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- Herausgeber: Booksell-Verlag
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
In 'The Fourth Dimension,' Charles Howard Hinton masterfully explores the concept of higher spatial dimensions, venturing beyond the limits of conventional three-dimensional thinking. Written in the late 19th century, Hinton's work combines scientific inquiry with philosophical reflection, utilizing a unique blend of mathematical precision and imaginative narrative. Through engaging metaphors and thought experiments, he attempts to elucidate the complexities of multi-dimensional spaces, urging readers to expand their perceptions and consider the implications of dimensions beyond their tangible experience. Hinton's literary style is both accessible and intellectually stimulating, reflecting the era's burgeoning interest in geometry and pioneering developments in physics. Charles Howard Hinton, often regarded as a precursor to later theorists of higher dimensions, was both a mathematician and a writer deeply influenced by the scientific landscapes of his time. His explorations of dimensionality were not solely theoretical; they were also shaped by his personal experiences and philosophical inquiries into the nature of reality. Hinton's background in mathematics, coupled with his artistic inclinations, allowed him to frame complex concepts in a manner that resonates with both scholars and lay readers. 'In The Fourth Dimension' is a compelling read for anyone fascinated by the interplay of mathematics, philosophy, and science. Hinton's innovative ideas invite readers to reconsider the fabric of reality itself, making this work not only a significant scholarly contribution but also a profound intellectual adventure that is as relevant today as it was over a century ago. In this enriched edition, we have carefully created added value for your reading experience: - A succinct Introduction situates the work's timeless appeal and themes. - The Synopsis outlines the central plot, highlighting key developments without spoiling critical twists. - A detailed Historical Context immerses you in the era's events and influences that shaped the writing. - A thorough Analysis dissects symbols, motifs, and character arcs to unearth underlying meanings. - Reflection questions prompt you to engage personally with the work's messages, connecting them to modern life. - Hand‐picked Memorable Quotes shine a spotlight on moments of literary brilliance. - Interactive footnotes clarify unusual references, historical allusions, and archaic phrases for an effortless, more informed read.
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Veröffentlichungsjahr: 2022
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The Fourth Dimension
Table of Contents
Introduction
To imagine space beyond the space we inhabit is to test the very boundaries of perception and reason. Charles Howard Hinton’s The Fourth Dimension invites readers into that test with a blend of rigor and wonder, asking not merely what a higher dimension might be, but how a mind like ours could ever come to picture it. Approached as both challenge and guide, the book encourages you to treat imagination as a disciplined instrument. It proposes that thinking differently about space can reorder familiar habits of seeing, and that careful practice can stretch common intuition into unfamiliar, enlightening forms.
Hinton, a British mathematician and writer active from the late nineteenth into the early twentieth century, worked at the crossroads of popular science and speculative literature. The Fourth Dimension belongs to the tradition often called the scientific romance: a genre that uses narrative and thought experiment to explore scientific ideas for general readers. First developed in essays and stories across the 1880s–1900s and gathered under this title, his exploration of higher space reflects an era fascinated by geometry, psychology, and epistemology. Rather than presenting a textbook, Hinton offers a sustained attempt to render an abstract concept mentally graspable through method, analogy, and patient instruction.
The book’s premise is disarmingly simple: if we can understand how a two-dimensional being might infer a third dimension, we can use similar reasoning to reach toward a fourth. From that starting point, Hinton builds a sequence of thought experiments, stepwise visualizations, and brief speculative sketches designed to cultivate new habits of attention. The narrative voice moves between careful pedagogy and imaginative conjecture, encouraging active participation. The mood is exploratory and earnest, with an undercurrent of intellectual adventure. Readers are not handed a finished system; they are invited into a practice that steadily refines intuition about forms that cannot be directly seen.
Hinton’s method emphasizes disciplined visualization. He often begins with familiar figures and processes—lines, planes, solids, transformations—and then extends them by analogy to higher-dimensional counterparts. The resulting explanations are precise yet flexible, making room for illustrative vignettes when instruction alone might prove dry. The pacing assumes diligence: ideas recur from new angles, each pass consolidating a skill before advancing. Expect a hands-on experience that rewards sketching, re-reading, and reflection. The structure resembles a training regimen for imagination, balancing method with play, and insisting that conceptual breakthroughs are earned incrementally, as earlier insights are woven into more demanding exercises.
