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Digitized by tine Internet Arciiive 
 
 in 2010 with funding from 
 
 Boston Library Consortium IVIember Libraries 
 
 http://www.archive.org/details/fourthdimensionbOOgran 
 
THE FOURTH DIMENSION 
 AND THE BIBLE 
 
 BY ^ 
 
 WILLIAM ANTHONY GRANVILLE, Ph.D., 
 LL.D> 
 
 PRESIDENT OF GETTYSBURG COLLEGE. 
 
 Formerly Instructor of Mathematics in Yale University; Author of 
 Granville's Differential and Integral Calculus, Granville's Plane 
 Trigonometry, Granville's Spherical Trigonometry, and Gran- 
 ville's Logarithmic Tables; loint Author of Smith and Gran- 
 ville's Elements of Analysis; Inventor of Polar Co- 
 ordinate Plotting Paper, Yale System of Mathemat- 
 ical Note Books, and Granville' s Transparent 
 Combined Ruler and Protractor. 
 
 BOSTON 
 RICHARD G. BADGER 
 
 THE GORHAM PRESS 
 
Copyright, 1922, by William A. Granville 
 
 All Rights Reserved 
 
 .^7 
 
 Made in the United States of America 
 
 The Gorham Press, Boston, U. S. A. 
 
 J30STON mUM^E MBRARl 
 
 r1AY 11 1972 
 477046 
 
PREFACE 
 
 That philosophy and the physical sciences, 
 when called to the defence of Christianity, have 
 so often proven to be broken reeds has been a 
 keen disappointment to many sincere seekers after 
 the truth. A little reflection, however, will make 
 clear that in the very nature of things such fail- 
 ures must inevitably result, because structures 
 resting on shifting sand can never serve as strong 
 buttresses. No two systems of philosophy agree 
 and every philosopher is continually tinkering 
 with his own system. There is no final authority 
 in philosophy to which an appeal can be made. 
 Such being the case it is obviously not to be ex- 
 pected that philosophy can furnish us with proofs 
 of the truths of Christianity whose validity will be 
 generally recognized. And the sciences have also 
 fallen short for similar reasons. The foundations 
 of our physical sciences are far from stable. To 
 doubt the principles of the conservation of energy, 
 the conservation of matter, or the Indivisibility of 
 the atom, would have been considered as rank 
 scientific heresy a few years ago; today all physi- 
 cists are familiar with phenomena which contradict 
 
iv Preface 
 
 one or more of them. The Einstein Theory of 
 Relativity is now opening up new fields for scien- 
 tific investigation which promises results that will 
 practically revolutionize many current fundamen- 
 tal scientific concepts. In fact, the foundation 
 principles on which the physical sciences rest have 
 been changing so rapidly of late that practically 
 all the science text-books now used in our schools 
 are more or less obsolete. The physical sciences 
 as well as philosophy are in a constant state of 
 flux and flow as well as of growth, and this must 
 necessarily continue to be the situation until the 
 ultimate truth is reached, If it is ever attained, in 
 these domains of knowledge. As all will readily 
 agree that the limits of these regions have not yet 
 been reached and, what is more, that these limits 
 are not even measurably in sight, it is evident 
 that we cannot reasonably expect that much con- 
 structive light will be thrown on Christianity now 
 or in the near future by either philosophy or the 
 physical sciences. 
 
 To satisfy completely human reason any confir- 
 mation of our Christian beliefs must come from a 
 source whose authority no one can question. Is 
 there then any department of human knowledge 
 whose foundations rest, not on shifting sand, but 
 on the bed rock of absolute truth? Pure mathe- 
 matics satisfies this condition ; it is the only exact 
 science that God has revealed to man and the 
 
Preface v 
 
 truths which it contains are the only truths that 
 can be absolutely established thru pure reason. 
 Because pure mathematics reveals absolute truth 
 it is part of God Himself, for God is the essence 
 of all truth. It was Emerson who said, "nature 
 geometrizes," and as nature is the handiwork of 
 God we may say that God geometrizes. In view 
 of this it is rather surprising that so few attempts 
 should have been made in the past to throw light 
 from mathematics on the fundamentals of Chris- 
 tianity. This is practically a virgin field for 
 theological research. 
 
 Pure mathematics is the vanguard of all the 
 sciences, it has marched centuries in advance of 
 them all. For instance, Calculus was discovered 
 nearly two hundred years before the science of 
 Chemistry was far enough advanced to allow the 
 use of Calculus as one of its instruments of re- 
 search. Practically the same thing is true of all the 
 other physical sciences, including engineering. It 
 may be, therefore, that the science of theology is 
 not yet far enough advanced to admit of the ready 
 use of mathematics as an instrument of research. 
 Today mathematicians are forging ahead into 
 realms of abstract thought and pure logic as never 
 before; judging from the past, some of the truths 
 which they are revealing now may not be put to 
 what we are pleased to call practical use for 
 centuries to come. 
 
vi Preface 
 
 That mathematics will ultimately prove to be 
 a valuable aid in solving many of the perplexing 
 problems connected with our Christian religion is 
 the firm belief of many sound thinkers. Professor 
 C. J. Keyser has pointed out many striking cor- 
 respondences between the transcendental proper- 
 ties of some elementary mathematical concepts 
 and those transcendental attributes of God and 
 His creation which are intimately associated with 
 our religious behefs. Concerning the possibility 
 and the probability of future investigations along 
 this and other lines shedding new light from 
 mathematical sources on current religious thought 
 he has the following to say in Hibhert Journal, 
 Vol. VII : "Over and above the humbler role of 
 mathematics as a metrical and computatory art, 
 over and above her unrivalled value as a standard 
 of exactitude and as an instrument in every field 
 of experimental and observational research, even 
 beyond her justly famed disciplinary and emanci- 
 pating power, releasing the faculties from the 
 fickle dominion of sense by winning their allegiance 
 to the things of the spirit, inuring them to the 
 austerities of reason, the stern demands of rigor- 
 ous thought, giving the mental enlargement, the 
 peaceful perspective, the poise and the elevation 
 that come at length from continued contemplation 
 of the universe under the aspects of the infinite 
 and the eternal — my conviction that above and 
 
Preface vii 
 
 beyond these services, which by common consent 
 of the competent are peculiarly her own, mathe- 
 matics will yet further demonstrate her human 
 significance by the shedding of light more and 
 more copious as the years go by on ultimate prob- 
 lems of philosophy and theology, is not a passing 
 fancy, or a momentary whim." 
 
 The only motive of the author in writing this 
 book is his sincere desire to throw some light, 
 even though very dim, on some of the questions 
 connected with our Christian beliefs. This is not, 
 however, an endeavor to construct a system of 
 Christian theology on mathematical foundations. 
 The author's chief aim will be to point out the 
 remarkable agreement which exists between nu- 
 merous Bible passages and some of the concepts 
 which follow quite naturally from the mathemat- 
 ical hypothesis of higher spaces. In making this 
 attempt the author is well aware that he is on a 
 "no man's land" exposed to fire from the mathe- 
 matical trenches on the one side and the theological 
 trenches on the other. But "fools step in where 
 angels fear to tread" and sometimes they survive 
 to tell the tale. 
 
 The scripture passages quoted in this text are 
 not employed for exegetical purposes but to show 
 that the conclusions arrived at in no way con- 
 tradict the divine word. 
 
 While there is quite an extensive bibliography 
 
vill Preface 
 
 dealing with the subject of higher spaces, the liter- 
 ature bearing on their relation to Bible state- 
 ments is practically nil. After a thoro search only 
 scattered references, each embracing only a few 
 phrases or sentences, have been found which 
 might be construed as supporting some of the 
 conclusions arrived at in this text. The author 
 is therefore taking a voyage on practically un- 
 charted seas. No doubt it may seem presump- 
 tuous on the part of the author, who aspires to 
 be a mathematician and is no theologian at all, 
 to poach thus on theological preserves; but the 
 case is perhaps no worse than if a theologian who 
 is not also a mathematician should undertake the 
 task. 
 
 A sincere effort has been made to present the 
 matter in popular form. To understand the text it 
 is not necessary for the reader to be either a pro- 
 fessional mathematician or a profound theologian. 
 No extended excursion into the theory of higher 
 spaces will be attempted; the author has limited 
 himself to those fundamental concepts only which 
 are necessary for the illumination of the applica- 
 tions to be made. No proofs will be given of the 
 mathematical propositions hereinafter stated; but 
 the reader can rely on them absolutely, no matter 
 how startling the conclusions may appear. They 
 have all been proven by mathematicians of the 
 highest repute and the validity of these proofs 
 
Preface ix 
 
 are not more in dispute than is the proposition 
 that 2X2 = 4. 
 
 The use of the term "Fourth Dimension" as 
 part of the title of this book does not mean that 
 we shall confine our discussion to that space ex- 
 clusively, important as it is ; it is so used because 
 in the populai^ mind this term represents the 
 mathematical concept of higher spaces in general. 
 
 William Anthony Granville. 
 
 Gettysburg College, 
 
 Gettysburg, Pa. 
 
CONTENTS 
 
 CHAPTER PAGE 
 
 I The Concept of Space ii 
 
 II Geometric Units in Each Space i8 
 
 III On the Existence of Higher Spaces 24 
 
 IV About the Supposed Inhabitants of Other Spaces 
 
 Than Our Own. Flatland. Lineland . . 36 
 
 V Motion of a Point Perpendicular to Each Space 47 
 
 VI Rotation of Symmetrical Configurations ... 54 
 
 VII Slctions of Bodies in One Space Made by Spaces 
 OF a Lower Order. Conservation of Matter 
 
 and Energy 69 
 
 VIII The Illusory and the Real 82 
 
 IX Spaces as Heavens and Hells 86 
 
 X Creation of Our Material Universe. Evolution 91 
 
 XI Raising of the Dead. Sacrament of the Lord's 
 
 Supper 99 
 
 XII Description of Heaven. Miracles 106 
 
 XIII Conclusion no 
 
THE FOURTH DIMENSION 
 AND THE BIBLE 
 
THE FOURTH DIMENSION 
 AND THE BIBLE 
 
 CHAPTER I 
 
 THE CONCEPT OF SPACE 
 (a.) Zero-Dimensional Spaces. A* 
 
 A point, as A, may be conceived of as a space 
 of no dimensions. Every point in that space 
 coincides with A and therefore is at no distance 
 from it. A point cannot move about in that 
 space, hence its degree of freedom of motion is 
 zero. Since we know that there is an infinite 
 number of points there is also an infinite number 
 of such zero-dimensional spaces. 
 
 (b) One-Dimensional Spaces. 
 
 Let O be a fixed point on a straight line. To 
 determine the position of any point on that line, 
 as A, all we need to know is its distance to the 
 
12 The Fourth Dimension and the Bible 
 
 right or left of O expressed in terms of some unit 
 of length, a positive number to indicate that it 
 is so far, say, to the right of O, or a negative 
 number when it is to the left of O. A single num- 
 ber will therefore suffice to determine the posi- 
 tion of any point on the straight line; hence the 
 straight line is said to be one-dimensional. The 
 straight line may also be conceived of as a space 
 of one dimension; any point in it has only one 
 degree of freedom of motion, it can move only 
 back and forth in the line. 
 
 The circumference of a circle may be consid- 
 ered as a curved one-dimensional space, as well 
 as every other curved line, whether it returns into 
 itself or not, because (as in the case of the 
 straight line) the position of any point, as A, may 
 be determined by a single number representing 
 its distance measured along the curve from a fixed 
 point (as O •) on that curve. Obviously there is 
 an infinite number of one-dimensional spaces be- 
 cause the number of straight and curved lines is 
 infinite. 
 
The Concept of Space 
 
 13 
 
 (c) Two-Dimensional Spaces. 
 
 y. 
 
 B 
 
 
 
 Let us now consider the plane of the two mutu- 
 ally perpendicular lines OX and OY intersecting 
 at O. Any point in this plane, as A, may be 
 determined by its perpendicular distances BA and 
 CA from OY and OX. Calling distances from 
 OY and OX measured to the right and upward 
 as positive, and distances to the left and down- 
 ward as negative, the position of any point in the 
 plane will be determined by the two numbers rep- 
 resenting these distances; hence the plane is said 
 to be two-dimensional. This plane may also be 
 conceived of as a space of two dimensions; any 
 point in it has two degrees of freedom of motion, 
 it can move to the right or left, and up or down. 
 A point in the plane can move from one position 
 to any other in that plane by moving in a path 
 which is always parallel to one of two fixed 
 mutually perpendicular lines in that plane. Take, 
 for instance, the top of a rectangular table. Any 
 
14 The Fourth Dimension and the Bible 
 
 article on it, as an Ink stand, may be moved from 
 one spot to any other along a path that will always 
 be parallel to one of the edges of the table top. 
 
 The surface of a sphere may be conceived of as 
 a curved two-dimensional space. The position of 
 any point on the surface of the earth Is deter- 
 mined when we know the two numbers indicating 
 its geographical latitude and longitude. By mov- 
 ing along parallels of latitude and lines of 
 longitude we may travel from any one place to 
 any other. Not only the surfaces of all spheres 
 but also an unlimited number of other curved 
 surfaces may be conceived of as curved two- 
 dimensional spaces. Evidently there Is an Infinite 
 number of two-dimensional spaces. 
 
 (d) Three-Dimensional Spaces. 
 
 f^^f 
 
The Concept of Space 15 
 
 The space in which we move and have our be- 
 ing is three-dimensional ; we say that bodies in it 
 have length, breadth and thickness. Consider 
 three mutually perpendicular planes meeting at O. 
 The position of any point in our space, as A, is 
 then determined by the three numbers represent- 
 ing its perpendicular distances BA, CA, DA, with 
 the proper signs, from the three planes. A point 
 in our space has three degrees of freedom, it 
 can move to the right or left, forward or back- 
 ward, and up or down. For instance, an article 
 in a rectangular shaped room may be moved from 
 one place to any other by moving it parallel to 
 the directions of the length, breadth, and height 
 of the room. The reader can reach no place 
 which may not be reached by going north or 
 south, east or west, and upward or downward. 
 
 The space we live in is the only three-dimen- 
 sional space of which we have any knowledge 
 gained by experience. We are not warranted, 
 however, in making the statement that no other 
 three-dimensional spaces exist; such a proposition 
 cannot be proven. Curved lines have given us 
 the conception of curved one-dimensional spaces, 
 and curved surfaces the conception of curved two- 
 dimensional spaces. We may also conceive of 
 the existence of curved three-dimensional spaces 
 although in this case we have no corresponding 
 
1 6 The Fourth Dimension and the Bible 
 
 visual Images on which to lean; we cannot con- 
 struct in our minds the picture of a curved three- 
 dimensional space. From analogy it is evident 
 that there may be an infinite number of three- 
 dimensional spaces, both curved and not curved. 
 
 (e) Spaces of Order Higher than the Third, 
 also called Hyper-Spaces. 
 
 In like manner we define a four-dimensional 
 space to be one in which it requires four numbers 
 to determine the position of any point. A point 
 in such a space will have four degrees of free- 
 dom. 
 
 A five-dimensional space would be one in which 
 it requires five numbers to determine the position 
 of a point; a point in it would have five degrees 
 of freedom. Similarly we define spaces of 6, 7, 
 8, . . . n, dimensions. 
 
 It is impossible for us to form a mental picture 
 of, say, the fourth dimension, the idea is wholly 
 intangible. Nevertheless, it is not meaningless 
 or an absurdity, but a useful mathematical con- 
 cept which has led to the development of a geom- 
 etry of four dimensions Involving no contradic- 
 tions. We reach this new concept of higher 
 spaces not at a single leap but slowly and gradu- 
 ally by climbing a ladder whose lowest rungs are 
 the lower dimensional spaces with which we are 
 
The Concept of Space 17 
 
 in a certain sense familiar. Guided by analogy 
 with dimensions of a lower order we shall thus 
 try to arrive step by step to at least a partial and 
 symbolic idea of the meaning of higher dimen- 
 sions. Of these we will be chiefly concerned with 
 the fourth dimension. Some claim that the Bible 
 recognizes space of four dimensions in Job 
 11:8,9 ^"d Eph. 3:18 where length, breadth, 
 depth and height, i. e., four dimensions, are men- 
 tioned. 
 
CHAPTER II 
 
 GEOMETRIC UNITS IN EACH SPACE 
 
 That there exist intimate relations closely ap- 
 proaching a sort of interdependence between the 
 spaces of various orders is forcibly suggested by 
 the following discussion showing how the unit in 
 each space may be generated by the unit of the 
 space of the next lower order moving in a new 
 direction not contained within itself. 
 
