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    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                           x


      _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 Psychogenetic
  Development--The Scope and Permissibility of
  Mathetic License--Kosmic Unitariness Underlying
  Diversity                                                23

      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                                       44

      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           69

      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                                    92

      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
  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                     118


      _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 Methods of Interpreting
  Intellectual Notions--The Tuitional and the
  Intuitional Faculties--The Illusionary Character of
  the Phenomenal--Consciousness and the Degrees of
  Realism                                                 161

      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                              203

      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    242

      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--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                         284

      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 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                                               327

  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 establish
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."

  [1] Vide _Popular Science Monthly_, vol. 78, p. 554, 1911.

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_.

    A +--------------------------------+ B
      |                                |
      |                                |
      |                                |
      |                                |
      |                                |
      |                                |
    C +--------------------------------+ D

    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, obtus 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."

  [2] Vide _New York Mathematical Society Bulletin_, Vol. III,
  1893-4, p. 79, G. B. HALSTEAD on _Lambert's Non-Euclidean Geometry_.

  [3] _Prolegomena_, KANT, p. 37, Trans. by J. P. MAHAFFY and J.
  H. BERNARD.

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.

  [4] In 1754 D'ALEMBERT (1717-1783) published an article in the
  famous old _Encyclopedia_ edited by DIDEROT and himself on _Dimension_.
  In this article the idea of the fourth dimension is dwelt upon at
  length. The view which he expressed in this article, of course, served
  greatly to popularize the conception among the learned men of the day,
  and owing to the close relationship existing between D'ALEMBERT and LA
  GRANGE, it is not surprising that the latter should have been very much
  enamored of the idea.

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:

  [Illustration: 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]

  [5] Vide _Nature_, Vol. VIII, pp. 14-17; 36, 37 (1873); also
  _Mathematical Papers_, pp. 65-71.

_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:

[Illustration: FIG. 3.]

[Illustration: 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]

  [6] _Mathematics_, by C. J. KEYSER, Adrian Professor of
  Mathematics, Columbia University.

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]

  [7] The science of pure mathematics is perhaps indebted to no one in so
  great a degree as to GEORGE BRUCE HALSTEAD, formerly of the University
  of Texas, whose labors in connection with the popular exposition of
  the non-Euclidean geometry have been most untiring and effectual. Vide
  _Popular Astronomy_, Vol. VII and VIII, 1900, Dr. G. B. HALSTEAD.

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.

[Illustration: 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."

  [8] NOTE.--_M_ may be any point on the line _BA_ indefinitely
  produced.

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.

  [9] Vide _Non-Euclidean Geometry_, p. 91.

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.

  [10] Vide _Nature_, Vol. XLV, 1892.

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.


           D
           +
     A     |
      \    |
       \   |
        \  |
         \ |
          \|
           +-------------------------
           B                        C

     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]

  [11] Vide _Monist_, Vol. XVI, 1896, Mathematical
  Emancipations.

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."

  [12] Vide _Monist_, Vol. XIX, p. 402 (1909).

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.

  [13] _Philosophical Review_, Vol. VII (1898).

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.

  [14] Vide _Foundations of Mathematics_, p. 107.

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]

  [15] Vide _Foundations of Mathematics_, p. 42.

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.

  [16] Vide _Science_, Vol. VII, p. 2, No. 158, 1898.

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, A^2 its area, and A^3 the
volume of a cube with edge equal to A; what may A^4, A^5 or A^{_n_th}
represent in our space? The conclusion, with respect to the quantity
A^4, 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
A^3 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.

    A -------------------------------- B

    FIG. 7.


    A +------------------------------+ B
      |                              |
      |                              |
      |                              |
      |                              |
      |                              |
      |                              |
    C +------------------------------+ D

    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.

[Illustration: FIG. 9.]

[Illustration: FIG. 10.--The Tesseract.]

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._

[Illustration: FIG. 11.--Tetrahedron.]

[Illustration: FIG. 12.--Hypertetrahedron.]

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."

  [17] Vide _Foundations of Mathematics_, pp. 93-94.

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."

  [18] Vide _Philosophical Review_, Vol. V, 1896, p. 352, et.
  seq.

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."

  [19: Vide _Monist_, Vol. XVI, 1896, Mathematical
  Emancipations.]

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.

      D             C            D1
      +-------------+.............
      |             |            :
      |             |            :
      |             |            :
      |             |            :
      |             |            :
      +-------------+.............
      A             B            A1

    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.

[Illustration: 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; by 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.

[Illustration: FIG. 15.]

[Illustration: FIG. 16.--Plane Rotation]

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."

  [20] Vide _Fourth Dimension, Simply Explained_, edited by
  H. P. MANNING, p. 28.

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."

  [21] Vide _Fourth Dimension_, p. 75, C. H. HINTON.

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 C_{5}H_{12}O 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."

  [22] Q. v., p. 242, edited by H. P. Manning.

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.

  [23] Vide _Science_, Vol. VII, 158, 1898, p. 4.

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
triangle [Illustration], the square [Illustration] and the circle
[Illustration], 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.

  [24] See Fig. 18.

[Illustration: 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.

[Illustration: 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.

  [25] Vide _Recherche_, Chap. VII, also Philosophical Review,
  V. 15, p. 401.--MALEBRANCHE.

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.

  [26] _Kathekosis_ (from Chaos-Theos-Kosmos) is to evolution
  what "chaogeny" is to involution. It is the end of evolution, but also
  the beginning of involution, and in the latter function is known as
  "chaogeny." See diagram No. 17.

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.

[Illustration: 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 picture 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.

  [27] See _The Germ Plasm; A Theory of Heredity_, by A. WEISSMAN.

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.

[Illustration: FIG. 20.--Kosmos and Consciousness]

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.

[Illustration: 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] Let us suppose that the _graph_ may also represent
the Thinker's outlook into the world of spatiality. It then appears
that, because of that movement of consciousness in its pursuit of
life which, as it expands, makes the objective world to appear to be
diminished in proportion to the extent of its expansion, it is quite
natural, under such circumstances, that parallel lines drawn anywhere
in the limits of the objective world should seem to come to a point in
the ultimate extension of themselves. While this _graph_ is not meant
to depict such a view, it may be found nevertheless, to be a true
delineation of the topography of that state of mind into which the
metageometrician brings himself when he visualizes space as _curved_;
for there is no doubt but that a state of intellectual ecstasy, such
as that in which the mind of the metageometrician must be functioning
in order to perceive space in that form, is quite different from the
normal and, therefore, in need of a different topographical survey.
But, if we grant that in the creational aspects of space there is
conceivable an ever-present tendency to convolution, or a rolling back
upon itself, it is imaginable that parallel lines inscribed either upon
its surface or in its texture need not necessarily meet but maintain
their parallelism regardless of the complexity of the convolutions. The
convergence of parallel lines is much like a tangent in the outgrowth
of the idea from the notion of space-curvature. The more a tangential
line is extended the farther away from the circumference it becomes
and consequently less in agreement therewith. The more subsidiary
propositions or corollaries are multiplied the more remote from the
truth the determinations become and especially is this true of the
hypothesis of space curvature.

  [28] Figure 20.

In the notion of the manifoldness of space, by virtue of which it is
conceived as existing in a series of superimposable and generable
manifolds of varying degrees of complexity, are discernible traces
of that intuitional intimation which underlies the assertion that
because of the necessary phenomenalization of reality for the purpose
of manifestation to the intellect it appears to exist in a series of
separate degrees, each one more refined and subtle than the preceding
one and requiring a more highly developed species of consciousness for
its comprehension. In other words, that intuitional glimpse of the
essential character of reality, as viewed by the human consciousness,
which impinged upon the minds of RIEMANN and BELTRAMI leading them
to postulate as a corollary proposition to space-curvature, its
manifoldness, is nothing more nor less than the intuition that the
universum of spatiality cannot otherwise present itself to the
intellect, owing to its peculiar adaptation to the sensuous, except by
a series of continuous degrees which are perceptible only in proportion
as the understanding is magnified to conform with it. After all,
however, it is not improbable that the very objectivism of the universe
in manifestation subsists in just the manner in which this intuitive
glimpse implies and that the wisdom and utilitariness of the kosmogonic
process which engendered spatiality are clearly demonstrated in that
arrangement of the contents of the kosmos which presents the grossest
elements of phenomena first to the intellect in its most impotent
state while reserving the less crass for that time when the Thinker
shall have evolved a cognitive organ adaptable to its presentations.
Those metageometricians who cling to the idea of the manifoldness of
space, based as we have shown upon the pseudo-interpretation of a
rather vague hint arising out of an unquestionably true intuition, have
allowed themselves to fall into the unconscious error of magnifying
the importance of the mere insinuation as to the space-nature to such
an extent as wholly to obscure in their own minds and in the minds of
those who think after them whatever of the true vision that may have
been grasped by them. Furthermore, it is indubitably true that that
same peculiarity of arrangement by which impalpable and invisible
forces really subtend gross matter producing that subtle schematism in
virtue of which the visible is subjoined to the invisible, the sensuous
to the non-sensuous, spirit to matter, etc., also characterizes the
appearance of spatiality to the human understanding. While there is a
superficial semblance of separate and discrete manifolds into which
space may be divided there are, in reality, no such sharp lines of
demarkation between the subtle and the gross, between the visible
and the invisible or between spirit and matter, each of these being
capable of reduction, by insensible degrees, into the other regardless
as to whether the reductional process originates on the side of the
most refined or on that of the grossest. Accordingly, there are no
reasonable grounds upon which the notion of a space-manifold may be
justified except as a metageometrical near-truth.

In addition to the foregoing, there are yet other very fundamental
considerations which would seem to debar the totality of analytical
conclusions as to the nature of space from any claim to ultimate
reliability and trustworthiness. These are _first_: the fact that
analyses are absolutely incapable of dealing with life; that being
the direct product of a sort of mechanical consistency which marks
the intellectual operations it has adaptability only for dealing
with fragments or disconnected parts, and that without any reference
whatsoever to the current of life or the flow of reality which has
produced the parts. This fact is clearly shown in that attitude of the
understanding which inevitably leads it to the declaration that a line
is an infinite series of points, a plane an infinite series of lines,
and a cube, an infinite series of planes, and so on, indefinitely.
To do this, to look upon all phenomena as a series of parts similar
to each other and piled, one upon the other, or juxtaposed in the
manner which they are discovered in the sensible world, is the natural
tendency of the intellect and this tendency finds its most facile
expression in analytics. Inadaptability of this sort is especially
observable in all problems of arithmetical analysis in which the vital
element is a factor. When these analyses are carried to their logical
conclusion, as has been shown in the chapter on "The Fourth Dimension,"
invariably they end in an evident absurdity. But it is at their very
conclusion where the life-element is encountered, where reality is
approached, that they break down. The failure of analysis, then, to
encompass life, to fit into its requirements and to satisfy its natural
outcome seems clearly to establish the basis of the perisophical
nature of the entirety of analytical claims, especially that species
of analysis which seeks the remoter fields of the conceptual for its
determinations. _Second_: the close connection which has been seen to
subsist between space and life as joint products of the same movement
makes it obvious that the same ultimate rule of interpretation must be
applied to both in order to insure correct and dependable judgments
regarding them. How different would be the intellectual attitude
towards space if it were considered in the same light as vitality,
provided one really understood anything about vitality! Moreover, as
it appears certain that the path of the intellect does not run in
the same direction as the path which life makes, but in an inverse
direction, it is clear that the judgments of the former, as to the
action and essence of the latter, must necessarily be ultimately
unreliable. It can readily be seen, however, that should the intellect
be focused so as to follow the path of life, to attach itself to the
very stream of life, it would have necessarily to neglect materiality.
And such an adjustment would, of course, obviate the need of a
material life at all for humanity. In fact, a physical life with an
intellect would be impossible under such conditions. It is well to
recognize the suitability of the present schematism and not to become
unwisely restive because of it; but it is also fitting that we should
discriminate between that which is possible for the intellect chained
to materiality and that which is impossible for it, in such a state,
when foraying in a territory foreign to its nature, and beyond its
powers to master.

The predominating tendency in the intellect to account for the
universe of life, mind and matter upon a strictly mechanical basis
is undoubtedly due to the constitution of the intellect which does
not admit of its direct consideration of the vital essence of things.
We are bound ineluctably to the surface of things. All our knowledge
is therefore superficial. We are even bound to the surface of ideas,
and cannot penetrate to the interior of these realities. Our art
is the reproductions of superfices; our philosophies are the husks
of eternalities; our religions, the habiliments of relations, and
while it cannot be doubted that this arrangement is pre-eminently
the best possible one for the present stage of man's evolution, it
is nevertheless worth while to note that it is this very restricted
activity of the intellect which shuts out from man's consciousness
those very elements about which he is most concerned when he goes into
the field of philosophy in search of a solution to his unanswerable
queries. But some progress most surely is made when the mind is
enabled to see its plight and recognize what are the difficulties and
limitations that lie in its path of ultimate attainment.

It is believed that the mechanistic, or true, character of the
intellect reached its zenith in the mind of LAGRANGE when he succeeded
in reducing the entirety of physics to certain mechanical laws and
formulæ which he embodied in his "_Mecanique Analytique_" This work
is undoubtedly the capstone of intellectual endeavors and stands
as a monument which marks the culmination of the present stage of
intellectual development. In thus placing the _Mecanique_ at the apex
of intellectual endeavors it is not thereby meant to be implied that
the intellect shall not make more progress nor that other formulæ,
equally as marvelous as those which LAGRANGE discovered, may be
devised, nor that other laws, heretofore undreamed of, may be found;
but what is maintained is the fact that while there will be growth and
development these will run along other channels, perhaps in the realm
of the intuitable, and not any longer, especially so notably as now,
in an opposite direction against the current of life and reality; and
further, that there will be a gradual turning away from mechanics to
biogenetics, from diversity to unity, from the purely intellectual
to the intuitional, and withal a final getting rid of the bonds of
illusion, of that thralldom of mechanics, whereupon will slowly arise
the obsolescence of all those disparities which may now be recognized
in our knowledge and in the applications of the intellect to the data
of the objective world.

Because the intellect is unsuited to deal with reality, and because of
its peculiar adaptation for diversity, for multiplicity, due to its
mechanistic _modus vivendi_, there has grown up a voluminous catalogue
of systems of philosophy. These embody such a multitudinous array of
beliefs, ideas, conceptions, theories and conjectures and constitute
a movement in human thought which oscillates between the empiricist
on the one hand and the transcendentalist on the other; between the
idealist and the realist, leaning sometimes towards the Platonic,
the Cartesian and the Kantian and at other times towards SPINOZA,
ARISTOTLE, SPENCER and SOCRATES, always terminating by multiplying the
number of diverse beliefs rather than unifying them that the conclusion
is unavoidable that so marked a lack of unanimity is indicative of
a profound mental prestriction. It was, therefore, inevitable that
mathematics should fall under the same spell and brook no let nor
hindrance until it had succeeded in devising several diverse systems
of geometry which it has done for the mere joy of doing something, of
following its instinctive aptitudes. There is no other basis for the
heterogeneity of our philosophies, our mathematics, indeed our beliefs
than this mechanical, and hence, radically illusionary character of the
intellect in consequence of which we have had to be satisfied with mere
glimpses, hints, intimations and faint glimmerings of reality, of life,
and of those kosmic movements which, if we had the ability to trace
them from their source outward, would lead us unerringly to a truer and
deeper knowledge of those things that under the present schematism must
remain for us a closed book.

The criterion of truth for us, constituted as we are and wedged
in between the stream of life and its shore of materiality, must
be that which relates our knowledge both to the stream and to the
shore. It must be so that all predicates which purport to approach
it shall exhibit a dual reference--one that relates to materiality
and another that relates to vitality, and yet a third that shall
combine these two relations into one. All assertions, therefore, which
pertain exclusively to either of these elements--to materiality or to
life--are necessarily partial, fragmentary and perisophical in nature.
Mathematics, because it relates to matter and the mechanical forces set
up by matter acting against matter cannot be said to agree with such
a criterion; art, because it relates to snapshots or static views of
matter is even more remote in its agreement; philosophy, as it has been
known in the past and is known to-day, because it seeks to deal with
a vitality fashioned after the image of materiality has failed when
posited alongside of this criterion; and thus, the intellectual toil of
millions of years has been in vain in so far as it has not succeeded
even in raising a corner of the cover which hides reality from our view.

