It is necessary then to take all the consequences of our assumptions and

see if they contain no contradiction.

If these consequences were finite in number, this would be easy; but

they are infinite in number; they are the whole of mathematics, or at

least all arithmetic.

What then is to be done? Perhaps strictly we could repeat the reasoning

of number III.

But as we have said, this reasoning is complete induction, and it is

precisely the principle of complete induction whose justification would

be the point in question.

VI

_The Logic of Hilbert_

I come now to the capital work of Hilbert which he communicated to the

Congress of Mathematicians at Heidelberg, and of which a French

translation by M. Pierre Boutroux appeared in _l'Enseignement

mathématique_, while an English translation due to Halsted appeared in

_The Monist_.[13] In this work, which contains profound thoughts, the

author's aim is analogous to that of Russell, but on many points he

diverges from his predecessor.

[13] 'The Foundations of Logic and Arithmetic,' _Monist_, XV.,

338-352.

"But," he says (_Monist_, p. 340), "on attentive consideration we become

aware that in the usual exposition of the laws of logic certain

fundamental concepts of arithmetic are already employed; for example,

the concept of the aggregate, in part also the concept of number.

"We fall thus into a vicious circle and therefore to avoid paradoxes a

partly simultaneous development of the laws of logic and arithmetic is

requisite."

We have seen above that what Hilbert says of the principles of logic _in

the usual exposition_ applies likewise to the logic of Russell. So for

Russell logic is prior to arithmetic; for Hilbert they are

'simultaneous.' We shall find further on other differences still

greater, but we shall point them out as we come to them. I prefer to

follow step by step the development of Hilbert's thought, quoting

textually the most important passages.

"Let us take as the basis of our consideration first of all a

thought-thing 1 (one)" (p. 341). Notice that in so doing we in no wise

imply the notion of number, because it is understood that 1 is here only

a symbol and that we do not at all seek to know its meaning. "The taking

of this thing together with itself respectively two, three or more

times...." Ah! this time it is no longer the same; if we introduce the

words 'two,' 'three,' and above all 'more,' 'several,' we introduce the

notion of number; and then the definition of finite whole number which

we shall presently find, will come too late. Our author was too

circumspect not to perceive this begging of the question. So at the end

of his work he tries to proceed to a truly patching-up process.

Hilbert then introduces two simple objects 1 and =, and considers all

the combinations of these two objects, all the combinations of their

combinations, etc. It goes without saying that we must forget the

ordinary meaning of these two signs and not attribute any to them.

Afterwards he separates these combinations into two classes, the class

of the existent and the class of the non-existent, and till further

orders this separation is entirely arbitrary. Every affirmative

statement tells us that a certain combination belongs to the class of

the existent; every negative statement tells us that a certain

combination belongs to the class of the non-existent.

VII

Note now a difference of the highest importance. For Russell any object

whatsoever, which he designates by _x_, is an object absolutely

undetermined and about which he supposes nothing; for Hilbert it is one

of the combinations formed with the symbols 1 and =; he could not

conceive of the introduction of anything other than combinations of

objects already defined. Moreover Hilbert formulates his thought in the

neatest way, and I think I must reproduce _in extenso_ his statement (p.

348):

"In the assumptions the arbitraries (as equivalent for the concept

'every' and 'all' in the customary logic) represent only those

thought-things and their combinations with one another, which at this

stage are laid down as fundamental or are to be newly defined.

Therefore in the deduction of inferences from the assumptions, the

arbitraries, which occur in the assumptions, can be replaced only by

such thought-things and their combinations.

"Also we must duly remember, that through the super-addition and making

fundamental of a new thought-thing the preceding assumptions undergo an

enlargement of their validity, and where necessary, are to be subjected

to a change in conformity with the sense."

The contrast with Russell's view-point is complete. For this philosopher

we may substitute for _x_ not only objects already known, but anything.

Russell is faithful to his point of view, which is that of

comprehension. He starts from the general idea of being, and enriches it

more and more while restricting it, by adding new qualities. Hilbert on

the contrary recognizes as possible beings only combinations of objects

already known; so that (looking at only one side of his thought) we

might say he takes the view-point of extension.

VIII

Let us continue with the exposition of Hilbert's ideas. He introduces

two assumptions which he states in his symbolic language but which

signify, in the language of the uninitiated, that every quality is equal

to itself and that every operation performed upon two identical

quantities gives identical results.

So stated, they are evident, but thus to present them would be to

misrepresent Hilbert's thought. For him mathematics has to combine only

pure symbols, and a true mathematician should reason upon them without

preconceptions as to their meaning. So his assumptions are not for him

what they are for the common people.

He considers them as representing the definition by postulates of the

symbol (=) heretofore void of all signification. But to justify this

definition we must show that these two assumptions lead to no

contradiction. For this Hilbert used the reasoning of our number III,

without appearing to perceive that he is using complete induction.

IX

The end of Hilbert's memoir is altogether enigmatic and I shall not lay

stress upon it. Contradictions accumulate; we feel that the author is

dimly conscious of the _petitio principii_ he has committed, and that he

seeks vainly to patch up the holes in his argument.

What does this mean? At the point of proving that the definition of the

whole number by the assumption of complete induction implies no

contradiction, Hilbert withdraws as Russell and Couturat withdrew,

because the difficulty is too great.

X

_Geometry_

Geometry, says M. Couturat, is a vast body of doctrine wherein the

principle of complete induction does not enter. That is true in a

certain measure; we can not say it is entirely absent, but it enters

very slightly. If we refer to the _Rational Geometry_ of Dr. Halsted

(New York, John Wiley and Sons, 1904) built up in accordance with the

principles of Hilbert, we see the principle of induction enter for the

first time on page 114 (unless I have made an oversight, which is quite

possible).[14]

[14] Second ed., 1907, p. 86; French ed., 1911, p. 97. G. B. H.

So geometry, which only a few years ago seemed the domain where the

reign of intuition was uncontested, is to-day the realm where the

logicians seem to triumph. Nothing could better measure the importance

of the geometric works of Hilbert and the profound impress they have

left on our conceptions.

But be not deceived. What is after all the fundamental theorem of

geometry? It is that the assumptions of geometry imply no contradiction,

and this we can not prove without the principle of induction.

How does Hilbert demonstrate this essential point? By leaning upon

analysis and through it upon arithmetic and through it upon the

principle of induction.

And if ever one invents another demonstration, it will still be

necessary to lean upon this principle, since the possible consequences

of the assumptions, of which it is necessary to show that they are not

contradictory, are infinite in number.

XI

_Conclusion_

Our conclusion straightway is that the principle of induction can not be

regarded as the disguised definition of the entire world.

Here are three truths: (1) The principle of complete induction; (2)

Euclid's postulate; (3) the physical law according to which phosphorus

melts at 44° (cited by M. Le Roy).

These are said to be three disguised definitions: the first, that of the

whole number; the second, that of the straight line; the third, that of

phosphorus.

I grant it for the second; I do not admit it for the other two. I must

explain the reason for this apparent inconsistency.

First, we have seen that a definition is acceptable only on condition

that it implies no contradiction. We have shown likewise that for the

first definition this demonstration is impossible; on the other hand, we

have just recalled that for the second Hilbert has given a complete

proof.

As to the third, evidently it implies no contradiction. Does this mean

that the definition guarantees, as it should, the existence of the

object defined? We are here no longer in the mathematical sciences, but

in the physical, and the word existence has no longer the same meaning.

It no longer signifies absence of contradiction; it means objective

existence.

You already see a first reason for the distinction I made between the

three cases; there is a second. In the applications we have to make of

these three concepts, do they present themselves to us as defined by

these three postulates?

The possible applications of the principle of induction are innumerable;

take, for example, one of those we have expounded above, and where it is

sought to prove that an aggregate of assumptions can lead to no

contradiction. For this we consider one of the series of syllogisms we

may go on with in starting from these assumptions as premises. When we

have finished the _n_th syllogism, we see we can make still another and

this is the _n_ + 1th. Thus the number _n_ serves to count a series of

successive operations; it is a number obtainable by successive

additions. This therefore is a number from which we may go back to

unity by _successive subtractions_. Evidently we could not do this if we

had _n_ = _n_ - 1, since then by subtraction we should always obtain

again the same number. So the way we have been led to consider this

number _n_ implies a definition of the finite whole number and this

definition is the following: A finite whole number is that which can be

obtained by successive additions; it is such that _n_ is not equal to

_n_ - 1.

That granted, what do we do? We show that if there has been no

contradiction up to the _n_th syllogism, no more will there be up to the

_n_ + 1th, and we conclude there never will be. You say: I have the

right to draw this conclusion, since the whole numbers are by definition

those for which a like reasoning is legitimate. But that implies another

definition of the whole number, which is as follows: A whole number is

that on which we may reason by recurrence. In the particular case it is

that of which we may say that, if the absence of contradiction up to the

time of a syllogism of which the number is an integer carries with it

the absence of contradiction up to the time of the syllogism whose

number is the following integer, we need fear no contradiction for any

of the syllogisms whose number is an integer.

The two definitions are not identical; they are doubtless equivalent,

but only in virtue of a synthetic judgment _a priori_; we can not pass

from one to the other by a purely logical procedure. Consequently we

have no right to adopt the second, after having introduced the whole

number by a way that presupposes the first.

On the other hand, what happens with regard to the straight line? I have

already explained this so often that I hesitate to repeat it again, and

shall confine myself to a brief recapitulation of my thought. We have

not, as in the preceding case, two equivalent definitions logically

irreducible one to the other. We have only one expressible in words.

Will it be said there is another which we feel without being able to

word it, since we have the intuition of the straight line or since we

represent to ourselves the straight line? First of all, we can not

represent it to ourselves in geometric space, but only in representative

space, and then we can represent to ourselves just as well the objects

which possess the other properties of the straight line, save that of

satisfying Euclid's postulate. These objects are 'the non-Euclidean

straights,' which from a certain point of view are not entities void of

sense, but circles (true circles of true space) orthogonal to a certain

sphere. If, among these objects equally capable of representation, it is

the first (the Euclidean straights) which we call straights, and not the

latter (the non-Euclidean straights), this is properly by definition.

And arriving finally at the third example, the definition of phosphorus,

we see the true definition would be: Phosphorus is the bit of matter I

see in yonder flask.

XII

And since I am on this subject, still another word. Of the phosphorus

example I said: "This proposition is a real verifiable physical law,

because it means that all bodies having all the other properties of

phosphorus, save its point of fusion, melt like it at 44°." And it was

answered: "No, this law is not verifiable, because if it were shown that

two bodies resembling phosphorus melt one at 44° and the other at 50°,

it might always be said that doubtless, besides the point of fusion,

there is some other unknown property by which they differ."

