The Mystery of Space - Forgotten Books

THE

MYST ERY OF SPACE

fl Study of the Hype rspa ce Movementin the Light of the Evolution of

New‘P sy chic Fa culties

and

fln Inq uiry into the Genesis and

Essential N a ture of Spa c e

BY

ROB E RT T . BROWN E

LONDONKEGAN PAUL ,

TRENCH,TRUBNER Co. , LTD .

NEW YORK : E. P. DUTTON CO

TO

THE CHERISHED MEMORY

ganglia Es Ian;

WHOSE WIFELY DEVOTION,

SM ATHETIC ASSISTANCE AND ENCOURAGEMENTDURING THE EARLY LABORS ON THE TEXT

WERE A CONSTANT SOURCE OF

INSPIRATION AND FORTITUDE To THE AUTHOR,THI S BOOK IS DEDICATED

AFFECTIONATELY

PREFACE

MATHEMATICS i s the b iometer o f intel lectual evolution. Hence

,the determination o f the status quo

of the intell ect a t any time can be a ccompl ished mosts ati s fa ctorily by applying to it the rigorous measureo f the mathemati cal method. The intellect has butone t rue divining rod and that i s mathemati cs . Byday and by night i t points the way unerringly, so longa s i t le ads through materi al i ty ; but , falteringly, bl indly,fatally

,when that way veers into the territo ry o f

vi tal i ty and spi ri tual ity.

Wherefore , when we have wi shed to ascertain thereal sta tus o f the intellect, a s well as its l im i tations ,tendencies, possib il i ti e s , we have turned to its conductin the field o f mathesi s where i t i s l ea st trammeled inits ingressive and egress ive motivations because o f thenatural and easy accommodation which is otiered fo rintellectual movement. Whethe r there a re s igns o fmoribundi ty o r symptomati c evidences o f markedgrowth o r o f a te rmination o f intellectua l regnancy,or whateve r may be the occasion for the examination ,no sure r index than the mathemati cal may be foundfor the pu rpose . Ful l logical j ust ification i s , there fore ,cla imed fo r the choice o f m athemati cal evidences totest the a ssumption that a new era o f conscious mental

vi i i PREFACE

conquest i s opening for the vangua rd o f humani ty and

sequently for the hum an fam i ly.

The treatm ent o f these evidences has fallen logically into two divi s i ons , namely, the fi rst, a bri e fand elementary revi ew o f the principl es of the nonEucl idean geometry and the i r bea rings upon the question o f space a s the subj ect o f mathemati cal s tudy ;and second

,the consideration o f space a s a psycho

logi cal , vita l and dynamic o r crea tive phenomenon .

In Pa rt I an effo rt ha s b een made to trace thegrowth o f the notion o f hyperspace and to Show thatit i s a symbol o f a new epoch of intellectual expansion,an actual se izure o f a new domain o f awareness bythe mind. And for thi s purpos e a cri ti cal examination o f the fundamental question of dimensi onal ity isentered upon from which i t develops that the sta tusof thi s prim a ry norm of mathem atica l thought exhib itsa rel atively inchoate cha racte r because o f i ts insufliciency a s a defin i t ive quantity, and further, because o fits rathe r sup e rfoetat ed aspect when util ized a s apanace a for matheti c symptom s . Also , i t ha s beenfound necessa ry to survey the fi eld o f the four-spacewhi ch ha s b een acco rded such a prom inent pl ace inthe mathemati ca l thought o f the day. The readershould find in the chapte r devoted to it adequate materi al fo r thought and suffi cient comprehensi on o f itsmeanings a s a mathematical contrivance .In Pa rt II an endeavo r is made to interp ret the

evidences offe red by high mathem ati c cons iderationsin the l ight o f the l a rge r psychogeneti c movement.For this undertaking the qual i ty o f awareness has beenstudied with the Vi ew to establ ishing its priority as adete rmining agency in the consideration o f space in

PREFACE ix

aspects broade r and less restri cted than those embraced in the mathematical premises. Wherefore , i tappea rs that there are massive impl i cat ions a risingout o f the hyperspace p ropaganda which have hithertobeen neglected in the prel im inari es incident to thefabri cation o f its structure . A very b rie f, and moreor l ess symbol ic , sketch o f the genesis o f space ha sserved to dem onstrate its essenti al nature a s a consubstantive o f materi a l i ty, vita l i ty and intel l ectual i ty,the three maj o r obj ective p rocesse s . Consequently i tbecom es imperat ively necessa ry that any View o f sp acewhich neglects its kosmi c intent and purpose shouldbe regarded as gravely fragmenta ry and insuffi ci ent .It is only by l inking up the two a spects o f space , themathemati cal and the psychological , in such a manne rthat the one shall supplement the other, that we shal lbe abl e to arrive at a truly sati s facto ry understandingo f its nature .

In Chapter IX attenti on i s invited to some ofthe extremiti es o f mathemati cal l aws where in i t isshown that, because mathematical goods a re stri ctlyo f intellectual texture and fabri c, va in i s the hope o freaching any rel i abl e certa inty with respect to manyVital quest ions, even regarding space itsel f, by meanso f the mathem atical method . The intelle ct

,and

,

there fore , mathemati cs encounte r the most formidabl estri cture when effo rt i s made to maneuve r in the fi eldo f vital i ty o r real i sm . In addition , i t i s shown that,when pushed to the u tmost l ogi cal l imits

,metageometry

proves not only futile , but emphasiz es th e need fora sharp turning o f the path o f se a rch from the intellectual or materi a l to the sp i ritual o r intui tional .Indeed , i t becomes pa infully certa in tha t the Golden

x PREFACE

Fleece of profounde r knowledge will be discoverednever by an expedit ion whose bark has i ts s a ils setfor the winds o f mathematical sea s . But, contrarily,a new ba rk , moored at the furthermost shore o f thesea o f intell ectual i ty with sa il s se t fo r the winds whichcome from the re alm of intu itional perception, mustb e se ized . Whereupon , by the stra ightest l ine , weshall , a t the l ast, l and upon the shore o f real i sm , o ftruth all inclusive .M athemati cal evidences have been used in these

discussi ons because they, of al l l ines o f knowledge ,afford a more just exemp lification of intell ectu al evolution. The science o f mathem ati cs is the m ea sure ofthe qual i ty o f intell e ctual growth and, there fo re , i tsdata , its postul ates, hypotheses and advances cl e a rlymark the stages o f the intell ectual m ovem ent .

Chapte r X is the na tural and logica l sequence o fthe inqu i ry into the question o f spati a li ty. The conclusions reached therein and the obvious inferenceswhich shoul d be drawn from the a rguments p resentedflow inevitably not only from the evidence s o f mathematical data but o f the comm on observations o f l i fe .And while we di scl a im any intention of demandingacceptance o f them as final , autho ritative decl arations,we shall b e satisfi ed i f the readers o f this volume beinci ted to solve fo r them selves the probl ems whichthese queri e s na turally suggest. Happy indeed shallbe the outcome i f there be any who , following the pathsketched herein , shall find the solution o f the Mystery of Space and apply its meanings to the enhancement o f the values of the in tu itive l i fe .In conclusi on, the autho r esteems i t a speci al obl i

gation of grat i tude that he Should here a cknowledge

PREFACE xi

the debt which he owes to all o f hi s fr i ends who havein any way assi sted o r encouraged him in the com

p letion of th is work . Among those whom he i s perm itt ed to thank in this way is M r . JAMES RINDFLEISCHwho , having ve ry kindly prepared the i llustrations fo rthe photo-engrave rs , i s deserving of sp ecial mention .

ROBERT T . BROWNE .

New York City,1919.

CONTENTS

PAGE

PREFACEINTRODUCTION : Exp lanatory Notes

PAR T ON E

CHAPTER I

THE PROLOGUE

the Variability of Psychic Powers—The Discovery ofthe Fourth Dim ension Marks a Distinct Stage inPsychogenesis— T he non-Methodical Character of

Discove ries—T he T hree Periods of PsychogeneticDeve lopment—The Scop e and Permissibility of

Mathetic License—Kosm ic Unitariness UnderlyingDiversity

CHAPTER I I

HISTORICAL SKETCH OF THE HYPERSPACE MOVEMENT

Egyp t the Birthplace of Geome try—Precursors : NasirEddin , Christoph Clavius, Saccheri, Lambert , LaGrange , Kant—Influence of the M ecanique And lytique—The Paral lel Postulate the Root and Substanceof the Non-Euc lidean Geometry—The Three GreatPeriods : The Form ative , Determ inative and Elaborativ e—Riemann and the Prop erties of AnalyticSpaces

xiv CONTENTS

CHAPTER I I I

ESSENTIALS OF THE NON-EUCLIDEAN GEOMETRYPAGE

The Non-Euclidean Geometry Concerned with Conceptual Space Entirely—Outcome of Failures at Solvingthe Parallel-Postulate—The Basis of the Non

Euclidean Geom e try—Space Curvature and Mani

foldness—Some Elements of the Non-EuclideanGeometry—Certainty, Necessity and Universality as

Bulwarks ofGeometry—Som e Consequences of Effortsat Solving the Parallel-Postulate—The Final Issueof the Non-Euclidean Geometry—Extended Con

sciousness

CHAPTER IV

D IMENSIONALITY

Arbitrary Character of Dimensionality—Various Definitions of Dim ension—Real Space and Geom etric SpaceDifferentiated—The Finity of Sp ace—Digerence Between the Purely Formal and the Actual—Space as

Dynamic App earance -The A Priori and the A

Posteriori as Defined by Paul Carus

CHAPTER V

THE FOURTH D IMENSION

Ideal and the Representative Nature of Objects inthe Sensible World—T he Fluxional , the Basis of

Mental D ifi erences—Natural Symbols and ArtificialSymbols —Use of Analogies to Prove the Existence of

a Fourth Dim ension—The Generation of a Hype r

Cube or T esseract—Possibilities in the World of the

Fourth Dimension—Som e Logical Diffi culties Inhering in the Four-Space Concep tion—T he Fallacy of

the Plane-Rotation Hypothesis—C. H. Hinton and

Major Ellis on the Fourth Dimension

CONTENTS xv

PAR T TWO

SPATIALITY ! AN INQUIRY INTO THE ESSENTIAL NATUREOF SPACE AS DISTINGUISHED FROM THE MATHE

MATICAL INTERPRETATION

CHAPTER VI

CONSCIOU SNESS THE NORM OF SPACE DETERMINATIONS

PAGE

Realism Is Determined by Awareness—Succession of De

grees of Realism—Suffi ciency of T ridim ensionalityThe Insuffi ciency of Self-consistency as a Norm of

T ruth—General Forw ard Movement in the Evolu

tion of Consciousness Imp lied in the Hyperspace Concep t—The Hypothetical Nature of Our Know ledgeHyperspace the Symbol of a More Extensive Realmof Awareness —Variations in the Methods of Interp ret ing Inte l lectual Notions—The T uitional and the

Intuitional Facul ties—T he I l lusionary Character of

the Phenomenal—Consciousness and the Degrees of

Realism

CHAPTER VI I

THE GENES IS AND NATURE OF SPACE

Symbology of Mathematical Know ledge—Manifestationand Non-manifestation Defined—The Pyknon and

Pyknosis—The Kosmic Engenderm ent of Space—O n

the Consubstantiality of Spatiality, Intel lectuality,Materiality, Vitality and Kosmic Geome trism

Chaos-Theos-Kosmos—Chaogeny and Chaomor

phogeny—N. Malebranche on God and the WorldThe Space-Mind—Space and Mind Are One—TheKosmic Pentoglyph

xv i CONTENTS

CHAPTER VIII

THE MYSTERY OF SPACE

Thinker and the Ego—Increscent Automatism ofthe Intellect—The EgOpsyche and the Omnipsyche

Kosm ic Order or G eometrism—Life as EngenderingElement—The Mystery of Space Stated—Kathekosand Kathekotic Consciousness—Function of the Ideal—The Path of Search for an Understanding of the

Nature and Extent of Space Must Proceed in an In

CHAPTER IX

METAGEOMETRICAL NEAR-TRUTHS

Realism Is Psychological and Vital—The Impermanence ofFacts—Ou the T endency of the Intellect to Fragmentate—The Intellect and Logic—The Passage of

Space—Kosmomet er and Zoome ter, Instrum ents forthe Measurement of the Passage of Space and the

Flow of Life—The Disposal of Life and the Power

to Create—Space , a Dynam ic , Creative Process

Numbers and Kosmogony—Kosm ic Significance of

the Circle and the Pi-p roportion—Mechanical T end

ence of the Intellect and Its Inap titude for the Understanding oi Life—The Criterion of T ruth

CHAPTER X

MEDIA OF NEW PERCEPTIVE FACULTIES

The Spiritualization of Matter Is the End of EvolutionSequence and Design in the Evolution of Human

Faculties—The Upsp ringing Intuition—Evidences of

Supernormal Pow ers of Percep tion and the Possibilityof Attainm ent—The Influence and Place of the Pitui

tary Body and the Pineal Gland in the Evolution of

Additional Faculties—The Skep tical Attitude of Em

p irical Science and the Need for a More LiberalPosture—The General Resul ts of Pituitarial Awakening Upon Man and the Theory of Knowledge .

BIBLIOGRAPHYINDEx

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2 THE MYSTERY OF SPACE

In 1854, the succeeding year, he was expelled from thiscollege in company w ith five of his colleagues who w ere ac

cused of p romoting “disorders

”against the Abbot LEONARDI,

rector of the college. The expulsion brought him many hardships and disappointments, and for two years he drifted alongmerely existing as his family was too poor to have him matt ieulated at another university. But in 1856, he went to Veronawhere he succeeded in securing employment as secretary to the

engineer, DIDAY, in the Government service of LombardyVenice. On January 10, 185 7 , he was dismissed from thisposition for political reasons but as the annexation of Lom

bardy to Piedmont occurred soon thereafter, he became againattached to the ofiice of DIDAY, his former emp loyer, when itwas 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, hisformer tutor, and also LUIGI CREMONA . Through the influenceOf these two men he was designated (October 18, 1862)

“Pro

fessore straordinario” in the University of Bologna. His workon Surfaces of Constant Negativ e Curvature, as the pseu

dosphere , and his app lication Of the exp ression given by LOBACHEVSKI for the angle of parallelism , very definitelysecure for him a p lace among the foremost workers in the fieldof 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 centerof gravity Of the centers of its inscribed and escribed circles.”

He died in the year 1900. (Vide Amer . Math. M0 Vol. IX ,

P.

BOLYAI , JANOS ( 1802 was born at Kleansenburg, Hungary. He is said to have inherited his mathematicalgenius from his father, BOLYAI FARKAS ( 1 7 75-18 who was

born at Bolya, Hungary. Being a very spirited youth, hisprogress in his studies was most remarkable . He comp letedthe curriculum at the Latin school when only twelve years of

INTRODUCTION 3

age. Was graduated from the Philosophical Curriculum as a

resul t of two years Of study and then entered the VienneseAcademy Of Engineers. Was appointed lieutenant at T emes

varlin, 1823 , whence on November 3 , 1823 , he wrote hisfather : “I have discovered such magnificent things that I ammyself astonished at them . I t would be damage eternal ifthey were lost. When you see them , my father, you yourse lfwill acknow ledge 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 beenpreserv ed at the Marcos Vasarhely, Hungary. The mathemati

cal concep tions 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. HOUEL, towhom belongs the credit of popularizing the works both of

BOLYAI and LOBACHEVSKI. (Vide Science, 11. s., Vol. 35 ,

No. 906,

CAYLEY, ARTHUR, born at Richmond, Surrey, England, August 16, 182 1 ; studied at King

’

s College school ;entered T rinity College , Cambridge , already a well equippedmathematician at the age of seventeen. When but twenty-oneyears of age he took two of the highest honors in the Uni

versity of Cambridge . He was Senior Wrangler and FirstSmith’s Prizeman. He published his first pape r in 184 1 and

this was followed by eight hundred memoirs.For fourteen years he p racticed as Conveyancer. In 1863

Lady SADLER’

S various trusts were consolidated, and a new

Sadlerian professorship of Pure Mathematics was created forthe express purpose of affording a p lace for Cayley. Mean

while , as early as 1852 he was a fellow of the Royal Society ;in 1858 he joined SYLVESTER and STOKES in publishing the

Quar terly Journal of Pure and Applied Mathematics.

He was for a considerable time p rincipal adviser as to the


merits of all mathematical papers which were presented forpublication to the Royal Socie ty , the Astronomical Society, theMathematical 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 know ledge of mathematies. (Vide Rev iew of Reviews, Vol. I I , 1895 , Sketch, t eprinted 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 worldp lasm into the first faint signs of orderliness ; the beginningsof the movement of life or the Creative Logos in p reparationof 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 boundsof the universe ; also, the origination and characte rization of all

forms as to tendence , purpose and limitations.

CONCEPTUALIZAT ION—The act of concep tualizing,the formulation of concep ts ; the process by which the Thinkerarrives at concep ts ; the logical p rocedure by which the con

sistency of a scheme of thought is established.

CONSTRUCTION,IDEAL—A purely formal concep

tion ; a theory, hypothesis ; a logical de term ination not meces

sarlly based upon facts, but possessing virtue because of con

sistency ; a self-consistent scheme of thought.

COSMOS—Whenever the term cosmos appears in the

text spelled as here Shown, it refers to phenom ena pe rtaining tothe earth or the solar system ; when spelled

“kosmos” reference

is made to the universe as a whole.

INTRODUCTION 5

CRITERION OF TRUTH—De fined in the text as a

four-fold standard of reference , embracing the follow ing ele

ments, namely, the causal , the sustentative , relational and developmental. Lacking any one of these , no view of truth ismore than fragmentary. App lied to space , it contemp lates an

inquiry into the genesis or causal aspect , an accounting for theduration aspect, a recognition of its relation to the totality of

objects, and lastly, a prophecy of its telestic or perfective cul

mination. This test has been app lied to the study of space as

sketched in the text and the conclusions reached are an out

come of the inquiry directed along these lines.

CURVATURE OF SPACE—A doctrine formulated byRIEMANN and which maintains that space is curved, and con

sequently, all lines drawn therein are curved lines. Pro

fessor PICKERING ap tly describes the resul ts of movements ina curved space by pointing out that if we go far enough eastwe arrive at the w est ; north, we arrive at the south ; towardsthe zenith, we arrive at the nadir

,and vice v ersa.

DEIFORM—The basic idea indicated is that the universeis the form or body of the supreme deity, since He is not onlyimmanent in the kosmos, but sustains it by His life ; that inorder to create a manifested universe , it was necessary to limit ,or sacrifice, in a measure , His own illimitability. Viewed inthis light, the kosmos assumes an added significance.

D IMENS ION—(L . Dimensio, to measure ) , measurement ; a system of Space m easurement . The Euclidean geome

try recognizes three dim ensions 01 coOrdinates as being necessary

to establish a point position ; w itness, the corners Of a cube toform which three of the edges come together at a point. These

edges represent coOrdinates. For the purposes of me tageome try,the term dim ension has been variously defined, as, direction,extent, a system of space measurement , or a system of COOrdi

nates. Regarded as a series of coordinates, it became possibleto postulate a system which required four coOrdinates to estab


lish the position of a point, as in the hypercube. There maybe five , six, seven, eight , or any num ber of such coOrdinate sys

tems according to the kind of space involved in the calculations.

Determinations based upon the logical necessities of the various coOrdinate systems have been found to be self-consistentthroughout and, therefore , valid for metageometrical purposes.Much dep ends upon the definition ; for , after the definition hasbeen once determined it remains then merely to make inferencesand conclusions conform to the intent of the definition.

DIVERSITY—Philosophically, the idea indicated has ref

erence to all dissimilarities, differences, inequalities, divergenttendencies, movements and characteristics to be noted in the

universum of life ; the antithesis of kosm ic unity ; the naturaloutcome of life in seeking expression ; the resul t of the fragmentative tendency of life .

DUODIM (duo, two ; dim, abbreviation of dimen

sion)—A hypothetical being supposed to be possessed with a

consciousness adap ted only to two dimensions ; a dweller in“Flatland” or two-space whose scope of motility is limited totwo directions, as on the face Of a p lane ; a term invented byhyperspace advocates for the purpose of establishing by analogysome of the characteristics Of the four space and also itsrationality.

DUOPYKNON (duo, two—pyknon, p rimary unit in the

process of kosmic involution, a oondensation) —Secondaryphase in the elaboration of chaos into kosmic order. DUOPYK

NOSIS (duo, two, secondary—pyknosis, process of condensationand origination) -The second period of the involutionarymovement of life during which the duadic p lane Of the kosmosis being established ; the second, in the series of sev en distinctphases, of space

-

genesis ; dual differentiation of kosmic p lasm .

Duopyknosis contemp lates that , in the passage of the kosmos~to-be from the p lane of non-manifestation to the p lane of mani

festation, there are seven distinct , though interdependent and

INTRODUCTION 7

interrelational , stages through which life passes, and that , of

these , it is the second. I t relates to the p lane 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 emo

tional or affect ive me chanism , the nerve-systems (cerebro-spinal

and sympathetic ) and the brain ; the objective or sense-derivedconsciousness of the Thinker which is elaborated from the

total mass of pe rcep tions transmitted through the senses ; the

medium of self-consciousness ; the intellectual consciousness as

distinguished from the intuitional or omnipsychic consciousness

The notion of the egopsychic consciousness is based upon

data already empirically determ ined from the mass of ev i

dences everywhere observable . I t se ems to be apparent thatthere is a consciousness, a seat of know ledge , in man the con

tent of which is unknown to the sense-consciousness. Dreams,premonitions, intuitions, impressions and the totality of all suchphenomena substantiate this view . Furthermore

,it is agreed

that the source of the intuition is not identical w ith that Of

the intellect. The egopsychic consciousness, accordingly, is

purely intellectual.

FLUXION, PSYCHIC—The difference between a mentalimage and an obj ect ; an image is the representation of cer

tain salient or cardinal characteristics Of an Object , sufficientfor identification ; but an image is not congruent, in every t e

spect , w ith the object. Thus when we p erceive an Object ,although as BERGSON contends w e perceive it in the p lace whereit is and not in the brain, it is the image of the obj ect whichtakes its p lace in memory and not the object itse lf. There is,

of course , a marked disparity be tween 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 p lace in memory,and neither could it do so if it possessed any of the p ropertiesof the real Obj ect. Consciousness is such that all due allowanecs are made for these conditions and the mind is able to

retain more or less exact know ledge of these properties in the

image ; but there is a difference , small though it may be. This

difference is the p sychic fluxional.

FOHAT (Skt )—A term app lied to the Creative Logoswho is said to be the generating element in the differentiationof chaos into kosmic orderliness ; the supreme deity in the rOle

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 orcondition ; 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 Chap ter V ) a space in which four coordinates ( fourlines drawn p erp endicular to each other ) are necessary and

sufficient to establish the position of a point, as, a hypercube.

GAUSS, CHARLES FREDERICK , born at Brunswick,Ap ril 30, 1 7 77 . His father, being a bricklayer, had intendedthat he should follow the same occupation. SO, in 1 784,

Charles was sent to the Biitner Public School in Brunswick, inorder that he m ight be taught the ordinary elements of education. But during his attendance at this school , his unusual intelligence and ap titude attracted the attention and friendshipof Professor BARTELS who later became the Professor OfMathematics at Dorpat. In 1 792, through the kindly representations of Professor BARTELS to the Duke of Brunswick, young


KANT closely adhered to this notion in his discussions of spaceas an aspect of divine intelligence. He believed that the intellect merely rediscovers thi°s latent geome trism when it turnsto the study of materiality , and this belief is Shared by BERGSON,

the foremost metaphysician of the present time.

HYPERSPACE (hyper, above , beyond, transcendingspace )—That species of space constructed by the intellect forconvenience of measurement an idealized construction ; a purelyarbitrary, conventional mathematical determination ; the fourthdimension ; any space that requires more than three coOrdinates

to fix a point position in it , as, a five space , an n-space.

INTUITOGRAPH- The means by which the omni

psychic consciousness transmits intuitional imp ressions to the

egopsychic or intellectual consciousness. An intuitogram is a

direct cognition, an intuition ; a primary truth p rojected intothe egOpsychic consciousness by the Thinker. I t is recognizedthat, under the necessities of the present schematism of things,it is exce edingly difficul t to propagate an intuition, especiallywith the same degree of ease as concep ts are p ropagable ; ye t ,this is believed to be a condition which will be overcome as theevolution of the higher faculties proceeds.

INVOLUT ION—Process of enfolding, involving ; antithesis of evolution ; philosophically, the doctrine of involutionmaintains that , during the process of kosmic pyknosis (spacegenesis) , all that is to be expressed, developed and perfectedas a result of the evolutionary movement was first involved,enfolded or deposited as latent archetypal tendencies and radi

cles in the original world-p lasm ; that , as the involutionarymovement proceeded through the various phases of space

genesis, these became more and more phenomenal until at lastthey terminated in the e laboration of a manifest universe : eachstage, accordingly, of the involutionary procedure became the

basic substructure of a p lane of specialized substance or ma

INTRODUCTION I t

teriality and consciousness. Thus it appears that evolutionreally begins where involution ends (vide Fig. and the

two opposing processes constitute the dualism of life as gen

crating element. This np tion 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 dev ised for the

express purpose of p roviding a convenient symbol to convey theidea embodied in the triglyph, Chaos-Theos-Kosmos, and is

composed of the first three letters in each one of the terms ofthe triglyph ; hence , symbolizes the triunity and interaction iavolved in the resolution of chaos into an orderly kosmos bythe will of the Creative Logos. Thus,

“kathekos

”embodies a

quadrup licate notion, name ly, chaos, Creative Logos, manifestedkosmos, 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 comprehen

siveness.

By referring to figure 1 7 , it will be seen that Kathekosdivides into two kinds—involutionary, or that which pertains toinvolution, 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’

étre

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 archetypalstate 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 aphase of the world age when the kosmos has reached ul timateperfection, embodying the perfected results of the possibilitieswhich inhered in the chaogenic period or in involutionarykathekos. Thus, kathekos is dual in nature , on the one hand

rep resenting kosmic potency, and on the other, kosmic perfec


tion of these potencies. It is Alpha, as re lated to involution,and Omega, as related to evolution.

KATHEKOSITY—A derivative , signifying creative ac

tivity and all that it imp lies ; the state of consciousness or coguition corresponding there to.

KLEIN , FELIX ( 1849 born at Dusseldorf ; studiedat Bonn , and when only seventeen years of age was made as

sistant to the noted PLUCKER in the Physical Institute. He

took his doctorate degree in 1868, then w ent to Berlin, and

later to GOt tingen where be assisted in editing PLUCKER’

S

works. He entered the GOt tingen faculty in 187 1 became Professor of Mathematics at Erlangen in 1872 ; and subsequentlyheld professorships at Munich, 1875 ; Leipzig, 1 880, and GOt

tingen, 1886. NO one else in Germany has exerted so greatinfluence upon American mathematics as he.

KOSMOS—See Cosmos.

LA GRANGE , JOSEPH LOUIS, born at Turin, January25 , 1 736 ; died at Paris, Ap ril 10, 1 8 13 ; regarded as the greatest mathematician Since the time of New ton. It may be in

teresting to note that LA GRANGE remarked that mechanics is

really a branch of pure mathematics analogous to a geom etryof four dimensions, namely, time , and the three coordinates of

the point in space . (Vide Ball'

s Account of the H istory ofMathematics. )

LIE , SOPHUS, a noted mathematician,referred to as

the“

great comparative anatomist of geometric theories, creatorof the doctrines of Contact T ransformations and Infinite Continuous Groups, and revolutionizer of the Theory of Differential

LOGOS—The supreme deity of the phenomenal universe ;Creator ; Fohar; a p lanetary god ; the deity of a solar system .

INTRODUCTION 13

MANVANTARA (Skt . )—A world age ; the periods of

involution and evolution combined ; the stage during whichthe universe is in manifestation ; a Day of Brahma.

MATHESIS (Gr. mathein, to learn)—Eri1dition ; profound learning ; the realm of metaphysical concep tions ; thefield of higher mathematics ; the sphere of conceivability ; thetheoretical.

MENTOGRAPH—A cognitive factor consisting of a

complete percep tion fused or in coalescence w ith a memory

image . Pure memory, of itself, is w ithout utility as an aid to

cognition ; but , when nourished or supp lemented by the sub.

stance of p ercep tion it becom es the basis of intellectual consciousness.

METAGEOMETRY (Gr. Meta,beyond, transcending

geometry) —Commonly, any kind of geom etry that differsfrom the Euclidean, as the non-Euclidean ; a geom etry basedupon the assump tion that the angular sum of a triangle isgreater or less than two right angles ; the highest form of

geometry ; a system of idealized mathematical constructions.

Sometimes called “

pangeomet ry”

; designated by GAU SS as

“Astral Geome try the geometry of hyperspace. I t consistsof 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 coordinate w ith the omnipsyche

and as such is the medium of kosmic consciousness.

MORPHOGENY (Gr. Morphe, form,vehicle , body

geny, evolution) -The evolution of forms, the p roduction of

individual bodies or vehicles for life, including organs and


faculties. Morphogenic—a derivative ; pertaining to mor

phogeny ; a kosmic process (vide figs. 1 7 and

N-DIMENSIONALITY—Quality of concep tual space byvirtue of which it may be regarded as possessing an indefinitenumber of dimensions.

NEAR-TRUTH—Any statement or view which is basedupon partial know ledge ; p redicates concerning a class or genus

derived from limited acquaintance w ith particulars of the classor genus ; statements based upon logical determinations inhering in idealized constructions and app lied to concre te or objec

tive conditions ; an abstraction view ed as a reality ; the app lication of the qualities of abstractions to realities.

NEUROGRAM—Psychologically, a movement received bythe afferent nerves in the form of a stimulation and transmittedthrough the brain and efferent nerves as either a reflex or vol

untary action ; a nerve impulse ; a p ercep tion ; a primary unitof intellectual consciousness ; cf. Intuitogram.

NEWCOMB , SIMON ( 1835 born at Wallace ,Nova Scotia ; educated in his father’s school and came to the

United States in 1 853 . Began, in 1 854, teaching in Mary

land ; was appointed computer on Nautical Almanac at Cam ~

bridge in 1857 ; was graduated at Law rence Scientific Schoolin 1858 ; appointed Professor of Mathematics in the U . S. Navyin 1 861 . He supervised the construction of the 26-inch

equatorial telescope at the Naval O bse rvatory, and was secre

tary of the T ransit of Venus Commission ; was a member ofnearly all of the Imp erial and Royal societies of Europ e and

of the various societies in the United States, receiving the

Cop ley Medal in 1 874 ; the Huygens, 1 878 ; the Royal Society,1 890 , and the Bruce Medal in 1 898 ; held the presidency of

the following learned societies,v iz; Am erican Association for

the Advancement of Science , 1 87 7 ; Society for Psychical Re

INTRODUCTION 1 5

search, 1885- 1886 ; Ame rican Mathematical Society, 18971 898 ; the Astronom ical and Astrophysical Society of America

from its foundation in 1899 . He rendered notable service inpopularizing 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 p rem ise excep t that which naturally,and not artificially and conventionally, conforms to what isactually perceived ; if so, the re should be justification for challenging the w isdom and utility of the p resent schematism of

things. There is an inherent conformity of space w ith intellectand intellect w ith space , and because of this natural complementarity of part w ith part and whole w ith whole , space cannotbe otherw ise than the intellectuality normally conceives it to

be, p rovided, of course , that the cognitive movement is free

and untrammeled by arbitrary hindrances. Consciousness,

therefore , is the norm or standard of reference for all questionsarising out of a consideration of spatiality.

OMNIPSYCHE—A term used to denote the Thinker’scognitive apparatus ; the universal soul manifesting in indi

v iduals ; the consciousness of the Thinker in virtue of which

he is at—one w ith 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 w ay be said justly to take its rise from

sense-experience or from any bodily p rocess. If divine , theneternal , and the refore , persistent . Broadly, the doctrine of

ev olution recognizes the passage Of life from form to form ,

adding a little to each successive form and inevitably pushingeach to a higher degree of p erfection. Now , what is it thatpasses from form to form ? Is it undifferentiated life or is it a

sp ecialized form of life ? From every evidence , it would bejudged that the life that ensouls an individual form is a specialized princip le, i.e . ,

lim ited to the execution of a given purpose.


If life as a specialized princip le , limited to the execution of a

given purpose in each form , passes on, it must p reserve , at least,the sublimated results Obtained during its residence in each in

dividual form . I t would thus become a sort of reservoir con

taining all these transmuted results. T he omnipsyche , w ithinthe meaning of the tex t, is p recisely this specialized life princip le.

PARALLEL-POSTULATE—Variously referred to as the

KIth, XIIth and XIIIth axiom of the Elem ents of Euclid

stated by MANN ING, in Non-Euclidean Geometry, p . 9 1 , in the

follow ing form :“If two lines are cut by a third, and the sum

of the interior angles on the sam e Side of the cutting line isless than two right angles, the lines will meet on that side

when sufficiently p roduced. This ce lebrated postulate hasp roven to be the most fruitful ever devised ; for it embodiesin itse lf the possibility of three geom etries based respectivelyupon the follow ing assump tions, nam ely : I . That there existsa triangle , the sum of whose angles is congruent to a straightangle , the Euclidean ; I I . That there e xists a triangle the sum

of whose angles is less than a straight angle , the LOBACHEVSKIAN ; I I I . That there exists a triangle the sum of whose anglesis greater than a straight angle , the CAYLEY-KLEIN . Speakingof the content of the last two named, EDWARD MOFFATWEYER,

1says :

“Hypothetical realms,wherein the dimensions

Of space are assumed to be greater in num ber than three , yieldstrange geom etries, which are only card castles, p roducts of a

sort of intellectual p lay in the construction of which the law sof logic supp ly the rules Of the game . The character of each isdetermined by whatsoever assump tion its builder lays downat the start.

PASSAGE OF SPACE—A phrase connoting the movement of Space from Chaos to perfect order, a p rocess believedto be infinite. The genesis of space necessarily implies an

‘Vide Popular Science Monthly, v ol. 78, p . 554 , 191 1 .


cerned in the genesrs ofmentality ; SEXTOPYKNOSIS, kosmicsensibility ; SEPTOPYKNOSIS, kosmic mate riality. These

seven phases of duration constitute the mom of space genesis or

kosmogenesis, and incidentally depose the substructure of kos

mic materiality, sensibility, intellectuality and sp irituality, as

well as the higher trinity of kosm ic modes. T he ramificationsof these p rincip les are innumerable and omnip resent. (See

Chap ter VII . )

QUARTODIM—A hypothetical be ing assum ed to have a

consciousness adap ted to hyperspace or the fourth dimension,and whose scope of action is encompassed w ithin a space whichrequires four coOrdinat es, as the four-space .

REALITY (Realism )—Life ; the harmony existing among

the parts to maintain their equilibrium in the whole ; the p rinciple of integrity subsisting among parts ; kosmic vitality.

RIEMANN , GEORGE FREDERICH BERNHARD,

was born Sep tember 1 7 , 1 826, in the village of Breselenz, nearDannenburg, in Hanover. Until he was eight years of age

his father w as his sole tutor, but even at this age he exhibitedgreat pow ers of arithmetical calculation. In the Sp ring of 1840

young RIEMANN w as sent to the Hanover Lyceum where he

remained for two years, leaving in 1 842 for the Gymnasium at

Luneburg. Here , under the direction of Professor SCHMALFUSS, he learned very rapidly, and is said to have required onlyone week thoroughly to familiarize himself w ith LEGENDRE’STheory of Numbers.

On Ap ril 12, 1846 (Easter ) , he entered the University of

GOt tingen as a student of Theology in accordance w ith hisfather’s w ishes. His passion for mathematics, how ever

,was

so aroused by the lectures of GAUSS that he begged his fatherto be allow ed to devote himself entirely to the studies of his

choice . For two years he studied under JACOBI at Berlin. He

then returned to GOt tingen, and was graduated, his thesis being

INTRODUCTION 19

a dissertation on the foundations of a general theory of func

tions of a variable comp lex magnitude . In 1 854 he qualifiedas a teacher by giving a lecture on the Hypothesis on which

Geometry is Founded. In 185 7 he became“Professor Extraor

dinarius,”

and in 1859 was elected Corresponding Memberof the Academy of Sciences of Berlin and in 1860 a m ember ofthe Academy of Sciences of GOttingen.

After four years of failing health, during which he visitedMessina, Palermo , Nap les, Rome , Florence , Pisa and Milan,he died at Lago Maggiore , July 20, 1866, in full possession of

his faculties and conscious of his app roaching end.

SCHWEIKART , FERDINAND KARL ( 1 780studied from 1 796 to 1 798 in Marburg, attending the mathematical lectures of J . K. F. HAUFF. In 18 12 he became pro

fessor in Charkov , a position which he held for four years. In

18 16 he became a tutor in the City of Marburg where he re

mained until 1820 when be transferred his labors to KOnigsberg.

It was during his tutorship at Charkov , Marburg and KOnigs

berg that he , entire ly alone and w ithout the slightest suggestionfrom any man, deve lop ed and taught a non-Euclidean geometryto the students under his care . For copy of his treatise on

non-Euclidean geom etry, see Historical Ske tch of the Hyper

space Mov em ent , Chap ter I I .

SCOPOGRAPHIC IMPRESSIONS—Sight p ercep tionsfused w ith an associated memory-image , and forming the basisof action on external phenom ena.

SENSOGRAPHIC IMPRESSIONS- Percep tions or impulses transmitted through the nerve s of a sense-organ ; any

impression acting through the media of the Senses.

SENSIBLE WORLD—The world of the senses ; thatwhich responds to the senses ; the domain of percep tion ; thephenomenal world ; world of percep tual space .


SPACE-CURVATURE (see Curvature of Space ) .

SPACE-GENESIS—The process of spatial engenderment ;the movement of life as engendering agent in bringing intomanifestation 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 con

sciousness do not otherw ise admit of the empirical establishment of the notion of its detailed p rocedure.

SPAT IALITY—Space as a dynamic, creative movement ;kosmic order, as opposed to disorder ; the path of the engender

ing movement of life ; the p lace Of life . Spatiality, materiality,intellectuality and geometricity or the latent geometrism of the

kosmos are thought of as being consubstantial and interde

pendent ; but , of these , spatiality is regarded as the substanceout of which the latter three are elaborated.

SUPERCONCEPTUAL—The purely intuitional ; an act

of cognition p erform ed w ithout the detailed w ork of concep

tion derived from sense-data ; concep tion of intuitions and theirinter-relations ; the Thinker’s consciousness freed from intellectual Characterization.

SUPERPERCEPT ION—Percep tion of concep tual relations ; a state of cognition wherein, instead of receiving perceptsor images from the external w orld, then elaborating them intoconcep ts, the Thinker apprehends composite images or con

cep ts at first hand. I t is a pow er which the liberated m ind of

the future w ill possess ow ing to the grow ing automatism of

the intellect and the more facile expression of the intuitionalconsciousness.

TESSERACT (Gr. T essera, four, cube , tessella)—Ahypercube (see Chap ter V. )

INTRODUCTION 2!

TH INKER (Skt . Manu, thinker ) —The real , spiritualman, as differentiated from his p ercep tive vehicles —mind, emotions and physical body ; the omnipsychic intelligence who t e

ceiv es, classifies, interp rets and p rese rves percep ts ; the manipulator of concep ts ; in fine , the higher, Sp iritual man.

The Thinker uses the various percep tive instrumentalitiesas so many tentacles or antennae by which he contacts the sen

sible world and makes the necessary adap tations to environment. He is the pure intelligence which is the source of all

cognitive motivation ; Opposed to ego, because the egopsychic

instrumentality is essential ly an individualizing, separativeagency ; while the Thinker ’s omnipsychic inte lligence is the

basis of his unity w ith the universal intelligence . This concep

tion of the Thinker imp lies that , as a spiritual intelligence , heis within and w ithout the body, filling it as the ocean fills thesponge , encompassing, envelop ing 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 p liability of his vehicles.

TRANSFINITY—A state or condition that is incomp rehensible 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 t ransfinite rather than infinite ine xtent. But space t ransfinite should be distinguished from

space finite though unbounded.

”For, there would seem to

be little worthy of Choice betw een a“finite , unbounded space

”

and an infinite one . The absence of boundary would naturallysuggest 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 inextent. I t would seem to do violence to common sense , if not


to logical necessuy, to view space both as unbounded and

finite in extent, ye t 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 per

ceptual space by virtue of which it is necessary and sufficient tohave three coOrdinates, and only three , to establish the positionof a point.

UNODIM—A hypothetical being assumed to have a con

sciousness limited to linear or one -space .

ZONES OF AFFINITY—Regions in the domain of in

tellectuality wherein m inds, possessing a common differential ,rate of vibration or quality, adhere to certain tenets fromchoice . Schools of philosophy, religions, and all those majordivisions of intellectual effort which divide and subdivide int ellectual allegiance are believ ed to take their rise in this p roperry of intellectuality in virtue of which all m inds having a

sim ilar coeflicient gravitate towards a common agreement , es

pecially where the movement is voluntary and untrammeled.

PAR T ONE

THE ESSENTIALS OF THE GEOMETRY OFHYPERSPACE AND THE IR

SIGNIFICATION8

CHAPTER I

THE PROLOGUE

On the Variability of Psychic Pow ers—The Discovery Of the

Fourth Dim ension Marks a Distinct Stage in Psycho

genesis—The Non-Me thodical Character of DiscoveriesT he Three Periods of Psychogenetic Developm ent—TheScope and Perm issibility of Mathetic License—KosmicUnitariness Underlying Diversity.

IN presenting thi s volume to the publi c p ro foundapologi e s a re made to the pro fess ional mathemati ci anfor the temeri ty which i s shown thereby. All technical di s cussion o f the problem s pertinent to thegeometry o f hypersp ace , however , has been ca re ful lyavo ided . The reader i s , there fore , re ferred to theb ibl iography publ i shed at the end o f thi s volume formatte r rel ating to thi s a spect o f the subj ect. Thea im rather has been to outl ine bri efly the progress o fmathematical thought which ha s l ed up to the ideaof the multipl e dimensi onal i ty o f space ; to state the

23


cardinal p rincipl es of the Non-Eucl idean geometryand to Offe r an interp retation o f the m etageometri calconcept in the l ight o f the evolutionary na ture o fhuman faculti e s and materi al cha racterist i cs and

p roperties .The onus o f thi s treatise is, there fo re , to di s

tinguish between what is commonly known as sensibl espace and that other speci e s Of spa ce known a s geometri c Spaces . Al so to show that the notion whichha s been styl ed hyp erspace i s nothing more nor lessthan an evidence o f the fa int , e a rly outcropp ings inthe human m ind o f a fa culty which , in the course o ftim e , will becom e the norm al possess ion o f the enti rehuman race . Thus the weight o f al l presentati ons wil lb e to give currency to the bel i e f , very strongly held ,that hum ani ty, now in its infancy, i s yet to evolvefacul ti es and cap ab il it i es, both mental and sp iri tu al ,to a degree hitherto vi ewed a s inconce ivabl e .

On this View i t must appea r that the faculty o fthought including the powers o f imagination and concep tualization a re not psychological invari ants , but , onthe other hand , a re true vari ants . They a re , consequently, answerable to the principl e o f evolut ion j usta s all vi ta l phenom ena a re . Some have thought thatno matte r what ide a may com e into the mind o f thehuman race o r a t what tim e the idea may be bornthe mind always ha s been able to conceive i t . Thatis, many bel ieve that the nature o f mind i s such thatno matter how complex an i de a may b e there ha salways b een in the mind the power o f conce iving it.But this View cannot be sa id to have the support o fany trustworthy testimony. I f so , then the m ind mustat once be recogni zed as fully matured and capabl e


sat ionalists, l ed by LEIBN ITZ, propounded a s thei rfundamental premise thi s dictum : There is no thingin the intellect which has not first been in the senses

excep t the intellect itse lf,” and thi s ha s never been

ga insa i d by any school that could di sp rove it. Theintuitional i st does not deny it : he merely cl a ims thatwe a re the recip i ents Of anothe r fo rm of knowl edge ,the intu it ional , which , instea d o f be ing derived fromsense-experi ence , i s p roj ected into the intell ectua l consciousness from another source which we designatethe Thinker. Thus , from the two forms o f consciousness, come into the a rea o f awareness truthsthat sp ring from enti rely different sources . From theone source a steady stre am of impressions flow con

stituting the substance o f intellectual consciousness ;from the other only a drop , eve ry now and then , fallsinto the grea t inrush ing mass so as to add a dimphosphorescence to an otherwi se unilluminated pool .Obviously, when there i s a l a ck of sensuous data fromwhich a certa in concept may be elabo rated there canb e no conception based upon them , and as the va ri etyand qual i ty o f concepts are in exact proportion to thevari e ty and quali ty o f sense-experi ence there can b eno demand fo r a parti cul a r sp eci es o f notions suchas might b e el aborated out o f the absent o r nonex i stent perception . Hence , the power of conce ivingsp rings forth from sense-experi ence . Sense-experi enceis essenti ally a mass of pe rceptions : these , creating ademand for additi onal adaptations, consp ire , as i f, toevoke the power o r facul ty to meet the dem and , andconsequently, an added conceptual i zation i s made .

Progress in human thought is made in a manne r

THE PROLOGUE 27

s imilar fo that which prevai l s in the development ofothe r na tural processe s , such as, the power o f speechin the child. In the developm ent o f th is faculty the rea re certa in definit e stages which appe a r in due sequence .The child is not gi fted with the powe r o f speech a tonce . I t comes , by gradual and sometimes pa infulgrowth , into a ful l use o f thi s fa cul ty. Now, muchthe same principl e holds t rue in the evolution o f themind in the human speci es. I t i s an establ ishedbiologi c principl e that the ontogeneti c processe s manifested in the individual a re but a recap i tul a tion o f thephylogeneti c processes which are observabl e in theprogress Of the enti re speci e s . The view becomeseven more cogent when note i s taken o f the fact thatthe foetus , during emb ryogenesis, pa sses success ivelythrough stages o f growth which have been shown tobe analogous, i f not i denti cal , with thos e stages throughwhich the hum an speci es has developed, namely, themineral , vegetal and animal .Where fore i t may be sa i d that the fourth dim en

si oua l concept marks a distinct stage in psychogenesi so r evolution o f mind . It requi red , a s wil l be Shownin Chapte r I I , nearly two thousand years for i t togerminate , take root and com e to ful l fru it ion . Fori t was not unti l the ea rly yea rs o f the nineteenth centu ry that mathemati ci ans , taking inspi ration fromR IEMANN ( 1 826-1 866 ) fully recognized the concepta s a metaphysi ca l possib il i ty , or even the ide a wasconc eived at al l . Serious doubt i s ente rta ined a s tothe possib il ity Of i ts conception by any human mindbefore thi s date , that is, the time when i t was actuallyborn . Pri or to that t ime , mathematical thought wastaking upon i tsel f that shape and tendence whi ch would


eventually lead to the di scovery of hype rspace ; but itcould not have reached the zenith o f its upward strivings a t one bound . That would have been unnatural .

Such i s the consti tution o f the mind tha t althoughit i s the quantity which bridge s the cha sm between thetwo stages o f man

’

s evolution when he merely thinksand when he real ly knows i t i s enti rely unde r thedoma in of l aw and must Observe the times and sea sons ,as i t were , in the performance o f i ts functions . Thescope o f psychogenes i s is very broad , perhaps unlimit ed ; but i ts various stage s a re ve ry cl ea rly definednotwithstanding the breadth o f its scop e of moti l ity.And while the distance from moneron to man , orfrom feel ing to thinking i s vast , the gul f which sep arates m an , the Thinker, from man , the knower, isvaste r s ti ll . Who , there fo re , can say what a re thede l ights yet in sto re for the mind as i t approaches ,by slow paces , th e goal whereat i t wil l not need tost rI

'

Iggle through the devious paths o f perce iving, conceiv ing, ana lyz ing, comparing , general i zing , in ferringand j udging ; but will be abl e to know defini tely, absolutely and instantaneously? That some such consum

m ation a s thi s shal l crown the l abors o f mental evoluti on seems only natural and logi cal .It may be thought by some that the cha racte r and

content Of revel ati onal impress ions constitute a vari ation from the requ i rements o f the law above re ferredto , but a l i ttl e thought wil l expose the fall acy o f thisview. The nature o f a reveal ed message i s such a sto make i t thoroughly am enable to the restricti onsimposed by the evolutiona ry a spects o f mind in general .That thi s i s true becomes appa rent upon an examination o f the fou r ca rdinal characte ri sti cs o f such

THE PROLOGUE 29

impressions . Fi rst, we have to conside r the indefinitecha racte r o f an apocalypti c ideograph which i s dueto i ts symboli c nature. This i s a feature which rel i evesthe impressi on Of any pragmati c value whatsoeve r,especi ally for the period embracing its p romulgation .

Then , such crypti c messages may or may not be un

derstood by the recipient in which l atte r ca se i t i snonpropagable . Second , the necessi ty o f previousexperience in the mind of the recip i ent in o rde r thathe may be able to inte rpret to hi s own mind the psychicimpingement. The basi s which such experi ence affordsmust necessari ly be present in order that the re m ay

be an adequate medium of mental qual i ti e s and powersin which the ideogram may be preserved. A thi rdcharacteristic is that revelations quite invar i ably presuppose a contempl ative attitude o f mind which , inthe very nature of the ca se , superinduces a state o fpreparedness in the mind for the proper enterta inmento f the concept involved . This fact p roves qu i te conelusively that revel ati onal impression s a re not exceptions to the gene ra l rule . Lastly, a dissati s fa ct ionwith the conditions with which the symbol i sm deal so r to which i t perta ins i s a lso a prerequ i s i te . Thi scondition i s re ally that whi ch call s forth the crypti cannunci ation , and yet , preceding i t i s a long seri e s o fcause s which have produced both the conditions andthe revol t which the revelato r feels a t the i r p resence .In view o f the foregoing , i t would appear that oh

jections b ased upon the alleged nonconformity o f thereveal ed or inspired cannot be ente rtained a s i t mustbe mani fest that i t, too , fall s within the scope o f thel aws of m enta l growth .

D i scoveri es , whethe r o f philosophi cal o r m echani cal


nature , or whether Of ethica l or purely mathemati caltendence , a re never the results o f a del iberate ,methodical o r purposive refl ection . For instance , le tus take L IE’S “transformation groups ,

” mathemati ccontrivance s used in the solution o f certa in theorems.

Now, i t ought to be obvious that these matheticmachinations were not discovered by SOPHUS LIE a sa consequence of any methodic o r purpose ful imtention on hi s p art. That is, he did not set out del ibe rately to di scove r trans formation groups .” For backo f the “groups” l ay the enti re range o f analyt ic lnv estigations ; the mathemati cal thought Of more thana thousand years furni shed the substructure upon whichLIE buil t the conception Of hi s groups .” Simil a rly,i t may be sa id with equal a ssurance that no matte rhow great the intensi ty o f thought, nor how purposeful , nor o f how long duration the se ri e s o f concentrated abstra ctions which l ed up to the inventi on o fthe printing press , the l inotype o r multiplex p rintingp ress o f our day could not have b een produced ab

rup tly, nor by use o f the mental dynam i cs o f thehuman mind o f remoter days. Its p roducti on hadto follow the path outl a id by the l aws Of psychogenes i sand awa i t the development o f those powers whichal one could give it b i rth . The whole question re

solve s i tsel f, there fore , into the idea Of the completesubservi ency o f the mind, i n all matte rs o f speci almom ent , to the l aws a forementioned. The sup ersess i on o f the l aw o f its own l i fe by the mind i s well-n ighunthinkable , i f not quite so .

I f we now vi ew the hi story of the mind a s manifested in the human speci e s , three grea t epochs whichdivi de the scope o f mental evoluti on into more or

THE PROLOGUE 3 1

l ess well-defined Stages present themselves. Thesea re : first , the forma tiv e stage ; second , the de termi

na tiv e stage ; third, the stage Of fre edom,or the

elabora tiv e stage .In all Of the e arly race s o f men , through every

step which even preceded the genus homo, the gene ri c

mind was being formul ated. It was be ing given Shape ,outl ine and di rection . Al l Of the fi rst stage , theformativ e

,was devoted to organiz ation and di rection .

Those el ementa ry sensations which constituted thebasi s o f mind in the primitive man were acco rdinglystrongly determinative of what the mind Should be inthese l a tter days . To thi s genera l resul t were con

tributed the effects o f the activi ty o f cell s , nerves,bones , fibers , muscle s and the blood .

The formativ e period naturally cove red a veryextens ive a re a in the histo ry o f mind o r p sychogeneti cdevelopment. I t was followed closely, but almost insensibly, by the de terminativ e period during which allthe l atent powers, capaciti e s and facult i e s which werethe di rect products Of the formativ e period were beinguti l iz ed in meeting the demands o f the l aw o f nec essity. The making o f provisions aga inst domesti cwant, aga inst the attacks o f external foe s ; the com

bating o f di seases , physi cal inefficiency, the weather,wild beasts , the a speriti es of tribal enmiti e s ; as wella s furthering the production o f a rt, music, sculpture ,the various b ranches o f handiwork , l iterature , philosophies, rel igions and the effectuation o f all thosethings which now appear a s the result o f the mentalactivity of the present-day man make up the essenceand purpose o f the dete rminative period .

Signs o f the dawn of the elaborativ e stage , also


called the stage o f freedom,have been mani fest now

for upwards o f three centuri es and i t i s , there fore ,in i ts beginnings . It i s not fully upon us. Not yetcan we fully real ize what i t may mean , nor can weunerringly forecast i ts ul timate outcome ; but we feelthat it

'

is even now here in all the glori e s o f i tsmatutinal freshness . And the mind i s beginning to befree from the grinding necess i ti e s o f the constructiveperiod having al re ady freed itsel f from the restri ctivehandicaps o f the primeval formul ation peri od. Al

ready the upgrowing rej uvenescences so common a tthe beginning of a new period a re commencing toShow themse lves in eve ry departm ent o f human ac

tiv ity in the almost un iversal desi re for greate r freedom . And this i s p a rticul a rly noti ceabl e in the manypol iti ca l upheaval s which , from time to time , a recoming to the surface a s well a s in the countl ess othe ra spects of the wide-spread rena iss ance . Pe rhap s thetime may come , neve r qui te ful ly, when there will beno longe r any necessi ty to provide aga inst the externa lexigenci es o f l i fe ; perhaps, the tim e will never bewhen the mind shall no more be bound by the l awo f sel f-preservation , not even when it has atta inedunto the imm ortal i ty Of absolute knowledge ; yet, i t isintu itively felt that i t must come to pa ss that the mindshal l be vastly free r than i t i s to-day. And with thisnew freedom must come l ib eration from the necessi t i esof the elem entary probl ems of mere physi cal existence .The infe rence i s , there fore , drawn that the fourth

dimensional concept, and all tha t i t connotes of hyperspace or spaces o f n-dimensi onal i ty a re some of theevidences tha t this stage of freedom is dawn ing. Andthe m ind , j oyous at the prospect o f unbounded l iberty


and multipl i city” i s qu ite real to him and subsi stsunder a re ign o f l aw the penalti e s o f which , while nota s auste re and unreasonabl e a s some which we find inour tridimens ional world , a re neverthele ss quite a spalpable and a s much to be feared . For the orthodoxyo f mathemati cs i s a s cold and intolerant a s ever therel igious fanati c can b e . But the real i ty and even theactual i ty which may be imputed to the domain o fmathesi s i s o f an enti rely different quality from thatwhich we expe ri ence in our world o f triune dim en

sionality and i t i s a regrettabl e erro r o f judgment toidenti fy them . I t ought , there fore , neve r b e expected,no r i s i t logically re a sonable to assume that the en

ti ti e s which inhabi t the matheti c realm o f the analystshould be submissive to the l aws o f sensibl e space ;nor that the conditions which may be found there incan eve r be made con fo rmable to the conditions whichexi st in perceptual sp ace .It was PLATO’S bel i e f that i dea s alone possessed

real ity and what we regard a s actual and real i s ona ccount o f its ephemeral i ty and evanescence not realbut i llusi ona ry. This Vi ew has been sha red by a numberof eminent thinkers who followed , with some ostentation , the l ea d establ ished by PLATO. For a considerable period o f t ime thi s school o f thinke rs hadmany adherents ; but the principl e s at length fell intodi s repute owing to the absurditi e s indulged in by someo f the l ess ca re ful followers . The real ism ,

or forthat matte r, the actuali o f idea s cannot be denied ;yet i t i s a real i sm which i s nei ther to be compa redwith the physi cal re al i ty o f sense-impressions nor i tsphenomena . The character and pecul i a rity of ide asare in a clas s apart from simil a r notions o f perceptual

THE PROLOGUE 35

space: content. I t i s a s i f we were considering thepotenti al iti e s o f the spi ri t world and the entiti e sthere in in connection with incarnate entit i es which inthe very nature o f the case i s not allowable. Furthermore , it is unreasonable to suppose that

‘

the condit ionson a highe r pl ane than the physical can be made responsible to a simil a r set o f conditi ons on the physicalpl ane .There a re certa in a stronomers who base the i r

specul ations a s to the habitabil i ty o f othe r pl anetsupon the absurd hypothesi s that the conditions o f l i feupon all pl anets must b e the same a s those on thee arth , forgetting that the extent o f the univers e and

the scope o f m oti l i ty o f l i fe itsel f are o f such a naturea s to adm i t o f endl ess vari ations and adaptations.There i s a real i sm of ideas and a re al ism of percep tual space . Yet thi s i s no re ason why the twoshould be identified. On the other hand

,owing to

the diversi ty in the universe , every cons ideration wouldna turally l ead to the a ssumption that they are dissimil a r; To invest ideas , notions , impl i cations andinferences with a real ity need not logically or otherwise affect the re al ity of a stone , a fig

,or even of a

sense-imp ressi on .

To a being on the spi ri t l evels our grossest real iti es must appea r a s non-existent. They a re neithe rpalpable no r contactabl e in any manne r within theo rdinary range o f physi cal possib il it ies . For us hi sgravest experiences can have no real i ty whatsoeve r ; fo rno matte r how real an experi ence may be to him i t i saltogethe r beyond ou r powers o f perception

,and

there fore , to us non-exi stent al so. I t should,however,

be stated that the sta te of our knowl edge about a

36 THE MYSTERY,OF SPACE

given condition can in no way affect i ts exi stence . I tmerely establ i shes the fact that two o r m ore real it i e smay exist independent o f one another and furthe r thatthe gamut o f re al ism in the un ive rse is infini te and

app roaches a final sta te when its occlus ion into abso

lute be ing follows as a logica l sequence .Recurring to the cons idera tion o f the real i ty o f

sp i ri t-re alm s a s compa red with that o f sensible space ,i t comes to View that our ideal i sm , that is, the i deal i smwhich i s a qual i ty o f conceptual i za tion , may be re

ga rded a s i dentical with the i r real i sm , a t l ea st a sbeing on the same plane a s i t. Stated differently, thethings tha t are idea l to us and which constitute thedata o f our consciousness may be a s real to them asthe commonest obj ect o f sense-knowledge is to us.

What, there fo re , appears to u s a s the most etherealand ideal i sti c may have qu ite a real i sti c cha racte r fo rthem .

Ult imately, howeve r, and in the final deeps o fanalys i s i t will b e found undoubtedly tha t both ou rre al ism and our ide al i sm a s well a s s imil a r qual it i e so f the sp i ri t world a re in all essent i al cons iderationsquite illusiona ry. Al l knowledge ga ined in a conditionShort o f divin i ty i tsel f is s adly rel ative . Even mathem atical knowledge fall s fa r Short o f the absolute

,the

fondest cl a ims o f the o rthodox mathematici an to thecontra ry notwithstanding. I t ha s been s a id frequentlythat a mathematical fact i s an absolute fact and thatits verity, necess i ty and ce rt a inty cannot be questi onedanywhere in the unive rs e whether on Jupi te r

,Nep

tune , Fomalhaut, Canopus o r Spica . But having so

decl a red , the fact o f the Sheer relativity o f ourknowledge i s not disturbed thereby nor controve rted.

THE PROLOGUE 37

Happily,ne ither distance nor a l ack of distance can

in any way affect the qual ity o f human knowledge ,mathematical know l edge not excepted. That can onlybe affected by conditions which cause i t to approachperfection and nothing but evolut ion can do that.In the l ight of resul ts obta ined in analyti c invest i

gations the question o f the fl exib il i ty o f mathematica lappl i cations becomes evident and one inste ad of be ingconvinced o f the vaunted invariabi l ity o f the l awsobta ining in the world Of mathesi s is, on the othe rhand , made aware o f the rem a rkable and seem inglyunrestra ined faci l i ty with which these l aws may bemade to apply to any condit ions or set of assumptionswithin the range o f the m ind’s powers of conception .

Mathematici ans have deified the definition and en

dowed i t with omnific powers imputing unto i t all theattributes o f divinity—immutab il i ty, inva ri ance , andsempiternity. In this they have erred grievously al

though , pe rhaps , necessarily. Matheti c conclusi onsa re enti rely conditional and depend for the i r certaintyupon the imputed certitude of other proposi tions whichin turn are dependent, in ever increasing and endlessly complex rel ations, upon previously assumedpostul ates. These facts make it exceedingly difficultto understand the a ttitude o f mind which has obscuredthe utter mutabil i ty and consequent ult imate unrelia

bility o f the fine-spun theori es of analyti c machinations.

