THE FOURTH DIMENSION
The Ideal and the Representative Nature of Objects in the
Sensible World--The Psychic Fluxional the Basis of Mental
Differences--Natural and Artificial Symbols--Use of Analogies to
Prove the Existence of a Fourth Dimension--The Generation of a
Hypercube or Tesseract--Possibilities in the World of the Fourth
Dimension--Some Logical Difficulties Inhering in the Four-Space
Conception--The Fallacy of the Plane-Rotation Hypothesis--C. H.
HINTON and Major ELLIS on the Fourth Dimension.
The world of mathesis is truly a marvelous domain. Vast are its
possibilities and vaster still its sweep of conceivability. It is the
kingdom of the mind where, in regal freedom, it may perform feats
which it is impossible to actualize in the phenomenal universe. In
fact, there is no necessity to consider the limitations imposed by
the actualities of the sensuous world. Logic is the architect of
this region, and for it there is no limit to the admissibility of
hypotheses. These may be multiplied at will, and legitimately so. The
chief error lies in the attempt to make them appear as actual facts of
the physical world.
Mathematicians, speculating upon the possibilities of mathetic
constructions and forgetting the necessary distinctions which should
be recognized as differentiating the two worlds, in their enthusiasm
have been led into the error of postulating as qualities of the
phenomenal world the characteristics of the conceptual. Accordingly,
a great deal of confusion as to the proper limits and restrictions of
these conceptions has arisen and there still may be found those who are
enthusiastically endeavoring to push the actualities of the physical
over into the conceptual. But in assuming any attitude towards mathetic
propositions, especially with a view to demonstrating their actuality,
very careful discrimination as to the essential qualities and their
connotations should be made. Hence, before taking up a brief study of
the fourth dimension proper, it is deemed fitting to indicate some of
the fundamental distinctions which every student of these questions
should be able to make with reference to the data which he meets.
All objects of the sensible world have both an essential or ideal
nature and a representative or sensuous nature. That is, they may be
studied from the standpoint of the ideal as well as the sensuous.
The representative nature is that which we recognize as the mode of
appearance to our senses which, as KANT held, is not the essential
or ideal character of the thing itself. For there is quite as much
difference between the sensuous percept and the real thing itself as
between an object and its shadow. In fact, a concept viewed in this
light, may be seen to have all the characteristics of an ordinary
shadow; for instance, the shadow of a tree. View it as the sun is
rising; it will then be seen to appear very much elongated, becoming
less in length and more distinct in outline as the sun rises to a
position directly overhead. The elongation may again be seen when the
sun is setting. Throughout the day as the sun assumes different angles
with reference to the tree the proportions and definiteness of the
shadow vary accordingly. Thus the angularity of the sun, the intensity
and fullness of the light and the shape and size of the tree operate to
determine the character of the shadow.
Much the same thing is true of a sensuous representation. If we examine
carefully our ideas of geometric quantities and magnitudes, it will be
found that the concepts themselves are not identical with the objects
of the physical world, but mere mental shadows of them. The angularity
of consciousness, or the distinctness of one's state of awareness,
being analogous to similar attitudes in the solar influence are the
main determinants of the character of the mental shadow or concept.
Wherefore mathematical "spaces" or magnitudes are not sensuous things
and have therefore no more real existence than a shadow, and strictly
speaking not as much; for a shadow may be seen, while such magnitudes
can only be conceived. It may be urged that since we can conceive of
such things they must have existence of some kind. And so they have,
but it is an existence of a different kind from that which we recognize
as belonging to things in the sensible world. They have a conceptual
existence, but not a sensuous one. Therein lies the great difference.
To be sure, a shadow is a more or less true representation of the thing
to which it pertains. That this is true can be established empirically.
Similarly, the degree of congruity between objects and concepts
likewise may be determined. If this were not true we should be very
much disappointed with what we find in the phenomenal world and could
never be quite sure that the mentograph existing in our minds was a
faithful representation of the thing which we might be examining. But
really the foundation for such a disappointment is present in every
concept, every percept with which the mind deals. This disappointment,
although in actual experience is reduced to an almost negligible
quantity, is due to the failure of sensuous objects to conform wholly
to the specific details of the mental shadow or mentograph. This
lack of congruence between the mental picture and the object itself
is necessary for obvious reasons. It is markedly observable in the
early efforts of a child in learning distances, weights, resistances,
temperatures and the like. No inconsiderable time is required for the
child to be able correctly to harmonize his sense-deliveries with
actual conditions. Otherwise, the child would never make any of the
ludicrous mistakes of judgment of which it is guilty when trying to
get its bearings in the world of the senses. In the course of time
the child gradually learns by experience that certain things are
true of objects, distances, temperatures, resistances, etc., and
that certain things are not true of them. He learns these things by
actually contacting various objects. He is then competent to render
correct judgments, within certain limits, as to the conditions which
he finds in the sensible world. And the allowances, equations and
corrections which his motor, sensory and psychic mechanisms learn to
make in childhood serve for all subsequent time. And this is important
to remember; for the mature mind is apt to forget or overlook the
adaptations which the child-mind has made in its growth.
If there were no such differences between the concept and the thing
itself, actual physical contact would not be necessary. For one could
rely wholly upon the sense-deliveries and each sense might operate
entirely independently of all the others as there would be no necessity
to correct the delivery of one by those of the others. This, of course,
raises the question as to the necessity of sense-experience at all
under conditions where there would be no disparity between the thing
itself and the ideal representation of it in the mind. The absence of
this variable quantity would open to the mind the possibility of really
knowing the essential nature of objects in the phenomenal world, a
condition of affairs which is admittedly now without the range of the
powers of the mind.
At any rate, the essential "thingness" of objects can never be
comprehended by the mind until the diminution of this disparity between
the object of sense and the mental picture of it which exists in the
consciousness has proceeded to such a limit as either completely to
have obliterated it or to such an extent that the psychic fluxion is so
slight as not to matter.
It is believed that the results of mental evolution, as the mind
approaches the transfinite as a limit, will operate to minimize the
fluxional quantity which subsists between all objects of sense and
their ideal representation as data of consciousness. The conclusion
that the mind of early men who lived hundreds of thousands and perhaps
millions of years ago on this planet consumed a much longer time in
learning the adjustments between the objects which it contacted in the
sensuous world and the elementary representations which were registered
in its youthful consciousness than is to-day required for similar
processes seems to be demanded, and substantiated as well, by what is
known of the phyletic development of the mind in the human race.
