CHAPTER XIII

ELECTRODYNAMICS

The history of electrodynamics is particularly instructive from our

point of view.

Ampère entitled his immortal work, 'Théorie des phénomènes

électrodynamiques, _uniquement_ fondée sur l'expérience.' He therefore

imagined that he had made _no_ hypothesis, but he had made them, as we

shall soon see; only he made them without being conscious of it.

His successors, on the other hand, perceived them, since their attention

was attracted by the weak points in Ampère's solution. They made new

hypotheses, of which this time they were fully conscious; but how many

times it was necessary to change them before arriving at the classic

system of to-day which is perhaps not yet final; this we shall see.

I. AMPERE'S THEORY.--When Ampère studied experimentally the mutual

actions of currents, he operated and he only could operate with closed

currents.

It was not that he denied the possibility of open currents. If two

conductors are charged with positive and negative electricity and

brought into communication by a wire, a current is established going

from one to the other, which continues until the two potentials are

equal. According to the ideas of Ampère's time this was an open current;

the current was known to go from the first conductor to the second, it

was not seen to return from the second to the first.

So Ampère considered as open currents of this nature, for example, the

currents of discharge of condensers; but he could not make them the

objects of his experiments because their duration is too short.

Another sort of open current may also be imagined. I suppose two

conductors, _A_ and _B_, connected by a wire _AMB_. Small conducting

masses in motion first come in contact with the conductor _B_, take

from it an electric charge, leave contact with _B_ and move along the

path _BNA_, and, transporting with them their charge, come into contact

with _A_ and give to it their charge, which returns then to _B_ along

the wire _AMB_.

Now there we have in a sense a closed circuit, since the electricity

describes the closed circuit _BNAMB_; but the two parts of this current

are very different. In the wire _AMB_, the electricity is displaced

through a fixed conductor, like a voltaic current, overcoming an ohmic

resistance and developing heat; we say that it is displaced by

conduction. In the part _BNA_, the electricity is carried by a moving

conductor; it is said to be displaced by convection.

If then the current of convection is considered as altogether analogous

to the current of conduction, the circuit _BNAMB_ is closed; if, on the

contrary, the convection current is not 'a true current' and, for

example, does not act on the magnet, there remains only the conduction

current _AMB_, which is open.

For example, if we connect by a wire the two poles of a Holtz machine,

the charged rotating disc transfers the electricity by convection from

one pole to the other, and it returns to the first pole by conduction

through the wire.

But currents of this sort are very difficult to produce with appreciable

intensity. With the means at Ampère's disposal, we may say that this was

impossible.

To sum up, Ampère could conceive of the existence of two kinds of open

currents, but he could operate on neither because they were not strong

enough or because their duration was too short.

Experiment therefore could only show him the action of a closed current

on a closed current, or, more accurately, the action of a closed current

on a portion of a current, because a current can be made to describe a

closed circuit composed of a moving part and a fixed part. It is

possible then to study the displacements of the moving part under the

action of another closed current.

On the other hand, Ampère had no means of studying the action of an open

current, either on a closed current or another open current.

1. _The Case of Closed Currents._--In the case of the mutual action of

two closed currents, experiment revealed to Ampère remarkably simple

laws.

I recall rapidly here those which will be useful to us in the sequel:

1º _If the intensity of the currents is kept constant_, and if the two

circuits, after having undergone any deformations and displacements

whatsoever, return finally to their initial positions, the total work of

the electrodynamic actions will be null.

In other words, there is an _electrodynamic potential_ of the two

circuits, proportional to the product of the intensities, and depending

on the form and relative position of the circuits; the work of the

electrodynamic actions is equal to the variation of this potential.

2º The action of a closed solenoid is null.

3º The action of a circuit _C_ on another voltaic circuit _C'_ depends

only on the 'magnetic field' developed by this circuit. At each point in

space we can in fact define in magnitude and direction a certain force

called _magnetic force_, which enjoys the following properties:

(_a_) The force exercised by _C_ on a magnetic pole is applied to that

pole and is equal to the magnetic force multiplied by the magnetic mass

of that pole;

(_b_) A very short magnetic needle tends to take the direction of the

magnetic force, and the couple to which it tends to reduce is

proportional to the magnetic force, the magnetic moment of the needle

and the sine of the dip of the needle;

(_c_) If the circuit _C_ is displaced, the work of the electrodynamic

action exercised by _C_ on _C'_ will be equal to the increment of the

'flow of magnetic force' which passes through the circuit.

2. _Action of a Closed Current on a Portion of Current._--Ampère not

having been able to produce an open current, properly so called, had

only one way of studying the action of a closed current on a portion of

current.

This was by operating on a circuit _C_ composed of two parts, the one

fixed, the other movable. The movable part was, for instance, a movable

wire [alpha][beta] whose extremities [alpha] and [beta] could slide

along a fixed wire. In one of the positions of the movable wire, the end

[alpha] rested on the _A_ of the fixed wire and the extremity [beta] on

the point _B_ of the fixed wire. The current circulated from [alpha] to

[beta], that is to say, from _A_ to _B_ along the movable wire, and then

it returned from _B_ to _A_ along the fixed wire. _This current was

therefore closed._

In a second position, the movable wire having slipped, the extremity

[alpha] rested on another point _A'_ of the fixed wire, and the

extremity [beta] on another point _B'_ of the fixed wire. The current

circulated then from [alpha] to [beta], that is to say from _A'_ to _B'_

along the movable wire, and it afterwards returned from _B'_ to _B_,

then from _B_ to _A_, then finally from _A_ to _A'_, always following

the fixed wire. The current was therefore also closed.

If a like current is subjected to the action of a closed current _C_,

the movable part will be displaced just as if it were acted upon by a

force. Ampère _assumes_ that the apparent force to which this movable

part _AB_ seems thus subjected, representing the action of the _C_ on

the portion [alpha][beta] of the current, is the same as if

[alpha][beta] were traversed by an open current, stopping at [alpha] and

[beta], in place of being traversed by a closed current which after

arriving at [beta] returns to [alpha] through the fixed part of the

circuit.

This hypothesis seems natural enough, and Ampère made it unconsciously;

nevertheless _it is not necessary_, since we shall see further on that

Helmholtz rejected it. However that may be, it permitted Ampère, though

he had never been able to produce an open current, to enunciate the laws

of the action of a closed current on an open current, or even on an

element of current.

The laws are simple:

1º The force which acts on an element of current is applied to this

element; it is normal to the element and to the magnetic force, and

proportional to the component of this magnetic force which is normal to

the element.

2º The action of a closed solenoid on an element of current is null.

But the electrodynamic potential has disappeared, that is to say that,

when a closed current and an open current, whose intensities have been

maintained constant, return to their initial positions, the total work

is not null.

3. _Continuous Rotations._--Among electrodynamic experiments, the most

remarkable are those in which continuous rotations are produced and

which are sometimes called _unipolar induction_ experiments. A magnet

may turn about its axis; a current passes first through a fixed wire,

enters the magnet by the pole _N_, for example, passes through half the

magnet, emerges by a sliding contact and reenters the fixed wire.

The magnet then begins to rotate continuously without being able ever to

attain equilibrium; this is Faraday's experiment.

How is it possible? If it were a question of two circuits of invariable

form, the one _C_ fixed, the other _C'_ movable about an axis, this

latter could never take on continuous rotation; in fact there is an

electrodynamic potential; there must therefore be necessarily a position

of equilibrium when this potential is a maximum.

Continuous rotations are therefore possible only when the circuit _C'_

is composed of two parts: one fixed, the other movable about an axis, as

is the case in Faraday's experiment. Here again it is convenient to draw

a distinction. The passage from the fixed to the movable part, or

inversely, may take place either by simple contact (the same point of

the movable part remaining constantly in contact with the same point of

the fixed part), or by a sliding contact (the same point of the movable

part coming successively in contact with diverse points of the fixed

part).

It is only in the second case that there can be continuous rotation.

This is what then happens: The system tends to take a position of

equilibrium; but, when at the point of reaching that position, the

sliding contact puts the movable part in communication with a new point

of the fixed part; it changes the connections, it changes therefore the

conditions of equilibrium, so that the position of equilibrium fleeing,

so to say, before the system which seeks to attain it, rotation may take

place indefinitely.

Ampère assumes that the action of the circuit on the movable part of

_C'_ is the same as if the fixed part of _C'_ did not exist, and

therefore as if the current passing through the movable part were open.

He concludes therefore that the action of a closed on an open current,

or inversely that of an open current on a closed current, may give rise

to a continuous rotation.

But this conclusion depends on the hypothesis I have enunciated and

which, as I said above, is not admitted by Helmholtz.

4. _Mutual Action of Two Open Currents._--In what concerns the mutual

actions of two open currents, and in particular that of two elements of

current, all experiment breaks down. Ampère has recourse to hypothesis.

He supposes:

1º That the mutual action of two elements reduces to a force acting

along their join;

2º That the action of two closed currents is the resultant of the mutual

actions of their diverse elements, which are besides the same as if

these elements were isolated.

What is remarkable is that here again Ampère makes these hypotheses

unconsciously.

However that may be, these two hypotheses, together with the experiments

on closed currents, suffice to determine completely the law of the

mutual action of two elements. But then most of the simple laws we have

met in the case of closed currents are no longer true.

In the first place, there is no electrodynamic potential; nor was there

any, as we have seen, in the case of a closed current acting on an open

current.

Next there is, properly speaking, no magnetic force.

And, in fact, we have given above three different definitions of this

force:

1º By the action on a magnetic pole;

2º By the director couple which orientates the magnetic needle;

3º By the action on an element of current.

But in the case which now occupies us, not only these three definitions

are no longer in harmony, but each has lost its meaning, and in fact:

1º A magnetic pole is no longer acted upon simply by a single force

applied to this pole. We have seen in fact that the force due to the

action of an element of current on a pole is not applied to the pole,

but to the element; it may moreover be replaced by a force applied to

the pole and by a couple;

2º The couple which acts on the magnetic needle is no longer a simple

director couple, for its moment with respect to the axis of the needle

is not null. It breaks up into a director couple, properly so called,

and a supplementary couple which tends to produce the continuous

rotation of which we have above spoken;

3º Finally the force acting on an element of current is not normal to

this element.

In other words, _the unity of the magnetic force has disappeared_.

Let us see in what this unity consists. Two systems which exercise the

same action on a magnetic pole will exert also the same action on an

indefinitely small magnetic needle, or on an element of current placed

at the same point of space as this pole.

Well, this is true if these two systems contain only closed currents;

this would no longer be true if these two systems contained open

currents.

It suffices to remark, for instance, that, if a magnetic pole is placed

at _A_ and an element at _B_, the direction of the element being along

the prolongation of the sect _AB_, this element which will exercise no

action on this pole will, on the other hand, exercise an action either

on a magnetic needle placed at the point _A_, or on an element of

current placed at the point _A_.

