The Fourth Dimension
By Charles H. Hinton
[This selection includes excerpts of The Fourth Dimension
(1904) including material from Chapters 1, 4, and 5. Copy-text:
pp 120-141, Speculations on the Fourth Dimension, Selected Writings
of Charles H. Hinton, Copyright 1980 by Dover Publications, Inc.,
ISBN 0-486-23916-0, LC 79-54399.]
There is nothing more indefinite, and at the same time more
real, than that which we indicate when we speak of the "higher."
In our social life we see it evidenced in a greater complexity
of relations. But this complexity is not all. There is, at the
same time, a contact with, an apprehension of, something more
fundamental, more real.
With the greater development of man there comes a consciousness
of something more than all the forms in which it shows itself.
There is a readiness to give up all the visible and tangible
for the sake of those principles and values of which the visible
and tangible are the representation. The physical life of civilized
man and of a mere savage are practically the same, but the civilized
man has discovered a depth in his existence, which makes him
feel that that which appears all to the savage is a mere externality
and appurtenage to his true being.
Now, this higher--how shall we apprehend it? It is generally
embraced by our religious faculties, by our idealizing tendency.
But the higher existence has two sides. It has a being as well
as qualities. And in trying to realize it through our emotions
we are always taking the subjective view. Our attention is always
fixed on what we feel, what we think. Is there any way of apprehending
the higher after the purely objective method of a natural science?
I think that there is.
Plato, in a wonderful allegory, speaks of some men living in
such a condition that they were practically reduced to be the
denizens of a shadow world. They were chained, and perceived
but the shadows of themselves and all real objects projected
on a wall, towards which their faces were turned. All movements
to them were but movements on the surface, all shapes but the
shapes of outlines with no substantiality.
Plato uses this illustration to portray the relation between
true being and the illusions of the sense world. He says that
just as a man liberated from his chains could learn and discover
that the world was solid and real, and could go back and tell
his bound companions of this greater higher reality, so the philosopher
who has been liberated, who has gone into the thought of the
ideal world, into the world of ideas greater and more real than
the things of sense, can come and tell his fellow men of that
which is more true than the visible sun--more noble than Athens,
the visible state.
Now, I take Plato's suggestion; but literally, not metaphorically.
He imagines a world which is lower than this world, in that shadow
figures and shadow motions are its constituents; and to it he
contrasts the real world. As the real world is to this shadow
world, so is the higher world to our world. I accept his analogy.
As our world in three dimensions is to a shadow or plane world,
so is the higher world to our three-dimensional world. That is,
the higher world is four-dimensional; the higher being is, so
far as its existence is concerned apart from its qualities, to
be sought through the conception of an actual existence spatially
higher than that which we realize with our senses.
Here you will observe I necessarily leave out all that gives
its charm and interest to Plato's writings. All those conceptions
of the beautiful and good which live immortally in his pages.
All that I keep from his great storehouse of wealth is this
one thing simply--a world spatially higher than this world, a
world which can only be approached through the stocks and stones
of it, a world which must be apprehended laboriously, patiently,
through the material things of it, the shapes, the movements,
the figures of it.
We must learn to realize the shapes of objects in this world
of the higher man; we must become familiar with the movements
that objects make in his world, so that we can learn something
about his daily experience, his thoughts of material objects,
The means for the prosecution of this enquiry are given in
the conception of space itself.
It often happens that that which we consider to be unique
and unrelated gives us, within itself, those relations by means
of which we are able to see it as related to others, determining
and determined by them.
Thus, on the earth is given that phenomenon of weight by means
of which Newton brought the earth into its true relation to the
sun and other planets. Our terrestrial globe was determined in
regard to other bodies of the solar system by means of a relation
which subsisted on the earth itself.
And so space itself bears within it relations of which we
can determine it as related to other space. For within space
are given the conceptions of point and line, line and plane,
which really involve the relation of space to a higher space.
Where one segment of a straight line leaves off and another
begins is a point, and the straight line itself can be generated
by the motion of the point.
One portion of a plane is bounded from another by a straight
line, and the plane itself can be generated by the straight line
moving in a direction not contained in itself.