Beneath its geometric program lies a set of philosophical questions. What is the relation between appearance and reality when perception is bounded by our senses? How can symbols, diagrams, and language extend thought without misleading it? Where is the boundary between legitimate inference and unfounded speculation? Hinton treats the fourth dimension both as a mathematical possibility and as a lens on epistemology, showing how perspective governs what we take to be real. The dimensional analogy doubles as a meditation on intellectual humility and aspiration, suggesting that careful methods can carry us beyond habitual limits while reminding us how provisional our grasp remains.
Although developed over a century ago, the book’s concerns feel contemporary. Written around the turn of the twentieth century, it captures a moment when science and literature cooperated to expand public imagination. Today, questions about modeling unseen structures animate fields from theoretical physics and higher-dimensional mathematics to data visualization and immersive technologies. Hinton’s emphasis on mental modeling, analogy, and verification through consequences rather than direct sight remains pertinent. The Fourth Dimension matters not because it settles debates, but because it equips readers to think systematically about the unfamiliar—an ability as valuable in modern inquiry as it was in Hinton’s day.
Approach this work as both exercise and exploration. Read slowly, let analogies do steady work, and measure progress by the clarity with which you can manipulate ideas, not by speed. The reward is a cultivated flexibility of mind: the sense that methods can be learned for imagining what cannot be directly observed. Hinton supplies tools rather than doctrines, and their usefulness grows with practice. If you come prepared to question your assumptions about space, perception, and explanation, The Fourth Dimension becomes a companion in disciplined curiosity, demonstrating how new ways of seeing can illuminate not just geometry but the habits that shape understanding itself.
Synopsis
The Fourth Dimension by Charles Howard Hinton is a collection of essays that introduces and develops the idea of space with four dimensions. Written for a general but diligent reader, it combines elementary geometry, illustrative analogies, and exercises in visualization. Hinton begins by motivating the subject and defining a fourth dimension as a direction independent from length, breadth, and height, approached by extending the logic used to ascend from a line to a plane and to a solid. He frames the inquiry as a disciplined training of the mind rather than a speculative novelty, proposing that a consistent higher spatial picture can illuminate problems of perception, physics, and philosophical outlook.
He proceeds by the method of dimensional analogy. Points generate lines by motion, lines sweep planes, planes bound solids; from this pattern, a solid may be regarded as bounding a higher solid in four-space. Perpendicularity and measurement generalize, and properties such as boundaries, symmetry, and continuity follow regular rules. Hinton emphasizes that proofs in three dimensions often have counterparts one step higher, allowing readers to infer plausible properties of four-dimensional figures. He clarifies terminology and assumptions, distinguishes between mathematical necessity and illustrative convenience, and establishes the central thesis that the fourth dimension does not violate known geometry but extends it by a consistent, incremental step.
With the foundations set, Hinton introduces projection and section as the main tools for thinking four-dimensionally. Just as a three-dimensional object casts a two-dimensional shadow or yields a sequence of planar cross-sections, a four-dimensional body can be studied through its three-dimensional shadows and slices. The familiar thought experiments of beings confined to a plane supply analogies for our situation. He describes how a passage of a higher object through our space would appear as a succession of changing three-dimensional forms, including sudden growth, shrinkage, or apparent interpenetration without fracture, and how a higher observer could survey concealed interiors and separated regions at once.
To develop a workable mental model, Hinton proposes systematic exercises using colored cubes and ordered arrays. By memorizing the positions and labels of blocks in a cubical lattice, the reader trains to apprehend complex spatial structure at a glance. He extends this regimen to the net and projection of the hypercube, or tesseract, and introduces new directional terms, such as ana and kata, to mark the two additional opposite directions in four-space. The exercises culminate in assembling, mentally, the faces, edges, and cells of the tesseract and tracking how they correspond under projection, laying a concrete basis for further reasoning.