That was not quite what I meant to say. I should have written, "All

bodies possessing such and such properties finite in number (to wit, the

properties of phosphorus stated in the books on chemistry, the

fusion-point excepted) melt at 44°."

And the better to make evident the difference between the case of the

straight and that of phosphorus, one more remark. The straight has in

nature many images more or less imperfect, of which the chief are the

light rays and the rotation axis of the solid. Suppose we find the ray

of light does not satisfy Euclid's postulate (for example by showing

that a star has a negative parallax), what shall we do? Shall we

conclude that the straight being by definition the trajectory of light

does not satisfy the postulate, or, on the other hand, that the straight

by definition satisfying the postulate, the ray of light is not

straight?

Assuredly we are free to adopt the one or the other definition and

consequently the one or the other conclusion; but to adopt the first

would be stupid, because the ray of light probably satisfies only

imperfectly not merely Euclid's postulate, but the other properties of

the straight line, so that if it deviates from the Euclidean straight,

it deviates no less from the rotation axis of solids which is another

imperfect image of the straight line; while finally it is doubtless

subject to change, so that such a line which yesterday was straight will

cease to be straight to-morrow if some physical circumstance has

changed.

Suppose now we find that phosphorus does not melt at 44°, but at 43.9°.

Shall we conclude that phosphorus being by definition that which melts

at 44°, this body that we did call phosphorus is not true phosphorus,

or, on the other hand, that phosphorous melts at 43.9°? Here again we

are free to adopt the one or the other definition and consequently the

one or the other conclusion; but to adopt the first would be stupid

because we can not be changing the name of a substance every time we

determine a new decimal of its fusion-point.

XIII

To sum up, Russell and Hilbert have each made a vigorous effort; they

have each written a work full of original views, profound and often well

warranted. These two works give us much to think about and we have much

to learn from them. Among their results, some, many even, are solid and

destined to live.

But to say that they have finally settled the debate between Kant and

Leibnitz and ruined the Kantian theory of mathematics is evidently

incorrect. I do not know whether they really believed they had done it,

but if they believed so, they deceived themselves.

CHAPTER V

THE LATEST EFFORTS OF THE LOGISTICIANS

I

The logicians have attempted to answer the preceding considerations. For

that, a transformation of logistic was necessary, and Russell in

particular has modified on certain points his original views. Without

entering into the details of the debate, I should like to return to the

two questions to my mind most important: Have the rules of logistic

demonstrated their fruitfulness and infallibility? Is it true they

afford means of proving the principle of complete induction without any

appeal to intuition?

II

_The Infallibility of Logistic_

On the question of fertility, it seems M. Couturat has naïve illusions.

Logistic, according to him, lends invention 'stilts and wings,' and on

the next page: "_Ten years ago_, Peano published the first edition of

his _Formulaire_." How is that, ten years of wings and not to have

flown!

I have the highest esteem for Peano, who has done very pretty things

(for instance his 'space-filling curve,' a phrase now discarded); but

after all he has not gone further nor higher nor quicker than the

majority of wingless mathematicians, and would have done just as well

with his legs.

On the contrary I see in logistic only shackles for the inventor. It is

no aid to conciseness--far from it, and if twenty-seven equations were

necessary to establish that 1 is a number, how many would be needed to

prove a real theorem? If we distinguish, with Whitehead, the individual

_x_, the class of which the only member is _x_ and which shall be called

[iota]_x_, then the class of which the only member is the class of which

the only member is _x_ and which shall be called [mu]_x_, do you think

these distinctions, useful as they may be, go far to quicken our pace?

Logistic forces us to say all that is ordinarily left to be understood;

it makes us advance step by step; this is perhaps surer but not quicker.

It is not wings you logisticians give us, but leading-strings. And then

we have the right to require that these leading-strings prevent our

falling. This will be their only excuse. When a bond does not bear much

interest, it should at least be an investment for a father of a family.

Should your rules be followed blindly? Yes, else only intuition could

enable us to distinguish among them; but then they must be infallible;

for only in an infallible authority can one have a blind confidence.

This, therefore, is for you a necessity. Infallible you shall be, or not

at all.

You have no right to say to us: "It is true we make mistakes, but so do

you." For us to blunder is a misfortune, a very great misfortune; for

you it is death.

Nor may you ask: Does the infallibility of arithmetic prevent errors in

addition? The rules of calculation are infallible, and yet we see those

blunder _who do not apply these rules_; but in checking their

calculation it is at once seen where they went wrong. Here it is not at

all the case; the logicians _have applied_ their rules, and they have

fallen into contradiction; and so true is this, that they are preparing

to change these rules and to "sacrifice the notion of class." Why change

them if they were infallible?

"We are not obliged," you say, "to solve _hic et nunc_ all possible

problems." Oh, we do not ask so much of you. If, in face of a problem,

you would give _no_ solution, we should have nothing to say; but on the

contrary you give us _two_ of them and those contradictory, and

consequently at least one false; this it is which is failure.

Russell seeks to reconcile these contradictions, which can only be done,

according to him, "by restricting or even sacrificing the notion of

class." And M. Couturat, discovering the success of his attempt, adds:

"If the logicians succeed where others have failed, M. Poincaré will

remember this phrase, and give the honor of the solution to logistic."

But no! Logistic exists, it has its code which has already had four

editions; or rather this code is logistic itself. Is Mr. Russell

preparing to show that one at least of the two contradictory reasonings

has transgressed the code? Not at all; he is preparing to change these

laws and to abrogate a certain number of them. If he succeeds, I shall

give the honor of it to Russell's intuition and not to the Peanian

logistic which he will have destroyed.

III

_The Liberty of Contradiction_

I made two principal objections to the definition of whole number

adopted in logistic. What says M. Couturat to the first of these

objections?

What does the word _exist_ mean in mathematics? It means, I said, to be

free from contradiction. This M. Couturat contests. "Logical existence,"

says he, "is quite another thing from the absence of contradiction. It

consists in the fact that a class is not empty." To say: _a_'s exist,

is, by definition, to affirm that the class _a_ is not null.

And doubtless to affirm that the class _a_ is not null, is, by

definition, to affirm that _a_'s exist. But one of the two affirmations

is as denuded of meaning as the other, if they do not both signify,

either that one may see or touch _a_'s which is the meaning physicists

or naturalists give them, or that one may conceive an _a_ without being

drawn into contradictions, which is the meaning given them by logicians

and mathematicians.

For M. Couturat, "it is not non-contradiction that proves existence, but

it is existence that proves non-contradiction." To establish the

existence of a class, it is necessary therefore to establish, by an

_example_, that there is an individual belonging to this class: "But, it

will be said, how is the existence of this individual proved? Must not

this existence be established, in order that the existence of the class

of which it is a part may be deduced? Well, no; however paradoxical may

appear the assertion, we never demonstrate the existence of an

individual. Individuals, just because they are individuals, are always

considered as existent.... We never have to express that an individual

exists, absolutely speaking, but only that it exists in a class." M.

Couturat finds his own assertion paradoxical, and he will certainly not

be the only one. Yet it must have a meaning. It doubtless means that the

existence of an individual, alone in the world, and of which nothing is

affirmed, can not involve contradiction; in so far as it is all alone it

evidently will not embarrass any one. Well, so let it be; we shall admit

the existence of the individual, 'absolutely speaking,' but nothing

more. It remains to prove the existence of the individual 'in a class,'

and for that it will always be necessary to prove that the affirmation,

"Such an individual belongs to such a class," is neither contradictory

in itself, nor to the other postulates adopted.

"It is then," continues M. Couturat, "arbitrary and misleading to

maintain that a definition is valid only if we first prove it is not

contradictory." One could not claim in prouder and more energetic terms

the liberty of contradiction. "In any case, the _onus probandi_ rests

upon those who believe that these principles are contradictory."

Postulates are presumed to be compatible until the contrary is proved,

just as the accused person is presumed innocent. Needless to add that I

do not assent to this claim. But, you say, the demonstration you require

of us is impossible, and you can not ask us to jump over the moon.

Pardon me; that is impossible for you, but not for us, who admit the

principle of induction as a synthetic judgment _a priori_. And that

would be necessary for you, as for us.

To demonstrate that a system of postulates implies no contradiction, it

is necessary to apply the principle of complete induction; this mode of

reasoning not only has nothing 'bizarre' about it, but it is the only

correct one. It is not 'unlikely' that it has ever been employed; and it

is not hard to find 'examples and precedents' of it. I have cited two

such instances borrowed from Hilbert's article. He is not the only one

to have used it, and those who have not done so have been wrong. What I

have blamed Hilbert for is not his having recourse to it (a born

mathematician such as he could not fail to see a demonstration was

necessary and this the only one possible), but his having recourse

without recognizing the reasoning by recurrence.

IV

_The Second Objection_

I pointed out a second error of logistic in Hilbert's article. To-day

Hilbert is excommunicated and M. Couturat no longer regards him as of

the logistic cult; so he asks if I have found the same fault among the

orthodox. No, I have not seen it in the pages I have read; I know not

whether I should find it in the three hundred pages they have written

which I have no desire to read.

Only, they must commit it the day they wish to make any application of

mathematics. This science has not as sole object the eternal

contemplation of its own navel; it has to do with nature and some day it

will touch it. Then it will be necessary to shake off purely verbal

definitions and to stop paying oneself with words.

To go back to the example of Hilbert: always the point at issue is

reasoning by recurrence and the question of knowing whether a system of

postulates is not contradictory. M. Couturat will doubtless say that

then this does not touch him, but it perhaps will interest those who do

not claim, as he does, the liberty of contradiction.

We wish to establish, as above, that we shall never encounter

contradiction after any number of deductions whatever, provided this

number be finite. For that, it is necessary to apply the principle of

induction. Should we here understand by finite number every number to

which by definition the principle of induction applies? Evidently not,

else we should be led to most embarrassing consequences. To have the

right to lay down a system of postulates, we must be sure they are not

contradictory. This is a truth admitted by _most_ scientists; I should

have written _by all_ before reading M. Couturat's last article. But

what does this signify? Does it mean that we must be sure of not meeting

contradiction after a _finite_ number of propositions, the _finite_

number being by definition that which has all properties of recurrent

nature, so that if one of these properties fails--if, for instance, we

come upon a contradiction--we shall agree to say that the number in

question is not finite? In other words, do we mean that we must be sure

not to meet contradictions, on condition of agreeing to stop just when

we are about to encounter one? To state such a proposition is enough to

condemn it.

So, Hilbert's reasoning not only assumes the principle of induction, but

it supposes that this principle is given us not as a simple definition,

but as a synthetic judgment _a priori_.

To sum up:

A demonstration is necessary.

The only demonstration possible is the proof by recurrence.

This is legitimate only if we admit the principle of induction and if we

regard it not as a definition but as a synthetic judgment.

