considerations: We have, let us suppose, a straight line on which a
point is situated, and the problem is proposed to determine the position
of the point on the line in an unequivocal manner. The simplest way to
solve this is, to state how far the point is removed in the one or the
other direction from some given fixed point; just as in a thermometer
the position of the surface of the mercury is given by a statement of
its distance in the direction of cold or heat from a predetermined fixed
point—the point of freezing water. To state, therefore, the position
of a point on a straight line, the sole datum necessary is a single
number, for beforehand we have fixed upon some standard line, like the
centimetre, and some definite point to which we give the value zero,
and have also previously decided in what direction from the zero-point,
points must be situated whose position is expressed by positive numbers,
and also in what direction those must lie whose position is expressed by
negative numbers. This last mentioned fact, that a _single_ number is
sufficient to determine the place of a point in a straight line, is the
real reason why we attribute to the straight line or to any part of it a
single dimension.
More generally, we call every totality or system, of infinitely numerous
things, _one_-dimensional, in which _one_ number is all that is requisite
to determine and distinguish any particular one of these things amidst
the entire totality. Thus, time is one-dimensional. We, as inhabitants
of the earth, have naturally chosen as our unit of time, the period of
the rotation of the earth about its axis, namely, the day, or a definite
portion of a day. The zero-point of time is regarded in Christian
countries as the year of the birth of Christ, and the positive direction
of time is the time _subsequent_ to the birth of Christ. These data
fixed, all that is necessary to establish and distinguish any definite
point of time amid the infinite totality of all the points of time, _is
a single number_. Of course this number need not be a whole number, but
may be made up of the sum of a whole number and a fraction in whose
numerator and denominator we may have numbers as great as we please. We
may, therefore, also say that the totality of all conceivable numerical
magnitudes, or of only such as are greater than one definite number and
smaller than some other definite number, is one-dimensional.
We shall add here a few additional examples of one-dimensioned magnitudes
presented by geometry. First, the circumference of a circle is a
one-dimensional magnitude, as is every curved line, whether it returns
into itself or not. Further, the totality of all equilateral triangles
which stand on the same base is one-dimensional, or the totality of
all circles that can be described through two fixed points. Also, the
totality of all conceivable cubes will be seen to be one-dimensional,
provided they are distinguished, not with respect to position, but with
respect to magnitude.
In conformity with the fundamental ideas by which we define the notion
of a one-dimensional manifoldness, it will be seen that the attribute
_two_-dimensional must be applied to all totalities of things in
which _two_ numbers are necessary (and sufficient) to distinguish
any determinate individual thing amid the totality. The simplest
two-dimensioned complex which we know of is the plane. To determine
accurately the position of a point in a plane, the simplest way is to
take two axes at right angles to each other, that is, fixed straight
lines, and then to specify the distances by which the point in question
is removed from each of these axes.
This method of determining the position of a point in a plane suggested
to the celebrated philosopher and mathematician Descartes the fundamental
idea of analytical geometry, a branch of mathematics in which by the
simple artifice of ascribing to every point in a plane two numerical
values, determined by its distances from the two axes above referred
to, planimetrical considerations are transformed into algebraical. So,
too, all kinds of curves that graphically represent the dependence of
things on time, make use of the fact that the totality of the points in
a plane is two-dimensional. For example, to represent in a graphical
form the increase of the population of a city, we take a horizontal axis
to represent the time, and a perpendicular one to represent the numbers
which are the measures of the population. Any two lines, then, whose
lengths practical considerations determine, are taken as the unit of
time, which we may say is a year, and as the unit of population, which
we will say is one thousand. Some definite year, say 1850, is fixed
upon as the zero point. Then, from all the equally distant points on
the horizontal axis, which points stand for the years, we proceed in
directions parallel to the other axis, that is, in the perpendicular
direction, just so much upwards as the numbers which stand for the
population of that year require. The terminal points so reached, or the
curve which runs through these terminal points, will then present a
graphic picture of the rates of increase of the population of the town
in the different years. The rectangular axes of Descartes are employed
in a similar way for the construction of barometer curves, which specify
for the different localities of a country the amount of variation of the
atmospheric pressure during any period of time. Immediately next to the
plane the surface of the earth will be recognised as a two-dimensional
aggregate of points. In this case geographical latitude and longitude
supply the two numbers that are requisite accurately to determine the
position of a point. Also, the totality of all the possible straight
lines that can be drawn through any point in space is two-dimensional, as
we shall best understand if we picture to ourselves a plane which is cut
in a point by each of these straight lines and then remember that by such
a construction every point on the plane will belong to some one line and,
_vice versa_, a line to every point, whence it follows that the totality
of all the straight lines which pass through the point assigned are of
the same dimensions as the totality of the points of the imagined plane.