Again, two portions of solid space are limited with regard
to each other by a plane; and the plane, moving in a direction
not contained in itself, can generate solid space.
Thus, going on, we may say that space is that which limits
two portions of higher space from each other, and that our space
will generate the higher space by moving in a direction not contained
Another indication of the nature of four-dimensional space
can be gained by considering the problem of the arrangement of
If I have a number of swords of varying degrees of brightness,
I can represent them in respect of this quality by points arranged
along a straight line.
If I place a sword at A, figure 22, and regard it as having
a certain brightness, then the other swords can be arranged in
a series along the line, as at A, B, C, etc., according to their
degrees of brightness.
If now I take account of another quality, say length, they
can be arranged in a plane. Starting from A, B, C, I can find
points to represent different degrees of length along such lines
as AF, BD, CE, drawn from A and B and C (see fig. 23). Points
on these lines represent different degrees of length with the
same degree of brightness. Thus the whole plane is occupied by
points representing all conceivable varieties of brightness and
Bringing in a third quality, say sharpness, I can draw, as
in figure 24, any number of upright lines. Let distances along
these upright lines represent degrees of sharpness, thus the
points F and G will represent swords of certain definite degrees
of the three qualities mentioned, and the whole of space will
serve to represent all conceivable degrees of these three qualities.
If now I bring in a fourth quality, such as weight, and try
to find a means of representing it as I did the other three qualities,
I find a difficulty. Every point in space is taken up by some
conceivable combination of the three qualities already taken.
To represent four qualities in the same way as that in which
I have represented three, I should need another dimension of
Thus we may indicate the nature of four-dimensional space
by saying that it is a kind of space which would give positions
representative of four qualities, as three-dimensional space
gives positions representative of three qualities.
A Chapter in the History of Four Space
Parmenides, and the Asiatic thinkers with whom he is in close
affinity, propound a theory of existence which is in close accord
with a conception of a possible relation between a higher and
a lower dimensional space. This theory, prior and in marked contrast
to the main stream of thought, which we shall afterwards describe,
forms a closed circle by itself. It is one which in all ages
has had a strong attraction for pure intellect, and is the natural
mode of thought for those who refrain from projecting their own
volition into nature under the guise of causality.
According to Parmenides of the school of Flea, the all is
one, unmoving and unchanging. The permanent amid the transient--that
foothold for thought, that solid ground for feeling, on the discovery
of which depends all our life--is no phantom; it is the image
amidst deception of true being, the eternal, the unmoved, the
Thus says Parmenides.
But how explain the shifting scene, these mutations of things!
"Illusion," answers Parmenides. Distinguishing between
truth and error, he tells of the true doctrine of the one--the
false opinion of a changing world. He is no less memorable for
the manner of his advocacy than for the cause he advocates. It
is as if from his firm foothold of being he could play with the
thoughts under the burden of which others labored, for from him
springs that fluency of supposition and hypothesis which forms
the texture of Plato's dialectic.
Can the mind conceive a more delightful intellectual picture
than that of Parmenides, pointing to the one, the true, the unchanging,
and yet on the other hand ready to discuss all manner of false
opinion, forming a cosmogony too, false "but mine own"
after the fashion of the time?
In support of the true opinion he proceeded by the negative
way of showing the self-contradictions in the ideas of change
and motion. It is doubtful if his criticism, save in minor points,
has ever been successfully refuted. To express his doctrine in
the ponderous modern way we must make the statement that motion
is phenomenal not real.
Let us represent his doctrine.
Imagine a sheet of still water into which a slanting stick
is being lowered with a motion vertically downwards. Let 1, 2,
3 (fig. 25), be three consecutive positions of the stick. A,
B, C, will be three consecutive positions of the meeting of the
stick with the surface of the water. As the stick passes down,
the meeting will move from A on to B and C.
Suppose now all the water to be removed except a film. At
the meeting of the film and the stick there will be an interruption
of the film. If we suppose the film to have a property, like
that of a soap bubble, of closing up round any penetrating object
then as the stick goes vertically downwards the interruption
in the film will move on.
