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*Science and Hypothesis*

## Chapter 5: Experiment and Geometry

### Henri Poincaré

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1. I have on several occasions in the preceding pages tried to show how the principles of geometry are not experimental facts, and that in particular Euclid's postulate cannot be proved by experiment. However convincing the reasons already given may appear to me, I feel I must dwell upon them, because there is a profoundly false conception deeply rooted in many minds.

2. Think of a material circle, measure its radius and circumference, and see if the ratio of the two
lengths is equal to . What have we done? We have made an experiment on the properties of the
matter with which this *roundness has *been realised, and of which the measure we used is made.

*3. Geometry and *Astronomy. — The same question may also be asked in another way. If
Lobatschewsky's geometry is true, the parallax of a very distant star will be finite. If Riemann's is
true, it will be negative. These are the results which seem within the reach of experiment, and it
is hoped that astronomical observations may enable us to decide between the two geometries.
But

(73) what we call a straight line in astronomy is simply the path of a ray of light. If, therefore, we were to discover negative parallaxes, or to prove that all parallaxes are higher than a certain limit, we should have a choice between two conclusions: we could give up Euclidean geometry, or modify the laws of optics, and suppose that light is not rigorously propagated in a straight line. It is needless to add that every one would look upon this solution as the more advantageous. Euclidean geometry, therefore, has nothing to fear from fresh experiments.

4. Can we maintain that certain phenomena which are possible in Euclidean space would be impossible in non-Euclidean space, so that experiment in establishing these phenomena would directly contradict the non-Euclidean hypothesis? I think that such a question cannot be seriously asked. To me it is exactly equivalent to the following, the absurdity of which is obvious: — There are lengths which can be expressed in metres and centimetres, but cannot be measured in toises, feet, and inches; so that experiment, by ascertaining the existence of these lengths, would directly contradict this hypothesis, that there are toises divided into six feet. Let us look at the question a little more closely. I assume that the straight line in Euclidean space possesses any two properties, which I shall call A and B; that in non-Euclidean space it still possesses the property A, but no longer possesses the property B ; and, finally, I

(74) assume that in both Euclidean and non-Euclidean space the straight line is the only line that possesses the property A. If this were so, experiment would be able to decide between the hypotheses of Euclid and Lobatschewsky. It would be found that some concrete object, upon which we can experiment — for example, a pencil of rays of light —possesses the property A. We should conclude that it is rectilinear, and we should then endeavour to find out if it does, or does not, possess the property B. But it is not so. There exists no property which can, like this property A, be an absolute criterion enabling us to recognise the straight line, and to distinguish it from every other line. Shall we say, for instance, " This property will be the following: the straight line is a line such that a figure of which this line is a part can move without the mutual distances of its points varying, and in such a way that all the points in this straight line remain fixed" ? Now, this is a property which in either Euclidean or non-Euclidean space belongs to the straight line, and belongs to it alone. But how can we ascertain by experiment if it belongs to any particular concrete object ? Distances must be measured, and how shall we know that any concrete magnitude which I have measured with my material instrument really represents the abstract distance? We have only removed the difficulty a little farther. In reality, the property that I have just enunciated is not a property of the straight line

75) alone; it is a property of the straight line and of distance. For it to serve as an absolute criterion, we must be able to show, not only that it does not also belong to any other line than the straight line and to distance, but also that it does not belong to any other line than the straight line, and to any other magnitude than distance. Now, that is not true, and if we are not convinced by these considerations, I challenge any one to give me a concrete experiment which can be interpreted in the Euclidean system, and which cannot be interpreted in the system of Lobatschewsky. As I am well aware that this challenge will never be accepted, I may conclude that no experiment will ever be in contradiction with Euclid's postulate; but, on the other hand, no experiment will ever be in contradiction with Lobatschewsky's postulate.

5. But it is not sufficient that the Euclidean (or non-Euclidean) geometry can ever be directly contradicted by experiment. Nor could it happen that it can only agree with experiment by a violation of the principle of sufficient reason, and of that of the relativity of space. Let me explain myself. Consider any material system whatever. We have to consider on the one hand the " state " of the various bodies of this system—for example, their temperature, their electric potential, etc.; and on the other hand their position in space. And among the data which enable us to define this position we distinguish the mutual distances of these bodies that define their relative positions,