A near-truth is any variation from this standard, this norm or
criterion. It may be either logical, cognitive, scientific or even
metaphysical. To define: a logical truth is a predicate based upon
and involving the coherency and consistency of thoughts themselves; a
cognitive truth is the conformity of knowledge with so much of reality
as is known; scientific truth is the conformity of thoughts to things
and conditions. All of these are obviously near-truths. Then, too, a
near-truth may be defined as an assertion based upon the criterion of
truth but falling within the category of cognitive truths owing to
insufficiency of data or vision. Such indeed are those metageometrical
predicates--_n_-dimensionality, space-flexure, space-manifoldness and
all other assertions based upon these in general and specifically. Any
recognition of truth must clearly embrace both the vital and material
aspects of its subject in order to be adequately inclusive, that is, it
should include the causative, the sustentative, the relational and the
developmental factors. These four factors are considered necessary and
sufficient to determine the conformity of any view to the criterion of
truth for when we are cognizant of the cause of a subject, understand
the sustentative factors which keep it in existence, are conversant
with its relations to other subjects and can follow its developmental
variations until we come to its final status, why then, our knowledge
is both sufficient and ultimate so far as that subject is concerned.
Is it asking too much of mathematics or of philosophy or any system of
thought that it conform to these standards or to this criterion before
we shall accept it as final? Or shall we be satisfied with less than
this? Let us hope not.

In the foregoing presentation stress has been placed upon the
fragmentary, and therefore, illusionary character of the intellect in
order to arrive at an understanding of the difficulties under which
real knowledge has to be acquired and to indicate the inanity of all
attempts to resolve the riddle of space by means of mathematics though
regarded as the most typical exemplification of the mechanistic nature
of the intellect. And further, to show that, on account of the radical
incongruity which estranges life, the producer of spatiality, from the
intellect which returns again to scrutinize the passage of life in
its outward expressions, no hope of ever gaining the true viewpoint
by means of the intellect need be entertained. But in doing so, it is
deemed fitting that a note of warning should be sounded against any
abortive attempts that may be made to obscure or distort the results of
such a close discrimination lest the true import of the examination be
lost for, if we emphasize the vanity of the intellect in the pursuit of
that which it is by nature unsuited to attain we also equally stress
the wise utilitarianism which limits it to the performance of the
tasks assigned while at the same time reserving for the function of
more highly evolved powers, and indeed, for the intuition itself, the
solution of the riddle of spatiality. And if we declare the futility
of the mathematical method in all endeavors aimed at unveiling the
mysteries of life and mind, and of that movement which has its roots
set in eternal duration from which it proceeds in an endless continuity
of purpose and promise, we do also recognize that in the science of
mathematics the intellect shall, as in no other method of cognition,
most fully fulfill the kosmic intent of its existence; and moreover, in
the pursuit thereof it shall push the frontiers of its possibilities
outward until it can be said almost to be able to make disposition of
life itself--at least to that point where, when the intuition shall
have come into its own, the passage from the mechanics of matter to the
dynamics of life, shall be comparatively easy and natural.




  CHAPTER X

  THE MEDIA OF NEW PERCEPTIVE FACULTIES

  The Spiritualization of Matter 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 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.


Evolution is a continuous process and the primal impetus back of the
great on-flowing ocean of life acts infinitively. It is not terminated
when life has succeeded in perfecting a form for the perfection of
forms in themselves is not the end of vital activity. _The end of
evolution is the complete spiritualization of matter._ So that it does
not matter how perfect a form may be either subjectively or in its
adaptation to environments; it does not matter how faultless a medium
for the ensouling life it may be, there is ever the eternal necessity
that life must drive it back over the path of its genesis until it
shall be transmuted into pure spirit. Adaptation succeeds adaptation
and with each there is a change in the form and this process continues
until there is a more or less perfect congruence between form and
juxtaposed environmental conditions. But no sooner than agreement
has been attained under one set of conditions new conditions arise
and require a new setting, new adaptative movements. Thus there is a
continuous proceeding from stage to stage, going from the grossest to
the subtlest and most refined, always the form is being pushed onward
and upward by life. But adaptation is not undergone for the benefit
of the form, but more truly for the informing principle. It is the
progression of the life-element which constitutes the adaptation of
form to form and to their peculiar environs. The form is a tool or
instrument of life which it discards the moment it fails to respond
to its requirements. Thus forms are constantly being assumed and as
constantly being relinquished. But no effort of life is lost regardless
as to whether the action is performed in one or another form. The
totality of matter is perpetually being acted upon by the totality of
life. Every appulse of life against matter means an added push in the
direction of spiritualization. The totality of such appulses of life
against matter may seem infinitely small in the visible results which
they produce in the process of spiritualization; but with each there
is an eternal gain in that movement that shall end in the complete
transmutation of materiality into spirituality. This action of life in
metamorphosing matter, the nether pole of the great pair of opposites,
into spirituality, its copolar factor, in its outward, visible effects,
is what we vaguely call evolution. And such it is; for life is merely
unfolding that which it has enfolded. Matter, having been involved as a
phase of kosmic involution, is now being evolved.

In the genesis of the kosmos there appear to be three great undulations
in the universal current of life. The first of these prepares the
field by depositing that elemental essence which is to become the
world-plasm; the second precipitates the universum of materiality,
spatiality and intellectuality, not as we now know them, of course,
but as potencies; the third great undulation in the current of life
effects the endowment of the world-plasm with those tendencies that
are to build around themselves forms appropriate to their fulfillment.
This ensoulment of the world-plasm with tendencies and the consequent
segmentation of it into separate forms by these tendencies constitute
the primary stages of that procedure of life which results ultimately
in the up-raisement of matter and its final exaltation into pure
spirit. Hence, the entire mass of materiality is besieged on all sides
by the sum-total of life and the former is being raised slowly and
irresistibly to heights that are immeasurably more sublime than its
present degree of grossness.

It appears paradoxical, therefore, that life, although in all respects
vastly superior to matter, should become the apparent vassal of
materiality and give itself up to all the strict rules of imprisonment
which are imposed upon it by the properties and qualities which we
observe in matter. It seems so subject to every whim and fancy of
matter that one is inclined to think that matter and not life is the
chief designer of universal destiny. This is not a condition to be
wondered at so much, for the reason that this apparent vassalage, this
seeming enslavement of life by matter, is due to that superior and most
marvelous adaptability of life which it enjoys in contradistinction to
the relative unpliability of matter, and due also to the fact that life
is kinetic and matter, being a mere deposit of life, is static. Life is
mobility while matter is immobility and thus in possessing a greater
range of freedom is, of course, correspondingly superior; but in this
adaptation of itself to the labyrinthine cavities and multiformed
interstices in matter it exhibits but a seeming serfdom which is really
not a serfdom but a mastery. It is as if a man had taken lumber,
hardware and stone and built a house wherein he might dwell--life has
merely used matter, molded and fashioned it so as to make for itself
a medium, a dwelling-place wherein it operates, not as a slave but as
a master possessing unlimited freedom of motility. In the production
of a form life stamps upon it, once for all time, the path of its
engendering action. It leaves its finger-prints upon the mold which
it makes for itself. So that if we would know where life _has been_
or where it _is_ we should look for its finger-prints (organization);
we should observe the sinuosities which mark its pathway, remembering
always that it is life that has formed the intricacies and complexities
of the form into which it pours itself so accommodatingly in order that
it may raise that form, develop and transmute it into something higher
and better.

When we speak of _form_ it must not be understood thereby that
reference is made only to the gross physical form, but to the entire
range of vital assumptions or vehicles which life ensouls for purposes
of manifestation. This range we believe to cover the whole path of
kosmogenesis seriating from the densest to the most subtle. Our chief
concern, however, is the immediate effect which the totality of life's
operations will have upon humanity or the form which it ensouls as
the human organism. For it is impossible that humanity shall escape
either the general or the specific results of the exalting power which
life exerts over materiality and its appurtenances. It is, of course,
impossible here to go into the various implications of this general
forward movement of the universum of materiality or even to outline
briefly the divergent lines of operation into which a satisfactory
exposition of this view would naturally lead. And then to do so would
be inappropriate in a volume of this kind. So we shall have to be
content at this juncture to limit our study to a consideration of what
we believe to be some of the immediate indications of this vast and
most far-reaching phenomenon.

In the chapter on the "Genesis and Nature of Space" it is shown that
the material universe is engendered at the same time and by the same
movement or process as the universum of spatiality and intellectuality
and that as the passage from chaos to kosmos proceeds the function of
this movement is changed gradually from engenderment to exaltation
wherein materiality is transmuted into spirituality. It is, of
course, obvious that as materiality is exalted so are spatiality and
intellectuality; and that as the one becomes more and more refined,
capable of answering to higher and yet higher requirements so do
all the others. For, at work in all and through all of these, is
the current of life which pervades them, engendering, sustaining
and elevating as it proceeds. So that as matter has evolved added
characteristics and properties, each answering to a given need
and arising out of the necessities inhering in the stage at which
it appeared, so has the intellect evolved faculties to correspond
therewith. In other words, the evolution of faculties for the
expression of the human intellect has proceeded synchronously with
the evolution of material qualities. And whenever a new faculty or an
additional scope of motility is achieved by humanity there is always
found a set of kosmic conditions which answers thereto. The cardinal
principles of the doctrine of evolution are not, therefore, adverse to
the conclusion that the organs of sense-perception--hearing, touch,
sight, taste and smell--have not been endowed upon the human race or
attained by it at one time; but rather that each answering to a newly
acquired need and opening a wider scope of motility for the intellect
has been evolved separately and in due order. It would also seem that
the quality of consciousness, as it has been manifested in the various
stages of life through which it has passed, and especially the mineral,
vegetal and lower animal, has not always been of the same degree of
efficiency. Nor has it enjoyed the same kind of freedom which it now
enjoys in the highly evolved _genus homo_. It is equally apparent that
matter itself has not always been in possession of the same qualities
and characteristics which it now exhibits; but that it, too, has
gone through various stages of evolution bringing forward into each
new stage the transmuted results of each preceding one as a basis
for further evolution and expansion. The innumerable archæological
evidences which support this view make it unnecessary to do more than
state the facts, as they appear to be substantiated by indubitable
testimonies. Furthermore, it is believed that the outstanding
implications of these phenomena will not be successfully controverted
by those who are disinclined to see such implications in the
evolutionary process. In a previous chapter we have briefly sketched
the characteristics which mark the upspringing of a new faculty showing
how, at first, it appears as an abnormality which exhibits itself in
a very few individuals only, and that in a more or less indefinite
manner; and how later the number of individuals in which it appears
gradually increases, the definiteness of the faculty, at the same time,
appearing more marked; then, like a tidal wave, it recurs in a still
larger number of persons until, at last after a long period of time
usually several thousands of years, it becomes universal exhibiting
itself in every individual and appearing as a hereditary characteristic
of the entire human race. It is, therefore, not without assurance as to
the ultimate soundness of this view that we make the assertions which
follow this brief introduction.

It has already been stated that for a very obvious reason, namely,
the satisfaction of the needs of our present humanity, the intuition
is for the time dominated by the intellect and held in subjugation by
it so that all of man's external operations are governed and dictated
almost entirely by the intellectuality, allowing the intuition only
rare moments when it can come to the fore at all. This is the rule in
the evolution of faculties and characteristics. The higher faculty,
although potentially present in every way, is ever held in abeyance
while the lower is brought, under the rigors of its own evolution, to a
point where its joint operation with the higher may be executed with
the least possible friction and retardation as also with the greatest
possible coördination and coöperation. Accordingly, notwithstanding the
fact that materiality must possess in potentiality all the qualities
which it will at any time reveal, it is nevertheless necessary that
these qualities shall come forth gradually and in due order. Similarly,
humanity has come into possession of its various faculties of mind, and
powers of physiological functions, by insensible degrees, the higher
always being held in abeyance until the lower is fully developed.
Those faculties which are to bestow added powers, additional freedom
and a greater scope of motility are the ones which appear later than
those which are truly primitive in character. These facts have been
amply demonstrated by the science of embryology wherein it is shown
that _ontogeny_ is a recapitulation of _phylogeny_. That is, the
history of the development of the individual is a recapitulation of
the development of the species. Thus the various stages of development
through which the human embryo passes while _in utero_ are but a
repetition of similar stages through which the entire human species has
passed in its phylogenetic development. Wherefore, it is certain that
humanity has not attained, at one and the same time, all the powers of
mind and body which it now possesses; that the childhood of the human
race represented a time when it had but few faculties or organs of
sense-perception--indeed a time when the higher sense-organs of smell,
taste and sight were entirely lacking although residing in potentiality
therein.

It is undoubtedly true that the earth has passed through a similar
evolution with respect to its own material characteristics, that
its childhood was, in all points, analogous to the childhood of
humanity; that the air, earth and water were wholly absent, except in
potentiality, during the nebulous youth of its genesis. It is even
probable that there are at work to-day processes which in the future
shall culminate in the evolution of newer, higher and more complicately
organized species of plants, animals and minerals. Every year brings
fresh evidences that crystallize the conviction that the earth has been
the scene for the appearance of many strange orders of animal life.
Fossiliferous strata are continually yielding incontestable testimonies
of changing flora and fauna. We count the animal and vegetal life of
to-day as being more highly developed than that of any other previous
age, and it is well that this is so, for simplicity of organization and
primality of manifestation are always succeeded by complexity and a
greater scope of adaptability.

We have said that the whole of that movement of the intellect which
has brought forth the metageometrical creations of hyperspaces, the
curvature of space and its manifoldness together with the entire
assemblage of mathetic contrivances are merely the early evidences of
the appearance in the human race of a new faculty, a new medium of
perception whereby the Thinker shall acquire a still greater range of
motility than that now offered by the intellect. Attention has been
called also to the fact that this phenomenon has been manifested not
alone in the field of mathematics, but in art, religion, politics
and also in science in which we have only to witness the marvelous
strides already made in the discovery of radio-active substances,
the ROENTGEN, BECQUEREL, LEONARD and other kinds of rays. It Is quite
confidently believed that these forward movements in every branch of
intellectual pursuit, these combined efforts of the intellect, in
peering into the occult side of material things, are in response to the
evolutionary needs of the Thinker, and in addition, are the evidences,
and shall in time be the cause, of the development of an additional set
of faculties. Function, or the performance of acts, determines faculty
or the power of action and ultimately the organ itself. Thus the mere
wish to perform aroused by desire and vitalized by the will actually
terminates, in the course of time, in the genesis of a faculty, or the
power to perform. The constant upreaching yearnings of the Thinker
through his intellect for greater freedom and a larger scope of action,
the desire to peer into the mysteries of life and mind, the infantile
out-feelings of the mentality after some safer and surer basis for its
theory of knowledge cannot fail in producing not only the faculty or
power to satisfy these cravings but the very organ or medium by virtue
of which the satisfaction may be attained.

It is not strange that in mathematics the intellect should have found
first the clue to the existence of a higher sphere of intellectual
research wherein it might become the creator of the various entities
which peopled the new found domain; it is not strange that the
mathematician should, in this instance, have assumed the role of
the prophet proclaiming by various mathetic contrivances (although
unconsciously) that the human race is nearing that time when it shall
actually be able to function consciously in some higher sphere;
neither is it to be wondered at that the voice of the prophet is heard
and respected throughout the earth; for, indeed the mathematician is
a spokesman who, as a rule, is unmoved by sudden outbursts of passion
and ecstatic frenzies of emotions but calmly and dispassionately
verifies his conclusions, tests them for consistency and having found
them to satisfy the most rigorous mathetic requirements hesitates not
to propound them. For this cause humanity respects the mathematician,
and when he speaks listens to his voice. It is well, too, that this
is so; for the history of mathematics is clearly the history of the
development of the intellect. So exact a determinator of the quality
of intellectual efficiency is it that its reign may be said to be an
absolute monarchism whose lines of dominance extend to the minutest
desire or appetency. It has always been the guide of the intellect,
going before, as it were, blazing the trail, pushing back the frontiers
of the intellect's domain and clearing away the _debris_ so that the
intellect with its retinue of servitors might have an easy path of
progress.

Mathematics, however, has not the aptitude to serve the intuition as
it serves the intellect. So the path into which the intuition would
lead humanity the mathematician, because of his training and peculiar
functions, is unprepared to enter. It is for this reason that when
mathematics leads the intellect up to that point where it encounters
life it fails, it becomes confused and its dictatorship becomes a
mockery, its decrees remain unexecuted and futile. In taking this view
we have certainly no desire to offend the mathematician or to detract
from the glory of his monarchistic rulership over the intellectual
progress of the race; for, in truth, mathematics is the diadem of gold
wherewith man has crowned his intellect. Yet it is well, yea imperative
in the light of recent developments in the realm of hyperspace, that
a careful discrimination should be made as between the sphere of the
intellect and that to which the intuition shall attain.