If we pass a spiral through the film, the intersection will
give a point moving in a circle shown by the dotted lines (fig.
26). Suppose now the spiral to be still and the film to move
vertically upwards the whole spiral will be represented in the
film of the consecutive positions of the point of intersection.
In the film the permanent existence of the spiral is experienced
as a time series--the record of traversing the spiral is a point
moving in a circle. If now we suppose a consciousness connected
with the film in such a way that the intersection of the spiral
with the film gives rise to a conscious experience, we see that
we shall have in the film a point moving in a circle, conscious
of its motion, knowing nothing of that real spiral the record
of the successive intersections of which by the film is the motion
of the point.
It is easy to imagine complicated structures of the nature
of the spiral, structures consisting of filaments, and to suppose
also that these structures are distinguishable from each other
at every section. If we consider the intersections of these filaments
with the film as it passes to be the atoms constituting a filmar
universe, we shall have in the film a world of apparent motion;
we shall have bodies corresponding to the filamentary structure,
and the positions of these structures with regard to one another
will give rise to bodies in the film moving amongst one another.
This mutual motion is apparent merely. The reality is of permanent
structures stationary, and all the relative motions accounted
for by one steady movement of the film as a whole.
Thus we can imagine a plane world, in which all the variety
of motion is the phenomenon of structures consisting of filamentary
atoms traversed by a plane of consciousness. Passing to four
dimensions and our space, we can conceive that all things and
movements in our world are the reading off of a permanent reality
by a space of consciousness. Each atom at every moment is not
what it was, but a new part of that endless line which is itself.
And all this system successively revealed in the time which is
but the succession of consciousness, separate as it is in parts,
in its entirety is one vast unity. Representing Parmenides' doctrine
thus, we gain a firmer hold on it than if we merely let his words
rest, grand and massive, in our minds. And we have gained the
means also of representing phases of that Eastern thought to
which Parmenides was no stranger. Modifying his uncompromising
doctrine, let us suppose, to go back to the plane of consciousness
and the structure of filamentary atoms, that these structures
are themselves moving--are acting, living. Then, in the transverse
motion of the film, there would be two phenomena of motion, one
due to the reading off in the film of the permanent existences
as they are in themselves, and another phenomenon of motion due
to the modification of the record of the things themselves, by
their proper motion during the process of traversing them.
Thus a conscious being in the plane would have, as it were,
a twofold experience. In the complete traversing of the structure,
the Intersection of which with the film gives his conscious all,
the main and principal movements and actions which he went through
would be the record of his higher self as it existed unmoved
and unacting. Slight modifications and deviations from these
movements and actions would represent the activity and self-determination
of the complete being, of his higher self.
It is admissible to suppose that the consciousness in the
plane has a share in that volition by which the complete existence
determines itself. Thus the motive and will, the initiative and
life, of the higher being, would be represented in the case of
the being in the film by an initiative and a will capable, not
of determining any great things or important movements in his
existence, but only of small and relatively insignificant activities.
In all the main features of his life his experience would be
representative of one state of the higher being whose existence
determines his as the film passes on. But in his minute and apparently
unimportant actions he would share in that will and determination
by which the whole of the being he really is acts and lives.
An alteration of the higher being would correspond to a different
life history for him. Let us now make the supposition that film
after film traverses these higher structures, that the life of
the real being is read off again and again in successive waves
of consciousness. There would be a succession of lives in the
different advancing planes of consciousness, each differing from
the preceding, and differing in virtue of that will and activity
which in the preceding had not been devoted to the greater and
apparently most significant things in life, but the minute and
apparently unimportant. In all great things the being of the
film shares in the existence of his higher self as it is at any
one time. In the small things he shares in that volition by which
the higher being alters and changes, acts and lives.
Thus we gain the conception of a life changing and developing
as a whole, a life in which our separation and cessation and
fugitiveness are merely apparent, but which in its events and
course alters, changes, develops; and the power of altering and
changing this whole neélies in the will and power the
limited being has of directing, guiding, altering himself in
the minute things of his existence.