(76) and the conditions which define the absolute position of the system and its absolute
orientation in space. The law of the phenomena which will be produced in this system will
depend on the state of these bodies, and on their mutual distances; but because of the relativity
and the inertia of space, they will not depend on the absolute position and orientation of the
system. In other words, the state of the bodies and their mutual distances at any moment will
solely depend on the state of the same bodies and on their mutual distances at the initial moment,
but will in no way depend on the absolute initial position of the system and of its absolute initial
orientation. This is what we shall call, for the sake of abbreviation, *the law of relativity.*

So far I have spoken as a Euclidean geometer. But I have said that an experiment, whatever it may be, requires an interpretation on the Euclidean hypothesis; it equally requires one on the non-Euclidean hypothesis. Well, we have made a series of experiments. We have interpreted them on the Euclidean hypothesis, and we have recognised that these experiments thus interpreted do not violate this "law of relativity." We now interpret them on the non-Euclidean hypothesis. This is always possible, only the non-Euclidean distances of our different bodies in this new interpretation will not generally be the same as the Euclidean distances in the primitive interpretation. Will our experiment interpreted in this new manner

(77) be still in agreement with our " law of relativity," and if this agreement had not taken place, would we not still have the right to say that experiment has proved the falsity of non-Euclidean geometry? It is easy to see that this is an idle fear. In fact, to apply the law of relativity in all its rigour, it must be applied to the entire universe; for if we were to consider only a part of the universe, and if the absolute position of this part were to vary, the distances of the other bodies of the universe would equally vary; their influence on the part of the universe considered might therefore increase or diminish, and this might modify the laws of the phenomena which take place in it. But if our system is the entire universe, experiment is powerless to give us any opinion on its position and its absolute orientation in space. All that our instruments, however perfect they may be, can let us know will be the state of the different parts of the universe, and their mutual distances. Hence, our law of relativity may be enunciated as follows: — The readings that we can make with our instruments at any given moment will depend only on the readings that we were able to make on the same instruments at the initial moment. Now such an enunciation is independent of all interpretation by experiments. If the law is true in the Euclidean interpretation, it will be also true in the non-Euclidean interpretation. Allow me to make a short digression on this point. I have spoken above of the data which define the position

78) of the different bodies of the system. I might also have spoken of those which define their velocities. I should then have to distinguish the velocity with which the mutual distances of the different bodies are changing, and on the other hand the velocities of translation and rotation of the system; that is to say, the velocities with which its absolute position and orientation are changing. For the mind to be fully satisfied, the law of relativity would have to be enunciated as follows: — The state of bodies and their mutual distances at any given moment, as well as the velocities with which those distances are changing at that moment, will depend only on the state of those bodies, on their mutual distances at the initial moment, and on the velocities with which those distances were changing at the initial moment. But they will not depend on the absolute initial position of the system nor on its absolute orientation, nor on the velocities with which that absolute position and orientation were changing at the initial moment. Unfortunately, the law thus enunciated does not agree with experiments — at least, as they are ordinarily interpreted. Suppose a man were translated to a planet, the sky of which was constantly covered with a thick curtain of clouds, so that he could never see the other stars. On that planet he would live as if it were isolated in space. But he would notice that it revolves, either by measuring its ellipticity (which is ordinarily done by means of astronomical observations, but which

79) could be done by purely geodesic means), or by 'repeating the experiment of Foucault's pendulum. The absolute rotation of this planet might be clearly shown in this way. Now, here is a fact which shocks the philosopher, but which the physicist is compelled to accept. We know that from this fact Newton concluded the existence of absolute space. I myself cannot accept this way of looking at it. I shall explain why in Part III., but for the moment it is not my intention to discuss this difficulty. I must therefore resign myself, in the enunciation of the law of relativity, to including velocities of every kind among the data which define the state of the bodies. However that may be, the difficulty is the same for both Euclid's geometry and for Lobatschewsky's.

I need not therefore trouble about it further, and I have only mentioned it incidentally. To sum up, whichever way we look at it, it is impossible to discover in geometric empiricism a rational meaning.

6. Experiments only teach us the relations of bodies to one another. They do not and cannot give us the relations of bodies and space, nor the mutual relations of the different parts of space. "Yes!" you reply, " a single experiment is not enough, because it only gives us one equation with several unknowns; but when I have made enough experiments I shall have enough equations to calculate all my unknowns." If I know the height of the main-mast, that is not sufficient to enable

(80) me to calculate the age of the captain. When you have measured every fragment of wood in a ship you will have many equations, but you will be no nearer knowing the captain's age. All your measurements bearing on your fragments of wood can tell you only what concerns those fragments; and similarly, your experiments, however numerous they may be, referring only to the relations of bodies with one another, will tell you nothing about the mutual relations of the different parts of space.

