The mystery of space _ a study of the hyperspace movement in the light of the evolution of new psychic faculties and an inquiry into the genesis and essential nature of space Rober

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Title: The mystery of space
a study of the hyperspace movement in the light of the
evolution of new psychic faculties and an inquiry into the
genesis and essential nature of space

Author: Robert T. Browne

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Most recently updated: October 24, 2024

Language: English

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*** START OF THE PROJECT GUTENBERG EBOOK THE MYSTERY
OF SPACE ***

THE
MYSTERY OF SPACE

The domain of the senses, in nature, is almost
infinitely small in comparison with the vast
region accessible to thought which lies beyond
them.— Tyndall, "On Radiant Heat."

THE
MYSTERY OF SPACE

A Study of the Hyperspace Movement
in the Light of the Evolution of
New Psychic Faculties
and
An Inquiry into the Genesis and
Essential Nature of Space

BY
ROBERT T. BROWNE

NEW YORK,
E. P. DUTTON & COMPANY
681 Fifth Avenue

Copyright 1919, by
E. P. DUTTON & COMPANY
All Rights Reserved

Printed in the United States of America

TO
THE CHERISHED MEMORY OF

Mylie De Prè
WHOSE WIFELY DEVOTION,
SYMPATHETIC ASSISTANCE AND ENCOURAGEMENT
DURING THE EARLY LABORS ON THE TEXT
WERE A CONSTANT SOURCE OF
INSPIRATION AND FORTITUDE TO THE AUTHOR,
THIS BOOK IS DEDICATED
AFFECTIONATELY

PREFACE

Mathematics is the biometer of intellectual evolution. Hence, the
determination of the status quo of the intellect at any time can be
accomplished most satisfactorily by applying to it the rigorous measure of
the mathematical method. The intellect has but one true divining rod and
that is mathematics. By day and by night it points the way unerringly, so
long as it leads through materiality; but, falteringly, blindly, fatally, when
that way veers into the territory of vitality and spirituality.
Wherefore, when we have wished to ascertain the real status of the intellect,
as well as its limitations, tendencies, possibilities, we have turned to its
conduct in the field of mathesis where it is least trammeled in its ingressive
and egressive motivations because of the natural and easy accommodation
which is offered for intellectual movement. Whether there are signs of
moribundity or symptomatic evidences of marked growth or of a
termination of intellectual regnancy, or whatever may be the occasion for
the examination, no surer index than the mathematical may be found for the
purpose. Full logical justification is, therefore, claimed for the choice of
mathematical evidences to test the assumption that a new era of conscious
mental conquest is opening for the vanguard of humanity and sequently for
the human family.
The treatment of these evidences has fallen logically into two divisions,
namely, the first, a brief and elementary review of the principles of the non-
Euclidean geometry and their bearings upon the question of space as the
subject of mathematical study; and second, the consideration of space as a
psychological, vital and dynamic or creative phenomenon.
In Part I an effort has been made to trace the growth of the notion of
hyperspace and to show that it is a symbol of a new epoch of intellectual
expansion, an actual seizure of a new domain of awareness by the mind.

And for this purpose a critical examination of the fundamental question of
dimensionality is entered upon from which it develops that the status of this
primary norm of mathematical thought exhibits a relatively inchoate
character because of its insufficiency as a definitive quantity, and further,
because of its rather superfoetated aspect when utilized as a panacea for
mathetic symptoms. Also, it has been found necessary to survey the field of
the four-space which has been accorded such a prominent place in the
mathematical thought of the day. The reader should find in the chapter
devoted to it adequate material for thought and sufficient comprehension of
its meanings as a mathematical contrivance.
In Part II an endeavor is made to interpret the evidences offered by high
mathematic considerations in the light of the larger psychogenetic
movement. For this undertaking the quality of awareness has been studied
with the view to establishing its priority as a determining agency in the
consideration of space in aspects broader and less restricted than those
embraced in the mathematical premises. Wherefore, it appears that there are
massive implications arising out of the hyperspace propaganda which have
hitherto been neglected in the preliminaries incident to the fabrication of its
structure. A very brief, and more or less symbolic, sketch of the genesis of
space has served to demonstrate its essential nature as a consubstantive of
materiality, vitality and intellectuality, the three major objective processes.
Consequently it becomes imperatively necessary that any view of space
which neglects its kosmic intent and purpose should be regarded as gravely
fragmentary and insufficient. It is only by linking up the two aspects of
space, the mathematical and the psychological, in such a manner that the
one shall supplement the other, that we shall be able to arrive at a truly
satisfactory understanding of its nature.
In Chapter IX attention is invited to some of the extremities of
mathematical laws wherein it is shown that, because mathematical goods
are strictly of intellectual texture and fabric, vain is the hope of reaching
any reliable certainty with respect to many vital questions, even regarding
space itself, by means of the mathematical method. The intellect, and,
therefore, mathematics encounter the most formidable stricture when effort
is made to maneuver in the field of vitality or realism. In addition, it is
shown that, when pushed to the utmost logical limits, metageometry proves
not only futile, but emphasizes the need for a sharp turning of the path of

search from the intellectual or material to the spiritual or intuitional. Indeed,
it becomes painfully certain that the Golden Fleece of profounder
knowledge will be discovered never by an expedition whose bark has its
sails set for the winds of mathematical seas. But, contrarily, a new bark,
moored at the furthermost shore of the sea of intellectuality with sails set
for the winds which come from the realm of intuitional perception, must be
seized. Whereupon, by the straightest line, we shall, at the last, land upon
the shore of realism, of truth all inclusive.
Mathematical evidences have been used in these discussions because they,
of all lines of knowledge, afford a more just exemplification of intellectual
evolution. The science of mathematics is the measure of the quality of
intellectual growth and, therefore, its data, its postulates, hypotheses and
advances clearly mark the stages of the intellectual movement.
Chapter X is the natural and logical sequence of the inquiry into the
question of spatiality. The conclusions reached therein and the obvious
inferences which should be drawn from the arguments presented flow
inevitably not only from the evidences of mathematical data but of the
common observations of life. And while we disclaim any intention of
demanding acceptance of them as final, authoritative declarations, we shall
be satisfied if the readers of this volume be incited to solve for themselves
the problems which these queries naturally suggest. Happy indeed shall be
the outcome if there be any who, following the path sketched herein, shall
find the solution of the Mystery of Space and apply its meanings to the
enhancement of the values of the intuitive life.
In conclusion, the author esteems it a special obligation of gratitude that he
should here acknowledge the debt which he owes to all of his friends who
have in any way assisted or encouraged him in the completion of this work.
Among those whom he is permitted to thank in this way is Mr. James
Rindfleisch who, having very kindly prepared the illustrations for the
photo-engravers, is deserving of special mention.
Robert T. Browne.
New York City,
1919.

CONTENTS

PAGE
Preface vii
Introduction: Explanatory Notes 1

PART ONE

CHAPTER I
The Prologue
On the Variability of Psychic Powers—The Discovery of the Fourth
Dimension Marks a Distinct Stage in Psychogenesis—The non-
Methodical Character of Discoveries—The Three Periods of 23
Psychogenetic Development—The Scope and Permissibility of
Mathetic License—Kosmic Unitariness Underlying Diversity

CHAPTER II
Historical Sketch of the Hyperspace Movement
Egypt the Birthplace of Geometry—Precursors: Nasir-Eddin,
Christoph Clavius, Saccheri, Lambert, La Grange, Kant—Influence
of the Mecanique Analytique—The Parallel Postulate the Root and
44
Substance of the Non-Euclidean Geometry—The Three Great
Periods: The Formative, Determinative and Elaborative—Riemann
and the Properties of Analytic Spaces

CHAPTER III
Essentials of the Non-Euclidean Geometry
The Non-Euclidean Geometry Concerned with Conceptual Space 69
Entirely—Outcome of Failures at Solving the Parallel-Postulate—
The Basis of the Non-Euclidean Geometry—Space Curvature and

Manifoldness—Some Elements of the Non-Euclidean Geometry—
Certainty, Necessity and Universality as Bulwarks of Geometry—
Some Consequences of Efforts at Solving the Parallel-Postulate—
The Final Issue of the Non-Euclidean Geometry—Extended
Consciousness

CHAPTER IV
Dimensionality
Arbitrary Character of Dimensionality—Various Definitions of
Dimension—Real Space and Geometric Space Differentiated—The
Finity of Space—Difference Between the Purely Formal and the 92
Actual—Space as Dynamic Appearance—The A Priori and the A
Posteriori as Defined by Paul Carus

CHAPTER V
The Fourth Dimension
The Ideal and the Representative Nature of Objects in the Sensible
World—The Fluxional, the Basis of Mental Differences—Natural
Symbols and Artificial Symbols—Use of Analogies to Prove the
Existence of a Fourth Dimension—The Generation of a Hypercube
118
or Tesseract—Possibilities in the World of the Fourth Dimension—
Some Logical Difficulties Inhering in the Four-Space Conception—
The Fallacy of the Plane-Rotation Hypothesis—C. H. Hinton and
Major Ellis on the Fourth Dimension

PART TWO
SPATIALITY: AN INQUIRY INTO THE ESSENTIAL NATURE
OF SPACE AS DISTINGUISHED FROM THE MATHEMATICAL
INTERPRETATION

CHAPTER VI
Conciousness the Norm of Space Determinations
Realism Is Determined by Awareness—Succession of Degrees of 161
Realism—Sufficiency of Tridimensionality—The Insufficiency of
Self-consistency as a Norm of Truth—General Forward Movement
in the Evolution of Consciousness Implied in the Hyperspace

Concept—The Hypothetical Nature of Our Knowledge—Hyperspace
the Symbol of a More Extensive Realm of Awareness—Variations in
the Methods of Interpreting Intellectual Notions—The Tuitional and
the Intuitional Faculties—The Illusionary Character of the
Phenomenal—Consciousness and the Degrees of Realism

CHAPTER VII
The Genesis and Nature of Space
Symbology of Mathematical Knowledge—Manifestation and Non-
manifestation Defined—The Pyknon and Pyknosis—The Kosmic
Engenderment of Space—On the Consubstantiality of Spatiality,
Intellectuality, Materiality, Vitality and Kosmic Geometrism— 203
Chaos-Theos-Kosmos—Chaogeny and Chaomorphogeny—N.
Malebranche on God and the World—The Space-Mind—Space and
Mind Are One—The Kosmic Pentoglyph

CHAPTER VIII
The Mystery of Space
The Thinker and the Ego—Increscent Automatism of the Intellect—
The Egopsyche and the Omnipsyche—Kosmic Order or Geometrism
—Life as Engendering Element—The Mystery of Space Stated—
242
Kathekos and Kathekotic Consciousness—Function of the Ideal—
The Path of Search for an Understanding of the Nature and Extent of
Space Must Proceed in an Inverse Direction

CHAPTER IX
Metageometrical Near-truths
Realism Is Psychological and Vital—The Impermanence of Facts—
On the Tendency of the Intellect to Fragmentate—The Intellect and
Logic—The Passage of Space—Kosmometer and Zoometer,
Instruments for the Measurement of the Passage of Space and the
Flow of Life—The Disposal of Life and the Power to Create— 284
Space, a Dynamic, Creative Process—Numbers and Kosmogony—
Kosmic Significance of the Circle and the Pi-proportion—
Mechanical Tendence of the Intellect and Its Inaptitude for the
Understanding of Life—The Criterion of Truth

CHAPTER X
Media of New Perceptive Faculties
The Spiritualization of Matter Is the End of Evolution—Sequence
and Design in the Evolution of Human Faculties—The Upspringing
Intuition—Evidences of Supernormal Powers of Perception and the
Possibility of Attainment—The Influence and Place of the Pituitary
327
Body and the Pineal Gland in the Evolution of Additional Faculties
—The Skeptical Attitude of Empirical Science and the Need for a
More Liberal Posture—The General Results of Pituitarial Awakening
Upon Man and the Theory of Knowledge
Bibliography 359
Index 367

THE MYSTERY OF SPACE

INTRODUCTION

Explanatory Notes
The following interpretation of words, phrases and notions occurring in the
text, and also biographical sketches which the scope and purpose of the
book itself make it impracticable to elaborate, are appended with the view
to facilitating its perusal.
AT-ONE-MENT (state of unity, unitariness); denotes the ultimate state of
oneness towards which all evolutionary movement tends; applied to
consciousness, indicates the final expansion of consciousness wherein it
coincides with the universal consciousness in extent and quality of
comprehension. As applied to things, denotes the unification of all
movements, tendencies, and evolutions as a singularity; the end of all
evolutionary activity (vide p. 270).
BELTRAMI, EUGENIO, was born at Cremona, November 16, 1835; there
he attended the elementary schools, the gymnasium and the lyceum,
excepting the scholastic year 1848-49 when he was at the Gymnasium of
Venice, now known as Marco Polo. He finished his lyceal studies in the
summer of 1853, and in the following autumn (November) became a
student in the Mathematical Faculty of the University of Pavia, after having
obtained a scholarship there on the Castiglioni Foundation in the Collegio
Ghisleri.
In 1854, the succeeding year, he was expelled from this college in company
with five of his colleagues who were accused of promoting "disorders"
against the Abbot Leonardi, rector of the college. The expulsion brought
him many hardships and disappointments, and for two years he drifted
along merely existing as his family was too poor to have him matriculated
at another university. But in 1856, he went to Verona where he succeeded in
securing employment as secretary to the engineer, Diday, in the

Government service of Lombardy-Venice. On January 10, 1857, he was
dismissed from this position "for political reasons"; but as the annexation of
Lombardy to Piedmont occurred soon thereafter, he became again attached
to the office of Diday, his former employer, when it was transferred to
Milan as a consequence of political changes.
At Milan Beltrami took up his mathematical education in real earnest as he
now had access to Professor Brioschi, his former tutor, and also Luigi
Cremona. Through the influence of these two men he was designated
(October 18, 1862) "Professore straordinario" in the University of Bologna.
His work on Surfaces of Constant Negative Curvature, as the pseudosphere,
and his application of the expression given by Lobachevski (q.v.) for the
angle of parallelism, very definitely secure for him a place among the
foremost workers in the field of the non-Euclidean geometry. He postulated
a theorem, known as Beltrami's Theorem, which he stated as follows: "The
center of a circle circumscribing a triangle is the center of gravity of the
centers of its inscribed and escribed circles." He died in the year 1900.
(Vide Amer. Math. Mo., Vol. IX, p. 59.)
BOLYAI, JANOS (1802-1860), was born at Kleansenburg, Hungary. He is
said to have inherited his mathematical genius from his father, Bolyai
Farkas (1775-1856), who was born at Bolya, Hungary. Being a very
spirited youth, his progress in his studies was most remarkable. He
completed the curriculum at the Latin school when only twelve years of
age. Was graduated from the Philosophical Curriculum as a result of two
years of study and then entered the Viennese Academy of Engineers. Was
appointed lieutenant at Temesvárlin, 1823, whence on November 3, 1823,
he wrote his father: "I have discovered such magnificent things that I am
myself astonished at them. It would be damage eternal if they were lost.
When you see them, my father, you yourself will acknowledge it. Now, I
cannot say more, only so much: that from nothing I have created another
wholly new world." This letter was written in the Magyar language and has
been preserved at the Marcos Vásárhely, Hungary. The mathematical
conceptions formulated by him became the appendix of the Tentamen, a
book which his father had written on the Theory of Parallels.
His Science Absolute of Space was translated into the French in 1868 by the
French mathematician, J. Hoüel, to whom belongs the credit of

popularizing the works both of Bolyai and Lobachevski. (Vide Science, n.
s., Vol. 35, No. 906, 1912.)
CAYLEY, ARTHUR, born at Richmond, Surrey, England, August 16, 1821;
studied at King's College school; entered Trinity College, Cambridge,
already a well equipped mathematician at the age of seventeen. When but
twenty-one years of age he took two of the highest honors in the University
of Cambridge. He was Senior Wrangler and First Smith's Prizeman. He
published his first paper in 1841 and this was followed by eight hundred
memoirs.
For fourteen years he practiced as Conveyancer. In 1863 Lady Sadler's
various trusts were consolidated, and a new Sadlerian professorship of Pure
Mathematics was created for the express purpose of affording a place for
Cayley. Meanwhile, as early as 1852 he was a fellow of the Royal Society;
in 1858 he joined Sylvester and Stokes in publishing the Quarterly
Journal of Pure and Applied Mathematics.
He was for a considerable time principal adviser as to the merits of all
mathematical papers which were presented for publication to the Royal
Society, the Astronomical Society, the Mathematical Society and the
Cambridge Philosophical Society. He is said to have been the "most learned
and erudite of mathematicians," and much of the material, therefore, which
now constitutes the basis of the non-Euclidean geometry is due to his
laborious efforts and comprehensive knowledge of mathematics. (Vide
Review of Reviews, Vol. II, 1895, Sketch, reprinted from Monist.)
CHAOGENY (Gr. Chaos, disorder, geny—generating, evolution); the
evolution of chaos into order. A kosmic process involving the elaboration of
the original, formless world-plasm into the first faint signs of orderliness;
the beginnings of the movement of life or the Creative Logos in preparation
of the field of evolution.
CHAOMORPHOGENY (Gr. Chaos, disorder—Morphe, form—geny,
becoming, generating); evolution of the space-form, the universe; the
establishment of the metes and bounds of the universe; also, the origination
and characterization of all forms as to tendence, purpose and limitations.

CONCEPTUALIZATION—The act of conceptualizing, the formulation of
concepts; the process by which the Thinker arrives at concepts; the logical
procedure by which the consistency of a scheme of thought is established.
CONSTRUCTION, IDEAL—A purely formal conception; a theory,
hypothesis; a logical determination not necessarily based upon facts, but
possessing virtue because of consistency; a self-consistent scheme of
thought.
COSMOS—Whenever the term "cosmos" appears in the text spelled as here
shown, it refers to phenomena pertaining to the earth or the solar system;
when spelled "kosmos" reference is made to the universe as a whole.
CRITERION OF TRUTH—Defined in the text as a four-fold standard of
reference, embracing the following elements, namely, the causal, the
sustentative, relational and developmental. Lacking any one of these, no
view of truth is more than fragmentary. Applied to space, it contemplates an
inquiry into the genesis or causal aspect, an accounting for the duration
aspect, a recognition of its relation to the totality of objects, and lastly, a
prophecy of its telestic or perfective culmination. This test has been applied
to the study of space as sketched in the text and the conclusions reached are
an outcome of the inquiry directed along these lines.
CURVATURE OF SPACE—A doctrine formulated by Riemann and which
maintains that space is curved, and consequently, all lines drawn therein are
curved lines. Professor Pickering aptly describes the results of movements
in a curved space by pointing out that if we go far enough east we arrive at
the west; north, we arrive at the south; towards the zenith, we arrive at the
nadir, and vice versa.
DEIFORM—The basic idea indicated is that the universe is the form or
body of the supreme deity, since He is not only immanent in the kosmos,
but sustains it by His life; that in order to create a manifested universe, it
was necessary to limit, or sacrifice, in a measure, His own illimitability.
Viewed in this light, the kosmos assumes an added significance.
DIMENSION—(L. Dimensio, to measure), measurement; a system of space
measurement. The Euclidean geometry recognizes three dimensions or
coördinates as being necessary to establish a point position; witness, the

corners of a cube to form which three of the edges come together at a point.
These edges represent coördinates. For the purposes of metageometry, the
term dimension has been variously defined, as, direction, extent, a system
of space measurement, or a system of coördinates. Regarded as a series of
coördinates, it became possible to postulate a system which required four
coördinates to estabish the position of a point, as in the hypercube. There
may be five, six, seven, eight, or any number of such coördinate systems
according to the kind of space involved in the calculations. Determinations
based upon the logical necessities of the various coördinate systems have
been found to be self-consistent throughout and, therefore, valid for
metageometrical purposes. Much depends upon the definition; for, after the
definition has been once determined it remains then merely to make
inferences and conclusions conform to the intent of the definition.
DIVERSITY—Philosophically, the idea indicated has reference to all
dissimilarities, differences, inequalities, divergent tendencies, movements
and characteristics to be noted in the universum of life; the antithesis of
kosmic unity; the natural outcome of life in seeking expression; the result of
the fragmentative tendency of life.
DUODIM (duo, two; dim, abbreviation of dimension)—A hypothetical
being supposed to be possessed with a consciousness adapted only to two
dimensions; a dweller in "Flatland" or two-space whose scope of motility is
limited to two directions, as on the face of a plane; a term invented by
hyperspace advocates for the purpose of establishing by analogy some of
the characteristics of the four space and also its rationality.
DUOPYKNON (duo, two—pyknon, primary unit in the process of kosmic
involution, a condensation)—Secondary phase in the elaboration of chaos
into kosmic order. DUOPYKNOSIS (duo, two, secondary—pyknosis,
process of condensation and origination)—The second period of the
involutionary movement of life during which the duadic plane of the
kosmos is being established; the second, in the series of seven distinct
phases, of space-genesis; dual differentiation of kosmic plasm.
Duopyknosis contemplates that, in the passage of the kosmos-to-be from
the plane of non-manifestation to the plane of manifestation, there are seven
distinct, though interdependent and interrelational, stages through which
life passes, and that, of these, it is the second. It relates to the plane of non-

manifestation and is, therefore, beyond the ken of the intellectuality, being a
symbol.
EGOPSYCHE (Ego, the self-conscious I—psyche, soul)—The mental,
emotional and physical mechanisms of man, the Thinker. These include the
purely mental system, the emotional or affective mechanism, the nerve-
systems (cerebro-spinal and sympathetic) and the brain; the objective or
sense-derived consciousness of the Thinker which is elaborated from the
total mass of perceptions transmitted through the senses; the medium of
self-consciousness; the intellectual consciousness as distinguished from the
intuitional or omnipsychic consciousness (q.v.).
The notion of the egopsychic consciousness is based upon data already
empirically determined from the mass of evidences everywhere observable.
It seems to be apparent that there is a consciousness, a seat of knowledge, in
man the content of which is unknown to the sense-consciousness. Dreams,
premonitions, intuitions, impressions and the totality of all such phenomena
substantiate this view. Furthermore, it is agreed that the source of the
intuition is not identical with that of the intellect. The egopsychic
consciousness, accordingly, is purely intellectual.
FLUXION, PSYCHIC—The difference between a mental image and an
object; an image is the representation of certain salient or cardinal
characteristics of an object, sufficient for identification; but an image is not
congruent, in every respect, with the object. Thus when we perceive an
object, although as Bergson contends we perceive it in the place where it is
and not in the brain, it is the image of the object which takes its place in
memory and not the object itself. There is, of course, a marked disparity
between this memory-image and the object. Even if the image possessed
one of the properties of the object, as, size, it could not take its place in
memory, and neither could it do so if it possessed any of the properties of
the real object. Consciousness is such that all due allowances are made for
these conditions and the mind is able to retain more or less exact knowledge
of these properties in the image; but there is a difference, small though it
may be. This difference is the psychic fluxional.
FOHAT (Skt)—A term applied to the Creative Logos who is said to be the
generating element in the differentiation of chaos into kosmic orderliness;

the supreme deity in the rôle of Creator.
FORM, PURE—An abstraction arrived at by subtracting the last vestige of
materiality or substantiality from an idea and viewing the remains as a pure
unsubstantial form or idealization; the shell or frame-work of a material
object or condition; existing in idea or thought only; a mental conception
regarded as a type or norm; a purely hypothetical construction.
FOUR SPACE—Often referred to as the fourth dimension (vide Chapter
V); a space in which four coördinates (four lines drawn perpendicular to
each other) are necessary and sufficient to establish the position of a point,
as, a hypercube.
GAUSS, CHARLES FREDERICK, born at Brunswick, April 30, 1777. His
father, being a bricklayer, had intended that he should follow the same
occupation. So, in 1784, Charles was sent to the Bütner Public School in
Brunswick, in order that he might be taught the ordinary elements of
education. But during his attendance at this school, his unusual intelligence
and aptitude attracted the attention and friendship of Professor Bartels who
later became the Professor of Mathematics at Dorpat. In 1792, through the
kindly representations of Professor Bartels to the Duke of Brunswick,
young Gauss was sent to the Collegium Carolinum. This greatly displeased
his father as he saw in this move the frustration of his plans for Charles. In
1794, however, Gauss entered the University of Göttingen still undecided
whether he should make mathematics or philology his life work. While
residing at Göttingen, he made his celebrated discoveries in analysis and
these turned his attention definitely to the field of mathematics.
He completed his studies at Göttingen and returned to Brunswick in 1798,
residing at Helmstadt where he had access to the Library in the preparation
of his Disquisitiones Arithmeticae which was published in 1801. He
received his doctorate degree (Ph.D.) on July 16, 1799. His next notable
work was the invention of a method by which he calculated the elements of
the orbit of the planet Ceres which had been discovered by Piazzi, January
1, 1800, and who had left no record of his calculations by which other
astronomers could locate the planet. Gauss also calculated an ephemeris of
Ceres' motion by means of which De Zach rediscovered the planet
December 31, 1800.

His Theoria Motus Corporum Coelestium in Conicis Sectionibus Solem
Ambientium, in which the author gives a "complete system of formulæ and
processes for computing the movements of a heavenly body revolving in a
conic section" is an outgrowth of his early researches and brought him
lasting fame.
Through the influence of his friend Olbers, he was appointed, July 9, 1807,
first director of the new Göttingen Observatory, and Professor of
Astronomy in the University, a position which he held until the end of his
life. He died February 23, 1855. (Vide Astronomical Society Notices, Vol.
16, p. 80, 1856; also Nature, Vol. XV, pp. 533-537, 1877.)
GEOMETRISM—Of geometrical quality; a notion derived from Plato's
declaration: "God geometrizes." It was his belief that the creative acts of the
deity are executed in accordance with geometric design and laws; that in the
totality of such acts there necessarily inheres a latent geometric quality.
Kant closely adhered to this notion in his discussions of space as an aspect
of divine intelligence. He believed that the intellect merely rediscovers this
latent geometrism when it turns to the study of materiality, and this belief is
shared by Bergson, the foremost metaphysician of the present time.
HYPERSPACE (hyper, above, beyond, transcending—space)—That
species of space constructed by the intellect for convenience of
measurement; an idealized construction; a purely arbitrary, conventional
mathematical determination; the fourth dimension; any space that requires
more than three coördinates to fix a point position in it, as, a five space, an
n-space.
INTUITOGRAPH—The means by which the omnipsychic consciousness
transmits intuitional impressions to the egopsychic or intellectual
consciousness. An intuitogram is a direct cognition, an intuition; a primary
truth projected into the egopsychic consciousness by the Thinker. It is
recognized that, under the necessities of the present schematism of things, it
is exceedingly difficult to propagate an intuition, especially with the same
degree of ease as concepts are propagable; yet, this is believed to be a
condition which will be overcome as the evolution of the higher faculties
proceeds.

INVOLUTION—Process of enfolding, involving; antithesis of evolution;
philosophically, the doctrine of involution maintains that, during the process
of kosmic pyknosis (space-genesis), all that is to be expressed, developed
and perfected as a result of the evolutionary movement was first involved,
enfolded or deposited as latent archetypal tendencies and radicles in the
original world-plasm; that, as the involutionary movement proceeded
through the various phases of space-genesis, these became more and more
phenomenal until at last they terminated in the elaboration of a manifest
universe: each stage, accordingly, of the involutionary procedure became
the basic substructure of a plane of specialized substance or materiality and
consciousness. Thus it appears that evolution really begins where involution
ends (vide Fig. 18), and the two opposing processes constitute the dualism
of life as generating element. This notion has been symbolized in the
Lingam yoni of Hellenistic philosophies, also in Yang and Yin of Chinese
philosophy, which represent the original pair of opposites.
KATHEKOS—A purely arbitrary term devised for the express purpose of
providing a convenient symbol to convey the idea embodied in the triglyph,
Chaos-Theos-Kosmos, and is composed of the first three letters in each one
of the terms of the triglyph; hence, symbolizes the triunity and interaction
involved in the resolution of chaos into an orderly kosmos by the will of the
Creative Logos. Thus, "kathekos" embodies a quadruplicate notion, namely,
chaos, Creative Logos, manifested kosmos, and the creational activity of the
Logos in the transmutation of disorder into order. The justification for this
term, therefore, resides in its convenience, brevity and comprehensiveness.
By referring to figure 17, it will be seen that Kathekos divides into two
kinds—involutionary, or that which pertains to involution, and evolutionary,
or that that pertains to evolution. It thus comprises the beginning and the
end of the world age or cycle and pertains to non-manifestation. The raison
d'être of this differentiation is embodied in the notion that, on the
involutionary arc of the cycle, the chaogenic period represents a phase of
the world age when space-genesis is in an archetypal state wherein are
involved all possibilities that are to become manifest in the kosmos, and on
the evolutionary arc, the kathekotic period which is parallel to the
chaogenic and represents a phase of the world age when the kosmos has
reached ultimate perfection, embodying the perfected results of the
possibilities which inhered in the chaogenic period or in involutionary

kathekos. Thus, kathekos is dual in nature, on the one hand representing
kosmic potency, and on the other, kosmic perfection of these potencies. It is
Alpha, as related to involution, and Omega, as related to evolution.
KATHEKOSITY—A derivative, signifying creative activity and all that it
implies; the state of consciousness or cognition corresponding thereto.
KLEIN, FELIX (1849—), born at Dusseldorf; studied at Bonn, and when
only seventeen years of age was made assistant to the noted Plücker in the
Physical Institute. He took his doctorate degree in 1868, then went to
Berlin, and later to Göttingen where he assisted in editing Plücker's works.
He entered the Göttingen faculty in 1871; became Professor of Mathematics
at Erlangen in 1872; and subsequently held professorships at Munich, 1875;
Leipzig, 1880, and Göttingen, 1886. No one else in Germany has exerted so
great influence upon American mathematics as he.
KOSMOS—See Cosmos.
LA GRANGE, JOSEPH LOUIS, born at Turin, January 25, 1736; died at
Paris, April 10, 1813; regarded as the greatest mathematician since the time
of Newton. It may be interesting to note that La Grange remarked that
mechanics is really a branch of pure mathematics analogous to a geometry
of four dimensions, namely, time, and the three coördinates of the point in
space. (Vide Ball's Account of the History of Mathematics.)
LIE, SOPHUS, a noted mathematician, referred to as the "great
comparative anatomist of geometric theories, creator of the doctrines of
Contact Transformations and Infinite Continuous Groups, and
revolutionizer of the Theory of Differential Equations."
LOGOS—The supreme deity of the phenomenal universe; Creator; Fohat; a
planetary god; the deity of a solar system.
MANVANTARA (Skt.)—A world age; the periods of involution and
evolution combined; the stage during which the universe is in
manifestation; a Day of Brahma.
MATHESIS (Gr. mathein, to learn)—Erudition; profound learning; the
realm of metaphysical conceptions; the field of higher mathematics; the
sphere of conceivability; the theoretical.

MENTOGRAPH—A cognitive factor consisting of a complete perception
fused or in coalescence with a memory-image. Pure memory, of itself, is
without utility as an aid to cognition; but, when nourished or supplemented
by the substance of perception it becomes the basis of intellectual
consciousness.
METAGEOMETRY (Gr. Meta, beyond, transcending—geometry)—
Commonly, any kind of geometry that differs from the Euclidean, as the
non-Euclidean; a geometry based upon the assumption that the angular sum
of a triangle is greater or less than two right angles; the highest form of
geometry; a system of idealized mathematical constructions. Sometimes
called "pangeometry"; designated by Gauss as "Astral Geometry"; the
geometry of hyperspace. It consists of results arrived at by geometers in
seeking a proof of the parallel-postulate.
META-SELF—The higher self in man; the universal self; the one self of
which all individual selves are but fragments or parts. In man, it is
coördinate with the omnipsyche (q.v.) and as such is the medium of kosmic
consciousness.
MORPHOGENY (Gr. Morphe, form, vehicle, body—geny, evolution)—
The evolution of forms, the production of individual bodies or vehicles for
life, including organs and faculties. Morphogenic—a derivative; pertaining
to morphogeny; a kosmic process (vide figs. 17 and 18).
N-DIMENSIONALITY—Quality of conceptual space by virtue of which it
may be regarded as possessing an indefinite number of dimensions.
NEAR-TRUTH—Any statement or view which is based upon partial
knowledge; predicates concerning a class or genus derived from limited
acquaintance with particulars of the class or genus; statements based upon
logical determinations inhering in idealized constructions and applied to
concrete or objective conditions; an abstraction viewed as a reality; the
application of the qualities of abstractions to realities.
NEUROGRAM—Psychologically, a movement received by the afferent
nerves in the form of a stimulation and transmitted through the brain and
efferent nerves as either a reflex or voluntary action; a nerve impulse; a
perception; a primary unit of intellectual consciousness; cf. Intuitogram.

NEWCOMB, SIMON (1835-1909), born at Wallace, Nova Scotia; educated
in his father's school and came to the United States in 1853. Began, in 1854,
teaching in Maryland; was appointed computer on Nautical Almanac at
Cambridge in 1857; was graduated at Lawrence Scientific School in 1858;
appointed Professor of Mathematics in the U. S. Navy in 1861. He
supervised the construction of the 26-inch equatorial telescope at the Naval
Observatory, and was secretary of the Transit of Venus Commission; was a
member of nearly all of the Imperial and Royal societies of Europe and of
the various societies in the United States, receiving the Copley Medal in
1874; the Huygens, 1878; the Royal Society, 1890, and the Bruce Medal in
1898; held the presidency of the following learned societies, viz: American
Association for the Advancement of Science, 1877; Society for Psychical
Research, 1885-1886; American Mathematical Society, 1897-1898; the
Astronomical and Astrophysical Society of America from its foundation in
1899. He rendered notable service in popularizing the doctrine of
hyperspace.
NORM—An authoritative standard; model or type; standard of reference.
The choice of a norm for spatial determinations cannot abide in any premise
except that which naturally, and not artificially and conventionally,
conforms to what is actually perceived; if so, there should be justification
for challenging the wisdom and utility of the present schematism of things.
There is an inherent conformity of space with intellect and intellect with
space, and because of this natural complementarity of part with part and
whole with whole, space cannot be otherwise than the intellectuality
normally conceives it to be, provided, of course, that the cognitive
movement is free and untrammeled by arbitrary hindrances. Consciousness,
therefore, is the norm or standard of reference for all questions arising out
of a consideration of spatiality.
OMNIPSYCHE—A term used to denote the Thinker's cognitive apparatus;
the universal soul manifesting in individuals; the consciousness of the
Thinker in virtue of which he is at-one with the universal consciousness; the
medium of kosmic consciousness; the source of the intuition, cf.
Egopsyche. The divinity in man (which is taken for granted), or his highest
self can in no way be said justly to take its rise from sense-experience or
from any bodily process. If divine, then eternal, and therefore, persistent.
Broadly, the doctrine of evolution recognizes the passage of life from form

to form, adding a little to each successive form and inevitably pushing each
to a higher degree of perfection. Now, what is it that passes from form to
form? Is it undifferentiated life or is it a specialized form of life? From
every evidence, it would be judged that the life that ensouls an individual
form is a specialized principle, i.e., limited to the execution of a given
purpose. If life as a specialized principle, limited to the execution of a given
purpose in each form, passes on, it must preserve, at least, the sublimated
results obtained during its residence in each individual form. It would thus
become a sort of reservoir containing all these transmuted results. The
omnipsyche, within the meaning of the text, is precisely this specialized life
principle.
PARALLEL-POSTULATE—Variously referred to as the XIth, XIIth and
XIIIth axiom of the Elements of Euclid; stated by Manning, in Non-
Euclidean Geometry, p. 91, in the following form: "If two lines are cut by a
third, and the sum of the interior angles on the same side of the cutting line
is less than two right angles, the lines will meet on that side when
sufficiently produced." This celebrated postulate has proven to be the most
fruitful ever devised; for it embodies in itself the possibility of three
geometries based respectively upon the following assumptions, namely: I.
That there exists a triangle, the sum of whose angles is congruent to a
straight angle, the Euclidean; II. That there exists a triangle the sum of
whose angles is less than a straight angle, the Lobachevskian; III. That
there exists a triangle the sum of whose angles is greater than a straight
angle, the Cayley-Klein. Speaking of the content of the last two named,
Edward Moffat Weyer[1] says: "Hypothetical realms, wherein the
dimensions of space are assumed to be greater in number than three, yield
strange geometries, which are only card castles, products of a sort of
intellectual play in the construction of which the laws of logic supply the
rules of the game. The character of each is determined by whatsoever
assumption its builder lays down at the start."
PASSAGE OF SPACE—A phrase connoting the movement of space from
chaos to perfect order, a process believed to be infinite. The genesis of
space necessarily implies an elaboration, a procedure, by which the
metamorphosis from disorder to kosmic order is made, and this movement
is referred to as the "passage of space," a phenomenon thought to be
measurable by means of a suitable instrumentality.

PERISOPHISM—See Near-Truth.
PSEUDOSPHERE—A surface of constant negative curvature; basis of
Beltrami's metageometrical calculations; surface resembling a champagne
glass or common spool. The assumption that space is pseudospherical has
given rise to the notion of space-curvature and various other conceptions.
PSYCHOGENY (Gr. Psyche, Soul—geny)—History of the evolution of the
soul or the development of the senso-mechanism in organisms. Ernst
Haeckel has traced the psychogeny of man through twenty-two different
stages from the moneron to the anthropoid apes, and man.
PRALAYA (Skt.)—Kosmic quietude; the period during which the universe
is not in manifestation; gestatory period; kosmic inactivity; opposed to
manvantara (q.v.); figuratively, the kosmic womb; world egg.
PYKNON (Gr. pyknon, hard)—The principle of kosmic condensation; the
primary basis of space-genesis; the initiation of the process by virtue of
which chaos is elaborated into kosmic order. PYKNOSIS—The process of
spatial engenderment. There are seven of these processes, each indicating a
phase of duration, namely: MONOPYKNOSIS, the primary phase;
DUOPYKNOSIS, secondary; TRIPYKNOSIS, tertiary. These three pertain
to the plane of non-manifestation, the pralayic or gestatory duration-phase.
The results arrived at during these duration-phases are concentrated in the
Quartopyknotic which corresponds to the causal plane of manifestation or
pure kosmic spirituality. QUINTOPYKNOSIS, a process concerned in the
genesis of mentality; SEXTOPYKNOSIS, kosmic sensibility;
SEPTOPYKNOSIS, kosmic materiality. These seven phases of duration
constitute the scope of space genesis or kosmogenesis, and incidentally
depose the substructure of kosmic materiality, sensibility, intellectuality and
spirituality, as well as the higher trinity of kosmic modes. The ramifications
of these principles are innumerable and omnipresent. (See Chapter VII.)
QUARTODIM—A hypothetical being assumed to have a consciousness
adapted to hyperspace or the fourth dimension, and whose scope of action is
encompassed within a space which requires four coördinates, as the four-
space.

REALITY (Realism)—Life; the harmony existing among the parts to
maintain their equilibrium in the whole; the principle of integrity subsisting
among parts; kosmic vitality.
RIEMANN, GEORGE FREDERICH BERNHARD, was born September
17, 1826, in the village of Breselenz, near Dannenburg, in Hanover. Until
he was eight years of age his father was his sole tutor, but even at this age
he exhibited great powers of arithmetical calculation. In the Spring of 1840
young Riemann was sent to the Hanover Lyceum where he remained for
two years, leaving in 1842 for the Gymnasium at Luneburg. Here, under the
direction of Professor Schmalfuss, he learned very rapidly, and is said to
have required only one week thoroughly to familiarize himself with
Legendre's Theory of Numbers.
On April 12, 1846 (Easter), he entered the University of Göttingen as a
student of Theology in accordance with his father's wishes. His passion for
mathematics, however, was so aroused by the lectures of Gauss that He
begged his father to be allowed to devote himself entirely to the studies of
his choice. For two years he studied under Jacobi at Berlin. He then
returned to Göttingen, and was graduated, his thesis being a dissertation on
the foundations of a general theory of functions of a variable complex
magnitude. In 1854 he qualified as a teacher by giving a lecture on the
"Hypothesis on which Geometry is Founded." In 1857 he became
"Professor Extraordinarius," and in 1859 was elected Corresponding
Member of the Academy of Sciences of Berlin and in 1860 a member of the
Academy of Sciences of Göttingen.
After four years of failing health, during which he visited Messina,
Palermo, Naples, Rome, Florence, Pisa and Milan, he died at Lago
Maggiore, July 20, 1866, in full possession of his faculties and conscious of
his approaching end.
SCHWEIKART, FERDINAND KARL (1780-1857), studied from 1796 to
1798 in Marburg, attending the mathematical lectures of J. K. F. Hauff. In
1812 he became professor in Charkov, a position which he held for four
years. In 1816 he became a tutor in the City of Marburg where he remained
until 1820 when he transferred his labors to Königsberg. It was during his
tutorship at Charkov, Marburg and Königsberg that he, entirely alone and

without the slightest suggestion from any man, developed and taught a non-
Euclidean geometry to the students under his care. For copy of his treatise
on non-Euclidean geometry, see Historical Sketch of the Hyperspace
Movement, Chapter II.
SCOPOGRAPHIC IMPRESSIONS—Sight perceptions fused with an
associated memory-image, and forming the basis of action on external
phenomena.
SENSOGRAPHIC IMPRESSIONS—Perceptions or impulses transmitted
through the nerves of a sense-organ; any impression acting through the
media of the senses.
SENSIBLE WORLD—The world of the senses; that which responds to the
senses; the domain of perception; the phenomenal world; world of
perceptual space.
SPACE-CURVATURE (see Curvature of Space).
SPACE-GENESIS—The process of spatial engenderment; the movement of
life as engendering agent in bringing into manifestation the kosmos; the
story of the appearance of the organized kosmos. The genesis of space can
only be symbolized, as has been done in the text, for the limitations of
human consciousness do not otherwise admit of the empirical establishment
of the notion of its detailed procedure.
SPATIALITY—Space as a dynamic, creative movement; kosmic order, as
opposed to disorder; the path of the engendering movement of life; the
place of life. Spatiality, materiality, intellectuality and geometricity or the
latent geometrism of the kosmos are thought of as being consubstantial and
interdependent; but, of these, spatiality is regarded as the substance out of
which the latter three are elaborated.
SUPERCONCEPTUAL—The purely intuitional; an act of cognition
performed without the detailed work of conception derived from sense-data;
conception of intuitions and their inter-relations; the Thinker's
consciousness freed from intellectual characterization.
SUPERPERCEPTION—Perception of conceptual relations; a state of
cognition wherein, instead of receiving percepts or images from the external

world, then elaborating them into concepts, the Thinker apprehends
composite images or concepts at first hand. It is a power which the liberated
mind of the future will possess owing to the growing automatism of the
intellect and the more facile expression of the intuitional consciousness.
TESSERACT (Gr. Tessera, four, cube, tessella)—A hypercube (see Chapter
V.)
THINKER (Skt. Manu, thinker)—The real, spiritual man, as differentiated
from his perceptive vehicles—mind, emotions and physical body; the
omnipsychic intelligence who receives, classifies, interprets and preserves
percepts; the manipulator of concepts; in fine, the higher, spiritual man.
The Thinker uses the various perceptive instrumentalities as so many
tentacles or antennae by which he contacts the sensible world and makes
the necessary adaptations to environment. He is the pure intelligence which
is the source of all cognitive motivation; opposed to ego, because the
egopsychic instrumentality is essentially an individualizing, separative
agency; while the Thinker's omnipsychic intelligence is the basis of his
unity with the universal intelligence. This conception of the Thinker implies
that, as a spiritual intelligence, he is within and without the body, filling it
as the ocean fills the sponge, encompassing, enveloping it and, at the same
time, originating the totality of activities which manifest in and through the
body. He is limited, therefore, in his manifestations in the sensible world
only by the pliability of his vehicles.
TRANSFINITY—A state or condition that is incomprehensible to finite
intelligence; that which transcends the finite, yet is not infinite; less than
infinity and greater than finity. Space is referred to as being transfinite
rather than infinite in extent. But space transfinite should be distinguished
from space "finite though unbounded." For, there would seem to be little
worthy of choice between a "finite, unbounded space" and an infinite one.
The absence of boundary would naturally suggest an infinite extent. And
although Riemann who is the author of the "unbounded" space arbitrarily
determined that such a space should be a manifold possessing a measure of
curvature which could be determined either by counting or actual
measurement, he undoubtedly knew, nevertheless, that while each manifold
might be an "unbounded" space the totality of such manifolds, infinite in

number, must also be infinite in extent. It would seem to do violence to
common sense, if not to logical necessity, to view space both as
"unbounded" and finite in extent, yet there would be no such difficulty in
the recognition of space as being both transfinite and finite; because it is
conceivable that the extent and character of space finite should transcend a
finite intellectuality, and yet not be infinite.
TRIDIM—A being whose scope of consciousness is limited to a space of
three dimensions, as ordinary human beings. TRIDIMENSIONALITY—
That quality possessed by perceptual space by virtue of which it is
necessary and sufficient to have three coördinates, and only three, to
establish the position of a point.
UNODIM—A hypothetical being assumed to have a consciousness limited
to linear or one-space.
ZONES OF AFFINITY—Regions in the domain of intellectuality wherein
minds, possessing a common differential, rate of vibration or quality, adhere
to certain tenets from choice. Schools of philosophy, religions, and all those
major divisions of intellectual effort which divide and subdivide intellectual
allegiance are believed to take their rise in this property of intellectuality in
virtue of which all minds having a similar coefficient gravitate towards a
common agreement, especially where the movement is voluntary and
untrammeled.

PART ONE
THE ESSENTIALS OF THE GEOMETRY OF
HYPERSPACE AND THEIR
SIGNIFICATIONS

CHAPTER I

The Prologue

On the Variability of Psychic Powers—The Discovery of the Fourth
Dimension Marks a Distinct Stage in Psychogenesis—The Non-
Methodical Character of Discoveries—The Three Periods of
Psychogenetic Development—The Scope and Permissibility of
Mathetic License—Kosmic Unitariness Underlying Diversity.

In presenting this volume to the public profound apologies are made to the
professional mathematician for the temerity which is shown thereby. All
technical discussion of the problems pertinent to the geometry of
hyperspace, however, has been carefully avoided. The reader is, therefore,
referred to the bibliography published at the end of this volume for matter
relating to this aspect of the subject. The aim rather has been to outline
briefly the progress of mathematical thought which has led up to the idea of
the multiple dimensionality of space; to state the cardinal principles of the
Non-Euclidean geometry and to offer an interpretation of the
metageometrical concept in the light of the evolutionary nature of human
faculties and material characteristics and properties.
The onus of this treatise is, therefore, to distinguish between what is
commonly known as sensible space and that other species of space known
as geometric spaces. Also to show that the notion which has been styled
hyperspace is nothing more nor less than an evidence of the faint, early
outcroppings in the human mind of a faculty which, in the course of time,
will become the normal possession of the entire human race. Thus the
weight of all presentations will be to give currency to the belief, very
strongly held, that humanity, now in its infancy, is yet to evolve faculties
and capabilities, both mental and spiritual, to a degree hitherto viewed as
inconceivable.

On this view it must appear that the faculty of thought including the powers
of imagination and conceptualization are not psychological invariants, but,
on the other hand, are true variants. They are, consequently, answerable to
the principle of evolution just as all vital phenomena are. Some have
thought that no matter what idea may come into the mind of the human race
or at what time the idea may be born the mind always has been able to
conceive it. That is, many believe that the nature of mind is such that no
matter how complex an idea may be there has always been in the mind the
power of conceiving it. But this view cannot be said to have the support of
any trustworthy testimony. If so, then the mind must at once be recognized
as fully matured and capable during every epoch of human evolution, no
less in the first than in the latest, which, of course, is absurd. It is
undoubtedly more reasonable and correct to believe that the powers of
conceptualization are matters of evolutionary concern. For instance, the
assertion that the mind was incapable of conceiving, in the realm of
theology, a non-anthropomorphic god, or, in the field of biology, the
doctrine of evolution, or, in the domain of invention, the wireless telegraph,
or, in mathematics, the concept of hyperspace before the actual time of
these conceptions, cannot be successfully controverted.
In fact, it may be laid down as one of the first principles of psychogenesis
that the mind rarely, if ever, conceives an idea until it has previously
developed the power of conceptualizing it and giving it expression in the
terms of prior experience. As in the growth of the body there are certain
processes which require the full development of the organ of expression
before they can be safely executed so in the phyletic development of
faculties there are certain ideas, conceptions and scopes of mental vision
which cannot be visualized or conceptualized until the basis for such
mentation has been laid by the appearance of previously developed faculties
of expression. And especially is this true of the intellect. Inasmuch as the
entire content of the intellect is constituted of sense-derived knowledge,
with the exception of intuitions which are not of intellectual origin though
dependent upon the intellect for interpretation, there can be no doubt as to
the necessity of there being first deposed in the intellect a sense-derived
basis for intellection before it can become manifest. The Sensationalists, led
by Leibnitz, propounded as their fundamental premise this dictum: "There
is nothing in the intellect which has not first been in the senses except the

intellect itself," and this has never been gainsaid by any school that could
disprove it. The intuitionalist does not deny it: he merely claims that we are
the recipients of another form of knowledge, the intuitional, which, instead
of being derived from sense-experience, is projected into the intellectual
consciousness from another source which we designate the Thinker. Thus,
from the two forms of consciousness, come into the area of awareness
truths that spring from entirely different sources. From the one source a
steady stream of impressions flow constituting the substance of intellectual
consciousness; from the other only a drop, every now and then, falls into
the great inrushing mass so as to add a dim phosphorescence to an
otherwise unilluminated pool. Obviously, when there is a lack of sensuous
data from which a certain concept may be elaborated there can be no
conception based upon them, and as the variety and quality of concepts are
in exact proportion to the variety and quality of sense-experience there can
be no demand for a particular species of notions such as might be
elaborated out of the absent or non-existent perception. Hence, the power of
conceiving springs forth from sense-experience. Sense-experience is
essentially a mass of perceptions: these, creating a demand for additional
adaptations, conspire, as if, to evoke the power or faculty to meet the
demand, and consequently, an added conceptualization is made.
Progress in human thought is made in a manner similar to that which
prevails in the development of other natural processes, such as, the power
of speech in the child. In the development of this faculty there are certain
definite stages which appear in due sequence. The child is not gifted with
the power of speech at once. It comes, by gradual and sometimes painful
growth, into a full use of this faculty. Now, much the same principle holds
true in the evolution of the mind in the human species. It is an established
biologic principle that the ontogenetic processes manifested in the
individual are but a recapitulation of the phylogenetic processes which are
observable in the progress of the entire species. The view becomes even
more cogent when note is taken of the fact that the foetus, during
embryogenesis, passes successively through stages of growth which have
been shown to be analogous, if not identical, with those stages through
which the human species has developed, namely, the mineral, vegetal and
animal.