 Starting with zero-dimensional space (a point) 
 we may consider the point itself as the unit. 
 
 •A POINT 
 
 Zero-dimensional 
 Space Unit 
 
 If the point A moves the distance of, say, one 
 inch to the right it will generate the unit line AB. 
 
 LINE 
 
 , , Bounded by the points A and B 
 
 One-dimensional Space (zero-dimensional units) 
 
 Unit 
 
 If the line AB moves upward one Inch In a 
 i8 
 
Geometric Units in Each Space 
 
 19 
 
 direction perpendicular to itself it will generate 
 the unit square ABCD. 
 
 SQUARE 
 
 Bounded by the four unit lines 
 AB, BC, CD, DA. 
 
 (one-dimensional units) 
 
 Two-dimensional Space Unit 
 
 If the square ABCD moves backward one inch 
 in a direction perpendicular to itself it will gener- 
 ate the unit cube ABCDEFGH. 
 
 CUBE 
 
 Bounded by the six unit 
 squares ABCD, EFGH, 
 ABFE, DCGH, AEHD, 
 BFGC • (two-dimensional 
 units). 
 
 Three-dimensional Space Unit 
 
 If the cube ABCDEFGH moves away one inch 
 in a direction perpendicular to itself it will gener- 
 ate the corresponding unit in four-dimensional 
 
20 The Fourth Dimension and the Bible 
 
 space. This unit has been given the name tesser- 
 act. Here we meet our first serious difficulty. 
 We are familiar with the idea of motion perpen- 
 dicular to a line, and of motion perpendicular to 
 a square (or, what amounts to the same thing, of 
 motion perpendicular to two mutually perpendicu- 
 lar lines lying in the plane of the square), but we 
 cannot conceive of a motion perpendicular to a 
 cube (i. e., a single motion perpendicular to all 
 the faces of the cube, or, as it may also be stated, 
 perpendicular to any three adjacent edges of the 
 cube). Not one of our senses calls for a fourth 
 direction perpendicular to the other three, expe- 
 rience leaves us satisfied with three dimensions. 
 We cannot visualize the tesseract but it is possible 
 for us to construct a figure in our space which will 
 symbolize the tesseract, that is, a figure having 
 many of the properties of the tesseract. Such a 
 figure is the following which we may conceive of 
 as having been generated by the original cube 
 ABCDEFGH expanding at a uniform rate into 
 the cube A'B'CD'ET'G'H^ the directions AA', 
 BB', CC, DD^ EE^ FF^ GG^ HH', being sup- 
 posed perpendicular to our space and equal in 
 length to an edge of 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 
 
Geometric Units in Each Space 21 
 
 TESSERACT 
 
 (Symbolical) 
 
 D 
 
 /i' 
 
 Bounded by the eight unit 
 cubes ABCDEFGH, A'B' 
 C'D'E'F'G'H', A'B'BAE' 
 F'FE, DCC'D'HGG'H', 
 
 BB'C'CFF'G'G, A'ADD' 
 E'EHH' A'B'C'D'ABCD, 
 EFGHE'F'G'H' (three-di- 
 mensional units). 
 
 Four-dimensional Space Unit 
 
 cubes. The faces (squares) generated, ABB'A', 
 BFF'B',DCCD', etc., and the cubes generated, 
 A'B'BAE'FTE, BBT'CFF^G^G, etc., necessarily 
 appear distorted in the above figure. The num- 
 ber of the various elements of the tesseract are 
 indicated in the following table which may be 
 easily verified by the reader. 
 
 Number in Number Number in Number in 
 Initial Cube. Generated. Final Cube. Tesseract. 
 
 Corners 8 + o + 8 = i6 
 
 Edges 13 + 8 + 12 = 32 
 
 Faces 6 + i3 + 6 = 24 
 
 Cubes I + 6 + I = 8 
 
 The statements made opposite the figures on 
 pp. 18, 19, 20 indicate that each space unit is 
 bounded by space units of the next lower order. 
 
22 The Fourth Dimension and the Bible 
 
 The following table indicates the number of each 
 of the various elements that are associated with 
 the boundaries of the various space units. 
 
 Points. Lines. Squares. Cubes. 
 
 Zero-dimensional unit (point) . , i o o o 
 
 One-dimensional unit (line) .... 3 1 o o 
 
 Two-dimensional unit (square) .4 4 i o 
 
 Three-dimensional unit (cube) . 8 12 6 i 
 
 Four-dimensional unit (tesseract) 16 32 24 8 
 
 Since there is an infinite number of points in a 
 line, an infinite number of lines in a square, and 
 an infinite number of squares in a cube, it is near 
 at hand to assume that a space of any order con- 
 tains an infinite number of lower dimensional 
 spaces. The following definition of space has 
 therefore been suggested: 
 
 Space is that which separates two portions of 
 the next higher space from each other. 
 
 Let us illustrate what is meant. 
 
 The point A (zero-dimensional 
 space) divides the line (one- 
 dimensional space) into two parts. 
 
 The line AB (one-dimensional 
 space) divides the plane CE (two- 
 R dimensional space) into two parts. 
 
 D 
 
Geometric Units in Each Space 23 
 
 d 
 
 The plane AB (two-dimen- 
 sional space) divides the solid 
 CD (three-dimensional space) 
 into two parts. 
 
 Analogy now steps in and suggests that we carry 
 this one step further by making the statement that 
 our three-dimensional space probably divides 
 four-dimensional space into two parts. And so on 
 for the other higher-dimensional spaces. 
 
CHAPTER III 
 
 ON THE EXISTENCE OF HIGHER 
 SPACES 
 
 It is to be hoped that by this time there has 
 dawned in the mind of the reader at least a faint 
 idea of the concept of higher spaces, an inkling 
 of what is meant by, say, the fourth dimension. 
 Additional discussions from other viewpoints 
 tending to show the probability of the existence 
 of space of four dimensions will be given in later 
 chapters of this book. 
 
 We experience no difficulty in forming a geo- 
 metric conception of zero-dimensional space 
 (point), of one-dimensional space (line), of two- 
 dimensional space (plane), and of three-dimen- 
 sional space (our universe) ; but when at first we 
 try to give a geometric interpretation to space of 
 four, five, six and higher dimensions, we are at 
 sea. We find nothing tangible on which to lean 
 in our fund of experiences accumulated in dealing 
 with the material things in our world, things of 
 which we have become conscious thru our 
 senses. We are therefore tempted to at once 
 
 24 
 
On the Existence of Higher Spaces 25 
 
 arrive at the conclusion that no spaces of four, 
 five, six, etc., dimensions exist. Such a conclusion, 
 however, would be premature because, as has al- 
 ready been stated, the proposition that higher 
 spaces do not exist cannot be proven. 
 
 Firstly, to dismiss the concept of the existence 
 of higher-dimensional spaces with the statement 
 that such a thing is incomprehensible is begging 
 the question, unless we are also willing to abandon 
 the idea of the existence of our own space of three 
 dimensions, because Its properties are also In the 
 main unfathomable mysteries to us. For in- 
 stance, "It may properly be claimed that a three- 
 dimensional solid of infinite length, infinite 
 breadth, and Infinite thickness would embrace in- 
 finite space; but is it possible to picture or com- 
 prehend what infinite space Is? Can a finite mind 
 picture a space with no beginning and no ending; 
 limitless space In which our vast solar system is 
 a mere dot, In which the known stellar universe 
 Is probably also comparatively a mere dot, al- 
 tho it is actually so vast in extent that the 
 light from some of Its component stars which 
 started towards us generations ago or centuries 
 ago is only now reaching us? If our space is 
 limitless, the idea is Incomprehensible, and if it 
 is limited, its limits are Incomprehensible. Our 
 space is limited or it is limitless; in either case 
 
26 The Fourth Dimension and the Bible 
 
 the idea is incomprehensible. Thus the mere 
 statement that a concept is incomprehensible does 
 not prove its non-existence." (Gunnell.) Obvi- 
 ously, it would be absurd for us to doubt the ex- 
 istence of our own space simply because many of 
 its properties are incomprehensible to us. The 
 human mind cannot fully encompass the idea of 
 infinity and yet this is one of the most useful 
 mathematical concepts employed daily without 
 hesitation in the teaching of elementary as well 
 as advanced mathematics. 
 
 Secondly, only an ignoramus would claim that 
 nothing exists which does not lie within his realm 
 ■of understanding, knowledge and experience. 
 One hundred years ago the results now achieved 
 thru the telephone, the phonograph, or the 
 wireless telegraph and telephone would have been 
 inconceivable, would have appeared as miracles 
 even to the wisest men of that day. 
 
 Thirdly, probably few readers of this book 
 would hesitate to make the assertion that one- 
 dimensional spaces (lines) and two-dimensional 
 spaces (planes) do actually exist. But as a mat- 
 ter of fact, the line as a one-dimensional space 
 and the plane as a two-dimensional space has no 
 more of a real existence to us than has four- 
 dimensional space. No one has ever seen or 
 heard or felt or tasted or smelled either a line 
 
On the Existence of Higher Spaces 27 
 
 or a plane. All our sense perceptions and the ex- 
 periences due to them are three-dimensional be- 
 cause all material things are three-dimensional. 
 Our senses can perceive only material things and 
 those only thru our sensory nerves which are 
 also material and therefore three-dimensional. 
 Even the shadow on a wall cannot truly be said 
 to be two-dimensional because what we really ob- 
 serve is that fewer rays of light enter our eyes 
 from the region covered by the shadow, while 
 from the region surrounding the shadow many 
 rays of light do enter our eyes. And these rays 
 of light are reflected from material particles 
 (pigment) on the surface of the wall, that is, 
 from particles of matter (three-dimensional). 
 Our eyes communicate to our brains only three- 
 dimensional knowledge; ''for the comprehension 
 of anything which has only one or two dimensions 
 a purely intellectual act of abstraction must he 
 added to that of perception." Lines drawn on 
 paper, strings, rods and other objects which sug- 
 gest to us the geometric idea of a line are really 
 three-dimensional forms of matter in which two 
 dimensions are very small as compared with the 
 third, its length. Similarly the plane surfaces of 
 our experience (sheets of paper, blackboard, 
 etc.) are all three-dimensional forms in which the 
 thickness is very small a§ compared with the 
 
28 The Fourth Dimension and the Bible 
 
 length and breadth; but these helps have been 
 found useful to us in visualizing the geometric 
 idea of a plane. Thus lines drawn on paper or 
 on a blackboard have been found very helpful in 
 teaching plane geometry, but they are not a part 
 of plane geometry itself. If all such material aids 
 were discarded, or should cease to exist, plane 
 geometry as we know it would still remain, and 
 every theorem proven in the plane geometry text- 
 books used in our schools today would still hold 
 true. Four-dimensional geometry exists in the 
 same sense that plane geometry exists, both have 
 a real existence. The difficulty we encounter in 
 our attempts to form mental images of geometric 
 figures of four dimensions consists in the fact 
 that we have no material aids to our imagination, 
 while figures in plane and solid geometry are 
 readily conceived of as corresponding to many 
 material objects in the world with which we are 
 familiar. 
 
 Geometry itself, therefore, throws no light on 
 the real nature of any space, not even on the 
 nature of the three-dimensional space in which 
 we live. In fact, it has not even been proven 
 that the solid geometry taught in our schools 
 and the space in which we live are in exact cor- 
 respondence; all we can say is that as far as our 
 observations have extended, the one does appear 
 
On the Existence of Higher Spaces 29 
 
 to fit the other. The study of space is an 
 empirical science, its conclusions must be ar- 
 rived at thru observation and experiments, while 
 geometry is a science built up by pure reason, it 
 is a branch of pure mathematics. To the mathe- 
 matician the geometry of four dimensions, as well 
 as the geometry of two dimensions, has just as 
 real an existence as the geometry of three dimen- 
 sions, the geometry relating to the space of our 
 experience. Many extended treatises have been 
 written on the geometry of higher spaces by 
 mathematicians of the highest repute and the 
 validity of the results obtained by them are not 
 questioned. There is no doubt about the existence 
 of geometry of four dimensions, but it has not 
 been proven that there exists a corresponding 
 space of four dimensions. Therefore if such a 
 thing as four-dimensional space does exist, we 
 already have at hand a geometry to fit it. It is 
 important that the reader should clearly perceive 
 that the concept of space is one thing, while the 
 geometry of space is quite another. 
 
 The geometry of four dimensions having then 
 a real existence in the same sense that geometry 
 of one, two, or three dimensions exists, the ques- 
 tion naturally arises, does there exist a four- 
 dimensional space which fits this geometry in the 
 same sense that the three-dimensional space of 
 
30 The Fourth Dimension and the Bible 
 
 our experience fits our geometry of three dimen- 
 sions? This question is one which we cannot 
 escape, it will continue to haunt the thoughtful 
 student, it is the most interesting question con- 
 nected with the subject. "The possibility that 
 we are a part of a four-dimensional space with 
 physical limitations which confine us to a three- 
 dimensional space, and with limitations of our 
 senses which prevent us from perceiving anything 
 outside of this space — this possibility excites the 
 interest of all who are inclined to abstract specu- 
 lation. Attempts may be made to discover 
 physical proofs of such a space, to build up 
 theories on its basis that will explain discoveries 
 of modern physics as yet but little understood, or 
 by it to account for various mysterious phenom- 
 ena. Most of us are satisfied that no real proofs 
 of the existence of space of four dimensions will 
 be found along these lines. Even a workable 
 hypothesis based on the (assumed) existence of 
 four-dimensional space, tho it might serve 
 temporarily better than any other hypothesis, 
 would hardly justify a belief in this existence. 
 But we do say that the existence of space of four 
 dimensions can never be disproved by showing 
 that it is absurd or inconsistent; for such is not 
 the case. Nor, on the other hand, will the most 
 elaborate development of analogies of different 
 
On the Existence of Higher Spaces 31 
 
 kinds ever prove that It does exist." (Manning.) 
 Kant not only recognized the possibility of the 
 existence of spaces of more than three dimensions, 
 but he inferred their very probable real existence. 
 "If It Is possible," he says, "that there are de- 
 velopments of other dimensions of space, It Is 
 also very probable that God has somewhere pro- 
 duced them. For his works have all the grandeur 
 and variety that can possibly be comprised." Or, 
 as a mathematician might put It, why should God 
 have created geometry of, say, four dimensions, 
 If there exists nowhere a space that will fit it? 
 God does nothing that Is futile or unnecessary. 
 
 The following quotation from "Creation Ex 
 NIhilo" (Gruber) will aid the reader in maintain- 
 ing the proper attitude of mind towards what has 
 gone before as well as prepare him for what 
 shall follow. 
 
 "Who can limit the possible existence, within 
 and beyond the physical universe, even of other 
 beings altogether inconceivable by us? Such 
 transcendent entities, as also God and the human 
 soul, because not governed by physical laws, could 
 In no way affect our physical sensorlum. Not be- 
 ing limited by time and (our) space relations, 
 they might co-exist with, and around, us and 
 thruout, as well as beyond, the physical uni- 
 verse — and this without in the least affecting 
 
32 The Fourth Dimension and the Bible 
 
 human experience and consciousness. ... As 
 experience, thru its proper avenues of ap- 
 proach, and consciousness, have their necessary 
 limitations, as sources of knowledge, in their 
 limited points of contact with self and nature, so 
 their bodyings forth in the pronouncements of 
 physical science do not exhaust reality. Hence, 
 the great whole of reality can never come within 
 their compass. And this is even true of physical 
 reality. But, as to the transcendent hyperphysical 
 or spiritual realities, consciousness and experience, 
 unaided — and therefore physical science — would 
 forever have to remain without a point of contact, 
 and therefore in total ignorance. No one should, 
 therefore, attempt either to set a limit to existence 
 by his limited experience or make his finite reason 
 the measure of the immeasurably complex uni- 
 verse of the infinite and eternal God." 
 
 "A person born without the sense of sight can- 
 not see light and color. He cannot even form, 
 any real conception of them. To him all is dark- 
 ness. And, as far as he could by his own powers 
 discover, both light and color would have no ex- 
 istence or would belong to the same category as 
 darkness. The man born without the sense of 
 hearing can neither hear sound nor can he even 
 form any correct conception of sound and music. 
 And thus, to a (congenital) Wind and deaf indi- 
 
On the Existence of Higher Spaces 33 
 
 vidual, even this very tangible physical world is 
 an entity altogether different from the reality. 
 Many of its marvelous phenomena of beauty are 
 to him totally non-existent. And if, perchance, 
 some explanation or revelation to him of these 
 things were attempted by a seeing and hearing 
 personality, these phenomena would yet in a sense 
 be utterly inconceivable by him. He can only 
 by touching, etc., acquire some indefinite idea of 
 grosser forms and movements. But he could not 
 perceive even any efect of those subtle marvelous 
 vibrations that produce light and color and 
 music." 
 
