itself to be of no especial value to mathematics. And the reason is that
in the operation of involution the law of commutation does not hold
good. In addition, the numbers to be added may be interchanged and the
introduction of multiplication is therefore of great value. So, also,
in multiplication the numbers which are combined, that is, the factors,
may be changed about in any way, and thus the introduction of involution
is of value. But in involution the base and the exponent cannot be
interchanged, and consequently the introduction of any higher operation
is almost valueless.
But with the introduction of the idea of point-aggregates of multiple
dimensions the case is wholly different. The innovation in question
has proved itself to be not only of great importance to research, but
the progress of science has irresistibly forced investigators to the
introduction of this idea, as we shall now set forth in detail.
In the first place, algebra, especially the algebraical theory of
systems of equations, derives much advantage from the notion of multiple
dimensioned spaces. If we have only three unknown quantities, _x_, _y_,
_z_, the algebraical questions which arise from the possible problems of
this class admit, as we have above seen, of geometrical representation
to the eye. Owing to this possibility of geometrical representation,
some certain simple geometrical ideas like “moving,” “lying in,”
“intersecting,” and so forth, may be translated into algebraical
events. Now, no reason exists why algebra should stop at three variable
quantities; it must in fact take into consideration any number of
variable quantities.
For purposes of brevity and greater evidentness, therefore, it is quite
natural to employ geometrical forms of speech in the consideration of
more than three variables. But when we do this, we assume, perhaps
without really intending to do so, the idea of a space of more than
three dimensions. If we have four variable quantities, _x_, _y_,
_z_, _u_, we arrive, by conceiving attributed to each of these four
quantities every possible numerical magnitude, at a four-dimensioned
manifoldness of numerical quantities, which we may just as well regard as
a four-dimensioned aggregate of points. Two equations which exist on this
supposition between _x_, _y_, _z_, and _u_, define two three-dimensioned
aggregates of points, which intersect, as we may briefly say, in a
two-dimensioned aggregate of points, that is, in a surface; and so on.
In a somewhat different manner the determination of the contents of a
square or a cube by the involution of a number which stands for the
length of its sides, leads to the notion of four-dimensioned structures,
and, consequently, to the notion of a four-dimensioned point-space. When
we note that _a_² stands for the contents of a square, and _a_³ for
the contents of a cube, we naturally inquire after the contents of a
structure which is produced from the cube as the cube is produced from
the square and which also will have the contents _a_⁴. We cannot, it
is true, clearly picture to ourselves a structure of this description,
but we can, nevertheless, establish its properties with mathematical
exactness.[80] It is bounded by 8 cubes just as the cube is bounded by 6
squares; it has 16 corners, 24 squares, and 32 edges, so that from every
corner 4 edges, 6 squares, and 4 cubes proceed, and from every edge 3
squares and 3 cubes.
Yet despite the great service to algebra of this idea of
multiple-dimensioned space, it must be conceded that the conception
although convenient is yet not indispensable. It is true, algebra is in
need of the idea of multiple dimensions, but it is not so absolutely in
need of the idea of _point_-aggregates of multiple dimensions.
This notion is, however, necessary and serviceable for a profound
comprehension of geometry. The system of geometrical knowledge which
Euclid of Alexandria created about three hundred years before Christ,
supplied during a period of more than two thousand years a brilliant
example of a body of conclusions and truths which were mutually
consistent and logical. Up to the present century the idea of elementary
geometry was indissolubly bound up with the name of Euclid, so that in
England where people adhered longest to the rigid deductive system of
the Grecian mathematician, the task of “learning geometry” and “reading
Euclid” were until a few years ago identical. Every proposition of this
Euclidian system rests on other propositions, as one building-stone in a
house rests upon another. Only the very lowest stones, the foundations,
were without supports. These are the axioms or fundamental propositions,
truths on which all other truths are, directly or indirectly, founded,
but which themselves are assumed without demonstration as self-evident.
But the spirit of mathematical research grew in time more and more
critical, and finally asked, whether these axioms might not possibly
admit of demonstration. Especially was a rigid proof sought for the
eleventh axiom of Euclid, which treats of parallels.
After centuries of fruitless attempts to prove Euclid’s eleventh axiom,
Gauss, and with him Bolyai and Lobatschewsky, Riemann and Helmholtz,
finally stated the decisive reasons why any attempt to prove the axiom
of the parallels must necessarily be futile. These reasons consist of
the fact that though this axiom holds good enough in the world-space
such as we do and can conceive it, yet three-dimensioned spaces are
ideally conceivable though not capable of mental representation, where
the axiom does not hold good. The axiom was thus shown to be a mere
fact of _observation_, and from that time on there could no longer be
any thought of a deductive demonstration of it. In view of the intimate
connection, which both in an historical and epistemological point of view
exists between the extension of the concept of space and the critical
examination of the axioms of Euclid, we must enter at somewhat greater
length into the discussion of the last mentioned propositions.