The apriori ty o f al l mathemati ca l knowledge i sopen to se rious questioning . And although there i sno hesi tancy in admitting the basi c agreement o f themost p rim a ry facts o f mathematical knowledge withthe essenti a l characte r o f the intell ect the existenceo f well-defined l imits fo r such congruence cannot be

38 THE MYSTERY FO SPACE

gainsaid. The subjunctive quality of geometri c andanalyti cal proposi ti ons is made apparent by an examination o f the possib i l i ti e s fall ing within the scopeo f pe rmiss ib il i ty offered by matheti c l i cense . For instance , privi leged to p roceed acco rding to the analyti cmethod i t i s allowabl e to reconst ruct the sequence o fvalues in our o rdina ry system of enumeration so a sto admi t o f the specification o f a new value fo r say,the enti re s eri e s o f odd numbers . Thi s value mightbe a ssumed to b e a plus-or—minus one , dependent uponits postu re in the s eri e s . That i s , al l odd numbersin the seri es b eginning with the digi t 3 , and continuing,5 , 7 , 9 , I I , 1 3 , 1 5 , 1 7 , 1 9 , n

,coul d be assumed

to have only a pl ace value whi ch might b e rega rdeda s a constant-vari able . The seri es o f even numbers ,2 , 4 , 6, 8 , 10 , 1 2 , I4 , 1 6, u

,may b e assumed

to reta in the i r p resent sequence values . Under thissystem the digi t 1 would have an absolute value ; al lothe r odd numbers would have a constant-va ri abl evalue ; constant, because always no more nor less than1 dependent upon thei r pl ace in the operati ons and

whethe r the i r values were to be appl i ed by additiono r sub traction to o r from one o f the values in theeven numbe r se ri es ; vari able , because thei r valueswould be determinabl e by the i r appl i cation and algeb ra ie use.There would, o f course , be util i ta ri an obj ection to

a system of thi s kind ; but under the conditions o f asuppositi ona ry hypothes is, i t would be sel f-consi stentthroughout, and i f given universa l a ssent would sui tour purposes equally a s well a s our present system .

But the fact that thi s can be done under the mathemati c method veri ly p rove s the viol abil i ty o f mathe

THE PROLOGUE 39

matical l aws and completely negatives the assumptionthat the sum of any two digits , a s say 2 plus 2 equal s

4 , i s necessarily and unavo idably immutable . Forit can be seen that the sum-value o f al l numbers maybe made dependent upon the a ssumed value which maybe assigned to them or to any colle ction thereo f.Furthe rmore , i t i s a matter of h i stori ca l knowledgetha t i t wa s the custom o f anci ent races o f men toa ccount fo r values by an enti rely different method fromwhat we use to-day. The l atte r i s a re sul t o f ev olution and whil e experi ence teaches that i t i s by far themost conveni ent, i t i s nevertheles s true that e a rl i e rm en managed at l ea st fa i rly well on a different b a s i s .Then , too , the fa ct o f the uti l i ty and universal applicability o f ou r pre sent system , based upon universala ssent, does not obvi ate the conclusi on that any othersystem , cons istent in i tsel f, might be made to serve ourpurposes as well .It ought to b e sa id , howeve r , in justi ce to th e rather

uti l i ta rian results obta ined by LA GRANGE,HELM

HOLTZ , FECHNER, and others who strove to make useo f thei r discove rie s in analysis in solving mechanical ,physiologica l and othe r p roblems o f more or l esspragmati c import tha t, in so fa r a s thi s i s tru e , mathematical knowledge must be recognized a s be ing consi stent with the necessi ti es of a priori requi rements .But even these resul ts may not be rega rded a s transcending the scope o f the most fundamental p rinciple so f sense-experience . I t wil l be discovered finally

,

perhaps, that the energy spent in elaborating compl icate seri es o f analyt i c cu rios iti es has been m isapp ro

p riated. I t will then be necessa ry to turn the attentiondefini tely to the study of that which l i es not at the


term inus o f the intell ect’s modus v iv endi,but which

i s both the origin o f the intell ect and its ete rnal sustaine r—the intuition , o r l i fe i tsel f. Thi s can resultin nothing les s than the complete sp i ritual ization o fman’s menta l outlook and the consequent inevitabl erecogni tion o f the underlying and ever-susta iningone-ness o f a ll vi ta l mani festations.One o f the curios itie s o f the tendency in man’s

mind to speci al iz e in analytics , whether in the fi eld o fpure mathemati cs o r metaphysi cs , i s the fact tha t i talmost inva ri ably l eads to an a ttempt to account fo rcosmic o rigins on the bas i s o f pa ralogic theori es . Thi sin times past ha s given ri se to the theo ry o f the purelymechanical o rigin o f the universe a s well a s manyother fantasti c fall a ci e s the chi e f e rro r o f which l ayin the fa ilure to distingu i sh b etween the real ism ofmental concepts and that o f the sens ibl e world. Inspi te o f this, howeve r, one is bound to appreci a te thebenefici a l effects o f analyt i c ope rations becaus e theyserve a s invigo rants to mental growth . It could not,there fore , he wished that there were no such thing a sanalytics ; fo r the equil ibri a-restoring prope rty o f themind may a t all times be rel i ed upon to minimize thedange r of excesses in e i ther di rection . Just a s thetide flowing in flows out aga in , thereby restoring theo cean’s equil ib rium , so the mind ascending in one

generation b eyond the sa fe ty mark ha s i ts equil ibriumrestored in the next by a rel inqui shment o f the fol l i e sof the former.The fou r-space i s one o f the curios itie s o f analytics ;

yet i t need not be a menac e to the sane contemplation o f the va ri egated p roducts o f analysis. Sa fetyhere abides in the re stra int which should chara cteri z e


cherri e s ; on a cob many gra ins o f corn . We havelea rned to a ssign to each o f thes e quantiti e s in thei rsummation a sequence value . But thi s is an emp i ri calnotion and cannot be sa i d to inhere in the mind itsel f.Let us take , for instance , the musta rd seed. I f i twere true that in one o f these seeds there exi sted all

the subsequent s eeds which appea r in the musta rdplant a s sep ara te and identifi abl e quantit i es, and notin essence , then there would perhaps be warrant fo rthe notion that dive rsi ty, as the calcul abl e elem ent, isan a priori conception . But , a s this is not the case ands ince divers i ty is purely empi rical and perta ins only tothe eflforescence o f the one l i fe i t is man i festly absurdto take that Vi ew .

Unde r the most charitabl e allowances , there fore ,there can b e but two quanti ti es—unity and diversi ty ;yet not two , fo r these are one . Unity is the one

quantity and divers ity is the divisi on o f uni ty into at ransfinity o f pa rts . Uni ty i s infini te , absolute andall-inclusive . D ive rsi ty i s fini te a lthough i t may beadmitted to b e t ransfinite , o r greate r than any assignable value . Uni ty alone is incomprehensibl e . In orde rto understand something of its nature we divide i t intoa dive rsity o f parts ; and because we fa il to understandthe t ransfinity o f the multitude o f parts we mistakenlyca ll them infini te .When analysi s shall have p roceeded fa r enough

into the abysmal mysteri e s o f dive rs ity ; when themathemati cal mind Shall have been overcom e by theoverwhelm ing perpl exi ty o f the maz e o f diverse parts ,i t shal l then fall asl e ep and upon awaking shall findthat wonderfully simple thing—unity. I t i s the onequantity that is endowed with a m agnitude which is

THE PROLOGUE 43

both inconceivable and i rresolvabl e . The one ineluctable fact in the universe i s the incomprehensib il i tyand all-inclus ivity o f one-ness. It i s incomprehensibl e ,inconce ivabl e and infini te at the present stage o f minddevelopment . But the goal o f mind i s to understandthe essenti al cha racte r o f unity, of l i fe . Its evolutionwil l then stop , fo r i t will have reached the p ri ze o fdivini ty itsel f whereupon the intelle ct exal ted by anduni ted with the intu ition shall a lso become one wi ththe divine consciousness .

CHAPTER II

HISTORICAL SKETCH OF THE HYPERSPACEMOVEMENT

Egyp t the Birthplace of Geometry—Precursors : NASIR-EDDIN,CHRISTOPH CLAVIUS, SACCHERI, LAMBERT, LA GRANGE,KANT—Influence of the M ecanique Analytique—TheParallel-Postulate the Root and Substance of the Non

Euclidean Geom etry—The Three Great Pe riods : T heFormative , Determ inative and Elaborative—RIEMANN and

the Prop erties of Analytic Spaces.

THE evolution o f the ide a o f a fourth dimensi onof space covers a long period o f yea rs. The ea rl i estknown record o f the beginnings of the study of sp acei s found in a hi e rati c papyrus which fo rms a part o fthe Rhind Coll ection in the British Museum and whichhas been deciphered by E ISENLOHR. It i s bel i evedto b e a copy o f an older m anuscript o f date 3400 B . C .

,

and i s entitled “D irections for Knowing All D ark

Things.

” The copy is sa i d to have b een made byAHMES

,an Egypti an p ri est b eween 1 700 and 1 100

B . C . It begins by giving the dimensi ons o f barns ;then follows the cons ideration o f va rious rectil inealfigures , ci rcl es , pyramids, and the value o f p i (n) ,Although many of the solutions given in the manuscript have been found to be incorrect in minor particulars, the fact rem a ins that Egypt is really thebirth-p l ace of geometry. And thi s fact i s buttressed

44

SKETCH or HYPERSPACE MOVEMENT 45

by the knowledge that THALES,long be fo re he

founded the Ioni an School which was the beginningo f G reek influence in the s tudy o f mathematics , i sfound s tudying geometry and astronomy in Egypt .The concept o f hyperspace began to germinate in

the l atte r part o f the fi rst century, B . C. For i t wasat thi s da te that GEMINOS o f RHODES ( B . C. 70 ) began to think seriously o f the mathematical l abyrinthinto which EUCLID’S parall el-postulate most certa inlywould lead i f an attempt at demonstrating its certitudewere made . He recognized the difficulti es whichwould engage the attention o f those who might ventureto delve into the mysterious possib il i ti e s o f the problem . There is no doubt , too , but that EUCLID himsel f was aware , ln some measure a t l east, of thesedifliculties ; fo r hi s own atti tude towards thi s postul ateseems to have been one o f noncomm ittance . It i s ,there fore , not strange that the astronomer, PTOLEMY(A. D . 87 should be found seeking to prove thepostul ate by a consideration o f the poss ib i l i ti e s o finterstell a r tri angl es. H is rese arches , however , broughthim no rel i e f from the general di ssati s faction whichhe fel t wi th respect to the val idity o f the p robl emitsel f.Fo r nearly one thousand years a fter the a ttempts

at solving the postul ate by GEMINOS and PTOLEMY,the

field o f mathemati cs l ay undi sturbed. For i t was a tthi s t ime that there a rose a strange phenomenon

,

more commonly known a s the “Dark Ages,

” whichput an effectual check to further resea rch or independent investigations . Mathemati ci ans throughoutthi s long lapse oi time were content to accept EUCLIDa s the one incontrove rtibl e , unimpeachable authority,


and even such investigations a s were made di d nothave a rebell ious tendence , but were ma inly endeavorsto substanti ate hi s cl a ims .Accordingly, i t was not unti l about the fi rst hal f

o f the thi rteenth century that any real advance wasmade . At th i s t im e there appea red an Arab , NASIREDDIN ( 1 20 1 - 1 274 ) who attempted to make an improvement on the p roblem of pa ra ll el i sm . H is workon EUCLID was printed in Rome in 1 594 A. D .

,about

three hundred and twenty yea rs a fte r h is demi se andwas communica ted in 1 65 1 by JOHN WALLIS ( 1 6 1 61 703 ) to the mathematici ans o f Oxford Universi ty.

Al though hi s calculations and conclus ions were re

sp ectfully rece ived by the Oxford author iti es nodefini te results were regarded a s a ccompl i shed bywhat he had done . It is bel i eved , however , tha t hi sWork reopened speculation upon the problem andse rved a s a basi s , however sl ight, for the greater workthat wa s to be done by those who followed him duringthe next succeeding eight hundred years .About twen ty yea rs be fo re the print ing o f the

work o f NAS IR-EDDIN,CHRISTOPH CLAVIUS ( 1 574 )

deduced the axiom of parallel s from the a ssumptionthat a l ine whose points a re al l equidi stant from astraight l ine is i tsel f straight . In hi s cons ideration o fthe pa rallel-pos tul ate he i s s a i d to have rega rded i ta s EUCLID’S XIIIth axiom . La ter BOLYA I spoke o fi t a s the KIth and l ate r still , TODHUNTER tre ated i ta s the XIIth . Hence , there does not s eem to havebeen any general unanimity o f opinion a s to the exactsta tus o f the pa rallel-postulate , and especially i s th i st ru e in Vi ew o f the unce rta inty now known to haveexisted i n EUCLID’S mind concerning i t.

SKETCH OF HYPERSPACE MOVEMENT 47

G IROLAMO SACCHERI ( 1 667 a lea rnedJesuit, born at San Remo , came next upon the stage .And so important was his work that i t wi ll perpe tuatethe memory o f his name in the histo ry o f mathematics .He was a teacher o f grammar in the Jesui t Collegiodi B rera where TOMMASO CEVA

,a brothe r o f G10

VANNI,the well-known mathemati cian , was teache r o f

mathematics. His a ssoci ation with the CEVA brothe rswas especi a lly benefici al to h im . He made use o fCEVA’

S very ingeni ous methods in his fi rst publ ishedbook , 1 693 , entitl ed Solutions of Six Geom e trical

Problems Proposed by Count Roger Ve ntimiglia.

FIG . 1 .

SACCHERI attacked the problem o f paral l el s inquite a new way. Examining a quadri l ateral , ABC

'

D,

i n whi ch the angle s A and B a re right angles and thesides AC and BD a re equal , be determ ined to showthat the angles C and D a re equal . He also soughtto prove that they a re e ither right angles , obtuse o racute . He undertook to prove the falsi ty o f the l a tte rtwo proposi t ions ( that they a re e i ther obtuse o r acute ) ,leaving as the only poss ibi l ity that they must be rightangles. In doing so , he found that hi s a ssumptionsl ed him into contraditions which he expe ri enced difficul ty in expla ining .

His l abo rs in connect ion with the solution o f theproblems p roposed by COUNT VENTIMIGLIA

,includ


ing hi s work on the question of parall els , l ed di rectlyinto the fi eld o f metageometri ca l resea rches, and perhaps to him a s to no othe r who had preceded him ,

or at l ea st to him in a l arge r degree , belongs the creditfo r a continued renewal o f interest in that seri e s o finvestigations which resulted in the formul ation o f thenon-Eucl idean geometry.

The l a st publ i shed work o f SACCHERI was a reci talo f hi s endeavors a t demonstrating the parallel-postul a te. This rece ived the “ Imprim a tur o f the In

quisition, July 13 , 1 7 33 ; the Provinci al Company ofJesus took posse ss ion o f the book for perusa l onAugust but unfortunately within two monthsa fte r i t had been revi ewed by these authorit i e s ,SACCHERI pa ssed away.

All efforts whi ch had been made prio r to the workof SACCHERI were based upon the a ssumption thatthere must b e an equivalent postul ate which , i f i t couldbe demonstra ted , would l ead to a dire ct , posi tive p roo fo f EUCLID’s propos ition . Al though these and allother attempts at reaching such a proof have signallyfa iled and although i t may correctly be sa id that theenti re hi sto ry o f demonstrations a iming at the solution o f the famous postulate ha s been one long seri esOf utte r fa ilu res , i t can b e a sserted with equal cert itudethat i t has proven to be one o f the most fruitful p roblems in the histo ry o f mathemati cal thought. For outo f these fa ilures has been built a superst ructure o fanalyti ca l inve stigations which surpass es the mosts anguine expectations of those who had l abored andfa i led .

In 1 7 66 JOHN LAMBERT ( 1 7 28-1 7 7 7 ) wrote apape r upon the Theory of Parallels dated Sept.


developments o f the mathem ati ca l idea o f space thathe very fully appreci ated the m arvelous scope o fanalyti c spaces . His conception o f sp ace , therefore ,must have had a profound influence upon the mathemati c thought o f the day causing i t to undergo a rapidreconstruction a t the hands o f geometers who cameafte r him .

Under the maste rly influence of LA GRANGE

( 1 7 3 6- 1 8 1 3 ) the idea of diffe rent spa ces began to takedefinit e shape and direction ; the geom etry o f hyperspace began to crystall i ze ; and the fi eld Of mathesisprepa red for the growt h of a conception the comprehension o f whi ch was destined to b e the profoundest undertaking eve r a ttempted by the hum anmind . Unl ike m ost great m en whom the world learnsta rdily to admire , LA GRANGE l ived to see his tal entsand genius fully recogni zed by his compeers ; fo r hewas the rec ip i ent o f many honors both from his

countrym en and hi s admire rs in fore ign l ands . Hespent twenty yea rs i n Prussi a where he went upon theinvitation of FREDERICK the G reat who in the Royalsummons re ferred to him sel f as the “greate st kingin Europ e” and to LA GRANGE as the “greatestmathemati ci an” in Europe . In Prussi a the M e canique

Analytique and a long seri es of mem o i rs which werepubl i shed in the Berl in and Turin Transactions wereproduced. LA GRANGE did not exhibi t any markedtaste fo r mathemati cs unti l he was 1 7 years o f age .Soon therea fte r he came into posse ssion o f a memoirby HALLEY qui te by accident and this so a rou sed hi sl atent genius tha t within one yea r a fte r he had re

viewed HALLEY’S memoi r he became an accompl ishedmathematic i an .

SKETCH OF HYPERSPACE MOVEMENT 5 :

He created the calculus of va ri ations , solved mosto f the problems proposed by Fermat, adding a numbe r o f theorems o f his own contrivance ; ra ised thetheo ry o f diffe renti a l equations to the posi tion o f asci ence rathe r than a series o f ingeni ous methods fo rthe solution o f speci a l p robl ems and furnished a solution fo r the famous isope rim etri ca l problem whichhad baffled the skil l o f the foremost m athem atici ansfo r nea rly hal f a century. All these stupendous taskshe performed by the tim e he reached the age o fnineteen .

The M e canique Analytique is his greatest andmost comprehensive work. In thi s he established thel aw o f vi rtual work from which , by the a id o f hi scal culus o f va ri ati ons , be deduced the whole o fmechani cs, including both sol ids and l iqu i ds. It washis obj ect in the Analytique to Show that the whole subject of mechanics i s impl i ci tly embraced in a singleprinciple , and to l ay down certa in formul a e from whichany particul a r resul t can be obta ined . He frequentlym ade the assertion that he had , in the M ecanz

'

que

Analytique , transform ed mechani cs which he persistently defined a s a

“geometry o f four dimensi ons’"

into a branch o f analyti cs and had shown the so-calledm echani cal principl es to be the simple results o f thec alculus. Hence , there can be no doubt but that LA

‘In 1754 D

’ALEMBERT ( 17 17-1 783 ) published an article in

the famous old Encyclopedia edited by DIDEROT and him se lf on

Dim ension. In this article the idea of the fourth dim ension is

dwe lt upon at length. The view which he e xp ressed in this article ,of course

, served greatly to p opularize the concep tion among the

learned m en o f the day , and owing to the Close re lationship existing between D ’ALEMBERT and LA GRANGE, it is not surp rising thatthe latter should have been very much enamored of the idea.


GRANGE not only completed the foundation , but prov ided most o f the materi al in hi s analyses and other“abstract re sults o f gre at general i ty” which he ob

tained in h i s numerous cal culations , fo r the superstructu re subsequently known as the geometry o fhypersp ace

,and in which the fourth dimensional con

cept occup i es a very fundamenta l pl ace .It i s a s i f for nea rly seventeen hundred years work

men , such a s GEMINOS, o f RHODES, PTOLEMY, SACCHERI

,NASIR-EDDIN

,LAMBERT

,CLAVIUS

,and hun

dred o f others who struggled with the p robl em ofparall el s , had made more or le ss sporadi c attempts a tthe excavation o f the l and whereon a marvelouslyintrica te building was to b e constructed. There i s nohi stori cal evidence to show that any o f them eve rdreamed that the resul ts o f thei r l abors would beutil i zed in the manner in which they have been used.

Then came KANT with the wonderfully penetratingsea rchlight o f hi s masterful intell ect who from theelevation which he occup ied saw tha t the s ite had greatposs ib i l iti es , but he had not the mathemati cal tal entto undertake the work o f a ctual , methodical construetion . Indeed hi s task was o f a different sort. However, he succeeded in opening the way fo r LA GRANGEand others who followed him . LA GRANGE imm edi

ately se i zed upon the ide a which for more than a

thousand years ha d be en impinging upon the mindso f mathematici ans va inly seeking lodgment and beganthe el abora tion o f a plan in acco rdance with whichminds bette r ski lled in the pragmati c appl i cation o fabstract principl es than his could complete the workbegun . Unfortunately, on account o f hi s intense dev otion and loyalty to the study o f pure mathematics ,


and when he had reached the summit o f hi s greatnesswhere he stood “without a riva l as the foremost l ivingmathemati ci an ,

” his heal th became seriously affected,causing him to suffe r constant a ttacks o f pro foundmelancholi a from which he died on Apri l 10 , 1 8 1 3 .

We come now to one o f the most remarkabl eperiods in the histo ry o f mental development . During the six hundred years be tween the b i rth o f NASIREDDIN and the death o f LA GRANGE the enti re worldo f mathesi s wa s being reconstituted . Since there hadbeen gradually go ing on an inte rnal p roces s which ,when completed, foreve r would l iberate the mindfrom the na rrow confines o f consciousness l imited tothe three-space , i t is not surprising that we should find ,in the mathemati ca l thought o f the time , an absolutelyepoch-making departure . The innumerabl e attemptsa t the solution o f the pa rall el-postul ate , all fa ilures i nthe sense that they did not prove , have intensifi edgreatly the este em in which the neve r-dying elementsof EUCLID a re held tod ay. And despi te the fact thatthere may com e a t ime when his axioms and conclus ions may be found to b e incongruent with the facts

o f sensuous real ity ; and though all o f hi s fundamenta lconceptions o f space in general , his theorems , proposit ions and postul ates may have to give way be fore thesea rching gla re o f a deepe r knowledge because o fsome revealed fault, the perfection o f hi s work inthe realm of pure mathematics will remain forever amaste r p i ece demanding the undiminished admirationo f mankind.

The pa ralle l-postulate,a s stated by EUCLID i n his

Elements of Geome try, reads a s follows


I f a stra ight l ine meet two stra ight l ine s so a sto make the two inte ri o r angles on the sam e si deo f i t taken togethe r l ess than two right angles,these straight l ines being continually produced , shalla t length m eet upon tha t si de on which a re theangles which a re l ess than two right angles.

”

On thi s postul ate hang al l the l aw and theprophets” of the non-Eucl idean G eometry. In i ta re the vi rtu al elem ents o f three possibl e geometrie s .Furthermore , i t is both the warp and the woof o fthe loom o f p resent-day metageom etri cal resea rches.

I t is the golden egg l a id by the god SEB at the beginning of a n ew l i fe cycl e in psychogenesis. Itsp rogeny a re numerous—hyperspaces , sects, stra ights,digons, equidistantials, pola rs , p lanars, coplanars, in

vari ants, quate rni ons, compl ex va ri able s , groups andmany others . A wonderfully interesting breed, fullof meaning and pregnant wi th the power o f finalemancipations fo r the human intellect !When the conclusi ons which were systemati cal ly

formulated as a resul t o f the investigations along thel ines o f hypotheses which controverted the parallelpostul ate were examined it was found that they fellinto three ma in divisi ons, nam ely : the syntheti c o rhyperbol i c ; the analyti c o r R IEMANN IAN and theel lipti c or CAYLEY-KLEIN . These divisi ons or groupsa re based upon the three poss ibil i ti e s which inhere inthe conception taken o f the sum of the angles re ferredto in the above postul a te a s to whether i t is equal to ,greate r o r less than two right angles .The assumption that the angula r sum is congruent

to a s tra ight angle is call ed the Eucl idean or parabolic hypothes i s and i s to be di stinguished from the


syntheti c o r hyperbol ic hypothes i s e stabl i shed byGAUSS

,LOBACHEVSK1 and BOLYA1 and which assumes

that the angula r sum is l ess than a stra ight angle .The ell ipti c or CAYLEY-KLEIN hypothesi s assumes thatthe angula r sum is greate r than a stra ight angle.LOBACHEVSKI

, however, not s atisfi ed with the statement o f the parall el-postul ate a s given by EUCLID andwhich had caused the age-long controversy, substi tutedfor i t the following

All stra ight l ines which , i n a pl ane , radi atefrom a given point, can , with respect to any otherstra ight l ine , 1n the same plane , be divided into twocla sses—the inte rsecting and the non- intersecting.

The boundary l ine Of the one and the other cl assis call ed pa rall el to the given l ine .

This i s but anothe r way o f saying about the samething that EUCLID had decl a red before , and yet, curiously enough i t afforded just the l ibe rty that LOBACHEVSK I needed to enabl e him to elaborate hi s theory .

Fo r the purposes o f this sketch the field o f thedevelopment o f non-Eucl idean geometry is dividedinto three periods to be known as : ( 1 ) the formativ eperiod in which mathemati cal thought was beingformulated for the new departure ; ( 2 ) the de terminativ e period during which the mathema ti cal i dea s weregiven di rection

,purpose and a general tendence ; (3 )

the e labora tiv e period during which the results o f theformer periods were el abo rated into definite kinds o fgeometri es and attempts made at popula riz ing thehypotheses .


The Formative Period

CHARLES FREDERICH GAuss ( 1 7 7 7-1 855 ) bysome has been regarded a s the most influent i al mathematician that figured in the formulation of the nonEucl i dean geometry ; but close r examination into hisefforts at inve stigating the p roperti es o f a tri angleshows that whi le hi s resea rche s l ed to the establ ishment o f the theorem that a regula r polygon of seventeen s ides (or o f any number which i s prime , andalso one more than a power o f two ) can be inscribed ,unde r the Eucl idean restr ictions a s to means, in aci rcl e , and also that the common spheri cal angl e onthe surface o f a sphere i s closely connected with theconsti tution o f the are a inclosed thereby, he cannotj ustly be ' designated a s the le ader o f those who formul ated the syntheti c school . And thi s , too , fo r thesimpl e rea son ' that, a s he himsel f admits i n one ofhis l e tte rs to TAURINUS

,he had not “publ ished any

thing on the subj ect .” In thi s s ame lett er he informsTAURINUS that he had pondered the subj ect for morethan thirty years and expressed the bel i e f that the recoul d not b e any one who had “concerned himsel f moreexhaustively with thi s s econd part ( that the sum ofthe angles o f a tri angle cannot be more than 1 80 degrees) than he had.

Writing from GOtt ingen to TAURINUS, November8 , 1 824 , and comment ing upon the geometri c value o fthe sum o f the angl es o f a tri angle , he s ays

Your presentation o f the demonstration thatthe sum of the angle s o f a pl ane tr1angl e cannot begreate r than 1 80 degrees does , indeed, l e ave some


KART,and that between SCHWEIKART and GERL1NG

,

there had grown up a general dissatisfaction in theminds o f mathematic i ans o f this period with Eucl ideangeometry and especi a lly the parall el-postul ate and its

connotations . BESSEL expresses this general discontent in one o f hi s l ette rs to GAUSS , dated Februa ry 10 ,1 829, in whi ch he says :

Through that whi ch LAMBERT sa i d and whatSCHWEIKART disclosed orally, i t has becom e cl ea rto me that our geometry i s incomplete , and shoul drece ive a corre ction , which i s hypotheti ca l , and if

the sum of the angle s o f the pl ane tri angle is equalto 1 80 degre es , vani shes .

”

The opinion o f l eading mathemati ci ans at thi s timeseems to have been crystall i z ing very rap idly. Un

consciously the m en of this fo rmative period wereadducing evidence which would give form and tendenceto the developments in the field o f mathesi s a t a l ate rdate . They appea r to have been reaching out fo rthat which , igni s fatuus-l ike , was a lways with in e asyreach , but not quite apprehensibl e.A bolde r student than GAUSS was FERDINAND

CARL SCHWEIKART ( 1 7 80- 1 857 ) who also ha s beencredi ted with the founding o f the non-Eucl ideangeometry. In fact , i f j udged by the same standardsa s GAUSS

,he would b e calle d the “fathe r o f the

geometry o f hyperspace” ; fo r he real ly publ ished thefi rst treati se on the subj ect . Thi s was in the natureo f an inclosure which he inse rted between the l eave sof a book he loaned to GERLING . He also asked thati t b e shown to GAUSS that he might give hi s j udgmentas to i ts meri ts .


SCHWEIKART’S treatise , dated Ma rburg, Decem

ber, 1 8 1 8 , i s he re quoted in full

The re i s a two-fold geometry—a geometry inthe na rrower sense , the Eucl idean , and an astralsci ence o f magnitude .

“The tri angle s o f the latte r have the pecul i a rity

that the sum of the three angle s i s not equal to tworight angl e s.

Th1s pre sumed , i t can be most rigorouslyproven : (a ) That the sum o f the three angles 1n

the tri angle i s l e ss than two right angles.

“

(b ) That this sum becomes ever smalle r, themore content the angle incloses. ( c ) That the al titude o f an i sosceles right-angled tri angle indeed everincrea se s , the m o re one lengthens the si de ; that it,howeve r

,cannot surpass a certa in l in e which I call

the constant .”

Squares have consequently the following form

FIG . 2.

I f thi s constant we re fo r us the radius of theearth ( so that every line drawn 1n the universe , from


one fixed star to anothe r, di stant 90° from the fi rst,

would be a tangent to the surface of the ea rth ) i twould be infin itely great in compari son with thesp aces which occur in da ily l i fe .”

The above , being the first publ i shed , not printed ,trea tise on the new geometry occupi es a uni que placein the hi sto ry o f highe r mathematics . I t gave addit ional strength to the formative tendenci e s whichchara cter i zed thi s p eriod and marked SCHWEIKART as

a constructive and original thinker.The na scent a sp ects o f thi s stage rece ived a frui t

ful contribution when N ICOLA I LOBACHEVSKI ( 1 7 931 847 ) created hi s Imaginary Geom e try and . JANosBOLYA1 ( 1 802-1 860 ) publ i shed a s an appendix to hisfather’s Tentamen

,hi s Science Absolute of Space .

LOBACHEVSKI and BOLYA1 have b een call ed the“Creators o f the Non-Eucl ide an G eometry .

” And thi sappel l at ion seems ri chly to be deserved by thesepionee rs . Their work gave j ust the impe tus mostneeded to fix the sta tus o f the new l ine o f re sea rche swhich led to such remarkabl e discoverie s in the morerecent yea rs . The Imaginary G e ome try and theScience Absolute of Sp ace were transl ated by theFrench mathemati ci an

, J . HOUEL in 1 868 and by himelevated out o f the i r fo rty-fiv e years o f obscurity and

non-effectiveness to a posi tion where they became ava ilable fo r the mathemati cal publ i c. To BOLYA1 andLOBACHEVSKI

,consequently, belong the honor o f

starting the movement which resulted in the development of metageometry and hence that which hasproved to be the gateway o f a new mathemati cal freedom .

SKETCH OF HYPERSPACE MOVEMENT 6 1

GAUSS,SCHWEIKART

,LOBACHEVSKI

,WOLFGANG

and JANOS BOLYA1 were the principal figures o f theformative period and the value o f thei r work withrespect to the formulation o f principle s upon whichwas constructed the Temple o f M etageometry cannotb e overestimated.

The Determinative Period

Thi s period i s characte ri zed chiefly by its closerelationship to the theory o f surfaces . R IEMANN’SHabil i ta tion Lecture on The Hypo theses Which Constitute the Bases of Geome try marks the beginning ofthis epoch . In this dissertation , R IEMANN not onlypromulgated the system upon which GAUSS had spentmore than thi rty yea rs of his l i fe in el aborating , fo rhe was a di scipl e o f GAUSS ; but he disclosed his own

vi ews with respect to space which he regarded as aparticul a r case o f mani fold . H is work conta ins twofundamental concepts, namely, the manifold and themeasure of curv ature o f a continuous mani fold , possessed o f what he cal led flatness in the smallest p arts.

The conception o f the measure o f curva ture i s extended by R IEMANN from surfaces to spaces and anew kind o f space , fini te , but unbounded , i s shownto b e possibl e . He showed that the dimensi ons o fany space a re determined by the number o f measurements nece ssary to establ ish the posi ti on o f a point inthat space . Conce iving, therefore , that space is a manifold of finite

, but unbounded , extensi on, he establ ishedthe fact that the passage from one el em ent o f a manifold to anothe r may be ei ther discrete o r continuous andthat the mani fold i s di sc rete o r continuous according


to the manner of pa ssage . Where the mani fold i sregarded as discrete two portions of i t can be compared

, as to magnitude , by counting ; where continuous,by mea surement. I f the whole mani fold be caused topass ove r into another mani fold each o f i ts elementspass ing through a one-dim ensional mani fold , a twodimensional mani fold is thus generated. In this way,a mani fold o f n-dimens ions can be generated . On theother hand , a mani fold o f n-dimensi ons can be an

alyzed into one o f one dimension and one o f (n- I )dimension s .To RIEMANN

,then , i s due the credi t fo r fi rst

promulgating the i de a that space be ing a speci al caseo f mani fold i s generable , and there fore , finite . Hela id the foundation fo r the establ ishm ent o f a speci alkind o f geometry known as the “ell ipti c.” Space , as

viewed by him , possessed the following prope rt i es ,viz . : generab il i ty, divisib i l ity, measurab il i ty, ponderab il i ty, finity and flexity.

These a re the six pill a rs upon which rests thestructure o f hyperspace analyses . 3

Generability is that p roperty o f geometri c spaceby vi rtue of which it may b e generated

,o r con

structed, by the movement o f a l ine , plane , surfaceo r sol i d in a direct ion without itsel f. D iv isibility i sthat p roperty o f geometri c space by vi rtue o f whichi t may be segmented or divided into separate pa rtsand superposed, or inse rted , upon or between eachother . M easurability i s tha t p roperty by vi rtue o f

‘ Vide Nature, Vol. V III, pp . 14- 1 7 ; 36, 37 also Mathe

matical Papers, pp . 65- 7 1 .

SKETCH OE HYPERSPACE MOVEMENT 63

which geometri c space i s determined to be a mani foldo f e ithe r a posit ive o r negative curva ture

,a lso by

which its extent may be measured . Ponde rability i sthat property o f geometri c space by virtue o f which i tmay be rega rded a s a quantity which can be manipu

l ated, assorted, shelved or otherwise disposed o f.Finity i s that property by vi rtue o f which geometri cSpace i s l imited to the scope o f the individual consciousness o f a unodim , a duodim o r a t ridim and byvi rtue o f which i t is finite in extent. Flexity i s thatproperty by vi rtue o f which geom etri c space is t e

garded a s possess ing curvature , and in consequence o fwhich p rogress through i t is made in a curved

,rather

than a geodeti c l ine , also by vi rtu e o f which i t may beflexed without disruption or dil atation .

RIEMANN who thus prepared the way for entranceinto a veritable l abyrinth o f hyperspace s is, the re fore ,correctly styled “The fathe r o f metageom etry,

” andthe fourth dimensi on is hi s eldest born . He died whilebut forty years o f age and neve r l ived long enoughfully to elaborate hi s theory with respect to i ts ap

pl i cation to the measure o f curvature o f space . Thiswas l e ft for his very energeti c discipl e , EUGEN IOBELTRAMI ( 1 83 5- 1 900 ) who was born nine ye arsa fte r R IEMANN and l ived thi rty-four yea rs longerthan he . His l abors ma rk the characterist i c standpoint of the determinative period . BELTRAMI’S mathematical investigations were devoted ma inly to thenon-Eucl idean geometry. These led him to the ratherrem arkable conclusion that the proposi tions embodiedtherein rel ate to figures lying upon surfaces of con

stant negative curva ture .


BELTRAM I sought to show that such surfaces pa rtake of the nature o f the -pseudosphere , and in doingso , made use o f the fol lowing i llustration

F111. 3 .

F113 . 4 .

I f the plane figure oabb i s made to revolve uponits axis of symm etry AB the two a rcs, ab and ab willdescribe a pseudospheri cal concave-convex surface like


The Elaborative Pe riod

The elaborative stage include s the work of all

those who , working upon the bases l a i d down byLOBACHEVSKI

,BOLYA1

,SCHWEIKART and R IEMANN

,

have sought to ampl i fy the conclusi ons reached bythem . Among those whose investigations have greatlymultipl i ed the appli cat ions o f hype rspace conceptionsa re HOUEL ( 1 866 ) and FLYE ST . MARIE ( 1 8 7 1 ) ofFrance ; HELMHOLTZ FRISCHAUFKLEIN and BALTZER ( 1 87 7 ) o f G ermany ;BELTRAM I ( 1 8 7 2 ) o f Italy ; DE T ILLY ( 1 879 ) o f Belgium ; CLIFFORD and CAYLEY ( 1 82 1 ) of England ;NEWCOMB ( 1 83 5 ) and HALSTEAD of Ameri ca .These have been most active in popula riz ing the

subj ect o f non-Eucl idean geometry and incidental ly theidea o f the fourth dim ensi on. The great mass o fnon-pro fessional mathematical re aders

,there fore , owe

these men an imm ea surabl e debt o f grati tude fo r thework that they have done in the matte r o f renderingthe conceptions which consti tute the fabri c o f metageometry understandabl e and thinkable . A glance atthe b ibl i ography appended at the end of thi s volumewill give some idea o f the enorm ous amount o f l aborthat ha s been expended in an effort to transl a te themost abstract mathemati ca l p rinciple s into a l anguagethat could eas ily be comprehended by the average int elligent person.

The chara cteri sti c standpoint o f thi s period is thepopula r comprehension o f the hyperspace concept andthe consequent menta l l iberation which fol lows . Forthe re i s no doubt but that unheard o f possib il i ti e s o f


thought have been revealed by investigations into thenature of space . An enti rely new world ha s be enopened to Vi ew and only a beginning has been madeat the explo ration o f i ts extent and resource s.One of the notable incidents o f the ea rly years o f

thi s period i s the position taken by FELIX KLEIN whostands in about the same relation to CAYLEY a sBELTRAMI does to R IEMANN

,in that he a ssumed the

task o f completing the work o f his predecesso r.KLEIN held that there a re only two kinds o f RIEMANNIAN space

—the ell iptical and the spherical . Orin other words , that there a re only two possibl e kindso f space in which the proposi tions announced byR IEMANN could apply. SOPHUS LIE

,called the “great

comparative anatomist o f geometric theories ,” carri ed

hi s classific ations to a final conclus ion in connectionwith spaces o f all kinds and decided that the re a repossibl e only four kinds o f three dimensional spaces .But whethe r men wi th peering , microscop ic , histo

logica l vi s ion shall establ i sh the exi stence of one o rmany spaces, and rega rdless o f the mathemati c rigorwith which they shal l demonstrate the sel f-consi stencyo f the doctrines which they hold , the fact remains thatthe hypothese s thus ma inta ined , while they may berega rded as tru e descriptions o f the spaces concerned,a re , neverthele ss , incompatibl e . All o f them cannotbe val id. I t will p erhaps be found that none o f themare val id , especi ally obj ectively so . The only truevi ew, there fore , o f these systems o f hyperspaces i sthat which a ss igns them to the i r rightful place in theinfinitely vast world o f pure mathesi s where thei rval idi ty may go unchall enged and the i r exi stence unquestioned ; for in that domain o f unconfined menta


t ion , in that realm o f divine intuitability, the marvelous wonderl and of ideas and notions , one i s not onlydi sincl ined to doubt the i r logical a ctual i ty, but i s quitewill ing to accede thei r cl a ims.

CHAPTER III

ESSENTIALS OF THE NON-EUCLIDEAN GEOMETRY

Non-Euclidean Geometry Concerned with Concep tualSpace Entirely—Outcome of Failures at Solving theParallel-Postulate—The Basis of the Non-EuclideanGeometry—Space Curvature and Manifoldness—SomeElem ents of the Non-Euclidean Geometry—Ce rtainty,Necessity and Universality as Bulwarks of GeometrySome Consequences of Efforts at Solving the ParallelPostulate—The Final Issue of the Non-Euclidean Geomet ty—Extended Consciousness.

THE term non-Eucl idean i s used to designateany system of geometry which i s not strictly Eucl ideanin content .It i s intere sting to note how the te rm came to be

used . It appears to have been employed fi rst byGAUSS . He did not strike upon it suddenly, however,a s in the correspondence between him and WACHTERin 1 8 1 6 he used the designation “anti-Eucl idean” andthen , l ate r, fol lowing SCHWEIKART, he adopted thel atte r’s terminology and call ed i t “Astral G eometry .

”

This he found in SCHWEIKART’S fi rst publ ished

treatise known by that name and which made its ap

p earance a t Ma rburg in December, 1 8 1 8 . Finally,in hi s correspondence with TAURINUS i n 1 824 , GAUSSfi rst used the express ion “

non-Eucl idean” to designatethe system which he had el aborated and continued to

69


us e i t in hi s correspondence with SCHUMACHER in1 83 1 .

“Non-Legendrean, semi-Eucl idean and nonArchimedean" a re titl e s used by M . DEHN to denoteal l kinds o f geom etri e s which represented va ri ationsfrom the hypotheses l a i d down by LEGENDRE EUCLIDand ARCH IMEDES .The semi-Eucl idean i s a system of geometry in

which the sum o f the angles o f a tri angle i s s a id tobe equal to two right angles , but in which one maydraw an infini ty o f pa rallel s to a stra ight l ine througha given point . The non-Eucl idean geometry embracesal l the resul ts obta ined as a consequence o f effortsmade at finding a satisfacto ry proof of the parall elpos tul ate and is, there fore , ba sed upon a conceptiono f space which i s a t va ri ance with that held byEUCLID . According to the Ion i an school space i s aninfinite continuum possessing uni formity throughoutits enti re extent. The non-Euclideans mainta in thatspace i s not an infinite extension ; but a fini te thoughunbounded mani fold capable o f be ing generated bythe movement o f a point, l ine o r pl ane in a di rect ionwithout i tsel f . It i s al so held that space i s cu rvedand exi sts in the Shape o f a sphere or pseudosphereand i s consequently ell ipti cal .The inappl icab i l i ty o f EUCLID’S parallel-postulate

to l ines drawn upon the surface o f a sphere suggestedthe poss ib il i ty o f a sp ace in which the postulate couldapply to al l poss ibl e surfa ces o r that space i tsel f maybe spheri cal in which case the postulate would be inval idated altogethe r. Hence , it i s quite na tural thatmathematici ans finding themselves unabl e to prove thepostul ate with due matheti c preci s ion should turn their

H H

THE NON-EUCLIDEAN GEOMETRY 7 1

a ttention to the conceptually possibl e . In thi s vi rtualabandonment o f the perceptual for the conceptual l ie sthe fundamenta l diffe rence between the Eucl idean andthe non-Eucl idean geometrie s . It may be sa id to thecredit o f the Euclideans tha t they have sought to makethei r geometri c conceptions conform a s closely a spossibl e to the actual na ture o f things in the sensuousworld whil e at the same time they must have perceiv ed tha t at best the i r spati a l notions were only approximations to the sensuous a ctual i ty of obj ects inspace .On the other hand , non-Euclideans make no pre

tense at discovering any congruency between the irnotions and things as they actually a re . The attitudeo f the m etageom e t ricians i n this respect i s very aptlydescribed by CASSIUS JACKSON KEYSER who says :

He constructs in thought a summitl ess hie ra rchy o f hyp erspaces , an endless se ri es o f o rderlyworlds, worlds that a re possibl e and logi cally actual ,and he i s content not to know i f any of them beothe rwise actua l o r actual ized .

” 1

The non-Eucl idean is, the re fore , not concernedabout the appl i cabil i ty o f ensembles , notions andproposi ti ons to real , perceptual space conditions . I ti s suffi ci ent fo r him to know that h i s creations a rethinkable . As soon a s he can re solve the nebulos i tyo f hi s consciousness into the conceptual “sta r-forms”

of definite idea s and notions, he sits down to the feastwhi ch he finds provided by sup erfoetated hypothesesfabri cated in the deeps o f mind and logi cal actual iti es‘Mathematics, by C. J. KEYSER, Adrian Professor of Mathematics,

Co lumbia University.


imperturbed and unmindful o f the weal o f perceptualSpace in i ts homogeneity o f form and dimens ional ity.Fundamentally, the non-Eucl idean geometry i s con

structed upon the basi s o f conceptual space almostenti rely. Knowledge o f its content i s a ccordinglyderived from a sup e rp ercep tual representation o f relations and inte rrel ations subsi sting between andamong notions , i dea s , proposi tions and magnitudesa ri s ing out o f a conceptual cons ideration thereof. Inother words , representations o f the non-Eucl ideanmagnitudes, cannot b e sa i d to be stri ctly perceptualin the same sense that three-space magni tudes a reperce ived ; fo r three-space m agni tudes a re really senseobj ects whi le hyperspace magnitudes a re not sense ob

jects. They a re fa r removed from the sensuous worldand in orde r to conceive them one must raiSe hi sconsciousness from the sensuous plane to the conceptual pl ane and become aware o f a cl a ss of perceptionswhich a re not perceptions in the strict sense of theword, but sup erp ercep tions ; because they a re rep re

sentations o f concepts rather than p recepts .Notions o f p ercep tual space a re constitute d o f the

tripl e p resentations a rising out of the visual , tactualand motor sensations which a re fused together in the i rfinal del ive ry to the consciousness . The synthesi s o fthese thre e sense-del iveri e s i s accomplished by equilibrating the i r respective differences and by correcting the perceptions o f one sense by those o f anotherin such a way a s to obta in a completely rel i able percep tion of the obj ect . This i s the manner in whi chthe characteri sti cs o f Eucl idean space a re establ ished.

The characte ri sti cs o f non-Eucl idean space a re nota r r ived at exactly in thi s way. Being beyond the scop e


lyt ical pl ea sure s qui te unrestra inedly. The differencebetween the two mental processe s—that which leadsfrom the sens ibl e world to conception and that whichveers into the fi elds b eyond—i s so grea t that i t ishardly permi ssibl e to view the results a rrived at inthe outcome o f the sepa rate processe s a s beingidenti cal .To illustrate this diffe rence, l et u s draw an analogy.

The mine r digs the i ron ore out o f the ground. Thei ron i s sepa rated from the extraneous mater i al anddel ivered to the furnaces where the metal i s meltedand turned out a s p ig i ron. It i s further t re ated, andsteel , o f va rious grades , ca st i ron and other kinds o fi ron a re produced . The trea tment o f the i ron oreup to thi s stage i s s imila r to the treatment o f senseimpressions by the Thinke r. Steel , cast i ron , et cetera ,a re s imil a r to mental concepts. Late r, the steel andother products a re conve rted into instruments andnumerous arti cles . This represents the sup e rp ercep tualprocess. Trafficking in i ron ore products , such a sinst ruments o f p reci s ion , watch springs , and the l ike ,represents a stage stil l fa rther removed from theprim a ry treatment o f the ore and i s sim i l a r to thatto which concepts a re trea ted when the m etageom etri

cian manipul ates them in the construction o f conceptualspace-forms . Perception i s the deal ing with raw i ronore while conception i s analogous to the productiono f the fini shed p roduct.

Sup erp ercep tion would be analogous to the t rafficking in the fini shed product a s such and without anyre ference to the source o r the preceding processes .Thus the notions and j udgments o f the non-Euclidean

geometry a re a rrived at as a resul t o f a tripl e p roces s


of perception, conception and sup erpercep tion the latterbeing merely sup e rconceiv ed as formal space-notions.But i t i s obvious that the more complex the processesby which judgments purpo rting to relate to perceptualthings a re derived the more l ikely are those j udgmentsto b e at vari ance with the na ture o f the things themselves .In view o f the foregoing, the dangers re sulting

from identi fying the p roducts of the two processesa re very obvious indeed. But the diffe rence betweenthe two procedures i s the difference between Eucl ideanand non-Euclidean geometrie s o r the difference between pe rceptual space notions and conceptual spacenotions . Hence , i t i s not understood just how or whyi t ha s occurred to anyone that the two notions couldbe made congruent. M agnitudes in pe rceptual , sen

sible space a re things apart from those that may besa id to exi st in mathemati cal space or that space whosequali ti e s and properti es have no existence outsi de o fthe mind which ha s conce ived them . It i s bel i eved tobe quite imposs ibl e to approach the s tudy of metageometri cal p roposit ions with a cl ea r , open mind without previously unde rstanding the fundamental distinctions whi ch exist between them .

I t follows , the re fore , a s a l ogical conclus ion thatgeometri c space Of! whatsoever nature i s a purelyformal construct ion o f the intell ect, and for this re ason i s completely under the sovereignty o f the intellecthowever whimsical its demands may be. Being thusthe crea ture o f the intell e ct, i ts poss ib il i tie s a re l imi tedonly by the l im ita ti ons of the intell ect itsel f. Per

cep tual space , being nei the r the creature o f the intellectno r necessarily an a priori notion res ident in the mental


substructure , but existing enti rely independent o f theintellect or its app rehension thereo f, cannot be ex

p ect ed to conform to the purely formal restr i ct ionsimposed by the mind except in so fa r a s those t e

str icti ons may be determined by the nature o f percep tual spa ce . And fo r that matte r , i t should not beforgotten that, as yet, we have no means o f det ermining whether o r not the testimony of the intell ect i sthoroughly credibl e simply because there i s no otherstandard by which we may prove its t estimony. It ispossible to j usti fy the del iveri es of the eye by the senseo f touch , or vice ve rsa . It i s a l so poss ibl e to p roveall ou r sense-deliveri es by one or the othe r o f thesenses . But we have no such good fortune with thedel iveri e s o f the intel lect. We have simply to accepti ts test im ony as final ; because we cannot do any bette r.But i f i t were poss ibl e to correct the t estimony of

the intel lect by som e othe r faculty o r powe r whichby na ture might be more accurate than the intell ectit shoul d b e found that the intellect i tsel f i s s adlyl imited.

The poss ibl e cu rva ture o f space is a notion whichal so characte riz es the content o f the non-Eucl ideangeometry. It is upon this notion that the quest ion o fthe finity and unboundedness o f space , in the mathem atical sense , rests. In the curved space , the stra ighte st l ine i s a curved l ine which returns upon i tsel f .Progressi on e astward brings one to the west ; pro

gression northward brings one to the south , e t cetera .

On this View space i s fini te , but may not be regardeda s possessing boundarie s .

Spa ce-curvature , re inforced by the idea that spaceis a lso a mani fold is the enabl ing cl ause o f meta

THE NON-EUCLIDEAN GEOMETRY 77

geometry and without them the analyst da res not proceed. Here aga in, we are led to the confession thathowever fantasti c these two notions may seem andevidently are , there is nevertheless to be recognized inthem a dim glimpse” of a veritabl e real ity—a sl ightforeshadowing o f the revelation of some great kosm i cmyste ry .

The mani foldness of space i s the fi at o f analys i s .It i s the inevi table outcome of the analyst’s methodof procedure . H is education , tra ining and view ofthings in general inhibi t hi s a rriving at any other t e

sul t and he may be pa rdoned with good grace for hismanufacture o f the space-m ani fold. For by it perhaps a bette r appreci ation o f that wonderful extensi onof consciousness in the na ture o f which is involved theexpl anation o f the perplexing probl ems which themani fold and other metageom etri cal expedients fa intlyadumbrate may be gained .

I t is pert inent, i n the l ight o f the above , to examine into some o f the relative merits of the threeformal bulwarks o f geometrical knowledge . Thesea re cer tainty, ne cessity and univ ersality.

G eometri c certa inty is derived solely from thenature o f the prem ises upon which it is based. I f thepremises be contradictory , i t is, of course , defect ive .But i f the premises a re non-contradictory o r sel fevident , then the certa inty of geom etri c notions andconclusions is valid . Anothe r consideration o f prim eimportance in this connection is the definition. Fromi t a ll prem ises proceed . Hence , the defini t ion i s evenmore important than the p remi se ; fo r i t is the persisting determ inant o f all geom etri c conclus ions whil ethe premise is dependent upon the l im i tations of the


definition. The determinative cha racte r o f the definition has l ed to i ts apotheosis ; but this, adm i ttedly, hasbeen necessary in order to give stabi l i ty and permanency to the conclus ions which followed. But inspite o f thi s i t would appea r that the certa inty o fgeometri c conclus ions i s not a qual i ty to be reckoneda s absolute o r final .With the same certa inty that i t can be said the

sum of the angles o f the tri angle is equal to two rightangles it may be a sserted that that sum is al so greate ro r le ss than two right angles . Certa inty which i sbased upon the inhe rent congruity o f definit ions,prem ises and proposit ions i s an enti rely diffe rentmatte r from that certa inty which a rises out o f the real ,ab iding val idi ty o f a scheme o f thought . But thi sdifference i s not l essened by the fact that the l atteri s dependent, in a measure , upon the correct systematization of our spati a l experi ence s by means o fmethodical processes. Eucl idean geometry, accordingly, is not so certa in in its appl ications as i t i s util ita ri an ; but non-Eucl idean geometry i s even l ess certa inthan the former and consequently more l acking in itsutil i ta ri an possib i l i ti es.

The necessi ty o f geometri ca l determinations i smerely the n ecessi ty which inheres in logical infe rence so r deductions . These may o r may not be val id.

Inasmuch a s the necessariness o f deductions i s p rimarilybased upon the conditional certa inty o f premi se s anddefin it ions it appears that thi s qual ity i s in no waypecul i a r to geometry whethe r Eucl idean o r nonEucl idean . In l ike manne r, the universal i ty o f geometri c j udgments may not p roperly be rega rded a sa pecul iarity o f geometry ; but i s expl icable upon the


ba si s o f the formal characte r o f the assumptionswhich unde rl i e i t. The chie f va lue

,then

,of non

Eucl idean geom etry seems to abide in the fact thati t cl a rifies our understanding as to the complex processe s by which i t i s possibl e to organize and sys

t ematize our spati al experi ence s for a ss imilation and

use in othe r branche s of knowledge .With the above statement o f the ca se of the

non-Eucl idean geometry it i s now thought permissibl eto state briefly some o f the elem ents thereo f .1

Below will be found some of the el ements ob

tained a s a consequence of efforts made both atproving and disproving the parall el-postulate ofEucl id

“I f two po ints dete rmine a l ine i t i s called astra ight .

“If two stra ights make with a transversal equal

alte rnate angle s they have a common perpendi cular.”“A pi ece o f a stra ight is called a sect.I f two equal coplana r sects a re erected per

p endicular to a stra ight, i f they do not m eet, then thesect j oining thei r extrem i ti es makes equal angles w iththem and i s b isected by a perpendicul a r e rected midway between thei r feet .

“The sum of the angl es o f a rectil ine al tri angl ei s a stra ight angle , in the hypothes i s o f the right

( angle ) is greater than a stra ight angle in the hypo

‘The science of pure mathematics is p erhaps indebted to no one

in so great a degree as to GEORGE BRUCE HALSTEAD, formerly of

the Unive rsity 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. VI I and V III ,1900, Dr . G. B . HALSTEAD.


thesi s o f the ob tuse ( angle ) ; is l ess than a stra ightangle in the hypothes is of the acute

“The hypothes i s o f right i s Eucl idean ; the hypo

thesi s of the a cute i s BOLYA1-LOBACHEVSKIAN ; thehypothes i s o f obtuse i s R IEMANN IAN .

”

“ I f one stra ight is pa rallel to a second the secondis pa rall el to the fi rst .

“Pa rall el s continually approach each other .The perpendiculars e rected at the middl e point

of the s ide s o f a tr i angl e a re all pa ral lel , i f two a reparallel .”

“I f the foot o f a perpendicul a r sli des on a stra ighti ts extremity describes a curve cal led an equidistantcurve , o r an equidistant i al .

”

“An equidistanti al will sl i de on its trace .In the hypothes i s o f the obtuse a st ra ight i s o f

fini te s iz e and returns into itsel f.”“Two stra ights a lways interse ct .”

Two stra ights perpendicul a r to a thi rd stra ightinters ect a t a point hal f a stra ight from the thi rd ei therway

A pol e i s hal f a st ra ight from i ts pol a r .A pol a r i s the locus o f coplana r points hal f a

stra ight from its pole . There fore , i f the pol e o f onestra ight l i e s on anothe r stra ight the pol e of th i s secondstra ight i s on the fi rst stra ight.

“The cross o f two stra ights i s the pol e o f the j oinof thei r pole s .”

“Any two st ra ights inclose a pl ane figure

,a digon .

“Two digons a re congruent i f thei r angle s a reequal.”

“The equidi stanti al is a ci rcl e with cente r at the

pole s o f its basal stra ight.”


The pa rall el-postul ate is s tated in the nonEucl idean geometry as follows

“I f a stra ight l ine m eeting two stra ight l ines makethose angle s which are inward and upon the same sideo f i t less than two right angl es the two st ra ight l inesbeing produced indefinitely will meet e ach othe r onth i s s ide where the angles a re l e ss than two rightangles.”

It is stated by MANN ING‘ i n the following lan

guage“I f two l ine s a re cut by a thi rd and the sum o f

the inter io r angles on the same s ide o f the cuttingl ine is less than two right angl es the l ine will meeton tha t s ide when suffi ci ently produced.

I t i s rather s ignificant that in thi s postul ate whichi s re ally a defini ti on o f space should b e found groundsfor such diverse interpretations a s to i ts nature . Ofcourse

,the moment the mind seeks to understand the

infini te by interpreting i t in the unmodified terms of theappa rently unchangeabl e finit e i t entangles i tsel f intoinsurmountabl e difficulti e s . As a drowning m an grasp sa fte r straws so the mind , immersed in endless abysseso f infin ity, fa i l s to conduct itsel f in a seemly manner ;but gasps , st ruggle s and flounders and i s happy i f i tcan , i n the depths o f i ts perplexity , discove r a way oflogical escape . The pure mathemati cian has a hankering a fte r the logi cal ly consistent in al l hi s pursuits ; tohim it i s the “Holy G ra i l” o f his highest aspirations .He seeks it a s the devotee seeks immortal i ty. I t i s tohim a phi losopher’s stone , the el ixi r o f perpetual youth ,the eternal cri terion o f a ll knowledge .Fa ilures to demonstrate the cel ebrated postula te o f

‘Vide Non-Euclidean Geometry, p . 91 .


EUCLID l ed, a s a matter o f course , to the substi tutiono f various othe r postul ates more or less equivalent toi t in that e ach o f them may be deduced from theother without the a id o f any new hypothesi s .Among those who sought proof by a restatement

o f the problem a re the followingI . PTOLEMY : The internal angl es which two par

allel s make with a transve rsal on the same s ide aresupplementa ry.

2 . CLAVIUS : Two parallel stra ight l ines a re equidi stant.

3 . PROCLUS : I f a stra ight l ine intersect s one o ftwo paral lels i t a lso inte rsects the othe r.

4 . WALLIS : A tri angle being given anothe r triangl e can be constructed s im i l a r to the given one andof any si z e whatever.

5 . BOLYA1 (W. ) Through three points not lyingon a stra ight l ine a sphere can always be drawn .

6 . LORENZ : Through a point between the l inesbounding an angle a stra ight l ine can always be drawnwhich will intersect these two l ines .

7 . SACCHERI : The sum of the angles o f a t ri

angle i s equal to two right angle s .There were

,o f course , many other statements and

substitutions used by mathematici ans in thei r en

deav ors s at isfactorily to establ ish the truth o f thepa rallel-postul ate . That thei r l abors should haveterminated

,fi rst

,by doubting it, then by denying, and

final ly, by bui lding up a system of geom etri e s whichaltogether ignores the postul ate is just what mightnatural ly b e expected o f these men who have givento the world the non-Eucl idean geom etry . In doingwhat they did many, i f not all of them , were not aware


i n any measure o f the proportions o f the imposingsuperstructure that would be bui lt upon thei r apparentfa ilures. All o f them undoubtedly must have sensedthe vague adumbrations forecast by the unfolding myste ries which they sought to l ay bare ; al l o f them musthave fel t as they executed the e a rly ta sks of thosecrepuscul a r days o f pure mathemati cs that the waywhich they were travel ing would l ead to the innershrine o f a higher knowledge and a wider freedom ;they may have be en l ed by divine intu it ion to strikeout on thi s n ew path and yet they could not haveknown how fully thei r dre ams would be real i zed by themathemati ci ans o f the twenti eth century. I f so , theywere truly gods and mathesis is the i r kingdom .

The analyst p roceeds upon a basi s enti rely a t variance with that which guides the ordinary invest igatorin the formulation o f hi s conclusi ons . The empi ri ca lsci enti st in a rriving at hi s theori es o r hypothese s i sgoverned at all t ime s by the degree o f conformitywhich hi s postul ate s exhib it to the actual phenomenaof nature . H e endeavors to ascerta in just how faro r in what degree his hypothesi s i s congruent withthings found in natu re . I f the diss i dence i s found topredom inate he abandons his theo ry and makes anotherstatement and again sets out to determine the degreeo f conform ity. I f he then finds that the natura l phenom ena agree with his theory he accepts i t as for thet ime being finally settl ing the question . In all thingsh e i s l imi ted by the answer which nature give s to hi sque rie s . Not so with the exponent o f pure mathem atics. For him the truth o f hypotheses and postul ates is not dependent upon the fact that physi calnature conta ins phenomena which answer to them . The


r ived at a s a resul t o f logical p rocesses should , i t s eems ,be rega rded a s j udgments a p osteriori, i.e . , the resul tso f empi ri cal op erations . Confessedly, the fact s ad

duced in course o f experimentation serve a s guides inchoosing among all o f the many poss ibl e logical conv ent i ons ; but our choice remain s untrammeled except bythe compulsion a ri s ing out o f a fea r o f incons i stency.

The real cri terion then of a ll geometri es i s ne ithe rtruth , confo rmabil i ty nor necess ity, but consi stency andconveni ence .The diffi culty wi th the non-Euclideans resolved

itsel f into the question as to whethe r i t is more consi stent , a s well a s conveni ent, to establ ish a proo f o fthe postul ate by taking advantage o f the support tobe found in other postul ate s o r whethe r, by seeking ademonstrati on based upon the del iveri es o f sense-ex

p erience as to the nature o f space and its prope rti e s ,a stil l more consi stent conclus ion might be reached .

