Forgotten Books › en › download › The... · PREFACE That philosophy and the physical...

THE FOURTH D IMENSION

AND TH E BIB LE

BY V/

WILLIAM ANTHONY GRANVILLE , PH .D . ,

PRESIDENT OF GETTYSBURG COLLEGE .F ormerly Instructor of Mathematics in Yale University ; Author ofGranville’s D ifi

‘erential and Integral Calculus, Granv ille’s P lane

Trigonometry , Granville’s Spherical Trigonometry , andlle’s Logarithmic Tables; 'oint Author of Smith and Granville’s E lements of Analysis ; Inventor of Po lar Co

ordinate P lotting Paper, Yale Sy stem of Mathemat

ical Note B ooks, and Granmlle’

s TransparentCombined Ruler and Protractor.

BOSTONRICHARD G . BADGER

TH E GORHAM PRESS

Made m the United States of America

Gorham Press, Boston, U. S . A.

477046

PREFACE

That philosophy and the physi cal sciences ,when called t o the defence of Christianity, haveso oft en proven to be b roken reeds. has. be en akeen disappointment to many sincere seekers. afterthe truth . A li ttle reflection , howeve r , will makeclear that in the very nature o f things such fa ilures must inevitably result, b ecause s tructuresresting on shi fting sand can neve r serve as. strongbuttresses . No two systems of philosophy agreeand every philosophe r i s continually tinkeringwith his own system . There is no final authorityin philosophy to which an appeal can be made .Such being the case i t is obviously not to be ex

pected that philosophy can furnish us with proofsof the truths of Chris tianity whose validity will b egenerally recognized . And the sciences have alsofallen short for similar reasons . The foundationsof our physical sciences a re far from stable . To

doubt the principles of the conservation of energy,the conservation ofmatter

,or the indivisibility of

the atom , would have been considered as rankscientifi c heresy a few years ago ; today all physicists are familia r with phenomena which contradict

m

Preface

one or more o f them . The E instein Theory o fRela tivity is now opening up new fields for scientific investigation which promises results that willp racti cally revolutionize many current fundamen

tal sci entifi c concepts . In fact, the foundationprinciples on which the physi cal sci ences res t havebeen changing so rap idly o f late that p racti callyall the sci ence text-books now used in our s choo lsare more or le ss obsole te . The physi cal sci encesas well as philosophy are in a constant state of

flux and flow as well as of growth , and this mustnecess ari ly continue to be the situation until theultimate truth is reached , i f it i s ever atta ined, inthese domains of knowledge . As. all will readilyagree that the limits of these regions have not yet‘been reached and , what is more , tha t these limitsare not even measurably in sight

,it i s evident

tha t we cannot reasonably expect that much constructive light will be thrown on Chri sti anity nowor in the near future by either philosophy or thephysi cal s ci ences .To s ati s fy completely human reason any confir

mation of our Christian belie fs must come from asource whose authori ty no one can question . Isthere then any dep artment of human knowledgewhose foundations res t, not on shi fting sand , buton the bed rock of absolute truth' Pure mathematies satisfies this condition ; i t is the only exactscience tha t God ha s revealed to man and the

Preface

truths which it contains are the only truths thatcan be absolutely estab l ished thru pure reason .

Because pure mathematics reveals absolute truthi t i s part of God Himself , for God is the essenceof all truth . It was Emerson who said ,

“naturegeometrizes , and as nature is the handiwork of

God we may say that G od geometriz es . In Viewo f this i t i s rather surprising that so few attempt sshould have b een made in the past to throw lightfrom mathemati cs on the fundamentals of Chri stianity. This is practi cally a virgin field fortheological research .

Pure mathematics i s the vanguard o f all thesci ences , i t ha s marched centuries in advance o fthem all . For instance , Calculus was discoverednearly two hundred years before the sci ence of

Chemistry was far enough advanced to allow theuse o f Calculus as. one o f i ts instruments o f re

search . Practi cally the same thing is true o f all theother physi cal sciences , including engineering. Itmay he , therefore, that the science of theology isnot yet far enough advanced to admit of the readyuse of mathematics a s an instrument o f research .

Today mathematici ans are forging ahead into ’

realms of ab stract thought and pure logic as neverbefore ; judging from. the past , some of the truthswhich they are revealing now may not be put towhat we are pleased to call practi cal use forcenturi es to come .

Preface

Tthat mathematics will ultimat ely prove to bea valuable a id in solving many o f the perplexingproblems connected with our Christian religion i sthe firm belief ofmany sound thinkers. Pro fessorC . J . Keyser has pointed out many striking correspondences between the transcendental prop erti es. of some elementary mathematical conceptsand those transcendental attributes of God andHis cre ation which are intimately associ a ted withreligious belie fs . C oncerning the possib ility

and the probability of future inves tigations alongthi s and other lines shedding new light frommathematical sources on current religious thoughthe has the following to s ay in H ibbert 'ournal

,

Vol. VII: “Over and above the humbler role of

mathematics as. a metri cal and computatory art ,over and above her unrivalled value as a s tandardof exactitude and as. an instrument in every fieldof experimental and observational research, evenbeyond her justly famed disciplinary and emaneip a ting power, rele asing the fa culties from thefickle dominion o f sense by winning thei r allegi anceto the things o f the sp ir it , inuring them to theausteriti es of reason , the stern demands o f rigorous thought

,giving the mental enlargement, the

peaceful perspective , the poise and the elevationthat come at length from continued contemplationof the universe under the aspe cts o f the infiniteand the ete rnal—my convi ction that above and

beyond these services, which by common consentof the competent are peculiarly her own, mathematies will yet further demonstrate her humansignifi cance by the shedding of light more andmore copious as the ye ars go by on ultimate problems o f philosophy and theology, is not a passingfancy, or a momentary whim .

”

The only mo tive o f the author in wri ting thisbook is his sincere desi re to throw some light,even though very dim , on some o f the questionsconnected wi th our Christi an b elie fs . This i s not ,however , an endeavor to construct a system ofChris tian theology on mathematical foundations .The author’s chie f aim will b e to p oint out theremarkabl e agreement which exists between nu

merous B ible passages and some of the concept swhich follow qui te naturally f rom the mathemat

i cal hypothesis of higher sp aces . In making thisat tempt the author is well aware that he i s on a“no man ’s land” exposed to fire from the mathematical trenches. on the one s ide and the theologicaltrenches on the other . But

“ fools step in whereangels fear to tread” and sometimes they surviveto tell the t ale .The scrip ture passages quoted in this text arenot emp loyed for exege ti cal purposes but to showthat the conclusions a rrived at in no way con

tradict the divine wo rd .

While there is quite an extenswe bibliography

Preface

dealing with the subj ect of higher spaces , the lite rature bearing on thei r relation to B ible statements is p racti cally nil . After a thoro search onlys ca tte red re ferences, each embracing only a few

phrases or sentences , have been found whichmight be construed as supporting some of theconclusions arrived at in this text. The authoris therefore taking a voyage on practi cally un

charted seas . No doubt it may seem presumptuous on the p art of the author, who aspi res tob e a mathemati ci an and i s no theologian at all ,to poach thus on theologi cal p reserves ; but thecase i s perhaps no worse than i f a theolog i an whoi s not also a mathemati ci an should undertake thetask.

A sincere effort has been made to present thematter in popular form . To unders tand the text i ti s not necessary for the re ader to be either a professional mathemati cian or a pro found theologi an .

N0 extended excursion into the theory o f higherspaces will be attempted ; the author has limitedhimself to those fundamental concep ts only whichare necessary for the i llumination of the appli cations to be made . No proofs wi ll b e given o f themathematical propositions hereinafter sta ted ; butthe reader can rely on them absolutely, no matterhow startling the conclusions may appear. Theyhave all been proven by mathemati ci ans o f thehighest repute and the validity of these proo fs

Preface

are not more in dispute than is the proposi tionms z z r

The use of the term Fourth Dimension aspart of the title of this book does not mean thatwe shall confine our discussion to that sp ace exelusively, important a s it i s ; i t is so used becausein the popular'mind this term represents themathematical concept of higher spaces in general .

WILLIAM ANTHONY GRANVILLE .

Gettysburg Co llege ,

Gettysburg, Pa .

CONTENTS

CHAPTER

I THE CONCEPT OF SPACE

II G EOMETRIC UN ITS IN EACH SPACEIII ON THE EXISTENCE OF HIGHER SPACESIV ABOUT THE SUPPOSED INHAB ITANTS OF OTHER SPACES

THAN OUR OWN. FLATLAND. LINELAND

V MOTION OF A POINT PERPENDICULAR To EACH SPACEVI ROTATION OF SYMMETRICAL CONFIGURATIONSVII SLCTIONS OF B ODIES IN ONE SPACE MADE BY SPACES

OF A LOWER ORDER. CONSERVATION OF MATTERAND ENERGY

THE ILLUSORY AND THE REALSPACES As HEAVEN S AND HELLS

CREATION OF OUR MATERIAL UN IVERSE . EVOLUTIONRAISING OF THE DEAD. SACRAMENT OF THE LORD’SSUPPER

XII DESCRIPTION OF HEAVEN . MIRACLES

XIII CONCLUSION

47

54

THE FOURTH DIMENSIONAND THE BIBLE

CHAPTER I

THE CONCEPT OF SPACE

(a) 'ero-D imensional Spaces. A '

A point , as A , may be conceived o f a s a spaceof no dimensions . Every point in that spacecoincides with A and there fore is at no distancefrom it . A point cannot move about in thatspace , hence its degree o f freedom of motion i sz ero . Since we know that there is an infinitenumber Of points there is also an infinite numbero f such zero-dimensional spaces .

(b ) One-D imensional Spaces.

Let O be a fixed point on a straight line . Todetermine the position of any point on that line ,as A , all we need to know is its dis tance to the

I I

1 2 The Fourth D imension and the B ible

right or le ft o f O expressed in terms of some unito f length , a positive number to indicate that iti s so far, say, to the right o f O , or a negativenumber when it i s to the left o f O . A single number will there fore suffice to determine the position o f any point on the straight line ; hence thestraight line i s s a id to be one-dimensional . .The

stra ight line may also be conceived o f a s a spaceOf one dimension ; any point in it has only onedegree o f freedom of motion , i t can move onlyback and forth in the line .

The circumference o f a circle may be considered as a curved one-dimensional space , as wellas every other curved line, whether it returns intoitself or not, because (as in the case o f thes traight line ) the position o f any point, as A , maybe determined by a single number representingits distance measured along the curve from a fixedpoint (as O ) on that curve . Obvi ously there isan infinite number of one-dimensional spaces because the number of stra ight and curved lines isinfinite .

https://www.forgottenbooks.com/join

14 The Fourth D imension and the Bible

article on it, as an ink stand , may be moved fromone spot to any other along a path that will alwaysbe p arallel to one of the edges of the table top .

The surface o f a sphere may be conceived o f asa curved two-dimensional space . The position ofany point on the surface o f the earth is determined when we know the two numbers indicatingits geographical l atitude and longitude . By moving along parallels o f lati tude and l ines o flongitude we may travel from any one place toany other . Not only the surfaces o f all spheresbut also an unlimited number o f other curvedsurfaces may be conceived of as curved twodimensional spaces . Evidently there is an infinitenumber of two-dimensional spaces .

(d ) Three-D imensional Spaces.

The Concep t of Space

The sp ace in which we move and have our being is three-dimensional ; we say that bodies in ithave length , breadth and thickness . Considerthree mutually perpendicular planes meeting at O .

The pos ition of any point in our space , as A , i sthen determined by the three numbers representing its perpendicular distances BA , CA , DA , withthe proper s igns , from the three planes . A pointin our space has three degrees of freedom, i tcan move to the right or le ft, forward or backward , and up or down .

“

For instance , an articlein a rectangular shaped room may be moved fromone place to any other by moving it parallel tothe directions of the length , breadth , and heighto f the room . The reader can reach no placewhich may not be reached by going north orsouth , east or west, and upward or downward.

The space we live in is the only three-dimensional space o f which we have any knowledgegained by experience . We are not warranted ,however , in making the statement that no otherthree-dimensional spaces exis t ; such a propositioncannot be proven . Curved lines have given usthe conception of curved one-dimensional spaces ,and curved surfaces the conception of curved twodimensional spaces . We may also conceive o fthe existence of curved three-dimensional spacesalthough in thi s case we have no corresponding


Visual images on which to lean ; we cannot construct in our minds the picture o f a curved threedimensional space . From analogy it i s evidentthat there may be an infinite number of threedimensional spaces , both curved and not curved .

(e ) Spaces of Order H igher than the Third,

also called Hyper-Spaces.