5. _Induction._--We know that the discovery of electrodynamic induction

soon followed the immortal work of Ampère.

As long as it is only a question of closed currents there is no

difficulty, and Helmholtz has even remarked that the principle of the

conservation of energy is sufficient for deducing the laws of induction

from the electrodynamic laws of Ampère. But always on one condition, as

Bertrand has well shown; that we make besides a certain number of

hypotheses.

The same principle again permits this deduction in the case of open

currents, although of course we can not submit the result to the test of

experiment, since we can not produce such currents.

If we try to apply this mode of analysis to Ampère's theory of open

currents, we reach results calculated to surprise us.

In the first place, induction can not be deduced from the variation of

the magnetic field by the formula well known to savants and practicians,

and, in fact, as we have said, properly speaking there is no longer a

magnetic field.

But, further, if a circuit _C_ is subjected to the induction of a

variable voltaic system _S_, if this system _S_ be displaced and

deformed in any way whatever, so that the intensity of the currents of

this system varies according to any law whatever, but that after these

variations the system finally returns to its initial situation, it seems

natural to suppose that the _mean_ electromotive force induced in the

circuit _C_ is null.

This is true if the circuit _C_ is closed and if the system _S_ contains

only closed currents. This would no longer be true, if one accepts the

theory of Ampère, if there were open currents. So that not only

induction will no longer be the variation of the flow of magnetic force,

in any of the usual senses of the word, but it can not be represented by

the variation of anything whatever.

II. THEORY OF HELMHOLTZ.--I have dwelt upon the consequences of Ampère's

theory, and of his method of explaining open currents.

It is difficult to overlook the paradoxical and artificial character of

the propositions to which we are thus led. One can not help thinking

'that can not be so.'

We understand therefore why Helmholtz was led to seek something else.

Helmholtz rejects Ampère's fundamental hypothesis, to wit, that the

mutual action of two elements of current reduces to a force along their

join. He assumes that an element of current is not subjected to a single

force, but to a force and a couple. It is just this which gave rise to

the celebrated polemic between Bertrand and Helmholtz.

Helmholtz replaces Ampère's hypothesis by the following: two elements

always admit of an electrodynamic potential depending solely on their

position and orientation; and the work of the forces that they exercise,

one on the other, is equal to the variation of this potential. Thus

Helmholtz can no more do without hypothesis than Ampère; but at least he

does not make one without explicitly announcing it.

In the case of closed currents, which are alone accessible to

experiment, the two theories agree.

In all other cases they differ.

In the first place, contrary to what Ampère supposed, the force which

seems to act on the movable portion of a closed current is not the same

as would act upon this movable portion if it were isolated and

constituted an open current.

Let us return to the circuit _C'_, of which we spoke above, and which

was formed of a movable wire [alpha][beta] sliding on a fixed wire. In

the only experiment that can be made, the movable portion [alpha][beta]

is not isolated, but is part of a closed circuit. When it passes from

_AB_ to _A'B'_, the total electrodynamic potential varies for two

reasons:

1º It undergoes a first increase because the potential of _A'B'_ with

respect to the circuit _C_ is not the same as that of _AB_;

2º It takes a second increment because it must be increased by the

potentials of the elements _AA'_, _BB'_ with respect to _C_.

It is this _double_ increment which represents the work of the force to

which the portion _AB_ seems subjected.

If, on the contrary, [alpha][beta] were isolated, the potential would

undergo only the first increase, and this first increment alone would

measure the work of the force which acts on _AB_.

In the second place, there could be no continuous rotation without

sliding contact, and, in fact, that, as we have seen _à propos_ of

closed currents, is an immediate consequence of the existence of an

electrodynamic potential.

In Faraday's experiment, if the magnet is fixed and if the part of the

current exterior to the magnet runs along a movable wire, that movable

part may undergo a continuous rotation. But this does not mean to say

that if the contacts of the wire with the magnet were suppressed, and an

_open_ current were to run along the wire, the wire would still take a

movement of continuous rotation.

I have just said in fact that an _isolated_ element is not acted upon in

the same way as a movable element making part of a closed circuit.

Another difference: The action of a closed solenoid on a closed current

is null according to experiment and according to the two theories. Its

action on an open current would be null according to Ampère; it would

not be null according to Helmholtz. From this follows an important

consequence. We have given above three definitions of magnetic force.

The third has no meaning here since an element of current is no longer

acted upon by a single force. No more has the first any meaning. What,

in fact, is a magnetic pole? It is the extremity of an indefinite linear

magnet. This magnet may be replaced by an indefinite solenoid. For the

definition of magnetic force to have any meaning, it would be necessary

that the action exercised by an open current on an indefinite solenoid

should depend only on the position of the extremity of this solenoid,

that is to say, that the action on a closed solenoid should be null. Now

we have just seen that such is not the case.

On the other hand, nothing prevents our adopting the second definition,

which is founded on the measurement of the director couple which tends

to orientate the magnetic needle.

But if it is adopted, neither the effects of induction nor the

electrodynamic effects will depend solely on the distribution of the

lines of force in this magnetic field.

III. DIFFICULTIES RAISED BY THESE THEORIES.--The theory of Helmholtz is

in advance of that of Ampère; it is necessary, however, that all the

difficulties should be smoothed away. In the one as in the other, the

phrase 'magnetic field' has no meaning, or, if we give it one, by a more

or less artificial convention, the ordinary laws so familiar to all

electricians no longer apply; thus the electromotive force induced in a

wire is no longer measured by the number of lines of force met by this

wire.

And our repugnance does not come alone from the difficulty of renouncing

inveterate habits of language and of thought. There is something more.

If we do not believe in action at a distance, electrodynamic phenomena

must be explained by a modification of the medium. It is precisely this

modification that we call 'magnetic field.' And then the electrodynamic

effects must depend only on this field.

All these difficulties arise from the hypothesis of open currents.

IV. MAXWELL'S THEORY.--Such were the difficulties raised by the dominant

theories when Maxwell appeared, who with a stroke of the pen made them

all vanish. To his mind, in fact, all currents are closed currents.

Maxwell assumes that if in a dielectric the electric field happens to

vary, this dielectric becomes the seat of a particular phenomenon,

acting on the galvanometer like a current, and which he calls _current

of displacement_.

If then two conductors bearing contrary charges are put in communication

by a wire, in this wire during the discharge there is an open current of

conduction; but there are produced at the same time in the surrounding

dielectric, currents of displacement which close this current of

conduction.

We know that Maxwell's theory leads to the explanation of optical

phenomena, which would be due to extremely rapid electrical

oscillations.

At that epoch such a conception was only a bold hypothesis, which could

be supported by no experiment.

At the end of twenty years, Maxwell's ideas received the confirmation of

experiment. Hertz succeeded in producing systems of electric

oscillations which reproduce all the properties of light, and only

differ from it by the length of their wave; that is to say as violet

differs from red. In some measure he made the synthesis of light.

It might be said that Hertz has not demonstrated directly Maxwell's

fundamental idea, the action of the current of displacement on the

galvanometer. This is true in a sense. What he has shown in sum is that

electromagnetic induction is not propagated instantaneously as was

supposed; but with the speed of light.

But to suppose there is no current of displacement, and induction is

propagated with the speed of light; or to suppose that the currents of

displacement produce effects of induction, and that the induction is

propagated instantaneously, _comes to the same thing_.

This can not be seen at the first glance, but it is proved by an

analysis of which I must not think of giving even a summary here.

V. ROWLAND'S EXPERIMENT.--But as I have said above, there are two kinds

of open conduction currents. There are first the currents of discharge

of a condenser or of any conductor whatever.

There are also the cases in which electric discharges describe a closed

contour, being displaced by conduction in one part of the circuit and by

convection in the other part.

For open currents of the first sort, the question might be considered as

solved; they were closed by the currents of displacement.

For open currents of the second sort, the solution appeared still more

simple. It seemed that if the current were closed, it could only be by

the current of convection itself. For that it sufficed to assume that a

'convection current,' that is to say a charged conductor in motion,

could act on the galvanometer.

But experimental confirmation was lacking. It appeared difficult in fact

to obtain a sufficient intensity even by augmenting as much as possible

the charge and the velocity of the conductors. It was Rowland, an

extremely skillful experimenter, who first triumphed over these

difficulties. A disc received a strong electrostatic charge and a very

great speed of rotation. An astatic magnetic system placed beside the

disc underwent deviations.

The experiment was made twice by Rowland, once in Berlin, once in

Baltimore. It was afterwards repeated by Himstedt. These physicists even

announced that they had succeeded in making quantitative measurements.

In fact, for twenty years Rowland's law was admitted without objection

by all physicists. Besides everything seemed to confirm it. The spark

certainly does produce a magnetic effect. Now does it not seem probable

that the discharge by spark is due to particles taken from one of the

electrodes and transferred to the other electrode with their charge? Is

not the very spectrum of the spark, in which we recognize the lines of

the metal of the electrode, a proof of it? The spark would then be a

veritable current of convection.

On the other hand, it is also admitted that in an electrolyte the

electricity is carried by the ions in motion. The current in an

electrolyte would therefore be also a current of convection; now, it

acts on the magnetic needle.

The same for cathode rays. Crookes attributed these rays to a very

subtile matter charged with electricity and moving with a very great

velocity. He regarded them, in other words, as currents of convection.

Now these cathode rays are deviated by the magnet. In virtue of the

principle of action and reaction, they should in turn deviate the

magnetic needle. It is true that Hertz believed he had demonstrated that

the cathode rays do not carry electricity, and that they do not act on

the magnetic needle. But Hertz was mistaken. First of all, Perrin

succeeded in collecting the electricity carried by these rays,

electricity of which Hertz denied the existence; the German scientist

appears to have been deceived by effects due to the action of X-rays,

which were not yet discovered. Afterwards, and quite recently, the

action of the cathode rays on the magnetic needle has been put in

evidence.

Thus all these phenomena regarded as currents of convection, sparks,

electrolytic currents, cathode rays, act in the same manner on the

galvanometer and in conformity with Rowland's law.

VI. THEORY OF LORENTZ.--We soon went farther. According to the theory of

Lorentz, currents of conduction themselves would be true currents of

convection. Electricity would remain inseparably connected with certain

material particles called _electrons_. The circulation of these

electrons through bodies would produce voltaic currents. And what would

distinguish conductors from insulators would be that the one could be

traversed by these electrons while the others would arrest their

movements.

The theory of Lorentz is very attractive. It gives a very simple

explanation of certain phenomena which the earlier theories, even

Maxwell's in its primitive form, could not explain in a satisfactory

way; for example, the aberration of light, the partial carrying away of

luminous waves, magnetic polarization and the Zeeman effect.

