time, actions in the dielectric material manifest. Maxwell called this action "Displacement' and it existed as a form of electric current. It is a response to a change in the electric strain within the dielectric, or insulating body in general. Such displacement can also exist in free space. Here now exists the companion to Faraday's Law of Electro-Magnetic Induction and its Electro-Motive Force.
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The European theoreticians were horrified. Two most fundamental of their dictums had been violated upon by the British. The first violation is related to the notion of the electro-static potential. It was reasoned that a potential is incapable of manifestation within a conductive metallic body. However, Faraday's E.M.F. acted as a potential did, and this inside the metallic body. The second violation relates to the notion of current flow. It was reasoned that a current is incapable of manifesting within an insulating dielectric body. However, Maxwell's displacement current acted much like a conduction current, this inside an non-conductive, or insulating body. The European noses went high, and they rejected Maxwell's work as a mere "Paper Theory" and it was incapable of verification. The E.M.F. and displacement are reactions to the time rate of variation of electricity, the Europeans knew only of the static condition.
Maxwell's advancement at this point in his work defines the first primordial quadrapolar aspects of electricity:
I) The Circulatory Magnetism II) The Electro-Motive Force III) The Strain of Dielectricity IV) Displacement Current
Maxwell presents a beautiful complimentary symmetrical electrical relationship and this would be continued throughout his work, endearing the minds of later important researchers.
Maxwell's genius created a synthesis of all things magnetic and dielectric, as well as their interactions, into a pair of fundamental field equations. This was a scientific achievement of singular grandeur. He finally identified light as electromagnetism through the formulation of his idea of a transverse electromagnetic wave. This wave represents a complete quadrapolar electric configuration, the first to draw attention to itself in the study of electricity. The notion of the propagation of electricity through space without interconnection here presented itself to Maxwell. Here exists one of the most important advances in the knowledge of electricity,
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expressing a quadrapolar relation and the existence of electric waves. Heaviside would say, "The Heaven-Sent Maxwell".
The space geometry of electromagnetism consists of three mutual perpendiculars, this existing in the same manner as the tri-dimensional space of mathematics. The electromagnetic field idea of Maxwell was a natural home for the quaternion system of Hamilton. Ultimately, quaternions would serve to hinder the effort, forcing electricity into a form which it manifests only in certain configurations. Maxwell's ideas needed expression in a symbolic representation that did not exist. Quaternions did not serve electricity, electricity served quaternions.
As much as it was, his elegant theory was none the less still regarded as a "Paper Theory". No experimental verification had yet been established. No means existed in 1873 to generate the requisite frequencies, nor did any device that could detect these waves exist. This situation remained for 15 years.
In 1888, one of their own in Germany broke rank with the prevailing thought. This was Heinrich Hertz, he demonstrated the transmission of electricity through empty space across the room of his laboratory. In a rapid succession of experiments, Hertz continued to verify the theory of Maxwell until it became a proven fact. Then the world loved it, Maxwell was a genius, Hertz was a hero. The accolades rang loud!
Along with Heinrich Hertz and his research into electric waves came those of Nikola Tesla. Tesla carried the Maxwell idea even farther, perfecting the high frequency devices required to study electric waves, this far beyond the ability of his contemporaries. He noted certain discrepancies in the Hertz Wave Theory, not surprising since the apparatus of Hertz was primitive. Hertz was a physicist, Tesla was an electrical engineer and could create much more refined apparatus. Tesla travelled to Germany to work these matters out with Hertz, the untimely death of Hertz at age 33 ended any further work. Tesla continued on and wireless communication was
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soon a reality. However, Tesla was an American and the credit would go to others deemed more worthy. Nikola Tesla, with his very refined instruments and apparatus would discover aspects of Maxwell's theories that the quaternions kept from view. This was in the realm of electro-static waves, and Maxwell's dielectric displacement. These waves were unlike those of electro-magnetism, no tridimensional geometry existed; propagation was in a single coordinate system, not understood by mathematics. Public demonstrations of these waves were considered beyond awesome, some ran in fear. The absence of a tri-polar geometry and any known representation would throw Tesla back into the wilderness of the European potentials and currents, but his work would not find friends in either camp, and it was uniquely American. Tesla never wrote on the theoretical basis of his discoveries, but he did refer to Maxwell as an "Electrical Poet" and held him in high regard. Public statements would indicate that Nikola Tesla derived his ideas from the German theoretician Herman Von Helmholtz, who wrote on the subject of longitudinal waves. To this day, however, the understanding of Nikola Tesla remains an enigma.