They had furthe r perplexity, however , when i t cameto a deci si on a s to whethe r the organi c world i s p roduced and ma intained in Eucl i dean sp ace or in a purelyconceptual space whi ch alone can be apprehended bythe mind’s powers o f representation . Unwill ing toa dmit the exi stence o f the world in Eucl idean space ,they turned thei r attention to the examination o f thep rope rti e s o f another kind o f space so-c all ed whichunl ike the space o f the Ioni an school could be made toanswer not only al l the purposes o f pl ane and sol i dfigures , but of sphe rics a s wel l . And so, the mani fol dspace was invented by R IEMANN and l ate r underwentsome remarkabl e improvements at the hands o f hi sdi scipl e , BELTRAMI . But i t may be sa id here , parentheti cally

,that the truth o f the whol e matte r i s that


our world i s ne ither in Euclide an nor non-Eucl ideanspace , both o f which , in the l ast analys i s , a re conceptual abstractions . Al though i t may not b e deni ed thatthe Eucli dean space is the more compatible .The problem of devis ing a space , i f only a very

l imited portion , in which could be demonstrated thea ssumed alte rnat ive hypothes i s and its consequence slogically developed , occasioned no inconsiderabl e concern fo r the non-Eucl idean investigators ; but neitherLOBACHEVSKI, BOLYA1 nor R IEMANN were to bebaffl ed by the diffi cul ti e s which they met. These onlyc ited them to more l aborious toi l . H aving succeededin mentally constructing the part icul a r kind of spacewhich wa s adaptabl e to thei r rigorous mathetic re

quirem ents i t immedi a tely occurred to them that al l thequal it i e s o f the l imited space thus devi sed mightlogically b e ampl ified and extended to the enti re worldo f space and that what i s true of figures constructedin the segmented portion o f space which they usedfor experimental purposes i s also true o f figures drawnanywhere in the universe o f thi s space a s al l l inesdrawn in the fini te , bounded portion could be extendedindefinitely and all magnitudes simil arly tre ated . Fromthese results , i t wa s but a s ingle step to the conclusionwhich followed—that e ither an enti rely new worldo f sp ace had been di scovered o r tha t our notion o fthe space in which the organi c worl d was produced iswholly wrong and needs revi si on . But notwithstanding the insurmountable obstacl e s which stood in theway o f the investigators who made the a ttempt todiscover the homology which might exi st between thecha racteri sti cs o f the newly fabricated space and thephenomenal world

,investigations were carri ed for


ward with almost am az ing reckle ssness and loyalty tothe m atheti c spi ri t unt i l i t was di scove red that all effortsto trace out any defini te l ines o f co rrespondence werefuti l e. Then the pol i cy o f ignoring the question o fconformabil ity was adopted and ha s sin ce been pursued with unchangeabl e regularity by the analyti calinvest iga to r.Among the resul ts obta ined by the non-Euclideans

in thei r pro found rese a rches into the nature o f hyperspace a re these : I . It was found that the angul a r sumo f a tri angle , be ing ordina ri ly a ssumed to b e a va ri ablequant ity, is e ithe r less or grea te r than two right angl esso that a stri ctly Eucl idean rectangl e could not be constructed. 2 . The angle sums o f two tri angl es of equala re a a re equal . 3 . No two tri angles not equal canhave the same angle s so that simil a r tri angles a re impossibl e unl ess they a re o f the s ame size . 4 . I f twoequal perpendicul ars a re e rected to the same l ine , thei rdi stance ap art increases with thei r l ength . 5 . A l ineeve ry point o f which i s equal ly distant from a givenstra ight l ine i s a curved line . 6 . Any two l ine s whichdo not meet, even at infini ty, have one common per

p endicular which measures the i r minimum di stance .

7 . Lines which me e t a t infinity are parallel. But it isappa rent that these re sults have not followed upon anymathemati cal consequence o f other supporting postul ate s o r axioms such a s would pl ace them on a coOrdinate basi s wi th those used a s a support fo r the parallelpostul ate ; fo r they are based upon the envi sagement o fan enti rely new principl e o f space-perception and belong to a wholly different set o f space qual iti es .The final issue then o f the non-Eucl idean geometry

i s neither in the uti l ity o f its processes and conclus ions


ing of the diurnal flower o f the mind’s immortal i ty and

the outp‘

uttmg of the o rgan o f consciousness wherewith the infini te stretches o f hypersp aces, the low-lyingvalleys o f reals and imagina ri es and the upli ftinghill s o f finites and infinite s shall b e divested o f thei rmyste rie s and stand out in thei r unitarines s no longe rdraped in the vei l o f the inscrutable and the incomp rehensible .

The fourth dimension, regarded by some a s a newscope o f motion fo r obj ects in space , by others a s anew and strange direction o f spati a l extent and byothe rs still as the doorway of the templ e o f ex egesi swherein an expl anation may be found for the enti reconge rie s o f mysteri es and supermysteri es which nowperplex the human mind , may also be sa id to bethe key to the non-Eucl idean geom etry . But i t reallycompl icates the s ituation ; for one ha s to be capabl e ofprolonged abstract thought even to envisage is a s aconceptual possib i l ity. POINCARE‘5 s ays : “Any onewho should dedicate his l i fe to i t could , perhaps,eventu ally imagine the fourth dim ensi on,

” implyingthereby that a l i fetime o f prolonged abstract thoughti s necessary to bring the mind to that point o f ecsta sywhere it could even so much a s imagine this additionaldimens ion. Nevertheless by i t ( the fourth dimension )was the non-Eucl idean geometry made and wi thout i twas not any of the hyperspaces made that were made .It i s the Vi ew which geometers have taken of spacein general that has made the fourth dimensi on poss ibl e , and not only the fourth , but dimensions o f al ldegrees . The basi s o f the non-Eucl idean geometry

“Vide Nature , Vol. XLV , 1892 .


may be found then in the notion o f space which hasbeen predom inant in the minds of the investigators.

Finally, i t should b e pointed out that the nonEucl idean geometry, though a consi stent system ofpostul ates, has been constructed upon a misconceptionbased upon the identification o f real , perceptual spacewith systems o f space-measurem ents . Hyperspaceswhich are not spaces a t al l should not be confoundedwith real space . But they constitute the substance o fnon-Eucl idean geometry ; they a re its blood and sinews .Thei r study is inte resting, because o f the possib i l i ti eso f speculat ion which i t offe rs . No mind that hasthought deeply upon the intri caci es o f the fourth dim ension, o r hyp erspace , remains the same a fter theprocess. It i s bound to experi ence a ce rtain sense ofhum i l ity, and ye t some pride bo rn o f a knowledge thati t ha s b een in the presence o f a great mystery and hasdelved into the fea rful deeps o f kosm i c mind. To themind that has thus been anointed by the sacred chrismof the inne r mysteri es of creat ive mental i ty therealways come that sti llness and calm such as chara cteri z e the a fte rmath of refl ection upon the incomp rehensible and the transfinite .

CHAPTER IV

D IMENSIONALITY

Arbitrary Character of Dimensionality—Various Definitions ofDimension—Real Space and Geometric Space Differentiated—The Finity of Space—Difi erence Between the

Purely Formal and the Actual—Space as Dynam ic Appear

ance—The A Priori and the A Posteriori as Defined byPAUL CARDS.

IN previous chapters we have traced the growthand development o f the non-Eucl ide an geometry showing that the so-ca lled fourth dimension i s an aspectthereof. I t i s now deemed fitting that we shouldenter into a more deta i le d study o f the questi on o fdim ensi onal i ty with a vi ew to examin ing some of thediffi cul ti e s which encompass i t.The question o f dimens ion i s as old a s geometry

i tsel f. Without i t geometri c conclusi ons a re void andmeaningles s . Yet the conception o f dimensi onal i tyitsel f i s pu rely conventional . In its appl ication to spacethe re i s involved a great deal o f confusi on because o fthe infe renti al cha racter o f its defini t ion . For instance ,commonly we measure a body in sp ace and a rbi trarilyass ign three el ements to determine its posit ion. Thesimplest standard for thi s purpose i s the cub e havingthree o f i ts edges term inating at one o f its corners .

Thus because it i s found that the enti re volume o fa cube is actua lly comprehended wi th in the di re ctions

92


space identi ca l with the notion o f spati a l extensity. Inother words, we take bodi es in sp ace and by examiningthei r characte risti cs and properti es a rrive a t an al

leged apode icti c j udgment of space . It i s by meanso f thi s conventional norm of geometric knowledge thatva ri ous other spaces , notably the one two four andn-space , have been devi sed . I t would appea r that i fsome more absolute standard of measurement o r definiti on of space were adopted the confusi on which now

cl ings to the conception o f dim ension could be obvi ated.

For i f i t be true that three and only three el ements arenecessa ry to dete rmine a point-posi ti on in our spaceand that in thi s determination we also find the numberof dimensions of spa ce , then it may also be true thatn-coOrdinates would just a s truly determine the dimensionality o f an n-space

,which is granted. But then

the n-space would be just a s l egitimate a s the threespace ; for i t i s dete rm ined by exactly the same standards . I t i s both quantita tively and qual i tat ively thesame . I f , howeve r , on account o f the exigenci es thatmight a ri se , we a re forced to seek solace in the notionof an n-space whithe r Shall we turn for it ? It cannotbe found ; fo r i t i s imperceptible , uninhab itabl e , nonexi stent, and there fore , absolutely and purely an ab

straction. Consequently, there must be somethingradically wrong with the defini t ion o f space o r withits dete rminants .The purely a rb itra ry characte r o f dimensional i ty i s

very aptly described by CASSIUs JACKSON KEYSER,who says

The dimensional ity of a given space i snot unique

,but depends upon the cho ice o f the

geometri c entity for prima ry or gene ratmg element.

DIMENSIONALITY 95

A space being given, i ts dimensi onal ity i s not therewith determined , but depends upon the wil l o f theinvestigator who by a p roper choice o f generatingelement endows the space with any dim ensional ityh e pl ea ses. That fact is o f cardinal s ignificance forsci ence and philo sophy.

” 1

It i s a fa ct o f ca rdina l significance fo r sci ence ;because i t emphasizes the necessity for some morerational procedure than that o f the geometri ci an in a rr iving at an absolutely unique method of determiningthe dimension and essenti a l nature o f rea l space . Itssignifi cance for philosophy l ies in the need of a logical ,rigidly exclusive and absolutely pecul i a r standard ofspace defin i tion . The definition o f perceptual spaceshould be such a s rigorously inhibits its inclusi on a s apa rti cula r in any general cl ass . The necessi ty for thi si s wa rranted by its unive rsal ity and uniqueness.The l ines o f demarkation between what i s recog

nized a s perceptua l space and what ha s been calledgeometric o r conceptua l space should be very sharplydrawn . So tha t when reference is made to e i ther therewill b e no doubt a s to which i s meant. And then , too ,conceptual space i s no space a t all , properly speaking .

I t i s merely a system of space-measurement. And a ssuch ha s no logical right to be put in the same categorya s perceptual space .

Rea l space i s unique . Geometri c space belongs toa cl a ss whose membe rs a re capabl e o f indefinite multipl i cation . I t . is certa inly most i llogi ca l to identi fythem . Pe rceptual space , figuratively speaking , i s aquanti ty ; analytic space i s the foot-rul e , the ya rd-stick ,the kilometer

,by which i t i s mea sured and apporti oned .

‘Vide Monis t, Vol. XVI, 1896, Mathematical Emancipations.


It is logically imposs ibl e to p redi cate the same conclusion for both o f them . That i s , to do so cause s aprofound fracture o f the fundam ental norms of logic.Such conclusions being thus i l l egitim ate i t i s rather surprising that an e rro r o f th is na tu re should have beenmade . It is perhaps a ccountable for on the grounds ofthe geomete r’s complete insouciance as to how hi spostulate s shall s tand in thei r rel ation to things in thephenomenal world .

It is agreed that a s conven i ent as i s Eucl id’s systemof space-measurement i t i s not by any means con

gruent with the extensi on o f real space obj ects. I tdoes , however, approxim ate congruity with these oh

jects a s nea rly a s poss ibl e . How then could it b e ex

p ected that a system of space-measurement so fa rrem oved from this primary congruence a s the nonEucl idean system is should exhib it more obvious s ignso f correspondence ? But the advocate s o f the n-dim en

sionality of sp ace have i ll a tively a sserted the identi tyof space and its dimensi ons. Acco rdingly, there i s notrecogni zed any distincti on between the i r conception o fspace i tsel f and i ts qual i tative peculi a r ities. They usethe te rms inte rchangeably. So that dimensi on meansspace and v ice v ersa . In thi s lack of di scriminationmay be found the source o f much o f the confus ionwhich attaches to the conception o f space .I f i t were a rguable that the rel ation b etween sp ace

and its dim ensi ons is the sam e as tha t between matte rand its prope rties then the restri ct ion o f this rel at ion tothree and only three directions o f extent would be disallowed ; for the reason that i f, as is commonly done ,dim ensi on be made to mean di rect i on o f extent, therewould be an unlimited number of di rections of extent


common a hitherto unknown property o r extens ion invi rtue o f which i t may be vi ewed as having an un

l im ited number of dimensi ons. To accept the l a tterVi ew is equ ivalent to saying that, in the above figure

,

the three l ines ab,bc and db have formed a tripl e

entente by which they have mutual ly and severally acqui red a new domain , hyperspace , and in which , because of the va st resources of the region , they a re ableto p erform wondrous things'.Let us exam ine bri efly the va rious current defini

ti ons o f dimension . I t i s a ssumed by not a,few that

dimens ion is ‘the same a s direction. But can we grantthi s wholly to be true ? I f so , then a mere chil d maysee that there a re and must necessa rily be a s manydimens ions a s there a re di rections . Primari ly, the re areS ix di recti ons o f space and an unl im ited numbe r o fsubsidi a ry di rect ions. On thi s Vi ew i t i s not necessaryto invent a new doma in o f space i f the obj ect bemerely to di scove r and uti l i ze a greate r number ofdimens ions than ha s heretofo re been allowed. Forthe identification of the term dimensi on wi th di rectiona l ready makes ava ilable an almost infini te number o fdimensions . But thi s Vi ew is obj ected to by the adv ocates , fo r i t i s contra ry to the hypothes i s o f n-dimen

sionality.

D imension a lso means extent . This i s parti allytrue . I t cannot be wholly true . For, i f i t were , then,space would have only one dimension whi ch i s al so notallowable under the hypothes i s . Then the definitionleave s out of account the ide a that space i s at the sametime a di re ct ion or collect ion o f di rections . The termextension i s generic and when appl i ed to sp ace meansextension in al l possibl e di rections and not in any one

DIMENS IONAL ITY 99

di rection . So that i t is not permissibl e to say tha tsp ace extends in this direction or that because it extends in all di rections s imultaneously and equally.Geom ete rs cla im that space is a system of coordi

nate s necessa ry for the establ ishment o f a point-posit ion in it . This View, howeve r, i dentifies space witha system of space-measurement and is there fore faulty.

According to this View there may be a s many spacesas there are systems o f space-measurement and thelatte r may be l imi tl ess . But i f the total i ty o f spacesa re to be Vi ewed as one space then we shall have onesp ace with an indefinite number o f dim ensions ; also anindefinite number o f space measurements which wouldbe confusing . Much , i f not all , of such a system’sutil i ty and conveni ence would b e unava il able o r usel ess . That, too , would be in viol at ion o f the avowedpurpose o f these investigations which is to enhance the

'

util i ty and conveni ence o f mathematic operations .Now it i s evident tha t space is neithe r di rection ,

extension,a system of space-measurement nor a system

of mani folds whose dimensions a re generabl e . And

yards—by which i t is apportioned . And because we

find that the fabri c of space l ends i tsel f accommodatingly to ou r conventional no rms of measurement i snot suffici ent reason fo r identi fying it with these no rms.

Here we have the source of a ll e rro r in mathemati calconclusi ons about the na ture o f space ; because all suchconclusions are b ased not upon the intrins i c na ture ofspace

,but upon a rt ifici al fo rm s which we choose to

impose upon i t fo r ou r own conveni ence . But i t should

b e remembered that the i rregul ariti e s whi ch we note

100 THE MYSTERY OF“

SPACE

a re nof In space itsel f but inhere in the forms which weuse . For these purposes space is extremely elasti c andaccommodates i tsel f to the shape and scope o f anyconstruction we may decide to try upon i t. In thi srespect i t i s l ike water whi ch ha s no regard fo r theshape , s iz e o r kind o f vessel into which i t may beposited. There i s one thing certa in that j udging fromthe above cons iderations there ha s been not yet anyabsolute , all-sat i s fying defin ition devi sed fo r sp ace bymathematici ans.The best defini tions hitherto constructed a re pu rely

a rtifici al and a rb itrary determinations. I t is ratheranomalous that there should be so l ittle unanimi tyabout what i s the most fundamental consi deration o fmathemati cal conclusi ons which a re supposed to be soce rta in , so neces sa ry and universal a s to b e incontrovertibl e . Confessedly, i t i s a condit ion which ra i se saga in the question a s to just what a re the l imits o fmathematical ce rta inty and necessi ty and j ust how farwe shall depend upon the val idi ty o f mathemati cs todetermine fo r u s absolutely certa in conclusi ons aboutthe nature o f space . In vi ew o f the uncerta inty noted,are we justified in fol lowing too closely the mathemati cle ad even in matters o f logi c , to say nothing of ourconception o f space ? I t se ems that we shal l havenecessa rily, on account o f the recogniz ed l imitations o fmathem ati cs in thi s matte r, to turn to some moretenabl e source fo r the norms of our knowledge concern ing space . For in the l ight o f the rathe r indefensible posi tion which m e tageom etricians have involved themselve s there appears to be no hope in thi sdi rection .

It i s undoubtedly s a fe r not to rely altogether upon


gruence between mathemati c and perceptual space tosuch an extent as to cause i t to encroach upon forb iddente rritory. In doing this they have erred gri evously,causing se rious offense to the more sensi tive spi ri t o fthe high-caste mathemati ci ans among whom arenone more truly conservative than PAUL CARUS,2

who says

Metageom e tricians a re a hot-headed race anddi splay som etimes al l the cha racteri s ti cs o f sectari anfanati cs . To them i t i s quit e cl e a r there may be twostra ight l ines through one and the sam e point whichdo not co incide and yet a re both pa ralle l to a thi rdinc .

”

To the s tudent who ha s ca re ful ly followed the dev elopm ent of the non-Eucl ide an geometry and thenotion of hyperspa ce the above characteri za tion isnone too severe no r i ll-deserved. Nothing could morevividly yet correctly portray the impious tacti cs o f them etageom etrician and establ ish hi s perceptua l obl iquitymore surely than the mere fact, mentioned byCARUS

,that he can with evident lack ,of menta l per

turbation procla im that two stra ight l ines , noncoincident with each other, may pass through a point andyet b e parallel to a third l ine. But thi s is a mere trifl e ,a bagatel l e , to the many other infractions o f whichhe i s guilty. The wonder i s that he i s abl e to securesuch obsequi ous acceptance o f his offe rings a s many ofthe most s erious minded mathem atici an s a re incl inedto give . IS i t to be wondered at that , despite thepro fuse p rotestati ons o f the advocate s , many whot ake up the study of the question o f hyp erspace should

’ Vide Monis t, Vol. XIX, p . 402

DIMENSIONALITY 103

experi ence a deep revuls ion from the posture assumedby m etageom e tricians with respect to these querie s ?

L inked with the ide a o f dimensi onal i ty i s thenotion that space i s infini te . This i s a conception whichhas it s roots imbedded in the depths o f antiquity.

Primitive man , looking up into the heavens at whatappeared to him as a neve r ending extensi on , wasawed by i ts vastness ; but the minds o f the mostl e a rned o f the present-day m en a re not free from thisinnate dread o f infinity. I t perm eate s the thought l i feo f al l a l ike and none seems to be abl e to rise above it.M athematici ans , philosophers , scientists all share inthe general bel i e f that space i s without l imi t

, unend

ing i n extent and eternally existent. RIEMANN,whose

thought l i fe found i ts most conveni ent mode of expression by means o f pure mathematics, was the fi rst in thehi sto ry o f human thought to surm ise that space i s notinfinite but l imi ted even though unbounded. But hi sconception has been much viti ated on account o f itsentanglement with an idealized construction by whichspace i s regarded as a thing to be manipulated andgenerated by act o f thought. Were i t not fo r this hisconception would indeed mark the beginning of a newera in psychogenes is. As i t is, when all the nonsensi ca leffusi ons have b een cleared away from our space concep tions and men come really to understand somethingof the essenti al nature o f spa ce thi s new e ra will findi ts true beginnings in the mind o f R IEMANN . Al thoughi t must be sa id, a s i s the cas e with al l progress ivemovements

,the l ate r development o f a rationale for

thi s conclusi on will va ry greatly from his original concep tion. For he had in mind a sp ace that i s generableand the re fo re a logical constructi on while ultimately


the mind will swing back to a consideration o f realSpace.

Al ready men a re beginning to see a new l ight.Already they a re beginning to take a new vi ew of sp acein general . The depa rture i s e speci ally noti ceabl ein the atti tude a ssumed by HIRAM M . STANLEY.

‘

He s ays :

I f we seek the most sati s factory unde rstanding o f space we shal l l ook neither to mathem ati csnor Psychology but to Physi cs . The trend ofPhys ics , say with such a representative a s OSTWALD,i s to make things the expressi on o f force ; the constitution and appearance of things a re determinedby dynamism ; and we may best inte rp ret Space a s amode o f thi s dynamic appea rance .”

Space , a s a mode of dynamic appearance i s a Sl ightimprovement upon the old idea o f a pure va cu ity ; forin the l ight o f what we now know about space content much o f the dignity o f that v 1ew ha s been lost.Men now know that sp ace is not an empty void. Theyknow that the atmosphe re fi lls a great deal o f space.They also have extended the ir conception in thi s direction to include the ethe r and occult ism goes further andpostul ates four kinds of ether—the chemical , l i fe , l ightand psychographi c ethe rs . But it does not stop here .I t postul ate s a serie s o f grades o f fine r matter thanthe physi cal which fi l ls space and permeates its enti reextent even to identifi ca tion with i ts essenti al nature .STANLEY continues :

Everything does not, a s commonly conce ived,fall 1nto some pre-exi stent space convenient for It ;

‘Philosophical Review,Vol. V II


whole . Yet it i s transfinite in the sense that i t transcends the comprehension o f finite minds o r processes .It i s finite because i t i s in manifestation. Everythingthat i s in mani festation i s finite . The infinite i s not inmani festat ion . Infinity ha s to be l imi ted always tobecome mani fest . The De i ty has l imited His being ino rder that there may be a mani fe sted universe . Al lthings , a ll appearance s a re finite ; because they a rephenomena connected with mani festation .

Thi s question may be viewed from another standpoint. Al l things in mani fe stat ion o r exi stence a repol a r in thei r constitution. For instance : there cannotbe a “here” without a “the re .” There cannot be anuppe r” without a lower.” Right i s copola r withwrong ; good i s copola r with evil ; night with day ;mani festati on with non-mani fe station ; t ruth wi thfal si ty ; infin ity with finity and so on , throughout thewhol e gamut o f the pa i rs o f oppos i te s . What i s thelogical in ference ? Space i s p a i red with a l ack ofspace . There cannot be what we cal l space withoutthere being at th e same time the possibil ity, a t l e a st,o f the lack of space or space lessness. Thi s i s a conclusion that i s rigorously logical and incontrovertibl e .But i t has been urged that i t i s impossibl e for the

mind to imagine a condition where there i s no space .I t even has been a sse rt ed th at i t i s contra ry to theconsti tution o f the mind i tsel f to imagine “no space .”

But whether im aginabl e o r not ha s no effect whateve rupon the val idity o f the conception . Nei ther, i t i s sa id,can we imagine a fourth dimensi on but the mind ha scome dange rously nea r to imagining i t. The di stancefrom excogi tating upon , discus sing and describ ing theproperti es of four-space to imagining it i s not so grea t

DIMENSIONALITY 107

a fter all. Truly it i s diffi cult indeed, i t s eems , to b eabl e to describe a thing yet not be able to imagine ormake a mental image o f i t . There i s an evident fa ll acyhere . Eithe r the description o f four-space i s no description a t all o r i t i s a true del ineation of an ideal izedconstruction which is well within the mind’s powers o fimagination . Indeed the question o f imaginabil i ty i snot dete rminative in i tsel f for what the mind may nowbe unable to imagine , because o f its more o r lessnebulous character , and owing to its in fancy may inthe course of time be eas ily accompl ished.

The unive rse i s a compacted p lenum . It i s chockfull o f mind, o f l i fe , o f energy and m atter. Thesefour a re bas ically one . They exist, of course , in va rying degrees o f tenui ty and intens i ty and answer to a

wide range o f vib rations . Together, in thei r manifestation o f action and intera ction, in the i r dynamicapp earance , i f you please , they consti tute space . I f thesewere rem oved wi th all that thei r exi stence impl i es therewould re sult a condition o f spacelessness in whi ch noone o f the appea rances which we now perceive wouldbe possibl e . Even sheer extensi ty would be non

existent. Al l scope o f motil i ty would be l acking. D im ension, coOrdinates, di rection , space-rel ations—all

would be imposs ibl e .A stra ight l ine i s an ideal constru ct ion o f the mind .

It does not exist in na ture . I t can never be a ctual izedin the phenomenal unive rse . Between the ideal andthe real

,o r a ctual , there is a kosm i c chasm. It

broadens o r narrows according a s the phenomenal ap

p earance approaches o r recedes from the ideal . What ,the refore

,can b e postul ated o f the one wil l not apply

W i th equal force to the other . They a re not congru


ent and can never be in the a ctual i zed universe . Themoment the actu al becomes i dentified with the ideali t ce ases to be the actu al . The universe does not exi stas pure form ,

nei ther does space . As purely formalconstructions o f the intell ect these can have no percep tible exi stence . The phenomenal o r sensibl e maynot be judged by exactly the same standard a s theformal. The phenomenal or sensibl e represents thingsas they appea r to the senses , or, so fa r a s the actual i zeduniverse i s concerned , as they r eally are . The formalrepresents things a s they a re made to appea r by themind. I t cannot be actual ized . It may be sa id thatthe purely formal is the l imit o f evolution. The phenom enal may approach the ideal a s a l imit , but cannever becom e ful ly congruent with i t. The diferencebe tween the ideal and the actual is a dynamic one ;

i t is by vi rtue o f this diffe rence that the universe isheld in mani festation. Evolution i s the decrement o fthi s diffe rence b etween the purely formal and thea ctual . So long then a s a kosmic diffe renti al is maintained the phenomenal continues to be mani fest : wheni t is finally reduced to nothing i t goes out o f manifestation. The phenomenal is finite ; the ideal infinite .Where fore , i t is undoubtedly improper to re fe r to

space as be ing infini te . The term really i s inappli cable . T ransfinity is much better and more accurate .Space i s t ransfinite because i ts scope i s greate r thanany fini te scop e o f motil i ty can encompass , because i texceeds finite comprehens ibi l i ty.

RIEMANN’S notion that space i s l imited gainsweight in the l ight o f the foregoing considerations .But he coul d not conceive o f the l imitabil ity and un

boundedness o f space a s such in its pure essence ; but

1 10 THE MYSTERY OF SPACE

Space , says PAUL CARUS , is the possibil i tyo f motion in all di rect ions.

” 4 To be sure , i t i s admitted that space offers opportuni ty for motion in alldi rections. But is space this opportunity of moti l ity?Or is possib il ity of motion space ? The possib il ity o fmotion must rest in the thing that moves . It impl iesa poten cy in the moving enti ty, not in space . I f i t i smeant that sp ace i s the potency that resi des in the moving element i t i s sti l l more diffi cult to understand theconnota tion . But even granting thi s vi ew, a re wenot compell ed to recogni z e the dynam ism of spaceas a necessary inference ? Anothe r defini t ion whichCARUS gives i s tha t space is a “

pure form of exten

sion.

” I f i t b e granted that space i s a pure form ofextension we should have to conclude that i t ha s noactual existence ; fo r pure form does not exi st exceptas an i deal ized construction . I t cannot b e found innature . Pure form i s ideal. Impure o r n atural formi s actual . There fo re the space in which we l ive andin which the unive rs e exi sts cannot be a “pure formbecause l i fe cannot exi st in the purely formal . It isusel ess to talk about space a s mere form so long as

i t mainta ins l i fe . The difliculty which this phase o f thequestion presents i s another evi dence o f the inade

quacy o f our definit ions .I t i s a l so found to be imposs ibl e to concur in

CARUS’ conception o f knowledge a p riori. H is notionof the a priori vari es somewhat from the Kanti an vi ew .

H e defines i t a s an “i deal i zed construction ,” the “mind

made ,” “abstract thought ,

” and places i t in the samecategory a s a concept. This i s undoubtedly born ofhi s des i re to get r id o f KANT’S “ innate idea s which

‘V ide Foundations of Mathematics, p . 107 .

DIMENS IONAL ITY 1 1 1

seem to be di stasteful to him . But in doing so it

appears that the real a p riori has been overlooked.

Let us examine fo r a moment this important question .

The a posteriori connotates all knowledge gainedthrough the senses, or sense experience . All knowledge there fo re whose origin can be traced to the sensesi s knowledge a posteriori. Now , knowledge a priorishould be just the opposi te o f this. I t Should indicatesuch knowledge a s that which does not have its originin the sense s , o r which is not dependent upon the ordina ry avenues of sense-experience . Abstract thought i sa s truly experi ence as smell ing, see ing o r hearing. I ti s by traversing its scope o f moti l ity that the mind findsout what the norms o f logic a re . It could not remainqui escent and discove r them . I t has to b e active , examining, comparing and j udging . Alm ost the enti rerange o f thought, its enti re scope , is characteri zed bythe a posterioristic m e thod. In fact, al l thought i sa p osterioristic. D espite the fact that, in thinking inthe abstract

,i t i s necessa ry mentally to rem ove al l ele

ments of concreteness, all materi al i ty and all actual ity,the conclusions reached have to b e re ferred to thestanda rds mainta ined by the actual , the concrete andthe materi a l . Then we do not start with the abstractin our thinking. We begin with the concrete and bymentally removing al l physical qual i ti es a rrive at the

abstra ct.The mind ha s a consti tution . It acts in a given

way because i t i s i ts na tu re so to act. Not be

cause it has l ea rned to act in that manne r . It performs certa in functions intu itively without p revious in

struction o r experi ence fo r the same reason that water

dampens o r heat warms . It i s na tu ral fo r it to do so.


This naturalness , thi s performance o f function without be ing taught o r without experi ence constitute thep rincipl e o f apriority in the mind . Aprioriness i s ap rincipl e o f mind partaking o f the very nature andessence o f mind . It i s the very mainspring of men

tal i ty. Pe rception and conception a re processes whichthe mind performs intu it ively. The mind perceivesand conceive s because i t i s impossibl e for the normalmind to do otherwise . We take a vi ew upon a givenquestion ; we assume certa in mental a tti tudes o faffi rmation , negation o r indifference because we havele arned to do so by vi rtu e o f the tuitiona l capabil i tyo f mind . These describe the a posteriori. That is,all knowl edge obta ined as a result o f volunta ry menta lprocesse s constitutes the mass o f knowledge a pos

teriori. The a priori i s what the mind is by na turethe a posteriori is what the mind becomes. I t i s themind-content.The a priori i s not a mental construction ; i t i s an

essent i a l p rincipl e o f mind. I t should not be identified with the “purely formal ,

” a s i s done by PAULCARUS ?He s ays

The a priori i s identi cal with the purely formalwhich originates in our mind by abstract ion . Whenwe l imit our a ttention to the p urely rel ational , drop

pl ingal l other fea tu res out of s ight, we produce a

eld o f abstraction in which we can construct purelyformal combination , such a s numbers , or the ideasof typ es and speci es . Thus we create a world o fpure thought which has the advantage o f being appl icabl e to any purely formal cons idera tion and we'Vide Foundations of Mathematics, p . 42.


congruence fo r al l o f the possib i l i ti es held out by thepurely formal .As an ete rna l p rincipl e o f mind , the a priori i s in

agreement with the divine mind o f the kosmos. In itsap osteriority the mind is o f diverse tendences , qual iti esand characterist ics. Ap riorily, i t a cts in unison withthe eternal purpose o f l i fe and the universal m ind.

In its ap ost eriority, i t o ften goes awry. In i ts apriorityi t can never be insane ; insani ty is a symptom of themorb id a p oste riori.The mind in man acts the same as mind in the

vegetal and low e r anim a l kingdom s. M etabol ism and

katabol ism , indeed al l cell-activi ty, a re a priori pe rform ances o f the m ind. G rowth and all its phenom ena, the cycl i cism of natural processes, and everyactivi ty connected therewith belong to the catego ry o fthe a p riori. Cells mul tiply, divide , bui l d up and tea rdown tissues and they do i t intui tively. Most certainly these functions a re performed without any as

sistance from th e intell ect . Al l the myri ad activiti esin na ture with which the intellect in m an has not thesl ightest conce rn, truly a cting in a ccord with som eprimordial impetus, are a ctivi ti es a p riori.

Now what is the att itude o f the intellect, i n thel ight o f the a priori, towards space and the questi ono f dimensi onal ity? It is evident that no matte r whatthi s attitude m ay be it is in agreem ent with the constitution o f things and o f the unive rse . And i f so , i ti s right and without illusi on. It i s also evident thatwhatever notion a p ost eriori the intel lect may entertain with respect to these questions i s unavoidablyl i abl e to the i l lusi ona ry drawbacks comm on to conelusions based upon l imited expe ri ence . The geometri c

DIMENSIONALITY 1 1 5

View of space belongs to the ca tegory o f the a post eriori. Hence i t is subj e ct to the usual imposi tion o fe rro r.

T ers ely stated , KANT’S view o f space i s that i t i s

a fo rm of intui tion, a form a priori, a transcendenta lform . As such he cons idered i t to be a na tive form ofpercept ion not belonging to the category o f sensedel ive ri e s . Acco rdingly, space i s a form of intuit iona ri s ing out o f and inhering in the constitution of mind.

I t i s a notion which constitutes the universal ande ternal p rerequi s i te o f mind and is, there fore , intrinsically necessa ry to a ll phases o f m entation. Now,this be ing tru e just what may be sa id to be the rel ationo f dimensi onal i ty to thi s a p riori fo rm of space whichi s found to exi st in the mind as an eternal aspect o fi ts nature ? Does the mind intu itively measure it s contents or i ts operations by the empirica l standard o fspace-measurement known as dim ensi on ? Is the a ttitude o f the mind towards the obj e ct ively real one o f discrimination a p riori a s to the di rection or dim ensi on inwhich a percept may originate ? In other words, doesthe mind habitual ly and intui tively re fe r i ts data to asystem of coOrdinates for final determ ination? Thereis no othe r answer but that the m ind makes no suchrefe rence and i s dependent upon no kind o f coOrdinatesystem in any o f its operations a p riori. As a formof intui tion

,the space notion is present in the mind

as a scope o f existence , of moti l i ty, o f being and ofsheer room iness. The notion o f di recti on or dim en

si on, being an a rt ifici al const ruction, does not ente r intothis fo rm of in tuition a t all . It is only when the mindcomes to el abo rate upon i t s pe rceptive performance sand poss ib i l it i e s that the questions o f relations , posi


t i ons and di rect ions a ri se . But thi s l a tte r i s a matte rsep arate and di st in ct from the state o f awarenesswhich embodies the notion of space .D imension i s an arbitrary norm constru ct ed by the

mind fo r the dete rmination o f various posi t ions insp ace . It i s an accident o r by-product o f the p rocesso f el aborative cognition , a conveni ent and appropri atemeans o f mea surement fo r obj ects in space and thei rspace-relations . But it i s no more a priori than a footrule o r a square . But being purely an empi ri ca l p roducti t may be sa id to b e an a spect o f p sychogenesi s becausei t relate s to the evolutiona ry aspect o f mind. Theassumption may there fore be al lowed that the mindmay, in the course o f i ts evolution , find i t conveni entand approp riate to devise an additional ordinate o rdimensi on to sati s fy the necessi ties o f i ts more com

plex ramifications into the na ture o f things and to determine the i r greatly increa sed space-rel ations . It may

be even possibl e fo r the mind to function normally ina spa ce o f four dimensions . But thi s would simplybe a new adj ustment, not a change in the essenti a lna ture of mind. I t would be like the seri es o f adjustments to envi ronments which man has made in the onward movement of civil iz ation . There has been noserious change in the manhood p er se o f man . Thatha s remained the same ; there has been merely a compl i cation o f envi ronmental influences. Simil a rly, in thea cqui sition o f four-dimensional powers, granting thatsuch an acqui siti on is possible , there i s nothing to beadded to the aprioriness o f mind itself. I s i t not,there fore , logi cal to a ssume that the discove ry o f afourth coc

’

irdinate and the consequent concep tualiza

ti on o f the same , point to the development in the mind

CHAPTER V

THE FOURTH D IMEN SION

Ideal and the Representative Nature of Objects in theSensible World—The Psychic Flux ional the Basis ofMental Differences—Natural and Artificial SymbolsUse of Analogies to Prove the Existence of a FourthDimension—The Generation of a Hyp ercube or T esseract—Possibilities in the World of the Fourth Dimension

Some Logical Difficulties Inhering in the Four-Space Con

cep tion—The Fallacy of the Plane-Rotation HypothesisC. H. HINTON andMajor ELLIS on the Fourth Dim ension.

THE world o f mathesis i s t ruly a marvelous domain . Vast a re i ts possib il it i es and vaste r sti ll i tssweep o f conce ivabi l ity. It i s the kingdom o f themind where, in regal fre edom , i t may perform featswhich it i s imposs ibl e to actual i z e in the phenomenaluniverse . In fact, there is no necessi ty to consi de r thelim itati ons imposed by the actual iti e s o f the sensuousworld. Logi c i s the a rchitect o f thi s region, and forit there i s no l imi t to the admissib il i ty o f hypotheses .These may be multipl i ed at wi ll , and legitimately so .

The chi e f e rror l i es i n the a ttempt to make themappear a s a ctual facts o f the physi cal world .

M athematici ans, spe cul ating upon the possib i l i t ie so f matheti c constructions and fo rgetting the necessa rydi stinctions which should be recogn ized a s differentiating the two worlds, in the i r enthusi a sm have been

1 1 8

THE FOURTH DIMENSION 1 19

led into the e rror o f postulating a s qual iti es o f thephenomenal world the characteri sti cs of the conceptual .Accordingly, a great deal o f confus ion a s to the properl imits and restricti ons o f these conceptions has ari senand there still may be found those who a re enthusiastically endeavoring to push the actual itie s o f thephysi ca l ove r into the conceptual. But in assumingany a tt itude towards matheti c p ropositions , especi allywith a vi ew to demonstrating the i r actual i ty, very careful di scrimination a s to the essenti al qual itie s and thei rconnotations should be made. Hence , before takingup a bri e f study o f the fourth dimension p roper, i t i sdeemed fi tting to indicate som e of the fundamental distinctions which every student o f these questions shouldbe abl e to make with re ference to the da ta which hemeets .All obj ects o f the sensibl e world have both an os

sential or ideal nature and a representative o r sensuousnature . That i s , they may be studied from the standpoint of the ideal a s well a s the sensuous . The rep t esentativ e nature is that which we recognize a s the modeof appearance to ou r senses which , a s KANT held , i snot the essential o r i deal chara cte r o f the thing itsel f.For there i s quite a s much difference between the sensuous pe rcept and the real thing i tsel f a s between anobj e ct and its shadow. In fact, a concept vi ewed inthi s l ight

,may b e seen to have all the cha racteri stics

o f an ordina ry shadow ; fo r instance , the Shadow of atree . Vi ew i t as the sun i s ri sing ; i t will then b eseen to appea r very much elongated, becoming l ess inl ength and more dist inct in outl ine a s the sun ri ses toa posi tion di rectly overhead. The elongation mayagain be seen when the sun is setting. Throughout the


day as the sun assumes different angles with referenceto the tree the proportions and definiteness of theshadow vary accordingly. Thus the angul a rity o f thesun, the intensi ty and fullnes s o f the l ight and the shapeand siz e o f th e tree operate to determine the cha racter o f the shadow.

Much the same thing i s true of a sensuous representation. I f we examine ca re fully our ide as of geometri c quanti ti e s and magni tudes , i t wil l b e found thatthe concepts themselves a re not i denti cal with theobj ects o f the physi cal world, but mere mental shadowso f them . The angula ri ty o f consciousness , or the distinctness of one’s state o f awa reness, being analogousto s imilar attitudes in the sol ar influence are the maindeterminants o f the characte r o f the mental shadowor concept. Where fore mathemati cal “spaces” or

magnitude s are not sensuous things and have therefo re no more real exi stence than a shadow, andstri ctly speaking not as much ; fo r a shadow may beseen , whil e such magnitude s can only be conce ived. I tmay be u rged that s ince we can conce ive o f such thingsthey must have existence o f some kind. And so theyhave , but i t i s an existence o f a different kind from thatwhich we recogni z e a s belonging to things in the sens ible world. They have a conceptual exi stence , but nota sensuous one . There in l i e s the gre a t diffe rence.To b e sure , a shadow i s a more o r les s true rep

resentation o f the thing to which i t p e rtains. Thatthi s i s t rue can b e establ i shed empi ri cally. Simil a rly,the degree o f congru i ty b etween obj ects and conceptsl ikewi s e may be determined. I f this were not truewe should be ve ry much di sappointed with what wefind in the phenom enal world and could neve r be


cept and the thing i tsel f, actual physi cal contact wouldnot be necessary. For one coul d rely wholly upon thesense-del iveri e s and e ach sense might operate enti relyindependently o f al l the others a s there would be nonecessity to corre ct the del ive ry o f one by those o f theothers . This , o f course , ra i ses the question as to thenecess i ty o f sense-exp eri ence a t al l unde r conditionswhere there would b e no dispa rity between the thingi tsel f and the ideal rep resentation o f i t in the mind.

The absence of thi s va ri abl e quanti ty would open tothe mind the possib i l i ty o f really knowing the essent i a l na tu re o f obj ects in the phenom ena l world , a condition o f affa i rs which i s a dmittedly now without therange o f the powers o f the mind .

At any rate , the essenti a l thingness” o f obj ects

can neve r b e comprehended by the mind unti l thediminution o f thi s dispa rity between the obj e ct o f senseand the mental p i cture o f i t which exi st s in the consciousness has p roce eded to such a l imi t a s e ithe r com

ple tely to have obl i terated i t o r to such an extentthat the psychic fluxion is so sl ight a s not to matter.I t i s bel i eved that the results o f mental evolution ,

a s the mind approaches the t ransfinite a s a l im i t, willoperate to minim i z e the fluxional quantity which sub

s ists between a ll obj ects o f sense and thei r i deal rep resentation a s data o f consciousness. The conclusi onthat the mind o f e a rly men who l ived hundreds o fthousands and perhap s mill ions o f years ago on thi spl anet consumed a much longer time in learning theadj ustments between the obj ects which i t contact ed inthe sensuous world and the el ementa ry representationswhi ch were registered in i ts youthful consciousness

than i s to-day requi red fo r s imila r p rocesse s s eems to

THE FOURTH DIMENSION 123

be demanded , and substanti a ted a s well , by what i sknown of the phyleti c development o f the mind in thehuman race .In view o f the above , i t i s thought that the dura

tion o f such simpl e menta l processes served not only top rolong the physi ca l l i fe o f the man o f those ea rlydays, but may also account fo r the pueril i ty and incap aci ty of the mind at that stage . Not that the slowmental p rocesse s were active causat ive agenci es inl engthening the l i fe o f man , but that they togetherwith the crass physi cal ity o f man necess ita ted a longerphysi ca l l i fe . This , perhaps in a larger sense thanany other consi deration, a ccounts for the fundamentaldiscrep ancies in the mind o f the p rimitive man in compari son with the effi ci ency o f the mind o f the presentday man . In vi ew o f the potenti al cha racte r o f mindand in the l ight o f the w ell graduated scal e o f its accomplishm ents, i t i s undoubtedly sa fe to conclude thatthe qual i ty o f mental cap aci ti es is propo rtional to thepsychic fluxional which may exist at any t ime betweenthe ideal and the essenti al o r re al . M enta l differencesand potenti al iti e s in general may be due to the magnitude o f the psychic fluxional or diffe renti a l that exi s tsbetween the conceptual and the perceptual universe .In some minds i t may be greater than in othe rs. Thechasm between things-in-themselves and the menta lnotion perta ining thereto may va ry in a di re ct ratio tothe individual mind’s place in psychogenes i s , and therefore

,be the key to all mental differences in this respect.Most certa in i t is that there may be ma rked fluctua

tions in the judicial approach o f minds towardsany psychic end. In other words, there i s not only a

fluxional o r differenti al between the obj e ct and its rep


resentation, but also a differenti a l between the ap

pro ach of one mind and another in the judici al determination of notions conce rning ideas . In this way,differences o f opin ions a s to the right and wrong o fj udgments a ri s e. Indeed, there s eem to be zoneso f affi nity for minds o f s imilar characte ri s ti cs , orminds that have the same degree o f differenti al ; so

that , i n choosing among the many poss ibl e j udgments p redicabl e upon a specie s o f data , al l thoseminds having the s ame degree o f psychi c differenti al discove r a speci a l affin i ty o r agreement amongthemselves. Hence , we have cults, schools o f thought,and various other sect ional bodi e s that find a ba si s o fagreement for the i r operations in thi s way. The outcome o f thi s remarkabl e intellectu al phenomenon i stha t there a re a s many diffe rent kinds o f j udgments a sthere a re zone s o f affinity among minds . Vari ous systems o f philosophy owe thei r exi stence to these cons ide rations , and the considerations themselves flowfrom the fa ct that all intell ectual ope rations a re essentially superfi ci al ; because there i s no means by whichthey may penetra te to the ste ady flowing st ream ofreal ity which pervade s and susta ins obj ects in the sen

‘

s ibl e wo rl d.

In View, there fore , of the foregoing and withspeci al re fe rence to geometri c constructions , i t is necessary in approaching a study o f the four-space that i t beunderstood at the outset tha t the fourth dimension canneither be actual i zed nor made obj ect ively poss ibl eeven in the sl ightest degree in the perceptua l world ;because i t belongs to the world o f pure thought andexists there a s an “extra personal affa i r,

” s ep arate anddi stinct from the world o f the senses .


rea soning will , there fo re , fai l ; because the basi s o fsuch a rguments i s i tsel f faulty. In the sentence : “Theman loves hi s bottl e ,

” the thing meant i s not the bottl e ,but what the bottl e conta ins . For the purpose o f thefigure the bottle s ign ifie s its contents . There i s nomore re al connection between the bottl e and what i tconta ins than between any word and the obj ect fo rwhich it stands . Words a re s a i d to be symbol s o fidea s . But they a re not natural symbol s ; they a reconventional symbols , made fo r the purpose o f cataloguing, indexing and systematiz ing ou r knowledge .Words can be divorced from idea s and obj ects , o rrathe r have never had any real connection with them .

There a re two classe s o f natural sym bols , nam elyobj ects and ideas. These , obj ects and ideas , symbol iz ereal i ti e s . Real i ti e s a re impercept ibl e and incomp rehensible to the intell ect which ha s apti tude only fo ra slight comprehension o f the symbol s o f re al it i e s .For instance , a tre e i s a natural symbol . I t repre sentsan a ctual ity which i s imperceptibl e to the intel l ect . Theintel l ect can deal only with the sensibl e symbol . It i sa na tura l symbol ; because i t i s poss ibl e di rectly to tracea l iving connection between the tree and the tre e

reality. That i s , i t would b e poss ibl e so to trace outthe vital conn ection between the tre e and its re al i tyi f the intelle ct had aptitude fo r such tracery. But , inreal i ty, s ince i t ha s no such aptitude , i t rema ins fo r thework of that highe r fa culty than the intell ect whichrecognizes both the connection and the intell e ct’s inab il ity to trace i t. Furthe r, an obj ect i s call ed anatural symbol because i t i s the b ridge between sensuous rep resentation and real i ty. It i s a s i f one couldbegin at the surface o f an obj ect and by a subtl e p roc

THE FOURTH D IMENSION 127

ess o f el imination and excorti cation a rrive a t the hea rto f the universum of real i ty . No such consummationmay be reached by deal ing with words which havemerely an a rtifici al relationship with the obj e cts whichthey s igni fy. Aga in , i dea s , that i s , i dea s that a re universal in appl i cat ion and have thei r roots in the greatocean o f real i ty, a re n atural symbol s ; because i f i t werepossibl e to handle an ide a with the physical handsi t would be possible to a rrive a t the hea rt o f that

;

which it symbol i zed without ever los ing our connection with the idea i tsel f. In other words, i deas andobj ects , unl ike words , can neve r be divorced from thatwhich they sym bol i z e . Both , being of the same class ,a re the opposi te pol e s o f re al iti e s . This then i s thediffe rence between natural symbol s and artifici al symbol s—th at a natural symbol , such a s obj ects and ideas ,i s copola r with real ity wherea s an artifici al symbol ,such a s words , geometri c constructions and the l ikenot only l acks thi s copolarity but i s itsel f a symbolo f natural symbol s .It is, therefore , inconce ivabl e that because the

algeb ra i c quanti ty A3 has been arbitra rily decreed tobe a representation o f the volume of a cube , eve rysuch quantity in the algebra i c series shal l actuallyrepresent some obj ect o r set o f obj e cts in the physicalworld . Even i f i t b e granted that such may be theca se , i s i t not certa in that there i s a l imi t to thingsin the obj ective universe ? Yet the re may not be anyl imi t to algebra i c o r mathemati cal dete rminations .The materi al universe i s l imited and conditioned ; theworld o f mathesis i s unl imited and unconditioned saveby its own l imita tions and conditions . It i s i rrational

128 THE MYSTERY OF SPACE i

to expect tha t phys ical phenomena shall justi fy all

mathemati cal p redicate s .There is p erhaps no single mathematical desider

atum or consideration which may be said to be the

na tural symbolism of realities; for the whole of mathe

matical conclusions is a mass of ar tificial and arbitrary

but concordant symbols of the crasser or ne ther p ole

of the antip odes of realism . I t i s exceedingly dangerous , there fo re , to predicate upon such a fa r-fetchedsymbol ism as mathem ati cs furnishes anything purporting to deal -with ultim ate real itie s . And those whoinsist upon doing so a re e ither bl ind themselves tothese l imi tations or a re madly endeavoring to be fogthe minds o f others who a re dependent upon themfor leadership in questions of mathem atical import.Analogies have been unsparingly used in efforts to

popul ari ze the four-space conception and much of theviolence which has been done to the notion i s due tothis vagary. The mathemati cal publ i cist, in trying togive a mental p i cture o f the fourth dimens i on , exam inesthe appearance s o f th ree dimensi onal be ings a s theymight appea r to a two dimensi onal be ing or duodim .

He imagines a race o f beings endowed with al l thehuman facult i e s except that they l ive in a l and of buttwo dimensi ons—l ength and breadth. He thinks o fthem as shadows o f three dimensi onal be ings to whomthere a re no such conceptions a s “up and “down .

”

They can see nothing nor sense anything in any waythat i s without thei r pl ane . They can move in anydirection within the pl ane in which they l ive , but canhave no idea o f any movem ent tha t might carry themwithout that pl ane . A house fo r such beings mightbe simply a seri e s o f rectangles. One of them might


strongly sealed chests o f trea sure s would be e asi ly andenti rely a t h i s disposal . No ci ty pol ice , Scotland Ya rddetective nor gendarme could have any terrors fo rhim . DRS. JEKYLL and MESSRs. HYDE might aboundeverywhere without fea r o f detect ion. Obj ects a swell a s persons might b e made to pass into o r out o fclos ed rooms “without penetrating the walls ,

” thusmaking escap e e asy fo r the impri soned. No tridimen

sional state , condition o r system of a rrangements ,accordingly, woul d be sa fe from the ravages o f evil lyincl ined four dimensi onal entiti e s . Obj ects that nowa re l imited to a po int o r l ine rotation could in thefourth dimension rotate about a plane and thusfurthe r incre ase the perplexiti e s o f our engineeringand mechani ca l p robl ems ; four l ines coul d be erectedperpendicul a r to each othe r wherea s in three spaceonly three such l ines can b e e rected ; the right handcould be maneuve red into the fourth dim ensi on andbe recove red as a l e ft hand ; the mysteri e s o f growth ,decay and death would find a satisfactory explanationon the basis o f the fourth dim ensi onal hypothesi s andmany, i f not all , of the perplexing problems o fphysi ology, chemistry, physi cs, a stronomy, anthro

pology and psychology would yi eld up thei r mysteri esto the skill o f the fourth dimensional operator.Marvelous possib i l i ti es the se and much to be desi red !But the most remarkabl e thing about these so-call edpossib i l i ti es i s thei r impossib il ity. I t i s thi s kind oferrati c re asoning that ha s b rought the conception o f afou rth dimension into general disrepute with the p opul a r mind . It is to be regretted

,too

,fo r the notion

i s a perfectly l egi timate one in the domain o f mathesiswhere i t o riginated and rightly belongs .


It i s not to be wondered at that metageometricians

and others should at fi rst surmise that, in the fourspa ce , they had found the key to the deep mysteri eso f nature in all b ranches o f inqui ry . For so vast wasthe doma in and so marvelous were the possibil i ti e swhich the new movement revealed that it was to beexpected that those who were privileged to get thefi rst gl impses the reo f would not be able toreal iz e fullythei r s ignificance . But the stound of thei r minds andthe attendant magnification of the el ements which theydi scove red were but incidents in the l a rger and morecomprehensive process of adjustment to the great outstanding facts of psychogenesis which i s only fa intlyfo reshadowed in the so-call ed hyperdimensional . The

whole scope o f inqu i ry connected with hyperspace i snot an end in itsel f . It i s merely a means to an end.

And that i s the preparation o f the human mind forthe inborning o f a new facul ty and consequently morel argely extended powers o f cognition . Metageo

metri ca l discoveri es a re there fo re the excrescences o fa deepe r, more significant world process o f mental unfoldm ent . They belong to the matutinal phenomenaincident to thi s new stage o f mental evolution . A l lsuch investigati ons a re but the prel im ina ry exerciseswhich give bi rth to new tendenci es which are destinedto flowe r forth into additional facul ti e s and capacitie s .So that i t is well that the evolutionary aspect o f thequestion b e not overlooked ; fo r there i s danger of thison account of the magni tude and kosmic importanceo f i ts scope o f motil ity.

A geometric l ine i s sa id to be a space o f one dimen

si on . A plane i s a space o f two dimensions and acube , a sp ace o f three dimensions . In figure 7 below,


the l ine ab is sa id to be one dimensional ; because onlyone coOrdinate i s necessary to locate a point-posi tionin i t. The plane , abcd, figure 8 , i s s a id to be two dim ensional becaus e two coOrdinates, ab and db a re requi red to locate a po int, a s the point b. The cubeabcdefgh, figure 9 , is sa id to b e tridimensional , because ,i n orde r to lo cate the point b

,fo r instance , i t i s neces

s a ry to have three coOrdinat es, ab,be and gb. The

tesseract i s sa i d to b e four dimensi onal , because , ino rder to lo cate the point b

,in the tesseract, i t i s nece s

sa ry to have four coOrdinates, ab, bc, bb' and h’

b,fig

ure 10 .

FIG . 7 .

FIG. 8 .

I t wil l b e noted that in figures 8 , 9 and 10, theel ement o f pe rp endicul ari ty enters a s a necess a ry determination . In figure 8 , the l ines ab and bd a re p er

p endicular to e ach o the r. Simil a rly, i n Fig. 10 , l inesab

,bc

,bb

'

and h’

b a re perpendi cul a r to one another .That i s , at thei r intersections, they m ake right angl es.

Simil a rly, figures representing any number o f dim en

si ons may be constructed.


The line ab represents a one-space . An enti tyl iving in a one space i s call ed a “unodim .

” The pl ane ,abcd

,repre sents a two-space , and entiti e s l iving in such

a sp ace a re call ed duodims. The cube , abcdefgh,represents a three-space and enti ti e s inhabiting sucha space are call ed tridims. Figure 10 represents afour-space , and i ts inhab itants a re cal led quar todims.

Each o f the above-ment i oned spaces i s s a i d to havecerta in l im itations pecul i a r to itsel f.The fourth dimens ion i s s a id to l i e in a di rection

a t right angles to each o f ou r three-space directions .This , o f course , gives r i se to the possib il ity o f gencrating a new kind of volume , the hype rvolume . Thehypercube o r tesse ract i s described by moving thegenerating cube in the di rect ion in whi ch the fourthdimens ion extends . For instance , i f the cube , Fig.

9 , were moved in a di rection at right angle s to eacho f its s ide s a di stance equal to one o f it s s ide s , a figureo f four dimensions , the tesse ract, would result.The init i al cube , when moved in a direc

t ion at right angl es to each o f its face s , generates thehypercube

,Fig. 10 .

-The l ines , aa'

,bb

'

,cc

'

,dd

'

,cc

'

,

ff'

, gg'

,hh

’

,a re a ssumed to be perpendi cul ar to the

l ines meeting at the po ints , a, b, c, d, e , f, g, h. Hencea’

b’

,b'

d,dd

'

,d’

a’

,ef, fg, gg

’

, g’

e,represent the final

cube resulting from the hyperspa ce movement. Counting the number o f cubes that compose the hypercub ewe find tha t there a re eight. The gene rating cube ,

and the final cube, a

'

h’

,b’

d,dd

'

,d'

a'

, ef, fg,gg

'

, g'

e, make two cubes ; and each fa ce generate s a

cub e making eight in a ll . A tesse ract , there fore , i s afigure bounded by e ight cubes.


To find the diffe rent el ements o f a tesseract, thefollowing rule s wil l apply

1 . To find the number of lines : Multiply the number of l ine s in the generating cube by two , and add al ine fo r e ach point or corne r in i t. E.g. , 2 X 1 2

24 8 3 2.

2. To find the number of planes, faces or squares

Multiply the number of pl anes in the generating cubeby 2 and add a pl ane fo r e ach l ine in i t. E.g . , 2 X 6

1 2 24 .

3 . To find the number of cubes in a hyp ercube

Multiply the number o f cubes in the generating cube ,one , by two and add a cube fo r ea ch plane in it. E.g. ,

2 X 1 6 8 .

4 . To find the number of p oints or corne rs : Multiply the numbe r of corners in the generating cube by2 . E.g . , 2 X 8 1 6.

In a pl ane there may be three points each equallydistant from one another . These may be j o ined , forming an equil ateral tri angl e in which there a re threeverti ce s o r points , three l ines o r sides and one sur

face .In th re e-space there may be four points e ach equi

di stant from the others . At the vertices of a regulartetrahedron may be found such points. The tetrahedron has four po ints , one at each vertex, 6 l ines and

4. equil ateral tr i angles, a s in Fig. 1 1 .

In four-space , we have 5 points each equidi stantfrom all the rest, gl g the hypertetrahedron . Thi sfour dimensional figure may be generated by movingthe tet rahedron in the di rection o f the fourth dimen

si on , as i n Fig. 1 2 . I f a pl ane be passed through each


o f the six edges o f the tetrahedron and the new vertexthere wi ll be six new pl ane s o r faces , making 1 0 inall , counting the original four. From the new vertexthere is also a tetrahedron resting upon each baseo f the o riginal tetrahedron so that there a re five tetrahedra in all . A hyp er te trahedron is a four-dimensioual figure consisting of fiv e te trahe dra, ten faces,1 0 lines and 5 points.

FIG. 1 1 .—Tetrahedron. h e . 1 2.

—Hypertetrahedron.

PAUL CARUS2 suggests the use o f mi rrors so

arranged that they give e ight representations o f a cub ewhen placed at the i r point o f inte rsection . H e says

I f we build up three mi rrors a t right angl es andpl ace any obj ect In the inte rsecting corner we shallsee the obj ect not once , but e ight tim es. The bodyi s reflected below and the obj ect thus doubl ed i smi rro red not only on both upright s ides but in addition in the corne r beyond , appearing in e ither o f theupright mi rro rs coincidingly in the sam e pl ace .Thus the total multipli cation o f our t ridimensionalboundari es o f a four dimensi onal complex is rende red e ight-fold .

“We must now bea r in mind that th is rep t esentatIOn o f a fourth dimension suffers from all the’ Vide Foundations of Mathematics, pp . 93

-

94.


establ i shed . Not at all . For there is no imaginabl econdi tion o f tridimensi ona l mechanics in which anobj ect may be s a id to have an obj ective existencesim il a r to that represented by the mi rrored cube .But there are di screpanci e s in thi s representation

which wel l might be conside red . They have vi rtuallythe force o f inval idating somewhat the conceptionwhi ch the analogy is designed to illustrate . For instance , in the cas e o f the mi rrored obj ect pl aced atthe point o f inte rsecti on o f the three mi rrors bu i l t upat right angles to each other. Upon examination ofsuch a construction it i s found that the reflection o fthe obj ect in the m i rro rs has not any perceptibl e connection with the obj ect i tsel f. And thi s , too , despitethe fact that they a re rega rded a s boundari e s o f thehypercube ; especi al ly i s this true when it is noted thatthese refl ections a re call e d upon to play the part o freal , palpabl e boundari es . I f a fourth dimensionalobj ect were really l ike the mi rro r-representation i twould be open

.

to serious obj ections from al l vi ewpo ints. The replacem ent of any of the boundari e srequ i red in the analogy would necess a ri ly mean thereplacement o f the hypercube itsel f. In other words,i f the real cube be removed from its positi on a t theintersection o f the mi rrors no reflection will be se en,and hence no boundari e s and no hypercube . The

analogy whil e adm i ttedly possessing some sl ight valuein the di re ction meant

,i s neverthel ess valuele ss so fa r

as a deta i le d representation i s concerned. So theanalogy fal l s down ; but once aga in i s the questionr a ised a s to whether the so-c alled fourth dim ens ion canbe est abl ished or proven a t al l upon purely m a them atical grounds. I t a l so emphasi ze s the necessi ty for

THE FOURTH D IMENS ION 139

a cl ea rer conception of the meaning of dimension and

space .The logical diffi cul tie s which beset the hyperspace

conception a re dwelt upon at length by JAMES H .

HYSLOP. He says :3

The supposit ion that there a re three dim en

s ions instea d o f one , or that there a re only threedimensi ons i s purely a rbitra ry, though convenientfor ce rta in p racti ca l purposes. Here the suppos ition expresses only differences o f di rections from ana ssumed point . Thus what would be sa id to l i ein a plane in one rel at ion would l i e in the thirddim ensi on in anothe r . There i s nothing to determine absolutely what i s the first, second, or thi rddim ension. I f the plane horizontal to the sensoriumbe called pl ane dimens ion , the pl ane verti cal to itwil l b e called sol id , o r the thi rd dim ension, but achange o f posi t ion will change the nam es of thesedimensi on s without involving the sl ightest qual ita tivechange o r diffe rence in m eaning .