The question might be asked, In what way and to what extent in this
case is the specification of _two_ numbers requisite and sufficient
to determine amid all the rays which pass through the specified point
a definite individual ray? To get a clear idea of the problem here
involved, let us imagine the ray produced far into the heavens, where
some quite definite point will correspond to it. Now, the position of
a point in the heavens depends, as does the position of a point on all
spherical surfaces, on two numbers. In the heavens these two numbers are
ordinarily supplied by the two angles called altitude, or the distance
above the plane of the horizon, and azimuth, or the angular distance
between the circle on which the altitude is measured and the meridian
of the observer. It will be seen thus that the totality of all the
luminous rays that an eye, conceived as a point, can receive from the
outer world is two-dimensional, and also that a luminous point emits
a two-dimensional group of luminous rays. It will also be observed,
in connection with this example, that the two-dimensional totality of
all the rays that can be drawn through a point in space is something
different from the totality of the rays that pass through a point but
are required to lie in a given plane. Such a group of objects as the
last-named one, is a one-dimensional totality.
Now that we have sufficiently discussed the attributes that are
characteristic of one and two-dimensional aggregates, we may, without any
further investigation of the subject, propose the following definition,
that, generally, _an n-dimensional totality of infinitely numerous
things is such, with respect to which the specification of n numbers
is necessary and sufficient to indicate a definite individual amid the
totality of all the infinitely numerous individuals of the group_.
Accordingly, the point-aggregate made up of the world-space which we
inhabit, is a three-dimensional totality. To get true bearings in this
space and to define any determinate point in it, we have therefore to lay
through any point which we take as our zero-point three axes at right
angles to each other, one running from right to left, one backwards
and forwards, and one upwards and downwards. We then join each two of
these axes by a plane and are enabled thus to specify the position of
every point in space by the three perpendicular distances by which the
point in question is removed in a positive or negative sense from
these three planes. It is customary to denote the numbers which are the
measures of these three distances by _x_, _y_, and _z_, the positive
_x_, positive _y_, and positive _z_ ordinarily being reckoned in the
right hand, the forward, and the upward directions from the origin.
If now, with direct reference to this fundamental axial system, any
particular specification of _x_, _y_, and _z_ be made, there will, by
such an operation, be cut out and isolated from the three-dimensional
manifoldness of all the points of space a totality of less dimensions.
If, for example, _z_ is equal to seven units or measures, this is
equivalent to a statement that only the two-dimensional totality of the
points is meant, which constitute the plane that can be laid at right
angles to the upward-passing _z_-axis at a distance of seven measures
from the zero-point. Consequently, every imaginable equation between _x_,
_y_, and _z_ isolates and defines a two-dimensional aggregate of points.
If two different equations obtain between _x_, _y_, and _z_, two such
two-dimensional totalities will be isolated from among all the points
of space. But as these last must have some one-dimensional totality in
common, we may say that the co-existence of two equations between _x_,
_y_, and _z_ defines a one-dimensional totality of points, that is to say
a straight line, a line curved in a plane, or even, perhaps, one curved
in space. It is evident from this that the introduction of the three axes
of reference forms a bridge between the theory of space and the theory of
equations involving three variable quantities, _x_, _y_, _z_. The reason
that the theory of space cannot thus be brought into connection with
algebra in general, that is, with the theory of indefinitely numerous
equations, but only with the algebra of three quantities, _x_, _y_, _z_,
is simply to be sought in the fact that space, as we picture it, can only
have three dimensions.
We have now only to supply a few additional examples of _n_-dimensional
totalities. All particles of air are four-dimensional in magnitude
when in addition to their position in space we also consider the
variable densities which they assume, as they are expressed by
the different heights of the barometer in the different parts of
the atmosphere. Similarly, all conceivable spheres in space are
four-dimensional magnitudes, for their centres form a three-dimensional
point-aggregate, and around each centre there may be additionally
conceived a one-dimensional totality of spheres, the radii of which can
be expressed by every numerical magnitude from zero to infinity. Further,
if we imagine a measuring stick of invariable length to assume every
conceivable position in space, the positions so obtained will constitute
a five-dimensional aggregate. For, in the first place, one of the
extremities of the measuring stick may be conceived to assume a position
at every point of space, and this determines for one extremity alone
of the stick a three-dimensional totality of positions; and secondly,
as we have seen above, there proceeds from every such position of this
extremity a two-dimensional totality of directions, and by conceiving the
measuring-stick to be placed lengthwise in every one of these directions
we shall obtain all the conceivable positions which the second extremity
can assume, and consequently, the dimensions must be 3 plus 2 or 5.