Wherefore it may be said that the fourth dimensional concept marks a
distinct stage in psychogenesis or evolution of mind. It required, as will be
shown in Chapter II, nearly two thousand years for it to germinate, take root
and come to full fruition. For it was not until the early years of the
nineteenth century that mathematicians, taking inspiration from Riemann
(1826-1866) fully recognized the concept as a metaphysical possibility, or
even the idea was conceived at all. Serious doubt is entertained as to the
possibility of its conception by any human mind before this date, that is, the
time when it was actually born. Prior to that time, mathematical thought
was taking upon itself that shape and tendence which would eventually lead
to the discovery of hyperspace; but it could not have reached the zenith of
its upward strivings at one bound. That would have been unnatural.
Such is the constitution of the mind that although it is the quantity which
bridges the chasm between the two stages of man's evolution when he
merely thinks and when he really knows it is entirely under the domain of
law and must observe the times and seasons, as it were, in the performance
of its functions. The scope of psychogenesis is very broad, perhaps
unlimited; but its various stages are very clearly defined notwithstanding
the breadth of its scope of motility. And while the distance from moneron to
man, or from feeling to thinking is vast, the gulf which separates man, the
Thinker, from man, the knower, is vaster still. Who, therefore, can say what
are the delights yet in store for the mind as it approaches, by slow paces, the
goal whereat it will not need to struggle through the devious paths of
perceiving, conceiving, analyzing, comparing, generalizing, inferring and
judging; but will be able to know definitely, absolutely and instantaneously?
That some such consummation as this shall crown the labors of mental
evolution seems only natural and logical.
It may be thought by some that the character and content of revelational
impressions constitute a variation from the requirements of the law above
referred to, but a little thought will expose the fallacy of this view. The
nature of a revealed message is such as to make it thoroughly amenable to
the restrictions imposed by the evolutionary aspects of mind in general.
That this is true becomes apparent upon an examination of the four cardinal
characteristics of such impressions. First, we have to consider the indefinite
character of an apocalyptic ideograph which is due to its symbolic nature.
This is a feature which relieves the impression of any pragmatic value

whatsoever, especially for the period embracing its promulgation. Then,
such cryptic messages may or may not be understood by the recipient in
which latter case it is nonpropagable. Second, the necessity of previous
experience in the mind of the recipient in order that he may be able to
interpret to his own mind the psychic impingement. The basis which such
experience affords must necessarily be present in order that there may be an
adequate medium of mental qualities and powers in which the ideogram
may be preserved. A third characteristic is that revelations quite invariably
presuppose a contemplative attitude of mind which, in the very nature of the
case, superinduces a state of preparedness in the mind for the proper
entertainment of the concept involved. This fact proves quite conclusively
that revelational impressions are not exceptions to the general rule. Lastly, a
dissatisfaction with the conditions with which the symbolism deals or to
which it pertains is also a prerequisite. This condition is really that which
calls forth the cryptic annunciation, and yet, preceding it is a long series of
causes which have produced both the conditions and the revolt which the
revelator feels at their presence. In view of the foregoing, it would appear
that objections based upon the alleged nonconformity of the revealed or
inspired cannot be entertained as it must be manifest that it, too, falls within
the scope of the laws of mental growth.
Discoveries, whether of philosophical or mechanical nature, or whether of
ethical or purely mathematical tendence, are never the results of a
deliberate, methodical or purposive reflection. For instance, let us take Lie's
"transformation groups," mathematic contrivances used in the solution of
certain theorems. Now, it ought to be obvious that these mathetic
machinations were not discovered by Sophus Lie as a consequence of any
methodic or purposeful intention on his part. That is, he did not set out
deliberately to discover "transformation groups." For back of the "groups"
lay the entire range of analytic investigations; the mathematical thought of
more than a thousand years furnished the substructure upon which Lie built
the conception of his "groups." Similarly, it may be said with equal
assurance that no matter how great the intensity of thought, nor how
purposeful, nor of how long duration the series of concentrated abstractions
which led up to the invention of the printing press, the linotype or multiplex
printing press of our day could not have been produced abruptly, nor by use
of the mental dynamics of the human mind of remoter days. Its production

had to follow the path outlaid by the laws of psychogenesis and await the
development of those powers which alone could give it birth. The whole
question resolves itself, therefore, into the idea of the complete
subserviency of the mind, in all matters of special moment, to the laws
aforementioned. The supersession of the law of its own life by the mind is
well-nigh unthinkable, if not quite so.
If we now view the history of the mind as manifested in the human species,
three great epochs which divide the scope of mental evolution into more or
less well-defined stages present themselves. These are: first, the formative
stage; second, the determinative stage; third, the stage of freedom, or the
elaborative stage.
In all of the early races of men, through every step which even preceded the
genus homo, the generic mind was being formulated. It was being given
shape, outline and direction. All of the first stage, the formative, was
devoted to organization and direction. Those elementary sensations which
constituted the basis of mind in the primitive man were accordingly
strongly determinative of what the mind should be in these latter days. To
this general result were contributed the effects of the activity of cells,
nerves, bones, fibers, muscles and the blood.
The formative period naturally covered a very extensive area in the history
of mind or psychogenetic development. It was followed closely, but almost
insensibly, by the determinative period during which all the latent powers,
capacities and faculties which were the direct products of the formative
period were being utilized in meeting the demands of the law of necessity.
The making of provisions against domestic want, against the attacks of
external foes; the combating of diseases, physical inefficiency, the weather,
wild beasts, the asperities of tribal enmities; as well as furthering the
production of art, music, sculpture, the various branches of handiwork,
literature, philosophies, religions and the effectuation of all those things
which now appear as the result of the mental activity of the present-day
man make up the essence and purpose of the determinative period.
Signs of the dawn of the elaborative stage, also called the stage of freedom,
have been manifest now for upwards of three centuries and it is, therefore,
in its beginnings. It is not fully upon us. Not yet can we fully realize what it

may mean, nor can we unerringly forecast its ultimate outcome; but we feel
that it is even now here in all the glories of its matutinal freshness. And the
mind is beginning to be free from the grinding necessities of the
constructive period having already freed itself from the restrictive
handicaps of the primeval formulation period. Already the upgrowing
rejuvenescences so common at the beginning of a new period are
commencing to show themselves in every department of human activity in
the almost universal desire for greater freedom. And this is particularly
noticeable in the many political upheavals which, from time to time, are
coming to the surface as well as in the countless other aspects of the wide-
spread renaissance. Perhaps the time may come, never quite fully, when
there will be no longer any necessity to provide against the external
exigencies of life; perhaps, the time will never be when the mind shall no
more be bound by the law of self-preservation, not even when it has
attained unto the immortality of absolute knowledge; yet, it is intuitively
felt that it must come to pass that the mind shall be vastly freer than it is to-
day. And with this new freedom must come liberation from the necessities
of the elementary problems of mere physical existence.
The inference is, therefore, drawn that the fourth dimensional concept, and
all that it connotes of hyperspace or spaces of n-dimensionality are some of
the evidences that this stage of freedom is dawning. And the mind, joyous at
the prospect of unbounded liberty which these concepts offer, cannot
restrain itself but has already begun to revel in the sunlit glories of a newer
day. What the end shall be; what effect this new liberty will have on man's
spiritual and economic life; and what it may mean in the upward strivings
of the Thinker for that sublime perpetuity which is always the property of
immediate knowledge no one can hope, at the present time, to fathom. It is,
however, believed with Keyser that "it is by the creation of hyperspaces
that the rational spirit secures release from limitation"; for, as he says, "in
them it lives ever joyously, sustained by an unfailing sense of infinite
freedom."
The elevating influence of abstract thinking, such as excogitation upon
problems dealing with entities inhabiting the domain of mathesis is, without
doubt, incalculable in view of the fact that it is only through this kind of
thought that the spirit is enabled to reach its highest possibilities. This is
undoubtedly the philosophy of those religious and occult exercises known

as "meditations," and this perhaps was the main idea in the mind of the
Hebrew poet when he exclaimed: "Let the words of my mouth and the
meditation of my heart be acceptable in thy sight, O Lord, my strength and
my Redeemer." The principal, if not the only, value possessed by the
"summitless hierarchies of hyperspaces" which the mathematician
constructs in the world of pure thought is the enrichening and ennobling
influence which they exert upon the mind. But admittedly this unbounded
domain of mathetic territory which he explores and which he finds "peopled
with ideas, ensembles, propositions, relations and implications in endless
variety and multiplicity" is quite real to him and subsists under a reign of
law the penalties of which, while not as austere and unreasonable as some
which we find in our tridimensional world, are nevertheless quite as
palpable and as much to be feared. For the orthodoxy of mathematics is as
cold and intolerant as ever the religious fanatic can be. But the reality and
even the actuality which may be imputed to the domain of mathesis is of an
entirely different quality from that which we experience in our world of
triune dimensionality and it is a regrettable error of judgment to identify
them. It ought, therefore, never be expected, nor is it logically reasonable to
assume that the entities which inhabit the mathetic realm of the analyst
should be submissive to the laws of sensible space; nor that the conditions
which may be found therein can ever be made conformable to the
conditions which exist in perceptual space.
It was Plato's belief that ideas alone possessed reality and what we regard
as actual and real is on account of its ephemerality and evanescence not real
but illusionary. This view has been shared by a number of eminent thinkers
who followed, with some ostentation, the lead established by Plato. For a
considerable period of time this school of thinkers had many adherents; but
the principles at length fell into disrepute owing to the absurdities indulged
in by some of the less careful followers. The realism, or for that matter, the
actuality of ideas cannot be denied; yet it is a realism which is neither to be
compared with the physical reality of sense-impressions nor its phenomena.
The character and peculiarity of ideas are in a class apart from similar
notions of perceptual space content. It is as if we were considering the
potentialities of the spirit world and the entities therein in connection with
incarnate entities which in the very nature of the case is not allowable.
Furthermore, it is unreasonable to suppose that the conditions on a higher

plane than the physical can be made responsible to a similar set of
conditions on the physical plane.
There are certain astronomers who base their speculations as to the
habitability of other planets upon the absurd hypothesis that the conditions
of life upon all planets must be the same as those on the earth, forgetting
that the extent of the universe and the scope of motility of life itself are of
such a nature as to admit of endless variations and adaptations. There is a
realism of ideas and a realism of perceptual space. Yet this is no reason why
the two should be identified. On the other hand, owing to the diversity in
the universe, every consideration would naturally lead to the assumption
that they are dissimilar. To invest ideas, notions, implications and inferences
with a reality need not logically or otherwise affect the reality of a stone, a
fig, or even of a sense-impression.
To a being on the spirit levels our grossest realities must appear as non-
existent. They are neither palpable nor contactable in any manner within the
ordinary range of physical possibilities. For us his gravest experiences can
have no reality whatsoever; for no matter how real an experience may be to
him it is altogether beyond our powers of perception, and therefore, to us
non-existent also. It should, however, be stated that the state of our
knowledge about a given condition can in no way affect its existence. It
merely establishes the fact that two or more realities may exist independent
of one another and further that the gamut of realism in the universe is
infinite and approaches a final state when its occlusion into absolute being
follows as a logical sequence.
Recurring to the consideration of the reality of spirit-realms as compared
with that of sensible space, it comes to view that our idealism, that is, the
idealism which is a quality of conceptualization, may be regarded as
identical with their realism, at least as being on the same plane as it. Stated
differently, the things that are ideal to us and which constitute the data of
our consciousness may be as real to them as the commonest object of sense-
knowledge is to us. What, therefore, appears to us as the most ethereal and
idealistic may have quite a realistic character for them.
Ultimately, however, and in the final deeps of analysis it will be found
undoubtedly that both our realism and our idealism as well as similar

qualities of the spirit world are in all essential considerations quite
illusionary. All knowledge gained in a condition short of divinity itself is
sadly relative. Even mathematical knowledge falls far short of the absolute,
the fondest claims of the orthodox mathematician to the contrary
notwithstanding. It has been said frequently that a mathematical fact is an
absolute fact and that its verity, necessity and certainty cannot be
questioned anywhere in the universe whether on Jupiter, Neptune,
Fomalhaut, Canopus or Spica. But having so declared, the fact of the sheer
relativity of our knowledge is not disturbed thereby nor controverted.
Happily, neither distance nor a lack of distance can in any way affect the
quality of human knowledge, mathematical knowledge not excepted. That
can only be affected by conditions which cause it to approach perfection
and nothing but evolution can do that.
In the light of results obtained in analytic investigations the question of the
flexibility of mathematical applications becomes evident and one instead of
being convinced of the vaunted invariability of the laws obtaining in the
world of mathesis is, on the other hand, made aware of the remarkable and
seemingly unrestrained facility with which these laws may be made to
apply to any conditions or set of assumptions within the range of the mind's
powers of conception. Mathematicians have deified the definition and
endowed it with omnific powers imputing unto it all the attributes of
divinity—immutability, invariance, and sempiternity. In this they have erred
grievously although, perhaps, necessarily. Mathetic conclusions are entirely
conditional and depend for their certainty upon the imputed certitude of
other propositions which in turn are dependent, in ever increasing and
endlessly complex relations, upon previously assumed postulates. These
facts make it exceedingly difficult to understand the attitude of mind which
has obscured the utter mutability and consequent ultimate unreliability of
the fine-spun theories of analytic machinations.
The apriority of all mathematical knowledge is open to serious questioning.
And although there is no hesitancy in admitting the basic agreement of the
most primary facts of mathematical knowledge with the essential character
of the intellect the existence of well-defined limits for such congruence
cannot be gainsaid. The subjunctive quality of geometric and analytical
propositions is made apparent by an examination of the possibilities falling
within the scope of permissibility offered by mathetic license. For instance,

privileged to proceed according to the analytic method it is allowable to
reconstruct the sequence of values in our ordinary system of enumeration so
as to admit of the specification of a new value for say, the entire series of
odd numbers. This value might be assumed to be a plus-or-minus one,
dependent upon its posture in the series. That is, all odd numbers in the
series beginning with the digit 3, and continuing, 5, 7, 9, 11, 13, 15, 17, 19,
... n, could be assumed to have only a place value which might be regarded
as a constant-variable. The series of even numbers, 2, 4, 6, 8, 10, 12, 14, 16,
... n, may be assumed to retain their present sequence values. Under this
system the digit 1 would have an absolute value; all other odd numbers
would have a constant-variable value; constant, because always no more
nor less than 1 dependent upon their place in the operations and whether
their values were to be applied by addition or subtraction to or from one of
the values in the even number series; variable, because their values would
be determinable by their application and algebraic use.
There would, of course, be utilitarian objection to a system of this kind; but
under the conditions of a suppositionary hypothesis, it would be self-
consistent throughout, and if given universal assent would suit our purposes
equally as well as our present system. But the fact that this can be done
under the mathematic method verily proves the violability of mathematical
laws and completely negatives the assumption that the sum of any two
digits, as say 2 plus 2 equals 4, is necessarily and unavoidably immutable.
For it can be seen that the sum-value of all numbers may be made
dependent upon the assumed value which may be assigned to them or to
any collection thereof. Furthermore, it is a matter of historical knowledge
that it was the custom of ancient races of men to account for values by an
entirely different method from what we use to-day. The latter is a result of
evolution and while experience teaches that it is by far the most convenient,
it is nevertheless true that earlier men managed at least fairly well on a
different basis. Then, too, the fact of the utility and universal applicability
of our present system, based upon universal assent, does not obviate the
conclusion that any other system, consistent in itself, might be made to
serve our purposes as well.
It ought to be said, however, in justice to the rather utilitarian results
obtained by La Grange, Helmholtz, Fechner, and others who strove to
make use of their discoveries in analysis in solving mechanical,

physiological and other problems of more or less pragmatic import that, in
so far as this is true, mathematical knowledge must be recognized as being
consistent with the necessities of a priori requirements. But even these
results may not be regarded as transcending the scope of the most
fundamental principles of sense-experience. It will be discovered finally,
perhaps, that the energy spent in elaborating complicate series of analytic
curiosities has been misappropriated. It will then be necessary to turn the
attention definitely to the study of that which lies not at the terminus of the
intellect's modus vivendi, but which is both the origin of the intellect and its
eternal sustainer—the intuition, or life itself. This can result in nothing less
than the complete spiritualization of man's mental outlook and the
consequent inevitable recognition of the underlying and ever-sustaining
one-ness of all vital manifestations.
One of the curiosities of the tendency in man's mind to specialize in
analytics, whether in the field of pure mathematics or metaphysics, is the
fact that it almost invariably leads to an attempt to account for cosmic
origins on the basis of paralogic theories. This in times past has given rise
to the theory of the purely mechanical origin of the universe as well as
many other fantastic fallacies the chief error of which lay in the failure to
distinguish between the realism of mental concepts and that of the sensible
world. In spite of this, however, one is bound to appreciate the beneficial
effects of analytic operations because they serve as invigorants to mental
growth. It could not, therefore, be wished that there were no such thing as
analytics; for the equilibria-restoring property of the mind may at all times
be relied upon to minimize the danger of excesses in either direction. Just as
the tide flowing in flows out again, thereby restoring the ocean's
equilibrium, so the mind ascending in one generation beyond the safety
mark has its equilibrium restored in the next by a relinquishment of the
follies of the former.
The four-space is one of the curiosities of analytics; yet it need not be a
menace to the sane contemplation of the variegated products of analysis.
Safety here abides in the restraint which should characterize all discussion
and application of the concept. If enthusiasts would be content not to
transport the so-called fourth dimensional space out of the sphere of
hyperspace and cease trying to speculate upon the results of its interposal
into three space conditions, which is in every way a constructual

impossibility, there could not be any possible objection to its due
consideration. This would obviate the danger of calling into question either
the sincerity or perspicacity of those whose enthusiasm tempts them to
transgress the limits of propriety in their behavior towards the inquiry.
There is but one life, one mind, one extension, one quantity, one quality,
one being, one state, one condition, one mood, one affection, one desire,
one feeling, one consciousness. There is also but one number and that is
unity. All so-called integers are but fractional parts of this kosmic unity. The
idea represented by the word two really connotates two parts of unity and
the same is true of a decillion, or any number of parts. These are merely the
infinitesimals of unity and they grow less in size and consequence as the
divisions increase in number. The analysis of unity into an infinity of parts
is purely an a posteriori procedure. That it is an inherent mind-process is a
fallacy. All our common quantities, as the mile, kilometer, yard, foot, inch,
gallon, quart, are conventional and arbitrary and susceptible of wide
variations. As the basis of all physical phenomena is unity; it is only in the
ephemeral manifestations of sensuous objects that they appear as separate
and distinct quantities.
We see on a tree many leaves, many apples or cherries; on a cob many
grains of corn. We have learned to assign to each of these quantities in their
summation a sequence value. But this is an empirical notion and cannot be
said to inhere in the mind itself. Let us take, for instance, the mustard seed.
If it were true that in one of these seeds there existed all the subsequent
seeds which appear in the mustard plant as separate and identifiable
quantities, and not in essence, then there would perhaps be warrant for the
notion that diversity, as the calculable element, is an a priori conception.
But, as this is not the case and since diversity is purely empirical and
pertains only to the efflorescence of the one life it is manifestly absurd to
take that view.
Under the most charitable allowances, therefore, there can be but two
quantities—unity and diversity; yet not two, for these are one. Unity is the
one quantity and diversity is the division of unity into a transfinity of parts.
Unity is infinite, absolute and all-inclusive. Diversity is finite although it
may be admitted to be transfinite, or greater than any assignable value.
Unity alone is incomprehensible. In order to understand something of its

nature we divide it into a diversity of parts; and because we fail to
understand the transfinity of the multitude of parts we mistakenly call them
infinite.
When analysis shall have proceeded far enough into the abysmal mysteries
of diversity; when the mathematical mind shall have been overcome by the
overwhelming perplexity of the maze of diverse parts, it shall then fall
asleep and upon awaking shall find that wonderfully simple thing—unity. It
is the one quantity that is endowed with a magnitude which is both
inconceivable and irresolvable. The one ineluctable fact in the universe is
the incomprehensibility and all-inclusivity of one-ness. It is
incomprehensible, inconceivable and infinite at the present stage of mind
development. But the goal of mind is to understand the essential character
of unity, of life. Its evolution will then stop, for it will have reached the
prize of divinity itself whereupon the intellect exalted by and united with
the intuition shall also become one with the divine consciousness.

CHAPTER II

Historical Sketch of the Hyperspace Movement

Egypt the Birthplace of Geometry—Precursors: Nasir-Eddin,
Christoph Clavius, Saccheri, Lambert, La Grange, Kant—Influence
of the Mecanique Analytique—The Parallel-Postulate the Root and
Substance of the Non-Euclidean Geometry—The Three Great Periods:
The Formative, Determinative and Elaborative—Riemann and the
Properties of Analytic Spaces.

The evolution of the idea of a fourth dimension of space covers a long
period of years. The earliest known record of the beginnings of the study of
space is found in a hieratic papyrus which forms a part of the Rhind
Collection in the British Museum and which has been deciphered by
Eisenlohr. It is believed to be a copy of an older manuscript of date 3400
B. C., and is entitled "Directions for Knowing All Dark Things" The copy is
said to have been made by Ahmes, an Egyptian priest between 1700 and
1100 B. C. It begins by giving the dimensions of barns; then follows the
consideration of various rectilineal figures, circles, pyramids, and the value
of pi ([Greek: p]). Although many of the solutions given in the manuscript
have been found to be incorrect in minor particulars, the fact remains that
Egypt is really the birth-place of geometry. And this fact is buttressed by the
knowledge that Thales, long before he founded the Ionian School which
was the beginning of Greek influence in the study of mathematics, is found
studying geometry and astronomy in Egypt.
The concept of hyperspace began to germinate in the latter part of the first
century, B. C. For it was at this date that Geminos of Rhodes (B. C. 70)
began to think seriously of the mathematical labyrinth into which Euclid's
parallel-postulate most certainly would lead if an attempt at demonstrating
its certitude were made. He recognized the difficulties which would engage

the attention of those who might venture to delve into the mysterious
possibilities of the problem. There is no doubt, too, but that Euclid himself
was aware, in some measure at least, of these difficulties; for his own
attitude towards this postulate seems to have been one of noncommittance.
It is, therefore, not strange that the astronomer, Ptolemy (A. D. 87-165),
should be found seeking to prove the postulate by a consideration of the
possibilities of interstellar triangles. His researches, however, brought him
no relief from the general dissatisfaction which he felt with respect to the
validity of the problem itself.
For nearly one thousand years after the attempts at solving the postulate by
Geminos and Ptolemy, the field of mathematics lay undisturbed. For it was
at this time that there arose a strange phenomenon, more commonly known
as the "Dark Ages," which put an effectual check to further research or
independent investigations. Mathematicians throughout this long lapse of
time were content to accept Euclid as the one incontrovertible,
unimpeachable authority, and even such investigations as were made did
not have a rebellious tendence, but were mainly endeavors to substantiate
his claims.
Accordingly, it was not until about the first half of the thirteenth century
that any real advance was made. At this time there appeared an Arab, Nasir-
Eddin (1201-1274) who attempted to make an improvement on the problem
of parallelism. His work on Euclid was printed in Rome in 1594 A. D.,
about three hundred and twenty years after his demise and was
communicated in 1651 by John Wallis (1616-1703) to the mathematicians
of Oxford University. Although his calculations and conclusions were
respectfully received by the Oxford authorities no definite results were
regarded as accomplished by what he had done. It is believed, however, that
his work reopened speculation upon the problem and served as a basis,
however slight, for the greater work that was to be done by those who
followed him during the next succeeding eight hundred years.
About twenty years before the printing of the work of Nasir-Eddin,
Christoph Clavius (1574) deduced the axiom of parallels from the
assumption that a line whose points are all equidistant from a straight line is
itself straight. In his consideration of the parallel-postulate he is said to have
regarded it as Euclid's XIIIth axiom. Later Bolyai spoke of it as the XIth

and later still, Todhunter treated it as the XIIth. Hence, there does not seem
to have been any general unanimity of opinion as to the exact status of the
parallel-postulate, and especially is this true in view of the uncertainty now
known to have existed in Euclid's mind concerning it.
Girolamo Saccheri (1667-1733), a learned Jesuit, born at San Remo, came
next upon the stage. And so important was his work that it will perpetuate
the memory of his name in the history of mathematics. He was a teacher of
grammar in the Jesuit Collegio di Brera where Tommaso Ceva, a brother of
Giovanni, the well-known mathematician, was teacher of mathematics. His
association with the Ceva brothers was especially beneficial to him. He
made use of Ceva's very ingenious methods in his first published book,
1693, entitled Solutions of Six Geometrical Problems Proposed by Count
Roger Ventimiglia.

Fig. 1.

Saccheri attacked the problem of parallels in quite a new way. Examining a
quadrilateral, ABCD, in which the angles A and B are right angles and the
sides AC and BD are equal, he determined to show that the angles C and D
are equal. He also sought to prove that they are either right angles, obtuse or
acute. He undertook to prove the falsity of the latter two propositions (that
they are either obtuse or acute), leaving as the only possibility that they
must be right angles. In doing so, he found that his assumptions led him
into contradictions which he experienced difficulty in explaining.
His labors in connection with the solution of the problems proposed by
Count Ventimiglia, including his work on the question of parallels, led
directly into the field of metageometrical researches, and perhaps to him as
to no other who had preceded him, or at least to him in a larger degree,
belongs the credit for a continued renewal of interest in that series of

investigations which resulted in the formulation of the non-Euclidean
geometry.
The last published work of Saccheri was a recital of his endeavors at
demonstrating the parallel-postulate. This received the "Imprimatur" of the
Inquisition, July 13, 1733; the Provincial Company of Jesus took possession
of the book for perusal on August 16, 1733; but unfortunately within two
months after it had been reviewed by these authorities, Saccheri passed
away.
All efforts which had been made prior to the work of Saccheri were based
upon the assumption that there must be an equivalent postulate which, if it
could be demonstrated, would lead to a direct, positive proof of Euclid's
proposition. Although these and all other attempts at reaching such a proof
have signally failed and although it may correctly be said that the entire
history of demonstrations aiming at the solution of the famous postulate has
been one long series of utter failures, it can be asserted with equal certitude
that it has proven to be one of the most fruitful problems in the history of
mathematical thought. For out of these failures has been built a
superstructure of analytical investigations which surpasses the most
sanguine expectations of those who had labored and failed.
In 1766 John Lambert (1728-1777) wrote a paper upon the Theory of
Parallels dated Sept. 5, 1766, first published in 1786, from the papers left
by F. Bernoulli, which contained the following assertions:[2]
1. The parallel-axiom needs proof, since it does not hold for geometry on
the surface of the sphere.
2. In order to make intuitive a geometry in which the triangle's sum is less
than two right angles, we need an "imaginary" sphere (the pseudosphere).
3. In a space in which the triangle's sum is different from two right angles
there is an absolute measure (a natural unit for length).
At this time Immanuel Kant (1724-1804), the noted German metaphysician,
was in the midst of his philosophical labors. And it is believed that it was he
who first suggested the idea of different spaces. Below is given a statement
taken from his Prolegomena[3] which corroborates this view.

"That complete space (which is itself no longer the boundary of
another space) has three dimensions, and that space in general cannot
have more, is based on the proposition that not more than three lines
can intersect at right angles in one point.... That we can require a line
to be drawn to infinity, a series of changes to be continued (for
example, spaces passed through by motion) in indefinitum,
presupposes a representation of space and time which can only attach
to intuition."

His differentiation between space in general and space which may be
considered as the "boundary of another space" shows, in the light of the
subsequent developments of the mathematical idea of space that he very
fully appreciated the marvelous scope of analytic spaces. His conception of
space, therefore, must have had a profound influence upon the mathematic
thought of the day causing it to undergo a rapid reconstruction at the hands
of geometers who came after him.
Under the masterly influence of La Grange (1736-1813) the idea of
different spaces began to take definite shape and direction; the geometry of
hyperspace began to crystallize; and the field of mathesis prepared for the
growth of a conception the comprehension of which was destined to be the
profoundest undertaking ever attempted by the human mind. Unlike most
great men whom the world learns tardily to admire, La Grange lived to see
his talents and genius fully recognized by his compeers; for he was the
recipient of many honors both from his countrymen and his admirers in
foreign lands. He spent twenty years in Prussia where he went upon the
invitation of Frederick the Great who in the Royal summons referred to
himself as the "greatest king in Europe" and to La Grange as the "greatest
mathematician" in Europe. In Prussia the Mecanique Analytique and a long
series of memoirs which were published in the Berlin and Turin
Transactions were produced. La Grange did not exhibit any marked taste
for mathematics until he was 17 years of age. Soon thereafter he came into
possession of a memoir by Halley quite by accident and this so aroused his
latent genius that within one year after he had reviewed Halley's memoir
he became an accomplished mathematician.
He created the calculus of variations, solved most of the problems proposed
by Fermat, adding a number of theorems of his own contrivance; raised the

theory of differential equations to the position of a science rather than a
series of ingenious methods for the solution of special problems and
furnished a solution for the famous isoperimetrical problem which had
baffled the skill of the foremost mathematicians for nearly half a century.
All these stupendous tasks he performed by the time he reached the age of
nineteen.
The Mecanique Analytique is his greatest and most comprehensive work. In
this he established the law of virtual work from which, by the aid of his
calculus of variations, he deduced the whole of mechanics, including both
solids and liquids. It was his object in the Analytique to show that the whole
subject of mechanics is implicitly embraced in a single principle, and to lay
down certain formulae from which any particular result can be obtained. He
frequently made the assertion that he had, in the Mecanique Analytique,
transformed mechanics which he persistently defined as a "geometry of four
dimensions"[4] into a branch of analytics and had shown the so-called
mechanical principles to be the simple results of the calculus. Hence, there
can be no doubt but that La Grange not only completed the foundation, but
provided most of the material in his analyses and other "abstract results of
great generality" which he obtained in his numerous calculations, for the
superstructure subsequently known as the geometry of hyperspace, and in
which the fourth dimensional concept occupies a very fundamental place.
It is as if for nearly seventeen hundred years workmen, such as Geminos, of
Rhodes, Ptolemy, Saccheri, Nasir-Eddin, Lambert, Clavius, and hundred
of others who struggled with the problem of parallels, had made more or
less sporadic attempts at the excavation of the land whereon a marvelously
intricate building was to be constructed. There is no historical evidence to
show that any of them ever dreamed that the results of their labors would be
utilized in the manner in which they have been used. Then came Kant with
the wonderfully penetrating searchlight of his masterful intellect who from
the elevation which he occupied saw that the site had great possibilities, but
he had not the mathematical talent to undertake the work of actual,
methodical construction. Indeed his task was of a different sort. However,
he succeeded in opening the way for La Grange and others who followed
him. La Grange immediately seized upon the idea which for more than a
thousand years had been impinging upon the minds of mathematicians
vainly seeking lodgment and began the elaboration of a plan in accordance

with which minds better skilled in the pragmatic application of abstract
principles than his could complete the work begun. Unfortunately, on
account of his intense devotion and loyalty to the study of pure
mathematics, and when he had reached the summit of his greatness where
he stood "without a rival as the foremost living mathematician," his health
became seriously affected, causing him to suffer constant attacks of
profound melancholia from which he died on April 10, 1813.
We come now to one of the most remarkable periods in the history of
mental development. During the six hundred years between the birth of
Nasir-Eddin and the death of La Grange the entire world of mathesis was
being reconstituted. Since there had been gradually going on an internal
process which, when completed, forever would liberate the mind from the
narrow confines of consciousness limited to the three-space, it is not
surprising that we should find, in the mathematical thought of the time, an
absolutely epoch-making departure. The innumerable attempts at the
solution of the parallel-postulate, all failures in the sense that they did not
prove, have intensified greatly the esteem in which the never-dying
elements of Euclid are held to-day. And despite the fact that there may
come a time when his axioms and conclusions may be found to be
incongruent with the facts of sensuous reality; and though all of his
fundamental conceptions of space in general, his theorems, propositions and
postulates may have to give way before the searching glare of a deeper
knowledge because of some revealed fault, the perfection of his work in the
realm of pure mathematics will remain forever a master piece demanding
the undiminished admiration of mankind.
The parallel-postulate, as stated by Euclid in his Elements of Geometry,
reads as follows:

"If a straight line meet two straight lines so as to make the two interior
angles on the same side of it taken together less than two right angles,
these straight lines being continually produced, shall at length meet
upon that side on which are the angles which are less than two right
angles."

On this postulate hang all the "law and the prophets" of the non-Euclidean
Geometry. In it are the virtual elements of three possible geometries.

Furthermore, it is both the warp and the woof of the loom of present-day
metageometrical researches. It is the golden egg laid by the god Seb at the
beginning of a new life cycle in psychogenesis. Its progeny are numerous—
hyperspaces, sects, straights, digons, equidistantials, polars, planars,
coplanars, invariants, quaternions, complex variables, groups and many
others. A wonderfully interesting breed, full of meaning and pregnant with
the power of final emancipations for the human intellect!
When the conclusions which were systematically formulated as a result of
the investigations along the lines of hypotheses which controverted the
parallel-postulate were examined it was found that they fell into three main
divisions, namely: the synthetic or hyperbolic; the analytic or Riemannian
and the elliptic or Cayley-Klein. These divisions or groups are based upon
the three possibilities which inhere in the conception taken of the sum of the
angles referred to in the above postulate as to whether it is equal to, greater
or less than two right angles.
The assumption that the angular sum is congruent to a straight angle is
called the Euclidean or parabolic hypothesis and is to be distinguished from
the synthetic or hyperbolic hypothesis established by Gauss, Lobachevski
and Bolyai and which assumes that the angular sum is less than a straight
angle. The elliptic or Cayley-Klein hypothesis assumes that the angular
sum is greater than a straight angle. Lobachevski, however, not satisfied
with the statement of the parallel-postulate as given by Euclid and which
had caused the age-long controversy, substituted for it the following:

"All straight lines which, in a plane, radiate from a given point, can,
with respect to any other straight line, in the same plane, be divided
into two classes—the intersecting and the non-intersecting. The
boundary line of the one and the other class is called parallel to the
given line."

This is but another way of saying about the same thing that Euclid had
declared before, and yet, curiously enough it afforded just the liberty that
Lobachevski needed to enable him to elaborate his theory.
For the purposes of this sketch the field of the development of non-
Euclidean geometry is divided into three periods to be known as: (1) the

formative period in which mathematical thought was being formulated for
the new departure; (2) the determinative period during which the
mathematical ideas were given direction, purpose and a general tendence;
(3) the elaborative period during which the results of the former periods
were elaborated into definite kinds of geometries and attempts made at
popularizing the hypotheses.
The Formative Period
Charles Frederich Gauss (1777-1855) by some has been regarded as the
most influential mathematician that figured in the formulation of the non-
Euclidean geometry; but closer examination into his efforts at investigating
the properties of a triangle shows that while his researches led to the
establishment of the theorem that a regular polygon of seventeen sides (or
of any number which is prime, and also one more than a power of two) can
be inscribed, under the Euclidean restrictions as to means, in a circle, and
also that the common spherical angle on the surface of a sphere is closely
connected with the constitution of the area inclosed thereby, he cannot
justly be designated as the leader of those who formulated the synthetic
school. And this, too, for the simple reason that, as he himself admits in one
of his letters to Taurinus, he had not "published anything on the subject." In
this same letter he informs Taurinus that he had pondered the subject for
more than thirty years and expressed the belief that there could not be any
one who had "concerned himself more exhaustively with this second part
(that the sum of the angles of a triangle cannot be more than 180 degrees)"
than he had.
Writing from Göttingen to Taurinus, November 8, 1824, and commenting
upon the geometric value of the sum of the angles of a triangle, he says:

"Your presentation of the demonstration that the sum of the angles of a
plane triangle cannot be greater than 180 degrees does, indeed, leave
something to be desired in point of geometrical precision. But this
could be supplied, and there is no doubt that the impossibility in
question admits of the most rigorous demonstration. But the case is
quite different with the second part, namely, that the sum of the angles
cannot be smaller than 180 degrees; this is the real difficulty, the rock
upon which all endeavors are wrecked.... The assumption that the sum

of the three angles is smaller than 180 degrees leads to a new geometry
entirely different from our Euclidean—a geometry which is throughout
consistent with itself, and which I have elaborated in a manner entirely
satisfactory to myself, so that I can solve every problem in it with the
exception of the determining of a constant which is not a priori
obtainable."

It appears from this correspondence that Gauss had in the privacy of his
own study elaborated a complete non-Euclidean geometry, and had so
thoroughly familiarized himself with its characteristics and possibilities that
the solution of every problem embraced within it was very clear to him
except that of the determination of a constant. He concluded the above letter
by saying:

"All my endeavors to discover contradiction or inconsistencies in this
non-Euclidean geometry have been in vain, and the only thing in it that
conflicts with our reason is the fact that if it were true there would
necessarily exist in space a linear magnitude quite determinate in
itself; yet unknown to us."

Judging from the correspondence between Gauss and Gerling (1788-1857),
Bessel (1784-1846), Schumacher and Taurinus, the nephew of
Schweikart, and that between Schweikart and Gerling, there had grown up
a general dissatisfaction in the minds of mathematicians of this period with
Euclidean geometry and especially the parallel-postulate and its
connotations. Bessel expresses this general discontent in one of his letters
to Gauss, dated February 10, 1829, in which he says:

"Through that which Lambert said and what Schweikart disclosed
orally, it has become clear to me that our geometry is incomplete, and
should receive a correction, which is hypothetical, and if the sum of
the angles of the plane triangle is equal to 180 degrees, vanishes."

The opinion of leading mathematicians at this time seems to have been
crystallizing very rapidly. Unconsciously the men of this formative period
were adducing evidence which would give form and tendence to the
developments in the field of mathesis at a later date. They appear to have

been reaching out for that which, ignis fatuus-like, was always within easy
reach, but not quite apprehensible.
A bolder student than Gauss was Ferdinand Carl Schweikart (1780-1857)
who also has been credited with the founding of the non-Euclidean
geometry. In fact, if judged by the same standards as Gauss, he would be
called the "father of the geometry of hyperspace"; for he really published
the first treatise on the subject. This was in the nature of an inclosure which
he inserted between the leaves of a book he loaned to Gerling. He also
asked that it be shown to Gauss that he might give his judgment as to its
merits.
Schweikart's treatise, dated Marburg, December, 1818, is here quoted in
full:

"There is a two-fold geometry—a geometry in the narrower sense, the
Euclidean, and an astral science of magnitude.
"The triangles of the latter have the peculiarity that the sum of the
three angles is not equal to two right angles.
"This presumed, it can be most rigorously proven: (a) That the sum of
the three angles in the triangle is less than two right angles.
"(b) That this sum becomes ever smaller, the more content the angle
incloses. (c) That the altitude of an isosceles right-angled triangle
indeed ever increases, the more one lengthens the side; that it,
however, cannot surpass a certain line which I call the constant."
Squares have consequently the following form:

Fig. 2.

"If this constant were for us the radius of the earth (so that every line
drawn in the universe, from one fixed star to another, distant 90° from
the first, would be a tangent to the surface of the earth) it would be
infinitely great in comparison with the spaces which occur in daily
life."

The above, being the first published, not printed, treatise on the new
geometry occupies a unique place in the history of higher mathematics. It
gave additional strength to the formative tendencies which characterized
this period and marked Schweikart as a constructive and original thinker.
The nascent aspects of this stage received a fruitful contribution when
Nicolai Lobachevski (1793-1847) created his Imaginary Geometry and
Janos Bolyai (1802-1860) published as an appendix to his father's
Tentamen, his Science Absolute of Space. Lobachevski and Bolyai have
been called the "Creators of the Non-Euclidean Geometry." And this
appellation seems richly to be deserved by these pioneers. Their work gave
just the impetus most needed to fix the status of the new line of researches
which led to such remarkable discoveries in the more recent years. The
Imaginary Geometry and the Science Absolute of Space were translated by
the French mathematician, J. Hoüel in 1868 and by him elevated out of
their forty-five years of obscurity and non-effectiveness to a position where

they became available for the mathematical public. To Bolyai and
Lobachevski, consequently, belong the honor of starting the movement
which resulted in the development of metageometry and hence that which
has proved to be the gateway of a new mathematical freedom.
Gauss, Schweikart, Lobachevski, Wolfgang and Janos Bolyai were the
principal figures of the formative period and the value of their work with
respect to the formulation of principles upon which was constructed the
Temple of Metageometry cannot be overestimated.
The Determinative Period
This period is characterized chiefly by its close relationship to the theory of
surfaces. Riemann's Habilitation Lecture on The Hypotheses Which
Constitute the Bases of Geometry marks the beginning of this epoch. In this
dissertation, Riemann not only promulgated the system upon which Gauss
had spent more than thirty years of his life in elaborating, for he was a
disciple of Gauss; but he disclosed his own views with respect to space
which he regarded as a particular case of manifold. His work contains two
fundamental concepts, namely, the manifold and the measure of curvature
of a continuous manifold, possessed of what he called flatness in the
smallest parts. The conception of the measure of curvature is extended by
Riemann from surfaces to spaces and a new kind of space, finite, but
unbounded, is shown to be possible. He showed that the dimensions of any
space are determined by the number of measurements necessary to establish
the position of a point in that space. Conceiving, therefore, that space is a
manifold of finite, but unbounded, extension, he established the fact that the
passage from one element of a manifold to another may be either discrete or
continuous and that the manifold is discrete or continuous according to the
manner of passage. Where the manifold is regarded as discrete two portions
of it can be compared, as to magnitude, by counting; where continuous, by
measurement. If the whole manifold be caused to pass over into another
manifold each of its elements passing through a one-dimensional manifold,
a two-dimensional manifold is thus generated. In this way, a manifold of n-
dimensions can be generated. On the other hand, a manifold of n-
dimensions can be analyzed into one of one dimension and one of (n-1)
dimensions.

To Riemann, then, is due the credit for first promulgating the idea that space
being a special case of manifold is generable, and therefore, finite. He laid
the foundation for the establishment of a special kind of geometry known as
the "elliptic." Space, as viewed by him, possessed the following properties,
viz.: generability, divisibility, measurability, ponderability, finity and flexity.
These are the six pillars upon which rests the structure of hyperspace
analyses.[5]
Generability is that property of geometric space by virtue of which it may
be generated, or constructed, by the movement of a line, plane, surface or
solid in a direction without itself. Divisibility is that property of geometric
space by virtue of which it may be segmented or divided into separate parts
and superposed, or inserted, upon or between each other. Measurability is
that property by virtue of which geometric space is determined to be a
manifold of either a positive or negative curvature, also by which its extent
may be measured. Ponderability is that property of geometric space by
virtue of which it may be regarded as a quantity which can be manipulated,
assorted, shelved or otherwise disposed of. Finity is that property by virtue
of which geometric space is limited to the scope of the individual
consciousness of a unodim, a duodim or a tridim and by virtue of which it is
finite in extent. Flexity is that property by virtue of which geometric space
is regarded as possessing curvature, and in consequence of which progress
through it is made in a curved, rather than a geodetic line, also by virtue of
which it may be flexed without disruption or dilatation.
Riemann who thus prepared the way for entrance into a veritable labyrinth
of hyperspaces is, therefore, correctly styled "The father of metageometry,"
and the fourth dimension is his eldest born. He died while but forty years of
age and never lived long enough fully to elaborate his theory with respect to
its application to the measure of curvature of space. This was left for his
very energetic disciple, Eugenio Beltrami (1835-1900) who was born nine
years after Riemann and lived thirty-four years longer than he. His labors
mark the characteristic standpoint of the determinative period. Beltrami's
mathematical investigations were devoted mainly to the non-Euclidean
geometry. These led him to the rather remarkable conclusion that the
propositions embodied therein relate to figures lying upon surfaces of
constant negative curvature.

Beltrami sought to show that such surfaces partake of the nature of the
pseudosphere, and in doing so, made use of the following illustration:

Fig. 3.

Fig. 4.

If the plane figure aabb is made to revolve upon its axis of symmetry AB
the two arcs, ab and ab will describe a pseudospherical concave-convex
surface like that of a solid anchor ring. Above and below, toward aa and bb,
the surface will turn outward with ever-increasing flexure till it becomes
perpendicular to the axis and ends at the edge with one curvature infinite.
Or, the half of a pseudospherical surface may be rolled up into the shape of
a champagne glass, as in Fig. 4. In this way, the two straightest lines of the
pseudospherical surface may be indefinitely produced, giving a kind of
space (pseudospherical) in which the axiom of parallels does not hold true.
The determinative period marks the most important stage in the
development of non-Euclidean geometry and certainly the most significant
in the evolution of the idea of hyperspaces and multiple dimensionality.
Riemann and Beltrami are chief among those whose labors characterize the
scope of this period. Their work gave direction and general outline for later
developments and all subsequent researches along these lines have been
conducted in strict conformity with the principles laid down by these
pioneer constructionists. They laid out the field and designated its confines
beyond which no adventurer has since dared to pass.
The great importance of the work of Riemann at this time may be seen
further in the fact that it not only marked the beginning of a new epoch in
geometry; but his pronouncement of the hypothesis that space is
unbounded, though finite, is really the first time in the history of human
thought that expression was ever given to the idea that space may yet be
only of limited extent. Before that time the minds of all men seemed to have
been unanimous in the consideration of space as an illimitable and infinite
quantity.
The Elaborative Period
The elaborative stage includes the work of all those who, working upon the
bases laid down by Lobachevski, Bolyai, SchweikarT and Riemann, have
sought to amplify the conclusions reached by them. Among those whose
investigations have greatly multiplied the applications of hyperspace
conceptions are Hoüel (1866) and Flye St. Marie (1871) of France;
Helmholtz (1868), Frischauf (1872), Klein (1849), and Baltzer (1877) of

Germany; Beltrami (1872) of Italy; De Tilly (1879) of Belgium; Clifford
and Cayley (1821) of England; Newcomb (1835) and Halstead of America.
These have been most active in popularizing the subject of non-Euclidean
geometry and incidentally the idea of the fourth dimension. The great mass
of non-professional mathematical readers, therefore, owe these men an
immeasurable debt of gratitude for the work that they have done in the
matter of rendering the conceptions which constitute the fabric of
metageometry understandable and thinkable. A glance at the bibliography
appended at the end of this volume will give some idea of the enormous
amount of labor that has been expended in an effort to translate the most
abstract mathematical principles into a language that could easily be
comprehended by the average intelligent person.
The characteristic standpoint of this period is the popular comprehension of
the hyperspace concept and the consequent mental liberation which follows.
For there is no doubt but that unheard of possibilities of thought have been
revealed by investigations into the nature of space. An entirely new world
has been opened to view and only a beginning has been made at the
exploration of its extent and resources.
One of the notable incidents of the early years of this period is the position
taken by Felix Klein who stands in about the same relation to Cayley as
Beltrami does to Riemann, in that he assumed the task of completing the
work of his predecessor. Klein held that there are only two kinds of
Riemannian space—the elliptical and the spherical. Or in other words, that
there are only two possible kinds of space in which the propositions
announced by Riemann could apply. Sophus Lie, called the "great
comparative anatomist of geometric theories," carried his classifications to
a final conclusion in connection with spaces of all kinds and decided that
there are possible only four kinds of three dimensional spaces.
But whether men with peering, microscopic, histological vision shall
establish the existence of one or many spaces, and regardless of the
mathematic rigor with which they shall demonstrate the self-consistency of
the doctrines which they hold, the fact remains that the hypotheses thus
maintained, while they may be regarded as true descriptions of the spaces
concerned, are, nevertheless, incompatible. All of them cannot be valid. It

will perhaps be found that none of them are valid, especially objectively so.
The only true view, therefore, of these systems of hyperspaces is that which
assigns them to their rightful place in the infinitely vast world of pure
mathesis where their validity may go unchallenged and their existence
unquestioned; for in that domain of unconfined mentation, in that realm of
divine intuitability, the marvelous wonderland of ideas and notions, one is
not only disinclined to doubt their logical actuality, but is quite willing to
accede their claims.

CHAPTER III

Essentials of the Non-Euclidean Geometry

The Non-Euclidean Geometry Concerned with Conceptual Space
Entirely—Outcome of Failures at Solving the Parallel-Postulate—The
Basis of the Non-Euclidean Geometry—Space Curvature and
Manifoldness—Some Elements of the Non-Euclidean Geometry—
Certainty, Necessity and Universality as Bulwarks of Geometry—
Some Consequences of Efforts at Solving the Parallel-Postulate—The
Final Issue of the Non-Euclidean Geometry—Extended
Consciousness.

The term "non-Euclidean" is used to designate any system of geometry
which is not strictly Euclidean in content.
It is interesting to note how the term came to be used. It appears to have
been employed first by Gauss. He did not strike upon it suddenly, however,
as in the correspondence between him and Wachter in 1816 he used the
designation "anti-Euclidean" and then, later, following Schweikart, he
adopted the latter's terminology and called it "Astral Geometry." This he
found in Schweikart's first published treatise known by that name and
which made its appearance at Marburg in December, 1818. Finally, in his
correspondence with Taurinus in 1824, Gauss first used the expression
"non-Euclidean" to designate the system which he had elaborated and
continued to use it in his correspondence with Schumacher in 1831.
"Non-Legendrean," "semi-Euclidean" and "non-Archimedean" are titles
used by M. Dehn to denote all kinds of geometries which represented
variations from the hypotheses laid down by Legendre, Euclid and
Archimedes.