Transferring our conceptions to those of an existence in a
higher dimensionality traversed by a space of consciousness,
we have an illustration of a thought which has found frequent
and varied expression. When, however, we ask ourselves what degree
of truth there lies in it, we must admit that, as far as we can
see, it is merely symbolical. The true path in the investigation
of a higher dimensionality lies in another direction.
The significance of the Parmenidean doctrine lies in this:
that here, as again and again, we find that those conceptions
which man introduces of himself, which he does not derive from
the mere record of his outward experience, have a striking and
significant correspondence to the conception of a physical existence
in a world of a higher space. How close we come to Parmenides'
thought by this manner of representation it is impossible to
say. What I want to point out is the adequateness of the illustration,
not only to give a static model of his doctrine, but one capable
as it were, of a plastic modification into a correspondence into
kindred forms of thought. Either one of two things must be true-that
four-dimensional conceptions give a wonderful power of representing
the thought of the East, or that the thinkers of the East must
have been looking at and regarding four-dimensional existence.
And from the numerical idealism of Pythagoras there is but
a step to the more rich and full idealism of Plato. That which
is apprehended by the sense of touch we put as primary and real,
and the other senses we say are merely concerned with appearances.
But Plato took them all as valid, as giving qualities of existence.
That the qualities were not permanent in the world as given to
the senses forced him to attribute to them a different kind of
permanence. He formed the conception of a world of ideas, in
which all that really is, all that affects us and gives the rich
and wonderful wealth of our experience, is not fleeting and transitory,
but eternal. And of this real and eternal we see in the things
about us the fleeting and transient images.
And this world of ideas was no exclusive one, wherein was
no place for the innermost convictions of the soul and its most
authoritative assertions. Therein existed justice beauty-the
one, the good, all that the soul demanded to be. The world of
ideas, Plato's wonderful creation preserved for man, for his
deliberate investigation and their sure development, all that
the rude incomprehensible changes of a harsh experience scatters
Plato believed in the reality of ideas. He meets us fairly
and squarely. Divide a line into two parts, he says (fig. 27);
one to represent the real objects in the world, the other to
represent the transitory appearances, such as the image in still
water, the glitter of the sun on a bright surface, the shadows
on the clouds.
Take another line and divide it into two parts (fig. 28),
one representing our ideas, the ordinary occupants of our minds,
such as whiteness, equality, and the other representing our true
knowledge, which is of eternal principles, such as beauty, goodness.
Then as A is to B, so is A' to B'.
That is, the soul can proceed, going away from real things
to a region of perfect certainty, where it beholds what is, not
the scattered reflections; beholds the sun, not the glitter on
the sands; true being, not chance opinion.
Now, this is to us, as it was to Aristotle, absolutely inconceivable
from a scientific point of view. We can understand that a being
is known in the fullness of his relations; it is in his relations
to his circumstances that a man's character is known; it is in
his acts under his conditions that his character exists. We cannot
grasp or conceive any principle of individuation apart from the
fullness of the relations to the surroundings.
But suppose now that Plato is talking about the higher man--the
four-dimensional being that is limited in our external experience
to a three-dimensional world. Do not his words begin to have
a meaning? Such a being would have a consciousness of motion
which is not as the ! motion he can see with the eyes of the
body. He, in his own being, knows a reality to which the outward
matter of this too solid earth is flimsy superficiality. He too
knows a mode of being, the fullness of relations, in which can
only be represented in the limited world of sense, as the painter
unsubstantially portrays the depths of woodland, plains, and
air. Thinking of such a being in man, was not Plato's line well
It is noteworthy that, if Plato omitted his doctrine of the
independent origin of ideas, he would present exactly the four-dimensional
argument; a real thing as we think it is an idea. A plane being's
idea of a square object is the idea of an abstraction, namely,
a geometrical square. Similarly our idea of a solid thing is
an abstraction, for in our idea there is not the four-dimensional
thickness which is necessary, however slight, to give reality.
The argument would then run, as a shadow is to a solid object,
so is the solid object to the reality. Thus A and B' would be
In the allegory which I have already alluded to, Plato in
almost as many words shows forth the relation between existence
in a superficies and in solid space. And he uses this relation
to point to the conditions of a higher being.