V

_The Cantor Antinomies_

Now to examine Russell's new memoir. This memoir was written with the

view to conquer the difficulties raised by those Cantor antinomies to

which frequent allusion has already been made. Cantor thought he could

construct a science of the infinite; others went on in the way he

opened, but they soon ran foul of strange contradictions. These

antinomies are already numerous, but the most celebrated are:

1. The Burali-Forti antinomy;

2. The Zermelo-König antinomy;

3. The Richard antinomy.

Cantor proved that the ordinal numbers (the question is of transfinite

ordinal numbers, a new notion introduced by him) can be ranged in a

linear series; that is to say that of two unequal ordinals one is always

less than the other. Burali-Forti proves the contrary; and in fact he

says in substance that if one could range _all_ the ordinals in a linear

series, this series would define an ordinal greater than _all_ the

others; we could afterwards adjoin 1 and would obtain again an ordinal

which would be _still greater_, and this is contradictory.

We shall return later to the Zermelo-König antinomy which is of a

slightly different nature. The Richard antinomy[15] is as follows:

Consider all the decimal numbers definable by a finite number of words;

these decimal numbers form an aggregate _E_, and it is easy to see that

this aggregate is countable, that is to say we can _number_ the

different decimal numbers of this assemblage from 1 to infinity. Suppose

the numbering effected, and define a number _N_ as follows: If the _n_th

decimal of the _n_th number of the assemblage _E_ is

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

the _n_th decimal of _N_ shall be:

1, 2, 3, 4, 5, 6, 7, 8, 1, 1

[15] _Revue générale des sciences_, June 30, 1905.

As we see, _N_ is not equal to the _n_th number of _E_, and as _n_ is

arbitrary, _N_ does not appertain to _E_ and yet _N_ should belong to

this assemblage since we have defined it with a finite number of words.

We shall later see that M. Richard has himself given with much sagacity

the explanation of his paradox and that this extends, _mutatis

mutandis_, to the other like paradoxes. Again, Russell cites another

quite amusing paradox: _What is the least whole number which can not be

defined by a phrase composed of less than a hundred English words_?

This number exists; and in fact the numbers capable of being defined by

a like phrase are evidently finite in number since the words of the

English language are not infinite in number. Therefore among them will

be one less than all the others. And, on the other hand, this number

does not exist, because its definition implies contradiction. This

number, in fact, is defined by the phrase in italics which is composed

of less than a hundred English words; and by definition this number

should not be capable of definition by a like phrase.

VI

_Zigzag Theory and No-class Theory_

What is Mr. Russell's attitude in presence of these contradictions?

After having analyzed those of which we have just spoken, and cited

still others, after having given them a form recalling Epimenides, he

does not hesitate to conclude: "A propositional function of one

variable does not always determine a class." A propositional function

(that is to say a definition) does not always determine a class. A

'propositional function' or 'norm' may be 'non-predicative.' And this

does not mean that these non-predicative propositions determine an empty

class, a null class; this does not mean that there is no value of x

satisfying the definition and capable of being one of the elements of

the class. The elements exist, but they have no right to unite in a

syndicate to form a class.

But this is only the beginning and it is needful to know how to

recognize whether a definition is or is not predicative. To solve this

problem Russell hesitates between three theories which he calls

A. The zigzag theory;

B. The theory of limitation of size;

C. The no-class theory.

According to the zigzag theory "definitions (propositional functions)

determine a class when they are very simple and cease to do so only when

they are complicated and obscure." Who, now, is to decide whether a

definition may be regarded as simple enough to be acceptable? To this

question there is no answer, if it be not the loyal avowal of a complete

inability: "The rules which enable us to recognize whether these

definitions are predicative would be extremely complicated and can not

commend themselves by any plausible reason. This is a fault which might

be remedied by greater ingenuity or by using distinctions not yet

pointed out. But hitherto in seeking these rules, I have not been able

to find any other directing principle than the absence of

contradiction."

This theory therefore remains very obscure; in this night a single

light--the word zigzag. What Russell calls the 'zigzaginess' is

doubtless the particular characteristic which distinguishes the argument

of Epimenides.

According to the theory of limitation of size, a class would cease to

have the right to exist if it were too extended. Perhaps it might be

infinite, but it should not be too much so. But we always meet again the

same difficulty; at what precise moment does it begin to be too much

so? Of course this difficulty is not solved and Russell passes on to the

third theory.

In the no-classes theory it is forbidden to speak the word 'class' and

this word must be replaced by various periphrases. What a change for

logistic which talks only of classes and classes of classes! It becomes

necessary to remake the whole of logistic. Imagine how a page of

logistic would look upon suppressing all the propositions where it is a

question of class. There would only be some scattered survivors in the

midst of a blank page. _Apparent rari nantes in gurgite vasto._

Be that as it may, we see how Russell hesitates and the modifications to

which he submits the fundamental principles he has hitherto adopted.

Criteria are needed to decide whether a definition is too complex or too

extended, and these criteria can only be justified by an appeal to

intuition.

It is toward the no-classes theory that Russell finally inclines. Be

that as it may, logistic is to be remade and it is not clear how much of

it can be saved. Needless to add that Cantorism and logistic are alone

under consideration; real mathematics, that which is good for something,

may continue to develop in accordance with its own principles without

bothering about the storms which rage outside it, and go on step by step

with its usual conquests which are final and which it never has to

abandon.

VII

_The True Solution_

What choice ought we to make among these different theories? It seems to

me that the solution is contained in a letter of M. Richard of which I

have spoken above, to be found in the _Revue générale des sciences_ of

June 30, 1905. After having set forth the antinomy we have called

Richard's antinomy, he gives its explanation. Recall what has already

been said of this antinomy. _E_ is the aggregate of _all_ the numbers

definable by a finite number of words, _without introducing the notion

of the aggregate E itself_. Else the definition of _E_ would contain a

vicious circle; we must not define _E_ by the aggregate _E_ itself.

Now we have defined _N_ with a finite number of words, it is true, but

with the aid of the notion of the aggregate _E_. And this is why _N_ is

not part of _E_. In the example selected by M. Richard, the conclusion

presents itself with complete evidence and the evidence will appear

still stronger on consulting the text of the letter itself. But the same

explanation holds good for the other antinomies, as is easily verified.

Thus _the definitions which should be regarded as not predicative are

those which contain a vicious circle_. And the preceding examples

sufficiently show what I mean by that. Is it this which Russell calls

the 'zigzaginess'? I put the question without answering it.

VIII

_The Demonstrations of the Principle of Induction_

Let us now examine the pretended demonstrations of the principle of

induction and in particular those of Whitehead and of Burali-Forti.

We shall speak of Whitehead's first, and take advantage of certain new

terms happily introduced by Russell in his recent memoir. Call

_recurrent class_ every class containing zero, and containing _n_ + 1 if

it contains _n_. Call _inductive number_ every number which is a part of

_all_ the recurrent classes. Upon what condition will this latter

definition, which plays an essential rôle in Whitehead's proof, be

'predicative' and consequently acceptable?

In accordance with what has been said, it is necessary to understand by

_all_ the recurrent classes, all those in whose definition the notion of

inductive number does not enter. Else we fall again upon the vicious

circle which has engendered the antinomies.

Now _Whitehead has not taken this precaution_. Whitehead's reasoning is

therefore fallacious; it is the same which led to the antinomies. It was

illegitimate when it gave false results; it remains illegitimate when by

chance it leads to a true result.

A definition containing a vicious circle defines nothing. It is of no

use to say, we are sure, whatever meaning we may give to our definition,

zero at least belongs to the class of inductive numbers; it is not a

question of knowing whether this class is void, but whether it can be

rigorously deliminated. A 'non-predicative' class is not an empty

class, it is a class whose boundary is undetermined. Needless to add

that this particular objection leaves in force the general objections

applicable to all the demonstrations.

IX

Burali-Forti has given another demonstration.[16] But he is obliged to

assume two postulates: First, there always exists at least one infinite

class. The second is thus expressed:

u[epsilon]K(K - [iota][Lambda]) · [inverted c] · u < v'u.

The first postulate is not more evident than the principle to be proved.

The second not only is not evident, but it is false, as Whitehead has

shown; as moreover any recruit would see at the first glance, if the

axiom had been stated in intelligible language, since it means that the

number of combinations which can be formed with several objects is less

than the number of these objects.

[16] In his article 'Le classi finite,' _Atti di Torino_, Vol. XXXII.

X

_Zermelo's Assumption_

A famous demonstration by Zermelo rests upon the following assumption:

In any aggregate (or the same in each aggregate of an assemblage of

aggregates) we can always choose _at random_ an element (even if this

assemblage of aggregates should contain an infinity of aggregates). This

assumption had been applied a thousand times without being stated, but,

once stated, it aroused doubts. Some mathematicians, for instance

M. Borel, resolutely reject it; others admire it. Let us see what,

according to his last article, Russell thinks of it. He does not speak

out, but his reflections are very suggestive.

And first a picturesque example: Suppose we have as many pairs of shoes

as there are whole numbers, and so that we can number _the pairs_ from

one to infinity, how many shoes shall we have? Will the number of shoes

be equal to the number of pairs? Yes, if in each pair the right shoe is

distinguishable from the left; it will in fact suffice to give the

number 2_n_ - 1 to the right shoe of the _n_th pair, and the number 2_n_

to the left shoe of the _n_th pair. No, if the right shoe is just like

the left, because a similar operation would become impossible--unless we

admit Zermelo's assumption, since then we could choose _at random_ in

each pair the shoe to be regarded as the right.

XI

_Conclusions_

A demonstration truly founded upon the principles of analytic logic will

be composed of a series of propositions. Some, serving as premises, will

be identities or definitions; the others will be deduced from the

premises step by step. But though the bond between each proposition and

the following is immediately evident, it will not at first sight appear

how we get from the first to the last, which we may be tempted to regard

as a new truth. But if we replace successively the different expressions

therein by their definition and if this operation be carried as far as

possible, there will finally remain only identities, so that all will

reduce to an immense tautology. Logic therefore remains sterile unless

made fruitful by intuition.

This I wrote long ago; logistic professes the contrary and thinks it has

proved it by actually proving new truths. By what mechanism? Why in

applying to their reasonings the procedure just described--namely,

replacing the terms defined by their definitions--do we not see them

dissolve into identities like ordinary reasonings? It is because this

procedure is not applicable to them. And why? Because their definitions

are not predicative and present this sort of hidden vicious circle which

I have pointed out above; non-predicative definitions can not be

substituted for the terms defined. Under these conditions _logistic is

not sterile, it engenders antinomies_.