In view of the above, it is thought that the duration of such simple
mental processes served not only to prolong the physical life of
the man of those early days, but may also account for the puerility
and incapacity of the mind at that stage. Not that the slow mental
processes were active causative agencies in lengthening the life
of man, but that they together with the crass physicality of man
necessitated a longer physical life. This, perhaps in a larger
sense than any other consideration, accounts for the fundamental
discrepancies in the mind of the primitive man in comparison with the
efficiency of the mind of the present-day man. In view of the potential
character of mind and in the light of the well graduated scale of its
accomplishments, it is undoubtedly safe to conclude that the quality of
mental capacities is proportional to the psychic fluxional which may
exist at any time between the ideal and the essential or real. Mental
differences and potentialities in general may be due to the magnitude
of the psychic fluxional or differential that exists between the
conceptual and the perceptual universe. In some minds it may be greater
than in others. The chasm between things-in-themselves and the mental
notion pertaining thereto may vary in a direct ratio to the individual
mind's place in psychogenesis, and therefore, be the key to all mental
differences in this respect.
Most certain it is that there may be marked fluctuations in the
judicial approach of minds towards any psychic end. In other words,
there is not only a fluxional or differential between the object and
its representation, but also a differential between the approach
of one mind and another in the judicial determination of notions
concerning ideas. In this way, differences of opinions as to the
right and wrong of judgments arise. Indeed, there seem to be zones
of affinity for minds of similar characteristics, or minds that have
the same degree of differential; so that, in choosing among the many
possible judgments predicable upon a species of data, all those minds
having the same degree of psychic differential discover a special
affinity or agreement among themselves. Hence, we have cults, schools
of thought, and various other sectional bodies that find a basis
of agreement for their operations in this way. The outcome of this
remarkable intellectual phenomenon is that there are as many different
kinds of judgments as there are zones of affinity among minds. Various
systems of philosophy owe their existence to these considerations, and
the considerations themselves flow from the fact that all intellectual
operations are essentially superficial; because there is no means by
which they may penetrate to the steady flowing stream of reality which
pervades and sustains objects in the sensible world.
In view, therefore, of the foregoing and with special reference to
geometric constructions, it is necessary in approaching a study of
the four-space that it be understood at the outset that the fourth
dimension can neither be actualized nor made objectively possible even
in the slightest degree in the perceptual world; because it belongs
to the world of pure thought and exists there as an "extra personal
affair," separate and distinct from the world of the senses.
As says SIMON NEWCOMB:[16]
"The experience of the race and all the refinements of modern
science may be regarded as showing quite conclusively that, within
the limits of our experience, there is no motion of material
masses, in the direction of a fourth dimension, no physical agency
which we can assume to have its origin in regions to which matter
cannot move, when it has three degrees of freedom."
There is, however, no logical objection to the study of the fourth
dimension as a purely hypothetical question, if by pursuit of the same
an improvement of methods of research and of the outlook upon the field
of the actual may be gained. Hence, it is with this attitude of mind
that we approach the consideration of the fourth dimension.
[16] Vide _Science_, Vol. VII, p. 2, No. 158, 1898.
Various efforts have been made to render the conception of a fourth
dimension of space thinkable. The student of space has reasoned: "We
say that there are three dimensions of space. Why should we stop here?
May there not be spaces of four dimensions and more?" Or he has said:
"If 'A' may represent the side of a square, A^2 its area, and A^3 the
volume of a cube with edge equal to A; what may A^4, A^5 or A^{_n_th}
represent in our space? The conclusion, with respect to the quantity
A^4, has been that it should represent a space of four dimensions."
Algebraic quantities, however, represent neither objects in space nor
space qualities except in a purely conventional manner. All efforts to
justify the objective existence of a fourth dimension based upon such
reasoning will, therefore, fail; because the basis of such arguments
is itself faulty. In the sentence: "The man loves his bottle," the
thing meant is not the bottle, but what the bottle contains. For the
purpose of the figure the bottle signifies its contents. There is no
more real connection between the bottle and what it contains than
between any word and the object for which it stands. Words are said
to be symbols of ideas. But they are not natural symbols; they are
conventional symbols, made for the purpose of cataloguing, indexing
and systematizing our knowledge. Words can be divorced from ideas
and objects, or rather have never had any real connection with them.
There are two classes of natural symbols, namely; _objects_ and
_ideas_. These, objects and ideas, symbolize realities. Realities
are imperceptible and incomprehensible to the intellect which has
aptitude only for a slight comprehension of the symbols of realities.
For instance, a tree is a natural symbol. It represents an actuality
which is imperceptible to the intellect. The intellect can deal only
with the sensible symbol. It is a natural symbol; because it is
possible directly to trace a living connection between the tree and
the _tree-reality_. That is, it would be possible so to trace out the
vital connection between the tree and its reality if the intellect
had aptitude for such tracery. But, in reality, since it has no such
aptitude, it remains for the work of that higher faculty than the
intellect which recognizes both the connection and the intellect's
inability to trace it. Further, an object is called a natural symbol
because it is the bridge between sensuous representation and reality.
It is as if one could begin at the surface of an object and by a
subtle process of elimination and excortication arrive at the heart
of the universum of reality. No such consummation may be reached by
dealing with words which have merely an artificial relationship with
the objects which they signify. Again, ideas, that is, ideas that are
universal in application and have their roots in the great ocean of
reality, are natural symbols; because if it were possible to handle
an idea with the physical hands it would be possible to arrive at the
heart of that which it symbolized without ever losing our connection
with the idea itself. In other words, ideas and objects, unlike words,
can never be divorced from that which they symbolize. Both, being of
the same class, are the opposite poles of realities. This then is
the difference between natural symbols and artificial symbols--that
a natural symbol, such as objects and ideas, is copolar with reality
whereas an artificial symbol, such as words, geometric constructions
and the like not only lacks this copolarity but is itself a symbol of
natural symbols.
It is, therefore, inconceivable that because the algebraic quantity
A^3 has been arbitrarily decreed to be a representation of the volume
of a cube, every such quantity in the algebraic series shall actually
represent some object or set of objects in the physical world. Even
if it be granted that such may be the case, is it not certain that
there is a limit to things in the objective universe? Yet there may
not be any limit to algebraic or mathematical determinations. The
material universe is limited and conditioned; the world of mathesis is
unlimited and unconditioned save by its own limitations and conditions.
It is irrational to expect that physical phenomena shall justify all
mathematical predicates.
_There is perhaps no single mathematical desideratum or consideration
which may be said to be the natural symbolism of realities; for the
whole of mathematical conclusions is a mass of artificial and arbitrary
but concordant symbols of the crasser or nether pole of the antipodes
of realism._ It is exceedingly dangerous, therefore, to predicate
upon such a far-fetched symbolism as mathematics furnishes anything
purporting to deal with ultimate realities. And those who insist upon
doing so are either blind themselves to these limitations or are madly
endeavoring to befog the minds of others who are dependent upon them
for leadership in questions of mathematical import.