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3.4 Oliver Heaviside
OLIVER HEAVISIDE 1850-1925 Figure 60
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Electro-magnetic telegraphy was invented by a joint effort of Samuel Morse and Joseph Henry, in 1838. The first telegraph line went into service in 1844, but little was yet known about the theory of propagation, nor the theory of circuits. No theoretical basis existed for the electro-magnetic telegraph, but it worked anyway.

SAMUEL MORSE 1791-1872

Joseph Henry 1797-1878

Figure 61

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MORSE TELEGRAPH SET and FIRST LONG DISTANCE TELEGRAPH LINE
Figure 62
Samuel Morse was a painter by profession and knew little about electricity. None the less, he began tinkering with wire coils and voltaic piles, this bringing in the efforts of Henry. Morse developed the first binary code, this in a digital format known as the Morse Code. Henry worked on devising the electro-magnets for the telegraphic sounders. Joseph Henry is known as the American Faraday. Not only did he independently discover the law of electro-magnetic induction, but years before Hertz, Henry demonstrated the propagation of electric waves through free space, and solid walls. His
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work, as an American, was ignored abroad. The electro-magnetic telegraph was ahead of the understanding of electricity that existed in its time. Theory would have to follow the telegraph, like it or not.
The tremendous success of this telegraph system led to the consideration of extension through undersea cables and in 1851 one became operational between England and France. This opened the fantastic possibility that a trans-continental submarine cable could be constructed and put into use. This would allow for "Instant" communication between England and America.
The cable between England and France exhibited puzzling problems, the signals arrived weak and distorted. Transmission was much slower, and more garbled than an equivalent distance of open wire line. No one knew why. In 1855 Sir William Thomson, known as Lord Kelvin, demonstrated in theory that trans-continental transmission is a mathematical impossibility. Using the diffusion of heat theory, Kelvin arrived at a capacitance-resistance propagation factor, then known as the "KR Law". This was to be one of the greatest mistakes in electrical theory. Fortunately, the notions of Kelvin did not discourage Cyrus Field, who did not place much stock in physics. He started laying cable across the Atlantic Ocean in 1857. The distortion was severe, but it worked! However, Lord Kelvin was the physicist of highest esteem so the KR Law remained The Law.
Major advancements in telegraphy were carried out in England by Sir Charles Wheatstone, where he is sometimes called the "Father of Telegraphy". He was also the uncle of Oliver Heaviside. Oliver Heaviside was born in London on May 13, 1850, the date when Maxwell would begin his theoretical treatment of Faraday's work. Wheatstone stimulated Heaviside's interests in telegraphic theory and Oliver Heaviside presented his first paper on the subject at age 22. He began work under the employ of the Great Northern Telegraph Company where he gained much valuable experience. He left his position for reasons unknown in 1874 and began his undivided study of Maxwell's "Treatise on Electricity and Magnetism". He became fascinated, if not obsessed, with Maxwell's field equations. This
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would lead to his profound advancement of space algebra and its application to electrical engineering. Heaviside's work carried forth Maxwell's theory into innumerable applications, providing an astonishing clarity without undo modification.
Maxwell's theory at the time of his death in 1879 was still a "Paper Theory", there was no experimental evidence of electromagnetic wave propagation. Oliver Heaviside's first paper on the subject, On Induction Between Parallel Wires, was published in 1881. He demonstrated, in terms of mathematics, the existence of travelling electromagnetic waves. These waves consisted of both magnetic and dielectric induction in the complimentary-symmetrical form of Maxwell. Kelvin's KR Law was based upon a heat like diffusion of electro-static force and ignored the magnetic induction. Here would exist a conflict with established law. Heaviside remarks on this are numerous.
To quote:
"Now very few (if any) un-mathematical electricians can understand this fact; many of them neither understand it nor believe it. Even many who do believe it simply because they are told so, and not because they can in the least feel positive about its truth of their knowledge. As an eminent practitioner remarked, after prolonged skepticism, "When Sir William Thompson (Lord Kelvin) says so, who can doubt it? What a world of worldly wisdom lay in that remark".