Some objections still remained. The phenomena of an electric system

seemed to depend on the absolute velocity of translation of the center

of gravity of this system, which is contrary to the idea we have of the

relativity of space. Supported by M. Crémieu, M. Lippmann has presented

this objection in a striking form. Imagine two charged conductors with

the same velocity of translation; they are relatively at rest. However,

each of them being equivalent to a current of convection, they ought to

attract one another, and by measuring this attraction we could measure

their absolute velocity.

"No!" replied the partisans of Lorentz. "What we could measure in that

way is not their absolute velocity, but their relative velocity _with

respect to the ether_, so that the principle of relativity is safe."

Whatever there may be in these latter objections, the edifice of

electrodynamics, at least in its broad lines, seemed definitively

constructed. Everything was presented under the most satisfactory

aspect. The theories of Ampère and of Helmholtz, made for open currents

which no longer existed, seemed to have no longer anything but a purely

historic interest, and the inextricable complications to which these

theories led were almost forgotten.

This quiescence has been recently disturbed by the experiments of M.

Crémieu, which for a moment seemed to contradict the result previously

obtained by Rowland.

But fresh researches have not confirmed them, and the theory of Lorentz

has victoriously stood the test.

The history of these variations will be none the less instructive; it

will teach us to what pitfalls the scientist is exposed, and how he may

hope to escape them.

* * * * *

THE VALUE OF SCIENCE

* * * * *

TRANSLATOR'S INTRODUCTION

1. _Does the Scientist create Science?_--Professor Rados of Budapest in

his report to the Hungarian Academy of Science on the award to Poincaré

of the Bolyai prize of ten thousand crowns, speaking of him as

unquestionably the most powerful investigator in the domain of

mathematics and mathematical physics, characterized him as the intuitive

genius drawing the inspiration for his wide-reaching researches from the

exhaustless fountain of geometric and physical intuition, yet working

this inspiration out in detail with marvelous logical keenness. With his

brilliant creative genius was combined the capacity for sharp and

successful generalization, pushing far out the boundaries of thought in

the most widely different domains, so that his works must be ranked with

the greatest mathematical achievements of all time. "Finally," says

Rados, "permit me to make especial mention of his intensely interesting

book, 'The Value of Science,' in which he in a way has laid down the

scientist's creed." Now what is this creed?

Sense may act as stimulus, as suggestive, yet not to awaken a dormant

depiction, or to educe the conception of an archetypal form, but rather

to strike the hour for creation, to summon to work a sculptor capable of

smoothing a Venus of Milo out of the formless clay. Knowledge is not a

gift of bare experience, nor even made solely out of experience. The

creative activity of mind is in mathematics particularly clear. The

axioms of geometry are conventions, disguised definitions or unprovable

hypotheses precreated by auto-active animal and human minds. Bertrand

Russell says of projective geometry: "It takes nothing from experience,

and has, like arithmetic, a creature of the pure intellect for its

object. It deals with an object whose properties are logically deduced

from its definition, not empirically discovered from data." Then does

the scientist create science? This is a question Poincaré here dissects

with a master hand.

The physiologic-psychologic investigation of the space problem must

give the meaning of the words _geometric fact_, _geometric reality_.

Poincaré here subjects to the most successful analysis ever made the

tridimensionality of our space.

2. _The Mind Dispelling Optical Illusions._--Actual perception of

spatial properties is accompanied by movements corresponding to its

character. In the case of optical illusions, with the so-called false

perceptions eye-movements are closely related. But though the perceived

object and its environment remain constant, the sufficiently powerful

mind can, as we say, dispel these illusions, the perception itself being

creatively changed. Photo-graphs taken at intervals during the presence

of these optical illusions, during the change, perhaps gradual and

unconscious, in the perception, and after these illusions have, as the

phrase is, finally disappeared, show quite clearly that changes in

eye-movements corresponding to those internally created in perception

itself successively occur. What is called accuracy of movement is

created by what is called correctness of perception. The higher creation

in the perception is the determining cause of an improvement, a

precision in the motion. Thus we see correct perception in the

individual helping to make that cerebral organization and accurate motor

adjustment on which its possibility and permanence seem in so far to

depend. So-called correct perception is connected with a long-continued

process of perceptual education motived and initiated from within. How

this may take place is here illustrated at length by our author.

3. _Euclid not Necessary._--Geometry is a construction of the intellect,

in application not certain but convenient. As Schiller says, when we see

these facts as clearly as the development of metageometry has compelled

us to see them, we must surely confess that the Kantian account of space

is hopelessly and demonstrably antiquated. As Royce says in 'Kant's

Doctrine of the Basis of Mathematics,' "That very use of intuition which

Kant regarded as geometrically ideal, the modern geometer regards as

scientifically defective, because surreptitious. No mathematical

exactness without explicit proof from assumed principles--such is the

motto of the modern geometer. But suppose the reasoning of Euclid

purified of this comparatively surreptitious appeal to intuition.

Suppose that the principles of geometry are made quite explicit at the

outset of the treatise, as Pieri and Hilbert or Professor Halsted or Dr.

Veblen makes his principles explicit in his recent treatment of

geometry. Then, indeed, geometry becomes for the modern mathematician a

purely rational science. But very few students of the logic of

mathematics at the present time can see any warrant in the analysis of

geometrical truth for regarding just the Euclidean system of principles

as possessing any discoverable necessity." Yet the environmental and

perhaps hereditary premiums on Euclid still make even the scientist

think Euclid most convenient.

4. _Without Hypotheses, no Science._--Nobody ever observed an

equidistantial, but also nobody ever observed a straight line.

Emerson's Uriel

"Gave his sentiment divine

Against the being of a line.

Line in Nature is not found."

Clearly not, being an eject from man's mind. What is called 'a knowledge

of facts' is usually merely a subjective realization that the old

hypotheses are still sufficiently elastic to serve in some domain; that

is, with a sufficiency of conscious or unconscious omissions and

doctorings and fudgings more or less wilful. In the present book we see

the very foundation rocks of science, the conservation of energy and the

indestructibility of matter, beating against the bars of their cages,

seemingly anxious to take wing away into the empyrean, to chase the once

divine parallel postulate broken loose from Euclid and Kant.

5. _What Outcome?_--What now is the definite, the permanent outcome?

What new islets raise their fronded palms in air within thought's

musical domain? Over what age-gray barriers rise the fragrant floods of

this new spring-tide, redolent of the wolf-haunted forest of

Transylvania, of far Erdély's plunging river, Maros the bitter, or broad

mother Volga at Kazan? What victory heralded the great rocket for which

young Lobachevski, the widow's son, was cast into prison? What severing

of age-old mental fetters symbolized young Bolyai's cutting-off with

his Damascus blade the spikes driven into his door-post, and strewing

over the sod the thirteen Austrian cavalry officers? This book by the

greatest mathematician of our time gives weightiest and most charming

answer.

GEORGE BRUCE HALSTED.

INTRODUCTION

The search for truth should be the goal of our activities; it is the

sole end worthy of them. Doubtless we should first bend our efforts to

assuage human suffering, but why? Not to suffer is a negative ideal more

surely attained by the annihilation of the world. If we wish more and

more to free man from material cares, it is that he may be able to

employ the liberty obtained in the study and contemplation of truth.

But sometimes truth frightens us. And in fact we know that it is

sometimes deceptive, that it is a phantom never showing itself for a

moment except to ceaselessly flee, that it must be pursued further and

ever further without ever being attained. Yet to work one must stop, as

some Greek, Aristotle or another, has said. We also know how cruel the

truth often is, and we wonder whether illusion is not more consoling,

yea, even more bracing, for illusion it is which gives confidence. When

it shall have vanished, will hope remain and shall we have the courage

to achieve? Thus would not the horse harnessed to his treadmill refuse

to go, were his eyes not bandaged? And then to seek truth it is

necessary to be independent, wholly independent. If, on the contrary, we

wish to act, to be strong, we should be united. This is why many of us

fear truth; we consider it a cause of weakness. Yet truth should not be

feared, for it alone is beautiful.

When I speak here of truth, assuredly I refer first to scientific truth;

but I also mean moral truth, of which what we call justice is only one

aspect. It may seem that I am misusing words, that I combine thus under

the same name two things having nothing in common; that scientific

truth, which is demonstrated, can in no way be likened to moral truth,

which is felt. And yet I can not separate them, and whosoever loves the

one can not help loving the other. To find the one, as well as to find

the other, it is necessary to free the soul completely from prejudice

and from passion; it is necessary to attain absolute sincerity. These

two sorts of truth when discovered give the same joy; each when

perceived beams with the same splendor, so that we must see it or close

our eyes. Lastly, both attract us and flee from us; they are never

fixed: when we think to have reached them, we find that we have still to

advance, and he who pursues them is condemned never to know repose. It

must be added that those who fear the one will also fear the other; for

they are the ones who in everything are concerned above all with

consequences. In a word, I liken the two truths, because the same

reasons make us love them and because the same reasons make us fear

them.

If we ought not to fear moral truth, still less should we dread

scientific truth. In the first place it can not conflict with ethics.

Ethics and science have their own domains, which touch but do not

interpenetrate. The one shows us to what goal we should aspire, the

other, given the goal, teaches us how to attain it. So they can never

conflict since they can never meet. There can no more be immoral science

than there can be scientific morals.

But if science is feared, it is above all because it can not give us

happiness. Of course it can not. We may even ask whether the beast does

not suffer less than man. But can we regret that earthly paradise where

man brute-like was really immortal in knowing not that he must die? When

we have tasted the apple, no suffering can make us forget its savor. We

always come back to it. Could it be otherwise? As well ask if one who

has seen and is blind will not long for the light. Man, then, can not be

happy through science, but to-day he can much less be happy without it.

But if truth be the sole aim worth pursuing, may we hope to attain it?

It may well be doubted. Readers of my little book 'Science and

Hypothesis' already know what I think about the question. The truth we

are permitted to glimpse is not altogether what most men call by that

name. Does this mean that our most legitimate, most imperative

aspiration is at the same time the most vain? Or can we, despite all,

approach truth on some side? This it is which must be investigated.

In the first place, what instrument have we at our disposal for this

conquest? Is not human intelligence, more specifically the intelligence

of the scientist, susceptible of infinite variation? Volumes could be

written without exhausting this subject; I, in a few brief pages, have

only touched it lightly. That the geometer's mind is not like the

physicist's or the naturalist's, all the world would agree; but

mathematicians themselves do not resemble each other; some recognize

only implacable logic, others appeal to intuition and see in it the only

source of discovery. And this would be a reason for distrust. To minds

so unlike can the mathematical theorems themselves appear in the same

light? Truth which is not the same for all, is it truth? But looking at

things more closely, we see how these very different workers collaborate

in a common task which could not be achieved without their cooperation.

And that already reassures us.

Next must be examined the frames in which nature seems enclosed and

which are called time and space. In 'Science and Hypothesis' I have

already shown how relative their value is; it is not nature which

imposes them upon us, it is we who impose them upon nature because we

find them convenient. But I have spoken of scarcely more than space, and

particularly quantitative space, so to say, that is of the mathematical

relations whose aggregate constitutes geometry. I should have shown that

it is the same with time as with space and still the same with

'qualitative space'; in particular, I should have investigated why we

attribute three dimensions to space. I may be pardoned then for taking

up again these important questions.