Of the axioms which Euclid premises to his geometry, only the following
three are really geometrical axioms:
_Eighth axiom_: Magnitudes which coincide with one another are equal to
one another.
_Eleventh axiom_: If a straight line meet two straight lines so as to
make the two interior angles on the same side of it taken together less
than two right angles, these straight lines, being continually produced,
shall at length meet on that side on which are the angles which are less
than two right angles.
_Twelfth axiom_: Two straight lines cannot inclose a [finite] space.
The numerous proofs which in the course of time were adduced in
demonstration of these axioms, especially of the eleventh, all turn out
on close examination to be pseudo-proofs. Legendre drew attention to the
fact that either of the following axioms might be substituted for the
eleventh:
_a_) Through a point there can be drawn to a straight line, within the
plane which joins the point with the line, one and one line only which
shall not intersect the first (parallels) however far the two lines may
be produced;
_b_) If two parallel lines are cut by a third straight line, the interior
alternate angles will be equal.
_c_) The sum of the angles of a triangle is equal to two right angles,
that is, to the angle of a straight line or 180°.
By the aid of any one of these three assertions, the eleventh axiom of
Euclid may be proved, and, _vice versa_, by the aid of the latter each
of the three assertions may be proved, of course with the help of the
other two axioms, eight and twelve. The perception that the eleventh
axiom does not admit of demonstration without the employment of one of
the foregoing substitutes may best be gained from the consideration of
congruent figures. Every reader will remember from his first instruction
in geometry that the congruence of two triangles is demonstrated by the
superposition of one triangle on the other and by then ascertaining
whether the two completely coincide, no assumptions being made in the
determination except those above mentioned.
[Illustration: Fig. 1.]
In the case of triangles which are congruent as are I and II in
the preceding cut, this coincidence may be effected by the simple
_displacement_ of one of the triangles; so that even a two-dimensional
being, supposed to be endowed with powers of reasoning, but only capable
of picturing to itself motions within a plane, also might convince itself
that the two triangles I and II could be made to coincide. But a being
of this description could not convince itself in like manner of the
congruence of triangles I and III. It would discover the equality of
the three sides and the three angles, but it could never succeed in so
superposing the two triangles on each other as to make them coincide. A
three-dimensioned being, however, can do this very easily. It has simply
to turn triangle I about one of its sides and to shove the triangle,
thus brought into the position of its reflection in a mirror, into the
position of triangle III. Similarly, triangles II and III may be made to
coincide by moving either out of the plane of the paper around one of
its sides as axis and turning it until it again falls in the plane of
the paper. The triangle thus turned over can then be brought into the
position of the other.
Later on we shall revert to these two kinds of congruence: “congruence
by displacement” and “congruence by circumversion.” For the present we
will start from the fact that it is always possible within the limits of
a plane to take a triangle out of one position and bring it into another
without altering its sides and angles. The question is, whether this
is only possible in the plane, or whether it can also be done on other
surfaces.
We find that there are certain surfaces in which this is possible, and
certain others in which it is not. For instance, it is impossible to move
the triangle drawn on the surface of an egg into some other position on
the egg’s surface without a distension or contraction of some of the
triangle’s parts. On the other hand, it is quite possible to move the
triangle drawn on the surface of a sphere into any other position on the
sphere’s surface without a distension or contraction of its parts. The
mathematical reason of this fact is, that the surface of a sphere, like
the plane, has everywhere the same curvature, but that the surface of an
egg at different places has different curvatures. Of a plane we say that
it has everywhere the curvature zero; of the surface of a sphere we say
it has everywhere a positive curvature, which is greater in proportion
as the radius is smaller. There are surfaces also which have a constant
negative curvature; these surfaces exhibit at every point in directions
proceeding from the same side a partly concave and a partly convex
structure, somewhat like the centre of a saddle. There is no necessity
of our entering in any detail into the character and structure of the
last-mentioned surfaces.
Intimately related with the plane, however, are all those surfaces,
which, like the plane, have the curvature zero; in this category belong
especially cylindrical surfaces and conical surfaces. A sheet of paper
of the form of the sector of a circle may, for example, be readily bent
into the shape of a conical surface. If two congruent triangles, now,
be drawn on the sheet of paper, which may by displacement be translated
the one into the other, these triangles will, it is plain, also remain
congruent on the conical surface; that is, on the conical surface also
we may displace the one into the other; for though a bending of the
figures will take place, there will be no distension or contraction.
Similarly, there are surfaces which, like the sphere, have everywhere a
constant positive curvature. On such surfaces also every figure can be
transferred into some other position without distension or contraction
of its parts. Accordingly, on all surfaces thus related to the plane or
sphere, the assumption which underlies the eighth axiom of Euclid, that
it is possible to transfer into any new position any figure drawn on such
surfaces without distortion, holds good.
