### Geometry and Space

**Contents**

Geometric space and representative space

Solid bodies and geometry

The law of homogeneity

The number of dimensions

The Non-Euclidean world

The four-dimensional World

Conclusions

Notes

In an article I published in the *Revue générale des
sciences, *t. ii., p. 774, on the subject of non-Euclidean geometry,
I wrote the following sentences:

Beings with a mind made like ours, with the same senses that we have,
but without any prior education, could receive from a properly selected
world impressions such that they would be led to construct a geometry
different from Euclid's and to localize the phenomena of that exterior
world in a non-Euclidean space or even in a four-dimensional space.

For us, educated in this present world, if we were suddenly transported
into that new world , we would have no difficulty in relating its
phenomena to our Euclidean space.

A man who devoted his life to it could perhaps succeed in picturing to
himself a fourth dimension. (1)

I did not follow this up with any further clarification and it must have
astonished several readers; it seems to me therefore that it is
necessary to develop my thought and that I owe some explanations to the
public.

*Geometric space and representative
space.* We often say that the images of
external objects are localized in space, and even that only on that
condition can they be formed. We also say that this space, which thus
serves as a *frame *prepared in advance for our sensations and
representations, is identical with that of the geometricians and that it
possesses all the same properties.

To all the good people who [think that way], the sentence cited above
must indeed have seemed quite extraordinary. But we ought to see whether
they might not be under some illusion that a thorough analysis might
dispel.

First, what are, strictly speaking, the properties of space? I mean the
space that is the object of geometry and which I call *geometric space.
*Here are some of the more essential:

- It is continuous;
- It is infinite;
- It has three dimensions;
- It is homogeneous, that is to say that all its
points are identical to one another;
- It is isotropic, that is to say that all the
straight lines passing through a given point are identical to one
another.

**. . . . .**

*Solid bodies and geometry.*
Among the objects that surround us, there are some that frequently
undergo displacements susceptible to correction by a *correlative *
movement of our own bodies; they are the *solid* *bodies.*

Other objects, whose form is variable, only exceptionally undergo
similar displacements (change of position without change of form). When
a body moves and in so doing *changes its form*, we can no longer
by appropriate movements bring our sense organs back into the same *
relative *situation with respect to that body; we can no longer,
consequently, re-establish the original set of impressions.

It is only later on and after a series of new experiences that we learn
to separate bodies of variable form into smaller elements such that each
of them moves in approximate accordance with the laws of motion of solid
bodies. We thus distinguish "deformations" from other changes of state;
in these deformations each element undergoes a simple change of position,
which can be corrected, but the modification of the ensemble is more
profound and can no longer be corrected by a correlative movement.

Such an idea is already very complex and could only have appeared
relatively late. It could moreover not have arisen if the observation of
solid bodies bad not already taught us to distinguish changes of
position.

*If, therefore, there were no solid bodies in nature, there would be
no geometry.*

Another remark also merits a moment of attention. Let us imagine a solid
body first occupying position *a* and passing afterward to position
*b*; in its first position, it would cause us to receive a set of
impressions *A*, and in its second position a set of impressions *
B*. Now let there be a second solid body having qualities entirely
different from the first, for example a different color. Now let us
suppose that it passes from position *a*, where we receive the set
of impressions *A'*, to position *b*, where we receive the set
of impressions *B'*.

In general set *A* will have nothing in common with set *A'* ,
nor set *B* with set *B'*. The passage from set *A* to
set *B* and that from set *A"* to set *B'* are therefore
two changes which in themselves have in general nothing in common.

And yet we consider both these changes as displacements, and what is
more, we consider them as the same displacement. How can that be?

It is simply because we can correct both by the same correlative
movement of our body.

It is therefore the correlative movement which constitutes the *only
link* between two phenomena that it would not otherwise have occurred
to us to compare.

On the other hand, our bodies, thanks to the number of their joints and
muscles, can go through a host of different movements; but not all are
capable of "correcting" a modification of external objects; those only
are capable of it in which our whole body, or at the very least, all the
sense organs involved, move *en* *bloc, *that is to say, like
a solid body without varying their relative positions.

In summary:

- We are first led to distinguish two categories of
phenomena: Some, involuntary, not accompanied by muscular sensations,
are attributed by us to exterior objects; they are external changes.
Others, whose qualities are opposite and which we attribute to the
movements of our own body, are internal changes.
- We notice that certain changes of each of these
categories can be corrected by a correlative change in the other
category.
- We distinguish, among external changes, those that
have also a correlative in the other category; we call these
displacements; and likewise among the internal changes we distinguish
those that have a correlative in the first category.