Finally, to find out how many dimensions the totality of all the possible
positions of a square, invariable in magnitude, possesses, we first give
one of its corners all conceivable positions in space, and we thus obtain
three dimensions. One definite point in space now being fixed for the
position of one corner of the square, we imagine drawn through this point
all possible lines, and on each we lay off the length of the side of
the square and thus obtain two additional dimensions. Through the point
obtained for the position of the second corner of the square we must
now conceive all the possible directions drawn that are perpendicular
to the line thus fixed, and we must lay off once more on each of these
directions the side of the square. By this last determination the
dimensions are only increased by one, for only one one-dimensional
totality of perpendicular directions is possible to one straight line
in one of its points. Three corners of the square are now fixed and
therewith the position of the fourth also is uniquely determined.
Accordingly, the totality of all equal squares which only differ from one
another by their position in space, constitutes a manifoldness of six
dimensions.
II.
THE INTRODUCTION OF THE NOTION OF FOUR-DIMENSIONAL POINT-AGGREGATES,
PERMISSIBLE.
In the preceding section it was shown that we can conceive not only
of manifoldnesses of one, two, and three dimensions, but also of
manifoldnesses of _any_ number of dimensions. But it was at the same time
indicated that our world-space, that is, the totality of all conceivable
_points_ that differ only in respect of position, cannot in agreement
with our notions of things possess more than three dimensions. But the
question now arises, whether, if the progress of science tends in such
a direction, it is permissible to extend the notion of space by the
introduction of point-aggregates of more than three dimensions, and to
engage in the study of the properties of such creations, although we know
that notwithstanding the fact that we may conceptually establish and
explore such aggregates of points, yet we cannot picture to ourselves
these creations as we do the spatial magnitudes which surround us, that
is, the regular three-dimensional aggregates of points.
To show the reader clearly that this question must be answered in the
affirmative, that the extension of our notion of space is permissible,
although it leads to things which we cannot perceive by our senses,
I may call the reader’s attention to the fact that in arithmetic
we are accustomed from our youth upwards to extensions of ideas,
which, accurately viewed, as little admit of graphic conception as a
four-dimensional space, that is, a point-aggregate of four dimensions. By
his senses man first reaches only the idea of whole numbers—the results
of counting. The observation of primitive peoples[79] and of children
clearly proves that the essential decisive factors of counting are these
three: First, we abstract, in the counting of things, completely from
the individual and characteristic attributes of these things, that is,
we consider them as homogeneous. Second, we associate individually with
the things which we count other homogeneous things. These other things
are even now, among uncivilised peoples, the ten fingers of the two
hands. They may, however, be simple strokes, or, as in the case of dice
and dominoes, black points on a white background. Third, we substitute
for the result of this association some concise symbol or word; for
example, the Romans substituted for three things counted, three strokes
placed side by side, namely: III; but for greater numbers of things
they employed abbreviated signs. The Aztecs, the original inhabitants
of Mexico, had time enough, it seems, to express all the numbers up to
nineteen by equal circles placed side by side. They had abbreviated signs
only for the numbers 20, 400, 8000, and so forth. In speaking, some one
same sound might be associated with the things counted; but this method
of counting is nowadays employed only by clocks: the languages of men
since prehistoric times have fashioned concise words for the results of
the association in question. From the notion of number, thus fixed as
the result of counting, man reached the notion of the addition of two
numbers, and thence the notion that is the inverse of the last process,
the notion of subtraction. But at this point it clearly appears that
not every problem which may be propounded is soluble; for there is no
number which can express the result of the subtraction of a number from
one which is equally large or from one which is smaller than itself. The
primary school pupil who says that 8 from 5 “won’t go” is perfectly right
from his point of view. For there really does not exist any result of
counting which added to eight will give five.
If humanity had abided by this point of view and had rested content with
the opinion that the problem “5 minus 8” is not solvable, the science of
arithmetic would never have received its full development, and humanity
would not have advanced as far in civilisation as it has. Fortunately,
men said to themselves at this crisis: “If 5 minus 8 won’t go, we’ll
_make it go_; if 5 minus 8 does not possess an intelligible meaning, we
will simply give it one.” As a fact, things which have not a meaning
always afford men a pleasing opportunity of investing them with one. The
question is, then, what significance is the problem “5 minus 8” to be
invested with?
The most natural and, therefore, the most advantageous solution
undoubtedly is to abide by the original notion of subtraction as the
inverse of addition, and to make the significance of 5 minus 8 such,
that for 5 minus 8 plus 8 we shall get our original minuend 5. By such a
method all the rules of computation which apply to real differences will
also hold good for unreal differences, such as 5 minus 8. But it then
clearly appears that all forms expressive of differences in which the
number that stands before the minus direction is less by an equal amount
than that which follows it may be regarded as equal; so that the simplest
course seems to be to introduce as the common characteristic of all equal
differential forms of this description a common sign, which will indicate
at the same time the difference of the two numbers thus associated. Thus
it came about, that for 5 minus 8, as well as for every differential form
which can be regarded as equal thereto the sign “-3” was introduced. But
in calling differential forms of this description numbers, the notion of
number was extended and a new domain was opened up, namely, the domain of
negative numbers.