The eleventh axiom in its turn also holds good on all surfaces of
constant curvature, whether the curvature be zero or positive; only in
such instances instead of “straight” line we must say “shortest” line. On
the surface of a sphere, namely, two shortest lines, that is, arcs of two
great circles, always intersect, no matter whether they are produced in
the direction of the side at which the third arc of a great circle makes
with them angles less than two right angles, or, in the direction of the
other side, where this arc makes with them angles of more than two right
angles. On the plane, however, two straight lines intersect only on the
side where a third straight line that meets them makes with them interior
angles less than two right angles.
The twelfth axiom of Euclid, finally, only holds good on the plane and
on the surfaces related to it, but not on the sphere or other surfaces
which, like the sphere, have a constant positive curvature. This also
accounts for the fact that one of the three postulates which we regarded
as substitutes for the eleventh axiom, though valid for the plane, is not
true for the surface of a sphere; namely, the postulate that defines
the sum of the angles of a triangle. This sum in a plane triangle is two
right angles; in a spherical triangle it is more than two right angles,
the spherical triangle being greater, the greater the excess the sum
of its angles is above two right angles. It will be seen, from these
considerations, that in geometries in which curved surfaces and not fixed
planes are studied, the axioms of Euclid are either all or partially
false.
The axioms of geometry thus having been revealed as facts of experience,
the question suggested itself whether in the same way in which it was
shown that different two-dimensional geometries were possible, also
different three-dimensional systems of geometry might not be developed;
and consequently what the relations were in which these might stand to
the geometry of the space given by our senses and representable to our
mind. As a fact, a three-dimensional geometry can be developed, which
like the geometry of the surface of an egg will exclude the axiom that a
figure or body can be transferred from any one part of space to any other
and yet remain congruent to itself. Of a three-dimensional space in which
such a geometry can be developed we say, that it has no constant measure
of curvature.
The space which is representable to us, and which we shall henceforth
call the _space of experience_, possesses, as our experiences without
exception confirm, the especial property that every bodily thing can be
transferred from any one part of it to any other without suffering in the
transference any distension or any contraction. The space of experience,
therefore, has a constant measure of curvature. The question, however,
whether this measure of curvature is zero or positive, that is, whether
the space of experience possesses the properties which in two-dimensioned
structures a plane possesses, or whether it is the three-dimensioned
analogon of the surface of a sphere is one which future experience alone
can answer. If the space of experience has a constant positive measure
of curvature which is different from zero, be the difference ever so
slight, a point which should move forever onward in a straight line, or,
more accurately expressed, in a shortest line, would sometime, though
perhaps after having traversed a distance which to us is inconceivable,
ultimately have to arrive from the opposite direction at the place from
which it set out, just as a point which moves forever onward in the
same direction on the surface of a sphere must ultimately arrive at its
starting point, the distance it traverses being longer the greater the
radius of the sphere or the smaller its curvature.
It will seem, at first blush, almost incredible, that the space of
experience even _can_ have this property. But an example, which is the
historical analogon of this modern transformation of our conceptions,
will render the idea less marvellous. Let us transport ourselves back to
the age of Homer. At that time people believed that the earth was a great
disc surrounded on all sides by oceans which were conceived to be in
all directions infinitely great. Indeed, for the primitive man, who has
never journeyed far from the place of his birth, this is the most natural
conception. But imagine now that some scholar had come, and had informed
the Homeric hero Ulysses that if he would travel forever on the earth in
the same direction he would ultimately come back to the point from which
he started; surely Ulysses would have gazed with as much astonishment
upon this scholar as we now look upon the mathematician who tells us that
it is possible that a point which moves forever onward in space in the
same direction may ultimately arrive at the place from which it started.
But despite the fact that Ulysses would have regarded the assertion of
the scholar as false because contradictory to his familiar conceptions,
that scholar, nevertheless, would have been right; for the earth is not
a plane but a spherical surface. So also the mathematician might be
right who bases this more recent strange view on the possible fact that
the space of experience may have a measure of curvature which is not
exactly zero but slightly greater than zero. If this were really the
case, the _volume_ of the space of experience, though very large, would,
nevertheless, be finite; just as the real spherical surface of the earth
as contrasted with the Homeric plane surface is finite, having so and so
many square miles. When the objection is here made that a finiteness of
space is totally at variance with our modes of thought and conceptions,
two ideas, “infinitely great” and “unlimited,” are confounded. All that
is at variance with our practical conceptions is that space can anywhere
have a limit; not that it may possibly be of tremendous but finite
magnitude.