It is the belief in the existence of the actual infinite which has given

birth to those non-predicative definitions. Let me explain. In these

definitions the word 'all' figures, as is seen in the examples cited

above. The word 'all' has a very precise meaning when it is a question

of a finite number of objects; to have another one, when the objects are

infinite in number, would require there being an actual (given complete)

infinity. Otherwise _all_ these objects could not be conceived as

postulated anteriorly to their definition, and then if the definition of

a notion _N_ depends upon _all_ the objects _A_, it may be infected with

a vicious circle, if among the objects _A_ are some indefinable without

the intervention of the notion _N_ itself.

The rules of formal logic express simply the properties of all possible

classifications. But for them to be applicable it is necessary that

these classifications be immutable and that we have no need to modify

them in the course of the reasoning. If we have to classify only a

finite number of objects, it is easy to keep our classifications without

change. If the objects are _indefinite_ in number, that is to say if one

is constantly exposed to seeing new and unforeseen objects arise, it may

happen that the appearance of a new object may require the

classification to be modified, and thus it is we are exposed to

antinomies. _There is no actual (given complete) infinity._ The

Cantorians have forgotten this, and they have fallen into contradiction.

It is true that Cantorism has been of service, but this was when applied

to a real problem whose terms were precisely defined, and then we could

advance without fear.

Logistic also forgot it, like the Cantorians, and encountered the same

difficulties. But the question is to know whether they went this way by

accident or whether it was a necessity for them. For me, the question is

not doubtful; belief in an actual infinity is essential in the Russell

logic. It is just this which distinguishes it from the Hilbert logic.

Hilbert takes the view-point of extension, precisely in order to avoid

the Cantorian antinomies. Russell takes the view-point of comprehension.

Consequently for him the genus is anterior to the species, and the

_summum genus_ is anterior to all. That would not be inconvenient if the

_summum genus_ was finite; but if it is infinite, it is necessary to

postulate the infinite, that is to say to regard the infinite as actual

(given complete). And we have not only infinite classes; when we pass

from the genus to the species in restricting the concept by new

conditions, these conditions are still infinite in number. Because they

express generally that the envisaged object presents such or such a

relation with all the objects of an infinite class.

But that is ancient history. Russell has perceived the peril and takes

counsel. He is about to change everything, and, what is easily

understood, he is preparing not only to introduce new principles which

shall allow of operations formerly forbidden, but he is preparing to

forbid operations he formerly thought legitimate. Not content to adore

what he burned, he is about to burn what he adored, which is more

serious. He does not add a new wing to the building, he saps its

foundation.

The old logistic is dead, so much so that already the zigzag theory and

the no-classes theory are disputing over the succession. To judge of the

new, we shall await its coming.

BOOK III

THE NEW MECHANICS

CHAPTER I

MECHANICS AND RADIUM

I

_Introduction_

The general principles of Dynamics, which have, since Newton, served as

foundation for physical science, and which appeared immovable, are they

on the point of being abandoned or at least profoundly modified? This is

what many people have been asking themselves for some years. According

to them, the discovery of radium has overturned the scientific dogmas we

believed the most solid: on the one hand, the impossibility of the

transmutation of metals; on the other hand, the fundamental postulates

of mechanics.

Perhaps one is too hasty in considering these novelties as finally

established, and breaking our idols of yesterday; perhaps it would be

proper, before taking sides, to await experiments more numerous and more

convincing. None the less is it necessary, from to-day, to know the new

doctrines and the arguments, already very weighty, upon which they rest.

In few words let us first recall in what those principles consist:

_A._ The motion of a material point isolated and apart from all exterior

force is straight and uniform; this is the principle of inertia: without

force no acceleration;

_B._ The acceleration of a moving point has the same direction as the

resultant of all the forces to which it is subjected; it is equal to the

quotient of this resultant by a coefficient called _mass_ of the moving

point.

The mass of a moving point, so defined, is a constant; it does not

depend upon the velocity acquired by this point; it is the same whether

the force, being parallel to this velocity, tends only to accelerate or

to retard the motion of the point, or whether, on the contrary, being

perpendicular to this velocity, it tends to make this motion deviate

toward the right, or the left, that is to say to _curve_ the trajectory;

_C._ All the forces affecting a material point come from the action of

other material points; they depend only upon the _relative_ positions

and velocities of these different material points.

Combining the two principles _B_ and _C_, we reach the _principle of

relative motion_, in virtue of which the laws of the motion of a system

are the same whether we refer this system to fixed axes, or to moving

axes animated by a straight and uniform motion of translation, so that

it is impossible to distinguish absolute motion from a relative motion

with reference to such moving axes;

_D._ If a material point _A_ acts upon another material point _B_, the

body _B_ reacts upon _A_, and these two actions are two equal and

directly opposite forces. This is _the principle of the equality of

action and reaction_, or, more briefly, the _principle of reaction_.

Astronomic observations and the most ordinary physical phenomena seem to

have given of these principles a confirmation complete, constant and

very precise. This is true, it is now said, but it is because we have

never operated with any but very small velocities; Mercury, for example,

the fastest of the planets, goes scarcely 100 kilometers a second. Would

this planet act the same if it went a thousand times faster? We see

there is yet no need to worry; whatever may be the progress of

automobilism, it will be long before we must give up applying to our

machines the classic principles of dynamics.

How then have we come to make actual speeds a thousand times greater

than that of Mercury, equal, for instance, to a tenth or a third of the

velocity of light, or approaching still more closely to that velocity?

It is by aid of the cathode rays and the rays from radium.

We know that radium emits three kinds of rays, designated by the three

Greek letters [alpha], [beta], [gamma]; in what follows, unless the

contrary be expressly stated, it will always be a question of the [beta]

rays, which are analogous to the cathode rays.

After the discovery of the cathode rays two theories appeared. Crookes

attributed the phenomena to a veritable molecular bombardment; Hertz, to

special undulations of the ether. This was a renewal of the debate which

divided physicists a century ago about light; Crookes took up the

emission theory, abandoned for light; Hertz held to the undulatory

theory. The facts seem to decide in favor of Crookes.

It has been recognized, in the first place, that the cathode rays carry

with them a negative electric charge; they are deviated by a magnetic

field and by an electric field; and these deviations are precisely such

as these same fields would produce upon projectiles animated by a very

high velocity and strongly charged with electricity. These two

deviations depend upon two quantities: one the velocity, the other the

relation of the electric charge of the projectile to its mass; we cannot

know the absolute value of this mass, nor that of the charge, but only

their relation; in fact, it is clear that if we double at the same time

the charge and the mass, without changing the velocity, we shall double

the force which tends to deviate the projectile, but, as its mass is

also doubled, the acceleration and deviation observable will not be

changed. The observation of the two deviations will give us therefore

two equations to determine these two unknowns. We find a velocity of

from 10,000 to 30,000 kilometers a second; as to the ratio of the charge

to the mass, it is very great. We may compare it to the corresponding

ratio in regard to the hydrogen ion in electrolysis; we then find that a

cathodic projectile carries about a thousand times more electricity than

an equal mass of hydrogen would carry in an electrolyte.

To confirm these views, we need a direct measurement of this velocity to

compare with the velocity so calculated. Old experiments of J. J.

Thomson had given results more than a hundred times too small; but they

were exposed to certain causes of error. The question was taken up again

by Wiechert in an arrangement where the Hertzian oscillations were

utilized; results were found agreeing with the theory, at least as to

order of magnitude; it would be of great interest to repeat these

experiments. However that may be, the theory of undulations appears

powerless to account for this complex of facts.

The same calculations made with reference to the [beta] rays of radium

have given velocities still greater: 100,000 or 200,000 kilometers or

more yet. These velocities greatly surpass all those we know. It is true

that light has long been known to go 300,000 kilometers a second; but it

is not a carrying of matter, while, if we adopt the emission theory for

the cathode rays, there would be material molecules really impelled at

the velocities in question, and it is proper to investigate whether the

ordinary laws of mechanics are still applicable to them.

II

_Mass Longitudinal and Mass Transversal_

We know that electric currents produce the phenomena of induction, in

particular _self-induction_. When a current increases, there develops an

electromotive force of self-induction which tends to oppose the current;

on the contrary, when the current decreases, the electromotive force of

self-induction tends to maintain the current. The self-induction

therefore opposes every variation of the intensity of the current, just

as in mechanics the inertia of a body opposes every variation of its

velocity.

_Self-induction is a veritable inertia._ Everything happens as if the

current could not establish itself without putting in motion the

surrounding ether and as if the inertia of this ether tended, in

consequence, to keep constant the intensity of this current. It would be

requisite to overcome this inertia to establish the current, it would be

necessary to overcome it again to make the current cease.

A cathode ray, which is a rain of projectiles charged with negative

electricity, may be likened to a current; doubtless this current

differs, at first sight at least, from the currents of ordinary

conduction, where the matter does not move and where the electricity

circulates through the matter. This is a _current of convection_, where

the electricity, attached to a material vehicle, is carried along by the

motion of this vehicle. But Rowland has proved that currents of

convection produce the same magnetic effects as currents of conduction;

they should produce also the same effects of induction. First, if this

were not so, the principle of the conservation of energy would be

violated; besides, Crémieu and Pender have employed a method putting in

evidence _directly_ these effects of induction.

If the velocity of a cathode corpuscle varies, the intensity of the

corresponding current will likewise vary; and there will develop effects

of self-induction which will tend to oppose this variation. These

corpuscles should therefore possess a double inertia: first their own

proper inertia, and then the apparent inertia, due to self-induction,

which produces the same effects. They will therefore have a total

apparent mass, composed of their real mass and of a fictitious mass of

electromagnetic origin. Calculation shows that this fictitious mass

varies with the velocity, and that the force of inertia of

self-induction is not the same when the velocity of the projectile

accelerates or slackens, or when it is deviated; therefore so it is with

the force of the total apparent inertia.

The total apparent mass is therefore not the same when the real force

applied to the corpuscle is parallel to its velocity and tends to

accelerate the motion as when it is perpendicular to this velocity and

tends to make the direction vary. It is necessary therefore to

distinguish the _total longitudinal mass_ from the _total transversal

mass_. These two total masses depend, moreover, upon the velocity. This

follows from the theoretical work of Abraham.

In the measurements of which we speak in the preceding section, what is

it we determine in measuring the two deviations? It is the velocity on

the one hand, and on the other hand the ratio of the charge to the

_total transversal mass_. How, under these conditions, can we make out

in this total mass the part of the real mass and that of the fictitious

electromagnetic mass? If we had only the cathode rays properly so

called, it could not be dreamed of; but happily we have the rays of

radium which, as we have seen, are notably swifter. These rays are not

all identical and do not behave in the same way under the action of an

electric field and a magnetic field. It is found that the electric

deviation is a function of the magnetic deviation, and we are able, by

receiving on a sensitive plate radium rays which have been subjected to

the action of the two fields, to photograph the curve which represents

the relation between these two deviations. This is what Kaufmann has

done, deducing from it the relation between the velocity and the ratio

of the charge to the total apparent mass, a ratio we shall call

[epsilon].