Is mathematical analysis, then, whose principal object is the study of

these empty frames, only a vain play of the mind? It can give to the

physicist only a convenient language; is this not a mediocre service,

which, strictly speaking, could be done without; and even is it not to

be feared that this artificial language may be a veil interposed between

reality and the eye of the physicist? Far from it; without this language

most of the intimate analogies of things would have remained forever

unknown to us; and we should forever have been ignorant of the internal

harmony of the world, which is, we shall see, the only true objective

reality.

The best expression of this harmony is law. Law is one of the most

recent conquests of the human mind; there still are people who live in

the presence of a perpetual miracle and are not astonished at it. On the

contrary, we it is who should be astonished at nature's regularity. Men

demand of their gods to prove their existence by miracles; but the

eternal marvel is that there are not miracles without cease. The world

is divine because it is a harmony. If it were ruled by caprice, what

could prove to us it was not ruled by chance?

This conquest of law we owe to astronomy, and just this makes the

grandeur of the science rather than the material grandeur of the objects

it considers. It was altogether natural, then, that celestial mechanics

should be the first model of mathematical physics; but since then this

science has developed; it is still developing, even rapidly developing.

And it is already necessary to modify in certain points the scheme from

which I drew two chapters of 'Science and Hypothesis.' In an address at

the St. Louis exposition, I sought to survey the road traveled; the

result of this investigation the reader shall see farther on.

The progress of science has seemed to imperil the best established

principles, those even which were regarded as fundamental. Yet nothing

shows they will not be saved; and if this comes about only imperfectly,

they will still subsist even though they are modified. The advance of

science is not comparable to the changes of a city, where old edifices

are pitilessly torn down to give place to new, but to the continuous

evolution of zoologic types which develop ceaselessly and end by

becoming unrecognizable to the common sight, but where an expert eye

finds always traces of the prior work of the centuries past. One must

not think then that the old-fashioned theories have been sterile and

vain.

Were we to stop there, we should find in these pages some reasons for

confidence in the value of science, but many more for distrusting it; an

impression of doubt would remain; it is needful now to set things to

rights.

Some people have exaggerated the rôle of convention in science; they

have even gone so far as to say that law, that scientific fact itself,

was created by the scientist. This is going much too far in the

direction of nominalism. No, scientific laws are not artificial

creations; we have no reason to regard them as accidental, though it be

impossible to prove they are not.

Does the harmony the human intelligence thinks it discovers in nature

exist outside of this intelligence? No, beyond doubt a reality

completely independent of the mind which conceives it, sees or feels it,

is an impossibility. A world as exterior as that, even if it existed,

would for us be forever inaccessible. But what we call objective reality

is, in the last analysis, what is common to many thinking beings, and

could be common to all; this common part, we shall see, can only be the

harmony expressed by mathematical laws. It is this harmony then which is

the sole objective reality, the only truth we can attain; and when I add

that the universal harmony of the world is the source of all beauty, it

will be understood what price we should attach to the slow and difficult

progress which little by little enables us to know it better.

PART I

THE MATHEMATICAL SCIENCES

CHAPTER I

INTUITION AND LOGIC IN MATHEMATICS

I

It is impossible to study the works of the great mathematicians, or even

those of the lesser, without noticing and distinguishing two opposite

tendencies, or rather two entirely different kinds of minds. The one

sort are above all preoccupied with logic; to read their works, one is

tempted to believe they have advanced only step by step, after the

manner of a Vauban who pushes on his trenches against the place

besieged, leaving nothing to chance. The other sort are guided by

intuition and at the first stroke make quick but sometimes precarious

conquests, like bold cavalrymen of the advance guard.

The method is not imposed by the matter treated. Though one often says

of the first that they are _analysts_ and calls the others _geometers_,

that does not prevent the one sort from remaining analysts even when

they work at geometry, while the others are still geometers even when

they occupy themselves with pure analysis. It is the very nature of

their mind which makes them logicians or intuitionalists, and they can

not lay it aside when they approach a new subject.

Nor is it education which has developed in them one of the two

tendencies and stifled the other. The mathematician is born, not made,

and it seems he is born a geometer or an analyst. I should like to cite

examples and there are surely plenty; but to accentuate the contrast I

shall begin with an extreme example, taking the liberty of seeking it in

two living mathematicians.

M. Méray wants to prove that a binomial equation always has a root, or,

in ordinary words, that an angle may always be subdivided. If there is

any truth that we think we know by direct intuition, it is this. Who

could doubt that an angle may always be divided into any number of equal

parts? M. Méray does not look at it that way; in his eyes this

proposition is not at all evident and to prove it he needs several

pages.

On the other hand, look at Professor Klein: he is studying one of the

most abstract questions of the theory of functions: to determine whether

on a given Riemann surface there always exists a function admitting of

given singularities. What does the celebrated German geometer do? He

replaces his Riemann surface by a metallic surface whose electric

conductivity varies according to certain laws. He connects two of its

points with the two poles of a battery. The current, says he, must pass,

and the distribution of this current on the surface will define a

function whose singularities will be precisely those called for by the

enunciation.

Doubtless Professor Klein well knows he has given here only a sketch;

nevertheless he has not hesitated to publish it; and he would probably

believe he finds in it, if not a rigorous demonstration, at least a kind

of moral certainty. A logician would have rejected with horror such a

conception, or rather he would not have had to reject it, because in his

mind it would never have originated.

Again, permit me to compare two men, the honor of French science, who

have recently been taken from us, but who both entered long ago into

immortality. I speak of M. Bertrand and M. Hermite. They were scholars

of the same school at the same time; they had the same education, were

under the same influences; and yet what a difference! Not only does it

blaze forth in their writings; it is in their teaching, in their way of

speaking, in their very look. In the memory of all their pupils these

two faces are stamped in deathless lines; for all who have had the

pleasure of following their teaching, this remembrance is still fresh;

it is easy for us to evoke it.

While speaking, M. Bertrand is always in motion; now he seems in combat

with some outside enemy, now he outlines with a gesture of the hand the

figures he studies. Plainly he sees and he is eager to paint, this is

why he calls gesture to his aid. With M. Hermite, it is just the

opposite; his eyes seem to shun contact with the world; it is not

without, it is within he seeks the vision of truth.

Among the German geometers of this century, two names above all are

illustrious, those of the two scientists who founded the general theory

of functions, Weierstrass and Riemann. Weierstrass leads everything back

to the consideration of series and their analytic transformations; to

express it better, he reduces analysis to a sort of prolongation of

arithmetic; you may turn through all his books without finding a figure.

Riemann, on the contrary, at once calls geometry to his aid; each of his

conceptions is an image that no one can forget, once he has caught its

meaning.

More recently, Lie was an intuitionalist; this might have been doubted

in reading his books, no one could doubt it after talking with him; you

saw at once that he thought in pictures. Madame Kovalevski was a

logician.

Among our students we notice the same differences; some prefer to treat

their problems 'by analysis,' others 'by geometry.' The first are

incapable of 'seeing in space,' the others are quickly tired of long

calculations and become perplexed.

The two sorts of minds are equally necessary for the progress of

science; both the logicians and the intuitionalists have achieved great

things that others could not have done. Who would venture to say whether

he preferred that Weierstrass had never written or that there had never

been a Riemann? Analysis and synthesis have then both their legitimate

rôles. But it is interesting to study more closely in the history of

science the part which belongs to each.

II

Strange! If we read over the works of the ancients we are tempted to

class them all among the intuitionalists. And yet nature is always the

same; it is hardly probable that it has begun in this century to create

minds devoted to logic. If we could put ourselves into the flow of ideas

which reigned in their time, we should recognize that many of the old

geometers were in tendency analysts. Euclid, for example, erected a

scientific structure wherein his contemporaries could find no fault. In

this vast construction, of which each piece however is due to intuition,

we may still to-day, without much effort, recognize the work of a

logician.

It is not minds that have changed, it is ideas; the intuitional minds

have remained the same; but their readers have required of them greater

concessions.

What is the cause of this evolution? It is not hard to find. Intuition

can not give us rigor, nor even certainty; this has been recognized more

and more. Let us cite some examples. We know there exist continuous

functions lacking derivatives. Nothing is more shocking to intuition

than this proposition which is imposed upon us by logic. Our fathers

would not have failed to say: "It is evident that every continuous

function has a derivative, since every curve has a tangent."

How can intuition deceive us on this point? It is because when we seek

to imagine a curve we can not represent it to ourselves without width;

just so, when we represent to ourselves a straight line, we see it under

the form of a rectilinear band of a certain breadth. We well know these

lines have no width; we try to imagine them narrower and narrower and

thus to approach the limit; so we do in a certain measure, but we shall

never attain this limit. And then it is clear we can always picture

these two narrow bands, one straight, one curved, in a position such

that they encroach slightly one upon the other without crossing. We

shall thus be led, unless warned by a rigorous analysis, to conclude

that a curve always has a tangent.

I shall take as second example Dirichlet's principle on which rest so

many theorems of mathematical physics; to-day we establish it by

reasoning very rigorous but very long; heretofore, on the contrary, we

were content with a very summary proof. A certain integral depending on

an arbitrary function can never vanish. Hence it is concluded that it

must have a minimum. The flaw in this reasoning strikes us immediately,

since we use the abstract term _function_ and are familiar with all the

singularities functions can present when the word is understood in the

most general sense.

But it would not be the same had we used concrete images, had we, for

example, considered this function as an electric potential; it would

have been thought legitimate to affirm that electrostatic equilibrium

can be attained. Yet perhaps a physical comparison would have awakened

some vague distrust. But if care had been taken to translate the

reasoning into the language of geometry, intermediate between that of

analysis and that of physics, doubtless this distrust would not have

been produced, and perhaps one might thus, even to-day, still deceive

many readers not forewarned.

Intuition, therefore, does not give us certainty. This is why the

evolution had to happen; let us now see how it happened.

It was not slow in being noticed that rigor could not be introduced in

the reasoning unless first made to enter into the definitions. For the

most part the objects treated of by mathematicians were long ill

defined; they were supposed to be known because represented by means of

the senses or the imagination; but one had only a crude image of them

and not a precise idea on which reasoning could take hold. It was there

first that the logicians had to direct their efforts.

So, in the case of incommensurable numbers. The vague idea of

continuity, which we owe to intuition, resolved itself into a

complicated system of inequalities referring to whole numbers.

By that means the difficulties arising from passing to the limit, or

from the consideration of infinitesimals, are finally removed. To-day in

analysis only whole numbers are left or systems, finite or infinite, of

whole numbers bound together by a net of equality or inequality

relations. Mathematics, as they say, is arithmetized.