Having supplied methods, Hinton discusses four-dimensional motions and transformations. He explains that rotations about higher axes would alter three-dimensional bodies in characteristic ways, accounting for effects that are impossible under ordinary movements. A solid might reverse handedness, pass through itself, or exchange inside and outside without cutting when guided along the fourth direction. Linked forms could be disengaged by a displacement in the new dimension. These scenarios are presented to illustrate kinematic consequences rather than to claim actual occurrences, emphasizing the logical difference between projection effects in our space and genuine higher movements in an expanded spatial framework.
The book then explores possible bearings on physical theory. Hinton surveys contemporary ideas about matter and medium, asking how electrical, optical, or gravitational phenomena might be reinterpreted if four-space underlies observables. He outlines how forces inverse in distance, distributions of matter, and field lines could be modeled within an extended geometry, and how apparent anomalies might become regular from a higher viewpoint. While careful to separate established results from conjecture, he argues that adopting four-dimensional assumptions can generate testable questions, simplify descriptions, or reveal hidden symmetries, even if decisive empirical confirmation remains beyond the immediate scope of the essays.
Interwoven with the geometric program are reflections on perception and the cultivation of character. Hinton maintains that training the imagination to conceive four-space fosters a more objective stance toward personal experience. He proposes that widening the mental horizon weakens narrow self-reference, encourages impartial judgment, and supports an ethical disposition oriented to larger wholes. Essays on casting out the self use spatial analogies to frame detachment not as denial but as reorganization of attention. The fourth dimension thus functions as a guiding image for expanding awareness, aligning intellectual discipline with a broadened sense of responsibility and coherence in life.
Pedagogically, Hinton outlines a graduated course of practice. Beginning with careful study of simple forms and their projections, the reader proceeds to memorize arrays, visualize rotations, and manipulate composite structures without external aids. He stresses firmness in naming, consistent orientation, and daily repetition, warning against vague picturing or fatigue. The aim is a stable internal apparatus for spatial reasoning, applicable beyond the particular models used. Throughout, he notes the limits of analogy, recommends corroborating geometric theorems wherever possible, and treats imaginative skill as a complement to analytic proof rather than a replacement, binding intuition to method.
The collection closes by restating its central message: the fourth dimension offers a coherent extension of spatial thought that can be approached by ordinary minds through disciplined practice. By linking analogy, projection, and exercise, Hinton assembles a toolkit for grasping higher forms, suggests avenues for scientific reinterpretation, and associates the effort with ethical clarification. The narrative advances from definition, through technique, to implication, without demanding adherence to any speculative system. Readers are invited to test the constructions, adopt what proves useful, and regard four-dimensional thinking as both a practical mental art and a framework for enlarged understanding.
Historical Context
Charles Howard Hinton’s The Fourth Dimension (issued in 1904, compiling essays first published from 1880 onward) is anchored in the late Victorian and early Edwardian scientific milieu rather than in a fictional place. Its intellectual setting stretches from London’s colleges and journals to the laboratories and lecture halls of the United States. The work grows out of Britain’s high-era of industrial modernity and imperial confidence, when mathematics and physics were rapidly professionalizing. It also reflects transatlantic networks: Hinton’s years in Britain, a late‑1880s sojourn in Japan, and a 1890s–1900s career in the United States. The book’s “place” is thus the international arena of mathematical research, popular science, and pedagogy circa 1880–1904.
A central historical foundation is the nineteenth‑century revolution in geometry. Carl F. Gauss privately considered non‑Euclidean possibilities in the 1820s, but public breakthroughs came with Nikolai I. Lobachevsky’s papers (Kazan, 1829–1830) and János Bolyai’s appendix to his father’s Tentamen (Marosvásárhely/Târgu Mureș, 1832). Bernhard Riemann’s 1854 Habilitationsvortrag at the University of Göttingen reformulated geometry as the study of manifolds with variable curvature. These events displaced Euclid’s Fifth Postulate from its privileged status and opened rigorous speculation about spaces beyond three dimensions. Hinton’s book translates this upheaval for educated readers, arguing that a fourth spatial dimension is a legitimate mathematical extension with cognitive and explanatory power.