Analogies have been unsparingly used in efforts to popularize the
four-space conception and much of the violence which has been done
to the notion is due to this vagary. The mathematical publicist, in
trying to give a mental picture of the fourth dimension, examines the
appearances of three dimensional beings as they might appear to a two
dimensional being or _duodim_. He imagines a race of beings endowed
with all the human faculties except that they live in a land of but
two dimensions--length and breadth. He thinks of them as shadows of
three dimensional beings to whom there are no such conceptions as "up"
and "down." They can see nothing nor sense anything in any way that
is without their plane. They can move in any direction within the
plane in which they live, but can have no idea of any movement that
might carry them without that plane. A house for such beings might be
simply a series of rectangles. One of them might be as safe behind
a line as a _tridim_ or three dimensional being would be behind a
stone wall. A bank safe for the _unodim_ would be a mere circle. A
_duodim_ in any two dimensional prison might be rescued by a tridim
without the opening of doors or the breaking of walls. An action of
a _tridim_ performed so as to contact their plane would be to them a
miracle, absolutely unaccountable upon the basis of any known fact to
the _unodim_ or _duodim_. A _tridim_ might go into a house where lived
a family of _duodims_, appear and disappear without being detected
or its ever being discovered how he accomplished such a marvelous
feat. Our miracles, after the same fashion, are said to be the antics
of some four dimensional being who has similar access to our three
dimensional world and whose actions are similarly inexplicable to us.
So the analogies have been multiplied. But the temptation to apply the
consequences of such reasoning to actual three-space conditions has
been so great that many have yielded to it and have consequently sought
actually to explain physical phenomena upon the basis of the fourth
dimension.
The utilitarian side of the question of hyperspace has not been
neglected either. And so, early in the development of the hypothesis
and its various connotations, the attention of investigators was turned
to this aspect of the inquiry. Strange possibilities were revealed as
a result. For instance, it was found that an expert fourth dimensional
operator is possessed of extraordinary advantages over ordinary
tridimensional beings. Operating from his mysterious hiding place in
hyperspace, he could easily appear and disappear in so mysterious a
manner that even the most strongly sealed chests of treasures would
be easily and entirely at his disposal. No city police, Scotland Yard
detective nor gendarme could have any terrors for him. DRS. JEKYLL
and MESSRS. HYDE might abound everywhere without fear of detection.
Objects as well as persons might be made to pass into or out of
closed rooms "without penetrating the walls," thus making escape easy
for the imprisoned. No tridimensional state, condition or system of
arrangements, accordingly, would be safe from the ravages of evilly
inclined four dimensional entities. Objects that now are limited to a
point or line rotation could in the fourth dimension rotate about a
plane and thus further increase the perplexities of our engineering
and mechanical problems; four lines could be erected perpendicular
to each other whereas in three space only three such lines can be
erected; the right hand could be maneuvered into the fourth dimension
and be recovered as a left hand; the mysteries of growth, decay and
death would find a satisfactory explanation on the basis of the
fourth dimensional hypothesis and many, if not all, of the perplexing
problems of physiology, chemistry, physics, astronomy, anthropology
and psychology would yield up their mysteries to the skill of the
fourth dimensional operator. Marvelous possibilities these and much
to be desired! But the most remarkable thing about these so-called
possibilities is their impossibility. It is this kind of erratic
reasoning that has brought the conception of a fourth dimension into
general disrepute with the popular mind. It is to be regretted, too,
for the notion is a perfectly legitimate one in the domain of mathesis
where it originated and rightly belongs.
It is not to be wondered at that metageometricians and others should
at first surmise that, in the four-space, they had found the key to
the deep mysteries of nature in all branches of inquiry. For so vast
was the domain and so marvelous were the possibilities which the new
movement revealed that it was to be expected that those who were
privileged to get the first glimpses thereof would not be able to
realize fully their significance. But the stound of their minds and
the attendant magnification of the elements which they discovered
were but incidents in the larger and more comprehensive process of
adjustment to the great outstanding facts of psychogenesis which is
only faintly foreshadowed in the so-called hyperdimensional. The whole
scope of inquiry connected with hyperspace is not an end in itself.
It is merely a means to an end. And that is the preparation of the
human mind for the inborning of a new faculty and consequently more
largely extended powers of cognition. Metageometrical discoveries are
therefore the excrescences of a deeper, more significant world process
of mental unfoldment. They belong to the matutinal phenomena incident
to this new stage of mental evolution. All such investigations are but
the preliminary exercises which give birth to new tendencies which are
destined to flower forth into additional faculties and capacities. So
that it is well that the evolutionary aspect of the question be not
overlooked; for there is danger of this on account of the magnitude and
kosmic importance of its scope of motility.
A geometric line is said to be a space of one dimension. A plane is a
space of two dimensions and a cube, a space of three dimensions. In
figure 7 below, the line _ab_ is said to be one dimensional; because
only one coördinate is necessary to locate a point-position in it.
The plane, _abcd_, figure 8, is said to be two dimensional because
two coördinates, _ab_ and _db_ are required to locate a point, as the
point _b_. The cube _abcdefgh_, figure 9, is said to be tridimensional,
because, in order to locate the point _b_, for instance, it is
necessary to have three coördinates, _ab_, _bc_ and _gb_. The tesseract
is said to be four dimensional, because, in order to locate the point
_b_, in the tesseract, it is necessary to have four coördinates, _ab_,
_bc_, _bb'_ and _h'b_, figure 10.
A -------------------------------- B
FIG. 7.
A +------------------------------+ B
| |
| |
| |
| |
| |
| |
C +------------------------------+ D
FIG. 8.
It will be noted that in figures 8, 9 and 10, the element of
perpendicularity enters as a necessary determination. In figure 8, the
lines _ab_ and _bd_ are perpendicular to each other. Similarly, in Fig.
10, lines _ab_, _bc_, _bb'_ and _h'b_ are perpendicular to one another.
That is, at their intersections, they make right angles. Similarly,
figures representing any number of dimensions may be constructed.
[Illustration: FIG. 9.]
[Illustration: FIG. 10.--The Tesseract.]
The line _ab_ represents a one-space. An entity living in a one space
is called a "unodim." The plane, _abcd_, represents a two-space,
and entities living in such a space are called _duodims_. The cube,
_abcdefgh_, represents a three-space and entities inhabiting such a
space are called _tridims_. Figure 10 represents a four-space, and its
inhabitants are called _quartodims_. Each of the above-mentioned spaces
is said to have certain limitations peculiar to itself.