III

A first question presents itself. Is this evolution ended? Have we

finally attained absolute rigor? At each stage of the evolution our

fathers also thought they had reached it. If they deceived themselves,

do we not likewise cheat ourselves?

We believe that in our reasonings we no longer appeal to intuition; the

philosophers will tell us this is an illusion. Pure logic could never

lead us to anything but tautologies; it could create nothing new; not

from it alone can any science issue. In one sense these philosophers are

right; to make arithmetic, as to make geometry, or to make any science,

something else than pure logic is necessary. To designate this something

else we have no word other than _intuition_. But how many different

ideas are hidden under this same word?

Compare these four axioms: (1) Two quantities equal to a third are equal

to one another; (2) if a theorem is true of the number 1 and if we prove

that it is true of _n_ + 1 if true for _n_, then will it be true of all

whole numbers; (3) if on a straight the point _C_ is between _A_ and _B_

and the point _D_ between _A_ and _C_, then the point _D_ will be

between _A_ and _B_; (4) through a given point there is not more than

one parallel to a given straight.

All four are attributed to intuition, and yet the first is the

enunciation of one of the rules of formal logic; the second is a real

synthetic _a priori_ judgment, it is the foundation of rigorous

mathematical induction; the third is an appeal to the imagination; the

fourth is a disguised definition.

Intuition is not necessarily founded on the evidence of the senses; the

senses would soon become powerless; for example, we can not represent to

ourselves a chiliagon, and yet we reason by intuition on polygons in

general, which include the chiliagon as a particular case.

You know what Poncelet understood by the _principle of continuity_. What

is true of a real quantity, said Poncelet, should be true of an

imaginary quantity; what is true of the hyperbola whose asymptotes are

real, should then be true of the ellipse whose asymptotes are imaginary.

Poncelet was one of the most intuitive minds of this century; he was

passionately, almost ostentatiously, so; he regarded the principle of

continuity as one of his boldest conceptions, and yet this principle did

not rest on the evidence of the senses. To assimilate the hyperbola to

the ellipse was rather to contradict this evidence. It was only a sort

of precocious and instinctive generalization which, moreover, I have no

desire to defend.

We have then many kinds of intuition; first, the appeal to the senses

and the imagination; next, generalization by induction, copied, so to

speak, from the procedures of the experimental sciences; finally, we

have the intuition of pure number, whence arose the second of the axioms

just enunciated, which is able to create the real mathematical

reasoning. I have shown above by examples that the first two can not

give us certainty; but who will seriously doubt the third, who will

doubt arithmetic?

Now in the analysis of to-day, when one cares to take the trouble to be

rigorous, there can be nothing but syllogisms or appeals to this

intuition of pure number, the only intuition which can not deceive us.

It may be said that to-day absolute rigor is attained.

IV

The philosophers make still another objection: "What you gain in rigor,"

they say, "you lose in objectivity. You can rise toward your logical

ideal only by cutting the bonds which attach you to reality. Your

science is infallible, but it can only remain so by imprisoning itself

in an ivory tower and renouncing all relation with the external world.

From this seclusion it must go out when it would attempt the slightest

application."

For example, I seek to show that some property pertains to some object

whose concept seems to me at first indefinable, because it is intuitive.

At first I fail or must content myself with approximate proofs; finally

I decide to give to my object a precise definition, and this enables me

to establish this property in an irreproachable manner.

"And then," say the philosophers, "it still remains to show that the

object which corresponds to this definition is indeed the same made

known to you by intuition; or else that some real and concrete object

whose conformity with your intuitive idea you believe you immediately

recognize corresponds to your new definition. Only then could you affirm

that it has the property in question. You have only displaced the

difficulty."

That is not exactly so; the difficulty has not been displaced, it has

been divided. The proposition to be established was in reality composed

of two different truths, at first not distinguished. The first was a

mathematical truth, and it is now rigorously established. The second was

an experimental verity. Experience alone can teach us that some real and

concrete object corresponds or does not correspond to some abstract

definition. This second verity is not mathematically demonstrated, but

neither can it be, no more than can the empirical laws of the physical

and natural sciences. It would be unreasonable to ask more.

Well, is it not a great advance to have distinguished what long was

wrongly confused? Does this mean that nothing is left of this objection

of the philosophers? That I do not intend to say; in becoming rigorous,

mathematical science takes a character so artificial as to strike every

one; it forgets its historical origins; we see how the questions can be

answered, we no longer see how and why they are put.

This shows us that logic is not enough; that the science of

demonstration is not all science and that intuition must retain its rôle

as complement, I was about to say as counterpoise or as antidote of

logic.

I have already had occasion to insist on the place intuition should hold

in the teaching of the mathematical sciences. Without it young minds

could not make a beginning in the understanding of mathematics; they

could not learn to love it and would see in it only a vain logomachy;

above all, without intuition they would never become capable of applying

mathematics. But now I wish before all to speak of the rôle of intuition

in science itself. If it is useful to the student it is still more so to

the creative scientist.

V

We seek reality, but what is reality? The physiologists tell us that

organisms are formed of cells; the chemists add that cells themselves

are formed of atoms. Does this mean that these atoms or these cells

constitute reality, or rather the sole reality? The way in which these

cells are arranged and from which results the unity of the individual,

is not it also a reality much more interesting than that of the isolated

elements, and should a naturalist who had never studied the elephant

except by means of the microscope think himself sufficiently acquainted

with that animal?

Well, there is something analogous to this in mathematics. The logician

cuts up, so to speak, each demonstration into a very great number of

elementary operations; when we have examined these operations one after

the other and ascertained that each is correct, are we to think we have

grasped the real meaning of the demonstration? Shall we have understood

it even when, by an effort of memory, we have become able to repeat this

proof by reproducing all these elementary operations in just the order

in which the inventor had arranged them? Evidently not; we shall not yet

possess the entire reality; that I know not what, which makes the unity

of the demonstration, will completely elude us.

Pure analysis puts at our disposal a multitude of procedures whose

infallibility it guarantees; it opens to us a thousand different ways on

which we can embark in all confidence; we are assured of meeting there

no obstacles; but of all these ways, which will lead us most promptly to

our goal? Who shall tell us which to choose? We need a faculty which

makes us see the end from afar, and intuition is this faculty. It is

necessary to the explorer for choosing his route; it is not less so to

the one following his trail who wants to know why he chose it.

If you are present at a game of chess, it will not suffice, for the

understanding of the game, to know the rules for moving the pieces. That

will only enable you to recognize that each move has been made

conformably to these rules, and this knowledge will truly have very

little value. Yet this is what the reader of a book on mathematics would

do if he were a logician only. To understand the game is wholly another

matter; it is to know why the player moves this piece rather than that

other which he could have moved without breaking the rules of the game.

It is to perceive the inward reason which makes of this series of

successive moves a sort of organized whole. This faculty is still more

necessary for the player himself, that is, for the inventor.

Let us drop this comparison and return to mathematics. For example, see

what has happened to the idea of continuous function. At the outset this

was only a sensible image, for example, that of a continuous mark traced

by the chalk on a blackboard. Then it became little by little more

refined; ere long it was used to construct a complicated system of

inequalities, which reproduced, so to speak, all the lines of the

original image; this construction finished, the centering of the arch,

so to say, was removed, that crude representation which had temporarily

served as support and which was afterward useless was rejected; there

remained only the construction itself, irreproachable in the eyes of the

logician. And yet if the primitive image had totally disappeared from

our recollection, how could we divine by what caprice all these

inequalities were erected in this fashion one upon another?

Perhaps you think I use too many comparisons; yet pardon still another.

You have doubtless seen those delicate assemblages of silicious needles

which form the skeleton of certain sponges. When the organic matter has

disappeared, there remains only a frail and elegant lace-work. True,

nothing is there except silica, but what is interesting is the form this

silica has taken, and we could not understand it if we did not know the

living sponge which has given it precisely this form. Thus it is that

the old intuitive notions of our fathers, even when we have abandoned

them, still imprint their form upon the logical constructions we have

put in their place.

This view of the aggregate is necessary for the inventor; it is equally

necessary for whoever wishes really to comprehend the inventor. Can

logic give it to us? No; the name mathematicians give it would suffice

to prove this. In mathematics logic is called _analysis_ and analysis

means _division_, _dissection_. It can have, therefore, no tool other

than the scalpel and the microscope.

Thus logic and intuition have each their necessary rôle. Each is

indispensable. Logic, which alone can give certainty, is the instrument

of demonstration; intuition is the instrument of invention.

VI

But at the moment of formulating this conclusion I am seized with

scruples. At the outset I distinguished two kinds of mathematical minds,

the one sort logicians and analysts, the others intuitionalists and

geometers. Well, the analysts also have been inventors. The names I have

just cited make my insistence on this unnecessary.

Here is a contradiction, at least apparently, which needs explanation.

And first, do you think these logicians have always proceeded from the

general to the particular, as the rules of formal logic would seem to

require of them? Not thus could they have extended the boundaries of

science; scientific conquest is to be made only by generalization.

In one of the chapters of 'Science and Hypothesis,' I have had occasion

to study the nature of mathematical reasoning, and I have shown how this

reasoning, without ceasing to be absolutely rigorous, could lift us from

the particular to the general by a procedure I have called _mathematical

induction_. It is by this procedure that the analysts have made science

progress, and if we examine the detail itself of their demonstrations,

we shall find it there at each instant beside the classic syllogism of

Aristotle. We, therefore, see already that the analysts are not simply

makers of syllogisms after the fashion of the scholastics.

Besides, do you think they have always marched step by step with no

vision of the goal they wished to attain? They must have divined the way

leading thither, and for that they needed a guide. This guide is, first,

analogy. For example, one of the methods of demonstration dear to

analysts is that founded on the employment of dominant functions. We

know it has already served to solve a multitude of problems; in what

consists then the rôle of the inventor who wishes to apply it to a new

problem? At the outset he must recognize the analogy of this question

with those which have already been solved by this method; then he must

perceive in what way this new question differs from the others, and

thence deduce the modifications necessary to apply to the method.

But how does one perceive these analogies and these differences? In the

example just cited they are almost always evident, but I could have

found others where they would have been much more deeply hidden; often a

very uncommon penetration is necessary for their discovery. The

analysts, not to let these hidden analogies escape them, that is, in

order to be inventors, must, without the aid of the senses and

imagination, have a direct sense of what constitutes the unity of a

piece of reasoning, of what makes, so to speak, its soul and inmost

life.

When one talked with M. Hermite, he never evoked a sensuous image, and

yet you soon perceived that the most abstract entities were for him like

living beings. He did not see them, but he perceived that they are not

an artificial assemblage and that they have some principle of internal

unity.

But, one will say, that still is intuition. Shall we conclude that the

distinction made at the outset was only apparent, that there is only one

sort of mind and that all the mathematicians are intuitionalists, at

least those who are capable of inventing?