 "Such an individual lives in a world of mar- 
 velous beauty, but he beholds it not, nor can he 
 even form any proper conception of it. But, 
 surely, it would be an almost unpardonable pre- 
 sumption on his part to deny the existence of the 
 glorious rainbow in the heavens and to argue with 
 an entranced auditor against the existence of the 
 majestic symphonies of a Beethoven. For these 
 things that lie beyond his limited physical senses, 
 he must needs accept the testimony of those who 
 have the necessary senses of perception to know 
 their reality." 
 
 "Like the blind and deaf individual, we stand 
 amid the wonders of nature. Tho we can 
 perceive, in light and color and music, a minute 
 
34 The Fourth Dimension and the Bible 
 
 fraction of the efect produced by waves of ether 
 and waves of air, yet these physical waves them- 
 selves lie totally beyond even our natural sense- 
 organs. Like that blind and deaf individual, who 
 thru his sense of touch, etc., can form some idea 
 of his physical environment, so we with our 
 limited sense-organs can acquire some knowledge 
 of the surrounding universe. But, as in his case, 
 our limited senses permit of but a very partial 
 knowledge, and beyond their range there are 
 realities even in physical nature concerning which 
 we can only speculate and which we may never 
 know. We are like children watching a game, 
 from a point some steps away, thru a small crack 
 or knot hole in a boarded enclosure. All appears 
 fragmentary and partial. Immeasurably the 
 greater part of the universe, from the infinitesi- 
 mally small to the universal whole, in ten thou- 
 sand marvels, lies beyond the range of the whole 
 outfit of our physical senses, or is a physically 
 intangible reality. Thus, the visible light or 
 color spectrum constitutes the record of but a 
 minute fraction of the whole range of the mys- 
 teriously wonderful perpetual dance of the 
 imponderable ether, it constitutes but one of the 
 octaves of the many-octaved key-board of vibra- 
 tions. And yet, the other octaves no less truly 
 exist as with our own invented tools of investiga- 
 
On the Existence of Higher Spaces 35 
 
 tion we are more and more discovering from their 
 effects; but these octaves themselves we cannot 
 perceive. If our eyes could be so adjusted as to 
 enable us to behold the whole range of this funda- 
 mental reality, vistas of visions hitherto incon- 
 ceivable would lie before our astonished gaze. 
 . . . And so might we conceive of superadded 
 transcendent spiritual faculties, if not confined 
 within our physical organism. And thus the 
 transcendent glories of spiritual realities, of an- 
 gelic beings, and even of God, might entrance the 
 astonished perceiving personality." 
 
 In what follows in this text, higher dimensional 
 spaces, and in particular the fourth, will often be 
 referred to as if they really existed. The reader 
 should clearly understand that this is done only 
 to save words by avoiding a needless repetition 
 of the statement that their existence is only as- 
 sumed. 
 
CHAPTER IV 
 
 ABOUT THE SUPPOSED INHABITANTS 
 OF OTHER SPACES THAN OUR OWN 
 
 As we are now going to make several excur- 
 sions into the realm of pure thought and abstract 
 speculation it will be necessary for the reader to 
 approach the subject with an open and receptive 
 mind. He will need to occasionally cut loose from 
 the sense perception moorings which anchor him 
 to our material world, he will often have to 
 leave experience behind and reason wholly from 
 analogy. A truly sympathetic attitude will add 
 much to the benefits that may be derived from 
 the discussions which follow. 
 
 Suppose that one, two, and four-dimensional 
 spaces do exist, in the same sense that the three- 
 dimensional space of our experience exists, and 
 let us imagine that in each of these spaces beings 
 exist, each individual being restricted to his own 
 particular space in his movements, his percep- 
 tions, and his understanding, just as we are re- 
 stricted to a three-dimensional space. Then let 
 us by analogy try to conceive what the attitude 
 
 36 
 
Supposed Inhabitants of Other Spaces 2>1 
 
 of a four-dimensional being living in a four- 
 dimensional space would be towards us living in 
 space of three dimensions by considering what 
 our attitude would be towards beings living in 
 space of one or two dimensions just as if we were 
 really conscious of their existence. For the sake 
 of brevity we shall designate space of two 
 dimensions as Flatland and its inhabitants as 
 Flatlanders, and the space of one dimension as 
 Lineland and its inhabitants as Linelanders. Cor- 
 responding to the space in which they live Line- 
 landers would have one dimension only (length) 
 and Flatlanders two dimensions only (length and 
 breadth). 
 
 FLATLAND 
 
 Let us consider Flatland. By imagining two- 
 dimensional beings (Flatlanders) living in a plane 
 (Flatland) and able to glide about in that plane 
 but absolutely unable to move out of it or to 
 perceive anything of a third dimension, just as 
 we are unable to perceive anything of a fourth 
 dimension, we may obtain a vivid idea of our 
 relation to a possible four-dimensional space. A 
 consideration of what would be the attitude of 
 Flatlanders towards any conception of space of 
 three dimensions makes clearer what should be 
 our attitude towards the conception of a space of 
 
38 The Fourth Dimension and the Bible 
 
 four dimensions. In doing this we may suppose 
 that what we call two-dimensional matter is 
 really three-dimensional, but with a very slight 
 thickness In the third dimension which Flatland- 
 ers are unable to recognize and of which they are 
 entirely unconscious. Thus we may Imagine 
 Flatland In the nature of a soap film (formed In 
 the closed loop of a wire) In which Flatlanders 
 live. Or, If we are willing to accept the statement 
 of some philosophers and physicists that a par- 
 ticle of our matter has no real substance (In the 
 sense In which we ordinarily use that term) but 
 consists only of a bundle of forces, attractive and 
 repellant, then there Is no difficulty In thinking of 
 such forces as lying entirely in a plane (or In a 
 line). All bodies In our space are bounded by 
 surfaces, in Flatland all bodies have one-dimen- 
 sional boundaries (lines and curves). To the skin 
 (surface) of a human being there would corre- 
 spond the bounding contour (perimeter) of a 
 Flatlander. To other Flatlanders he would be 
 exposed only along his perimeter, his interior 
 could be reached by them only thru an opening 
 in this perimeter. A Flatlander surgeon oper- 
 ating on a Flatlander would have to make an 
 incision in the perimeter of his patient just as one 
 of our surgeons must make an incision In the skin 
 of a patient to remove his appendix. To one of 
 
Supposed Inhabitants of Other Spaces 39 
 
 our surgeons gifted with two-dimensional vision, 
 however, all the interior organs of a Flatlander 
 would be exposed and he could operate without 
 cutting his Flatlander patient open. Not only 
 could we see (if we had two-dimensional vision) 
 the insides of Flatlanders but we could also see 
 the insides of their houses and all their closed 
 compartments. To illustrate let us assume that 
 the plane ABCD on this page is part of Flatland 
 and that a Flatlander inhabiting it is triangular 
 shaped (Fig. a). 
 
 As there is no "upward" or "downward," no 
 "above" or "below" in Flatland, houses and 
 barns would have no roofs or floors, only the lines 
 of their sides (perimeters) would be there. 
 Figure (b) may be taken to represent the house 
 of a Flatlander. He could enter his house only 
 thru the opening (door) EF. There would be 
 no difference between doors and windows (Fig. c). 
 To get to the other side of the line GH the 
 Flatlander would have to move around one end 
 of the line. The line would form an insuperable 
 
40 The Fourth Dimension and the Bible 
 
 barrier to him, he could not cross it or see over 
 or under it, just as in our space a stone wall of 
 infinite height and depth would hinder our motion 
 and obstruct our line of sight. However, a three- 
 dimensional being (man) with two-dimensional 
 powers could lift the Flatlander up out of the 
 plane, carry him across the line and then set him 
 down again in the plane. To his Flatlander 
 neighbors the Flatlander during this operation 
 would suddenly vanish and remain invisible until 
 he reappeared in their world on the other side 
 of the line. This performance would appear as 
 a miracle to the Flatlanders because not only 
 would it be impossible for them to see anything 
 not in their plane but all their experiences being 
 confined to that plane they could not possibly ex- 
 plain the phenomenon. In the same manner the 
 Flatlander might be placed in or taken out of 
 a closed room (Fig. d) to the utter mystification 
 of his friends. A robber from our space with 
 two-dimensional powers could therefore enter 
 their houses and steal their money and jewelry 
 from their safes without breaking locks or doors. 
 There would be nothing in Flatland correspond- 
 ing to our tubes and it would be impossible to 
 tie knots or thread needles there. As a Flatlander 
 could not "flip" a coin, to him it would always be 
 "heads-up" or always "tails-up." Story books 
 
Supposed Inhabitants of Other Spaces 41 
 
 have been written about life in Flatland, sup- 
 posedly by Flatland authors, in which the whole 
 gamut of our human experiences are duplicated 
 in so far as the restrictions under which they are 
 supposed to live will allow. There are descrip- 
 tions of life and industry in Flatland as well as 
 tales of love and war, and the love stories close 
 with the regulation "they married and lived 
 happily ever after." "An Episode in Flatland" 
 and "Scientific Romances" by C. H. Hinton; 
 "The Fairyland of Geometry" by Simon New- 
 comb; "Flatland" by E. A. Abbott, which has 
 also been dramatized and acted. 
 
 LINELAND 
 
 Let us now consider Lineland (one-dimensional 
 space) and its supposed inhabitants, the Line- 
 landers. 
 
 A '"-" ' I "- 1 lag-— oas— ■^a — — — — fl 
 
 " D € £ 
 
 A Linelander (as C) would thruout life be re- 
 stricted to his line (of which AB is a part) which 
 he would conceive of as extending indefinitely in 
 both directions. He could slide freely back and 
 forth along this line but it would be impossible 
 for him to move out of it. He would not be 
 conscious of nor could he conceive of a second 
 dimension; the concept of a plane surface would 
 
42 The Fourth Dimension and the Bible 
 
 He entirely outside of his mental realm, just as 
 the existence of our space would be inconceivable 
 to a Flatlander. Bodies in Lineland would not 
 be bounded by surfaces as in our space or by con- 
 tours (perimeters) as in Flatland, but by two 
 points, their two extremities. If a Linelander 
 encountered another similar being (as D) in his 
 space neither could pass the other for to do so 
 would mean that at least one of them would have 
 to move out of the line, i.e., out of their world. 
 Thruout life a Linelander (as C) could have only 
 two neighbors (as D and E), one on each side 
 of him. His experiences would be extremely 
 limited, to human beings life in Lineland would 
 be, to state it mildly, very dull indeed. 
 
 Let us now suppose that one of these Line- 
 landers (as C) in some way becomes conscious 
 of the existence of a second dimension and some- 
 how becomes endowed with the power to move 
 out of the line AB into a plane containing that 
 line as, for instance, the plane of page 41. The 
 instant he moved out of the line AB (his world) 
 he would become invisible to his Lineland neigh- 
 bors D and E. The vanishing of C, apparently 
 into nowhere, would be utterly incomprehensible 
 to D and E ; they could not account for it in any 
 way, it would appear to them as a supernatural 
 phenomenon, i.e., as a miracle. Possessing the 
 
Supposed Inhabitants of Other Spaces 43 
 
 freedom to move about in the plane (his new 
 world) the Linelander C could then move back 
 into the line AB (his old world). If he should 
 return to a point in the line between his old Line- 
 land neighbors D and E his appearance would be 
 in 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 his 
 disappearance. 
 
 The happenings described in the last paragraph 
 may also be imagined as occurring when the Line- 
 lander C has himself no conception of a second 
 dimension and has no power to move out of the 
 line AB (his world). We may conceive of a 
 situation in which the Linelander C thru some 
 influence is in the power of a Flatlander or other 
 higher-dimensional being. This latter higher- 
 dimensional being might then take the Linelander 
 C out of the line AB and replace him at will, and 
 the attendant phenomena would appear the same 
 as before to the two Linelanders D and E. While 
 this was taking place the Flatlander, or other 
 higher-dimensional being who was exercising his 
 power over C, would be invisible to the Line- 
 landers D and E, or, if not invisible, he would 
 appear to them only as a one-dimensional being 
 like themselves. 
 
 A Linelander, therefore, who in some manner 
 
44 The Fourth Dimension and the Bible 
 
 has strayed off his line (one-dimensional space) 
 into a plane (two-dimensional space) will remain 
 invisible to his Lineland neighbors until he re- 
 turns to his own world (the line). 
 
 Similarly, a Flatlander who has moved out of 
 his plane (two-dimensional space) into space of 
 three dimensions (our space) will remain in- 
 visible to his Flatland neighbors until he returns 
 to his own world (the plane). 
 
 Continuing, the next step, as suggested by 
 analogy, would then be the following. A man 
 (three-dimensional being) who has been trans- 
 lated from our space into a higher-dimensional 
 space will remain invisible to earthy beings until 
 he returns again to our space. We have Bible 
 records describing the bodily disappearance of 
 human beings from our space in a manner 
 analogous to the above. One case is that of 
 Enoch who did not see death but was taken bodily 
 out of our world by God. Gen. 5 :24, Heb. 1 1 :^. 
 Another example is the bodily translation of 
 Elijah into heaven. II Kings 2:11. This again 
 suggests that space of four dimensions may be 
 our heaven. Or, since the plural form of the 
 word heaven occurs very often in the Bible, it may 
 be that "the heavens" include not only space of 
 four dimensions but also all the other higher- 
 dimensional spaces. We have no record of a 
 
Supposed Inhabitants of Other Spaces 45 
 
 reappearance in our space of Enoch, but Elijah 
 (and Moses) did appear to the disciples at the 
 transfiguration of Christ. Math. 17 :3, Mark 9 :4, 
 Luke 9:30. That neither Enoch nor Elijah did 
 possess the inherent power to translate themselves 
 from earth to a higher realm is evident from the 
 Bible statements that they were taken by God 
 (the highest-dimensional being). On two differ- 
 ent occasions before his crucifixion Christ passed 
 unseen thru threatening multitudes, since his time 
 had not yet come. John 8:59, 10:39. Did he 
 make himself invisible each time by entering a 
 higher space? Was such a space Christ's abode 
 in the interval between his resurrection and 
 ascension and during which time he was seen in 
 bodily form by his followers on ten recorded 
 occasions? Each time both his appearance and 
 his disappearance were miraculous, each one par- 
 took of the nature of a visit from a higher to a 
 lower realm, from a higher to a lower order of 
 space. 
 
 The Bible abounds with numerous accounts of 
 celestial visitors (angels, archangels, prophets, 
 and God himself) revealing themselves to in- 
 habitants of our earth. In view of our discussion 
 it is near at hand for us to think of them as 
 coming down, from the higher dimensional spaces 
 
46 The Fourth Dimension and the Bible 
 
 in which they dwell and, after having fulfilled 
 their missions in our three-dimensional space, to 
 again ascend on high. Did Paul have a higher- 
 dimensional space in mind when he wrote, "I 
 knew a man, whether in the body, or out of the 
 body, I cannot tell, how that he was caught up 
 into paradise, and heard unspeakable words." 
 II Cor. 12:2, 3. Read again the Revelations of 
 St. John, read them with the added light of the 
 mathematical hypothesis of higher spaces. It will 
 be found that much of the fog with which our 
 limited space perceptions has enveloped it has 
 vanished and many passages will take on a new 
 meaning; the eyes of the soul will penetrate 
 deeper than ever before into the dim and distant 
 vistas which veil for us the present as well as 
 the great unknown hereafter. 
 
CHAPTER V 
 
 MOTION OF A POINT PERPENDICULAR 
 TO EACH SPACE 
 
 In Chapter II the concept of the motion of a 
 point perpendicular to a unit in each space was 
 partially developed. We will now lead up to an 
 extension of this idea by making use of the two 
 following obvious propositions. 
 
 (a) A point (as A) starting from the center 
 of a line (as BC) and moving of in a direction 
 perpendicular to the line will never approach any 
 portion of the line and will move away at the same 
 rate 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 
 
48 The Fourth Dimension and the Bible 
 
 dicular to its plane (as BC) will never approach 
 any portion of the circumference of the circle 
 hut will move away at the same rate 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 never 
 approach any portion of the surface of the sphere 
 but will move away at the same rate from all 
 points on this surface. 
 