The fourth dimension is said to lie in a direction at right angles to
each of our three-space directions. This, of course, gives rise to the
possibility of generating a new kind of volume, the hypervolume. The
hypercube or tesseract is described by moving the generating cube in
the direction in which the fourth dimension extends. For instance, if
the cube, Fig. 9, were moved in a direction at right angles to each
of its sides a distance equal to one of its sides, a figure of four
dimensions, the tesseract, would result.
The initial cube, _abcc'e'fhh'_, when moved in a direction at right
angles to each of its faces, generates the hypercube, Fig. 10. The
lines, _aa'_, _bb'_, _cc'_, _dd'_, _ee'_, _ff'_, _gg'_, _hh'_, are
assumed to be perpendicular to the lines meeting at the points, _a_,
_b_, _c_, _d_, _e_, _f_, _g_, _h_. Hence _a'b'_, _b'd_, _dd'_, _d'a'_,
_ef_, _fg_, _gg'_, _g'e_, represent the final cube resulting from
the hyperspace movement. Counting the number of cubes that compose
the hypercube we find that there are eight. The generating cube,
_abcc'e'f'hh'_, and the final cube, _a'b'_, _b'd_, _dd'_, _d'a'_, _ef_,
_fg_, _gg'_, _g'e_, make two cubes; and each face generates a cube
making eight in all. A tesseract, therefore, is a figure bounded by
eight cubes.
To find the different elements of a tesseract, the following rules will
apply:
1. _To find the number of lines_: Multiply the number of lines in the
generating cube by two, and add a line for each point or corner in it.
E.g., 2 × 12 = 24 + 8 = 32.
2. _To find the number of planes, faces or squares_: Multiply the
number of planes in the generating cube by 2 and add a plane for each
line in it. E.g., 2 × 6 + 12 = 24.
3. _To find the number of cubes in a hypercube_: Multiply the number of
cubes in the generating cube, one, by two and add a cube for each plane
in it. E.g., 2 × 1 + 6 = 8.
4. _To find the number of points or corners_: Multiply the number of
corners in the generating cube by 2. E.g., 2 × 8 = 16.
* * * * *
In a plane there may be three points each equally distant from one
another. These may be joined, forming an equilateral triangle in which
there are three vertices or points, three lines or sides and one
surface.
In three-space there may be four points each equidistant from the
others. At the vertices of a regular tetrahedron may be found such
points. The tetrahedron has four points, one at each vertex, 6 lines
and 4 equilateral triangles, as in Fig. 11.
In four-space, we have 5 points each equidistant from all the rest,
giving the hypertetrahedron. This four dimensional figure may be
generated by moving the tetrahedron in the direction of the fourth
dimension, as in Fig. 12. If a plane be passed through each of the
six edges of the tetrahedron and the new vertex there will be six new
planes or faces, making 10 in all, counting the original four. From
the new vertex there is also a tetrahedron resting upon each base of
the original tetrahedron so that there are five tetrahedra in all.
_A hypertetrahedron is a four-dimensional figure consisting of five
tetrahedra, ten faces, 10 lines and 5 points._
[Illustration: FIG. 11.--Tetrahedron.]
[Illustration: FIG. 12.--Hypertetrahedron.]
PAUL CARUS[17] suggests the use of mirrors so arranged that they
give eight representations of a cube when placed at their point of
intersection. He says:
"If we build up three mirrors at right angles and place any object
in the intersecting corner we shall see the object not once, but
eight times. The body is reflected below and the object thus
doubled is mirrored not only on both upright sides but in addition
in the corner beyond, appearing in either of the upright mirrors
coincidingly in the same place. Thus the total multiplication of
our tridimensional boundaries of a four dimensional complex is
rendered eight-fold.
"We must now bear in mind that this representation of a fourth
dimension suffers from all the faults of the analogous figure
of a cube in two dimensional space. The several figures are not
eight independent bodies but are mere boundaries and the four
dimensional space is conditioned by their interrelation. It is
that unrepresentable something which they inclose, or in other
words, of which they are assumed to be boundaries. If we were
four dimensional beings we could naturally and easily enter into
the mirrored space and transfer tridimensional bodies or parts
of them into those other objects reflected here in the mirrors
representing the boundaries of the four dimensional object. While
thus on the one hand the mirrored pictures would be as real as the
original object, they would not take up the space of our three
dimensions, and in this respect, our method of representing the
fourth dimension by mirrors would be quite analogous to the cube
pictured on a plane surface, for the space to which we (being
limited to our tridimensional space-conception), would naturally
relegate the seven additional mirrored images is unoccupied and if
we should make the trial, we would find it empty."
[17] Vide _Foundations of Mathematics_, pp. 93-94.
The utility of such a representation as that which CARUS outlines in
the above is granted, i.e., so far as the purpose which it serves in
giving a general idea of what a four-space object might be imagined
to be like, but the illustration does not demonstrate the existence
of a fourth dimension. It only shows what might be if there were a
four-space in which objects could exist and be examined. We, of course,
have no right to assume that because it can be shown by analogous
reasoning that certain characteristics of the fourth dimensional object
can be represented in three-space the possible existence of such an
object is thereby established. Not at all. For there is no imaginable
condition of tridimensional mechanics in which an object may be said to
have an objective existence similar to that represented by the mirrored
cube.
But there are discrepancies in this representation which well might be
considered. They have virtually the force of invalidating somewhat the
conception which the analogy is designed to illustrate. For instance,
in the case of the mirrored object placed at the point of intersection
of the three mirrors built up at right angles to each other. Upon
examination of such a construction it is found that the reflection of
the object in the mirrors has not any perceptible connection with the
object itself. And this, too, despite the fact that they are regarded
as boundaries of the hypercube; especially is this true when it is
noted that these reflections are called upon to play the part of real,
palpable boundaries. If a fourth dimensional object were really like
the mirror-representation it would be open to serious objections from
all viewpoints. The replacement of any of the boundaries required in
the analogy would necessarily mean the replacement of the hypercube
itself. In other words, if the real cube be removed from its position
at the intersection of the mirrors no reflection will be seen, and
hence no boundaries and no hypercube. The analogy while admittedly
possessing some slight value in the direction meant, is nevertheless
valueless so far as a detailed representation is concerned. So the
analogy falls down; but once again is the question raised as to whether
the so-called fourth dimension can be established or proven at all upon
purely mathematical grounds. It also emphasizes the necessity for a
clearer conception of the meaning of dimension and space.