No, our distinction corresponds to something real. I have said above

that there are many kinds of intuition. I have said how much the

intuition of pure number, whence comes rigorous mathematical induction,

differs from sensible intuition to which the imagination, properly so

called, is the principal contributor.

Is the abyss which separates them less profound than it at first

appeared? Could we recognize with a little attention that this pure

intuition itself could not do without the aid of the senses? This is the

affair of the psychologist and the metaphysician and I shall not discuss

the question. But the thing's being doubtful is enough to justify me in

recognizing and affirming an essential difference between the two kinds

of intuition; they have not the same object and seem to call into play

two different faculties of our soul; one would think of two

search-lights directed upon two worlds strangers to one another.

It is the intuition of pure number, that of pure logical forms, which

illumines and directs those we have called _analysts_. This it is which

enables them not alone to demonstrate, but also to invent. By it they

perceive at a glance the general plan of a logical edifice, and that too

without the senses appearing to intervene. In rejecting the aid of the

imagination, which, as we have seen, is not always infallible, they can

advance without fear of deceiving themselves. Happy, therefore, are

those who can do without this aid! We must admire them; but how rare

they are!

Among the analysts there will then be inventors, but they will be few.

The majority of us, if we wished to see afar by pure intuition alone,

would soon feel ourselves seized with vertigo. Our weakness has need of

a staff more solid, and, despite the exceptions of which we have just

spoken, it is none the less true that sensible intuition is in

mathematics the most usual instrument of invention.

Apropos of these reflections, a question comes up that I have not the

time either to solve or even to enunciate with the developments it would

admit of. Is there room for a new distinction, for distinguishing among

the analysts those who above all use pure intuition and those who are

first of all preoccupied with formal logic?

M. Hermite, for example, whom I have just cited, can not be classed

among the geometers who make use of the sensible intuition; but neither

is he a logician, properly so called. He does not conceal his aversion

to purely deductive procedures which start from the general and end in

the particular.

CHAPTER II

THE MEASURE OF TIME

I

So long as we do not go outside the domain of consciousness, the notion

of time is relatively clear. Not only do we distinguish without

difficulty present sensation from the remembrance of past sensations or

the anticipation of future sensations, but we know perfectly well what

we mean when we say that of two conscious phenomena which we remember,

one was anterior to the other; or that, of two foreseen conscious

phenomena, one will be anterior to the other.

When we say that two conscious facts are simultaneous, we mean that they

profoundly interpenetrate, so that analysis can not separate them

without mutilating them.

The order in which we arrange conscious phenomena does not admit of any

arbitrariness. It is imposed upon us and of it we can change nothing.

I have only a single observation to add. For an aggregate of sensations

to have become a remembrance capable of classification in time, it must

have ceased to be actual, we must have lost the sense of its infinite

complexity, otherwise it would have remained present. It must, so to

speak, have crystallized around a center of associations of ideas which

will be a sort of label. It is only when they thus have lost all life

that we can classify our memories in time as a botanist arranges dried

flowers in his herbarium.

But these labels can only be finite in number. On that score,

psychologic time should be discontinuous. Whence comes the feeling that

between any two instants there are others? We arrange our recollections

in time, but we know that there remain empty compartments. How could

that be, if time were not a form pre-existent in our minds? How could we

know there were empty compartments, if these compartments were revealed

to us only by their content?

II

But that is not all; into this form we wish to put not only the

phenomena of our own consciousness, but those of which other

consciousnesses are the theater. But more, we wish to put there physical

facts, these I know not what with which we people space and which no

consciousness sees directly. This is necessary because without it

science could not exist. In a word, psychologic time is given to us and

must needs create scientific and physical time. There the difficulty

begins, or rather the difficulties, for there are two.

Think of two consciousnesses, which are like two worlds impenetrable one

to the other. By what right do we strive to put them into the same mold,

to measure them by the same standard? Is it not as if one strove to

measure length with a gram or weight with a meter? And besides, why do

we speak of measuring? We know perhaps that some fact is anterior to

some other, but not _by how much_ it is anterior.

Therefore two difficulties: (1) Can we transform psychologic time, which

is qualitative, into a quantitative time? (2) Can we reduce to one and

the same measure facts which transpire in different worlds?

III

The first difficulty has long been noticed; it has been the subject of

long discussions and one may say the question is settled. _We have not a

direct intuition of the equality of two intervals of time._ The persons

who believe they possess this intuition are dupes of an illusion. When I

say, from noon to one the same time passes as from two to three, what

meaning has this affirmation?

The least reflection shows that by itself it has none at all. It will

only have that which I choose to give it, by a definition which will

certainly possess a certain degree of arbitrariness. Psychologists could

have done without this definition; physicists and astronomers could not;

let us see how they have managed.

To measure time they use the pendulum and they suppose by definition

that all the beats of this pendulum are of equal duration. But this is

only a first approximation; the temperature, the resistance of the air,

the barometric pressure, make the pace of the pendulum vary. If we

could escape these sources of error, we should obtain a much closer

approximation, but it would still be only an approximation. New causes,

hitherto neglected, electric, magnetic or others, would introduce minute

perturbations.

In fact, the best chronometers must be corrected from time to time, and

the corrections are made by the aid of astronomic observations;

arrangements are made so that the sidereal clock marks the same hour

when the same star passes the meridian. In other words, it is the

sidereal day, that is, the duration of the rotation of the earth, which

is the constant unit of time. It is supposed, by a new definition

substituted for that based on the beats of the pendulum, that two

complete rotations of the earth about its axis have the same duration.

However, the astronomers are still not content with this definition.

Many of them think that the tides act as a check on our globe, and that

the rotation of the earth is becoming slower and slower. Thus would be

explained the apparent acceleration of the motion of the moon, which

would seem to be going more rapidly than theory permits because our

watch, which is the earth, is going slow.

IV

All this is unimportant, one will say; doubtless our instruments of

measurement are imperfect, but it suffices that we can conceive a

perfect instrument. This ideal can not be reached, but it is enough to

have conceived it and so to have put rigor into the definition of the

unit of time.

The trouble is that there is no rigor in the definition. When we use the

pendulum to measure time, what postulate do we implicitly admit? _It is

that the duration of two identical phenomena is the same_; or, if you

prefer, that the same causes take the same time to produce the same

effects.

And at first blush, this is a good definition of the equality of two

durations. But take care. Is it impossible that experiment may some day

contradict our postulate?

Let me explain myself. I suppose that at a certain place in the world

the phenomenon [alpha] happens, causing as consequence at the end of a

certain time the effect [alpha]'. At another place in the world very

far away from the first, happens the phenomenon [beta], which causes as

consequence the effect [beta]'. The phenomena [alpha] and [beta] are

simultaneous, as are also the effects [alpha]' and [beta]'.

Later, the phenomenon [alpha] is reproduced under approximately the same

conditions as before, and _simultaneously_ the phenomenon [beta] is also

reproduced at a very distant place in the world and almost under the

same circumstances. The effects [alpha]' and [beta]' also take place.

Let us suppose that the effect [alpha]' happens perceptibly before the

effect [beta]'.

If experience made us witness such a sight, our postulate would be

contradicted. For experience would tell us that the first duration

[alpha][alpha]' is equal to the first duration [beta][beta]' and that

the second duration [alpha][alpha]' is less than the second duration

[beta][beta]'. On the other hand, our postulate would require that the

two durations [alpha][alpha]' should be equal to each other, as likewise

the two durations [beta][beta]'. The equality and the inequality deduced

from experience would be incompatible with the two equalities deduced

from the postulate.

Now can we affirm that the hypotheses I have just made are absurd? They

are in no wise contrary to the principle of contradiction. Doubtless

they could not happen without the principle of sufficient reason seeming

violated. But to justify a definition so fundamental I should prefer

some other guarantee.

V

But that is not all. In physical reality one cause does not produce a

given effect, but a multitude of distinct causes contribute to produce

it, without our having any means of discriminating the part of each of

them.

Physicists seek to make this distinction; but they make it only

approximately, and, however they progress, they never will make it

except approximately. It is approximately true that the motion of the

pendulum is due solely to the earth's attraction; but in all rigor every

attraction, even of Sirius, acts on the pendulum.

Under these conditions, it is clear that the causes which have produced

a certain effect will never be reproduced except approximately. Then we

should modify our postulate and our definition. Instead of saying: 'The

same causes take the same time to produce the same effects,' we should

say: 'Causes almost identical take almost the same time to produce

almost the same effects.'

Our definition therefore is no longer anything but approximate. Besides,

as M. Calinon very justly remarks in a recent memoir:[7]

One of the circumstances of any phenomenon is the velocity of

the earth's rotation; if this velocity of rotation varies, it

constitutes in the reproduction of this phenomenon a

circumstance which no longer remains the same. But to suppose

this velocity of rotation constant is to suppose that we know

how to measure time.

[7] _Etude sur les diverses grandeurs_, Paris, Gauthier-Villars, 1897.

Our definition is therefore not yet satisfactory; it is certainly not

that which the astronomers of whom I spoke above implicitly adopt, when

they affirm that the terrestrial rotation is slowing down.

What meaning according to them has this affirmation? We can only

understand it by analyzing the proofs they give of their proposition.

They say first that the friction of the tides producing heat must

destroy _vis viva_. They invoke therefore the principle of _vis viva_,

or of the conservation of energy.

They say next that the secular acceleration of the moon, calculated

according to Newton's law, would be less than that deduced from

observations unless the correction relative to the slowing down of the

terrestrial rotation were made. They invoke therefore Newton's law. In

other words, they define duration in the following way: time should be

so defined that Newton's law and that of _vis viva_ may be verified.

Newton's law is an experimental truth; as such it is only approximate,

which shows that we still have only a definition by approximation.

If now it be supposed that another way of measuring time is adopted, the

experiments on which Newton's law is founded would none the less have

the same meaning. Only the enunciation of the law would be different,

because it would be translated into another language; it would evidently

be much less simple. So that the definition implicitly adopted by the

astronomers may be summed up thus: Time should be so defined that the

equations of mechanics may be as simple as possible. In other words,

there is not one way of measuring time more true than another; that

which is generally adopted is only more _convenient_. Of two watches, we

have no right to say that the one goes true, the other wrong; we can

only say that it is advantageous to conform to the indications of the

first.

The difficulty which has just occupied us has been, as I have said,

often pointed out; among the most recent works in which it is

considered, I may mention, besides M. Calinon's little book, the

treatise on mechanics of Andrade.

VI

The second difficulty has up to the present attracted much less

attention; yet it is altogether analogous to the preceding; and even,

logically, I should have spoken of it first.

Two psychological phenomena happen in two different consciousnesses;

when I say they are simultaneous, what do I mean? When I say that a

physical phenomenon, which happens outside of every consciousness, is

before or after a psychological phenomenon, what do I mean?