We have thus defined, thanks to this reciprocity, a
particular class of phenomena that we call displacements. *It is the
laws of these phenomena that are the object of geometry.*

*The law of homogeneity.*
The first of these laws is that of homogeneity. Let us suppose that, by
an external change *a, *we were to pass from a set of impressions
*A* to set *B*, then that this change were corrected by a
voluntary correlative movement *b* such that we were brought back
to set *A*. Let us suppose now that another external change *a'*
caused us again to pass from set A to set *B.*

Experience teaches us that this change *a'* is, like *a, *
susceptible to correction by a voluntary movement *b'*, and that
this voluntary movement *b'* corresponds to the same muscular
sensations as the movement *b* which corrected a. It is this fact
that we normally have in mind when we say that *space is homogeneous
and isotropic.*

We can also say that a movement that occurs once can be repeated a
second time, a third time, and so on, without its properties varying.

Those readers who know the article I wrote in this journal on the nature
of mathematical reasoning will perhaps remember the importance that I
attribute to the possibility of indefinitely repeating a single
operation.

It is from this repetition that mathematical reasoning draws its
strength; it is thanks to the law of homogeneity which it has taken from
geometric facts.

To be complete, we should annex to the law of homogeneity a host of
other analogous laws into the details of which I do not wish to enter,
but which mathematicians subsume into a single word by saying that
displacements form a "group".

*The number of dimensions.*
I feel more difficulty in explaining my thought concerning the origin of
the notion of point and the number of dimensions: it is markedly
different from opinions generally accepted and it is not easy to state
it in ordinary language.

We understand displacements in terms of the passage from a set of
impressions *A* to a different set *b*; but among these
displacements we distinguish some such that the initial set *A* and
the final set *B* conserve certain common qualities. I do not wish
to go into further detail nor to seek to determine in exactly what these
common qualities consist.

I am satisfied to note that we are led to distinguish certain special
displacements such that we may say they leave fixed one of the points of
space.

That is the origin of the idea of point.

The set of all displacements constitutes what we call a group; the set
of those displacements that leave fixed a point of space constitutes a
partial or subgroup.

It is in the relation of this group and subgroup that we must seek the
explanation of the fact that space has three dimensions.

The group total is of the order 6, that is to say that any such
displacement can be considered as a combination of six elementary and
independent movements.

The subgroup is of the order 3, that is to say that any displacement
belonging to this subgroup, or, in other words, any displacement which
leaves fixed a point of space, can be considered a combination of three
elementary and independent movements.

The difference 6 - 3 represents the number of dimensions.

*The Non-Euclidean world.*
If geometric space were a frame imposed on *each* of our sensations,
considered individually, it would be impossible to represent an image
stripped of that frame, and we could change nothing in our geometry.

But that is not the case; geometry is only the résumé of the laws in
accordance with which images succeed one another. Nothing prevents our
imagining a series of representations, in every way similar to our
ordinary representations, but succeeding one another according to laws
different from those to which we are accustomed.

We can therefore conceive that beings whose education took place in a
milieu where these laws were inoperative could have a geometry very
different from ours.

Let us imagine, for example, a world enclosed in a great sphere and
subject to the following laws:

The temperature is not uniform; it is maximal in the
center, and diminishes in proportion as we move away, reducing to
absolute zero when we get to the sphere in which the world is enclosed.

I will further specify the precise law according to
which the temperature varies. Let *R* be the radius of the limiting
sphere; let *r* be the distance from a given point to the center of
this sphere. The absolute temperature will be proportional to *R*²-
*r*².

I will suppose, in addition, that in this world, all bodies have the
same coefficient of expansion such that the length of any rule will be
proportional to the absolute temperature.

Finally I will suppose that an object transported from one point to
another whose temperature is different, immediately assumes calorific
equilibrium in its new environment.

Nothing in these hypotheses is contradictory or unimaginable.

A mobile object will then become smaller and smaller as it approaches
the limiting sphere.

First let us observe that although this world is limited from the point
of view of our habitual geometry, it will appear infinite to its
inhabitants.

Indeed, when they wish to approach the limiting sphere, they get colder
and become smaller and smaller. The steps they take are therefore also
smaller and smaller, so that they can never reach the limiting sphere.

If, for us, geometry is only the study of the laws by which invariable
solids move, for these imaginary beings, it would be a study of the laws
by which solids move *deformed by those differences of temperature*
I have just mentioned.

True, in our world natural solids also submit to variations of form and
volume due to beating and cooling. But we neglect these variations in
laying the foundation of geometry, for in addition to their being very
weak, they are irregular and consequently seem to us to be accidental.

In this hypothetical world, this would not be the case, -and these
variations would follow regular and simple laws.

Moreover, the diverse solid particles making up the bodies of the
inhabitants would also undergo the same variations ,of form and volume.

I will add one further hypothesis. I will assume that light crosses
diversely refracting media in such a way that the index of refraction is
inversely proportional to *R²* - *r²*. It is easy to see that
in these conditions, light rays would be not rectilinear but circular.

To justify the foregoing, it remains to be shown that certain changes
occurring in the position of exterior objects could be *corrected *
by correlative movements of the sensory beings who inhabit this
imaginary world; and in such a way as to restore the original group of
impressions felt by these sensory beings.