Technical advances in higher‑dimensional geometry made Hinton’s program tangible. The Swiss mathematician Ludwig Schläfli developed the theory of polytopes in four and higher dimensions around 1852–1853 (published posthumously in 1901), giving concrete models—analogues of polygons and polyhedra in 4D. Hermann Grassmann’s Ausdehnungslehre (1844) and William Rowan Hamilton’s discovery of quaternions in Dublin in 1843 supplied algebraic tools for reasoning beyond 3D. In 1888 Hinton coined the term “tesseract” and proposed pedagogical “colored cubes” (notably sets of 81 cubes) to train spatial imagination. The Fourth Dimension draws on this lineage, presenting exercises and thought experiments that mirror Schläfli’s and Grassmann’s abstractions in accessible, didactic form.
The British mathematical culture of the era also shaped the work. Cambridge figures such as Arthur Cayley and J. J. Sylvester advanced projective and invariant theory; Peter Guthrie Tait’s Treatise on Quaternions (1867) popularized higher‑dimensional algebra; and Felix Klein’s Erlangen Program (1872) unified geometries via transformation groups. Physicists and philosophers—James Clerk Maxwell (Treatise, 1873) and Hermann von Helmholtz (1868 essay on geometric axioms)—linked geometry to physical cognition. Hinton’s early essays appeared in the Dublin University Magazine in 1880 and later in Scientific Romances (1884; 1896). The Fourth Dimension consolidates that tradition, presenting higher space as a natural outgrowth of British and continental debates about the foundations of geometry and perception.
A powerful social movement intersecting Hinton’s project was the late‑Victorian surge of spiritualism and occult inquiry. The Theosophical Society (founded New York, 1875) and the Society for Psychical Research (London, 1882) sought scientific frameworks for phenomena such as clairvoyance, often invoking “higher planes” or extra dimensions. Public lectures and periodicals debated whether a fourth dimension might explain apparitions or telepathy. Hinton acknowledged the cultural allure of hyperspace yet insisted on rigorous, mathematical visualization. The Fourth Dimension thus positions the concept as a disciplined cognitive tool—appropriating the era’s fascination with the unseen while resisting its drift into unverifiable mysticism.
Personal events connected Hinton to broader currents of modernization and social reform. He married Mary Ellen Boole, daughter of logician George Boole, in 1880. After a widely reported bigamy conviction in England in 1886, he left for Japan, where the Meiji state—guided by the 1872 Gakusei (Education Order) and institutions such as the 1877 Imperial University of Tokyo—was importing Western science. Hinton taught during the late 1880s in this reformist climate. The encounter with Meiji educational policy and precision pedagogy reinforced his view that disciplined training (like his cube systems) could reshape perception. The Fourth Dimension carries that conviction, presenting higher‑space reasoning as a learnable, modern skill.
Hinton’s subsequent American career unfolded amid the Gilded Age and Progressive Era emphasis on applied science. By the mid‑1890s he was in U.S. academic and governmental circles; in 1897 he unveiled a gunpowder‑powered pitching machine for Princeton’s baseball team, emblematic of experimental ingenuity. He later worked in Washington, D.C., including at the U.S. Patent Office, and died there in 1907 after collapsing at a scientific gathering. The 1904 issuance of The Fourth Dimension in the United States met an audience attuned to efficiency, standardization, and technical optimism. The book’s practical visual drills and concrete models aligned with American laboratory culture and engineering pedagogy.
The Fourth Dimension functions as a critique of period certainties by challenging the limits of empirical common sense and social habit. It exposes how Victorian authority—rooted in everyday 3D intuition, class‑coded educational access, and disciplinary silos—can hinder discovery. By insisting that citizens can train themselves to perceive higher‑space relations, Hinton implicitly contests elite monopolies on advanced knowledge and the complacency of materialist dogma. His careful separation of mathematical hyperspace from occult claims also rebukes the era’s credulity and sensationalism. The book advocates disciplined imagination as a civic virtue, urging institutions to value cognitive reform as much as mechanical progress.
The Fourth Dimension
PREFACE
I have endeavoured to present the subject of the higher dimensionality of space in a clear manner, devoid of mathematical subtleties and technicalities. In order to engage the interest of the reader, I have in the earlier chapters dwelt on the perspective the hypothesis of a fourth dimension opens, and have treated of the many connections there are between this hypothesis and the ordinary topics of our thoughts.