The intuition, long held in abeyance until the intellect should be
fully crowned and reach the zenith of its powers, is now coming to the
front. It will be many centuries perhaps before it shall have grown to
such proportions as those already attained by the intellect; perhaps
a few thousand years may pass before the intuition shall have evolved
to that point where it may labor as coadjutor to the intellect; but
undoubtedly the time will come when it, too, shall reward the Thinker's
labors with that which shall be more precious than the crown of gold
which the intellect has won. Then, the intellect, grown old and
decrepit with years of reigning shall become dim and crystal-shaped and
finally pass into automatism or reflexive movements where without the
urge of volitional impulses it will perform with exactness, precision
and utter loyalty the tasks which it has learned so well to execute
in the days of its forgotten glory. Mankind will then be free. A new
freedom, wherein the erstwhile lightning flashes of intuition will
become fused into one glorious sheen of all-revealing light, shall come
to men and thus the race resplendent will walk the earth enshrined in
the majesty of divine powers attained as a result of millions of years
of aspiration.

That there are supersensuous realms so far above the range of our
senses as to be entirely beyond their ken needs now no proof or
argument; for the scientist has demonstrated, by the invention of
instruments of extreme delicacy and precision, that such a world does
really exist. Already we know of stars so distant that, though light
traverses in the brief space of an hour six hundred million miles,
they might have ceased to shine before the pyramids were built and
yet be visible to us in the skies. If the human eye were as sensitive
as the spectroscope many thousand tints and shades might be added to
the world of color; if they possessed the magnifying powers of the
microscope we should live in constant terror and awe of the monstrous
entities that teem in the water which we drink and in the air which
we breathe; and if our ears could detect the microphonic vibrations
which register in the delicate apparatus of some microphones the dead,
vacuum-stillness of nature's great silences would appear as a babel of
voices by the seaside. The sense of touch, responding to the same range
of vibrations as the micrometer, would reveal actually the interstices
between particles of the densest elements; and gold, silver, platinum
and mercury would seem but honeycombs of matter. But, to the
forward-looking there is no element of absurdity in the expectation
that all these senses shall, one day, be able to dispense with the
artificial aid of physical apparatus and perform, with even greater
precision and faithfulness, the task which they now perform so crudely
and ineffectively. There are without doubt vibrations of taste and
smell which are so far above the range of these senses that they have
no effect upon them whatsoever. Notwithstanding the fact, however, that
the galvanometer, microscope, the microphone, the spectroscope and the
telescope have extended thus the sphere of sense-knowledge there are
yet subtler vibrations to which these delicate instruments do not and
ought not be expected to respond. But to say, as do many empiricists,
that since these phenomena cannot be detected by scientific instruments
they do not, therefore, exist seems to be expecting too much of
material means as well as exposing oneself unnecessarily to criticism
on the grounds of extreme materialistic appetences.

There is indeed need of a more liberal attitude among men of science
towards the world of the unseen. Intolerance of the data which it
offers will for a time perhaps preserve the aloofness of scientific
dogmatism inviolate but there will most surely come a reaction against
the dogmatism of science and men will seek freedom and attain it
despite their fetters. Sir OLIVER LODGE, in his book, the _Survival
of Man_,[29] says: "Man's outlook upon the universe is entering upon
a new phase. Simultaneously with the beginning of a revolutionary
increase in his powers of physical locomotion--which will soon be
extended to a third dimension and no longer limited to a solid or
liquid surface--his power of reciprocal mental intercourse is also in
process of being enlarged; for there are signs that it will some day be
no longer limited to contemporary denizens of earth, but will permit a
utilization of knowledge and powers superior to his own, even to the
extent of ultimately attaining trustworthy information concerning other
conditions of existence."

  [29] See pp. 338, 341.

It is the author's good fortune that he has for a period extending over
several years been able to verify the conclusions which Sir OLIVER
LODGE expresses in the above, and thus to satisfy his own mind that the
process by which man's mental powers are "being enlarged" is indeed
demonstrable by actual observation and experimental methods.

LODGE continues:

  "The boundary between the two states--the known and the unknown,
  is still substantial, but it is wearing thin in places, and like
  excavators engaged in boring a tunnel from opposite ends amid the
  roar of water and other noises, we are beginning to hear now and
  again the strokes of the pick-axes of our comrades on the other
  side."

CAMILLE FLAMMARION[30] cites 436 cases of psychic manifestations
examined by himself and which establish beyond any reasonable doubt
that there are certain perceptive faculties, namely, clairvoyance and
clairaudience, that crop out in certain individuals, sometimes in
abnormal conditions and sometimes normally, the very unusual character
of which proves their rudimentary nature and the potency of their
maturescence in the humanity of the future. Among the cases cited by
FLAMMARION are 186 instances of manifestations from the dying received
by persons who were awake; 70 cases were manifestations received by
persons asleep; 57 were observations of direct transmission of thought
without the aid of sight, hearing or touch or other physical means; 49
were cases of sight at a distance or clairvoyance by persons awake,
in dreams or in somnambulism and 74 cases of premonitory dreams or
predictions of the future. Indeed, there are few persons now living
who have not had similar experiences, if not exactly like these, of
the same nature. These examples, of course, may be greatly multiplied
in every country in the world, and it is unnecessary to enumerate
them further; for, when once the existence of such faculties has been
demonstrated in persons, either in a normal or an abnormal condition,
their presence can no longer be questioned by the fair-minded. It is,
then, only a question of evolution before they will appear in the
normal way and their universalization, as transmissible characters, be
an accomplished fact. When we are brought face to face with this sort
of phenomenon which seems to be increasing rapidly the conclusion is
inevitably forced upon us that since evolution must be a continuous
process and matter destined to yield higher and more refined powers and
humanity to come into a far more extensive scope of motility because
of the opening avenues of knowledge, it is not impossible that these
acuter senses, these new faculties are now existing in the human
race in a rudimentary stage and are designed to become the universal
possession of all. That this is to be the almost immediate outcome
of the perpetual exalting power which life exercises not alone over
materiality but over human organs and faculties as well, seems to be
the one big, outstanding implication of the evolutionary process. The
presence of such functions as the ability to sense the invisible and
the inaudible, to answer to vibrations far subtler than anything in the
scope of our external sense-organs, certainly indicates the existence
of rudimentary faculties which make these functions possible. _Back
of these vague, indefinite functions, back of every supernormal or
abnormal manifestation of man's mentality and back of all that class of
phenomena which take their rise out of supersensuous areas must lie,
in ever increasing potency, faculties and organs, however rudimentary,
which are the source of these manifestations._ Life, that ineluctable
agent of creation, which is incessantly pushing outward the confines
of the intellect's scope of motility, never wearying, never tiring
nor sleeping, has long ago, in the dim and distant past of man's
evolution, laid the foundations; and in fact, with one stroke of its
creative hand, has molded the organs which are to become the active
media of these new faculties. And now, these incipient demonstrations,
these infantile struggles which we see now and again outputting from
them, are but the specializing processes through which, in their later
development, these organs are proceeding. These are the outward signs
which should tell us that life is breaking up these organs into special
parts, assigning to each a certain division of labor and making of
each a perfect coördinate of all the others. It is, by these very
dispread exhibitions, cutting up, specializing and by slow degrees
determining the function, character and general tendence of the organs
of expression wherewith these manifestations shall be centralized and
put into effective operation. In doing this, it is but following its
accustomed procedure, the procedure which it adopted when it produced
the eye, the ear, the heart and the spleen. We shall, therefore, gauge
our understanding of the purport and end of evolution; in fact, we
shall determine our exact intellectual comprehension of life itself, by
the attitude which we adopt towards it and the mode of its appearance.
Much depends, accordingly, upon the posture which we assume towards
life--whether we shall say the totality of life's creative powers has
been dissipated in the bringing of the human body to its present degree
of perfection; whether we shall say that it is neither necessary nor
possible for life to produce other organs and faculties which shall
respond to the unseen world about us revealing its glories in a way
far more perfect than do our external sense-organs reveal the wonders
of the world of sensation; or, whether we shall conclude from these
most palpable evidences that life has yet other powers and faculties
which it designs to bestow upon the human mind and other organs and
capabilities with which it shall endow the human body so that man,
in his evolution, shall be enabled to rise to still higher spheres
while yet incarnate. There may be, and undoubtedly are, those who,
for various reasons prefer to take the former positions and there are
certainly those who like LODGE, FLAMMARION, HUDSON, CROOKES and a host
of others, preferring the latter view, would rather believe in the
strength of the great mass of corroborative testimonies that we are
even to-day in the midst of the matutinal hours of a newer, a better
and a far more efficient era of human evolution than any through which
we have hitherto come.

  [30] See _Unknown_, p. 485, et. seq.

Already, recent scientific investigations and the results obtained
therefrom have begun to turn the attention of medical authorities to
the activities of two very small organs situated in the mid-brain and
known as the _pineal gland_ and the _pituitary body_. These organs,
and especially the _pineal gland_ hitherto supposed to be a vestige
of the past, are now beginning to be recognized as rudimentary organs
belonging to the future evolution of humanity. Dr. CHARLES DE M.
SAJOUS, who is an authority on the _pituitary body_, believes that it
has no active internal secretions but is an "epithelio-nervous organ"
which controls, through nerves leading to the adrenals and thyroid
bone, the processes of general oxygenation, metabolism and nutrition.
Little is known of the functions of the pineal gland except that it is
an ovoid, reddish organ attached to the posterior cerebral commissure
projecting downward and backward between the anterior pair of the
_corpora quadrigemina_. It is otherwise known as the "_conarium_" the
"_pinus_" or "_epiphysis_." Situated at the base of the brain, it
is held in position by a fold of the _pia mater_ while its base is
connected with the cerebrum by two pedicles. It contains amylaceous
and gritty, calcareous particles constituting the brain sand. There
are, however, marked structural resemblances between the _pineal
gland_ and the _pituitary body_ and their formation is perhaps the
most interesting feature of the development of the _thalamencephalon_
or mid-brain. The _hypophysis cerebri_ or _pituitary body_ is a small,
ovoid, pale, reddish mass varying in weight from five to ten grains
and situated at the basal extremity of the brain in a depression of
the cranium known as the _sella turcica_, a configuration very much
like a Turkish saddle in shape. It is a composite, ductless gland and
consists of two divisions, an anterior and a posterior, connected
by an intermedial portion--all of which are attached to the base of
the cerebrum by the _infundibulum_. The anterior lobe is larger than
the posterior and very vascular, springing in its development from
the buccal cavity of the embryo; the posterior lobe is situated in
a depression of the anterior and is a brain-process. The _pituitary
body_ itself is lodged in a cavity of the _sphenoid_ bone called the
_pituitary fossa_. This is a most remarkable position, for the reason
that the _sphenoid, or wedge-shaped_, bone which lies at the base of
the skull articulates from behind with the occipital and in front with
the _frontal_ and _ethmoid_ bones and by lateral processes with the
_frontal_, _parietal_ and _temporal_ bones. From this position it binds
together all the bones of the cranium, and moreover, articulates with
many bones of the face. It is upon the upper surface of the _sphenoid_
bone which occupies such a prominent and commanding position in the
cranium, in a deep depression, that the _pituitary gland_ is located.

Each nasal chamber is lined by a mucous membrane called the _pituitary_
or _Schneiderian_. This membrane is prolonged into the meatuses and air
sinuses which open into the nasal chambers. The _pituitary_ membrane is
thick and soft and diminishes the size of the meatuses and air sinuses.
It is covered by a ciliated columnar epithelium and contains numerous
racemose glands for the secretion of mucous or _pituita_. It is also
vascular and the veins which ramify it have a plexiform or net-work
like arrangement. It divides into two membranes--a respiratory, which
is concerned in breathing, and an olfactory region. The respiratory
region corresponds to the floor of the nose, to the inferior turbinated
bone and to the lower third of the nasal septum. The olfactory region
is the seat and distribution of the olfactory nerve and corresponds
to the base of the nose, to the superior and middle turbinals and the
upper two-thirds of the nasal septum.

Recent developments prove that this gland has a profound influence
over the animal economy. It is believed by some that the _pituitary
body_ actually destroys certain substances which have a toxic influence
on the nervous system; others believe that it secretes material
media for the proper action of the trophic or nutritive apparatus;
still others believe that it influences blood-pressure. It is known,
however, from experimentation, that its removal in dogs, cats, mice
and guinea pigs causes a fall of temperature, lassitude, muscular
twitchings, dyspnoea or difficult breathing, and even speedy death.
Hypertrophy of the gland is directly associated with certain diseases,
such as _giantism_ and _acromegaly_. The latter is a disease which
causes a general enlargement of the bones of the head, feet and hands,
usually occurring between the ages of twenty and forty years, and most
frequently in females. The fact that these diseases are so closely
associated with a hypertrophic condition of the pituitary gland has led
to the conclusion that perhaps the giants or Cyclops of ancient times
were cases of _giantism_ or _acromegaly_. This view, while interesting
from the standpoint of the functions of the _pituitary_ gland, is not
necessarily a correct one; for the age of giants, when men attained
to a much larger stature than at present, can be accounted for on
other grounds, namely; that the early mesozoic man, on account of his
having to live among animals, trees and other vegetation of such huge
size, had naturally to be fitted with a frame proportional to other
animals in order that he might successfully cope with his environing
conditions. Nature thus wisely fitted him for the conditions which she
had prepared for the scenes of his life.

The facts adduced in the foregoing description are purely empirical
and may be verified by any who seek to establish their correctness or
incorrectness. But we are about to introduce a species of testimony
which while it may also be verified will not be found so easy of
verification as the above-mentioned physiological facts, and not by
the same means; yet they are nevertheless deserving of a place here.
It is the liberal attitude that we must adopt towards all phenomena,
excluding none that give promise of the widening and deepening of
our knowledge and an explanation of much that has seemed heretofore
unaccountable.

We have noted how subtle is the physical connection between these two
bodies, the _pineal gland_ and the _pituitary body_; we have seen how
profound is the effect which the latter has been demonstrated, in a
measure, to have over the entire bodily economy; but there is even
other testimony to the effect that those gifted with the inner vision
can observe the "pulsating aura" in each body, a movement which is not
unlike the pulsations of the heart and which never ceases throughout
life. In the development of clairvoyance it is known that this motion
becomes intensified, the auric vibrations becoming stronger and more
pronounced. The _pituitary body_ is the _energizer_ of the _pineal
gland_ and, as its pulsating arc rises more and more until it contacts
the _pineal gland_, it awakens and arouses it into a renewed activity
in much the same manner as current electricity excites nervous tissue.
When the _pineal gland_ is thus aroused clairvoyant perception is
said to become possible. These are facts which cannot be proved by
the materialistic man of science nor can they be demonstrated to the
layman who has to depend alone upon sense-deliveries for his knowledge.
This is true for the reason that, in the first place, it is necessary
that he shall either feel in his own mid-brain the energizing activity
of these two organs and have his entire nerve-body shaken from crown
to toe by the down rushing currents of that subtle energy with which
the _pituitary body_ floods it or be himself the perceiver of its
activities. Nevertheless attention is here called to these phenomena
and the conclusions drawn therefrom are offered as a means of denoting
the probable line of investigations which will establish the directions
which we should pursue and the source whence we shall find outcropping
the new faculties and their organs of expression.

We confess to a knowledge of the fact that men of empirical science
have long maintained a rather skeptical, if not contemptuous, attitude
towards all these phenomena but it is also felt that there is far more
of discredit in their attitude than of credit; for, in so doing, they
have voluntarily adopted measures by means of which the knowledge
that they so eagerly seek is shut out from their attainment. In vain,
then, is appeal made to the intellect to remove the barriers which it
unconsciously interposes between itself and the goal of its pursuit;
in vain do we appeal to the materialist to give ear to testimony the
data of which cannot be made amenable to his knife and scalpel neither
to the microscope nor microphone; in sheer vanity is he adjured to
look _within_--into the interior of life, of mind and the things which
he handles with his instruments--for the answers to his queries, for
the path which leads into the wake of life and consciousness. Because
his utter loyalty and devotion to the _modus vivendi_ of the intellect
will not permit this; but, after all, it is not wholly wise to allure
him away unbetimes from his search after truth through superficialities
nor to inveigle him into giving up his tenacious prosecution of the
physically determinable. We would not have it so; for, perchance, he,
too, one fine day, in the quiet of his laboratory shall come upon
the data which may substantiate in his own mind the long settled
conclusions of the occultist who, frequently and not without cause,
grows impatient at the scientist's obstinate delay. These two workers,
the empiricist and the occultist, must ultimately come together as
collaborators--the one working upon the form, the vehicle, physical
matter and the other seeking to understand the life, the interior
forces which produce, the creative element. They cannot remain always
aloof from one another; for they, too, are as men digging a tunnel from
opposite ends. Finally, the partition will break and thus will dawn a
new day for the knowledge of humanity and men will see the rationale,
the truth and good sense of coöperation in this respect.