One might suppose there are several species of rays, each characterized

by a fixed velocity, by a fixed charge and by a fixed mass. But this

hypothesis is improbable; why, in fact, would all the corpuscles of the

same mass take always the same velocity? It is more natural to suppose

that the charge as well as the _real_ mass are the same for all the

projectiles, and that these differ only by their velocity. If the ratio

[epsilon] is a function of the velocity, this is not because the real

mass varies with this velocity; but, since the fictitious

electromagnetic mass depends upon this velocity, the total apparent

mass, alone observable, must depend upon it, though the real mass does

not depend upon it and may be constant.

The calculations of Abraham let us know the law according to which the

_fictitious_ mass varies as a function of the velocity; Kaufmann's

experiment lets us know the law of variation of the _total_ mass.

The comparison of these two laws will enable us therefore to determine

the ratio of the real mass to the total mass.

Such is the method Kaufmann used to determine this ratio. The result is

highly surprising: _the real mass is naught_.

This has led to conceptions wholly unexpected. What had only been proved

for cathode corpuscles was extended to all bodies. What we call mass

would be only semblance; all inertia would be of electromagnetic origin.

But then mass would no longer be constant, it would augment with the

velocity; sensibly constant for velocities up to 1,000 kilometers a

second, it then would increase and would become infinite for the

velocity of light. The transversal mass would no longer be equal to the

longitudinal: they would only be nearly equal if the velocity is not too

great. The principle _B_ of mechanics would no longer be true.

III

_The Canal Rays_

At the point where we now are, this conclusion might seem premature. Can

one apply to all matter what has been proved only for such light

corpuscles, which are a mere emanation of matter and perhaps not true

matter? But before entering upon this question, a word must be said of

another sort of rays. I refer to the _canal rays_, the _Kanalstrahlen_

of Goldstein.

The cathode, together with the cathode rays charged with negative

electricity, emits canal rays charged with positive electricity. In

general, these canal rays not being repelled by the cathode, are

confined to the immediate neighborhood of this cathode, where they

constitute the `chamois cushion,' not very easy to perceive; but, if the

cathode is pierced with holes and if it almost completely blocks up the

tube, the canal rays spread _back_ of the cathode, in the direction

opposite to that of the cathode rays, and it becomes possible to study

them. It is thus that it has been possible to show their positive charge

and to show that the magnetic and electric deviations still exist, as

for the cathode rays, but are much feebler.

Radium likewise emits rays analogous to the canal rays, and relatively

very absorbable, called [alpha] rays.

We can, as for the cathode rays, measure the two deviations and thence

deduce the velocity and the ratio [epsilon]. The results are less

constant than for the cathode rays, but the velocity is less, as well as

the ratio [epsilon]; the positive corpuscles are less charged than the

negative; or if, which is more natural, we suppose the charges equal and

of opposite sign, the positive corpuscles are much the larger. These

corpuscles, charged the ones positively, the others negatively, have

been called _electrons_.

IV

_The Theory of Lorentz_

But the electrons do not merely show us their existence in these rays

where they are endowed with enormous velocities. We shall see them in

very different rôles, and it is they that account for the principal

phenomena of optics and electricity. The brilliant synthesis about to be

noticed is due to Lorentz.

Matter is formed solely of electrons carrying enormous charges, and, if

it seems to us neutral, this is because the charges of opposite sign of

these electrons compensate each other. We may imagine, for example, a

sort of solar system formed of a great positive electron, around which

gravitate numerous little planets, the negative electrons, attracted by

the electricity of opposite name which charges the central electron. The

negative charges of these planets would balance the positive charge of

this sun, so that the algebraic sum of all these charges would be

naught.

All these electrons swim in the ether. The ether is everywhere

identically the same, and perturbations in it are propagated according

to the same laws as light or the Hertzian oscillations _in vacuo_. There

is nothing but electrons and ether. When a luminous wave enters a part

of the ether where electrons are numerous, these electrons are put in

motion under the influence of the perturbation of the ether, and they

then react upon the ether. So would be explained refraction, dispersion,

double refraction and absorption. Just so, if for any cause an electron

be put in motion, it would trouble the ether around it and would give

rise to luminous waves, and this would explain the emission of light by

incandescent bodies.

In certain bodies, the metals for example, we should have fixed

electrons, between which would circulate moving electrons enjoying

perfect liberty, save that of going out from the metallic body and

breaking the surface which separates it from the exterior void or from

the air, or from any other non-metallic body.

These movable electrons behave then, within the metallic body, as do,

according to the kinetic theory of gases, the molecules of a gas within

the vase where this gas is confined. But, under the influence of a

difference of potential, the negative movable electrons would tend to go

all to one side, and the positive movable electrons to the other. This

is what would produce electric currents, and _this is why these bodies

would be conductors_. On the other hand, the velocities of our electrons

would be the greater the higher the temperature, if we accept the

assimilation with the kinetic theory of gases. When one of these movable

electrons encounters the surface of the metallic body, whose boundary it

can not pass, it is reflected like a billiard ball which has hit the

cushion, and its velocity undergoes a sudden change of direction. But

when an electron changes direction, as we shall see further on, it

becomes the source of a luminous wave, and this is why hot metals are

incandescent.

In other bodies, the dielectrics and the transparent bodies, the movable

electrons enjoy much less freedom. They remain as if attached to fixed

electrons which attract them. The farther they go away from them the

greater becomes this attraction and tends to pull them back. They

therefore can make only small excursions; they can no longer circulate,

but only oscillate about their mean position. This is why these bodies

would not be conductors; moreover they would most often be transparent,

and they would be refractive, since the luminous vibrations would be

communicated to the movable electrons, susceptible of oscillation, and

thence a perturbation would result.

I can not here give the details of the calculations; I confine myself to

saying that this theory accounts for all the known facts, and has

predicted new ones, such as the Zeeman effect.

V

_Mechanical Consequences_

We now may face two hypotheses:

1º The positive electrons have a real mass, much greater than their

fictitious electromagnetic mass; the negative electrons alone lack real

mass. We might even suppose that apart from electrons of the two signs,

there are neutral atoms which have only their real mass. In this case,

mechanics is not affected; there is no need of touching its laws; the

real mass is constant; simply, motions are deranged by the effects of

self-induction, as has always been known; moreover, these perturbations

are almost negligible, except for the negative electrons which, not

having real mass, are not true matter.

2º But there is another point of view; we may suppose there are no

neutral atoms, and the positive electrons lack real mass just as the

negative electrons. But then, real mass vanishing, either the word

_mass_ will no longer have any meaning, or else it must designate the

fictitious electromagnetic mass; in this case, mass will no longer be

constant, the transversal _mass_ will no longer be equal to the

longitudinal, the principles of mechanics will be overthrown.

First a word of explanation. We have said that, for the same charge, the

_total_ mass of a positive electron is much greater than that of a

negative. And then it is natural to think that this difference is

explained by the positive electron having, besides its fictitious mass,

a considerable real mass; which takes us back to the first hypothesis.

But we may just as well suppose that the real mass is null for these as

for the others, but that the fictitious mass of the positive electron is

much the greater since this electron is much the smaller. I say

advisedly: much the smaller. And, in fact, in this hypothesis inertia is

exclusively electromagnetic in origin; it reduces itself to the inertia

of the ether; the electrons are no longer anything by themselves; they

are solely holes in the ether and around which the ether moves; the

smaller these holes are, the more will there be of ether, the greater,

consequently, will be the inertia of the ether.

How shall we decide between these two hypotheses? By operating upon the

canal rays as Kaufmann did upon the [beta] rays? This is impossible; the

velocity of these rays is much too slight. Should each therefore decide

according to his temperament, the conservatives going to one side and

the lovers of the new to the other? Perhaps, but, to fully understand

the arguments of the innovators, other considerations must come in.

CHAPTER II

MECHANICS AND OPTICS

I

_Aberration_

You know in what the phenomenon of aberration, discovered by Bradley,

consists. The light issuing from a star takes a certain time to go

through a telescope; during this time, the telescope, carried along by

the motion of the earth, is displaced. If therefore the telescope were

pointed in the _true_ direction of the star, the image would be formed

at the point occupied by the crossing of the threads of the network when

the light has reached the objective; and this crossing would no longer

be at this same point when the light reached the plane of the network.

We would therefore be led to mis-point the telescope to bring the image

upon the crossing of the threads. Thence results that the astronomer

will not point the telescope in the direction of the absolute velocity

of the light, that is to say toward the true position of the star, but

just in the direction of the relative velocity of the light with

reference to the earth, that is to say toward what is called the

apparent position of the star.

The velocity of light is known; we might therefore suppose that we have

the means of calculating the _absolute_ velocity of the earth. (I shall

soon explain my use here of the word absolute.) Nothing of the sort; we

indeed know the apparent position of the star we observe; but we do not

know its true position; we know the velocity of the light only in

magnitude and not in direction.

If therefore the absolute velocity of the earth were straight and

uniform, we should never have suspected the phenomenon of aberration;

but it is variable; it is composed of two parts: the velocity of the

solar system, which is straight and uniform; the velocity of the earth

with reference to the sun, which is variable. If the velocity of the

solar system, that is to say if the constant part existed alone, the

observed direction would be invariable. This position that one would

thus observe is called the _mean_ apparent position of the star.

Taking account now at the same time of the two parts of the velocity of

the earth, we shall have the actual apparent position, which describes a

little ellipse around the mean apparent position, and it is this ellipse

that we observe.

Neglecting very small quantities, we shall see that the dimensions of

this ellipse depend only upon the ratio of the velocity of the earth

with reference to the sun to the velocity of light, so that the relative

velocity of the earth with regard to the sun has alone come in.

But wait! This result is not exact, it is only approximate; let us push

the approximation a little farther. The dimensions of the ellipse will

depend then upon the absolute velocity of the earth. Let us compare the

major axes of the ellipse for the different stars: we shall have,

theoretically at least, the means of determining this absolute velocity.

That would be perhaps less shocking than it at first seems; it is a

question, in fact, not of the velocity with reference to an absolute

void, but of the velocity with regard to the ether, which is taken _by

definition_ as being absolutely at rest.

Besides, this method is purely theoretical. In fact, the aberration is

very small; the possible variations of the ellipse of aberration are

much smaller yet, and, if we consider the aberration as of the first

order, they should therefore be regarded as of the second order: about a

millionth of a second; they are absolutely inappreciable for our

instruments. We shall finally see, further on, why the preceding theory

should be rejected, and why we could not determine this absolute

velocity even if our instruments were ten thousand times more precise!