He imagines a number of men prisoners, chained so that they
look at the wall of a cavern in which they are confined, with
their backs to the road and the light. Over the road pass men
and women, figures and processions, but of all this pageant all
that the prisoners behold is the shadow of it on the wall whereon
they gaze. Their own shadows and the shadows of the things in
the world are all that they see, and identifying themselves with
their shadows related as shadows to a world of shadows, they
live in a kind of dream.
Plato imagines one of their number to pass out from amongst
them into the real space world, and then returning to tell them
of their condition.
Here he presents most plainly the relation between existence
in a plane world and existence in a three-dimensional world.
And he uses this illustration as a type of the manner in which
we are to proceed to a higher state from the three-dimensional
life we know.
It must have hung upon the weight of a shadow which path he
took! Whether the one we shall follow toward the higher solid
and the four-dimensional existence, or the one which makes ideas
the higher realities, and the direct perception of them the contact
with the truer world.
The theories which are generally connected with the names
of Lobatchewsky and Bolyai bear a singular and curious relation
to the subject of higher space.
In order to show what this relation is, I must ask the reader
to be at the pains to count carefully the sets of points by which
I shall estimate the volumes of certain figures.
No mathematical processes beyond this simple one of counting
will be necessary.
Let us suppose we have before us in figure 29 a plane covered
with points at regular intervals, so placed that every four determine
Now it is evident that as four points determine a square,
so four squares meet in a point.
Thus, considering a point inside a square as belonging to it,
we may say that a point on the corner of a square belongs to
it and to four others equally: belongs a quarter of it to each
Thus the square ACDE (fig. 31) contains one point, and has
four points at the four corners. Since one-fourth of each of
these four belongs to the square, the four together count as
one point, and the point value of the square is two points--the
one inside and the four at the corner make two points belonging
to it exclusively.
Now the area of this square is two unit squares, as can be
seen by drawing two diagonals in figure 32.
We also notice that the square in question is equal to the
sum of the squares on the sides AB, BC, of the right-angled triangle
ABC. Thus we recognize the proposition that the square on the
hypotenuse is equal to the sum of the squares on the two sides
of a right-angled triangle.
Now suppose we set ourselves the question of determining whereabouts,
in the ordered system of points, the end of a line would wEcome
when it turned about a point keeping one extremity fixed at the
We can solve this problem in a particular case. If we can
find a square lying slantwise amongst the dots which is equal
to one which goes regularly, we shall know that the two sides
are equal, and that the slanting side is equal to the straight-way
side. Thus the volume and shape of a figure remaining unchanged
will be the test of its having rotated about the point, so that
we can say that its side in its first position would turn into
its side in the second position.
Now, such a square can be found in the one whose side is five
units in length.
In figure 33, in the square on AB, there are
9 points interior 9
4 at the corners 1
4 sides with 3 on each side, 6
considered as 1 1/2, on each
side, because belonging
equally to two squares
The total is 16. There are 9 points in the square on BC. In
the square on AC there are--
24 points inside 24
4 at the corners 1
or 25 altogether.
Hence we see again that the square on the hypotenuse is equal
to the squares on the sides.
Now take the square AFHC, which is larger than the square on
AB. It contains 25 points.
16 inside 16
16 on the sides, counting as 8
4 on the corners 1
making 25 altogether.
If two squares are equal we conclude the sides are equal. Hence,
the line AF turning round A would move so that it would after
a certain turning coincide with AC.
This is preliminary, but it involves all the mathematical difficulties
that will present themselves.
There are two alterations of a body by which its volume is not
One is the one we have just considered, rotation, the other is
what is called shear.
Consider a book, or heap of loose pages. They can be slid so
that each one slips over the preceding one, and the whole assumes
the shape b in figure 34.
The deformation is not shear alone, but shear accompanied
Shear can be considered as produced in another way.
Take the square ABCD (fig. 35), and suppose that it is pulled
out from along one of its diagonals both ways, and proportionately
compressed along the other diagonal. It will assume the shape
in figure 36.
This compression and expansion along two lines at right angles
is what is called shear; it is equivalent to the sliding illustrated
above combined with a turning round.