In like manner we define a four-dimensionalspace to be one In Wt h It requlres four numbersto determine the position of any point. A pointin such a space will have four degrees of freedom .

A five-dimensional space would be one in whichit requires five numbers to determine the positionof a point ; a point in it would have five degreeso f freedom . Similarly we define spaces of 6, 7,8 , 11 , dimensions .It i s imposs ible for us to form a mental p icture

o f, s ay, the fourth dimension, the idea is whollyintangible . Nevertheless , i t i s not meaninglessor an absurdity, but a use ful mathematical concept which has led to the development of a geometry o f four dimens ions involving no contradic

tions . We reach thi s new concep t o f higherspaces not at a single leap but slowly and gradually by climbing a ladder whose lowest rungs arethe lower dimensional spaces with which we are

The Concep t of Space

in a certa in sense famili ar . Guided by analogywith dimensions of a lower order we shall thustry to arrive step by step to at least a partial andsymbolic idea of the meaning of higher dimensions . Of these we will be chiefly concerned withthe fourth dimension . Some claim that the B iblerecognizes space o f four dimensions in Job

9 and Eph. where length , breadth ,depth and height, i . e . , four dimensions , are mentioned.

CHAPTER II

GEOMETRIC UNITS IN EACH SPACE

That there exist intimate relations closely ap

proaching a sort of interdependence between thespaces o f various orders i s forcibly suggested bythe following discussion showing how the unit ine ach sp ace may be generated by the uni t of thespace of the next lower order moving in a newdirection not Contained within itself .Starting with zero-dimensional space (a point)

we may consider the point i tsel f a s the unit.

A POINT

'ero-dimensionalSpace Uni t

I f the point A moves the distance o f, s ay, oneinch to the right it will generate the unit line

LINE

A BOne-dimensional Space

Unit

If the l ine AB

Bounded by the points A and B

(zero-dimensiona l units)

moves upward one inch in a18

Geome tric Units in Each Space

direction perpendicular to itself it will generatethe unit square ABCD ;

S'UARE

Bounded by the four unit linesAB ,

BC,CD, DA.

(one-dimensional units)

Two-dimensiona l Space Unit

If the square ABCD moves backward one inchin a direction perpendicular to itself i t will generthe uni t cube ABCDEFGH .

CUBE

Bounded by the six unitsquares ABCD

,EFGH,

ABFE, DCGH

,AEHD ,

BFGC (two-dimensionalunits).

Three-dimensional Space Unit

I f the cube ABCDEFGH moves away one inchin a direction perpendicular to itself it will generate the corresponding unit in four-dimensional

20 The Fourth D imension and the B ible

Space . This unit has been given the name tesseract.

. Here we meet our fi rs t serious diffi culty .

We are famili a r wi th the idea of motion perpendicular to a line , and o f motion perpendicular toa square (or , what amounts to the same thing, ofmotion perpendicular to two mutually perpendicular lines lying in the plane o f the square ) , but wecannot conceive of a motion perpendicular to acube (i . e . , a s ingle motion perpendicular to allthe face s of the cube , or , as i t may also be s tated,perpendicular to any three adj acent edges of thecube ) . Not one of our senses calls for a fourthdirection perp endicular to the other three , expe

rience leaves us satisfied with three dimensions .We cannot visualize the tesseract but it i s possiblefor us to construct a figure in our space which willsymbolize the tesseract, that is , a figure havingmany of the properties o f the tesseract . Such afigure is the following which we

’

may conce ive ofa s having been generated by the original cubeABCDEFGH expanding at a uniform rate intothe cube the directions AA’

,

BB’

, CC’

, DD’

,EE’

, FF’

, GG’

, HH’

, being supposed perpendicular to our space and equal inlength to an edge o f the cube .In generating a tesseract by the motion of a

cube , the latter’s corners generate edges , its edges

generate faces (squares ) and its faces generate

2 2 The'

Fourth D imension and the B ible

The following table indicates the number of eachof the various elements that are associa ted withthe boundaries of the various sp ace units .

Points. Lines. Squares. Cubes.'ero-dimensiona l unit (point) I o o o

One-dimensional unit ( l ine ) 2 1 o o

Two-dimensiona l unit (square ) 4. 4 I o

Three-dimensional unit (cub e) 8 1 2 6 1

Four-dimensiona l unit (tesser act) I6 32 24. 8

Since there is an infin ite number o f points in aline , an infinite numb er of l ines in a square , andan infinite number of squares in a cube , i t i s nearat hand to assume that a space of any order conta ins an infinite number o f lower dimenSIOnal

spaces . The following definition o f space hasthere fore been suggestedSpace is that which separa tes two portions of

the next higher space from each o ther.

Let us illustrate what is meant .

The point A (zero-dimensiona lspace) divide s the l ine (onedimensional space ) into two parts.

The l ine AB (one-dimensionalspace ) divides the p l ane CE (two

5 dimensional space) into two parts.

Geometric Units in Each Space

The pl ane AB (two-dimehsioual space) divides the sol idCD (three-dimensiona l sp ace)into two parts.

Analogy now steps in and suggests that we carrythis one step further by making the statement thatour three-dimensional space probably divides

four-dimensiona l space into two parts. And so onfor the other higher-dimensional spaces .

CHAPTER III

ON THE EXISTENCE OF HIGHERSPACES

It is to be hoped that by this time there hasdawned in the mind o f the reader at least a faintidea of the concept o f h igher spaces , an inklingof what i s meant by, s ay, the fourth dimension .

Additional discuss ions from other vi ewpointstending to show the prob ability of the existenceo f space o f four dimensions will be given in laterchapters o f this book .

We experience no difiiculty in forming a geometric conception o f zero-dimensional space

(point) , of one-dimensional space (l ine ), of twodimensional space (plane) , and of three-dimen

sional space (our universe ) but when at first wetry to give a geometric interpretation to space o ffour , five , six and higher dimensions , we are atsea . We find nothing tangible on which to leanin our fund of experiences accumulated in dealingwith the material things in our world

,things of

which we have b ecome conscious thru our

senses . We are there fore tempted to a t once24

On the Existence of H igher Spaces 2 5

arrive at the conclusion that no spaces o f four ,five , s ix, etc . , dimensions exist . Such a conclus ion ,however, would b e premature because, as has already b een st ated , the proposit ion that higherspaces do not exist cannot be proven .

Firs tly, to dismiss the concept o f the existenceo f higher-dimensional spaces with the statementthat such a thing is incomprehensible is b eggingthe question , unless we are also willing to ab andonthe idea of the existence of our own space of threedimensions , because its properties are also in themain unfathomable mysteries to us . For in

stance ,“it may properly be claimed that a three

dimensional solid of infinite length , infinitebreadth , and infinite thickness would embrace infinite space ; but is it possib le to picture or comprehend what infinite space i s' Can a finite mindpicture a space with no b eginning and no ending ;limitless space In which our vast solar system isa mere dot , in which the known stellar universeis p robably also comparatively a mere dot, al

tho i t i s actually so vast in extent that thelight from some of itS '

component stars whichstarted towards us generations ago or centuriesago i s only now reaching us' I f our space islimitless , the idea 18 IncomprehenSIble , and i f iti s limited , i ts limits are incomprehensible . Our

space is limited or i t i s limitless ; in e ither case

26 The Fourth D imension. and the Bible

the idea i s incomprehensible . Thus the merestatement that a concept is incomprehensible doesnot prove its non-existence .” (Gunnell . ) Obvi

ously, i t would be absurd for us to doubt the existence of our own sp ace simply because many of

its properties are incomprehensible to us . The

human mind cannot fully encompass the ide a ofinfinity and yet thi s i s one of the most usefulmathematical concepts employed daily withouthes ita tion in the teaching of elementary as wellas advanced mathematics .Secondly, only an ignoramus would cla im that

nothing exists which does not l ie within his realmof understanding, knowledge and experience .One hundred years ago the results now achievedthru the telephone , the phonograph , or thewireless telegraph and telephone would have beeninconceivable , would have appeared as miracleseven to the wises t men of that day.

Thirdly, probably few readers o f this bookwould hesitate to make the assertion that onedimensional spaces (lines ) and two-dimensionalspaces (planes ) do actually exist . But a s a matter of fact, the line as a one-dimensional sp aceand the plane as a two-dimensional space has nomore o f a real existence to us than has fourdimensional space . No one has ever seen orheard or felt or tasted or smelled either a l ine

On the Existence of Higher Spaces 27

or a plane . All our sense perceptions and the ex

periences due to them are three-dimensional because all material things are three-dimensional .Our senses can perceive only material things andthose only thru our sensory nerves which arealso material and therefore three-dimenSIOnal.Even the Shadow on a wall cannot truly be saidto be two-dimensional because what we really ohserve is that fewer rays o f light enter our eyesfrom the region covered by the shadow, whilefrom the region surrounding

t the shadow manyrays of light do enter our eyes . And these raysof light ‘ are reflected from material particles

(pigment) on the surface of the wall , that is ,from particles o f matter (three-dimensional) .

Our eyes communicate to our brains only threedimensional knowledge ;

“

for the comprehensionof anything which has only one or two dimensions

a purely intellectual act of abstraction must be

added to tha t of percep tion.

” Lines drawn onpaper, strings , rods and other obj ects which suggest to us the geometric idea of a l ine are reallythree-dimensional forms of matter in which twodimensions are very small as compared with thethird , i ts length . Similarly the plane surfaces ofour experience (sheets o f paper , blackboard,etc . ) are all three-dimensional forms in which thethickness is very small as compared with t he

2 8 The Fourth Dimension and the B ible

length and breadth ; but these helps have b eenfound useful to us in Visualiz ing the geometricidea o f a plane . Thus l ines drawn on paper oron a blackboard have been found very help ful inteaching plane geometry, but they are no t a partof plane geometry itself. I f all such material a idswere discarded , or should cease to exist, planegeometry as we know it would still remain , andevery theorem proven in the plane geometry textbooks used in our schools today would still holdtrue . Four-dimensional geome try exists in the

same sense tha t plane geometry exists,both have

a real existence . The difliculty we encounter inour attempts to form mental images of geometricfigures of four dimensions consists in the factthat we have no material a ids to our imagination ,while figures in plane and solid geometry arereadily

'

conceived of a s corresponding to manymaterial obj ects in the world with which we arefamili ar .Geometry itsel f, there fore , throws no light on

the real nature of any sp ace , not even on the

nature of the three-dimensional space in whichwe live . In fact, i t has not even been proventhat the solid geometry taught in our schoolsand the space in which we live are in exact correspondence ; all we can say is that as far as ourobservations have extended , the one does appear


our experience fits our geometry of three dimen

sions' This question is one which we cannotescape , i t will continue to haunt the thoughtfulstudent, i t i s the most interesting question con

nected with the subj ect “The poss ibil ity thatwe are a part o f a four-dimensional space withphysical l imitations which confine us to a threedimensional space , and with limitations o f oursenses which prevent us from perceiving anythingouts ide of this space— this poss ibili ty excites theinterest o f all who are inclined to abstract speculation . Attempts may be made to discoverphysical proo fs . of such a space , to build up

theori es on its basi s that will expla in discoverieso f modern physics a s yet but l ittle understood , orby it to account for various mysterious phenomena . Most o f us are satisfied that no re al proo fso f the existence o f space of four dimensions willbe found along thes e l ines . Even a workablehypothesis b ased on the (assumed ) existence offour-dimensional space, tho i t might servetemporarily better than any other hypothes is ,would hardly justi fy a beli e f in this existence .But we do s ay that the existence of space o f fourdimensions can never be disproved by showingthat it i s absurd or inconsistent ; for such is notthe case . Nor , on the other hand , will the mostelaborate development of analogi es o f different

On the Existence of Higher Spaces 3 1

kinds ever prove that it does exist . (Manning )Kant no t only recognized the possib ili ty of theexistence of spaces o f more than three dimensions ,but he inferred their very probab le real existence .If it i s poss ible ,

” he says , that there are dev elopments of other dimensions o f Space , it i salso very probable that God has somewhere produced them . For his works have all the grandeurand variety that can possibly be compri sed .

”Or

,

as a mathematician might put i t,why should God

have created geometry of , say, four dimensions,i f there exists nowhere a space that will fit it'God does nothing that is futile or unnecessary.