A lack of mathematical knowledge will prove of no disadvantage to the reader, for I have used no mathematical processes of reasoning. I have taken the view that the space which we ordinarily think of, the space of real things (which I would call permeable matter), is different from the space treated of by mathematics. Mathematics will tell us a great deal about space, just as the atomic theory will tell us a great deal about the chemical combinations of bodies. But after all, a theory is not precisely equivalent to the subject with regard to which it is held. There is an opening, therefore, from the side of our ordinary space perceptions for a simple, altogether rational, mechanical, and observational way of treating this subject of higher space, and of this opportunity I have availed myself.
The details introduced in the earlier chapters, especially in Chapters VIII., IX., X., may perhaps be found wearisome. They are of no essential importance in the main line of argument, and if left till Chapters XI. and XII. have been read, will be found to afford interesting and obvious illustrations of the properties discussed in the later chapters.
My thanks are due to the friends who have assisted me in designing and preparing the modifications of my previous models, and in no small degree to the publisher of this volume, Mr. Sonnenschein, to whose unique appreciation of the line of thought of this, as of my former essays, their publication is owing. By the provision of a coloured plate, in addition to the other illustrations, he has added greatly to the convenience of the reader.
C. Howard Hinton.
THE FOURTH DIMENSION
CHAPTER IFOUR-DIMENSIONAL SPACE
There is nothing more indefinite, and at the same time more real, than that which we indicate when we speak of the “higher.” In our social life we see it evidenced in a greater complexity of relations. But this complexity is not all. There is, at the same time, a contact with, an apprehension of, something more fundamental, more real.
With the greater development of man there comes a consciousness of something more than all the forms in which it shows itself. There is a readiness to give up all the visible and tangible for the sake of those principles and values of which the visible and tangible are the representation. The physical life of civilised man and of a mere savage are practically the same, but the civilised man has discovered a depth in his existence, which makes him feel that that which appears all to the savage is a mere externality and appurtenage to his true being.
Now, this higher—how shall we apprehend it? It is generally embraced by our religious faculties, by our idealising tendency. But the higher existence has two sides. It has a being as well as qualities. And in trying to realise it through our emotions we are always taking the subjective view. Our attention is always fixed on what we feel, what we think. Is there any way of apprehending the higher after the purely objective method of a natural science? I think that there is.
Plato, in a wonderful allegory, speaks of some men living in such a condition that they were practically reduced to be the denizens of a shadow world. They were chained, and perceived but the shadows of themselves and all real objects projected on a wall, towards which their faces were turned. All movements to them were but movements on the surface, all shapes but the shapes of outlines with no substantiality.
Plato uses this illustration to portray the relation between true being and the illusions of the sense world. He says that just as a man liberated from his chains could learn and discover that the world was solid and real, and could go back and tell his bound companions of this greater higher reality, so the philosopher who has been liberated, who has gone into the thought of the ideal world, into the world of ideas greater and more real than the things of sense, can come and tell his fellow men of that which is more true than the visible sun—more noble than Athens, the visible state.
Now, I take Plato’s suggestion; but literally, not metaphorically. He imagines a world which is lower than this world, in that shadow figures and shadow motions are its constituents; and to it he contrasts the real world. As the real world is to this shadow world, so is the higher world to our world. I accept his analogy. As our world in three dimensions is to a shadow or plane world, so is the higher world to our three-dimensional world. That is, the higher world is four-dimensional; the higher being is, so far as its existence is concerned apart from its qualities, to be sought through the conception of an actual existence spatially higher than that which we realise with our senses.
Here you will observe I necessarily leave out all that gives its charm and interest to Plato’s writings. All those conceptions of the beautiful and good which live immortally in his pages.
All that I keep from his great storehouse of wealth is this one thing simply—a world spatially higher than this world, a world which can only be approached through the stocks and stones of it, a world which must be apprehended laboriously, patiently, through the material things of it, the shapes, the movements, the figures of it.
We must learn to realise the shapes of objects in this world of the higher man; we must become familiar with the movements that objects make in his world, so that we can learn something about his daily experience, his thoughts of material objects, his machinery.
The means for the prosecution of this enquiry are given in the conception of space itself.