 This proposition in the Geometry of Four 
 Dimensions was first proven by the great astrono- 
 mer and mathematician Simon Newcomb, and it 
 reveals some of the things, extraordinary to us, 
 which may be done in four-dimensional space if 
 such a space exists. We may also state the propo- 
 sition as follows. Just as in Flatland a point can 
 move In a direction perpendicular to a line, a 
 motion Inconceivable to LInelanders, and in our 
 space a point can move in a direction perpendicu- 
 lar to two mutually perpendicular lines, a motion 
 
Motion of a Point Perpendicular 49 
 
 Inconceivable to Flatlanders, so In space of four 
 dimensions a point can move in a direction per- 
 pendicular to three mutually perpendicular lines, 
 a motion which is inconceivable to us. Assuming 
 that the above sphere Is replaced by a material 
 hollow 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 of 
 four dimensions the grain of wheat may be placed 
 inside of the globe, or transferred from the inside 
 to the outside of the globe, without breaking the 
 globe or interfering with its structure in any 
 manner. This would mean that in space of four 
 dimensions any of our material objects can be 
 taken out of, or placed inside of, one of our 
 closed rooms or boxes without penetrating the 
 walls (sides). A four-dimensional robber, there- 
 fore, could enter our houses without hindrance 
 and steal the money and jewelry in our safes with- 
 out breaking locks or doors, and none of our jails 
 could hold him If caught. This may be where 
 the money we miss has gone, and perhaps the 
 articles we so often lose have accidentally rolled 
 into four-dimensional space ! A four-dimensional 
 man could eat one of our soft boiled eggs for 
 breakfast without breaking the shell and he could 
 drink from one of our bottles without removing 
 the cork. He could extract the pulp and nut from 
 
50 The Fourth Dimension and the Bible 
 
 one of our plums without breaking the skin of 
 the plum ; in fact, he could remove the kernel from 
 the nut without breaking the skin of the plum or 
 the shell of the nut. He could take off his stock- 
 ings without removing his shoes, and he could 
 take out the inner tube of one of our automobile 
 tires, repair it and then replace it, without re- 
 moving the outer casing. Just as one of our sur- 
 geons could operate on a Flatlander without 
 making an incision in his perimeter so a four- 
 dimensional surgeon could operate on one of us 
 without cutting us open. It is quite certain that 
 if such a surgeon should locate among us he would 
 very soon have a large and lucrative practical 
 All of our space, including the interior of the 
 densest solid, is open to inspection and manipula- 
 tion from the fourth dimension. 
 
 — C5 — 
 
 Fig. a 
 
 An ordinary knot (Fig. a) cannot be untied in 
 our space if the two ends of the cord are fixed (as 
 at A and B). In four-dimensional space, how- 
 ever, this can be done without unfastening the 
 ends, in fact, a knot could be tied or untied on an 
 endless string. Our knots would therefore be 
 
Motion of a Point Perpendicular 51 
 
 useless in four-dimensional space. Evidently our 
 chains would also be useless because the links in 
 them could be separated without breaking. 
 Corresponding to this in Flatland would be a cord 
 fastened at A and B (Fig. a) and making a loop 
 around, say, C. This loop could not be unwound 
 by a Flatlander, but one of us could do it by 
 simply lifting that part of the cord composing the 
 
 F/g. a Fig. b 
 
 loop into the third dimension, stretching it out, 
 and then replacing it in the plane (Fig. b). 
 
 As has already been stated, it has been mathe- 
 matically proven, beyond the shadow of a doubt, 
 that if there exists a space of four dimensions 
 then in that space material objects of our space 
 may be placed in or taken out of any of our 
 hermetically sealed closed compartments without 
 hindrance. It has never been satisfactorily 
 proven, however, that this can be done by a 
 human being; our nearest approach to it is to 
 think of it being done. We can think of the kernel 
 
52 The Fourth Dimension and the Bible 
 
 of a nut being extracted without breaking the shell 
 of the nut, in spite of the fact that we do not 
 understand how it can be done. We can think of 
 ourselves as entering or leaving closed compart- 
 ments; their material walls do not in the least 
 hinder our imagining ourselves doing it. In the 
 Bible, however, we have records telling of two 
 distinct occasions when Christ in the body did 
 appear before his disciples assembled behind 
 closed doors for fear of the Jews. John 20: 
 19-23, 26-29. Oi^ the first appearance Thomas 
 was absent but on the second he was present and 
 we read "then saith he to Thomas, reach hither 
 thy finger, and behold my hands; and reach hither 
 thy hand and thrust it into my side; and be not 
 faithless but believing. And Thomas said unto 
 Him, my Lord and my God." That Christ ap- 
 peared to his disciples on this remarkable occasion 
 clothed in his material body is further emphasized 
 in Luke 24: 36-43. "Behold my hands and my 
 feet that it is I myself : handle me, and see ; for a 
 spirit hath not flesh and bones, as ye see me have. 
 . . . And they gave Him a piece of a broiled fish, 
 and of an honeycomb. And he took it, and did 
 eat before them." Christ, considered as a higher- 
 dimensional being, certainly 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 
 
Motion of a Point Perpendicular 53 
 
 space of three dimensions but which is possible 
 in our hypothetical space of four dimensions by 
 those who may dwell there. The following re- 
 marks on John 20:19 are quoted from a Pulpit 
 Commentary on St. John. "It is more than pos- 
 sible — nay, it is entirely presumable — that the 
 spiritual body becomes possessed of additional 
 senses, of which we have no conception or experi- 
 ence; and, therefore, the spirit clothed with such 
 body Is alive to properties of matter and dimen- 
 sions of space and active forces all of which 
 would be supernatural to us. Our Lord before 
 his passion gave numerous proofs of the domi- 
 nance of his spirit over the body; his repeated 
 escapes from his enemies, his transfiguration 
 glory, his superiority to gravitation, in walking 
 upon the sea and hushing its storms. So that he, 
 on this occasion. Is revealing to the world some 
 of the functions of spiritual corporeity. He is 
 manifesting the kind of life which will eventually 
 be the condition of all the redeemed." 
 
CHAPTER VI 
 
 ROTATION OF SYMMETRICAL 
 CONFIGURATIONS 
 
 Let ABC and A'B'C be two lines In Llneland 
 LL', the points A,B?C, and A',B',C' respectively 
 
 i . ,^ _____„» L' 
 
 A be' ~7.' 1' c 
 
 being arranged in the same order and so located 
 that AB = A'B^ AC = A'C, BC -= B'C. One 
 line may then be made to coincide with the other 
 so that A coincides with A', B with B' and C with 
 C by simply sliding one of them along LL'. This 
 could be done by a Linelander. 
 
 \ 
 \ 
 
 L' 
 
 h B K C B' A' 
 
 Let us now consider the two lines ABC and 
 CB'A' where, as before, AB = A'B', AC = 
 
 54 
 
Rotation of Symmetrical Configurations 55 
 
 A'C, BC = B'C, but in which the correspond- 
 ing points are arranged in the reverse order. In 
 this case the one line cannot, by sliding it along 
 LL", be made to coincide with the other so that A 
 shall coincide with A', B with B', and C with C, 
 and a Linelander could not conceive of any other 
 way in which it might be done. A Flatlander, 
 however, or one of us, would at once observe that 
 the corresponding points could be brought into 
 coincidence by rotating one of the lines about O in 
 the plane of the page, 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 point 
 O. Roughly stated, the line ABC may be taken 
 up into Flatland, turned over, and put down again 
 on the line C'B'A'' so that the point A coincides 
 with A^ B with B' and C with C\ While this 
 rotation was taking place the moving line, as in 
 the position A'^B^'C", would be invisible to the 
 Linelanders living in LL' because it would be 
 outside of their world; the whole process would 
 appear to them as a miracle. 
 
56 The Fourth Dimension and the Bible 
 
 Let ABC and A'B'C be two triangles in the 
 plane of this page, such that AB = A'B', BC = 
 B'C^ CA = C'A', the corresponding sides of the 
 two triangles being arranged in the same order. 
 One triangle may then be made to coincide with 
 the other by simply sliding it along in the plane of 
 the page until A coincides with A', B with B', 
 and C with C\ This could be done by a Flat- 
 lander. 
 
 Y 
 Cj 
 
 Let us now consider the two triangles ABC and 
 A'B'C^ in which as before AB = A'B', BC = 
 B'C, and CA = C'A', but with their correspond- 
 ing sides arranged in the reverse order. In this 
 case the one triangle cannot, by sliding it along 
 in the plane, be made to coincide with the other, 
 and a Flatlander could not conceive of any other 
 way in which it might be done. The reader should 
 verify the first statement in the last sentence by 
 cutting a triangle out of cardboard to fit, say ABC, 
 and then, by sliding it along the page, try to 
 make it fit on A'B'C^ A three-dimensional 
 mathematician would observe, however, that the 
 
Rotation of Symmetrical Configurations 57 
 
 two triangles ABC and A'B'C are symmetrical 
 with respect to the line XY (corresponding points 
 on the two triangles being located on the same 
 perpendicular to XY and at equal distances from 
 it) and that they can be brought into coincidence 
 by rotating one (with the plane in which it lies) 
 about the axis of symmetry XY thru three-dimen- 
 sional space until it coincides with the other. 
 That is, one triangle must be taken up into our 
 space, turned over, and put down on the other. 
 To Flatlanders this operation would be an un- 
 solvable mystery. 
 
 r/ff.a 
 
 rig.b 
 
 If one of us should take a Flatlander (as 
 Fig. a) and turn him over thru our space back 
 into his plane (as Fig. b) he would become a sort 
 of mirror reflection of his former self. His heart 
 would now be found on his right side instead of 
 his left, and if he was right handed before he 
 would now be left handed. While he was being 
 turned over in our space he would be invisible to 
 his Flatland neighbors; first he would apparently 
 
58 The Fourth Dimension and the Bible 
 
 vanish before their eyes into nowhere and then 
 as suddenly he would reappear like an apparition. 
 Before taking the next step in our discussion 
 let us digress a trifle to point out that geometrical 
 propositions relating to configurations in a par- 
 ticular space may sometimes be proven with 
 greater ease by operating in a higher space than if 
 we are restricted to the original space. A case in 
 point is the plane geometry proposition that two 
 triangles are equal if the three sides of one are 
 equal respectively to the three sides of the other. 
 In practically all the plane geometry text-books 
 used in our schools the proof of this proposition 
 requires a rotation of one of these triangles about 
 one of its sides. But this means that the pupil is 
 abandoning the space of his plane geometry to 
 operate in the space of solid geometry (three 
 dimensions). In like manner there are many 
 theorems in both plane and solid geometry which 
 can be proven with greater ease by using geometry 
 of four dimensions than by using the methods 
 found in our school books. In fact both plane 
 and solid geometry have been greatly enriched by 
 the discovery thru the geometry of four dimen- 
 sions of a number of new theorems. Just as solid 
 geometry throws a new light on plane geometry 
 and enables us to solve problems in the plane 
 with greater ease than if we confined ourselves 
 
Rotation of Symmetrical Configurations 59 
 
 to the plane, so it has been found that geometry 
 of four dimensions illuminates both plane and 
 solid geometry, and has made easy or rendered 
 possible the solution of many problems both in 
 the plane and in our three-dimensional space 
 which, without the concepts of geometry of four 
 dimensions, would have been very difficult of solu- 
 tion to say the least. Schubert has illustrated this 
 by means of some very interesting examples. 
 We now return to our line of discussion. 
 
 Let ABCD and A'B'CD' be two tetrahedrons 
 in our space having all the faces of one equal to 
 the corresponding faces of the other and arranged 
 in the same order. By superimposing one on the 
 other we can make one coincide with the other 
 without difficulty. 
 
 /I 
 
 1/ 
 
6o The Fourth Dimension and the Bible 
 
 Let us now consider the two tetrahedrons 
 ABCD and A'B'C^D' in which, as before, all the 
 faces of one are equal to the corresponding faces 
 of the other, but arranged in the reverse order. 
 No amount of manipulation will now enable us 
 to bring the two tetrahedrons into coincidence, 
 and no human being can conceive of any way in 
 which it can be done. This task is as impossible 
 to us as was the task of the Linelander to bring 
 the symmetrical lines ABC and CB'A' into coin- 
 cidence (p. 54) or the task of the Flatlander to 
 make the two symmetrical triangles ABC and 
 A''B''C'' coincide (p. ^6). Mathematicians of the 
 highest standing, however, have proven that it 
 can be done in four dimensional space by rotating 
 one of the tetrahedrons about their plane of sym- 
 metry XY (corresponding points of the two tetra- 
 hedrons being located on the same perpendicular 
 to XY and at equal distances from it), a kind of 
 rotation that is impossible in our space of three 
 dimensions. Or, stating it roughly, one of the 
 tetrahedrons must be taken up into space of four 
 dimensions, turned over, and then brought back 
 into concidence with the other. One of these 
 tetrahedrons corresponds to the mirror image of 
 the other, the mirror being in the plane of XY. 
 While this rotation is taking place the moving 
 
Rotation of Symmetrical Configurations 6i 
 
 tetrahedron would be invisible to us; just as the 
 rotating line was invisible to the Linelander (p. 
 54) and the rotating triangle was invisible to the 
 Flatlander (p. 56). 
 
 When you look at your image in a mirror, the 
 right and left sides of your body appear to have 
 exchanged places. A mole on your right cheek 
 is seen on the left cheek of your image. Your 
 image is symmetrical to your body with respect 
 to the plane of the mirror. You may go behind 
 the mirror to the place where your image ap- 
 peared but you cannot take the exact position of 
 the image, or rather, you cannot make your body 
 coincide in every detail with that position. Turn 
 and twist about as much as you please In our 
 space you cannot make your right and left sides 
 exchange places. But a four-dimensional being 
 could do that very thing to you by rotating you 
 about a plane In his space. And, what is more, 
 no part of your body would be dislocated during 
 the operation, you would not experience any tear- 
 ing or wrenching of the tissues of your body. 
 After this "reversal" or "twist" In space of four 
 dimensions, during which you would be invisible 
 to all other human beings, everything on you 
 would also be changed from right to left and 
 vice-versa, even the seams and buttons on your 
 clothes, every dimple and wrinkle on your face 
 
62 The Fourth Dimension and the Bible 
 
 and every hair on your head. Your point of view 
 would be completely turned around, so that every- 
 one 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 type 
 form. The hands of a clock would appear to 
 you as going backward and the sun to rise in the 
 west and set in the east; the whole world would 
 appear to you as a looking-glass world. When 
 you look into a mirror you see a picture of the 
 actual "reversal" or "twist" that can be imparted 
 to material bodies in the space of four dimensions, 
 but which cannot be brought about in our space 
 of three dimensions. 
 
 Our two hands appear to be at least approxi- 
 mately of the same size, and shape but we can 
 never manipulate one so that it shall occupy the 
 same space that has been occupied by the other; 
 we cannot put a right glove on the left hand, or 
 vice-versa. The reason for this is that our two 
 hands are symmetrical with respect to some plane 
 in our space and, like the two symmetrical tetra- 
 hedrons shown on p. 59, they cannot be made to 
 coincide by any operation entirely confined to our 
 space of three dimensions. For the same reason 
 we cannot wear a left shoe on the right foot, or 
 vice-versa. A four-dimensional man, however, 
 can change a right shoe into a left shoe by rotat- 
 
Rotation of Symmetrical Configurations 63 
 
 ing It about a plane In his space. If a one armed 
 man in our space should In some way become 
 endowed with the power to rotate bodies about 
 planes In four-dlmenslonal space he could use both 
 of an ordinary pair of gloves because he could 
 change either one Into the other, so that he would 
 have either two rights or two lefts. 
 
 Instead of rotating material bodies about a 
 plane In space of four dimensions as described 
 above It has been proven that a curved surface 
 may also be used as the "axis" of rotation by 
 allowing for a slight degree of distortion. This 
 leads to very interesting results If the rotation is 
 applied to flexible material surfaces like a curved 
 sheet of rubber. The two surfaces of the sheet 
 of rubber may be made to exchange places by 
 rotating them about the sheet itself in space of 
 four dimensions. During the rotation there 
 would be no Interference betw^een portions of the 
 surfaces so that it may be done with surfaces that 
 are closed (like a sealed hollow rubber ball) as 
 well as with surfaces that are open (like a sheet 
 of rubber). 
 
 This means that a hollow rubber ball (Fig. a) 
 could be turned Inside out without tearing just as 
 we may turn a rubber band or the section of an 
 inner tire tube (Fig. b) Inside out. In both cases 
 
64 The Fourth Dimension and the Bible 
 
 the inside and outside surfaces would exchange 
 places. 
 