The semi-Euclidean is a system of geometry in which the sum of the angles
of a triangle is said to be equal to two right angles, but in which one may
draw an infinity of parallels to a straight line through a given point. The
non-Euclidean geometry embraces all the results obtained as a consequence
of efforts made at finding a satisfactory proof of the parallel-postulate and
is, therefore, based upon a conception of space which is at variance with
that held by Euclid. According to the Ionian school space is an infinite
continuum possessing uniformity throughout its entire extent. The non-
Euclideans maintain that space is not an infinite extension; but a finite
though unbounded manifold capable of being generated by the movement
of a point, line or plane in a direction without itself. It is also held that space
is curved and exists in the shape of a sphere or pseudosphere and is
consequently elliptical.
The inapplicability of Euclid's parallel-postulate to lines drawn upon the
surface of a sphere suggested the possibility of a space in which the
postulate could apply to all possible surfaces or that space itself may be
spherical in which case the postulate would be invalidated altogether.
Hence, it is quite natural that mathematicians finding themselves unable to
prove the postulate with due mathetic precision should turn their attention
to the conceptually possible. In this virtual abandonment of the perceptual
for the conceptual lies the fundamental difference between the Euclidean
and the non-Euclidean geometries. It may be said to the credit of the
Euclideans that they have sought to make their geometric conceptions
conform as closely as possible to the actual nature of things in the sensuous
world while at the same time they must have perceived that at best their
spatial notions were only approximations to the sensuous actuality of
objects in space.
On the other hand, non-Euclideans make no pretense at discovering any
congruency between their notions and things as they actually are. The
attitude of the metageometricians in this respect is very aptly described by
Cassius Jackson Keyser who says:

"He constructs in thought a summitless hierarchy of hyperspaces, an
endless series of orderly worlds, worlds that are possible and logically
actual, and he is content not to know if any of them be otherwise actual
or actualized."[6]

The non-Euclidean is, therefore, not concerned about the applicability of
ensembles, notions and propositions to real, perceptual space conditions. It
is sufficient for him to know that his creations are thinkable. As soon as he
can resolve the nebulosity of his consciousness into the conceptual "star-
forms" of definite ideas and notions, he sits down to the feast which he
finds provided by superfoetated hypotheses fabricated in the deeps of mind
and logical actualities imperturbed and unmindful of the weal of perceptual
space in its homogeneity of form and dimensionality.
Fundamentally, the non-Euclidean geometry is constructed upon the basis
of conceptual space almost entirely. Knowledge of its content is accordingly
derived from a superperceptual representation of relations and interrelations
subsisting between and among notions, ideas, propositions and magnitudes
arising out of a conceptual consideration thereof. In other words,
representations of the non-Euclidean magnitudes, cannot be said to be
strictly perceptual in the same sense that three-space magnitudes are
perceived; for three-space magnitudes are really sense objects while
hyperspace magnitudes are not sense objects. They are far removed from
the sensuous world and in order to conceive them one must raise his
consciousness from the sensuous plane to the conceptual plane and become
aware of a class of perceptions which are not perceptions in the strict sense
of the word, but superperceptions; because they are representations of
concepts rather than precepts.
Notions of perceptual space are constituted of the triple presentations
arising out of the visual, tactual and motor sensations which are fused
together in their final delivery to the consciousness. The synthesis of these
three sense-deliveries is accomplished by equilibrating their respective
differences and by correcting the perceptions of one sense by those of
another in such a way as to obtain a completely reliable perception of the
object. This is the manner in which the characteristics of Euclidean space
are established.
The characteristics of non-Euclidean space are not arrived at exactly in this
way. Being beyond the scope of the visual, tactile and motor sense
apprehensions, it cannot be said to represent judgments derived from any
consideration or elaboration of the deliveries presented through these
media. Yet, the substance of metageometry, or the science of the

measurement of hyperspaces, may not be regarded as an a priori
substructure upon which the system is founded. That is, the conceptual
space of non-Euclidean geometry is not presented to the consciousness as
an a priori notion. On the other hand, the a posterioristic quality of
metageometric spaces marks the entire scope of motility of the notions
appertaining thereto.
The notions, therefore, of conceptual space are derivable only from the
perception of concepts, or, otherwise consist of judgments concerning
interconceptual relations. The process of apperception involved in the
recognition of relations which may be methodically determined is much
removed from the primary procedure of perceiving sense-impressions and
fusing them into final deliveries to the consciousness for conceptualization
or the elaboration into concepts or general notions. It is a procedure which
is in every way superconceptual and extra-sensuous. The metageometrician
or analyst in no way relies upon sense-deliveries for the data of his
constructions; for, if he did, he should, then, be reduced to the necessity of
confining his conclusions to the sphere of motility imposed by the sensible
world with the result that we should be able to verify empirically all his
postulations. But, contrarily, he goes to the extra-sensuous, and there in the
realm of pure conceptuality, he finds the requisite freedom for his theories;
thus, environed by a sort of intellectual anarchism, he pursues analytical
pleasures quite unrestrainedly. The difference between the two mental
processes—that which leads from the sensible world to conception and that
which veers into the fields beyond—is so great that it is hardly permissible
to view the results arrived at in the outcome of the separate processes as
being identical.
To illustrate this difference, let us draw an analogy. The miner digs the iron
ore out of the ground. The iron is separated from the extraneous material
and delivered to the furnaces where the metal is melted and turned out as
pig iron. It is further treated, and steel, of various grades, cast iron and other
kinds of iron are produced. The treatment of the iron ore up to this stage is
similar to the treatment of sense-impressions by the Thinker. Steel, cast
iron, et cetera, are similar to mental concepts. Later, the steel and other
products are converted into instruments and numerous articles. This
represents the superperceptual process. Trafficking in iron ore products,
such as instruments of precision, watch springs, and the like, represents a

stage still farther removed from the primary treatment of the ore and is
similar to that to which concepts are treated when the metageometrician
manipulates them in the construction of conceptual space-forms. Perception
is the dealing with raw iron ore while conception is analogous to the
production of the finished product.
Superperception would be analogous to the trafficking in the finished
product as such and without any reference to the source or the preceding
processes. Thus the notions and judgments of the non-Euclidean geometry
are arrived at as a result of a triple process of perception, conception and
superperception the latter being merely superconceived as formal space-
notions. But it is obvious that the more complex the processes by which
judgments purporting to relate to perceptual things are derived the more
likely are those judgments to be at variance with the nature of the things
themselves.
In view of the foregoing, the dangers resulting from identifying the
products of the two processes are very obvious indeed. But the difference
between the two procedures is the difference between Euclidean and non-
Euclidean geometries or the difference between perceptual space notions
and conceptual space notions. Hence, it is not understood just how or why it
has occurred to anyone that the two notions could be made congruent.
Magnitudes in perceptual, sensible space are things apart from those that
may be said to exist in mathematical space or that space whose qualities and
properties have no existence outside of the mind which has conceived them.
It is believed to be quite impossible to approach the study of
metageometrical propositions with a clear, open mind without previously
understanding the fundamental distinctions which exist between them.
It follows, therefore, as a logical conclusion that geometric space of
whatsoever nature is a purely formal construction of the intellect, and for
this reason is completely under the sovereignty of the intellect however
whimsical its demands may be. Being thus the creature of the intellect, its
possibilities are limited only by the limitations of the intellect itself.
Perceptual space, being neither the creature of the intellect nor necessarily
an a priori notion resident in the mental substructure, but existing entirely
independent of the intellect or its apprehension thereof, cannot be expected
to conform to the purely formal restrictions imposed by the mind except in

so far as those restrictions may be determined by the nature of perceptual
space. And for that matter, it should not be forgotten that, as yet, we have no
means of determining whether or not the testimony of the intellect is
thoroughly credible simply because there is no other standard by which we
may prove its testimony. It is possible to justify the deliveries of the eye by
the sense of touch, or vice versa. It is also possible to prove all our sense-
deliveries by one or the other of the senses. But we have no such good
fortune with the deliveries of the intellect. We have simply to accept its
testimony as final; because we cannot do any better. But if it were possible
to correct the testimony of the intellect by some other faculty or power
which by nature might be more accurate than the intellect it should be found
that the intellect itself is sadly limited.
The possible curvature of space is a notion which also characterizes the
content of the non-Euclidean geometry. It is upon this notion that the
question of the finity and unboundedness of space, in the mathematical
sense, rests. In the curved space, the straightest line is a curved line which
returns upon itself. Progression eastward brings one to the west; progression
northward brings one to the south, et cetera. On this view space is finite, but
may not be regarded as possessing boundaries.
Space-curvature, reinforced by the idea that space is also a manifold is the
enabling clause of metageometry and without them the analyst dares not
proceed. Here again, we are led to the confession that however fantastic
these two notions may seem and evidently are, there is nevertheless to be
recognized in them a "dim glimpse" of a veritable reality—a slight
foreshadowing of the revelation of some great kosmic mystery.
The manifoldness of space is the fiat of analysis. It is the inevitable
outcome of the analyst's method of procedure. His education, training and
view of things in general inhibit his arriving at any other result and he may
be pardoned with good grace for his manufacture of the space-manifold.
For by it perhaps a better appreciation of that wonderful extension of
consciousness in the nature of which is involved the explanation of the
perplexing problems which the manifold and other metageometrical
expedients faintly adumbrate may be gained.

It is pertinent, in the light of the above, to examine into some of the relative
merits of the three formal bulwarks of geometrical knowledge. These are
certainty, necessity and universality.
Geometric certainty is derived solely from the nature of the premises upon
which it is based. If the premises be contradictory, it is, of course, defective.
But if the premises are non-contradictory or self-evident, then the certainty
of geometric notions and conclusions is valid. Another consideration of
prime importance in this connection is the definition. From it all premises
proceed. Hence, the definition is even more important than the premise; for
it is the persisting determinant of all geometric conclusions while the
premise is dependent upon the limitations of the definition. The
determinative character of the definition has led to its apotheosis; but this,
admittedly, has been necessary in order to give stability and permanency to
the conclusions which followed. But in spite of this it would appear that the
certainty of geometric conclusions is not a quality to be reckoned as
absolute or final.
With the same certainty that it can be said the sum of the angles of the
triangle is equal to two right angles it may be asserted that that sum is also
greater or less than two right angles. Certainty which is based upon the
inherent congruity of definitions, premises and propositions is an entirely
different matter from that certainty which arises out of the real, abiding
validity of a scheme of thought. But this difference is not lessened by the
fact that the latter is dependent, in a measure, upon the correct
systematization of our spatial experiences by means of methodical
processes. Euclidean geometry, accordingly, is not so certain in its
applications as it is utilitarian; but non-Euclidean geometry is even less
certain than the former and consequently more lacking in its utilitarian
possibilities.
The necessity of geometrical determinations is merely the necessity which
inheres in logical inferences or deductions. These may or may not be valid.
Inasmuch as the necessariness of deductions is primarily based upon the
conditional certainty of premises and definitions it appears that this quality
is in no way peculiar to geometry whether Euclidean or non-Euclidean. In
like manner, the universality of geometric judgments may not properly be
regarded as a peculiarity of geometry; but is explicable upon the basis of the

formal character of the assumptions which underlie it. The chief value, then,
of non-Euclidean geometry seems to abide in the fact that it clarifies our
understanding as to the complex processes by which it is possible to
organize and systematize our spatial experiences for assimilation and use in
other branches of knowledge.
With the above statement of the case of the non-Euclidean geometry it is
now thought permissible to state briefly some of the elements thereof.[7]
Below will be found some of the elements obtained as a consequence of
efforts made both at proving and disproving the parallel-postulate of Euclid:
"If two points determine a line it is called a straight."
"If two straights make with a transversal equal alternate angles they have a
common perpendicular."
"A piece of a straight is called a sect."
"If two equal coplanar sects are erected perpendicular to a straight, if they
do not meet, then the sect joining their extremities makes equal angles with
them and is bisected by a perpendicular erected midway between their feet."
"The sum of the angles of a rectilineal triangle is a straight angle, in the
hypothesis of the right (angle); is greater than a straight angle in the
hypothesis of the obtuse (angle); is less than a straight angle in the
hypothesis of the acute (angle)."
"The hypothesis of right is Euclidean; the hypothesis of the acute is Bolyai-
Lobachevskian; the hypothesis of obtuse is Riemannian."
"If one straight is parallel to a second the second is parallel to the first."
"Parallels continually approach each other."
"The perpendiculars erected at the middle point of the sides of a triangle are
all parallel, if two are parallel."
"If the foot of a perpendicular slides on a straight its extremity describes a
curve called an equidistant curve, or an equidistantial."

"An equidistantial will slide on its trace."
"In the hypothesis of the obtuse a straight is of finite size and returns into
itself."
"Two straights always intersect."
"Two straights perpendicular to a third straight intersect at a point half a
straight from the third either way."
"A pole is half a straight from its polar."
"A polar is the locus of coplanar points half a straight from its pole.
Therefore, if the pole of one straight lies on another straight the pole of this
second straight is on the first straight."
"The cross of two straights is the pole of the join of their poles."
"Any two straights inclose a plane figure, a digon."
"Two digons are congruent if their angles are equal."
"The equidistantial is a circle with center at the poles of its basal straight."
A typical postulate based upon the Bolyai hypothesis of the acute angle is
the following:
"From any point P drop PC, a perpendicular to any given straight line AB.
If D move off indefinitely on the ray CB, the sect will approach as limit PF
copunctal with AB at infinity.

Fig. 5.

PD is said to be at P the parallel to AB toward B. PF makes with PC an
angle CPF which is called the angle of parallelism for the perpendicular
PC. It is less than a right angle by an amount which is the limit of the
deficiency of the triangle PCD. On the other side of PC, an equal angle of
parallelism gives the parallel P to BA towards AM.[8] Thus at any point
there are two parallels to a straight. A straight has, therefore, two separate
points at infinity."
"Straights through P which make with PC an angle greater than the angle of
parallelism and less than its supplement do not meet the straight AB at all
not even at infinity."
The parallel-postulate is stated in the non-Euclidean geometry as follows:
"If a straight line meeting two straight lines make those angles which are
inward and upon the same side of it less than two right angles the two
straight lines being produced indefinitely will meet each other on this side
where the angles are less than two right angles."

It is stated by Manning[9] in the following language:
"If two lines are cut by a third and the sum of the interior angles on the
same side of the cutting line is less than two right angles the line will meet
on that side when sufficiently produced."
It is rather significant that in this postulate which is really a definition of
space should be found grounds for such diverse interpretations as to its
nature. Of course, the moment the mind seeks to understand the infinite by
interpreting it in the unmodified terms of the apparently unchangeable finite
it entangles itself into insurmountable difficulties. As a drowning man
grasps after straws so the mind, immersed in endless abysses of infinity,
fails to conduct itself in a seemly manner; but gasps, struggles and
flounders and is happy if it can, in the depths of its perplexity, discover a
way of logical escape. The pure mathematician has a hankering after the
logically consistent in all his pursuits; to him it is the "Holy Grail" of his
highest aspirations. He seeks it as the devotee seeks immortality. It is to him
a philosopher's stone, the elixir of perpetual youth, the eternal criterion of
all knowledge.

Failures to demonstrate the celebrated postulate of Euclid led, as a matter
of course, to the substitution of various other postulates more or less
equivalent to it in that each of them may be deduced from the other without
the aid of any new hypothesis.
Among those who sought proof by a restatement of the problem are the
following:
1. Ptolemy: The internal angles which two parallels make with a transversal
on the same side are supplementary.
2. Clavius: Two parallel straight lines are equidistant.
3. Proclus: If a straight line intersects one of two parallels it also intersects
the other.
4. Wallis: A triangle being given another triangle can be constructed
similar to the given one and of any size whatever.
5. Bolyai (W.): Through three points not lying on a straight line a sphere
can always be drawn.
6. Lorenz: Through a point between the lines bounding an angle a straight
line can always be drawn which will intersect these two lines.
7. Saccheri: The sum of the angles of a triangle is equal to two right angles.
There were, of course, many other statements and substitutions used by
mathematicians in their endeavors satisfactorily to establish the truth of the
parallel-postulate. That their labors should have terminated, first, by
doubting it, then by denying, and finally, by building up a system of
geometries which altogether ignores the postulate is just what might
naturally be expected of these men who have given to the world the non-
Euclidean geometry. In doing what they did many, if not all of them, were
not aware in any measure of the proportions of the imposing superstructure
that would be built upon their apparent failures. All of them undoubtedly
must have sensed the vague adumbrations forecast by the unfolding
mysteries which they sought to lay bare; all of them must have felt as they
executed the early tasks of those crepuscular days of pure mathematics that
the way which they were traveling would lead to the inner shrine of a

higher knowledge and a wider freedom; they may have been led by divine
intuition to strike out on this new path and yet they could not have known
how fully their dreams would be realized by the mathematicians of the
twentieth century. If so, they were truly gods and mathesis is their kingdom.
The analyst proceeds upon a basis entirely at variance with that which
guides the ordinary investigator in the formulation of his conclusions. The
empirical scientist in arriving at his theories or hypotheses is governed at all
times by the degree of conformity which his postulates exhibit to the actual
phenomena of nature. He endeavors to ascertain just how far or in what
degree his hypothesis is congruent with things found in nature. If the
dissidence is found to predominate he abandons his theory and makes
another statement and again sets out to determine the degree of conformity.
If he then finds that the natural phenomena agree with his theory he accepts
it as for the time being finally settling the question. In all things he is
limited by the answer which nature gives to his queries. Not so with the
exponent of pure mathematics. For him the truth of hypotheses and
postulates is not dependent upon the fact that physical nature contains
phenomena which answer to them. The sole determining factor for him is
whether or not he is able to state with rational consistency the assumed first
principles and then logically develop their consequences. If he can do this,
that is, if he can state his hypotheses with consistency and develop their
consequences into a logical system of thought, he is quite satisfied and well
pleased with his performances. But the fact that this is true is of vital
significance for all who seek clearly to understand the essential character of
hyperspatiality.
It appears, therefore, that the science of consequences is the radical essence
of pure geometry. The metageometrician enjoys unlimited freedom in the
choice of his postulates and suffers curtailment only when it comes to the
question of consistency. He is at liberty to formulate as many systems of
geometry as the barriers of consistency will permit and these are practically
innumerable. So long then as the laws of compatibility remain inviolate his
multiplication of postulate-systems may proceed indefinitely. Is it strange
then that under conditions where an investigator has such unbridled liberty
he should be found indulging in mathetic excesses?

Kant held that the axioms of geometry are synthetic judgments a priori; but
it appears that in the strictest sense this is not the case. It depends upon the
type of mind which is taken as a standard of reference. If it be the
uncultivated mind, it is certain that to it the relations expressed by an axiom
would never appear spontaneously. If on the other hand, the standard be that
of a cultivated mind it is also equally certain that to it these relations would
be discovered only after methodical operations. All judgments arrived at as
a result of logical processes should, it seems, be regarded as judgments a
posteriori, i.e., the results of empirical operations. Confessedly, the facts
adduced in course of experimentation serve as guides in choosing among all
of the many possible logical conventions; but our choice remains
untrammeled except by the compulsion arising out of a fear of
inconsistency. The real criterion then of all geometries is neither truth,
conformability nor necessity, but consistency and convenience.
The difficulty with the non-Euclideans resolved itself into the question as to
whether it is more consistent, as well as convenient, to establish a proof of
the postulate by taking advantage of the support to be found in other
postulates or whether, by seeking a demonstration based upon the deliveries
of sense-experience as to the nature of space and its properties, a still more
consistent conclusion might be reached. They had further perplexity,
however, when it came to a decision as to whether the organic world is
produced and maintained in Euclidean space or in a purely conceptual space
which alone can be apprehended by the mind's powers of representation.
Unwilling to admit the existence of the world in Euclidean space, they
turned their attention to the examination of the properties of another kind of
space so-called which unlike the space of the Ionian school could be made
to answer not only all the purposes of plane and solid figures, but of
spherics as well. And so, the manifold space was invented by Riemann and
later underwent some remarkable improvements at the hands of his disciple,
Beltrami. But it may be said here, parenthetically, that the truth of the
whole matter is that our world is neither in Euclidean nor non-Euclidean
space, both of which, in the last analysis, are conceptual abstractions.
Although it may not be denied that the Euclidean space is the more
compatible.
The problem of devising a space, if only a very limited portion, in which
could be demonstrated the assumed alternative hypothesis and its

consequences logically developed, occasioned no inconsiderable concern
for the non-Euclidean investigators; but neither Lobachevski, Bolyai nor
Riemann were to be baffled by the difficulties which they met. These only
cited them to more laborious toil. Having succeeded in mentally
constructing the particular kind of space which was adaptable to their
rigorous mathetic requirements it immediately occurred to them that all the
qualities of the limited space thus devised might logically be amplified and
extended to the entire world of space and that what is true of figures
constructed in the segmented portion of space which they used for
experimental purposes is also true of figures drawn anywhere in the
universe of this space as all lines drawn in the finite, bounded portion could
be extended indefinitely and all magnitudes similarly treated. From these
results, it was but a single step to the conclusion which followed—that
either an entirely new world of space had been discovered or that our notion
of the space in which the organic world was produced is wholly wrong and
needs revision. But notwithstanding the insurmountable obstacles which
stood in the way of the investigators who made the attempt to discover the
homology which might exist between the characteristics of the newly
fabricated space and the phenomenal world, investigations were carried
forward with almost amazing recklessness and loyalty to the mathetic spirit
until it was discovered that all efforts to trace out any definite lines of
correspondence were futile. Then the policy of ignoring the question of
conformability was adopted and has since been pursued with unchangeable
regularity by the analytical investigator.
Among the results obtained by the non-Euclideans in their profound
researches into the nature of hyperspace are these: 1. It was found that the
angular sum of a triangle, being ordinarily assumed to be a variable
quantity, is either less or greater than two right angles so that a strictly
Euclidean rectangle could not be constructed. 2. The angle sums of two
triangles of equal area are equal. 3. No two triangles not equal can have the
same angles so that similar triangles are impossible unless they are of the
same size. 4. If two equal perpendiculars are erected to the same line, their
distance apart increases with their length. 5. A line every point of which is
equally distant from a given straight line is a curved line. 6. Any two lines
which do not meet, even at infinity, have one common perpendicular which
measures their minimum distance. 7. Lines which meet at infinity are

parallel. But it is apparent that these results have not followed upon any
mathematical consequence of other supporting postulates or axioms such as
would place them on a coördinate basis with those used as a support for the
parallel-postulate; for they are based upon the envisagement of an entirely
new principle of space-perception and belong to a wholly different set of
space qualities.
The final issue then of the non-Euclidean geometry is neither in the utility
of its processes and conclusions nor in the increscent inclination towards a
new outlook upon the world of mathesis; but resides solely in the
possibilities yet to be developed in that vast domain of analytical thought
which it has discovered and opened to view. To say that it sheds any light
upon the nature of the universe is perhaps to take the radical view; yet it
cannot be doubted that the researches incident to the formulation of the
non-Euclidean geometry have greatly extended the scope of consciousness.
Whether the extension is valid and normal or simply a hypertrophic
excrescence of mental feverishness; whether by virtue of it we shall more
closely approach an understanding of the true nature of the mind of the
Infinite, or shall all fall into insanity, are certainly debatable questions. It
nevertheless appears evident that humanity has gained something of real,
abiding permanence by this new departure. If that something be merely an
extended consciousness or an awakening to the fact that there are stages of
awareness beyond the strictly sensuous, and every observable evidence
points to this, then there has only begun the process by which the faculty of
conscious functioning in this new world shall become the normal
possession of the human species. But this new world cannot be said to be of
mathematical import; for it is doubtful if mathematical laws such as have
been devised up to the present time, would obtain therein. So that if
anything, it must be psychological and vital.
On this view the worlds of hyperspace inlaid with analytic manifoldnesses
and constant curvatures are but the primal excitants which will finally
awaken in the mind the faculty of awareness in the new domain of
psychological content. Then will come the blooming of the diurnal flower
of the mind's immortality and the outputting of the organ of consciousness
wherewith the infinite stretches of hyperspaces, the low-lying valleys of
reals and imaginaries and the uplifting hills of finites and infinites shall be

divested of their mysteries and stand out in their unitariness no longer
draped in the veil of the inscrutable and the incomprehensible.
The fourth dimension, regarded by some as a new scope of motion for
objects in space, by others as a new and strange direction of spatial extent
and by others still as the doorway of the temple of exegesis wherein an
explanation may be found for the entire congeries of mysteries and
supermysteries which now perplex the human mind, may also be said to be
the key to the non-Euclidean geometry. But it really complicates the
situation; for one has to be capable of prolonged abstract thought even to
envisage is as a conceptual possibility. Poincaré[10] says: "Any one who
should dedicate his life to it could, perhaps, eventually imagine the fourth
dimension," implying thereby that a lifetime of prolonged abstract thought
is necessary to bring the mind to that point of ecstasy where it could even so
much as imagine this additional dimension. Nevertheless by it (the fourth
dimension) was the non-Euclidean geometry made and without it was not
any of the hyperspaces made that were made. It is the view which
geometers have taken of space in general that has made the fourth
dimension possible, and not only the fourth, but dimensions of all degrees.
The basis of the non-Euclidean geometry may be found then in the notion
of space which has been predominant in the minds of the investigators.
Finally, it should be pointed out that the non-Euclidean geometry, though a
consistent system of postulates, has been constructed upon a misconception
based upon the identification of real, perceptual space with systems of
space-measurements. Hyperspaces which are not spaces at all should not be
confounded with real space. But they constitute the substance of non-
Euclidean geometry; they are its blood and sinews. Their study is
interesting, because of the possibilities of speculation which it offers. No
mind that has thought deeply upon the intricacies of the fourth dimension,
or hyperspace, remains the same after the process. It is bound to experience
a certain sense of humility, and yet some pride born of a knowledge that it
has been in the presence of a great mystery and has delved into the fearful
deeps of kosmic mind. To the mind that has thus been anointed by the
sacred chrism of the inner mysteries of creative mentality there always
come that stillness and calm such as characterize the aftermath of reflection
upon the incomprehensible and the transfinite.

CHAPTER IV

Dimensionality

Arbitrary Character of Dimensionality—Various Definitions of
Dimension—Real Space and Geometric Space Differentiated—The
Finity of Space—Difference Between the Purely Formal and the
Actual—Space as Dynamic Appearance—The A Priori and the A
Posteriori as Defined by Paul Carus.

In previous chapters we have traced the growth and development of the
non-Euclidean geometry showing that the so-called fourth dimension is an
aspect thereof. It is now deemed fitting that we should enter into a more
detailed study of the question of dimensionality with a view to examining
some of the difficulties which encompass it.
The question of dimension is as old as geometry itself. Without it geometric
conclusions are void and meaningless. Yet the conception of dimensionality
itself is purely conventional. In its application to space there is involved a
great deal of confusion because of the inferential character of its definition.
For instance, commonly we measure a body in space and arbitrarily assign
three elements to determine its position. The simplest standard for this
purpose is the cube having three of its edges terminating at one of its
corners.

Fig. 6.

Thus because it is found that the entire volume of a cube is actually
comprehended within the directions indicated by the lines ab, bc and db it is
determined that the three coördinates of the point b are necessary and
sufficient to establish the dimensions of the cube and consequently of the
space in which it rests. The conception may be stated in this way: If a
collection of elements, say points or lines, be of such a nature or order that
it is sufficient to know a certain definite number of facts about it in order to
be able to distinguish every one of the elements from all the others, then the
assemblage or collection of elements is said to be of the same number of
dimensions as there are elements necessary to its determination. In the
above figure there are three elements, namely, the lines ab, bc, and db,
which are necessary and sufficient for the determination of the position of
the point b. In this way geometers have determined that our space is
tridimensional; but it is obvious that this conclusion is based not upon any
examination of space itself but upon the measurement of bodies in space.
Upon this view it is seen that conclusions based upon such a procedure
render our notion of the extension of bodies in space identical with the
notion of spatial extensity. In other words, we take bodies in space and by
examining their characteristics and properties arrive at an alleged apodeictic
judgment of space. It is by means of this conventional norm of geometric
knowledge that various other spaces, notably the one-, two-, four-and n-
space, have been devised. It would appear that if some more absolute
standard of measurement or definition of space were adopted the confusion
which now clings to the conception of dimension could be obviated. For if
it be true that three and only three elements are necessary to determine a
point-position in our space and that in this determination we also find the
number of dimensions of space, then it may also be true that n-coördinates

would just as truly determine the dimensionality of an n-space, which is
granted. But then the n-space would be just as legitimate as the three-space;
for it is determined by exactly the same standards. It is both quantitatively
and qualitatively the same. If, however, on account of the exigencies that
might arise, we are forced to seek solace in the notion of an n-space whither
shall we turn for it? It cannot be found; for it is imperceptible,
uninhabitable, non-existent, and therefore, absolutely and purely an
abstraction. Consequently, there must be something radically wrong with
the definition of space or with its determinants.
The purely arbitrary character of dimensionality is very aptly described by
Cassius Jackson Keyser, who says:

"... The dimensionality of a given space is not unique, but depends
upon the choice of the geometric entity for primary or generating
element. A space being given, its dimensionality is not therewith
determined, but depends upon the will of the investigator who by a
proper choice of generating element endows the space with any
dimensionality he pleases. That fact is of cardinal significance for
science and philosophy."[11]

It is a fact of "cardinal significance" for science; because it emphasizes the
necessity for some more rational procedure than that of the geometrician in
arriving at an absolutely unique method of determining the dimension and
essential nature of real space. Its significance for philosophy lies in the need
of a logical, rigidly exclusive and absolutely peculiar standard of space
definition. The definition of perceptual space should be such as rigorously
inhibits its inclusion as a particular in any general class. The necessity for
this is warranted by its universality and uniqueness.
The lines of demarkation between what is recognized as perceptual space
and what has been called geometric or conceptual space should be very
sharply drawn. So that when reference is made to either there will be no
doubt as to which is meant. And then, too, conceptual space is no space at
all, properly speaking. It is merely a system of space-measurement. And as
such has no logical right to be put in the same category as perceptual space.

Real space is unique. Geometric space belongs to a class whose members
are capable of indefinite multiplication. It is certainly most illogical to
identify them. Perceptual space, figuratively speaking, is a quantity;
analytic space is the foot-rule, the yard-stick, the kilometer, by which it is
measured and apportioned. It is logically impossible to predicate the same
conclusion for both of them. That is, to do so causes a profound fracture of
the fundamental norms of logic. Such conclusions being thus illegitimate it
is rather surprising that an error of this nature should have been made. It is
perhaps accountable for on the grounds of the geometer's complete
insouciance as to how his postulates shall stand in their relation to things in
the phenomenal world.
It is agreed that as convenient as is Euclid's system of space-measurement it
is not by any means congruent with the extension of real space objects. It
does, however, approximate congruity with these objects as nearly as
possible. How then could it be expected that a system of space-
measurement so far removed from this primary congruence as the non-
Euclidean system is should exhibit more obvious signs of correspondence?
But the advocates of the n-dimensionality of space have illatively asserted
the identity of space and its dimensions. Accordingly, there is not
recognized any distinction between their conception of space itself and its
qualitative peculiarities. They use the terms interchangeably. So that
dimension means space and vice versa. In this lack of discrimination may
be found the source of much of the confusion which attaches to the
conception of space.
If it were arguable that the relation between space and its dimensions is the
same as that between matter and its properties then the restriction of this
relation to three and only three directions of extent would be disallowed; for
the reason that if, as is commonly done, dimension be made to mean
direction of extent, there would be an unlimited number of directions of
extent and they would all be perceptible. But this is really another
fundamental fault. Non-Euclideans have stretched the meaning of the term
dimension so that it not only includes the idea of direction but an entirely
new class of qualities—the fourth dimension. And despite this reformation
of the original conception, they demand that it shall be called space.

We have just shown that the generic concept of dimensionality is that three
and only three coördinates are necessary and sufficient for its
determination. Granting that this is true, are we not compelled consequently
to see that we have, by adding a fourth or n-dimensions, involved ourselves
into a more complex situation than before? For by postulating a fourth
dimension either we have created a new world whose dimensions are four
in number or we have explicitly admitted that the three dimensions have a
fourth. Aside from the logical difficulties which beset these conclusions
there is also set up a condition which is at variance with the most
elementary requirements of common sense.
Thus far mathematical thought has not served to clarify our notions of space
nor to shed any new light upon the vital processes which are alleged to have
their explanation in the new discovery. Simply stated, metageometricians
have brought us to the place where we must either recognize that the fourth
dimension is another sphere lying dangerously near the earth in which space
extends in four primary directions and in which four coördinates are
necessary for its determination or we are driven to the other horn of the
dilemma where we are brought face to face with the conclusion that the
three perceptual space dimensions have in common a hitherto unknown
property or extension in virtue of which it may be viewed as having an
unlimited number of dimensions. To accept the latter view is equivalent to
saying that, in the above figure, the three lines ab, bc and db have formed a
triple entente by which they have mutually and severally acquired a new
domain, hyperspace, and in which, because of the vast resources of the
region, they are able to perform wondrous things.
Let us examine briefly the various current definitions of dimension. It is
assumed by not a few that dimension is the same as direction. But can we
grant this wholly to be true? If so, then a mere child may see that there are
and must necessarily be as many dimensions as there are directions.
Primarily, there are six directions of space and an unlimited number of
subsidiary directions. On this view it is not necessary to invent a new
domain of space if the object be merely to discover and utilize a greater
number of dimensions than has heretofore been allowed. For the
identification of the term dimension with direction already makes available
an almost infinite number of dimensions. But this view is objected to by the
advocates, for it is contrary to the hypothesis of n-dimensionality.

Dimension also means extent. This is partially true. It cannot be wholly
true. For, if it were, then, space would have only one dimension which is
also not allowable under the hypothesis. Then the definition leaves out of
account the idea that space is at the same time a direction or collection of
directions. The term extension is generic and when applied to space means
extension in all possible directions and not in any one direction. So that it is
not permissible to say that space extends in this direction or that because it
extends in all directions simultaneously and equally.
Geometers claim that space is a system of coördinates necessary for the
establishment of a point-position in it. This view, however, identifies space
with a system of space-measurement and is therefore faulty. According to
this view there may be as many spaces as there are systems of space-
measurement and the latter may be limitless. But if the totality of spaces are
to be viewed as one space then we shall have one space with an indefinite
number of dimensions; also an indefinite number of space measurements
which would be confusing. Much, if not all, of such a system's utility and
convenience would be unavailable or useless. That, too, would be in
violation of the avowed purpose of these investigations which is to enhance
the utility and convenience of mathematic operations.
Now it is evident that space is neither direction, extension, a system of
space-measurement nor a system of manifolds whose dimensions are
generable. And this is so for the same reason that a piece of cloth is not the
elements of measurement—inches, feet, yards—by which it is apportioned.
And because we find that the fabric of space lends itself accommodatingly
to our conventional norms of measurement is not sufficient reason for
identifying it with these norms. Here we have the source of all error in
mathematical conclusions about the nature of space; because all such
conclusions are based not upon the intrinsic nature of space, but upon
artificial forms which we choose to impose upon it for our own
convenience. But it should be remembered that the irregularities which we
note are not in space itself but inhere in the forms which we use. For these
purposes space is extremely elastic and accommodates itself to the shape
and scope of any construction we may decide to try upon it. In this respect
it is like water which has no regard for the shape, size or kind of vessel into
which it may be posited. There is one thing certain that judging from the

above considerations there has been not yet any absolute, all-satisfying
definition devised for space by mathematicians.
The best definitions hitherto constructed are purely artificial and arbitrary
determinations. It is rather anomalous that there should be so little
unanimity about what is the most fundamental consideration of
mathematical conclusions which are supposed to be so certain, so necessary
and universal as to be incontrovertible. Confessedly, it is a condition which
raises again the question as to just what are the limits of mathematical
certainty and necessity and just how far we shall depend upon the validity
of mathematics to determine for us absolutely certain conclusions about the
nature of space. In view of the uncertainty noted, are we justified in
following too closely the mathematic lead even in matters of logic, to say
nothing of our conception of space? It seems that we shall have necessarily,
on account of the recognized limitations of mathematics in this matter, to
turn to some more tenable source for the norms of our knowledge
concerning space. For in the light of the rather indefensible position which
metageometricians have involved themselves there appears to be no hope in
this direction.
It is undoubtedly safer not to rely altogether upon the purely abstract, even
in the world of mathesis, for any absolute criterion of knowledge. It is
perhaps well that we should expunge the word absolute from our
vocabularies. It is really a misnomer and has no meaning in the lexicon of
nature. There is in reality no absolute in the sense of final absolution from
all conditions or restrictions.
In the ultimate analysis there is unquestionably no hue, tone, quality,
condition nor any imaginable posture of life, being or manifestation that is
absolved from every other one of its class or from the totality. All these are
relational and interdependent. There is no room for the absolute. In fact, it is
a quality which cannot in any way be ascribed to any aspect of kosmic
manifestation. It has existence only in the mind and has been devised for
the purpose of marking the limits of its scope. All being is relative; all life is
relative and is destined to change its qualities as it evolves. All knowledge
is also relative and what is true of one state may not be true of another;
what is true of one life may not be true of another life; the limitations of one
degree of knowledge may not have any bearings upon another degree. The

norms of one will not satisfy the conditions of another stage of
manifestation. It is always within limits that the criterion of knowledge will
be found to satisfy a given set of conditions. Hence within certain limits
mathematical conclusions will maintain their validity. Error is committed
by pushing the validity of these limits to a position without the sphere of
limitations. This seems to be the crux of the whole matter. Mathematicians,
notably non-Euclideans, have sought to extend the comparatively small
sphere of limits of congruence between mathematic and perceptual space to
such an extent as to cause it to encroach upon forbidden territory. In doing
this they have erred grievously, causing serious offense to the more
sensitive spirit of the high-caste mathematicians among whom are none
more truly conservative than Paul Carus,[12] who says:

"Metageometricians are a hot-headed race and display sometimes all
the characteristics of sectarian fanatics. To them it is quite clear there
may be two straight lines through one and the same point which do not
coincide and yet are both parallel to a third line."

To the student who has carefully followed the development of the non-
Euclidean geometry and the notion of hyperspace the above
characterization is none too severe nor ill-deserved. Nothing could more
vividly yet correctly portray the impious tactics of the metageometrician
and establish his perceptual obliquity more surely than the mere fact,
mentioned by Carus, that he can with evident lack of mental perturbation
proclaim that two straight lines, noncoincident with each other, may pass
through a point and yet be parallel to a third line. But this is a mere trifle, a
bagatelle, to the many other infractions of which he is guilty. The wonder is
that he is able to secure such obsequious acceptance of his offerings as
many of the most serious minded mathematicians are inclined to give. Is it
to be wondered at that, despite the profuse protestations of the advocates,
many who take up the study of the question of hyperspace should
experience a deep revulsion from the posture assumed by
metageometricians with respect to these queries?
Linked with the idea of dimensionality is the notion that space is infinite.
This is a conception which has its roots imbedded in the depths of antiquity.
Primitive man, looking up into the heavens at what appeared to him as a
never ending extension, was awed by its vastness; but the minds of the most

learned of the present-day men are not free from this innate dread of
infinity. It permeates the thought life of all alike and none seems to be able
to rise above it. Mathematicians, philosophers, scientists all share in the
general belief that space is without limit, unending in extent and eternally
existent. Riemann, whose thought life found its most convenient mode of
expression by means of pure mathematics, was the first in the history of
human thought to surmise that space is not infinite but limited even though
unbounded. But his conception has been much vitiated on account of its
entanglement with an idealized construction by which space is regarded as a
thing to be manipulated and generated by act of thought. Were it not for this
his conception would indeed mark the beginning of a new era in
psychogenesis. As it is, when all the nonsensical effusions have been
cleared away from our space conceptions and men come really to
understand something of the essential nature of space this new era will find
its true beginnings in the mind of Riemann. Although it must be said, as is
the case with all progressive movements, the later development of a
rationale for this conclusion will vary greatly from his original conception.
For he had in mind a space that is generable and therefore a logical
construction while ultimately the mind will swing back to a consideration of
real space.
Already men are beginning to see a new light. Already they are beginning
to take a new view of space in general. The departure is especially
noticeable in the attitude assumed by Hiram M. Stanley.[13] He says:

"If we seek the most satisfactory understanding of space we shall look
neither to mathematics nor Psychology but to Physics. The trend of
Physics, say with such a representative as Ostwald, is to make things
the expression of force; the constitution and appearance of things are
determined by dynamism; and we may best interpret space as a mode
of this dynamic appearance."

Space, as a mode of dynamic appearance is a slight improvement upon the
old idea of a pure vacuity; for in the light of what we now know about
space content much of the dignity of that view has been lost. Men now
know that space is not an empty void. They know that the atmosphere fills a
great deal of space. They also have extended their conception in this
direction to include the ether and occultism goes further and postulates four

kinds of ether—the chemical, life, light and psychographic ethers. But it
does not stop here. It postulates a series of grades of finer matter than the
physical which fills space and permeates its entire extent even to
identification with its essential nature.
Stanley continues:

"Everything does not, as commonly conceived, fall into some pre-
existent space convenient for it; but everything makes its own
spaciousness by its own defensive and offensive force, and the totality
of all appearance is space in general."

According to Stanley, not only do physical, perceptual objects, by their
"offensive and defensive force" make their own space but the appearance of
that in which no physical object is makes room for itself by its own
dynamic force. In other words, that which we call "pure extensity" is by
virtue of its dynamism the cause of its own existence.
At first hand there appears to be little worthy of serious consideration in this
view of Stanley; yet, if carried to its logical conclusion, the merit of the
hypothesis becomes apparent. Accordingly, interstellar distances which are
commonly said to be even without air or life of any kind are really an
appearance possessed of a dynamism peculiar to itself. And this very force-
appearance, constituting space, is that which makes it perceivable. For
instance, let us say the space that exists between the earth and the moon, is
not really empty nor does it have an existence prior to itself, but is a mode
of dynamic appearance which is the cause of its own existence. Its dynamic
character makes it to appear perceptible to our senses. Logically, if the
dynamism were removed there would remain neither space nor the
appearance of space. If this were true, and it is worthy of serious thought,
then space is certainly finite, as in its totality, according to Stanley's view,
it would have to be regarded as a "phenomenon of the inner and finite life
of the infinite."
It is believed that we may go a step further and unqualifiedly assert that
space is finite, even denying its infinity as a "general mode of the activity of
the whole." Yet it is transfinite in the sense that it transcends the
comprehension of finite minds or processes. It is finite because it is in

manifestation. Everything that is in manifestation is finite. The infinite is
not in manifestation. Infinity has to be limited always to become manifest.
The Deity has limited His being in order that there may be a manifested
universe. All things, all appearances are finite; because they are phenomena
connected with manifestation.
This question may be viewed from another standpoint. All things in
manifestation or existence are polar in their constitution. For instance: there
cannot be a "here" without a "there." There cannot be an "upper" without a
"lower." Right is copolar with wrong; good is copolar with evil; night with
day; manifestation with non-manifestation; truth with falsity; infinity with
finity and so on, throughout the whole gamut of the pairs of opposites.
What is the logical inference? Space is paired with a lack of space. There
cannot be what we call space without there being at the same time the
possibility, at least, of the lack of space or spacelessness. This is a
conclusion that is rigorously logical and incontrovertible.
But it has been urged that it is impossible for the mind to imagine a
condition where there is no space. It even has been asserted that it is
contrary to the constitution of the mind itself to imagine "no space." But
whether imaginable or not has no effect whatever upon the validity of the
conception. Neither, it is said, can we imagine a fourth dimension but the
mind has come dangerously near to imagining it. The distance from
excogitating upon, discussing and describing the properties of four-space to
imagining it is not so great after all. Truly it is difficult indeed, it seems, to
be able to describe a thing yet not be able to imagine or make a mental
image of it. There is an evident fallacy here. Either the description of four-
space is no description at all or it is a true delineation of an idealized
construction which is well within the mind's powers of imagination. Indeed
the question of imaginability is not determinative in itself; for what the
mind may now be unable to imagine, because of its more or less nebulous
character, and owing to its infancy may in the course of time be easily
accomplished.
The universe is a compacted plenum. It is chock-full of mind, of life, of
energy and matter. These four are basically one. They exist, of course, in
varying degrees of tenuity and intensity and answer to a wide range of
vibrations. Together, in their manifestation of action and interaction, in their

dynamic appearance, if you please, they constitute space. If these were
removed with all that their existence implies there would result a condition
of spacelessness in which no one of the appearances which we now
perceive would be possible. Even sheer extensity would be non-existent. All
scope of motility would be lacking. Dimension, coördinates, direction,
space-relations—all would be impossible.
A straight line is an ideal construction of the mind. It does not exist in
nature. It can never be actualized in the phenomenal universe. Between the
ideal and the real, or actual, there is a kosmic chasm. It broadens or narrows
according as the phenomenal appearance approaches or recedes from the
ideal. What, therefore, can be postulated of the one will not apply with
equal force to the other. They are not congruent and can never be in the
actualized universe. The moment the actual becomes identified with the
ideal it ceases to be the actual. The universe does not exist as pure form,
neither does space. As purely formal constructions of the intellect these can
have no perceptible existence. The phenomenal or sensible may not be
judged by exactly the same standard as the formal. The phenomenal or
sensible represents things as they appear to the senses, or, so far as the
actualized universe is concerned, as they really are. The formal represents
things as they are made to appear by the mind. It cannot be actualized. It
may be said that the purely formal is the limit of evolution. The phenomenal
may approach the ideal as a limit, but can never become fully congruent
with it. The difference between the ideal and the actual is a dynamic one; it
is by virtue of this difference that the universe is held in manifestation.
Evolution is the decrement of this difference between the purely formal and
the actual. So long then as a kosmic differential is maintained the
phenomenal continues to be manifest: when it is finally reduced to nothing
it goes out of manifestation. The phenomenal is finite; the ideal infinite.
Wherefore, it is undoubtedly improper to refer to space as being infinite.
The term really is inapplicable. Transfinity is much better and more
accurate. Space is transfinite because its scope is greater than any finite
scope of motility can encompass, because it exceeds finite
comprehensibility.
Riemann's notion that space is limited gains weight in the light of the
foregoing considerations. But he could not conceive of the limitability and

unboundedness of space as such in its pure essence; but was compelled, by
his own limitations, to make an idealized construction in which he could
actualize his conception. And for real, dynamic space, he substituted his
ideal construction and proceeded upon that basis. And of course, his view
while it had no reference to perceptual space nevertheless possessed an
illative relation thereto and should be recognized as construable in that
light.
The process of squaring the circle recognized as a geometric impossibility
is significant of the fluxional nature of the universal residuum perpetually
maintained between the archetypal and the manifested kosmos. It seems
that there is a profound truth embodied in this problem. There is a lesson
that may be learned by mathematicians, philosophers, scientists and
thinkers in general. There is an element of eternal necessity and universality
about it which is truly symbolic of the finity of the universe and the infinity
of the archetypal. Just as a square or a series of polygonal figures inscribed
in a circle cannot be made to coincide exactly with the circle so cannot the
actual be made to coincide with the ideal. The circumference of the circle is
the unapproachable limit of inscribed squares. If it were possible so to
multiply squares thus inscribed that a figure coincident with the
circumference of a circle might be constructed, such a figure would not be a
square but a circle. The manifested universe is like that—the process of
inscribing squares within a circle. It is ever becoming, evolving, developing,
but never quite attains. Infinity is a process. But no single stage in that
process is infinite. Each is finite and their totality makes the infinity of the
process. The universe manifested to the senses or the intellect is finite.

"Space," says Paul Carus, "is the possibility of motion in all directions."[14]
To be sure, it is admitted that space offers opportunity for motion in all
directions. But is space this opportunity of motility? Or is possibility of
motion space? The possibility of motion must rest in the thing that moves. It
implies a potency in the moving entity, not in space. If it is meant that space
is the potency that resides in the moving element it is still more difficult to
understand the connotation. But even granting this view, are we not
compelled to recognize the dynamism of space as a necessary inference?
Another definition which Carus gives is that space is a "pure form of
extension." If it be granted that space is a pure form of extension we should
have to conclude that it has no actual existence; for pure form does not exist

except as an idealized construction. It cannot be found in nature. Pure form
is ideal. Impure or natural form is actual. Therefore the space in which we
live and in which the universe exists cannot be a "pure form" because life
cannot exist in the purely formal. It is useless to talk about space as mere
form so long as it maintains life. The difficulty which this phase of the
question presents is another evidence of the inadequacy of our definitions.
It is also found to be impossible to concur in Carus' conception of
knowledge a priori. His notion of the a priori varies somewhat from the
Kantian view. He defines it as an "idealized construction," the "mind
made," "abstract thought," and places it in the same category as a concept.
This is undoubtedly born of his desire to get rid of Kant's "innate ideas"
which seem to be distasteful to him. But in doing so it appears that the real
a priori has been overlooked. Let us examine for a moment this important
question. The a posteriori connotates all knowledge gained through the
senses, or sense experience. All knowledge therefore whose origin can be
traced to the senses is knowledge a posteriori. Now, knowledge a priori
should be just the opposite of this. It should indicate such knowledge as that
which does not have its origin in the senses, or which is not dependent upon
the ordinary avenues of sense-experience. Abstract thought is as truly
experience as smelling, seeing or hearing. It is by traversing its scope of
motility that the mind finds out what the norms of logic are. It could not
remain quiescent and discover them. It has to be active, examining,
comparing and judging. Almost the entire range of thought, its entire scope,
is characterized by the a posterioristic method. In fact, all thought is a
posterioristic. Despite the fact that, in thinking in the abstract, it is
necessary mentally to remove all elements of concreteness, all materiality
and all actuality, the conclusions reached have to be referred to the
standards maintained by the actual, the concrete and the material. Then we
do not start with the abstract in our thinking. We begin with the concrete
and by mentally removing all physical qualities arrive at the abstract.
The mind has a constitution. It acts in a given way because it is its nature so
to act. Not because it has learned to act in that manner. It performs certain
functions intuitively without previous instruction or experience for the same
reason that water dampens or heat warms. It is natural for it to do so. This
naturalness, this performance of function without being taught or without
experience constitute the principle of apriority in the mind. Aprioriness is a

principle of mind partaking of the very nature and essence of mind. It is the
very mainspring of mentality. Perception and conception are processes
which the mind performs intuitively. The mind perceives and conceives
because it is impossible for the normal mind to do otherwise. We take a
view upon a given question; we assume certain mental attitudes of
affirmation, negation or indifference because we have learned to do so by
virtue of the tuitional capability of mind. These describe the a posteriori.
That is, all knowledge obtained as a result of voluntary mental processes
constitutes the mass of knowledge a posteriori. The a priori is what the
mind is by nature: the a posteriori is what the mind becomes. It is the mind-
content.
The a priori is not a mental construction; it is an essential principle of mind.
It should not be identified with the "purely formal," as is done by Paul
Carus:[15]
He says:

"The a priori is identical with the purely formal which originates in
our mind by abstraction. When we limit our attention to the purely
relational, dropping all other features out of sight, we produce a field
of abstraction in which we can construct purely formal combination,
such as numbers, or the ideas of types and species. Thus we create a
world of pure thought which has the advantage of being applicable to
any purely formal consideration and we work out systems of numbers
which, when counting, we can use as standards of reference for our
experience in practical life."