The logical difficulties which beset the hyperspace conception are
dwelt upon at length by JAMES H. HYSLOP. He says:[18]
"The supposition that there are three dimensions instead of one,
or that there are only three dimensions is purely arbitrary,
though convenient for certain practical purposes. Here the
supposition expresses only differences of directions from an
assumed point. Thus what would be said to lie in a plane in one
relation would lie in the third dimension in another. There
is nothing to determine absolutely what is the first, second,
or third dimension. If the plane horizontal to the sensorium
be called plane dimension, the plane vertical to it will be
called solid, or the third dimension, but a change of position
will change the names of these dimensions without involving the
slightest qualitative change or difference in meaning.
"Moreover, we usually select three lines or planes terminating
vertically at the same point, the lines connecting the three
surfaces of a cube with the same point, as the representative of
what is meant by three dimensions, and reduce all other lines
and planes to these. But interesting facts are observable here.
1. If the vertical relation between two lines be necessary for
defining a dimension, then all other lines than the specified ones
are either not in any dimension at all, or they are outside the
three given dimensions. This is denied by all parties, which only
shows that a vertical relation to other lines is not necessary to
the determination of a dimension. 2. If lines outside the three
vertically intersecting lines still lie in dimension or are
reducible to the other dimensions they may lie in more than one
dimension at the same time which after all is a fact. This only
shows that qualitatively all three dimensions are the same and
that any line outside of another can only represent a dimension in
the sense of _direction_ from a given point or line, and we are
entitled to assume as many dimensions as we please, all within
three dimensions.
"This mode of treatment shows the source of the illusion about
the 'fourth dimension.' The term in its generic import denotes
commensurable quality and denotes only one such quality, so that
the property supposed to determine non-Euclidean geometry must
be qualitatively different from this, if its figures involve the
necessary qualitative differentiation from Euclidean mathematics.
But this would shut out the idea of 'dimension' as its basis
which is contrary to the supposition. On the other hand, the term
has a specific meaning which as different qualitatively from the
generic includes a right to use the generic term to describe them
differentially, but if used only quantitatively, that is, to
express direction as it, in fact, does in these cases, involves
the admission of the actual, not a supposititious, existence of
a fourth dimension which again is contrary to the supposition
of the non-Euclidean geometry. Stated briefly, dimension as
commensurable quality makes the existence of the fourth dimension
a transcendental problem, but as mere direction, an empirical
problem. And the last conception satisfies all the requirements
of the case because it conforms to the purely quantitative
differences which exist between Euclidean and non-Euclidean
geometry as the very language about 'surfaces,' 'triangles,' etc.,
in spite of the prefix 'pseudo,' necessarily implies."
[18] Vide _Philosophical Review_, Vol. V, 1896, p. 352, et.
seq.
Thus it would seem that those who have been most diligent in
constructing the hyperspace conception have been the least careful
of the logical difficulties which beset the elaboration of their
assumptions. Yet it sometimes requires the illogical, the absurd and
the aberrant to bring us to a right conception of the truth, and when
we come to a comparison of the two, truth and absurdity, we are the
more surprised that error could have gained so great foothold in face
of so overwhelming evidences to the contrary.
The entire situation is, accordingly, aptly set forth by HYSLOP when he
says, continuing:
"There are either a confusion of the abstract with the concrete
or of quantitative with qualitative logic, ... so that all
discussion about a fourth dimension is simply an extended mass
of equivocations turning upon the various meanings of the
term 'dimension.' This when once discovered, either makes the
controversy ridiculous or the claim for non-Euclidean properties a
mere truism, but effectually explodes the logical claims for a new
dimensional quality of space as a piece of mere jugglery in which
the juggler is as badly deceived as his spectators. It simply
forces mathematics to transcend its own functions as defined by
its own advocates and to assume the prerogatives of metaphysics."
Shall we, therefore, assent to the imperialistic policy of
mathematicians who would fain usurp the preserves of the metaphysician
in order that they may exploit a superfoetated hypothesis? It is not
believed that the harshness of HYSLOP'S judgment in this respect
is undeserved. It is, however, regretted that the notions of
mathematicians have been so inchoate as to justify this rather caustic,
though appropriate criticism. For it does appear that the moment the
mathematician deserts the province of his restricted sphere of motility
and enters the realm of the transcendental, that moment he loses his
way and becomes an inexperienced mariner on an uncharted sea.
It is interesting to note that CASSIUS JACKSON KEYSER,[19] while
recognizing the purely arbitrary character of the so-called
dimensionality of space, nevertheless lends himself to the view that
"if we think of the line as generating element we shall find that our
space has four dimensions. That fact may be seen in various ways, as
follows:
"A line is determined by any two of its points. Every line pierces
every plane. By joining the points of one plane to all the points
of another, all the lines of space are obtained. To determine
a line, it is, then, enough to determine two of its points,
one in the one plane and one in the other. For each of these
determinations two data, as before explained, are necessary and
sufficient. The position of the line is thus seen to depend upon
four independent variables, and the four dimensionality of our
space _in lines_ is obvious."
[19: Vide _Monist_, Vol. XVI, 1896, Mathematical
Emancipations.]
Similarly he argues for the four dimensionality of space in spheres:
"We may view our space as an assemblage of its spheres. To
distinguish a sphere from all other spheres, we need to know four
and but four independent facts about it, as say, three that shall
determine its center and one its size. Hence our space is four
dimensional also in spheres. In circles, its dimensionality is
six; in surfaces of second order (those that are pierced by a
straight line in two points), nine; and so on ad infinitum."
The view taken by KEYSER is a typical one. It is the mathematical
view and is characterized by a certain lack of restraint which is
found to be peculiar to the whole scheme of thought relating to
hyperspace. It is clear that the kind of space that will permit
of such radical changes in its nature as to be at one time three
dimensional, at another time four dimensional, then six, nine and
even _n_-dimensional is not the kind of space in which the objective
world is known to exist. Indeed, it is not the kind of space that
really exists at all. In the first place, a line cannot generate
perceptual space. Neither can a circle, nor a sphere nor any other
geometrical construction. It is, therefore, not permissible, except
mathematically, to view our space either as "an assemblage of its
spheres," its circles or its surfaces; for obviously perceptual space
is not a geometrical construction even though the intellect naturally
finds inhering in it a sort of latent geometrism which is kosmical.
For there is a wide difference between that kosmic order which is
space and the finely elaborated abstraction which the geometer
deceives himself into identifying with space. There is absolutely
neither perceptible nor imperceptible means by which perceptual space
in anywise can be affected by an act of will, ideation or movement.