In pure shear a body is compressed and extended in two directions
at right angles to each other, so that its volume remains unchanged.
Now we know that our material bodies resist shear--shear does
violence to the internal arrangement of their particles, but
they turn as wholes without such internal resistance.
But there is an exception. In a liquid shear and rotation
take place equally easily, there is no more resistance against
a shear than there is against a rotation.
Now, suppose all bodies were to be reduced to the liquid state,
in which they yield to shear and to rotation equally easily,
and then were to be reconstructed as solids, but in such a way
that shear and rotation had interchanged places.
That is to say, let us suppose that when they had become solids
again they would shear without offering any internal resistance,
but a rotation would do violence to their internal arrangement.
That is, we should have a world in which shear would have
taken the place of rotation.
A shear does not alter the volume of a body: thus an inhabitant
living in such a world would look on a body sheared as we look
on a body rotated. He would say that it was of the same shape,
but had turned a bit round.
Let us imagine a Pythagoras in this world going to work to
investigate, as is his wont.
Figure 37 represents a square unsheared. Figure 38 represents
a square sheared. It is not the figure into which the square
in figure 37 would turn, but the result of shear on some square
not drawn. It is a simple slanting placed figure, taken now as
we took a simple slanting placed square before. Now, since bodies
in this world of shear offer no internal resistance to shearing,
and keep their volume when sheared, an inhabitant accustomed
to them would not consider that they altered their shape under
shear. He would call ACDE as much a square as the square in figure
37. We will call such figures shear squares. Counting the dots
in ACDF, we find
2 inside 2
4 at corners 1
or a total of 3.
Now, the square on the side AB has 4 points, that on BC has
1 point. Here the shear square on the hypotenuse has not 5 points
but 3; it is not the sum of the squares on the sides, but the
This relation always holds. Look at figure 39.
Shear square on hypotenuse
7 internal 7
4 at corners 1
Square on one side--which the reader can draw for himself--
4 internal 4
8 on sides 4
4 at corners 1
The square on the other side is 1. Hence in this case again
the difference is equal to the shear square on the hypotenuse,
9 - 1 = 8.
Thus in a world of shear the square on the hypotenuse would
be equal to the difference of the squares on the sides of a right-angled
In figure 40 another shear square is drawn on which the above
relation can be tested.
What now would be the position a line on turning by shear
would take up?
We must settle this in the same way as previously with our
Since a body sheared remains the same, we must find two equal
bodies, one in the straight way, one in the slanting way, which
have the same volume. Then the side of one will by turning become
the side of the other, for the two figures are each what the
other becomes by a shear turning.
We can solve the problem in a particular case--
In the figure ACDE (fig. 41) there are
15 inside 15
4 at corners 1
a total of 16.
Now in the square ABCF, there are 16--
9 inside 9
12 on sides 6
4 at corners 1
Hence the square on AB would, by the shear turning, become
the shear square ACDE.
And hence the inhabitant of this world would say that the
line AB turned into the line AC. These two lines would be to
him two lines of equal length, one turned a little way round
from the other.
That is, putting shear in place of rotation, we get a different
kind of figure, as the result of the shear rotation, from what
we got with our ordinary rotation. And as a consequence we get
a position for the end of a line of invariable length when it
turns by the shear rotation, different from the position which
it would assume on turning by our rotation.
A real material rod in the shear world would, on turning about
A, pass from the position AB to the position AC. We say that
its length alters when it becomes AC, but this transformation
of AB would seem to an inhabitant of the shear world like a turning
of AB without altering in length.
If now we suppose a communication of ideas that takes place
between one of ourselves and an inhabitant of the shear world,
there would evidently be a difference between his views of distance
We should say that his line AB increased in length in turning
to AC. He would say that our line AF (fig. 33) decreased in length
in turning to AC. He would think that what we called an equal
line was in reality a shorter one.
We should say that a rod turning round would have its extremities
in the positions we call at equal distances. So would he--but
the positions would be different. He could, like us, appeal to
the properties of matter. His rod to him alters as little as
ours does to us.
Now, is there any standard to which we could appeal, to say
which of the two is right in this argument? There is no standard.