7. Will you say that if the experiments have reference to the bodies, they at least have reference to the geometrical properties of the bodies. First, what do you understand by the geometrical properties of bodies ? I assume that it is a question of the relations of the bodies to space. These properties therefore are not reached by experiments which only have reference to the relations of bodies to one another, and that is enough to show that it is not of those properties that there can be a question. Let us therefore begin by making ourselves clear as to the sense of the phrase: geometrical properties of bodies. When I say that a body is composed of several parts, I presume that I am thus enunciating a geometrical property, and that will be true even if I agree to give the improper name of points to the very small parts I am considering. When I say that this or that part of a certain body is in contact with this or that part of another body, I am

(81) enunciating a proposition which concerns the mutual relations of the two bodies, and not
their relations with space. I assume that you will agree with me that these are not geometrical
properties. I am sure that at least you will grant that these properties are independent of all
knowledge of metrical geometry. Admitting this, I suppose that we have a solid body formed of
eight thin iron rods, *oa, ob, oc, od, oe, of, og, oh, *connected at one of their extremities, o. And let
us take a second solid body — for example, a piece of wood, on which are marked three little
spots of ink which I shall call . I now suppose that we find that we can bring into contact
ß with *ago; *by that I mean with a, and at the same time ß with *g, *and with o. Then we can
successively bring into contact with *bgo, cgo, dgo, ego, fgo, *then with *aho, bho, cho, dho,
eho, fho; *and then ay successively with *ab, bc, cd, de, ef, fa. *Now these are observations that
can be made without having any idea beforehand as to the form or the metrical properties of
space. They have no reference whatever to the "geometrical properties of bodies." These
observations will not be possible if the bodies on which we experiment move in a group having
the same structure as the Lobatschewskian group (I mean according to the same laws as solid
bodies in Lobatschewsky's geometry). They therefore suffice to prove that these bodies move
according to the Euclidean group; or at least that they do not move according

82) to the Lobatschewskian group. That they may be compatible with the Euclidean group is
easily seen; for we might make them so if the body aßy were an invariable solid of our ordinary
geometry in the shape of a right-angled triangle, and if the points *abcdefgh *were the vertices of a
polyhedron formed of two regular hexagonal pyramids of our ordinary geometry having *abcdef
*as their common base, and having the one g and the other *h *as their vertices. Suppose now,
instead of the previous observations, we note that we can as before apply aly successively to *ago,
bgo, ego, dgo, ego, fgo, aho, bho, cho, dho, eho, f ho, *and then that we can apply
aß (and no
longer ) successively to *ab, bc, cd, de, ef, *and *fa. *These are observations that could be made if
non-Euclidean geometry were true. If the bodies , *oabcdefgh *were invariable solids, if the
former were a right angled triangle, and the latter a double regular hexagonal pyramid of suitable
dimensions. These new verifications are therefore impossible if the bodies move according to the
Euclidean group; but they become possible if we suppose the bodies to move according to the
Lobatschewskian group. They would therefore suffice to show, if we carried them out, that the
bodies in question do not move according to the Euclidean group. And so, without making any
hypothesis on the form and the nature of space, on the relations of the bodies and space, and
without attributing to bodies any geometrical property, I have made observations

83) which have enabled me to show in one case that the bodies experimented upon move
according to a group, the structure of which is Euclidean, and in the other case, that they move in
a group, the structure of which is Lobatschewskian. It cannot be said that all the first
observations would constitute an experiment proving that space is Euclidean, and the second an
experiment proving that space is non-Euclidean ; in fact, it might be imagined (note that I use the
word *imagined) *that there are bodies moving in such a manner as to render possible the second
series of observations: and the proof is that the first mechanic who came our way could construct
it if he would only take the trouble. But you must not conclude, however, that space is non-Euclidean. In the same way, just as ordinary solid bodies would continue to exist when the
mechanic had constructed the strange bodies I have just mentioned, he would have to conclude
that space is both Euclidean and non-Euclidean. Suppose, for instance, that we have a large
sphere of radius R, and that its temperature decreases from the centre to the surface of the sphere
according to the law of which I spoke when I was describing the nonEuclidean world. We might
have bodies whose dilatation is negligeable, and which would behave as ordinary invariable
solids; and, on the other hand, we might have very dilatable bodies, which would behave as non-Euclidean solids. We might have two double pyramids *oabcdefgh *and

84) *o'a'b'c'd'e'f'g'h', *and two triangles and ' ' '. The first double pyramid would be
rectilinear, and the second curvilinear. The triangle would consist of undilatable matter,
and the other of very dilatable matter. We might therefore make our first observations with the
double pyramid *o'a'h' *and the triangle ' ' '.