It will now be asked if we cannot determine by actual observation
whether the measure of curvature of experiential space is exactly zero
or slightly different therefrom. The theorem of the sum of the angles of
a triangle and the conclusions which follow from this theorem do indeed
supply us with a means of ascertaining this fact. And the results of
observation have been, that _the measure of curvature of space is in all
probability exactly equal to zero or if it is slightly different from
zero it is so little so that the technical means of observation at our
command and especially our telescopes are not competent to determine the
amount of the deviation_. More, we cannot with any certainty say.
All these reflections, to which the criticism of the hypotheses that
underlie geometry long ago led investigators, compel us to institute a
comparison between the space of experience and other three-dimensioned
aggregates of points (spaces), which we cannot mentally represent but can
in thought and word accurately define and investigate. As soon, however,
as we are fully involved in the task of accurately investigating the
properties of three-dimensional aggregates of points, we similarly find
ourselves forced to regard such aggregates as the component elements
of a manifoldness of more than three dimensions. In this way the exact
criticism of even ordinary geometry leads us to the abstract assumption
of a space of more than three dimensions. And as the extension of
every idea gives a clearer and more translucent form to the idea as it
originally stood, here too the idea of multiple-dimensioned aggregates
of points and the investigation of their properties has thrown a new
light on the truths of ordinary geometry and placed its properties in
clearer relief. Amid the numerous examples which show how the notion of a
space of multiple-dimensions has been of great service to science in the
investigation of three-dimensioned space, we shall give one a place here
which is within the comprehension of non-mathematicians.
Imagine in a plane two triangles whose angles are denoted by pairs of
numbers—namely, by 1-2, 1-3, 1-4, and 2-5, 3-5, 4-5. (See Fig. 2.) Let
the two triangles so lie that the three lines which join the angles
1-2 and 2-5, 1-3 and 3-5, and 1-4 and 4-5 intersect at a point, which
we will call 1-5. If now we cause the sides of the triangles which are
opposite to these angles to intersect, it will be found that the points
of intersection so obtained possess the peculiar property of lying all
in one and the same straight line. The point of intersection of the
connection 1-3 and 1-4 with the connection 4-5 and 3-5 may appropriately
be called 3-4. Similarly, the point of intersection 2-4 is produced by
the meeting of 4-5, 2-5 and 1-2, 1-4; and the point of intersection
2-3, by the meeting of 1-3, 1-2 and 3-5, 2-5. The statement, that the
three points of intersection 3-4, 2-4, 2-3, thus obtained, lie in one
straight line, can be proved by the principles of plane geometry only
with difficulty and great circumstantiality. But by resorting to the
three-dimensional space of experience, in which the plane of the drawing
lies, the proposition may be rendered almost self-evident.
[Illustration: Fig. 2.]
To begin with, imagine any five points in space which may be denoted by
the numbers 1, 2, 3, 4, 5; then imagine all the possible ten straight
lines of junction drawn between each two of these points, namely, 1-2,
1-3 ... 4-5; and finally, also, all the ten planes of junction of every
three points described, namely, the plane 1-2-3, 1-2-4, ... 3-4-5. A
spatial figure will thus be obtained, whose ten straight lines will meet
some interposed plane in ten points whose relative positions are exactly
those of the ten points above described. Thus, for example, on this plane
the points 1-2, 1-3, and 2-3 will lie in a straight line, for through the
three spatial points 1, 2, 3, a plane can be drawn which will cut the
plane of a drawing in a straight line. The reason, therefore, that the
three points 3-4, 2-4, 2-3, also must ultimately lie in a straight line,
consists in the simple fact that the plane of the three points 2, 3, 4,
must cut the plane of the drawing in a straight line. The figure here
considered consists of ten points of which sets of three so lie ten times
in a straight line that conversely from every point also three straight
lines proceed.
[Illustration: Fig. 3.]
Now, just as this figure is a section of a complete three-dimensional
pentagon, so another remarkable figure, of similar properties, may be
obtained by the section of a figure of four-dimensioned space. Imagine,
namely, six points, 1, 2, 3, 4, 5, 6, situated in this four-dimensioned
space, and every three of them connected by a plane, and every four of
them by a three-dimensioned space. We shall obtain thus twenty planes and
fifteen three-dimensioned spaces which will cut the plane in which the
figure is to be produced in twenty points and fifteen rays which so lie
that each point sends out three rays and every ray contains four points.
(See Fig. 3.) Figures of this description, which are so composed of
points and rays that an equal number of rays proceed from every point and
an equal number of points lie in every ray, are called _configurations_.
Other configurations may, of course, be produced, by taking a different
number of points and by assuming that the points taken lie in a space of
different or even higher dimensions. The author of this article was the
first to draw attention to configurations derived from spaces of higher
dimensions. As we see, then, the notion of a space of more than three
dimensions has performed important work in the investigations of common
plane geometry.