It can be said with confidence that whatever in the future may be
learned as to the physiologic functions of the _pituitary body_ and
the _pineal gland_, it suffices to know that it is life which they
express and that, too, in a far superior manner than any of the other
sense organs. The _modus_ of these two glands differs in a very
marked way from that of the organs of sight, hearing, taste, smell and
feeling. For these latter are designed for contact with the external,
objective world of sensations, their growth and evolution being
dependent upon stimuli received from without while with the former the
case is far different, in fact, just the opposite. The mode of life
of the _pituitary body_ and the _pineal gland_, instead of receiving
sustenance and impetus from external stimuli, is rather dependent
upon impacts received from the Thinker's own consciousness and made
to impinge upon them by an exclusively interior process. Situated in
the mid-brain, safely secluded from all external interference, they
are naturally limited to stimuli which come from within, or it may
be said, they are responsive to excitations that are more spiritual
than those which come through the external sense-organs. If, as has
been said they control the internal processes of metabolism (anabolism
and katabolism), oxygenation, nutrition, and other important internal
movements, none of which can be said to be under the control of the
intellect, is it not, therefore, justly assumed that their response
is directed towards stimuli which arise interiorly or upon a plane
higher than the intellectual? It is a matter of scientific knowledge
that those persons gifted with clairvoyance, and commonly known as
"sensitives" are far more responsive to nervous excitation than those
who are not so gifted. This would seem to imply that, on account of
the superactivity of these two organs, the entire nerve-body has, in
consequence, become more delicately and subtly organized. They seem
to act as a switchboard for the regulation of the flow of the current
of life through the body. Not only do they come more nearly to an
adequate expression of the physiologic function of life, but, as their
energization means an enlargement of the scope of perception by giving
the Thinker's active consciousness access to hitherto unapproachable
realities and by penetrating the outer mask which life ensouls and also
laying bare a domain of unlimited knowledge the manifestation of which
is far more real than anything the senses can disclose, it is evident
that they constitute, in their collaborative functions, a more highly
adaptable medium for the expression of the Thinker's consciousness. And
if so, for the kosmic consciousness which is the source of all forms of
consciousness, they furnish a specializing and _adaptizing_ agency.

Now, in all those cases of inspirations, revelations, telepathic
communications, clairaudience, clairvoyance, dreams, visions, etc.,
wherein the Thinker is enabled to perceive facts and verities which
are then presented to his consciousness in a manner clearly without
the province of the common sense-organs, it must be apparent that
these manifestations are apprehended by a perceptual mechanism which
is entirely independent of external sense presentations but which is
an interior and subtler form of psychic activity. Sounds which are
heard by so-called "sensitives" and objects which are perceived by eyes
that are keener than those organs said to have been evolved from the
"medusa" cannot be heard by other persons nor perceived by them in any
way. Thus it would seem that there are inner organs of perception which
respond to these finer vibrations and which enable the person so gifted
to apprehend them.

There are those who, presumably basing their assertions upon actual
observation and knowledge, unqualifiedly assert that in order "to
gain contact with the inner worlds all that remains to be done is the
awakening of the _pituitary body_ and the _pineal gland_. When this
is accomplished man will again possess the faculty of perception in
the higher worlds, but on a grander scale than formerly (when humanity
was in its infancy and exercised a lower form of psychic power only);
because it will be in connection with the voluntary nervous system, and
therefore, under the control of the will. Through this inner perceptive
faculty all avenues of knowledge will be opened to him and he will
have at his service a means of acquiring information compared with
which all other methods of investigation are but child's play."[31]
It is the lack of this ability to see, with our physical eyes, as it
were, by the "Roentgen rays," to penetrate the inwardness of things
that has baffled and confounded men for so long a time and which has
eventually led certain mathematicians and others to conjecture such
strange, and in many cases, illogical possibilities for the denizens
of four-space. This inability together with the desire to fathom the
innermost complexities of solids and to handle, albeit with unholy
hands, the supersensuous, the mysterious and the unapproachable
identity of "things-in-themselves" have induced the more zealous among
them to contrive some kind of hypothesis which would, at least, offer
an explanation of these phenomena. It has driven them to wrestle
with metaphysical possibilities in a vain endeavor to grasp that
which, _ignis-fatuus_ like, ever evades their slightest intellectual
approach. But why this prolonged struggle, why this intellectual
maneuvering and sophistry? "We can calculate, compute, excogitate,"
says PAUL CARUS,[32] "and describe all the characteristics of
four-dimensional space, so long as we remain in the realm of abstract
thought and do not venture to make use of our motility and execute
our plans in an actualized construction of motion; but as soon as we
make an _a priori_ construction of the scope of our motility, we find
out the incompatibility of the whole scheme." Thus mathematicians are
forced to relinquish all hopes of transforming the world of life into
a sort of four-space dwelling place where everything is done according
to the laws of mathematics. But whether they shall accept it or not
there is a wider, truer and more rational view which recognizes all
metageometrical investigations, as well as all kindred phenomena, as
universal evidences indeed, as the very causes which, in the future
humanity, will actually awaken and cause to be accelerated in their
development these little inner sense-organs, the _pineal gland_ and the
_pituitary body_, whose perfect development promises to provide for
the Thinker's consciousness an avenue of expression such as humanity
has possessed never before. And too, it is not without full knowledge
of the fact that it has been customary, among certain scientists or
perhaps all of them, to regard these bodies, at least the _pineal
gland_, as vestigal organs belonging to the past of human evolution,
that we make these assertions. Yet, as man proceeds in the perfection
of mechanical science, in the development of instruments of precision
that aid his external senses, responds more and more to the subtle
vibrations teeming everywhere in the atmosphere about him, and comes,
in the course of time, naturally to possess a more sensitively keyed
nervous mechanism, a finer body and higher spiritual aspirations,
there will be a corresponding widening of his scope of vision and the
attainment of larger powers of perception which must inevitably, in the
very nature of things, tend towards a deeper and truer knowledge.

  [31] _Rosicrucian Cosmo-Conception_, p. 477, MAX HEINDEL.

  [32] FOUNDATIONS OF MATHEMATICS, p. 90.

In view of the foregoing, it is believed that the general results of
this pituitarial awakening which may be expected as humanity continues
to evolve should be seen in the marked effects which will be wrought
in the entire metabolistic area of the human body whereby a gradual
intensification and sensitization of the whole neural mechanism
will raise the peculiar efficiency of all the senses whether purely
physiologic or psychic. For there are undoubtedly notes so delicate in
their intensity that they transcend the grasp of the audital nerves;
scents and fragrances so subtle in their excelling purity that it is
beyond the powers of our present olfactorial contrivances to detect
them; colors and other external stimuli so sublimely supersensuous that
a nervous mechanism perhaps ten-fold more delicate and responsive than
ours is required to apprehend them. All these, and more than at present
is conceivable, will come, with the aid of pituitarial stimulation,
within the purview of a more highly developed humanity of the future.
And because mathematics have led a movement into the very camp of
the intellectuals--logic-bound and tethered by the severest rigors
of mathesis--whereby the intolerant intellect has been compelled,
by rules of its own making, to recognize the existence of the
supersensuous, and by looking into the glaring light of the sun of the
intuitable to gain strength of vision and boldness to press forward,
a great and far-reaching service has been wrought for humanity. And
in the tower of hyperspace mathematics have erected a monument to the
intellect which, as long as the human race remains, will mark the great
turning point in man's path to the highest life.

What if it were possible that the scientist, when he had carried
instruments to their utmost precision and penetration, should suddenly,
or otherwise, be endowed with a clear-perceptivity of sight, hearing
and smell, so that he could with his own powers of vision, feeling and
hearing take up the task where the microscope, the microphone and the
micrometer left off and delve into depths far too unfathomable for his
appliances, perceiving the innermost realities of things and processes?
What if it were possible for him, with these added powers, to see and
examine without the aid of the magnifying lens the electron, the atom
and the molecule? What if the cell, the bacterium, and other invisible
forms of life would then deliver up their secrets to his knowing mind?
What if he could sense with his own inner vision, the ultra-violet and
the infra-red rays; what indeed, if spirit itself, the innermost sheath
of life, should be visible and palpable to him and he could note the
internal processes, the action and movements of the infinitesimals
of life? Think you not that such direct contact, such immediate and
incontrovertible knowledge would be far superior to any advantage which
his manufactured devices now bestow? It is even so.

Thus will react upon man's perceptive apparatus the flood of light
which the awakened intuition will shed upon them and thus will man rise
higher, driven on by the current of life with the mass of materiality,
to a point of complete spiritualization and take additional steps in
that direction which leads to Raja Yoga or the Royal Union with the
divine life of the universe.

Before this step is taken, however, and before the passage from
mechanics to biogenetics is made, as made it must be, man must win a
complete mastery over matter. But this he will do; for more and more
he is learning to put all those forms of labor which are so exacting
as to leave him no time for the development of his higher powers into
the hands of machinery. He will not be free until he has done this
well-nigh completely. This is the task of the intellect and with it
man must win his way to these higher faculties which are destined to
succeed the intellect whereupon he will be ushered out of a life bound
and restricted by mechanics to a life of unimaginable freedom, the
_intuitive life_.

The outcome of these new faculties of perception and the development of
the intuition will be the springing up of a new species of art that,
turning away from appearances and sinking beneath or rising above,
superficialities, will seek to portray in newly found colors, the
plastic essence of things so that we shall have an art which pertains
to the real, superseding that which pertains to the phenomenal.
Language and the need of it will pass away; for man will have outgrown
the use of signs and symbols in his communion with his fellows and will
use the language of the intuition--direct and instantaneous cognition.
Philosophy will be regenerated, re-created. Speculation will give
way to truth and there shall be but one philosophy and that shall be
the _knowledge of the real_. Mathematics, the royal insignia of the
intellectual life, because it can deal only with immobilities, with
segments and parts and has no aptitude for the continuous flow, will
yield its kingdom to a higher form of kinetics which will serve the
intuitive faculty as mathematics now serve the intellect. Science will
then be no longer empirical in its method; but a system of direct and
incontrovertible truths. Religion will rise to meet these changes which
will come in the path of human evolution; and faith will surrender its
place to knowledge. Ethics, recast in a new mold, will deal with the
new aspect of man's relation to his fellowmen. Man, for whose highest
good these ultimate changes will come, will be a new creature, a higher
and better man; and humanity shall evolve a new race. There shall,
indeed, be "a new heaven and a new earth."


  THE END.




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     Space_.


  OPEN COURT (Magazine):
    Vol. XVI, pp. 513-521.
    Vide December, 1913 edition, p. 757.


  PHILOSOPHY OF MATHEMATICS, by PAUL CARUS.

  PHILOSOPHICAL REVIEW:
    Vol. V, 1896, p. 175, F. C. S. SCHILLER.
    Idem, p. 357, J. H. HYSLOP.
    Vol. VII, p. 615.
    Vol. X, pp. 113, 229, 314, 375, 488, 579, 583.
    Vol. XII, p. 493.

  POPULAR SCIENCE MONTHLY:
    No. 66 (1905), pp. 639-646.
    Vol. LXVII, pp. 639, et seq.
    Vol. LXVIII, pp. 21, et seq.
    Vol. LXXVIII. p. 554 (1911), SAMUEL MOFFAT WEYER.

  POPULAR ASTRONOMY:
    Vols. VII and VIII (1900), G. B. HALSTED.
    Vol. XVIII, pp. 42, et seq. (1910), W. H. PICKERING.
    Vol. LXXV, p. 179, W. W. PAYNE, on _Attraction and Figure of the
     Earth_.
    Vide No. 84 (1901), pp. 187-190.

  POPULARIZATION, THE, OF NON-EUCLIDEAN GEOMETRY:
    _Amer. Math. Mo._, Vol. VIII, pp. 31-35.

  PSYCHOLOGICAL BULLETIN:
    Vol. VIII, p. 22 (1911).


  QUEEN'S QUARTERLY:
    Vol. XX, pp. 431-446, A. L. CLARK (1913).


  REVUE GENERALE DES SCIENCES:
    No. 23, trans. _Nature_, Vol. XLV, 1892.

  ROYAL ASTRONOMICAL SOCIETY PROCEEDINGS:
    Vol. XVI, pp. 80-85.

  ROYAL SOCIETY PROCEEDINGS, LONDON:
    Obituary Notices of Deceased Fellows, Vol. XVI, p. 69 (1867-68).

  ROYAL SOCIETY OF N. S. W. JOURNAL AND PROCEEDINGS:
    Vol. XXXV, p. 243 (1902), G. H. KNIBBS.


  SCIENCE (Magazine):
    Vol. III (new series), No. 58, 1896.
    Vol. VI, pp. 487-491 (1897).
    Vol. VII, pp. 1-7, _Philosophy of Hyperspace_, S. NEWCOMB.
    Idem, p. 861 (1898).
    Vol. XIII, pp. 462-465.
    Vol. XIV (new series), pp. 711-712, G. B. HALSTED.
    Vol. XIX, pp. 401-413.
    Vol. XX, pp. 353-456, JOSIAH ROYCE.
    Vol. XXII, pp. 270-271.
    Vol. XXXV, No. 906 (1912).
    Vol. XXXVII, pp. 885, et seq., C. J. KEYSER, on _Concerning the
     Figure of the Universe of Space_.

  SCIENCE ET L'HYPOTHESE, R. POINCARÉ.

  SCIENTIFIC AMERICAN SUPPLEMENT:
    Vol. LXVI, pp. 14-16 (1908).

  SCIENTIFIC ROMANCES, C. H. HINTON.


  THOUGHT, NEW ERA OF, C. H. HINTON.


  WESTMINSTER REVIEW:
    Vol. CLII, pp. 675, et seq.


  ZOLLNER, J. C. F., _On Space of Four Dimensions_:
    _Quar. Jour. of Science_, Vol. VIII (1878), pp. 227-237.