One might imagine some other means, and in fact, so one has. The

velocity of light is not the same in water as in air; could we not

compare the two apparent positions of a star seen through a telescope

first full of air, then full of water? The results have been negative;

the apparent laws of reflection and refraction are not altered by the

motion of the earth. This phenomenon is capable of two explanations:

1º It might be supposed that the ether is not at rest, but that it is

carried along by the body in motion. It would then not be astonishing

that the phenomena of refraction are not altered by the motion of the

earth, since all, prisms, telescopes and ether, are carried along

together in the same translation. As to the aberration itself, it would

be explained by a sort of refraction happening at the surface of

separation of the ether at rest in the interstellar spaces and the ether

carried along by the motion of the earth. It is upon this hypothesis

(bodily carrying along of the ether) that is founded the _theory of

Hertz_ on the electrodynamics of moving bodies.

2º Fresnel, on the contrary, supposes that the ether is at absolute rest

in the void, at rest almost absolute in the air, whatever be the

velocity of this air, and that it is partially carried along by

refractive media. Lorentz has given to this theory a more satisfactory

form. For him, the ether is at rest, only the electrons are in motion;

in the void, where it is only a question of the ether, in the air, where

this is almost the case, the carrying along is null or almost null; in

refractive media, where perturbation is produced at the same time by

vibrations of the ether and those of electrons put in swing by the

agitation of the ether, the undulations are _partially_ carried along.

To decide between the two hypotheses, we have Fizeau's experiment,

comparing by measurements of the fringes of interference, the velocity

of light in air at rest or in motion. These experiments have confirmed

Fresnel's hypothesis of partial carrying along. They have been repeated

with the same result by Michelson. _The theory of Hertz must therefore

be rejected._

II

_The Principle of Relativity_

But if the ether is not carried along by the motion of the earth, is it

possible to show, by means of optical phenomena, the absolute velocity

of the earth, or rather its velocity with respect to the unmoving ether?

Experiment has answered negatively, and yet the experimental procedures

have been varied in all possible ways. Whatever be the means employed

there will never be disclosed anything but relative velocities; I mean

the velocities of certain material bodies with reference to other

material bodies. In fact, if the source of light and the apparatus of

observation are on the earth and participate in its motion, the

experimental results have always been the same, whatever be the

orientation of the apparatus with reference to the orbital motion of the

earth. If astronomic aberration happens, it is because the source, a

star, is in motion with reference to the observer.

The hypotheses so far made perfectly account for this general result,

_if we neglect very small quantities of the order of the square of the

aberration_. The explanation rests upon the notion of _local time_,

introduced by Lorentz, which I shall try to make clear. Suppose two

observers, placed one at _A_, the other at _B_, and wishing to set their

watches by means of optical signals. They agree that _B_ shall send a

signal to _A_ when his watch marks an hour determined upon, and _A_ is

to put his watch to that hour the moment he sees the signal. If this

alone were done, there would be a systematic error, because as the light

takes a certain time _t_ to go from _B_ to _A_, _A_'s watch would be

behind _B_'s the time _t_. This error is easily corrected. It suffices

to cross the signals. _A_ in turn must signal _B_, and, after this new

adjustment, _B_'s watch will be behind _A_'s the time _t_. Then it will

be sufficient to take the arithmetic mean of the two adjustments.

But this way of doing supposes that light takes the same time to go from

_A_ to _B_ as to return from _B_ to _A_. That is true if the observers

are motionless; it is no longer so if they are carried along in a common

translation, since then _A_, for example, will go to meet the light

coming from _B_, while _B_ will flee before the light coming from _A_.

If therefore the observers are borne along in a common translation and

if they do not suspect it, their adjustment will be defective; their

watches will not indicate the same time; each will show the _local time_

belonging to the point where it is.

The two observers will have no way of perceiving this, if the unmoving

ether can transmit to them only luminous signals all of the same

velocity, and if the other signals they might send are transmitted by

media carried along with them in their translation. The phenomenon each

observes will be too soon or too late; it would be seen at the same

instant only if the translation did not exist; but as it will be

observed with a watch that is wrong, this will not be perceived and the

appearances will not be altered.

It results from this that the compensation is easy to explain so long as

we neglect the square of the aberration, and for a long time the

experiments were not sufficiently precise to warrant taking account of

it. But the day came when Michelson imagined a much more delicate

procedure: he made rays interfere which had traversed different courses,

after being reflected by mirrors; each of the paths approximating a

meter and the fringes of interference permitting the recognition of a

fraction of a thousandth of a millimeter, the square of the aberration

could no longer be neglected, and _yet the results were still negative_.

Therefore the theory required to be completed, and it has been by the

_Lorentz-Fitzgerald hypothesis_.

These two physicists suppose that all bodies carried along in a

translation undergo a contraction in the sense of this translation,

while their dimensions perpendicular to this translation remain

unchanged. _This contraction is the same for all bodies_; moreover, it

is very slight, about one two-hundred-millionth for a velocity such as

that of the earth. Furthermore our measuring instruments could not

disclose it, even if they were much more precise; our measuring rods in

fact undergo the same contraction as the objects to be measured. If the

meter exactly fits when applied to a body, if we point the body and

consequently the meter in the sense of the motion of the earth, it will

not cease to exactly fit in another orientation, and that although the

body and the meter have changed in length as well as orientation, and

precisely because the change is the same for one as for the other. But

it is quite different if we measure a length, not now with a meter, but

by the time taken by light to pass along it, and this is just what

Michelson has done.

A body, spherical when at rest, will take thus the form of a flattened

ellipsoid of revolution when in motion; but the observer will always

think it spherical, since he himself has undergone an analogous

deformation, as also all the objects serving as points of reference. On

the contrary, the surfaces of the waves of light, remaining rigorously

spherical, will seem to him elongated ellipsoids.

What happens then? Suppose an observer and a source of light carried

along together in the translation: the wave surfaces emanating from the

source will be spheres having as centers the successive positions of the

source; the distance from this center to the actual position of the

source will be proportional to the time elapsed after the emission, that

is to say to the radius of the sphere. All these spheres are therefore

homothetic one to the other, with relation to the actual position _S_ of

the source. But, for our observer, because of the contraction, all these

spheres will seem elongated ellipsoids, and all these ellipsoids will

moreover be homothetic, with reference to the point _S_; the

excentricity of all these ellipsoids is the same and depends solely upon

the velocity of the earth. _We shall so select the law of contraction

that the point S may be at the focus of the meridian section of the

ellipsoid._

This time the compensation is _rigorous_, and this it is which explains

Michelson's experiment.

I have said above that, according to the ordinary theories, observations

of the astronomic aberration would give us the absolute velocity of the

earth, if our instruments were a thousand times more precise. I must

modify this statement. Yes, the observed angles would be modified by the

effect of this absolute velocity, but the graduated circles we use to

measure the angles would be deformed by the translation: they would

become ellipses; thence would result an error in regard to the angle

measured, and _this second error would exactly compensate the first_.

This Lorentz-Fitzgerald hypothesis seems at first very extraordinary;

all we can say for the moment, in its favor, is that it is only the

immediate translation of Michelson's experimental result, if we _define_

lengths by the time taken by light to run along them.

However that may be, it is impossible to escape the impression that the

principle of relativity is a general law of nature, that one will never

be able by any imaginable means to show any but relative velocities, and

I mean by that not only the velocities of bodies with reference to the

ether, but the velocities of bodies with regard to one another. Too many

different experiments have given concordant results for us not to feel

tempted to attribute to this principle of relativity a value comparable

to that, for example, of the principle of equivalence. In any case, it

is proper to see to what consequences this way of looking at things

would lead us and then to submit these consequences to the control of

experiment.

III

_The Principle of Reaction_

Let us see what the principle of the equality of action and reaction

becomes in the theory of Lorentz. Consider an electron _A_ which for any

cause begins to move; it produces a perturbation in the ether; at the

end of a certain time, this perturbation reaches another electron _B_,

which will be disturbed from its position of equilibrium. In these

conditions there can not be equality between action and reaction, at

least if we do not consider the ether, but only the electrons, _which

alone are observable_, since our matter is made of electrons.

In fact it is the electron _A_ which has disturbed the electron _B_;

even in case the electron _B_ should react upon _A_, this reaction could

be equal to the action, but in no case simultaneous, since the electron

_B_ can begin to move only after a certain time, necessary for the

propagation. Submitting the problem to a more exact calculation, we

reach the following result: Suppose a Hertz discharger placed at the

focus of a parabolic mirror to which it is mechanically attached; this

discharger emits electromagnetic waves, and the mirror reflects all

these waves in the same direction; the discharger therefore will radiate

energy in a determinate direction. Well, the calculation shows that _the

discharger recoils_ like a cannon which has shot out a projectile. In

the case of the cannon, the recoil is the natural result of the equality

of action and reaction. The cannon recoils because the projectile upon

which it has acted reacts upon it. But here it is no longer the same.

What has been sent out is no longer a material projectile: it is energy,

and energy has no mass: it has no counterpart. And, in place of a

discharger, we could have considered just simply a lamp with a reflector

concentrating its rays in a single direction.

It is true that, if the energy sent out from the discharger or from the

lamp meets a material object, this object receives a mechanical push as

if it had been hit by a real projectile, and this push will be equal to

the recoil of the discharger and of the lamp, if no energy has been lost

on the way and if the object absorbs the whole of the energy. Therefore

one is tempted to say that there still is compensation between the

action and the reaction. But this compensation, even should it be

complete, is always belated. It never happens if the light, after

leaving its source, wanders through interstellar spaces without ever

meeting a material body; it is incomplete, if the body it strikes is not

perfectly absorbent.

Are these mechanical actions too small to be measured, or are they

accessible to experiment? These actions are nothing other than those due

to the _Maxwell-Bartholi_ pressures; Maxwell had predicted these

pressures from calculations relative to electrostatics and magnetism;

Bartholi reached the same result by thermodynamic considerations.

This is how the _tails of comets_ are explained. Little particles detach

themselves from the nucleus of the comet; they are struck by the light

of the sun, which pushes them back as would a rain of projectiles coming

from the sun. The mass of these particles is so little that this

repulsion sweeps it away against the Newtonian attraction; so in moving

away from the sun they form the tails.

The direct experimental verification was not easy to obtain. The first

endeavor led to the construction of the _radiometer_. But this

instrument _turns backward_, in the sense opposite to the theoretic

sense, and the explanation of its rotation, since discovered, is wholly

different. At last success came, by making the vacuum more complete, on

the one hand, and on the other by not blackening one of the faces of the

paddles and directing a pencil of luminous rays upon one of the faces.

The radiometric effects and the other disturbing causes are eliminated

by a series of pains-taking precautions, and one obtains a deviation

which is very minute, but which is, it would seem, in conformity with

the theory.