Thus Carus definitely links up the a priori to a factor which is nothing
more nor less than a mental by-product. For such is the category in which
would be placed both the process of abstraction and its results. It is
therefore exceedingly difficult to understand why so cursory a consideration
should have been given to the principle of apriority than which no other
element of mind is more essentially a part of the mind itself.
The formal is symbolic. It signifies an informing quantity. Pure form itself
is but a negation of that which formerly filled it. Then, too, the formal is
purely artificial because it is a mental construction. Essentially there is as

much difference between the purely formal and the a priori as between
creator and creature, as between potter and clay. The one is the builder, the
other is the material; the one the knower and the other the known. Thus, the
only reason that the formal is found to be answerable to the a priori at all is
due to the fact that it is construable only upon the basis of the a priori. But
being so is not sufficient warrant for its identification with the a priori. The
formal merely represents the totality of possibilities in the universe as
viewed by the mind; but as the number of possibilities open to the mind is,
on account of its nature and purpose limited, it is not to be supposed that it
(the mind) shall measure up to all the possibilities offered by the formal.
Moreover, it is certain that no sane mind cherishes the hope that there shall
ever be found in the universe of life and form a congruence for all of the
possibilities held out by the purely formal.
As an eternal principle of mind, the a priori is in agreement with the divine
mind of the kosmos. In its aposteriority the mind is of diverse tendences,
qualities and characteristics. Apriorily, it acts in unison with the eternal
purpose of life and the universal mind. In its aposteriority, it often goes
awry. In its apriority it can never be insane; insanity is a symptom of the
morbid a posteriori.
The mind in man acts the same as mind in the vegetal and lower animal
kingdoms. Metabolism and katabolism, indeed all cell-activity, are a priori
performances of the mind. Growth and all its phenomena, the cyclicism of
natural processes, and every activity connected therewith belong to the
category of the a priori. Cells multiply, divide, build up and tear down
tissues and they do it intuitively. Most certainly these functions are
performed without any assistance from the intellect. All the myriad
activities in nature with which the intellect in man has not the slightest
concern, truly acting in accord with some primordial impetus, are activities
a priori.
Now what is the attitude of the intellect, in the light of the a priori, towards
space and the question of dimensionality? It is evident that no matter what
this attitude may be it is in agreement with the constitution of things and of
the universe. And if so, it is right and without illusion. It is also evident that
whatever notion a posteriori the intellect may entertain with respect to these
questions is unavoidably liable to the illusionary drawbacks common to

conclusions based upon limited experience. The geometric view of space
belongs to the category of the a posteriori. Hence it is subject to the usual
imposition of error.
Tersely stated, Kant's view of space is that it is a form of intuition, a form a
priori, a transcendental form. As such he considered it to be a native form
of perception not belonging to the category of sense-deliveries.
Accordingly, space is a form of intuition arising out of and inhering in the
constitution of mind. It is a notion which constitutes the universal and
eternal prerequisite of mind and is, therefore, intrinsically necessary to all
phases of mentation. Now, this being true just what may be said to be the
relation of dimensionality to this a priori form of space which is found to
exist in the mind as an eternal aspect of its nature? Does the mind
intuitively measure its contents or its operations by the empirical standard
of space-measurement known as dimension? Is the attitude of the mind
towards the objectively real one of discrimination a priori as to the
direction or dimension in which a percept may originate? In other words,
does the mind habitually and intuitively refer its data to a system of
coördinates for final determination? There is no other answer but that the
mind makes no such reference and is dependent upon no kind of coördinate
system in any of its operations a priori. As a form of intuition, the space
notion is present in the mind as a scope of existence, of motility, of being
and of sheer roominess. The notion of direction or dimension, being an
artificial construction, does not enter into this form of intuition at all. It is
only when the mind comes to elaborate upon its perceptive performances
and possibilities that the questions of relations, positions and directions
arise. But this latter is a matter separate and distinct from the state of
awareness which embodies the notion of space.
Dimension is an arbitrary norm constructed by the mind for the
determination of various positions in space. It is an accident or by-product
of the process of elaborative cognition, a convenient and appropriate means
of measurement for objects in space and their space-relations. But it is no
more a priori than a foot rule or a square. But being purely an empirical
product it may be said to be an aspect of psychogenesis because it relates to
the evolutionary aspect of mind. The assumption may therefore be allowed
that the mind may, in the course of its evolution, find it convenient and
appropriate to devise an additional ordinate or dimension to satisfy the

necessities of its more complex ramifications into the nature of things and
to determine their greatly increased space-relations. It may be even possible
for the mind to function normally in a space of four dimensions. But this
would simply be a new adjustment, not a change in the essential nature of
mind. It would be like the series of adjustments to environments which man
has made in the onward movement of civilization. There has been no
serious change in the manhood per se of man. That has remained the same;
there has been merely a complication of environmental influences.
Similarly, in the acquisition of four-dimensional powers, granting that such
an acquisition is possible, there is nothing to be added to the aprioriness of
mind itself. Is it not, therefore, logical to assume that the discovery of a
fourth coördinate and the consequent conceptualization of the same, point
to the development in the mind of a greatly extended faculty, more keenly
penetrative powers of cognition and a further diversification of its
environments than it has hitherto enjoyed? Indeed, it seems so.

CHAPTER V

The Fourth Dimension

The Ideal and the Representative Nature of Objects in the Sensible
World—The Psychic Fluxional the Basis of Mental Differences—
Natural and Artificial Symbols—Use of Analogies to Prove the
Existence of a Fourth Dimension—The Generation of a Hypercube or
Tesseract—Possibilities in the World of the Fourth Dimension—Some
Logical Difficulties Inhering in the Four-Space Conception—The
Fallacy of the Plane-Rotation Hypothesis—C. H. Hinton and Major
Ellis on the Fourth Dimension.

The world of mathesis is truly a marvelous domain. Vast are its possibilities
and vaster still its sweep of conceivability. It is the kingdom of the mind
where, in regal freedom, it may perform feats which it is impossible to
actualize in the phenomenal universe. In fact, there is no necessity to
consider the limitations imposed by the actualities of the sensuous world.
Logic is the architect of this region, and for it there is no limit to the
admissibility of hypotheses. These may be multiplied at will, and
legitimately so. The chief error lies in the attempt to make them appear as
actual facts of the physical world.
Mathematicians, speculating upon the possibilities of mathetic
constructions and forgetting the necessary distinctions which should be
recognized as differentiating the two worlds, in their enthusiasm have been
led into the error of postulating as qualities of the phenomenal world the
characteristics of the conceptual. Accordingly, a great deal of confusion as
to the proper limits and restrictions of these conceptions has arisen and
there still may be found those who are enthusiastically endeavoring to push
the actualities of the physical over into the conceptual. But in assuming any
attitude towards mathetic propositions, especially with a view to

demonstrating their actuality, very careful discrimination as to the essential
qualities and their connotations should be made. Hence, before taking up a
brief study of the fourth dimension proper, it is deemed fitting to indicate
some of the fundamental distinctions which every student of these questions
should be able to make with reference to the data which he meets.
All objects of the sensible world have both an essential or ideal nature and a
representative or sensuous nature. That is, they may be studied from the
standpoint of the ideal as well as the sensuous. The representative nature is
that which we recognize as the mode of appearance to our senses which, as
Kant held, is not the essential or ideal character of the thing itself. For there
is quite as much difference between the sensuous percept and the real thing
itself as between an object and its shadow. In fact, a concept viewed in this
light, may be seen to have all the characteristics of an ordinary shadow; for
instance, the shadow of a tree. View it as the sun is rising; it will then be
seen to appear very much elongated, becoming less in length and more
distinct in outline as the sun rises to a position directly overhead. The
elongation may again be seen when the sun is setting. Throughout the day
as the sun assumes different angles with reference to the tree the
proportions and definiteness of the shadow vary accordingly. Thus the
angularity of the sun, the intensity and fullness of the light and the shape
and size of the tree operate to determine the character of the shadow.
Much the same thing is true of a sensuous representation. If we examine
carefully our ideas of geometric quantities and magnitudes, it will be found
that the concepts themselves are not identical with the objects of the
physical world, but mere mental shadows of them. The angularity of
consciousness, or the distinctness of one's state of awareness, being
analogous to similar attitudes in the solar influence are the main
determinants of the character of the mental shadow or concept. Wherefore
mathematical "spaces" or magnitudes are not sensuous things and have
therefore no more real existence than a shadow, and strictly speaking not as
much; for a shadow may be seen, while such magnitudes can only be
conceived. It may be urged that since we can conceive of such things they
must have existence of some kind. And so they have, but it is an existence
of a different kind from that which we recognize as belonging to things in
the sensible world. They have a conceptual existence, but not a sensuous
one. Therein lies the great difference.

To be sure, a shadow is a more or less true representation of the thing to
which it pertains. That this is true can be established empirically. Similarly,
the degree of congruity between objects and concepts likewise may be
determined. If this were not true we should be very much disappointed with
what we find in the phenomenal world and could never be quite sure that
the mentograph existing in our minds was a faithful representation of the
thing which we might be examining. But really the foundation for such a
disappointment is present in every concept, every percept with which the
mind deals. This disappointment, although in actual experience is reduced
to an almost negligible quantity, is due to the failure of sensuous objects to
conform wholly to the specific details of the mental shadow or mentograph.
This lack of congruence between the mental picture and the object itself is
necessary for obvious reasons. It is markedly observable in the early efforts
of a child in learning distances, weights, resistances, temperatures and the
like. No inconsiderable time is required for the child to be able correctly to
harmonize his sense-deliveries with actual conditions. Otherwise, the child
would never make any of the ludicrous mistakes of judgment of which it is
guilty when trying to get its bearings in the world of the senses. In the
course of time the child gradually learns by experience that certain things
are true of objects, distances, temperatures, resistances, etc., and that certain
things are not true of them. He learns these things by actually contacting
various objects. He is then competent to render correct judgments, within
certain limits, as to the conditions which he finds in the sensible world. And
the allowances, equations and corrections which his motor, sensory and
psychic mechanisms learn to make in childhood serve for all subsequent
time. And this is important to remember; for the mature mind is apt to
forget or overlook the adaptations which the child-mind has made in its
growth.
If there were no such differences between the concept and the thing itself,
actual physical contact would not be necessary. For one could rely wholly
upon the sense-deliveries and each sense might operate entirely
independently of all the others as there would be no necessity to correct the
delivery of one by those of the others. This, of course, raises the question as
to the necessity of sense-experience at all under conditions where there
would be no disparity between the thing itself and the ideal representation
of it in the mind. The absence of this variable quantity would open to the

mind the possibility of really knowing the essential nature of objects in the
phenomenal world, a condition of affairs which is admittedly now without
the range of the powers of the mind.
At any rate, the essential "thingness" of objects can never be comprehended
by the mind until the diminution of this disparity between the object of
sense and the mental picture of it which exists in the consciousness has
proceeded to such a limit as either completely to have obliterated it or to
such an extent that the psychic fluxion is so slight as not to matter.
It is believed that the results of mental evolution, as the mind approaches
the transfinite as a limit, will operate to minimize the fluxional quantity
which subsists between all objects of sense and their ideal representation as
data of consciousness. The conclusion that the mind of early men who lived
hundreds of thousands and perhaps millions of years ago on this planet
consumed a much longer time in learning the adjustments between the
objects which it contacted in the sensuous world and the elementary
representations which were registered in its youthful consciousness than is
to-day required for similar processes seems to be demanded, and
substantiated as well, by what is known of the phyletic development of the
mind in the human race.
In view of the above, it is thought that the duration of such simple mental
processes served not only to prolong the physical life of the man of those
early days, but may also account for the puerility and incapacity of the mind
at that stage. Not that the slow mental processes were active causative
agencies in lengthening the life of man, but that they together with the crass
physicality of man necessitated a longer physical life. This, perhaps in a
larger sense than any other consideration, accounts for the fundamental
discrepancies in the mind of the primitive man in comparison with the
efficiency of the mind of the present-day man. In view of the potential
character of mind and in the light of the well graduated scale of its
accomplishments, it is undoubtedly safe to conclude that the quality of
mental capacities is proportional to the psychic fluxional which may exist at
any time between the ideal and the essential or real. Mental differences and
potentialities in general may be due to the magnitude of the psychic
fluxional or differential that exists between the conceptual and the
perceptual universe. In some minds it may be greater than in others. The

chasm between things-in-themselves and the mental notion pertaining
thereto may vary in a direct ratio to the individual mind's place in
psychogenesis, and therefore, be the key to all mental differences in this
respect.
Most certain it is that there may be marked fluctuations in the judicial
approach of minds towards any psychic end. In other words, there is not
only a fluxional or differential between the object and its representation, but
also a differential between the approach of one mind and another in the
judicial determination of notions concerning ideas. In this way, differences
of opinions as to the right and wrong of judgments arise. Indeed, there seem
to be zones of affinity for minds of similar characteristics, or minds that
have the same degree of differential; so that, in choosing among the many
possible judgments predicable upon a species of data, all those minds
having the same degree of psychic differential discover a special affinity or
agreement among themselves. Hence, we have cults, schools of thought,
and various other sectional bodies that find a basis of agreement for their
operations in this way. The outcome of this remarkable intellectual
phenomenon is that there are as many different kinds of judgments as there
are zones of affinity among minds. Various systems of philosophy owe their
existence to these considerations, and the considerations themselves flow
from the fact that all intellectual operations are essentially superficial;
because there is no means by which they may penetrate to the steady
flowing stream of reality which pervades and sustains objects in the
sensible world.
In view, therefore, of the foregoing and with special reference to geometric
constructions, it is necessary in approaching a study of the four-space that it
be understood at the outset that the fourth dimension can neither be
actualized nor made objectively possible even in the slightest degree in the
perceptual world; because it belongs to the world of pure thought and exists
there as an "extra personal affair," separate and distinct from the world of
the senses.

As says Simon Newcomb:[16]

"The experience of the race and all the refinements of modern science
may be regarded as showing quite conclusively that, within the limits

of our experience, there is no motion of material masses, in the
direction of a fourth dimension, no physical agency which we can
assume to have its origin in regions to which matter cannot move,
when it has three degrees of freedom."

There is, however, no logical objection to the study of the fourth dimension
as a purely hypothetical question, if by pursuit of the same an improvement
of methods of research and of the outlook upon the field of the actual may
be gained. Hence, it is with this attitude of mind that we approach the
consideration of the fourth dimension.
Various efforts have been made to render the conception of a fourth
dimension of space thinkable. The student of space has reasoned: "We say
that there are three dimensions of space. Why should we stop here? May
there not be spaces of four dimensions and more?" Or he has said: "If 'A'
may represent the side of a square, A2 its area, and A3 the volume of a cube
with edge equal to A; what may A4, A5 or Anth represent in our space? The
conclusion, with respect to the quantity A4, has been that it should represent
a space of four dimensions."
Algebraic quantities, however, represent neither objects in space nor space
qualities except in a purely conventional manner. All efforts to justify the
objective existence of a fourth dimension based upon such reasoning will,
therefore, fail; because the basis of such arguments is itself faulty. In the
sentence: "The man loves his bottle," the thing meant is not the bottle, but
what the bottle contains. For the purpose of the figure the bottle signifies its
contents. There is no more real connection between the bottle and what it
contains than between any word and the object for which it stands. Words
are said to be symbols of ideas. But they are not natural symbols; they are
conventional symbols, made for the purpose of cataloguing, indexing and
systematizing our knowledge. Words can be divorced from ideas and
objects, or rather have never had any real connection with them. There are
two classes of natural symbols, namely; objects and ideas. These, objects
and ideas, symbolize realities. Realities are imperceptible and
incomprehensible to the intellect which has aptitude only for a slight
comprehension of the symbols of realities. For instance, a tree is a natural
symbol. It represents an actuality which is imperceptible to the intellect.
The intellect can deal only with the sensible symbol. It is a natural symbol;

because it is possible directly to trace a living connection between the tree
and the tree-reality. That is, it would be possible so to trace out the vital
connection between the tree and its reality if the intellect had aptitude for
such tracery. But, in reality, since it has no such aptitude, it remains for the
work of that higher faculty than the intellect which recognizes both the
connection and the intellect's inability to trace it. Further, an object is called
a natural symbol because it is the bridge between sensuous representation
and reality. It is as if one could begin at the surface of an object and by a
subtle process of elimination and excortication arrive at the heart of the
universum of reality. No such consummation may be reached by dealing
with words which have merely an artificial relationship with the objects
which they signify. Again, ideas, that is, ideas that are universal in
application and have their roots in the great ocean of reality, are natural
symbols; because if it were possible to handle an idea with the physical
hands it would be possible to arrive at the heart of that which it symbolized
without ever losing our connection with the idea itself. In other words, ideas
and objects, unlike words, can never be divorced from that which they
symbolize. Both, being of the same class, are the opposite poles of realities.
This then is the difference between natural symbols and artificial symbols—
that a natural symbol, such as objects and ideas, is copolar with reality
whereas an artificial symbol, such as words, geometric constructions and
the like not only lacks this copolarity but is itself a symbol of natural
symbols.

It is, therefore, inconceivable that because the algebraic quantity A3 has
been arbitrarily decreed to be a representation of the volume of a cube,
every such quantity in the algebraic series shall actually represent some
object or set of objects in the physical world. Even if it be granted that such
may be the case, is it not certain that there is a limit to things in the
objective universe? Yet there may not be any limit to algebraic or
mathematical determinations. The material universe is limited and
conditioned; the world of mathesis is unlimited and unconditioned save by
its own limitations and conditions. It is irrational to expect that physical
phenomena shall justify all mathematical predicates.
There is perhaps no single mathematical desideratum or consideration
which may be said to be the natural symbolism of realities; for the whole of
mathematical conclusions is a mass of artificial and arbitrary but

concordant symbols of the crasser or nether pole of the antipodes of
realism. It is exceedingly dangerous, therefore, to predicate upon such a far-
fetched symbolism as mathematics furnishes anything purporting to deal
with ultimate realities. And those who insist upon doing so are either blind
themselves to these limitations or are madly endeavoring to befog the minds
of others who are dependent upon them for leadership in questions of
mathematical import.
Analogies have been unsparingly used in efforts to popularize the four-
space conception and much of the violence which has been done to the
notion is due to this vagary. The mathematical publicist, in trying to give a
mental picture of the fourth dimension, examines the appearances of three
dimensional beings as they might appear to a two dimensional being or
duodim. He imagines a race of beings endowed with all the human faculties
except that they live in a land of but two dimensions—length and breadth.
He thinks of them as shadows of three dimensional beings to whom there
are no such conceptions as "up" and "down." They can see nothing nor
sense anything in any way that is without their plane. They can move in any
direction within the plane in which they live, but can have no idea of any
movement that might carry them without that plane. A house for such
beings might be simply a series of rectangles. One of them might be as safe
behind a line as a tridim or three dimensional being would be behind a
stone wall. A bank safe for the unodim would be a mere circle. A duodim in
any two dimensional prison might be rescued by a tridim without the
opening of doors or the breaking of walls. An action of a tridim performed
so as to contact their plane would be to them a miracle, absolutely
unaccountable upon the basis of any known fact to the unodim or duodim. A
tridim might go into a house where lived a family of duodims, appear and
disappear without being detected or its ever being discovered how he
accomplished such a marvelous feat. Our miracles, after the same fashion,
are said to be the antics of some four dimensional being who has similar
access to our three dimensional world and whose actions are similarly
inexplicable to us. So the analogies have been multiplied. But the
temptation to apply the consequences of such reasoning to actual three-
space conditions has been so great that many have yielded to it and have
consequently sought actually to explain physical phenomena upon the basis
of the fourth dimension.

The utilitarian side of the question of hyperspace has not been neglected
either. And so, early in the development of the hypothesis and its various
connotations, the attention of investigators was turned to this aspect of the
inquiry. Strange possibilities were revealed as a result. For instance, it was
found that an expert fourth dimensional operator is possessed of
extraordinary advantages over ordinary tridimensional beings. Operating
from his mysterious hiding place in hyperspace, he could easily appear and
disappear in so mysterious a manner that even the most strongly sealed
chests of treasures would be easily and entirely at his disposal. No city
police, Scotland Yard detective nor gendarme could have any terrors for
him. Drs. Jekyll and Messrs. Hyde might abound everywhere without fear
of detection. Objects as well as persons might be made to pass into or out of
closed rooms "without penetrating the walls," thus making escape easy for
the imprisoned. No tridimensional state, condition or system of
arrangements, accordingly, would be safe from the ravages of evilly
inclined four dimensional entities. Objects that now are limited to a point or
line rotation could in the fourth dimension rotate about a plane and thus
further increase the perplexities of our engineering and mechanical
problems; four lines could be erected perpendicular to each other whereas
in three space only three such lines can be erected; the right hand could be
maneuvered into the fourth dimension and be recovered as a left hand; the
mysteries of growth, decay and death would find a satisfactory explanation
on the basis of the fourth dimensional hypothesis and many, if not all, of the
perplexing problems of physiology, chemistry, physics, astronomy,
anthropology and psychology would yield up their mysteries to the skill of
the fourth dimensional operator. Marvelous possibilities these and much to
be desired! But the most remarkable thing about these so-called possibilities
is their impossibility. It is this kind of erratic reasoning that has brought the
conception of a fourth dimension into general disrepute with the popular
mind. It is to be regretted, too, for the notion is a perfectly legitimate one in
the domain of mathesis where it originated and rightly belongs.
It is not to be wondered at that metageometricians and others should at first
surmise that, in the four-space, they had found the key to the deep mysteries
of nature in all branches of inquiry. For so vast was the domain and so
marvelous were the possibilities which the new movement revealed that it
was to be expected that those who were privileged to get the first glimpses

thereof would not be able to realize fully their significance. But the stound
of their minds and the attendant magnification of the elements which they
discovered were but incidents in the larger and more comprehensive process
of adjustment to the great outstanding facts of psychogenesis which is only
faintly foreshadowed in the so-called hyperdimensional. The whole scope
of inquiry connected with hyperspace is not an end in itself. It is merely a
means to an end. And that is the preparation of the human mind for the
inborning of a new faculty and consequently more largely extended powers
of cognition. Metageometrical discoveries are therefore the excrescences of
a deeper, more significant world process of mental unfoldment. They
belong to the matutinal phenomena incident to this new stage of mental
evolution. All such investigations are but the preliminary exercises which
give birth to new tendencies which are destined to flower forth into
additional faculties and capacities. So that it is well that the evolutionary
aspect of the question be not overlooked; for there is danger of this on
account of the magnitude and kosmic importance of its scope of motility.
A geometric line is said to be a space of one dimension. A plane is a space
of two dimensions and a cube, a space of three dimensions. In figure 7
below, the line ab is said to be one dimensional; because only one
coördinate is necessary to locate a point-position in it. The plane, abcd,
figure 8, is said to be two dimensional because two coördinates, ab and db
are required to locate a point, as the point b. The cube abcdefgh, figure 9, is
said to be tridimensional, because, in order to locate the point b, for
instance, it is necessary to have three coördinates, ab, bc and gb. The
tesseract is said to be four dimensional, because, in order to locate the point
b, in the tesseract, it is necessary to have four coördinates, ab, bc, bb' and
h'b, figure 10.

Fig. 7.

Fig. 8.

It will be noted that in figures 8, 9 and 10, the element of perpendicularity
enters as a necessary determination. In figure 8, the lines ab and bd are
perpendicular to each other. Similarly, in Fig. 10, lines ab, bc, bb' and h'b
are perpendicular to one another. That is, at their intersections, they make
right angles. Similarly, figures representing any number of dimensions may
be constructed.

Fig. 9.

Fig. 10.

The line ab represents a one-space. An entity living in a one space is called
a "unodim." The plane, abcd, represents a two-space, and entities living in
such a space are called duodims. The cube, abcdefgh, represents a three-
space and entities inhabiting such a space are called tridims. Figure 10
represents a four-space, and its inhabitants are called quartodims. Each of
the above-mentioned spaces is said to have certain limitations peculiar to
itself.
The fourth dimension is said to lie in a direction at right angles to each of
our three-space directions. This, of course, gives rise to the possibility of
generating a new kind of volume, the hypervolume. The hypercube or
tesseract is described by moving the generating cube in the direction in
which the fourth dimension extends. For instance, if the cube, Fig. 9, were
moved in a direction at right angles to each of its sides a distance equal to
one of its sides, a figure of four dimensions, the tesseract, would result.
The initial cube, abcc'e'fhh', when moved in a direction at right angles to
each of its faces, generates the hypercube, Fig. 10. The lines, aa', bb', cc',
dd', ee', ff', gg', hh', are assumed to be perpendicular to the lines meeting at
the points, a, b, c, d, e, f, g, h. Hence a'b', b'd, dd', d'a', ef, fg, gg', g'e,
represent the final cube resulting from the hyperspace movement. Counting
the number of cubes that compose the hypercube we find that there are
eight. The generating cube, abcc'e'f'hh', and the final cube, a'b', b'd, dd',
d'a', ef, fg, gg', g'e, make two cubes; and each face generates a cube making
eight in all. A tesseract, therefore, is a figure bounded by eight cubes.
To find the different elements of a tesseract, the following rules will apply:
1. To find the number of lines: Multiply the number of lines in the
generating cube by two, and add a line for each point or corner in it. E.g., 2
× 12 = 24 + 8 = 32.
2. To find the number of planes, faces or squares: Multiply the number of
planes in the generating cube by 2 and add a plane for each line in it. E.g., 2
× 6 + 12 = 24.
3. To find the number of cubes in a hypercube: Multiply the number of
cubes in the generating cube, one, by two and add a cube for each plane in
it. E.g., 2 × 1 + 6 = 8.

4. To find the number of points or corners: Multiply the number of corners
in the generating cube by 2. E.g., 2 × 8 = 16.

In a plane there may be three points each equally distant from one another.
These may be joined, forming an equilateral triangle in which there are
three vertices or points, three lines or sides and one surface.
In three-space there may be four points each equidistant from the others. At
the vertices of a regular tetrahedron may be found such points. The
tetrahedron has four points, one at each vertex, 6 lines and 4 equilateral
triangles, as in Fig. 11.
In four-space, we have 5 points each equidistant from all the rest, giving the
hypertetrahedron. This four dimensional figure may be generated by
moving the tetrahedron in the direction of the fourth dimension, as in Fig.
12. If a plane be passed through each of the six edges of the tetrahedron and
the new vertex there will be six new planes or faces, making 10 in all,
counting the original four. From the new vertex there is also a tetrahedron
resting upon each base of the original tetrahedron so that there are five
tetrahedra in all. A hypertetrahedron is a four-dimensional figure consisting
of five tetrahedra, ten faces, 10 lines and 5 points.

Fig. 11.

Fig. 12.

Paul Carus[17] suggests the use of mirrors so arranged that they give eight
representations of a cube when placed at their point of intersection. He says:

"If we build up three mirrors at right angles and place any object in the
intersecting corner we shall see the object not once, but eight times.
The body is reflected below and the object thus doubled is mirrored
not only on both upright sides but in addition in the corner beyond,
appearing in either of the upright mirrors coincidingly in the same
place. Thus the total multiplication of our tridimensional boundaries of
a four dimensional complex is rendered eight-fold.
"We must now bear in mind that this representation of a fourth
dimension suffers from all the faults of the analogous figure of a cube
in two dimensional space. The several figures are not eight
independent bodies but are mere boundaries and the four dimensional
space is conditioned by their interrelation. It is that unrepresentable
something which they inclose, or in other words, of which they are
assumed to be boundaries. If we were four dimensional beings we
could naturally and easily enter into the mirrored space and transfer
tridimensional bodies or parts of them into those other objects

reflected here in the mirrors representing the boundaries of the four
dimensional object. While thus on the one hand the mirrored pictures
would be as real as the original object, they would not take up the
space of our three dimensions, and in this respect, our method of
representing the fourth dimension by mirrors would be quite analogous
to the cube pictured on a plane surface, for the space to which we
(being limited to our tridimensional space-conception), would
naturally relegate the seven additional mirrored images is unoccupied
and if we should make the trial, we would find it empty."

The utility of such a representation as that which Carus outlines in the
above is granted, i.e., so far as the purpose which it serves in giving a
general idea of what a four-space object might be imagined to be like, but
the illustration does not demonstrate the existence of a fourth dimension. It
only shows what might be if there were a four-space in which objects could
exist and be examined. We, of course, have no right to assume that because
it can be shown by analogous reasoning that certain characteristics of the
fourth dimensional object can be represented in three-space the possible
existence of such an object is thereby established. Not at all. For there is no
imaginable condition of tridimensional mechanics in which an object may
be said to have an objective existence similar to that represented by the
mirrored cube.
But there are discrepancies in this representation which well might be
considered. They have virtually the force of invalidating somewhat the
conception which the analogy is designed to illustrate. For instance, in the
case of the mirrored object placed at the point of intersection of the three
mirrors built up at right angles to each other. Upon examination of such a
construction it is found that the reflection of the object in the mirrors has
not any perceptible connection with the object itself. And this, too, despite
the fact that they are regarded as boundaries of the hypercube; especially is
this true when it is noted that these reflections are called upon to play the
part of real, palpable boundaries. If a fourth dimensional object were really
like the mirror-representation it would be open to serious objections from
all viewpoints. The replacement of any of the boundaries required in the
analogy would necessarily mean the replacement of the hypercube itself. In
other words, if the real cube be removed from its position at the intersection
of the mirrors no reflection will be seen, and hence no boundaries and no

hypercube. The analogy while admittedly possessing some slight value in
the direction meant, is nevertheless valueless so far as a detailed
representation is concerned. So the analogy falls down; but once again is
the question raised as to whether the so-called fourth dimension can be
established or proven at all upon purely mathematical grounds. It also
emphasizes the necessity for a clearer conception of the meaning of
dimension and space.
The logical difficulties which beset the hyperspace conception are dwelt
upon at length by James H. Hyslop. He says:[18]

"The supposition that there are three dimensions instead of one, or that
there are only three dimensions is purely arbitrary, though convenient
for certain practical purposes. Here the supposition expresses only
differences of directions from an assumed point. Thus what would be
said to lie in a plane in one relation would lie in the third dimension in
another. There is nothing to determine absolutely what is the first,
second, or third dimension. If the plane horizontal to the sensorium be
called plane dimension, the plane vertical to it will be called solid, or
the third dimension, but a change of position will change the names of
these dimensions without involving the slightest qualitative change or
difference in meaning.
"Moreover, we usually select three lines or planes terminating
vertically at the same point, the lines connecting the three surfaces of a
cube with the same point, as the representative of what is meant by
three dimensions, and reduce all other lines and planes to these. But
interesting facts are observable here. 1. If the vertical relation between
two lines be necessary for defining a dimension, then all other lines
than the specified ones are either not in any dimension at all, or they
are outside the three given dimensions. This is denied by all parties,
which only shows that a vertical relation to other lines is not necessary
to the determination of a dimension. 2. If lines outside the three
vertically intersecting lines still lie in dimension or are reducible to the
other dimensions they may lie in more than one dimension at the same
time which after all is a fact. This only shows that qualitatively all
three dimensions are the same and that any line outside of another can
only represent a dimension in the sense of direction from a given point

or line, and we are entitled to assume as many dimensions as we
please, all within three dimensions.
"This mode of treatment shows the source of the illusion about the
'fourth dimension.' The term in its generic import denotes
commensurable quality and denotes only one such quality, so that the
property supposed to determine non-Euclidean geometry must be
qualitatively different from this, if its figures involve the necessary
qualitative differentiation from Euclidean mathematics. But this would
shut out the idea of 'dimension' as its basis which is contrary to the
supposition. On the other hand, the term has a specific meaning which
as different qualitatively from the generic includes a right to use the
generic term to describe them differentially, but if used only
quantitatively, that is, to express direction as it, in fact, does in these
cases, involves the admission of the actual, not a supposititious,
existence of a fourth dimension which again is contrary to the
supposition of the non-Euclidean geometry. Stated briefly, dimension
as commensurable quality makes the existence of the fourth dimension
a transcendental problem, but as mere direction, an empirical problem.
And the last conception satisfies all the requirements of the case
because it conforms to the purely quantitative differences which exist
between Euclidean and non-Euclidean geometry as the very language
about 'surfaces,' 'triangles,' etc., in spite of the prefix 'pseudo,'
necessarily implies."

Thus it would seem that those who have been most diligent in constructing
the hyperspace conception have been the least careful of the logical
difficulties which beset the elaboration of their assumptions. Yet it
sometimes requires the illogical, the absurd and the aberrant to bring us to a
right conception of the truth, and when we come to a comparison of the
two, truth and absurdity, we are the more surprised that error could have
gained so great foothold in face of so overwhelming evidences to the
contrary.
The entire situation is, accordingly, aptly set forth by Hyslop when he says,
continuing:

"There are either a confusion of the abstract with the concrete or of
quantitative with qualitative logic, ... so that all discussion about a
fourth dimension is simply an extended mass of equivocations turning
upon the various meanings of the term 'dimension.' This when once
discovered, either makes the controversy ridiculous or the claim for
non-Euclidean properties a mere truism, but effectually explodes the
logical claims for a new dimensional quality of space as a piece of
mere jugglery in which the juggler is as badly deceived as his
spectators. It simply forces mathematics to transcend its own functions
as defined by its own advocates and to assume the prerogatives of
metaphysics."

Shall we, therefore, assent to the imperialistic policy of mathematicians
who would fain usurp the preserves of the metaphysician in order that they
may exploit a superfoetated hypothesis? It is not believed that the harshness
of Hyslop's judgment in this respect is undeserved. It is, however, regretted
that the notions of mathematicians have been so inchoate as to justify this
rather caustic, though appropriate criticism. For it does appear that the
moment the mathematician deserts the province of his restricted sphere of
motility and enters the realm of the transcendental, that moment he loses his
way and becomes an inexperienced mariner on an uncharted sea.

It is interesting to note that Cassius Jackson Keyser,[19] while recognizing
the purely arbitrary character of the so-called dimensionality of space,
nevertheless lends himself to the view that "if we think of the line as
generating element we shall find that our space has four dimensions. That
fact may be seen in various ways, as follows:

"A line is determined by any two of its points. Every line pierces every
plane. By joining the points of one plane to all the points of another, all
the lines of space are obtained. To determine a line, it is, then, enough
to determine two of its points, one in the one plane and one in the
other. For each of these determinations two data, as before explained,
are necessary and sufficient. The position of the line is thus seen to
depend upon four independent variables, and the four dimensionality
of our space in lines is obvious."

Similarly he argues for the four dimensionality of space in spheres:

"We may view our space as an assemblage of its spheres. To
distinguish a sphere from all other spheres, we need to know four and
but four independent facts about it, as say, three that shall determine its
center and one its size. Hence our space is four dimensional also in
spheres. In circles, its dimensionality is six; in surfaces of second order
(those that are pierced by a straight line in two points), nine; and so on
ad infinitum."

The view taken by Keyser is a typical one. It is the mathematical view and
is characterized by a certain lack of restraint which is found to be peculiar
to the whole scheme of thought relating to hyperspace. It is clear that the
kind of space that will permit of such radical changes in its nature as to be
at one time three dimensional, at another time four dimensional, then six,
nine and even n-dimensional is not the kind of space in which the objective
world is known to exist. Indeed, it is not the kind of space that really exists
at all. In the first place, a line cannot generate perceptual space. Neither can
a circle, nor a sphere nor any other geometrical construction. It is, therefore,
not permissible, except mathematically, to view our space either as "an
assemblage of its spheres," its circles or its surfaces; for obviously
perceptual space is not a geometrical construction even though the intellect
naturally finds inhering in it a sort of latent geometrism which is kosmical.
For there is a wide difference between that kosmic order which is space and
the finely elaborated abstraction which the geometer deceives himself into
identifying with space. There is absolutely neither perceptible nor
imperceptible means by which perceptual space in anywise can be affected
by an act of will, ideation or movement. Just why mathematicians persist in
vagarizing upon the generability of space by movement of lines, circles,
planes, etc., is confessedly not easily understood especially when the
natural outcome of such procedure is self-stultification. It is far better to
recognize, as a guiding principle in all mathematical disquisitions
respecting the nature of space that the possibilities found to inhere in an
idealized construction cannot be objectified in kosmic, sensible space. The
line of demarkation should be drawn once for all, and all metageometrical
calculations and theories should be prefaced by the remark that: "if
objective space were amenable to the peculiarities of an idealized
construction such and such a result would be possible," or words to that
effect. This mode of procedure would serve to clarify many if not all of the

hyperspace conceptions for the non-mathematician as well as for the
metageometricians themselves, especially those who are unwilling to
recognize the utter impossibility of their constructions as applied to
perceptual space. We should then cease to have the spectacle of otherwise
well-demeanored men committing the error of trying to realize abstractions
or abstractionizing realities. Herein is the crux of the whole matter, that
mathematicians, rather than be content with realities as they find them in
the kosmos, should seek to reduce them to abstractions, or, on the other
hand, make their abstractions appear to be realities.
Keyser proceeds to show how the concept of the generability of hyperspace
may be conceived by beginning with the point, moving it in a direction
without itself and generating a line; beginning with the line, treating it
similarly, and generating a plane; taking the plane, moving it in a direction
at right angles to itself and generating a cube; finally, using the cube as
generating element and constructing a four-space figure, the tesseract. Now,
as a matter of fact, a point being intangible cannot be moved in any
direction neither can a point-portion of sensible space be removed.
Nevertheless, we quite agree with him when he asserts:

"Certainly there is naught of absurdity in supposing that under suitable
stimulation the human mind may, in the course of time, speedily
develop a spatial intuition of four or more dimensions." (The italics in
the above quotation are ours.)

Here we have a tacit implication that the notion which geometers have
heretofore designated as "dimension" really is a matter of consciousness, of
intuition, and therefore, determinable only by the limitations of
consciousness and the deliveries of our intuitive cognitions. As a more
detailed discussion of this phase of the subject shall be entered into when
we come to a consideration of Chapter VI on "Consciousness as the Norm
of Space Determinations" further comment is deferred until then.
Now, as it appears certain that what geometers are accustomed to call
"dimension" is both relative and interchangeable in meaning—the one
becoming the other according as it is viewed—the conclusion very naturally
follows that neither constructive nor symbolic geometry is based upon
dimension as commensurable quality. The real basis of the non-Euclidean

geometry is dimension as direction. For whatever else may be said of the
fourth dimension so-called it is certainly unthinkable, even to the
metageometricians, when it is absolved from direction although no specific
direction can be assigned to it. It is agreed perhaps among all non-Euclidean
publicists that the fourth dimension must lie in a "direction which is at right
angles to all the three dimensions." But if they are asked how this direction
may be ascertained or even imagined they are nonplused because they
simply do not know. The difficulty in this connection seems to hinge about
the question of identifying the conditions of the world of phantasy with
those of the world of sense. There are distortions, ramifications,
submersibles, duplex convolutions and other mathetic acrobatics which can
be performed in the realm of the conceptual the execution of which could
never be actualized in the objective world. Because these antics are possible
in the premises of the mathematical imagination is scarce justification for
the attempts at reproduction in an actualized and phenomenal universe.
One of the proudest boasts of the fourth dimensionist is that hyperspace
offers the possibility of a new species of rotation, namely, rotation about a
plane. He refers to the fact that in the so-called one-space, rotation can take
place only about a point. For instance in Figure 7, the line ab represents a
one-space in which rotation can take place only about one of the two points
a and b. In Figure 8 which represents a two-space, rotation may take place
about the line ab or the line cd, etc., or, in other words, the plane abcd can
be rotated on the axial line ab in the direction of the third dimension. In
tridimensional space only two kinds of rotation are possible, namely,
rotation about a point and about a line. In the fourth dimension it is claimed
that rotation can take place about a plane. For example, the cube in Figure
9, by manipulation in the direction of the fourth dimension, can be made to
rotate about the side abgf.
A very ingenious argument is used to show how rotation about a plane is
thinkable and possible in hyperspace. But with this, as with the entire fabric
of hyperspace speculations, dependence is placed almost entirely upon
analogous and symbolic conceptions for evidence as to the consistency and
rationality of the conclusions arrived at.

Fig. 13.

It is urged that inasmuch as the rotation about the line bc in Figure 13
would be incomprehensible or unimaginable to a plane being for the reason
that such a rotation involves a movement of the plane into the third
dimension, a dimension of which the plane being has no knowledge, in like
manner rotation about a plane is also unimaginable or incomprehensible to
a tridim or a three dimensional being. It is shown, however, that the plane
being, by making use of the possibilities of an "assumed" tridimension,
could arrive at a rational explanation of line rotation.

Fig. 14.

Figure 14 offers an illustration by means of which a two dimensional
mathematician could demonstrate the possibility of line rotation. He is
already acquainted with rotation about a point; for it is the only possible
rotation that is observable in his two dimensional world. By conceiving of a
line as an infinity or succession of points extending in the same direction;
imagining the movement of his plane in the direction of the third dimension
thereby generating a cube and at the same time assuming that the lines thus
generated were merely successions of points extending in the same
direction, he could demonstrate that the entire cube Figure 14, could be
rotated about the line BHX used as an axis. For upon this hypothesis it
would be arguable that a cube is a succession of planes piled one upon the
other and limited only by the length of the cube which would be extending
in the, to him, unknown direction of the third dimension. He could very
logically conclude that as a plane can rotate about a point, a succession of
planes constituting a tridimensional cube, could also be conceived as
rotating about a line which would be a succession of points under the
condition of the hypothesis. His demonstration, therefore, that the cube,

Figure 14, can be made to rotate around the line BHX would be thoroughly
rational. He could thus prove line-rotation without even being able to
actualize in his experience such a rotation.
Analogously, it is sought by metageometricians to prove in like manner the
possibility of rotation about a plane. Thus in Figure 16 is shown a cube
which has been rotated about one of its faces and changed from its initial
position to the position it would occupy when the rotation had been
completed or its final position attained.

Fig. 15.

Fig. 16.

The gist of the arguments put forward as a basis for plane-rotation is briefly
stated thus: The face cefg is conceived as consisting of an infinity of lines.
A cube, as in Figure 15, is imagined or assumed to be sected into an infinity
of such lines, each line being the terminus of one of the planes which make
up the cube. Each one of the constituting planes is thought of as rotating
about its line-boundary which intersects the side of the cube. The process is
continued indefinitely until the entire series of planes is rotated, one by one,
around the series of lines which constitute the axial plane. Hence, in order
that the cube, Figure 16, may change from its initial position to its final
position each one of the infinitesimal planes of which the cube is assumed
to be composed must be made to rotate about each one of the infinitesimal
lines of which the plane used as an axis is composed. In this way, it is
shown that the entire cube has been made to rotate about its face, cefg. This
concisely, is the "quod erat demonstrandum" of the metageometrician who
sets out to prove rotation about a plane. Thus it is made to appear that in
order that tridimensional beings may be enabled to conceive of four-space
rotation, as in Figures 15 and 16, in which the rotation must also be thought
of as taking place in the direction of the fourth dimension, they must adopt
the same tactics that a two dimensional being would use to understand some
of the possibilities of the tridimensional world.
It is, of course, unwise to assume that because a thing can be shown to be
possible by analogical reasoning its actuality is thereby established. This
consideration cannot be too emphatically insisted upon; for many have been
led into the error by relying too confidentially upon results based upon this
line of argumentation. There is a vast difference between mentally doing
what may be assumed to be possible, the hypothetical, and the doing of
what is actually possible, the practical.
In the first place, plane-rotation in the actual universe is a structural
impossibility. The very nature and constitution of material bodies will not
admit of such contortion as that required by the rotation of a body, say a
cube, about one of its faces. Let us examine some of the results of plane
rotation. 1. The rotation must take place in the direction of the fourth
dimension. Now, as it is utterly impossible for any one, whether layman or
metageometrician, even to imagine or conceive, in any way that is practical,

the direction of the fourth dimension it is also impossible for one to move
or rotate a plane, surface, line or any other body in that direction. We are in
the very beginning of the process of plane-rotation so-called confronted
with a physical impossibility. 2. Plane rotation necessarily involves the
orbital diversion of every particle in the cube. This alone is sufficient to
prohibit such a rotation; for it is obvious that the moment a particle or any
series of particles is diverted from its established orbital path disruption of
that portion of the cube must necessarily follow. This upon the assumption
that the particles of matter are in motion and revolving in their corpuscular
orbits. 3. Plane-rotation necessitates a radical change in the absolute motion
of each individual particle, electron, atom or molecule of matter in the cube
and a consequent retardation or acceleration of this motion. This upon the
hypothesis that the particles of matter are vibrating at the rate of absolute
motion. 4. It presupposes a reconstitution of each atom, molecule or particle
in the cube, changing the path of intra-corpuscular rotation either from a
right to left direction or from a left to right direction, as the case may be.
The particles of matter in the cube will be acted upon in much the same
manner as the particles in a glove when it is maneuvered in the fourth
dimension. In describing this phenomenon, Manning says:[20]

"Every part by itself, in its own place is turned over with only a slight
possible stretching and slight changing of positions of the different
particles of matter which go to make up the glove."

The slight stretching and slight changing of the positions of the particles
referred to would be of small consequence if applied to ponderable bodies.
But when used in connection with particles of matter which are themselves
of very infinitesimal size means far more—enough, as we have said, to
militate severely against the integrity of the cube. It is not deemed
necessary to go further into the physical aspects of plane-rotation as it is
believed sufficient has been said to negative the assumption from a purely
structural viewpoint.
Among the vagaries of hyperspace publicists none is perhaps more notable
than the view taken by C. H. HINTON:[21]

"If it could be shown that the electric current in the negative direction
were exactly alike the electric current in the positive direction, except

for a reversal of the components of the motion in three dimensional
space, then the dissimilarity of the discharge from the positive and
negative poles would be an indication of the one-sidedness of our
space. The only cause of difference in the two discharges would be due
to a component in the fourth dimension, which directed in one
direction transverse to our space, met with a different resistance to that
which it met when directed in the opposite direction."

To be sure. And with equal certainty it might be said that if the moon were
made of green cheese it might well be the ambition of the world's chefs to
be able at some time to flavor macaroni with it, thus serving a rare dish.
Even so, if there were an actual, objective fourth dimension to our space we
might be able to shove into it all the perplexing problems of life and let it
solve them for us. But the fact that the fourth dimensional hypothesis is
itself a mere supposition seems to have been overlooked or rather
completely ignored by Hinton. Or else, ought it not be an obvious folly to
hope to construct a rational explanation of perplexing physical conditions
upon the basis of a purely suppositionary, and therefore unproven,
hypothesis?
The recognized domain of the four-space, mathematically considered, is
according to the most generous allowance very small, so small, in fact, that
the disposition of some to crowd into it the essential content of the
manifested universe is a matter of profound amazement. Then, too, it cannot
be denied that there is no appreciable urgency or necessity for having
recourse to a purely hypothetical construction for explicatory data regarding
a phenomenon which has not been shown to be without the scope of
ordinary scientific methods of procedure to unravel.
The claim of certain spiritualists, notably Zollner of Leipsig, that the
phenomena of spiritism is accountable for on the grounds that the fourth
dimension affords a residential area for discarnate beings whence spiritistic
forayers may impose their presence upon unprotected three dimensional
beings is no less fatuous than the original supposition itself. For upon this
latter is built the entire fabric of meaningless speculations so gleefully
indulged in by those who glibly proclaim the reality of the four-space.
Indeed, clearer second thought will reveal that, when the pendulum of

erratic thinking and trafficking in mental constructions swings back,
hyperspaces, after all, are but the ignes fatuii of mathetic obscurantism.
Then, why should it be deemed necessary to discover some more
mysterious realm of four dimensional proportions in which the spirits of the
dead may find a habitation? Are the spiritualists, too, reduced to the
necessity of further mystifying their already adequately mysterious
phenomena? If there were not quite enough of physicality upon the basis of
which all the antics of these entities can be explained, and that
satisfactorily, one would, as a matter of course, be inclined to lend some
credence to these claims; but as it is clear that all organized beings have
some power, if no more than that which maintains their organization, and as
it ought also be an acceptable fact that such a being is directed by mind; and
further, that owing to the nature of a spirit body it can penetrate solid matter
or matter of any other degree of density below the coefficient of spirit
matter, it ought likewise be unnecessary to go without the province of
strictly tridimensional mechanics for an explanation of spiritistic
phenomena.
Equally unnecessary and uncalled for is the attempt of certain others who
lean toward the view of speculative chemists to account for the none too
securely established hypothesis that eight different alcohols, each having
the formula C5H12O may be produced without variation. This is said to be
due to the fact that certain of the component atoms, notably the carbon
atoms, take a fourth dimensional position in the compound and thus
produce the unusual spectacle of eight alcohols from one formula. Have
chemists actually exhausted all purely physical means of reaching an
understanding of the carbon compounds and are therefore compelled to
resort to questionable means in order to make additional progress in their
field? It is incredible. Hence the more facetious appears the mathematical
extravaganza in which originates the tendence among the more sanguine
advocates to make of the fourth dimension a sort of "jack of all trades," a
veritable "Aladdin's lamp" wherewith all kosmic profundities may be
illuminated and made plain. Not until the perfection of instruments of
precision has been reached, and not until human ingenuity has been
exhausted in its efforts to produce more refined methods of research should
it be permissible even to venture into untried and more or less debatable
fields in search of a relief which after all is unobtainable.