The following quotation from “Creation Ex

Nihilo (Grub er ) will a id the reader in maintaining the proper attitude of mind towards what hasgone before as well a s prepare him for whatshall follow

“Who can limit the poss ible existence , withinand beyond the physical universe , even o f o therb eings altogether inconceivable by us' Suchtranscendent entities , as also God and the humansoul , b ecause not governed by physical laws , couldin no way affect our physical sensorium . Not b e

ing limited by time and (our ) space relations ,they might co exist with , and around , us andthruout , as well as beyond , the physical universe— and this without in the least affecting


human experi ence and consciousness . AS

experience , thru its prop er avenues of ap

proach , and consciousness , have their necessarylimitations , as sources o f knowledge , in theirlimited po ints of contact with self and nature , sothei r bodyings forth in the pronouncements o fphysical science do not exhaust reality. H ence ,the great whole of reality can never come withinthei r compass . And this i s even true of physicalreality. But , as to the transcendent hyperphysicalor spiritual realiti es , consciousness and experience ,unaided— and therefore physi cal s ci ence—wouldforever have to remain without a point of contact,and therefore in total ignorance . No one should,there fore , a ttempt either to set a limi t to existenceby his limited experience or make his finite reasonthe measure o f the immeasurab ly complex universe o f the infinite and eternal God .

”

“A person born without the sense of sight cannot see light and color . He cannot even formany real concep tion of them . To him all i s darkness . And

,a s far a s he could by his own powers

discover , both light and color would have no existence or would belong to the Same category asdarkness . The man born without the sense o fhearing can neither hear sound nor can he evenform any correct concep tion of sound and music .And thus , to a (congeni tal) blind and deaf indi

On the Existence of H igher Spaces 3 3

Vidual , even this very tangible physical world isan entity altogether different from the reality.

Many of i ts marvelous phenomena of beauty areto him totally non-existent. And i f , perchance ,s

‘

ome explanation or revelation to him of thesethings were attempted by a seeing and hearingpersonality, these phenomena would yet in a sensehe utterly inconceivable by him . He can onlyby touching, etc . , acquire some indefinite idea o f

grosser forms and movements . But he could notperceive even any efiect of those subtle marvelousVibrations that produce light and color andmusic .”

“Such an individual lives in a world of mar

velous b eauty, but he beholds it not, nor can heeven form any proper conception of it . But ,

surely, i t would b e an almost unpardonab le presumption on his part to deny the existence of theglorious rainbow in the heavens and to argue withan entranced auditor against the existence of themaj estic symphonies of a B eethoven . For thesethings that lie beyond his limited physical senses ,he must needs accept the testimony of those whohave the necessary senses of percep tion to knowtheir reality .

”

“Like the blind and dea f individual , we standamid the wonders of nature . Tho we can

perceive , in light and color and music , a minute

34. The Fourth D imension and the B ible

fraction o f the efiect produced by waves o f etherand waves o f a ir , yet these phys ical waves themselves l ie totally beyond even our natural senseorgans . Like that blind and dea f individual , whothru his sense o f touch , etc . , can form some ideaof hi s physical environment, SO we with ourlimited sense-organs can acquire some knowledgeof the surrounding universe . But , as in his case ,our limited senses p ermit o f but a Very partialknowledge , and beyond their range there arerealiti es even in physical na ture concerning whichwe can only speculate and which we may neverknow. We are like children watching a game ,from a point some steps away, thru a small crackor knot hole in a boarded enclosure . All appearsfragmentary and parti al . Immeasurably the

greater part o f the universe , from the infinitesimally small to the universal whole , in ten thousand marvels , l i es beyond the range o f the wholeoutfi t o f our physical senses , or is a physicallyintangible reali ty. Thus , the vis ible light orcolor sp ectrum consti tutes the record of but aminute fraction o f the whole range of the mysteriously wonderful perpetual dance o f theimponderable ether , i t consti tutes but one of theoctaves o f the many-octaved key-board of Vibrations . And yet , the. other octaves no less trulyexist a s with our own invented tools of investiga

On the Existence of H igher Spa ces 3 5

tion we are more and more discovering from theireffects ; but these octaves themselv es we cannotperce ive . If our eyes could be SO adjusted as toenable us to b ehold the whole range of this fundamental reali ty

,vistas o f vis ions hitherto incon

ceivable would lie be fore our astonished gaze .And so might we conceive of superadded

transcendent spiritua l faculties, i f not confinedwithin our physical organism . And thus thetranscendent glories o f spiritual realities , of an

gelic beings , and even of God , might entrance theastonished perceiving personality .

In what follows in this text , higher dimensionalspaces , and in particular the fourth , will often bere ferred to as i f they really exi sted . The readershould clearly understand that this is done onlyto save words by avoiding a needless repetitionO f the statement that their existence is only assumed .

CHAPTER IV

ABOUT THE SUPPOSED INHABITANTSOF OTHER SPACES THAN OUR OWN

AS we are now going to make several excursions into the realm o f pure thought and abstractspeculation it will be necessary for the reader toapproach the subj ect with an open and receptivemind . He will need to occas ionally cut loose fromthe sense perception moorings which anchor himto our material world , he will o ften have to

leave experience b ehind and reason wholly fromanalogy . A truly sympathetic attitude Will addmuch to the benefits that may be derived fromthe discussions which follow .

Suppose tha t one , two , and four-dimensionalspaces do exist , in the same sense that the threedimensional space of our experience exists , andlet us imagine that in each of these spaces beingsexist , each individual being restricted to his ownparticular space in his movements , his perceptions , and his understanding, j ust as we are re

stricted to a three-dimensional space . Then letus by analogy try to conceive what the attitude

36

3 8 The F ourth D imension and the B ible

four dimensions . In doing this we may supposethat what we call two-dimensional matter i sre ally three-dimensional , but with a very slightthickness in ' the third dimens ion which Flatlanders are unable to recognize and of which they areentirely unconscious . Thus we may imagineFla tland in the nature o f a soap film (formed inthe closed loop of a wire ) in which Flatlanderslive . Or

, i f we are willing to accept the statementof some philosophers and physici s ts that a partiele of our matte r has no real subs tance (in thesense in which we ordinarily use that term ) butconsists only o f a b undle o f forces , attractive andrepellant , then there i s no diffi culty in thinking ofsuch forces as lying entirely in a plane (or in aline ) . All bodies in our spa ce are bounded bysurfaces , in Flatland all bodies have one-dimen

sional boundaries (l ines and curves ) . To the skin

(surface ) of a human b eing there would correspond the bounding contour (perimeter ) o f aFla tlander . To other Flatlanders he would beexposed only along his perimeter , his interiorcould be reached by them only thru an openingin this perimeter . A Fla tlander surgeon operating on a Flatlander would have to make anincis ion in the perimeter o f his patient just a s oneo f our surgeons must make an meiSion in the Skinof a patient to remove his appendix . To one of

Supposed Inhabitants of Other Spaces 3 9

our surgeons gi fted with two-dimensional vision,however , all the interior organs o f a Flatlanderwould be exposed and he could operate withoutcutting his Flatlander patient open . Not onlycould we see ( i f we had two-dimensional Vis ion)the insides of Flatlanders but we could also seethe insides of their houses and all their closedcompartments . To illustrate let us assume thatthe plane ABCD on this page is part of Flatlandand that a Flatlander inhabiting it is triangularshaped (Fig . a ) .

As there is no upward or downward , no

above or below in Flatland , houses andbarns would have no roo fs or floors , only the linesof the ir Sides (perimeters ) would be there .Figure (b ) may be taken to represent the houseof a Fla tl ander . He could enter his house onlythru the opening (door ) EF. There would beno difference between doors and windows (Fig. c )To get to the other side of the line GH theFlatlander would have to move around one endof the line . The l ine would form an insuperable

40 The‘

F ourth D imension and the B ible

barrier to him , he could not cro ss i t or see overor under it, j ust a s in our space a stone wall o finfinite height and depth would hinder our motionand obstruct our line o f s ight. However , a threedimensional b eing (man) with two-dimensionalpowers could li ft the Flatlander up out of theplane , carry him across the l ine and then set himdown aga in in the plane . To his Flatlanderneighbors the Flatlander during this operationwould suddenly vanish and remain invis ible untilhe reappeared in thei r world on the other s ideo f the line . This per formance would appear asa miracle to the Flatlanders because not onlywould it be impossible for them to see anythingnot in their plane but all their experi ences beingconfined to tha t plane they could not possibly explain the phenomenon . In the same manner theFlatlander might be placed in or taken out ofa closed room (Fig . (I) to the utter mystificationo f his fri ends . A robber from our space withtwo-dimensional powers could therefore entertheir houses and steal thei r money and j ewelryfrom their sa fes without breaking locks or doors .There would be nothing in Flatland corresponding to our tubes and i t would be impossible toti e knots or thread needles there . As a Fla tlandercould not “fl ip” a co in , to him it would always be“heads-up or always “ta ils-up .

”Story books

Supposed Inhabitants of Other Spaces 4 1

have been written about li fe in Flatland, sup

posedly by Flatland authors , in which the wholegamut o f our human experiences are duplicatedin so far as the restrictions under which they aresupposed to live will allow . There are descriptions o f li fe and industry in Flatland as well astales o f love and war , and the love stories closewith the regula tion they married and livedhappily ever a fter .” An Episode in Flatlandand “

Scientific Romances” by C. H . H inton ;“The Fairyland of Geometry” by Simon Newcomb ;

“Fl atl and” by E . A . Abbott, which has

also been dramatiz ed and acted .

LINELAND

Let us now consider Lineland one-dimensionalspace ) and its supposed inhabitant s , the Linelanders .

D C E'

A Linelander (as C) would thruout l ife be restricted to his line (of which AB i s a part) whichhe would conceive of as extending indefinitely inboth directions . He could slide freely back andforth along this line but it would be impossiblefor him to move out of i t. He would not beconscious of nor could he conceive o f a seconddimension ; the concept of a plane surface would


l i e entirely outs ide o f his mental realm , j ust asthe existence o f our space would be inconceivableto a Fla tlander. Bodies in Lineland would notbe bounded by surfaces as in our space or by contours (perimeters ) as in Flatland , but by twopoints , their two extremities . I f a Linelanderencountered another s imilar being (a s D ) in hisspace neither could p ass the other for to do sowould mean that at least one o f them would haveto move out of the line , i.e . , out o f their world .

Thruout li fe a Linelander (as C) could have onlytwo neighbors (as D and E ) , one on e ach s ideo f him . His experi ences would be extremelylimited , to human beings li fe in Lineland wouldbe , to state it mildly, very dull indeed.

Let us now suppose that one o f these Linelanders (as C) in some way becomes consciouso f the exis tence of a second dimension and somehow becomes endowed with the power to moveout o f the line AB into a plane conta ining thatline as , for instance , the plane of p age 4 1 . Theinstant he moved out o f the line AB (his world)he would become invis ible to his Lineland neighbors D and E The vanishing of C, apparentlyinto nowhere , would be utterly incomprehensibleto D and E ; they could not account for it in anyway, it would appear to them as a supernaturalphenomenon , i.e . , a s a miracle . Possess ing the

Supposed Inhabitants of Other Spaces 43

freedom to move about in the plane (his newworld ) the Linelander C could then move backinto the line AB (his old world) . I f he shouldreturn to a point in the line between his old Lineland neighbors D and E his appearance would bein the nature of an apparition to both of them .

His reappearance , apparently out of nowhere ,would be as deep a mystery to them as was hisdisappearance .The happenings described in the last paragraphmay also be imagined as occurring when the Linelander C has himself no conception o f a seconddimension and has no power to move out of theline AB (his world) . We may conceive of aSituation in which the Linelander C thru someinfluence is in the power o f a Flatlander or otherhigher-dimensional b eing . This la tter higherdimensional being might then take the LinelanderC out of the line AB and replace him at will , andthe attendant phenomena would appear the sameas b efore to the two Linelanders D and E . Whilethis was taking place the Flatlander , or otherhigher-dimensional being who was exercis ing hispower over C , would be invis ible to the Linelanders D and E , or, i f not invisible , he wouldappear to them only as a one-dimensional beinglike themselves .A Linelander, there fore , who in some manner


has strayed off his line (one-dimensional space )into a plane (two -dimensional space ) will remaininvisible to his Lineland neighbors until he returns to his own world (the l ine ) .

Similarly , a Fl atl ander who has moved out ofhis pl ane (two-dimensional sp ace ) into space ofthree dimensions (our space ) will remain invis ible to his Flatland neighbors until he returnsto his own world (the plane ) .

Continuing , the next step , as suggested byanalogy, would then be the following. A man

(three-dimensional being) who has been translated ‘ from our space into a higher-dimensionalspace will remain invisible to earthy beings untilhe returns again to our space . We have B iblerecords describing the bodily disappearance ofhuman beings from our space in a manneranalogous to the above .

'

One case i s that ofEnoch who did not see death but was taken bodilyout of our world by God. Gen . 5 :24, H eb . 1 1 :5 .