Just why mathematicians persist in vagarizing upon the generability
of space by movement of lines, circles, planes, etc., is confessedly
not easily understood especially when the natural outcome of such
procedure is self-stultification. It is far better to recognize, as
a guiding principle in all mathematical disquisitions respecting the
nature of space that the possibilities found to inhere in an idealized
construction cannot be objectified in kosmic, sensible space. The line
of demarkation should be drawn once for all, and all metageometrical
calculations and theories should be prefaced by the remark that: "if
objective space were amenable to the peculiarities of an idealized
construction such and such a result would be possible," or words to
that effect. This mode of procedure would serve to clarify many if
not all of the hyperspace conceptions for the non-mathematician as
well as for the metageometricians themselves, especially those who are
unwilling to recognize the utter impossibility of their constructions
as applied to perceptual space. We should then cease to have the
spectacle of otherwise well-demeanored men committing the error of
trying to realize abstractions or abstractionizing realities. Herein
is the crux of the whole matter, that mathematicians, rather than be
content with realities as they find them in the kosmos, should seek
to reduce them to abstractions, or, on the other hand, make their
abstractions appear to be realities.
KEYSER proceeds to show how the concept of the generability of
hyperspace may be conceived by beginning with the point, moving it in
a direction without itself and generating a line; beginning with the
line, treating it similarly, and generating a plane; taking the plane,
moving it in a direction at right angles to itself and generating a
cube; finally, using the cube as generating element and constructing
a four-space figure, the tesseract. Now, as a matter of fact, a
point being intangible cannot be moved in any direction neither can
a point-portion of sensible space be removed. Nevertheless, we quite
agree with him when he asserts:
"Certainly there is naught of absurdity in supposing that _under
suitable stimulation the human mind may, in the course of time,
speedily develop a spatial intuition of four or more dimensions_."
(The italics in the above quotation are ours.)
Here we have a tacit implication that the notion which geometers
have heretofore designated as "dimension" really is a matter of
consciousness, of intuition, and therefore, determinable only by the
limitations of consciousness and the deliveries of our intuitive
cognitions. As a more detailed discussion of this phase of the subject
shall be entered into when we come to a consideration of Chapter VI on
"Consciousness as the Norm of Space Determinations" further comment is
deferred until then.
Now, as it appears certain that what geometers are accustomed to call
"dimension" is both relative and interchangeable in meaning--the one
becoming the other according as it is viewed--the conclusion very
naturally follows that neither constructive nor symbolic geometry
is based upon dimension as commensurable quality. The real basis of
the non-Euclidean geometry is dimension as direction. For whatever
else may be said of the fourth dimension so-called it is certainly
unthinkable, even to the metageometricians, when it is absolved from
direction although no specific direction can be assigned to it. It
is agreed perhaps among all non-Euclidean publicists that the fourth
dimension must lie in a "direction which is at right angles to all the
three dimensions." But if they are asked how this direction may be
ascertained or even imagined they are nonplused because they simply do
not know. The difficulty in this connection seems to hinge about the
question of identifying the conditions of the world of phantasy with
those of the world of sense. There are distortions, ramifications,
submersibles, duplex convolutions and other mathetic acrobatics which
can be performed in the realm of the conceptual the execution of which
could never be actualized in the objective world. Because these antics
are possible in the premises of the mathematical imagination is scarce
justification for the attempts at reproduction in an actualized and
phenomenal universe.
One of the proudest boasts of the fourth dimensionist is that
hyperspace offers the possibility of a new species of rotation, namely,
_rotation about a plane_. He refers to the fact that in the so-called
one-space, rotation can take place only about a point. For instance
in Figure 7, the line _ab_ represents a one-space in which rotation
can take place only about one of the two points _a_ and _b_. In
Figure 8 which represents a two-space, rotation may take place about
the line _ab_ or the line _cd_, etc., or, in other words, the plane
_abcd_ can be rotated on the axial line _ab_ in the direction of the
third dimension. In tridimensional space only two kinds of rotation
are possible, namely, rotation about a point and about a line. In the
fourth dimension it is claimed that rotation can take place about a
plane. For example, the cube in Figure 9, by manipulation in the
direction of the fourth dimension, can be made to rotate about the side
_abgf_.
A very ingenious argument is used to show how rotation about a plane
is thinkable and possible in hyperspace. But with this, as with the
entire fabric of hyperspace speculations, dependence is placed almost
entirely upon analogous and symbolic conceptions for evidence as to the
consistency and rationality of the conclusions arrived at.
D C D1
+-------------+.............
| | :
| | :
| | :
| | :
| | :
+-------------+.............
A B A1
FIG. 13.
It is urged that inasmuch as the rotation about the line _bc_ in Figure
13 would be incomprehensible or unimaginable to a plane being for the
reason that such a rotation involves a movement of the plane into the
third dimension, a dimension of which the plane being has no knowledge,
in like manner rotation about a plane is also unimaginable or
incomprehensible to a tridim or a three dimensional being. It is shown,
however, that the plane being, by making use of the possibilities of an
"assumed" tridimension, could arrive at a rational explanation of line
rotation.
[Illustration: FIG. 14.]
Figure 14 offers an illustration by means of which a two dimensional
mathematician could demonstrate the possibility of line rotation.
He is already acquainted with rotation about a point; for it is the
only possible rotation that is observable in his two dimensional
world. By conceiving of a line as an infinity or succession of points
extending in the same direction; by imagining the movement of his
plane in the direction of the third dimension thereby generating a
cube and at the same time assuming that the lines thus generated were
merely successions of points extending in the same direction, he could
demonstrate that the entire cube Figure 14, could be rotated about
the line _BHX_ used as an axis. For upon this hypothesis it would be
arguable that a cube is a succession of planes piled one upon the other
and limited only by the length of the cube which would be extending
in the, to him, unknown direction of the third dimension. He could
very logically conclude that as a plane can rotate about a point, a
succession of planes constituting a tridimensional cube, could also
be conceived as rotating about a line which would be a succession
of points under the condition of the hypothesis. His demonstration,
therefore, that the cube, Figure 14, can be made to rotate around
the line _BHX_ would be thoroughly rational. He could thus prove
line-rotation without even being able to actualize in his experience
such a rotation.
Analogously, it is sought by metageometricians to prove in like manner
the possibility of rotation about a plane. Thus in Figure 16 is shown a
cube which has been rotated about one of its faces and changed from its
initial position to the position it would occupy when the rotation had
been completed or its final position attained.
[Illustration: FIG. 15.]