Notwithstanding the fact that all attempts at accounting for physical
phenomena on the basis of n-dimensionality (which is itself by all the
standards of objective reference a non-existent quantity and therefore
irreconcilable with perceptual space requirements) are to be characterized
simply as a senseless dalliance with otherwise deeply profound questions,
many have fallen into a complete forgetfulness of the logical barriers
inhering in and hedging about the query and have committed other and less
excusable errors in the premises. Take, for instance, the suggestion that the
action of a tartrate upon a beam of polarized light is due to the assumption
of a fourth dimensional direction by some component in the acid. This for
the reason that experimentation has shown that tartaric acid, in one form,
will turn the plane of polarized light to the right while in another form will
turn it to the left. It is not believed, however, that there is any warrant for
such an assumption. There is also another kind of tartrate which seems to be
neutral in that it has no effect whatever upon the beam of light, turning it
neither to the right nor to the left nor having other visible or determinable
effect upon it. Indeed, it is not clear how it is hoped to prove such a case by
constituting as a norm a hypothesis which is essentially indemonstrable. A
more logical procedure would be first to establish the objective,
discoverable posture of four-space; show the actual movement of matter
and entities therein; locate it by empirical methods of research, and then,
basing our assertions upon apodeictic evidences, assume a new attitude
toward these phenomena because of the support found in established and
verifiable facts. Some hope of gaining a respectful hearing might then be
entertained; but at least to do so now appears to be quite untimely.
Major Wilmot E. Ellis, Coast Artillery Corps, United States Army, in The
Fourth Dimension Simply Explained,[22] remarks:

"... in the ether, if anywhere, we should expect to find some fourth
dimensional characteristics. Gravitation, electricity, magnetism and
light are known to be due to stresses in, or motions of, the infinitesimal
particles of the ether. The real nature of these phenomena has never
been fully explained by three dimensional mathematical analysis.
Indeed, the unexplained residuum would seem to indicate that so far
we have merely been considering the three dimensional aspects of four
dimensional processes. As one illustration of many, it has been shown

both mathematically and experimentally that no more than five
corpuscles may have an independent grouping in an atom."

The weakness of this view may be due to the fact that at that time Major
Ellis was emphasizing in his own mind the necessity of simplifying the
conception so as to make it of easy comprehension rather than the
establishment of any fealty to truth or the spirit of mathesis in his
examination of the problem. What therefore of reality the student fails to
find in his view may be attributed to the sacrifice which the writer (Major
Ellis) felt himself called upon to make for the sake of simplicity. Hence a
certain expressed connivance at his position is allowable. But, on the other
hand, if such were not the conscious intent of Major Ellis it is not
understood how it should appear that "the unexplained residuum would
seem to indicate that so far we have merely been considering the three
dimensional aspects of four dimensional processes." Contrarily, it has yet to
be proved that three dimensional space does not afford ample scope of
motility for all observable or recognizable physical processes and that there
is no necessity for reference to hyperspace phenomena for an explanation of
the "unexplained residuum." It is, of course, understood that many of the
possibilities predicated for hyperspace are purely nonsensical so far as their
actual realization is concerned. Our concern is, therefore, not with that class
of predicates, but with those wherein reside some slight show of probability
of their response to the conditions of n-dimensionality either as a system of
space-measurement or a so-called space or series of spaces.
Major Ellis concludes his simple study of four-space by proposing the
following query:

"May not birth be an unfolding through the ether into the symmetrical
life-cell, and death, the reverse process of a folding-up into four
dimensional unity?"

It is confessed that there seems to be nothing to warrant the giving of an
affirmative reply to this query. It is, perhaps, sentimentally speaking a very
beautiful thing to contemplate death as a painless, unconscious involvement
into a glorious one-ness with all life, and birth, as the reverse of all this. But
where is the utility of such a dream if it be merely a dream and impossible
of realization?

Simon Newcomb,[23] at one time one of the outstanding figures in the early
development of the fourth dimensional hypothesis, openly declared that
"there is no proof that the molecule may not vibrate in a fourth dimension.
There are facts which seem to indicate at least the possibility of molecular
motion or change of some sort not expressible in terms of time and the three
coördinates in space."
Of course, there is no proof that a molecule may not at times be ensconced
in a four-space neither is there proof nor probability that it is so hidden.
Indeed, there is no proof that there is such a thing as a molecule for that
matter.
In all of the foregoing proposals it is assumed that the fourth dimension
really exists and that it lies just beneath the surface of the visible, palpable
limits of the material universe; that lying in close juxtaposition to all that
we are able to see, to hear or sense in any way is this mysterious, eternally
prolific, all-powerful something, hyperspace, ever-ready to nourish and
sustain the forms which have the nether parts firmly encysted in one or the
other of her n-dimensional berths. Thus it would seem that while yet
functioning in a strictly tridimensional atmosphere, some one, more
reckless than the rest, should at last stumble upon some up-lying portion of
it and be instantly transformed into a mathetic fay of etherealized four-
dimensional stuff.

PART TWO
SPATIALITY
AN INQUIRY INTO THE ESSENTIAL NATURE OF SPACE AS
DISTINGUISHED FROM THE MATHEMATICAL INTERPRETATION

CHAPTER VI

Consciousness the Norm of Space Determinations

Realism Is Determined by Awareness—Succession of Degrees of
Realism—Sufficiency of Tridimensionality—The Insufficiency of
Self-Consistency as a Norm of Truth—General Forward Movement in
the Evolution of Consciousness Implied in the Hyperspace Concept—
The Hypothetical Nature of Our Knowledge—Hyperspace the Symbol
of a More Extensive Realm of Awareness—Variations in the Method
of Interpreting Intellectual Notions—The Tuitional and the Intuitional
Faculties—The Illusionary Character of the Phenomenal—
Consciousness and the Degrees of Realism.

Things have value for us only to the extent to which we can become aware
of their being. The appraisement of all objects, conditions, states or qualities
is determined directly by the degree or quality of awareness with which we
apprehend them. Those elements which are without the intellect's scope of
awareness have no interest and hence no value so far as the individual
intellect is concerned. And this is true of all degrees and states of
consciousness from the lowest to the highest, from the human to the divine.
There enter into all conscious determinations three factors, namely: (a) the
scope, or totality, of adaptations which an organism can make in the
sensible world, (b) the power of consciousness to make adaptations and (c)
environment. These three are interdependent. The totality of adaptations
depends primarily, of course, upon the quality of conscious powers or
faculties, and also, in a lesser degree, upon opportunities afforded by
environment. Faculties of consciousness are derived directly from the
influences exerted upon the organism by his environment and the results of
the struggle to overcome them. Environment is of two kinds, artificial and
natural. The artificial environment is such as has been modified by our

conscious action upon external phenomena. The residue is natural. And thus
the scope of adaptability becomes an unvarying witness to the quality of
consciousness manifesting through a given organism.
The universe is so constructed that the essential character of its various
states and qualities is a fixed quantity for a given scope of consciousness
and varies only as the sphere of consciousness varies. States of existence or
scopes of adaptation which are registering upon a higher plane or in a more
subtle sphere of existence than that in which we may at any time be
functioning can only appear evidential to us when the mechanism of our
consciousness becomes congruently adjusted therewith. So that the focus of
consciousness must always be a variable quantity adaptable, under proper
conditions, to any plane in the kosmos. Consciousness, then, becomes the
sphere of limits both of knowledge and adaptability. But lest we seem to
admit implicitly part of the contentions which mathematical publicists have
made in postulating the unodim and duodim consciousness, it is necessary
carefully to differentiate between the results arrived at as a result of the two
procedures. In the first place, analysts assume the existence of a unodim
and duodim plane of consciousness and proceed to construct thereon an
analogy designed to show the feasibility of another assumption, the fourth
dimension. While, in laying the foundation of consciousness upon a
tridimensional plane we do not start with an assumption, but with a fact.
Therein lies the difference. Enormous advantages inhere in a procedure
based upon facts, but in a series of planes built upon assumptions no such
advantages are discovered. For however much the series of hypothetical
planes may be extended or elaborated there must inhere necessarily
throughout the series an assumptional value which vitiates the conclusions
no less than the premises. The sanity and integrity of intellectual operations
depend almost entirely upon the differentiation which we make between the
necessities arising out of assumptions and those which spring up
empirically from established facts. No procedure is necessary to establish
the value of such a differentiation, nevertheless it may be suggested that it is
allowable, under the rules of logic, to make any assumption whatsoever so
long as care is taken to see that the conclusions embody in themselves the
characteristics of the original premise. For instance, it is permissible to
assume that space is curved. Under such an assumption, it is only necessary
that the constructions which follow shall be self-consistent. But the case is

different when we come to deal with spatiality and vitality. These are
quantities which cannot, in the last analysis, be made to conform to the
rules of the game of logic.
Thus, when it is intimated that realism lends itself to an apparent division
into degrees, and that each degree has a corresponding state of
consciousness, it is by no means to be inferred that such apparent divisions
are of mathematical import. For, in reality, i.e., when the consciousness has
expanded so as to become congruent with the limits of even the space mind
(vide Fig. 20), there appear to be no divisions in realism. It is only because
of the fragmentariness of our outlook upon the kosmos that realism appears
to be divided into various planes; for all of these planes are one. The
divisions exist for relative knowledge, but not for complete knowledge;
they exist for a finite intelligence, but not for a transfinite intelligence. That
is why we view realism as a series of planes. It is because we discover that,
as we proceed, as our consciousness expands and we take in more and more
of the vital activities of the kosmos and understand better the causes
underlying that which we contact, we have passed from a state of lesser
knowledge to one of greater knowledge. And so we say we have passed
from one degree of realism to another, whereas, really we have not passed
from one degree of realism to another degree. Instead, it is our
consciousness that has expanded.
If now, we conceive reality to be a scale extending from one extremity to
another (that is, from supreme consciousness to entire unconsciousness,
from final knowledge to total ignorance), and the intellectual consciousness
as the indicator which traverses the scale denoting at all times the precise
degree of our comprehension of reality, and hence the degree of expansion
of consciousness, we shall constitute a similitude closely approximating the
real status quo of humanity with respect to the sensible and supersensible
worlds. The quantity or force which causes the indicator to move along the
scale is the quality of awareness. And this varies directly as the scope of
adaptability varies. Realism is homogeneous throughout its extent; but the
scale marked upon it registers from naught to unity. And between these
every conceivable degree of awareness may be registered. The indicator
moves only as the scope widens, and thus is shown a change in the quality
of awareness. For, however paradoxical it may seem, the wider the scope of

knowledge the better its quality: the more one knows, the more complete
and of higher quality becomes that which he knows.
The intellect is of scientific tendence, studiously rejecting all phenomena
which do not yield to its senso-mechanisms. Even intuitions suffer the
humility of rejection and do not escape the limitations which the intellect
imposes upon them. This is so, because, as yet, there is no adequate
perceptive and conceptive apparatus for the propagation and classification
of intuitions, as apart from concepts. The outcome of these proscriptions is
that intuitions—free, mobile, and more or less formless in themselves, must
first be rehabilitated and vestured in garments a la intellect to conform to
the prevailing mode. But intuitions thus treated are no longer intuitions, but
empirical concepts. True intuitions are like aqueous vapor—amorphous,
permeating, diffusive: axioms or empirical concepts are like cakes of ice—
formal, inflexible and conforming to the shape of the mold into which they
are poured. Because of this—the scientific tendence of the intellect and the
consequent necessity of reforming so much of the data which constitute its
substructure, of pressing, condensing and reshaping it to suit its own ready-
made patterns—it can be perceived how profound is the influence of the
intellectual consciousness in determining the character of the totality of data
which the sensible world, and for that matter, the supersensible, offer us.
The intellect is the only means at hand for the interpretation of the meaning
and significance of the world of phenomena. Consequently, whatever
meaning or significance we are led to attach to that part of the universe
which we contact, in any way, is dictated by the intellectual consciousness.
There is no escape from the decisions of the intellect so long as the present
scheme of things endures.
Thus, by whatever standard of reference the matter may be determined, it
remains indisputably established that the intellectual consciousness is the
sole determinant of the phenomenal value of everything within our scope of
awareness or adaptability. And whatever the fault, incongruity or
discrepancy that may be revealed by a more intimate knowledge of the
genesis and character of the appearance of the sensible world, it will be
found to be due to the peculiar cut and mode of the intellect and not to
things themselves. The value, qualitative or existential, which the intellect
irrevocably assigns to objects and conditions in the world of the senses is
the exclusive norm not only by which these are judged, but also, by which

our action upon them and their action upon us are determined. Images or
objects which do not act upon us and upon which we cannot act have no
interest for us. But as an integral part of the totality of images or objects in
the sensible world, we must inevitably act upon all that is outside of
ourselves, and these, in turn, must react upon us. On the other hand, there
must be objects and images in the universe of life and form upon which,
because of their inherent nature and on account of the lack of our interest in
them and their interest in us, we can neither act nor become the object of
their action.
But herein is a mystery. For, either we act upon and are recipients of the
action of the totality of images or objects in both the sensible and
supersensible worlds, or we are so placed in the grand scheme of things that
both ourselves and the sphere of our interests and possible actions are
closed circuits, hermetically sealed and non-communicative with the other
like spheres, which do not and cannot act upon us. There is yet a third
possibility—that we are so fashioned, in the entirety of our being, that some
part of us is exactly congruent with some part of every sphere of possible
actions and interests in the kosmos, and therefore, each of us has being or
consciousness of a kind that is keyed to and registering in the totality of
such spheres; and that, at present, because our interests and possible actions
are limited to the plane of sensibility, we are conscious only there. And
further, that although those spheres of our consciousness which are fixed to
register in other planes do not answer to the lowest on which we now
operate, having a character of which we are unaware, they nevertheless
cannot be said not to exist, because of the lack of communication between
them. Among these three possible choices, we have no hesitancy in
expressing a decided preference for the last mentioned—that the range of
our being is co-extensive with the range of reality, and like a pendulum, we
oscillate, at long intervals, between two kosmic extremities—nescience and
omniscience.
The intellectual consciousness is the touch-stone of realism. It is like a
spreading light which, as it expands, reveals previously darkened corners
and conditions, only it has power both to reveal and to bring into
manifestation. In its present state, man's consciousness is like a searchlight.
It illumines and takes cognizance of everything that falls within its scope of
motility and is consequently able to study in detail that which it reveals. But

that which is beyond its scope is as if it never existed so far as the
individual consciousness is concerned. It is not reasonable to predict that
the same characteristics that are observable in any given state shall persist
throughout all the various scopes through which the consciousness must
proceed in its evolutionary expansion. For the scale of kosmic realism is
one grand panorama extending from the grossest to the most subtle and
refined. While in general the thread of realism may pervade the entire scale
it is nevertheless marked by many and diverse changes in its characteristics
as it is followed from one stage to another. So that the realistic character of
one stage may vary greatly from that which next preceded it or from that
which will succeed it. It would appear, therefore, that in passing from one
stage of realism to another there need not remain anything but the mere fact
of reality in its connection with ultimate reality; for it is obvious that in
every condition of realism which may be encountered in the kosmos there
must be a basic thread of ultimate reality running through the whole. The
entire gamut of realism may accordingly be traversed without the danger of
being diverted from the golden thread of realism which thus permeates all.
It is always the phenomena of realism with which we are concerned and
which we are trying to understand rather than realism itself. It is this that
confounds us. If it were not for the phenomena, which is the way realism or
life presents itself to our consciousness, we should experience no trouble in
discovering the reality, all other things being equal. For the former ever
obscures the latter. It is the supreme task of mental evolution to break
through the clouds of phenomena in the search for the eternal substratum of
reality which runs through the sensible universe of things.
The first view of conditions that the mind takes upon awakening to
consciousness in any new sphere of cognition is necessarily hazy and
inchoate. There is more or less of astonishment, wonder and bewilderment
upon first becoming aware of a new scope of realism. In this state it is
natural that the mind should overlook or ignore much that is essential and
perhaps all that is so even escaping the true import of the phenomena which
it senses. It is reasonable, too, that in such a state the main outlines of what
is really seen may be greatly distorted and exaggerated so that it is well-
nigh impossible to secure a correct comprehension of the character of a new
scope of realism from any early survey. It is not until later years, after much
study and circumspection that the mind, becoming used to the new

conditions, begins to get correct impressions and to make valid judgments
as to that which it discerns. And even then, it not infrequently happens that
the resultant view of things in general is found to be in need of revision and
correction. Hence, after everything is sifted down to the ultimate allowance
for the illusion incident to too much enthusiasm and wonder we have only a
very small residuum of truth upon which to build and this latter we often
find to be the single thread of reality which runs through all the phenomena
and which is, therefore, the only quantity that remains worthy of much
consideration.
Thus it is with religion. The path of progress over which our religious
conceptions have come need not be outlined here, but to any one at all
acquainted with the history of religious thought and ideals it at once must
be patent that it has been one continuous surrender of the old for the new, of
one degree of realism for another newer and higher degree; that always it
has been the phenomena, the flora of the ideals which have had to give way,
while nothing was left but the roots of realism from which they have
sprung. It has been the same with scientific knowledge. Facts have been
collected and hypotheses proposed to synthesize them and yet these have
had to give way for others, and others still, until the data of scientific
knowledge to-day are quite different from what they were in earlier days.
And yet permeating the scientific knowledge of all times has been the
golden thread of reality, and of all facts and systems of facts which man has
successively assumed and surrendered nothing has remained but the reality;
indeed, nothing could so remain, but reality. So it is with air phenomena
with which consciousness has to deal. This perhaps is due to the fact that
the mind interprets phenomena in accordance with the quality of its
awareness, and as consciousness is a variable quantity, its standards of
interpretation will likewise vary. Each new scope of awareness, after this
manner, yields higher and more exact standards of interpretation. And then,
progressing in awareness from the segment to the whole a fuller view of the
phenomena as well as of reality itself is gained and also a more
comprehensive judgment of the relations which exist between the segment
and the whole. In other words, as the scope of consciousness widens it
becomes more and more apparent that what was first thought to be a
separate segment is in reality identified with the whole in an indissoluble
manner. For the Thinker is then not only aware of the segment as such, but

he is also conscious of the fact that it has definite relations with the entirety
and that what he needs is merely a more extended consciousness.
In denying the existence of the four-space or spaces of n-dimensionality as
described and defined by geometricians, we do not thereby deny the
existence of a plane of consciousness which is as much unlike the
conditions of the tridimensional world as it is said to be unlike the four-
dimensional world; but what we do deny is that such a higher plane of
existence has necessarily to be conditioned by such characteristics as the
metageometricians have proposed. It is maintained that there is no basis in
consciousness for a world of four dimensions; that the consciousness has no
tendency for action in four-space. Neither has matter nor life any inclination
or potency to behave in a four-dimensional manner. It is indeed more
rational to suppose that there is a higher plane, in fact, a series of higher
planes, in which the thread of realism is continuous, not broken as it
necessarily would have to be in extending to hyperspace, nor curved as in a
manifold; that this series of subtler and finer planes of consciousness are
merely an elongation of our three dimensional scope of realism. It,
therefore, remains only to master the phenomena of each in just the same
manner as we have, in a measure, mastered the phenomena of
tridimensionality. For it is easily conceivable that the quality of
consciousness is such that it may adapt itself to a far wider range of
possibilities than may be discovered in hyperspace and still be a tri-space
quantity.
It is believed, however, that in all the new and higher planes of
consciousness tridimensionality is the norm both of the phenomena and of
the reality peculiar to them. And that, being such, does not change or vary,
but is a fixed quantity regardless of the plane of consciousness.
Furthermore, it is believed that the highest state of consciousness in the
entire kosmos could easily exist, and does so exist, upon the basis of three-
space as the norm of its extent.
A sharp line of demarkation should be drawn between the reality which is
life and consciousness and that which belongs to the realm of phantasy. For
it is the prerogative of the intellect to create, out of the remains and deposits
which it finds in the pathway of life, whatsoever it wills. This it does
continuously; but it scarcely can be expected that such creations shall be

endowed with the same dynamic character as that which life bestows upon
its creations. The creations of the one are merely dead carcasses while those
of the other are vital and real. Between them the same marked difference
exists as between the growing tree and the lumber which the builder
converts into a house. The organization which we witness when we look
upon a building made of the dead body of a tree is not the same kind of
organization as that which we see when we view the living, growing, vital
tree. The dead tree is a deposit of life cast off by it when it passed on.
Whatever the intellect can do in disposing of the remains of the tree-life is
conventional and artificial. If it convert it into an edifice it will then bestow
upon it a sort of consistency which is quite sufficient for all purposes. But
the consistency which holds the organization of an edifice together is not
the kind of consistency which holds a living tree together. In fact, there is a
consistency that is not consistent. Such is the consistency of metageometry.
It is self-consistent and yet inconsistent with the consistency of the kosmos
and its norm of being which is consciousness.
Self-consistency is one thing and kosmic consistency is quite another. It
does not necessarily follow that because a given scheme of thought is
consistent in all its parts that it is also consistent with universal truth or with
life. This very vital fact was overlooked by Gauss and all those who
followed in his wake when he discovered that his Astral Geometry was
consistent throughout in all its parts. There is only one norm of truth and
that is kosmic consistency. It matters little that a thing shall be self-
consistent; it matters much whether it is consistent with the universal
standard. It has been shown to be logically possible to elaborate at least two
different systems of geometry, namely, the geometry of the acute angle and
that of the obtuse, which, while each of them is self-consistent throughout,
are nevertheless inconsistent with each other and with the geometry of the
right angle (Euclidean). This, it would seem, appears to be sufficient for the
invalidation of either one or both of the non-Euclidean systems of
geometric thought. Indeed, if it can be shown that the Euclidean geometry
is more representative of the true approach to the norm of space-genesis and
of creation so far as its mode of manifestation is concerned, and
consequently true of the norm set up by consciousness, the rejection of both
systems of non-Euclidean geometry seems to be thoroughly warranted. But
this is obvious and requires no demonstration nor comment to make it clear.

We have only to ask ourselves whether it has ever occurred to us that
consciousness has either a tendency to or adaptability for action in a
curvilinear manner; or, if when we contemplate ideas or idea-relations we
have the impression of perceiving a curvilinear or manifold tendence in
them either of a positive or negative nature, and also whether it has been
observed that our thought processes naturally assume four-dimensional
attitudes. If we find that such a query must be answered negatively, and
indeed we must so find, then, we have no basis for the assumption that any
one of the systems of non-Euclidean geometry is valid either for the present
status of consciousness or for a future existence, since it is true that the
future is but an elongation of the present. Evolution is to bring no radical
changes in the norms of reality; it has merely to deepen and widen and
make more intense, efficient and comprehensive the present scope of our
consciousness and thereby, while the Thinker is passing from one degree of
realism to another, to bring him into a clearer conception of what his own
limited scope of motility means to the whole.
The four-space is a mathetic divertisement. That is, it cannot be said to lie
in the direction of a straight line which proceeds either in a forward or
lateral direction. Neither does it lie in a plane which is vertical or horizontal
to the sensorium. It is, therefore, a fractural departure from any conceivable
tridimensional direction, a geometric anomaly. Evolution, despite the minor
aspects of its movement, undoubtedly proceeds in a straight line and not by
a zigzag nor discontinuous line and hence it is irrational to assume that it
will, when it passes on to the next advanced stage, emerge into the realm of
the four-space. For the so-called hyperspace of geometry cannot by any
standards of reference be said to lie in the plane of any straight line which
can be described in three-space. If life is to evolve more efficient forms and
if the forms are to evolve into more perfect organizations and mind and
consciousness to become more intense and comprehensive expressions of
the divine mind of the kosmos it is certainly not in the domain of
hyperspace that these shall find the substructure of their higher
development; but, if at all, it shall be found, as in all times past, in the realm
of perceptual space where bodies are said to have three and only three
possibilities of motion.
What then is the significance of the more than a thousand years of
mathematical labors; of all that has been said and done in an endeavor to

bring into the popular consciousness a conception of hyperspace? Is it a
question of "Love's Labour's Lost?" Or is it a mere prostitution of
mathematical talent? To answer these queries is the burden of this treatise
and it is hoped that as the text continues the reader may be able to arrive at
his own conclusions as to the relative value of the work of the
mathematicians in this respect and be able to judge for himself the true
significance of it all.
The specific value of consciousness as a determinative factor in space-
measurement has been recognized by all who have sought to arrive at a
logical justification for the conception of four-dimensionality by analogous
reasoning. The existence of the unodim with consciousness limited to a line
or point has been assumed and it has been shown how greatly such a being
would be handicapped by his limited area of consciousness, it having been
proposed to confine his consciousness to one dimension. An unodim would,
of course, be entirely unaware of any other dimension than that in which he
could consciously function. So that with respect to his own consciousness
no other dimension would be necessary for the continuance of his life
processes. He might live his life without any knowledge even of any
limitations or barriers to other things higher than those of his plane. He
would be content to exist in the one-space and enjoy the benefits which it
offered. He could have no notion of the two-space, but it has been allowed
that a super-unodim, an unodim metageometrician, if you please, could
reason out a mental conception of what the two-space might be. Passing on
to a space of two dimensions, the domain of the duodim, a greater freedom
of movement is allowed and instead of being able to function in only one
dimension the inhabitants of this plane would find themselves able to move
about in at least two directions. Consciousness would accordingly enjoy a
more comprehensive scope. But in a manner similar to that used by the
unodim metageometrician it is held that the duodim could get a conception
of the three-space by analogous reasoning and so gradually become
conscious of a higher degree of spatiality than his own. In the conscious
reasoning of both, however, is the condition of perpendicularity. That is, it
must be assumed by both the unodim and the duodim that the new
dimension must lie in a plane perpendicular to their space. So, the unodim
would postulate that the two-space must lie in a direction at right angles to
his space, and yet he would not be able to indicate the direction owing to

his ignorance of any experience that would acquaint him with the new
space as well as the want of possibility of motion therein. Similarly, the
duodim would arrive at a conception of three-space. Thus, it has been
argued that tridims, or people living in our tridimensional world, could, by
using a like line of argument or reasoning, arrive at a conception or
understanding of the four-space, which, of course, must also lie in a
direction at right angles to three-space.
The implications of this mode of thought show how thoroughly the
mathematician recognizes the limitations which consciousness imposes
upon our knowledge of the world and the subtler conditions about us. But,
moreover, it is even obvious to all who stop to think about it; for it can
readily be seen how little those things which do not enter our scope of
awareness affect us either physically, mentally or spiritually. But no one can
be so bold as to deny utterly that anything exists but what is found in our
consciousnesses. It is even true that in the great centers of population where
people are compelled to live many families in the same house, it is the usual
thing for these individual families to live in complete forgetfulness of all the
others in the house and live their lives so completely that it would be
exceedingly difficult to measure the effect the one has upon the other. The
mathematician, as is shown by the hyperspace movement, recognizes that
there are planes of superconsciousness the nature and character of which
persons confined to limited areas of consciousness can have no knowledge
and may only arrive at that knowledge by serious thought and
contemplation. In other words, they tacitly admit the existence of higher
planes of consciousness as well as the necessity of elevating the personal
consciousness in order to comprehend them. Although it was not expressly
allowable in the analogy of the unodim, it is nevertheless one of the
strongest implications of the process of reasoning that the unodim could
have easily raised the plane of his consciousness by continuing his
researches until he, too, became conscious of the three-space,
mathematically, as well as the two-space. For it was not necessary for him
to raise the plane of consciousness in order to contact the two-space. He had
need only to widen it. But in order to comprehend the mathematical three-
space it would be necessary for him to elevate his consciousness.
The fundamental error in the foregoing line of thought rests in the fact that
awareness in the human family has not developed in the manner outlined.

The human species has not come into conscious relations with the three-
space by outgrowing the one-space and the two-space in succession. The
fact of the matter is that when consciousness first dawned it must have
encompassed all three dimensions simultaneously and equally and there is
nothing to indicate that its rise was otherwise. Then, specifically there is no
evidence that the evolution of consciousness has proceeded in a rectangular
manner. Indeed, there is undoubtedly no warrant for the assumption that it
has progressed in ways that are mathematically determinable at all. The
question very naturally rises in view of the above as to the relative value of
mathematical knowledge in the scheme of psychogenesis. Can
mathematical knowledge or laws be said actually and finally to settle once
for all time any question in which consciousness or life enters as a factor?
Upon the response to this question hinges unanswerably the decision as to
the category which mathematical knowledge should by right occupy in the
entire schematism of life. If it can be successfully maintained that final
judicative power abides in mathematics in the determination of these
questions, then it would be useless to struggle against the fiat of
mathematics and mathematicians; verily, we should be compelled to accept
nolens volens all that mathematicians have devised about hyperspace and its
connotations. If, on the other hand, it can be shown that no such judicative
power inheres in mathematical knowledge we shall then be able to establish
for mathematics a true category and to dispose of the hyperspace movement
in a manner that shall at once be logical and necessary.
That the discovery of hyperspace by the mathematician is merely an aspect
of a general forward movement in the evolution of consciousness can be
shown by a brief correlative study. Hyperspace is the artificial symbol of a
higher and more extensive realm of awareness. For it cannot be doubted
that to be able to think in the terms of hyperspace, to study the various
relations and interrelations upspringing from the original premises, actually
to become conscious in the hyperspatial realm thus constructed, requires a
different species or quality of consciousness than that required for ordinary
thinking. The period covering the rise of artificial spatiality is
contemporaneous with the rise of the phenomena identified with the
spiritual life of Swedenborg; for during the same time he began a series of
visions which revealed to him great knowledge of the unseen and
supersensuous realities of life and existence. His consciousness was being

flooded with the light from so-called celestial spheres and he was gradually
becoming conscious of a "new dimension," a new space, a higher world that
is altogether unlike the world of the senses. During this period, too, Dante,
the great kosmic seer began to jot down the results of his "hyperspace"
experiences, after which he wrote his Divina Commedia in which he
describes more or less minutely some of the characteristics of the
hyperspace domain which was revealed to his consciousness as he saw and
interpreted it. Both Swedenborg and Dante being deeply religious and
pious-minded had their reports of the new world colored by their own
mental experiences and proclivities. Plato had at an earlier day set down
what he conceived to be the ethical and civic characteristics of the new age,
the Utopia of mankind living on a higher plane. It was during these days
that the doctrine of evolution was born, although it remained for Darwin to
formulate and buttress it with a stupendous congeries of facts. Martin
Luther, the great religious reformer, likewise contacted the radiating light-
glow of a higher consciousness into which the race was coming but of
which only the foremost were able to get advance glimpses. Kant, one of
the peerless leaders of the vanguard of humanity, at this time also,
conceived and wrote his Critique of Pure Reason which is likewise an
evidence of the upliftment of his consciousness on the side of pure
intellectuality and the commencement of a general period of illumination.
And then, later, but embraced within the same period, artists began to get
glimpses of this higher consciousness which showed itself in a new and
strange departure in art. In rapid succession new schools sprang up and
came to be known as the "cubist," "post-impressionist," "futurist,"
"orphimist," the "synchromist" and the "vorticist." Art really began the
search for the "plastic essence of the world" trying to portray its conception
of the "real image of the spirit" of the world. Color acquired a new kind of
splendor and painting gave birth to a new intrinsic beauty of material and
sheer loveliness of texture. All of which were evidences of an intellectual
up-reaching in response to the sharp appulsions from above. Darwin's mind,
being of scientific bent, saw and interpreted everything in the terms of
materialistic science; but there is no doubt but that the expansion of the area
of awareness which his mind experienced in his great conception of
evolution as a continuous process and all that it implies thereby was a result
of the universal appulsion of the human intellect against the new domain of

consciousness. And Kant's conception of space in general perhaps may be
said to have been the seed-thought for the metageometrician.
But thus it will be noted that in all the cases mentioned in the foregoing
there is always present the personal element of the investigator, and that the
reports of each of these have been colored and characterized by their
individual consciousness and experiences. That all reports would agree with
respect to details connected with the new domain of consciousness could
scarcely be expected owing to the wonder and bewilderment which seize
the intellect under such circumstances. No implication that the
mathematicians have been unduly excited by what they have discovered
after years of patient research in this direction is indicated by the foregoing
observations; but it cannot be denied that the enthusiasm of the moment and
the consequent minimization of all other phenomena but the special line
being investigated serve very effectively to obscure the mental vision of the
more partisan. It perhaps is sufficient that the investigator should set down
in as orderly manner as possible the things which he conceives, and that he
should interpret them according to the standards of his own intellect. More
than this cannot be expected. Moreover, it usually suffices that the future
investigator, far removed from the beclouding influences of partisanship,
who successfully raises his consciousness to that higher plane shall be able
to synthesize the findings of all and thereby with the aid which comes to
him from a more advantageous position arrive at sounder views and a more
reliable judgment.
It will thus be seen that the metageometrician's method of interpretation is
no more entitled to final credence and general acceptance than that of the
spiritualist, Swedenborg, or the occult seer, Dante. For in their best moods
and at their highest points of mental efficiency these have only succeeded in
vaguely symbolizing what they have conceived of the realities of the
supersensuous realm in terms of their own experiences. Is there any more
cogent reason, then, for accepting the analyst's conception of a world of
hyperspace peopled with ensembles, propositions, spaces of n-
dimensionality and other mathetic contrivances than the Inferno of Dante,
inhabited by hideous shapes and repellent denizens, the remains of ill-spent
earth-lives or Swedenborg's Celestial Realm, wherein dwell numerous
beings of celestial character performing various tasks in the work of the
world? These observations should not lead the reader to come to the

conclusion that the visions of Dante and swedenborg are deemed to be
more worthy of credence than mathematical knowledge when that
knowledge is limited to the sphere where it rightfully belongs; but the
proper view is that which would make it appear that it is the way these
widely differing workers interpret what they have seen; that it is the
adaptation of the unseen realms to the peculiarities of the mentalities which
observe them. The mathematicians have simply portrayed as well as they
could their conception of the new stage of consciousness and its contents,
and following the modus vivendi of all intellects have interpreted these
things in the terms of mathematics, merely because mathematics constituted
the best available symbology at hand for the purpose. Similarly, the painter
sees a new world of color; the politician, a new era of political freedom; the
religious enthusiast, a new religious conception; the scientist, a new
condition of matter and energy, and so on, to the most ordinary mind, each
sees something new while at the same time is necessarily limited to the
confines of his own mentality when he comes to interpret what he sees and
conceives. Hence, there would appear to be only one way to regard all these
advances and that is by synthesizing them, by correlating, and by tracing
them to a common source, and finally by seeing them as one general
forward movement of intellectual evolution.
Man, the Thinker, who in essence is a pure intelligence, has two mental
mechanisms or organs of consciousness. One of these is the brain-
consciousness or the egoic. It is so called because the brain is its organ of
expression and impression. It manifests through the brain and uses it to
further the various objective cognitive processes. The brain-consciousness
is a child of the physical body and its life is intimately identified with the
life of the body. This consciousness may be called the a posterioristic
mechanism or organ of the Thinker and is therefore his means of
interpreting the phenomena of the objective world. Cell-consciousness is a
phase of the ergonic functions of the a posterioristic mechanism. The other
organ of consciousness is an aspect of the intelligence of the Thinker
himself and perhaps may be said to be the active, organized portion of that
intelligence. It is separate and distinct from the a posterioristic
consciousness yet sustaining a substructural relationship with it, being the
source of the egoic or brain-consciousness. It may be called the a priori
consciousness. Its roots are buried deep within the heart of the space-mind

and it is therefrom nourished and developed by what it receives in the way
of intelligence. It is the intuitional faculty; knows without being taught;
conceives without reason; interprets according to the norms of the space-
mind or the divine mind of the kosmos. It always resides on a higher plane
than that of the brain-mind or consciousness, only at rare intervals being
able to contact it with flashes of its own intelligence as intuitions.
The a priori consciousness being the intuitive faculty of the Thinker is,
therefore, a phase of his mental life on a higher plane than the sensuous. All
its conceptions constitute the a priori knowledge of the brain mind so-
called. The a priori faculty of man's higher consciousness gives the
character possessed by that form of knowledge known to philosophy as the
a priori. So that the a priori has a more substantial basis than has hitherto
been surmised. It is not only that which may be said to transcend experience
but that which is the organ of contact with the supersensuous realities as
well as of expression through the brain-consciousness.
The mind's method of apprehending objective phenomena is not a direct
process but an indirect process by virtue of which neurograms or nerve-
impacts registered in the brain are interpreted. External sense-impressions
are, of course, conveyed to the cortical area by means of appropriate
vibrations which traverse the lines of the neural mechanism. These are
recorded in the brain areas just as a telegraphic communication is registered
in the apparatus of the receiving end, and in being so, they make terminal
registrations which man, the Thinker, interprets after a psychic code which
has been built up empirically. That is, he comes to know that certain rates of
vibration and certain peculiarities therein mean certain things when referred
to the sensorium. He then interprets according to this experience the
symbolism of all neurographical impressions. But it is obvious that under
such circumstances, where the interpreter is far removed from the thing
itself and finds it necessary to interpret rates of vibration or symbols in
order to arrive at a knowledge of the intelligence which is conveyed to him,
the chances of inadequate conception are very great indeed. In fact, it is not
possible through the use of neurographical symbols alone to get any
complete notion of the phenomena considered. And thus there stands
between the Thinker and absolute knowledge a barrier which prevents his
arriving at a state of certitude in his knowledge of the world of sensible
objects. It is, however, a barrier which will always remain, checking ever

his approach to finality in his understanding of the universum of
appearances.
A markedly different condition obtains in the realm of the a priori or
intuitional for the reason that the barriers which inhere in the
neurographical or a posterioristic method are absent and the Thinker has a
more direct approach to the objects of cognition. Hence the chance for error
is very small indeed. This will account, therefore, for the superiority of the
intuitional over the rational or the perceptual. Indeed, it is doubtful whether
the purely rational possesses any value whatsoever when its modus vivendi
is unsanctioned by the intuitional.
Else why can we not be certain that the results of our rational processes are
correct at all times? Is it not because we lack the power to perceive whether
our premises are correct in the first place? Quite truly. For if the Thinker
can intuit the necessity and certitude of any given premise it follows that the
consequences of that premise are true. It would, therefore, appear that the
more the intuitional faculty is developed the clearer will be our perceptions
not alone of abstract values but of objective things themselves. Further, it is
doubtlessly true that the more the space-mind is developed in the human
race the deeper will become our perceptions of the essential be-ness of
things so that whatever may be the presentations of the space-mind to the
brain-mind they will be by far more accurate than the impressions we
receive through the latter as a medium of apprehension. It is but natural,
however, that in the present more or less chaotic condition in which the
faculty of the intuitional is found it should be difficult even to interpret its
presentations accurately. It is perhaps due to the fact that we are unused to
its deliveries and mode of registration as well as to the fact that it has been
overshadowed by the intellectual or rational faculty. But the mere fact that it
is present and functioning, even if but rudimentarily, is evidence of its
potentiality and the possibility of its future development to a still higher
degree of efficiency.
There is no doubt but that the original impulse which resulted in the
creation of the faculty of perceptibility in the Thinker also marked out the
metes and bounds of our entire range of perceivability which includes not
only the intuitional but something higher still. There is no doubting either
the obvious fact that these metes and bounds cannot have been other than

rudimentary or general lines of denotation, and that the work of their further
elaboration and refinement is a matter of evolutionary detail. For if we
assume that the general principles of evolution are true we immediately
recognize the cogency of this view. That which we now call the hand has
not always been the perfect instrument that it is nor has the ear always been
so keenly adjusted as at present. It has required undoubtedly many million
of years for the eye to reach its present degree of complexity and
adaptability. Yet in all these cases the different organs existed in potentiality
from the beginning; the metes and bounds of the hand, the ear and the eye
were laid out primordially. Evolution has specialized and adjusted them to
environments and needs. Thus it will be seen that while the intuitional
faculty was designed for manifestation from the beginning it has
nevertheless required ages for its appearance even in the most rudimentary
fashion.
Almost the entire content of human knowledge is based upon assumptions
or hypotheses; in fact, is but a mass of these, and especially is this true of
mathematics, science and philosophy. Of course, there are certain minor
observable facts which by reason of the seeming permanence of their
existence have been eliminated from the category of assumptions. But even
this elimination when it is traced to its depths may be found to be
erroneous, and perhaps after all, when we have really begun to know
something of the reality of things, may have again to be placed in this
category. And then, too, the hypothetical nature of our knowledge is due
largely to the Thinker's method of contacting the objective world which is
the subject of his knowledge. It is because it is necessary for him to
interpret the neurographical symbols which sense-impressions make in the
brain matter according to a psychic code that renders his knowledge of
things in general hypothetical. His interpretations are based upon an
assumed value which experience has taught him to give to each neurogram.
But even when his interpretations leave nothing to be desired in respect to
their accuracy of apprehension of what the neurographical vibration implies
there is that further barrier to his cognition of reality which is due to his
remote removal from the object itself and the consequent extreme difficulty,
if not present impossibility, of identifying his consciousness with the
essence of the objects which he contemplates.

When the Thinker's consciousness is presented with a neurograph of say, a
cube, it is not the cube itself which he contemplates or observes; it is the
neurograph or psychic symbol which the sense-impressions make in the
brain. His consciousness deals not with the object but with the symbols. It is
true that when he verifies one neurograph by another, as the scopographic
or sight impressions by the tactographic or touch impressions it is found
that the delivery thus determined is a true enough representation. It is also
true that the Thinker, as a rule, does not accept a neurograph as valid until it
has been verified by at least one or more presentations through his outer
sense organs. It occurs, therefore, that all such deliveries are verified and
corrected by one or more sense witnesses before final acceptance by the
Thinker; but even then it cannot be said that his notions thus gained are in
all respects correct and true to the standards set up by the brain-
consciousness not to mention higher forms of consciousness. And then,
when we consider that in addition to the numerous chances for error which
naturally inhere in this method of cognition it must also be apparent that the
Thinker's approach to the reality of things is much impeded by his
separation therefrom, the unreliability of our ordinary methods of cognition
is much emphasized.
But aside from the egoic or brain-consciousness there is the higher
consciousness of the Thinker himself. For the brain-consciousness is merely
his method of regarding and comprehending the neurographical deliveries,
the psychic code by which he systematizes and organizes his cognitions or
impressions of the sensible world. This higher consciousness constitutes the
faculty a priori for the Thinker on a higher plane of existence, and because
it deals with elements altogether unlike those which make up the data of
brain-consciousness is, accordingly, less liable to error in its judgments of
the supersensuous presentations than is the objective or brain-mind. In fact,
it is difficult to conceive of a state or conditions wherein, supported as this
view contemplates, the intuition should err in judgment. Viewed from the
standpoint of external impedimenta, this condition may be said to be due to
the absence of sensuous barriers which would otherwise prevent the near
approach of the Thinker's consciousness to the essence of things which is
the object of his consciousness on this higher plane. Directly, however, it is
undoubtedly due to the fact that, following the lead of life itself, yea, as the
veritable handmaid of life, it cannot err where life is concerned. When

dealing with notions a priori or intuitograms the Thinker is relieved of the
onerous necessities and limitations incident to the examination and
determination of neurographic symbols registered in the brain cortex and so
is free to study, to examine and judge at first hand the impressions which
are received from his own plane of intuition. The difference is about the
same as that which should exist between the methods of communication
between two telegraphic operators when in one instance they would have to
depend upon the deliveries conveyed over the wires, while in the other,
when they stood face to face with each other, they could communicate by
direct conversation. In the one case the method of communication is direct
and simple, while in the other it is indirect, circuitous and complex. It can,
therefore, be readily seen that in all cases where the approach is made in a
direct, simple manner the probability of error is much less than in cases
where the intellectual approach is less direct and more complicated. Hence
in drawing conclusions as to the relative importance of the two mechanisms
of consciousness, the a posterioristic and the a priori, it is necessary to bear
in mind the comparative superiority of the one over the other as a means of
cognition. It matters little that the intuitional faculty is not so well
developed as the tuitional because it is but natural that inasmuch as the
Thinker's needs are subserved in the sensuous realm by the tuitional
consciousness it should, from more active use, gain somewhat over the
intuitional in facility of expression and general utility. And the more so,
because the two faculties serve different purposes; one is attuned to receive
impressions from a subtler plane while the other is fitted for contact with
the phenomenal universe; one is related to the conceptual while the other is
related to the perceptual. They differ not only in function, but in nature as
well. There is, of course, a natural barrier consisting of the inherent
limitations of each faculty which prevents the full mergence or unification
of the two states of consciousness so that there exists a state of
consciousness the data of which the brain-mind is unaware, it being able
only at rare intervals even to receive so much as slight impressions from it
in the nature of intuitional flashes or inspirations and the like. Viewed in
this light it would appear that the cognitions which are most truthworthy are
those which are presented by the intuitional faculty because they are nearer
to the essential reality of things; they have to do more specifically with the
nature of that which appears while the tuitional mind can only regard that
which is the appearance. Herein lies the whole difference.

The natural outcome of this division of labor between the tuitional and the
intuitional is the establishment of the fact of man's relationship both to the
phenomenal and the real; that in his psychic nature must reside the faculty
of apprehending the real and that he shall one day awaken into activity this
now latent faculty whereby he may make a direct and naked contact with
reality.
If it be true that, as Plato said, God does geometrize, and that the divine
geometry, as will appear, is based upon a system, an alphabet which taken
together are the point . , the line ——, , the square , and the circle ,
then, we have in this geometric alphabet the very secret of the divine
geometry. With these, and in the kosmic laboratory of chaogeny, the
Creative Logos has measured off the limits and confines of space; with
them He has traced out its dimensions the archeological evidences of which
we may view in the space-mind itself; and with them he has established the
manner of its appearance to the Thinker. In dimensions, three, and yet not
three, but one, Space, the eternal progenitor of all forms and energies,
having received the divine fiat in the beginning that thus far it should
extend and no further, persists in faithful obedience to the law of its being—
tridimensionality. It must be so because it is thus sanctioned by the highest
faculty in man that can render judgment thereon. If tridimensionality inhere
in the space-mind, as the law of its being and in the intuitional
consciousness as the norm of its essential nature and as the easiest and
simplest expression of the tuitional mind, how can it be gainsaid that these
considerations obviate the necessity of the mathetical hyperspace?
If the reality of things is hidden from us and if we are, therefore, unable to
perceive their real essences it is because our mode of thought and our
consciousness have obscured our vision and limited us to this state of
paucity of perception. It is not because reality is itself a hidden, inscrutable
quantity nor that its modus vivendi is "unknowable"; but because we being
multiformly limited, "cabined, cribbed and confined" are resultantly lacking
in the power to discern that which otherwise would be most obvious to us.
It may well be set down as axiomatic that when, in the process of our
thinking, we arrive at the inscrutable, the unknowable and the infinite, it is
evident that our thought processes are dealing with a form of realism which
is higher and beyond the possibilities of our loftiest thought-reaches. And in
order to symbolize to itself this condition the intellect poses such terms as

"inscrutable," "unknowable" and the "infinite" simply because that is the
best it can do. Hence when it is said that space is infinite it is apparent that
the mind recognizes that when it contemplates space it is dealing with
something whose degree of realism transcends its powers of
comprehension. Infinity is a relative term, and in fact, decreases in extensity
in the proportion that the consciousness expands and comprehends. It is not
unlikely that should the intellect one day discover that it had awakened into
union with the space-mind it would immediately reject its preconceived
notion of the infinity of space. But we need not wait until the coming of this
far off event in the path of psychogenesis; for we can here and now perceive
with what must be a higher faculty than the intellect the verity of this
conclusion.
But certain it is that the intellect, in the pride and arrogance of its traditional
heritage, will not without a great struggle yield the ground and prestige it
has held for an aeon of time; and in vain does the intuition serve notice of
dispossessal in these premises; but however stubbornly fought the battle,
however tenaciously held the position time will discover the weakening of
the intellect's hand. Death for the intellect may ensue as a result of the
conflict but it will be a death wherefrom it will arise, quickened, revivified
and uplifted by its disposer, the intuition, upon the remains of its dead self
to a higher and grander state than it has ever enjoyed before.
Space is not static. It is dynamic, potential and kinetic. It is a process, a
becoming. Its duration as a process is never ending. Its extensity is limited
and finite. The so-called infinity of space is one of the capital illusions of
the intellect which can only be removed by an expansion of the
consciousness, by a mergence of the individual consciousness with the
space-consciousness. In the ever-widening circle of the individual
consciousness lesser realities give way to greater as the darkness recedes
from the light—the lesser appearing in comparison with the greater, as the
consciousness broadens, as matter to spirit, as night to day or as limitation
to non-limitation. Thus the most solid facts and conditions of our limited
life are but the shadows of the deeper realities which shall be revealed to
the Thinker in the days of his larger and more glorious life of freedom from
limitations.