 Fig a 
 
 rig.b 
 
 Not only in the animal kingdom do we find this 
 wonderful symmetry (right and left handedness) 
 exemplified in the bodies of all creatures, as well 
 as in most of the interior organs of those bodies, 
 but it also appears very often in the vegetable 
 kingdom as shown by the symmetry of the two 
 sides of a leaf, and in flowers, bulbs, nuts, fruits, 
 etc. In fact, the right and left rotation (twist) 
 which is characteristically four-dimensional is of 
 very frequent occurrence in nature. A beam of 
 polarized light (whose wave-vibrations are all in 
 one plane) is rotated either to the right or to the 
 left on passing thru certain organic substances 
 (sugars, starches) just as we might hold a ribbon 
 by the ends and give it a twist to the right or left. 
 Two forms of sugar found in honey called dex- 
 trose and levulose owe their names to the fact 
 that one rotates a polarized beam to the right 
 (dextra, "right hand") and the other to the left 
 (laeva, "left hand"). In chemical constitution 
 
Rotation of Symmetrical Configurations 65 
 
 they are exactly the same, they are both repre- 
 sented by the same formula, and yet they rotate 
 the plane of polarization of a beam of light in 
 opposite directions. Such substances are called 
 "isomeric." A molecule of each of the two sub- 
 stances contains the same number of each kind 
 of atom, the difference in the effect on light being 
 accounted for by a different arrangement or group- 
 ing of the atoms in the two molecules. 
 
 The following discussion will explain what rela- 
 tion this has to a possible four-dimensional space. 
 
 In a plane three points (as the vertices of the 
 
 r/^. a F'ig. b 
 
 triangle In Fig. a), and no more, can be located 
 so that the three distances between them shall 
 be independent of each other, i.e., any one of the 
 distances may change without changing the others. 
 
 In our space of three dimensions four points 
 (as the vertices of the tetrahedron in Fig. b), and 
 no more, can be located so that the six distances 
 between them shall be independent. 
 
 In space of four dimensions five points, and 
 
66 The Fourth Dimension and the Bible 
 
 no more, can be located so that the ten distances 
 between them shall be independent, something that 
 cannot be done in our space. 
 
 If then a molecule consists of five atoms we 
 cannot in our space alter the distance between two 
 of them without at least altering some second dis- 
 tance. "But if we imagine the centers of the 
 atoms placed in four-dimensional space, this can 
 be done; all the ten distances which may be con- 
 ceived to exist between the five points will then 
 be independent of one another. To reach the 
 same result in the case of six atoms we must as- 
 sume a five-dimensional space; and so on. Now, 
 if the independence of all possible distances be- 
 tween the atoms of a molecule is absolutely re- 
 quired by theoretical chemical research, the 
 science is really compelled, if it deals with mole- 
 cules of more than four atoms, to make use of the 
 idea of a space of more than three dimensions. 
 This idea is, in this case, simply an instrument 
 of research, just as are, also, the ideas of mole- 
 cules and atoms — means designed to embrace in 
 a perspicuous and systematic form the phenomena 
 of chemistry and to discover the conditions under 
 which new phenomena can be evoked. Whether 
 or not a four-dimensioned space really exists is 
 a question whose insolubility cannot prevent re- 
 search from making use of the idea, exactly as 
 
Rotation of Symmetrical Configurations 67 
 
 chemistry has not been prevented from making 
 use of the notion of atom, altho no one really 
 knows whether the things we call atoms exist or 
 not." (Schubert.) 
 
 For example, at least eight different alcohols 
 have been discovered having the same formula 
 C5 H12 O. Chemists account for these different 
 compounds by supposing that there is a different 
 arrangement or grouping of the five carbon, or 
 C-atoms, in each compound, something which 
 would be possible in space of four dimensions but 
 not possible in our space. In other words, should 
 it become necessary for the explanation of the 
 structure of this molecule to assume that the five 
 carbon atoms in it are equidistant from each other 
 (the simplest possible case) we would also have 
 to assume the existence of four-dimensional space, 
 because in that space, and not in our space, is it 
 possible to locate five points equidistant from each 
 other. 
 
 There are two varieties of tartaric acid which 
 crystaUize into symmetrical forms, forms bearing 
 the relation of object to mirror-image as here 
 shown. 
 
68 The Fourth Dimension and the Bible 
 
 Investigators have found that one of these 
 crystals apparently changes into the other with- 
 out chemical resolution and reconstitution and 
 without any manifestation of force. If it could be 
 proven without the shadow of a doubt that this 
 is what actually happens, then this would point 
 to the existence of space of four dimensions, be- 
 cause in that space, and not in ours, is it possible 
 for a right-handed shape to change into its sym- 
 metrical left-handed shape by a simple movement. 
 Certain snails, exactly alike in other respects, 
 differ similarly in that some are coiled to the right 
 and others to the left. An astonishing phenome- 
 non connected with these snails is the fact that 
 their juices have the property of rotating a polar- 
 ized beam of light to the right or to the left 
 corresponding to the direction in which the snails 
 themselves are coiled. This suggests that their 
 external form is the expression of an internal 
 difference due to a right or left twist of their 
 atoms caused by a four-dimensional force. It 
 has been suggested that such a four-dimensional 
 twist runs thru all living forms, and that the life 
 force is, in part at least, four-dimensional. 
 
CHAPTER VII 
 
 SECTIONS OF BODIES IN ONE SPACE 
 MADE BY SPACES OF A LOWER 
 ORDER. CONSERVATION OF MAT- 
 TER AND ENERGY 
 
 Suppose we pass a circular cone downward per- 
 pendicularly thru Flatland XY (see figure). The 
 cross sections of the cone made by the plane of 
 
 Flatland will be circles (as AB). The Flatland- 
 ers living in the plane would see at first only a 
 point as the apex of the cone entered their plane, 
 then they would see a circle gradually increasing 
 to the size of the base of the cone, when suddenly 
 every trace of the cone would vanish. The Flat- 
 landers would have no conception whatever of 
 the cone as we know it, indeed, they would not 
 suspect that the two-dimensional object (circle) 
 
 69 
 
70 The Fourth Dimension and the Bible 
 
 which they had seen had anything to do with a 
 three-dimensional solid at all, even if they had an 
 inkling of the existence of our space. The phe- 
 nomenon would appear to them as the creation of 
 a new two-dimensional body (circle) which con- 
 tinued to grow larger and larger and then sud- 
 denly vanished. The alpha and omega of this 
 phenomenon would be deep mysteries to them, 
 just as birth and death are to us. In like manner 
 It may be that each one of us is in reality a four- 
 (or higher-) dimensional being, our personality 
 (as we are known to ourselves and others) being 
 the section of our real, our higher self, made by 
 the three-dimensional space of our experience. 
 Like the cone passing thru Flatland, we arrive into 
 this three-dimensional world from some unknown 
 shore, i.e., are born; we grow up thru childhood, 
 youth, middle age, old age ; we die, and our souls, 
 our real, our higher selves, pass into the great 
 unknown hereafter. And thruout life we have 
 been thought of by ourselves and our fellowmen 
 as three-dimensional beings only. Perhaps all liv- 
 ing things, as they appear to us, are the sections 
 of four- (or higher-) dimensional creatures made 
 by our space. "Viewing human life from this 
 standpoint, the conclusion may be reached that I, 
 as I write this, am merely that section of my four- 
 dimensional self that happens to be passing thru 
 
Sections of Bodies in One Space 71 
 
 this world at this moment, and that the whole 
 of me, from my birth to my death, is a four- 
 dimensional entity; that the past and the future 
 are past and future only in a three-dimensional 
 sense, and that in a four-dimensional sense the 
 past and future are present — that is, both what 
 was and will be, is." (Taylor.) 
 
 Suppose now that a particular Flatlander had 
 In some way acquired the Intellectual power to 
 sum up, in his mind, in a moment of time, all the 
 cross sections (circles) of the cone he had seen. 
 He would then obtain 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, time 
 being the third dimension. In like manner, by 
 a concentrated mental effort, we may in a moment 
 of time assemble all the impressions we have re- 
 ceived of some friend or relative with whom we 
 have been intimately associated from his, or her, 
 birth to death. The mental image thus obtained 
 would include all the years of his or her life, the 
 whole span of life of that individual would be 
 compressed into a moment of time; and this 
 mental image would have four dimensions, time 
 being the fourth dimension. Mental vision has 
 therefore been called four-dimensional by some 
 writers. All of us have in our dreams experienced 
 this crowding together of a long series of occur- 
 
72 The Fourth Dimension and the Bible 
 
 rences into an instant of time. There are no re- 
 strictions of time, or restrictions of the space of 
 our experience, in dreamland. "The unthinkable 
 velocity of time in dreams may be inferred from 
 the fact that between the movement of impact of 
 an impression at the sense periphery and its re- 
 ception at the center of consciousness — moments 
 so closely compacted that we think of them as 
 simultaneous — a coherent series of representa- 
 tions take place, involving what seem to be pro- 
 tracted periods for their unfoldment. Every 
 reader will easily call to mind dream experiences 
 of this character, in which the long-delayed 
 denouement was suggested and prepared for by 
 some extraneous sense impression, showing that 
 the entire dream drama unfolded within the time 
 it took that impression to travel from the skin to 
 the brain." (Bragdon). Perhaps dreamland is 
 located in four-dimensional space. 
 
 A man who has experienced the sensations of 
 a dying person as, for instance, a man who be- 
 lieved he was drowning, tells us that in an instant 
 the whole panorama of his past life was spread 
 out before him in a vivid momentary perspective. 
 "Memory is a carrying forward of the past into 
 the present, and the fact that we can recall a past 
 event without mentally rehearsing all the inter- 
 mediate happenings in Inverse order, shows that 
 
Sections of Bodies in One Space 73 
 
 in the time aspect of memory tliere is simultaneity 
 as well as sequence." If our mental vision is four- 
 dimensional it points to the possibility that our 
 mental or spiritual self is four- (or higher-) 
 dimensional. If time is but the way in which we 
 perceive the fourth dimension, then our spiritual 
 selves, being higher-dimensional, are independent 
 of time, above time, outside of time, that is, 
 eternal. In this higher space our grammatical 
 tenses would then have no meaning. Perhaps this 
 explains why these tenses as used in the Bible 
 sometimes 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 is 
 past." Ps. 90:4. That God, the highest-dimen- 
 sional being is not, like we, restricted by time 
 limitations is clear. "One day is with the Lord as 
 a thousand years, and a thousand years as one 
 day." II Peter 3 :8. "Behold, thou hast made 
 my days as an handbreadth; and mine age is as 
 nothing before thee." Ps. 39:5. 
 
 Another interesting illustration similar to that 
 of the cone passing thru Flatland (p. 69) is that 
 of a wire bent in the form of a circular helix 
 (coiled wire spring) which is passed thru Flat- 
 land so that its axis is perpendicular to the plane 
 of Flatland (see figure). The cross section of 
 the wire made by the plane (as A) will be a very 
 
74 The Fourth Dimension and the Bible 
 
 small ellipse (two-dimensional). If the wire is 
 passed downward the Flatlanders would see this 
 eUipse travel in the circle BC in one direction, 
 while if the wire is passed upward it would travel 
 in the opposite direction. The "birth" of this 
 ellipse as the wire entered the plane, its "death" 
 when it left the plane, and its behavior in the 
 interim — all would be inscrutable mysteries to 
 the Flatlanders living in that plane. 
 
 It is evident that if the cone to which we re- 
 ferred on p. 69 is passed obliq^uely thru the plane, 
 then the sections will no longer be circles but 
 ellipses, parabolas, hyperbolas, or straight lines, 
 all depending on the angle which the axis of the 
 cone makes with the plane. To further illustrate 
 that the kind of section made in a given case will 
 depend not only on the shape of the solid but also 
 on how the solid enters the plane, let us assume 
 that the same cube passes thru the plane several 
 times, each time with a new orientation with re- 
 spect to the plane. 
 
Sections of Bodies in One Space 75 
 
 F/j- a 
 
 Fig.b 
 
 Fig. c 
 
 If a face of the cube is always parallel to the 
 plane, then all the sections made will be equal 
 squares (Fig. a). 
 
 If an edge of the cube (and no face) is always 
 parallel to the plane then all the sections made will 
 be rectangles except the first and last which would 
 be lines, i.e., the top and bottom edges (Fig. b). 
 
 If no edge or face is parallel to the plane, i.e., 
 the cube is passed thru obliquely, the first section 
 will be a point (a corner of the cube), then they 
 would be triangles, then polygons, then again tri- 
 angles, and finally the last one would again be 
 a point (Fig. c). 
 
 Since we live in space of three dimensions the 
 above phenomena are fully understood by us, but 
 to Flatlanders viewing the appearance, behavior, 
 and disappearance of these sections In their plane, 
 it would all be utterly Inexplicable because they 
 have knowledge and experience only of two- 
 dimensional bodies. Least of all would they sus- 
 pect that sections so unlike were produced by the 
 
76 The Fourth Dimension and the Bible 
 
 passing of the same higher-dimensional body thru 
 their world. If then, as it has been suggested, 
 the personality of a man is the section of his real, 
 his higher-dimensional self, made by the space of 
 our experience, it makes it clearer to us why the 
 personalities of all the people we know have such 
 characteristic differences. Both as to mind and 
 body they all seem to have been cast in different 
 moulds. All this suggests that the personality of 
 a man may depend not only on the character of 
 his higher-dimensional self but that it is also influ- 
 enced by the manner in which that higher self was 
 projected into our world. 
 
 In the preceding paragraphs of this chapter the 
 reader naturally thought of the cone, helix, and 
 cube as substantial solids, while the sections 
 (points, circles, triangles, lines, etc.) of these 
 solids made by the planes thru which they were 
 passing impressed him with their changing, fleet- 
 ing, and evanescent character. Passing now to the 
 next higher order of space, analogy suggests the 
 above relations as one explanation of the transi- 
 tory character of our lives; they may be merely 
 the sections of our spiritual, our higher-dimen- 
 sional selves made by the three-dimensional space 
 of our experience. Many passages in the Bible 
 emphasize the ephemeral character of our earthly 
 existence, while others point out the higher, the 
 
Sections of Bodies in One Space 77 
 
 divinely transcendental origin and nature of man. 
 "As for man, his days are as grass; as a flower of 
 the field, so he flourisheth. For the wind passeth 
 over it, and it is gone ; and the place thereof shall 
 know it no more." Ps. 103:15, 16. "For what 
 is your life? It is even a vapour, that appeareth 
 for a little time and then vanisheth away." James 
 4:14. "Man that is born of a woman is of few 
 days, and full of trouble. He cometh forth like 
 a flower, and is cut down : he fleeth also as a 
 shadow, and continueth not." Job 14:1, 2. 
 "Mine age is departed, and is removed from me 
 like a shepherd's tent." Is. 38:12. "So God 
 created man in his own image, in the image of 
 God created he Him." Gen. i :27. "For thou 
 hast made him a little lower than the angels, and 
 hast crowned him with honor and glory." Ps. 
 8 :5. "For in him we live and move and have our 
 being." Acts 17 :28. 
 
 In this connection it is of interest to refer to 
 the method employed by the biologist to deter- 
 mine the interior structure and composition of 
 solids. For instance, by cutting an apple into a 
 large number of thin slices of uniform thickness 
 and arranging them in the proper sequence on a 
 flat surface a clear concept of the structure of the 
 apple may be obtained. That is, he mentally sums 
 up the total of the impressions on his mind made 
 
78 The Fourth Dimension and the Bible 
 
 by these slices and thus obtains a mental image 
 of the Interior as well as the exterior of the apple. 
 Thus, if the apple happens to be worm-eaten it is 
 easy to trace the passage made by the worm, as 
 well as to locate the worm, by noting the relative 
 locations of the series of worm holes in the slices. 
 This method is of the greatest utility to the 
 biologist in the study of the structure and compo- 
 sition of the embryos or organs of animate beings. 
 For instance, the embryo of a frog is cut into a 
 large number of sections by means of the micro- 
 tome, an instrument constructed for this purpose, 
 so thin that they are practically transparent under 
 the microscope. These slices, which are prac- 
 tically plane sections of the embryo, are then 
 mounted on glass plates by means of a perfectly 
 transparent adhesive and studied under the 
 microscope. These plates, called slides, when 
 consecutively arranged in a series, will vividly pro- 
 tray the most minute detail of the structure of the 
 embryo. By means of these slides the biologist 
 Is able to mentally reassemble all these sections In 
 their proper order and thus to mentally recon- 
 struct the embryo. 
 