[Illustration: FIG. 16.--Plane Rotation]
The gist of the arguments put forward as a basis for plane-rotation
is briefly stated thus: The face _cefg_ is conceived as consisting of
an infinity of lines. A cube, as in Figure 15, is imagined or assumed
to be sected into an infinity of such lines, each line being the
terminus of one of the planes which make up the cube. Each one of the
constituting planes is thought of as rotating about its line-boundary
which intersects the side of the cube. The process is continued
indefinitely until the entire series of planes is rotated, one by
one, around the series of lines which constitute the axial plane.
Hence, in order that the cube, Figure 16, may change from its initial
position to its final position each one of the infinitesimal planes
of which the cube is assumed to be composed must be made to rotate
about each one of the infinitesimal lines of which the plane used as
an axis is composed. In this way, it is shown that the entire cube
has been made to rotate about its face, _cefg_. This concisely, is
the "QUOD ERAT DEMONSTRANDUM" of the metageometrician who sets out to
prove rotation about a plane. Thus it is made to appear that in order
that tridimensional beings may be enabled to conceive of four-space
rotation, as in Figures 15 and 16, in which the rotation must also be
thought of as taking place in the direction of the fourth dimension,
they must adopt the same tactics that a two dimensional being would use
to understand some of the possibilities of the tridimensional world.
It is, of course, unwise to assume that because a thing can be
shown to be possible by analogical reasoning its actuality is
thereby established. This consideration cannot be too emphatically
insisted upon; for many have been led into the error by relying too
confidentially upon results based upon this line of argumentation.
There is a vast difference between mentally doing what may be assumed
to be possible, the hypothetical, and the doing of what is actually
possible, the practical.
In the first place, plane-rotation in the actual universe is a
structural impossibility. The very nature and constitution of material
bodies will not admit of such contortion as that required by the
rotation of a body, say a cube, about one of its faces. Let us examine
some of the results of plane rotation. 1. The rotation must take
place in the direction of the fourth dimension. Now, as it is utterly
impossible for any one, whether layman or metageometrician, even to
imagine or conceive, in any way that is practical, the direction of
the fourth dimension it is also impossible for one to move or rotate
a plane, surface, line or any other body in that direction. We are
in the very beginning of the process of plane-rotation so-called
confronted with a physical impossibility. 2. Plane rotation necessarily
involves the orbital diversion of every particle in the cube. This
alone is sufficient to prohibit such a rotation; for it is obvious
that the moment a particle or any series of particles is diverted from
its established orbital path disruption of that portion of the cube
must necessarily follow. This upon the assumption that the particles
of matter are in motion and revolving in their corpuscular orbits. 3.
Plane-rotation necessitates a radical change in the absolute motion
of each individual particle, electron, atom or molecule of matter in
the cube and a consequent retardation or acceleration of this motion.
This upon the hypothesis that the particles of matter are vibrating
at the rate of absolute motion. 4. It presupposes a reconstitution
of each atom, molecule or particle in the cube, changing the path of
intra-corpuscular rotation either from a right to left direction or
from a left to right direction, as the case may be. The particles of
matter in the cube will be acted upon in much the same manner as the
particles in a glove when it is maneuvered in the fourth dimension. In
describing this phenomenon, MANNING says:[20]
"Every part by itself, in its own place is turned over with only a
slight possible stretching and slight changing of positions of the
different particles of matter which go to make up the glove."
[20] Vide _Fourth Dimension, Simply Explained_, edited by
H. P. MANNING, p. 28.
The slight stretching and slight changing of the positions of the
particles referred to would be of small consequence if applied to
ponderable bodies. But when used in connection with particles of matter
which are themselves of very infinitesimal size means far more--enough,
as we have said, to militate severely against the integrity of the
cube. It is not deemed necessary to go further into the physical
aspects of plane-rotation as it is believed sufficient has been said to
negative the assumption from a purely structural viewpoint.
Among the vagaries of hyperspace publicists none is perhaps more
notable than the view taken by C. H. HINTON:[21]
"If it could be shown that the electric current in the negative
direction were exactly alike the electric current in the positive
direction, except for a reversal of the components of the
motion in three dimensional space, then the dissimilarity of
the discharge from the positive and negative poles would be an
indication of the one-sidedness of our space. The only cause of
difference in the two discharges would be due to a component in
the fourth dimension, which directed in one direction transverse
to our space, met with a different resistance to that which it met
when directed in the opposite direction."
[21] Vide _Fourth Dimension_, p. 75, C. H. HINTON.
To be sure. And with equal certainty it might be said that if the
moon were made of green cheese it might well be the ambition of the
world's chefs to be able at some time to flavor macaroni with it,
thus serving a rare dish. Even so, if there were an actual, objective
fourth dimension to our space we might be able to shove into it all the
perplexing problems of life and let it solve them for us. But the fact
that the fourth dimensional hypothesis is itself a mere supposition
seems to have been overlooked or rather completely ignored by HINTON.
Or else, ought it not be an obvious folly to hope to construct a
rational explanation of perplexing physical conditions upon the basis
of a purely suppositionary, and therefore unproven, hypothesis?
The recognized domain of the four-space, mathematically considered,
is according to the most generous allowance very small, so small, in
fact, that the disposition of some to crowd into it the essential
content of the manifested universe is a matter of profound amazement.
Then, too, it cannot be denied that there is no appreciable urgency or
necessity for having recourse to a purely hypothetical construction
for explicatory data regarding a phenomenon which has not been shown
to be without the scope of ordinary scientific methods of procedure to
unravel.
The claim of certain spiritualists, notably ZOLLNER of Leipsig, that
the phenomena of spiritism is accountable for on the grounds that the
fourth dimension affords a residential area for discarnate beings
whence spiritistic forayers may impose their presence upon unprotected
three dimensional beings is no less fatuous than the original
supposition itself. For upon this latter is built the entire fabric of
meaningless speculations so gleefully indulged in by those who glibly
proclaim the reality of the four-space. Indeed, clearer second thought
will reveal that, when the pendulum of erratic thinking and trafficking
in mental constructions swings back, hyperspaces, after all, are but
the _ignes fatuii_ of mathetic obscurantism.
Then, why should it be deemed necessary to discover some more
mysterious realm of four dimensional proportions in which the spirits
of the dead may find a habitation? Are the spiritualists, too, reduced
to the necessity of further mystifying their already adequately
mysterious phenomena? If there were not quite enough of physicality
upon the basis of which all the antics of these entities can be
explained, and that satisfactorily, one would, as a matter of course,
be inclined to lend some credence to these claims; but as it is clear
that all organized beings have some power, if no more than that which
maintains their organization, and as it ought also be an acceptable
fact that such a being is directed by mind; and further, that owing to
the nature of a spirit body it can penetrate solid matter or matter of
any other degree of density below the coefficient of spirit matter, it
ought likewise be unnecessary to go without the province of strictly
tridimensional mechanics for an explanation of spiritistic phenomena.