We should say that, with a change of position, the configuration
and shape of his objects altered. He would say that the configuration
and shape of our objects altered in what we called merely a change
of position. Hence distance independent of position is inconceivable,
or practically, distance is solely a property of matter.
There is no principle to which either party in this controversy
could appeal. There is nothing to connect the definition of distance
with our ideas rather than with his, except the behavior of an
actual piece of matter. For the study of the processes which
go on in our world the definition of distance given by taking
the sum of the squares is of paramount importance to us. But
as a question of pure space without making any unnecessary assumptions,
the shear world is just as possible and just as interesting as
It was the geometry of such conceivable worlds that Lobatchewsky
and Bolyai studied.
This kind of geometry has evidently nothing to do directly
with four-dimensional space.
But a connection arises in this way. It is evident that, instead
of taking a simple shear as I have done, and defining it as that
change of the arrangement of the particles of a solid which they
will undergo without offering any resistance due to their mutual
action, I might take a complex motion, composed of a shear and
a rotation together, or some other kind of deformation.
Let us suppose such an alteration picked out and defined as
the one which means simple rotation; then the type, according
to which all bodies will alter by this rotation, is fixed.
Looking at the movements of this kind, we should say that
the objects were altering their shape as well as rotating. But
to the inhabitants of that world they would seem to be unaltered,
and our figures in their motions would seem to them to alter.
In such a world the features of geometry are different. We
have seen one such difference in the case of our illustration
of the world of shear, where the square on the hypotenuse was
equal to the difference, not the sum, of the squares on the sides.
In our illustration we have the same laws of parallel lines
as in our ordinary rotation world, but in general the laws of
parallel lines are different.
In one of these worlds of a different constitution of matter,
through one point there can be two parallels to a given line,
in another of them there can be none; that is, although a line
be drawn parallel to another it will meet it after a time.
Now it was precisely in this respect of parallels that Lobatchewsky
and Bolyai discovered these different worlds. They did not think
of them as worlds of matter, but they discovered that space did
not necessarily mean that our law of parallels is true. They
made the distinction between laws of space and laws of matter,
although that is not the form in which they stated their results.
The way in which they were led to these results was the following.
Euclid had stated the existence of parallel lines as a postulate--putting
frankly this unproved proposition--that one line and only one
parallel to a given straight line can be drawn, as a demand,
as something that must be assumed. The words of his ninth postulate
are these: "if a straight line meeting two other straight
lines makes the interior angles on the same side of it equal
to two right angles, the two straight lines will never meet."
The mathematicians of later ages did not like this bald assumption,
and not being able to prove the proposition they called it an
axiom--the eleventh axiom.
Many attempts were made to prove the axiom; no one doubted
of its truth, but no means could be found to demonstrate it.
At last an Italian, Sacchieri, unable to find a proof, said:
"Let us suppose it not true." He deduced the results
of there being possibly two parallels to one given line through
a given point, but feeling the waters too deep for the human
reason, he devoted the latter half of his book to disproving
what he had assumed in the first part.
Then Bolyai and Lobatchewsky with firm step entered on the
forbidden path. There can be no greater evidence of the indomitable
nature of the human spirit, or of its manifest destiny to conquer
all those limitations which bind it down within the sphere of
sense than this grand assertion of Bolyai and Lobatchewsky.
Take a line AB and a point C. We say and see and know that through
C can only be drawn one line parallel to AB.
But Bolyai said: "I will draw two." Let CD be parallel
to AB, that is, not meet AB however far produced, and let lines
beyond CD also not meet AB; let there be a certain region between
CD and CE, in which no line drawn meets AB. CE and CD produced
backwards through C will give a similar region on the other side
Nothing so triumphantly, one may almost say so insolently, ignoring
of sense had ever been written before. Men had struggled against
the limitations of the body, fought them, despised them, conquered
them. But no one had ever thought simply as if the body, the
bodily eyes, the organs of vision, all this vast experience of
space, had never existed. The age-long contest of the soul with
the body, the struggle for mastery, had come to a culmination.
Bolyai and Lobatchewsky simply thought as if the body was not.