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The KR Law was enforced by the head of the British Telephone and Telegraph System, William Preece.
WILLIAM HENRY PREECE 1834-1913 Figure 63
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In 1885 Heaviside began regular publication of his papers on electromagnetic propagation in telegraph lines, emphasizing the importance of the magnetic component. This gave rise to the Telegraph Equation" the foundation of transmission theory. Oliver Heaviside made a most remarkable theoretical discovery, when the rate of magnetic dissipation was made equal to the rate of dielectric dissipation, the distortion on the line vanished. Moreover, the long line equation reduced to a very simple algebraic expression. This so angered William Preece that Heaviside was banned from any further publishing on the subject. The KR Law had become Admiralty Law! In America the work of Heaviside laid the basis for long distance telephony and led to the great success of the American Telephone and Telegraph Company, and its creation of the "Bell System"
The Bell System, 1885-1986 Figure 64
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With his "Telegraph Equation" Oliver Heaviside made a most important advance in the quadrapolar understanding of electricity. Needless to say, it mirrors the quadrapolar form shown earlier in Maxwell. The electromagnetic wave in wire line propagation can be represented in terms of four constants exhibited by the wire line structure. These are:
I) Resistance, representing the rate of magnetic decay II) Inductance, representing the ability to contain magnetism III) Conductance, representing the rate of dielectric decay IV) Capacitance, representing the ability to contain dielectricity
In general, the resistance, R, is a property of the so-called conductors, the conductance, G, is a property of the so-called insulators. The magnetic inductance and the dielectric capacitance, L, and, C, respectively are both a property of the space bounded by the conductors, and containing the insulators. LC then represents the propagation, or velocity, of the wave, RG represents the consumption of the wave in the course of its propagation, its dissipation.
Heaviside would coin many new words in the course of his writings, such as Inductance, or Reactance, for example. These were extensions of the commonly used word, Resistance. His terminology horrified the literary critics, of which James Swineburne wrote:
"A prolific inventor of new terms", or "A murderous hatred of the Queen's English, with words like Leakance, Reactance, and etc".
Nevertheless, these words of Heaviside have today become the language of electrical engineering. In fact, the work of Oliver Heaviside is the basis for almost the entirety of electrical engineering. Steinmetz, Kennelly, and others would continue from Heaviside's work. It is very striking how something which was so vigorously opposed, is now in such common acceptance as to be regarded as indispensible.
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3.5 Quaternions and Maxwell
MAXWELLIAN QUATERNION EXPRESSION Figure 65
Maxwell was bogged down in the quaternion system of Hamilton and Tait. When Heaviside studied Maxwell he clearly saw the need for a more suitable form of mathematical expression, even Maxwell himself began the task by modification of the quaternion notation. Heaviside had decided to take on the task of "Removing the Baggage from Maxwell". His dislike for the quaternion, as well as those who professed it few, finally becoming hostile to them. In the journals of the time considerable conflict existed on these subjects of space algebra and Heaviside kept the fires burning hot, particularly with his biting parables. One such parable opens in volume three of his "Electromagnetic Theory", Chapter IX:
Adagio...Andante..Alegro Moderato
The following story is true: There was a little boy, and his father said, "Do try to be like other people. Don't frown". And he tried and tried, but could not. So his father beat him with a strap; and then he was eaten up by lions.
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Reader, if young, take warning by his sad life and death. For though it may be an honour to be different from other people, if Carlyle's dictum about the 80 million still be true, yet other people don't like it. So if you are different, you had better hide it, and pretend to be wooden headed. Until you can make your fortune. For most wooden headed people worship money; and, really, I do not see what else they can do. In particular, if you are going to write a book, remember the wooden headed. So be rigorous; that will cover a multitude of sins. And do not frown. Heaviside completely rejected the quaternion method and began the development of his own vector algebra methodology exclusively. He transformed all prior modes of expression and representation, eliminating all confusing symbolisms. Establishing his new basis for space algebra he took over where Maxwell had left off, but kept his faith in the soundness of Maxwell's theory of electromagnetic waves. Heaviside set out to expound the Maxwellian understanding in very clear and systematic terms. The result was the now well known Heaviside Equations shown in the figure.