Another example is the bodily translation OfEl ij ah into heaven . II Kings This againsuggests that space of four dimensions may beour heaven . Or, Since the plural form of theword heaven occurs very often in the B ible , i t maybe that “ the heavens” include not only space o ffour dimensions but also all the other higherdimensional spaces . We have no record o f a


in which they dwell and , a fter having fulfilledtheir missions in our three-dimensional space , toagain ascend on high . Did Paul have a higherdimensional space in mind when he wrote ,

“Iknew a man ,

“ whether in the body, or out o f thebody, I cannot tell , how that he was caught upinto paradise , and heard unspeakable words .II Cor. 3 . Re ad again the Revelations ofSt . John , read them with the added light of themathematical hypothes is o f higher spaces . I t willbe found that much of the fog with which ourlimited space p erceptions has enveloped it hasvanished and many passages will take on a newmeaning ; the eyes of the soul will p enetratedeeper than ever before into the dim and distantvi sta s which veil for us the present a s well asthe great unknown herea fter .

CHAPTER V

MOTION OF A POINT PERPENDICULARTO EACH SPACE

In Chapter II the concept of the motion of apoint p erpendi cular to a unit in each space waspartially developed . We will now lead up to anextension o f this idea by making use o f the twofollowing obvious propositions .

(a ) A point (as 11) starting from the center

of a line (as BC) and moving of in a direction

perpendicular to the line will nev er approach any

portion of the line andwill mov e away a t the same

ra te from its two end points B and C.

(b ) A point (as A) starting from the center

of a circle and moving of in a direction perpen

47


dicular to its plane (as BC) will nev er approachany portion of the circumference of the circle

but will move away a t the same ra te from all

points on this circumference .

Advancing another step we get

(c ) A point (as A) starting from the center

of a sphere and moving of in a direction perpen

dicular to our three-dimensional space will nev erapproach any portion of the surface of the spherebut will move away a t the same ra te from all

points on this surface .

This proposi tion in the Geometry o f FourD imensions. was first proven by the great astronomer and mathematician Simon Newcomb , and itreveals some o f the things , extraordinary to us ,which may be done in four-dimenSIonal space i fsuch a space exists . We may also state the propositiou as follows . Just as in Flatland a point canmove in a direction perpendicular to a line , amotion inconceivable to Linelanders, and in our

space a point can move in a direction p erpendicular to two mutually perpendi cular l ines , a motion

Mo tion of a Point Perpendicular 49“

inconceivable to Flatlanders , so in space of fourdimensions a point can move in a direction per

pendicular to three mutually perpendicular lines ,a motion which i s inconceivable to us . Assumingthat the above sphere i s replaced by a materialhollow sphere (as a hollow sealed glass globe )and the point A is replaced by a grain of wheat,the above proposition asserts that in space offour dimensions the grain of wheat may be placedinside of the globe , or trans ferred from the insideto the outside of the glob e , without breaking theglob e or interfering with its structure in any

manner . This would mean that in space o f fourdimensions any of our material Ob j ects can betaken out of , or placed inside of , one of ourclosed rooms or boxes without penetrating thewalls (sides ) . A four-dimensional robb er , therefore , could enter our houses without hindranceand steal the money and j ewelry in our safes without breaking locks or doors , and none o f our j ailscould hold him i f caught. This may be wherethe money we miss has gone , and perhaps thearticles we so often lose have accidentally rolledinto four-dimensional space ' A four-dimensionalman could eat one of our soft boiled eggs forbreakfast without breaking the shell and he coulddrink from one of our bottles without removingthe cork. He could extract the pulp and nut from


one o f our plums without breaking the Skin o fthe plum ; in fact , he could remove the kernel fromthe nut without breaking the skin o f the plum orthe shell o f the nut. He could take off his stockings without removing his shoes , and he couldtake out the inner tube of one o f our automobiletires , repair it and then replace it, without re

moving the outer casing . Just as one o f our surgeons could operate on a Flatlander withoutmaking an incis ion in his perimeter so a fourdimensional surgeon could operate on one o f uswithout cutting us open . It is quite certa in thati f such a surgeon should locate among us he wouldvery soon have a la rge and lucrative practi ce 'All of our space , including the interior o f thedensest solid , i s open to inspection and manipulation from the fourth dimension .

An ordinary knot (Fig . a ) cannot be unti ed inour space i f the two ends of the cord are fixed (asat A and B ) . In four-dimensional space , however , this can be done without unfastening theends , in fact, a knot could be ti ed or untied on anendless string . Our knots would there fore be

Mo tion of a Point Perpendicular 5 1

useless in four-dimensional space . Evidently ourchains would also be useless because the l inks inthem could be sep arated without breaking.

Corresponding to this in Flatland would be a cordfastened at A and B (Fig . a ) and making a looparound , say, C. This loop could not be unwoundby a Flatlander , but one o f us could do it bySimply li fting that part of the cord composing the

loop into the third dimension, stretching i t out,and then replacing it in the plane (Fig. b ) .

As has already been stated , it has been mathematically proven , beyond the shadow of a doubt,that i f there exists a space o f four dimensionsthen in that space material ob j ects o f our spacemay be placed in or taken out of any of ourhermetically s ealed closed compartments withouthindrance . It has never been satisfactorilyproven , however , that this can be done by ahuman being ; our nearest approach to i t i s tothink of it being done . We can think of the kernel


o f a nut being extracted without breaking the shello f the nut, in sp ite o f the fact that we do notunderstand how i t can be done . We can think ofourselves a s entering or leaving closed compartments ; theirmateri al walls do not in the le asthinder our imagining ours elves doing it . In theB ible , however , we have records telling o f twodistinct occas ions when Christ in the body didappear before his disciples assembled behindclosed doors for fear o f the Jews . John 20 :

1 9-23 , 26-29 . On the first appearance Thomaswas absent but on the second he was present andwe read “ then saith he to Thomas , reach hitherthy finger , and behold my hands ; and reach hitherthy hand and thrust it into my side ; and be notfa ithless but beli eving . And Thomas sa id untoH im, my Lord and my God . That Christ appeared to his disciples on this remarkable occasionclothed in his material body is further emphasizedin Luke 24 : 3 6-43 .

“Behold my hands and my

feet tha t it i s I myself : handle me , and see ; for asp irit hath not flesh and bones , as ye see me have .

And they gave Him a piece o f a broiled fish,

and o f an honeycomb . And he took it , and dideat be fore them .

”Christ, considered as a higher

dimensional being , certa inly had the power to ap

pear in his body as described above , or to do

anything else which cannot be done by us in our

CHAPTER VI

ROTATION OF SYMMETRICALCONFIGURATIONS

Let ABC and A’B’

C’ be two lines In Lineland

LL’

, the points A ,B ,C, and respectively

LI

A m m a n-rm

B C A'

5'

C’

being arranged in the same order and so locatedthat AB A’B’

, AC A’

C’

, BC B’

C’. One

l ine may then be made to co incide with the otherso that A coincides with A’

, B with B’ and C with

C’ by simply sliding one of them along LL’ Thiscould be done by a Linelander.

ss

Le t us now consider the two l ines ABC andC

’

B’A’ where , as b efore , AC

54

R o ta tion of Symme trical Configura tions 5 5

A’C

’

,BC B’C’

, but in which the corresponding points are arranged in the reverse order . Inthis case the one l ine cannot , by Sliding it alongLL’

, be made to coincide with the other so that Ashall coincide with A’

,B with B ’

,and C with C’

,

and a Linelander could not conceive of any otherway in which it might be done . A Flatlander ,however , or one of us , would at once observe thatthe corresponding points could b e brought intocoincidence by rotating one of the lines about O inthe plane of the p age , O being the center of CC

’

The points A , B , C and A’

,B ’

, C’ respectively are

said to be symmetrical with respect to the pointO . Roughly s tated , the line ABC may be taken l

up into Flatland, turned over , and put down againon the line C’B ’A’ so that the point A coincideswith A’

, B with B’ and C with C’

. While thisrotation was taking place the moving line , as inthe position would be invisible to theLinelanders living in LL’ because it would beoutside O f their world ; the whole process wouldappear to them as a miracle .


Let ABC and A’B’C’ be two triangles in theplane of this page , such that AB BC

B’C’

, CA C’A’

,the corresponding Sides of the

two triangles b eing arranged in the same order .

One triangle may then be made to coincide withthe other by s imply sliding it along in the plane ofthe page until A coincides with A’

,B with B’

,

and C with C’

. This could be done by a Flatlander .

Let us now consider the two triangles ABC andin which as before AB BC

B’C’

, and CA 2 C’

A’

, but with their corresponding sides arranged in the reverse order . In thiscase the one tri angle cannot, by Sliding it alongin the plane , be made to coincide with the other,and a Fl a tlander could not conceive o f any otherway in which it might be done . The reader shouldveri fy the first statement in the last sentence bycutting a tri angle out of cardboard to fi t, s ay ABC,

and then , by sliding it along the page , try tomake it fit on A’B’C’ A three-dimensionalmathematician would observe , however , that the

R o ta tion of Symmetrical Configura tions 57

two triangles ABC and A’B’C’ are symmetricalwith respect to the line XY (corresponding pointson the two triangles being located on the sameperpendicular to XY and at equal distances fromit ) and that they can be brought into coincidenceby rotating one (with the plane in which it lies )about the axis o f symmetry XY thru three-dimen

sioual space until it coincides with the other .That is , one triangle must be taken up into ourspace , turned over , and put down on the other .To Flatlanders this operation would be an un

solvable mystery.

If one of us should take a Flatlander (asFig . a ) and turn him over thru our space backinto his plane (as Fig. b ) he would become a sortof mirror reflection o f his former self. H is heartwould now be found on his right s ide ins tead o fhis left, and i f he was right handed before hewould now b e left handed . While he was beingturned over in our space he would be invisible tohis Flatland neighbors ; first he would apparently


vanish before their eyes into nowhere and thenas suddenly he would reappear like an apparition .

B efore taking the next step in our discussion

let us digress a trifle to point out that geometricalpropositions relating to configurations in a particular space may sometimes be proven withgreater ease by operating in a higher space than i fwe are restricted to the original space . A case inpoint i s the plane geometry proposi tion that twotriangles are equal i f the three s ides of one areequal respectively to the three s ides of the other .In practically all the plane geometry text-booksused in our schools the proof of this propositionrequires a ro ta tion of one of these triangles aboutone of its sides. But thi s means that the pupil isabandoning the space of his plane geometry to

operate in the space o f solid geometry (threedimensions ) . In like manner there are manytheorems in both plane and solid geometry whichcan be proven with greater ease by using geometryof four dimens ions than by using the methodsfound in our school books . In fact both planeand solid geometry have been greatly enriched bythe discovery thru the geometry of four dimen

s ions O f a number. o f new theorems . Just as solidgeometry throws a new light on plane geometryand enables us to solve problems in the planewith greater ease than i f we confined ourselves

R o ta tion of Symmetrical Configura tions 59

to the plane , so it has been found that geometryof four dimensions illuminates both plane andsolid geometry, and has made easy or renderedpossible the solution of many problems

l

bo th inthe plane and in our three-dimensional spacewhich , without the concepts of geometry of fourdimensions , would have b een very diflicult of solution to say the least . Schubert has illustrated thisby means of some very interesting examples .We now return to our line of discuss ion .

Let ABCD and be two tetrahedronsin our space having all the faces of one equal tothe

‘

corresponding faces of the other and arrangedin the same order . By superimposing one on theother we can make one coincide with the otherwithout difliculty .


Let us now consider the two tetrahedronsABCD and in which , a s before , all thefaces o f one are equal to the corresponding faceso f the other , but arranged in the revers e order .N0 amount o f manipulation will now enable usto bring the two tetrahedrons into coincidence ,and no human being can conceive o f any way inwhich it can b e done . This task i s as imposs ibleto us as was the task of the Linelander to bring

the symmetrical lines ABC and C’B’A’ into coin

cidence (p . 54) or the task of the Fl atlander tomake the two symmetrical tri angles ABC and

A’B ’C’ coinci de (p . Mathematici ans of thehighest standing, however , have proven that

'

it

can be done in four dimensional space by rotating

one of the tetrahedrons about their plane of symmetry XY (corresponding points o f the two tetra

hedrons being located on the same perpendicularto XY and at equal distances from it) , a kind of

rotation that i s impossible in our space o f three

dimensions . Or, stating it roughly, one o f thetetrahedrons must be taken up into space of fourdimensions , turned over , and then brought backinto concidence with the other . One of these

te trahedrons corresponds to the mirror image o fthe other , the mirror b eing in the plane of XY.