  INDEX


  A

  Abeyance, higher faculties held in, until lower are fully developed, 334

  -- intuition kept in, 338, _ff_

  Ability, to sense the inaudible and the invisible, 342

  Absolute, no room for the, 101

  Abstract thought, invigorating power of, 33, 40

  Abstractions, realizing, 144, 294

  Absurdity, analysis inevitably ends in an, 319-20

  Acrobatics, mathetic, 146

  Acromegaly, 347

  Action, automatic, of the intellect, 253-4

  -- engendering, of life on the form, 330

  Activities, space, as physical and chemical phenomena, 229-30

  Actualities of the physical world pushed over into the conceptual, 119

  Actuality imperceptible to the intellect, 126

  Adaptability of life, 330

  -- -- mathematical laws, 37-39

  Adaptation, purpose of, 327-8

  Adaptations made by an organism, 162

  Additional freedom bestowed by higher faculties, 334

  Affinity, zones of, 22, 124

  Agency, interpretative, intellect as an, 166-7

  Agent, ego as an, of the Thinker, 243, _ff_

  -- engendering, life as an, 331

  -- of creation, life as an, 343

  Ahmes, an Egyptian priest, 44

  Alchemy of psychogenesis, 225

  Alcohols, eight different, from one formula, 155

  Algebraic quantities and space, 125-7

  Allowability of the rules of logic, 163

  Aloofness, the, of scientific dogmatism, 340

  Alphabet of space-genesis, 237

  -- the geometric, 193

  Amenability of mind to the laws of evolution, 28-31

  Analogies, use of, to popularize the four-space, 128-9

  Analogy, difference between two processes illustrated by, 74

  Analysis incapable of dealing with life, 319

  -- the manifold, the fiat of, 77

  Analyst and the manufacture of the space-manifold, 77

  -- disregards the conformity of the nature of things, 84

  Analytics and the mechanical origin of the universe, 40-1

  -- perisophical nature of, 320, _ff_

  -- the four-space, a curiosity of, 40

  Anchorage, fourth dimension denied, 265

  Angularity of consciousness, 120

  Answer of the senses to new needs, 332

  -- the, of consciousness to realism, 97

  Anti-Euclidean geometry, 69

  Apotheosis of the definition, 78

  Apparatus, man's perceptive, 357

  Apparent vassalage of life, 329-30

  Appearance, dynamic, space as, 104-7

  Appetences, materialistic, of empiricists, 340

  Appulses of life against matter, 328

  Appurtenances of materiality, 331

  _A priori_, 185, _ff_

  -- -- knowledge, 110-5

  Aprioriness, 112, 116

  Apriority, the principle of, 113, 223

  Arbitrariness of the common mensurative quantities, 41

  Archeological evidences, 332

  Archimedes, 70

  Aristotle, 322

  Arrangement of the contents of the kosmos, 317-8

  Art and the criterion of truth, 323

  -- -- -- higher consciousness, 181

  -- evidences of new faculties in the field of, 335

  Assemblage of mathetic contrivances, 335

  Assumptions, category of, 189

  -- procedure based upon, disadvantages of, 163

  At-one-ment, 224, 252, 270

  -- defined as the end of evolutionary activity, 1

  Attachment of the consciousness to the order of becoming, 305

  Attainment, the difficulty of, by the intellectual method, 206

  -- of space-consciousness, 225

  Attitude, the need of a more liberal, 340

  -- towards things spatial, 306

  Audital nerves, notes that transcend the grasp of, 355

  Aura, pulsating, 348

  Automatism of the intellect, 253, _ff_; 338

  Avenues of knowledge opened to inner perceptive faculties, 353

  Awakening of the faculty of awareness in a new domain, 90

  -- -- -- pituitary body and the pineal gland, 353

  -- pituitarial, general results of, 355-8

  Awareness, a determinant of conception, 120

  -- as a gauge of the existence of things, 161-202

  -- degrees of, 165

  -- hyperspace, a symbol of a more extensive, 180

  -- progress in, 171

  -- Thinker's sphere of, 274, 283


  B

  Baltzer, 66

  Barrier, the, to the Thinker's certitude, 186-8

  Barriers, freedom determined by absence of, 280

  Becoming, endless labyrinth of, 220

  -- the kosmos, in a state of, 265

  -- the order of, 305

  Becomings, infinity of, 234

  Becquerel rays, 336

  Being, identification of consciousness with, 205

  -- kathekotic, 210

  Being, the interiority of, 290

  -- the numericity of, 309

  -- the range of, co-extensive with reality, 168

  Beings, discarnate, 154

  Beltrami, Eugenio, 1-2, 63-5, 66-7

  -- -- and the manifold, 317

  -- -- and the pseudosphere, 64-5

  Besiegement of matter by life, 329-30

  Bessel, on geometry as incomplete, 58

  Bewilderment, new realities cause, 169-70

  -- of the mind, 182

  Biogenetics, passage from mechanics to, 357

  -- turning to, from mechanics, 322

  Birth, as a fourth dimensional process, 159

  Blind men and the elephant, 257

  Bolyai, Janos, 2-3, 46, 60, 66, 83, 87

  Boundaries of a hypercube, 136-8

  Brain, the physics of, 292

  Brain-consciousness, 184, 186

  -- standards of, 190

  Brotherhood, and the tendencies of the Thinker's consciousness, 301

  -- the keynote of the intuition, 252

  Bulwarks, the formal, of geometry, 77-9


  C

  Carpet, the sensible world as a, 196

  Carus, Paul, on four-dimensional space, 254

  -- -- on metageometricians, 102

  -- -- on space, 110

  -- -- on the representation of the tesseract, 136-7

  Cayley, 3-4, 54-5, 66-7

  Cell-activity as a performance _a priori_, 114

  Cell-colony, 250, 301

  Cell-consciousness as an aposterioristic phenomenon, 185

  Centers, pyknotic, 232

  Certainty, geometric, the basis of, 77-8

  -- of mathetic conclusions, provisional, 37

  Chaogenic period of involution, 208, _ff_

  Chaogeny, 231, 233

  -- the evolution of chaos into order, 4

  -- the laboratory of, 193

  Chaomorphogeny, defined, 4, 231-2

  Chaos, as egg-plasm, 265

  -- duration of, 268

  -- movement of life in, 307

  -- shores of, 307

  Chaos-geometricity, 307

  -- -intellectuality, 307

  -- -materiality, 307

  -- -spatiality, 307

  -- -Theos-Kosmos, 215, 233-4, 237

  -- -- -- a triglyph symbolizing kathekos, 11-2

  Character of the universe, fixed by consciousness, 162

  Characteristics, changes in the, of reality, 169

  -- fourth dimensional, of the ether, 157

  -- minds of similar, fall into zones of affinity, 124

  -- of non-Euclidean space, 72-3

  Chasm, kosmic, between the real and the ideal, 107

  Chemists, speculative, and the ourth dimension, 155

  Chrism, sacred, of creative mentality, 91

  Circle, significance of process of squaring, 109

  Circle, the symbology of, 309

  Circuits, closed, our interests as, 167

  Clairaudience, 352

  Clairvoyance, 352

  Clairvoyant perception, 349

  Clause, enabling, of metageometry, 76-7

  Clavius, Christoph, 46, 52, 83

  Clifford, 66

  Code, psychic, for systematizing cognitions, 190-1

  Cognition, instantaneous, organ of, 258

  -- intimacies of direct, 205

  -- the method of, 186-8

  Cognitions, intuitive, 145, 192-3

  -- psychic code for systematizing, 190-1

  Commensurable quality, dimension as, 140

  Communal consciousness, 270, _ff_

  Communalization and the intuition, 301

  Conarium, the, 345

  Conceivability and evolution of mind, 24

  -- ultimate range of, 278

  Conceiving, power of, derived from sense-experience, 26

  Concept, as a shadow, 119

  -- the hyperspace, gradual rise of, 27-8

  Conception and awareness, 120

  -- every, based upon prior experience, 25

  -- freedom, the, of the mind from, 28

  Conceptions, symbolic, dependence placed upon, 147

  Concepts, and the Thinker, 244

  -- intuitograms as, 248

  -- quality of, dependent upon sense-experience, 26

  Concepts, the perception of, 255

  Conceptual, related to the objective, 291

  -- the, and mathematicians, 71

  Conceptualization, the act of, 4

  -- the power of, dependent upon evolution of mind, 25

  Conclusions, mathetic, complex dependence of, 37

  Concrete hewn into shape by the intellect, 295

  Congruency, the, between intellect and the universe, 319, _ff_

  Congruity, the, between concepts and objects, 120

  Connection, the, traced out between reality and object, 126

  Co-originality of things, 303

  Consciousness and kosmos, graph representing, 271

  -- -- the character of the sensible world, 166

  -- -- -- wake of life, 205

  -- -- time, 224

  -- angularity of, 120

  -- _a priori_, 185

  -- a replica of, and judgments, 205

  -- as a scale, 165

  -- -- barrier to ultimate knowledge, 207

  -- -- determinant of dimension, 145

  -- -- life, 295

  -- -- variable quantity, 171

  -- attachment of, to the order of becoming, 305

  -- dawn of, and dimension, 179

  -- degrees of, 162

  -- deprived of the validity of notions, 314

  -- evolution of, 250

  -- expansion of, 164, 272

  -- extended, 89-90

  -- focus of, 290

  Consciousness and the planes of the kosmos, 163

  -- fragmentary view of, 260

  -- freedom of, in the _genus homo_, 332

  -- habitation of, 278

  -- higher, and art, 181

  -- -- planes of, 178

  -- history of, 298-9

  -- identification of, with objects, 189-90

  -- intuitional, 255

  -- keyed to the entire range of reality, 167-8

  -- kinds of, 270, _ff_

  -- kosmic, Elysian fields of, 235

  -- of the primitive man, 253

  -- organ of, 184, _ff_; 90

  -- psychics of, 292

  -- raised from the sensuous to the conceptual, 72

  -- recipient of truths from dual sources, 26

  -- sensible world as instrument of, 199

  -- triple presentation of notions of space to, 72

  -- turned inward, 283

  -- unification of the states of, 192

  -- youthful, of mankind, 122

  Consequences, the science of, 85

  Considerations, mathematical, and reality, 128

  Consistency, kosmic, 174

  -- self, of hyperspatial hypotheses, 67

  -- the criterion of geometry, 86

  -- -- inconsistence of, 173

  -- -- metageometrician curtailed by, 85

  Construction, idealized, cannot be objectified, 144, 276

  -- ideal, the meaning of, 4

  -- mental, trafficking in, 154

  Consubstantiality of intellectuality and spatiality, 331, _ff_

  Continuity of the psychic plasm, 260

  Contrivances, mathetic, the passing of, 241

  Convergence of parallel lines, 315

  Convolutions, duplex, 146

  Coördinates, as spatial determinants, 203

  -- of hyperspace, 94, 97, 99, 115-6

  -- systems of, 301, _ff_

  -- three, 132, 232

  Copolarity of ideas and objects, 127

  _Corpora quadrigemina_, 345

  Corpuscular orbits of particles, 152

  Cosmos, significance of, 4

  Creation, impregnated screen of, 232

  Creations, the, of the intellect, 173

  Criterion of geometry as consistency and convenience, 86

  -- -- truth, 323-4

  Crookes, 344

  Cube, as succession of planes, 148

  -- generation of the, 144

  -- illustrated, 133

  -- the generating, 134

  Curiosities, analytic, energy spent in elaborating misappropriated, 39

  Current, electric, as component in the fourth dimension, 153

  Curvature of space, the doctrine of, formulated by Riemann, 5

  -- -- -- valid element in the notion of, 305

  -- the measure of, 61

  Curved space and metageometricians, 316


  D

  D'Alembert, _note_, 51

  Dante, 181

  Darwin, 181

  Day, the Great Kosmic, 213

  De Tilly, 66

  Death, as fourth dimensional unity, 159

  -- of the intellect, 195

  Decrement of the diametrical ratio to the circumference of circle, 312

  Deeps, fearful, of kosmic mind, 71, 91

  Definition, apotheosis of, 77-8

  -- as an arbitrary determination, 100

  -- the deification of, 37

  Degrees of realism, 164-5

  Deiform, basic idea of the, 5

  Demarkation between reality and phantasy, the line of, 173

  Deposits of life, 173

  Descartes, 322

  Designs, cut out in materiality by life, 264

  Details, the power to dispose of innumerable, 303

  Determinations, geometrical, the necessity of, inheres in logical
   deductions, 78

  -- the factors of conscious, 162

  Determinative period of mental development, 31

  Development of the intuitive faculty, future, 188

  Diacritics of life, 286

  Diameter, the ratio of, to the circumference, 311-2

  Difference between concept and thing, 121-2

  -- between mathematical and perceptual space, 120

  -- between the ideal and the actual is dynamic, 108

  Differential, among minds, 124

  Difficulty of imagining the fourth dimension, 90

  -- of propagating an intuition, 315

  Difficulties in the acquisition of real knowledge, 325

  -- of hyperspace, the logical, 139-41

  Dilemma of metageometricians, 97-8

  Dimension, a distinct stage in psychogenesis, 27

  -- -- system of space measurement, 5

  -- and analysis, 163

  -- -- D'Alembert, _note_, 51

  -- -- the action of a tartrate in, 156

  -- as an arbitrary contrivance, 262

  -- -- -- assemblage of elements, 93

  -- -- direction, 140

  -- -- extent, 98

  -- -- lying near the surface of things, 159-60

  -- -- space, 96

  -- current definitions of, 98-100

  -- denied legitimate anchorage, 265

  -- determined by consciousness, 145

  -- does not explain spiritualism, 155

  -- evolution of the fourth, 44

  -- fourth, direction of, 134

  -- as "jack of all trades," 156

  -- impossible to actualize, 124

  -- key, to non-Euclidean geometry, 90

  -- movement of a plane into, 147-8

  -- no motion of material masses into, 125

  -- not _a priori_, 116

  -- proof of the existence of the fourth, 136

  Dimensionality, and the intellect, 200

  -- as dependent upon the will of the investigator, 95

  Dimensionality, conception of, purely conventional, 92

  -- logical difficulties which beset, 97

  -- of perceptual space, 72

  -- -- space, the four, 142-3

  Dimensions, four, no basis in consciousness for, 172

  -- piled upon one another, 300

  -- vanity, the, of segmenting space into many, 299

  Direction as dimension, 99

  Disappointment with the sensible world, 120-1

  Discoveries, metageometrical, as excrescences of mental unfoldment, 131

  -- never the result of methodic reflection, 29-30

  Disorder, the edict of, and space, 230

  Disorderliness, the fringe of, 277

  Dispossessal of the intellect by the intuition, 195

  Dissimilarities, besetment of, 291

  Distinctions, between conceptual and perceptual space, fundamental, 73-5

  Diversity, a result of the fragmentative tendency of life, 6

  -- as transfinite quantity, 42

  Divinity, prize of, won by the intellect, 43

  Duadic phase of evolution, 210-1

  Duodim, defined, 6; 128, 129, 134, 177

  -- consciousness, 163

  Duopyknon, 212, 213

  -- the meaning of, 6

  Duopyknosis, 208-10, 218

  -- as a stage in the evolution of space, 6

  Duration and space, 224

  -- eternal, 234

  Dynamic appearance, space as, 104, 107

  Dynamism, of the intellect, functional, 303


  E

  Earth, the nebulous youth of, 335

  -- a new, 358

  East effloresces as West, 306

  Ego, the, 270

  -- and percepts, 244

  -- as sovereign, 244

  -- Thinker's dependence upon, 245

  -- -- treatment of, 246

  Egopsyche, 265

  -- as an agency of self-consciousness, 258

  -- defined, 7

  Egos, compared to choppy sea, 256-7

  Egypt, birth-place of geometry, 44

  Eisenlohr, 44

  Elaborative period of the non-Euclidean geometry, 66-70

  -- -- of mental development, 31-32

  Element of evolution, basal, 259

  Elements of the non-Euclidean geometry, 79-80

  Elephant and blind men, 257

  Ellis, Wilmot E., on ether as four-dimensional, 157-9

  Embrace of direct cognition, 205

  Engenderment of space, 260

  Ensoulment of the world-plasm, 329

  Entities, hyperspace, and the phenomenal world, 128-130

  Enumeration, reformation of the system of, 38

  Enveilment of consciousness, 273-4

  Environment, artificial and natural, 162

  Epiphysis, the, 345

  Equidistantial, described by hyperspatial perpendicular, 80

  Essence, elemental, as world-plasm, 329

  Ether as possessing fourth dimensional characteristics, 157

  Ethics recast in a new mold, 358

  Euclid, 46, 70, 83, 263

  -- and the parallel-postulate, 45

  -- never-dying elements of, 53

  Evidences, mathematical, exemplifies intellectual evolution, x

  -- the vanity of fragmentary, 204

  Evolution, a continuous process, 327

  -- and the norms of reality, 175

  -- basal element of, 259

  -- commencement of, 232-3

  -- intellectual, forward movement of, 184

  -- kosmic, vicissitudes of, 215

  -- laws of, govern mind, 28

  -- mental, results of, 122

  -- of faculties, synchronous with evolution of matter, 332

  -- -- material characteristics of the earth, 335

  -- -- organs, time required for, 188

  -- preparation of the field of, 218

  Exaltation of matter into spirit, 329

  Examples of new perceptive evidences, 341-3

  Existence on a higher plane, states of, 162

  Experience, corroborative testimony of, denied hyperspace, 263

  -- prior, and conception, 25

  Experiences, spatial, systematization of, 78

  Extension, space as an unbounded, 61

  Extra-spatiality, degraded into spatiality, 261

  Extravaganza, mathematical, and the fourth dimension, 156

  Eyes, as Roentgen rays, 353


  F

  Fact-mass, 289

  Factors of conscious determinations, 162

  -- four, of the criterion of truth, 324-5

  Facts as facets of truth, 284

  -- attempts to reform, 304

  -- logic as symbolism of, 287

  -- -- the modeler of, 288

  Faculties, dual derivation of, 162

  -- evolution of, synchronous with evolution of material
   qualities, 332, _ff_

  -- extended, 239

  -- foreshadowed by the hyperdimensional, 131

  -- higher, man must win his way to, 357

  -- new faculties evidenced by four-space, 24-29

  -- rudimentary nature of, 341, _ff_

  -- the source of, 349

  Faculty and the intellect, 247, _ff_

  -- as transmissible character, 251

  -- determined by function, 336

  -- greatly extended, 117

  -- higher than the intellect, 126

  -- I-making, self-consciousness as, 243

  -- intuitive, 185

  -- of perception in higher worlds, 353

  -- -- awareness, the awakening of, 89-91

  -- overshadowed by the intellect, 188

  -- of perception in higher worlds, 353

  -- rudimentary condition of, 192, _ff_

  Faculty, outcropping of, 249-51

  Failure of efforts to justify the objective existence of
   four-space, 125-6

  Failures at solving the parallel-postulate, outcome of, 48, 83

  Faith, dispossessed by knowledge, 358

  Fay, mathetic, 160

  Fechner, 39

  Finity and unboundedness of space, 76

  Flammarion, Camille, 341-2, 344

  Flexity, as property of hyperspace, 63

  Flexure, space as a, 305

  Fluxional, between sense objects and ideal representations, 122-3

  Fluxion, psychic, as difference between memory image and
   object, 7-8; 122

  Focus of consciousness, 163, 290

  -- -- the intellect, 310

  Fohat, and the creation of morphons, 219

  -- as creative energy, 213

  -- Creator, 8

  Fohatic energy, 226

  Form, as vehicle of life, 330

  -- definition of, 8

  -- driven back over the path of its genesis, 327

  -- pure, cannot exist in nature, 294

  -- the idealty of, 110

  -- the universe not a pure, 108

  Formative period of mental development, 31

  -- -- of non-Euclidean geometry, 55

  Formula, eight different alcohols from one, 155

  Four-dimensionality, justification of, 176

  Four-space, a curiosity of analytics, 40

  -- and Riemann, 27

  -- as a divertisement, 175

  -- consciousness does not act in the, 172

  -- existence of, denied, 171-2

  -- movement of matter in, 157

  -- reality of, glibly proclaimed, 154

  -- the, 8, 240

  -- -- denizens of, 353

  -- -- domain of, 154

  -- -- study of, 124

  -- use of analogies to popularize, 128

  Fourth dimension, analogical reasoning of, 177-9

  -- as a transcendental problem, 140

  -- -- an attitude of the intellect, 200

  -- electric current as a component in, 153

  -- imaginability of, 106-7

  -- Simon Newcomb on, 125

  Fragmentariness of the intellectual method, 164

  Fragmentary view of the universe, 260

  Fragmentation, tendency to, 296

  -- harks back to cell-division, 301

  Freedom, a new, 338

  -- determined by absence of bonds, 280

  -- mathematical, 60

  -- mental, 66-7

  -- now dawning for the mind, 32

  -- of consciousness in _genus homo_, 332

  -- regal freedom of the mind, 118

  -- three degrees of, 125

  -- unrealizable, for the Thinker, 255

  Fringe, chaogenetic, 308, 309

  -- of disorderliness, 257

  -- -- kathekosity, 229

  Frischauf, 66

  Full, the universe as a, 310

  Function determines faculty, 336

  Functions back of latent faculties, indefinite, 343

  -- cellular and histologic, 253

  Functioning, instinctive, of the intellect, 289

  Fundamentals, totality of, kosmic, 265


  G

  Gamut of realism, 169

  Gauge, awareness as a, 161-202

  Gauss, Charles Frederick, 8-9, 56-8, 59, 174

  -- -- -- as formulator of the non-Euclidean geometry, 57

  Geminos of Rhodes, 45, 52

  Generability, as property of space, 62

  -- of hyperspace, Keyser on, 144

  -- -- space by lines, 143

  Generation, the, of the hypercube, 134

  -- of the hypertetrahedron, 135-6

  Genesis, of space, 211, 227, _ff_

  -- -- the earth, its nebulous youth, 335

  -- -- -- form, 327

  -- -- -- sensible world, 167

  _Genus homo_, freedom of consciousness in, 332

  Geometricity, 266

  Geometries, non-Euclidean, based upon a negation of the latent
   geometrism, 262-3