HEAVISIDE VECTOR EXPRESSION Figure 66
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The historical scribes took a dim view of Heaviside, so today this set of equations has been given the name "Maxwell's Equations", but where are the quaternions? And the parrots chatter on.
Ultimately Heaviside had backed himself into the same corner as Maxwell, his expressions still bore the arch type of transverse electromagnetism in a tri-dimensional space. The theoreticians were married to "Three Dimensional Space", a mathematical fiction, but written in stone. Heaviside could not help but denounce anything outside transverse electromagnetism.
So what lay still uncovered in the writings of James Clerk Maxwell, buried in quaternions? Are his ideas the totality of all things? His work is primordial, born of Leyden jars, and milligram balances, this compounded with no means of expression. Is it wise to place too much stock in Maxwell? Heaviside states this in the very beginning of his book, to quote:
"It is by no means to be concluded that Maxwell spells finality. It cannot be even said that the Hertzian waves prove Maxwell's Theory".
So what is left to be discovered?
3.6 Lost In Space
The subject of space algebra was one of controversy in the days of Oliver Heaviside. The conflicts sometimes became severe. The reality was that the math did not represent the actual physical process, it is as if the physical reality had become subordinate to mathematical theory. Edwin Armstrong best put this to words: "They substitute words for reality, and then talk about the words". When space algebra degenerated into Einstein-Minkowsky Relativity, Nikola Tesla made a strong statement; "Today's scientists have substituted mathematics for experiments and they wander off through equation after equation and eventually build a structure which has no relation to reality."
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Oliver Heaviside was a strong opponent of the physical becoming subservient to the mathematical, and worked this way in his space algebra. As for the quaternions, he states: "Quaternion was, I think, defined by an American school girl to be an ancient religious ceremony". The resulting situation with regard to space algebra is that it is derived from two principle formats, the quaternion system of Hamilton and Tait, and the vector system of Gibbs and Heaviside. And off to the side is the "Ausdehnungslehre" of Grassman.
ALEXANDER MACFARLANE 1851-1913 Figure 67
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A synthesis was derived from the methods of all these mathematical formats by Alexander MacFarlane. This is presented in his paper The Principles of the Algebra of Physics. MacFarlane begins with a discussion on the relative merits of each format, emphasizing the distinction between versor, vector, and tensor. Further on is his presentation of a more unified format with a particular emphasis on the versor form of expression. His paper leads to some rather remarkable results and provides the ground work for a new, more general, form of space algebra. This is where to begin. To quote Heaviside: "There is a time for all things; for shouting, for gentle speaking, for silence, for the washing of pet and the writing of books. Let now the pots go black, and set to work. It is hard to make a beginning, but it must be done". 3.7 Engineering Reality
500kv Electric Arc - Vacuum Switch Failure Figure 68
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From the position of the electrical scientist, if such an entity even exists in our day, the question is, what do we have to work with? To start, the "Ore" from which to build is almost all Maxwellian, and since his electromagnetic theory aided and abetted the Special Theory of Relativity, is he a good place to start? Then there is the Helmholtz-Tesla approach, but little information exists here to work with. To compound the situation the space algebra that exists is useless in representing anything but transverse waves. Finally, the situation is rendered hopeless by the social operative that the right thing to do when something is not understood is to suppress it, or make certain to destroy it if necessary. Then heard is Amen! The objective here is to derive a quadrapolar representation electricity, this for any dimension, time or space, and etc. Everything seems against this effort as far as established theory provides. Heaviside's approach bitingly emphasizes again and again - go back to the physical, the real world, and derive from this the mathematical. He considered mathematics an experimental science. Just as one would configure an idea into an assemblage of resistors, condensers, and transistors, he would configure his ideas into an assemblage of operators, functions, and variables. Academic mathematicians despised Heaviside for this, but his "Magic Equations" could work wonders.
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Transformers Figure 69
A fundamental electrical engineering situation is that of travelling waves in transformer windings. This physical structure is incapable of a rigorous expression in any form of existing mathematics. The Heaviside-Maxwell Equations are of no use in this application. Approximation and misrepresentations must be resorted to. However, the geometry of the transformer windings is self evident, and an intrinsic quadrapolar form presents itself, and the elegant symmetry found by Maxwell and Heaviside is buried within the winding structure in a more sophisticated form.
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Figure 70
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