While thi s rotation i s taking place the moving


and every hair on your head . Your point of Viewwould be completely turned around , so that everyone else would appear to you to be reversed . The

letters on this page would seem reversed to you ,i.e . , as they really do appear on the printer

’s typeform . The hands of a clock would appear toyou as gomg backward and the sun to ris e in thewest and set in the east ; the whole world wouldappear to you as a looking-glass world . Whenyou look into a mirror you see a p icture of theactual “ reversal or “ twist” that can be impartedto material bodies in the space o f four dimensions ,but which cannot be brought about in our spaceo f three dimensions .Our two hands appear to be at least approxi

mately of the same size, and shape but we cannever manipulate one so that it shall occupy thes ame sp ace that has been occupied by the other ;we cannot put a right glove on the le ft hand , orVice-Versa . The reason i for this i s that our twohands are symmetrical with respect to some planein our space and , like the two symmetrical tetrahedrons shown on p . 59 , they cannot be made tocoincide by any operation entirely confined to ourspace of three dimensions . For the same reasonwe cannot wear a le ft shoe on the right foot

,or

Vice-vers a . A four-dimensional man,however ,

can change a right shoe into a left shoe by rotat

R o ta tion of Symme trical Configura tions 63

ing it about a plane in his space . I f a one armedman in our space should in some way becomeendowed with the power to rota te bodies aboutplanes in four-dimensional space he could use bothof an ordinary pair of gloves because he couldchange either one into the other , so that he wouldhave either two rights or two lefts .Instead o f rotating material bodies about a

plane in space o f four dimensions as described

above it has been proven that a curved surfacemay also be used as the axis” of rotation byallowing for a slight degree o f distortion . Thisleads to very interesting results i f the rotation isapplied to fl exible material surfaces like a curved

sheet o f rubber . The two surfaces of the sheetof rubber may be made to exchange places byrotating them about the sheet itself in space offour dimensions . During the rotation there

would be no interference between portions o f thesurfaces so that it may be done with surfaces thatare closed (like a sealed hollow rubber ball ) as

well as with sur faces that are open (like a sheetof rubber ) .

This means that a hollow rubber ball (Fig. a )could be turned ins ide out without tearing just aswe may turn a rubber band or the section of aninner tire tube (Fig . b ) inside out . In both cases


the inside and outs ide surfaces would exchangeplaces .

Not only in the animal kingdom do we find thiswonderful symmetry (right and le ft handedness )exemplified in the bodies of all creatures , as wella s in most o f the interior organs of those bodies ,but it also appears very o ften in the vegetablekingdom as shown by the“ symmetry of the twosides of a lea f , and in flowers , bulbs , nuts , fruits ,etc . In fact , the right and le ft rotation (twi s t)which i s characteristically four-dimensional is ofvery frequent occurrence in nature . A beam ofpolarized light (whose wave-vibrations are all in

one plane ) i s rotated either to the right or to thele ft on pass ing thru certa in organic substances

(sugars , starches ) j ust as we might hold a r ibbonby the ends and give it a twi st to the right or left .Two forms of sugar found in honey called dextrose and levulose owe their names to the factthat one rotates a polarized beam to the right

(dextra ,“right hand” ) and the other to the left

(laeva,“le ft In chemical constitution

R ota tion of Symmetrical Configura tions 65

they are exactly the same , they are both repre

sented by the same formula , and yet they rotatethe plane of polarization o f a beam of light inopposite directions . Such sub stances are called“ i someric .” A molecule o f each of the two sub

stances contains the same number of each kindof atom , the difference in the effect on light beingaccounted for by a different arrangement or grouping of the atoms in the two molecules .The following discussion will explain what relation this has to a possible four-dimensional space .In a plane three points (as the vertices o f the

triangle in Fig . a ) , and no more , can be locatedso that the three distances b etween them shallbe independent of each other , i.e . , any one of thedistances may change without changing the others .In our space o f three dimensions four points

(as the vertices of the tetrahedronm Fig. b ) , andno more , can be located so that the six di stancesbetween them shall be independent .In space of four dimensions five points , and


no more , can be located so that the ten distancesbetween them shall be independent, something thatcannot be done in our space .I f then a molecule consists o f five atoms we

cannot in our space alter the distance between twoof them Without at leas t altering some second distance . “

But i f we imagine the centers o f theatoms placed in four-dimensional space , this canbe done ; all the ten distances which may be conceived to exist between the five points will thenbe independent of one another . To reach thesame result in the case of s ix atoms we must assume

‘

a five-dimensional space ; and so on. Now,

i f the independence of all poss ible distances between the atoms o f a molecule is absolutely re

quired by theoretical chemical research , thescience is really compelled , i f it deals with molecules o f more than four atoms , to make use o f theidea o f a space o f more than three dimensions .Thi s idea i s , in this case , simply an instrumentof research , just as are , also , the ide as of molecules and atoms— means designed to embrace ina persp icuous and systematic form the phenome

o f chemistry and to discover the. conditions underwhich new phenomena can be evoked . Whetheror not a four-dimensioned space really exists ISa question whose insolubility cannot prevent research from making use of the idea , exactly as

R ota tion of Symme trical Configura tions 67

chemistry has not been prevented from makinguse o f the notion of atom , altho no one reallyknows whether the things we call atoms exist ornot .” (Schubert . )For example , at least eight different alcohols

have been discovered having the same formulaC5 H I2 O . Chemists account for thes e differentcompounds by supposing that there is a differentarrangement or grouping of the five carbon , or

C-atoms , in each compound , something whichwould be possible in space o f four dimensions butnot possible in our spac e . In other words , shouldit become necessary for the explanation of thestructure of this molecule to assume that the fivecarbon atoms in it are equidistant from each other

(the simplest possible case ) we would also haveto assume the existence of four-dimensional space ,because in that space , and not in our space , i s itpossible to loca te five points equidistant from eachother .There are two vari eties of tartaric acid whichcrystallize into symmetrical forms , forms bearingthe relation o f obj ect to mirror-image as hereshown.


Investigators have found that one o f thesecrys tals apparently changes into the other with;

out chemical resolution and reconstrtution andwithout any manifestation of force . If it could beproven without the shadow of a doubt that thi si s what actually happens , then this would pointto the existence of space

'

o f four dimensions , because in that space , and not in ours , i s it poss iblefor a right-handed shape to change into its symmetrical le ft-handed shape bya s imple movement.Certain snails , exactly alike in other respects ,differ s imilarly in that some are coiled to the rightand Others to the le ft. An astonishing phenomenon connected with these snails i s the fact thattheir juices have the property of rotating a polariz ed beam of light to the right or to the leftcorresponding to the direction in which the snailsthemselves are coiled. This suggests that theirexternal form is the expression of an internaldifference due to a right or le ft twist o f theiratoms caused by a four-dimensional force . I thas been suggested that such a four-dimens ionaltwist runs thru all living forms , and that the li feforce is , in p art at least , four-dimensional .


which they had seen had anything to do with athree-dimensional solid at all , even i f they had aninkling o f the existence of our space . The phenomenon would appear to them as the creation ofa new two-dimens1onal body (circle ) Which coritinued to grow larger and larger and then suddenly vanished . The alpha and omega of thisphenomenon would be deep mysteries to them ,

just as b irth and death are to us . In like mannerit may be that each one o f us is in reality a four

(or higher dimensional being, our personality

(as we are known to ourselves and others ) beingthe

‘

section o f our real , our higher s elf , made bythe three-dimensional sp ace o f our experience .Like the cone pass ing thru Flatland, we arrive intothis three-dimensional world from some unknownshore , i.e . , a re born ; we grow up thru childhood,youth , middle age , old age ; we die , and our souls ,our real , our higher selves , pass into the greatunknown herea fter . And thruout li fe we havebeen thought o f by ourselves and our fellowmenas three-dimensional beings only. Perhaps all living things , as they appear to us , are the sectionso f four (or higher dimensional creatures madeby our space .

“Vi ewing human li fe from thisstandpoint, the conclusion may be reached that I ,as I wri te this , am merely that section o f my fourdimensional self that happens to be passing thru

Sections of B odies in One Space 71

this world at this moment , and that the wholeof me , from my b i rth to my death , i s a fourdimensional entity ; that the past and the futureare past and future only in a three-dimensionalsense , and that in a four-dimensional sense thepast and future are present— that is , both whatwas and will be , i s .

”

(Taylor . )Suppose now that a particular Flatlander hadin some way acquired the intellectual power tosum up , in his mind , in a moment of time , all thecross sections (circles ) of the cone he had seen .

He would then obta in a mental picture of the cone ,not as we see it , but as a body, not of two dimen

sions , but as a body having three dimensions , timeb eing the third dimension . In like manner

,by

a concentrated mental effort , we may in a momentof time assemble all the impressions we have received of some friend or relative with whom wehave been intimately associated from his , or her ,birth to death . The mental image thus obtainedwould include all the years o f his or her li fe , thewhole span o f life of that individual would becompressed into a moment of time ; and thismental image would have four dimensions , timebeing the fourth dimension . Mental vision hastherefore been called four-dimensional by somewriters . All of us have in our dreams experi encedthis crowding together of a long series o f occur

72 The Fourth Dimension and the B ible

rences into an instant o f time . There a re no t e

strictions of time , or restrictions o f the space o four experience , in dreamland .

“The unthinkab le ‘

velocity of time in dreams may be inferred fromthe fact that between the movement of impact ofan impression at the , sense periphery and its reception at the center of consciousness—momentsso closely compacted that we think of them ass imultaneous— a coherent series o f representations take place , involving what seem to be protracted periods for their unfoldment. Everyreader will e asily call to mind dream experienceso f this character , in which the long-delayedd'nouement was suggested and prepared for bysome extraneous s ense impression , showing thatthe entire dream drama unfolded within the timeit took that impress ion to travel from the skin tothe bra in .

”

(B ragdon) . Perhaps dreamland islocated in four-dimensional space .A man who has experi enced the sensations o fa dying person as , for instance , a man who believed he was drowning, tells us that in an instantthe whole panorama of his past li fe was spreadout before him in a vivid momentary persp ective .“Memory is a carrying forward of the past intothe present, and the fact that we can recall a pastevent without mentally rehearsing all the intermediate happenings in inverse order , shows that

Sections of B oclies‘ in One Space

in the time aspect o f memory there is s imultaneityas well as sequence . I f our mental Vision is fourdimensional i t po ints to the possib ility that ourmental or sp iritual self is four (or higherdimensional . If time is but the way in which weperceive the fourth dimension , then our spiritualselves, b eing higher-dimensional, are independentof time , above time , outs ide o f time , that is ,eternal . In this higher space our grammaticaltenses would then have no meaning . Perhaps thisexplains why these tenses as used in the B iblesometimes appear “mixed” to us . Jesus said “

be

fore Abraham was , I am .

” “For a thousand

years in thy sight are but as yesterday when it i spast .” Ps. That God, the highest-dimen

sional being is not, like we , restricted by timelimi tations is clear . “

One day is with the Lord asa thousand years , and a thousand years as oneday.

” II Peter “Behold , thou hast mademy days as an handbreadth ; and mine age is asnothing before thee . Ps.

Another interesting illustration similar to thatof the cone passing thru Fl atland (p . 69 ) i s thatof a wire bent in the form of a circula r helix

(coiled wire spring) which i s passed thru Flatland so that its axis i s perpendicular to the planeo f Flatland (see figure ) . The cross section o fthe wire made by the plane (as A ) will be a very

74 The Fourth Dimension and the Bible

small ellipse (two-dimensional) . the wire ispassed downward the Flatlanders would see thisellipse travel in the ci rcle BC in one di rection ,while i f the wire is p assed upward 1t would travelin the opposite direction . The

“birth” of thisellipse as the wire entered the plane , i ts

“death”

when it le ft the plane , and its behavior in theinterim— all would be inscrutable myster ies tothe Flatlanders living in that plane .It i s evident that i f the cone to which we t e

ferred on p . 69 i s p assed obliquely thru the plane ,then the sections will no longer be circles butellipses , parabolas , hyperbolas , or stra ight lines ,all depending on the angle which the axis o f thecone makes with the plane . To further illustratethat the kind of section made in a given case willdepend not only on the shape of the solid but alsoon how the

‘

solid enters the plane , let us assumethat the same cube passes thru the plane severaltimes , e ach time with a new orientation with re

spect to the plane .


If a face of the cube is always parallel to theplane . then all the sections made will be equalsquares (Fig. a ) .

I f an edge of the cube (and no face ) i s alwaysparallel to the plane then all the sections made willbe rectangles except the first and last which wouldbe lines , 1.e . , the top and bottom edges (Fig. b ) .