And now it will appear that the whole fabric of our knowledge shall have to
be reduced to the bare warp and woof; for nothing is real but these. It is as
if the Thinker, using the tuitional mind, had been in all times past studying
the design woven in the surface of a very thick plush carpet. There are the
warp and woof, the long vertical threads which make the plush and the
intricate design appearing on the surface. Our knowledge may be likened to
the design. It represents the contents of our knowledge. We have not even
so much as begun the study of the nature of the vertical threads as they
appear beneath the surface to say nothing of beginning the study of the
warp and woof. The warp and the woof are the realism of the kosmos; the
vertical threads are the roots and stem of the phenomenal world; the design
is our sensible world as it appears to the intellect. The life of the intellect
has been spent in contemplating this design; while of the hands which wove
the carpet, of the mind which directed the hands and of the spirit which
vitalized all, it knows nothing nor indeed can it know anything. Where shall
we say are those hands, that mind and that spirit which made the carpet
possible and an actuality? In vain do we search among the remains, among
the soft, glistening threads of the carpet or among the intricacies of the
design. For they are not there. They have passed on. The intellect looks at
the design or at the vertical threads and because it is unable to follow them
to their source, it decides that they are infinite, inscrutable and unknowable.
But not so. All that is required are eyes to see and a mind (or shall we say a
mind vitalized by the intuition) trained to discern the threads as they point
upward with their termini firmly rooted in the warp and woof of the fabric.
But we must first master the design, and then turning to the threads, master
them. Then shall the doors of kosmic reality swing wide and the Thinker
shall be ushered into the eternal palace of kosmic realism wherein he shall
find the great secret, the heart, the purpose, the beginning and the end, the
very nature of things-in-themselves.
The nature of every degree or condition of realism is so constituted that its
qualities, characteristics and limitations are exactly adequate for the
satisfaction and fulfillment of all the requirements and needs of every
possible state of normal consciousness. So that each degree of reality and
each state of normal consciousness is sufficient and complete in itself and
mutually satisfies the necessities of each other. The substratum of reality or
life which extends from the heart of the kosmos to the extreme limits of the

phenomenal universe exists in degrees, not discrete, but continuous. And
these merge into one another by insensible stages. Such is the imperceptible
continuity of the whole as each degree is gradually immerged into the other
that only the limitations of consciousness itself make it to appear as if it
were discontinuous. For every stage of realism there is a state of
consciousness which answers to it completely and sufficiently. So both the
state of consciousness and that of reality, manifesting at any given stage,
seem to be complete and final for that stage. Realism or life and
consciousness possess only a relative finality fashioned upon the necessities
and requirements for any given state of being. Consciousness alone fixes
the apparent limits of life; it also determines the state of our knowledge of
life. And thus when the Thinker is confined to any stage of reality and
congruent degree of consciousness it appears that what he there finds is
ample for all his purposes. Accordingly he is convinced that that stage is the
final consideration of his scope of motility. It is only when he is able to
raise his consciousness to a point where he can contact higher realities that
he becomes aware that there are higher stages in which his consciousness
may manifest. This peculiarity of the Thinker's consciousness is accentuated
when he allows himself to become wholly engrossed with a study of the
phenomena of that stage in which he can consciously function. Hence it
constantly occurs that men exhausting the study of the phenomenal find
themselves floundering upon the beach of the outlying shores of
consciousness where in sheer desperation they fall into the illusion that they
have indeed reached the limits of manifested life and that beyond those
limits there is no organized being. Unconscious are they that in ever
widening circles the fertile lands extend and await the awakening of their
consciousness when they may till the fallow ground of this new domain and
begin again the search for the ultimately real.
With respect to the present powers of consciousness, it cannot be
successfully controverted that the concept of tridimensionality of space is
sufficient for all purposes. It must be so for it is not only an aspect of the
phenomena of space but of reality as well. This fact is attested by the nature
of mind that answers to the nature of space. Tridimensionality characterizes
the entire extent of consciousness and life, and therefore, of space itself.
This characterization may be traced to the very doors of the heart of space
where the three become one. Nor would this conception be in the least

vitiated if it were allowed that the mass of the phenomena of the
supersensuous world, lying in close proximity to the sensuous world, does
present itself to the consciousness in a four-dimensional manner and that
the phenomena of a still higher plane present themselves in a five or n-
dimensional manner to that state of consciousness which may be congruent
with them; because then we should be making allowances for the changes
in phenomena and their mode of presentation to the consciousness which by
no means implies a corresponding change in reality or life. All phenomena
are fashioned by the intellect. The phenomenal world is just what the
intellect interprets it to be. It is that and nothing more. Its qualities,
attributes and characteristics are such as the consciousness gives to it. It
exists only for the purposes of the evolving consciousness. And, as an
instrument of consciousness, its existence is strictly subject to the
evolutionary needs thereof. In that moment that the immediate needs of the
consciousness shall no longer be able to find satisfaction in the phenomena
of any plane of nature, in that moment the phenomena of that plane
disappear, recede and are swallowed up in the maelstrom of eternal reality.
In the gradual expansion of consciousness as it passes through the infinite
series of grades of awareness meantime becoming deeper, broader and more
comprehensive as it proceeds, there may be observed running through all
these planes and orders that which is neither the phenomena of the various
planes nor the consciousness; but which must be the substructural basis of
both, remaining the same, unchanged and unchangeable. That is the thread
of reality, the passage of life itself which is the eternal basis of all. Now it is
to this reality, life, that the space-mind is related and in which its roots, its
heart and the very center of its being are at one with the divine mind of the
kosmos.
The question of dimensionality is solely a concern of the objective or brain-
mind which is the intellect. It is one of the ways in which the intellect
endeavors to understand phenomena. It is an arbitrary contrivance devised
by the intellect for its convenience in studying the world of things. Without
it, as obviously appears, the intellect would not be able to go very far in its
consideration of the minor problems which inhere in material things. The
fourth dimension is but another attitude, another contrivance, which the
intellect has devised in order that it may study from another angle the
evanescent phenomena of the world of appearances. Having apparently

exhausted the possibilities of motion in three dimensions, and being driven
on to the acquirement of more picturesque views by the very necessity of its
continued growth, it has betaken itself to another higher mountain peak,
called "hyperspace" where with larger lenses and higher powered
instruments it is beginning to scan the landscapes of a new intellectual
realm of consciousness. Yet the celestial wonders of this new-found realm
of consciousness remain in undisturbed forgetfulness or neglect. But it is
not by a scrutiny of mathetic landscapes nor by a study of the celestial
wonders that the Thinker shall one day realize the object of his eagerly
pushed quest after the real; for he shall find it, if at all, in the temple of the
kosmic mind which is not made by the intellect nor meted and bounded by
geometric systems of space-measurement.
In all the learned pother incident to the mastery of the phenomenal, the
furniture of the world of the senses, it is as if the self in man, the Thinker,
sat secluded in a six-walled tenement, and hence six times removed from
the subject of his study, and endeavored to interpret that which appeared to
his vision. And thus, thinking that what he sees is the only reality, he
remains in inglorious nescience of the reality of that upon which he himself
stands, unconscious that the tunnel-shaped aperture through which he peers
leads not outward, but backward and within to the habitation of the real of
which he himself is a part. Men are deeply and well-nigh hopelessly
concerned with appearances, with static views of life, with instantaneous
exposures. Life, reality and all the eternal verities pass on and assume
countless postures, attitudes, moods, tenses and nuances. The intellect is
content to occupy itself with a single tense or mood. Indeed, it has no
aptitude or power to consider more than one at a single time. It thus misses
the continuity, the ceaselessness, of life. What is more, every singularity,
every attitude, mood or tense which the intellect grasps for consideration is
immediately remade so as to fit its own moods and tenses. And upon each
and every nuance the intellect immediately imposes its own form—actually
and literally rehabilitates them with its own habiliments. Unfortunately, this
peculiarity occludes the intellect from any approach to the true nature of
that phase which it can grasp.
Hyperspace is one of the illusions of the phenomenal; it is the dress which
the intellect has superimposed upon a single nuance; it is a mask which is
an exact replica of the mood of the intellect. Yet through this mask the

intellect grandly hopes to approach reality. The period through which the
mind is now passing is a repetition of the evil days of scholasticism when
men set out to determine the exact number of celestial beings that could be
perched upon the extremity of a needle point. It is a time when men's minds
easily assume grotesque and hideous shapes and their thoughts become the
embodiment of fantastic entities. The exclusive occupation of such minds
becomes the fabrication of mathetic monstrosities which rapidly deliquesce
upon the first approach of the real or the appearance of the first ray of
intuition which may escape through the dim and misty condition of the
intellectual over-hangings. It will not be ever thus; for the Thinker will one
day pass from a study of the arrangement of phenomena in space and by
well-ordered steps come once again to himself. And then through the maze
of it all set out upon the true path——the tridimensionality of space
following which he will inevitably approach the citadel of the real, the
kosmic space-mind.

CHAPTER VII

The Genesis and Nature of Space

Symbology of Mathematical Knowledge—Manifestation and Non-
manifestation Defined—The Pyknon and Pyknosis—The Kosmic
Engenderment of Space—On the Consubstantiality of Spatiality,
Intellectuality, Materiality, Vitality and Kosmic Geometrism—Chaos-
Theos-Kosmos—Chaogeny and Chaomorphogeny—N. Malebranche
On God and the World—The Space-Mind—Space and Mind Are One
—The Kosmic Pentoglyph.

Geometry is concerned primarily with a study of the measurement of
magnitudes in space. Three coördinates are necessary and sufficient for all
of its determinations. Metageometry comprehends the study of the
measurement of magnitudes in conceptual space. For its purposes four or n-
coördinates are necessary and sufficient. Perceptual space is that form of
extension in which the physical universe is recognized to have been created
and in which it now exists. Conceptual space is an idealized conception
belonging to the domain of mathesis and has no actual, physical existence
outside of the mind. Mathematical space represents the idealism of
perceptual space.
Geometrical magnitudes may be defined as symbols of physical objects and
geometry as a treatise on the symbology of forms in space. In fact, all
cognitive processes are simply efforts at interpreting the symbolism of
sense-deliveries; and the difference between mere knowledge and wisdom,
which is the essence of all knowledge, is the difference between the
understanding of a symbol and the comprehension of the essential nature of
the thing symbolized. So long as knowledge of space is limited to the
understanding of a symbol or symbols by which it is presented to the
consciousness so long will it fall short of the comprehension of the essential

nature of space. In vain have we sought in times past to understand space by
studying relations, positions and the characteristics of forms in space; in
vain have we based our conclusions as to its real nature upon the
fragmentary evidences which our senses present to our consciousnesses. It
is as if one had busied himself with one of the meshes in a great net and
confined his entire attention to what he found there, meanwhile remaining
in complete ignorance of the nature of the net, how it came to be there, of
what it is made and how great its extent may be.
There is ever a marked difference between a symbol and the thing which it
symbolizes. Words are the symbols of ideas; ideas, as they exist in the
mind, are the symbols of eternal verities as they exist in the consciousness
of the Logos of the universe. There may be a wide diversity of symbolic
forms which represent one single idea; as, for instance, the variety of word
forms which represent the idea of deity in the various languages. Likewise
there may be a multiplicity of ideas which represent a single verity. But
neither is the idea nor the word the real thing in itself. That quality of a life-
aspect which we call its thingness has an essential nature which cannot
become the object of consciousness except by virtue of its representation
through ideas and their symbolisms, and even then, the thing which we
conceive is not the nature of a quality of the life-aspect but an idea of it—a
symbol which stands for that idea. In order, therefore, for the mind to arrive
at an understanding of an eternal verity, such as space, it must first be able
to synthesize all of the representative ideas and then abstract from their
compositeness a notion of its essential nature. But this can be done only by
identifying the consciousness with the essential being of the object
considered. In other words, the consciousness and the intrinsic being of
forms, principles, forces and processes must embrace each other in the
intimacies of direct cognition; the life which is consciousness and that life
which is essential being, being coeval, coördinate and mutually responsive,
must in so close a contact as here intimated reach an understanding of the
realism shared by both. That is, the human consciousness, following in the
wake of life and consisting of a specialized aspect of life itself, will, by
such an intimate approach to the life-principle of forms, readily understand;
for it has only to recognize a replica of itself in rendering its judgment. But
it is not claimed that such a state of recognition by the consciousness of life
itself can be attained at all by ordinary means, neither is it believed that it is

the next stage in conscious evolution. However, it is not doubted but that
such an exaltation of the consciousness is possible, yea practical; but the
difficulties which beset the path of attainment in this direction are so great
that it may as well be considered unattainable. The mere fact of these
difficulties, however, only re-emphasizes the insufficiency of the
intellectual method. The identification of consciousness with essential being
is a procedure which cannot be accomplished by an act of will directly and
immediately. Because it is a process, a series of unfoldments, an adjustment
of the focus of consciousness to the kosmic essentialities which constitute
the substructure of the manifested universe. In the very nature of things, a
kosmic essentiality cannot be viewed as being in manifestation especially in
the same degree as ordinary physical objects are manifest. The former is a
state, a potentiality, a dynamic force, an existence which should be thought
of as an extra-kosmic affair dwelling on the plane of unity or kosmic
origins; while the latter are the exact opposite of this. The one can be seen,
felt and sensed while the other is the roots which are not seen but lie buried
deeply in the heart of the universal plasm of being and beyond the ken of
sensuous apperception. The term manifestation is both relative and flexible
in its use. It is relative because it will apply equally to all stages of
cognition. A thing is in manifestation when it is presentable to the ordinary
means of cognition belonging to any stage of conscious functioning; it is
not in manifestation when it is beyond the scope of the Thinker's
schematism of cognitive powers. Its flexibility is seen in its ready yieldance
to the entire range of implications inhering in the process of cognition,
fitting the simplest as well as the highest and most complex.
Great is the gulf which is interposed between manifestation and non-
manifestation; and yet the two, in essence, are one. They are linked together
as the stem of a flower is joined to its roots. Likewise one is visible,
palpable while the other is invisible, impalpable though no less real and
abiding. As the thin crust of earth separates the stem, leaves and flowers
from the roots so the limitations of man's consciousness separate the
manifest from the unmanifest. Similarly, as when the surrounding earth is
removed from the roots and they are laid bare revealing their continuity and
unity with the outputting stem and flowers, so, when the limitations of
consciousness are removed by the subtle process of expansion to which the
consciousness is amenable so that it can encompass the erstwhile

unmanifest, it, too, will reveal the eternal unity of the kosmic polars—
manifestation and unmanifestation.
There is but one barrier to ultimate knowledge and that is the human
consciousness however paradoxical this may seem. The unutterable
darkness which shuts out the so-called "unknowable" from our cognition is
the limitation of man's upreaching consciousness. These limitations
constitute the difference between the human intellect and the mind of the
Logos. Nevertheless the outlying frontiers of man's consciousness gradually
are being pushed farther and farther without. Every new idea gained, each
new emotion indulged, each new conception conquered, and every mental
foray which the Thinker makes into the realm of the conceptual, every
exploration into the abysmal labyrinth of man's inner nature are the self's
expeditionary forces which are gradually annihilating the frontier barriers of
consciousness and thus approaching more closely upon the Ultima Thule of
man's spiritual possibilities.
Space is in manifestation. It exists and has being whether it is viewed as an
object, an entity or the mere possibility of motion. That it offers an
opportunity of motion and renders it possible for objects to move freely
from point to point cannot be denied and yet this fact has no bearing
whatever upon the essential nature of space. The very fact of its appearance,
its manifestation, makes it obvious that it is the nether pole of that eternal
pair of opposites—manifestation and non-manifestation, being and non-
being, which are essentially one.
It will be seen from figure 17 that the period of involution embraces seven
separate stages, the monopyknon, duopyknon, tripyknon, etc., being the unit
principle or engendering elements of the respective stages. Involution
comprises all creative activity from the first faint stirrings of the void and
formless chaos until the universe has actually become manifest and dense
physical matter has appeared. It is divided into two cardinal periods,
namely, the chaogenic period during which primordial chaos is given its
character and directive tendencies for the world age. It is a phase of
duration wherein the fiat of kosmic order is promulgated, and consist of
three stages, monopyknosis, duopyknosis, tripyknosis[24] gradually,
insensibly, gradating into the fourth or quartopyknotic; the
chaomorphogenic period is likewise divided into three stages—

quintopkynosis, sextopyknosis and septopyknosis, developing out of the
fourth gradually. The quartopyknotic stage is the stage of metamorphosis or
transmutation wherein the transition from non-manifestation to
manifestation is completed; it is also the stage of kosmic causation, because
from it spring the matured causes or "vital impetus" which engender all that
follow.

Fig. 17.—Involution and Evolution

The close of the involutionary phase of the world age is marked by the final
deposition of dense physical matter and this is closely followed by the
beginnings of the evolutionary movement which, like the involutionary
movement, is divided into two cardinal periods, namely, the morphogenic

(during which are produced, in turn, insensible forms, sensible forms and
spiritual forms) and the kathekotic period which marks the perfection, the
consummation of the evolutionary movement. These two cardinal periods
of the evolutionary phase of duration and the two cardinal periods of the
involutionary phase complete the kosmic age, the "Great Day of Brahma."
The concentric circles, beginning with the dot and ending with the seven
concentric circles, and designated as a, b, c, d, e, f, g, are representations of
the constitution of the respective units corresponding to each of the seven
subdivisions. They symbolize the seven degrees of condensation or
pyknosis which comprise the genesis of space, on the one hand, and on the
other, the stages of unfoldment. Because, during involution all potencies,
powers and characters were being infolded, involved; but during evolution,
these are being unfolded, expressed, evolved.
The figure 18 is another view of these two major movements, involution
and evolution. The genesis of space is here shown symbolized by the
Kosmic Egg. The seven stages of involution are referred to as, the
monopyknotic, duopyknotic, tripyknotic, quartopyknotic, quintopyknotic,
sextopyknotic and the septopyknotic; while the corresponding stages of
evolution are referred to as, the physical, the sentient, mental, causative or
spiritual, the triadic, duadic and monadic, indicating that the principle of
physicality is succeeded by the principles of sentience, mentality,
spirituality, and the three forms of kathekotic being. This symbolism, it
should be stated, is designed with respect to the universe and man and has
no reference to other possible evolutions than the human and
contemporaneous animal, plant and mineral evolutions.

Fig. 18.—The Genesis of Space

To follow the ramifications of the symbolism above would involve a survey
of all branches of knowledge, and indeed, would be out of place in this
book. Only the widest general outlines can be suggested here and it is
believed that this is sufficient to enable the reader to grasp the magnitude of
the symbol and to understand its purpose and intent.
It would appear, therefore, that if it is possible for the intellect to traverse by
means of a study of kosmic symbolism, used as a standard of reference, the
entire length of the bridge which engages the antipodes into an eternal
unity, something may be gained in the way of a more definite and clearer
understanding of the essential nature of space in its relation to
kosmogenesis.
In the diagram, figure 17, is shown a table which represents the stages of
space-genesis. It will be noted that the whole scheme is divided into seven
stages. It is not an arbitrary division simply but a symbolic one and
represents fullness, completeness, entirety. The names given, namely,
"monopyknotic, duopyknotic," etc., represent the symbolic characteristics
of each stage in its relation to the universe in the process of becoming. The
terms "monadic," "duadic," "triadic," etc., are representative of the seven
planes of matter in the universe.

A pyknon is a kosmic principle and represents the typal aspect of
kosmogenesis. It is a generic term and may be identified in its relation to
the various stages by the prefix. The monopyknon belongs to the ulterior
pole of the antipodes on the side of non-manifestation. So do the duopyknon
and the tripyknon. Pyknosis is a process of kosmic condensation, or
limitation for purposes of manifestation. It is a stage in the descent of the
kosmic Spirit-Life, a degradation of non-manifestation into manifestation,
and is, therefore, the cardinal causative principle of creation. The term
pyknon being generic is applicable alike to a particle of matter, a state of
being, a condition of existence, a process or a principle. The monopyknon
is, accordingly, the primary aspect of the process of kosmic pyknosis. It is
the archetype and therefore all inclusive and omnipotential. But whether
regarded as a singularity or as a whole it should never be divorced in
thought from the primal act of creation. It represents the first act of material
differentiation in the being of the creative Logos on the plane of non-
manifestation. It is the beginning of every great planetary or kosmic
manvantara or period of manifestation. During either a planetary, solar or
kosmic pralaya, or gestatory period, the kosmic plasm is in a quiescent,
undifferentiated condition. This undifferentiated plasm when acted upon by
the will of the Creative Logos, Fohat, as He is sometimes called in Eastern
Philosophies, begins to become conditioned, begins to differentiate. The
primal act or stage in such a process is the formation or appearance of a
monopyknon. It then becomes the characteristic aspect of that stage.
Monopyknons are the quiescent, unawakened, though potential and
archetypal principles peculiar to the monopyknotic period of space-genesis
which are ultimately to become, on the physical plane, singularities of life
of whatsoever kind. Thus the lineage of every single life-form or principle
in the universe runs unbroken back from the present of the Now to the
present of monopyknosis. So great is the design of the kosmos that the entity
which is now man or the atom was started on its journey to this culmination
at the break of the Great Kosmic Day when the omni-pregnant wheels of
monopyknosis first began to turn. Duopyknons and tripyknons constitute
the two remaining stages on the plane of non-manifestation. And their
correspondences in every stage of involution or the descent of spirit into
matter are eternal and kosmic. Likewise the lineage of the dual and triple

aspects of all life forms on the path of evolution may be traced rearward to
the duopyknotic and tripyknotic stages of kosmogenesis.
The metamorphosis by which the monopyknon becomes a duopyknon
contrives the differentiation of the pristine plasm of kosmic being so that
the first becomes the ensouling or vitalizing principle of the second; and, in
turn, the second becomes the vitalizing or inner principle of the third; the
third of the fourth and the fourth of the fifth and so on throughout the series
until the last is reached which is the septopyknon. The septopyknon is,
therefore, a seven-principled form. It is both unitary and septenary—unitary
in the sense that the seven are really one and septenary for the reason that
each of the seven principles, in the course of evolution, becomes a separate
process specially adapted to functioning upon its peculiar plane of matter.
Thus it is seen that the utmost significance attaches to this septenary
pyknosis of the kosmic plasm of life. The implications of this conception
are, of course, too vast and multifarious to be set down here. We shall have
to dismiss it with one observation only, and that is: every single appearance
of life and form in the totality of such appearances is rooted in kosmic
pyknosis where it has received its inner vitalizing force, its form and the law
of its mode and manner of appearance together with the metes and bounds
of its existence.
These processes, monopyknosis and duopyknosis, are to be regarded as
taking place, each in its own period, everywhere throughout the Body of
Being of the Logos but on the plane of non-manifestation. They are states of
preparation for manifestation analogous to the germinative period of the
seed or the egg. They represent the first stirrings of the kosmic plasm and
contain the promise and potency of all that is to succeed them. There is one
other stage, coördinate with these two, and that is the tripyknotic which
completes the unmanifest trinity and constitutes the archetypal vehicle
whence proceeds the manifested universe. The ensouling principle of the
tripyknon is the duopyknotic principle. But when the descent has reached
the tripyknotic stage the three have been merged into one and the
characteristics which were peculiar to each as a separate pyknon are then
fused into a single quality having three aspects which are mutually
interdependent and coördinate.

The unmanifest trinity, now complete as a result of the triple pyknotic
process, is the imperishable and ever sustaining radix, the all-mother of the
manifested universe. It is the Golden Egg laid by the god Seb at the
beginning of a great life-cycle. It is also Chaos-Theos-Kosmos, i.e., kosmic
disorder, divine will or generating element, and kosmic order or space. In it,
as in an egg, resides in kosmic potency all that the universe is to become in
any "Great Day of Brahma" or any Great Kosmic Life Cycle. Its eldest born
is SPACE, physiological and perceptual, and the latter is the eternal father-
mother of the universe. Space is, therefore, the male-female principle of
manifestation; in its kosmic womb all forms are created, developed, evolved
and sustained. Into it again, at the close of the Great Day, all existences,
forms and all principles matured and ripened by the vicissitudes of kosmic
evolution will be inhaled with the return of the Great Breath of Life. The
unmanifest trinity is the archetype, and therefore, the pattern or model for
the manifest, embodying in potency all that the manifest may ever become.
Forth from the unmanifest proceeds space as a dynamic process endued
with the potentiality for generating all that the manifested universe contains.
The terms "unmanifest," "unconditioned" and "unlimited" have a special
meaning here and are used in the same sense as the mathematical term
"transfinite," and therefore, imply a transcending of any finite or assignable
degree or quality of manifestation. Hence they should be distinguished from
the term "absolute" which has a different implication. So that although the
triple process outlined above may not be viewed except as a
characterization of the plane of non-manifestation, and hence of the
primordial activity of the Creative Logos, there is nothing in the symbolism
to warrant the identification of this process in any way with the Logos in
absolution. For on this view Absolute Being is, in a large measure,
sacrificed when the monopyknotic process is begun and the monopyknon
(kosmic principle) begins to appear. Absolute Being, while it may not be
defined, delineated or described may be symbolized by the ideograph:
"action in inaction"; "being in non-being"; "manifestation in non-
manifestation"; for these are symbols merely and do not describe or
delineate.
We have observed the subtle connection which exists between
manifestation and non-manifestation and have seen how that, as the roots of
a plant sustain the outer growth of stem and flowers, the former being the

matrix out of which proceeds the latter, and that in like manner does the
unmanifest sustain the manifest; it should, therefore, be clear that the Body
of Being of the Unmanifest Logos, in a similar manner, is the basis and
cause of all that is manifest or in existence; and yet it is more, it is
essentially all that is and all that the all is to be in manifestation at any time
throughout the immeasurable process of kosmogenesis. It is the Self of the
Universe, the meta-self of the great diversity of selves. The Self, however,
although it cannot be said to exist except as a simple, homogeneous
quantity is nevertheless, in the very nature of things, a triunity, and in
essence, not only the basal element of the All-Space and the concrete forms
which exist therein, but is also identical in essence and substance with
space.
A stage has now been reached in the description of our symbolism when it
may be assumed, upon the basis of the foregoing, that the meta-self has
become manifest, i.e., in potential and dynamic appearance, as a result of
the triple process of pyknosis hereinbefore outlined. The meta-self may then
be identified with the Supreme Manifest Deity; for there is ever a subtle
identification of the manifest with the unmanifest. Out of their action and
interaction the universe is made manifest and phenomenal and is thereby
sustained. The reciprocity of action between these two kosmic polars, is the
metamorphotic key to creation; it is the symbol of the procedure of Creative
Will in the act of creating. It is the transitional process whereby the passage
is made from non-being to being; from the unconditioned to the
conditioned; from undifferentiation to differentiation and is reflected and
symbolized in every natural process wherein matter is transformed from
one state to another, or life and mind and spirit diffused, centralized and
organized into ever new and higher forms and expressions.
Another important notion to be gained in this connection is the fact that it
appears as a logical sequence to the foregoing that the being of the
manifested Logos must necessarily fill all space, yea more, is that space in
every conceivable essentiality. His limitations are the limitations of space.
His qualities, properties and attributes are the qualities, properties and
attributes of space and are only different from the original spatial character
when manifested through a diversity of forms by whose very inner
constitution and exterior form the modifications are accomplished. All
matter in the universe, all energy, and indeed, all manifestations or

emanations of whatsoever character, hue, tone or quality are, in reality, His
being and nothing but His being. There is, therefore, no form nor ensouling
principle whether of life, mind or sheer dynamism which can exist outside
of His being and be, even in the slightest degree, absolved from an eternal
identity therewith. Once this idea is grasped and its varied implications
noted it then is no longer conceivable that any other order or schematism
can be possible in our universe, and that, too, despite the multiform
conceptions peculiar to the varied systems of philosophy.
The matutinal dawn of creation came at the close of the tripyknotic
movement which resulted in the elaboration of materials, the initiation of
principles, processes and types, and the final preparation of the field of
evolution. The three processes or aspects of non-manifestation projected in
preparation for manifestation, namely, monopyknosis, duopyknosis and
tripyknosis represent the earliest stages of germinal development. When
these had closed, the Great Kosmic Egg began to germinate; the first faint,
indescribable signs of manifested life began to appear. Involution set in.
The fourth or quartopyknotic stage, though only slightly differentiated, or
rather representing that period of kosmic involution when that which is to
become the manifested universe first begins to fall under the sway of
kosmic order, is nevertheless the basis of all great world processes. It is just
midway between the poles—manifestation and non-manifestation. During
this stage the life elements are receiving the imprints of character, being
endowed with directive tendencies and stored with such dynamism as will
persist throughout the Great Life Cycle in which they are to manifest. It is
here that begins the movement of involution, the storing away of those
elements and factors, no more and no less, that are to show forth on the
upward path of evolution; it is here that matter begins to assume form;
electrons, ions and atoms created, or, that those minute processes which on
the evolutionary side are to culminate in these are originated. This is the
metamorphotic stage. It is the laboratory of the universe wherein Fohat, the
Creative Logos, prepares the materials out of which and in which the vast
diversity of morphons or forms is created. Quartopyknosis, accordingly, is
the first active step, on the plane of manifestation, which results in the
appearance of perceptual space and consequently of physical matter itself as
well as all the other grades of matter in the kosmos. Space, brought into
existence by the act of the Creative Logos in imposing limitation upon His

being, is in its primordial form composed of quartopyknons or
quadruplicate principles and tendencies which act in unison and to the
accomplishment of a single end or purpose. On this plane or during the
continuance of this period of space-creation, the roots of universal law and
order are produced. In it are planted the principles of good and evil and a
sharp line of demarkation established between all the conceivable pairs of
opposites which exist. It accounts for the duality of life and form. Male-
female; father-mother; positive-negative; Rajah-Tamas (action-inaction)—
all these find in this process their eternal origination. It is the stage of
harmony, bliss, ideality, perfection, perfect equilibrium and balance. Here,
innumerable ages before they actually appear, the glow-worm and the daisy,
the amoeba and the dynosaur, man and the planetary gods alike abode their
time awaiting the toppling of the scales of kosmic potency when all would
be plunged headlong into the endless labyrinth of becoming.
The quartopyknotic process is similar in all details to the three preceding
processes, these latter being prototypes of all succeeding stages of
involution. The quintopyknon, accordingly, symbolizes the quintuplicative
action of life in its descending movement toward the creation of matter in
its densest form. That is, it is a five-fold principle acting in unison and
kosmic consistency, infolding in the universal wherewithal that which is to
become mental matter on the side of evolution. Just, as may be seen in the
diagram, Figure 17, the quartopyknotic process symbolizes, on the
involutionary side of the life current that which is to become on the
evolutionary side, spiritual essence, so the quintopyknotic deposits the
seeds of that which is to become mental matter. During both the
quartopyknotic and the quintopyknotic processes all the potencies and
promises, residing on the plane of non-manifestation and destined to show
forth as spirit and mind are brought into a fuller and more marked degree of
manifestation and become the seeds of spirituality and mentality which are
to ripen and be ladened with the fruitage thereof many ages hence.
The reader should bear in mind that the processes here described are
thought of as taking place at the beginning and as having their roots planted
in eternal duration; that they refer to a period long before the universe even
resembled anything like its present aspect; at a time even before there were
individual minds to perceive it, before even the gods—solar, planetary,
super-solar and super-planetary were in existence to take part in the

matutinal ceremonials of creation's vast hour of stillness. The mind must
accustom itself to go back of appearances, back of time, back of space itself
and discern the foundations of time, space and appearances being laid and
to perceive that which is no less than the action of the Supreme Deity
Himself in brooding over the primordial formlessness which is Himself, and
from which will gradually evolve the universe of qualities, conditions and
appearances.
The quintopyknon is, therefore, the base of the mind-principle in the
kosmos. All the qualities of mind whether in man or in the planetary gods,
whether in the moneron or the tyrannosaur, in the mountain or the oak,
reside in kosmic potency in the quintopyknon.

Nicholas Malebranche,[25] in one of his very lucid moments, beheld the
essential character of the symbology of space and was led to the conclusion
that God is space itself. To him it was equally certain that all our ideas of
space, geometrical or purely physiological, as well as our notions of the
great suprasensual domain of ideas, exist in the kosmic deiform, or body of
the Logos of Being. He saw "all things in God." God did not create ideas;
they are a part of God Himself; God did not create mind; it is a part of
Himself; no kind of matter did He create; it is a veritable part of Himself
and indissoluble from Himself. The great outstanding implication of this
philosophy is that our consciousness of God is but a part of God's
consciousness of Himself; our consciousness of self and the not-self is but
God's consciousness of these things. There is no existence of anything,
either of the self or the not-self, except in this consciousness. It is
refreshing, therefore, to note that although the approach is made from
another and entirely different direction, almost the same conclusion as to
the ultimate resolution of all chaogenetic elements into what is the very
systasis or consistence of the great kosmic deiform, is reached.
But a marvelous vision comes with the dawn of this truth upon the lower
mind. It establishes clearly the truth of Kant's notion when he said: "Since
everything we conceive is conceived as being in space, there is nothing
which comes before our mind from which the idea of space can be derived;
it is equally present in the most rudimentary perception and the most
complete." The mind cannot get away from the conception of space,
because, out of the very essence of space, as a result of the quintopyknotic

process, it was produced, created and organized. The idea of space is,
therefore, not derived from things in space nor from their relations in space.
It sprang up with self-consciousness. As soon as the Thinker became
conscious of himself he became aware of space. The very state of self-
consciousness implies space. The self in man is a specialized aspect of
space. Indeed, it is a projection of space. The moment the self can say: "I
am," it also can complete the declaration by saying: "I am Space." When the
self looks out from his six-walled cabin of imprisonment into the immensity
of what we call space he looks out into that which is himself and his
immensity; he perceives the source and the ever-present sustenance of his
being and recognizes his identity therewith, provided he does not allow
himself to become entangled in the philosophical difficulties which the
intellect is prone to throw around the simple, yet marvelously complex,
notion of self-consciousness.
This should settle, once for all, the question of apriority. The a priori
inheres in quintopyknosis or kosmic psychogenesis. It is the essential nature
of mind; it is the mind's lines of organization; it is the law of the mind's
being and action. All mental perception originates from things in space. No
thought of any detail, of any state or condition, whether limited or
unlimited, related or isolated can be conceived except it be of things in
space. And this is so, because mind and space are one. It is not so with our
conception of time. Time is merely an aspect of consciousness in its
limitation and does not inhere in the mind in the same manner that does
space. In fact, it is not a part of the mind's nature as it has been shown that
space is. It would, therefore, seem to be a grave mistake so to coördinate the
two notions. Space is the progenitor of mind and is continually identified
with mind. Time is the child of consciousness. That is, it is one of the
illusions of consciousness which the ego will shed as his consciousness
expands. Duration alone is a coördinate of space.
The mind now recognizes space as something apart and separate from itself
only because of its unconsciousness of the identity existing between it and
space. Just so, it is not by mind alone that the at-one state of consciousness
shall be attained; for although in one form or another it is able to gain some
knowledge of the apparent oneness of all life it cannot directly realize this
oneness. In order to do this fully it must be able consciously to identify
itself with the life, feel what it feels and experience what it experiences and

otherwise come into a conscious relationship with the root and source of
life. Space-consciousness is a simple, direct cognitive process; while time-
consciousness is a complex, and therefore, indirect process. The former
cannot be analyzed. That is, no analysis is necessary to its sufficient
comprehension; the latter must always be analyzed and categorized for its
sufficient apprehension. Every moment of time whether past, present or
future, when presented to the consciousness, is determined by its
relationship to some other moment of time. Space is indivisible; time is
divisible. Space is an intuitional concept; time is an intellectual concept.
Time belongs to phenomena. Space is the root and source of phenomena.
Time is the leaves of a tree while space is the life of the tree.
Space-consciousness, in its relation to the present status of mind-
development, is itself an illusion; for despite the fact that the Thinker's
apperception of it as a state is simple, direct and fundamental, it is only so
because of the inability to realize to itself the unity of the seeming two. The
attainment of space-consciousness or the space mind, which contrives the
understanding of the identity of mind and space also annihilates the
consciousness of space as a separate notion from the mind. Once the
Thinker's consciousness has arisen to that state where it perceives its unity
with space all sense of separateness is lost. Just as when two molecules of
hydrogen uniting with one molecule of oxygen to form a new compound
lose their identity in the new realization of unity, so does the consciousness
when by the alchemy of psychogenesis it becomes identified with space,
not only lose its identity as such, but also any consciousness whatever that
space exists as something separate and distinct from itself. Imagine the
whole of the duration aspect of kosmogenesis crowded into an infinitesimal
instant and the bulk of all matter, suns, stars, worlds and planets, condensed
into a space less than the magnitude of an hydrogen ion and in this way a
symbol of what it may mean to attain unto absolute knowledge or unto the
space-mind, may be obtained.
Recurring to the process of quintopyknosis, it may be noted that the
quintopyknon or five-fold kosmic principle of life which we have seen to be
identical to the seeds of mental matter brought into existence by the
reaction of Fohatic energy or the Will of the Creative Logos upon the
substance of the quartopyknotic stage, is, symbolically speaking, more
dense and compact than the pyknons of the preceding stages. It is ensouled

by the quartopyknon. It is a rather complex state of ensoulment consisting
of four condensations or pyknoses.
The next stage in the process of kosmic involution which is also concerned
with the preparation of the evolutionary field is that of sextopyknosis and
implies the senary condensation of the original world-stuff with the view to
the formation of emotional or "astral" matter. Identically the same process
of ensoulment or involution obtains upon the plane of sextopyknosis as
have been observed to obtain upon the preceding planes of involution.
Involution must necessarily precede evolution. That which has not been
involved, enfolded or ensouled cannot be evolved or unfolded. Whatever
potencies, powers and capabilities or qualities and characteristics that may
appear at any time in the universe of life and form must have first been
involved or enfolded or else they could not have been evolved. Space itself
is an evolution. It is a process of becoming, of unfolding, of flowering forth.
As it evolves more and more there will appear new and added
characteristics and qualities of life and form. New possibilities will arise
and in the end a supernal vision of a glorified universe will burst into view.
The scheme of space-genesis is completed during the septopyknotic process
wherein the basal elements of dense physical matter and its various
gradations are produced and given character, form and direction. But this
completion means merely a temporary estopment in the process of
kosmogenesis which actually results in the formation of physical matter in
its crassest state. It does not mean a final arrest of the entire process which
is conceived of as continuing only in a regressive manner back to a
kathekotic[26] condition wherein it embodies the fruitage of the entire
scheme. The septopyknon, accordingly, is a seven-ply pyknon in which are
embodied, in varying degrees of manifestation and phanerobiogenic (life-
exhibiting) quality, the essentialities of all that has preceded on all planes
and during all stages of space-genesis. That is to say—in the physical life of
the universe is confined the essence of all the series of grades of life in the
kosmos. In man's physical body are wrapped up all the glories attainable in
his long, almost unending pilgrimage of evolution; in it are stored all the
possibilities of the spirit; all powers, all qualities, all characteristics, ever
intended for man's attainment are in the physical. But they must be evolved,
they must be unfolded and expressed. The physical must be glorified,
spiritualized, deified. For by the way of the glorification and spiritualization

of the flesh man may attain unto oneness with the divinity in himself and
consequently with the divine life of the world.
To summarize: The genesis of space embraces seven stages, namely, the
monopyknotic, the duopyknotic and the tripyknotic which belong to the
plane of non-manifestation and are the primordial world-stuff and together
make up the unmanifest body of the Logos of Being. These become the
seed-germ of the universe of spatiality. The quartopyknotic is the fourth
stage in the process of space-genesis and is the metamorphotic or crucial
stage during which non-manifestation is metamorphosed into manifestation.
In it the unmanifest becomes the manifest. It corresponds to the plane of
pure spirit, and indeed, embodies within itself all the qualities which
spirituality is to show forth during the life of the kosmos. The
quintopyknotic is the fifth stage and corresponds to the mental plane,
embodying in itself all qualities of mentality in the universe and furnishing
the basis and essence of that which is to become the kosmic mind in
manifestation. The sextopyknotic is the sixth process and symbolizes the
sixth stage which embodies all the characteristics and properties of
emotional matter in the universe and is the basal element of the plastic
essence of sentient existence in the kosmos. The septopyknotic is the
seventh and final stage corresponding to the physical plane of the kosmos
and contains in its seven-fold constitution the seeds and potencies of all the
preceding stages, as well as all the characteristics and properties which
physical matter is destined to show forth during the manvantara or world
age. These seven processes result in the dynamic appearance of space, the
mother and container of all things, and complete the involutionary aspects
of kosmogenesis. Evolution began where involution ceased and will end for
this manvantara when the last vestige of those powers, capabilities and
potencies which were involved shall have been evolved unto kosmic
perfection.
The measure of the Great Kosmic Space-form was sealed at the close of the
involutionary movement of the Great Life Wave. Then its metes and bounds
were fixed by the fringe of kathekosity which circumscribed it.
If it be true that the reader found it extremely difficult to grant the
connotations of the symbolism when the mental or quintopyknotic stage
was reached when illative cognizance was given to the fact that space is

also composed of mental matter, it may be still more difficult to grant the
claim that physical matter is also essentially a part of space. But this is the
implication. And, therefore, it follows that all matter, all energy, all life and
all mind wherever it may be found in the Great Space-Form is space itself
and nothing but space. Hence, it appears that space is indeed the dynamism
of the universe. In its kosmic womb the great world egg was formed of its
own substance solely and in it still the universe of form persists and evolves
withal.
If it be suitable for the physicist to talk of gravitation, electricity, magnetism
and force let him do so, for these terms serve the present category of human
knowledge; but the human mind will not lament the day when it comes to
recognize that these things, these forces, these aspects are nothing more
than space-activities and space-phenomena. If a planet's place be preserved
in space it is because space, vital, dynamic, creative space, sustains it and
from its gentle, yet eternally firm grasp there is no escape. All that the
planets, suns and worlds are and all that they may ever become in this
manvantara or world age have been derived from space, yea, are of the very
essence of spatiality. If the chemist choose to talk of chemism, negative and
positive, of combining properties and dissociative phenomena let him also
become aware that these phenomena are but the external aspects of the
inner and ephemeral life-processes of space-forms and that ultimately these,
too, may be traced back into an eternal originality within the bosom of the
all-mother, spatiality.
Dense physical matter, such as constitutes the physicality of celestial bodies
in its ultimate dissociation would, accordingly, be resolved into the original
chaogenetic formlessness which marked the chaogeny of non-manifestation
although it would naturally be orderly and progressive passing through the
seven stages, septopyknosis, sextopyknosis, quintopyknosis, etc., until the
end had been reached, meanwhile exhibiting in each plane the phenomena
peculiar to the dissociative processes thereof. On this view space is a
plenum of matter of varying degrees of intensity, ranging from the densest
physical to the most tenuous and formless matter of the highest levels of the
manifested universe. But as neither the dense material forms nor the other
grades of matter have an eternally enduring quality, being alike subject to
mutation, space likewise falls under the law of becoming whereby it, too,
must yield to the edict of kosmic disorder.

Some may be inclined to argue that since space and mind are one and the
same thing it must necessarily follow that whatever possibilities of
measurement may be found to exist in the mind would logically be found to
exist in space; and that since all the necessary conditions of hyperspatial
operations are proved to be existent in the mind the case of the
hyperspatiality of perceptual space is proved thereby. In other words, if the
fourth dimension can be proved to be mentally construable it is also
possible in perceptual space. But these hypotheses are not granted, and
neither will they be acceptable to those minds who choose to take that view
when it is known that there is a marked difference between the mind that is
purely intellectual and mind that is purely intuitional or mind a priori. The
intellect is fashioned for matter only; it is so constructed as to fit squarely
into every nook and cranny, every groove and interstice in matter; yet for
the generating element, life, it has no aptitude nor suitable congruence.
The attainment of the space-mind or kosmic consciousness would then
imply a mastery of all fundamental possibilities pertaining to all degrees of
matter. Thus by becoming conscious in the matter of all the planes one
makes a certain definite approach to this ultimate state of consciousness
until all the barriers between ordinary self-consciousness and the
consciousness of the space-mind have been entirely obliterated.
Pyknosis, in all of its septenary aspects, is concerned primarily with
involution or the preparation of the chaogenetic elements for the work of
kosmic evolution. It may be thought of as being divided into two great
divisions, namely: chaogeny and chaomorphogeny. It is concerned with the
organization of chaos, the establishment of kosmic geometrism in the
formless, void, arupic substance and preparation for evolution. Chaogeny,
of course, is that kosmic process by virtue of which space itself becomes
manifest and in which there is no established order. Chaomorphogeny (from
Chaos + Morphe + Geny) signifies the activities of the creative Logos in
laying the foundations in primordial space-matter of the various star-forms,
including nebulæ, worlds, planets, suns, etc., of which Canopus, Jupiter,
Fomalhaut and Sirius and our own sun are examples, giving direction and
general tendence to their varied life-processes. Both these processes are
concerned with the preparation of the field and its consequent fertilization
in anticipation of its cultivation and harvest. These two constitute kosmic
involution or the great life wave's passage on the downward arc of the Great

World Egg or Circle. It is during the chaomorphogenic cycle that the
constitution of the universe of manifestation is promulgated; when laws for
its government during that manvantara are sketched out in the world of
nascent spatiality; when the archetype of every imaginable or possible form
is projected upon the impregnated screen of creation, then folded in, pushed
toward the center, involved, awaiting that time when the life wave begins its
passage upon the upward arc and evolution ensues, calling forth all that has
been enfolded in the bosom of the pyknotic centers of manifestation. It is
easily conceivable that here during the troublous times of the
chaomorphogenic enfoldment the now known six directions of space were
among the eternal edicts of space-genesis and that that law which now
makes it appear that three coördinates, and only three, are sufficient for the
determination of a point position in space was imprinted in the very nature
of that which was to become space.
The kosmic field having been prepared as a result of the chaomorphogenic
activities, lowly and scarcely organized forms begin to appear and the
ascent upon the upward arc of the Great Cycle commences. Evolution
begins. Its scope is likewise divided into two great stages, namely: (a)
Morphogeny, the purpose of which is the development of life-forms or
pyknons which are to appear on the various planets, stars, worlds and suns
of the universe. It embraces the whole span of the life-aspect on the
evolutionary side of manifestation. In this aspect is included also every
conceivable adaptation of the universal principle of life from the beginning
of its movement to the end. The universe is now functioning in and
progressing through the vicissitudes of this stage. That is, all the present
observable adaptations which the life-pyknon or principle is making, has
made or will make, are embraced within the scope of what is here
designated as the morphogenetic aspect of evolution. (b) The second and
last division of the arc of evolution is called Kathekos, thus symbolizing the
syncretism of the trinitarian aspects of kosmogenesis, Chaos-Theos-
Kosmos, perfected and united as a result of the labors of manifestation. In
this final summation of the labors of the life-wave as it has progressed from
involution through all the devious manifestations of evolution are embodied
the perfection and ultimate elaboration and expression of all the pyknotic
tendencies which were established during the entire scope of space-genesis.

Thus it will be seen that the first three stages of space-genesis, called
chaogeny, encompass the first three pyknotic processes or are analogous
thereto while the latter called chaomorphogeny, the organization and
ensoulment of space-forms, embraces the latter four, quartopyknosis,
quintopyknosis, sextopyknosis and septopyknosis. This division obtains on
the involutionary side of the great life-cycle. The upward arc of the Great
Kosmic Egg or Cycle is also divided into two great stages, namely,
Morphogeny (manifestation of life through the various forms which it
assumes) and Kathekos, or the kathekotic plane of perfected triunity which
is represented by the evolutionary union of Chaos-Theos-Kosmos. Kathekos
would, therefore, symbolize the ultimate elaboration of Chaos into a well-
ordered kosmos wherein are expressed all the possibilities which inhered in
the archetypal plan of the Creative Logos or Theos and in which all had
reached the ultimate perfection in the body of being of the Logos Himself.
But the kathekotic plane is to be distinguished from the original Chaos-
Theos-Kosmos represented as functionating upon the plane of non-
manifestation during chaogeny. Kathekos symbolizes the perfected
manifestations of the triune aspects of the Creative Logos through the
perfected forms resulting from the labors of kosmic evolution, while Chaos-
Theos-Kosmos symbolized, as a triune glyph, the Unmanifest Trinity in the
primordial beginnings of space-genesis. One is the seed; the other the fully
matured plant; one the egg; while the other is the full grown bird; one the
root; the other the fruitage; one Alpha, the other Omega; one the beginning,
the other the end. The end, however, is reached only that, in due time, the
entire scheme may be commenced again, once more utilizing the results of
the preceding scheme of evolution as the basis of the ensuing one. Thus
after every Kosmic Day, commences the Kosmic Night. The succession of
kosmic days and nights is infinite. This infinity of becomings in the life of
the kosmos is a necessary outcome of eternal duration.
The above, thus briefly set down, is the symbolism of space-genesis. It is
commended to the reader as a basis for the conception that the real,
essential, perceptual space is something far more wonderful, more
fundamental than either the geometrician or the metageometrician has ever
dreamed of, and yet the latter's consciousness is undoubtedly being
appulsed by the fingers of a new species of conceptualizations which, one
day in the not too distant future, will arouse in it the faint hungerings after

the realization of the real space-nature. These mathetic appetites thus
brought into being will finally lead the human mind into the Elysian fields
of kosmic consciousness where for another million years, perhaps, it may
feed upon the mysteries and hypermysteries to be found in the granaries of
the Space-Mind.
The study of space in its wider and deeper meanings is necessary in order
that a clearer understanding of its true significance, as the subject of
geometric researches, may be gained. It is confessed, however, that there is
neither direct evidence nor implicative authority for any assumption that the
view herein outlined affords any justification for the notion of the n-
dimensionality of space. For, although the line of reasoning indulged in
must lead inevitably to the conclusion that the worlds of spatiality,
materiality, intellectuality and spirituality, essentially and fundamentally
one so far as origins and qualities are concerned, were alike engendered by
the same generating element, life; and that spatiality being the primal basis
of the others is, nevertheless, under the exigencies of this aspect of the
kosmos, highly susceptible to the mensurative requirements of the grossest,
there appears to be no necessity for calling upon extraneous considerations
for assistance in our efforts to comprehend the various connotations of the
symbolism. Then, too, it is easily conceivable that under conditions where
these elements, spatiality, intellectuality and materiality, are not only co-
extensive but interpenetrative, there is no justification for the assumption
that they must exist in layers or manifoldnesses or in discrete degrees,
separated from one another as if they were constituted of different
substances and occupied different spheres. For every single point in
perceptual space is a focus for lines drawn through every conceivable grade
of materiality, spatiality or intellectuality in the kosmos. And the same
system of coördinates which is necessary and sufficient for the localization
of a point in our space is also sufficient for the location of a point anywhere
in the entire world of spatiality, intellectuality or spirituality. In fact, the
external, visible worlds of materiality and spatiality are nothing more than
the mass-termini of lines extending from divinity to physicality; from
primordial originality to kosmic modernity and it is intellectually
conceivable that progression back over the grooves made by these mass-
termini of lines would lead directly and unerringly to originality itself. In
spite of the manifold pyknoses which we have shown to characterize the

symbolism of space-genesis it is a very simple matter; for the entire scheme
could and must have proceeded along strictly tridimensional lines.
Tridimensionality must have inhered in the primeval archetype of space or
else it could not appear as an outstanding fact of perceptual space now; for
all that we can now observe in space as characteristics must have first been
included, enfolded, involved, before it could have been evolved. Hence, it is
to be remembered that we are to-day dealing with the expressions of
tendencies and principles which inhered in the manifested universe as
potentialities in the very beginning.
The alphabet of space-genesis consists of five characters, namely, the point,
the line, the triangle, the square and the circle. These are the
pentagrammaton of space, of intellectuality, materiality and of spirituality.
They constitute the basis of kosmic geometrism. With these all geometrical
figures may be constructed; with them all magnitudes may be delineated
and projected. They describe every conceivable activity of the Creative
Logos and designate the bounds of the entire scope of motility of
kosmogenesis.
In figure 19, are shown the dot, the line, triangle, square and the circle
which together form the kosmic pentoglyph. The point symbolizes kosmic
inertia, inactivity or the beginning of motion; the line is the first aspect of
motion, the beginning of creation; the triple aspect of kosmogenesis is
symbolized by the triangle, Chaos-Theos-Kosmos, the Unmanifest Trinity;
the square emblematizes kosmic being in evolution; while the circle is the
syncretism of all these and stands for the perfected kosmos, or the kosmos
in process of perfection. Very truly did Plato remark: "God geometrizes";
for the pentagrammaton—the point-line-triangle-square-circle—is the
deity's way of manifesting Himself. But there is here no need for space-
curvature nor for triangles whose value is greater or less than 180 degrees;
there is no need even for the mathematical fourth dimension.