In the further development of the science of arithmetic, through the
operation of division viewed as the inverse of multiplication, a second
extension of the idea of number was reached, namely, the notion of
fractional numbers as the outcome of divisions that had led to numbers
hitherto undefined. We find, thus, that the science of arithmetic
throughout its whole development has strictly adhered to the principle
of conformity and consistency and has invested every association of
two numbers, which before had no significance, by the introduction of
new numbers, with a real significance, such that similar operations in
conformity with exactly the same rules could be performed with the new
numbers, viewed as the results of this association, as with the numbers
which were before known and perfectly defined. Thus the science proceeded
further on its way and reached the notions of irrational, imaginary, and
complex numbers.
The point in all this, which the reader must carefully note, is, that
all the numbers of arithmetic, with the exception of the positive whole
numbers, are artificial products of human thought, invented to make the
language of arithmetic more flexible, and to accelerate the progress of
science. All these numbers lack the attributes of representability.
No man in the world can picture to himself “minus three trees.” It is
possible, of course, to know that when three trees of a garden have been
cut down and carried away, that three are missing, and by substituting
for “missing” the inverse notion of “added,” we may say, perhaps, that
“minus three trees” are added. But this is quite different from the feat
of imagining a negative number of trees. We can only picture to ourselves
a number of trees that results from actual counting, that is, a positive
whole number. Yet, notwithstanding all this, people had not the slightest
hesitation in extending the notion of number. Exactly so must it be
permitted us in geometry to extend the notion of space, even though such
an extension can only be mentally defined and can never be brought within
the range of human powers of representation.
In mathematics, in fact, the extension of any notion is admissible,
provided such extension does not lead to contradictions with itself or
with results which are well established. Whether such extensions are
necessary, justifiable, or important for the advancement of science
is a different question. It must be admitted, therefore, that the
mathematician is justified in the extension of the notion of space as a
point-aggregate of three dimensions, and in the introduction of space or
point-aggregates of more than three dimensions, and in the employment
of them as means of research. Other sciences also operate with things
which they do not know exist, and which, though they are sufficiently
defined, cannot be perceived by our senses. For example, the physicist
employs the ether as a means of investigation, though he can have no
sensory knowledge of it. The ether is nothing more than a means which
enables us to comprehend mechanically the effects known as action at a
distance and to bring them within the range of a common point of view.
Without the assumption of a material which penetrates everything, and
by means of whose undulations impulses are transmitted to the remotest
parts of space, the phenomena of light, of heat, of gravitation, and of
electricity would be a jumble of isolated and unconnected mysteries. The
assumption of an ether, however, comprises in a systematic scheme all
these isolated events, facilitates our mental control of the phenomena
of nature, and enables us to produce these phenomena at will. But it
must not be forgotten in such reflections that the ether itself is
even a greater problem for man, and that the ether-hypothesis does not
solve the difficulties of phenomena, but only puts them in a unitary
conceptual shape. Notwithstanding all this, physicists have never had
the least hesitation in employing the ether as a means of investigation.
And as little do reasons exist why the mathematicians should hesitate to
investigate the properties of a four-dimensioned point-aggregate, with
the view of acquiring thus a convenient means of research.
III.
THE INTRODUCTION OF THE IDEA OF FOUR-DIMENSIONED POINT-AGGREGATES OF
SERVICE TO RESEARCH.
From the concession that the mathematician has the right to define
and investigate the properties of point-aggregates of more than three
dimensions, it does not necessarily follow that the introduction of an
idea of this description is of value to science. Thus, for example, in
arithmetic, the introduction of operations which spring from involution,
as involution and its two inverse operations proceed from multiplication,
is undoubtedly permitted. Just as for “_a_ times _a_ times _a_” we write
the abbreviated symbol “_a_³,” (which we read, _a_ to the third power,)
and investigate in detail the operation of involution thus defined, so we
might also introduce some shorthand symbol for “_a_ to the _a_ᵗʰ power to
the _a_ᵗʰ power” and thus reach an operation of the fourth degree, which
would regard _a_ as a passive number and the number 3, or any higher
number, as the active number, that is, as the number which indicates how
often _a_ is taken as the base of a power whose exponent may be _a_, or
“_a_ to the _a_ᵗʰ,” or “_a_ to the _a_ᵗʰ to the _a_ᵗʰ power.”