“Moreover , we usually select three l ines o rplanes terminat ing verti cally at the sam e po int , thel ine s connecting the thre e su rfa ces of a cube withthe same point, a s the representat ive o f what i smeant by thre e dimensions, and reduce all otherl ines and planes to these . But interesting facts a reobservabl e here . I . I f the vertica l rel a tion betweentwo l ines b e necessa ry fo r defining a dim ension , thenall other l ines than the specified ones are e ithe r notin any dim ensi on at all , o r they a re outside theth ree given dim ensions. This i s deni ed by all

pa rti es,which only shows that a verti ca l relation

to other l ines is not necessa ry to the dete rminationof a dimensi on 2 . I f l ines outsi de the thrge v er

ti ca lly inte rsect ing l ines stil l l i e in dimens10n or' Vide Philosophical Review, Vol. V , 1896, p . 352, et . seq .


are reducibl e to the other dimens ions they may l i ein more than one dim ensi on a t the same tim ewhich a fter al l i s a fact. This only shows thatqual i tat ively all three dimensi ons a re the sameand that any l ine outside o f another can only rep resent a dim ension in the sense of dire ction from agiven point o r l ine , and we a re entitled to assum eas many dimensi ons a s we plea se , all within thre edim ensi ons .

“This mode o f treatment Shows the source o fthe i llusion about the ‘fourth dim ensi on .

’ The termin its generi c import denote s commensurable qual i tyand denotes only one such . qual i ty, so that thep roperty supposed to dete rm ine non-Eucl ideangeometry must be qual itatively different from this,i f its figure s involve the nece ssa ry qual i tat ive di fferentiation from Eucl idean mathemati cs . But thi swould shut out the ide a o f ‘dim ension’ a s its ba s i swhich i s contrary to the supposit ion. On the otherhand , the term has a specific meaning which a sdiffe rent qualitatively from the generi c includes aright to use the generi c term to describ e themdiffe renti a lly, but i f used only quantita tively, thatis, to express di recti on a s i t , in fact, does in thesecases , involve s the admission o f the actual , not asupposi tit ious , exi stence of a fourth dimensionwhich aga in i s contra ry to the supposi ti on o f thenon-Eucl i dean geometry. Stated bri efly, dimensi onas commensurabl e qual ity makes the exi stence o fthe fourth dimension a transcendental p roblem , buta s mere di rection , an emp iri cal p roblem . And thel ast conception sat isfi e s al l the requi rements o f thecase because i t confo rms to the purely quanti tativedifferences which exist between Eucl idean and non

Eucli dean geometry as the ve ry l anguage about‘surfaces ,

’ ‘tri angles,’ etc . , in sp ite o f the prefix

‘pseudo ,’ necessarily impl i e s .”


notions o f mathemati ci ans have been so inchoate a s tojust i fy thi s rather causti c , though appropri ate cri ti ci sm .

For i t doe s appea r that the moment the mathemati ci andeserts the province o f hi s re stri cted sphere o f motil ityand enters the realm o f the transcendental , that momenthe loses hi s way and becomes an inexper ienced marine ron an uncharted sea.

I t i s inte re sting to note that CASS IUS JACKSONKEYSER,

‘ whil e recogni z ing the pu rely a rbi tra ry chara cte r of the so-call ed dimensi onal ity o f space , nevertheless l ends himse l f to the Vi ew that “ i f we think o fthe l ine a s generating element we shall find that ourspace ha s four dimensi ons . That fact may be seen inva rious ways , a s follows

A l ine i s determined by any two o f i ts points .Every l ine p i e rces every pl ane . By j oining thepoints o f one plane to a ll the points o f anothe r, allthe l ines o f space are obta ined . To determine al ine , i t i s , then , enough to determine two of itspo ints, one in the one plane and one in the other.Fo r each o f these dete rminations two data , a sbefo re expl a ined , a re necessa ry and suffi ci ent. The

positi on o f the l ine is thus seen to depend upon fourindependent vari abl es, and the four dimensi onal ityof our space in lines is obvious .

Simila rly he argues for the four dimensional ity o fspace in spheres

We may view our space as an assemblage o fi ts spheres. To di stinguish a sphe re from all othe rspheres, we need to know four and but four independent fa cts about it , a s say, three that shall de‘Vide Monist, Vol. XVI, 1896, Mathematical Emancipations.


te rmine its cente r and one i ts si ze . Hence ou r spaceis four dim ensional also in s heres. In ci rcles , itsdimensi onal ity is six ; in surfaces of second orde r( those that a re

(pi ercedby a stra ight l ine in two

points ) , nine ; an so on ad infinitum .

The view taken by KEYSER i s a typical one .I t i s the mathematical vi ew and i s chara cte rizedby a certa in l ack o f restra int which i s found to bepecul i a r to the whole schem e o f thought rel ating tohyperspace . It i s clea r that the kind o f space thatwill permit o f such radical changes in its nature a sto be at one time thre e dimensional , at anothe r timefour dimensional , then six , nine and even n-dimen

sional i s not the kind of space in which the obj ectiveworld i s known to exist. Indeed , i t is not the kind ofspace that real ly exists a t all . In the first pl ace , a l inecannot generate perceptua l space . Neithe r can a ci rcl e ,nor a sphere nor any other geometrical construction.

It i s , there fore , not perm i ssibl e , except mathemati cally,to Vi ew our space e i ther as

“an assemblage o f itsspheres , its ci rcl es o r its surfaces ; fo r obviously percep tual space i s not a geometri cal const ruction eventhough the intellect naturally finds inhering in i t aso rt o f l atent geometrism which is kosmical. For therei s a wide difference between that kosmic o rder whichi s space and the finely elaborated abstract ion which thegeometer dece ives himsel f into identi fying with space .There i s absolutely nei the r perceptibl e nor impercep tible means by which perceptua l space in anywisecan be affected by an act o f will , i deation or movement.Just why mathemati ci ans persi st in v agarizing uponthe generabil i ty o f sp ace by movement o f l ines , ci rcles,planes

,e tc. , is confessedly not e asily understood e spe


cially when the natural outcome of such p rocedure issel f-stultification. It i s fa r bette r to recogni ze , a s aguiding p rincipl e i n al l mathemati cal di squisitions t esp ecting the nature o f space that the poss ib il i ti e s foundto inhere in an ideal i z ed construction cannot be ob

jectified in kosmic, sensible space . The l ine o f demarkation should be drawn once for all , and allmetageometri cal cal culations and theo ri e s should bepre faced by the remark that : “ i f obj ective sp ace wereamenabl e to the peculi a rit i e s o f an ideal iz ed construct ion such and such a resul t would b e poss ibl e ,

” orwords to that effect . Thi s mode of procedure wouldserve to cl a ri fy many i f not all o f the hype rspaceconceptions fo r the non-mathemati ci an a s well a s forthe m etageome tricians themselve s , e speci ally thosewho a re unwill ing to recognize the utter imposs ibil i tyo f the i r constructi ons a s appl i ed to pe rceptual space .We should then ce a se to have the spectacl e o f otherwi se well-demeanored men committing the erro r o ftrying to real i z e abstract ions or abstractionizing real it i e s . Herein i s the crux of the whol e matte r , thatmathematici ans , rather than be content with real i ti e sa s they find them in the kosmos , should seek to reducethem to abstracti ons , or , on the othe r hand , make theirabstractions appea r to be real i ti e s .KEYSER proceeds to Show how the concept o f

the gene rab il i ty o f hypersp ace may be conceived bybeginning with the po int, moving it in a di rectionwithout i tsel f and generating a l ine ; beginningwith the l ine , treating i t sim i l a rly, and generatinga plane ; taking the plane , moving i t in a di rection atright angle s to itsel f and generating a cube ; finally,using the cube a s generating el ement and constructing


i s agreed perhap s among al l non-Eucl idean publ ici ststhat the fourth dimens ion must l i e in a “di rection whichi s a t right angl e s to al l the three dimens ions .” Buti f they are a sked how thi s di rect ion may be ascerta inedo r even imagined they are nonplused because theys imply do not know. The diffi culty in thi s connect ionse em s to hinge about the question o f i denti fying theconditions o f the worl d o f phanta sy with those o f theworld o f sense . There a re distortions, ramifications ,subme rs ibl es, duplex convolutions and othe r mathe ti ca crobati cs which can be performed in the realm of theconceptu al the execution o f whi ch coul d neve r beactual i z ed in the obj ective world . Because these anti csa re poss ibl e in the premises o f the mathematicalimagination is sca rce j ustifi cati on fo r the a ttempts a trep roduction in an actual i z ed and phenomenal universe .One of the p roudest boa sts o f the fourth dim en

sionist is that hyperspace offe rs the possib i l ity of anew speci e s o f rotation , namely, ro tation about a

plane . He refers to the fact that in the so-c all ed onesp ace , rotation can take pl ace only about a point. Forinstance in Figu re 7 , the l ine ab represents a one-spacei n which rotation can take place only about one o fthe two po ints a and b . In Figure 8 which rep resentsa two-space , rotation may take place about the l ineab o r the l ine cd

,etc. , o r, in othe r words , the pl ane

abcd can be rotated on the axi al l ine ab in the direct i on o f the thi rd dimension . In tridimensional Spaceonly two kinds of rota tion a re poss ibl e , namely, rotat ion about a point and about a l ine . In the fourthdimension i t i s cl a imed tha t rotation can take placeabout a pl ane . Fo r example , the cube in Figure 9 , by


manipulation in the di rection o f the fourth dimension,can be made to rotate about the si de abgf.

A very ingenious a rgument i s used to show howrotation about a plane i s thinkable and poss ibl e inhypersp ace . But with thi s , as with the enti re fab ri co f hyperspace speculations , dependence i s pl aced al

most enti rely upon analogous and symboli c conceptionsfo r evidence as to the consi stency and rational i ty o fthe conclusions a rrived at.

FIG . 1 3 .

It is urged that ina smuch as the rotation aboutthe l ine he in Figure 13 would be incomprehensibleo r un imaginabl e to a plane being fo r the reasontha t such a rotation involves a movement o f theplane into the thi rd dimens ion, a dimension o f whichthe plane being has no knowledge , in l ike mannerrota tion about a plane is also unim aginabl e or incomprehensibl e to a tridim or a three dimensional being.

It is shown , howeve r, that the plan e be ing, by makinguse o f the possib il i t i e s o f an “assum ed” t ridim ension,

could a rrive at a rational explanation o f l ine rotation .


Figure 14 offe rs an illustra tion by means of whicha two dimensi onal mathematici an could demonstratethe possib il ity o f l ine rotation. He i s a lready ac

quainted with rotation about a point ; for i t is the onlypossibl e rota tion that i s bserv able in hi s two dim en

sional world . By conceiving of a l ine as an infini tyo r successi on o f points extending in the same direc

F1G . 14 .

tion ; by imagin ing the movement o f his p lane i n thedi rection o f the thi rd dimension thereby generatinga cube and at the s ame time a ssuming that the l inesthus generated were merely successi ons o f pointsextending in the same di rection , he could demonstratethat the enti re cube Figure 14 , could be rota ted aboutthe l ine BHX used a s an axis. For upon thi s hypothes is i t would be a rguabl e that a cube i s a successiono f planes p il ed one upon the othe r and l imited only


the cub e i s a ssumed to b e composed must be made torotate about e ach one o f the infinitesimal l ine s of

FIG . 1 5 .

Init ial Position Final Posit ion

FIG . 1 6 .

—Plane Rotation

which the plane used a s an ax i s i s composed . In thi sway, i t i s shown that the enti re cube has been made

to rotate about its face , cefg . Thi s concisely, i s the

THE FOURTH DIMENSION 1 5 1

quod erat demonstrandum of the m etageomet rician

who sets out to prove rotation about a plane . Thusi t i s made to appea r that in order that tridimensionalbe ings may be enabled to conceive of four-space rotation , a s in Figures 1 5 and 1 6 , in which the rotationmust also be thought of a s taking place in the direct ion o f the fourth dimension , they must adopt the sam etacti cs tha t a two dim ensi onal being would use tounderstand some of the poss ibi l i ti es of the t ridimen

sional world.

It i s , o f course , unwi se to assume that because athing can b e shown to be possibl e by analogical reasoning i ts actual i ty i s thereby e stabl ished . This cons ideration cannot be too emphatically insisted upon ;for many have b een l ed into the error by relying tooconfidenti ally upon results based upon thi s l ine ofa rgumentation . There is a vast diffe rence be tweenmentally doing what may be assum ed to be possibl e ,the hypotheti cal , and the doing o f what is actuallypossibl e , the practi cal .In the fi rst place , plane-rotation in the actual uni

vers e is a structural impossib il ity. The very na tureand consti tution o f materia l bodies will not adm i t o fsuch conto rtion a s that requi red by the rotation ofa body

, say a cube , about one of its faces. Let usexamine some of the results o f plane rotation . 1 . The

rotation must take place in the di rection o f the fourthdim ensi on . Now , as i t i s utterly impossibl e for anyone

,whethe r layman or m e tageom e t rician, even to

im agine o r conceive , in any way that is practi cal , thedi recti on o f the fourth dim ensi on i t is also impossibl efo r one to move or rotate a plane , surface , l ine or

any othe r body in that direction . We are in the very


beginning o f the p rocess of plane-rotation so-cal ledconfronted with a physi ca l impossib i l i ty. 2 . Pl anerotation necess arily involves the orb ital divers ion o fevery particl e in the cube . Thi s alone i s suflicient toprohib i t such a rotation ; fo r i t i s obvious that themoment a parti cle or any seri e s o f parti cl e s i s dive rt edfrom its e stabl i shed orb ital path di sruption o f thatportion o f the cube must necessa ri ly follow . Thi supon the a ssumpti on that the particle s o f matter a rein motion and revolving in thei r corpuscula r orbits .

3 . Pl ane-rotation necess itate s a radical change in theabsolute motion o f e ach individual p articl e , el ectron ,atom o r molecul e o f matte r in the cube and a consequent reta rdation o r a ccele rati on of this motion . Thisupon the hypothesis that the parti cles o f matte r a revibrating at the rate o f absolute motion . 4 . It presupposes a reconsti tution o f e ach atom , molecule orparticl e in the cube , changing the path o f intra-co rpuscular rotation e ithe r from a right to l e ft direction o rfrom a le ft t o right di rect ion , a s the case may be .The particle s o f matter in the cube will be acted uponin much the same manne r a s the part i cle s in a glovewhen it i s maneuvered in the fourth dimension . Indescr ibing thi s phenomenon , MANN ING s ays z

“

Eve ry part by itsel f, in i ts own place i s tu rnedove r with only a sl ight possibl e stretching and sl ightchanging of posi ti ons o f the different p arti cl es o fmatte r which go to make up the glove .

The sl ight stretching and sl ight changing o f theposi tions o f the particl es re fe rred to would be o f small

“Vide Four th Dim ension, Simply Explained, edited by H. P.

MANNING, p . 28 .


l i fe and le t i t solve them fo r us . But the fact thatthe fourth dimensiona l hypothes i s i s i tsel f a mere supposi ti on seems to have b een overlooked o r rathe rcompletely ignored by H INTON . Or el se , oughti t not be an obvious folly to hope to construct a ra

tional explanation o f p erpl exing physical condi tionsupon the basi s o f a purely supposit ionary, and therefo re unproven, hypothes i s ?The recogni zed doma in o f the four-Space , mathe

matically consi dered , i s acco rding to the most gene rousal lowance very small , so small , in fact, that the di spos i tion of some to crowd into i t the e ssent i a l contento f the mani fe sted unive rse i s a matte r o f pro foundamazement . Then , too , i t cannot be denied that therei s no appreci abl e urgency o r necess i ty fo r having t e

cours e to a purely hypotheti cal construction fo r ex

plicatory data regarding a phenomenon which ha s notbeen shown to be without the scop e o f o rdinarysci entific methods o f procedure to unravel .The cla im of certa in sp i ri tu al ists , notably ZOLL

NER of Le ipsig , tha t the phenomena o f sp i ritismi s a ccountable fo r on the grounds tha t the fourthdimensi on affords a resi denti a l a re a fo r di sca rnate beings whence spi ri tist i c fo rayers may imposethei r p resence upon unprotected three dimens ionalbeings is no l es s fatuous than the o riginal suppos itioni tsel f. Fo r upon thi s l atte r i s built the enti re fabri co f meaningles s speculations so glee fully indulged inby those who gl ibly procl aim the real ity o f the fourspace . Indeed , cl e a re r second thought wi ll reveal that,when the pendulum of e rrati c thinking and traffi ckingin menta l constructions swings back, hyperspaces, a fterall, a re but the ignes fatuii o f m atheti c obscuranti sm .

THE FOURTH D IMENS ION 155

Then, why should i t be deemed necessary to di scover some more mysterious realm of four dim en

sional proportions in which the spi rits of the dead mayfind a habita tion ? Are the spi ritual i sts, too , reducedto the necess ity o f further mysti fying the i r a l readyadequately mysterious phenomena ? I f there were notquite enough of physi cal i ty upon the basi s o f whi chal l the ant i cs o f these enti ties can be explained, andthat satisfactori ly, one would , as a matter of course ,be incl ined to l end some credence to these cla ims ; buta s i t i s cl e a r that all organi zed beings have somepower , i f no more than that which mainta ins the i rorgan i z ation , and as i t ought also be an acceptable facttha t such a being i s di rected by mind ; and further,that owing to the nature o f a spi rit body it can penetra te sol i d matter o r matte r o f any other degree ofdensi ty below the coeffi cient o f spi ri t matte r, it oughtl ikewi se be unnecessary to go wi thout the provinceo f stri ctly tridimens ional mechanics for an explanationo f spi ri tisti c phenomena .Equally unnecessa ry and uncal led for i s the attempt

o f certa in others who l e an toward the view of sp ecul a tive chemists to account fo r the none too securelyestabl i shed hypothesis tha t eight diffe rent al cohol s ,e ach having the fo rmul a C,H , ,O may be producedwithout vari a tion . This is sa id to be due to the factthat certain o f the component atoms , notably theca rbon atoms

,take a fourth dimensi onal posit ion in

the compound and thus produce the unusual spectacleo f e ight al cohols from one formul a . Have chemi stsactually exhausted all purely physi cal means o f reaching an understanding of the ca rbon compounds and

a re there fore compell ed to resort to questionable


means in o rde r to make addi tional p rogre ss in thei rfi eld? I t i s incredibl e . Hence the more facetiousappea rs the mathematical extravaganza in whichorigina te s the tendence am ong the more sanguine ad

v ocates to make o f the fou rth dimension a sort o f “j acko f all trades ,

” a veri tabl e “Al addin’s l amp” wherewitha ll kosmic pro funditie s may be i lluminated and madepla in. Not until the perfection o f instruments o f precision has b een reached , and not until human ingenuityha s been exhausted in its efforts to p roduce more re

fined methods of rese a rch should i t b e permissibl eeven to venture into untri ed and more or less debatabl e fi elds in s ea rch o f a rel i e f which a fte r a ll i sunobta inable .Notwi thstanding the fa ct that all a ttempts at ac

counting fo r physi cal phenomena on the ba si s o fn-dimensional i ty (which i s i tsel f by all the standa rds o fobj ective re ference a non-exi stent quantity and there forei rreconci l abl e with p erceptual Space requi rements ) a reto be characte ri z ed simply as a sensel ess dall i ance withotherwise deeply p ro found questions , many have fal l eninto a complete f orgetfulness o f the logi ca l b a rri ersinhe ring in and hedging about the query and havecommitted other and l e s s excusabl e e rrors in thepremises . Take , for instance , the suggest IOn that thea ction o f a ta rtrate upon a beam of pola ri zed l ightis due to the a ssumption o f a fourth dimensi onaldi r ect ion by some component in the a cid . This for there ason that expe rimentation ha s shown that ta rta ri ca cid , in one form , will turn the pl ane o f pola ri zedl ight to the right whil e in anothe r form will turn i tto the le ft. I t i s not b el i eved

,however

,that there i s

any warrant fo r such an a ssumption . There i s al so


no more than five corpuscl e s may have an independent groupmg In an atom .

”

The weakness o f thi s vi ew may b e due to the factthat a t that t ime MAJOR ELLIS was emphasi z ing inhi s own mind the necessi ty o f simpl i fying the conception so a s to make i t o f e a sy comprehens ion ratherthan the establ i shment o f any fealty to truth o r thesp i ri t o f mathes is in hi s exam ination o f the p roblem .

What there fo re o f real ity the student fa i l s to find inhi s Vi ew may be attributed to the s acrifice which thewri te r (MAJOR ELLIS ) fel t himsel f called upon tomake fo r the sake o f s impl i ci ty. Hence a certa in expressed connivance a t h i s pos i tion i s a llowable . But,on the othe r hand , i f such were not the consciousintent o f MAJOR ELLIS i t i s not unde rstood how i tshould appea r tha t “the unexpl a ined resi duum wouldseem to indi ca te that so fa r we have merely be enconsi dering the three dimensi onal aspects o f fourdimensional processes .” Contra rily, i t ha s yet to bep roved that three dimensi onal space doe s not affordampl e scope o f moti l i ty fo r all obse rvabl e o r recognizable physical p rocesses and that there i s no necessiry for re fe rence to hype rspace phenomena for anexplanation o f the “unexpl a ined resi duum .

” I t i s,o f

course , understood that m any o f the possib il i ti e spredicated fo r hyperspace a re pu rely nonsensi ca l sofa r a s thei r a ctual real iz ation i s concerned . Our concern i s , there fore , not with that cl a ss o f p redicates ,but with those wherein resi de some slight Show ofp robabil ity o f the i r response to th e conditions o fn-dimensional ity ei the r a s a system of space-measurement o r a so-call ed space o r seri es o f spaces .


MAJOR ELLIS concludes hi s s imple study of foursp ace by proposing the following query :

M ay not b i rth be an unfolding through theether into the symm etrica l l i fe-cell

,and death

,the

reverse proces s o f a folding-up into four dimen

SIOnal unity ?”

I t i s confessed that there seems to b e nothing towarrant the giving o f an aflirmativ e reply to thi sque ry. It i s , perhaps , sentimentally speaking a verybeauti ful thing to contemplate death as a painle ss, unconscious involvement into a glorious one-ness with alll i fe , and bi rth , a s the reverse of all this. But wherei s the uti l i ty o f such a dream i f i t be merely a dreamand impossibl e o f real i zation?

S IMON NEWCOMB,

8 a t one time one o f the outstanding figures in the e arly development o f thefourth dimensi onal hypothesis, openly decl a red that“there i s no p roo f that the molecul e may not vibratein a fourth dimension . There a re facts which seemto indicate a t le ast the possib il ity of molecul a r motiono r change o f some sort not expressible in te rms oftime and the three coOrdinate s in space .”

Of course,there i s no proof that a molecul e may

not at time s be ensconced in a four-space neither isthe re proo f nor probabil i ty that i t i s so hidden . Inde ed

,there is no proof that there is such a thing as

a molecule for that matte r.In al l o f the foregoing proposal s i t i s a ssumed

that the fourth dimens ion really exi sts and that i t l i e sj ust beneath the surface o f the vi sibl e , palpabl e l imi ts

‘Vide Science, Vol. V II , 158 , 1898, p . 4 .


o f the mate ri al unive rse ; that lying in close juxtaposition to all that w e a re abl e to se e , to hea r o r sensein any way i s thi s myste rious , ete rnally p rol ific, allpowerful something , hyperspace , ever-ready to nourishand susta in the fo rms which have the nethe r p a rtsfi rmly encysted in one o r the other of he r n-dim en

sional be rths . Thus it would seem that while yetfunctioning in a stri ctly tridimensional atmosphere ,some one , more reckless than the rest, should at l asts tumble upon some up -lying po rt ion o f i t and be instantly t ransfo rmed into a mathetic fay of ethe real i zedfour-dimensional stuff.


o f awareness with which we apprehend them . Thoseelements which a re without the intel l ect’s scope ofawareness have no inte rest and hence no value so fara s the individual intel l ect i s concerned . And th i s ist rue o f all degrees and states o f consciousness fromthe lowest to the highest, from the human to thedivine .There enter into all conscious determinations three

factors,namely : (a ) the scope , o r total ity, of adapta

tions which an organi sm can make in the sensiblewo rld

, (b ) the powe r o f consciousness to make adaptations and ( c ) envi ronment . These three a re inte rdependent. The total i ty o f adaptations dependsprimari ly, o f course , upon the qual i ty o f consciouspowers or faculti es, and al so , in a l esser degree , uponopportuniti es afforded by envi ronment . Faculti e s o fconsciousnes s a re derived di rectly from the influencesexerted upon the o rganism by his envi ronment and

the results of the struggl e to overcome them . En

v ironm ent is of two kinds, a rtific i a l and natu ral . Thea rt ifici al envi ronm ent i s such as has been modifi ed byour conscious action upon exte rnal phenomena . Theresi due i s natural . And thus the scope o f adaptabil i tybecom e s an unvarying witness to the qual ity o f consciousness mani festing through a given o rganism .

The universe is so constructed that the essenti alcha racte r o f i ts va rious states and qual iti e s i s a fixedquantity for a given scope o f consciousness and va ri e sonly a s the sphere o f consciousness va ries . States o fexistence or scopes o f adaptati on which are registe ringupon a higher pl ane o r i n a more subtle sphere o fexistence than that in which we may a t any time befunctioning can only appea r evidenti al to us when the

NORM OF SPACE DETERM INATIONS 163

mechani sm o f ou r consciousness becom es congruentlyadj usted therewi th . So that the focus of consciousnessmust always be a vari ab le quantity adaptable

,under

proper conditions , to any plane in the kosmos. Con

sciousness, then , becomes the sphere o f l im its bothof knowledge and adaptab il i ty. But l est we seem toadm i t impl icitly pa rt of the contentions which mathematical publ ic ists have m ade in postul ating the unodimand duodim consciousness , i t .is necessa ry carefully todiffe renti ate between the results a rr ived at a s a resulto f the two procedures . In the fi rst pl ace , analystsassume the existence o f a unodim and duodim planeo f consciousness and proceed to construct thereon ananalogy desi gned to Show the feasib i l ity of anothera ssumption , the fourth dimensi on. While , in l ayingthe foundation o f consciousness upon a tridim ensionalpl ane we do not start with an assump tion, but witha fa ct . Therein l i e s the difference . Enormous ad

vantages inhere in a procedure based upon facts , butin a seri es o f planes bu il t upon assumptions no suchadvantages a re discovered. For however much these ries o f hypotheti cal pl anes may be extended orel aborated there must inhere necessarily throughoutthe seri es an assump tional value which vitiates theconclus ions no l ess than the premises. The s anityand integrity o f intel lectual operations depend almostenti rely upon the differenti ation which we make between the necessit i es a rising out of a ssumptions and

those which spring up empiri cally from establ i shedfacts. No procedure is necessa ry to es tabl ish thevalue o f such a diffe renti ation, neverthel e ss i t may besuggested that i t is all owable , under the rul es of logic,to make any assumption whatsoeve r so long as ca re


is taken to see that the conclusions embody in themselves the cha racte ri st i cs of the original p remise .For instance , i t is permi ss ibl e to assume that spacei s curved. Under such an a ssumption , i t i s only necessary that the constructions which follow shall b esel f-consi stent . But the ca se i s different when we cometo deal with spatiality and vi tal i ty. These a re quantit i e s which cannot, in the last analysi s , be made toconform to the rule s o f the game o f logi c.Thus , when it i s intimated that real i sm lends i tsel f

to an appa rent division into degrees , and that ea chdegree has a corresponding state o f consciousness, i t

i s by no means to be inferred that such apparent divis ions a re o f mathemati cal impo rt. For, i n rea l i ty,i.e . , when the consciousness has expanded so a s tobecome congruent with the l imits o f even the spacemind (vide Fig. there appea r to b e no divisi onsin real i sm . It i s only because o f the fragmentarinesso f ou r outlook upon the kosmos that re al i sm appearsto b e divided into va rious pl anes ; for all o f thesepl ane s a re one . The divisions exi st fo r rel ative knowledge , but not fo r complete knowledge ; they ex ist fo ra finite intell igence , but not for a t ransfinite intelli

gence . That i s why we View real i sm as a seri es o fplanes . It i s b ecause we di scover that, a s we proceed ,as ou r consciousness expands and we take in more andmore o f the vital act ivit i es o f the kosmos and fInderstand bette r the cause s underlying that which weconta ct , we have passed from a state o f lesser knowledge to one o f greate r knowledge . And so we say

we have passed from one degree o f real i sm to an

othe r, whereas , really we have not p assed from one


these proscriptions is that intu it i ons—free , mobil e , andmore o r l es s forml ess in themselves, must fi rst b erehab il i tated and vestured in ga rm ents a la intellect

to conform to the preva il ing mode . But intuitionsthus trea ted are no longer intuit ions , but empi ricalconcepts . T rue intu i ti ons a re l ike aqueous vaporamorphous , permeating, diffusive : axioms or emp i ricalconcept s a re l ike cakes o f i ce—formal , infl exible andconforming to the shape o f the mold into whi ch theyare poured. Because o f this—the sci entific tendenceo f the intel lect and the c onsequent necessi ty of re

forming SO much o f the data which constitute its sub

structure, of press ing, condensing and re shapmg It to

sui t i t s own ready-made patte rns—it can b e perceivedhow pro found i s the influence of the intell ectual consciousness in dete rmining the characte r o f the total i tyo f data which the sensibl e world , and fo r that matte r,the supersensibl e , offe r us. The intellect is the onlym e ans at hand fo r the interpretation o f the meaningand significance o f the world o f phenomena . Conse

quently, whatever meaning or signifi cance we a re l edto attach to that p a rt o f the universe which we contact, in any way, is dictated by the intellectual consciousness. There i s no escape from the decisi ons o fthe intell ect so long a s the present scheme o f thingsendures .Thus, by whateve r standard of re ference the

matte r may b e determined,i t rema ins indisputably

establ ished that the intell ectual consciousness i s thesol e dete rminant o f the phenomenal value o f eve rything within ou r scop e o f awareness or adaptab il i ty.

And whatever the fault , incongru i ty or discrepancythat may be revealed by a more intimate knowledge

NORM OF SPACE DETERMINATIONS 167

o f the genesi s and characte r o f the appea rance o fthe sens ibl e world , i t will be found to be due to thepecul i a r cut and mode o f the intellect and not tothings themselves. The value , qual itative or existenti al , which the intelle ct i rrevocably assigns to obj ectsand conditions in the world o f the senses i s the exclusiv e norm not only by which these a re judged , butalso , by which our action upon them and thei r actionupon us a re determined . Images or obj ects which donot act upon us and upon which we cannot act haveno interest fo r us . But as an integral pa rt of thetotal ity o f im ages o r obj ects in the sensibl e world ,we must inevitably act upon al l that is outs ide o fourselves , and these , in turn, must react upon us. Onthe othe r hand , there must be obj ects and images inthe un ivers e o f l i fe and form upon which , because ofthe i r inherent nature and on account o f the lack ofour interest in them and the i r inte rest in us, we can

ne ithe r act nor become the obj ect o f thei r action.

But here in is a mystery. For, e ither we act uponand are recip ients of the action of the total ity o fimages o r obj ects in both the sensible and supersensibl e worlds, or we are so placed in the grandscheme o f things that both ourselves and the sphe reo f our inte rests and possibl e actions a re closed ci rcu its

,hermeti cally sealed and non-communicative with

the other l ike spheres , which do not and cannot actupon us . There is yet a third possib il ity—that we areso fashioned

,in the enti rety o f our being , that some

part o f us is exactly congruent with some part ofeve ry sphere o f possibl e actions and interests in thekosmos

,and there fore , each o f us ha s being or con

sciousness o f a kind that is keyed to and registering


in the total i ty of such spheres ; and that , at p resent ,because our interests and poss ible actions are l imitedto the plane of sensib i l i ty, we are conscious only there .And further, that although those spheres of our consciousness which are fixed to regi ster in othe r planesdo not answer to the lowest on which we now operate ,having a characte r o f which we . a re unaware , theyneverthele s s cannot be sa i d not to exi st , because o fthe l ack of communi cati on between them . Amongthese three poss ibl e cho ices , we hav e

’

no hesitancy inexpress ing a decided pre fe rence fo r the l a st mentioned—that the range o f our be ing i s co-extensive withthe range o f real ity, and l ike a pendulum , we oscil l ate ,a t long interval s , between two kosmic extremiti esnesci ence and omnisci ence .The intellectua l consciousness i s the touch-stone of

real i sm . It i s l ike a sp re ading l ight which , a s i t expands , reveal s previously da rkened corners and conditions, only i t ha s power both to reveal and to bringinto mani fe station. In its p resent state , man

’s consciousness i s l ike a se archl ight. I t i llumine s and takescognizance o f everything that fall s with in i ts scope o fmotil i ty and i s consequently abl e to study in deta i lthat which i t reveal s . But that whi ch i s beyond i tss cope is a s i f i t never exi sted so far a s the individualconsciousness i s concerned . It i s not reasonabl e topredi ct tha t the same characte ri sti cs that a re observ

able in any given state shall pers i s t throughout al lthe various scopes through which the consciousnessmust proceed in its evoluti onary expansion . For thescal e o f kosmi c real i sm i s one grand panorama extending from the grosse st to the most subtl e and re

fined . While in genera l the thread of real i sm may


overlook or ignore much that i s essenti a l and perhapsall that i s so even escap ing the true import o f thephenomena which it senses. I t i s rea sonable , too , thatin such a state the main outl ines of what i s real lyseen may be greatly distorted and exaggerated so thati t i s well-nigh imposs ibl e to secure a correct comprehension of the characte r o f a new scope o f re alismfrom any early survey. It i s not until l ater yea rs,a fte r much study and ci rcumspection that the mind,becoming used to the new condi ti ons, begins to getcorrect impressi ons and to make val id judgments a sto that which it disce rns. And even then, i t not infrequently happens that the resul tant vi ew of thingsin general i s found to b e in need o f revisi on and correction . Hence , a fte r eve rything i s si fted down tothe ultimate allowance fo r the i l lusi on incident to toomuch enthusi a sm and wonder we have only a verysmall res iduum o f truth upon which to build and thisl atte r we often find to be the s ingle thre ad o f r eal i tywhich runs through all the phenomena and which is,there fo re , the only quantity that rema ins wort hy ofmuch consideration .

Thus i t i s wi th rel igion . The path o f p rogressover which o ur rel igious conceptions have come neednot be outl ined here , but to any one a t al l a cqua intedwith the histo ry o f rel igious thought and ideal s ita t once must be p atent that i t ha s been one continuoussurrende r o f the ol d fo r the new

,o f one degree o f

real ism for anothe r newer and higher degree ; thatalways i t ha s been the phenomena

,the flora o f the

ideal s which have had to give way,while nothing wa s

le ft but the roots o f real ism from which they haveSprung . It has been the same with sci entific knowledge .

NORM OF SPACE DETERM INATIONS 17 1

Facts have been coll ected and hypotheses proposed tosynthesi ze them and yet these have had to give way forothers, and othe rs sti ll , unti l the data of sci entific knowledge to-day are qu ite different from what they werein ea rl i e r days . And yet permeating the sci entificknowl edge o f all times has been the golden threadof real i ty, and of all facts and systems of facts whichman has success ively assumed and surrendered nothinghas remained but the re al i ty ; indeed, nothing couldso rem ain, but real i ty. So i t is with all phenomenawith which consciousness has to deal . This perhapsi s due to the fact that the mind inte rp rets phenomenain accordance with the qual ity of its awareness, anda s consciousness i s a vari able quanti ty , i ts standa rds ofinterp retation will l ikew ise va ry . Each new scope ofawa reness, a fte r thi s m anne r , yields higher and moreexact standards of interpretation . And then , progressing in awareness from the segment to the wholea full e r View of the phenomena as well a s o f reali tyi tsel f i s ga ined and also a m ore comprehensive judgment o f the rel ations which exi st between the segmentand the whole . In other words, as the scope ofconsciousness widens i t becom es more and more ap

pa rent that what was fi rst though t to be a separatesegment i s in real ity identified with the whole in anindissolubl e manner. For the Thinker is then notonly aware o f the segment as such , but he is alsoconscious of the fact that i t ha s definite rel ations wi ththe enti rety and that what he needs is merely a moreextended consciousness .In denying the exi stence o f the four-space or

spaces o f n-dim ensional ity a s described and definedby geometri c i ans, we do not the reby deny the existence


o f a pl ane of consciousness which i s as much unl ikethe conditi ons o f the tridimensi onal world as i t i s sa idto be unl ike the four-dimensional world ; but what wedo deny i s that such a higher plane o f exi stence ha snecessarily to be conditi oned by such cha racte risti csa s the m etageom e tricians have proposed. I t is maintained tha t there i s no basi s in consciousness fo r aworld o f fou r dimensions ; that the consciousness hasno tendency for action in fou r-space . Neither hasmatte r nor l i fe any incl ination or potency to behavein a four-dimensional manner. I t i s indeed more ra

tional to suppose that there i s a h ighe r pl ane , in fact,a seri e s o f highe r pl anes , in which the thread o fre al i sm i s continuous , not broken a s i t necessari lywould have to be in extending to hyperspace , nor

curved a s i n a mani fold ; that thi s s erie s o f sub tl erand finer pl anes of consciousness a re merely an elongation o f our three dimensional scop e o f re al i sm . I t,there fore , rema ins only to maste r the phenomena o feach in j ust the same manner as we have , in a measure ,mastered the phenomena o f tridimensional ity. For iti s e a sily conce ivable tha t the qual i ty o f consciousnes si s such that i t may adap t i tsel f to a fa r wide r rangeo f poss ib il i ti e s than may be di scovered in hypersp aceand stil l b e a t ri-sp ace quantity.I t i s beli eved

,however

,that in all the new and

highe r pl anes of consciousness tridimensional i ty is thenorm both of the phenom ena and o f the real i typecul i a r to them . And that

,being such , does not

change o r vary, but i s a fixed quantity rega rdl ess o fthe plane o f consciousness . Furthe rmore , i t is bel i evedthat the highest state o f consciousnes s in th e enti re


s i stency i s quite another . I t does not necessa ri ly follow that b ecause a given scheme of thought is con

s istent in all i ts p a rts that it i s a lso consistent withuniversal t ruth o r with l i fe . This very vita l fact wasoverlooked by GAUSS and all those who followed inhis wake when he di scovered that hi s Astral Geom

atry was consi stent throughout in all its p a rts . Thereis only one norm of truth and that i s kosmi c con

si stency. I t matters l i ttl e that a th ing shall b e sel fconsistent ; i t matters much whether i t i s consi s tentwith the universal standard . It has been shown tobe logically poss ibl e to elaborate at l e ast two differentsyst ems o f geometry, namely, the geometry o f thea cute angl e and that o f the obtuse , whi ch , whil e eacho f them i s sel f-consistent throughout, a re neverthel essinconsi stent with each other and with the geometryo f the right angl e (Eucl ide an ) . This , i t would seem ,

appears to be suffi ci ent fo r the inval idation o f ei therone o r bo th o f the non-Eucl idean system s o f geometri c thought. Indeed , if i t can be shown that theEucl i de an geometry is more representative o f the tru eapproach to the norm o f space-genesis and of creationso fa r a s its mode of mani festation i s concerned, andconsequently true o f the norm set up by consciousness ,the rej ection o f both systems o f non-Eucl ideangeom etry seems to be thoroughly warranted . But thisis obvious and requi res no demonstration nor comment to make i t cl e a r. We have only to ask ourselveswhethe r i t has eve r occurred to u s that consciousnessha s e ither a tendency to o r adaptab il i ty fo r a ctionin a cu rvil ine a r manne r ; o r, i f when we contempl ateidea s o r i de a-rel ations we have the impressi on o f pe rceiv ing a curvi l inea r or m ani fol d tendence in them

NORM OF SPACE DETERMINATIONS 17 5

e ithe r o f a posit ive o r negative nature,and also

whether i t has been obse rved that our thought processes na turally a ssume four-dim ensiona l attitudes.

I fwe find that such a query must b e answered negatively

,

and indeed we must so find, then , we have no basi sfor the assumption that any one o f the systems of nonEuclidean geometry i s val i d e ithe r for the presentstatus o f consciousness o r fo r a future exi stence

,s ince

i t i s t rue that the futu re is but an elonga tion of thepresent. Evolution i s to bring no radical changes inthe no rms o f re al i ty ; i t has merely to deepen andwiden and make more intense

, eflicient and comprehensiv e the p resent scope o f our consciousness andthereby, while the Thinke r i s passing from one degreeo f real ism to another, to bring him into a clearer concep tion o f what hi s own l imited scope of motil itymeans to the whole .The four-space is a m atheti c divertisement . That

i s , i t cannot be sa id to l i e in the di rection o f a straightl ine which p roceeds e ither in a fo rward or l atera ldirection . Neither does i t l i e in a plane which isvertical or horizonta l to the sensorium . I t is, therefore , a fractura l departure from any conceivable tridimensional di re ct ion, a geometri c anomaly. Ev olu

tion, despite the mino r aspects of it s movement,undoubtedly p roceeds in a stra ight l ine and not by a

z igzag nor discontinuous l ine and hence i t is i rrationalto a ssume that i t wil l

,when it passes on to the next ad

v anced stage , em erge into the realm of the four-space .For the so-cal led hyperspace o f geometry cannot byany standa rds o f re fe rence be sa id to l i e in the pl aneo f any stra ight l ine which can be described in threesp ace . I f l i fe i s to evolve more effi ci ent forms and


if the forms a re to evolve into more p erfecf organi zat ions and mind and consciousness to become moreintense and comprehensive expre ssions of the divinemind of the kosmos it i s certa inly not in the domaino f hyperspace that these sha ll find the substructure ofthe i r higher development ; but , i f a t all , i t shall befound, a s i n all times p a st, i n the realm o f percep tualspace where bodi e s a re s a id to have three and onlythre e poss ib il it i e s o f motion .

What then i s the s ignificance o f the more than athousand years o f mathemati cal l abo rs ; o f all tha thas b een sai d and done in an endeavor to b ring intothe popul a r consciousness a conception of hyperspace ?I s i t a question o f “L ov e’s L abour’s L ost?” Or i s i ta mere p rosti tution o f mathemati cal tal ent? To an

swer these que ri e s i s the burden o f thi s treatise andi t i s hoped that a s th e text continues the reade r maybe abl e to arrive at hi s own conclusi ons a s to therel ative value o f the work of the mathemati ci ans inthi s respect and be abl e to judge for himsel f the trues ignificance o f i t a ll .The specific value o f consciousness a s a dete r

minativ e facto r i n sp ace-measurement has been recognized by all who have sought to a rr ive a t a logicaljustification for the conception o f four-dimensional ityby analogous rea soning. The existence o f the unodimwith consciousness l imi ted to a l in e o r point ha s beena ssumed and i t ha s been shown how gre atly such abeing would be handicapped by his l imited a re a o fconsciousness , i t h aving been proposed to confine hi sconsciousness to one dimens ion . An unodim would , o fcourse , be enti rely unaware o f any other dimensi onthan that in which he could consciously function . So


o r peopl e l iving in our tridimensional world , could,by using a l ike l ine o f a rgument o r reasoning, arrivea t a conception o r understanding of the four-space ,which

,of course , must al so l i e in a direction at right

angle s to three-space .The impl i cati ons o f thi s mode o f thought show

how thoroughly the mathematici an recognize s thelimitations whi ch consciousness imposes upon ourknowledge o f the worl d and the subtle r conditionsabout us. But, moreove r, i t i s even obvious to al l whostop to think about i t ; fo r i t can readily be seen howl ittl e those things which do not ente r our scope o fawareness affect us e ither physi cally, mentally o r spi ritually. But no one can be so bold as to deny utterlythat anything exists but what i s found in our con

sciousnesses. I t i s even true that in the gre at cente rso f population where people are compell ed to l ive manyfam i l i e s in the same house , i t is the usual thing fo rthese indivi dual famil i es to l ive in complete fo rgetfulness o f all the others in the house and l ive the i r l ivesso compl etely that it would be exceedingly diffi cul t tomeasure the effect the one ha s upon the other . Themathem ati ci an , as is Shown by the hyperspace movement, recogni ze s that there a re pl anes o f sup ercon

sciousness the na ture and character o f which pe rsonsconfined to limited area s o f consciousness can have noknowledge and may only a rrive a t that knowledge byserious thought and contempl ation . In othe r words,they taci tly admit the exi stence o f higher pl anes o fconsciousness as well as the necessity o f elevating thepersonal consciousness in orde r to comp rehend them .

Although i t was not exp ressly allowabl e in the analogyof the unodim, i t i s nevertheless one o f the strongest

NORM OF SPACE DETERMINATIONS 179

impl i cations o f the proces s of reasoning that theunodim could have easi ly ra ised the plane of his con

sciousness by continuing his researches unti l he,too

,

becam e conscious of the three-space,mathemati ca lly

,

as well as the two-space . For it was not necess a ryfo r him to ra i se the pl ane o f consciousness i n orderto contact the two-space . He had need only to widenit. But in o rde r to comprehend the mathem ati ca lthree-space i t would be necessa ry for him to elevatehis consciousness .The fundamental e rror in the foregoing l ine of

thought rests in the fact that awareness in the hum anfam i ly ha s not developed in the manner outl ined .

The human speci es has not com e into conscious rel ations with the three-space by outgrowing the one-spaceand the two-spa ce in succession. The fact o f thematter i s that when consciousness fi rst dawned i t musthave encompassed al l three dim ensi ons simultaneouslyand equally and there i s nothing to indicate that itsrise was otherwise . Then

,Specifically there is no

evidence that the evolution o f consciousness has proce eded in a rectangul a r manner . Indeed , there is un

doubtedly no warrant fo r the assumption that i t hasprogressed in ways that a re mathemat i cally de te rm inable at all . The question very naturally ri ses in viewo f the above as to the rel at ive value o f mathematicalknowledge in the schem e of psychogenesis. Can

mathemati cal knowledge or l aws be sa id a ctually and

finally to settl e once fo r al l tim e any question in whichconsciousness o r l i fe ente rs as a facto r? Upon theresponse to this question hinges unanswerably the decision as to the catego ry which m a them atical knowledge should by right occupy in the enti re schemati sm


of l i fe . I f it can b e successfully mainta ined that finalj udi cative powe r ab ide s in mathemati cs in the dete rmination o f these questions , then i t woul d be uselessto struggl e against the fi at o f mathem ati cs and mathem aticians ; verily, we should be compell ed to acceptnolens v olens al l tha t mathemati ci ans have devisedabout hype rspace and i ts connotations . I f , on theothe r hand, i t c an be shown that no such j udicativepower inhe res in mathemati ca l knowledge we shal lthen b e abl e to establ ish fo r mathemati cs a true category and to dispose o f the hyperspace movement ina manne r that shall a t once b e logi cal and nece ssa ry .

That the discovery o f hyperspace by the mathematician i s merely an aspect o f a general fo rwardmovement in the evolution o f consciousness can beshown by a bri e f co rrelative study. Hyperspace i s thea rtifici al symbol o f a higher and more extensive re almof awareness . For i t cannot b e doubted that to beabl e to think in the te rms o f hype rsp ace , to study thevarious rel ations and inte rrel at ions upspringing fromthe original p remises, actually to become conscious inthe hyperspati a l realm thus constructed, requi res adiffe rent specie s o r qual i ty o f consci ousness than thatrequi red fo r o rdina ry thinking . The peri od coveringthe ri s e o f a rtificial sp ati a l i ty i s contemporaneous withthe rise o f the phenomena identifi ed with the sp iri tuall i fe o f SWEDENBORG ; for during the same tim e hebegan a se ri es o f visi ons which revealed to him greatknowledge o f the unseen and sup ersensuous real iti e so f l i fe and ex istence . H i s consciousness was beingflooded with the l ight from so-call ed cel esti a l spheresand he wa s gradually b ecoming conscious o f a newdimension ,

”a new sp ace , a higher world that i s a lto


“vorti ci st.” Art really began the sea rch for thepl a sti c e ssence o f the world” trying to po rt ray itsconception o f the re al im age o f the spirit” o f theworld. Colo r a cqui red a new kind o f splendor and

pa inting gave b i rth to a new intrins i c b eauty o f mater i a l and shee r lovel iness of texture . Al l o f whi chwe re evidence s o f an intell ectual up-reaching in re

sponso to the Sha rp appulsions from above . DARWIN’S

mind , being o f scientifi c b ent, saw and interpretedeve rything in the te rms o f materi a l ist i c sci ence ; butthere i s no doubt but that the expans ion o f the a reao f awareness which his mind experi enced in his greatconception o f evolution as a continuous process andal l that i t impl i es thereby wa s a resul t o f the universa lappulsi on o f the human intellect against the newdoma in o f consci ousness . And KANT’

S conception o fspace i n general p erhaps may b e said to have been theseed-thought fo r the m etageom etrician.

But thus i t will b e noted that in all the cases mentioned in the fo rego ing there i s a lways present thepersonal el ement o f the investigato r, and that the re

ports o f e ach o f these have been colored and charact e rized by the i r individual consciousness and exp e

riences. That al l reports would agree with respectto deta il s connected with the new doma in o f consciousne ss could sca rcely be expected owing to the wonderand bewi lderment whi ch se i ze the intellect under suchci rcumstances. No impli cation that the mathem ati

cians have been unduly excited by what they have discove red a fte r yea rs o f pati ent res ea rch in this direction is indicated by the fo regoing observations ; buti t cannot b e deni ed that the enthusi a sm o f the momentand the consequent minimization of al l othe r phenom

OF SPACE DETERMINATIONS 183

e speci al l ine being investigated serve veryto obscure the mental visi on of the moreIt perhaps i s suffi cient that the investigato rdown in a s orderly manne r a s possible thech he conce ives, and that he should interpretrding to the standards of his own intellect .this cannot be expected. Moreover

,i t

that the future investigato r, fa r re

beclouding influence s o f p a rtisanship ,ra ises his consciousness to that

shall be able to synthesi ze the findingsthereby with the a id which com es to himre advantageous posi tion a rrive at soundera more rel iable judgm ent.

in thus be seen tha t the m e tageom et rician’

s

of inte rpretation is no m ore entitled to finaland general a cceptance than that of theist , SWEDENBORG , or the occul t seer, DANTE .

the i r best moods and at thei r highest pointsta l effi ci ency these have only succeeded insymbol iz ing what they have conce ived o f theo f the supersensuous realm in terms of thei r> e rience s. Is there any more cogent reason,1 accepting the analyst’s conception o f a worldrsp ace peopled with ensemble s , propositions,sf n-dim ensional i ty and other matheti c cons than the Infe rno of DANTE, inhabited byShapes and repellent denizens, the remains o fearth-l iv es o r SWEDENBORG’S Celestial Realm

,

dwell num erous beings of celesti a l character1ing various tasks in the work of the world ?b serv ations should not l ead the reade r to com econclusion that the vi si ons o f DANTE and


SWEDENBORG are deemed to be more worthy ofcredence than mathematical knowledge when thatknowledge is l imited to the sphere where i t rightfullybelongs ; but the proper Vi ew is that which would makei t appea r that i t

’

is the way these widely differingworkers inte rpret what they have seen ; tha t i t i s theadaptati on o f the unse en realms to the pecul i a ri ti e so f the menta l iti e s which observe them . The mathem aticians have simply portrayed a s well a s they coul dthe i r conception o f the new stage o f consciousness andi ts contents , and following the modus v iv endi of allintellects have interpreted thes e things in the terms ofmathemati cs, merely because mathemati cs constitutedthe best ava i l abl e symbology at hand fo r the purpose .Simi la rly, the pa inte r sees a new world o f colo r ; thepol iti ci an , a new era o f political freedom ; the rel igiousenthusi a st, a new religious conception ; the sc i entist, anew condit ion o f matte r and energy, and so on , tothe most ordina ry mind

,e ach see s something new

whil e a t the same time is necessarily l imited to theconfines o f h i s own mental i ty when he com e s to interp ret what he sees and conceives . Hence , the rewould appear to be only one way to regard all theseadvances and that i s by synthesi z ing them , by correl ating , and by tracing them to a common source , and

finally by seeing them a s one general forward movement o f intellectual evolution .

Man, the Thinke r, who in e ssence i s a pure int elligence , ha s two mental mechani sm s o r organs o fconsciousness . One of these is the b ra in-consciousnesso r the egoic. It is so called because the bra in is i tso rgan o f express ion and impression . It mani feststhrough the bra in and uses i t to further the va rious


basi s than ha s hithe rto been surmi sed. It i s not onlythat whi ch m ay be sa id to t ranscend experi ence butthat which is the organ of contact with the sup ersensuous re al it i e s a s well a s o f express ion through thebra in-consc i ousness .The mind’s method o f apprehending obj ect ive phe

nom ena i s not a di rect p rocess but an indi rect process byvirtu e o f whi ch neurograms or nerve- impacts registeredin the b ra in a re inte rpreted. External sense- impressi onsa re , o f course , conveyed to the corti cal a re a by meansof appropri ate vib rations which trave rse the l ines o fthe neural mechani sm . These are recorded in thebra in a rea s just a s a telegraphi c communi cation i sregi ste red in the appa ratus o f the rece iving end, and

in being so , they make terminal registrati ons whichm an, the Thinke r, inte rp rets a fte r a psychi c code whichhas been bui l t up empi ri cally. That i s

,he comes to

know that certa in rates o f vib ra tion and certa inpecul i a ri ti e s the rein mean certa in things when re

ferred to the sensorium . He then inte rprets according to this experi ence the symbol i sm o f all neurographi cal impressi ons . But it is obvious that unde rsuch ci rcum stances , where the inte rprete r is fa r re

moved from the thing i tsel f and finds it necessa ry tointerp ret ra te s o f vib ration o r symbols in orde r toa rrive at a knowledge o f the intel l igence which i s conv eyed to him , the chance s o f inadequate conceptiona re very grea t indeed. In fact, i t i s not possibl ethrough the use o f neurographical symbols alone toget any complete notion of the phenomena conside red.

And thus there stands between the Thinker and abso

lute knowledge a ba rri e r which prevents his a rrivingat a state o f ce rt itude in his knowledge o f the world

RM OF SPACE DETERMINATIONS 1 87

ibl e obj ects. I t i s , howeve r , a barri er whi chways rem a in, che cking ever h i s approach toin hi s understanding of the universum o f apes.

markedly different condition obta ins in the realma priori o r intuitional for the reason that thes which inhere in the neurographical o r a

oristic method a re absent and the Thinker ha sdi rect approach to the obj ects o f cognition .

the chance fo r erro r is very small indeed . Thiscount , there fore , fo r the superio ri ty o f the inll. over the rationa l o r the perceptual . Indeed,oubtful whether the purely rational possesses, lue whatsov er when its modus v iv endi i s unned by the intu itional .e why can we no t b e certain that the results o ftional p rocesses a re correct at all times? Is i tcause we lack the power to perce ive whether“

em ises a re corre ct in the fi rst pl ace ? Qui teFo r i f the Thinke r can intui t the necessi ty andie o f any given p rem ise i t follows that theuences o f that p remise a re tru e I t would,) re , appea r that the more the intu itional facultyloped the cl ea re r will b e our perceptions notof abst ract values but o f obj ective things themFurthe r, i t is doubtlessly true that the moreace-mind i s developed in the human race thewil l become our perceptions o f the essenti al

1 o f things so that whatever may be the p resenta)f the space-m ind to the bra in-mind they will bem ore a ccurate than the impressions we rece iveh the l atter as a medium of apprehensi on. Itnatural , however, that in the present more or


l ess chaoti c condition in which the facul ty o f the

intui ti onal i s found i t should be diffi cult even to inte rpret its p resenta ti ons accurately. I t i s perhap s dueto the fact that we a re unused to its del iveri es andmode o f regi stration a s well a s to the fact that it hasbeen overshadowed by the intel lectual o r rationalfaculty. But the mere fact that it i s p resent and functioning, even i f but rudimenta ri ly, i s evidence o f i tspotenti al i ty and the possib i l i ty o f i ts future development to a sti l l higher degree o f effi ci ency.

There i s no doubt but that the original impulsewhich re sulted in the crea tion o f the faculty o f percep tibility in the Thinke r also marked out the mete sand bounds o f ou r enti re range o f perce ivabi l ity whi chincludes not only the intuitional but something highe rsti ll . There i s no doubting e ithe r the obviousfact that these metes and bounds cannot have beenothe r than rudimenta ry or general l ines o f denotation ,and that th e work o f the i r further el abora tion and

refinement i s a matte r o f evolutionary deta il . For i fwe assume that the general principl e s o f evolution a retrue we immedi ately recogn ize the cogency o f thi sview. That whi ch we now call the hand has notalways been the perfect instrument that i t i s no r ha sthe e a r always been so keenly adj usted a s a t p resent.It ha s requi red undoubtedly many mil lion o f yea rsfo r the eye to re ach i ts present degree o f complexityand adaptab il ity. Ye t in all these ca se s the diffe rentorgans exi sted in potent i al i ty from the beginning ; themete s and bounds o f the hand , the e a r and the eyewere l a id out primordi ally. Evolution has speci al izedand adjusted them to envi ronments and needs. Thusi t will be s een that whil e the intui tional fa culty wa s


with a neurograph of say, a cube , i t i s not the cub eitsel f which he contemplates o r observes ; i t i s theneurograph o r psychic symbol which the sense-impressi ons make in the bra in. His consciousness deal s notwith the obj ect but with the symbol s . It i s true thatwhen he verifi es on e neurograph by another, a s thescop ographic or s ight impressi ons by the tactographicor touch impre ssions i t i s found that the del ivery thusdetermined i s a true enough rep resentat ion . I t i s a lsotrue that the Thinker , as a rule , does not accept aneurograph a s vali d unti l i t ha s been verifi ed by atl ea st one or more p resentat ions through his outersense o rgans . I t occurs, there fore , tha t all such del ive ri es a re ver ifi ed and co rrected by one or moresens e witnesse s be fore final acceptance by the Thinker ;but even then it cannot b e sa id that his notions thu sga ined a re i n a ll respects correct

'

and tru e to thestandards set up by the b ra in-consciousness not tomention highe r fo rms o f consciousness . And then ,when we consi der that in addition to the numerouschances fo r e rro r which naturally inhere in thi s methodo f cognition i t must al so be appa rent that the Thinker’sapproach to the real i ty o f things i s much impeded byhi s separati on there from , th e unrel iabi l i ty o f our

ordinary methods o f cogni t ion i s much emphas ized.

But aside from the egoic or bra in-consciousnessthere i s the higher consciousness o f the Thinke r himsel f. For the bra in-consciousness is merely hi s methodo f rega rding and comprehending the neurographical

del iveri es, the psychi c code by which he systemati zesand organi zes his cogni ti ons or impress ions o f thesensibl e world. Thi s higher consciousness consti tutesthe faculty a priori fo r the Thinker on a higher plane

RM OF SPACE DETERMINATIONS 19 1

tence , and because it de al s with el ements altounlike those which make up the data o f bra inusness is, accordingly, l ess l i able to erro r in i tso f the supersensuous presentations than i s

ectiv e o r b ra in-mind. In fact , i t i s diflicult toe o f a state or conditions where in , supportedvi ew contemplates , the intu ition should e rr int. V i ewed from the standpoint o f externalenta , this condition may be sa i d to be due tonce o f sensuous ba rri ers which would otherent the nea r approach of the Thinker’s conto the essence o f things which i s the obj ect

consciousness on thi s highe r pl ane . D i rectly,ter, i t i s undoubtedly due to the fact that, follow1e l e ad o f l i fe i tsel f, yea , a s the veri tabl e handof l i fe , i t cannot err where l i fe is concerned .

1 deal ing with notions a priori or intuitograms

hinke r is rel i eved of the onerous necess iti e s andtions incident to the examination and determina1f neurographi c symbols registe red in the bra inand so i s free to study, to examine and judget hand the impress ions which are rece ived fromtn plane o f intuiti on . The difference i s about theas that which should exi st between the methodsmmunication between two telegraphi c operatorsin one instance they would have to depend uponeliv eries conveyed ove r the wi res, whil e in thewhen they stood face to face with each othe r,oul d communi cate by di rect conve rsation . In thease the method of communication i s direct andwhil e in the othe r i t is indi rect , ci rcu itous andex . It can , there fore , be re adily seen that inses where the approach i s made in a di rect , s imple


manner the probabi l ity o f error i s much less than incases where the intellectual approach i s l ess di rect andmore compl i cated. Hence in drawing conclus ions a sto the rel ative importance o f the two mechani sms ofconsciousness , the a p oste rioristic and the a p riori, i tis necessa ry to bea r in mind the comparative supe riori tyof the one ove r the o ther a s a means o f cogni tion.

It matte rs l i ttl e that the intuit i onal faculty i s not sowel l developed as the tui ti onal because it is but na turaltha t inasmuch a s the Thinker’s ne eds a re subservedin the sensuous realm by the tui tional consciousness i tshoul d , from more active use , ga in somewhat overthe intuiti onal in facil i ty of expressi on and generaluti l i ty. And the more so , because the two facult i e sserve diffe rent purposes ; one i s attuned to rece ive impressi on s from a subtl er pl ane whil e the other i s fi ttedfor contact with the phenomenal universe ; one i s rel ated to the conceptual whi le the other i s rel ated tothe percep tual . They diffe r not only in function , butin nature as well . There is, o f course , a natu ral ba rri e r consi sting o f the inherent l imi tations o f eachfacul ty whi ch prevents the ful l mergence o r unifi cationo f the two state s o f consciousness so that there existsa state o f consciousness the da ta o f whi ch the bra inmind i s unaware, i t be ing abl e only a t ra re inte rval seven to rece ive so much a s sl ight impressi ons from itin the na ture o f intu itional fl ashes o r insp i rati ons andthe l ike . Vi ewed in this l ight i t would appea r thatthe cogn it ions which a re most truthworthy a re thosewhich are presented by the intu i ti onal facul ty becausethey are nea re r to the essenti al real ity o f things ; theyhave to do more specifically with the na ture o f thatwhich app ears while the tu i tional mind can only regard


as the norm of its essenti al na tu re and as the ea si estand simpl est express ion o f the tuiti onal m ind , how canit b e gainsa id that these considerations obvi ate the

necess ity o f the mathe tical hypersp ace ?I f the re al i ty o f things i s hi dden from us and i f

we are,there fore , unabl e to perce ive thei r real es sences

i t i s b ecaus e our mode o f thought and our consciousness have obscured our vi sion and l imited us to thisstate o f p aucity o f p erception . I t i s not because real i tyi s itsel f a hi dden , inscrutabl e quanti ty nor that itsmo‘dus v iv endi i s “unknowabl e but because we being multiformly l imited ,

“cab ined, cribbed and con

fined” a re resultantly l a cking in the powe r to di scernthat which otherwise would b e most obvious to us . Itmay well b e set down a s axiomati c that when , in theprocess o f our thinking , we a rrive at the inscrutable,the unknowabl e and the infinite , i t i s evident that ourthought p roces se s a re deal ing with a fo rm of real i smwhich i s higher and beyond the possib il i ti e s o f our

lo fti est thought-reaches . And in o rde r to symbol izeto i tsel f thi s condition the intellect poses such te rms as“ inscrutabl e ,

” “unknowable” and the “ infini te” s implybecause that i s the best i t can do . Hence when i t i ssa id that sp ace i s infin i te i t is apparent that the mindrecognize s that when i t contempl a te s sp ace i t i s dealing with something whose degree o f rea l i sm transcendsi ts powers o f comprehens ion. Infinity i s a rel ativeterm , and in fa ct, decrea ses in extensi ty in the proportion that the consciousness expands and comprehends.I t i s not unl ikely that should the intellect one daydi scove r that i t had awakened into union with thespace-mind it would immediately rej ect its preconce ivednotion o f the infinity of space. But we need not wai t

DETERMINATIONS 105

ing of thi s far off event in the path o ffo r we can here and now perce ive witha higher faculty than the intel le ct the

f thi s conclusion.

certa in i t i s that the intelle ct,in the pri de and

cc o f its traditi onal heri tage , will not withoutst ruggle yi eld the ground and prestige i t ha san aeon o f time ; and in va in does the intu ie not i ce o f dispossessal in these premises ; but

r stubbornly fought the battl e , however tenaheld the posi ti on t ime wi ll di scover the weakf the intel lect’s hand. Death for the intellect

nsue a s a resul t o f the confl i ct but i t wi ll be a deathfrom it wi ll a ri se , quickened, rev iv ified and up

by i ts disposer , the intuiti on , upon the remainsdead sel f to a highe r and grander state than it

ver enj oyed before .