  -- three possible, 54

  Geometrism, engenderment of, 262

  -- established by life, 264

  -- kosmic geometrism, establishment of, 231

  -- latent, 257

  -- native geometrism of space, 262

  -- rediscovered by the intellect, 9-10

  Geometrism, the basis of, 237, 261

  Geometry, anti-Euclidean, 69

  -- and the study of magnitudes in space, 203

  -- artificial, 262-4

  -- a two-fold, 59

  -- breakdown of, 266

  -- determinative period of the non-Euclidean, 61-4

  -- diverse systems of, 323

  -- Euclid's Elements of, 53-4

  -- formal bulwarks of, 77-9

  -- _Imaginary Geometry, The_, 60

  -- natural geometry, 261

  -- Plato and the divine geometry, 193

  -- possible systems of, 174

  -- radical essence of pure, 85

  Geometry, non-Euclidean, at variance with the parallel-postulate, 83-5

  -- -- based upon a misconception, 91

  -- -- determined by qualitative differences, 140

  -- -- elaborative period of, 66-70

  -- -- first published treatise on, 60

  -- -- formative period of, 55

  -- -- the final issue of, 88-9

  -- -- growth and development of, 92

  -- -- invalidation of, 174

  -- -- key to, 90

  -- Lambert's Non-Euclidean geometry, _note_, 51

  -- -- popularization of, 66-7

  -- -- Schweikart's treatise on, 59

  -- -- self-consistence of, 57

  -- -- some elements of, 79-80

  -- -- superperceptual knowledge of, 72

  Geometry, non-Legendrean, 70

  -- symbolic, and commensurable quality, 145

  Gerling, as correspondent of Gauss, 58

  Germ-plasm, continuity of, 260

  Giantism, 347

  Glimpse, a, of the reality of space, 314

  Glorification of the flesh, 227

  Groups, transformation, discovery of, 30

  Guide-posts to a new domain, 273, 277

  Gulf, interposed between manifestation and non-manifestation, 206


  H

  Halley, memoir of, mastered by La Grange, 50

  Halstead, G. B., _note_, 49, 66; _note_, 79

  Heindel, Max, _note_, 353

  Helmholtz, 39, 66

  Hewer, the, of the concrete, the intellect as, 295

  Hinton, C. H., 154

  -- on the fourth dimension, 153

  History, the, of mind, three great epochs of, 31-32

  Homogeneity, the, of realism, 165

  Hoüel, J., 60, 66

  Hudson, 344

  Humanity and the exalting power of life, 331, _ff_

  Hypercube, boundaries of, and mirrors, 136-9

  -- the generation of, 134, 145

  Hyperdimensional, as a prophecy of new faculties, 131

  Hyperspace, 175

  -- a monument to the intellect, 356

  -- and the involved procedure of arriving at a recognition of its
   relations, 73

  -- and the passage thither, 293, _ff_

  Hyperspace as a figurative mountain-peak, 200

  -- -- -- movement, 44-68

  -- -- an all-powerful something, 160

  -- -- -- idealized construction, 10

  -- -- evidence of new faculties, 24

  -- -- _ignes fatuii_, 154

  -- -- illusion, 202

  -- a symbol of higher consciousness, 180-4

  -- concept of, as an evolutionary quantity, 44-5

  -- confounded with real space, 91

  -- creation of, 335

  -- denied the corroborative testimony of experience, 263

  -- direction of, baffling to mathematicians, 146

  -- discovery of, a sign of mental evolution, 180

  -- domain of, a fairy-land, 239

  -- magnitudes of, the non-sensuous, 72

  -- mysterious hiding place in, 129

  -- the logical difficulties of, 139-141

  -- the six pillars of, 62-3

  Hyperspatiality as the toys of childhood, 241

  Hypertetrahedron, the, 135

  Hypertrophy of the pituitary body, 347

  Hypervolume, the, 134

  Hypothesis, a superfoetated, 141

  -- four-space, utility of, 129; 154

  Hypotheses, admissibility of, 118

  -- incompatibility of the non-Euclidean, 67; 71

  -- Riemannian, 80

  -- solution of, Bolyai, 31

  Hyslop, James H., on the logical difficulties of hyperspace, 139-141


  I

  Icosahedron, examination of the, 285

  Ideal, and the real, kosmic chasm between, 107

  -- perceptual value of, 275, _ff_

  Ideas and words as symbols, 126-7

  -- Malebranche on, 222

  -- realism of, 24-35

  -- the symbolism of, 205

  Identification of consciousness with being, 205-6

  -- with the objects of study, 189-90

  Identity of things-in-themselves, 353

  _Ignes fatuii_ and hyperspaces, 154

  Illustration of plane-rotation, 148-150

  -- -- the tesseract, 133

  Images, totality of, recoils upon us, 167

  Imaginability of the fourth dimension, 90, 106-7

  Imagination, premises of, the mathematical, 146

  Impossibility of plane-rotation, the structural, 151-3

  Impressions, the symbolism of, neurographic, 186

  Impulse, the satisfaction of the original creative, 310-1

  Incomprehensibility of reality to the intellect, 126

  Incongruity, life estranged by a radical, 325

  Individual as space, 223

  Ineptitude of intellectual determinations for vitality, 314

  Infinite, interpreted in the terms of the finite, 82

  Infinitesimals of unity, numbers as, 41

  Infinity as a process, 109

  -- of becomings, 234

  Infinity of parallels through a given point, 70

  -- -- space, a capital illusion, 195

  -- the concept of, 277

  -- -- innate dread of, 103

  -- -- relativity of, 194

  Influence of abstract thought, 33

  -- -- Kant on the non-Euclidean geometry, 49-50

  -- -- La Grange, 51-2

  -- -- the intellect, 315

  Infundibulum, the, 345

  Inner organs of perception, 352

  Innermost, the, realities of things, 356

  _Insouciance_ of the geometer, 96, 294

  Instant-exposure and intellect, 311

  Instrument, intellect likened to a color-bearing, 313

  -- for the measurement of the passage of space, 297

  -- of consciousness, the sensible world as, 199

  -- -- life, form as an, 328

  Integers, as fractional parts of unity, 41

  Intellect and its domination of the intuition, 333

  -- -- -- final union with the space-mind, 194

  -- -- -- topography, 312

  -- -- spatiality, 263

  -- -- the deposits of life, 173

  -- -- -- designs cut by life in materiality, 264

  -- -- -- dictum of Sensationalists, 26

  -- -- -- instant-exposure, 311

  -- -- -- intuitive faculty, 247, _ff_

  -- -- -- prize of divinity, 43

  -- as a color-bearing instrument, 313-4

  -- -- -- searchlight, 168

  Intellect as a fashioner of phenomena, 199, _ff_

  -- -- hewer of the concrete, 295

  -- -- sole interpretative agency, 166-7

  -- automatism of, 253, 338

  -- cannot seize life, 282

  -- crowned by a diadem of gold, 338

  -- dominated by the intuition, 250

  -- fashioned for matter only, 231

  -- follows in the grooves of logic, 294

  -- hyperspace as a monument to, 356

  -- in the field of vitality, ix

  -- its aptitude for starts and stops, 292, 302

  -- -- instinctive tendency to fragmentate, 296

  -- quality, determined by mathematics, 337

  -- makes for individuality, 252

  -- misses the ceaselessness of life, 201

  -- _modus vivendi_ of, its influence upon knowledge, 184

  -- the constitution of, 291

  -- -- cut and mode of, 167

  -- -- focus of, 310

  -- -- illusion of, 246

  -- -- illusionary character of, 323

  -- -- incomprehensibility of reality to, 126

  -- -- instinctive functioning of, 289

  -- -- judgments of, 302

  -- -- moods of, 202

  -- -- predominating tendency in, 320-1

  -- -- scientific tendence of, 165-6

  -- -- struggle of, against dispossessal, 195

  -- unsuited to deal with reality, 322, _ff_

  Intellectuality and reality, 304

  -- -- spatiality, consubstantial, 331

  Intellectuality as co-extensive with spatiality and materiality, 236

  -- the source of, 260

  -- -- Thinker makes his own, 242

  Intelligence and automatism, 253, _ff_

  -- the Thinker as a pure, 243

  -- transfinite intelligence and the degrees of realism, 164

  Intent, the kosmic, of the intellect, 326

  Interests, the sphere of our, as closed circuits, 167

  Interior, the great, 290

  Interiority of being, 290

  Interpretation, the standards of, vary as consciousness varies, 171

  Interstices of materiality, 264

  Intuition and brotherhood, 252

  -- -- communalization, 301

  -- -- the riddle of spatiality, 325

  -- as dispossessor of the intellect, 195

  -- cannot be served by mathematics, 337

  -- dominated by the intellect, 333

  -- held in abeyance, 338, _ff_

  -- its domination of the intellect, 250

  -- the development of a spatial, not absurd, 145

  -- -- need of a sympathetic attitude towards, 249

  -- -- results of the development of, 357

  Intuitional consciousness, 255

  -- the superiority of, over the rational, 187

  Intuitions and the lead of life, 191

  -- -- -- Thinker, 27

  -- free, mobile and formless, 166

  -- the conceptualization of, 248

  -- -- humility of, 165

  -- -- nature of, 185

  Intuitograms, as concepts, 248

  Intuitograph as means of contacting the egopsychic consciousness by
   the Thinker, 10

  -- as super-concepts, 255

  -- the difficulty of transmitting, 315

  Invariability, the vaunted, of the laws of mathesis, 37

  Invariants, psychological, 24

  Investigations, metageometrical, and the new sense-organs, 354

  Involution, as antithesis of evolution, 10-11

  -- kosmic involution, 226

  -- of matter, 328

  -- the movement of, 219

  -- -- seven stages of, 208-12

  Ions, creation of, 219

  -- magnitude of a hydrogen ion, 225


  J

  Judgments _a priori_, Kant on, 85-6

  -- and the faculty _a priori_, 190-1

  -- -- zones of affinity, 124

  -- based upon a replica of consciousness, 205

  -- no trustworthy, can be predicated upon fragmentary knowledge, 282

  -- of the intellect, 302

  -- the lessening of error in, 256

  -- -- more complex the more at variance with the nature of things, 75

  -- -- synthesis of, 257-8

  -- valid judgments long delayed, 170-1

  Judicative power of mathematics, 180

  Justification for a multi-dimensional quality in space, 262

  -- of four-dimensionality, 176

  Justification of sense-deliveries by one another, 76

  -- -- the existence of the fourth dimension, 125


  K

  Kant, 85, 181, 182, 322

  -- and the faculty of thinking, 261

  -- -- -- idea of space, 322-3

  -- influence of, on the hyperspace movement, 49-50

  -- on the nature of things, 119

  -- -- space as an intuition, 115

  Kathekos, 233-4

  -- as chaos, 266

  -- -- symbol of Chaos-Theos-Kosmos, 11-12

  -- symbology of, 234

  Kathekosis, _note_, 227

  Kathekosity, fringe of, 229

  -- rock-bound coast of, 267

  -- significance of, 12

  Kathekotic consciousness, 272

  -- period, 209

  Key to the mysteries of nature, the fourth dimension as, 131

  Keyser, Cassius Jackson, and freedom of the mind, 33

  -- -- -- on attitude of metageometricians, 71

  -- -- -- -- dimensionality, 94-5

  -- -- -- -- four-dimensionality of space, 142-3

  -- -- -- -- generability of hyperspace, 144

  Klein, Felix, 12, 54-5, 66-7

  Knowledge, all, relative, 101

  -- barrier to the certitude of the Thinker's, 186-7

  -- fabric of, 196

  -- hypothetical nature of, 189, _ff_

  -- immeasurable realm of, laid bare by the telescope, 298

  Knowledge, mathematical, apriority of, questioned, 37

  -- nature of the non-Euclidean, superperceptual, 72

  -- real, difficulties of acquiring, 325

  -- related to the stream of life and to the shore of materiality, 323

  -- relative, degrees exist for, 164

  -- sphere of, 184

  -- systematization of, 126

  -- ultimate, consciousness as a barrier to, 207

  -- unification of, 256

  Kosmogenesis, the latent geometrism of, 264

  -- -- scope of, 237

  Kosmometer, the, 297

  Kosmos, and consciousness, graph representing, 271

  -- arrangements of the contents of, 317-8

  -- in a state of becoming, 265

  -- magnitude of, 308

  -- moods of, 311

  -- space as the consistence of, 239

  -- _see_ Cosmos, 12


  L

  Labor, division of, between the tuitional and the intuitional
   faculties, 193-4