 CONSERVATION OF MATTER AND ENERGY 
 
 Again making use of a cube passing obliquely 
 downward thru a plane (as in next figure) we note 
 
Sections of Bodies in One Space 79 
 
 that the section of the cube made by the plane is 
 continually changing in magnitude. Starting as 
 a point (when the lowest corner of the cube en- 
 ters the plane) the section changes into an increas- 
 ing triangle, then into polygons of more than 
 three sides, then into a decreasing triangle which 
 finally contracts into a point (as the highest cor- 
 ner of the cube emerges from the plane). 
 
 The Flatlanders living in that plane would ob- 
 serve that the area of this section is continually 
 changing, i.e., this area does not remain constant. 
 This suggests the possibility that the total mass 
 of our material universe (considered as the sec- 
 tion of a higher-dimensional configuration passing 
 thru our space of three dimensions) may in like 
 manner be continually changing, i.e., the total 
 mass of our material universe may not be constant 
 and hence the principle of the conservation of 
 matter may not hold true. And this may apply 
 not only to our material universe as a whole but 
 also to ail of its parts from the smallest atom 
 to the largest planets and suns. To illustrate 
 
8o The Fourth Dimension and the Bible 
 
 again, suppose that instead of the cube an auto- 
 mobile is passing downward thru the plane. The 
 plane section of the automobile will be of very 
 great complexity, and not only the whole of the 
 section but also all of its many parts would be 
 continually changing in form and area. And not 
 only that, but during the interval between the 
 instant when a section of the automobile first ap- 
 peared in the plane and the instant the last trace 
 of it vanished the particular sections of hundreds 
 of the individual parts of the automobile would 
 suddenly appear (be created), pass thru a con- 
 tinuous series of changes, and then as suddenly 
 vanish (be annihilated). For instance, we can 
 visualize the section of some particular bolt in the 
 automobile as it is passing thru the plane. 
 
 May it not be possible that the whole of our 
 material universe as well as all its component parts 
 are similarly changing in form and total mass so 
 that there may be additions to or subtractions 
 from this total which we as three-dimensional 
 beings cannot account for? Astronomers have 
 observed the sudden appearance in the heavens 
 of stars and comets, and also the apparent van- 
 ishing of such celestial bodies, all in a manner 
 suggesting that they were either entering or leav- 
 ing our universe. For these and other reasons 
 there has been for some time a suspicion in the 
 
Sections of Bodies in One Space 8i 
 
 minds of many scientists that the total mass of 
 our material universe is not constant. Since every 
 atom of matter is a reservoir of energy it follows 
 that if the principle of the conservation of matter 
 is not true, then the principle of the conservation 
 of energy also fails. Numerous instances are re- 
 corded in the Bible where new matter or new 
 energy was apparently added to our material 
 universe by supernatural means. The manna fall- 
 ing in the wilderness, the ever replenished oil in 
 the widow's cruse, and the loaves and fishes which 
 fed the five thousand, all this food may have 
 been the three-dimensional sections of higher- 
 dimensional supplies projected at those particular 
 times thru our space by some higher agency. Also 
 the energy which divided the waters of the Red 
 Sea, which caused the walls of Jericho to fall, 
 and by means of which Jesus stilled the storm at 
 sea and walked on its waters, this energy was 
 in some to us mysterious way added to the already 
 existing energy in our universe. When God 
 translated Enoch and Elijah bodily into heaven 
 matter and energy was subtracted from our 
 material universe. 
 
CHAPTER VIII 
 
 THE ILLUSORY AND THE REAL 
 
 In the case of the cube passing thru a plane it 
 is evident that the cube itself is the real, the true 
 individual, while the sections (squares, triangles, 
 etc.) of it seen by the Flatlanders living in that 
 plane are but illusory, inadequate images of the 
 cube. However, it is only thru these countless 
 sections that the cube is manifested to the Flat- 
 landers. But the "birth" of a section when the 
 cube enters the plane is not the "birth" of the 
 cube, and the "death" of a section when the cube 
 leaves the plane is not the "death" of the cube; 
 this "birth" and this "death" are mere illusions 
 as far as the cube itself is concerned. And so, 
 because man is a higher-dimensional being his 
 birth, that is, his entrance into this world, is 
 not his beginning, nor will his death, that is, his 
 passing out of this world, be his end. 
 
 The classical allegory of the chained shadow- 
 watchers given by Plato in the seventh book of 
 his "Republic" vividly portrays the relation be- 
 tween true being and the illusions of the sense 
 
 82 
 
The Illusory and the Real 83 
 
 world. Plato imagines a group of prisoners 
 chained just inside the entrance to a large cavern. 
 All movement is impossible to them, they cannot 
 even turn their heads, hence they always look 
 upon the opposite wall of the cavern. A strong 
 light streams in thru the mouth of the cavern. 
 The prisoners never see anything but their own 
 shadows on the wall together with the shadows 
 of all persons and objects moving behind them. 
 In the course of time they would identify them- 
 selves with their shadows, they would refer all 
 their experiences to these shadows, they would 
 practically become the denizens of a shadow 
 world. All movements observed by them would 
 be the movements of shadows on the wall, the 
 movements of shapes with outlines but with no 
 substantiality. Plato thus conceived of a pos- 
 sible state in which man is reduced by his limited 
 consciousness to less than he really is and thus 
 cleared the way for the concept that the normal 
 man in an analogous manner is reduced by his 
 consciousness to less than he really is. Man is 
 thus but the shadow cast on our world by his real, 
 his higher self dwelling in supernal regions. As 
 the real world of the chained prisoners was to 
 their shadow world, so is the supernal world to 
 the world of our experience. 
 
 Let us suppose the surface of a smooth wet 
 
84 The Fourth Dimension and the Bible 
 
 sidewalk to be part of Flatland. A man walking 
 on it would make tracks as shown above. The 
 Flatlanders living in the plane of the surface of 
 the sidewalk would see only the bottoms of the 
 heels and soles of his shoes. First the heel and 
 sole of one foot would appear to them apparently 
 out of nowhere and then vanish just as the heel 
 and sole of the other foot appeared and in turn 
 vanished. This would be repeated again and 
 again to the great mystification of the Flatlanders. 
 To them it would seem as if two two-dimensional 
 bodies (the bottoms of the heel and sole of one 
 shoe) were traveling together for some unknown 
 reason accompanied by their twins (the bottoms 
 of the heel and sole of the other shoe). The 
 Flatlanders could not conceive of each heel and 
 sole being connected, that is, as belonging to the 
 same shoe (a three-dimensional body), and much 
 less could they conceive of these shoes as belong- 
 ing to the same three-dimensional being (the 
 man). Suppose, however, that some Flatlander 
 of superior intellect had an inkling of the third 
 dimension such as we have of a fourth dimension, 
 and that he suspected that the two pairs of impres- 
 
The Illusory and the Real 85 
 
 sions made on the sidewalk really belonged to the 
 same three-dimensional being. As only the bot- 
 toms of the man's heels and soles would come 
 within his range of observation he would prob- 
 ably state as his opinion that this three-dimen- 
 sional being (the man) had no intelligence or 
 feeling and therefore was of little or no account. 
 He would judge the man from what he had seen 
 of the bottoms of his heels and soles ! By analogy 
 then, if man is a higher-dimensional being of 
 whom we can observe his three-dimensional 
 attributes only, it may be that, comparatively 
 speaking, we have no clearer or fuller conception 
 of what man actually Is in the sight of God than 
 had the Flatlander when he tried to describe a 
 three-dimensional being (the man) from what he 
 had seen of the bottoms of his heels and soles. 
 
CHAPTER IX 
 
 SPACES AS HEAVENS AND HELLS 
 
 Perhaps the higher-dimensional spaces are the 
 heavens to which we ascend at death if we have 
 so lived on earth as to deserve such promotion; 
 and the lower-dimensional spaces are the hells to 
 which we will be consigned in case our lives have 
 fallen below a certain standard. Thus at death 
 the soul of a good man (his higher-dimensional 
 self) shakes off the limitations of understanding, 
 knowledge and movement which our three-dimen- 
 sional space imposed on him, and enters space of 
 four dimensions. Instead of being restricted to 
 three degrees of freedom of motion he will then 
 be able to move in four independent directions. 
 Such would be his increased freedom of action 
 that, if he could look back, his former life in 
 our space would probably appear to him as life 
 in a prison cell would now seem to us. Unlimited 
 vistas of new knowledge would open up to him, 
 and his powers of understanding and possibilities 
 of action would have increased to an extent far 
 beyond the present limits of human conception. 
 
 86 
 
spaces as Heavens and Hells 87 
 
 Coming from the restrictions imposed by our 
 space, the space of four dimensions would indeed 
 seem heaven to him ! On the other hand, the soul 
 of the wicked man would at death be condemned 
 to live in Flatland, the space of two dimensions. 
 He could move only in two independent directions, 
 all his knowledge and experience of a third dimen- 
 sion would be blotted out of his mind, and life in 
 Flatland would certainly be hell to him. If during 
 his existence in Flatland the fallen man did not 
 mend his ways but continued to deteriorate in 
 character, then at death he would again be de- 
 moted, he would be consigned to Lineland, an 
 experience that would abound with horrors. He 
 could then move in one direction only and his fund 
 of knowledge and his powers of understanding 
 would be still further diminished. If after this 
 awful experience he still persisted in his evil ways 
 he would at death be consigned to space of zero 
 dimensions, that is, to a point. There no motion 
 at all would be possible, all his knowledge and 
 understanding would have vanished, he would be 
 in the deepest pit of hell. "For a fire is kindled 
 in mine anger, and shall burn unto the lowest 
 hell." Deut. 32 :22. 
 
 But let us turn back to the more pleasant pros- 
 pect, the assumption that our earthly friend has 
 so lived that at death he is translated into space 
 
88 The Fourth Dimension and the Bible 
 
 of four dimensions, that is, the first heaven for 
 us. "And I saw a new heaven and a new 
 earth: for the first heaven and the first earth 
 were passed away." Rev. 21 :i. Happily during 
 his life in space of four dimensions our friend 
 continues his striving after that which is highest 
 and noblest and best, and so when he has finished 
 his course there he Is promoted to space of five 
 dimensions, that is, our second heaven, a place 
 of far surpassing knowledge and freedom and 
 glory. Continuing in growth of character he 
 would then be advanced to space of six dimen- 
 sions, that Is, our third heaven. "I knew a man 
 in Christ above fourteen years ago (whether in 
 the body I cannot tell; or whether out of the 
 body, I cannot tell: God knoweth), such an one 
 caught up into the third heaven." II Cor. 12:2. 
 Not stopping here we can Imagine that the 
 noblest and greatest individuals will, after such 
 successive existences, be translated to still higher 
 spaces (heavens). The higher the order of space 
 the greater and more glorious would be the state 
 of the blessed who dwell there, until the space of 
 Infinite dimensions is reached where God has his 
 dwelling place. There are no restrictions to 
 motion and no limitations of power, knowledge 
 or understanding. God Is omnipotent, omniscient, 
 omnipresent; the very personification of justice 
 
spaces as Heavens and Hells 89 
 
 and mercy, wisdom and love. "God is absolutely 
 infinite, consisting of infinite attributes each ex- 
 pressing eternal and infinite essentiality." 
 
 The Bible has made the above concept of more 
 than one heaven familiar to us all. "In my 
 Father's house are many mansions." John 14:2. 
 "The heaven, even the heavens, are the Lord's." 
 Ps. 115 :i6. "When I consider thy heavens." Ps. 
 8 :3. "When he prepared the heavens." Prov. 
 8 :27. "Therefore will I shake the heavens." Isa. 
 13.3. "For, behold, I create new heavens." Isa. 
 65:17. Also Hag. 2:6, Ezek. 1:1, Matt. 3:16, 
 Mark 1:10, II Cor. 5:1, II Peter 3:13. We 
 have referred to higher and lower orders of 
 spaces. That there are also various orders of 
 heavens may be inferred from scripture passages. 
 "Behold, the heaven and the heaven of heavens 
 is the Lord's thy God." Deut. 10:14. "Behold, 
 the heaven and the heaven of heavens cannot con- 
 tain thee." I King 8 :27. 
 
 We find nothing in the hypothesis of higher 
 spaces that we have been developing which con- 
 tradicts the reality of sin or the fact of the divine 
 redemption of a fallen race. Whether or not 
 the inhabitants of other spaces than ours have 
 passed or are passing thru the experiences of an 
 Adam's fall and a Christ's atonement we do not 
 know, but the presumption is strong that at least 
 
90 The Fourth Dimension and the Bible 
 
 some of the dwellers in higher space (heavens) 
 did disobey God and therefore incurred the in- 
 evitable punishment which is sure to overtake the 
 sinner, whether he be man or angel. "Behold, he 
 putteth no trust in his saints : yea, the heavens 
 are not clean in his sight." Job 15 :i5. "For if 
 God spared not the angels that sinned but cast 
 them down to hell, and delivered them into chains 
 of darkness, to be reserved unto judgment." II 
 Peter 2 14. "And the angels which kept not their 
 first estate, but left their own habitation, he hath 
 reserved in everlasting chains under darkness unto 
 the judgment of the great day." Jude 6. "And 
 he said unto them, I beheld Satan as lightning fall 
 from heaven." Luke 10:18. 
 
CHAPTER X 
 
 CREATION OF OUR MATERIAL UNL 
 VERSE. EVOLUTION 
 
 Is the theory of higher spaces in accord with 
 the Bible account of creation? Let us approach 
 this subject by making use of the concept em- 
 ployed in the last chapter, the assumption that 
 bodies in any given space are the sections of 
 higher-dimensional bodies made by that space. 
 For example, let the plane AB represent a portion 
 of a two-dimensional space (Flatland). 
 
 Assume figure (a) to be a solid circular cylinder 
 (three-dimensional body) with its axis always 
 perpendicular to the plane and moving towards 
 
 91 
 
92 The Fourth Dimension and the Bible 
 
 and completely thru the plane and assume the 
 rest of the figures as successive stages of the same 
 cylinder as it is passing thru the plane. Before 
 the cylinder reaches the plane, Flatlanders living 
 in that plane would know absolutely nothing 
 about it, the cylinder would be non-existent as far 
 as they are concerned. The instant the lower base 
 of the cylinder coincided with the plane, as in fig- 
 ure (Z?), the circle CD, the section of the cylinder 
 made by the plane, would suddenly appear before 
 the astonished Flatlanders, coming apparently 
 out of nowhere. To them this phenomenon would 
 be the creation 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 material {three-dimensional) universe out 
 of nothing. They could not even imagine an ex- 
 planation of the mystery, they could simply state 
 that where there was nothing before there was 
 now a circle; a real two-dimensional body had 
 been created apparently out of nothing. 
 
 During the time the cylinder is passing thru 
 their plane the Flatlanders would behold the cir- 
 cular sections made by the plane as a continuing 
 entity, a circle, (as EF, GH, etc.) until its upper 
 base KL, figure (/), coincided with the plane. 
 Then the instant the cylinder passed out of the 
 plane the circle would suddenly vanish, apparently 
 
Creation of Our Material Universe 93 
 
 into nowhere. To the Flatlanders the circle would 
 cease to exist, where there had been something 
 before there was now nothing. They could no 
 more account for the annihilation of the circle 
 than we can conceive of the possibility of the an- 
 nihilation of our material universe, it would be 
 and remain an unsolvable mystery to them. 
 