In 1572, Tycho Brahe noticed in the heavens a new star. An immense

conflagration had happened in some far distant heavenly body; but it had

happened long before; at least two hundred years were necessary for the

light from that star to reach our earth. This conflagration therefore

happened before the discovery of America. Well, when I say that; when,

considering this gigantic phenomenon, which perhaps had no witness,

since the satellites of that star were perhaps uninhabited, I say this

phenomenon is anterior to the formation of the visual image of the isle

of Española in the consciousness of Christopher Columbus, what do I

mean?

A little reflection is sufficient to understand that all these

affirmations have by themselves no meaning. They can have one only as

the outcome of a convention.

VII

We should first ask ourselves how one could have had the idea of putting

into the same frame so many worlds impenetrable to one another. We

should like to represent to ourselves the external universe, and only by

so doing could we feel that we understood it. We know we never can

attain this representation: our weakness is too great. But at least we

desire the ability to conceive an infinite intelligence for which this

representation could be possible, a sort of great consciousness which

should see all, and which should classify all _in its time_, as we

classify, _in our time_, the little we see.

This hypothesis is indeed crude and incomplete, because this supreme

intelligence would be only a demigod; infinite in one sense, it would be

limited in another, since it would have only an imperfect recollection

of the past; and it could have no other, since otherwise all

recollections would be equally present to it and for it there would be

no time. And yet when we speak of time, for all which happens outside of

us, do we not unconsciously adopt this hypothesis; do we not put

ourselves in the place of this imperfect god; and do not even the

atheists put themselves in the place where god would be if he existed?

What I have just said shows us, perhaps, why we have tried to put all

physical phenomena into the same frame. But that can not pass for a

definition of simultaneity, since this hypothetical intelligence, even

if it existed, would be for us impenetrable. It is therefore necessary

to seek something else.

VIII

The ordinary definitions which are proper for psychologic time would

suffice us no more. Two simultaneous psychologic facts are so closely

bound together that analysis can not separate without mutilating them.

Is it the same with two physical facts? Is not my present nearer my past

of yesterday than the present of Sirius?

It has also been said that two facts should be regarded as simultaneous

when the order of their succession may be inverted at will. It is

evident that this definition would not suit two physical facts which

happen far from one another, and that, in what concerns them, we no

longer even understand what this reversibility would be; besides,

succession itself must first be defined.

IX

Let us then seek to give an account of what is understood by

simultaneity or antecedence, and for this let us analyze some examples.

I write a letter; it is afterward read by the friend to whom I have

addressed it. There are two facts which have had for their theater two

different consciousnesses. In writing this letter I have had the visual

image of it, and my friend has had in his turn this same visual image in

reading the letter. Though these two facts happen in impenetrable

worlds, I do not hesitate to regard the first as anterior to the second,

because I believe it is its cause.

I hear thunder, and I conclude there has been an electric discharge; I

do not hesitate to consider the physical phenomenon as anterior to the

auditory image perceived in my consciousness, because I believe it is

its cause.

Behold then the rule we follow, and the only one we can follow: when a

phenomenon appears to us as the cause of another, we regard it as

anterior. It is therefore by cause that we define time; but most often,

when two facts appear to us bound by a constant relation, how do we

recognize which is the cause and which the effect? We assume that the

anterior fact, the antecedent, is the cause of the other, of the

consequent. It is then by time that we define cause. How save ourselves

from this _petitio principii_?

We say now _post hoc, ergo propter hoc_; now _propter hoc, ergo post

hoc_; shall we escape from this vicious circle?

X

Let us see, not how we succeed in escaping, for we do not completely

succeed, but how we try to escape.

I execute a voluntary act _A_ and I feel afterward a sensation _D_,

which I regard as a consequence of the act _A_; on the other hand, for

whatever reason, I infer that this consequence is not immediate, but

that outside my consciousness two facts _B_ and _C_, which I have not

witnessed, have happened, and in such a way that _B_ is the effect of

_A_, that _C_ is the effect of _B_, and _D_ of _C_.

But why? If I think I have reason to regard the four facts _A_, _B_,

_C_, _D_, as bound to one another by a causal connection, why range

them in the causal order _A B C D_, and at the same time in the

chronologic order _A B C D_, rather than in any other order?

I clearly see that in the act _A_ I have the feeling of having been

active, while in undergoing the sensation _D_ I have that of having been

passive. This is why I regard _A_ as the initial cause and _D_ as the

ultimate effect; this is why I put _A_ at the beginning of the chain and

_D_ at the end; but why put _B_ before _C_ rather than _C_ before _B_?

If this question is put, the reply ordinarily is: we know that it is _B_

which is the cause of _C_ because we always see _B_ happen before _C_.

These two phenomena, when witnessed, happen in a certain order; when

analogous phenomena happen without witness, there is no reason to invert

this order.

Doubtless, but take care; we never know directly the physical phenomena

_B_ and _C_. What we know are sensations _B'_ and _C'_ produced

respectively by _B_ and _C_. Our consciousness tells us immediately that

_B'_ precedes _C'_ and we suppose that _B_ and _C_ succeed one another

in the same order.

This rule appears in fact very natural, and yet we are often led to

depart from it. We hear the sound of the thunder only some seconds after

the electric discharge of the cloud. Of two flashes of lightning, the

one distant, the other near, can not the first be anterior to the

second, even though the sound of the second comes to us before that of

the first?

XI

Another difficulty; have we really the right to speak of the cause of a

phenomenon? If all the parts of the universe are interchained in a

certain measure, any one phenomenon will not be the effect of a single

cause, but the resultant of causes infinitely numerous; it is, one often

says, the consequence of the state of the universe a moment before. How

enunciate rules applicable to circumstances so complex? And yet it is

only thus that these rules can be general and rigorous.

Not to lose ourselves in this infinite complexity, let us make a simpler

hypothesis. Consider three stars, for example, the sun, Jupiter and

Saturn; but, for greater simplicity, regard them as reduced to material

points and isolated from the rest of the world. The positions and the

velocities of three bodies at a given instant suffice to determine their

positions and velocities at the following instant, and consequently

at any instant. Their positions at the instant t determine their

positions at the instant _t_ + _h_ as well as their positions at the

instant _t_ - _h_.

Even more; the position of Jupiter at the instant _t_, together with

that of Saturn at the instant _t_ + _a_, determines the position of

Jupiter at any instant and that of Saturn at any instant.

The aggregate of positions occupied by Jupiter at the instant _t_ + _e_

and Saturn at the instant _t_ + _a_ + _e_ is bound to the aggregate of

positions occupied by Jupiter at the instant _t_ and Saturn at the

instant _t_ + _a_, by laws as precise as that of Newton, though more

complicated. Then why not regard one of these aggregates as the cause of

the other, which would lead to considering as simultaneous the instant

_t_ of Jupiter and the instant _t_ + _a_ of Saturn?

In answer there can only be reasons, very strong, it is true, of

convenience and simplicity.

XII

But let us pass to examples less artificial; to understand the

definition implicitly supposed by the savants, let us watch them at work

and look for the rules by which they investigate simultaneity.

I will take two simple examples, the measurement of the velocity of

light and the determination of longitude.

When an astronomer tells me that some stellar phenomenon, which his

telescope reveals to him at this moment, happened, nevertheless, fifty

years ago, I seek his meaning, and to that end I shall ask him first how

he knows it, that is, how he has measured the velocity of light.

He has begun by _supposing_ that light has a constant velocity, and in

particular that its velocity is the same in all directions. That is a

postulate without which no measurement of this velocity could be

attempted. This postulate could never be verified directly by

experiment; it might be contradicted by it if the results of different

measurements were not concordant. We should think ourselves fortunate

that this contradiction has not happened and that the slight

discordances which may happen can be readily explained.

The postulate, at all events, resembling the principle of sufficient

reason, has been accepted by everybody; what I wish to emphasize is that

it furnishes us with a new rule for the investigation of simultaneity,

entirely different from that which we have enunciated above.

This postulate assumed, let us see how the velocity of light has been

measured. You know that Roemer used eclipses of the satellites of

Jupiter, and sought how much the event fell behind its prediction. But

how is this prediction made? It is by the aid of astronomic laws; for

instance Newton's law.

Could not the observed facts be just as well explained if we attributed

to the velocity of light a little different value from that adopted, and

supposed Newton's law only approximate? Only this would lead to

replacing Newton's law by another more complicated. So for the velocity

of light a value is adopted, such that the astronomic laws compatible

with this value may be as simple as possible. When navigators or

geographers determine a longitude, they have to solve just the problem

we are discussing; they must, without being at Paris, calculate Paris

time. How do they accomplish it? They carry a chronometer set for Paris.

The qualitative problem of simultaneity is made to depend upon the

quantitative problem of the measurement of time. I need not take up the

difficulties relative to this latter problem, since above I have

emphasized them at length.

Or else they observe an astronomic phenomenon, such as an eclipse of the

moon, and they suppose that this phenomenon is perceived simultaneously

from all points of the earth. That is not altogether true, since the

propagation of light is not instantaneous; if absolute exactitude were

desired, there would be a correction to make according to a complicated

rule.

Or else finally they use the telegraph. It is clear first that the

reception of the signal at Berlin, for instance, is after the sending of

this same signal from Paris. This is the rule of cause and effect

analyzed above. But how much after? In general, the duration of the

transmission is neglected and the two events are regarded as

simultaneous. But, to be rigorous, a little correction would still have

to be made by a complicated calculation; in practise it is not made,

because it would be well within the errors of observation; its theoretic

necessity is none the less from our point of view, which is that of a

rigorous definition. From this discussion, I wish to emphasize two

things: (1) The rules applied are exceedingly various. (2) It is

difficult to separate the qualitative problem of simultaneity from the

quantitative problem of the measurement of time; no matter whether a

chronometer is used, or whether account must be taken of a velocity of

transmission, as that of light, because such a velocity could not be

measured without _measuring_ a time.

XIII

To conclude: We have not a direct intuition of simultaneity, nor of the

equality of two durations. If we think we have this intuition, this is

an illusion. We replace it by the aid of certain rules which we apply

almost always without taking count of them.

But what is the nature of these rules? No general rule, no rigorous

rule; a multitude of little rules applicable to each particular case.

These rules are not imposed upon us and we might amuse ourselves in

inventing others; but they could not be cast aside without greatly

complicating the enunciation of the laws of physics, mechanics and

astronomy.

We therefore choose these rules, not because they are true, but because

they are the most convenient, and we may recapitulate them as follows:

"The simultaneity of two events, or the order of their succession, the

equality of two durations, are to be so defined that the enunciation of

the natural laws may be as simple as possible. In other words, all these

rules, all these definitions are only the fruit of an unconscious

opportunism."

CHAPTER III

THE NOTION OF SPACE

1. _Introduction_

In the articles I have heretofore devoted to space I have above all

emphasized the problems raised by non-Euclidean geometry, while leaving

almost completely aside other questions more difficult of approach, such

as those which pertain to the number of dimensions. All the geometries I

considered had thus a common basis, that tridimensional continuum which

was the same for all and which differentiated itself only by the figures

one drew in it or when one aspired to measure it.