Fig. 19.—Kosmic Pentoglyph

It cannot be believed, however, that metageometricians are really in earnest
in what they suggest of hyperspace and n-dimensionality; it cannot be
believed that they are entirely satisfied with what they have found of the so-
called hyperspatial, and yet, some of them are fanatically patriotic over the
new-found domain; some are even intolerant. But there are others who look
upon the fabric of metageometry as a stepping stone to space-realities, a
mile-post on the path to the realization of a higher consciousness, the
consciousness of the space-mind or kosmic consciousness. And may this
not, after all, be the goal of the human intellect, now slightly distraught by
the exuberances of youth and the joys of a new mental freedom? The work
of the future mathematicians will be the destruction of the tumorous
inconsistences to be found in the various non-Euclidean systems of
geometrical thought, the elimination of the novelties and the nonsensicals,
the synthesizing of those elements which are sanctioned by the space-mind
and the building thereon a sane interpretation of space-phenomena in the
light of illuminations received from greatly extended faculties and a
participation in that larger consciousness into which the human race seems
slowly to be immerging.

The domain of hyperspace is but the fairy-land of mathesis, peopled with
goblins, gnomes, kobolds, elves and fays which are the spaces, dimensions,
propositions, ensembles and theorems of the metageometrician. But like the
fairies and nature spirits of the unseen about us, they have their bases in the
real, objective world of facts however difficult it may be to establish their
direct connection with it. As the gay, invisible sprites of phantom-land
represent intelligent natural forces at work in the furtherance of the
evolution of forms, so the impalpable things of mathesis are emblems of
kosmic forces at work in the upbuilding of structures of higher
consciousness which shall be towers of vision for the human soul whence it
may view the hill-crests of infinite knowledge and the low-lying plains of
kosmic mysteries.
Finally, it has been noted that space is the very consistence of the kosmos; it
is the life, the form and both the outer and the inner manifestation of the
combined life and form; it is reality, also illusion; it is concrete, also ideal.
We have noted also that mind is consubstantial with space and that space
gives it its inner life and nature as well as nourishes its outer growth and
development. In fact, we have seen that space and mind are one essentially
and that they exist as aspects of the same thing, life. In whatever way, then,
that the mind normally views space that is the natural way. All attempts to
deviate from the natural way are, therefore, unsanctioned by the nature of
things. So long as geometry remains true to the nature of mind and space so
long will it be valid universally and possessed of kosmic necessity and
invariance. It behaves most unseemly when it departs from its fealty to the
nature of things per se.
Both the outlook of the mind upon the objective world as well as its inlook
upon its own states of consciousness or the subjective world are
tridimensional. Its growth is tridimensional, its nature is likewise
tridimensional, and there is not even the slightest tendence either to
perceive, conceive or perform in a four-dimensional manner,
mathematically speaking. Trace out the biologic development of each
mental faculty, from the mind of the moneron to the mind of the most
highly developed man and it will be found that everywhere and always,
without variation or exception, the nature of each of these has been to
express itself tridimensionally and naturally. There is not even the slightest
sign of so much as a germinal appetence for the four-space; it would,

therefore, seem almost a prostitution of mental faculty to divert mental
energy into the seemingly useless channel of present-day metageometrical
researches; yet, it must be admitted that even though the end sought cannot
be attained, the final results of the intellectual delvings into the dread
homogeneity of kosmic origins and the consequent realization of the
awesome coevalism of mind and space whence shall arise the recognition of
the wondrous unitariness of all existences, will be that we shall come upon
that thrice mysterious contrivance—the Heart of Divinity, the Kosmic
Space-Center in which abide the roots of the Great All in a marvelously
indescribable unity and infinite originality.
We may conclude, then, that hyperspatiality and all its appurtenances are
but the toys of the childhood of humanity. But, as the years pass and the
days of maturity come on apace, it, toy-like, will also be discarded. And the
mind will seize then upon the seriousness of reality just as the matured
youth responds to the stern realities of life and manhood responsibilities.
But no one can say that the toys of childhood are wholly useless; no one can
say that the joys which they bring are entirely fatuous and unreal nor shall
we attempt to intimate that mathetic contrivances are without utility,
without purpose and significance in the life of the growing mind of
humanity. But they, too, will pass away.

CHAPTER VIII

The Mystery of Space

The Thinker and the Ego—Increscent Automatism of the Intellect—
The Egopsyche and the Omnipsyche—Kosmic Order or Geometrism
—Life as Engendering Element—The Mystery of Space Stated—
Kathekos and Kathekotic Consciousness—Function of the Ideal—The
Path of Search for an Understanding of the Nature and Extent of Space
Must Proceed in an Inverse Direction.

The fragmentariness of the Thinker's outlook upon the universe of spatiality
is due to the inhibitive action set up by the constrictive bonds which his
complicate mechanism of intellectuality interposes between himself and
reality. The Thinker, who stands back of and uses the various media of
objective consciousness, such as the neural mechanisms, brain, emotions,
his individualized life-force and the mind which together make up the
instruments with which he contacts the sensuous domain, by adapting his
consciousness to these means, as the artisan utilizes his tools, constitutes his
own intellectuality. The intellectuality, then, is the totality of media by
which consciousness effects its entrance into the sensuous world and by
which it receives impressions therefrom. In other words, it is the sum of all
those qualities, operations, processes and mechanisms which are recognized
as constituting the modus vivendi of man's intellectuality, and these are, in
reality, nothing more than the ego himself.
Many have been inclined to regard that which has been called the ego as the
highest sovereign power in the state of manhood. He has been looked upon
as the final consideration in the constitution of the human being. But the
ego is an evolutionary product and the concomitant of self-consciousness
which is the I-making faculty in man's psychic life. It is that quality of
consciousness which makes man conceive of himself as a separate,

detached and independent being. It is a purely intellectual or tuitional
product, and, as such, is to be differentiated from the intuitional or life-
quality which is the essence of man's real selfhood. With respect to the
Thinker, the ego occupies precisely the same status as the agent to his
principal. As the agent is the representative of the principal in all matters
which come within the scope of his prescribed jurisdiction so is the ego the
agent of the Thinker who is a spiritual intelligence. Accordingly, from an
ethical viewpoint the Thinker is responsible for the acts of the agent and can
in no wise escape the penalties accruing as a result of the agent's violations.
Just as a commercial firm sends out a representative for the collection of
data concerning certain phases of its business or it may be of any business
or the entire world market so the Thinker projects his own consciousness
into the mechanisms which are in their totality the egoic life. That is, he
sends out his agent, the ego, into life and into the objective world of facts
and demands that he shall convey to him, from all points of the territory
which he is expected to cover, reports of his findings. Of course, these
reports which are transmitted by the ego (the intellectual mechanism of the
Thinker) are more or less well prepared summations of his individual
observations and deductions. These are the percepts which the ego presents
to the Thinker's consciousness. Concepts are formed by the Thinker in his
treatment of these sense-presentations. It very frequently happens that the
ego transmits reports which, for one reason or another, give very imperfect
knowledge of the matter which his reports are designed to cover. Often it is
necessary that additional and supplemental reports be made about the same
thing, and even then, it is well-nigh impossible, if not quite so, for him fully
to cover every detail of the matter under consideration and in no case is it
possible for him to do more than report on the superficialities of the
question under scrutiny. If the ego, in his operations, be imagined to be
hampered by similar circumstances and difficulties as those which would
ordinarily beset a commercial attaché it will then be clear that his reports
must ever be fragmentary because of the inaccessibility of much of the data
which would be necessary for a full report, and further, because of the
inadequacy of his methods and means of gathering data due to the inherent
limitations of his capabilities, endurance and perspicacity and innumerable
other limitations and difficulties which must be faced in all search for the
real. So that, while the sufficiency of the means which the ego enjoys at this
stage for all practical purposes is granted no hesitancy is entertained when it

comes to a discovery of the reals of knowledge in declaring their
insufficiency.
Then, too, when it is remembered that these egoic reports are in the nature
of neurographical communications which are similar to telegraphic
despatches and must pass through several stations, as ganglia, etc., often
being relayed from one to another, it will be quite apparent that much, even
of the original quality of the missives forwarded, will have been lost or
radically changed in some way before it is finally delivered for the
inspection of the Thinker himself. It not infrequently happens, even in
perfectly normal beings, that the ego in filing, recording, transcribing,
interpreting, translating and otherwise preparing these data for the Thinker's
use, lets a cog slip, misplaces some of the data, loses or destroys fragments
of it and so is unable to maintain a complete portfolio of his materials.
As the Thinker is entirely dependent upon his agent, the ego, for the
trustworthiness of his information covering the matter of the sensuous
world it is obvious that at best his information is very fragmentary indeed,
and necessarily so when it is considered that the modus operandi of his
agent and the difficulty of his operations are so complicate as to magnify
the obstructions in the way to complete freedom in this regard.
To continue the similitude of principal and agent it may be asserted that it is
also true that the commercial house that sends out its attaches frequently
will send letters containing directions as to procedure, sometimes censuring
for past delinquencies and sometimes commending for praiseworthy deeds;
and this, too, in addition to the original instructions which were given at the
outset. It even comes to pass that the home office, because of some
meretricious accomplishment, as the marked increase of efficiency shown
by the agent's close application to his duties and the consequent success of
his operations, confers certain favors upon the agent or removes some of the
restrictions which were originally imposed, gives an increase in salary or
promotes the agent to a higher and more lucrative office with larger powers
and greater authority. This is analogous to what the Thinker does for the
ego. For he not only receives reports from the ego, but often, in the shape of
intuitions, gives additional information as to the proper manner of doing
things, sheds more light upon some obscure operation, commends for duty
well performed, condemns for failures or for wrong-doing, rewards arduous

toil with greater powers of vision, keener insight, greater capabilities; in
fact, promotes the ego in the sight of other egos by marking him out as an
exceptional ego. But the curious aspect of this procedure is that, in time and
after the ego has been repeatedly commended and promoted and otherwise
favored by the Thinker, he begins to think that he owns the firm, that he is
the life and main support of the whole corporation. He becomes arrogant,
self-willed and finally falls into the illusion that he alone is responsible for
the phenomenal success of the firm. This is the source of that illusion of the
intellect which makes itself think that it, the ego, is all there is to man, that
his instruments of operation in the objective world are the only kind of
instruments that may be used; that his method of gathering data about
things is the only safe and sure method; and so it develops that the
intellectuality is the source of man's separateness, his individuality and his
apparent aloofness from other men and things. It is, of course, needless to
point out that in this way the intellect comes to be the tyrant of man, ruling
with a rigid monopoly and as an all-exclusive autocracy.
From the above implications it would appear that the intellect and the
intuitive faculty are two separate and distinct processes, and so they are.
One is the inverse of the other. The tendence of the egoic life or the intellect
is for the external while the intuition is an internal process. The intellect
acts from without towards the interior while the intuition acts from within
outward. The intellect is the product of the intuition which is another term
for the consciousness of the Thinker on his own plane. Just as the child
lives a separate and distinct, though dependent, life from the parents so the
intellect has a modus vivendi which is distinct and separate from that of the
Thinker, and yet it is in all points dependent upon the life of the Thinker.
Here again, we find an analogy in the relation of the child to the parent. As
some children are more amenable to the will of the parent than others, so, in
some persons, the intellect is more amenable to the action of the intuition
than in others. Yet it is a certain fact that the more the outward life is
governed by the intuition, i.e., the more the intellect responds to the
intuitive faculty of the Thinker, the higher the order of the life of the ego
and the more accurate his decisions and judgments. In fact, it assuredly may
be asserted that the place of every individual in the scale of evolution is
determined in a very large measure by the degree of agreement between the
intuition and the intellect or by the ease with which the intuition may

operate through the intellect as a medium. At least, the quality of one's life
may be determined directly by these considerations.
The Thinker being himself a pure spiritual intelligence, living upon the
plane of spirit and therefore unhampered by the difficulties which the ego
meets in his operations in the objective sensorium, and possessed of far
greater knowledge, is correspondingly free from the limitations of the ego
and very naturally closer to kosmic realities. Hence, he is better situated for
the procurement of correct notions of relations, essentialities and the like. It
is believed, therefore, that in the proportion that these two processes, the
intellectual and the intuitional, are brought, in the course of evolution, to a
closer and more rigid agreement, in the proportion that the Thinker is able
to transmit the intuitograms in the shape of concepts or that the intuition is
made more and more conceptual, in just that proportion is humanity
becoming perfect and its evolution complete. The difficulty found to inhere
in the conceptualization of intuitions so that they may be propagated from
man to man seems not to lie in the Thinker himself, but more essentially in
the ego, in the intellectuality and its complicate schematism or plan of
action. It would appear, therefore, that the only way of escaping or
transcending this difficulty is for the ego so to refine his vehicles or so
facilitate his plan of action by eliminating the numerous relays or sub-
stations intervening between the consciousness of the Thinker and that
which may be said to be his own that the transmission of intuitograms may
be accomplished with the greatest ease and clearness. While no attempt will
be made to indicate the probable line of action which the ego or objective
man will adopt for this purpose, it is believed that it may be said without
pedanticism that the only true method of attaining unto this much desired
state of things is, first of all, by assuming a sympathetic attitude not only
towards the question of the intuition itself but to all phenomena which are
an outgrowth of, or incident to, the manifestations of the intuitive faculty
through the intellectuality, and second, by the practice of prolonged abstract
thought, this latter procedure effecting a suspension of the intellectuality
temporarily at the same time allowing it to experience an undisturbed
contact with the intuitional consciousness, thereby laying the basis for
future recognition of its nature and quality. It would seem that these two
conditions are absolutely necessary in order that a more congruent
relationship may be promoted between these two cognitive faculties.

Ordinarily, it would appear that the philosopher who is undoubtedly inured
to the necessities of continuous abstraction or the mathematician whose
most common tasks naturally fall in this category would be among all men
most apt to develop to the point of conceptualizing intuitograms readily, yet
it seems that this is not the case. And there is good reason for it. The mind
of the philosopher and the mathematician is intellectual rather than
intuitional and is, therefore, wedded to matter, to the action and reaction of
matter against matter and hence operating in a direction at variance with the
trend of an intuitional mind. And this condition is undoubtedly due to a lack
of a sympathetic attitude towards this species of consciousness. At any rate,
it is thought that a too great anxiety in this respect need not be entertained
by humanity at all, for the reason that in the case of a faculty, the
rudimentary outcroppings of which are so marked and universally
observable and existing in greater or lesser degrees in various human
beings, there is ample evidence for the belief that it is being carefully and
duly promoted by a well-directed evolution of psychic faculties and powers,
so that at the proper time, determinable by the state of perfection reached by
the intellectuality or the ego in the operation of his cognitive processes, the
much desired agreement of these two faculties will have been realized and
the conceptualization of intuitograms into propagable conceptions an
accomplished fact. Until this goal shall have been reached and the intuition
shall have overshadowed the intellect as the intellect now overshadows the
intuition; or the consciousness of the ego, derived from the interplay of the
Thinker's consciousness among the various elements which constitute the
ego himself, shall have been merged with that of the Thinker, the outlook
must remain fragmentary, only becoming a well-ordered whole as the
barriers of dissidence are broken down in succession.
The evolution of consciousness, from the simple, undifferentiated moneron
to the differentiated cell and from that to the cell-colony and from the cell-
colony to the organism, traversing in successive paces through all the stages
of lower life—mineral, vegetable and animal—to the stages of the simple,
communal consciousness of the higher animals, to the self or individual
consciousness of the human being, each requiring millions of years for its
perfection before a more advanced stage is entered, has been one
continuous relinquishment of the lower and less complicate for the higher
and more complex expression of itself through the given media. When a

newer and higher stage of consciousness is being entered by humanity its
appearance or manifestation is first made in the most advanced of the race
and that only in a dim, vague way. This rudimentary condition persists for
some time, perhaps many thousands of years, then the faculty becomes
more general in appearance, the number of advanced individuals increases,
and consequently, as in the case of the intuitive faculty, it becomes
universally prevalent in all humanity; becomes transmissible as so-called
"acquired characters," and then appears as the normal faculty of the entire
human family cropping out in each individual. Thus, in passing from the
few advanced ones in the beginning to that stage where it becomes the
common possession of all, a faculty requires many thousands of years for
its perfection, and especially has this been true in the past history of the
development of human faculties. But it is believed that the sweep of the life
current as it proceeds from form to form, from faculty to faculty, gains in
momentum as it proceeds, so that in these latter years due to the already
highly developed vehicular mechanisms at its disposal not so great a period
of time as formerly is required for the out-bringing of a new faculty. It
might well be that while in the past hundreds of thousands of years were
necessary in the perfection of organs and faculties, in these latter days only
a few thousand, perhaps hundreds, may be necessary and that in the days of
the future not even so many years may be required to universalize a faculty.
And especially does this appear to be true in a state of affairs where so large
a number of persons are beginning consciously to take their evolution in
hand and by volitional activities are supplying greatly increased impetus to
their psychic processes which under ordinary, natural methods would be
considerably slower in their development. It is quite obvious that all
cultural efforts when applied to the betterment of a given plant, animal or
faculty result in a corresponding hastening of the process of growth far in
excess of what that growth would be under normal, natural conditions. All
the present faculties possessed by man are remarkably susceptible to
cultural influences; in fact, the standing edict of ethical and social law is
that the human faculties must be cultivated as highly as possible, thereby
giving the spirit a more perfect medium of expression. These observations,
therefore, lead irresistibly and unavoidably to the conclusion that the time
for the upspringing of the intuitional faculty in the human organism is even
now upon us, that undoubtedly in certain very advanced ones it has already

reached a notable degree of perfection and is rather more general than
would appear in the absence of careful investigation.
Now, just as the intellect has made for individuality, has emphasized the
separateness of the Thinker's existence from that of other thinkers, has
developed self-consciousness to a very high degree, even pushing it far over
into the domain of the higher consciousness to the temporary obscuration of
the latter, so the intuitional will make for union, for the brotherhood of man,
for co-operation and for the common weal. Through it man will come
gradually into the consciousness that fundamentally, in his inner nature, in
every respect of vital concern, he is at-one with his fellowmen and not only
with the apparent units of life but with all life as expressed in whatsoever
form throughout the universe. Then, too, he will be closer to the reality of
things, of actions and natural processes; in fine, he will have begun the
development of the space-mind which will bring him to the knowledge that
he is one with space also and, therefore, with the divine life of the world.
One of the peculiarities of the vital force which shows itself in the
consciousness as man's intellect, is its growing automatism, or that
tendency which enables the consciousness to perform its functions
automatically and thus allow opportunity for the development of newer and
higher faculties. Actions, oft repeated, tend to become automatic. This is
also true of thought and consciousness. It is one of the beneficent results of
abstract thought that it develops, or tends to develop, a kind of automatism
whereby a marked saving in time and energy is effected. This affords
opportunity for other things. It is undoubtedly true that in the days of the
truly primitive man his consciousness was more completely engaged in the
execution of the ergonic functions of cells, organs and tissues; that all those
processes which are now said to be involuntary and reflexive were at one
time, in the distant past of man's evolution, the results of conscious
volitions. This is a condition which must have preceded even the
development of the intellect itself. Indeed, there could be no intellect in a
state where the entire modicum of consciousness was being utilized in the
performance of cellular and histologic functions.
The rise of the intellect must have been in direct ratio to the development of
automatism among the cells, tissues and organs, so that as these came
gradually to perform their special labors reflexively the intellect began to be

formulated and to grow, at first only incipiently, then more and more
completely until it reached its present state. At the present stage of its
evolution, a great deal of the labor of the intellect is beginning to fall into a
kind of increscent automatism, although only rudimentarily, in many
instances. Yet, as a result of this tendency, quite the whole of the
phenomena of perception is characterized by a sort of automatic action. And
the mind perceives without conscious volition. Many of the steps of
conceptualization are automatic, in part, if not wholly. Certain it is that
impulses once set in operation whether consciously or unconsciously
continue to act along the same line until exhausted or until the end has been
attained. Consequently, it is a proven fact that often serious mathematical
and philosophic problems have been solved by the mind long after any
conscious effort to solve them had ceased. Often solutions have been
arrived at during sleep. Many such cases might be cited, but the
phenomenon is now so common that almost every one can cite some
experience in his own life that will substantiate the claim.
There is no doubt but that these phenomena are evidences of a reflexive
development in the intellect. The time will come undoubtedly, and
necessarily so if the intellect is to give way to a higher faculty, which shall
be as much above the intellect in its grasp of things as the intellect is now
above the simple consciousness of the lower animal, when quite the entirety
of our intellectual processes will become automatic or self-performing.
What then remains of the egoic schematism, after its transmutation or
elevation as the organ of the intuitional consciousness will be utilized as the
organ of the Thinker's involuntary cognitive processes. This will mean that
all of that laborious ideation which is now the abstract thought of the
Thinker will be performed automatically, leaving the higher aspect of the
egoic consciousness free to conceptualize or intuitograph the intuitions.
Perceptualization then will be replaced by conceptualization. This latter will
occupy about the same status as the former does now. And necessarily,
perception will become more complex. In other words, while we now
perceive simple percepts which are again arranged into concepts making a
composite picture of the object, we shall then be taking in the composite
picture of the object at first hand, thereby dispensing with the rather slow
process of perception as it now operates. We shall still be perceiving, but
what we perceive will be concepts rather than percepts, as at present.

The increased powers of intellection gained as a result of the increscent
automatism in the intellect, the flowering forth of the intuitive faculty and
the general enhancement of the intellect throughout all its processes will
enable it to entertain concepts or composite pictures of things just as readily
and as perfectly as it can at present deal with a single percept. Concepts will
be replaced by super-concepts or intuitographs. Increased perspicacity will
enable the Thinker to manipulate the concepts and intuitographs with the
same ease and readiness and withal the mind will have attained unto an
almost unrealizable freedom in its search after truth.
The outcome of this new adjustment which, of course, will not spring up at
once, but by insensible degrees, will be the clarification and unification of
our knowledge. It will mean also the simplification of it; the obviation of
diversities of opinions, the springing up of a new and winnowed system of
philosophy which shall be the true one; further, it will imply the lessening
of the probability of error in our judgments and conclusions; the removal of
illusion to a much larger degree than to-day is possible and the realization
by every one of something of the essence of things, of causes and effects, of
actions, operations, natural forces and laws; in fact, a condition of mind
which will present to the consciousness the simple truth above every
conceivable phase of kosmic life which may come within the scope of the
Thinker's observation.
The further implications of this view are that there is a difference between
the Thinker and the intellectuality. The Thinker is eternal and partakes,
therefore, of the very essence of primordial originality while the mentality
is an artificial process, the resultant of the adaptation of the Thinker's
consciousness to his vehicular contrivances of objective cognition and the
interplay of his life among them.
If the appearance of a choppy sea disturbed by the passage of a brisk breeze
over its surface be imagined, a similitude of the great ocean of life may be
envisaged. The wavelet crests symbolize the egos; the base of the wavelet
which is one with the great sea of water represents the Thinker which is one
with the divine life and consciousness of the kosmos. Just as wavelet crests
are continually springing up and falling back into the sea, so are egos
continually being cast forth and reabsorbed into the universality of life only
to be recast, as a wavelet crest or ego, upon the surface of the moving ocean

of life. And so, in this respect, the universum of life and consciousness
which are essentially one is in a constant state of ever-becoming, un-
becoming and re-becoming.
Another implication is that, on account of the diversity and complexity of
the means of contact with the external world, it is not possible for the ego to
arrive at more than a fragmentary understanding of even the latent
geometrism of life, mind and materiality. In our examination of the
sensuous world, we are very much like the three blind men set to examining
an elephant. One set to scrutinizing his trunk by means of his sense of
feeling. When asked for his judgment as to what the elephant was he
declared it was a snake; a second who began with the legs found it to be
like huge pillars; and a third who caught hold of the elephant's tail and
declared the elephant to be like a rope. Each one of the blind men described
what he was able to perceive. To each what he felt was all there was upon
which he could render judgment. And so, artists, philosophers,
mathematicians, musicians, mechanicians, religious seers, metaphysicians
and all other types of mind, are just so many blind men set to the
examination of an elephant, or the sensuous world. Each one confidently
believes his view to be correct; each one is satisfied with the deliveries of
his senses. Yet no one of them is wholly correct, no one of them has seen
every phase and aspect of the problem. Does it not, therefore, appear to be
the more reasonable and urgent that the view which synthesizes the
judgments of all the possible examiners thereby constructing a composite
idea of the entire mass of judgments is the more reliable and the more
correct?
Referring again to the dual intelligence, the ego and the Thinker, which
together constitute man, it is deemed necessary, in order to present the
concept of this duality to the mind of the reader in the way that shall enable
him easily to recall it, to designate the egoic intelligence as the egopsyche,
and the Thinker's intelligence as the omnipsyche.
The egopsyche is the I-making faculty, the faculty of self-consciousness and
the synthesis of all those psychic states and functions known as the intellect
or mind and includes the ethical aspect of man's nature. The omnipsyche is
the organism of kosmic consciousness, the space-mind, or man's higher self
and that which connects with or allies him to all life; it is the basis of

human unity and of unity with divinity, just as the egopsyche is the basis of
separation and individuality; it is the organ of direct and instantaneous
cognition and the permanent essence which has persisted through every
form which the being, man, has ever assumed and through every stage of
human evolution. In it are stored up the memories of the Thinker's past, the
secrets of life, mind, being, reality, and the history of life from the
beginning; in it also the plan of action for the future of the life-wave as it
passes from plane to plane, from stage to stage, and from form to form. It is
the spark from the flame that is never quite free from its source; it is the
continuous spark, the prolonged ray which does not go out and cannot be
extinguished. It is that in man which when full union therewith has been
attained makes him a god in full consciousness.
The omnipsyche is really a neglected and overlooked factor in the doctrine
of evolution. Evolutionists, while they claim life to be continuous and that
man has come through all the kingdoms of nature in succession and has
spent millions of years in the perfection of his various organs, faculties and
stages of consciousness, make no ample allowance for what is in reality the
basal element in evolution—a continuous, persisting, permanent life-force
which does not lose its identity from the beginning to the end of the
process. This fact—that that spark of life which set out upon the
evolutionary journey as a moneron has glowed steadily from that stage to
manhood, maintaining meantime its original purposiveness and intent—
seems to be the most obvious consideration of the whole doctrine, yet it has
been more or less completely ignored. The elementary requirements of
evolution would seem to establish clearly the necessity for some such
eternally persisting principle as the omnipsyche which is capable of such
subtle adaptations to every conceivable form of life and in which should be
gathered up the evolutionary results of every life-cycle. For this purpose the
omnipsyche or unifying principle in man was designed from the beginning
and it is that which constitutes the basis of his intellectual nature while in a
far larger sense it is the divinity in man himself. It is indeed strange that so
important a factor as the omnipsyche should have been omitted by
evolutionists. Yet it can be accounted for upon the grounds of the purely
mechanistic character of all intellectual attempts at solving the problems of
vital manifestations. But so long as men rely upon mechanical explanations
of such phenomena so long will they be prone to overlook the very

essentialities of the problems which they devoutly wish to solve. The
continuity of the physical germ-plasm of the human species,[27] now quite
generally admitted, would suggest, it seems, an analogous condition to the
continuity of the psychic plasm called the omnipsyche, the only difference
being that the omnipsyche is an intelligent factor while the physical plasm
is a medium of transmission though non-intelligent. The omnipsyche is,
therefore, the psychic reservoir of evolution into which are stored the
transmuted psychics of moneron, amoeba, jellyfish and every other form
which it has ensouled and acts as the storeroom of man's psychic operations
as well as the source of his intellectuality.
We turn now from the study of a sketch of the mechanism of man's
consciousness which gives at its best only a fragmentary view of the
universe of spatiality to a consideration of space itself in the light of its
interrelational bearings upon the question of intellectuality.
In the chapter on the "Genesis and Nature of Space" we have, in tracing out
the engenderment of space, proved it to be basically one with matter (and
indeed the progenitor of matter), also with life and consciousness. Further,
it has been shown that all the characteristics of materiality are due to the
adaptation of consciousness to it and that out of this adaptation grew the
intellectuality. A close approximation to this view was maintained by Kant
when he discovered that our faculty of thinking or the intellect only finds
again in matter the mathematical order or properties which our faculty of
perceiving or consciousness has deposed there. It appears, therefore, that
when the intellect approaches matter or spatiality it finds always a ready
yieldance to its demands simply because intellectuality has previously
established therein the delineation or map of the path over which it
necessarily must traverse in its examination of the object of its pursuit. In
other words, the kosmic mind in engendering materiality and spatiality has
set up therein a kosmic order or geometrism. Both motor and intellectual
progress, therefore, can be made through the world of spatiality because of
the immanence of this kosmic geometrism which lies latent in the very
fabric of the world of substance fashioning both the character and the nature
of the intellect as well as of space itself. So that there is a perfect congruity
subsisting between spatiality and intellectuality. Accordingly it is
impossible for either one or the other to transcend the grim grasp of the
mathematical order which binds them in such lasting and fundamental

agreement. Extra-spatiality may degrade itself into spatiality, and indeed in
the very nature of the case, does so degrade itself, yet spatiality can never
raise itself beyond the limits set by its engendering parent. Materiality may
become more and more spatialized and consciousness more and more
intellectualized, but they must proceed hand-in-hand one not superseding
the other.
Being the essence of the natural geometry which is everywhere immanent
in the universum of matter, space becomes an organized and ordered
extension, in fact is the totality of such organized and ordered extension,
which conforms to the latent geometrism the engenderment of which it is
the sole cause in the last analysis. Does it not appear then that all that mass
of artificial geometry which has sprung up as a result of departures made
from this natural geometry is utterly baseless and most certainly lacking in
the kosmic agreement which spatiality lends to our primary conceptions?
Of course, it is admittedly possible to devise certain conventional forms of
logic and endow them with all the evidences of a rigid consistency but
which, because of their purely artificial character, will fall far short of any
real conformity to the potential geometrism which has been established in
spatiality. And this fact is of utmost significance for all those who seek to
find justification either logically or naturally for the existence of a multi-
dimensional quality in space; for, if a clear, discriminative conception as to
the categorical relationship, each to each, of the two kinds of geometry be
carried in mind, it will not be easy to confound them neither will it be
difficult to discern where the one ends and the other begins.
Now, the fourth dimension and the entirety of those mathematical
speculations touching upon the question of hyperspace, dimensionality,
space-curvature and the manifoldness of space are purely conventional and
arbitrary contrivances and do not meet with any agreement or authority in
the native geometrism which we find inhering in space and which the
intellect recognizes there. This conclusion seems to be obvious for the
reason that, in the first place, the non-Euclidean geometries have been
constructed upon the basis of a negation of the latent geometrism of space
and intellectuality; and if so, is it reasonable to expect that either they or
any of their conclusions should accord with the nature of that form of
geometry so admirably delineated by Euclid? Obviously not. It is a matter
of historical knowledge that the whole of the artificial non-Euclidean

geometries consists of those purely conventional results which investigators
arrived at when they denied or controverted the norms supplied by the
natural geometry. When metageometricians found that they could neither
prove nor disprove the Euclidean parallel-postulate they then set upon the
examination of idealized constructions which negatived the postulate. The
results, thus obtained, although self-consistent enough, were compiled into
systems of geometry which naturally were at variance with each other and
with this inherent geometrism which is found in spatiality and answered to
by the intellect both normally and logically.
Furthermore, there is another consideration which to us seems to be equally
if not more forbidding, in its objections to the coördination of the two
systems of geometry, and that is the fact that the geometry of hyperspace is
denied the corroborative testimony of experience and this is true of
practically the whole of its data. Indeed, there is perhaps no single element
in its entire constitution which claims the authority of experience. This is
undoubtedly the weakest point in the structure of the hyperspatial
geometries. Contrarily, such is not the case with the natural geometry; for,
in this, the intellect in retracing its steps over the path laid out by that
movement which has at the same time created both the intellect and
spatiality, finds an orderly and commodious arrangement into which it
naturally and easily falls. So exact is this agreement of the intellect with the
kosmic order that if it were possible to remove the whole of spatiality and
materiality there would still be left the frame work which is this latent
geometrism of kosmogenesis. But the fact that the intellect naturally fills all
the interstices of materiality and spatiality, fitting snugly into all of them as
if molded for just that purpose, by no means warrants the assumption that it
would or does also fit the engendering factor which has created these
interstices. The frame work, the order or the geometrism of the kosmos has
been established by life acting consciously upon the universum of
materiality. And in order to establish this geometrism life had to be mobile,
active, creative. It could not remain static, immobile, and accomplish it.
Being mobile, dynamic, creative, it passes on. It is like a fashioning tool
which the cabinet makers use in cutting out designs upon a piece of wood.
It moves, and keeps moving until the design is finished, and then it is ready
for more designing. Life is like that. It cuts out the designs in materiality,
fashions the form, molds the material, and passes on to other forms. The

intellect fits into these designs gracefully. But what it finds is not life itself,
only the design which life has made. Hence, as there is neither an empirical
spatiality nor materiality in conformity with which the artificial geometry of
the analyst may be said to exist, and as it may not be said to conform to the
path which life has made in passing through either of these, it is absurd to
predicate it upon the same basis as the natural geometry. And so, we are
forced, in the light of these considerations to deny the validity and hence
the acceptability of the non-Euclidean geometries as either reasonable or
warrantable substitutes for the Euclidean, and denying which we also
formally ignore the claims of the fourth dimension, as mathematically
designed, to any legitimate anchorage in either our vital or intellectual
movements.
It has been shown that the flow of life, as it describes that movement which
we call evolution, engenders simultaneously and consubstantially spatiality,
materiality and intellectuality, and these, in turn, the natural order or
geometrism everywhere immanent in the universe; and that automatically,
one out of the other and each out of the all, these constitute the totality of
kosmic fundamentals. Also we have sketched the mechanism of man's
consciousness and discovered how, in its evolutionary development it has
divided into two aspects, the egopsychic and the omnipsychic, and these
two factors ally him definitely and adequately to the world of the senses and
to the world of supersensuous cognitions. And thus we have cleared up
some of the misconceptions which had to be confronted and made more
easy the approach to the central idea, thereby conserving the substantiating
influence which a general and more comprehensive view of the whole
would naturally give.
The totality of kosmic order is space. It is circumscribed by an orderless
envelope of chaos just as the germ of an egg is surrounded by the egg-
plasm. The organized kosmos is the germ, kernel or central, nucleated
mass, enduring in a state of becoming. Involutionary kathekos or
primordial chaos is the egg-plasm which nourishes the germ or the kosmos
and is that out of which the germ evolves. Kathekos or chaos is the
unmanifest, unorganized, unconditioned, unlimited and undifferentiated
plasm. Space is the manifest, limited, finite, organized germ that, feeding
upon the enveloping chaos, exists in a perpetual state of alternate

manifestation and non-manifestation—appearing, disappearing and
reappearing indefinitely.
The appearance of the kosmos as an orderly elaboration of the involutionary
phase of kosmogenesis, in so far as kosmic order may be said to be an
accomplished fact, marked the turning point in that procedure whose
function it was to make manifest a universe possessing certain definite
characteristics of orderliness; but the kosmos, as it now stands, may not be
thought of as having attained unto a state of ultimate orderliness. The idea
meant to be conveyed is that between the point of becoming and the
actually pyknosed, or solidified stage in the process of creation there is a
more or less well defined line of demarkation cutting off that which is
spatiality from that which is non-spatiality. Beyond the limits of spatiality is
an absence of geometric order. Here geometry breaks down, becomes
impotent, because it is an intellectual construction; at least, it is not so
apparent as in the manifested kosmos. It is a state about which it is utterly
futile to predicate anything; because no words can describe it. The most that
may be said is that it is absence of geometric order as it inheres in space.
And if so, all those movements comprehended under the general notions of
spatiality, materiality, intellectuality and geometricity have both their
extensive and detensive or inverse movements nullified in their approach to
it. Involutionary Kathekos, therefore, may be said to be the primordial
wilderness of disorder which outskirts the well laid-out and carefully
planned garden of the spatial universe. We may excogitate upon some of the
obvious functions of this kathekotic world-plasm; but in doing so we must
leave off all attempts at a description of its appearance, its magnitude,
extent or other qualities, and think only of its kosmic function. We cannot
say that there is back of it a spatiality nor can we say that it is a spatiality;
for whatever may be its extent or volume, it suffices that it may not be said
to be space. It is chaos. Space is order, organization, geometricity. It cannot
be said that there is a latent geometrism in chaos; because geometric order
is found only in spatiality and is that which distinguishes spatiality from
kathekosity or non-spatiality. Chaos is the lack of spatiality. This, of course,
implies that it is impenetrable to the intellectuality or to vitality. All inverse
movement such as is discovered as taking place in spatiality and which
results in the phenomenalization of space runs aground when it strikes
against the rock-bound coast of kathekosity. We can only say that it is both

the point of origin for the evolving universe of life and form and its
terminus. It is the nebulosity out of which the whole came and into which
all is ultimately occluded.
A great and far-reaching error is made in all our thinking with respect to the
kosmogonic processes when we postulate the complete absorption of chaos
as an early act of kosmogony. Customarily, we think of kosmic chaos as a
primordial condition whose existence was done away as soon as the
universe came into active manifestation. This because it has been
exceedingly difficult, if not quite impossible, for those whose privilege it
was to determine the trend of philosophic thought to free themselves from
the bondage of a dogma which owed its existence to a traditional or
legendary interpretation of facts that ought never have been so interpreted.
Chaos IS and EVER SHALL BE, so long as the universe itself lacks
completion, fullness or perfection in purpose, extent and possibility. It is
undoubtedly being diminished, however, in proportion as the kosmos is
approaching absolute perfection. And when the last vestige of chaos
disappears from the outerskirts of the maturing kosmos there shall appear a
glorified universe of indescribable qualities.
Space being a perception a priori cannot be determined wholly by purely
objective methods. The yard-stick, the telescope and the light-year are
objects which belong exclusively to the phenomenal and with them alone
never can we arrive at a true conception of the nature of space. We can no
more demonstrate the nature of space by the use of objective instruments
and movements than we can measure the spirit in a balance. Certainly, then,
it cannot be hoped that by taking the measurement of space-distances in
light-years, or studying the nature of material bodies, we shall be able to
fathom this most objectively incomprehensible and ineluctable thing which
we call space. It is such that every Thinker must, in his own inner
consciousness, come into the realization of that awfully mysterious
something which is the nature of space both as to existence and extent by
his own subjective efforts unaided, uncharted and alone. When we measure,
weigh, apportion and otherwise try to determine a thing we are dealing with
the phenomenal which is no more the thing itself than a shadow is the
object which casts it.

What does it matter that metageometricians shall be able to demonstrate
that space exhibits itself to the senses in a four- or n-dimensional manner?
Granting that they may be able to do this, if merely for the sake of the
discussion, when they have finished, it will not be space that they have
determined, but the phenomena of space, its arborescence, while space itself
remain indeterminate and unapproachable by phenomenal methods. If there
are curvature, manifoldness and n-dimensionality these are not properties of
space, but of intellectuality in its cultured state and when it is, therefore,
removed from the native state of conception. Scientists may be able to
weigh the human body, count every cell, name and describe every nerve,
muscle and fiber; they may even be able to know it in every conceivable
part and from every physical angle and relationship, and yet know nothing
of the life which vitalizes that body and makes it appear the phenomenal
thing that it is. So it is not by instruments which man may devise that we
shall be able to determine the true nature and purpose of space. We must
adopt other methods and means and assume other angles of approach than
the purely objective in order to comprehend space which, being the sole
inherent aspect of consciousness, can be understood best by applying the
measures which the latter provides for its understanding. It would appear,
therefore, that the best study of space is the consciousness itself, knowing
which we shall know space.
The universum of space, including the phenomenal universe, and its
relation to consciousness may be likened to a conical funnel whose base
represents the phenomenal world of the senses and whose apex or smallest
point represents ultimate reality.
In Figure 20, we have endeavored to symbolize graphically this conception
of space. The base marked "Sensorium" represents the sensible world. That
marked "Realism" symbolizes the ultimate plane of reality, the inner
essence of the world, the plane of "things-in-themselves."
The cone arising from the base "sensorium" symbolizes the objective world
as compared with consciousness; the subverted cone, with apex in the
sensorium, represents the evolving human consciousness.
The successive bases have the following symbology: Self-consciousness,
Communal Consciousness, Mikrocosmic Consciousness, Makrokosmic or

Universal Consciousness, the Plane of the Space-Mind Consciousness,
Divine Consciousness, Kathekotic Consciousness, or the Plane of Final
Union with the Manifest Logos.
Self-consciousness is that form of consciousness which enables the ego to
become aware of himself as distinguished from other selves or the Not-self;
the Omnipsychic or Communal Consciousness is that form of
consciousness from which arises the realization by the Thinker of his
oneness with all other thinkers and with other forms of life. Mikrocosmic
consciousness denotes a still higher form of consciousness, as that which
enables the Thinker to become conscious of his living identity with the life
of the world or the planet on which he lives. It represents a stage in the
expansion of consciousness when it becomes one with the consciousness of
the planet upon which it may be functioning. Makrokosmic consciousness
accomplishes the awareness of the Thinker's unity with the life of the
kosmos or universe. The space-mind and the consciousness which
constitutes it enable the Thinker to comprehend the originality and the
terminality of kosmic processes. It is archetypal so far as the life-cycle of
the universe is concerned because the beginning, the intermediate portion
and the ending of the kosmos are encompassed within it. Divine
consciousness is that form of consciousness which arises upon the
unification of the Thinker's consciousness with that of the manifest deity; it
is, in fact, omniscience. The kathekotic consciousness belongs to the
ultimate plane of reality; to kosmic origins and chaogeny, and therefore,
pertains to the plane of non-manifestation.

Fig. 20.—Kosmos and Conciousness

The implications are that in comparison with the sensorium, the Thinker's
consciousness is a mere point in space. It is, in reality, so small and
insignificant that the extensity of the physical world or universe seems

unlimited, unfathomable in meaning and infinite in extent. But as his
consciousness expands, as it passes, in evolutionary succession from one
plane of reality to another and higher one, the illimitability, the
incomprehensibility and infinity of the universe grow ever smaller and
smaller, until the plane of divine consciousness is reached. Then the
previously incomprehensible dwindles into insignificance, lost in the real
illimitability, infinity and unfathomability of consciousness itself. Kosmic
psychogenesis, as exhibited and specialized for the purposes of the
evolution of the Thinker, can have no other destiny than the flowering forth
as the ne plus ultra of manifestation which is nothing short of unification
with the highest form of consciousness existent in the kosmos.
It is not to be considered really that the scope of space is diminished but
that the growing, expanding consciousness of the Thinker will so reduce the
relative extension of it that illimitability will be swallowed up in its
extensity. Consciousness, in becoming infinite in comprehension,
annihilates the imaginary infinity of space. Accordingly, that which now
appears to be beyond mental encompassment undergoes a corresponding
diminution in every respect as the consciousness expands and becomes
more comprehensive. The mystery of space decreases as the scope of
consciousness increases. As the Thinker's consciousness expands the
extensity of the manifested universe decreases. Thus the mystery of every
aspect of kosmic life lessens, and fades away, as the intimacy of our
knowledge concerning it becomes more and more complete. There is no
mystery where knowledge is. Mysteriousness is a symbol of ignorance or
unconsciousness, and that which we do not understand acts as a Flaming
Sword keeping the way of the Temple of Reality lest ignorance break in and
despoil the treasures thereof.
Figure 21 is a graph showing a sectional view of consciousness on all
planes represented as seven concentric circles. This describes the analogous
enveilment of the consciousness when it ensouls a physical body or when
bound to the purely objective world of the senses. The overcoming of the
barriers of reality, represented by the circumscribing circles is the work of
the Thinker who is forever seeking to expand and to know. For only at its
center, as symbolized here, is the consciousness at one with the highest
aspect of kosmic consciousness and there alone is the mystery of space
despoiled of its habiliments.

Fig. 21.—Septenary Enveilment of Consciousness

Accordingly, as consciousness or the Thinker is more and more divested of
carnal barriers and illusions there develops a gradual recognition of the
unitariness of spatial extent and magnitude; there arises the certain
knowledge that space has but one dimension and that dimension is sheer
extension. The Thinker's sphere of awareness is represented as if it begins
as a point in space and develops into a line which divides into two lines, the
boundaries of the space cones. Thus it may be perceived that the ancients
had a similar conception in mind when they symbolized kosmogenesis with
the dot (.), the line, and the circle with diameter inscribed, which together
represent the universe in manifestation.
We realize the impossibility of adequately depicting the full significance of
the inverse ratio existing between the extensity of space and the increscent
inclusivity of consciousness by means of graphs; for neither words nor
diagrams can portray the scope and meaning of the conception in its
entirety. Yet they aid the intellect to grasp a ray of light, an intimation of
what the Thinker sees and understands interiorly.