It often happens that that which we consider to be unique and unrelated gives us, within itself, those relations by means of which we are able to see it as related to others, determining and determined by them.
Thus, on the earth is given that phenomenon of weight by means of which Newton brought the earth into its true relation to the sun and other planets. Our terrestrial globe was determined in regard to other bodies of the solar system by means of a relation which subsisted on the earth itself.
And so space itself bears within it relations of which we can determine it as related to other space. For within space are given the conceptions of point and line, line and plane, which really involve the relation of space to a higher space.
Where one segment of a straight line leaves off and another begins is a point, and the straight line itself can be generated by the motion of the point.
One portion of a plane is bounded from another by a straight line, and the plane itself can be generated by the straight line moving in a direction not contained in itself.
Again, two portions of solid space are limited with regard to each other by a plane; and the plane, moving in a direction not contained in itself, can generate solid space.
Thus, going on, we may say that space is that which limits two portions of higher space from each other, and that our space will generate the higher space by moving in a direction not contained in itself.
Another indication of the nature of four-dimensional space[1] can be gained by considering the problem of the arrangement of objects.
If I have a number of swords of varying degrees of brightness, I can represent them in respect of this quality by points arranged along a straight line.
If I place a sword at A, fig. 1, and regard it as having a certain brightness, then the other swords can be arranged in a series along the line, as at A, B, C, etc., according to their degrees of brightness.
If now I take account of another quality, say length, they can be arranged in a plane. Starting from A, B, C, I can find points to represent different degrees of length along such lines as AF, BD, CE, drawn from A and B and C. Points on these lines represent different degrees of length with the same degree of brightness. Thus the whole plane is occupied by points representing all conceivable varieties of brightness and length.
Bringing in a third quality, say sharpness, I can draw, as in fig. 3, any number of upright lines. Let distances along these upright lines represent degrees of sharpness, thus the points F and G will represent swords of certain definite degrees of the three qualities mentioned, and the whole of space will serve to represent all conceivable degrees of these three qualities.
If now I bring in a fourth quality, such as weight, and try to find a means of representing it as I did the other three qualities, I find a difficulty. Every point in space is taken up by some conceivable combination of the three qualities already taken.
To represent four qualities in the same way as that in which I have represented three, I should need another dimension of space.
Thus we may indicate the nature of four-dimensional space by saying that it is a kind of space which would give positions representative of four qualities, as three-dimensional space gives positions representative of three qualities.
CHAPTER IITHE ANALOGY OF A PLANE WORLD
At the risk of some prolixity I will go fully into the experience of a hypothetical creature confined to motion on a plane surface. By so doing I shall obtain an analogy which will serve in our subsequent enquiries, because the change in our conception, which we make in passing from the shapes and motions in two dimensions to those in three, affords a pattern by which we can pass on still further to the conception of an existence in four-dimensional space.
A piece of paper on a smooth table affords a ready image of a two-dimensional existence. If we suppose the being represented by the piece of paper to have no knowledge of the thickness by which he projects above the surface of the table, it is obvious that he can have no knowledge of objects of a similar description, except by the contact with their edges. His body and the objects in his world have a thickness of which however, he has no consciousness. Since the direction stretching up from the table is unknown to him he will think of the objects of his world as extending in two dimensions only. Figures are to him completely bounded by their lines, just as solid objects are to us by their surfaces. He cannot conceive of approaching the centre of a circle, except by breaking through the circumference, for the circumference encloses the centre in the directions in which motion is possible to him. The plane surface over which he slips and with which he is always in contact will be unknown to him; there are no differences by which he can recognise its existence.
But for the purposes of our analogy this representation is deficient.
A being as thus described has nothing about him to push off from, the surface over which he slips affords no means by which he can move in one direction rather than another. Placed on a surface over which he slips freely, he is in a condition analogous to that in which we should be if we were suspended free in space. There is nothing which he can push off from in any direction known to him.
Let us therefore modify our representation. Let us suppose a vertical plane against which particles of thin matter slip, never leaving the surface. Let these particles possess an attractive force and cohere together into a disk; this disk will represent the globe of a plane being[2]. He must be conceived as existing on the rim[1q].