In this continuum, primitively amorphous, we may imagine a network of

lines and surfaces, we may then convene to regard the meshes of this net

as equal to one another, and it is only after this convention that this

continuum, become measurable, becomes Euclidean or non-Euclidean space.

From this amorphous continuum can therefore arise indifferently one or

the other of the two spaces, just as on a blank sheet of paper may be

traced indifferently a straight or a circle.

In space we know rectilinear triangles the sum of whose angles is equal

to two right angles; but equally we know curvilinear triangles the sum

of whose angles is less than two right angles. The existence of the one

sort is not more doubtful than that of the other. To give the name of

straights to the sides of the first is to adopt Euclidean geometry; to

give the name of straights to the sides of the latter is to adopt the

non-Euclidean geometry. So that to ask what geometry it is proper to

adopt is to ask, to what line is it proper to give the name straight?

It is evident that experiment can not settle such a question; one would

not ask, for instance, experiment to decide whether I should call _AB_

or _CD_ a straight. On the other hand, neither can I say that I have not

the right to give the name of straights to the sides of non-Euclidean

triangles because they are not in conformity with the eternal idea of

straight which I have by intuition. I grant, indeed, that I have the

intuitive idea of the side of the Euclidean triangle, but I have equally

the intuitive idea of the side of the non-Euclidean triangle. Why should

I have the right to apply the name of straight to the first of these

ideas and not to the second? Wherein does this syllable form an

integrant part of this intuitive idea? Evidently when we say that the

Euclidean straight is a _true_ straight and that the non-Euclidean

straight is not a true straight, we simply mean that the first intuitive

idea corresponds to a _more noteworthy_ object than the second. But how

do we decide that this object is more noteworthy? This question I have

investigated in 'Science and Hypothesis.'

It is here that we saw experience come in. If the Euclidean straight is

more noteworthy than the non-Euclidean straight, it is so chiefly

because it differs little from certain noteworthy natural objects from

which the non-Euclidean straight differs greatly. But, it will be said,

the definition of the non-Euclidean straight is artificial; if we for a

moment adopt it, we shall see that two circles of different radius both

receive the name of non-Euclidean straights, while of two circles of the

same radius one can satisfy the definition without the other being able

to satisfy it, and then if we transport one of these so-called straights

without deforming it, it will cease to be a straight. But by what right

do we consider as equal these two figures which the Euclidean geometers

call two circles with the same radius? It is because by transporting one

of them without deforming it we can make it coincide with the other. And

why do we say this transportation is effected without deformation? It is

impossible to give a good reason for it. Among all the motions

conceivable, there are some of which the Euclidean geometers say that

they are not accompanied by deformation; but there are others of which

the non-Euclidean geometers would say that they are not accompanied by

deformation. In the first, called Euclidean motions, the Euclidean

straights remain Euclidean straights and the non-Euclidean straights do

not remain non-Euclidean straights; in the motions of the second sort,

or non-Euclidean motions, the non-Euclidean straights remain

non-Euclidean straights and the Euclidean straights do not remain

Euclidean straights. It has, therefore, not been demonstrated that it

was unreasonable to call straights the sides of non-Euclidean triangles;

it has only been shown that that would be unreasonable if one continued

to call the Euclidean motions motions without deformation; but it has at

the same time been shown that it would be just as unreasonable to call

straights the sides of Euclidean triangles if the non-Euclidean motions

were called motions without deformation.

Now when we say that the Euclidean motions are the _true_ motions

without deformation, what do we mean? We simply mean that they are _more

noteworthy_ than the others. And why are they more noteworthy? It is

because certain noteworthy natural bodies, the solid bodies, undergo

motions almost similar.

And then when we ask: Can one imagine non-Euclidean space? That means:

Can we imagine a world where there would be noteworthy natural objects

affecting almost the form of non-Euclidean straights, and noteworthy

natural bodies frequently undergoing motions almost similar to the

non-Euclidean motions? I have shown in 'Science and Hypothesis' that to

this question we must answer yes.

It has often been observed that if all the bodies in the universe were

dilated simultaneously and in the same proportion, we should have no

means of perceiving it, since all our measuring instruments would grow

at the same time as the objects themselves which they serve to measure.

The world, after this dilatation, would continue on its course without

anything apprising us of so considerable an event. In other words, two

worlds similar to one another (understanding the word similitude in the

sense of Euclid, Book VI.) would be absolutely indistinguishable. But

more; worlds will be indistinguishable not only if they are equal or

similar, that is, if we can pass from one to the other by changing the

axes of coordinates, or by changing the scale to which lengths are

referred; but they will still be indistinguishable if we can pass from

one to the other by any 'point-transformation' whatever. I will explain

my meaning. I suppose that to each point of one corresponds one point of

the other and only one, and inversely; and besides that the coordinates

of a point are continuous functions, _otherwise altogether arbitrary_,

of the corresponding point. I suppose besides that to each object of the

first world corresponds in the second an object of the same nature

placed precisely at the corresponding point. I suppose finally that this

correspondence fulfilled at the initial instant is maintained

indefinitely. We should have no means of distinguishing these two worlds

one from the other. The relativity of space is not ordinarily understood

in so broad a sense; it is thus, however, that it would be proper to

understand it.

If one of these universes is our Euclidean world, what its inhabitants

will call straight will be our Euclidean straight; but what the

inhabitants of the second world will call straight will be a curve which

will have the same properties in relation to the world they inhabit and

in relation to the motions that they will call motions without

deformation. Their geometry will, therefore, be Euclidean geometry, but

their straight will not be our Euclidean straight. It will be its

transform by the point-transformation which carries over from our world

to theirs. The straights of these men will not be our straights, but

they will have among themselves the same relations as our straights to

one another. It is in this sense I say their geometry will be ours. If

then we wish after all to proclaim that they deceive themselves, that

their straight is not the true straight, if we still are unwilling to

admit that such an affirmation has no meaning, at least we must confess

that these people have no means whatever of recognizing their error.

2. _Qualitative Geometry_

All that is relatively easy to understand, and I have already so often

repeated it that I think it needless to expatiate further on the matter.

Euclidean space is not a form imposed upon our sensibility, since we can

imagine non-Euclidean space; but the two spaces, Euclidean and

non-Euclidean, have a common basis, that amorphous continuum of which I

spoke in the beginning. From this continuum we can get either Euclidean

space or Lobachevskian space, just as we can, by tracing upon it a

proper graduation, transform an ungraduated thermometer into a

Fahrenheit or a Réaumur thermometer.

And then comes a question: Is not this amorphous continuum, that our

analysis has allowed to survive, a form imposed upon our sensibility? If

so, we should have enlarged the prison in which this sensibility is

confined, but it would always be a prison.

This continuum has a certain number of properties, exempt from all idea

of measurement. The study of these properties is the object of a science

which has been cultivated by many great geometers and in particular by

Riemann and Betti and which has received the name of analysis situs. In

this science abstraction is made of every quantitative idea and, for

example, if we ascertain that on a line the point _B_ is between the

points _A_ and _C_, we shall be content with this ascertainment and

shall not trouble to know whether the line _ABC_ is straight or curved,

nor whether the length _AB_ is equal to the length _BC_, or whether it

is twice as great.

The theorems of analysis situs have, therefore, this peculiarity, that

they would remain true if the figures were copied by an inexpert

draftsman who should grossly change all the proportions and replace the

straights by lines more or less sinuous. In mathematical terms, they are

not altered by any 'point-transformation' whatsoever. It has often been

said that metric geometry was quantitative, while projective geometry

was purely qualitative. That is not altogether true. The straight is

still distinguished from other lines by properties which remain

quantitative in some respects. The real qualitative geometry is,

therefore, analysis situs.

The same questions which came up apropos of the truths of Euclidean

geometry, come up anew apropos of the theorems of analysis situs. Are

they obtainable by deductive reasoning? Are they disguised conventions?

Are they experimental verities? Are they the characteristics of a form

imposed either upon our sensibility or upon our understanding?

I wish simply to observe that the last two solutions exclude each other.

We can not admit at the same time that it is impossible to imagine space

of four dimensions and that experience proves to us that space has three

dimensions. The experimenter puts to nature a question: Is it this or

that? and he can not put it without imagining the two terms of the

alternative. If it were impossible to imagine one of these terms, it

would be futile and besides impossible to consult experience. There is

no need of observation to know that the hand of a watch is not marking

the hour 15 on the dial, because we know beforehand that there are only

12, and we could not look at the mark 15 to see if the hand is there,

because this mark does not exist.

Note likewise that in analysis situs the empiricists are disembarrassed

of one of the gravest objections that can be leveled against them, of

that which renders absolutely vain in advance all their efforts to apply

their thesis to the verities of Euclidean geometry. These verities are

rigorous and all experimentation can only be approximate. In analysis

situs approximate experiments may suffice to give a rigorous theorem

and, for instance, if it is seen that space can not have either two or

less than two dimensions, nor four or more than four, we are certain

that it has exactly three, since it could not have two and a half or

three and a half.

Of all the theorems of analysis situs, the most important is that which

is expressed in saying that space has three dimensions. This it is that

we are about to consider, and we shall put the question in these terms:

When we say that space has three dimensions, what do we mean?

3. _The Physical Continuum of Several Dimensions_

I have explained in 'Science and Hypothesis' whence we derive the notion

of physical continuity and how that of mathematical continuity has

arisen from it. It happens that we are capable of distinguishing two

impressions one from the other, while each is indistinguishable from a

third. Thus we can readily distinguish a weight of 12 grams from a

weight of 10 grams, while a weight of 11 grams could be distinguished

from neither the one nor the other. Such a statement, translated into

symbols, may be written:

_A_ = _B_, _B_ = _C_, _A_ < _C_.

This would be the formula of the physical continuum, as crude experience

gives it to us, whence arises an intolerable contradiction that has

been obviated by the introduction of the mathematical continuum. This is

a scale of which the steps (commensurable or incommensurable numbers)

are infinite in number but are exterior to one another, instead of

encroaching on one another as do the elements of the physical continuum,

in conformity with the preceding formula.

The physical continuum is, so to speak, a nebula not resolved; the most

perfect instruments could not attain to its resolution. Doubtless if we

measured the weights with a good balance instead of judging them by the

hand, we could distinguish the weight of 11 grams from those of 10 and

12 grams, and our formula would become:

_A_ < _B_, _B_ < _C_, _A_ < _C_.

But we should always find between _A_ and _B_ and between _B_ and _C_

new elements _D_ and _E_, such that

_A_ = _D_, _D_ = _B_, _A_ < _B_;

_B_ = _E_, _E_ = _C_, _B_ < _C_,

and the difficulty would only have receded and the nebula would always

remain unresolved; the mind alone can resolve it and the mathematical

continuum it is which is the nebula resolved into stars.