The struggle for dominion, the strife and combat of the soul
were over; they had mastered, and the Hungarian drew his line.
Can we point out any connection, as in the case of Parmenides,
between these speculations and higher space? Can we suppose it
was any inner perception by the soul of a motion not known to
the senses, which resulted in this theory so free from the bonds
of sense? No such supposition appears to be possible.
Practically, however, metageometry had a great influence in
bringing the higher space to the front as a working hypothesis.
This can be traced to the tendency the mind has to move in the
direction of least resistance. The results of the new geometry
could not be neglected, the problem of parallels had occupied
a place too prominent in the development of mathematical thought
for its final solution to be neglected. But this utter independence
of all mechanical considerations, this perfect cutting loose
from the familiar intuitions, was so difficult that almost any
other hypothesis was more easy of acceptance, and when Beltrami
showed that the geometry of Lobatchewsky and Bolyai was the geometry
of shortest lines drawn on certain curved surfaces, the ordinary
definitions of measurement being retained, attention was drawn
to the theory of a higher space. An illustration of Beltrami's
theory is furnished by the simple consideration of hypothetical
beings living on a spherical surface (fig. 44).
Let ABCD be the equator of a globe, and AP, BP, meridian lines
drawn to the pole, P. The lines AB, AP, BP would seem to be perfectly
straight to a person moving on the surface of the sphere, and
unconscious of its curvature. Now AP and BP both make right angles
with AB. Hence they satisfy the definition of parallels. Yet
they meet in P. Hence a being living on a spherical surface,
and unconscious of its curvature, would find that parallel lines
would meet. He would also find that the angles in a triangle
were greater than two right angles. In the triangle PAB, for
instance, the angles at A and B are right angles, so the three
angles of the triangle PAB are greater than two right angles.
Now in one of the systems of metageometry (for after Lobatchewsky
had shown the way it was found that other systems were possible
besides his), the angles of a triangle are greater than two right
Thus a being on a sphere would form conclusions about his
space which are the same as he would form if he lived on a plane,
the matter in which had such properties as are presupposed by
one of these systems of geometry. Beltrami also discovered a
certain surface on which there could be drawn more than one "straight"
line through a point which would not meet another given line.
I use the word straight as equivalent to the line having the
property of giving the shortest path between any two points on
it. Hence, without giving up the ordinary methods of measurement,
it was possible to find conditions in which a plane being would
necessarily have an experience corresponding to Lobatchewsky's
geometry. And by the consideration of a higher space, and a solid
curved in such a higher space, it was possible to account for
a similar experience in a space of three dimensions.
Now, it is far more easy to conceive of a higher dimensionality
to space than to imagine that a rod in rotating does not move
so that its end describes a circle. Hence, a logical conception
having been found harder than that of a four-dimensional space,
thought turned to the latter as a simple explanation of the possibilities
to which Lobatchewsky had awakened it. Thinkers became accustomed
to deal with the geometry of higher space--it was Kant, says
Veronese, who first used the expression of "different spaces"--and
with familiarity the inevitableness of the conception made itself
From this point it is but a small step to adapt the ordinary
mechanical conceptions to a higher spatial existence, and then
the recognition of its objective existence could be delayed no
longer. Here, too, as in so many cases, it turns out that the
order and connection of our ideas is the order and connection
What is the significance of Lobatchewsky's and Bolyai's work?
It must be recognized as something totally different from
the conception of a higher space; it is applicable to spaces
of any number of dimensions. By immersing the conception of distance
in matter to which it properly belongs, it promises to be of
the greatest aid in analysis; for the effective distance of any
two particles is the product of complex material conditions and
cannot be measured by hard and fast rules. Its ultimate significance
is altogether unknown. It is a cutting loose from the bonds of
sense, not coincident with the recognition of a higher dimensionality,
but indirectly contributory thereto.
Thus, finally, we have come to accept what Plato held in the
hollow of his hand; what Aristotle's doctrine of the relativity
of substance implies. The vast universe, too, has its higher,
and in recognizing it we find that the directing being within
us no longer stands inevitably outside our systematic knowledge.
Uncle Taz Library
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