 Continuing this line of reasoning one step fur- 
 ther we may think of our material universe as 
 the section of a higher-dimensional body (aggre- 
 gate) made by our space of three dimensions. 
 Just as the circle, in what has gone before, was 
 a two-dimensional body created by the entrance 
 of a cylinder (three-dimensional body) into two- 
 dimensional space (Flatland) so we may conceive 
 of our material universe as being created by the 
 entrance of a higher-dimensional body (aggre- 
 gate) into our three-dimensional space. As human 
 beings restricted to the knowledge and experi- 
 ences of a three-dimensional space we can have 
 no knowledge of how this can be brought about 
 or understand the real nature of the higher-dimen- 
 sional body (aggregate) in question. All that 
 we can say is that where there was nothing before 
 we now behold our material universe, that is, our 
 material universe was created out of nothing. 
 "In the beginning God created the heaven and 
 the earth." Gen. 1:1. "Through faith we under- 
 
94 The Fourth Dimension and the Bible 
 
 stand that the worlds were framed by the word 
 of God, so that things which are seen were not 
 made of things which do appear.'* Heb. 11:3. 
 Just as Flatlanders (p. 95) continued to behold 
 as a continuing entity the circular section of the 
 cylinder made by their space (plane) while the 
 cylinder was passing thru the plane, so we today 
 behold our material universe as a continuing en- 
 tity, an aggregate of planets, stars, and nebulas, 
 which aggregate we may conceive of as being the 
 section of a higher-dimensional configuration of 
 surpassing greatness and variety formed by the 
 passing of this transcendent configuration thru our 
 space of three dimensions. If, in our illustration, 
 instead of a circular cylinder some irregular 
 shaped solid had been passed thru the plane then 
 the section seen by the Flatlander would have 
 been, not a circle, but a continually changing irre- 
 gular shaped plane figure ( two-dimensional body) . 
 And so, because every part of our material uni- 
 verse from the smallest particle of matter on 
 earth to the most distant star is in a state of 
 continual change we may assume that these 
 changes are merely three-dimensional manifes- 
 tations of a transcendental configuration of in- 
 finite richness in its composition and form caused 
 by the passing of the latter thru our space of 
 tliree dimensions. According to this view the 
 
Creation of Our Material Universe 95 
 
 events of the six days of creation as recorded in 
 the Bible, as well as all other events which have 
 since taken place and are at this moment taking 
 place, are three-dimensional sections (images, 
 manifestations) of some higher-dimensional ag- 
 gregate (heaven) in which God dwells. 
 
 In the case of the cylinder and the plane we 
 pointed out on pp. 92 and 97 that as the cylinder 
 passed out of the plane the circle which the Flat- 
 landers had observed suddenly vanished before 
 their eyes after its coincidence with the upper base 
 of the cylinder, and was seen by them no more. 
 No trace whatever of the circle remained in the 
 two-dimensional world of the Flatlanders, to them 
 it appeared as the total annihilation of a body 
 (the circle) in their space ( plane), a phenomenon 
 which they could in no way account for. The 
 Flatlanders might well record it in their annals 
 as the end of the circle. And so to us (three- 
 dimensional beings) the end, the total annihi- 
 lation, of our material three-dimensional uni- 
 verse may be considered as coming to pass when 
 the higher-dimensional (transcendent) configura- 
 tion to which we have referred passes out of our 
 three-dimensional space. Numerous Bible pas- 
 sages indicate unmistakably that not only did our 
 universe have a beginning (was created) but that 
 it will also in the fullness of time have an end 
 
g6 The Fourth Dimension and the Bible 
 
 (be annihilated) Ps. 75:3, Mat. 13:40, II Pet. 
 3:10, 13, Rev. 21:1. The Bible does not, and 
 could not, tell us how something can become noth- 
 ing any more than it can tell us how nothing can 
 become something because it was written for hu- 
 man (three-dimensional) beings wholly incapable 
 of comprehending such (higher-dimensional) 
 mysteries. 
 
 EVOLUTION 
 
 Let us assume a solid circular cone (three-di- 
 mensional body) passing thru a plane, as AB, 
 
 a 
 
 V- 
 
 V 
 
 (two-dimensional space) the axis of the cone being 
 always perpendicular to the plane. 
 
 Before the cone reaches any part of the plane, 
 as in the position fig. («), the cone is non-exist- 
 ent to the Flatlanders living in the plane. The 
 instant the apex of the cone enters the plane (fig. 
 (Z?), the point C (the section of the cone made 
 by the plane) will appear to the Flatlanders in 
 
Evolution 97 
 
 that plane coming apparently from nowhere. To 
 them it will be the creation of a point out of 
 nothing. Then as the cone continues to move 
 thru the plane this section (point) will be seen to 
 develop into a circle (as DE) with a continually 
 increasing radius. In other words, the Flat- 
 landers would behold the evolution of a point into 
 a circle. And this evolutionary process would 
 continue until the circular section has grown into 
 the circle FG, the base of the cone. As the cone 
 passes out of the plane the circle will vanish, ap- 
 parently into nowhere. A Flatlander living in 
 the plane AB and observing all that had taken 
 place might well describe it all as the creation, the 
 development by evolution, and the annihilation 
 of a circle; and this without having any concep- 
 tion whatever of the existence of the cone itself 
 or its relation to the phenomenon he had wit- 
 nessed. 
 
 May it not be that all evolutionary processes 
 in our material three-dimensional universe are 
 in like manner three-dimensional sections (images, 
 manifestations) of the elements of some higher 
 dimensional (transcendent) aggregate of infinite 
 complexity? In this concept there is no conflict 
 between the doctrine of instantaneous creation 
 and the theory of evolution, each is a part of the 
 process involved. Both of them, however, ope- 
 
98 The Fourth Dimension and the Bible 
 
 rate In a manner incomprehensible to three-dimen- 
 sional beings, both are miracles to us. To God 
 instantaneous creation and evolutionary processes 
 are equally possible, he can and does operate with 
 either or both according to his Infinite wisdom. 
 To deny the possibility of Instantaneous creation 
 Is to deny the power of God and to cast over- 
 board the concept of evolution is to deny the 
 existence of the numberless examples of evolution- 
 ary processes which are developing today before 
 our very eyes or are unfolded to our minds. 
 
CHAPTER XI 
 
 RAISING OF THE DEAD. SACRAMENT 
 OF THE LORD'S SUPPER 
 
 The soul of man, being higher-dimensional, 
 can leave the human body, which is three-dimen- 
 sional, without doing any violence to the body or 
 leaving any trace on It of a point of emergence. 
 Investigators have often tried to photograph the 
 departing soul of a dying man, and to discover 
 any lesions on his body caused thereby, but always 
 without success. No image of the soul, which is 
 higher-dimensional, can be secured by material 
 (three-dimensional) means; nor can the human 
 body imprison the soul any more than (as we 
 have pointed out) it would be possible to imprison 
 a four-dimensional being in a three-dimensional 
 prison cell or to confine a three-dimensional being 
 within two-dimensional (Flatland) barriers. 
 
 Likewise it follows that a soul may enter a 
 human body without hindrance as far as spacial 
 considerations are concerned, just as a three-di- 
 mensional being may freely enter a closed com- 
 partment in Flatland. The raising of the dead 
 
 99 
 
100 The Fourth Dimension and the Bible 
 
 involves the re-entrance of souls into their for- 
 mer corporeal habitations. What the forces are 
 which can bring this about is a deep mystery to 
 us. We know, however, that Jesus had the power 
 to perform this miracle, and that thru him, also 
 Paul and Peter raised the dead. Jesus restored 
 to life the daughter of Jairus (Mat. 9:16: Mark 
 5 14: Luke 8 :54), the son of the widow of Nain 
 (Luke 7:15) and Lazarus (John 11), who had 
 been four days in his grave. Paul did the same 
 for Eutychus (Acts 20:10) and Peter the same 
 for Tabitha (Acts 9:40). Likewise Elijah (I 
 King 17:17, 23) and Elisha (II King 4:32, 37) 
 raised the dead. The dead came out of their 
 graves on the first Good Friday, later entering 
 the Holy City and appearing unto many. (Mat. 
 27:52). 
 
 That we have no record of the experiences of 
 Lazarus and the others who were raised from the 
 dead, covering the interval between their deaths 
 and their coming to life again, has been a keen 
 disappointment to many Bible readers. In fact, 
 this seeming omission of a description of the de- 
 tails of their existence during this death interval 
 has been used by Bible scoffers not only to cast 
 doubt on the truth of these miracles but also to 
 discredit all Christian beliefs. Where was the 
 immortal part of Lazarus during the four days 
 
Raising of the Dead loi 
 
 In which his body was in the grave, what did he 
 do, what were his experiences? A message from 
 Mars would be of trivial interest to dwellers on 
 earth compared with the importance of the mes- 
 sage that Lazarus should have brought back from 
 the regions beyond the grave. It would have an- 
 swered the great burning question of past, pres- 
 ent and future ages, the question beside which 
 all other questions vanish into insignificance. That 
 Lazarus did not answer this question is evident 
 because if he had it would have been recorded. 
 The people of his day were no less consumed by 
 interest in the answer to this question than we 
 are today. If Lazarus had related his experi- 
 ences during those four fateful days his narrative 
 certainly would have been handed down to us in 
 some form. 
 
 The reason why Lazarus did not answer this 
 question, did not relate his experiences between 
 his death and his return from the grave, was be- 
 cause he could not. Granting that he had a con- 
 scious existence during this four-day period it 
 may have been that of his higher-dimensional 
 being (his soul, his spiritual self) in a space of 
 higher dimensions, a space radically different 
 from our space of three dimensions. The im- 
 pressions he received were therefore the impres- 
 sions received in a space totally foreign to us, 
 
102 The Fourth Dimension and the Bible 
 
 Impressions which could not be described to us 
 because we have no words, no language which 
 will convey such a description to human (three- 
 dimensional) beings. What Is more, any such 
 impressions that Lazarus may have been con- 
 scious of before Jesus raised him from the grave 
 vanished the instant he came to life, that is, the 
 Instant he again became a mortal, a three-dimen- 
 sional being. On being again clothed with his 
 material body his spirit at once became subject to 
 the restrictions and limitations imposed on be- 
 ings living in our space of three dimensions. 
 
 It seems futile therefore to expect that we shall 
 ever receive a message thru human agencies from 
 those who have departed to the great beyond de- 
 scribing their state of existence because such a 
 message can convey no meaning whatever to us. 
 The Bible contains no promise of any such knowl- 
 edge but on the contrary asserts that this must re- 
 main a sealed book to mortals. This does not, 
 however. Interpose any obstacles whatever to di- 
 vine revelations to man, either In the past or In 
 the future, because God is not limited in any man- 
 ner by the restrictions of any space. 
 
 THE SACRAMENT OF THE LORD^S SUPPER 
 
 All the different shades of doctrine held by 
 Christian believers regarding the sacrament of 
 
Sacrament of the Lord's Supper 103 
 
 the Lord's Supper may be roughly divided into 
 three distinct groups. 
 
 Group I. Doctrines which include transubstan- 
 tiation as a fundamental principle. These assert 
 that the bread and wine are literally transformed 
 into the body and blood of Jesus. According to 
 this principle of transubstantiation the believer 
 partakes, not of the bread, but of the actual ma- 
 terial human body of Jesus, and in drinking the 
 cup drinks, not wine, but the actual material human 
 blood of Jesus. 
 
 Group II. Doctrines which hold that the bread 
 and wine only represent or symbolize the body 
 and blood of Jesus. According to these the be- 
 liever partakes only of the material bread and 
 wine, the act being merely a remembrance, a cele- 
 bration commemorating the last supper of Jesus 
 with his disciples. 
 
 Group III. Doctrines which assert that the be- 
 liever in partaking of the bread and wine also 
 by the same act partakes not of the material 
 body and blood of Jesus but of the spiritual body 
 and blood of a resurrected, a glorified Jesus. 
 
 The first of these three groups of doctrines is 
 naturally repellant to us because of the grossness 
 of the act involved, namely, the consumption of 
 human flesh and blood. It is the human body and 
 blood of Jesus, into which the bread and wine 
 
I04 The Fourth Dimension and the Bible 
 
 has been transformed, of which the believer is 
 supposed to partake. 
 
 On the other hand, the second of these three 
 groups of doctrines does not entirely satisfy those 
 who are spiritually minded because it minimizes 
 the sacrament by making it merely a commemora- 
 tion ceremony. This viewpoint is difficult to rec- 
 oncile with the words of Jesus, "this is my body 
 . . . this is my blood." 
 
 Defenders of the doctrine of transubstantiation 
 invariably attempt to justify their belief in it by 
 quoting the above words of Jesus. In so doing 
 they are trying to support their case on the wholly 
 unwarranted assumption that Jesus spoke of his 
 material, his human body and blood. Jesus knew 
 of his coming suffering, death and resurrection. 
 He spoke of the future beyond Golgotha when 
 he said "this do, as oft as ye drink it, in remem- 
 brance of me." In this future there would be 
 no more a material, a human body and blood of 
 Jesus, but only his spiritual, his glorified body 
 and blood. 
 
 The third of the above groups of doctrines 
 therefore is sanctioned by the words of Jesus. 
 It also commends itself to human reason. It is 
 therefore near at hand to assert that Jesus is 
 present in the sacrament of the Lord's Supper 
 as a higher-dimensional being. He descends 
 
Sacrament of the Lord's Supper 105 
 
 from heaven (higher-dimensional space) to com- 
 mune with the believer in our world (three-dimen- 
 sional space). We cannot detect his presence 
 thru our material senses because he is a higher- 
 dimensional being while we are laboring under the 
 restrictions and limitations imposed on three-di- 
 mensional beings. There is no necessity for the 
 spacial presence (in the ordinary sense) of Jesus 
 in order to fulfill his promise to be with us. How 
 much greater and glorious as well as more satis- 
 fying to the believer is not this presence of the 
 spiritual, the glorified (higher-dimensional) body 
 and blood of Jesus than would be the presence of 
 merely his human, his material body and blood! 
 
CHAPTER XII 
 
 DESCRIPTION OF HEAVEN. MIRACLES 
 
 It is evident that we as three-dimensional be- 
 ings cannot have a true conception of what heaven 
 is; being a higher space it is beyond our powers 
 of understanding. If an accurate description of 
 heaven were in some way available we would 
 not understand an iota of it, it would be mean- 
 ingless to us. "For since the beginning of the 
 world men have not heard, nor perceived by the 
 ear, neither hath the eye seen, O God, besides 
 thee, what he hath prepared for him that wait- 
 eth for him." Isa. 64:4. "Eye hath not seen, 
 nor ear heard, neither have entered into the heart 
 of man, the things which God hath prepared for 
 them that love Him." I Cor. 2 :<). Christ spoke 
 to his Apostles of the impossibility of their hav- 
 ing any image or notion of the place to which 
 when he disappeared he would go and whence he 
 would return. Attempts to describe heaven al- 
 ways result in a description in terms of our three- 
 dimensional environment. In this connection it 
 
 106 
 
Description of Heaven 
 
 107 
 
 is significant to note that the Bible usually tells 
 what heaven is not rather than what heaven is. 
 The adjectives are generally negative. We should 
 never lose sight of the fact that the Bible was 
 written for beings living in a three-dimensional 
 space, beings whose knowledge and experiences 
 are wholly limited to that space. Dr. P. Anstadt 
 in his book "Recognition in Heaven" gives the 
 following summary of Bible statements describ- 
 ing heaven : 
 
 Negative Features, or 
 the Things that Will 
 Not Be There 
 
 Indestructible. 
 
 Undefilable. 
 
 Unchangeable. 
 
 No crying. 
 
 No tears 
 
 No pain. 
 
 No sorrow. 
 
 No death. 
 
 No burning sun. 
 
 No cold or heat. 
 
 No night. 
 
 No hunger. 
 
 No thirst. 
 
 No bad men. 
 
 No sin. 
 
 No curse. 
 
 Positive Features, or 
 the Things that Will 
 Be There. 
 
 The city of our God — 
 
 the heavenly Jerusalem. 
 
 Beautiful waters. 
 
 Delicious fruits. 
 
 Sure healing for the na- 
 tions. 
 
 Populous with happy 
 people. 
 
 Beautiful garments. 
 
 Devout worship. 
 
 Enchanting music. 
 
 A just ruler. 
 
 An eternal kingdom. 
 
 The grandest capitol. 
 
 Many mansions. 
 
io8 The Fourth Dimension and the Bible 
 
 MIRACLES 
 
 One reason why some very good people hesi- 
 tate about accepting the miracles of the Bible 
 at their face value is that they assume that such 
 miracles can only be brought about by violating 
 or in some way interfering with the laws of 
 nature, the laws governing our three-dimensional 
 universe. This difficulty vanishes if instead we 
 look upon the miracles of the Bible as the per- 
 fectly logical results of the working out of laws 
 connected with higher spaces, laws which include 
 the lower, the so-called natural laws of our ma- 
 terial universe, just as our space is contained 
 within the higher spaces. We may then consider 
 the miracles performed by Jesus, the apostles and 
 the prophets to consist of momentary revelations 
 by God to mortals of the operations of some of 
 those higher laws of which we are as yet igno- 
 rant. It is not at all probable that the apostles 
 and prophets themselves understood these higher 
 laws, and it was not necessary that they should, 
 any more than that it is necessary for the operator 
 who sends a message by wireless to understand 
 the theory of wireless telegraphy or to have actual 
 knowledge of the medium thru which it is trans- 
 mitted. These higher laws may be the laws of 
 four-dimensional or still higher spaces or they 
 
Miracles 109 
 
 may be laws connected with realms of thought 
 and being of which we now have no conception 
 whatever. To insist that God can make wine out 
 of water only by pouring fermented grape juice 
 into it is ridiculous ! If he were thus limited he 
 would not be God. The rending of the veil of 
 the Temple on the crucifixion of Christ and the 
 speaking with many tongues on the giving out of 
 the Holy Ghost, these may all be revelations to 
 man of higher-dimensional phenomena. 
 