I f no edge or face 1s parallel to the plane , i.e. ,

the cub e i s passed thru obliquely, the first sectionwill b e a point (a corner of the cube ) , then theywould be triangles , then polygons , then aga in triangles , and finally the last one would again bea p oint (Fig. c ) .

Since we live in space o f three dimensions theabove phenomena are fully understood by us , butto Flatlanders viewing the appearance , behavior,and disappearance of these sections in their plane ,i t would all be utterly inexplicable because theyhave knowledge and experience only of twodimensional bodies . Least of all would they sus

pect that sections'

so unlike were produced by the


pass ing of the same higher-dimensional body thrutheir world .

‘ I f then , as it has been suggested,the personality o f a man is the section of his real ,his higher-dimens1onal self , made by the space ofour experience , i t makes it clearer to us why thepersonaliti es o f all the people we know have suchcharacteristic differences . Both as to mind andbody they all s eem to have been cast in differentmoulds . All this suggests that the personality ofa man may depend not only on the character of

his higher-dimensional self but that it is also influenced by the manner in which that higher self wasproj ected into our world .

In the preceding paragraphs of thi s chapter thereader naturally thought of the cone , helix, andcube as substanti al solids , while the sections

(points , circles , triangles , lines , etc . ) of thesesolids made by the planes thru which they werepassing impressed him with the ir changing, fl eeting, and evanescent character . Pass ing now to thenext higher order o f space , analogy suggests theabove relations as one explanation of the transitory character o f our lives ; they may be merelythe sections of our spiritual , our higher-dimen

sional selves made by the three-dimensional spaceo f our experience . Many passages in the B ibleemphasize the ephemeral character o f our earthlyexistence , while others point out the higher , the


by these slices and thus obtains a mental imageo f the interior as well as the exterior o f the apple .Thus , 1f the apple happens to be worm-eaten it i seasy to trace the passage made by the worm , aswell as to locate the worm , by noting the r elativelocations o f the seri es o f worm holes in the slices .This method is of the greates t utili ty to thebiologist in the study of the structure and composition of the embryos or organs of ammate beings .For instance , the embryo of a frog is cut into alarge number o f sections by means of the microtome , an instrument constructed for this purpose ,so thin that they are p racti cally transparent underthe microscope . These slices , which are praetically plane sections of the embryo , are thenmounted on glass pla tes by means o f a perfectlytransparent adhesive and studied under themicroscope . These plates , called slides , whenconsecutively arranged in a series , will vividly protray the most minute deta il o f the structure o f theembryo . By means o f these slides the biologisti s able to mentally reassemble all these sections intheir proper order and thus to mentally reconstruct the embryo .

CON SERVATION ”

OF MATTER AND ENERGY

Again making use o f a cube passing obliquelydownward thru a plane (as in next figure ) we note


that the section of the cube made by the plane i scontlnually changing in magnitude . Starting asa point (when the lowest corner of the cub e enters the plane ) the section changes into an increasing tri angle , then into polygons of more thanthree s ides , then into a decreasing tr1angle whichfinally contracts into a point (as the highest corner of the cube emerges from the plane ) .

The Flatlanders living in that plane would observe that the area of this section is continuallychanging, i.e . , this are a does not remain constant.This suggests the possib ility that the total masso f our material universe (considered as the section o f a higher-dimensional configuration passingthru our space of three dimensions ) may in likemanner be continually changing

, i.e . , the totalmass of our material universe may not be constantand hence the principle of the conservation o fmatter may not hold true . And this may applynot only to our material universe as a whole butalso to all o f its parts from the smallest atomto the largest planets and suns . To illustrate

80 The Fourth Dimension and the Bible

again , suppose that instead of the cube an automobile i s pass ing downward thru the plane . Theplane section of the automobile will be -

of verygrea t complexity , and not only the whole of thesection but also all of i ts many parts would becontinually changing in form /

and area . And notonly that , but during the interval between theinstant when a section of the automobile first appeared in the plane and the instant the last traceof i t vanished the particular sections o f hundredso f the individual parts o f the automobile wouldsuddenly appear (be created), pass thru a continuous series of changes , and then as suddenlyvanish (be annihilated ) . For instance , we canVisualiz e the section of some particular bolt in theautomobile as i t i s p assing thru the plane .May it not be possible that the whole of our

materi al universe as well as all its component partsare similarly changing in form and total mass sothat there may be additions to or subtractionsfrom this total which we as three-dimensionalbeings cannot account for' Astronomers haveobserved the sudden appearance in the heavenso f stars and comets , and also the apparent vanishing of such celesti al bodies , all in a mannersuggesting that they were either entering or leaving our universe . For these and other reasonsthere has been for some time a susp 1c1on 1n the

Sections of B odies in One Space

minds of many scientists that the total mass ofour material universe is not

'

constant. Since everyatom of matter is a reservoir of energy it followsthat i f the principle of the conservation o f matteris not true , then the principle of the conservationof energy also fails . Numerous instances are recorded in the B ible where new matter or newenergy was apparently added to our materialuniverse by supernatural means . The manna fall1ng ln the wilderness , the ever replenished oil inthe widow

’s cruse , and the loaves and fishes whichfed the five thousand , all this food may havebeen the three-dimensional sections of higherdimensional supplies proj ected at those particulartimes thru our space by some higher agency . Alsothe energy which divided the waters of the RedSea , which caused the walls of Jericho to fall ,and by means of which Jesus stilled the storm atsea and walked on its waters , this energy wasin some to us mysterious way added to the alreadyexisting energy in our universe . When Godtranslated Enoch and El1jah bodily into heavenmatter and energy was subtracted from ourmaterial universe .

CHAPTER VIII

THE ILLUSORY AND THE REAL

In the case of the cube passing thru a plane iti s evident that the cube itself i s the re al , the trueindividual , while the sections (squares , triangles ,etc . ) o f it s een by the Fla tlanders l iving in thatplane are but illusory, inadequate images of thecube . However , i t i s only thru thes e countlesssections that the cube i s mani fested to the Flatlanders . But the “b irth” of a section when thecube enters the plane is not the “birth” o f thecube , and the

“death” of a section when the cubeleaves the plane i s not the “death” of the cube ;thi s “birth and this “death are mere illusionsas far a s the cube itself i s concerned . And so ,

because man is a higher-dimensional being hisbirth , that is , his entrance into thi s world, i snot his beginning, nor will his death , that is , hispass ing out of this world, be his end .

The classical allegory o f the chained shadowwatchers given by Plato in the seventh book ofhis “

Repub l ic” vividly portrays the relation between true being and the illus ions of the sense

82

The Illusory and

)

the R eal

world . Plato imagines a group of prisonerschained just ins ide the entrance to a large cavern .

All movement is impossible to them , they cannoteven turn their heads , hence they always lookupon the opposite wall of the cavern . A stronglight streams in thru the mouth of the cavern .

The prisoners never see anything but their ownshadows on the wall together with the shadowso f all persons and obj ects moving b ehind them .

In the course o f time they would identi fy themselves with thei r shadows , they would refer alltheir experiences to these shadows

,they would

practically b ecome the denizens o f a shadowworld .

” All movements observed by them wouldbe the movements o f shadows on the wall , themovements of shapes with outlines but with nosubstantiality. Plato thus conceived of a possible state in which man is reduced by his limitedconsciousness to less than he really is and thuscleared the way for the concept tha t the normalman in an analogous manner i s reduced by hisconsciousness to less than he really is . Man i sthus but the shadow cast on our world by his real ,his higher self dwelling“ in supernal regions . Asthe real world o f the chained prisoners was totheir shadow world , so i s the supernal world tothe world of our experience .Let us suppose the surface of a smooth wet

84 The Fourth Dimension and the B ible

-4

s idewalk to be part o f Fl atland . A man walkingon it would make tracks as shown above . The

Fla tlanders living in the plane of the surface o fthe s idewalk would see only the bottoms of theheels and soles o f his shoes . Fi rst the heel andsole of one foot would appear to them apparentlyout o f nowhere and then vanish just as the heeland sole of the other foot appeared and in turnvanished . This would be repeated again andagainto the great mystification of the Fl atlanders .To them i t would seem as if two two-dimensionalbodies (the bottoms o f the heel and sole o f oneshoe ) were traveling together forsome unknownreason accompanied by their twins (the bottomsof the heel and sole of the other shoe ) . The

Flatlanders could not conceive of each heel andsole being connected , that is , as belonging to thes ame shoe (a three-dimensional body) , and muchless could they conceive o f these shoes as belonging to the same three-dimensional being (theman ) . Suppose , however, that some Fl atlandero f superio r intellect had an inkl ing of the thirddimension such as we have o f a fourth dimension ,and that he suspected that the two pairs o f impres

CHAPTER IX

SPACESASHEAVENSAND HELLS

Perhaps the higher-dimens ional spaces are theheavens to which we ascend at death i f we haveso lived on earth as to deserve such promotion ;and the lower-dimens ional spaces are the hells towhich we will be consigned in case our l ives havefallen b elow a certa in standard . Thus at deaththe soul of a good man - (his higher-dimensionalsel f) shakes off the limitations of understanding,knowledge and movement which our three-dimen

sional space imposed on him , and enters space offour dimensions . Instead o f being restricted tothree degrees o f freedom of motion he will thenbe able to move in four independent directions .Such would be his increased freedom of actionthat , i f he could look back, his former li fe inour space would prob ably appear to him as li fein a pri son cell would now seem to us . Unlimitedv i sta s o f new knowledge would open up to him ,

and his powers o f understanding and possib i litieso f action would have increased to an extent farbeyond the present limits o f human conception .

86

Spaces as H eavens and H ells

Coming from the restrictions imposed by ourspace , the space of four dimensions would indeedseem heaven to him ' On the other hand , the soulo f the wicked man would at death be condemnedto live in Flatland , the space of two dimensions .He could moveonly in two independent direct1ons,all his knowledge and experience of a thi rd dimen

s ion would be blotted out o f his mind , and li fe inFlatland would certa inly be hell to him . I f duringhis existence in Flatland the fallen man did notmend hi s ways but continued to deteriorate incharacter , then at death he would again be demoted , he would be consigned to Lineland, anexperience that would abound with horrors . He

could then move in one direction only and his fundof knowledge and his powers of understandingwould be still further diminished . If after thisawful experience he still persisted in his evil wayshe would at death be consigned to sp ace of z erodimensions , that is , to a point . There no motionat all would be possible , all his knowledge andunderstanding would have vanished , he would bein the deepest pit of hell .

“For a fire is kindled

in mine anger, and shall burn unto the lowesthell .” Deut.But let us turn back to the more pleasant pros

pect, the assumption that our earthly friend hasso lived that at death he is translated into space


o f four dimens ions , that 13 , the first heaven forus . “And I s aw a new heaven and a newearth : for the first heaven and the fi rs t earth

'

were passed away.

”Rev . 2 1 : 1 . H appily during

his li fe in spa ce. o f four dimensions our friendcontinues his s triving after that which is highestand noblest and best, and so when he has finishedhis course there he is promoted to space of fivedimensions , that is , our s econd heaven , a placeo f far surpass ing knowledge and freedom andglory. Continuing in growth of character hewould then b e advanced to spa ce of six dimens ions , that is , our third heaven .

“ I knew a manin Christ above fourteen years ago (whether inthe body I cannot tell ; or whether out of thebody, I cannot tell : God knoweth ) , such an onecaught up into the third heav en.

” II Cor.

Not stopp ing here we can imagine that thenoblest and greatest individuals will, a fter suchsuccessive existences , be translated to still higherspaces (heavens ) . The higher the order o f spacethe greater and more glorious would be the state

o f the blessed who dwe ll there , until the space ofinfinite dimensions is reached where God has hisdwelling place . There are no restrictions

,to

motion and no l imitations of power , knowledgeor understanding. God is omnipotent , omniscient ,omnipresent ; the very personification of j ustice

Spaces as H eavens and H ells 89

and mercy, wisdom and love . God i s absolutelyinfinite , consisting of infinite attributes each express ing eternal and infinite essentiality.

The B ible has made the above concep t of morethan one heaven familiar to us all . “ In myFather’s house are many mansions . John“The heaven , even the heavens , are the Lord

’s .”

Ps. 1 1 5 : 1 6.

“When I consider thy heavens .” Ps.

When he prepared the heavens .” Prov .

“Therefore will I shake the heavens . Isa .For, b ehold , I create new heavens . Isa .

Also Hag. Ezek . Matt .Mark II Cor. II Peter Wehave referred to higher and lower orders ofspaces . That there are also various orders of

heavens may be inferred from scripture passages .Behold , the heaven and the heaven of heavensi s the Lord’s thy God .