The same effects of the Maxwell-Bartholi pressure are forecast likewise

by the theory of Hertz of which we have before spoken, and by that of

Lorentz. But there is a difference. Suppose that the energy, under the

form of light, for example, proceeds from a luminous source to any body

through a transparent medium. The Maxwell-Bartholi pressure will act,

not alone upon the source at the departure, and on the body lit up at

the arrival, but upon the matter of the transparent medium which it

traverses. At the moment when the luminous wave reaches a new region of

this medium, this pressure will push forward the matter there

distributed and will put it back when the wave leaves this region. So

that the recoil of the source has for counterpart the forward movement

of the transparent matter which is in contact with this source; a little

later, the recoil of this same matter has for counterpart the forward

movement of the transparent matter which lies a little further on, and

so on.

Only, is the compensation perfect? Is the action of the Maxwell-Bartholi

pressure upon the matter of the transparent medium equal to its reaction

upon the source, and that whatever be this matter? Or is this action by

so much the less as the medium is less refractive and more rarefied,

becoming null in the void?

If we admit the theory of Hertz, who regards matter as mechanically

bound to the ether, so that the ether may be entirely carried along by

matter, it would be necessary to answer yes to the first question and no

to the second.

There would then be perfect compensation, as required by the principle

of the equality of action and reaction, even in the least refractive

media, even in the air, even in the interplanetary void, where it would

suffice to suppose a residue of matter, however subtile. If on the

contrary we admit the theory of Lorentz, the compensation, always

imperfect, is insensible in the air and becomes null in the void.

But we have seen above that Fizeau's experiment does not permit of our

retaining the theory of Hertz; it is necessary therefore to adopt the

theory of Lorentz, and consequently _to renounce the principle of

reaction_.

IV

_Consequences of the Principle of Relativity_

We have seen above the reasons which impel us to regard the principle of

relativity as a general law of nature. Let us see to what consequences

this principle would lead, should it be regarded as finally

demonstrated.

First, it obliges us to generalize the hypothesis of Lorentz and

Fitzgerald on the contraction of all bodies in the sense of the

translation. In particular, we must extend this hypothesis to the

electrons themselves. Abraham considered these electrons as spherical

and indeformable; it will be necessary for us to admit that these

electrons, spherical when in repose, undergo the Lorentz contraction

when in motion and take then the form of flattened ellipsoids.

This deformation of the electrons will influence their mechanical

properties. In fact I have said that the displacement of these charged

electrons is a veritable current of convection and that their apparent

inertia is due to the self-induction of this current: exclusively as

concerns the negative electrons; exclusively or not, we do not yet know,

for the positive electrons. Well, the deformation of the electrons, a

deformation which depends upon their velocity, will modify the

distribution of the electricity upon their surface, consequently the

intensity of the convection current they produce, consequently the laws

according to which the self-induction of this current will vary as a

function of the velocity.

At this price, the compensation will be perfect and will conform to the

requirements of the principle of relativity, but only upon two

conditions:

1º That the positive electrons have no real mass, but only a fictitious

electromagnetic mass; or at least that their real mass, if it exists, is

not constant and varies with the velocity according to the same laws as

their fictitious mass;

2º That all forces are of electromagnetic origin, or at least that they

vary with the velocity according to the same laws as the forces of

electromagnetic origin.

It still is Lorentz who has made this remarkable synthesis; stop a

moment and see what follows therefrom. First, there is no more matter,

since the positive electrons no longer have real mass, or at least no

constant real mass. The present principles of our mechanics, founded

upon the constancy of mass, must therefore be modified. Again, an

electromagnetic explanation must be sought of all the known forces, in

particular of gravitation, or at least the law of gravitation must be so

modified that this force is altered by velocity in the same way as the

electromagnetic forces. We shall return to this point.

All that appears, at first sight, a little artificial. In particular,

this deformation of electrons seems quite hypothetical. But the thing

may be presented otherwise, so as to avoid putting this hypothesis of

deformation at the foundation of the reasoning. Consider the electrons

as material points and ask how their mass should vary as function of the

velocity not to contravene the principle of relativity. Or, still

better, ask what should be their acceleration under the influence of an

electric or magnetic field, that this principle be not violated and that

we come back to the ordinary laws when we suppose the velocity very

slight. We shall find that the variations of this mass, or of these

accelerations, must be _as if_ the electron underwent the Lorentz

deformation.

V

_Kaufmann's Experiment_

We have before us, then, two theories: one where the electrons are

indeformable, this is that of Abraham; the other where they undergo the

Lorentz deformation. In both cases, their mass increases with the

velocity, becoming infinite when this velocity becomes equal to that of

light; but the law of the variation is not the same. The method employed

by Kaufmann to bring to light the law of variation of the mass seems

therefore to give us an experimental means of deciding between the two

theories.

Unhappily, his first experiments were not sufficiently precise for that;

so he decided to repeat them with more precautions, and measuring with

great care the intensity of the fields. Under their new form _they are

in favor of the theory of Abraham_. Then the principle of relativity

would not have the rigorous value we were tempted to attribute to it;

there would no longer be reason for believing the positive electrons

denuded of real mass like the negative electrons. However, before

definitely adopting this conclusion, a little reflection is necessary.

The question is of such importance that it is to be wished Kaufmann's

experiment were repeated by another experimenter.[17] Unhappily, this

experiment is very delicate and could be carried out successfully only

by a physicist of the same ability as Kaufmann. All precautions have

been properly taken and we hardly see what objection could be made.

[17] At the moment of going to press we learn that M. Bucherer has

repeated the experiment, taking new precautions, and that he

has obtained, contrary to Kaufmann, results confirming the

views of Lorentz.

There is one point however to which I wish to draw attention: that is to

the measurement of the electrostatic field, a measurement upon which all

depends. This field was produced between the two armatures of a

condenser; and, between these armatures, there was to be made an

extremely perfect vacuum, in order to obtain a complete isolation. Then

the difference of potential of the two armatures was measured, and the

field obtained by dividing this difference by the distance apart of the

armatures. That supposes the field uniform; is this certain? Might there

not be an abrupt fall of potential in the neighborhood of one of the

armatures, of the negative armature, for example? There may be a

difference of potential at the meeting of the metal and the vacuum, and

it may be that this difference is not the same on the positive side and

on the negative side; what would lead me to think so is the electric

valve effects between mercury and vacuum. However slight the probability

that it is so, it seems that it should be considered.

VI

_The Principle of Inertia_

In the new dynamics, the principle of inertia is still true, that is to

say that an _isolated_ electron will have a straight and uniform motion.

At least this is generally assumed; however, Lindemann has made

objections to this view; I do not wish to take part in this discussion,

which I can not here expound because of its too difficult character. In

any case, slight modifications to the theory would suffice to shelter it

from Lindemann's objections.

We know that a body submerged in a fluid experiences, when in motion,

considerable resistance, but this is because our fluids are viscous; in

an ideal fluid, perfectly free from viscosity, the body would stir up

behind it a liquid hill, a sort of wake; upon departure, a great effort

would be necessary to put it in motion, since it would be necessary to

move not only the body itself, but the liquid of its wake. But, the

motion once acquired, it would perpetuate itself without resistance,

since the body, in advancing, would simply carry with it the

perturbation of the liquid, without the total vis viva of the liquid

augmenting. Everything would happen therefore as if its inertia was

augmented. An electron advancing in the ether would behave in the same

way: around it, the ether would be stirred up, but this perturbation

would accompany the body in its motion; so that, for an observer carried

along with the electron, the electric and magnetic fields accompanying

this electron would appear invariable, and would change only if the

velocity of the electron varied. An effort would therefore be necessary

to put the electron in motion, since it would be necessary to create the

energy of these fields; on the contrary, once the movement acquired, no

effort would be necessary to maintain it, since the created energy would

only have to go along behind the electron as a wake. This energy,

therefore, could only augment the inertia of the electron, as the

agitation of the liquid augments that of the body submerged in a perfect

fluid. And anyhow, the negative electrons at least have no other inertia

except that.

In the hypothesis of Lorentz, the vis viva, which is only the energy of

the ether, is not proportional to _v_^{2}. Doubtless if _v_ is very

slight, the vis viva is sensibly proportional to _v_^{2}, the quantity

of motion sensibly proportional to _v_, the two masses sensibly constant

and equal to each other. But _when the velocity tends toward the

velocity of light, the vis viva, the quantity of motion and the two

masses increase beyond all limit_.

In the hypothesis of Abraham, the expressions are a little more

complicated; but what we have just said remains true in essentials.

So the mass, the quantity of motion, the vis viva become infinite when

the velocity is equal to that of light.

Thence results that _no body can attain in any way a velocity beyond

that of light_. And in fact, in proportion as its velocity increases,

its mass increases, so that its inertia opposes to any new increase of

velocity a greater and greater obstacle.

A question then suggests itself: let us admit the principle of

relativity; an observer in motion would not have any means of perceiving

his own motion. If therefore no body in its absolute motion can exceed

the velocity of light, but may approach it as nearly as you choose, it

should be the same concerning its relative motion with reference to our

observer. And then we might be tempted to reason as follows: The

observer may attain a velocity of 200,000 kilometers; the body in its

relative motion with reference to the observer may attain the same

velocity; its absolute velocity will then be 400,000 kilometers, which

is impossible, since this is beyond the velocity of light. This is only

a seeming, which vanishes when account is taken of how Lorentz evaluates

local time.

VII

_The Wave of Acceleration_

When an electron is in motion, it produces a perturbation in the ether

surrounding it; if its motion is straight and uniform, this perturbation

reduces to the wake of which we have spoken in the preceding section.

But it is no longer the same, if the motion be curvilinear or varied.

The perturbation may then be regarded as the superposition of two

others, to which Langevin has given the names _wave of velocity_ and

_wave of acceleration_. The wave of velocity is only the wave which

happens in uniform motion.

As to the wave of acceleration, this is a perturbation altogether

analogous to light waves, which starts from the electron at the instant

when it undergoes an acceleration, and which is then propagated by

successive spherical waves with the velocity of light. Whence follows:

in a straight and uniform motion, the energy is wholly conserved; but,

when there is an acceleration, there is loss of energy, which is

dissipated under the form of luminous waves and goes out to infinity

across the ether.

However, the effects of this wave of acceleration, in particular the

corresponding loss of energy, are in most cases negligible, that is to

say not only in ordinary mechanics and in the motions of the heavenly

bodies, but even in the radium rays, where the velocity is very great

without the acceleration being so. We may then confine ourselves to

applying the laws of mechanics, putting the force equal to the product

of acceleration by mass, this mass, however, varying with the velocity

according to the laws explained above. We then say the motion is

_quasi-stationary_.

It would not be the same in all cases where the acceleration is great,

of which the chief are the following:

1º In incandescent gases certain electrons take an oscillatory motion of

very high frequency; the displacements are very small, the velocities

are finite, and the accelerations very great; energy is then

communicated to the ether, and this is why these gases radiate light of

the same period as the oscillations of the electron;

2º Inversely, when a gas receives light, these same electrons are put in

swing with strong accelerations and they absorb light;

3º In the Hertz discharger, the electrons which circulate in the

metallic mass undergo, at the instant of the discharge, an abrupt

acceleration and take then an oscillatory motion of high frequency.