>ace i s not sta tic. It i s dynami c, potenti a l andc. I t i s a process , a becoming. I ts duration a sces s i s never ending. I ts extens ity i s l im ite d andThe so-call ed infini ty o f space is one o f the capi

us ions o f the intelle ct whi ch can only be removed

1 expansion o f the consciousness , by a mergence

1c individua l consciousness with the space-con;ness . In the ever-widening ci rcl e o f the indi

l consciousnes s l esse r real iti es give way to greater

.e da rkness recedes from the l ight—the l esser1ring in comparison with the greate r, a s the con:ness b roadens , a s matte r to spi ri t, a s night totl

‘ a s l imitation to non-l imita tion . Thus the mostfa cts and condit ions o f our l imited l i fe a re but

hadows of the deeper real iti e s which shall be


reveal ed to the Thinker in the days o f his l a rger andmore glorious l i fe o f freedom from limitations .And now it will appear that the whole fabri c o f

our knowl edge Shall have to be reduced to the barewarp and woof ; for nothing i s real but these . I ti s a s i f the Thinker, using the tui ti onal mind, hadbeen in all times p ast studying the design wovenin the surface o f a very thick plush carpet. Therea re the warp and woo f, the long verti cal threadswhich make the plush and the intr i cate designappearing on the surface . Our knowledge may b el ikened to the design . It rep resents the contents o fou r knowledge . We have not even so much a s begunthe study o f the nature o f the verti ca l threads a s theyappea r b eneath the surface to say nothing of beginning the study o f the warp and woof. The warp andthe woof a re the real ism of the kosmos ; the verticalthreads a re the roots and stem of the phenomenalworl d ; the design is ou r s ensibl e wo rld a s i t appearsto the intell ect. The l i fe o f the intellect ha s beenspent in contempl ating thi s design ; while o f the handswhich wove the ca rpet , o f the mind which di rectedthe hands and of the sp ir i t whi ch vita l i zed all , i t knowsnothing no r indeed can it know anything. Where shallwe say a re those hands , tha t mind and that sp i ri twhi ch made the carpet possibl e and an a ctual i ty? Inva in do we se a rch among the remains , am ong the so ft ,gl istening threads o f the carp et or among the int ricacies o f the design . For they a re not the re . Theyhave passed on . The intell ect looks a t the design o ra t the verti ca l th reads and because it i s unabl e to follow them to the i r source , i t decides that they a re iafinite , inscrutabl e and unknowable . But not so . All


fashioned upon the necess iti es and requi rements forany given sta te o f be ing . Consciousness a lone fixesthe appa rent l im i ts o f l i fe ; i t also dete rmines the stateo f our knowledge o f l i fe . And thus when the Thinkeris confined to any stage o f real ity and congruent degreeo f consciousness i t appea rs that what he there findsi s ample fo r all hi s purposes . Accordingly he i s conv inced that that stage i s the final consideration o fhis scope o f motil i ty. It i s only when he i s able tora ise hi s consciousness to a point where he can contacthighe r re al i ti e s that he becomes aware that there arehighe r stages in which his consciousness may mani fest.This pecul i a ri ty o f the Thinker’s consciousness i s accentuated when he allows him sel f to become whollyengrossed with a study of the phenomena of that stagein which he can consci ously function . Hence i t constantly occurs that men exhausting the study o f thephenomenal find themselves floundering upon the beacho f the outlying shores o f consci ousness where in Sheerdesperation they fall into the i llusi on that they haveindeed reached the l imits o f mani fested l i fe and thatbeyond those l im its there i s no organ i zed being. Un

conscious a re they that in eve r widening circl e s thefe rtil e l ands extend and awai t the awakening of the i rconsciousness when they may til l the fallow groundo f thi s new doma in and begin again the sea rch for theultimately re al .With respect to the p resent powers o f consci ous

ness , i t cannot be success fully controverted that theconcept o f tridimensional i ty o f space i s suffi ci ent foral l purpo se s . It must b e so fo r i t is not only an aspecto f the phenomena o f space but o f real ity a s well . Thisfact is atte sted by the nature o f mind that answers

RM OF SPACE DETERMINATIONS 190

na ture of space . Tridimensional i ty charactere enti re extent of consciousness and l i fe

,and

re , of space i tsel f . This characte riz ation may be0 the ve ry doors of the hea rt o f space where theecome one . Nor would this conception be i nt viti a ted i f i t were allowed that the mass o f thea o f the supersensuous world, lying in close

ty to the sensuous world, does present itsel fonsciousness in a four-dim ensi onal manner andphenom en a o f a sti l l higher plane present thema five or n-dimensional manner to that state ofsness which may be congruent with them ; be

we should be making allowances for thes In phenomena and thei r mode o f p resentationconsciousness which by no means impl i e s a cording change in real i ty o r l i fe . All phenomenaihioned by the intel lect. The phenomenal worldwhat the intell ect interp rets i t to be . It i s that>thing more . I ts qual iti es, a ttribute s and charti cs a re such as the consciousness gives to it . Itonly fo r the purposes o f the evolving consciousAnd

,a s an instrument of consciousness , i ts ex

i s stri ctly subj ect to the evolutiona ry needsf. In that moment that the immedi ate needsconsciousness shall no longer be abl e to findct ion in the phenomena of any plane o f na ture ,moment the phenomena o f that pl ane disap

°

ecede and a re swallowed up in the mael stromnal real i ty.

the gradual expansion o f consciousness as i tthrough the infini te seri e s o f grades o f aware

Ieantime becoming deepe r, broade r and more

hensiv e as i t p roceeds , there may be observed


running through all these pl anes and orders that whichi s ne ithe r the phenomena of the va rious planes northe consciousness ; but which must b e the substructuralbas i s o f both , remaining the same , unchanged and unchangeabl e . That -i s the thread of re al i ty , the passageo f l i fe i tsel f which is the eternal basi s o f all . Nowit i s to thi s reality, l i fe , that the space-mind i s rel atedand in which its roots , i ts he a rt and the very cente ro f i ts be ing a re a t one with the divine mind of thekosmos .The question o f dimensional i ty i s sol ely a conce rn

o f the obj ect ive o r b ra in-mind which i s the intell ect.It i s one of the ways in which the intellect endeavorsto understand phenomena . I t i s an a rb itra ry cont riv ance devi sed by the intell ect fo r its convenience instudying the worl d o f things. Without i t, as ob

v iously appears , the intell ect would not be abl e to gove ry fa r in its consi de ration o f the minor problemswhich inhere in materi a l things . The fourth dim en

s ion i s but anothe r a tt i tude , another contr ivance , whichthe intelle ct has devised in o rde r that it may studyfrom anothe r angle the evanescent phenomena of theworld o f appea rance s . H aving apparently exhaustedthe poss ib il it i e s o f motion in three dimensions , andbeing driven on to the acqui rement o f more p icturesquevi ews by the very necessi ty o f i ts continued growth , i th a s betaken itself to another highe r mounta in peak ,called “hyperspace” where wi th l a rge r l enses andhigher powered instruments i t i s beginning to scanthe l andscape s o f a new intelle ctual realm o f consciousness . Yet the celesti a l wonders o f this new-foundrealm o f consciousness rem a in in undisturbed forgetfulness or neglect . But i t is not by a scrutiny o f


iarity occludes the intellect from any approach to thetrue na ture of that phase which it can grasp .

Hyperspace i s one o f the il lus ions of the phenomenal ; i t i s the dres s which the intellect has superimposed upon a s ingl e nuance ; i t i s a mask which i san exact repl i ca o f the mood o f the intellect. Yetthrough thi s mask the intell ect grandly hopes to ap

pro ach real i ty. The period through which the mindi s now passing is a repeti tion o f the evi l days ofschola sticism when m en set out to determine the exactnumber o f cel esti a l be ings that could be pe rched uponthe extremi ty o f a needle point. I t i s a time whenmen’s minds e a s ily a ssume grotesque and hideousshapes and the i r thoughts become the embodiment o ffantasti c enti ti e s . The exclusive occupation o f suchminds becomes the fab ri cat ion o f matheti c monstrosi

ti e s whi ch rapi dly del iquesce upon the fi rst appro acho f the re al o r the appearance o f the fi rst ray o f intu iti on which may escape through the dim and misty condit ion o f the intell ectu a l over-hangings . I t will not beever thus ; fo r the Thinker wil l one day pa ss from astudy o f the a rrangement o f phenomena in spa ce and

by well-ordered steps come once aga in to himsel f .And then through the maze o f i t a ll se t out upon thetrue path the t ridim ensionalty o f sp ace followingwhich he wil l inevi tably approach the ci tadel o f there al , the kosmic space-mind.

CHAPTER VII

THE GENESIS AND NATURE OF SPACE

gy of Mathematical Knowledge—Manifestation andanifestation D efined—The Pyknon and PyknosisKosmic Engenderment of Space—On the Consubstany of Spatiality, Intel lectual ity, Materiality, VitalityKosmic G eometrism—Chaos-Theos-Kosmos—Chaoand Chaomorphogeny—N. MALEBRANCHE On God

I the World—The Space-Mind—Space and Mind Aree—The Kosmic Pentoglyph.

OMETRY i s conce rned primarily with a study of:asurem ent of magni tudes in space . Three co'

es a re necessary and suffi ci ent fo r all of i tsinations M etageometry comprehends theif the mea surement of magni tudes in conceptualFor its purposes four or n-coordinate s a re

ry and suffici ent. Percep tual space i s that formrnsion in which the physi cal unive rs e is recog0 have b een created and in whi ch i t now exists .itual space i s an ideal i zed conception belongingdoma in o f mathesi s and has no actual , physicalcc outs ide o f the mind . Mathemati cal Space:nts the ideal i sm o f perceptual space .om etrical magni tudes may be defined a s symbolssical obj ects and geometry a s a treatise on theogy o f forms in space . In fact, al l cognitive:es a re s imply effo rts a t inte rpreting the sym


bolism o f sense-del iveri es ; and the diffe rence betweenmere knowledge and wisdom , which is the essence o fall knowledge , i s the difference between the understanding of a symbol and the comp rehens ion o f theessenti al n ature o f the thing symbol iz ed. So long as

knowledge o f space is l im ited to the understanding o fa symbol or symbols by which i t is presented to theconsciousnes s so long wil l i t fa ll short o f the comprehension o f the essenti al nature o f space . In va in havewe sought in t imes past to understand space bystudying rel ations, posit ions and the characteri sti cs o fforms in space ; i n va in have we based ou r conclus ionsa s to i ts rea l na ture upon the fragmenta ry evidenceswhi ch ou r senses p resent to ou r consciousnesse s . I t isa s i f one had busi ed himsel f wi th one of the meshes ina gre a t net and confined his enti re a ttention to whathe found there , meanwhil e remaining in complete ignorance o f the nature o f the net, how it came to b ethere , o f what i t i s made and how great i t s extentmay be .There i s ever a marked difference between a symbol

and the thing which i t symbol i ze s . Words a re thesymbol s o f idea s ; i dea s , a s they exi st in the mind, a rethe symbol s o f ete rnal verit ie s a s they exi s t in the consciousness o f the Logos o f the universe . There maybe a wide divers ity o f symbol i c forms which representone s ingle i de a ; a s , for instance , th e var iety o f wordforms which represent the i de a o f deity in the va riouslanguages . L ikewise there may be a multipl i ci ty o fidea s which represent a s ingl e veri ty. But neither i sthe idea nor the word the real thing in itsel f. Thatqual i ty o f a l i fe-aspect which we cal l its thingness hasan essenti al na ture which cannot become the obj ect


emphasi zes the insufliciency o f the intell ectu al method.

The identification o f consciousnes s with ess enti a l be ingis

“

a procedure whi ch cannot b e a ccompl i shed by anact o f will di rectly and immedi ately. Because i t i sa process , a seri e s o f unfoldments , an adj ustment o fthe focus o f consciousness to the kosmic e ssenti al i ti e swhich constitute the substructu re o f the mani festeduniverse . In the very nature o f things , a kosmi ce s senti al i ty cannot b e viewed a s be ing in mani festationespeci al ly in the s ame degre e a s o rdina ry phys icalobj ects a re mani fest . The former i s a state , a potentiality, a dynamic force , an existence which should b ethought o f a s an extra-kosmic affa ir dwell ing on thepl ane o f unity o r kosmic o rigins ; while the l a tte r a rethe exact Opposi te o f th i s . The one can be seen , feltand sensed whil e the other i s the roots which a re notse en but l i e buri ed deeply in the heart o f the universal pl asm of b eing and beyond the ken of sensuousapperception . The te rm manifesta tion i s both rel ative and flexibl e in its u se . I t i s rel a tive because i twi ll apply equally to all stages o f cognition . A thingi s in mani festation when i t i s pres entabl e to theordina ry mean s o f cogni ti on belonging to any stageo f conscious functioning ; i t i s not in mani fe sta tion wheni t i s beyond the scop e o f the Thinke r’s schemati smo f cogni tive powers . Its fl exibi l i ty is seen in i ts readyyi eldance to the enti re range o f impl i cations inheringin the p roces s o f cogniti on , fitting the s implest a s wella s the highest and most complex.

G reat i s the gul f which i s interposed betweenmani fe station and non-m an i festation ; and yet the two ,in essence , a re one . They are l inked togethe r a s thestem of a flower i s j oined to its roots. Likewise one

ENESIS AND NATURE OF SPACE 207

le , palpable whil e the other is invisible , ime though no less real and abiding . As the thinea rth separates the stern , l eave s and flowersroots so the l imi tations of man’s consciousrate the mani fest from the unmani fest.a s when the surrounding earth i s removedroots and they are l a i d bare reveal ing thei rand unity with the outputt ing stem and

0 , when the l im i ta tions o f consciousness a reby the subtl e p rocess o f expansion to whichousness is am enabl e so tha t i t can encompassunm ani fest, i t , too , will reveal the eternalkosmic polars—mani festation and un

sta tion .

Iere i s but one barri e r to ultimate knowledge and1 the human consciousness howeve r pa radoxicalay seem. The unutterable darkness which shuts.e so-call ed unknowable” from our cognitionl imitation o f man’s upreaching consciousness.

l imitat ions constitute the difference between the1 intell ect and the mind o f the Logos. Neverthe outlying fronti ers o f man’s consciousness

ally a re being pushed farther and fa rthe r withEvery new idea ga ined , each new emotion in

1, e ach new conception conque red, and everyI foray which the Thinker makes into the realmconceptual

,every explo ration into the abys

abyrinth o f man’s inne r nature a re the self

’

s

ltionary forces which a re gradually annihil ating°

ontier b arrie rs o f consciousness and thus ap

Iing more closely upon the Ultima Thule ofspi ritual possibi l iti e s .a ce is i n mani fe station . It exi sts and has being


whether i t is viewed a s an obj ect , an entity or themere poss ibil ity o f motion . That i t offers an oppor

tunity o f motion and renders i t possibl e fo r obj ectsto move freely from point to po int cannot be deni edand yet this fact ha s no bearing whatever upon thee ss enti al na ture o f space . The very fact o f i ts ap

p earance , i t s mani festation , makes i t obvious that i ti s the nether pole o f that eternal p a i r o f opposi tesmani fe station and non-mani fe station , be ing and nonbeing , which a re essentially one .I t wil l be s een from figure 1 7 that the p eriod

o f involut i on embraces s even sep arate stages, themonopyknon, duopyknon, tripyknon, etc . , be ing theuni t principl e o r engendering elements of the respectivestage s . Involution compri se s al l creative activi ty fromthe fi rst fa int st i rrings o f the void and formless chaosunti l the univers e ha s actually b ecome mani fe st anddense phys i cal matte r ha s appea red. It i s dividedinto two cardinal p eriods , n amely, the chaogenic

period during which primordi al chaos i s given i tscharacte r and di rective tendenci e s fo r the world age .I t i s a phase o f duration wherein the fi at o f kosmico rder i s promulgated , and consi st o f three stages ,m onOpyknosis, duopyknosis, tripyknosis

1 gradually,insensibly, gradating into the fourth o r quar topyknatie ; the chaomorphogenic period i s l ikewi se dividedinto three stages—quintopkynosis, sextopyknosis andsep topyknosis, developing out o f the fourth gradually.The quartopyknotic stage i s the stage o f metamorphos is or transmutation wherein the trans itionfrom non-mani festation to mani festa tion i s completed ;it is also the stage o f kosmi c causa tion , because from

’ See Fig. 18 .


of the evolutionary movement whi ch , l ik e the involutionary movement, i s divi ded into two ca rdinal periods,namely, the morphogeni c (during which a re produced, i n turn , insensibl e forms, sensibl e fo rms andsp i ri tual fo rms ) and the kathekotic period whichmarks the perfection , the consummation o f the evolutionary movement. These two ca rdinal p eriods o fthe evoluti onary phase o f duration and the two ca rdinalperiods o f the involut ionary phase complete the kosmi cage , the

“Grea t Day of Brahma .” The concentri c

c i rcl e s , beginning with the dot and ending with thes even concentri c ci rcle s , and des ignated a s a , b , c , d,e , f, g, a re representations o f the consti tution o f therespective units co rresponding to e ach o f the sevensubdivisi ons. They symbol iz e the seven degrees o fcondensation or pyknosi s which comprise the genes i so f space , on the one hand, and on the other, the stage so f unfoldment. Because , during involut ion all potenci e s , powers and cha ra cte rs were be ing infolded , involved ; but during evolution , these a re being unfolded,expressed , evolved.

The figure 1 8 is anothe r view o f these two maj o rmovements , involution and evolution . The genes is ofspace is here shown sym bol i zed by the Kosm i c Egg.

The seven stages o f involut ion a re re fe rred to a s , themonopyknotic, duopyknotic, tripyknotic, quartopyk

notic, quintopyknotic, sextopyknotic and the sep topyknotic ; whil e the co rresponding stages o f evolution a rere ferred to as

, the physi ca l , the senti ent, mental ,c ausative o r Spi ri tu al , the tri adic, duadi c and monadic ,i ndica ting that the principl e o f physi ca l i ty i s succeededby the principl es of senti ence , mental ity, spi ri tual ity,and the three fo rms o f kathekotic being . This sym

ENESIS AND NATURE OF SPACE 2 1 1

i t Should be stated , is designed with respect toiv erse and man and ha s no re fe rence to other[C evolutions than the hum an and cont empora

animal,plant and mineral evolut ions.

follow the ramifications o f the symbol ismwould involve a survey of a ll branche s of

1opyknosrs Nomadic

:k no 5 15

>pyknosxs Physrc a l

(I here and i t i s bel ieved that thi s i s suffi ci ent to

e the reade r to grasp the magnitude o f the

01 and to understand its purpose and intent .

twould appear, there fore , th at i f i t i s possibl e fo r

i t ellect to traverse by means of a study of kosmic

tolism ,used as a standard of reference , the entire


length of the bridge which engages the antipodes intoan eternal unity, something may be ga ined in the wayof a more defin i te and cle arer understanding o f thee ssenti a l n ature o f sp ace in i ts rel ation to kosmogenesi s .In the diagram , figure 1 7 , i s shown a table which

rep resents the stages o f space-genesi s . It wil l b e notedthat the whol e scheme is divided into seven stages .I t i s not an a rb itrary divisi on s imply but a symbol i cone and represents ful lness, completeness , enti re ty.

The names given , namely, monopyknotic, duopyk

notic,” etc. , represent the symbol i c characteri sti cs o f

e ach stage in i ts rel ation to the universe in the p rocesso f becoming. The terms “monadic,

” “duadic ,” “tr i

adic ,” e tc . , a re representative o f th e seven planes o f

matte r in the universe .A pyknon i s a kosmic p rificiple and represents the

typal a spect o f kosm ogenesis. It i s a gene ri c termand may be identified in i ts rel a tion to the variousstage s by the prefix . The monopyknon belongs to theulte rio r pole o f the antipodes on the s ide o f non

man i festation. SO do the duopyknon and the tripyknon. Pyknosi s i s a process o f kosmi c condensation,o r l imi tation fo r purposes o f mani fe station . I t i s astage in the descent o f the kosm i c Sp i rit-L i fe , a degradation o f non-mani festati on into mani festation, and i s ,there fo re , the cardinal causative principl e o f creation .

The term pyknon be ing generi c is appl i cable al ike toa particl e o f matte r

,a state o f be ing, a condi tion o f

existence , a process or a principl e . The monopyknon

i s , accordinglyt the primary a spect o f the process ofkosmic pyknosis. It is the a rchetype and there foreal l inclus ive and omnipot ential. But whethe r regarded


a spects of all l i fe forms on the path o f evoluti on maybe traced rearward to the duopyknotic and tripyknoticstages o f kosmogenesis.

The metamo rphos i s by whi ch the monopyknon be

comes a duopyknon contrives the differenti ation o f thepristine pl asm of kosmic being so tha t the fi rst b ecomesthe ensoul ing or Vi tal iz ing principl e o f the second ;and , in tu rn, the second becomes the Vital i z ing o rinner principl e o f the thi rd ; the th ird o f the fourthand the fourth o f the fi fth and so on throughout theseri es unti l the l a st i s reached which i s the sep to

pyknon. The sep topyknon is, there fo re , a sevenprincipled form . I t i s both unita ry and septena ryunita ry in the sense that the seven a re really one andseptena ry for the rea son that each o f the sevenprinciples , in the course o f evoluti on , becomes a sep arate p roce ss sp eci ally adapted to functioning upon itspecul i a r pl ane o f matte r. Thus i t i s seen that theutmost significance a ttaches to thi s septena ry pyknosi so f the kosmic pl a sm of l i fe . The impl ications o fthi s conception are , o f course , too vast and multifarious to be set down here . We shall have to dismissi t with one ob servation only, and that i s : ev ery

single app earance of life and form in the to tality ofsuch app earances is roo ted in kosmic pyknosis whereit has re ce iv ed its inner v italizing force , its form and

the law of its mode and manner of app earance to

ge ther with the me tes and bounds of its existence .

These p rocesses , monopyknosis and duopyknosis,a re to b e regarded a s taking place , each in its ownperiod , everywhere throughout the Body o f Being o fthe Logos but on the plane o f non-m ani festation .

They a re states o f preparation for mani festation

ESIS AND NATURE OF SPACE 215

s to the ge rminative period of the seed org. They represent the fi rst sti rrings of thepla sm and conta in the promise and potencythat i s to succeed them . There i s one othercoOrdinate with these two , and that i s theoti c which complete s the unmani fest trinity andthe a rchetypal vehicle whence proceeds theuniverse . The ensoul ing principle o f the

n i s the duopyknotic principl e . But when theha s reached the t ripyknotic stage the threeen merged into one and the characteristi cse re pecul i a r to each as a separate pyknon a reed into a singl e qual i ty having three aspects

1 a re mutually interdependent and coOrdinate .

“

he unmani fest trinity, now complete a s a resul t.e tripl e pyknotic p rocess, i s the imperishabl e andsusta ining radix , the all-mothe r o f the mani fested:rse . I t i s the Golden Egg l a id by the god SebIe beginning o f a great l i fe-cycle . It i s alsots-Theos-Kosmos

,i. e . , kosmic disorder , divine will

enerating element, and kosmic o rder o r space .a s in an egg , resides i n kosmic potency al l that

universe i s to become in any“G reat Day o f

ma” o r any Great Kosmic Li fe Cycle . Its e ldesti s SPACE , physiological and perceptual , and ther i s the eternal father-mother o f the universe .

e is,therefore

,the male -female principle of

festa tion; in its kosmic womb all forms are

f

ed,dev elop ed, ev olv ed and sustained. Into it

1, a t the close o f the G reat D ay, all exi stences,1s and all p rincipl es matured and ripened by thesitudes o f kosmi c evolut ion will b e inhal ed withre tu rn o f the Great B reath o f L i fe . The un


mani fest trini ty i s the a rchetyp e , and there fore , thepattern or model for the mani fe st, embodying inpotency all that the mani fe st may ever become . Fo rthfrom the unmani fest proceeds space a s a dynamicprocess endued with the potenti a l ity fo r generatingal l that the mani fested universe conta ins .The terms “

unmani fest,” “unconditi oned and

unl imited” have a special meaning here and a re u sedin the same sense as the mathematical term “ transfinite ,

” and there fore , imply a transcending o f anyfini t e o r a ssignabl e degree o r qual i ty o f mani festation.

Hence they shoul d be distingu ished from the term“absolute” whi ch has a different impl i cation . So thatal though the tripl e p rocess outl ined above may notbe vi ewed except a s a characteri zation o f the planeo f non-mani festat ion , and hence o f the primordia la ctivi ty o f the Creative Logos, there i s nothing in thesymbol i sm to wa rrant the i dentification o f thi s processin any way with the Logos in ab solution . For on thi svi ew Absolute Being i s , in a l a rge measure , sa crificedwhen the monopyknotic process i s begun and themonopyknon (kosmic principl e ) begins to appea r.Absolute Be ing , whil e i t m ay not be defined, deline atedo r described may b e symbol iz ed by the i deograph :“action in inaction” ;

“be ing in non-being “manifestation in non-mani festation” ; for these a re symbol smerely and do not describe or del ineate .We have obse rved the subtl e connection which

exi sts between mani festat ion and non-mani festationand have seen how that , a s the roots o f a pl ant susta inthe oute r growth o f stem and flowers

,the former

being the matrix out o f which proceeds the l a tte r , andthat in l ike manne r does the unmani fest susta in the


mind and Spi ri t diffused, central ized and organi zed intoever new and higher forms and expressi ons.

Another important notion to be ga ined in thi sconnection i s the fact tha t it appea rs a s a logicalsequence to the foregoing that the being o f the manifested Logos must necessa rily fil l a l l Space , yea more ,is that space in every conceivabl e essenti al ity. H isl im i ta tions a re the l imitations o f space . H is qual iti es,properties and attributes are the qual i ti es, properti e sand attributes of sp ace and a re only diffe rent from theoriginal sp ati a l character when mani fested through adiversi ty o f forms by whose very inne r constitution andexterio r form the modifications are a ccompl ished . All

mat ter in the univ erse,all energy, and indeed

,all

manifesta tions or emana tions of whatsoev er character ,hue

,tone or quality are

,in reality, H is being and

no thing but H is being . There is, therefore , no formnor ensouling principle whe ther of life , mind or sheer

dynamism which can exist outside of H is being and

be,ev en in the slightest degre e , absolv ed from an

e ternal identity ther ewith. Once thi s i dea i s graspedand its va ri ed impl i cat ions noted it then i s no longe rconce ivabl e that any other orde r o r schematism canbe possible in our universe , and that , too , despite themulti form conceptions p ecul i a r to the varied systemso f philosophy.The ma tutina l dawn o f creation came at the close

o f the tripyknotic movement which resulted in theel abora ti on o f materi al s , the ini ti ati on o f principl es ,process es and types , and the final preparation o f thefi eld o f evolution . The three p rocesse s o r a spects o fnon-mani festation proj ected in prepa ration fo r manifestation, namely, m onopyknosis, duopyknosis and

NATURE OF SPACE 2 19

ep resent the earl i est stages o f germinalWhen these had closed , the G re atbegan to germinate ; the fi rst fa int , ins igns o f mani fested l i fe began to appea r.se t in . The fourth o r quartopyknotic

gh only sl ightly diffe renti ated,o r rathe r

ting that period o f kosm i c involution whench i s to become the mani fested univers e fi rstto fal l unde r the sway o f kosmic orde r

, is

heless the basis of all great world processes . I tmidway between the poles—m ani festat ion and

ation . During this stage the l i fe elem entsg the imprints o f characte r, being endowed

di rective tendenci es and stored with such dynama s wil l persist throughout the Grea t Li fe Cyclehi ch they a re to mani fest . It is here that beginsnov em ent o f involution , the storing away o f thoseents and facto rs, no more and no l e ss, that are10w forth on the upward path o f evolution ; i t i sthat matter begins to assume form ; electrons , i onsatoms created, or, tha t those minute p rocessesh on the evolutiona ry s ide a re to culm inate ina re originated . This is the metamorphoticI t i s the l aboratory of the universe wherein

ll,the Creative Logos, prepares the material s

o f which and in which.the vast diversity o f

>hons o r forms is created . Quartopyknosis, ac

ingly, i s the fi rst active step , on the plane o ffestation, which resul ts in the appearance o f per1al space and consequently o f physi cal matter itas wel l a s all the other grade s o f matte r in the1os. Space , brought into existence by the a ct o fC reative Logos in imposing l imi tation upon H is


being, i s in i ts primordi al form composed of quarto

pyknons o r quadrupl i cate p rincipl es and tendenci e swhich act in un ison and to the accompl i shment o f as ingle end or purpose . On thi s pl ane o r during thecontinuance o f thi s p eriod o f space-creation, the rootso f universal law and o rder a re produced. In i t a repl anted the p rinciple s o f good and evil and a sharpl ine o f demarkation establ i shed between al l the conceiv able pa i rs o f opposites whi ch exi st. I t accountsfor the dual i ty o f l i fe and form . M al e-female ;father-mothe r ; posi tive-negative ; Raj ah-Tamas ( action-inact ion)—al l these find in thi s p rocess thei rete rnal o rigination . It i s the stage o f harmony, bl i ss ,ideali ty, pe rfecti on, perfect equi l ib rium and balance .Here , innumerabl e age s be fore they actually appea r,the glow-worm and the da isy, the amoeba and thedynosaur , man and the pl aneta ry gods al ike abodethe i r tim e awai ting the toppl ing o f the scal es o f kosmicpotency when all woul d be plunged headlong into theendless l abyrinth o f becoming.

The quartopyknotic process i s s imil a r in all de tai l sto the three preceding processes , these l a tte r b eingprototype s o f al l succeeding stages o f involution .

The quintopyknon, accordingly, symbol ize s the quintupl i cative a ction o f l i fe in i ts descending movementtoward the crea ti on o f matter i n i ts densest fo rm .

That is, i t i s a fiv e-fold principl e a cting in uni son andkosm i c consi stency

,infolding in the universal where

withal that which i s to b ecome mental matte r on theS ide o f evolution. Just, a s may be seen in the di agram , Figu re 1 7 , the quartopyknotic process sym

bolizes, on the involuti onary si de o f the l i fe currentthat which is to become on the evolutiona ry s ide ,


or the o ak, reside in kosmic potency in the quinto

pyknon.

N ICHOLAS MALEBRANCHE,

2 i n one o f his ve rylucid m oments , beheld the essenti a l cha racte r o f thesym bology o f space and wa s l ed to the conclusi on thatGod is space i tsel f. To him it was equally certain thatal l ou r ide a s o f space , geometri cal o r purely physi ologi cal , as well a s our notions of the great suprasensua l domain o f idea s , exist in the kosmic de i fo rm ,

o r body o f the Logos of Being . He saw“all things

in God.

”God did not cre ate ideas ; they are a p art

o f God Himsel f ; God did not cre ate mind ; i t i s a pa rto f H imsel f ; no kind o f matte r did He create ; i t i s averi tabl e part o f H imsel f and indissolubl e from Hims el f . The grea t outstanding impl i cation of thisphilosophy i s tha t our consciousness o f God i s buta part o f God’s consciousness o f H imsel f ; our consciousness o f sel f and the not-sel f is but God’s consciousne ss o f these things. There is no existence o fanything , e i the r o f the sel f or the not-sel f, except inthi s consciousness . I t i s re freshing , there fore , to notethat a lthough the appro ach i s made from another andenti rely different di rection , almost th e s ame conclusiona s to the ultimate resolution o f all chaogene tic el ements into what i s the very systas i s o r consi stence o fthe great kosmic de i form , i s r eached.

But a marvelous visi on comes with the dawn o fthi s truth upon the lower mind . It establ ishes cl e arlythe t ruth o f KANT’S notion when he s a id : “

Sinceeverything we conce ive i s conce ived a s b eing in sp ace ,there i s nothing which comes befo re our mind from

‘Vide Recherche , Chap . V I I , also Philosophical Review, V. 15,

p . 401 .—MALEBRANCHE.

ESIS AND NATURE OF SPACE 223

i dea o f space can be derived ; i t i s equallythe most rudim enta ry pe rception and the

l ete .” The mind cannot get away from theo f space , because , out o f the ve ry essencea resul t o f the quintopyknotic process , i td, created and organized. The ide a o f

there fore , not derived from things in spacethei r rel at ions in space . I t sprang up with

iousness . As soon a s the Thinke r becameo f himsel f he became aware of space . Thee of sel f-consciousness impl i es space. Theman is a speci al i zed a spect o f space. Indeed ,proj ection o f space . The moment the sel f canI am ,

” i t al so can complete the decl aration by“I am Space .

” When the sel f looks out fromx-walled cab in o f impri sonment into the im

y of what we cal l space he looks out into thati s h imsel f and his imm ensi ty ; he perce ives theand the ever-present sustenance o f his beingcogn izes his i dentity therewith , provided he doeslow himsel f to become entangled in the philoal diffi culities which the intell ect i s prone toa round the simple , ye t marvelously compl ex ,o f sel f-consciousness .

Iis should settl e , once fo r all , the question o f'ity. The a priori inheres in quintOpyknosis orc psychogenesis. It is the essential na ture o fi t i s the mind’s l ine s of organiz ation ; it i s thef the mind’s being and action . All mental pern ori ginate s from things in space . No thought

7 deta i l , o f any state or condit i on , whether l imr unl imited , rel ated o r i solated can be conceivedi t b e o f things in space . And thi s i s so , because


mind and space are,one . I t i s not so with our con

cep tion o f time . Time i s merely an aspect o f consciousness in it s l imita tion and does not inhere in themind in the s ame manne r that doe s sp ace . In fact

,

i t i s not a p a rt o f the mind’s na ture a s i t h as b eenshown tha t space i s . It would , there fore , seem to bea grave mi stake so to coOrdinate the two notions .Space is the progenito r o f mind and i s continuallyi dentified wi th mind . T ime i s the child o f consciousness . That i s , i t i s one o f the i llus ions o f consciousness which the ego will shed a s h i s consciousness expands . Duration alone i s a coOrdinate o f sp ace .The mind now recognizes space a s something apa rt

and sep a rate from itsel f only b ecause o f i ts unconsciousness of the i denti ty existing between i t and space .Just so , i t is not by m ind alone that the at-one stateo f consciousness shall be atta ined ; for al though in oneform or another i t is abl e to ga in som e knowledge o fthe apparent oneness o f all l i fe i t cannot di rectlyreal iz e thi s oneness. In order to do thi s fully i t mustb e able consciously to identi fy i tsel f with the l i fe , feelwha t i t feel s and experi ence what i t experi ences andothe rwi se come into a conscious relationship with theroot and source o f l i fe . Space-consciousness i s as imple , di rect cognitive process ; while time-consciousness i s a complex

, and there fo re , indirect process.The form er cannot be analyzed. That is, no analys i si s necessa ry to its suffi ci ent comprehensi on ; the l atte rmust always be analyzed and categori zed for

'

its suffi

ci ent apprehension . Eve ry moment o f tim e whetherpast, present o r future , when presented to the con

sciousness, is dete rmined by its rel ationship to som eo ther moment o f tim e . Space i s indivi sibl e ; tim e i s


to the seeds o f mental matter brought into exi stenceby the reaction o f Fohatic energy o r the Will o f theCreative Logos upon the substance o f the quartopyknotie stage

, is, symbol i cally speaking, more dense andcompact than the pyknons of the preceding stages . I ti s ensoul ed by the quartopyknon. It i s a rather complex state o f ensoulment consisting o f four condensations o r pyknoses .Th e next stage in the proces s o f kosmic involuti on

which i s a l so concerned with the p repa ration o f theevolutiona ry field i s tha t o f sextopyknosis and impli e sthe sena ry condensati on o f the origina l wo rld-stuffwith the vi ew to the formation o f emotional o r “a stral”

matter. Identi cally the same proces s o f ensoulmento r involution obta ins upon the pl ane o f sextopyknosisa s have been observed to obta in upon the p recedingplanes o f involut ion . Involution must necessa ri ly precede evolution . That which ha s not be en involved,enfolded or ensouled cannot be evolved o r unfolded .

Whateve r potencie s , powers and capab il i ti e s or qual iti e s and cha racteri sti cs that may appea r a t any timein the univers e o f l i fe and form must have first beeninvolved or enfolded or else they could not have beenevolved. Space i tsel f is an evolution . It is a processo f becoming, o f unfolding, o f flowering fo rth . As itevolves more and more there wil l appea r new andadded characteri sti cs and qual i ti e s o f l i fe and fo rm .

New possib il i ti es will a ri se and in the end a supernalvisi on o f a glo rifi ed univers e wil l burst into vi ew.

The scheme o f space-genesi s is completed duringthe sep topyknotic process wherein the ba sal elementso f dense physi ca l matte r and i ts va rious gradationsa re produced and given cha racte r

,form and di rection

.

NESIS AND NATURE OF SPACE 227

completion means merely a temporary estopthe p rocess of kosmogenesis which actuallyin the fo rm ation of physical matte r in itsstate . I t does not mean a final arrest o f therocess which i s conce ived o f a s continuing only

sive manne r back to a kathekotica conditiont embodi es the fruitage o f the enti re schem e .opyknon, accordingly, is a seven-ply pyknona re embodied , in varying degrees of maniand phane robiogenic ( l i fe-exhibiting ) qual ity,tialities o f all that has preceded on all planesng al l stage s o f space-genesis. That i s tohe physi cal l i fe of the universe i s confined theo f all the seri es o f grades of l i fe in the kosmos .

i’

s physi cal body a re wrapped up all the glories.bl e in hi s long, almost unending pilgrimage1lution ; i n it a re stored all the possibil iti es o firi t ; al l powers, all qual iti es, al l cha racteristi cs ,1tended for man’s attainm ent a re in the physical .ey must b e evolved , they must be unfolded and

sed . The physi cal must be glorified, spiritual

le ified. For by the way of the glorification andalization of the fl esh man may atta in unto oneith the divini ty in himsel f and consequently withfine l i fe o f the world .

1 summari ze : The genes is of Space embracesstages, namely, the monopyknotic, the duopykand the t ripyknotic which belong to the planen-mani festation and a re the primordial world

hekosis ( from Chaos-Theos-Kosmos) is to evolution whatmy

”is to invo lution. It is the end of evo lution but also the

1g of invo lution, and in the latter function 15 known as

my .

”See diagram No. 17 .


stuff and together make up the unmani fe st body of theLogo s of Being . These become the seed-germ of theunive rse o f spati al i ty. The quartopyknotic i s thefourth stage in the p roces s o f sp ace-genesis and i s theme tamorphotic o r cruci a l stage during which nonmani festation i s metamorphosed into mani festation .

In i t the unmani fe st be comes the mani fest. I t corresponds to the pl ane o f pure spi rit , and indeed , embodies within itsel f a ll the quali ti e s whi ch spi ritual i tyis to show forth during the l i fe o f the kosmos . The

quintopyknotic i s the fi fth stage and corresponds tothe mental pl ane , embodying in itsel f al l qual iti es o fmental ity in the universe and furni shing the basi s and

essence o f that which i s to become the kosmic mindin mani festation . The sextopyknotic i s the s ixthproces s and symbol iz e s the s ixth stage which embodie sall the characteri st i cs and properti e s o f emotional matte r i n the unive rse and i s the basal e lement o f thepla sti c essence o f senti ent exi stence in the kosmos . Thesep topyknotic i s the seventh and final stage corresp onding to the physi ca l pl ane o f the kosm os andconta ins i n i ts seven-fold constitution the seeds andpotenci es o f all the preceding stages

,a s well a s al l

the characteristi cs and properti es which physi ca l matte ris destined to show forth during the manv antara orWorld age. These seven processes re sul t in thedynamic appearance o f space , the mothe r and container o f all things , and compl ete the involuti onarya spects o f kosm ogene sis. Evolution began where involution cea sed and will end for thi s manv antara whenthe l ast vestige o f those powe rs , capab il i ti es and

potenci e s which were involved shall h ave been evolvedunto kosmic perfection.


chemi st choose to talk o f chemi sm , negative and positive , of combining properti es and dissoci at ive phenomena le t h im also become aware that these phenom enaare but the external a spects o f the inne r and ephemerall i fe-process es o f sp ace-forms and that ultimately these ,too , may be traced back into an eterna l or iginal itywithin the bosom of the al l-mothe r, spati al i ty.

D ense physi cal matte r , such a s consti tute s thephys i cal i ty of celesti al bodie s in i ts ultimate di ssoci ati on would , accordingly, be resolved into the o riginalchaogenetic formlessness whi ch marked the chaogenyo f non-mani fe station although i t would naturally beo rderly and progress ive p ass ing through the sevenstages, sep topyknosis, sextopyknosis, quintopyknosis,

e tc . , until the end had been reached , meanwhil e exhibiting in e ach plane the phenomena peculi a r to thedi ssoci at ive processes thereo f. On thi s Vi ew space i sa p lenum of matter o f varying degrees o f intensi ty,ranging from the densest physi ca l to the most tenuousand formless matter o f the highest levels of the manifested un iverse . But a s neithe r the dense materialforms nor the other grades o f matte r have an eternallyenduring qual i ty, being al ike subj ect to mutation , spacel ikewise falls unde r the l aw of becom ing whereby i t,too , must yi el d to the edict o f kosmic disorder.

Some may be incl ined to a rgue that since space andmind a re one and the same thing it must necessa ri lyfoll ow that whatever possib il i ti e s o f measurement maybe found to exist in the mind would logical ly b e foundto exi st in space ; and that s ince a ll the necessary conditions o f hypersp ati al ope rations are proved to b eexi stent in the mind the ca se o f the hyp ersp atiality ofperceptual sp ace i s proved thereby. In othe r words ,

IS AND NATURE OF SPACE 23 1

dimension can be p roved to be mentallyis also possibl e in perceptual space . But

ses a re not granted , and nei ther wil l theyto those minds who choose to take thatis known that there i s a marked difference

n the mind that is purely intellectual and mindpurely intuitional o r mind a priori. The inis fa shioned for matte r only ; i t is so constructedsqua rely into eve ry nook and cranny, everynd intersti ce in matter ; yet for the generatingl i fe , i t has no aptitude nor suitabl e con

atta inment o f the sp ace-mind or kosmic consuess would then imply a mastery o f all fundaal poss ibil iti e s pe rtaining to all degrees of matter.by becoming conscious in the matte r of all the3 one makes a ce rta in defini te approach to thisa te state o f consciousness unti l all the barri erszen o rdinary sel f-consciousness and the consciousof the sp ace-mind have been entirely obl itera ted.

'

yknosis, i n al l o f i ts septena ry aspects , i s cond primari ly with involution o r the preparation ofchaogenetic el ements fo r the work of kosmiclt l

’

l . I t may be thought of a s being divided intogreat divisions , namely : chaogeny and chaovhogeny. It is concerned with the o rganiza tionlaos , the establi shment o f kosmi c geometrism in'

ormless, vo id , arup ic substance and preparationevolut ion . Chaogeny, of course , i s that kosmicss by vi rtue o f which space i tsel f becomes maniand in which there i s no establ i shed order.morphogeny ( from Chaos+M orphe+Geny ) sig

the activiti e s o f the creative Logos in l aying the


foundations in primordial space-matte r o f the varioussta r-fo rms , including nebulae, worlds, planets, suns ,etc . , of which Canopus, Jupiter , Fomalhaut and Si riusand our own sun a re exampl es , giving di rection and

general tendence to thei r vari ed l i fe-processes . Boththese p rocesses a re concerned with the preparation o fthe fi eld and i ts consequent fertil i z ati on in anti cipationo f its cultivation and harvest. These two consti tutekosmic involution or the great l i fe wave’s p assage onthe downward a rc o f the G reat World Egg or C i rcl e .I t i s during the chaom orphogenic cycle that the constitution of the un ivers e o f man i festation is p romulgated ; when l aws fo r its government during thatmanv antara a re sketched out in the world o f nascentspati al i ty ; when the a rchetype o f every imaginabl e o rpossibl e form i s p roj ected upon the impregnatedscreen o f cre ati on , then folded in , pushed toward thecente r, involved, awai ting that time when the l i fe wavebegins i ts passage upon the upward arc and evoluti onensues , call ing forth al l that ha s been enfolded in thebosom o f the pyknotic

'

cent ers o f mani festation. I t i sea s ily conceivable that here during the troublous timesof the chaomorphogenetic enfoldment the now knownsix di recti ons o f sp ace were among the ete rna l edictso f space-genes i s and that that l aw which now makesi t appea r that thre e coordinate s , and only three , a resuffi ci ent fo r the determination o f a point position inspace wa s imprinted in the very nature o f that whichwas to become space .The kosmic fi eld having been prepared a s a result

o f th e chaomorphogenic activit i es, lowly and sca rcelyo rgan ized forms begin to appear and the a scent uponthe upward arc o f the G reat Cycle commences . Evo


G reat Kosmic Egg o r Cycle i s a lso divided into twogre at stages, namely, Morphogeny (man i festation o fl i fe through the var ious forms which i t assumes ) andKa thekos

,or the kathekotic plan e o f perfected triunity

which i s represented by the evolutiona ry union o fChaos-Theos-Kosmos. Ka thekos would , there fore ,symbolize the ultimate el aboration o f Chaos into awell-ordered kosmos wherein a re expressed al l theposs ib il it i e s which inhered in the a rchetypal pl an o fthe Crea tive Logos or Theo : and in whi ch all hadre ached the ultimate perfection in the body o f b eingo f the Logos H im sel f. But the kathekotic plane isto be di stingu i shed from the original Chaos-TheosKosmos represented as functionating upon the pl ane o fnon-mani festat ion during chaogeny. Kathekos sym

bolizes the perfected mani festations o f the triunea spects of the Creative Logo s through the perfe ctedforms resulting from the l abors o f kosmic evolution ,while Chaos-Theos-Kosmos symbol i zed, a s a triune

glyph, the Unmani fest Trini ty in the prim o rdi al beginnings o f sp ace-genesi s . One is the seed ; the otherthe ful ly ma tured plant ; one the egg ; while the othe ri s the full grown bi rd ; one the root ; the other thefruitage ; one Alpha , the other Omega ; one the beginning , the other the end. The end , howeve r, i sreached only that, i n due t ime , the enti re scheme maybe commenced again , once more util iz ing the resultso f the p receding scheme o f evolution a s the basi so f the ensuing one . Thus a fter every Kosmi c D ay,commences the Kosmi c Night. The succession o fkosmi c days and nights i s infini te . This infini ty o fbecomings i n the l i fe o f the kosmo s i s a necessa ryoutcome of eternal duration .

AND NATURE OF SPACE 235

e , thus bri efly set down, i s the symbol ismsis . I t i s commended to the re ader a s aconception that the real , essenti al , peri s something fa r more wonderful

,more

than e ither the geometri ci an o r the metaan ha s ever dreamed o f, and yet the l atte r

’sess is undoubtedly being appulsed by the

o f a new speci es o f conceptual izations which ,ay i n the not too distant future , will a rouse infa int hunge rings a fte r the reali zation o f the realna ture . These matheti c appetites thus broughteing will finally le ad the human m ind into then fields o f kosmic consciousness where fo r an

mill ion yea rs , perhaps , i t may feed upon theri e s and hyp ermysteries to be found in the

tri e s o f the Space-M ind.

“

he study o f space in its wider and deepe r meanis necess a ry in orde r that a cl ea re r understanding5 true s ignificance , a s the subj ect o f geometri crches, may b e ga ined . It is confessed , howeve r ,there is neithe r di rect evidence nor impl i cativeprity fo r any assumption that the vi ew herein outaffords any j ustification for the notion o f the

tensionality o f space . For , a lthough the l ine o fn ing indulged in must l ead inevitably to the con

)n that the worlds o f spati al i ty, materi al i ty , in

tuality and spi ri tual ity, essenti ally and fundaal ly one so fa r a s o rigins and qual i ti es a re cond, were al ike engendered by the same generatingent , l i fe ; and that spati al i ty being the primal bas i se others i s

,neve rthel ess , under the exigenci es o f

a spect o f the kosmo s , highly susceptibl e to the

urative requi rements o f the grosse st, there ap

236 T HE MYSTERY OF SPACE

pears to be no necess i ty for cal l ing upon extraneou scons iderations fo r a ss istance in our efforts to comprehend the various connotations o f the symbolism . Then , too , i t i s e asi ly conceivabl e tha t unde rconditi ons where these el ements, sp a ti al ity, intellectua l i ty and mate ri a l ity , a re not only co-extensive butinterpenetrative , there i s no j ustificati on fo r the a ssumption that they must exi st in l ayers o r mani foldnesse s or in di screte degrees , separated from oneanother a s i f they were consti tuted o f different substances and occup i ed different spheres . For everysingle point in perceptual space i s a focus fo r l inesdrawn through every conceivabl e grade o f materi al i ty,sp ati al i ty or intellectual ity in the kosmos . And thesame system of coordinates which i s necessary andsuffi ci ent fo r the l ocal ization o f a po int in our sp acei s also suflicient fo r the location o f a po int anywherein the enti re world o f spat i al i ty, intel lectual i ty o rsp i ritual i ty. In fact, the externa l , vi sibl e worlds o fmateri a l ity and spati al i ty a re nothing more than themass-termini o f l ine s extending from divini ty tophysi cal ity ; from primordi al o riginal i ty to kosmi cmoderni ty and i t i s intell e ctually conceivabl e that pro

gression back ove r the grooves made by these masste rmini o f l ines would l ead di rectly and unerringly too riginal i ty i tsel f. In spite o f the mani fold pyknose swhich we have shown to cha racte ri z e the symbol i smof space-genesi s it i s a very s impl e matte r ; fo r theenti re scheme could and must have p roceeded alongstri ctly tridimensi onal l ines . Tridimensional ity musthave inhe red in the primeval a rchetyp e o f sp ace or elsei t could not appea r a s an outstanding fact o f perceptualspace now ; for all that we can now observe in space


I t cannot be bel i eved, however, that m etageometri

clan s a re really in e arnest in what they suggest o fhype rspace and n-dimensional i ty ; i t cannot b e bel i evedthat they are enti rely s ati sfied with what they havefound of the so-called hyperspati al , and yet, some o fthem are fanati cally p atrioti c ove r the new-founddomain ; some a re even intolerant. But there a re

FIG. 1 9 .

—Kosmic Pentoglyph

others who look upon the fab ri c o f metageometry a sa stepping stone to space-real iti e s , a mil e-post on thepath to the re al iz ation o f a higher consciousness , theconsciousness of the space-mind or kosmic consciousness . And may thi s not, a fte r all, be the goal o f thehuman intell ect , now sl ightly distraught by the exuberances o f youth and the j oys o f a new mental freedom ? The work of the futu re mathemati ci ans willb e the de st ruction o f the tum orous inconsi stences to

SPACE 239

nd in the various non-Euclidean systems o frical thought , the el imination of the novelti es

sensicals, the synthesiz ing of those el ementsanctioned by the space-mind and the bu i lda sane inte rpretation of space-phenomenao f i lluminations rece ived from greatly exlt i es and a pa rt icipation in that la rger conwhich the human race seems slowly

merging.

domain o f hyperspace i s but the fa iry-l and ofpeopled with gobl ins , gnomes , kobolds, elveswhich are the Spaces , dimensions, proposi

ensembles and theo rems of the m etageom etri

But l ike the fa i ri es and nature sp i rits of the1 about us , they have thei r bases in the real ,iv e world o f facts however diffi cult it may beablish thei r di rect connection wi th it . As theiv isible sprite s o f phantom-l and represent intelna tural forces at work in the furtherance ofolution o f forms , so the impalpable things ofsi s a re emblems of kosm i c fo rces a t work in theding o f st ructu res o f higher consciousness whi ch) 6 towers o f vision for the human soul whencer vi ew the hill-crests o f infini te knowledge and

w-lying pla ins o f kosmic mysterie s .nally, i t ha s been noted that space i s the very:ence o f th e kosmos ; i t i s the l i fe , the form andhe oute r and the inner mani festation o f the comli fe and form ; i t i s real i ty, also illusi on ; it is

te,al so ideal . We have noted al so that m ind

substanti al with space and that space gives i te r l i fe and nature a s well as nouri shes its outer

h and development. In fact , we have seen that

240 THE MYSTERY or SPACE

sp ace and mind are one essenti ally and that they exi stas aspects o f the same thing , l i fe . In whatever way,

then, that the mind normally Vi ews sp ace that i s thenatural way. All attempts to devi ate from the naturalway a re , there fore , unsanctioned by the natu re o fthings . So long a s geometry rema ins true to thenatu re o f mind and space so long will i t be val id univ ersally and possessed o f kosmic necessi ty and invari ance . I t behaves most unseemly when i t depa rtsfrom its fe alty to the nature o f things p er se .

Both the outlook of the mind upon the obj ectiveworld a s wel l as i t s inlook upon its own states ofconsciousness o r the subj ective world a re tridim en

sional. Its growth i s tridimensi onal , its na ture i sl ikewi s e tridimens ional , and there i s not even thesl ightest tendence e ither to perce ive , conceive o r pe rform in a four-dimensi onal manner , mathemati callyspeaking. Trace out the b iologic developm ent o f eachmental facul ty, from the mind o f the mone ron to themind of the most highly developed man and it willbe found that eve rywhere and always

,without vari a

t ion o r exception , the natu re o f e ach o f thes e has beento exp ress i tsel f tridimensi onally and naturally. Therei s not even the sl ightest sign o f so much as a germinalappetence for the four-space ; i t would , there fore ,seem almost a prostitution o f mental facul ty to dive rtmental energy into the seemingly useles s channel o fp resent-day metageometri ca l re sea rche s ; yet, i t mustb e admitted that even though the end sought cannotbe atta ined , the final results o f the intellectual delv ingsinto the dread homogenei ty o f kosmi c origins and theconsequent real i z ation o f the awesome coev alism ofmind and space whence shal l a ri se the recognition of

CHAPTER VII I

THE MYSTERY or SPACE

The Thinker and the Ego —Increscent Automatism of the Intellect—The EgOpsyche and the Omnip syche

—KosmicOrder or Geom et rism—Life as Engendering ElementThe Mystery of Space Stated—Kathekos and Kathekot ic

Consciousness—Function of the Ideal—The Path of Search

for an Understanding of the Nature and Extent of SpaceMust Proceed in an Inverse Direction.

THE fragmenta riness o f the Thinker’s outlookupon the univers e o f spati al i ty i s due to the inhib itivea ction set up by the constri ctive bonds which his compli cate mechani sm o f intell ectual ity inte rposes betweenhimsel f and real i ty . The Thinker , who stands ba cko f and uses the vari ous medi a o f obj e ct ive consci ousness, such as the neura l mechanisms , brain , emotions ,his individual i zed l i fe-force and the mind which together make up the instrum ents with whi ch he contacts the sensuous doma in,

by adapting hi s consciousness to these means

, as the arti san uti l iz e s his tool s ,constitutes hi s own intell ectual i ty . The intell ectual i ty,then, i s the total i ty o f medi a by whi ch consciousnesseffects i ts entrance into the sensuous world and bywhich it receives impressi ons there from . In otherwords, i t is the sum o f al l those qual iti es , ope ra tions ,processes and mechanism s which are recogni z ed as

consti tu ting the modus v iv endi of man’

s intel lectual i ty,

THE MYSTERY or SPACE 243

se a re , in real i ty, nothing more than the ego

ny have been incl ined to regard that which hasll ed the ego as the highest sovere ign power ino f manhood. He has been looked upon a sconsi deration in the constitution of the hum anBut the ego is an evolutionary product andcom i tant o f sel f-consciousness which is thefaculty in man’s psychic l i fe . It is thatconsciousness which m akes m an conce ive o fa separate , detached and independent bea purely intell ectual or tuitional product ,h , is to be diffe renti ated from the intu itional

-qual i ty which i s the essence of man’s real sel fWith respect to the Thinker , the ego occupie sly the same status as the agent to his principal .agent i s the representative o f the principal in

ltters which come within the scop e of his prei j uri sdi ct ion so is the ego the agent o f thee r who i s a sp i ritua l intell igence . Accordingly,an ethi cal viewpoint the Thinke r i s respons ibl ee acts o f the agent and can in no wi se escape theies a ccruing a s a result o f the agent’s violat ions .s a commerci a l firm sends out a representativete collection o f data concerning ce rt a in phasesbusiness or it may be of any busine ss or theworld ma rket so the Thinker proj ects his ownousness into the mechanisms which a re in thei ry the egoic l i fe . That is, he sends out his agent ,0 ,into l i fe and into the obj ective world o f facts

emands tha t he shal l convey to him , from al lo f the territory which he i s expected to cover,

,s o f hi s findings . Of course , these reports which


are transmitted by the ego ( the intel lectual mechanismof the Thinker ) a re more or l es s well p repared sum

mations o f hi s individual observations and deductions.

These a re the percepts which the ego p resents to theThinker’s consc iousness. Concepts a re formed by theThinker in his treatment o f these sense-presentations .I t very frequently happens that the ego transmits reports which , for one reason o r another, give veryimperfect knowledge o f the matte r which his reportsa re designed to cover. Often it i s necessa ry that additional and supplem enta l reports be made about thesame thing , and even then, i t is well-nigh impossibl e ,i f not quite so , fo r him fully to cover every detai l o fthe matte r under considerat ion and in no ca se 18 it

poss ibl e fo r him to do more than report on the superficialities o f the questi on under scrutiny. I f the ego ,in hi s ope rations

,be im agined to be hampered by

simil ar ci rcumstances and difficulti es as those whichwould ordinarily beset a commerci al atta ché i t willthen be clea r that hi s reports must ever be fragmenta rybecause o f the inaccessib i l i ty o f much o f the data whichwould be necessa ry for a full repo rt , and further, because o f the inadequacy o f hi s methods and means o fgathering data due to the inherent l im i tations o f hi scapabi l i ti e s , endurance and perspi caci ty and innumer

abl e other l imitations and difficulti es which must befaced in all se a rch fo r the real . So that , whil e thesuffici ency o f the means which the ego enj oys a t thi sstage fo r al l p racti cal purposes i s granted no hes itancy is enterta ined when i t comes to a di scovery o fthe real s o f knowledge in decl aring thei r insuffici ency.

Then , too , when i t i s remembered that these egoicreports a re in the nature o f neurographical communi


the agent’s close appl i cat ion to hi s dut i es and the consequent success of his operations , confers ce rta in favorsupon the agent o r rem oves some of the restrictionswhich were o riginal ly imposed , give s an increase insal ary or promotes the agent to a higher and morelucrative offi ce with l arger powers and greater autho ri ty. Thi s i s analogous to what the Thinker does fo rthe ego . For he not only receive s reports from theego

,but o ften , in the shap e o f intu iti ons , gives addi

tional information as to the proper manne r o f do ingthings, sheds more l ight upon some obscure operation , commends fo r duty well performed , condemnsfor fa i lure s or for wrong-doing , rewards a rduous to ilwith greater powers of visi on , keener ins ight, grea tercapab il i ti e s ; in fact, promotes the ego in the sight o fother egos by marking him out a s an exceptional ego .

But the curi ous a spect o f this procedure i s tha t, in timeand a fte r the ego has been repeatedly commendedand promoted and otherwi se favored by the Thinker,he begins to think that he owns the fi rm , tha t he i sthe l i fe and main support o f the whole co rporation .

H e becomes a rrogant , sel f-wil led and finally falls intothe i llusion that he alone i s re sponsibl e for the phenom enal succes s o f the fi rm . Thi s i s the source o f thati llus ion o f the intellect which makes itsel f think that i t,the ego , i s al l there i s to man , tha t hi s inst ruments ofoperation in the obj ect ive world are the only kindo f instruments that may be used ; tha t h i s method ofgathering data about things i s the only sa fe and suremethod ; and so i t develops that the intell ectual i ty isthe source of man’s separateness , hi s individual i ty andhi s apparent aloo fness from othe r men and things. Itis, of course , needless to point out that in thi s way


llect comes to be the tyrant of m an, rul ing withmonopoly and as an all-exclusive autocracy.

m the above impl ications i t woul d appea r thatllect and the intu itive facul ty a re two separatetinct processes , and so they are . One i s theo f the othe r. The tendence o f the egoic l i feintellect i s for the exte rnal whi le the intui tionternal process . The intellect acts from withouts the inter io r while the intuition acts fromoutward . The intelle ct is the product of theion which i s another term fo r the consciousnesse Thinker on hi s own plane . Just a s the chi lda sepa rate and di st inct , though dependent , l i fethe parents so the intell ect has a modus v iv endi

1 is di stinct and separate from that o f the Thinker,yet i t is in all points dependent upon the l i fe o fFhinker. Here again, we find an analogy in theion o f the child to the pa rent. As som e childrennore amenabl e to the will o f the parent than'

S, so , i n some persons, the intellect is more amento the action of the intuition than in others . Yeta certa in fact that the more the outward l i fe i srned by the intu iti on,

i.e .,the more the intell ect

mds to the intui tive faculty of the Thinker, ther the orde r o f the l i fe o f the ego and the moreat e hi s deci s i ons and judgments. In fact, i t aslly may be a sserted that the pl ace o f every inua l in the scal e o f evolution i s determined in al a rge measure by the degree of agreement be1 the intu iti on and the intelle ct o r by the ease with1 the intuit ion may operate through the intelle ctmedium . At least , the qual ity of one

’

s l i fe may

: t ermined directly by these conside rations.