In this connection it is interesting to note the function of the ideal in the
evolution and expansion of consciousness. The ideal has no perceptual
value; it has no status in the world of the senses. It is unapproachable either
in thought or action, and therefore, lies beyond the grasp of both the
intellectuality and the vitality. It is indescribable, inconceptible and
searchless; for the moment that we describe, define, or approach the ideal,
either intellectually or vitally, in that moment it ceases to be ideal, but
actual. It flees from even the slightest approach; it never remains the same;
it cannot be attained, at least its attainment causes it to lose its idealty. It is
then no longer the ideal. It is like an ignis fatuus; the closer we come to it
the farther away it recedes. It hangs suspended before the mind like the
luscious grapes which hung before the mouth of the hungry Tantalus. As the
grapes and the water receded from his reach at every effort he made to seize
them so the ideal remains eternally unseizable and unattainable. Whatever,
therefore, is in our thought processes, or in our knowledge, that may be said
to be ideal, does not really exist. The ideal is a phantom growing out of the
nature and essence of the intellectuality. Its purpose is to lead merely the
mind on; to allure it, to tantalize, and compel it to grow by exertion, by the
struggle to attain, by the desire to overcome. In this respect, it serves well
its function in the economy of intellectual evolution. It is a mysterious
aspect of the original and eternal desire to live which is the kosmic urge
present in all organized being and has its roots hidden in the divine purpose
of creation.
Idealized constructions, then, are like Arabian feasts conjured up by a
famishing mentality. They are like the dreams of a starving man in which he
actualizes in phantom-stuff the choicest viands in abundant supply for his
imaginary delectation. The mind that is satisfied never idealizes, never
makes an idealized construction. It is only when an "aching void" is felt,
when a longing for the realization of that which it has not arises within
itself, when a feeling of distinct lack, a want, a hankering after something
not in its reach, takes possession of the mind that it begins to idealize. That
is why some minds are without ideals. It is because they are satisfied with
what they have and can understand. They feel no hungering for better and
grander things; they have no desire to understand the unknown and the
mysterious; hence they do not idealize; they make no attempt to represent

unto themselves a picture of that which is beyond them. Such minds are
dormant, hibernant, asleep, unfeeling and unresponsive to the divine urge.
But the ideal is neither obtainable objectively nor subjectively, neither
phenomenally nor really, so that when we come upon the ideal in our mode
of thinking we have arrived not at a finality or end, but at that which is
designed to lead us on to something higher, to nobler accomplishments and
more extensive conquests. When we have devised idealized constructions,
therefore, we should not therewith be content but should scrutinize them,
examine and study them for their implications; for thereby we may discover
the path and the guide-posts to a new domain, a new ideal, following which
we shall, in time, come to a point in our search for the real where the
fluxional is at a minimum; we shall reach that something which will admit
of no further struggle—the last chasm between the phenomenal and the real
—and standing on the bridge, consciousness, which engages the twain, shall
have a complete view of that Sacred and Imperishable Land of Kosmic
Realism where like a fleeting cloud of sheerest vapor shall be seen the
phantom-ideal deliquescing and disappearing in the cold, thin air of the real
and the eternal.
Since space is judged to be infinite by the intellect occluded in such clouds
of illusion and hampered by such constrictive bonds of limitations, as it
now endures, we have no right to conclude that the concept of infinity
would still linger before the mind's eyes when the illusionary veil is
removed; in fact, there is ample reason to believe, nay for the assertion, that
the recession of the veil will reveal just the opposite of this illusion, namely
that space is finite, and even bounded by the fringe of chaogenetic
disorderliness. Either we perceive the real or we do not; either the pure
thingness of all objects can be perceived or it cannot be perceived. If not,
granting that there is such a thing as the real, it must be within the ultimate
range of conceivability. It also seems reasonable that realism exists
somewhere, and if so, must be sought in a direction inverse to that in which
we find the phenomenal and the approach thereto must necessarily be
gradual, continuous and direct and not by abrupt breaks, by twists and turns.
The phenomenal must lie at the terminus of the real, and vice versa. So that
by retracing the path blazed out by the real in coming to phenomenalization
we shall perhaps find that which casts our shadowy world, just as by tracing

a shadow in a direction inverse to that in which it extends we may find the
object which projects it.
It is not out and beyond that we shall find the end of space; it is not by
counting tens of thousands of light-years that the supposed limits of space
shall be attained. The path of search must project in an opposite direction—
not star-ward but Thinker-ward, toward the subtle habitation of the
consciousness itself. We err greatly when we think that by measuring
distances we shall encompass space; for that which we measure and
determine is but the clouds caused by the vapor of reality. It is, therefore,
not without, but within, in an inverse direction that the search must proceed.
Going back over the life-stream, beginning where it strikes against the
shores of solid objectivity, deeper and deeper still, past the innermost mile-
stone of the self-consciousness, back into the very heart of the
imperturbable interior of being where the Thinker's castle opens its doors to
the Great Kosmic Self, from that open door-way we may step out into that
great mystery of space—limited, yet not limited, multi-dimensioned, and
yet having only one dimension, veritably real and fundamental, the Father-
Mother of all phenomena. Here the great mystery of mysteries is revealed
as the citadel of the universal and the ultimate real. In this citadel, the plane
of kosmic consciousness, space loses its spaciousness and time its
timeliness, diversity its multiplicity and oneness alone reigns supreme.
But the movement towards the center and circumference of space, after this
manner, requires aid neither from the notion of space-curvature nor that of
the space-manifold, except, indeed, only in so far as a state of
consciousness or a degree of realism may be said to be a tridimensional
manifold. The feeling that space is single-pointed, and yet ubiquitously
centered, has been indulged by mathematicians and others in a more or less
modified form; but they have imagined it in the terms of an indefinite
proceeding outward until in some manner unaccountable alike to all we
come back to the point of origin. It has been expressed by Pickering when
he says that if we go far enough east we shall arrive at the west; far enough
north we shall come to the south; far enough into the zenith we shall come
to the nadir. But this conception is based upon a notion of space which is
the exclusive result of mathematical determinations and subject to all the
restrictions of mathetic rigorousness. It requires that we shall allow space to
be curved. This we decline to do for the reason that it is both unnecessary

and contrary to the most fundamental affirmations of the a priori faculty of
the Thinker's cognitive apparatus. It would seem to be necessary only that
we should extend our consciousness backward, revert it into the direction
whence life came to find that which we seek. By extension of consciousness
is meant the ability to function consciously upon the various superkosmic
spheres or planes just as we do on the physical. Yet it should be quite as
easy to devise an idealized construction which would imagify the results of
this ingressive movement of the consciousness as to represent the results of
a progressive outward movement star-ward. Having done so the
examination of them could be conducted along lines similar to those
followed in the scrutiny of objective results.
What would it mean to the Thinker if he were able to identify his
consciousness with the ether in all its varying degrees; what would it mean
if he were able to identify his consciousness with life and with the pure
mind-matter of the kosmos; and lastly, with the spiritual essence of the
universe? What if his various vehicles of awareness were available for his
purposes of cognition? What, indeed, if he could traverse consciously the
entire gamut of realism and consciousness from man to the divine
consciousness? Does it not appear reasonable that as he assumed each of
these various vestures of consciousness, in succession, he would gradually
and finally, come to a full understanding of reality itself? It seems so. This
view is even more cogent when it is considered that the limitations, and
consequent obscuration of consciousness are proportional to the number of
vehicles or barriers through which the Thinker is required to act in
contacting the phenomenal universe. Common sense suggests that freedom
of motility is determined by the presence or absence (more particularly the
latter) of bonds and barriers; that the less the number of such barriers the
greater the scope of motility and consequently greater the knowledge.
Plato evidently had this in mind when he imagined the life of men spent in
cave-walled prisons in which their bodies were so fixed that they were
compelled to sit in one prescribed position, and therefore, be unable to see
anything except the shadows of persons or objects as they passed by. He
conceived that men thus conditioned would, in time, suffer the diminution
of their scope of consciousness to such an extent as to reduce it to
identification with the shadows on the walls. Their consciousness would be
mere shadow-consciousnesses the entire data of which would be

shadowgraphs. So that for them the only reality would be the shadows
which they constantly saw. A similar thing really happens to man's
consciousness limited to the plane of the objective world. Things which are
not objective do not appear as real to him, if they do appear at all. It is not
that there are no other realities than those which appear to the egopsychic
consciousness or that fall within its scope; but that this form of
consciousness is incompetent to judge of the nature and appearance of those
realities which do not answer to the limitations under which it exists. And
so, with men whose data of consciousness or whose outlook upon the world
of facts, or rather life, are confined to the narrow bounds of mathematic
rigor and exclusiveness, there may appear to exist no realities which may
not be defined in the terms of mathematics. Similarly, to the empiricist,
used to measurements of magnitudes, weights, and rates of motion, there
may also appear to be no realities which are not amenable to the mold of his
empirical contrivances—the balance, the chromatic and the scalpel. All of
these are shadow men constricted to the metes and bounds of shadows
which they observe only because they are ignorant of the realities which lie
without their plane.
Life has so many ways of exhibiting its remains to the intellect; and these
remains have so many facets or viewpoints from which they may be
studied, that nothing short of a panoramic view of all the modes of
exhibition and of all the facets and angles of appearances will suffice to
present a trustworthy and comprehensive view of the whole. Then, life itself
is so illusive, so unseizable by the intellect that the testimony of all
investigators are required to summarize its modes of appearance. And,
therefore, eventual contentment shall be secured only when the mass of
diverse testimonies is reduced to the lowest common divisor, and for this
purpose the operations of every class of investigators must be viewed as the
work of specialists upon separate phases, facets and angles of life's remains.
And so it is manifestly absurd for the empiricists, by taking note of the
dimension, extent, quality and character of the shadows, or one single class
of angles, to hope to predicate any trustworthy judgments about either the
realities which cast the shadows or underlie the angles; because whatever
notion or conception they may be able to gain must of necessity be merely
fragmentary and entirely inadequate. Despite this fact, however, we still
have the spectacle of men who, studying the sensible universum of space-

content, endeavor either to make it appear as a finality in itself, or that the
world of the real must necessarily be conformable to the precise standards
which they arbitrarily set up in their examination of the objective world. It
can be said with assurance that we shall never be able even so much as to
approach a true understanding of the unseen, real world until we shall have
changed our mental attitude towards it and ceased to expect that it shall
necessarily be fashioned and ordered in exactly the same way as the world
of our senses, or that it shall be understood by applying the same methods
of procedure as those which we use in our examination of the phenomenal,
sensuous world. It is a matter of logical necessity that, as there are no
senses which can respond to the real, as there are no organs which vibrate in
accord with the rates of vibration of the real, there can be no reasonable
hope of understanding it by means of sensuous contrivances and standards.
Let the consciousness, therefore, be turned not outward, but inward where
is situated the temple of divine life; let there be taken away the outward
sheaths which enshrine the pure intelligence of the Thinker; let him grow
and expand his sphere of awareness; let there be an exploration of the
abysmal deeps of mind, of life and consciousness; for buried deeply in
man's own inner nature is the answer to all queries which may vex his
impuissant intellectuality.

CHAPTER IX

Metageometrical Near-Truths

Realism is Psychological and Vital—The Impermanence of Facts—On
the Tendency of the Intellect to Fragmentate—The Intellect and Logic
—The Passage of Space, the Kosmometer and Zoometer, Instruments
for the Measurement of the Passage of Space and the Flow of Life—
The Disposal of Life and the Power to Create—Space a Dynamic,
Creative Process—Numbers and Kosmogony—The Kosmic
Significance of the Circle and the Pi-Proportion—Mechanical
Tendence of the Intellect and its Inaptitude for the Understanding of
Life—The Criterion of Truth.

Kosmic truth has many facets. The rays of light which we see darting from
its surface do not always come from the core. Often they are reflections of
rays whose light stops short at the superfice; and these, in turn, are
reflections of deeper realities. Thus the reflected light may be traced to its
source by following the lead of external reflections. It is now known that
moonlight, and perhaps, in many cases, starlight, are reflections of sunlight,
if not of our sun, some other in the universe. But it is only at certain times
and under certain conditions that we can see the sun which is the source of
the other kinds of light. The stars which owe their light to suns are so many
facets of sunlight. The moon is a facet of sunlight also. Facts are facets of
truth. They are so many faces of eternal truth. They represent the many
ways reality exhibits itself, or rather its effects, to the consciousness. When
we, therefore, become aware of facts we have not in virtue thereof become
aware of the reality which produces the facts. We have come to know only
something of the termini of realism while the complexities and internal
ramifications which lie between realism itself and these termini altogether
elude our cognition.

Let us examine briefly an icosahedron, for instance. An icosahedron is a
figure comprehended under twenty equal sides. These various sides are so
many faces by means of which the figure presents itself to the
consciousness. These faces, however, are not the real object. The figure
may be examined by viewing it from any one of its sides; yet, by simply
examining a single face, or any number of faces, less the total number, we
arrive at no satisfactory knowledge of the magnitude or its substance. We
must first become conscious of all the faces, holding them in mind as a
composite picture, before we can even begin to have anything like a
complete notion of the icosahedron. Then by continuing the examination we
may find that the magnitude is composed of wood fiber or stone or metal, as
the case may be. In this way we might carry the examination to indefinite
limits and finally arrive at a very comprehensive knowledge of the
icosahedron and yet be unaware altogether of the forces which have been at
work in the production of the magnitude or of the reality which lies back of
it.
Realism is psychological and vital. In essence it is mind, spirit, life. Yet
these three are one. Mind is the outward vehicle of life; spirit is the form or
the interior vehicle which life assumes in order to express itself. Realism,
then, is life. Is the logician dealing with reality when he collects and
coördinates the various modes of interpretation by which we learn to
understand the symbolism of life? Obviously not. The data of logic are
simply a collection of rules for interpreting concepts. It is a compendium of
indices for the Book of Life. It is no more the book itself than a table of
contents is a book. But logic occupies about the same category as does an
index to a volume. A book, however, is more than its conventional contents.
It is the thought that is symbolized therein. The book of life, accordingly, is
the sum total of life's expressions; but it is not life itself. That is the subtle,
evasive something which the contents of the book of life symbolize. Nature,
both in her palpable and impalpable aspects, may be said to be the book of
life wherein are recorded the movements, the expressions, and the diacritics
of life. The whole is a magnitude of many facets (little faces). We shall have
to know all the faces before we can say that we have a comprehensive
knowledge of nature. For so long as we have only a fragmentary knowledge
of the whole, so long even as we have merely a superficial knowledge of
any aspect of nature, just so long will our knowledge be in vain. Just as it

frequently happens that, on account of the partial view of things, we are led
to make incorrect judgments concerning them, so when we come to make
assertions about life or nature in general, we are apt to fall into the error of
rendering judgments upon insufficient data. And it is not at all likely that
judgments thus arrived at can possess true validity because it may happen,
and undoubtedly does always so happen under the present limitations of
human knowledge, that the very elements which are ignored or neglected in
forming a judgment possess enough of virtue to alter the intrinsic value of
determinations based upon otherwise insufficient data. Hence it develops
not infrequently that our judgments repeatedly have to be changed in
proportion as our data are made more and more comprehensive. Men
searching eagerly for the truth sometimes allow themselves to be carried
away by the enthusiasm of the moment which arises upon the discovery of a
new facet of truth; but if all searchers were to bear in mind the fact that
reality presents itself to consciousness in myriad ways and that there are
innumerable facets all leading eventually back to the source of all they not
so easily would be induced to jump to the conclusion that they had covered
the entire ground. For when we have discovered a million facts, or many
millions of them, about nature we may say that we have only merely begun
and that what we have found is not to be compared with the totality even of
the directly observable phases of nature.
Logic, therefore, deals with the symbolism existing between and among
facets of truth, and not directly with truth itself, although the conclusions
reached by the logicians may be true enough from an intrinsic standpoint.
Logic is not truth, however; it is merely the consistence of relations and
inter-relations between facts and among groups of facts. Truth is not
established by logic; it stands in no need of the light of logic for its
revelation; indeed, more apt than not is logic to obscure truth. Truth is its
own proof; it is self-evident. Logic is a mere modeler of facts; it is static,
immobile, fixed. All truly logical processes need a starting point, a
foundation, a premise, a base. Truth, being eternal, mobile, dynamic, vital,
needs no starting point; needs no foundation because it is itself
fundamental; it requires no premise because no premise is comprehensive
enough to encompass it. There is only one way of arriving at truth and that
is not to arrive at all—just to recognize it without procedure. The fact that
facts are, and the fact of their relations and inter-relations, their sequence

and implications, can be arrived at only by logical processes. Life, in its
passage through the universum of spatiality, carefully diacriticizes between
the realm of facts and the domain of truth, marking each off from the other
by unmistakable signs and barriers. Truth is perceived as an axiomatic, self-
evident principle and no amount of logic could prove or establish its verity.
Facts are intellectual creatures; truth is intuitional, vital. The intellect
conceives the consistence of facts while the intuition recognizes truth—is
truth, and therefore, follows in the wake of life as consciousness.
There is no permanence in facts and the intellectual recognition of their
consistence. The discovery of a single new fact may destroy the consistence
of a whole mass of previously correlated facts. Thus is revealed the miracle
power of logic over facts. It can take a mass of facts, related or unrelated,
mold them into hypotheses, endow them with a sort of interior consistency,
and make these hypotheses take the posture of truth. Hence logic is often an
effective mask which the intellect commonly imposes upon its material; but
it does so instinctively and can no more escape the rigorous compulsion of
this instinctive functioning than water can escape its liquidity. Wherefore,
we conclude that true permanence abides alone in truth because truth is
duration itself. For the foundations of the whole structure of facts in
religion, science, art and philosophy which man has toilfully built up in the
last million years might easily be destroyed or overturned by the discovery
of some great fact or by appreciating the true value of truth. Let us suppose
it should suddenly be realized by men that they are really and truly gods
capable of creating and possessing all the other virtues, powers and
capabilities which we are accustomed to impute to supreme divinity; and
suppose that the fact of their omnipotence and divine omniscience always
had been obvious but that men were so engaged upon details and the non-
essentials of life and matter that they had not noticed nor realized it before,
would not this realization make a vast difference in the character of our
knowledge and the attitude which we would necessarily assume thereafter
towards matter, life and the problems which they present? Would not it
completely revolutionize our arts, our sciences and our philosophies? How
much, then, of the facts of these would be left when the light of
omniscience had been turned on—when truth itself could be perceived and
interiorly realized? Not much, to be sure. We should undoubtedly have to
dispense with the entirety of our fact-mass, for it should then be entirely

useless and meaningless in the light of the resplendent omniscience of truth.
As at present constituted consciousness is focused upon the material plane
for the purposes of superficial observances. But if the focus of
consciousness should be changed so as to reveal conditions upon what must
be a higher and more interior level, the aspect of things would be entirely
changed and the whole of our theory of knowledge would have to be
reconstituted. It is conceivable, yea obvious, that the stern reality of being is
far removed into the Great Interior of that which is; and there is a point in
the path to the interiority of being where there is no illusion, no appearance,
indeed, nothing but the cold, illuminating body of reality itself. It must
appear also that along the journey interior-ward there are many apparent
levels or planes, each of which requires a new focus. It is unreasonable,
then, to suppose that the conclusions arrived at as a result of purely logical
processes, confined to the lowest levels of reality, are pertinent and valid for
the entirety of realism which is neither of mathematical nor logical import.
For instance, if we take the purely axiomatic assertion: x equals x, the
intellect is at once certain that this is so, and cannot be otherwise, and yet a
proposition of this kind is purely conceptual, conventional and arbitrary. x
may also equal 1, 2, 3, 4, or any other quantity. Then, if each x in the above
equation be replaced, by say, a horse, there immediately arises a difficulty.
For it is not possible to find two horses which are in all respects mutually
equal. So that as soon as we pass from the conceptual into the actual,
whether on the side of objective reality or that of absolute reality, the
validity of the axiom is immediately exposed to serious questioning. The
truth of the matter is that on both sides of the conceptual it is always found
that there is a variance from the standards set up by the conceptual, this
variance being more marked on the side nearest to absolute reality than on
the side of objectivism. Objectively, the conformity of the sensible with the
conceptual is of such approximation as to lend trustworthy utility to the
conceptual in its application to the sensuous. Thus by simply eliminating
the vital factors from our equations we are enabled to proceed in a
reasonably safe manner with our judgments. Really, however, no such
approximate congruence can be found; for on the side of reality we are
dealing with an indivisible something—something that is eternally and
absolutely unitary in its constitution while when we transfer the scene of
our observations to the objective world we discover a contrary situation.
Here we are everywhere beset by diversities, multiplicities and

dissimilarities. This is so because the intellect naturally tends toward the
objective where it finds a most comfortable atmosphere for its operations.
The conceptual is related to the objective as a train of cars is related to the
railway. That is to say, the constitution of the intellect is such that it finds its
most facile expression in the objective world and is about as comfortable in
the domain of realism as the same train of cars would be on the ocean.
The intellectuality is designed to deal with facets of truth; it is made to
manipulate segments, parts, fractions, and cannot chart its way through a
continuum such as reality. Being constitutionally a conventionality of the
Thinker's own contrivance, and arising out of the subtle adaptation of his
vehicles to the environment afforded by the sensuous world, it can only find
congruence in that conventionality which is the instrumentality of a higher
intellectuality expressed in a diversity of forms, into which reality divides
itself for manifestation. The human intellect is, therefore, the bridge over
which is made the passage from the individual consciousness to the All-
Consciousness; simultaneously, the medium whereby the physics of the
brain are converted into the psychics of unconsciousness. It may be likened
to a pair of specially constructed tongs which are so formed as to fit exactly
the objects which a higher intellectuality has made. It is without the
province of the intellect to take note of what intervenes between physics
and psychics; it is always oblivious of interstices while taking cognizance
of objects or things. In this respect, the intellect is much like a steerage
passenger on board an ocean liner who sees only his port of departure and
port of arrival, knowing nothing in the meantime of what happens during
the voyage, nothing of what the other passengers on the upper decks may
experience and taking no part in any of the passing show until he lands. So
that the passage of the intellect from fact to fact is an altogether
uninteresting voyage; it may as well be made unconsciously, and to all
intents and purposes, is so made.
Accordingly, the advocates of n-dimensionality find it quite impossible to
predicate anything whatsoever of the passage, say, from tridimensionality to
quartodimensionality. They find themselves at ease in tridimensionality and
have even contrived to find pleasant environs in the four-space having made
therein such idealized constructions as will afford ample hospitality to the
intellect. But the questions as to how the passage from the three-space to
the four-space is to be made and how the intellect shall demean itself during

the passage have been completely ignored and, therefore, left unanswered.
What, then, shall be said of an explorer who says he has found a new land
and yet can give no intimation as to how one may proceed to arrive at the
new land, what changes are to be made en route, nor the slightest
suggestion as to the direction one should take in setting out for it? It is not
likely that the report of such an explorer, in practical life, would be taken
seriously; and yet, there are those who, relying utterly upon similar reports
made by certain enthusiastic analysts, dare to place credence in their
asseverations. Not only have they given wide credence to these reports, but
have, indeed, sought to rehabilitate their own territory in accordance with
the strange descriptions given by unhappy analytical explorers. Now the
question of greatest concern, granting for the nonce that there is such a
domain as hyperspace, is the passage. How shall we make the passage? Or,
is the passage possible? In vain do we interrogate the analyst; for he does
not know, nor does he confess to know. Evidently it is impossible for him to
know by means of the intellect alone; for the intellect not being fitted to
take cognizance of the "passage," but only the starts and stops, has no
aptitude for such questions. Hence, what seems to be the most important
phase of the entire question will have to remain utterly inscrutable until the
intellect nourished by the intuition shall be aroused from its lethargy and
brought to a certain high point of illumination where it, too, may take note
of the passage.
Space is the path which life makes in its downward sweep through all the
stages of pyknosis or kosmic condensation by virtue of which it
accomplishes the engenderment of materiality as also the path marked out
by it in its upward swing whereby it accomplishes the spiritualization of
matter. It is the kosmic order which life establishes by means of its
outgoings and incomings. When we look out into space we perceive that
which is a dynamic appearance of life itself, and not a pure form. Nothing
that is a pure form can exist in nature and in as much as space is not only
indissoluble from nature but partakes of its very essence it cannot be said to
be a pure form. The intellect, however, prone to follow the grooves laid out
by pure logic, never fails to seek to make everything that it contacts
conform to these logical necessities. But, if the analyst were to make careful
discrimination as to the respective categories—that into which life falls and
that in which the intellect is forced by its nature to proceed—he not so

easily would be led into the fault of attempting to shape realities upon
models which being strictly conventional were not meant for such uses. But
neither the logician nor the mathematician can be condemned for such
generosity if such condemnation were justifiable. For they everywhere and
at all times insist upon realizing abstractions and abstractionizing realities,
and they do this with an insouciance that is at times surprising. Yet it is in
this very vagary that is discovered the true nature of the intellect. There is a
sort of dual tendence observed in the method of the intellect's operation. A
polarity is maintained throughout: the abstractive and the concretional. It
vacillates continually between the abstract and the concrete and no sooner
has it found a concrete than it begins to set up an abstract for it; and vice
versa—as soon as it is has constructed an abstract it immediately seeks
either its concrete or sets out to hew some other concrete into such shape as
will fit it. And between these two extremes numerous excuses are found for
exercising this peculiar characteristic, and that too, without regard to
consequences. It would seem that the intellect, in thus functioning, was
really engaged at a sort of sensuous play out of which it derived an intense
and not altogether unselfish pleasure.
Of course, it must be granted that diversity has its specific and withal
necessary uses in that it affords a field for the operation of human
intellectuality and represents the adaptation of the kosmic intellect to the
human for the purposes of evolution. This adaptation while necessary for
the intellectual development is, however, not an end in itself. It is merely a
means to a higher purpose. In fact, if we regard materiality as a deposit of
life, carried by it as a kind of impedimentum, and consciousness, which is
life, as being identical with the intellectuality which makes these
adaptations, there should be no grounds for the statement that the one is
adaptable to the other at all. And as this is really the view which we assume
it would perhaps be more strict to regard the adaptation as subsisting
between the human intellect and materiality both of which having been
constructed by kosmic intellectuality. Pursuant to the diversity of uses to
which materiality lends itself there arises in the intellect a supreme
tendency to segment, to break up into separate parts, to multiply and
diversify. It is not content unless it is at this favorite and natural pastime. It
delights in taking a whole and dividing it into innumerable parts. This it will
do again and again; because all its muscles, sinews and nerves are molded

in that mold and can no more cease in their tendency to fragmentation than
can the muscles of a dancing mouse cease in their circular twirling of the
mouse's body. Yet, in this it is but creating a well-nigh endless task for itself
—which task must be performed to the uttermost. But in its performance,
that is, in the intellect's complete understanding of the diversity of parts, in
the knowledge of their relations and inter-relations and in their synthesis, it
may arrive at that one ineluctable something which is called unity. And so
doing, become ultimately free.
In view of the foregoing, it is not surprising that the intellect should have,
finally, fallen upon the notion of n-dimensionality. It has come to that as
naturally as it has performed its most common task. Left alone and
unhampered in its movements, it has simply followed the lead of the Great
Highway through the domain of materiality. And now it has arrived at a
stage where it thinks it has succeeded in fractionalizing space. Time has
long ago yielded to fragmentation, been divided into minute parts and each
part carefully measured. Space, not having a visible indicator like time to
denote its passage or parts, suffered a long and tedious delay before it could
boast of a measurer. As the sun-dial measured time in the past and became
the forerunner of the modern clock so n-dimensionality measures space for
the mathematician. What more practical instrument for this purpose may yet
be devised is not ours to prophesy; yet it is not to be despaired of that some
one shall find a suitable means for this purpose. Seriously, however, it is not
without possibility that should some subtle mind devise an instrument for
marking the passage of space as we have for denoting the passage of time a
great stride forward would be accomplished in the evolution of the human
intellect. For the general outcome of the intellect's attention being turned to
the passage of space would undoubtedly be to recognize not only its
dynamism but its becoming-ness, as a process of kosmogenesis. Because
such an instrument would have to be so constructed as to take note of the
movement of life, and for this reason, it would have to be extremely
sensitive necessarily and keyed to the subtleties of vitality and not to
materiality. Mathematics shall have failed utterly in the utilitarian aspects of
this phase of its latest diversion if it do not justify its claims by crowning its
work in the field of hyperspace with a "Kosmometer," an instrument
devised for the measurement of the movement of space or a "Zoometer," an
instrument devised for the measurement of the passage of life. We should

like to encourage inventive minds to turn their attention life-ward and
space-ward with the end in view of constructing such an instrument. When
once we have learned accurately to measure life we shall then be able to
dispose of it—to create. It is not doubted that if ever humanity is to arrive at
that point in its evolution where it can understand life; if ever it is to attain
unto the supreme mastery both of vitality and materiality and to come to the
ultimate attainment of divine consciousness (all of which we confidently
believe to be in store for humanity) it must be accomplished after this
manner: first, by syncretizing materiality with vitality, and then, by
intuitionally recognizing the truth of the implications of the syncretism.
The history of consciousness in the human family is identical with the
history of man's conquest over matter and physical forces. And this is
clearly indicated in the incidentals contingent upon the toilsome rise of the
genus homo from the earliest caveman whose status denoted a
comparatively negligible transcendence of material forces, to the present-
day man who has gained a markedly notable conquest over these forces.
Always consciousness seeks the means of adequately expressing itself in
the sensible world. And to this end it engenders faculties, organs and
processes in the bodily mechanism, and, in matter, devises instruments of
application whereupon and wherewith it may test, analyze, combine and
recombine the forces and materials it finds. The unlimited range of
expressions lying open to the consciousness makes it necessary continually
to devise higher and higher grades of appliances to meet its needs as it
expands. It will not be gainsaid that the telescope has served actually to lay
bare to the consciousness an immeasurable realm of knowledge nor that the
microscope, turning its attention in an opposite direction, marvelously has
enlarged and enrichened our knowledge of the world about us. And similar
declaration may be made anent almost every invention, discovery and
conquest which man has made over natural phenomena. Thus, by externally
applying mechanical implements to the subject of his consciousness, man
has extended actually his consciousness, his sphere of knowledge; has
greatly enhanced its quality, and, in the process, has urged the intellect to
endeavors that have wrought its present unequaled mastery of things. Nor
have the spiritual aspects of our advance along these lines been the least
notable. For these have enjoyed the essence of all that has been gained in
the process and have, therefore, kept pace with the onward movement of the

intellectual consciousness. But heretofore no advance has been made as a
result of methodic or reflexive determinations. That is, men did not set out
from the beginning, equipped with foreknowledge of what their efforts
would bring, to develop the present quality of human consciousness. They
simply worked on, their attention being absorbed by the problems that lay
nearest and demanded earliest consideration. So the advance has come as a
resultant of man's close application to his ever-present needs—shelter,
clothing, food, protection and other preservative measures—and it has come
naturally and inevitably and without prepense. Nevertheless, if man,
knowing what to expect from the syncretization of matter and mind, after
this fashion, should set out deliberately to accelerate the intensification,
expansion and growth of his consciousness, there is no doubt but that the
consequence would be most far-reaching and satisfactory.
But the path that leads to this grand consummation does not lie in the
direction of diversity; it lies in the opposite direction. In vain, then, does the
intellect fractionalize in the hope that by doing so it shall come to the solid
substructure of life; in vain does the analyst segment space into any number
of parts or orders; in vain does he ask how many and how much; for by
answering none of these queries will he find the satisfaction which he
vaguely seeks.
If it be true that it is not by analysis but by synthesis that the true norm of
life, and therefore, of reality shall be found it is futile to entertain serious
hope of finding it in any other way. As a perisophism or near-truth, then, n-
dimensionality takes foremost rank. And this is so for the reason that when
we proceed in the direction of multiple dimensions, that is, one dimension
piled upon another dimension or inserted between two others we are
traveling in a direction which, the more we multiply our dimensions, leads
us farther and farther away from the truth. This is a simple truism. If we
take, for instance, a wooden ball and cut it up into four quarters, and divide
each one of these quarters into eighths, into sixteenths, thirty-seconds,
sixty-fourths, etc., indefinitely, we shall have a multiplicity of parts, each
one unlike the original ball. But from no examination of the multipartite
segments can we derive anything like an adequate conception of the
original ball. Something, of course, can be learned, but not enough to
enable the rendering of a correct judgment as to the nature, size, shape and
general appearance of the ball. But this is precisely what happens when the

analyst divides space into many dimensions. He cuts it up into n-
dimensional parts and the more minutely he divides it into parts the more
remote will each part be in its similarity to the original shape and form of
space, and the farther away from the true conception of the nature of space
he is led thereby.
Now, n-dimensionality or that phase of metageometry which regards space
as being divisible into any number of dimensions or systems of coördinates
is a direct and inevitable product of that tendency of the intellect to
individuate and to singularize phenomena. Biologically speaking, it is a
peculiarity which harks back to the time when life was manifested through
the cell-colony and when the individual cells began, because of increasing
consciousness, to detach themselves from the colony and set out for
themselves, and thus each intellect recapitulates in its modus vivendi the
salient tendencies of phylogenesis. Let it suffice, then, to point out that this
universal tendency to segment and fragmentate which rigorously
characterizes intellectual operations upon every phenomenon with which it
deals is a culmination of the primordial tendency among cells to divide,
inasmuch as this phase of cell life must be the work of the kosmic intellect.
The natural inference is that from the extreme of individualization there
shall be a gradual turning, whether of the intellect per se or of the intellect
joined to the intuition does not matter, towards that other extreme of
communalization. And from this latter shall grow up, as one of the
inevitable and ineluctable tendencies of the Thinker's consciousness a
torrentious movement in human society towards coöperation, brotherhood,
mutuality and union in everything. So that whereas in the past and at the
present time the intellect has been developing under the dominant note of
individuality it will then be coming gradually under another dominant note
—communality. The result of this development will be the unification of all
things, and instead of many dimensions of space, many measures of time,
and a general diversification of all phenomena, we shall come to the only
true notion of these things and realize pragmatically the true value and
extent of unity in the universe.
It is admitted that the intellect, in treating objects singly and dealing only
with the starts and stops of a movement, is withal loyal to the kosmic order,
design and purpose which have priorly characterized manifested
phenomena by segmentation. And in this loyalty it has been following

merely a natural lead which, while admitting of the widest development and
experience, nevertheless at the same time beneficently obscures the
underlying reality in order that in its adaptation to the sensuous world the
intellect might have the greatest freedom for the development suited to the
given stage of its evolution. But in thus admitting the natural congruence
between the intellectuality and the phenomenal or sensuous we do not
thereby unite with those who already believe that this kosmic agreement is
the ne plus ultra of psychogenesis. On the other hand, it is maintained that
this is merely a phase of psychogenesis which shall be outgrown in just the
same measure as other phases have been outgrown. And notwithstanding
the fact that judgments of the intellect with respect to inter-factual relations
or the ens of facts themselves are as valid as its judicial determination of
self-consciousness, no more and no less, we are, by the very rigor and
exclusiveness of this logical necessity and inherent limitation, led to view
the intellect's interpretation of phenomena as partial and fragmentary; for
the reason that the necessitous confinement of its understanding and
interpretative powers to fact-relations quite effectively inhibits the use of
these powers for the contemplation of the deeper causative agencies which
have operated to produce the phenomena. But it is apparent that just as the
transmuted results of other phases of psychogenesis are now being utilized
as a basis for the efficient operation of the intellect in the sensuous world,
thereby enabling the attainment of a very high mastery over matter, so will
the functional dynamism acquired by it in the pursuit and comprehension of
diversity serve well when, in later days, it has acquired the power to deal
directly with reality, to create and dispose of life just as the kosmic intellect
has and is now using it in the execution of the infinite process of becoming
through which creation is proceeding. It would seem that the necessary
prerequisite to the development of any higher functional capability is that
the intellect should be capable of disposing of innumerable details, indeed
the totality of kosmic detail, before it can come wholly into the power and
capacity to understand and manipulate life. Furthermore, it appears that the
acquirement of this power quite necessarily has been delayed awaiting that
time when, dominated by the intuition, the intellect shall have attained the
requisite managerial ability for marshaling an exceedingly large number of
details.

The supreme tendency of life is expression. And this expression, singularly
enough, reaches its most perfect phenomenalization by means of that
movement which results in the multiplication of forms. Despite the fact,
therefore, that the comprehension of reality involves a gradual turning away
from the exclusive occupation of organizing a multitude of separate and
apparently unrelated facts to a monistic view which at once recognizes the
unitariness and co-originality of all things, of life, mind and form, the
intellect will need the training and development which come from the
mastery of diversity. It is, then, not difficult to perceive the wise
utilitarianism of the present schematism of things as shown in the universal
tendency in the intellect to devote itself exclusively to parts or segments of
truth.
Whenever an individual intellectuality, on account of prolonged thought
and the consequent inurement of the mind to higher and higher vibrations of
the kosmic intellect, brings itself to such a high point of sensitiveness that it
can receive so much as an intimation of some great truth, it begins to sense,
in a more or less vague way, something of the substance and general
tendence of the underlying reality of that which foreshadows its appearance.
Then, confounded by the multiformal characteristics of kosmic truth
because of the fact that it presents itself in such numerous ways and forms,
men often are induced to attempt the reformation of all facts, or a great
mass of kindred facts, in accordance with the newly-found fact or principle.
They forget evidently that no fact in the universe can be at variance with
any other fact and still be a fact. So that in the totality of facts every
separate and distinct fact must be congruent with every other fact forming a
beautiful, harmonious and symmetrical whole; but often the whole is made
to suffer in the attempt at making it conform to the substance of a mere
intimation. Moreover, it is conceivable that even the totality of facts may
lack a rigid conformity with reality in all its parts and that having
compassed the entire mass of facts one may fall short of the understanding
of realism.
This is practically what has happened in the mind of the metageometrician
who having received an intimation as to the real nature of space as that
whose center is everywhere and yet nowhere and whose nature is
psychological and vital rather than mathematical and logical, misses the
great outstanding facts and clings to the intimations which he experiences

as to the nature of space. He, therefore, concludes that the form of space is
that of a flexure or curve. There is a valid element in the notion of the
curvature of space but not enough of truth wholly to validate the notion.
Since the very reality of space is a matter which can be determined only by
the conformance of the consciousness with it in such a manner as to render
the conception of it entirely unintelligible to the intellect except in so far as
it may be able to identify itself with the space-process, there is much room
for the serious questioning of the mathematic conclusion upon the grounds
of its fragmentariness if not entirely upon the basis of its invalidity.
Wherefore it may be seen that any search for either the center or the
extreme outer limits which proceeds in a manner conformable to the
external indications of the intellectual order is vain, indeed. Although it is
undoubtedly true that the attainment of a central or frontier position in
space does not involve any lineal progression whatsoever, the same being
attainable, not by progression nor by overcoming distances, but by a subtle
adjustment, yea, a sort of attachment of the consciousness to the order of
becoming which binds the appearance of space, wherever one may be, it is
nevertheless difficult and painful for the intellect to grasp the totality of this
truth at one sweep. Indeed, it is not possible for it, alone and unaided by the
intuition, to grasp it at all. Hence, the mathematician who depends entirely
upon the deliveries of the intellect which conform, in their passage from the
conceptual to the written or spoken word, to all the rigors of mathetic
requirements, fails utterly in perceiving the magnitude of this conception
and all its connotations; he fails because his prejudices and the woof and
warp of his intellectual habits prevent his assuming a sympathetic attitude
toward it and thereby precluding at the start any calm consideration of it.
And not only is this true of the mathematician but of all those whose
endeavors are confined to the plane of purely sensuous and logical data. It
would, therefore, appear that our entire attitude towards things spatial must
be changed before we can even begin to perceive the reality which is really
the object of all researches in this domain. But, on the surface, there is after
all little difference between the ultimate facts involved in these two totally
different conceptions. Mathematically speaking, all progression eastward
would terminate at the west, and vice versa; and the same would be true
regardless of the point from which progression might originate. Always the
terminus would be the opposite of the starting point. Then, too, it might be
said that if we sought the space-center we should arrive at the

circumference. The difficulty with this view is that there is a very remote,
though important, connection between it and the truth of the matter. But the
partiality of this view, and the absence of either experience or intuition to
intimate a more reasonable view, serve effectively to buttress it as a
hypothesis acceptable to many. Thus it is ever more difficult to supplant a
near-truth than it is to gain credence for the whole truth. On the other hand,
according to the view which we maintain here, it is quite true that the
seeking of the kosmic space-center will reveal the circumference; that the
search for the nadir will uncover the zenith; the east effloresces as the west,
and a northward journey will wind up at the south, etc., but in quite a
different manner from that which the mathematician has in mind when he
postulates the curvature of space. Our view involves no space curvature nor
any other spatial distortion. It deals with space as reality, as a dynamic
process, a flux which, like the sea, is continually casting itself upon the
shores of chaos and falling back upon itself only to be recast against the
rock-bound coast of its chaotic limits. Now, that which falls back upon itself
and rolls in a recurrent movement upon its own surface is life which, in its
recession is the natural and kosmic limitations of itself, generates matter in
all its varied expressions. Space, in its extensity, cannot transcend life; for it
is the path which life makes in its out-coming, its manifestation. Of the
chaotic fringe which circumscribes the manifested universe it is absurd to
say that it is vital or psychological in any sense of these terms. For
notwithstanding the fact that out of its very substance are engendered life,
intellectuality, spatiality and materiality, it is nevertheless none of these in
its primary essence. It is Chaos-Kosmos; because from its content the
kosmos is evolved, and it still remains; it is chaos-spatiality; chaos-
materiality; chaos-intellectuality; chaos-geometricity; because these are
engendered by the movement of life in chaos while at the same time there
remains a residuum of the chaogenetic substance which constitutes the
limitations of all these subsequent processes. In this sense, the chaogenetic
fringe becomes the limits of the manifested universe so that it would appear
that all those major processes outlined above are finite manifestations of the
eternal chaos. But none of those possibilities of motion which are found in
these major movements of the kosmos can be logically said to exist in
chaos. It is the embodiment of everything that is the opposite of those
qualities which may be found in them, that is, in materiality, vitality,
spatiality, intellectuality and geometricity.

Apropos to this phase of the discussion let us examine briefly one of its
most significant implications, both mathematical and kosmic, which arises
out of the fact that space is an engendered product of life that is bound by
the fringe of chaos which sustains and limits it. The chaotic fringe plus
manifesting kosmos constitute the absolute magnitude of the kosmos. The
manifestation factor is complemented by the chaos factor and together the
two define the full universe. Kosmogony is the universal movement of all
kosmic elements or factors in diminishing the chaotic complement and
reducing it to kosmic order or geometrism. It is undoubtedly impossible to
determine mathematically the exact volume of either complement or the
ratio of the one to the other; yet it is conceivable that the chaotic fringe is
greater in extent than the ordered portion of the kosmic uni-circle or
universe. It is even conceivable that the difference, upon the basis of the
meaning of the Pythagorean Tetragrammaton and the view outlined in the
Chapter on the "Mystery of Space," is as seven to three wherefrom the
conclusion might be drawn that the universe has yet seven complete stages
more or less of evolution before the close of the Great Cycle of
Manifestation when the fringe of chaos shall have been totally used up in
the work of creation. But for those who may experience impatience at the
infinitude of the process when viewed in this light the terms may be
reversed and the difference may be conceived as the ratio of three to seven
wherefrom the conclusion would follow that the kosmogonic process is
seven-tenths complete, as it will not vary the seeming infinitude either way
it may be determined. The notion, despite its speculative character, offers an
explanation of otherwise inexplicable conditions, and, on account of its
profound connotations, may even be found to be productive of the highest
good in its equilibrating influence upon our mode of thinking.
In any event, there does appear to be a subtle relation subsisting between
true numbers and kosmogony. Number is a phase of the kosmogonic
movement, a measurer of the intellect and the establisher of the geometrism
of space, answering tentatively to the numericity of pure being. In fact,
being actually expresses number and number itself is an evolution and not a
thing posited once for all as a pure, invariable form in the universe. It is,
like the kosmos, in a state of becoming and there may yet appear to our
cognitive powers a whole series of new numbers pure in itself and
altogether conformable to the conditions reigning at the time.

The symbology of the circle, in all times recognized as the true symbol of
the kosmos in eternity, of eternity itself, of the archetypal, of space,
duration and Ultimate Perfection, is replete with profound significations.
But it should be understood that the circle is a symbol of the perfected
universe and not the universe in a state of evolution. It symbolizes
perfection, completion and the ultimate union of the manifesting with the
archetypal which results in the crowning deed of Perfection. The circle is,
therefore, not a symbol of the universe as it now stands; it does not
represent a snapshot view of the kosmos but the universe as a full. It cannot
be a full until it has attained the ne plus ultra of completion; for a kosmic
full is that state to be attained by the manifested kosmos upon the
termination of all the fundamental processes now in operation. But it is this
state that the circle really represents, and by virtue of which it possesses its
intrinsic qualities and also in virtue of which the intellect recognizes these
qualities. The properties of understanding and recognizance in the intellect
are veritably fixed by the status quo of the universe during every stage. That
is, the focus of the intellect, like the focus of a chromatic lens, is adjusted
by the fiat of the nature and eternal fitness of things to correspond exactly
with every state through which the kosmos itself passes. This is one of the
obvious implications of the phanerobiogenic behavior of the kosmos and is
necessarily resident in the notion of the genesis of space and intellectuality
as consubstantial and coördinate factors.
Wherefore the more cogent is the reason for the belief that the inherent
qualities of the kosmogonic fundamentals; as, vitality, materiality,
spatiality, intellectuality and geometricity, are true variants, and that their
variability is proportional to the progress of these major movements toward
the ultimate satisfaction of the original creative impulse. May it not be,
therefore, that the indeterminate character of the ratio of the diameter to the
circumference (3.1415926 ...), is due to causes far more profound than the
crudity of our micrometers or the mere supposed fact of the circle's
peculiarity? May it not also be true that the pi proportion shall become a
whole number, and in its integration, keep apace with the perfecting process
of the kosmos, diminishing, by retrogression to one or increasing, by
progression, to ten which, after all, is essentially unity, being the perfect
numeral? It is not without the utmost assurance that these queries will be
categorically questioned by the orthodox, creed-loyal, strictly intellectual

type that we sketch these implications, but it is felt to be an urgent duty to
remind all such that the most effective barrier to realization in the field of
philosophy is an intolerant attitude towards all lines of thought which
suggest the impermanence of conditions as we find them in the kosmos at
the present time. The fact is that our lives are so distressingly short that we
have neither time nor opportunity to watch the changing moods of the
kosmos nor discern the gradual reduction of mere appearance to the firm
basis of reality, and accordingly, the intellect tenaciously clings to those
notions which it derives from the instant-exposure which the lens of
intellectual conceivability allots to it. Once the view is taken it is
immediately invested with everlastingness. This everlastingness is then
imputed to the kosmos in that particular pose, attitude or state. Always the
intellect beholds in that passing view, snatched from the fleeting panorama
of eternal duration, a picture of itself which it mistakes for the reality of the
not-self.
The inclination of the axis of the earth toward the plane of its orbit is
approximately twenty-three and one-half degrees. No well-informed
astronomer, however, doubts now the fact that this ecliptic angle is being
gradually lessened; because, as a result of centuries of observation, it has
been found to be decreasing at the rate of about 46.3 seconds per century.
Yet no intellect is able to perceive in any given lifetime the actual
decrement of this angle. It is only by careful measurements after centuries
of waiting that a difference can be discovered at all. Thus it may even be so
with the ratio of the diameter to the circumference of a circle, the only
difference being that it has not yet been determined whether there is a
decrement or an increase in the size of the ratio.
The pi proportion is, then, a register or measurer of the slow, measured
approach of the manifesting kosmos to the standard of ultimate perfection.
Therefore, and in view of these considerations, we may not hesitate to
confirm our belief in the validity of the notion that it actually and literally
expresses the key to the evolutionary status of kosmogony. The
mathematical determination which limits it as an unchangeable, inelastic
quantity is, consequently, only partially true and leads to the inclusion of
this quantity under the category of mathematical near-truths, for such it
appears to be in spite of its rigorous establishment.

The formal topography into which the intellect spreads when seeking the
ideal and the abstract is not a condition which is derivable from the real
essence of life or matter, but, on the other hand, is a product of the intellect
itself partaking of its nature rather than of the nature of reality. There is,
therefore, a very important distinction to be made between all deliveries of
the intellect and the realism both of the objects and conditions to which the
intellectual deliveries pertain. One of the most marked peculiarities of the
human intellect is the fact that it always unavoidably stamps its own nature
and features upon every datum which passes through it to the
consciousness. The utmost importance attaches to this phenomenon, for the
reason that it points to the necessity of carefully scrutinizing intellectual
deliveries and the making of allowances for those ever-present
characteristics which the intellect superimposes upon its data. Perhaps the
inherent colorific quality which it imposes upon our knowledge would be
better understood if a similitude were indulged at this juncture. The intellect
may be likened to a color-bearing instrument which, when it has once
handled an object, leaves forever its own color transfused into every cell
and fiber of the object so that when the same object is presented to the
consciousness for purposes of cognition it bears always the same peculiar
marks and colorations which the intellect, in its manipulation of it, places
thereon. In this respect the intellect may also be said to be like a potter who
has but one mold and that of a peculiar formation. Hence, whatever wares it
presents to the consciousness will invariably be found to be molded in
conformity with that particular mold. If it were possible to view reality or
the essential nature of things the difficulty which now the intellect lays in
the path of direct and uncolored cognition would be obviated; for then there
no longer would be any necessity of viewing things as they are colored or
molded by the intellect. The intuition, being a process of pure
consciousness, will, when it has arisen to a position where it may dominate
the intellect as the intellect now dominates it, so modify this tendency
which we see so ineradicably bound up in the very nature of the intellect
that the apparently insurmountable difficulties which it has interposed
between mere perception and a direct cognitive operation will be quite
completely overcome. Thus, in the above, is discovered another obstacle
which posits itself between the notion of space as reality and the intellectual
determination of it which the mathematician examines and to which his
consciousness is necessarily limited. Furthermore, it may be perceived also

how easily the mind may be deluded into thinking that the intellectual
notion which it entertains of space is necessarily correct, when obversely, it
is simply examining a concept which has been remade by the intellect into a
form which is not at all unlike its own peculiar nature, and therefore, as
much short of reality as the intellect itself is. Similarly, if the mathematical
mind succeed in catching a glimpse of the reality of space in the form of an
intimation, which, in itself though fragmentary, is nevertheless true, its
consciousness is finally deprived of the true validity thereof simply because
of the behavior of the intellect in its manipulation of it. The importance of
these intellectual difficulties cannot be over-estimated for they furnish the
grounds for the ineptitude of intellectual determinations made in a sphere of
motility to which the intellect is a stranger. And this fact will appear more
evident when it is perceived that quite the entire content of human
knowledge has been thoroughly vitiated by them. So that only in those very
rare moments which (in a highly sensitive mentality) enable the intuition to
gain a momentary ascendancy over the intellect is it possible for the
Thinker to catch hold of realism itself, and project the truth of what he sees
into the lower, intellectual consciousness. But so small is that portion of our
knowledge which owes its origin to the intuition that when compared with
the totality of that which we seem to understand it is well-nigh negligible.
And then, when it is considered that at present there is no way of
conceptualizing adequately the intuitograph so as to make it propagable the
insignificance of this form of knowledge is even more notable. It can now
be seen in how large a measure the notion of the curvature of space is
merely an intellectual translation of a true intuition into the terms of the
intellect which, in the very nature of the case, can only approximate the
truth because of its colorific habits.
A similar declaration may be made of that other datum of metageometrical
knowledge which postulates the ultimate convergence of parallel lines. In
fact, what has been said as to the perisophical nature of the notion of space-
curvature will apply with equal force to the idea of parallel convergence
since the latter is a derivative of the former. But there is yet another
consideration, apart from the colorific influence of the intellect, which,
although it partakes of the nature of this quality, is nevertheless a near-truth
of quite a different order. This may be better understood by referring to the
graph showing the inverse ratio of objective space to the consciousness.[28]