  -- mathematical labors, significance of, 176

  La Grange, Joseph Louis, 12, 39, 50-3, 321, 322

  -- -- -- -- and the parallel-postulate, 51-2

  Lambert, John, 58

  -- -- and the theory of parallels, 48-9

  Language, the passing of, 357

  Legendre, 70

  Leibnitz' dictum, 26

  Leonard rays, 336

  License, mathematic, permissibility of, 38

  Lie, Sophus, 12, 67

  -- -- and transformation groups, 30

  Life, analysis incapable of dealing with, 319

  -- and consciousness as one, 224

  -- -- form rooted in pyknosis, 214, 328

  -- -- the fourth dimension, 172

  -- -- -- inaptitude of mathematics for, 179

  -- -- -- power to create, 297

  -- as agent of creation, 343

  -- -- creative agent, 264

  -- -- expression, 303

  -- -- vassal of materiality, 329-30

  -- causative agencies in prolonging, 123

  -- current of, as engendering element, 331

  -- deposits of, and the intellect, 173

  -- estranged by a radical incongruity, 325

  -- exalting power of, and humanity, 331, _ff_; 342

  -- exhibition of its remains, 282

  -- flow of, 265, 297

  -- form as an instrument of, 328

  -- indescribable signs of, 219

  -- infinitive action of, 327

  -- intellect has no aptitude for, 231

  -- intuitive, values of, x

  -- larger life of the Thinker, 196

  -- lead of, followed by intuition, 191

  -- limits of, fixed by consciousness, 198, _ff_

  -- most solid facts of, as shadows, 195

  -- motility of, admits of endless variations, 35

  Life, movement of chaos in, 307

  -- passage of, 201

  -- passage through spatiality, 288

  -- power to manipulate, 303

  -- recurrent movement of, 307

  -- totality of egoic, 243

  -- undulations in the current of, 329

  -- uniqueness of, 41

  -- wake of, and consciousness, 205, 288

  Life-cycle, evolutionary results of the, 259

  Life-stream, the, 278

  Light, polarized, and the fourth dimension, 156

  -- consciousness as a spreading, 168

  Light-years and space, 278

  Limits, the sphere of, consciousness as, 163

  Line, as generating element, 144

  -- the straight, a curved, 76

  Lineage of every principle runs back to monopyknosis, 213

  Lines, perpendicularity of, in four space, 130

  Lobachevski, 55, 60, 66, 87

  Lodge, Sir Oliver, 340-1; 344

  Logic as architect, 118

  -- conventional forms of, 262

  -- data of, 286

  -- intellect follows in the grooves laid out by, 294

  -- miracle power of, over facts, 288-9

  -- rules of, allowability under, 163

  -- -- -- the game of, 164

  -- as symbolism of facts, 287

  Logos and the limits of space, 193

  -- being of, 218

  -- body of being of, 214

  -- consciousness of, 204

  -- creative, 12, 213

  -- the, 216-7

  Lorenz, 83

  Luther, Martin, 181


  M

  Magnitudes, geometry, a study of, 203

  -- non-sensuous, of hyperspace, 72

  Makrokosmic consciousness, 270

  Malebranche, N., 222; see _note_

  Manifestation and non-manifestation, 206

  Manifold, finite, though unbounded, 70

  -- manufactured by the analyst, 77

  -- the, 61-2, 172

  Manifoldness, analytical manifoldnesses, as mental excitants, 89-90

  -- as a conventional construction, 262

  -- -- an intuition, 318

  -- of space, as a near-truth, 317

  Manning, 82, 157

  -- and maneuvers in the fourth dimension, 152

  Manvantara, 213, 228-9, 232

  -- as evolution and involution combined, 13

  Mask, logic as a, 288-9

  Mass-termini, of lines, 236

  Mastery of life over matter, 330

  -- of the sensible world, 196-7

  Materiality as a deposit of life, 295

  -- -- consubstantial with spatiality, 331, _ff_

  -- becoming spatialized, 261

  -- characteristics of, 260

  -- engendered by kosmic mind, 261

  -- engenderment of, 294

  -- interpenetrative with spirituality, 236

  -- interstices of, 264

  -- neglect of, 320

  Materiality, shore of, 323

  -- transmuted into spirituality, 328

  Mathematicians and the definition, 37

  -- -- -- limitations of consciousness, 178

  -- -- -- phenomenal world, 118-9

  -- as prophets, 336

  -- respected by humanity, 337

  -- the gods of mathesis, 84

  Mathematics, a determinant of the quality of the intellect, vii

  -- and the criterion of truth, 323

  -- -- -- kosmic intent of the intellect, 326

  -- as symbology, 184

  -- Euclidean, 140

  -- fails when it encounters life, 337

  -- its inaptitude for life, 179

  -- -- kingdom yielded to kinetics, 358

  -- possesses no judicative power over life, 180

  -- the orthodoxy of, 34

  Mathesis, and conceptional space, 203

  -- conduct of the intellect in the field of, vii

  -- definition of, 13

  -- domain of, as origin of fourth dimension, 130

  -- gods of, 84

  -- marvelous domain of, 118

  -- realm of, not submissive to laws of sensible space, 34

  -- things of, as emblems of kosmic forces, 239

  -- world of, 67, 127

  Matter and mind, syncretization of, 299

  -- appulses of life against, 328

  -- as a deposit of life, 330

  -- honeycombs of, 339

  Matter metamorphosed by life, 328

  -- mind as wedded to, 249

  -- movement of, in four-space, 157

  -- qualities of, and faculties, 332

  -- seven planes of, 212

  -- spiritualization of, as end of evolution, 327

  -- totality of, acted upon by totality of life, 328

  -- unity of, with space, 260

  -- universum of, 261

  Matutinal ceremonials of creation, 221

  Measurability, as property of space, 62-3

  Measurement of hyperspaces, the science of, 73

  Measurements, space determined by the number of, 61

  -- systems of space, 91

  Measurer of space, _n_-dimensionality as a, 296-7

  _Mecanique Analytique_, 50-1, 321

  Mechanics of matter, the passage from, to the dynamics of life, 326

  -- to biogenetics, the passage from 357

  -- the turning from, to biogenetics, 322, _ff_

  Mechanism, the doctrine of, due to analytics, 40

  Memories, stored in the omnipsyche, 258

  Men as gods, 289

  Mental evolution, determinative period of, 31

  -- -- elaborative period of, 31-32

  -- -- formative period of, 31

  Mentality, inner mysteries of creative, 91

  -- infantile out-feelings of the, 336

  -- the principle of, 210, _ff_

  Mentalities, the adaptation of phenomena to, 184

  Mentograph, the, 121

  -- the basis of intellectual consciousness, 13

  Mesozoic man and his environments, 347-8

  Metageometrical investigations and the new sense-organs, 354

  Metageometricians and hyperspace, 238

  -- -- proof of rotation about a plane, 149-151

  -- -- the curved space, 316

  -- -- -- fourth dimension, 145

  -- -- -- key to the mysteries of nature, 131

  -- baffled by the direction of hyperspace, 146

  -- dilemma of, 97-8

  -- eschew sense-data, 73-4

  -- Keyser on the attitude of, 71

  -- perceptual obliquity of, 102

  -- -- seed thought of, 182

  -- turned to idealized constructions, 263

  Metageometry, as a stepping stone, 238

  -- defined, 13

  -- Riemann, the father of, 63

  -- the fabric of, 66

  Metamorphosis of matter by life, 328

  -- of the monopyknon, 214

  Metamorphotic stage of space-genesis, 228

  Metaphysician, the preserves of, usurped by mathematicians, 141

  Meta-self as medium of kosmic consciousness, 13, 217

  Methods of the ego, 344

  -- scientific, and the ample explication of phenomena, 154

  Mikrokosmic consciousness, 270

  Mind, amenability of, to the evolutionary movement, 28-32

  -- and freedom from conception, 28

  -- -- -- -- problems of physical existence, 32

  -- -- the new freedom, 33

  -- as vehicle of life, 285-6

  -- -- wedded to matter, 249

  -- coevalism of, and space, 240

  -- consubstantial with space, 239-40

  -- divine, of the kosmos, 176

  -- evolution of, and conceivability, 24

  -- of the duodim, 128

  -- powers of, not attained simultaneously, 334

  -- profound influence of hyperspace on the, 91

  -- space as progenitor of, 224

  -- three great epochs of the history of, 30-31

  -- tuitional, the limitations of, 192-3

  -- the unity of, with space, 224

  Mind-principle, the quintopyknon as the basis of, 221, _ff_

  Mirrors, three, and the hypercube, 136-8

  Mobility, life as, 264

  Molecule, the four-dimensional, Newcomb on, 159

  Monadic phase of evolution, 210-1

  Moneron and man, the gulf between, 28

  Monopyknon, the, 213, 216

  Monopyknosis, defined, 17, 208-10, 218

  Monstrosities, mathetic, 202

  Moods of the kosmos, 311

  Morphogeny, 13, 233-4

  Morphons, the creation of, 219

  Motility, the scope of, 131

  Movement, forward, in knowledge, 336

  -- hyperspace as an evolutionary, 44-68

  -- -- influence of Kant on, 49-50

  -- of life in chaos, 307

  Multi-dimensional quality of space, justification for, 262

  Multiplication of hypotheses, 118

  Mutuality, intuition as promoter of, 301, _ff_

  Mysteries, inner, of creative mentality, 91

  -- kosmic, low-lying plains of, 239

  --- of life and mind, 326

  -- -- the sciences and the four-space operator, 131

  Mystery, kosmic, the revelation of, foreshadowed in manifold, 77

  -- of reaction, 167-8

  -- -- space decreases as copes of consciousness increases, 273, _ff_


  N

  _n_-Coördinates, 203

  _n_-Dimensionality, as a phase of geometry, 300

  -- -- property of intellectuality, 269

  -- -- quality of conceptual space, 14

  -- irreconcilable with perceptual space, 156

  -- no justification for, 235

  -- not surprising that the intellect fell upon, 296, _ff_

  -- predicates concerning, 158

  _n_-Space, imperceptibility of, 94

  Nasir-Eddin, 46, 53

  Natural symbols, ideas as, 126

  Nature, representative, of things, 119

  -- the vacuum-stillness of, 339

  _Naught_ to _unity_, consciousness registers from, 165

  Near-truth, 300, _ff_

  -- as variation from criterion of truth, 324

  -- based upon partial knowledge, 14

  -- space-curvature as a, 306

  -- space-manifold as a, 318

  Nebulosity, 267

  Necessity, as a bulwark of geometry, 77-8

  Nerve-body, the, 349

  Neurogram, 14

  Neurograph, the Thinker's scrutiny of, 190

  Neurographical communications, 244

  Newcomb, Simon, 14-5, 66

  -- -- and the four-dimensional molecule, 159

  -- -- on the fourth dimension, 125

  Non-Euclideans and the meaning of dimension, 97

  -- -- things as they actually are, 71

  -- dilemma of, 86

  -- meaning of the term, 69

  -- results obtained by, 88

  Non-manifestation, as antithesis of manifestation, 106

  -- and manifestation, 206

  Non-methodic advancement of human consciousness, 299

  Non-spatiality, cut off from spatiality, 266

  Norm, as a value assigned by the intellect, 167

  -- consciousness as a, 15; 161-202

  -- of life, found by synthesis, 300

  -- -- the natural geometry, 263

  Note, the dominant, of communality, 301

  Notions of space, triple presentations to the consciousness, 72

  Number, as a phase of kosmogenesis, 309

  -- numbers as infinitesimals of unity, 41

  Numericity of being, 309


  O

  Objects and ideas, 126-7

  -- passage of, into the fourth dimension, 130

  -- of study, identification of consciousness with, 189

  Obliquity, perceptual, of metageometricians, 102

  Obscurantism, mathetic, 154

  Occultist and the scientist, 350

  Omnipsyche, 265

  -- as agency of kosmic consciousness, 258

  -- -- neglected factor of evolution, 259

  -- defined, 15

  One-space represented by a line, 134

  Ontogeny recapitulates phylogeny, 27, 334

  Order, kosmic, the difference between, and hyperspace, 143

  -- mathematical, discovered by the intellect, 261

  -- the fiat of, 208

  -- the totality of, 265

  Orderliness, 266

  Organ, determined by function to be performed, 336

  Organs broken up by life into special parts, 343

  -- the evolution of, 188

  Orthodoxy of mathematics, 34

  Outcome of new adjustments, 256

  Out-feelings of the Thinker, infantile, 336

  Outlook, the spiritualization of our mental, 40


  P

  Papyrus, a hieratic, 44

  Parallel-postulate and the surface of a sphere, 70

  -- as basis of non-Euclidean geometry, 54

  -- -- stated by Euclid, 55

  -- -- -- -- Lobachevski, 55

  -- -- -- -- Manning, 82

  -- failures to demonstrate, 48, 82-3

  -- in the Elements of Euclid, 16

  -- Saccheri's proof of, 47

  Parallels, convergence of, 80

  -- infinity of, through a given point, 70

  -- meet at infinity, 88

  -- theory of, 47-9

  Passage from mechanics to biogenetics, 357

  -- from mechanics to dynamics, 326

  -- from three-space to four-space, 292-3

  -- of mathetic contrivances, 241

  -- of space, 16, 297

  Pathway of life, sinuosities in, 330

  Patterns of the intellect, ready-made, 166

  Pentagrammaton of space, 237

  Perceptibility in the Thinker, the faculty of, 188

  Perception, domain of, and the sensible world, 19

  -- inner organs of, 352

  -- replaced by conception, 255

  Percepts, and the ego, 244

  Period, chaomorphogenic, 208, _ff_

  -- determinative, 61-5

  -- kathekotic, 210

  Perisophical nature of analytics, 320, _ff_

  Perisophism, a, 300

  Permissibility of mathetic license, unlimited, 38

  Perpendicularity, 132, 177

  -- of lines in the four-space, 130

  Phantasy and reality, line of demarkation between, 173

  -- the world of, 146

  Phantom-ideal, the, 277

  Phase of the world age, evolutionary, 209

  Phenomena, efforts to explain, on the basis of the fourth dimension, 129

  -- fashioned by the intellect, 199

  -- fragmentary interpretation of, 302

  -- mind's method of apprehending, 186

  -- physical phenomena amply explained by scientific methods, 154

  Phenomenal, the inverse of realism, 278

  Philosophy and the criterion of truth, 323-4

  -- regeneration of, 358

  -- systems of, and zones of affinity, 124

  Phosphorescence, dim, added to the unilluminated pool of
   sense-consciousness by the intuition, 26