”Deut. “

Behold,the heaven and the heaven of heavens cannot conta in thee .” I KingWe find nothing in the hypothesis o f higherspaces that we have been developing which contradicts the reality of s in or the fact o f the divineredemption of a fallen race . Whether or notthe inhabitants of other spaces than ours havepassed or are passing thru the experiences o f anAdam’s fall and a Christ’s atonement we do not

know, but the presumption i s strong that a t least

90 The Fourth D imension and the Bible

some o f the dwellers in higher space (heavens )did disobey ‘

' God and there fore incurred the inevitable punishment which is sure to overtake thesinner , whether he be man or angel .

“Behold, he

putteth no trust in his sa ints : y ea , the heavensare not

‘

clean in his s ight .” Job “For i f

God spared not the angels that sinned but castthem down to hell , and delivered them into chainso f darkness , to be reserved unto judgment .

” IIPe ter And the angels which kept not theirfirst estate , but le ft their own habitation , he hathreserved in everlasting chains under darkness untothe judgment of the great day.

” Jude 6.

“Andhe said unto them , I beheld Satan as lightning fallfrom heaven . Luke

CHAPTER X

CREATION OF OUR MATERIAL UNI

VERSE . EVOLUTION

Is the theory of higher spaces in accord withthe B ible account o f creation' Let us approachthis subj ect by making use o f the concept employed in the last chapter , the assumption thatbodies in any given space are the sections ofhigher-dimensional bodie s made by tha t space .For example , let the plane AB represent a portionof a two-dimensional space (Flatland) .

Assume figure (a ) to be a solid ci rcular cylinder

(three-dimens1onal body) with its axis alwaysperpendicular to the plane and moving towards

9 1


and completely thru the plane and assume therest of the figures as success ive stages of the samecylinder as it i s p ass ing thru the plane . B eforethe cylinder reaches the plane , Flatlanders livingin that plane would know absolutely nothingabout i t, the cylinder would be non-exi stent as fara s they are concerned . The instant the lower baseo f the cylinder coincided with the plane , as in figure (b) , the circle CD , the section of the cylindermade by the plane , would suddenly appear beforethe astonished Flatlanders , coming apparentlyout o f nowhere . To them this phenomenon wouldbe the crea tion of a circle (two-dimensional body )out of nothing . They could no more account for

this miracle than we can account for the creation

of our ma terial (three-dimensional) universe out

of no thing . They could not even imagine an explanation o f the mystery, they could simply statethat where there was nothing before there wasnow a circle ; a real two-dimensional body hadbeen created apparently out o f nothing.

During the time the cylinder 1s passing thrutheir plane the Flatlanders would behold the ci r’

cular s ections made by the plane as a cont1nu1ngenti ty, a circle , (as EF, GH , etc . ) until its upperbase KL, figure (f) , coincided with the plane .Then the instant the cylinder p assed out of the

plane the circle would suddenly vanish , apparently


stand that the worlds were framed by the wordof God , so tha t things which are seen were no t

made of things . which do appear.

” H eb .

Just as Flatlanders (p . 9 5 ) continued to beholdas a continuing entity the circular section of thecylinder made by thei r space (plane ) while thecylinder was pass ing thru the plane , so we todaybehold our material universe as a cont1nu1ng eu

tity, an aggregate of planets , stars , and nebulae,which aggregate we may conceive o f as being thesection of a higher-dimensional configuration ofsurpassing greatness and variety formed by thepassmg o f this transcendent configuration thru ourspace of three dimensions . I f , in our illustration ,instead o f a circular cylinder some irregularshaped solid had been passed thru the plane thenthe section seen by the Fla tlander would haveb een , not a cir cle , but a continually changing irre

gular shaped plane figure (two-dimensional body) .

And so , because every part o f our material universe from the smallest particle of matter onearth to the most dis tant star i s in a state ofcontinual change we may assume that thesechanges are merely three-dimensiona l manifestations o f a transcendental configuration o f infinite richness in its composition and form causedby the pass ing of the latter thru our space ofthree dimensions . According to this View the

Crea tion of Our Ma terial Universe 9 5

events of the six days of creation as recorded inthe B ible , as well as all other events which havesince taken place and are at this moment takingplace , are three-dimensional sections ( images ,manifestations ) of some higher-dimensional ag

gregate (heaven) in which God dwells .In the cas e of the cylinder and the plane we

pointed out on pp . 9 2 and 97that as the cylinderpassed out of the plane the ci rcle which the Flatl anders had observed suddenly vanished beforetheir eyes after i ts coincidence with the upper b aseof the cylinder , and was seen 'by them no more .No trace whatever of the circle remained in thetwo-dimensional world of the Fl atlanders , to themit appeared as the total annihilation of a body

(the circle ) in their space ( plane ) , a phenomenonwhich they could in no way account for . The

Fl atlanders might well record it in their annalsas the ‘ end of the circle . And so to us (threedimensional beings ) the end , the total annihi

lation ,“ of our material three-dimensional uni

verse may be considered as coming to pass whenthe higher-dimens1onal (transcendent) configura

tion to which we have referred passes out of ourthree-dimensional space . Numerous B ible passages indicate unmistakably that not only did ouruniverse have a b eginning (was created) but thatit will also in the fullness o f time have an end


(be annihilated ) Ps. Mat . II Pet.1 3 , Rev . The Bible does not , and

could not, tell us how something can become nothing any more than it can tell us how nothing canbecome something because it was written for human (three-dimensional ) beings wholly incapableo f comprehending such (higher-dimehsional)mysteries .

EVOLUTION

Let us assume a solid circular c’

one (three-dimensional body) passing thru a plane , as AB ,

(two-dimensional space ) the axis of the cone beingalways perpendicular to the plane .B efore the cone reaches any part of the plane ,

as in the position fig . (a ) , the cone is non-existent to the Flatlanders l iving in the plane . The

instant the apex of the cone ente rs the plane (fig.

(b ) , the point C (the section of the cone madeby the plane) will appear to the Flatlanders in


rate in a manner incomprehensible to three-dimensional beings , both are miracles to us . To God

instantaneous creation and evolut1onary processesare equally poss ible , he can and does operate witheither or both according to his infinite wisdom .

To deny the poss ibil ity o f instantaneous creationi s to deny the power of God and to cast overboard the concept o f evolution i s to deny theexi s tence of the numberless examples o f evolutionary processes which are developing today be foreour very eyes or are unfolded to our minds .

CHAPTER XI

RAISING OF THE DEAD . SACRAMENTOF THE LORD ’

S SUPPER

The soul of man , being higher-dimensional ,can leave the human body, which i s three-dimen

sioual , without doing any violence to the body orleaving any trace on it o f a point o f emergence .Investigators have often tried to photograph thedeparting soul o f a dying man , and to discoverany les ions on his body caused thereby, but alwayswithout success . No image of the soul , which ishigher-dimensional , can be secured by material

(three-dimensional) means ; nor can the humanbody imprison the soul any more than (a s wehave pointed out) it would be possible to imprisona four-dimensional being in a three-dimensionalprison cell or to confine a three-dimensional beingwithin two-dimensional (Flatland) barriers .Likewise i t follows that a soul may enter ahuman body without hindrance as far as sp acialconsiderations are concerned , j ust as a three-dimensional being may freely enter a closed com

partment in Flatland . The rais ing of the dead99

1 00 The Fourth D imension and the Bible

1nvolves the re-entrance o f souls into their former corporeal habitations . What the forces arewhich can bring this about is a deep mystery tous . We know, however , that Jesus had the powerto perform this miracle , and that thru him , alsoPaul and Peter rais ed the dead . Jesus restoredto li fe the daughter o f Jairus (Mat. Mark

Luke the son of the widow of Nain

(Luke and Lazarus (John who hadbeen four days in his grave . Paul did the samefor Eutychus (Acts and Peter the samefor Tabitha (Acts Likewise El ij ah (IKing 23 ) and Eli sha (II King 37)ra ised the dead. The dead came out of theirgraves on the first Good Friday, later enteringthe Holy City and appearing unto many. (Mat.

That we have no record of the exper1ences ofLazarus and the others who were ra is ed from thedead , covering the interval between their deathsand their coming to l i fe aga in , has been a keendis appointment to many B ible re aders . In fact,this seeming omiss ion o f a description o f the deta il s o f their existence during this death intervalhas been used by B ible scoflers not only to castdoubt on the truth o f these miracles but also todiscredit all Christian beli e fs . Where was theimmortal part o f Laz arus during the four days

1 0 2 The Fourth Dimension and the B ible

imp ressions which could not be described to usbecause we have no words , no l anguage whichwill convey such a description to human (threedimensional ) beings . What is more , any suchimpressions that Lazarus may have been conscious of before Jesus raised him from the gravevanished the instant he came to l i fe , that is, theinstant he again became a mortal , a three-dimen

sional being . On being again clothed with hismateri al body his spiri t at once became sub j ect tothe restrictions and limitations imposed on beings living in our space of three dimensions .It seems futile therefore to expect that we shall

ever receive a message thru human agencies fromthose who have departed to the great beyond describing their state of existence because such amessage can convey no meaning whatever to us .The B ible contains no promise of any such knowledge but on the contrary asserts that this must remaina sealed book to mortals . This does not ,however, interpose any obstacles whatever to divine revelations to man , either in the past or inthe future , because God is not limited in any manner by the restrictions o f any space .

THE SACRAMENT or THE LORD’s SUPPER

All the different shades o f doctrine held byChris tian believers regarding the sacrament o f

Sacrament of the Lord’s Supper

the Lord’s Supper may be'

roughly divided intothree distinct groups .Group I Doctrines which include transubstan

tiation as a fundamental principle . These assertthat the bread and wine are li terally transformedinto the body and blood of Jesus . According tothis. principle o f transubstanti at ion the believerpartakes , not of the bread , but of the a ctual material human body of Jesus , and in drinking thecup drinks , not wine , but the actual material humanblood of Jesus .Group II. Doctrines which hold that the breadand wine only represent or symbolize the bodyand blood of Jesus . According to these the bel iever partakes only of the material bread andwine , the act being merely a remembrance , a celebration commemorating the last supper of Jesuswith his disciples .Group III. Doctrines which assert that the be

liever in partaking of the bread and wine alsoby the same act partakes not o f the materialbody and blood of Jesus but of the sp iritua l bodyand blood of a resurrected, a glorified Jesus .The first of these three groups of doctrines isnaturally repellant to us becauseof the grossnesso f the act involved , namely, the consumption of

human flesh and blood . It is the human body andblood o f Jesus , into which the bread and wine


has been transformed , of which the beli ever i ssupposed to partake .On the other hand, the second of these three

groups of doctrines does not entirely satis fy thosewho are sp iri tually minded because it minimizesthe sacrament by making it merely a commemo ration ceremony. This viewpoint is difficult to reconcile with the words of Jesus ,

“ this is my bodythis is my blood .

”

D efenders of the doctrine of transubstant1at1oninvari ably attempt to justi fy their belie f in i t byquoting the above words of Jesus . In so doingthey are trying to support their case on the whollyunwarranted assumption that Jesus spoke o f hismateri al , his human body and blood . Jesus knewo f his coming suffering, death and resurrection .

He spoke o f the future beyond Golgotha whenhe sa id “ this do , a s o ft as ye drink it, in remembrance of me .” In this future there

’

would beno more a material , a human body and blood ofJesus , but only his sp iritual , his glorified bodyand blood .

The third o f the above groups of doctrinesthere fore i s sanctioned by the words of Jesus .It also commends itsel f to human reason . It istherefore near at hand to assert that Jesus ispresent in the sacrament o f the Lord’s Suppera s a higher-dimensional b eing. He descends

CHAPTER XII

DESCRIPTION OF HEAVEN . MIRACLES

It i s evident that we as three-dimensional beings cannot have a true conception of what heavenis ; being a higher space it i s beyond our powersof understanding . I f an accurate description ofheaven were in some way available we wouldnot understand an iota o f it , i t would be meaningless to us

“For s ince the beginning of the

world men have not heard , nor p erce ived by theear , neither hath the eye seen , O God, besidesthee , what he hath prepared for him tha t waiteth for him ” Isa . Eye hath not seen,nor ear heard , neither have entered into the hearto f man , the things which God hath prepared forthem that love H im.

” I Cor. Chris t spoketo his Apostles o f the impossib ili ty o f their having any image or notion of the place to whichwhen he disappeared he would go and whence hewould return . Attempts to describe heaven al

ways result in a description in terms of our threedimensional environment. In this connection i t

1 06

D escription of H eav en 1 07

i s significant to note that the B ible usually tellswhat heaven is not rather than what heaven is .The adj ectives are generally negative . We shouldnever lose s ight of the fact that the B ible waswritten for beings l iving in a three-dimensionalspace , beings whose knowledge and experiencesare wholly limited to that space . Dr. P. Anstadt

in his book “Recognition in Heaven gives the

following summary of B ible statements describing heaven

Nega tive Fea tures, or Positive Fea tures,

or

the Things tha t Will the Things tha t WillNo t B e There B e There.