Let us suppose that an object moves, while changing shape, not as an
invariable solid, but as a solid undergoing unequal dilations in exact
accord with the law of temperature that I proposed above. Permit me, in
the interest of succinct language, to call such a movement *a non-Euclidean
displacement*. . .

If a sensory being were in the area, his impressions would be modified
by the displacement of the object, but he could re-establish them by
making suitable movements. It suffices that finally the object and the
sensory being, considered as forming a single body, should have
undergone one of those particular displacements that I have just called
non-Euclidean.

Although from the point of view of our habitual geometry bodies are
deformed in this displacement and their diverse parts are no longer in
the same relative position, nevertheless we shall see that the
impressions of that sensory being have again become the same.

Indeed, although the mutual distances of the diverse parts could and did
vary, nevertheless the parts originally in contact have returned into
contact. Therefore the tactile impressions have not changed.

Furthermore, taking into account the hypothesis set forth above as to
the refraction and curve of light rays, the visual impressions will also
have remained the same.

These imaginary beings will thus be led like us to classify the
phenomena they witness, and to distinguish among them those "changes of
position" that are susceptible to correction by a voluntary correlative
movement.

If they were to found a geometry, it would not be like ours, the study
of the movements of our invariable solids, it would be that of the
changes of position that they will thus have distinguished, and which
are none other than the "non-Euclidean displacements"; *it will be
non-Euclidean geometry.*

Thus, beings like us, educated in such a world, would not have the same
geometry as we.

*The four-dimensional World.*
In the same way as a non-Euclidean world, we can represent a world
having four dimensions.

The sense of sight, even with a single eye, joined to muscular
sensations relative to movements of the eyeball could suffice to our
knowing a three-dimensional world.

The images of exterior objects come and paint themselves on the retina,
which is a two-dimensional tabula; these are perspectives.

But, since these objects are mobile, and since the same is true of our
eye, we see successively different perspectives of the same body, taken
from several different points of view. We notice at the same time that
the passage from one perspective to another is often accompanied by
muscular sensations.

If the passage from perspective *A* to perspective *B* and
that from perspective *A' *to perspective *B' *are accompanied
by the same muscular sensations, we consider them to be operations of
the same kind.

Studying afterward the laws according to which these operations combine,
we recognize that they form a group having the same structure as that of
the movements of invariable solids.

Now, we have seen that it is from the properties of this group that we
have deduced the notions of geometric and of three dimensions.

Thus we understand how the idea of a three-dimensional space was able to
arise from the spectacle of these perspectives, even though each of them
has only two dimensions, because *they succeed one another in
accordance with certain laws.*

In the same way that we can make on a plane the perspective of a figure
having three dimensions, we can make that of a four-dimensional figure
on a tabula with three (or two) dimensions. It is only a game for the
geometrician.

We can even take from a single figure several perspectives from several
different viewpoints.

We can easily represent these perspectives since they have only three
dimensions.

Let us imagine that the diverse perspectives of a single object succeed
one another; that the passage from one to the other is accompanied by
muscular sensations.

We will naturally consider two of these passages as two operations of
the same kind when they are associated with the same muscular sensations.

Then nothing prevents our imagining that these operations might combine
following any law that we wish, for example in such a way as to form a
group having the same structure as that of an invariable four-dimensional
solid.

In all this there is nothing we cannot represent and nevertheless these
sensations are precisely those that a being furnished with a two-dimensional
retina would feel if be could move in four-dimensional space.

It is in this sense that we may say that we can represent the fourth
dimension.

*Conclusions.*
We say that experience plays an indispensable role in the genesis of
geometry; but it would be an error to conclude that geometry is an
experimental science, even in part.

If it were experimental it would be only approximate and provisional.
And what a crude approximation!

Geometry would only be the study of the movements of solids; but in
reality it is not concerned with natural solids, it has as its object
certain ideal solids, absolutely invariable, which are only a simplified
and very distant image of the natural ones.

The notion of ideal bodies is drawn entirely from our minds and
experience is only an occasion that invites us to construct such a
notion.

The object of geometry is the study of a particular "group"; but the
general concept of group pre-exists in our minds, at least potentially.
It is imposed on us, not as a form of sensibility, but as a form of
understanding.

Still, among all possible groups, we must choose that one which will be,
so to speak, the *standard *to which we will refer natural
phenomena.

Experience guides us in this choice which it does not impose on us; it
makes us recognize not what is the truest geometry, but what is the most
convenient. It will be noticed that I have been able to describe the
fantastic worlds that I imagined above *without ceasing to use the
language of ordinary geometry.*

Indeed, we should not have to change anything if we were to be
transported to such a world.

Beings who were educated there would no doubt find it more convenient to
create a geometry different from ours, better adapted to their
impressions. As for us, in the face of the same impressions, it is
certain that we would find it more convenient not to change our habits.

(1) This
selection first appeared as an article, *"L'Espace et la géométrie*,"
in *Revue de métaphysique et de morale, *1895 t.iii, pp. 631-46. It
was especially translated for this volume by William Ryding of the
Department of French, Columbia University.