Let 1 represent this vertical disk of flat matter and 2 the plane being on it, standing upon its rim as we stand on the surface of our earth. The direction of the attractive force of his matter will give the creature a knowledge of up and down, determining for him one direction in his plane space. Also, since he can move along the surface of his earth, he will have the sense of a direction parallel to its surface, which we may call forwards and backwards.
He will have no sense of right and left—that is, of the direction which we recognise as extending out from the plane to our right and left.
The distinction of right and left is the one that we must suppose to be absent, in order to project ourselves into the condition of a plane being.
Let the reader imagine himself, as he looks along the plane, fig. 4, to become more and more identified with the thin body on it, till he finally looks along parallel to the surface of the plane earth, and up and down, losing the sense of the direction which stretches right and left. This direction will be an unknown dimension to him.
Our space conceptions are so intimately connected with those which we derive from the existence of gravitation that it is difficult to realise the condition of a plane being, without picturing him as in material surroundings with a definite direction of up and down. Hence the necessity of our somewhat elaborate scheme of representation, which, when its import has been grasped, can be dispensed with for the simpler one of a thin object slipping over a smooth surface, which lies in front of us.
It is obvious that we must suppose some means by which the plane being is kept in contact with the surface on which he slips. The simplest supposition to make is that there is a transverse gravity[3], which keeps him to the plane. This gravity must be thought of as different to the attraction exercised by his matter, and as unperceived by him.
At this stage of our enquiry I do not wish to enter into the question of how a plane being could arrive at a knowledge of the third dimension, but simply to investigate his plane consciousness.
It is obvious that the existence of a plane being must be very limited. A straight line standing up from the surface of his earth affords a bar to his progress. An object like a wheel which rotates round an axis would be unknown to him, for there is no conceivable way in which he can get to the centre without going through the circumference. He would have spinning disks, but could not get to the centre of them. The plane being can represent the motion from any one point of his space to any other, by means of two straight lines drawn at right angles to each other.
Let AX and AY be two such axes. He can accomplish the translation from A to B by going along AX to C, and then from C along CB parallel to AY.
The same result can of course be obtained by moving to D along AY and then parallel to AX from D to B, or of course by any diagonal movement compounded by these axial movements.
By means of movements parallel to these two axes he can proceed (except for material obstacles) from any one point of his space to any other.
If now we suppose a third line drawn out from A at right angles to the plane it is evident that no motion in either of the two dimensions he knows will carry him in the least degree in the direction represented by AZ.
The lines AZ and AX determine a plane. If he could be taken off his plane, and transferred to the plane AXZ, he would be in a world exactly like his own. From every line in his world there goes off a space world exactly like his own.
From every point in his world a line can be drawn parallel to AZ in the direction unknown to him. If we suppose the square in fig. 7 to be a geometrical square from every point of it, inside as well as on the contour, a straight line can be drawn parallel to AZ. The assemblage of these lines constitute a solid figure, of which the square in the plane is the base. If we consider the square to represent an object in the plane being’s world then we must attribute to it a very small thickness, for every real thing must possess all three dimensions. This thickness he does not perceive, but thinks of this real object as a geometrical square. He thinks of it as possessing area only, and no degree of solidity. The edges which project from the plane to a very small extent he thinks of as having merely length and no breadth—as being, in fact, geometrical lines.
With the first step in the apprehension of a third dimension there would come to a plane being the conviction that he had previously formed a wrong conception of the nature of his material objects. He had conceived them as geometrical figures of two dimensions only. If a third dimension exists, such figures are incapable of real existence. Thus he would admit that all his real objects had a certain, though very small thickness in the unknown dimension, and that the conditions of his existence demanded the supposition of an extended sheet of matter, from contact with which in their motion his objects never diverge.
Analogous conceptions must be formed by us on the supposition of a four-dimensional existence. We must suppose a direction in which we can never point extending from every point of our space. We must draw a distinction between a geometrical cube and a cube of real matter. The cube of real matter we must suppose to have an extension in an unknown direction, real, but so small as to be imperceptible by us. From every point of a cube, interior as well as exterior, we must imagine that it is possible to draw a line in the unknown direction. The assemblage of these lines would constitute a higher solid. The lines going off in the unknown direction from the face of a cube would constitute a cube starting from that face. Of this cube all that we should see in our space would be the face.