The Thinke r being himsel f a pure spi ritual intel

ligence , l iving upon the plane of spi r i t and there foreunhampered by the diffi culti e s whi ch the ego meets inhi s operations in the obj ective sensorium ,

and pos

sessed o f fa r greate r knowledge , i s co rrespondinglyfree from the l im i tat ions o f the ego and very naturallycloser to kosmic re al i ti e s . Hence , he i s better si tua tedfo r the procurement o f correct not ions o f rel ations ,essenti ali ti e s and the l ike . I t i s bel i eved, there fore ,that in the proportion that these two p rocesses , theintellectual and the intu itional , a re b rought , in thecourse o f evolution , to a closer and more rigi d agreement, in the proportion that the Thinker i s abl e totransmit the intuitogram s in the shape o f concepts orthat the intui t ion i s made more and more conceptual ,in j ust that p roportion i s human ity becoming p erfectand its evolution complete . The diffi cul ty found toinhere in the conceptual iz ation o f intui tions so thatthey may be propagated from man to man seems notto l i e in the Thinker himsel f

,but more essenti al ly in

the ego , in the intellectual i ty and i ts compl i cateschematism or pl an o f action . It would appear ,there fore , that the only way of escap ing or t ranscending th i s diffi culty is fo r the ego so to refine hi s vehicl e so r so facil i tate hi s pl an o f a ction by el iminating thenumerous rel ays o r sub-stations intervening be tweenthe consci ousness of the Thinker and that which maybe sa id to b e hi s own that the transmissi on o f intuitogram s may be accompl i shed with the greatest easeand clearness . Whil e no attempt will b e m ade to indicate the p robabl e l ine o f action whi ch the ego o robj ec tive m an will adopt for thi s purpose , i t is beli evedthat i t may be sa id without pedanti ci sm that the only


unive rsally obse rvabl e and existing in gre ater or l esserdegrees in various human beings, there is ample evidence fo r the bel i e f that i t i s b e ing care fully and dulyp romoted by a well-di rected evolution o f psychi cfaculti es and powers, so that at the p rope r time , determinable by the state o f perfect ion reached by theintell ectual ity o r the ego in the operation o f hi s cognitiv e processe s , the much desi red agreement o f thesetwo faculti es will have been re al i zed and the conceptualization o f intuitogram s into propagable conceptionsan accompl ished fact . Unti l thi s goal shall have b eenreached and the intuiti on shall have overshadowed theintell ect as the intellect now overshadows the intuiti on ;o r the consc iousness o f the ego , derived from the interplay o f the Thinke r

’s consciousnes s among thevari ous el ements which constitute the ego himsel f, shallhave been merged with that o f th e Thinke r, the outlook must remain fragmenta ry, only becoming a wellordered whole a s the ba rri ers o f di ss idence a re brokendown in success ion .

The evolution o f consciousness , from the simple ,undifferenti ated mone ron to the diffe rentiated cell andfrom that to the cel l-colony and from the cell-colonyto the o rgan ism , traversing in successive pacesthrough all the stages o f lower l i fe—mineral , v ege

tabl e and animal—to the stages o f the simple , communal consciousness o f the highe r animals, to thesel f or individual consciousness o f the hum an be ing,e ach requ iring mill ions o f yea rs fo r i ts perfection before a more advanced stage is entered , ha s been onecontinuous rel inqui shment o f the lowe r and l ess compli cate fo r the highe r and more compl ex expressiono f itsel f through the given medi a . When a newer and

THE MYSTERY OF SPACE 25 :

stage o f consciousness is being entered by huty its appea rance or mani festation i s fi rst madee most advanced of the race and that only in avague way. This rudimenta ry condition persistsome tim e , perhaps many thousands o f years , thenl ty becomes more general in appearance

,the

o f advanced individuals increases, and conse

a s in the ca se o f the intuitive facul ty, i t bes universally prevalent in all humani ty ; becomesmissible as so-cal l ed “acquired cha racters

,

” andappea rs a s the normal fa culty of the entirean family cropping out in e ach individual . Thus,oassing from the few advanced ones in the beginto that stage where i t becomes the common

;ession o f all , a faculty requi res many thousandsea rs for i ts perfection , and especi ally ha s this beenin the past histo ry of the development o f humanulties. But i t is bel ieved that the sweep of thecurrent as i t p roceeds from form to form , from[lty to faculty, gains in mom entum as i t proceeds,hat in these l a tte r years due to the alre ady highlyslop ed Vehicul a r mechanisms at i ts di sposal not

great a period of time as formerly i s requi red fo rout-bringing o f a new faculty. It might well bewhil e in the past hundreds of thousands of yearse necessa ry in the perfection of organs and faculin these l atte r days only a few thousand , perhapsdreds , may be necessa ry and that in the days o ffuture not even so many yea rs may be requi rediniv ersalize a facul ty. And especially does thi sear to b e true in a state o f affa i rs where so l a rgeimber o f persons a re beginning consciously to taker evolution in hand and by vol iti onal activi ti e s a re

252'

THE MYSTERY or SPACE

supplying greatly increa sed impetus to thei r psychi cprocesses which under ordina ry, natural methodswoul d be cons iderably slower in the i r development.It is qui te obvious that al l cultu ral efforts when appli edto the betterment o f a given pl ant, animal or fa cultyresult in a corresponding hastening of the p rocess o fgrowth far in exces s o f what tha t growth would beunder normal , na tural conditions . Al l the presentfacul ti es possessed by man are remarkably susceptibl eto cultural influences ; in fact , the standing edict o fethica l and soci a l l aw i s that the human faculti e s mustbe cultivated a s highly a s possibl e , thereby giving thespi rit a more perfect medium o f expressi on. Theseobse rvations , there fore , l e a d i rresistibly and unav oid

ably to the conclusi on that the time fo r the upspringingo f the intui ti ona l fa culty in the human organism iseven now upon us , that undoubtedly in ce rta in ve ryadvanced ones i t h as al ready reached a notable degreeo f perfe ct ion and i s rathe r more general than woul dappea r in the absence of ca re ful investigation .

Now,just as the intell ect ha s made for individu

al i ty,ha s emphasi zed the separateness o f the Thinker’s

existence from that o f other thinke rs , ha s developedsel f-consciousness to a very high degree , even pushingi t fa r ove r into the domain o f the higher consciousnessto the tempora ry obscuration o f the l atte r, so the intui tional wi ll make for union , fo r the brotherhoodof man , for co-operation and for the common weal .Through i t man will come gradually into the consciousness that fundamentally, in his inne r nature , inevery respect o f vital conce rn , he i s at-one wi th hi sfellowm en and not only wi th the appa rent uni ts ofl i fe but with al l l i fe as expressed in whatso ever form


gradually to perfo rm thei r speci a l l abo rs reflex

iv ely the inte ll ect began to be formulated and togrow, a t fi rst only incip i ently, then more and morecompletely until i t reached its present state . At thep resent stage o f i ts evolution , a great deal o f thel abo r o f the intel lect i s beginning to fa ll into a kindo f increscent automati sm , although only rudimentarily,i n many instances . Yet, a s a resul t o f thi s tendency,quite the whole o f the phenomena of perception i scharacteri zed by a so rt o f automati c action . And themind perce ive s without conscious voli tion. Many ofthe steps o f conceptual i zation a re automatic , i n part ,i f not wholly. Ce rta in i t i s that impul se s once setin operation whethe r consciously o r unconsciously continue to a ct along the s ame l ine until exhausted orunti l the end has been atta ined. Consequently, i t i sa p roven fact that o ften se rious mathematical andphilosophi c p roblems have b een solved by the mindlong after any conscious effo rt to solve them hadceased. Often solutions have been a rrived at duringsl eep . M any such cases might be ci ted, but thephenomenon i s now so comm on that almost eve ry onecan cite some experi ence in his own li fe that will substantiate the cl a im .

There i s no doub t but that these phenomenaa re evidences o f a refl exive development in theintell ect. The tim e will come undoubtedly, and necessarily so i f the intellect i s to give way to a higherfa cul ty , which shall b e a s much above the intelle ct ini ts grasp o f things a s the intellect is now above thesimple consci ousness o f the lower anim al , when qu i tethe enti rety o f our intell ectual p rocesses will becom eautomati c or s el f-perfo rming . What then remains o f

THE MYSTERY OF SPACE 255

egoic schematism , a fte r its transmutation or el eon a s the o rgan of the intuitional consciousnessbe uti l iz ed as the organ of the Thinker’s invol

.ry cogni t ive processes . This will mean that al lhat l aborious ideation which i s now the abstracttght o f the Thinker wil l be perfo rmed autoleally, l e aving the highe r aspect o f the egoic con

i sness free to conceptual ize or intuitograph theitions. Percep tualization then will be repl aced byep tualization. This la tte r will occupy about thee sta tus as the former does now. And neces

.y, perception will become more complex . In othe rwhil e we now perce ive s impl e percepts whicha in a rranged into concepts making a compositethe obj ect, we shall then be taking in thep ictu re o f the obj ect a t fi rst hand , thereby

nsing with the ra the r Slow process o f perceptionnow operates . We shall sti ll be perce iving, butwe perce ive will b e concepts rathe r than peras a t p resent.he increased powers o f intel lection gained a st o f th e increscent automa tism in the inthe flowering forth o f the intu itive facultygenera l enhancement o f the intellect throughits processes will enabl e i t to ente rta in con

r compos ite p icture o f things j ust a s readilyperfectly a s i t can a t present deal with a s ingleConcepts wil l b e repl aced by supe r-concepts

tographs. Increa sed persp icacity will enableuker to manipulate the concepts and intui to

the same ea se and readiness and withall have a tta ined unto an almost unrealiz

fte r truth .


The outcome of th i s new adjustment which , ofcourse , will not sp ring up at once , but by insensibl edegrees , wil l b e the cl arification and unification o f ourknowledge . It will mean also the s implification o f i t ;the obvi ation o f divers i ti es of opinions , the springingup of a new and winnowed system o f philosophywhich shall be the t rue one ; further , i t will imply thel essening o f ' the probabil i ty o f error in our j udgmentsand conclusions ; the removal o f i llusi on to a muchl a rger degree than to-day i s poss ibl e and the real i zati on by every one o f something o f the essence o f things ,o f cause s and effects , o f a ct ions , operations , na tura lforces and l aws ; i n fact, a condition o f mind whichwil l p resent to the consciousness the simple truth aboveevery conce ivabl e phase o f kosmic l i fe which may comewithin the scope o f the Thinker’s obse rvation .

The further impli cations o f thi s vi ew a re that the rei s a diffe rence between the Thinker and the intellectuality. The Thinker i s e ternal and p artakes , therefore , o f the ve ry essence o f primordi al o riginal itywhil e the mental ity i s an artifici al p rocess, the re

sultant o f the adaptation o f the Thinker’s consciousnessto hi s vehi cul a r contrivance s o f obj ect ive cogniti on andthe interpl ay o f hi s l i fe among them .

I f the appea rance o f a choppy sea di sturb ed by thepassage o f a b risk breez e over i ts surface be imagined ,a s imil i tude o f the grea t ocean o f l i fe may be en

vi saged. The wavelet crests symbol i ze the egos ; thebas e o f the wavelet which i s one with the grea t seao f water represents the Thinker which i s one withthe divine l i fe and consciousness o f the kosmos . Justa s wavelet crests a re continually

'

sp ringing up andfall ing back into the sea, so are egos continually being


examiners thereby constructing a composi te ide a ofthe enti re mass o f j udgments i s the more rel i able andthe more co rrect ?

Re fe rring aga in to the dual intell igence , the egoand the Thinker, whi ch togethe r consti tute man , i t i sdeemed nece ssa ry, in o rde r to present the concept ofthi s dual i ty to the mind o f the reader in the way thatshall enabl e him easily to recall i t, to designate theegoic intell igence a s the egopysche, and the Thinker

’sintel l igence a s the omnipsyche .

The egop syche i s the I-making faculty, the fa cultyo f sel f-consciousness and the synthe si s o f all thosepsychic s tates and functions known as the intellect ormind and includes the ethi cal a spect of man’s nature .The omnip syche i s the o rgan ism o f kosmic consci ousness , the space-mind, or man

’s higher sel f and thatwhich connects with o r al l i es him to all l i fe ; i t i s thebasi s o f human uni ty and o f uni ty with divini ty, justa s the egop syche i s the b asis o f sepa ration and indiv iduality ; it i s the o rgan of direct and instantaneouscognit ion and the permanent essence which has pers isted through every fo rm which the being, man , hasever a ssumed and through every stage o f human ev olution . In i t a re stored up the memori es o f the Thinker

’

s

past, the secrets o f l i fe , mind , being, real i ty, and thehi story of l i fe from the beginning ; in i t also the pl ano f a ction fo r the future o f the ‘ l i fe-wave a s i t passesfrom plane to plane , from stage to stage , and fromform to form . It is the spark from the fl ame thati s never quite free from i ts source ; i t is the continuousSpark , the prolonged ray which does not go out and

cannot be extingui shed . It is tha t in man which when


union therewith has been attained makes him ain full consciousness.

he omnip syche i s really a neglected and overd facto r in the doct rine o f evolution . Evolu

ts , while they cl a im l i fe to be continuous andn has come through all the kingdom s of naturesi on and has spent mill ions of years in the

eetion o f hi s va rious organs , faculti es and stagesonsciousne ss, make no ample allowance fo r whatreal ity the basa l el ement in evolution—a continupe rsisting, permanent l i fe-force which does notits i dentity from the beginning to the end of theess . This fact—that that spark of l i fe which se t

upon the evolutiona ry j ourney a s a moneron hasved steadily from that stage to manhood , maining meantime its original purposiveness and intentems to be the most obvious conside ration of thesle doctrine , yet it has been more or l ess comely ignored . The elementa ry requi rements oflution woul d seem to establ ish clearly the necessi tysom e such eternally persisting p rincipl e as thelip syche which i s capable o f such subtl e adaptationsvery conceivable form of l i fe and in which should

gathe red up the evoluti onary results o f eve ry l i fee. For thi s purpose the omnip syche o r uni fyingciple in man was designed from the beginning andthat which constitutes the basis o f his intelle ctua lire while in a fa r l a rge r sense i t i s the divini tytan him sel f. I t is indeed strange that so importantctor a s the omnipsyche should have been om i ttedevoluti onists . Yet i t can be accounted for upongrounds o f the purely mechanisti c characte r o f allllectual a ttempts a t solving the problem s o f vital


mani fe stations . But so long a s m en rely uponmechani ca l expl anations o f such phenomena so longwill they be p rone to overlook the very essenti al it ieso f the p robl ems which they devoutly wish to solve .The continuity of the physi cal germ-pla sm of thehuman Speci es,1 now qui te generally admi tted , wouldsuggest, i t seem s , an analogous condit ion to the cont inuity of the psychi c pl a sm call ed the omnip syche , theonly difference being that the omnipsyche i s an in

telligent factor while the physi ca l pla sm is a mediumof transmission though non-intell igent. The omnipsyche i s , there fore , the psychi c reservoi r of evolutioninto whi ch a re stored the transmuted psychics o fmoneron, amoeba , j ellyfish and every o the r formwhich it ha s ensouled and a cts as the storeroom o fman’s p sychi c op era tions a s well a s the source o f hisintel le ctual i ty.

We turn now from the study of a sketch o f themechani sm o f man’s consciousne ss which gives at itsbest only a fragmentary Vi ew o f the universe o f spatiality to a consi deration o f space i tsel f in the l ightof i ts interrelational bearings upon the question o fintell ectu al i ty.

In the chapte r on the Genes i s and Nature o fSpace” we have , in tracing out the engenderment o fspace , proved i t to b e basi cally one with ma tte r ( andindeed the progeni to r o f matte r ) , al so with l i fe andconsciousness . Further,

‘it has been shown that a ll thecharacteristi cs of materi al i ty a re due to the adaptationo f consciousness to i t and that out o f thi s adaptati ongrew the intellectual i ty. A close approximation to

‘See The Germ Plasm ; A Theory of Heredity, by A.


in fact i s the total ity o f such o rganized and or

dered extension , which conforms to the l atent geo

m e trism the engenderm ent o f which it is the sole causein the l a st analys i s . Does i t not appea r then that al lthat mass o f a rtifici a l geometry which has sp rung upa s a result o f departures made from thi s naturalgeom etry is utterly basel ess and most ce rta inly l a ckingin the kosmic agreement which spati al i ty l ends to ourprima ry conceptions ? Of cou rse , i t i s a dmittedly possibl e to devi s e certa in conventional forms o f logic andendow them with al l the evidences o f a rigi d consi stency but whi ch , because o f thei r purely a rt ifici alcharacte r

,will fall fa r short o f any real confo rmi ty to

the potenti a l geom etrism which has been establ ished inspati al ity . And this fact i s o f utmost S i gnifi cance fo rall those who s eek to find j ustification eithe r logical lyor na tu ral ly fo r the existence o f a mul ti-dimensionalqual ity in space ; for , i f a cl ea r, discriminative concep tion a s to the categori cal rel ationship , each to each ,o f the two kinds o f geom etry be carri ed in mind , i twill not be easy to confound them neither will i t bedifficul t to discern where the one ends and the othe rbegins .Now, the fourth dimensi on and the enti rety o f

those mathem atica l specul ations touching upon thequestion o f hyperspace , dimensi onal i ty, space-curvatureand the mani foldness o f space a re purely convent ionaland arb itrary contrivances and do not mee t with any

agreement o r authority in the native geom et rism whichwe find inhering in spa ce and which the intellect recognizes there . This conclusi on seems to b e obvious forthe rea son that, in the fi rst pl ace

,the non-Eucl idean

geometri es have been constructed upon the basi s of a


ation o f the l atent geom etrism of space and intelJality ; and i f so , is i t reasonable to expe ct thate r they or any of thei r conclusions should accordl the nature of that form of geometry so admi rablyneated by EUCLID ? Obviously not . It is a ma tte rlistorical knowledge that the whole o f the artifici a lE ucl idean geom etri es consists of those purely con"ional results which investigators a rrived at whendeni ed o r controverted the norms suppl ied by the

iral geom etry. When m etageom etricians foundthey could neithe r prove nor disprove the

lidean parall el-postulate they then set upon thenination o f ideal ized constructions which negatived

late . The results, thus obta ined , althoughtent enough , were compi led into systems of

ett y whi ch natura lly were a t vari ance with eachand with thi s inherent geom e trism which is foundatiality and answered to by the intellect botha lly and logically.

rthermore , there is anothe r consideration whichseems to be equally i f not more forb idding, in

j ections to the cobrdination of the two systemsom etry, and that i s the fact that the geom etry

ace i s deni ed the corroborative testimony ofience and thi s i s t rue o f practi cally the whol edata . Indeed , there i s perhaps no singl e el emententi re constitution which cla ims the authori ty of

rience . Thi s is undoubtedly the weakest po int instructure o f the hyperspati al geom etr i es . COD-3

ly, such i s not the ca se wi th the natural geometry ;in thi s , the intelle ct in retracing i ts steps over thel a i d out by that movement which has a t the samecreated both the intellect and spati al i ty, finds an


o rde rly and commodious a rrangement into whi ch i tnaturally and easily fal ls . So exact is thi s agreemento f the intellect with the kosm i c o rder that i f i t werepossibl e to remove the

'

whole o f spati al ity and materiality there would stil l b e left the frame work whichi s this l atent geom etrism of ko sm ogene sis. But the factthat the intell ect naturally fi ll s all the inte rsti ce s o f mate riality and spati al i ty, fi tt ing snugly into all o f them a si f molded for just tha t purpose , by no me ans warrantsthe a ssumption that i t would o r does also fi t the en

gendering facto r which has cre ated these inte rsti ces .The frame work , the order or the geom etrism of thekosmos ha s been establ i shed by l i fe acting consciouslyupon the universum o f materi al i ty. And in order toestabl ish thi s geom et rism l i fe had to be mobil e , active ,creative . It could not remain stati c , immobil e , andaccompli sh i t. Being mobile , dynam i c , creative , i tp asse s on . It i s l ike a fa shioning tool which the cabinetmakers use in cutting out designs upon a p i ece o fwood . It moves , and keeps moving unti l the desi gnis finished , and then i t i s ready fo r more designing .

Li fe i s l ike that . It cuts out the designs in materi al i ty,fashions the fo rm , molds the materi al , and passes onto other forms . The intel lect fi ts into these designsgrace fully. But what i t finds is not l i fe i tsel f, onlythe design which l i fe ha s made . Hence , a s there i sne ither an empi ri ca l sp ati al ity nor materi al i ty in confo rmi ty with which the a rtifici al geometry o f theanalyst m ay be sa id to exist, and a s i t may not bes a id to conform to the path which l i fe ha s made inpassing through e ither o f these , i t i s absurd to predicate i t upon the same ba sis a s the natural geometry.And so , we are forced , in the l ight of these considera


is that out of which the germ ev olv es. Kathekos or

chaos is the unmanifest,unorganized, unconditioned

,

unlimited and undifferentia ted plasm . Space is the

manifest,limited

,finite

,organized germ that

,fe eding

upon the env elop ing chaos,exists in a p erp e tual state

of alternate manifestation and non-manifestationapp earing, disapp earing and reapp earing indefinitely.

The appearance o f the kosmos a s an orderlyel abo ration o f the involutionary phase o f kosmogene

sis, in so fa r a s kosmi c order may be sa i d to be anaccompl i shed fact, marked the turn ing point in thatp rocedure whose funct ion i t was to make mani fest auniverse possessing certa in definite chara cteri sti cs o forderl iness ; but the kosmos, as i t now stands , may notbe thought o f a s having atta ined unto a state o f ultimate orderl ine ss . The idea meant to be conveyed i sthat between the point o f b ecoming and the actually

pyknosed, or sol idified stage in the p roces s o f creationthere i s a more o r les s well defined l in e o f demarka ti oncutting off that whi ch i s spati al i ty from that which i snon-spatial i ty. Beyond the l imits o f spati al i ty i s anabsence o f geometri c o rde r. Here geometry bre aksdown , becomes impotent, because i t i s an intelle ctualconstruction ; a t le a st, i t i s not so apparent a s in themani fested kosmos . It i s a state about which i t i sutterly futi l e to predicate anything ; because no wordscan describe i t. The most that may be s a id i s tha tit is absence o f geometri c o rde r a s i t inhere s in sp ace .And i f so , al l those movements comprehended underthe general notions o f spati al i ty

,material i ty, intellec

tuality and geom e tricity have both the i r extensive anddetensiv e o r inverse movements null ifi ed in their ap

p roach to i t. Involutiona ry Ka thekos,there fore , may


sa id to be the primordi al wilderness o f di sorderich outski rts the well l a id-out and care fully pl anned‘den of the spati a l universe . We may excogitatein some of the obvious functi ons of thi s kathekoticrld—pl asm ; but in doing so we must l eave off all:mpts a t a description of i ts appearance

,its magni

e, extent o r other qual i ties, and think only o f itsmic function . We cannot say that there i s ba ckit a spati al i ty no r can we say that i t i s a spati al ity ;whatever may be i ts extent o r volume

,i t suflices

t i t may not be sa i d to be space. It i s chaos. Spacerde r, organi zation , geom et ricity. It cannot be sa idt there i s a l atent geom etrism in chaos ; becausemetri c orde r i s found only in spati al ity and is thatch distinguishes spati al ity from kathekosity or nontiality. Chaos i s the l ack o f spati al i ty. This , oft se , impl i es that i t is impenetr able to the inteluality or to vital ity. Al l inverse movement such8 discove red as taking place in spati al i ty and whichlltS in the phenomenalization o f space runs agroundn i t strikes aga inst the rock-bound coast o f katheity. We can only say that i t i s both the point o f

gin fo r the evolving universe o f l i fe and fo rm andterminus . It is the nebulosity out o f which the)lC cam e and into which al l i s ul timately occluded .

A great and fa r-reaching error is made in al l ourking wi th respect to the kosmogonic processes:n we postulate the complete absorption of chaosin e arly a ct o f kosmogony. Customari ly, we thinkcosmi c chaos a s a primordi al condition whose exlCC was done away a s soon as the universe came intove mani festat ion . Thi s because i t has been exceed

y diffi cult, i f not quite imposs ibl e , fo r those whose


privi lege i t was to determine the trend o f phi losophi cthought to free themselves from the bondage o f adogma which owed its exi stence to a tradi tional o rl egenda ry inte rpretation o f facts that ought neve rhave been so interpreted. Chaos Is and EVER SHALL BE

,

so long a s the univers e i tsel f l acks compl etion,fullne ss

o r pe rfe ction in purpose , extent and poss ibi l i ty. I ti s undoubtedly being diminished , however, in p roportion a s the kosmos is approaching absolute perfect ion .

And when the last vestige o f chaos di sappears fromthe out erskirts o f the maturing kosmos there shall app e a r a glorified univ erse of indescrib able qual iti e s.

Space being a perception a priori cannot b e det ermined wholly by purely obj ect ive methods . Theya rd-sti ck, the telescope and the l ight-year a re obj ectswhich belong exclusively to the phenomenal and wi ththem alone neve r can we arrive at a true conceptionof the nature o f space . We can no more demonstratethe n ature o f space by the use o f obj ective instrumentsand movements than we can mea sure the spi ri t in abal ance . Certainly, then , i t cannot be hoped that bytaking the measurement o f space-distances in l ightyears

,or studying the na ture o f materi a l bodies , we

shal l be abl e to fathom thi s most obj ectively incomprehensible and inelu ctabl e thing which we cal l space .I t i s such that eve ry Thinker must, in hi s own inne rconsciousness , come into the re al i zation o f that awfullymyste rious something which i s the na tu re o f space bothas to existence and extent by hi s own subj ect ive effortsuna ided , uncha rted and alone . When we measure ,weigh , apportion and otherwise try to determ ine athing we are deal ing with the phenomenal which i s no


universe,and its rel ation to consciousness may b e

l ikened to a coni cal funnel whose bas e represents thephenomenal world o f the senses and whose ap ex o rsmallest point represents ultimate re al i ty.

In Figure 20 , we have endeavored to symbol iz e

graphica lly thi s conception o f sp ace . The base markedSenso rium” rep resents the s ensibl e world. Thatmarked “

Real i sm” symbol i ze s the ul timate pl ane o fre al ity, the inne r ess ence o f the world , the plane o f“things-in-themselve s ."

The cone a ri sing from the b ase s ensorium symbolizes the obj ective world a s compared with consci ousness ; the subverted cone , with apex in the sensorium ,

rep resents the evolving human consciousness .The success ive bases have the following symbology

Sel f-c onsciousness , Communal Consciousne ss , M ikrocosmic Consciousness , Makrokosmic or Unive rsa lConsci ousness , the Pl ane o f the Space-M ind Consciousness , D ivine Consci ousness , Kathekotic Consciousness ,o r the Pl ane o f Final Uni on wi th the M ani fe st Logos .

Sel f-consciousness i s that form of consciousnesswhich enables the ego to become aware o f h imsel f asdi stingu i shed from othe r selve s o r the Not-sel f ; theOmnipsychic o r Communal Consciousness i s that formo f consciousness from which ari se s the real iz ation bythe Thinke r o f hi s oneness with al l o ther thinkers andwith others fo rms o f l i fe : Mikrocosmic consciousnessdenotes a sti ll highe r fo rm o f consciousness , a s thatwhi ch enable s the Thinker to become conscious of

hi s l iving ident ity with the l i fe of the world o r theplanet on which he l ives . It represents a stage in thcexpansion o f consciousness when it b ecomes one wit]the consciousness o f the pl anet upon which i t ma]


REALISM

[Fm ao .

—Kosmos and Consciousness


be function ing . Makrokosmic consci ousness accom

plishes the awareness o f the Thinker’s unity with the

l i fe o f the kosmos o r univers e . The space-mind andthe consciousness which const i tute s i t enabl e theThinke r to comprehend the original ity and the te rminality o f kosmi c processe s . I t i s a rchetypal so fa ra s the l i fe-cycl e o f the universe is concerned becausethe beginning , the inte rmedi ate portion and the endingo f the kosmos are encompassed within it. D ivine consciousness i s that form o f consciousness whi ch ari se supon the unificati on o f the Thinker’s consci ousness withthat o f the mani fe st de ity ; i t i s , in fact , omniscience.The kathekotic consciousnes s b elongs to the ult imatepl ane o f real i ty ; to kosmic origins and chaogeny, and

there fore , p erta in s to the plane o f non-mani festation .

The impl i ca ti ons a re that i n compari son with the

senso rium,the Thinke r’s consci ousness is a mere po int

in space . It i s , in re al ity, so smal l and insignificantthat the extensi ty o f the physi cal worl d o r unive rseseems unl imited

,unfa thomabl e in meaning and infinite in

extent. But a s hi s consci ousness expands , a s i t passes ,in evolutionary success ion from one plane o f real i tyto anothe r and higher one , the i ll imitab il ity, the in

comprehens ibi l ity and infini ty o f the universe grow

ever small er and small e r, unti l the plane o f divine

consciousnes s i s reached . Then the p reviously incom

prehensibl e dwindle s into insignificance , lost in the realill imi tabil ity, infini ty and unfathomabil i ty o f con

sciousness i tsel f . Kosmi c psychogenesi s , a s exhib i tedand speci al i zed fo r the purposes o f the evolution o fthe Thinke r

, can have no other destiny than the flower

ing forth a s the ne p lus ultra of mani fe station which


kosmic consc iousness and the re alone i s the myste ryo f sp ace despoil ed of its habil iments .Accordingly

,a s consciousness or the Thinker is

more and more divested o f ca rnal b a rri ers and i l lus ions there develops a gradual re cognit ion o f the uni

tariness of sp ati al extent and magnitude ; there a ri se s

Comm a! S ac o.

Consciousne ss onscrousness

FIG . 21 .

- Septenary Enveilment of Consciousness

the certa in knowl edge that space ha s but one dimen

s ion and that dimens ion is sheer extension. TheThinker’s sphe re o f awareness i s rep res ented as i f i tb egins as a po int in space and develops into a l ine whichdivides into two l ines , the boundaries of the space cones.

Thus i t may be perceived that the anci ents had a simil a r conception in mind when they symbol i zed kosmogenesi s with the dot the l ine , and the ci rcl e with

THE MYSTERY or SPACE 27 5

te r inscribed , which together represent the uniin mani festation .

reali ze the imposs ibil i ty o f adequately depictful l significance o f the inverse ratio exi stingthe extensi ty o f space and the increscent inof consciousness by means of graphs ; forwords nor diagram s can portray the scope andg o f the conception in its enti r ety. Yet theye intellect to grasp a ray of l ight

,an intimation

a t the Thinke r sees and understands interio rly.thi s connection i t i s interesting to note the funco f the ideal in the evolution and expansion ofciousness. The ideal ha s no perceptual value ; i tno status in the world of the senses. I t is unapachable e ither in thought o r action, and there fo re ,beyond the grasp o f both the intellectual ity and

vital ity. I t is indescribable , inconceptible and'

chless ; for the mom ent tha t we describe , define,approach the ideal , e i ther intelle ctually o r vitally,fiat mom ent i t ceases to be i deal , but actual . It fl eesn even the sl ightest approach ; i t neve r remainssame ; i t cannot be atta ined , at leas t its atta inm entses i t to lose its idealty. It is then no longer thell. It i s l ike an ignis fatuus; the close r we com et the fa rthe r away i t recedes. It hangs suspendedare the mind l ike the luscious grapes which hungare the mouth of the hungry Tantalus. As the

pe s and the wate r receded from his reach a t everyrt he made to seiz e them so the ideal remainsn al ly unse izable and unatta inable . Whateve r,°

efore , i s in our thought processes , o r in our knowle,that may be sa id to be ideal

,does not really

t . The ideal is a phantom growing out of the


nature and essence o f the intell ectual ity. Its purposei s to l ead merely the mind on ; to al lure i t, to tantal ize ,and compel i t to grow by exertion, by the struggl eto a tta in, by the desi re to ove rcom e . In thi s respect,i t se rves well i ts function in the economy of intell ectual evolution . It is a mysterious aspect of theo riginal and ete rna l desi re to l ive which i s the kosm i curge present in all o rgani zed being and has its rootsh idden in the divine purpose o f cre at ion .

Ideal i zed constructions, then, a re l ike Arab ianfe asts conjured up by a famishing menta l i ty. Theya re l ike the dre ams o f a sta rving man in whi ch hea ctual i ze s in phantom-stuff the cho icest vi ands inabundant supply fo r his imagina ry dele ct ati on . Themind that is s atisfied never ideal i ze s , neve r makes anideal ized construction . I t is only when an “ a chingvoid” is felt , when a longing for the real i z ation o fthat which i t has not a ri ses within i tsel f, when afeel ing of dist inct l ack , a want, a hanke ring a fte rsomething not in i ts reach , takes possess ion o f themind that i t begins to i deal i ze . That i s why someminds a re without ideals. I t is because they a re s ati sfied with what they have and can understand. Theyfeel no hungering for bette r and grander things ; theyhave no desi re to understand the unknown and themysterious ; hence they do not ideal iz e ; they make noattempt to rep resent unto them selves a p icture o f thatwhich is beyond them . Such minds a re dormant ,hibem ant , asleep , unfeel ing and unresponsive to thedivine urge .But the ideal i s ne ither obta inable obj ectively nor

subj e ct ively, neither phenomenally nor re ally, so thatwhen we come upon the i deal in our mode o f think


range of conce ivab il ity. I t al so seems reasonable thatreal i sm exi sts somewhere , and i f so , must be soughtin a di re ct ion invers e to that in which we find thephenomenal and the approach thereto must necessa ri lybe gradual , continuous and di rect and not by abruptbreaks , by twi sts and turns . The phenomenal must l i eat the te rminus o f the re al , and v ice v ersa. So thatby retra cing the p ath bl azed out by the real in comingto phenomenal iz ation we shall perhap s find that whi chcasts our shadowy world, j ust a s by tracing a shadowin a di rection inverse to that in whi ch i t extends wemay find the obj ect which proj ects i t.

It is not out and beyond that we shall find the end

of space ; it is not by counting tens of thousands oflight

-

years that the supp osed limits of space shall be

at tained. The path of search must proje ct in an op

posite direction—not star-ward but Thinker-ward, to

ward the subtle habita tion of the consciousness itself.

We err greatly when we th ink that by mea suring distance s we shall encompass space ; for that which wemeasure and dete rmin e i s but the clouds caused bythe vapor o f real i ty. It is, there fore , not without,but within , i n an inverse di rect ion that the se archmust proce ed. Going back over the l i fe-stream , be

ginning where i t str ikes aga inst the shores o f sol idobj ectivi ty, deepe r and deepe r still , past the innermostmile-stone o f the s el f-consciousness , back into the ve ryheart of the imperturbable interio r o f be ing whe re theThinke r’s castl e opens i ts doors to the G reat KosmicSel f, from that open door-way we may step out intothat grea t mystery o f space—l imi ted , yet not l imi ted,multi-dimensioned , and yet having only one dimensi on,veri tably re al and fundamenta l , the Father-Mother o f


phenomena . Here the grea t mystery of myste ri esrevealed a s the citadel o f the universal and theimat e re al . In this ci tadel , the pl ane o f kosm i c con

ess, space lose s its spaciousness and t ime i tsess, diversi ty its multipl ici ty and oneness a lonesupreme .the movement towa rds the center and ci rcumof space , a fte r this manner, requi res a id nei therthe notion o f space-curvature nor that of the

-mani fold , except, indeed , only in so fa r as ao f consciousness o r a degree o f real ism may beto be a tridim ensi ona l mani fold . The feel ingspace i s s ingle-pointed , and yet ubiquitously cen

’

cd, has been indulged by ma thematici ans and othersa more o r l ess modified form ; but they have

iagined it in the term s o f an indefini te proceedingitward unti l in som e manne r unaccountabl e al ike tol we come back to the point o f origin . It has beenpressed by PICKERING when he says that i f we gor enough east we shall a rrive at the west ; fa rough no rth we shal l com e to the south ; far enough:0 the zenith we shall come to the nadi r. But thisncep tion i s based upon a notion of space which ise ex clusive result o f mathem atical determinati ons andbject to al l the restri ctions o f matheti c rigorous3 8 . I t requ i re s that we Shall allow space to berv ed. This we decl in e to do for the reason that itboth unnecessa ry and contrary to the most fundaental affirmations o f the a p riori faculty o f thehinker

’

s cognitive appara tus . I t woul d s eem to be

cessary only that we should extend our consciousnessckward, reve rt i t into the di rection whence l i fe camefind that which we seek . By extension o f conscious


ness i s meant the abil ity to function consciously uponthe various sup erkosmic spheres or pl anes j ust as wedo on the physi cal . Ye t i t should be quite a s e a sy todevi s e an ideal i zed constructi on which would imagify

the results o f thi s ingress ive movement o f the consciousness a s to rep resent the resul ts o f a progress iveou tward movement star-ward . Having done so theexam in ation o f them could b e conducted along l ine ssimi l a r to those followed in the scrutiny o f obj ectivere sults.What would i t mean to the Thinke r i f he were

able to identi fy hi s consciousness with the ethe r inall i ts va rying degree s ; what would it mean i f he wereable to i denti fy hi s consciousness with l i fe and withthe pure mind-matte r o f the kosmos ; and l astly, withthe spi ri tu al e ssence o f the unive rs e ? What i f hi svari ous vehicles o f awareness were ava i l abl e for hispurposes o f cognit ion ? What, indeed, i f he couldtravers e consciously the enti re gamut o f re al i sm andconsciousness from man to the divine consciousness ?Does i t not appear rea sonable that a s he a ssumed eacho f these va rious vestures o f consciousness , in succession ,he would gradual ly and finally, come to a full unde rstanding of real ity i tsel f? I t seems so . This vi ew i seven more cogent when i t i s considered that thel imitat ions, and consequent ob scuration o f consciousness a re proporti onal to the number o f vehicle s o rba rr i ers through which the Thinke r i s requi red to actin contacting the phenomenal unive rse . Common sensesuggest s that freedom of motil ity i s determined by thepresence o r absence (more parti cul arly the latter ) ofbonds and ba rri ers ; that the l ess the number o f such


trivances—the bal ance , the chromati c and the scalpel .Al l o f these a re shadow men constr i cted to the m etesand bounds o f shadows which they obse rve only because they a re ignorant o f the real i ti e s whi ch l i e without the i r pl ane .L i fe ha s so many ways o f exhib iting its rem a ins

to the intel lect ; and these remains have so many facetso r vi ewpoints from which they may b e studi ed , thatnothing short o f a panorami c vi ew of al l the modesof exhib it ion and o f all the facets and angles o f app earances will suffi ce to present a trustwo rthy and

comprehens ive Vi ew o f the whol e . Then , l i fe i tsel fi s so i l lusive , so unse iz abl e by the intel l ect that thete stimony o f all investigators a re requi red to summariz e its modes o f appearance . And, therefore ,eventu al contentment shal l b e secu red only when themass o f diverse testimonie s i s reduced to the lowestcommon divisor, and for thi s purpose the operations o feve ry cl ass o f investigators must b e vi ewed a s the worko f speci al i sts upon separate pha ses , facets and angl eso f l i fe ’s remains .And so i t i s mani festly absurd for the empiri cists ,

by taking note o f the dim ensi on, extent, qual ity and

chara cte r of the shadows , o r one single cl ass o f angles ,to hope to predicate any trustworthy judgments aboute i ther the real it i e s which cast the shadows o r underl i ethe angl es ; because whateve r notion o r conceptionthey may be abl e to ga in must o f necessi ty be merelyfragmentary and enti rely inadequate . Despite thi sfact, howeve r, we stil l have the spectacl e o f m en who ,

studying the sensibl e universum of space-content, en

deav or e i the r to make i t appea r as a final i ty in i tsel f,or that the world o f the rea l must necessa rily be con


)rm able to the preci se standards which they arbiarily se t up in thei r exam ination of the obj ectiveorld. I t can be sa id with assurance that we shall:ver be able even so much as to approach a trueide rstanding o f the unseen, real world unti l we shalllv e changed ou r mental a ttitude towards i t and

ased to expect that i t shall necessarily be fashioned1d ordered in exactly the sam e way as the world of11

‘ senses , or that it shall be understood by applyingre same methods o f procedure a s those which wee in our examination o f the phenomenal , sensuousorld. I t i s a matter of logical necessi ty that, as

ere a re no senses which can respond to the real ,the re a re no organs which vibrate in accord with

ie rate s o f vibration o f the real , there can be no~

asonable hope o f understanding i t by means ofnsuous contr ivances and standards.

Le t the consciousness, there fore , be turned not1tward, but inward where is si tuated the temple o fvine l i fe ; let there be taken away the outward sheathshich enshrine the pure intell igence o f the Thinker ;t him grow and expand his sphere o f awareness ; l etre re be an explorat ion o f the abysmal deeps of mind ,f l i fe and consciousness ; fo r buri ed deeply in man

’

s

wn inne r nature is the answer to all queri es which

.ay vex hi s impui ssant intellectual ity.

CHAPTER IX

METAGEOMETRICAL NEAR-TRUTHS

Realism is Psychological and Vital—The Impermanence ofFacts—On the T endency of the Intellect to FragmentateThe Intellect and Logic—The Passage of Space , the Kosmom e ter and Zoom et er, Instrum ents for the Measure

m ent of the Passage of Space and the Flow of Lifeb The

Disposal of Life and the Pow er to Create—Space a Dynam ic , Creative Process—Numbers and Kosmogony—TheKosm ic Significance of the Circle and the Pi-ProportionMechanical T endence of the Inte l lect and its Inap titudefor the Understanding of Life—The Criterion of Truth.

KOSMIC truth has many facets . The rays o f l ightwhich we see darting from its surface do not a lwayscom e from the co re . Often they are refl ections o frays whose l ight stops Short at the sup e rfice ; andthese , in turn , a re reflections o f deepe r real i ti es . Thusthe refl ected l ight may be traced to i ts source byfollowing the lead o f externa l refl ections. It i s nowknown that moonl ight, and perhaps, in many cases,starl ight , a re refl ections o f sunl ight , i f not o f our sun,

some other in the universe . But i t i s only a t certaint imes and under certa in conditions that we can see

the sun which is the source o f the othe r kinds o fl ight . The sta rs which owe the i r l ight to suns a re somany fa cets of sunl ight. The moon i s a facet ofsunl ight also . Facts a re fa cets of truth . They a re

284


is the outward vehicl e of l i fe ; sp i ri t i s the fo rm orthe inte ri or vehicl e which l i fe a ssumes in orde r to express itsel f. Real i sm , then , i s l i fe . I s the logici andealing wi th real i ty when he coll ects and coé rdinates

the various modes o f interp retation by which we l ea rnto understand the symbol i sm of l i fe ? Obviously not .’

The data o f logic a re s imply a coll ection of rules fo rinte rpreting concepts . I t i s a compendium of indi cesfor the Book o f Li fe . I t is no more the book itsel fthan a table o f contents i s a book . But logic occup i e sabout the same ca tegory a s does an index to a volume .A book , howeve r, i s more than i ts conventional contents . I t i s the thought tha t i s symbol i zed therein .

The book o f l i fe , a ccordingly, i s the sum total o fl i fe’s express ions ; but i t is not l i fe i tsel f. That isthe subtl e , evasive something which the contents o fthe book of l i fe symbol i ze . Nature , both in he r pal

p able and impalpable a spects , may be sa id to be thebook of l i fe wherein a re recorded the movements

,the

expressi ons, and the di acri ti cs of l i fe . The whol e i sa magni tude o f many facets ( l i ttl e faces ) . We shallhave to know al l the faces before we can say that wehave a comprehensive knowledge o f nature . For solong as we have only a fragmentary knowledge o fthe whole , so long even a s we have merely a superficial knowledge o f any asp ect o f nature , j ust so longwil l our knowledge be in va in . Just a s i t frequentlyhappens that, on account o f the parti al vi ew o f things ,we are led to make incorrect j udgments conce rningthem , so when we come to make a ssertions about l i feo r nature in gene ral , w e a re apt to fall into the e rro ro f rendering judgments upon insuffi ci ent data . Andi t is not a t al l l ikely that j udgments thus a rrived at

METAGEOMETRICAL NEAR-TRUTHS 287

l posses s true val idi ty because i t may happen , and

loubtedly does always so happen unde r the presentitations o f human knowledge , that the very elementsich a re ignored or neglected in forming a j udgnt posses s enough of virtue to alte r the intrinsi cue of determinations based upon othe rwise insuffiIt data . Hence it develops not infrequently thatjudgments repeatedly have to be changed in prot i on a s our data are made more and more comhensiv e . M en searching eagerly for the truthie tim es allow themselves to be ca rried away by thehusiasm o f the moment which ari se s upon the disery of a new facet o f truth ; but i f all searchers°

e to bea r in mind the fact that re al ity presentsIf to consciousness i n myri ad ways and that thereinnumerable facets al l l eading eventually back tosource o f al l they not so easi ly would be induced

jump to the conclusion that they had covered theire ground. For when we have discovered a mil1 facts , o r many mill ions o f them , about nature we

y say that we have only merely begun and thatat we have found is not to be compared with theility even o f the di rectly observabl e phases ofure .Logi c

,there fore , deals with the symbol i sm exist

between and among facets o f truth , and notctly with truth itsel f, although the conclusi onsched by the logici ans may be true enough from an

°insic standpoint. Logic i s not truth , however ; i tmerely the consistence of rel ations and inter-rela18 between facts and among groups o f facts. T ruthrot established by logic ; i t stands in no need of theIt of logi c fo r i ts revel ation ; indeed, more apt than


not i s logi c to obscure truth . Truth i s its own proofi t is sel f-evident. Logi c i s a mere model er o f factsi t i s sta ti c, immob ile , fixed. Al l truly logi cal p rocesseneed a start ing point , a foundation , a premise , a baseTruth , be ing eternal , mobil e , dynamic , vital , needno starting point ; needs no foundation because i t ii tse l f fundamental ; i t requi res no premi se because inpremi se i s comprehensive enough to encompass itThere i s only one way of a rr iving a t truth and thais not to a rr ive at all—j ust to recogni ze i t withouprocedure . The fact tha t facts a re , and the fact 0thei r rel ations and inte r-rel ations , the i r sequence ant

impl i cations, can be a rrived at only by logical p roce sses . L i fe , in i ts p assage through the un iversum ospati al ity, ca re fully diacriticizes between the realm ofacts and the doma in o f truth , marking each off fronthe other by unmistakabl e signs and barri e rs . Trutl

i s p e rce ived a s an axiomati c , sel f-evi dent principl e an :no amount o f logi c could prove or establ i sh its veri tyFacts are intellectual cre atures ; truth i s intu i tionalvi ta l . The intel lect conce ives the consistence o f factwhil e the intuition recognize s truth—i s truth , ant

there fore , fol lows in the wake o f l i fe a s consciousnessThere i s no p ermanence in facts and the intef

l ectual recogniti on o f the i r consistence . The di scovero f a single new fact may destroy the consi stence oa whole mass o f p reviously correl ated facts . Thu

i s reveal ed the mi racl e power o f logi c over facts . 1

can take a mass o f facts , related o r unrel ated , molthem into hypotheses, endow them with a sort o f itte rior consistency, and make these hypotheses take thposture o f truth . Hence logi c is o ften an effectiv

mask which the intellect commonly imposes upon it


servances. But i f the focus o f consciousness shoulbe changed so a s to reveal conditions upon what musbe a higher and more interio r l evel , the aspect 0

things would be enti rely changed and the whole 0

our theory of knowledge would have to be recox

st ituted. I t i s conce ivabl e , yea obvious , tha t the sterreal ity o f be ing is fa r removed into the G re a t Interioof that which i s ; and there i s a point in the path tthe inter iority of‘ be ing where there i s no i llusion , nappea rance , indeed , nothing but the cold , illum inatinbody of rea li ty itsel f. I t must appea r al so tha t alonthe j ourney inter ior-ward there a re many app arer

l evel s o r planes , each o f which requi re s a new focu:It i s unrea sonable , then , to suppose that the conchs ions a rrived at a s a resul t o f purely logi cal p rocesseconfined to the lowest l evel s o f real ity, a re p e rtinerand val id for the enti re ty o f real ism which i s me i theof mathemati cal nor logica l import . For instance , iwe take the purely axiomati c a ssertion : x equal sthe intell ect i s a t once certa in that thi s i s so , and cannot be otherwi se , and yet a proposi ti on o f thi s kini s purely conceptual

,conventional and a rbi trary.

may also equal 1 , 2 , 3 , 4 , o r any other quantity. Theni f e ach x in the above equation be replaced, bsay, a horse , there immedi ately a rise s a difliculty. Fc

i t i s not possibl e to find two horses which are in arespects mutually equal . So tha t as soon a s we pa :from the conceptual into the a ctual , whether on tis ide o f obj ective real i ty o r that o f absolute re al i ty, tlval idi ty o f the axiom i s immedi ately exposed f

se rious questioning. The truth o f the matter i s thon both si des o f the conceptual i t i s always found ththe re i s a vari ance from the standards set up by tl

METAGEOMETRICAL NEAR-TRUTHS 29 1

ep tual, this vari ance being m ore marked on thenea rest to absolute real ity than on the side o f

ectiv ism . Obj ectively, the conform i ty of the sensiwith the concep tua l is of such approximation as

l end trustworthy util i ty to the conceptual in itson to the sensuous . Thus by simply el imina

the vita l fa ctors from our equations we are en

d to proceed in a reasonably sa fe manne r withjudgments . Really, however, no such approximate

can be found ; for on the S ide o f real ityl ing with an indivisible som ething—som ei s eternally and absolutely unita ry in itswhile when we transfer the scene o f our

;erv ations to the obj ective world we discover a conry s i tu ation . Here we are everywhere beset bye rsities, multipl ici ti e s and dissimila ri tie s . This i sbecause the intellect naturally tends toward the ohtive where i t finds a most comfortabl e a tmosphereits operations . The conceptual i s related to the

ectiv e a s a train of cars i s rel ated to the ra ilway.at i s to say, the constitution o f the intellect i s sucht i t finds its most facil e expression in the obj ectiverld and i s about a s com fortable in the domain o flism as the same tra in of cars would be on thean .

The intellectual i ty i s designed to deal with facetst ruth ; i t i s made to manipul ate segm ents, parts,ctions, and cannot cha rt i ts way through a con

Jum such as real i ty . Being consti tutional ly a con

itionality o f the Thinker’

s own contrivance , andsing out of the subtl e adaptation o f his vehicl esthe environment afforded by the sensuous world , i t

1 only find congruence in that conventional i ty which


is the instrumental i ty o f a higher intellectual i ty expressed in a diversity o f forms, into which real itydivides i tsel f fo r mani fe station. The hum an intellecti s , there fore , the b ridge ove r whi ch i s made the passage from the indivi dual consciousness to the Al lConsciousness ; s imultaneously, the medium wherebythe physi cs o f the b ra in a re converted into the p sychicso f unconsc iousness . I t may be l ikened to a pa i r ofspeci ally constructed tongs which a re so formed as

to fi t exa ctly the obj ects which a higher intell ectu al i tyha s made. It i s without the province o f the intell ectto take note o f wha t inte rvenes between physi cs andpsychi cs ; i t is always obl ivious o f intersti ces whiletaking cogni z ance o f obj ects o r things . In thi s re

spect, the intellect is much l ike a stee rage passengeron boa rd an ocean l iner who s ee s only his port ofdeparture and port o f a rrival , knowing nothing in themeantime of what happens during the voyage , nothingof what the othe r p assengers on the upper decks may

experi ence and taking no part in any of the passingshow unti l he l ands. So tha t the passage o f the intellect from fact to fact i s an altogether uninte restingvoyage ; i t may as well be made unconsciously, and

to all intents and purposes , i s so made .Accordingly, the advocate s o f n-dim ensionalitj

find i t qui te imposs ib l e to predicate anything whatsoeve r of the passage , s ay, from tridimensionali ty tc

quartodim ensionality. They find them selves at e ase irtridimensional ity and have even contrived to fine

pl easant envi rons i n th e four-space having mad <there in such ideal i zed constructions a s wil l afforc

ample hospital i ty to the intell ect . But the question :a s to how the p assage from the three-space to the


ward sweep through all the stage s o f pyknosi s orkosmic condensation by vi rtue o f which i t a ccomplishesthe engenderment o f materi al i ty as also the pathmarked out by it in its upward swing whereby it accompl ishes the sp iri tu al i z ation of matte r. I t i s thekosmi c orde r which l i fe establ ishes by means o f itsoutgoings and incomings . When we look out intospace we p erce ive that whi ch is a dynami c app ea ranceo f l i fe i tsel f, and not a pure form . Nothing that isa pure form can exi st in nature and in a s much as

space i s not only indi ssoluble from nature but pa rtakes of i ts very essence i t cannot b e s a id to be a purefo rm . The intell ect, howeve r, prone to follow the

grooves l a id out by pure logi c, neve r fa i ls to seekto make eve rything that i t conta ct s conform to theselogical necessi ti es. But, i f the analyst were to makeca reful di scrim ination a s to the respective categories—that into which l i fe falls and that in which the intellect i s forced by its nature to proceed—he not soe as ily would b e l ed into the faul t o f attempting to

shape re al i t i e s upon model s which be ing stri ctly conv entional were not meant fo r such uses. But ne i therthe logici an nor the mathemati ci an can b e condemnedfo r such generosi ty i f such condemnation were justifiable . For they everywhere and at al l times insistupon realizing abstractions and abstractionizing real ities , and they do thi s with an insouciance that i s attimes surprising. Yet i t is i n thi s ve ry vaga ry thatis discovered the t rue na ture o f the intel l ect. Therei s a sort o f dual tendence observed in the method of

the intell ect’s operation . A pol ari ty is mainta inedthroughout : the abstractive and the concretional . It

vaci l l ates continually between the abstract and the


nerete and no sooner has i t found a concrete thanbegins to se t up an abstract fo r i t ; and v ice v ersa

soon as it is has constructed an abstract i t imm edi

seeks e ithe r its concrete o r sets out to hewothe r concrete into such shape as wi ll fi t i t.between these two extrem es num erous excusesfound fo r exercising this pecul i a r characteristi c,that too , without regard to consequences . Itseem that the intellect, in thus funct ioning, wasengaged at a sort of sensuous play out of which

derived an intense and not altogether unselfishasure .Of course , i t must be granted that divers ity hasspecific and withal necessary uses in that i t affords

field for the operation o f hum an intellectual ity andpresents the adaptati on o f the kosm i c intelle ct toe hum an fo r the purposes o f evolution. This adaption while necessa ry fo r the intellectual developent is, howeve r, not an end in i tsel f. It i s merelymeans to a higher purpose . In fact, i f we regardate riality a s a deposi t o f l i fe , ca rri ed by i t a s a kindimpedim entum , and consciousness , which is l i fe , asing i denti ca l with the intel lectual i ty which makesese adaptations, there should be no grounds for theatem ent that the one i s adaptable to the other at1. And a s this is re ally the vi ew whi ch we a ssumewould perhaps b e more strict to regard the adapta)n as subsisting between the human intell ect and

at eriality both o f which having been constructed by) sm ic intell ectual ity. Pursuant to the dive rsi ty o f:es to which materi al ity l ends i tsel f there a rises ine intellect a suprem e tendency to segment, to breakinto separate parts

,to multiply and diversi fy. It


i s not content unless i t i s at thi s favori te and naturalpastime. I t del ights in taking a whol e and dividingi t into innumerable p arts . This it wil l do again andagain ; because a ll i ts muscl es , S inews and nerves a remolded in that mold and can no more ceas e in theirtendency to fragmentation than can the muscl es o f adancing mouse cea se in thei r ci rcular twirl ing o f themouse’s body. Yet, in thi s i t i s but cre ating a wel lnigh endless t ask for i tsel f—which task must b e performed to the uttermost. But in its p erform ance , thati s , i n the intell ect

’s complete understanding of thedivers ity of pa rts , in the knowledge o f the i r rel ationsand inte r-rel ations and in the i r synthesi s , i t may arrivea t that one ineluctabl e something which i s called unity.

‘And so doing, become ultimately free .In vi ew of the foregomg, it IS not su rpri s ing that

the intell ect should have , finally, fallen upon the not ion o f n-dimensional i ty. It has com e to that as

naturally a s i t h a s perform ed i ts most common task.

Le ft alone and unhampered in i ts movements, i t ha ss imply followed the le ad of the Gre at H ighwaythrough the doma in o f materi al i ty. And now i t ha sa rrived at a stage where i t thinks i t has succeeded infractional i z ing space. Time has long ago yi elded tofragm entation , been divided into minute parts and eachpa rt ca re fully mea sured. Space , not having a visibl eindicator l ike tim e to denote i ts p assage o r parts

,

suffered a long and tedious del ay before it coul d boasto f a measure r. As the sun-di al measured tim e in thepast and became the fore runner o f the modern clockso n-dim ens i onal i ty measures space for the m athemati

e i an . What more practi cal instrument fo r thi s pu rposemay yet be devised i s not ours to p rophesy ; yet i t i s


be accompl ished a fte r thi s manner : fi rst , by syncretiz

ing materi a l i ty with vi tal i ty, and then , by intu it ionallyrecogn iz ing the truth o f the impl i ca ti ons o f the syn

creti sm .

The hi story of consciousness in the human familyi s identi ca l with the hi story o f man’s conques t ove rmatter and physi ca l forces . And thi s i s cl e a rly indicated in the incidental s contingent upon the to ilsomeri se o f the genus homo from the ea rl i est cavem anwhose status denoted a comparatively negligibl e t ranscendence o f materi al forces , to the present-day manwho ha s ga ined a markedly notable conquest ove rthese forces . Always consciousness s eeks the meanso f adequately expressing itsel f in the sensible world .

And to thi s end i t engenders facul ti es, o rgans and

processe s in the bodily mechani sm , and, i n matter,devi ses instruments o f appl ication whereupon and

wherewith i t may test, analyze , combine and recom

bine the forces and materi al s i t finds. The unlim

ited range of exp ressi ons lying open to the consciousness makes i t necessary cont inua lly to devi se higherand higher grades o f appl iances to meet its needs a si t expands. It will not be ga insa i d that the tel escopehas se rved actually to l ay ba re to the consciousnessan immeasurabl e realm of knowledge nor that themicroscope , turning its attention in an opposite di rcetion , m a rvelously has enl a rged and enrichened our

knowledge o f the world about us . And sim i l a r declaration may be made anent almost every invention ,discove ry and conquest which man ha s made ove rnatura l phenomena . Thus , by externally applyingmechanical implem ents to the subj ect of hi s consciousness , man has extended actually his consciousness, his


re o f knowledge ; has greatly enhanced i ts qual i ty,in the process, ha s urged the intelle ct to en

ors that have wrought its present unequaledtery of th ings. Nor have the sp iri tual a spects o fadvance along these l ines been the least notable .these have enj oyed the essence o f all that hasga ined in the process and have

,therefore

,kept

with the onward movement o f the intell ectua liousness. But heretofore no advance has beena s a result o f methodi c or reflexive dete rminaThat i s , men did not set out from the beginning,ed with foreknowledge o f what thei r effortsbring, to develop the present qual i ty o f human

nsciousness. They simply worked on, their attensn being absorbed by the problem s that lay nearest(1 demanded earl i est consideration. So the advance3 come as a resultant o f man’s close appl icat ion toever-present needs—shelter, clothing, food , protion and othe r p reservative measures—and i t hasme na turally and inevitably and without prepense .ev e rtheless, i f man , knowing what to expect fromc syncretization o f matter and mind , a fter thisShion

,should set out del iberately to accelerate the

:ensification, expansion and growth o f hi s consciousss, there i s no doubt but that the consequence wouldmost fa r-reaching and sati s facto ry.

But the path that l eads to this grand consumma

in does not l i e in the di rection o f diversity ; i t l i e sthe opposi te di rect i on . In vain, then, does the

“

ellect fractional ize in the hope that by doing so i tall come to the sol id substructu re o f l i fe ; in vainies the analyst segment space into any number of.rts or orde rs ; in va in does he ask how many and


how much ; for by answering none o f these que ri eswill he find the satisfaction which he vaguely seeks .I f i t be true that i t i s not by analysi s but by syn

thesi s that the tru e norm of l i fe , and there fo re , ofreal ity shall be found it i s futi l e to enterta in se rioushop e o f finding it in any other way. As a p eris0phismo r near-truth

,then , n-dimensi onal ity takes foremost

rank. And thi s i s so fo r the rea son that when weproceed in the di rection o f multipl e dimensions , thati s , one dimens ion p iled upon another dimension or

ins erted between two others we are t ravel ing in adirection which , the more we multiply our dim ens ions ,l e ads us farther and fa rther away from the truth .

Thi s i s a s imple trui sm . I f we take , fo r instance , awooden b all and cut i t up into four quarte rs , and

divide each one o f these qua rte rs into e ighths , intosixteenths , thirty-seconds , sixty-fourths , etc. , indefinit ely, we shal l have a multipl ici ty of parts , e ach one

unl ike the original b all . But from no examination of

the multip a rti te segments can we derive anything l ikean adequate conception o f the original ball . Something, of course , can be l ea rned, but not enough toenabl e the rendering o f a correct judgment as to thena ture , si z e , Shape and gene ral appearance o f the ball .But thi s i s p recisely what happens when the analystdivides space into many dimens ions . H e cuts i t upinto n-dimensional p arts and the more minutely hedivide s i t into p arts the more remote will each partbe in its s imi l a ri ty to the original shape and form of

space , and the fa rther away from the true concep tiorof the nature o f space he i s l ed thereby.