 Swedenborg's involved descriptions of "heav- 
 enly" forms, motions, and mechanics, become 
 somewhat more intelligible when interpreted in 
 terms of higher spaces even if we are not willing, 
 as the author is not, to agree with his conclusions. 
 Helmholtz continually kept the possibility of 
 physical higher spaces before him in his dynami- 
 cal studies, reasonings which are among the class- 
 ics of physical science. Kelvin also felt the pres- 
 sure of the reality of higher spaces so strongly 
 that he declared himself ready to accept them 
 as an explanation of physical phenomena when 
 these could be more consistently explained by such 
 a concept. 
 
CHAPTER XIII 
 
 CONCLUSION 
 
 The new vistas of thought which have been 
 opened to us by the higher space hypothesis have 
 revealed unlimited realms of being and doing, 
 all of which are interrelated in a manner far sur- 
 passing the powers of ingenuity of even a super- 
 man. In the light of this vision, how small and 
 mean, how low and limited, yes, how contempt- 
 ible, does not the materialist appear as he bur- 
 rows into the dust in his attempts to discover the 
 source of the glorious rays of truth and light 
 which radiate from God! As we, in this study 
 of higher spaces, have caught a momentary 
 glimpse of the sublimity and vastness, the glory 
 and grandeur, the symmetry and order, of what 
 must be only a very small part, perhaps only an 
 infinitesimal part, of creation; how preposterous 
 does it not seem, yes, how ridiculous even to sup- 
 pose that all should have come into being thru a 
 succession of fortuitous circumstances, by blind 
 chance, and not thru a supreme intelligence with 
 no limitations of knowledge or power. 
 
Conclusion 1 1 1 
 
 How pitiful and weak is man with his limita- 
 tions of thought and action in the presence of 
 these infinite possibilities ! And yet there are men 
 presumptuous enough to claim that human reason 
 and understanding can account for it all. No 
 matter how wise or powerful a man may be he 
 is but one of billions of humans, to whom this 
 our earth has been or is the dwelling place. And 
 our earth is only one of the smallest of the plan- 
 ets revolving around our sun, every one of which 
 may be inhabited by beings, not necessarily with 
 bodies like ours, because they would have to be 
 adapted to their own physical environment, but 
 perhaps with souls like ours. Our sun, however, 
 is only one of the smaller of the suns, fixed stars, 
 apparently without number, which dot our heav- 
 ens. In some parts of the sky there appear clus- 
 ters of them, as in the Milky Way, so numerous 
 that they appear only as patches of light. Some 
 of these suns are so far away that it takes cen- 
 turies for the light from them to reach us. A few 
 years ago a star of very large magnitude suddenly 
 appeared in our sky. It then grew gradually 
 smaller until now it is not at all conspicuous. As- 
 tronomers told us that this new star really rep- 
 resented a conflagration in the heavens, two large 
 suns had collided, but the collision had taken 
 place over one hundred years before, at the very 
 
112 The Fourth Dimension and the Bible 
 
 time when Napoleon was walking the world like 
 a Colossus, and the flash of light caused by the 
 impact never reached earth until in our day. In 
 fact, there may be suns so far from us and cre- 
 ated so long ago that light from them has not 
 yet reached the earth. Each of these countless 
 suns may have many more and larger planets 
 revolving about them than is the case with our 
 sun, and each planet may be inhabited by beings 
 with souls. All of this vast infinitude that we 
 have been contemplating exists, however, simply 
 in our space of three dimensions. There may 
 exist an infinite number of other three-dimen- 
 sional spaces, each one comprising a universe as 
 great as or greater than our own. But enfolding 
 these three-dimensional spaces as in a garment 
 may be a four-dimensional space, and that may 
 be only one of an infinite number of other pos- 
 sible four-dimensional spaces. Again, these four- 
 dimensional spaces may be enfolded by five-di- 
 mensional spaces; and so on thru spaces of six, 
 seven, etc., dimensions up to space of infinite di- 
 mensions, where God himself dwells subject to no 
 space limitations. 
 
 But now man, at the thought of these possi- 
 bilities, stands aghast, stupified; he exclaims, "it 
 can't be true, it is all too vast, too stupendous to 
 lie within the realms of the possible." Even 
 
Conclusion 113 
 
 some who believe in God are inclined to think that 
 this concept exceeds the bounds of the attainable. 
 But by thus placing a limit to what God can do 
 we are virtually denying that he is God. Instead 
 of being restricted to those possibilities which 
 may be humanly conceived, God is able to put our 
 unbelief to shame by exhibitions of power of 
 which we have now not the slightest conception. 
 
INDEX 
 
 Abbott, 41 
 
 Abraham, 73 
 
 Adam's fall, 89 
 
 Alcohols, 67 
 
 Analogy, 17, 23, 76, 85 
 
 An Episode in Flatland, 41 
 
 Angels, 45, 77 
 
 Animal kingdom, 64 
 
 Annihilation of the universe, 
 
 93 
 Anstadt, 107 
 Apostles, 106 
 Apparition, 43, 58 
 Archangels, 45 
 Astronomers, 80, iii 
 Atoms, 66 
 
 Atonement, Christ's, 89 
 Automobile, plane section of. 
 
 Axis of rotation, 57, 63 
 
 Beethoven, 33 
 
 Bible, 17, 44, 45, 52, 73, 77, 81, 
 89, 90, 91, 95> 100, 102, 
 107, 108 
 Biologist, 77 
 Birth, 70, 82 
 Blind, 33 
 
 Blood of Jesus, 103 
 Bodies, celestial, 80 
 Body of Jesus, 103 
 Boundaries, Flatland, 38 
 
 Lineland, 42 
 
 of space units, 22 
 Bragdon, 72 
 
 Carbon, 67 
 
 Celestial bodies, 80 
 
 Chained shadow watchers of 
 
 Plato, 82 
 Chemistry, 66 
 Christ, 45, 52, 88, 106, 109 
 
 atonement, 89 
 Circle, circumference of, 12 
 Cone, Flatland sections of, 69, 
 
 73, 96 
 Concept of space, iiflF. 
 Configurations, higher-dimen- 
 sional, 79 
 
 rotation of symmetrical, 54 
 Conservation of energy, 81 
 
 of matter, 81 
 Creation, Ex-Nihilo, 31 
 
 instantaneous, 97 
 
 of the universe, 91 
 Crucifixion of Christ, 45, 109 
 Cruse, widow's, 81 
 Crystals, 68 
 Cube, 19 
 
 Flatland sections of a, 75, 79 
 
 motion perpendicular to a, 
 20, 48 
 Curved spaces, 12, 14, 15 
 Cylinder, Flatland sections of 
 a, 91 
 
 Daughter of Jairus, 100 
 Dead, raising of the, 99 
 Deaf, 33 
 Death, 70, 82 
 Definition of space, 22 
 Dextrose, 64 
 
 "5 
 
ii6 
 
 Index 
 
 Dimensions of space, iiff 
 
 geometry of four, 29 
 
 rotation, reversal, twist in, 
 61 
 
 theorems in four, 58 
 
 time as fourth, 71 
 Distorted, 21, 63 
 Dreams and dreamland, 72 
 
 Earth, first, 88 
 
 Elements of the tesseract, 21 
 Elijah, 44, 45, 81, loo 
 Elisha, 100 
 
 Embryos, plane sections of, 78 
 Empirical Science, 29 
 Energy, conservation of, 81 
 Enoch, 44, 45, 81. 
 Eternal, 73 
 Eutychus, 100 
 Evanescent, 76 
 Evolution, 96 
 
 Existence of higher spaces, 23, 
 25 
 
 First earth, heaven, 88 
 Fishes, the loaves and, 81 
 Five-dimensional space, 16 
 Flatland, 37, 48, 55, 69!?, 84, 
 
 87 
 boundaries, 38 
 houses, 39 
 life in, 37ff 
 
 sections of a helix, 73 
 sections of a cone, 69, 73, 96 
 sections of a cube, 75, 79 
 sections of a cylinder, 91 
 
 Flatlanders, 37ff, SS^) 92 
 
 Foot, 84 
 
 Four-dimensional spaces, 16 
 space unit, 20 
 
 Four dimensions, geometry of, 
 
 life in space of, 48 
 rotation, reversal, twist in, 
 
 61 
 theorems in, 58 
 
 Fourth dimension, time as the, 
 
 7^ 
 Freedom of motion of points, 
 
 iiflF 
 
 Generation of space units, i8ff 
 Geographical latitude and 
 
 longitude, 14 
 Geometric units in each space, 
 
 i8ff 
 Geometry not the same as 
 
 space, 28 
 Geometry of four dimensions, 
 
 29, 58 
 theorems in plane and solid, 
 
 58 
 Glass globe, 49 
 Grammatical tenses, 73 
 Grave, loi 
 Gravitation, 53 
 Gruber, 31 
 God, 31, 35, 44, 45, 73, 77, 
 
 85, 88, 93, 95, 98, 102, 106, 
 
 109 
 Good Friday, 100 
 Gunnell, 26 
 
 Heaven, 44, 86flF, 89 
 
 description of, 106 
 
 first, second, third, 88 
 
 of heavens, 89 
 
 recognition in, 107 
 Heels, 84 
 
 Helix, Flatland sections of, 73 
 Hells, 86ff 
 Helmholtz, 109 
 
 Higher-dimensional configura- 
 tions 79 
 Higher spaces, 16 
 
 existence of, 23, 25 
 
 divided by lower spaces, 
 23 
 Hinton, 41 
 Holy City, 100 
 Holy Ghost, 109 
 Hyper-spaces, 16 
 
Index 
 
 117 
 
 Illusory and real, 82 
 Image in a mirror, 61 
 
 mental, 71 
 Imagination, 28, 52 
 Incomprehensible, 25, 26 
 Infinite number of points, ii 
 
 number of one-, two-, and 
 three-dimensional spaces, 
 
 I2ff. 
 
 space, 25 
 Infinity, 25 
 
 Inhabitants of other spaces, 36 
 Ink stand, 14 
 Instantaneous creation, 97 
 Isomeric substances, 65 
 
 Jairus, daughter of, icxj 
 
 Jericho, 81 
 
 Jesus, 73, 8r, 100, i02ff 
 
 body and blood of, 103 
 Jews, 52 
 
 Kant, 31 
 Kelvin, 109 
 
 Kingdoms, animal and vege- 
 table, 64 
 
 Ladder, 16 
 
 Latitude, geographic, 14 
 
 parallels of, 14 
 Laws, natural, 108 
 Lazarus, looff 
 Left-handed, 57 
 Levulose, 64 
 Life, transitory character of, 
 
 in Flatland, 37 
 
 in four-dimensional space, 
 
 in Lineland, 41 
 Limits of space, 25 
 Line, motion perpendicular to 
 
 a, 19, 47 
 Lineland, 37!? 
 life in, 41 
 boundaries, 42 
 
 Linelanders, 375, 48, 54, 55 
 Lines, rotation of symmetrical, 
 
 of longitude, 14 
 Loaves and fishes, 81 
 Longitude, geographical, 14 
 Looking glass world, 62 
 Lord's Supper, 102 
 
 Manna in the wilderness, 81 
 Manning, 31 
 Mars, message from, loi 
 Mass of the universe, 79 
 Materialist, no 
 Mathematics, pure, 29 
 Matter, conservation of, 79 
 Memory, 72 
 Mental image, 71 
 
 vision, 71, 73 
 Message from Mars, loi 
 Microscope, 78 
 Milky Way, in 
 Miracles, 40, 45, 108 
 Mirror image, 57, 61 
 Molecules, 66 
 Moses, 45 
 Motion of points, 11 if 
 
 perpendicular to a line or 
 square, 19, 47 
 
 perpendicular to a cube, 20, 
 
 Nain, son of the widow of, 100 
 
 N-dimensional space, 16 
 
 Napoleon, 112 
 
 Natural laws, 108 
 
 Nebulae, 94 
 
 Newcomb, 41, 48 
 
 Number of points, 11 
 
 Oil in the widow's cruse, 81 
 One-dimensional spaces, iiff 
 space units, 18 
 
 Parallels of latitude, 14 
 Paul, 46, 100 
 
ii8 
 
 Index 
 
 Personality, 76 
 
 Peter, 100 
 
 Physics, 30 
 
 Plane geometry, theorems in, 
 
 Plane sections of an automo- 
 bile, 80 
 of an embryo, 78 
 Planets, 79, 94 
 Plato, 82 
 
 Points, freedom of motion of, 
 I iff 
 moving perpendicular to 
 spaces, 47ff 
 Polarization of light, 64 
 Prophets, 45, 108 
 Pure mathematics, 29 
 
 Raising of the dead, 99 
 Real, the illusory and the, 82 
 Recognition in heaven, 107 
 Red Sea, 81 
 
 Rending of the veil, 109 
 Revelations of St. John, 46 
 Reversal in space of four di- 
 mensions, 61 
 Right handed, 57 
 Robber, 49 
 
 Rotation of symmetrical lines, 
 triangles, tetrahedrons, 
 
 . 54ff 
 
 in space of four dimensions, 
 61 
 Rubber band or ball, 63 
 
 Sacrament of the Lord's Sup- 
 per, 102 
 Satan, 90 
 Saul, 99 
 
 Schubert, 59, 67 
 Scientific romances, 41 
 Second heaven, 88 
 Sections of embryos, 78 
 Sense perceptions, 27 
 Shadow watchers of Plato, 82 
 Solar system, 25 
 
 Soles, 84 
 
 Solid geometry, theorems in, 
 
 58 
 Son of the Widow of Nain, 
 
 100 
 Soul, 99 
 Space defined, 22 
 
 boundaries of, 22 
 Space, zero-, one-, two-, three-, 
 four-, five-, higher-, or 
 hyper-, iiff 
 curved, i2ff 
 
 life in four-dimensional, 48 
 limitations, 30 
 limits, 25 
 units, i8ff 
 Spaces as heavens and hells, 
 86ff 
 existence of higher, 25 
 higher spaces divided by 
 
 lower, 23 
 inhabitants of other, 36 
 Spectrum, 34 
 Sphere, surface of, 14, 48 
 Spiritual, 35,^ 53, 76, 103 ^ 
 Square, motion perpendicular 
 
 to a, 19, 47 
 Stars, 94, III 
 Suns, 79, III 
 Supernatural, 42, 53 
 Swedenborg, 109 
 Symbolic, 17, 20 
 Symmetrical configurations, 79 
 
 rotation of, 54, 56, 60 
 Symmetry in animal and vege- 
 table kingdoms, 64 
 
 Tabitha, 100 
 Table top, 14 
 Tartaric acid, 67 
 Taylor, 71 
 Telegraph, 26 
 Telephone, 26 
 
 Temple, rending of the veil of, 
 109 
 
Index 
 
 119 
 
 Tenses, grammatical, 73 
 Tesseract, 20, 21 
 Tetrahedrons, 59, 60, 65 
 The Fairyland of Geometry, 
 
 41 
 
 Theorems in plane, solid and 
 four-dimensional geome- 
 try, 58 
 
 Third heaven, 88 
 
 Thomas, 52 
 
 Three-dimensional spaces, 14 
 space units, 19 
 
 Time as the fourth dimension, 
 
 71 
 Tire tube, 64 
 Transitory character of life, 
 
 76 , . . 
 Transubstantiation, doctrine 
 
 of, 103 
 
 Triangles, 56, 57 
 
 Twist in space of four dimen- 
 sions, 6 1 
 
 Two-dimensional spaces, 13 
 unit, 19 
 vision, 39 
 
 Units of space, i8ff 
 
 boundaries of space, 22 
 
 Universe, annihilation of, 93 
 creation of, 91 
 mass of, 79 
 
 Veil of the Temple, rending 
 
 of, 109 
 Vision, mental, 71, 73 
 Visualizing, 28 
 
 Wilderness, manna in the, 81 
 Widow's cruse, 81 
 Widow of Nain, son of, 100 
 Wireless telegraph, 26, 108 
 telephone, 26 
 
 Zero-dimensional space, 11 
 space unit, 18 
 

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