Indestructible . The city of our GodUndefilable . the heavenly Jerusalem .

Unchangeable . B eauti ful waters .No crying . Delicious frui ts .No tears Sure healing for the naNo pain . tions .No sorrow. Populous with happyNo death . people .No burning sun . B eaut i ful :garments .No cold or heat. D evout worshipNo night. Enchanting mus1c .No hunger . A just ruler .No thirst . An eternal kingdom .

No bad men . The grandest capitol .No sin. Many mansions .No curse .

1 0 8 The Fourth Dimension and the B ible

MIRACLES

One reason why some very good people hes itate about accepting the miracles of the B ibleat their face value is that they assume that suchmiracles can only be brought about by violatingor in some way interfering with the laws ofnature , the laws governing our three-dimens1onal

universe . This difficulty vanishes i f instead welook upon the miracles of the B ible as the perfectly logical results of the working out o f lawsconnected with higher spaces , laws which includethe lower , the so-called natural laws o f our material universe , just as our space is containedwithin the higher spaces . We may then considerthe miracles performed by Jesus , the apostles andthe prophets to consis t o f momentary revelationsby God to mortals o f the operations o f some ofthose higher l aws of which we are as

'yet igno

rant. It i s not at all probable that the apostlesand prophets themselves understood these higherlaws , and it was. not necessary that they should,any more than that it i s necess ary for the ope ratorwho sends a message by wireless to understandthe theory o f wireless telegraphy or to have actualknowledge o f the medium thru which it i s transmitted. These higher laws may be the laws o ffour-dimensional or still higher spaces o r they

CHAPTER XIII

CONCLUSION

The new vistas o f thought which have beenopened to us by the higher space hypothes is haverevealed unlimited realms of being and doing,all o f which are interrelated in a manner far surpass ing the powers o f ingenuity of even a superman . In the light o f this vis ion , how small andmean , how low and limited , yes , how contemptible , does not the materialist appea r as he burrows into the dust in his attempts to discover thesource o f the glorious rays o f truth and lightwhich radi ate from God ' As we , in this studyof higher spaces , have caught a momentaryglimpse of the sublimity and vastness , the gloryand grandeur , the symmetry and order , o f whatmust be only a very small part, perhaps only aninfinitesimal part, o f creation ; how preposterousdoes i t not seem , yes, how ridi culous even to suppose that all should have come into being thru asuccession o f fortuitous circumstances , by blindchance , and not thru a supreme intelligence withno limitations o f knowledge or power .

1 10

Conclusion

How piti ful and weak is man with his limitations o f thought and action in the presence ofthese infinite poss ib ili ties ' And yet there are menpresumptuous enough to cla1m that human reasonand understanding can account for it all . No

matter how wise or powerful a man may be heis b ut one of billions of humans , to whom thisour earth has been or is the dwelling place . Andour earth is only one of the smallest o f the planets revolving around our sun , every one of whichmay be inhab ited by beings , no t necessarily withbodies like ours , b ecause they would have to beadapted to their own physical environment, butperhaps with souls like ours . Our sun , however ,i s only one of the smaller of the suns , fixed stars ,apparently without number , which dot our heave ns. In some parts of the sky there appear clusters o f them , as in the Milky Way, so numerousthat they appear only as patches o f light . someof these suns are so far away that it takes centur1es for the light from them to reach us . A fewyears ago a star of very large magnitude suddenlyappeared in our sky . It then grew graduallysmaller until now it is not at all consp icuous . Astronomers told us that this new star re ally represented a conflagration in the heavens , two largesuns had collided , but the collis ion had takenplace over one hundred years before , at the very

1 1 2 The Fourth D imension and the B ible

time when Napoleon was walking the world likea Colossus , and the flash of light caused by theimpact never reached earth until in our day. In

fact , there may be suns so far from us and crea ted so long ago that light from them has notyet reached the earth . Each of these countlesssuns may have many more and larger planetsrevolving about them than is the case with oursun , and each planet may be inhab ited by beingswith souls . All o f this vast infinitude tha t wehave been contemplating exists , however , simplyin our space of three dimensions . There mayexist an infinite number of other three-dimensional spaces , each one comprising a universe asgre at as or gre ate r than our own . But enfoldingthese three-dimensional spaces as in a garmentmay b e a four-dimens ional space , and that maybe only one o f an infinite number of other possible four-dimensional spaces . Again , these fourdimensional spaces may be enfolded by five-dimensional spaces ; and so on thru spaces of six ,

seven,etc . , dimensions up to space of infinite di

mensions, where God himself dwells subj ect to nospace limitations .But now man , at the thought o f these poss i

bilities, stands aghast , stupified ; he exclaims ,“ it

can’t be true , it is all too vast , to o stupendous tolie within the realms of the possible .” Even

INDEX

Abbott, 41Abraham

, 73Adam’s fal l

,89

A lcohol s, 67Analogy, 17, 23 , 76, 85An Episode in Fl atland

, 41

Angels, 45, 77

Animal kingdom, 64

Annihi l ation of the universe,

9 3Anstadt, 107Apostles, 1 06

Apparition, 43 , 58

Archangel s, 45

Astronomers, 80 , 1 1 1

Atoms,66

Atonement, Chri st’s, 89Automobi le

,p l ane section of

,

80

Axis of rotation, 57, 63

Beethoven, 33

Bib le, 17, 44, 4s, 52, 7s, 77, 8 1 ,3 9, 9 1 , 9 5, 100 , 102,

1 07, 1 0 8

Biologist, 77

B irth, 70 , 82

B l ind, 3 3Blood of Jesus

,103

Bodies,cele stial

, 80

B ody of Jesus,10 3

B oundaries,Flatl and, 38

Lineland, 42

of space units, 22B ragdon, 72

Carbon , 67Celesti al bodies

, 80

Chained shadow watchers ofPl ato

, 82

Chemistry, 66Christ

, 45, 52, 88 , 106, 10 9atonement

, 89Ci rcl e, circumference of, 1 2Cone, Flatl and sections of, 69,

73 , 96Concept of space

, rrff.

Configurations,higher-dimen

sional, 79

rotation of symmetrica l, 54

Conservation of energy, 8 1of matter, 8 1Creation

,Ex-Nihilo

, 3 1

instantaneous, 97

of the universe, 9 1

Crucifixion of Christ, 45, 1 0 9

Cruse, widow’s, 8 1

Crystals, 68Cube , 1 9Flatl and sections of a

, 75, 79motion perpendicul ar to a,20 , 48

Curved spaces, 1 2, 14, 1 5Cylinder, Flatl and sect ions of

a, 9 1

Daughter of Jai rus, 100

Dead,raising of the, 99

Deaf, 3 3

Death, 70 , 82Definition of space

, 22

Dextrose, 64

Index

D1mens1ons of space, 1 1 ffgeometry of four

, 29rotation

,reversal , twist

61

theorems in four, 58

time as fourth, 71D istorted

, 2 1 , 63Dreams and dreamland, 72

Earth, first,88

Elements of the tesseract, 2 1Elijah, 44, 45, 8 1 , 100

Elisha,1 00

Embryos,pl ane secti ons of, 78

Empirical Science, 29Energy, conservation of, 8 1Enoch

, 44, 45 , 8 1 .

Eternal, 73

Eutychus, 1 00

Evanescent, 76Evolution

, 96

Existence of higher spaces, 23 ,25

First earth,heaven, 88

Fishes,the lo aves and, 8 1

Five-dimensional space, 16

Fl atland, 37, 48 , 5 5, 693 . 84,

87boundari es, 38houses, 3 9l ife in

, 373sections of a he l ix, 73sections of a cone

, 69, 73 , 96

sections of a cube, 75 , 79sections of a cylinder, 9 1

Fl atl anders, 37E, 5sd, 92

Foot,84

Four-dimensiona l spaces, 16space unit, 20

Four dimensions, geometry of,

l ife in space of, 48

rotation,reversal , tW1st m,

61

theorems in, 58

Fourth dimension, time as the,71

Freedom of motion of points,1 1 ff

Generation of space units, 1 83Geographical latitude and

longitude, 14Geometric units in each space,

1 83

Geometry not the s ame asspace, 28

Geometry of four dimensions,29 , 58

theorems in p lane and sol id,58

Gl ass globe, 49

Grammatical tenses, 73Grave , 1 0 1

Gravitation, 53

Gruber, 3 1

G od, 3 1 , 3 5, 44, 4s, 73 , 77,8 5, 88, 93 , 9s, 98. 1 02. m6,1 0 9

Good Friday,100

Gunne l l , 26

Heaven, 44, 863 , 89description of

,106

first,second

,third, 88

of heavens,89

recognition in, 107Hee l s

,84

Helix, Flatland sections of, 73,

Hel ls, 863

Helmholtz, 10 9Higher-dimensiona l configura

tions 79Higher spaces, 16existence of, 23, 25divided by lower spaces,23

Hinton,Holy City

,1 00

Holy Ghost, 1 09Hyper-spaces, 16

Index

Persona l ity, 76Peter, 1 00

Physics, 30

Plane geometry,theorems in,

58

P lane sections of an automobi l e

,80

of an embryo, 78

Planets, 79, 94Plato

,82

Points,fr eedom of motion of,

1 1 3

moving perpendicul ar tospaces, 473

Po l arization of light, 64

Prophets, 45 , 1 0 8

Pure mathematics, 29

Raising of the dead, 99Real

,the i l lusory and the, 82

Recognition in heaven, 1 07Red Sea

, 8 1

Rending of the veil, 1 0 9

Reve l ations of St. John, 46Reversal in sp ace of four di

,mensions, 61

Right handed, 57Robber

, 49Rotation of symmetrical l ines,

tri angles,

tetrahedrons,543

in space of four dimensions,61

Rubber band or ba l l,63

Sacrament of the Lord’s Supper, 1 02

Satan, 90

Sau l, 9 9

Schubert, 59 , 67

Scientific romances, 41Second heaven, 88Sections of embryos, 78Sense perceptions, 27Shadow watchers of P lato, 82So lar system, 25

Sol es, 84Sol id geometry , theorems in,

58

Son of the Widow of Nain,

1 00

Soul, 99

Space defined, 22

boundaries of, 22Space, zero one two three

four fiv e higher orhyper 1 1 3

curved, 1 z li

l ife in four-dimensional, 48

l imitations, 30

l imits, 2 5

units,1 8tf

Space s as heavens and hel l s,86t

’

r'

existence of higher, 25

higher spaces divided bylower , 2 3

inhabitants of other, 36

Spectrum, 34

Sphere,surface of, 14, 48

Spiritua l, 3 5, 53 , 76. 1 0 3Square, motion perpendicu l ar

to a,1 9 , 47

Stars, 94, 1 1 1

Suns, 79 , 1 1 1Supernatural , 42, 53Swedenborg

, 10 9Symbolic, 17, 20

Symmetrical configurations, 79rotation of

, 54, 56, 60

Symmetry 1n animal and vagetab l e kingdoms, 64

Tabitha,1 00

Tabl e top,14

Tartaric acid,67

Taylor, 71

Te l egraph, 26Te lephone, 26Temple

,rending of the ve i l of,

10 9

Index

Tenses, grammatical, 73Tesseract

, 20 , 2 1

Tetrahedrons, 59 , 60 , 65

The Fairyland of Geometry,

41

Theorems in p l ane,sol id and

four-dimensional geometry, 58

Thi rd heaven, 88Thomas

, 52

Three-dimensional spaces, 14

space units, 1 9Time as the fourth dimension,

71Tire tube, 64Transitory characte r of l ife,

76

Transubstantiation, doctrme

of,1 0 3

Triangles, 56, 57Twist in space of four dimen

sions, 61

Two-dimensional spaces, 1 3unit

,1 9

v 1s1on, 39

Units of space,1 83

boundaries of space, 22Universe

,annihi lation of, 9 3

creation of, 9 1

mass of, 79

Veil of the Templ e, t ending

of,1 0 9

V1s1on, mental , 71 , 73Visualizing

, 28

Wil derness, manna in the, 8 1Widow’s cruse, 8 1Widow of Nain, son of, 100

Wireless tel egraph , 26, 10 8

te lephone, 26

'ero-dimensional space, 1 1

space umt, 1 8

Forgotten Books › en › download › The... · PREFACE That philosophy and the physical...

Documents

Transcript of Forgotten Books › en › download › The... · PREFACE That philosophy and the physical...