Now,n-dimensional i ty or that phase o f meta

geometry which rega rds space a s being divisibl e into any


these things and re al ize pragmati cally the true v aand extent o f un i ty in the un ive rs e .I t is admitted that the intell ect, in treating obj i

singly and dealing only with the sta rts and stopsa movement , i s withal loyal to the kosmic o rder , desand purpose whi ch have priorly chara cteri zed m :fested phenomena by segmentation . And in thi s 1al ty i t has been fol lowing merely a natural le ad whiwhil e admitt ing o f the widest development and

p erience , neverthel es s a t the same time beneficerobscures the underlying real ity in order that inadaptation to the sensuous world the intellect mihave the greatest freedom fo r the development suito the given stage o f its evoluti on . But in thusmitting the natural congru ence between the inteltuality and the phenomenal or s ensuous we dothereby uni te wi th those who al ready bel i eve thatkosmic agre ement i s the ne plus ultra of psychogeneOn the other hand , i t i s ma inta ined that this i s mea phas e o f psychogenesi s which shall b e outgrow:just the same mea sure as othe r phases have beengrown . And notwithstanding the fact that judgm l

o f the intell ect with respect to inte r-factual rela t .or the ens o f facts themselve s a re a s vali d a s i ts jcial determination o f sel f-consciousness, no moreno less , we a re , by the very rigo r and exclusiv enes

thi s logical necess i ty and inherent l imitation , leevi ew the intel l ect’s interpretation o f phenomenapartial and fragmenta ry ; fo r the reason tha t the nesitous confinem ent o f i ts unde rstanding and intertativ e powers to fact-rel ations qui te effectively inhfthe use o f thes e powers fo r the contemplation o fdeeper causative agenci es which have operated to


duce the phenomena . But i t i s appa rent that just a sthe transmuted resul ts of other phases of psychogenes isa re now be ing util iz ed as a basis fo r the effi ci entop e ration o f the intellect in the sensuous world

,thereby

enabling the atta inment o f a very high mastery overmatter, so wi ll the functiona l dynamism acquired byi t in the pursui t and comprehension of diversi ty se rvewell when, in l ate r days, i t has acqui red the powerto deal di rectly with real i ty, fo crea te and dispose ofl i fe j ust as the kosm i c intellect has and i s now usingit in the execution o f the infini te process of be comingth rough which creation is proceeding. I t would seemthat the necessa ry prerequisi te to the developm ent o fany higher functional capabil ity is that the intellectshould b e capable o f disposing o f innum e rable deta ils ,indeed the total i ty o f kosm i c deta il , before i t can

come wholly into the powe r and capacity to understand and manipulate l i fe . Furthermore , i t appearsthat the acqui rem ent o f this power quite necessarilyhas been delayed awaiting that tim e when, dom inatedby the intui tion, the intellect shall have atta ined therequisi te manageri al abi l i ty fo r marshaling an exce edingly l a rge numbe r o f deta ils .The suprem e tendency o f l i fe is express ion. And thi s

expression, singularly enough , reaches i ts most perfect

phenomenal iz ation by means o f that movement whichresults in the mul tipl i cation o f form s . Despite the fact ,the re fore

,that the comprehensi on o f re al i ty involves

a gradual turning away from the exclusive occupationof organi z ing a multi tude o f separate and apparently

unrel ated facts to a monisti c vi ew which at oncerecognizes the uni ta riness and co-original i ty o f all

things , o f l i fe , mind and form , the intelle ct wil l need


the tra ining and development which come frommastery o f divers ity. I t is, then, not diffi cult to

ceiv e the wise util i ta ri anism of the p resent schem ao f things as shown in the universal tendency inintel lect to devote itsel f exclus ively to pa rts or segrro f truth.

Wheneve r an indivi dual intellectual i ty, on ace

o f prolonged thought and the consequent inure1o f the mind to higher and highe r vibrations ofkosm i c intell ect, brings itsel f to such a high po irs ensi tiveness that i t can rece ive so much a s an intti on of some great truth , i t begins to sense , i n a 1

or l es s vague way, som ething o f the substancegene ral tendence o f th e underlying real ity o fwhich foreshadows its appea rance. Then , confouby the multiform al cha racte ri sti cs o f kosmic trutlcause o f the fact that i t pres ents i tself in such nuous ways and form s

, men often a re induced to at t l

the re formation o f al l facts , o r a gre at maskindred facts, in accordance with the newly-foundo r principl e . They forget evi dently that no fathe unive rse can b e at vari ance with any otherand stil l be a fact . So that in the total i ty o fevery sep a rate and di st inct fact must be congwith every othe r fact fo rming a beauti ful , ha rm01and symmetri ca l whol e ; but often the whol e is 1

to suffe r in the attempt a t making i t confo rm t t

substance o f a mere int im ation . Moreover , i t isceiv able that even the total i ty o f facts may larigi d conformi ty with rea l i ty in a ll its parts and

having compassed the enti re mass o f facts one

fall Short o f the understanding of real ism .

This is practically what has happened in the

306THE MYSTERY or SPACE

i t a t all . Hence , the mathemati ci an who dependsti t ely upon the del iveri es of the intell ect which 1

form ,in thei r p assage from the conceptual to

written or spoken word , to al l the rigors of mathrequi rements , fa i l s utterly in perce iving the magnito f thi s concepti on and al l its connotations ; he fbecause his prejudice s and the woof and warp ofintell ectual habits prevent hi s a ssuming a sympatha tt i tude toward i t and thereby precluding at the 5any calm consideration o f i t. And not only i strue of the mathemati ci an but o f al l those whosede av o rs are confined to the pl ane o f purely sensuand logical data . I t would , the re fore , appea r 1our enti re atti tude towards things Spati al mustchanged be fore we can even begin to p erce ivereal ity which is really the obj ect o f al l researchethi s dom a in. But , on the surfa ce , there i s a fterl i ttl e diffe rence between the ultim a te facts inv oli n these two totally diffe rent conceptions . Mamatically sp eaking , all progressi on eastwa rd we

termina te a t the west, and v ice v e rsa ; and the s

would b e tru e regardl ess o f the point from which 1

gression might o riginate . Always the terminus Wt

be the opposi te o f the starting point. Then , tocmight b e sa i d that i f we sought the space-centershould a rrive a t the ci rcum fe rence . The diffi cul ty 1

this vi ew is tha t there i s a ve ry remote , thoughportant, connection between i t and the truth o fmatte r. But the pa rti a l i ty o f this vi ew,

and thesence o f e ithe r experience o r intu i tion to intima '

more reasonabl e vi ew, serve effectively to buttre :as a hypothesi s a cceptable to many. Thus it i s t

more diffi cult to supplant a near-truth than i t i


n credence for the whol e truth . On the other hand ,ording to the vi ew which we m ainta in here

,i t i s

true that the seeking of the kosm i c space-cente rrevea l the ci rcumference ; that the sea rch for thei r wil l uncover the zeni th ; the e ast eflloresces a se west, and a northward j ourney will wind up ate south , etc . , but in qui te a diffe rent manner fromat which the mathemati ci an has in mind when hestulates the curvature of space . Our view involvesspace curva ture nor any othe r spati al disto rtion.

deals with space as r eality, as a dynamic process,ich

, like the sea,is continually casting itse lf

the shor es of chaos and falling back upon itselfanly to be r ecast against the rock-bound coast of itsrhao tic limits. Now, that which falls back upon i tsel f1nd rolls in a recurrent movem ent up on i ts own sur

race i s life which , in its recession is the natural and:osm i c l im itations o f i tsel f, gene rates matter in allts va ried expressions. Space , in its extensi ty, cannotranscend l i fe ; fo r i t i s the path which l i fe makes ints out-coming, i ts mani festation. Of the chaoti c fringevhich ci rcumscribes the mani fested unive rse i t i s ab.urd to say tha t i t i s vi ta l o r psychological in any.ense o f these terms . For notwithstanding the facthat out o f its very substance a re engende red li fe ,ntellectuality, spati al i ty and materi al i ty , i t i s neverheless none o f these in i ts primary essence . It is:haos-Kosmos ; because from its content the kosmosS evolved , and i t sti ll rema ins ; i t is chaos-spa tial ityhaos-m a teri al ity chaos-intellectuality ; chaos-geom etri cty ; because these a re engendered by the movementif l i fe in chaos whil e a t the same t ime there remainsresi duum of the chaogene tic substance which consti


tutes the l im itations o f all these subsequent p roce :In this sense , the chaogene tic fringe becom es the l iof the man i feste d universe so that i t would apj

that al l those maj o r processes outl ined above are fimani fe stat ions o f the eternal chaos. But none of tlpossib i l iti e s o f motion which are found in these mmovements o f the kosmos can be logically s a ieexi st in chaos. It is the embodim ent o f ev erytl

that is the opposite o f those qual i ti es which ma!found in them , tha t is, in materi al i ty, vi tal i ty,tiality, intell ectual ity and geom e tricity.

Apropos to this phase o f the discussi on letexam ine briefly one of its m ost S ignificant impli catiboth mathem atica l and kosmic, which a ri se s out offact that space i s an engendered product o f l i feis bound by the fringe of chaos which susta insl imi ts i t. The chaoti c fringe plus mani festing kosconstitute the absolute magnitude o f the kosm os .mani festation facto r i s complemented by the cl

factor and together th e two define the full univiKosmogony is the un ive rsal movement o f all koelem ent s o r factors in dim inishing the chaoti c 1

plement and reducing i t to kosmic orde r o r geom e tr

It is undoubtedly impossibl e to dete rmine m e

matically the exact volume o f e i the r complem en

the ratio o f the one to the other ; yet i t is conceiv

that the chaoti c fringe is greate r in extent thano rdered portion o f the kosmic uni-ci rcl e o r univiI t is even conceivabl e that the difference , uponbasis o f the meaning of the Pythagorean Te trag1

maton and the Vi ew outl ined in the Chapter on“Mystery o f Space ,

”is as seven to three wheref

the conclusion might be drawn that the universe


nifications. But it should be understood thatci rcl e is a symbol o f the p erfe cted univ erse andthe univers e in a state o f evolution . It symbol

p erfection , completion and the ultimate union o fmani fe sting with the a rchetypal which resul ts incrowning deed of Pe rfection . The ci rcl e is, therefnot a symbol o f the unive rs e a s i t now stands ; i tnot represent a snapshot Vi ew of the kosmos butuniverse a s a full. I t cannot be a full unti l i ta tt a ined the ne plus ultra o f compl etion ; for a kosfull i s tha t state to b e atta ined by the m anife

kosmos upon the termination o f all the fundame

processe s now in ope ration . But i t i s this statethe ci rcl e re ally represents, and by vi rtue o f whicpossesse s its intrinsi c qual i ti e s and also in virtuewhich the intellect recogni zes these qual i ti e s .prope rti es o f understanding and recognizance inintellect a re veri tably fixed by the status quo ofuniverse during eve ry stage . That is, the focus ofintell e ct, l ike the focu s o f a chromati c l ens, i s a dj uby the flat o f the nature and ete rnal fi tness o f thto correspond exactly with every state through wthe kosmos itsel f passes . This i s one of the obvimpl i ca tions o f the phanerobiogenic behavior o fkosmos and i s necessa rily resi dent in the no tior

the genesi s o f space and intell ectual i ty as consube

ti a l and coordinate facto rs .Whe refor e the more cogent is the r eason for

belief that the inherent qualities of the kosmog

fundamentals ; as,v itality, materiality, spatiality,

tellectuality and geome tricity, are true v ariants,

that their v ariability is prop or tional to the p rogof these major m ov em ents toward the ultimate s

METAGEOMETRICAL NEAR-TRUTHS 3 1 1

faction of the original crea tiv e impulse . May it notb e , the re fore , that the indete rminate cha ra cter o fthe ratio o f the diam ete r to the ci rcumference

i s due to causes fa r m ore profound than the crudity o f our m i crometers or themere supposed fact o f the ci rcl e’s pecul i a rity? May i tnot also be true that the pi propor tion shal l b ecomea whole number, and in its integration , keep apacewith the perfecting process o f the kosmos, dim inishing,by retrogression to one or increasing , by progression,to ten which , a fte r all , is essent i ally uni ty

,being the

perfe ct num e ral ? I t i s not without the utmost as

surance tha t these queri es wil l be categori cally questioned by the o rthodox , creed-loyal , stri ctly intellectual type that we sketch these impl i cations, but i t isfel t to be an urgent duty to remind all such that themost effective barri e r to real iz ati on in the field o fphilosophy is an intolerant a tti tude towards a ll l ineso f thought which suggest the impermanence o f conditions a s we find them in the kosmos at the presentt im e . The fa ct is that our l ives a re so distressinglyShort that we have ne ithe r time nor opportuni ty towatch the changing moods of the kosmos nor discernthe gradual reduction of mere appea rance to the fi rmbasi s o f real i ty, and accordingly, the intellect tenaciously cl ings to those notions which i t derives fromthe instant-exposure which the lens o f intelle ctual con

ceiv ability allots to i t. Once the view i s taken i t i simm ediately invested with everl astingness . Thiseverl ast ingness is then imputed to the kosm os in thatparticul a r pose , a tti tude or state . Always the intellect beholds in that passing vi ew, snatched from the

fl eeting panorama o f eternal duration , a pi cture of

3 1 2 THE MYSTERY 01? SPACE

i tsel f which it mistakes fo r the re al i ty o f the not-selThe inclination o f the axis o f the earth towa 1

the pl ane o f i ts orb it is approximately tWCIItY-thl'

t

and one-hal f degrees . No well-inform ed astronomehowever, doubts now th e fact that thi s ecl ipti c angi s be ing gradually l essened ; because , a s a result 1

centurie s o f observ atiori, i t has been found to b e dcre asing at the rate o f about seconds per centurYe t no intellect is able to pe rce ive in any given l i ftime the actual decrement o f this angle . It i s on

'

by care ful measurem ents a fte r centurie s o f waitir

that a diffe rence can be discovered a t al l . Thusmay even be so with the ra tio o f the di am ete r 1

the ci rcumference o f a ci rcle , the only diffe rence beirthat i t ha s not yet been determined whether therea decreme nt o r an incre ase in the S ize of the ra ti o .

The pi proportion i s , then , a registe r o r m easur i

o f the slow, measured approach o f the manifestir

kosmos to the standard o f ul timate perfection . Therfore , and in vi ew of these consi derations, we m :not hesitate to confi rm our bel i e f in the validity t

the notion that i t a ctually and l i te ral ly expresses tikey to the evolutionary sta tu s o f kosm ogony. Tlmathematical dete rmination which l imits i t a s an uchangeable

,inelasti c quanti ty is, consequently, on

pa rti ally true and le ads to the inclus ion o f thi s quati ty under the category of mathemati cal near-truthfo r such it appears to b e in sp ite o f its rigorous e stalishm ent .

The formal topography into which the intellesp reads when seeking the idea l and the abstractnot a condition which i s derivabl e from the real esence o f l i fe or matte r, but, on the other hand , i s


no longer would be any necessi ty of vi ewing thin;a s they are colo red o r molded by the intellect. Tl

intuition , being a process o f pure consciousness , wi'

when i t has a ri sen to a pos i ti on where i t may dominathe intelle ct a s the intelle ct now dominate s i t,modi fy thi s tendency which we see so ineradicab

bound up in the very nature o f the intel lect that tlapparently insurmountabl e diffi cul ti es which i t ha s iterposed between mere perception and a di rect conitiv e operation will b e quite completely ov ercom

Thus , i n the above , i s discovered another obstac

which pos it s i tsel f between the notion o f spa cereal ity and the intell ectual determination o f i t whi<the mathemati ci an examines and to which hi s cosciousness i s necessa rily l im i ted . Furtherm ore , i t m :be perce ived also how easi ly the mind may be deludeinto thinking that the intell ectual notion which i t et ertains o f space i s necessari ly correct, when obv erseli t i s S imply exam in ing a concept which ha s been 1

made by the intellect into a form which is not a t aunl ike i ts own pecul i ar nature , and there fore , a s muisho rt o f real ity a s the inte ll ect i tsel f i s . Sim ilarl

i f the mathemati cal mind succeed in catching a gl impo f the real i ty o f space in the form of an intimatio

which , i n i tsel f though fragmentary, is nev erthele

true , i t s consciousnes s i s finally deprived of tl

true vali dity thereo f s imply b ecause o f the behav ico f the intell e ct in i ts manipul a ti on o f i t. The irportance o f these intellectual diffi cul ti es cannot lover-estimated fo r they furni sh the grounds fo r tlineptitude o f intell ectual determinations made insphere o f moti l i ty to whi ch the intellect is a strange‘And thi s fact will appea r more evident when i t


perce ived that quite the enti re content o f humanknowledge has been thoroughly viti ated by them . So

that only in those very rare moments. which ( in ahighly sensitive mental i ty ) enable the intuiti on to gaina mom entary a scendancy ove r the intell ect i s i t p ossibl e for the Thinke r to catch hold o f re al ism itsel f,and proj ect the t ruth o f what he sees into the lower

,

intellectua l consciousness . But so sm all is that portiono f our knowledge which owes its origin to the intuition tha t when compa red with the total ity o f thatwhich we seem to understand it i s well-nigh negligibl e .And then , when i t i s cons ide red that a t present therei s no way o f conceptual iz ing adequately the intuitograph so a s to make it p ropagabl e the insignificanceo f thi s fo rm of knowledge i s even more notable . Itcan now be seen in how large a measure the notiono f the curvatu re o f space is merely an intellectualtransl ation o f a true intui tion into the te rms of theintellect which , i n the ve ry nature o f the case , canonly approximate the truth because o f i ts colorific

habits .A simil a r declara ti on may be made of that other

datum of metageometri cal knowledge which postulate sthe ultimate convergence o f pa rallel l ines. In fact ,what has been s a id a s to the p erisophical nature o fthe notion o f space-curvature will apply with equalforce to the idea o f paral lel convergence s ince thel atte r is a derivative o f the former. But there i s yetanother consi deration, apart from the colorific in

fluence o f the intell ect, which , al though i t pa rtakeso f the nature o f thi s quality, i s nevertheless a nea rtruth o f quite a different o rder. Thi s

[may be bette r

understood by referring to the graph showing the


invers e ratio o f obj ective space to the conscior

ness 1 Let u s suppose that the graph may al

rep resent the Thinker’s outlook into the worldspati al i ty. I t then appears that , becaus e o f tha t mos

ment o f consciousness in i ts pursui t o f l i fe whi ch ,i t expands, makes the obj ect ive world to appea rb e diminished in proportion to the extent o f i ts e

pansion, i t i s qui t e natural , under such ci rcumstancethat p a rall e l lines drawn anywhere in the l imitsthe obj ect ive world Should seem to come to a poii n the ultimate extensi on o f themselves. Whil e ti

graph i s not meant to dep i ct such a Vi ew, i t mayfound neve rthele ss , to b e a true del ineation o f t?topography o f that s ta te o f mind into which the m et

geometri ci an brings himsel f when he vi sual i ze s Sp aa s curv ed; fo r there i s no doubt but that a stateintell ectual ecsta sy, such a s that in which the mindthe m e tageom etrician must be functioning in o rde rperce ive sp ace in that form , i s qui te diffe rent frothe normal and, there fore , in ne ed o f a different to;graphical survey. But , i f we grant that i n the cretional a spects o f space there is conceivable an ev e

present tendency to convolution , o r a roll ing back upi tsel f, i t i s imaginable that p aralle l l ines inscribe i ther upon its surface or i n i ts texture need not necisarily meet but mainta in thei r pa ra ll el i sm regardle

of the complexity o f the convolutions . The cc

vergence o f p a rall el l ine s i s much l ike a tangentthe outgrowth of the idea from the notion o f sp acurvature. The more a tangenti a l l ine i s extended tfa rther away from the ci rcumference i t becomes a .

consequently les s in agreement therewith . The mc‘Figure 20.

3 18 THE MYSTERY or SPACE

fi rst to the inte l l ect in its most impotent state whrese rving the les s crass for that time when t

Thinker shall have evolved a cogni tive organ adaable to its presentations. Those m e tageom e trici:

who cl ing to the idea o f the mani foldness o f spabased as we have shown upon the pseudo-interp retatio f a rather vague hint a ri sing out of an unquestiona

’

tru e intui tion , have allowed them selves to fall into 1unconscious e rro r o f magni fying the importancethe mere insinuation as to the space-nature to suchextent a s wholly to obscure in the i r own minds 3

in the minds o f those who think a fte r them what e 1

of the true vi si on that may have b een graspedthem . Furthermore , i t i s indubitably true that t]s ame pecul ia rity of a rrangement by which imp alpaand invisibl e forces really subtend gross matter pducing tha t subtl e schem atism in vi rtue o f whichvisibl e i s subj oined to the invisibl e , the sensuousthe non-s ensuous, spi ri t to matter, etc. , also ch

acterizes the appearance o f spati al i ty to the hununderstanding. Whil e the re is a superfici al semblao f separate and di screte mani folds into which sp

may be divided there a re , in re al i ty, no such sh:

lines o f demarkation b etween the subtl e andgross, between the visibl e and the invi s ibl e or betwspi ri t and matter , e ach o f these being capable o fduction , by insensibl e degrees , into the othe r regal es s as to whethe r the reductional p rocess origins

on the S i de of the most refined o r on that o fgrossest. Accordingly

,there a re no reasons

grounds upon which the notion o f a space-manifmay be j ust ified except a s a metageometri ca l netruth .


In addi tion to the forego ing,there a re yet other

ery fundamental considerations which would seemdeba r the total i ty o f analytical conclusions as tonatu re o f space from any cl a im to ultimate re

iability and trustworthines s . These a re first : the facthat analyses are absolutely incapable o f dealing withi fe ; tha t being the di rect product of a so rt o f

anical consistency which marks the intellectuali t has adaptabil i ty only fo r deal ing withor di sconnected pa rt s , and that without any

rence whatsoever to the current o f l i fe o r theo f real i ty which has produced the pa rts. Thisi s cl e a rly shown in tha t a ttitude o f the under

standing which inevitably leads it to the decla rationthat a l ine is an infini te se ri es of points, a plane an

infinite series o f l ines, and a cube , an infinite se ri es ofplanes, and so on, indefini tely. To do this, to lookupon all phenom ena a s a seri es of parts S im i l a r toe ach other and pil ed , one upon the other , or juxtaposedin the manne r which they are discovered in the sen

sibl e world , i s the natural tendency o f the intellectand thi s tendency finds its m ost facil e expression inanalytics. Inadaptabil i ty o f this sort is especi ally oh

se rvable in all problems of arithm etical analysis inwhich the vital elem ent is a factor . When theseanalyses a re ca rried to thei r logical conclusi on , as hasbeen Shown in the chapte r on “The Fourth Dim en

sion , invari ably they end in an evident absurdity.

But i t is a t thei r very conclusi on where the l i fe-elem entis encountered, where real ity is approached , that theybreak down . The fa i lure o f analysis, then , to en

compass l i fe , to fi t into its requi rements and to satisfy.ts natural outcome seems cl ea rly to establish the basi s


of the p erisophical nature o f the enti re ty of analycla ims, especi ally that speci es o f analys i s whi ch 3

the remote r fi elds o f the conceptu al for its de terrrt i ons . Second : the close connection which has 1seen to subsi st between sp ace and l i fe a s j oint proc

'

of the same movement makes i t obvious that the Sultimate rule o f interpretation must b e appliec

both in. orde r to insure correct and dependable jments regarding them . How different would be thtellectual attitude towards space i f i t we re considi n the s ame l ight a s vital i ty, provided one reallyderstood anyth ing about vital i ty ! Moreover, 8

appears certa in that the path o f the intell ect doesrun in the same di rection a s the path which l i fe ma

but i n an inverse di rection , i t i s cle a r that the jments of the former, a s to the action and essenethe l atte r, must necessari ly be ultimately unreli:

It can readily be seen , howeve r, that shoul d thetellect be focused so a s to follow the path of l i feattach itsel f to the very stream of l i fe , i t would 1necessari ly to neglect materi al i ty. And such an

justrnent would , of course , obvi ate the need e

materi a l l i fe at all for humanity. In fact, a phyl i fe with an intell ect would be impossibl e undercondit i ons . I t i s well to re cognize the suitabilit

the present schemati sm and not to become unw

restive because o f it ; but i t is al so fi tt ing that we shdiscriminate be tween that which i s possible fo rintell ect chained to m ate ri al i ty and that which is ins ible for it , in such a state , when foraying in a terrifo reign to its na ture , and beyond its powers to m a

The predominating tendency in the intelleca ccount for the universe o f l i fe , mind and matte r 1


ous as those which LAGRANGE di scovered, maydevised , nor that other l aws, he reto fo re undrea1

of, may be found ; but wha t i s mainta ined i s thetha t whil e there will b e growth and development tlwill run along other channels , perhaps in the reo f the intu itabl e , and not any longe r, e speci allynotably a s now, in an opposi te direction aga instcurrent o f l i fe and real i ty ; and furthe r, that t]will b e a gradual turning away from mechanicsb i ogeneti cs, from dive rsi ty to unity, from the puintellectual to the intuit ional , and withal a final getrid o f the bonds of i llusi on , o f that thralldommechani cs , whereupon will Slowly a ri se the olescence o f al l those dispa riti e s which may now

recogni zed in ou r knowledge and in the applicat

of the intell ect to the data o f the obj ective world.

Because the intellect i s unsui ted to deal with 1

ity, and becaus e o f i ts pecul i a r adaptation fo r diverfor multipl ici ty, due to its mechani sti c modus v iv e

there has grown up a voluminous catalogue o f systo f philosophy. These embody such a multitudiea rray o f bel i e fs, i de a s , conceptions , theori es and

jectures and constitute a movement in human thoewhich oscill a te s between the empi rici s t on thehand and the transcendental i st on the other ; be tvthe ideal ist and the real i st, l e an ing sometimes tow:the Pl atoni c, the Cartesi an and the Kanti an ane

other times towards SPINOZA , ARISTOTLE , SPENand SOCRATES

,always termina ting by multiplying

number o f dive rse bel i e fs rathe r than uni fying t.tha t the conclusion i s unavoidabl e that so markel ack o f unan imity is indi cative o f a profound meprest ri cti on . It was, there fore , inevi table that ma


maties should fall unde r the sam e spell and brookno l et nor hindrance unti l i t had succeeded in devisingseveral diverse systems of geometry which i t has donefor the mere j oy of doing something

,of following its

inst inctive apti tudes . There i s no othe r basis fo r thehete rogene ity of our philosophies

, our mathematics ,indeed our bel i e fs than this mechani cal , and hence ,radically illus iona ry chara cter of the intell ect in consequence of which we have had to be satisfied withmere gl impses , hints , intimations and fa int gl immerings o f real i ty, of l i fe , and of those kosm i c movement swhich , i f we had the abil ity to trace them from thei rsource outward , would lead us une rringly to a truerand deepe r knowledge o f those th ings that under thepresent schematism must remain for uS a closed book .

The criterion o f truth for us, constituted as weare and wedged in between the stream of l i fe and i tsShore of materi al i ty, must be that which relate s ourknowledge both to the stream and to the Shore. Itmust be so that all predica tes which purport to ap

proach i t sha ll exhibi t a dual re ference—one that rel ate s to materi al i ty and anothe r that rel ates to vital ity,and yet a thi rd that Shall combine these two rel ationsinto one . Al l a sserti ons , there fore , which perta in exelusively to either o f these elements—4 0 mate ri al ityor to l i fe—a re necessari ly p arti al , fragmenta ry andp e risophical in nature . Mathemati cs, because it re

l a te s to matter and the mechanical forces set up bymatte r acting aga inst matte r cannot be s a i d to agreewith such a cri te rion ; a rt, because i t rel ates to snapshots or stati c vi ews o f matter i s even more remotein i ts agreement ; philosophy, as i t has been known inthe past and i s known to-day, because i t seeks to deal


with a vi tal ity fashi oned a fter the image o f mriality ha s fa iled when posi ted alongside o fcrite rion ; and thus , the intellectual toi l o f mill ionsyea rs has been in va in in so fa r a s i t ha s not succe e e

even in ra i s ing a corner o f the cover which hire al i ty from our view .

A near-truth i s any variation from thi s standsthi s no rm or cri te rion . I t may be e ither logicogni tive , sci entifi c o r even metaphysi cal . To defia logical tru th IS a predicate b ased upon and inv olvthe coherency and consi stency o f thoughts them sels

a cogni tive truth IS the conformity o f knowledge vso much o f real ity as i s known ; sci entifi c truth i sconformi ty o f thoughts to things and conditions .o f these a re obviously nea r-truths . Then , too , a metruth may be defined a s an assertion b ased uponcriterion o f truth but fall ing within the catego rycognitive truths owing to insuffi ci ency o f data o r v isfiSuch indeed a re those metageometri cal predi ca ten-dimens ional i ty, sp ace-flexure , sp ace-mani foldnessal l othe r a sserti ons based upon these in generalspecifically. Any recogniti on o f truth must cleembrace both the vi ta l and materi al a spects o fsubj ect in orde r to be adequately inclus ive , that isshoul d include the causative , the sustentative , the rtional and the developmental factors . These ffactors a re considered necessa ry and suffi ci ent totermine the conformity o f any vi ew to the criteriortruth fo r when we a re cogni zant o f the cause 0

subj ect, understand the sustentative factors which I:i t in exi stence , a re conve rsant with its rel ations to ofsubj ects and can follow its developmental va ri atiunti l we come to its final status , why then , our kne


a imed at unvei l ing the mys te ries o f l ife and mind ,of that movement which ha s its roots set i n eteeduration from whi ch it proceeds in an endl ess e

tinuity of pu rpose and p romise , we do al so recog1that in th e Sci ence o f mathemati cs the intell e ct St,as i n no other method of cognition , most fully futhe kosmic intent o f its existence ; and moreoverthe pursui t thereo f i t Shal l push the fronti e rs o fpossib i l i ti e s outward unti l i t can be sa i d almost toabl e to make di spositi on o f l i fe itsel f—at l ea sttha t point where , when the 1ntu1t10n shall have ceinto i ts own, the p assage from the mechani cs o f mato the dynamics o f l i fe, Shall be comparatively e

and natu ral.

CHAPTER X

THE MEDIA or NEW PERCEPTIVE FACULTIES

The Spiritualization ofMatter the End ofEvolution—Sequenceand Design in the Evolution of Human Faculties—T heUpsp ringing Intuition—Evidences of Supernormal Pow ers

of Percep tion and the Possibility of Attainm ent—The Influence and Place of the Pituitary Body and the Pineal

Gland in the Evolution of Additional Faculties—TheSkep tical Attitude of Emp irical Science and the Need

for a More Libe ral Posture—The General Resul ts of

Pituitarial Awakening Upon Man and the Theory of

Know ledge .

EVOLUTION i s a continuous process and the prim alimpetus back of the grea t on-flowing ocean o f l i fe a ctsinfinitiv ely. It i s not terminated when l i fe has succeeded in pe rfecting a form for the perfection o fforms in themselves is not the end o f vital activity.

The end of ev olution is the comple te sp iritualiza tion

of matter . So that i t does not matte r how perfecta fo rm may be eithe r subj ectively o r in i ts adaptationto envi ronments ; i t doe s not matter how faultless amedium for the ensouling l i fe it may be , there i s everthe eternal necessi ty that l i fe must drive i t back overthe path o f its genesi s until i t Shall be transmuted intopure sp iri t. Adaptation succeeds adaptation and witheach there i s a change in the form and this processcontinues unti l there i s a more o r le ss perfect con


gruence between form and j uxtaposed environmer

conditions . But no sooner than agreement has b e

a tta ined under one set o f conditions new conditie

a ri se and requi re a new setting, new adaptative mo

ments . Thus there i s a continuous proceeding frstage to stage , going from the grossest to the subtland most refined , always the form i s being pus]onward and upwa rd by l i fe. But adaptation i sundergone fo r the benefi t o f the form , but more trfo r the informing principle . It i s the progressionthe l i fe-el ement which consti tute s the adaptationfo rm to fo rm and to thei r p ecul i ar envi rons .fo rm i s a tool o r inst rument o f l i fe whi ch it discathe moment i t fa ils to respond to i ts requirem e1

Thus forms a re constantly be ing a ssumed and a s cstantly being rel inquished. But no effort o f l i fe i s lrega rdl ess a s to whether the act ion is performedone o r anothe r form . The total ity o f matte r i s l?

p etually be ing acted upon by the total i ty o f l i fe . Evappulse o f l i fe aga inst matte r means an added p

'

in the direct ion o f sp i ritual i zation . The total itysuch appul ses o f l i fe aga inst matter may seem infinitsm all in the visibl e results which they p roduce inprocess o f spi ritu al i z ation ; but with each there i seternal ga in in that movem ent that sha ll end incomplete transmutati on o f materi al i ty into spi ri tualThi s action o f l i fe in metamo rphosing matte r,nethe r pole o f the great p a i r o f opposi te s , into sp i ral ity, i ts copol a r facto r, in i ts outward, vi sibl e effei s wha t w e vaguely call evolution . And such i tfor l i fe i s merely unfolding that whi ch i t ha s enfolcfM atte r, having been involved a s a phase o f kos:i nvolution, is now be ing evolved.


in contradi stinction to the rel ative unpl i ab il i tymatte r, and due a lso to the fa ct that l i fe i s kinetic zmatte r , being a mere deposit o f l i fe , is stat i c . I.i s mob il i ty whi l e matte r is immobil i ty and thuspossessing a greate r range o f freedom i s , of COD]co rrespondingly superio r ; but in thi s adaptationi tsel f to the l abyrinth ine caviti e s and multi formedt erstices in matter i t exhibi ts but a seeming serfd

which i s really not a se rfdom but a maste ry. It

a s i f a man had taken lumbe r, hardware and stc

and built a hous e wherein he might dwell—l i fe lmerely us ed mat te r, molded and fashioned i t soto make fo r i tsel f a medium , a dwell ing-place whe ti t ope rate s , not a s a sl ave but a s a maste r posse ssunl imi ted freedom o f motil i ty. In the product iona form li fe stamps upon it, once fo r a ll time, the po f i ts engendering action . I t l e ave s i ts finger-p riupon the mold which i t makes fo r itsel f. So thaiwe would know where l i fe has been or where i t isshould look fo r i ts finger-prints ( o rgani zation ) ;shoul d obse rve the s inuosi ti es which mark its pzway, remembering always that i t is l i fe that has fornthe intrica ci es and complexi ti es o f the form into wbi t pours i tsel f so accommodatingly in o rde r that i t nra i se that fo rm , develop and transmute i t into 801

th ing higher and bette r.When we speak o f form i t must not b e underst <

thereby that refe rence i s made only to the grphysi cal fo rm

, but to the enti re range o f vi ta l a s surti ons o r vehicl e s which l i fe ensoul s for purpose smani festat ion . This range we bel i eve to coverwhol e path o f kosmogenesis se ri a ting from the den :to the most subtle . Our chi e f concern , however.

NEW PERCEPTIVE FACULTIES 331

the immedi ate effect whi ch the totali ty of l i fe’s operations will have upon hum ani ty o r the form which itensouls a s the human organism . For it i s impossiblethat humanity shal l escape e ither the gene ral or thespecific re sults o f th e exalting power which l i fe exertsover materi al i ty and i ts appurtenances. It i s

,o f

course , impossibl e here to go into the various impli cations o f thi s general forward movement o f the uni

v ersum of mate ri al ity o r even to outl ine bri efly thedivergent l ine s o f operation into which a sati s facto ryexposi tion o f thi s vi ew would natu rally l ead . And

then to do so would be inappropri ate in a volume ofthi s kind . So w e shall have to be content a t thisj uncture to l imi t our s tudy to a consi deration o f whatwe beli eve to b e some o f the immedi ate indications ofthis vast and m ost far-reaching phenomenon .

In the chapter on the “Genesi s and Nature o f

Space i t is shown tha t the materi al universe i s en

gendered at the same time and by the s ame movement o r p rocess a s the universum o f spati al i ty andintell e ctuali ty and tha t a s the passage from chaos tokosmos proceeds the function of this movement ischanged gradually from engenderment to exaltationwhere in m ateri ality is transmuted into spi ritual i ty . Itis, o f course , obvious tha t as materi al ity i s exaltedso a re sp ati al i ty and intellectuali ty ; and that as theone becomes more and more refined , capable o f an

swering to higher and ye t h ighe r requi rements so doall the o thers . For, a t work in all and through all o fthese

,i s the cu rrent o f l i fe which pervades them , en

gendering,sustaining and el evating as i t proceeds.

So tha t a s matter has evolved added characteri sti csand properti es, each answering to a given need and


ari s ing out o f the nece ssi ti e s inhering in the stagewhich it appea red , so has the intellect evolved fatti es to correspond therewith . In other words, the e

‘

lution of facul ti e s for the exp ression o f the hurrintelle ct ha s p roceeded synchronously with the ev e

tion o f materi al qual iti e s. And whenever a 11

faculty o r an addit ional scope o f moti l i ty i s achiesby humanity there i s always found a set o f kosrconditions which answers thereto . The ca rdinal p rciples of the doctrine o f evolution a re not, therefoa dve rs e to the conclusion that the organs o f sen

perception—hea ring,touch , s ight , taste and smell

have not been endowed upon the human race ortained by i t a t one time ; but rathe r that

‘

each amswing to a newly acqui red need and opening a wi<scope o f moti l i ty fo r the intell ect ha s been CV0l\

sep arately and in due orde r. It would al so seem t]the qual i ty o f consciousnes s , a s i t has b een manifeS‘

i n the vari ous stages o f l i fe through whi ch i t has p assand especi ally the mine ral , vegetal and lower anirrhas not always been o f the same degree o f e ffi cienNor has i t enj oyed the same kind of freedom wh

i t now enj oys in the highly evolved genus homo .

i s equally apparent that matter i tsel f has not a lwbeen in possess ion o f the s ame qual iti e s and ch

act eristics which i t now exhib its ; but that i t, too , I

gone through variou s stages o f evolution bringfo rwa rd into each new stage the t ransmuted resultse ach preceding one as a basi s for furthe r ev olutand expans ion. The innumerabl e a rchae ologi cal 1

dences which support thi s vi ew make i t unnecessto do more than state the facts, as they appearbe substanti ated by indubitabl e testimoni e s . Furtl


the l ea st poss ibl e fri ct ion and retardation as also ithe greatest possibl e coordination and coop erat

Acco rdingly, notwithstanding the fa ct that materiz

must posses s in potenti al i ty a ll the qual iti e s whicwil l a t any time reveal , i t is neverthele s s necessa rythese qual i ti e s shal l come forth gradual ly and ino rder. Simil a rly, humanity has come into posse so f i ts va rious faculti e s o f mind , and powers o f phjlogi cal funct ions

,by insensibl e degrees , the hig

always being held in abeyance until the lower i s 1developed . Those faculti e s which are to bes

added powers , addit ional freedom and a greate r s<of moti l i ty are th e one s which appea r l ate r th an t]which a re truly primit ive in cha ra cte r. These fhave b een amply demonstrated by th e sci ence o fbryology wherei n i t i s Shown that ontogeny i s acapitul ation o f phylogeny. That i s , the hi sto ry ofdevelopment o f the individual i s a recapi tulation ofdevelopment o f the speci es . Thus the various stof development through which the human empasse s whil e in utero a re but a repeti tion o f sinstages through which the ent i re human speci e spassed in its phylogeneti c development . Wherei t i s ce rta in that humani ty has not atta ined , a t onethe same time , all the powers o f mind and body vit now possesses ; that the ch ildhood o f the humanrepresented a time when i t had but few facultie

organs o f sense-perception—indeed a time wher.higher sense-organs o f smell , taste and sightenti rely l acking although resi ding in potent‘

therein .

It i s undoubtedly true that the ea rth ha s pthrough a simil a r evolution with respect to i ts


material cha racte ri sti cs , tha t its childhood was , in al lpo ints, analogous to the childhood of humani ty ; thatthe a i r, ea rth and wate r were wholly absent

,except

in potent i al i ty, during the nebulous youth of i ts genesi s .It i s even p robabl e that there a re a t work to dayprocesse s which in the future shall culm inate in theevoluti on o f newe r, higher and more complicately or

ganized species o f pl ants, animals and minerals. Everyyea r brings fresh evidences that crystall ize the con

v iction that the e arth has been the scene for the ap

p earance o f many strange orders o f anim a l l i fe .Fossil i fe rous strata a re continually yielding inconte stable testimonie s of changing flora and fauna . Wecount the animal and vegetal l i fe of to-day a s beingmore highly developed than that of any other previousage , and i t is wel l tha t thi s i s so , for simpl ici ty ofo rgani zation and primal ity o f mani festa tion a re alwayssucceeded by complexity and a greate r scope o fadaptabil i ty.

We have sa i d that the whole of that movemento f the intellect whi ch has brought forth the metageometri cal creations of hyperspaces, the curvature o fspace and its mani foldness together with the enti rea ssemblage o f matheti c contrivance s a re merely theea rly evidences o f the appearance in the human raceo f a new facul ty

,a new medium of perception whereby

the Thinker shall a cqui re a still greate r range o fmotil ity than tha t now offered by the intelle ct . Attention has been call ed also to the fact that this phenomenon has been mani fested not alone in the field o fmathemati cs , but in a rt, rel igion , pol i ti cs and also insci ence in which w e have only to witness the m arvelousstrides already made in the di scovery o f radio-active


substances , the ROENTGEN , BECQUEREL, LEOIand othe r kinds o f rays. I t i s quite confidentli

liev ed that these forward movements in eve ry bro f intell ectual pursui t, these combined efforts of

i ntellect , in pe ering into the occult S i de o f mat

things , a re in response to the evolutionary - ncccthe Thinke r, and in a ddition , a re the evidences ,shal l in time b e the caus e , o f the development caddit ional set of facultie s . Funct ion , or the perfance o f acts, dete rmines fa culty o r the power of aand ultimately the organ itsel f. Thus the mereto perform a roused by desi re and vital i z ed bywill a ctually te rminates, in the course o f time , iigenesi s o f a faculty , or the power to perfo rm .

constant uprea ching yea rnings o f the Thinke r thrhi s intell ect fo r gre ate r fre edom and a la rgero f action , the desi re to peer into the mysteri e s oand mind , the infanti l e out-feel ings o f the men

a fter some sa fe r and surer basis for i ts the01knowledge cannot fa il in producing not onlyfa cul ty or powe r to sati sfy these cravings but theo rgan o r medium by vi rtue of which the satisfzmay be atta ined .

It i s not strange that in mathematics the in!should have found fi rst the clue to the existena higher sphere o f intellectual rese arch whenm ight become the creato r o f the va rious enti ti e speopled the new found domain ; i t i s not strangethe mathemati ci an should, in thi s instance , ha\sum ed the rol e o f the prophet procla im ing by v:matheti c contrivance s ( although unconsciously)the hum an race is nea ring that time when i ta ctually be abl e to function consciously in some t


race ; for, in truth , mathema t i cs i s the diadem ofwherewith man has crowned hi s intel lect. Yetwell , yea imperative in the l ight o f recent dev elopnin the re alm o f hypersp ace , that a ca re ful di scrimt ion should b e ma de a s between the sphere ofintell ect and that to which the intu iti on Shall atta in .

The intuiti on , l ong hel d in abeyan ce unti l thtellect should be fully crowned and reach the l l

o f its powers , i s now coming to the front. It wimany centuri es p erhap s befo re i t shall have growsuch proportions as those a lready atta ined by thtellect ; perhaps a few thousand years may pass bethe intuition sha ll have evolved to tha t point wi t may l abor a s coadj uto r to the intel lect ; butdoubtedly the t ime wil l com e when i t, too , Shall refthe Thinke r’s l abors with that which Shal l be r

precious than the crown o f gold which the intchas won . Then , the intell ect, grown old and decwith yea rs o f re igning shall become dim and cry

shaped and finally pass into autom atism or reflcmovements where without the urge o f vol itionalpulse s i t will p e rform with exactness, precisionutt e r loyal ty the ta sks whi ch i t ha s le a rned so

to execute i n the days o f its forgotten glory. l‘

kind will then b e free . A new freedom , whereir

erstwhil e l ightn ing flashe s o f intu ition wil l betfused into one glorious sheen o f all-reveal ing 1shall come to m en and thus the race resplendentwalk the ea rth enshrined in the maj esty o f dpowers atta ined as a resul t o f mi ll ions o f yeara spi ra tion .

That there a re supersensuous realm s so fa r athe range o f our sense s as to be enti rely beyond


ken needs now no proof o r argum ent ; for the sci ent isthas demonstrated , by the invention of instruments ofextreme deli ca cy and precision, that such a worlddoes really exist. Already we know of sta rs so distant tha t, though l ight traverses in the brie f space o fan hour six hundred m i ll ion miles, they m ight haveceased to shine before the pyramids were bui lt and

yet be visible to us in the Skies. I f the hum an eyewere as sensi tive as the spectroscope many thousandtints and shades might be added to the world o fcolor ; i f they possessed the magni fying powers o f themicroscope we should l ive in constant terror and aweo f the m onst rous entiti es tha t teem in the wate r whichwe drink and in the a i r which we breathe ; and i f oure ars could detect the microphoni c vibrations whi chregi ste r in the del icate appa ratus o f som e microphonesthe dead , vacuum-sti llness o f nature’s grea t si lenceswoul d appea r a s a babel o f voi ces by the seaside .The sense o f touch , responding to the sam e range ofvibrations as the microm eter, would reveal actually theinterstices between parti cl es of the densest el em ents ;and gold, s ilve r, pl atinum and mercury would seembut honeycombs of matter. But , to the fo rwardlooking there is no elem ent o f absurdity in the ex

p ectation tha t al l these senses shall , one day, be ableto dispense with the a rtifici al a id of physi cal ap

paratus and perform ,with even gre ate r precision and

fa ithfulness , the task which they now perform so

crudely and ineffectively. There are without doubtvibrations o f taste and sm el l which are so far abovethe range o f these sense s tha t they have no effect uponthem whatsoever. Notwithstanding the fact, howeve r,that the galvanomete r, microscope , the microphone,


the spectroscope and the tel escope have extendedthe sphere o f sense-knowledge there are yet sulvibrations to which these del i cate inst ruments doand ought not be expected to respond . But toa s do many empiri cists, tha t since these phenoncannot be dete cted by sci entifi c instruments the)not, there fore , exist s eems to be exp ecting too n

of materi al means a s well a s exposing onesel fnecessa ri ly to cri ti cism on the grounds o f ext:materi a l i sti c appetence s .There i s indeed ne ed o f a more l ibe ral atti

among men o f s ci ence towards the world o f theseen . Intole rance o f the data which it ofle rs willa time perhaps p rese rve the aloofness o f

'

sci e1dogmati sm inviolate but there wil l most surely c

a reaction aga inst the dogm atism of sci ence andwill seek freedom and atta in i t despite thei r fetSir OLIVER LODGE

,in hi s book , the Surviv al ofM

s ays : “M an’

s outlook upon the unive rse i s ent tupon a new phase . Simultaneously with the begino f a revolutionary incre ase in hi s powers o f phylocomotion—which will soon be extended to a ldimensi on and no longe r l imi ted to a sol i d o r Ifsurface—his power o f recip rocal mental int ercouralso i n proces s o f being enl a rged ; for the re a re athat i t wil l some day b e no longe r l imited totemporary deni z en s o f ea rth , but will permit a utition o f knowledge and powers superio r to hi s 1

even to the extent o f ultimately atta ining trustwcinfo rmation concerning othe r conditions o f exi stenIt i s the author’s good fortune that he ha s 1

period extending over several yea rs been abl e to v

See p p . 338 , 34 1 .


future . Indeed , there a re few persons now l iving vhave not had s imil a r experi ences

,i f not exactly l

these , o f the s ame nature . These example s , of coumay be greatly multipl i ed in eve ry count ry inworld , and i t i s unnecessary to enumerate th

fu rther ; fo r, when once the existence o f such faculhas been demonstrated in persons, e ither in a nonor an abnormal conditi on . the i r p resence can no lonbe questioned by the fa i r-minded. It i s , then, onl

question o f evolution be fore they will appea r innormal way and thei r universal i z ation, a s transm ible cha racte rs , be an a ccompl i shed fact . Whenare brought face to fa ce wi th this sort o f phenom e i

which se ems to be increasing rap idly the conclusi s inevi tably forced upon us tha t since evolution rr

b e a continuous p rocess and matte r destined to yihigher and more refined powers and hum ani ty to ccinto a fa r m ore extensive scope o f motil i ty becao f the opening avenues o f knowledge , i t i s notpossibl e tha t these a cute r senses , these new facul

a re now existing in the hum an race in a rudimentstage and a re desi gned to becom e the unive rsal Isess ion o f all . That thi s i s to be the almost imm i

ate outcome o f the pe rpetual exalting power wll i fe exerci se s not alone ove r materi al ity but 0

human organs and faculti e s a s well , seems to beone big, outstanding impl i cation o f the evolutionp rocess . The presence o f such functions a s the abito sense the invisible and the inaudibl e , to answervibrations fa r subtl e r than anything in the scopeour external sense-organs, ce rta inly indi cates theist ence o f rudimentary fa cult ies which make th

functions poss ibl e . Back of these v ague , indefi:

NEW PERCEPTIVE FACULTIES343

functions,back of ev ery sup ernormal or abnormal

manifestation of man’

s mentality and back of all thatclass of phenomena which take their rise out of sup ersensuous areas must lie

,in ev e r increasing po tency,

faculties and organs, howev er rudimentary, which are

the source of these manifestations . Li fe,tha t in

eluctable agent o f creation, which i s incessantly pushing outward the confines of the intel le ct ’s scope ofmotil i ty, never wearying, neve r tiring no r sle eping, haslong ago , i n the dim and distant past of m an’

s ev olu

t ion , l a id the foundations ; and in fact, with one strokeo f i ts cre ative hand , has molded the o rgans which areto become the active m edi a o f these new facul ti es.

And now, these incipient demonstrations , these infantile struggles which we se e now and aga in out

putting from them , a re but the Speci al i zing processesthrough whi ch , in the i r l ate r developm ent, theseorgans a re proceeding. These a re the outwa rd signswhich should tel l u s that l i fe is breaking up theseorgans into specia l pa rts, assign ing to each a certa indivi s ion of l abo r and making of each a perfect coOrdinate o f all the othe rs . It i s , by these ve ry dispreadexhib itions, cutting up, speci al iz ing and by slow degrees dete rmining the functi on , characte r and gene raltendence o f the organs of expression wherewith thesemani fe sta tions shall be central ized and put into ef

fectiv e ope ration . In doing thi s , i t is but followingits accustomed procedure , the procedure which i tadopted when i t produced the eye , the ea r, the hea rtand the sple en . We sha ll , there fore , gauge our un

de rstanding of the purport and end o f evoluti on ; infact

,we shal l dete rmine our exact intellectual com

prehensi on o f l i fe i tsel f, by the a tti tude which we


adopt towa rds i t and the mode o f i ts app earan

Much depends, accordingly, upon the posture whwe a ssum e towards l i fe—whethe r we shall say 1

total i ty o f l i fe ’s creat ive powers has been dissip ati n the bringing o f the hum an body to i ts pres‘degree o f perfection ; whether we shall say tha t i tn either necessa ry nor poss ibl e for l i fe to p rodl

o ther o rgans and facult i es which shall respond to tunseen world about u s reveal ing i ts glori e s in a wfa r more pe rfe ct than do our externa l sense-orgareveal the wonders o f the worl d o f sensation ; c

whethe r we shal l conclude from these mo st palpalevidence s that l i fe ha s yet othe r powers and facultwhich i t designs to bestow upon the human mind aothe r o rgans and capabil i ti e s with which i t shall endithe hum an body so tha t man , in his evolution , sh

be enabled to rise to st i l l higher spheres whil e I

inca rnate . There may be , and undoubtedly a re , thcwho , fo r various reasons prefe r to take the forrrposi t ions and there a re certa inly those who liLODGE

,FLAMMARION

, HUDsoN,CROOKES and a ht

o f others , pre fe rring the l atte r Vi ew, would ra tkbel ieve in the strength o f the great m ass o f corrobortive testimonie s tha t w e a re even to-day in the m ito f the matutinal hours o f a newer, a bette r and

fa r more effi ci ent e ra o f human evolution than athrough which we have hi the rto come .Al re ady, recent sci entifi c investigations and t

results obta ined there from have begun to turn thetenti on o f medica l authori ti es to the activiti es of Uvery small organs si tuated in the mid-bra in and knowa s the pineal gland and the p ituitary body. The

organs, and especi ally the pine al gland hitherto su


poste ri o r and very va scul a r, sp ringing in its deve]

ment from the buccal cavity o f the embryo ; theterior lob e i s s i tuated in a dep ression of the anterio ri s a bra in-process. The p ituitary body i tsel f is lodin a cavity o f the sphenoid bone called the pituitfossa. Thi s i s a most rema rkable posit i on , fo rreason that the sphenoid, or wedge-shaped, bone wll i e s at the base o f the skull a rti cul ates from bel

with the occipital and in front with the frontale thmoid bones and by l a tera l processes withfrontal

, parie tal and temp oral bones . From thi s ption i t b inds togethe r al l the bone s o f the craniand moreove r, a rti cul a tes with many bones o f the f.I t i s upon the uppe r surface o f the sphenoid bwhi ch occupie s such a prominent and comfmanc

posi tion in the cranium , in a deep depress ion , that

pituitary gland i s l ocated .

Each nasal chambe r i s l ined by a mucous mbrane called the pituitary o r Schneiderian. This mbrane i s p rolonged into the mea tuses and a i r Simwhich open into the nasal chambers . The p ituiim embrance i s thick and so ft and dim ini shes theo f the mea tuse s and ai r s inuses . I t i s covered bci l i ated columna r epithel ium and conta ins numerracemos e gl ands for the secretion o f mucous

pituita. I t i s al so va scul a r and the veins whi ch ran

i t have a plexi fo rm or net-work l ike a rrangement.divide s into two membranes—a respi rato ry, whic]concerned In breathing, and an ol factory region .

resp i rato ry region corresponds to the floor o f the Into the inferior turb inated bone and to the lower tiof the nasal septum . The ol factory region is theand distribution o f the ol facto ry nerve and co


sponds to the base of the nose , to the superior and

middle turbinals and the upper two-thi rds of the nasalseptum .

Recent developments prove that this gl and has apro found influence over the anim al economy. I t i sbel i eved by some that the p ituitary body actually destroys ce rta in substances which have a toxi c influenceon the nervous system ; others beli eve that i t secretesmateri al medi a fo r the prope r action of the trophicor nutri tive appa ratus ; stil l others beli eve that i t influences blood-pressure . It i s known, however, fromexperimentation , that its removal in dogs , cats , miceand guine a p igs causes a fall o f tempera ture , l assitude ,muscul a r twitchings, dyspnoea o r diffi cul t b reathing,and even speedy death . Hypertrophy of the glandi s di rectly a ssoci a ted with certa in diseases, such as

giantism and acrom egaly. The latte r is a diseasewhich causes a general enl a rgem ent of the bones o fthe head

,fee t and hands, usually occurring between

the age s o f twenty and forty years, and m ost frequentlyin females. The fact that these diseases a re so closelyassoci ated with a hypertrophi c condition o f thep i tu i ta ry gland has led to the conclusi on that perhapsthe gi ants o r Cyclops o f anci ent tim es were cases of

giantism or acrom egaly. This view, while interestingfrom the standpoint o f the funct ions of the pituitarygland , is not necessa rily a correct one ; for the ageo f gi ants, when men attained to a much l a rger staturethan at present, can be accounted fo r on othe r grounds ,nam ely ; that the early mesozoic man, on account o fhi s having to l ive among animals, trees and othervegetation o f such huge S i ze , had na turally to be fi ttedwith a frame proportional to othe r animals in order


that he might successfully cop e with his environcondit ions. Natu re thus wisely fi tted him forcondi tions whi ch she had prepa red for the scene shi s l i fe .The facts adduced in the foregoing description

purely empi rica l and may be verified by any who St

to establ ish thei r correctnes s o r inco rrectness . l

we are about to introduce a speci e s o f testim ony whwhile i t may also be verified wil l not be founde asy o f verification a s the above-mentioned phylogica l facts , and not by the same means ; yet t]a re neverthel es s deserving o f a pla ce here . It i sl iberal atti tude that we must adopt towardsphenomena , excluding none that give promise ofwidening and deepen ing o f our knowledge and an

planat ion o f much that has seemed heretofo re uncountabl e .We have noted how subtl e is the physical conr

ti on between these two bodies, the p ineal glandthe p ituitary body; we have seen how profound i seffect which the l atter has been demonstrated, itmeasure , to have ove r the enti re bodi ly economy ;the re is even othe r testimony to the effect tha t thgi fted with the inner visi on can observe the “

pulsat

aura” i n each body, a movement which i s not unlthe pulsa tions o f the hea rt and which never cea

throughout l i fe . In the developm ent o f clairv oyai t i s known that thi s motion becomes intensified ,auri c vib rati ons becoming stronge r and more Pnounced. The p ituitary body is the energizer o f

pineal gland and, as its pulsating,a rc ri ses more a

more unti l it contacts the p ineal gland, i t awakensa rouses i t into a renewed activi ty in much the sa


sheer vanity i s he adjured to look within—into theterior of l i fe , of mind and the things which he hanwi th hi s instruments—for the answers to his que 1fo r the path which l eads into the wake of l i feconsciousness. Because his utter loyalty and devoto the m odus v iv endi o f the intell ect wil l not p erthi s ; but, a fter all , i t i s not wholly wi se to al lu reaway unbetim es from hi s s ea rch a fte r truth throsup e rficialities no r to inve igl e him into giving uptenacious p rosecution o f the physi cally determinaWe would not have i t so ; fo r, perchance , he ,one fine day, i n the qu i et o f hi s l aborato ry shal l ccupon the data which may substanti ate in hi s own 11the long settled conclus ions o f the occult i st who ,quently and not without cause , grows impati ent a ts ci enti st’s obstinate del ay. These two workers,empi ri ci st and the occult i st, must ul timately comegethe r a s coll abora tors—the one working uponfo rm , the vehi cl e , physical m atter and the othe r si

ing to understand the l i fe , the interi o r fo rces w]produce

,the creative element . They cannot ren

a lways aloo f from one anothe r ; fo r they, too , aremen digging a tunnel from opposi te ends . Fin :the partit ion wil l b re ak and thus will dawn a newfor the knowledge o f humanity and men wil l seera tiona le , the truth and good sense o f coéip era

'

i n thi s respect.I t can b e sa i d with confidence that whateve r in

future may be l ea rned a s to the phys iologi c functio f the pituitary body and the pineal gland, i t sufficeknow that i t is l i fe whi ch they express and that,in a far superio r manne r than any of the other 3 :organs. The modus o f the se two glands differs i

NEW PERCEPTIVE FACULTIES 35 1

very marked way from that o f the organs o f s ight,

hearing , taste , smell and feeling . For these latter aredesigned for contact wi th the external , obj ective worldo f sensa tions, the i r growth and evolution being dependent upon stimul i rece ived from without whil e withthe former the case is fa r diffe rent

, in fa ct , just theopposite . The mode o f l i fe o f the pituitary body andthe pine al gland, instead of rece iving sustenance and

impetus from external stimul i , i s rather dependent uponimpacts rece ived from the Thinker’s own consciousness and made to impinge upon them by an exclusivelyinterior process . Situated in the mid-brain, sa felysecluded from all external interference , they a rena turally l imited to stimul i which come from within, orit may be sa id , they are responsive to excitations thata re more spi ri tual than those which com e through theexte rnal sense-organs . I f, as has been sa id they con

trol the inte rnal p rocesses of metabol ism ( anabol ismand katabol ism ) , oxygenation, nutrition, and otherimportant internal movem ents, none of which can besai d to be under the control o f the intell ect, is i tnot

,the re fo re

,j ustly assum ed that thei r response is

di rected towards stimul i whi ch ari se interio rly o r upona pl ane highe r than the intell ectual ? It i s a mattero f sci entifi c knowledge that those persons gi fted withcl a i rvoyance

,and commonly known as “

sensitives” a refa r more re sponsive to nervous exci tation than thosewho are not so gi fted. This would seem to implythat

,on account o f the superactivi ty o f these two

organs,the enti re ne rve-body has , in consequence , be

come more del i cately and subtly o rgani zed . Theyseem to act a s a switchboard for the regulation of theflow of the current of l i fe through the body. Not

352. THE MYSTERY OF SPACE

only do they come more nearly to an adequatepress ion o f the physi ologi c function o f l i fe , butthe i r energization means an enl a rgem ent o f the so f perception by giving the Thinker’s a ctive consciness access to hitherto unapproachabl e real iti es anpenetrating the oute r mask which l i fe ensouls andlaying ba re a dom a in o f unl im i ted knowledgemani festat ion o f which is fa r more re al than anyt

the senses can di sclose , i t i s evident that they constiin the i r col laborative functions , a more highly ac

abl e medium fo r the expression o f the Thinker’ssciousness. And i f so , fo r the kosmic consciouswhich is the source o f all fo rms o f consciousness ,furnish a speci al i z ing and adap tizing agency.

Now, i n a ll those cases o f insp i rations , rev elattelep athic communications, cl a i raudience , clairv oy:dreams , visi ons, e tc . , where in the Thinker i s enato perce ive facts and veri ti e s which are then preseto hi s consc iousness in a manner cl e arly withoutprovince of the comm on sense-organs, i t must bepa rent tha t these mani festations a re app rehende‘

a perceptual mechanism which is enti rely indep eno f external s ense p re sentations but which i s an intnand subtl e r form of psychi c activi ty. Sounds vs

a re hea rd by so-called “s ensi t ives” and obj ects wa re perce ived by eyes that a re keener than those o rsa id to have been evolved from the “medusa” cabe heard by othe r person s nor perce ived by the )any way. Thus i t would seem that there a re io rgans o f p erception which respond to thesevibrations and which enabl e the p erson so gi fteap p rehend them .