Thence results that a part of the energy radiates under the form of

Hertzian waves;

4º In an incandescent metal, the electrons enclosed in this metal are

impelled with great velocity; upon reaching the surface of the metal,

which they can not get through, they are reflected and thus undergo a

considerable acceleration. This is why the metal emits light. The

details of the laws of the emission of light by dark bodies are

perfectly explained by this hypothesis;

5º Finally when the cathode rays strike the anticathode, the negative

electrons, constituting these rays, which are impelled with very great

velocity, are abruptly arrested. Because of the acceleration they thus

undergo, they produce undulations in the ether. This, according to

certain physicists, is the origin of the Röntgen rays, which would only

be light rays of very short wave-length.

CHAPTER III

THE NEW MECHANICS AND ASTRONOMY

I

_Gravitation_

Mass may be defined in two ways:

1º By the quotient of the force by the acceleration; this is the true

definition of the mass, which measures the inertia of the body.

2° By the attraction the body exercises upon an exterior body, in virtue

of Newton's law. We should therefore distinguish the mass coefficient of

inertia and the mass coefficient of attraction. According to Newton's

law, there is rigorous proportionality between these two coefficients.

But that is demonstrated only for velocities to which the general

principles of dynamics are applicable. Now, we have seen that the mass

coefficient of inertia increases with the velocity; should we conclude

that the mass coefficient of attraction increases likewise with the

velocity and remains proportional to the coefficient of inertia, or, on

the contrary, that this coefficient of attraction remains constant? This

is a question we have no means of deciding.

On the other hand, if the coefficient of attraction depends upon the

velocity, since the velocities of two bodies which mutually attract are

not in general the same, how will this coefficient depend upon these two

velocities?

Upon this subject we can only make hypotheses, but we are naturally led

to investigate which of these hypotheses would be compatible with the

principle of relativity. There are a great number of them; the only one

of which I shall here speak is that of Lorentz, which I shall briefly

expound.

Consider first electrons at rest. Two electrons of the same sign repel

each other and two electrons of contrary sign attract each other; in the

ordinary theory, their mutual actions are proportional to their electric

charges; if therefore we have four electrons, two positive _A_ and

_A'_, and two negative _B_ and _B'_, the charges of these four being the

same in absolute value, the repulsion of _A_ for _A'_ will be, at the

same distance, equal to the repulsion of _B_ for _B'_ and equal also to

the attraction of _A_ for _B'_, or of _A'_ for _B_. If therefore _A_ and

_B_ are very near each other, as also _A'_ and _B'_, and we examine the

action of the system _A_ + _B_ upon the system _A'_ + _B'_, we shall

have two repulsions and two attractions which will exactly compensate

each other and the resulting action will be null.

Now, material molecules should just be regarded as species of solar

systems where circulate the electrons, some positive, some negative, and

_in such a way that the algebraic sum of all the charges is null_. A

material molecule is therefore wholly analogous to the system _A_ + _B_

of which we have spoken, so that the total electric action of two

molecules one upon the other should be null.

But experiment shows us that these molecules attract each other in

consequence of Newtonian gravitation; and then we may make two

hypotheses: we may suppose gravitation has no relation to the

electrostatic attractions, that it is due to a cause entirely different,

and is simply something additional; or else we may suppose the

attractions are not proportional to the charges and that the attraction

exercised by a charge +1 upon a charge -1 is greater than the mutual

repulsion of two +1 charges, or two -1 charges.

In other words, the electric field produced by the positive electrons

and that which the negative electrons produce might be superposed and

yet remain distinct. The positive electrons would be more sensitive to

the field produced by the negative electrons than to the field produced

by the positive electrons; the contrary would be the case for the

negative electrons. It is clear that this hypothesis somewhat

complicates electrostatics, but that it brings back into it gravitation.

This was, in sum, Franklin's hypothesis.

What happens now if the electrons are in motion? The positive electrons

will cause a perturbation in the ether and produce there an electric and

magnetic field. The same will be the case for the negative electrons.

The electrons, positive as well as negative, undergo then a mechanical

impulsion by the action of these different fields. In the ordinary

theory, the electromagnetic field, due to the motion of the positive

electrons, exercises, upon two electrons of contrary sign and of the

same absolute charge, equal actions with contrary sign. We may then

without inconvenience not distinguish the field due to the motion of the

positive electrons and the field due to the motion of the negative

electrons and consider only the algebraic sum of these two fields, that

is to say the resulting field.

In the new theory, on the contrary, the action upon the positive

electrons of the electromagnetic field due to the positive electrons

follows the ordinary laws; it is the same with the action upon the

negative electrons of the field due to the negative electrons. Let us

now consider the action of the field due to the positive electrons upon

the negative electrons (or inversely); it will still follow the same

laws, but _with a different coefficient_. Each electron is more

sensitive to the field created by the electrons of contrary name than to

the field created by the electrons of the same name.

Such is the hypothesis of Lorentz, which reduces to Franklin's

hypothesis for slight velocities; it will therefore explain, for these

small velocities, Newton's law. Moreover, as gravitation goes back to

forces of electrodynamic origin, the general theory of Lorentz will

apply, and consequently the principle of relativity will not be

violated.

We see that Newton's law is no longer applicable to great velocities and

that it must be modified, for bodies in motion, precisely in the same

way as the laws of electrostatics for electricity in motion.

We know that electromagnetic perturbations spread with the velocity of

light. We may therefore be tempted to reject the preceding theory upon

remembering that gravitation spreads, according to the calculations of

Laplace, at least ten million times more quickly than light, and that

consequently it can not be of electromagnetic origin. The result of

Laplace is well known, but one is generally ignorant of its

signification. Laplace supposed that, if the propagation of gravitation

is not instantaneous, its velocity of spread combines with that of the

body attracted, as happens for light in the phenomenon of astronomic

aberration, so that the effective force is not directed along the

straight joining the two bodies, but makes with this straight a small

angle. This is a very special hypothesis, not well justified, and, in

any case, entirely different from that of Lorentz. Laplace's result

proves nothing against the theory of Lorentz.

II

_Comparison with Astronomic Observations_

Can the preceding theories be reconciled with astronomic observations?

First of all, if we adopt them, the energy of the planetary motions will

be constantly dissipated by the effect of the _wave of acceleration_.

From this would result that the mean motions of the stars would

constantly accelerate, as if these stars were moving in a resistant

medium. But this effect is exceedingly slight, far too much so to be

discerned by the most precise observations. The acceleration of the

heavenly bodies is relatively slight, so that the effects of the wave of

acceleration are negligible and the motion may be regarded as _quasi

stationary_. It is true that the effects of the wave of acceleration

constantly accumulate, but this accumulation itself is so slow that

thousands of years of observation would be necessary for it to become

sensible. Let us therefore make the calculation considering the motion

as quasi-stationary, and that under the three following hypotheses:

A. Admit the hypothesis of Abraham (electrons indeformable) and retain

Newton's law in its usual form;

B. Admit the hypothesis of Lorentz about the deformation of electrons

and retain the usual Newton's law;

C. Admit the hypothesis of Lorentz about electrons and modify Newton's

law as we have done in the preceding paragraph, so as to render it

compatible with the principle of relativity.

It is in the motion of Mercury that the effect will be most sensible,

since this planet has the greatest velocity. Tisserand formerly made an

analogous calculation, admitting Weber's law; I recall that Weber had

sought to explain at the same time the electrostatic and electrodynamic

phenomena in supposing that electrons (whose name was not yet invented)

exercise, one upon another, attractions and repulsions directed along

the straight joining them, and depending not only upon their distances,

but upon the first and second derivatives of these distances,

consequently upon their velocities and their accelerations. This law of

Weber, different enough from those which to-day tend to prevail, none

the less presents a certain analogy with them.

Tisserand found that, if the Newtonian attraction conformed to Weber's

law there resulted, for Mercury's perihelion, secular variation of 14",

_of the same sense as that which has been observed and could not be

explained_, but smaller, since this is 38".

Let us recur to the hypotheses A, B and C, and study first the motion of

a planet attracted by a fixed center. The hypotheses B and C are no

longer distinguished, since, if the attracting point is fixed, the field

it produces is a purely electrostatic field, where the attraction varies

inversely as the square of the distance, in conformity with Coulomb's

electrostatic law, identical with that of Newton.

The vis viva equation holds good, taking for vis viva the new

definition; in the same way, the equation of areas is replaced by

another equivalent to it; the moment of the quantity of motion is a

constant, but the quantity of motion must be defined as in the new

dynamics.

The only sensible effect will be a secular motion of the perihelion.

With the theory of Lorentz, we shall find, for this motion, half of what

Weber's law would give; with the theory of Abraham, two fifths.

If now we suppose two moving bodies gravitating around their common

center of gravity, the effects are very little different, though the

calculations may be a little more complicated. The motion of Mercury's

perihelion would therefore be 7" in the theory of Lorentz and 5".6 in

that of Abraham.

The effect moreover is proportional to (_n_^{3})(_a_^{2}), where _n_ is

the star's mean motion and a the radius of its orbit. For the planets,

in virtue of Kepler's law, the effect varies then inversely as

sqrt(_a_^{5}); it is therefore insensible, save for Mercury.

It is likewise insensible for the moon though _n_ is great, because _a_

is extremely small; in sum, it is five times less for Venus, and six

hundred times less for the moon than for Mercury. We may add that as to

Venus and the earth, the motion of the perihelion (for the same angular

velocity of this motion) would be much more difficult to discern by

astronomic observations, because the excentricity of their orbits is

much less than for Mercury.

To sum up, _the only sensible effect upon astronomic observations would

be a motion of Mercury's perihelion, in the same sense as that which has

been observed without being explained, but notably slighter_.

That can not be regarded as an argument in favor of the new dynamics,

since it will always be necessary to seek another explanation for the

greater part of Mercury's anomaly; but still less can it be regarded as

an argument against it.

III

_The Theory of Lesage_

It is interesting to compare these considerations with a theory long

since proposed to explain universal gravitation.

Suppose that, in the interplanetary spaces, circulate in every

direction, with high velocities, very tenuous corpuscles. A body

isolated in space will not be affected, apparently, by the impacts of

these corpuscles, since these impacts are equally distributed in all

directions. But if two bodies _A_ and _B_ are present, the body _B_ will

play the rôle of screen and will intercept part of the corpuscles which,

without it, would have struck _A_. Then, the impacts received by _A_ in

the direction opposite that from _B_ will no longer have a counterpart,

or will now be only partially compensated, and this will push _A_ toward

_B_.

Such is the theory of Lesage; and we shall discuss it, taking first the

view-point of ordinary mechanics.