And then the experiment would seem to show, first, that Euclidean geometry is true, and then
that it is false. Hence, *experiments have reference not to space but to bodies.*

SUPPLEMENT.

8. To round the matter off, I ought to speak of a very delicate question, which will require
considerable development; but I shall confine myself to summing up what I have written in the
*Revue de Métaphysique et de Morale *and in the *Monist. *When we say that space has three
dimensions, what do we mean ? We have seen the importance of these " internal changes "which
are revealed to us by our muscular sensations. They may serve to characterise the different
attitudes of our body. Let us take arbitrarily as our origin one of these attitudes, A. When we pass
from this initial attitude to another attitude B we experience a series of muscular sensations, and
this series S of muscular sensations will define B. Observe, however, that h e shall often look
upon two series 5 and S' as defining the same attitude B (since the

(85) initial and final attitudes A and B remaining the same, the intermediary attitudes of the
corresponding sensations may differ). How then can we recognise the equivalence of these two
series ? Because they may serve to compensate for the same external change, or more generally,
because, when it is a question of compensation for an external change, one of the series may be
replaced by the other. Among these series we have distinguished those which can alone
compensate for an external change, and which we have called "displacements." As we cannot
distinguish two displacements which are very close together, the aggregate of these
displacements presents the characteristics of a physical continuum. Experience teaches us that
they are the characteristics of a physical continuum of six dimensions; but we do not know as yet
how many dimensions space itself possesses, so we must first of all answer another question.
What is a point in space ? Every one thinks he knows, but that is an illusion. What we see when
we try to represent to ourselves a point in space is a black spot on white paper, a spot of chalk on
a blackboard, always an object. The question should therefore be understood as
follows:—What do
I mean when I say the object B is at the point which a moment before was occupied
by the
object A ? Again, what criterion will enable me to recognise it ? I mean that *although I have *not
*moved (my muscular sense tells me this), my *finger, which just now touched the object A, is

86) now touching the object B. I might have used other criteria — for instance, another finger or the sense of sight — but the first criterion is sufficient. I know that if it answers in the affirmative all other criteria will give the same answer. I know it from experiment. I cannot know it à priori. For the same reason I say that touch cannot be exercised at a distance; that is another way of enunciating the same experimental fact. If I say, on the contrary, that sight is exercised at a distance, it means that the criterion furnished by sight may give an affirmative answer while the others reply in the negative.

To sum up. For each attitude of my body my finger determines a point, and it is that and that only which defines a point in space. To each attitude corresponds in this way a point. But it often happens that the same point corresponds to several different attitudes (in this case we say that our finger has not moved, but the rest of our body has). We distinguish, therefore, among changes of attitude those in which the finger does not move. How are we led to this ? It is because we often remark that in these changes the object which is in touch with the finger remains in contact with it. Let us arrange then in the same class all the attitudes which are deduced one from the other by one of the changes that we have thus distinguished. To all these attitudes of the same class will correspond the same point in space. Then to each class will correspond a point, and to

(87) each point a class. Yet it may be said that what we get from this experiment is not the point, but the class of changes, or, better still, the corresponding class of muscular sensations. Thus, when we say that space has three dimensions, we merely mean that the aggregate of these classes appears to us with the characteristics of a physical continuum of three dimensions. Then if, instead of defining the points in space with the aid of the first finger, I use, for example, another finger, would the results be the same ? That is by no means à priori evident. But, as we have seen, experiment has shown us that all our criteria are in agreement, and this enables us to answer in the affirmative. If we recur to what we have called displacements, the aggregate of which forms, as we have seen, a group, we shall be brought to distinguish those in which a finger does not move; and by what has preceded, those are the displacements which characterise a point in space, and their aggregate will form a sub-group of our group. To each sub-group of this kind, then, will correspond a point in space. We might be tempted to conclude that experiment has taught us the number of dimensions of space; but in reality our experiments have referred not to space, but to our body and its relations with neighbouring objects. What is more, our experiments are exceeding crude. In our mind the latent idea of a certain number of groups pre-existed : these are the groups with which Lie's theory is con-

88) -cerned. Which shall we choose to form a kind of standard by which to compare natural phenomena? And when this group is chosen, which of the sub-groups shall we take to characterise a point in space ? Experiment has guided us by showing us what choice adapts itself best to the properties of our body; but there its rôle ends.