Equally unnecessary and uncalled for is the attempt of certain others
who lean toward the view of speculative chemists to account for the
none too securely established hypothesis that eight different alcohols,
each having the formula C_{5}H_{12}O may be produced without variation.
This is said to be due to the fact that certain of the component atoms,
notably the carbon atoms, take a fourth dimensional position in the
compound and thus produce the unusual spectacle of eight alcohols from
one formula. Have chemists actually exhausted all purely physical means
of reaching an understanding of the carbon compounds and are therefore
compelled to resort to questionable means in order to make additional
progress in their field? It is incredible. Hence the more facetious
appears the mathematical extravaganza in which originates the tendence
among the more sanguine advocates to make of the fourth dimension a
sort of "jack of all trades," a veritable "Aladdin's lamp" wherewith
all kosmic profundities may be illuminated and made plain. Not until
the perfection of instruments of precision has been reached, and not
until human ingenuity has been exhausted in its efforts to produce more
refined methods of research should it be permissible even to venture
into untried and more or less debatable fields in search of a relief
which after all is unobtainable.
Notwithstanding the fact that all attempts at accounting for physical
phenomena on the basis of _n_-dimensionality (which is itself by all
the standards of objective reference a non-existent quantity and
therefore irreconcilable with perceptual space requirements) are to be
characterized simply as a senseless dalliance with otherwise deeply
profound questions, many have fallen into a complete forgetfulness of
the logical barriers inhering in and hedging about the query and have
committed other and less excusable errors in the premises. Take, for
instance, the suggestion that the action of a tartrate upon a beam
of polarized light is due to the assumption of a fourth dimensional
direction by some component in the acid. This for the reason that
experimentation has shown that tartaric acid, in one form, will turn
the plane of polarized light to the right while in another form will
turn it to the left. It is not believed, however, that there is any
warrant for such an assumption. There is also another kind of tartrate
which seems to be neutral in that it has no effect whatever upon the
beam of light, turning it neither to the right nor to the left nor
having other visible or determinable effect upon it. Indeed, it is
not clear how it is hoped to prove such a case by constituting as a
norm a hypothesis which is essentially indemonstrable. A more logical
procedure would be first to establish the objective, discoverable
posture of four-space; show the actual movement of matter and entities
therein; locate it by empirical methods of research, and then, basing
our assertions upon apodeictic evidences, assume a new attitude toward
these phenomena because of the support found in established and
verifiable facts. Some hope of gaining a respectful hearing might then
be entertained; but at least to do so now appears to be quite untimely.
MAJOR WILMOT E. ELLIS, Coast Artillery Corps, United States Army, in
_The Fourth Dimension Simply Explained_,[22] remarks:
"... in the ether, if anywhere, we should expect to find some
fourth dimensional characteristics. Gravitation, electricity,
magnetism and light are known to be due to stresses in, or
motions of, the infinitesimal particles of the ether. The real
nature of these phenomena has never been fully explained by
three dimensional mathematical analysis. Indeed, the unexplained
residuum would seem to indicate that so far we have merely been
considering the three dimensional aspects of four dimensional
processes. As one illustration of many, it has been shown both
mathematically and experimentally that no more than five
corpuscles may have an independent grouping in an atom."
[22] Q. v., p. 242, edited by H. P. Manning.
The weakness of this view may be due to the fact that at that
time MAJOR ELLIS was emphasizing in his own mind the necessity of
simplifying the conception so as to make it of easy comprehension
rather than the establishment of any fealty to truth or the spirit
of mathesis in his examination of the problem. What therefore of
reality the student fails to find in his view may be attributed to the
sacrifice which the writer (MAJOR ELLIS) felt himself called upon to
make for the sake of simplicity. Hence a certain expressed connivance
at his position is allowable. But, on the other hand, if such were not
the conscious intent of MAJOR ELLIS it is not understood how it should
appear that "the unexplained residuum would seem to indicate that so
far we have merely been considering the three dimensional aspects of
four dimensional processes." Contrarily, it has yet to be proved that
three dimensional space does not afford ample scope of motility for
all observable or recognizable physical processes and that there is
no necessity for reference to hyperspace phenomena for an explanation
of the "unexplained residuum." It is, of course, understood that many
of the possibilities predicated for hyperspace are purely nonsensical
so far as their actual realization is concerned. Our concern is,
therefore, not with that class of predicates, but with those wherein
reside some slight show of probability of their response to the
conditions of n-dimensionality either as a system of space-measurement
or a so-called space or series of spaces.
MAJOR ELLIS concludes his simple study of four-space by proposing the
following query:
"May not birth be an unfolding through the ether into the
symmetrical life-cell, and death, the reverse process of a
folding-up into four dimensional unity?"
It is confessed that there seems to be nothing to warrant the giving
of an affirmative reply to this query. It is, perhaps, sentimentally
speaking a very beautiful thing to contemplate death as a painless,
unconscious involvement into a glorious _one-ness_ with all life, and
birth, as the reverse of all this. But where is the utility of such a
dream if it be merely a dream and impossible of realization?
SIMON NEWCOMB,[23] at one time one of the outstanding figures in
the early development of the fourth dimensional hypothesis, openly
declared that "there is no proof that the molecule may not vibrate in a
fourth dimension. There are facts which seem to indicate at least the
possibility of molecular motion or change of some sort not expressible
in terms of time and the three coördinates in space."
Of course, there is no proof that a molecule may not at times be
ensconced in a four-space neither is there proof nor probability that
it is so hidden. Indeed, there is no proof that there is such a thing
as a molecule for that matter.
[23] Vide _Science_, Vol. VII, 158, 1898, p. 4.
In all of the foregoing proposals it is assumed that the fourth
dimension really exists and that it lies just beneath the surface of
the visible, palpable limits of the material universe; that lying
in close juxtaposition to all that we are able to see, to hear or
sense in any way is this mysterious, eternally prolific, all-powerful
something, hyperspace, ever-ready to nourish and sustain the forms
which have the nether parts firmly encysted in one or the other of her
_n_-dimensional berths. Thus it would seem that while yet functioning
in a strictly tridimensional atmosphere, some one, more reckless
than the rest, should at last stumble upon some up-lying portion of
it and be instantly transformed into a mathetic fay of etherealized
four-dimensional stuff.
_PART TWO_
SPATIALITY
AN INQUIRY INTO THE ESSENTIAL NATURE OF SPACE AS DISTINGUISHED
FROM THE MATHEMATICAL INTERPRETATION