CHAPTER I

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.