There are those who,presumably basing 1


But why thi s p rolonged struggl e , why thi s intelle imaneuvering and sophistry? “We can calcu

compute,excogitate ,

” s ays PAUL CARUS,4“and

scribe all the characte ri st i cs o f four-dimensional 31so long a s we remain in the realm of abstra ct thoand do not venture to make use of our motil i tyexecute ou r plans in an actual iz ed construct ior

motion ; but as soon a s we make an a priori

struction o f the scope o f ou r motil i ty, we findthe incompatib il i ty of the whol e scheme .mathematici ans a re fo rced to rel inquish all hopetransforming the world o f l i fe into a sort o f 1space dwelling place where everything i s done accing to the l aws o f mathem atics. But wheth ershall a ccept i t o r not there is a wide r, true r and r

rational vi ew which recognizes al l m etageom et

i nvestigat ions, as well as al l kindred phenomenauniversal evidences indeed , a s the ve ry cause s W]i n the future humani ty, will a ctu ally awakencause to be a ccelerated in thei r developm ent tl i ttl e inner sense-organs, the p ineal gland and

pituitary body, whose perfect development p rorto provi de fo r the Thinker’s consciousness an avof expression such a s hum ani ty has possessed n

be fore . And too , i t i s not without ful l knowledgthe fact that i t has been customary, among CCJ

sci enti sts o r p erhap s all o f them , to regard tbodi es, at l ea s t the p ineal gland, a s vest igal o rbelonging to the past o f hum an evolution, tha tmake these assertions . Yet, a s man proceeds inperfecti on o f mechani cal sci ence , i n the dev elopro f in struments o f preci s ion that a i d hi s ext ernal SC]

‘Foundations of Mathematics, p . 90.


responds more and more to the subtle vibrations teem o

ing everywhere in the atmosphere about him, and

comes , in the course o f time , naturally to possess amore sensi tively keyed ne rvous mechanism,

a finerbody and higher spi ritual aspirations, there will b ea corresponding widen ing of hi s scope of visi on and

the atta inment o f large r powers of perception whichmust inevitably, i n the ve ry nature of things, tendtowards a deeper and true r knowledge .In view of the foregoing , it is bel ieved that the

general re sults of this p ituitarial awakening whichmay be expected as humani ty continues to evolveshould be seen in the marked effects which will bewrought in the enti re m etabolistic a rea o f the humanbody whereby a gradual intensification and sensitiza

tion o f the whol e neural mechani sm will ra ise thepecul iar effici ency o f al l the senses whethe r purelyphysi ologi c o r psychic. For there are undoubtedlynotes so del ica te in the i r intensi ty that they transcendthe grasp o f the audital nerves ; scents and fragrancesso subtl e in thei r excell ing purity that i t is beyond thepow ers o f our present olfactorial contrivances to det ect them ; colors and other external stimul i so sub

lim ely supersensuous that a nervous mechanism perhaps ten-fold more del icate and responsive than oursis requi red to apprehend them . Al l these , and morethan at p resent i s conceivable , will com e , with thea id of p ituitarial stimu lation, within the purview o fa more highly developed humani ty o f the futu re . Andbecause mathematics have l ed a movement into thevery camp of the intellectual s—logic-bound and

tethered by the seve rest rigors o f mathesis -wherebythe intol erant intellect ha s been compelled , by rules


o f its own making, to recogni ze the exi stence o fsupersensuous, and by looking into the gl a ring lof the sun of the in tu itab le to ga in strength of v iand boldnes s to press forwa rd, a gre at andreaching servi ce ha s been wrought for humanity.in the tower o f hyperspace mathemati cs have ema monument to the intel lect which , as long as

human ra ce rema ins, will mark the great tu rning pin man’s path to the highest l i fe .What i f i t were possible that the sci enti st , w

he had ca rri ed instruments to the i r utmost p reciand penetration , should suddenly, o r otherwise ,endowed with a cl ear-perceptivi ty o f s ight, hearingsmell , so tha t he could with his own powers o f visfeel ing and hearing take up the task where the miscop e , th e microphone and the micromete r l e ftand delve into depths fa r too unfa thomable forappl i ances , perce iving the innermost real iti e sthings and processe s ? What i f i t were possibl eh im , with thes e added powers , to se e and exar.

without the ai d o f the magni fying l ens the el ectthe a tom and the molecul e ? What i f the cell ,bacterium , and othe r invisibl e forms of l i fe Wt

then del ive r up the i r secrets to hi s knowing m iWhat i f he could sense with hi s own inne r visi on ,ultra-viole t and the infra-red rays ; what indeedsp irit i tsel f, the innermost sheath o f l i fe , shoulc

visibl e and palpable to him and he could noteinte rnal p rocesse s , the action and movements o finfinitesimals o f l i fe ? Think you not tha t such dicontact, such immedi ate and incontrovertible kni

edge would be fa r superio r to any advantage whi s manufactured devices now bestow? I t is even


the intu i ti on—di rect and instantaneous cognit

Philosophy will b e regenerated, re-cre ated. Spection will g ive way to truth and there shall b e butphilosophy and that shal l b e the knowledge of the r

Mathematics,the royal ins igni a o f the intell ectual

becaus e i t can deal only with immobil it i e s , withments and parts and ha s no aptitude for the contou s flow, will yi eld it s kingdom to .a higher fo rmkineti cs whi ch wil l serve the intu it ive fa cul ty a s mam atics now serve the intell ect. Sci ence wil l thenno longe r empi ri cal i n i ts method ; but a systemdirect and incontrovertibl e truths . Rel igion willto meet thes e changes which wil l come in the p atlhuman evolution ; and fa i th wil l surrender i ts pl accknowledge . Ethi cs , recast in a new m old , willwith the new aspect o f man’s re lation to hi s fellmen . M an , for whose highest good these ultinchanges will come , will be a new creature , a bigand bette r man ; and humanity shall evolve a 1

race . There shall , indeed, be“a new heaven an

new earth.

”

THE END.

B I B L I O G R A P H Y

AMERICAN MATHEMATICAL MONTHLYVol. VII I , p . 16 1 , P. BARBARIN

, for Le Mathematiche,T rans. by GEO. B . HALSTED. See also pp . 3 1

-

35 .

Vol. VII I , 190 1 , p . 2 1 7 , Supp l . Report on Non-EuclideanGeometry, GEO. BRUCE HALSTED .

Vol. IX , p . 59 , GEO. B . HALSTED.

Idem , pp . 1 53- 159 .

Vol. XI , pp . 85 -86, Simon’

s C laim for Gauss.AMERICAN MATHEMATICAL SOCIETY BULLETIN, Lan

caster, Pa.

Vol. I I , series 1895 .

Vol. XX , 19 14, pp . 409-

4 12, R. C. ARCHIBALD.

ANNALS OF MATHEMATICS, 2d series, 1902

Vol. XI I , F. S. WOOD.

ASTRONOMICAL SOCIETY NOTICESVol. XVI , p . 80, 1 856.

BALL’S SHORT ACCOUNT OF THE HISTORY OF MATHEMATICS.

BOSTON COLLOQU IUM , for 1903 , N. Y. 1905 , Forms of Non

Euclidean Spaces, Vols. I-I I I , pp . 3 1 , e t seq . , 189 1 .

BULLETIN NEw YORK MATHEMATICAL SOCIETYVol. I I , pp . 1-144.

Vol. I I I , pp . 1 , et seq . , 189 1 , G. B. MCCLINTOCK.

CAMBRIDGE MATHEMATICAL JOURNALVol. IV, 1845 , CAYLEY, on Analyt tcal Geometry

n-Dimensions.

CANADIAN INSTITUTE TRANSACTIONSVol. VII I , pp . 3 15

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339 , 1909 , AUGUST KIRSCHEAUM.

359

360 B IBLIOGRAPHY

CAYLEY, A , Vide Science, Vol. IX , pp . 59-63Amer . Math. M o., Vol. I I , pp . 102- 106.

Amer . Math. Mo., Vol. VI , pp . 59 65 .

Review of Rev iews, June , 1895 , pp . 693-694.

CLASS BOOK OF NON -EUCLIDEAN GEOMETRY :Vide Science, XI I I , pp . 462-

465 .

CORNHILL MAGAZ INEVol. XXXVII , pp . 647

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Dimension.

EASY NON-EUCLIDMonist , XIX , pp . 399-402.

EDUCATIONAL REVIEWNovember, 1902, issue , pp . 346, et seq .

EUCLID’S PARALLEL-POSTULATE, by JOHN W. WITHERS.

FLATLAND, by A. SQUARE.

FOUNDATIONS OF GEOMETRYScience, Vol. VI , pp . 487-49 1 .

FOUNDATIONS OF MATHEMATICS, by PAUL CARUS.FOURTH DIMENSION :

By HERMAN SCHUBERT,C. H . H INTON,CLAUDE BRAGDON .

FOURTH DIMENSION, SIMPLY EXPLAINEDEdited by H . P. MANN ING.

FOURTH DIMENSIONCornhill Magazine, n.s. , Vol. XXXVII , pp . 647-665C. H . BRYAN.

FOURTH DIMENSION THE RIDDLE OFH. A. BRUCE, In Scientific American Supplement ,

LXVI , p . 146.

GEOMETRY, FOUNDATIONS OF, by O. VEBLENPopular Science M onthly, Vol. LXVII I , pp . 2 1 , et

GEOMETRY. LOEACHEVSKI’S :Theory of Parallels, trans. by GEO. BRUCE HALSTED

GEOMETRY, NON-EUCMDEAN :

American Mathematical Monthly, Vol. VII , pp . 123

362 B IBLIOGRAPHY

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247-249 ; 269 270 ; 307-308 ( 1897 )

Vol. V, pp . 1 -2 ; 67-69 ; 127 ; 2 18 ; 290

Rev iew of Reviews, Vol. I I , 1895 , pp . 693-694.

Space , New Ideas About , Popular Science Monthly,

XI. P 36+

HELMHOLTZ, H. , On Origin and Meaning of Geom

Axioms :Vide Mind, Part I , No. 3 , July, 1 876. For rep lies

discussions of the above , vide Nature , JEVONS,IV, p . 48 1 ; J. L. TUPPER, Nature , Vol. V, p .

HELMHOLTz, in Academy , Vol. I I I , p . 52 ; MPart I I , Ap ril , 1878.

HERMATHENA, Vol. XI I I, pp . 49 1 , e t seq .,1906 .

H IGHER SPACE, A PRIMER OF, by CLAUDE BRAGDON.

INGLEBY, C. MA Non-Mathematical C riticism of a Part of Profi

Clifford’s Lecture on“Curved Space , Macm il

Magazine, October, 1 872 ; also Nature, Vol.

1873 .

JOURNAL OF MATHEMATICS, AMERICANVol. I , pp . 262, 276, 384, 386 ; HALSTED

JOUR. PHILOS ., PSYCHOL . AND SCIENTIFIC METHODSVol. VI , pp . 6 1 7 , e t seq .

KANT, EMMANUEL, trans. by FRED. PAULSEN.

KANT, EMMANUEL, Prolegomena, trans. by J. P. MAH.‘

and J. H. BERNARD, pp . 35 -39.

LAMBERT, BIOGRAPHY OFAmerican Math. Mo., Vol. I I , pp . 209-21 1.

LIE, SOPHUS :Science, Vol. IX , pp . 447

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448.

LOBACHEVSKI, N

Open Cour t , Vol. X I I , pp . 4 1 1-

4 15 ,Amer. Math. Mo., Vol. I I , pp . 137-139 .

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MATHEMATICS, FOUNDATIONS OF : PAUL CARUS.MATHEMATICAL PAPERS OF CHICAGO CONGRESS, p . 92.

MATHEMATICAL PAPERS, pp . 55-

7 1 .

MATHEMATICS (a brochure )By C. J .

KEYSER, Adrian Professor of Mathematics, Colum bia Univ.

MATHEMATICS, HIGHEREdited by MANSFIELD MERRIAM and ROBERT S. WOOD,

1902, pp . 508 , e t seq .

MATHEMATICS , SPIRITUAL SIGN IFICANCE OFC. J.

KEYSER, rep rinted from Co lumbia Univ. Quan ,

December, 1 9 1 1 .

MEMORIAL VOLUME OF SCIENTIFIC PAPERSPub. by Univ. Of St . Andrews, pp . 3

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45 , D.

SOMERVILLEMIND (Magazine )

Vol. I , 1876 , pp . 303 , et seq . J. N. P. LAND.

Vol. I I , 1 87 7 , p . 40 , J . N. P. LAND.

Vol. IX , pp . 522, ct seq . 1900 .

MONIST (Magazine )Vol. I I , pp . 50 , 32 1 , PAUL CARUS .Vol. IV ,

1 894, Articles by G. B . HALSTED and J. DEL

BOEU E.

Vol. VI , 1 895 , HERMAN SCHUBERT.

Vol. VII , G. B. HALSTED, on Saccheri.

Vol. XII I , pp . 80- 102 ; 2 1 8-234.

Vol. XIV, Artic le by E. MACH.

Vol. XV, p . 1 27 , PAUL CARUS .Vol. XVI , 1906 , p . 433 , DAVID P. ABBOTT.

Idem ,1906 , p . 65 , C. J .

KEYSER, on Mathemat ical

Emancipatioas.

NATURE (Magazine )

Vol. VIII , pp . 8 , 14- 1 7 , 36

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3 7

Vol. XV . Pp 533-53 7

Vol. XLV, p . 404

364 B IBLIOGRAPHY

NEWCOMB’S PHILOSOPHY OF HYPERSPACEScience, Vol. VI I , p . 2 12.

NEW YORK MATH . SOC. BULLETINVol. I I I , p . 79 F. S. WOOD on Forms of !

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OPEN COURT (Magazine )Vol. XVI , pp . 5 13

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Vide December, 19 1 3 edition, p . 757 .

PHILOSOPHY OF MATHEMATICS, by PAUL CARUS.PHILOSOPHICAL REVIEW

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Vol. XII, p . 493 .

POPULAR SCIENCE MONTHLYNo. 66 pp . 639-646.

Vol. LXVII , pp . 639 , et seq .

Vol. LXVII I , pp . 2 1 , e t seq .

Vol. LXXVII I . p . 554 SAMUEL MOF

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POPULAR ASTRONOMYVols. VII and VI I I G. B . HALSTED.

Vol. XVIII , pp . 42 , et seq . ( 19 10 W. H. PICKERINGVol. LXXV, p . 1 79 , W. W. PAYNE, on At traction

Figure of the Ear th.

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POPULARIZATION, T HE, OF NON -EUCLIDEAN GEOMETPYAm er . Math. M o Vol. VII I

, pp . 3 1-

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PSYCHOLOGICAL BULLETINVol. VII I , p . 22

QUEEN’

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446, A. L. CLARK

REVUE GENERALE DES SCIENCESNo. 23 , trans. Nature, Vol. XLV, 1892.

INDEX

Abeyance , higher faculties he ld in,until lower are fully developed,334

intuition kep t in , 338 , fiAbility , to sense the inaudible and

the invisible , 342Absolute , no room for the , 101

Abstract thought , invigoratingPower of. 33 . 4°

Abstractions , realizing , 144 , 294Absurdity , analysis inevitably ends

in an , 3 19—20

Acrobatics , mathetic , 146Acromegaly. 347Action, automatic, of the inte llect,

253-

4

engendering, of life on the form ,

33°

Activities , space , as physical andchemical phenomena ,

229-

30

Actualities of the physical worldpushed over into the concep

tual , 1 19Actuat impercep tible to the in

tellect , 1 26

Adap tability of life , 330

mathematica l laws, 3 7- 39Adap tation , purpose of, 3 27

—8Adap tations made by an organism ,

1 62

Additional freedom bestowed by

higher faculties, 334Afi nity , zones of, 22, 1 24

Agency , interpretative , inte llect uan

, 166-

7

Agent, ego as an, of the Thinker,

—engendering, life as an, 33 1

of creation, life as an, 343

Ahmes, an Egyp tian p riest , 44Alchemy of psychogenesis , 225Alcohols , eight different, from one

formula 1 55

Algebraic quantities and space ,

1 25-

7

Allowability of the rules of logic, 163Aloofness, the , of scientifi c dogma

tism. 34°

Alphabet of space-genesis , 23 7the geometric , 193

Amenability of mind to the laws ofevolution, 28—3 1

Analogies , use of,to popularize the

four-space , 1 28-9

Analogy , difi erence be tween two

proces ses illustrated by , 74Analysis incapable of dea ling with

life , 3 19the manifold, the fiat of, 77

Analyst and the manufacture of the

space-manifold, 7 7

disregards the conformity of the

nature of things , 84Analytics and the m echanical origin

of the universe , 40—1perisophical nature of, 3 20 , jthe four-space , a curiosity of, 40

368 INDEX

Anchorage , fourth dimension denied,265

Angularity of consciousness, 1 20

Answer of the senses to new needs,

332

the , of consciousness to realism ,

IO7

Anti-Euclidean geometry , 69Apotheosis of the definition , 78

Apparatus, man’s percep tive , 3 57

Apparent vassalage of life , 3 29—30Appearance , dynamic, space as

,

Appetences, materialistic, of emp iricists

, 340

Appulses of life against matter,

3 28

Appurtenances ofmateriality , 33 1A Prion!knowledge , 1 10—5

Aprioriness, 1 1 2, 1 1 6

Ap riority ,the p rincip le of, 1 1 3 , 223

Arbitrariness of the common men

surativ e quantities, 4 1

Archeological evidences, 33 2Archimedes

, 70

Aristotle, 3 22

Arrangement of the contents of thekosmos

, 3 1 7—8

Art and the criterion of truth, 3 23

higher consciousness, 1 8 1

evidences of new faculties in thefie ld of, 335

Assemblage of mathetic contriv

ances, 335

Assum p tions , category of, 1 89

p rocedure based upon, disadvan

tages of, 1 63At -one-ment

,224 , 25 2, 270

defined as the end of evolutionaryactivity , 1

Attachment of the consciousness to

the order of becoming , 305

Attainm ent, the diffi culty of

,by the

intellectual method , 206of space

-consciousness, 225

Attitude,the need of a more liberal ,

340

towards things spatial, 306Audital nerves

,notes that tran

scend the grasp of, 355

Aura, pulsating, 348

Automatism of the intellect, 253 , fi ;338

Avenues of knowledge Opened to

inner percep tive faculties, 3 53Awakening of the faculty of aware

ness in a new domain, 90

p ituitary body and the p in

eal gland, 353p ituitarial, general results of

,

355- 8

Awareness,a determinant of concep

tion,1 20

as a gauge of the existence of

things, 1 61 - 202

degrees of, 1 65

hyp erspace , a symbol of a moreextensive

,1 80

p rogress in,1 7 1

Thinker’s sphere of, 274, 283

Baltzer, 66Barrier, the , to the Thinker’s certi

tude , 1 86—8Barriers, freedom determined by

absence of, 280Becoming, endless labyrinth of, 220

the kosmos, in a state of,265

the order of, 305

Becomings, infinity of, 234

Becquere l rays, 336Being, identification of consciousness

with , 205

kathekotic, 210

370 INDEX

Circle , the symbology Of, 309

Circuits, closed, our interests as, 1 67Clairaudience , 3 5 2Clairvoyance , 3 52Clairvoyant percep tion, 349

Clause , enabling, of metageometry ,76—7

Clavius, Christoph , 46 , 52, 83C lifford, 66Code , p sychic, for systematizing cog

nitions, 190—1

Cognition, instantaneous, organ of,

258

intimacies of direct , 205the method of

,1 86—8

Cognitions, intuitive , 145 , 1 92—3psychic code for systematizing,

1 90- 1

Commensurable qua lity , dimensionas, 1 40

Communal consciousness, 270, fiCommunalization and the intuition ,

301

Conarium ,the

, 345

Conceivability and evolution of

mind,24

ultimate range of, 278Conceiving, power of, derived from

sense -experience , 26Concep t , as a shadow,

1 19- the hyperspace , gradual rise of,

27—8

Concep tion and awareness, 1 20

every , based upon p rior experi

ence , 25

freedom, the , of the mind from ,

28

Concep tions , symbolic, dependencePlaced upon. 1 47

Concep ts, and the Thinker, 244intuitogram s as

,248

quality of,dependent upon sense

experience , 26

Concep ts, the percep tion of, 255

Concep tual , re lated to the objective ,29 1

the , and mathematicians, 7 1Concep tualization, the act of, 4the power of

, dependent upon

evolution ofmind, 25

Conclusions, mathetic, comp lex dependence of, 3 7

Concrete hewn into shape by the

inte llect,295

Congruency , the , between intellectand the universe ,

Congruity, the , be tween concep tsand Objects, 1 20

Connection, the , traced out betweenreality and object, 1 26

Cov riginality of things, 303

Consciousness and kosmos, graphrep resenting, 27 1

the character of the sensibleworld, 1 66

wake of life , 205time , 224

angularity Of, 1 20

a priori, 1 85a rep lica of, and judgments, 205as a scale , 165

barrier to ultimate knowledge,207

determinant Ofdimension, 145life. 295variable quantity, 1 7 1

attachment of, to the order of

becoming, 305dawn of, and dimension, 1 79

degrees of, 1 62

deprived of the validity of no

tions, 3 14evolution of, 250expansion of, 1 64, 272

extended, 89—90focus of, 290

INDEX

Consciousness and the p lanes Of thekosmos, 1 63

fragmentary View of, 260

freedom of, in the genus homo, 332

habitation of, 278

higher, and art , 1 8 1

p lanes of, 1 78history of, 298

—9

identification of, with objects,1 89—90

intuitional , 255keyed to the entire range of

reality , 167- 8kinds Of,kosmic, Elysian fields of, 235of the p rim itive man, 253organ of, 1 84 , fi ; 90

p sychics of, 292

raised from the sensuous to the

concep tual, 7 2recipient of truths from dual

sources, 26

sensible world as instrument of,199

trip le presentation of notions Of

space to, 72

turned inward, 283unification of the states of, 192youthful , ofmankind, 1 22

Consequences , the science of, 85

Considerations, mathematical , and

reality , 1 28Consistency , kosmic, 1 74

se lf, of hyp erspatial hypotheses,67

the criterion of geometry, 86inconsistence of, 1 73me tageometrician curtailed by,85

Construction, idealized, cannot beobjectified , 144 , 276

ideal , the meaning of, 4

mental, trafficking in, 1 54

D

D’Alembert , note , 5 1

Dante , 18 1.

37 I

Consubstantiality of intellectualityand spatiality ,

Continuity of the p sychic p lasm ,

260

Contrivances, mathetic, the passingof, 24 1

Convergence of paralle l lines, 3 15Convolutions, dup lex, 146Coordinates, as spatial determi

nants, 203of hyperspace. 94. 97. 99 , 1 1 5

- 6

systems of, 301 , jthree , 1 32, 232

Copolarity Of ideas and Objects, 1 27Corpora quadrigemina , 345Corpuscular orbits ofparticles, 1 52Cosmos, significance of‘, 4Creation, imp regnated screen of, 232

Creations, the , of the intellect, 1 73Criterion of geometry as consistency

and convenience, 86truth. 3 23—4

Crookes, 344Cube , as succession of p lanes, 148generation of the

, 144

illustrated, 133the generating, 134

Curiosities, analytic, energy Spentin elaborating misappropri

ated , 39Current, electric, as component in

the fourth dim ension, 1 53

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

valid e lement in the notionof. 305

the measure of, 61

Curved space and me tageome tri

cians, 3 16

372 INDEX

Darwin, 1 81

Day , the Great Kosmic, 213De Til ly , 66D eath , as fourth dimensional unity,

1 59- oi the intellect, 1 95Decrement of the diametrical ratio

to the circumference of circle ,3 1 2

Deeps, fearful, of kosmic mind, 7 1 ,9 1

Definition, apo theosis of, 7 7—8—as an arbitrary determination,

1 00

the deification of, 3 7

Degrees of realism ,1 64

-

5

D eiform , basic idea of the , 5

D emarkation lbe tween reality and

phantasy , the line of, 1 73Deposits of life, 1 73D escartes , 3 22D esigns, cut out in materiality by

life,264

D etails, the power to dispose of in

numerable , 303D eterminations, geometrical , the

necessity of, inheres in logicaldeductions, 78

the factors Of conscious, 1 62D eterminative period of mental

development , 3 1Developm ent of the intuitive fac

ulty , future , 1 88

D iacritics of life,286

Diameter, the ratio of, to the cir

cumference, 3 1 1

—2Difference between concep t and

thing, 1 21—2—between mathematical and per

cep tual space , 1 20

between the ideal and the actualis dynamic, 108

Differential , among minds, 1 24

Diffi culty of imagining the fOI

dimension, 90

of propagating an intuition,Diffi culties in the acquisition of

knowledge , 3 25of hyp erspace , the logical, I

41

Dilemma ofm e tageometricians, gDimension, a distinct stage in j

chogenesis, 27

system of space m easurem

5

and analysis, 1 63D

’Alembert , note, 5 1

the action of a tartrate1 56

as an arbitrary contrivance , 2assemblage of e lements

direction, 140extent, 98lyingnear the surface ofthiJ1 59—60

Space , 96

current definitions of, 98—100denied legitimate anchorage ,

determined by consciousness,

does not exp lain sp iritualism,

evolution of the fourth , 44fourth , direction of, 134

as“ jack ofall trades,” 1 56

impossible to actualize , 1 24key, to non-Euclidean geome9°

movement of a p lane into, 14nomotion ofmaterial masses ii

1 25

not a priori, 1 1 6

proof of the existence of

fourth , 1 36Dimensionality , and the intel]

200

as dep endent upon the will ofinvestigator, 95

374 INDEX

Equidistantial, described by hyperspatial perpendicular, 80

Essence , e lemental , as world-

p lasm ,

3 29

Ether as possessing fourth dimensional characteristics, 1 5 7

Ethics recast in a new mold, 358Euclid

, 46, 70, 83 , 263

and the parallel-postulate , 45never-dying e lements of, 53

Evidences, mathematical , exemp lifies inte llectual evolution, x

the vanity of fragmentary , 204Evolution, a continuous process, 3 27

and the norms of reality , 1 75basal element of, 259commencement of, 232- 3intellectual , forward movement

of, 1 84

kosmic, vicissitudes of, 21 5laws of, govern mind, 28mental , results Of, 1 22of faculties, synchronous with

evolution ofmatter, 332material characteristics of the

earth , 335organs, time required for, 1 88

p reparation of the field of, 21 8Exaltation of matter into Spirit,

329

Examp les of new percep tive evi

deuces, 341—3

Existence on a higher p lane, statesof, 1 62

Experience , corroborative testimonyof, denied hyp erspace , 263

prior, and concep tion, 25

Experiences, spatial , systematiza

tion of, 78

Extension, space as an unbounded,6 1

tiality , 261

Extravaganza, mathematical , I

the fourth dimension, 1 56

Eyes, as Roentgen rays, 353

F

Fact-mass, 289Factors of conscious determinatic

162

four, of the criterion ofm

3 24-

s

Facts as facets of truth, 284

attemp ts to reform , 304

logic as symbolism Of,287

the modeler of, 288Faculties, dual derivation of, 1 62

evolution of, synchronous v1

evolution ofmaterial qualit33 2. ff

extended, 239foreshadowed by the hype

mensional, 13 1

higher, man must win his way357

new faculties evidenced by fospace , 24

—29rudimentary nature of, 341 , fithe source of, 349

Faculty and the intellect, 247 , fias transmissible character, 25 1determined by function, 336greatly extended, 1 1 7

—higher than the inte llect, 1 26I-making, self-consciousness

243

intuitive , 1 85

ofpercep tion in higherworlds, ,

- awareness , the awakemng89—9 1

overshadowed by the intello

1 88

of percep tion in higher wor

353

rudimentary condition of, 192

INDEX

Faculty, outcropping of, 249—51Failure of efl

'

orts to justify the Ob

jectiv e existence of four-space ,1 25—6

Failures at solving the paralle l-postulate , outcome of

, 48 , 83

Faith , dispossessed by knowledge ,358

Fay, mathetic, 1 60Fechner, 39Finity and unboundedness Of space,

76

Flammarion, Camille , 341 - 2, 344Flexity , as prope rty of hyp erspace ,

63

Flexure, space as a, 305

Fluxional, between sense objects andideal representations, 1 22—3

Fluxion, psychic , as difference be

tween memory image and

object, 7—8 ; 1 22

Focus of consciousness, 1 63 , 290the intellect, 3 1 0

Fohat, and the creation Ofmorphons,219

as creative energy , 213Creator, 8

Fohatic energy , 226

Form , as vehicle of life , 330definition of, 8

driven back over the path Of its

genesis, 3 27

pure , cannot exist in nature ,294

the idealty of, 1 10

the universe not a pure , 108Formative pe riod of mental devel

opment , 3 1

of non-Euclidean geometry , 55Formula ,

eight different alcoholsfrom one , 1 55

Four-dimensionality , justification Of,1 76

375

Four-space , a curiosity of analytics,40

and Riemann, 27as a divertisement

,1 75

consciousness does not act in the ,1 72

existence of, denied, 1 7 1—2

movement ofmat te r in, 1 5 7

reality of, glibly p roclaimed, 1 54the

, 8 , 240

denizenc Of 353

domain of, 1 54

study of, 1 24

use of analogies to popularize ,1 28

Fourth dim ension, analogical reasoning of, 1 7 7

-

9—'

as a transcendental problem , 140

an attitude of the inte llect, 200—e lectric current as a component

in, 1 53

imaginability of, 106—7

Simon Newcomb on, 1 25Fragmentariness of the intellectual

me thod, 164Fragmentary view of the universe ,

260

Fragmentation, tendency to, 296harks back to ce ll—division, 30 1

Freedom , a new, 338

determined by absence of bonds,280

—’

mathematical, 60mental , 66—7

—.

now dawning for the mind, 32

of consciousness in genus homo,

332

regal freedom Of the mind, 1 18

three degrees of, 1 25unrealizable , for the Thinker, 255

Fringe , chaogene tic , 308 , 309of disorderliness, 25 7kathekosity , 229

375 INDEX

Frischauf, 66Ful l, the universe as a, 3 10Function determines faculty , 336Functions back of latent faculties,

indefinite , 343ce llular and histologic, 253

Functioning, instinctive , Of the ia

tellect , 289

Fundamentals, totality of,kosmic,

265

GGamut of realism ,

1 69

Gauge , awareness as a , 1 6 1—202Gauss, Charles Frederick , 8—9 , 56- 8 ,

592 1 74

as formulator of the non

Euclidean geometry , 5 7G eminos of Rhodes, 45 , 5 2Generability , as p rope rty of space , 62

of hyperspace , Keyser on, 144space by lines, 143

Generation, the , of the hypercube ,134

of the hypertetrahedron, 1 35—6Genesis, of space , 2 1 1 ,

the earth , its nebulous youth ,335

form , 3 27

sensible world, 1 67Genus homo, freedom of conscious

ness in , 33 2

Geometricity , 266

Geometries,non-Euclidean, based

upon a negation of the latentgeome trism ,

262—3three possible , 54

Geometrism , engenderment of, 262established by life , 264kosmic geometrism , establishment of

,23 1

latent, 25 7native geometrism of space , 262

rediscovered by the inte llect, 9—10

Geometrism, the basis of, 23 7 , 261Geometry , anti-Euclidean,

69

and the study of magnitudes inspace , 203

artificial, 262- 4a two-fold, 59breakdown of, 266

determinative period of the non

Euclidean, 6 1—4diverse systems of, 3 23Euclid’s Elements of, 53—4formal bulwarks of

, 7 7—9

—Imaginary Geometry, The , 60

natural geom etry , 261Plato and the divine geometry ,

1 93

possible systems of, 1 74radica l essence Of pure , 85

Geometry , non-Euclidean, at variance with the parallel-postulate , 83—5based upon a misconcep tion,9 1

determined by qualitative differences, 140

e laborative pe riod of,66—70

first published treatise on,60

formative period of, 55

the final issue of, 88-9growth and development of, 92invalidation of, 1 74

key to, 90

Lambert’s Non-Euclidean ge

om etry , note, 5 1

popularization of, 66—7

Schweikart’s treatise on, 59

se lf-consistence of, 5 7

some e lements of, 79—80superpercep tual knowledge of,72

Geom etry , non-Legendrean, 70

- symbolic , and comm ensurablequality. 145

378 INDEX

I

Icosahedron, examination of the ,

285

Ideal , and the real , kosmic chasm

between,107

pe rcep tual value of, 275 ,flIdeas and words as symbols, 1 26—7Malebranche on, 222realism of, 24

—3 5

the symbolism of, 205

Identifi cation of consciousness withbeing, 205—6

with the objects of study , 1 89—90Identity of things-ih -them se lves, 3 53Ignesfatuii and hyp erspaces, 1 54Illustration of p lane-rotation, 148

1 50

the tesseract, 1 33Images, totality of

,recoils upon us

,

1 67

Imaginability of the fourth dim en

sion, 90 , 106—7

Imagination, p remises of, the math

ematical,1 46

Impossibility of p lane -rotation, thestructural, 1 5 1

—3

Imp ressions, the symbolism of, neu

rographic, 1 86

Impulse , the satisfaction of the orig

inal creative , 3 1 0—1Incomp rehensibility of reality to

the intellec t, 1 26Incongruity , life estranged by a rad

ical , 3 25Individual as space , 223Inep titude of inte llectual determin

ations for Vitality , 3 14Infinite , interp reted in the terms of

the finite , 8 2Infinitesin mls of unity, numbers as,

4 1

Infinity as a p rocess, 109

of becomings, 234

Infinity of parallels through a.

given point, 70space , a capital il lusion, 195

the concep t of, 27 7innate dread of, 1 03

relativity of, 194

Influence of abstract thought , 33Kant on the non-Euclideangeometry ,

La Grange , 5 1—2the intellect, 3 15

Infundibulum,the , 345

Inner organs of percep tion, 352Innermost, the , realities of things,

356

Insouciance of the geometer, 96 , 294

Instant- exp osure and inte llect, 3 1 1Instrum ent , inte llect likened to a

color—bearing, 3 1 3for the measurement of the pas

sage oi space , 297

of consciousness, the sensibleworld as, 1 99

life , form as an, 3 28

Integers, as fractional parts ofunity,4 1

Inte llec t and its domination of the

intuition, 333

final union with the spacemind, 1 94

topography. 3 1 2spatiality , 263

the deposits Of life , 1 73designs cut by life in materiality , 264

dictum of Sensationalists,26

instant-exposure , 3 1 1intuitive faculty , 247 , fip rize of divinity , 43

as a color—bearing instrum ent,3 13

-

4

searchlight, 1 68

INDEX

Inte llect as a fashioner of phenomena, 199 .fi

r

hewer of the concrete , 295sole interpretative agency,

166—7automatism of, 253 , 338

cannot seize life , 282crowned by a diadem of gold, 338dominated by the intuition, 250fashioned for matter only , 23 1follows in the grooves of logic,

hyp erspace as amonument to, 356in the field ofvitality , ixits ap titude for starts and stops,

instinctive tendency to frag

mentate , 296- quality , determined by mathes

matics 33 7

makes for individuality , 25 2

misses the cease lessness of life , 20 1modus vivendi of, its influence

upon knowledge , 1 84the constitution of, 29 1

cut and mode of, 167

focus of, 3 10

illusion of, 246

illusionary character of, 3 23incomprehensibility of realityto, 1 26

instinctive functioning Of, 289

p redominating tendency in,

3 20—1

scientific tendence of, 1 65—6struggle of, against dispossessal, 1 95

unsuited to deal with reality ,

3 22.flIntellectua lity and reality , 304

spatiality , consubstantial , 33 1

379

Intellectuality as coextensive withspatiality and materiality , 236

the source of, 260

Thinker makes his own , 242

Intelligence and automatism , 253 , jthe Thinker as a pure , 243transfinite inte lligence and the

degrees of realism ,164

Intent, the kosmic, of the intellect,326

Interests, the sphere of our, as

closed circuits, 1 67Interior

,the great , 290

Interiority of being , 290Interpretation, the standards of,

vary as consciousness varies,1 7 1

Interstices of materiality , 264

Intuition and brotherhood, 25 2communalization, 301

—the riddle of spatiality , 3 25as dispossessor of the inte llect, 195cannot be served by mathematics ,33 7

dominated by the inte llect, 333held in abeyance , 338 , fiits domination of the inte llect,

2so

the deve lopment of a spatial ,not absurd, 145

need of a sym pathetic attitude towards, 249

results of the deve lopment of,35 7

Intuitiona l consciousness, 255—the superiority of, over the ta

tional, 1 87

Intuitions and the lead of life , 191Thinker, 27

free , mobile and formless, 166the concep tualization of, 248

—humility of, 165

nature of, 185

380 INDEX

Xtuitograms, as concepts, 248 Justification of sense-deliveriltuitograph as means of contacting one another, 76

the egopsychic consciousness the existence of the fom

by the Thinker, ro mension, 1 25rcuperconcep ts, 255

tot difiiculty of transmitting,

uniq llty, the vaunted, of the

wake bfmathesis, 3 7288 p sychological , 24

Life-cyclens,metageometrical, andthe , v sense -organs, 354

Life-streams antithesis of ev olu

Light, polo—urr: involution, 226

CO .i tter, 3 28movem ent of, 219seven stages of, 208—1 2

Ions, creation of,21 9 ;

magnitude of a hydrogen ion, 225

J

Judgments a priori, Kant on, 85—6and the faculty a priori, 190—r—

zones of affinity , 1 24based upon a rep lica of conscious

ness,205

no trustworthy , can be p redicatedupon fragmentary knowledge ,282

of the inte llect , 302the lessening of error in, 256

more comp lex the more at

variance with the nature of

things, 75synthesis of, 25 7—8

valid judgments long de layed,r 7o—r

Indicative power of mathematics,1 80

Justification for a multi-dim ensionalquality in space , 262

of four-dimensionality , 1 76

K

Kant, 85 , 1 8 1 , 1 82, 3 22and the faculty of thinking,—idea of space , 3 22

—3

influence of, on the hype )

movem ent, 49—50on the nature of things, 1 19

space as an intuition, nKathekos, 233

—4

as chaos, 266—

symbo l of Chaos-Theosmos, 1 1—1 2

symbology of, 234

Kathekosis, note, 227

Kathekosity , fringe of, 229

rock-bound coast of, 267significance of, 1 2

Kathekotic consciousness, 272

period ,209

Key to the mysteries of naturefourth dim ension as, 1 3 1

Keyser, Cassius Jackson, and

dom of the m ind, 33

on attitude of met

m etricians, 7 1dim ensionality , 94four-dim ensiona lity

space , 142—3

generability of h

space , 144

K lein ,Fe lix , 1 2 , 54—5 , 66—7

Knowledge , all, relative , rorbarrier to the certitude 01

Thinker’s, 1 86—7fabric of, 1 96hypothetical nature of, 1 89 ,immeasurable realm of, laidby the telescope , 298

382 INDEX

" ( Life movement of chaos in, 307passage of, 20 1

passage through spatiality, 288rcpower to manipulate , 303tot ecurrent movement of, 30 7undt

‘ality of egoic, 243

Uniqfit ilations in the current of, 3 29wake oneness of, 4 1

288 p10f, and consciousness, 205 ,Life cyclens

the, y 5. evolutionary results of

Life—streams 259Light, p om—1 1 , the , 278

n: lhvolarized, and the fourth dico i t teri ension, 1 56

moxnsciousness as a Spreading,

sev ' 1 68

ans. Light-years and space , 278Limits, the sphere of, consciousness

as,1 63

Line , as generating element,144

the straight, a curved, 76Lineage of every p rincip le runs

back to monopyknosis, 213Lines, perpendicularity of, in four

space , 1 30

Lobachev ski, 55 , 60 ,66

, 8 7

Lodge , Sir Oliver, 340—1 ; 344Logic as architect, 1 1 8

conventional forms of, 262data of, 286

intellect follows in the grooveslaid out by , 294

miracle power of, over facts, 288—9rules of, allowability under, 163

the gam e of, 1 64

as symbolism of facts, 28 7Logos and the limits of space, 193being of, 21 8body of being of

,214

consciousness of, 204

creative , 1 2, 213the, 21 6

—7

Lorenz, 83Luther, Martin, 1 81

Magnitudes, geometry, a study of,203

non-sensuous, of hyperspace , 72

Makrokosmic consciousness, 270

Malebranch e , N. , 222; see note

Manifestation and non-manifesta

tion,206

Manifold, finite, though unbounded,70

manufactured by the analyst, 7 7the , 61

—2,1 72

Manifoldness, analytica l manifoldnesses

, as mental excitants,

as a conventional construction,262

an intuition, 3 1 8of space , as a near-truth, 3 1 7Manning, 82, 1 5 7

and maneuvers in the fourth dimension, 1 5 2

Manvantara , 213 , 228—9 , 23 2as evolution and involution com

bined, 13

Mask , logic as a, 288—9Mass-termini, of lines, 236Mastery of life over matter, 330

of the sensible world, 1 96—7Materiality as a deposit of life , 295

consubstantia l with spatiality,33 1 , fi

becoming spatialized, 261

characteristics of, 260engendered by kosmic mind, 261engenderment of

,294

interpenetrative with spirituality,236

interstices of, 264neglect of, 3 20

INDEX

ateriality , shore of, 3 23

transmuted into spirituality, 3 28ematicians and the definition,

3 7

limitations ofconsciousness,1 78

phenomenal world, 1 1 8—9as prophets , 336respected by humanity , 33 7the gods of mathesis, 84

of the

quality of the inte llect, viiand the criterion of truth , 323—kosmic intent of the inte llect, 3 26

as symbology , 1 84Euclidean, 140fails when it encounters life , 33 7its inap titude for life , 1 79—kingdom yielded to kinetics,

358

possesses no judicative power overlife , 1 80

the orthodoxy of, 34

Mathesis, and concep tional space ,203

conduct of the intellect in the

field of, vii

definition of, 1 3

domain of, as origin of fourthdimension, 1 30

gods of, 84

marvelous domain of, 1 1 8

realm of, not submissive to lawsof sensible space , 34

things of, as emblems of kosmicforces , 239

world of, 67 , 1 27

Matter and mind, syncretization of,

299

appulses of life against, 328as a deposit of life , 330honeycombs of, 339

Matter metamorphosed by life ,3 28

mind as wedded to, 249

movement of, in four~ space , 1 5 7qualities of, and faculties, 332sev en p lanes of, 21 2spiritualization of, as end of evo

lution, 327

totality of, acted upon by totalityof life , 328

unity of, with space , 260universum of, 26 1

Matutinal ceremonials of creation,221

Measurability , as property of space ,62—3

Measurement of hyp erspaces, the

science of, 73

Measurements, space determined bythe number of, 61

systems of space , 9 1Measurer of space , n-dimensionality

as a, 296-

7

Mecanique Analytique , 50—1 , 3 21

Mechanics of matter, the passage

from , to the dynamics of life ,

3 26—to biogenetics, the passage from

357

the turning from , to biogenetics,

Mechanism , the doctrine of, due to

Memories, stored in the omnipsyche ,258

Men as gods, 289

Mental evolution, determinativeperiod of, 3 1

elaborative period of, 3 1—32

formative period of, 3 1

Mentality, inner mysteries of crea

tive , 9 1infantile out-feelings of the , 336

384 INDEX

Mentality , the princip le of, 210 , fiMentalities, the adap tation of phe

nomena to, 1 84

Mentograph , the , 1 21

the basis of inte llectual consciousness, 1 3

Mesozoic man and his environments, 347—8

Metageometrica l investigations andthe new sense-organs, 354

Metageometricians and hyp erspace ,

238

proof of rotation about a

p lane , 149—1 5 1the curved space , 3 1 6

fourth dimension, 1 45—key to the mysteries of

nature , 1 3 1baffled by the direction ofhyper

space , 1 46

dilemma of, 9 7

- 8

eschew sense-data, 73-

4

Keyser on the attitude of, 7 1pe rcep tual obliquity of, 102

—seed thought of, 1 82

turned to idealized constructions,263

Metageometry, as a stepping stone ,238

defined, 13Riemann , the father of, 63the fabric of, 66Metamorphosis of matter by life ,

3 28

of the monopyknon, 214

Metamorphotic stage of space

genesis, 228

Metaphysician, the p reserves of,

usurped by mathematicians,14 1

Meta-se lf as medium of kosmic consciousness, 1 3 , 21 7

Methods of the ego, 344

Methods, scientific, and the a

exp lication of phenomena

Mikrokosmic consciousness, 27<

Mind, am enability of,to the e

tionary movement, 28—3 :and freedom from concep tio

p roblems of phi

existence , 3 2the new freedom , 33

as vehicle of life , 285—6wedded to matter, 249

coev alism of, and space , 24C

consubstantial with space , 2;divine , of the kosmos, 1 76

evolution of, and conceiv al

24

of the duodim , 1 28

powers of, not attained s

taneously , 334

p rofound influence of hyper

on the , 9 1

space as p rogenitor of, 224three great epochs of the hi

of, 30-

3 1

tuitional , the limitations of, 1the unity of, with space , 224

Mind-

p rincip le , the quintopy

as the basis of, 221 ,flMirrors, three , and the hyper

1 36—8

Mobility , life as, 264Molecule , the four-dim ensi

Newcomb on , 1 59

Monadic phase of evolution,Moneron and man, the gulfbe t

28

Monopyknon, the , 213 , 21 6

Monopyknosis, defined, 1 7 ; 20

21 8

Monstrosities, mathetic, 202

Moods of the kosmos, 3 1 1

Morphogeny, 1 3 , 233—4

Morphons, the creation of, 21

1

386 INDEX

Number, as a phase of kosmogen

esis, 309

numbers as infinitesimals ofunity ,41

Numericity of being, 309

OObjects and ideas, 1 26—7passage of

,into the fourth di

mension, 130

of study, identification of con

sciousness with , 1 89Obliquity, pe rcep tual , of me tageo

metricians, 102Obscurantism ,

mathetic, 1 54

Occultist and the scientist, 350Omnip syche , 265

as agency of kosmic conscious

ness,258

neglected factor of evolution,259

defined, 1 5One-space repre sented by a line , 1 34Ontogeny recapitulates phylogeny ,

Order, kosmic, the diflerence be

tween, and hyp erspace , 143mathematical, discovered by theintellect, 261

the flat of, 208

the totat of, 265

Orderliness, 266Organ, determined by function to be

perform ed, 336

Organs broken up by life into specialparts, 343

the evolution of, 1 88

Orthodoxy ofmathematics, 34Outcome of new adjustments, 256Out -fee lings of the Thinker, infan

tile , 336Outlook, the spiritualization of our

mental , 40

P

Papyrus, a hieratic, 44Paralle l-postulate and the surf:

a sphere , 70—as basis of non-Euclidean 1

e try , 54

stated by Euclid, 55Lobachev ski, 55

Manning, 82

failures to demonstrate , 48 ,in the Elements of Euclid, ISaccheri

’

s p roof of, 47

Parallels, convergence of, 80

infinity of, through a given 170

mee t at infinity , 88theory of, 47

-

9

Passage from mechanics to

genetics, 3 57from mechanics to dynamic

:

from three-space to four-s

292—3

of mathetic contrivances, 2.

of space , 1 6, 297

Pathway of life , sinuosities inPatterns of the inte llect, ready-1

1 66

Pentagramm aton of space , 23 7

Percep tibility in the Thinkerfaculty of, 1 88

Percep tion, domain of, and thc

sible world, 19inner organs of, 352

rep laced by concep tion, 255Percep ts, and the ego, 244Period , chaomorphogenic, 208 ,

determinative , 6 1—5kathekotic, 21 0

Perisophical nature of anal

3 20 ,flPerisophism , a, 300

Permissibility of mathetic liunlimited, 38

INDEX

Perpendicularity , 132, 1 77—

of lines in the four-space , 1 30Phantasy and reality , line of de

markation between,1 73

the world of,146

Phantom -ideal , the , 27 7Phase of the world age , evolutionary ,

209

Phenomena , eflorts to explain, on

the basis of the fourth dimension, 1 29

fashioned by the intellect, 199fragmentary interpre tation of, 302mind

’

s me thod of apprehending,

1 86

physical phenomena amp ly ex

p lained by scientific me thods,1 54

Phenomenal , the inverse of rea lism ,

278

Philosophy and the criterion of

truth, 3 23- 4regeneration of, 358

systems of, and zones of affinity ,1 24

Phosphorescence , dim , added to the

unillum inated pool of sense

consciousness by the intuition,26

Phylogeny and ontogeny , 334represented in ontogeny , 27

Physical as embodiment of all possibilities, 227

phenomena as space-activities ,

229

Physicality , the princip le of, 2 10 ,flPhysics of the brain, 292

Pi proportion, significance of, 3 1 1 - 2Pickering on space

-curvature , 279

the ) 344 ) jnot a vestigal organ, 354

Pituitarial awakening, general re

sults of, 355—8

Pituitary body and the Thinker, 35 1as an organ, 344

—352

C . De M . Sajous on, 345

hypertrophy of, 347

location of, 246

the ,

Plane , as generating e lement, 144movement of, into the fourth dimension, 147

realism appears to be divided into ,1 64

rotation about a, 130 , 146—1 5 1the seven p lanes of matter, 21 2

Planes , a cube as a succession of,

148—9

Plasm ,kosmic, strivings of, 21 5

p sychic, 260

the difl'

erentiation of, 214

Plato , 3 22and the divine geometry, 1 93 , 23 7

shadow consciousness, 28 1

on God as geome trizer, 1 8 1

ideas, 34

Play, sensuous, of the intellect, 295Plenum ,

space as a, 230

the universe as a, 107Poincairé, 90

Point, an infinity ofparalle ls througha given, 70

as generating e lement, 144position of, 1 3 2

succession of points, a line as,

148

Polarity between concrete and ab

stract,Ponderability , as p roperty of hyper

space , 63

Popularization of the fourth dimension, 1 28

of the non-Euclidean geometry ,

66- 7

388 INDEX

Possibilities in the world of hype r

space , 1 29

of four-space , the marve lous, 130Postulate-systems, the multip lica

tion of, 85

Powers of mind, not attained at oneand the same time

, 334

Pralaya, 21 3

as kosmic quietude , 1 7Predicates, all mathematical , not

justified by the phenomenal ,1 28

Princip le , ensouling, 2 14—5Printing p ress, the invention of, and

the laws of p sychogenesis, 30

Problem , the fourth dim ension as a

transcendental, 140Procedure , the involved, of arriving

at hyp erspace notions, 73

Process, evolution a continuous, 3 27space as a dynamic, 307

Proclus, 83Profundities, kosmic, and hyp er

space , 1 56

Progression eastward terminates at

the west, 306Proofof rotation about a p lane , 146

1 5 1

Propagation of intuitographs, dif

ficulty of, 3 1 5

Propositions , geometric, subjunctivequality of, 38

Pseudosphere and the shape of space ,

70

as basis of Be ltrami’s calculations, 1 7

the nature of, 64-

5

Psychics of consciousness, 292

transmuted, 260Psychogenesis, alchemy of

,225

kosmic, 272laws of, and the printing p ress,30 , 1 1 6

Psychogenesis, mind’s p lace in,

outgrown phase of, 302outstanding facts of, 13 1

Psychogeny , defined , 1 7Ptolemy , 45 , 52, 83Publicist, the mathematical , am

duodim , 1 28

the non-Euclidean, 1 46Pyknon as a kosmic p rincip le ,

basis of space-genesis, 1 7Pyknosis, 2 1 2

and involution, 210 , 23 1as a metamorphotic p rocess,the seven p rocesses of, 1 7 , 21

Q

Quality , dimension as comm eI

able,140

Quantities, algebraic , cannotresent space , 1 25—7

Quantity , consciousness as a

space , 1 72

Quartodim , 1 8 , 1 34

Quartodim ensionality , 292

Quartopyknosis, 1 7 , 208—10Quartopyknotic, p rincip le as

of sp irituality , 228Quintopyknon ,

220

Quintopyknosis and the a priori

meaning of, 1 7 ; 208—10

Quintopyknotic p rincip le as ba:

mentality , 228

R

Race,humanity to evolve a new

resp lendent, 33 8Rajah-Tamas, 220

Rajah Yoga, 3 57Rational, valueless when um

tioned by the intuition, 12Reaction , as a mystery , 1 67—8Real, Thinker as a part of the ,

390 INDEX

Riemann, G. F . B ., as inventor of

the manifold, 86on the bases of geometry , 61 - 4Roentgen rays, 336

Rotation about a line , 147- 8about a p lane , 146—15 1

illustrated, 148- 50intra-corpuscular, 1 52

Rudimentary organs, vague functions lying back of, 343

Rule in the evolution of faculties, 333

S

Saccheri, Girolamo, 4 7—8 , 5 2, 83Sajous, C . De M. , on the pituitary

body, 345Schematism , the suitability of the

present, 3 20- 1Schweikart , Ferdinand Karl, 19, 66,

69—

on the non-Euclidean geometry,58

- 60

Science Absolute of Space, 60Science and rea lity , 1 70- 1

no longer empirical , 358skep tical attitude of, 349

Scientist,m ethod of the , opposed

to the analyst, 84-

5

Scopographic impressions, 19 , 190

Screen, the impregnated , of creation,23 2

Sea , choppy, and egos, 256-7

Search , the path of, for spatial understanding, 278

Seb , the god , 2 1 5

Sects, cop lanar, 79Selfand the immensity of space, 223

of the universe , 21 7the kosmic , 278

Se lf-consciousness, 270

as I-making faculty, 243determination of, 302

Se lf-consistency , 1 73Sella turcica , the , 345

Semi-Euclidean geom etry, the ,Sensationalists on the intellect

,

Sense -data , spurned by the n

geom etrician, 73

Sense-delivery , one , justified b)other , 76

symbolism of, 203-

4

Sense -knowledge , the sphere of

tended, 339- 40Senses, the answer of, to new 11

332

Sensible world, as domain of

cep tion, 1 9

Sensographic impressions , as

cep tions, 1 9

Senso-mechanisms of the inte1 65

Sentience , the princip le of, 210

Sep topyknon, the , 227

Sep topyknosis, as kosm ic matiity , 1 8 , 208

- 10

Sep topyknotic, the , p rincip le , 2Seven p lanes ofmatter, 21 2Sextopyknon, the , 209

Sextopyknosis, defined as kc

sensibility , 1 8 , 208- 10Sextopyknotic, as emotional

ciple , 228

Shadow likened to a sense p res

tion, 1 1 9Shadowgraphs, 28 1

Silences, nature ’s great, 33 9Similitude of agent and prim

243—6

Sinuosities in the pathway of lifeSocrates, 3 22Solitariness of intellectual

mony , 76

Space and edict of disorder , 234four-dimensional entities,

130

INDEX

Space and quartopyknons, 219- 20

se lf-consciousness , 223the idealized construction, 144

as a dynamic process, 216

all-mother , 229—30an assemblage of spheres, 143becoming, 1 95boundary ofanother space , 4950

consistence oi the kosmos,239

curved,1 64

dynamic appearance , 228 , 30 7

e ldest born of kathekosis,

21 5

engendered product of life,

evolution, 226, jfinite extension, 62, 106generable quantity , 1 03infinite continum , 70

intuitional concept, 225kosmic order, 265 , 267nether pole of non-manifes

tation,20 7

- 8

path of life , 293- 4pseudosphere , 70

sheer roominess, 1 15

system of coerdinates, 99

:unbounded extension,6 1

best study of, the consciousness,

269

concep tual, as basis of non-Eucli

dean geome try , 7 2

fundamentally distinct frompercep tual , 72- 75

e lliptical , 67engenderment of, 260essential nature of, 21 2fabric of, the , lends itself to mea

surement, 99- 100four 8

fractionalized space , 296

foundations of, 22 1

Space , four-dimensional , 4 1Paul Carus on

, 3 54

fourth dimension of, efl'

orts at

making it thinkable , 1 25generability of, by lines , 143genesis of,

geometric , purely formal con

struction, 75—6limits of and the Creativ e Logos ,

193

manifoldness of, 77mind and, the coev alism of, 240

mystery of, 278 , ffnature of, answered to by mind,

198-

9

nether pole of non-manifestation,207

- 8

non-Euclidean, the characteris

tics of, 72—3

not a pure form , 1 10

passage of, 16

—.path of search for, must beThinker-ward, 278

perceptual , irreconcilable with thefourth dimension, 1 56

possible curvature of, and the

non-Euclidean geometry , 76- 7primeval , and tridimensionality ,

236

p roblem of devrsrng a, 8 7—8

psychological nature of, 305real , confounded with hyper

space , OI

Riemann on the curvature of, 5the unity of,

with matter, 260Space

-activities , as chemism and

physicism , 229-

30

Space-center, the , 306

-7

Space-consciousness, 225

as a direct process, 224mergence of, with the individual

consciousness, 19s

392 INDEX

Space-curvature, as an arbitraryconstruction, 262

as near- truth , 306nature of, 279no need for, 23 7

translation of, in terms of the

intellect, 3 1 5Space

-forms, the construction of, 74

Space—genesis, 20 ,

alphabet of, 23 7comp letion of, 226

norm of, 1 74

pyknon as a basis of, 1 7symbolism of, 208 , j ,

235

Space-manifold, the , 279

Space~ measurement , and reality ,Euclid’s system of, 96

systems of, 201Space

-mind and archeological ev idences, 1 93

knowledge , 1 87the Thinker, 253 , 272

attainment of, 225 , 23 1granaries of, 235

the realization of, 238

Space-

process, the , 305

Space-rea lities, hyperspace as a

stepping stone to, 238Spark , the omnipsyche as a , 258

Spatiality and the intellect , 263—4consubstantial with materiality ,

20,

cut -ofl from non-spatiality , 266

interpenetrative with materiality,236

nonconformity of, with logic , 1 64riddle of, and the intuition, 3 25rise of, 1 80

Thinker’s outlook in, 3 16

world of nascent , 23 2Spencer, Herbert, 3 22Spinoza, 3 22

Spiritism , the phenomena of, 1 54

Spm tualists, the claim of, regardingthe fourth dim ension, 1 54

-

5

Sp irituality, materiality transmutedinto, 3 28

princip le of, 210,fl'

seeds of, 221

Sp iritualization of the flesh , 227man

’

s mental outlook , 40—matter, the end of evolution,

3 27

Squares, the form of, in the non

Euclidean geometry , 59St . Marie , Flye , 66Stage of pyknosis, 208 , fi

'

quartopyknotic, 21 9 , fiStanley , 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, 79Stuff, e therealized four-dimensional ,

1 60

Sum , angular, 79triangular, less than two right

angles, 59Superconcep ts, as intuitographs, 255

Sup erconcep tual , as the intuitional ,20

Superp ercep tion and automatism of

the intellect, 20Superp ercep tual, the , 74

Sup erunodim , the , 1 7 7

Swedenborg, 1 80 , 1 84Symbol , p sychic, and the brain,

1 90

words as symbols of ideas, 1 26-

7

Symbolism of life,286

neurographical impressions ,

1 86

realities, 1 28sense-de liveries, 204space

-

genesis, 208 , fi

394

Thinker,his schematism of cogui

tive powers, 206sphere of awareness, 283

the difference between, and the

inte llectuality , 256evolutionary needs of, 336

-

upreaching yearnings of, 336

Thinker-ward, space must be soughtin a direction, 278

Thinking, abstract, e levating influence of, 33

Thought, p rolonged abstract, benefi ts oi

,249

Time, as an aspect of consciousness,224

divested of time liness, 279Todhunter, 46Tool, fashioning, life as a, 264Topography of the intellect, 3 1 2Touchstone , consciousness as a, 1 68Tracery of connection between ideas

and objects, 1 26Traflicking in mental constructions,

1 54

Transcendental , the realm of,and

mathematicians, 142Transfinite as a limit, 1 22 ; 2 1 6

Transfinity , 21- 2

Tree-reality , 1 26Triadic pha se of evolution, 21 0—1Triangles, angular sum of, 78 , 88

Tridim , the , 22, 1 29 , 1 34 , 147, 1 77

Tridimension, the , 1 4 7

Tridimensionality and p rimevalspace , 236

and the space-mind, 1 93

a quality of percep tual space , 22mastery of the phenomena of, 1 72the sufliciency of, 1 98

Tripyknon, the , 21 2, 213

Tripyknosis, 1 7 , 208- 1 0

,219

Truth compared with facts, 288criterion of, 5 , 3 23- 4

INDEX

Truth, facets of, 284kinds of, 3 24

- lcg1c does not illuminate , 287- 8Tuitive , the , and the intuitive facul

ties,1 92—3

U

Ultima Thule , the , 207Unbounded, the , as a finite exten

sion, 70

Undulations, three , in the current oflife , 329

Unfoldment, mental, and me ta

geom etrical discoveries, 13 1Uni-circle , the , 308Unification of all knowledge , 256Uniqueness of real space , 95Unitariness of all existences, 241Unity , as end of analysis, 42

death as a fourth dimensional ,1 59

kosmic , integers as fractionalparts of, 4 1

ofm ind and space , 224, 230

the new realization of, 225

Universality , geom etric, based upon

the formal character of as

sump tions, 7 7—8

Universe , a glorified, 226, 268as a full , 308—p lenum , 107

and the seven p lanes of matter,21 2

character of, fixed by conscious

ness , 1 62

—lim ited and conditioned, 1 27

not a pure form ,1 08

the perfected , and the circle , 3 10- theory of the mechanical ori

gin of, due to analytics, 40unity in the , 301 - 2Universum of appearance , 1 87

of life and consciousness, 25 7

INDEX

Universum ofreality, 1 27space , 269

Unknowable as a symbol , 194the darkness which shuts out

the , 207

Unodim , the , 1 29 , 1 34, 1 76—7

consciousness, 163

defined, 22Upraisement of matter , 3 29Up springing of a new faculty , 333

V

Vacuum -stillness of nature , 339Validity of ma thematical conclu

sions, 101

Value assigned by the intellect tothe sensible world, 167

assump tional, vitiating influenceof, 1 63

—oi the ideal , 275Vanity of fragmentary evidences,

204

of intellectual method,segmenting space into manydimensions, 299

Vassal , life as the , of materiality,329

Vehicle of life , form as, 330

mind as, 286

Vicissitudes of kosmic evolution,21 5

Vision, the inner, 356

395

Y

Yoga , Rajah , 357Youth of the earth , the nebulous,

335

Z

Zollner, and the claims of the spiritualists , r54

Zones ofAffinity, 22, 1 24Zoometer, the, 297

Wachter, 69Wallis

, John, 46 , 83

Weismann , note, 260

Words, as symbols of ideas, 1 26- 7 ,204

World and the child mind, 1 21as instrument of consciousness,

199 ; 298

fabric of, and geometrism , 261—impossibility of objectifying the

fourth dimension in the per

cep tual, 1 24—of phantasy, conditions of, iden

tifi ed with sensible rea lm , 146

phenomenal , and hyperspace en

ties , 1 28- 130sensible , as a carpet, 196the domain of a perception, 19genesis of, 167

World-

p lasm , kathekotic, 267

as elemental essence. 3 29

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