Again, just as the plane being can represent any motion in his space by two axes, so we can represent any motion in our three-dimensional space by means of three axes. There is no point in our space to which we cannot move by some combination of movements on the directions marked out by these axes.
On the assumption of a fourth dimension we have to suppose a fourth axis, which we will call AW. It must be supposed to be at right angles to each and every one of the three axes AX, AY, AZ. Just as the two axes, AX, AZ, determine a plane which is similar to the original plane on which we supposed the plane being to exist, but which runs off from it, and only meets it in a line; so in our space if we take any three axes such as AX, AY, and AW, they determine a space like our space world. This space runs off from our space, and if we were transferred to it we should find ourselves in a space exactly similar to our own.
We must give up any attempt to picture this space in its relation to ours, just as a plane being would have to give up any attempt to picture a plane at right angles to his plane.
Such a space and ours run in different directions from the plane of AX and AY. They meet in this plane but have nothing else in common, just as the plane space of AX and AY and that of AX and AZ run in different directions and have but the line AX in common.
Omitting all discussion of the manner on which a plane being might be conceived to form a theory of a three-dimensional existence, let us examine how, with the means at his disposal, he could represent the properties of three-dimensional objects.
There are two ways in which the plane being can think of one of our solid bodies. He can think of the cube, fig. 8, as composed of a number of sections parallel to his plane, each lying in the third dimension a little further off from his plane than the preceding one. These sections he can represent as a series of plane figures lying in his plane, but in so representing them he destroys the coherence of them in the higher figure. The set of squares, A, B, C, D, represents the section parallel to the plane of the cube shown in figure, but they are not in their proper relative positions.
The plane being can trace out a movement in the third dimension by assuming discontinuous leaps from one section to another. Thus, a motion along the edge of the cube from left to right would be represented in the set of sections in the plane as the succession of the corners of the sections A, B, C, D. A point moving from A through BCD in our space must be represented in the plane as appearing in A, then in B, and so on, without passing through the intervening plane space.
In these sections the plane being leaves out, of course, the extension in the third dimension; the distance between any two sections is not represented. In order to realise this distance the conception of motion can be employed.
Let fig. 9 represent a cube passing transverse to the plane. It will appear to the plane being as a square object, but the matter of which this object is composed will be continually altering. One material particle takes the place of another, but it does not come from anywhere or go anywhere in the space which the plane being knows.
The analogous manner of representing a higher solid in our case, is to conceive it as composed of a number of sections, each lying a little further off in the unknown direction than the preceding.
We can represent these sections as a number of solids. Thus the cubes A, B, C, D, may be considered as the sections at different intervals in the unknown dimension of a higher cube. Arranged thus their coherence in the higher figure is destroyed, they are mere representations.
A motion in the fourth dimension from A through B, C, etc., would be continuous, but we can only represent it as the occupation of the positions A, B, C, etc., in succession. We can exhibit the results of the motion at different stages, but no more.
In this representation we have left out the distance between one section and another; we have considered the higher body merely as a series of sections, and so left out its contents. The only way to exhibit its contents is to call in the aid of the conception of motion.
If a higher cube passes transverse to our space, it will appear as a cube isolated in space, the part that has not come into our space and the part that has passed through will not be visible. The gradual passing through our space would appear as the change of the matter of the cube before us. One material particle in it is succeeded by another, neither coming nor going in any direction we can point to. In this manner, by the duration of the figure, we can exhibit the higher dimensionality of it; a cube of our matter, under the circumstances supposed, namely, that it has a motion transverse to our space, would instantly disappear. A higher cube would last till it had passed transverse to our space by its whole distance of extension in the fourth dimension.
As the plane being can think of the cube as consisting of sections, each like a figure he knows, extending away from his plane, so we can think of a higher solid as composed of sections, each like a solid which we know, but extending away from our space.
Thus, taking a higher cube, we can look on it as starting from a cube in our space and extending in the unknown dimension.
Take the face A and conceive it to exist as simply a face, a square with no thickness. From this face the cube in our space extends by the occupation of space which we can see.