  Phylogeny and ontogeny, 334

  -- represented in ontogeny, 27

  Physical as embodiment of all possibilities, 227

  -- phenomena as space-activities, 229

  Physicality, the principle of, 210, _ff_

  Physics of the brain, 292

  _Pi_ proportion, significance of, 311-2

  Pickering on space-curvature, 279

  Pineal gland, energized by the pituitary body, 348-9

  -- -- the, 344, _ff_

  -- -- not a vestigal organ, 354

  Pituitarial awakening, general results of, 355-8

  Pituitary body and the Thinker, 351

  -- -- as an organ, 344-352

  -- -- C. De M. Sajous on, 345

  -- -- hypertrophy of, 347

  -- -- location of, 246

  -- -- the, 344, _ff_

  Plane, as generating element, 144

  -- movement of, into the fourth dimension, 147

  -- realism appears to be divided into, 164

  -- rotation about a, 130, 146-151

  -- the seven planes of matter, 212

  Planes, a cube as a succession of, 148-9

  Plasm, kosmic, strivings of, 215

  -- psychic, 260

  -- the differentiation of, 214

  Plato, 322

  -- and the divine geometry, 193, 237

  -- -- -- shadow consciousness, 281

  -- on God as geometrizer, 181

  -- -- ideas, 34

  Play, sensuous, of the intellect, 295

  Plenum, space as a, 230

  -- the universe as a, 107

  Poincairé, 90

  Point, an infinity of parallels through a given, 70

  -- as generating element, 144

  -- position of, 132

  -- succession of points, a line as, 148

  Polarity between concrete and abstract, 294-5

  Ponderability, as property of hyperspace, 63

  Popularization of the fourth dimension, 128

  -- of the non-Euclidean geometry, 66-7

  Possibilities in the world of hyperspace, 129

  -- of four-space, the marvelous, 130

  Postulate-systems, the multiplication of, 85

  Powers of mind, not attained at one and the same time, 334

  Pralaya, 213

  -- as kosmic quietude, 17

  Predicates, all mathematical, not justified by the phenomenal, 128

  Principle, ensouling, 214-5

  Printing press, the invention of, and the laws of psychogenesis, 30

  Problem, the fourth dimension as a transcendental, 140

  Procedure, the involved, of arriving at hyperspace notions, 73

  Process, evolution a continuous, 327

  -- space as a dynamic, 307

  Proclus, 83

  Profundities, kosmic, and hyperspace, 156

  Progression eastward terminates at the west, 306

  Proof of rotation about a plane, 146-151

  Propagation of intuitographs, difficulty of, 315

  Propositions, geometric, subjunctive quality of, 38

  Pseudosphere and the shape of space, 70

  -- as basis of Beltrami's calculations, 17

  -- the nature of, 64-5

  Psychics of consciousness, 292

  -- transmuted, 260

  Psychogenesis, alchemy of, 225

  -- kosmic, 272

  -- laws of, and the printing press, 30, 116

  Psychogenesis, mind's place in, 123

  -- outgrown phase of, 302

  -- outstanding facts of, 131

  Psychogeny, defined, 17

  Ptolemy, 45, 52, 83

  Publicist, the mathematical, and the duodim, 128

  -- the non-Euclidean, 146

  Pyknon as a kosmic principle, 212

  -- -- basis of space-genesis, 17

  Pyknosis, 212

  -- and involution, 210, 231

  -- as a metamorphotic process, 214

  -- the seven processes of, 17, 294


  Q

  Quality, dimension as commensurable, 140

  Quantities, algebraic, cannot represent space, 125-7

  Quantity, consciousness as a tri-space, 172

  Quartodim, 18, 134

  Quartodimensionality, 292

  Quartopyknosis, 17, 208-10

  Quartopyknotic, principle as basis of spirituality, 228

  Quintopyknon, 220

  Quintopyknosis and the _a priori_, 223

  -- meaning of, 17, 208-10

  Quintopyknotic principle as basis of mentality, 228


  R

  Race, humanity to evolve a new, 358

  -- resplendent, 338

  Rajah-Tamas, 220

  Rajah Yoga, 357

  Rational, valueless when unsanctioned by the intuition, 187

  Reaction, as a mystery, 167-8

  Real, Thinker as a part of the, 201

  Realism and its degrees, 164-5, 279

  -- -- the Thinker, 315

  -- as life, 286

  -- consciousness as touch-stone of, 168

  -- found in the direction inverse to the phenomenal world, 278

  -- homogeneity of, 165

  -- illusionary nature of, 36

  -- infinite gamut of, 36

  -- its degrees and the states of consciousness, 197

  -- new scope of, bewilders the mind, 170-1

  -- not of mathematical import, 290

  -- of concepts and the sensible world, 40

  -- -- the domain of mathesis, 33-4

  -- psychological quality of, 285

  -- the thread of, 169

  -- three dimensional scope of, 172

  Realities, abstractionizing, 144, 294

  -- incomprehensible to the intellect, 126

  -- innermost, of things, 356

  -- lesser give way to greater, 195

  -- natural symbolism of, 128

  -- our, non-existent to beings on spirit-levels, 35

  -- shaped upon conventional models, 294

  -- supersensuous, 180

  Reality, as life and consciousness, 173

  -- barrier to the cognition of, 189, _ff_, 273

  -- comprehension of, 165, 303

  -- current of, 322

  -- defined as life, 18

  -- flow of, 319

  -- flowing stream of, 124

  -- myriad ways of presenting itself, 287

  -- naked contact with, 193

  -- not an inscrutable quantity, 194

  -- obscured, 302

  -- of the four-space glibly asserted, 154

  -- thread of, 200

  -- universum of, 127

  Realms, supersensuous, 338

  Re-becoming, life and consciousness as a, 257

  Reciprocity of consciousness and realism, 197-8

  -- of the manifest and the unmanifest, 217

  Recoil of images upon us, 167-8

  Reflexive development of the intellect, 254

  Relativity of all knowledge, 101

  Religion and realism, 170

  -- the changes in, 358

  Remains of life exhibited to the intellect, 283

  Remaking of moods by the intellect, 201

  Replica of consciousness as basis of judgments, 205

  Representation, sensuous, compared with a shadow, 119-120

  Research, improvement in the methods of, 125

  Reservoir, psychic, of evolution, 260

  Residuum, the unexplained, 157-8

  Revelational impressions, 29

  Riddle of spatiality and the intuition, 325

  Riemann, G. F. B., 18, 19, 54, 66-7, 87

  -- and a limited space, 108-9

  -- -- the curvature of space, 5

  -- -- -- determinative period, 65

  -- -- finite space, 103

  -- -- -- four-space, 27

  -- -- -- manifold, 317

  -- -- -- as inventor of the manifold, 86

  -- on the bases of geometry, 61-4

  Roentgen rays, 336

  Rotation about a line, 147-8

  -- about a plane, 146-151

  -- -- -- -- illustrated, 148-50

  -- intra-corpuscular, 152

  Rudimentary organs, vague functions lying back of, 343

  Rule in the evolution of faculties, 333


  S

  Saccheri, Girolamo, 47-8, 52, 83

  Sajous, C. De M., on the pituitary body, 345

  Schematism, the suitability of the present, 320-1

  Schweikart, Ferdinand Karl, 19, 66, 69

  -- on the non-Euclidean geometry, 58-60

  _Science Absolute of Space_, 60

  Science and reality, 170-1

  -- no longer empirical, 358

  -- skeptical attitude of, 349

  Scientist, method of the, opposed to the analyst, 84-5

  Scopographic impressions, 19, 190

  Screen, the impregnated, of creation, 232

  Sea, choppy, and egos, 256-7

  Search, the path of, for spatial understanding, 278

  Seb, the god, 215

  Sects, coplanar, 79

  Self and the immensity of space, 223

  -- of the universe, 217

  -- the kosmic, 278

  Self-consciousness, 270

  -- and space, 223

  -- as I-making faculty, 243

  -- determination of, 302

  Self-consistency, 173

  _Sella turcica_, the, 345

  Semi-Euclidean geometry, the, 70

  Sensationalists on the intellect, 26

  Sense-data, spurned by the metageometrician, 73

  Sense-delivery, one, justified by the other, 76

  -- symbolism of, 203-4

  Sense-knowledge, the sphere of, extended, 339-40

  Senses, the answer of, to new needs, 332

  Sensible world, as domain of perception, 19

  Sensographic impressions, as perceptions, 19

  Senso-mechanisms of the intellect, 165

  Sentience, the principle of, 210, _ff_

  Septopyknon, the, 227

  Septopyknosis, as kosmic materiality, 18, 208-10

  Septopyknotic, the, principle, 228

  Seven planes of matter, 212

  Sextopyknon, the, 209

  Sextopyknosis, defined as kosmic sensibility, 18, 208-10

  Sextopyknotic, as emotional principle, 228

  Shadow likened to a sense presentation, 119

  Shadowgraphs, 281

  Silences, nature's great, 339

  Similitude of agent and principle, 243-6

  Sinuosities in the pathway of life, 330

  Socrates, 322

  Solitariness of intellectual testimony, 76

  Space and edict of disorder, 230

  -- -- four-dimensional entities, 128-130

  Space and quartopyknons, 219-20

  -- -- self-consciousness, 223

  -- -- the idealized construction, 144

  -- as a dynamic process, 216

  -- -- all-mother, 229-30

  -- -- an assemblage of spheres, 143

  -- -- becoming, 195

  -- -- boundary of another space, 49-50

  -- -- consistence of the kosmos, 239

  -- -- curved, 164

  -- -- dynamic appearance, 228, 307

  -- -- eldest born of kathekosis, 215

  -- -- engendered product of life, 308, _ff_

  -- -- evolution, 226, _ff_

  -- -- finite extension, 62, 106

  -- -- generable quantity, 103

  -- -- infinite continuum, 70

  -- -- intuitional concept, 225

  -- -- kosmic order, 265, 267

  -- -- nether pole of non-manifestation, 207-8

  -- -- path of life, 293-4

  -- -- pseudosphere, 70

  -- -- sheer roominess, 115

  -- -- system of coördinates, 99

  -- -- unbounded extension, 61

  -- best study of, the consciousness, 269

  -- conceptual, as basis of non-Euclidean geometry, 72

  -- -- fundamentally distinct from perceptual, 72-75

  -- elliptical, 67

  -- engenderment of, 260

  -- essential nature of, 212

  -- fabric of, the, lends itself to measurement, 99-100

  -- four-, 8

  -- fractionalized space, 296

  -- foundations of, 221

  -- four-dimensional, 41

  -- Paul Carus on, 354

  -- fourth dimension of, efforts at making it thinkable, 125

  -- generability of, by lines, 143

  -- genesis of, 227, _ff_

  -- geometric, purely formal construction, 75-6

  -- limits of and the Creative Logos, 193

  -- manifoldness of, 77

  -- mind and, the coevalism of, 240

  -- mystery of, 278, _ff_

  -- nature of, answered to by mind, 198-9

  -- nether pole of non-manifestation, 207-8

  -- non-Euclidean, the characteristics of, 72-3

  -- not a pure form, 110

  -- passage of, 16

  -- path of search for, must be Thinker-ward, 278

  -- perceptual, irreconcilable with the fourth dimension, 156

  -- possible curvature of, and the non-Euclidean geometry, 76-7

  -- primeval, and tridimensionality, 236

  -- problem of devising a, 87-8

  -- psychological nature of, 305

  -- real, confounded with hyperspace, 91

  -- Riemann on the curvature of, 5

  -- the unity of, 224

  -- -- -- -- with matter, 260

  Space-activities, as chemism and physicism, 229-30

  Space-center, the, 306-7

  Space-consciousness, 225

  -- as a direct process, 224

  -- mergence of, with the individual consciousness, 195

  Space-curvature, as an arbitrary construction, 262

  -- as near-truth, 306

  -- nature of, 279

  -- no need for, 237

  -- translation of, in terms of the intellect, 315

  Space-forms, the construction of, 74

  Space-genesis, 20, 232, 233

  -- alphabet of, 237

  -- completion of, 226

  -- norm of, 174

  -- pyknon as a basis of, 17

  -- symbolism of, 208, _ff_, 235

  Space-manifold, the, 279

  Space-measurement, and reality, Euclid's system of, 96

  -- systems of, 201

  Space-mind and archeological evidences, 193

  -- -- knowledge, 187

  -- -- the Thinker, 253, 272

  -- attainment of, 225, 231

  -- granaries of, 235

  -- the realization of, 238

  Space-process, the, 305

  Space-realities, hyperspace as a stepping stone to, 238

  Spark, the omnipsyche as a, 258

  Spatiality and the intellect, 263-4

  -- consubstantial with materiality, 20, 331, _ff_

  -- cut-off from non-spatiality, 266

  -- interpenetrative with materiality, 236

  -- nonconformity of, with logic, 164

  -- riddle of, and the intuition, 325

  --, rise of, 180

  -- Thinker's outlook in, 316

  -- world of nascent, 232

  Spencer, Herbert, 322

  Spinoza, 322

  Spiritism, the phenomena of, 154

  Spiritualists, the claim of, regarding the fourth dimension, 154-5

  Spirituality, materiality transmuted into, 328

  -- principle of, 210, _ff_

  -- seeds of, 221

  Spiritualization of the flesh, 227

  -- -- man's mental outlook, 40

  -- -- matter, the end of evolution, 327

  Squares, the form of, in the non-Euclidean geometry, 59

  St. Marie, Flye, 66

  Stage of pyknosis, 208, _ff_

  --, quartopyknotic, 219, _ff_

  Stanley, Hiram M., on space as dynamism, 104-7

  States, the two, of consciousness; one unaware of the other, 192

  Straight, a, determined by two points, 79

  Stuff, etherealized four-dimensional, 160

  Sum, angular, 79

  -- triangular, less than two right angles, 59

  Superconcepts, as intuitographs, 255

  Superconceptual, as the intuitional, 20

  Superperception and automatism of the intellect, 20

  Superperceptual, the, 74

  Superunodim, the, 177

  Swedenborg, 180, 184

  Symbol, psychic, and the brain, 190

  -- words as symbols of ideas, 126-7

  Symbolism of life, 286

  -- -- neurographical impressions, 186

  -- -- realities, 128

  -- -- sense-deliveries, 204

  -- -- space-genesis, 208, _ff_

  Symbology, depicting planes of consciousness, 270

  -- of mathematical knowledge, 203, _ff_

  Synchronous evolution of faculties and properties of matter, 332

  Syncretization of mind and matter, 299

  Synthesis and the norm of reality, 300

  Synthetic school of geometry, 56


  T

  Tactographic impressions, 190

  Tartrate and a component in the fourth dimension, 156-7

  Task of mental evolution, the supreme, 169

  Taurinus, 56-7, 69

  Telescope, its effect upon consciousness, 298

  Tendence, intellect as scientific, 165-6

  Tendencies, destined to flower as faculties, 89-91, 332

  Tendency to fragmentate, the intellect and, 296

  Tesseract as represented by Carus, 136-8

  -- illustrated, 133

  -- the, 20, 145

  -- the elements of, 135

  -- the four coördinates of, 132

  Testimony of the intellect, incapable of comparison, 76

  Tetragrammaton, the, 308

  Tetrahedron, the, 135

  Thalamencephalon, the, 345

  Thales, 44

  Theory of knowledge, 290

  Thingness of objects and the impossibility of comprehending, 122

  Things, the dual nature of, 119

  -- -- value of, 161

  Things-in-themselves, the identity of, 353

  Thinkability of the fourth dimension, 125

  Thinker as a pure intelligence, 184, 248

  -- -- engenderer of the intellect, 291

  -- -- principal to the ego, 243, _ff_

  -- -- six times removed from the sensible world, 201

  -- -- the source of the intuition, 26

  -- and concepts, 244

  -- -- his approach to the kosmic mind, 202, 223

  -- -- -- larger life, 196

  -- -- -- treatment of the ego, 246

  -- -- -- of sense-impressions, 74-5

  -- -- realism, 315

  -- -- space, 193

  -- -- a segment of reality, 171

  -- the, and the amplitude of the degrees of realism, 198, 270

  -- and the barriers of consciousness, 207

  -- -- -- brotherhood of man, 301

  -- -- -- degrees of realism, 175

  -- -- -- manipulation of concepts, 255

  -- -- -- nature of space, 268

  -- -- -- neurograph, 190

  -- -- -- new faculty, 335

  -- -- -- perceptive faculty, 188

  -- -- -- pituitary body, 351

  -- defined as the spiritual man, 21

  -- his consciousness and the ether, 280

  -- -- dependence upon the ego, 245

  -- -- method of contacting the external world, 189-191

  -- -- outlook upon the universe of spatiality, 242

  Thinker, his schematism of cognitive powers, 206

  -- -- sphere of awareness, 283

  -- the difference between, and the intellectuality, 256

  -- -- evolutionary needs of, 336

  -- -- upreaching yearnings of, 336

  Thinker-ward, space must be sought in a direction, 278

  Thinking, abstract, elevating influence of, 33

  Thought, prolonged abstract, benefits of, 249

  Time, as an aspect of consciousness, 224

  -- divested of timeliness, 279

  Todhunter, 46

  Tool, fashioning, life as a, 264

  Topography of the intellect, 312

  Touchstone, consciousness as a, 168

  Tracery of connection between ideas and objects, 126

  Trafficking in mental constructions, 154

  Transcendental, the realm of, and mathematicians, 142

  Transfinite as a limit, 122, 216

  Transfinity, 21-2

  Tree-reality, 126

  Triadic phase of evolution, 210-1

  Triangles, angular sum of, 78, 88

  Tridim, the, 22, 129, 134, 147, 177

  Tridimension, the, 147

  Tridimensionality and primeval space, 236

  -- and the space-mind, 193

  -- a quality of perceptual space, 22

  -- mastery of the phenomena of, 172

  -- the sufficiency of, 198

  Tripyknon, the, 212, 213

  Tripyknosis, 17, 208-10, 219

  Truth compared with facts, 288

  -- criterion of, 5, 323-4

  -- facets of, 284

  -- kinds of, 324

  -- logic does not illuminate, 287-8

  Tuitive, the, and the intuitive faculties, 192-3


  U

  Ultima Thule, the, 207

  Unbounded, the, as a finite extension, 70

  Undulations, three, in the current of life, 329

  Unfoldment, mental, and metageometrical discoveries, 131

  Uni-circle, the, 308

  Unification of all knowledge, 256

  Uniqueness of real space, 95

  Unitariness of all existences, 241

  Unity, as end of analysis, 42

  -- death as a fourth dimensional, 159

  -- kosmic, integers as fractional parts of, 41

  -- of mind and space, 224, 230

  -- the new realization of, 225

  Universality, geometric, based upon the formal character of
   assumptions, 77-8

  Universe, a glorified, 226, 268

  -- as a full, 308

  -- -- -- plenum, 107

  -- and the seven planes of matter, 212

  -- character of, fixed by consciousness, 162

  -- limited and conditioned, 127

  -- not a pure form, 108

  -- the perfected, and the circle, 310

  -- -- theory of the mechanical origin of, due to analytics, 40

  -- unity in the, 301-2

  Universum of appearance, 187

  -- of life and consciousness, 257

  Universum of reality, 127

  -- -- space, 269

  Unknowable as a symbol, 194

  -- the darkness which shuts out the, 207

  Unodim, the, 129, 134, 176-7

  -- -- consciousness, 163

  -- defined, 22

  Up-raisement of matter, 329

  Upspringing of a new faculty, 333


  V

  Vacuum-stillness of nature, 339

  Validity of mathematical conclusions, 101

  Value assigned by the intellect to the sensible world, 167

  -- assumptional, vitiating influence of, 163

  -- -- of the ideal, 275

  Vanity of fragmentary evidences, 204

  -- of intellectual method, 325, _ff_

  -- -- segmenting space into many dimensions, 299

  Vassal, life as the, of materiality, 329

  Vehicle of life, form as, 330

  -- -- -- mind as, 286

  Vicissitudes of kosmic evolution, 215

  Vision, the inner, 356


  W

  Wachter, 69

  Wallis, John, 46, 83

  Weissman, _note_, 260

  Words, as symbols of ideas, 126-7, 204

  World and the child mind, 121

  -- as instrument of consciousness, 199, 298

  -- fabric of, and geometrism, 261

  -- impossibility of objectifying the fourth dimension in the
   perceptual, 124

  -- of phantasy, conditions of, identified with sensible realm, 146

  -- phenomenal, and hyperspace entities, 128-130

  -- sensible, as a carpet, 196

  -- the domain of a perception, 19

  -- -- genesis of, 167

  World-plasm, kathekotic, 267

  -- as elemental essence, 329


  Y

  Yoga, Rajah, 357

  Youth of the earth, the nebulous, 335


  Z

  Zollner, and the claims of the spiritualists, 154

  Zones of Affinity, 22, 124

  Zoometer, the, 297




  Transcriber's Notes

  Variations in spelling, punctuation and hyphenation have been retained
  except in obvious cases of typographical errors. Inconsistencies
  between the text and index have been resolved in favour of the text.

  Duplication of the sub title (The Mystery of Space) on consecutive
  pages has been removed.

  Italics are shown thus _italic_.

  In the five cases where illustrations could be realistically rendered
  using only text symbols this has been done.





End of Project Gutenberg's The Mystery of Space, by Robert T. Browne